Properties

Label 78.2.k.a
Level $78$
Weight $2$
Character orbit 78.k
Analytic conductor $0.623$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,2,Mod(11,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.k (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{7}) q^{2} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} - \beta_{7}) q^{2} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{12} - 6 \beta_{11} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} - 24 q^{10} - 24 q^{13} + 8 q^{16} - 16 q^{19} - 24 q^{21} - 8 q^{28} + 24 q^{30} + 16 q^{31} - 24 q^{33} + 24 q^{34} + 24 q^{36} + 16 q^{37} + 48 q^{39} + 24 q^{45} + 24 q^{46} + 24 q^{49} - 8 q^{52} - 24 q^{55} - 24 q^{57} - 24 q^{60} - 24 q^{61} - 24 q^{63} - 48 q^{66} + 32 q^{67} - 48 q^{69} - 24 q^{72} + 56 q^{73} - 16 q^{76} - 96 q^{79} + 24 q^{81} - 48 q^{82} - 24 q^{85} + 48 q^{87} - 16 q^{91} - 24 q^{93} - 24 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1548 \nu^{15} - 13188 \nu^{14} + 76652 \nu^{13} - 318997 \nu^{12} + 1019601 \nu^{11} - 2665197 \nu^{10} + \cdots - 204955 ) / 17095 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + \cdots + 142555 ) / 17095 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + \cdots + 5945 ) / 65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1548 \nu^{15} + 26338 \nu^{14} - 168702 \nu^{13} + 849994 \nu^{12} - 3008933 \nu^{11} + \cdots + 979490 ) / 17095 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15207 \nu^{15} - 93144 \nu^{14} + 525827 \nu^{13} - 1795712 \nu^{12} + 5254311 \nu^{11} + \cdots + 281130 ) / 17095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15977 \nu^{15} - 98393 \nu^{14} + 556086 \nu^{13} - 1902441 \nu^{12} + 5567337 \nu^{11} + \cdots + 610830 ) / 17095 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} + \cdots - 1883725 ) / 17095 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15207 \nu^{15} - 151267 \nu^{14} + 932688 \nu^{13} - 4151403 \nu^{12} + 14099264 \nu^{11} + \cdots - 2984015 ) / 17095 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49663 \nu^{15} - 358139 \nu^{14} + 2089429 \nu^{13} - 8006091 \nu^{12} + 25109764 \nu^{11} + \cdots - 1090565 ) / 17095 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 46358 \nu^{15} - 350841 \nu^{14} + 2068305 \nu^{13} - 8157610 \nu^{12} + 26035188 \nu^{11} + \cdots - 2064105 ) / 17095 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 46358 \nu^{15} + 373985 \nu^{14} - 2230313 \nu^{13} + 9095994 \nu^{12} - 29559388 \nu^{11} + \cdots + 3439595 ) / 17095 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 54184 \nu^{15} + 399016 \nu^{14} - 2335837 \nu^{13} + 9056462 \nu^{12} - 28575749 \nu^{11} + \cdots + 1312670 ) / 17095 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} + \cdots - 370592 ) / 3419 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + \cdots + 1951760 ) / 17095 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + 7 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 8 \beta_{12} - 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21 \beta_{15} - 46 \beta_{14} + 44 \beta_{13} - 19 \beta_{12} + 44 \beta_{11} + 19 \beta_{10} + \cdots + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{15} - 43 \beta_{14} - 5 \beta_{13} - 70 \beta_{12} + 55 \beta_{11} + 49 \beta_{10} - 30 \beta_{9} + \cdots - 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 321 \beta_{15} + 267 \beta_{14} - 365 \beta_{13} + 17 \beta_{12} - 213 \beta_{11} - 28 \beta_{10} + \cdots - 307 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 256 \beta_{15} + 446 \beta_{14} - 108 \beta_{13} + 544 \beta_{12} - 528 \beta_{11} - 304 \beta_{10} + \cdots + 267 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3026 \beta_{15} - 1225 \beta_{14} + 2701 \beta_{13} + 906 \beta_{12} + 525 \beta_{11} - 297 \beta_{10} + \cdots + 2725 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 274 \beta_{15} - 4001 \beta_{14} + 2070 \beta_{13} - 3737 \beta_{12} + 4287 \beta_{11} + 1922 \beta_{10} + \cdots - 932 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23673 \beta_{15} + 1926 \beta_{14} - 17346 \beta_{13} - 14155 \beta_{12} + 4640 \beta_{11} + \cdots - 22758 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10280 \beta_{15} + 31947 \beta_{14} - 24436 \beta_{13} + 21968 \beta_{12} - 30552 \beta_{11} + \cdots - 3138 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 160655 \beta_{15} + 46695 \beta_{14} + 88659 \beta_{13} + 152123 \beta_{12} - 96769 \beta_{11} + \cdots + 171575 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 160979 \beta_{15} - 225841 \beta_{14} + 234065 \beta_{13} - 95851 \beta_{12} + 187408 \beta_{11} + \cdots + 106657 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 914760 \beta_{15} - 806121 \beta_{14} - 233687 \beta_{13} - 1368388 \beta_{12} + 1121513 \beta_{11} + \cdots - 1132093 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(-1\) \(-\beta_{9} + \beta_{14}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 1.74530i
0.500000 + 1.33108i
0.500000 0.331082i
0.500000 + 2.74530i
0.500000 0.410882i
0.500000 2.00333i
0.500000 0.589118i
0.500000 + 1.00333i
0.500000 + 0.410882i
0.500000 + 2.00333i
0.500000 + 0.589118i
0.500000 1.00333i
0.500000 + 1.74530i
0.500000 1.33108i
0.500000 + 0.331082i
0.500000 2.74530i
−0.965926 0.258819i −1.73022 + 0.0795432i 0.866025 + 0.500000i 2.76293 2.76293i 1.69185 + 0.370982i −0.657464 2.45369i −0.707107 0.707107i 2.98735 0.275255i −3.38389 + 1.95369i
11.2 −0.965926 0.258819i 1.73022 0.0795432i 0.866025 + 0.500000i −0.313444 + 0.313444i −1.69185 0.370982i −0.0745867 0.278362i −0.707107 0.707107i 2.98735 0.275255i 0.383889 0.221638i
11.3 0.965926 + 0.258819i −0.933998 + 1.45865i 0.866025 + 0.500000i 0.313444 0.313444i −1.27970 + 1.16721i −0.0745867 0.278362i 0.707107 + 0.707107i −1.25529 2.72474i 0.383889 0.221638i
11.4 0.965926 + 0.258819i 0.933998 1.45865i 0.866025 + 0.500000i −2.76293 + 2.76293i 1.27970 1.16721i −0.657464 2.45369i 0.707107 + 0.707107i −1.25529 2.72474i −3.38389 + 1.95369i
41.1 −0.258819 + 0.965926i −0.0795432 1.73022i −0.866025 0.500000i 2.02097 + 2.02097i 1.69185 + 0.370982i 3.46723 0.929042i 0.707107 0.707107i −2.98735 + 0.275255i −2.47517 + 1.42904i
41.2 −0.258819 + 0.965926i 0.0795432 + 1.73022i −0.866025 0.500000i 0.428520 + 0.428520i −1.69185 0.370982i −0.735180 + 0.196991i 0.707107 0.707107i −2.98735 + 0.275255i −0.524827 + 0.303009i
41.3 0.258819 0.965926i −1.45865 0.933998i −0.866025 0.500000i −2.02097 2.02097i −1.27970 + 1.16721i 3.46723 0.929042i −0.707107 + 0.707107i 1.25529 + 2.72474i −2.47517 + 1.42904i
41.4 0.258819 0.965926i 1.45865 + 0.933998i −0.866025 0.500000i −0.428520 0.428520i 1.27970 1.16721i −0.735180 + 0.196991i −0.707107 + 0.707107i 1.25529 + 2.72474i −0.524827 + 0.303009i
59.1 −0.258819 0.965926i −0.0795432 + 1.73022i −0.866025 + 0.500000i 2.02097 2.02097i 1.69185 0.370982i 3.46723 + 0.929042i 0.707107 + 0.707107i −2.98735 0.275255i −2.47517 1.42904i
59.2 −0.258819 0.965926i 0.0795432 1.73022i −0.866025 + 0.500000i 0.428520 0.428520i −1.69185 + 0.370982i −0.735180 0.196991i 0.707107 + 0.707107i −2.98735 0.275255i −0.524827 0.303009i
59.3 0.258819 + 0.965926i −1.45865 + 0.933998i −0.866025 + 0.500000i −2.02097 + 2.02097i −1.27970 1.16721i 3.46723 + 0.929042i −0.707107 0.707107i 1.25529 2.72474i −2.47517 1.42904i
59.4 0.258819 + 0.965926i 1.45865 0.933998i −0.866025 + 0.500000i −0.428520 + 0.428520i 1.27970 + 1.16721i −0.735180 0.196991i −0.707107 0.707107i 1.25529 2.72474i −0.524827 0.303009i
71.1 −0.965926 + 0.258819i −1.73022 0.0795432i 0.866025 0.500000i 2.76293 + 2.76293i 1.69185 0.370982i −0.657464 + 2.45369i −0.707107 + 0.707107i 2.98735 + 0.275255i −3.38389 1.95369i
71.2 −0.965926 + 0.258819i 1.73022 + 0.0795432i 0.866025 0.500000i −0.313444 0.313444i −1.69185 + 0.370982i −0.0745867 + 0.278362i −0.707107 + 0.707107i 2.98735 + 0.275255i 0.383889 + 0.221638i
71.3 0.965926 0.258819i −0.933998 1.45865i 0.866025 0.500000i 0.313444 + 0.313444i −1.27970 1.16721i −0.0745867 + 0.278362i 0.707107 0.707107i −1.25529 + 2.72474i 0.383889 + 0.221638i
71.4 0.965926 0.258819i 0.933998 + 1.45865i 0.866025 0.500000i −2.76293 2.76293i 1.27970 + 1.16721i −0.657464 + 2.45369i 0.707107 0.707107i −1.25529 + 2.72474i −3.38389 1.95369i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.k.a 16
3.b odd 2 1 inner 78.2.k.a 16
4.b odd 2 1 624.2.cn.d 16
12.b even 2 1 624.2.cn.d 16
13.c even 3 1 1014.2.g.c 16
13.e even 6 1 1014.2.g.d 16
13.f odd 12 1 inner 78.2.k.a 16
13.f odd 12 1 1014.2.g.c 16
13.f odd 12 1 1014.2.g.d 16
39.h odd 6 1 1014.2.g.d 16
39.i odd 6 1 1014.2.g.c 16
39.k even 12 1 inner 78.2.k.a 16
39.k even 12 1 1014.2.g.c 16
39.k even 12 1 1014.2.g.d 16
52.l even 12 1 624.2.cn.d 16
156.v odd 12 1 624.2.cn.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.k.a 16 1.a even 1 1 trivial
78.2.k.a 16 3.b odd 2 1 inner
78.2.k.a 16 13.f odd 12 1 inner
78.2.k.a 16 39.k even 12 1 inner
624.2.cn.d 16 4.b odd 2 1
624.2.cn.d 16 12.b even 2 1
624.2.cn.d 16 52.l even 12 1
624.2.cn.d 16 156.v odd 12 1
1014.2.g.c 16 13.c even 3 1
1014.2.g.c 16 13.f odd 12 1
1014.2.g.c 16 39.i odd 6 1
1014.2.g.c 16 39.k even 12 1
1014.2.g.d 16 13.e even 6 1
1014.2.g.d 16 13.f odd 12 1
1014.2.g.d 16 39.h odd 6 1
1014.2.g.d 16 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(78, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 300 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + 2 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 30 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 151807041 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 60 T^{14} + \cdots + 18974736 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 33871089681 \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 375769)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 65697655057281 \) Copy content Toggle raw display
$43$ \( (T^{8} - 18 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 41006250000 \) Copy content Toggle raw display
$53$ \( (T^{8} + 156 T^{6} + \cdots + 522729)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 43489065701376 \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 47961)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots + 676)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5143987297296 \) Copy content Toggle raw display
$73$ \( (T^{8} - 28 T^{7} + \cdots + 16834609)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + \cdots + 312)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 36804120336 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} - 8 T^{7} + \cdots + 43264)^{2} \) Copy content Toggle raw display
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