↑

Properties

Label 670.2.h.b
Level $670$
Weight $2$
Character orbit 670.h
Analytic conductor $5.350$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [670,2,Mod(29,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.h (of order \(6\), degree \(2\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3}) q^{2} + ( - \beta_{9} - \beta_{4}) q^{3} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{8} - \beta_1) q^{6} + (2 \beta_{6} + \beta_{5}) q^{7} + \beta_{4} q^{8} + ( - \beta_{2} + 3 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{3}) q^{2} + ( - \beta_{9} - \beta_{4}) q^{3} + ( - \beta_{8} + 1) q^{4} + ( - \beta_{4} + 2) q^{5} + (\beta_{11} + \beta_{8} - \beta_1) q^{6} + (2 \beta_{6} + \beta_{5}) q^{7} + \beta_{4} q^{8} + ( - \beta_{2} + 3 \beta_1 - 1) q^{9} + (\beta_{8} + 2 \beta_{4} + 2 \beta_{3}) q^{10} + (2 \beta_{11} + 2 \beta_{10} + \cdots - 2) q^{11}+ \cdots + ( - 12 \beta_{11} - 16 \beta_{8} + 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} + 24 q^{5} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 24 q^{5} + 6 q^{6} - 8 q^{9} + 6 q^{10} - 8 q^{11} - 8 q^{14} - 12 q^{15} - 6 q^{16} - 12 q^{19} + 12 q^{20} - 4 q^{21} + 12 q^{24} + 36 q^{25} + 4 q^{26} + 8 q^{29} + 12 q^{30} - 18 q^{31} + 12 q^{34} + 4 q^{35} - 4 q^{36} + 16 q^{39} + 12 q^{40} + 38 q^{41} + 8 q^{44} - 16 q^{45} + 16 q^{46} + 14 q^{49} + 24 q^{50} + 24 q^{51} - 30 q^{54} - 16 q^{55} - 4 q^{56} - 72 q^{59} - 6 q^{60} + 4 q^{61} - 12 q^{64} + 4 q^{65} - 64 q^{66} + 40 q^{69} - 16 q^{70} + 6 q^{71} - 22 q^{74} - 48 q^{75} - 24 q^{76} - 44 q^{79} - 12 q^{80} + 140 q^{81} + 4 q^{84} + 12 q^{85} + 8 q^{86} - 60 q^{89} - 4 q^{90} + 56 q^{91} - 24 q^{94} - 24 q^{95} + 6 q^{96} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + \cdots + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} + \cdots + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + \cdots - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + \cdots - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} + \cdots + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + \cdots - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + \cdots - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + \cdots - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} + \cdots + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + \cdots - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} + \cdots + 612 ) / 230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/670\mathbb{Z}\right)^\times\).

\(n\) \(471\) \(537\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.550552 − 0.147520i
−1.16746 + 0.312819i
1.98293 − 0.531325i
−0.531325 − 1.98293i
0.312819 + 1.16746i
−0.147520 − 0.550552i
1.98293 + 0.531325i
−1.16746 − 0.312819i
0.550552 + 0.147520i
−0.147520 + 0.550552i
0.312819 − 1.16746i
−0.531325 + 1.98293i
−0.866025 − 0.500000i − 0.675131i 0.500000 + 0.866025i 2.00000 + 1.00000i −0.337565 + 0.584680i −1.11480 + 0.643629i − 1.00000i 2.54420 −1.23205 − 1.86603i
29.2 −0.866025 − 0.500000i 0.460811i 0.500000 + 0.866025i 2.00000 + 1.00000i 0.230406 − 0.399074i 4.22565 − 2.43968i − 1.00000i 2.78765 −1.23205 − 1.86603i
29.3 −0.866025 − 0.500000i 3.21432i 0.500000 + 0.866025i 2.00000 + 1.00000i 1.60716 − 2.78368i −1.37880 + 0.796052i − 1.00000i −7.33185 −1.23205 − 1.86603i
29.4 0.866025 + 0.500000i − 3.21432i 0.500000 + 0.866025i 2.00000 − 1.00000i 1.60716 − 2.78368i 1.37880 − 0.796052i 1.00000i −7.33185 2.23205 + 0.133975i
29.5 0.866025 + 0.500000i − 0.460811i 0.500000 + 0.866025i 2.00000 − 1.00000i 0.230406 − 0.399074i −4.22565 + 2.43968i 1.00000i 2.78765 2.23205 + 0.133975i
29.6 0.866025 + 0.500000i 0.675131i 0.500000 + 0.866025i 2.00000 − 1.00000i −0.337565 + 0.584680i 1.11480 − 0.643629i 1.00000i 2.54420 2.23205 + 0.133975i
439.1 −0.866025 + 0.500000i − 3.21432i 0.500000 − 0.866025i 2.00000 − 1.00000i 1.60716 + 2.78368i −1.37880 − 0.796052i 1.00000i −7.33185 −1.23205 + 1.86603i
439.2 −0.866025 + 0.500000i − 0.460811i 0.500000 − 0.866025i 2.00000 − 1.00000i 0.230406 + 0.399074i 4.22565 + 2.43968i 1.00000i 2.78765 −1.23205 + 1.86603i
439.3 −0.866025 + 0.500000i 0.675131i 0.500000 − 0.866025i 2.00000 − 1.00000i −0.337565 − 0.584680i −1.11480 − 0.643629i 1.00000i 2.54420 −1.23205 + 1.86603i
439.4 0.866025 − 0.500000i − 0.675131i 0.500000 − 0.866025i 2.00000 + 1.00000i −0.337565 − 0.584680i 1.11480 + 0.643629i − 1.00000i 2.54420 2.23205 − 0.133975i
439.5 0.866025 − 0.500000i 0.460811i 0.500000 − 0.866025i 2.00000 + 1.00000i 0.230406 + 0.399074i −4.22565 − 2.43968i − 1.00000i 2.78765 2.23205 − 0.133975i
439.6 0.866025 − 0.500000i 3.21432i 0.500000 − 0.866025i 2.00000 + 1.00000i 1.60716 + 2.78368i 1.37880 + 0.796052i − 1.00000i −7.33185 2.23205 − 0.133975i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.c even 3 1 inner
335.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.h.b ✓ 12
5.b even 2 1 inner 670.2.h.b ✓ 12
67.c even 3 1 inner 670.2.h.b ✓ 12
335.i even 6 1 inner 670.2.h.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.h.b ✓ 12 1.a even 1 1 trivial
670.2.h.b ✓ 12 5.b even 2 1 inner
670.2.h.b ✓ 12 67.c even 3 1 inner
670.2.h.b ✓ 12 335.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 11T_{3}^{4} + 7T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} + 11 T^{4} + 7 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} - 28 T^{10} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{6} + 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{12} - 72 T^{10} + \cdots + 6250000 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + 40 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 64 T^{10} + \cdots + 29986576 \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 9 T^{5} + \cdots + 100489)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 51 T^{10} + \cdots + 707281 \) Copy content Toggle raw display
$41$ \( (T^{6} - 19 T^{5} + \cdots + 34225)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 176 T^{4} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 193877776 \) Copy content Toggle raw display
$53$ \( (T^{6} + 35 T^{4} + \cdots + 361)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 18 T^{2} + \cdots + 104)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} - 2 T^{5} + \cdots + 17956)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 90458382169 \) Copy content Toggle raw display
$71$ \( (T^{6} - 3 T^{5} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 52 T^{10} + \cdots + 4477456 \) Copy content Toggle raw display
$79$ \( (T^{6} + 22 T^{5} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 147 T^{10} + \cdots + 625 \) Copy content Toggle raw display
$89$ \( (T + 5)^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 655360000 \) Copy content Toggle raw display
show more
show less