↑

Properties

Label 1670.2.a.j
Level $1670$
Weight $2$
Character orbit 1670.a
Self dual yes
Analytic conductor $13.335$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1670,2,Mod(1,1670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1670, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1670 = 2 \cdot 5 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1670.a (trivial)

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: yes
Analytic conductor: \(13.3350171376\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 16x^{7} + 2x^{6} + 67x^{5} + 19x^{4} - 74x^{3} - 12x^{2} + 27x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{5} + 1) q^{3} + q^{4} - q^{5} + ( - \beta_{5} + 1) q^{6} + ( - \beta_{8} + \beta_{7} + \beta_{3} + 1) q^{7} + q^{8} + (\beta_{7} + \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{5} + 1) q^{3} + q^{4} - q^{5} + ( - \beta_{5} + 1) q^{6} + ( - \beta_{8} + \beta_{7} + \beta_{3} + 1) q^{7} + q^{8} + (\beta_{7} + \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{9}+ \cdots + (\beta_{8} - 8 \beta_{7} - \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} - 9 q^{5} + 7 q^{6} + 11 q^{7} + 9 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} - 9 q^{5} + 7 q^{6} + 11 q^{7} + 9 q^{8} + 16 q^{9} - 9 q^{10} + q^{11} + 7 q^{12} + 13 q^{13} + 11 q^{14} - 7 q^{15} + 9 q^{16} + 6 q^{17} + 16 q^{18} + 15 q^{19} - 9 q^{20} - 4 q^{21} + q^{22} + 10 q^{23} + 7 q^{24} + 9 q^{25} + 13 q^{26} + 16 q^{27} + 11 q^{28} + 5 q^{29} - 7 q^{30} + 9 q^{31} + 9 q^{32} + 25 q^{33} + 6 q^{34} - 11 q^{35} + 16 q^{36} + 11 q^{37} + 15 q^{38} + 7 q^{39} - 9 q^{40} + 8 q^{41} - 4 q^{42} + 9 q^{43} + q^{44} - 16 q^{45} + 10 q^{46} + 14 q^{47} + 7 q^{48} + 20 q^{49} + 9 q^{50} + 10 q^{51} + 13 q^{52} - 31 q^{53} + 16 q^{54} - q^{55} + 11 q^{56} + 3 q^{57} + 5 q^{58} + 3 q^{59} - 7 q^{60} + 17 q^{61} + 9 q^{62} + 17 q^{63} + 9 q^{64} - 13 q^{65} + 25 q^{66} + 16 q^{67} + 6 q^{68} - 26 q^{69} - 11 q^{70} + 23 q^{71} + 16 q^{72} + 13 q^{73} + 11 q^{74} + 7 q^{75} + 15 q^{76} - 29 q^{77} + 7 q^{78} - 2 q^{79} - 9 q^{80} + 53 q^{81} + 8 q^{82} - 13 q^{83} - 4 q^{84} - 6 q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 16 q^{89} - 16 q^{90} + 20 q^{91} + 10 q^{92} - 2 q^{93} + 14 q^{94} - 15 q^{95} + 7 q^{96} + 31 q^{97} + 20 q^{98} - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 16x^{7} + 2x^{6} + 67x^{5} + 19x^{4} - 74x^{3} - 12x^{2} + 27x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{8} + 69\nu^{7} + 52\nu^{6} - 499\nu^{5} + 802\nu^{4} + 429\nu^{3} - 2767\nu^{2} - 148\nu + 1139 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43 \nu^{8} + 183 \nu^{7} - 1024 \nu^{6} - 3203 \nu^{5} + 4673 \nu^{4} + 12774 \nu^{3} - 3101 \nu^{2} + \cdots + 1483 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53 \nu^{8} - 189 \nu^{7} + 1253 \nu^{6} + 3235 \nu^{5} - 5842 \nu^{4} - 12162 \nu^{3} + 4270 \nu^{2} + \cdots - 1901 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -92\nu^{8} + 102\nu^{7} + 1478\nu^{6} - 413\nu^{5} - 6196\nu^{4} - 579\nu^{3} + 6196\nu^{2} - 65\nu - 623 ) / 393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 172 \nu^{8} + 54 \nu^{7} + 2917 \nu^{6} + 1415 \nu^{5} - 12404 \nu^{4} - 9831 \nu^{3} + 12797 \nu^{2} + \cdots - 4360 ) / 393 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -226\nu^{8} + 336\nu^{7} + 3289\nu^{6} - 1792\nu^{5} - 11957\nu^{4} - 81\nu^{3} + 8813\nu^{2} + 71\nu - 565 ) / 393 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 334 \nu^{8} + 507 \nu^{7} + 4819 \nu^{6} - 2704 \nu^{5} - 17351 \nu^{4} - 231 \nu^{3} + 12635 \nu^{2} + \cdots - 1621 ) / 393 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} - 3\beta_{7} + \beta_{4} + \beta_{3} + 9\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{8} - 18\beta_{7} - 3\beta_{6} - 7\beta_{5} - \beta_{4} - 3\beta_{3} - 12\beta_{2} + 23\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 46\beta_{8} - 63\beta_{7} - 6\beta_{6} - \beta_{5} + 5\beta_{4} + \beta_{3} - 14\beta_{2} + 116\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 233 \beta_{8} - 281 \beta_{7} - 49 \beta_{6} - 52 \beta_{5} - 16 \beta_{4} - 48 \beta_{3} - 145 \beta_{2} + \cdots + 455 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 783 \beta_{8} - 1033 \beta_{7} - 136 \beta_{6} - 36 \beta_{5} + 4 \beta_{4} - 84 \beta_{3} - 324 \beta_{2} + \cdots + 1454 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3382 \beta_{8} - 4213 \beta_{7} - 709 \beta_{6} - 471 \beta_{5} - 214 \beta_{4} - 673 \beta_{3} + \cdots + 6248 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.06232
−1.29885
0.847850
0.536194
−2.60225
0.175617
−1.64683
2.20106
3.84952
1.00000 −3.12101 1.00000 −1.00000 −3.12101 3.48158 1.00000 6.74069 −1.00000
1.2 1.00000 −1.01180 1.00000 −1.00000 −1.01180 0.685292 1.00000 −1.97626 −1.00000
1.3 1.00000 −0.518776 1.00000 −1.00000 −0.518776 4.40255 1.00000 −2.73087 −1.00000
1.4 1.00000 −0.373039 1.00000 −1.00000 −0.373039 −2.72824 1.00000 −2.86084 −1.00000
1.5 1.00000 1.14789 1.00000 −1.00000 1.14789 1.95530 1.00000 −1.68236 −1.00000
1.6 1.00000 2.15109 1.00000 −1.00000 2.15109 3.22060 1.00000 1.62717 −1.00000
1.7 1.00000 2.37811 1.00000 −1.00000 2.37811 −1.46773 1.00000 2.65541 −1.00000
1.8 1.00000 2.97197 1.00000 −1.00000 2.97197 4.34343 1.00000 5.83261 −1.00000
1.9 1.00000 3.37557 1.00000 −1.00000 3.37557 −2.89278 1.00000 8.39445 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(167\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1670.2.a.j ✓ 9
5.b even 2 1 8350.2.a.w 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1670.2.a.j ✓ 9 1.a even 1 1 trivial
8350.2.a.w 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 7T_{3}^{8} + 3T_{3}^{7} + 74T_{3}^{6} - 150T_{3}^{5} - 58T_{3}^{4} + 265T_{3}^{3} + 16T_{3}^{2} - 127T_{3} - 36 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1670))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 7 T^{8} + \cdots - 36 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 11 T^{8} + \cdots + 3328 \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} + \cdots - 88416 \) Copy content Toggle raw display
$13$ \( T^{9} - 13 T^{8} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{9} - 6 T^{8} + \cdots + 17406 \) Copy content Toggle raw display
$19$ \( T^{9} - 15 T^{8} + \cdots + 12448 \) Copy content Toggle raw display
$23$ \( T^{9} - 10 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{9} - 5 T^{8} + \cdots - 1631376 \) Copy content Toggle raw display
$31$ \( T^{9} - 9 T^{8} + \cdots - 160872 \) Copy content Toggle raw display
$37$ \( T^{9} - 11 T^{8} + \cdots + 1758576 \) Copy content Toggle raw display
$41$ \( T^{9} - 8 T^{8} + \cdots - 4272 \) Copy content Toggle raw display
$43$ \( T^{9} - 9 T^{8} + \cdots + 1174400 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} + \cdots - 9460992 \) Copy content Toggle raw display
$53$ \( T^{9} + 31 T^{8} + \cdots - 21471088 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} + \cdots - 1783344 \) Copy content Toggle raw display
$61$ \( T^{9} - 17 T^{8} + \cdots - 65885712 \) Copy content Toggle raw display
$67$ \( T^{9} - 16 T^{8} + \cdots - 2489856 \) Copy content Toggle raw display
$71$ \( T^{9} - 23 T^{8} + \cdots + 17965824 \) Copy content Toggle raw display
$73$ \( T^{9} - 13 T^{8} + \cdots + 638154 \) Copy content Toggle raw display
$79$ \( T^{9} + 2 T^{8} + \cdots - 4780032 \) Copy content Toggle raw display
$83$ \( T^{9} + 13 T^{8} + \cdots + 244864 \) Copy content Toggle raw display
$89$ \( T^{9} - 16 T^{8} + \cdots - 7973946 \) Copy content Toggle raw display
$97$ \( T^{9} - 31 T^{8} + \cdots + 132525392 \) Copy content Toggle raw display
show more
show less