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Properties

Label 867.2.h.j
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,2,Mod(688,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.688");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.h (of order \(8\), degree \(4\), not 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.1963501163244660295991296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1889x^{8} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{7} q^{3} + (\beta_{9} + 3 \beta_{8}) q^{4} + ( - \beta_{15} - \beta_{14}) q^{5} - \beta_{11} q^{6} + ( - \beta_{13} + 5 \beta_{12}) q^{8} + \beta_{12} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{7} q^{3} + (\beta_{9} + 3 \beta_{8}) q^{4} + ( - \beta_{15} - \beta_{14}) q^{5} - \beta_{11} q^{6} + ( - \beta_{13} + 5 \beta_{12}) q^{8} + \beta_{12} q^{9} + ( - 4 \beta_{2} + 2 \beta_1) q^{10} + (\beta_{11} - \beta_{10}) q^{11} + ( - \beta_{15} - 2 \beta_{14}) q^{12} + (\beta_{9} - 2 \beta_{8}) q^{13} + ( - \beta_{5} - 2 \beta_{3}) q^{15} + 3 \beta_{4} q^{16} + \beta_{4} q^{18} + 3 \beta_{5} q^{19} + ( - 6 \beta_{7} + 4 \beta_{6}) q^{20} + 4 \beta_{14} q^{22} + ( - \beta_{11} + 5 \beta_{10}) q^{23} + ( - 4 \beta_{2} + \beta_1) q^{24} + (3 \beta_{13} - 3 \beta_{12}) q^{25} + (2 \beta_{13} + 2 \beta_{12}) q^{26} - \beta_{2} q^{27} + ( - 4 \beta_{15} + 2 \beta_{14}) q^{29} + ( - 2 \beta_{9} - 6 \beta_{8}) q^{30} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{31} + ( - \beta_{5} - 5 \beta_{3}) q^{32} + ( - \beta_{4} - 1) q^{33} + ( - \beta_{5} - 3 \beta_{3}) q^{36} - 2 \beta_{6} q^{37} + 12 \beta_{8} q^{38} + ( - \beta_{15} + 3 \beta_{14}) q^{39} + (6 \beta_{11} + 8 \beta_{10}) q^{40} + (\beta_{2} - \beta_1) q^{41} - 3 \beta_{13} q^{43} + (2 \beta_{2} - 2 \beta_1) q^{44} + (\beta_{11} + \beta_{10}) q^{45} + (4 \beta_{15} - 4 \beta_{14}) q^{46} + (2 \beta_{9} + 8 \beta_{8}) q^{47} + 3 \beta_{6} q^{48} - 7 \beta_{3} q^{49} + ( - 3 \beta_{4} + 12) q^{50} + 2 q^{52} + ( - 4 \beta_{5} - 6 \beta_{3}) q^{53} + \beta_{6} q^{54} + (\beta_{9} + 4 \beta_{8}) q^{55} + ( - 3 \beta_{11} + 3 \beta_{10}) q^{57} + ( - 16 \beta_{2} + 2 \beta_1) q^{58} + (2 \beta_{13} - 4 \beta_{12}) q^{59} + (4 \beta_{13} - 10 \beta_{12}) q^{60} + (4 \beta_{2} - 2 \beta_1) q^{61} - 8 \beta_{10} q^{62} + (\beta_{9} - 3 \beta_{8}) q^{64} + ( - \beta_{7} - \beta_{6}) q^{65} + 4 \beta_{3} q^{66} - 4 q^{67} + (\beta_{4} + 5) q^{69} + ( - 4 \beta_{7} - 4 \beta_{6}) q^{71} + ( - \beta_{9} - 5 \beta_{8}) q^{72} + (4 \beta_{15} + 2 \beta_{14}) q^{73} + ( - 2 \beta_{11} - 8 \beta_{10}) q^{74} - 3 \beta_1 q^{75} + ( - 6 \beta_{13} + 12 \beta_{12}) q^{76} + ( - 4 \beta_{2} - 2 \beta_1) q^{78} + ( - 6 \beta_{11} + 6 \beta_{10}) q^{79} + (6 \beta_{15} + 12 \beta_{14}) q^{80} - \beta_{8} q^{81} + (4 \beta_{7} - 2 \beta_{6}) q^{82} + (2 \beta_{5} - 4 \beta_{3}) q^{83} - 12 q^{86} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{87} - 4 \beta_{6} q^{88} + ( - 2 \beta_{9} + 2 \beta_{8}) q^{89} + (2 \beta_{15} + 4 \beta_{14}) q^{90} + (6 \beta_{2} - 2 \beta_1) q^{92} - 2 \beta_{13} q^{93} + ( - 8 \beta_{13} + 16 \beta_{12}) q^{94} + ( - 9 \beta_{2} + 3 \beta_1) q^{95} + (\beta_{11} + 4 \beta_{10}) q^{96} + (2 \beta_{15} - 8 \beta_{14}) q^{97} + ( - 7 \beta_{9} - 7 \beta_{8}) q^{98} + ( - \beta_{7} - \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{16} - 8 q^{18} - 8 q^{33} + 216 q^{50} + 32 q^{52} - 64 q^{67} + 72 q^{69} - 192 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1889x^{8} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 1165\nu ) / 1764 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 2929\nu^{2} ) / 7056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 724 ) / 441 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{10} - 7589\nu^{2} ) / 7056 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 2929\nu^{3} ) / 7056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{11} + 7589\nu^{3} ) / 28224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{12} + 2145\nu^{4} ) / 12544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -29\nu^{12} - 49661\nu^{4} ) / 112896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -29\nu^{13} - 49661\nu^{5} ) / 451584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{13} + 2145\nu^{5} ) / 12544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 65\nu^{14} + 126881\nu^{6} ) / 1806336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 181\nu^{14} + 325525\nu^{6} ) / 1806336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -181\nu^{15} - 325525\nu^{7} ) / 7225344 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\nu^{15} + 126881\nu^{7} ) / 1806336 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{9} + 29\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{11} + 36\beta_{10} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -65\beta_{13} + 181\beta_{12} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 181\beta_{15} + 260\beta_{14} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 441\beta_{4} - 724 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1764\beta_{2} - 1165\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2929\beta_{5} - 7589\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11716\beta_{7} - 7589\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -19305\beta_{9} - 49661\beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -49661\beta_{11} - 77220\beta_{10} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 126881\beta_{13} - 325525\beta_{12} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -325525\beta_{15} - 507524\beta_{14} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1
−1.44269 + 0.597580i
1.44269 − 0.597580i
2.36657 − 0.980264i
−2.36657 + 0.980264i
−1.44269 − 0.597580i
1.44269 + 0.597580i
2.36657 + 0.980264i
−2.36657 − 0.980264i
−0.980264 + 2.36657i
0.980264 − 2.36657i
0.597580 − 1.44269i
−0.597580 + 1.44269i
−0.980264 − 2.36657i
0.980264 + 2.36657i
0.597580 + 1.44269i
−0.597580 − 1.44269i
−1.10418 + 1.10418i −0.382683 + 0.923880i − 0.438447i −0.518807 − 0.214897i −0.597580 − 1.44269i 0 −1.72424 − 1.72424i −0.707107 − 0.707107i 0.810145 − 0.335573i
688.2 −1.10418 + 1.10418i 0.382683 − 0.923880i − 0.438447i 0.518807 + 0.214897i 0.597580 + 1.44269i 0 −1.72424 − 1.72424i −0.707107 − 0.707107i −0.810145 + 0.335573i
688.3 1.81129 − 1.81129i −0.382683 + 0.923880i − 4.56155i 3.29045 + 1.36295i 0.980264 + 2.36657i 0 −4.63972 − 4.63972i −0.707107 − 0.707107i 8.42865 − 3.49126i
688.4 1.81129 − 1.81129i 0.382683 − 0.923880i − 4.56155i −3.29045 − 1.36295i −0.980264 − 2.36657i 0 −4.63972 − 4.63972i −0.707107 − 0.707107i −8.42865 + 3.49126i
712.1 −1.10418 − 1.10418i −0.382683 − 0.923880i 0.438447i −0.518807 + 0.214897i −0.597580 + 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i 0.810145 + 0.335573i
712.2 −1.10418 − 1.10418i 0.382683 + 0.923880i 0.438447i 0.518807 − 0.214897i 0.597580 − 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i −0.810145 − 0.335573i
712.3 1.81129 + 1.81129i −0.382683 − 0.923880i 4.56155i 3.29045 − 1.36295i 0.980264 − 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i 8.42865 + 3.49126i
712.4 1.81129 + 1.81129i 0.382683 + 0.923880i 4.56155i −3.29045 + 1.36295i −0.980264 + 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i −8.42865 − 3.49126i
733.1 −1.81129 − 1.81129i −0.923880 + 0.382683i 4.56155i −1.36295 − 3.29045i 2.36657 + 0.980264i 0 4.63972 − 4.63972i 0.707107 − 0.707107i −3.49126 + 8.42865i
733.2 −1.81129 − 1.81129i 0.923880 − 0.382683i 4.56155i 1.36295 + 3.29045i −2.36657 − 0.980264i 0 4.63972 − 4.63972i 0.707107 − 0.707107i 3.49126 − 8.42865i
733.3 1.10418 + 1.10418i −0.923880 + 0.382683i 0.438447i 0.214897 + 0.518807i −1.44269 − 0.597580i 0 1.72424 − 1.72424i 0.707107 − 0.707107i −0.335573 + 0.810145i
733.4 1.10418 + 1.10418i 0.923880 − 0.382683i 0.438447i −0.214897 − 0.518807i 1.44269 + 0.597580i 0 1.72424 − 1.72424i 0.707107 − 0.707107i 0.335573 − 0.810145i
757.1 −1.81129 + 1.81129i −0.923880 − 0.382683i − 4.56155i −1.36295 + 3.29045i 2.36657 − 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i −3.49126 − 8.42865i
757.2 −1.81129 + 1.81129i 0.923880 + 0.382683i − 4.56155i 1.36295 − 3.29045i −2.36657 + 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i 3.49126 + 8.42865i
757.3 1.10418 − 1.10418i −0.923880 − 0.382683i − 0.438447i 0.214897 − 0.518807i −1.44269 + 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i −0.335573 − 0.810145i
757.4 1.10418 − 1.10418i 0.923880 + 0.382683i − 0.438447i −0.214897 + 0.518807i 1.44269 − 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i 0.335573 + 0.810145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 688.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.j 16
17.b even 2 1 inner 867.2.h.j 16
17.c even 4 2 inner 867.2.h.j 16
17.d even 8 4 inner 867.2.h.j 16
17.e odd 16 1 51.2.a.b ✓ 2
17.e odd 16 1 867.2.a.f 2
17.e odd 16 2 867.2.d.c 4
17.e odd 16 4 867.2.e.f 8
51.i even 16 1 153.2.a.e 2
51.i even 16 1 2601.2.a.t 2
68.i even 16 1 816.2.a.m 2
85.o even 16 1 1275.2.b.d 4
85.p odd 16 1 1275.2.a.n 2
85.r even 16 1 1275.2.b.d 4
119.p even 16 1 2499.2.a.o 2
136.q odd 16 1 3264.2.a.bl 2
136.s even 16 1 3264.2.a.bg 2
187.m even 16 1 6171.2.a.p 2
204.t odd 16 1 2448.2.a.v 2
221.y odd 16 1 8619.2.a.q 2
255.be even 16 1 3825.2.a.s 2
357.be odd 16 1 7497.2.a.v 2
408.bg odd 16 1 9792.2.a.cz 2
408.bm even 16 1 9792.2.a.cy 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b ✓ 2 17.e odd 16 1
153.2.a.e 2 51.i even 16 1
816.2.a.m 2 68.i even 16 1
867.2.a.f 2 17.e odd 16 1
867.2.d.c 4 17.e odd 16 2
867.2.e.f 8 17.e odd 16 4
867.2.h.j 16 1.a even 1 1 trivial
867.2.h.j 16 17.b even 2 1 inner
867.2.h.j 16 17.c even 4 2 inner
867.2.h.j 16 17.d even 8 4 inner
1275.2.a.n 2 85.p odd 16 1
1275.2.b.d 4 85.o even 16 1
1275.2.b.d 4 85.r even 16 1
2448.2.a.v 2 204.t odd 16 1
2499.2.a.o 2 119.p even 16 1
2601.2.a.t 2 51.i even 16 1
3264.2.a.bg 2 136.s even 16 1
3264.2.a.bl 2 136.q odd 16 1
3825.2.a.s 2 255.be even 16 1
6171.2.a.p 2 187.m even 16 1
7497.2.a.v 2 357.be odd 16 1
8619.2.a.q 2 221.y odd 16 1
9792.2.a.cy 2 408.bm even 16 1
9792.2.a.cz 2 408.bg odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{8} + 49T_{2}^{4} + 256 \) Copy content Toggle raw display
\( T_{5}^{16} + 25889T_{5}^{8} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 49 T^{4} + 256)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 25889 T^{8} + 256 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 1889 T^{8} + 65536 \) Copy content Toggle raw display
$13$ \( (T^{4} + 21 T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 3969 T^{4} + 1679616)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$29$ \( (T^{8} + 21381376)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$41$ \( T^{16} + 25889 T^{8} + 256 \) Copy content Toggle raw display
$43$ \( (T^{8} + 3969 T^{4} + 1679616)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 132 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 22816 T^{4} + 7311616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 2576 T^{4} + 4096)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 47988992 T^{8} + 16777216 \) Copy content Toggle raw display
$67$ \( (T + 4)^{16} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 281474976710656 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 53459728531456 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{8} + 6928 T^{4} + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 52 T^{2} + 64)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1099511627776 \) Copy content Toggle raw display
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