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Properties

Label 867.2.e.j
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,2,Mod(616,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.616");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.e (of order \(4\), degree \(2\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.722204136308736.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{8} + 69x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{7}) q^{2} + \beta_{3} q^{3} + (\beta_{6} + 2 \beta_{2} - 1) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{11} - \beta_{10}) q^{6} + ( - \beta_{11} + \beta_{10} - \beta_{8}) q^{7} + (\beta_{9} + 2 \beta_{7} + 3 \beta_{4}) q^{8} - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{7}) q^{2} + \beta_{3} q^{3} + (\beta_{6} + 2 \beta_{2} - 1) q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{11} - \beta_{10}) q^{6} + ( - \beta_{11} + \beta_{10} - \beta_{8}) q^{7} + (\beta_{9} + 2 \beta_{7} + 3 \beta_{4}) q^{8} - \beta_{7} q^{9} + (\beta_{11} - 2 \beta_{8}) q^{10} + ( - 3 \beta_{11} + \beta_{10} + \beta_{8}) q^{11} + ( - \beta_{5} - \beta_{3} + 2 \beta_1) q^{12} + (\beta_{6} + \beta_{2} + 5) q^{13} + (\beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{14} + ( - \beta_{7} + \beta_{4}) q^{15} + ( - 3 \beta_{6} - 3 \beta_{2} + 1) q^{16} + (\beta_{2} - 1) q^{18} + ( - 3 \beta_{9} - \beta_{7} - 3 \beta_{4}) q^{19} + (2 \beta_{3} - \beta_1) q^{20} + ( - \beta_{6} - 2 \beta_{2} + 1) q^{21} + (\beta_{5} + 6 \beta_{3} - 2 \beta_1) q^{22} + (3 \beta_{11} + \beta_{10} - 2 \beta_{8}) q^{23} + (\beta_{11} + 2 \beta_{10} - 3 \beta_{8}) q^{24} + (\beta_{9} + 2 \beta_{7} + 2 \beta_{4}) q^{25} + (5 \beta_{9} - 4 \beta_{7} + 2 \beta_{4}) q^{26} - \beta_{10} q^{27} + (2 \beta_{11} - 7 \beta_{10} + 5 \beta_{8}) q^{28} + ( - 2 \beta_{5} + 2 \beta_{3} - 3 \beta_1) q^{29} + ( - 2 \beta_{6} - \beta_{2}) q^{30} + ( - \beta_{5} - 4 \beta_{3} - \beta_1) q^{31} + 3 \beta_{9} q^{32} + (\beta_{6} - 2 \beta_{2} + 1) q^{33} + (\beta_{6} + \beta_{2} - 2) q^{35} + ( - \beta_{9} + \beta_{7} + \beta_{4}) q^{36} + ( - 5 \beta_{5} + \beta_{3}) q^{37} + (3 \beta_{6} + 4 \beta_{2} + 2) q^{38} + (5 \beta_{3} + \beta_1) q^{39} + (4 \beta_{11} - 3 \beta_{10} - 2 \beta_{8}) q^{40} + ( - \beta_{11} - 6 \beta_{10} - \beta_{8}) q^{41} + (\beta_{9} - 4 \beta_{7} - 3 \beta_{4}) q^{42} + (8 \beta_{7} + 2 \beta_{4}) q^{43} + ( - 7 \beta_{10} + 5 \beta_{8}) q^{44} + ( - \beta_{10} - \beta_{8}) q^{45} + (\beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{46} + (5 \beta_{6} + \beta_{2}) q^{47} + (\beta_{3} - 3 \beta_1) q^{48} + ( - 2 \beta_{9} + 5 \beta_{4}) q^{49} + ( - 3 \beta_{6} - 5 \beta_{2} + 2) q^{50} + (3 \beta_{6} + 7 \beta_{2} - 2) q^{52} + ( - \beta_{9} - 8 \beta_{7}) q^{53} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{54} + (5 \beta_{6} + \beta_{2}) q^{55} + ( - 5 \beta_{5} - 8 \beta_{3} + 11 \beta_1) q^{56} + ( - 3 \beta_{11} - \beta_{10} + 3 \beta_{8}) q^{57} + (2 \beta_{11} - 3 \beta_{10} + 8 \beta_{8}) q^{58} + (3 \beta_{9} + 3 \beta_{7} - 3 \beta_{4}) q^{59} + ( - 2 \beta_{7} - \beta_{4}) q^{60} + (4 \beta_{11} + 3 \beta_{10} - 3 \beta_{8}) q^{61} + ( - 4 \beta_{11} + 4 \beta_{10} + 3 \beta_{8}) q^{62} + (\beta_{5} + \beta_{3} - 2 \beta_1) q^{63} + ( - 3 \beta_{6} - 3 \beta_{2} - 4) q^{64} + (\beta_{5} + 7 \beta_{3} + 5 \beta_1) q^{65} + (\beta_{9} - 6 \beta_{7} - \beta_{4}) q^{66} + ( - 4 \beta_{6} - 6) q^{67} + ( - 2 \beta_{6} + \beta_{2} + 1) q^{69} + ( - 2 \beta_{9} + 3 \beta_{7} + 2 \beta_{4}) q^{70} + (4 \beta_{5} - 5 \beta_{3} - 4 \beta_1) q^{71} + ( - 3 \beta_{6} - 2 \beta_{2} + 2) q^{72} + ( - \beta_{5} - 5 \beta_{3} - \beta_1) q^{73} + (\beta_{11} + 4 \beta_{10} + 5 \beta_{8}) q^{74} + (\beta_{11} + 2 \beta_{10} - 2 \beta_{8}) q^{75} + ( - 4 \beta_{9} + \beta_{7} + \beta_{4}) q^{76} + (7 \beta_{7} + 5 \beta_{4}) q^{77} + (5 \beta_{11} - 4 \beta_{10} - 2 \beta_{8}) q^{78} + (2 \beta_{11} - \beta_{10} - 4 \beta_{8}) q^{79} + ( - 3 \beta_{5} - 5 \beta_{3} + \beta_1) q^{80} - q^{81} + ( - 6 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{82} + (6 \beta_{9} - 3 \beta_{4}) q^{83} + (5 \beta_{6} + 7 \beta_{2} - 7) q^{84} + ( - 4 \beta_{6} - 12 \beta_{2} + 10) q^{86} + ( - 2 \beta_{9} - 2 \beta_{7} - 5 \beta_{4}) q^{87} + ( - 5 \beta_{5} + 13 \beta_1) q^{88} + ( - 2 \beta_{6} - 7 \beta_{2}) q^{89} + ( - \beta_{5} - \beta_1) q^{90} + ( - 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8}) q^{91} + (3 \beta_{11} + 2 \beta_{10} - \beta_{8}) q^{92} + ( - \beta_{9} + 4 \beta_{7} - 2 \beta_{4}) q^{93} + ( - 3 \beta_{7} + 6 \beta_{4}) q^{94} + ( - 6 \beta_{11} + 2 \beta_{10} + 5 \beta_{8}) q^{95} + 3 \beta_{11} q^{96} + (2 \beta_{5} + 5 \beta_{3}) q^{97} + ( - 12 \beta_{6} - 12 \beta_{2} + 9) q^{98} + (3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 60 q^{13} + 12 q^{16} - 12 q^{18} + 12 q^{21} + 12 q^{33} - 24 q^{35} + 24 q^{38} + 24 q^{50} - 24 q^{52} - 48 q^{64} - 72 q^{67} + 12 q^{69} + 24 q^{72} - 12 q^{81} - 84 q^{84} + 120 q^{86} + 108 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{8} + 69x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 27\nu^{4} + 100 ) / 53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 27\nu^{5} + 153\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} + 27\nu^{6} + 153\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} - 82\nu^{5} - 235\nu ) / 53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{8} - 55\nu^{4} - 82 ) / 53 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{10} - 109\nu^{6} - 441\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{11} - 109\nu^{7} - 441\nu^{3} ) / 53 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{10} - 218\nu^{6} - 829\nu^{2} ) / 53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -18\nu^{11} - 327\nu^{7} - 1270\nu^{3} ) / 53 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -35\nu^{11} - 627\nu^{7} - 2387\nu^{3} ) / 53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{9} + 19\beta_{7} + 6\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6\beta_{11} - 21\beta_{10} + 28\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -27\beta_{6} - 55\beta_{2} + 62 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -27\beta_{5} - 82\beta_{3} + 117\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 90\beta_{9} - 207\beta_{7} - 109\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -109\beta_{11} + 308\beta_{10} - 297\beta_{8} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
616.1
1.32893 − 1.32893i
−1.32893 + 1.32893i
−1.08335 + 1.08335i
1.08335 − 1.08335i
−0.245576 + 0.245576i
0.245576 − 0.245576i
−0.245576 − 0.245576i
0.245576 + 0.245576i
−1.08335 − 1.08335i
1.08335 + 1.08335i
1.32893 + 1.32893i
−1.32893 − 1.32893i
− 2.53209i −0.707107 + 0.707107i −4.41147 0.621819 − 0.621819i 1.79046 + 1.79046i −3.11938 − 3.11938i 6.10607i − 1.00000i −1.57450 − 1.57450i
616.2 − 2.53209i 0.707107 − 0.707107i −4.41147 −0.621819 + 0.621819i −1.79046 − 1.79046i 3.11938 + 3.11938i 6.10607i − 1.00000i 1.57450 + 1.57450i
616.3 − 1.34730i −0.707107 + 0.707107i 0.184793 −1.79046 + 1.79046i 0.952682 + 0.952682i 0.130668 + 0.130668i − 2.94356i − 1.00000i 2.41228 + 2.41228i
616.4 − 1.34730i 0.707107 − 0.707107i 0.184793 1.79046 − 1.79046i −0.952682 − 0.952682i −0.130668 − 0.130668i − 2.94356i − 1.00000i −2.41228 − 2.41228i
616.5 0.879385i −0.707107 + 0.707107i 1.22668 −0.952682 + 0.952682i −0.621819 − 0.621819i 0.867395 + 0.867395i 2.83750i − 1.00000i −0.837775 − 0.837775i
616.6 0.879385i 0.707107 − 0.707107i 1.22668 0.952682 − 0.952682i 0.621819 + 0.621819i −0.867395 − 0.867395i 2.83750i − 1.00000i 0.837775 + 0.837775i
829.1 − 0.879385i −0.707107 − 0.707107i 1.22668 −0.952682 − 0.952682i −0.621819 + 0.621819i 0.867395 − 0.867395i − 2.83750i 1.00000i −0.837775 + 0.837775i
829.2 − 0.879385i 0.707107 + 0.707107i 1.22668 0.952682 + 0.952682i 0.621819 − 0.621819i −0.867395 + 0.867395i − 2.83750i 1.00000i 0.837775 − 0.837775i
829.3 1.34730i −0.707107 − 0.707107i 0.184793 −1.79046 − 1.79046i 0.952682 − 0.952682i 0.130668 − 0.130668i 2.94356i 1.00000i 2.41228 − 2.41228i
829.4 1.34730i 0.707107 + 0.707107i 0.184793 1.79046 + 1.79046i −0.952682 + 0.952682i −0.130668 + 0.130668i 2.94356i 1.00000i −2.41228 + 2.41228i
829.5 2.53209i −0.707107 − 0.707107i −4.41147 0.621819 + 0.621819i 1.79046 − 1.79046i −3.11938 + 3.11938i − 6.10607i 1.00000i −1.57450 + 1.57450i
829.6 2.53209i 0.707107 + 0.707107i −4.41147 −0.621819 − 0.621819i −1.79046 + 1.79046i 3.11938 − 3.11938i − 6.10607i 1.00000i 1.57450 − 1.57450i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 616.6
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.e.j 12
17.b even 2 1 inner 867.2.e.j 12
17.c even 4 2 inner 867.2.e.j 12
17.d even 8 1 867.2.a.i ✓ 3
17.d even 8 1 867.2.a.j yes 3
17.d even 8 2 867.2.d.d 6
17.e odd 16 8 867.2.h.l 24
51.g odd 8 1 2601.2.a.y 3
51.g odd 8 1 2601.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.i ✓ 3 17.d even 8 1
867.2.a.j yes 3 17.d even 8 1
867.2.d.d 6 17.d even 8 2
867.2.e.j 12 1.a even 1 1 trivial
867.2.e.j 12 17.b even 2 1 inner
867.2.e.j 12 17.c even 4 2 inner
867.2.h.l 24 17.e odd 16 8
2601.2.a.y 3 51.g odd 8 1
2601.2.a.z 3 51.g odd 8 1

Hecke kernels

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

\( T_{2}^{6} + 9T_{2}^{4} + 18T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{12} + 45T_{5}^{8} + 162T_{5}^{4} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 9 T^{4} + 18 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 45 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{12} + 381 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} + 693 T^{8} + \cdots + 10556001 \) Copy content Toggle raw display
$13$ \( (T^{3} - 15 T^{2} + \cdots - 109)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 57 T^{4} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 1341 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 2058346161 \) Copy content Toggle raw display
$31$ \( T^{12} + 3090 T^{8} + \cdots + 1874161 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1568239201 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 547981281 \) Copy content Toggle raw display
$43$ \( (T^{6} + 216 T^{4} + \cdots + 179776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 63 T - 171)^{4} \) Copy content Toggle raw display
$53$ \( (T^{6} + 198 T^{4} + \cdots + 239121)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 189 T^{4} + \cdots + 210681)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 8181 T^{8} + \cdots + 25411681 \) Copy content Toggle raw display
$67$ \( (T^{3} + 18 T^{2} + 60 T - 8)^{4} \) Copy content Toggle raw display
$71$ \( T^{12} + 24723 T^{8} + \cdots + 6765201 \) Copy content Toggle raw display
$73$ \( T^{12} + 5277 T^{8} + \cdots + 62742241 \) Copy content Toggle raw display
$79$ \( T^{12} + 3891 T^{8} + \cdots + 1874161 \) Copy content Toggle raw display
$83$ \( (T^{6} + 378 T^{4} + \cdots + 210681)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 117 T - 153)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + 6243 T^{8} + \cdots + 28398241 \) Copy content Toggle raw display
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