Properties

Label 693.2.i.i
Level $693$
Weight $2$
Character orbit 693.i
Analytic conductor $5.534$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(100,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.i (of order \(3\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.10423593216.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{7} + \beta_{3} + 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{7} + \beta_{3} + 2 \beta_{2} - 2) q^{8} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} - 2) q^{10} + (\beta_{4} - 1) q^{11} - \beta_{7} q^{13} + (\beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{14}+ \cdots + (4 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{4} + 4 q^{5} + 2 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 4 q^{4} + 4 q^{5} + 2 q^{7} - 24 q^{8} - 10 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} - 12 q^{16} + 2 q^{17} - 4 q^{22} + 4 q^{23} - 4 q^{25} - 4 q^{26} - 22 q^{28} - 16 q^{29} + 12 q^{31} + 26 q^{32} - 32 q^{34} + 2 q^{35} + 4 q^{37} + 8 q^{38} + 6 q^{40} - 4 q^{41} + 36 q^{43} - 4 q^{44} + 14 q^{46} + 12 q^{47} - 4 q^{49} - 4 q^{50} + 6 q^{52} - 12 q^{53} - 8 q^{55} - 48 q^{56} + 4 q^{58} + 12 q^{59} - 2 q^{61} + 52 q^{62} + 112 q^{64} - 4 q^{65} - 28 q^{67} - 48 q^{68} - 32 q^{70} - 24 q^{71} - 6 q^{73} - 16 q^{74} - 36 q^{76} - 4 q^{77} - 2 q^{79} + 16 q^{80} + 12 q^{82} + 24 q^{83} + 36 q^{85} + 36 q^{86} + 12 q^{88} + 8 q^{89} + 12 q^{91} + 32 q^{92} - 20 q^{94} + 34 q^{95} - 88 q^{97} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
−0.758290 + 1.31340i
−0.276205 + 0.478401i
0.643668 1.11487i
1.39083 2.40898i
−0.758290 1.31340i
−0.276205 0.478401i
0.643668 + 1.11487i
1.39083 + 2.40898i
−0.758290 + 1.31340i 0 −0.150007 0.259820i −1.16659 + 2.02059i 0 −2.28580 1.33233i −2.57816 0 −1.76922 3.06438i
100.2 −0.276205 + 0.478401i 0 0.847422 + 1.46778i 0.795012 1.37700i 0 0.886763 + 2.49272i −2.04107 0 0.439172 + 0.760669i
100.3 0.643668 1.11487i 0 0.171383 + 0.296844i 1.95872 3.39260i 0 −0.234193 2.63537i 3.01593 0 −2.52153 4.36742i
100.4 1.39083 2.40898i 0 −2.86880 4.96890i 0.412855 0.715087i 0 2.63323 0.257073i −10.3967 0 −1.14842 1.98912i
298.1 −0.758290 1.31340i 0 −0.150007 + 0.259820i −1.16659 2.02059i 0 −2.28580 + 1.33233i −2.57816 0 −1.76922 + 3.06438i
298.2 −0.276205 0.478401i 0 0.847422 1.46778i 0.795012 + 1.37700i 0 0.886763 2.49272i −2.04107 0 0.439172 0.760669i
298.3 0.643668 + 1.11487i 0 0.171383 0.296844i 1.95872 + 3.39260i 0 −0.234193 + 2.63537i 3.01593 0 −2.52153 + 4.36742i
298.4 1.39083 + 2.40898i 0 −2.86880 + 4.96890i 0.412855 + 0.715087i 0 2.63323 + 0.257073i −10.3967 0 −1.14842 + 1.98912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.i.i 8
3.b odd 2 1 231.2.i.e 8
7.c even 3 1 inner 693.2.i.i 8
7.c even 3 1 4851.2.a.bt 4
7.d odd 6 1 4851.2.a.bu 4
21.g even 6 1 1617.2.a.x 4
21.h odd 6 1 231.2.i.e 8
21.h odd 6 1 1617.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 3.b odd 2 1
231.2.i.e 8 21.h odd 6 1
693.2.i.i 8 1.a even 1 1 trivial
693.2.i.i 8 7.c even 3 1 inner
1617.2.a.x 4 21.g even 6 1
1617.2.a.z 4 21.h odd 6 1
4851.2.a.bt 4 7.c even 3 1
4851.2.a.bu 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 8T_{2}^{6} + 21T_{2}^{4} - 4T_{2}^{3} + 28T_{2}^{2} + 12T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} + 20T_{5}^{6} - 24T_{5}^{5} + 108T_{5}^{4} - 176T_{5}^{3} + 352T_{5}^{2} - 240T_{5} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 8 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 225 \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + \cdots + 7921 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots + 216225 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} + 12 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} + \cdots + 1948816 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} - 44 T^{2} + \cdots + 60)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 18 T^{3} + \cdots - 1385)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 39087504 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + \cdots + 62001 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{8} + 28 T^{7} + \cdots + 2131600 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots - 699)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} + \cdots + 294328336 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 150544 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 145154304 \) Copy content Toggle raw display
$97$ \( (T^{4} + 44 T^{3} + \cdots + 8501)^{2} \) Copy content Toggle raw display
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