Properties

Label 693.2.i.g
Level $693$
Weight $2$
Character orbit 693.i
Analytic conductor $5.534$
Analytic rank $0$
Dimension $6$
CM no
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{2} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{2} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 3) q^{8} + ( - \beta_{5} - 3 \beta_{4} + 3) q^{10} + (\beta_{4} - 1) q^{11} + ( - \beta_{2} - 4) q^{13} + (\beta_{5} + 3 \beta_{4} + \beta_{2} - \beta_1 - 3) q^{14} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{16} + ( - \beta_{5} + \beta_{4} - 1) q^{17} + (\beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{19} + (2 \beta_{3} - \beta_{2} - 5) q^{20} - \beta_{3} q^{22} + (\beta_{5} + 4 \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{4} + 3 \beta_1 - 2) q^{25} + (3 \beta_{5} - 3 \beta_{3} - \beta_{2} - \beta_1) q^{26} + (\beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{28} + (\beta_{3} - 3 \beta_{2} + 2) q^{29} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_1 + 2) q^{31} - 3 \beta_1 q^{32} + (\beta_{2} - 3) q^{34} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} - 3 \beta_{2} - 1) q^{35} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{37} + ( - 6 \beta_{5} - 3 \beta_{4} - \beta_1 + 3) q^{38} + (4 \beta_{5} - 4 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{40} + ( - 4 \beta_{3} + \beta_{2} + 2) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{43} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{44} + ( - 5 \beta_{5} - 3 \beta_{4} + \beta_1 + 3) q^{46} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{47} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_1 - 3) q^{49} + (\beta_{3} - 3 \beta_{2}) q^{50} + ( - 4 \beta_{5} - \beta_{4} + 4 \beta_1 + 1) q^{52} + (2 \beta_{5} - 6 \beta_{4} - \beta_1 + 6) q^{53} + (\beta_{3} + \beta_{2} + 1) q^{55} + ( - 4 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{56} + ( - 4 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{58} + (6 \beta_{5} - 3 \beta_{4} - \beta_1 + 3) q^{59} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3}) q^{61} + (4 \beta_{2} - 3) q^{62} + (\beta_{3} + \beta_{2} - 2) q^{64} + ( - 4 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{65} + ( - 6 \beta_{4} - 2 \beta_1 + 6) q^{67} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{68} + ( - 3 \beta_{4} + 4 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{70} + (3 \beta_{3} + 4 \beta_{2} - 1) q^{71} + ( - 5 \beta_{4} + 5 \beta_1 + 5) q^{73} + ( - 2 \beta_{5} - 6 \beta_{4} + 4 \beta_1 + 6) q^{74} + (6 \beta_{3} + 3 \beta_{2} - 12) q^{76} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{77} + ( - \beta_{5} + \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{79} + ( - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_1 + 2) q^{80} + ( - 5 \beta_{5} - 12 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{82} + (\beta_{3} - 2 \beta_{2} + 3) q^{83} + (\beta_{2} + 4) q^{85} + ( - 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{86} + (\beta_{5} + 3 \beta_{4} - 3) q^{88} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{89} + (2 \beta_{5} + 7 \beta_{4} - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 7) q^{91} + (7 \beta_{3} + 4 \beta_{2} - 7) q^{92} + ( - 3 \beta_{4} + \beta_1 + 3) q^{94} + (4 \beta_{5} + 6 \beta_{4} + \beta_1 - 6) q^{95} + (3 \beta_{2} + 4) q^{97} + (3 \beta_{5} - 3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 2 q^{5} + 2 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} - 2 q^{5} + 2 q^{7} + 18 q^{8} + 9 q^{10} - 3 q^{11} - 22 q^{13} - 12 q^{14} - 2 q^{16} - 3 q^{17} + 11 q^{19} - 28 q^{20} + 12 q^{23} - 3 q^{25} + q^{26} + 13 q^{28} + 18 q^{29} + 3 q^{31} - 3 q^{32} - 20 q^{34} - 9 q^{35} + 4 q^{37} + 8 q^{38} + 3 q^{40} + 10 q^{41} + 4 q^{43} - 4 q^{44} + 10 q^{46} - 3 q^{47} - 24 q^{49} + 6 q^{50} + 7 q^{52} + 17 q^{53} + 4 q^{55} - 3 q^{56} + 13 q^{58} + 8 q^{59} + 24 q^{61} - 26 q^{62} - 14 q^{64} + 15 q^{65} + 16 q^{67} + 5 q^{68} - 27 q^{70} - 14 q^{71} + 20 q^{73} + 22 q^{74} - 78 q^{76} + 2 q^{77} - 3 q^{79} + 9 q^{80} - 41 q^{82} + 22 q^{83} + 22 q^{85} - 21 q^{86} - 9 q^{88} + q^{89} - 15 q^{91} - 50 q^{92} + 10 q^{94} - 17 q^{95} + 18 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 12\beta_{4} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} + 3\beta_{4} + 6\beta_{3} - 17\beta_{2} - 17\beta_1 \) 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
0.356769 + 0.617942i
−0.956115 1.65604i
1.09935 + 1.90412i
0.356769 0.617942i
−0.956115 + 1.65604i
1.09935 1.90412i
−1.24543 + 2.15715i 0 −2.10220 3.64112i 1.10220 1.90907i 0 −1.10220 + 2.40523i 5.49086 0 2.74543 + 4.75523i
100.2 0.328310 0.568650i 0 0.784425 + 1.35866i −1.78442 + 3.09071i 0 1.78442 + 1.95341i 2.34338 0 1.17169 + 2.02943i
100.3 0.917122 1.58850i 0 −0.682224 1.18165i −0.317776 + 0.550404i 0 0.317776 2.62660i 1.16576 0 0.582878 + 1.00958i
298.1 −1.24543 2.15715i 0 −2.10220 + 3.64112i 1.10220 + 1.90907i 0 −1.10220 2.40523i 5.49086 0 2.74543 4.75523i
298.2 0.328310 + 0.568650i 0 0.784425 1.35866i −1.78442 3.09071i 0 1.78442 1.95341i 2.34338 0 1.17169 2.02943i
298.3 0.917122 + 1.58850i 0 −0.682224 + 1.18165i −0.317776 0.550404i 0 0.317776 + 2.62660i 1.16576 0 0.582878 1.00958i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.3
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.g 6
3.b odd 2 1 77.2.e.b 6
7.c even 3 1 inner 693.2.i.g 6
7.c even 3 1 4851.2.a.bo 3
7.d odd 6 1 4851.2.a.bn 3
12.b even 2 1 1232.2.q.k 6
21.c even 2 1 539.2.e.l 6
21.g even 6 1 539.2.a.i 3
21.g even 6 1 539.2.e.l 6
21.h odd 6 1 77.2.e.b 6
21.h odd 6 1 539.2.a.h 3
33.d even 2 1 847.2.e.d 6
33.f even 10 4 847.2.n.d 24
33.h odd 10 4 847.2.n.e 24
84.j odd 6 1 8624.2.a.ck 3
84.n even 6 1 1232.2.q.k 6
84.n even 6 1 8624.2.a.cl 3
231.k odd 6 1 5929.2.a.w 3
231.l even 6 1 847.2.e.d 6
231.l even 6 1 5929.2.a.v 3
231.z odd 30 4 847.2.n.e 24
231.be even 30 4 847.2.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 3.b odd 2 1
77.2.e.b 6 21.h odd 6 1
539.2.a.h 3 21.h odd 6 1
539.2.a.i 3 21.g even 6 1
539.2.e.l 6 21.c even 2 1
539.2.e.l 6 21.g even 6 1
693.2.i.g 6 1.a even 1 1 trivial
693.2.i.g 6 7.c even 3 1 inner
847.2.e.d 6 33.d even 2 1
847.2.e.d 6 231.l even 6 1
847.2.n.d 24 33.f even 10 4
847.2.n.d 24 231.be even 30 4
847.2.n.e 24 33.h odd 10 4
847.2.n.e 24 231.z odd 30 4
1232.2.q.k 6 12.b even 2 1
1232.2.q.k 6 84.n even 6 1
4851.2.a.bn 3 7.d odd 6 1
4851.2.a.bo 3 7.c even 3 1
5929.2.a.v 3 231.l even 6 1
5929.2.a.w 3 231.k odd 6 1
8624.2.a.ck 3 84.j odd 6 1
8624.2.a.cl 3 84.n even 6 1

Hecke kernels

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

\( T_{2}^{6} + 5T_{2}^{4} - 6T_{2}^{3} + 25T_{2}^{2} - 15T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} + 11T_{5}^{4} - 4T_{5}^{3} + 59T_{5}^{2} + 35T_{5} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{4} - 6 T^{3} + 25 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 11 T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + 14 T^{4} - 23 T^{3} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} + 11 T^{2} + 36 T + 35)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + 11 T^{4} + 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} - 11 T^{5} + 101 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + 101 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$29$ \( (T^{3} - 9 T^{2} - 20 T + 53)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + 53 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + 52 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$41$ \( (T^{3} - 5 T^{2} - 80 T + 109)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 2 T^{2} - 25 T + 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + 11 T^{4} + 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$53$ \( T^{6} - 17 T^{5} + 215 T^{4} + \cdots + 441 \) Copy content Toggle raw display
$59$ \( T^{6} - 8 T^{5} + 221 T^{4} + \cdots + 1750329 \) Copy content Toggle raw display
$61$ \( T^{6} - 24 T^{5} + 404 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + 188 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$71$ \( (T^{3} + 7 T^{2} - 86 T - 419)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 20 T^{5} + 375 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + 47 T^{4} + \cdots + 19881 \) Copy content Toggle raw display
$83$ \( (T^{3} - 11 T^{2} + 16 T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - T^{5} + 9 T^{4} + 2 T^{3} + 67 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$97$ \( (T^{3} - 9 T^{2} - 12 T + 47)^{2} \) Copy content Toggle raw display
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