Properties

Label 693.2.m.d
Level $693$
Weight $2$
Character orbit 693.m
Analytic conductor $5.534$
Analytic rank $0$
Dimension $4$
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(64,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.m (of order \(5\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 0.587785i
0.500000 1.53884i 0 −0.500000 0.363271i −0.190983 0.587785i 0 0.809017 + 0.587785i 1.80902 1.31433i 0 −1.00000
190.1 0.500000 0.363271i 0 −0.500000 + 1.53884i −1.30902 0.951057i 0 −0.309017 + 0.951057i 0.690983 + 2.12663i 0 −1.00000
379.1 0.500000 + 1.53884i 0 −0.500000 + 0.363271i −0.190983 + 0.587785i 0 0.809017 0.587785i 1.80902 + 1.31433i 0 −1.00000
631.1 0.500000 + 0.363271i 0 −0.500000 1.53884i −1.30902 + 0.951057i 0 −0.309017 0.951057i 0.690983 2.12663i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.m.d 4
3.b odd 2 1 231.2.j.b 4
11.c even 5 1 inner 693.2.m.d 4
11.c even 5 1 7623.2.a.bo 2
11.d odd 10 1 7623.2.a.z 2
33.f even 10 1 2541.2.a.x 2
33.h odd 10 1 231.2.j.b 4
33.h odd 10 1 2541.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.b 4 3.b odd 2 1
231.2.j.b 4 33.h odd 10 1
693.2.m.d 4 1.a even 1 1 trivial
693.2.m.d 4 11.c even 5 1 inner
2541.2.a.p 2 33.h odd 10 1
2541.2.a.x 2 33.f even 10 1
7623.2.a.z 2 11.d odd 10 1
7623.2.a.bo 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 9 T^{3} + 41 T^{2} + 99 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 9 T^{3} + 61 T^{2} + 209 T + 361 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 64 T^{2} - 192 T + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} + 76 T^{2} + 434 T + 961 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T + 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + 106 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$53$ \( T^{4} + 25 T^{3} + 310 T^{2} + \cdots + 9025 \) Copy content Toggle raw display
$59$ \( T^{4} + T^{3} + 16 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$61$ \( T^{4} - 19 T^{3} + 241 T^{2} + \cdots + 6241 \) Copy content Toggle raw display
$67$ \( (T^{2} - 22 T + 116)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - T^{3} + 141 T^{2} + 1159 T + 3721 \) Copy content Toggle raw display
$73$ \( T^{4} + 11 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$79$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$83$ \( T^{4} - 14 T^{3} + 76 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$89$ \( (T^{2} - 13 T - 19)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} + 279 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
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