Properties

Label 261.2.k.a
Level $261$
Weight $2$
Character orbit 261.k
Analytic conductor $2.084$
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] = [261,2,Mod(82,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.k (of order \(7\), degree \(6\), 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} - \zeta_{14} + 1) q^{2} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{4} + (2 \zeta_{14}^{5} + 2 \zeta_{14}^{3} + \zeta_{14} - 1) q^{5} + ( - \zeta_{14}^{5} + 2 \zeta_{14}^{4} - 2 \zeta_{14} + 1) q^{7} + ( - \zeta_{14}^{5} - 2 \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} - \zeta_{14} + 1) q^{2} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{4} + (2 \zeta_{14}^{5} + 2 \zeta_{14}^{3} + \zeta_{14} - 1) q^{5} + ( - \zeta_{14}^{5} + 2 \zeta_{14}^{4} - 2 \zeta_{14} + 1) q^{7} + ( - \zeta_{14}^{5} - 2 \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14} + 2) q^{8} + ( - \zeta_{14}^{5} + \zeta_{14}^{4} + 2 \zeta_{14}^{2} + \zeta_{14} + 2) q^{10} + (\zeta_{14}^{4} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{2} + 2 \zeta_{14} + 1) q^{11} + ( - 3 \zeta_{14}^{4} + 4 \zeta_{14}^{3} - 2 \zeta_{14}^{2} + 4 \zeta_{14} - 3) q^{13} + ( - 3 \zeta_{14}^{5} + 2 \zeta_{14}^{4} + \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 3 \zeta_{14}) q^{14} + (2 \zeta_{14}^{4} - 3 \zeta_{14}^{3} - 3 \zeta_{14} + 2) q^{16} + ( - 2 \zeta_{14}^{5} + 2 \zeta_{14}^{4} - 2 \zeta_{14}^{3} + 2 \zeta_{14}^{2}) q^{17} + ( - 2 \zeta_{14}^{5} + 2 \zeta_{14}^{4} + \zeta_{14}^{2} + 1) q^{19} + (\zeta_{14}^{5} + \zeta_{14}^{3} - \zeta_{14}^{2} + \zeta_{14} - 1) q^{20} + ( - 4 \zeta_{14}^{5} + 4 \zeta_{14}^{4} - 3 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 4 \zeta_{14} + 4) q^{22} + (\zeta_{14}^{5} - 3 \zeta_{14}^{4} - \zeta_{14}^{3} - 3 \zeta_{14}^{2} + \zeta_{14}) q^{23} + (4 \zeta_{14}^{5} - 4 \zeta_{14}^{4} - 7 \zeta_{14}^{2} + 3 \zeta_{14} - 7) q^{25} + (2 \zeta_{14}^{5} - 4 \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} + 4 \zeta_{14} - 2) q^{26} + ( - \zeta_{14}^{5} - \zeta_{14}^{4} + \zeta_{14}^{3} + \zeta_{14}^{2} - 2) q^{28} + (2 \zeta_{14}^{5} - 6 \zeta_{14}^{4} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{2} - 3) q^{29} + (2 \zeta_{14}^{5} - 6 \zeta_{14}^{4} + 2 \zeta_{14}^{3} + \zeta_{14} - 1) q^{31} + ( - 3 \zeta_{14}^{5} + 2 \zeta_{14}^{4} + \zeta_{14}^{3} - \zeta_{14}^{2} - 2 \zeta_{14} + 3) q^{32} - 2 \zeta_{14}^{4} q^{34} + ( - \zeta_{14}^{4} - \zeta_{14}^{3} - \zeta_{14}^{2}) q^{35} + ( - 3 \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 2 \zeta_{14} + 3) q^{37} + ( - 3 \zeta_{14}^{5} - \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 2 \zeta_{14} + 1) q^{38} + (3 \zeta_{14}^{4} + 2 \zeta_{14}^{3} + 5 \zeta_{14}^{2} + 2 \zeta_{14} + 3) q^{40} + (2 \zeta_{14}^{5} - 2 \zeta_{14}^{2} - 4) q^{41} + (3 \zeta_{14}^{5} - 3 \zeta_{14}^{4} + \zeta_{14}^{3} - 3 \zeta_{14}^{2} + 3 \zeta_{14}) q^{43} + ( - 3 \zeta_{14}^{5} + 5 \zeta_{14}^{4} - 4 \zeta_{14}^{3} + 5 \zeta_{14}^{2} - 3 \zeta_{14}) q^{44} + (4 \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} - 4 \zeta_{14}^{2} - 1) q^{46} + ( - \zeta_{14}^{4} + 6 \zeta_{14}^{2} - 1) q^{47} + (\zeta_{14}^{5} + 3 \zeta_{14}^{3} + 4 \zeta_{14}^{2} - 4 \zeta_{14} - 3) q^{49} + (11 \zeta_{14}^{5} + 4 \zeta_{14}^{3} - 4 \zeta_{14}^{2} + 4 \zeta_{14} - 4) q^{50} + (\zeta_{14}^{5} + 3 \zeta_{14}^{4} - 6 \zeta_{14}^{3} + 3 \zeta_{14}^{2} + \zeta_{14}) q^{52} + ( - 2 \zeta_{14}^{5} + 4 \zeta_{14}^{4} - 2 \zeta_{14}^{3} - \zeta_{14} + 1) q^{53} + (7 \zeta_{14}^{5} - 3 \zeta_{14}^{4} + 6 \zeta_{14}^{3} - 6 \zeta_{14}^{2} + 3 \zeta_{14} - 7) q^{55} + (3 \zeta_{14}^{4} - 2 \zeta_{14} + 2) q^{56} + (7 \zeta_{14}^{5} - 5 \zeta_{14}^{4} + 3 \zeta_{14}^{3} - 4 \zeta_{14}^{2} + 7 \zeta_{14} - 7) q^{58} + (2 \zeta_{14}^{5} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{3} - 2 \zeta_{14}^{2} + 8) q^{59} + ( - 3 \zeta_{14}^{5} + 4 \zeta_{14}^{4} - 2 \zeta_{14}^{3} + 2 \zeta_{14}^{2} - 4 \zeta_{14} + 3) q^{61} + (5 \zeta_{14}^{5} - 5 \zeta_{14}^{4} - 4 \zeta_{14}^{2} + 7 \zeta_{14} - 4) q^{62} + ( - 2 \zeta_{14}^{5} - 3 \zeta_{14}^{4} - \zeta_{14}^{3} - 3 \zeta_{14}^{2} - 2 \zeta_{14}) q^{64} + (3 \zeta_{14}^{5} - \zeta_{14}^{4} + 4 \zeta_{14}^{3} - 4 \zeta_{14}^{2} + \zeta_{14} - 3) q^{65} + ( - 8 \zeta_{14}^{5} - 7 \zeta_{14}^{3} + 4 \zeta_{14}^{2} - 4 \zeta_{14} + 7) q^{67} + (2 \zeta_{14}^{5} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{2} - 2 \zeta_{14} + 2) q^{68} + (2 \zeta_{14}^{5} - 2 \zeta_{14}^{2} - 1) q^{70} + ( - 3 \zeta_{14}^{4} + 6 \zeta_{14}^{2} - 3) q^{71} + ( - 7 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 5 \zeta_{14}^{3} + 3 \zeta_{14}^{2} - 7 \zeta_{14}) q^{73} + (\zeta_{14}^{4} - \zeta_{14}^{3} - 3 \zeta_{14}^{2} - \zeta_{14} + 1) q^{74} + ( - \zeta_{14}^{3} + 3 \zeta_{14}^{2} - \zeta_{14}) q^{76} + ( - 5 \zeta_{14}^{5} + 5 \zeta_{14}^{4} + \zeta_{14}^{2} - 6 \zeta_{14} + 1) q^{77} + (6 \zeta_{14}^{5} + 3 \zeta_{14}^{3} - 3) q^{79} + ( - 6 \zeta_{14}^{5} + \zeta_{14}^{4} - 5 \zeta_{14}^{3} + 5 \zeta_{14}^{2} - \zeta_{14} + 6) q^{80} + (4 \zeta_{14}^{5} + 4 \zeta_{14}^{3} + 2 \zeta_{14} - 2) q^{82} + ( - 4 \zeta_{14}^{5} + 4 \zeta_{14}^{4} - \zeta_{14}^{2} - 4 \zeta_{14} - 1) q^{83} + (2 \zeta_{14}^{5} + 2 \zeta_{14}^{4} + 2 \zeta_{14}^{3} + 2 \zeta_{14} - 2) q^{85} + (2 \zeta_{14}^{5} + 3 \zeta_{14}^{4} - 3 \zeta_{14}^{3} - 2 \zeta_{14}^{2} + 3) q^{86} + ( - \zeta_{14}^{5} - \zeta_{14}^{4} + \zeta_{14}^{3} + \zeta_{14}^{2} + 7) q^{88} + ( - 6 \zeta_{14}^{5} - 6 \zeta_{14}^{3} - \zeta_{14} + 1) q^{89} + (9 \zeta_{14}^{5} - 9 \zeta_{14}^{4} - 5 \zeta_{14}^{2} + 12 \zeta_{14} - 5) q^{91} + ( - 3 \zeta_{14}^{5} + 4 \zeta_{14}^{4} - 3 \zeta_{14}^{3} - 2 \zeta_{14} + 2) q^{92} + (2 \zeta_{14}^{5} - 8 \zeta_{14}^{4} + \zeta_{14}^{3} - \zeta_{14}^{2} + 8 \zeta_{14} - 2) q^{94} + (6 \zeta_{14}^{5} + 5 \zeta_{14}^{3} - 3 \zeta_{14}^{2} + 3 \zeta_{14} - 5) q^{95} + ( - 8 \zeta_{14}^{5} + 8 \zeta_{14}^{4} + 5 \zeta_{14}^{2} - 8 \zeta_{14} + 5) q^{97} + ( - 6 \zeta_{14}^{4} + 7 \zeta_{14}^{3} + 3 \zeta_{14}^{2} + 7 \zeta_{14} - 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 2 q^{4} - q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 2 q^{4} - q^{5} + q^{7} + 7 q^{8} + 9 q^{10} + 11 q^{11} - 5 q^{13} - 9 q^{14} + 4 q^{16} - 8 q^{17} + q^{19} - 2 q^{20} + 6 q^{22} + 7 q^{23} - 24 q^{25} - 4 q^{26} - 12 q^{28} - 6 q^{29} + 5 q^{31} + 13 q^{32} + 2 q^{34} + q^{35} + 11 q^{37} - 2 q^{38} + 14 q^{40} - 20 q^{41} + 13 q^{43} - 20 q^{44} - 11 q^{47} - 22 q^{49} - q^{50} - 10 q^{52} - 3 q^{53} - 17 q^{55} + 7 q^{56} - 16 q^{58} + 56 q^{59} + 3 q^{61} - 3 q^{62} + q^{64} - 5 q^{65} + 19 q^{67} + 12 q^{68} - 2 q^{70} - 21 q^{71} - 25 q^{73} + 6 q^{74} - 5 q^{76} - 11 q^{77} - 9 q^{79} + 18 q^{80} - 2 q^{82} - 17 q^{83} - 8 q^{85} + 16 q^{86} + 42 q^{88} - 7 q^{89} + 5 q^{91} + 8 q^{94} - 13 q^{95} + q^{97} - 19 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(-\zeta_{14}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
0.222521 0.974928i
−0.623490 + 0.781831i
0.900969 0.433884i
−0.623490 0.781831i
0.900969 + 0.433884i
0.222521 + 0.974928i
1.12349 + 1.40881i 0 −0.277479 + 1.21572i −0.222521 0.279032i 0 0.900969 + 3.94740i 1.22252 0.588735i 0 0.143104 0.626980i
136.1 −0.400969 + 0.193096i 0 −1.12349 + 1.40881i 0.623490 0.300257i 0 0.222521 + 0.279032i 0.376510 1.64960i 0 −0.192021 + 0.240787i
181.1 0.277479 + 1.21572i 0 0.400969 0.193096i −0.900969 3.94740i 0 −0.623490 0.300257i 1.90097 + 2.38374i 0 4.54892 2.19064i
190.1 −0.400969 0.193096i 0 −1.12349 1.40881i 0.623490 + 0.300257i 0 0.222521 0.279032i 0.376510 + 1.64960i 0 −0.192021 0.240787i
199.1 0.277479 1.21572i 0 0.400969 + 0.193096i −0.900969 + 3.94740i 0 −0.623490 + 0.300257i 1.90097 2.38374i 0 4.54892 + 2.19064i
226.1 1.12349 1.40881i 0 −0.277479 1.21572i −0.222521 + 0.279032i 0 0.900969 3.94740i 1.22252 + 0.588735i 0 0.143104 + 0.626980i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.k.a 6
3.b odd 2 1 29.2.d.a 6
12.b even 2 1 464.2.u.f 6
15.d odd 2 1 725.2.l.b 6
15.e even 4 2 725.2.r.b 12
29.d even 7 1 inner 261.2.k.a 6
29.d even 7 1 7569.2.a.r 3
29.e even 14 1 7569.2.a.p 3
87.d odd 2 1 841.2.d.d 6
87.f even 4 2 841.2.e.d 12
87.h odd 14 1 841.2.a.f 3
87.h odd 14 2 841.2.d.a 6
87.h odd 14 2 841.2.d.c 6
87.h odd 14 1 841.2.d.d 6
87.j odd 14 1 29.2.d.a 6
87.j odd 14 1 841.2.a.e 3
87.j odd 14 2 841.2.d.b 6
87.j odd 14 2 841.2.d.e 6
87.k even 28 2 841.2.b.c 6
87.k even 28 4 841.2.e.b 12
87.k even 28 4 841.2.e.c 12
87.k even 28 2 841.2.e.d 12
348.s even 14 1 464.2.u.f 6
435.w odd 14 1 725.2.l.b 6
435.bj even 28 2 725.2.r.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.d.a 6 3.b odd 2 1
29.2.d.a 6 87.j odd 14 1
261.2.k.a 6 1.a even 1 1 trivial
261.2.k.a 6 29.d even 7 1 inner
464.2.u.f 6 12.b even 2 1
464.2.u.f 6 348.s even 14 1
725.2.l.b 6 15.d odd 2 1
725.2.l.b 6 435.w odd 14 1
725.2.r.b 12 15.e even 4 2
725.2.r.b 12 435.bj even 28 2
841.2.a.e 3 87.j odd 14 1
841.2.a.f 3 87.h odd 14 1
841.2.b.c 6 87.k even 28 2
841.2.d.a 6 87.h odd 14 2
841.2.d.b 6 87.j odd 14 2
841.2.d.c 6 87.h odd 14 2
841.2.d.d 6 87.d odd 2 1
841.2.d.d 6 87.h odd 14 1
841.2.d.e 6 87.j odd 14 2
841.2.e.b 12 87.k even 28 4
841.2.e.c 12 87.k even 28 4
841.2.e.d 12 87.f even 4 2
841.2.e.d 12 87.k even 28 2
7569.2.a.p 3 29.e even 14 1
7569.2.a.r 3 29.d even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2T_{2}^{5} + 4T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 15 T^{4} - 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} + 15 T^{4} + 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 11 T^{5} + 79 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$13$ \( T^{6} + 5 T^{5} + 25 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - T^{5} + T^{4} - 15 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{6} - 7 T^{5} + 49 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} - 13 T^{4} + \cdots + 24389 \) Copy content Toggle raw display
$31$ \( T^{6} - 5 T^{5} - 3 T^{4} - 139 T^{3} + \cdots + 6889 \) Copy content Toggle raw display
$37$ \( T^{6} - 11 T^{5} + 79 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} + 24 T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 13 T^{5} + 85 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( T^{6} + 11 T^{5} + 65 T^{4} + \cdots + 57121 \) Copy content Toggle raw display
$53$ \( T^{6} + 3 T^{5} - 5 T^{4} + 41 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{3} - 28 T^{2} + 252 T - 728)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + 37 T^{4} - 13 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{6} - 19 T^{5} + 235 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{6} + 21 T^{5} + 189 T^{4} + \cdots + 35721 \) Copy content Toggle raw display
$73$ \( T^{6} + 25 T^{5} + 373 T^{4} + \cdots + 175561 \) Copy content Toggle raw display
$79$ \( T^{6} + 9 T^{5} + 81 T^{4} - 27 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$83$ \( T^{6} + 17 T^{5} + 219 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$89$ \( T^{6} + 7 T^{5} + 119 T^{4} + \cdots + 8281 \) Copy content Toggle raw display
$97$ \( T^{6} - T^{5} + 211 T^{4} - 1849 T^{3} + \cdots + 169 \) Copy content Toggle raw display
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