Properties

Label 234.3.bb.b
Level $234$
Weight $3$
Character orbit 234.bb
Analytic conductor $6.376$
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] = [234,3,Mod(19,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 234.bb (of order \(12\), 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: \(6.37603818603\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (2 \zeta_{12}^{2} - 4) q^{10} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{11} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 1) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( - 13 \zeta_{12}^{2} + 6 \zeta_{12} - 13) q^{17} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{20} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + (21 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 21 \zeta_{12} - 4) q^{23} + 19 \zeta_{12}^{3} q^{25} + ( - 15 \zeta_{12}^{3} - \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{26} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + (16 \zeta_{12}^{3} + 27 \zeta_{12}^{2} + 16 \zeta_{12}) q^{29} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + (2 \zeta_{12}^{3} + 21 \zeta_{12}^{2} - \zeta_{12} - 21) q^{35} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + ( - 7 \zeta_{12}^{3} + 34 \zeta_{12}^{2} - 17) q^{38} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + (26 \zeta_{12}^{3} - 26 \zeta_{12}^{2} - 13 \zeta_{12} + 13) q^{41} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + (10 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 10) q^{44} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + ( - 33 \zeta_{12}^{3} - 33) q^{47} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + ( - 19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 19 \zeta_{12}) q^{50} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + (22 \zeta_{12}^{3} - 44 \zeta_{12} + 42) q^{53} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + (14 \zeta_{12}^{2} + 2 \zeta_{12} + 14) q^{56} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + (13 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{59} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + ( - 64 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 64 \zeta_{12} + 20) q^{62} + 8 \zeta_{12}^{3} q^{64} + (7 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 2 \zeta_{12} + 25) q^{65} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + ( - 26 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 26 \zeta_{12}) q^{68} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + ( - 25 \zeta_{12}^{3} - 29 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{71} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + (14 \zeta_{12}^{3} - 19 \zeta_{12}^{2} - 7 \zeta_{12} + 19) q^{74} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + ( - 15 \zeta_{12}^{3} - 64 \zeta_{12}^{2} + 32) q^{77} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 8) q^{80} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + (25 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 46 \zeta_{12} - 25) q^{83} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + (59 \zeta_{12}^{3} - 40 \zeta_{12}^{2} - 40 \zeta_{12} + 59) q^{86} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + (56 \zeta_{12}^{3} + 56 \zeta_{12}^{2} - 43 \zeta_{12} - 13) q^{89} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + (4 \zeta_{12}^{3} - 8 \zeta_{12} - 42) q^{92} + 66 \zeta_{12}^{2} q^{94} + (7 \zeta_{12}^{2} - 51 \zeta_{12} + 7) q^{95} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + (14 \zeta_{12}^{3} + 18 \zeta_{12}^{2} - 32 \zeta_{12} - 32) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 22 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 22 q^{7} - 8 q^{8} - 12 q^{10} + 6 q^{11} - 12 q^{13} + 4 q^{14} + 8 q^{16} - 78 q^{17} - 58 q^{19} + 12 q^{20} + 18 q^{22} - 12 q^{23} + 14 q^{26} - 44 q^{28} + 54 q^{29} - 128 q^{31} + 8 q^{32} - 24 q^{34} - 42 q^{35} + 40 q^{37} + 120 q^{43} - 36 q^{44} + 54 q^{46} - 132 q^{47} + 42 q^{49} - 38 q^{50} + 40 q^{52} + 168 q^{53} + 6 q^{55} + 84 q^{56} - 42 q^{58} - 6 q^{59} + 78 q^{61} + 60 q^{62} + 120 q^{65} + 86 q^{67} + 24 q^{68} + 84 q^{70} - 42 q^{71} - 136 q^{73} + 38 q^{74} + 116 q^{76} + 192 q^{79} + 24 q^{80} - 78 q^{82} - 192 q^{83} - 42 q^{85} + 156 q^{86} + 12 q^{88} + 60 q^{89} + 38 q^{91} - 168 q^{92} + 132 q^{94} + 42 q^{95} + 280 q^{97} - 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(\zeta_{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0.366025 + 1.36603i 0 −1.73205 + 1.00000i −1.73205 + 1.73205i 0 −2.03590 + 7.59808i −2.00000 2.00000i 0 −3.00000 1.73205i
37.1 0.366025 1.36603i 0 −1.73205 1.00000i −1.73205 1.73205i 0 −2.03590 7.59808i −2.00000 + 2.00000i 0 −3.00000 + 1.73205i
145.1 −1.36603 0.366025i 0 1.73205 + 1.00000i 1.73205 1.73205i 0 −8.96410 + 2.40192i −2.00000 2.00000i 0 −3.00000 + 1.73205i
163.1 −1.36603 + 0.366025i 0 1.73205 1.00000i 1.73205 + 1.73205i 0 −8.96410 2.40192i −2.00000 + 2.00000i 0 −3.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.3.bb.b 4
3.b odd 2 1 26.3.f.a 4
12.b even 2 1 208.3.bd.c 4
13.f odd 12 1 inner 234.3.bb.b 4
39.d odd 2 1 338.3.f.d 4
39.f even 4 1 338.3.f.c 4
39.f even 4 1 338.3.f.f 4
39.h odd 6 1 338.3.d.e 4
39.h odd 6 1 338.3.f.c 4
39.i odd 6 1 338.3.d.d 4
39.i odd 6 1 338.3.f.f 4
39.k even 12 1 26.3.f.a 4
39.k even 12 1 338.3.d.d 4
39.k even 12 1 338.3.d.e 4
39.k even 12 1 338.3.f.d 4
156.v odd 12 1 208.3.bd.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 3.b odd 2 1
26.3.f.a 4 39.k even 12 1
208.3.bd.c 4 12.b even 2 1
208.3.bd.c 4 156.v odd 12 1
234.3.bb.b 4 1.a even 1 1 trivial
234.3.bb.b 4 13.f odd 12 1 inner
338.3.d.d 4 39.i odd 6 1
338.3.d.d 4 39.k even 12 1
338.3.d.e 4 39.h odd 6 1
338.3.d.e 4 39.k even 12 1
338.3.f.c 4 39.f even 4 1
338.3.f.c 4 39.h odd 6 1
338.3.f.d 4 39.d odd 2 1
338.3.f.d 4 39.k even 12 1
338.3.f.f 4 39.f even 4 1
338.3.f.f 4 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 36 \) acting on \(S_{3}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} + 78 T^{3} + 2499 T^{2} + \cdots + 221841 \) Copy content Toggle raw display
$19$ \( T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} - 381 T^{2} + \cdots + 184041 \) Copy content Toggle raw display
$29$ \( T^{4} - 54 T^{3} + 2955 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{4} + 1521 T^{2} + 39546 T + 257049 \) Copy content Toggle raw display
$43$ \( T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041 \) Copy content Toggle raw display
$47$ \( (T^{2} + 66 T + 2178)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 84 T + 312)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 1305 T^{2} + \cdots + 84681 \) Copy content Toggle raw display
$61$ \( T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 \) Copy content Toggle raw display
$67$ \( T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401 \) Copy content Toggle raw display
$71$ \( T^{4} + 42 T^{3} + 3357 T^{2} + \cdots + 154449 \) Copy content Toggle raw display
$73$ \( T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844 \) Copy content Toggle raw display
$79$ \( (T^{2} - 96 T - 1584)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192 T^{3} + 18432 T^{2} + \cdots + 2056356 \) Copy content Toggle raw display
$89$ \( T^{4} - 60 T^{3} + 5661 T^{2} + \cdots + 21594609 \) Copy content Toggle raw display
$97$ \( T^{4} - 280 T^{3} + \cdots + 20043529 \) Copy content Toggle raw display
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