Properties

Label 234.3.bb.c
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 78)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{5} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{5} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{10} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 8 \zeta_{12} + 2) q^{11} + 13 \zeta_{12} q^{13} + (5 \zeta_{12}^{3} - 10 \zeta_{12} + 5) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{2} + 12 \zeta_{12} - 4) q^{17} + (12 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{19} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{20} + ( - 4 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 2 \zeta_{12} + 18) q^{22} + (18 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 18 \zeta_{12} - 16) q^{23} + (\zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{25} + (13 \zeta_{12}^{3} + 13) q^{26} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10) q^{28} + (10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{29} + ( - 29 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 29) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + (16 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{34} + (20 \zeta_{12}^{3} - 10 \zeta_{12}) q^{35} + (23 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 13 \zeta_{12} - 36) q^{37} + ( - 26 \zeta_{12}^{3} + 24 \zeta_{12}^{2} - 12) q^{38} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 12) q^{40} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 20 \zeta_{12} + 28) q^{41} + (5 \zeta_{12}^{2} + 24 \zeta_{12} + 5) q^{43} + ( - 20 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 20) q^{44} + (16 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 26 \zeta_{12} - 26) q^{46} + ( - 18 \zeta_{12}^{3} - 30 \zeta_{12}^{2} - 30 \zeta_{12} - 18) q^{47} + ( - \zeta_{12}^{3} + 25 \zeta_{12}^{2} + \zeta_{12} - 50) q^{49} + ( - 11 \zeta_{12}^{3} - 11 \zeta_{12}^{2} - 13 \zeta_{12} + 24) q^{50} + 26 \zeta_{12}^{2} q^{52} + (46 \zeta_{12}^{3} - 92 \zeta_{12} - 18) q^{53} + ( - 24 \zeta_{12}^{3} - 60 \zeta_{12}^{2} - 24 \zeta_{12}) q^{55} + ( - 10 \zeta_{12}^{2} + 10 \zeta_{12} - 10) q^{56} + (20 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{58} + ( - 2 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 44 \zeta_{12} - 44) q^{59} + (50 \zeta_{12}^{3} + 36 \zeta_{12}^{2} - 25 \zeta_{12} - 36) q^{61} + ( - 53 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 53 \zeta_{12} + 10) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( - 26 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 52 \zeta_{12} + 52) q^{65} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 31 \zeta_{12} - 43) q^{67} + ( - 8 \zeta_{12}^{3} + 24 \zeta_{12}^{2} - 8 \zeta_{12}) q^{68} + (10 \zeta_{12}^{3} + 20 \zeta_{12}^{2} - 20 \zeta_{12} - 10) q^{70} + ( - 22 \zeta_{12}^{3} + 46 \zeta_{12}^{2} + 68 \zeta_{12} - 68) q^{71} + ( - 59 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 23 \zeta_{12} - 59) q^{73} + (72 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 36 \zeta_{12} - 10) q^{74} + ( - 14 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 38 \zeta_{12} - 24) q^{76} + ( - 30 \zeta_{12}^{3} + 80 \zeta_{12}^{2} - 40) q^{77} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 24) q^{79} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 16 \zeta_{12} + 8) q^{80} + (28 \zeta_{12}^{2} + 12 \zeta_{12} + 28) q^{82} + (40 \zeta_{12}^{3} - 52 \zeta_{12}^{2} + 52 \zeta_{12} - 40) q^{83} + ( - 24 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{85} + (19 \zeta_{12}^{3} + 10 \zeta_{12}^{2} + 10 \zeta_{12} + 19) q^{86} + ( - 36 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 36 \zeta_{12} + 8) q^{88} + ( - 64 \zeta_{12}^{3} - 64 \zeta_{12}^{2} + 104 \zeta_{12} - 40) q^{89} + ( - 65 \zeta_{12}^{3} + 65 \zeta_{12}^{2} - 65) q^{91} + (16 \zeta_{12}^{3} - 32 \zeta_{12} - 36) q^{92} + ( - 30 \zeta_{12}^{3} - 66 \zeta_{12}^{2} - 30 \zeta_{12}) q^{94} + (10 \zeta_{12}^{2} - 42 \zeta_{12} + 10) q^{95} + (71 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 84 \zeta_{12} + 84) q^{97} + (50 \zeta_{12}^{3} - 26 \zeta_{12}^{2} - 24 \zeta_{12} - 24) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 12 q^{5} - 10 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 12 q^{5} - 10 q^{7} + 8 q^{8} + 12 q^{10} - 12 q^{11} + 20 q^{14} + 8 q^{16} - 24 q^{17} + 62 q^{19} + 24 q^{20} + 36 q^{22} - 48 q^{23} + 52 q^{26} - 20 q^{28} + 106 q^{31} - 8 q^{32} + 48 q^{34} - 98 q^{37} + 48 q^{40} + 96 q^{41} + 30 q^{43} + 72 q^{44} - 84 q^{46} - 132 q^{47} - 150 q^{49} + 74 q^{50} + 52 q^{52} - 72 q^{53} - 120 q^{55} - 60 q^{56} + 60 q^{58} - 84 q^{59} - 72 q^{61} + 30 q^{62} + 156 q^{65} - 148 q^{67} + 48 q^{68} - 180 q^{71} - 190 q^{73} - 20 q^{74} - 124 q^{76} + 96 q^{79} + 48 q^{80} + 168 q^{82} - 264 q^{83} + 48 q^{85} + 96 q^{86} + 24 q^{88} - 288 q^{89} - 130 q^{91} - 144 q^{92} - 132 q^{94} + 60 q^{95} + 310 q^{97} - 148 q^{98}+O(q^{100}) \) 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 1.26795 1.26795i 0 −2.50000 + 9.33013i 2.00000 + 2.00000i 0 −2.19615 1.26795i
37.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.26795 + 1.26795i 0 −2.50000 9.33013i 2.00000 2.00000i 0 −2.19615 + 1.26795i
145.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i 4.73205 4.73205i 0 −2.50000 + 0.669873i 2.00000 + 2.00000i 0 8.19615 4.73205i
163.1 1.36603 0.366025i 0 1.73205 1.00000i 4.73205 + 4.73205i 0 −2.50000 0.669873i 2.00000 2.00000i 0 8.19615 + 4.73205i
\(n\): 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.c 4
3.b odd 2 1 78.3.l.a 4
13.f odd 12 1 inner 234.3.bb.c 4
39.h odd 6 1 1014.3.f.d 4
39.i odd 6 1 1014.3.f.e 4
39.k even 12 1 78.3.l.a 4
39.k even 12 1 1014.3.f.d 4
39.k even 12 1 1014.3.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.a 4 3.b odd 2 1
78.3.l.a 4 39.k even 12 1
234.3.bb.c 4 1.a even 1 1 trivial
234.3.bb.c 4 13.f odd 12 1 inner
1014.3.f.d 4 39.h odd 6 1
1014.3.f.d 4 39.k even 12 1
1014.3.f.e 4 39.i odd 6 1
1014.3.f.e 4 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 12T_{5}^{3} + 72T_{5}^{2} - 144T_{5} + 144 \) 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
$5$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 180 T^{2} + \cdots + 24336 \) Copy content Toggle raw display
$13$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + 96 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$19$ \( T^{4} - 62 T^{3} + 986 T^{2} + \cdots + 14884 \) Copy content Toggle raw display
$23$ \( T^{4} + 48 T^{3} + 636 T^{2} + \cdots + 17424 \) Copy content Toggle raw display
$29$ \( T^{4} + 300 T^{2} + 90000 \) Copy content Toggle raw display
$31$ \( T^{4} - 106 T^{3} + 5618 T^{2} + \cdots + 1868689 \) Copy content Toggle raw display
$37$ \( T^{4} + 98 T^{3} + 5882 T^{2} + \cdots + 3587236 \) Copy content Toggle raw display
$41$ \( T^{4} - 96 T^{3} + 3600 T^{2} + \cdots + 1218816 \) Copy content Toggle raw display
$43$ \( T^{4} - 30 T^{3} - 201 T^{2} + \cdots + 251001 \) Copy content Toggle raw display
$47$ \( T^{4} + 132 T^{3} + 8712 T^{2} + \cdots + 685584 \) Copy content Toggle raw display
$53$ \( (T^{2} + 36 T - 6024)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 84 T^{3} + 4068 T^{2} + \cdots + 16353936 \) Copy content Toggle raw display
$61$ \( T^{4} + 72 T^{3} + 5763 T^{2} + \cdots + 335241 \) Copy content Toggle raw display
$67$ \( T^{4} + 148 T^{3} + 8501 T^{2} + \cdots + 6723649 \) Copy content Toggle raw display
$71$ \( T^{4} + 180 T^{3} + \cdots + 33315984 \) Copy content Toggle raw display
$73$ \( T^{4} + 190 T^{3} + \cdots + 13830961 \) Copy content Toggle raw display
$79$ \( (T^{2} - 48 T + 429)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 264 T^{3} + \cdots + 21678336 \) Copy content Toggle raw display
$89$ \( T^{4} + 288 T^{3} + \cdots + 137170944 \) Copy content Toggle raw display
$97$ \( T^{4} - 310 T^{3} + 27389 T^{2} + \cdots + 8162449 \) Copy content Toggle raw display
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