Properties

Label 234.6.b.b
Level $234$
Weight $6$
Character orbit 234.b
Analytic conductor $37.530$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,6,Mod(181,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.181");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 234.b (of order \(2\), degree \(1\), 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: \(37.5298138362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 i q^{2} - 16 q^{4} + 51 i q^{5} - 105 i q^{7} + 64 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 i q^{2} - 16 q^{4} + 51 i q^{5} - 105 i q^{7} + 64 i q^{8} + 204 q^{10} + 120 i q^{11} + ( - 117 i - 598) q^{13} - 420 q^{14} + 256 q^{16} + 1101 q^{17} + 1170 i q^{19} - 816 i q^{20} + 480 q^{22} - 1050 q^{23} + 524 q^{25} + (2392 i - 468) q^{26} + 1680 i q^{28} + 4104 q^{29} - 9624 i q^{31} - 1024 i q^{32} - 4404 i q^{34} + 5355 q^{35} - 8709 i q^{37} + 4680 q^{38} - 3264 q^{40} - 9480 i q^{41} + 9995 q^{43} - 1920 i q^{44} + 4200 i q^{46} - 2943 i q^{47} + 5782 q^{49} - 2096 i q^{50} + (1872 i + 9568) q^{52} + 750 q^{53} - 6120 q^{55} + 6720 q^{56} - 16416 i q^{58} - 40938 i q^{59} - 57920 q^{61} - 38496 q^{62} - 4096 q^{64} + ( - 30498 i + 5967) q^{65} - 22812 i q^{67} - 17616 q^{68} - 21420 i q^{70} + 63741 i q^{71} - 58866 i q^{73} - 34836 q^{74} - 18720 i q^{76} + 12600 q^{77} + 63202 q^{79} + 13056 i q^{80} - 37920 q^{82} + 55458 i q^{83} + 56151 i q^{85} - 39980 i q^{86} - 7680 q^{88} - 104778 i q^{89} + (62790 i - 12285) q^{91} + 16800 q^{92} - 11772 q^{94} - 59670 q^{95} - 160452 i q^{97} - 23128 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 408 q^{10} - 1196 q^{13} - 840 q^{14} + 512 q^{16} + 2202 q^{17} + 960 q^{22} - 2100 q^{23} + 1048 q^{25} - 936 q^{26} + 8208 q^{29} + 10710 q^{35} + 9360 q^{38} - 6528 q^{40} + 19990 q^{43} + 11564 q^{49} + 19136 q^{52} + 1500 q^{53} - 12240 q^{55} + 13440 q^{56} - 115840 q^{61} - 76992 q^{62} - 8192 q^{64} + 11934 q^{65} - 35232 q^{68} - 69672 q^{74} + 25200 q^{77} + 126404 q^{79} - 75840 q^{82} - 15360 q^{88} - 24570 q^{91} + 33600 q^{92} - 23544 q^{94} - 119340 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
4.00000i 0 −16.0000 51.0000i 0 105.000i 64.0000i 0 204.000
181.2 4.00000i 0 −16.0000 51.0000i 0 105.000i 64.0000i 0 204.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.b.b 2
3.b odd 2 1 26.6.b.a 2
12.b even 2 1 208.6.f.b 2
13.b even 2 1 inner 234.6.b.b 2
39.d odd 2 1 26.6.b.a 2
39.f even 4 1 338.6.a.a 1
39.f even 4 1 338.6.a.c 1
156.h even 2 1 208.6.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 3.b odd 2 1
26.6.b.a 2 39.d odd 2 1
208.6.f.b 2 12.b even 2 1
208.6.f.b 2 156.h even 2 1
234.6.b.b 2 1.a even 1 1 trivial
234.6.b.b 2 13.b even 2 1 inner
338.6.a.a 1 39.f even 4 1
338.6.a.c 1 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2601 \) acting on \(S_{6}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2601 \) Copy content Toggle raw display
$7$ \( T^{2} + 11025 \) Copy content Toggle raw display
$11$ \( T^{2} + 14400 \) Copy content Toggle raw display
$13$ \( T^{2} + 1196 T + 371293 \) Copy content Toggle raw display
$17$ \( (T - 1101)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1368900 \) Copy content Toggle raw display
$23$ \( (T + 1050)^{2} \) Copy content Toggle raw display
$29$ \( (T - 4104)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 92621376 \) Copy content Toggle raw display
$37$ \( T^{2} + 75846681 \) Copy content Toggle raw display
$41$ \( T^{2} + 89870400 \) Copy content Toggle raw display
$43$ \( (T - 9995)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8661249 \) Copy content Toggle raw display
$53$ \( (T - 750)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1675919844 \) Copy content Toggle raw display
$61$ \( (T + 57920)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 520387344 \) Copy content Toggle raw display
$71$ \( T^{2} + 4062915081 \) Copy content Toggle raw display
$73$ \( T^{2} + 3465205956 \) Copy content Toggle raw display
$79$ \( (T - 63202)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3075589764 \) Copy content Toggle raw display
$89$ \( T^{2} + 10978429284 \) Copy content Toggle raw display
$97$ \( T^{2} + 25744844304 \) Copy content Toggle raw display
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