Properties

Label 234.6.b.a
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} - 16 q^{4} + 34 \beta q^{5} - 41 \beta q^{7} - 32 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} - 16 q^{4} + 34 \beta q^{5} - 41 \beta q^{7} - 32 \beta q^{8} - 272 q^{10} + 195 \beta q^{11} + (169 \beta + 507) q^{13} + 328 q^{14} + 256 q^{16} - 1738 q^{17} + 537 \beta q^{19} - 544 \beta q^{20} - 1560 q^{22} - 2104 q^{23} - 1499 q^{25} + (1014 \beta - 1352) q^{26} + 656 \beta q^{28} + 1690 q^{29} + 715 \beta q^{31} + 512 \beta q^{32} - 3476 \beta q^{34} + 5576 q^{35} - 4426 \beta q^{37} - 4296 q^{38} + 4352 q^{40} + 3380 \beta q^{41} - 16916 q^{43} - 3120 \beta q^{44} - 4208 \beta q^{46} - 12579 \beta q^{47} + 10083 q^{49} - 2998 \beta q^{50} + ( - 2704 \beta - 8112) q^{52} - 38214 q^{53} - 26520 q^{55} - 5248 q^{56} + 3380 \beta q^{58} + 10643 \beta q^{59} - 5458 q^{61} - 5720 q^{62} - 4096 q^{64} + (17238 \beta - 22984) q^{65} - 22271 \beta q^{67} + 27808 q^{68} + 11152 \beta q^{70} - 8895 \beta q^{71} - 15532 \beta q^{73} + 35408 q^{74} - 8592 \beta q^{76} + 31980 q^{77} - 45360 q^{79} + 8704 \beta q^{80} - 27040 q^{82} - 62273 \beta q^{83} - 59092 \beta q^{85} - 33832 \beta q^{86} + 24960 q^{88} - 9372 \beta q^{89} + ( - 20787 \beta + 27716) q^{91} + 33664 q^{92} + 100632 q^{94} - 73032 q^{95} + 60744 \beta q^{97} + 20166 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 544 q^{10} + 1014 q^{13} + 656 q^{14} + 512 q^{16} - 3476 q^{17} - 3120 q^{22} - 4208 q^{23} - 2998 q^{25} - 2704 q^{26} + 3380 q^{29} + 11152 q^{35} - 8592 q^{38} + 8704 q^{40} - 33832 q^{43} + 20166 q^{49} - 16224 q^{52} - 76428 q^{53} - 53040 q^{55} - 10496 q^{56} - 10916 q^{61} - 11440 q^{62} - 8192 q^{64} - 45968 q^{65} + 55616 q^{68} + 70816 q^{74} + 63960 q^{77} - 90720 q^{79} - 54080 q^{82} + 49920 q^{88} + 55432 q^{91} + 67328 q^{92} + 201264 q^{94} - 146064 q^{95}+O(q^{100}) \) 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 68.0000i 0 82.0000i 64.0000i 0 −272.000
181.2 4.00000i 0 −16.0000 68.0000i 0 82.0000i 64.0000i 0 −272.000
\(n\): 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.a 2
3.b odd 2 1 26.6.b.b 2
12.b even 2 1 208.6.f.a 2
13.b even 2 1 inner 234.6.b.a 2
39.d odd 2 1 26.6.b.b 2
39.f even 4 1 338.6.a.b 1
39.f even 4 1 338.6.a.e 1
156.h even 2 1 208.6.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.b 2 3.b odd 2 1
26.6.b.b 2 39.d odd 2 1
208.6.f.a 2 12.b even 2 1
208.6.f.a 2 156.h even 2 1
234.6.b.a 2 1.a even 1 1 trivial
234.6.b.a 2 13.b even 2 1 inner
338.6.a.b 1 39.f even 4 1
338.6.a.e 1 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4624 \) 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
$5$ \( T^{2} + 4624 \) Copy content Toggle raw display
$7$ \( T^{2} + 6724 \) Copy content Toggle raw display
$11$ \( T^{2} + 152100 \) Copy content Toggle raw display
$13$ \( T^{2} - 1014 T + 371293 \) Copy content Toggle raw display
$17$ \( (T + 1738)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1153476 \) Copy content Toggle raw display
$23$ \( (T + 2104)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1690)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2044900 \) Copy content Toggle raw display
$37$ \( T^{2} + 78357904 \) Copy content Toggle raw display
$41$ \( T^{2} + 45697600 \) Copy content Toggle raw display
$43$ \( (T + 16916)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 632924964 \) Copy content Toggle raw display
$53$ \( (T + 38214)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 453093796 \) Copy content Toggle raw display
$61$ \( (T + 5458)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1983989764 \) Copy content Toggle raw display
$71$ \( T^{2} + 316484100 \) Copy content Toggle raw display
$73$ \( T^{2} + 964972096 \) Copy content Toggle raw display
$79$ \( (T + 45360)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 15511706116 \) Copy content Toggle raw display
$89$ \( T^{2} + 351337536 \) Copy content Toggle raw display
$97$ \( T^{2} + 14759334144 \) Copy content Toggle raw display
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