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Properties

Label 234.4.h.h
Level $234$
Weight $4$
Character orbit 234.h
Analytic conductor $13.806$
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] = [234,4,Mod(55,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.h (of order \(3\), degree \(2\), 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: \(13.8064469413\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{217})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{3} + 3) q^{5} + (23 \beta_{2} - \beta_1) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 2) q^{2} - 4 \beta_{2} q^{4} + (\beta_{3} + 3) q^{5} + (23 \beta_{2} - \beta_1) q^{7} - 8 q^{8} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 6) q^{10} + ( - 7 \beta_{3} + \beta_{2} - 7 \beta_1 + 6) q^{11} + ( - 2 \beta_{3} + 6 \beta_{2} + \cdots - 9) q^{13}+ \cdots + (480 \beta_{2} - 90 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} + 14 q^{5} + 45 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} + 14 q^{5} + 45 q^{7} - 32 q^{8} + 14 q^{10} + 5 q^{11} - 35 q^{13} + 180 q^{14} - 32 q^{16} - 130 q^{17} - 3 q^{19} - 28 q^{20} - 10 q^{22} + 33 q^{23} - 234 q^{25} - 20 q^{26} + 180 q^{28} - 198 q^{29} + 560 q^{31} + 64 q^{32} - 520 q^{34} + 266 q^{35} + 702 q^{37} - 12 q^{38} - 112 q^{40} + 242 q^{41} + 93 q^{43} - 40 q^{44} - 66 q^{46} - 448 q^{47} - 435 q^{49} - 234 q^{50} + 100 q^{52} + 1070 q^{53} - 742 q^{55} - 360 q^{56} + 396 q^{58} - 389 q^{59} - 654 q^{61} + 560 q^{62} + 256 q^{64} + 203 q^{65} + 107 q^{67} - 520 q^{68} + 1064 q^{70} - 569 q^{71} - 1246 q^{73} - 1404 q^{74} - 12 q^{76} - 1294 q^{77} + 1520 q^{79} - 112 q^{80} - 484 q^{82} + 3528 q^{83} + 413 q^{85} + 372 q^{86} - 40 q^{88} - 871 q^{89} - 1539 q^{91} - 264 q^{92} - 448 q^{94} + 98 q^{95} + 879 q^{97} + 870 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 55\nu^{2} - 55\nu + 2916 ) / 2970 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 109 ) / 55 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 54\beta_{2} + \beta _1 - 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 55\beta_{3} - 109 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
3.93273 − 6.81169i
−3.43273 + 5.94566i
3.93273 + 6.81169i
−3.43273 − 5.94566i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.86546 0 7.56727 − 13.1069i −8.00000 0 −3.86546 − 6.69517i
55.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 10.8655 0 14.9327 − 25.8642i −8.00000 0 10.8655 + 18.8195i
217.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −3.86546 0 7.56727 + 13.1069i −8.00000 0 −3.86546 + 6.69517i
217.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 10.8655 0 14.9327 + 25.8642i −8.00000 0 10.8655 − 18.8195i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.4.h.h 4
3.b odd 2 1 26.4.c.b ✓ 4
12.b even 2 1 208.4.i.d 4
13.c even 3 1 inner 234.4.h.h 4
39.d odd 2 1 338.4.c.j 4
39.f even 4 2 338.4.e.f 8
39.h odd 6 1 338.4.a.g 2
39.h odd 6 1 338.4.c.j 4
39.i odd 6 1 26.4.c.b ✓ 4
39.i odd 6 1 338.4.a.h 2
39.k even 12 2 338.4.b.e 4
39.k even 12 2 338.4.e.f 8
156.p even 6 1 208.4.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.b ✓ 4 3.b odd 2 1
26.4.c.b ✓ 4 39.i odd 6 1
208.4.i.d 4 12.b even 2 1
208.4.i.d 4 156.p even 6 1
234.4.h.h 4 1.a even 1 1 trivial
234.4.h.h 4 13.c even 3 1 inner
338.4.a.g 2 39.h odd 6 1
338.4.a.h 2 39.i odd 6 1
338.4.b.e 4 39.k even 12 2
338.4.c.j 4 39.d odd 2 1
338.4.c.j 4 39.h odd 6 1
338.4.e.f 8 39.f even 4 2
338.4.e.f 8 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 7T_{5} - 42 \) acting on \(S_{4}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7 T - 42)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 45 T^{3} + \cdots + 204304 \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + \cdots + 7033104 \) Copy content Toggle raw display
$13$ \( T^{4} + 35 T^{3} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} + 130 T^{3} + \cdots + 567009 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$23$ \( T^{4} - 33 T^{3} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{4} + 198 T^{3} + \cdots + 3956121 \) Copy content Toggle raw display
$31$ \( (T^{2} - 280 T - 11648)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 14965362889 \) Copy content Toggle raw display
$41$ \( T^{4} - 242 T^{3} + \cdots + 49829481 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1007681536 \) Copy content Toggle raw display
$47$ \( (T^{2} + 224 T - 157584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 535 T + 71502)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1237069584 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 2639596129 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 45137551936 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 3764558736 \) Copy content Toggle raw display
$73$ \( (T^{2} + 623 T + 84826)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 760 T + 19408)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1764 T + 707616)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 18913400676 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 10397065156 \) Copy content Toggle raw display
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