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Properties

Label 3311.1.h.o
Level $3311$
Weight $1$
Character orbit 3311.h
Self dual yes
Analytic conductor $1.652$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
CM discriminant -3311
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3311,1,Mod(3310,3311)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3311.3310");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3311.h (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: yes
Analytic conductor: \(1.65240425683\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{4} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} + (\beta_{5} + \beta_{3} + \beta_1) q^{6} - q^{7} + ( - \beta_{4} - \beta_{2} + 1) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_1 q^{3} + ( - \beta_{4} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} + (\beta_{5} + \beta_{3} + \beta_1) q^{6} - q^{7} + ( - \beta_{4} - \beta_{2} + 1) q^{8} + (\beta_{2} + 1) q^{9} - \beta_1 q^{10} - q^{11} + (\beta_{5} - \beta_1) q^{12} - \beta_{3} q^{13} + (\beta_{4} + \beta_{2}) q^{14} + ( - \beta_{4} - \beta_{2} - 1) q^{15} + ( - \beta_{4} - \beta_{2} + 1) q^{16} - \beta_{5} q^{17} + ( - \beta_{4} - 2 \beta_{2} - 1) q^{18} + \beta_{3} q^{20} + \beta_1 q^{21} + (\beta_{4} + \beta_{2}) q^{22} + q^{23} + (\beta_{5} + \beta_{3}) q^{24} + (\beta_{4} + 1) q^{25} + ( - \beta_{5} + \beta_1) q^{26} + ( - \beta_{3} - \beta_1) q^{27} + (\beta_{4} - 1) q^{28} - \beta_{4} q^{29} + (\beta_{2} + 2) q^{30} + ( - \beta_{4} + 1) q^{32} + \beta_1 q^{33} + ( - \beta_{5} - \beta_{3}) q^{34} + ( - \beta_{5} - \beta_1) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{4} + 2 \beta_{2}) q^{39} + \beta_{5} q^{40} + \beta_{5} q^{41} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{42} + q^{43} + (\beta_{4} - 1) q^{44} + (\beta_{3} + \beta_1) q^{45} + ( - \beta_{4} - \beta_{2}) q^{46} + (\beta_{5} + \beta_{3}) q^{48} + q^{49} - q^{50} + (\beta_{4} - 1) q^{51} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{52} - \beta_{2} q^{53} + (\beta_{3} + 2 \beta_1) q^{54} + ( - \beta_{5} - \beta_1) q^{55} + (\beta_{4} + \beta_{2} - 1) q^{56} + ( - \beta_{4} - \beta_{2} + 1) q^{58} + ( - \beta_{4} - 2 \beta_{2}) q^{60} + ( - \beta_{2} - 1) q^{63} + ( - \beta_{4} - \beta_{2}) q^{64} + (\beta_{4} - \beta_{2}) q^{65} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{66} + (\beta_{4} + \beta_{2}) q^{67} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{68} - \beta_1 q^{69} + \beta_1 q^{70} + ( - \beta_{4} - \beta_{2}) q^{72} + ( - \beta_{5} - \beta_1) q^{75} + q^{77} + (\beta_{4} - \beta_{2} - 3) q^{78} + \beta_{5} q^{80} + (\beta_{4} + 2 \beta_{2} + 1) q^{81} + (\beta_{5} + \beta_{3}) q^{82} - \beta_1 q^{83} + ( - \beta_{5} + \beta_1) q^{84} + (\beta_{2} - 1) q^{85} + ( - \beta_{4} - \beta_{2}) q^{86} + \beta_{5} q^{87} + (\beta_{4} + \beta_{2} - 1) q^{88} + ( - \beta_{3} - 2 \beta_1) q^{90} + \beta_{3} q^{91} + ( - \beta_{4} + 1) q^{92} + (\beta_{5} - \beta_1) q^{96} + ( - \beta_{4} - \beta_{2}) q^{98} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{4} - 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{4} - 6 q^{7} + 6 q^{8} + 6 q^{9} - 6 q^{11} - 6 q^{15} + 6 q^{16} - 6 q^{18} + 6 q^{23} + 6 q^{25} - 6 q^{28} + 12 q^{30} + 6 q^{32} + 12 q^{36} + 6 q^{43} - 6 q^{44} + 6 q^{49} - 6 q^{50} - 6 q^{51} - 6 q^{56} + 6 q^{58} - 6 q^{63} + 6 q^{77} - 18 q^{78} + 6 q^{81} - 6 q^{85} - 6 q^{88} + 6 q^{92} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 4\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 11\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times\).

\(n\) \(904\) \(1893\) \(2927\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3310.1
1.96962
−1.96962
0.684040
−0.684040
1.28558
−1.28558
−1.53209 −1.96962 1.34730 1.28558 3.01763 −1.00000 −0.532089 2.87939 −1.96962
3310.2 −1.53209 1.96962 1.34730 −1.28558 −3.01763 −1.00000 −0.532089 2.87939 1.96962
3310.3 −0.347296 −0.684040 −0.879385 1.96962 0.237565 −1.00000 0.652704 −0.532089 −0.684040
3310.4 −0.347296 0.684040 −0.879385 −1.96962 −0.237565 −1.00000 0.652704 −0.532089 0.684040
3310.5 1.87939 −1.28558 2.53209 −0.684040 −2.41609 −1.00000 2.87939 0.652704 −1.28558
3310.6 1.87939 1.28558 2.53209 0.684040 2.41609 −1.00000 2.87939 0.652704 1.28558
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3310.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3311.h odd 2 1 CM by \(\Q(\sqrt{-3311}) \)
7.b odd 2 1 inner
473.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.h.o ✓ 6
7.b odd 2 1 inner 3311.1.h.o ✓ 6
11.b odd 2 1 3311.1.h.p yes 6
43.b odd 2 1 3311.1.h.p yes 6
77.b even 2 1 3311.1.h.p yes 6
301.c even 2 1 3311.1.h.p yes 6
473.d even 2 1 inner 3311.1.h.o ✓ 6
3311.h odd 2 1 CM 3311.1.h.o ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.o ✓ 6 1.a even 1 1 trivial
3311.1.h.o ✓ 6 7.b odd 2 1 inner
3311.1.h.o ✓ 6 473.d even 2 1 inner
3311.1.h.o ✓ 6 3311.h odd 2 1 CM
3311.1.h.p yes 6 11.b odd 2 1
3311.1.h.p yes 6 43.b odd 2 1
3311.1.h.p yes 6 77.b even 2 1
3311.1.h.p yes 6 301.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

\( T_{2}^{3} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 6T_{3}^{4} + 9T_{3}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - 3 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} - 6 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( (T - 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$43$ \( (T - 1)^{6} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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