Properties

Label 215.1.d.a
Level $215$
Weight $1$
Character orbit 215.d
Self dual yes
Analytic conductor $0.107$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -215
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [215,1,Mod(214,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("215.214");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 215.d (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: \(0.107298977716\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.9938375.1
Artin image: $D_7$
Artin field: Galois closure of 7.1.9938375.1

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + ( - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{8} + ( - \beta_{2} + \beta_1) q^{9} - \beta_1 q^{10} - \beta_1 q^{11} + (\beta_1 + 1) q^{12} + (\beta_1 - 1) q^{14} + \beta_{2} q^{15} + \beta_1 q^{16} - q^{18} + (\beta_{2} + 1) q^{20} + (\beta_{2} - \beta_1) q^{21} + (\beta_{2} + 2) q^{22} + ( - \beta_1 - 1) q^{24} + q^{25} + (\beta_{2} - \beta_1) q^{27} - q^{28} + ( - \beta_{2} - 1) q^{30} + ( - \beta_{2} + \beta_1 - 1) q^{31} - q^{32} + ( - \beta_{2} - 1) q^{33} + ( - \beta_{2} + \beta_1 - 1) q^{35} + \beta_{2} q^{36} - \beta_1 q^{37} + ( - \beta_{2} - 1) q^{40} + \beta_{2} q^{41} + q^{42} + q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + ( - \beta_{2} + \beta_1) q^{45} + (\beta_{2} + 1) q^{48} + ( - \beta_1 + 1) q^{49} - \beta_1 q^{50} + q^{54} - \beta_1 q^{55} + q^{56} + \beta_{2} q^{59} + (\beta_1 + 1) q^{60} + (\beta_1 - 1) q^{62} + ( - \beta_{2} + 1) q^{63} + (\beta_{2} + \beta_1 + 1) q^{66} + (\beta_1 - 1) q^{70} - \beta_{2} q^{72} + \beta_{2} q^{73} + (\beta_{2} + 2) q^{74} + \beta_{2} q^{75} + (\beta_1 - 1) q^{77} - \beta_1 q^{79} + \beta_1 q^{80} - \beta_{2} q^{81} + ( - \beta_{2} - 1) q^{82} - \beta_{2} q^{84} - \beta_1 q^{86} + (\beta_{2} + \beta_1 + 1) q^{88} - q^{90} + (\beta_{2} - \beta_1) q^{93} - \beta_{2} q^{96} + (\beta_{2} - \beta_1 + 2) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - q^{11} + 4 q^{12} - 2 q^{14} - q^{15} + q^{16} - 3 q^{18} + 2 q^{20} - 2 q^{21} + 5 q^{22} - 4 q^{24} + 3 q^{25} - 2 q^{27} - 3 q^{28} - 2 q^{30} - q^{31} - 3 q^{32} - 2 q^{33} - q^{35} - q^{36} - q^{37} - 2 q^{40} - q^{41} + 3 q^{42} + 3 q^{43} - 3 q^{44} + 2 q^{45} + 2 q^{48} + 2 q^{49} - q^{50} + 3 q^{54} - q^{55} + 3 q^{56} - q^{59} + 4 q^{60} - 2 q^{62} + 4 q^{63} + 3 q^{66} - 2 q^{70} + q^{72} - q^{73} + 5 q^{74} - q^{75} - 2 q^{77} - q^{79} + q^{80} + q^{81} - 2 q^{82} + q^{84} - q^{86} + 3 q^{88} - 3 q^{90} - 2 q^{93} + q^{96} + 4 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/215\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(87\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
1.80194
0.445042
−1.24698
−1.80194 1.24698 2.24698 1.00000 −2.24698 −0.445042 −2.24698 0.554958 −1.80194
214.2 −0.445042 −1.80194 −0.801938 1.00000 0.801938 1.24698 0.801938 2.24698 −0.445042
214.3 1.24698 −0.445042 0.554958 1.00000 −0.554958 −1.80194 −0.554958 −0.801938 1.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
215.d odd 2 1 CM by \(\Q(\sqrt{-215}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.1.d.a 3
3.b odd 2 1 1935.1.h.b 3
4.b odd 2 1 3440.1.p.b 3
5.b even 2 1 215.1.d.b yes 3
5.c odd 4 2 1075.1.c.c 6
15.d odd 2 1 1935.1.h.a 3
20.d odd 2 1 3440.1.p.a 3
43.b odd 2 1 215.1.d.b yes 3
129.d even 2 1 1935.1.h.a 3
172.d even 2 1 3440.1.p.a 3
215.d odd 2 1 CM 215.1.d.a 3
215.g even 4 2 1075.1.c.c 6
645.d even 2 1 1935.1.h.b 3
860.b even 2 1 3440.1.p.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.1.d.a 3 1.a even 1 1 trivial
215.1.d.a 3 215.d odd 2 1 CM
215.1.d.b yes 3 5.b even 2 1
215.1.d.b yes 3 43.b odd 2 1
1075.1.c.c 6 5.c odd 4 2
1075.1.c.c 6 215.g even 4 2
1935.1.h.a 3 15.d odd 2 1
1935.1.h.a 3 129.d even 2 1
1935.1.h.b 3 3.b odd 2 1
1935.1.h.b 3 645.d even 2 1
3440.1.p.a 3 20.d odd 2 1
3440.1.p.a 3 172.d even 2 1
3440.1.p.b 3 4.b odd 2 1
3440.1.p.b 3 860.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(215, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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