Properties

Label 3071.1.d.c
Level $3071$
Weight $1$
Character orbit 3071.d
Self dual yes
Analytic conductor $1.533$
Analytic rank $0$
Dimension $9$
Projective image $D_{19}$
CM discriminant -3071
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3071,1,Mod(3070,3071)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3071, 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("3071.3070");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3071 = 37 \cdot 83 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3071.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: \(1.53262865380\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\Q(\zeta_{38})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{19}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{19} - \cdots)\)

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{4} q^{3} + ( - \beta_1 + 1) q^{4} + \beta_{6} q^{5} + (\beta_{6} - \beta_{5}) q^{6} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{2} + \beta_{4} q^{3} + ( - \beta_1 + 1) q^{4} + \beta_{6} q^{5} + (\beta_{6} - \beta_{5}) q^{6} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + (\beta_{8} + 1) q^{9} + (\beta_{4} - \beta_{3}) q^{10} + \beta_{6} q^{11} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{12} + \beta_{8} q^{13} + ( - \beta_1 + 2) q^{14} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{15} + (\beta_{2} - \beta_1 + 1) q^{16} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{18} - \beta_{3} q^{19} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{20} + (\beta_{6} - \beta_{5}) q^{21} + (\beta_{4} - \beta_{3}) q^{22} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4}) q^{24} + ( - \beta_{7} + 1) q^{25} + (\beta_{2} - \beta_1) q^{26} + ( - \beta_{7} + \beta_{4}) q^{27} + ( - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{28}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - q^{13} + 17 q^{14} - 2 q^{15} + 7 q^{16} - 3 q^{18} - q^{19} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 8 q^{25} - 2 q^{26} - 2 q^{27} - 3 q^{28} + 15 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 5 q^{36} + 9 q^{37} - 2 q^{38} - 2 q^{39} - 4 q^{40} - q^{41} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - 5 q^{48} + 8 q^{49} - 3 q^{50} - 3 q^{52} - 4 q^{54} + 17 q^{55} + 15 q^{56} - 2 q^{57} - 6 q^{60} - 3 q^{63} + 6 q^{64} - 2 q^{65} + 15 q^{66} - 4 q^{70} + 13 q^{72} - q^{74} - 3 q^{75} - 3 q^{76} - 2 q^{77} - 4 q^{78} - q^{79} - 5 q^{80} + 7 q^{81} - 2 q^{82} + 9 q^{83} - 6 q^{84} - 2 q^{86} - 4 q^{88} - q^{89} - 6 q^{90} - 2 q^{91} - 2 q^{95} - 6 q^{96} - q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{38} + \zeta_{38}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3071\mathbb{Z}\right)^\times\).

\(n\) \(334\) \(2740\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3070.1
−1.89163
−1.09390
0.165159
1.35456
1.97272
1.75895
0.803391
−0.490971
−1.57828
−1.97272 0.490971 2.89163 −0.803391 −0.968550 −1.97272 −3.73167 −0.758948 1.58487
3070.2 −1.75895 −1.35456 2.09390 1.89163 2.38261 −1.75895 −1.92411 0.834841 −3.32729
3070.3 −1.35456 1.89163 0.834841 −1.75895 −2.56234 −1.35456 0.223718 2.57828 2.38261
3070.4 −0.803391 −1.97272 −0.354563 0.490971 1.58487 −0.803391 1.08824 2.89163 −0.394442
3070.5 −0.165159 1.57828 −0.972723 1.09390 −0.260667 −0.165159 0.325812 1.49097 −0.180666
3070.6 0.490971 −0.803391 −0.758948 −1.97272 −0.394442 0.490971 −0.863592 −0.354563 −0.968550
3070.7 1.09390 −0.165159 0.196609 1.57828 −0.180666 1.09390 −0.878826 −0.972723 1.72648
3070.8 1.57828 1.09390 1.49097 −0.165159 1.72648 1.57828 0.774890 0.196609 −0.260667
3070.9 1.89163 −1.75895 2.57828 −1.35456 −3.32729 1.89163 2.98553 2.09390 −2.56234
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3070.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3071.d odd 2 1 CM by \(\Q(\sqrt{-3071}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3071.1.d.c 9
37.b even 2 1 3071.1.d.d yes 9
83.b odd 2 1 3071.1.d.d yes 9
3071.d odd 2 1 CM 3071.1.d.c 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3071.1.d.c 9 1.a even 1 1 trivial
3071.1.d.c 9 3071.d odd 2 1 CM
3071.1.d.d yes 9 37.b even 2 1
3071.1.d.d yes 9 83.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3071, [\chi])\):

\( T_{2}^{9} + T_{2}^{8} - 8T_{2}^{7} - 7T_{2}^{6} + 21T_{2}^{5} + 15T_{2}^{4} - 20T_{2}^{3} - 10T_{2}^{2} + 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{9} + T_{3}^{8} - 8T_{3}^{7} - 7T_{3}^{6} + 21T_{3}^{5} + 15T_{3}^{4} - 20T_{3}^{3} - 10T_{3}^{2} + 5T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{9} \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( T^{9} \) Copy content Toggle raw display
$37$ \( (T - 1)^{9} \) Copy content Toggle raw display
$41$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{9} \) Copy content Toggle raw display
$53$ \( T^{9} \) Copy content Toggle raw display
$59$ \( T^{9} \) Copy content Toggle raw display
$61$ \( T^{9} \) Copy content Toggle raw display
$67$ \( T^{9} \) Copy content Toggle raw display
$71$ \( T^{9} \) Copy content Toggle raw display
$73$ \( T^{9} \) Copy content Toggle raw display
$79$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( (T - 1)^{9} \) Copy content Toggle raw display
$89$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{9} + T^{8} - 8 T^{7} - 7 T^{6} + 21 T^{5} + \cdots + 1 \) Copy content Toggle raw display
show more
show less