Properties

Label 1710.2.d.c
Level $1710$
Weight $2$
Character orbit 1710.d
Analytic conductor $13.654$
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] = [1710,2,Mod(1369,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.1369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.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: no
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{8}^{2} q^{2} - q^{4} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - 3 \zeta_{8}^{2} - \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{2} q^{2} - q^{4} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} - 3 \zeta_{8}^{2} - \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{10} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{11} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8}) q^{13} + (\zeta_{8}^{3} - \zeta_{8} - 3) q^{14} + q^{16} + \zeta_{8}^{2} q^{17} + q^{19} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{20} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{22} + (3 \zeta_{8}^{3} - 5 \zeta_{8}^{2} + 3 \zeta_{8}) q^{23} + ( - 3 \zeta_{8}^{2} - 4) q^{25} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 3) q^{26} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} + \zeta_{8}) q^{28} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 3) q^{29} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 2) q^{31} - \zeta_{8}^{2} q^{32} + q^{34} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} + 6 \zeta_{8} + 3) q^{35} + (6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{37} - \zeta_{8}^{2} q^{38} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{40} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{41} + ( - 3 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 3 \zeta_{8}) q^{43} + (\zeta_{8}^{3} - \zeta_{8}) q^{44} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} - 5) q^{46} + (6 \zeta_{8}^{3} - 6 \zeta_{8} - 4) q^{49} + (4 \zeta_{8}^{2} - 3) q^{50} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 2 \zeta_{8}) q^{52} + ( - 6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 6 \zeta_{8}) q^{53} + (3 \zeta_{8}^{2} - 1) q^{55} + ( - \zeta_{8}^{3} + \zeta_{8} + 3) q^{56} + ( - 2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 2 \zeta_{8}) q^{58} + ( - 7 \zeta_{8}^{3} + 7 \zeta_{8} - 3) q^{59} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 10) q^{61} + ( - 3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 3 \zeta_{8}) q^{62} - q^{64} + (3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 6 \zeta_{8} - 6) q^{65} + (3 \zeta_{8}^{3} - 9 \zeta_{8}^{2} + 3 \zeta_{8}) q^{67} - \zeta_{8}^{2} q^{68} + ( - 6 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3 \zeta_{8} + 1) q^{70} + ( - \zeta_{8}^{3} + \zeta_{8} + 12) q^{71} + (6 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 6 \zeta_{8}) q^{73} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8}) q^{74} - q^{76} + ( - 3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 3 \zeta_{8}) q^{77} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8} - 2) q^{79} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{80} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{82} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 6 \zeta_{8}) q^{83} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{85} + (3 \zeta_{8}^{3} - 3 \zeta_{8} + 6) q^{86} + (\zeta_{8}^{3} + \zeta_{8}) q^{88} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{89} + ( - 9 \zeta_{8}^{3} + 9 \zeta_{8} + 13) q^{91} + ( - 3 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 3 \zeta_{8}) q^{92} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{95} + ( - 4 \zeta_{8}^{3} + 6 \zeta_{8}^{2} - 4 \zeta_{8}) q^{97} + (6 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 6 \zeta_{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 12 q^{14} + 4 q^{16} + 4 q^{19} - 16 q^{25} + 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 12 q^{35} - 20 q^{46} - 16 q^{49} - 12 q^{50} - 4 q^{55} + 12 q^{56} - 12 q^{59} + 40 q^{61} - 4 q^{64} - 24 q^{65} + 4 q^{70} + 48 q^{71} - 4 q^{76} - 8 q^{79} + 24 q^{86} + 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
1.00000i 0 −1.00000 −0.707107 + 2.12132i 0 4.41421i 1.00000i 0 2.12132 + 0.707107i
1369.2 1.00000i 0 −1.00000 0.707107 2.12132i 0 1.58579i 1.00000i 0 −2.12132 0.707107i
1369.3 1.00000i 0 −1.00000 −0.707107 2.12132i 0 4.41421i 1.00000i 0 2.12132 0.707107i
1369.4 1.00000i 0 −1.00000 0.707107 + 2.12132i 0 1.58579i 1.00000i 0 −2.12132 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.d.c 4
3.b odd 2 1 190.2.b.a 4
5.b even 2 1 inner 1710.2.d.c 4
5.c odd 4 1 8550.2.a.bn 2
5.c odd 4 1 8550.2.a.cb 2
12.b even 2 1 1520.2.d.e 4
15.d odd 2 1 190.2.b.a 4
15.e even 4 1 950.2.a.f 2
15.e even 4 1 950.2.a.g 2
60.h even 2 1 1520.2.d.e 4
60.l odd 4 1 7600.2.a.v 2
60.l odd 4 1 7600.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 3.b odd 2 1
190.2.b.a 4 15.d odd 2 1
950.2.a.f 2 15.e even 4 1
950.2.a.g 2 15.e even 4 1
1520.2.d.e 4 12.b even 2 1
1520.2.d.e 4 60.h even 2 1
1710.2.d.c 4 1.a even 1 1 trivial
1710.2.d.c 4 5.b even 2 1 inner
7600.2.a.v 2 60.l odd 4 1
7600.2.a.bg 2 60.l odd 4 1
8550.2.a.bn 2 5.c odd 4 1
8550.2.a.cb 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{4} + 22T_{7}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 34T_{13}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 22T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 34T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 86T^{2} + 49 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 108T^{2} + 324 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 162T^{2} + 3969 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 89)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 82)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 198T^{2} + 3969 \) Copy content Toggle raw display
$71$ \( (T^{2} - 24 T + 142)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 162T^{2} + 3969 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
$89$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
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