Properties

Label 1352.1.g.c
Level $1352$
Weight $1$
Character orbit 1352.g
Self dual yes
Analytic conductor $0.675$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1352.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.674735897080\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.2471326208.1
Artin image: $D_7$
Artin field: Galois closure of 7.1.2471326208.1

$q$-expansion

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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{11} - \beta_1 q^{12} + q^{16} + ( - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{2} + 1) q^{18} + \beta_{2} q^{19} + ( - \beta_{2} + \beta_1 - 1) q^{22} - \beta_1 q^{24} + q^{25} + ( - \beta_{2} - 1) q^{27} + q^{32} + (\beta_1 - 1) q^{33} + ( - \beta_{2} + \beta_1 - 1) q^{34} + (\beta_{2} + 1) q^{36} + \beta_{2} q^{38} + \beta_{2} q^{41} + \beta_{2} q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{44} - \beta_1 q^{48} + q^{49} + q^{50} + (\beta_1 - 1) q^{51} + ( - \beta_{2} - 1) q^{54} + ( - \beta_{2} - 1) q^{57} - \beta_1 q^{59} + q^{64} + (\beta_1 - 1) q^{66} - \beta_1 q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{68} + (\beta_{2} + 1) q^{72} - \beta_1 q^{73} - \beta_1 q^{75} + \beta_{2} q^{76} + \beta_1 q^{81} + \beta_{2} q^{82} + \beta_{2} q^{83} + \beta_{2} q^{86} + ( - \beta_{2} + \beta_1 - 1) q^{88} - \beta_1 q^{89} - \beta_1 q^{96} + ( - \beta_{2} + \beta_1 - 1) q^{97} + q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 3 q^{8} + 2 q^{9} - q^{11} - q^{12} + 3 q^{16} - q^{17} + 2 q^{18} - q^{19} - q^{22} - q^{24} + 3 q^{25} - 2 q^{27} + 3 q^{32} - 2 q^{33} - q^{34} + 2 q^{36} - q^{38} - q^{41} - q^{43} - q^{44} - q^{48} + 3 q^{49} + 3 q^{50} - 2 q^{51} - 2 q^{54} - 2 q^{57} - q^{59} + 3 q^{64} - 2 q^{66} - q^{67} - q^{68} + 2 q^{72} - q^{73} - q^{75} - q^{76} + q^{81} - q^{82} - q^{83} - q^{86} - q^{88} - q^{89} - q^{96} - q^{97} + 3 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) 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 \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
1.80194
0.445042
−1.24698
1.00000 −1.80194 1.00000 0 −1.80194 0 1.00000 2.24698 0
339.2 1.00000 −0.445042 1.00000 0 −0.445042 0 1.00000 −0.801938 0
339.3 1.00000 1.24698 1.00000 0 1.24698 0 1.00000 0.554958 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.1.g.c yes 3
8.d odd 2 1 CM 1352.1.g.c yes 3
13.b even 2 1 1352.1.g.b 3
13.c even 3 2 1352.1.n.b 6
13.d odd 4 2 1352.1.h.a 6
13.e even 6 2 1352.1.n.c 6
13.f odd 12 4 1352.1.p.c 12
104.h odd 2 1 1352.1.g.b 3
104.m even 4 2 1352.1.h.a 6
104.n odd 6 2 1352.1.n.b 6
104.p odd 6 2 1352.1.n.c 6
104.u even 12 4 1352.1.p.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.1.g.b 3 13.b even 2 1
1352.1.g.b 3 104.h odd 2 1
1352.1.g.c yes 3 1.a even 1 1 trivial
1352.1.g.c yes 3 8.d odd 2 1 CM
1352.1.h.a 6 13.d odd 4 2
1352.1.h.a 6 104.m even 4 2
1352.1.n.b 6 13.c even 3 2
1352.1.n.b 6 104.n odd 6 2
1352.1.n.c 6 13.e even 6 2
1352.1.n.c 6 104.p odd 6 2
1352.1.p.c 12 13.f odd 12 4
1352.1.p.c 12 104.u even 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{3} + T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 2T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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