Properties

Label 1344.3.f.e
Level $1344$
Weight $3$
Character orbit 1344.f
Analytic conductor $36.621$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.6213475300\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (2 \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{7} - 3 q^{9} + ( - \beta_{3} - 6) q^{11} + ( - 8 \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} + 6) q^{15} + ( - 2 \beta_{2} - 11 \beta_1) q^{17} + ( - 2 \beta_{2} + 8 \beta_1) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{21} + (3 \beta_{3} - 6) q^{23} + (4 \beta_{3} + 7) q^{25} + 3 \beta_{2} q^{27} - 30 q^{29} + (12 \beta_{2} + 12 \beta_1) q^{31} + (6 \beta_{2} + 3 \beta_1) q^{33} + (3 \beta_{3} - 10 \beta_{2} + 8 \beta_1 - 6) q^{35} + ( - 12 \beta_{3} + 20) q^{37} + (2 \beta_{3} - 24) q^{39} + (14 \beta_{2} - 7 \beta_1) q^{41} + ( - 10 \beta_{3} - 32) q^{43} + ( - 6 \beta_{2} + 3 \beta_1) q^{45} + ( - 28 \beta_{2} - 4 \beta_1) q^{47} + ( - 8 \beta_{3} - 20 \beta_{2} + 2 \beta_1 - 5) q^{49} + ( - 11 \beta_{3} - 6) q^{51} + (4 \beta_{3} + 54) q^{53} - 6 \beta_{2} q^{55} + (8 \beta_{3} - 6) q^{57} + ( - 28 \beta_{2} + 20 \beta_1) q^{59} + ( - 4 \beta_{2} + 4 \beta_1) q^{61} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 6) q^{63} + ( - 12 \beta_{3} + 60) q^{65} + ( - 4 \beta_{3} + 44) q^{67} + (6 \beta_{2} - 9 \beta_1) q^{69} + ( - 19 \beta_{3} + 30) q^{71} + (4 \beta_{2} + 26 \beta_1) q^{73} + ( - 7 \beta_{2} - 12 \beta_1) q^{75} + ( - 4 \beta_{3} + 18 \beta_{2} + 15 \beta_1 - 6) q^{77} + ( - 24 \beta_{3} - 32) q^{79} + 9 q^{81} + (32 \beta_{2} + 20 \beta_1) q^{83} + (20 \beta_{3} - 54) q^{85} + 30 \beta_{2} q^{87} + ( - 54 \beta_{2} - 21 \beta_1) q^{89} + ( - 14 \beta_{3} + 28 \beta_{2} - 28 \beta_1) q^{91} + (12 \beta_{3} + 36) q^{93} + ( - 18 \beta_{3} + 60) q^{95} + (44 \beta_{2} + 10 \beta_1) q^{97} + (3 \beta_{3} + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 12 q^{9} - 24 q^{11} + 24 q^{15} - 12 q^{21} - 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} + 80 q^{37} - 96 q^{39} - 128 q^{43} - 20 q^{49} - 24 q^{51} + 216 q^{53} - 24 q^{57} + 24 q^{63} + 240 q^{65} + 176 q^{67} + 120 q^{71} - 24 q^{77} - 128 q^{79} + 36 q^{81} - 216 q^{85} + 144 q^{93} + 240 q^{95} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.707107 + 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 1.73205i 0 1.01461i 0 2.24264 6.63103i 0 −3.00000 0
769.2 0 1.73205i 0 5.91359i 0 −6.24264 + 3.16693i 0 −3.00000 0
769.3 0 1.73205i 0 5.91359i 0 −6.24264 3.16693i 0 −3.00000 0
769.4 0 1.73205i 0 1.01461i 0 2.24264 + 6.63103i 0 −3.00000 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.3.f.e 4
4.b odd 2 1 1344.3.f.f 4
7.b odd 2 1 inner 1344.3.f.e 4
8.b even 2 1 336.3.f.c 4
8.d odd 2 1 42.3.c.a 4
24.f even 2 1 126.3.c.b 4
24.h odd 2 1 1008.3.f.g 4
28.d even 2 1 1344.3.f.f 4
40.e odd 2 1 1050.3.f.a 4
40.k even 4 2 1050.3.h.a 8
56.e even 2 1 42.3.c.a 4
56.h odd 2 1 336.3.f.c 4
56.k odd 6 1 294.3.g.b 4
56.k odd 6 1 294.3.g.c 4
56.m even 6 1 294.3.g.b 4
56.m even 6 1 294.3.g.c 4
168.e odd 2 1 126.3.c.b 4
168.i even 2 1 1008.3.f.g 4
168.v even 6 1 882.3.n.a 4
168.v even 6 1 882.3.n.d 4
168.be odd 6 1 882.3.n.a 4
168.be odd 6 1 882.3.n.d 4
280.n even 2 1 1050.3.f.a 4
280.y odd 4 2 1050.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 8.d odd 2 1
42.3.c.a 4 56.e even 2 1
126.3.c.b 4 24.f even 2 1
126.3.c.b 4 168.e odd 2 1
294.3.g.b 4 56.k odd 6 1
294.3.g.b 4 56.m even 6 1
294.3.g.c 4 56.k odd 6 1
294.3.g.c 4 56.m even 6 1
336.3.f.c 4 8.b even 2 1
336.3.f.c 4 56.h odd 2 1
882.3.n.a 4 168.v even 6 1
882.3.n.a 4 168.be odd 6 1
882.3.n.d 4 168.v even 6 1
882.3.n.d 4 168.be odd 6 1
1008.3.f.g 4 24.h odd 2 1
1008.3.f.g 4 168.i even 2 1
1050.3.f.a 4 40.e odd 2 1
1050.3.f.a 4 280.n even 2 1
1050.3.h.a 8 40.k even 4 2
1050.3.h.a 8 280.y odd 4 2
1344.3.f.e 4 1.a even 1 1 trivial
1344.3.f.e 4 7.b odd 2 1 inner
1344.3.f.f 4 4.b odd 2 1
1344.3.f.f 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} + 36T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 36T^{2} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} + 42 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$17$ \( T^{4} + 1476 T^{2} + 509796 \) Copy content Toggle raw display
$19$ \( T^{4} + 792 T^{2} + 138384 \) Copy content Toggle raw display
$23$ \( (T^{2} + 12 T - 126)^{2} \) Copy content Toggle raw display
$29$ \( (T + 30)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2592 T^{2} + 186624 \) Copy content Toggle raw display
$37$ \( (T^{2} - 40 T - 2192)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 1764 T^{2} + 86436 \) Copy content Toggle raw display
$43$ \( (T^{2} + 64 T - 776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4896 T^{2} + \cdots + 5089536 \) Copy content Toggle raw display
$53$ \( (T^{2} - 108 T + 2628)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 9504 T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 288T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} - 88 T + 1648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 60 T - 5598)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8208 T^{2} + \cdots + 16064064 \) Copy content Toggle raw display
$79$ \( (T^{2} + 64 T - 9344)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 10944 T^{2} + \cdots + 451584 \) Copy content Toggle raw display
$89$ \( T^{4} + 22788 T^{2} + \cdots + 37234404 \) Copy content Toggle raw display
$97$ \( T^{4} + 12816 T^{2} + \cdots + 27123264 \) Copy content Toggle raw display
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