Properties

Label 4368.2.h.l
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1092)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + q^{9} + 2 \beta_1 q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{2} + \beta_1) q^{15} - 2 q^{17} + (\beta_{2} + \beta_1) q^{19} - \beta_{2} q^{21} + (3 \beta_{3} - 4) q^{23} + (3 \beta_{3} - 3) q^{25} + q^{27} + ( - \beta_{3} + 6) q^{29} + (\beta_{2} + \beta_1) q^{31} + 2 \beta_1 q^{33} + (\beta_{3} - 2) q^{35} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{39} - 10 \beta_{2} q^{41} + (\beta_{3} - 4) q^{43} + ( - \beta_{2} + \beta_1) q^{45} + (3 \beta_{2} + \beta_1) q^{47} - q^{49} - 2 q^{51} + ( - \beta_{3} + 2) q^{53} + (4 \beta_{3} - 12) q^{55} + (\beta_{2} + \beta_1) q^{57} + (12 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{3} - 6) q^{61} - \beta_{2} q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} + 8) q^{65} + (10 \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{3} - 4) q^{69} + 6 \beta_1 q^{71} + (9 \beta_{2} + \beta_1) q^{73} + (3 \beta_{3} - 3) q^{75} + (2 \beta_{3} - 2) q^{77} + (3 \beta_{3} - 4) q^{79} + q^{81} + (\beta_{2} + 3 \beta_1) q^{83} + (2 \beta_{2} - 2 \beta_1) q^{85} + ( - \beta_{3} + 6) q^{87} + ( - \beta_{2} - 3 \beta_1) q^{89} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{91} + (\beta_{2} + \beta_1) q^{93} + (\beta_{3} - 4) q^{95} + (11 \beta_{2} + 3 \beta_1) q^{97} + 2 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 6 q^{13} - 8 q^{17} - 10 q^{23} - 6 q^{25} + 4 q^{27} + 22 q^{29} - 6 q^{35} - 6 q^{39} - 14 q^{43} - 4 q^{49} - 8 q^{51} + 6 q^{53} - 40 q^{55} - 28 q^{61} + 26 q^{65} - 10 q^{69} - 6 q^{75} - 4 q^{77} - 10 q^{79} + 4 q^{81} + 22 q^{87} + 6 q^{91} - 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
2.56155i
1.56155i
1.56155i
2.56155i
0 1.00000 0 3.56155i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 0.561553i 0 1.00000i 0 1.00000 0
337.3 0 1.00000 0 0.561553i 0 1.00000i 0 1.00000 0
337.4 0 1.00000 0 3.56155i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.l 4
4.b odd 2 1 1092.2.e.e 4
12.b even 2 1 3276.2.e.e 4
13.b even 2 1 inner 4368.2.h.l 4
28.d even 2 1 7644.2.e.i 4
52.b odd 2 1 1092.2.e.e 4
156.h even 2 1 3276.2.e.e 4
364.h even 2 1 7644.2.e.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1092.2.e.e 4 4.b odd 2 1
1092.2.e.e 4 52.b odd 2 1
3276.2.e.e 4 12.b even 2 1
3276.2.e.e 4 156.h even 2 1
4368.2.h.l 4 1.a even 1 1 trivial
4368.2.h.l 4 13.b even 2 1 inner
7644.2.e.i 4 28.d even 2 1
7644.2.e.i 4 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 36T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 18 T^{2} + 78 T + 169 \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 5 T - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 11 T + 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 276 T^{2} + 10816 \) Copy content Toggle raw display
$61$ \( (T^{2} + 14 T + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( T^{4} + 324 T^{2} + 20736 \) Copy content Toggle raw display
$73$ \( T^{4} + 153T^{2} + 4624 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$89$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$97$ \( T^{4} + 257T^{2} + 2704 \) Copy content Toggle raw display
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