Properties

Label 4368.2.h.o
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + \beta_{4} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + \beta_{4} q^{7} + q^{9} + (3 \beta_{4} - \beta_1) q^{11} + (2 \beta_{4} + \beta_{3} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{15} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{17} + ( - 2 \beta_{4} + 2 \beta_1) q^{19} - \beta_{4} q^{21} + 2 \beta_{2} q^{23} + ( - 3 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 3) q^{25} - q^{27} + ( - 2 \beta_{5} - 2 \beta_{3} - 2) q^{29} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{31} + ( - 3 \beta_{4} + \beta_1) q^{33} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{35} + ( - 2 \beta_{4} - 2 \beta_1) q^{37} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{39} + (\beta_{5} + 5 \beta_{4} - \beta_{3} + 3 \beta_1) q^{41} + ( - 2 \beta_{5} - 2 \beta_{3} + 4) q^{43} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{45} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{47} - q^{49} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{51} - 6 q^{53} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 4) q^{55} + (2 \beta_{4} - 2 \beta_1) q^{57} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_1) q^{59} + (3 \beta_{5} + 3 \beta_{3} - 8 \beta_{2} + 4) q^{61} + \beta_{4} q^{63} + (4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{65} + ( - 6 \beta_{4} - 2 \beta_1) q^{67} - 2 \beta_{2} q^{69} + (4 \beta_{5} + \beta_{4} - 4 \beta_{3} + 5 \beta_1) q^{71} + (3 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{73} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 3) q^{75} + (\beta_{2} - 3) q^{77} + (\beta_{5} + \beta_{3} + 6) q^{79} + q^{81} + (\beta_{5} - 5 \beta_{4} - \beta_{3} - 5 \beta_1) q^{83} + (2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3}) q^{85} + (2 \beta_{5} + 2 \beta_{3} + 2) q^{87} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} - 5 \beta_1) q^{89} + ( - \beta_{5} - \beta_{2} - 2) q^{91} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 4 \beta_1) q^{93} + ( - 2 \beta_{5} - 2 \beta_{3} - 4) q^{95} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{97} + (3 \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{23} - 10 q^{25} - 6 q^{27} - 4 q^{29} + 4 q^{35} + 2 q^{39} + 32 q^{43} - 6 q^{49} - 36 q^{53} + 16 q^{55} + 28 q^{61} + 20 q^{65} + 4 q^{69} + 10 q^{75} - 20 q^{77} + 32 q^{79} + 6 q^{81} + 4 q^{87} - 8 q^{91} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) 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
0.403032 + 0.403032i
1.45161 1.45161i
−0.854638 + 0.854638i
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 0.403032i
0 −1.00000 0 4.15633i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 1.52543i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 0.630898i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 0.630898i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 1.52543i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 4.15633i 0 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
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.o 6
4.b odd 2 1 273.2.c.b 6
12.b even 2 1 819.2.c.c 6
13.b even 2 1 inner 4368.2.h.o 6
28.d even 2 1 1911.2.c.h 6
52.b odd 2 1 273.2.c.b 6
52.f even 4 1 3549.2.a.k 3
52.f even 4 1 3549.2.a.q 3
156.h even 2 1 819.2.c.c 6
364.h even 2 1 1911.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 4.b odd 2 1
273.2.c.b 6 52.b odd 2 1
819.2.c.c 6 12.b even 2 1
819.2.c.c 6 156.h even 2 1
1911.2.c.h 6 28.d even 2 1
1911.2.c.h 6 364.h even 2 1
3549.2.a.k 3 52.f even 4 1
3549.2.a.q 3 52.f even 4 1
4368.2.h.o 6 1.a even 1 1 trivial
4368.2.h.o 6 13.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{6} + 20T_{5}^{4} + 48T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 44T_{11}^{4} + 384T_{11}^{2} + 400 \) Copy content Toggle raw display
\( T_{17}^{3} - 16T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 20 T^{4} + 48 T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + 27 T^{4} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 20 T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 52 T - 40)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400 \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{6} + 132 T^{4} + 464 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{3} - 16 T^{2} + 32 T + 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$53$ \( (T + 6)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 40 T^{4} + 80 T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} - 172 T + 2392)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{6} + 332 T^{4} + 30784 T^{2} + \cdots + 547600 \) Copy content Toggle raw display
$73$ \( T^{6} + 272 T^{4} + 6720 T^{2} + \cdots + 43264 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + 72 T - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 408 T^{4} + 49616 T^{2} + \cdots + 1567504 \) Copy content Toggle raw display
$89$ \( T^{6} + 212 T^{4} + 6832 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$97$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
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