Properties

Label 3744.1.eh.a
Level $3744$
Weight $1$
Character orbit 3744.eh
Analytic conductor $1.868$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -39
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.eh (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{16}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{32}^{3} q^{2} + \zeta_{32}^{6} q^{4} + ( -\zeta_{32} - \zeta_{32}^{11} ) q^{5} -\zeta_{32}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{32}^{3} q^{2} + \zeta_{32}^{6} q^{4} + ( -\zeta_{32} - \zeta_{32}^{11} ) q^{5} -\zeta_{32}^{9} q^{8} + ( \zeta_{32}^{4} + \zeta_{32}^{14} ) q^{10} + ( \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{11} + \zeta_{32}^{2} q^{13} + \zeta_{32}^{12} q^{16} + ( \zeta_{32} - \zeta_{32}^{7} ) q^{20} + ( 1 - \zeta_{32}^{2} ) q^{22} + ( \zeta_{32}^{2} - \zeta_{32}^{6} + \zeta_{32}^{12} ) q^{25} -\zeta_{32}^{5} q^{26} -\zeta_{32}^{15} q^{32} + ( -\zeta_{32}^{4} + \zeta_{32}^{10} ) q^{40} + ( -\zeta_{32}^{9} - \zeta_{32}^{15} ) q^{41} + ( \zeta_{32}^{4} - \zeta_{32}^{8} ) q^{43} + ( -\zeta_{32}^{3} + \zeta_{32}^{5} ) q^{44} + ( \zeta_{32}^{7} + \zeta_{32}^{9} ) q^{47} + \zeta_{32}^{8} q^{49} + ( -\zeta_{32}^{5} + \zeta_{32}^{9} - \zeta_{32}^{15} ) q^{50} + \zeta_{32}^{8} q^{52} + ( -1 + \zeta_{32}^{8} - \zeta_{32}^{10} - \zeta_{32}^{14} ) q^{55} + ( \zeta_{32} - \zeta_{32}^{11} ) q^{59} + ( -\zeta_{32}^{6} - \zeta_{32}^{14} ) q^{61} -\zeta_{32}^{2} q^{64} + ( -\zeta_{32}^{3} - \zeta_{32}^{13} ) q^{65} + ( -\zeta_{32}^{3} + \zeta_{32}^{5} ) q^{71} + ( -\zeta_{32}^{6} + \zeta_{32}^{10} ) q^{79} + ( \zeta_{32}^{7} - \zeta_{32}^{13} ) q^{80} + ( -\zeta_{32}^{2} + \zeta_{32}^{12} ) q^{82} + ( -\zeta_{32}^{7} + \zeta_{32}^{13} ) q^{83} + ( -\zeta_{32}^{7} + \zeta_{32}^{11} ) q^{86} + ( \zeta_{32}^{6} - \zeta_{32}^{8} ) q^{88} + ( \zeta_{32}^{11} - \zeta_{32}^{13} ) q^{89} + ( -\zeta_{32}^{10} - \zeta_{32}^{12} ) q^{94} -\zeta_{32}^{11} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 16q^{22} - 16q^{55} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{32}^{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
0.980785 + 0.195090i
−0.195090 + 0.980785i
0.195090 0.980785i
−0.980785 0.195090i
0.980785 0.195090i
−0.195090 0.980785i
0.195090 + 0.980785i
−0.980785 + 0.195090i
−0.555570 0.831470i
−0.831470 + 0.555570i
0.831470 0.555570i
0.555570 + 0.831470i
−0.555570 + 0.831470i
−0.831470 0.555570i
0.831470 + 0.555570i
0.555570 0.831470i
−0.831470 0.555570i 0 0.382683 + 0.923880i −0.425215 1.02656i 0 0 0.195090 0.980785i 0 −0.216773 + 1.08979i
883.2 −0.555570 + 0.831470i 0 −0.382683 0.923880i −0.636379 1.53636i 0 0 0.980785 + 0.195090i 0 1.63099 + 0.324423i
883.3 0.555570 0.831470i 0 −0.382683 0.923880i 0.636379 + 1.53636i 0 0 −0.980785 0.195090i 0 1.63099 + 0.324423i
883.4 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0.425215 + 1.02656i 0 0 −0.195090 + 0.980785i 0 −0.216773 + 1.08979i
1819.1 −0.831470 + 0.555570i 0 0.382683 0.923880i −0.425215 + 1.02656i 0 0 0.195090 + 0.980785i 0 −0.216773 1.08979i
1819.2 −0.555570 0.831470i 0 −0.382683 + 0.923880i −0.636379 + 1.53636i 0 0 0.980785 0.195090i 0 1.63099 0.324423i
1819.3 0.555570 + 0.831470i 0 −0.382683 + 0.923880i 0.636379 1.53636i 0 0 −0.980785 + 0.195090i 0 1.63099 0.324423i
1819.4 0.831470 0.555570i 0 0.382683 0.923880i 0.425215 1.02656i 0 0 −0.195090 0.980785i 0 −0.216773 1.08979i
2755.1 −0.980785 + 0.195090i 0 0.923880 0.382683i 0.360480 0.149316i 0 0 −0.831470 + 0.555570i 0 −0.324423 + 0.216773i
2755.2 −0.195090 0.980785i 0 −0.923880 + 0.382683i 1.81225 0.750661i 0 0 0.555570 + 0.831470i 0 −1.08979 1.63099i
2755.3 0.195090 + 0.980785i 0 −0.923880 + 0.382683i −1.81225 + 0.750661i 0 0 −0.555570 0.831470i 0 −1.08979 1.63099i
2755.4 0.980785 0.195090i 0 0.923880 0.382683i −0.360480 + 0.149316i 0 0 0.831470 0.555570i 0 −0.324423 + 0.216773i
3691.1 −0.980785 0.195090i 0 0.923880 + 0.382683i 0.360480 + 0.149316i 0 0 −0.831470 0.555570i 0 −0.324423 0.216773i
3691.2 −0.195090 + 0.980785i 0 −0.923880 0.382683i 1.81225 + 0.750661i 0 0 0.555570 0.831470i 0 −1.08979 + 1.63099i
3691.3 0.195090 0.980785i 0 −0.923880 0.382683i −1.81225 0.750661i 0 0 −0.555570 + 0.831470i 0 −1.08979 + 1.63099i
3691.4 0.980785 + 0.195090i 0 0.923880 + 0.382683i −0.360480 0.149316i 0 0 0.831470 + 0.555570i 0 −0.324423 0.216773i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3691.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
416.be odd 8 1 inner
1248.cp even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.eh.a 16
3.b odd 2 1 inner 3744.1.eh.a 16
13.b even 2 1 inner 3744.1.eh.a 16
32.h odd 8 1 inner 3744.1.eh.a 16
39.d odd 2 1 CM 3744.1.eh.a 16
96.o even 8 1 inner 3744.1.eh.a 16
416.be odd 8 1 inner 3744.1.eh.a 16
1248.cp even 8 1 inner 3744.1.eh.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.1.eh.a 16 1.a even 1 1 trivial
3744.1.eh.a 16 3.b odd 2 1 inner
3744.1.eh.a 16 13.b even 2 1 inner
3744.1.eh.a 16 32.h odd 8 1 inner
3744.1.eh.a 16 39.d odd 2 1 CM
3744.1.eh.a 16 96.o even 8 1 inner
3744.1.eh.a 16 416.be odd 8 1 inner
3744.1.eh.a 16 1248.cp even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3744, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$13$ \( ( 1 + T^{8} )^{2} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$43$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{4} \)
$47$ \( ( 2 + 16 T^{2} + 20 T^{4} + 8 T^{6} + T^{8} )^{2} \)
$53$ \( T^{16} \)
$59$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$61$ \( ( 16 + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( ( 2 - 4 T^{2} + T^{4} )^{4} \)
$83$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$89$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$97$ \( T^{16} \)
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