Properties

Label 1216.3.e.c
Level $1216$
Weight $3$
Character orbit 1216.e
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -7 q^{5} -11 q^{7} -23 q^{9} +O(q^{10})\) \( q + \beta q^{3} -7 q^{5} -11 q^{7} -23 q^{9} + 3 q^{11} + 2 \beta q^{13} -7 \beta q^{15} -17 q^{17} -19 q^{19} -11 \beta q^{21} -2 q^{23} + 24 q^{25} -14 \beta q^{27} + 7 \beta q^{29} -\beta q^{31} + 3 \beta q^{33} + 77 q^{35} + 7 \beta q^{37} -64 q^{39} + 7 \beta q^{41} -21 q^{43} + 161 q^{45} + 5 q^{47} + 72 q^{49} -17 \beta q^{51} -\beta q^{53} -21 q^{55} -19 \beta q^{57} + 6 \beta q^{59} -23 q^{61} + 253 q^{63} -14 \beta q^{65} + 7 \beta q^{67} -2 \beta q^{69} -16 \beta q^{71} + 39 q^{73} + 24 \beta q^{75} -33 q^{77} + 17 \beta q^{79} + 241 q^{81} -6 q^{83} + 119 q^{85} -224 q^{87} + 21 \beta q^{89} -22 \beta q^{91} + 32 q^{93} + 133 q^{95} -30 \beta q^{97} -69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{5} - 22q^{7} - 46q^{9} + O(q^{10}) \) \( 2q - 14q^{5} - 22q^{7} - 46q^{9} + 6q^{11} - 34q^{17} - 38q^{19} - 4q^{23} + 48q^{25} + 154q^{35} - 128q^{39} - 42q^{43} + 322q^{45} + 10q^{47} + 144q^{49} - 42q^{55} - 46q^{61} + 506q^{63} + 78q^{73} - 66q^{77} + 482q^{81} - 12q^{83} + 238q^{85} - 448q^{87} + 64q^{93} + 266q^{95} - 138q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.41421i
1.41421i
0 5.65685i 0 −7.00000 0 −11.0000 0 −23.0000 0
1025.2 0 5.65685i 0 −7.00000 0 −11.0000 0 −23.0000 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.e.c 2
4.b odd 2 1 1216.3.e.d 2
8.b even 2 1 304.3.e.f 2
8.d odd 2 1 152.3.e.a 2
19.b odd 2 1 inner 1216.3.e.c 2
24.f even 2 1 1368.3.o.a 2
24.h odd 2 1 2736.3.o.b 2
76.d even 2 1 1216.3.e.d 2
152.b even 2 1 152.3.e.a 2
152.g odd 2 1 304.3.e.f 2
456.l odd 2 1 1368.3.o.a 2
456.p even 2 1 2736.3.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.a 2 8.d odd 2 1
152.3.e.a 2 152.b even 2 1
304.3.e.f 2 8.b even 2 1
304.3.e.f 2 152.g odd 2 1
1216.3.e.c 2 1.a even 1 1 trivial
1216.3.e.c 2 19.b odd 2 1 inner
1216.3.e.d 2 4.b odd 2 1
1216.3.e.d 2 76.d even 2 1
1368.3.o.a 2 24.f even 2 1
1368.3.o.a 2 456.l odd 2 1
2736.3.o.b 2 24.h odd 2 1
2736.3.o.b 2 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{2} + 32 \)
\( T_{5} + 7 \)
\( T_{7} + 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 32 + T^{2} \)
$5$ \( ( 7 + T )^{2} \)
$7$ \( ( 11 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 128 + T^{2} \)
$17$ \( ( 17 + T )^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( 1568 + T^{2} \)
$31$ \( 32 + T^{2} \)
$37$ \( 1568 + T^{2} \)
$41$ \( 1568 + T^{2} \)
$43$ \( ( 21 + T )^{2} \)
$47$ \( ( -5 + T )^{2} \)
$53$ \( 32 + T^{2} \)
$59$ \( 1152 + T^{2} \)
$61$ \( ( 23 + T )^{2} \)
$67$ \( 1568 + T^{2} \)
$71$ \( 8192 + T^{2} \)
$73$ \( ( -39 + T )^{2} \)
$79$ \( 9248 + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 14112 + T^{2} \)
$97$ \( 28800 + T^{2} \)
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