Properties

Label 304.3.e.c
Level $304$
Weight $3$
Character orbit 304.e
Analytic conductor $8.283$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.28340003655\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} -5 q^{7} + q^{9} +O(q^{10})\) \( q + \beta q^{3} - q^{5} -5 q^{7} + q^{9} -5 q^{11} + 6 \beta q^{13} -\beta q^{15} -25 q^{17} -19 q^{19} -5 \beta q^{21} + 10 q^{23} -24 q^{25} + 10 \beta q^{27} -15 \beta q^{29} -15 \beta q^{31} -5 \beta q^{33} + 5 q^{35} + 9 \beta q^{37} -48 q^{39} + 15 \beta q^{41} -5 q^{43} - q^{45} -5 q^{47} -24 q^{49} -25 \beta q^{51} + 9 \beta q^{53} + 5 q^{55} -19 \beta q^{57} + 30 \beta q^{59} + 95 q^{61} -5 q^{63} -6 \beta q^{65} + 39 \beta q^{67} + 10 \beta q^{69} -25 q^{73} -24 \beta q^{75} + 25 q^{77} + 15 \beta q^{79} -71 q^{81} + 130 q^{83} + 25 q^{85} + 120 q^{87} + 45 \beta q^{89} -30 \beta q^{91} + 120 q^{93} + 19 q^{95} -6 \beta q^{97} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 10q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{5} - 10q^{7} + 2q^{9} - 10q^{11} - 50q^{17} - 38q^{19} + 20q^{23} - 48q^{25} + 10q^{35} - 96q^{39} - 10q^{43} - 2q^{45} - 10q^{47} - 48q^{49} + 10q^{55} + 190q^{61} - 10q^{63} - 50q^{73} + 50q^{77} - 142q^{81} + 260q^{83} + 50q^{85} + 240q^{87} + 240q^{93} + 38q^{95} - 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.41421i
1.41421i
0 2.82843i 0 −1.00000 0 −5.00000 0 1.00000 0
113.2 0 2.82843i 0 −1.00000 0 −5.00000 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
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 304.3.e.c 2
3.b odd 2 1 2736.3.o.h 2
4.b odd 2 1 38.3.b.a 2
8.b even 2 1 1216.3.e.i 2
8.d odd 2 1 1216.3.e.j 2
12.b even 2 1 342.3.d.a 2
19.b odd 2 1 inner 304.3.e.c 2
20.d odd 2 1 950.3.c.a 2
20.e even 4 2 950.3.d.a 4
57.d even 2 1 2736.3.o.h 2
76.d even 2 1 38.3.b.a 2
152.b even 2 1 1216.3.e.j 2
152.g odd 2 1 1216.3.e.i 2
228.b odd 2 1 342.3.d.a 2
380.d even 2 1 950.3.c.a 2
380.j odd 4 2 950.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 4.b odd 2 1
38.3.b.a 2 76.d even 2 1
304.3.e.c 2 1.a even 1 1 trivial
304.3.e.c 2 19.b odd 2 1 inner
342.3.d.a 2 12.b even 2 1
342.3.d.a 2 228.b odd 2 1
950.3.c.a 2 20.d odd 2 1
950.3.c.a 2 380.d even 2 1
950.3.d.a 4 20.e even 4 2
950.3.d.a 4 380.j odd 4 2
1216.3.e.i 2 8.b even 2 1
1216.3.e.i 2 152.g odd 2 1
1216.3.e.j 2 8.d odd 2 1
1216.3.e.j 2 152.b even 2 1
2736.3.o.h 2 3.b odd 2 1
2736.3.o.h 2 57.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 \(S_{3}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{2} + 8 \)
\( T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 8 + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 5 + T )^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 288 + T^{2} \)
$17$ \( ( 25 + T )^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( ( -10 + T )^{2} \)
$29$ \( 1800 + T^{2} \)
$31$ \( 1800 + T^{2} \)
$37$ \( 648 + T^{2} \)
$41$ \( 1800 + T^{2} \)
$43$ \( ( 5 + T )^{2} \)
$47$ \( ( 5 + T )^{2} \)
$53$ \( 648 + T^{2} \)
$59$ \( 7200 + T^{2} \)
$61$ \( ( -95 + T )^{2} \)
$67$ \( 12168 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 25 + T )^{2} \)
$79$ \( 1800 + T^{2} \)
$83$ \( ( -130 + T )^{2} \)
$89$ \( 16200 + T^{2} \)
$97$ \( 288 + T^{2} \)
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