Properties

Label 304.3.e.d
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: \(\Q(\sqrt{-13}) \)
Defining polynomial: \(x^{2} + 13\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 4 q^{5} + 5 q^{7} -4 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 4 q^{5} + 5 q^{7} -4 q^{9} + 10 q^{11} + \beta q^{13} + 4 \beta q^{15} + 15 q^{17} + ( 6 - 5 \beta ) q^{19} + 5 \beta q^{21} -35 q^{23} -9 q^{25} + 5 \beta q^{27} + 5 \beta q^{29} + 10 \beta q^{31} + 10 \beta q^{33} + 20 q^{35} -6 \beta q^{37} -13 q^{39} + 10 \beta q^{41} + 20 q^{43} -16 q^{45} -10 q^{47} -24 q^{49} + 15 \beta q^{51} -21 \beta q^{53} + 40 q^{55} + ( 65 + 6 \beta ) q^{57} -5 \beta q^{59} -40 q^{61} -20 q^{63} + 4 \beta q^{65} -11 \beta q^{67} -35 \beta q^{69} -30 \beta q^{71} + 105 q^{73} -9 \beta q^{75} + 50 q^{77} + 10 \beta q^{79} -101 q^{81} + 40 q^{83} + 60 q^{85} -65 q^{87} + 5 \beta q^{91} -130 q^{93} + ( 24 - 20 \beta ) q^{95} + 34 \beta q^{97} -40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} + 10q^{7} - 8q^{9} + O(q^{10}) \) \( 2q + 8q^{5} + 10q^{7} - 8q^{9} + 20q^{11} + 30q^{17} + 12q^{19} - 70q^{23} - 18q^{25} + 40q^{35} - 26q^{39} + 40q^{43} - 32q^{45} - 20q^{47} - 48q^{49} + 80q^{55} + 130q^{57} - 80q^{61} - 40q^{63} + 210q^{73} + 100q^{77} - 202q^{81} + 80q^{83} + 120q^{85} - 130q^{87} - 260q^{93} + 48q^{95} - 80q^{99} + O(q^{100}) \)

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
3.60555i
3.60555i
0 3.60555i 0 4.00000 0 5.00000 0 −4.00000 0
113.2 0 3.60555i 0 4.00000 0 5.00000 0 −4.00000 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.d 2
3.b odd 2 1 2736.3.o.d 2
4.b odd 2 1 19.3.b.b 2
8.b even 2 1 1216.3.e.h 2
8.d odd 2 1 1216.3.e.g 2
12.b even 2 1 171.3.c.b 2
19.b odd 2 1 inner 304.3.e.d 2
20.d odd 2 1 475.3.c.b 2
20.e even 4 2 475.3.d.b 4
57.d even 2 1 2736.3.o.d 2
76.d even 2 1 19.3.b.b 2
76.f even 6 2 361.3.d.b 4
76.g odd 6 2 361.3.d.b 4
76.k even 18 6 361.3.f.d 12
76.l odd 18 6 361.3.f.d 12
152.b even 2 1 1216.3.e.g 2
152.g odd 2 1 1216.3.e.h 2
228.b odd 2 1 171.3.c.b 2
380.d even 2 1 475.3.c.b 2
380.j odd 4 2 475.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 4.b odd 2 1
19.3.b.b 2 76.d even 2 1
171.3.c.b 2 12.b even 2 1
171.3.c.b 2 228.b odd 2 1
304.3.e.d 2 1.a even 1 1 trivial
304.3.e.d 2 19.b odd 2 1 inner
361.3.d.b 4 76.f even 6 2
361.3.d.b 4 76.g odd 6 2
361.3.f.d 12 76.k even 18 6
361.3.f.d 12 76.l odd 18 6
475.3.c.b 2 20.d odd 2 1
475.3.c.b 2 380.d even 2 1
475.3.d.b 4 20.e even 4 2
475.3.d.b 4 380.j odd 4 2
1216.3.e.g 2 8.d odd 2 1
1216.3.e.g 2 152.b even 2 1
1216.3.e.h 2 8.b even 2 1
1216.3.e.h 2 152.g odd 2 1
2736.3.o.d 2 3.b odd 2 1
2736.3.o.d 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 \(S_{3}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{2} + 13 \)
\( T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 13 + T^{2} \)
$5$ \( ( -4 + T )^{2} \)
$7$ \( ( -5 + T )^{2} \)
$11$ \( ( -10 + T )^{2} \)
$13$ \( 13 + T^{2} \)
$17$ \( ( -15 + T )^{2} \)
$19$ \( 361 - 12 T + T^{2} \)
$23$ \( ( 35 + T )^{2} \)
$29$ \( 325 + T^{2} \)
$31$ \( 1300 + T^{2} \)
$37$ \( 468 + T^{2} \)
$41$ \( 1300 + T^{2} \)
$43$ \( ( -20 + T )^{2} \)
$47$ \( ( 10 + T )^{2} \)
$53$ \( 5733 + T^{2} \)
$59$ \( 325 + T^{2} \)
$61$ \( ( 40 + T )^{2} \)
$67$ \( 1573 + T^{2} \)
$71$ \( 11700 + T^{2} \)
$73$ \( ( -105 + T )^{2} \)
$79$ \( 1300 + T^{2} \)
$83$ \( ( -40 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( 15028 + T^{2} \)
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