Properties

Label 304.2.i.d
Level $304$
Weight $2$
Character orbit 304.i
Analytic conductor $2.427$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} -3 q^{11} -2 \zeta_{6} q^{13} -4 \zeta_{6} q^{15} + ( -2 + 2 \zeta_{6} ) q^{17} + ( 2 - 5 \zeta_{6} ) q^{19} + 6 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + 5 q^{27} + 4 \zeta_{6} q^{29} + 10 q^{31} + ( -3 + 3 \zeta_{6} ) q^{33} + 2 q^{37} -2 q^{39} + ( -9 + 9 \zeta_{6} ) q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 8 q^{45} -12 \zeta_{6} q^{47} -7 q^{49} + 2 \zeta_{6} q^{51} + 2 \zeta_{6} q^{53} + ( -12 + 12 \zeta_{6} ) q^{55} + ( -3 - 2 \zeta_{6} ) q^{57} + ( -1 + \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -8 q^{65} + 9 \zeta_{6} q^{67} + 6 q^{69} + ( -6 + 6 \zeta_{6} ) q^{71} + ( 9 - 9 \zeta_{6} ) q^{73} -11 q^{75} + ( -4 + 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 5 q^{83} + 8 \zeta_{6} q^{85} + 4 q^{87} + 18 \zeta_{6} q^{89} + ( 10 - 10 \zeta_{6} ) q^{93} + ( -12 - 8 \zeta_{6} ) q^{95} + ( -1 + \zeta_{6} ) q^{97} -6 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + q^{3} + 4q^{5} + 2q^{9} - 6q^{11} - 2q^{13} - 4q^{15} - 2q^{17} - q^{19} + 6q^{23} - 11q^{25} + 10q^{27} + 4q^{29} + 20q^{31} - 3q^{33} + 4q^{37} - 4q^{39} - 9q^{41} - 4q^{43} + 16q^{45} - 12q^{47} - 14q^{49} + 2q^{51} + 2q^{53} - 12q^{55} - 8q^{57} - q^{59} + 8q^{61} - 16q^{65} + 9q^{67} + 12q^{69} - 6q^{71} + 9q^{73} - 22q^{75} - 4q^{79} - q^{81} + 10q^{83} + 8q^{85} + 8q^{87} + 18q^{89} + 10q^{93} - 32q^{95} - q^{97} - 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 2.00000 3.46410i 0 0 0 1.00000 + 1.73205i 0
273.1 0 0.500000 + 0.866025i 0 2.00000 + 3.46410i 0 0 0 1.00000 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.i.d 2
3.b odd 2 1 2736.2.s.a 2
4.b odd 2 1 152.2.i.b 2
8.b even 2 1 1216.2.i.b 2
8.d odd 2 1 1216.2.i.f 2
12.b even 2 1 1368.2.s.a 2
19.c even 3 1 inner 304.2.i.d 2
19.c even 3 1 5776.2.a.e 1
19.d odd 6 1 5776.2.a.j 1
57.h odd 6 1 2736.2.s.a 2
76.f even 6 1 2888.2.a.a 1
76.g odd 6 1 152.2.i.b 2
76.g odd 6 1 2888.2.a.d 1
152.k odd 6 1 1216.2.i.f 2
152.p even 6 1 1216.2.i.b 2
228.m even 6 1 1368.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.b 2 4.b odd 2 1
152.2.i.b 2 76.g odd 6 1
304.2.i.d 2 1.a even 1 1 trivial
304.2.i.d 2 19.c even 3 1 inner
1216.2.i.b 2 8.b even 2 1
1216.2.i.b 2 152.p even 6 1
1216.2.i.f 2 8.d odd 2 1
1216.2.i.f 2 152.k odd 6 1
1368.2.s.a 2 12.b even 2 1
1368.2.s.a 2 228.m even 6 1
2736.2.s.a 2 3.b odd 2 1
2736.2.s.a 2 57.h odd 6 1
2888.2.a.a 1 76.f even 6 1
2888.2.a.d 1 76.g odd 6 1
5776.2.a.e 1 19.c even 3 1
5776.2.a.j 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5}^{2} - 4 T_{5} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 19 + T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 16 - 4 T + T^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 81 + 9 T + T^{2} \)
$43$ \( 16 + 4 T + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( 4 - 2 T + T^{2} \)
$59$ \( 1 + T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 81 - 9 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( 81 - 9 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( ( -5 + T )^{2} \)
$89$ \( 324 - 18 T + T^{2} \)
$97$ \( 1 + T + T^{2} \)
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