Newform orbit 304.4.a.d

• Label: 304.4.a.d
• Level: $304$
• Weight: $4$
• Character orbit: 304.a
• Self dual: yes
• Analytic conductor: $17.937$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(17.9365806417\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{177}) \)
Defining polynomial:	\( x^{2} - x - 44 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^3 + (-2*b + 6) * q^5 + (-b - 28) * q^7 + (b + 17) * q^9 + (-2*b - 4) * q^11 + (-7*b + 10) * q^13 + (-4*b + 88) * q^15 + (-15*b - 18) * q^17 + 19 * q^19 + (29*b + 44) * q^21 + (-13*b + 84) * q^23 + (-20*b + 87) * q^25 + (9*b - 44) * q^27 + (29*b - 54) * q^29 + 16*b * q^31 + (6*b + 88) * q^33 + (52*b - 80) * q^35 + (-16*b + 198) * q^37 + (-3*b + 308) * q^39 + (6*b - 398) * q^41 + (-48*b - 124) * q^43 + (-30*b + 14) * q^45 + (-40*b + 120) * q^47 + (57*b + 485) * q^49 + (33*b + 660) * q^51 + (9*b + 194) * q^53 + 152 * q^55 - 19*b * q^57 + (71*b - 136) * q^59 + (44*b - 362) * q^61 + (-46*b - 520) * q^63 + (-48*b + 676) * q^65 + (43*b + 448) * q^67 + (-71*b + 572) * q^69 + (-22*b - 192) * q^71 + (-b + 62) * q^73 + (-67*b + 880) * q^75 + (62*b + 200) * q^77 + (-58*b - 24) * q^79 + (8*b - 855) * q^81 + (6*b - 1116) * q^83 + (-24*b + 1212) * q^85 + (25*b - 1276) * q^87 + (-10*b - 430) * q^89 + (193*b + 28) * q^91 + (-16*b - 704) * q^93 + (-38*b + 114) * q^95 + (-76*b - 894) * q^97 + (-40*b - 156) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^3 + 10 * q^5 - 57 * q^7 + 35 * q^9 - 10 * q^11 + 13 * q^13 + 172 * q^15 - 51 * q^17 + 38 * q^19 + 117 * q^21 + 155 * q^23 + 154 * q^25 - 79 * q^27 - 79 * q^29 + 16 * q^31 + 182 * q^33 - 108 * q^35 + 380 * q^37 + 613 * q^39 - 790 * q^41 - 296 * q^43 - 2 * q^45 + 200 * q^47 + 1027 * q^49 + 1353 * q^51 + 397 * q^53 + 304 * q^55 - 19 * q^57 - 201 * q^59 - 680 * q^61 - 1086 * q^63 + 1304 * q^65 + 939 * q^67 + 1073 * q^69 - 406 * q^71 + 123 * q^73 + 1693 * q^75 + 462 * q^77 - 106 * q^79 - 1702 * q^81 - 2226 * q^83 + 2400 * q^85 - 2527 * q^87 - 870 * q^89 + 249 * q^91 - 1424 * q^93 + 190 * q^95 - 1864 * q^97 - 352 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

7.15207
−6.15207

0

−7.15207

0

−8.30413

0

−35.1521

0

24.1521

0

1.2

0

6.15207

0

18.3041

0

−21.8479

0

10.8479

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(19\)	\( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.4.a.d		2
4.b	odd	2	1		38.4.a.b	✓	2
8.b	even	2	1		1216.4.a.l		2
8.d	odd	2	1		1216.4.a.j		2
12.b	even	2	1		342.4.a.k		2
20.d	odd	2	1		950.4.a.h		2
20.e	even	4	2		950.4.b.g		4
28.d	even	2	1		1862.4.a.b		2
76.d	even	2	1		722.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.4.a.b	✓	2	4.b	odd	2	1
304.4.a.d		2	1.a	even	1	1	trivial
342.4.a.k		2	12.b	even	2	1
722.4.a.i		2	76.d	even	2	1
950.4.a.h		2	20.d	odd	2	1
950.4.b.g		4	20.e	even	4	2
1216.4.a.j		2	8.d	odd	2	1
1216.4.a.l		2	8.b	even	2	1
1862.4.a.b		2	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T - 44 \)
$5$	\( T^{2} - 10T - 152 \)
$7$	\( T^{2} + 57T + 768 \)
$11$	\( T^{2} + 10T - 152 \)