Newform orbit 672.4.a.h

• Label: 672.4.a.h
• Level: $672$
• Weight: $4$
• Character orbit: 672.a
• Self dual: yes
• Analytic conductor: $39.649$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(39.6492835239\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{137}) \)
Defining polynomial:	\( x^{2} - x - 34 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 3 * q^3 + (-b + 5) * q^5 - 7 * q^7 + 9 * q^9 + (b - 11) * q^11 + (2*b + 12) * q^13 + (3*b - 15) * q^15 + (b + 15) * q^17 + (4*b - 16) * q^19 + 21 * q^21 + (3*b - 41) * q^23 + (-10*b + 37) * q^25 - 27 * q^27 + (20*b + 18) * q^29 + (8*b - 56) * q^31 + (-3*b + 33) * q^33 + (7*b - 35) * q^35 + (-14*b + 24) * q^37 + (-6*b - 36) * q^39 + (-b - 35) * q^41 + (-20*b - 20) * q^43 + (-9*b + 45) * q^45 + (-34*b - 210) * q^47 + 49 * q^49 + (-3*b - 45) * q^51 + (18*b + 88) * q^53 + (16*b - 192) * q^55 + (-12*b + 48) * q^57 + (2*b - 202) * q^59 + (40*b - 78) * q^61 - 63 * q^63 + (-2*b - 214) * q^65 + (-30*b + 2) * q^67 + (-9*b + 123) * q^69 + (5*b - 407) * q^71 + (-46*b - 108) * q^73 + (30*b - 111) * q^75 + (-7*b + 77) * q^77 + (-22*b - 790) * q^79 + 81 * q^81 + (-4*b - 232) * q^83 + (-10*b - 62) * q^85 + (-60*b - 54) * q^87 + (87*b - 579) * q^89 + (-14*b - 84) * q^91 + (-24*b + 168) * q^93 + (36*b - 628) * q^95 + (62*b - 880) * q^97 + (9*b - 99) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 22 * q^11 + 24 * q^13 - 30 * q^15 + 30 * q^17 - 32 * q^19 + 42 * q^21 - 82 * q^23 + 74 * q^25 - 54 * q^27 + 36 * q^29 - 112 * q^31 + 66 * q^33 - 70 * q^35 + 48 * q^37 - 72 * q^39 - 70 * q^41 - 40 * q^43 + 90 * q^45 - 420 * q^47 + 98 * q^49 - 90 * q^51 + 176 * q^53 - 384 * q^55 + 96 * q^57 - 404 * q^59 - 156 * q^61 - 126 * q^63 - 428 * q^65 + 4 * q^67 + 246 * q^69 - 814 * q^71 - 216 * q^73 - 222 * q^75 + 154 * q^77 - 1580 * q^79 + 162 * q^81 - 464 * q^83 - 124 * q^85 - 108 * q^87 - 1158 * q^89 - 168 * q^91 + 336 * q^93 - 1256 * q^95 - 1760 * q^97 - 198 * 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

6.35235
−5.35235

0

−3.00000

0

−6.70470

0

−7.00000

0

9.00000

0

1.2

0

−3.00000

0

16.7047

0

−7.00000

0

9.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( +1 \)
\(7\)	\( +1 \)

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	672.4.a.h	✓	2
3.b	odd	2	1		2016.4.a.i		2
4.b	odd	2	1		672.4.a.m	yes	2
8.b	even	2	1		1344.4.a.bm		2
8.d	odd	2	1		1344.4.a.be		2
12.b	even	2	1		2016.4.a.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
672.4.a.h	✓	2	1.a	even	1	1	trivial
672.4.a.m	yes	2	4.b	odd	2	1
1344.4.a.be		2	8.d	odd	2	1
1344.4.a.bm		2	8.b	even	2	1
2016.4.a.i		2	3.b	odd	2	1
2016.4.a.j		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(672))\):

\( T_{5}^{2} - 10T_{5} - 112 \)

\( T_{11}^{2} + 22T_{11} - 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( T^{2} - 10T - 112 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 22T - 16 \)