Newform orbit 546.4.a.k

• Label: 546.4.a.k
• Level: $546$
• Weight: $4$
• Character orbit: 546.a
• Self dual: yes
• Analytic conductor: $32.215$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.2150428631\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{1401}) \)
Defining polynomial:	\( x^{2} - x - 350 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1401})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + 3 * q^3 + 4 * q^4 + (-b + 3) * q^5 + 6 * q^6 - 7 * q^7 + 8 * q^8 + 9 * q^9 + (-2*b + 6) * q^10 + (b - 9) * q^11 + 12 * q^12 - 13 * q^13 - 14 * q^14 + (-3*b + 9) * q^15 + 16 * q^16 + (b + 67) * q^17 + 18 * q^18 + (b + 85) * q^19 + (-4*b + 12) * q^20 - 21 * q^21 + (2*b - 18) * q^22 + (-b + 147) * q^23 + 24 * q^24 + (-5*b + 234) * q^25 - 26 * q^26 + 27 * q^27 - 28 * q^28 + (b - 97) * q^29 + (-6*b + 18) * q^30 + (8*b + 60) * q^31 + 32 * q^32 + (3*b - 27) * q^33 + (2*b + 134) * q^34 + (7*b - 21) * q^35 + 36 * q^36 + (7*b - 239) * q^37 + (2*b + 170) * q^38 - 39 * q^39 + (-8*b + 24) * q^40 + (8*b + 104) * q^41 - 42 * q^42 + (-9*b + 151) * q^43 + (4*b - 36) * q^44 + (-9*b + 27) * q^45 + (-2*b + 294) * q^46 + (10*b - 204) * q^47 + 48 * q^48 + 49 * q^49 + (-10*b + 468) * q^50 + (3*b + 201) * q^51 - 52 * q^52 + (8*b + 110) * q^53 + 54 * q^54 + (11*b - 377) * q^55 - 56 * q^56 + (3*b + 255) * q^57 + (2*b - 194) * q^58 + (42*b - 44) * q^59 + (-12*b + 36) * q^60 + (-3*b + 523) * q^61 + (16*b + 120) * q^62 - 63 * q^63 + 64 * q^64 + (13*b - 39) * q^65 + (6*b - 54) * q^66 + (-46*b - 18) * q^67 + (4*b + 268) * q^68 + (-3*b + 441) * q^69 + (14*b - 42) * q^70 + (-26*b - 420) * q^71 + 72 * q^72 + (-35*b - 177) * q^73 + (14*b - 478) * q^74 + (-15*b + 702) * q^75 + (4*b + 340) * q^76 + (-7*b + 63) * q^77 - 78 * q^78 + (22*b - 450) * q^79 + (-16*b + 48) * q^80 + 81 * q^81 + (16*b + 208) * q^82 + (60*b + 66) * q^83 - 84 * q^84 + (-65*b - 149) * q^85 + (-18*b + 302) * q^86 + (3*b - 291) * q^87 + (8*b - 72) * q^88 + (-10*b - 310) * q^89 + (-18*b + 54) * q^90 + 91 * q^91 + (-4*b + 588) * q^92 + (24*b + 180) * q^93 + (20*b - 408) * q^94 + (-83*b - 95) * q^95 + 96 * q^96 + (-16*b - 194) * q^97 + 98 * q^98 + (9*b - 81) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 5 * q^5 + 12 * q^6 - 14 * q^7 + 16 * q^8 + 18 * q^9 + 10 * q^10 - 17 * q^11 + 24 * q^12 - 26 * q^13 - 28 * q^14 + 15 * q^15 + 32 * q^16 + 135 * q^17 + 36 * q^18 + 171 * q^19 + 20 * q^20 - 42 * q^21 - 34 * q^22 + 293 * q^23 + 48 * q^24 + 463 * q^25 - 52 * q^26 + 54 * q^27 - 56 * q^28 - 193 * q^29 + 30 * q^30 + 128 * q^31 + 64 * q^32 - 51 * q^33 + 270 * q^34 - 35 * q^35 + 72 * q^36 - 471 * q^37 + 342 * q^38 - 78 * q^39 + 40 * q^40 + 216 * q^41 - 84 * q^42 + 293 * q^43 - 68 * q^44 + 45 * q^45 + 586 * q^46 - 398 * q^47 + 96 * q^48 + 98 * q^49 + 926 * q^50 + 405 * q^51 - 104 * q^52 + 228 * q^53 + 108 * q^54 - 743 * q^55 - 112 * q^56 + 513 * q^57 - 386 * q^58 - 46 * q^59 + 60 * q^60 + 1043 * q^61 + 256 * q^62 - 126 * q^63 + 128 * q^64 - 65 * q^65 - 102 * q^66 - 82 * q^67 + 540 * q^68 + 879 * q^69 - 70 * q^70 - 866 * q^71 + 144 * q^72 - 389 * q^73 - 942 * q^74 + 1389 * q^75 + 684 * q^76 + 119 * q^77 - 156 * q^78 - 878 * q^79 + 80 * q^80 + 162 * q^81 + 432 * q^82 + 192 * q^83 - 168 * q^84 - 363 * q^85 + 586 * q^86 - 579 * q^87 - 136 * q^88 - 630 * q^89 + 90 * q^90 + 182 * q^91 + 1172 * q^92 + 384 * q^93 - 796 * q^94 - 273 * q^95 + 192 * q^96 - 404 * q^97 + 196 * q^98 - 153 * 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

19.2150
−18.2150

2.00000

3.00000

4.00000

−16.2150

6.00000

−7.00000

8.00000

9.00000

−32.4299

1.2

2.00000

3.00000

4.00000

21.2150

6.00000

−7.00000

8.00000

9.00000

42.4299

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)
\(13\)	\( +1 \)

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	546.4.a.k	✓	2
3.b	odd	2	1		1638.4.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.4.a.k	✓	2	1.a	even	1	1	trivial
1638.4.a.l		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 5T_{5} - 344 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} - 5T - 344 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 17T - 278 \)