Newform orbit 294.4.a.k

• Label: 294.4.a.k
• Level: $294$
• Weight: $4$
• Character orbit: 294.a
• Self dual: yes
• Analytic conductor: $17.347$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(17.3465615417\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 7 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 7\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 3 * q^3 + 4 * q^4 + (b - 6) * q^5 - 6 * q^6 - 8 * q^8 + 9 * q^9 + (-2*b + 12) * q^10 + (-6*b - 2) * q^11 + 12 * q^12 + (-3*b - 24) * q^13 + (3*b - 18) * q^15 + 16 * q^16 + (b - 42) * q^17 - 18 * q^18 + (2*b + 36) * q^19 + (4*b - 24) * q^20 + (12*b + 4) * q^22 + (6*b - 154) * q^23 - 24 * q^24 + (-12*b + 9) * q^25 + (6*b + 48) * q^26 + 27 * q^27 + (18*b - 40) * q^29 + (-6*b + 36) * q^30 + (6*b - 192) * q^31 - 32 * q^32 + (-18*b - 6) * q^33 + (-2*b + 84) * q^34 + 36 * q^36 + (-12*b + 268) * q^37 + (-4*b - 72) * q^38 + (-9*b - 72) * q^39 + (-8*b + 48) * q^40 + (-5*b - 378) * q^41 + (24*b + 200) * q^43 + (-24*b - 8) * q^44 + (9*b - 54) * q^45 + (-12*b + 308) * q^46 + (10*b - 156) * q^47 + 48 * q^48 + (24*b - 18) * q^50 + (3*b - 126) * q^51 + (-12*b - 96) * q^52 + (-24*b - 26) * q^53 - 54 * q^54 + (34*b - 576) * q^55 + (6*b + 108) * q^57 + (-36*b + 80) * q^58 + (-2*b - 432) * q^59 + (12*b - 72) * q^60 + (13*b - 708) * q^61 + (-12*b + 384) * q^62 + 64 * q^64 + (-6*b - 150) * q^65 + (36*b + 12) * q^66 + (24*b + 72) * q^67 + (4*b - 168) * q^68 + (18*b - 462) * q^69 + (-30*b - 762) * q^71 - 72 * q^72 + (-83*b - 372) * q^73 + (24*b - 536) * q^74 + (-36*b + 27) * q^75 + (8*b + 144) * q^76 + (18*b + 144) * q^78 + (84*b + 488) * q^79 + (16*b - 96) * q^80 + 81 * q^81 + (10*b + 756) * q^82 + (-136*b + 156) * q^83 + (-48*b + 350) * q^85 + (-48*b - 400) * q^86 + (54*b - 120) * q^87 + (48*b + 16) * q^88 + (29*b - 54) * q^89 + (-18*b + 108) * q^90 + (24*b - 616) * q^92 + (18*b - 576) * q^93 + (-20*b + 312) * q^94 + (24*b - 20) * q^95 - 96 * q^96 + (125*b + 372) * q^97 + (-54*b - 18) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 - 12 * q^5 - 12 * q^6 - 16 * q^8 + 18 * q^9 + 24 * q^10 - 4 * q^11 + 24 * q^12 - 48 * q^13 - 36 * q^15 + 32 * q^16 - 84 * q^17 - 36 * q^18 + 72 * q^19 - 48 * q^20 + 8 * q^22 - 308 * q^23 - 48 * q^24 + 18 * q^25 + 96 * q^26 + 54 * q^27 - 80 * q^29 + 72 * q^30 - 384 * q^31 - 64 * q^32 - 12 * q^33 + 168 * q^34 + 72 * q^36 + 536 * q^37 - 144 * q^38 - 144 * q^39 + 96 * q^40 - 756 * q^41 + 400 * q^43 - 16 * q^44 - 108 * q^45 + 616 * q^46 - 312 * q^47 + 96 * q^48 - 36 * q^50 - 252 * q^51 - 192 * q^52 - 52 * q^53 - 108 * q^54 - 1152 * q^55 + 216 * q^57 + 160 * q^58 - 864 * q^59 - 144 * q^60 - 1416 * q^61 + 768 * q^62 + 128 * q^64 - 300 * q^65 + 24 * q^66 + 144 * q^67 - 336 * q^68 - 924 * q^69 - 1524 * q^71 - 144 * q^72 - 744 * q^73 - 1072 * q^74 + 54 * q^75 + 288 * q^76 + 288 * q^78 + 976 * q^79 - 192 * q^80 + 162 * q^81 + 1512 * q^82 + 312 * q^83 + 700 * q^85 - 800 * q^86 - 240 * q^87 + 32 * q^88 - 108 * q^89 + 216 * q^90 - 1232 * q^92 - 1152 * q^93 + 624 * q^94 - 40 * q^95 - 192 * q^96 + 744 * q^97 - 36 * 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

−1.41421
1.41421

−2.00000

3.00000

4.00000

−15.8995

−6.00000

0

−8.00000

9.00000

31.7990

1.2

−2.00000

3.00000

4.00000

3.89949

−6.00000

0

−8.00000

9.00000

−7.79899

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	294.4.a.k	yes	2
3.b	odd	2	1		882.4.a.bi		2
4.b	odd	2	1		2352.4.a.bn		2
7.b	odd	2	1		294.4.a.j	✓	2
7.c	even	3	2		294.4.e.n		4
7.d	odd	6	2		294.4.e.o		4
21.c	even	2	1		882.4.a.bc		2
21.g	even	6	2		882.4.g.bd		4
21.h	odd	6	2		882.4.g.y		4
28.d	even	2	1		2352.4.a.cd		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
294.4.a.j	✓	2	7.b	odd	2	1
294.4.a.k	yes	2	1.a	even	1	1	trivial
294.4.e.n		4	7.c	even	3	2
294.4.e.o		4	7.d	odd	6	2
882.4.a.bc		2	21.c	even	2	1
882.4.a.bi		2	3.b	odd	2	1
882.4.g.y		4	21.h	odd	6	2
882.4.g.bd		4	21.g	even	6	2
2352.4.a.bn		2	4.b	odd	2	1
2352.4.a.cd		2	28.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5}^{2} + 12T_{5} - 62 \)

\( T_{11}^{2} + 4T_{11} - 3524 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} + 12T - 62 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T - 3524 \)