Newform orbit 315.4.a.k

• Label: 315.4.a.k
• Level: $315$
• Weight: $4$
• Character orbit: 315.a
• Self dual: yes
• Analytic conductor: $18.586$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.5856016518\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q + (b + 1) * q^2 + (2*b + 1) * q^4 - 5 * q^5 - 7 * q^7 + (-5*b + 9) * q^8 + (-5*b - 5) * q^10 + (-20*b + 8) * q^11 + (-2*b - 38) * q^13 + (-7*b - 7) * q^14 + (-12*b - 39) * q^16 + (2*b + 62) * q^17 + (-16*b - 48) * q^19 + (-10*b - 5) * q^20 + (-12*b - 152) * q^22 + (34*b + 8) * q^23 + 25 * q^25 + (-40*b - 54) * q^26 + (-14*b - 7) * q^28 + (54*b - 94) * q^29 + (18*b - 60) * q^31 + (-11*b - 207) * q^32 + (64*b + 78) * q^34 + 35 * q^35 + (-66*b - 66) * q^37 + (-64*b - 176) * q^38 + (25*b - 45) * q^40 + (-80*b - 50) * q^41 + (62*b - 268) * q^43 + (-4*b - 312) * q^44 + (42*b + 280) * q^46 + (42*b + 464) * q^47 + 49 * q^49 + (25*b + 25) * q^50 + (-78*b - 70) * q^52 + (-64*b - 442) * q^53 + (100*b - 40) * q^55 + (35*b - 63) * q^56 + (-40*b + 338) * q^58 + (204*b - 52) * q^59 + (-42*b - 234) * q^61 + (-42*b + 84) * q^62 + (-122*b + 17) * q^64 + (10*b + 190) * q^65 + (38*b - 844) * q^67 + (126*b + 94) * q^68 + (35*b + 35) * q^70 + (150*b + 68) * q^71 + (322*b + 254) * q^73 + (-132*b - 594) * q^74 + (-112*b - 304) * q^76 + (140*b - 56) * q^77 + (-232*b - 216) * q^79 + (60*b + 195) * q^80 + (-130*b - 690) * q^82 + (84*b + 292) * q^83 + (-10*b - 310) * q^85 + (-206*b + 228) * q^86 + (-220*b + 872) * q^88 + (-112*b + 702) * q^89 + (14*b + 266) * q^91 + (50*b + 552) * q^92 + (506*b + 800) * q^94 + (80*b + 240) * q^95 + (46*b - 594) * q^97 + (49*b + 49) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8 - 10 * q^10 + 16 * q^11 - 76 * q^13 - 14 * q^14 - 78 * q^16 + 124 * q^17 - 96 * q^19 - 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 - 108 * q^26 - 14 * q^28 - 188 * q^29 - 120 * q^31 - 414 * q^32 + 156 * q^34 + 70 * q^35 - 132 * q^37 - 352 * q^38 - 90 * q^40 - 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 + 928 * q^47 + 98 * q^49 + 50 * q^50 - 140 * q^52 - 884 * q^53 - 80 * q^55 - 126 * q^56 + 676 * q^58 - 104 * q^59 - 468 * q^61 + 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 + 188 * q^68 + 70 * q^70 + 136 * q^71 + 508 * q^73 - 1188 * q^74 - 608 * q^76 - 112 * q^77 - 432 * q^79 + 390 * q^80 - 1380 * q^82 + 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 + 1404 * q^89 + 532 * q^91 + 1104 * q^92 + 1600 * q^94 + 480 * q^95 - 1188 * q^97 + 98 * q^98

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

−1.82843

0

−4.65685

−5.00000

0

−7.00000

23.1421

0

9.14214

1.2

3.82843

0

6.65685

−5.00000

0

−7.00000

−5.14214

0

−19.1421

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( +1 \)
\(7\)	\( +1 \)

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	315.4.a.k		2
3.b	odd	2	1		105.4.a.e	✓	2
5.b	even	2	1		1575.4.a.q		2
7.b	odd	2	1		2205.4.a.bb		2
12.b	even	2	1		1680.4.a.bo		2
15.d	odd	2	1		525.4.a.l		2
15.e	even	4	2		525.4.d.l		4
21.c	even	2	1		735.4.a.o		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.e	✓	2	3.b	odd	2	1
315.4.a.k		2	1.a	even	1	1	trivial
525.4.a.l		2	15.d	odd	2	1
525.4.d.l		4	15.e	even	4	2
735.4.a.o		2	21.c	even	2	1
1575.4.a.q		2	5.b	even	2	1
1680.4.a.bo		2	12.b	even	2	1
2205.4.a.bb		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 7 \)
$3$	\( T^{2} \)
$5$	\( (T + 5)^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 16T - 3136 \)