Newform orbit 105.4.a.e

• Label: 105.4.a.e
• Level: $105$
• Weight: $4$
• Character orbit: 105.a
• Self dual: yes
• Analytic conductor: $6.195$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.19520055060\)
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:	yes
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 - 3 * q^3 + (-2*b + 1) * q^4 + 5 * q^5 + (-3*b + 3) * q^6 - 7 * q^7 + (-5*b - 9) * q^8 + 9 * q^9 + (5*b - 5) * q^10 + (-20*b - 8) * q^11 + (6*b - 3) * q^12 + (2*b - 38) * q^13 + (-7*b + 7) * q^14 - 15 * q^15 + (12*b - 39) * q^16 + (2*b - 62) * q^17 + (9*b - 9) * q^18 + (16*b - 48) * q^19 + (-10*b + 5) * q^20 + 21 * q^21 + (12*b - 152) * q^22 + (34*b - 8) * q^23 + (15*b + 27) * q^24 + 25 * q^25 + (-40*b + 54) * q^26 - 27 * q^27 + (14*b - 7) * q^28 + (54*b + 94) * q^29 + (-15*b + 15) * q^30 + (-18*b - 60) * q^31 + (-11*b + 207) * q^32 + (60*b + 24) * q^33 + (-64*b + 78) * q^34 - 35 * q^35 + (-18*b + 9) * q^36 + (66*b - 66) * q^37 + (-64*b + 176) * q^38 + (-6*b + 114) * q^39 + (-25*b - 45) * q^40 + (-80*b + 50) * q^41 + (21*b - 21) * q^42 + (-62*b - 268) * q^43 + (-4*b + 312) * q^44 + 45 * q^45 + (-42*b + 280) * q^46 + (42*b - 464) * q^47 + (-36*b + 117) * q^48 + 49 * q^49 + (25*b - 25) * q^50 + (-6*b + 186) * q^51 + (78*b - 70) * q^52 + (-64*b + 442) * q^53 + (-27*b + 27) * q^54 + (-100*b - 40) * q^55 + (35*b + 63) * q^56 + (-48*b + 144) * q^57 + (40*b + 338) * q^58 + (204*b + 52) * q^59 + (30*b - 15) * q^60 + (42*b - 234) * q^61 + (-42*b - 84) * q^62 - 63 * q^63 + (122*b + 17) * q^64 + (10*b - 190) * q^65 + (-36*b + 456) * q^66 + (-38*b - 844) * q^67 + (126*b - 94) * q^68 + (-102*b + 24) * q^69 + (-35*b + 35) * q^70 + (150*b - 68) * q^71 + (-45*b - 81) * q^72 + (-322*b + 254) * q^73 + (-132*b + 594) * q^74 - 75 * q^75 + (112*b - 304) * q^76 + (140*b + 56) * q^77 + (120*b - 162) * q^78 + (232*b - 216) * q^79 + (60*b - 195) * q^80 + 81 * q^81 + (130*b - 690) * q^82 + (84*b - 292) * q^83 + (-42*b + 21) * q^84 + (10*b - 310) * q^85 + (-206*b - 228) * q^86 + (-162*b - 282) * q^87 + (220*b + 872) * q^88 + (-112*b - 702) * q^89 + (45*b - 45) * q^90 + (-14*b + 266) * q^91 + (50*b - 552) * q^92 + (54*b + 180) * q^93 + (-506*b + 800) * q^94 + (80*b - 240) * q^95 + (33*b - 621) * q^96 + (-46*b - 594) * q^97 + (49*b - 49) * q^98 + (-180*b - 72) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^6 - 14 * q^7 - 18 * q^8 + 18 * q^9 - 10 * q^10 - 16 * q^11 - 6 * q^12 - 76 * q^13 + 14 * q^14 - 30 * q^15 - 78 * q^16 - 124 * q^17 - 18 * q^18 - 96 * q^19 + 10 * q^20 + 42 * q^21 - 304 * q^22 - 16 * q^23 + 54 * q^24 + 50 * q^25 + 108 * q^26 - 54 * q^27 - 14 * q^28 + 188 * q^29 + 30 * q^30 - 120 * q^31 + 414 * q^32 + 48 * q^33 + 156 * q^34 - 70 * q^35 + 18 * q^36 - 132 * q^37 + 352 * q^38 + 228 * q^39 - 90 * q^40 + 100 * q^41 - 42 * q^42 - 536 * q^43 + 624 * q^44 + 90 * q^45 + 560 * q^46 - 928 * q^47 + 234 * q^48 + 98 * q^49 - 50 * q^50 + 372 * q^51 - 140 * q^52 + 884 * q^53 + 54 * q^54 - 80 * q^55 + 126 * q^56 + 288 * q^57 + 676 * q^58 + 104 * q^59 - 30 * q^60 - 468 * q^61 - 168 * q^62 - 126 * q^63 + 34 * q^64 - 380 * q^65 + 912 * q^66 - 1688 * q^67 - 188 * q^68 + 48 * q^69 + 70 * q^70 - 136 * q^71 - 162 * q^72 + 508 * q^73 + 1188 * q^74 - 150 * q^75 - 608 * q^76 + 112 * q^77 - 324 * q^78 - 432 * q^79 - 390 * q^80 + 162 * q^81 - 1380 * q^82 - 584 * q^83 + 42 * q^84 - 620 * q^85 - 456 * q^86 - 564 * q^87 + 1744 * q^88 - 1404 * q^89 - 90 * q^90 + 532 * q^91 - 1104 * q^92 + 360 * q^93 + 1600 * q^94 - 480 * q^95 - 1242 * q^96 - 1188 * q^97 - 98 * q^98 - 144 * 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

−3.82843

−3.00000

6.65685

5.00000

11.4853

−7.00000

5.14214

9.00000

−19.1421

1.2

1.82843

−3.00000

−4.65685

5.00000

−5.48528

−7.00000

−23.1421

9.00000

9.14214

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.e	✓	2	1.a	even	1	1	trivial
315.4.a.k		2	3.b	odd	2	1
525.4.a.l		2	5.b	even	2	1
525.4.d.l		4	5.c	odd	4	2
735.4.a.o		2	7.b	odd	2	1
1575.4.a.q		2	15.d	odd	2	1
1680.4.a.bo		2	4.b	odd	2	1
2205.4.a.bb		2	21.c	even	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 \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).

Hecke characteristic polynomials

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