Newform orbit 765.4.a.i

• Label: 765.4.a.i
• Level: $765$
• Weight: $4$
• Character orbit: 765.a
• Self dual: yes
• Analytic conductor: $45.136$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(45.1364611544\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 2) * q^2 + (4*b - 1) * q^4 - 5 * q^5 + (9*b + 1) * q^7 + (-b - 6) * q^8 + (-5*b - 10) * q^10 + (13*b + 19) * q^11 + (28*b - 40) * q^13 + (19*b + 29) * q^14 + (-40*b - 7) * q^16 + 17 * q^17 + (-6*b - 90) * q^19 + (-20*b + 5) * q^20 + (45*b + 77) * q^22 + (101*b + 21) * q^23 + 25 * q^25 + (16*b + 4) * q^26 + (-5*b + 107) * q^28 + (46*b + 108) * q^29 + (141*b + 35) * q^31 + (-79*b - 86) * q^32 + (17*b + 34) * q^34 + (-45*b - 5) * q^35 + (-106*b + 176) * q^37 + (-102*b - 198) * q^38 + (5*b + 30) * q^40 + (142*b - 172) * q^41 + (-166*b - 44) * q^43 + (63*b + 137) * q^44 + (223*b + 345) * q^46 + (8*b + 174) * q^47 + (18*b - 99) * q^49 + (25*b + 50) * q^50 + (-188*b + 376) * q^52 + (64*b + 82) * q^53 + (-65*b - 95) * q^55 + (-55*b - 33) * q^56 + (200*b + 354) * q^58 + (-66*b + 718) * q^59 + (268*b - 38) * q^61 + (317*b + 493) * q^62 + (76*b - 353) * q^64 + (-140*b + 200) * q^65 + (308*b - 22) * q^67 + (68*b - 17) * q^68 + (-95*b - 145) * q^70 + (-119*b + 755) * q^71 + (18*b + 544) * q^73 + (-36*b + 34) * q^74 + (-354*b + 18) * q^76 + (184*b + 370) * q^77 + (-437*b - 479) * q^79 + (200*b + 35) * q^80 + (112*b + 82) * q^82 + (-238*b - 124) * q^83 - 85 * q^85 + (-376*b - 586) * q^86 + (-97*b - 153) * q^88 + (-742*b + 102) * q^89 + (-332*b + 716) * q^91 + (-17*b + 1191) * q^92 + (190*b + 372) * q^94 + (30*b + 450) * q^95 + (436*b + 506) * q^97 + (-63*b - 144) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 - 2 * q^4 - 10 * q^5 + 2 * q^7 - 12 * q^8 - 20 * q^10 + 38 * q^11 - 80 * q^13 + 58 * q^14 - 14 * q^16 + 34 * q^17 - 180 * q^19 + 10 * q^20 + 154 * q^22 + 42 * q^23 + 50 * q^25 + 8 * q^26 + 214 * q^28 + 216 * q^29 + 70 * q^31 - 172 * q^32 + 68 * q^34 - 10 * q^35 + 352 * q^37 - 396 * q^38 + 60 * q^40 - 344 * q^41 - 88 * q^43 + 274 * q^44 + 690 * q^46 + 348 * q^47 - 198 * q^49 + 100 * q^50 + 752 * q^52 + 164 * q^53 - 190 * q^55 - 66 * q^56 + 708 * q^58 + 1436 * q^59 - 76 * q^61 + 986 * q^62 - 706 * q^64 + 400 * q^65 - 44 * q^67 - 34 * q^68 - 290 * q^70 + 1510 * q^71 + 1088 * q^73 + 68 * q^74 + 36 * q^76 + 740 * q^77 - 958 * q^79 + 70 * q^80 + 164 * q^82 - 248 * q^83 - 170 * q^85 - 1172 * q^86 - 306 * q^88 + 204 * q^89 + 1432 * q^91 + 2382 * q^92 + 744 * q^94 + 900 * q^95 + 1012 * q^97 - 288 * 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

−1.73205
1.73205

0.267949

0

−7.92820

−5.00000

0

−14.5885

−4.26795

0

−1.33975

1.2

3.73205

0

5.92820

−5.00000

0

16.5885

−7.73205

0

−18.6603

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( +1 \)
\(17\)	\( -1 \)

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	765.4.a.i		2
3.b	odd	2	1		85.4.a.d	✓	2
12.b	even	2	1		1360.4.a.m		2
15.d	odd	2	1		425.4.a.e		2
15.e	even	4	2		425.4.b.e		4
51.c	odd	2	1		1445.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.4.a.d	✓	2	3.b	odd	2	1
425.4.a.e		2	15.d	odd	2	1
425.4.b.e		4	15.e	even	4	2
765.4.a.i		2	1.a	even	1	1	trivial
1360.4.a.m		2	12.b	even	2	1
1445.4.a.i		2	51.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(765))\):

\( T_{2}^{2} - 4T_{2} + 1 \)

\( T_{11}^{2} - 38T_{11} - 146 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 4T + 1 \)
$3$	\( T^{2} \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} - 2T - 242 \)
$11$	\( T^{2} - 38T - 146 \)