Newform orbit 693.4.a.i

• Label: 693.4.a.i
• Level: $693$
• Weight: $4$
• Character orbit: 693.a
• Self dual: yes
• Analytic conductor: $40.888$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(40.8883236340\)
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 77)
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 + (-3*b + 2) * q^5 + 7 * q^7 + (-5*b + 9) * q^8 + (-b - 22) * q^10 + 11 * q^11 + (-2*b - 40) * q^13 + (7*b + 7) * q^14 + (-12*b - 39) * q^16 + (31*b - 24) * q^17 + (3*b - 48) * q^19 + (b - 46) * q^20 + (11*b + 11) * q^22 + (18*b + 28) * q^23 + (-12*b - 49) * q^25 + (-42*b - 56) * q^26 + (14*b + 7) * q^28 + (-90*b - 2) * q^29 + (-71*b + 30) * q^31 + (-11*b - 207) * q^32 + (7*b + 224) * q^34 + (-21*b + 14) * q^35 + (-12*b - 190) * q^37 + (-45*b - 24) * q^38 + (-37*b + 138) * q^40 + (-89*b - 56) * q^41 - 316 * q^43 + (22*b + 11) * q^44 + (46*b + 172) * q^46 + (97*b + 302) * q^47 + 49 * q^49 + (-61*b - 145) * q^50 + (-82*b - 72) * q^52 + (58*b - 338) * q^53 + (-33*b + 22) * q^55 + (-35*b + 63) * q^56 + (-92*b - 722) * q^58 + (-62*b + 258) * q^59 + (30*b - 148) * q^61 + (-41*b - 538) * q^62 + (-122*b + 17) * q^64 + (116*b - 32) * q^65 + (80*b - 576) * q^67 + (-17*b + 472) * q^68 + (-7*b - 154) * q^70 + (-146*b + 532) * q^71 + (173*b + 84) * q^73 + (-202*b - 286) * q^74 - 93*b * q^76 + 77 * q^77 + (266*b - 184) * q^79 + (93*b + 210) * q^80 + (-145*b - 768) * q^82 + (125*b - 864) * q^83 + (134*b - 792) * q^85 + (-316*b - 316) * q^86 + (-55*b + 99) * q^88 + (-116*b - 762) * q^89 + (-14*b - 280) * q^91 + (74*b + 316) * q^92 + (399*b + 1078) * q^94 + (150*b - 168) * q^95 + (-108*b + 702) * q^97 + (49*b + 49) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 14 * q^7 + 18 * q^8 - 44 * q^10 + 22 * q^11 - 80 * q^13 + 14 * q^14 - 78 * q^16 - 48 * q^17 - 96 * q^19 - 92 * q^20 + 22 * q^22 + 56 * q^23 - 98 * q^25 - 112 * q^26 + 14 * q^28 - 4 * q^29 + 60 * q^31 - 414 * q^32 + 448 * q^34 + 28 * q^35 - 380 * q^37 - 48 * q^38 + 276 * q^40 - 112 * q^41 - 632 * q^43 + 22 * q^44 + 344 * q^46 + 604 * q^47 + 98 * q^49 - 290 * q^50 - 144 * q^52 - 676 * q^53 + 44 * q^55 + 126 * q^56 - 1444 * q^58 + 516 * q^59 - 296 * q^61 - 1076 * q^62 + 34 * q^64 - 64 * q^65 - 1152 * q^67 + 944 * q^68 - 308 * q^70 + 1064 * q^71 + 168 * q^73 - 572 * q^74 + 154 * q^77 - 368 * q^79 + 420 * q^80 - 1536 * q^82 - 1728 * q^83 - 1584 * q^85 - 632 * q^86 + 198 * q^88 - 1524 * q^89 - 560 * q^91 + 632 * q^92 + 2156 * q^94 - 336 * q^95 + 1404 * 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

10.4853

0

7.00000

23.1421

0

−19.1716

1.2

3.82843

0

6.65685

−6.48528

0

7.00000

−5.14214

0

−24.8284

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(7\)	\( -1 \)
\(11\)	\( -1 \)

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	693.4.a.i		2
3.b	odd	2	1		77.4.a.b	✓	2
12.b	even	2	1		1232.4.a.m		2
15.d	odd	2	1		1925.4.a.l		2
21.c	even	2	1		539.4.a.d		2
33.d	even	2	1		847.4.a.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.4.a.b	✓	2	3.b	odd	2	1
539.4.a.d		2	21.c	even	2	1
693.4.a.i		2	1.a	even	1	1	trivial
847.4.a.c		2	33.d	even	2	1
1232.4.a.m		2	12.b	even	2	1
1925.4.a.l		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):

\( T_{2}^{2} - 2T_{2} - 7 \)

\( T_{5}^{2} - 4T_{5} - 68 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T - 7 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 4T - 68 \)
$7$	\( (T - 7)^{2} \)
$11$	\( (T - 11)^{2} \)