Newform orbit 1682.4.a.c

• Label: 1682.4.a.c
• Level: $1682$
• Weight: $4$
• Character orbit: 1682.a
• Self dual: yes
• Analytic conductor: $99.241$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(99.2412126297\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + (6*b - 5) * q^5 + (2*b + 2) * q^6 + (-8*b - 8) * q^7 + 8 * q^8 + (2*b - 20) * q^9 + (12*b - 10) * q^10 + (3*b + 45) * q^11 + (4*b + 4) * q^12 + (8*b - 25) * q^13 + (-16*b - 16) * q^14 + (b + 31) * q^15 + 16 * q^16 + (-16*b + 22) * q^17 + (4*b - 40) * q^18 + (4*b - 54) * q^19 + (24*b - 20) * q^20 + (-16*b - 56) * q^21 + (6*b + 90) * q^22 + (-78*b - 14) * q^23 + (8*b + 8) * q^24 + (-60*b + 116) * q^25 + (16*b - 50) * q^26 + (-45*b - 35) * q^27 + (-32*b - 32) * q^28 + (2*b + 62) * q^30 + (-109*b - 33) * q^31 + 32 * q^32 + (48*b + 63) * q^33 + (-32*b + 44) * q^34 + (-8*b - 248) * q^35 + (8*b - 80) * q^36 + (-4*b - 20) * q^37 + (8*b - 108) * q^38 + (-17*b + 23) * q^39 + (48*b - 40) * q^40 + (80*b - 152) * q^41 + (-32*b - 112) * q^42 + (-53*b + 65) * q^43 + (12*b + 180) * q^44 + (-130*b + 172) * q^45 + (-156*b - 28) * q^46 + (99*b + 257) * q^47 + (16*b + 16) * q^48 + (128*b + 105) * q^49 + (-120*b + 232) * q^50 + (6*b - 74) * q^51 + (32*b - 100) * q^52 + (52*b - 479) * q^53 + (-90*b - 70) * q^54 + (255*b - 117) * q^55 + (-64*b - 64) * q^56 + (-50*b - 30) * q^57 + (250*b - 90) * q^59 + (4*b + 124) * q^60 + (12*b - 514) * q^61 + (-218*b - 66) * q^62 + (144*b + 64) * q^63 + 64 * q^64 + (-190*b + 413) * q^65 + (96*b + 126) * q^66 + (-20*b - 456) * q^67 + (-64*b + 88) * q^68 + (-92*b - 482) * q^69 + (-16*b - 496) * q^70 + (34*b + 398) * q^71 + (16*b - 160) * q^72 + (-176*b + 428) * q^73 + (-8*b - 40) * q^74 + (56*b - 244) * q^75 + (16*b - 216) * q^76 + (-384*b - 504) * q^77 + (-34*b + 46) * q^78 + (361*b + 159) * q^79 + (96*b - 80) * q^80 + (-134*b + 235) * q^81 + (160*b - 304) * q^82 + (-38*b - 914) * q^83 + (-64*b - 224) * q^84 + (212*b - 686) * q^85 + (-106*b + 130) * q^86 + (24*b + 360) * q^88 + (-72*b + 472) * q^89 + (-260*b + 344) * q^90 + (136*b - 184) * q^91 + (-312*b - 56) * q^92 + (-142*b - 687) * q^93 + (198*b + 514) * q^94 + (-344*b + 414) * q^95 + (32*b + 32) * q^96 + (-100*b - 184) * q^97 + (256*b + 210) * q^98 + (30*b - 864) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9 - 20 * q^10 + 90 * q^11 + 8 * q^12 - 50 * q^13 - 32 * q^14 + 62 * q^15 + 32 * q^16 + 44 * q^17 - 80 * q^18 - 108 * q^19 - 40 * q^20 - 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 - 100 * q^26 - 70 * q^27 - 64 * q^28 + 124 * q^30 - 66 * q^31 + 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 - 40 * q^37 - 216 * q^38 + 46 * q^39 - 80 * q^40 - 304 * q^41 - 224 * q^42 + 130 * q^43 + 360 * q^44 + 344 * q^45 - 56 * q^46 + 514 * q^47 + 32 * q^48 + 210 * q^49 + 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 - 234 * q^55 - 128 * q^56 - 60 * q^57 - 180 * q^59 + 248 * q^60 - 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 + 252 * q^66 - 912 * q^67 + 176 * q^68 - 964 * q^69 - 992 * q^70 + 796 * q^71 - 320 * q^72 + 856 * q^73 - 80 * q^74 - 488 * q^75 - 432 * q^76 - 1008 * q^77 + 92 * q^78 + 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 - 448 * q^84 - 1372 * q^85 + 260 * q^86 + 720 * q^88 + 944 * q^89 + 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 + 828 * q^95 + 64 * q^96 - 368 * q^97 + 420 * q^98 - 1728 * 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

−2.44949
2.44949

2.00000

−1.44949

4.00000

−19.6969

−2.89898

11.5959

8.00000

−24.8990

−39.3939

1.2

2.00000

3.44949

4.00000

9.69694

6.89898

−27.5959

8.00000

−15.1010

19.3939

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(29\)	\( +1 \)

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	1682.4.a.c		2
29.b	even	2	1		58.4.a.c	✓	2
87.d	odd	2	1		522.4.a.j		2
116.d	odd	2	1		464.4.a.e		2
145.d	even	2	1		1450.4.a.g		2
232.b	odd	2	1		1856.4.a.i		2
232.g	even	2	1		1856.4.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.4.a.c	✓	2	29.b	even	2	1
464.4.a.e		2	116.d	odd	2	1
522.4.a.j		2	87.d	odd	2	1
1450.4.a.g		2	145.d	even	2	1
1682.4.a.c		2	1.a	even	1	1	trivial
1856.4.a.i		2	232.b	odd	2	1
1856.4.a.l		2	232.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 2T - 5 \)
$5$	\( T^{2} + 10T - 191 \)
$7$	\( T^{2} + 16T - 320 \)
$11$	\( T^{2} - 90T + 1971 \)