Newform orbit 1617.4.a.m

• Label: 1617.4.a.m
• Level: $1617$
• Weight: $4$
• Character orbit: 1617.a
• Self dual: yes
• Analytic conductor: $95.406$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(95.4060884793\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^2 + 3 * q^3 + (3*b - 3) * q^4 + (3*b - 2) * q^5 + (3*b + 3) * q^6 + (-5*b + 1) * q^8 + 9 * q^9 + (4*b + 10) * q^10 - 11 * q^11 + (9*b - 9) * q^12 + (13*b + 6) * q^13 + (9*b - 6) * q^15 + (-33*b + 5) * q^16 + (4*b + 54) * q^17 + (9*b + 9) * q^18 + (23*b + 24) * q^19 + (-6*b + 42) * q^20 + (-11*b - 11) * q^22 + (8*b - 32) * q^23 + (-15*b + 3) * q^24 + (-3*b - 85) * q^25 + (32*b + 58) * q^26 + 27 * q^27 + (65*b - 38) * q^29 + (12*b + 30) * q^30 + (-26*b + 168) * q^31 + (-21*b - 135) * q^32 - 33 * q^33 + (62*b + 70) * q^34 + (27*b - 27) * q^36 + (107*b - 86) * q^37 + (70*b + 116) * q^38 + (39*b + 18) * q^39 + (-2*b - 62) * q^40 + (-86*b + 22) * q^41 + (120*b - 76) * q^43 + (-33*b + 33) * q^44 + (27*b - 18) * q^45 - 16*b * q^46 + (163*b - 132) * q^47 + (-99*b + 15) * q^48 + (-91*b - 97) * q^50 + (12*b + 162) * q^51 + (18*b + 138) * q^52 + (58*b + 54) * q^53 + (27*b + 27) * q^54 + (-33*b + 22) * q^55 + (69*b + 72) * q^57 + (92*b + 222) * q^58 + (-53*b + 32) * q^59 + (-18*b + 126) * q^60 + (80*b + 178) * q^61 + (116*b + 64) * q^62 + (87*b - 259) * q^64 + (31*b + 144) * q^65 + (-33*b - 33) * q^66 + (-95*b - 16) * q^67 + (162*b - 114) * q^68 + (24*b - 96) * q^69 + (-56*b + 496) * q^71 + (-45*b + 9) * q^72 + (-317*b + 322) * q^73 + (128*b + 342) * q^74 + (-9*b - 255) * q^75 + (72*b + 204) * q^76 + (96*b + 174) * q^78 + (124*b - 176) * q^79 + (-18*b - 406) * q^80 + 81 * q^81 + (-150*b - 322) * q^82 + (30*b + 116) * q^83 + (166*b - 60) * q^85 + (164*b + 404) * q^86 + (195*b - 114) * q^87 + (55*b - 11) * q^88 + (216*b - 130) * q^89 + (36*b + 90) * q^90 + (-96*b + 192) * q^92 + (-78*b + 504) * q^93 + (194*b + 520) * q^94 + (95*b + 228) * q^95 + (-63*b - 405) * q^96 + (-18*b + 1142) * q^97 - 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^2 + 6 * q^3 - 3 * q^4 - q^5 + 9 * q^6 - 3 * q^8 + 18 * q^9 + 24 * q^10 - 22 * q^11 - 9 * q^12 + 25 * q^13 - 3 * q^15 - 23 * q^16 + 112 * q^17 + 27 * q^18 + 71 * q^19 + 78 * q^20 - 33 * q^22 - 56 * q^23 - 9 * q^24 - 173 * q^25 + 148 * q^26 + 54 * q^27 - 11 * q^29 + 72 * q^30 + 310 * q^31 - 291 * q^32 - 66 * q^33 + 202 * q^34 - 27 * q^36 - 65 * q^37 + 302 * q^38 + 75 * q^39 - 126 * q^40 - 42 * q^41 - 32 * q^43 + 33 * q^44 - 9 * q^45 - 16 * q^46 - 101 * q^47 - 69 * q^48 - 285 * q^50 + 336 * q^51 + 294 * q^52 + 166 * q^53 + 81 * q^54 + 11 * q^55 + 213 * q^57 + 536 * q^58 + 11 * q^59 + 234 * q^60 + 436 * q^61 + 244 * q^62 - 431 * q^64 + 319 * q^65 - 99 * q^66 - 127 * q^67 - 66 * q^68 - 168 * q^69 + 936 * q^71 - 27 * q^72 + 327 * q^73 + 812 * q^74 - 519 * q^75 + 480 * q^76 + 444 * q^78 - 228 * q^79 - 830 * q^80 + 162 * q^81 - 794 * q^82 + 262 * q^83 + 46 * q^85 + 972 * q^86 - 33 * q^87 + 33 * q^88 - 44 * q^89 + 216 * q^90 + 288 * q^92 + 930 * q^93 + 1234 * q^94 + 551 * q^95 - 873 * q^96 + 2266 * q^97 - 198 * 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

−1.56155
2.56155

−0.561553

3.00000

−7.68466

−6.68466

−1.68466

0

8.80776

9.00000

3.75379

1.2

3.56155

3.00000

4.68466

5.68466

10.6847

0

−11.8078

9.00000

20.2462

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	1617.4.a.m		2
7.b	odd	2	1		231.4.a.h	✓	2
21.c	even	2	1		693.4.a.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.4.a.h	✓	2	7.b	odd	2	1
693.4.a.g		2	21.c	even	2	1
1617.4.a.m		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{2} - 3T_{2} - 2 \)

\( T_{5}^{2} + T_{5} - 38 \)

Hecke characteristic polynomials

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