Newform orbit 176.4.a.i

• Label: 176.4.a.i
• Level: $176$
• Weight: $4$
• Character orbit: 176.a
• Self dual: yes
• Analytic conductor: $10.384$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	176.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.3843361610\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^3 + (2*b + 1) * q^5 + (b - 10) * q^7 + (2*b + 22) * q^9 + 11 * q^11 + (-5*b + 40) * q^13 + (3*b + 97) * q^15 + (3*b - 62) * q^17 + (-15*b - 36) * q^19 + (-9*b + 38) * q^21 + (9*b + 49) * q^23 + (4*b + 68) * q^25 + (-3*b + 91) * q^27 + (-14*b + 72) * q^29 + (-7*b + 17) * q^31 + (11*b + 11) * q^33 + (-19*b + 86) * q^35 + (-2*b + 27) * q^37 + (35*b - 200) * q^39 + (-b + 268) * q^41 + (4*b + 30) * q^43 + (46*b + 214) * q^45 + (30*b + 136) * q^47 + (-20*b - 195) * q^49 + (-59*b + 82) * q^51 + (-14*b - 246) * q^53 + (22*b + 11) * q^55 + (-51*b - 756) * q^57 + (33*b - 317) * q^59 + (46*b + 420) * q^61 + (2*b - 124) * q^63 + (75*b - 440) * q^65 + (5*b - 377) * q^67 + (58*b + 481) * q^69 + (-19*b + 339) * q^71 + (-117*b - 200) * q^73 + (72*b + 260) * q^75 + (11*b - 110) * q^77 + (-164*b - 158) * q^79 + (34*b - 647) * q^81 + (-30*b - 234) * q^83 + (-121*b + 226) * q^85 + (58*b - 600) * q^87 + (-82*b - 921) * q^89 + (90*b - 640) * q^91 + (10*b - 319) * q^93 + (-87*b - 1476) * q^95 + (36*b + 1097) * q^97 + (22*b + 242) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 2 * q^3 + 2 * q^5 - 20 * q^7 + 44 * q^9 + 22 * q^11 + 80 * q^13 + 194 * q^15 - 124 * q^17 - 72 * q^19 + 76 * q^21 + 98 * q^23 + 136 * q^25 + 182 * q^27 + 144 * q^29 + 34 * q^31 + 22 * q^33 + 172 * q^35 + 54 * q^37 - 400 * q^39 + 536 * q^41 + 60 * q^43 + 428 * q^45 + 272 * q^47 - 390 * q^49 + 164 * q^51 - 492 * q^53 + 22 * q^55 - 1512 * q^57 - 634 * q^59 + 840 * q^61 - 248 * q^63 - 880 * q^65 - 754 * q^67 + 962 * q^69 + 678 * q^71 - 400 * q^73 + 520 * q^75 - 220 * q^77 - 316 * q^79 - 1294 * q^81 - 468 * q^83 + 452 * q^85 - 1200 * q^87 - 1842 * q^89 - 1280 * q^91 - 638 * q^93 - 2952 * q^95 + 2194 * q^97 + 484 * 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

−1.73205
1.73205

0

−5.92820

0

−12.8564

0

−16.9282

0

8.14359

0

1.2

0

7.92820

0

14.8564

0

−3.07180

0

35.8564

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(11\)	\( -1 \)

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	176.4.a.i		2
3.b	odd	2	1		1584.4.a.bc		2
4.b	odd	2	1		11.4.a.a	✓	2
8.b	even	2	1		704.4.a.n		2
8.d	odd	2	1		704.4.a.p		2
11.b	odd	2	1		1936.4.a.w		2
12.b	even	2	1		99.4.a.c		2
20.d	odd	2	1		275.4.a.b		2
20.e	even	4	2		275.4.b.c		4
28.d	even	2	1		539.4.a.e		2
44.c	even	2	1		121.4.a.c		2
44.g	even	10	4		121.4.c.f		8
44.h	odd	10	4		121.4.c.c		8
52.b	odd	2	1		1859.4.a.a		2
60.h	even	2	1		2475.4.a.q		2
132.d	odd	2	1		1089.4.a.v		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.4.a.a	✓	2	4.b	odd	2	1
99.4.a.c		2	12.b	even	2	1
121.4.a.c		2	44.c	even	2	1
121.4.c.c		8	44.h	odd	10	4
121.4.c.f		8	44.g	even	10	4
176.4.a.i		2	1.a	even	1	1	trivial
275.4.a.b		2	20.d	odd	2	1
275.4.b.c		4	20.e	even	4	2
539.4.a.e		2	28.d	even	2	1
704.4.a.n		2	8.b	even	2	1
704.4.a.p		2	8.d	odd	2	1
1089.4.a.v		2	132.d	odd	2	1
1584.4.a.bc		2	3.b	odd	2	1
1859.4.a.a		2	52.b	odd	2	1
1936.4.a.w		2	11.b	odd	2	1
2475.4.a.q		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 47 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(176))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T - 47 \)
$5$	\( T^{2} - 2T - 191 \)
$7$	\( T^{2} + 20T + 52 \)
$11$	\( (T - 11)^{2} \)