Newform orbit 936.2.w.k

• Label: 936.2.w.k
• Level: $936$
• Weight: $2$
• Character orbit: 936.w
• Analytic conductor: $7.474$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.47399762919\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 40 q^{16} - 8 q^{19} - 32 q^{22} + 24 q^{28} + 8 q^{34} + 16 q^{40} - 8 q^{46} + 24 q^{52} - 24 q^{58} + 40 q^{67} - 24 q^{70} + 56 q^{76} + 104 q^{91} - 64 q^{94} - 32 q^{97}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
307.1	−1.40147	+	0.189453i		0		1.92821	−	0.531025i	−0.612096	+	0.612096i		0		−1.94766	−	1.94766i	−2.60172	+	1.10952i		0		0.741868	−	0.973796i
307.2	−1.38609	−	0.280631i		0		1.84249	+	0.777959i	−1.86097	+	1.86097i		0		1.43671	+	1.43671i	−2.33554	−	1.59538i		0		3.10171	−	2.05722i
307.3	−1.20976	−	0.732456i		0		0.927017	+	1.77218i	−0.232047	+	0.232047i		0		−1.32779	−	1.32779i	0.176583	−	2.82291i		0		0.450684	−	0.110756i
307.4	−1.20231	−	0.744615i		0		0.891098	+	1.79052i	1.32005	−	1.32005i		0		3.23172	+	3.23172i	0.261868	−	2.81628i		0		−2.57005	+	0.604182i
307.5	−1.18191	+	0.776588i		0		0.793821	−	1.83571i	−2.44864	+	2.44864i		0		2.97330	+	2.97330i	0.487370	+	2.78612i		0		0.992484	−	4.79565i
307.6	−1.07748	+	0.915988i		0		0.321932	−	1.97392i	1.17044	−	1.17044i		0		0.308113	+	0.308113i	1.46121	+	2.42175i		0		−0.189019	+	2.33324i
307.7	−0.915988	+	1.07748i		0		−0.321932	−	1.97392i	1.17044	−	1.17044i		0		−0.308113	−	0.308113i	2.42175	+	1.46121i		0		0.189019	+	2.33324i
307.8	−0.776588	+	1.18191i		0		−0.793821	−	1.83571i	−2.44864	+	2.44864i		0		−2.97330	−	2.97330i	2.78612	+	0.487370i		0		−0.992484	−	4.79565i
307.9	−0.744615	−	1.20231i		0		−0.891098	+	1.79052i	−1.32005	+	1.32005i		0		−3.23172	−	3.23172i	2.81628	−	0.261868i		0		2.57005	+	0.604182i
307.10	−0.732456	−	1.20976i		0		−0.927017	+	1.77218i	0.232047	−	0.232047i		0		1.32779	+	1.32779i	2.82291	−	0.176583i		0		−0.450684	−	0.110756i
307.11	−0.280631	−	1.38609i		0		−1.84249	+	0.777959i	1.86097	−	1.86097i		0		−1.43671	−	1.43671i	1.59538	+	2.33554i		0		−3.10171	−	2.05722i
307.12	−0.189453	+	1.40147i		0		−1.92821	−	0.531025i	−0.612096	+	0.612096i		0		1.94766	+	1.94766i	1.10952	−	2.60172i		0		−0.741868	−	0.973796i
307.13	0.189453	−	1.40147i		0		−1.92821	−	0.531025i	0.612096	−	0.612096i		0		1.94766	+	1.94766i	−1.10952	+	2.60172i		0		−0.741868	−	0.973796i
307.14	0.280631	+	1.38609i		0		−1.84249	+	0.777959i	−1.86097	+	1.86097i		0		−1.43671	−	1.43671i	−1.59538	−	2.33554i		0		−3.10171	−	2.05722i
307.15	0.732456	+	1.20976i		0		−0.927017	+	1.77218i	−0.232047	+	0.232047i		0		1.32779	+	1.32779i	−2.82291	+	0.176583i		0		−0.450684	−	0.110756i
307.16	0.744615	+	1.20231i		0		−0.891098	+	1.79052i	1.32005	−	1.32005i		0		−3.23172	−	3.23172i	−2.81628	+	0.261868i		0		2.57005	+	0.604182i
307.17	0.776588	−	1.18191i		0		−0.793821	−	1.83571i	2.44864	−	2.44864i		0		−2.97330	−	2.97330i	−2.78612	−	0.487370i		0		−0.992484	−	4.79565i
307.18	0.915988	−	1.07748i		0		−0.321932	−	1.97392i	−1.17044	+	1.17044i		0		−0.308113	−	0.308113i	−2.42175	−	1.46121i		0		0.189019	+	2.33324i
307.19	1.07748	−	0.915988i		0		0.321932	−	1.97392i	−1.17044	+	1.17044i		0		0.308113	+	0.308113i	−1.46121	−	2.42175i		0		−0.189019	+	2.33324i
307.20	1.18191	−	0.776588i		0		0.793821	−	1.83571i	2.44864	−	2.44864i		0		2.97330	+	2.97330i	−0.487370	−	2.78612i		0		0.992484	−	4.79565i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
13.d	odd	4	1	inner
24.f	even	2	1	inner
39.f	even	4	1	inner
104.m	even	4	1	inner
312.w	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	936.2.w.k	✓	48
3.b	odd	2	1	inner	936.2.w.k	✓	48
8.d	odd	2	1	inner	936.2.w.k	✓	48
13.d	odd	4	1	inner	936.2.w.k	✓	48
24.f	even	2	1	inner	936.2.w.k	✓	48
39.f	even	4	1	inner	936.2.w.k	✓	48
104.m	even	4	1	inner	936.2.w.k	✓	48
312.w	odd	4	1	inner	936.2.w.k	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
936.2.w.k	✓	48	1.a	even	1	1	trivial
936.2.w.k	✓	48	3.b	odd	2	1	inner
936.2.w.k	✓	48	8.d	odd	2	1	inner
936.2.w.k	✓	48	13.d	odd	4	1	inner
936.2.w.k	✓	48	24.f	even	2	1	inner
936.2.w.k	✓	48	39.f	even	4	1	inner
936.2.w.k	✓	48	104.m	even	4	1	inner
936.2.w.k	✓	48	312.w	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{24} + 212T_{5}^{20} + 10880T_{5}^{16} + 159232T_{5}^{12} + 716800T_{5}^{8} + 361472T_{5}^{4} + 4096 \)

\( T_{7}^{24} + 836T_{7}^{20} + 203520T_{7}^{16} + 13320192T_{7}^{12} + 269926400T_{7}^{8} + 1673249792T_{7}^{4} + 59969536 \)