Newform orbit 77.3.g.a

• Label: 77.3.g.a
• Level: $77$
• Weight: $3$
• Character orbit: 77.g
• Analytic conductor: $2.098$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	77.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.09809803557\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 6 q^{3} - 32 q^{4} - 2 q^{7} + 12 q^{8} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−1.93286	+	3.34781i	3.55762	−	2.05400i	−5.47190	−	9.47761i	4.92403	+	2.84289i			15.8803i	1.06313	+	6.91880i	26.8428			3.93779	−	6.82046i	−19.0349	+	10.9898i
12.2	−1.61097	+	2.79028i	−0.352611	+	0.203580i	−3.19046	−	5.52604i	−5.16482	−	2.98191i		−	1.31185i	−1.07785	−	6.91652i	7.67118			−4.41711	+	7.65066i	16.6407	−	9.60754i
12.3	−1.55589	+	2.69488i	−4.59110	+	2.65067i	−2.84159	−	4.92179i	−0.0791900	−	0.0457204i		−	16.4966i	1.79525	+	6.76588i	5.23772			9.55214	−	16.5448i	0.246422	−	0.142272i
12.4	−1.14074	+	1.97582i	−0.855469	+	0.493905i	−0.602579	−	1.04370i	8.55576	+	4.93967i		−	2.25367i	−5.42968	−	4.41798i	−6.37638			−4.01212	+	6.94919i	−19.5198	+	11.2698i
12.5	−0.817922	+	1.41668i	3.98619	−	2.30143i	0.662007	+	1.14663i	−1.09565	−	0.632573i			7.52956i	5.87066	−	3.81253i	−8.70926			6.09315	−	10.5536i	1.79231	−	1.03479i
12.6	−0.670448	+	1.16125i	−0.543665	+	0.313885i	1.10100	+	1.90699i	−0.745054	−	0.430157i		−	0.841776i	3.55353	+	6.03096i	−8.31624			−4.30295	+	7.45293i	0.999041	−	0.576796i
12.7	0.0259384	−	0.0449266i	−2.84313	+	1.64148i	1.99865	+	3.46177i	−7.11509	−	4.10790i			0.170310i	−6.93587	−	0.945372i	0.414874			0.888927	−	1.53967i	−0.369108	+	0.213105i
12.8	0.138470	−	0.239837i	3.15733	−	1.82289i	1.96165	+	3.39768i	1.78380	+	1.02988i		−	1.00966i	−6.99973	−	0.0614542i	2.19427			2.14583	−	3.71668i	0.494005	−	0.285214i
12.9	0.525577	−	0.910326i	−1.41494	+	0.816919i	1.44754	+	2.50721i	1.91762	+	1.10714i			1.71741i	6.61862	−	2.27901i	7.24778			−3.16529	+	5.48244i	2.01571	−	1.16377i
12.10	0.964862	−	1.67119i	−4.29122	+	2.47754i	0.138081	+	0.239163i	6.02719	+	3.47980i			9.56193i	−4.98921	+	4.90997i	8.25182			7.77639	−	13.4691i	11.6308	−	6.71506i
12.11	1.10476	−	1.91349i	3.24216	−	1.87186i	−0.440972	−	0.763786i	−6.93366	−	4.00315i		−	8.27181i	1.18425	+	6.89910i	6.88938			2.50775	−	4.34356i	−15.3200	+	8.84501i
12.12	1.34512	−	2.32982i	1.05603	−	0.609700i	−1.61869	−	2.80366i	0.149278	+	0.0861856i		−	3.28048i	−0.670635	−	6.96780i	2.05161			−3.75653	+	6.50650i	0.401593	−	0.231860i
12.13	1.76583	−	3.05850i	−4.08349	+	2.35760i	−4.23630	−	7.33749i	−6.97443	−	4.02669i			16.6525i	6.96245	−	0.724053i	−15.7957			6.61659	−	11.4603i	−24.6313	+	14.2209i
12.14	1.85828	−	3.21864i	0.976290	−	0.563661i	−4.90643	−	8.49818i	4.75021	+	2.74254i		−	4.18977i	−1.94492	+	6.72438i	−21.6039			−3.86457	+	6.69364i	17.6545	−	10.1928i
45.1	−1.93286	−	3.34781i	3.55762	+	2.05400i	−5.47190	+	9.47761i	4.92403	−	2.84289i		−	15.8803i	1.06313	−	6.91880i	26.8428			3.93779	+	6.82046i	−19.0349	−	10.9898i
45.2	−1.61097	−	2.79028i	−0.352611	−	0.203580i	−3.19046	+	5.52604i	−5.16482	+	2.98191i			1.31185i	−1.07785	+	6.91652i	7.67118			−4.41711	−	7.65066i	16.6407	+	9.60754i
45.3	−1.55589	−	2.69488i	−4.59110	−	2.65067i	−2.84159	+	4.92179i	−0.0791900	+	0.0457204i			16.4966i	1.79525	−	6.76588i	5.23772			9.55214	+	16.5448i	0.246422	+	0.142272i
45.4	−1.14074	−	1.97582i	−0.855469	−	0.493905i	−0.602579	+	1.04370i	8.55576	−	4.93967i			2.25367i	−5.42968	+	4.41798i	−6.37638			−4.01212	−	6.94919i	−19.5198	−	11.2698i
45.5	−0.817922	−	1.41668i	3.98619	+	2.30143i	0.662007	−	1.14663i	−1.09565	+	0.632573i		−	7.52956i	5.87066	+	3.81253i	−8.70926			6.09315	+	10.5536i	1.79231	+	1.03479i
45.6	−0.670448	−	1.16125i	−0.543665	−	0.313885i	1.10100	−	1.90699i	−0.745054	+	0.430157i			0.841776i	3.55353	−	6.03096i	−8.31624			−4.30295	−	7.45293i	0.999041	+	0.576796i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.3.g.a	✓	28
7.c	even	3	1		539.3.d.a		28
7.d	odd	6	1	inner	77.3.g.a	✓	28
7.d	odd	6	1		539.3.d.a		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.3.g.a	✓	28	1.a	even	1	1	trivial
77.3.g.a	✓	28	7.d	odd	6	1	inner
539.3.d.a		28	7.c	even	3	1
539.3.d.a		28	7.d	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(77, [\chi])\).