Newform orbit 161.4.g.a

• Label: 161.4.g.a
• Level: $161$
• Weight: $4$
• Character orbit: 161.g
• Analytic conductor: $9.499$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	161.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.49930751092\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 6 q^{3} - 168 q^{4} + 60 q^{8} + 348 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
45.1	−2.75749	−	4.77611i	4.01923	+	2.32051i	−11.2075	+	19.4119i	−6.66496	−	11.5441i		−	25.5950i	14.1806	+	11.9126i	79.4978			−2.73050	−	4.72937i	−36.7571	+	63.6651i
45.2	−2.75749	−	4.77611i	4.01923	+	2.32051i	−11.2075	+	19.4119i	6.66496	+	11.5441i		−	25.5950i	−14.1806	−	11.9126i	79.4978			−2.73050	−	4.72937i	36.7571	−	63.6651i
45.3	−2.55109	−	4.41862i	−6.15725	−	3.55489i	−9.01611	+	15.6164i	−5.07712	−	8.79383i			36.2754i	−13.7630	+	12.3927i	51.1862			11.7745	+	20.3940i	−25.9044	+	44.8677i
45.4	−2.55109	−	4.41862i	−6.15725	−	3.55489i	−9.01611	+	15.6164i	5.07712	+	8.79383i			36.2754i	13.7630	−	12.3927i	51.1862			11.7745	+	20.3940i	25.9044	−	44.8677i
45.5	−2.21778	−	3.84131i	−1.03344	−	0.596658i	−5.83713	+	10.1102i	−6.89492	−	11.9424i			5.29303i	−3.08073	−	18.2622i	16.2974			−12.7880	−	22.1495i	−30.5829	+	52.9711i
45.6	−2.21778	−	3.84131i	−1.03344	−	0.596658i	−5.83713	+	10.1102i	6.89492	+	11.9424i			5.29303i	3.08073	+	18.2622i	16.2974			−12.7880	−	22.1495i	30.5829	−	52.9711i
45.7	−2.08232	−	3.60669i	7.07999	+	4.08763i	−4.67214	+	8.09239i	−2.30193	−	3.98706i		−	34.0471i	0.631017	−	18.5095i	5.59849			19.9175	+	34.4981i	−9.58672	+	16.6047i
45.8	−2.08232	−	3.60669i	7.07999	+	4.08763i	−4.67214	+	8.09239i	2.30193	+	3.98706i		−	34.0471i	−0.631017	+	18.5095i	5.59849			19.9175	+	34.4981i	9.58672	−	16.6047i
45.9	−1.99796	−	3.46057i	−0.769658	−	0.444362i	−3.98369	+	6.89995i	−3.47219	−	6.01401i			3.55127i	18.4818	+	1.19366i	−0.130385			−13.1051	−	22.6987i	−13.8746	+	24.0315i
45.10	−1.99796	−	3.46057i	−0.769658	−	0.444362i	−3.98369	+	6.89995i	3.47219	+	6.01401i			3.55127i	−18.4818	−	1.19366i	−0.130385			−13.1051	−	22.6987i	13.8746	−	24.0315i
45.11	−1.47330	−	2.55183i	5.42339	+	3.13120i	−0.341223	+	0.591016i	−9.39166	−	16.2668i		−	18.4528i	−15.6862	+	9.84599i	−21.5619			6.10881	+	10.5808i	−27.6735	+	47.9318i
45.12	−1.47330	−	2.55183i	5.42339	+	3.13120i	−0.341223	+	0.591016i	9.39166	+	16.2668i		−	18.4528i	15.6862	−	9.84599i	−21.5619			6.10881	+	10.5808i	27.6735	−	47.9318i
45.13	−1.37813	−	2.38698i	−7.49964	−	4.32992i	0.201539	−	0.349075i	−1.19426	−	2.06852i			23.8687i	−13.7597	−	12.3964i	−23.1610			23.9964	+	41.5630i	−3.29169	+	5.70137i
45.14	−1.37813	−	2.38698i	−7.49964	−	4.32992i	0.201539	−	0.349075i	1.19426	+	2.06852i			23.8687i	13.7597	+	12.3964i	−23.1610			23.9964	+	41.5630i	3.29169	−	5.70137i
45.15	−1.10496	−	1.91385i	−3.82983	−	2.21115i	1.55811	−	2.69873i	−8.16385	−	14.1402i			9.77297i	−1.90650	+	18.4219i	−24.5660			−3.72161	−	6.44602i	−18.0415	+	31.2488i
45.16	−1.10496	−	1.91385i	−3.82983	−	2.21115i	1.55811	−	2.69873i	8.16385	+	14.1402i			9.77297i	1.90650	−	18.4219i	−24.5660			−3.72161	−	6.44602i	18.0415	−	31.2488i
45.17	−0.923513	−	1.59957i	2.40467	+	1.38834i	2.29425	−	3.97375i	−2.57123	−	4.45351i		−	5.12858i	18.4570	−	1.52994i	−23.2513			−9.64505	−	16.7057i	−4.74914	+	8.22575i
45.18	−0.923513	−	1.59957i	2.40467	+	1.38834i	2.29425	−	3.97375i	2.57123	+	4.45351i		−	5.12858i	−18.4570	+	1.52994i	−23.2513			−9.64505	−	16.7057i	4.74914	−	8.22575i
45.19	−0.475042	−	0.822798i	−3.34199	−	1.92950i	3.54867	−	6.14648i	−8.95281	−	15.5067i			3.66637i	3.46541	−	18.1932i	−14.3438			−6.05407	−	10.4860i	−8.50593	+	14.7327i
45.20	−0.475042	−	0.822798i	−3.34199	−	1.92950i	3.54867	−	6.14648i	8.95281	+	15.5067i			3.66637i	−3.46541	+	18.1932i	−14.3438			−6.05407	−	10.4860i	8.50593	−	14.7327i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
23.b	odd	2	1	inner
161.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	161.4.g.a	✓	92
7.d	odd	6	1	inner	161.4.g.a	✓	92
23.b	odd	2	1	inner	161.4.g.a	✓	92
161.g	even	6	1	inner	161.4.g.a	✓	92

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
161.4.g.a	✓	92	1.a	even	1	1	trivial
161.4.g.a	✓	92	7.d	odd	6	1	inner
161.4.g.a	✓	92	23.b	odd	2	1	inner
161.4.g.a	✓	92	161.g	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(161, [\chi])\).