Newform orbit 61.6.g.a

• Label: 61.6.g.a
• Level: $61$
• Weight: $6$
• Character orbit: 61.g
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78341300859\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 5 q^{2} + 15 q^{3} + 465 q^{4} - 122 q^{5} + 445 q^{6} + 230 q^{7} - 5 q^{8} - 2841 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−10.5062	−	3.41368i	5.93230	−	18.2577i	72.8388	+	52.9205i	−13.6170	−	9.89333i	−124.652	+	171.569i	11.9097	−	3.86970i	−376.825	−	518.654i	−101.562	−	73.7890i	109.290	+	150.425i
3.2	−9.91809	−	3.22258i	−4.15654	+	12.7925i	62.0949	+	45.1146i	83.4540	+	60.6329i	82.4498	−	113.482i	−36.0235	+	11.7047i	−274.326	−	377.577i	50.2196	+	36.4867i	−632.310	−	870.300i
3.3	−9.73128	−	3.16189i	−8.97793	+	27.6312i	58.8118	+	42.7293i	−79.6286	−	57.8536i	174.733	−	240.500i	151.602	−	49.2586i	−244.753	−	336.873i	−486.290	−	353.310i	591.962	+	814.766i
3.4	−8.16596	−	2.65328i	−0.536847	+	1.65224i	33.7545	+	24.5241i	−24.9431	−	18.1222i	8.76774	−	12.0678i	−94.2813	+	30.6339i	−49.0699	−	67.5390i	194.149	+	141.058i	155.601	+	214.166i
3.5	−7.28026	−	2.36550i	4.76030	−	14.6507i	21.5180	+	15.6337i	0.187196	+	0.136006i	−69.3124	+	95.4003i	234.418	−	76.1672i	24.3073	+	33.4561i	4.60899	+	3.34863i	−1.04111	−	1.43297i
3.6	−6.77906	−	2.20265i	7.89712	−	24.3049i	15.2154	+	11.0547i	63.4760	+	46.1180i	−107.070	+	147.369i	−180.523	+	58.6556i	55.2733	+	76.0771i	−331.770	−	241.045i	−328.726	−	452.452i
3.7	−6.75325	−	2.19426i	−3.58180	+	11.0236i	14.9030	+	10.8277i	7.31209	+	5.31254i	48.3775	−	66.5859i	39.0804	−	12.6980i	56.6744	+	78.0057i	87.8999	+	63.8630i	−37.7232	−	51.9216i
3.8	−5.35057	−	1.73851i	7.62230	−	23.4590i	−0.282341	−	0.205133i	−74.6601	−	54.2438i	−81.5673	+	112.268i	−109.581	+	35.6050i	106.973	+	147.235i	−295.635	−	214.792i	305.171	+	420.032i
3.9	−3.94487	−	1.28177i	−7.73018	+	23.7910i	−11.9695	−	8.69633i	−11.5628	−	8.40083i	60.9891	−	83.9443i	−211.654	+	68.7706i	114.089	+	157.031i	−309.667	−	224.986i	34.8457	+	47.9609i
3.10	−3.71354	−	1.20660i	−7.25729	+	22.3356i	−13.5540	−	9.84758i	60.5656	+	44.0035i	53.9005	−	74.1877i	178.785	−	58.0907i	111.894	+	154.009i	−249.622	−	181.361i	−171.818	−	236.487i
3.11	−2.95760	−	0.960982i	2.65010	−	8.15618i	−18.0646	−	13.1247i	60.4383	+	43.9110i	−15.6759	+	21.5760i	11.3775	−	3.69676i	99.3080	+	136.686i	137.091	+	99.6024i	−136.555	−	187.951i
3.12	−2.28313	−	0.741832i	−0.718823	+	2.21231i	−21.2262	−	15.4217i	−84.4577	−	61.3621i	3.28233	−	4.51774i	78.0713	−	25.3669i	82.1753	+	113.105i	192.214	+	139.651i	147.307	+	202.751i
3.13	−0.681471	−	0.221423i	1.96123	−	6.03604i	−25.4732	−	18.5073i	−2.70697	−	1.96673i	−2.67304	+	3.67912i	−118.598	+	38.5349i	26.7388	+	36.8027i	164.004	+	119.156i	1.40924	+	1.93965i
3.14	0.580441	+	0.188597i	−6.01418	+	18.5098i	−25.5872	−	18.5902i	−24.9177	−	18.1038i	−6.98176	+	9.60957i	84.3182	−	27.3966i	−22.8252	−	31.4163i	−109.850	−	79.8104i	−11.0489	−	15.2076i
3.15	0.632478	+	0.205504i	8.01843	−	24.6782i	−25.5307	−	18.5492i	−4.43129	−	3.21952i	10.1430	−	13.9606i	128.232	−	41.6652i	−24.8443	−	34.1952i	−348.127	−	252.929i	−2.14107	−	2.94692i
3.16	3.45632	+	1.12303i	−0.707000	+	2.17592i	−15.2036	−	11.0460i	37.1500	+	26.9911i	−4.88724	+	6.72671i	112.143	−	36.4374i	−108.499	−	149.337i	192.356	+	139.755i	98.0907	+	135.010i
3.17	3.76517	+	1.22338i	−4.80371	+	14.7843i	−13.2087	−	9.59666i	75.2721	+	54.6884i	−36.1736	+	49.7887i	−178.072	+	57.8590i	−112.457	−	154.783i	1.09088	+	0.792570i	216.508	+	297.998i
3.18	3.92707	+	1.27598i	−5.88917	+	18.1250i	−12.0948	−	8.78737i	−23.8009	−	17.2924i	−46.2544	+	63.6637i	16.7659	−	5.44756i	−113.951	−	156.840i	−97.2426	−	70.6509i	−71.4031	−	98.2779i
3.19	4.95441	+	1.60978i	5.19281	−	15.9818i	−3.93379	−	2.85807i	−30.3233	−	22.0312i	51.4546	−	70.8211i	−161.593	+	52.5048i	−112.872	−	155.356i	−31.8620	−	23.1491i	−114.769	−	157.965i
3.20	6.57445	+	2.13617i	2.40553	−	7.40345i	12.7716	+	9.27911i	−56.3574	−	40.9460i	31.6300	−	43.5350i	142.497	−	46.2999i	−65.8790	−	90.6746i	147.567	+	107.213i	−283.051	−	389.586i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.g	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.6.g.a	✓	104
61.g	even	10	1	inner	61.6.g.a	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.6.g.a	✓	104	1.a	even	1	1	trivial
61.6.g.a	✓	104	61.g	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(61, [\chi])\).