Newform orbit 61.6.k.a

• Label: 61.6.k.a
• Level: $61$
• Weight: $6$
• Character orbit: 61.k
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	61.k (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78341300859\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 7 q^{2} + 12 q^{3} - 413 q^{4} + 224 q^{5} - 481 q^{6} + 293 q^{7} - 10 q^{8} - 3324 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−10.4446	−	1.09777i	−4.00228	−	2.90783i	76.5844	+	16.2785i	−23.2197	+	25.7881i	38.6102	+	34.7648i	20.4513	−	45.9345i	−462.404	−	150.244i	−67.5283	−	207.831i	270.830	−	243.857i
4.2	−9.62649	−	1.01178i	18.6532	+	13.5523i	60.3448	+	12.8267i	42.3701	−	47.0568i	−165.853	−	149.334i	67.1837	−	150.897i	−273.346	−	88.8156i	89.1845	+	274.482i	−455.487	+	410.122i
4.3	−9.20096	−	0.967060i	19.2665	+	13.9979i	52.4217	+	11.1426i	−61.2982	+	68.0785i	−163.733	−	147.426i	−66.5866	+	149.556i	−189.992	−	61.7322i	100.165	+	308.276i	629.838	−	567.109i
4.4	−9.03754	−	0.949883i	−0.162257	−	0.117887i	49.4741	+	10.5160i	59.4396	−	66.0144i	1.35443	+	1.21953i	−90.7608	+	203.852i	−160.573	−	52.1733i	−75.0787	−	231.068i	−599.894	+	540.147i
4.5	−8.33541	−	0.876087i	−22.1822	−	16.1163i	37.4108	+	7.95190i	−7.81324	+	8.67749i	170.779	+	153.770i	−22.2929	+	50.0706i	−49.7919	−	16.1784i	157.224	+	483.886i	72.7288	−	65.4853i
4.6	−7.19703	−	0.756438i	−4.70637	−	3.41938i	19.9242	+	4.23503i	−1.88871	+	2.09762i	31.2853	+	28.1694i	50.2645	−	112.896i	80.0476	+	26.0090i	−64.6334	−	198.921i	15.1798	−	13.6680i
4.7	−5.13780	−	0.540005i	13.8945	+	10.0950i	−5.19529	−	1.10429i	11.7189	−	13.0152i	−65.9361	−	59.3692i	−57.6269	+	129.432i	183.320	+	59.5644i	16.0587	+	49.4235i	−67.2377	+	60.5411i
4.8	−5.06867	−	0.532739i	1.57893	+	1.14716i	−5.89312	−	1.25262i	−65.9752	+	73.2729i	−7.39194	−	6.65574i	13.9334	−	31.2948i	184.312	+	59.8865i	−73.9141	−	227.484i	373.442	−	336.248i
4.9	−4.50872	−	0.473885i	−14.5934	−	10.6027i	−11.1968	−	2.37995i	65.2304	−	72.4457i	60.7728	+	54.7201i	58.7776	−	132.017i	187.328	+	60.8667i	25.4578	+	78.3512i	−328.436	+	295.725i
4.10	−4.35232	−	0.457447i	15.4713	+	11.2405i	−12.5673	−	2.67126i	2.84228	−	3.15667i	−62.1940	−	55.9997i	56.0556	−	125.903i	186.662	+	60.6502i	37.9195	+	116.704i	−13.8145	+	12.4386i
4.11	−2.76110	−	0.290203i	−12.5180	−	9.09485i	−23.7613	−	5.05062i	−34.9164	+	38.7786i	31.9240	+	28.7445i	−78.0905	+	175.394i	148.635	+	48.2945i	−1.10738	−	3.40818i	107.661	−	96.9387i
4.12	−0.483465	−	0.0508142i	−6.40810	−	4.65576i	−31.0696	−	6.60404i	22.8450	−	25.3719i	2.86151	+	2.57652i	−18.8782	+	42.4012i	29.4802	+	9.57869i	−55.7035	−	171.438i	−12.3340	+	11.1056i
4.13	0.755164	+	0.0793710i	7.60063	+	5.52218i	−30.7367	−	6.53330i	41.2745	−	45.8400i	5.30143	+	4.77343i	−18.3252	+	41.1590i	−45.8019	−	14.8819i	−47.8160	−	147.163i	34.8074	−	31.3407i
4.14	1.08436	+	0.113971i	−19.5548	−	14.2074i	−30.1379	−	6.40600i	−26.0609	+	28.9436i	−19.5853	−	17.6347i	54.2479	−	121.843i	−65.1335	−	21.1631i	105.449	+	324.539i	−31.5583	+	28.4152i
4.15	1.75880	+	0.184857i	24.1245	+	17.5275i	−28.2415	−	6.00292i	−19.7902	+	21.9792i	39.1900	+	35.2869i	−19.8694	+	44.6274i	−102.383	−	33.2663i	199.689	+	614.579i	−38.8699	+	34.9986i
4.16	2.13149	+	0.224029i	7.91149	+	5.74804i	−26.8077	−	5.69815i	−54.5486	+	60.5824i	15.5755	+	14.0243i	74.8547	−	168.127i	−121.090	−	39.3446i	−45.5393	−	140.156i	−129.842	+	116.910i
4.17	5.05657	+	0.531467i	−23.6118	−	17.1550i	−6.01430	−	1.27838i	62.1455	−	69.0196i	−110.277	−	99.2942i	−95.4098	+	214.294i	−184.471	−	59.9381i	188.133	+	579.013i	350.925	−	315.974i
4.18	5.50960	+	0.579082i	3.38497	+	2.45933i	−1.28037	−	0.272151i	−38.0583	+	42.2680i	17.2257	+	15.5101i	−70.6663	+	158.719i	−175.498	−	57.0229i	−69.6814	−	214.457i	−234.163	+	210.841i
4.19	5.76452	+	0.605875i	14.0911	+	10.2377i	1.56185	+	0.331982i	65.4894	−	72.7334i	75.0253	+	67.5531i	79.5828	−	178.746i	−167.600	−	54.4567i	18.6551	+	57.4146i	421.582	−	379.595i
4.20	5.89321	+	0.619401i	−7.47378	−	5.43002i	3.04550	+	0.647341i	10.3534	−	11.4986i	−40.6812	−	36.6295i	18.1681	−	40.8062i	−162.794	−	52.8949i	−48.7188	−	149.941i	68.1369	−	61.3507i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.k	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.6.k.a	✓	200
61.k	even	30	1	inner	61.6.k.a	✓	200

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.6.k.a	✓	200	1.a	even	1	1	trivial
61.6.k.a	✓	200	61.k	even	30	1	inner

Hecke kernels

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