Newform orbit 58.6.e.a

• Label: 58.6.e.a
• Level: $58$
• Weight: $6$
• Character orbit: 58.e
• Analytic conductor: $9.302$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	58.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.30226154915\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 192 q^{4} + 10 q^{5} + 128 q^{6} - 76 q^{7} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−3.12733	+	2.49396i	−23.4565	+	5.35380i	3.56033	−	15.5988i	−55.1948	−	69.2121i	60.0041	−	75.2427i	−5.80191	−	25.4198i	27.7686	+	57.6620i	302.611	−	145.730i	345.224	+	78.7951i
5.2	−3.12733	+	2.49396i	−17.0123	+	3.88295i	3.56033	−	15.5988i	44.8463	+	56.2355i	43.5192	−	54.5713i	32.0566	+	140.449i	27.7686	+	57.6620i	55.4065	−	26.6824i	−280.498	−	64.0219i
5.3	−3.12733	+	2.49396i	−13.5333	+	3.08889i	3.56033	−	15.5988i	16.7279	+	20.9761i	34.6195	−	43.4114i	−46.0754	−	201.870i	27.7686	+	57.6620i	−45.3265	+	21.8281i	−104.627	−	23.8804i
5.4	−3.12733	+	2.49396i	7.89870	−	1.80283i	3.56033	−	15.5988i	−37.9018	−	47.5273i	−20.2056	+	25.3371i	20.0433	+	87.8156i	27.7686	+	57.6620i	−159.796	+	76.9538i	237.062	+	54.1079i
5.5	−3.12733	+	2.49396i	8.30761	−	1.89616i	3.56033	−	15.5988i	−0.700969	−	0.878987i	−21.2517	+	26.6488i	−30.5870	−	134.011i	27.7686	+	57.6620i	−153.514	+	73.9286i	4.38432	+	1.00069i
5.6	−3.12733	+	2.49396i	22.9429	−	5.23657i	3.56033	−	15.5988i	23.0782	+	28.9391i	−58.6902	+	73.5952i	25.1378	+	110.136i	27.7686	+	57.6620i	280.021	−	134.851i	−144.346	−	32.9460i
5.7	3.12733	−	2.49396i	−18.4730	+	4.21634i	3.56033	−	15.5988i	52.0924	+	65.3218i	−47.2557	+	59.2567i	−54.3606	−	238.169i	−27.7686	−	57.6620i	104.538	−	50.3430i	325.820	+	74.3662i
5.8	3.12733	−	2.49396i	−13.9505	+	3.18410i	3.56033	−	15.5988i	3.71404	+	4.65726i	−35.6866	+	44.7496i	8.60147	+	37.6855i	−27.7686	−	57.6620i	−34.4588	+	16.5945i	23.2300	+	5.30211i
5.9	3.12733	−	2.49396i	−2.71736	+	0.620220i	3.56033	−	15.5988i	−38.2099	−	47.9137i	−6.95127	+	8.71661i	7.80945	+	34.2155i	−27.7686	−	57.6620i	−211.936	+	102.063i	−238.989	−	54.5478i
5.10	3.12733	−	2.49396i	4.38688	−	1.00128i	3.56033	−	15.5988i	49.6492	+	62.2581i	11.2221	−	14.0720i	45.5124	+	199.403i	−27.7686	−	57.6620i	−200.693	+	96.6488i	310.539	+	70.8784i
5.11	3.12733	−	2.49396i	16.5554	−	3.77867i	3.56033	−	15.5988i	−21.2057	−	26.5911i	42.3504	−	53.1057i	−28.5190	−	124.950i	−27.7686	−	57.6620i	40.8684	−	19.6812i	−132.634	−	30.2729i
5.12	3.12733	−	2.49396i	29.0514	−	6.63080i	3.56033	−	15.5988i	27.9842	+	35.0911i	74.3163	−	93.1897i	15.6388	+	68.5181i	−27.7686	−	57.6620i	581.082	−	279.834i	175.031	+	39.9498i
9.1	−1.73553	−	3.60388i	−17.1495	+	13.6763i	−9.97584	+	12.5093i	36.6221	−	17.6363i	79.0510	+	38.0690i	80.7882	+	101.305i	62.3954	+	14.2413i	52.9921	−	232.174i	−127.118	−	101.373i
9.2	−1.73553	−	3.60388i	−15.8644	+	12.6514i	−9.97584	+	12.5093i	−68.5577	+	33.0157i	73.1273	+	35.2162i	−41.7236	−	52.3197i	62.3954	+	14.2413i	37.5474	−	164.506i	237.969	+	189.774i
9.3	−1.73553	−	3.60388i	−4.31476	+	3.44091i	−9.97584	+	12.5093i	29.5933	−	14.2514i	19.8890	+	9.57805i	−88.0394	−	110.398i	62.3954	+	14.2413i	−47.2953	+	207.214i	−102.721	−	81.9169i
9.4	−1.73553	−	3.60388i	6.43863	−	5.13464i	−9.97584	+	12.5093i	79.2587	−	38.1690i	−29.6791	−	14.2927i	22.4740	+	28.1815i	62.3954	+	14.2413i	−38.9811	+	170.787i	−275.113	−	219.395i
9.5	−1.73553	−	3.60388i	8.20211	−	6.54097i	−9.97584	+	12.5093i	−42.6495	+	20.5389i	−37.8079	−	18.2073i	107.602	+	134.929i	62.3954	+	14.2413i	−29.5822	+	129.608i	148.039	+	118.057i
9.6	−1.73553	−	3.60388i	22.6803	−	18.0869i	−9.97584	+	12.5093i	−1.67707	+	0.807636i	−104.545	−	50.3464i	−84.2002	−	105.584i	62.3954	+	14.2413i	133.186	−	583.526i	5.82124	+	4.64228i
9.7	1.73553	+	3.60388i	−23.2649	+	18.5532i	−9.97584	+	12.5093i	−67.3930	+	32.4547i	−107.240	−	51.6442i	143.617	+	180.090i	−62.3954	−	14.2413i	142.964	−	626.368i	−233.926	−	186.549i
9.8	1.73553	+	3.60388i	−11.9779	+	9.55204i	−9.97584	+	12.5093i	−6.92485	+	3.33483i	−55.2124	−	26.5889i	−95.2215	−	119.404i	−62.3954	−	14.2413i	−1.84442	+	8.08094i	−24.0366	−	19.1686i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.e	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	58.6.e.a	✓	72
29.e	even	14	1	inner	58.6.e.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.6.e.a	✓	72	1.a	even	1	1	trivial
58.6.e.a	✓	72	29.e	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(58, [\chi])\).