Newform orbit 61.8.g.a

• Label: 61.8.g.a
• Level: $61$
• Weight: $8$
• Character orbit: 61.g
• Analytic conductor: $19.055$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0554865545\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 5 q^{2} - 57 q^{3} + 1969 q^{4} + 414 q^{5} - 275 q^{6} - 2650 q^{7} - 5 q^{8} - 22677 q^{9}+O(q^{10}) \)

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	−20.3459	−	6.61080i	−6.09895	+	18.7706i	266.700	+	193.769i	45.0268	+	32.7139i	248.178	−	341.587i	1375.42	−	446.901i	−2535.77	−	3490.19i	1454.18	+	1056.52i	−699.848	−	963.259i
3.2	−19.9812	−	6.49230i	−19.0229	+	58.5463i	253.546	+	184.212i	−30.3590	−	22.0571i	760.200	−	1046.33i	−1613.15	+	524.145i	−2289.52	−	3151.25i	−1296.48	−	941.949i	463.409	+	637.828i
3.3	−18.8795	−	6.13431i	16.2847	−	50.1192i	215.250	+	156.388i	104.780	+	76.1270i	−614.893	+	846.327i	−701.080	+	227.795i	−1610.95	−	2217.28i	−477.419	−	346.865i	−1511.20	−	2079.99i
3.4	−17.3893	−	5.65013i	20.1040	−	61.8736i	166.910	+	121.267i	−411.808	−	299.196i	−699.188	+	962.349i	838.874	−	272.567i	−841.634	−	1158.41i	−1654.85	−	1202.32i	5470.57	+	7529.59i
3.5	−16.6686	−	5.41595i	−6.86780	+	21.1369i	144.955	+	105.316i	−300.368	−	218.230i	228.953	−	315.127i	−112.472	+	36.5444i	−527.180	−	725.601i	1369.72	+	995.158i	3824.78	+	5264.36i
3.6	−15.0100	−	4.87704i	2.76472	−	8.50895i	97.9598	+	71.1720i	331.822	+	241.083i	−82.9969	+	114.235i	402.722	−	130.852i	64.1484	+	88.2927i	1704.56	+	1238.44i	−3804.87	−	5236.96i
3.7	−14.8998	−	4.84124i	−24.9745	+	76.8636i	95.0121	+	69.0304i	206.652	+	150.142i	744.229	−	1024.34i	466.590	−	151.604i	97.2293	+	133.825i	−3514.96	−	2553.77i	−2352.20	−	3237.53i
3.8	−12.5539	−	4.07900i	28.4897	−	87.6823i	37.4072	+	27.1779i	178.313	+	129.552i	−715.312	+	984.543i	884.658	−	287.443i	634.370	+	873.135i	−5107.21	−	3710.60i	−1710.07	−	2353.71i
3.9	−11.5690	−	3.75898i	−16.0335	+	49.3462i	16.1568	+	11.7386i	−182.253	−	132.415i	370.983	−	510.614i	377.381	−	122.619i	772.410	+	1063.13i	−408.648	−	296.900i	1610.74	+	2216.99i
3.10	−10.9747	−	3.56588i	9.44017	−	29.0538i	4.17344	+	3.03218i	−212.097	−	154.098i	−207.205	+	285.193i	−1195.80	+	388.538i	833.197	+	1146.80i	1014.31	+	736.940i	1778.20	+	2447.49i
3.11	−10.1918	−	3.31152i	−9.08969	+	27.9752i	−10.6471	−	7.73556i	328.341	+	238.553i	185.281	−	255.017i	−1338.15	+	434.791i	889.155	+	1223.82i	1069.33	+	776.915i	−2556.41	−	3518.60i
3.12	−8.20636	−	2.66641i	10.5229	−	32.3862i	−43.3196	−	31.4735i	21.6178	+	15.7063i	−172.709	+	237.714i	390.476	−	126.873i	920.766	+	1267.33i	831.188	+	603.893i	−135.524	−	186.533i
3.13	−5.03852	−	1.63712i	1.58678	−	4.88360i	−80.8476	−	58.7392i	−144.612	−	105.067i	−15.9900	+	22.0084i	1552.29	−	504.371i	709.779	+	976.927i	1747.99	+	1269.99i	556.624	+	766.128i
3.14	−4.77671	−	1.55205i	23.4437	−	72.1524i	−83.1461	−	60.4091i	−61.3885	−	44.6014i	−223.968	+	308.265i	−852.732	+	277.069i	681.285	+	937.708i	−2887.04	−	2097.56i	224.012	+	308.326i
3.15	−4.44127	−	1.44306i	−25.3536	+	78.0303i	−85.9117	−	62.4185i	−295.123	−	214.419i	225.204	−	309.967i	−224.464	+	72.9327i	642.825	+	884.773i	−3676.60	−	2671.20i	1001.30	+	1378.17i
3.16	−2.97734	−	0.967395i	−11.0874	+	34.1235i	−95.6255	−	69.4760i	161.188	+	117.110i	66.0219	−	90.8713i	−852.980	+	277.150i	453.031	+	623.543i	727.835	+	528.803i	−366.618	−	504.607i
3.17	0.288102	+	0.0936099i	−15.8546	+	48.7955i	−103.480	−	75.1826i	207.325	+	150.630i	−9.13549	+	12.5739i	922.834	−	299.847i	−45.5662	−	62.7164i	−360.313	−	261.783i	45.6301	+	62.8045i
3.18	0.345795	+	0.112356i	16.0407	−	49.3680i	−103.447	−	75.1588i	434.105	+	315.396i	11.0936	−	15.2690i	715.750	−	232.561i	−54.6823	−	75.2638i	−410.582	−	298.305i	114.675	+	157.836i
3.19	2.31143	+	0.751028i	−6.27067	+	19.2991i	−98.7755	−	71.7646i	−315.111	−	228.941i	−28.9884	+	39.8991i	−940.509	+	305.590i	−357.268	−	491.738i	1436.18	+	1043.45i	−556.414	−	765.838i
3.20	3.02163	+	0.981788i	17.8559	−	54.9549i	−95.3878	−	69.3033i	244.017	+	177.289i	107.908	−	148.523i	−1203.33	+	390.986i	−459.222	−	632.065i	−931.890	−	677.058i	563.270	+	775.274i

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	61.8.g.a	✓	136
61.g	even	10	1	inner	61.8.g.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.8.g.a	✓	136	1.a	even	1	1	trivial
61.8.g.a	✓	136	61.g	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(61, [\chi])\).