Newform orbit 84.9.l.a

• Label: 84.9.l.a
• Level: $84$
• Weight: $9$
• Character orbit: 84.l
• Analytic conductor: $34.220$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(34.2198032451\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 3 q^{2} - 2592 q^{3} - 93 q^{4} - 2256 q^{7} + 4500 q^{8} + 69984 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−15.9385	−	1.40203i	−40.5000	−	23.3827i	252.069	+	44.6923i	−136.605	−	236.607i	612.724	+	429.466i	824.317	−	2255.06i	−3954.92	−	1065.73i	1093.50	+	1894.00i	1845.55	+	3962.68i
67.2	−15.8978	−	1.80515i	−40.5000	−	23.3827i	249.483	+	57.3961i	569.802	+	986.925i	601.653	+	444.843i	−2329.07	−	583.315i	−3862.63	−	1362.83i	1093.50	+	1894.00i	−7277.06	−	16718.6i
67.3	−15.8905	−	1.86845i	−40.5000	−	23.3827i	249.018	+	59.3813i	−610.376	−	1057.20i	599.877	+	447.235i	−1727.74	+	1667.25i	−3846.07	−	1408.88i	1093.50	+	1894.00i	7723.87	+	17940.0i
67.4	−15.4075	+	4.31388i	−40.5000	−	23.3827i	218.781	−	132.932i	13.0207	+	22.5525i	724.873	+	185.556i	1734.00	+	1660.74i	−2797.41	+	2991.94i	1093.50	+	1894.00i	−297.905	−	291.308i
67.5	−14.1122	+	7.53960i	−40.5000	−	23.3827i	142.309	−	212.801i	8.66226	+	15.0035i	747.841	+	24.6276i	−1920.30	−	1441.27i	−403.861	+	4076.04i	1093.50	+	1894.00i	−235.364	−	146.422i
67.6	−13.9872	−	7.76906i	−40.5000	−	23.3827i	135.284	+	217.335i	309.042	+	535.277i	384.820	+	641.705i	2329.27	+	582.496i	−203.753	−	4090.93i	1093.50	+	1894.00i	−164.040	−	9888.00i
67.7	−11.6912	−	10.9231i	−40.5000	−	23.3827i	17.3706	+	255.410i	−80.2127	−	138.932i	218.084	+	715.759i	−1952.50	−	1397.34i	2586.79	−	3175.80i	1093.50	+	1894.00i	−579.790	+	2500.47i
67.8	−10.6919	+	11.9031i	−40.5000	−	23.3827i	−27.3664	−	254.533i	−484.456	−	839.103i	711.348	−	232.069i	1146.53	−	2109.57i	3322.32	+	2395.70i	1093.50	+	1894.00i	15167.7	+	3205.09i
67.9	−10.1818	+	12.3422i	−40.5000	−	23.3827i	−48.6620	−	251.332i	−88.3352	−	153.001i	700.958	−	261.783i	−2198.03	+	966.170i	3597.47	+	1958.42i	1093.50	+	1894.00i	2787.79	+	467.571i
67.10	−9.78919	−	12.6559i	−40.5000	−	23.3827i	−64.3435	+	247.782i	−422.737	−	732.201i	100.533	+	741.461i	2385.11	−	275.739i	3765.77	−	1611.26i	1093.50	+	1894.00i	−5128.41	+	12517.8i
67.11	−9.61537	+	12.7885i	−40.5000	−	23.3827i	−71.0892	−	245.932i	588.651	+	1019.57i	688.451	−	293.099i	−315.896	+	2380.13i	3828.63	+	1455.60i	1093.50	+	1894.00i	−18698.9	−	2275.64i
67.12	−6.43741	−	14.6479i	−40.5000	−	23.3827i	−173.120	+	188.589i	478.743	+	829.208i	−81.7912	+	743.762i	544.790	−	2338.38i	3876.86	+	1321.81i	1093.50	+	1894.00i	9064.25	−	12350.5i
67.13	−5.68194	−	14.9571i	−40.5000	−	23.3827i	−191.431	+	169.971i	−192.195	−	332.891i	−119.619	+	738.623i	−1086.12	+	2141.29i	3629.98	+	1897.49i	1093.50	+	1894.00i	−3887.06	+	4766.15i
67.14	−4.92103	+	15.2244i	−40.5000	−	23.3827i	−207.567	−	149.840i	153.283	+	265.494i	555.290	−	501.523i	2212.02	−	933.681i	3302.67	−	2422.72i	1093.50	+	1894.00i	−4796.30	+	1027.14i
67.15	−1.91595	−	15.8849i	−40.5000	−	23.3827i	−248.658	+	60.8692i	320.516	+	555.150i	−293.835	+	688.137i	−1356.87	+	1980.84i	1443.32	+	3833.28i	1093.50	+	1894.00i	8204.40	−	6155.00i
67.16	−1.68189	+	15.9114i	−40.5000	−	23.3827i	−250.342	−	53.5224i	−315.647	−	546.717i	440.167	−	605.083i	580.656	+	2329.73i	1272.66	−	3893.27i	1093.50	+	1894.00i	9229.89	−	4102.86i
67.17	2.29163	−	15.8350i	−40.5000	−	23.3827i	−245.497	−	72.5760i	−333.396	−	577.459i	−463.077	+	587.735i	2387.34	+	255.757i	−1711.83	+	3721.14i	1093.50	+	1894.00i	−9908.10	+	3956.02i
67.18	2.40390	+	15.8184i	−40.5000	−	23.3827i	−244.443	+	76.0517i	325.875	+	564.432i	272.518	−	696.854i	−949.633	−	2205.22i	−1790.63	−	3683.86i	1093.50	+	1894.00i	−8145.02	+	6511.65i
67.19	3.45820	−	15.6218i	−40.5000	−	23.3827i	−232.082	−	108.047i	−48.5211	−	84.0409i	−505.337	+	551.821i	−518.930	−	2344.25i	−2490.47	+	3251.89i	1093.50	+	1894.00i	−1480.67	+	467.356i
67.20	5.11612	+	15.1600i	−40.5000	−	23.3827i	−203.651	+	155.121i	−583.927	−	1011.39i	147.278	−	733.608i	−1772.76	−	1619.30i	−3393.53	−	2293.72i	1093.50	+	1894.00i	12345.3	−	14026.7i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	84.9.l.a	✓	64
4.b	odd	2	1		84.9.l.b	yes	64
7.c	even	3	1		84.9.l.b	yes	64
28.g	odd	6	1	inner	84.9.l.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.9.l.a	✓	64	1.a	even	1	1	trivial
84.9.l.a	✓	64	28.g	odd	6	1	inner
84.9.l.b	yes	64	4.b	odd	2	1
84.9.l.b	yes	64	7.c	even	3	1