Newform orbit 64.12.i.a

• Label: 64.12.i.a
• Level: $64$
• Weight: $12$
• Character orbit: 64.i
• Analytic conductor: $49.174$
• Analytic rank: $0$
• Dimension: $696$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	64.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(49.1739635558\)
Analytic rank:	\(0\)
Dimension:	\(696\)
Relative dimension:	\(87\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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	−45.2527	+	0.440829i	491.613	+	328.486i	2047.61	−	39.8974i	12129.7	+	2412.76i	−22391.6	−	14648.1i	16326.7	−	39416.1i	−92642.3	+	2708.11i	65989.7	+	159313.i	−549967.	−	103837.i
5.2	−45.2506	+	0.618640i	31.0263	+	20.7311i	2047.23	−	55.9877i	36.1815	+	7.19694i	−1416.78	−	918.901i	2148.77	−	5187.58i	−92604.0	+	3799.98i	−67258.4	−	162376.i	−1641.69	−	303.283i
5.3	−45.0781	−	3.99526i	−414.704	−	277.096i	2016.08	+	360.198i	−555.080	−	110.412i	17587.0	+	14147.8i	11728.9	−	28316.1i	−89441.8	−	24291.8i	27405.8	+	66163.5i	24580.8	+	7194.87i
5.4	−44.8708	+	5.88307i	196.675	+	131.414i	1978.78	−	527.956i	−9168.52	−	1823.73i	−9598.10	−	4739.61i	6070.55	−	14655.6i	−85683.4	+	35331.1i	−46379.8	−	111971.i	422128.	+	27893.3i
5.5	−44.7300	−	6.87241i	251.048	+	167.745i	1953.54	+	614.805i	4987.75	+	992.126i	−10076.5	−	9228.52i	−26028.6	+	62838.7i	−83156.6	−	40925.8i	−32904.6	−	79438.6i	−216284.	−	78655.6i
5.6	−43.8882	+	11.0377i	−688.424	−	459.990i	1804.34	−	968.845i	−1066.56	−	212.153i	35290.9	+	12589.5i	2768.54	−	6683.85i	−68495.4	+	62436.5i	194546.	+	469675.i	49151.2	−	2461.37i
5.7	−43.4865	−	12.5269i	−328.106	−	219.234i	1734.15	+	1089.50i	−2763.05	−	549.605i	11521.9	+	13643.9i	−20531.0	+	49566.3i	−61764.1	−	69102.3i	−8200.82	−	19798.5i	113271.	+	58512.9i
5.8	−43.3106	−	13.1223i	578.347	+	386.439i	1703.61	+	1136.67i	−6322.09	−	1257.54i	−19977.5	−	24326.1i	9582.02	−	23133.0i	−58868.5	−	71585.1i	117359.	+	283329.i	257311.	+	137425.i
5.9	−42.7314	+	14.9005i	−204.839	−	136.869i	1603.95	−	1273.44i	7992.41	+	1589.79i	10792.5	+	2796.42i	25176.3	−	60780.9i	−49564.4	+	78315.4i	−44565.3	−	107590.i	−365216.	+	51156.6i
5.10	−41.9528	+	16.9694i	−286.773	−	191.616i	1472.08	−	1423.83i	−12415.3	−	2469.57i	15282.5	+	3172.44i	−22986.3	+	55493.8i	−37596.2	+	84714.0i	−22269.0	−	53762.2i	562766.	−	107076.i
5.11	−41.7341	+	17.5005i	−352.212	−	235.341i	1435.46	−	1460.73i	10280.1	+	2044.85i	18817.8	+	3657.83i	−32327.6	+	78045.7i	−34344.2	+	86083.8i	876.945	+	2117.13i	−464818.	+	94568.0i
5.12	−41.4648	−	18.1294i	−464.453	−	310.338i	1390.65	+	1503.46i	12956.6	+	2577.24i	13632.2	+	21288.3i	1595.21	−	3851.18i	−30406.2	−	87552.3i	51616.0	+	124612.i	−490520.	−	341760.i
5.13	−41.3116	+	18.4757i	585.993	+	391.548i	1365.30	−	1526.52i	−2823.72	−	561.673i	−31442.4	−	5348.82i	−12308.6	+	29715.5i	−28198.9	+	88287.9i	122286.	+	295226.i	127030.	−	28966.7i
5.14	−40.9426	−	19.2796i	−393.843	−	263.158i	1304.60	+	1578.71i	−12048.3	−	2396.55i	11051.4	+	18367.5i	26739.2	−	64554.2i	−22976.8	−	89788.6i	18069.3	+	43623.1i	447084.	+	330407.i
5.15	−38.5940	+	23.6326i	341.547	+	228.214i	930.998	−	1824.16i	6128.98	+	1219.13i	−18575.0	−	736.060i	−12167.5	+	29374.9i	7178.65	+	92403.5i	−3218.78	−	7770.81i	−265353.	+	97792.7i
5.16	−38.3367	−	24.0478i	269.590	+	180.134i	891.405	+	1843.83i	−10751.7	−	2138.64i	−6003.35	−	13388.8i	−25719.5	+	62092.4i	10166.5	−	92122.6i	−27560.9	−	66538.0i	360754.	+	340543.i
5.17	−37.6615	+	25.0921i	−155.339	−	103.794i	788.771	−	1890.01i	2148.51	+	427.365i	8454.70	+	11.2552i	−1850.61	+	4467.77i	17718.2	+	90972.5i	−54434.3	−	131416.i	−91639.5	+	37815.5i
5.18	−37.5203	+	25.3027i	220.345	+	147.230i	767.549	−	1898.73i	−7435.73	−	1479.06i	−11992.7	+	51.2080i	32138.2	−	77588.4i	19244.2	+	90662.0i	−40916.0	−	98780.0i	316415.	−	132649.i
5.19	−37.0359	−	26.0066i	52.1464	+	34.8431i	695.311	+	1926.36i	5140.30	+	1022.47i	−1025.14	−	2646.60i	7620.13	−	18396.6i	24346.6	−	89426.9i	−66286.0	−	160029.i	−163785.	−	171550.i
5.20	−36.6811	−	26.5047i	253.768	+	169.562i	643.003	+	1944.44i	624.950	+	124.310i	−4814.28	−	12945.8i	29697.1	−	71695.2i	27950.8	−	88366.8i	−32144.5	−	77603.6i	−19629.0	−	21123.9i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
64.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	64.12.i.a	✓	696
64.i	even	16	1	inner	64.12.i.a	✓	696

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
64.12.i.a	✓	696	1.a	even	1	1	trivial
64.12.i.a	✓	696	64.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(64, [\chi])\).