Newform orbit 322.3.p.a

• Label: 322.3.p.a
• Level: $322$
• Weight: $3$
• Character orbit: 322.p
• Analytic conductor: $8.774$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	322.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.77386451240\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q + 64 q^{4} + 16 q^{6} + 112 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.25700	−	0.648030i	−3.89257	−	3.06115i	1.16011	+	1.62915i	−5.21143	+	5.46559i	2.90925	+	6.37038i	−6.68556	+	2.07445i	−0.402527	−	2.79964i	3.65962	+	15.0852i	10.0926	−	3.49310i
11.2	−1.25700	−	0.648030i	−3.85387	−	3.03072i	1.16011	+	1.62915i	5.16879	−	5.42087i	2.88033	+	6.30705i	4.35506	+	5.48028i	−0.402527	−	2.79964i	3.54525	+	14.6137i	−10.0101	+	3.46452i
11.3	−1.25700	−	0.648030i	−3.73796	−	2.93957i	1.16011	+	1.62915i	3.97965	−	4.17374i	2.79370	+	6.11735i	−4.89473	−	5.00416i	−0.402527	−	2.79964i	3.20948	+	13.2297i	−7.70714	+	2.66747i
11.4	−1.25700	−	0.648030i	−3.02934	−	2.38230i	1.16011	+	1.62915i	−1.89902	+	1.99164i	2.26409	+	4.95767i	6.80071	+	1.65840i	−0.402527	−	2.79964i	1.37973	+	5.68733i	3.67772	−	1.27287i
11.5	−1.25700	−	0.648030i	−2.00962	−	1.58038i	1.16011	+	1.62915i	1.20550	−	1.26430i	1.50196	+	3.28884i	−6.90349	+	1.15834i	−0.402527	−	2.79964i	−0.580867	−	2.39437i	−2.33462	+	0.808021i
11.6	−1.25700	−	0.648030i	−1.41063	−	1.10934i	1.16011	+	1.62915i	−4.46910	+	4.68706i	1.05429	+	2.30857i	1.67113	−	6.79760i	−0.402527	−	2.79964i	−1.36257	−	5.61657i	8.65504	−	2.99554i
11.7	−1.25700	−	0.648030i	−0.755926	−	0.594467i	1.16011	+	1.62915i	3.32345	−	3.48553i	0.564969	+	1.23711i	1.11071	−	6.91132i	−0.402527	−	2.79964i	−1.90380	−	7.84756i	−6.43631	+	2.22763i
11.8	−1.25700	−	0.648030i	0.0985099	+	0.0774690i	1.16011	+	1.62915i	−4.68358	+	4.91199i	−0.0736249	−	0.161216i	1.08599	+	6.91525i	−0.402527	−	2.79964i	−2.11813	−	8.73104i	9.07039	−	3.13929i
11.9	−1.25700	−	0.648030i	0.383835	+	0.301851i	1.16011	+	1.62915i	−1.43428	+	1.50423i	−0.286873	−	0.628164i	5.66810	+	4.10764i	−0.402527	−	2.79964i	−2.06562	−	8.51458i	2.77768	−	0.961364i
11.10	−1.25700	−	0.648030i	1.22290	+	0.961696i	1.16011	+	1.62915i	5.14971	−	5.40086i	−0.913976	−	2.00133i	6.81593	−	1.59470i	−0.402527	−	2.79964i	−1.55122	−	6.39420i	−9.97313	+	3.45173i
11.11	−1.25700	−	0.648030i	1.24577	+	0.979686i	1.16011	+	1.62915i	2.88072	−	3.02121i	−0.931073	−	2.03877i	−3.79398	+	5.88266i	−0.402527	−	2.79964i	−1.52967	−	6.30537i	−5.57891	+	1.93088i
11.12	−1.25700	−	0.648030i	2.57257	+	2.02309i	1.16011	+	1.62915i	−2.00945	+	2.10745i	−1.92271	−	4.21014i	−3.84021	−	5.85259i	−0.402527	−	2.79964i	0.403390	+	1.66279i	3.89157	−	1.34689i
11.13	−1.25700	−	0.648030i	2.74918	+	2.16198i	1.16011	+	1.62915i	−0.940848	+	0.986733i	−2.05470	−	4.49917i	−6.99974	−	0.0597914i	−0.402527	−	2.79964i	0.762015	+	3.14107i	1.82208	−	0.630629i
11.14	−1.25700	−	0.648030i	3.56094	+	2.80035i	1.16011	+	1.62915i	0.809396	−	0.848871i	−2.66140	−	5.82765i	4.21811	−	5.58637i	−0.402527	−	2.79964i	2.71647	+	11.1975i	−1.56751	+	0.542520i
11.15	−1.25700	−	0.648030i	4.36265	+	3.43082i	1.16011	+	1.62915i	5.58209	−	5.85433i	−3.26058	−	7.13968i	−0.862587	+	6.94665i	−0.402527	−	2.79964i	5.14030	+	21.1886i	−10.8105	+	3.74155i
11.16	−1.25700	−	0.648030i	4.36393	+	3.43183i	1.16011	+	1.62915i	−6.08549	+	6.38228i	−3.26154	−	7.14178i	2.75429	+	6.43536i	−0.402527	−	2.79964i	5.14458	+	21.2063i	11.7854	−	4.07897i
11.17	1.25700	+	0.648030i	−4.41158	−	3.46931i	1.16011	+	1.62915i	−0.984942	+	1.03298i	−3.29716	−	7.21977i	−5.92223	−	3.73192i	0.402527	+	2.79964i	5.30414	+	21.8640i	−1.90748	+	0.660184i
11.18	1.25700	+	0.648030i	−3.80389	−	2.99141i	1.16011	+	1.62915i	3.89919	−	4.08935i	−2.84298	−	6.22525i	4.30373	+	5.52068i	0.402527	+	2.79964i	3.39920	+	14.0117i	7.55132	−	2.61354i
11.19	1.25700	+	0.648030i	−3.42677	−	2.69484i	1.16011	+	1.62915i	−2.69686	+	2.82839i	−2.56112	−	5.60808i	6.97589	−	0.580476i	0.402527	+	2.79964i	2.35876	+	9.72294i	−5.22285	+	1.80765i
11.20	1.25700	+	0.648030i	−2.29767	−	1.80691i	1.16011	+	1.62915i	6.45892	−	6.77392i	−1.71725	−	3.76025i	−6.66189	−	2.14924i	0.402527	+	2.79964i	−0.107458	−	0.442949i	12.5086	−	4.32926i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
23.d	odd	22	1	inner
161.p	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	322.3.p.a	✓	640
7.c	even	3	1	inner	322.3.p.a	✓	640
23.d	odd	22	1	inner	322.3.p.a	✓	640
161.p	odd	66	1	inner	322.3.p.a	✓	640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
322.3.p.a	✓	640	1.a	even	1	1	trivial
322.3.p.a	✓	640	7.c	even	3	1	inner
322.3.p.a	✓	640	23.d	odd	22	1	inner
322.3.p.a	✓	640	161.p	odd	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).