Newform orbit 768.2.z.a

• Label: 768.2.z.a
• Level: $768$
• Weight: $2$
• Character orbit: 768.z
• Analytic conductor: $6.133$
• Analytic rank: $0$
• Dimension: $2048$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.z (of order \(64\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13251087523\)
Analytic rank:	\(0\)
Dimension:	\(2048\)
Relative dimension:	\(64\) over \(\Q(\zeta_{64})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{64}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2048 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.41359	+	0.0418894i	−0.595699	+	0.803208i	1.99649	−	0.118429i	−0.327704	+	0.361566i	0.808431	−	1.16036i	0.00690361	+	0.0129157i	−2.81726	+	0.251042i	−0.290285	−	0.956940i	0.448095	−	0.524835i
13.2	−1.41350	+	0.0449561i	0.595699	−	0.803208i	1.99596	−	0.127091i	−0.976967	+	1.07792i	−0.805911	+	1.16211i	1.24169	+	2.32303i	−2.81557	+	0.269373i	−0.290285	−	0.956940i	1.33248	−	1.56756i
13.3	−1.40260	−	0.180850i	0.595699	−	0.803208i	1.93459	+	0.507320i	−1.16541	+	1.28583i	−0.980789	+	1.01885i	0.519267	+	0.971481i	−2.62171	−	1.06144i	−0.290285	−	0.956940i	1.86715	−	1.59275i
13.4	−1.40141	−	0.189837i	−0.595699	+	0.803208i	1.92792	+	0.532081i	−2.39330	+	2.64060i	0.987300	−	1.01254i	0.550061	+	1.02909i	−2.60081	−	1.11166i	−0.290285	−	0.956940i	3.85529	−	3.24623i
13.5	−1.31235	+	0.527010i	0.595699	−	0.803208i	1.44452	−	1.38324i	2.40204	−	2.65024i	−0.358467	+	1.36803i	−1.04282	−	1.95097i	−1.16673	+	2.57657i	−0.290285	−	0.956940i	−1.75561	+	4.74394i
13.6	−1.28906	+	0.581660i	0.595699	−	0.803208i	1.32334	−	1.49959i	−2.89087	+	3.18959i	−0.300698	+	1.38188i	−1.59573	−	2.98541i	−0.833617	+	2.70279i	−0.290285	−	0.956940i	1.87125	−	5.79307i
13.7	−1.28767	+	0.584731i	−0.595699	+	0.803208i	1.31618	−	1.50588i	−0.794581	+	0.876685i	0.297403	−	1.38259i	−1.79620	−	3.36045i	−0.814267	+	2.70868i	−0.290285	−	0.956940i	0.510531	−	1.59349i
13.8	−1.27841	−	0.604702i	0.595699	−	0.803208i	1.26867	+	1.54612i	2.84678	−	3.14094i	−1.24725	+	0.666609i	1.05684	+	1.97721i	−0.686944	−	2.74374i	−0.290285	−	0.956940i	−5.53870	+	2.29396i
13.9	−1.27105	−	0.620023i	−0.595699	+	0.803208i	1.23114	+	1.57616i	2.05611	−	2.26857i	1.25517	−	0.651571i	1.47679	+	2.76288i	−0.587592	−	2.76672i	−0.290285	−	0.956940i	−4.01999	+	1.60863i
13.10	−1.23955	+	0.680812i	−0.595699	+	0.803208i	1.07299	−	1.68781i	−0.0613270	+	0.0676639i	0.191568	−	1.40118i	1.39357	+	2.60719i	−0.180950	+	2.82263i	−0.290285	−	0.956940i	0.0299517	−	0.125625i
13.11	−1.22972	+	0.698414i	0.595699	−	0.803208i	1.02444	−	1.71771i	−0.583087	+	0.643337i	−0.171574	+	1.40377i	0.568107	+	1.06285i	−0.0601003	+	2.82779i	−0.290285	−	0.956940i	0.267720	−	1.19836i
13.12	−1.22570	−	0.705459i	0.595699	−	0.803208i	1.00466	+	1.72935i	0.365682	−	0.403468i	−1.29678	+	0.564246i	−1.13087	−	2.11570i	−0.0114149	−	2.82840i	−0.290285	−	0.956940i	−0.732845	+	0.236555i
13.13	−1.18968	−	0.764631i	−0.595699	+	0.803208i	0.830679	+	1.81933i	−0.610959	+	0.674089i	1.32285	−	0.500070i	2.04708	+	3.82983i	0.402876	−	2.79959i	−0.290285	−	0.956940i	1.24228	−	0.334793i
13.14	−1.05561	−	0.941111i	−0.595699	+	0.803208i	0.228622	+	1.98689i	−0.573397	+	0.632646i	1.38473	−	0.287254i	−0.583115	−	1.09093i	1.62855	−	2.31254i	−0.290285	−	0.956940i	1.20067	−	0.128197i
13.15	−1.04992	+	0.947452i	0.595699	−	0.803208i	0.204669	−	1.98950i	1.31140	−	1.44691i	0.135563	+	1.40770i	1.32278	+	2.47475i	1.67007	+	2.28273i	−0.290285	−	0.956940i	−0.00599155	+	2.76162i
13.16	−0.944846	+	1.05227i	−0.595699	+	0.803208i	−0.214534	−	1.98846i	1.33230	−	1.46997i	−0.282345	−	1.38574i	0.822805	+	1.53936i	2.29509	+	1.65304i	−0.290285	−	0.956940i	0.287980	+	2.79083i
13.17	−0.931240	+	1.06433i	−0.595699	+	0.803208i	−0.265584	−	1.98229i	2.54694	−	2.81011i	−0.300137	−	1.38200i	−2.20759	−	4.13011i	2.35713	+	1.56332i	−0.290285	−	0.956940i	0.619069	+	5.32767i
13.18	−0.860470	−	1.12232i	0.595699	−	0.803208i	−0.519184	+	1.93144i	−1.93867	+	2.13899i	−1.41403	+	0.0225731i	−1.40064	−	2.62041i	2.61442	−	1.07925i	−0.290285	−	0.956940i	4.06879	+	0.335261i
13.19	−0.806589	−	1.16164i	−0.595699	+	0.803208i	−0.698828	+	1.87394i	2.26883	−	2.50327i	1.41352	+	0.0441314i	−2.04067	−	3.81782i	2.74051	−	0.699709i	−0.290285	−	0.956940i	−4.73792	−	0.616462i
13.20	−0.797967	−	1.16758i	0.595699	−	0.803208i	−0.726497	+	1.86338i	1.05514	−	1.16417i	−1.41316	−	0.0545948i	1.83096	+	3.42548i	2.75538	−	0.638674i	−0.290285	−	0.956940i	−2.20122	−	0.302995i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
256.m	even	64	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.2.z.a	✓	2048
256.m	even	64	1	inner	768.2.z.a	✓	2048

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
768.2.z.a	✓	2048	1.a	even	1	1	trivial
768.2.z.a	✓	2048	256.m	even	64	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(768, [\chi])\).