Newform orbit 744.2.o.e

• Label: 744.2.o.e
• Level: $744$
• Weight: $2$
• Character orbit: 744.o
• Analytic conductor: $5.941$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 744 = 2^{3} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	744.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.94086991038\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{4} - 32 q^{7} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
557.1	−1.40181	−	0.186864i	−1.24985	+	1.19911i	1.93016	+	0.523898i	0.453631			1.97613	−	1.44738i	2.34134			−2.60783	−	1.09509i	0.124269	−	2.99743i	−0.635905	−	0.0847673i
557.2	−1.40181	−	0.186864i	1.24985	−	1.19911i	1.93016	+	0.523898i	0.453631			−1.97613	+	1.44738i	2.34134			−2.60783	−	1.09509i	0.124269	−	2.99743i	−0.635905	−	0.0847673i
557.3	−1.40181	+	0.186864i	−1.24985	−	1.19911i	1.93016	−	0.523898i	0.453631			1.97613	+	1.44738i	2.34134			−2.60783	+	1.09509i	0.124269	+	2.99743i	−0.635905	+	0.0847673i
557.4	−1.40181	+	0.186864i	1.24985	+	1.19911i	1.93016	−	0.523898i	0.453631			−1.97613	−	1.44738i	2.34134			−2.60783	+	1.09509i	0.124269	+	2.99743i	−0.635905	+	0.0847673i
557.5	−1.34962	−	0.422518i	−0.637359	+	1.61052i	1.64296	+	1.14048i	2.38521			1.54067	−	1.90430i	−4.12791			−1.73550	−	2.23339i	−2.18755	−	2.05296i	−3.21913	−	1.00780i
557.6	−1.34962	−	0.422518i	0.637359	−	1.61052i	1.64296	+	1.14048i	2.38521			−1.54067	+	1.90430i	−4.12791			−1.73550	−	2.23339i	−2.18755	−	2.05296i	−3.21913	−	1.00780i
557.7	−1.34962	+	0.422518i	−0.637359	−	1.61052i	1.64296	−	1.14048i	2.38521			1.54067	+	1.90430i	−4.12791			−1.73550	+	2.23339i	−2.18755	+	2.05296i	−3.21913	+	1.00780i
557.8	−1.34962	+	0.422518i	0.637359	+	1.61052i	1.64296	−	1.14048i	2.38521			−1.54067	−	1.90430i	−4.12791			−1.73550	+	2.23339i	−2.18755	+	2.05296i	−3.21913	+	1.00780i
557.9	−1.32933	−	0.482574i	−1.03404	−	1.38952i	1.53424	+	1.28300i	3.64345			0.704034	+	2.34613i	2.15013			−1.42038	−	2.44592i	−0.861527	+	2.87363i	−4.84335	−	1.75824i
557.10	−1.32933	−	0.482574i	1.03404	+	1.38952i	1.53424	+	1.28300i	3.64345			−0.704034	−	2.34613i	2.15013			−1.42038	−	2.44592i	−0.861527	+	2.87363i	−4.84335	−	1.75824i
557.11	−1.32933	+	0.482574i	−1.03404	+	1.38952i	1.53424	−	1.28300i	3.64345			0.704034	−	2.34613i	2.15013			−1.42038	+	2.44592i	−0.861527	−	2.87363i	−4.84335	+	1.75824i
557.12	−1.32933	+	0.482574i	1.03404	−	1.38952i	1.53424	−	1.28300i	3.64345			−0.704034	+	2.34613i	2.15013			−1.42038	+	2.44592i	−0.861527	−	2.87363i	−4.84335	+	1.75824i
557.13	−1.30178	−	0.552593i	−1.50847	−	0.851189i	1.38928	+	1.43871i	−2.01938			1.49334	+	1.94163i	−0.359027			−1.01352	−	2.64060i	1.55095	+	2.56798i	2.62880	+	1.11590i
557.14	−1.30178	−	0.552593i	1.50847	+	0.851189i	1.38928	+	1.43871i	−2.01938			−1.49334	−	1.94163i	−0.359027			−1.01352	−	2.64060i	1.55095	+	2.56798i	2.62880	+	1.11590i
557.15	−1.30178	+	0.552593i	−1.50847	+	0.851189i	1.38928	−	1.43871i	−2.01938			1.49334	−	1.94163i	−0.359027			−1.01352	+	2.64060i	1.55095	−	2.56798i	2.62880	−	1.11590i
557.16	−1.30178	+	0.552593i	1.50847	−	0.851189i	1.38928	−	1.43871i	−2.01938			−1.49334	+	1.94163i	−0.359027			−1.01352	+	2.64060i	1.55095	−	2.56798i	2.62880	−	1.11590i
557.17	−1.23502	−	0.689003i	−1.72001	+	0.203883i	1.05055	+	1.70187i	1.58364			2.26472	+	0.933293i	−1.93355			−0.124858	−	2.82567i	2.91686	−	0.701361i	−1.95582	−	1.09113i
557.18	−1.23502	−	0.689003i	1.72001	−	0.203883i	1.05055	+	1.70187i	1.58364			−2.26472	−	0.933293i	−1.93355			−0.124858	−	2.82567i	2.91686	−	0.701361i	−1.95582	−	1.09113i
557.19	−1.23502	+	0.689003i	−1.72001	−	0.203883i	1.05055	−	1.70187i	1.58364			2.26472	−	0.933293i	−1.93355			−0.124858	+	2.82567i	2.91686	+	0.701361i	−1.95582	+	1.09113i
557.20	−1.23502	+	0.689003i	1.72001	+	0.203883i	1.05055	−	1.70187i	1.58364			−2.26472	+	0.933293i	−1.93355			−0.124858	+	2.82567i	2.91686	+	0.701361i	−1.95582	+	1.09113i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	2	1	inner
31.b	odd	2	1	inner
93.c	even	2	1	inner
248.g	odd	2	1	inner
744.o	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	744.2.o.e	✓	96
3.b	odd	2	1	inner	744.2.o.e	✓	96
8.b	even	2	1	inner	744.2.o.e	✓	96
24.h	odd	2	1	inner	744.2.o.e	✓	96
31.b	odd	2	1	inner	744.2.o.e	✓	96
93.c	even	2	1	inner	744.2.o.e	✓	96
248.g	odd	2	1	inner	744.2.o.e	✓	96
744.o	even	2	1	inner	744.2.o.e	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
744.2.o.e	✓	96	1.a	even	1	1	trivial
744.2.o.e	✓	96	3.b	odd	2	1	inner
744.2.o.e	✓	96	8.b	even	2	1	inner
744.2.o.e	✓	96	24.h	odd	2	1	inner
744.2.o.e	✓	96	31.b	odd	2	1	inner
744.2.o.e	✓	96	93.c	even	2	1	inner
744.2.o.e	✓	96	248.g	odd	2	1	inner
744.2.o.e	✓	96	744.o	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(744, [\chi])\):

T5^24 - 81*T5^22 + 2828*T5^20 - 55840*T5^18 + 688478*T5^16 - 5529626*T5^14 + 29327188*T5^12 - 101952520*T5^10 + 225374953*T5^8 - 296510253*T5^6 + 201730776*T5^4 - 50774464*T5^2 + 4121088

T13^24 - 147*T13^22 + 9302*T13^20 - 331286*T13^18 + 7290264*T13^16 - 102510864*T13^14 + 921455328*T13^12 - 5153667664*T13^10 + 16872936608*T13^8 - 28859815968*T13^6 + 21465865856*T13^4 - 6134766464*T13^2 + 558835200