Newform orbit 740.2.be.a

• Label: 740.2.be.a
• Level: $740$
• Weight: $2$
• Character orbit: 740.be
• Analytic conductor: $5.909$
• Analytic rank: $0$
• Dimension: $304$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.90892974957\)
Analytic rank:	\(0\)
Dimension:	\(304\)
Relative dimension:	\(76\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 304 * q + 4 * q^2 - 12 * q^4 - 8 * q^8 - 152 * q^9 + 16 * q^13 - 24 * q^14 + 20 * q^16 - 60 * q^18 - 16 * q^22 - 24 * q^24 - 32 * q^29 - 56 * q^32 - 32 * q^37 + 40 * q^38 + 20 * q^42 + 12 * q^44 - 8 * q^46 + 104 * q^49 + 4 * q^50 - 40 * q^52 + 40 * q^54 + 4 * q^56 - 48 * q^57 - 12 * q^58 - 28 * q^60 - 16 * q^61 - 60 * q^62 + 24 * q^65 - 52 * q^66 - 200 * q^68 - 196 * q^72 + 20 * q^74 - 40 * q^76 + 96 * q^77 - 48 * q^78 - 152 * q^81 + 32 * q^82 - 112 * q^84 + 12 * q^88 - 8 * q^89 + 140 * q^92 - 16 * q^93 - 28 * q^94 - 132 * q^96 + 48 * q^97 - 36 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.41421		0.000708265i	−1.00731	−	1.74471i	2.00000	+	0.00200328i	0.965926	+	0.258819i	1.42331	+	2.46810i	−2.33505	+	1.34814i	−2.82842	−	0.00424959i	−0.529335	+	0.916835i	−1.36584	−	0.366709i
51.2	−1.41335	+	0.0493513i	0.328872	+	0.569623i	1.99513	−	0.139502i	0.965926	+	0.258819i	−0.492923	−	0.788847i	−2.40655	+	1.38942i	−2.81294	+	0.295627i	1.28369	−	2.22341i	−1.37797	−	0.318133i
51.3	−1.41162	+	0.0855720i	0.839608	+	1.45424i	1.98535	−	0.241591i	−0.965926	−	0.258819i	−1.30965	−	1.98100i	−0.586292	+	0.338496i	−2.78190	+	0.510926i	0.0901155	−	0.156085i	1.38567	+	0.282699i
51.4	−1.40586	+	0.153475i	−1.10372	−	1.91170i	1.95289	−	0.431528i	−0.965926	−	0.258819i	1.84508	+	2.51819i	−0.0394290	+	0.0227643i	−2.67927	+	0.906387i	−0.936403	+	1.62190i	1.39768	+	0.215619i
51.5	−1.39498	−	0.232435i	−0.140643	−	0.243602i	1.89195	+	0.648485i	−0.965926	−	0.258819i	0.139574	+	0.372510i	3.36739	−	1.94416i	−2.48850	−	1.34438i	1.46044	−	2.52955i	1.28729	+	0.585563i
51.6	−1.38367	+	0.292354i	−1.60937	−	2.78750i	1.82906	−	0.809039i	0.965926	+	0.258819i	3.04176	+	3.38647i	4.09048	−	2.36164i	−2.29428	+	1.65417i	−3.68012	+	6.37416i	−1.41218	−	0.0757269i
51.7	−1.34883	−	0.425031i	−1.46817	−	2.54294i	1.63870	+	1.14659i	−0.965926	−	0.258819i	0.899483	+	4.05402i	−2.48215	+	1.43307i	−1.72299	−	2.24306i	−2.81103	+	4.86885i	1.19287	+	0.759652i
51.8	−1.34447	+	0.438647i	1.60937	+	2.78750i	1.61518	−	1.17949i	0.965926	+	0.258819i	−3.38647	−	3.04176i	−4.09048	+	2.36164i	−1.65417	+	2.29428i	−3.68012	+	6.37416i	−1.41218	+	0.0757269i
51.9	−1.31917	−	0.509705i	0.769115	+	1.33215i	1.48040	+	1.34477i	0.965926	+	0.258819i	−0.335590	−	2.14935i	1.23671	−	0.714015i	−1.26746	−	2.52855i	0.316923	−	0.548926i	−1.14230	−	0.833762i
51.10	−1.29425	+	0.570018i	1.10372	+	1.91170i	1.35016	−	1.47549i	−0.965926	−	0.258819i	−2.51819	−	1.84508i	0.0394290	−	0.0227643i	−0.906387	+	2.67927i	−0.936403	+	1.62190i	1.39768	−	0.215619i
51.11	−1.26529	+	0.631704i	−0.839608	−	1.45424i	1.20190	−	1.59857i	−0.965926	−	0.258819i	1.98100	+	1.30965i	0.586292	−	0.338496i	−0.510926	+	2.78190i	0.0901155	−	0.156085i	1.38567	−	0.282699i
51.12	−1.24867	+	0.663937i	−0.328872	−	0.569623i	1.11838	−	1.65808i	0.965926	+	0.258819i	0.788847	+	0.492923i	2.40655	−	1.38942i	−0.295627	+	2.81294i	1.28369	−	2.22341i	−1.37797	+	0.318133i
51.13	−1.22439	+	0.707720i	1.00731	+	1.74471i	0.998265	−	1.73305i	0.965926	+	0.258819i	−2.46810	−	1.42331i	2.33505	−	1.34814i	0.00424959	+	2.82842i	−0.529335	+	0.916835i	−1.36584	+	0.366709i
51.14	−1.16908	−	0.795771i	−0.450967	−	0.781098i	0.733498	+	1.86064i	0.965926	+	0.258819i	−0.0943580	+	1.27203i	2.84609	−	1.64319i	0.623124	−	2.75893i	1.09326	−	1.89358i	−0.923284	−	1.07124i
51.15	−1.16119	−	0.807234i	1.22756	+	2.12620i	0.696746	+	1.87471i	0.965926	+	0.258819i	0.290903	−	3.45986i	−1.51707	+	0.875879i	0.704274	−	2.73934i	−1.51381	+	2.62200i	−0.912700	−	1.08027i
51.16	−1.13569	−	0.842742i	0.902444	+	1.56308i	0.579572	+	1.91418i	−0.965926	−	0.258819i	0.292378	−	2.53570i	−0.124486	+	0.0718719i	0.954949	−	2.66234i	−0.128811	+	0.223107i	0.878872	+	1.10796i
51.17	−1.09187	+	0.898785i	0.140643	+	0.243602i	0.384370	−	1.96272i	−0.965926	−	0.258819i	−0.372510	−	0.139574i	−3.36739	+	1.94416i	1.34438	+	2.48850i	1.46044	−	2.52955i	1.28729	−	0.585563i
51.18	−1.06502	−	0.930442i	−1.41047	−	2.44300i	0.268554	+	1.98189i	−0.965926	−	0.258819i	−0.770890	+	3.91421i	2.56631	−	1.48166i	1.55802	−	2.36063i	−2.47883	+	4.29347i	0.787918	+	1.17439i
51.19	−0.994365	−	1.00560i	−0.766676	−	1.32792i	−0.0224764	+	1.99987i	−0.965926	−	0.258819i	−0.573007	+	2.09141i	−3.13458	+	1.80975i	2.03343	−	1.96600i	0.324416	−	0.561906i	0.700214	+	1.22870i
51.20	−0.955608	+	1.04250i	1.46817	+	2.54294i	−0.173628	−	1.99245i	−0.965926	−	0.258819i	−4.05402	−	0.899483i	2.48215	−	1.43307i	2.24306	+	1.72299i	−2.81103	+	4.86885i	1.19287	−	0.759652i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
37.g	odd	12	1	inner
148.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	740.2.be.a	✓	304
4.b	odd	2	1	inner	740.2.be.a	✓	304
37.g	odd	12	1	inner	740.2.be.a	✓	304
148.l	even	12	1	inner	740.2.be.a	✓	304

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
740.2.be.a	✓	304	1.a	even	1	1	trivial
740.2.be.a	✓	304	4.b	odd	2	1	inner
740.2.be.a	✓	304	37.g	odd	12	1	inner
740.2.be.a	✓	304	148.l	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).