Newform orbit 740.2.n.a

• Label: 740.2.n.a
• Level: $740$
• Weight: $2$
• Character orbit: 740.n
• Analytic conductor: $5.909$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.90892974957\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(108\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 216 * q - 8 * q^6 - 12 * q^8 - 16 * q^10 - 16 * q^12 + 8 * q^16 + 20 * q^20 - 16 * q^21 + 4 * q^22 + 16 * q^26 - 4 * q^28 - 24 * q^30 - 20 * q^32 + 16 * q^33 + 32 * q^36 - 12 * q^38 - 36 * q^40 - 40 * q^42 + 40 * q^46 - 36 * q^48 + 24 * q^50 - 20 * q^52 - 16 * q^56 - 32 * q^57 - 16 * q^58 + 20 * q^60 - 64 * q^61 + 4 * q^62 - 32 * q^65 + 52 * q^68 - 24 * q^70 + 40 * q^72 - 16 * q^73 - 24 * q^76 + 48 * q^77 + 24 * q^78 - 56 * q^80 - 136 * q^81 - 4 * q^82 + 48 * q^85 - 88 * q^86 - 64 * q^88 + 120 * q^90 + 56 * q^92 + 16 * q^96 - 80 * q^97 - 64 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
223.1	−1.41316	+	0.0545886i	−0.951440	+	0.951440i	1.99404	−	0.154285i	−0.914889	+	2.04034i	1.29260	−	1.39647i	−1.13976	−	1.13976i	−2.80947	+	0.326881i			1.18953i	1.18151	−	2.93327i
223.2	−1.41301	−	0.0583967i	0.962036	−	0.962036i	1.99318	+	0.165030i	2.04543	−	0.903457i	−1.41554	+	1.30318i	−1.74495	−	1.74495i	−2.80674	−	0.349584i			1.14897i	−2.94296	+	1.15715i
223.3	−1.41119	+	0.0924716i	0.888719	−	0.888719i	1.98290	−	0.260990i	1.22558	+	1.87028i	−1.17197	+	1.33633i	0.0171664	+	0.0171664i	−2.77411	+	0.551667i			1.42036i	−1.90247	−	2.52599i
223.4	−1.40602	−	0.152053i	−0.119582	+	0.119582i	1.95376	+	0.427577i	−1.64048	+	1.51948i	0.186316	−	0.149951i	3.40042	+	3.40042i	−2.68200	−	0.898254i			2.97140i	2.53758	−	1.88697i
223.5	−1.40438	−	0.166475i	2.15691	−	2.15691i	1.94457	+	0.467589i	−1.12739	−	1.93106i	−3.38820	+	2.67006i	0.558475	+	0.558475i	−2.65308	−	0.980396i		−	6.30456i	1.26181	+	2.89963i
223.6	−1.40119	−	0.191494i	−1.30753	+	1.30753i	1.92666	+	0.536639i	2.18104	−	0.493016i	2.08247	−	1.58171i	2.86607	+	2.86607i	−2.59685	−	1.12088i		−	0.419248i	−3.15046	+	0.273152i
223.7	−1.39465	+	0.234390i	−0.532261	+	0.532261i	1.89012	−	0.653787i	0.164080	−	2.23004i	0.617563	−	0.867076i	−0.702605	−	0.702605i	−2.48283	+	1.35483i			2.43340i	0.293865	+	3.14859i
223.8	−1.39119	+	0.254135i	−1.00358	+	1.00358i	1.87083	−	0.707102i	−2.22383	+	0.233612i	1.14113	−	1.65122i	−3.06916	−	3.06916i	−2.42298	+	1.45916i			0.985641i	3.03441	−	0.890154i
223.9	−1.38440	−	0.288867i	−2.33229	+	2.33229i	1.83311	+	0.799813i	1.44400	−	1.70729i	3.90254	−	2.55510i	−2.92128	−	2.92128i	−2.30672	−	1.63678i		−	7.87919i	−2.49226	+	1.94645i
223.10	−1.35300	−	0.411587i	1.00904	−	1.00904i	1.66119	+	1.11375i	−2.23607		0.000144217i	−1.78053	+	0.949915i	−0.740784	−	0.740784i	−1.78918	−	2.19062i			0.963694i	3.02533	+	0.920531i
223.11	−1.34504	+	0.436894i	−2.03010	+	2.03010i	1.61825	−	1.17528i	−2.00579	−	0.988345i	1.84362	−	3.61750i	2.09201	+	2.09201i	−1.66313	+	2.28779i		−	5.24262i	3.12966	+	0.453044i
223.12	−1.33397	+	0.469611i	1.90038	−	1.90038i	1.55893	−	1.25289i	2.19518	+	0.425649i	−1.64260	+	3.42748i	2.19059	+	2.19059i	−1.49119	+	2.40341i		−	4.22286i	−3.12819	+	0.463080i
223.13	−1.32143	−	0.503803i	−1.04444	+	1.04444i	1.49237	+	1.33148i	−1.72951	−	1.41732i	1.90634	−	0.853962i	0.225601	+	0.225601i	−1.30126	−	2.51132i			0.818304i	1.57137	+	2.74423i
223.14	−1.31922	+	0.509560i	0.532040	−	0.532040i	1.48070	−	1.34445i	−0.341068	−	2.20990i	−0.430773	+	0.972985i	2.43768	+	2.43768i	−1.26829	+	2.52813i			2.43387i	1.57602	+	2.74156i
223.15	−1.31184	+	0.528270i	−2.09162	+	2.09162i	1.44186	−	1.38601i	0.921582	+	2.03732i	1.63893	−	3.84881i	0.220841	+	0.220841i	−1.15931	+	2.57992i		−	5.74972i	−2.28523	−	2.18580i
223.16	−1.26502	−	0.632247i	0.507201	−	0.507201i	1.20053	+	1.59960i	1.27860	−	1.83444i	−0.962293	+	0.320940i	2.44835	+	2.44835i	−0.507338	−	2.78255i			2.48549i	−2.77727	+	1.51221i
223.17	−1.25796	+	0.646169i	1.96497	−	1.96497i	1.16493	−	1.62571i	−1.98542	+	1.02864i	−1.20215	+	3.74157i	0.969540	+	0.969540i	−0.414954	+	2.79782i		−	4.72224i	1.83291	−	2.57690i
223.18	−1.25640	−	0.649201i	−0.333967	+	0.333967i	1.15708	+	1.63131i	1.91259	+	1.15845i	0.636408	−	0.202784i	−2.67537	−	2.67537i	−0.394699	−	2.80075i			2.77693i	−1.65091	−	2.69713i
223.19	−1.20660	+	0.737648i	1.78917	−	1.78917i	0.911751	−	1.78009i	1.06004	−	1.96883i	−0.839029	+	3.47858i	−2.99225	−	2.99225i	0.212963	+	2.82040i		−	3.40226i	0.173260	+	3.15753i
223.20	−1.20163	+	0.745715i	−1.00581	+	1.00581i	0.887819	−	1.79214i	2.14883	−	0.618474i	0.458562	−	1.95866i	−1.33145	−	1.33145i	0.269600	+	2.81555i			0.976691i	−2.12089	+	2.34559i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	740.2.n.a	✓	216
4.b	odd	2	1	inner	740.2.n.a	✓	216
5.c	odd	4	1	inner	740.2.n.a	✓	216
20.e	even	4	1	inner	740.2.n.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
740.2.n.a	✓	216	1.a	even	1	1	trivial
740.2.n.a	✓	216	4.b	odd	2	1	inner
740.2.n.a	✓	216	5.c	odd	4	1	inner
740.2.n.a	✓	216	20.e	even	4	1	inner

Hecke kernels

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