Newform orbit 416.2.bi.a

• Label: 416.2.bi.a
• Level: $416$
• Weight: $2$
• Character orbit: 416.bi
• Analytic conductor: $3.322$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bi (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 4 q^{2} - 8 q^{3} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 4 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
99.1	−1.41416	−	0.0124808i	0.185333	−	0.447434i	1.99969	+	0.0352996i	−0.172454	+	0.416340i	−0.267675	+	0.630429i	−2.63339			−2.82744	−	0.0748769i	1.95547	+	1.95547i	0.249073	−	0.586618i
99.2	−1.40664	+	0.146174i	−0.514362	+	1.24178i	1.95727	−	0.411228i	−0.618863	+	1.49407i	0.542006	−	1.82192i	4.58865			−2.69306	+	0.864551i	0.843873	+	0.843873i	0.652123	−	2.19207i
99.3	−1.39321	−	0.242829i	1.24651	−	3.00935i	1.88207	+	0.676624i	0.701430	−	1.69340i	−2.46741	+	3.88996i	2.55030			−2.45781	−	1.39970i	−5.38106	−	5.38106i	−1.38845	+	2.18894i
99.4	−1.37040	+	0.349306i	−0.598132	+	1.44402i	1.75597	−	0.957376i	0.944556	−	2.28036i	0.315273	−	2.18781i	−3.99101			−2.07196	+	1.92536i	0.393895	+	0.393895i	−0.497872	+	3.45493i
99.5	−1.32900	−	0.483486i	0.890323	−	2.14943i	1.53248	+	1.28511i	−1.52443	+	3.68031i	−2.22246	+	2.42613i	−1.19925			−1.41534	−	2.44884i	−1.70605	−	1.70605i	3.80535	−	4.15409i
99.6	−1.29332	−	0.572126i	0.174306	−	0.420812i	1.34534	+	1.47988i	1.36963	−	3.30658i	−0.466191	+	0.444519i	1.57593			−0.893278	−	2.68366i	1.97462	+	1.97462i	−3.66315	+	3.49286i
99.7	−1.29190	−	0.575332i	−0.680922	+	1.64389i	1.33799	+	1.48654i	−1.07264	+	2.58958i	1.82546	−	1.73198i	−0.894378			−0.873285	−	2.69024i	−0.117403	−	0.117403i	2.87561	−	2.72834i
99.8	−1.26784	+	0.626573i	0.685368	−	1.65463i	1.21481	−	1.58878i	−1.01561	+	2.45191i	0.167810	+	2.52723i	1.75642			−0.544694	+	2.77548i	−0.146735	−	0.146735i	−0.248669	−	3.74497i
99.9	−1.26157	+	0.639092i	0.429027	−	1.03576i	1.18312	−	1.61252i	1.40552	−	3.39324i	0.120699	+	1.58087i	2.38266			−0.462046	+	2.79043i	1.23258	+	1.23258i	0.395421	+	5.17907i
99.10	−1.24057	−	0.678953i	−1.17354	+	2.83317i	1.07805	+	1.68458i	0.191443	−	0.462185i	3.37944	−	2.71798i	−1.90349			−0.193645	−	2.82179i	−4.52833	−	4.52833i	−0.551302	+	0.443394i
99.11	−1.21725	+	0.719929i	−1.29743	+	3.13227i	0.963404	−	1.75267i	−0.636854	+	1.53750i	−0.675714	−	4.74682i	−1.05850			0.0890924	+	2.82702i	−6.00647	−	6.00647i	−0.331680	−	2.33002i
99.12	−1.21550	+	0.722878i	1.19450	−	2.88378i	0.954896	−	1.75732i	0.304891	−	0.736071i	0.632701	+	4.36873i	−4.22727			0.109649	+	2.82630i	−4.76805	−	4.76805i	0.161494	+	1.11510i
99.13	−1.13109	−	0.848900i	−0.351936	+	0.849649i	0.558737	+	1.92037i	0.415158	−	1.00228i	1.11934	−	0.662272i	0.957261			0.998218	−	2.64642i	1.52328	+	1.52328i	−1.32042	+	0.781243i
99.14	−0.988333	+	1.01153i	−0.636993	+	1.53784i	−0.0463956	−	1.99946i	0.127261	−	0.307235i	−0.926012	−	2.16424i	1.40083			2.06837	+	1.92920i	0.162135	+	0.162135i	0.185002	+	0.432379i
99.15	−0.945408	−	1.05176i	0.941327	−	2.27256i	−0.212406	+	1.98869i	0.238989	−	0.576971i	−3.28013	+	1.15845i	−1.21092			2.29244	−	1.65672i	−2.15713	−	2.15713i	−0.832779	+	0.294114i
99.16	−0.942649	+	1.05424i	−0.164830	+	0.397935i	−0.222826	−	1.98755i	−1.54999	+	3.74200i	−0.264140	−	0.548882i	−4.22581			2.30539	+	1.63865i	1.99014	+	1.99014i	−2.48385	−	5.16144i
99.17	−0.822394	+	1.15051i	0.571616	−	1.38000i	−0.647338	−	1.89234i	−0.0774044	+	0.186871i	1.11761	+	1.79255i	2.98479			2.70952	+	0.811481i	0.543659	+	0.543659i	−0.151339	−	0.242736i
99.18	−0.761769	−	1.19151i	0.766663	−	1.85089i	−0.839415	+	1.81532i	−0.354084	+	0.854833i	−2.78938	+	0.496460i	4.51134			2.80242	−	0.382678i	−0.716698	−	0.716698i	1.28828	−	0.229290i
99.19	−0.717929	−	1.21843i	0.172072	−	0.415418i	−0.969155	+	1.74950i	−0.252815	+	0.610350i	−0.629694	+	0.0885829i	−3.40059			2.82743	−	0.0751652i	1.97836	+	1.97836i	0.925174	−	0.130150i
99.20	−0.557499	−	1.29969i	−0.709273	+	1.71234i	−1.37839	+	1.44915i	−1.29428	+	3.12466i	2.62093	−	0.0327901i	1.96257			2.65190	+	0.983581i	−0.307710	−	0.307710i	4.78265	−	0.0598350i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
416.bi	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.bi.a	yes	216
13.d	odd	4	1		416.2.bd.a	✓	216
32.h	odd	8	1		416.2.bd.a	✓	216
416.bi	even	8	1	inner	416.2.bi.a	yes	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.bd.a	✓	216	13.d	odd	4	1
416.2.bd.a	✓	216	32.h	odd	8	1
416.2.bi.a	yes	216	1.a	even	1	1	trivial
416.2.bi.a	yes	216	416.bi	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).