Newform orbit 615.2.r.a

• Label: 615.2.r.a
• Level: $615$
• Weight: $2$
• Character orbit: 615.r
• Analytic conductor: $4.911$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 615 = 3 \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	615.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.91079972431\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) 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.

\(\operatorname{Tr}(f)(q) = \) 160 * q - 8 * q^6 - 8 * q^7 - 24 * q^10 - 14 * q^15 - 152 * q^16 + 4 * q^18 + 8 * q^24 - 8 * q^25 + 10 * q^30 - 16 * q^31 - 12 * q^33 - 16 * q^34 - 8 * q^37 + 24 * q^39 + 24 * q^40 + 12 * q^42 + 32 * q^43 + 96 * q^49 + 8 * q^51 - 32 * q^52 - 20 * q^55 + 24 * q^58 + 46 * q^60 - 72 * q^63 - 56 * q^66 - 36 * q^69 + 16 * q^70 + 36 * q^72 - 16 * q^75 + 48 * q^76 + 44 * q^78 + 8 * q^79 + 32 * q^81 - 64 * q^82 + 24 * q^84 - 100 * q^85 + 28 * q^87 - 216 * q^88 - 40 * q^90 - 96 * q^94 + 44 * q^96 + 24 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−1.93141	−	1.93141i	1.72970	−	0.0901717i			5.46066i	−2.22866	+	0.181832i	−3.51491	−	3.16660i	−2.36236			6.68393	−	6.68393i	2.98374	−	0.311940i	4.65564	+	3.95326i
32.2	−1.90761	−	1.90761i	−0.486774	−	1.66224i			5.27794i	0.0903349	+	2.23424i	−2.24234	+	4.09948i	2.32577			6.25303	−	6.25303i	−2.52610	+	1.61827i	4.08974	−	4.43438i
32.3	−1.88681	−	1.88681i	0.935589	+	1.45763i			5.12009i	1.55532	−	1.60654i	0.984982	−	4.51554i	2.86502			5.88701	−	5.88701i	−1.24934	+	2.72748i	−5.96583	+	0.0966353i
32.4	−1.85429	−	1.85429i	−1.62955	+	0.587012i			4.87676i	2.23239	+	0.128162i	4.11013	+	1.93316i	0.659233			5.33433	−	5.33433i	2.31083	−	1.91312i	−3.90185	−	4.37714i
32.5	−1.84151	−	1.84151i	0.917450	−	1.46911i			4.78231i	2.06963	−	0.846535i	−4.39487	+	1.01589i	−2.38837			5.12365	−	5.12365i	−1.31657	−	2.69567i	−5.37015	−	2.25234i
32.6	−1.81403	−	1.81403i	−1.40852	−	1.00800i			4.58142i	−0.973951	−	2.01281i	0.726548	+	4.38365i	0.972545			4.68279	−	4.68279i	0.967855	+	2.83959i	−1.88453	+	5.41809i
32.7	−1.75032	−	1.75032i	−0.00348917	+	1.73205i			4.12722i	0.957415	+	2.02073i	3.03774	−	3.02552i	−4.74925			3.72330	−	3.72330i	−2.99998	−	0.0120868i	1.86114	−	5.21270i
32.8	−1.70813	−	1.70813i	−0.0275900	+	1.73183i			3.83543i	−2.18111	−	0.492700i	3.00532	−	2.91107i	0.802352			3.13516	−	3.13516i	−2.99848	−	0.0955623i	2.88403	+	4.56722i
32.9	−1.58028	−	1.58028i	−1.69085	−	0.375547i			2.99458i	−1.64966	+	1.50951i	2.07854	+	3.26548i	−2.25110			1.57171	−	1.57171i	2.71793	+	1.26999i	4.99238	+	0.221462i
32.10	−1.53653	−	1.53653i	1.68928	−	0.382539i			2.72185i	2.01492	+	0.969582i	−3.18341	−	2.00785i	3.61671			1.10915	−	1.10915i	2.70733	−	1.29243i	−1.60620	−	4.58578i
32.11	−1.51359	−	1.51359i	−1.03003	+	1.39249i			2.58193i	−0.0650452	+	2.23512i	3.66672	−	0.548616i	3.82040			0.880802	−	0.880802i	−0.878066	−	2.86862i	3.48152	−	3.28461i
32.12	−1.51198	−	1.51198i	−1.44263	+	0.958557i			2.57215i	−0.441633	−	2.19202i	3.63053	+	0.731902i	−4.88662			0.865083	−	0.865083i	1.16234	−	2.76568i	−2.64655	+	3.98203i
32.13	−1.50851	−	1.50851i	1.47784	+	0.903314i			2.55120i	0.936795	−	2.03037i	−0.866683	−	3.59200i	−2.95622			0.831486	−	0.831486i	1.36805	+	2.66992i	−4.47600	+	1.64967i
32.14	−1.50520	−	1.50520i	1.27783	−	1.16925i			2.53128i	−1.05186	+	1.97322i	−3.68336	−	0.163428i	0.392544			0.799687	−	0.799687i	0.265692	−	2.98821i	4.55336	−	1.38683i
32.15	−1.45876	−	1.45876i	0.498902	−	1.65864i			2.25597i	−0.723075	−	2.11593i	−3.14735	+	1.69179i	−1.48908			0.373406	−	0.373406i	−2.50219	−	1.65500i	−2.03185	+	4.14143i
32.16	−1.29733	−	1.29733i	1.71357	+	0.252374i			1.36614i	−1.49068	−	1.66670i	−1.89565	−	2.55048i	3.24596			−0.822328	+	0.822328i	2.87261	+	0.864918i	−0.228361	+	4.09616i
32.17	−1.29283	−	1.29283i	−0.726807	−	1.57218i			1.34283i	1.82412	+	1.29328i	−1.09293	+	2.97221i	−3.45365			−0.849605	+	0.849605i	−1.94350	+	2.28534i	−0.686282	−	4.03028i
32.18	−1.22901	−	1.22901i	−1.08296	−	1.35174i			1.02094i	1.33308	−	1.79524i	−0.330327	+	2.99227i	2.11054			−1.20328	+	1.20328i	−0.654385	+	2.92776i	−3.84474	+	0.567996i
32.19	−1.20133	−	1.20133i	1.39118	+	1.03181i			0.886366i	−1.33262	+	1.79558i	−0.431722	−	2.91079i	−0.826459			−1.33784	+	1.33784i	0.870750	+	2.87085i	3.75799	−	0.556170i
32.20	−1.15430	−	1.15430i	−1.72925	−	0.0985233i			0.664797i	2.22385	−	0.233461i	1.88234	+	2.10979i	0.533046			−1.54122	+	1.54122i	2.98059	+	0.340742i	−2.83646	−	2.29749i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
205.f	odd	4	1	inner
615.r	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	615.2.r.a	yes	160
3.b	odd	2	1	inner	615.2.r.a	yes	160
5.c	odd	4	1		615.2.k.a	✓	160
15.e	even	4	1		615.2.k.a	✓	160
41.c	even	4	1		615.2.k.a	✓	160
123.f	odd	4	1		615.2.k.a	✓	160
205.f	odd	4	1	inner	615.2.r.a	yes	160
615.r	even	4	1	inner	615.2.r.a	yes	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
615.2.k.a	✓	160	5.c	odd	4	1
615.2.k.a	✓	160	15.e	even	4	1
615.2.k.a	✓	160	41.c	even	4	1
615.2.k.a	✓	160	123.f	odd	4	1
615.2.r.a	yes	160	1.a	even	1	1	trivial
615.2.r.a	yes	160	3.b	odd	2	1	inner
615.2.r.a	yes	160	205.f	odd	4	1	inner
615.2.r.a	yes	160	615.r	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(615, [\chi])\).