Newform orbit 300.2.n.a

• Label: 300.2.n.a
• Level: $300$
• Weight: $2$
• Character orbit: 300.n
• Analytic conductor: $2.396$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	300.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.39551206064\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.40873	+	0.124406i	0.673811	−	1.59561i	1.96905	−	0.350510i	−2.20342	−	0.380687i	−0.750715	+	2.33161i		−	0.0822219i	−2.73025	+	0.738736i	−2.09196	−	2.15028i	3.15139	+	0.262166i
11.2	−1.40679	+	0.144680i	−1.60959	+	0.639710i	1.95814	−	0.407070i	2.17904	+	0.501802i	2.17180	−	1.13282i		−	2.67645i	−2.69580	+	0.855967i	2.18154	−	2.05934i	−3.13805	−	0.390668i
11.3	−1.39445	+	0.235585i	−1.04145	−	1.38397i	1.88900	−	0.657024i	1.89530	−	1.18653i	1.77830	+	1.68454i			4.72716i	−2.47934	+	1.36121i	−0.830762	+	2.88268i	−2.36338	+	2.10106i
11.4	−1.39367	−	0.240196i	−0.354401	+	1.69541i	1.88461	+	0.669506i	−0.834949	+	2.07433i	0.901146	−	2.27770i			4.41884i	−2.46571	−	1.38574i	−2.74880	−	1.20171i	1.66189	−	2.69038i
11.5	−1.38528	−	0.284622i	1.64016	+	0.556662i	1.83798	+	0.788561i	1.45398	+	1.69881i	−2.11364	−	1.23796i		−	2.78321i	−2.32167	−	1.61550i	2.38025	+	1.82603i	−1.53064	−	2.76716i
11.6	−1.31876	−	0.510760i	−1.38129	−	1.04500i	1.47825	+	1.34714i	−1.62369	+	1.53741i	1.28785	+	2.08361i		−	1.57078i	−1.26139	−	2.53158i	0.815942	+	2.88691i	2.92650	−	1.19816i
11.7	−1.29913	−	0.558792i	1.06350	−	1.36711i	1.37550	+	1.45189i	1.51671	−	1.64305i	−2.14555	+	1.18178i		−	1.88972i	−0.975659	−	2.65482i	−0.737954	−	2.90782i	−2.88853	+	1.28702i
11.8	−1.25692	+	0.648193i	1.55000	−	0.772971i	1.15969	−	1.62945i	0.411823	+	2.19782i	−1.44720	+	1.97626i			3.34784i	−0.401439	+	2.79979i	1.80503	−	2.39622i	−1.94224	−	2.49554i
11.9	−1.24841	+	0.664442i	1.66858	+	0.464581i	1.11703	−	1.65899i	0.345709	−	2.20918i	−2.39175	+	0.528689i		−	0.309856i	−0.292213	+	2.81329i	2.56833	+	1.55038i	1.03629	+	2.98766i
11.10	−1.24461	+	0.671529i	−1.27306	+	1.17444i	1.09810	−	1.67158i	−2.10329	−	0.759058i	0.795787	−	2.31662i		−	0.738888i	−0.244189	+	2.81787i	0.241361	−	2.99028i	3.12750	−	0.467689i
11.11	−1.21555	−	0.722800i	−1.65104	+	0.523516i	0.955121	+	1.75720i	−0.573135	−	2.16137i	2.38532	+	0.557011i			1.39878i	0.109105	−	2.82632i	2.45186	−	1.72869i	−0.865563	+	3.04151i
11.12	−1.03327	−	0.965583i	0.539210	+	1.64598i	0.135297	+	1.99542i	−1.86451	−	1.23434i	1.03218	−	2.22140i		−	3.63805i	1.78694	−	2.19245i	−2.41850	+	1.77506i	0.734688	+	3.07575i
11.13	−1.02327	+	0.976179i	0.339606	+	1.69843i	0.0941493	−	1.99778i	2.10329	+	0.759058i	−2.00548	−	1.40643i			0.738888i	1.85385	+	2.13617i	−2.76934	+	1.15359i	−2.89320	+	1.27647i
11.14	−1.01770	+	0.981981i	−1.62298	−	0.604914i	0.0714277	−	1.99872i	−0.345709	+	2.20918i	2.24573	−	0.978119i			0.309856i	1.89002	+	2.10424i	2.26816	+	1.96353i	−1.81755	−	2.58776i
11.15	−1.00488	+	0.995099i	−0.799639	−	1.53642i	0.0195579	−	1.99990i	−0.411823	−	2.19782i	2.33243	+	0.748191i		−	3.34784i	1.97045	+	2.02912i	−1.72115	+	2.45716i	2.60088	+	1.79873i
11.16	−0.923787	−	1.07080i	1.20611	+	1.24310i	−0.293236	+	1.97839i	2.16410	−	0.562738i	0.216925	−	2.43987i			3.72853i	2.38935	−	1.51361i	−0.0905939	+	2.99863i	−2.60175	−	1.79747i
11.17	−0.884246	−	1.10368i	0.846655	−	1.51102i	−0.436219	+	1.95185i	0.448288	+	2.19067i	−2.41633	+	0.401675i			2.41810i	2.53994	−	1.24447i	−1.56635	−	2.55862i	2.02140	−	2.43186i
11.18	−0.776415	−	1.18202i	−0.846655	+	1.51102i	−0.794359	+	1.83548i	0.448288	+	2.19067i	2.44341	−	0.172411i		−	2.41810i	2.78634	−	0.486145i	−1.56635	−	2.55862i	2.24137	−	2.23076i
11.19	−0.732928	−	1.20947i	−1.20611	−	1.24310i	−0.925633	+	1.77291i	2.16410	−	0.562738i	−0.619499	+	2.36986i		−	3.72853i	2.82270	−	0.179888i	−0.0905939	+	2.99863i	−2.26674	−	2.20497i
11.20	−0.654964	+	1.25340i	1.65603	−	0.507508i	−1.14204	−	1.64187i	−1.89530	+	1.18653i	−0.448527	+	2.40807i		−	4.72716i	2.80592	−	0.356077i	2.48487	−	1.68090i	−0.245847	−	3.15271i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
25.d	even	5	1	inner
75.j	odd	10	1	inner
100.j	odd	10	1	inner
300.n	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.2.n.a	✓	224
3.b	odd	2	1	inner	300.2.n.a	✓	224
4.b	odd	2	1	inner	300.2.n.a	✓	224
12.b	even	2	1	inner	300.2.n.a	✓	224
25.d	even	5	1	inner	300.2.n.a	✓	224
75.j	odd	10	1	inner	300.2.n.a	✓	224
100.j	odd	10	1	inner	300.2.n.a	✓	224
300.n	even	10	1	inner	300.2.n.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.2.n.a	✓	224	1.a	even	1	1	trivial
300.2.n.a	✓	224	3.b	odd	2	1	inner
300.2.n.a	✓	224	4.b	odd	2	1	inner
300.2.n.a	✓	224	12.b	even	2	1	inner
300.2.n.a	✓	224	25.d	even	5	1	inner
300.2.n.a	✓	224	75.j	odd	10	1	inner
300.2.n.a	✓	224	100.j	odd	10	1	inner
300.2.n.a	✓	224	300.n	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).