Newform orbit 300.2.r.a

• Label: 300.2.r.a
• Level: $300$
• Weight: $2$
• Character orbit: 300.r
• 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.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 6 q^{4} - 7 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41059	−	0.101112i	−1.18137	+	1.26664i	1.97955	+	0.285257i	−0.243022	+	2.22282i	1.79450	−	1.66726i	−4.47645			−2.76350	−	0.602540i	−0.208752	−	2.99273i	0.567561	−	3.11093i
59.2	−1.41029	−	0.105278i	1.45041	−	0.946744i	1.97783	+	0.296946i	−2.23180	−	0.138020i	−2.14516	+	1.18249i	−1.71470			−2.75806	−	0.627002i	1.20735	−	2.74632i	3.13296	+	0.429608i
59.3	−1.40665	+	0.146030i	1.00033	+	1.41398i	1.95735	−	0.410828i	−1.66487	−	1.49272i	−1.61361	−	1.84290i	−0.571505			−2.69332	+	0.863724i	−0.998661	+	2.82890i	2.55988	+	1.85662i
59.4	−1.37885	−	0.314289i	1.03587	−	1.38816i	1.80244	+	0.866714i	1.94623	−	1.10099i	−1.86458	+	1.58850i	2.63015			−2.21290	−	1.76156i	−0.853964	−	2.87589i	−3.02959	+	0.906413i
59.5	−1.37668	+	0.323633i	−1.44316	−	0.957755i	1.79052	−	0.891082i	2.16605	+	0.555182i	2.29674	+	0.851472i	0.146941			−2.17660	+	1.80621i	1.16541	+	2.76438i	−3.16164	−	0.0633046i
59.6	−1.32465	+	0.495274i	1.60045	+	0.662243i	1.50941	−	1.31213i	0.462356	+	2.18774i	−2.44803	−	0.0845826i	2.28606			−1.34958	+	2.48569i	2.12287	+	2.11977i	−1.69599	−	2.66901i
59.7	−1.30486	−	0.545281i	0.0653839	+	1.73082i	1.40534	+	1.42303i	2.14307	+	0.638169i	0.858463	−	2.29413i	4.06778			−1.05782	−	2.62317i	−2.99145	+	0.226335i	−2.44843	−	2.00130i
59.8	−1.29016	−	0.579209i	−1.71582	−	0.236546i	1.32903	+	1.49455i	−0.101169	−	2.23378i	2.07668	+	1.29900i	−0.903444			−0.849014	−	2.69799i	2.88809	+	0.811741i	−1.16330	+	2.94053i
59.9	−1.28571	−	0.589029i	−0.760326	−	1.55625i	1.30609	+	1.51464i	−1.70605	+	1.44547i	0.0608822	+	2.44873i	1.06519			−0.787085	−	2.71671i	−1.84381	+	2.36651i	3.04491	−	0.853547i
59.10	−1.26137	+	0.639492i	−0.689827	+	1.58875i	1.18210	−	1.61327i	1.00094	−	1.99953i	−0.145871	−	2.44514i	−0.923130			−0.459388	+	2.79087i	−2.04828	−	2.19193i	0.0161381	+	3.16224i
59.11	−1.19645	+	0.753994i	−0.269355	−	1.71098i	0.862987	−	1.80423i	−1.27460	−	1.83722i	1.61234	+	1.84401i	4.55385			0.327858	+	2.80936i	−2.85490	+	0.921722i	2.91025	+	1.23711i
59.12	−1.12183	+	0.861108i	1.64941	−	0.528616i	0.516985	−	1.93203i	1.99646	−	1.00704i	−1.39516	+	2.01334i	−3.00727			1.08372	+	2.61258i	2.44113	−	1.74381i	−1.37251	+	2.84890i
59.13	−1.06241	+	0.933425i	0.154412	−	1.72515i	0.257435	−	1.98336i	−0.778418	+	2.09620i	1.44625	+	1.97695i	−3.14870			1.57782	+	2.34744i	−2.95231	−	0.532768i	−1.12965	−	2.95362i
59.14	−0.957505	−	1.04076i	0.760326	+	1.55625i	−0.166367	+	1.99307i	−1.70605	+	1.44547i	0.891666	−	2.28143i	−1.06519			2.23361	−	1.73523i	−1.84381	+	2.36651i	3.13794	+	0.391539i
59.15	−0.949542	−	1.04803i	1.71582	+	0.236546i	−0.196739	+	1.99030i	−0.101169	−	2.23378i	−1.38134	−	2.02285i	0.903444			2.27271	−	1.68369i	2.88809	+	0.811741i	−2.24500	+	2.22710i
59.16	−0.932419	+	1.06329i	−1.73059	+	0.0711045i	−0.261191	−	1.98287i	−1.84864	−	1.25798i	1.53803	−	1.90643i	−2.80461			2.35192	+	1.57114i	2.98989	−	0.246105i	3.06132	−	0.792685i
59.17	−0.921818	−	1.07250i	−0.0653839	−	1.73082i	−0.300505	+	1.97730i	2.14307	+	0.638169i	−1.79603	+	1.66562i	−4.06778			2.39766	−	1.50042i	−2.99145	+	0.226335i	−1.29108	−	2.88671i
59.18	−0.724994	−	1.21424i	−1.03587	+	1.38816i	−0.948767	+	1.76064i	1.94623	−	1.10099i	2.43656	+	0.251385i	−2.63015			2.82569	−	0.124420i	−0.853964	−	2.87589i	−2.74787	−	1.56499i
59.19	−0.723120	+	1.21536i	−1.44187	+	0.959691i	−0.954195	−	1.75770i	1.84864	+	1.25798i	−0.123723	−	2.44636i	2.80461			2.82623	+	0.111339i	1.15799	−	2.76750i	−2.86569	+	1.33709i
59.20	−0.559437	+	1.29886i	1.13894	+	1.30492i	−1.37406	−	1.45326i	0.778418	−	2.09620i	−2.33207	+	0.749304i	3.14870			2.65627	−	0.971702i	−0.405623	+	2.97245i	2.28719	+	2.18375i

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.e	even	10	1	inner
75.h	odd	10	1	inner
100.h	odd	10	1	inner
300.r	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.r.a	✓	224
3.b	odd	2	1	inner	300.2.r.a	✓	224
4.b	odd	2	1	inner	300.2.r.a	✓	224
12.b	even	2	1	inner	300.2.r.a	✓	224
25.e	even	10	1	inner	300.2.r.a	✓	224
75.h	odd	10	1	inner	300.2.r.a	✓	224
100.h	odd	10	1	inner	300.2.r.a	✓	224
300.r	even	10	1	inner	300.2.r.a	✓	224

    

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

Hecke kernels

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