Newform orbit 375.3.n.a

• Label: 375.3.n.a
• Level: $375$
• Weight: $3$
• Character orbit: 375.n
• Analytic conductor: $10.218$
• Analytic rank: $0$
• Dimension: $1960$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	375.n (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.2180099135\)
Analytic rank:	\(0\)
Dimension:	\(1960\)
Relative dimension:	\(98\) over \(\Q(\zeta_{50})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{50}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1960 q - 20 q^{3} - 40 q^{4} - 20 q^{6} - 30 q^{7} - 20 q^{9}+O(q^{10}) \)

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	−2.63475	−	2.80572i	1.77288	−	2.42010i	−0.679014	+	10.7926i	−4.70343	+	1.69638i	−11.4612	+	1.40214i	−1.93024	+	5.94067i	20.2076	−	16.7172i	−2.71378	−	8.58111i	17.1519	+	8.72698i
11.2	−2.61946	−	2.78944i	−2.19993	−	2.03968i	−0.668249	+	10.6215i	−0.118276	−	4.99860i	0.0730753	+	11.4794i	2.92184	−	8.99250i	19.5849	−	16.2020i	0.679422	+	8.97432i	−13.6335	+	13.4235i
11.3	−2.58057	−	2.74803i	−2.67379	+	1.36045i	−0.641153	+	10.1908i	−4.31457	+	2.52676i	10.6385	+	3.83692i	0.474894	−	1.46157i	18.0407	−	14.9245i	5.29835	−	7.27513i	18.0776	+	5.33606i
11.4	−2.56145	−	2.72767i	−2.99228	−	0.215016i	−0.627979	+	9.98144i	4.96770	+	0.567433i	7.07809	+	8.71270i	−3.95457	+	12.1709i	17.3021	−	14.3135i	8.90754	+	1.28678i	−11.1767	−	15.0037i
11.5	−2.56061	−	2.72677i	2.11204	+	2.13056i	−0.627402	+	9.97227i	−4.62568	−	1.89819i	0.401447	−	11.2146i	1.82226	−	5.60835i	17.2699	−	14.2869i	−0.0785819	+	8.99966i	6.66863	+	17.4737i
11.6	−2.44585	−	2.60457i	−1.43725	−	2.63331i	−0.550419	+	8.74866i	−0.127436	+	4.99838i	−3.34334	+	10.1841i	1.73594	−	5.34267i	13.1207	−	10.8544i	−4.86865	+	7.56943i	13.3303	−	11.8934i
11.7	−2.44413	−	2.60273i	2.81789	−	1.02932i	−0.549291	+	8.73072i	4.67778	+	1.76590i	−9.56634	−	4.81841i	3.50123	−	10.7757i	13.0620	−	10.8058i	6.88099	−	5.80104i	−6.83693	−	16.4911i
11.8	−2.44217	−	2.60064i	−0.0329874	+	2.99982i	−0.548006	+	8.71031i	4.70750	−	1.68507i	7.88202	−	7.24027i	0.573301	−	1.76444i	12.9953	−	10.7507i	−8.99782	−	0.197913i	−15.8788	−	8.12729i
11.9	−2.43456	−	2.59255i	2.50755	+	1.64688i	−0.543037	+	8.63133i	1.87227	+	4.63623i	−1.83516	−	10.5104i	−2.09868	+	6.45908i	12.7380	−	10.5378i	3.57557	+	8.25926i	7.46147	−	16.1411i
11.10	−2.33180	−	2.48312i	2.96025	−	0.486738i	−0.477407	+	7.58816i	1.43814	−	4.78871i	−8.11135	−	6.21567i	−1.04802	+	3.22548i	9.45698	−	7.82350i	8.52617	−	2.88173i	−15.2444	+	7.59526i
11.11	−2.31902	−	2.46951i	−1.77020	+	2.42207i	−0.469443	+	7.46158i	−1.96020	−	4.59974i	10.0864	−	1.24531i	−0.617832	+	1.90149i	9.07408	−	7.50673i	−2.73281	−	8.57507i	−6.81334	+	15.5076i
11.12	−2.19548	−	2.33795i	−0.0689295	−	2.99921i	−0.394708	+	6.27371i	−0.299373	−	4.99103i	−6.86065	+	6.74585i	−2.40171	+	7.39170i	5.64941	−	4.67360i	−8.99050	+	0.413468i	−11.0115	+	11.6576i
11.13	−2.11038	−	2.24733i	−2.58624	+	1.52032i	−0.345612	+	5.49334i	3.02192	+	3.98346i	8.87462	+	2.60366i	2.49002	−	7.66349i	3.57310	−	2.95593i	4.37723	−	7.86383i	2.57474	−	15.1979i
11.14	−2.02984	−	2.16156i	−0.690048	+	2.91956i	−0.300930	+	4.78315i	−2.96625	+	4.02509i	7.71150	−	4.43466i	−4.02727	+	12.3947i	1.81091	−	1.49812i	−8.04767	−	4.02927i	14.7215	−	1.75856i
11.15	−1.99017	−	2.11932i	1.57105	−	2.55574i	−0.279562	+	4.44350i	−4.43101	−	2.31649i	−8.54308	+	1.75679i	3.71601	−	11.4367i	1.01317	−	0.838169i	−4.06357	−	8.03040i	3.90908	+	14.0009i
11.16	−1.94228	−	2.06832i	0.867162	+	2.87194i	−0.254328	+	4.04242i	−2.80949	+	4.13603i	4.25582	−	7.37169i	2.72977	−	8.40136i	0.110214	−	0.0911767i	−7.49606	+	4.98087i	14.0115	−	2.22241i
11.17	−1.94166	−	2.06765i	−0.868109	−	2.87165i	−0.254002	+	4.03724i	3.79553	+	3.25484i	−4.25201	+	7.37071i	−1.33678	+	4.11418i	0.0988333	−	0.0817620i	−7.49277	+	4.98582i	−0.639723	−	14.1676i
11.18	−1.93480	−	2.06035i	−2.16522	−	2.07650i	−0.250438	+	3.98060i	−4.99995	+	0.0215771i	−0.0890504	+	8.47870i	−2.56934	+	7.90760i	−0.0251001	+	0.0207646i	0.376333	+	8.99213i	9.71835	+	10.2599i
11.19	−1.91027	−	2.03423i	2.05456	−	2.18605i	−0.237803	+	3.77977i	−0.0240579	+	4.99994i	−8.37169	−	0.00350468i	−0.262179	+	0.806905i	−0.457465	+	0.378448i	−0.557594	−	8.98271i	10.2170	−	9.50231i
11.20	−1.82395	−	1.94231i	−2.77613	−	1.13714i	−0.194610	+	3.09323i	4.39492	−	2.38425i	2.85485	+	7.46619i	1.53535	−	4.72532i	−1.84904	+	1.52966i	6.41383	+	6.31370i	−12.6471	−	4.18755i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
125.g	even	25	1	inner
375.n	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	375.3.n.a	✓	1960
3.b	odd	2	1	inner	375.3.n.a	✓	1960
125.g	even	25	1	inner	375.3.n.a	✓	1960
375.n	odd	50	1	inner	375.3.n.a	✓	1960

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
375.3.n.a	✓	1960	1.a	even	1	1	trivial
375.3.n.a	✓	1960	3.b	odd	2	1	inner
375.3.n.a	✓	1960	125.g	even	25	1	inner
375.3.n.a	✓	1960	375.n	odd	50	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(375, [\chi])\).