Newform orbit 375.3.p.a

• Label: 375.3.p.a
• Level: $375$
• Weight: $3$
• Character orbit: 375.p
• 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.p (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.

\(\operatorname{Tr}(f)(q) = \) \( 1960 q - 20 q^{3} - 40 q^{4} - 20 q^{6} - 50 q^{7} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
14.1	−1.68926	−	3.58987i	−2.97064	−	0.418686i	−7.48385	+	9.04642i	−1.07499	+	4.88307i	3.51517	+	11.3715i	−5.06561	−	1.64592i	29.7464	+	7.63757i	8.64940	+	2.48753i	19.3455	−	4.38973i
14.2	−1.63384	−	3.47209i	2.18513	−	2.05553i	−6.83628	+	8.26365i	3.29115	−	3.76408i	−10.7071	−	4.22858i	8.66426	+	2.81519i	24.9946	+	6.41751i	0.549617	−	8.98320i	−18.4465	−	5.27727i
14.3	−1.60103	−	3.40235i	−2.47973	+	1.68847i	−6.46303	+	7.81246i	−1.74931	−	4.68401i	9.71490	+	5.73363i	3.55579	+	1.15535i	22.3599	+	5.74104i	3.29812	−	8.37391i	−13.1360	+	13.4510i
14.4	−1.60052	−	3.40129i	2.98667	+	0.282520i	−6.45738	+	7.80563i	−3.91956	−	3.10436i	−3.81930	−	10.6107i	−11.0932	−	3.60439i	22.3206	+	5.73095i	8.84036	+	1.68759i	−4.28549	+	18.3001i
14.5	−1.57597	−	3.34910i	0.726770	+	2.91064i	−6.18309	+	7.47407i	4.55261	−	2.06731i	8.60264	−	7.02109i	−4.43138	−	1.43984i	20.4354	+	5.24692i	−7.94361	+	4.23073i	−14.0984	−	11.9891i
14.6	−1.54647	−	3.28642i	−0.460374	−	2.96447i	−5.85931	+	7.08268i	−4.96189	+	0.616133i	−9.03053	+	6.09745i	9.50012	+	3.08678i	18.2660	+	4.68991i	−8.57611	+	2.72953i	9.69831	+	15.3541i
14.7	−1.51029	−	3.20953i	2.10800	−	2.13456i	−5.47042	+	6.61260i	2.32986	+	4.42400i	−10.0346	−	3.54190i	−4.36643	−	1.41874i	15.7426	+	4.04200i	−0.112663	−	8.99929i	10.6802	−	14.1593i
14.8	−1.50136	−	3.19056i	1.39183	+	2.65759i	−5.37590	+	6.49835i	−4.53777	+	2.09968i	6.38957	−	8.43074i	5.28133	+	1.71601i	15.1431	+	3.88808i	−5.12561	+	7.39785i	13.5120	+	11.3256i
14.9	−1.44716	−	3.07538i	−1.48924	−	2.60426i	−4.81398	+	5.81910i	−0.275188	−	4.99242i	−5.85390	+	8.34878i	−5.80085	−	1.88481i	11.6942	+	3.00257i	−4.56431	+	7.75675i	−14.9553	+	8.07116i
14.10	−1.43795	−	3.05579i	−1.66173	+	2.49773i	−4.72047	+	5.70606i	3.65114	+	3.41602i	10.0220	+	1.48631i	9.55210	+	3.10367i	11.1399	+	2.86023i	−3.47729	−	8.30111i	5.18850	−	16.0692i
14.11	−1.41748	−	3.01230i	−0.576403	−	2.94411i	−4.51499	+	5.45769i	4.47350	+	2.23334i	−8.05148	+	5.90951i	−2.87033	−	0.932627i	9.94189	+	2.55264i	−8.33552	+	3.39399i	0.386378	−	16.6412i
14.12	−1.36395	−	2.89853i	2.91878	−	0.693340i	−3.99145	+	4.82483i	−2.44924	+	4.35904i	−5.99073	−	7.51450i	4.18201	+	1.35882i	7.01797	+	1.80191i	8.03856	−	4.04742i	15.9754	+	1.15369i
14.13	−1.34405	−	2.85626i	−0.467457	+	2.96336i	−3.80203	+	4.59587i	−1.72095	+	4.69450i	9.09240	−	2.64773i	−10.1985	−	3.31371i	6.00705	+	1.54235i	−8.56297	−	2.77049i	15.7217	−	1.39417i
14.14	−1.31954	−	2.80417i	−2.97010	−	0.422525i	−3.57249	+	4.31840i	4.93662	−	0.793613i	2.73434	+	8.88620i	−2.47848	−	0.805308i	4.81656	+	1.23668i	8.64295	+	2.50988i	−8.73951	−	12.7959i
14.15	−1.30305	−	2.76913i	2.86785	+	0.880583i	−3.42043	+	4.13460i	4.99762	−	0.154135i	−1.29852	−	9.08890i	4.17855	+	1.35769i	4.04924	+	1.03967i	7.44915	+	5.05076i	−6.93899	−	13.6382i
14.16	−1.23326	−	2.62081i	−1.92814	+	2.29832i	−2.79802	+	3.38222i	−4.93283	−	0.816795i	8.40136	+	2.21885i	−3.97383	−	1.29118i	1.09293	+	0.280617i	−1.56458	−	8.86296i	3.94280	+	13.9353i
14.17	−1.22657	−	2.60660i	2.68490	+	1.33841i	−2.74020	+	3.31233i	−1.31693	−	4.82345i	0.195465	−	8.64012i	7.81433	+	2.53903i	0.833924	+	0.214115i	5.41734	+	7.18696i	−10.9575	+	9.34904i
14.18	−1.17810	−	2.50359i	−2.13166	−	2.11093i	−2.33036	+	2.81692i	0.289532	+	4.99161i	−2.77361	+	7.82369i	11.0564	+	3.59246i	−0.922199	−	0.236780i	0.0879080	+	8.99957i	12.1559	−	6.60549i
14.19	−1.15116	−	2.44634i	−2.86760	−	0.881397i	−2.10970	+	2.55019i	−4.99959	+	0.0638640i	1.14487	+	8.02975i	−1.66437	−	0.540787i	−1.80761	−	0.464116i	7.44628	+	5.05499i	5.91156	+	12.1572i
14.20	−1.13177	−	2.40512i	2.66218	+	1.38303i	−1.95403	+	2.36202i	3.10930	+	3.91564i	0.313403	−	7.96815i	−11.4743	−	3.72823i	−2.40593	−	0.617737i	5.17443	+	7.36378i	5.89860	−	11.9098i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
125.h	even	50	1	inner
375.p	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.p.a	✓	1960
3.b	odd	2	1	inner	375.3.p.a	✓	1960
125.h	even	50	1	inner	375.3.p.a	✓	1960
375.p	odd	50	1	inner	375.3.p.a	✓	1960

    

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

Hecke kernels

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