Newform orbit 225.4.u.a

• Label: 225.4.u.a
• Level: $225$
• Weight: $4$
• Character orbit: 225.u
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $704$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(88\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1.13719	−	5.35004i	−4.50950	+	2.58155i	−20.0214	+	8.91409i	2.25903	−	10.9497i	18.9395	+	21.1903i	24.9371	−	14.3974i	44.7393	+	61.5784i	13.6712	−	23.2830i	−61.1505	+	0.366009i
4.2	−1.12585	−	5.29669i	−5.17983	+	0.411540i	−19.4791	+	8.67264i	6.78358	+	8.88724i	8.01150	+	26.9726i	−21.8175	+	12.5963i	42.4038	+	58.3638i	26.6613	−	4.26341i	39.4357	−	45.9362i
4.3	−1.11854	−	5.26231i	4.92317	+	1.66204i	−19.1324	+	8.51831i	11.0988	−	1.34760i	3.23940	−	27.7663i	−14.9230	+	8.61579i	40.9287	+	56.3335i	21.4753	+	16.3650i	−19.5060	−	56.8982i
4.4	−1.10981	−	5.22123i	−1.99679	−	4.79717i	−18.7212	+	8.33520i	−7.33503	+	8.43785i	−22.8311	+	15.7496i	18.8450	−	10.8801i	39.1967	+	53.9496i	−19.0257	+	19.1578i	52.1964	+	28.9335i
4.5	−1.07574	−	5.06097i	0.736845	−	5.14364i	−17.1478	+	7.63470i	−5.20342	−	9.89568i	−26.8245	+	1.80408i	−29.1252	+	16.8154i	32.7558	+	45.0845i	−25.9141	−	7.58014i	−44.4842	+	36.9795i
4.6	−1.06743	−	5.02188i	−0.0785966	+	5.19556i	−16.7715	+	7.46716i	−8.45497	+	7.31529i	26.1754	−	5.15121i	−9.64326	+	5.56754i	31.2598	+	43.0254i	−26.9876	−	0.816706i	45.7616	+	34.6513i
4.7	−1.06231	−	4.99779i	5.10707	−	0.958021i	−16.5410	+	7.36455i	−10.9332	+	2.33771i	−10.2133	−	24.5064i	−5.12017	+	2.95613i	30.3522	+	41.7762i	25.1644	−	9.78537i	23.2979	+	52.1585i
4.8	−1.04294	−	4.90666i	2.73456	−	4.41839i	−15.6792	+	6.98082i	10.5280	−	3.76300i	−24.5315	−	8.80939i	16.9358	−	9.77791i	27.0170	+	37.1857i	−12.0444	−	24.1647i	−29.4439	−	47.7329i
4.9	−1.01603	−	4.78006i	3.24528	+	4.05810i	−14.5083	+	6.45951i	−5.06856	−	9.96543i	16.1007	−	19.6358i	11.0356	−	6.37141i	22.6385	+	31.1592i	−5.93634	+	26.3393i	−42.4855	+	34.3532i
4.10	−0.943007	−	4.43650i	−5.11723	−	0.902178i	−11.4849	+	5.11340i	−11.1329	−	1.02923i	0.823077	+	23.5534i	−6.46246	+	3.73110i	12.1882	+	16.7756i	25.3722	+	9.23331i	5.93219	+	50.3615i
4.11	−0.939504	−	4.42002i	−1.03009	+	5.09303i	−11.3455	+	5.05136i	9.01568	+	6.61193i	23.4791	−	0.231910i	13.1856	−	7.61268i	11.7378	+	16.1557i	−24.8778	−	10.4925i	20.7546	−	46.0614i
4.12	−0.922831	−	4.34158i	4.70291	−	2.20967i	−10.6893	+	4.75919i	1.62060	+	11.0623i	−13.9335	−	18.3789i	8.42574	−	4.86461i	9.65541	+	13.2895i	17.2347	−	20.7838i	46.5321	−	17.2446i
4.13	−0.912762	−	4.29421i	−3.93731	−	3.39081i	−10.2987	+	4.58529i	5.02582	−	9.98705i	−10.9670	+	20.0026i	−4.48355	+	2.58858i	8.44680	+	11.6260i	4.00486	+	26.7013i	−47.4739	−	12.4661i
4.14	−0.849046	−	3.99445i	0.343429	−	5.18479i	−7.92637	+	3.52905i	5.82974	+	9.54013i	−21.0020	+	3.03032i	−17.1647	+	9.91002i	1.62379	+	2.23496i	−26.7641	−	3.56121i	33.1578	−	31.3866i
4.15	−0.822434	−	3.86925i	−4.35066	−	2.84108i	−6.98632	+	3.11051i	9.64418	+	5.65595i	−7.41470	+	19.1704i	13.6923	−	7.90527i	−0.819642	−	1.12814i	10.8566	+	24.7211i	13.9526	−	41.9674i
4.16	−0.814957	−	3.83407i	−3.33680	+	3.98318i	−6.72759	+	2.99532i	−9.34277	−	6.14107i	17.9912	+	9.54742i	−7.03825	+	4.06354i	−1.46471	−	2.01600i	−4.73148	−	26.5822i	−15.9313	+	40.8255i
4.17	−0.801643	−	3.77143i	5.10602	−	0.963611i	−6.27271	+	2.79279i	−2.58409	−	10.8776i	−7.72740	−	18.4845i	11.3837	−	6.57239i	−2.56924	−	3.53626i	25.1429	−	9.84043i	−38.9527	+	18.4657i
4.18	−0.792549	−	3.72865i	2.16415	+	4.72403i	−5.96634	+	2.65639i	9.75102	−	5.46969i	15.8991	−	11.8134i	−4.93704	+	2.85040i	−3.29151	−	4.53038i	−17.6329	+	20.4470i	−28.1227	−	32.0232i
4.19	−0.770399	−	3.62444i	−4.10935	+	3.18013i	−5.23470	+	2.33064i	9.67594	−	5.60144i	14.6920	+	12.4441i	−20.0051	+	11.5499i	−4.94384	−	6.80461i	6.77352	−	26.1366i	−27.7564	−	30.7545i
4.20	−0.728639	−	3.42798i	−4.82414	+	1.93071i	−3.91175	+	1.74162i	−5.26207	+	9.86462i	10.1335	+	15.1303i	23.0240	−	13.2929i	−7.65893	−	10.5416i	19.5447	−	18.6280i	37.6498	+	10.8505i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
25.e	even	10	1	inner
225.u	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.4.u.a	✓	704
9.c	even	3	1	inner	225.4.u.a	✓	704
25.e	even	10	1	inner	225.4.u.a	✓	704
225.u	even	30	1	inner	225.4.u.a	✓	704

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.4.u.a	✓	704	1.a	even	1	1	trivial
225.4.u.a	✓	704	9.c	even	3	1	inner
225.4.u.a	✓	704	25.e	even	10	1	inner
225.4.u.a	✓	704	225.u	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(225, [\chi])\).