Newform orbit 225.3.t.a

• Label: 225.3.t.a
• Level: $225$
• Weight: $3$
• Character orbit: 225.t
• Analytic conductor: $6.131$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13080594811\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 9 q^{2} - 8 q^{3} - 115 q^{4} + 24 q^{5} + 2 q^{6} - 8 q^{7} + 4 q^{9}+O(q^{10}) \)

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	−3.86927	+	0.406676i	−0.573457	−	2.94468i	10.8932	−	2.31543i	1.17580	+	4.85978i	3.41639	+	11.1605i	−0.327733	+	0.567650i	−26.4066	+	8.58001i	−8.34229	+	3.37730i	−6.52586	−	18.3256i
11.2	−3.86655	+	0.406391i	−2.97887	+	0.355458i	10.8725	−	2.31101i	0.860154	−	4.92546i	11.3735	−	2.58498i	6.32059	−	10.9476i	−26.3095	+	8.54846i	8.74730	−	2.11772i	−1.32417	+	19.3941i
11.3	−3.80513	+	0.399936i	2.85485	+	0.921851i	10.4065	−	2.21197i	4.17041	−	2.75820i	−11.2318	−	2.36601i	−5.47021	+	9.47469i	−24.1582	+	7.84946i	7.30038	+	5.26350i	−14.7659	+	12.1632i
11.4	−3.52310	+	0.370293i	2.42465	+	1.76666i	8.36255	−	1.77751i	−4.99290	+	0.266443i	−9.19646	−	5.32630i	2.18608	−	3.78640i	−15.3274	+	4.98018i	2.75781	+	8.56706i	17.4918	−	2.78754i
11.5	−3.44533	+	0.362119i	−1.43204	+	2.63614i	7.82660	−	1.66359i	−3.44542	+	3.62341i	3.97927	−	9.60096i	−0.718272	+	1.24408i	−13.1838	+	4.28367i	−4.89850	−	7.55014i	10.5585	−	13.7315i
11.6	−3.35051	+	0.352153i	1.50296	−	2.59637i	7.18934	−	1.52814i	−4.37978	−	2.41196i	−4.12136	+	9.22843i	1.04298	−	1.80649i	−10.7335	+	3.48753i	−4.48224	−	7.80446i	15.5239	+	6.53895i
11.7	−3.23081	+	0.339572i	−2.62988	+	1.44351i	6.41026	−	1.36254i	4.61772	+	1.91746i	8.00648	−	5.55676i	−5.57558	+	9.65718i	−7.88924	+	2.56337i	4.83254	−	7.59253i	−15.5701	−	4.62690i
11.8	−3.07924	+	0.323641i	−2.61774	−	1.46542i	5.46440	−	1.16149i	−3.93412	−	3.08589i	8.53492	+	3.66518i	−5.84635	+	10.1262i	−4.67165	+	1.51791i	4.70509	+	7.67217i	13.1128	+	8.22896i
11.9	−3.00210	+	0.315533i	2.95156	−	0.536932i	5.00044	−	1.06288i	2.83930	+	4.11563i	−8.69145	+	2.54324i	4.71844	−	8.17258i	−3.19287	+	1.03743i	8.42341	−	3.16958i	−9.82246	−	11.4596i
11.10	−2.89454	+	0.304228i	−0.147636	+	2.99637i	4.37320	−	0.929552i	4.72463	−	1.63641i	−0.484240	−	8.71800i	2.89464	−	5.01366i	−1.30345	+	0.423518i	−8.95641	−	0.884743i	−13.1778	+	6.17401i
11.11	−2.88946	+	0.303695i	1.87165	−	2.34455i	4.34417	−	0.923382i	3.40556	−	3.66090i	−4.69603	+	7.34291i	0.431094	−	0.746676i	−1.21918	+	0.396135i	−1.99387	−	8.77636i	−8.72844	+	11.6123i
11.12	−2.75888	+	0.289970i	0.583664	+	2.94267i	3.61473	−	0.768334i	−1.09506	−	4.87861i	−2.46354	−	7.94923i	−0.887137	+	1.53657i	0.803404	−	0.261042i	−8.31867	+	3.43507i	4.43577	+	13.1420i
11.13	−2.59778	+	0.273038i	−2.78816	−	1.10733i	2.76131	−	0.586935i	−2.53042	+	4.31242i	7.54536	+	2.11533i	3.41870	−	5.92136i	2.92395	−	0.950048i	6.54763	+	6.17483i	5.39602	−	11.8936i
11.14	−2.56945	+	0.270060i	−1.83152	−	2.37604i	2.61656	−	0.556166i	4.99561	+	0.209407i	5.34767	+	5.61049i	1.20300	−	2.08366i	3.25570	−	1.05784i	−2.29109	+	8.70350i	−12.8925	+	0.811055i
11.15	−2.18441	+	0.229590i	1.86111	+	2.35293i	0.806324	−	0.171389i	0.734719	+	4.94572i	−4.60562	−	4.71246i	−3.17310	+	5.49596i	6.63376	−	2.15544i	−2.07257	+	8.75811i	−2.74041	−	10.6348i
11.16	−1.77758	+	0.186831i	−2.15945	+	2.08249i	−0.787712	+	0.167433i	−2.55015	−	4.30078i	3.44952	−	4.10524i	−0.529466	+	0.917062i	8.16850	−	2.65411i	0.326453	−	8.99408i	5.33661	+	7.16853i
11.17	−1.76163	+	0.185155i	−0.0259598	−	2.99989i	−0.843539	+	0.179300i	−4.02846	+	2.96168i	0.601174	+	5.27988i	−2.29756	+	3.97949i	8.19135	−	2.66153i	−8.99865	+	0.155753i	6.54828	−	5.96326i
11.18	−1.75610	+	0.184574i	1.28142	−	2.71256i	−0.862770	+	0.183387i	3.81059	+	3.23719i	−1.74964	+	5.00004i	−4.62754	+	8.01513i	8.19866	−	2.66391i	−5.71592	−	6.95185i	−7.28927	−	4.98150i
11.19	−1.65244	+	0.173678i	2.99904	−	0.0758159i	−1.21221	+	0.257663i	−1.30778	−	4.82594i	−4.94256	+	0.646148i	−3.85160	+	6.67116i	8.27922	−	2.69008i	8.98850	−	0.454750i	2.99918	+	7.74743i
11.20	−1.63901	+	0.172266i	2.97857	−	0.357917i	−1.25593	+	0.266955i	−4.88848	+	1.05011i	−4.82024	+	1.09974i	0.672987	−	1.16565i	8.28198	−	2.69098i	8.74379	−	2.13216i	7.83135	−	2.56325i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
25.d	even	5	1	inner
225.t	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.3.t.a	✓	464
9.d	odd	6	1	inner	225.3.t.a	✓	464
25.d	even	5	1	inner	225.3.t.a	✓	464
225.t	odd	30	1	inner	225.3.t.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.3.t.a	✓	464	1.a	even	1	1	trivial
225.3.t.a	✓	464	9.d	odd	6	1	inner
225.3.t.a	✓	464	25.d	even	5	1	inner
225.3.t.a	✓	464	225.t	odd	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(225, [\chi])\).