Newform orbit 225.3.x.a

• Label: 225.3.x.a
• Level: $225$
• Weight: $3$
• Character orbit: 225.x
• Analytic conductor: $6.131$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13080594811\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(58\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 8 q^{2} - 14 q^{3} - 10 q^{4} - 8 q^{5} - 12 q^{6} - 8 q^{7} - 16 q^{8} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−2.51131	−	3.10121i	−2.37510	−	1.83274i	−2.47919	+	11.6637i	−4.45894	−	2.26226i	0.280908	+	11.9683i	7.82954	+	2.09792i	28.1752	−	14.3560i	2.28216	+	8.70584i	4.18206	+	19.5094i
13.2	−2.39531	−	2.95797i	2.99721	−	0.129320i	−2.18039	+	10.2579i	0.996446	−	4.89970i	−7.56179	−	8.55589i	−6.10048	−	1.63462i	21.9999	−	11.2095i	8.96655	−	0.775200i	−16.8800	+	8.78888i
13.3	−2.37770	−	2.93621i	1.06484	+	2.80466i	−2.13625	+	10.0503i	1.94075	+	4.60798i	5.70320	−	9.79523i	−1.60340	−	0.429629i	21.1234	−	10.7629i	−6.73222	+	5.97304i	8.91549	−	16.6548i
13.4	−2.23416	−	2.75896i	0.755096	−	2.90342i	−1.78873	+	8.41532i	0.0648426	+	4.99958i	−9.69741	+	4.40342i	−8.97463	−	2.40474i	14.5611	−	7.41926i	−7.85966	−	4.38472i	13.6488	−	11.3488i
13.5	−2.06492	−	2.54997i	0.552560	−	2.94867i	−1.40678	+	6.61838i	4.95326	−	0.682042i	−8.66001	+	4.67977i	8.45676	+	2.26598i	8.08727	−	4.12067i	−8.38936	−	3.25864i	−11.9673	−	11.2223i
13.6	−2.03605	−	2.51431i	−2.20067	+	2.03888i	−1.34462	+	6.32594i	−3.36026	−	3.70252i	9.60706	+	1.38192i	−13.0248	−	3.48998i	7.11235	−	3.62392i	0.685940	−	8.97382i	−2.46763	+	15.9873i
13.7	−2.01370	−	2.48671i	0.363939	+	2.97784i	−1.29711	+	6.10241i	1.07408	−	4.88327i	6.67218	−	6.90150i	9.13770	+	2.44844i	6.38276	−	3.25218i	−8.73510	+	2.16750i	−14.3062	+	7.16254i
13.8	−2.00044	−	2.47034i	−1.70954	+	2.46525i	−1.26917	+	5.97096i	−3.88822	+	3.14352i	9.50985	−	0.708432i	4.38514	+	1.17500i	5.96010	−	3.03682i	−3.15491	−	8.42891i	15.5437	+	3.31679i
13.9	−1.95475	−	2.41391i	−2.99126	−	0.228865i	−1.17428	+	5.52456i	1.21817	+	4.84934i	5.29469	+	7.66800i	0.0185560	+	0.00497207i	4.56092	−	2.32390i	8.89524	+	1.36919i	9.32464	−	12.4198i
13.10	−1.88803	−	2.33152i	−2.53077	−	1.61096i	−1.03969	+	4.89135i	4.35459	−	2.45714i	1.02217	+	8.94208i	−6.02581	−	1.61461i	2.67477	−	1.36287i	3.80959	+	8.15396i	−13.9504	−	5.51367i
13.11	−1.84346	−	2.27648i	2.55116	+	1.57847i	−0.952387	+	4.48063i	−4.96038	+	0.628232i	−1.10959	−	8.71753i	3.30239	+	0.884874i	1.51571	−	0.772291i	4.01684	+	8.05388i	10.5744	+	10.1341i
13.12	−1.83098	−	2.26107i	2.04676	−	2.19335i	−0.928316	+	4.36738i	−4.92462	−	0.864912i	−8.70690	−	0.611883i	−0.399002	−	0.106912i	1.20532	−	0.614139i	−0.621576	−	8.97851i	7.06127	+	12.7186i
13.13	−1.75378	−	2.16573i	2.99709	+	0.131996i	−0.783024	+	3.68384i	4.65885	+	1.81526i	−4.97037	−	6.72240i	5.95805	+	1.59646i	−0.580693	+	0.295878i	8.96515	+	0.791206i	−4.23921	−	13.2734i
13.14	−1.45990	−	1.80282i	0.453324	+	2.96555i	−0.287223	+	1.35128i	4.88955	−	1.04512i	4.68456	−	5.14666i	−9.18780	−	2.46186i	−5.41239	+	2.75775i	−8.58899	+	2.68871i	−9.02240	−	7.28923i
13.15	−1.33618	−	1.65005i	−2.89557	+	0.784641i	−0.105629	+	0.496948i	0.831842	−	4.93032i	5.16371	+	3.72941i	9.81243	+	2.62923i	−6.60607	+	3.36596i	7.76868	−	4.54397i	−9.24676	+	5.21523i
13.16	−1.30275	−	1.60876i	−0.852534	−	2.87631i	−0.0593127	+	0.279044i	−2.09692	−	4.53904i	−3.51667	+	5.11865i	−1.50124	−	0.402256i	−6.85167	+	3.49110i	−7.54637	+	4.90431i	−4.57048	+	9.28670i
13.17	−1.21615	−	1.50182i	2.74216	+	1.21678i	0.0552088	−	0.259737i	−1.16333	+	4.86278i	−1.50749	−	5.59800i	−11.2846	−	3.02370i	−7.34461	+	3.74227i	6.03889	+	6.67321i	8.71778	−	4.16676i
13.18	−1.14635	−	1.41562i	−1.14916	−	2.77118i	0.141774	−	0.666994i	−1.93109	+	4.61204i	−2.60560	+	4.80352i	13.0663	+	3.50110i	−7.59885	+	3.87181i	−6.35886	+	6.36906i	8.74261	−	2.55330i
13.19	−1.00635	−	1.24274i	−1.88326	+	2.33524i	0.299984	−	1.41131i	4.11042	+	2.84684i	4.79732	−	0.00967335i	2.58466	+	0.692557i	−7.75506	+	3.95140i	−1.90669	−	8.79571i	−0.598647	−	7.97311i
13.20	−0.919820	−	1.13588i	2.48692	−	1.67786i	0.387485	−	1.82297i	−0.264765	+	4.99299i	−4.19338	−	1.28152i	2.85419	+	0.764777i	−7.63631	+	3.89090i	3.36957	−	8.34542i	5.91498	−	4.29191i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
25.f	odd	20	1	inner
225.x	odd	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.3.x.a	✓	928
9.c	even	3	1	inner	225.3.x.a	✓	928
25.f	odd	20	1	inner	225.3.x.a	✓	928
225.x	odd	60	1	inner	225.3.x.a	✓	928

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.3.x.a	✓	928	1.a	even	1	1	trivial
225.3.x.a	✓	928	9.c	even	3	1	inner
225.3.x.a	✓	928	25.f	odd	20	1	inner
225.3.x.a	✓	928	225.x	odd	60	1	inner

Hecke kernels

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