Newform orbit 825.2.cu.a

• Label: 825.2.cu.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.cu
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.cu (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(116\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 8 q^{3} - 20 q^{4} - 2 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−0.436173	−	2.75389i	−1.34992	+	1.08523i	−5.49153	+	1.78431i	−1.56654	−	1.59560i	3.57740	+	3.24417i	3.87412	−	0.613600i	4.77739	+	9.37615i	0.644546	−	2.92994i	−3.71082	+	5.01004i
53.2	−0.429419	−	2.71125i	1.27608	+	1.17116i	−5.26434	+	1.71049i	−0.134670	+	2.23201i	2.62732	−	3.96269i	−1.87681	+	0.297258i	4.40571	+	8.64669i	0.256775	+	2.98899i	6.10935	−	0.593345i
53.3	−0.428168	−	2.70335i	−1.65546	−	0.509380i	−5.22265	+	1.69694i	2.15942	−	0.580427i	−0.668218	+	4.69337i	−3.17827	+	0.503388i	4.33841	+	8.51462i	2.48106	+	1.68651i	−2.49369	−	5.58915i
53.4	−0.425641	−	2.68739i	0.829589	−	1.52045i	−5.13878	+	1.66969i	−0.131866	−	2.23218i	−4.43916	−	1.58226i	1.07411	−	0.170123i	4.20387	+	8.25056i	−1.62356	−	2.52270i	−5.94260	+	1.30448i
53.5	−0.419093	−	2.64605i	−0.0355665	−	1.73169i	−4.92381	+	1.59984i	0.558806	+	2.16512i	−4.56722	+	0.819847i	3.00481	−	0.475916i	3.86429	+	7.58409i	−2.99747	+	0.123180i	5.49481	−	2.38601i
53.6	−0.405141	−	2.55796i	1.72974	+	0.0895197i	−4.47690	+	1.45463i	2.21377	−	0.315015i	−0.471799	−	4.46086i	−0.127344	+	0.0201693i	3.18313	+	6.24725i	2.98397	+	0.309691i	−1.70268	−	5.53510i
53.7	−0.394486	−	2.49068i	−1.03132	−	1.39153i	−4.14578	+	1.34705i	−2.16965	+	0.540952i	−3.05903	+	3.11765i	−0.695321	+	0.110128i	2.70083	+	5.30068i	−0.872739	+	2.87025i	2.20323	+	5.19051i
53.8	−0.391039	−	2.46892i	−1.05823	+	1.37119i	−4.04056	+	1.31286i	1.96200	+	1.07263i	3.79916	+	2.07649i	1.11874	−	0.177190i	2.55168	+	5.00795i	−0.760309	−	2.90206i	1.88103	−	5.26348i
53.9	−0.390296	−	2.46423i	0.114350	+	1.72827i	−4.01798	+	1.30552i	0.335862	−	2.21070i	4.21423	−	0.956322i	−3.71490	+	0.588382i	2.51994	+	4.94566i	−2.97385	+	0.395257i	−5.57876	+	0.0351845i
53.10	−0.383893	−	2.42381i	−1.71587	+	0.236234i	−3.82536	+	1.24293i	−1.30555	+	1.81536i	1.23129	+	4.06824i	−2.72999	+	0.432389i	2.25296	+	4.42168i	2.88839	−	0.810690i	4.90128	+	2.46750i
53.11	−0.381281	−	2.40731i	1.22867	+	1.22081i	−3.74766	+	1.21769i	−2.00282	−	0.994343i	2.47040	−	3.42326i	−0.137276	+	0.0217424i	2.14722	+	4.21416i	0.0192571	+	2.99994i	−1.63006	+	5.20053i
53.12	−0.368019	−	2.32358i	1.39525	−	1.02629i	−3.36148	+	1.09221i	0.0386766	+	2.23573i	−2.89815	−	2.86429i	−4.37796	+	0.693400i	1.63886	+	3.21645i	0.893463	−	2.86387i	5.18068	−	0.912661i
53.13	−0.364117	−	2.29894i	0.588151	+	1.62913i	−3.25045	+	1.05614i	2.23382	+	0.100244i	3.53113	−	1.94532i	4.48472	−	0.710310i	1.49812	+	2.94023i	−2.30816	+	1.91635i	−0.582915	−	5.17193i
53.14	−0.358809	−	2.26543i	−1.34068	−	1.09663i	−3.10131	+	1.00768i	1.78318	−	1.34918i	−2.00328	+	3.43069i	3.41507	−	0.540894i	1.31299	+	2.57690i	0.594826	+	2.94044i	−3.69629	−	3.55556i
53.15	−0.355232	−	2.24285i	−0.299632	−	1.70594i	−3.00205	+	0.975426i	−1.76159	−	1.37725i	−3.71971	+	1.27803i	−2.39451	+	0.379253i	1.19231	+	2.34004i	−2.82044	+	1.02231i	−2.46318	+	4.44021i
53.16	−0.350043	−	2.21009i	1.47447	−	0.908806i	−2.85984	+	0.929219i	−1.84636	+	1.26133i	−2.52467	−	2.94059i	2.94003	−	0.465654i	1.02299	+	2.00773i	1.34814	−	2.68002i	3.43396	+	3.63909i
53.17	−0.346377	−	2.18694i	1.73122	−	0.0536266i	−2.76062	+	0.896979i	0.514492	−	2.17607i	−0.716933	−	3.76750i	0.718240	−	0.113758i	0.907401	+	1.78088i	2.99425	−	0.185679i	−4.93715	−	0.371420i
53.18	−0.323281	−	2.04112i	−1.51215	+	0.844630i	−2.15953	+	0.701674i	1.14449	−	1.92098i	2.21284	+	2.81342i	−0.343906	+	0.0544693i	0.253938	+	0.498382i	1.57320	−	2.55442i	−4.29093	−	1.71501i
53.19	−0.316722	−	1.99970i	−0.0379228	+	1.73164i	−1.99639	+	0.648666i	−2.20733	+	0.357327i	3.47477	−	0.472613i	1.47834	−	0.234147i	0.0911140	+	0.178821i	−2.99712	−	0.131337i	1.41366	+	4.30084i
53.20	−0.316611	−	1.99900i	−0.369698	+	1.69214i	−1.99365	+	0.647778i	1.36483	+	1.77123i	3.49963	+	0.203279i	−3.41702	+	0.541203i	0.0884406	+	0.173574i	−2.72665	−	1.25116i	3.10857	−	3.28908i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
275.bh	odd	20	1	inner
825.cu	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.cu.a	yes	928
3.b	odd	2	1	inner	825.2.cu.a	yes	928
11.c	even	5	1		825.2.ci.a	✓	928
25.f	odd	20	1		825.2.ci.a	✓	928
33.h	odd	10	1		825.2.ci.a	✓	928
75.l	even	20	1		825.2.ci.a	✓	928
275.bh	odd	20	1	inner	825.2.cu.a	yes	928
825.cu	even	20	1	inner	825.2.cu.a	yes	928

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.ci.a	✓	928	11.c	even	5	1
825.2.ci.a	✓	928	25.f	odd	20	1
825.2.ci.a	✓	928	33.h	odd	10	1
825.2.ci.a	✓	928	75.l	even	20	1
825.2.cu.a	yes	928	1.a	even	1	1	trivial
825.2.cu.a	yes	928	3.b	odd	2	1	inner
825.2.cu.a	yes	928	275.bh	odd	20	1	inner
825.2.cu.a	yes	928	825.cu	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).