Newform orbit 825.2.s.a

• Label: 825.2.s.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.s
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.s (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(116\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 460 q^{4} + 5 q^{6} + 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
479.1		−	2.80066i	−0.0642985	−	1.73086i	−5.84370			1.82822	+	1.28748i	−4.84754	+	0.180078i	−0.904194	+	0.656935i			10.7649i	−2.99173	+	0.222583i	3.60579	−	5.12023i
479.2		−	2.75060i	−1.72450	−	0.161553i	−5.56583			−2.15940	+	0.580503i	−0.444368	+	4.74342i	2.13118	−	1.54840i			9.80818i	2.94780	+	0.557196i	1.59673	+	5.93966i
479.3		−	2.69008i	−0.101884	+	1.72905i	−5.23652			−1.13491	+	1.92665i	4.65128	+	0.274076i	−0.124451	+	0.0904187i			8.70650i	−2.97924	−	0.352326i	5.18283	+	3.05301i
479.4		−	2.68273i	1.70274	+	0.317270i	−5.19702			−1.87945	−	1.21147i	0.851147	−	4.56800i	−3.46106	+	2.51461i			8.57672i	2.79868	+	1.08046i	−3.25005	+	5.04205i
479.5		−	2.66370i	1.70038	+	0.329722i	−5.09528			2.23396	−	0.0970432i	0.878280	−	4.52929i	0.265440	−	0.192853i			8.24490i	2.78257	+	1.12130i	−0.258494	−	5.95060i
479.6		−	2.64285i	0.897345	−	1.48148i	−4.98466			−0.854664	−	2.06629i	−3.91532	−	2.37155i	3.40417	−	2.47328i			7.88801i	−1.38955	−	2.65879i	−5.46089	+	2.25875i
479.7		−	2.56347i	0.528333	+	1.64950i	−4.57140			−0.995537	−	2.00223i	4.22846	−	1.35437i	2.04460	−	1.48549i			6.59171i	−2.44173	+	1.74298i	−5.13265	+	2.55203i
479.8		−	2.50632i	−0.0878203	+	1.72982i	−4.28165			2.00594	−	0.988026i	4.33549	+	0.220106i	−2.22481	+	1.61642i			5.71853i	−2.98458	−	0.303827i	−2.47631	−	5.02754i
479.9		−	2.47101i	1.63197	−	0.580248i	−4.10589			−0.478657	+	2.18424i	−1.43380	−	4.03260i	1.34664	−	0.978389i			5.20367i	2.32662	−	1.89389i	5.39727	+	1.18277i
479.10		−	2.42479i	−1.72564	+	0.148938i	−3.87961			1.45361	+	1.69912i	0.361143	+	4.18430i	−4.07843	+	2.96315i			4.55766i	2.95564	−	0.514024i	4.12001	−	3.52471i
479.11		−	2.40284i	−1.38510	−	1.03995i	−3.77364			1.46974	−	1.68519i	−2.49884	+	3.32817i	0.173424	−	0.126000i			4.26178i	0.836995	+	2.88087i	−4.04923	−	3.53156i
479.12		−	2.36050i	−1.18380	−	1.26436i	−3.57196			−2.17626	−	0.513716i	−2.98453	+	2.79437i	−2.13662	+	1.55234i			3.71061i	−0.197222	+	2.99351i	−1.21263	+	5.13706i
479.13		−	2.29383i	−1.46053	+	0.931057i	−3.26165			1.20941	+	1.88078i	2.13568	+	3.35020i	1.51845	−	1.10322i			2.89401i	1.26627	−	2.71966i	4.31418	−	2.77418i
479.14		−	2.26768i	0.908227	−	1.47483i	−3.14238			−2.09336	+	0.786026i	−3.34444	−	2.05957i	−0.138458	+	0.100595i			2.59055i	−1.35025	−	2.67896i	1.78246	+	4.74708i
479.15		−	2.25534i	−1.70050	−	0.329073i	−3.08657			1.96863	−	1.06043i	−0.742173	+	3.83522i	1.93447	−	1.40548i			2.45059i	2.78342	+	1.11918i	−2.39163	−	4.43993i
479.16		−	2.25213i	1.09105	+	1.34522i	−3.07209			1.77024	+	1.36611i	3.02961	−	2.45718i	3.77369	−	2.74175i			2.41448i	−0.619229	+	2.93540i	3.07665	−	3.98681i
479.17		−	2.22225i	−1.27729	+	1.16984i	−2.93841			−0.454533	−	2.18938i	2.59967	+	2.83847i	1.19984	−	0.871733i			2.08538i	0.262964	−	2.98845i	−4.86536	+	1.01009i
479.18		−	2.18154i	−0.599442	−	1.62501i	−2.75911			−0.0497216	+	2.23552i	−3.54503	+	1.30771i	3.91040	−	2.84107i			1.65603i	−2.28134	+	1.94820i	4.87686	+	0.108469i
479.19		−	2.11925i	1.16832	−	1.27868i	−2.49121			0.713406	−	2.11921i	−2.70984	−	2.47597i	−4.07113	+	2.95785i			1.04100i	−0.270036	−	2.98782i	−4.49113	−	1.51189i
479.20		−	2.10539i	1.43748	+	0.966257i	−2.43265			−0.225784	+	2.22464i	2.03434	−	3.02645i	−2.36262	+	1.71654i			0.910901i	1.13270	+	2.77795i	4.68373	+	0.475362i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
275.bd	odd	10	1	inner
825.s	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.s.a	✓	464
3.b	odd	2	1	inner	825.2.s.a	✓	464
11.d	odd	10	1		825.2.bu.a	yes	464
25.e	even	10	1		825.2.bu.a	yes	464
33.f	even	10	1		825.2.bu.a	yes	464
75.h	odd	10	1		825.2.bu.a	yes	464
275.bd	odd	10	1	inner	825.2.s.a	✓	464
825.s	even	10	1	inner	825.2.s.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.s.a	✓	464	1.a	even	1	1	trivial
825.2.s.a	✓	464	3.b	odd	2	1	inner
825.2.s.a	✓	464	275.bd	odd	10	1	inner
825.2.s.a	✓	464	825.s	even	10	1	inner
825.2.bu.a	yes	464	11.d	odd	10	1
825.2.bu.a	yes	464	25.e	even	10	1
825.2.bu.a	yes	464	33.f	even	10	1
825.2.bu.a	yes	464	75.h	odd	10	1

Hecke kernels

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