Newform orbit 828.2.u.a

• Label: 828.2.u.a
• Level: $828$
• Weight: $2$
• Character orbit: 828.u
• Analytic conductor: $6.612$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	828.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.61161328736\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(10\) over \(\Q(\zeta_{22})\)
Twist minimal:	no (minimal twist has level 92)
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 7 q^{2} - 11 q^{4} + 22 q^{5} + 10 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.28274	+	0.595470i		0		1.29083	−	1.52766i	1.54295	−	2.40088i		0		0.249719	+	1.73683i	−0.746118	+	2.72824i		0		−0.549549	+	3.99848i
19.2	−1.14497	−	0.830084i		0		0.621920	+	1.90085i	1.32878	−	2.06761i		0		−0.647613	−	4.50425i	0.865782	−	2.69266i		0		−3.23770	+	1.26436i
19.3	−0.908675	−	1.08366i		0		−0.348619	+	1.96938i	−1.74032	+	2.70799i		0		0.439673	+	3.05800i	2.45091	−	1.41175i		0		4.51592	−	0.574777i
19.4	−0.579968	+	1.28982i		0		−1.32727	−	1.49611i	−0.782273	+	1.21724i		0		0.0556744	+	0.387224i	2.69949	−	0.844251i		0		−1.11633	−	1.71495i
19.5	−0.223917	−	1.39637i		0		−1.89972	+	0.625343i	−1.74032	+	2.70799i		0		−0.439673	−	3.05800i	1.29859	+	2.51270i		0		4.17106	+	1.82378i
19.6	0.122461	−	1.40890i		0		−1.97001	−	0.345071i	1.32878	−	2.06761i		0		0.647613	+	4.50425i	−0.727420	+	2.73329i		0		−2.75034	−	2.12532i
19.7	0.542021	+	1.30622i		0		−1.41243	+	1.41600i	0.235454	−	0.366374i		0		−0.319944	−	2.22526i	−2.61517	−	1.07744i		0		0.606186	+	0.108973i
19.8	0.632227	+	1.26503i		0		−1.20058	+	1.59957i	0.235454	−	0.366374i		0		0.319944	+	2.22526i	−2.78253	−	0.507470i		0		0.612332	+	0.0662238i
19.9	1.29004	−	0.579478i		0		1.32841	−	1.49510i	1.54295	−	2.40088i		0		−0.249719	−	1.73683i	0.847324	−	2.69853i		0		0.599213	−	3.99134i
19.10	1.35458	+	0.406342i		0		1.66977	+	1.10085i	−0.782273	+	1.21724i		0		−0.0556744	−	0.387224i	1.81452	+	2.16968i		0		−1.55427	+	1.33098i
199.1	−1.37581	+	0.327342i		0		1.78570	−	0.900718i	2.73647	−	2.37117i		0		−2.32465	−	1.49396i	−2.16193	+	1.82375i		0		−2.98868	+	4.15804i
199.2	−1.28305	−	0.594804i		0		1.29242	+	1.52632i	0.722435	−	0.625994i		0		2.35722	+	1.51489i	−0.750365	−	2.72708i		0		−1.29926	+	0.373471i
199.3	−1.04458	+	0.953336i		0		0.182300	−	1.99167i	−1.69342	+	1.46736i		0		0.867633	+	0.557594i	1.70831	+	2.25426i		0		0.370032	−	3.14718i
199.4	−0.936105	−	1.06005i		0		−0.247417	+	1.98464i	−1.65610	+	1.43502i		0		−3.14077	−	2.01845i	2.33542	−	1.59555i		0		3.07148	+	0.412224i
199.5	−0.0506424	−	1.41331i		0		−1.99487	+	0.143146i	1.03293	−	0.895042i		0		2.14768	+	1.38023i	0.303335	+	2.81211i		0		−1.31728	−	1.41452i
199.6	0.446765	−	1.34179i		0		−1.60080	−	1.19893i	1.03293	−	0.895042i		0		−2.14768	−	1.38023i	−2.32390	+	1.61230i		0		−0.739479	−	1.78585i
199.7	0.733683	+	1.20901i		0		−0.923420	+	1.77406i	−1.69342	+	1.46736i		0		−0.867633	−	0.557594i	−2.82236	+	0.185173i		0		−3.01649	−	0.970793i
199.8	1.19684	−	0.753380i		0		0.864836	−	1.80335i	−1.65610	+	1.43502i		0		3.14077	+	2.01845i	−0.323539	−	2.80986i		0		−0.900968	+	2.96516i
199.9	1.22786	+	0.701692i		0		1.01526	+	1.72315i	2.73647	−	2.37117i		0		2.32465	+	1.49396i	0.0374674	+	2.82818i		0		5.02383	−	0.991290i
199.10	1.39865	−	0.209235i		0		1.91244	−	0.585292i	0.722435	−	0.625994i		0		−2.35722	−	1.51489i	2.55237	−	1.21877i		0		0.879454	−	1.02670i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
23.d	odd	22	1	inner
92.h	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	828.2.u.a		100
3.b	odd	2	1		92.2.h.a	✓	100
4.b	odd	2	1	inner	828.2.u.a		100
12.b	even	2	1		92.2.h.a	✓	100
23.d	odd	22	1	inner	828.2.u.a		100
69.g	even	22	1		92.2.h.a	✓	100
92.h	even	22	1	inner	828.2.u.a		100
276.j	odd	22	1		92.2.h.a	✓	100

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
92.2.h.a	✓	100	3.b	odd	2	1
92.2.h.a	✓	100	12.b	even	2	1
92.2.h.a	✓	100	69.g	even	22	1
92.2.h.a	✓	100	276.j	odd	22	1
828.2.u.a		100	1.a	even	1	1	trivial
828.2.u.a		100	4.b	odd	2	1	inner
828.2.u.a		100	23.d	odd	22	1	inner
828.2.u.a		100	92.h	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^50 - 11*T5^49 + 52*T5^48 - 132*T5^47 + 218*T5^46 - 242*T5^45 - 1138*T5^44 + 11968*T5^43 - 47615*T5^42 + 92829*T5^41 - 57135*T5^40 - 25476*T5^39 + 424295*T5^38 - 6730790*T5^37 + 24208384*T5^36 - 14417403*T5^35 - 101489372*T5^34 - 26393609*T5^33 + 2120209142*T5^32 - 4827604947*T5^31 + 1092568940*T5^30 + 15508213160*T5^29 + 19308294170*T5^28 - 190316356241*T5^27 + 331068544576*T5^26 - 16397645011*T5^25 - 289166526484*T5^24 - 1840917690997*T5^23 + 6396584076267*T5^22 - 5097598798325*T5^21 - 486704841227*T5^20 - 10240487462106*T5^19 + 25574483019316*T5^18 - 6597567587281*T5^17 - 17212647277122*T5^16 - 6912562596137*T5^15 + 28191968577003*T5^14 - 13145173163177*T5^13 + 22788121962870*T5^12 - 36439428971550*T5^11 + 16617789956559*T5^10 + 4201052610986*T5^9 + 920927365160*T5^8 + 1300659986769*T5^7 - 5267502897921*T5^6 + 804889982567*T5^5 + 944128304339*T5^4 - 395386221120*T5^3 + 20602414569*T5^2 + 62232178767*T5 + 9912543971