Newform orbit 414.3.l.c

• Label: 414.3.l.c
• Level: $414$
• Weight: $3$
• Character orbit: 414.l
• Analytic conductor: $11.281$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.l (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.2806829445\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{4}+O(q^{10}) \)

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	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	−5.16660	+	8.03938i		0		−10.8325	+	1.55748i	2.71386	+	0.796860i		0		13.3773	+	1.92336i
19.2	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	−1.30307	+	2.02762i		0		4.20587	−	0.604713i	2.71386	+	0.796860i		0		3.37389	+	0.485092i
19.3	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	2.90981	−	4.52775i		0		0.515290	−	0.0740875i	2.71386	+	0.796860i		0		−7.53403	−	1.08323i
19.4	−0.587486	−	1.28641i		0		−1.30972	+	1.51150i	3.55986	−	5.53925i		0		−0.310301	+	0.0446146i	2.71386	+	0.796860i		0		−9.21713	−	1.32522i
19.5	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	−3.55986	+	5.53925i		0		−0.310301	+	0.0446146i	−2.71386	−	0.796860i		0		−9.21713	−	1.32522i
19.6	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	−2.90981	+	4.52775i		0		0.515290	−	0.0740875i	−2.71386	−	0.796860i		0		−7.53403	−	1.08323i
19.7	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	1.30307	−	2.02762i		0		4.20587	−	0.604713i	−2.71386	−	0.796860i		0		3.37389	+	0.485092i
19.8	0.587486	+	1.28641i		0		−1.30972	+	1.51150i	5.16660	−	8.03938i		0		−10.8325	+	1.55748i	−2.71386	−	0.796860i		0		13.3773	+	1.92336i
37.1	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	−8.16225	+	1.17355i		0		−2.09488	−	0.956699i	−1.85223	+	2.13758i		0		10.6080	−	4.84451i
37.2	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	−2.79643	+	0.402066i		0		9.94408	+	4.54131i	−1.85223	+	2.13758i		0		3.63436	−	1.65976i
37.3	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	3.02216	−	0.434521i		0		−4.90669	−	2.24081i	−1.85223	+	2.13758i		0		−3.92772	+	1.79373i
37.4	−1.35693	+	0.398430i		0		1.68251	−	1.08128i	7.93652	−	1.14110i		0		7.86197	+	3.59044i	−1.85223	+	2.13758i		0		−10.3146	+	4.71054i
37.5	1.35693	−	0.398430i		0		1.68251	−	1.08128i	−7.93652	+	1.14110i		0		7.86197	+	3.59044i	1.85223	−	2.13758i		0		−10.3146	+	4.71054i
37.6	1.35693	−	0.398430i		0		1.68251	−	1.08128i	−3.02216	+	0.434521i		0		−4.90669	−	2.24081i	1.85223	−	2.13758i		0		−3.92772	+	1.79373i
37.7	1.35693	−	0.398430i		0		1.68251	−	1.08128i	2.79643	−	0.402066i		0		9.94408	+	4.54131i	1.85223	−	2.13758i		0		3.63436	−	1.65976i
37.8	1.35693	−	0.398430i		0		1.68251	−	1.08128i	8.16225	−	1.17355i		0		−2.09488	−	0.956699i	1.85223	−	2.13758i		0		10.6080	−	4.84451i
109.1	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	−5.16660	−	8.03938i		0		−10.8325	−	1.55748i	2.71386	−	0.796860i		0		13.3773	−	1.92336i
109.2	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	−1.30307	−	2.02762i		0		4.20587	+	0.604713i	2.71386	−	0.796860i		0		3.37389	−	0.485092i
109.3	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	2.90981	+	4.52775i		0		0.515290	+	0.0740875i	2.71386	−	0.796860i		0		−7.53403	+	1.08323i
109.4	−0.587486	+	1.28641i		0		−1.30972	−	1.51150i	3.55986	+	5.53925i		0		−0.310301	−	0.0446146i	2.71386	−	0.796860i		0		−9.21713	+	1.32522i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
23.d	odd	22	1	inner
69.g	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	414.3.l.c	✓	80
3.b	odd	2	1	inner	414.3.l.c	✓	80
23.d	odd	22	1	inner	414.3.l.c	✓	80
69.g	even	22	1	inner	414.3.l.c	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
414.3.l.c	✓	80	1.a	even	1	1	trivial
414.3.l.c	✓	80	3.b	odd	2	1	inner
414.3.l.c	✓	80	23.d	odd	22	1	inner
414.3.l.c	✓	80	69.g	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^80 - 220*T5^78 + 26666*T5^76 - 2658876*T5^74 + 266330727*T5^72 - 22552666444*T5^70 + 1348308660076*T5^68 - 65825219147468*T5^66 + 3906904713800841*T5^64 - 253631656039866060*T5^62 + 15202614809755695196*T5^60 - 849566955043652240472*T5^58 + 43835285058145689649053*T5^56 - 2010964031276220247242972*T5^54 + 79832657020070090570298312*T5^52 - 2828907605298154811771874688*T5^50 + 84470450935047521676423661007*T5^48 - 1728461752666443048292827596544*T5^46 + 56319036143591994027908171528316*T5^44 - 168333966835524475957696755336922*T5^42 + 20872289292808359446187949246259958*T5^40 + 170847626190250260292788210817420562*T5^38 - 3298069142906442646603265594891106000*T5^36 - 73957410795669642716009655202287751282*T5^34 + 429233302326681366951505376274020901977*T5^32 + 6077415531549199520523063819710687038466*T5^30 + 2848030126159402593766092414229777558208*T5^28 - 363448627738133187951579260199052975947614*T5^26 - 472954302515757076938539661175586100480026*T5^24 + 648274077587380232941892612093680948194616*T5^22 + 72681741079452556982523518232388492141909206*T5^20 - 47693020773206017242939284945295058752396754*T5^18 - 11695151030936576776070918891766957885955430*T5^16 + 104985985899529094975090650923022987443118652*T5^14 + 108734794571687262477722268473684476272811180*T5^12 - 180692999736421239373624742161767124571480752*T5^10 + 150514668281493025451299154536401813888924377*T5^8 - 2130209639837855697287114498008720042094302*T5^6 + 214436571177560594710418627856851421024195*T5^4 - 526292453808223749867633727391717433574*T5^2 + 11127644038183113430360543931061959041