Newform orbit 80.3.t.a

• Label: 80.3.t.a
• Level: $80$
• Weight: $3$
• Character orbit: 80.t
• Analytic conductor: $2.180$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	80.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17984211488\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} - 4 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{6} - 8 q^{8} + 108 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.96193	+	0.388354i	−4.95045			3.69836	−	1.52385i	−3.04373	−	3.96683i	9.71244	−	1.92253i	7.61189	+	7.61189i	−6.66415	+	4.42596i	15.5069			7.51212	+	6.60061i
53.2	−1.94569	+	0.462923i	4.38426			3.57140	−	1.80141i	1.54952	−	4.75384i	−8.53040	+	2.02957i	−3.84157	−	3.84157i	−6.11493	+	5.15826i	10.2217			−0.814216	+	9.96680i
53.3	−1.92883	+	0.528784i	−2.05195			3.44078	−	2.03987i	1.09885	+	4.87776i	3.95786	−	1.08504i	−6.87250	−	6.87250i	−5.55803	+	5.75399i	−4.78950			−4.69877	−	8.82732i
53.4	−1.89674	−	0.634321i	2.77329			3.19527	+	2.40629i	−2.05735	+	4.55712i	−5.26021	−	1.75915i	5.39242	+	5.39242i	−4.53426	−	6.59094i	−1.30888			6.79294	−	7.33867i
53.5	−1.55338	−	1.25977i	−0.119786			0.825960	+	3.91379i	−3.18046	−	3.85807i	0.186072	+	0.150902i	−4.73972	−	4.73972i	3.64745	−	7.12012i	−8.98565			0.0801655	+	9.99968i
53.6	−1.35261	−	1.47324i	−4.50609			−0.340894	+	3.98545i	4.65788	+	1.81773i	6.09498	+	6.63857i	−1.52625	−	1.52625i	6.33263	−	4.88854i	11.3048			−3.62234	−	9.32087i
53.7	−1.20488	+	1.59633i	0.390820			−1.09653	−	3.84677i	4.99627	+	0.192979i	−0.470891	+	0.623877i	6.36907	+	6.36907i	7.46189	+	2.88447i	−8.84726			−6.32797	+	7.74318i
53.8	−0.993712	+	1.73567i	−0.616720			−2.02507	−	3.44951i	−4.57731	−	2.01201i	0.612842	−	1.07042i	−3.63369	−	3.63369i	7.99953	−	0.0870298i	−8.61966			8.04071	−	5.94533i
53.9	−0.800895	−	1.83264i	3.83124			−2.71713	+	2.93550i	4.99784	+	0.146974i	−3.06842	−	7.02129i	1.69668	+	1.69668i	7.55586	+	2.62850i	5.67842			−3.73339	−	9.27695i
53.10	−0.419008	+	1.95562i	5.73309			−3.64886	−	1.63884i	−0.266013	+	4.99292i	−2.40221	+	11.2117i	−3.79616	−	3.79616i	4.73384	−	6.44909i	23.8683			−9.65277	−	2.61229i
53.11	−0.284962	−	1.97960i	−2.50699			−3.83759	+	1.12822i	−4.36488	+	2.43881i	0.714398	+	4.96283i	7.18571	+	7.18571i	3.32699	+	7.27538i	−2.71500			6.07168	+	7.94573i
53.12	0.0957584	+	1.99771i	−4.94472			−3.98166	+	0.382594i	3.69037	−	3.37360i	−0.473499	−	9.87810i	−3.22480	−	3.22480i	−1.14559	−	7.91755i	15.4503			7.09285	+	7.04922i
53.13	0.565399	−	1.91842i	−1.96075			−3.36065	−	2.16934i	2.39640	−	4.38831i	−1.10861	+	3.76154i	−2.51657	−	2.51657i	−6.06181	+	5.22058i	−5.15546			−7.06369	−	7.07844i
53.14	0.635873	+	1.89622i	−1.50709			−3.19133	+	2.41151i	−3.37127	+	3.69249i	−0.958316	−	2.85778i	1.28182	+	1.28182i	−6.60205	−	4.51807i	−6.72868			−9.14550	−	4.04473i
53.15	0.715976	−	1.86745i	4.59062			−2.97476	−	2.67410i	−4.31900	−	2.51918i	3.28677	−	8.57276i	1.15913	+	1.15913i	−7.12361	+	3.64063i	12.0738			−7.79675	+	6.26185i
53.16	0.884585	+	1.79374i	3.32036			−2.43502	+	3.17343i	−1.05740	−	4.88691i	2.93714	+	5.95587i	9.08173	+	9.08173i	−7.84630	−	1.56062i	2.02480			7.83050	−	6.21959i
53.17	1.46337	+	1.36329i	1.90859			0.282885	+	3.98998i	4.97175	−	0.530759i	2.79297	+	2.60196i	−8.62025	−	8.62025i	−5.02554	+	6.22446i	−5.35728			7.99907	+	6.00124i
53.18	1.51908	−	1.30092i	1.70661			0.615196	−	3.95241i	2.37894	+	4.39780i	2.59247	−	2.22016i	0.332763	+	0.332763i	−4.20725	−	6.80434i	−6.08749			9.33499	+	3.58579i
53.19	1.65576	−	1.12181i	−5.30326			1.48309	−	3.71489i	−4.70173	+	1.70109i	−8.78093	+	5.94924i	−7.26221	−	7.26221i	−1.71176	−	7.81472i	19.1246			−5.87666	+	8.09104i
53.20	1.86073	+	0.733258i	−3.80597			2.92466	+	2.72880i	2.60487	+	4.26786i	−7.08190	−	2.79076i	5.17093	+	5.17093i	3.44111	+	7.22210i	5.48540			1.71753	+	9.85140i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.t	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	80.3.t.a	yes	44
4.b	odd	2	1		320.3.t.a		44
5.b	even	2	1		400.3.t.b		44
5.c	odd	4	1		80.3.i.a	✓	44
5.c	odd	4	1		400.3.i.b		44
8.b	even	2	1		640.3.t.b		44
8.d	odd	2	1		640.3.t.a		44
16.e	even	4	1		80.3.i.a	✓	44
16.e	even	4	1		640.3.i.b		44
16.f	odd	4	1		320.3.i.a		44
16.f	odd	4	1		640.3.i.a		44
20.e	even	4	1		320.3.i.a		44
40.i	odd	4	1		640.3.i.b		44
40.k	even	4	1		640.3.i.a		44
80.i	odd	4	1		400.3.t.b		44
80.i	odd	4	1		640.3.t.b		44
80.j	even	4	1		320.3.t.a		44
80.q	even	4	1		400.3.i.b		44
80.s	even	4	1		640.3.t.a		44
80.t	odd	4	1	inner	80.3.t.a	yes	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.3.i.a	✓	44	5.c	odd	4	1
80.3.i.a	✓	44	16.e	even	4	1
80.3.t.a	yes	44	1.a	even	1	1	trivial
80.3.t.a	yes	44	80.t	odd	4	1	inner
320.3.i.a		44	16.f	odd	4	1
320.3.i.a		44	20.e	even	4	1
320.3.t.a		44	4.b	odd	2	1
320.3.t.a		44	80.j	even	4	1
400.3.i.b		44	5.c	odd	4	1
400.3.i.b		44	80.q	even	4	1
400.3.t.b		44	5.b	even	2	1
400.3.t.b		44	80.i	odd	4	1
640.3.i.a		44	16.f	odd	4	1
640.3.i.a		44	40.k	even	4	1
640.3.i.b		44	16.e	even	4	1
640.3.i.b		44	40.i	odd	4	1
640.3.t.a		44	8.d	odd	2	1
640.3.t.a		44	80.s	even	4	1
640.3.t.b		44	8.b	even	2	1
640.3.t.b		44	80.i	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(80, [\chi])\).