Newform orbit 61.2.i.a

• Label: 61.2.i.a
• Level: $61$
• Weight: $2$
• Character orbit: 61.i
• Analytic conductor: $0.487$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.i (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.487087452330\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{2} - 4 q^{3} - 10 q^{4} + 2 q^{5} + q^{6} + q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−2.17004	−	0.966162i	−1.76760	+	1.28423i	2.43733	+	2.70693i	3.68353	−	0.782959i	5.07653	−	1.07905i	−0.302464	+	2.87776i	−1.20568	−	3.71070i	0.548089	−	1.68684i	−8.74987	−	1.85984i
12.2	−1.94561	−	0.866241i	1.86941	−	1.35820i	1.69676	+	1.88444i	−1.20434	+	0.255990i	−4.81367	+	1.02318i	0.404506	−	3.84862i	−0.352600	−	1.08519i	0.722916	−	2.22491i	2.56491	+	0.545189i
12.3	0.0266816	+	0.0118794i	1.08919	−	0.791346i	−1.33769	−	1.48566i	0.565916	−	0.120289i	0.0384622	−	0.00817539i	−0.331787	+	3.15674i	−0.0360937	−	0.111085i	−0.366935	+	1.12931i	0.0165285	+	0.00351325i
12.4	1.50629	+	0.670642i	−1.69101	+	1.22859i	0.480878	+	0.534070i	1.40283	−	0.298182i	−3.37108	+	0.716546i	0.125217	−	1.19136i	−0.652866	−	2.00931i	0.423022	−	1.30193i	2.31304	+	0.491653i
15.1	−0.261670	−	2.48962i	1.09846	−	0.798080i	−4.17345	+	0.887094i	2.01896	+	2.24228i	−2.27435	−	2.52592i	−2.42543	+	1.07987i	1.75345	+	5.39656i	−0.357362	+	1.09985i	5.05413	−	5.61318i
15.2	−0.0842909	−	0.801975i	−0.246407	+	0.179025i	1.32024	−	0.280625i	−0.519162	−	0.576588i	0.164343	+	0.182522i	−0.0513167	+	0.0228477i	−0.834716	−	2.56899i	−0.898385	+	2.76494i	−0.418649	+	0.464956i
15.3	0.145470	+	1.38405i	−2.12014	+	1.54037i	0.0618510	−	0.0131468i	−0.176149	−	0.195633i	−2.44038	−	2.71031i	−0.373885	+	0.166464i	0.887298	+	2.73082i	1.19520	−	3.67845i	0.245143	−	0.272259i
15.4	0.283167	+	2.69415i	0.768086	−	0.558047i	−5.22198	+	1.10997i	−0.799461	−	0.887892i	1.72096	+	1.91132i	3.76417	−	1.67592i	−2.79486	−	8.60169i	−0.648512	+	1.99591i	2.16573	−	2.40529i
16.1	−1.92853	−	0.409922i	0.379526	+	1.16806i	1.72411	+	0.767623i	0.174208	+	1.65748i	−0.253114	−	2.40822i	1.99411	+	2.21469i	0.179808	+	0.130638i	1.20672	−	0.876735i	0.343472	−	3.26792i
16.2	−1.18602	−	0.252095i	−0.507067	−	1.56059i	−0.484011	−	0.215496i	−0.0538027	−	0.511898i	0.207971	+	1.97872i	−2.89428	−	3.21442i	2.48160	+	1.80299i	0.248718	−	0.180704i	−0.0652363	+	0.620682i
16.3	0.873313	+	0.185628i	0.508502	+	1.56501i	−1.09887	−	0.489250i	−0.0637708	−	0.606739i	0.153571	+	1.46113i	−0.455761	−	0.506174i	−2.31346	−	1.68083i	0.236374	−	0.171736i	0.0569361	−	0.541711i
16.4	1.30584	+	0.277564i	−0.880961	−	2.71132i	−0.198923	−	0.0885664i	0.308047	+	2.93087i	−0.397826	−	3.78506i	2.02506	+	2.24905i	−2.39527	−	1.74027i	−4.14811	+	3.01378i	−0.411246	+	3.91274i
22.1	−1.70355	−	1.89199i	−0.826723	+	2.54439i	−0.468466	+	4.45716i	−2.63869	+	1.17482i	6.22233	−	2.77036i	−0.0544423	−	0.0115721i	5.11156	−	3.71377i	−3.36341	−	2.44366i	6.71789	+	2.99100i
22.2	−0.592778	−	0.658346i	−0.289378	+	0.890614i	0.127022	−	1.20854i	2.60246	−	1.15869i	0.757869	−	0.337425i	0.0927128	+	0.0197067i	−2.30434	+	1.67420i	1.71760	+	1.24791i	−2.30550	−	1.02648i
22.3	−0.321184	−	0.356711i	0.520127	−	1.60079i	0.184973	−	1.75990i	−3.28207	+	1.46127i	−0.738075	+	0.328612i	3.82105	+	0.812190i	−1.46385	+	1.06355i	0.135065	+	0.0981308i	1.57540	+	0.701413i
22.4	1.05291	+	1.16938i	0.0959745	−	0.295379i	−0.0497630	+	0.473463i	−1.01852	+	0.453476i	0.446463	−	0.198778i	−4.83747	−	1.02824i	1.94001	−	1.40950i	2.34901	+	1.70666i	−1.60270	−	0.713568i
25.1	−1.70355	+	1.89199i	−0.826723	−	2.54439i	−0.468466	−	4.45716i	−2.63869	−	1.17482i	6.22233	+	2.77036i	−0.0544423	+	0.0115721i	5.11156	+	3.71377i	−3.36341	+	2.44366i	6.71789	−	2.99100i
25.2	−0.592778	+	0.658346i	−0.289378	−	0.890614i	0.127022	+	1.20854i	2.60246	+	1.15869i	0.757869	+	0.337425i	0.0927128	−	0.0197067i	−2.30434	−	1.67420i	1.71760	−	1.24791i	−2.30550	+	1.02648i
25.3	−0.321184	+	0.356711i	0.520127	+	1.60079i	0.184973	+	1.75990i	−3.28207	−	1.46127i	−0.738075	−	0.328612i	3.82105	−	0.812190i	−1.46385	−	1.06355i	0.135065	−	0.0981308i	1.57540	−	0.701413i
25.4	1.05291	−	1.16938i	0.0959745	+	0.295379i	−0.0497630	−	0.473463i	−1.01852	−	0.453476i	0.446463	+	0.198778i	−4.83747	+	1.02824i	1.94001	+	1.40950i	2.34901	−	1.70666i	−1.60270	+	0.713568i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.i	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.2.i.a	✓	32
3.b	odd	2	1		549.2.bl.b		32
4.b	odd	2	1		976.2.bw.c		32
61.i	even	15	1	inner	61.2.i.a	✓	32
61.i	even	15	1		3721.2.a.j		16
61.k	even	30	1		3721.2.a.l		16
183.t	odd	30	1		549.2.bl.b		32
244.u	odd	30	1		976.2.bw.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.2.i.a	✓	32	1.a	even	1	1	trivial
61.2.i.a	✓	32	61.i	even	15	1	inner
549.2.bl.b		32	3.b	odd	2	1
549.2.bl.b		32	183.t	odd	30	1
976.2.bw.c		32	4.b	odd	2	1
976.2.bw.c		32	244.u	odd	30	1
3721.2.a.j		16	61.i	even	15	1
3721.2.a.l		16	61.k	even	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).