Newform orbit 616.2.bw.a

• Label: 616.2.bw.a
• Level: $616$
• Weight: $2$
• Character orbit: 616.bw
• Analytic conductor: $4.919$
• Analytic rank: $0$
• Dimension: $96$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$616 = 2^{3} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	616.bw (of order $15$, degree $8$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.91878476451$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$12$ over $\Q(\zeta_{15})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 96 * q - 2 * q^3 - 6 * q^7 + 18 * q^9 - 2 * q^11 + 18 * q^13 - 2 * q^15 - 9 * q^17 + 11 * q^19 - 14 * q^21 + 24 * q^25 + 16 * q^27 + 34 * q^29 + 2 * q^31 - 17 * q^33 + 40 * q^35 + 6 * q^37 - 6 * q^39 - 10 * q^41 + 44 * q^43 + 72 * q^45 + 58 * q^49 - 6 * q^51 - 17 * q^53 + 64 * q^55 - 40 * q^57 + 13 * q^59 - q^61 + 81 * q^63 - 12 * q^65 - 18 * q^67 + 8 * q^69 - 42 * q^71 - 24 * q^73 + 19 * q^75 + 8 * q^77 + 16 * q^79 - 70 * q^81 - 92 * q^83 + 8 * q^85 - 38 * q^87 + 60 * q^89 - 27 * q^91 - 81 * q^93 + 24 * q^95 + 50 * q^97 - 40 * q^99

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
9.1		0		−2.58064	−	1.14897i		0		0.790449	+	0.168015i		0		−2.48787	+	0.900269i		0		3.33215	+	3.70073i		0
9.2		0		−2.48929	−	1.10830i		0		4.09501	+	0.870421i		0		2.53615	−	0.753612i		0		2.96084	+	3.28835i		0
9.3		0		−2.16291	−	0.962991i		0		−3.79797	−	0.807283i		0		−1.08776	−	2.41180i		0		1.74345	+	1.93630i		0
9.4		0		−1.63732	−	0.728981i		0		−1.76997	−	0.376220i		0		2.36549	+	1.18510i		0		0.142008	+	0.157716i		0
9.5		0		−1.46878	−	0.653941i		0		0.236112	+	0.0501871i		0		−1.54893	+	2.14495i		0		−0.277730	−	0.308450i		0
9.6		0		−0.258235	−	0.114974i		0		0.279761	+	0.0594650i		0		2.23433	−	1.41697i		0		−1.95393	−	2.17005i		0
9.7		0		0.171513	+	0.0763627i		0		3.90687	+	0.830432i		0		−2.12786	−	1.57232i		0		−1.98381	−	2.20324i		0
9.8		0		1.09424	+	0.487186i		0		2.77006	+	0.588795i		0		1.03405	+	2.43531i		0		−1.04739	−	1.16324i		0
9.9		0		1.13046	+	0.503313i		0		−1.64736	−	0.350157i		0		−1.84936	+	1.89205i		0		−0.982776	−	1.09148i		0
9.10		0		1.65728	+	0.737868i		0		−2.98976	−	0.635493i		0		−1.95619	−	1.78138i		0		0.194734	+	0.216274i		0
9.11		0		2.03125	+	0.904372i		0		−0.801161	−	0.170292i		0		2.64018	+	0.171552i		0		1.30071	+	1.44458i		0
9.12		0		2.68534	+	1.19559i		0		2.46693	+	0.524361i		0		−0.432037	−	2.61024i		0		3.77421	+	4.19168i		0
25.1		0		−2.92414	−	0.621545i		0		0.424764	−	4.04136i		0		2.33990	+	1.23485i		0		5.42364	+	2.41476i		0
25.2		0		−2.51342	−	0.534245i		0		−0.226421	+	2.15425i		0		2.52533	−	0.789122i		0		3.29125	+	1.46536i		0
25.3		0		−2.25684	−	0.479705i		0		−0.100588	+	0.957030i		0		−2.09175	−	1.62005i		0		2.12255	+	0.945022i		0
25.4		0		−1.33634	−	0.284048i		0		−0.00194692	+	0.0185237i		0		−0.886493	+	2.49282i		0		−1.03551	−	0.461039i		0
25.5		0		−0.227792	−	0.0484186i		0		0.298475	−	2.83980i		0		1.34842	−	2.27635i		0		−2.69109	−	1.19815i		0
25.6		0		−0.0275529	−	0.00585654i		0		0.0415095	−	0.394937i		0		0.188898	+	2.63900i		0		−2.73991	−	1.21989i		0
25.7		0		0.337058	+	0.0716438i		0		−0.393273	+	3.74174i		0		−2.63607	−	0.226126i		0		−2.63216	−	1.17191i		0
25.8		0		1.04060	+	0.221187i		0		−0.00568803	+	0.0541180i		0		2.19956	−	1.47036i		0		−1.70670	−	0.759873i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.c	even	5	1	inner
77.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	616.2.bw.a	✓	96
7.c	even	3	1	inner	616.2.bw.a	✓	96
11.c	even	5	1	inner	616.2.bw.a	✓	96
77.m	even	15	1	inner	616.2.bw.a	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
616.2.bw.a	✓	96	1.a	even	1	1	trivial
616.2.bw.a	✓	96	7.c	even	3	1	inner
616.2.bw.a	✓	96	11.c	even	5	1	inner
616.2.bw.a	✓	96	77.m	even	15	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^96 + 2*T3^95 - 25*T3^94 - 66*T3^93 + 230*T3^92 + 756*T3^91 - 361*T3^90 - 961*T3^89 - 11916*T3^88 - 69405*T3^87 + 202768*T3^86 + 1002444*T3^85 - 1964600*T3^84 - 6949772*T3^83 + 7106029*T3^82 - 3204351*T3^81 + 71887920*T3^80 + 664869796*T3^79 - 1021014030*T3^78 - 6929164276*T3^77 + 7021819093*T3^76 + 26121305157*T3^75 - 37148419283*T3^74 + 140030248896*T3^73 + 31899870888*T3^72 - 2562656531206*T3^71 + 2410069336174*T3^70 + 17011887575956*T3^69 - 18549602478383*T3^68 - 27159719966126*T3^67 + 38137278196388*T3^66 - 399853852024578*T3^65 + 298479617466729*T3^64 + 3361765982833462*T3^63 - 2415664191955729*T3^62 - 10444422181196814*T3^61 + 6867856639558058*T3^60 - 9271185248954383*T3^59 + 19473168604581592*T3^58 + 285722205427863400*T3^57 - 248587925826018843*T3^56 - 1625641524954275665*T3^55 + 728592202930254269*T3^54 + 3437040437187534535*T3^53 + 1626672347169209268*T3^52 + 12838664324242910877*T3^51 - 13438302114147551604*T3^50 - 100190051742681258329*T3^49 + 13292991893062701424*T3^48 + 192419035985003106953*T3^47 + 109249951070567697706*T3^46 + 330572032632837409407*T3^45 - 217757624555776033699*T3^44 - 2415372796409861249767*T3^43 - 453295939052907015357*T3^42 + 4533011865322353914806*T3^41 + 2118109625096555647219*T3^40 - 894808656189981731446*T3^39 + 1250381180949585013507*T3^38 - 21459549847964591014263*T3^37 - 8666098539012234094812*T3^36 + 62173420359442762280183*T3^35 + 11758907197800073383398*T3^34 - 66921818158450453127550*T3^33 - 4558932521618927539445*T3^32 - 100648508759309572556649*T3^31 + 91965919554807838404199*T3^30 + 315600306718240456355853*T3^29 - 141182739165109489861033*T3^28 - 279017147430015776593619*T3^27 - 132190421514329883561170*T3^26 + 69068501583339597237140*T3^25 + 396284644726527335861092*T3^24 + 122353395349843445712622*T3^23 - 33696102372246633430086*T3^22 - 737480766898861201771946*T3^21 + 388922484997562087193691*T3^20 + 119208615705928824346756*T3^19 - 21543784257564870595553*T3^18 - 17363593739420226532784*T3^17 + 573033088595210648804*T3^16 + 192472003371923744176*T3^15 - 124156521806061542171*T3^14 + 97235942003092285058*T3^13 + 20203155103416685778*T3^12 - 8952976773819236823*T3^11 + 852117356230127758*T3^10 + 478795782399508638*T3^9 - 50933382642881033*T3^8 + 5356957319526192*T3^7 - 444331426971744*T3^6 - 559875279338082*T3^5 + 143422297090209*T3^4 + 2919809632014*T3^3 - 82735190505*T3^2 + 1764003582*T3 + 96059601