LMFDB - Embedded newform 1680.2.ba.c.911.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(911,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.911"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-10,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 640 x^{12} - 1656 x^{11} + 3522 x^{10} - 6148 x^{9} + \cdots + 13 $ x^16 - 8x^15 + 48x^14 - 196x^13 + 640x^12 - 1656x^11 + 3522x^10 - 6148x^9 + 8856x^8 - 10460x^7 + 10032x^6 - 7680x^5 + 4586x^4 - 2062x^3 + 659x^2 - 134*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.15
Root			$0.500000 + 0.0342497i$ of defining polynomial
Character	$\chi$	$=$	1680.911
Dual form			1680.2.ba.c.911.16

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.69926 - 0.335422i) q^{3} -1.00000i q^{5} -1.00000i q^{7} +(2.77498 - 1.13994i) q^{9} +4.49036 q^{11} +1.59797 q^{13} +(-0.335422 - 1.69926i) q^{15} -0.506137i q^{17} -0.377701i q^{19} +(-0.335422 - 1.69926i) q^{21} -2.84053 q^{23} -1.00000 q^{25} +(4.33306 - 2.86785i) q^{27} +1.04311i q^{29} -0.219074i q^{31} +(7.63030 - 1.50617i) q^{33} -1.00000 q^{35} -2.40449 q^{37} +(2.71538 - 0.535996i) q^{39} +3.00524i q^{41} -1.31398i q^{43} +(-1.13994 - 2.77498i) q^{45} -8.43265 q^{47} -1.00000 q^{49} +(-0.169770 - 0.860060i) q^{51} +2.04125i q^{53} -4.49036i q^{55} +(-0.126689 - 0.641814i) q^{57} +6.65739 q^{59} +7.40099 q^{61} +(-1.13994 - 2.77498i) q^{63} -1.59797i q^{65} -11.9407i q^{67} +(-4.82681 + 0.952777i) q^{69} +3.89564 q^{71} +12.3491 q^{73} +(-1.69926 + 0.335422i) q^{75} -4.49036i q^{77} -15.2574i q^{79} +(6.40107 - 6.32663i) q^{81} -0.697682 q^{83} -0.506137 q^{85} +(0.349882 + 1.77252i) q^{87} +2.30345i q^{89} -1.59797i q^{91} +(-0.0734824 - 0.372265i) q^{93} -0.377701 q^{95} +5.53949 q^{97} +(12.4607 - 5.11874i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 10 q^{9} - 12 q^{11} + 2 q^{15} + 2 q^{21} - 8 q^{23} - 16 q^{25} + 12 q^{27} + 12 q^{33} - 16 q^{35} - 8 q^{37} + 14 q^{39} + 8 q^{45} - 16 q^{47} - 16 q^{49} + 34 q^{51} + 12 q^{57} - 32 q^{59}+ \cdots + 50 q^{99}+O(q^{100}) $ 16 * q - 10 * q^9 - 12 * q^11 + 2 * q^15 + 2 * q^21 - 8 * q^23 - 16 * q^25 + 12 * q^27 + 12 * q^33 - 16 * q^35 - 8 * q^37 + 14 * q^39 + 8 * q^45 - 16 * q^47 - 16 * q^49 + 34 * q^51 + 12 * q^57 - 32 * q^59 + 8 * q^61 + 8 * q^63 - 20 * q^69 + 32 * q^71 + 48 * q^73 - 22 * q^81 - 72 * q^83 - 12 * q^85 + 44 * q^87 + 28 * q^93 - 8 * q^95 - 8 * q^97 + 50 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.69926	−	0.335422i	0.981069	−	0.193656i
$3$	1.69926	−	0.335422i	0.981069	−	0.193656i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.77498	−	1.13994i	0.924995	−	0.379980i
$10$		0			0
$11$	4.49036			1.35389			0.676947	−	0.736031i	$-0.263302\pi$
$11$	4.49036			1.35389			0.676947	+	0.736031i	$0.263302\pi$
$12$		0			0
$13$	1.59797			0.443198			0.221599	−	0.975138i	$-0.428872\pi$
$13$	1.59797			0.443198			0.221599	+	0.975138i	$0.428872\pi$
$14$		0			0
$15$	−0.335422	−	1.69926i	−0.0866056	−	0.438748i
$16$		0			0
$17$		−	0.506137i		−	0.122756i	−0.998115	−	0.0613782i	$-0.980450\pi$
$17$		−	0.506137i		−	0.122756i	0.998115	−	0.0613782i	$-0.0195496\pi$
$18$		0			0
$19$		−	0.377701i		−	0.0866506i	−0.999061	−	0.0433253i	$-0.986205\pi$
$19$		−	0.377701i		−	0.0866506i	0.999061	−	0.0433253i	$-0.0137952\pi$
$20$		0			0
$21$	−0.335422	−	1.69926i	−0.0731951	−	0.370809i
$22$		0			0
$23$	−2.84053			−0.592292			−0.296146	−	0.955143i	$-0.595701\pi$
$23$	−2.84053			−0.592292			−0.296146	+	0.955143i	$0.595701\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	4.33306	−	2.86785i	0.833899	−	0.551918i
$28$		0			0
$29$			1.04311i			0.193700i	0.995299	+	0.0968502i	$0.0308768\pi$
$29$			1.04311i			0.193700i	−0.995299	+	0.0968502i	$0.969123\pi$
$30$		0			0
$31$		−	0.219074i		−	0.0393469i	−0.999806	−	0.0196735i	$-0.993737\pi$
$31$		−	0.219074i		−	0.0393469i	0.999806	−	0.0196735i	$-0.00626266\pi$
$32$		0			0
$33$	7.63030	−	1.50617i	1.32826	−	0.262190i
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$	−2.40449			−0.395296			−0.197648	−	0.980273i	$-0.563330\pi$
$37$	−2.40449			−0.395296			−0.197648	+	0.980273i	$0.563330\pi$
$38$		0			0
$39$	2.71538	−	0.535996i	0.434808	−	0.0858280i
$40$		0			0
$41$			3.00524i			0.469340i	0.972075	+	0.234670i	$0.0754009\pi$
$41$			3.00524i			0.469340i	−0.972075	+	0.234670i	$0.924599\pi$
$42$		0			0
$43$		−	1.31398i		−	0.200380i	−0.994968	−	0.100190i	$-0.968055\pi$
$43$		−	1.31398i		−	0.200380i	0.994968	−	0.100190i	$-0.0319451\pi$
$44$		0			0
$45$	−1.13994	−	2.77498i	−0.169932	−	0.413670i
$46$		0			0
$47$	−8.43265			−1.23003			−0.615014	−	0.788516i	$-0.710850\pi$
$47$	−8.43265			−1.23003			−0.615014	+	0.788516i	$0.710850\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.169770	−	0.860060i	−0.0237725	−	0.120433i
$52$		0			0
$53$			2.04125i			0.280388i	0.990124	+	0.140194i	$0.0447726\pi$
$53$			2.04125i			0.280388i	−0.990124	+	0.140194i	$0.955227\pi$
$54$		0			0
$55$		−	4.49036i		−	0.605480i
$56$		0			0
$57$	−0.126689	−	0.641814i	−0.0167804	−	0.0850103i
$58$		0			0
$59$	6.65739			0.866718			0.433359	−	0.901221i	$-0.357328\pi$
$59$	6.65739			0.866718			0.433359	+	0.901221i	$0.357328\pi$
$60$		0			0
$61$	7.40099			0.947600			0.473800	−	0.880632i	$-0.342882\pi$
$61$	7.40099			0.947600			0.473800	+	0.880632i	$0.342882\pi$
$62$		0			0
$63$	−1.13994	−	2.77498i	−0.143619	−	0.349615i
$64$		0			0
$65$		−	1.59797i		−	0.198204i
$66$		0			0
$67$		−	11.9407i		−	1.45879i	−0.684091	−	0.729397i	$-0.739801\pi$
$67$		−	11.9407i		−	1.45879i	0.684091	−	0.729397i	$-0.260199\pi$
$68$		0			0
$69$	−4.82681	+	0.952777i	−0.581080	+	0.114701i
$70$		0			0
$71$	3.89564			0.462327			0.231164	−	0.972915i	$-0.425747\pi$
$71$	3.89564			0.462327			0.231164	+	0.972915i	$0.425747\pi$
$72$		0			0
$73$	12.3491			1.44535			0.722675	−	0.691188i	$-0.242912\pi$
$73$	12.3491			1.44535			0.722675	+	0.691188i	$0.242912\pi$
$74$		0			0
$75$	−1.69926	+	0.335422i	−0.196214	+	0.0387312i
$76$		0			0
$77$		−	4.49036i		−	0.511724i
$78$		0			0
$79$		−	15.2574i		−	1.71659i	−0.513155	−	0.858296i	$-0.671523\pi$
$79$		−	15.2574i		−	1.71659i	0.513155	−	0.858296i	$-0.328477\pi$
$80$		0			0
$81$	6.40107	−	6.32663i	0.711230	−	0.702959i
$82$		0			0
$83$	−0.697682			−0.0765805			−0.0382903	−	0.999267i	$-0.512191\pi$
$83$	−0.697682			−0.0765805			−0.0382903	+	0.999267i	$0.512191\pi$
$84$		0			0
$85$	−0.506137			−0.0548983
$86$		0			0
$87$	0.349882	+	1.77252i	0.0375113	+	0.190034i
$88$		0			0
$89$			2.30345i			0.244165i	0.992520	+	0.122083i	$0.0389573\pi$
$89$			2.30345i			0.244165i	−0.992520	+	0.122083i	$0.961043\pi$
$90$		0			0
$91$		−	1.59797i		−	0.167513i
$92$		0			0
$93$	−0.0734824	−	0.372265i	−0.00761977	−	0.0386021i
$94$		0			0
$95$	−0.377701			−0.0387513
$96$		0			0
$97$	5.53949			0.562450			0.281225	−	0.959642i	$-0.409259\pi$
$97$	5.53949			0.562450			0.281225	+	0.959642i	$0.409259\pi$
$98$		0			0
$99$	12.4607	−	5.11874i	1.25235	−	0.514453i
$100$		0			0
$101$			14.4480i			1.43763i	0.695200	+	0.718816i	$0.255316\pi$
$101$			14.4480i			1.43763i	−0.695200	+	0.718816i	$0.744684\pi$
$102$		0			0
$103$		−	9.18152i		−	0.904683i	−0.891845	−	0.452341i	$-0.850589\pi$
$103$		−	9.18152i		−	0.904683i	0.891845	−	0.452341i	$-0.149411\pi$
$104$		0			0
$105$	−1.69926	+	0.335422i	−0.165831	+	0.0327338i
$106$		0			0
$107$	−15.4174			−1.49046			−0.745229	−	0.666809i	$-0.767660\pi$
$107$	−15.4174			−1.49046			−0.745229	+	0.666809i	$0.767660\pi$
$108$		0			0
$109$	−15.3203			−1.46741			−0.733707	−	0.679465i	$-0.762212\pi$
$109$	−15.3203			−1.46741			−0.733707	+	0.679465i	$0.762212\pi$
$110$		0			0
$111$	−4.08587	+	0.806520i	−0.387813	+	0.0765515i
$112$		0			0
$113$			19.7155i			1.85468i	0.374218	+	0.927341i	$0.377911\pi$
$113$			19.7155i			1.85468i	−0.374218	+	0.927341i	$0.622089\pi$
$114$		0			0
$115$			2.84053i			0.264881i
$116$		0			0
$117$	4.43435	−	1.82159i	0.409956	−	0.168406i
$118$		0			0
$119$	−0.506137			−0.0463975
$120$		0			0
$121$	9.16334			0.833031
$122$		0			0
$123$	1.00802	+	5.10669i	0.0908904	+	0.460455i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			3.05575i			0.271154i	0.990767	+	0.135577i	$0.0432888\pi$
$127$			3.05575i			0.271154i	−0.990767	+	0.135577i	$0.956711\pi$
$128$		0			0
$129$	−0.440738	−	2.23280i	−0.0388048	−	0.196587i
$130$		0			0
$131$	−2.61320			−0.228317			−0.114158	−	0.993463i	$-0.536417\pi$
$131$	−2.61320			−0.228317			−0.114158	+	0.993463i	$0.536417\pi$
$132$		0			0
$133$	−0.377701			−0.0327509
$134$		0			0
$135$	−2.86785	−	4.33306i	−0.246825	−	0.372931i
$136$		0			0
$137$			9.11266i			0.778547i	0.921122	+	0.389273i	$0.127274\pi$
$137$			9.11266i			0.778547i	−0.921122	+	0.389273i	$0.872726\pi$
$138$		0			0
$139$		−	12.3984i		−	1.05162i	−0.850603	−	0.525808i	$-0.823763\pi$
$139$		−	12.3984i		−	1.05162i	0.850603	−	0.525808i	$-0.176237\pi$
$140$		0			0
$141$	−14.3293	+	2.82850i	−1.20674	+	0.238202i
$142$		0			0
$143$	7.17548			0.600044
$144$		0			0
$145$	1.04311			0.0866255
$146$		0			0
$147$	−1.69926	+	0.335422i	−0.140153	+	0.0276651i
$148$		0			0
$149$			16.0780i			1.31716i	0.752510	+	0.658581i	$0.228843\pi$
$149$			16.0780i			1.31716i	−0.752510	+	0.658581i	$0.771157\pi$
$150$		0			0
$151$		−	10.8950i		−	0.886622i	−0.896368	−	0.443311i	$-0.853804\pi$
$151$		−	10.8950i		−	0.886622i	0.896368	−	0.443311i	$-0.146196\pi$
$152$		0			0
$153$	−0.576966	−	1.40452i	−0.0466450	−	0.113549i
$154$		0			0
$155$	−0.219074			−0.0175965
$156$		0			0
$157$	15.8489			1.26488			0.632442	−	0.774608i	$-0.282053\pi$
$157$	15.8489			1.26488			0.632442	+	0.774608i	$0.282053\pi$
$158$		0			0
$159$	0.684681	+	3.46862i	0.0542987	+	0.275080i
$160$		0			0
$161$			2.84053i			0.223865i
$162$		0			0
$163$			23.6635i			1.85346i	0.375722	+	0.926732i	$0.377395\pi$
$163$			23.6635i			1.85346i	−0.375722	+	0.926732i	$0.622605\pi$
$164$		0			0
$165$	−1.50617	−	7.63030i	−0.117255	−	0.594018i
$166$		0			0
$167$	−22.8687			−1.76964			−0.884818	−	0.465937i	$-0.845717\pi$
$167$	−22.8687			−1.76964			−0.884818	+	0.465937i	$0.845717\pi$
$168$		0			0
$169$	−10.4465			−0.803575
$170$		0			0
$171$	−0.430557	−	1.04812i	−0.0329255	−	0.0801514i
$172$		0			0
$173$			7.58403i			0.576603i	0.957540	+	0.288302i	$0.0930906\pi$
$173$			7.58403i			0.576603i	−0.957540	+	0.288302i	$0.906909\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	11.3126	−	2.23303i	0.850311	−	0.167845i
$178$		0			0
$179$	6.83244			0.510681			0.255340	−	0.966851i	$-0.417812\pi$
$179$	6.83244			0.510681			0.255340	+	0.966851i	$0.417812\pi$
$180$		0			0
$181$	−5.36693			−0.398921			−0.199460	−	0.979906i	$-0.563919\pi$
$181$	−5.36693			−0.398921			−0.199460	+	0.979906i	$0.563919\pi$
$182$		0			0
$183$	12.5762	−	2.48246i	0.929662	−	0.183508i
$184$		0			0
$185$			2.40449i			0.176782i
$186$		0			0
$187$		−	2.27274i		−	0.166199i
$188$		0			0
$189$	−2.86785	−	4.33306i	−0.208605	−	0.315184i
$190$		0			0
$191$	0.142457			0.0103078			0.00515391	−	0.999987i	$-0.498359\pi$
$191$	0.142457			0.0103078			0.00515391	+	0.999987i	$0.498359\pi$
$192$		0			0
$193$	−10.1022			−0.727176			−0.363588	−	0.931560i	$-0.618448\pi$
$193$	−10.1022			−0.727176			−0.363588	+	0.931560i	$0.618448\pi$
$194$		0			0
$195$	−0.535996	−	2.71538i	−0.0383834	−	0.194452i
$196$		0			0
$197$		−	10.1619i		−	0.724008i	−0.932176	−	0.362004i	$-0.882093\pi$
$197$		−	10.1619i		−	0.724008i	0.932176	−	0.362004i	$-0.117907\pi$
$198$		0			0
$199$			20.0613i			1.42211i	0.703137	+	0.711054i	$0.251782\pi$
$199$			20.0613i			1.42211i	−0.703137	+	0.711054i	$0.748218\pi$
$200$		0			0
$201$	−4.00519	−	20.2904i	−0.282504	−	1.43118i
$202$		0			0
$203$	1.04311			0.0732119
$204$		0			0
$205$	3.00524			0.209895
$206$		0			0
$207$	−7.88243	+	3.23804i	−0.547867	+	0.225059i
$208$		0			0
$209$		−	1.69602i		−	0.117316i
$210$		0			0
$211$			9.77698i			0.673075i	0.941670	+	0.336537i	$0.109256\pi$
$211$			9.77698i			0.673075i	−0.941670	+	0.336537i	$0.890744\pi$
$212$		0			0
$213$	6.61971	−	1.30668i	0.453575	−	0.0895324i
$214$		0			0
$215$	−1.31398			−0.0896127
$216$		0			0
$217$	−0.219074			−0.0148717
$218$		0			0
$219$	20.9843	−	4.14215i	1.41799	−	0.279901i
$220$		0			0
$221$		−	0.808794i		−	0.0544054i
$222$		0			0
$223$			3.77035i			0.252481i	0.992000	+	0.126241i	$0.0402912\pi$
$223$			3.77035i			0.252481i	−0.992000	+	0.126241i	$0.959709\pi$
$224$		0			0
$225$	−2.77498	+	1.13994i	−0.184999	+	0.0759960i
$226$		0			0
$227$	−7.03702			−0.467064			−0.233532	−	0.972349i	$-0.575028\pi$
$227$	−7.03702			−0.467064			−0.233532	+	0.972349i	$0.575028\pi$
$228$		0			0
$229$	−20.3487			−1.34468			−0.672341	−	0.740242i	$-0.734711\pi$
$229$	−20.3487			−1.34468			−0.672341	+	0.740242i	$0.734711\pi$
$230$		0			0
$231$	−1.50617	−	7.63030i	−0.0990984	−	0.502037i
$232$		0			0
$233$			4.86976i			0.319029i	0.987196	+	0.159514i	$0.0509929\pi$
$233$			4.86976i			0.319029i	−0.987196	+	0.159514i	$0.949007\pi$
$234$		0			0
$235$			8.43265i			0.550085i
$236$		0			0
$237$	−5.11767	−	25.9263i	−0.332428	−	1.68410i
$238$		0			0
$239$	27.6916			1.79122			0.895611	−	0.444838i	$-0.146739\pi$
$239$	27.6916			1.79122			0.895611	+	0.444838i	$0.146739\pi$
$240$		0			0
$241$	24.4442			1.57459			0.787294	−	0.616578i	$-0.211482\pi$
$241$	24.4442			1.57459			0.787294	+	0.616578i	$0.211482\pi$
$242$		0			0
$243$	8.75501	−	12.8977i	0.561634	−	0.827386i
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$		−	0.603557i		−	0.0384034i
$248$		0			0
$249$	−1.18554	+	0.234018i	−0.0751308	+	0.0148303i
$250$		0			0
$251$	−18.3504			−1.15827			−0.579133	−	0.815233i	$-0.696609\pi$
$251$	−18.3504			−1.15827			−0.579133	+	0.815233i	$0.696609\pi$
$252$		0			0
$253$	−12.7550			−0.801901
$254$		0			0
$255$	−0.860060	+	0.169770i	−0.0538591	+	0.0106314i
$256$		0			0
$257$		−	11.2438i		−	0.701371i	−0.936493	−	0.350685i	$-0.885949\pi$
$257$		−	11.2438i		−	0.701371i	0.936493	−	0.350685i	$-0.114051\pi$
$258$		0			0
$259$			2.40449i			0.149408i
$260$		0			0
$261$	1.18908	+	2.89461i	0.0736023	+	0.179172i
$262$		0			0
$263$	2.94134			0.181371			0.0906853	−	0.995880i	$-0.471094\pi$
$263$	2.94134			0.181371			0.0906853	+	0.995880i	$0.471094\pi$
$264$		0			0
$265$	2.04125			0.125393
$266$		0			0
$267$	0.772629	+	3.91417i	0.0472841	+	0.239543i
$268$		0			0
$269$			3.79117i			0.231152i	0.993299	+	0.115576i	$0.0368714\pi$
$269$			3.79117i			0.231152i	−0.993299	+	0.115576i	$0.963129\pi$
$270$		0			0
$271$			17.9389i			1.08971i	0.838530	+	0.544855i	$0.183415\pi$
$271$			17.9389i			1.08971i	−0.838530	+	0.544855i	$0.816585\pi$
$272$		0			0
$273$	−0.535996	−	2.71538i	−0.0324399	−	0.164342i
$274$		0			0
$275$	−4.49036			−0.270779
$276$		0			0
$277$	−22.1475			−1.33072			−0.665358	−	0.746525i	$-0.731721\pi$
$277$	−22.1475			−1.33072			−0.665358	+	0.746525i	$0.731721\pi$
$278$		0			0
$279$	−0.249732	−	0.607928i	−0.0149510	−	0.0363957i
$280$		0			0
$281$			16.4121i			0.979065i	0.871985	+	0.489532i	$0.162832\pi$
$281$			16.4121i			0.979065i	−0.871985	+	0.489532i	$0.837168\pi$
$282$		0			0
$283$			16.4262i			0.976436i	0.872722	+	0.488218i	$0.162353\pi$
$283$			16.4262i			0.976436i	−0.872722	+	0.488218i	$0.837647\pi$
$284$		0			0
$285$	−0.641814	+	0.126689i	−0.0380178	+	0.00750443i
$286$		0			0
$287$	3.00524			0.177394
$288$		0			0
$289$	16.7438			0.984931
$290$		0			0
$291$	9.41305	−	1.85807i	0.551803	−	0.108922i
$292$		0			0
$293$		−	22.3301i		−	1.30454i	−0.757989	−	0.652268i	$-0.773818\pi$
$293$		−	22.3301i		−	1.30454i	0.757989	−	0.652268i	$-0.226182\pi$
$294$		0			0
$295$		−	6.65739i		−	0.387608i
$296$		0			0
$297$	19.4570	−	12.8777i	1.12901	−	0.747238i
$298$		0			0
$299$	−4.53910			−0.262503
$300$		0			0
$301$	−1.31398			−0.0757366
$302$		0			0
$303$	4.84619	+	24.5510i	0.278406	+	1.41042i
$304$		0			0
$305$		−	7.40099i		−	0.423780i
$306$		0			0
$307$		−	7.60545i		−	0.434066i	−0.976164	−	0.217033i	$-0.930362\pi$
$307$		−	7.60545i		−	0.434066i	0.976164	−	0.217033i	$-0.0696379\pi$
$308$		0			0
$309$	−3.07969	−	15.6018i	−0.175197	−	0.887556i
$310$		0			0
$311$	1.48281			0.0840822			0.0420411	−	0.999116i	$-0.486614\pi$
$311$	1.48281			0.0840822			0.0420411	+	0.999116i	$0.486614\pi$
$312$		0			0
$313$	25.4496			1.43849			0.719247	−	0.694754i	$-0.244487\pi$
$313$	25.4496			1.43849			0.719247	+	0.694754i	$0.244487\pi$
$314$		0			0
$315$	−2.77498	+	1.13994i	−0.156353	+	0.0642283i
$316$		0			0
$317$		−	11.2766i		−	0.633354i	−0.948533	−	0.316677i	$-0.897433\pi$
$317$		−	11.2766i		−	0.633354i	0.948533	−	0.316677i	$-0.102567\pi$
$318$		0			0
$319$			4.68394i			0.262250i
$320$		0			0
$321$	−26.1982	+	5.17134i	−1.46224	+	0.288636i
$322$		0			0
$323$	−0.191169			−0.0106369
$324$		0			0
$325$	−1.59797			−0.0886397
$326$		0			0
$327$	−26.0331	+	5.13875i	−1.43964	+	0.284174i
$328$		0			0
$329$			8.43265i			0.464907i
$330$		0			0
$331$			22.1430i			1.21709i	0.793519	+	0.608545i	$0.208247\pi$
$331$			22.1430i			1.21709i	−0.793519	+	0.608545i	$0.791753\pi$
$332$		0			0
$333$	−6.67243	+	2.74098i	−0.365647	+	0.150205i
$334$		0			0
$335$	−11.9407			−0.652392
$336$		0			0
$337$	−28.1418			−1.53298			−0.766491	−	0.642255i	$-0.777999\pi$
$337$	−28.1418			−1.53298			−0.766491	+	0.642255i	$0.777999\pi$
$338$		0			0
$339$	6.61302	+	33.5019i	0.359170	+	1.81957i
$340$		0			0
$341$		−	0.983723i		−	0.0532716i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	0.952777	+	4.82681i	0.0512958	+	0.259867i
$346$		0			0
$347$	−25.5141			−1.36967			−0.684835	−	0.728698i	$-0.740126\pi$
$347$	−25.5141			−1.36967			−0.684835	+	0.728698i	$0.740126\pi$
$348$		0			0
$349$	3.55066			0.190063			0.0950313	−	0.995474i	$-0.469705\pi$
$349$	3.55066			0.190063			0.0950313	+	0.995474i	$0.469705\pi$
$350$		0			0
$351$	6.92413	−	4.58275i	0.369582	−	0.244609i
$352$		0			0
$353$			33.0204i			1.75750i	0.477283	+	0.878750i	$0.341622\pi$
$353$			33.0204i			1.75750i	−0.477283	+	0.878750i	$0.658378\pi$
$354$		0			0
$355$		−	3.89564i		−	0.206759i
$356$		0			0
$357$	−0.860060	+	0.169770i	−0.0455192	+	0.00898516i
$358$		0			0
$359$	−19.4832			−1.02828			−0.514142	−	0.857705i	$-0.671890\pi$
$359$	−19.4832			−1.02828			−0.514142	+	0.857705i	$0.671890\pi$
$360$		0			0
$361$	18.8573			0.992492
$362$		0			0
$363$	15.5709	−	3.07359i	0.817262	−	0.161321i
$364$		0			0
$365$		−	12.3491i		−	0.646380i
$366$		0			0
$367$			12.9394i			0.675432i	0.941248	+	0.337716i	$0.109654\pi$
$367$			12.9394i			0.675432i	−0.941248	+	0.337716i	$0.890346\pi$
$368$		0			0
$369$	3.42579	+	8.33949i	0.178340	+	0.434137i
$370$		0			0
$371$	2.04125			0.105977
$372$		0			0
$373$	−14.7091			−0.761609			−0.380805	−	0.924656i	$-0.624353\pi$
$373$	−14.7091			−0.761609			−0.380805	+	0.924656i	$0.624353\pi$
$374$		0			0
$375$	0.335422	+	1.69926i	0.0173211	+	0.0877495i
$376$		0			0
$377$			1.66686i			0.0858477i
$378$		0			0
$379$		−	2.04347i		−	0.104966i	−0.998622	−	0.0524831i	$-0.983286\pi$
$379$		−	2.04347i		−	0.104966i	0.998622	−	0.0524831i	$-0.0167136\pi$
$380$		0			0
$381$	1.02497	+	5.19252i	0.0525106	+	0.266021i
$382$		0			0
$383$	19.0507			0.973447			0.486723	−	0.873556i	$-0.338192\pi$
$383$	19.0507			0.973447			0.486723	+	0.873556i	$0.338192\pi$
$384$		0			0
$385$	−4.49036			−0.228850
$386$		0			0
$387$	−1.49786	−	3.64627i	−0.0761404	−	0.185351i
$388$		0			0
$389$			20.2842i			1.02845i	0.857656	+	0.514224i	$0.171920\pi$
$389$			20.2842i			1.02845i	−0.857656	+	0.514224i	$0.828080\pi$
$390$		0			0
$391$			1.43770i			0.0727076i
$392$		0			0
$393$	−4.44052	+	0.876525i	−0.223994	+	0.0442149i
$394$		0			0
$395$	−15.2574			−0.767683
$396$		0			0
$397$	−17.5290			−0.879756			−0.439878	−	0.898057i	$-0.644978\pi$
$397$	−17.5290			−0.879756			−0.439878	+	0.898057i	$0.644978\pi$
$398$		0			0
$399$	−0.641814	+	0.126689i	−0.0321309	+	0.00634240i
$400$		0			0
$401$		−	24.2810i		−	1.21253i	−0.795261	−	0.606267i	$-0.792666\pi$
$401$		−	24.2810i		−	1.21253i	0.795261	−	0.606267i	$-0.207334\pi$
$402$		0			0
$403$		−	0.350075i		−	0.0174385i
$404$		0			0
$405$	−6.32663	−	6.40107i	−0.314373	−	0.318072i
$406$		0			0
$407$	−10.7970			−0.535190
$408$		0			0
$409$	25.9536			1.28332			0.641662	−	0.766988i	$-0.278245\pi$
$409$	25.9536			1.28332			0.641662	+	0.766988i	$0.278245\pi$
$410$		0			0
$411$	3.05659	+	15.4848i	0.150770	+	0.763808i
$412$		0			0
$413$		−	6.65739i		−	0.327589i
$414$		0			0
$415$			0.697682i			0.0342479i
$416$		0			0
$417$	−4.15868	−	21.0681i	−0.203652	−	1.03171i
$418$		0			0
$419$	22.8542			1.11650			0.558250	−	0.829673i	$-0.311473\pi$
$419$	22.8542			1.11650			0.558250	+	0.829673i	$0.311473\pi$
$420$		0			0
$421$	−37.7870			−1.84162			−0.920812	−	0.390007i	$-0.872473\pi$
$421$	−37.7870			−1.84162			−0.920812	+	0.390007i	$0.872473\pi$
$422$		0			0
$423$	−23.4005	+	9.61271i	−1.13777	+	0.467386i
$424$		0			0
$425$			0.506137i			0.0245513i
$426$		0			0
$427$		−	7.40099i		−	0.358159i
$428$		0			0
$429$	12.1930	−	2.40681i	0.588685	−	0.116202i
$430$		0			0
$431$	17.3083			0.833712			0.416856	−	0.908973i	$-0.363132\pi$
$431$	17.3083			0.833712			0.416856	+	0.908973i	$0.363132\pi$
$432$		0			0
$433$	−17.2683			−0.829863			−0.414932	−	0.909853i	$-0.636194\pi$
$433$	−17.2683			−0.829863			−0.414932	+	0.909853i	$0.636194\pi$
$434$		0			0
$435$	1.77252	−	0.349882i	0.0849856	−	0.0167755i
$436$		0			0
$437$			1.07287i			0.0513225i
$438$		0			0
$439$			38.0720i			1.81708i	0.417803	+	0.908538i	$0.362800\pi$
$439$			38.0720i			1.81708i	−0.417803	+	0.908538i	$0.637200\pi$
$440$		0			0
$441$	−2.77498	+	1.13994i	−0.132142	+	0.0542829i
$442$		0			0
$443$	−27.6089			−1.31174			−0.655870	−	0.754874i	$-0.727698\pi$
$443$	−27.6089			−1.31174			−0.655870	+	0.754874i	$0.727698\pi$
$444$		0			0
$445$	2.30345			0.109194
$446$		0			0
$447$	5.39292	+	27.3207i	0.255076	+	1.29223i
$448$		0			0
$449$		−	11.2427i		−	0.530576i	−0.964169	−	0.265288i	$-0.914533\pi$
$449$		−	11.2427i		−	0.530576i	0.964169	−	0.265288i	$-0.0854670\pi$
$450$		0			0
$451$			13.4946i			0.635436i
$452$		0			0
$453$	−3.65442	−	18.5135i	−0.171700	−	0.869838i
$454$		0			0
$455$	−1.59797			−0.0749142
$456$		0			0
$457$	12.4653			0.583100			0.291550	−	0.956556i	$-0.405829\pi$
$457$	12.4653			0.583100			0.291550	+	0.956556i	$0.405829\pi$
$458$		0			0
$459$	−1.45152	−	2.19313i	−0.0677514	−	0.102366i
$460$		0			0
$461$			2.78102i			0.129525i	0.997901	+	0.0647625i	$0.0206290\pi$
$461$			2.78102i			0.129525i	−0.997901	+	0.0647625i	$0.979371\pi$
$462$		0			0
$463$		−	5.10682i		−	0.237334i	−0.992934	−	0.118667i	$-0.962138\pi$
$463$		−	5.10682i		−	0.237334i	0.992934	−	0.118667i	$-0.0378621\pi$
$464$		0			0
$465$	−0.372265	+	0.0734824i	−0.0172634	+	0.00340766i
$466$		0			0
$467$	−5.70181			−0.263848			−0.131924	−	0.991260i	$-0.542116\pi$
$467$	−5.70181			−0.263848			−0.131924	+	0.991260i	$0.542116\pi$
$468$		0			0
$469$	−11.9407			−0.551372
$470$		0			0
$471$	26.9315	−	5.31609i	1.24094	−	0.244952i
$472$		0			0
$473$		−	5.90025i		−	0.271294i
$474$		0			0
$475$			0.377701i			0.0173301i
$476$		0			0
$477$	2.32691	+	5.66444i	0.106542	+	0.259357i
$478$		0			0
$479$	28.8974			1.32036			0.660179	−	0.751109i	$-0.270481\pi$
$479$	28.8974			1.32036			0.660179	+	0.751109i	$0.270481\pi$
$480$		0			0
$481$	−3.84232			−0.175195
$482$		0			0
$483$	0.952777	+	4.82681i	0.0433529	+	0.219627i
$484$		0			0
$485$		−	5.53949i		−	0.251535i
$486$		0			0
$487$		−	27.1360i		−	1.22965i	−0.788663	−	0.614826i	$-0.789226\pi$
$487$		−	27.1360i		−	1.22965i	0.788663	−	0.614826i	$-0.210774\pi$
$488$		0			0
$489$	7.93724	+	40.2104i	0.358935	+	1.81838i
$490$		0			0
$491$	−34.3384			−1.54967			−0.774835	−	0.632164i	$-0.782167\pi$
$491$	−34.3384			−1.54967			−0.774835	+	0.632164i	$0.782167\pi$
$492$		0			0
$493$	0.527956			0.0237780
$494$		0			0
$495$	−5.11874	−	12.4607i	−0.230070	−	0.560066i
$496$		0			0
$497$		−	3.89564i		−	0.174743i
$498$		0			0
$499$			20.3244i			0.909843i	0.890532	+	0.454921i	$0.150333\pi$
$499$			20.3244i			0.909843i	−0.890532	+	0.454921i	$0.849667\pi$
$500$		0			0
$501$	−38.8600	+	7.67068i	−1.73614	+	0.342700i
$502$		0			0
$503$	25.6193			1.14231			0.571153	−	0.820843i	$-0.306496\pi$
$503$	25.6193			1.14231			0.571153	+	0.820843i	$0.306496\pi$
$504$		0			0
$505$	14.4480			0.642929
$506$		0			0
$507$	−17.7513	+	3.50398i	−0.788363	+	0.155617i
$508$		0			0
$509$			43.0798i			1.90948i	0.297441	+	0.954740i	$0.403867\pi$
$509$			43.0798i			1.90948i	−0.297441	+	0.954740i	$0.596133\pi$
$510$		0			0
$511$		−	12.3491i		−	0.546291i
$512$		0			0
$513$	−1.08319	−	1.63660i	−0.0478240	−	0.0722579i
$514$		0			0
$515$	−9.18152			−0.404586
$516$		0			0
$517$	−37.8656			−1.66533
$518$		0			0
$519$	2.54385	+	12.8873i	0.111663	+	0.565688i
$520$		0			0
$521$			14.0521i			0.615634i	0.951446	+	0.307817i	$0.0995984\pi$
$521$			14.0521i			0.615634i	−0.951446	+	0.307817i	$0.900402\pi$
$522$		0			0
$523$		−	42.0750i		−	1.83981i	−0.392142	−	0.919905i	$-0.628266\pi$
$523$		−	42.0750i		−	1.83981i	0.392142	−	0.919905i	$-0.371734\pi$
$524$		0			0
$525$	0.335422	+	1.69926i	0.0146390	+	0.0741619i
$526$		0			0
$527$	−0.110882			−0.00483008
$528$		0			0
$529$	−14.9314			−0.649190
$530$		0			0
$531$	18.4741	−	7.58902i	0.801710	−	0.329336i
$532$		0			0
$533$			4.80229i			0.208010i
$534$		0			0
$535$			15.4174i			0.666553i
$536$		0			0
$537$	11.6101	−	2.29175i	0.501013	−	0.0988964i
$538$		0			0
$539$	−4.49036			−0.193414
$540$		0			0
$541$	29.8322			1.28259			0.641293	−	0.767296i	$-0.278398\pi$
$541$	29.8322			1.28259			0.641293	+	0.767296i	$0.278398\pi$
$542$		0			0
$543$	−9.11982	+	1.80019i	−0.391369	+	0.0772534i
$544$		0			0
$545$			15.3203i			0.656248i
$546$		0			0
$547$		−	33.0089i		−	1.41136i	−0.708531	−	0.705679i	$-0.750642\pi$
$547$		−	33.0089i		−	1.41136i	0.708531	−	0.705679i	$-0.249358\pi$
$548$		0			0
$549$	20.5376	−	8.43669i	0.876525	−	0.360069i
$550$		0			0
$551$	0.393984			0.0167843
$552$		0			0
$553$	−15.2574			−0.648811
$554$		0			0
$555$	0.806520	+	4.08587i	0.0342349	+	0.173435i
$556$		0			0
$557$			24.2862i			1.02904i	0.857479	+	0.514519i	$0.172029\pi$
$557$			24.2862i			1.02904i	−0.857479	+	0.514519i	$0.827971\pi$
$558$		0			0
$559$		−	2.09971i		−	0.0888081i
$560$		0			0
$561$	−0.762327	−	3.86198i	−0.0321855	−	0.163053i
$562$		0			0
$563$	−3.80335			−0.160292			−0.0801461	−	0.996783i	$-0.525539\pi$
$563$	−3.80335			−0.160292			−0.0801461	+	0.996783i	$0.525539\pi$
$564$		0			0
$565$	19.7155			0.829439
$566$		0			0
$567$	−6.32663	−	6.40107i	−0.265694	−	0.268820i
$568$		0			0
$569$			22.4847i			0.942608i	0.881971	+	0.471304i	$0.156217\pi$
$569$			22.4847i			0.942608i	−0.881971	+	0.471304i	$0.843783\pi$
$570$		0			0
$571$			5.59266i			0.234045i	0.993129	+	0.117023i	$0.0373350\pi$
$571$			5.59266i			0.234045i	−0.993129	+	0.117023i	$0.962665\pi$
$572$		0			0
$573$	0.242072	−	0.0477832i	0.0101127	−	0.00199617i
$574$		0			0
$575$	2.84053			0.118458
$576$		0			0
$577$	−4.75638			−0.198011			−0.0990053	−	0.995087i	$-0.531566\pi$
$577$	−4.75638			−0.198011			−0.0990053	+	0.995087i	$0.531566\pi$
$578$		0			0
$579$	−17.1664	+	3.38852i	−0.713410	+	0.140822i
$580$		0			0
$581$			0.697682i			0.0289447i
$582$		0			0
$583$			9.16596i			0.379615i
$584$		0			0
$585$	−1.82159	−	4.43435i	−0.0753137	−	0.183338i
$586$		0			0
$587$	−17.0701			−0.704559			−0.352279	−	0.935895i	$-0.614593\pi$
$587$	−17.0701			−0.704559			−0.352279	+	0.935895i	$0.614593\pi$
$588$		0			0
$589$	−0.0827447			−0.00340944
$590$		0			0
$591$	−3.40854	−	17.2678i	−0.140209	−	0.710303i
$592$		0			0
$593$			24.5169i			1.00679i	0.864057	+	0.503394i	$0.167915\pi$
$593$			24.5169i			1.00679i	−0.864057	+	0.503394i	$0.832085\pi$
$594$		0			0
$595$			0.506137i			0.0207496i
$596$		0			0
$597$	6.72900	+	34.0894i	0.275400	+	1.39519i
$598$		0			0
$599$	32.3347			1.32116			0.660580	−	0.750755i	$-0.270310\pi$
$599$	32.3347			1.32116			0.660580	+	0.750755i	$0.270310\pi$
$600$		0			0
$601$	−0.757011			−0.0308791			−0.0154396	−	0.999881i	$-0.504915\pi$
$601$	−0.757011			−0.0308791			−0.0154396	+	0.999881i	$0.504915\pi$
$602$		0			0
$603$	−13.6117	−	33.1354i	−0.554312	−	1.34938i
$604$		0			0
$605$		−	9.16334i		−	0.372543i
$606$		0			0
$607$			32.9204i			1.33620i	0.744072	+	0.668100i	$0.232892\pi$
$607$			32.9204i			1.33620i	−0.744072	+	0.668100i	$0.767108\pi$
$608$		0			0
$609$	1.77252	−	0.349882i	0.0718260	−	0.0141779i
$610$		0			0
$611$	−13.4752			−0.545146
$612$		0			0
$613$	−28.5463			−1.15297			−0.576487	−	0.817106i	$-0.695577\pi$
$613$	−28.5463			−1.15297			−0.576487	+	0.817106i	$0.695577\pi$
$614$		0			0
$615$	5.10669	−	1.00802i	0.205922	−	0.0406474i
$616$		0			0
$617$		−	25.9719i		−	1.04559i	−0.852458	−	0.522795i	$-0.824889\pi$
$617$		−	25.9719i		−	1.04559i	0.852458	−	0.522795i	$-0.175111\pi$
$618$		0			0
$619$		−	28.0578i		−	1.12774i	−0.825864	−	0.563869i	$-0.809312\pi$
$619$		−	28.0578i		−	1.12774i	0.825864	−	0.563869i	$-0.190688\pi$
$620$		0			0
$621$	−12.3082	+	8.14621i	−0.493912	+	0.326896i
$622$		0			0
$623$	2.30345			0.0922859
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.568881	−	2.88198i	−0.0227189	−	0.115095i
$628$		0			0
$629$			1.21700i			0.0485251i
$630$		0			0
$631$			3.49217i			0.139021i	0.997581	+	0.0695105i	$0.0221437\pi$
$631$			3.49217i			0.139021i	−0.997581	+	0.0695105i	$0.977856\pi$
$632$		0			0
$633$	3.27941	+	16.6136i	0.130345	+	0.660333i
$634$		0			0
$635$	3.05575			0.121264
$636$		0			0
$637$	−1.59797			−0.0633140
$638$		0			0
$639$	10.8103	−	4.44079i	0.427650	−	0.175675i
$640$		0			0
$641$		−	18.2734i		−	0.721755i	−0.932613	−	0.360877i	$-0.882477\pi$
$641$		−	18.2734i		−	0.721755i	0.932613	−	0.360877i	$-0.117523\pi$
$642$		0			0
$643$		−	20.1319i		−	0.793926i	−0.917835	−	0.396963i	$-0.870064\pi$
$643$		−	20.1319i		−	0.793926i	0.917835	−	0.396963i	$-0.129936\pi$
$644$		0			0
$645$	−2.23280	+	0.440738i	−0.0879163	+	0.0173540i
$646$		0			0
$647$	−7.95277			−0.312655			−0.156328	−	0.987705i	$-0.549966\pi$
$647$	−7.95277			−0.312655			−0.156328	+	0.987705i	$0.549966\pi$
$648$		0			0
$649$	29.8941			1.17345
$650$		0			0
$651$	−0.372265	+	0.0734824i	−0.0145902	+	0.00288000i
$652$		0			0
$653$		−	42.1791i		−	1.65059i	−0.564699	−	0.825297i	$-0.691008\pi$
$653$		−	42.1791i		−	1.65059i	0.564699	−	0.825297i	$-0.308992\pi$
$654$		0			0
$655$			2.61320i			0.102106i
$656$		0			0
$657$	34.2685	−	14.0772i	1.33694	−	0.549204i
$658$		0			0
$659$	−9.31028			−0.362677			−0.181339	−	0.983421i	$-0.558043\pi$
$659$	−9.31028			−0.362677			−0.181339	+	0.983421i	$0.558043\pi$
$660$		0			0
$661$	29.9209			1.16379			0.581895	−	0.813264i	$-0.302312\pi$
$661$	29.9209			1.16379			0.581895	+	0.813264i	$0.302312\pi$
$662$		0			0
$663$	−0.271287	−	1.37435i	−0.0105359	−	0.0533755i
$664$		0			0
$665$			0.377701i			0.0146466i
$666$		0			0
$667$		−	2.96298i		−	0.114727i
$668$		0			0
$669$	1.26466	+	6.40681i	0.0488945	+	0.247702i
$670$		0			0
$671$	33.2331			1.28295
$672$		0			0
$673$	−25.2562			−0.973555			−0.486777	−	0.873526i	$-0.661828\pi$
$673$	−25.2562			−0.973555			−0.486777	+	0.873526i	$0.661828\pi$
$674$		0			0
$675$	−4.33306	+	2.86785i	−0.166780	+	0.110384i
$676$		0			0
$677$			12.9463i			0.497568i	0.968559	+	0.248784i	$0.0800310\pi$
$677$			12.9463i			0.497568i	−0.968559	+	0.248784i	$0.919969\pi$
$678$		0			0
$679$		−	5.53949i		−	0.212586i
$680$		0			0
$681$	−11.9577	+	2.36037i	−0.458222	+	0.0904497i
$682$		0			0
$683$	8.94085			0.342112			0.171056	−	0.985261i	$-0.445282\pi$
$683$	8.94085			0.342112			0.171056	+	0.985261i	$0.445282\pi$
$684$		0			0
$685$	9.11266			0.348177
$686$		0			0
$687$	−34.5778	+	6.82541i	−1.31923	+	0.260406i
$688$		0			0
$689$			3.26187i			0.124267i
$690$		0			0
$691$			41.0870i			1.56302i	0.623890	+	0.781512i	$0.285551\pi$
$691$			41.0870i			1.56302i	−0.623890	+	0.781512i	$0.714449\pi$
$692$		0			0
$693$	−5.11874	−	12.4607i	−0.194445	−	0.473342i
$694$		0			0
$695$	−12.3984			−0.470297
$696$		0			0
$697$	1.52106			0.0576144
$698$		0			0
$699$	1.63343	+	8.27501i	0.0617819	+	0.312989i
$700$		0			0
$701$		−	8.30601i		−	0.313714i	−0.987621	−	0.156857i	$-0.949864\pi$
$701$		−	8.30601i		−	0.313714i	0.987621	−	0.156857i	$-0.0501361\pi$
$702$		0			0
$703$			0.908181i			0.0342527i
$704$		0			0
$705$	2.82850	+	14.3293i	0.106527	+	0.539672i
$706$		0			0
$707$	14.4480			0.543374
$708$		0			0
$709$	−9.90380			−0.371945			−0.185972	−	0.982555i	$-0.559544\pi$
$709$	−9.90380			−0.371945			−0.185972	+	0.982555i	$0.559544\pi$
$710$		0			0
$711$	−17.3925	−	42.3391i	−0.652271	−	1.58784i
$712$		0			0
$713$			0.622288i			0.0233049i
$714$		0			0
$715$		−	7.17548i		−	0.268348i
$716$		0			0
$717$	47.0553	−	9.28838i	1.75731	−	0.346881i
$718$		0			0
$719$	17.7191			0.660812			0.330406	−	0.943839i	$-0.392814\pi$
$719$	17.7191			0.660812			0.330406	+	0.943839i	$0.392814\pi$
$720$		0			0
$721$	−9.18152			−0.341938
$722$		0			0
$723$	41.5370	−	8.19911i	1.54478	−	0.304928i
$724$		0			0
$725$		−	1.04311i		−	0.0387401i
$726$		0			0
$727$		−	25.5246i		−	0.946655i	−0.880886	−	0.473328i	$-0.843053\pi$
$727$		−	25.5246i		−	0.946655i	0.880886	−	0.473328i	$-0.156947\pi$
$728$		0			0
$729$	10.5509	−	24.8531i	0.390774	−	0.920487i
$730$		0			0
$731$	−0.665055			−0.0245979
$732$		0			0
$733$	34.9292			1.29014			0.645070	−	0.764124i	$-0.276828\pi$
$733$	34.9292			1.29014			0.645070	+	0.764124i	$0.276828\pi$
$734$		0			0
$735$	0.335422	+	1.69926i	0.0123722	+	0.0626782i
$736$		0			0
$737$		−	53.6182i		−	1.97505i
$738$		0			0
$739$		−	1.68178i		−	0.0618652i	−0.999521	−	0.0309326i	$-0.990152\pi$
$739$		−	1.68178i		−	0.0618652i	0.999521	−	0.0309326i	$-0.00984772\pi$
$740$		0			0
$741$	−0.202446	−	1.02560i	−0.00743705	−	0.0376764i
$742$		0			0
$743$	−22.5532			−0.827397			−0.413698	−	0.910414i	$-0.635763\pi$
$743$	−22.5532			−0.827397			−0.413698	+	0.910414i	$0.635763\pi$
$744$		0			0
$745$	16.0780			0.589053
$746$		0			0
$747$	−1.93606	+	0.795315i	−0.0708366	+	0.0290991i
$748$		0			0
$749$			15.4174i			0.563340i
$750$		0			0
$751$		−	40.0442i		−	1.46123i	−0.682789	−	0.730616i	$-0.739233\pi$
$751$		−	40.0442i		−	1.46123i	0.682789	−	0.730616i	$-0.260767\pi$
$752$		0			0
$753$	−31.1821	+	6.15513i	−1.13634	+	0.224305i
$754$		0			0
$755$	−10.8950			−0.396509
$756$		0			0
$757$	16.1907			0.588459			0.294230	−	0.955735i	$-0.404937\pi$
$757$	16.1907			0.588459			0.294230	+	0.955735i	$0.404937\pi$
$758$		0			0
$759$	−21.6741	+	4.27831i	−0.786721	+	0.155293i
$760$		0			0
$761$		−	33.4166i		−	1.21135i	−0.795711	−	0.605676i	$-0.792903\pi$
$761$		−	33.4166i		−	1.21135i	0.795711	−	0.605676i	$-0.207097\pi$
$762$		0			0
$763$			15.3203i			0.554631i
$764$		0			0
$765$	−1.40452	+	0.576966i	−0.0507806	+	0.0208603i
$766$		0			0
$767$	10.6383			0.384128
$768$		0			0
$769$	−50.3186			−1.81454			−0.907268	−	0.420553i	$-0.861836\pi$
$769$	−50.3186			−1.81454			−0.907268	+	0.420553i	$0.861836\pi$
$770$		0			0
$771$	−3.77143	−	19.1062i	−0.135825	−	0.688093i
$772$		0			0
$773$		−	30.8312i		−	1.10892i	−0.832210	−	0.554461i	$-0.812925\pi$
$773$		−	30.8312i		−	1.10892i	0.832210	−	0.554461i	$-0.187075\pi$
$774$		0			0
$775$			0.219074i			0.00786938i
$776$		0			0
$777$	0.806520	+	4.08587i	0.0289338	+	0.146580i
$778$		0			0
$779$	1.13508			0.0406686
$780$		0			0
$781$	17.4928			0.625942
$782$		0			0
$783$	2.99148	+	4.51986i	0.106907	+	0.161527i
$784$		0			0
$785$		−	15.8489i		−	0.565673i
$786$		0			0
$787$			6.72636i			0.239769i	0.992788	+	0.119884i	$0.0382524\pi$
$787$			6.72636i			0.239769i	−0.992788	+	0.119884i	$0.961748\pi$
$788$		0			0
$789$	4.99810	−	0.986589i	0.177937	−	0.0351235i
$790$		0			0
$791$	19.7155			0.701004
$792$		0			0
$793$	11.8266			0.419975
$794$		0			0
$795$	3.46862	−	0.684681i	0.123019	−	0.0242831i
$796$		0			0
$797$			0.0624056i			0.00221052i	0.999999	+	0.00110526i	$0.000351815\pi$
$797$			0.0624056i			0.00221052i	−0.999999	+	0.00110526i	$0.999648\pi$
$798$		0			0
$799$			4.26808i			0.150994i
$800$		0			0
$801$	2.62580	+	6.39204i	0.0927780	+	0.225852i
$802$		0			0
$803$	55.4518			1.95685
$804$		0			0
$805$	2.84053			0.100116
$806$		0			0
$807$	1.27164	+	6.44220i	0.0447639	+	0.226776i
$808$		0			0
$809$		−	52.5906i		−	1.84899i	−0.381197	−	0.924494i	$-0.624488\pi$
$809$		−	52.5906i		−	1.84899i	0.381197	−	0.924494i	$-0.375512\pi$
$810$		0			0
$811$			18.6372i			0.654439i	0.944948	+	0.327220i	$0.106112\pi$
$811$			18.6372i			0.654439i	−0.944948	+	0.327220i	$0.893888\pi$
$812$		0			0
$813$	6.01710	+	30.4829i	0.211029	+	1.06908i
$814$		0			0
$815$	23.6635			0.828895
$816$		0			0
$817$	−0.496292			−0.0173631
$818$		0			0
$819$	−1.82159	−	4.43435i	−0.0636517	−	0.154949i
$820$		0			0
$821$			33.7327i			1.17728i	0.808396	+	0.588639i	$0.200336\pi$
$821$			33.7327i			1.17728i	−0.808396	+	0.588639i	$0.799664\pi$
$822$		0			0
$823$		−	5.09754i		−	0.177689i	−0.996046	−	0.0888445i	$-0.971683\pi$
$823$		−	5.09754i		−	0.177689i	0.996046	−	0.0888445i	$-0.0283174\pi$
$824$		0			0
$825$	−7.63030	+	1.50617i	−0.265653	+	0.0524380i
$826$		0			0
$827$	−9.75310			−0.339149			−0.169574	−	0.985517i	$-0.554239\pi$
$827$	−9.75310			−0.339149			−0.169574	+	0.985517i	$0.554239\pi$
$828$		0			0
$829$	47.2069			1.63956			0.819782	−	0.572676i	$-0.194095\pi$
$829$	47.2069			1.63956			0.819782	+	0.572676i	$0.194095\pi$
$830$		0			0
$831$	−37.6344	+	7.42876i	−1.30552	+	0.257701i
$832$		0			0
$833$			0.506137i			0.0175366i
$834$		0			0
$835$			22.8687i			0.791405i
$836$		0			0
$837$	−0.628272	−	0.949264i	−0.0217163	−	0.0328113i
$838$		0			0
$839$	−22.0695			−0.761923			−0.380962	−	0.924591i	$-0.624407\pi$
$839$	−22.0695			−0.761923			−0.380962	+	0.924591i	$0.624407\pi$
$840$		0			0
$841$	27.9119			0.962480
$842$		0			0
$843$	5.50498	+	27.8885i	0.189602	+	0.960530i
$844$		0			0
$845$			10.4465i			0.359370i
$846$		0			0
$847$		−	9.16334i		−	0.314856i
$848$		0			0
$849$	5.50971	+	27.9124i	0.189093	+	0.957952i
$850$		0			0
$851$	6.83005			0.234131
$852$		0			0
$853$	11.1304			0.381099			0.190550	−	0.981678i	$-0.438973\pi$
$853$	11.1304			0.381099			0.190550	+	0.981678i	$0.438973\pi$
$854$		0			0
$855$	−1.04812	+	0.430557i	−0.0358448	+	0.0147247i
$856$		0			0
$857$			31.5913i			1.07914i	0.841941	+	0.539570i	$0.181413\pi$
$857$			31.5913i			1.07914i	−0.841941	+	0.539570i	$0.818587\pi$
$858$		0			0
$859$		−	54.7698i		−	1.86872i	−0.356327	−	0.934361i	$-0.615971\pi$
$859$		−	54.7698i		−	1.86872i	0.356327	−	0.934361i	$-0.384029\pi$
$860$		0			0
$861$	5.10669	−	1.00802i	0.174036	−	0.0343533i
$862$		0			0
$863$	31.2131			1.06251			0.531253	−	0.847213i	$-0.321721\pi$
$863$	31.2131			1.06251			0.531253	+	0.847213i	$0.321721\pi$
$864$		0			0
$865$	7.58403			0.257865
$866$		0			0
$867$	28.4521	−	5.61625i	0.966286	−	0.190738i
$868$		0			0
$869$		−	68.5112i		−	2.32408i
$870$		0			0
$871$		−	19.0810i		−	0.646535i
$872$		0			0
$873$	15.3720	−	6.31469i	0.520264	−	0.213720i
$874$		0			0
$875$	1.00000			0.0338062
$876$		0			0
$877$	0.564288			0.0190546			0.00952732	−	0.999955i	$-0.496967\pi$
$877$	0.564288			0.0190546			0.00952732	+	0.999955i	$0.496967\pi$
$878$		0			0
$879$	−7.48999	−	37.9446i	−0.252631	−	1.27984i
$880$		0			0
$881$		−	56.5866i		−	1.90645i	−0.302259	−	0.953226i	$-0.597741\pi$
$881$		−	56.5866i		−	1.90645i	0.302259	−	0.953226i	$-0.402259\pi$
$882$		0			0
$883$		−	55.6710i		−	1.87348i	−0.350027	−	0.936739i	$-0.613828\pi$
$883$		−	55.6710i		−	1.87348i	0.350027	−	0.936739i	$-0.386172\pi$
$884$		0			0
$885$	−2.23303	−	11.3126i	−0.0750626	−	0.380271i
$886$		0			0
$887$	27.0471			0.908152			0.454076	−	0.890963i	$-0.349969\pi$
$887$	27.0471			0.908152			0.454076	+	0.890963i	$0.349969\pi$
$888$		0			0
$889$	3.05575			0.102487
$890$		0			0
$891$	28.7431	−	28.4089i	0.962931	−	0.951732i
$892$		0			0
$893$			3.18502i			0.106583i
$894$		0			0
$895$		−	6.83244i		−	0.228383i
$896$		0			0
$897$	−7.71312	+	1.52251i	−0.257533	+	0.0508352i
$898$		0			0
$899$	0.228518			0.00762152
$900$		0			0
$901$	1.03315			0.0344194
$902$		0			0
$903$	−2.23280	+	0.440738i	−0.0743028	+	0.0146668i
$904$		0			0
$905$			5.36693i			0.178403i
$906$		0			0
$907$		−	14.4312i		−	0.479179i	−0.970874	−	0.239590i	$-0.922987\pi$
$907$		−	14.4312i		−	0.479179i	0.970874	−	0.239590i	$-0.0770129\pi$
$908$		0			0
$909$	16.4699	+	40.0930i	0.546271	+	1.32980i
$910$		0			0
$911$	−15.2814			−0.506294			−0.253147	−	0.967428i	$-0.581466\pi$
$911$	−15.2814			−0.506294			−0.253147	+	0.967428i	$0.581466\pi$
$912$		0			0
$913$	−3.13284			−0.103682
$914$		0			0
$915$	−2.48246	−	12.5762i	−0.0820675	−	0.415757i
$916$		0			0
$917$			2.61320i			0.0862955i
$918$		0			0
$919$			11.5120i			0.379746i	0.981809	+	0.189873i	$0.0608077\pi$
$919$			11.5120i			0.379746i	−0.981809	+	0.189873i	$0.939192\pi$
$920$		0			0
$921$	−2.55103	−	12.9236i	−0.0840594	−	0.425849i
$922$		0			0
$923$	6.22513			0.204903
$924$		0			0
$925$	2.40449			0.0790593
$926$		0			0
$927$	−10.4664	−	25.4786i	−0.343761	−	0.836827i
$928$		0			0
$929$			46.9026i			1.53883i	0.638752	+	0.769413i	$0.279451\pi$
$929$			46.9026i			1.53883i	−0.638752	+	0.769413i	$0.720549\pi$
$930$		0			0
$931$			0.377701i			0.0123787i
$932$		0			0
$933$	2.51968	−	0.497366i	0.0824905	−	0.0162830i
$934$		0			0
$935$	−2.27274			−0.0743265
$936$		0			0
$937$	7.72755			0.252448			0.126224	−	0.992002i	$-0.459714\pi$
$937$	7.72755			0.252448			0.126224	+	0.992002i	$0.459714\pi$
$938$		0			0
$939$	43.2455	−	8.53634i	1.41126	−	0.278573i
$940$		0			0
$941$		−	8.77099i		−	0.285926i	−0.989728	−	0.142963i	$-0.954337\pi$
$941$		−	8.77099i		−	0.285926i	0.989728	−	0.142963i	$-0.0456630\pi$
$942$		0			0
$943$		−	8.53648i		−	0.277986i
$944$		0			0
$945$	−4.33306	+	2.86785i	−0.140955	+	0.0932911i
$946$		0			0
$947$	−8.37854			−0.272266			−0.136133	−	0.990691i	$-0.543467\pi$
$947$	−8.37854			−0.272266			−0.136133	+	0.990691i	$0.543467\pi$
$948$		0			0
$949$	19.7335			0.640576
$950$		0			0
$951$	−3.78240	−	19.1618i	−0.122653	−	0.621365i
$952$		0			0
$953$			51.8327i			1.67902i	0.543341	+	0.839512i	$0.317159\pi$
$953$			51.8327i			1.67902i	−0.543341	+	0.839512i	$0.682841\pi$
$954$		0			0
$955$		−	0.142457i		−	0.00460980i
$956$		0			0
$957$	1.57110	+	7.95923i	0.0507863	+	0.257286i
$958$		0			0
$959$	9.11266			0.294263
$960$		0			0
$961$	30.9520			0.998452
$962$		0			0
$963$	−42.7831	+	17.5749i	−1.37867	+	0.566344i
$964$		0			0
$965$			10.1022i			0.325203i
$966$		0			0
$967$		−	8.69391i		−	0.279577i	−0.990181	−	0.139789i	$-0.955358\pi$
$967$		−	8.69391i		−	0.279577i	0.990181	−	0.139789i	$-0.0446423\pi$
$968$		0			0
$969$	−0.324846	+	0.0641222i	−0.0104356	+	0.00205990i
$970$		0			0
$971$	42.7949			1.37335			0.686677	−	0.726962i	$-0.259069\pi$
$971$	42.7949			1.37335			0.686677	+	0.726962i	$0.259069\pi$
$972$		0			0
$973$	−12.3984			−0.397473
$974$		0			0
$975$	−2.71538	+	0.535996i	−0.0869617	+	0.0171656i
$976$		0			0
$977$		−	38.7722i		−	1.24043i	−0.784431	−	0.620216i	$-0.787045\pi$
$977$		−	38.7722i		−	1.24043i	0.784431	−	0.620216i	$-0.212955\pi$
$978$		0			0
$979$			10.3433i			0.330574i
$980$		0			0
$981$	−42.5135	+	17.4642i	−1.35735	+	0.557588i
$982$		0			0
$983$	−13.4573			−0.429220			−0.214610	−	0.976700i	$-0.568848\pi$
$983$	−13.4573			−0.429220			−0.214610	+	0.976700i	$0.568848\pi$
$984$		0			0
$985$	−10.1619			−0.323786
$986$		0			0
$987$	2.82850	+	14.3293i	0.0900320	+	0.456106i
$988$		0			0
$989$			3.73240i			0.118684i
$990$		0			0
$991$		−	37.7507i		−	1.19919i	−0.800303	−	0.599595i	$-0.795328\pi$
$991$		−	37.7507i		−	1.19919i	0.800303	−	0.599595i	$-0.204672\pi$
$992$		0			0
$993$	7.42725	+	37.6268i	0.235697	+	1.19405i
$994$		0			0
$995$	20.0613			0.635986
$996$		0			0
$997$	−11.3781			−0.360348			−0.180174	−	0.983635i	$-0.557666\pi$
$997$	−11.3781			−0.360348			−0.180174	+	0.983635i	$0.557666\pi$
$998$		0			0
$999$	−10.4188	+	6.89572i	−0.329637	+	0.218171i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.ba.c.911.15	✓	16
3.2	odd	2		1680.2.ba.d.911.1	yes	16
4.3	odd	2		1680.2.ba.d.911.2	yes	16
12.11	even	2	inner	1680.2.ba.c.911.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.ba.c.911.15	✓	16	1.1	even	1	trivial
1680.2.ba.c.911.16	yes	16	12.11	even	2	inner
1680.2.ba.d.911.1	yes	16	3.2	odd	2
1680.2.ba.d.911.2	yes	16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.69926 - 0.335422i) q^{3} -1.00000i q^{5} -1.00000i q^{7} +(2.77498 - 1.13994i) q^{9} +4.49036 q^{11} +1.59797 q^{13} +(-0.335422 - 1.69926i) q^{15} -0.506137i q^{17} -0.377701i q^{19} +(-0.335422 - 1.69926i) q^{21} -2.84053 q^{23} -1.00000 q^{25} +(4.33306 - 2.86785i) q^{27} +1.04311i q^{29} -0.219074i q^{31} +(7.63030 - 1.50617i) q^{33} -1.00000 q^{35} -2.40449 q^{37} +(2.71538 - 0.535996i) q^{39} +3.00524i q^{41} -1.31398i q^{43} +(-1.13994 - 2.77498i) q^{45} -8.43265 q^{47} -1.00000 q^{49} +(-0.169770 - 0.860060i) q^{51} +2.04125i q^{53} -4.49036i q^{55} +(-0.126689 - 0.641814i) q^{57} +6.65739 q^{59} +7.40099 q^{61} +(-1.13994 - 2.77498i) q^{63} -1.59797i q^{65} -11.9407i q^{67} +(-4.82681 + 0.952777i) q^{69} +3.89564 q^{71} +12.3491 q^{73} +(-1.69926 + 0.335422i) q^{75} -4.49036i q^{77} -15.2574i q^{79} +(6.40107 - 6.32663i) q^{81} -0.697682 q^{83} -0.506137 q^{85} +(0.349882 + 1.77252i) q^{87} +2.30345i q^{89} -1.59797i q^{91} +(-0.0734824 - 0.372265i) q^{93} -0.377701 q^{95} +5.53949 q^{97} +(12.4607 - 5.11874i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{9} - 12 q^{11} + 2 q^{15} + 2 q^{21} - 8 q^{23} - 16 q^{25} + 12 q^{27} + 12 q^{33} - 16 q^{35} - 8 q^{37} + 14 q^{39} + 8 q^{45} - 16 q^{47} - 16 q^{49} + 34 q^{51} + 12 q^{57} - 32 q^{59}+ \cdots + 50 q^{99}+O(q^{100}) \) 16 * q - 10 * q^9 - 12 * q^11 + 2 * q^15 + 2 * q^21 - 8 * q^23 - 16 * q^25 + 12 * q^27 + 12 * q^33 - 16 * q^35 - 8 * q^37 + 14 * q^39 + 8 * q^45 - 16 * q^47 - 16 * q^49 + 34 * q^51 + 12 * q^57 - 32 * q^59 + 8 * q^61 + 8 * q^63 - 20 * q^69 + 32 * q^71 + 48 * q^73 - 22 * q^81 - 72 * q^83 - 12 * q^85 + 44 * q^87 + 28 * q^93 - 8 * q^95 - 8 * q^97 + 50 * q^99

Introduction

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