LMFDB - Embedded newform 2640.2.k.d.1871.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2640,2,Mod(1871,2640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2640.1871"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.195105024.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1871.3
Root			$1.30512 - 1.13871i$ of defining polynomial
Character	$\chi$	$=$	2640.1871
Dual form			2640.2.k.d.1871.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.30512 - 1.13871i) q^{3} -1.00000i q^{5} -4.27743i q^{7} +(0.406663 + 2.97231i) q^{9} -1.00000 q^{11} +5.46410 q^{13} +(-1.13871 + 1.30512i) q^{15} +5.74153i q^{17} +6.07434i q^{19} +(-4.87076 + 5.58254i) q^{21} +6.79849 q^{23} -1.00000 q^{25} +(2.85387 - 4.34229i) q^{27} +7.16509i q^{29} +0.853867i q^{31} +(1.30512 + 1.13871i) q^{33} -4.27743 q^{35} +4.61023 q^{37} +(-7.13129 - 6.22205i) q^{39} +10.2626i q^{41} +12.0190i q^{43} +(2.97231 - 0.406663i) q^{45} -5.09075 q^{47} -11.2964 q^{49} +(6.53796 - 7.49337i) q^{51} -5.85544i q^{53} +1.00000i q^{55} +(6.91693 - 7.92772i) q^{57} +7.88924 q^{59} +9.49790 q^{61} +(12.7138 - 1.73947i) q^{63} -5.46410i q^{65} +6.24363i q^{67} +(-8.87282 - 7.74153i) q^{69} -2.83228 q^{71} -3.88608 q^{73} +(1.30512 + 1.13871i) q^{75} +4.27743i q^{77} -9.35334i q^{79} +(-8.66925 + 2.41746i) q^{81} -1.77532 q^{83} +5.74153 q^{85} +(8.15898 - 9.35128i) q^{87} +6.05538i q^{89} -23.3723i q^{91} +(0.972310 - 1.11440i) q^{93} +6.07434 q^{95} -7.62665 q^{97} +(-0.406663 - 2.97231i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 6 q^{9} - 8 q^{11} + 16 q^{13} + 2 q^{15} - 14 q^{21} + 8 q^{23} - 8 q^{25} + 8 q^{27} + 4 q^{33} - 12 q^{35} + 24 q^{37} - 20 q^{39} + 8 q^{45} - 24 q^{47} + 4 q^{49} + 18 q^{51} + 8 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 6 * q^9 - 8 * q^11 + 16 * q^13 + 2 * q^15 - 14 * q^21 + 8 * q^23 - 8 * q^25 + 8 * q^27 + 4 * q^33 - 12 * q^35 + 24 * q^37 - 20 * q^39 + 8 * q^45 - 24 * q^47 + 4 * q^49 + 18 * q^51 + 8 * q^57 + 28 * q^61 + 36 * q^63 + 16 * q^69 + 44 * q^71 - 24 * q^73 + 4 * q^75 + 2 * q^81 + 56 * q^83 - 4 * q^85 + 44 * q^87 - 8 * q^93 + 8 * q^95 - 72 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.30512	−	1.13871i	−0.753510	−	0.657437i
$3$	−1.30512	−	1.13871i	−0.753510	−	0.657437i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	4.27743i		−	1.61672i	−0.588692	−	0.808358i	$-0.700357\pi$
$7$		−	4.27743i		−	1.61672i	0.588692	−	0.808358i	$-0.299643\pi$
$8$		0			0
$9$	0.406663	+	2.97231i	0.135554	+	0.990770i
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$	5.46410			1.51547			0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410			1.51547			0.757735	+	0.652563i	$0.226306\pi$
$14$		0			0
$15$	−1.13871	+	1.30512i	−0.294015	+	0.336980i
$16$		0			0
$17$			5.74153i			1.39253i	0.717787	+	0.696263i	$0.245155\pi$
$17$			5.74153i			1.39253i	−0.717787	+	0.696263i	$0.754845\pi$
$18$		0			0
$19$			6.07434i			1.39355i	0.717291	+	0.696774i	$0.245382\pi$
$19$			6.07434i			1.39355i	−0.717291	+	0.696774i	$0.754618\pi$
$20$		0			0
$21$	−4.87076	+	5.58254i	−1.06289	+	1.21821i
$22$		0			0
$23$	6.79849			1.41758			0.708791	−	0.705418i	$-0.249241\pi$
$23$	6.79849			1.41758			0.708791	+	0.705418i	$0.249241\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	2.85387	−	4.34229i	0.549227	−	0.835673i
$28$		0			0
$29$			7.16509i			1.33052i	0.746610	+	0.665262i	$0.231680\pi$
$29$			7.16509i			1.33052i	−0.746610	+	0.665262i	$0.768320\pi$
$30$		0			0
$31$			0.853867i			0.153359i	0.997056	+	0.0766795i	$0.0244318\pi$
$31$			0.853867i			0.153359i	−0.997056	+	0.0766795i	$0.975568\pi$
$32$		0			0
$33$	1.30512	+	1.13871i	0.227192	+	0.198225i
$34$		0			0
$35$	−4.27743			−0.723017
$36$		0			0
$37$	4.61023			0.757918			0.378959	−	0.925413i	$-0.376282\pi$
$37$	4.61023			0.757918			0.378959	+	0.925413i	$0.376282\pi$
$38$		0			0
$39$	−7.13129	−	6.22205i	−1.14192	−	0.996325i
$40$		0			0
$41$			10.2626i			1.60275i	0.598164	+	0.801373i	$0.295897\pi$
$41$			10.2626i			1.60275i	−0.598164	+	0.801373i	$0.704103\pi$
$42$		0			0
$43$			12.0190i			1.83287i	0.400179	+	0.916437i	$0.368948\pi$
$43$			12.0190i			1.83287i	−0.400179	+	0.916437i	$0.631052\pi$
$44$		0			0
$45$	2.97231	−	0.406663i	0.443086	−	0.0606217i
$46$		0			0
$47$	−5.09075			−0.742563			−0.371281	−	0.928520i	$-0.621081\pi$
$47$	−5.09075			−0.742563			−0.371281	+	0.928520i	$0.621081\pi$
$48$		0			0
$49$	−11.2964			−1.61377
$50$		0			0
$51$	6.53796	−	7.49337i	0.915497	−	1.04928i
$52$		0			0
$53$		−	5.85544i		−	0.804307i	−0.915572	−	0.402154i	$-0.868262\pi$
$53$		−	5.85544i		−	0.804307i	0.915572	−	0.402154i	$-0.131738\pi$
$54$		0			0
$55$			1.00000i			0.134840i
$56$		0			0
$57$	6.91693	−	7.92772i	0.916170	−	1.05005i
$58$		0			0
$59$	7.88924			1.02709			0.513546	−	0.858062i	$-0.328332\pi$
$59$	7.88924			1.02709			0.513546	+	0.858062i	$0.328332\pi$
$60$		0			0
$61$	9.49790			1.21608			0.608041	−	0.793906i	$-0.291956\pi$
$61$	9.49790			1.21608			0.608041	+	0.793906i	$0.291956\pi$
$62$		0			0
$63$	12.7138	−	1.73947i	1.60179	−	0.219153i
$64$		0			0
$65$		−	5.46410i		−	0.677738i
$66$		0			0
$67$			6.24363i			0.762781i	0.924414	+	0.381391i	$0.124555\pi$
$67$			6.24363i			0.762781i	−0.924414	+	0.381391i	$0.875445\pi$
$68$		0			0
$69$	−8.87282	−	7.74153i	−1.06816	−	0.931971i
$70$		0			0
$71$	−2.83228			−0.336130			−0.168065	−	0.985776i	$-0.553752\pi$
$71$	−2.83228			−0.336130			−0.168065	+	0.985776i	$0.553752\pi$
$72$		0			0
$73$	−3.88608			−0.454832			−0.227416	−	0.973798i	$-0.573028\pi$
$73$	−3.88608			−0.454832			−0.227416	+	0.973798i	$0.573028\pi$
$74$		0			0
$75$	1.30512	+	1.13871i	0.150702	+	0.131487i
$76$		0			0
$77$			4.27743i			0.487458i
$78$		0			0
$79$		−	9.35334i		−	1.05233i	−0.850382	−	0.526167i	$-0.823629\pi$
$79$		−	9.35334i		−	1.05233i	0.850382	−	0.526167i	$-0.176371\pi$
$80$		0			0
$81$	−8.66925	+	2.41746i	−0.963250	+	0.268606i
$82$		0			0
$83$	−1.77532			−0.194867			−0.0974336	−	0.995242i	$-0.531063\pi$
$83$	−1.77532			−0.194867			−0.0974336	+	0.995242i	$0.531063\pi$
$84$		0			0
$85$	5.74153			0.622756
$86$		0			0
$87$	8.15898	−	9.35128i	0.874735	−	1.00256i
$88$		0			0
$89$			6.05538i			0.641869i	0.947101	+	0.320935i	$0.103997\pi$
$89$			6.05538i			0.641869i	−0.947101	+	0.320935i	$0.896003\pi$
$90$		0			0
$91$		−	23.3723i		−	2.45008i
$92$		0			0
$93$	0.972310	−	1.11440i	0.100824	−	0.115558i
$94$		0			0
$95$	6.07434			0.623214
$96$		0			0
$97$	−7.62665			−0.774369			−0.387185	−	0.922002i	$-0.626552\pi$
$97$	−7.62665			−0.774369			−0.387185	+	0.922002i	$0.626552\pi$
$98$		0			0
$99$	−0.406663	−	2.97231i	−0.0408711	−	0.298728i
$100$		0			0
$101$		−	6.66877i		−	0.663567i	−0.943355	−	0.331784i	$-0.892350\pi$
$101$		−	6.66877i		−	0.663567i	0.943355	−	0.331784i	$-0.107650\pi$
$102$		0			0
$103$		−	12.9800i		−	1.27896i	−0.768809	−	0.639478i	$-0.779150\pi$
$103$		−	12.9800i		−	1.27896i	0.768809	−	0.639478i	$-0.220850\pi$
$104$		0			0
$105$	5.58254	+	4.87076i	0.544801	+	0.475338i
$106$		0			0
$107$	2.66877			0.258000			0.129000	−	0.991645i	$-0.458823\pi$
$107$	2.66877			0.258000			0.129000	+	0.991645i	$0.458823\pi$
$108$		0			0
$109$	−12.2436			−1.17273			−0.586364	−	0.810048i	$-0.699441\pi$
$109$	−12.2436			−1.17273			−0.586364	+	0.810048i	$0.699441\pi$
$110$		0			0
$111$	−6.01690	−	5.24974i	−0.571099	−	0.498283i
$112$		0			0
$113$			7.88924i			0.742157i	0.928601	+	0.371079i	$0.121012\pi$
$113$			7.88924i			0.742157i	−0.928601	+	0.371079i	$0.878988\pi$
$114$		0			0
$115$		−	6.79849i		−	0.633962i
$116$		0			0
$117$	2.22205	+	16.2410i	0.205428	+	1.50148i
$118$		0			0
$119$	24.5590			2.25132
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	11.6861	−	13.3939i	1.05370	−	1.20769i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			21.9082i			1.94404i	0.234902	+	0.972019i	$0.424523\pi$
$127$			21.9082i			1.94404i	−0.234902	+	0.972019i	$0.575477\pi$
$128$		0			0
$129$	13.6861	−	15.6861i	1.20500	−	1.38109i
$130$		0			0
$131$	2.89441			0.252886			0.126443	−	0.991974i	$-0.459644\pi$
$131$	2.89441			0.252886			0.126443	+	0.991974i	$0.459644\pi$
$132$		0			0
$133$	25.9825			2.25297
$134$		0			0
$135$	−4.34229	−	2.85387i	−0.373724	−	0.245622i
$136$		0			0
$137$			11.4831i			0.981064i	0.871423	+	0.490532i	$0.163197\pi$
$137$			11.4831i			0.981064i	−0.871423	+	0.490532i	$0.836803\pi$
$138$		0			0
$139$			4.53590i			0.384730i	0.981323	+	0.192365i	$0.0616157\pi$
$139$			4.53590i			0.384730i	−0.981323	+	0.192365i	$0.938384\pi$
$140$		0			0
$141$	6.64403	+	5.79691i	0.559528	+	0.488188i
$142$		0			0
$143$	−5.46410			−0.456931
$144$		0			0
$145$	7.16509			0.595028
$146$		0			0
$147$	14.7431	+	12.8633i	1.21599	+	1.06095i
$148$		0			0
$149$			6.05538i			0.496076i	0.968750	+	0.248038i	$0.0797858\pi$
$149$			6.05538i			0.496076i	−0.968750	+	0.248038i	$0.920214\pi$
$150$		0			0
$151$		−	9.28575i		−	0.755664i	−0.925874	−	0.377832i	$-0.876670\pi$
$151$		−	9.28575i		−	0.755664i	0.925874	−	0.377832i	$-0.123330\pi$
$152$		0			0
$153$	−17.0656	+	2.33487i	−1.37967	+	0.188763i
$154$		0			0
$155$	0.853867			0.0685842
$156$		0			0
$157$	22.8297			1.82200			0.911002	−	0.412401i	$-0.135310\pi$
$157$	22.8297			1.82200			0.911002	+	0.412401i	$0.135310\pi$
$158$		0			0
$159$	−6.66767	+	7.64204i	−0.528781	+	0.606053i
$160$		0			0
$161$		−	29.0800i		−	2.29183i
$162$		0			0
$163$			6.83228i			0.535146i	0.963538	+	0.267573i	$0.0862216\pi$
$163$			6.83228i			0.535146i	−0.963538	+	0.267573i	$0.913778\pi$
$164$		0			0
$165$	1.13871	−	1.30512i	0.0886487	−	0.101603i
$166$		0			0
$167$	−8.31797			−0.643664			−0.321832	−	0.946797i	$-0.604299\pi$
$167$	−8.31797			−0.643664			−0.321832	+	0.946797i	$0.604299\pi$
$168$		0			0
$169$	16.8564			1.29665
$170$		0			0
$171$	−18.0548	+	2.47021i	−1.38069	+	0.188901i
$172$		0			0
$173$		−	5.10971i		−	0.388484i	−0.980954	−	0.194242i	$-0.937775\pi$
$173$		−	5.10971i		−	0.388484i	0.980954	−	0.194242i	$-0.0622247\pi$
$174$		0			0
$175$			4.27743i			0.323343i
$176$		0			0
$177$	−10.2964	−	8.98358i	−0.773924	−	0.675247i
$178$		0			0
$179$	5.44830			0.407225			0.203613	−	0.979052i	$-0.434732\pi$
$179$	5.44830			0.407225			0.203613	+	0.979052i	$0.434732\pi$
$180$		0			0
$181$	−7.62665			−0.566885			−0.283442	−	0.958989i	$-0.591476\pi$
$181$	−7.62665			−0.566885			−0.283442	+	0.958989i	$0.591476\pi$
$182$		0			0
$183$	−12.3959	−	10.8154i	−0.916329	−	0.799496i
$184$		0			0
$185$		−	4.61023i		−	0.338951i
$186$		0			0
$187$		−	5.74153i		−	0.419862i
$188$		0			0
$189$	−18.5738	−	12.2072i	−1.35105	−	0.887944i
$190$		0			0
$191$	−5.75637			−0.416516			−0.208258	−	0.978074i	$-0.566779\pi$
$191$	−5.75637			−0.416516			−0.208258	+	0.978074i	$0.566779\pi$
$192$		0			0
$193$	−1.57127			−0.113103			−0.0565513	−	0.998400i	$-0.518010\pi$
$193$	−1.57127			−0.113103			−0.0565513	+	0.998400i	$0.518010\pi$
$194$		0			0
$195$	−6.22205	+	7.13129i	−0.445570	+	0.510683i
$196$		0			0
$197$			15.3723i			1.09523i	0.836730	+	0.547615i	$0.184464\pi$
$197$			15.3723i			1.09523i	−0.836730	+	0.547615i	$0.815536\pi$
$198$		0			0
$199$		−	13.4763i		−	0.955310i	−0.878547	−	0.477655i	$-0.841487\pi$
$199$		−	13.4763i		−	0.955310i	0.878547	−	0.477655i	$-0.158513\pi$
$200$		0			0
$201$	7.10971	−	8.14867i	0.501480	−	0.574763i
$202$		0			0
$203$	30.6481			2.15108
$204$		0			0
$205$	10.2626			0.716770
$206$		0			0
$207$	2.76469	+	20.2072i	0.192159	+	1.40450i
$208$		0			0
$209$		−	6.07434i		−	0.420171i
$210$		0			0
$211$		−	13.2304i		−	0.910816i	−0.890283	−	0.455408i	$-0.849493\pi$
$211$		−	13.2304i		−	0.910816i	0.890283	−	0.455408i	$-0.150507\pi$
$212$		0			0
$213$	3.69646	+	3.22516i	0.253277	+	0.220984i
$214$		0			0
$215$	12.0190			0.819686
$216$		0			0
$217$	3.65235			0.247938
$218$		0			0
$219$	5.07180	+	4.42514i	0.342720	+	0.299023i
$220$		0			0
$221$			31.3723i			2.11033i
$222$		0			0
$223$		−	18.8364i		−	1.26138i	−0.776035	−	0.630689i	$-0.782772\pi$
$223$		−	18.8364i		−	1.26138i	0.776035	−	0.630689i	$-0.217228\pi$
$224$		0			0
$225$	−0.406663	−	2.97231i	−0.0271109	−	0.198154i
$226$		0			0
$227$	6.15183			0.408311			0.204156	−	0.978938i	$-0.434555\pi$
$227$	6.15183			0.408311			0.204156	+	0.978938i	$0.434555\pi$
$228$		0			0
$229$	14.3765			0.950026			0.475013	−	0.879979i	$-0.342443\pi$
$229$	14.3765			0.950026			0.475013	+	0.879979i	$0.342443\pi$
$230$		0			0
$231$	4.87076	−	5.58254i	0.320473	−	0.367304i
$232$		0			0
$233$		−	28.1528i		−	1.84435i	−0.386772	−	0.922175i	$-0.626410\pi$
$233$		−	28.1528i		−	1.84435i	0.386772	−	0.922175i	$-0.373590\pi$
$234$		0			0
$235$			5.09075i			0.332084i
$236$		0			0
$237$	−10.6508	+	12.2072i	−0.691842	+	0.792943i
$238$		0			0
$239$	−25.8132			−1.66972			−0.834860	−	0.550463i	$-0.814451\pi$
$239$	−25.8132			−1.66972			−0.834860	+	0.550463i	$0.814451\pi$
$240$		0			0
$241$	9.59382			0.617992			0.308996	−	0.951063i	$-0.400007\pi$
$241$	9.59382			0.617992			0.308996	+	0.951063i	$0.400007\pi$
$242$		0			0
$243$	14.0672	+	6.71673i	0.902410	+	0.430878i
$244$		0			0
$245$			11.2964i			0.721699i
$246$		0			0
$247$			33.1908i			2.11188i
$248$		0			0
$249$	2.31701	+	2.02159i	0.146834	+	0.128113i
$250$		0			0
$251$	−1.59697			−0.100800			−0.0504000	−	0.998729i	$-0.516050\pi$
$251$	−1.59697			−0.100800			−0.0504000	+	0.998729i	$0.516050\pi$
$252$		0			0
$253$	−6.79849			−0.427417
$254$		0			0
$255$	−7.49337	−	6.53796i	−0.469253	−	0.409423i
$256$		0			0
$257$		−	15.8892i		−	0.991144i	−0.868567	−	0.495572i	$-0.834959\pi$
$257$		−	15.8892i		−	0.991144i	0.868567	−	0.495572i	$-0.165041\pi$
$258$		0			0
$259$		−	19.7199i		−	1.22534i
$260$		0			0
$261$	−21.2969	+	2.91377i	−1.31824	+	0.180358i
$262$		0			0
$263$	−11.1354			−0.686639			−0.343319	−	0.939219i	$-0.611551\pi$
$263$	−11.1354			−0.686639			−0.343319	+	0.939219i	$0.611551\pi$
$264$		0			0
$265$	−5.85544			−0.359697
$266$		0			0
$267$	6.89534	−	7.90298i	0.421988	−	0.483655i
$268$		0			0
$269$			18.7035i			1.14037i	0.821515	+	0.570187i	$0.193129\pi$
$269$			18.7035i			1.14037i	−0.821515	+	0.570187i	$0.806871\pi$
$270$		0			0
$271$			15.7564i			0.957131i	0.878052	+	0.478566i	$0.158843\pi$
$271$			15.7564i			0.957131i	−0.878052	+	0.478566i	$0.841157\pi$
$272$		0			0
$273$	−26.6144	+	30.5036i	−1.61077	+	1.84616i
$274$		0			0
$275$	1.00000			0.0603023
$276$		0			0
$277$	19.5052			1.17195			0.585976	−	0.810328i	$-0.300711\pi$
$277$	19.5052			1.17195			0.585976	+	0.810328i	$0.300711\pi$
$278$		0			0
$279$	−2.53796	+	0.347236i	−0.151943	+	0.0207885i
$280$		0			0
$281$		−	29.7322i		−	1.77367i	−0.462085	−	0.886836i	$-0.652898\pi$
$281$		−	29.7322i		−	1.77367i	0.462085	−	0.886836i	$-0.347102\pi$
$282$		0			0
$283$			13.5780i			0.807129i	0.914951	+	0.403565i	$0.132229\pi$
$283$			13.5780i			0.807129i	−0.914951	+	0.403565i	$0.867771\pi$
$284$		0			0
$285$	−7.92772	−	6.91693i	−0.469598	−	0.409724i
$286$		0			0
$287$	43.8975			2.59119
$288$		0			0
$289$	−15.9652			−0.939127
$290$		0			0
$291$	9.95367	+	8.68457i	0.583495	+	0.509099i
$292$		0			0
$293$		−	3.88924i		−	0.227212i	−0.993526	−	0.113606i	$-0.963760\pi$
$293$		−	3.88924i		−	0.227212i	0.993526	−	0.113606i	$-0.0362401\pi$
$294$		0			0
$295$		−	7.88924i		−	0.459329i
$296$		0			0
$297$	−2.85387	+	4.34229i	−0.165598	+	0.251965i
$298$		0			0
$299$	37.1476			2.14830
$300$		0			0
$301$	51.4102			2.96324
$302$		0			0
$303$	−7.59382	+	8.70353i	−0.436253	+	0.500005i
$304$		0			0
$305$		−	9.49790i		−	0.543848i
$306$		0			0
$307$			6.62014i			0.377831i	0.981993	+	0.188916i	$0.0604972\pi$
$307$			6.62014i			0.377831i	−0.981993	+	0.188916i	$0.939503\pi$
$308$		0			0
$309$	−14.7805	+	16.9404i	−0.840833	+	0.963706i
$310$		0			0
$311$	−24.1698			−1.37054			−0.685272	−	0.728287i	$-0.740317\pi$
$311$	−24.1698			−1.37054			−0.685272	+	0.728287i	$0.740317\pi$
$312$		0			0
$313$	6.81429			0.385166			0.192583	−	0.981281i	$-0.438313\pi$
$313$	6.81429			0.385166			0.192583	+	0.981281i	$0.438313\pi$
$314$		0			0
$315$	−1.73947	−	12.7138i	−0.0980081	−	0.716344i
$316$		0			0
$317$		−	26.6697i		−	1.49792i	−0.662614	−	0.748961i	$-0.730553\pi$
$317$		−	26.6697i		−	1.49792i	0.662614	−	0.748961i	$-0.269447\pi$
$318$		0			0
$319$		−	7.16509i		−	0.401168i
$320$		0			0
$321$	−3.48306	−	3.03896i	−0.194405	−	0.169618i
$322$		0			0
$323$	−34.8760			−1.94055
$324$		0			0
$325$	−5.46410			−0.303094
$326$		0			0
$327$	15.9794	+	13.9420i	0.883662	+	0.770994i
$328$		0			0
$329$			21.7753i			1.20051i
$330$		0			0
$331$		−	12.5062i		−	0.687404i	−0.939079	−	0.343702i	$-0.888319\pi$
$331$		−	12.5062i		−	0.687404i	0.939079	−	0.343702i	$-0.111681\pi$
$332$		0			0
$333$	1.87481	+	13.7030i	0.102739	+	0.750922i
$334$		0			0
$335$	6.24363			0.341126
$336$		0			0
$337$	26.6945			1.45414			0.727070	−	0.686563i	$-0.240881\pi$
$337$	26.6945			1.45414			0.727070	+	0.686563i	$0.240881\pi$
$338$		0			0
$339$	8.98358	−	10.2964i	0.487921	−	0.559223i
$340$		0			0
$341$		−	0.853867i		−	0.0462395i
$342$		0			0
$343$			18.3775i			0.992290i
$344$		0			0
$345$	−7.74153	+	8.87282i	−0.416790	+	0.477697i
$346$		0			0
$347$	6.29227			0.337787			0.168893	−	0.985634i	$-0.445981\pi$
$347$	6.29227			0.337787			0.168893	+	0.985634i	$0.445981\pi$
$348$		0			0
$349$	−11.3070			−0.605251			−0.302625	−	0.953110i	$-0.597863\pi$
$349$	−11.3070			−0.605251			−0.302625	+	0.953110i	$0.597863\pi$
$350$		0			0
$351$	15.5938	−	23.7267i	0.832336	−	1.26644i
$352$		0			0
$353$			4.29542i			0.228622i	0.993445	+	0.114311i	$0.0364661\pi$
$353$			4.29542i			0.228622i	−0.993445	+	0.114311i	$0.963534\pi$
$354$		0			0
$355$			2.83228i			0.150322i
$356$		0			0
$357$	−32.0523	−	27.9656i	−1.69639	−	1.48010i
$358$		0			0
$359$	14.7067			0.776189			0.388094	−	0.921620i	$-0.373133\pi$
$359$	14.7067			0.776189			0.388094	+	0.921620i	$0.373133\pi$
$360$		0			0
$361$	−17.8976			−0.941977
$362$		0			0
$363$	−1.30512	−	1.13871i	−0.0685009	−	0.0597670i
$364$		0			0
$365$			3.88608i			0.203407i
$366$		0			0
$367$			12.9421i			0.675571i	0.941223	+	0.337786i	$0.109678\pi$
$367$			12.9421i			0.675571i	−0.941223	+	0.337786i	$0.890322\pi$
$368$		0			0
$369$	−30.5036	+	4.17341i	−1.58795	+	0.217259i
$370$		0			0
$371$	−25.0462			−1.30034
$372$		0			0
$373$	−9.84060			−0.509527			−0.254764	−	0.967003i	$-0.581998\pi$
$373$	−9.84060			−0.509527			−0.254764	+	0.967003i	$0.581998\pi$
$374$		0			0
$375$	1.13871	−	1.30512i	0.0588029	−	0.0673960i
$376$		0			0
$377$			39.1508i			2.01637i
$378$		0			0
$379$			35.2805i			1.81224i	0.423024	+	0.906119i	$0.360969\pi$
$379$			35.2805i			1.81224i	−0.423024	+	0.906119i	$0.639031\pi$
$380$		0			0
$381$	24.9472	−	28.5928i	1.27808	−	1.46485i
$382$		0			0
$383$	−25.4692			−1.30141			−0.650707	−	0.759329i	$-0.725527\pi$
$383$	−25.4692			−1.30141			−0.650707	+	0.759329i	$0.725527\pi$
$384$		0			0
$385$	4.27743			0.217998
$386$		0			0
$387$	−35.7241	+	4.88766i	−1.81596	+	0.248454i
$388$		0			0
$389$		−	19.6781i		−	0.997717i	−0.866683	−	0.498859i	$-0.833753\pi$
$389$		−	19.6781i		−	0.997717i	0.866683	−	0.498859i	$-0.166247\pi$
$390$		0			0
$391$			39.0337i			1.97402i
$392$		0			0
$393$	−3.77754	−	3.29590i	−0.190552	−	0.166256i
$394$		0			0
$395$	−9.35334			−0.470618
$396$		0			0
$397$	−19.1773			−0.962481			−0.481241	−	0.876589i	$-0.659814\pi$
$397$	−19.1773			−0.962481			−0.481241	+	0.876589i	$0.659814\pi$
$398$		0			0
$399$	−33.9103	−	29.5867i	−1.69764	−	1.48119i
$400$		0			0
$401$		−	36.1640i		−	1.80595i	−0.429697	−	0.902973i	$-0.641380\pi$
$401$		−	36.1640i		−	1.80595i	0.429697	−	0.902973i	$-0.358620\pi$
$402$		0			0
$403$			4.66562i			0.232411i
$404$		0			0
$405$	2.41746	+	8.66925i	0.120124	+	0.430779i
$406$		0			0
$407$	−4.61023			−0.228521
$408$		0			0
$409$	−9.29121			−0.459421			−0.229710	−	0.973259i	$-0.573778\pi$
$409$	−9.29121			−0.459421			−0.229710	+	0.973259i	$0.573778\pi$
$410$		0			0
$411$	13.0759	−	14.9867i	0.644987	−	0.739241i
$412$		0			0
$413$		−	33.7456i		−	1.66051i
$414$		0			0
$415$			1.77532i			0.0871472i
$416$		0			0
$417$	5.16509	−	5.91988i	0.252936	−	0.289898i
$418$		0			0
$419$	−24.4873			−1.19628			−0.598141	−	0.801391i	$-0.704094\pi$
$419$	−24.4873			−1.19628			−0.598141	+	0.801391i	$0.704094\pi$
$420$		0			0
$421$	5.28491			0.257571			0.128785	−	0.991673i	$-0.458892\pi$
$421$	5.28491			0.257571			0.128785	+	0.991673i	$0.458892\pi$
$422$		0			0
$423$	−2.07022	−	15.1313i	−0.100658	−	0.735709i
$424$		0			0
$425$		−	5.74153i		−	0.278505i
$426$		0			0
$427$		−	40.6266i		−	1.96606i
$428$		0			0
$429$	7.13129	+	6.22205i	0.344302	+	0.300403i
$430$		0			0
$431$	−19.2236			−0.925969			−0.462985	−	0.886366i	$-0.653222\pi$
$431$	−19.2236			−0.925969			−0.462985	+	0.886366i	$0.653222\pi$
$432$		0			0
$433$	−37.1013			−1.78297			−0.891487	−	0.453046i	$-0.850337\pi$
$433$	−37.1013			−1.78297			−0.891487	+	0.453046i	$0.850337\pi$
$434$		0			0
$435$	−9.35128	−	8.15898i	−0.448360	−	0.391193i
$436$		0			0
$437$			41.2963i			1.97547i
$438$		0			0
$439$			16.5094i			0.787949i	0.919121	+	0.393975i	$0.128900\pi$
$439$			16.5094i			0.787949i	−0.919121	+	0.393975i	$0.871100\pi$
$440$		0			0
$441$	−4.59382	−	33.5763i	−0.218753	−	1.59887i
$442$		0			0
$443$	−2.29227			−0.108909			−0.0544544	−	0.998516i	$-0.517342\pi$
$443$	−2.29227			−0.108909			−0.0544544	+	0.998516i	$0.517342\pi$
$444$		0			0
$445$	6.05538			0.287053
$446$		0			0
$447$	6.89534	−	7.90298i	0.326139	−	0.373798i
$448$		0			0
$449$			5.33754i			0.251894i	0.992037	+	0.125947i	$0.0401969\pi$
$449$			5.33754i			0.251894i	−0.992037	+	0.125947i	$0.959803\pi$
$450$		0			0
$451$		−	10.2626i		−	0.483246i
$452$		0			0
$453$	−10.5738	+	12.1190i	−0.496801	+	0.569400i
$454$		0			0
$455$	−23.3723			−1.09571
$456$		0			0
$457$	5.35693			0.250587			0.125293	−	0.992120i	$-0.460013\pi$
$457$	5.35693			0.250587			0.125293	+	0.992120i	$0.460013\pi$
$458$		0			0
$459$	24.9314	+	16.3856i	1.16370	+	0.764812i
$460$		0			0
$461$		−	3.16509i		−	0.147413i	−0.997280	−	0.0737064i	$-0.976517\pi$
$461$		−	3.16509i		−	0.147413i	0.997280	−	0.0737064i	$-0.0234828\pi$
$462$		0			0
$463$		−	14.9472i		−	0.694653i	−0.937744	−	0.347327i	$-0.887089\pi$
$463$		−	14.9472i		−	0.694653i	0.937744	−	0.347327i	$-0.112911\pi$
$464$		0			0
$465$	−1.11440	−	0.972310i	−0.0516789	−	0.0450898i
$466$		0			0
$467$	−25.9529			−1.20095			−0.600477	−	0.799642i	$-0.705023\pi$
$467$	−25.9529			−1.20095			−0.600477	+	0.799642i	$0.705023\pi$
$468$		0			0
$469$	26.7067			1.23320
$470$		0			0
$471$	−29.7954	−	25.9964i	−1.37290	−	1.19785i
$472$		0			0
$473$		−	12.0190i		−	0.552632i
$474$		0			0
$475$		−	6.07434i		−	0.278710i
$476$		0			0
$477$	17.4042	−	2.38119i	0.796883	−	0.109027i
$478$		0			0
$479$	36.2163			1.65476			0.827382	−	0.561640i	$-0.189829\pi$
$479$	36.2163			1.65476			0.827382	+	0.561640i	$0.189829\pi$
$480$		0			0
$481$	25.1908			1.14860
$482$		0			0
$483$	−33.1138	+	37.9529i	−1.50673	+	1.72691i
$484$		0			0
$485$			7.62665i			0.346308i
$486$		0			0
$487$			28.7720i			1.30378i	0.758313	+	0.651891i	$0.226024\pi$
$487$			28.7720i			1.30378i	−0.758313	+	0.651891i	$0.773976\pi$
$488$		0			0
$489$	7.78001	−	8.91693i	0.351824	−	0.403237i
$490$		0			0
$491$	1.14350			0.0516056			0.0258028	−	0.999667i	$-0.491786\pi$
$491$	1.14350			0.0516056			0.0258028	+	0.999667i	$0.491786\pi$
$492$		0			0
$493$	−41.1386			−1.85279
$494$		0			0
$495$	−2.97231	+	0.406663i	−0.133595	+	0.0182781i
$496$		0			0
$497$			12.1149i			0.543427i
$498$		0			0
$499$		−	1.02316i		−	0.0458030i	−0.999738	−	0.0229015i	$-0.992710\pi$
$499$		−	1.02316i		−	0.0458030i	0.999738	−	0.0229015i	$-0.00729042\pi$
$500$		0			0
$501$	10.8559	+	9.47178i	0.485007	+	0.423168i
$502$		0			0
$503$	−35.8299			−1.59758			−0.798788	−	0.601613i	$-0.794525\pi$
$503$	−35.8299			−1.59758			−0.798788	+	0.601613i	$0.794525\pi$
$504$		0			0
$505$	−6.66877			−0.296756
$506$		0			0
$507$	−21.9996	−	19.1946i	−0.977036	−	0.852463i
$508$		0			0
$509$			8.04107i			0.356414i	0.983993	+	0.178207i	$0.0570297\pi$
$509$			8.04107i			0.356414i	−0.983993	+	0.178207i	$0.942970\pi$
$510$		0			0
$511$			16.6224i			0.735334i
$512$		0			0
$513$	26.3765	+	17.3353i	1.16455	+	0.765374i
$514$		0			0
$515$	−12.9800			−0.571967
$516$		0			0
$517$	5.09075			0.223891
$518$		0			0
$519$	−5.81849	+	6.66877i	−0.255404	+	0.292726i
$520$		0			0
$521$			38.3816i			1.68153i	0.541402	+	0.840764i	$0.317894\pi$
$521$			38.3816i			1.68153i	−0.541402	+	0.840764i	$0.682106\pi$
$522$		0			0
$523$			4.49273i			0.196453i	0.995164	+	0.0982266i	$0.0313170\pi$
$523$			4.49273i			0.196453i	−0.995164	+	0.0982266i	$0.968683\pi$
$524$		0			0
$525$	4.87076	−	5.58254i	0.212578	−	0.243642i
$526$		0			0
$527$	−4.90250			−0.213556
$528$		0			0
$529$	23.2194			1.00954
$530$		0			0
$531$	3.20826	+	23.4493i	0.139227	+	1.01761i
$532$		0			0
$533$			56.0758i			2.42891i
$534$		0			0
$535$		−	2.66877i		−	0.115381i
$536$		0			0
$537$	−7.11067	−	6.20405i	−0.306848	−	0.267725i
$538$		0			0
$539$	11.2964			0.486570
$540$		0			0
$541$	−7.41997			−0.319009			−0.159505	−	0.987197i	$-0.550990\pi$
$541$	−7.41997			−0.319009			−0.159505	+	0.987197i	$0.550990\pi$
$542$		0			0
$543$	9.95367	+	8.68457i	0.427153	+	0.372691i
$544$		0			0
$545$			12.2436i			0.524460i
$546$		0			0
$547$			30.8067i			1.31720i	0.752493	+	0.658600i	$0.228851\pi$
$547$			30.8067i			1.31720i	−0.752493	+	0.658600i	$0.771149\pi$
$548$		0			0
$549$	3.86244	+	28.2307i	0.164845	+	1.20486i
$550$		0			0
$551$	−43.5232			−1.85415
$552$		0			0
$553$	−40.0082			−1.70132
$554$		0			0
$555$	−5.24974	+	6.01690i	−0.222839	+	0.255403i
$556$		0			0
$557$		−	32.5496i		−	1.37917i	−0.724204	−	0.689585i	$-0.757793\pi$
$557$		−	32.5496i		−	1.37917i	0.724204	−	0.689585i	$-0.242207\pi$
$558$		0			0
$559$			65.6728i			2.77766i
$560$		0			0
$561$	−6.53796	+	7.49337i	−0.276033	+	0.316370i
$562$		0			0
$563$	24.1190			1.01649			0.508247	−	0.861211i	$-0.330294\pi$
$563$	24.1190			1.01649			0.508247	+	0.861211i	$0.330294\pi$
$564$		0			0
$565$	7.88924			0.331903
$566$		0			0
$567$	10.3405	+	37.0821i	0.434260	+	1.55730i
$568$		0			0
$569$		−	36.6224i		−	1.53529i	−0.640874	−	0.767646i	$-0.721428\pi$
$569$		−	36.6224i		−	1.53529i	0.640874	−	0.767646i	$-0.278572\pi$
$570$		0			0
$571$		−	29.2304i		−	1.22325i	−0.791147	−	0.611626i	$-0.790516\pi$
$571$		−	29.2304i		−	1.22325i	0.791147	−	0.611626i	$-0.209484\pi$
$572$		0			0
$573$	7.51274	+	6.55485i	0.313849	+	0.273833i
$574$		0			0
$575$	−6.79849			−0.283516
$576$		0			0
$577$	6.33859			0.263879			0.131940	−	0.991258i	$-0.457880\pi$
$577$	6.33859			0.263879			0.131940	+	0.991258i	$0.457880\pi$
$578$		0			0
$579$	2.05069	+	1.78923i	0.0852239	+	0.0743577i
$580$		0			0
$581$			7.59382i			0.315045i
$582$		0			0
$583$			5.85544i			0.242508i
$584$		0			0
$585$	16.2410	−	2.22205i	0.671483	−	0.0918703i
$586$		0			0
$587$	40.9783			1.69136			0.845678	−	0.533693i	$-0.179196\pi$
$587$	40.9783			1.69136			0.845678	+	0.533693i	$0.179196\pi$
$588$		0			0
$589$	−5.18667			−0.213713
$590$		0			0
$591$	17.5046	−	20.0627i	0.720045	−	0.825267i
$592$		0			0
$593$		−	10.9895i		−	0.451284i	−0.974210	−	0.225642i	$-0.927552\pi$
$593$		−	10.9895i		−	0.451284i	0.974210	−	0.225642i	$-0.0724480\pi$
$594$		0			0
$595$		−	24.5590i		−	1.00682i
$596$		0			0
$597$	−15.3457	+	17.5882i	−0.628056	+	0.719836i
$598$		0			0
$599$	−6.20984			−0.253727			−0.126864	−	0.991920i	$-0.540491\pi$
$599$	−6.20984			−0.253727			−0.126864	+	0.991920i	$0.540491\pi$
$600$		0			0
$601$	37.4534			1.52776			0.763878	−	0.645361i	$-0.223293\pi$
$601$	37.4534			1.52776			0.763878	+	0.645361i	$0.223293\pi$
$602$		0			0
$603$	−18.5580	+	2.53905i	−0.755741	+	0.103398i
$604$		0			0
$605$		−	1.00000i		−	0.0406558i
$606$		0			0
$607$			31.5011i			1.27859i	0.768962	+	0.639294i	$0.220773\pi$
$607$			31.5011i			1.27859i	−0.768962	+	0.639294i	$0.779227\pi$
$608$		0			0
$609$	−39.9994	−	34.8995i	−1.62086	−	1.41420i
$610$		0			0
$611$	−27.8164			−1.12533
$612$		0			0
$613$	35.8025			1.44605			0.723025	−	0.690822i	$-0.242751\pi$
$613$	35.8025			1.44605			0.723025	+	0.690822i	$0.242751\pi$
$614$		0			0
$615$	−13.3939	−	11.6861i	−0.540093	−	0.471231i
$616$		0			0
$617$		−	24.8143i		−	0.998986i	−0.866318	−	0.499493i	$-0.833520\pi$
$617$		−	24.8143i		−	0.998986i	0.866318	−	0.499493i	$-0.166480\pi$
$618$		0			0
$619$			28.4386i			1.14305i	0.820586	+	0.571523i	$0.193647\pi$
$619$			28.4386i			1.14305i	−0.820586	+	0.571523i	$0.806353\pi$
$620$		0			0
$621$	19.4020	−	29.5210i	0.778574	−	1.18464i
$622$		0			0
$623$	25.9014			1.03772
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−6.91693	+	7.92772i	−0.276236	+	0.316603i
$628$		0			0
$629$			26.4698i			1.05542i
$630$		0			0
$631$		−	29.9771i		−	1.19337i	−0.802476	−	0.596684i	$-0.796485\pi$
$631$		−	29.9771i		−	1.19337i	0.802476	−	0.596684i	$-0.203515\pi$
$632$		0			0
$633$	−15.0656	+	17.2672i	−0.598804	+	0.686309i
$634$		0			0
$635$	21.9082			0.869400
$636$		0			0
$637$	−61.7246			−2.44562
$638$		0			0
$639$	−1.15178	−	8.41842i	−0.0455638	−	0.333027i
$640$		0			0
$641$			31.9181i			1.26069i	0.776316	+	0.630344i	$0.217086\pi$
$641$			31.9181i			1.26069i	−0.776316	+	0.630344i	$0.782914\pi$
$642$		0			0
$643$		−	28.9778i		−	1.14277i	−0.820681	−	0.571386i	$-0.806406\pi$
$643$		−	28.9778i		−	1.14277i	0.820681	−	0.571386i	$-0.193594\pi$
$644$		0			0
$645$	−15.6861	−	13.6861i	−0.617641	−	0.538892i
$646$		0			0
$647$	6.16255			0.242275			0.121137	−	0.992636i	$-0.461346\pi$
$647$	6.16255			0.242275			0.121137	+	0.992636i	$0.461346\pi$
$648$		0			0
$649$	−7.88924			−0.309680
$650$		0			0
$651$	−4.76675	−	4.15898i	−0.186824	−	0.163003i
$652$		0			0
$653$		−	17.5600i		−	0.687177i	−0.939120	−	0.343588i	$-0.888357\pi$
$653$		−	17.5600i		−	0.687177i	0.939120	−	0.343588i	$-0.111643\pi$
$654$		0			0
$655$		−	2.89441i		−	0.113094i
$656$		0			0
$657$	−1.58033	−	11.5506i	−0.0616544	−	0.450634i
$658$		0			0
$659$	−43.2277			−1.68391			−0.841957	−	0.539545i	$-0.818596\pi$
$659$	−43.2277			−1.68391			−0.841957	+	0.539545i	$0.818596\pi$
$660$		0			0
$661$	9.62981			0.374556			0.187278	−	0.982307i	$-0.440033\pi$
$661$	9.62981			0.374556			0.187278	+	0.982307i	$0.440033\pi$
$662$		0			0
$663$	35.7241	−	40.9445i	1.38741	−	1.59015i
$664$		0			0
$665$		−	25.9825i		−	1.00756i
$666$		0			0
$667$			48.7118i			1.88613i
$668$		0			0
$669$	−21.4493	+	24.5837i	−0.829276	+	0.950461i
$670$		0			0
$671$	−9.49790			−0.366662
$672$		0			0
$673$	38.1099			1.46903			0.734515	−	0.678592i	$-0.237410\pi$
$673$	38.1099			1.46903			0.734515	+	0.678592i	$0.237410\pi$
$674$		0			0
$675$	−2.85387	+	4.34229i	−0.109845	+	0.167135i
$676$		0			0
$677$			0.356702i			0.0137092i	0.999977	+	0.00685459i	$0.00218190\pi$
$677$			0.356702i			0.0137092i	−0.999977	+	0.00685459i	$0.997818\pi$
$678$		0			0
$679$			32.6224i			1.25193i
$680$		0			0
$681$	−8.02886	−	7.00517i	−0.307666	−	0.268439i
$682$		0			0
$683$	34.5509			1.32205			0.661026	−	0.750363i	$-0.270121\pi$
$683$	34.5509			1.32205			0.661026	+	0.750363i	$0.270121\pi$
$684$		0			0
$685$	11.4831			0.438745
$686$		0			0
$687$	−18.7630	−	16.3707i	−0.715854	−	0.624582i
$688$		0			0
$689$		−	31.9947i		−	1.21890i
$690$		0			0
$691$			5.20151i			0.197875i	0.995094	+	0.0989375i	$0.0315444\pi$
$691$			5.20151i			0.197875i	−0.995094	+	0.0989375i	$0.968456\pi$
$692$		0			0
$693$	−12.7138	+	1.73947i	−0.482959	+	0.0660770i
$694$		0			0
$695$	4.53590			0.172056
$696$		0			0
$697$	−58.9229			−2.23187
$698$		0			0
$699$	−32.0580	+	36.7427i	−1.21254	+	1.38974i
$700$		0			0
$701$		−	5.08716i		−	0.192139i	−0.995375	−	0.0960697i	$-0.969373\pi$
$701$		−	5.08716i		−	0.192139i	0.995375	−	0.0960697i	$-0.0306272\pi$
$702$		0			0
$703$			28.0041i			1.05620i
$704$		0			0
$705$	5.79691	−	6.64403i	0.218324	−	0.250229i
$706$		0			0
$707$	−28.5252			−1.07280
$708$		0			0
$709$	20.9364			0.786284			0.393142	−	0.919478i	$-0.371388\pi$
$709$	20.9364			0.786284			0.393142	+	0.919478i	$0.371388\pi$
$710$		0			0
$711$	27.8010	−	3.80366i	1.04262	−	0.142648i
$712$		0			0
$713$			5.80500i			0.217399i
$714$		0			0
$715$			5.46410i			0.204346i
$716$		0			0
$717$	33.6893	+	29.3939i	1.25815	+	1.09773i
$718$		0			0
$719$	16.4968			0.615228			0.307614	−	0.951511i	$-0.400469\pi$
$719$	16.4968			0.615228			0.307614	+	0.951511i	$0.400469\pi$
$720$		0			0
$721$	−55.5210			−2.06771
$722$		0			0
$723$	−12.5211	−	10.9246i	−0.465663	−	0.406291i
$724$		0			0
$725$		−	7.16509i		−	0.266105i
$726$		0			0
$727$		−	27.7183i		−	1.02801i	−0.857786	−	0.514007i	$-0.828161\pi$
$727$		−	27.7183i		−	1.02801i	0.857786	−	0.514007i	$-0.171839\pi$
$728$		0			0
$729$	−10.7109	−	24.7846i	−0.396700	−	0.917949i
$730$		0			0
$731$	−69.0072			−2.55232
$732$		0			0
$733$	1.95137			0.0720753			0.0360377	−	0.999350i	$-0.488526\pi$
$733$	1.95137			0.0720753			0.0360377	+	0.999350i	$0.488526\pi$
$734$		0			0
$735$	12.8633	−	14.7431i	0.474472	−	0.543808i
$736$		0			0
$737$		−	6.24363i		−	0.229987i
$738$		0			0
$739$			24.0272i			0.883854i	0.897051	+	0.441927i	$0.145705\pi$
$739$			24.0272i			0.883854i	−0.897051	+	0.441927i	$0.854295\pi$
$740$		0			0
$741$	37.7948	−	43.3179i	1.38843	−	1.59132i
$742$		0			0
$743$	32.3559			1.18702			0.593511	−	0.804826i	$-0.297741\pi$
$743$	32.3559			1.18702			0.593511	+	0.804826i	$0.297741\pi$
$744$		0			0
$745$	6.05538			0.221852
$746$		0			0
$747$	−0.721958	−	5.27681i	−0.0264151	−	0.193068i
$748$		0			0
$749$		−	11.4155i		−	0.417112i
$750$		0			0
$751$		−	20.8855i		−	0.762122i	−0.924550	−	0.381061i	$-0.875559\pi$
$751$		−	20.8855i		−	0.762122i	0.924550	−	0.381061i	$-0.124441\pi$
$752$		0			0
$753$	2.08424	+	1.81849i	0.0759538	+	0.0662696i
$754$		0			0
$755$	−9.28575			−0.337943
$756$		0			0
$757$	2.91156			0.105822			0.0529112	−	0.998599i	$-0.483150\pi$
$757$	2.91156			0.105822			0.0529112	+	0.998599i	$0.483150\pi$
$758$		0			0
$759$	8.87282	+	7.74153i	0.322063	+	0.281000i
$760$		0			0
$761$		−	0.444093i		−	0.0160984i	−0.999968	−	0.00804919i	$-0.997438\pi$
$761$		−	0.444093i		−	0.0160984i	0.999968	−	0.00804919i	$-0.00256216\pi$
$762$		0			0
$763$			52.3712i			1.89597i
$764$		0			0
$765$	2.33487	+	17.0656i	0.0844173	+	0.617008i
$766$		0			0
$767$	43.1076			1.55653
$768$		0			0
$769$	−44.5418			−1.60622			−0.803110	−	0.595831i	$-0.796823\pi$
$769$	−44.5418			−1.60622			−0.803110	+	0.595831i	$0.796823\pi$
$770$		0			0
$771$	−18.0933	+	20.7373i	−0.651614	+	0.746836i
$772$		0			0
$773$			10.7354i			0.386125i	0.981186	+	0.193063i	$0.0618421\pi$
$773$			10.7354i			0.386125i	−0.981186	+	0.193063i	$0.938158\pi$
$774$		0			0
$775$		−	0.853867i		−	0.0306718i
$776$		0			0
$777$	−22.4554	+	25.7368i	−0.805582	+	0.923304i
$778$		0			0
$779$	−62.3384			−2.23351
$780$		0			0
$781$	2.83228			0.101347
$782$		0			0
$783$	31.1129	+	20.4482i	1.11188	+	0.730759i
$784$		0			0
$785$		−	22.8297i		−	0.814825i
$786$		0			0
$787$		−	12.2702i		−	0.437384i	−0.975794	−	0.218692i	$-0.929821\pi$
$787$		−	12.2702i		−	0.437384i	0.975794	−	0.218692i	$-0.0701790\pi$
$788$		0			0
$789$	14.5330	+	12.6800i	0.517389	+	0.451422i
$790$		0			0
$791$	33.7456			1.19986
$792$		0			0
$793$	51.8975			1.84293
$794$		0			0
$795$	7.64204	+	6.66767i	0.271035	+	0.236478i
$796$		0			0
$797$		−	54.8882i		−	1.94424i	−0.234485	−	0.972120i	$-0.575340\pi$
$797$		−	54.8882i		−	1.94424i	0.234485	−	0.972120i	$-0.424660\pi$
$798$		0			0
$799$		−	29.2287i		−	1.03404i
$800$		0			0
$801$	−17.9985	+	2.46250i	−0.635945	+	0.0870081i
$802$		0			0
$803$	3.88608			0.137137
$804$		0			0
$805$	−29.0800			−1.02494
$806$		0			0
$807$	21.2980	−	24.4103i	0.749724	−	0.859283i
$808$		0			0
$809$		−	44.5315i		−	1.56564i	−0.622246	−	0.782822i	$-0.713780\pi$
$809$		−	44.5315i		−	1.56564i	0.622246	−	0.782822i	$-0.286220\pi$
$810$		0			0
$811$		−	6.03642i		−	0.211968i	−0.994368	−	0.105984i	$-0.966201\pi$
$811$		−	6.03642i		−	0.211968i	0.994368	−	0.105984i	$-0.0337992\pi$
$812$		0			0
$813$	17.9420	−	20.5639i	0.629253	−	0.721208i
$814$		0			0
$815$	6.83228			0.239324
$816$		0			0
$817$	−73.0072			−2.55420
$818$		0			0
$819$	69.4697	−	9.50464i	2.42747	−	0.332119i
$820$		0			0
$821$			26.9598i			0.940904i	0.882426	+	0.470452i	$0.155909\pi$
$821$			26.9598i			0.940904i	−0.882426	+	0.470452i	$0.844091\pi$
$822$		0			0
$823$		−	33.3881i		−	1.16384i	−0.813247	−	0.581918i	$-0.802302\pi$
$823$		−	33.3881i		−	1.16384i	0.813247	−	0.581918i	$-0.197698\pi$
$824$		0			0
$825$	−1.30512	−	1.13871i	−0.0454384	−	0.0396449i
$826$		0			0
$827$	0.782685			0.0272166			0.0136083	−	0.999907i	$-0.495668\pi$
$827$	0.782685			0.0272166			0.0136083	+	0.999907i	$0.495668\pi$
$828$		0			0
$829$	12.3121			0.427616			0.213808	−	0.976876i	$-0.431413\pi$
$829$	12.3121			0.427616			0.213808	+	0.976876i	$0.431413\pi$
$830$		0			0
$831$	−25.4565	−	22.2108i	−0.883077	−	0.770484i
$832$		0			0
$833$		−	64.8585i		−	2.24721i
$834$		0			0
$835$			8.31797i			0.287855i
$836$		0			0
$837$	3.70773	+	2.43682i	0.128158	+	0.0842289i
$838$		0			0
$839$	−27.5264			−0.950318			−0.475159	−	0.879900i	$-0.657609\pi$
$839$	−27.5264			−0.950318			−0.475159	+	0.879900i	$0.657609\pi$
$840$		0			0
$841$	−22.3385			−0.770293
$842$		0			0
$843$	−33.8564	+	38.8039i	−1.16608	+	1.33648i
$844$		0			0
$845$		−	16.8564i		−	0.579878i
$846$		0			0
$847$		−	4.27743i		−	0.146974i
$848$		0			0
$849$	15.4615	−	17.7209i	0.530636	−	0.608180i
$850$		0			0
$851$	31.3426			1.07441
$852$		0			0
$853$	38.9503			1.33363			0.666817	−	0.745222i	$-0.267656\pi$
$853$	38.9503			1.33363			0.666817	+	0.745222i	$0.267656\pi$
$854$		0			0
$855$	2.47021	+	18.0548i	0.0844793	+	0.617461i
$856$		0			0
$857$			40.2185i			1.37384i	0.726735	+	0.686918i	$0.241037\pi$
$857$			40.2185i			1.37384i	−0.726735	+	0.686918i	$0.758963\pi$
$858$		0			0
$859$		−	0.210799i		−	0.00719238i	−0.999994	−	0.00359619i	$-0.998855\pi$
$859$		−	0.210799i		−	0.00719238i	0.999994	−	0.00359619i	$-0.00114471\pi$
$860$		0			0
$861$	−57.2914	−	49.9866i	−1.95248	−	1.70354i
$862$		0			0
$863$	47.6867			1.62327			0.811637	−	0.584163i	$-0.198577\pi$
$863$	47.6867			1.62327			0.811637	+	0.584163i	$0.198577\pi$
$864$		0			0
$865$	−5.10971			−0.173735
$866$		0			0
$867$	20.8364	+	18.1797i	0.707641	+	0.617416i
$868$		0			0
$869$			9.35334i			0.317290i
$870$		0			0
$871$			34.1158i			1.15597i
$872$		0			0
$873$	−3.10148	−	22.6688i	−0.104969	−	0.767222i
$874$		0			0
$875$	4.27743			0.144603
$876$		0			0
$877$	−31.8619			−1.07590			−0.537949	−	0.842977i	$-0.680801\pi$
$877$	−31.8619			−1.07590			−0.537949	+	0.842977i	$0.680801\pi$
$878$		0			0
$879$	−4.42873	+	5.07591i	−0.149377	+	0.171206i
$880$		0			0
$881$			2.60310i			0.0877008i	0.999038	+	0.0438504i	$0.0139625\pi$
$881$			2.60310i			0.0877008i	−0.999038	+	0.0438504i	$0.986038\pi$
$882$		0			0
$883$		−	46.0875i		−	1.55097i	−0.631367	−	0.775484i	$-0.717506\pi$
$883$		−	46.0875i		−	1.55097i	0.631367	−	0.775484i	$-0.282494\pi$
$884$		0			0
$885$	−8.98358	+	10.2964i	−0.301980	+	0.346109i
$886$		0			0
$887$	−1.92926			−0.0647781			−0.0323890	−	0.999475i	$-0.510312\pi$
$887$	−1.92926			−0.0647781			−0.0323890	+	0.999475i	$0.510312\pi$
$888$		0			0
$889$	93.7107			3.14296
$890$		0			0
$891$	8.66925	−	2.41746i	0.290431	−	0.0809878i
$892$		0			0
$893$		−	30.9229i		−	1.03480i
$894$		0			0
$895$		−	5.44830i		−	0.182117i
$896$		0			0
$897$	−48.4820	−	42.3005i	−1.61877	−	1.41237i
$898$		0			0
$899$	−6.11803			−0.204048
$900$		0			0
$901$	33.6192			1.12002
$902$		0			0
$903$	−67.0964	−	58.5415i	−2.23283	−	1.94814i
$904$		0			0
$905$			7.62665i			0.253518i
$906$		0			0
$907$		−	8.69500i		−	0.288713i	−0.989526	−	0.144356i	$-0.953889\pi$
$907$		−	8.69500i		−	0.288713i	0.989526	−	0.144356i	$-0.0461111\pi$
$908$		0			0
$909$	19.8216	−	2.71194i	0.657443	−	0.0899494i
$910$		0			0
$911$	−32.8805			−1.08938			−0.544690	−	0.838637i	$-0.683353\pi$
$911$	−32.8805			−1.08938			−0.544690	+	0.838637i	$0.683353\pi$
$912$		0			0
$913$	1.77532			0.0587546
$914$		0			0
$915$	−10.8154	+	12.3959i	−0.357546	+	0.409795i
$916$		0			0
$917$		−	12.3806i		−	0.408844i
$918$		0			0
$919$			2.01072i			0.0663276i	0.999450	+	0.0331638i	$0.0105583\pi$
$919$			2.01072i			0.0663276i	−0.999450	+	0.0331638i	$0.989442\pi$
$920$		0			0
$921$	7.53844	−	8.64005i	0.248400	−	0.284699i
$922$		0			0
$923$	−15.4759			−0.509395
$924$		0			0
$925$	−4.61023			−0.151584
$926$		0			0
$927$	38.5806	−	5.27848i	1.26715	−	0.173368i
$928$		0			0
$929$		−	0.490011i		−	0.0160767i	−0.999968	−	0.00803837i	$-0.997441\pi$
$929$		−	0.490011i		−	0.0160767i	0.999968	−	0.00803837i	$-0.00255872\pi$
$930$		0			0
$931$		−	68.6180i		−	2.24887i
$932$		0			0
$933$	31.5445	+	27.5225i	1.03272	+	0.901046i
$934$		0			0
$935$	−5.74153			−0.187768
$936$		0			0
$937$	8.21749			0.268454			0.134227	−	0.990951i	$-0.457145\pi$
$937$	8.21749			0.268454			0.134227	+	0.990951i	$0.457145\pi$
$938$		0			0
$939$	−8.89345	−	7.75952i	−0.290227	−	0.253222i
$940$		0			0
$941$			14.9191i			0.486350i	0.969982	+	0.243175i	$0.0781890\pi$
$941$			14.9191i			0.486350i	−0.969982	+	0.243175i	$0.921811\pi$
$942$		0			0
$943$			69.7701i			2.27203i
$944$		0			0
$945$	−12.2072	+	18.5738i	−0.397100	+	0.604206i
$946$		0			0
$947$	9.05748			0.294329			0.147164	−	0.989112i	$-0.452985\pi$
$947$	9.05748			0.294329			0.147164	+	0.989112i	$0.452985\pi$
$948$		0			0
$949$	−21.2340			−0.689284
$950$		0			0
$951$	−30.3692	+	34.8071i	−0.984788	+	1.12870i
$952$		0			0
$953$		−	28.0852i		−	0.909769i	−0.890550	−	0.454884i	$-0.849681\pi$
$953$		−	28.0852i		−	0.909769i	0.890550	−	0.454884i	$-0.150319\pi$
$954$		0			0
$955$			5.75637i			0.186272i
$956$		0			0
$957$	−8.15898	+	9.35128i	−0.263743	+	0.302284i
$958$		0			0
$959$	49.1179			1.58610
$960$		0			0
$961$	30.2709			0.976481
$962$		0			0
$963$	1.08529	+	7.93241i	0.0349730	+	0.255618i
$964$		0			0
$965$			1.57127i			0.0505810i
$966$		0			0
$967$		−	3.17403i		−	0.102070i	−0.998697	−	0.0510349i	$-0.983748\pi$
$967$		−	3.17403i		−	0.102070i	0.998697	−	0.0510349i	$-0.0162520\pi$
$968$		0			0
$969$	45.5172	+	39.7137i	1.46222	+	1.27579i
$970$		0			0
$971$	−32.8522			−1.05428			−0.527139	−	0.849779i	$-0.676735\pi$
$971$	−32.8522			−1.05428			−0.527139	+	0.849779i	$0.676735\pi$
$972$		0			0
$973$	19.4020			0.621999
$974$		0			0
$975$	7.13129	+	6.22205i	0.228384	+	0.199265i
$976$		0			0
$977$			33.4997i			1.07175i	0.844297	+	0.535875i	$0.180018\pi$
$977$			33.4997i			1.07175i	−0.844297	+	0.535875i	$0.819982\pi$
$978$		0			0
$979$		−	6.05538i		−	0.193531i
$980$		0			0
$981$	−4.97903	−	36.3919i	−0.158968	−	1.16190i
$982$		0			0
$983$	48.6909			1.55300			0.776499	−	0.630119i	$-0.216994\pi$
$983$	48.6909			1.55300			0.776499	+	0.630119i	$0.216994\pi$
$984$		0			0
$985$	15.3723			0.489802
$986$		0			0
$987$	24.7959	−	28.4194i	0.789261	−	0.904598i
$988$		0			0
$989$			81.7107i			2.59825i
$990$		0			0
$991$		−	21.0990i		−	0.670231i	−0.942177	−	0.335116i	$-0.891225\pi$
$991$		−	21.0990i		−	0.670231i	0.942177	−	0.335116i	$-0.108775\pi$
$992$		0			0
$993$	−14.2410	+	16.3221i	−0.451925	+	0.517966i
$994$		0			0
$995$	−13.4763			−0.427228
$996$		0			0
$997$	49.6326			1.57188			0.785940	−	0.618303i	$-0.212180\pi$
$997$	49.6326			1.57188			0.785940	+	0.618303i	$0.212180\pi$
$998$		0			0
$999$	13.1570	−	20.0190i	0.416269	−	0.633372i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.k.d.1871.3	✓	8
3.2	odd	2		2640.2.k.f.1871.5	yes	8
4.3	odd	2		2640.2.k.f.1871.6	yes	8
12.11	even	2	inner	2640.2.k.d.1871.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2640.2.k.d.1871.3	✓	8	1.1	even	1	trivial
2640.2.k.d.1871.4	yes	8	12.11	even	2	inner
2640.2.k.f.1871.5	yes	8	3.2	odd	2
2640.2.k.f.1871.6	yes	8	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:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.30512 - 1.13871i) q^{3} -1.00000i q^{5} -4.27743i q^{7} +(0.406663 + 2.97231i) q^{9} -1.00000 q^{11} +5.46410 q^{13} +(-1.13871 + 1.30512i) q^{15} +5.74153i q^{17} +6.07434i q^{19} +(-4.87076 + 5.58254i) q^{21} +6.79849 q^{23} -1.00000 q^{25} +(2.85387 - 4.34229i) q^{27} +7.16509i q^{29} +0.853867i q^{31} +(1.30512 + 1.13871i) q^{33} -4.27743 q^{35} +4.61023 q^{37} +(-7.13129 - 6.22205i) q^{39} +10.2626i q^{41} +12.0190i q^{43} +(2.97231 - 0.406663i) q^{45} -5.09075 q^{47} -11.2964 q^{49} +(6.53796 - 7.49337i) q^{51} -5.85544i q^{53} +1.00000i q^{55} +(6.91693 - 7.92772i) q^{57} +7.88924 q^{59} +9.49790 q^{61} +(12.7138 - 1.73947i) q^{63} -5.46410i q^{65} +6.24363i q^{67} +(-8.87282 - 7.74153i) q^{69} -2.83228 q^{71} -3.88608 q^{73} +(1.30512 + 1.13871i) q^{75} +4.27743i q^{77} -9.35334i q^{79} +(-8.66925 + 2.41746i) q^{81} -1.77532 q^{83} +5.74153 q^{85} +(8.15898 - 9.35128i) q^{87} +6.05538i q^{89} -23.3723i q^{91} +(0.972310 - 1.11440i) q^{93} +6.07434 q^{95} -7.62665 q^{97} +(-0.406663 - 2.97231i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 6 q^{9} - 8 q^{11} + 16 q^{13} + 2 q^{15} - 14 q^{21} + 8 q^{23} - 8 q^{25} + 8 q^{27} + 4 q^{33} - 12 q^{35} + 24 q^{37} - 20 q^{39} + 8 q^{45} - 24 q^{47} + 4 q^{49} + 18 q^{51} + 8 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 6 * q^9 - 8 * q^11 + 16 * q^13 + 2 * q^15 - 14 * q^21 + 8 * q^23 - 8 * q^25 + 8 * q^27 + 4 * q^33 - 12 * q^35 + 24 * q^37 - 20 * q^39 + 8 * q^45 - 24 * q^47 + 4 * q^49 + 18 * q^51 + 8 * q^57 + 28 * q^61 + 36 * q^63 + 16 * q^69 + 44 * q^71 - 24 * q^73 + 4 * q^75 + 2 * q^81 + 56 * q^83 - 4 * q^85 + 44 * q^87 - 8 * q^93 + 8 * q^95 - 72 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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