LMFDB - Embedded newform 1120.2.e.a.1119.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1120,2,Mod(1119,1120)] 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("1120.1119"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1119.16
Character	$\chi$	$=$	1120.1119
Dual form			1120.2.e.a.1119.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.06386i q^{3} +(1.39003 + 1.75152i) q^{5} +(-2.46063 - 0.972270i) q^{7} -6.38723 q^{9} -5.46601i q^{11} +1.22883 q^{13} +(-5.36640 + 4.25886i) q^{15} -0.813283 q^{17} -6.68583 q^{19} +(2.97890 - 7.53902i) q^{21} -3.51731 q^{23} +(-1.13563 + 4.86933i) q^{25} -10.3780i q^{27} -8.09020 q^{29} -9.00726 q^{31} +16.7471 q^{33} +(-1.71740 - 5.66132i) q^{35} +4.61599i q^{37} +3.76496i q^{39} -2.59512i q^{41} -1.61833 q^{43} +(-8.87844 - 11.1873i) q^{45} +5.97579i q^{47} +(5.10938 + 4.78479i) q^{49} -2.49178i q^{51} -1.51536i q^{53} +(9.57381 - 7.59792i) q^{55} -20.4844i q^{57} +10.1846 q^{59} +3.16168i q^{61} +(15.7166 + 6.21011i) q^{63} +(1.70811 + 2.15231i) q^{65} +13.1445 q^{67} -10.7765i q^{69} +12.4755i q^{71} +7.65228 q^{73} +(-14.9189 - 3.47940i) q^{75} +(-5.31443 + 13.4498i) q^{77} +2.19667i q^{79} +12.6350 q^{81} +7.88172i q^{83} +(-1.13049 - 1.42448i) q^{85} -24.7872i q^{87} +1.19849i q^{89} +(-3.02369 - 1.19475i) q^{91} -27.5970i q^{93} +(-9.29351 - 11.7103i) q^{95} -0.813283 q^{97} +34.9126i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 8 q^{21} - 16 q^{25} + 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q - 48 * q^9 + 8 * q^21 - 16 * q^25 + 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 + 32 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$			3.06386i			1.76892i	0.466617	+	0.884460i	$0.345473\pi$
$3$			3.06386i			1.76892i	−0.466617	+	0.884460i	$0.654527\pi$
$4$		0			0
$5$	1.39003	+	1.75152i	0.621641	+	0.783303i
$5$	1.39003	+	1.75152i	0.621641	+	0.783303i
$6$		0			0
$7$	−2.46063	−	0.972270i	−0.930030	−	0.367483i
$7$	−2.46063	−	0.972270i	−0.930030	−	0.367483i
$8$		0			0
$9$	−6.38723			−2.12908
$10$		0			0
$11$		−	5.46601i		−	1.64806i	−0.566544	−	0.824032i	$-0.691720\pi$
$11$		−	5.46601i		−	1.64806i	0.566544	−	0.824032i	$-0.308280\pi$
$12$		0			0
$13$	1.22883			0.340816			0.170408	−	0.985374i	$-0.445491\pi$
$13$	1.22883			0.340816			0.170408	+	0.985374i	$0.445491\pi$
$14$		0			0
$15$	−5.36640	+	4.25886i	−1.38560	+	1.09963i
$16$		0			0
$17$	−0.813283			−0.197250			−0.0986250	−	0.995125i	$-0.531444\pi$
$17$	−0.813283			−0.197250			−0.0986250	+	0.995125i	$0.531444\pi$
$18$		0			0
$19$	−6.68583			−1.53383			−0.766917	−	0.641746i	$-0.778210\pi$
$19$	−6.68583			−1.53383			−0.766917	+	0.641746i	$0.778210\pi$
$20$		0			0
$21$	2.97890	−	7.53902i	0.650048	−	1.64515i
$22$		0			0
$23$	−3.51731			−0.733410			−0.366705	−	0.930337i	$-0.619514\pi$
$23$	−3.51731			−0.733410			−0.366705	+	0.930337i	$0.619514\pi$
$24$		0			0
$25$	−1.13563	+	4.86933i	−0.227126	+	0.973865i
$26$		0			0
$27$		−	10.3780i		−	1.99724i
$28$		0			0
$29$	−8.09020			−1.50231			−0.751156	−	0.660125i	$-0.770503\pi$
$29$	−8.09020			−1.50231			−0.751156	+	0.660125i	$0.770503\pi$
$30$		0			0
$31$	−9.00726			−1.61775			−0.808876	−	0.587980i	$-0.799923\pi$
$31$	−9.00726			−1.61775			−0.808876	+	0.587980i	$0.799923\pi$
$32$		0			0
$33$	16.7471			2.91529
$34$		0			0
$35$	−1.71740	−	5.66132i	−0.290294	−	0.956938i
$36$		0			0
$37$			4.61599i			0.758864i	0.925220	+	0.379432i	$0.123881\pi$
$37$			4.61599i			0.758864i	−0.925220	+	0.379432i	$0.876119\pi$
$38$		0			0
$39$			3.76496i			0.602875i
$40$		0			0
$41$		−	2.59512i		−	0.405290i	−0.979252	−	0.202645i	$-0.935046\pi$
$41$		−	2.59512i		−	0.405290i	0.979252	−	0.202645i	$-0.0649538\pi$
$42$		0			0
$43$	−1.61833			−0.246793			−0.123397	−	0.992357i	$-0.539379\pi$
$43$	−1.61833			−0.246793			−0.123397	+	0.992357i	$0.539379\pi$
$44$		0			0
$45$	−8.87844	−	11.1873i	−1.32352	−	1.66771i
$46$		0			0
$47$			5.97579i			0.871658i	0.900029	+	0.435829i	$0.143545\pi$
$47$			5.97579i			0.871658i	−0.900029	+	0.435829i	$0.856455\pi$
$48$		0			0
$49$	5.10938	+	4.78479i	0.729912	+	0.683541i
$50$		0			0
$51$		−	2.49178i		−	0.348919i
$52$		0			0
$53$		−	1.51536i		−	0.208151i	−0.994569	−	0.104076i	$-0.966812\pi$
$53$		−	1.51536i		−	0.208151i	0.994569	−	0.104076i	$-0.0331884\pi$
$54$		0			0
$55$	9.57381	−	7.59792i	1.29093	−	1.02450i
$56$		0			0
$57$		−	20.4844i		−	2.71323i
$58$		0			0
$59$	10.1846			1.32593			0.662963	−	0.748652i	$-0.269299\pi$
$59$	10.1846			1.32593			0.662963	+	0.748652i	$0.269299\pi$
$60$		0			0
$61$			3.16168i			0.404811i	0.979302	+	0.202406i	$0.0648760\pi$
$61$			3.16168i			0.404811i	−0.979302	+	0.202406i	$0.935124\pi$
$62$		0			0
$63$	15.7166	+	6.21011i	1.98010	+	0.782400i
$64$		0			0
$65$	1.70811	+	2.15231i	0.211865	+	0.266962i
$66$		0			0
$67$	13.1445			1.60586			0.802931	−	0.596072i	$-0.203273\pi$
$67$	13.1445			1.60586			0.802931	+	0.596072i	$0.203273\pi$
$68$		0			0
$69$		−	10.7765i		−	1.29734i
$70$		0			0
$71$			12.4755i			1.48057i	0.672293	+	0.740285i	$0.265310\pi$
$71$			12.4755i			1.48057i	−0.672293	+	0.740285i	$0.734690\pi$
$72$		0			0
$73$	7.65228			0.895631			0.447816	−	0.894126i	$-0.352202\pi$
$73$	7.65228			0.895631			0.447816	+	0.894126i	$0.352202\pi$
$74$		0			0
$75$	−14.9189	−	3.47940i	−1.72269	−	0.401767i
$76$		0			0
$77$	−5.31443	+	13.4498i	−0.605636	+	1.53275i
$78$		0			0
$79$			2.19667i			0.247145i	0.992336	+	0.123572i	$0.0394351\pi$
$79$			2.19667i			0.247145i	−0.992336	+	0.123572i	$0.960565\pi$
$80$		0			0
$81$	12.6350			1.40389
$82$		0			0
$83$			7.88172i			0.865131i	0.901603	+	0.432565i	$0.142392\pi$
$83$			7.88172i			0.865131i	−0.901603	+	0.432565i	$0.857608\pi$
$84$		0			0
$85$	−1.13049	−	1.42448i	−0.122619	−	0.154506i
$86$		0			0
$87$		−	24.7872i		−	2.65747i
$88$		0			0
$89$			1.19849i			0.127040i	0.997981	+	0.0635198i	$0.0202326\pi$
$89$			1.19849i			0.127040i	−0.997981	+	0.0635198i	$0.979767\pi$
$90$		0			0
$91$	−3.02369	−	1.19475i	−0.316969	−	0.125244i
$92$		0			0
$93$		−	27.5970i		−	2.86167i
$94$		0			0
$95$	−9.29351	−	11.7103i	−0.953494	−	1.20146i
$96$		0			0
$97$	−0.813283			−0.0825764			−0.0412882	−	0.999147i	$-0.513146\pi$
$97$	−0.813283			−0.0825764			−0.0412882	+	0.999147i	$0.513146\pi$
$98$		0			0
$99$			34.9126i			3.50885i
$100$		0			0
$101$		−	1.76504i		−	0.175628i	−0.996137	−	0.0878142i	$-0.972012\pi$
$101$		−	1.76504i		−	0.175628i	0.996137	−	0.0878142i	$-0.0279882\pi$
$102$		0			0
$103$			10.0600i			0.991242i	0.868539	+	0.495621i	$0.165060\pi$
$103$			10.0600i			0.991242i	−0.868539	+	0.495621i	$0.834940\pi$
$104$		0			0
$105$	17.3455	−	5.26188i	1.69275	−	0.513507i
$106$		0			0
$107$	−13.1445			−1.27073			−0.635366	−	0.772211i	$-0.719151\pi$
$107$	−13.1445			−1.27073			−0.635366	+	0.772211i	$0.719151\pi$
$108$		0			0
$109$	0.128572			0.0123149			0.00615747	−	0.999981i	$-0.498040\pi$
$109$	0.128572			0.0123149			0.00615747	+	0.999981i	$0.498040\pi$
$110$		0			0
$111$	−14.1427			−1.34237
$112$		0			0
$113$			2.04955i			0.192806i	0.995342	+	0.0964028i	$0.0307337\pi$
$113$			2.04955i			0.192806i	−0.995342	+	0.0964028i	$0.969266\pi$
$114$		0			0
$115$	−4.88917	−	6.16063i	−0.455917	−	0.574482i
$116$		0			0
$117$	−7.84881			−0.725622
$118$		0			0
$119$	2.00119	+	0.790730i	0.183448	+	0.0724861i
$120$		0			0
$121$	−18.8773			−1.71611
$122$		0			0
$123$	7.95109			0.716926
$124$		0			0
$125$	−10.1073	+	4.77944i	−0.904022	+	0.427486i
$126$		0			0
$127$	−5.81155			−0.515691			−0.257846	−	0.966186i	$-0.583013\pi$
$127$	−5.81155			−0.515691			−0.257846	+	0.966186i	$0.583013\pi$
$128$		0			0
$129$		−	4.95834i		−	0.436557i
$130$		0			0
$131$	16.1544			1.41142			0.705709	−	0.708501i	$-0.250628\pi$
$131$	16.1544			1.41142			0.705709	+	0.708501i	$0.250628\pi$
$132$		0			0
$133$	16.4513	+	6.50043i	1.42651	+	0.563659i
$134$		0			0
$135$	18.1772	−	14.4257i	1.56445	−	1.24157i
$136$		0			0
$137$		−	5.90852i		−	0.504799i	−0.967623	−	0.252400i	$-0.918780\pi$
$137$		−	5.90852i		−	0.504799i	0.967623	−	0.252400i	$-0.0812197\pi$
$138$		0			0
$139$	−3.82751			−0.324645			−0.162323	−	0.986738i	$-0.551899\pi$
$139$	−3.82751			−0.324645			−0.162323	+	0.986738i	$0.551899\pi$
$140$		0			0
$141$	−18.3090			−1.54189
$142$		0			0
$143$		−	6.71679i		−	0.561686i
$144$		0			0
$145$	−11.2456	−	14.1701i	−0.933898	−	1.17676i
$146$		0			0
$147$	−14.6599	+	15.6544i	−1.20913	+	1.29116i
$148$		0			0
$149$	−7.10275			−0.581880			−0.290940	−	0.956741i	$-0.593968\pi$
$149$	−7.10275			−0.581880			−0.290940	+	0.956741i	$0.593968\pi$
$150$		0			0
$151$		−	10.7144i		−	0.871924i	−0.899965	−	0.435962i	$-0.856408\pi$
$151$		−	10.7144i		−	0.871924i	0.899965	−	0.435962i	$-0.143592\pi$
$152$		0			0
$153$	5.19462			0.419960
$154$		0			0
$155$	−12.5204	−	15.7764i	−1.00566	−	1.26719i
$156$		0			0
$157$	5.12741			0.409212			0.204606	−	0.978844i	$-0.434409\pi$
$157$	5.12741			0.409212			0.204606	+	0.978844i	$0.434409\pi$
$158$		0			0
$159$	4.64286			0.368202
$160$		0			0
$161$	8.65479	+	3.41977i	0.682093	+	0.269516i
$162$		0			0
$163$	−13.3674			−1.04701			−0.523507	−	0.852021i	$-0.675377\pi$
$163$	−13.3674			−1.04701			−0.523507	+	0.852021i	$0.675377\pi$
$164$		0			0
$165$	23.2790	+	29.3328i	1.81226	+	2.28356i
$166$		0			0
$167$			20.3403i			1.57398i	0.616966	+	0.786990i	$0.288362\pi$
$167$			20.3403i			1.57398i	−0.616966	+	0.786990i	$0.711638\pi$
$168$		0			0
$169$	−11.4900			−0.883845
$170$		0			0
$171$	42.7039			3.26565
$172$		0			0
$173$	−7.38042			−0.561123			−0.280561	−	0.959836i	$-0.590521\pi$
$173$	−7.38042			−0.561123			−0.280561	+	0.959836i	$0.590521\pi$
$174$		0			0
$175$	7.52866	−	10.8775i	0.569113	−	0.822259i
$176$		0			0
$177$			31.2043i			2.34546i
$178$		0			0
$179$		−	14.6037i		−	1.09153i	−0.837937	−	0.545767i	$-0.816239\pi$
$179$		−	14.6037i		−	1.09153i	0.837937	−	0.545767i	$-0.183761\pi$
$180$		0			0
$181$		−	11.2568i		−	0.836711i	−0.908283	−	0.418355i	$-0.862607\pi$
$181$		−	11.2568i		−	0.836711i	0.908283	−	0.418355i	$-0.137393\pi$
$182$		0			0
$183$	−9.68693			−0.716079
$184$		0			0
$185$	−8.08499	+	6.41637i	−0.594420	+	0.471741i
$186$		0			0
$187$			4.44541i			0.325081i
$188$		0			0
$189$	−10.0902	+	25.5364i	−0.733954	+	1.85750i
$190$		0			0
$191$			8.19284i			0.592813i	0.955062	+	0.296406i	$0.0957883\pi$
$191$			8.19284i			0.592813i	−0.955062	+	0.296406i	$0.904212\pi$
$192$		0			0
$193$			17.6355i			1.26943i	0.772747	+	0.634715i	$0.218882\pi$
$193$			17.6355i			1.26943i	−0.772747	+	0.634715i	$0.781118\pi$
$194$		0			0
$195$	−6.59439	+	5.23340i	−0.472234	+	0.374772i
$196$		0			0
$197$			6.60429i			0.470536i	0.971931	+	0.235268i	$0.0755968\pi$
$197$			6.60429i			0.470536i	−0.971931	+	0.235268i	$0.924403\pi$
$198$		0			0
$199$	6.35712			0.450644			0.225322	−	0.974284i	$-0.427657\pi$
$199$	6.35712			0.450644			0.225322	+	0.974284i	$0.427657\pi$
$200$		0			0
$201$			40.2730i			2.84064i
$202$		0			0
$203$	19.9070	+	7.86585i	1.39720	+	0.552075i
$204$		0			0
$205$	4.54541	−	3.60730i	0.317465	−	0.251945i
$206$		0			0
$207$	22.4658			1.56148
$208$		0			0
$209$			36.5448i			2.52786i
$210$		0			0
$211$		−	9.97949i		−	0.687016i	−0.939150	−	0.343508i	$-0.888385\pi$
$211$		−	9.97949i		−	0.687016i	0.939150	−	0.343508i	$-0.111615\pi$
$212$		0			0
$213$	−38.2232			−2.61901
$214$		0			0
$215$	−2.24953	−	2.83454i	−0.153417	−	0.193314i
$216$		0			0
$217$	22.1635	+	8.75748i	1.50456	+	0.594497i
$218$		0			0
$219$			23.4455i			1.58430i
$220$		0			0
$221$	−0.999385			−0.0672259
$222$		0			0
$223$			7.18795i			0.481341i	0.970607	+	0.240671i	$0.0773673\pi$
$223$			7.18795i			0.481341i	−0.970607	+	0.240671i	$0.922633\pi$
$224$		0			0
$225$	7.25352	−	31.1015i	0.483568	−	2.07343i
$226$		0			0
$227$			3.43607i			0.228060i	0.993477	+	0.114030i	$0.0363760\pi$
$227$			3.43607i			0.228060i	−0.993477	+	0.114030i	$0.963624\pi$
$228$		0			0
$229$		−	4.87652i		−	0.322250i	−0.986934	−	0.161125i	$-0.948488\pi$
$229$		−	4.87652i		−	0.322250i	0.986934	−	0.161125i	$-0.0515122\pi$
$230$		0			0
$231$	−41.2083	−	16.2827i	−2.71131	−	1.07132i
$232$		0			0
$233$			10.4530i			0.684802i	0.939554	+	0.342401i	$0.111240\pi$
$233$			10.4530i			0.684802i	−0.939554	+	0.342401i	$0.888760\pi$
$234$		0			0
$235$	−10.4667	+	8.30653i	−0.682772	+	0.541858i
$236$		0			0
$237$	−6.73028			−0.437179
$238$		0			0
$239$		−	16.9379i		−	1.09562i	−0.836601	−	0.547812i	$-0.815461\pi$
$239$		−	16.9379i		−	1.09562i	0.836601	−	0.547812i	$-0.184539\pi$
$240$		0			0
$241$		−	17.7741i		−	1.14493i	−0.819928	−	0.572466i	$-0.805987\pi$
$241$		−	17.7741i		−	1.14493i	0.819928	−	0.572466i	$-0.194013\pi$
$242$		0			0
$243$			7.57786i			0.486120i
$244$		0			0
$245$	−1.27844	+	15.6002i	−0.0816765	+	0.996659i
$246$		0			0
$247$	−8.21574			−0.522755
$248$		0			0
$249$	−24.1485			−1.53035
$250$		0			0
$251$	−15.6509			−0.987873			−0.493937	−	0.869498i	$-0.664443\pi$
$251$	−15.6509			−0.987873			−0.493937	+	0.869498i	$0.664443\pi$
$252$		0			0
$253$			19.2256i			1.20871i
$254$		0			0
$255$	4.36440	−	3.46366i	0.273309	−	0.216903i
$256$		0			0
$257$	21.2065			1.32283			0.661414	−	0.750021i	$-0.269957\pi$
$257$	21.2065			1.32283			0.661414	+	0.750021i	$0.269957\pi$
$258$		0			0
$259$	4.48799	−	11.3582i	0.278870	−	0.705766i
$260$		0			0
$261$	51.6739			3.19854
$262$		0			0
$263$	−13.0849			−0.806847			−0.403423	−	0.915013i	$-0.632180\pi$
$263$	−13.0849			−0.806847			−0.403423	+	0.915013i	$0.632180\pi$
$264$		0			0
$265$	2.65418	−	2.10640i	0.163045	−	0.129395i
$266$		0			0
$267$	−3.67200			−0.224723
$268$		0			0
$269$		−	0.357794i		−	0.0218151i	−0.999941	−	0.0109075i	$-0.996528\pi$
$269$		−	0.357794i		−	0.0218151i	0.999941	−	0.0109075i	$-0.00347204\pi$
$270$		0			0
$271$	0.151601			0.00920910			0.00460455	−	0.999989i	$-0.498534\pi$
$271$	0.151601			0.00920910			0.00460455	+	0.999989i	$0.498534\pi$
$272$		0			0
$273$	3.66055	−	9.26416i	0.221547	−	0.560692i
$274$		0			0
$275$	26.6158	+	6.20735i	1.60499	+	0.374318i
$276$		0			0
$277$		−	29.1822i		−	1.75339i	−0.481047	−	0.876695i	$-0.659743\pi$
$277$		−	29.1822i		−	1.75339i	0.481047	−	0.876695i	$-0.340257\pi$
$278$		0			0
$279$	57.5314			3.44431
$280$		0			0
$281$	23.8389			1.42211			0.711054	−	0.703137i	$-0.248218\pi$
$281$	23.8389			1.42211			0.711054	+	0.703137i	$0.248218\pi$
$282$		0			0
$283$		−	14.6344i		−	0.869922i	−0.900449	−	0.434961i	$-0.856762\pi$
$283$		−	14.6344i		−	0.869922i	0.900449	−	0.434961i	$-0.143238\pi$
$284$		0			0
$285$	35.8788	−	28.4740i	2.12528	−	1.68665i
$286$		0			0
$287$	−2.52316	+	6.38564i	−0.148937	+	0.376932i
$288$		0			0
$289$	−16.3386			−0.961092
$290$		0			0
$291$		−	2.49178i		−	0.146071i
$292$		0			0
$293$	−11.4646			−0.669771			−0.334886	−	0.942259i	$-0.608698\pi$
$293$	−11.4646			−0.669771			−0.334886	+	0.942259i	$0.608698\pi$
$294$		0			0
$295$	14.1570	+	17.8386i	0.824250	+	1.03860i
$296$		0			0
$297$	−56.7261			−3.29158
$298$		0			0
$299$	−4.32217			−0.249957
$300$		0			0
$301$	3.98211	+	1.57345i	0.229525	+	0.0906924i
$302$		0			0
$303$	5.40784			0.310672
$304$		0			0
$305$	−5.53773	+	4.39483i	−0.317090	+	0.251647i
$306$		0			0
$307$		−	13.8117i		−	0.788273i	−0.919052	−	0.394136i	$-0.871044\pi$
$307$		−	13.8117i		−	0.788273i	0.919052	−	0.394136i	$-0.128956\pi$
$308$		0			0
$309$	−30.8224			−1.75343
$310$		0			0
$311$	24.0932			1.36620			0.683099	−	0.730325i	$-0.260632\pi$
$311$	24.0932			1.36620			0.683099	+	0.730325i	$0.260632\pi$
$312$		0			0
$313$	−21.6899			−1.22599			−0.612994	−	0.790088i	$-0.710035\pi$
$313$	−21.6899			−1.22599			−0.612994	+	0.790088i	$0.710035\pi$
$314$		0			0
$315$	10.9694	+	36.1601i	0.618058	+	2.03739i
$316$		0			0
$317$		−	12.5279i		−	0.703638i	−0.936068	−	0.351819i	$-0.885563\pi$
$317$		−	12.5279i		−	0.703638i	0.936068	−	0.351819i	$-0.114437\pi$
$318$		0			0
$319$			44.2211i			2.47591i
$320$		0			0
$321$		−	40.2730i		−	2.24782i
$322$		0			0
$323$	5.43747			0.302549
$324$		0			0
$325$	−1.39549	+	5.98357i	−0.0774080	+	0.331909i
$326$		0			0
$327$			0.393926i			0.0217842i
$328$		0			0
$329$	5.81008	−	14.7042i	0.320320	−	0.810668i
$330$		0			0
$331$		−	28.8936i		−	1.58814i	−0.607828	−	0.794069i	$-0.707959\pi$
$331$		−	28.8936i		−	1.58814i	0.607828	−	0.794069i	$-0.292041\pi$
$332$		0			0
$333$		−	29.4834i		−	1.61568i
$334$		0			0
$335$	18.2713	+	23.0229i	0.998269	+	1.25788i
$336$		0			0
$337$		−	12.7694i		−	0.695592i	−0.937570	−	0.347796i	$-0.886930\pi$
$337$		−	12.7694i		−	0.695592i	0.937570	−	0.347796i	$-0.113070\pi$
$338$		0			0
$339$	−6.27954			−0.341058
$340$		0			0
$341$			49.2337i			2.66616i
$342$		0			0
$343$	−7.92019	−	16.7413i	−0.427650	−	0.903944i
$344$		0			0
$345$	18.8753	−	14.9797i	1.01621	−	0.806481i
$346$		0			0
$347$	29.6898			1.59383			0.796916	−	0.604091i	$-0.206463\pi$
$347$	29.6898			1.59383			0.796916	+	0.604091i	$0.206463\pi$
$348$		0			0
$349$			29.5579i			1.58220i	0.611688	+	0.791099i	$0.290491\pi$
$349$			29.5579i			1.58220i	−0.611688	+	0.791099i	$0.709509\pi$
$350$		0			0
$351$		−	12.7528i		−	0.680692i
$352$		0			0
$353$	−21.0795			−1.12195			−0.560973	−	0.827834i	$-0.689573\pi$
$353$	−21.0795			−1.12195			−0.560973	+	0.827834i	$0.689573\pi$
$354$		0			0
$355$	−21.8511	+	17.3413i	−1.15973	+	0.920383i
$356$		0			0
$357$	−2.42268	+	6.13135i	−0.128222	+	0.324506i
$358$		0			0
$359$			14.8898i			0.785854i	0.919570	+	0.392927i	$0.128538\pi$
$359$			14.8898i			0.785854i	−0.919570	+	0.392927i	$0.871462\pi$
$360$		0			0
$361$	25.7003			1.35265
$362$		0			0
$363$		−	57.8372i		−	3.03567i
$364$		0			0
$365$	10.6369	+	13.4031i	0.556761	+	0.701550i
$366$		0			0
$367$			25.1814i			1.31446i	0.753691	+	0.657229i	$0.228272\pi$
$367$			25.1814i			1.31446i	−0.753691	+	0.657229i	$0.771728\pi$
$368$		0			0
$369$			16.5756i			0.862894i
$370$		0			0
$371$	−1.47334	+	3.72874i	−0.0764920	+	0.193587i
$372$		0			0
$373$		−	23.2586i		−	1.20428i	−0.798389	−	0.602142i	$-0.794314\pi$
$373$		−	23.2586i		−	1.20428i	0.798389	−	0.602142i	$-0.205686\pi$
$374$		0			0
$375$	−14.6435	−	30.9672i	−0.756189	−	1.59914i
$376$		0			0
$377$	−9.94146			−0.512011
$378$		0			0
$379$		−	1.86285i		−	0.0956884i	−0.998855	−	0.0478442i	$-0.984765\pi$
$379$		−	1.86285i		−	0.0956884i	0.998855	−	0.0478442i	$-0.0152351\pi$
$380$		0			0
$381$		−	17.8058i		−	0.912216i
$382$		0			0
$383$			5.15731i			0.263526i	0.991281	+	0.131763i	$0.0420638\pi$
$383$			5.15731i			0.263526i	−0.991281	+	0.131763i	$0.957936\pi$
$384$		0			0
$385$	−30.9448	+	9.38734i	−1.57709	+	0.478423i
$386$		0			0
$387$	10.3367			0.525442
$388$		0			0
$389$	10.6749			0.541240			0.270620	−	0.962686i	$-0.412771\pi$
$389$	10.6749			0.541240			0.270620	+	0.962686i	$0.412771\pi$
$390$		0			0
$391$	2.86057			0.144665
$392$		0			0
$393$			49.4949i			2.49669i
$394$		0			0
$395$	−3.84750	+	3.05344i	−0.193589	+	0.153635i
$396$		0			0
$397$	21.1047			1.05922			0.529608	−	0.848243i	$-0.322339\pi$
$397$	21.1047			1.05922			0.529608	+	0.848243i	$0.322339\pi$
$398$		0			0
$399$	−19.9164	+	50.4046i	−0.997067	+	2.52339i
$400$		0			0
$401$	−3.97650			−0.198577			−0.0992886	−	0.995059i	$-0.531657\pi$
$401$	−3.97650			−0.198577			−0.0992886	+	0.995059i	$0.531657\pi$
$402$		0			0
$403$	−11.0684			−0.551355
$404$		0			0
$405$	17.5630	+	22.1304i	0.872714	+	1.09967i
$406$		0			0
$407$	25.2310			1.25066
$408$		0			0
$409$		−	24.1166i		−	1.19249i	−0.802803	−	0.596244i	$-0.796659\pi$
$409$		−	24.1166i		−	1.19249i	0.802803	−	0.596244i	$-0.203341\pi$
$410$		0			0
$411$	18.1029			0.892949
$412$		0			0
$413$	−25.0606	−	9.90221i	−1.23315	−	0.487256i
$414$		0			0
$415$	−13.8050	+	10.9558i	−0.677659	+	0.537801i
$416$		0			0
$417$		−	11.7270i		−	0.574271i
$418$		0			0
$419$	5.64109			0.275585			0.137793	−	0.990461i	$-0.455999\pi$
$419$	5.64109			0.275585			0.137793	+	0.990461i	$0.455999\pi$
$420$		0			0
$421$	14.9114			0.726738			0.363369	−	0.931645i	$-0.381626\pi$
$421$	14.9114			0.726738			0.363369	+	0.931645i	$0.381626\pi$
$422$		0			0
$423$		−	38.1687i		−	1.85583i
$424$		0			0
$425$	0.923587	−	3.96014i	0.0448005	−	0.192095i
$426$		0			0
$427$	3.07400	−	7.77971i	0.148761	−	0.376487i
$428$		0			0
$429$	20.5793			0.993577
$430$		0			0
$431$			17.1329i			0.825264i	0.910898	+	0.412632i	$0.135390\pi$
$431$			17.1329i			0.825264i	−0.910898	+	0.412632i	$0.864610\pi$
$432$		0			0
$433$	−18.2171			−0.875460			−0.437730	−	0.899106i	$-0.644217\pi$
$433$	−18.2171			−0.875460			−0.437730	+	0.899106i	$0.644217\pi$
$434$		0			0
$435$	43.4152	−	34.4550i	2.08160	−	1.65199i
$436$		0			0
$437$	23.5161			1.12493
$438$		0			0
$439$	−29.5281			−1.40930			−0.704650	−	0.709555i	$-0.748896\pi$
$439$	−29.5281			−1.40930			−0.704650	+	0.709555i	$0.748896\pi$
$440$		0			0
$441$	−32.6348	−	30.5615i	−1.55404	−	1.45531i
$442$		0			0
$443$	−4.42623			−0.210296			−0.105148	−	0.994457i	$-0.533532\pi$
$443$	−4.42623			−0.210296			−0.105148	+	0.994457i	$0.533532\pi$
$444$		0			0
$445$	−2.09918	+	1.66594i	−0.0995104	+	0.0789730i
$446$		0			0
$447$		−	21.7618i		−	1.02930i
$448$		0			0
$449$	13.2925			0.627313			0.313657	−	0.949536i	$-0.398446\pi$
$449$	13.2925			0.627313			0.313657	+	0.949536i	$0.398446\pi$
$450$		0			0
$451$	−14.1850			−0.667944
$452$		0			0
$453$	32.8274			1.54236
$454$		0			0
$455$	−2.11039	−	6.95679i	−0.0989367	−	0.326139i
$456$		0			0
$457$		−	12.2949i		−	0.575130i	−0.957761	−	0.287565i	$-0.907154\pi$
$457$		−	12.2949i		−	0.575130i	0.957761	−	0.287565i	$-0.0928458\pi$
$458$		0			0
$459$			8.44023i			0.393956i
$460$		0			0
$461$		−	23.4068i		−	1.09016i	−0.838383	−	0.545082i	$-0.816499\pi$
$461$		−	23.4068i		−	1.09016i	0.838383	−	0.545082i	$-0.183501\pi$
$462$		0			0
$463$	−21.7828			−1.01233			−0.506165	−	0.862437i	$-0.668937\pi$
$463$	−21.7828			−1.01233			−0.506165	+	0.862437i	$0.668937\pi$
$464$		0			0
$465$	48.3366	−	38.3606i	2.24155	−	1.77893i
$466$		0			0
$467$			32.6245i			1.50968i	0.655909	+	0.754840i	$0.272285\pi$
$467$			32.6245i			1.50968i	−0.655909	+	0.754840i	$0.727715\pi$
$468$		0			0
$469$	−32.3439	−	12.7800i	−1.49350	−	0.590128i
$470$		0			0
$471$			15.7097i			0.723863i
$472$		0			0
$473$			8.84582i			0.406731i
$474$		0			0
$475$	7.59262	−	32.5555i	0.348373	−	1.49375i
$476$		0			0
$477$			9.67896i			0.443169i
$478$		0			0
$479$	10.7215			0.489879			0.244939	−	0.969538i	$-0.421232\pi$
$479$	10.7215			0.489879			0.244939	+	0.969538i	$0.421232\pi$
$480$		0			0
$481$			5.67226i			0.258633i
$482$		0			0
$483$	−10.4777	+	26.5170i	−0.476752	+	1.20657i
$484$		0			0
$485$	−1.13049	−	1.42448i	−0.0513328	−	0.0646823i
$486$		0			0
$487$	−35.7679			−1.62080			−0.810398	−	0.585880i	$-0.800749\pi$
$487$	−35.7679			−1.62080			−0.810398	+	0.585880i	$0.800749\pi$
$488$		0			0
$489$		−	40.9557i		−	1.85208i
$490$		0			0
$491$		−	22.9944i		−	1.03772i	−0.854859	−	0.518860i	$-0.826356\pi$
$491$		−	22.9944i		−	1.03772i	0.854859	−	0.518860i	$-0.173644\pi$
$492$		0			0
$493$	6.57962			0.296331
$494$		0			0
$495$	−61.1501	+	48.5296i	−2.74849	+	2.18125i
$496$		0			0
$497$	12.1296	−	30.6976i	0.544085	−	1.37697i
$498$		0			0
$499$			13.0307i			0.583336i	0.956520	+	0.291668i	$0.0942103\pi$
$499$			13.0307i			0.583336i	−0.956520	+	0.291668i	$0.905790\pi$
$500$		0			0
$501$	−62.3198			−2.78424
$502$		0			0
$503$		−	12.3987i		−	0.552832i	−0.961038	−	0.276416i	$-0.910853\pi$
$503$		−	12.3987i		−	0.552832i	0.961038	−	0.276416i	$-0.0891468\pi$
$504$		0			0
$505$	3.09150	−	2.45346i	0.137570	−	0.109178i
$506$		0			0
$507$		−	35.2037i		−	1.56345i
$508$		0			0
$509$			14.8884i			0.659918i	0.943995	+	0.329959i	$0.107035\pi$
$509$			14.8884i			0.659918i	−0.943995	+	0.329959i	$0.892965\pi$
$510$		0			0
$511$	−18.8294	−	7.44008i	−0.832964	−	0.329130i
$512$		0			0
$513$			69.3854i			3.06344i
$514$		0			0
$515$	−17.6203	+	13.9837i	−0.776442	+	0.616196i
$516$		0			0
$517$	32.6637			1.43655
$518$		0			0
$519$		−	22.6126i		−	0.992581i
$520$		0			0
$521$			40.6065i			1.77900i	0.456932	+	0.889502i	$0.348948\pi$
$521$			40.6065i			1.77900i	−0.456932	+	0.889502i	$0.651052\pi$
$522$		0			0
$523$		−	0.346454i		−	0.0151494i	−0.999971	−	0.00757469i	$-0.997589\pi$
$523$		−	0.346454i		−	0.0151494i	0.999971	−	0.00757469i	$-0.00241112\pi$
$524$		0			0
$525$	33.3270	+	23.0667i	1.45451	+	1.00672i
$526$		0			0
$527$	7.32545			0.319101
$528$		0			0
$529$	−10.6285			−0.462110
$530$		0			0
$531$	−65.0515			−2.82300
$532$		0			0
$533$		−	3.18896i		−	0.138129i
$534$		0			0
$535$	−18.2713	−	23.0229i	−0.789938	−	0.995367i
$536$		0			0
$537$	44.7437			1.93083
$538$		0			0
$539$	26.1537	−	27.9279i	1.12652	−	1.20294i
$540$		0			0
$541$	−14.3137			−0.615396			−0.307698	−	0.951484i	$-0.599559\pi$
$541$	−14.3137			−0.615396			−0.307698	+	0.951484i	$0.599559\pi$
$542$		0			0
$543$	34.4892			1.48007
$544$		0			0
$545$	0.178719	+	0.225196i	0.00765547	+	0.00964633i
$546$		0			0
$547$	39.7534			1.69973			0.849865	−	0.527000i	$-0.176683\pi$
$547$	39.7534			1.69973			0.849865	+	0.527000i	$0.176683\pi$
$548$		0			0
$549$		−	20.1944i		−	0.861874i
$550$		0			0
$551$	54.0897			2.30430
$552$		0			0
$553$	2.13575	−	5.40519i	0.0908215	−	0.229852i
$554$		0			0
$555$	−19.6588	−	24.7713i	−0.834471	−	1.05148i
$556$		0			0
$557$			33.2813i			1.41018i	0.709120	+	0.705088i	$0.249092\pi$
$557$			33.2813i			1.41018i	−0.709120	+	0.705088i	$0.750908\pi$
$558$		0			0
$559$	−1.98865			−0.0841110
$560$		0			0
$561$	−13.6201			−0.575041
$562$		0			0
$563$		−	16.3139i		−	0.687549i	−0.939052	−	0.343774i	$-0.888294\pi$
$563$		−	16.3139i		−	0.687549i	0.939052	−	0.343774i	$-0.111706\pi$
$564$		0			0
$565$	−3.58983	+	2.84894i	−0.151025	+	0.119856i
$566$		0			0
$567$	−31.0900	−	12.2846i	−1.30566	−	0.515905i
$568$		0			0
$569$	6.54541			0.274398			0.137199	−	0.990544i	$-0.456190\pi$
$569$	6.54541			0.274398			0.137199	+	0.990544i	$0.456190\pi$
$570$		0			0
$571$			39.5547i			1.65531i	0.561234	+	0.827657i	$0.310327\pi$
$571$			39.5547i			1.65531i	−0.561234	+	0.827657i	$0.689673\pi$
$572$		0			0
$573$	−25.1017			−1.04864
$574$		0			0
$575$	3.99435	−	17.1269i	0.166576	−	0.714242i
$576$		0			0
$577$	−40.4544			−1.68414			−0.842069	−	0.539369i	$-0.818663\pi$
$577$	−40.4544			−1.68414			−0.842069	+	0.539369i	$0.818663\pi$
$578$		0			0
$579$	−54.0326			−2.24552
$580$		0			0
$581$	7.66315	−	19.3940i	0.317921	−	0.804598i
$582$		0			0
$583$	−8.28298			−0.343046
$584$		0			0
$585$	−10.9101	−	13.7473i	−0.451076	−	0.568382i
$586$		0			0
$587$			29.6642i			1.22437i	0.790714	+	0.612185i	$0.209709\pi$
$587$			29.6642i			1.22437i	−0.790714	+	0.612185i	$0.790291\pi$
$588$		0			0
$589$	60.2210			2.48136
$590$		0			0
$591$	−20.2346			−0.832340
$592$		0			0
$593$	−6.50273			−0.267035			−0.133517	−	0.991046i	$-0.542627\pi$
$593$	−6.50273			−0.267035			−0.133517	+	0.991046i	$0.542627\pi$
$594$		0			0
$595$	1.39673	+	4.60425i	0.0572605	+	0.188756i
$596$		0			0
$597$			19.4773i			0.797153i
$598$		0			0
$599$			4.70631i			0.192295i	0.995367	+	0.0961474i	$0.0306520\pi$
$599$			4.70631i			0.192295i	−0.995367	+	0.0961474i	$0.969348\pi$
$600$		0			0
$601$		−	21.0706i		−	0.859490i	−0.902950	−	0.429745i	$-0.858604\pi$
$601$		−	21.0706i		−	0.859490i	0.902950	−	0.429745i	$-0.141396\pi$
$602$		0			0
$603$	−83.9572			−3.41900
$604$		0			0
$605$	−26.2400	−	33.0638i	−1.06681	−	1.34424i
$606$		0			0
$607$		−	8.45732i		−	0.343272i	−0.985160	−	0.171636i	$-0.945095\pi$
$607$		−	8.45732i		−	0.343272i	0.985160	−	0.171636i	$-0.0549053\pi$
$608$		0			0
$609$	−24.0999	+	60.9921i	−0.976575	+	2.47153i
$610$		0			0
$611$			7.34322i			0.297075i
$612$		0			0
$613$		−	39.7547i		−	1.60568i	−0.596198	−	0.802838i	$-0.703323\pi$
$613$		−	39.7547i		−	1.60568i	0.596198	−	0.802838i	$-0.296677\pi$
$614$		0			0
$615$	11.0523	+	13.9265i	0.445670	+	0.561570i
$616$		0			0
$617$			19.7088i			0.793446i	0.917938	+	0.396723i	$0.129853\pi$
$617$			19.7088i			0.793446i	−0.917938	+	0.396723i	$0.870147\pi$
$618$		0			0
$619$	40.1553			1.61398			0.806990	−	0.590566i	$-0.201095\pi$
$619$	40.1553			1.61398			0.806990	+	0.590566i	$0.201095\pi$
$620$		0			0
$621$			36.5026i			1.46480i
$622$		0			0
$623$	1.16525	−	2.94904i	0.0466849	−	0.118151i
$624$		0			0
$625$	−22.4207	−	11.0595i	−0.896828	−	0.442380i
$626$		0			0
$627$	−111.968			−4.47157
$628$		0			0
$629$		−	3.75410i		−	0.149686i
$630$		0			0
$631$		−	25.4148i		−	1.01175i	−0.862608	−	0.505873i	$-0.831170\pi$
$631$		−	25.4148i		−	1.01175i	0.862608	−	0.505873i	$-0.168830\pi$
$632$		0			0
$633$	30.5757			1.21528
$634$		0			0
$635$	−8.07823	−	10.1790i	−0.320575	−	0.403942i
$636$		0			0
$637$	6.27856	+	5.87968i	0.248765	+	0.232962i
$638$		0			0
$639$		−	79.6839i		−	3.15225i
$640$		0			0
$641$	−44.0099			−1.73829			−0.869143	−	0.494561i	$-0.835329\pi$
$641$	−44.0099			−1.73829			−0.869143	+	0.494561i	$0.835329\pi$
$642$		0			0
$643$			11.9892i			0.472807i	0.971655	+	0.236404i	$0.0759687\pi$
$643$			11.9892i			0.472807i	−0.971655	+	0.236404i	$0.924031\pi$
$644$		0			0
$645$	8.68462	−	6.89224i	0.341957	−	0.271382i
$646$		0			0
$647$		−	15.1947i		−	0.597366i	−0.954352	−	0.298683i	$-0.903453\pi$
$647$		−	15.1947i		−	0.597366i	0.954352	−	0.298683i	$-0.0965473\pi$
$648$		0			0
$649$		−	55.6693i		−	2.18521i
$650$		0			0
$651$	−26.8317	+	67.9059i	−1.05162	+	2.66144i
$652$		0			0
$653$			23.4965i			0.919490i	0.888051	+	0.459745i	$0.152059\pi$
$653$			23.4965i			0.919490i	−0.888051	+	0.459745i	$0.847941\pi$
$654$		0			0
$655$	22.4552	+	28.2948i	0.877395	+	1.10557i
$656$		0			0
$657$	−48.8768			−1.90687
$658$		0			0
$659$			20.6544i			0.804583i	0.915512	+	0.402292i	$0.131786\pi$
$659$			20.6544i			0.804583i	−0.915512	+	0.402292i	$0.868214\pi$
$660$		0			0
$661$			30.8321i			1.19923i	0.800289	+	0.599614i	$0.204679\pi$
$661$			30.8321i			1.19923i	−0.800289	+	0.599614i	$0.795321\pi$
$662$		0			0
$663$		−	3.06197i		−	0.118917i
$664$		0			0
$665$	11.4823	+	37.8506i	0.445263	+	1.46778i
$666$		0			0
$667$	28.4557			1.10181
$668$		0			0
$669$	−22.0229			−0.851453
$670$		0			0
$671$	17.2818			0.667155
$672$		0			0
$673$			26.7263i			1.03022i	0.857123	+	0.515111i	$0.172249\pi$
$673$			26.7263i			1.03022i	−0.857123	+	0.515111i	$0.827751\pi$
$674$		0			0
$675$	50.5338	+	11.7855i	1.94505	+	0.453625i
$676$		0			0
$677$	−31.9033			−1.22614			−0.613071	−	0.790028i	$-0.710066\pi$
$677$	−31.9033			−1.22614			−0.613071	+	0.790028i	$0.710066\pi$
$678$		0			0
$679$	2.00119	+	0.790730i	0.0767985	+	0.0303454i
$680$		0			0
$681$	−10.5276			−0.403420
$682$		0			0
$683$	8.92815			0.341626			0.170813	−	0.985303i	$-0.445361\pi$
$683$	8.92815			0.341626			0.170813	+	0.985303i	$0.445361\pi$
$684$		0			0
$685$	10.3489	−	8.21303i	0.395410	−	0.313804i
$686$		0			0
$687$	14.9410			0.570034
$688$		0			0
$689$		−	1.86212i		−	0.0709411i
$690$		0			0
$691$	14.7006			0.559238			0.279619	−	0.960111i	$-0.409792\pi$
$691$	14.7006			0.559238			0.279619	+	0.960111i	$0.409792\pi$
$692$		0			0
$693$	33.9445	−	85.9070i	1.28944	−	3.26334i
$694$		0			0
$695$	−5.32036	−	6.70395i	−0.201813	−	0.254295i
$696$		0			0
$697$			2.11057i			0.0799435i
$698$		0			0
$699$	−32.0267			−1.21136
$700$		0			0
$701$	17.7065			0.668766			0.334383	−	0.942437i	$-0.391472\pi$
$701$	17.7065			0.668766			0.334383	+	0.942437i	$0.391472\pi$
$702$		0			0
$703$		−	30.8617i		−	1.16397i
$704$		0			0
$705$	−25.4500	−	32.0685i	−0.958503	−	1.20777i
$706$		0			0
$707$	−1.71610	+	4.34311i	−0.0645405	+	0.163340i
$708$		0			0
$709$	−42.7222			−1.60447			−0.802234	−	0.597010i	$-0.796355\pi$
$709$	−42.7222			−1.60447			−0.802234	+	0.597010i	$0.796355\pi$
$710$		0			0
$711$		−	14.0306i		−	0.526189i
$712$		0			0
$713$	31.6813			1.18647
$714$		0			0
$715$	11.7646	−	9.33654i	0.439970	−	0.349167i
$716$		0			0
$717$	51.8954			1.93807
$718$		0			0
$719$	−26.2758			−0.979923			−0.489961	−	0.871744i	$-0.662989\pi$
$719$	−26.2758			−0.979923			−0.489961	+	0.871744i	$0.662989\pi$
$720$		0			0
$721$	9.78104	−	24.7539i	0.364265	−	0.921885i
$722$		0			0
$723$	54.4574			2.02529
$724$		0			0
$725$	9.18745	−	39.3938i	0.341214	−	1.46305i
$726$		0			0
$727$			21.4192i			0.794393i	0.917734	+	0.397196i	$0.130017\pi$
$727$			21.4192i			0.794393i	−0.917734	+	0.397196i	$0.869983\pi$
$728$		0			0
$729$	14.6875			0.543981
$730$		0			0
$731$	1.31616			0.0486800
$732$		0			0
$733$	25.3586			0.936642			0.468321	−	0.883558i	$-0.344859\pi$
$733$	25.3586			0.936642			0.468321	+	0.883558i	$0.344859\pi$
$734$		0			0
$735$	−47.7967	−	3.91696i	−1.76301	−	0.144479i
$736$		0			0
$737$		−	71.8482i		−	2.64656i
$738$		0			0
$739$		−	9.43697i		−	0.347145i	−0.984821	−	0.173572i	$-0.944469\pi$
$739$		−	9.43697i		−	0.347145i	0.984821	−	0.173572i	$-0.0555310\pi$
$740$		0			0
$741$		−	25.1719i		−	0.924711i
$742$		0			0
$743$	40.7813			1.49612			0.748061	−	0.663630i	$-0.230985\pi$
$743$	40.7813			1.49612			0.748061	+	0.663630i	$0.230985\pi$
$744$		0			0
$745$	−9.87305	−	12.4406i	−0.361720	−	0.455788i
$746$		0			0
$747$		−	50.3423i		−	1.84193i
$748$		0			0
$749$	32.3439	+	12.7800i	1.18182	+	0.466973i
$750$		0			0
$751$			14.1304i			0.515624i	0.966195	+	0.257812i	$0.0830016\pi$
$751$			14.1304i			0.515624i	−0.966195	+	0.257812i	$0.916998\pi$
$752$		0			0
$753$		−	47.9520i		−	1.74747i
$754$		0			0
$755$	18.7664	−	14.8933i	0.682980	−	0.542024i
$756$		0			0
$757$		−	15.5846i		−	0.566432i	−0.959056	−	0.283216i	$-0.908599\pi$
$757$		−	15.5846i		−	0.566432i	0.959056	−	0.283216i	$-0.0914013\pi$
$758$		0			0
$759$	−58.9046			−2.13810
$760$		0			0
$761$		−	10.6693i		−	0.386763i	−0.981124	−	0.193382i	$-0.938054\pi$
$761$		−	10.6693i		−	0.386763i	0.981124	−	0.193382i	$-0.0619455\pi$
$762$		0			0
$763$	−0.316368	−	0.125006i	−0.0114533	−	0.00452554i
$764$		0			0
$765$	7.22068	+	9.09847i	0.261064	+	0.328956i
$766$		0			0
$767$	12.5152			0.451896
$768$		0			0
$769$		−	10.6332i		−	0.383441i	−0.981450	−	0.191721i	$-0.938593\pi$
$769$		−	10.6332i		−	0.383441i	0.981450	−	0.191721i	$-0.0614068\pi$
$770$		0			0
$771$			64.9738i			2.33998i
$772$		0			0
$773$	−48.0043			−1.72659			−0.863297	−	0.504696i	$-0.831605\pi$
$773$	−48.0043			−1.72659			−0.863297	+	0.504696i	$0.831605\pi$
$774$		0			0
$775$	10.2289	−	43.8593i	0.367433	−	1.57547i
$776$		0			0
$777$	34.8000	+	13.7506i	1.24844	+	0.493298i
$778$		0			0
$779$			17.3506i			0.621648i
$780$		0			0
$781$	68.1912			2.44007
$782$		0			0
$783$			83.9599i			3.00048i
$784$		0			0
$785$	7.12726	+	8.98076i	0.254383	+	0.320537i
$786$		0			0
$787$			52.3628i			1.86653i	0.359185	+	0.933266i	$0.383055\pi$
$787$			52.3628i			1.86653i	−0.359185	+	0.933266i	$0.616945\pi$
$788$		0			0
$789$		−	40.0901i		−	1.42725i
$790$		0			0
$791$	1.99272	−	5.04318i	0.0708528	−	0.179315i
$792$		0			0
$793$			3.88516i			0.137966i
$794$		0			0
$795$	6.45371	+	8.13204i	0.228890	+	0.288414i
$796$		0			0
$797$	40.2475			1.42564			0.712820	−	0.701347i	$-0.247418\pi$
$797$	40.2475			1.42564			0.712820	+	0.701347i	$0.247418\pi$
$798$		0			0
$799$		−	4.86000i		−	0.171935i
$800$		0			0
$801$		−	7.65502i		−	0.270477i
$802$		0			0
$803$		−	41.8274i		−	1.47606i
$804$		0			0
$805$	6.04063	+	19.9126i	0.212904	+	0.701827i
$806$		0			0
$807$	1.09623			0.0385891
$808$		0			0
$809$	−16.1962			−0.569429			−0.284714	−	0.958612i	$-0.591899\pi$
$809$	−16.1962			−0.569429			−0.284714	+	0.958612i	$0.591899\pi$
$810$		0			0
$811$	−37.9985			−1.33431			−0.667154	−	0.744920i	$-0.732488\pi$
$811$	−37.9985			−1.33431			−0.667154	+	0.744920i	$0.732488\pi$
$812$		0			0
$813$			0.464484i			0.0162902i
$814$		0			0
$815$	−18.5811	−	23.4132i	−0.650866	−	0.820129i
$816$		0			0
$817$	10.8199			0.378540
$818$		0			0
$819$	19.3130	+	7.63115i	0.674851	+	0.266654i
$820$		0			0
$821$	−25.7149			−0.897456			−0.448728	−	0.893668i	$-0.648123\pi$
$821$	−25.7149			−0.897456			−0.448728	+	0.893668i	$0.648123\pi$
$822$		0			0
$823$	48.5297			1.69164			0.845819	−	0.533469i	$-0.179112\pi$
$823$	48.5297			1.69164			0.845819	+	0.533469i	$0.179112\pi$
$824$		0			0
$825$	−19.0185	+	81.5470i	−0.662137	+	2.83910i
$826$		0			0
$827$	10.4578			0.363654			0.181827	−	0.983331i	$-0.441799\pi$
$827$	10.4578			0.363654			0.181827	+	0.983331i	$0.441799\pi$
$828$		0			0
$829$			13.7971i			0.479195i	0.970872	+	0.239597i	$0.0770154\pi$
$829$			13.7971i			0.479195i	−0.970872	+	0.239597i	$0.922985\pi$
$830$		0			0
$831$	89.4102			3.10161
$832$		0			0
$833$	−4.15537	−	3.89139i	−0.143975	−	0.134829i
$834$		0			0
$835$	−35.6264	+	28.2736i	−1.23290	+	0.978450i
$836$		0			0
$837$			93.4772i			3.23104i
$838$		0			0
$839$	−40.8691			−1.41096			−0.705479	−	0.708731i	$-0.749268\pi$
$839$	−40.8691			−1.41096			−0.705479	+	0.708731i	$0.749268\pi$
$840$		0			0
$841$	36.4513			1.25694
$842$		0			0
$843$			73.0389i			2.51559i
$844$		0			0
$845$	−15.9714	−	20.1249i	−0.549434	−	0.692318i
$846$		0			0
$847$	46.4499	+	18.3538i	1.59604	+	0.630643i
$848$		0			0
$849$	44.8376			1.53882
$850$		0			0
$851$		−	16.2359i		−	0.556558i
$852$		0			0
$853$	−19.9104			−0.681721			−0.340860	−	0.940114i	$-0.610718\pi$
$853$	−19.9104			−0.681721			−0.340860	+	0.940114i	$0.610718\pi$
$854$		0			0
$855$	59.3598	+	74.7967i	2.03006	+	2.55799i
$856$		0			0
$857$	−46.2728			−1.58065			−0.790324	−	0.612689i	$-0.790088\pi$
$857$	−46.2728			−1.58065			−0.790324	+	0.612689i	$0.790088\pi$
$858$		0			0
$859$	36.6565			1.25070			0.625352	−	0.780343i	$-0.284955\pi$
$859$	36.6565			1.25070			0.625352	+	0.780343i	$0.284955\pi$
$860$		0			0
$861$	−19.5647	−	7.73060i	−0.666763	−	0.263458i
$862$		0			0
$863$	−25.1182			−0.855032			−0.427516	−	0.904008i	$-0.640611\pi$
$863$	−25.1182			−0.855032			−0.427516	+	0.904008i	$0.640611\pi$
$864$		0			0
$865$	−10.2590	−	12.9269i	−0.348817	−	0.439529i
$866$		0			0
$867$		−	50.0591i		−	1.70009i
$868$		0			0
$869$	12.0070			0.407310
$870$		0			0
$871$	16.1524			0.547303
$872$		0			0
$873$	5.19462			0.175811
$874$		0			0
$875$	29.5171	−	1.93344i	0.997862	−	0.0653622i
$876$		0			0
$877$		−	26.7977i		−	0.904896i	−0.891791	−	0.452448i	$-0.850551\pi$
$877$		−	26.7977i		−	0.904896i	0.891791	−	0.452448i	$-0.149449\pi$
$878$		0			0
$879$		−	35.1260i		−	1.18477i
$880$		0			0
$881$			38.7099i			1.30417i	0.758146	+	0.652084i	$0.226105\pi$
$881$			38.7099i			1.30417i	−0.758146	+	0.652084i	$0.773895\pi$
$882$		0			0
$883$	33.3515			1.12237			0.561183	−	0.827692i	$-0.310346\pi$
$883$	33.3515			1.12237			0.561183	+	0.827692i	$0.310346\pi$
$884$		0			0
$885$	−54.6548	+	43.3749i	−1.83720	+	1.45803i
$886$		0			0
$887$			30.9373i			1.03877i	0.854539	+	0.519386i	$0.173839\pi$
$887$			30.9373i			1.03877i	−0.854539	+	0.519386i	$0.826161\pi$
$888$		0			0
$889$	14.3001	+	5.65039i	0.479609	+	0.189508i
$890$		0			0
$891$		−	69.0629i		−	2.31370i
$892$		0			0
$893$		−	39.9531i		−	1.33698i
$894$		0			0
$895$	25.5787	−	20.2996i	0.855001	−	0.678541i
$896$		0			0
$897$		−	13.2425i		−	0.442155i
$898$		0			0
$899$	72.8705			2.43037
$900$		0			0
$901$			1.23242i			0.0410578i
$902$		0			0
$903$	−4.82084	+	12.2006i	−0.160428	+	0.406012i
$904$		0			0
$905$	19.7165	−	15.6473i	0.655397	−	0.520133i
$906$		0			0
$907$	−19.1073			−0.634449			−0.317224	−	0.948350i	$-0.602751\pi$
$907$	−19.1073			−0.634449			−0.317224	+	0.948350i	$0.602751\pi$
$908$		0			0
$909$			11.2737i			0.373926i
$910$		0			0
$911$		−	38.5483i		−	1.27716i	−0.769554	−	0.638582i	$-0.779521\pi$
$911$		−	38.5483i		−	1.27716i	0.769554	−	0.638582i	$-0.220479\pi$
$912$		0			0
$913$	43.0815			1.42579
$914$		0			0
$915$	−13.4651	−	16.9668i	−0.445144	−	0.560906i
$916$		0			0
$917$	−39.7500	−	15.7065i	−1.31266	−	0.518673i
$918$		0			0
$919$			14.5994i			0.481589i	0.970576	+	0.240795i	$0.0774080\pi$
$919$			14.5994i			0.481589i	−0.970576	+	0.240795i	$0.922592\pi$
$920$		0			0
$921$	42.3170			1.39439
$922$		0			0
$923$			15.3303i			0.504602i
$924$		0			0
$925$	−22.4768	−	5.24205i	−0.739031	−	0.172357i
$926$		0			0
$927$		−	64.2556i		−	2.11043i
$928$		0			0
$929$			18.4422i			0.605068i	0.953139	+	0.302534i	$0.0978326\pi$
$929$			18.4422i			0.605068i	−0.953139	+	0.302534i	$0.902167\pi$
$930$		0			0
$931$	−34.1605	−	31.9903i	−1.11956	−	1.04844i
$932$		0			0
$933$			73.8181i			2.41670i
$934$		0			0
$935$	−7.78622	+	6.17926i	−0.254636	+	0.202083i
$936$		0			0
$937$	3.27094			0.106857			0.0534285	−	0.998572i	$-0.482985\pi$
$937$	3.27094			0.106857			0.0534285	+	0.998572i	$0.482985\pi$
$938$		0			0
$939$		−	66.4549i		−	2.16867i
$940$		0			0
$941$		−	23.1015i		−	0.753086i	−0.926399	−	0.376543i	$-0.877113\pi$
$941$		−	23.1015i		−	0.753086i	0.926399	−	0.376543i	$-0.122887\pi$
$942$		0			0
$943$			9.12785i			0.297244i
$944$		0			0
$945$	−58.7531	+	17.8232i	−1.91124	+	0.579788i
$946$		0			0
$947$	−30.1476			−0.979666			−0.489833	−	0.871816i	$-0.662942\pi$
$947$	−30.1476			−0.979666			−0.489833	+	0.871816i	$0.662942\pi$
$948$		0			0
$949$	9.40334			0.305245
$950$		0			0
$951$	38.3838			1.24468
$952$		0			0
$953$			15.9517i			0.516725i	0.966048	+	0.258363i	$0.0831829\pi$
$953$			15.9517i			0.516725i	−0.966048	+	0.258363i	$0.916817\pi$
$954$		0			0
$955$	−14.3499	+	11.3883i	−0.464352	+	0.368517i
$956$		0			0
$957$	−135.487			−4.37968
$958$		0			0
$959$	−5.74468	+	14.5387i	−0.185505	+	0.469478i
$960$		0			0
$961$	50.1307			1.61712
$962$		0			0
$963$	83.9572			2.70548
$964$		0			0
$965$	−30.8888	+	24.5139i	−0.994347	+	0.789129i
$966$		0			0
$967$	−3.80814			−0.122461			−0.0612307	−	0.998124i	$-0.519503\pi$
$967$	−3.80814			−0.122461			−0.0612307	+	0.998124i	$0.519503\pi$
$968$		0			0
$969$			16.6596i			0.535185i
$970$		0			0
$971$	−41.7036			−1.33833			−0.669167	−	0.743112i	$-0.733349\pi$
$971$	−41.7036			−1.33833			−0.669167	+	0.743112i	$0.733349\pi$
$972$		0			0
$973$	9.41808	+	3.72137i	0.301930	+	0.119302i
$974$		0			0
$975$	−18.3328	−	4.27559i	−0.587120	−	0.136928i
$976$		0			0
$977$			55.5801i			1.77816i	0.457748	+	0.889082i	$0.348656\pi$
$977$			55.5801i			1.77816i	−0.457748	+	0.889082i	$0.651344\pi$
$978$		0			0
$979$	6.55095			0.209369
$980$		0			0
$981$	−0.821218			−0.0262195
$982$		0			0
$983$			57.6224i			1.83787i	0.394409	+	0.918935i	$0.370949\pi$
$983$			57.6224i			1.83787i	−0.394409	+	0.918935i	$0.629051\pi$
$984$		0			0
$985$	−11.5675	+	9.18016i	−0.368572	+	0.292504i
$986$		0			0
$987$	45.0516	+	17.8012i	1.43401	+	0.566620i
$988$		0			0
$989$	5.69217			0.181001
$990$		0			0
$991$		−	16.7139i		−	0.530934i	−0.964120	−	0.265467i	$-0.914474\pi$
$991$		−	16.7139i		−	0.530934i	0.964120	−	0.265467i	$-0.0855261\pi$
$992$		0			0
$993$	88.5260			2.80929
$994$		0			0
$995$	8.83659	+	11.1346i	0.280139	+	0.352991i
$996$		0			0
$997$	36.3449			1.15106			0.575528	−	0.817782i	$-0.304797\pi$
$997$	36.3449			1.15106			0.575528	+	0.817782i	$0.304797\pi$
$998$		0			0
$999$	47.9047			1.51564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.e.a.1119.16	yes	48
4.3	odd	2	inner	1120.2.e.a.1119.19	yes	48
5.4	even	2	inner	1120.2.e.a.1119.7	yes	48
7.6	odd	2	inner	1120.2.e.a.1119.3	✓	48
8.3	odd	2		2240.2.e.g.2239.4		48
8.5	even	2		2240.2.e.g.2239.7		48
20.19	odd	2	inner	1120.2.e.a.1119.4	yes	48
28.27	even	2	inner	1120.2.e.a.1119.8	yes	48
35.34	odd	2	inner	1120.2.e.a.1119.20	yes	48
40.19	odd	2		2240.2.e.g.2239.19		48
40.29	even	2		2240.2.e.g.2239.16		48
56.13	odd	2		2240.2.e.g.2239.20		48
56.27	even	2		2240.2.e.g.2239.15		48
140.139	even	2	inner	1120.2.e.a.1119.15	yes	48
280.69	odd	2		2240.2.e.g.2239.3		48
280.139	even	2		2240.2.e.g.2239.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.e.a.1119.3	✓	48	7.6	odd	2	inner
1120.2.e.a.1119.4	yes	48	20.19	odd	2	inner
1120.2.e.a.1119.7	yes	48	5.4	even	2	inner
1120.2.e.a.1119.8	yes	48	28.27	even	2	inner
1120.2.e.a.1119.15	yes	48	140.139	even	2	inner
1120.2.e.a.1119.16	yes	48	1.1	even	1	trivial
1120.2.e.a.1119.19	yes	48	4.3	odd	2	inner
1120.2.e.a.1119.20	yes	48	35.34	odd	2	inner
2240.2.e.g.2239.3		48	280.69	odd	2
2240.2.e.g.2239.4		48	8.3	odd	2
2240.2.e.g.2239.7		48	8.5	even	2
2240.2.e.g.2239.8		48	280.139	even	2
2240.2.e.g.2239.15		48	56.27	even	2
2240.2.e.g.2239.16		48	40.29	even	2
2240.2.e.g.2239.19		48	40.19	odd	2
2240.2.e.g.2239.20		48	56.13	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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.06386i q^{3} +(1.39003 + 1.75152i) q^{5} +(-2.46063 - 0.972270i) q^{7} -6.38723 q^{9} -5.46601i q^{11} +1.22883 q^{13} +(-5.36640 + 4.25886i) q^{15} -0.813283 q^{17} -6.68583 q^{19} +(2.97890 - 7.53902i) q^{21} -3.51731 q^{23} +(-1.13563 + 4.86933i) q^{25} -10.3780i q^{27} -8.09020 q^{29} -9.00726 q^{31} +16.7471 q^{33} +(-1.71740 - 5.66132i) q^{35} +4.61599i q^{37} +3.76496i q^{39} -2.59512i q^{41} -1.61833 q^{43} +(-8.87844 - 11.1873i) q^{45} +5.97579i q^{47} +(5.10938 + 4.78479i) q^{49} -2.49178i q^{51} -1.51536i q^{53} +(9.57381 - 7.59792i) q^{55} -20.4844i q^{57} +10.1846 q^{59} +3.16168i q^{61} +(15.7166 + 6.21011i) q^{63} +(1.70811 + 2.15231i) q^{65} +13.1445 q^{67} -10.7765i q^{69} +12.4755i q^{71} +7.65228 q^{73} +(-14.9189 - 3.47940i) q^{75} +(-5.31443 + 13.4498i) q^{77} +2.19667i q^{79} +12.6350 q^{81} +7.88172i q^{83} +(-1.13049 - 1.42448i) q^{85} -24.7872i q^{87} +1.19849i q^{89} +(-3.02369 - 1.19475i) q^{91} -27.5970i q^{93} +(-9.29351 - 11.7103i) q^{95} -0.813283 q^{97} +34.9126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 8 q^{21} - 16 q^{25} + 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q - 48 * q^9 + 8 * q^21 - 16 * q^25 + 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 + 32 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more