LMFDB - Embedded newform 1156.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1156,2,Mod(1,1156)] 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("1156.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1156 = 2^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1156.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,6,0,-6,0,0] 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:	yes
Analytic conductor:	$9.23070647366$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	1156.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87939 q^{3} +1.65270 q^{5} -3.18479 q^{7} +5.29086 q^{9} +5.06418 q^{11} -1.41147 q^{13} +4.75877 q^{15} +8.10607 q^{19} -9.17024 q^{21} -6.35504 q^{23} -2.26857 q^{25} +6.59627 q^{27} -1.29086 q^{29} +0.426022 q^{31} +14.5817 q^{33} -5.26352 q^{35} +5.34730 q^{37} -4.06418 q^{39} +7.04189 q^{41} -0.283119 q^{43} +8.74422 q^{45} -4.90167 q^{47} +3.14290 q^{49} -5.06418 q^{53} +8.36959 q^{55} +23.3405 q^{57} -4.50980 q^{59} +4.55438 q^{61} -16.8503 q^{63} -2.33275 q^{65} -2.07873 q^{67} -18.2986 q^{69} +14.1334 q^{71} +4.77332 q^{73} -6.53209 q^{75} -16.1284 q^{77} -13.9067 q^{79} +3.12061 q^{81} -9.49794 q^{83} -3.71688 q^{87} -9.10607 q^{89} +4.49525 q^{91} +1.22668 q^{93} +13.3969 q^{95} -4.36959 q^{97} +26.7939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{5} - 6 q^{7} + 6 q^{11} + 6 q^{13} + 3 q^{15} + 12 q^{19} - 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} + 12 q^{29} + 9 q^{31} + 12 q^{33} - 21 q^{35} + 15 q^{37} - 3 q^{39} + 18 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^5 - 6 * q^7 + 6 * q^11 + 6 * q^13 + 3 * q^15 + 12 * q^19 - 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 + 12 * q^29 + 9 * q^31 + 12 * q^33 - 21 * q^35 + 15 * q^37 - 3 * q^39 + 18 * q^41 - 9 * q^43 - 3 * q^45 - 3 * q^47 + 9 * q^49 - 6 * q^53 + 18 * q^55 + 27 * q^57 - 15 * q^59 + 3 * q^61 - 9 * q^63 + 12 * q^65 - 15 * q^67 - 15 * q^69 + 9 * q^71 + 21 * q^73 - 15 * q^75 - 30 * q^77 - 15 * q^79 + 15 * q^81 - 3 * q^83 - 3 * q^87 - 15 * q^89 - 3 * q^91 - 3 * q^93 + 12 * q^95 - 6 * q^97 + 24 * q^99

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
$3$	2.87939	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$3$	2.87939	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$4$	0	0
$5$	1.65270	0.739112	0.369556	−	0.929209i	$-0.379510\pi$
$5$	1.65270	0.739112	0.369556	+	0.929209i	$0.379510\pi$
$6$	0	0
$7$	−3.18479	−1.20374	−0.601869	−	0.798595i	$-0.705577\pi$
$7$	−3.18479	−1.20374	−0.601869	+	0.798595i	$0.705577\pi$
$8$	0	0
$9$	5.29086	1.76362
$10$	0	0
$11$	5.06418	1.52691	0.763454	−	0.645863i	$-0.223502\pi$
$11$	5.06418	1.52691	0.763454	+	0.645863i	$0.223502\pi$
$12$	0	0
$13$	−1.41147	−0.391472	−0.195736	−	0.980657i	$-0.562710\pi$
$13$	−1.41147	−0.391472	−0.195736	+	0.980657i	$0.562710\pi$
$14$	0	0
$15$	4.75877	1.22871
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	8.10607	1.85966	0.929830	−	0.367990i	$-0.119954\pi$
$19$	8.10607	1.85966	0.929830	+	0.367990i	$0.119954\pi$
$20$	0	0
$21$	−9.17024	−2.00111
$22$	0	0
$23$	−6.35504	−1.32512	−0.662558	−	0.749010i	$-0.730529\pi$
$23$	−6.35504	−1.32512	−0.662558	+	0.749010i	$0.730529\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	0	0
$27$	6.59627	1.26945
$28$	0	0
$29$	−1.29086	−0.239707	−0.119853	−	0.992792i	$-0.538242\pi$
$29$	−1.29086	−0.239707	−0.119853	+	0.992792i	$0.538242\pi$
$30$	0	0
$31$	0.426022	0.0765158	0.0382579	−	0.999268i	$-0.487819\pi$
$31$	0.426022	0.0765158	0.0382579	+	0.999268i	$0.487819\pi$
$32$	0	0
$33$	14.5817	2.53835
$34$	0	0
$35$	−5.26352	−0.889697
$36$	0	0
$37$	5.34730	0.879090	0.439545	−	0.898221i	$-0.355140\pi$
$37$	5.34730	0.879090	0.439545	+	0.898221i	$0.355140\pi$
$38$	0	0
$39$	−4.06418	−0.650789
$40$	0	0
$41$	7.04189	1.09976	0.549879	−	0.835244i	$-0.314674\pi$
$41$	7.04189	1.09976	0.549879	+	0.835244i	$0.314674\pi$
$42$	0	0
$43$	−0.283119	−0.0431752	−0.0215876	−	0.999767i	$-0.506872\pi$
$43$	−0.283119	−0.0431752	−0.0215876	+	0.999767i	$0.506872\pi$
$44$	0	0
$45$	8.74422	1.30351
$46$	0	0
$47$	−4.90167	−0.714983	−0.357491	−	0.933916i	$-0.616368\pi$
$47$	−4.90167	−0.714983	−0.357491	+	0.933916i	$0.616368\pi$
$48$	0	0
$49$	3.14290	0.448986
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.06418	−0.695618	−0.347809	−	0.937565i	$-0.613074\pi$
$53$	−5.06418	−0.695618	−0.347809	+	0.937565i	$0.613074\pi$
$54$	0	0
$55$	8.36959	1.12855
$56$	0	0
$57$	23.3405	3.09152
$58$	0	0
$59$	−4.50980	−0.587126	−0.293563	−	0.955940i	$-0.594841\pi$
$59$	−4.50980	−0.587126	−0.293563	+	0.955940i	$0.594841\pi$
$60$	0	0
$61$	4.55438	0.583128	0.291564	−	0.956551i	$-0.405824\pi$
$61$	4.55438	0.583128	0.291564	+	0.956551i	$0.405824\pi$
$62$	0	0
$63$	−16.8503	−2.12294
$64$	0	0
$65$	−2.33275	−0.289342
$66$	0	0
$67$	−2.07873	−0.253957	−0.126978	−	0.991905i	$-0.540528\pi$
$67$	−2.07873	−0.253957	−0.126978	+	0.991905i	$0.540528\pi$
$68$	0	0
$69$	−18.2986	−2.20289
$70$	0	0
$71$	14.1334	1.67733	0.838663	−	0.544650i	$-0.183337\pi$
$71$	14.1334	1.67733	0.838663	+	0.544650i	$0.183337\pi$
$72$	0	0
$73$	4.77332	0.558675	0.279337	−	0.960193i	$-0.409885\pi$
$73$	4.77332	0.558675	0.279337	+	0.960193i	$0.409885\pi$
$74$	0	0
$75$	−6.53209	−0.754261
$76$	0	0
$77$	−16.1284	−1.83800
$78$	0	0
$79$	−13.9067	−1.56463	−0.782314	−	0.622884i	$-0.785961\pi$
$79$	−13.9067	−1.56463	−0.782314	+	0.622884i	$0.785961\pi$
$80$	0	0
$81$	3.12061	0.346735
$82$	0	0
$83$	−9.49794	−1.04253	−0.521267	−	0.853394i	$-0.674541\pi$
$83$	−9.49794	−1.04253	−0.521267	+	0.853394i	$0.674541\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.71688	−0.398492
$88$	0	0
$89$	−9.10607	−0.965241	−0.482621	−	0.875830i	$-0.660315\pi$
$89$	−9.10607	−0.965241	−0.482621	+	0.875830i	$0.660315\pi$
$90$	0	0
$91$	4.49525	0.471230
$92$	0	0
$93$	1.22668	0.127201
$94$	0	0
$95$	13.3969	1.37450
$96$	0	0
$97$	−4.36959	−0.443664	−0.221832	−	0.975085i	$-0.571204\pi$
$97$	−4.36959	−0.443664	−0.221832	+	0.975085i	$0.571204\pi$
$98$	0	0
$99$	26.7939	2.69288
$100$	0	0
$101$	−8.43882	−0.839694	−0.419847	−	0.907595i	$-0.637916\pi$
$101$	−8.43882	−0.839694	−0.419847	+	0.907595i	$0.637916\pi$
$102$	0	0
$103$	−9.49525	−0.935595	−0.467798	−	0.883836i	$-0.654952\pi$
$103$	−9.49525	−0.935595	−0.467798	+	0.883836i	$0.654952\pi$
$104$	0	0
$105$	−15.1557	−1.47904
$106$	0	0
$107$	−4.61587	−0.446233	−0.223116	−	0.974792i	$-0.571623\pi$
$107$	−4.61587	−0.446233	−0.223116	+	0.974792i	$0.571623\pi$
$108$	0	0
$109$	12.9067	1.23624	0.618120	−	0.786084i	$-0.287895\pi$
$109$	12.9067	1.23624	0.618120	+	0.786084i	$0.287895\pi$
$110$	0	0
$111$	15.3969	1.46141
$112$	0	0
$113$	−8.11381	−0.763283	−0.381641	−	0.924310i	$-0.624641\pi$
$113$	−8.11381	−0.763283	−0.381641	+	0.924310i	$0.624641\pi$
$114$	0	0
$115$	−10.5030	−0.979409
$116$	0	0
$117$	−7.46791	−0.690409
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.6459	1.33145
$122$	0	0
$123$	20.2763	1.82825
$124$	0	0
$125$	−12.0128	−1.07446
$126$	0	0
$127$	−2.55943	−0.227113	−0.113556	−	0.993532i	$-0.536224\pi$
$127$	−2.55943	−0.227113	−0.113556	+	0.993532i	$0.536224\pi$
$128$	0	0
$129$	−0.815207	−0.0717750
$130$	0	0
$131$	3.98545	0.348211	0.174105	−	0.984727i	$-0.444297\pi$
$131$	3.98545	0.348211	0.174105	+	0.984727i	$0.444297\pi$
$132$	0	0
$133$	−25.8161	−2.23854
$134$	0	0
$135$	10.9017	0.938267
$136$	0	0
$137$	6.16250	0.526498	0.263249	−	0.964728i	$-0.415206\pi$
$137$	6.16250	0.526498	0.263249	+	0.964728i	$0.415206\pi$
$138$	0	0
$139$	−4.85710	−0.411974	−0.205987	−	0.978555i	$-0.566040\pi$
$139$	−4.85710	−0.411974	−0.205987	+	0.978555i	$0.566040\pi$
$140$	0	0
$141$	−14.1138	−1.18860
$142$	0	0
$143$	−7.14796	−0.597742
$144$	0	0
$145$	−2.13341	−0.177170
$146$	0	0
$147$	9.04963	0.746401
$148$	0	0
$149$	12.2422	1.00292	0.501459	−	0.865182i	$-0.332797\pi$
$149$	12.2422	1.00292	0.501459	+	0.865182i	$0.332797\pi$
$150$	0	0
$151$	12.7023	1.03370	0.516851	−	0.856076i	$-0.327104\pi$
$151$	12.7023	1.03370	0.516851	+	0.856076i	$0.327104\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.704088	0.0565537
$156$	0	0
$157$	−8.16519	−0.651653	−0.325827	−	0.945430i	$-0.605643\pi$
$157$	−8.16519	−0.651653	−0.325827	+	0.945430i	$0.605643\pi$
$158$	0	0
$159$	−14.5817	−1.15641
$160$	0	0
$161$	20.2395	1.59509
$162$	0	0
$163$	−24.4270	−1.91327	−0.956633	−	0.291295i	$-0.905914\pi$
$163$	−24.4270	−1.91327	−0.956633	+	0.291295i	$0.905914\pi$
$164$	0	0
$165$	24.0993	1.87612
$166$	0	0
$167$	−3.00681	−0.232674	−0.116337	−	0.993210i	$-0.537115\pi$
$167$	−3.00681	−0.232674	−0.116337	+	0.993210i	$0.537115\pi$
$168$	0	0
$169$	−11.0077	−0.846749
$170$	0	0
$171$	42.8881	3.27973
$172$	0	0
$173$	17.4757	1.32865	0.664325	−	0.747444i	$-0.268719\pi$
$173$	17.4757	1.32865	0.664325	+	0.747444i	$0.268719\pi$
$174$	0	0
$175$	7.22493	0.546153
$176$	0	0
$177$	−12.9855	−0.976046
$178$	0	0
$179$	−15.6604	−1.17052	−0.585258	−	0.810847i	$-0.699007\pi$
$179$	−15.6604	−1.17052	−0.585258	+	0.810847i	$0.699007\pi$
$180$	0	0
$181$	17.6655	1.31307	0.656533	−	0.754297i	$-0.272022\pi$
$181$	17.6655	1.31307	0.656533	+	0.754297i	$0.272022\pi$
$182$	0	0
$183$	13.1138	0.969401
$184$	0	0
$185$	8.83750	0.649746
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−21.0077	−1.52809
$190$	0	0
$191$	−3.75608	−0.271781	−0.135890	−	0.990724i	$-0.543389\pi$
$191$	−3.75608	−0.271781	−0.135890	+	0.990724i	$0.543389\pi$
$192$	0	0
$193$	0.724622	0.0521594	0.0260797	−	0.999660i	$-0.491698\pi$
$193$	0.724622	0.0521594	0.0260797	+	0.999660i	$0.491698\pi$
$194$	0	0
$195$	−6.71688	−0.481006
$196$	0	0
$197$	1.06149	0.0756280	0.0378140	−	0.999285i	$-0.487961\pi$
$197$	1.06149	0.0756280	0.0378140	+	0.999285i	$0.487961\pi$
$198$	0	0
$199$	12.9736	0.919674	0.459837	−	0.888003i	$-0.347908\pi$
$199$	12.9736	0.919674	0.459837	+	0.888003i	$0.347908\pi$
$200$	0	0
$201$	−5.98545	−0.422181
$202$	0	0
$203$	4.11112	0.288544
$204$	0	0
$205$	11.6382	0.812844
$206$	0	0
$207$	−33.6236	−2.33700
$208$	0	0
$209$	41.0506	2.83953
$210$	0	0
$211$	−0.223993	−0.0154203	−0.00771017	−	0.999970i	$-0.502454\pi$
$211$	−0.223993	−0.0154203	−0.00771017	+	0.999970i	$0.502454\pi$
$212$	0	0
$213$	40.6955	2.78841
$214$	0	0
$215$	−0.467911	−0.0319113
$216$	0	0
$217$	−1.35679	−0.0921050
$218$	0	0
$219$	13.7442	0.928748
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.7392	−0.786113	−0.393056	−	0.919514i	$-0.628582\pi$
$223$	−11.7392	−0.786113	−0.393056	+	0.919514i	$0.628582\pi$
$224$	0	0
$225$	−12.0027	−0.800179
$226$	0	0
$227$	−21.8229	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$227$	−21.8229	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$228$	0	0
$229$	−27.4861	−1.81633	−0.908166	−	0.418611i	$-0.862517\pi$
$229$	−27.4861	−1.81633	−0.908166	+	0.418611i	$0.862517\pi$
$230$	0	0
$231$	−46.4397	−3.05551
$232$	0	0
$233$	24.5868	1.61073	0.805366	−	0.592777i	$-0.201969\pi$
$233$	24.5868	1.61073	0.805366	+	0.592777i	$0.201969\pi$
$234$	0	0
$235$	−8.10101	−0.528452
$236$	0	0
$237$	−40.0428	−2.60106
$238$	0	0
$239$	11.1010	0.718065	0.359032	−	0.933325i	$-0.383107\pi$
$239$	11.1010	0.718065	0.359032	+	0.933325i	$0.383107\pi$
$240$	0	0
$241$	−2.99226	−0.192748	−0.0963742	−	0.995345i	$-0.530725\pi$
$241$	−2.99226	−0.192748	−0.0963742	+	0.995345i	$0.530725\pi$
$242$	0	0
$243$	−10.8033	−0.693035
$244$	0	0
$245$	5.19429	0.331851
$246$	0	0
$247$	−11.4415	−0.728005
$248$	0	0
$249$	−27.3482	−1.73312
$250$	0	0
$251$	−26.6013	−1.67906	−0.839530	−	0.543313i	$-0.817170\pi$
$251$	−26.6013	−1.67906	−0.839530	+	0.543313i	$0.817170\pi$
$252$	0	0
$253$	−32.1830	−2.02333
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.6536	0.976447	0.488224	−	0.872719i	$-0.337645\pi$
$257$	15.6536	0.976447	0.488224	+	0.872719i	$0.337645\pi$
$258$	0	0
$259$	−17.0300	−1.05819
$260$	0	0
$261$	−6.82976	−0.422751
$262$	0	0
$263$	25.3037	1.56029	0.780145	−	0.625599i	$-0.215145\pi$
$263$	25.3037	1.56029	0.780145	+	0.625599i	$0.215145\pi$
$264$	0	0
$265$	−8.36959	−0.514140
$266$	0	0
$267$	−26.2199	−1.60463
$268$	0	0
$269$	21.4243	1.30626	0.653130	−	0.757246i	$-0.273455\pi$
$269$	21.4243	1.30626	0.653130	+	0.757246i	$0.273455\pi$
$270$	0	0
$271$	2.91622	0.177148	0.0885739	−	0.996070i	$-0.471769\pi$
$271$	2.91622	0.177148	0.0885739	+	0.996070i	$0.471769\pi$
$272$	0	0
$273$	12.9436	0.783380
$274$	0	0
$275$	−11.4884	−0.692779
$276$	0	0
$277$	−27.3037	−1.64052	−0.820259	−	0.571993i	$-0.806171\pi$
$277$	−27.3037	−1.64052	−0.820259	+	0.571993i	$0.806171\pi$
$278$	0	0
$279$	2.25402	0.134945
$280$	0	0
$281$	−6.80335	−0.405854	−0.202927	−	0.979194i	$-0.565045\pi$
$281$	−6.80335	−0.405854	−0.202927	+	0.979194i	$0.565045\pi$
$282$	0	0
$283$	−1.17024	−0.0695638	−0.0347819	−	0.999395i	$-0.511074\pi$
$283$	−1.17024	−0.0695638	−0.0347819	+	0.999395i	$0.511074\pi$
$284$	0	0
$285$	38.5749	2.28498
$286$	0	0
$287$	−22.4270	−1.32382
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−12.5817	−0.737553
$292$	0	0
$293$	28.8384	1.68476	0.842380	−	0.538885i	$-0.181154\pi$
$293$	28.8384	1.68476	0.842380	+	0.538885i	$0.181154\pi$
$294$	0	0
$295$	−7.45336	−0.433952
$296$	0	0
$297$	33.4047	1.93834
$298$	0	0
$299$	8.96997	0.518747
$300$	0	0
$301$	0.901674	0.0519716
$302$	0	0
$303$	−24.2986	−1.39592
$304$	0	0
$305$	7.52704	0.430997
$306$	0	0
$307$	−14.3105	−0.816741	−0.408371	−	0.912816i	$-0.633903\pi$
$307$	−14.3105	−0.816741	−0.408371	+	0.912816i	$0.633903\pi$
$308$	0	0
$309$	−27.3405	−1.55535
$310$	0	0
$311$	21.1857	1.20133	0.600666	−	0.799500i	$-0.294902\pi$
$311$	21.1857	1.20133	0.600666	+	0.799500i	$0.294902\pi$
$312$	0	0
$313$	0.389185	0.0219981	0.0109990	−	0.999940i	$-0.496499\pi$
$313$	0.389185	0.0219981	0.0109990	+	0.999940i	$0.496499\pi$
$314$	0	0
$315$	−27.8485	−1.56909
$316$	0	0
$317$	−14.9932	−0.842102	−0.421051	−	0.907037i	$-0.638339\pi$
$317$	−14.9932	−0.842102	−0.421051	+	0.907037i	$0.638339\pi$
$318$	0	0
$319$	−6.53714	−0.366010
$320$	0	0
$321$	−13.2909	−0.741823
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.20203	0.177617
$326$	0	0
$327$	37.1634	2.05514
$328$	0	0
$329$	15.6108	0.860652
$330$	0	0
$331$	−6.48246	−0.356308	−0.178154	−	0.984003i	$-0.557013\pi$
$331$	−6.48246	−0.356308	−0.178154	+	0.984003i	$0.557013\pi$
$332$	0	0
$333$	28.2918	1.55038
$334$	0	0
$335$	−3.43552	−0.187702
$336$	0	0
$337$	5.23442	0.285137	0.142569	−	0.989785i	$-0.454464\pi$
$337$	5.23442	0.285137	0.142569	+	0.989785i	$0.454464\pi$
$338$	0	0
$339$	−23.3628	−1.26889
$340$	0	0
$341$	2.15745	0.116833
$342$	0	0
$343$	12.2841	0.663276
$344$	0	0
$345$	−30.2422	−1.62818
$346$	0	0
$347$	20.4492	1.09777	0.548886	−	0.835897i	$-0.315052\pi$
$347$	20.4492	1.09777	0.548886	+	0.835897i	$0.315052\pi$
$348$	0	0
$349$	6.73917	0.360740	0.180370	−	0.983599i	$-0.442271\pi$
$349$	6.73917	0.360740	0.180370	+	0.983599i	$0.442271\pi$
$350$	0	0
$351$	−9.31046	−0.496956
$352$	0	0
$353$	5.18748	0.276102	0.138051	−	0.990425i	$-0.455916\pi$
$353$	5.18748	0.276102	0.138051	+	0.990425i	$0.455916\pi$
$354$	0	0
$355$	23.3583	1.23973
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.9495	−1.42234	−0.711171	−	0.703019i	$-0.751835\pi$
$359$	−26.9495	−1.42234	−0.711171	+	0.703019i	$0.751835\pi$
$360$	0	0
$361$	46.7083	2.45833
$362$	0	0
$363$	42.1712	2.21341
$364$	0	0
$365$	7.88888	0.412923
$366$	0	0
$367$	−13.6450	−0.712261	−0.356131	−	0.934436i	$-0.615904\pi$
$367$	−13.6450	−0.712261	−0.356131	+	0.934436i	$0.615904\pi$
$368$	0	0
$369$	37.2576	1.93956
$370$	0	0
$371$	16.1284	0.837342
$372$	0	0
$373$	−17.6955	−0.916240	−0.458120	−	0.888890i	$-0.651477\pi$
$373$	−17.6955	−0.916240	−0.458120	+	0.888890i	$0.651477\pi$
$374$	0	0
$375$	−34.5895	−1.78619
$376$	0	0
$377$	1.82201	0.0938385
$378$	0	0
$379$	−20.1293	−1.03397	−0.516986	−	0.855994i	$-0.672946\pi$
$379$	−20.1293	−1.03397	−0.516986	+	0.855994i	$0.672946\pi$
$380$	0	0
$381$	−7.36959	−0.377555
$382$	0	0
$383$	−37.8753	−1.93534	−0.967668	−	0.252228i	$-0.918837\pi$
$383$	−37.8753	−1.93534	−0.967668	+	0.252228i	$0.918837\pi$
$384$	0	0
$385$	−26.6554	−1.35848
$386$	0	0
$387$	−1.49794	−0.0761446
$388$	0	0
$389$	18.9162	0.959091	0.479545	−	0.877517i	$-0.340802\pi$
$389$	18.9162	0.959091	0.479545	+	0.877517i	$0.340802\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	11.4757	0.578870
$394$	0	0
$395$	−22.9837	−1.15644
$396$	0	0
$397$	11.2344	0.563839	0.281920	−	0.959438i	$-0.409029\pi$
$397$	11.2344	0.563839	0.281920	+	0.959438i	$0.409029\pi$
$398$	0	0
$399$	−74.3346	−3.72139
$400$	0	0
$401$	30.4047	1.51834	0.759168	−	0.650894i	$-0.225606\pi$
$401$	30.4047	1.51834	0.759168	+	0.650894i	$0.225606\pi$
$402$	0	0
$403$	−0.601319	−0.0299538
$404$	0	0
$405$	5.15745	0.256276
$406$	0	0
$407$	27.0797	1.34229
$408$	0	0
$409$	−29.5895	−1.46310	−0.731552	−	0.681785i	$-0.761204\pi$
$409$	−29.5895	−1.46310	−0.731552	+	0.681785i	$0.761204\pi$
$410$	0	0
$411$	17.7442	0.875258
$412$	0	0
$413$	14.3628	0.706746
$414$	0	0
$415$	−15.6973	−0.770549
$416$	0	0
$417$	−13.9855	−0.684871
$418$	0	0
$419$	−18.5604	−0.906733	−0.453367	−	0.891324i	$-0.649777\pi$
$419$	−18.5604	−0.906733	−0.453367	+	0.891324i	$0.649777\pi$
$420$	0	0
$421$	12.2412	0.596601	0.298301	−	0.954472i	$-0.403580\pi$
$421$	12.2412	0.596601	0.298301	+	0.954472i	$0.403580\pi$
$422$	0	0
$423$	−25.9341	−1.26096
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.5047	−0.701934
$428$	0	0
$429$	−20.5817	−0.993695
$430$	0	0
$431$	−17.6682	−0.851047	−0.425523	−	0.904947i	$-0.639910\pi$
$431$	−17.6682	−0.851047	−0.425523	+	0.904947i	$0.639910\pi$
$432$	0	0
$433$	26.3114	1.26444	0.632222	−	0.774787i	$-0.282143\pi$
$433$	26.3114	1.26444	0.632222	+	0.774787i	$0.282143\pi$
$434$	0	0
$435$	−6.14290	−0.294530
$436$	0	0
$437$	−51.5144	−2.46427
$438$	0	0
$439$	19.6604	0.938342	0.469171	−	0.883107i	$-0.344553\pi$
$439$	19.6604	0.938342	0.469171	+	0.883107i	$0.344553\pi$
$440$	0	0
$441$	16.6287	0.791841
$442$	0	0
$443$	38.8631	1.84644	0.923220	−	0.384271i	$-0.125547\pi$
$443$	38.8631	1.84644	0.923220	+	0.384271i	$0.125547\pi$
$444$	0	0
$445$	−15.0496	−0.713421
$446$	0	0
$447$	35.2499	1.66726
$448$	0	0
$449$	−10.7452	−0.507095	−0.253548	−	0.967323i	$-0.581597\pi$
$449$	−10.7452	−0.507095	−0.253548	+	0.967323i	$0.581597\pi$
$450$	0	0
$451$	35.6614	1.67923
$452$	0	0
$453$	36.5749	1.71844
$454$	0	0
$455$	7.42932	0.348292
$456$	0	0
$457$	15.9145	0.744447	0.372224	−	0.928143i	$-0.378595\pi$
$457$	15.9145	0.744447	0.372224	+	0.928143i	$0.378595\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.0172363	−0.000802773 0	−0.000401386	−	1.00000i	$-0.500128\pi$
$461$	−0.0172363	−0.000802773 0	−0.000401386		1.00000i	$0.500128\pi$
$462$	0	0
$463$	−6.68180	−0.310530	−0.155265	−	0.987873i	$-0.549623\pi$
$463$	−6.68180	−0.310530	−0.155265	+	0.987873i	$0.549623\pi$
$464$	0	0
$465$	2.02734	0.0940157
$466$	0	0
$467$	38.1935	1.76738	0.883691	−	0.468070i	$-0.155050\pi$
$467$	38.1935	1.76738	0.883691	+	0.468070i	$0.155050\pi$
$468$	0	0
$469$	6.62031	0.305698
$470$	0	0
$471$	−23.5107	−1.08332
$472$	0	0
$473$	−1.43376	−0.0659245
$474$	0	0
$475$	−18.3892	−0.843754
$476$	0	0
$477$	−26.7939	−1.22681
$478$	0	0
$479$	−5.96080	−0.272356	−0.136178	−	0.990684i	$-0.543482\pi$
$479$	−5.96080	−0.272356	−0.136178	+	0.990684i	$0.543482\pi$
$480$	0	0
$481$	−7.54757	−0.344140
$482$	0	0
$483$	58.2772	2.65171
$484$	0	0
$485$	−7.22163	−0.327917
$486$	0	0
$487$	40.9454	1.85541	0.927707	−	0.373308i	$-0.121777\pi$
$487$	40.9454	1.85541	0.927707	+	0.373308i	$0.121777\pi$
$488$	0	0
$489$	−70.3346	−3.18064
$490$	0	0
$491$	−30.6100	−1.38141	−0.690705	−	0.723137i	$-0.742700\pi$
$491$	−30.6100	−1.38141	−0.690705	+	0.723137i	$0.742700\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	44.2823	1.99034
$496$	0	0
$497$	−45.0120	−2.01906
$498$	0	0
$499$	−5.35504	−0.239724	−0.119862	−	0.992791i	$-0.538245\pi$
$499$	−5.35504	−0.239724	−0.119862	+	0.992791i	$0.538245\pi$
$500$	0	0
$501$	−8.65776	−0.386800
$502$	0	0
$503$	3.48515	0.155395	0.0776975	−	0.996977i	$-0.475243\pi$
$503$	3.48515	0.155395	0.0776975	+	0.996977i	$0.475243\pi$
$504$	0	0
$505$	−13.9469	−0.620627
$506$	0	0
$507$	−31.6955	−1.40765
$508$	0	0
$509$	26.1198	1.15774	0.578870	−	0.815420i	$-0.303494\pi$
$509$	26.1198	1.15774	0.578870	+	0.815420i	$0.303494\pi$
$510$	0	0
$511$	−15.2020	−0.672498
$512$	0	0
$513$	53.4698	2.36075
$514$	0	0
$515$	−15.6928	−0.691509
$516$	0	0
$517$	−24.8229	−1.09171
$518$	0	0
$519$	50.3191	2.20876
$520$	0	0
$521$	19.1088	0.837170	0.418585	−	0.908178i	$-0.362526\pi$
$521$	19.1088	0.837170	0.418585	+	0.908178i	$0.362526\pi$
$522$	0	0
$523$	−14.5699	−0.637095	−0.318548	−	0.947907i	$-0.603195\pi$
$523$	−14.5699	−0.637095	−0.318548	+	0.947907i	$0.603195\pi$
$524$	0	0
$525$	20.8033	0.907933
$526$	0	0
$527$	0	0
$528$	0	0
$529$	17.3865	0.755935
$530$	0	0
$531$	−23.8607	−1.03547
$532$	0	0
$533$	−9.93944	−0.430525
$534$	0	0
$535$	−7.62866	−0.329816
$536$	0	0
$537$	−45.0925	−1.94588
$538$	0	0
$539$	15.9162	0.685560
$540$	0	0
$541$	−28.2003	−1.21242	−0.606212	−	0.795303i	$-0.707312\pi$
$541$	−28.2003	−1.21242	−0.606212	+	0.795303i	$0.707312\pi$
$542$	0	0
$543$	50.8658	2.18286
$544$	0	0
$545$	21.3310	0.913719
$546$	0	0
$547$	13.0692	0.558800	0.279400	−	0.960175i	$-0.409864\pi$
$547$	13.0692	0.558800	0.279400	+	0.960175i	$0.409864\pi$
$548$	0	0
$549$	24.0966	1.02842
$550$	0	0
$551$	−10.4638	−0.445773
$552$	0	0
$553$	44.2900	1.88340
$554$	0	0
$555$	25.4466	1.08015
$556$	0	0
$557$	34.1043	1.44505	0.722523	−	0.691347i	$-0.242982\pi$
$557$	34.1043	1.44505	0.722523	+	0.691347i	$0.242982\pi$
$558$	0	0
$559$	0.399615	0.0169019
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.99319	−0.126148	−0.0630740	−	0.998009i	$-0.520090\pi$
$563$	−2.99319	−0.126148	−0.0630740	+	0.998009i	$0.520090\pi$
$564$	0	0
$565$	−13.4097	−0.564151
$566$	0	0
$567$	−9.93851	−0.417378
$568$	0	0
$569$	−34.9709	−1.46606	−0.733028	−	0.680198i	$-0.761894\pi$
$569$	−34.9709	−1.46606	−0.733028	+	0.680198i	$0.761894\pi$
$570$	0	0
$571$	31.9914	1.33880	0.669400	−	0.742902i	$-0.266551\pi$
$571$	31.9914	1.33880	0.669400	+	0.742902i	$0.266551\pi$
$572$	0	0
$573$	−10.8152	−0.451812
$574$	0	0
$575$	14.4169	0.601224
$576$	0	0
$577$	18.8452	0.784537	0.392269	−	0.919851i	$-0.371690\pi$
$577$	18.8452	0.784537	0.392269	+	0.919851i	$0.371690\pi$
$578$	0	0
$579$	2.08647	0.0867106
$580$	0	0
$581$	30.2490	1.25494
$582$	0	0
$583$	−25.6459	−1.06214
$584$	0	0
$585$	−12.3422	−0.510289
$586$	0	0
$587$	11.2175	0.462996	0.231498	−	0.972835i	$-0.425637\pi$
$587$	11.2175	0.462996	0.231498	+	0.972835i	$0.425637\pi$
$588$	0	0
$589$	3.45336	0.142293
$590$	0	0
$591$	3.05644	0.125725
$592$	0	0
$593$	16.5422	0.679306	0.339653	−	0.940551i	$-0.389690\pi$
$593$	16.5422	0.679306	0.339653	+	0.940551i	$0.389690\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	37.3560	1.52888
$598$	0	0
$599$	−14.5585	−0.594844	−0.297422	−	0.954746i	$-0.596127\pi$
$599$	−14.5585	−0.594844	−0.297422	+	0.954746i	$0.596127\pi$
$600$	0	0
$601$	11.7460	0.479128	0.239564	−	0.970881i	$-0.422995\pi$
$601$	11.7460	0.479128	0.239564	+	0.970881i	$0.422995\pi$
$602$	0	0
$603$	−10.9982	−0.447883
$604$	0	0
$605$	24.2053	0.984086
$606$	0	0
$607$	34.2249	1.38915	0.694573	−	0.719422i	$-0.255593\pi$
$607$	34.2249	1.38915	0.694573	+	0.719422i	$0.255593\pi$
$608$	0	0
$609$	11.8375	0.479680
$610$	0	0
$611$	6.91859	0.279896
$612$	0	0
$613$	37.0205	1.49525	0.747623	−	0.664124i	$-0.231195\pi$
$613$	37.0205	1.49525	0.747623	+	0.664124i	$0.231195\pi$
$614$	0	0
$615$	33.5107	1.35128
$616$	0	0
$617$	39.5280	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$617$	39.5280	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$618$	0	0
$619$	−22.7547	−0.914587	−0.457293	−	0.889316i	$-0.651181\pi$
$619$	−22.7547	−0.914587	−0.457293	+	0.889316i	$0.651181\pi$
$620$	0	0
$621$	−41.9195	−1.68217
$622$	0	0
$623$	29.0009	1.16190
$624$	0	0
$625$	−8.51073	−0.340429
$626$	0	0
$627$	118.200	4.72047
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−8.36009	−0.332810	−0.166405	−	0.986057i	$-0.553216\pi$
$631$	−8.36009	−0.332810	−0.166405	+	0.986057i	$0.553216\pi$
$632$	0	0
$633$	−0.644963	−0.0256350
$634$	0	0
$635$	−4.22998	−0.167862
$636$	0	0
$637$	−4.43613	−0.175766
$638$	0	0
$639$	74.7779	2.95817
$640$	0	0
$641$	9.28817	0.366861	0.183430	−	0.983033i	$-0.441280\pi$
$641$	9.28817	0.366861	0.183430	+	0.983033i	$0.441280\pi$
$642$	0	0
$643$	45.8871	1.80961	0.904806	−	0.425824i	$-0.140016\pi$
$643$	45.8871	1.80961	0.904806	+	0.425824i	$0.140016\pi$
$644$	0	0
$645$	−1.34730	−0.0530497
$646$	0	0
$647$	−26.8411	−1.05523	−0.527617	−	0.849483i	$-0.676914\pi$
$647$	−26.8411	−1.05523	−0.527617	+	0.849483i	$0.676914\pi$
$648$	0	0
$649$	−22.8384	−0.896487
$650$	0	0
$651$	−3.90673	−0.153117
$652$	0	0
$653$	24.1317	0.944345	0.472172	−	0.881506i	$-0.343470\pi$
$653$	24.1317	0.944345	0.472172	+	0.881506i	$0.343470\pi$
$654$	0	0
$655$	6.58677	0.257366
$656$	0	0
$657$	25.2550	0.985290
$658$	0	0
$659$	34.3381	1.33762	0.668812	−	0.743432i	$-0.266803\pi$
$659$	34.3381	1.33762	0.668812	+	0.743432i	$0.266803\pi$
$660$	0	0
$661$	−0.213888	−0.00831929	−0.00415965	−	0.999991i	$-0.501324\pi$
$661$	−0.213888	−0.00831929	−0.00415965	+	0.999991i	$0.501324\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−42.6664	−1.65453
$666$	0	0
$667$	8.20346	0.317639
$668$	0	0
$669$	−33.8016	−1.30685
$670$	0	0
$671$	23.0642	0.890383
$672$	0	0
$673$	−25.4584	−0.981350	−0.490675	−	0.871343i	$-0.663250\pi$
$673$	−25.4584	−0.981350	−0.490675	+	0.871343i	$0.663250\pi$
$674$	0	0
$675$	−14.9641	−0.575968
$676$	0	0
$677$	3.82058	0.146837	0.0734185	−	0.997301i	$-0.476609\pi$
$677$	3.82058	0.146837	0.0734185	+	0.997301i	$0.476609\pi$
$678$	0	0
$679$	13.9162	0.534056
$680$	0	0
$681$	−62.8367	−2.40791
$682$	0	0
$683$	7.87164	0.301200	0.150600	−	0.988595i	$-0.451879\pi$
$683$	7.87164	0.301200	0.150600	+	0.988595i	$0.451879\pi$
$684$	0	0
$685$	10.1848	0.389141
$686$	0	0
$687$	−79.1430	−3.01949
$688$	0	0
$689$	7.14796	0.272315
$690$	0	0
$691$	0.166623	0.00633862	0.00316931	−	0.999995i	$-0.498991\pi$
$691$	0.166623	0.00633862	0.00316931	+	0.999995i	$0.498991\pi$
$692$	0	0
$693$	−85.3329	−3.24153
$694$	0	0
$695$	−8.02734	−0.304494
$696$	0	0
$697$	0	0
$698$	0	0
$699$	70.7948	2.67770
$700$	0	0
$701$	28.2841	1.06827	0.534137	−	0.845398i	$-0.320637\pi$
$701$	28.2841	1.06827	0.534137	+	0.845398i	$0.320637\pi$
$702$	0	0
$703$	43.3455	1.63481
$704$	0	0
$705$	−23.3259	−0.878506
$706$	0	0
$707$	26.8759	1.01077
$708$	0	0
$709$	−3.31727	−0.124583	−0.0622913	−	0.998058i	$-0.519841\pi$
$709$	−3.31727	−0.124583	−0.0622913	+	0.998058i	$0.519841\pi$
$710$	0	0
$711$	−73.5785	−2.75941
$712$	0	0
$713$	−2.70739	−0.101392
$714$	0	0
$715$	−11.8135	−0.441798
$716$	0	0
$717$	31.9641	1.19372
$718$	0	0
$719$	−9.18573	−0.342570	−0.171285	−	0.985222i	$-0.554792\pi$
$719$	−9.18573	−0.342570	−0.171285	+	0.985222i	$0.554792\pi$
$720$	0	0
$721$	30.2404	1.12621
$722$	0	0
$723$	−8.61587	−0.320428
$724$	0	0
$725$	2.92841	0.108758
$726$	0	0
$727$	45.7151	1.69548	0.847740	−	0.530412i	$-0.177963\pi$
$727$	45.7151	1.69548	0.847740	+	0.530412i	$0.177963\pi$
$728$	0	0
$729$	−40.4688	−1.49885
$730$	0	0
$731$	0	0
$732$	0	0
$733$	14.6996	0.542944	0.271472	−	0.962446i	$-0.412490\pi$
$733$	14.6996	0.542944	0.271472	+	0.962446i	$0.412490\pi$
$734$	0	0
$735$	14.9564	0.551674
$736$	0	0
$737$	−10.5270	−0.387768
$738$	0	0
$739$	13.2695	0.488127	0.244063	−	0.969759i	$-0.421520\pi$
$739$	13.2695	0.488127	0.244063	+	0.969759i	$0.421520\pi$
$740$	0	0
$741$	−32.9445	−1.21025
$742$	0	0
$743$	−21.3054	−0.781620	−0.390810	−	0.920471i	$-0.627805\pi$
$743$	−21.3054	−0.781620	−0.390810	+	0.920471i	$0.627805\pi$
$744$	0	0
$745$	20.2327	0.741268
$746$	0	0
$747$	−50.2523	−1.83863
$748$	0	0
$749$	14.7006	0.537147
$750$	0	0
$751$	−29.9614	−1.09331	−0.546654	−	0.837359i	$-0.684099\pi$
$751$	−29.9614	−1.09331	−0.546654	+	0.837359i	$0.684099\pi$
$752$	0	0
$753$	−76.5954	−2.79129
$754$	0	0
$755$	20.9932	0.764021
$756$	0	0
$757$	−0.956356	−0.0347594	−0.0173797	−	0.999849i	$-0.505532\pi$
$757$	−0.956356	−0.0347594	−0.0173797	+	0.999849i	$0.505532\pi$
$758$	0	0
$759$	−92.6674	−3.36361
$760$	0	0
$761$	27.1317	0.983522	0.491761	−	0.870730i	$-0.336353\pi$
$761$	27.1317	0.983522	0.491761	+	0.870730i	$0.336353\pi$
$762$	0	0
$763$	−41.1052	−1.48811
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.36547	0.229844
$768$	0	0
$769$	−25.9982	−0.937521	−0.468760	−	0.883325i	$-0.655299\pi$
$769$	−25.9982	−0.937521	−0.468760	+	0.883325i	$0.655299\pi$
$770$	0	0
$771$	45.0729	1.62326
$772$	0	0
$773$	−12.0564	−0.433640	−0.216820	−	0.976212i	$-0.569568\pi$
$773$	−12.0564	−0.433640	−0.216820	+	0.976212i	$0.569568\pi$
$774$	0	0
$775$	−0.966461	−0.0347163
$776$	0	0
$777$	−49.0360	−1.75916
$778$	0	0
$779$	57.0820	2.04518
$780$	0	0
$781$	71.5741	2.56112
$782$	0	0
$783$	−8.51485	−0.304296
$784$	0	0
$785$	−13.4946	−0.481644
$786$	0	0
$787$	−5.86215	−0.208963	−0.104482	−	0.994527i	$-0.533318\pi$
$787$	−5.86215	−0.208963	−0.104482	+	0.994527i	$0.533318\pi$
$788$	0	0
$789$	72.8590	2.59385
$790$	0	0
$791$	25.8408	0.918793
$792$	0	0
$793$	−6.42839	−0.228279
$794$	0	0
$795$	−24.0993	−0.854713
$796$	0	0
$797$	−40.9986	−1.45224	−0.726122	−	0.687566i	$-0.758679\pi$
$797$	−40.9986	−1.45224	−0.726122	+	0.687566i	$0.758679\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−48.1789	−1.70232
$802$	0	0
$803$	24.1729	0.853044
$804$	0	0
$805$	33.4499	1.17895
$806$	0	0
$807$	61.6887	2.17155
$808$	0	0
$809$	40.1976	1.41327	0.706636	−	0.707577i	$-0.250212\pi$
$809$	40.1976	1.41327	0.706636	+	0.707577i	$0.250212\pi$
$810$	0	0
$811$	−27.4029	−0.962246	−0.481123	−	0.876653i	$-0.659771\pi$
$811$	−27.4029	−0.962246	−0.481123	+	0.876653i	$0.659771\pi$
$812$	0	0
$813$	8.39693	0.294493
$814$	0	0
$815$	−40.3705	−1.41412
$816$	0	0
$817$	−2.29498	−0.0802911
$818$	0	0
$819$	23.7837	0.831071
$820$	0	0
$821$	7.39693	0.258155	0.129077	−	0.991635i	$-0.458798\pi$
$821$	7.39693	0.258155	0.129077	+	0.991635i	$0.458798\pi$
$822$	0	0
$823$	9.53890	0.332505	0.166253	−	0.986083i	$-0.446833\pi$
$823$	9.53890	0.332505	0.166253	+	0.986083i	$0.446833\pi$
$824$	0	0
$825$	−33.0797	−1.15169
$826$	0	0
$827$	31.6979	1.10224	0.551122	−	0.834425i	$-0.314200\pi$
$827$	31.6979	1.10224	0.551122	+	0.834425i	$0.314200\pi$
$828$	0	0
$829$	−28.1257	−0.976845	−0.488422	−	0.872607i	$-0.662427\pi$
$829$	−28.1257	−0.976845	−0.488422	+	0.872607i	$0.662427\pi$
$830$	0	0
$831$	−78.6177	−2.72722
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.96936	−0.171972
$836$	0	0
$837$	2.81016	0.0971332
$838$	0	0
$839$	−17.7442	−0.612599	−0.306299	−	0.951935i	$-0.599091\pi$
$839$	−17.7442	−0.612599	−0.306299	+	0.951935i	$0.599091\pi$
$840$	0	0
$841$	−27.3337	−0.942541
$842$	0	0
$843$	−19.5895	−0.674697
$844$	0	0
$845$	−18.1925	−0.625842
$846$	0	0
$847$	−46.6441	−1.60271
$848$	0	0
$849$	−3.36959	−0.115644
$850$	0	0
$851$	−33.9823	−1.16490
$852$	0	0
$853$	−18.3277	−0.627528	−0.313764	−	0.949501i	$-0.601590\pi$
$853$	−18.3277	−0.627528	−0.313764	+	0.949501i	$0.601590\pi$
$854$	0	0
$855$	70.8813	2.42409
$856$	0	0
$857$	−20.7098	−0.707432	−0.353716	−	0.935353i	$-0.615082\pi$
$857$	−20.7098	−0.707432	−0.353716	+	0.935353i	$0.615082\pi$
$858$	0	0
$859$	−20.8212	−0.710410	−0.355205	−	0.934788i	$-0.615589\pi$
$859$	−20.8212	−0.710410	−0.355205	+	0.934788i	$0.615589\pi$
$860$	0	0
$861$	−64.5758	−2.20074
$862$	0	0
$863$	−13.6031	−0.463054	−0.231527	−	0.972828i	$-0.574372\pi$
$863$	−13.6031	−0.463054	−0.231527	+	0.972828i	$0.574372\pi$
$864$	0	0
$865$	28.8821	0.982020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−70.4261	−2.38904
$870$	0	0
$871$	2.93407	0.0994171
$872$	0	0
$873$	−23.1189	−0.782455
$874$	0	0
$875$	38.2583	1.29337
$876$	0	0
$877$	−33.0591	−1.11633	−0.558164	−	0.829731i	$-0.688494\pi$
$877$	−33.0591	−1.11633	−0.558164	+	0.829731i	$0.688494\pi$
$878$	0	0
$879$	83.0369	2.80077
$880$	0	0
$881$	18.5107	0.623642	0.311821	−	0.950141i	$-0.399061\pi$
$881$	18.5107	0.623642	0.311821	+	0.950141i	$0.399061\pi$
$882$	0	0
$883$	−36.3432	−1.22305	−0.611523	−	0.791227i	$-0.709443\pi$
$883$	−36.3432	−1.22305	−0.611523	+	0.791227i	$0.709443\pi$
$884$	0	0
$885$	−21.4611	−0.721407
$886$	0	0
$887$	−44.9968	−1.51085	−0.755423	−	0.655238i	$-0.772568\pi$
$887$	−44.9968	−1.51085	−0.755423	+	0.655238i	$0.772568\pi$
$888$	0	0
$889$	8.15125	0.273384
$890$	0	0
$891$	15.8033	0.529432
$892$	0	0
$893$	−39.7333	−1.32962
$894$	0	0
$895$	−25.8821	−0.865142
$896$	0	0
$897$	25.8280	0.862372
$898$	0	0
$899$	−0.549935	−0.0183413
$900$	0	0
$901$	0	0
$902$	0	0
$903$	2.59627	0.0863983
$904$	0	0
$905$	29.1958	0.970502
$906$	0	0
$907$	28.8522	0.958020	0.479010	−	0.877809i	$-0.340996\pi$
$907$	28.8522	0.958020	0.479010	+	0.877809i	$0.340996\pi$
$908$	0	0
$909$	−44.6486	−1.48090
$910$	0	0
$911$	11.5621	0.383070	0.191535	−	0.981486i	$-0.438653\pi$
$911$	11.5621	0.383070	0.191535	+	0.981486i	$0.438653\pi$
$912$	0	0
$913$	−48.0993	−1.59185
$914$	0	0
$915$	21.6732	0.716495
$916$	0	0
$917$	−12.6928	−0.419154
$918$	0	0
$919$	−2.78880	−0.0919940	−0.0459970	−	0.998942i	$-0.514646\pi$
$919$	−2.78880	−0.0919940	−0.0459970	+	0.998942i	$0.514646\pi$
$920$	0	0
$921$	−41.2053	−1.35776
$922$	0	0
$923$	−19.9489	−0.656627
$924$	0	0
$925$	−12.1307	−0.398856
$926$	0	0
$927$	−50.2380	−1.65003
$928$	0	0
$929$	40.2540	1.32069	0.660346	−	0.750962i	$-0.270410\pi$
$929$	40.2540	1.32069	0.660346	+	0.750962i	$0.270410\pi$
$930$	0	0
$931$	25.4766	0.834961
$932$	0	0
$933$	61.0019	1.99711
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.7739	−1.07068	−0.535339	−	0.844637i	$-0.679816\pi$
$937$	−32.7739	−1.07068	−0.535339	+	0.844637i	$0.679816\pi$
$938$	0	0
$939$	1.12061	0.0365699
$940$	0	0
$941$	40.9121	1.33370	0.666848	−	0.745194i	$-0.267643\pi$
$941$	40.9121	1.33370	0.666848	+	0.745194i	$0.267643\pi$
$942$	0	0
$943$	−44.7515	−1.45731
$944$	0	0
$945$	−34.7196	−1.12943
$946$	0	0
$947$	15.4807	0.503055	0.251528	−	0.967850i	$-0.419067\pi$
$947$	15.4807	0.503055	0.251528	+	0.967850i	$0.419067\pi$
$948$	0	0
$949$	−6.73742	−0.218706
$950$	0	0
$951$	−43.1712	−1.39992
$952$	0	0
$953$	46.9677	1.52143	0.760717	−	0.649084i	$-0.224848\pi$
$953$	46.9677	1.52143	0.760717	+	0.649084i	$0.224848\pi$
$954$	0	0
$955$	−6.20769	−0.200876
$956$	0	0
$957$	−18.8229	−0.608460
$958$	0	0
$959$	−19.6263	−0.633766
$960$	0	0
$961$	−30.8185	−0.994145
$962$	0	0
$963$	−24.4219	−0.786985
$964$	0	0
$965$	1.19759	0.0385516
$966$	0	0
$967$	36.8881	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$967$	36.8881	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.0087	0.995116	0.497558	−	0.867431i	$-0.334230\pi$
$971$	31.0087	0.995116	0.497558	+	0.867431i	$0.334230\pi$
$972$	0	0
$973$	15.4688	0.495908
$974$	0	0
$975$	9.21987	0.295272
$976$	0	0
$977$	−13.9905	−0.447596	−0.223798	−	0.974636i	$-0.571846\pi$
$977$	−13.9905	−0.447596	−0.223798	+	0.974636i	$0.571846\pi$
$978$	0	0
$979$	−46.1147	−1.47383
$980$	0	0
$981$	68.2877	2.18026
$982$	0	0
$983$	−21.0888	−0.672629	−0.336315	−	0.941750i	$-0.609181\pi$
$983$	−21.0888	−0.672629	−0.336315	+	0.941750i	$0.609181\pi$
$984$	0	0
$985$	1.75433	0.0558975
$986$	0	0
$987$	44.9495	1.43076
$988$	0	0
$989$	1.79923	0.0572122
$990$	0	0
$991$	−14.2362	−0.452227	−0.226114	−	0.974101i	$-0.572602\pi$
$991$	−14.2362	−0.452227	−0.226114	+	0.974101i	$0.572602\pi$
$992$	0	0
$993$	−18.6655	−0.592332
$994$	0	0
$995$	21.4415	0.679741
$996$	0	0
$997$	9.52528	0.301669	0.150834	−	0.988559i	$-0.451804\pi$
$997$	9.52528	0.301669	0.150834	+	0.988559i	$0.451804\pi$
$998$	0	0
$999$	35.2722	1.11596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1156.2.a.f.1.3	yes	3
4.3	odd	2		4624.2.a.be.1.1		3
17.2	even	8		1156.2.e.g.905.6		12
17.3	odd	16		1156.2.h.h.757.6		24
17.4	even	4		1156.2.b.e.577.1		6
17.5	odd	16		1156.2.h.h.977.1		24
17.6	odd	16		1156.2.h.h.733.6		24
17.7	odd	16		1156.2.h.h.1001.1		24
17.8	even	8		1156.2.e.g.829.1		12
17.9	even	8		1156.2.e.g.829.6		12
17.10	odd	16		1156.2.h.h.1001.6		24
17.11	odd	16		1156.2.h.h.733.1		24
17.12	odd	16		1156.2.h.h.977.6		24
17.13	even	4		1156.2.b.e.577.6		6
17.14	odd	16		1156.2.h.h.757.1		24
17.15	even	8		1156.2.e.g.905.1		12
17.16	even	2		1156.2.a.e.1.1	✓	3
68.67	odd	2		4624.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1156.2.a.e.1.1	✓	3	17.16	even	2
1156.2.a.f.1.3	yes	3	1.1	even	1	trivial
1156.2.b.e.577.1		6	17.4	even	4
1156.2.b.e.577.6		6	17.13	even	4
1156.2.e.g.829.1		12	17.8	even	8
1156.2.e.g.829.6		12	17.9	even	8
1156.2.e.g.905.1		12	17.15	even	8
1156.2.e.g.905.6		12	17.2	even	8
1156.2.h.h.733.1		24	17.11	odd	16
1156.2.h.h.733.6		24	17.6	odd	16
1156.2.h.h.757.1		24	17.14	odd	16
1156.2.h.h.757.6		24	17.3	odd	16
1156.2.h.h.977.1		24	17.5	odd	16
1156.2.h.h.977.6		24	17.12	odd	16
1156.2.h.h.1001.1		24	17.7	odd	16
1156.2.h.h.1001.6		24	17.10	odd	16
4624.2.a.be.1.1		3	4.3	odd	2
4624.2.a.bf.1.3		3	68.67	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:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1156.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87939 q^{3} +1.65270 q^{5} -3.18479 q^{7} +5.29086 q^{9} +5.06418 q^{11} -1.41147 q^{13} +4.75877 q^{15} +8.10607 q^{19} -9.17024 q^{21} -6.35504 q^{23} -2.26857 q^{25} +6.59627 q^{27} -1.29086 q^{29} +0.426022 q^{31} +14.5817 q^{33} -5.26352 q^{35} +5.34730 q^{37} -4.06418 q^{39} +7.04189 q^{41} -0.283119 q^{43} +8.74422 q^{45} -4.90167 q^{47} +3.14290 q^{49} -5.06418 q^{53} +8.36959 q^{55} +23.3405 q^{57} -4.50980 q^{59} +4.55438 q^{61} -16.8503 q^{63} -2.33275 q^{65} -2.07873 q^{67} -18.2986 q^{69} +14.1334 q^{71} +4.77332 q^{73} -6.53209 q^{75} -16.1284 q^{77} -13.9067 q^{79} +3.12061 q^{81} -9.49794 q^{83} -3.71688 q^{87} -9.10607 q^{89} +4.49525 q^{91} +1.22668 q^{93} +13.3969 q^{95} -4.36959 q^{97} +26.7939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{5} - 6 q^{7} + 6 q^{11} + 6 q^{13} + 3 q^{15} + 12 q^{19} - 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} + 12 q^{29} + 9 q^{31} + 12 q^{33} - 21 q^{35} + 15 q^{37} - 3 q^{39} + 18 q^{41}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^5 - 6 * q^7 + 6 * q^11 + 6 * q^13 + 3 * q^15 + 12 * q^19 - 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 + 12 * q^29 + 9 * q^31 + 12 * q^33 - 21 * q^35 + 15 * q^37 - 3 * q^39 + 18 * q^41 - 9 * q^43 - 3 * q^45 - 3 * q^47 + 9 * q^49 - 6 * q^53 + 18 * q^55 + 27 * q^57 - 15 * q^59 + 3 * q^61 - 9 * q^63 + 12 * q^65 - 15 * q^67 - 15 * q^69 + 9 * q^71 + 21 * q^73 - 15 * q^75 - 30 * q^77 - 15 * q^79 + 15 * q^81 - 3 * q^83 - 3 * q^87 - 15 * q^89 - 3 * q^91 - 3 * q^93 + 12 * q^95 - 6 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more