LMFDB - Embedded newform 2116.2.a.i.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2116,2,Mod(1,2116)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2116.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2116.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$16.8963450677$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 19x^{8} + 15x^{7} + 129x^{6} - 62x^{5} - 387x^{4} + 47x^{3} + 447x^{2} + 106x - 23 $ x^10 - x^9 - 19x^8 + 15x^7 + 129x^6 - 62x^5 - 387x^4 + 47x^3 + 447x^2 + 106x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.47688$ of defining polynomial
Character	$\chi$	$=$	2116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78660 q^{3} -1.56806 q^{5} -0.164812 q^{7} +4.76513 q^{9} +O(q^{10})$	$q+2.78660 q^{3} -1.56806 q^{5} -0.164812 q^{7} +4.76513 q^{9} -6.37160 q^{11} -4.56766 q^{13} -4.36956 q^{15} +0.560844 q^{17} -4.23320 q^{19} -0.459265 q^{21} -2.54118 q^{25} +4.91870 q^{27} -5.63502 q^{29} +8.95162 q^{31} -17.7551 q^{33} +0.258436 q^{35} +5.93466 q^{37} -12.7282 q^{39} -3.51907 q^{41} -8.23443 q^{43} -7.47201 q^{45} -5.52419 q^{47} -6.97284 q^{49} +1.56285 q^{51} +1.87575 q^{53} +9.99106 q^{55} -11.7962 q^{57} +6.16349 q^{59} -1.42040 q^{61} -0.785351 q^{63} +7.16237 q^{65} +7.56650 q^{67} -2.00185 q^{71} +10.8343 q^{73} -7.08125 q^{75} +1.05012 q^{77} +8.71390 q^{79} -0.588949 q^{81} +0.811858 q^{83} -0.879438 q^{85} -15.7025 q^{87} +6.48798 q^{89} +0.752805 q^{91} +24.9446 q^{93} +6.63792 q^{95} -7.56444 q^{97} -30.3615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} - 12 q^{5} - q^{7} + 9 q^{9}+O(q^{10}) $ 10 * q + q^3 - 12 * q^5 - q^7 + 9 * q^9	$ 10 q + q^{3} - 12 q^{5} - q^{7} + 9 q^{9} - 10 q^{11} + 3 q^{13} + 3 q^{15} - 23 q^{17} - 11 q^{19} - 15 q^{21} + 14 q^{25} + 7 q^{27} - 6 q^{29} - 12 q^{31} - 25 q^{33} - 15 q^{35} - 6 q^{37} - 7 q^{39} - 8 q^{41} - 11 q^{43} - 39 q^{45} + 13 q^{47} - 3 q^{49} - 31 q^{51} - 29 q^{53} + 24 q^{55} - 18 q^{57} - 5 q^{59} - 27 q^{61} - 14 q^{63} - 39 q^{65} + 20 q^{71} + 49 q^{73} - 43 q^{75} - 40 q^{77} - 8 q^{79} - 38 q^{81} - 26 q^{83} + 37 q^{85} - 53 q^{87} - 60 q^{89} - 13 q^{91} - 7 q^{93} - 24 q^{95} - 27 q^{97} - 27 q^{99}+O(q^{100}) $ 10 * q + q^3 - 12 * q^5 - q^7 + 9 * q^9 - 10 * q^11 + 3 * q^13 + 3 * q^15 - 23 * q^17 - 11 * q^19 - 15 * q^21 + 14 * q^25 + 7 * q^27 - 6 * q^29 - 12 * q^31 - 25 * q^33 - 15 * q^35 - 6 * q^37 - 7 * q^39 - 8 * q^41 - 11 * q^43 - 39 * q^45 + 13 * q^47 - 3 * q^49 - 31 * q^51 - 29 * q^53 + 24 * q^55 - 18 * q^57 - 5 * q^59 - 27 * q^61 - 14 * q^63 - 39 * q^65 + 20 * q^71 + 49 * q^73 - 43 * q^75 - 40 * q^77 - 8 * q^79 - 38 * q^81 - 26 * q^83 + 37 * q^85 - 53 * q^87 - 60 * q^89 - 13 * q^91 - 7 * q^93 - 24 * q^95 - 27 * q^97 - 27 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.78660	1.60884	0.804421	−	0.594059i	$-0.202475\pi$
$3$	2.78660	1.60884	0.804421	+	0.594059i	$0.202475\pi$
$4$	0	0
$5$	−1.56806	−0.701258	−0.350629	−	0.936514i	$-0.614032\pi$
$5$	−1.56806	−0.701258	−0.350629	+	0.936514i	$0.614032\pi$
$6$	0	0
$7$	−0.164812	−0.0622931	−0.0311466	−	0.999515i	$-0.509916\pi$
$7$	−0.164812	−0.0622931	−0.0311466	+	0.999515i	$0.509916\pi$
$8$	0	0
$9$	4.76513	1.58838
$10$	0	0
$11$	−6.37160	−1.92111	−0.960555	−	0.278090i	$-0.910299\pi$
$11$	−6.37160	−1.92111	−0.960555	+	0.278090i	$0.910299\pi$
$12$	0	0
$13$	−4.56766	−1.26684	−0.633420	−	0.773808i	$-0.718349\pi$
$13$	−4.56766	−1.26684	−0.633420	+	0.773808i	$0.718349\pi$
$14$	0	0
$15$	−4.36956	−1.12821
$16$	0	0
$17$	0.560844	0.136025	0.0680123	−	0.997684i	$-0.478334\pi$
$17$	0.560844	0.136025	0.0680123	+	0.997684i	$0.478334\pi$
$18$	0	0
$19$	−4.23320	−0.971163	−0.485582	−	0.874191i	$-0.661392\pi$
$19$	−4.23320	−0.971163	−0.485582	+	0.874191i	$0.661392\pi$
$20$	0	0
$21$	−0.459265	−0.100220
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.54118	−0.508237
$26$	0	0
$27$	4.91870	0.946604
$28$	0	0
$29$	−5.63502	−1.04640	−0.523198	−	0.852211i	$-0.675261\pi$
$29$	−5.63502	−1.04640	−0.523198	+	0.852211i	$0.675261\pi$
$30$	0	0
$31$	8.95162	1.60776	0.803880	−	0.594792i	$-0.202766\pi$
$31$	8.95162	1.60776	0.803880	+	0.594792i	$0.202766\pi$
$32$	0	0
$33$	−17.7551	−3.09076
$34$	0	0
$35$	0.258436	0.0436836
$36$	0	0
$37$	5.93466	0.975652	0.487826	−	0.872941i	$-0.337790\pi$
$37$	5.93466	0.975652	0.487826	+	0.872941i	$0.337790\pi$
$38$	0	0
$39$	−12.7282	−2.03815
$40$	0	0
$41$	−3.51907	−0.549586	−0.274793	−	0.961503i	$-0.588609\pi$
$41$	−3.51907	−0.549586	−0.274793	+	0.961503i	$0.588609\pi$
$42$	0	0
$43$	−8.23443	−1.25574	−0.627869	−	0.778319i	$-0.716073\pi$
$43$	−8.23443	−1.25574	−0.627869	+	0.778319i	$0.716073\pi$
$44$	0	0
$45$	−7.47201	−1.11386
$46$	0	0
$47$	−5.52419	−0.805786	−0.402893	−	0.915247i	$-0.631995\pi$
$47$	−5.52419	−0.805786	−0.402893	+	0.915247i	$0.631995\pi$
$48$	0	0
$49$	−6.97284	−0.996120
$50$	0	0
$51$	1.56285	0.218842
$52$	0	0
$53$	1.87575	0.257654	0.128827	−	0.991667i	$-0.458879\pi$
$53$	1.87575	0.257654	0.128827	+	0.991667i	$0.458879\pi$
$54$	0	0
$55$	9.99106	1.34719
$56$	0	0
$57$	−11.7962	−1.56245
$58$	0	0
$59$	6.16349	0.802419	0.401209	−	0.915986i	$-0.368590\pi$
$59$	6.16349	0.802419	0.401209	+	0.915986i	$0.368590\pi$
$60$	0	0
$61$	−1.42040	−0.181864	−0.0909319	−	0.995857i	$-0.528985\pi$
$61$	−1.42040	−0.181864	−0.0909319	+	0.995857i	$0.528985\pi$
$62$	0	0
$63$	−0.785351	−0.0989449
$64$	0	0
$65$	7.16237	0.888382
$66$	0	0
$67$	7.56650	0.924395	0.462198	−	0.886777i	$-0.347061\pi$
$67$	7.56650	0.924395	0.462198	+	0.886777i	$0.347061\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.00185	−0.237575	−0.118788	−	0.992920i	$-0.537901\pi$
$71$	−2.00185	−0.237575	−0.118788	+	0.992920i	$0.537901\pi$
$72$	0	0
$73$	10.8343	1.26806	0.634030	−	0.773309i	$-0.281400\pi$
$73$	10.8343	1.26806	0.634030	+	0.773309i	$0.281400\pi$
$74$	0	0
$75$	−7.08125	−0.817673
$76$	0	0
$77$	1.05012	0.119672
$78$	0	0
$79$	8.71390	0.980390	0.490195	−	0.871613i	$-0.336926\pi$
$79$	8.71390	0.980390	0.490195	+	0.871613i	$0.336926\pi$
$80$	0	0
$81$	−0.588949	−0.0654388
$82$	0	0
$83$	0.811858	0.0891130	0.0445565	−	0.999007i	$-0.485813\pi$
$83$	0.811858	0.0891130	0.0445565	+	0.999007i	$0.485813\pi$
$84$	0	0
$85$	−0.879438	−0.0953885
$86$	0	0
$87$	−15.7025	−1.68349
$88$	0	0
$89$	6.48798	0.687724	0.343862	−	0.939020i	$-0.388265\pi$
$89$	6.48798	0.687724	0.343862	+	0.939020i	$0.388265\pi$
$90$	0	0
$91$	0.752805	0.0789155
$92$	0	0
$93$	24.9446	2.58663
$94$	0	0
$95$	6.63792	0.681036
$96$	0	0
$97$	−7.56444	−0.768053	−0.384026	−	0.923322i	$-0.625463\pi$
$97$	−7.56444	−0.768053	−0.384026	+	0.923322i	$0.625463\pi$
$98$	0	0
$99$	−30.3615	−3.05144
$100$	0	0
$101$	−3.01995	−0.300496	−0.150248	−	0.988648i	$-0.548007\pi$
$101$	−3.01995	−0.300496	−0.150248	+	0.988648i	$0.548007\pi$
$102$	0	0
$103$	−2.89437	−0.285191	−0.142596	−	0.989781i	$-0.545545\pi$
$103$	−2.89437	−0.285191	−0.142596	+	0.989781i	$0.545545\pi$
$104$	0	0
$105$	0.720156	0.0702800
$106$	0	0
$107$	−16.2291	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$107$	−16.2291	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$108$	0	0
$109$	−11.2351	−1.07613	−0.538065	−	0.842903i	$-0.680845\pi$
$109$	−11.2351	−1.07613	−0.538065	+	0.842903i	$0.680845\pi$
$110$	0	0
$111$	16.5375	1.56967
$112$	0	0
$113$	6.88981	0.648139	0.324070	−	0.946033i	$-0.394949\pi$
$113$	6.88981	0.648139	0.324070	+	0.946033i	$0.394949\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−21.7655	−2.01222
$118$	0	0
$119$	−0.0924339	−0.00847340
$120$	0	0
$121$	29.5973	2.69066
$122$	0	0
$123$	−9.80623	−0.884198
$124$	0	0
$125$	11.8250	1.05766
$126$	0	0
$127$	−6.15401	−0.546080	−0.273040	−	0.962003i	$-0.588029\pi$
$127$	−6.15401	−0.546080	−0.273040	+	0.962003i	$0.588029\pi$
$128$	0	0
$129$	−22.9460	−2.02029
$130$	0	0
$131$	−5.56623	−0.486324	−0.243162	−	0.969986i	$-0.578185\pi$
$131$	−5.56623	−0.486324	−0.243162	+	0.969986i	$0.578185\pi$
$132$	0	0
$133$	0.697683	0.0604968
$134$	0	0
$135$	−7.71282	−0.663814
$136$	0	0
$137$	3.23168	0.276101	0.138051	−	0.990425i	$-0.455916\pi$
$137$	3.23168	0.276101	0.138051	+	0.990425i	$0.455916\pi$
$138$	0	0
$139$	−4.67006	−0.396109	−0.198055	−	0.980191i	$-0.563462\pi$
$139$	−4.67006	−0.396109	−0.198055	+	0.980191i	$0.563462\pi$
$140$	0	0
$141$	−15.3937	−1.29638
$142$	0	0
$143$	29.1033	2.43374
$144$	0	0
$145$	8.83606	0.733795
$146$	0	0
$147$	−19.4305	−1.60260
$148$	0	0
$149$	−2.87677	−0.235674	−0.117837	−	0.993033i	$-0.537596\pi$
$149$	−2.87677	−0.235674	−0.117837	+	0.993033i	$0.537596\pi$
$150$	0	0
$151$	12.6374	1.02842	0.514210	−	0.857665i	$-0.328085\pi$
$151$	12.6374	1.02842	0.514210	+	0.857665i	$0.328085\pi$
$152$	0	0
$153$	2.67249	0.216058
$154$	0	0
$155$	−14.0367	−1.12745
$156$	0	0
$157$	15.9688	1.27445	0.637224	−	0.770678i	$-0.280082\pi$
$157$	15.9688	1.27445	0.637224	+	0.770678i	$0.280082\pi$
$158$	0	0
$159$	5.22695	0.414524
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.323837	0.0253649	0.0126824	−	0.999920i	$-0.495963\pi$
$163$	0.323837	0.0253649	0.0126824	+	0.999920i	$0.495963\pi$
$164$	0	0
$165$	27.8411	2.16742
$166$	0	0
$167$	0.826751	0.0639759	0.0319880	−	0.999488i	$-0.489816\pi$
$167$	0.826751	0.0639759	0.0319880	+	0.999488i	$0.489816\pi$
$168$	0	0
$169$	7.86350	0.604884
$170$	0	0
$171$	−20.1717	−1.54257
$172$	0	0
$173$	−18.4358	−1.40164	−0.700822	−	0.713336i	$-0.747183\pi$
$173$	−18.4358	−1.40164	−0.700822	+	0.713336i	$0.747183\pi$
$174$	0	0
$175$	0.418818	0.0316597
$176$	0	0
$177$	17.1752	1.29097
$178$	0	0
$179$	−16.4356	−1.22846	−0.614228	−	0.789129i	$-0.710532\pi$
$179$	−16.4356	−1.22846	−0.614228	+	0.789129i	$0.710532\pi$
$180$	0	0
$181$	5.86777	0.436148	0.218074	−	0.975932i	$-0.430023\pi$
$181$	5.86777	0.436148	0.218074	+	0.975932i	$0.430023\pi$
$182$	0	0
$183$	−3.95809	−0.292590
$184$	0	0
$185$	−9.30591	−0.684184
$186$	0	0
$187$	−3.57347	−0.261318
$188$	0	0
$189$	−0.810661	−0.0589669
$190$	0	0
$191$	−19.3993	−1.40369	−0.701844	−	0.712331i	$-0.747639\pi$
$191$	−19.3993	−1.40369	−0.701844	+	0.712331i	$0.747639\pi$
$192$	0	0
$193$	−11.1530	−0.802808	−0.401404	−	0.915901i	$-0.631478\pi$
$193$	−11.1530	−0.802808	−0.401404	+	0.915901i	$0.631478\pi$
$194$	0	0
$195$	19.9586	1.42927
$196$	0	0
$197$	14.1331	1.00694	0.503471	−	0.864012i	$-0.332056\pi$
$197$	14.1331	1.00694	0.503471	+	0.864012i	$0.332056\pi$
$198$	0	0
$199$	16.0986	1.14120	0.570599	−	0.821229i	$-0.306711\pi$
$199$	16.0986	1.14120	0.570599	+	0.821229i	$0.306711\pi$
$200$	0	0
$201$	21.0848	1.48721
$202$	0	0
$203$	0.928720	0.0651833
$204$	0	0
$205$	5.51812	0.385402
$206$	0	0
$207$	0	0
$208$	0	0
$209$	26.9723	1.86571
$210$	0	0
$211$	−7.65044	−0.526678	−0.263339	−	0.964703i	$-0.584824\pi$
$211$	−7.65044	−0.526678	−0.263339	+	0.964703i	$0.584824\pi$
$212$	0	0
$213$	−5.57834	−0.382221
$214$	0	0
$215$	12.9121	0.880598
$216$	0	0
$217$	−1.47534	−0.100152
$218$	0	0
$219$	30.1908	2.04011
$220$	0	0
$221$	−2.56174	−0.172322
$222$	0	0
$223$	−2.24296	−0.150200	−0.0750999	−	0.997176i	$-0.523928\pi$
$223$	−2.24296	−0.150200	−0.0750999	+	0.997176i	$0.523928\pi$
$224$	0	0
$225$	−12.1091	−0.807271
$226$	0	0
$227$	−18.2621	−1.21210	−0.606048	−	0.795428i	$-0.707246\pi$
$227$	−18.2621	−1.21210	−0.606048	+	0.795428i	$0.707246\pi$
$228$	0	0
$229$	17.3351	1.14554	0.572768	−	0.819718i	$-0.305870\pi$
$229$	17.3351	1.14554	0.572768	+	0.819718i	$0.305870\pi$
$230$	0	0
$231$	2.92625	0.192533
$232$	0	0
$233$	−15.4790	−1.01406	−0.507031	−	0.861928i	$-0.669257\pi$
$233$	−15.4790	−1.01406	−0.507031	+	0.861928i	$0.669257\pi$
$234$	0	0
$235$	8.66227	0.565064
$236$	0	0
$237$	24.2821	1.57729
$238$	0	0
$239$	20.9663	1.35620	0.678098	−	0.734971i	$-0.262804\pi$
$239$	20.9663	1.35620	0.678098	+	0.734971i	$0.262804\pi$
$240$	0	0
$241$	−14.9849	−0.965262	−0.482631	−	0.875824i	$-0.660319\pi$
$241$	−14.9849	−0.965262	−0.482631	+	0.875824i	$0.660319\pi$
$242$	0	0
$243$	−16.3973	−1.05188
$244$	0	0
$245$	10.9338	0.698537
$246$	0	0
$247$	19.3358	1.23031
$248$	0	0
$249$	2.26232	0.143369
$250$	0	0
$251$	−3.98093	−0.251274	−0.125637	−	0.992076i	$-0.540097\pi$
$251$	−3.98093	−0.251274	−0.125637	+	0.992076i	$0.540097\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.45064	−0.153465
$256$	0	0
$257$	−22.7245	−1.41751	−0.708757	−	0.705453i	$-0.750744\pi$
$257$	−22.7245	−1.41751	−0.708757	+	0.705453i	$0.750744\pi$
$258$	0	0
$259$	−0.978104	−0.0607764
$260$	0	0
$261$	−26.8516	−1.66207
$262$	0	0
$263$	18.0988	1.11602	0.558009	−	0.829835i	$-0.311565\pi$
$263$	18.0988	1.11602	0.558009	+	0.829835i	$0.311565\pi$
$264$	0	0
$265$	−2.94129	−0.180682
$266$	0	0
$267$	18.0794	1.10644
$268$	0	0
$269$	5.53448	0.337443	0.168721	−	0.985664i	$-0.446036\pi$
$269$	5.53448	0.337443	0.168721	+	0.985664i	$0.446036\pi$
$270$	0	0
$271$	25.2073	1.53124	0.765619	−	0.643295i	$-0.222433\pi$
$271$	25.2073	1.53124	0.765619	+	0.643295i	$0.222433\pi$
$272$	0	0
$273$	2.09777	0.126963
$274$	0	0
$275$	16.1914	0.976378
$276$	0	0
$277$	1.48028	0.0889416	0.0444708	−	0.999011i	$-0.485840\pi$
$277$	1.48028	0.0889416	0.0444708	+	0.999011i	$0.485840\pi$
$278$	0	0
$279$	42.6556	2.55372
$280$	0	0
$281$	−23.6217	−1.40915	−0.704577	−	0.709628i	$-0.748863\pi$
$281$	−23.6217	−1.40915	−0.704577	+	0.709628i	$0.748863\pi$
$282$	0	0
$283$	5.18400	0.308157	0.154078	−	0.988059i	$-0.450759\pi$
$283$	5.18400	0.308157	0.154078	+	0.988059i	$0.450759\pi$
$284$	0	0
$285$	18.4972	1.09568
$286$	0	0
$287$	0.579985	0.0342355
$288$	0	0
$289$	−16.6855	−0.981497
$290$	0	0
$291$	−21.0791	−1.23568
$292$	0	0
$293$	−28.7325	−1.67857	−0.839284	−	0.543693i	$-0.817026\pi$
$293$	−28.7325	−1.67857	−0.839284	+	0.543693i	$0.817026\pi$
$294$	0	0
$295$	−9.66474	−0.562703
$296$	0	0
$297$	−31.3400	−1.81853
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.35713	0.0782239
$302$	0	0
$303$	−8.41538	−0.483451
$304$	0	0
$305$	2.22728	0.127534
$306$	0	0
$307$	8.66052	0.494282	0.247141	−	0.968980i	$-0.420509\pi$
$307$	8.66052	0.494282	0.247141	+	0.968980i	$0.420509\pi$
$308$	0	0
$309$	−8.06545	−0.458828
$310$	0	0
$311$	4.55650	0.258375	0.129188	−	0.991620i	$-0.458763\pi$
$311$	4.55650	0.258375	0.129188	+	0.991620i	$0.458763\pi$
$312$	0	0
$313$	15.8380	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$313$	15.8380	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$314$	0	0
$315$	1.23148	0.0693859
$316$	0	0
$317$	1.00497	0.0564449	0.0282224	−	0.999602i	$-0.491015\pi$
$317$	1.00497	0.0564449	0.0282224	+	0.999602i	$0.491015\pi$
$318$	0	0
$319$	35.9041	2.01024
$320$	0	0
$321$	−45.2241	−2.52416
$322$	0	0
$323$	−2.37417	−0.132102
$324$	0	0
$325$	11.6073	0.643855
$326$	0	0
$327$	−31.3078	−1.73132
$328$	0	0
$329$	0.910454	0.0501949
$330$	0	0
$331$	−17.4211	−0.957549	−0.478775	−	0.877938i	$-0.658919\pi$
$331$	−17.4211	−0.957549	−0.478775	+	0.877938i	$0.658919\pi$
$332$	0	0
$333$	28.2794	1.54970
$334$	0	0
$335$	−11.8647	−0.648240
$336$	0	0
$337$	−15.7768	−0.859416	−0.429708	−	0.902968i	$-0.641383\pi$
$337$	−15.7768	−0.859416	−0.429708	+	0.902968i	$0.641383\pi$
$338$	0	0
$339$	19.1991	1.04275
$340$	0	0
$341$	−57.0362	−3.08868
$342$	0	0
$343$	2.30289	0.124345
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.600463	−0.0322346	−0.0161173	−	0.999870i	$-0.505131\pi$
$347$	−0.600463	−0.0322346	−0.0161173	+	0.999870i	$0.505131\pi$
$348$	0	0
$349$	−10.0570	−0.538338	−0.269169	−	0.963093i	$-0.586749\pi$
$349$	−10.0570	−0.538338	−0.269169	+	0.963093i	$0.586749\pi$
$350$	0	0
$351$	−22.4669	−1.19920
$352$	0	0
$353$	−1.02757	−0.0546922	−0.0273461	−	0.999626i	$-0.508706\pi$
$353$	−1.02757	−0.0546922	−0.0273461	+	0.999626i	$0.508706\pi$
$354$	0	0
$355$	3.13902	0.166602
$356$	0	0
$357$	−0.257576	−0.0136324
$358$	0	0
$359$	−10.4820	−0.553217	−0.276608	−	0.960983i	$-0.589210\pi$
$359$	−10.4820	−0.553217	−0.276608	+	0.960983i	$0.589210\pi$
$360$	0	0
$361$	−1.07999	−0.0568418
$362$	0	0
$363$	82.4758	4.32886
$364$	0	0
$365$	−16.9889	−0.889237
$366$	0	0
$367$	3.84260	0.200582	0.100291	−	0.994958i	$-0.468023\pi$
$367$	3.84260	0.200582	0.100291	+	0.994958i	$0.468023\pi$
$368$	0	0
$369$	−16.7688	−0.872949
$370$	0	0
$371$	−0.309146	−0.0160501
$372$	0	0
$373$	9.46679	0.490172	0.245086	−	0.969501i	$-0.421184\pi$
$373$	9.46679	0.490172	0.245086	+	0.969501i	$0.421184\pi$
$374$	0	0
$375$	32.9516	1.70161
$376$	0	0
$377$	25.7388	1.32562
$378$	0	0
$379$	−25.2570	−1.29736	−0.648681	−	0.761060i	$-0.724679\pi$
$379$	−25.2570	−1.29736	−0.648681	+	0.761060i	$0.724679\pi$
$380$	0	0
$381$	−17.1488	−0.878557
$382$	0	0
$383$	−0.0846897	−0.00432744	−0.00216372	−	0.999998i	$-0.500689\pi$
$383$	−0.0846897	−0.00432744	−0.00216372	+	0.999998i	$0.500689\pi$
$384$	0	0
$385$	−1.64665	−0.0839210
$386$	0	0
$387$	−39.2381	−1.99458
$388$	0	0
$389$	−18.9498	−0.960792	−0.480396	−	0.877052i	$-0.659507\pi$
$389$	−18.9498	−0.960792	−0.480396	+	0.877052i	$0.659507\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.5108	−0.782419
$394$	0	0
$395$	−13.6639	−0.687507
$396$	0	0
$397$	−30.8884	−1.55024	−0.775122	−	0.631812i	$-0.782311\pi$
$397$	−30.8884	−1.55024	−0.775122	+	0.631812i	$0.782311\pi$
$398$	0	0
$399$	1.94416	0.0973299
$400$	0	0
$401$	11.7151	0.585023	0.292511	−	0.956262i	$-0.405509\pi$
$401$	11.7151	0.585023	0.292511	+	0.956262i	$0.405509\pi$
$402$	0	0
$403$	−40.8879	−2.03677
$404$	0	0
$405$	0.923509	0.0458895
$406$	0	0
$407$	−37.8133	−1.87434
$408$	0	0
$409$	−9.78683	−0.483928	−0.241964	−	0.970285i	$-0.577792\pi$
$409$	−9.78683	−0.483928	−0.241964	+	0.970285i	$0.577792\pi$
$410$	0	0
$411$	9.00540	0.444204
$412$	0	0
$413$	−1.01582	−0.0499852
$414$	0	0
$415$	−1.27304	−0.0624912
$416$	0	0
$417$	−13.0136	−0.637277
$418$	0	0
$419$	1.80966	0.0884075	0.0442038	−	0.999023i	$-0.485925\pi$
$419$	1.80966	0.0884075	0.0442038	+	0.999023i	$0.485925\pi$
$420$	0	0
$421$	20.9157	1.01937	0.509684	−	0.860362i	$-0.329762\pi$
$421$	20.9157	1.01937	0.509684	+	0.860362i	$0.329762\pi$
$422$	0	0
$423$	−26.3235	−1.27989
$424$	0	0
$425$	−1.42521	−0.0691327
$426$	0	0
$427$	0.234100	0.0113289
$428$	0	0
$429$	81.0992	3.91550
$430$	0	0
$431$	−3.48982	−0.168099	−0.0840493	−	0.996462i	$-0.526785\pi$
$431$	−3.48982	−0.168099	−0.0840493	+	0.996462i	$0.526785\pi$
$432$	0	0
$433$	−24.1407	−1.16013	−0.580063	−	0.814571i	$-0.696972\pi$
$433$	−24.1407	−1.16013	−0.580063	+	0.814571i	$0.696972\pi$
$434$	0	0
$435$	24.6225	1.18056
$436$	0	0
$437$	0	0
$438$	0	0
$439$	29.4486	1.40550	0.702751	−	0.711436i	$-0.251955\pi$
$439$	29.4486	1.40550	0.702751	+	0.711436i	$0.251955\pi$
$440$	0	0
$441$	−33.2264	−1.58221
$442$	0	0
$443$	−10.8497	−0.515483	−0.257741	−	0.966214i	$-0.582978\pi$
$443$	−10.8497	−0.515483	−0.257741	+	0.966214i	$0.582978\pi$
$444$	0	0
$445$	−10.1736	−0.482273
$446$	0	0
$447$	−8.01640	−0.379162
$448$	0	0
$449$	39.2950	1.85445	0.927224	−	0.374508i	$-0.122188\pi$
$449$	39.2950	1.85445	0.927224	+	0.374508i	$0.122188\pi$
$450$	0	0
$451$	22.4221	1.05582
$452$	0	0
$453$	35.2154	1.65456
$454$	0	0
$455$	−1.18045	−0.0553401
$456$	0	0
$457$	27.2140	1.27302	0.636509	−	0.771269i	$-0.280378\pi$
$457$	27.2140	1.27302	0.636509	+	0.771269i	$0.280378\pi$
$458$	0	0
$459$	2.75862	0.128761
$460$	0	0
$461$	−42.2854	−1.96943	−0.984713	−	0.174187i	$-0.944270\pi$
$461$	−42.2854	−1.96943	−0.984713	+	0.174187i	$0.944270\pi$
$462$	0	0
$463$	−9.59753	−0.446035	−0.223018	−	0.974814i	$-0.571591\pi$
$463$	−9.59753	−0.446035	−0.223018	+	0.974814i	$0.571591\pi$
$464$	0	0
$465$	−39.1146	−1.81390
$466$	0	0
$467$	18.4041	0.851641	0.425821	−	0.904808i	$-0.359985\pi$
$467$	18.4041	0.851641	0.425821	+	0.904808i	$0.359985\pi$
$468$	0	0
$469$	−1.24705	−0.0575835
$470$	0	0
$471$	44.4986	2.05039
$472$	0	0
$473$	52.4665	2.41241
$474$	0	0
$475$	10.7573	0.493581
$476$	0	0
$477$	8.93817	0.409251
$478$	0	0
$479$	31.9699	1.46074	0.730371	−	0.683051i	$-0.239347\pi$
$479$	31.9699	1.46074	0.730371	+	0.683051i	$0.239347\pi$
$480$	0	0
$481$	−27.1075	−1.23600
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.8615	0.538603
$486$	0	0
$487$	−12.1448	−0.550336	−0.275168	−	0.961396i	$-0.588733\pi$
$487$	−12.1448	−0.550336	−0.275168	+	0.961396i	$0.588733\pi$
$488$	0	0
$489$	0.902404	0.0408081
$490$	0	0
$491$	20.0289	0.903891	0.451946	−	0.892045i	$-0.350730\pi$
$491$	20.0289	0.903891	0.451946	+	0.892045i	$0.350730\pi$
$492$	0	0
$493$	−3.16037	−0.142336
$494$	0	0
$495$	47.6087	2.13985
$496$	0	0
$497$	0.329929	0.0147993
$498$	0	0
$499$	2.65530	0.118868	0.0594338	−	0.998232i	$-0.481070\pi$
$499$	2.65530	0.118868	0.0594338	+	0.998232i	$0.481070\pi$
$500$	0	0
$501$	2.30382	0.102927
$502$	0	0
$503$	−12.5224	−0.558344	−0.279172	−	0.960241i	$-0.590060\pi$
$503$	−12.5224	−0.558344	−0.279172	+	0.960241i	$0.590060\pi$
$504$	0	0
$505$	4.73546	0.210725
$506$	0	0
$507$	21.9124	0.973164
$508$	0	0
$509$	33.3074	1.47632	0.738162	−	0.674624i	$-0.235694\pi$
$509$	33.3074	1.47632	0.738162	+	0.674624i	$0.235694\pi$
$510$	0	0
$511$	−1.78563	−0.0789914
$512$	0	0
$513$	−20.8218	−0.919307
$514$	0	0
$515$	4.53856	0.199993
$516$	0	0
$517$	35.1979	1.54800
$518$	0	0
$519$	−51.3730	−2.25503
$520$	0	0
$521$	−31.4292	−1.37694	−0.688468	−	0.725266i	$-0.741717\pi$
$521$	−31.4292	−1.37694	−0.688468	+	0.725266i	$0.741717\pi$
$522$	0	0
$523$	31.8029	1.39065	0.695323	−	0.718698i	$-0.255262\pi$
$523$	31.8029	1.39065	0.695323	+	0.718698i	$0.255262\pi$
$524$	0	0
$525$	1.16708	0.0509354
$526$	0	0
$527$	5.02046	0.218695
$528$	0	0
$529$	0	0
$530$	0	0
$531$	29.3698	1.27454
$532$	0	0
$533$	16.0739	0.696238
$534$	0	0
$535$	25.4483	1.10022
$536$	0	0
$537$	−45.7994	−1.97639
$538$	0	0
$539$	44.4281	1.91366
$540$	0	0
$541$	−8.39091	−0.360753	−0.180377	−	0.983598i	$-0.557732\pi$
$541$	−8.39091	−0.360753	−0.180377	+	0.983598i	$0.557732\pi$
$542$	0	0
$543$	16.3511	0.701694
$544$	0	0
$545$	17.6174	0.754645
$546$	0	0
$547$	4.80684	0.205526	0.102763	−	0.994706i	$-0.467232\pi$
$547$	4.80684	0.205526	0.102763	+	0.994706i	$0.467232\pi$
$548$	0	0
$549$	−6.76840	−0.288868
$550$	0	0
$551$	23.8542	1.01622
$552$	0	0
$553$	−1.43616	−0.0610716
$554$	0	0
$555$	−25.9318	−1.10075
$556$	0	0
$557$	46.1194	1.95414	0.977071	−	0.212913i	$-0.0682952\pi$
$557$	46.1194	1.95414	0.977071	+	0.212913i	$0.0682952\pi$
$558$	0	0
$559$	37.6121	1.59082
$560$	0	0
$561$	−9.95784	−0.420420
$562$	0	0
$563$	26.2009	1.10423	0.552117	−	0.833766i	$-0.313820\pi$
$563$	26.2009	1.10423	0.552117	+	0.833766i	$0.313820\pi$
$564$	0	0
$565$	−10.8037	−0.454513
$566$	0	0
$567$	0.0970660	0.00407639
$568$	0	0
$569$	−15.9451	−0.668455	−0.334228	−	0.942492i	$-0.608475\pi$
$569$	−15.9451	−0.668455	−0.334228	+	0.942492i	$0.608475\pi$
$570$	0	0
$571$	46.2746	1.93653	0.968266	−	0.249923i	$-0.0804053\pi$
$571$	46.2746	1.93653	0.968266	+	0.249923i	$0.0804053\pi$
$572$	0	0
$573$	−54.0582	−2.25831
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.69004	0.236880	0.118440	−	0.992961i	$-0.462211\pi$
$577$	5.69004	0.236880	0.118440	+	0.992961i	$0.462211\pi$
$578$	0	0
$579$	−31.0788	−1.29159
$580$	0	0
$581$	−0.133804	−0.00555113
$582$	0	0
$583$	−11.9515	−0.494981
$584$	0	0
$585$	34.1296	1.41108
$586$	0	0
$587$	−28.5385	−1.17791	−0.588956	−	0.808165i	$-0.700461\pi$
$587$	−28.5385	−1.17791	−0.588956	+	0.808165i	$0.700461\pi$
$588$	0	0
$589$	−37.8940	−1.56140
$590$	0	0
$591$	39.3833	1.62001
$592$	0	0
$593$	−1.23824	−0.0508485	−0.0254243	−	0.999677i	$-0.508094\pi$
$593$	−1.23824	−0.0508485	−0.0254243	+	0.999677i	$0.508094\pi$
$594$	0	0
$595$	0.144942	0.00594205
$596$	0	0
$597$	44.8603	1.83601
$598$	0	0
$599$	−9.25426	−0.378119	−0.189059	−	0.981966i	$-0.560544\pi$
$599$	−9.25426	−0.378119	−0.189059	+	0.981966i	$0.560544\pi$
$600$	0	0
$601$	−11.4591	−0.467426	−0.233713	−	0.972306i	$-0.575088\pi$
$601$	−11.4591	−0.467426	−0.233713	+	0.972306i	$0.575088\pi$
$602$	0	0
$603$	36.0553	1.46829
$604$	0	0
$605$	−46.4104	−1.88685
$606$	0	0
$607$	−11.9390	−0.484589	−0.242295	−	0.970203i	$-0.577900\pi$
$607$	−11.9390	−0.484589	−0.242295	+	0.970203i	$0.577900\pi$
$608$	0	0
$609$	2.58797	0.104870
$610$	0	0
$611$	25.2326	1.02080
$612$	0	0
$613$	−23.6082	−0.953525	−0.476763	−	0.879032i	$-0.658190\pi$
$613$	−23.6082	−0.953525	−0.476763	+	0.879032i	$0.658190\pi$
$614$	0	0
$615$	15.3768	0.620051
$616$	0	0
$617$	1.62245	0.0653174	0.0326587	−	0.999467i	$-0.489603\pi$
$617$	1.62245	0.0653174	0.0326587	+	0.999467i	$0.489603\pi$
$618$	0	0
$619$	34.7338	1.39607	0.698034	−	0.716065i	$-0.254059\pi$
$619$	34.7338	1.39607	0.698034	+	0.716065i	$0.254059\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.06930	−0.0428405
$624$	0	0
$625$	−5.83648	−0.233459
$626$	0	0
$627$	75.1609	3.00164
$628$	0	0
$629$	3.32842	0.132713
$630$	0	0
$631$	−16.1444	−0.642698	−0.321349	−	0.946961i	$-0.604136\pi$
$631$	−16.1444	−0.642698	−0.321349	+	0.946961i	$0.604136\pi$
$632$	0	0
$633$	−21.3187	−0.847342
$634$	0	0
$635$	9.64987	0.382943
$636$	0	0
$637$	31.8495	1.26192
$638$	0	0
$639$	−9.53905	−0.377359
$640$	0	0
$641$	−15.3027	−0.604422	−0.302211	−	0.953241i	$-0.597725\pi$
$641$	−15.3027	−0.604422	−0.302211	+	0.953241i	$0.597725\pi$
$642$	0	0
$643$	−34.0990	−1.34473	−0.672367	−	0.740218i	$-0.734722\pi$
$643$	−34.0990	−1.34473	−0.672367	+	0.740218i	$0.734722\pi$
$644$	0	0
$645$	35.9808	1.41674
$646$	0	0
$647$	25.9146	1.01881	0.509405	−	0.860527i	$-0.329866\pi$
$647$	25.9146	1.01881	0.509405	+	0.860527i	$0.329866\pi$
$648$	0	0
$649$	−39.2713	−1.54153
$650$	0	0
$651$	−4.11117	−0.161129
$652$	0	0
$653$	−42.0346	−1.64494	−0.822470	−	0.568808i	$-0.807405\pi$
$653$	−42.0346	−1.64494	−0.822470	+	0.568808i	$0.807405\pi$
$654$	0	0
$655$	8.72820	0.341039
$656$	0	0
$657$	51.6268	2.01415
$658$	0	0
$659$	24.2911	0.946246	0.473123	−	0.880997i	$-0.343127\pi$
$659$	24.2911	0.946246	0.473123	+	0.880997i	$0.343127\pi$
$660$	0	0
$661$	−27.4235	−1.06665	−0.533325	−	0.845910i	$-0.679058\pi$
$661$	−27.4235	−1.06665	−0.533325	+	0.845910i	$0.679058\pi$
$662$	0	0
$663$	−7.13855	−0.277238
$664$	0	0
$665$	−1.09401	−0.0424239
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−6.25023	−0.241648
$670$	0	0
$671$	9.05024	0.349380
$672$	0	0
$673$	32.5734	1.25561	0.627805	−	0.778370i	$-0.283953\pi$
$673$	32.5734	1.25561	0.627805	+	0.778370i	$0.283953\pi$
$674$	0	0
$675$	−12.4993	−0.481099
$676$	0	0
$677$	22.6803	0.871674	0.435837	−	0.900026i	$-0.356452\pi$
$677$	22.6803	0.871674	0.435837	+	0.900026i	$0.356452\pi$
$678$	0	0
$679$	1.24671	0.0478444
$680$	0	0
$681$	−50.8890	−1.95007
$682$	0	0
$683$	34.8071	1.33186	0.665928	−	0.746016i	$-0.268036\pi$
$683$	34.8071	1.33186	0.665928	+	0.746016i	$0.268036\pi$
$684$	0	0
$685$	−5.06748	−0.193618
$686$	0	0
$687$	48.3059	1.84299
$688$	0	0
$689$	−8.56777	−0.326406
$690$	0	0
$691$	30.0080	1.14156	0.570778	−	0.821104i	$-0.306642\pi$
$691$	30.0080	1.14156	0.570778	+	0.821104i	$0.306642\pi$
$692$	0	0
$693$	5.00394	0.190084
$694$	0	0
$695$	7.32294	0.277775
$696$	0	0
$697$	−1.97365	−0.0747573
$698$	0	0
$699$	−43.1337	−1.63146
$700$	0	0
$701$	−38.0457	−1.43697	−0.718484	−	0.695544i	$-0.755163\pi$
$701$	−38.0457	−1.43697	−0.718484	+	0.695544i	$0.755163\pi$
$702$	0	0
$703$	−25.1226	−0.947518
$704$	0	0
$705$	24.1383	0.909100
$706$	0	0
$707$	0.497724	0.0187188
$708$	0	0
$709$	17.9112	0.672668	0.336334	−	0.941743i	$-0.390813\pi$
$709$	17.9112	0.672668	0.336334	+	0.941743i	$0.390813\pi$
$710$	0	0
$711$	41.5228	1.55723
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−45.6358	−1.70668
$716$	0	0
$717$	58.4246	2.18191
$718$	0	0
$719$	−36.6302	−1.36608	−0.683038	−	0.730383i	$-0.739341\pi$
$719$	−36.6302	−1.36608	−0.683038	+	0.730383i	$0.739341\pi$
$720$	0	0
$721$	0.477028	0.0177654
$722$	0	0
$723$	−41.7569	−1.55296
$724$	0	0
$725$	14.3196	0.531817
$726$	0	0
$727$	−26.1006	−0.968019	−0.484009	−	0.875063i	$-0.660820\pi$
$727$	−26.1006	−0.968019	−0.484009	+	0.875063i	$0.660820\pi$
$728$	0	0
$729$	−43.9257	−1.62688
$730$	0	0
$731$	−4.61823	−0.170811
$732$	0	0
$733$	−44.0256	−1.62612	−0.813061	−	0.582178i	$-0.802201\pi$
$733$	−44.0256	−1.62612	−0.813061	+	0.582178i	$0.802201\pi$
$734$	0	0
$735$	30.4682	1.12384
$736$	0	0
$737$	−48.2107	−1.77586
$738$	0	0
$739$	−12.9786	−0.477425	−0.238713	−	0.971090i	$-0.576725\pi$
$739$	−12.9786	−0.477425	−0.238713	+	0.971090i	$0.576725\pi$
$740$	0	0
$741$	53.8812	1.97937
$742$	0	0
$743$	−47.7186	−1.75062	−0.875312	−	0.483558i	$-0.839344\pi$
$743$	−47.7186	−1.75062	−0.875312	+	0.483558i	$0.839344\pi$
$744$	0	0
$745$	4.51095	0.165268
$746$	0	0
$747$	3.86860	0.141545
$748$	0	0
$749$	2.67476	0.0977335
$750$	0	0
$751$	39.2711	1.43302	0.716512	−	0.697575i	$-0.245738\pi$
$751$	39.2711	1.43302	0.716512	+	0.697575i	$0.245738\pi$
$752$	0	0
$753$	−11.0933	−0.404261
$754$	0	0
$755$	−19.8163	−0.721188
$756$	0	0
$757$	48.9932	1.78069	0.890344	−	0.455289i	$-0.150464\pi$
$757$	48.9932	1.78069	0.890344	+	0.455289i	$0.150464\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.10393	0.0762673	0.0381336	−	0.999273i	$-0.487859\pi$
$761$	2.10393	0.0762673	0.0381336	+	0.999273i	$0.487859\pi$
$762$	0	0
$763$	1.85169	0.0670355
$764$	0	0
$765$	−4.19063	−0.151513
$766$	0	0
$767$	−28.1527	−1.01654
$768$	0	0
$769$	32.7844	1.18224	0.591119	−	0.806585i	$-0.298687\pi$
$769$	32.7844	1.18224	0.591119	+	0.806585i	$0.298687\pi$
$770$	0	0
$771$	−63.3239	−2.28056
$772$	0	0
$773$	−20.8052	−0.748310	−0.374155	−	0.927366i	$-0.622067\pi$
$773$	−20.8052	−0.748310	−0.374155	+	0.927366i	$0.622067\pi$
$774$	0	0
$775$	−22.7477	−0.817122
$776$	0	0
$777$	−2.72558	−0.0977798
$778$	0	0
$779$	14.8969	0.533738
$780$	0	0
$781$	12.7550	0.456409
$782$	0	0
$783$	−27.7170	−0.990523
$784$	0	0
$785$	−25.0401	−0.893718
$786$	0	0
$787$	9.96759	0.355306	0.177653	−	0.984093i	$-0.443150\pi$
$787$	9.96759	0.355306	0.177653	+	0.984093i	$0.443150\pi$
$788$	0	0
$789$	50.4340	1.79550
$790$	0	0
$791$	−1.13553	−0.0403746
$792$	0	0
$793$	6.48791	0.230392
$794$	0	0
$795$	−8.19618	−0.290689
$796$	0	0
$797$	−13.6089	−0.482052	−0.241026	−	0.970519i	$-0.577484\pi$
$797$	−13.6089	−0.482052	−0.241026	+	0.970519i	$0.577484\pi$
$798$	0	0
$799$	−3.09821	−0.109607
$800$	0	0
$801$	30.9160	1.09236
$802$	0	0
$803$	−69.0319	−2.43608
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.4224	0.542893
$808$	0	0
$809$	−21.6688	−0.761833	−0.380917	−	0.924609i	$-0.624391\pi$
$809$	−21.6688	−0.761833	−0.380917	+	0.924609i	$0.624391\pi$
$810$	0	0
$811$	10.4642	0.367446	0.183723	−	0.982978i	$-0.441185\pi$
$811$	10.4642	0.367446	0.183723	+	0.982978i	$0.441185\pi$
$812$	0	0
$813$	70.2427	2.46352
$814$	0	0
$815$	−0.507797	−0.0177873
$816$	0	0
$817$	34.8580	1.21953
$818$	0	0
$819$	3.58721	0.125347
$820$	0	0
$821$	−8.75392	−0.305514	−0.152757	−	0.988264i	$-0.548815\pi$
$821$	−8.75392	−0.305514	−0.152757	+	0.988264i	$0.548815\pi$
$822$	0	0
$823$	−21.8371	−0.761195	−0.380597	−	0.924741i	$-0.624282\pi$
$823$	−21.8371	−0.761195	−0.380597	+	0.924741i	$0.624282\pi$
$824$	0	0
$825$	45.1189	1.57084
$826$	0	0
$827$	−15.2620	−0.530713	−0.265357	−	0.964150i	$-0.585490\pi$
$827$	−15.2620	−0.530713	−0.265357	+	0.964150i	$0.585490\pi$
$828$	0	0
$829$	26.8540	0.932679	0.466340	−	0.884606i	$-0.345572\pi$
$829$	26.8540	0.932679	0.466340	+	0.884606i	$0.345572\pi$
$830$	0	0
$831$	4.12495	0.143093
$832$	0	0
$833$	−3.91067	−0.135497
$834$	0	0
$835$	−1.29640	−0.0448637
$836$	0	0
$837$	44.0303	1.52191
$838$	0	0
$839$	−1.85000	−0.0638692	−0.0319346	−	0.999490i	$-0.510167\pi$
$839$	−1.85000	−0.0638692	−0.0319346	+	0.999490i	$0.510167\pi$
$840$	0	0
$841$	2.75344	0.0949462
$842$	0	0
$843$	−65.8242	−2.26711
$844$	0	0
$845$	−12.3304	−0.424180
$846$	0	0
$847$	−4.87800	−0.167610
$848$	0	0
$849$	14.4457	0.495776
$850$	0	0
$851$	0	0
$852$	0	0
$853$	20.6173	0.705921	0.352961	−	0.935638i	$-0.385175\pi$
$853$	20.6173	0.705921	0.352961	+	0.935638i	$0.385175\pi$
$854$	0	0
$855$	31.6305	1.08174
$856$	0	0
$857$	−27.2657	−0.931377	−0.465689	−	0.884949i	$-0.654193\pi$
$857$	−27.2657	−0.931377	−0.465689	+	0.884949i	$0.654193\pi$
$858$	0	0
$859$	20.9009	0.713128	0.356564	−	0.934271i	$-0.383948\pi$
$859$	20.9009	0.713128	0.356564	+	0.934271i	$0.383948\pi$
$860$	0	0
$861$	1.61619	0.0550795
$862$	0	0
$863$	41.4092	1.40959	0.704793	−	0.709413i	$-0.251040\pi$
$863$	41.4092	1.40959	0.704793	+	0.709413i	$0.251040\pi$
$864$	0	0
$865$	28.9084	0.982915
$866$	0	0
$867$	−46.4956	−1.57907
$868$	0	0
$869$	−55.5215	−1.88344
$870$	0	0
$871$	−34.5612	−1.17106
$872$	0	0
$873$	−36.0455	−1.21996
$874$	0	0
$875$	−1.94891	−0.0658852
$876$	0	0
$877$	−20.9816	−0.708500	−0.354250	−	0.935151i	$-0.615264\pi$
$877$	−20.9816	−0.708500	−0.354250	+	0.935151i	$0.615264\pi$
$878$	0	0
$879$	−80.0658	−2.70055
$880$	0	0
$881$	−26.4852	−0.892309	−0.446154	−	0.894956i	$-0.647207\pi$
$881$	−26.4852	−0.892309	−0.446154	+	0.894956i	$0.647207\pi$
$882$	0	0
$883$	25.5041	0.858283	0.429141	−	0.903237i	$-0.358816\pi$
$883$	25.5041	0.858283	0.429141	+	0.903237i	$0.358816\pi$
$884$	0	0
$885$	−26.9317	−0.905300
$886$	0	0
$887$	23.3967	0.785585	0.392792	−	0.919627i	$-0.371509\pi$
$887$	23.3967	0.785585	0.392792	+	0.919627i	$0.371509\pi$
$888$	0	0
$889$	1.01426	0.0340171
$890$	0	0
$891$	3.75255	0.125715
$892$	0	0
$893$	23.3850	0.782550
$894$	0	0
$895$	25.7720	0.861465
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−50.4426	−1.68235
$900$	0	0
$901$	1.05200	0.0350473
$902$	0	0
$903$	3.78179	0.125850
$904$	0	0
$905$	−9.20103	−0.305852
$906$	0	0
$907$	−47.8303	−1.58818	−0.794089	−	0.607801i	$-0.792052\pi$
$907$	−47.8303	−1.58818	−0.794089	+	0.607801i	$0.792052\pi$
$908$	0	0
$909$	−14.3904	−0.477300
$910$	0	0
$911$	39.3713	1.30443	0.652214	−	0.758035i	$-0.273840\pi$
$911$	39.3713	1.30443	0.652214	+	0.758035i	$0.273840\pi$
$912$	0	0
$913$	−5.17283	−0.171196
$914$	0	0
$915$	6.20653	0.205181
$916$	0	0
$917$	0.917383	0.0302946
$918$	0	0
$919$	−24.8047	−0.818232	−0.409116	−	0.912482i	$-0.634163\pi$
$919$	−24.8047	−0.818232	−0.409116	+	0.912482i	$0.634163\pi$
$920$	0	0
$921$	24.1334	0.795222
$922$	0	0
$923$	9.14375	0.300970
$924$	0	0
$925$	−15.0811	−0.495862
$926$	0	0
$927$	−13.7921	−0.452990
$928$	0	0
$929$	17.8971	0.587185	0.293593	−	0.955931i	$-0.405149\pi$
$929$	17.8971	0.587185	0.293593	+	0.955931i	$0.405149\pi$
$930$	0	0
$931$	29.5174	0.967395
$932$	0	0
$933$	12.6971	0.415685
$934$	0	0
$935$	5.60343	0.183252
$936$	0	0
$937$	8.25458	0.269665	0.134833	−	0.990868i	$-0.456950\pi$
$937$	8.25458	0.269665	0.134833	+	0.990868i	$0.456950\pi$
$938$	0	0
$939$	44.1341	1.44026
$940$	0	0
$941$	−36.2508	−1.18174	−0.590870	−	0.806767i	$-0.701216\pi$
$941$	−36.2508	−1.18174	−0.590870	+	0.806767i	$0.701216\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.27117	0.0413510
$946$	0	0
$947$	18.6917	0.607400	0.303700	−	0.952768i	$-0.401778\pi$
$947$	18.6917	0.607400	0.303700	+	0.952768i	$0.401778\pi$
$948$	0	0
$949$	−49.4874	−1.60643
$950$	0	0
$951$	2.80045	0.0908110
$952$	0	0
$953$	−51.9908	−1.68415	−0.842074	−	0.539362i	$-0.818666\pi$
$953$	−51.9908	−1.68415	−0.842074	+	0.539362i	$0.818666\pi$
$954$	0	0
$955$	30.4194	0.984347
$956$	0	0
$957$	100.050	3.23417
$958$	0	0
$959$	−0.532621	−0.0171992
$960$	0	0
$961$	49.1316	1.58489
$962$	0	0
$963$	−77.3339	−2.49205
$964$	0	0
$965$	17.4885	0.562976
$966$	0	0
$967$	51.7211	1.66324	0.831618	−	0.555347i	$-0.187415\pi$
$967$	51.7211	1.66324	0.831618	+	0.555347i	$0.187415\pi$
$968$	0	0
$969$	−6.61585	−0.212532
$970$	0	0
$971$	35.9717	1.15439	0.577193	−	0.816608i	$-0.304148\pi$
$971$	35.9717	1.15439	0.577193	+	0.816608i	$0.304148\pi$
$972$	0	0
$973$	0.769682	0.0246749
$974$	0	0
$975$	32.3447	1.03586
$976$	0	0
$977$	−29.6376	−0.948191	−0.474095	−	0.880473i	$-0.657225\pi$
$977$	−29.6376	−0.948191	−0.474095	+	0.880473i	$0.657225\pi$
$978$	0	0
$979$	−41.3388	−1.32119
$980$	0	0
$981$	−53.5368	−1.70930
$982$	0	0
$983$	−3.85817	−0.123056	−0.0615282	−	0.998105i	$-0.519597\pi$
$983$	−3.85817	−0.123056	−0.0615282	+	0.998105i	$0.519597\pi$
$984$	0	0
$985$	−22.1616	−0.706127
$986$	0	0
$987$	2.53707	0.0807558
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−18.0209	−0.572452	−0.286226	−	0.958162i	$-0.592401\pi$
$991$	−18.0209	−0.572452	−0.286226	+	0.958162i	$0.592401\pi$
$992$	0	0
$993$	−48.5455	−1.54055
$994$	0	0
$995$	−25.2436	−0.800275
$996$	0	0
$997$	−50.0807	−1.58607	−0.793035	−	0.609176i	$-0.791500\pi$
$997$	−50.0807	−1.58607	−0.793035	+	0.609176i	$0.791500\pi$
$998$	0	0
$999$	29.1908	0.923556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2116.2.a.i.1.10		10
4.3	odd	2		8464.2.a.cd.1.1		10
23.5	odd	22		92.2.e.a.25.1	✓	20
23.14	odd	22		92.2.e.a.81.1	yes	20
23.22	odd	2		2116.2.a.j.1.10		10
69.5	even	22		828.2.q.a.577.2		20
69.14	even	22		828.2.q.a.541.2		20
92.51	even	22		368.2.m.d.209.2		20
92.83	even	22		368.2.m.d.81.2		20
92.91	even	2		8464.2.a.ce.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.2.e.a.25.1	✓	20	23.5	odd	22
92.2.e.a.81.1	yes	20	23.14	odd	22
368.2.m.d.81.2		20	92.83	even	22
368.2.m.d.209.2		20	92.51	even	22
828.2.q.a.541.2		20	69.14	even	22
828.2.q.a.577.2		20	69.5	even	22
2116.2.a.i.1.10		10	1.1	even	1	trivial
2116.2.a.j.1.10		10	23.22	odd	2
8464.2.a.cd.1.1		10	4.3	odd	2
8464.2.a.ce.1.1		10	92.91	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2116.a (trivial)

\(f(q)\)	\(=\)	\(q+2.78660 q^{3} -1.56806 q^{5} -0.164812 q^{7} +4.76513 q^{9} +O(q^{10})\)	\(q+2.78660 q^{3} -1.56806 q^{5} -0.164812 q^{7} +4.76513 q^{9} -6.37160 q^{11} -4.56766 q^{13} -4.36956 q^{15} +0.560844 q^{17} -4.23320 q^{19} -0.459265 q^{21} -2.54118 q^{25} +4.91870 q^{27} -5.63502 q^{29} +8.95162 q^{31} -17.7551 q^{33} +0.258436 q^{35} +5.93466 q^{37} -12.7282 q^{39} -3.51907 q^{41} -8.23443 q^{43} -7.47201 q^{45} -5.52419 q^{47} -6.97284 q^{49} +1.56285 q^{51} +1.87575 q^{53} +9.99106 q^{55} -11.7962 q^{57} +6.16349 q^{59} -1.42040 q^{61} -0.785351 q^{63} +7.16237 q^{65} +7.56650 q^{67} -2.00185 q^{71} +10.8343 q^{73} -7.08125 q^{75} +1.05012 q^{77} +8.71390 q^{79} -0.588949 q^{81} +0.811858 q^{83} -0.879438 q^{85} -15.7025 q^{87} +6.48798 q^{89} +0.752805 q^{91} +24.9446 q^{93} +6.63792 q^{95} -7.56444 q^{97} -30.3615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} - 12 q^{5} - q^{7} + 9 q^{9}+O(q^{10}) \) 10 * q + q^3 - 12 * q^5 - q^7 + 9 * q^9	\( 10 q + q^{3} - 12 q^{5} - q^{7} + 9 q^{9} - 10 q^{11} + 3 q^{13} + 3 q^{15} - 23 q^{17} - 11 q^{19} - 15 q^{21} + 14 q^{25} + 7 q^{27} - 6 q^{29} - 12 q^{31} - 25 q^{33} - 15 q^{35} - 6 q^{37} - 7 q^{39} - 8 q^{41} - 11 q^{43} - 39 q^{45} + 13 q^{47} - 3 q^{49} - 31 q^{51} - 29 q^{53} + 24 q^{55} - 18 q^{57} - 5 q^{59} - 27 q^{61} - 14 q^{63} - 39 q^{65} + 20 q^{71} + 49 q^{73} - 43 q^{75} - 40 q^{77} - 8 q^{79} - 38 q^{81} - 26 q^{83} + 37 q^{85} - 53 q^{87} - 60 q^{89} - 13 q^{91} - 7 q^{93} - 24 q^{95} - 27 q^{97} - 27 q^{99}+O(q^{100}) \) 10 * q + q^3 - 12 * q^5 - q^7 + 9 * q^9 - 10 * q^11 + 3 * q^13 + 3 * q^15 - 23 * q^17 - 11 * q^19 - 15 * q^21 + 14 * q^25 + 7 * q^27 - 6 * q^29 - 12 * q^31 - 25 * q^33 - 15 * q^35 - 6 * q^37 - 7 * q^39 - 8 * q^41 - 11 * q^43 - 39 * q^45 + 13 * q^47 - 3 * q^49 - 31 * q^51 - 29 * q^53 + 24 * q^55 - 18 * q^57 - 5 * q^59 - 27 * q^61 - 14 * q^63 - 39 * q^65 + 20 * q^71 + 49 * q^73 - 43 * q^75 - 40 * q^77 - 8 * q^79 - 38 * q^81 - 26 * q^83 + 37 * q^85 - 53 * q^87 - 60 * q^89 - 13 * q^91 - 7 * q^93 - 24 * q^95 - 27 * q^97 - 27 * q^99

Introduction

L-functions

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