LMFDB - Embedded newform 4104.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4104,2,Mod(1,4104)] 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("4104.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4104 = 2^{3} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2,0,-2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.7706049895$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.38266$ of defining polynomial
Character	$\chi$	$=$	4104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67708 q^{5} -2.38266 q^{7} -1.93618 q^{11} +1.82915 q^{13} +7.67301 q^{17} -1.00000 q^{19} -2.53473 q^{23} -2.18739 q^{25} +5.44241 q^{29} +3.29035 q^{31} +3.99593 q^{35} +5.35798 q^{37} +2.23060 q^{41} -9.45713 q^{43} +2.78004 q^{47} -1.32292 q^{49} +0.407345 q^{53} +3.24714 q^{55} +0.202106 q^{59} -12.4963 q^{61} -3.06763 q^{65} -1.81261 q^{67} -11.8357 q^{71} -6.22653 q^{73} +4.61326 q^{77} +9.47748 q^{79} -7.58884 q^{83} -12.8683 q^{85} +16.3539 q^{89} -4.35824 q^{91} +1.67708 q^{95} -4.72619 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} + 3 q^{17} - 4 q^{19} + q^{23} + 2 q^{25} - 2 q^{29} - 7 q^{31} - 3 q^{35} - q^{37} + 5 q^{41} - 11 q^{43} - 7 q^{47} - 14 q^{49} + 9 q^{53} - 14 q^{55} - 13 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 + 3 * q^17 - 4 * q^19 + q^23 + 2 * q^25 - 2 * q^29 - 7 * q^31 - 3 * q^35 - q^37 + 5 * q^41 - 11 * q^43 - 7 * q^47 - 14 * q^49 + 9 * q^53 - 14 * q^55 - 13 * q^59 - 11 * q^61 - 10 * q^65 - 18 * q^67 + 6 * q^71 - 2 * q^73 + 7 * q^77 - 17 * q^79 - 28 * q^83 - 27 * q^85 + 24 * q^89 - 19 * q^91 - 2 * q^95 - 16 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.67708	−0.750015	−0.375007	−	0.927022i	$-0.622360\pi$
$5$	−1.67708	−0.750015	−0.375007	+	0.927022i	$0.622360\pi$
$6$	0	0
$7$	−2.38266	−0.900562	−0.450281	−	0.892887i	$-0.648676\pi$
$7$	−2.38266	−0.900562	−0.450281	+	0.892887i	$0.648676\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.93618	−0.583780	−0.291890	−	0.956452i	$-0.594284\pi$
$11$	−1.93618	−0.583780	−0.291890	+	0.956452i	$0.594284\pi$
$12$	0	0
$13$	1.82915	0.507314	0.253657	−	0.967294i	$-0.418367\pi$
$13$	1.82915	0.507314	0.253657	+	0.967294i	$0.418367\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.67301	1.86098	0.930489	−	0.366319i	$-0.119382\pi$
$17$	7.67301	1.86098	0.930489	+	0.366319i	$0.119382\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.53473	−0.528527	−0.264263	−	0.964451i	$-0.585129\pi$
$23$	−2.53473	−0.528527	−0.264263	+	0.964451i	$0.585129\pi$
$24$	0	0
$25$	−2.18739	−0.437478
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.44241	1.01063	0.505315	−	0.862935i	$-0.331376\pi$
$29$	5.44241	1.01063	0.505315	+	0.862935i	$0.331376\pi$
$30$	0	0
$31$	3.29035	0.590964	0.295482	−	0.955348i	$-0.404520\pi$
$31$	3.29035	0.590964	0.295482	+	0.955348i	$0.404520\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.99593	0.675435
$36$	0	0
$37$	5.35798	0.880847	0.440423	−	0.897790i	$-0.354828\pi$
$37$	5.35798	0.880847	0.440423	+	0.897790i	$0.354828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.23060	0.348361	0.174181	−	0.984714i	$-0.444272\pi$
$41$	2.23060	0.348361	0.174181	+	0.984714i	$0.444272\pi$
$42$	0	0
$43$	−9.45713	−1.44220	−0.721099	−	0.692832i	$-0.756363\pi$
$43$	−9.45713	−1.44220	−0.721099	+	0.692832i	$0.756363\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.78004	0.405511	0.202756	−	0.979229i	$-0.435010\pi$
$47$	2.78004	0.405511	0.202756	+	0.979229i	$0.435010\pi$
$48$	0	0
$49$	−1.32292	−0.188988
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.407345	0.0559531	0.0279766	−	0.999609i	$-0.491094\pi$
$53$	0.407345	0.0559531	0.0279766	+	0.999609i	$0.491094\pi$
$54$	0	0
$55$	3.24714	0.437844
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.202106	0.0263119	0.0131560	−	0.999913i	$-0.495812\pi$
$59$	0.202106	0.0263119	0.0131560	+	0.999913i	$0.495812\pi$
$60$	0	0
$61$	−12.4963	−1.59998	−0.799991	−	0.600011i	$-0.795163\pi$
$61$	−12.4963	−1.59998	−0.799991	+	0.600011i	$0.795163\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.06763	−0.380493
$66$	0	0
$67$	−1.81261	−0.221446	−0.110723	−	0.993851i	$-0.535317\pi$
$67$	−1.81261	−0.221446	−0.110723	+	0.993851i	$0.535317\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.8357	−1.40464	−0.702321	−	0.711861i	$-0.747853\pi$
$71$	−11.8357	−1.40464	−0.702321	+	0.711861i	$0.747853\pi$
$72$	0	0
$73$	−6.22653	−0.728760	−0.364380	−	0.931250i	$-0.618719\pi$
$73$	−6.22653	−0.728760	−0.364380	+	0.931250i	$0.618719\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.61326	0.525730
$78$	0	0
$79$	9.47748	1.06630	0.533150	−	0.846021i	$-0.321008\pi$
$79$	9.47748	1.06630	0.533150	+	0.846021i	$0.321008\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.58884	−0.832984	−0.416492	−	0.909139i	$-0.636741\pi$
$83$	−7.58884	−0.832984	−0.416492	+	0.909139i	$0.636741\pi$
$84$	0	0
$85$	−12.8683	−1.39576
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.3539	1.73351	0.866755	−	0.498733i	$-0.166201\pi$
$89$	16.3539	1.73351	0.866755	+	0.498733i	$0.166201\pi$
$90$	0	0
$91$	−4.35824	−0.456868
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.67708	0.172065
$96$	0	0
$97$	−4.72619	−0.479872	−0.239936	−	0.970789i	$-0.577126\pi$
$97$	−4.72619	−0.479872	−0.239936	+	0.970789i	$0.577126\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.12738	0.410690	0.205345	−	0.978690i	$-0.434168\pi$
$101$	4.12738	0.410690	0.205345	+	0.978690i	$0.434168\pi$
$102$	0	0
$103$	1.85477	0.182756	0.0913779	−	0.995816i	$-0.470873\pi$
$103$	1.85477	0.182756	0.0913779	+	0.995816i	$0.470873\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.42769	0.234694	0.117347	−	0.993091i	$-0.462561\pi$
$107$	2.42769	0.234694	0.117347	+	0.993091i	$0.462561\pi$
$108$	0	0
$109$	−13.2431	−1.26846	−0.634228	−	0.773146i	$-0.718682\pi$
$109$	−13.2431	−1.26846	−0.634228	+	0.773146i	$0.718682\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.7393	−1.85692	−0.928460	−	0.371432i	$-0.878867\pi$
$113$	−19.7393	−1.85692	−0.928460	+	0.371432i	$0.878867\pi$
$114$	0	0
$115$	4.25095	0.396403
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−18.2822	−1.67593
$120$	0	0
$121$	−7.25121	−0.659201
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0539	1.07813
$126$	0	0
$127$	−2.10703	−0.186969	−0.0934844	−	0.995621i	$-0.529801\pi$
$127$	−2.10703	−0.186969	−0.0934844	+	0.995621i	$0.529801\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.59266	−0.663373	−0.331687	−	0.943390i	$-0.607618\pi$
$131$	−7.59266	−0.663373	−0.331687	+	0.943390i	$0.607618\pi$
$132$	0	0
$133$	2.38266	0.206603
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.1752	−1.63825	−0.819123	−	0.573618i	$-0.805539\pi$
$137$	−19.1752	−1.63825	−0.819123	+	0.573618i	$0.805539\pi$
$138$	0	0
$139$	−8.87010	−0.752352	−0.376176	−	0.926548i	$-0.622761\pi$
$139$	−8.87010	−0.752352	−0.376176	+	0.926548i	$0.622761\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.54156	−0.296160
$144$	0	0
$145$	−9.12738	−0.757988
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.04910	0.331715	0.165858	−	0.986150i	$-0.446961\pi$
$149$	4.04910	0.331715	0.165858	+	0.986150i	$0.446961\pi$
$150$	0	0
$151$	−16.1586	−1.31497	−0.657485	−	0.753467i	$-0.728380\pi$
$151$	−16.1586	−1.31497	−0.657485	+	0.753467i	$0.728380\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.51819	−0.443232
$156$	0	0
$157$	−1.27695	−0.101911	−0.0509557	−	0.998701i	$-0.516227\pi$
$157$	−1.27695	−0.101911	−0.0509557	+	0.998701i	$0.516227\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.03940	0.475971
$162$	0	0
$163$	−19.8660	−1.55603	−0.778014	−	0.628247i	$-0.783773\pi$
$163$	−19.8660	−1.55603	−0.778014	+	0.628247i	$0.783773\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.0081	1.47089	0.735447	−	0.677583i	$-0.236972\pi$
$167$	19.0081	1.47089	0.735447	+	0.677583i	$0.236972\pi$
$168$	0	0
$169$	−9.65422	−0.742632
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.05975	−0.232628	−0.116314	−	0.993212i	$-0.537108\pi$
$173$	−3.05975	−0.232628	−0.116314	+	0.993212i	$0.537108\pi$
$174$	0	0
$175$	5.21181	0.393976
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.78793	0.208380	0.104190	−	0.994557i	$-0.466775\pi$
$179$	2.78793	0.208380	0.104190	+	0.994557i	$0.466775\pi$
$180$	0	0
$181$	−7.73909	−0.575242	−0.287621	−	0.957744i	$-0.592864\pi$
$181$	−7.73909	−0.575242	−0.287621	+	0.957744i	$0.592864\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.98579	−0.660648
$186$	0	0
$187$	−14.8563	−1.08640
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.41210	0.102176	0.0510879	−	0.998694i	$-0.483731\pi$
$191$	1.41210	0.102176	0.0510879	+	0.998694i	$0.483731\pi$
$192$	0	0
$193$	13.9052	1.00092	0.500458	−	0.865761i	$-0.333165\pi$
$193$	13.9052	1.00092	0.500458	+	0.865761i	$0.333165\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.95365	−0.139192	−0.0695961	−	0.997575i	$-0.522171\pi$
$197$	−1.95365	−0.139192	−0.0695961	+	0.997575i	$0.522171\pi$
$198$	0	0
$199$	−8.55533	−0.606472	−0.303236	−	0.952916i	$-0.598067\pi$
$199$	−8.55533	−0.606472	−0.303236	+	0.952916i	$0.598067\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.9674	−0.910135
$204$	0	0
$205$	−3.74090	−0.261276
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.93618	0.133928
$210$	0	0
$211$	−9.23743	−0.635931	−0.317965	−	0.948102i	$-0.603000\pi$
$211$	−9.23743	−0.635931	−0.317965	+	0.948102i	$0.603000\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.8604	1.08167
$216$	0	0
$217$	−7.83979	−0.532200
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.0351	0.944101
$222$	0	0
$223$	19.2064	1.28616	0.643079	−	0.765800i	$-0.277657\pi$
$223$	19.2064	1.28616	0.643079	+	0.765800i	$0.277657\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.6280	−1.03727	−0.518633	−	0.854997i	$-0.673559\pi$
$227$	−15.6280	−1.03727	−0.518633	+	0.854997i	$0.673559\pi$
$228$	0	0
$229$	−13.7613	−0.909369	−0.454685	−	0.890653i	$-0.650248\pi$
$229$	−13.7613	−0.909369	−0.454685	+	0.890653i	$0.650248\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.53854	−0.166305	−0.0831526	−	0.996537i	$-0.526499\pi$
$233$	−2.53854	−0.166305	−0.0831526	+	0.996537i	$0.526499\pi$
$234$	0	0
$235$	−4.66237	−0.304139
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.6827	0.820377	0.410188	−	0.912001i	$-0.365463\pi$
$239$	12.6827	0.820377	0.410188	+	0.912001i	$0.365463\pi$
$240$	0	0
$241$	17.2202	1.10925	0.554625	−	0.832100i	$-0.312862\pi$
$241$	17.2202	1.10925	0.554625	+	0.832100i	$0.312862\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.21864	0.141744
$246$	0	0
$247$	−1.82915	−0.116386
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.7640	−0.995016	−0.497508	−	0.867460i	$-0.665751\pi$
$251$	−15.7640	−0.995016	−0.497508	+	0.867460i	$0.665751\pi$
$252$	0	0
$253$	4.90768	0.308543
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.2134	1.38563	0.692816	−	0.721114i	$-0.256370\pi$
$257$	22.2134	1.38563	0.692816	+	0.721114i	$0.256370\pi$
$258$	0	0
$259$	−12.7663	−0.793257
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.50649	−0.0928942	−0.0464471	−	0.998921i	$-0.514790\pi$
$263$	−1.50649	−0.0928942	−0.0464471	+	0.998921i	$0.514790\pi$
$264$	0	0
$265$	−0.683152	−0.0419657
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.3257	0.812481	0.406240	−	0.913766i	$-0.366840\pi$
$269$	13.3257	0.812481	0.406240	+	0.913766i	$0.366840\pi$
$270$	0	0
$271$	−5.78819	−0.351607	−0.175804	−	0.984425i	$-0.556252\pi$
$271$	−5.78819	−0.351607	−0.175804	+	0.984425i	$0.556252\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.23518	0.255391
$276$	0	0
$277$	1.00501	0.0603853	0.0301927	−	0.999544i	$-0.490388\pi$
$277$	1.00501	0.0603853	0.0301927	+	0.999544i	$0.490388\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.63179	0.0973447	0.0486723	−	0.998815i	$-0.484501\pi$
$281$	1.63179	0.0973447	0.0486723	+	0.998815i	$0.484501\pi$
$282$	0	0
$283$	23.3654	1.38893	0.694465	−	0.719526i	$-0.255641\pi$
$283$	23.3654	1.38893	0.694465	+	0.719526i	$0.255641\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.31477	−0.313721
$288$	0	0
$289$	41.8751	2.46324
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.0299	1.22858	0.614291	−	0.789080i	$-0.289442\pi$
$293$	21.0299	1.22858	0.614291	+	0.789080i	$0.289442\pi$
$294$	0	0
$295$	−0.338948	−0.0197343
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.63639	−0.268129
$300$	0	0
$301$	22.5332	1.29879
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.9573	1.20001
$306$	0	0
$307$	−28.7932	−1.64331	−0.821657	−	0.569982i	$-0.806950\pi$
$307$	−28.7932	−1.64331	−0.821657	+	0.569982i	$0.806950\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.5957	−1.73492	−0.867460	−	0.497507i	$-0.834249\pi$
$311$	−30.5957	−1.73492	−0.867460	+	0.497507i	$0.834249\pi$
$312$	0	0
$313$	−23.6138	−1.33473	−0.667365	−	0.744731i	$-0.732578\pi$
$313$	−23.6138	−1.33473	−0.667365	+	0.744731i	$0.732578\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.1408	−0.625729	−0.312864	−	0.949798i	$-0.601289\pi$
$317$	−11.1408	−0.625729	−0.312864	+	0.949798i	$0.601289\pi$
$318$	0	0
$319$	−10.5375	−0.589986
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.67301	−0.426938
$324$	0	0
$325$	−4.00106	−0.221939
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.62391	−0.365188
$330$	0	0
$331$	6.88300	0.378324	0.189162	−	0.981946i	$-0.439423\pi$
$331$	6.88300	0.378324	0.189162	+	0.981946i	$0.439423\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.03990	0.166088
$336$	0	0
$337$	−6.69362	−0.364625	−0.182312	−	0.983241i	$-0.558358\pi$
$337$	−6.69362	−0.364625	−0.182312	+	0.983241i	$0.558358\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.37070	−0.344993
$342$	0	0
$343$	19.8307	1.07076
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.7796	−1.33024	−0.665120	−	0.746737i	$-0.731619\pi$
$347$	−24.7796	−1.33024	−0.665120	+	0.746737i	$0.731619\pi$
$348$	0	0
$349$	16.8957	0.904407	0.452204	−	0.891915i	$-0.350638\pi$
$349$	16.8957	0.904407	0.452204	+	0.891915i	$0.350638\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.5184	−1.73078	−0.865391	−	0.501098i	$-0.832930\pi$
$353$	−32.5184	−1.73078	−0.865391	+	0.501098i	$0.832930\pi$
$354$	0	0
$355$	19.8495	1.05350
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.79778	0.305995	0.152998	−	0.988227i	$-0.451107\pi$
$359$	5.79778	0.305995	0.152998	+	0.988227i	$0.451107\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.4424	0.546581
$366$	0	0
$367$	−36.5712	−1.90900	−0.954501	−	0.298206i	$-0.903612\pi$
$367$	−36.5712	−1.90900	−0.954501	+	0.298206i	$0.903612\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.970566	−0.0503893
$372$	0	0
$373$	−26.3476	−1.36423	−0.682113	−	0.731247i	$-0.738939\pi$
$373$	−26.3476	−1.36423	−0.682113	+	0.731247i	$0.738939\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.95497	0.512707
$378$	0	0
$379$	−5.31477	−0.273001	−0.136501	−	0.990640i	$-0.543586\pi$
$379$	−5.31477	−0.273001	−0.136501	+	0.990640i	$0.543586\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.1526	0.723162	0.361581	−	0.932341i	$-0.382237\pi$
$383$	14.1526	0.723162	0.361581	+	0.932341i	$0.382237\pi$
$384$	0	0
$385$	−7.73683	−0.394305
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.91150	0.0969168	0.0484584	−	0.998825i	$-0.484569\pi$
$389$	1.91150	0.0969168	0.0484584	+	0.998825i	$0.484569\pi$
$390$	0	0
$391$	−19.4490	−0.983577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.8945	−0.799740
$396$	0	0
$397$	1.92935	0.0968312	0.0484156	−	0.998827i	$-0.484583\pi$
$397$	1.92935	0.0968312	0.0484156	+	0.998827i	$0.484583\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.0215	0.550390	0.275195	−	0.961388i	$-0.411258\pi$
$401$	11.0215	0.550390	0.275195	+	0.961388i	$0.411258\pi$
$402$	0	0
$403$	6.01853	0.299804
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.3740	−0.514221
$408$	0	0
$409$	−34.6473	−1.71320	−0.856599	−	0.515983i	$-0.827427\pi$
$409$	−34.6473	−1.71320	−0.856599	+	0.515983i	$0.827427\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.481550	−0.0236955
$414$	0	0
$415$	12.7271	0.624750
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.7622	0.672327	0.336164	−	0.941804i	$-0.390871\pi$
$419$	13.7622	0.672327	0.336164	+	0.941804i	$0.390871\pi$
$420$	0	0
$421$	14.6845	0.715680	0.357840	−	0.933783i	$-0.383513\pi$
$421$	14.6845	0.715680	0.357840	+	0.933783i	$0.383513\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.7839	−0.814137
$426$	0	0
$427$	29.7744	1.44088
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.11492	0.101872	0.0509360	−	0.998702i	$-0.483780\pi$
$431$	2.11492	0.101872	0.0509360	+	0.998702i	$0.483780\pi$
$432$	0	0
$433$	40.0421	1.92430	0.962151	−	0.272516i	$-0.0878558\pi$
$433$	40.0421	1.92430	0.962151	+	0.272516i	$0.0878558\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.53473	0.121252
$438$	0	0
$439$	−27.9907	−1.33592	−0.667961	−	0.744197i	$-0.732833\pi$
$439$	−27.9907	−1.33592	−0.667961	+	0.744197i	$0.732833\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.5985	−0.931155	−0.465578	−	0.885007i	$-0.654153\pi$
$443$	−19.5985	−0.931155	−0.465578	+	0.885007i	$0.654153\pi$
$444$	0	0
$445$	−27.4269	−1.30016
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.8645	0.607112	0.303556	−	0.952814i	$-0.401826\pi$
$449$	12.8645	0.607112	0.303556	+	0.952814i	$0.401826\pi$
$450$	0	0
$451$	−4.31884	−0.203366
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.30914	0.342658
$456$	0	0
$457$	6.15915	0.288113	0.144057	−	0.989569i	$-0.453985\pi$
$457$	6.15915	0.288113	0.144057	+	0.989569i	$0.453985\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.20272	0.382039	0.191019	−	0.981586i	$-0.438821\pi$
$461$	8.20272	0.382039	0.191019	+	0.981586i	$0.438821\pi$
$462$	0	0
$463$	3.92779	0.182540	0.0912699	−	0.995826i	$-0.470907\pi$
$463$	3.92779	0.182540	0.0912699	+	0.995826i	$0.470907\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.5404	0.626573	0.313287	−	0.949659i	$-0.398570\pi$
$467$	13.5404	0.626573	0.313287	+	0.949659i	$0.398570\pi$
$468$	0	0
$469$	4.31884	0.199426
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18.3107	0.841927
$474$	0	0
$475$	2.18739	0.100364
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.00883	−0.0460944	−0.0230472	−	0.999734i	$-0.507337\pi$
$479$	−1.00883	−0.0460944	−0.0230472	+	0.999734i	$0.507337\pi$
$480$	0	0
$481$	9.80054	0.446866
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.92621	0.359911
$486$	0	0
$487$	16.8298	0.762632	0.381316	−	0.924445i	$-0.375471\pi$
$487$	16.8298	0.762632	0.381316	+	0.924445i	$0.375471\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.89139	0.356134	0.178067	−	0.984018i	$-0.443016\pi$
$491$	7.89139	0.356134	0.178067	+	0.984018i	$0.443016\pi$
$492$	0	0
$493$	41.7597	1.88076
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.2005	1.26497
$498$	0	0
$499$	−18.1437	−0.812222	−0.406111	−	0.913824i	$-0.633115\pi$
$499$	−18.1437	−0.812222	−0.406111	+	0.913824i	$0.633115\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.5037	1.18174	0.590871	−	0.806766i	$-0.298784\pi$
$503$	26.5037	1.18174	0.590871	+	0.806766i	$0.298784\pi$
$504$	0	0
$505$	−6.92197	−0.308023
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.87867	−0.171919	−0.0859595	−	0.996299i	$-0.527396\pi$
$509$	−3.87867	−0.171919	−0.0859595	+	0.996299i	$0.527396\pi$
$510$	0	0
$511$	14.8357	0.656294
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.11060	−0.137070
$516$	0	0
$517$	−5.38266	−0.236729
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.39175	0.323838	0.161919	−	0.986804i	$-0.448232\pi$
$521$	7.39175	0.323838	0.161919	+	0.986804i	$0.448232\pi$
$522$	0	0
$523$	7.93686	0.347055	0.173527	−	0.984829i	$-0.444484\pi$
$523$	7.93686	0.347055	0.173527	+	0.984829i	$0.444484\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.2469	1.09977
$528$	0	0
$529$	−16.5752	−0.720659
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.08010	0.176729
$534$	0	0
$535$	−4.07145	−0.176024
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.56140	0.110327
$540$	0	0
$541$	−23.0254	−0.989938	−0.494969	−	0.868911i	$-0.664821\pi$
$541$	−23.0254	−0.989938	−0.494969	+	0.868911i	$0.664821\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.2097	0.951360
$546$	0	0
$547$	−16.3032	−0.697076	−0.348538	−	0.937295i	$-0.613322\pi$
$547$	−16.3032	−0.697076	−0.348538	+	0.937295i	$0.613322\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.44241	−0.231854
$552$	0	0
$553$	−22.5816	−0.960269
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.48814	0.274911	0.137456	−	0.990508i	$-0.456108\pi$
$557$	6.48814	0.274911	0.137456	+	0.990508i	$0.456108\pi$
$558$	0	0
$559$	−17.2985	−0.731648
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−45.5809	−1.92101	−0.960504	−	0.278267i	$-0.910240\pi$
$563$	−45.5809	−1.92101	−0.960504	+	0.278267i	$0.910240\pi$
$564$	0	0
$565$	33.1045	1.39272
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.0518	0.924461	0.462231	−	0.886760i	$-0.347049\pi$
$569$	22.0518	0.924461	0.462231	+	0.886760i	$0.347049\pi$
$570$	0	0
$571$	−12.7271	−0.532613	−0.266306	−	0.963888i	$-0.585803\pi$
$571$	−12.7271	−0.532613	−0.266306	+	0.963888i	$0.585803\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.54443	0.231219
$576$	0	0
$577$	−32.6821	−1.36057	−0.680286	−	0.732946i	$-0.738145\pi$
$577$	−32.6821	−1.36057	−0.680286	+	0.732946i	$0.738145\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.0817	0.750153
$582$	0	0
$583$	−0.788693	−0.0326643
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.9891	−0.783764	−0.391882	−	0.920016i	$-0.628176\pi$
$587$	−18.9891	−0.783764	−0.391882	+	0.920016i	$0.628176\pi$
$588$	0	0
$589$	−3.29035	−0.135576
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.8777	0.775215	0.387608	−	0.921824i	$-0.373302\pi$
$593$	18.8777	0.775215	0.387608	+	0.921824i	$0.373302\pi$
$594$	0	0
$595$	30.6608	1.25697
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.0679	−0.860810	−0.430405	−	0.902636i	$-0.641629\pi$
$599$	−21.0679	−0.860810	−0.430405	+	0.902636i	$0.641629\pi$
$600$	0	0
$601$	33.3513	1.36043	0.680214	−	0.733014i	$-0.261887\pi$
$601$	33.3513	1.36043	0.680214	+	0.733014i	$0.261887\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.1609	0.494410
$606$	0	0
$607$	17.7200	0.719234	0.359617	−	0.933100i	$-0.382907\pi$
$607$	17.7200	0.719234	0.359617	+	0.933100i	$0.382907\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.08511	0.205721
$612$	0	0
$613$	−22.0326	−0.889886	−0.444943	−	0.895559i	$-0.646776\pi$
$613$	−22.0326	−0.889886	−0.444943	+	0.895559i	$0.646776\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.0150	−0.966806	−0.483403	−	0.875398i	$-0.660599\pi$
$617$	−24.0150	−0.966806	−0.483403	+	0.875398i	$0.660599\pi$
$618$	0	0
$619$	−16.9309	−0.680511	−0.340255	−	0.940333i	$-0.610513\pi$
$619$	−16.9309	−0.680511	−0.340255	+	0.940333i	$0.610513\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38.9659	−1.56113
$624$	0	0
$625$	−9.27839	−0.371136
$626$	0	0
$627$	0	0
$628$	0	0
$629$	41.1119	1.63924
$630$	0	0
$631$	24.7788	0.986430	0.493215	−	0.869907i	$-0.335822\pi$
$631$	24.7788	0.986430	0.493215	+	0.869907i	$0.335822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.53367	0.140229
$636$	0	0
$637$	−2.41981	−0.0958763
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.57099	−0.220041	−0.110020	−	0.993929i	$-0.535092\pi$
$641$	−5.57099	−0.220041	−0.110020	+	0.993929i	$0.535092\pi$
$642$	0	0
$643$	29.7603	1.17363	0.586816	−	0.809720i	$-0.300381\pi$
$643$	29.7603	1.17363	0.586816	+	0.809720i	$0.300381\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.13579	0.319851	0.159925	−	0.987129i	$-0.448875\pi$
$647$	8.13579	0.319851	0.159925	+	0.987129i	$0.448875\pi$
$648$	0	0
$649$	−0.391313	−0.0153604
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.7771	−1.20440	−0.602200	−	0.798345i	$-0.705709\pi$
$653$	−30.7771	−1.20440	−0.602200	+	0.798345i	$0.705709\pi$
$654$	0	0
$655$	12.7335	0.497540
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.2017	−1.17649	−0.588245	−	0.808683i	$-0.700181\pi$
$659$	−30.2017	−1.17649	−0.588245	+	0.808683i	$0.700181\pi$
$660$	0	0
$661$	15.4162	0.599619	0.299809	−	0.953999i	$-0.403077\pi$
$661$	15.4162	0.599619	0.299809	+	0.953999i	$0.403077\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.99593	−0.154955
$666$	0	0
$667$	−13.7950	−0.534145
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.1950	0.934038
$672$	0	0
$673$	39.7875	1.53370	0.766848	−	0.641829i	$-0.221824\pi$
$673$	39.7875	1.53370	0.766848	+	0.641829i	$0.221824\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−40.2548	−1.54712	−0.773559	−	0.633724i	$-0.781526\pi$
$677$	−40.2548	−1.54712	−0.773559	+	0.633724i	$0.781526\pi$
$678$	0	0
$679$	11.2609	0.432154
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.7160	−0.563091	−0.281545	−	0.959548i	$-0.590847\pi$
$683$	−14.7160	−0.563091	−0.281545	+	0.959548i	$0.590847\pi$
$684$	0	0
$685$	32.1584	1.22871
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.745094	0.0283858
$690$	0	0
$691$	36.7769	1.39906	0.699529	−	0.714604i	$-0.253393\pi$
$691$	36.7769	1.39906	0.699529	+	0.714604i	$0.253393\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.8759	0.564276
$696$	0	0
$697$	17.1154	0.648293
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21.7861	−0.822849	−0.411425	−	0.911444i	$-0.634969\pi$
$701$	−21.7861	−0.822849	−0.411425	+	0.911444i	$0.634969\pi$
$702$	0	0
$703$	−5.35798	−0.202080
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.83416	−0.369852
$708$	0	0
$709$	9.31634	0.349883	0.174941	−	0.984579i	$-0.444026\pi$
$709$	9.31634	0.349883	0.174941	+	0.984579i	$0.444026\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.34013	−0.312340
$714$	0	0
$715$	5.93949	0.222124
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−47.5963	−1.77504	−0.887522	−	0.460766i	$-0.847575\pi$
$719$	−47.5963	−1.77504	−0.887522	+	0.460766i	$0.847575\pi$
$720$	0	0
$721$	−4.41929	−0.164583
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−11.9047	−0.442128
$726$	0	0
$727$	−26.9899	−1.00100	−0.500499	−	0.865737i	$-0.666850\pi$
$727$	−26.9899	−1.00100	−0.500499	+	0.865737i	$0.666850\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−72.5646	−2.68390
$732$	0	0
$733$	−5.37797	−0.198640	−0.0993199	−	0.995056i	$-0.531667\pi$
$733$	−5.37797	−0.198640	−0.0993199	+	0.995056i	$0.531667\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.50954	0.129276
$738$	0	0
$739$	23.8301	0.876604	0.438302	−	0.898828i	$-0.355580\pi$
$739$	23.8301	0.876604	0.438302	+	0.898828i	$0.355580\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.48500	0.0911659	0.0455830	−	0.998961i	$-0.485485\pi$
$743$	2.48500	0.0911659	0.0455830	+	0.998961i	$0.485485\pi$
$744$	0	0
$745$	−6.79069	−0.248792
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.78438	−0.211357
$750$	0	0
$751$	−6.68497	−0.243938	−0.121969	−	0.992534i	$-0.538921\pi$
$751$	−6.68497	−0.243938	−0.121969	+	0.992534i	$0.538921\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	27.0994	0.986248
$756$	0	0
$757$	51.8293	1.88377	0.941884	−	0.335940i	$-0.109054\pi$
$757$	51.8293	1.88377	0.941884	+	0.335940i	$0.109054\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.9844	−0.869434	−0.434717	−	0.900567i	$-0.643152\pi$
$761$	−23.9844	−0.869434	−0.434717	+	0.900567i	$0.643152\pi$
$762$	0	0
$763$	31.5538	1.14232
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.369681	0.0133484
$768$	0	0
$769$	−22.3077	−0.804435	−0.402218	−	0.915544i	$-0.631761\pi$
$769$	−22.3077	−0.804435	−0.402218	+	0.915544i	$0.631761\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.4796	−0.376926	−0.188463	−	0.982080i	$-0.560350\pi$
$773$	−10.4796	−0.376926	−0.188463	+	0.982080i	$0.560350\pi$
$774$	0	0
$775$	−7.19727	−0.258534
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.23060	−0.0799196
$780$	0	0
$781$	22.9161	0.820002
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.14155	0.0764351
$786$	0	0
$787$	34.8563	1.24249	0.621247	−	0.783615i	$-0.286626\pi$
$787$	34.8563	1.24249	0.621247	+	0.783615i	$0.286626\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	47.0322	1.67227
$792$	0	0
$793$	−22.8575	−0.811694
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.9584	0.919494	0.459747	−	0.888050i	$-0.347940\pi$
$797$	25.9584	0.919494	0.459747	+	0.888050i	$0.347940\pi$
$798$	0	0
$799$	21.3313	0.754647
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.0557	0.425436
$804$	0	0
$805$	−10.1286	−0.356986
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.3691	−1.70057	−0.850284	−	0.526325i	$-0.823570\pi$
$809$	−48.3691	−1.70057	−0.850284	+	0.526325i	$0.823570\pi$
$810$	0	0
$811$	−4.66530	−0.163821	−0.0819104	−	0.996640i	$-0.526102\pi$
$811$	−4.66530	−0.163821	−0.0819104	+	0.996640i	$0.526102\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	33.3170	1.16704
$816$	0	0
$817$	9.45713	0.330863
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0091	−0.349319	−0.174660	−	0.984629i	$-0.555883\pi$
$821$	−10.0091	−0.349319	−0.174660	+	0.984629i	$0.555883\pi$
$822$	0	0
$823$	44.0152	1.53427	0.767137	−	0.641483i	$-0.221681\pi$
$823$	44.0152	1.53427	0.767137	+	0.641483i	$0.221681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.4656	−0.885527	−0.442763	−	0.896638i	$-0.646002\pi$
$827$	−25.4656	−0.885527	−0.442763	+	0.896638i	$0.646002\pi$
$828$	0	0
$829$	−34.4477	−1.19642	−0.598208	−	0.801341i	$-0.704121\pi$
$829$	−34.4477	−1.19642	−0.598208	+	0.801341i	$0.704121\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10.1507	−0.351703
$834$	0	0
$835$	−31.8782	−1.10319
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.78186	−0.303184	−0.151592	−	0.988443i	$-0.548440\pi$
$839$	−8.78186	−0.303184	−0.151592	+	0.988443i	$0.548440\pi$
$840$	0	0
$841$	0.619835	0.0213736
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.1909	0.556985
$846$	0	0
$847$	17.2772	0.593651
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.5810	−0.465551
$852$	0	0
$853$	55.6217	1.90445	0.952225	−	0.305399i	$-0.0987897\pi$
$853$	55.6217	1.90445	0.952225	+	0.305399i	$0.0987897\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0030	1.02488	0.512441	−	0.858722i	$-0.328741\pi$
$857$	30.0030	1.02488	0.512441	+	0.858722i	$0.328741\pi$
$858$	0	0
$859$	17.5985	0.600455	0.300227	−	0.953868i	$-0.402937\pi$
$859$	17.5985	0.600455	0.300227	+	0.953868i	$0.402937\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.5133	1.82161	0.910807	−	0.412832i	$-0.135460\pi$
$863$	53.5133	1.82161	0.910807	+	0.412832i	$0.135460\pi$
$864$	0	0
$865$	5.13145	0.174475
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.3501	−0.622484
$870$	0	0
$871$	−3.31553	−0.112343
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−28.7203	−0.970923
$876$	0	0
$877$	15.4020	0.520087	0.260044	−	0.965597i	$-0.416263\pi$
$877$	15.4020	0.520087	0.260044	+	0.965597i	$0.416263\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.5213	1.36520	0.682599	−	0.730793i	$-0.260850\pi$
$881$	40.5213	1.36520	0.682599	+	0.730793i	$0.260850\pi$
$882$	0	0
$883$	−24.8520	−0.836336	−0.418168	−	0.908370i	$-0.637328\pi$
$883$	−24.8520	−0.836336	−0.418168	+	0.908370i	$0.637328\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.6022	−1.76621	−0.883105	−	0.469175i	$-0.844551\pi$
$887$	−52.6022	−1.76621	−0.883105	+	0.469175i	$0.844551\pi$
$888$	0	0
$889$	5.02035	0.168377
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.78004	−0.0930306
$894$	0	0
$895$	−4.67559	−0.156288
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.9074	0.597246
$900$	0	0
$901$	3.12556	0.104128
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.9791	0.431440
$906$	0	0
$907$	30.1644	1.00159	0.500796	−	0.865565i	$-0.333041\pi$
$907$	30.1644	1.00159	0.500796	+	0.865565i	$0.333041\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22.1955	0.735371	0.367685	−	0.929950i	$-0.380150\pi$
$911$	22.1955	0.735371	0.367685	+	0.929950i	$0.380150\pi$
$912$	0	0
$913$	14.6934	0.486279
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.0907	0.597409
$918$	0	0
$919$	15.2693	0.503688	0.251844	−	0.967768i	$-0.418963\pi$
$919$	15.2693	0.503688	0.251844	+	0.967768i	$0.418963\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.6493	−0.712594
$924$	0	0
$925$	−11.7200	−0.385351
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.97350	0.327220	0.163610	−	0.986525i	$-0.447686\pi$
$929$	9.97350	0.327220	0.163610	+	0.986525i	$0.447686\pi$
$930$	0	0
$931$	1.32292	0.0433568
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.9153	0.814818
$936$	0	0
$937$	55.9900	1.82911	0.914557	−	0.404457i	$-0.132539\pi$
$937$	55.9900	1.82911	0.914557	+	0.404457i	$0.132539\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.61377	0.183004	0.0915018	−	0.995805i	$-0.470833\pi$
$941$	5.61377	0.183004	0.0915018	+	0.995805i	$0.470833\pi$
$942$	0	0
$943$	−5.65396	−0.184118
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.8458	−1.61977	−0.809886	−	0.586587i	$-0.800471\pi$
$947$	−49.8458	−1.61977	−0.809886	+	0.586587i	$0.800471\pi$
$948$	0	0
$949$	−11.3892	−0.369710
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−15.4947	−0.501923	−0.250961	−	0.967997i	$-0.580747\pi$
$953$	−15.4947	−0.501923	−0.250961	+	0.967997i	$0.580747\pi$
$954$	0	0
$955$	−2.36821	−0.0766334
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45.6880	1.47534
$960$	0	0
$961$	−20.1736	−0.650762
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.3201	−0.750702
$966$	0	0
$967$	33.9976	1.09329	0.546645	−	0.837364i	$-0.315905\pi$
$967$	33.9976	1.09329	0.546645	+	0.837364i	$0.315905\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.1912	1.28980	0.644898	−	0.764268i	$-0.276900\pi$
$971$	40.1912	1.28980	0.644898	+	0.764268i	$0.276900\pi$
$972$	0	0
$973$	21.1345	0.677540
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.9812	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$977$	10.9812	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$978$	0	0
$979$	−31.6641	−1.01199
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.5963	0.656920	0.328460	−	0.944518i	$-0.393470\pi$
$983$	20.5963	0.656920	0.328460	+	0.944518i	$0.393470\pi$
$984$	0	0
$985$	3.27644	0.104396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.9712	0.762241
$990$	0	0
$991$	47.6854	1.51478	0.757388	−	0.652965i	$-0.226475\pi$
$991$	47.6854	1.51478	0.757388	+	0.652965i	$0.226475\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.3480	0.454863
$996$	0	0
$997$	−20.5402	−0.650516	−0.325258	−	0.945625i	$-0.605451\pi$
$997$	−20.5402	−0.650516	−0.325258	+	0.945625i	$0.605451\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4104.2.a.m.1.1	yes	4
3.2	odd	2		4104.2.a.i.1.4	✓	4
4.3	odd	2		8208.2.a.bz.1.1		4
12.11	even	2		8208.2.a.br.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.i.1.4	✓	4	3.2	odd	2
4104.2.a.m.1.1	yes	4	1.1	even	1	trivial
8208.2.a.br.1.4		4	12.11	even	2
8208.2.a.bz.1.1		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4104.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67708 q^{5} -2.38266 q^{7} -1.93618 q^{11} +1.82915 q^{13} +7.67301 q^{17} -1.00000 q^{19} -2.53473 q^{23} -2.18739 q^{25} +5.44241 q^{29} +3.29035 q^{31} +3.99593 q^{35} +5.35798 q^{37} +2.23060 q^{41} -9.45713 q^{43} +2.78004 q^{47} -1.32292 q^{49} +0.407345 q^{53} +3.24714 q^{55} +0.202106 q^{59} -12.4963 q^{61} -3.06763 q^{65} -1.81261 q^{67} -11.8357 q^{71} -6.22653 q^{73} +4.61326 q^{77} +9.47748 q^{79} -7.58884 q^{83} -12.8683 q^{85} +16.3539 q^{89} -4.35824 q^{91} +1.67708 q^{95} -4.72619 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} + 3 q^{17} - 4 q^{19} + q^{23} + 2 q^{25} - 2 q^{29} - 7 q^{31} - 3 q^{35} - q^{37} + 5 q^{41} - 11 q^{43} - 7 q^{47} - 14 q^{49} + 9 q^{53} - 14 q^{55} - 13 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 + 3 * q^17 - 4 * q^19 + q^23 + 2 * q^25 - 2 * q^29 - 7 * q^31 - 3 * q^35 - q^37 + 5 * q^41 - 11 * q^43 - 7 * q^47 - 14 * q^49 + 9 * q^53 - 14 * q^55 - 13 * q^59 - 11 * q^61 - 10 * q^65 - 18 * q^67 + 6 * q^71 - 2 * q^73 + 7 * q^77 - 17 * q^79 - 28 * q^83 - 27 * q^85 + 24 * q^89 - 19 * q^91 - 2 * q^95 - 16 * q^97

Introduction

L-functions

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