LMFDB - Embedded newform 819.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.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:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.81361 q^{2} +5.91638 q^{4} +1.28917 q^{5} -1.00000 q^{7} +11.0192 q^{8} +O(q^{10})$	\(q+2.81361 q^{2} +5.91638 q^{4} +1.28917 q^{5} -1.00000 q^{7} +11.0192 q^{8} +3.62721 q^{10} -4.20555 q^{11} -1.00000 q^{13} -2.81361 q^{14} +19.1708 q^{16} -1.62721 q^{17} -6.33804 q^{19} +7.62721 q^{20} -11.8328 q^{22} +2.71083 q^{23} -3.33804 q^{25} -2.81361 q^{26} -5.91638 q^{28} +4.33804 q^{29} -7.49472 q^{31} +31.9008 q^{32} -4.57834 q^{34} -1.28917 q^{35} +3.42166 q^{37} -17.8328 q^{38} +14.2056 q^{40} +7.62721 q^{41} -4.91638 q^{43} -24.8816 q^{44} +7.62721 q^{46} +1.08362 q^{47} +1.00000 q^{49} -9.39194 q^{50} -5.91638 q^{52} -1.75971 q^{53} -5.42166 q^{55} -11.0192 q^{56} +12.2056 q^{58} +4.57834 q^{59} -5.25443 q^{61} -21.0872 q^{62} +51.4147 q^{64} -1.28917 q^{65} +8.67609 q^{67} -9.62721 q^{68} -3.62721 q^{70} +6.78389 q^{71} +4.07306 q^{73} +9.62721 q^{74} -37.4983 q^{76} +4.20555 q^{77} -8.91638 q^{79} +24.7144 q^{80} +21.4600 q^{82} +4.33804 q^{83} -2.09775 q^{85} -13.8328 q^{86} -46.3416 q^{88} -4.54359 q^{89} +1.00000 q^{91} +16.0383 q^{92} +3.04888 q^{94} -8.17081 q^{95} +11.3275 q^{97} +2.81361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 3 q^{7} + 12 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 3 * q^7 + 12 * q^8	$ 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 3 q^{7} + 12 q^{8} - 2 q^{10} + 2 q^{11} - 3 q^{13} - 2 q^{14} + 18 q^{16} + 8 q^{17} - 7 q^{19} + 10 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{28} + q^{29} - 7 q^{31} + 36 q^{32} - 12 q^{34} - 3 q^{35} + 12 q^{37} - 26 q^{38} + 28 q^{40} + 10 q^{41} - q^{43} - 36 q^{44} + 10 q^{46} + 17 q^{47} + 3 q^{49} - 20 q^{50} - 4 q^{52} + 5 q^{53} - 18 q^{55} - 12 q^{56} + 22 q^{58} + 12 q^{59} + 10 q^{61} - 10 q^{62} + 58 q^{64} - 3 q^{65} + 2 q^{67} - 16 q^{68} + 2 q^{70} + 4 q^{71} - 5 q^{73} + 16 q^{74} - 30 q^{76} - 2 q^{77} - 13 q^{79} + 8 q^{80} + 24 q^{82} + q^{83} + 16 q^{85} - 14 q^{86} - 60 q^{88} + 13 q^{89} + 3 q^{91} + 6 q^{92} - 2 q^{94} + 15 q^{95} - 9 q^{97} + 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 3 * q^7 + 12 * q^8 - 2 * q^10 + 2 * q^11 - 3 * q^13 - 2 * q^14 + 18 * q^16 + 8 * q^17 - 7 * q^19 + 10 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 4 * q^28 + q^29 - 7 * q^31 + 36 * q^32 - 12 * q^34 - 3 * q^35 + 12 * q^37 - 26 * q^38 + 28 * q^40 + 10 * q^41 - q^43 - 36 * q^44 + 10 * q^46 + 17 * q^47 + 3 * q^49 - 20 * q^50 - 4 * q^52 + 5 * q^53 - 18 * q^55 - 12 * q^56 + 22 * q^58 + 12 * q^59 + 10 * q^61 - 10 * q^62 + 58 * q^64 - 3 * q^65 + 2 * q^67 - 16 * q^68 + 2 * q^70 + 4 * q^71 - 5 * q^73 + 16 * q^74 - 30 * q^76 - 2 * q^77 - 13 * q^79 + 8 * q^80 + 24 * q^82 + q^83 + 16 * q^85 - 14 * q^86 - 60 * q^88 + 13 * q^89 + 3 * q^91 + 6 * q^92 - 2 * q^94 + 15 * q^95 - 9 * q^97 + 2 * q^98

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 $

$2$	2.81361	1.98952	0.994760	−	0.102237i	$-0.0325999\pi$
$2$	2.81361	1.98952	0.994760	+	0.102237i	$0.0325999\pi$
$3$	0	0
$3$	0	0
$4$	5.91638	2.95819
$5$	1.28917	0.576534	0.288267	−	0.957550i	$-0.406921\pi$
$5$	1.28917	0.576534	0.288267	+	0.957550i	$0.406921\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	11.0192	3.89586
$9$	0	0
$10$	3.62721	1.14703
$11$	−4.20555	−1.26802	−0.634011	−	0.773324i	$-0.718592\pi$
$11$	−4.20555	−1.26802	−0.634011	+	0.773324i	$0.718592\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.81361	−0.751968
$15$	0	0
$16$	19.1708	4.79270
$17$	−1.62721	−0.394657	−0.197329	−	0.980337i	$-0.563227\pi$
$17$	−1.62721	−0.394657	−0.197329	+	0.980337i	$0.563227\pi$
$18$	0	0
$19$	−6.33804	−1.45405	−0.727024	−	0.686613i	$-0.759097\pi$
$19$	−6.33804	−1.45405	−0.727024	+	0.686613i	$0.759097\pi$
$20$	7.62721	1.70550
$21$	0	0
$22$	−11.8328	−2.52275
$23$	2.71083	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$23$	2.71083	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$24$	0	0
$25$	−3.33804	−0.667609
$26$	−2.81361	−0.551794
$27$	0	0
$28$	−5.91638	−1.11809
$29$	4.33804	0.805555	0.402777	−	0.915298i	$-0.368045\pi$
$29$	4.33804	0.805555	0.402777	+	0.915298i	$0.368045\pi$
$30$	0	0
$31$	−7.49472	−1.34609	−0.673046	−	0.739601i	$-0.735014\pi$
$31$	−7.49472	−1.34609	−0.673046	+	0.739601i	$0.735014\pi$
$32$	31.9008	5.63932
$33$	0	0
$34$	−4.57834	−0.785178
$35$	−1.28917	−0.217909
$36$	0	0
$37$	3.42166	0.562518	0.281259	−	0.959632i	$-0.409248\pi$
$37$	3.42166	0.562518	0.281259	+	0.959632i	$0.409248\pi$
$38$	−17.8328	−2.89286
$39$	0	0
$40$	14.2056	2.24609
$41$	7.62721	1.19117	0.595585	−	0.803292i	$-0.296920\pi$
$41$	7.62721	1.19117	0.595585	+	0.803292i	$0.296920\pi$
$42$	0	0
$43$	−4.91638	−0.749741	−0.374871	−	0.927077i	$-0.622313\pi$
$43$	−4.91638	−0.749741	−0.374871	+	0.927077i	$0.622313\pi$
$44$	−24.8816	−3.75105
$45$	0	0
$46$	7.62721	1.12457
$47$	1.08362	0.158062	0.0790310	−	0.996872i	$-0.474817\pi$
$47$	1.08362	0.158062	0.0790310	+	0.996872i	$0.474817\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−9.39194	−1.32822
$51$	0	0
$52$	−5.91638	−0.820455
$53$	−1.75971	−0.241714	−0.120857	−	0.992670i	$-0.538564\pi$
$53$	−1.75971	−0.241714	−0.120857	+	0.992670i	$0.538564\pi$
$54$	0	0
$55$	−5.42166	−0.731057
$56$	−11.0192	−1.47250
$57$	0	0
$58$	12.2056	1.60267
$59$	4.57834	0.596049	0.298024	−	0.954558i	$-0.403672\pi$
$59$	4.57834	0.596049	0.298024	+	0.954558i	$0.403672\pi$
$60$	0	0
$61$	−5.25443	−0.672760	−0.336380	−	0.941726i	$-0.609203\pi$
$61$	−5.25443	−0.672760	−0.336380	+	0.941726i	$0.609203\pi$
$62$	−21.0872	−2.67808
$63$	0	0
$64$	51.4147	6.42683
$65$	−1.28917	−0.159902
$66$	0	0
$67$	8.67609	1.05995	0.529976	−	0.848012i	$-0.322201\pi$
$67$	8.67609	1.05995	0.529976	+	0.848012i	$0.322201\pi$
$68$	−9.62721	−1.16747
$69$	0	0
$70$	−3.62721	−0.433535
$71$	6.78389	0.805099	0.402550	−	0.915398i	$-0.368124\pi$
$71$	6.78389	0.805099	0.402550	+	0.915398i	$0.368124\pi$
$72$	0	0
$73$	4.07306	0.476715	0.238358	−	0.971177i	$-0.423391\pi$
$73$	4.07306	0.476715	0.238358	+	0.971177i	$0.423391\pi$
$74$	9.62721	1.11914
$75$	0	0
$76$	−37.4983	−4.30135
$77$	4.20555	0.479267
$78$	0	0
$79$	−8.91638	−1.00317	−0.501586	−	0.865108i	$-0.667250\pi$
$79$	−8.91638	−1.00317	−0.501586	+	0.865108i	$0.667250\pi$
$80$	24.7144	2.76315
$81$	0	0
$82$	21.4600	2.36986
$83$	4.33804	0.476162	0.238081	−	0.971245i	$-0.423482\pi$
$83$	4.33804	0.476162	0.238081	+	0.971245i	$0.423482\pi$
$84$	0	0
$85$	−2.09775	−0.227533
$86$	−13.8328	−1.49163
$87$	0	0
$88$	−46.3416	−4.94003
$89$	−4.54359	−0.481620	−0.240810	−	0.970572i	$-0.577413\pi$
$89$	−4.54359	−0.481620	−0.240810	+	0.970572i	$0.577413\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	16.0383	1.67211
$93$	0	0
$94$	3.04888	0.314468
$95$	−8.17081	−0.838307
$96$	0	0
$97$	11.3275	1.15013	0.575066	−	0.818107i	$-0.304976\pi$
$97$	11.3275	1.15013	0.575066	+	0.818107i	$0.304976\pi$
$98$	2.81361	0.284217
$99$	0	0
$100$	−19.7491	−1.97491
$101$	−14.3033	−1.42323	−0.711616	−	0.702569i	$-0.752036\pi$
$101$	−14.3033	−1.42323	−0.711616	+	0.702569i	$0.752036\pi$
$102$	0	0
$103$	−19.6655	−1.93770	−0.968851	−	0.247645i	$-0.920343\pi$
$103$	−19.6655	−1.93770	−0.968851	+	0.247645i	$0.920343\pi$
$104$	−11.0192	−1.08052
$105$	0	0
$106$	−4.95112	−0.480896
$107$	−2.20555	−0.213219	−0.106609	−	0.994301i	$-0.533999\pi$
$107$	−2.20555	−0.213219	−0.106609	+	0.994301i	$0.533999\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−15.2544	−1.45445
$111$	0	0
$112$	−19.1708	−1.81147
$113$	2.17081	0.204212	0.102106	−	0.994774i	$-0.467442\pi$
$113$	2.17081	0.204212	0.102106	+	0.994774i	$0.467442\pi$
$114$	0	0
$115$	3.49472	0.325884
$116$	25.6655	2.38298
$117$	0	0
$118$	12.8816	1.18585
$119$	1.62721	0.149166
$120$	0	0
$121$	6.68665	0.607877
$122$	−14.7839	−1.33847
$123$	0	0
$124$	−44.3416	−3.98199
$125$	−10.7491	−0.961433
$126$	0	0
$127$	−0.745574	−0.0661590	−0.0330795	−	0.999453i	$-0.510531\pi$
$127$	−0.745574	−0.0661590	−0.0330795	+	0.999453i	$0.510531\pi$
$128$	80.8591	7.14700
$129$	0	0
$130$	−3.62721	−0.318128
$131$	14.5089	1.26764	0.633822	−	0.773479i	$-0.281485\pi$
$131$	14.5089	1.26764	0.633822	+	0.773479i	$0.281485\pi$
$132$	0	0
$133$	6.33804	0.549578
$134$	24.4111	2.10880
$135$	0	0
$136$	−17.9305	−1.53753
$137$	−4.41110	−0.376866	−0.188433	−	0.982086i	$-0.560341\pi$
$137$	−4.41110	−0.376866	−0.188433	+	0.982086i	$0.560341\pi$
$138$	0	0
$139$	18.9894	1.61066	0.805332	−	0.592825i	$-0.201987\pi$
$139$	18.9894	1.61066	0.805332	+	0.592825i	$0.201987\pi$
$140$	−7.62721	−0.644617
$141$	0	0
$142$	19.0872	1.60176
$143$	4.20555	0.351686
$144$	0	0
$145$	5.59247	0.464429
$146$	11.4600	0.948434
$147$	0	0
$148$	20.2439	1.66404
$149$	9.42166	0.771853	0.385926	−	0.922530i	$-0.373882\pi$
$149$	9.42166	0.771853	0.385926	+	0.922530i	$0.373882\pi$
$150$	0	0
$151$	−16.4111	−1.33552	−0.667758	−	0.744378i	$-0.732746\pi$
$151$	−16.4111	−1.33552	−0.667758	+	0.744378i	$0.732746\pi$
$152$	−69.8399	−5.66476
$153$	0	0
$154$	11.8328	0.953511
$155$	−9.66196	−0.776067
$156$	0	0
$157$	−9.51941	−0.759732	−0.379866	−	0.925042i	$-0.624030\pi$
$157$	−9.51941	−0.759732	−0.379866	+	0.925042i	$0.624030\pi$
$158$	−25.0872	−1.99583
$159$	0	0
$160$	41.1255	3.25126
$161$	−2.71083	−0.213643
$162$	0	0
$163$	−0.745574	−0.0583979	−0.0291989	−	0.999574i	$-0.509296\pi$
$163$	−0.745574	−0.0583979	−0.0291989	+	0.999574i	$0.509296\pi$
$164$	45.1255	3.52371
$165$	0	0
$166$	12.2056	0.947334
$167$	−3.59247	−0.277994	−0.138997	−	0.990293i	$-0.544388\pi$
$167$	−3.59247	−0.277994	−0.138997	+	0.990293i	$0.544388\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−5.90225	−0.452682
$171$	0	0
$172$	−29.0872	−2.21788
$173$	15.7250	1.19555	0.597773	−	0.801665i	$-0.296052\pi$
$173$	15.7250	1.19555	0.597773	+	0.801665i	$0.296052\pi$
$174$	0	0
$175$	3.33804	0.252332
$176$	−80.6238	−6.07725
$177$	0	0
$178$	−12.7839	−0.958193
$179$	−24.9547	−1.86520	−0.932601	−	0.360910i	$-0.882466\pi$
$179$	−24.9547	−1.86520	−0.932601	+	0.360910i	$0.882466\pi$
$180$	0	0
$181$	10.9411	0.813244	0.406622	−	0.913597i	$-0.366707\pi$
$181$	10.9411	0.813244	0.406622	+	0.913597i	$0.366707\pi$
$182$	2.81361	0.208558
$183$	0	0
$184$	29.8711	2.20212
$185$	4.41110	0.324311
$186$	0	0
$187$	6.84333	0.500434
$188$	6.41110	0.467578
$189$	0	0
$190$	−22.9894	−1.66783
$191$	−5.04888	−0.365324	−0.182662	−	0.983176i	$-0.558471\pi$
$191$	−5.04888	−0.365324	−0.182662	+	0.983176i	$0.558471\pi$
$192$	0	0
$193$	2.74557	0.197631	0.0988154	−	0.995106i	$-0.468495\pi$
$193$	2.74557	0.197631	0.0988154	+	0.995106i	$0.468495\pi$
$194$	31.8711	2.28821
$195$	0	0
$196$	5.91638	0.422599
$197$	−5.42166	−0.386277	−0.193139	−	0.981171i	$-0.561867\pi$
$197$	−5.42166	−0.386277	−0.193139	+	0.981171i	$0.561867\pi$
$198$	0	0
$199$	−20.4111	−1.44690	−0.723452	−	0.690374i	$-0.757446\pi$
$199$	−20.4111	−1.44690	−0.723452	+	0.690374i	$0.757446\pi$
$200$	−36.7824	−2.60091
$201$	0	0
$202$	−40.2439	−2.83155
$203$	−4.33804	−0.304471
$204$	0	0
$205$	9.83276	0.686750
$206$	−55.3311	−3.85510
$207$	0	0
$208$	−19.1708	−1.32926
$209$	26.6550	1.84376
$210$	0	0
$211$	−14.4842	−0.997130	−0.498565	−	0.866852i	$-0.666140\pi$
$211$	−14.4842	−0.997130	−0.498565	+	0.866852i	$0.666140\pi$
$212$	−10.4111	−0.715037
$213$	0	0
$214$	−6.20555	−0.424203
$215$	−6.33804	−0.432251
$216$	0	0
$217$	7.49472	0.508775
$218$	28.1361	1.90561
$219$	0	0
$220$	−32.0766	−2.16261
$221$	1.62721	0.109458
$222$	0	0
$223$	16.8469	1.12815	0.564076	−	0.825723i	$-0.309233\pi$
$223$	16.8469	1.12815	0.564076	+	0.825723i	$0.309233\pi$
$224$	−31.9008	−2.13146
$225$	0	0
$226$	6.10780	0.406285
$227$	21.9305	1.45558	0.727790	−	0.685800i	$-0.240548\pi$
$227$	21.9305	1.45558	0.727790	+	0.685800i	$0.240548\pi$
$228$	0	0
$229$	−5.25443	−0.347222	−0.173611	−	0.984814i	$-0.555544\pi$
$229$	−5.25443	−0.347222	−0.173611	+	0.984814i	$0.555544\pi$
$230$	9.83276	0.648353
$231$	0	0
$232$	47.8016	3.13833
$233$	−4.07306	−0.266835	−0.133417	−	0.991060i	$-0.542595\pi$
$233$	−4.07306	−0.266835	−0.133417	+	0.991060i	$0.542595\pi$
$234$	0	0
$235$	1.39697	0.0911281
$236$	27.0872	1.76323
$237$	0	0
$238$	4.57834	0.296770
$239$	2.37279	0.153483	0.0767414	−	0.997051i	$-0.475548\pi$
$239$	2.37279	0.153483	0.0767414	+	0.997051i	$0.475548\pi$
$240$	0	0
$241$	14.9164	0.960849	0.480424	−	0.877036i	$-0.340483\pi$
$241$	14.9164	0.960849	0.480424	+	0.877036i	$0.340483\pi$
$242$	18.8136	1.20938
$243$	0	0
$244$	−31.0872	−1.99015
$245$	1.28917	0.0823620
$246$	0	0
$247$	6.33804	0.403280
$248$	−82.5855	−5.24418
$249$	0	0
$250$	−30.2439	−1.91279
$251$	1.08719	0.0686228	0.0343114	−	0.999411i	$-0.489076\pi$
$251$	1.08719	0.0686228	0.0343114	+	0.999411i	$0.489076\pi$
$252$	0	0
$253$	−11.4005	−0.716746
$254$	−2.09775	−0.131625
$255$	0	0
$256$	124.676	7.79226
$257$	16.6167	1.03652	0.518259	−	0.855224i	$-0.326580\pi$
$257$	16.6167	1.03652	0.518259	+	0.855224i	$0.326580\pi$
$258$	0	0
$259$	−3.42166	−0.212612
$260$	−7.62721	−0.473020
$261$	0	0
$262$	40.8222	2.52200
$263$	27.0524	1.66813	0.834063	−	0.551670i	$-0.186009\pi$
$263$	27.0524	1.66813	0.834063	+	0.551670i	$0.186009\pi$
$264$	0	0
$265$	−2.26856	−0.139356
$266$	17.8328	1.09340
$267$	0	0
$268$	51.3311	3.13554
$269$	−7.52946	−0.459079	−0.229540	−	0.973299i	$-0.573722\pi$
$269$	−7.52946	−0.459079	−0.229540	+	0.973299i	$0.573722\pi$
$270$	0	0
$271$	16.8222	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$271$	16.8222	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$272$	−31.1950	−1.89147
$273$	0	0
$274$	−12.4111	−0.749782
$275$	14.0383	0.846542
$276$	0	0
$277$	17.4947	1.05116	0.525578	−	0.850745i	$-0.323849\pi$
$277$	17.4947	1.05116	0.525578	+	0.850745i	$0.323849\pi$
$278$	53.4288	3.20445
$279$	0	0
$280$	−14.2056	−0.848944
$281$	−27.4005	−1.63458	−0.817290	−	0.576227i	$-0.804524\pi$
$281$	−27.4005	−1.63458	−0.817290	+	0.576227i	$0.804524\pi$
$282$	0	0
$283$	20.0766	1.19343	0.596716	−	0.802453i	$-0.296472\pi$
$283$	20.0766	1.19343	0.596716	+	0.802453i	$0.296472\pi$
$284$	40.1361	2.38164
$285$	0	0
$286$	11.8328	0.699686
$287$	−7.62721	−0.450220
$288$	0	0
$289$	−14.3522	−0.844246
$290$	15.7350	0.923992
$291$	0	0
$292$	24.0978	1.41021
$293$	27.0524	1.58042	0.790210	−	0.612836i	$-0.209971\pi$
$293$	27.0524	1.58042	0.790210	+	0.612836i	$0.209971\pi$
$294$	0	0
$295$	5.90225	0.343642
$296$	37.7038	2.19149
$297$	0	0
$298$	26.5089	1.53562
$299$	−2.71083	−0.156771
$300$	0	0
$301$	4.91638	0.283376
$302$	−46.1744	−2.65704
$303$	0	0
$304$	−121.505	−6.96881
$305$	−6.77384	−0.387869
$306$	0	0
$307$	−20.1708	−1.15121	−0.575604	−	0.817728i	$-0.695233\pi$
$307$	−20.1708	−1.15121	−0.575604	+	0.817728i	$0.695233\pi$
$308$	24.8816	1.41776
$309$	0	0
$310$	−27.1849	−1.54400
$311$	6.57834	0.373023	0.186512	−	0.982453i	$-0.440282\pi$
$311$	6.57834	0.373023	0.186512	+	0.982453i	$0.440282\pi$
$312$	0	0
$313$	−10.7456	−0.607376	−0.303688	−	0.952772i	$-0.598218\pi$
$313$	−10.7456	−0.607376	−0.303688	+	0.952772i	$0.598218\pi$
$314$	−26.7839	−1.51150
$315$	0	0
$316$	−52.7527	−2.96757
$317$	−24.6066	−1.38204	−0.691022	−	0.722833i	$-0.742839\pi$
$317$	−24.6066	−1.38204	−0.691022	+	0.722833i	$0.742839\pi$
$318$	0	0
$319$	−18.2439	−1.02146
$320$	66.2822	3.70529
$321$	0	0
$322$	−7.62721	−0.425048
$323$	10.3133	0.573850
$324$	0	0
$325$	3.33804	0.185161
$326$	−2.09775	−0.116184
$327$	0	0
$328$	84.0455	4.64063
$329$	−1.08362	−0.0597418
$330$	0	0
$331$	30.6550	1.68495	0.842475	−	0.538736i	$-0.181098\pi$
$331$	30.6550	1.68495	0.842475	+	0.538736i	$0.181098\pi$
$332$	25.6655	1.40858
$333$	0	0
$334$	−10.1078	−0.553074
$335$	11.1849	0.611099
$336$	0	0
$337$	−6.50528	−0.354365	−0.177183	−	0.984178i	$-0.556698\pi$
$337$	−6.50528	−0.354365	−0.177183	+	0.984178i	$0.556698\pi$
$338$	2.81361	0.153040
$339$	0	0
$340$	−12.4111	−0.673086
$341$	31.5194	1.70687
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−54.1744	−2.92089
$345$	0	0
$346$	44.2439	2.37856
$347$	−24.3033	−1.30467	−0.652335	−	0.757931i	$-0.726210\pi$
$347$	−24.3033	−1.30467	−0.652335	+	0.757931i	$0.726210\pi$
$348$	0	0
$349$	17.7597	0.950655	0.475328	−	0.879809i	$-0.342330\pi$
$349$	17.7597	0.950655	0.475328	+	0.879809i	$0.342330\pi$
$350$	9.39194	0.502021
$351$	0	0
$352$	−134.160	−7.15077
$353$	−1.94056	−0.103286	−0.0516428	−	0.998666i	$-0.516446\pi$
$353$	−1.94056	−0.103286	−0.0516428	+	0.998666i	$0.516446\pi$
$354$	0	0
$355$	8.74557	0.464167
$356$	−26.8816	−1.42472
$357$	0	0
$358$	−70.2127	−3.71086
$359$	33.6272	1.77478	0.887388	−	0.461023i	$-0.152517\pi$
$359$	33.6272	1.77478	0.887388	+	0.461023i	$0.152517\pi$
$360$	0	0
$361$	21.1708	1.11425
$362$	30.7839	1.61797
$363$	0	0
$364$	5.91638	0.310103
$365$	5.25086	0.274842
$366$	0	0
$367$	−10.5783	−0.552185	−0.276092	−	0.961131i	$-0.589040\pi$
$367$	−10.5783	−0.552185	−0.276092	+	0.961131i	$0.589040\pi$
$368$	51.9688	2.70906
$369$	0	0
$370$	12.4111	0.645222
$371$	1.75971	0.0913595
$372$	0	0
$373$	−5.47002	−0.283227	−0.141614	−	0.989922i	$-0.545229\pi$
$373$	−5.47002	−0.283227	−0.141614	+	0.989922i	$0.545229\pi$
$374$	19.2544	0.995623
$375$	0	0
$376$	11.9406	0.615787
$377$	−4.33804	−0.223421
$378$	0	0
$379$	−5.42166	−0.278492	−0.139246	−	0.990258i	$-0.544468\pi$
$379$	−5.42166	−0.278492	−0.139246	+	0.990258i	$0.544468\pi$
$380$	−48.3416	−2.47987
$381$	0	0
$382$	−14.2056	−0.726819
$383$	−14.1461	−0.722833	−0.361416	−	0.932405i	$-0.617707\pi$
$383$	−14.1461	−0.722833	−0.361416	+	0.932405i	$0.617707\pi$
$384$	0	0
$385$	5.42166	0.276314
$386$	7.72496	0.393190
$387$	0	0
$388$	67.0177	3.40231
$389$	−0.313348	−0.0158874	−0.00794370	−	0.999968i	$-0.502529\pi$
$389$	−0.313348	−0.0158874	−0.00794370	+	0.999968i	$0.502529\pi$
$390$	0	0
$391$	−4.41110	−0.223079
$392$	11.0192	0.556551
$393$	0	0
$394$	−15.2544	−0.768507
$395$	−11.4947	−0.578362
$396$	0	0
$397$	20.6797	1.03788	0.518941	−	0.854810i	$-0.326326\pi$
$397$	20.6797	1.03788	0.518941	+	0.854810i	$0.326326\pi$
$398$	−57.4288	−2.87865
$399$	0	0
$400$	−63.9930	−3.19965
$401$	12.7456	0.636484	0.318242	−	0.948010i	$-0.396908\pi$
$401$	12.7456	0.636484	0.318242	+	0.948010i	$0.396908\pi$
$402$	0	0
$403$	7.49472	0.373339
$404$	−84.6238	−4.21019
$405$	0	0
$406$	−12.2056	−0.605751
$407$	−14.3900	−0.713285
$408$	0	0
$409$	−18.6514	−0.922252	−0.461126	−	0.887335i	$-0.652554\pi$
$409$	−18.6514	−0.922252	−0.461126	+	0.887335i	$0.652554\pi$
$410$	27.6655	1.36630
$411$	0	0
$412$	−116.349	−5.73209
$413$	−4.57834	−0.225285
$414$	0	0
$415$	5.59247	0.274524
$416$	−31.9008	−1.56407
$417$	0	0
$418$	74.9966	3.66820
$419$	−3.10831	−0.151851	−0.0759256	−	0.997113i	$-0.524191\pi$
$419$	−3.10831	−0.151851	−0.0759256	+	0.997113i	$0.524191\pi$
$420$	0	0
$421$	13.4005	0.653102	0.326551	−	0.945180i	$-0.394113\pi$
$421$	13.4005	0.653102	0.326551	+	0.945180i	$0.394113\pi$
$422$	−40.7527	−1.98381
$423$	0	0
$424$	−19.3905	−0.941685
$425$	5.43171	0.263477
$426$	0	0
$427$	5.25443	0.254279
$428$	−13.0489	−0.630741
$429$	0	0
$430$	−17.8328	−0.859972
$431$	−9.89220	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$431$	−9.89220	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$432$	0	0
$433$	0.578337	0.0277931	0.0138966	−	0.999903i	$-0.495576\pi$
$433$	0.578337	0.0277931	0.0138966	+	0.999903i	$0.495576\pi$
$434$	21.0872	1.01222
$435$	0	0
$436$	59.1638	2.83343
$437$	−17.1814	−0.821896
$438$	0	0
$439$	23.7350	1.13281	0.566405	−	0.824127i	$-0.308334\pi$
$439$	23.7350	1.13281	0.566405	+	0.824127i	$0.308334\pi$
$440$	−59.7422	−2.84810
$441$	0	0
$442$	4.57834	0.217769
$443$	3.86751	0.183751	0.0918754	−	0.995771i	$-0.470714\pi$
$443$	3.86751	0.183751	0.0918754	+	0.995771i	$0.470714\pi$
$444$	0	0
$445$	−5.85746	−0.277670
$446$	47.4005	2.24448
$447$	0	0
$448$	−51.4147	−2.42911
$449$	−1.68665	−0.0795980	−0.0397990	−	0.999208i	$-0.512672\pi$
$449$	−1.68665	−0.0795980	−0.0397990	+	0.999208i	$0.512672\pi$
$450$	0	0
$451$	−32.0766	−1.51043
$452$	12.8433	0.604099
$453$	0	0
$454$	61.7038	2.89590
$455$	1.28917	0.0604372
$456$	0	0
$457$	−3.83276	−0.179289	−0.0896445	−	0.995974i	$-0.528573\pi$
$457$	−3.83276	−0.179289	−0.0896445	+	0.995974i	$0.528573\pi$
$458$	−14.7839	−0.690806
$459$	0	0
$460$	20.6761	0.964028
$461$	11.2161	0.522386	0.261193	−	0.965287i	$-0.415884\pi$
$461$	11.2161	0.522386	0.261193	+	0.965287i	$0.415884\pi$
$462$	0	0
$463$	−25.8328	−1.20055	−0.600275	−	0.799794i	$-0.704942\pi$
$463$	−25.8328	−1.20055	−0.600275	+	0.799794i	$0.704942\pi$
$464$	83.1638	3.86078
$465$	0	0
$466$	−11.4600	−0.530873
$467$	−41.0872	−1.90129	−0.950644	−	0.310283i	$-0.899576\pi$
$467$	−41.0872	−1.90129	−0.950644	+	0.310283i	$0.899576\pi$
$468$	0	0
$469$	−8.67609	−0.400625
$470$	3.93051	0.181301
$471$	0	0
$472$	50.4494	2.32212
$473$	20.6761	0.950688
$474$	0	0
$475$	21.1567	0.970735
$476$	9.62721	0.441263
$477$	0	0
$478$	6.67609	0.305357
$479$	−2.43580	−0.111294	−0.0556472	−	0.998450i	$-0.517722\pi$
$479$	−2.43580	−0.111294	−0.0556472	+	0.998450i	$0.517722\pi$
$480$	0	0
$481$	−3.42166	−0.156014
$482$	41.9688	1.91163
$483$	0	0
$484$	39.5608	1.79822
$485$	14.6030	0.663090
$486$	0	0
$487$	−6.57834	−0.298093	−0.149046	−	0.988830i	$-0.547620\pi$
$487$	−6.57834	−0.298093	−0.149046	+	0.988830i	$0.547620\pi$
$488$	−57.8993	−2.62098
$489$	0	0
$490$	3.62721	0.163861
$491$	−28.3033	−1.27731	−0.638655	−	0.769493i	$-0.720509\pi$
$491$	−28.3033	−1.27731	−0.638655	+	0.769493i	$0.720509\pi$
$492$	0	0
$493$	−7.05892	−0.317918
$494$	17.8328	0.802334
$495$	0	0
$496$	−143.680	−6.45141
$497$	−6.78389	−0.304299
$498$	0	0
$499$	28.8222	1.29026	0.645129	−	0.764073i	$-0.276803\pi$
$499$	28.8222	1.29026	0.645129	+	0.764073i	$0.276803\pi$
$500$	−63.5960	−2.84410
$501$	0	0
$502$	3.05892	0.136526
$503$	−8.67609	−0.386848	−0.193424	−	0.981115i	$-0.561959\pi$
$503$	−8.67609	−0.386848	−0.193424	+	0.981115i	$0.561959\pi$
$504$	0	0
$505$	−18.4394	−0.820541
$506$	−32.0766	−1.42598
$507$	0	0
$508$	−4.41110	−0.195711
$509$	9.28917	0.411735	0.205868	−	0.978580i	$-0.433998\pi$
$509$	9.28917	0.411735	0.205868	+	0.978580i	$0.433998\pi$
$510$	0	0
$511$	−4.07306	−0.180181
$512$	189.072	8.35587
$513$	0	0
$514$	46.7527	2.06217
$515$	−25.3522	−1.11715
$516$	0	0
$517$	−4.55721	−0.200426
$518$	−9.62721	−0.422995
$519$	0	0
$520$	−14.2056	−0.622955
$521$	11.7250	0.513680	0.256840	−	0.966454i	$-0.417319\pi$
$521$	11.7250	0.513680	0.256840	+	0.966454i	$0.417319\pi$
$522$	0	0
$523$	0.676089	0.0295633	0.0147817	−	0.999891i	$-0.495295\pi$
$523$	0.676089	0.0295633	0.0147817	+	0.999891i	$0.495295\pi$
$524$	85.8399	3.74993
$525$	0	0
$526$	76.1149	3.31877
$527$	12.1955	0.531244
$528$	0	0
$529$	−15.6514	−0.680495
$530$	−6.38283	−0.277253
$531$	0	0
$532$	37.4983	1.62576
$533$	−7.62721	−0.330371
$534$	0	0
$535$	−2.84333	−0.122928
$536$	95.6032	4.12943
$537$	0	0
$538$	−21.1849	−0.913348
$539$	−4.20555	−0.181146
$540$	0	0
$541$	−0.578337	−0.0248647	−0.0124323	−	0.999923i	$-0.503957\pi$
$541$	−0.578337	−0.0248647	−0.0124323	+	0.999923i	$0.503957\pi$
$542$	47.3311	2.03304
$543$	0	0
$544$	−51.9094	−2.22560
$545$	12.8917	0.552219
$546$	0	0
$547$	−0.985867	−0.0421526	−0.0210763	−	0.999778i	$-0.506709\pi$
$547$	−0.985867	−0.0421526	−0.0210763	+	0.999778i	$0.506709\pi$
$548$	−26.0978	−1.11484
$549$	0	0
$550$	39.4983	1.68421
$551$	−27.4947	−1.17131
$552$	0	0
$553$	8.91638	0.379163
$554$	49.2233	2.09130
$555$	0	0
$556$	112.349	4.76465
$557$	−41.2333	−1.74711	−0.873556	−	0.486725i	$-0.838192\pi$
$557$	−41.2333	−1.74711	−0.873556	+	0.486725i	$0.838192\pi$
$558$	0	0
$559$	4.91638	0.207941
$560$	−24.7144	−1.04437
$561$	0	0
$562$	−77.0943	−3.25203
$563$	−6.91995	−0.291641	−0.145821	−	0.989311i	$-0.546582\pi$
$563$	−6.91995	−0.291641	−0.145821	+	0.989311i	$0.546582\pi$
$564$	0	0
$565$	2.79854	0.117735
$566$	56.4877	2.37436
$567$	0	0
$568$	74.7527	3.13655
$569$	29.5713	1.23970	0.619848	−	0.784722i	$-0.287194\pi$
$569$	29.5713	1.23970	0.619848	+	0.784722i	$0.287194\pi$
$570$	0	0
$571$	−19.4252	−0.812921	−0.406460	−	0.913668i	$-0.633237\pi$
$571$	−19.4252	−0.812921	−0.406460	+	0.913668i	$0.633237\pi$
$572$	24.8816	1.04035
$573$	0	0
$574$	−21.4600	−0.895722
$575$	−9.04888	−0.377364
$576$	0	0
$577$	−38.8222	−1.61619	−0.808095	−	0.589053i	$-0.799501\pi$
$577$	−38.8222	−1.61619	−0.808095	+	0.589053i	$0.799501\pi$
$578$	−40.3814	−1.67964
$579$	0	0
$580$	33.0872	1.37387
$581$	−4.33804	−0.179972
$582$	0	0
$583$	7.40054	0.306499
$584$	44.8816	1.85722
$585$	0	0
$586$	76.1149	3.14428
$587$	−24.0731	−0.993601	−0.496801	−	0.867865i	$-0.665492\pi$
$587$	−24.0731	−0.993601	−0.496801	+	0.867865i	$0.665492\pi$
$588$	0	0
$589$	47.5019	1.95728
$590$	16.6066	0.683683
$591$	0	0
$592$	65.5960	2.69598
$593$	−40.4741	−1.66207	−0.831036	−	0.556218i	$-0.812252\pi$
$593$	−40.4741	−1.66207	−0.831036	+	0.556218i	$0.812252\pi$
$594$	0	0
$595$	2.09775	0.0859994
$596$	55.7422	2.28329
$597$	0	0
$598$	−7.62721	−0.311900
$599$	2.49523	0.101953	0.0509763	−	0.998700i	$-0.483767\pi$
$599$	2.49523	0.101953	0.0509763	+	0.998700i	$0.483767\pi$
$600$	0	0
$601$	35.2333	1.43720	0.718598	−	0.695426i	$-0.244784\pi$
$601$	35.2333	1.43720	0.718598	+	0.695426i	$0.244784\pi$
$602$	13.8328	0.563781
$603$	0	0
$604$	−97.0943	−3.95071
$605$	8.62022	0.350462
$606$	0	0
$607$	4.89169	0.198547	0.0992737	−	0.995060i	$-0.468348\pi$
$607$	4.89169	0.198547	0.0992737	+	0.995060i	$0.468348\pi$
$608$	−202.189	−8.19983
$609$	0	0
$610$	−19.0589	−0.771673
$611$	−1.08362	−0.0438385
$612$	0	0
$613$	7.83276	0.316362	0.158181	−	0.987410i	$-0.449437\pi$
$613$	7.83276	0.316362	0.158181	+	0.987410i	$0.449437\pi$
$614$	−56.7527	−2.29035
$615$	0	0
$616$	46.3416	1.86716
$617$	−15.7350	−0.633468	−0.316734	−	0.948514i	$-0.602586\pi$
$617$	−15.7350	−0.633468	−0.316734	+	0.948514i	$0.602586\pi$
$618$	0	0
$619$	−41.0177	−1.64864	−0.824320	−	0.566124i	$-0.808442\pi$
$619$	−41.0177	−1.64864	−0.824320	+	0.566124i	$0.808442\pi$
$620$	−57.1638	−2.29575
$621$	0	0
$622$	18.5089	0.742137
$623$	4.54359	0.182035
$624$	0	0
$625$	2.83276	0.113311
$626$	−30.2338	−1.20839
$627$	0	0
$628$	−56.3205	−2.24743
$629$	−5.56777	−0.222002
$630$	0	0
$631$	−19.5960	−0.780106	−0.390053	−	0.920792i	$-0.627543\pi$
$631$	−19.5960	−0.780106	−0.390053	+	0.920792i	$0.627543\pi$
$632$	−98.2510	−3.90822
$633$	0	0
$634$	−69.2333	−2.74961
$635$	−0.961171	−0.0381429
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−51.3311	−2.03222
$639$	0	0
$640$	104.241	4.12049
$641$	20.8953	0.825313	0.412656	−	0.910887i	$-0.364601\pi$
$641$	20.8953	0.825313	0.412656	+	0.910887i	$0.364601\pi$
$642$	0	0
$643$	−9.15667	−0.361104	−0.180552	−	0.983565i	$-0.557788\pi$
$643$	−9.15667	−0.361104	−0.180552	+	0.983565i	$0.557788\pi$
$644$	−16.0383	−0.631998
$645$	0	0
$646$	29.0177	1.14169
$647$	6.24386	0.245472	0.122736	−	0.992439i	$-0.460833\pi$
$647$	6.24386	0.245472	0.122736	+	0.992439i	$0.460833\pi$
$648$	0	0
$649$	−19.2544	−0.755802
$650$	9.39194	0.368382
$651$	0	0
$652$	−4.41110	−0.172752
$653$	38.8222	1.51923	0.759615	−	0.650373i	$-0.225387\pi$
$653$	38.8222	1.51923	0.759615	+	0.650373i	$0.225387\pi$
$654$	0	0
$655$	18.7044	0.730840
$656$	146.220	5.70893
$657$	0	0
$658$	−3.04888	−0.118858
$659$	0.954695	0.0371896	0.0185948	−	0.999827i	$-0.494081\pi$
$659$	0.954695	0.0371896	0.0185948	+	0.999827i	$0.494081\pi$
$660$	0	0
$661$	2.28968	0.0890584	0.0445292	−	0.999008i	$-0.485821\pi$
$661$	2.28968	0.0890584	0.0445292	+	0.999008i	$0.485821\pi$
$662$	86.2510	3.35224
$663$	0	0
$664$	47.8016	1.85506
$665$	8.17081	0.316850
$666$	0	0
$667$	11.7597	0.455338
$668$	−21.2544	−0.822358
$669$	0	0
$670$	31.4700	1.21579
$671$	22.0978	0.853074
$672$	0	0
$673$	−44.6797	−1.72227	−0.861137	−	0.508373i	$-0.830247\pi$
$673$	−44.6797	−1.72227	−0.861137	+	0.508373i	$0.830247\pi$
$674$	−18.3033	−0.705017
$675$	0	0
$676$	5.91638	0.227553
$677$	37.0278	1.42309	0.711546	−	0.702639i	$-0.247995\pi$
$677$	37.0278	1.42309	0.711546	+	0.702639i	$0.247995\pi$
$678$	0	0
$679$	−11.3275	−0.434709
$680$	−23.1155	−0.886437
$681$	0	0
$682$	88.6832	3.39586
$683$	26.8605	1.02779	0.513894	−	0.857853i	$-0.328202\pi$
$683$	26.8605	1.02779	0.513894	+	0.857853i	$0.328202\pi$
$684$	0	0
$685$	−5.68665	−0.217276
$686$	−2.81361	−0.107424
$687$	0	0
$688$	−94.2510	−3.59329
$689$	1.75971	0.0670395
$690$	0	0
$691$	2.86802	0.109105	0.0545523	−	0.998511i	$-0.482627\pi$
$691$	2.86802	0.109105	0.0545523	+	0.998511i	$0.482627\pi$
$692$	93.0349	3.53666
$693$	0	0
$694$	−68.3799	−2.59567
$695$	24.4806	0.928602
$696$	0	0
$697$	−12.4111	−0.470104
$698$	49.9688	1.89135
$699$	0	0
$700$	19.7491	0.746448
$701$	42.9930	1.62382	0.811912	−	0.583780i	$-0.198427\pi$
$701$	42.9930	1.62382	0.811912	+	0.583780i	$0.198427\pi$
$702$	0	0
$703$	−21.6867	−0.817928
$704$	−216.227	−8.14936
$705$	0	0
$706$	−5.45998	−0.205489
$707$	14.3033	0.537931
$708$	0	0
$709$	3.49115	0.131113	0.0655564	−	0.997849i	$-0.479118\pi$
$709$	3.49115	0.131113	0.0655564	+	0.997849i	$0.479118\pi$
$710$	24.6066	0.923469
$711$	0	0
$712$	−50.0666	−1.87632
$713$	−20.3169	−0.760875
$714$	0	0
$715$	5.42166	0.202759
$716$	−147.641	−5.51762
$717$	0	0
$718$	94.6137	3.53095
$719$	−27.5194	−1.02630	−0.513150	−	0.858299i	$-0.671522\pi$
$719$	−27.5194	−1.02630	−0.513150	+	0.858299i	$0.671522\pi$
$720$	0	0
$721$	19.6655	0.732382
$722$	59.5663	2.21683
$723$	0	0
$724$	64.7316	2.40573
$725$	−14.4806	−0.537795
$726$	0	0
$727$	13.4983	0.500624	0.250312	−	0.968165i	$-0.419467\pi$
$727$	13.4983	0.500624	0.250312	+	0.968165i	$0.419467\pi$
$728$	11.0192	0.408397
$729$	0	0
$730$	14.7738	0.546804
$731$	8.00000	0.295891
$732$	0	0
$733$	−5.08362	−0.187768	−0.0938839	−	0.995583i	$-0.529928\pi$
$733$	−5.08362	−0.187768	−0.0938839	+	0.995583i	$0.529928\pi$
$734$	−29.7633	−1.09858
$735$	0	0
$736$	86.4777	3.18761
$737$	−36.4877	−1.34404
$738$	0	0
$739$	−53.1638	−1.95566	−0.977831	−	0.209394i	$-0.932851\pi$
$739$	−53.1638	−1.95566	−0.977831	+	0.209394i	$0.932851\pi$
$740$	26.0978	0.959372
$741$	0	0
$742$	4.95112	0.181761
$743$	46.1149	1.69179	0.845897	−	0.533347i	$-0.179066\pi$
$743$	46.1149	1.69179	0.845897	+	0.533347i	$0.179066\pi$
$744$	0	0
$745$	12.1461	0.444999
$746$	−15.3905	−0.563486
$747$	0	0
$748$	40.4877	1.48038
$749$	2.20555	0.0805890
$750$	0	0
$751$	−32.7774	−1.19606	−0.598032	−	0.801472i	$-0.704051\pi$
$751$	−32.7774	−1.19606	−0.598032	+	0.801472i	$0.704051\pi$
$752$	20.7738	0.757544
$753$	0	0
$754$	−12.2056	−0.444500
$755$	−21.1567	−0.769970
$756$	0	0
$757$	−30.1708	−1.09658	−0.548288	−	0.836289i	$-0.684720\pi$
$757$	−30.1708	−1.09658	−0.548288	+	0.836289i	$0.684720\pi$
$758$	−15.2544	−0.554066
$759$	0	0
$760$	−90.0354	−3.26593
$761$	1.81915	0.0659440	0.0329720	−	0.999456i	$-0.489503\pi$
$761$	1.81915	0.0659440	0.0329720	+	0.999456i	$0.489503\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−29.8711	−1.08070
$765$	0	0
$766$	−39.8016	−1.43809
$767$	−4.57834	−0.165314
$768$	0	0
$769$	−0.937507	−0.0338074	−0.0169037	−	0.999857i	$-0.505381\pi$
$769$	−0.937507	−0.0338074	−0.0169037	+	0.999857i	$0.505381\pi$
$770$	15.2544	0.549731
$771$	0	0
$772$	16.2439	0.584629
$773$	8.03831	0.289118	0.144559	−	0.989496i	$-0.453824\pi$
$773$	8.03831	0.289118	0.144559	+	0.989496i	$0.453824\pi$
$774$	0	0
$775$	25.0177	0.898662
$776$	124.819	4.48075
$777$	0	0
$778$	−0.881639	−0.0316083
$779$	−48.3416	−1.73202
$780$	0	0
$781$	−28.5300	−1.02088
$782$	−12.4111	−0.443820
$783$	0	0
$784$	19.1708	0.684672
$785$	−12.2721	−0.438011
$786$	0	0
$787$	25.3275	0.902827	0.451414	−	0.892315i	$-0.350920\pi$
$787$	25.3275	0.902827	0.451414	+	0.892315i	$0.350920\pi$
$788$	−32.0766	−1.14268
$789$	0	0
$790$	−32.3416	−1.15066
$791$	−2.17081	−0.0771850
$792$	0	0
$793$	5.25443	0.186590
$794$	58.1844	2.06489
$795$	0	0
$796$	−120.760	−4.28022
$797$	−40.8122	−1.44564	−0.722820	−	0.691036i	$-0.757155\pi$
$797$	−40.8122	−1.44564	−0.722820	+	0.691036i	$0.757155\pi$
$798$	0	0
$799$	−1.76328	−0.0623803
$800$	−106.486	−3.76486
$801$	0	0
$802$	35.8610	1.26630
$803$	−17.1294	−0.604485
$804$	0	0
$805$	−3.49472	−0.123173
$806$	21.0872	0.742765
$807$	0	0
$808$	−157.610	−5.54471
$809$	−25.4947	−0.896347	−0.448173	−	0.893947i	$-0.647925\pi$
$809$	−25.4947	−0.896347	−0.448173	+	0.893947i	$0.647925\pi$
$810$	0	0
$811$	−13.1567	−0.461993	−0.230997	−	0.972955i	$-0.574199\pi$
$811$	−13.1567	−0.461993	−0.230997	+	0.972955i	$0.574199\pi$
$812$	−25.6655	−0.900683
$813$	0	0
$814$	−40.4877	−1.41909
$815$	−0.961171	−0.0336683
$816$	0	0
$817$	31.1602	1.09016
$818$	−52.4777	−1.83484
$819$	0	0
$820$	58.1744	2.03154
$821$	23.1083	0.806486	0.403243	−	0.915093i	$-0.367883\pi$
$821$	23.1083	0.806486	0.403243	+	0.915093i	$0.367883\pi$
$822$	0	0
$823$	−17.4911	−0.609703	−0.304852	−	0.952400i	$-0.598607\pi$
$823$	−17.4911	−0.609703	−0.304852	+	0.952400i	$0.598607\pi$
$824$	−216.698	−7.54902
$825$	0	0
$826$	−12.8816	−0.448210
$827$	−39.3905	−1.36974	−0.684871	−	0.728665i	$-0.740141\pi$
$827$	−39.3905	−1.36974	−0.684871	+	0.728665i	$0.740141\pi$
$828$	0	0
$829$	10.8222	0.375871	0.187935	−	0.982181i	$-0.439820\pi$
$829$	10.8222	0.375871	0.187935	+	0.982181i	$0.439820\pi$
$830$	15.7350	0.546170
$831$	0	0
$832$	−51.4147	−1.78248
$833$	−1.62721	−0.0563796
$834$	0	0
$835$	−4.63130	−0.160273
$836$	157.701	5.45420
$837$	0	0
$838$	−8.74557	−0.302111
$839$	−4.57834	−0.158062	−0.0790309	−	0.996872i	$-0.525183\pi$
$839$	−4.57834	−0.158062	−0.0790309	+	0.996872i	$0.525183\pi$
$840$	0	0
$841$	−10.1814	−0.351082
$842$	37.7038	1.29936
$843$	0	0
$844$	−85.6938	−2.94970
$845$	1.28917	0.0443487
$846$	0	0
$847$	−6.68665	−0.229756
$848$	−33.7350	−1.15847
$849$	0	0
$850$	15.2827	0.524192
$851$	9.27555	0.317962
$852$	0	0
$853$	−54.6585	−1.87147	−0.935736	−	0.352700i	$-0.885263\pi$
$853$	−54.6585	−1.87147	−0.935736	+	0.352700i	$0.885263\pi$
$854$	14.7839	0.505894
$855$	0	0
$856$	−24.3033	−0.830670
$857$	−8.20555	−0.280296	−0.140148	−	0.990131i	$-0.544758\pi$
$857$	−8.20555	−0.280296	−0.140148	+	0.990131i	$0.544758\pi$
$858$	0	0
$859$	−3.20503	−0.109354	−0.0546772	−	0.998504i	$-0.517413\pi$
$859$	−3.20503	−0.109354	−0.0546772	+	0.998504i	$0.517413\pi$
$860$	−37.4983	−1.27868
$861$	0	0
$862$	−27.8328	−0.947988
$863$	21.7038	0.738807	0.369404	−	0.929269i	$-0.379562\pi$
$863$	21.7038	0.738807	0.369404	+	0.929269i	$0.379562\pi$
$864$	0	0
$865$	20.2721	0.689273
$866$	1.62721	0.0552949
$867$	0	0
$868$	44.3416	1.50505
$869$	37.4983	1.27204
$870$	0	0
$871$	−8.67609	−0.293978
$872$	110.192	3.73156
$873$	0	0
$874$	−48.3416	−1.63518
$875$	10.7491	0.363387
$876$	0	0
$877$	−38.5371	−1.30131	−0.650653	−	0.759375i	$-0.725505\pi$
$877$	−38.5371	−1.30131	−0.650653	+	0.759375i	$0.725505\pi$
$878$	66.7810	2.25375
$879$	0	0
$880$	−103.938	−3.50374
$881$	20.2056	0.680742	0.340371	−	0.940291i	$-0.389447\pi$
$881$	20.2056	0.680742	0.340371	+	0.940291i	$0.389447\pi$
$882$	0	0
$883$	5.49115	0.184792	0.0923959	−	0.995722i	$-0.470547\pi$
$883$	5.49115	0.184792	0.0923959	+	0.995722i	$0.470547\pi$
$884$	9.62721	0.323798
$885$	0	0
$886$	10.8816	0.365576
$887$	−45.7633	−1.53658	−0.768290	−	0.640102i	$-0.778892\pi$
$887$	−45.7633	−1.53658	−0.768290	+	0.640102i	$0.778892\pi$
$888$	0	0
$889$	0.745574	0.0250057
$890$	−16.4806	−0.552430
$891$	0	0
$892$	99.6727	3.33729
$893$	−6.86802	−0.229830
$894$	0	0
$895$	−32.1708	−1.07535
$896$	−80.8591	−2.70131
$897$	0	0
$898$	−4.74557	−0.158362
$899$	−32.5124	−1.08435
$900$	0	0
$901$	2.86342	0.0953943
$902$	−90.2510	−3.00503
$903$	0	0
$904$	23.9205	0.795583
$905$	14.1049	0.468863
$906$	0	0
$907$	6.67252	0.221557	0.110779	−	0.993845i	$-0.464666\pi$
$907$	6.67252	0.221557	0.110779	+	0.993845i	$0.464666\pi$
$908$	129.749	4.30588
$909$	0	0
$910$	3.62721	0.120241
$911$	−23.5330	−0.779684	−0.389842	−	0.920882i	$-0.627470\pi$
$911$	−23.5330	−0.779684	−0.389842	+	0.920882i	$0.627470\pi$
$912$	0	0
$913$	−18.2439	−0.603784
$914$	−10.7839	−0.356699
$915$	0	0
$916$	−31.0872	−1.02715
$917$	−14.5089	−0.479125
$918$	0	0
$919$	−28.1955	−0.930084	−0.465042	−	0.885289i	$-0.653961\pi$
$919$	−28.1955	−0.930084	−0.465042	+	0.885289i	$0.653961\pi$
$920$	38.5089	1.26960
$921$	0	0
$922$	31.5577	1.03930
$923$	−6.78389	−0.223294
$924$	0	0
$925$	−11.4217	−0.375542
$926$	−72.6832	−2.38852
$927$	0	0
$928$	138.387	4.54278
$929$	6.97582	0.228869	0.114435	−	0.993431i	$-0.463494\pi$
$929$	6.97582	0.228869	0.114435	+	0.993431i	$0.463494\pi$
$930$	0	0
$931$	−6.33804	−0.207721
$932$	−24.0978	−0.789348
$933$	0	0
$934$	−115.603	−3.78265
$935$	8.82220	0.288517
$936$	0	0
$937$	45.5960	1.48956	0.744779	−	0.667311i	$-0.232555\pi$
$937$	45.5960	1.48956	0.744779	+	0.667311i	$0.232555\pi$
$938$	−24.4111	−0.797051
$939$	0	0
$940$	8.26499	0.269574
$941$	52.8086	1.72151	0.860755	−	0.509019i	$-0.169992\pi$
$941$	52.8086	1.72151	0.860755	+	0.509019i	$0.169992\pi$
$942$	0	0
$943$	20.6761	0.673306
$944$	87.7704	2.85668
$945$	0	0
$946$	58.1744	1.89141
$947$	−8.73553	−0.283867	−0.141933	−	0.989876i	$-0.545332\pi$
$947$	−8.73553	−0.283867	−0.141933	+	0.989876i	$0.545332\pi$
$948$	0	0
$949$	−4.07306	−0.132217
$950$	59.5266	1.93130
$951$	0	0
$952$	17.9305	0.581131
$953$	−56.0036	−1.81413	−0.907067	−	0.420987i	$-0.861684\pi$
$953$	−56.0036	−1.81413	−0.907067	+	0.420987i	$0.861684\pi$
$954$	0	0
$955$	−6.50885	−0.210622
$956$	14.0383	0.454031
$957$	0	0
$958$	−6.85337	−0.221422
$959$	4.41110	0.142442
$960$	0	0
$961$	25.1708	0.811962
$962$	−9.62721	−0.310394
$963$	0	0
$964$	88.2510	2.84237
$965$	3.53951	0.113941
$966$	0	0
$967$	29.9094	0.961821	0.480911	−	0.876770i	$-0.340306\pi$
$967$	29.9094	0.961821	0.480911	+	0.876770i	$0.340306\pi$
$968$	73.6813	2.36821
$969$	0	0
$970$	41.0872	1.31923
$971$	−11.5194	−0.369676	−0.184838	−	0.982769i	$-0.559176\pi$
$971$	−11.5194	−0.369676	−0.184838	+	0.982769i	$0.559176\pi$
$972$	0	0
$973$	−18.9894	−0.608773
$974$	−18.5089	−0.593062
$975$	0	0
$976$	−100.732	−3.22434
$977$	36.4182	1.16512	0.582561	−	0.812787i	$-0.302051\pi$
$977$	36.4182	1.16512	0.582561	+	0.812787i	$0.302051\pi$
$978$	0	0
$979$	19.1083	0.610704
$980$	7.62721	0.243642
$981$	0	0
$982$	−79.6344	−2.54123
$983$	−17.6902	−0.564230	−0.282115	−	0.959381i	$-0.591036\pi$
$983$	−17.6902	−0.564230	−0.282115	+	0.959381i	$0.591036\pi$
$984$	0	0
$985$	−6.98944	−0.222702
$986$	−19.8610	−0.632504
$987$	0	0
$988$	37.4983	1.19298
$989$	−13.3275	−0.423789
$990$	0	0
$991$	−11.0589	−0.351298	−0.175649	−	0.984453i	$-0.556202\pi$
$991$	−11.0589	−0.351298	−0.175649	+	0.984453i	$0.556202\pi$
$992$	−239.087	−7.59104
$993$	0	0
$994$	−19.0872	−0.605409
$995$	−26.3133	−0.834189
$996$	0	0
$997$	−49.6727	−1.57315	−0.786575	−	0.617495i	$-0.788147\pi$
$997$	−49.6727	−1.57315	−0.786575	+	0.617495i	$0.788147\pi$
$998$	81.0943	2.56700
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.j.1.3		3
3.2	odd	2		273.2.a.d.1.1	✓	3
7.6	odd	2		5733.2.a.bc.1.3		3
12.11	even	2		4368.2.a.bq.1.2		3
15.14	odd	2		6825.2.a.bd.1.3		3
21.20	even	2		1911.2.a.n.1.1		3
39.38	odd	2		3549.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.d.1.1	✓	3	3.2	odd	2
819.2.a.j.1.3		3	1.1	even	1	trivial
1911.2.a.n.1.1		3	21.20	even	2
3549.2.a.t.1.3		3	39.38	odd	2
4368.2.a.bq.1.2		3	12.11	even	2
5733.2.a.bc.1.3		3	7.6	odd	2
6825.2.a.bd.1.3		3	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+2.81361 q^{2} +5.91638 q^{4} +1.28917 q^{5} -1.00000 q^{7} +11.0192 q^{8} +O(q^{10})\)	\(q+2.81361 q^{2} +5.91638 q^{4} +1.28917 q^{5} -1.00000 q^{7} +11.0192 q^{8} +3.62721 q^{10} -4.20555 q^{11} -1.00000 q^{13} -2.81361 q^{14} +19.1708 q^{16} -1.62721 q^{17} -6.33804 q^{19} +7.62721 q^{20} -11.8328 q^{22} +2.71083 q^{23} -3.33804 q^{25} -2.81361 q^{26} -5.91638 q^{28} +4.33804 q^{29} -7.49472 q^{31} +31.9008 q^{32} -4.57834 q^{34} -1.28917 q^{35} +3.42166 q^{37} -17.8328 q^{38} +14.2056 q^{40} +7.62721 q^{41} -4.91638 q^{43} -24.8816 q^{44} +7.62721 q^{46} +1.08362 q^{47} +1.00000 q^{49} -9.39194 q^{50} -5.91638 q^{52} -1.75971 q^{53} -5.42166 q^{55} -11.0192 q^{56} +12.2056 q^{58} +4.57834 q^{59} -5.25443 q^{61} -21.0872 q^{62} +51.4147 q^{64} -1.28917 q^{65} +8.67609 q^{67} -9.62721 q^{68} -3.62721 q^{70} +6.78389 q^{71} +4.07306 q^{73} +9.62721 q^{74} -37.4983 q^{76} +4.20555 q^{77} -8.91638 q^{79} +24.7144 q^{80} +21.4600 q^{82} +4.33804 q^{83} -2.09775 q^{85} -13.8328 q^{86} -46.3416 q^{88} -4.54359 q^{89} +1.00000 q^{91} +16.0383 q^{92} +3.04888 q^{94} -8.17081 q^{95} +11.3275 q^{97} +2.81361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 3 q^{7} + 12 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 3 * q^7 + 12 * q^8	\( 3 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 3 q^{7} + 12 q^{8} - 2 q^{10} + 2 q^{11} - 3 q^{13} - 2 q^{14} + 18 q^{16} + 8 q^{17} - 7 q^{19} + 10 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{28} + q^{29} - 7 q^{31} + 36 q^{32} - 12 q^{34} - 3 q^{35} + 12 q^{37} - 26 q^{38} + 28 q^{40} + 10 q^{41} - q^{43} - 36 q^{44} + 10 q^{46} + 17 q^{47} + 3 q^{49} - 20 q^{50} - 4 q^{52} + 5 q^{53} - 18 q^{55} - 12 q^{56} + 22 q^{58} + 12 q^{59} + 10 q^{61} - 10 q^{62} + 58 q^{64} - 3 q^{65} + 2 q^{67} - 16 q^{68} + 2 q^{70} + 4 q^{71} - 5 q^{73} + 16 q^{74} - 30 q^{76} - 2 q^{77} - 13 q^{79} + 8 q^{80} + 24 q^{82} + q^{83} + 16 q^{85} - 14 q^{86} - 60 q^{88} + 13 q^{89} + 3 q^{91} + 6 q^{92} - 2 q^{94} + 15 q^{95} - 9 q^{97} + 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 3 * q^7 + 12 * q^8 - 2 * q^10 + 2 * q^11 - 3 * q^13 - 2 * q^14 + 18 * q^16 + 8 * q^17 - 7 * q^19 + 10 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 4 * q^28 + q^29 - 7 * q^31 + 36 * q^32 - 12 * q^34 - 3 * q^35 + 12 * q^37 - 26 * q^38 + 28 * q^40 + 10 * q^41 - q^43 - 36 * q^44 + 10 * q^46 + 17 * q^47 + 3 * q^49 - 20 * q^50 - 4 * q^52 + 5 * q^53 - 18 * q^55 - 12 * q^56 + 22 * q^58 + 12 * q^59 + 10 * q^61 - 10 * q^62 + 58 * q^64 - 3 * q^65 + 2 * q^67 - 16 * q^68 + 2 * q^70 + 4 * q^71 - 5 * q^73 + 16 * q^74 - 30 * q^76 - 2 * q^77 - 13 * q^79 + 8 * q^80 + 24 * q^82 + q^83 + 16 * q^85 - 14 * q^86 - 60 * q^88 + 13 * q^89 + 3 * q^91 + 6 * q^92 - 2 * q^94 + 15 * q^95 - 9 * q^97 + 2 * q^98

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