LMFDB - Embedded newform 819.2.a.k.1.1

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:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36865$ 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)

$f(q)$	$=$	$q-2.61050 q^{2} +4.81471 q^{4} +3.81471 q^{5} +1.00000 q^{7} -7.34780 q^{8} +O(q^{10})$	\(q-2.61050 q^{2} +4.81471 q^{4} +3.81471 q^{5} +1.00000 q^{7} -7.34780 q^{8} -9.95830 q^{10} +4.73730 q^{11} +1.00000 q^{13} -2.61050 q^{14} +9.55201 q^{16} +5.22100 q^{17} +2.92259 q^{19} +18.3667 q^{20} -12.3667 q^{22} -3.33101 q^{23} +9.55201 q^{25} -2.61050 q^{26} +4.81471 q^{28} +0.922589 q^{29} -7.51941 q^{31} -10.2399 q^{32} -13.6294 q^{34} +3.81471 q^{35} +0.154821 q^{37} -7.62942 q^{38} -28.0297 q^{40} -6.36672 q^{41} -6.55201 q^{43} +22.8087 q^{44} +8.69560 q^{46} -9.03571 q^{47} +1.00000 q^{49} -24.9355 q^{50} +4.81471 q^{52} -8.55201 q^{53} +18.0714 q^{55} -7.34780 q^{56} -2.40842 q^{58} -3.95830 q^{59} +12.4420 q^{61} +19.6294 q^{62} +7.62729 q^{64} +3.81471 q^{65} -10.6620 q^{67} +25.1376 q^{68} -9.95830 q^{70} +6.58248 q^{71} -7.73517 q^{73} -0.404161 q^{74} +14.0714 q^{76} +4.73730 q^{77} +13.3646 q^{79} +36.4381 q^{80} +16.6203 q^{82} +1.40629 q^{83} +19.9166 q^{85} +17.1040 q^{86} -34.8087 q^{88} +1.96953 q^{89} +1.00000 q^{91} -16.0378 q^{92} +23.5877 q^{94} +11.1488 q^{95} -2.11001 q^{97} -2.61050 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8	$ 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29} + 3 q^{31} - 7 q^{32} - 30 q^{34} + 3 q^{35} + 10 q^{37} - 6 q^{38} - 14 q^{40} + 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} - 5 q^{47} + 4 q^{49} - 13 q^{50} + 7 q^{52} - 5 q^{53} + 10 q^{55} - 3 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{61} + 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} + 10 q^{68} - 4 q^{70} - 13 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 11 q^{79} + 42 q^{80} + 10 q^{82} - q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} + 5 q^{89} + 4 q^{91} - 34 q^{92} + 34 q^{94} - 13 q^{95} - 17 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - 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 $

$2$	−2.61050	−1.84590	−0.922951	−	0.384917i	$-0.874230\pi$
$2$	−2.61050	−1.84590	−0.922951	+	0.384917i	$0.874230\pi$
$3$	0	0
$3$	0	0
$4$	4.81471	2.40735
$5$	3.81471	1.70599	0.852995	−	0.521919i	$-0.174784\pi$
$5$	3.81471	1.70599	0.852995	+	0.521919i	$0.174784\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−7.34780	−2.59784
$9$	0	0
$10$	−9.95830	−3.14909
$11$	4.73730	1.42835	0.714175	−	0.699968i	$-0.246802\pi$
$11$	4.73730	1.42835	0.714175	+	0.699968i	$0.246802\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.61050	−0.697685
$15$	0	0
$16$	9.55201	2.38800
$17$	5.22100	1.26628	0.633139	−	0.774038i	$-0.281766\pi$
$17$	5.22100	1.26628	0.633139	+	0.774038i	$0.281766\pi$
$18$	0	0
$19$	2.92259	0.670488	0.335244	−	0.942131i	$-0.391181\pi$
$19$	2.92259	0.670488	0.335244	+	0.942131i	$0.391181\pi$
$20$	18.3667	4.10692
$21$	0	0
$22$	−12.3667	−2.63659
$23$	−3.33101	−0.694563	−0.347282	−	0.937761i	$-0.612895\pi$
$23$	−3.33101	−0.694563	−0.347282	+	0.937761i	$0.612895\pi$
$24$	0	0
$25$	9.55201	1.91040
$26$	−2.61050	−0.511961
$27$	0	0
$28$	4.81471	0.909895
$29$	0.922589	0.171321	0.0856603	−	0.996324i	$-0.472700\pi$
$29$	0.922589	0.171321	0.0856603	+	0.996324i	$0.472700\pi$
$30$	0	0
$31$	−7.51941	−1.35053	−0.675263	−	0.737577i	$-0.735970\pi$
$31$	−7.51941	−1.35053	−0.675263	+	0.737577i	$0.735970\pi$
$32$	−10.2399	−1.81018
$33$	0	0
$34$	−13.6294	−2.33743
$35$	3.81471	0.644804
$36$	0	0
$37$	0.154821	0.0254525	0.0127262	−	0.999919i	$-0.495949\pi$
$37$	0.154821	0.0254525	0.0127262	+	0.999919i	$0.495949\pi$
$38$	−7.62942	−1.23766
$39$	0	0
$40$	−28.0297	−4.43189
$41$	−6.36672	−0.994314	−0.497157	−	0.867661i	$-0.665623\pi$
$41$	−6.36672	−0.994314	−0.497157	+	0.867661i	$0.665623\pi$
$42$	0	0
$43$	−6.55201	−0.999172	−0.499586	−	0.866264i	$-0.666514\pi$
$43$	−6.55201	−0.999172	−0.499586	+	0.866264i	$0.666514\pi$
$44$	22.8087	3.43854
$45$	0	0
$46$	8.69560	1.28210
$47$	−9.03571	−1.31799	−0.658997	−	0.752146i	$-0.729019\pi$
$47$	−9.03571	−1.31799	−0.658997	+	0.752146i	$0.729019\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−24.9355	−3.52641
$51$	0	0
$52$	4.81471	0.667680
$53$	−8.55201	−1.17471	−0.587354	−	0.809330i	$-0.699830\pi$
$53$	−8.55201	−1.17471	−0.587354	+	0.809330i	$0.699830\pi$
$54$	0	0
$55$	18.0714	2.43675
$56$	−7.34780	−0.981891
$57$	0	0
$58$	−2.40842	−0.316241
$59$	−3.95830	−0.515327	−0.257663	−	0.966235i	$-0.582953\pi$
$59$	−3.95830	−0.515327	−0.257663	+	0.966235i	$0.582953\pi$
$60$	0	0
$61$	12.4420	1.59303	0.796517	−	0.604616i	$-0.206673\pi$
$61$	12.4420	1.59303	0.796517	+	0.604616i	$0.206673\pi$
$62$	19.6294	2.49294
$63$	0	0
$64$	7.62729	0.953411
$65$	3.81471	0.473156
$66$	0	0
$67$	−10.6620	−1.30257	−0.651286	−	0.758832i	$-0.725770\pi$
$67$	−10.6620	−1.30257	−0.651286	+	0.758832i	$0.725770\pi$
$68$	25.1376	3.04838
$69$	0	0
$70$	−9.95830	−1.19024
$71$	6.58248	0.781196	0.390598	−	0.920561i	$-0.372268\pi$
$71$	6.58248	0.781196	0.390598	+	0.920561i	$0.372268\pi$
$72$	0	0
$73$	−7.73517	−0.905333	−0.452667	−	0.891680i	$-0.649527\pi$
$73$	−7.73517	−0.905333	−0.452667	+	0.891680i	$0.649527\pi$
$74$	−0.404161	−0.0469828
$75$	0	0
$76$	14.0714	1.61410
$77$	4.73730	0.539865
$78$	0	0
$79$	13.3646	1.50363	0.751817	−	0.659372i	$-0.229178\pi$
$79$	13.3646	1.50363	0.751817	+	0.659372i	$0.229178\pi$
$80$	36.4381	4.07391
$81$	0	0
$82$	16.6203	1.83541
$83$	1.40629	0.154360	0.0771802	−	0.997017i	$-0.475408\pi$
$83$	1.40629	0.154360	0.0771802	+	0.997017i	$0.475408\pi$
$84$	0	0
$85$	19.9166	2.16026
$86$	17.1040	1.84437
$87$	0	0
$88$	−34.8087	−3.71062
$89$	1.96953	0.208770	0.104385	−	0.994537i	$-0.466713\pi$
$89$	1.96953	0.208770	0.104385	+	0.994537i	$0.466713\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−16.0378	−1.67206
$93$	0	0
$94$	23.5877	2.43289
$95$	11.1488	1.14385
$96$	0	0
$97$	−2.11001	−0.214239	−0.107119	−	0.994246i	$-0.534163\pi$
$97$	−2.11001	−0.214239	−0.107119	+	0.994246i	$0.534163\pi$
$98$	−2.61050	−0.263700
$99$	0	0
$100$	45.9901	4.59901
$101$	−0.850419	−0.0846198	−0.0423099	−	0.999105i	$-0.513472\pi$
$101$	−0.850419	−0.0846198	−0.0423099	+	0.999105i	$0.513472\pi$
$102$	0	0
$103$	−1.47460	−0.145296	−0.0726482	−	0.997358i	$-0.523145\pi$
$103$	−1.47460	−0.145296	−0.0726482	+	0.997358i	$0.523145\pi$
$104$	−7.34780	−0.720511
$105$	0	0
$106$	22.3250	2.16840
$107$	−2.62418	−0.253689	−0.126844	−	0.991923i	$-0.540485\pi$
$107$	−2.62418	−0.253689	−0.126844	+	0.991923i	$0.540485\pi$
$108$	0	0
$109$	−17.9166	−1.71610	−0.858049	−	0.513567i	$-0.828324\pi$
$109$	−17.9166	−1.71610	−0.858049	+	0.513567i	$0.828324\pi$
$110$	−47.1754	−4.49800
$111$	0	0
$112$	9.55201	0.902580
$113$	−0.922589	−0.0867899	−0.0433950	−	0.999058i	$-0.513817\pi$
$113$	−0.922589	−0.0867899	−0.0433950	+	0.999058i	$0.513817\pi$
$114$	0	0
$115$	−12.7068	−1.18492
$116$	4.44200	0.412429
$117$	0	0
$118$	10.3331	0.951242
$119$	5.22100	0.478608
$120$	0	0
$121$	11.4420	1.04018
$122$	−32.4798	−2.94059
$123$	0	0
$124$	−36.2038	−3.25119
$125$	17.3646	1.55314
$126$	0	0
$127$	17.4746	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$127$	17.4746	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$128$	0.568798	0.0502751
$129$	0	0
$130$	−9.95830	−0.873401
$131$	−0.967402	−0.0845223	−0.0422611	−	0.999107i	$-0.513456\pi$
$131$	−0.967402	−0.0845223	−0.0422611	+	0.999107i	$0.513456\pi$
$132$	0	0
$133$	2.92259	0.253421
$134$	27.8332	2.40442
$135$	0	0
$136$	−38.3629	−3.28959
$137$	−3.29628	−0.281620	−0.140810	−	0.990037i	$-0.544971\pi$
$137$	−3.29628	−0.281620	−0.140810	+	0.990037i	$0.544971\pi$
$138$	0	0
$139$	0.370581	0.0314323	0.0157161	−	0.999876i	$-0.494997\pi$
$139$	0.370581	0.0314323	0.0157161	+	0.999876i	$0.494997\pi$
$140$	18.3667	1.55227
$141$	0	0
$142$	−17.1836	−1.44201
$143$	4.73730	0.396153
$144$	0	0
$145$	3.51941	0.292271
$146$	20.1927	1.67116
$147$	0	0
$148$	0.745420	0.0612732
$149$	15.7425	1.28968	0.644840	−	0.764318i	$-0.276924\pi$
$149$	15.7425	1.28968	0.644840	+	0.764318i	$0.276924\pi$
$150$	0	0
$151$	10.2914	0.837505	0.418753	−	0.908100i	$-0.362467\pi$
$151$	10.2914	0.837505	0.418753	+	0.908100i	$0.362467\pi$
$152$	−21.4746	−1.74182
$153$	0	0
$154$	−12.3667	−0.996539
$155$	−28.6844	−2.30398
$156$	0	0
$157$	−11.4137	−0.910909	−0.455455	−	0.890259i	$-0.650523\pi$
$157$	−11.4137	−0.910909	−0.455455	+	0.890259i	$0.650523\pi$
$158$	−34.8883	−2.77556
$159$	0	0
$160$	−39.0623	−3.08815
$161$	−3.33101	−0.262520
$162$	0	0
$163$	−13.4746	−1.05541	−0.527706	−	0.849427i	$-0.676948\pi$
$163$	−13.4746	−1.05541	−0.527706	+	0.849427i	$0.676948\pi$
$164$	−30.6539	−2.39367
$165$	0	0
$166$	−3.67112	−0.284934
$167$	−19.1905	−1.48501	−0.742504	−	0.669842i	$-0.766362\pi$
$167$	−19.1905	−1.48501	−0.742504	+	0.669842i	$0.766362\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−51.9923	−3.98763
$171$	0	0
$172$	−31.5460	−2.40536
$173$	19.5124	1.48350	0.741752	−	0.670675i	$-0.233995\pi$
$173$	19.5124	1.48350	0.741752	+	0.670675i	$0.233995\pi$
$174$	0	0
$175$	9.55201	0.722064
$176$	45.2507	3.41090
$177$	0	0
$178$	−5.14146	−0.385369
$179$	16.5856	1.23967	0.619833	−	0.784734i	$-0.287201\pi$
$179$	16.5856	1.23967	0.619833	+	0.784734i	$0.287201\pi$
$180$	0	0
$181$	−2.81684	−0.209374	−0.104687	−	0.994505i	$-0.533384\pi$
$181$	−2.81684	−0.209374	−0.104687	+	0.994505i	$0.533384\pi$
$182$	−2.61050	−0.193503
$183$	0	0
$184$	24.4756	1.80436
$185$	0.590599	0.0434217
$186$	0	0
$187$	24.7334	1.80869
$188$	−43.5043	−3.17288
$189$	0	0
$190$	−29.1040	−2.11143
$191$	−15.1601	−1.09694	−0.548472	−	0.836169i	$-0.684790\pi$
$191$	−15.1601	−1.09694	−0.548472	+	0.836169i	$0.684790\pi$
$192$	0	0
$193$	0.0651962	0.00469293	0.00234646	−	0.999997i	$-0.499253\pi$
$193$	0.0651962	0.00469293	0.00234646	+	0.999997i	$0.499253\pi$
$194$	5.50818	0.395464
$195$	0	0
$196$	4.81471	0.343908
$197$	17.1415	1.22128	0.610639	−	0.791909i	$-0.290913\pi$
$197$	17.1415	1.22128	0.610639	+	0.791909i	$0.290913\pi$
$198$	0	0
$199$	6.44200	0.456661	0.228331	−	0.973584i	$-0.426673\pi$
$199$	6.44200	0.456661	0.228331	+	0.973584i	$0.426673\pi$
$200$	−70.1862	−4.96292
$201$	0	0
$202$	2.22002	0.156200
$203$	0.922589	0.0647531
$204$	0	0
$205$	−24.2872	−1.69629
$206$	3.84944	0.268203
$207$	0	0
$208$	9.55201	0.662313
$209$	13.8452	0.957691
$210$	0	0
$211$	−16.0266	−1.10332	−0.551659	−	0.834070i	$-0.686005\pi$
$211$	−16.0266	−1.10332	−0.551659	+	0.834070i	$0.686005\pi$
$212$	−41.1754	−2.82794
$213$	0	0
$214$	6.85042	0.468285
$215$	−24.9940	−1.70458
$216$	0	0
$217$	−7.51941	−0.510451
$218$	46.7713	3.16775
$219$	0	0
$220$	87.0086	5.86612
$221$	5.22100	0.351202
$222$	0	0
$223$	20.0266	1.34108	0.670540	−	0.741873i	$-0.266062\pi$
$223$	20.0266	1.34108	0.670540	+	0.741873i	$0.266062\pi$
$224$	−10.2399	−0.684183
$225$	0	0
$226$	2.40842	0.160206
$227$	−19.2171	−1.27549	−0.637743	−	0.770249i	$-0.720132\pi$
$227$	−19.2171	−1.27549	−0.637743	+	0.770249i	$0.720132\pi$
$228$	0	0
$229$	5.25884	0.347514	0.173757	−	0.984789i	$-0.444409\pi$
$229$	5.25884	0.347514	0.173757	+	0.984789i	$0.444409\pi$
$230$	33.1712	2.18724
$231$	0	0
$232$	−6.77900	−0.445063
$233$	26.6234	1.74416	0.872079	−	0.489365i	$-0.162771\pi$
$233$	26.6234	1.74416	0.872079	+	0.489365i	$0.162771\pi$
$234$	0	0
$235$	−34.4686	−2.24848
$236$	−19.0581	−1.24057
$237$	0	0
$238$	−13.6294	−0.883464
$239$	−4.29104	−0.277564	−0.138782	−	0.990323i	$-0.544319\pi$
$239$	−4.29104	−0.277564	−0.138782	+	0.990323i	$0.544319\pi$
$240$	0	0
$241$	7.52367	0.484642	0.242321	−	0.970196i	$-0.422091\pi$
$241$	7.52367	0.484642	0.242321	+	0.970196i	$0.422091\pi$
$242$	−29.8693	−1.92007
$243$	0	0
$244$	59.9046	3.83500
$245$	3.81471	0.243713
$246$	0	0
$247$	2.92259	0.185960
$248$	55.2511	3.50845
$249$	0	0
$250$	−45.3303	−2.86694
$251$	−11.3198	−0.714498	−0.357249	−	0.934009i	$-0.616285\pi$
$251$	−11.3198	−0.714498	−0.357249	+	0.934009i	$0.616285\pi$
$252$	0	0
$253$	−15.7800	−0.992079
$254$	−45.6174	−2.86229
$255$	0	0
$256$	−16.7394	−1.04621
$257$	24.8504	1.55013	0.775063	−	0.631884i	$-0.217718\pi$
$257$	24.8504	1.55013	0.775063	+	0.631884i	$0.217718\pi$
$258$	0	0
$259$	0.154821	0.00962013
$260$	18.3667	1.13906
$261$	0	0
$262$	2.52540	0.156020
$263$	−17.1762	−1.05913	−0.529565	−	0.848270i	$-0.677645\pi$
$263$	−17.1762	−1.05913	−0.529565	+	0.848270i	$0.677645\pi$
$264$	0	0
$265$	−32.6234	−2.00404
$266$	−7.62942	−0.467790
$267$	0	0
$268$	−51.3345	−3.13575
$269$	7.74640	0.472306	0.236153	−	0.971716i	$-0.424113\pi$
$269$	7.74640	0.472306	0.236153	+	0.971716i	$0.424113\pi$
$270$	0	0
$271$	−29.7008	−1.80420	−0.902099	−	0.431530i	$-0.857974\pi$
$271$	−29.7008	−1.80420	−0.902099	+	0.431530i	$0.857974\pi$
$272$	49.8710	3.02388
$273$	0	0
$274$	8.60494	0.519844
$275$	45.2507	2.72872
$276$	0	0
$277$	25.1488	1.51105	0.755523	−	0.655122i	$-0.227383\pi$
$277$	25.1488	1.51105	0.755523	+	0.655122i	$0.227383\pi$
$278$	−0.967402	−0.0580209
$279$	0	0
$280$	−28.0297	−1.67510
$281$	−8.40030	−0.501120	−0.250560	−	0.968101i	$-0.580615\pi$
$281$	−8.40030	−0.501120	−0.250560	+	0.968101i	$0.580615\pi$
$282$	0	0
$283$	−21.3912	−1.27157	−0.635787	−	0.771864i	$-0.719324\pi$
$283$	−21.3912	−1.27157	−0.635787	+	0.771864i	$0.719324\pi$
$284$	31.6927	1.88062
$285$	0	0
$286$	−12.3667	−0.731259
$287$	−6.36672	−0.375815
$288$	0	0
$289$	10.2588	0.603461
$290$	−9.18742	−0.539504
$291$	0	0
$292$	−37.2426	−2.17946
$293$	9.59895	0.560777	0.280388	−	0.959887i	$-0.409537\pi$
$293$	9.59895	0.560777	0.280388	+	0.959887i	$0.409537\pi$
$294$	0	0
$295$	−15.0998	−0.879142
$296$	−1.13760	−0.0661215
$297$	0	0
$298$	−41.0959	−2.38062
$299$	−3.33101	−0.192637
$300$	0	0
$301$	−6.55201	−0.377651
$302$	−26.8658	−1.54595
$303$	0	0
$304$	27.9166	1.60113
$305$	47.4626	2.71770
$306$	0	0
$307$	−15.1488	−0.864589	−0.432295	−	0.901732i	$-0.642296\pi$
$307$	−15.1488	−0.864589	−0.432295	+	0.901732i	$0.642296\pi$
$308$	22.8087	1.29965
$309$	0	0
$310$	74.8805	4.25293
$311$	4.37058	0.247833	0.123916	−	0.992293i	$-0.460455\pi$
$311$	4.37058	0.247833	0.123916	+	0.992293i	$0.460455\pi$
$312$	0	0
$313$	−1.49280	−0.0843783	−0.0421891	−	0.999110i	$-0.513433\pi$
$313$	−1.49280	−0.0843783	−0.0421891	+	0.999110i	$0.513433\pi$
$314$	29.7954	1.68145
$315$	0	0
$316$	64.3466	3.61978
$317$	−23.2129	−1.30377	−0.651883	−	0.758320i	$-0.726021\pi$
$317$	−23.2129	−1.30377	−0.651883	+	0.758320i	$0.726021\pi$
$318$	0	0
$319$	4.37058	0.244706
$320$	29.0959	1.62651
$321$	0	0
$322$	8.69560	0.484587
$323$	15.2588	0.849024
$324$	0	0
$325$	9.55201	0.529850
$326$	35.1754	1.94819
$327$	0	0
$328$	46.7814	2.58307
$329$	−9.03571	−0.498155
$330$	0	0
$331$	1.10402	0.0606822	0.0303411	−	0.999540i	$-0.490341\pi$
$331$	1.10402	0.0606822	0.0303411	+	0.999540i	$0.490341\pi$
$332$	6.77088	0.371600
$333$	0	0
$334$	50.0969	2.74118
$335$	−40.6725	−2.22218
$336$	0	0
$337$	−4.24237	−0.231096	−0.115548	−	0.993302i	$-0.536862\pi$
$337$	−4.24237	−0.231096	−0.115548	+	0.993302i	$0.536862\pi$
$338$	−2.61050	−0.141992
$339$	0	0
$340$	95.8926	5.20051
$341$	−35.6217	−1.92902
$342$	0	0
$343$	1.00000	0.0539949
$344$	48.1428	2.59569
$345$	0	0
$346$	−50.9372	−2.73840
$347$	−14.0336	−0.753362	−0.376681	−	0.926343i	$-0.622935\pi$
$347$	−14.0336	−0.753362	−0.376681	+	0.926343i	$0.622935\pi$
$348$	0	0
$349$	4.10575	0.219776	0.109888	−	0.993944i	$-0.464951\pi$
$349$	4.10575	0.219776	0.109888	+	0.993944i	$0.464951\pi$
$350$	−24.9355	−1.33286
$351$	0	0
$352$	−48.5096	−2.58557
$353$	16.0753	0.855601	0.427800	−	0.903873i	$-0.359289\pi$
$353$	16.0753	0.855601	0.427800	+	0.903873i	$0.359289\pi$
$354$	0	0
$355$	25.1102	1.33271
$356$	9.48272	0.502583
$357$	0	0
$358$	−43.2967	−2.28830
$359$	18.1510	0.957971	0.478985	−	0.877823i	$-0.341005\pi$
$359$	18.1510	0.957971	0.478985	+	0.877823i	$0.341005\pi$
$360$	0	0
$361$	−10.4585	−0.550446
$362$	7.35336	0.386484
$363$	0	0
$364$	4.81471	0.252359
$365$	−29.5074	−1.54449
$366$	0	0
$367$	−23.3955	−1.22123	−0.610616	−	0.791927i	$-0.709078\pi$
$367$	−23.3955	−1.22123	−0.610616	+	0.791927i	$0.709078\pi$
$368$	−31.8178	−1.65862
$369$	0	0
$370$	−1.54176	−0.0801522
$371$	−8.55201	−0.443998
$372$	0	0
$373$	−4.75164	−0.246031	−0.123015	−	0.992405i	$-0.539256\pi$
$373$	−4.75164	−0.246031	−0.123015	+	0.992405i	$0.539256\pi$
$374$	−64.5666	−3.33866
$375$	0	0
$376$	66.3926	3.42394
$377$	0.922589	0.0475158
$378$	0	0
$379$	−6.81258	−0.349939	−0.174969	−	0.984574i	$-0.555983\pi$
$379$	−6.81258	−0.349939	−0.174969	+	0.984574i	$0.555983\pi$
$380$	53.6784	2.75364
$381$	0	0
$382$	39.5753	2.02485
$383$	−2.25746	−0.115351	−0.0576754	−	0.998335i	$-0.518369\pi$
$383$	−2.25746	−0.115351	−0.0576754	+	0.998335i	$0.518369\pi$
$384$	0	0
$385$	18.0714	0.921005
$386$	−0.170195	−0.00866268
$387$	0	0
$388$	−10.1591	−0.515749
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−17.3912	−0.879511
$392$	−7.34780	−0.371120
$393$	0	0
$394$	−44.7478	−2.25436
$395$	50.9820	2.56518
$396$	0	0
$397$	24.9897	1.25420	0.627100	−	0.778939i	$-0.284242\pi$
$397$	24.9897	1.25420	0.627100	+	0.778939i	$0.284242\pi$
$398$	−16.8168	−0.842952
$399$	0	0
$400$	91.2409	4.56204
$401$	−27.6529	−1.38092	−0.690460	−	0.723370i	$-0.742592\pi$
$401$	−27.6529	−1.38092	−0.690460	+	0.723370i	$0.742592\pi$
$402$	0	0
$403$	−7.51941	−0.374568
$404$	−4.09452	−0.203710
$405$	0	0
$406$	−2.40842	−0.119528
$407$	0.733435	0.0363550
$408$	0	0
$409$	13.1488	0.650168	0.325084	−	0.945685i	$-0.394607\pi$
$409$	13.1488	0.650168	0.325084	+	0.945685i	$0.394607\pi$
$410$	63.4017	3.13119
$411$	0	0
$412$	−7.09976	−0.349780
$413$	−3.95830	−0.194775
$414$	0	0
$415$	5.36459	0.263337
$416$	−10.2399	−0.502053
$417$	0	0
$418$	−36.1428	−1.76780
$419$	5.69462	0.278200	0.139100	−	0.990278i	$-0.455579\pi$
$419$	5.69462	0.278200	0.139100	+	0.990278i	$0.455579\pi$
$420$	0	0
$421$	−23.0206	−1.12196	−0.560978	−	0.827831i	$-0.689575\pi$
$421$	−23.0206	−1.12196	−0.560978	+	0.827831i	$0.689575\pi$
$422$	41.8375	2.03662
$423$	0	0
$424$	62.8384	3.05170
$425$	49.8710	2.41910
$426$	0	0
$427$	12.4420	0.602111
$428$	−12.6347	−0.610719
$429$	0	0
$430$	65.2469	3.14648
$431$	−20.8045	−1.00212	−0.501058	−	0.865414i	$-0.667056\pi$
$431$	−20.8045	−1.00212	−0.501058	+	0.865414i	$0.667056\pi$
$432$	0	0
$433$	31.4808	1.51287	0.756436	−	0.654068i	$-0.226939\pi$
$433$	31.4808	1.51287	0.756436	+	0.654068i	$0.226939\pi$
$434$	19.6294	0.942242
$435$	0	0
$436$	−86.2632	−4.13126
$437$	−9.73517	−0.465696
$438$	0	0
$439$	22.3811	1.06819	0.534095	−	0.845425i	$-0.320653\pi$
$439$	22.3811	1.06819	0.534095	+	0.845425i	$0.320653\pi$
$440$	−132.785	−6.33028
$441$	0	0
$442$	−13.6294	−0.648285
$443$	−8.52465	−0.405018	−0.202509	−	0.979280i	$-0.564910\pi$
$443$	−8.52465	−0.405018	−0.202509	+	0.979280i	$0.564910\pi$
$444$	0	0
$445$	7.51319	0.356159
$446$	−52.2795	−2.47550
$447$	0	0
$448$	7.62729	0.360356
$449$	−18.7142	−0.883178	−0.441589	−	0.897218i	$-0.645585\pi$
$449$	−18.7142	−0.883178	−0.441589	+	0.897218i	$0.645585\pi$
$450$	0	0
$451$	−30.1610	−1.42023
$452$	−4.44200	−0.208934
$453$	0	0
$454$	50.1663	2.35442
$455$	3.81471	0.178836
$456$	0	0
$457$	−14.1366	−0.661283	−0.330641	−	0.943756i	$-0.607265\pi$
$457$	−14.1366	−0.661283	−0.330641	+	0.943756i	$0.607265\pi$
$458$	−13.7282	−0.641476
$459$	0	0
$460$	−61.1797	−2.85252
$461$	2.67636	0.124651	0.0623253	−	0.998056i	$-0.480148\pi$
$461$	2.67636	0.124651	0.0623253	+	0.998056i	$0.480148\pi$
$462$	0	0
$463$	−2.53162	−0.117655	−0.0588273	−	0.998268i	$-0.518736\pi$
$463$	−2.53162	−0.117655	−0.0588273	+	0.998268i	$0.518736\pi$
$464$	8.81258	0.409114
$465$	0	0
$466$	−69.5005	−3.21955
$467$	−2.00426	−0.0927460	−0.0463730	−	0.998924i	$-0.514766\pi$
$467$	−2.00426	−0.0927460	−0.0463730	+	0.998924i	$0.514766\pi$
$468$	0	0
$469$	−10.6620	−0.492326
$470$	89.9803	4.15048
$471$	0	0
$472$	29.0848	1.33874
$473$	−31.0388	−1.42717
$474$	0	0
$475$	27.9166	1.28090
$476$	25.1376	1.15218
$477$	0	0
$478$	11.2018	0.512357
$479$	14.6609	0.669872	0.334936	−	0.942241i	$-0.391285\pi$
$479$	14.6609	0.669872	0.334936	+	0.942241i	$0.391285\pi$
$480$	0	0
$481$	0.154821	0.00705925
$482$	−19.6405	−0.894602
$483$	0	0
$484$	55.0899	2.50409
$485$	−8.04907	−0.365489
$486$	0	0
$487$	38.2143	1.73165	0.865827	−	0.500344i	$-0.166793\pi$
$487$	38.2143	1.73165	0.865827	+	0.500344i	$0.166793\pi$
$488$	−91.4213	−4.13845
$489$	0	0
$490$	−9.95830	−0.449870
$491$	−21.3758	−0.964677	−0.482339	−	0.875985i	$-0.660213\pi$
$491$	−21.3758	−0.964677	−0.482339	+	0.875985i	$0.660213\pi$
$492$	0	0
$493$	4.81684	0.216939
$494$	−7.62942	−0.343264
$495$	0	0
$496$	−71.8255	−3.22506
$497$	6.58248	0.295264
$498$	0	0
$499$	27.0920	1.21281	0.606403	−	0.795158i	$-0.292612\pi$
$499$	27.0920	1.21281	0.606403	+	0.795158i	$0.292612\pi$
$500$	83.6054	3.73895
$501$	0	0
$502$	29.5503	1.31889
$503$	−24.8778	−1.10925	−0.554623	−	0.832102i	$-0.687137\pi$
$503$	−24.8778	−1.10925	−0.554623	+	0.832102i	$0.687137\pi$
$504$	0	0
$505$	−3.24410	−0.144361
$506$	41.1936	1.83128
$507$	0	0
$508$	84.1351	3.73289
$509$	−2.70297	−0.119807	−0.0599034	−	0.998204i	$-0.519079\pi$
$509$	−2.70297	−0.119807	−0.0599034	+	0.998204i	$0.519079\pi$
$510$	0	0
$511$	−7.73517	−0.342184
$512$	42.5607	1.88093
$513$	0	0
$514$	−64.8720	−2.86138
$515$	−5.62516	−0.247874
$516$	0	0
$517$	−42.8049	−1.88256
$518$	−0.404161	−0.0177578
$519$	0	0
$520$	−28.0297	−1.22918
$521$	33.4472	1.46535	0.732675	−	0.680579i	$-0.238272\pi$
$521$	33.4472	1.46535	0.732675	+	0.680579i	$0.238272\pi$
$522$	0	0
$523$	19.3198	0.844795	0.422397	−	0.906411i	$-0.361189\pi$
$523$	19.3198	0.844795	0.422397	+	0.906411i	$0.361189\pi$
$524$	−4.65776	−0.203475
$525$	0	0
$526$	44.8384	1.95505
$527$	−39.2588	−1.71014
$528$	0	0
$529$	−11.9044	−0.517582
$530$	85.1634	3.69926
$531$	0	0
$532$	14.0714	0.610073
$533$	−6.36672	−0.275773
$534$	0	0
$535$	−10.0105	−0.432791
$536$	78.3424	3.38387
$537$	0	0
$538$	−20.2220	−0.871832
$539$	4.73730	0.204050
$540$	0	0
$541$	−24.9554	−1.07292	−0.536459	−	0.843927i	$-0.680238\pi$
$541$	−24.9554	−1.07292	−0.536459	+	0.843927i	$0.680238\pi$
$542$	77.5340	3.33037
$543$	0	0
$544$	−53.4626	−2.29219
$545$	−68.3466	−2.92765
$546$	0	0
$547$	3.80037	0.162492	0.0812460	−	0.996694i	$-0.474110\pi$
$547$	3.80037	0.162492	0.0812460	+	0.996694i	$0.474110\pi$
$548$	−15.8706	−0.677960
$549$	0	0
$550$	−118.127	−5.03695
$551$	2.69635	0.114868
$552$	0	0
$553$	13.3646	0.568320
$554$	−65.6510	−2.78924
$555$	0	0
$556$	1.78424	0.0756686
$557$	−43.8792	−1.85922	−0.929610	−	0.368546i	$-0.879856\pi$
$557$	−43.8792	−1.85922	−0.929610	+	0.368546i	$0.879856\pi$
$558$	0	0
$559$	−6.55201	−0.277120
$560$	36.4381	1.53979
$561$	0	0
$562$	21.9290	0.925018
$563$	−12.3768	−0.521620	−0.260810	−	0.965390i	$-0.583990\pi$
$563$	−12.3768	−0.521620	−0.260810	+	0.965390i	$0.583990\pi$
$564$	0	0
$565$	−3.51941	−0.148063
$566$	55.8417	2.34720
$567$	0	0
$568$	−48.3667	−2.02942
$569$	26.6234	1.11611	0.558056	−	0.829803i	$-0.311547\pi$
$569$	26.6234	1.11611	0.558056	+	0.829803i	$0.311547\pi$
$570$	0	0
$571$	10.9983	0.460263	0.230132	−	0.973160i	$-0.426084\pi$
$571$	10.9983	0.460263	0.230132	+	0.973160i	$0.426084\pi$
$572$	22.8087	0.953680
$573$	0	0
$574$	16.6203	0.693719
$575$	−31.8178	−1.32689
$576$	0	0
$577$	30.4238	1.26656	0.633280	−	0.773923i	$-0.281708\pi$
$577$	30.4238	1.26656	0.633280	+	0.773923i	$0.281708\pi$
$578$	−26.7807	−1.11393
$579$	0	0
$580$	16.9449	0.703600
$581$	1.40629	0.0583428
$582$	0	0
$583$	−40.5134	−1.67789
$584$	56.8365	2.35191
$585$	0	0
$586$	−25.0581	−1.03514
$587$	−3.84207	−0.158579	−0.0792895	−	0.996852i	$-0.525265\pi$
$587$	−3.84207	−0.158579	−0.0792895	+	0.996852i	$0.525265\pi$
$588$	0	0
$589$	−21.9761	−0.905511
$590$	39.4179	1.62281
$591$	0	0
$592$	1.47886	0.0607806
$593$	23.8904	0.981061	0.490530	−	0.871424i	$-0.336803\pi$
$593$	23.8904	0.981061	0.490530	+	0.871424i	$0.336803\pi$
$594$	0	0
$595$	19.9166	0.816501
$596$	75.7958	3.10471
$597$	0	0
$598$	8.69560	0.355589
$599$	15.4206	0.630070	0.315035	−	0.949080i	$-0.397984\pi$
$599$	15.4206	0.630070	0.315035	+	0.949080i	$0.397984\pi$
$600$	0	0
$601$	−1.49280	−0.0608928	−0.0304464	−	0.999536i	$-0.509693\pi$
$601$	−1.49280	−0.0608928	−0.0304464	+	0.999536i	$0.509693\pi$
$602$	17.1040	0.697108
$603$	0	0
$604$	49.5503	2.01617
$605$	43.6479	1.77454
$606$	0	0
$607$	−4.44626	−0.180468	−0.0902340	−	0.995921i	$-0.528761\pi$
$607$	−4.44626	−0.180468	−0.0902340	+	0.995921i	$0.528761\pi$
$608$	−29.9271	−1.21370
$609$	0	0
$610$	−123.901	−5.01661
$611$	−9.03571	−0.365546
$612$	0	0
$613$	26.6640	1.07695	0.538474	−	0.842642i	$-0.319001\pi$
$613$	26.6640	1.07695	0.538474	+	0.842642i	$0.319001\pi$
$614$	39.5460	1.59595
$615$	0	0
$616$	−34.8087	−1.40248
$617$	−27.3495	−1.10105	−0.550525	−	0.834819i	$-0.685572\pi$
$617$	−27.3495	−1.10105	−0.550525	+	0.834819i	$0.685572\pi$
$618$	0	0
$619$	−9.24836	−0.371723	−0.185861	−	0.982576i	$-0.559508\pi$
$619$	−9.24836	−0.371723	−0.185861	+	0.982576i	$0.559508\pi$
$620$	−138.107	−5.54651
$621$	0	0
$622$	−11.4094	−0.457475
$623$	1.96953	0.0789076
$624$	0	0
$625$	18.4808	0.739233
$626$	3.89697	0.155754
$627$	0	0
$628$	−54.9535	−2.19288
$629$	0.808323	0.0322299
$630$	0	0
$631$	24.5134	0.975864	0.487932	−	0.872882i	$-0.337751\pi$
$631$	24.5134	0.975864	0.487932	+	0.872882i	$0.337751\pi$
$632$	−98.2003	−3.90620
$633$	0	0
$634$	60.5972	2.40662
$635$	66.6605	2.64534
$636$	0	0
$637$	1.00000	0.0396214
$638$	−11.4094	−0.451703
$639$	0	0
$640$	2.16980	0.0857689
$641$	−34.3972	−1.35861	−0.679304	−	0.733857i	$-0.737718\pi$
$641$	−34.3972	−1.35861	−0.679304	+	0.733857i	$0.737718\pi$
$642$	0	0
$643$	−7.69036	−0.303278	−0.151639	−	0.988436i	$-0.548455\pi$
$643$	−7.69036	−0.303278	−0.151639	+	0.988436i	$0.548455\pi$
$644$	−16.0378	−0.631979
$645$	0	0
$646$	−39.8332	−1.56722
$647$	−41.9123	−1.64774	−0.823872	−	0.566776i	$-0.808191\pi$
$647$	−41.9123	−1.64774	−0.823872	+	0.566776i	$0.808191\pi$
$648$	0	0
$649$	−18.7516	−0.736066
$650$	−24.9355	−0.978051
$651$	0	0
$652$	−64.8763	−2.54075
$653$	−34.2262	−1.33938	−0.669688	−	0.742642i	$-0.733572\pi$
$653$	−34.2262	−1.33938	−0.669688	+	0.742642i	$0.733572\pi$
$654$	0	0
$655$	−3.69036	−0.144194
$656$	−60.8149	−2.37442
$657$	0	0
$658$	23.5877	0.919545
$659$	−45.1685	−1.75951	−0.879757	−	0.475424i	$-0.842295\pi$
$659$	−45.1685	−1.75951	−0.879757	+	0.475424i	$0.842295\pi$
$660$	0	0
$661$	11.9839	0.466119	0.233059	−	0.972463i	$-0.425126\pi$
$661$	11.9839	0.466119	0.233059	+	0.972463i	$0.425126\pi$
$662$	−2.88203	−0.112013
$663$	0	0
$664$	−10.3331	−0.401004
$665$	11.1488	0.432333
$666$	0	0
$667$	−3.07315	−0.118993
$668$	−92.3968	−3.57494
$669$	0	0
$670$	106.176	4.10192
$671$	58.9415	2.27541
$672$	0	0
$673$	−29.5117	−1.13759	−0.568796	−	0.822479i	$-0.692591\pi$
$673$	−29.5117	−1.13759	−0.568796	+	0.822479i	$0.692591\pi$
$674$	11.0747	0.426581
$675$	0	0
$676$	4.81471	0.185181
$677$	31.0662	1.19397	0.596985	−	0.802252i	$-0.296365\pi$
$677$	31.0662	1.19397	0.596985	+	0.802252i	$0.296365\pi$
$678$	0	0
$679$	−2.11001	−0.0809747
$680$	−146.343	−5.61200
$681$	0	0
$682$	92.9904	3.56079
$683$	−5.79433	−0.221714	−0.110857	−	0.993836i	$-0.535360\pi$
$683$	−5.79433	−0.221714	−0.110857	+	0.993836i	$0.535360\pi$
$684$	0	0
$685$	−12.5744	−0.480441
$686$	−2.61050	−0.0996693
$687$	0	0
$688$	−62.5848	−2.38602
$689$	−8.55201	−0.325806
$690$	0	0
$691$	18.8617	0.717531	0.358766	−	0.933428i	$-0.383198\pi$
$691$	18.8617	0.717531	0.358766	+	0.933428i	$0.383198\pi$
$692$	93.9467	3.57132
$693$	0	0
$694$	36.6347	1.39063
$695$	1.41366	0.0536232
$696$	0	0
$697$	−33.2406	−1.25908
$698$	−10.7181	−0.405685
$699$	0	0
$700$	45.9901	1.73826
$701$	32.3180	1.22064	0.610318	−	0.792157i	$-0.291042\pi$
$701$	32.3180	1.22064	0.610318	+	0.792157i	$0.291042\pi$
$702$	0	0
$703$	0.452479	0.0170656
$704$	36.1328	1.36180
$705$	0	0
$706$	−41.9645	−1.57936
$707$	−0.850419	−0.0319833
$708$	0	0
$709$	4.72296	0.177374	0.0886872	−	0.996060i	$-0.471733\pi$
$709$	4.72296	0.177374	0.0886872	+	0.996060i	$0.471733\pi$
$710$	−65.5503	−2.46006
$711$	0	0
$712$	−14.4717	−0.542350
$713$	25.0472	0.938026
$714$	0	0
$715$	18.0714	0.675833
$716$	79.8548	2.98431
$717$	0	0
$718$	−47.3831	−1.76832
$719$	40.8902	1.52495	0.762474	−	0.647019i	$-0.223985\pi$
$719$	40.8902	1.52495	0.762474	+	0.647019i	$0.223985\pi$
$720$	0	0
$721$	−1.47460	−0.0549169
$722$	27.3018	1.01607
$723$	0	0
$724$	−13.5623	−0.504037
$725$	8.81258	0.327291
$726$	0	0
$727$	18.7292	0.694627	0.347313	−	0.937749i	$-0.387094\pi$
$727$	18.7292	0.694627	0.347313	+	0.937749i	$0.387094\pi$
$728$	−7.34780	−0.272328
$729$	0	0
$730$	77.0291	2.85098
$731$	−34.2080	−1.26523
$732$	0	0
$733$	−14.1100	−0.521165	−0.260583	−	0.965452i	$-0.583915\pi$
$733$	−14.1100	−0.521165	−0.260583	+	0.965452i	$0.583915\pi$
$734$	61.0738	2.25428
$735$	0	0
$736$	34.1093	1.25728
$737$	−50.5092	−1.86053
$738$	0	0
$739$	−49.9209	−1.83637	−0.918184	−	0.396154i	$-0.870345\pi$
$739$	−49.9209	−1.83637	−0.918184	+	0.396154i	$0.870345\pi$
$740$	2.84356	0.104531
$741$	0	0
$742$	22.3250	0.819577
$743$	−2.15868	−0.0791945	−0.0395972	−	0.999216i	$-0.512607\pi$
$743$	−2.15868	−0.0791945	−0.0395972	+	0.999216i	$0.512607\pi$
$744$	0	0
$745$	60.0532	2.20018
$746$	12.4042	0.454149
$747$	0	0
$748$	119.084	4.35415
$749$	−2.62418	−0.0958854
$750$	0	0
$751$	16.8372	0.614399	0.307199	−	0.951645i	$-0.400608\pi$
$751$	16.8372	0.614399	0.307199	+	0.951645i	$0.400608\pi$
$752$	−86.3092	−3.14737
$753$	0	0
$754$	−2.40842	−0.0877095
$755$	39.2588	1.42878
$756$	0	0
$757$	−17.6028	−0.639785	−0.319893	−	0.947454i	$-0.603647\pi$
$757$	−17.6028	−0.639785	−0.319893	+	0.947454i	$0.603647\pi$
$758$	17.7842	0.645953
$759$	0	0
$760$	−81.9193	−2.97153
$761$	−19.2179	−0.696648	−0.348324	−	0.937374i	$-0.613249\pi$
$761$	−19.2179	−0.696648	−0.348324	+	0.937374i	$0.613249\pi$
$762$	0	0
$763$	−17.9166	−0.648624
$764$	−72.9913	−2.64073
$765$	0	0
$766$	5.89310	0.212926
$767$	−3.95830	−0.142926
$768$	0	0
$769$	−14.0406	−0.506315	−0.253158	−	0.967425i	$-0.581469\pi$
$769$	−14.0406	−0.506315	−0.253158	+	0.967425i	$0.581469\pi$
$770$	−47.1754	−1.70008
$771$	0	0
$772$	0.313901	0.0112975
$773$	−22.4339	−0.806891	−0.403445	−	0.915004i	$-0.632187\pi$
$773$	−22.4339	−0.806891	−0.403445	+	0.915004i	$0.632187\pi$
$774$	0	0
$775$	−71.8255	−2.58005
$776$	15.5039	0.556558
$777$	0	0
$778$	15.6630	0.561546
$779$	−18.6073	−0.666676
$780$	0	0
$781$	31.1832	1.11582
$782$	45.3997	1.62349
$783$	0	0
$784$	9.55201	0.341143
$785$	−43.5398	−1.55400
$786$	0	0
$787$	−0.926847	−0.0330385	−0.0165193	−	0.999864i	$-0.505258\pi$
$787$	−0.926847	−0.0330385	−0.0165193	+	0.999864i	$0.505258\pi$
$788$	82.5311	2.94005
$789$	0	0
$790$	−133.089	−4.73508
$791$	−0.922589	−0.0328035
$792$	0	0
$793$	12.4420	0.441828
$794$	−65.2357	−2.31513
$795$	0	0
$796$	31.0164	1.09935
$797$	26.0154	0.921512	0.460756	−	0.887527i	$-0.347578\pi$
$797$	26.0154	0.921512	0.460756	+	0.887527i	$0.347578\pi$
$798$	0	0
$799$	−47.1754	−1.66895
$800$	−97.8118	−3.45817
$801$	0	0
$802$	72.1879	2.54904
$803$	−36.6438	−1.29313
$804$	0	0
$805$	−12.7068	−0.447857
$806$	19.6294	0.691417
$807$	0	0
$808$	6.24870	0.219829
$809$	44.8539	1.57698	0.788490	−	0.615048i	$-0.210863\pi$
$809$	44.8539	1.57698	0.788490	+	0.615048i	$0.210863\pi$
$810$	0	0
$811$	27.2511	0.956916	0.478458	−	0.878110i	$-0.341196\pi$
$811$	27.2511	0.956916	0.478458	+	0.878110i	$0.341196\pi$
$812$	4.44200	0.155884
$813$	0	0
$814$	−1.91463	−0.0671078
$815$	−51.4017	−1.80052
$816$	0	0
$817$	−19.1488	−0.669933
$818$	−34.3250	−1.20015
$819$	0	0
$820$	−116.936	−4.08357
$821$	−20.4003	−0.711975	−0.355988	−	0.934491i	$-0.615855\pi$
$821$	−20.4003	−0.711975	−0.355988	+	0.934491i	$0.615855\pi$
$822$	0	0
$823$	54.3509	1.89455	0.947276	−	0.320418i	$-0.103824\pi$
$823$	54.3509	1.89455	0.947276	+	0.320418i	$0.103824\pi$
$824$	10.8350	0.377457
$825$	0	0
$826$	10.3331	0.359536
$827$	−3.48272	−0.121106	−0.0605530	−	0.998165i	$-0.519286\pi$
$827$	−3.48272	−0.121106	−0.0605530	+	0.998165i	$0.519286\pi$
$828$	0	0
$829$	−29.8417	−1.03645	−0.518223	−	0.855246i	$-0.673406\pi$
$829$	−29.8417	−1.03645	−0.518223	+	0.855246i	$0.673406\pi$
$830$	−14.0043	−0.486095
$831$	0	0
$832$	7.62729	0.264429
$833$	5.22100	0.180897
$834$	0	0
$835$	−73.2063	−2.53341
$836$	66.6605	2.30550
$837$	0	0
$838$	−14.8658	−0.513530
$839$	43.0033	1.48464	0.742320	−	0.670045i	$-0.233725\pi$
$839$	43.0033	1.48464	0.742320	+	0.670045i	$0.233725\pi$
$840$	0	0
$841$	−28.1488	−0.970649
$842$	60.0953	2.07102
$843$	0	0
$844$	−77.1634	−2.65608
$845$	3.81471	0.131230
$846$	0	0
$847$	11.4420	0.393152
$848$	−81.6889	−2.80521
$849$	0	0
$850$	−130.188	−4.46542
$851$	−0.515711	−0.0176784
$852$	0	0
$853$	34.1023	1.16764	0.583820	−	0.811883i	$-0.301557\pi$
$853$	34.1023	1.16764	0.583820	+	0.811883i	$0.301557\pi$
$854$	−32.4798	−1.11144
$855$	0	0
$856$	19.2819	0.659043
$857$	45.7344	1.56226	0.781129	−	0.624370i	$-0.214644\pi$
$857$	45.7344	1.56226	0.781129	+	0.624370i	$0.214644\pi$
$858$	0	0
$859$	15.9861	0.545437	0.272719	−	0.962094i	$-0.412077\pi$
$859$	15.9861	0.545437	0.272719	+	0.962094i	$0.412077\pi$
$860$	−120.339	−4.10352
$861$	0	0
$862$	54.3100	1.84981
$863$	24.5096	0.834315	0.417157	−	0.908834i	$-0.363026\pi$
$863$	24.5096	0.834315	0.417157	+	0.908834i	$0.363026\pi$
$864$	0	0
$865$	74.4343	2.53084
$866$	−82.1807	−2.79261
$867$	0	0
$868$	−36.2038	−1.22884
$869$	63.3120	2.14771
$870$	0	0
$871$	−10.6620	−0.361269
$872$	131.648	4.45815
$873$	0	0
$874$	25.4137	0.859630
$875$	17.3646	0.587030
$876$	0	0
$877$	53.4913	1.80627	0.903136	−	0.429354i	$-0.141259\pi$
$877$	53.4913	1.80627	0.903136	+	0.429354i	$0.141259\pi$
$878$	−58.4258	−1.97177
$879$	0	0
$880$	172.618	5.81896
$881$	−26.1745	−0.881840	−0.440920	−	0.897546i	$-0.645348\pi$
$881$	−26.1745	−0.881840	−0.440920	+	0.897546i	$0.645348\pi$
$882$	0	0
$883$	23.1840	0.780202	0.390101	−	0.920772i	$-0.372440\pi$
$883$	23.1840	0.780202	0.390101	+	0.920772i	$0.372440\pi$
$884$	25.1376	0.845469
$885$	0	0
$886$	22.2536	0.747624
$887$	13.7008	0.460029	0.230015	−	0.973187i	$-0.426123\pi$
$887$	13.7008	0.460029	0.230015	+	0.973187i	$0.426123\pi$
$888$	0	0
$889$	17.4746	0.586079
$890$	−19.6132	−0.657435
$891$	0	0
$892$	96.4223	3.22846
$893$	−26.4077	−0.883699
$894$	0	0
$895$	63.2692	2.11486
$896$	0.568798	0.0190022
$897$	0	0
$898$	48.8534	1.63026
$899$	−6.93733	−0.231373
$900$	0	0
$901$	−44.6500	−1.48751
$902$	78.7354	2.62160
$903$	0	0
$904$	6.77900	0.225466
$905$	−10.7454	−0.357190
$906$	0	0
$907$	25.5621	0.848777	0.424388	−	0.905480i	$-0.360489\pi$
$907$	25.5621	0.848777	0.424388	+	0.905480i	$0.360489\pi$
$908$	−92.5249	−3.07055
$909$	0	0
$910$	−9.95830	−0.330114
$911$	−2.07643	−0.0687951	−0.0343976	−	0.999408i	$-0.510951\pi$
$911$	−2.07643	−0.0687951	−0.0343976	+	0.999408i	$0.510951\pi$
$912$	0	0
$913$	6.66202	0.220481
$914$	36.9036	1.22066
$915$	0	0
$916$	25.3198	0.836589
$917$	−0.967402	−0.0319464
$918$	0	0
$919$	17.8514	0.588863	0.294432	−	0.955673i	$-0.404870\pi$
$919$	17.8514	0.588863	0.294432	+	0.955673i	$0.404870\pi$
$920$	93.3672	3.07823
$921$	0	0
$922$	−6.98664	−0.230093
$923$	6.58248	0.216665
$924$	0	0
$925$	1.47886	0.0486245
$926$	6.60881	0.217179
$927$	0	0
$928$	−9.44724	−0.310121
$929$	37.0553	1.21575	0.607873	−	0.794034i	$-0.292023\pi$
$929$	37.0553	1.21575	0.607873	+	0.794034i	$0.292023\pi$
$930$	0	0
$931$	2.92259	0.0957840
$932$	128.184	4.19881
$933$	0	0
$934$	5.23212	0.171200
$935$	94.3509	3.08560
$936$	0	0
$937$	39.7540	1.29871	0.649354	−	0.760486i	$-0.275039\pi$
$937$	39.7540	1.29871	0.649354	+	0.760486i	$0.275039\pi$
$938$	27.8332	0.908786
$939$	0	0
$940$	−165.956	−5.41290
$941$	47.3530	1.54366	0.771832	−	0.635827i	$-0.219341\pi$
$941$	47.3530	1.54366	0.771832	+	0.635827i	$0.219341\pi$
$942$	0	0
$943$	21.2076	0.690614
$944$	−37.8097	−1.23060
$945$	0	0
$946$	81.0268	2.63441
$947$	−8.80446	−0.286106	−0.143053	−	0.989715i	$-0.545692\pi$
$947$	−8.80446	−0.286106	−0.143053	+	0.989715i	$0.545692\pi$
$948$	0	0
$949$	−7.73517	−0.251094
$950$	−72.8763	−2.36442
$951$	0	0
$952$	−38.3629	−1.24335
$953$	−7.72895	−0.250365	−0.125183	−	0.992134i	$-0.539952\pi$
$953$	−7.72895	−0.250365	−0.125183	+	0.992134i	$0.539952\pi$
$954$	0	0
$955$	−57.8312	−1.87137
$956$	−20.6601	−0.668196
$957$	0	0
$958$	−38.2722	−1.23652
$959$	−3.29628	−0.106442
$960$	0	0
$961$	25.5415	0.823920
$962$	−0.404161	−0.0130307
$963$	0	0
$964$	36.2243	1.16671
$965$	0.248705	0.00800608
$966$	0	0
$967$	14.3054	0.460030	0.230015	−	0.973187i	$-0.426122\pi$
$967$	14.3054	0.460030	0.230015	+	0.973187i	$0.426122\pi$
$968$	−84.0735	−2.70222
$969$	0	0
$970$	21.0121	0.674658
$971$	15.1692	0.486803	0.243402	−	0.969926i	$-0.421737\pi$
$971$	15.1692	0.486803	0.243402	+	0.969926i	$0.421737\pi$
$972$	0	0
$973$	0.370581	0.0118803
$974$	−99.7583	−3.19646
$975$	0	0
$976$	118.846	3.80417
$977$	8.25746	0.264180	0.132090	−	0.991238i	$-0.457831\pi$
$977$	8.25746	0.264180	0.132090	+	0.991238i	$0.457831\pi$
$978$	0	0
$979$	9.33026	0.298196
$980$	18.3667	0.586703
$981$	0	0
$982$	55.8016	1.78070
$983$	25.8525	0.824568	0.412284	−	0.911055i	$-0.364731\pi$
$983$	25.8525	0.824568	0.412284	+	0.911055i	$0.364731\pi$
$984$	0	0
$985$	65.3897	2.08349
$986$	−12.5744	−0.400449
$987$	0	0
$988$	14.0714	0.447671
$989$	21.8248	0.693988
$990$	0	0
$991$	−56.2100	−1.78557	−0.892785	−	0.450484i	$-0.851252\pi$
$991$	−56.2100	−1.78557	−0.892785	+	0.450484i	$0.851252\pi$
$992$	76.9981	2.44469
$993$	0	0
$994$	−17.1836	−0.545029
$995$	24.5744	0.779059
$996$	0	0
$997$	37.4789	1.18697	0.593484	−	0.804846i	$-0.297752\pi$
$997$	37.4789	1.18697	0.593484	+	0.804846i	$0.297752\pi$
$998$	−70.7237	−2.23872
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.k.1.1		4
3.2	odd	2		273.2.a.e.1.4	✓	4
7.6	odd	2		5733.2.a.bf.1.1		4
12.11	even	2		4368.2.a.br.1.1		4
15.14	odd	2		6825.2.a.bg.1.1		4
21.20	even	2		1911.2.a.s.1.4		4
39.38	odd	2		3549.2.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.4	✓	4	3.2	odd	2
819.2.a.k.1.1		4	1.1	even	1	trivial
1911.2.a.s.1.4		4	21.20	even	2
3549.2.a.w.1.1		4	39.38	odd	2
4368.2.a.br.1.1		4	12.11	even	2
5733.2.a.bf.1.1		4	7.6	odd	2
6825.2.a.bg.1.1		4	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)

\(f(q)\)	\(=\)	\(q-2.61050 q^{2} +4.81471 q^{4} +3.81471 q^{5} +1.00000 q^{7} -7.34780 q^{8} +O(q^{10})\)	\(q-2.61050 q^{2} +4.81471 q^{4} +3.81471 q^{5} +1.00000 q^{7} -7.34780 q^{8} -9.95830 q^{10} +4.73730 q^{11} +1.00000 q^{13} -2.61050 q^{14} +9.55201 q^{16} +5.22100 q^{17} +2.92259 q^{19} +18.3667 q^{20} -12.3667 q^{22} -3.33101 q^{23} +9.55201 q^{25} -2.61050 q^{26} +4.81471 q^{28} +0.922589 q^{29} -7.51941 q^{31} -10.2399 q^{32} -13.6294 q^{34} +3.81471 q^{35} +0.154821 q^{37} -7.62942 q^{38} -28.0297 q^{40} -6.36672 q^{41} -6.55201 q^{43} +22.8087 q^{44} +8.69560 q^{46} -9.03571 q^{47} +1.00000 q^{49} -24.9355 q^{50} +4.81471 q^{52} -8.55201 q^{53} +18.0714 q^{55} -7.34780 q^{56} -2.40842 q^{58} -3.95830 q^{59} +12.4420 q^{61} +19.6294 q^{62} +7.62729 q^{64} +3.81471 q^{65} -10.6620 q^{67} +25.1376 q^{68} -9.95830 q^{70} +6.58248 q^{71} -7.73517 q^{73} -0.404161 q^{74} +14.0714 q^{76} +4.73730 q^{77} +13.3646 q^{79} +36.4381 q^{80} +16.6203 q^{82} +1.40629 q^{83} +19.9166 q^{85} +17.1040 q^{86} -34.8087 q^{88} +1.96953 q^{89} +1.00000 q^{91} -16.0378 q^{92} +23.5877 q^{94} +11.1488 q^{95} -2.11001 q^{97} -2.61050 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8	\( 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29} + 3 q^{31} - 7 q^{32} - 30 q^{34} + 3 q^{35} + 10 q^{37} - 6 q^{38} - 14 q^{40} + 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} - 5 q^{47} + 4 q^{49} - 13 q^{50} + 7 q^{52} - 5 q^{53} + 10 q^{55} - 3 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{61} + 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} + 10 q^{68} - 4 q^{70} - 13 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 11 q^{79} + 42 q^{80} + 10 q^{82} - q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} + 5 q^{89} + 4 q^{91} - 34 q^{92} + 34 q^{94} - 13 q^{95} - 17 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 7 * q^4 + 3 * q^5 + 4 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^11 + 4 * q^13 - q^14 + 9 * q^16 + 2 * q^17 + 7 * q^19 + 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 - q^26 + 7 * q^28 - q^29 + 3 * q^31 - 7 * q^32 - 30 * q^34 + 3 * q^35 + 10 * q^37 - 6 * q^38 - 14 * q^40 + 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 - 5 * q^47 + 4 * q^49 - 13 * q^50 + 7 * q^52 - 5 * q^53 + 10 * q^55 - 3 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^61 + 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 + 10 * q^68 - 4 * q^70 - 13 * q^73 + 6 * q^74 - 6 * q^76 + 2 * q^77 + 11 * q^79 + 42 * q^80 + 10 * q^82 - q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 + 5 * q^89 + 4 * q^91 - 34 * q^92 + 34 * q^94 - 13 * q^95 - 17 * q^97 - q^98

Introduction

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