LMFDB - Embedded newform 855.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -4.42864 q^{7} +1.21432 q^{8} +O(q^{10})$	\(q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -4.42864 q^{7} +1.21432 q^{8} +0.311108 q^{10} +2.62222 q^{11} +0.474572 q^{13} +1.37778 q^{14} +3.42864 q^{16} -5.05086 q^{17} -1.00000 q^{19} +1.90321 q^{20} -0.815792 q^{22} +1.37778 q^{23} +1.00000 q^{25} -0.147643 q^{26} +8.42864 q^{28} +7.80642 q^{29} +1.24443 q^{31} -3.49532 q^{32} +1.57136 q^{34} +4.42864 q^{35} +4.47457 q^{37} +0.311108 q^{38} -1.21432 q^{40} +5.05086 q^{41} +12.0415 q^{43} -4.99063 q^{44} -0.428639 q^{46} +4.42864 q^{47} +12.6128 q^{49} -0.311108 q^{50} -0.903212 q^{52} -7.52543 q^{53} -2.62222 q^{55} -5.37778 q^{56} -2.42864 q^{58} +2.19358 q^{59} +3.67307 q^{61} -0.387152 q^{62} -5.76986 q^{64} -0.474572 q^{65} -1.65878 q^{67} +9.61285 q^{68} -1.37778 q^{70} -7.61285 q^{71} -3.80642 q^{73} -1.39207 q^{74} +1.90321 q^{76} -11.6128 q^{77} -13.4193 q^{79} -3.42864 q^{80} -1.57136 q^{82} +10.6222 q^{83} +5.05086 q^{85} -3.74620 q^{86} +3.18421 q^{88} +12.6637 q^{89} -2.10171 q^{91} -2.62222 q^{92} -1.37778 q^{94} +1.00000 q^{95} +17.8938 q^{97} -3.92396 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31} + 3 q^{32} + 18 q^{34} + 20 q^{37} + q^{38} + 3 q^{40} + 2 q^{41} - 4 q^{43} + 12 q^{44} + 12 q^{46} + 11 q^{49} - q^{50} + 4 q^{52} - 16 q^{53} - 8 q^{55} - 16 q^{56} + 6 q^{58} + 20 q^{59} - 2 q^{61} - 28 q^{62} - 11 q^{64} - 8 q^{65} + 2 q^{67} + 2 q^{68} - 4 q^{70} + 4 q^{71} + 2 q^{73} + 2 q^{74} - q^{76} - 8 q^{77} + 3 q^{80} - 18 q^{82} + 32 q^{83} + 2 q^{85} + 16 q^{86} - 4 q^{88} - 2 q^{89} + 20 q^{91} - 8 q^{92} - 4 q^{94} + 3 q^{95} + 20 q^{97} + 15 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * 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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.42864	−1.67387	−0.836934	−	0.547304i	$-0.815654\pi$
$7$	−4.42864	−1.67387	−0.836934	+	0.547304i	$0.815654\pi$
$8$	1.21432	0.429327
$9$	0	0
$10$	0.311108	0.0983809
$11$	2.62222	0.790628	0.395314	−	0.918546i	$-0.370636\pi$
$11$	2.62222	0.790628	0.395314	+	0.918546i	$0.370636\pi$
$12$	0	0
$13$	0.474572	0.131623	0.0658114	−	0.997832i	$-0.479036\pi$
$13$	0.474572	0.131623	0.0658114	+	0.997832i	$0.479036\pi$
$14$	1.37778	0.368228
$15$	0	0
$16$	3.42864	0.857160
$17$	−5.05086	−1.22501	−0.612506	−	0.790466i	$-0.709839\pi$
$17$	−5.05086	−1.22501	−0.612506	+	0.790466i	$0.709839\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.90321	0.425571
$21$	0	0
$22$	−0.815792	−0.173927
$23$	1.37778	0.287288	0.143644	−	0.989629i	$-0.454118\pi$
$23$	1.37778	0.287288	0.143644	+	0.989629i	$0.454118\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.147643	−0.0289552
$27$	0	0
$28$	8.42864	1.59286
$29$	7.80642	1.44962	0.724808	−	0.688951i	$-0.241928\pi$
$29$	7.80642	1.44962	0.724808	+	0.688951i	$0.241928\pi$
$30$	0	0
$31$	1.24443	0.223506	0.111753	−	0.993736i	$-0.464353\pi$
$31$	1.24443	0.223506	0.111753	+	0.993736i	$0.464353\pi$
$32$	−3.49532	−0.617890
$33$	0	0
$34$	1.57136	0.269486
$35$	4.42864	0.748577
$36$	0	0
$37$	4.47457	0.735615	0.367808	−	0.929902i	$-0.380109\pi$
$37$	4.47457	0.735615	0.367808	+	0.929902i	$0.380109\pi$
$38$	0.311108	0.0504684
$39$	0	0
$40$	−1.21432	−0.192001
$41$	5.05086	0.788811	0.394406	−	0.918936i	$-0.370951\pi$
$41$	5.05086	0.788811	0.394406	+	0.918936i	$0.370951\pi$
$42$	0	0
$43$	12.0415	1.83631	0.918155	−	0.396222i	$-0.129679\pi$
$43$	12.0415	1.83631	0.918155	+	0.396222i	$0.129679\pi$
$44$	−4.99063	−0.752366
$45$	0	0
$46$	−0.428639	−0.0631994
$47$	4.42864	0.645983	0.322992	−	0.946402i	$-0.395311\pi$
$47$	4.42864	0.645983	0.322992	+	0.946402i	$0.395311\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	−0.311108	−0.0439973
$51$	0	0
$52$	−0.903212	−0.125253
$53$	−7.52543	−1.03370	−0.516848	−	0.856077i	$-0.672895\pi$
$53$	−7.52543	−1.03370	−0.516848	+	0.856077i	$0.672895\pi$
$54$	0	0
$55$	−2.62222	−0.353579
$56$	−5.37778	−0.718637
$57$	0	0
$58$	−2.42864	−0.318896
$59$	2.19358	0.285579	0.142790	−	0.989753i	$-0.454393\pi$
$59$	2.19358	0.285579	0.142790	+	0.989753i	$0.454393\pi$
$60$	0	0
$61$	3.67307	0.470289	0.235144	−	0.971960i	$-0.424444\pi$
$61$	3.67307	0.470289	0.235144	+	0.971960i	$0.424444\pi$
$62$	−0.387152	−0.0491684
$63$	0	0
$64$	−5.76986	−0.721232
$65$	−0.474572	−0.0588635
$66$	0	0
$67$	−1.65878	−0.202652	−0.101326	−	0.994853i	$-0.532309\pi$
$67$	−1.65878	−0.202652	−0.101326	+	0.994853i	$0.532309\pi$
$68$	9.61285	1.16573
$69$	0	0
$70$	−1.37778	−0.164677
$71$	−7.61285	−0.903479	−0.451739	−	0.892150i	$-0.649196\pi$
$71$	−7.61285	−0.903479	−0.451739	+	0.892150i	$0.649196\pi$
$72$	0	0
$73$	−3.80642	−0.445508	−0.222754	−	0.974875i	$-0.571505\pi$
$73$	−3.80642	−0.445508	−0.222754	+	0.974875i	$0.571505\pi$
$74$	−1.39207	−0.161825
$75$	0	0
$76$	1.90321	0.218313
$77$	−11.6128	−1.32341
$78$	0	0
$79$	−13.4193	−1.50979	−0.754893	−	0.655848i	$-0.772311\pi$
$79$	−13.4193	−1.50979	−0.754893	+	0.655848i	$0.772311\pi$
$80$	−3.42864	−0.383334
$81$	0	0
$82$	−1.57136	−0.173528
$83$	10.6222	1.16594	0.582970	−	0.812494i	$-0.301891\pi$
$83$	10.6222	1.16594	0.582970	+	0.812494i	$0.301891\pi$
$84$	0	0
$85$	5.05086	0.547842
$86$	−3.74620	−0.403963
$87$	0	0
$88$	3.18421	0.339438
$89$	12.6637	1.34235	0.671175	−	0.741299i	$-0.265790\pi$
$89$	12.6637	1.34235	0.671175	+	0.741299i	$0.265790\pi$
$90$	0	0
$91$	−2.10171	−0.220319
$92$	−2.62222	−0.273385
$93$	0	0
$94$	−1.37778	−0.142108
$95$	1.00000	0.102598
$96$	0	0
$97$	17.8938	1.81684	0.908422	−	0.418054i	$-0.137288\pi$
$97$	17.8938	1.81684	0.908422	+	0.418054i	$0.137288\pi$
$98$	−3.92396	−0.396379
$99$	0	0
$100$	−1.90321	−0.190321
$101$	10.4286	1.03769	0.518844	−	0.854869i	$-0.326362\pi$
$101$	10.4286	1.03769	0.518844	+	0.854869i	$0.326362\pi$
$102$	0	0
$103$	−5.65878	−0.557576	−0.278788	−	0.960353i	$-0.589933\pi$
$103$	−5.65878	−0.557576	−0.278788	+	0.960353i	$0.589933\pi$
$104$	0.576283	0.0565092
$105$	0	0
$106$	2.34122	0.227399
$107$	−6.90321	−0.667359	−0.333679	−	0.942687i	$-0.608290\pi$
$107$	−6.90321	−0.667359	−0.333679	+	0.942687i	$0.608290\pi$
$108$	0	0
$109$	5.61285	0.537613	0.268807	−	0.963194i	$-0.413371\pi$
$109$	5.61285	0.537613	0.268807	+	0.963194i	$0.413371\pi$
$110$	0.815792	0.0777827
$111$	0	0
$112$	−15.1842	−1.43477
$113$	−13.8938	−1.30702	−0.653511	−	0.756917i	$-0.726705\pi$
$113$	−13.8938	−1.30702	−0.653511	+	0.756917i	$0.726705\pi$
$114$	0	0
$115$	−1.37778	−0.128479
$116$	−14.8573	−1.37946
$117$	0	0
$118$	−0.682439	−0.0628236
$119$	22.3684	2.05051
$120$	0	0
$121$	−4.12399	−0.374908
$122$	−1.14272	−0.103457
$123$	0	0
$124$	−2.36842	−0.212690
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.19850	−0.638763	−0.319382	−	0.947626i	$-0.603475\pi$
$127$	−7.19850	−0.638763	−0.319382	+	0.947626i	$0.603475\pi$
$128$	8.78568	0.776552
$129$	0	0
$130$	0.147643	0.0129492
$131$	2.10171	0.183627	0.0918136	−	0.995776i	$-0.470734\pi$
$131$	2.10171	0.183627	0.0918136	+	0.995776i	$0.470734\pi$
$132$	0	0
$133$	4.42864	0.384012
$134$	0.516060	0.0445808
$135$	0	0
$136$	−6.13335	−0.525931
$137$	−1.70471	−0.145644	−0.0728218	−	0.997345i	$-0.523200\pi$
$137$	−1.70471	−0.145644	−0.0728218	+	0.997345i	$0.523200\pi$
$138$	0	0
$139$	8.72393	0.739954	0.369977	−	0.929041i	$-0.379366\pi$
$139$	8.72393	0.739954	0.369977	+	0.929041i	$0.379366\pi$
$140$	−8.42864	−0.712350
$141$	0	0
$142$	2.36842	0.198753
$143$	1.24443	0.104065
$144$	0	0
$145$	−7.80642	−0.648288
$146$	1.18421	0.0980058
$147$	0	0
$148$	−8.51606	−0.700016
$149$	6.81579	0.558371	0.279186	−	0.960237i	$-0.409935\pi$
$149$	6.81579	0.558371	0.279186	+	0.960237i	$0.409935\pi$
$150$	0	0
$151$	−5.80642	−0.472520	−0.236260	−	0.971690i	$-0.575922\pi$
$151$	−5.80642	−0.472520	−0.236260	+	0.971690i	$0.575922\pi$
$152$	−1.21432	−0.0984943
$153$	0	0
$154$	3.61285	0.291132
$155$	−1.24443	−0.0999551
$156$	0	0
$157$	0.193576	0.0154491	0.00772453	−	0.999970i	$-0.497541\pi$
$157$	0.193576	0.0154491	0.00772453	+	0.999970i	$0.497541\pi$
$158$	4.17484	0.332132
$159$	0	0
$160$	3.49532	0.276329
$161$	−6.10171	−0.480882
$162$	0	0
$163$	−8.42864	−0.660182	−0.330091	−	0.943949i	$-0.607079\pi$
$163$	−8.42864	−0.660182	−0.330091	+	0.943949i	$0.607079\pi$
$164$	−9.61285	−0.750637
$165$	0	0
$166$	−3.30465	−0.256491
$167$	−12.4429	−0.962863	−0.481431	−	0.876484i	$-0.659883\pi$
$167$	−12.4429	−0.962863	−0.481431	+	0.876484i	$0.659883\pi$
$168$	0	0
$169$	−12.7748	−0.982675
$170$	−1.57136	−0.120518
$171$	0	0
$172$	−22.9175	−1.74744
$173$	22.1891	1.68701	0.843504	−	0.537123i	$-0.180489\pi$
$173$	22.1891	1.68701	0.843504	+	0.537123i	$0.180489\pi$
$174$	0	0
$175$	−4.42864	−0.334774
$176$	8.99063	0.677694
$177$	0	0
$178$	−3.93978	−0.295299
$179$	11.9081	0.890056	0.445028	−	0.895517i	$-0.353194\pi$
$179$	11.9081	0.890056	0.445028	+	0.895517i	$0.353194\pi$
$180$	0	0
$181$	17.6128	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$181$	17.6128	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$182$	0.653858	0.0484672
$183$	0	0
$184$	1.67307	0.123340
$185$	−4.47457	−0.328977
$186$	0	0
$187$	−13.2444	−0.968529
$188$	−8.42864	−0.614722
$189$	0	0
$190$	−0.311108	−0.0225701
$191$	0.266706	0.0192982	0.00964909	−	0.999953i	$-0.496929\pi$
$191$	0.266706	0.0192982	0.00964909	+	0.999953i	$0.496929\pi$
$192$	0	0
$193$	2.66815	0.192058	0.0960288	−	0.995379i	$-0.469386\pi$
$193$	2.66815	0.192058	0.0960288	+	0.995379i	$0.469386\pi$
$194$	−5.56691	−0.399681
$195$	0	0
$196$	−24.0049	−1.71464
$197$	5.34614	0.380897	0.190448	−	0.981697i	$-0.439006\pi$
$197$	5.34614	0.380897	0.190448	+	0.981697i	$0.439006\pi$
$198$	0	0
$199$	17.1240	1.21389	0.606944	−	0.794745i	$-0.292395\pi$
$199$	17.1240	1.21389	0.606944	+	0.794745i	$0.292395\pi$
$200$	1.21432	0.0858654
$201$	0	0
$202$	−3.24443	−0.228277
$203$	−34.5718	−2.42647
$204$	0	0
$205$	−5.05086	−0.352767
$206$	1.76049	0.122659
$207$	0	0
$208$	1.62714	0.112822
$209$	−2.62222	−0.181382
$210$	0	0
$211$	13.1526	0.905460	0.452730	−	0.891648i	$-0.350450\pi$
$211$	13.1526	0.905460	0.452730	+	0.891648i	$0.350450\pi$
$212$	14.3225	0.983672
$213$	0	0
$214$	2.14764	0.146810
$215$	−12.0415	−0.821223
$216$	0	0
$217$	−5.51114	−0.374120
$218$	−1.74620	−0.118268
$219$	0	0
$220$	4.99063	0.336468
$221$	−2.39700	−0.161239
$222$	0	0
$223$	−10.5161	−0.704207	−0.352104	−	0.935961i	$-0.614534\pi$
$223$	−10.5161	−0.704207	−0.352104	+	0.935961i	$0.614534\pi$
$224$	15.4795	1.03427
$225$	0	0
$226$	4.32248	0.287527
$227$	−6.90321	−0.458182	−0.229091	−	0.973405i	$-0.573575\pi$
$227$	−6.90321	−0.458182	−0.229091	+	0.973405i	$0.573575\pi$
$228$	0	0
$229$	18.0415	1.19222	0.596108	−	0.802905i	$-0.296713\pi$
$229$	18.0415	1.19222	0.596108	+	0.802905i	$0.296713\pi$
$230$	0.428639	0.0282637
$231$	0	0
$232$	9.47949	0.622359
$233$	12.3684	0.810282	0.405141	−	0.914254i	$-0.367222\pi$
$233$	12.3684	0.810282	0.405141	+	0.914254i	$0.367222\pi$
$234$	0	0
$235$	−4.42864	−0.288893
$236$	−4.17484	−0.271759
$237$	0	0
$238$	−6.95899	−0.451084
$239$	−24.8573	−1.60788	−0.803942	−	0.594708i	$-0.797268\pi$
$239$	−24.8573	−1.60788	−0.803942	+	0.594708i	$0.797268\pi$
$240$	0	0
$241$	9.05086	0.583017	0.291508	−	0.956568i	$-0.405843\pi$
$241$	9.05086	0.583017	0.291508	+	0.956568i	$0.405843\pi$
$242$	1.28300	0.0824746
$243$	0	0
$244$	−6.99063	−0.447529
$245$	−12.6128	−0.805805
$246$	0	0
$247$	−0.474572	−0.0301963
$248$	1.51114	0.0959573
$249$	0	0
$250$	0.311108	0.0196762
$251$	22.5718	1.42472	0.712361	−	0.701813i	$-0.247626\pi$
$251$	22.5718	1.42472	0.712361	+	0.701813i	$0.247626\pi$
$252$	0	0
$253$	3.61285	0.227138
$254$	2.23951	0.140519
$255$	0	0
$256$	8.80642	0.550401
$257$	−4.94470	−0.308442	−0.154221	−	0.988036i	$-0.549287\pi$
$257$	−4.94470	−0.308442	−0.154221	+	0.988036i	$0.549287\pi$
$258$	0	0
$259$	−19.8163	−1.23132
$260$	0.903212	0.0560148
$261$	0	0
$262$	−0.653858	−0.0403955
$263$	−9.37778	−0.578259	−0.289129	−	0.957290i	$-0.593366\pi$
$263$	−9.37778	−0.578259	−0.289129	+	0.957290i	$0.593366\pi$
$264$	0	0
$265$	7.52543	0.462283
$266$	−1.37778	−0.0844774
$267$	0	0
$268$	3.15701	0.192845
$269$	−19.7146	−1.20202	−0.601009	−	0.799242i	$-0.705234\pi$
$269$	−19.7146	−1.20202	−0.601009	+	0.799242i	$0.705234\pi$
$270$	0	0
$271$	−1.11108	−0.0674932	−0.0337466	−	0.999430i	$-0.510744\pi$
$271$	−1.11108	−0.0674932	−0.0337466	+	0.999430i	$0.510744\pi$
$272$	−17.3176	−1.05003
$273$	0	0
$274$	0.530350	0.0320396
$275$	2.62222	0.158126
$276$	0	0
$277$	5.52098	0.331724	0.165862	−	0.986149i	$-0.446959\pi$
$277$	5.52098	0.331724	0.165862	+	0.986149i	$0.446959\pi$
$278$	−2.71408	−0.162780
$279$	0	0
$280$	5.37778	0.321384
$281$	15.8064	0.942932	0.471466	−	0.881884i	$-0.343725\pi$
$281$	15.8064	0.942932	0.471466	+	0.881884i	$0.343725\pi$
$282$	0	0
$283$	14.2351	0.846187	0.423093	−	0.906086i	$-0.360944\pi$
$283$	14.2351	0.846187	0.423093	+	0.906086i	$0.360944\pi$
$284$	14.4889	0.859756
$285$	0	0
$286$	−0.387152	−0.0228928
$287$	−22.3684	−1.32037
$288$	0	0
$289$	8.51114	0.500655
$290$	2.42864	0.142615
$291$	0	0
$292$	7.24443	0.423948
$293$	−7.52543	−0.439640	−0.219820	−	0.975540i	$-0.570547\pi$
$293$	−7.52543	−0.439640	−0.219820	+	0.975540i	$0.570547\pi$
$294$	0	0
$295$	−2.19358	−0.127715
$296$	5.43356	0.315819
$297$	0	0
$298$	−2.12045	−0.122834
$299$	0.653858	0.0378136
$300$	0	0
$301$	−53.3274	−3.07374
$302$	1.80642	0.103948
$303$	0	0
$304$	−3.42864	−0.196646
$305$	−3.67307	−0.210319
$306$	0	0
$307$	−2.81135	−0.160452	−0.0802260	−	0.996777i	$-0.525564\pi$
$307$	−2.81135	−0.160452	−0.0802260	+	0.996777i	$0.525564\pi$
$308$	22.1017	1.25936
$309$	0	0
$310$	0.387152	0.0219888
$311$	13.8479	0.785243	0.392621	−	0.919700i	$-0.371568\pi$
$311$	13.8479	0.785243	0.392621	+	0.919700i	$0.371568\pi$
$312$	0	0
$313$	23.2444	1.31385	0.656926	−	0.753955i	$-0.271856\pi$
$313$	23.2444	1.31385	0.656926	+	0.753955i	$0.271856\pi$
$314$	−0.0602231	−0.00339858
$315$	0	0
$316$	25.5397	1.43672
$317$	−2.96343	−0.166443	−0.0832215	−	0.996531i	$-0.526521\pi$
$317$	−2.96343	−0.166443	−0.0832215	+	0.996531i	$0.526521\pi$
$318$	0	0
$319$	20.4701	1.14611
$320$	5.76986	0.322545
$321$	0	0
$322$	1.89829	0.105788
$323$	5.05086	0.281037
$324$	0	0
$325$	0.474572	0.0263245
$326$	2.62222	0.145231
$327$	0	0
$328$	6.13335	0.338658
$329$	−19.6128	−1.08129
$330$	0	0
$331$	−0.949145	−0.0521697	−0.0260849	−	0.999660i	$-0.508304\pi$
$331$	−0.949145	−0.0521697	−0.0260849	+	0.999660i	$0.508304\pi$
$332$	−20.2163	−1.10952
$333$	0	0
$334$	3.87109	0.211817
$335$	1.65878	0.0906289
$336$	0	0
$337$	2.28100	0.124254	0.0621269	−	0.998068i	$-0.480212\pi$
$337$	2.28100	0.124254	0.0621269	+	0.998068i	$0.480212\pi$
$338$	3.97433	0.216175
$339$	0	0
$340$	−9.61285	−0.521330
$341$	3.26317	0.176710
$342$	0	0
$343$	−24.8573	−1.34217
$344$	14.6222	0.788377
$345$	0	0
$346$	−6.90321	−0.371119
$347$	−12.3368	−0.662273	−0.331136	−	0.943583i	$-0.607432\pi$
$347$	−12.3368	−0.662273	−0.331136	+	0.943583i	$0.607432\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.37778	0.0736457
$351$	0	0
$352$	−9.16547	−0.488521
$353$	−2.56199	−0.136361	−0.0681806	−	0.997673i	$-0.521719\pi$
$353$	−2.56199	−0.136361	−0.0681806	+	0.997673i	$0.521719\pi$
$354$	0	0
$355$	7.61285	0.404048
$356$	−24.1017	−1.27739
$357$	0	0
$358$	−3.70471	−0.195800
$359$	−24.3368	−1.28445	−0.642223	−	0.766518i	$-0.721988\pi$
$359$	−24.3368	−1.28445	−0.642223	+	0.766518i	$0.721988\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.47949	−0.287996
$363$	0	0
$364$	4.00000	0.209657
$365$	3.80642	0.199237
$366$	0	0
$367$	−4.42864	−0.231173	−0.115587	−	0.993297i	$-0.536875\pi$
$367$	−4.42864	−0.231173	−0.115587	+	0.993297i	$0.536875\pi$
$368$	4.72393	0.246252
$369$	0	0
$370$	1.39207	0.0723705
$371$	33.3274	1.73027
$372$	0	0
$373$	−23.7003	−1.22715	−0.613577	−	0.789635i	$-0.710270\pi$
$373$	−23.7003	−1.22715	−0.613577	+	0.789635i	$0.710270\pi$
$374$	4.12045	0.213063
$375$	0	0
$376$	5.37778	0.277338
$377$	3.70471	0.190802
$378$	0	0
$379$	8.20342	0.421381	0.210691	−	0.977553i	$-0.432429\pi$
$379$	8.20342	0.421381	0.210691	+	0.977553i	$0.432429\pi$
$380$	−1.90321	−0.0976327
$381$	0	0
$382$	−0.0829744	−0.00424534
$383$	20.9131	1.06861	0.534304	−	0.845293i	$-0.320574\pi$
$383$	20.9131	1.06861	0.534304	+	0.845293i	$0.320574\pi$
$384$	0	0
$385$	11.6128	0.591846
$386$	−0.830082	−0.0422501
$387$	0	0
$388$	−34.0558	−1.72892
$389$	24.1017	1.22201	0.611003	−	0.791629i	$-0.290766\pi$
$389$	24.1017	1.22201	0.611003	+	0.791629i	$0.290766\pi$
$390$	0	0
$391$	−6.95899	−0.351931
$392$	15.3160	0.773576
$393$	0	0
$394$	−1.66323	−0.0837921
$395$	13.4193	0.675197
$396$	0	0
$397$	7.92687	0.397838	0.198919	−	0.980016i	$-0.436257\pi$
$397$	7.92687	0.397838	0.198919	+	0.980016i	$0.436257\pi$
$398$	−5.32741	−0.267039
$399$	0	0
$400$	3.42864	0.171432
$401$	32.5718	1.62656	0.813280	−	0.581873i	$-0.197680\pi$
$401$	32.5718	1.62656	0.813280	+	0.581873i	$0.197680\pi$
$402$	0	0
$403$	0.590573	0.0294185
$404$	−19.8479	−0.987470
$405$	0	0
$406$	10.7556	0.533790
$407$	11.7333	0.581598
$408$	0	0
$409$	36.3684	1.79830	0.899151	−	0.437638i	$-0.144185\pi$
$409$	36.3684	1.79830	0.899151	+	0.437638i	$0.144185\pi$
$410$	1.57136	0.0776040
$411$	0	0
$412$	10.7699	0.530593
$413$	−9.71456	−0.478022
$414$	0	0
$415$	−10.6222	−0.521424
$416$	−1.65878	−0.0813284
$417$	0	0
$418$	0.815792	0.0399017
$419$	31.6958	1.54844	0.774221	−	0.632915i	$-0.218142\pi$
$419$	31.6958	1.54844	0.774221	+	0.632915i	$0.218142\pi$
$420$	0	0
$421$	37.4005	1.82279	0.911395	−	0.411532i	$-0.135006\pi$
$421$	37.4005	1.82279	0.911395	+	0.411532i	$0.135006\pi$
$422$	−4.09187	−0.199189
$423$	0	0
$424$	−9.13828	−0.443794
$425$	−5.05086	−0.245002
$426$	0	0
$427$	−16.2667	−0.787201
$428$	13.1383	0.635063
$429$	0	0
$430$	3.74620	0.180658
$431$	−4.94914	−0.238392	−0.119196	−	0.992871i	$-0.538032\pi$
$431$	−4.94914	−0.238392	−0.119196	+	0.992871i	$0.538032\pi$
$432$	0	0
$433$	32.3827	1.55621	0.778107	−	0.628132i	$-0.216180\pi$
$433$	32.3827	1.55621	0.778107	+	0.628132i	$0.216180\pi$
$434$	1.71456	0.0823014
$435$	0	0
$436$	−10.6824	−0.511596
$437$	−1.37778	−0.0659084
$438$	0	0
$439$	10.0731	0.480764	0.240382	−	0.970678i	$-0.422727\pi$
$439$	10.0731	0.480764	0.240382	+	0.970678i	$0.422727\pi$
$440$	−3.18421	−0.151801
$441$	0	0
$442$	0.745724	0.0354705
$443$	13.9684	0.663657	0.331828	−	0.943340i	$-0.392335\pi$
$443$	13.9684	0.663657	0.331828	+	0.943340i	$0.392335\pi$
$444$	0	0
$445$	−12.6637	−0.600317
$446$	3.27163	0.154916
$447$	0	0
$448$	25.5526	1.20725
$449$	−24.5718	−1.15962	−0.579808	−	0.814753i	$-0.696873\pi$
$449$	−24.5718	−1.15962	−0.579808	+	0.814753i	$0.696873\pi$
$450$	0	0
$451$	13.2444	0.623656
$452$	26.4429	1.24377
$453$	0	0
$454$	2.14764	0.100794
$455$	2.10171	0.0985297
$456$	0	0
$457$	−3.51114	−0.164244	−0.0821220	−	0.996622i	$-0.526170\pi$
$457$	−3.51114	−0.164244	−0.0821220	+	0.996622i	$0.526170\pi$
$458$	−5.61285	−0.262271
$459$	0	0
$460$	2.62222	0.122261
$461$	−10.2034	−0.475221	−0.237610	−	0.971361i	$-0.576364\pi$
$461$	−10.2034	−0.475221	−0.237610	+	0.971361i	$0.576364\pi$
$462$	0	0
$463$	8.33677	0.387443	0.193721	−	0.981057i	$-0.437944\pi$
$463$	8.33677	0.387443	0.193721	+	0.981057i	$0.437944\pi$
$464$	26.7654	1.24255
$465$	0	0
$466$	−3.84791	−0.178251
$467$	42.7052	1.97616	0.988080	−	0.153940i	$-0.0491961\pi$
$467$	42.7052	1.97616	0.988080	+	0.153940i	$0.0491961\pi$
$468$	0	0
$469$	7.34614	0.339213
$470$	1.37778	0.0635525
$471$	0	0
$472$	2.66370	0.122607
$473$	31.5754	1.45184
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−42.5718	−1.95128
$477$	0	0
$478$	7.73329	0.353713
$479$	−41.4608	−1.89439	−0.947195	−	0.320658i	$-0.896096\pi$
$479$	−41.4608	−1.89439	−0.947195	+	0.320658i	$0.896096\pi$
$480$	0	0
$481$	2.12351	0.0968237
$482$	−2.81579	−0.128256
$483$	0	0
$484$	7.84882	0.356764
$485$	−17.8938	−0.812518
$486$	0	0
$487$	30.0370	1.36111	0.680554	−	0.732698i	$-0.261739\pi$
$487$	30.0370	1.36111	0.680554	+	0.732698i	$0.261739\pi$
$488$	4.46028	0.201907
$489$	0	0
$490$	3.92396	0.177266
$491$	−15.3461	−0.692562	−0.346281	−	0.938131i	$-0.612556\pi$
$491$	−15.3461	−0.692562	−0.346281	+	0.938131i	$0.612556\pi$
$492$	0	0
$493$	−39.4291	−1.77580
$494$	0.147643	0.00664278
$495$	0	0
$496$	4.26671	0.191581
$497$	33.7146	1.51230
$498$	0	0
$499$	−25.8479	−1.15711	−0.578556	−	0.815643i	$-0.696383\pi$
$499$	−25.8479	−1.15711	−0.578556	+	0.815643i	$0.696383\pi$
$500$	1.90321	0.0851142
$501$	0	0
$502$	−7.02227	−0.313419
$503$	4.40006	0.196189	0.0980945	−	0.995177i	$-0.468725\pi$
$503$	4.40006	0.196189	0.0980945	+	0.995177i	$0.468725\pi$
$504$	0	0
$505$	−10.4286	−0.464068
$506$	−1.12399	−0.0499672
$507$	0	0
$508$	13.7003	0.607851
$509$	27.2355	1.20719	0.603597	−	0.797290i	$-0.293734\pi$
$509$	27.2355	1.20719	0.603597	+	0.797290i	$0.293734\pi$
$510$	0	0
$511$	16.8573	0.745722
$512$	−20.3111	−0.897633
$513$	0	0
$514$	1.53833	0.0678530
$515$	5.65878	0.249356
$516$	0	0
$517$	11.6128	0.510732
$518$	6.16500	0.270874
$519$	0	0
$520$	−0.576283	−0.0252717
$521$	−38.5531	−1.68904	−0.844521	−	0.535522i	$-0.820115\pi$
$521$	−38.5531	−1.68904	−0.844521	+	0.535522i	$0.820115\pi$
$522$	0	0
$523$	18.1575	0.793971	0.396986	−	0.917825i	$-0.370056\pi$
$523$	18.1575	0.793971	0.396986	+	0.917825i	$0.370056\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	2.91750	0.127209
$527$	−6.28544	−0.273798
$528$	0	0
$529$	−21.1017	−0.917466
$530$	−2.34122	−0.101696
$531$	0	0
$532$	−8.42864	−0.365428
$533$	2.39700	0.103825
$534$	0	0
$535$	6.90321	0.298452
$536$	−2.01429	−0.0870041
$537$	0	0
$538$	6.13335	0.264428
$539$	33.0736	1.42458
$540$	0	0
$541$	−13.7748	−0.592224	−0.296112	−	0.955153i	$-0.595690\pi$
$541$	−13.7748	−0.592224	−0.296112	+	0.955153i	$0.595690\pi$
$542$	0.345665	0.0148476
$543$	0	0
$544$	17.6543	0.756923
$545$	−5.61285	−0.240428
$546$	0	0
$547$	42.9862	1.83796	0.918978	−	0.394308i	$-0.129016\pi$
$547$	42.9862	1.83796	0.918978	+	0.394308i	$0.129016\pi$
$548$	3.24443	0.138595
$549$	0	0
$550$	−0.815792	−0.0347855
$551$	−7.80642	−0.332565
$552$	0	0
$553$	59.4291	2.52718
$554$	−1.71762	−0.0729747
$555$	0	0
$556$	−16.6035	−0.704144
$557$	−14.2953	−0.605711	−0.302855	−	0.953037i	$-0.597940\pi$
$557$	−14.2953	−0.605711	−0.302855	+	0.953037i	$0.597940\pi$
$558$	0	0
$559$	5.71456	0.241700
$560$	15.1842	0.641650
$561$	0	0
$562$	−4.91750	−0.207432
$563$	−29.9541	−1.26241	−0.631207	−	0.775615i	$-0.717440\pi$
$563$	−29.9541	−1.26241	−0.631207	+	0.775615i	$0.717440\pi$
$564$	0	0
$565$	13.8938	0.584518
$566$	−4.42864	−0.186150
$567$	0	0
$568$	−9.24443	−0.387888
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−38.2351	−1.60009	−0.800044	−	0.599942i	$-0.795190\pi$
$571$	−38.2351	−1.60009	−0.800044	+	0.599942i	$0.795190\pi$
$572$	−2.36842	−0.0990285
$573$	0	0
$574$	6.95899	0.290463
$575$	1.37778	0.0574576
$576$	0	0
$577$	−16.5718	−0.689895	−0.344947	−	0.938622i	$-0.612103\pi$
$577$	−16.5718	−0.689895	−0.344947	+	0.938622i	$0.612103\pi$
$578$	−2.64788	−0.110137
$579$	0	0
$580$	14.8573	0.616915
$581$	−47.0420	−1.95163
$582$	0	0
$583$	−19.7333	−0.817270
$584$	−4.62222	−0.191269
$585$	0	0
$586$	2.34122	0.0967149
$587$	−7.94962	−0.328116	−0.164058	−	0.986451i	$-0.552458\pi$
$587$	−7.94962	−0.328116	−0.164058	+	0.986451i	$0.552458\pi$
$588$	0	0
$589$	−1.24443	−0.0512759
$590$	0.682439	0.0280956
$591$	0	0
$592$	15.3417	0.630540
$593$	17.0794	0.701368	0.350684	−	0.936494i	$-0.385949\pi$
$593$	17.0794	0.701368	0.350684	+	0.936494i	$0.385949\pi$
$594$	0	0
$595$	−22.3684	−0.917016
$596$	−12.9719	−0.531350
$597$	0	0
$598$	−0.203420	−0.00831848
$599$	−5.68598	−0.232323	−0.116161	−	0.993230i	$-0.537059\pi$
$599$	−5.68598	−0.232323	−0.116161	+	0.993230i	$0.537059\pi$
$600$	0	0
$601$	−33.2543	−1.35647	−0.678235	−	0.734845i	$-0.737255\pi$
$601$	−33.2543	−1.35647	−0.678235	+	0.734845i	$0.737255\pi$
$602$	16.5906	0.676181
$603$	0	0
$604$	11.0509	0.449653
$605$	4.12399	0.167664
$606$	0	0
$607$	−10.9032	−0.442548	−0.221274	−	0.975212i	$-0.571021\pi$
$607$	−10.9032	−0.442548	−0.221274	+	0.975212i	$0.571021\pi$
$608$	3.49532	0.141754
$609$	0	0
$610$	1.14272	0.0462674
$611$	2.10171	0.0850261
$612$	0	0
$613$	−47.6227	−1.92346	−0.961731	−	0.273995i	$-0.911655\pi$
$613$	−47.6227	−1.92346	−0.961731	+	0.273995i	$0.911655\pi$
$614$	0.874632	0.0352973
$615$	0	0
$616$	−14.1017	−0.568174
$617$	46.6450	1.87786	0.938928	−	0.344114i	$-0.111821\pi$
$617$	46.6450	1.87786	0.938928	+	0.344114i	$0.111821\pi$
$618$	0	0
$619$	−32.2163	−1.29488	−0.647442	−	0.762115i	$-0.724161\pi$
$619$	−32.2163	−1.29488	−0.647442	+	0.762115i	$0.724161\pi$
$620$	2.36842	0.0951179
$621$	0	0
$622$	−4.30819	−0.172743
$623$	−56.0830	−2.24692
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.23152	−0.289030
$627$	0	0
$628$	−0.368416	−0.0147014
$629$	−22.6004	−0.901138
$630$	0	0
$631$	−30.9719	−1.23297	−0.616486	−	0.787366i	$-0.711444\pi$
$631$	−30.9719	−1.23297	−0.616486	+	0.787366i	$0.711444\pi$
$632$	−16.2953	−0.648192
$633$	0	0
$634$	0.921948	0.0366152
$635$	7.19850	0.285664
$636$	0	0
$637$	5.98571	0.237162
$638$	−6.36842	−0.252128
$639$	0	0
$640$	−8.78568	−0.347285
$641$	22.1748	0.875854	0.437927	−	0.899011i	$-0.355713\pi$
$641$	22.1748	0.875854	0.437927	+	0.899011i	$0.355713\pi$
$642$	0	0
$643$	−6.23506	−0.245887	−0.122943	−	0.992414i	$-0.539233\pi$
$643$	−6.23506	−0.245887	−0.122943	+	0.992414i	$0.539233\pi$
$644$	11.6128	0.457610
$645$	0	0
$646$	−1.57136	−0.0618244
$647$	1.29481	0.0509042	0.0254521	−	0.999676i	$-0.491897\pi$
$647$	1.29481	0.0509042	0.0254521	+	0.999676i	$0.491897\pi$
$648$	0	0
$649$	5.75203	0.225787
$650$	−0.147643	−0.00579104
$651$	0	0
$652$	16.0415	0.628233
$653$	−30.2953	−1.18555	−0.592773	−	0.805370i	$-0.701967\pi$
$653$	−30.2953	−1.18555	−0.592773	+	0.805370i	$0.701967\pi$
$654$	0	0
$655$	−2.10171	−0.0821206
$656$	17.3176	0.676137
$657$	0	0
$658$	6.10171	0.237869
$659$	4.17484	0.162629	0.0813143	−	0.996689i	$-0.474088\pi$
$659$	4.17484	0.162629	0.0813143	+	0.996689i	$0.474088\pi$
$660$	0	0
$661$	−6.56199	−0.255232	−0.127616	−	0.991824i	$-0.540732\pi$
$661$	−6.56199	−0.255232	−0.127616	+	0.991824i	$0.540732\pi$
$662$	0.295286	0.0114766
$663$	0	0
$664$	12.8988	0.500569
$665$	−4.42864	−0.171735
$666$	0	0
$667$	10.7556	0.416457
$668$	23.6815	0.916266
$669$	0	0
$670$	−0.516060	−0.0199371
$671$	9.63158	0.371823
$672$	0	0
$673$	12.7413	0.491140	0.245570	−	0.969379i	$-0.421025\pi$
$673$	12.7413	0.491140	0.245570	+	0.969379i	$0.421025\pi$
$674$	−0.709636	−0.0273341
$675$	0	0
$676$	24.3131	0.935120
$677$	30.9260	1.18858	0.594291	−	0.804250i	$-0.297433\pi$
$677$	30.9260	1.18858	0.594291	+	0.804250i	$0.297433\pi$
$678$	0	0
$679$	−79.2454	−3.04116
$680$	6.13335	0.235203
$681$	0	0
$682$	−1.01520	−0.0388739
$683$	−34.3412	−1.31403	−0.657015	−	0.753877i	$-0.728181\pi$
$683$	−34.3412	−1.31403	−0.657015	+	0.753877i	$0.728181\pi$
$684$	0	0
$685$	1.70471	0.0651338
$686$	7.73329	0.295259
$687$	0	0
$688$	41.2859	1.57401
$689$	−3.57136	−0.136058
$690$	0	0
$691$	24.2163	0.921233	0.460616	−	0.887599i	$-0.347628\pi$
$691$	24.2163	0.921233	0.460616	+	0.887599i	$0.347628\pi$
$692$	−42.2306	−1.60537
$693$	0	0
$694$	3.83807	0.145691
$695$	−8.72393	−0.330917
$696$	0	0
$697$	−25.5111	−0.966303
$698$	3.11108	0.117756
$699$	0	0
$700$	8.42864	0.318573
$701$	−11.4064	−0.430812	−0.215406	−	0.976525i	$-0.569108\pi$
$701$	−11.4064	−0.430812	−0.215406	+	0.976525i	$0.569108\pi$
$702$	0	0
$703$	−4.47457	−0.168762
$704$	−15.1298	−0.570226
$705$	0	0
$706$	0.797056	0.0299976
$707$	−46.1847	−1.73695
$708$	0	0
$709$	−13.0223	−0.489062	−0.244531	−	0.969642i	$-0.578634\pi$
$709$	−13.0223	−0.489062	−0.244531	+	0.969642i	$0.578634\pi$
$710$	−2.36842	−0.0888851
$711$	0	0
$712$	15.3778	0.576307
$713$	1.71456	0.0642107
$714$	0	0
$715$	−1.24443	−0.0465391
$716$	−22.6637	−0.846982
$717$	0	0
$718$	7.57136	0.282561
$719$	−52.2163	−1.94734	−0.973670	−	0.227961i	$-0.926794\pi$
$719$	−52.2163	−1.94734	−0.973670	+	0.227961i	$0.926794\pi$
$720$	0	0
$721$	25.0607	0.933309
$722$	−0.311108	−0.0115782
$723$	0	0
$724$	−33.5210	−1.24580
$725$	7.80642	0.289923
$726$	0	0
$727$	14.0602	0.521465	0.260732	−	0.965411i	$-0.416036\pi$
$727$	14.0602	0.521465	0.260732	+	0.965411i	$0.416036\pi$
$728$	−2.55215	−0.0945889
$729$	0	0
$730$	−1.18421	−0.0438295
$731$	−60.8198	−2.24950
$732$	0	0
$733$	−4.48886	−0.165800	−0.0829000	−	0.996558i	$-0.526418\pi$
$733$	−4.48886	−0.165800	−0.0829000	+	0.996558i	$0.526418\pi$
$734$	1.37778	0.0508549
$735$	0	0
$736$	−4.81579	−0.177512
$737$	−4.34968	−0.160223
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	8.51606	0.313057
$741$	0	0
$742$	−10.3684	−0.380637
$743$	37.5669	1.37820	0.689098	−	0.724668i	$-0.258007\pi$
$743$	37.5669	1.37820	0.689098	+	0.724668i	$0.258007\pi$
$744$	0	0
$745$	−6.81579	−0.249711
$746$	7.37334	0.269957
$747$	0	0
$748$	25.2070	0.921658
$749$	30.5718	1.11707
$750$	0	0
$751$	52.5817	1.91873	0.959366	−	0.282163i	$-0.0910520\pi$
$751$	52.5817	1.91873	0.959366	+	0.282163i	$0.0910520\pi$
$752$	15.1842	0.553711
$753$	0	0
$754$	−1.15257	−0.0419740
$755$	5.80642	0.211317
$756$	0	0
$757$	5.73329	0.208380	0.104190	−	0.994557i	$-0.466775\pi$
$757$	5.73329	0.208380	0.104190	+	0.994557i	$0.466775\pi$
$758$	−2.55215	−0.0926982
$759$	0	0
$760$	1.21432	0.0440480
$761$	32.7338	1.18660	0.593299	−	0.804982i	$-0.297825\pi$
$761$	32.7338	1.18660	0.593299	+	0.804982i	$0.297825\pi$
$762$	0	0
$763$	−24.8573	−0.899894
$764$	−0.507598	−0.0183643
$765$	0	0
$766$	−6.50622	−0.235079
$767$	1.04101	0.0375887
$768$	0	0
$769$	10.1619	0.366449	0.183224	−	0.983071i	$-0.441347\pi$
$769$	10.1619	0.366449	0.183224	+	0.983071i	$0.441347\pi$
$770$	−3.61285	−0.130198
$771$	0	0
$772$	−5.07805	−0.182763
$773$	36.0785	1.29765	0.648827	−	0.760936i	$-0.275260\pi$
$773$	36.0785	1.29765	0.648827	+	0.760936i	$0.275260\pi$
$774$	0	0
$775$	1.24443	0.0447013
$776$	21.7288	0.780020
$777$	0	0
$778$	−7.49823	−0.268825
$779$	−5.05086	−0.180966
$780$	0	0
$781$	−19.9625	−0.714315
$782$	2.16500	0.0774201
$783$	0	0
$784$	43.2449	1.54446
$785$	−0.193576	−0.00690903
$786$	0	0
$787$	−30.5446	−1.08880	−0.544399	−	0.838826i	$-0.683242\pi$
$787$	−30.5446	−1.08880	−0.544399	+	0.838826i	$0.683242\pi$
$788$	−10.1748	−0.362464
$789$	0	0
$790$	−4.17484	−0.148534
$791$	61.5308	2.18778
$792$	0	0
$793$	1.74314	0.0619007
$794$	−2.46611	−0.0875190
$795$	0	0
$796$	−32.5906	−1.15514
$797$	27.9037	0.988399	0.494200	−	0.869348i	$-0.335461\pi$
$797$	27.9037	0.988399	0.494200	+	0.869348i	$0.335461\pi$
$798$	0	0
$799$	−22.3684	−0.791338
$800$	−3.49532	−0.123578
$801$	0	0
$802$	−10.1334	−0.357821
$803$	−9.98126	−0.352231
$804$	0	0
$805$	6.10171	0.215057
$806$	−0.183732	−0.00647168
$807$	0	0
$808$	12.6637	0.445508
$809$	25.6128	0.900500	0.450250	−	0.892903i	$-0.351335\pi$
$809$	25.6128	0.900500	0.450250	+	0.892903i	$0.351335\pi$
$810$	0	0
$811$	−6.01874	−0.211346	−0.105673	−	0.994401i	$-0.533700\pi$
$811$	−6.01874	−0.211346	−0.105673	+	0.994401i	$0.533700\pi$
$812$	65.7975	2.30904
$813$	0	0
$814$	−3.65032	−0.127944
$815$	8.42864	0.295242
$816$	0	0
$817$	−12.0415	−0.421278
$818$	−11.3145	−0.395602
$819$	0	0
$820$	9.61285	0.335695
$821$	6.20342	0.216501	0.108250	−	0.994124i	$-0.465475\pi$
$821$	6.20342	0.216501	0.108250	+	0.994124i	$0.465475\pi$
$822$	0	0
$823$	1.75605	0.0612119	0.0306059	−	0.999532i	$-0.490256\pi$
$823$	1.75605	0.0612119	0.0306059	+	0.999532i	$0.490256\pi$
$824$	−6.87157	−0.239382
$825$	0	0
$826$	3.02227	0.105158
$827$	53.2083	1.85024	0.925118	−	0.379680i	$-0.123966\pi$
$827$	53.2083	1.85024	0.925118	+	0.379680i	$0.123966\pi$
$828$	0	0
$829$	−26.9777	−0.936975	−0.468488	−	0.883470i	$-0.655201\pi$
$829$	−26.9777	−0.936975	−0.468488	+	0.883470i	$0.655201\pi$
$830$	3.30465	0.114706
$831$	0	0
$832$	−2.73822	−0.0949306
$833$	−63.7057	−2.20727
$834$	0	0
$835$	12.4429	0.430605
$836$	4.99063	0.172605
$837$	0	0
$838$	−9.86082	−0.340636
$839$	−46.9501	−1.62090	−0.810449	−	0.585810i	$-0.800777\pi$
$839$	−46.9501	−1.62090	−0.810449	+	0.585810i	$0.800777\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	−11.6356	−0.400989
$843$	0	0
$844$	−25.0321	−0.861641
$845$	12.7748	0.439466
$846$	0	0
$847$	18.2636	0.627546
$848$	−25.8020	−0.886044
$849$	0	0
$850$	1.57136	0.0538972
$851$	6.16500	0.211333
$852$	0	0
$853$	−46.4701	−1.59111	−0.795553	−	0.605883i	$-0.792820\pi$
$853$	−46.4701	−1.59111	−0.795553	+	0.605883i	$0.792820\pi$
$854$	5.06070	0.173174
$855$	0	0
$856$	−8.38271	−0.286515
$857$	7.79213	0.266174	0.133087	−	0.991104i	$-0.457511\pi$
$857$	7.79213	0.266174	0.133087	+	0.991104i	$0.457511\pi$
$858$	0	0
$859$	−37.4479	−1.27770	−0.638852	−	0.769330i	$-0.720590\pi$
$859$	−37.4479	−1.27770	−0.638852	+	0.769330i	$0.720590\pi$
$860$	22.9175	0.781480
$861$	0	0
$862$	1.53972	0.0524430
$863$	−47.7605	−1.62579	−0.812893	−	0.582413i	$-0.802109\pi$
$863$	−47.7605	−1.62579	−0.812893	+	0.582413i	$0.802109\pi$
$864$	0	0
$865$	−22.1891	−0.754453
$866$	−10.0745	−0.342346
$867$	0	0
$868$	10.4889	0.356015
$869$	−35.1882	−1.19368
$870$	0	0
$871$	−0.787212	−0.0266736
$872$	6.81579	0.230812
$873$	0	0
$874$	0.428639	0.0144989
$875$	4.42864	0.149715
$876$	0	0
$877$	−22.5763	−0.762347	−0.381173	−	0.924504i	$-0.624480\pi$
$877$	−22.5763	−0.762347	−0.381173	+	0.924504i	$0.624480\pi$
$878$	−3.13383	−0.105762
$879$	0	0
$880$	−8.99063	−0.303074
$881$	10.8988	0.367189	0.183594	−	0.983002i	$-0.441227\pi$
$881$	10.8988	0.367189	0.183594	+	0.983002i	$0.441227\pi$
$882$	0	0
$883$	−39.9782	−1.34537	−0.672687	−	0.739927i	$-0.734860\pi$
$883$	−39.9782	−1.34537	−0.672687	+	0.739927i	$0.734860\pi$
$884$	4.56199	0.153436
$885$	0	0
$886$	−4.34567	−0.145995
$887$	−49.6785	−1.66804	−0.834020	−	0.551734i	$-0.813966\pi$
$887$	−49.6785	−1.66804	−0.834020	+	0.551734i	$0.813966\pi$
$888$	0	0
$889$	31.8796	1.06921
$890$	3.93978	0.132062
$891$	0	0
$892$	20.0143	0.670128
$893$	−4.42864	−0.148199
$894$	0	0
$895$	−11.9081	−0.398045
$896$	−38.9086	−1.29985
$897$	0	0
$898$	7.64449	0.255100
$899$	9.71456	0.323999
$900$	0	0
$901$	38.0098	1.26629
$902$	−4.12045	−0.137196
$903$	0	0
$904$	−16.8716	−0.561140
$905$	−17.6128	−0.585471
$906$	0	0
$907$	−18.2779	−0.606909	−0.303454	−	0.952846i	$-0.598140\pi$
$907$	−18.2779	−0.606909	−0.303454	+	0.952846i	$0.598140\pi$
$908$	13.1383	0.436009
$909$	0	0
$910$	−0.653858	−0.0216752
$911$	44.7654	1.48314	0.741572	−	0.670873i	$-0.234080\pi$
$911$	44.7654	1.48314	0.741572	+	0.670873i	$0.234080\pi$
$912$	0	0
$913$	27.8537	0.921824
$914$	1.09234	0.0361315
$915$	0	0
$916$	−34.3368	−1.13452
$917$	−9.30772	−0.307368
$918$	0	0
$919$	−33.6316	−1.10940	−0.554702	−	0.832049i	$-0.687168\pi$
$919$	−33.6316	−1.10940	−0.554702	+	0.832049i	$0.687168\pi$
$920$	−1.67307	−0.0551595
$921$	0	0
$922$	3.17436	0.104542
$923$	−3.61285	−0.118918
$924$	0	0
$925$	4.47457	0.147123
$926$	−2.59364	−0.0852321
$927$	0	0
$928$	−27.2859	−0.895704
$929$	−13.0223	−0.427247	−0.213623	−	0.976916i	$-0.568527\pi$
$929$	−13.0223	−0.427247	−0.213623	+	0.976916i	$0.568527\pi$
$930$	0	0
$931$	−12.6128	−0.413369
$932$	−23.5397	−0.771069
$933$	0	0
$934$	−13.2859	−0.434729
$935$	13.2444	0.433139
$936$	0	0
$937$	28.5433	0.932468	0.466234	−	0.884662i	$-0.345611\pi$
$937$	28.5433	0.932468	0.466234	+	0.884662i	$0.345611\pi$
$938$	−2.28544	−0.0746223
$939$	0	0
$940$	8.42864	0.274912
$941$	−13.2257	−0.431145	−0.215573	−	0.976488i	$-0.569162\pi$
$941$	−13.2257	−0.431145	−0.215573	+	0.976488i	$0.569162\pi$
$942$	0	0
$943$	6.95899	0.226616
$944$	7.52098	0.244787
$945$	0	0
$946$	−9.82335	−0.319385
$947$	−3.66323	−0.119039	−0.0595194	−	0.998227i	$-0.518957\pi$
$947$	−3.66323	−0.119039	−0.0595194	+	0.998227i	$0.518957\pi$
$948$	0	0
$949$	−1.80642	−0.0586390
$950$	0.311108	0.0100937
$951$	0	0
$952$	27.1624	0.880339
$953$	−12.1704	−0.394238	−0.197119	−	0.980380i	$-0.563158\pi$
$953$	−12.1704	−0.394238	−0.197119	+	0.980380i	$0.563158\pi$
$954$	0	0
$955$	−0.266706	−0.00863041
$956$	47.3087	1.53007
$957$	0	0
$958$	12.8988	0.416740
$959$	7.54956	0.243788
$960$	0	0
$961$	−29.4514	−0.950045
$962$	−0.660640	−0.0212999
$963$	0	0
$964$	−17.2257	−0.554802
$965$	−2.66815	−0.0858907
$966$	0	0
$967$	8.52051	0.274001	0.137000	−	0.990571i	$-0.456254\pi$
$967$	8.52051	0.274001	0.137000	+	0.990571i	$0.456254\pi$
$968$	−5.00784	−0.160958
$969$	0	0
$970$	5.56691	0.178743
$971$	2.67259	0.0857676	0.0428838	−	0.999080i	$-0.486345\pi$
$971$	2.67259	0.0857676	0.0428838	+	0.999080i	$0.486345\pi$
$972$	0	0
$973$	−38.6351	−1.23859
$974$	−9.34476	−0.299425
$975$	0	0
$976$	12.5936	0.403112
$977$	20.3827	0.652101	0.326050	−	0.945352i	$-0.394282\pi$
$977$	20.3827	0.652101	0.326050	+	0.945352i	$0.394282\pi$
$978$	0	0
$979$	33.2070	1.06130
$980$	24.0049	0.766809
$981$	0	0
$982$	4.77430	0.152354
$983$	−10.3126	−0.328922	−0.164461	−	0.986384i	$-0.552588\pi$
$983$	−10.3126	−0.328922	−0.164461	+	0.986384i	$0.552588\pi$
$984$	0	0
$985$	−5.34614	−0.170342
$986$	12.2667	0.390652
$987$	0	0
$988$	0.903212	0.0287350
$989$	16.5906	0.527550
$990$	0	0
$991$	20.0919	0.638239	0.319120	−	0.947714i	$-0.396613\pi$
$991$	20.0919	0.638239	0.319120	+	0.947714i	$0.396613\pi$
$992$	−4.34968	−0.138102
$993$	0	0
$994$	−10.4889	−0.332687
$995$	−17.1240	−0.542867
$996$	0	0
$997$	25.0509	0.793369	0.396684	−	0.917955i	$-0.370161\pi$
$997$	25.0509	0.793369	0.396684	+	0.917955i	$0.370161\pi$
$998$	8.04149	0.254549
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.i.1.2		3
3.2	odd	2		95.2.a.a.1.2	✓	3
5.4	even	2		4275.2.a.bk.1.2		3
12.11	even	2		1520.2.a.p.1.1		3
15.2	even	4		475.2.b.d.324.4		6
15.8	even	4		475.2.b.d.324.3		6
15.14	odd	2		475.2.a.f.1.2		3
21.20	even	2		4655.2.a.u.1.2		3
24.5	odd	2		6080.2.a.bo.1.1		3
24.11	even	2		6080.2.a.by.1.3		3
57.56	even	2		1805.2.a.f.1.2		3
60.59	even	2		7600.2.a.bx.1.3		3
285.284	even	2		9025.2.a.bb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.2	✓	3	3.2	odd	2
475.2.a.f.1.2		3	15.14	odd	2
475.2.b.d.324.3		6	15.8	even	4
475.2.b.d.324.4		6	15.2	even	4
855.2.a.i.1.2		3	1.1	even	1	trivial
1520.2.a.p.1.1		3	12.11	even	2
1805.2.a.f.1.2		3	57.56	even	2
4275.2.a.bk.1.2		3	5.4	even	2
4655.2.a.u.1.2		3	21.20	even	2
6080.2.a.bo.1.1		3	24.5	odd	2
6080.2.a.by.1.3		3	24.11	even	2
7600.2.a.bx.1.3		3	60.59	even	2
9025.2.a.bb.1.2		3	285.284	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -4.42864 q^{7} +1.21432 q^{8} +O(q^{10})\)	\(q-0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -4.42864 q^{7} +1.21432 q^{8} +0.311108 q^{10} +2.62222 q^{11} +0.474572 q^{13} +1.37778 q^{14} +3.42864 q^{16} -5.05086 q^{17} -1.00000 q^{19} +1.90321 q^{20} -0.815792 q^{22} +1.37778 q^{23} +1.00000 q^{25} -0.147643 q^{26} +8.42864 q^{28} +7.80642 q^{29} +1.24443 q^{31} -3.49532 q^{32} +1.57136 q^{34} +4.42864 q^{35} +4.47457 q^{37} +0.311108 q^{38} -1.21432 q^{40} +5.05086 q^{41} +12.0415 q^{43} -4.99063 q^{44} -0.428639 q^{46} +4.42864 q^{47} +12.6128 q^{49} -0.311108 q^{50} -0.903212 q^{52} -7.52543 q^{53} -2.62222 q^{55} -5.37778 q^{56} -2.42864 q^{58} +2.19358 q^{59} +3.67307 q^{61} -0.387152 q^{62} -5.76986 q^{64} -0.474572 q^{65} -1.65878 q^{67} +9.61285 q^{68} -1.37778 q^{70} -7.61285 q^{71} -3.80642 q^{73} -1.39207 q^{74} +1.90321 q^{76} -11.6128 q^{77} -13.4193 q^{79} -3.42864 q^{80} -1.57136 q^{82} +10.6222 q^{83} +5.05086 q^{85} -3.74620 q^{86} +3.18421 q^{88} +12.6637 q^{89} -2.10171 q^{91} -2.62222 q^{92} -1.37778 q^{94} +1.00000 q^{95} +17.8938 q^{97} -3.92396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31} + 3 q^{32} + 18 q^{34} + 20 q^{37} + q^{38} + 3 q^{40} + 2 q^{41} - 4 q^{43} + 12 q^{44} + 12 q^{46} + 11 q^{49} - q^{50} + 4 q^{52} - 16 q^{53} - 8 q^{55} - 16 q^{56} + 6 q^{58} + 20 q^{59} - 2 q^{61} - 28 q^{62} - 11 q^{64} - 8 q^{65} + 2 q^{67} + 2 q^{68} - 4 q^{70} + 4 q^{71} + 2 q^{73} + 2 q^{74} - q^{76} - 8 q^{77} + 3 q^{80} - 18 q^{82} + 32 q^{83} + 2 q^{85} + 16 q^{86} - 4 q^{88} - 2 q^{89} + 20 q^{91} - 8 q^{92} - 4 q^{94} + 3 q^{95} + 20 q^{97} + 15 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * q^98

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