LMFDB - Embedded newform 5445.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5445,2,Mod(1,5445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,5,-3,0,0,-9,0,1,0,0,2,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90321 q^{2} +1.62222 q^{4} -1.00000 q^{5} +4.42864 q^{7} -0.719004 q^{8} -1.90321 q^{10} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} -7.05086 q^{19} -1.62222 q^{20} -8.85728 q^{23} +1.00000 q^{25} +1.18421 q^{26} +7.18421 q^{28} -7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} -4.42864 q^{35} -2.00000 q^{37} -13.4193 q^{38} +0.719004 q^{40} -0.193576 q^{41} -5.67307 q^{43} -16.8573 q^{46} +2.75557 q^{47} +12.6128 q^{49} +1.90321 q^{50} +1.00937 q^{52} +10.8573 q^{53} -3.18421 q^{56} -14.8573 q^{58} +4.85728 q^{59} -6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.24443 q^{67} -8.40990 q^{68} -8.42864 q^{70} -2.75557 q^{71} -4.23506 q^{73} -3.80642 q^{74} -11.4380 q^{76} -8.56199 q^{79} +4.61285 q^{80} -0.368416 q^{82} +0.133353 q^{83} +5.18421 q^{85} -10.7971 q^{86} -5.61285 q^{89} +2.75557 q^{91} -14.3684 q^{92} +5.24443 q^{94} +7.05086 q^{95} +7.24443 q^{97} +24.0049 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 9 q^{8} + q^{10} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} - 8 q^{19} - 5 q^{20} + 3 q^{25} - 10 q^{26} + 8 q^{28} - 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34}+ \cdots + 39 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 9 * q^8 + q^10 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 - 8 * q^19 - 5 * q^20 + 3 * q^25 - 10 * q^26 + 8 * q^28 - 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 - 6 * q^37 + 9 * q^40 - 14 * q^41 - 4 * q^43 - 24 * q^46 + 8 * q^47 + 11 * q^49 - q^50 + 30 * q^52 + 6 * q^53 + 4 * q^56 - 18 * q^58 - 12 * q^59 + 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - 4 * q^67 + 42 * q^68 - 12 * q^70 - 8 * q^71 + 14 * q^73 + 2 * q^74 - 48 * q^76 - 12 * q^79 - 13 * q^80 + 26 * q^82 + 2 * q^85 + 8 * q^86 + 10 * q^89 + 8 * q^91 - 16 * q^92 + 16 * q^94 + 8 * q^95 + 22 * q^97 + 39 * q^98

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 $

$2$	1.90321	1.34577	0.672887	−	0.739745i	$-0.265054\pi$
$2$	1.90321	1.34577	0.672887	+	0.739745i	$0.265054\pi$
$3$	0	0
$3$	0	0
$4$	1.62222	0.811108
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	−0.719004	−0.254206
$9$	0	0
$10$	−1.90321	−0.601848
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	8.42864	2.25265
$15$	0	0
$16$	−4.61285	−1.15321
$17$	−5.18421	−1.25736	−0.628678	−	0.777666i	$-0.716403\pi$
$17$	−5.18421	−1.25736	−0.628678	+	0.777666i	$0.716403\pi$
$18$	0	0
$19$	−7.05086	−1.61758	−0.808789	−	0.588100i	$-0.799876\pi$
$19$	−7.05086	−1.61758	−0.808789	+	0.588100i	$0.799876\pi$
$20$	−1.62222	−0.362738
$21$	0	0
$22$	0	0
$23$	−8.85728	−1.84687	−0.923435	−	0.383754i	$-0.874631\pi$
$23$	−8.85728	−1.84687	−0.923435	+	0.383754i	$0.874631\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.18421	0.232242
$27$	0	0
$28$	7.18421	1.35769
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	0	0
$31$	2.75557	0.494915	0.247457	−	0.968899i	$-0.420405\pi$
$31$	2.75557	0.494915	0.247457	+	0.968899i	$0.420405\pi$
$32$	−7.34122	−1.29776
$33$	0	0
$34$	−9.86665	−1.69212
$35$	−4.42864	−0.748577
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−13.4193	−2.17689
$39$	0	0
$40$	0.719004	0.113684
$41$	−0.193576	−0.0302315	−0.0151158	−	0.999886i	$-0.504812\pi$
$41$	−0.193576	−0.0302315	−0.0151158	+	0.999886i	$0.504812\pi$
$42$	0	0
$43$	−5.67307	−0.865135	−0.432568	−	0.901602i	$-0.642392\pi$
$43$	−5.67307	−0.865135	−0.432568	+	0.901602i	$0.642392\pi$
$44$	0	0
$45$	0	0
$46$	−16.8573	−2.48547
$47$	2.75557	0.401941	0.200971	−	0.979597i	$-0.435590\pi$
$47$	2.75557	0.401941	0.200971	+	0.979597i	$0.435590\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	1.90321	0.269155
$51$	0	0
$52$	1.00937	0.139974
$53$	10.8573	1.49136	0.745681	−	0.666303i	$-0.232124\pi$
$53$	10.8573	1.49136	0.745681	+	0.666303i	$0.232124\pi$
$54$	0	0
$55$	0	0
$56$	−3.18421	−0.425508
$57$	0	0
$58$	−14.8573	−1.95086
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	0	0
$61$	−6.85728	−0.877985	−0.438992	−	0.898491i	$-0.644664\pi$
$61$	−6.85728	−0.877985	−0.438992	+	0.898491i	$0.644664\pi$
$62$	5.24443	0.666043
$63$	0	0
$64$	−4.74620	−0.593275
$65$	−0.622216	−0.0771764
$66$	0	0
$67$	−1.24443	−0.152031	−0.0760157	−	0.997107i	$-0.524220\pi$
$67$	−1.24443	−0.152031	−0.0760157	+	0.997107i	$0.524220\pi$
$68$	−8.40990	−1.01985
$69$	0	0
$70$	−8.42864	−1.00742
$71$	−2.75557	−0.327026	−0.163513	−	0.986541i	$-0.552283\pi$
$71$	−2.75557	−0.327026	−0.163513	+	0.986541i	$0.552283\pi$
$72$	0	0
$73$	−4.23506	−0.495677	−0.247838	−	0.968801i	$-0.579720\pi$
$73$	−4.23506	−0.495677	−0.247838	+	0.968801i	$0.579720\pi$
$74$	−3.80642	−0.442488
$75$	0	0
$76$	−11.4380	−1.31203
$77$	0	0
$78$	0	0
$79$	−8.56199	−0.963299	−0.481650	−	0.876364i	$-0.659962\pi$
$79$	−8.56199	−0.963299	−0.481650	+	0.876364i	$0.659962\pi$
$80$	4.61285	0.515732
$81$	0	0
$82$	−0.368416	−0.0406848
$83$	0.133353	0.0146374	0.00731870	−	0.999973i	$-0.497670\pi$
$83$	0.133353	0.0146374	0.00731870	+	0.999973i	$0.497670\pi$
$84$	0	0
$85$	5.18421	0.562306
$86$	−10.7971	−1.16428
$87$	0	0
$88$	0	0
$89$	−5.61285	−0.594961	−0.297480	−	0.954728i	$-0.596146\pi$
$89$	−5.61285	−0.594961	−0.297480	+	0.954728i	$0.596146\pi$
$90$	0	0
$91$	2.75557	0.288862
$92$	−14.3684	−1.49801
$93$	0	0
$94$	5.24443	0.540922
$95$	7.05086	0.723402
$96$	0	0
$97$	7.24443	0.735561	0.367780	−	0.929913i	$-0.380118\pi$
$97$	7.24443	0.735561	0.367780	+	0.929913i	$0.380118\pi$
$98$	24.0049	2.42486
$99$	0	0
$100$	1.62222	0.162222
$101$	4.66370	0.464056	0.232028	−	0.972709i	$-0.425464\pi$
$101$	4.66370	0.464056	0.232028	+	0.972709i	$0.425464\pi$
$102$	0	0
$103$	−11.6128	−1.14425	−0.572124	−	0.820167i	$-0.693880\pi$
$103$	−11.6128	−1.14425	−0.572124	+	0.820167i	$0.693880\pi$
$104$	−0.447375	−0.0438688
$105$	0	0
$106$	20.6637	2.00704
$107$	2.62222	0.253499	0.126750	−	0.991935i	$-0.459546\pi$
$107$	2.62222	0.253499	0.126750	+	0.991935i	$0.459546\pi$
$108$	0	0
$109$	19.7146	1.88831	0.944156	−	0.329499i	$-0.106880\pi$
$109$	19.7146	1.88831	0.944156	+	0.329499i	$0.106880\pi$
$110$	0	0
$111$	0	0
$112$	−20.4286	−1.93032
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	8.85728	0.825946
$116$	−12.6637	−1.17580
$117$	0	0
$118$	9.24443	0.851019
$119$	−22.9590	−2.10465
$120$	0	0
$121$	0	0
$122$	−13.0509	−1.18157
$123$	0	0
$124$	4.47013	0.401429
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.1842	1.34738	0.673690	−	0.739014i	$-0.264708\pi$
$127$	15.1842	1.34738	0.673690	+	0.739014i	$0.264708\pi$
$128$	5.64941	0.499342
$129$	0	0
$130$	−1.18421	−0.103862
$131$	1.24443	0.108726	0.0543632	−	0.998521i	$-0.482687\pi$
$131$	1.24443	0.108726	0.0543632	+	0.998521i	$0.482687\pi$
$132$	0	0
$133$	−31.2257	−2.70761
$134$	−2.36842	−0.204600
$135$	0	0
$136$	3.72746	0.319627
$137$	0.488863	0.0417663	0.0208832	−	0.999782i	$-0.493352\pi$
$137$	0.488863	0.0417663	0.0208832	+	0.999782i	$0.493352\pi$
$138$	0	0
$139$	−17.8064	−1.51032	−0.755161	−	0.655540i	$-0.772441\pi$
$139$	−17.8064	−1.51032	−0.755161	+	0.655540i	$0.772441\pi$
$140$	−7.18421	−0.607176
$141$	0	0
$142$	−5.24443	−0.440103
$143$	0	0
$144$	0	0
$145$	7.80642	0.648288
$146$	−8.06022	−0.667069
$147$	0	0
$148$	−3.24443	−0.266691
$149$	−1.43801	−0.117806	−0.0589031	−	0.998264i	$-0.518760\pi$
$149$	−1.43801	−0.117806	−0.0589031	+	0.998264i	$0.518760\pi$
$150$	0	0
$151$	12.1748	0.990774	0.495387	−	0.868672i	$-0.335026\pi$
$151$	12.1748	0.990774	0.495387	+	0.868672i	$0.335026\pi$
$152$	5.06959	0.411198
$153$	0	0
$154$	0	0
$155$	−2.75557	−0.221333
$156$	0	0
$157$	18.4701	1.47408	0.737038	−	0.675851i	$-0.236224\pi$
$157$	18.4701	1.47408	0.737038	+	0.675851i	$0.236224\pi$
$158$	−16.2953	−1.29638
$159$	0	0
$160$	7.34122	0.580374
$161$	−39.2257	−3.09142
$162$	0	0
$163$	−10.1017	−0.791227	−0.395614	−	0.918417i	$-0.629468\pi$
$163$	−10.1017	−0.791227	−0.395614	+	0.918417i	$0.629468\pi$
$164$	−0.314022	−0.0245210
$165$	0	0
$166$	0.253799	0.0196986
$167$	−16.3368	−1.26418	−0.632089	−	0.774896i	$-0.717802\pi$
$167$	−16.3368	−1.26418	−0.632089	+	0.774896i	$0.717802\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	9.86665	0.756737
$171$	0	0
$172$	−9.20294	−0.701718
$173$	−9.18421	−0.698262	−0.349131	−	0.937074i	$-0.613523\pi$
$173$	−9.18421	−0.698262	−0.349131	+	0.937074i	$0.613523\pi$
$174$	0	0
$175$	4.42864	0.334774
$176$	0	0
$177$	0	0
$178$	−10.6824	−0.800683
$179$	25.3274	1.89306	0.946530	−	0.322617i	$-0.104563\pi$
$179$	25.3274	1.89306	0.946530	+	0.322617i	$0.104563\pi$
$180$	0	0
$181$	−13.6128	−1.01184	−0.505918	−	0.862582i	$-0.668846\pi$
$181$	−13.6128	−1.01184	−0.505918	+	0.862582i	$0.668846\pi$
$182$	5.24443	0.388743
$183$	0	0
$184$	6.36842	0.469486
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	4.47013	0.326017
$189$	0	0
$190$	13.4193	0.973536
$191$	−6.10171	−0.441504	−0.220752	−	0.975330i	$-0.570851\pi$
$191$	−6.10171	−0.441504	−0.220752	+	0.975330i	$0.570851\pi$
$192$	0	0
$193$	−18.3368	−1.31991	−0.659955	−	0.751305i	$-0.729425\pi$
$193$	−18.3368	−1.31991	−0.659955	+	0.751305i	$0.729425\pi$
$194$	13.7877	0.989898
$195$	0	0
$196$	20.4608	1.46148
$197$	−6.69535	−0.477024	−0.238512	−	0.971140i	$-0.576660\pi$
$197$	−6.69535	−0.477024	−0.238512	+	0.971140i	$0.576660\pi$
$198$	0	0
$199$	14.1017	0.999644	0.499822	−	0.866128i	$-0.333399\pi$
$199$	14.1017	0.999644	0.499822	+	0.866128i	$0.333399\pi$
$200$	−0.719004	−0.0508412
$201$	0	0
$202$	8.87601	0.624514
$203$	−34.5718	−2.42647
$204$	0	0
$205$	0.193576	0.0135199
$206$	−22.1017	−1.53990
$207$	0	0
$208$	−2.87019	−0.199012
$209$	0	0
$210$	0	0
$211$	10.6637	0.734120	0.367060	−	0.930197i	$-0.380364\pi$
$211$	10.6637	0.734120	0.367060	+	0.930197i	$0.380364\pi$
$212$	17.6128	1.20966
$213$	0	0
$214$	4.99063	0.341153
$215$	5.67307	0.386900
$216$	0	0
$217$	12.2034	0.828422
$218$	37.5210	2.54124
$219$	0	0
$220$	0	0
$221$	−3.22570	−0.216984
$222$	0	0
$223$	8.85728	0.593127	0.296564	−	0.955013i	$-0.404159\pi$
$223$	8.85728	0.593127	0.296564	+	0.955013i	$0.404159\pi$
$224$	−32.5116	−2.17227
$225$	0	0
$226$	11.4193	0.759599
$227$	13.3778	0.887915	0.443957	−	0.896048i	$-0.353574\pi$
$227$	13.3778	0.887915	0.443957	+	0.896048i	$0.353574\pi$
$228$	0	0
$229$	11.5111	0.760677	0.380339	−	0.924847i	$-0.375807\pi$
$229$	11.5111	0.760677	0.380339	+	0.924847i	$0.375807\pi$
$230$	16.8573	1.11154
$231$	0	0
$232$	5.61285	0.368502
$233$	−4.32693	−0.283467	−0.141733	−	0.989905i	$-0.545268\pi$
$233$	−4.32693	−0.283467	−0.141733	+	0.989905i	$0.545268\pi$
$234$	0	0
$235$	−2.75557	−0.179753
$236$	7.87955	0.512915
$237$	0	0
$238$	−43.6958	−2.83238
$239$	−3.34614	−0.216444	−0.108222	−	0.994127i	$-0.534516\pi$
$239$	−3.34614	−0.216444	−0.108222	+	0.994127i	$0.534516\pi$
$240$	0	0
$241$	1.34614	0.0867126	0.0433563	−	0.999060i	$-0.486195\pi$
$241$	1.34614	0.0867126	0.0433563	+	0.999060i	$0.486195\pi$
$242$	0	0
$243$	0	0
$244$	−11.1240	−0.712140
$245$	−12.6128	−0.805805
$246$	0	0
$247$	−4.38715	−0.279148
$248$	−1.98126	−0.125810
$249$	0	0
$250$	−1.90321	−0.120370
$251$	−22.7556	−1.43632	−0.718159	−	0.695879i	$-0.755015\pi$
$251$	−22.7556	−1.43632	−0.718159	+	0.695879i	$0.755015\pi$
$252$	0	0
$253$	0	0
$254$	28.8988	1.81327
$255$	0	0
$256$	20.2444	1.26528
$257$	6.85728	0.427745	0.213873	−	0.976862i	$-0.431392\pi$
$257$	6.85728	0.427745	0.213873	+	0.976862i	$0.431392\pi$
$258$	0	0
$259$	−8.85728	−0.550365
$260$	−1.00937	−0.0625983
$261$	0	0
$262$	2.36842	0.146321
$263$	−29.5812	−1.82406	−0.912028	−	0.410129i	$-0.865484\pi$
$263$	−29.5812	−1.82406	−0.912028	+	0.410129i	$0.865484\pi$
$264$	0	0
$265$	−10.8573	−0.666957
$266$	−59.4291	−3.64383
$267$	0	0
$268$	−2.01874	−0.123314
$269$	−8.48886	−0.517575	−0.258788	−	0.965934i	$-0.583323\pi$
$269$	−8.48886	−0.517575	−0.258788	+	0.965934i	$0.583323\pi$
$270$	0	0
$271$	−14.6637	−0.890757	−0.445378	−	0.895343i	$-0.646931\pi$
$271$	−14.6637	−0.890757	−0.445378	+	0.895343i	$0.646931\pi$
$272$	23.9140	1.45000
$273$	0	0
$274$	0.930409	0.0562081
$275$	0	0
$276$	0	0
$277$	−14.6035	−0.877438	−0.438719	−	0.898624i	$-0.644568\pi$
$277$	−14.6035	−0.877438	−0.438719	+	0.898624i	$0.644568\pi$
$278$	−33.8894	−2.03255
$279$	0	0
$280$	3.18421	0.190293
$281$	−0.193576	−0.0115478	−0.00577389	−	0.999983i	$-0.501838\pi$
$281$	−0.193576	−0.0115478	−0.00577389	+	0.999983i	$0.501838\pi$
$282$	0	0
$283$	−27.1842	−1.61593	−0.807967	−	0.589228i	$-0.799432\pi$
$283$	−27.1842	−1.61593	−0.807967	+	0.589228i	$0.799432\pi$
$284$	−4.47013	−0.265253
$285$	0	0
$286$	0	0
$287$	−0.857279	−0.0506036
$288$	0	0
$289$	9.87601	0.580942
$290$	14.8573	0.872449
$291$	0	0
$292$	−6.87019	−0.402047
$293$	−2.81579	−0.164500	−0.0822502	−	0.996612i	$-0.526211\pi$
$293$	−2.81579	−0.164500	−0.0822502	+	0.996612i	$0.526211\pi$
$294$	0	0
$295$	−4.85728	−0.282802
$296$	1.43801	0.0835825
$297$	0	0
$298$	−2.73683	−0.158540
$299$	−5.51114	−0.318717
$300$	0	0
$301$	−25.1240	−1.44812
$302$	23.1713	1.33336
$303$	0	0
$304$	32.5245	1.86541
$305$	6.85728	0.392647
$306$	0	0
$307$	24.4286	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$307$	24.4286	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$308$	0	0
$309$	0	0
$310$	−5.24443	−0.297864
$311$	−19.8796	−1.12727	−0.563633	−	0.826025i	$-0.690597\pi$
$311$	−19.8796	−1.12727	−0.563633	+	0.826025i	$0.690597\pi$
$312$	0	0
$313$	−15.7146	−0.888239	−0.444120	−	0.895967i	$-0.646483\pi$
$313$	−15.7146	−0.888239	−0.444120	+	0.895967i	$0.646483\pi$
$314$	35.1526	1.98377
$315$	0	0
$316$	−13.8894	−0.781340
$317$	−16.4889	−0.926107	−0.463053	−	0.886330i	$-0.653246\pi$
$317$	−16.4889	−0.926107	−0.463053	+	0.886330i	$0.653246\pi$
$318$	0	0
$319$	0	0
$320$	4.74620	0.265321
$321$	0	0
$322$	−74.6548	−4.16035
$323$	36.5531	2.03387
$324$	0	0
$325$	0.622216	0.0345143
$326$	−19.2257	−1.06481
$327$	0	0
$328$	0.139182	0.00768504
$329$	12.2034	0.672796
$330$	0	0
$331$	15.3461	0.843500	0.421750	−	0.906712i	$-0.361416\pi$
$331$	15.3461	0.843500	0.421750	+	0.906712i	$0.361416\pi$
$332$	0.216327	0.0118725
$333$	0	0
$334$	−31.0923	−1.70130
$335$	1.24443	0.0679905
$336$	0	0
$337$	−28.2351	−1.53806	−0.769031	−	0.639212i	$-0.779261\pi$
$337$	−28.2351	−1.53806	−0.769031	+	0.639212i	$0.779261\pi$
$338$	−24.0049	−1.30570
$339$	0	0
$340$	8.40990	0.456091
$341$	0	0
$342$	0	0
$343$	24.8573	1.34217
$344$	4.07896	0.219923
$345$	0	0
$346$	−17.4795	−0.939703
$347$	2.62222	0.140768	0.0703840	−	0.997520i	$-0.477578\pi$
$347$	2.62222	0.140768	0.0703840	+	0.997520i	$0.477578\pi$
$348$	0	0
$349$	−5.14272	−0.275284	−0.137642	−	0.990482i	$-0.543952\pi$
$349$	−5.14272	−0.275284	−0.137642	+	0.990482i	$0.543952\pi$
$350$	8.42864	0.450530
$351$	0	0
$352$	0	0
$353$	9.34614	0.497445	0.248722	−	0.968575i	$-0.419989\pi$
$353$	9.34614	0.497445	0.248722	+	0.968575i	$0.419989\pi$
$354$	0	0
$355$	2.75557	0.146250
$356$	−9.10525	−0.482577
$357$	0	0
$358$	48.2034	2.54763
$359$	10.7556	0.567657	0.283829	−	0.958875i	$-0.408395\pi$
$359$	10.7556	0.567657	0.283829	+	0.958875i	$0.408395\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	−25.9081	−1.36170
$363$	0	0
$364$	4.47013	0.234298
$365$	4.23506	0.221673
$366$	0	0
$367$	33.7975	1.76422	0.882108	−	0.471046i	$-0.156124\pi$
$367$	33.7975	1.76422	0.882108	+	0.471046i	$0.156124\pi$
$368$	40.8573	2.12983
$369$	0	0
$370$	3.80642	0.197887
$371$	48.0830	2.49634
$372$	0	0
$373$	−33.9496	−1.75784	−0.878922	−	0.476965i	$-0.841737\pi$
$373$	−33.9496	−1.75784	−0.878922	+	0.476965i	$0.841737\pi$
$374$	0	0
$375$	0	0
$376$	−1.98126	−0.102176
$377$	−4.85728	−0.250163
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	11.4380	0.586757
$381$	0	0
$382$	−11.6128	−0.594165
$383$	14.6351	0.747820	0.373910	−	0.927465i	$-0.378017\pi$
$383$	14.6351	0.747820	0.373910	+	0.927465i	$0.378017\pi$
$384$	0	0
$385$	0	0
$386$	−34.8988	−1.77630
$387$	0	0
$388$	11.7520	0.596619
$389$	5.61285	0.284583	0.142291	−	0.989825i	$-0.454553\pi$
$389$	5.61285	0.284583	0.142291	+	0.989825i	$0.454553\pi$
$390$	0	0
$391$	45.9180	2.32217
$392$	−9.06868	−0.458038
$393$	0	0
$394$	−12.7427	−0.641966
$395$	8.56199	0.430801
$396$	0	0
$397$	−12.7556	−0.640184	−0.320092	−	0.947387i	$-0.603714\pi$
$397$	−12.7556	−0.640184	−0.320092	+	0.947387i	$0.603714\pi$
$398$	26.8385	1.34529
$399$	0	0
$400$	−4.61285	−0.230642
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	1.71456	0.0854082
$404$	7.56553	0.376399
$405$	0	0
$406$	−65.7975	−3.26548
$407$	0	0
$408$	0	0
$409$	7.12399	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$409$	7.12399	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$410$	0.368416	0.0181948
$411$	0	0
$412$	−18.8385	−0.928108
$413$	21.5111	1.05849
$414$	0	0
$415$	−0.133353	−0.00654605
$416$	−4.56782	−0.223956
$417$	0	0
$418$	0	0
$419$	−15.6128	−0.762738	−0.381369	−	0.924423i	$-0.624547\pi$
$419$	−15.6128	−0.762738	−0.381369	+	0.924423i	$0.624547\pi$
$420$	0	0
$421$	7.89829	0.384939	0.192470	−	0.981303i	$-0.438350\pi$
$421$	7.89829	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$422$	20.2953	0.987959
$423$	0	0
$424$	−7.80642	−0.379113
$425$	−5.18421	−0.251471
$426$	0	0
$427$	−30.3684	−1.46963
$428$	4.25380	0.205615
$429$	0	0
$430$	10.7971	0.520680
$431$	34.3051	1.65242	0.826210	−	0.563362i	$-0.190492\pi$
$431$	34.3051	1.65242	0.826210	+	0.563362i	$0.190492\pi$
$432$	0	0
$433$	14.4701	0.695390	0.347695	−	0.937608i	$-0.386964\pi$
$433$	14.4701	0.695390	0.347695	+	0.937608i	$0.386964\pi$
$434$	23.2257	1.11487
$435$	0	0
$436$	31.9813	1.53162
$437$	62.4514	2.98746
$438$	0	0
$439$	19.3176	0.921977	0.460988	−	0.887406i	$-0.347495\pi$
$439$	19.3176	0.921977	0.460988	+	0.887406i	$0.347495\pi$
$440$	0	0
$441$	0	0
$442$	−6.13918	−0.292011
$443$	13.1240	0.623539	0.311770	−	0.950158i	$-0.399078\pi$
$443$	13.1240	0.623539	0.311770	+	0.950158i	$0.399078\pi$
$444$	0	0
$445$	5.61285	0.266074
$446$	16.8573	0.798215
$447$	0	0
$448$	−21.0192	−0.993064
$449$	32.3051	1.52457	0.762287	−	0.647240i	$-0.224077\pi$
$449$	32.3051	1.52457	0.762287	+	0.647240i	$0.224077\pi$
$450$	0	0
$451$	0	0
$452$	9.73329	0.457816
$453$	0	0
$454$	25.4608	1.19493
$455$	−2.75557	−0.129183
$456$	0	0
$457$	23.4608	1.09745	0.548724	−	0.836004i	$-0.315114\pi$
$457$	23.4608	1.09745	0.548724	+	0.836004i	$0.315114\pi$
$458$	21.9081	1.02370
$459$	0	0
$460$	14.3684	0.669931
$461$	−28.8671	−1.34448	−0.672238	−	0.740335i	$-0.734667\pi$
$461$	−28.8671	−1.34448	−0.672238	+	0.740335i	$0.734667\pi$
$462$	0	0
$463$	−19.3461	−0.899091	−0.449546	−	0.893257i	$-0.648414\pi$
$463$	−19.3461	−0.899091	−0.449546	+	0.893257i	$0.648414\pi$
$464$	36.0098	1.67172
$465$	0	0
$466$	−8.23506	−0.381482
$467$	−3.14272	−0.145428	−0.0727139	−	0.997353i	$-0.523166\pi$
$467$	−3.14272	−0.145428	−0.0727139	+	0.997353i	$0.523166\pi$
$468$	0	0
$469$	−5.51114	−0.254481
$470$	−5.24443	−0.241908
$471$	0	0
$472$	−3.49240	−0.160751
$473$	0	0
$474$	0	0
$475$	−7.05086	−0.323515
$476$	−37.2444	−1.70710
$477$	0	0
$478$	−6.36842	−0.291285
$479$	−24.8573	−1.13576	−0.567879	−	0.823112i	$-0.692236\pi$
$479$	−24.8573	−1.13576	−0.567879	+	0.823112i	$0.692236\pi$
$480$	0	0
$481$	−1.24443	−0.0567412
$482$	2.56199	0.116696
$483$	0	0
$484$	0	0
$485$	−7.24443	−0.328953
$486$	0	0
$487$	−11.5299	−0.522468	−0.261234	−	0.965275i	$-0.584129\pi$
$487$	−11.5299	−0.522468	−0.261234	+	0.965275i	$0.584129\pi$
$488$	4.93041	0.223189
$489$	0	0
$490$	−24.0049	−1.08443
$491$	16.3872	0.739542	0.369771	−	0.929123i	$-0.379436\pi$
$491$	16.3872	0.739542	0.369771	+	0.929123i	$0.379436\pi$
$492$	0	0
$493$	40.4701	1.82268
$494$	−8.34968	−0.375670
$495$	0	0
$496$	−12.7110	−0.570742
$497$	−12.2034	−0.547398
$498$	0	0
$499$	−25.3274	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$499$	−25.3274	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$500$	−1.62222	−0.0725477
$501$	0	0
$502$	−43.3087	−1.93296
$503$	19.0923	0.851285	0.425643	−	0.904891i	$-0.360048\pi$
$503$	19.0923	0.851285	0.425643	+	0.904891i	$0.360048\pi$
$504$	0	0
$505$	−4.66370	−0.207532
$506$	0	0
$507$	0	0
$508$	24.6321	1.09287
$509$	32.4514	1.43838	0.719191	−	0.694812i	$-0.244512\pi$
$509$	32.4514	1.43838	0.719191	+	0.694812i	$0.244512\pi$
$510$	0	0
$511$	−18.7556	−0.829698
$512$	27.2306	1.20343
$513$	0	0
$514$	13.0509	0.575649
$515$	11.6128	0.511723
$516$	0	0
$517$	0	0
$518$	−16.8573	−0.740666
$519$	0	0
$520$	0.447375	0.0196187
$521$	29.2257	1.28040	0.640200	−	0.768208i	$-0.278851\pi$
$521$	29.2257	1.28040	0.640200	+	0.768208i	$0.278851\pi$
$522$	0	0
$523$	−6.71408	−0.293586	−0.146793	−	0.989167i	$-0.546895\pi$
$523$	−6.71408	−0.293586	−0.146793	+	0.989167i	$0.546895\pi$
$524$	2.01874	0.0881889
$525$	0	0
$526$	−56.2993	−2.45477
$527$	−14.2854	−0.622284
$528$	0	0
$529$	55.4514	2.41093
$530$	−20.6637	−0.897574
$531$	0	0
$532$	−50.6548	−2.19616
$533$	−0.120446	−0.00521710
$534$	0	0
$535$	−2.62222	−0.113368
$536$	0.894751	0.0386473
$537$	0	0
$538$	−16.1561	−0.696539
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−27.9081	−1.19876
$543$	0	0
$544$	38.0584	1.63174
$545$	−19.7146	−0.844479
$546$	0	0
$547$	−41.3689	−1.76881	−0.884403	−	0.466724i	$-0.845434\pi$
$547$	−41.3689	−1.76881	−0.884403	+	0.466724i	$0.845434\pi$
$548$	0.793040	0.0338770
$549$	0	0
$550$	0	0
$551$	55.0420	2.34487
$552$	0	0
$553$	−37.9180	−1.61244
$554$	−27.7935	−1.18083
$555$	0	0
$556$	−28.8859	−1.22503
$557$	−20.7971	−0.881200	−0.440600	−	0.897704i	$-0.645234\pi$
$557$	−20.7971	−0.881200	−0.440600	+	0.897704i	$0.645234\pi$
$558$	0	0
$559$	−3.52987	−0.149298
$560$	20.4286	0.863268
$561$	0	0
$562$	−0.368416	−0.0155407
$563$	37.7275	1.59002	0.795012	−	0.606594i	$-0.207465\pi$
$563$	37.7275	1.59002	0.795012	+	0.606594i	$0.207465\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−51.7373	−2.17468
$567$	0	0
$568$	1.98126	0.0831320
$569$	−7.33630	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$569$	−7.33630	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$570$	0	0
$571$	−36.6450	−1.53354	−0.766772	−	0.641919i	$-0.778138\pi$
$571$	−36.6450	−1.53354	−0.766772	+	0.641919i	$0.778138\pi$
$572$	0	0
$573$	0	0
$574$	−1.63158	−0.0681010
$575$	−8.85728	−0.369374
$576$	0	0
$577$	4.22216	0.175771	0.0878853	−	0.996131i	$-0.471989\pi$
$577$	4.22216	0.175771	0.0878853	+	0.996131i	$0.471989\pi$
$578$	18.7961	0.781817
$579$	0	0
$580$	12.6637	0.525832
$581$	0.590573	0.0245011
$582$	0	0
$583$	0	0
$584$	3.04503	0.126004
$585$	0	0
$586$	−5.35905	−0.221380
$587$	−34.3684	−1.41854	−0.709268	−	0.704939i	$-0.750974\pi$
$587$	−34.3684	−1.41854	−0.709268	+	0.704939i	$0.750974\pi$
$588$	0	0
$589$	−19.4291	−0.800563
$590$	−9.24443	−0.380587
$591$	0	0
$592$	9.22570	0.379174
$593$	27.9398	1.14735	0.573675	−	0.819083i	$-0.305517\pi$
$593$	27.9398	1.14735	0.573675	+	0.819083i	$0.305517\pi$
$594$	0	0
$595$	22.9590	0.941227
$596$	−2.33276	−0.0955535
$597$	0	0
$598$	−10.4889	−0.428921
$599$	31.2257	1.27585	0.637924	−	0.770100i	$-0.279794\pi$
$599$	31.2257	1.27585	0.637924	+	0.770100i	$0.279794\pi$
$600$	0	0
$601$	8.75557	0.357147	0.178574	−	0.983927i	$-0.442852\pi$
$601$	8.75557	0.357147	0.178574	+	0.983927i	$0.442852\pi$
$602$	−47.8163	−1.94885
$603$	0	0
$604$	19.7502	0.803625
$605$	0	0
$606$	0	0
$607$	15.1842	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$607$	15.1842	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$608$	51.7619	2.09922
$609$	0	0
$610$	13.0509	0.528414
$611$	1.71456	0.0693636
$612$	0	0
$613$	−42.7239	−1.72560	−0.862802	−	0.505543i	$-0.831292\pi$
$613$	−42.7239	−1.72560	−0.862802	+	0.505543i	$0.831292\pi$
$614$	46.4929	1.87630
$615$	0	0
$616$	0	0
$617$	3.51114	0.141353	0.0706765	−	0.997499i	$-0.477484\pi$
$617$	3.51114	0.141353	0.0706765	+	0.997499i	$0.477484\pi$
$618$	0	0
$619$	17.5941	0.707167	0.353584	−	0.935403i	$-0.384963\pi$
$619$	17.5941	0.707167	0.353584	+	0.935403i	$0.384963\pi$
$620$	−4.47013	−0.179525
$621$	0	0
$622$	−37.8350	−1.51705
$623$	−24.8573	−0.995886
$624$	0	0
$625$	1.00000	0.0400000
$626$	−29.9081	−1.19537
$627$	0	0
$628$	29.9625	1.19564
$629$	10.3684	0.413416
$630$	0	0
$631$	15.8163	0.629636	0.314818	−	0.949152i	$-0.398057\pi$
$631$	15.8163	0.629636	0.314818	+	0.949152i	$0.398057\pi$
$632$	6.15610	0.244877
$633$	0	0
$634$	−31.3818	−1.24633
$635$	−15.1842	−0.602567
$636$	0	0
$637$	7.84791	0.310946
$638$	0	0
$639$	0	0
$640$	−5.64941	−0.223313
$641$	−25.8163	−1.01968	−0.509841	−	0.860269i	$-0.670296\pi$
$641$	−25.8163	−1.01968	−0.509841	+	0.860269i	$0.670296\pi$
$642$	0	0
$643$	18.1017	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$643$	18.1017	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$644$	−63.6325	−2.50747
$645$	0	0
$646$	69.5683	2.73713
$647$	47.0420	1.84941	0.924705	−	0.380684i	$-0.124311\pi$
$647$	47.0420	1.84941	0.924705	+	0.380684i	$0.124311\pi$
$648$	0	0
$649$	0	0
$650$	1.18421	0.0464485
$651$	0	0
$652$	−16.3872	−0.641770
$653$	−30.0830	−1.17724	−0.588619	−	0.808411i	$-0.700328\pi$
$653$	−30.0830	−1.17724	−0.588619	+	0.808411i	$0.700328\pi$
$654$	0	0
$655$	−1.24443	−0.0486240
$656$	0.892937	0.0348633
$657$	0	0
$658$	23.2257	0.905432
$659$	−10.2854	−0.400664	−0.200332	−	0.979728i	$-0.564202\pi$
$659$	−10.2854	−0.400664	−0.200332	+	0.979728i	$0.564202\pi$
$660$	0	0
$661$	−27.7146	−1.07797	−0.538986	−	0.842315i	$-0.681192\pi$
$661$	−27.7146	−1.07797	−0.538986	+	0.842315i	$0.681192\pi$
$662$	29.2070	1.13516
$663$	0	0
$664$	−0.0958814	−0.00372092
$665$	31.2257	1.21088
$666$	0	0
$667$	69.1437	2.67725
$668$	−26.5018	−1.02538
$669$	0	0
$670$	2.36842	0.0914999
$671$	0	0
$672$	0	0
$673$	9.86665	0.380331	0.190166	−	0.981752i	$-0.439097\pi$
$673$	9.86665	0.380331	0.190166	+	0.981752i	$0.439097\pi$
$674$	−53.7373	−2.06988
$675$	0	0
$676$	−20.4608	−0.786952
$677$	−5.65433	−0.217314	−0.108657	−	0.994079i	$-0.534655\pi$
$677$	−5.65433	−0.217314	−0.108657	+	0.994079i	$0.534655\pi$
$678$	0	0
$679$	32.0830	1.23123
$680$	−3.72746	−0.142942
$681$	0	0
$682$	0	0
$683$	−34.1847	−1.30804	−0.654020	−	0.756477i	$-0.726919\pi$
$683$	−34.1847	−1.30804	−0.654020	+	0.756477i	$0.726919\pi$
$684$	0	0
$685$	−0.488863	−0.0186785
$686$	47.3087	1.80625
$687$	0	0
$688$	26.1690	0.997684
$689$	6.75557	0.257367
$690$	0	0
$691$	−19.2257	−0.731380	−0.365690	−	0.930737i	$-0.619167\pi$
$691$	−19.2257	−0.731380	−0.365690	+	0.930737i	$0.619167\pi$
$692$	−14.8988	−0.566366
$693$	0	0
$694$	4.99063	0.189442
$695$	17.8064	0.675436
$696$	0	0
$697$	1.00354	0.0380118
$698$	−9.78769	−0.370469
$699$	0	0
$700$	7.18421	0.271538
$701$	−29.9081	−1.12961	−0.564807	−	0.825223i	$-0.691050\pi$
$701$	−29.9081	−1.12961	−0.564807	+	0.825223i	$0.691050\pi$
$702$	0	0
$703$	14.1017	0.531856
$704$	0	0
$705$	0	0
$706$	17.7877	0.669448
$707$	20.6539	0.776768
$708$	0	0
$709$	−15.3274	−0.575633	−0.287816	−	0.957686i	$-0.592929\pi$
$709$	−15.3274	−0.575633	−0.287816	+	0.957686i	$0.592929\pi$
$710$	5.24443	0.196820
$711$	0	0
$712$	4.03566	0.151243
$713$	−24.4068	−0.914043
$714$	0	0
$715$	0	0
$716$	41.0865	1.53548
$717$	0	0
$718$	20.4701	0.763938
$719$	−23.8163	−0.888197	−0.444098	−	0.895978i	$-0.646476\pi$
$719$	−23.8163	−0.888197	−0.444098	+	0.895978i	$0.646476\pi$
$720$	0	0
$721$	−51.4291	−1.91532
$722$	58.4563	2.17552
$723$	0	0
$724$	−22.0830	−0.820707
$725$	−7.80642	−0.289923
$726$	0	0
$727$	−32.9403	−1.22169	−0.610843	−	0.791752i	$-0.709169\pi$
$727$	−32.9403	−1.22169	−0.610843	+	0.791752i	$0.709169\pi$
$728$	−1.98126	−0.0734305
$729$	0	0
$730$	8.06022	0.298322
$731$	29.4104	1.08778
$732$	0	0
$733$	29.8666	1.10315	0.551575	−	0.834125i	$-0.314027\pi$
$733$	29.8666	1.10315	0.551575	+	0.834125i	$0.314027\pi$
$734$	64.3239	2.37424
$735$	0	0
$736$	65.0232	2.39679
$737$	0	0
$738$	0	0
$739$	5.06959	0.186488	0.0932440	−	0.995643i	$-0.470276\pi$
$739$	5.06959	0.186488	0.0932440	+	0.995643i	$0.470276\pi$
$740$	3.24443	0.119268
$741$	0	0
$742$	91.5121	3.35951
$743$	22.4385	0.823188	0.411594	−	0.911367i	$-0.364972\pi$
$743$	22.4385	0.823188	0.411594	+	0.911367i	$0.364972\pi$
$744$	0	0
$745$	1.43801	0.0526845
$746$	−64.6133	−2.36566
$747$	0	0
$748$	0	0
$749$	11.6128	0.424324
$750$	0	0
$751$	−6.63512	−0.242119	−0.121060	−	0.992645i	$-0.538629\pi$
$751$	−6.63512	−0.242119	−0.121060	+	0.992645i	$0.538629\pi$
$752$	−12.7110	−0.463523
$753$	0	0
$754$	−9.24443	−0.336662
$755$	−12.1748	−0.443088
$756$	0	0
$757$	8.75557	0.318227	0.159113	−	0.987260i	$-0.449136\pi$
$757$	8.75557	0.318227	0.159113	+	0.987260i	$0.449136\pi$
$758$	−38.0642	−1.38256
$759$	0	0
$760$	−5.06959	−0.183893
$761$	3.15257	0.114280	0.0571402	−	0.998366i	$-0.481802\pi$
$761$	3.15257	0.114280	0.0571402	+	0.998366i	$0.481802\pi$
$762$	0	0
$763$	87.3087	3.16079
$764$	−9.89829	−0.358108
$765$	0	0
$766$	27.8537	1.00640
$767$	3.02227	0.109128
$768$	0	0
$769$	28.9590	1.04429	0.522144	−	0.852857i	$-0.325132\pi$
$769$	28.9590	1.04429	0.522144	+	0.852857i	$0.325132\pi$
$770$	0	0
$771$	0	0
$772$	−29.7462	−1.07059
$773$	−29.1427	−1.04819	−0.524095	−	0.851660i	$-0.675596\pi$
$773$	−29.1427	−1.04819	−0.524095	+	0.851660i	$0.675596\pi$
$774$	0	0
$775$	2.75557	0.0989830
$776$	−5.20877	−0.186984
$777$	0	0
$778$	10.6824	0.382984
$779$	1.36488	0.0489018
$780$	0	0
$781$	0	0
$782$	87.3916	3.12512
$783$	0	0
$784$	−58.1811	−2.07790
$785$	−18.4701	−0.659227
$786$	0	0
$787$	11.2672	0.401632	0.200816	−	0.979629i	$-0.435641\pi$
$787$	11.2672	0.401632	0.200816	+	0.979629i	$0.435641\pi$
$788$	−10.8613	−0.386918
$789$	0	0
$790$	16.2953	0.579760
$791$	26.5718	0.944786
$792$	0	0
$793$	−4.26671	−0.151515
$794$	−24.2766	−0.861543
$795$	0	0
$796$	22.8760	0.810819
$797$	−41.9625	−1.48639	−0.743195	−	0.669075i	$-0.766690\pi$
$797$	−41.9625	−1.48639	−0.743195	+	0.669075i	$0.766690\pi$
$798$	0	0
$799$	−14.2854	−0.505383
$800$	−7.34122	−0.259551
$801$	0	0
$802$	−3.80642	−0.134409
$803$	0	0
$804$	0	0
$805$	39.2257	1.38252
$806$	3.26317	0.114940
$807$	0	0
$808$	−3.35322	−0.117966
$809$	−27.8064	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$809$	−27.8064	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$810$	0	0
$811$	−6.78415	−0.238224	−0.119112	−	0.992881i	$-0.538005\pi$
$811$	−6.78415	−0.238224	−0.119112	+	0.992881i	$0.538005\pi$
$812$	−56.0830	−1.96813
$813$	0	0
$814$	0	0
$815$	10.1017	0.353847
$816$	0	0
$817$	40.0000	1.39942
$818$	13.5585	0.474060
$819$	0	0
$820$	0.314022	0.0109661
$821$	3.62269	0.126433	0.0632164	−	0.998000i	$-0.479864\pi$
$821$	3.62269	0.126433	0.0632164	+	0.998000i	$0.479864\pi$
$822$	0	0
$823$	42.0642	1.46627	0.733134	−	0.680085i	$-0.238057\pi$
$823$	42.0642	1.46627	0.733134	+	0.680085i	$0.238057\pi$
$824$	8.34968	0.290875
$825$	0	0
$826$	40.9403	1.42449
$827$	30.8256	1.07191	0.535956	−	0.844246i	$-0.319951\pi$
$827$	30.8256	1.07191	0.535956	+	0.844246i	$0.319951\pi$
$828$	0	0
$829$	7.12399	0.247426	0.123713	−	0.992318i	$-0.460520\pi$
$829$	7.12399	0.247426	0.123713	+	0.992318i	$0.460520\pi$
$830$	−0.253799	−0.00880950
$831$	0	0
$832$	−2.95316	−0.102382
$833$	−65.3876	−2.26555
$834$	0	0
$835$	16.3368	0.565357
$836$	0	0
$837$	0	0
$838$	−29.7146	−1.02647
$839$	−3.34614	−0.115522	−0.0577608	−	0.998330i	$-0.518396\pi$
$839$	−3.34614	−0.115522	−0.0577608	+	0.998330i	$0.518396\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	15.0321	0.518041
$843$	0	0
$844$	17.2988	0.595450
$845$	12.6128	0.433895
$846$	0	0
$847$	0	0
$848$	−50.0830	−1.71986
$849$	0	0
$850$	−9.86665	−0.338423
$851$	17.7146	0.607247
$852$	0	0
$853$	26.4197	0.904595	0.452297	−	0.891867i	$-0.350605\pi$
$853$	26.4197	0.904595	0.452297	+	0.891867i	$0.350605\pi$
$854$	−57.7975	−1.97779
$855$	0	0
$856$	−1.88538	−0.0644411
$857$	38.7783	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$857$	38.7783	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$858$	0	0
$859$	−27.3087	−0.931760	−0.465880	−	0.884848i	$-0.654262\pi$
$859$	−27.3087	−0.931760	−0.465880	+	0.884848i	$0.654262\pi$
$860$	9.20294	0.313818
$861$	0	0
$862$	65.2899	2.22378
$863$	49.5308	1.68605	0.843024	−	0.537875i	$-0.180773\pi$
$863$	49.5308	1.68605	0.843024	+	0.537875i	$0.180773\pi$
$864$	0	0
$865$	9.18421	0.312272
$866$	27.5397	0.935838
$867$	0	0
$868$	19.7966	0.671940
$869$	0	0
$870$	0	0
$871$	−0.774305	−0.0262363
$872$	−14.1748	−0.480021
$873$	0	0
$874$	118.858	4.02044
$875$	−4.42864	−0.149715
$876$	0	0
$877$	4.50177	0.152014	0.0760070	−	0.997107i	$-0.475783\pi$
$877$	4.50177	0.152014	0.0760070	+	0.997107i	$0.475783\pi$
$878$	36.7654	1.24077
$879$	0	0
$880$	0	0
$881$	15.1240	0.509540	0.254770	−	0.967002i	$-0.418000\pi$
$881$	15.1240	0.509540	0.254770	+	0.967002i	$0.418000\pi$
$882$	0	0
$883$	−30.2480	−1.01793	−0.508963	−	0.860789i	$-0.669971\pi$
$883$	−30.2480	−1.01793	−0.508963	+	0.860789i	$0.669971\pi$
$884$	−5.23277	−0.175997
$885$	0	0
$886$	24.9777	0.839143
$887$	57.1941	1.92039	0.960194	−	0.279333i	$-0.0901135\pi$
$887$	57.1941	1.92039	0.960194	+	0.279333i	$0.0901135\pi$
$888$	0	0
$889$	67.2454	2.25534
$890$	10.6824	0.358076
$891$	0	0
$892$	14.3684	0.481090
$893$	−19.4291	−0.650171
$894$	0	0
$895$	−25.3274	−0.846602
$896$	25.0192	0.835833
$897$	0	0
$898$	61.4835	2.05173
$899$	−21.5111	−0.717437
$900$	0	0
$901$	−56.2864	−1.87517
$902$	0	0
$903$	0	0
$904$	−4.31402	−0.143482
$905$	13.6128	0.452506
$906$	0	0
$907$	−53.2641	−1.76861	−0.884303	−	0.466913i	$-0.845366\pi$
$907$	−53.2641	−1.76861	−0.884303	+	0.466913i	$0.845366\pi$
$908$	21.7017	0.720195
$909$	0	0
$910$	−5.24443	−0.173851
$911$	0.590573	0.0195665	0.00978327	−	0.999952i	$-0.496886\pi$
$911$	0.590573	0.0195665	0.00978327	+	0.999952i	$0.496886\pi$
$912$	0	0
$913$	0	0
$914$	44.6508	1.47692
$915$	0	0
$916$	18.6735	0.616991
$917$	5.51114	0.181994
$918$	0	0
$919$	−55.8707	−1.84300	−0.921502	−	0.388375i	$-0.873037\pi$
$919$	−55.8707	−1.84300	−0.921502	+	0.388375i	$0.873037\pi$
$920$	−6.36842	−0.209960
$921$	0	0
$922$	−54.9403	−1.80936
$923$	−1.71456	−0.0564354
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−36.8198	−1.20997
$927$	0	0
$928$	57.3087	1.88125
$929$	−15.3274	−0.502876	−0.251438	−	0.967873i	$-0.580903\pi$
$929$	−15.3274	−0.502876	−0.251438	+	0.967873i	$0.580903\pi$
$930$	0	0
$931$	−88.9314	−2.91461
$932$	−7.01921	−0.229922
$933$	0	0
$934$	−5.98126	−0.195713
$935$	0	0
$936$	0	0
$937$	27.8479	0.909752	0.454876	−	0.890555i	$-0.349684\pi$
$937$	27.8479	0.909752	0.454876	+	0.890555i	$0.349684\pi$
$938$	−10.4889	−0.342474
$939$	0	0
$940$	−4.47013	−0.145799
$941$	10.4157	0.339543	0.169772	−	0.985483i	$-0.445697\pi$
$941$	10.4157	0.339543	0.169772	+	0.985483i	$0.445697\pi$
$942$	0	0
$943$	1.71456	0.0558337
$944$	−22.4059	−0.729250
$945$	0	0
$946$	0	0
$947$	−8.47013	−0.275242	−0.137621	−	0.990485i	$-0.543946\pi$
$947$	−8.47013	−0.275242	−0.137621	+	0.990485i	$0.543946\pi$
$948$	0	0
$949$	−2.63512	−0.0855397
$950$	−13.4193	−0.435379
$951$	0	0
$952$	16.5076	0.535014
$953$	−8.71408	−0.282277	−0.141138	−	0.989990i	$-0.545076\pi$
$953$	−8.71408	−0.282277	−0.141138	+	0.989990i	$0.545076\pi$
$954$	0	0
$955$	6.10171	0.197447
$956$	−5.42816	−0.175559
$957$	0	0
$958$	−47.3087	−1.52847
$959$	2.16500	0.0699114
$960$	0	0
$961$	−23.4068	−0.755059
$962$	−2.36842	−0.0763608
$963$	0	0
$964$	2.18373	0.0703333
$965$	18.3368	0.590282
$966$	0	0
$967$	−44.2449	−1.42282	−0.711410	−	0.702777i	$-0.751943\pi$
$967$	−44.2449	−1.42282	−0.711410	+	0.702777i	$0.751943\pi$
$968$	0	0
$969$	0	0
$970$	−13.7877	−0.442696
$971$	57.1437	1.83383	0.916914	−	0.399085i	$-0.130672\pi$
$971$	57.1437	1.83383	0.916914	+	0.399085i	$0.130672\pi$
$972$	0	0
$973$	−78.8582	−2.52808
$974$	−21.9438	−0.703124
$975$	0	0
$976$	31.6316	1.01250
$977$	16.2480	0.519819	0.259909	−	0.965633i	$-0.416307\pi$
$977$	16.2480	0.519819	0.259909	+	0.965633i	$0.416307\pi$
$978$	0	0
$979$	0	0
$980$	−20.4608	−0.653595
$981$	0	0
$982$	31.1882	0.995256
$983$	−1.12399	−0.0358496	−0.0179248	−	0.999839i	$-0.505706\pi$
$983$	−1.12399	−0.0358496	−0.0179248	+	0.999839i	$0.505706\pi$
$984$	0	0
$985$	6.69535	0.213331
$986$	77.0232	2.45292
$987$	0	0
$988$	−7.11691	−0.226419
$989$	50.2480	1.59779
$990$	0	0
$991$	−53.6513	−1.70429	−0.852144	−	0.523307i	$-0.824698\pi$
$991$	−53.6513	−1.70429	−0.852144	+	0.523307i	$0.824698\pi$
$992$	−20.2292	−0.642279
$993$	0	0
$994$	−23.2257	−0.736674
$995$	−14.1017	−0.447054
$996$	0	0
$997$	35.7275	1.13150	0.565750	−	0.824577i	$-0.308587\pi$
$997$	35.7275	1.13150	0.565750	+	0.824577i	$0.308587\pi$
$998$	−48.2034	−1.52585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.z.1.3		3
3.2	odd	2		1815.2.a.m.1.1		3
11.10	odd	2		495.2.a.e.1.1		3
15.14	odd	2		9075.2.a.cf.1.3		3
33.32	even	2		165.2.a.c.1.3	✓	3
44.43	even	2		7920.2.a.cj.1.3		3
55.32	even	4		2475.2.c.r.199.2		6
55.43	even	4		2475.2.c.r.199.5		6
55.54	odd	2		2475.2.a.bb.1.3		3
132.131	odd	2		2640.2.a.be.1.3		3
165.32	odd	4		825.2.c.g.199.5		6
165.98	odd	4		825.2.c.g.199.2		6
165.164	even	2		825.2.a.k.1.1		3
231.230	odd	2		8085.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	33.32	even	2
495.2.a.e.1.1		3	11.10	odd	2
825.2.a.k.1.1		3	165.164	even	2
825.2.c.g.199.2		6	165.98	odd	4
825.2.c.g.199.5		6	165.32	odd	4
1815.2.a.m.1.1		3	3.2	odd	2
2475.2.a.bb.1.3		3	55.54	odd	2
2475.2.c.r.199.2		6	55.32	even	4
2475.2.c.r.199.5		6	55.43	even	4
2640.2.a.be.1.3		3	132.131	odd	2
5445.2.a.z.1.3		3	1.1	even	1	trivial
7920.2.a.cj.1.3		3	44.43	even	2
8085.2.a.bk.1.3		3	231.230	odd	2
9075.2.a.cf.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90321 q^{2} +1.62222 q^{4} -1.00000 q^{5} +4.42864 q^{7} -0.719004 q^{8} -1.90321 q^{10} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} -7.05086 q^{19} -1.62222 q^{20} -8.85728 q^{23} +1.00000 q^{25} +1.18421 q^{26} +7.18421 q^{28} -7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} -4.42864 q^{35} -2.00000 q^{37} -13.4193 q^{38} +0.719004 q^{40} -0.193576 q^{41} -5.67307 q^{43} -16.8573 q^{46} +2.75557 q^{47} +12.6128 q^{49} +1.90321 q^{50} +1.00937 q^{52} +10.8573 q^{53} -3.18421 q^{56} -14.8573 q^{58} +4.85728 q^{59} -6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.24443 q^{67} -8.40990 q^{68} -8.42864 q^{70} -2.75557 q^{71} -4.23506 q^{73} -3.80642 q^{74} -11.4380 q^{76} -8.56199 q^{79} +4.61285 q^{80} -0.368416 q^{82} +0.133353 q^{83} +5.18421 q^{85} -10.7971 q^{86} -5.61285 q^{89} +2.75557 q^{91} -14.3684 q^{92} +5.24443 q^{94} +7.05086 q^{95} +7.24443 q^{97} +24.0049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 3 q^{5} - 9 q^{8} + q^{10} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} - 8 q^{19} - 5 q^{20} + 3 q^{25} - 10 q^{26} + 8 q^{28} - 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34}+ \cdots + 39 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 9 * q^8 + q^10 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 - 8 * q^19 - 5 * q^20 + 3 * q^25 - 10 * q^26 + 8 * q^28 - 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 - 6 * q^37 + 9 * q^40 - 14 * q^41 - 4 * q^43 - 24 * q^46 + 8 * q^47 + 11 * q^49 - q^50 + 30 * q^52 + 6 * q^53 + 4 * q^56 - 18 * q^58 - 12 * q^59 + 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - 4 * q^67 + 42 * q^68 - 12 * q^70 - 8 * q^71 + 14 * q^73 + 2 * q^74 - 48 * q^76 - 12 * q^79 - 13 * q^80 + 26 * q^82 + 2 * q^85 + 8 * q^86 + 10 * q^89 + 8 * q^91 - 16 * q^92 + 16 * q^94 + 8 * q^95 + 22 * q^97 + 39 * q^98

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