LMFDB - Embedded newform 5445.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ 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)

$f(q)$	$=$	$q+2.74483 q^{2} +5.53407 q^{4} +1.00000 q^{5} -2.32331 q^{7} +9.70041 q^{8} +O(q^{10})$	$q+2.74483 q^{2} +5.53407 q^{4} +1.00000 q^{5} -2.32331 q^{7} +9.70041 q^{8} +2.74483 q^{10} -0.534070 q^{13} -6.37709 q^{14} +15.5578 q^{16} -2.42151 q^{17} +4.95558 q^{19} +5.53407 q^{20} +4.53407 q^{23} +1.00000 q^{25} -1.46593 q^{26} -12.8574 q^{28} +5.48965 q^{29} +1.04442 q^{31} +23.3026 q^{32} -6.64663 q^{34} -2.32331 q^{35} +7.48965 q^{37} +13.6022 q^{38} +9.70041 q^{40} -10.6022 q^{41} +4.32331 q^{43} +12.4452 q^{46} -6.76855 q^{47} -1.60221 q^{49} +2.74483 q^{50} -2.95558 q^{52} +4.53407 q^{53} -22.5371 q^{56} +15.0681 q^{58} +6.44523 q^{59} +6.79861 q^{61} +2.86675 q^{62} +32.8461 q^{64} -0.534070 q^{65} +0.721104 q^{67} -13.4008 q^{68} -6.37709 q^{70} -4.53407 q^{71} +1.06814 q^{73} +20.5578 q^{74} +27.4245 q^{76} +4.64663 q^{79} +15.5578 q^{80} -29.1012 q^{82} -13.6022 q^{83} -2.42151 q^{85} +11.8667 q^{86} -12.7148 q^{89} +1.24081 q^{91} +25.0919 q^{92} -18.5785 q^{94} +4.95558 q^{95} +4.75919 q^{97} -4.39779 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q - q^2 + 9 * q^4 + 3 * q^5 - q^7 + 9 * q^8	$ 3 q - q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 9 q^{8} - q^{10} + 6 q^{13} - 5 q^{14} + 13 q^{16} - 4 q^{17} + 4 q^{19} + 9 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 25 q^{28} - 2 q^{29} + 14 q^{31} + 27 q^{32} - 8 q^{34} - q^{35} + 4 q^{37} + 18 q^{38} + 9 q^{40} - 9 q^{41} + 7 q^{43} + 8 q^{46} + 15 q^{47} + 18 q^{49} - q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{56} + 30 q^{58} - 10 q^{59} + 3 q^{61} - 24 q^{62} + 29 q^{64} + 6 q^{65} + 19 q^{67} - 5 q^{70} - 6 q^{71} - 12 q^{73} + 28 q^{74} + 16 q^{76} + 2 q^{79} + 13 q^{80} - 27 q^{82} - 18 q^{83} - 4 q^{85} + 3 q^{86} - 11 q^{89} + 20 q^{91} + 34 q^{92} - 59 q^{94} + 4 q^{95} - 2 q^{97} - 36 q^{98}+O(q^{100}) $ 3 * q - q^2 + 9 * q^4 + 3 * q^5 - q^7 + 9 * q^8 - q^10 + 6 * q^13 - 5 * q^14 + 13 * q^16 - 4 * q^17 + 4 * q^19 + 9 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 25 * q^28 - 2 * q^29 + 14 * q^31 + 27 * q^32 - 8 * q^34 - q^35 + 4 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 7 * q^43 + 8 * q^46 + 15 * q^47 + 18 * q^49 - q^50 + 2 * q^52 + 6 * q^53 + 3 * q^56 + 30 * q^58 - 10 * q^59 + 3 * q^61 - 24 * q^62 + 29 * q^64 + 6 * q^65 + 19 * q^67 - 5 * q^70 - 6 * q^71 - 12 * q^73 + 28 * q^74 + 16 * q^76 + 2 * q^79 + 13 * q^80 - 27 * q^82 - 18 * q^83 - 4 * q^85 + 3 * q^86 - 11 * q^89 + 20 * q^91 + 34 * q^92 - 59 * q^94 + 4 * q^95 - 2 * q^97 - 36 * 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 $

$2$	2.74483	1.94089	0.970443	−	0.241332i	$-0.0775843\pi$
$2$	2.74483	1.94089	0.970443	+	0.241332i	$0.0775843\pi$
$3$	0	0
$3$	0	0
$4$	5.53407	2.76704
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.32331	−0.878130	−0.439065	−	0.898455i	$-0.644690\pi$
$7$	−2.32331	−0.878130	−0.439065	+	0.898455i	$0.644690\pi$
$8$	9.70041	3.42961
$9$	0	0
$10$	2.74483	0.867990
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.534070	−0.148124	−0.0740622	−	0.997254i	$-0.523596\pi$
$13$	−0.534070	−0.148124	−0.0740622	+	0.997254i	$0.523596\pi$
$14$	−6.37709	−1.70435
$15$	0	0
$16$	15.5578	3.88945
$17$	−2.42151	−0.587303	−0.293651	−	0.955913i	$-0.594870\pi$
$17$	−2.42151	−0.587303	−0.293651	+	0.955913i	$0.594870\pi$
$18$	0	0
$19$	4.95558	1.13689	0.568444	−	0.822722i	$-0.307546\pi$
$19$	4.95558	1.13689	0.568444	+	0.822722i	$0.307546\pi$
$20$	5.53407	1.23746
$21$	0	0
$22$	0	0
$23$	4.53407	0.945419	0.472709	−	0.881218i	$-0.343276\pi$
$23$	4.53407	0.945419	0.472709	+	0.881218i	$0.343276\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.46593	−0.287492
$27$	0	0
$28$	−12.8574	−2.42982
$29$	5.48965	1.01940	0.509701	−	0.860351i	$-0.329756\pi$
$29$	5.48965	1.01940	0.509701	+	0.860351i	$0.329756\pi$
$30$	0	0
$31$	1.04442	0.187583	0.0937915	−	0.995592i	$-0.470101\pi$
$31$	1.04442	0.187583	0.0937915	+	0.995592i	$0.470101\pi$
$32$	23.3026	4.11936
$33$	0	0
$34$	−6.64663	−1.13989
$35$	−2.32331	−0.392712
$36$	0	0
$37$	7.48965	1.23129	0.615646	−	0.788023i	$-0.288895\pi$
$37$	7.48965	1.23129	0.615646	+	0.788023i	$0.288895\pi$
$38$	13.6022	2.20657
$39$	0	0
$40$	9.70041	1.53377
$41$	−10.6022	−1.65579	−0.827894	−	0.560885i	$-0.810461\pi$
$41$	−10.6022	−1.65579	−0.827894	+	0.560885i	$0.810461\pi$
$42$	0	0
$43$	4.32331	0.659299	0.329650	−	0.944103i	$-0.393069\pi$
$43$	4.32331	0.659299	0.329650	+	0.944103i	$0.393069\pi$
$44$	0	0
$45$	0	0
$46$	12.4452	1.83495
$47$	−6.76855	−0.987294	−0.493647	−	0.869662i	$-0.664337\pi$
$47$	−6.76855	−0.987294	−0.493647	+	0.869662i	$0.664337\pi$
$48$	0	0
$49$	−1.60221	−0.228887
$50$	2.74483	0.388177
$51$	0	0
$52$	−2.95558	−0.409865
$53$	4.53407	0.622802	0.311401	−	0.950279i	$-0.399202\pi$
$53$	4.53407	0.622802	0.311401	+	0.950279i	$0.399202\pi$
$54$	0	0
$55$	0	0
$56$	−22.5371	−3.01165
$57$	0	0
$58$	15.0681	1.97854
$59$	6.44523	0.839098	0.419549	−	0.907733i	$-0.362188\pi$
$59$	6.44523	0.839098	0.419549	+	0.907733i	$0.362188\pi$
$60$	0	0
$61$	6.79861	0.870472	0.435236	−	0.900316i	$-0.356665\pi$
$61$	6.79861	0.870472	0.435236	+	0.900316i	$0.356665\pi$
$62$	2.86675	0.364077
$63$	0	0
$64$	32.8461	4.10576
$65$	−0.534070	−0.0662433
$66$	0	0
$67$	0.721104	0.0880968	0.0440484	−	0.999029i	$-0.485974\pi$
$67$	0.721104	0.0880968	0.0440484	+	0.999029i	$0.485974\pi$
$68$	−13.4008	−1.62509
$69$	0	0
$70$	−6.37709	−0.762208
$71$	−4.53407	−0.538095	−0.269048	−	0.963127i	$-0.586709\pi$
$71$	−4.53407	−0.538095	−0.269048	+	0.963127i	$0.586709\pi$
$72$	0	0
$73$	1.06814	0.125016	0.0625082	−	0.998044i	$-0.480090\pi$
$73$	1.06814	0.125016	0.0625082	+	0.998044i	$0.480090\pi$
$74$	20.5578	2.38979
$75$	0	0
$76$	27.4245	3.14581
$77$	0	0
$78$	0	0
$79$	4.64663	0.522787	0.261393	−	0.965232i	$-0.415818\pi$
$79$	4.64663	0.522787	0.261393	+	0.965232i	$0.415818\pi$
$80$	15.5578	1.73941
$81$	0	0
$82$	−29.1012	−3.21369
$83$	−13.6022	−1.49304	−0.746518	−	0.665365i	$-0.768276\pi$
$83$	−13.6022	−1.49304	−0.746518	+	0.665365i	$0.768276\pi$
$84$	0	0
$85$	−2.42151	−0.262650
$86$	11.8667	1.27962
$87$	0	0
$88$	0	0
$89$	−12.7148	−1.34776	−0.673881	−	0.738840i	$-0.735374\pi$
$89$	−12.7148	−1.34776	−0.673881	+	0.738840i	$0.735374\pi$
$90$	0	0
$91$	1.24081	0.130073
$92$	25.0919	2.61601
$93$	0	0
$94$	−18.5785	−1.91622
$95$	4.95558	0.508432
$96$	0	0
$97$	4.75919	0.483222	0.241611	−	0.970373i	$-0.422324\pi$
$97$	4.75919	0.483222	0.241611	+	0.970373i	$0.422324\pi$
$98$	−4.39779	−0.444244
$99$	0	0
$100$	5.53407	0.553407
$101$	6.57849	0.654584	0.327292	−	0.944923i	$-0.393864\pi$
$101$	6.57849	0.654584	0.327292	+	0.944923i	$0.393864\pi$
$102$	0	0
$103$	16.3564	1.61164	0.805822	−	0.592158i	$-0.201724\pi$
$103$	16.3564	1.61164	0.805822	+	0.592158i	$0.201724\pi$
$104$	−5.18070	−0.508009
$105$	0	0
$106$	12.4452	1.20879
$107$	−10.2108	−0.987111	−0.493556	−	0.869714i	$-0.664303\pi$
$107$	−10.2108	−0.987111	−0.493556	+	0.869714i	$0.664303\pi$
$108$	0	0
$109$	−12.0919	−1.15819	−0.579095	−	0.815260i	$-0.696594\pi$
$109$	−12.0919	−1.15819	−0.579095	+	0.815260i	$0.696594\pi$
$110$	0	0
$111$	0	0
$112$	−36.1456	−3.41544
$113$	10.8430	1.02003	0.510013	−	0.860167i	$-0.329641\pi$
$113$	10.8430	1.02003	0.510013	+	0.860167i	$0.329641\pi$
$114$	0	0
$115$	4.53407	0.422804
$116$	30.3801	2.82072
$117$	0	0
$118$	17.6910	1.62859
$119$	5.62593	0.515728
$120$	0	0
$121$	0	0
$122$	18.6610	1.68949
$123$	0	0
$124$	5.77988	0.519049
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.12192	−0.720704	−0.360352	−	0.932816i	$-0.617343\pi$
$127$	−8.12192	−0.720704	−0.360352	+	0.932816i	$0.617343\pi$
$128$	43.5515	3.84944
$129$	0	0
$130$	−1.46593	−0.128571
$131$	−5.79861	−0.506627	−0.253313	−	0.967384i	$-0.581520\pi$
$131$	−5.79861	−0.506627	−0.253313	+	0.967384i	$0.581520\pi$
$132$	0	0
$133$	−11.5134	−0.998336
$134$	1.97930	0.170986
$135$	0	0
$136$	−23.4897	−2.01422
$137$	−8.86675	−0.757537	−0.378769	−	0.925491i	$-0.623652\pi$
$137$	−8.86675	−0.757537	−0.378769	+	0.925491i	$0.623652\pi$
$138$	0	0
$139$	−16.8667	−1.43062	−0.715309	−	0.698808i	$-0.753714\pi$
$139$	−16.8667	−1.43062	−0.715309	+	0.698808i	$0.753714\pi$
$140$	−12.8574	−1.08665
$141$	0	0
$142$	−12.4452	−1.04438
$143$	0	0
$144$	0	0
$145$	5.48965	0.455891
$146$	2.93186	0.242642
$147$	0	0
$148$	41.4483	3.40703
$149$	−4.46593	−0.365863	−0.182932	−	0.983126i	$-0.558559\pi$
$149$	−4.46593	−0.365863	−0.182932	+	0.983126i	$0.558559\pi$
$150$	0	0
$151$	−11.5134	−0.936945	−0.468473	−	0.883478i	$-0.655196\pi$
$151$	−11.5134	−0.936945	−0.468473	+	0.883478i	$0.655196\pi$
$152$	48.0712	3.89909
$153$	0	0
$154$	0	0
$155$	1.04442	0.0838897
$156$	0	0
$157$	−5.91116	−0.471762	−0.235881	−	0.971782i	$-0.575798\pi$
$157$	−5.91116	−0.471762	−0.235881	+	0.971782i	$0.575798\pi$
$158$	12.7542	1.01467
$159$	0	0
$160$	23.3026	1.84223
$161$	−10.5341	−0.830201
$162$	0	0
$163$	23.4990	1.84058	0.920292	−	0.391231i	$-0.127951\pi$
$163$	23.4990	1.84058	0.920292	+	0.391231i	$0.127951\pi$
$164$	−58.6734	−4.58162
$165$	0	0
$166$	−37.3357	−2.89781
$167$	−14.2345	−1.10150	−0.550748	−	0.834671i	$-0.685658\pi$
$167$	−14.2345	−1.10150	−0.550748	+	0.834671i	$0.685658\pi$
$168$	0	0
$169$	−12.7148	−0.978059
$170$	−6.64663	−0.509773
$171$	0	0
$172$	23.9255	1.82430
$173$	6.44523	0.490022	0.245011	−	0.969520i	$-0.421208\pi$
$173$	6.44523	0.490022	0.245011	+	0.969520i	$0.421208\pi$
$174$	0	0
$175$	−2.32331	−0.175626
$176$	0	0
$177$	0	0
$178$	−34.8998	−2.61585
$179$	−16.9793	−1.26909	−0.634546	−	0.772885i	$-0.718813\pi$
$179$	−16.9793	−1.26909	−0.634546	+	0.772885i	$0.718813\pi$
$180$	0	0
$181$	4.48965	0.333713	0.166857	−	0.985981i	$-0.446638\pi$
$181$	4.48965	0.333713	0.166857	+	0.985981i	$0.446638\pi$
$182$	3.40582	0.252456
$183$	0	0
$184$	43.9823	3.24242
$185$	7.48965	0.550650
$186$	0	0
$187$	0	0
$188$	−37.4576	−2.73188
$189$	0	0
$190$	13.6022	0.986808
$191$	2.75919	0.199648	0.0998239	−	0.995005i	$-0.468172\pi$
$191$	2.75919	0.199648	0.0998239	+	0.995005i	$0.468172\pi$
$192$	0	0
$193$	11.8036	0.849642	0.424821	−	0.905277i	$-0.360337\pi$
$193$	11.8036	0.849642	0.424821	+	0.905277i	$0.360337\pi$
$194$	13.0631	0.937879
$195$	0	0
$196$	−8.86675	−0.633339
$197$	6.51035	0.463843	0.231922	−	0.972734i	$-0.425499\pi$
$197$	6.51035	0.463843	0.231922	+	0.972734i	$0.425499\pi$
$198$	0	0
$199$	23.9586	1.69838	0.849190	−	0.528087i	$-0.177090\pi$
$199$	23.9586	1.69838	0.849190	+	0.528087i	$0.177090\pi$
$200$	9.70041	0.685922
$201$	0	0
$202$	18.0568	1.27047
$203$	−12.7542	−0.895168
$204$	0	0
$205$	−10.6022	−0.740491
$206$	44.8955	3.12802
$207$	0	0
$208$	−8.30895	−0.576122
$209$	0	0
$210$	0	0
$211$	−16.8667	−1.16115	−0.580577	−	0.814205i	$-0.697173\pi$
$211$	−16.8667	−1.16115	−0.580577	+	0.814205i	$0.697173\pi$
$212$	25.0919	1.72332
$213$	0	0
$214$	−28.0267	−1.91587
$215$	4.32331	0.294848
$216$	0	0
$217$	−2.42651	−0.164722
$218$	−33.1901	−2.24791
$219$	0	0
$220$	0	0
$221$	1.29326	0.0869939
$222$	0	0
$223$	−25.0174	−1.67529	−0.837644	−	0.546216i	$-0.816068\pi$
$223$	−25.0174	−1.67529	−0.837644	+	0.546216i	$0.816068\pi$
$224$	−54.1393	−3.61733
$225$	0	0
$226$	29.7622	1.97975
$227$	−4.00936	−0.266111	−0.133055	−	0.991109i	$-0.542479\pi$
$227$	−4.00936	−0.266111	−0.133055	+	0.991109i	$0.542479\pi$
$228$	0	0
$229$	−17.3327	−1.14538	−0.572688	−	0.819774i	$-0.694099\pi$
$229$	−17.3327	−1.14538	−0.572688	+	0.819774i	$0.694099\pi$
$230$	12.4452	0.820614
$231$	0	0
$232$	53.2519	3.49616
$233$	−2.11256	−0.138398	−0.0691992	−	0.997603i	$-0.522044\pi$
$233$	−2.11256	−0.138398	−0.0691992	+	0.997603i	$0.522044\pi$
$234$	0	0
$235$	−6.76855	−0.441531
$236$	35.6684	2.32181
$237$	0	0
$238$	15.4422	1.00097
$239$	23.9349	1.54822	0.774110	−	0.633052i	$-0.218198\pi$
$239$	23.9349	1.54822	0.774110	+	0.633052i	$0.218198\pi$
$240$	0	0
$241$	−14.5134	−0.934889	−0.467444	−	0.884023i	$-0.654825\pi$
$241$	−14.5134	−0.934889	−0.467444	+	0.884023i	$0.654825\pi$
$242$	0	0
$243$	0	0
$244$	37.6240	2.40863
$245$	−1.60221	−0.102361
$246$	0	0
$247$	−2.64663	−0.168401
$248$	10.1313	0.643337
$249$	0	0
$250$	2.74483	0.173598
$251$	−28.8066	−1.81826	−0.909129	−	0.416514i	$-0.863252\pi$
$251$	−28.8066	−1.81826	−0.909129	+	0.416514i	$0.863252\pi$
$252$	0	0
$253$	0	0
$254$	−22.2933	−1.39880
$255$	0	0
$256$	53.8491	3.36557
$257$	27.6497	1.72474	0.862369	−	0.506280i	$-0.168980\pi$
$257$	27.6497	1.72474	0.862369	+	0.506280i	$0.168980\pi$
$258$	0	0
$259$	−17.4008	−1.08123
$260$	−2.95558	−0.183297
$261$	0	0
$262$	−15.9162	−0.983304
$263$	−3.13325	−0.193205	−0.0966024	−	0.995323i	$-0.530798\pi$
$263$	−3.13325	−0.193205	−0.0966024	+	0.995323i	$0.530798\pi$
$264$	0	0
$265$	4.53407	0.278526
$266$	−31.6022	−1.93766
$267$	0	0
$268$	3.99064	0.243767
$269$	−18.0030	−1.09766	−0.548832	−	0.835933i	$-0.684927\pi$
$269$	−18.0030	−1.09766	−0.548832	+	0.835933i	$0.684927\pi$
$270$	0	0
$271$	20.3327	1.23512	0.617561	−	0.786523i	$-0.288121\pi$
$271$	20.3327	1.23512	0.617561	+	0.786523i	$0.288121\pi$
$272$	−37.6734	−2.28428
$273$	0	0
$274$	−24.3377	−1.47029
$275$	0	0
$276$	0	0
$277$	−25.4897	−1.53152	−0.765762	−	0.643124i	$-0.777638\pi$
$277$	−25.4897	−1.53152	−0.765762	+	0.643124i	$0.777638\pi$
$278$	−46.2963	−2.77667
$279$	0	0
$280$	−22.5371	−1.34685
$281$	−18.2726	−1.09005	−0.545025	−	0.838420i	$-0.683480\pi$
$281$	−18.2726	−1.09005	−0.545025	+	0.838420i	$0.683480\pi$
$282$	0	0
$283$	−11.0538	−0.657079	−0.328539	−	0.944490i	$-0.606556\pi$
$283$	−11.0538	−0.657079	−0.328539	+	0.944490i	$0.606556\pi$
$284$	−25.0919	−1.48893
$285$	0	0
$286$	0	0
$287$	24.6323	1.45400
$288$	0	0
$289$	−11.1363	−0.655075
$290$	15.0681	0.884832
$291$	0	0
$292$	5.91116	0.345925
$293$	−27.1393	−1.58550	−0.792748	−	0.609550i	$-0.791350\pi$
$293$	−27.1393	−1.58550	−0.792748	+	0.609550i	$0.791350\pi$
$294$	0	0
$295$	6.44523	0.375256
$296$	72.6527	4.22285
$297$	0	0
$298$	−12.2582	−0.710098
$299$	−2.42151	−0.140040
$300$	0	0
$301$	−10.0444	−0.578951
$302$	−31.6022	−1.81850
$303$	0	0
$304$	77.0979	4.42187
$305$	6.79861	0.389287
$306$	0	0
$307$	11.7385	0.669951	0.334976	−	0.942227i	$-0.391272\pi$
$307$	11.7385	0.669951	0.334976	+	0.942227i	$0.391272\pi$
$308$	0	0
$309$	0	0
$310$	2.86675	0.162820
$311$	−10.6416	−0.603431	−0.301716	−	0.953398i	$-0.597559\pi$
$311$	−10.6416	−0.603431	−0.301716	+	0.953398i	$0.597559\pi$
$312$	0	0
$313$	−27.1807	−1.53634	−0.768172	−	0.640244i	$-0.778833\pi$
$313$	−27.1807	−1.53634	−0.768172	+	0.640244i	$0.778833\pi$
$314$	−16.2251	−0.915636
$315$	0	0
$316$	25.7148	1.44657
$317$	−4.97930	−0.279666	−0.139833	−	0.990175i	$-0.544657\pi$
$317$	−4.97930	−0.279666	−0.139833	+	0.990175i	$0.544657\pi$
$318$	0	0
$319$	0	0
$320$	32.8461	1.83615
$321$	0	0
$322$	−28.9142	−1.61132
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−0.534070	−0.0296249
$326$	64.5007	3.57236
$327$	0	0
$328$	−102.846	−5.67871
$329$	15.7255	0.866973
$330$	0	0
$331$	7.55477	0.415247	0.207624	−	0.978209i	$-0.433427\pi$
$331$	7.55477	0.415247	0.207624	+	0.978209i	$0.433427\pi$
$332$	−75.2756	−4.13128
$333$	0	0
$334$	−39.0712	−2.13788
$335$	0.721104	0.0393981
$336$	0	0
$337$	3.69105	0.201064	0.100532	−	0.994934i	$-0.467946\pi$
$337$	3.69105	0.201064	0.100532	+	0.994934i	$0.467946\pi$
$338$	−34.8998	−1.89830
$339$	0	0
$340$	−13.4008	−0.726761
$341$	0	0
$342$	0	0
$343$	19.9856	1.07912
$344$	41.9379	2.26114
$345$	0	0
$346$	17.6910	0.951077
$347$	15.7655	0.846338	0.423169	−	0.906051i	$-0.360918\pi$
$347$	15.7655	0.846338	0.423169	+	0.906051i	$0.360918\pi$
$348$	0	0
$349$	−10.5578	−0.565146	−0.282573	−	0.959246i	$-0.591188\pi$
$349$	−10.5578	−0.565146	−0.282573	+	0.959246i	$0.591188\pi$
$350$	−6.37709	−0.340870
$351$	0	0
$352$	0	0
$353$	24.5528	1.30681	0.653407	−	0.757007i	$-0.273339\pi$
$353$	24.5528	1.30681	0.653407	+	0.757007i	$0.273339\pi$
$354$	0	0
$355$	−4.53407	−0.240643
$356$	−70.3644	−3.72931
$357$	0	0
$358$	−46.6052	−2.46316
$359$	16.6704	0.879827	0.439914	−	0.898040i	$-0.355009\pi$
$359$	16.6704	0.879827	0.439914	+	0.898040i	$0.355009\pi$
$360$	0	0
$361$	5.55779	0.292515
$362$	12.3233	0.647699
$363$	0	0
$364$	6.86675	0.359915
$365$	1.06814	0.0559090
$366$	0	0
$367$	15.2789	0.797552	0.398776	−	0.917048i	$-0.369435\pi$
$367$	15.2789	0.797552	0.398776	+	0.917048i	$0.369435\pi$
$368$	70.5401	3.67716
$369$	0	0
$370$	20.5578	1.06875
$371$	−10.5341	−0.546902
$372$	0	0
$373$	17.8461	0.924033	0.462017	−	0.886871i	$-0.347126\pi$
$373$	17.8461	0.924033	0.462017	+	0.886871i	$0.347126\pi$
$374$	0	0
$375$	0	0
$376$	−65.6577	−3.38604
$377$	−2.93186	−0.150998
$378$	0	0
$379$	1.89244	0.0972082	0.0486041	−	0.998818i	$-0.484523\pi$
$379$	1.89244	0.0972082	0.0486041	+	0.998818i	$0.484523\pi$
$380$	27.4245	1.40685
$381$	0	0
$382$	7.57349	0.387493
$383$	−5.31698	−0.271685	−0.135842	−	0.990730i	$-0.543374\pi$
$383$	−5.31698	−0.271685	−0.135842	+	0.990730i	$0.543374\pi$
$384$	0	0
$385$	0	0
$386$	32.3988	1.64906
$387$	0	0
$388$	26.3377	1.33709
$389$	26.8273	1.36020	0.680100	−	0.733120i	$-0.261936\pi$
$389$	26.8273	1.36020	0.680100	+	0.733120i	$0.261936\pi$
$390$	0	0
$391$	−10.9793	−0.555247
$392$	−15.5421	−0.784994
$393$	0	0
$394$	17.8698	0.900266
$395$	4.64663	0.233797
$396$	0	0
$397$	31.5972	1.58582	0.792909	−	0.609340i	$-0.208565\pi$
$397$	31.5972	1.58582	0.792909	+	0.609340i	$0.208565\pi$
$398$	65.7622	3.29636
$399$	0	0
$400$	15.5578	0.777890
$401$	3.13628	0.156618	0.0783092	−	0.996929i	$-0.475048\pi$
$401$	3.13628	0.156618	0.0783092	+	0.996929i	$0.475048\pi$
$402$	0	0
$403$	−0.557793	−0.0277856
$404$	36.4058	1.81126
$405$	0	0
$406$	−35.0080	−1.73742
$407$	0	0
$408$	0	0
$409$	0.424538	0.0209921	0.0104960	−	0.999945i	$-0.496659\pi$
$409$	0.424538	0.0209921	0.0104960	+	0.999945i	$0.496659\pi$
$410$	−29.1012	−1.43721
$411$	0	0
$412$	90.5175	4.45947
$413$	−14.9743	−0.736837
$414$	0	0
$415$	−13.6022	−0.667706
$416$	−12.4452	−0.610178
$417$	0	0
$418$	0	0
$419$	−6.71174	−0.327890	−0.163945	−	0.986469i	$-0.552422\pi$
$419$	−6.71174	−0.327890	−0.163945	+	0.986469i	$0.552422\pi$
$420$	0	0
$421$	2.77988	0.135483	0.0677416	−	0.997703i	$-0.478421\pi$
$421$	2.77988	0.135483	0.0677416	+	0.997703i	$0.478421\pi$
$422$	−46.2963	−2.25367
$423$	0	0
$424$	43.9823	2.13597
$425$	−2.42151	−0.117461
$426$	0	0
$427$	−15.7953	−0.764388
$428$	−56.5070	−2.73137
$429$	0	0
$430$	11.8667	0.572265
$431$	−16.1964	−0.780153	−0.390076	−	0.920782i	$-0.627551\pi$
$431$	−16.1964	−0.780153	−0.390076	+	0.920782i	$0.627551\pi$
$432$	0	0
$433$	7.82733	0.376157	0.188079	−	0.982154i	$-0.439774\pi$
$433$	7.82733	0.376157	0.188079	+	0.982154i	$0.439774\pi$
$434$	−6.66035	−0.319707
$435$	0	0
$436$	−66.9172	−3.20475
$437$	22.4690	1.07484
$438$	0	0
$439$	20.7355	0.989650	0.494825	−	0.868993i	$-0.335232\pi$
$439$	20.7355	0.989650	0.494825	+	0.868993i	$0.335232\pi$
$440$	0	0
$441$	0	0
$442$	3.54977	0.168845
$443$	−5.30262	−0.251935	−0.125968	−	0.992034i	$-0.540204\pi$
$443$	−5.30262	−0.251935	−0.125968	+	0.992034i	$0.540204\pi$
$444$	0	0
$445$	−12.7148	−0.602738
$446$	−68.6684	−3.25154
$447$	0	0
$448$	−76.3117	−3.60539
$449$	30.0030	1.41593	0.707965	−	0.706247i	$-0.249613\pi$
$449$	30.0030	1.41593	0.707965	+	0.706247i	$0.249613\pi$
$450$	0	0
$451$	0	0
$452$	60.0061	2.82245
$453$	0	0
$454$	−11.0050	−0.516490
$455$	1.24081	0.0581702
$456$	0	0
$457$	37.5134	1.75480	0.877401	−	0.479758i	$-0.159276\pi$
$457$	37.5134	1.75480	0.877401	+	0.479758i	$0.159276\pi$
$458$	−47.5752	−2.22304
$459$	0	0
$460$	25.0919	1.16991
$461$	−15.7829	−0.735083	−0.367542	−	0.930007i	$-0.619800\pi$
$461$	−15.7829	−0.735083	−0.367542	+	0.930007i	$0.619800\pi$
$462$	0	0
$463$	−16.5672	−0.769941	−0.384970	−	0.922929i	$-0.625788\pi$
$463$	−16.5672	−0.769941	−0.384970	+	0.922929i	$0.625788\pi$
$464$	85.4069	3.96491
$465$	0	0
$466$	−5.79861	−0.268615
$467$	−3.70041	−0.171234	−0.0856172	−	0.996328i	$-0.527286\pi$
$467$	−3.70041	−0.171234	−0.0856172	+	0.996328i	$0.527286\pi$
$468$	0	0
$469$	−1.67535	−0.0773605
$470$	−18.5785	−0.856962
$471$	0	0
$472$	62.5214	2.87778
$473$	0	0
$474$	0	0
$475$	4.95558	0.227378
$476$	31.1343	1.42704
$477$	0	0
$478$	65.6971	3.00492
$479$	−1.26454	−0.0577781	−0.0288890	−	0.999583i	$-0.509197\pi$
$479$	−1.26454	−0.0577781	−0.0288890	+	0.999583i	$0.509197\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−39.8367	−1.81451
$483$	0	0
$484$	0	0
$485$	4.75919	0.216104
$486$	0	0
$487$	21.6022	0.978890	0.489445	−	0.872034i	$-0.337199\pi$
$487$	21.6022	0.978890	0.489445	+	0.872034i	$0.337199\pi$
$488$	65.9492	2.98538
$489$	0	0
$490$	−4.39779	−0.198672
$491$	−25.5371	−1.15247	−0.576237	−	0.817283i	$-0.695479\pi$
$491$	−25.5371	−1.15247	−0.576237	+	0.817283i	$0.695479\pi$
$492$	0	0
$493$	−13.2933	−0.598698
$494$	−7.26454	−0.326847
$495$	0	0
$496$	16.2488	0.729594
$497$	10.5341	0.472518
$498$	0	0
$499$	24.3614	1.09057	0.545283	−	0.838252i	$-0.316422\pi$
$499$	24.3614	1.09057	0.545283	+	0.838252i	$0.316422\pi$
$500$	5.53407	0.247491
$501$	0	0
$502$	−79.0692	−3.52903
$503$	12.4960	0.557169	0.278584	−	0.960412i	$-0.410135\pi$
$503$	12.4960	0.557169	0.278584	+	0.960412i	$0.410135\pi$
$504$	0	0
$505$	6.57849	0.292739
$506$	0	0
$507$	0	0
$508$	−44.9473	−1.99421
$509$	−24.4483	−1.08365	−0.541825	−	0.840491i	$-0.682266\pi$
$509$	−24.4483	−1.08365	−0.541825	+	0.840491i	$0.682266\pi$
$510$	0	0
$511$	−2.48163	−0.109781
$512$	60.7034	2.68274
$513$	0	0
$514$	75.8935	3.34752
$515$	16.3564	0.720749
$516$	0	0
$517$	0	0
$518$	−47.7622	−2.09855
$519$	0	0
$520$	−5.18070	−0.227189
$521$	0.0394184	0.00172695	0.000863476	−	1.00000i	$-0.499725\pi$
$521$	0.0394184	0.00172695	0.000863476		1.00000i	$0.499725\pi$
$522$	0	0
$523$	−21.0919	−0.922283	−0.461141	−	0.887327i	$-0.652560\pi$
$523$	−21.0919	−0.922283	−0.461141	+	0.887327i	$0.652560\pi$
$524$	−32.0899	−1.40185
$525$	0	0
$526$	−8.60024	−0.374988
$527$	−2.52907	−0.110168
$528$	0	0
$529$	−2.44221	−0.106183
$530$	12.4452	0.540586
$531$	0	0
$532$	−63.7158	−2.76243
$533$	5.66232	0.245263
$534$	0	0
$535$	−10.2108	−0.441449
$536$	6.99500	0.302138
$537$	0	0
$538$	−49.4152	−2.13044
$539$	0	0
$540$	0	0
$541$	−33.4402	−1.43771	−0.718854	−	0.695161i	$-0.755333\pi$
$541$	−33.4402	−1.43771	−0.718854	+	0.695161i	$0.755333\pi$
$542$	55.8097	2.39723
$543$	0	0
$544$	−56.4276	−2.41931
$545$	−12.0919	−0.517958
$546$	0	0
$547$	32.8066	1.40271	0.701355	−	0.712812i	$-0.252579\pi$
$547$	32.8066	1.40271	0.701355	+	0.712812i	$0.252579\pi$
$548$	−49.0692	−2.09613
$549$	0	0
$550$	0	0
$551$	27.2044	1.15895
$552$	0	0
$553$	−10.7956	−0.459075
$554$	−69.9647	−2.97251
$555$	0	0
$556$	−93.3418	−3.95857
$557$	−28.3327	−1.20049	−0.600247	−	0.799815i	$-0.704931\pi$
$557$	−28.3327	−1.20049	−0.600247	+	0.799815i	$0.704931\pi$
$558$	0	0
$559$	−2.30895	−0.0976583
$560$	−36.1456	−1.52743
$561$	0	0
$562$	−50.1550	−2.11566
$563$	−29.0361	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$563$	−29.0361	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$564$	0	0
$565$	10.8430	0.456169
$566$	−30.3407	−1.27531
$567$	0	0
$568$	−43.9823	−1.84546
$569$	−28.7098	−1.20358	−0.601788	−	0.798656i	$-0.705545\pi$
$569$	−28.7098	−1.20358	−0.601788	+	0.798656i	$0.705545\pi$
$570$	0	0
$571$	0.824301	0.0344959	0.0172480	−	0.999851i	$-0.494510\pi$
$571$	0.824301	0.0344959	0.0172480	+	0.999851i	$0.494510\pi$
$572$	0	0
$573$	0	0
$574$	67.6113	2.82204
$575$	4.53407	0.189084
$576$	0	0
$577$	−31.5421	−1.31311	−0.656557	−	0.754276i	$-0.727988\pi$
$577$	−31.5421	−1.31311	−0.656557	+	0.754276i	$0.727988\pi$
$578$	−30.5672	−1.27143
$579$	0	0
$580$	30.3801	1.26147
$581$	31.6022	1.31108
$582$	0	0
$583$	0	0
$584$	10.3614	0.428758
$585$	0	0
$586$	−74.4927	−3.07726
$587$	26.7735	1.10506	0.552531	−	0.833492i	$-0.313662\pi$
$587$	26.7735	1.10506	0.552531	+	0.833492i	$0.313662\pi$
$588$	0	0
$589$	5.17570	0.213261
$590$	17.6910	0.728329
$591$	0	0
$592$	116.522	4.78904
$593$	11.0731	0.454719	0.227360	−	0.973811i	$-0.426991\pi$
$593$	11.0731	0.454719	0.227360	+	0.973811i	$0.426991\pi$
$594$	0	0
$595$	5.62593	0.230641
$596$	−24.7148	−1.01236
$597$	0	0
$598$	−6.64663	−0.271801
$599$	11.9349	0.487646	0.243823	−	0.969820i	$-0.421598\pi$
$599$	11.9349	0.487646	0.243823	+	0.969820i	$0.421598\pi$
$600$	0	0
$601$	5.26454	0.214745	0.107372	−	0.994219i	$-0.465756\pi$
$601$	5.26454	0.214745	0.107372	+	0.994219i	$0.465756\pi$
$602$	−27.5702	−1.12368
$603$	0	0
$604$	−63.7158	−2.59256
$605$	0	0
$606$	0	0
$607$	35.8461	1.45495	0.727473	−	0.686136i	$-0.240695\pi$
$607$	35.8461	1.45495	0.727473	+	0.686136i	$0.240695\pi$
$608$	115.478	4.68325
$609$	0	0
$610$	18.6610	0.755561
$611$	3.61488	0.146242
$612$	0	0
$613$	6.55779	0.264867	0.132433	−	0.991192i	$-0.457721\pi$
$613$	6.55779	0.264867	0.132433	+	0.991192i	$0.457721\pi$
$614$	32.2201	1.30030
$615$	0	0
$616$	0	0
$617$	−17.7986	−0.716545	−0.358272	−	0.933617i	$-0.616634\pi$
$617$	−17.7986	−0.716545	−0.358272	+	0.933617i	$0.616634\pi$
$618$	0	0
$619$	44.8905	1.80430	0.902150	−	0.431422i	$-0.141988\pi$
$619$	44.8905	1.80430	0.902150	+	0.431422i	$0.141988\pi$
$620$	5.77988	0.232126
$621$	0	0
$622$	−29.2094	−1.17119
$623$	29.5404	1.18351
$624$	0	0
$625$	1.00000	0.0400000
$626$	−74.6063	−2.98187
$627$	0	0
$628$	−32.7128	−1.30538
$629$	−18.1363	−0.723141
$630$	0	0
$631$	21.9112	0.872270	0.436135	−	0.899881i	$-0.356347\pi$
$631$	21.9112	0.872270	0.436135	+	0.899881i	$0.356347\pi$
$632$	45.0742	1.79296
$633$	0	0
$634$	−13.6673	−0.542799
$635$	−8.12192	−0.322309
$636$	0	0
$637$	0.855693	0.0339038
$638$	0	0
$639$	0	0
$640$	43.5515	1.72152
$641$	−35.7335	−1.41139	−0.705694	−	0.708517i	$-0.749365\pi$
$641$	−35.7335	−1.41139	−0.705694	+	0.708517i	$0.749365\pi$
$642$	0	0
$643$	0.412150	0.0162536	0.00812681	−	0.999967i	$-0.497413\pi$
$643$	0.412150	0.0162536	0.00812681	+	0.999967i	$0.497413\pi$
$644$	−58.2963	−2.29720
$645$	0	0
$646$	−32.9379	−1.29592
$647$	16.5484	0.650586	0.325293	−	0.945613i	$-0.394537\pi$
$647$	16.5484	0.650586	0.325293	+	0.945613i	$0.394537\pi$
$648$	0	0
$649$	0	0
$650$	−1.46593	−0.0574985
$651$	0	0
$652$	130.045	5.09296
$653$	−11.0080	−0.430777	−0.215389	−	0.976528i	$-0.569102\pi$
$653$	−11.0080	−0.430777	−0.215389	+	0.976528i	$0.569102\pi$
$654$	0	0
$655$	−5.79861	−0.226570
$656$	−164.947	−6.44010
$657$	0	0
$658$	43.1637	1.68269
$659$	−42.9980	−1.67497	−0.837483	−	0.546464i	$-0.815974\pi$
$659$	−42.9980	−1.67497	−0.837483	+	0.546464i	$0.815974\pi$
$660$	0	0
$661$	19.4927	0.758177	0.379089	−	0.925360i	$-0.376238\pi$
$661$	19.4927	0.758177	0.379089	+	0.925360i	$0.376238\pi$
$662$	20.7365	0.805948
$663$	0	0
$664$	−131.947	−5.12054
$665$	−11.5134	−0.446470
$666$	0	0
$667$	24.8905	0.963763
$668$	−78.7746	−3.04788
$669$	0	0
$670$	1.97930	0.0764672
$671$	0	0
$672$	0	0
$673$	−6.80663	−0.262376	−0.131188	−	0.991357i	$-0.541879\pi$
$673$	−6.80663	−0.262376	−0.131188	+	0.991357i	$0.541879\pi$
$674$	10.1313	0.390242
$675$	0	0
$676$	−70.3644	−2.70632
$677$	1.02070	0.0392285	0.0196143	−	0.999808i	$-0.493756\pi$
$677$	1.02070	0.0392285	0.0196143	+	0.999808i	$0.493756\pi$
$678$	0	0
$679$	−11.0571	−0.424332
$680$	−23.4897	−0.900787
$681$	0	0
$682$	0	0
$683$	−2.09820	−0.0802853	−0.0401426	−	0.999194i	$-0.512781\pi$
$683$	−2.09820	−0.0802853	−0.0401426	+	0.999194i	$0.512781\pi$
$684$	0	0
$685$	−8.86675	−0.338781
$686$	54.8571	2.09445
$687$	0	0
$688$	67.2612	2.56431
$689$	−2.42151	−0.0922523
$690$	0	0
$691$	11.9950	0.456311	0.228156	−	0.973625i	$-0.426730\pi$
$691$	11.9950	0.456311	0.228156	+	0.973625i	$0.426730\pi$
$692$	35.6684	1.35591
$693$	0	0
$694$	43.2736	1.64264
$695$	−16.8667	−0.639792
$696$	0	0
$697$	25.6734	0.972449
$698$	−28.9793	−1.09688
$699$	0	0
$700$	−12.8574	−0.485963
$701$	−29.3594	−1.10889	−0.554445	−	0.832220i	$-0.687069\pi$
$701$	−29.3594	−1.10889	−0.554445	+	0.832220i	$0.687069\pi$
$702$	0	0
$703$	37.1156	1.39984
$704$	0	0
$705$	0	0
$706$	67.3931	2.53637
$707$	−15.2839	−0.574810
$708$	0	0
$709$	−37.5451	−1.41004	−0.705018	−	0.709189i	$-0.749061\pi$
$709$	−37.5451	−1.41004	−0.705018	+	0.709189i	$0.749061\pi$
$710$	−12.4452	−0.467061
$711$	0	0
$712$	−123.338	−4.62230
$713$	4.73546	0.177345
$714$	0	0
$715$	0	0
$716$	−93.9647	−3.51162
$717$	0	0
$718$	45.7572	1.70764
$719$	3.99500	0.148988	0.0744942	−	0.997221i	$-0.476266\pi$
$719$	3.99500	0.148988	0.0744942	+	0.997221i	$0.476266\pi$
$720$	0	0
$721$	−38.0011	−1.41523
$722$	15.2552	0.567739
$723$	0	0
$724$	24.8461	0.923396
$725$	5.48965	0.203881
$726$	0	0
$727$	−25.4990	−0.945706	−0.472853	−	0.881141i	$-0.656776\pi$
$727$	−25.4990	−0.945706	−0.472853	+	0.881141i	$0.656776\pi$
$728$	12.0364	0.446098
$729$	0	0
$730$	2.93186	0.108513
$731$	−10.4690	−0.387208
$732$	0	0
$733$	−36.2024	−1.33717	−0.668584	−	0.743637i	$-0.733099\pi$
$733$	−36.2024	−1.33717	−0.668584	+	0.743637i	$0.733099\pi$
$734$	41.9379	1.54796
$735$	0	0
$736$	105.656	3.89452
$737$	0	0
$738$	0	0
$739$	−6.06011	−0.222925	−0.111462	−	0.993769i	$-0.535553\pi$
$739$	−6.06011	−0.222925	−0.111462	+	0.993769i	$0.535553\pi$
$740$	41.4483	1.52367
$741$	0	0
$742$	−28.9142	−1.06147
$743$	−36.2869	−1.33124	−0.665619	−	0.746292i	$-0.731832\pi$
$743$	−36.2869	−1.33124	−0.665619	+	0.746292i	$0.731832\pi$
$744$	0	0
$745$	−4.46593	−0.163619
$746$	48.9843	1.79344
$747$	0	0
$748$	0	0
$749$	23.7228	0.866812
$750$	0	0
$751$	−15.7335	−0.574123	−0.287062	−	0.957912i	$-0.592678\pi$
$751$	−15.7335	−0.574123	−0.287062	+	0.957912i	$0.592678\pi$
$752$	−105.304	−3.84003
$753$	0	0
$754$	−8.04744	−0.293071
$755$	−11.5134	−0.419015
$756$	0	0
$757$	−25.1393	−0.913704	−0.456852	−	0.889543i	$-0.651023\pi$
$757$	−25.1393	−0.913704	−0.456852	+	0.889543i	$0.651023\pi$
$758$	5.19442	0.188670
$759$	0	0
$760$	48.0712	1.74372
$761$	15.4709	0.560821	0.280410	−	0.959880i	$-0.409529\pi$
$761$	15.4709	0.560821	0.280410	+	0.959880i	$0.409529\pi$
$762$	0	0
$763$	28.0932	1.01704
$764$	15.2695	0.552432
$765$	0	0
$766$	−14.5942	−0.527309
$767$	−3.44221	−0.124291
$768$	0	0
$769$	17.3220	0.624647	0.312323	−	0.949976i	$-0.398893\pi$
$769$	17.3220	0.624647	0.312323	+	0.949976i	$0.398893\pi$
$770$	0	0
$771$	0	0
$772$	65.3220	2.35099
$773$	22.6991	0.816429	0.408214	−	0.912886i	$-0.366152\pi$
$773$	22.6991	0.816429	0.408214	+	0.912886i	$0.366152\pi$
$774$	0	0
$775$	1.04442	0.0375166
$776$	46.1661	1.65726
$777$	0	0
$778$	73.6363	2.63999
$779$	−52.5401	−1.88245
$780$	0	0
$781$	0	0
$782$	−30.1363	−1.07767
$783$	0	0
$784$	−24.9269	−0.890245
$785$	−5.91116	−0.210978
$786$	0	0
$787$	3.10122	0.110547	0.0552734	−	0.998471i	$-0.482397\pi$
$787$	3.10122	0.110547	0.0552734	+	0.998471i	$0.482397\pi$
$788$	36.0287	1.28347
$789$	0	0
$790$	12.7542	0.453774
$791$	−25.1918	−0.895716
$792$	0	0
$793$	−3.63093	−0.128938
$794$	86.7288	3.07789
$795$	0	0
$796$	132.589	4.69948
$797$	19.0919	0.676268	0.338134	−	0.941098i	$-0.390204\pi$
$797$	19.0919	0.676268	0.338134	+	0.941098i	$0.390204\pi$
$798$	0	0
$799$	16.3901	0.579841
$800$	23.3026	0.823872
$801$	0	0
$802$	8.60855	0.303978
$803$	0	0
$804$	0	0
$805$	−10.5341	−0.371277
$806$	−1.53104	−0.0539287
$807$	0	0
$808$	63.8140	2.24497
$809$	−4.46896	−0.157120	−0.0785601	−	0.996909i	$-0.525032\pi$
$809$	−4.46896	−0.157120	−0.0785601	+	0.996909i	$0.525032\pi$
$810$	0	0
$811$	−4.86675	−0.170895	−0.0854473	−	0.996343i	$-0.527232\pi$
$811$	−4.86675	−0.170895	−0.0854473	+	0.996343i	$0.527232\pi$
$812$	−70.5826	−2.47696
$813$	0	0
$814$	0	0
$815$	23.4990	0.823135
$816$	0	0
$817$	21.4245	0.749550
$818$	1.16528	0.0407432
$819$	0	0
$820$	−58.6734	−2.04896
$821$	−17.0050	−0.593479	−0.296739	−	0.954959i	$-0.595899\pi$
$821$	−17.0050	−0.593479	−0.296739	+	0.954959i	$0.595899\pi$
$822$	0	0
$823$	−21.7479	−0.758082	−0.379041	−	0.925380i	$-0.623746\pi$
$823$	−21.7479	−0.758082	−0.379041	+	0.925380i	$0.623746\pi$
$824$	158.664	5.52731
$825$	0	0
$826$	−41.1019	−1.43012
$827$	20.7084	0.720103	0.360051	−	0.932932i	$-0.382759\pi$
$827$	20.7084	0.720103	0.360051	+	0.932932i	$0.382759\pi$
$828$	0	0
$829$	11.3090	0.392776	0.196388	−	0.980526i	$-0.437079\pi$
$829$	11.3090	0.392776	0.196388	+	0.980526i	$0.437079\pi$
$830$	−37.3357	−1.29594
$831$	0	0
$832$	−17.5421	−0.608163
$833$	3.87977	0.134426
$834$	0	0
$835$	−14.2345	−0.492604
$836$	0	0
$837$	0	0
$838$	−18.4226	−0.636397
$839$	43.5084	1.50208	0.751038	−	0.660259i	$-0.229554\pi$
$839$	43.5084	1.50208	0.751038	+	0.660259i	$0.229554\pi$
$840$	0	0
$841$	1.13628	0.0391821
$842$	7.63029	0.262957
$843$	0	0
$844$	−93.3418	−3.21296
$845$	−12.7148	−0.437401
$846$	0	0
$847$	0	0
$848$	70.5401	2.42236
$849$	0	0
$850$	−6.64663	−0.227977
$851$	33.9586	1.16409
$852$	0	0
$853$	−1.11559	−0.0381969	−0.0190985	−	0.999818i	$-0.506080\pi$
$853$	−1.11559	−0.0381969	−0.0190985	+	0.999818i	$0.506080\pi$
$854$	−43.3553	−1.48359
$855$	0	0
$856$	−99.0485	−3.38541
$857$	−7.80361	−0.266566	−0.133283	−	0.991078i	$-0.542552\pi$
$857$	−7.80361	−0.266566	−0.133283	+	0.991078i	$0.542552\pi$
$858$	0	0
$859$	30.5341	1.04181	0.520905	−	0.853615i	$-0.325595\pi$
$859$	30.5341	1.04181	0.520905	+	0.853615i	$0.325595\pi$
$860$	23.9255	0.815854
$861$	0	0
$862$	−44.4563	−1.51419
$863$	4.28959	0.146019	0.0730097	−	0.997331i	$-0.476740\pi$
$863$	4.28959	0.146019	0.0730097	+	0.997331i	$0.476740\pi$
$864$	0	0
$865$	6.44523	0.219145
$866$	21.4847	0.730078
$867$	0	0
$868$	−13.4285	−0.455792
$869$	0	0
$870$	0	0
$871$	−0.385120	−0.0130493
$872$	−117.296	−3.97214
$873$	0	0
$874$	61.6734	2.08613
$875$	−2.32331	−0.0785424
$876$	0	0
$877$	11.3644	0.383749	0.191875	−	0.981419i	$-0.438543\pi$
$877$	11.3644	0.383749	0.191875	+	0.981419i	$0.438543\pi$
$878$	56.9152	1.92080
$879$	0	0
$880$	0	0
$881$	11.3277	0.381639	0.190820	−	0.981625i	$-0.438885\pi$
$881$	11.3277	0.381639	0.190820	+	0.981625i	$0.438885\pi$
$882$	0	0
$883$	31.2883	1.05293	0.526467	−	0.850196i	$-0.323516\pi$
$883$	31.2883	1.05293	0.526467	+	0.850196i	$0.323516\pi$
$884$	7.15698	0.240715
$885$	0	0
$886$	−14.5548	−0.488977
$887$	28.2819	0.949614	0.474807	−	0.880090i	$-0.342518\pi$
$887$	28.2819	0.949614	0.474807	+	0.880090i	$0.342518\pi$
$888$	0	0
$889$	18.8698	0.632872
$890$	−34.8998	−1.16984
$891$	0	0
$892$	−138.448	−4.63558
$893$	−33.5421	−1.12244
$894$	0	0
$895$	−16.9793	−0.567556
$896$	−101.184	−3.38031
$897$	0	0
$898$	82.3531	2.74816
$899$	5.73349	0.191223
$900$	0	0
$901$	−10.9793	−0.365774
$902$	0	0
$903$	0	0
$904$	105.182	3.49829
$905$	4.48965	0.149241
$906$	0	0
$907$	33.6166	1.11622	0.558110	−	0.829767i	$-0.311527\pi$
$907$	33.6166	1.11622	0.558110	+	0.829767i	$0.311527\pi$
$908$	−22.1881	−0.736338
$909$	0	0
$910$	3.40582	0.112902
$911$	−7.53104	−0.249515	−0.124757	−	0.992187i	$-0.539815\pi$
$911$	−7.53104	−0.249515	−0.124757	+	0.992187i	$0.539815\pi$
$912$	0	0
$913$	0	0
$914$	102.968	3.40587
$915$	0	0
$916$	−95.9202	−3.16929
$917$	13.4720	0.444884
$918$	0	0
$919$	40.0424	1.32088	0.660439	−	0.750880i	$-0.270370\pi$
$919$	40.0424	1.32088	0.660439	+	0.750880i	$0.270370\pi$
$920$	43.9823	1.45005
$921$	0	0
$922$	−43.3213	−1.42671
$923$	2.42151	0.0797050
$924$	0	0
$925$	7.48965	0.246258
$926$	−45.4740	−1.49437
$927$	0	0
$928$	127.923	4.19929
$929$	−51.3407	−1.68443	−0.842217	−	0.539139i	$-0.818750\pi$
$929$	−51.3407	−1.68443	−0.842217	+	0.539139i	$0.818750\pi$
$930$	0	0
$931$	−7.93989	−0.260219
$932$	−11.6910	−0.382953
$933$	0	0
$934$	−10.1570	−0.332346
$935$	0	0
$936$	0	0
$937$	−35.8510	−1.17120	−0.585601	−	0.810599i	$-0.699142\pi$
$937$	−35.8510	−1.17120	−0.585601	+	0.810599i	$0.699142\pi$
$938$	−4.59855	−0.150148
$939$	0	0
$940$	−37.4576	−1.22173
$941$	0.607210	0.0197945	0.00989724	−	0.999951i	$-0.496850\pi$
$941$	0.607210	0.0197945	0.00989724	+	0.999951i	$0.496850\pi$
$942$	0	0
$943$	−48.0712	−1.56541
$944$	100.274	3.26363
$945$	0	0
$946$	0	0
$947$	26.3851	0.857401	0.428701	−	0.903447i	$-0.358972\pi$
$947$	26.3851	0.857401	0.428701	+	0.903447i	$0.358972\pi$
$948$	0	0
$949$	−0.570462	−0.0185180
$950$	13.6022	0.441314
$951$	0	0
$952$	54.5738	1.76875
$953$	35.3594	1.14540	0.572702	−	0.819764i	$-0.305895\pi$
$953$	35.3594	1.14540	0.572702	+	0.819764i	$0.305895\pi$
$954$	0	0
$955$	2.75919	0.0892852
$956$	132.457	4.28398
$957$	0	0
$958$	−3.47093	−0.112141
$959$	20.6002	0.665216
$960$	0	0
$961$	−29.9092	−0.964813
$962$	−10.9793	−0.353987
$963$	0	0
$964$	−80.3180	−2.58687
$965$	11.8036	0.379971
$966$	0	0
$967$	−3.86872	−0.124410	−0.0622048	−	0.998063i	$-0.519813\pi$
$967$	−3.86872	−0.124410	−0.0622048	+	0.998063i	$0.519813\pi$
$968$	0	0
$969$	0	0
$970$	13.0631	0.419432
$971$	−24.1076	−0.773648	−0.386824	−	0.922153i	$-0.626428\pi$
$971$	−24.1076	−0.773648	−0.386824	+	0.922153i	$0.626428\pi$
$972$	0	0
$973$	39.1868	1.25627
$974$	59.2943	1.89991
$975$	0	0
$976$	105.771	3.38566
$977$	−10.8143	−0.345980	−0.172990	−	0.984924i	$-0.555343\pi$
$977$	−10.8143	−0.345980	−0.172990	+	0.984924i	$0.555343\pi$
$978$	0	0
$979$	0	0
$980$	−8.86675	−0.283238
$981$	0	0
$982$	−70.0949	−2.23682
$983$	38.7084	1.23461	0.617304	−	0.786725i	$-0.288225\pi$
$983$	38.7084	1.23461	0.617304	+	0.786725i	$0.288225\pi$
$984$	0	0
$985$	6.51035	0.207437
$986$	−36.4877	−1.16200
$987$	0	0
$988$	−14.6466	−0.465971
$989$	19.6022	0.623314
$990$	0	0
$991$	61.4533	1.95213	0.976064	−	0.217485i	$-0.0697854\pi$
$991$	61.4533	1.95213	0.976064	+	0.217485i	$0.0697854\pi$
$992$	24.3377	0.772722
$993$	0	0
$994$	28.9142	0.917102
$995$	23.9586	0.759539
$996$	0	0
$997$	−38.3377	−1.21417	−0.607083	−	0.794638i	$-0.707661\pi$
$997$	−38.3377	−1.21417	−0.607083	+	0.794638i	$0.707661\pi$
$998$	66.8678	2.11666
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bb.1.3		3
3.2	odd	2		605.2.a.h.1.1	yes	3
11.10	odd	2		5445.2.a.bd.1.1		3
12.11	even	2		9680.2.a.cb.1.2		3
15.14	odd	2		3025.2.a.p.1.3		3
33.2	even	10		605.2.g.p.81.1		12
33.5	odd	10		605.2.g.o.366.3		12
33.8	even	10		605.2.g.p.251.3		12
33.14	odd	10		605.2.g.o.251.1		12
33.17	even	10		605.2.g.p.366.1		12
33.20	odd	10		605.2.g.o.81.3		12
33.26	odd	10		605.2.g.o.511.1		12
33.29	even	10		605.2.g.p.511.3		12
33.32	even	2		605.2.a.g.1.3	✓	3
132.131	odd	2		9680.2.a.bz.1.2		3
165.164	even	2		3025.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.g.1.3	✓	3	33.32	even	2
605.2.a.h.1.1	yes	3	3.2	odd	2
605.2.g.o.81.3		12	33.20	odd	10
605.2.g.o.251.1		12	33.14	odd	10
605.2.g.o.366.3		12	33.5	odd	10
605.2.g.o.511.1		12	33.26	odd	10
605.2.g.p.81.1		12	33.2	even	10
605.2.g.p.251.3		12	33.8	even	10
605.2.g.p.366.1		12	33.17	even	10
605.2.g.p.511.3		12	33.29	even	10
3025.2.a.p.1.3		3	15.14	odd	2
3025.2.a.u.1.1		3	165.164	even	2
5445.2.a.bb.1.3		3	1.1	even	1	trivial
5445.2.a.bd.1.1		3	11.10	odd	2
9680.2.a.bz.1.2		3	132.131	odd	2
9680.2.a.cb.1.2		3	12.11	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74483 q^{2} +5.53407 q^{4} +1.00000 q^{5} -2.32331 q^{7} +9.70041 q^{8} +O(q^{10})\)	\(q+2.74483 q^{2} +5.53407 q^{4} +1.00000 q^{5} -2.32331 q^{7} +9.70041 q^{8} +2.74483 q^{10} -0.534070 q^{13} -6.37709 q^{14} +15.5578 q^{16} -2.42151 q^{17} +4.95558 q^{19} +5.53407 q^{20} +4.53407 q^{23} +1.00000 q^{25} -1.46593 q^{26} -12.8574 q^{28} +5.48965 q^{29} +1.04442 q^{31} +23.3026 q^{32} -6.64663 q^{34} -2.32331 q^{35} +7.48965 q^{37} +13.6022 q^{38} +9.70041 q^{40} -10.6022 q^{41} +4.32331 q^{43} +12.4452 q^{46} -6.76855 q^{47} -1.60221 q^{49} +2.74483 q^{50} -2.95558 q^{52} +4.53407 q^{53} -22.5371 q^{56} +15.0681 q^{58} +6.44523 q^{59} +6.79861 q^{61} +2.86675 q^{62} +32.8461 q^{64} -0.534070 q^{65} +0.721104 q^{67} -13.4008 q^{68} -6.37709 q^{70} -4.53407 q^{71} +1.06814 q^{73} +20.5578 q^{74} +27.4245 q^{76} +4.64663 q^{79} +15.5578 q^{80} -29.1012 q^{82} -13.6022 q^{83} -2.42151 q^{85} +11.8667 q^{86} -12.7148 q^{89} +1.24081 q^{91} +25.0919 q^{92} -18.5785 q^{94} +4.95558 q^{95} +4.75919 q^{97} -4.39779 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q - q^2 + 9 * q^4 + 3 * q^5 - q^7 + 9 * q^8	\( 3 q - q^{2} + 9 q^{4} + 3 q^{5} - q^{7} + 9 q^{8} - q^{10} + 6 q^{13} - 5 q^{14} + 13 q^{16} - 4 q^{17} + 4 q^{19} + 9 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 25 q^{28} - 2 q^{29} + 14 q^{31} + 27 q^{32} - 8 q^{34} - q^{35} + 4 q^{37} + 18 q^{38} + 9 q^{40} - 9 q^{41} + 7 q^{43} + 8 q^{46} + 15 q^{47} + 18 q^{49} - q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{56} + 30 q^{58} - 10 q^{59} + 3 q^{61} - 24 q^{62} + 29 q^{64} + 6 q^{65} + 19 q^{67} - 5 q^{70} - 6 q^{71} - 12 q^{73} + 28 q^{74} + 16 q^{76} + 2 q^{79} + 13 q^{80} - 27 q^{82} - 18 q^{83} - 4 q^{85} + 3 q^{86} - 11 q^{89} + 20 q^{91} + 34 q^{92} - 59 q^{94} + 4 q^{95} - 2 q^{97} - 36 q^{98}+O(q^{100}) \) 3 * q - q^2 + 9 * q^4 + 3 * q^5 - q^7 + 9 * q^8 - q^10 + 6 * q^13 - 5 * q^14 + 13 * q^16 - 4 * q^17 + 4 * q^19 + 9 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 25 * q^28 - 2 * q^29 + 14 * q^31 + 27 * q^32 - 8 * q^34 - q^35 + 4 * q^37 + 18 * q^38 + 9 * q^40 - 9 * q^41 + 7 * q^43 + 8 * q^46 + 15 * q^47 + 18 * q^49 - q^50 + 2 * q^52 + 6 * q^53 + 3 * q^56 + 30 * q^58 - 10 * q^59 + 3 * q^61 - 24 * q^62 + 29 * q^64 + 6 * q^65 + 19 * q^67 - 5 * q^70 - 6 * q^71 - 12 * q^73 + 28 * q^74 + 16 * q^76 + 2 * q^79 + 13 * q^80 - 27 * q^82 - 18 * q^83 - 4 * q^85 + 3 * q^86 - 11 * q^89 + 20 * q^91 + 34 * q^92 - 59 * q^94 + 4 * q^95 - 2 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more