LMFDB - Embedded newform 9027.2.a.t.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9027,2,Mod(1,9027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9027, 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("9027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9027 = 3^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9027.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:	$72.0809579046$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 3009)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	9027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10017 q^{2} +2.41071 q^{4} +2.30678 q^{5} +4.51480 q^{7} +0.862553 q^{8} +O(q^{10})$	\(q+2.10017 q^{2} +2.41071 q^{4} +2.30678 q^{5} +4.51480 q^{7} +0.862553 q^{8} +4.84464 q^{10} +3.88210 q^{11} -2.10288 q^{13} +9.48184 q^{14} -3.00991 q^{16} +1.00000 q^{17} -4.56971 q^{19} +5.56098 q^{20} +8.15306 q^{22} +2.70687 q^{23} +0.321256 q^{25} -4.41640 q^{26} +10.8839 q^{28} +1.80504 q^{29} +1.68580 q^{31} -8.04642 q^{32} +2.10017 q^{34} +10.4147 q^{35} +8.51916 q^{37} -9.59715 q^{38} +1.98972 q^{40} -1.10866 q^{41} +8.34430 q^{43} +9.35860 q^{44} +5.68488 q^{46} -10.1035 q^{47} +13.3834 q^{49} +0.674692 q^{50} -5.06943 q^{52} -4.71276 q^{53} +8.95516 q^{55} +3.89426 q^{56} +3.79088 q^{58} +1.00000 q^{59} -3.91583 q^{61} +3.54047 q^{62} -10.8790 q^{64} -4.85089 q^{65} +13.1112 q^{67} +2.41071 q^{68} +21.8726 q^{70} +3.91855 q^{71} -3.15443 q^{73} +17.8917 q^{74} -11.0162 q^{76} +17.5269 q^{77} +6.51359 q^{79} -6.94321 q^{80} -2.32838 q^{82} +15.5265 q^{83} +2.30678 q^{85} +17.5244 q^{86} +3.34851 q^{88} +6.03654 q^{89} -9.49409 q^{91} +6.52547 q^{92} -21.2192 q^{94} -10.5413 q^{95} +13.4902 q^{97} +28.1075 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{2} + 34 q^{4} + 3 q^{5} + 19 q^{7} - 3 q^{8}+O(q^{10}) $ 24 * q - 2 * q^2 + 34 * q^4 + 3 * q^5 + 19 * q^7 - 3 * q^8	$ 24 q - 2 q^{2} + 34 q^{4} + 3 q^{5} + 19 q^{7} - 3 q^{8} + 10 q^{10} - 11 q^{11} + 21 q^{13} - 9 q^{14} + 50 q^{16} + 24 q^{17} + 13 q^{19} - q^{20} + 5 q^{22} - 18 q^{23} + 39 q^{25} + 7 q^{26} + 40 q^{28} + 10 q^{29} + 47 q^{31} - 18 q^{32} - 2 q^{34} - 5 q^{35} + 54 q^{37} - 5 q^{38} + 29 q^{40} + 16 q^{43} - 7 q^{44} - 8 q^{46} - 10 q^{47} + 55 q^{49} - 21 q^{50} + 68 q^{52} - 6 q^{53} + 15 q^{55} + 5 q^{56} + q^{58} + 24 q^{59} + 42 q^{61} + 7 q^{62} + 53 q^{64} + 9 q^{65} + 28 q^{67} + 34 q^{68} - 20 q^{70} - 54 q^{71} + 33 q^{73} + 17 q^{74} - 56 q^{76} + 23 q^{77} + 35 q^{79} - 3 q^{80} - 12 q^{82} + 17 q^{83} + 3 q^{85} - 36 q^{86} + 47 q^{88} - 15 q^{89} + 74 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{95} + 52 q^{97} + 48 q^{98}+O(q^{100}) $ 24 * q - 2 * q^2 + 34 * q^4 + 3 * q^5 + 19 * q^7 - 3 * q^8 + 10 * q^10 - 11 * q^11 + 21 * q^13 - 9 * q^14 + 50 * q^16 + 24 * q^17 + 13 * q^19 - q^20 + 5 * q^22 - 18 * q^23 + 39 * q^25 + 7 * q^26 + 40 * q^28 + 10 * q^29 + 47 * q^31 - 18 * q^32 - 2 * q^34 - 5 * q^35 + 54 * q^37 - 5 * q^38 + 29 * q^40 + 16 * q^43 - 7 * q^44 - 8 * q^46 - 10 * q^47 + 55 * q^49 - 21 * q^50 + 68 * q^52 - 6 * q^53 + 15 * q^55 + 5 * q^56 + q^58 + 24 * q^59 + 42 * q^61 + 7 * q^62 + 53 * q^64 + 9 * q^65 + 28 * q^67 + 34 * q^68 - 20 * q^70 - 54 * q^71 + 33 * q^73 + 17 * q^74 - 56 * q^76 + 23 * q^77 + 35 * q^79 - 3 * q^80 - 12 * q^82 + 17 * q^83 + 3 * q^85 - 36 * q^86 + 47 * q^88 - 15 * q^89 + 74 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^95 + 52 * q^97 + 48 * 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 $

$2$	2.10017	1.48504	0.742522	−	0.669822i	$-0.233630\pi$
$2$	2.10017	1.48504	0.742522	+	0.669822i	$0.233630\pi$
$3$	0	0
$3$	0	0
$4$	2.41071	1.20535
$5$	2.30678	1.03163	0.515813	−	0.856701i	$-0.327490\pi$
$5$	2.30678	1.03163	0.515813	+	0.856701i	$0.327490\pi$
$6$	0	0
$7$	4.51480	1.70643	0.853217	−	0.521555i	$-0.174648\pi$
$7$	4.51480	1.70643	0.853217	+	0.521555i	$0.174648\pi$
$8$	0.862553	0.304959
$9$	0	0
$10$	4.84464	1.53201
$11$	3.88210	1.17050	0.585248	−	0.810854i	$-0.300997\pi$
$11$	3.88210	1.17050	0.585248	+	0.810854i	$0.300997\pi$
$12$	0	0
$13$	−2.10288	−0.583234	−0.291617	−	0.956535i	$-0.594193\pi$
$13$	−2.10288	−0.583234	−0.291617	+	0.956535i	$0.594193\pi$
$14$	9.48184	2.53413
$15$	0	0
$16$	−3.00991	−0.752477
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.56971	−1.04836	−0.524181	−	0.851607i	$-0.675629\pi$
$19$	−4.56971	−1.04836	−0.524181	+	0.851607i	$0.675629\pi$
$20$	5.56098	1.24347
$21$	0	0
$22$	8.15306	1.73824
$23$	2.70687	0.564422	0.282211	−	0.959352i	$-0.408932\pi$
$23$	2.70687	0.564422	0.282211	+	0.959352i	$0.408932\pi$
$24$	0	0
$25$	0.321256	0.0642512
$26$	−4.41640	−0.866128
$27$	0	0
$28$	10.8839	2.05686
$29$	1.80504	0.335187	0.167594	−	0.985856i	$-0.446400\pi$
$29$	1.80504	0.335187	0.167594	+	0.985856i	$0.446400\pi$
$30$	0	0
$31$	1.68580	0.302779	0.151390	−	0.988474i	$-0.451625\pi$
$31$	1.68580	0.302779	0.151390	+	0.988474i	$0.451625\pi$
$32$	−8.04642	−1.42242
$33$	0	0
$34$	2.10017	0.360176
$35$	10.4147	1.76040
$36$	0	0
$37$	8.51916	1.40054	0.700271	−	0.713877i	$-0.253062\pi$
$37$	8.51916	1.40054	0.700271	+	0.713877i	$0.253062\pi$
$38$	−9.59715	−1.55686
$39$	0	0
$40$	1.98972	0.314603
$41$	−1.10866	−0.173144	−0.0865720	−	0.996246i	$-0.527591\pi$
$41$	−1.10866	−0.173144	−0.0865720	+	0.996246i	$0.527591\pi$
$42$	0	0
$43$	8.34430	1.27249	0.636247	−	0.771486i	$-0.280486\pi$
$43$	8.34430	1.27249	0.636247	+	0.771486i	$0.280486\pi$
$44$	9.35860	1.41086
$45$	0	0
$46$	5.68488	0.838190
$47$	−10.1035	−1.47375	−0.736877	−	0.676027i	$-0.763700\pi$
$47$	−10.1035	−1.47375	−0.736877	+	0.676027i	$0.763700\pi$
$48$	0	0
$49$	13.3834	1.91192
$50$	0.674692	0.0954158
$51$	0	0
$52$	−5.06943	−0.703003
$53$	−4.71276	−0.647347	−0.323674	−	0.946169i	$-0.604918\pi$
$53$	−4.71276	−0.647347	−0.323674	+	0.946169i	$0.604918\pi$
$54$	0	0
$55$	8.95516	1.20751
$56$	3.89426	0.520392
$57$	0	0
$58$	3.79088	0.497767
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−3.91583	−0.501371	−0.250685	−	0.968069i	$-0.580656\pi$
$61$	−3.91583	−0.501371	−0.250685	+	0.968069i	$0.580656\pi$
$62$	3.54047	0.449640
$63$	0	0
$64$	−10.8790	−1.35988
$65$	−4.85089	−0.601679
$66$	0	0
$67$	13.1112	1.60179	0.800893	−	0.598808i	$-0.204359\pi$
$67$	13.1112	1.60179	0.800893	+	0.598808i	$0.204359\pi$
$68$	2.41071	0.292341
$69$	0	0
$70$	21.8726	2.61427
$71$	3.91855	0.465046	0.232523	−	0.972591i	$-0.425302\pi$
$71$	3.91855	0.465046	0.232523	+	0.972591i	$0.425302\pi$
$72$	0	0
$73$	−3.15443	−0.369198	−0.184599	−	0.982814i	$-0.559099\pi$
$73$	−3.15443	−0.369198	−0.184599	+	0.982814i	$0.559099\pi$
$74$	17.8917	2.07987
$75$	0	0
$76$	−11.0162	−1.26365
$77$	17.5269	1.99738
$78$	0	0
$79$	6.51359	0.732837	0.366418	−	0.930450i	$-0.380584\pi$
$79$	6.51359	0.732837	0.366418	+	0.930450i	$0.380584\pi$
$80$	−6.94321	−0.776274
$81$	0	0
$82$	−2.32838	−0.257126
$83$	15.5265	1.70425	0.852126	−	0.523337i	$-0.175313\pi$
$83$	15.5265	1.70425	0.852126	+	0.523337i	$0.175313\pi$
$84$	0	0
$85$	2.30678	0.250206
$86$	17.5244	1.88971
$87$	0	0
$88$	3.34851	0.356953
$89$	6.03654	0.639872	0.319936	−	0.947439i	$-0.396339\pi$
$89$	6.03654	0.639872	0.319936	+	0.947439i	$0.396339\pi$
$90$	0	0
$91$	−9.49409	−0.995251
$92$	6.52547	0.680327
$93$	0	0
$94$	−21.2192	−2.18859
$95$	−10.5413	−1.08152
$96$	0	0
$97$	13.4902	1.36973	0.684863	−	0.728672i	$-0.259862\pi$
$97$	13.4902	1.36973	0.684863	+	0.728672i	$0.259862\pi$
$98$	28.1075	2.83928
$99$	0	0
$100$	0.774454	0.0774454
$101$	8.50500	0.846279	0.423140	−	0.906065i	$-0.360928\pi$
$101$	8.50500	0.846279	0.423140	+	0.906065i	$0.360928\pi$
$102$	0	0
$103$	−5.09592	−0.502116	−0.251058	−	0.967972i	$-0.580779\pi$
$103$	−5.09592	−0.502116	−0.251058	+	0.967972i	$0.580779\pi$
$104$	−1.81385	−0.177862
$105$	0	0
$106$	−9.89759	−0.961339
$107$	−17.5791	−1.69943	−0.849716	−	0.527241i	$-0.823227\pi$
$107$	−17.5791	−1.69943	−0.849716	+	0.527241i	$0.823227\pi$
$108$	0	0
$109$	−14.7562	−1.41339	−0.706696	−	0.707517i	$-0.749815\pi$
$109$	−14.7562	−1.41339	−0.706696	+	0.707517i	$0.749815\pi$
$110$	18.8073	1.79321
$111$	0	0
$112$	−13.5891	−1.28405
$113$	−16.8773	−1.58768	−0.793842	−	0.608124i	$-0.791922\pi$
$113$	−16.8773	−1.58768	−0.793842	+	0.608124i	$0.791922\pi$
$114$	0	0
$115$	6.24417	0.582272
$116$	4.35142	0.404019
$117$	0	0
$118$	2.10017	0.193336
$119$	4.51480	0.413871
$120$	0	0
$121$	4.07068	0.370061
$122$	−8.22390	−0.744557
$123$	0	0
$124$	4.06398	0.364956
$125$	−10.7929	−0.965342
$126$	0	0
$127$	0.935504	0.0830126	0.0415063	−	0.999138i	$-0.486784\pi$
$127$	0.935504	0.0830126	0.0415063	+	0.999138i	$0.486784\pi$
$128$	−6.75493	−0.597057
$129$	0	0
$130$	−10.1877	−0.893520
$131$	−10.7050	−0.935301	−0.467651	−	0.883913i	$-0.654899\pi$
$131$	−10.7050	−0.935301	−0.467651	+	0.883913i	$0.654899\pi$
$132$	0	0
$133$	−20.6313	−1.78896
$134$	27.5357	2.37872
$135$	0	0
$136$	0.862553	0.0739633
$137$	−18.3907	−1.57122	−0.785611	−	0.618721i	$-0.787651\pi$
$137$	−18.3907	−1.57122	−0.785611	+	0.618721i	$0.787651\pi$
$138$	0	0
$139$	8.06594	0.684144	0.342072	−	0.939674i	$-0.388871\pi$
$139$	8.06594	0.684144	0.342072	+	0.939674i	$0.388871\pi$
$140$	25.1067	2.12191
$141$	0	0
$142$	8.22962	0.690614
$143$	−8.16359	−0.682673
$144$	0	0
$145$	4.16383	0.345788
$146$	−6.62484	−0.548276
$147$	0	0
$148$	20.5372	1.68815
$149$	−7.59683	−0.622357	−0.311178	−	0.950352i	$-0.600724\pi$
$149$	−7.59683	−0.622357	−0.311178	+	0.950352i	$0.600724\pi$
$150$	0	0
$151$	16.1171	1.31159	0.655797	−	0.754937i	$-0.272333\pi$
$151$	16.1171	1.31159	0.655797	+	0.754937i	$0.272333\pi$
$152$	−3.94161	−0.319707
$153$	0	0
$154$	36.8094	2.96619
$155$	3.88879	0.312355
$156$	0	0
$157$	15.9310	1.27143	0.635715	−	0.771924i	$-0.280705\pi$
$157$	15.9310	1.27143	0.635715	+	0.771924i	$0.280705\pi$
$158$	13.6796	1.08829
$159$	0	0
$160$	−18.5614	−1.46740
$161$	12.2210	0.963149
$162$	0	0
$163$	−6.59354	−0.516446	−0.258223	−	0.966085i	$-0.583137\pi$
$163$	−6.59354	−0.516446	−0.258223	+	0.966085i	$0.583137\pi$
$164$	−2.67266	−0.208700
$165$	0	0
$166$	32.6082	2.53089
$167$	9.38923	0.726560	0.363280	−	0.931680i	$-0.381657\pi$
$167$	9.38923	0.726560	0.363280	+	0.931680i	$0.381657\pi$
$168$	0	0
$169$	−8.57789	−0.659838
$170$	4.84464	0.371567
$171$	0	0
$172$	20.1157	1.53380
$173$	−6.79701	−0.516767	−0.258384	−	0.966042i	$-0.583190\pi$
$173$	−6.79701	−0.516767	−0.258384	+	0.966042i	$0.583190\pi$
$174$	0	0
$175$	1.45041	0.109641
$176$	−11.6848	−0.880771
$177$	0	0
$178$	12.6777	0.950237
$179$	−3.19822	−0.239046	−0.119523	−	0.992831i	$-0.538136\pi$
$179$	−3.19822	−0.239046	−0.119523	+	0.992831i	$0.538136\pi$
$180$	0	0
$181$	−16.8898	−1.25541	−0.627704	−	0.778452i	$-0.716005\pi$
$181$	−16.8898	−1.25541	−0.627704	+	0.778452i	$0.716005\pi$
$182$	−19.9392	−1.47799
$183$	0	0
$184$	2.33482	0.172125
$185$	19.6519	1.44483
$186$	0	0
$187$	3.88210	0.283887
$188$	−24.3567	−1.77639
$189$	0	0
$190$	−22.1386	−1.60610
$191$	−26.9463	−1.94977	−0.974883	−	0.222718i	$-0.928507\pi$
$191$	−26.9463	−1.94977	−0.974883	+	0.222718i	$0.928507\pi$
$192$	0	0
$193$	16.4020	1.18065	0.590323	−	0.807167i	$-0.299001\pi$
$193$	16.4020	1.18065	0.590323	+	0.807167i	$0.299001\pi$
$194$	28.3318	2.03410
$195$	0	0
$196$	32.2635	2.30454
$197$	3.45189	0.245937	0.122968	−	0.992411i	$-0.460759\pi$
$197$	3.45189	0.245937	0.122968	+	0.992411i	$0.460759\pi$
$198$	0	0
$199$	8.65187	0.613315	0.306657	−	0.951820i	$-0.400789\pi$
$199$	8.65187	0.613315	0.306657	+	0.951820i	$0.400789\pi$
$200$	0.277100	0.0195940
$201$	0	0
$202$	17.8619	1.25676
$203$	8.14939	0.571975
$204$	0	0
$205$	−2.55744	−0.178620
$206$	−10.7023	−0.745664
$207$	0	0
$208$	6.32948	0.438870
$209$	−17.7400	−1.22710
$210$	0	0
$211$	−18.5611	−1.27780	−0.638900	−	0.769290i	$-0.720610\pi$
$211$	−18.5611	−1.27780	−0.638900	+	0.769290i	$0.720610\pi$
$212$	−11.3611	−0.780282
$213$	0	0
$214$	−36.9190	−2.52373
$215$	19.2485	1.31274
$216$	0	0
$217$	7.61107	0.516673
$218$	−30.9906	−2.09895
$219$	0	0
$220$	21.5883	1.45548
$221$	−2.10288	−0.141455
$222$	0	0
$223$	−17.0243	−1.14003	−0.570017	−	0.821633i	$-0.693063\pi$
$223$	−17.0243	−1.14003	−0.570017	+	0.821633i	$0.693063\pi$
$224$	−36.3280	−2.42727
$225$	0	0
$226$	−35.4452	−2.35778
$227$	9.96647	0.661498	0.330749	−	0.943719i	$-0.392699\pi$
$227$	9.96647	0.661498	0.330749	+	0.943719i	$0.392699\pi$
$228$	0	0
$229$	−6.02183	−0.397934	−0.198967	−	0.980006i	$-0.563759\pi$
$229$	−6.02183	−0.397934	−0.198967	+	0.980006i	$0.563759\pi$
$230$	13.1138	0.864699
$231$	0	0
$232$	1.55694	0.102218
$233$	−23.6217	−1.54751	−0.773755	−	0.633485i	$-0.781624\pi$
$233$	−23.6217	−1.54751	−0.773755	+	0.633485i	$0.781624\pi$
$234$	0	0
$235$	−23.3067	−1.52036
$236$	2.41071	0.156924
$237$	0	0
$238$	9.48184	0.614617
$239$	13.4612	0.870733	0.435367	−	0.900253i	$-0.356619\pi$
$239$	13.4612	0.870733	0.435367	+	0.900253i	$0.356619\pi$
$240$	0	0
$241$	13.0684	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$241$	13.0684	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$242$	8.54911	0.549557
$243$	0	0
$244$	−9.43992	−0.604329
$245$	30.8727	1.97239
$246$	0	0
$247$	9.60955	0.611441
$248$	1.45410	0.0923351
$249$	0	0
$250$	−22.6668	−1.43358
$251$	7.67180	0.484240	0.242120	−	0.970246i	$-0.422157\pi$
$251$	7.67180	0.484240	0.242120	+	0.970246i	$0.422157\pi$
$252$	0	0
$253$	10.5083	0.660653
$254$	1.96472	0.123277
$255$	0	0
$256$	7.57154	0.473222
$257$	16.0802	1.00305	0.501526	−	0.865142i	$-0.332772\pi$
$257$	16.0802	1.00305	0.501526	+	0.865142i	$0.332772\pi$
$258$	0	0
$259$	38.4623	2.38993
$260$	−11.6941	−0.725236
$261$	0	0
$262$	−22.4823	−1.38896
$263$	−19.1252	−1.17931	−0.589656	−	0.807654i	$-0.700737\pi$
$263$	−19.1252	−1.17931	−0.589656	+	0.807654i	$0.700737\pi$
$264$	0	0
$265$	−10.8713	−0.667820
$266$	−43.3292	−2.65669
$267$	0	0
$268$	31.6072	1.93072
$269$	27.8237	1.69644	0.848221	−	0.529643i	$-0.177674\pi$
$269$	27.8237	1.69644	0.848221	+	0.529643i	$0.177674\pi$
$270$	0	0
$271$	−9.03969	−0.549122	−0.274561	−	0.961570i	$-0.588532\pi$
$271$	−9.03969	−0.549122	−0.274561	+	0.961570i	$0.588532\pi$
$272$	−3.00991	−0.182502
$273$	0	0
$274$	−38.6235	−2.33333
$275$	1.24715	0.0752058
$276$	0	0
$277$	13.3912	0.804601	0.402301	−	0.915508i	$-0.368211\pi$
$277$	13.3912	0.804601	0.402301	+	0.915508i	$0.368211\pi$
$278$	16.9398	1.01598
$279$	0	0
$280$	8.98321	0.536850
$281$	−19.4798	−1.16207	−0.581033	−	0.813880i	$-0.697351\pi$
$281$	−19.4798	−1.16207	−0.581033	+	0.813880i	$0.697351\pi$
$282$	0	0
$283$	4.19744	0.249512	0.124756	−	0.992187i	$-0.460185\pi$
$283$	4.19744	0.249512	0.124756	+	0.992187i	$0.460185\pi$
$284$	9.44648	0.560545
$285$	0	0
$286$	−17.1449	−1.01380
$287$	−5.00539	−0.295459
$288$	0	0
$289$	1.00000	0.0588235
$290$	8.74475	0.513509
$291$	0	0
$292$	−7.60441	−0.445015
$293$	−22.1989	−1.29687	−0.648436	−	0.761269i	$-0.724577\pi$
$293$	−22.1989	−1.29687	−0.648436	+	0.761269i	$0.724577\pi$
$294$	0	0
$295$	2.30678	0.134306
$296$	7.34823	0.427107
$297$	0	0
$298$	−15.9546	−0.924226
$299$	−5.69223	−0.329190
$300$	0	0
$301$	37.6729	2.17143
$302$	33.8487	1.94778
$303$	0	0
$304$	13.7544	0.788868
$305$	−9.03298	−0.517227
$306$	0	0
$307$	−1.42930	−0.0815746	−0.0407873	−	0.999168i	$-0.512987\pi$
$307$	−1.42930	−0.0815746	−0.0407873	+	0.999168i	$0.512987\pi$
$308$	42.2522	2.40754
$309$	0	0
$310$	8.16711	0.463861
$311$	12.1203	0.687280	0.343640	−	0.939102i	$-0.388340\pi$
$311$	12.1203	0.687280	0.343640	+	0.939102i	$0.388340\pi$
$312$	0	0
$313$	−4.92301	−0.278265	−0.139133	−	0.990274i	$-0.544431\pi$
$313$	−4.92301	−0.278265	−0.139133	+	0.990274i	$0.544431\pi$
$314$	33.4577	1.88813
$315$	0	0
$316$	15.7024	0.883327
$317$	−20.4685	−1.14963	−0.574814	−	0.818284i	$-0.694925\pi$
$317$	−20.4685	−1.14963	−0.574814	+	0.818284i	$0.694925\pi$
$318$	0	0
$319$	7.00733	0.392335
$320$	−25.0955	−1.40288
$321$	0	0
$322$	25.6661	1.43032
$323$	−4.56971	−0.254265
$324$	0	0
$325$	−0.675563	−0.0374735
$326$	−13.8476	−0.766945
$327$	0	0
$328$	−0.956280	−0.0528017
$329$	−45.6155	−2.51487
$330$	0	0
$331$	−7.63062	−0.419417	−0.209709	−	0.977764i	$-0.567252\pi$
$331$	−7.63062	−0.419417	−0.209709	+	0.977764i	$0.567252\pi$
$332$	37.4298	2.05422
$333$	0	0
$334$	19.7190	1.07897
$335$	30.2447	1.65244
$336$	0	0
$337$	−15.2833	−0.832536	−0.416268	−	0.909242i	$-0.636662\pi$
$337$	−15.2833	−0.832536	−0.416268	+	0.909242i	$0.636662\pi$
$338$	−18.0150	−0.979888
$339$	0	0
$340$	5.56098	0.301587
$341$	6.54446	0.354402
$342$	0	0
$343$	28.8200	1.55613
$344$	7.19740	0.388058
$345$	0	0
$346$	−14.2749	−0.767422
$347$	−11.9854	−0.643409	−0.321705	−	0.946840i	$-0.604256\pi$
$347$	−11.9854	−0.643409	−0.321705	+	0.946840i	$0.604256\pi$
$348$	0	0
$349$	−1.42775	−0.0764255	−0.0382127	−	0.999270i	$-0.512166\pi$
$349$	−1.42775	−0.0764255	−0.0382127	+	0.999270i	$0.512166\pi$
$350$	3.04610	0.162821
$351$	0	0
$352$	−31.2370	−1.66494
$353$	−24.5749	−1.30799	−0.653994	−	0.756499i	$-0.726908\pi$
$353$	−24.5749	−1.30799	−0.653994	+	0.756499i	$0.726908\pi$
$354$	0	0
$355$	9.03925	0.479754
$356$	14.5523	0.771272
$357$	0	0
$358$	−6.71679	−0.354993
$359$	−15.5965	−0.823153	−0.411576	−	0.911375i	$-0.635022\pi$
$359$	−15.5965	−0.823153	−0.411576	+	0.911375i	$0.635022\pi$
$360$	0	0
$361$	1.88222	0.0990641
$362$	−35.4714	−1.86434
$363$	0	0
$364$	−22.8875	−1.19963
$365$	−7.27660	−0.380875
$366$	0	0
$367$	25.2605	1.31859	0.659294	−	0.751885i	$-0.270855\pi$
$367$	25.2605	1.31859	0.659294	+	0.751885i	$0.270855\pi$
$368$	−8.14743	−0.424714
$369$	0	0
$370$	41.2723	2.14564
$371$	−21.2772	−1.10466
$372$	0	0
$373$	−18.9136	−0.979308	−0.489654	−	0.871917i	$-0.662877\pi$
$373$	−18.9136	−0.979308	−0.489654	+	0.871917i	$0.662877\pi$
$374$	8.15306	0.421585
$375$	0	0
$376$	−8.71485	−0.449434
$377$	−3.79578	−0.195493
$378$	0	0
$379$	−36.2133	−1.86015	−0.930076	−	0.367368i	$-0.880259\pi$
$379$	−36.2133	−1.86015	−0.930076	+	0.367368i	$0.880259\pi$
$380$	−25.4121	−1.30361
$381$	0	0
$382$	−56.5918	−2.89549
$383$	15.1569	0.774483	0.387241	−	0.921978i	$-0.373428\pi$
$383$	15.1569	0.774483	0.387241	+	0.921978i	$0.373428\pi$
$384$	0	0
$385$	40.4308	2.06054
$386$	34.4471	1.75331
$387$	0	0
$388$	32.5210	1.65100
$389$	19.9540	1.01171	0.505853	−	0.862620i	$-0.331178\pi$
$389$	19.9540	1.01171	0.505853	+	0.862620i	$0.331178\pi$
$390$	0	0
$391$	2.70687	0.136892
$392$	11.5439	0.583056
$393$	0	0
$394$	7.24954	0.365227
$395$	15.0255	0.756013
$396$	0	0
$397$	11.1566	0.559934	0.279967	−	0.960010i	$-0.409677\pi$
$397$	11.1566	0.559934	0.279967	+	0.960010i	$0.409677\pi$
$398$	18.1704	0.910799
$399$	0	0
$400$	−0.966951	−0.0483476
$401$	−13.8699	−0.692631	−0.346315	−	0.938118i	$-0.612567\pi$
$401$	−13.8699	−0.692631	−0.346315	+	0.938118i	$0.612567\pi$
$402$	0	0
$403$	−3.54505	−0.176591
$404$	20.5031	1.02007
$405$	0	0
$406$	17.1151	0.849407
$407$	33.0722	1.63933
$408$	0	0
$409$	35.2301	1.74201	0.871007	−	0.491270i	$-0.163467\pi$
$409$	35.2301	1.74201	0.871007	+	0.491270i	$0.163467\pi$
$410$	−5.37106	−0.265258
$411$	0	0
$412$	−12.2848	−0.605228
$413$	4.51480	0.222159
$414$	0	0
$415$	35.8162	1.75815
$416$	16.9207	0.829603
$417$	0	0
$418$	−37.2571	−1.82230
$419$	−29.5716	−1.44467	−0.722333	−	0.691545i	$-0.756930\pi$
$419$	−29.5716	−1.44467	−0.722333	+	0.691545i	$0.756930\pi$
$420$	0	0
$421$	8.04388	0.392035	0.196017	−	0.980600i	$-0.437199\pi$
$421$	8.04388	0.392035	0.196017	+	0.980600i	$0.437199\pi$
$422$	−38.9815	−1.89759
$423$	0	0
$424$	−4.06500	−0.197414
$425$	0.321256	0.0155832
$426$	0	0
$427$	−17.6792	−0.855556
$428$	−42.3780	−2.04842
$429$	0	0
$430$	40.4251	1.94947
$431$	30.3068	1.45983	0.729913	−	0.683540i	$-0.239560\pi$
$431$	30.3068	1.45983	0.729913	+	0.683540i	$0.239560\pi$
$432$	0	0
$433$	22.6918	1.09050	0.545248	−	0.838275i	$-0.316435\pi$
$433$	22.6918	1.09050	0.545248	+	0.838275i	$0.316435\pi$
$434$	15.9845	0.767282
$435$	0	0
$436$	−35.5730	−1.70364
$437$	−12.3696	−0.591719
$438$	0	0
$439$	−15.4745	−0.738560	−0.369280	−	0.929318i	$-0.620396\pi$
$439$	−15.4745	−0.738560	−0.369280	+	0.929318i	$0.620396\pi$
$440$	7.72430	0.368242
$441$	0	0
$442$	−4.41640	−0.210067
$443$	−26.2278	−1.24612	−0.623059	−	0.782175i	$-0.714110\pi$
$443$	−26.2278	−1.24612	−0.623059	+	0.782175i	$0.714110\pi$
$444$	0	0
$445$	13.9250	0.660108
$446$	−35.7539	−1.69300
$447$	0	0
$448$	−49.1166	−2.32054
$449$	26.5724	1.25403	0.627014	−	0.779008i	$-0.284277\pi$
$449$	26.5724	1.25403	0.627014	+	0.779008i	$0.284277\pi$
$450$	0	0
$451$	−4.30393	−0.202664
$452$	−40.6863	−1.91372
$453$	0	0
$454$	20.9313	0.982353
$455$	−21.9008	−1.02673
$456$	0	0
$457$	−20.0503	−0.937911	−0.468956	−	0.883222i	$-0.655370\pi$
$457$	−20.0503	−0.937911	−0.468956	+	0.883222i	$0.655370\pi$
$458$	−12.6469	−0.590949
$459$	0	0
$460$	15.0529	0.701843
$461$	−26.1566	−1.21824	−0.609118	−	0.793079i	$-0.708477\pi$
$461$	−26.1566	−1.21824	−0.609118	+	0.793079i	$0.708477\pi$
$462$	0	0
$463$	19.2819	0.896106	0.448053	−	0.894007i	$-0.352118\pi$
$463$	19.2819	0.896106	0.448053	+	0.894007i	$0.352118\pi$
$464$	−5.43299	−0.252220
$465$	0	0
$466$	−49.6096	−2.29812
$467$	3.38575	0.156674	0.0783369	−	0.996927i	$-0.475039\pi$
$467$	3.38575	0.156674	0.0783369	+	0.996927i	$0.475039\pi$
$468$	0	0
$469$	59.1944	2.73334
$470$	−48.9480	−2.25780
$471$	0	0
$472$	0.862553	0.0397022
$473$	32.3934	1.48945
$474$	0	0
$475$	−1.46805	−0.0673586
$476$	10.8839	0.498861
$477$	0	0
$478$	28.2708	1.29308
$479$	0.702324	0.0320900	0.0160450	−	0.999871i	$-0.494892\pi$
$479$	0.702324	0.0320900	0.0160450	+	0.999871i	$0.494892\pi$
$480$	0	0
$481$	−17.9148	−0.816844
$482$	27.4457	1.25012
$483$	0	0
$484$	9.81321	0.446055
$485$	31.1191	1.41304
$486$	0	0
$487$	−11.3938	−0.516301	−0.258151	−	0.966105i	$-0.583113\pi$
$487$	−11.3938	−0.516301	−0.258151	+	0.966105i	$0.583113\pi$
$488$	−3.37761	−0.152897
$489$	0	0
$490$	64.8379	2.92908
$491$	37.2721	1.68207	0.841034	−	0.540983i	$-0.181948\pi$
$491$	37.2721	1.68207	0.841034	+	0.540983i	$0.181948\pi$
$492$	0	0
$493$	1.80504	0.0812948
$494$	20.1817	0.908016
$495$	0	0
$496$	−5.07411	−0.227834
$497$	17.6915	0.793571
$498$	0	0
$499$	18.4823	0.827383	0.413691	−	0.910417i	$-0.364239\pi$
$499$	18.4823	0.827383	0.413691	+	0.910417i	$0.364239\pi$
$500$	−26.0184	−1.16358
$501$	0	0
$502$	16.1121	0.719117
$503$	17.9752	0.801473	0.400736	−	0.916193i	$-0.368754\pi$
$503$	17.9752	0.801473	0.400736	+	0.916193i	$0.368754\pi$
$504$	0	0
$505$	19.6192	0.873043
$506$	22.0693	0.981099
$507$	0	0
$508$	2.25523	0.100060
$509$	−15.3785	−0.681639	−0.340819	−	0.940129i	$-0.610704\pi$
$509$	−15.3785	−0.681639	−0.340819	+	0.940129i	$0.610704\pi$
$510$	0	0
$511$	−14.2416	−0.630013
$512$	29.4114	1.29981
$513$	0	0
$514$	33.7710	1.48958
$515$	−11.7552	−0.517996
$516$	0	0
$517$	−39.2230	−1.72502
$518$	80.7774	3.54915
$519$	0	0
$520$	−4.18415	−0.183487
$521$	−8.23943	−0.360976	−0.180488	−	0.983577i	$-0.557768\pi$
$521$	−8.23943	−0.360976	−0.180488	+	0.983577i	$0.557768\pi$
$522$	0	0
$523$	−43.2960	−1.89320	−0.946600	−	0.322410i	$-0.895507\pi$
$523$	−43.2960	−1.89320	−0.946600	+	0.322410i	$0.895507\pi$
$524$	−25.8066	−1.12737
$525$	0	0
$526$	−40.1662	−1.75133
$527$	1.68580	0.0734348
$528$	0	0
$529$	−15.6728	−0.681428
$530$	−22.8316	−0.991742
$531$	0	0
$532$	−49.7361	−2.15633
$533$	2.33138	0.100983
$534$	0	0
$535$	−40.5511	−1.75318
$536$	11.3091	0.488478
$537$	0	0
$538$	58.4345	2.51929
$539$	51.9558	2.23790
$540$	0	0
$541$	4.64721	0.199799	0.0998995	−	0.994998i	$-0.468148\pi$
$541$	4.64721	0.199799	0.0998995	+	0.994998i	$0.468148\pi$
$542$	−18.9849	−0.815470
$543$	0	0
$544$	−8.04642	−0.344987
$545$	−34.0395	−1.45809
$546$	0	0
$547$	4.22818	0.180784	0.0903920	−	0.995906i	$-0.471188\pi$
$547$	4.22818	0.180784	0.0903920	+	0.995906i	$0.471188\pi$
$548$	−44.3345	−1.89388
$549$	0	0
$550$	2.61922	0.111684
$551$	−8.24849	−0.351398
$552$	0	0
$553$	29.4076	1.25054
$554$	28.1238	1.19487
$555$	0	0
$556$	19.4446	0.824636
$557$	−30.7888	−1.30456	−0.652282	−	0.757976i	$-0.726188\pi$
$557$	−30.7888	−1.30456	−0.652282	+	0.757976i	$0.726188\pi$
$558$	0	0
$559$	−17.5471	−0.742162
$560$	−31.3472	−1.32466
$561$	0	0
$562$	−40.9108	−1.72572
$563$	6.71743	0.283106	0.141553	−	0.989931i	$-0.454790\pi$
$563$	6.71743	0.283106	0.141553	+	0.989931i	$0.454790\pi$
$564$	0	0
$565$	−38.9323	−1.63790
$566$	8.81533	0.370536
$567$	0	0
$568$	3.37996	0.141820
$569$	−20.0294	−0.839676	−0.419838	−	0.907599i	$-0.637913\pi$
$569$	−20.0294	−0.839676	−0.419838	+	0.907599i	$0.637913\pi$
$570$	0	0
$571$	44.5126	1.86279	0.931396	−	0.364007i	$-0.118592\pi$
$571$	44.5126	1.86279	0.931396	+	0.364007i	$0.118592\pi$
$572$	−19.6800	−0.822863
$573$	0	0
$574$	−10.5122	−0.438769
$575$	0.869599	0.0362648
$576$	0	0
$577$	37.5728	1.56418	0.782089	−	0.623167i	$-0.214154\pi$
$577$	37.5728	1.56418	0.782089	+	0.623167i	$0.214154\pi$
$578$	2.10017	0.0873555
$579$	0	0
$580$	10.0378	0.416796
$581$	70.0989	2.90819
$582$	0	0
$583$	−18.2954	−0.757718
$584$	−2.72087	−0.112590
$585$	0	0
$586$	−46.6214	−1.92591
$587$	27.9033	1.15169	0.575847	−	0.817558i	$-0.304673\pi$
$587$	27.9033	1.15169	0.575847	+	0.817558i	$0.304673\pi$
$588$	0	0
$589$	−7.70363	−0.317423
$590$	4.84464	0.199451
$591$	0	0
$592$	−25.6419	−1.05388
$593$	29.9035	1.22799	0.613995	−	0.789310i	$-0.289561\pi$
$593$	29.9035	1.22799	0.613995	+	0.789310i	$0.289561\pi$
$594$	0	0
$595$	10.4147	0.426960
$596$	−18.3137	−0.750160
$597$	0	0
$598$	−11.9546	−0.488861
$599$	1.52281	0.0622203	0.0311101	−	0.999516i	$-0.490096\pi$
$599$	1.52281	0.0622203	0.0311101	+	0.999516i	$0.490096\pi$
$600$	0	0
$601$	−33.8978	−1.38272	−0.691359	−	0.722511i	$-0.742988\pi$
$601$	−33.8978	−1.38272	−0.691359	+	0.722511i	$0.742988\pi$
$602$	79.1193	3.22466
$603$	0	0
$604$	38.8537	1.58094
$605$	9.39017	0.381765
$606$	0	0
$607$	28.6840	1.16425	0.582124	−	0.813100i	$-0.302222\pi$
$607$	28.6840	1.16425	0.582124	+	0.813100i	$0.302222\pi$
$608$	36.7698	1.49121
$609$	0	0
$610$	−18.9708	−0.768104
$611$	21.2466	0.859544
$612$	0	0
$613$	39.0863	1.57868	0.789341	−	0.613955i	$-0.210422\pi$
$613$	39.0863	1.57868	0.789341	+	0.613955i	$0.210422\pi$
$614$	−3.00178	−0.121142
$615$	0	0
$616$	15.1179	0.609117
$617$	−9.41263	−0.378938	−0.189469	−	0.981887i	$-0.560677\pi$
$617$	−9.41263	−0.378938	−0.189469	+	0.981887i	$0.560677\pi$
$618$	0	0
$619$	35.1721	1.41369	0.706844	−	0.707370i	$-0.250118\pi$
$619$	35.1721	1.41369	0.706844	+	0.707370i	$0.250118\pi$
$620$	9.37473	0.376498
$621$	0	0
$622$	25.4547	1.02064
$623$	27.2538	1.09190
$624$	0	0
$625$	−26.5031	−1.06012
$626$	−10.3392	−0.413236
$627$	0	0
$628$	38.4049	1.53252
$629$	8.51916	0.339681
$630$	0	0
$631$	25.3838	1.01051	0.505256	−	0.862970i	$-0.331398\pi$
$631$	25.3838	1.01051	0.505256	+	0.862970i	$0.331398\pi$
$632$	5.61832	0.223485
$633$	0	0
$634$	−42.9874	−1.70725
$635$	2.15801	0.0856379
$636$	0	0
$637$	−28.1438	−1.11510
$638$	14.7166	0.582635
$639$	0	0
$640$	−15.5822	−0.615939
$641$	−19.3147	−0.762886	−0.381443	−	0.924392i	$-0.624573\pi$
$641$	−19.3147	−0.762886	−0.381443	+	0.924392i	$0.624573\pi$
$642$	0	0
$643$	−13.0669	−0.515307	−0.257654	−	0.966237i	$-0.582949\pi$
$643$	−13.0669	−0.515307	−0.257654	+	0.966237i	$0.582949\pi$
$644$	29.4612	1.16093
$645$	0	0
$646$	−9.59715	−0.377595
$647$	21.8728	0.859907	0.429954	−	0.902851i	$-0.358530\pi$
$647$	21.8728	0.859907	0.429954	+	0.902851i	$0.358530\pi$
$648$	0	0
$649$	3.88210	0.152386
$650$	−1.41880	−0.0556498
$651$	0	0
$652$	−15.8951	−0.622500
$653$	14.1947	0.555481	0.277740	−	0.960656i	$-0.410414\pi$
$653$	14.1947	0.555481	0.277740	+	0.960656i	$0.410414\pi$
$654$	0	0
$655$	−24.6942	−0.964881
$656$	3.33697	0.130287
$657$	0	0
$658$	−95.8003	−3.73468
$659$	2.63833	0.102775	0.0513875	−	0.998679i	$-0.483636\pi$
$659$	2.63833	0.102775	0.0513875	+	0.998679i	$0.483636\pi$
$660$	0	0
$661$	−5.86329	−0.228056	−0.114028	−	0.993478i	$-0.536375\pi$
$661$	−5.86329	−0.228056	−0.114028	+	0.993478i	$0.536375\pi$
$662$	−16.0256	−0.622852
$663$	0	0
$664$	13.3924	0.519726
$665$	−47.5920	−1.84554
$666$	0	0
$667$	4.88600	0.189187
$668$	22.6347	0.875762
$669$	0	0
$670$	63.5189	2.45395
$671$	−15.2016	−0.586852
$672$	0	0
$673$	20.3970	0.786248	0.393124	−	0.919485i	$-0.371394\pi$
$673$	20.3970	0.786248	0.393124	+	0.919485i	$0.371394\pi$
$674$	−32.0976	−1.23635
$675$	0	0
$676$	−20.6788	−0.795338
$677$	4.41467	0.169670	0.0848349	−	0.996395i	$-0.472964\pi$
$677$	4.41467	0.169670	0.0848349	+	0.996395i	$0.472964\pi$
$678$	0	0
$679$	60.9058	2.33735
$680$	1.98972	0.0763024
$681$	0	0
$682$	13.7445	0.526302
$683$	−44.8987	−1.71800	−0.859001	−	0.511974i	$-0.828915\pi$
$683$	−44.8987	−1.71800	−0.859001	+	0.511974i	$0.828915\pi$
$684$	0	0
$685$	−42.4233	−1.62091
$686$	60.5268	2.31092
$687$	0	0
$688$	−25.1156	−0.957522
$689$	9.91037	0.377555
$690$	0	0
$691$	−14.8725	−0.565777	−0.282889	−	0.959153i	$-0.591293\pi$
$691$	−14.8725	−0.565777	−0.282889	+	0.959153i	$0.591293\pi$
$692$	−16.3856	−0.622887
$693$	0	0
$694$	−25.1713	−0.955490
$695$	18.6064	0.705781
$696$	0	0
$697$	−1.10866	−0.0419936
$698$	−2.99851	−0.113495
$699$	0	0
$700$	3.49651	0.132156
$701$	19.4166	0.733356	0.366678	−	0.930348i	$-0.380495\pi$
$701$	19.4166	0.733356	0.366678	+	0.930348i	$0.380495\pi$
$702$	0	0
$703$	−38.9301	−1.46828
$704$	−42.2334	−1.59173
$705$	0	0
$706$	−51.6114	−1.94242
$707$	38.3984	1.44412
$708$	0	0
$709$	−29.8929	−1.12265	−0.561326	−	0.827595i	$-0.689709\pi$
$709$	−29.8929	−1.12265	−0.561326	+	0.827595i	$0.689709\pi$
$710$	18.9840	0.712455
$711$	0	0
$712$	5.20683	0.195134
$713$	4.56325	0.170895
$714$	0	0
$715$	−18.8316	−0.704263
$716$	−7.70996	−0.288135
$717$	0	0
$718$	−32.7553	−1.22242
$719$	−25.8668	−0.964668	−0.482334	−	0.875987i	$-0.660211\pi$
$719$	−25.8668	−0.964668	−0.482334	+	0.875987i	$0.660211\pi$
$720$	0	0
$721$	−23.0071	−0.856829
$722$	3.95298	0.147115
$723$	0	0
$724$	−40.7163	−1.51321
$725$	0.579879	0.0215362
$726$	0	0
$727$	−20.1442	−0.747105	−0.373553	−	0.927609i	$-0.621860\pi$
$727$	−20.1442	−0.747105	−0.373553	+	0.927609i	$0.621860\pi$
$728$	−8.18916	−0.303510
$729$	0	0
$730$	−15.2821	−0.565615
$731$	8.34430	0.308625
$732$	0	0
$733$	−18.9876	−0.701323	−0.350661	−	0.936502i	$-0.614043\pi$
$733$	−18.9876	−0.701323	−0.350661	+	0.936502i	$0.614043\pi$
$734$	53.0513	1.95816
$735$	0	0
$736$	−21.7806	−0.802844
$737$	50.8989	1.87488
$738$	0	0
$739$	37.0452	1.36273	0.681365	−	0.731944i	$-0.261387\pi$
$739$	37.0452	1.36273	0.681365	+	0.731944i	$0.261387\pi$
$740$	47.3749	1.74154
$741$	0	0
$742$	−44.6857	−1.64046
$743$	12.0107	0.440631	0.220316	−	0.975429i	$-0.429291\pi$
$743$	12.0107	0.440631	0.220316	+	0.975429i	$0.429291\pi$
$744$	0	0
$745$	−17.5243	−0.642039
$746$	−39.7217	−1.45431
$747$	0	0
$748$	9.35860	0.342184
$749$	−79.3660	−2.89997
$750$	0	0
$751$	1.49051	0.0543896	0.0271948	−	0.999630i	$-0.491343\pi$
$751$	1.49051	0.0543896	0.0271948	+	0.999630i	$0.491343\pi$
$752$	30.4107	1.10897
$753$	0	0
$754$	−7.97177	−0.290315
$755$	37.1788	1.35307
$756$	0	0
$757$	−14.3861	−0.522870	−0.261435	−	0.965221i	$-0.584196\pi$
$757$	−14.3861	−0.522870	−0.261435	+	0.965221i	$0.584196\pi$
$758$	−76.0540	−2.76241
$759$	0	0
$760$	−9.09246	−0.329818
$761$	−37.1200	−1.34560	−0.672800	−	0.739825i	$-0.734908\pi$
$761$	−37.1200	−1.34560	−0.672800	+	0.739825i	$0.734908\pi$
$762$	0	0
$763$	−66.6215	−2.41186
$764$	−64.9596	−2.35016
$765$	0	0
$766$	31.8321	1.15014
$767$	−2.10288	−0.0759306
$768$	0	0
$769$	39.6193	1.42871	0.714354	−	0.699784i	$-0.246720\pi$
$769$	39.6193	1.42871	0.714354	+	0.699784i	$0.246720\pi$
$770$	84.9115	3.06000
$771$	0	0
$772$	39.5405	1.42309
$773$	1.10324	0.0396807	0.0198404	−	0.999803i	$-0.493684\pi$
$773$	1.10324	0.0396807	0.0198404	+	0.999803i	$0.493684\pi$
$774$	0	0
$775$	0.541575	0.0194539
$776$	11.6360	0.417710
$777$	0	0
$778$	41.9067	1.50243
$779$	5.06626	0.181518
$780$	0	0
$781$	15.2122	0.544335
$782$	5.68488	0.203291
$783$	0	0
$784$	−40.2829	−1.43868
$785$	36.7493	1.31164
$786$	0	0
$787$	1.04346	0.0371953	0.0185977	−	0.999827i	$-0.494080\pi$
$787$	1.04346	0.0371953	0.0185977	+	0.999827i	$0.494080\pi$
$788$	8.32149	0.296441
$789$	0	0
$790$	31.5560	1.12271
$791$	−76.1977	−2.70928
$792$	0	0
$793$	8.23452	0.292416
$794$	23.4307	0.831526
$795$	0	0
$796$	20.8571	0.739261
$797$	−47.1482	−1.67008	−0.835038	−	0.550192i	$-0.814555\pi$
$797$	−47.1482	−1.67008	−0.835038	+	0.550192i	$0.814555\pi$
$798$	0	0
$799$	−10.1035	−0.357438
$800$	−2.58496	−0.0913922
$801$	0	0
$802$	−29.1292	−1.02859
$803$	−12.2458	−0.432145
$804$	0	0
$805$	28.1912	0.993609
$806$	−7.44519	−0.262246
$807$	0	0
$808$	7.33601	0.258080
$809$	20.8110	0.731675	0.365837	−	0.930679i	$-0.380783\pi$
$809$	20.8110	0.731675	0.365837	+	0.930679i	$0.380783\pi$
$810$	0	0
$811$	−36.4991	−1.28166	−0.640828	−	0.767685i	$-0.721409\pi$
$811$	−36.4991	−1.28166	−0.640828	+	0.767685i	$0.721409\pi$
$812$	19.6458	0.689432
$813$	0	0
$814$	69.4572	2.43447
$815$	−15.2099	−0.532779
$816$	0	0
$817$	−38.1310	−1.33403
$818$	73.9891	2.58697
$819$	0	0
$820$	−6.16525	−0.215300
$821$	−16.3589	−0.570930	−0.285465	−	0.958389i	$-0.592148\pi$
$821$	−16.3589	−0.570930	−0.285465	+	0.958389i	$0.592148\pi$
$822$	0	0
$823$	4.97004	0.173245	0.0866224	−	0.996241i	$-0.472393\pi$
$823$	4.97004	0.173245	0.0866224	+	0.996241i	$0.472393\pi$
$824$	−4.39550	−0.153125
$825$	0	0
$826$	9.48184	0.329916
$827$	−37.4464	−1.30214	−0.651070	−	0.759018i	$-0.725680\pi$
$827$	−37.4464	−1.30214	−0.651070	+	0.759018i	$0.725680\pi$
$828$	0	0
$829$	8.65185	0.300491	0.150246	−	0.988649i	$-0.451994\pi$
$829$	8.65185	0.300491	0.150246	+	0.988649i	$0.451994\pi$
$830$	75.2201	2.61093
$831$	0	0
$832$	22.8773	0.793127
$833$	13.3834	0.463709
$834$	0	0
$835$	21.6589	0.749538
$836$	−42.7660	−1.47909
$837$	0	0
$838$	−62.1053	−2.14539
$839$	−13.4693	−0.465013	−0.232507	−	0.972595i	$-0.574693\pi$
$839$	−13.4693	−0.465013	−0.232507	+	0.972595i	$0.574693\pi$
$840$	0	0
$841$	−25.7418	−0.887650
$842$	16.8935	0.582188
$843$	0	0
$844$	−44.7454	−1.54020
$845$	−19.7874	−0.680706
$846$	0	0
$847$	18.3783	0.631486
$848$	14.1850	0.487114
$849$	0	0
$850$	0.674692	0.0231417
$851$	23.0603	0.790496
$852$	0	0
$853$	−15.1364	−0.518261	−0.259130	−	0.965842i	$-0.583436\pi$
$853$	−15.1364	−0.518261	−0.259130	+	0.965842i	$0.583436\pi$
$854$	−37.1293	−1.27054
$855$	0	0
$856$	−15.1629	−0.518256
$857$	46.7584	1.59724	0.798618	−	0.601838i	$-0.205565\pi$
$857$	46.7584	1.59724	0.798618	+	0.601838i	$0.205565\pi$
$858$	0	0
$859$	−18.9518	−0.646629	−0.323314	−	0.946292i	$-0.604797\pi$
$859$	−18.9518	−0.646629	−0.323314	+	0.946292i	$0.604797\pi$
$860$	46.4025	1.58231
$861$	0	0
$862$	63.6494	2.16791
$863$	−0.683193	−0.0232562	−0.0116281	−	0.999932i	$-0.503701\pi$
$863$	−0.683193	−0.0232562	−0.0116281	+	0.999932i	$0.503701\pi$
$864$	0	0
$865$	−15.6792	−0.533110
$866$	47.6565	1.61943
$867$	0	0
$868$	18.3481	0.622774
$869$	25.2864	0.857783
$870$	0	0
$871$	−27.5712	−0.934216
$872$	−12.7280	−0.431026
$873$	0	0
$874$	−25.9783	−0.878728
$875$	−48.7276	−1.64729
$876$	0	0
$877$	2.91649	0.0984828	0.0492414	−	0.998787i	$-0.484320\pi$
$877$	2.91649	0.0984828	0.0492414	+	0.998787i	$0.484320\pi$
$878$	−32.4991	−1.09679
$879$	0	0
$880$	−26.9542	−0.908626
$881$	12.0223	0.405043	0.202522	−	0.979278i	$-0.435086\pi$
$881$	12.0223	0.405043	0.202522	+	0.979278i	$0.435086\pi$
$882$	0	0
$883$	−26.0110	−0.875340	−0.437670	−	0.899136i	$-0.644196\pi$
$883$	−26.0110	−0.875340	−0.437670	+	0.899136i	$0.644196\pi$
$884$	−5.06943	−0.170503
$885$	0	0
$886$	−55.0827	−1.85054
$887$	−45.9354	−1.54236	−0.771180	−	0.636617i	$-0.780333\pi$
$887$	−45.9354	−1.54236	−0.771180	+	0.636617i	$0.780333\pi$
$888$	0	0
$889$	4.22362	0.141656
$890$	29.2448	0.980289
$891$	0	0
$892$	−41.0407	−1.37414
$893$	46.1703	1.54503
$894$	0	0
$895$	−7.37759	−0.246606
$896$	−30.4972	−1.01884
$897$	0	0
$898$	55.8064	1.86228
$899$	3.04294	0.101488
$900$	0	0
$901$	−4.71276	−0.157005
$902$	−9.03898	−0.300965
$903$	0	0
$904$	−14.5576	−0.484178
$905$	−38.9611	−1.29511
$906$	0	0
$907$	−36.2034	−1.20211	−0.601057	−	0.799206i	$-0.705253\pi$
$907$	−36.2034	−1.20211	−0.601057	+	0.799206i	$0.705253\pi$
$908$	24.0262	0.797339
$909$	0	0
$910$	−45.9954	−1.52473
$911$	16.7461	0.554821	0.277411	−	0.960751i	$-0.410524\pi$
$911$	16.7461	0.554821	0.277411	+	0.960751i	$0.410524\pi$
$912$	0	0
$913$	60.2752	1.99482
$914$	−42.1089	−1.39284
$915$	0	0
$916$	−14.5169	−0.479651
$917$	−48.3310	−1.59603
$918$	0	0
$919$	29.7233	0.980480	0.490240	−	0.871587i	$-0.336909\pi$
$919$	29.7233	0.980480	0.490240	+	0.871587i	$0.336909\pi$
$920$	5.38593	0.177569
$921$	0	0
$922$	−54.9334	−1.80913
$923$	−8.24025	−0.271231
$924$	0	0
$925$	2.73683	0.0899865
$926$	40.4952	1.33076
$927$	0	0
$928$	−14.5241	−0.476776
$929$	12.5057	0.410298	0.205149	−	0.978731i	$-0.434232\pi$
$929$	12.5057	0.410298	0.205149	+	0.978731i	$0.434232\pi$
$930$	0	0
$931$	−61.1584	−2.00439
$932$	−56.9450	−1.86530
$933$	0	0
$934$	7.11064	0.232667
$935$	8.95516	0.292865
$936$	0	0
$937$	−16.0944	−0.525780	−0.262890	−	0.964826i	$-0.584676\pi$
$937$	−16.0944	−0.525780	−0.262890	+	0.964826i	$0.584676\pi$
$938$	124.318	4.05913
$939$	0	0
$940$	−56.1856	−1.83257
$941$	−48.8862	−1.59365	−0.796823	−	0.604213i	$-0.793488\pi$
$941$	−48.8862	−1.59365	−0.796823	+	0.604213i	$0.793488\pi$
$942$	0	0
$943$	−3.00101	−0.0977262
$944$	−3.00991	−0.0979641
$945$	0	0
$946$	68.0315	2.21190
$947$	36.5527	1.18780	0.593902	−	0.804537i	$-0.297587\pi$
$947$	36.5527	1.18780	0.593902	+	0.804537i	$0.297587\pi$
$948$	0	0
$949$	6.63340	0.215329
$950$	−3.08314	−0.100030
$951$	0	0
$952$	3.89426	0.126214
$953$	20.4996	0.664046	0.332023	−	0.943271i	$-0.392269\pi$
$953$	20.4996	0.664046	0.332023	+	0.943271i	$0.392269\pi$
$954$	0	0
$955$	−62.1593	−2.01143
$956$	32.4510	1.04954
$957$	0	0
$958$	1.47500	0.0476550
$959$	−83.0303	−2.68119
$960$	0	0
$961$	−28.1581	−0.908325
$962$	−37.6241	−1.21305
$963$	0	0
$964$	31.5040	1.01467
$965$	37.8360	1.21798
$966$	0	0
$967$	−36.6475	−1.17850	−0.589252	−	0.807949i	$-0.700577\pi$
$967$	−36.6475	−1.17850	−0.589252	+	0.807949i	$0.700577\pi$
$968$	3.51117	0.112853
$969$	0	0
$970$	65.3553	2.09843
$971$	50.7446	1.62847	0.814235	−	0.580535i	$-0.197157\pi$
$971$	50.7446	1.62847	0.814235	+	0.580535i	$0.197157\pi$
$972$	0	0
$973$	36.4161	1.16745
$974$	−23.9289	−0.766730
$975$	0	0
$976$	11.7863	0.377270
$977$	−27.8266	−0.890253	−0.445126	−	0.895468i	$-0.646841\pi$
$977$	−27.8266	−0.890253	−0.445126	+	0.895468i	$0.646841\pi$
$978$	0	0
$979$	23.4344	0.748968
$980$	74.4251	2.37742
$981$	0	0
$982$	78.2777	2.49794
$983$	−35.2725	−1.12502	−0.562509	−	0.826791i	$-0.690164\pi$
$983$	−35.2725	−1.12502	−0.562509	+	0.826791i	$0.690164\pi$
$984$	0	0
$985$	7.96276	0.253715
$986$	3.79088	0.120726
$987$	0	0
$988$	23.1658	0.737002
$989$	22.5869	0.718223
$990$	0	0
$991$	−15.8594	−0.503789	−0.251894	−	0.967755i	$-0.581054\pi$
$991$	−15.8594	−0.503789	−0.251894	+	0.967755i	$0.581054\pi$
$992$	−13.5647	−0.430679
$993$	0	0
$994$	37.1551	1.17849
$995$	19.9580	0.632711
$996$	0	0
$997$	−49.9654	−1.58242	−0.791210	−	0.611544i	$-0.790549\pi$
$997$	−49.9654	−1.58242	−0.791210	+	0.611544i	$0.790549\pi$
$998$	38.8160	1.22870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9027.2.a.t.1.20		24
3.2	odd	2		3009.2.a.j.1.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3009.2.a.j.1.5	✓	24	3.2	odd	2
9027.2.a.t.1.20		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9027 = 3^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9027.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10017 q^{2} +2.41071 q^{4} +2.30678 q^{5} +4.51480 q^{7} +0.862553 q^{8} +O(q^{10})\)	\(q+2.10017 q^{2} +2.41071 q^{4} +2.30678 q^{5} +4.51480 q^{7} +0.862553 q^{8} +4.84464 q^{10} +3.88210 q^{11} -2.10288 q^{13} +9.48184 q^{14} -3.00991 q^{16} +1.00000 q^{17} -4.56971 q^{19} +5.56098 q^{20} +8.15306 q^{22} +2.70687 q^{23} +0.321256 q^{25} -4.41640 q^{26} +10.8839 q^{28} +1.80504 q^{29} +1.68580 q^{31} -8.04642 q^{32} +2.10017 q^{34} +10.4147 q^{35} +8.51916 q^{37} -9.59715 q^{38} +1.98972 q^{40} -1.10866 q^{41} +8.34430 q^{43} +9.35860 q^{44} +5.68488 q^{46} -10.1035 q^{47} +13.3834 q^{49} +0.674692 q^{50} -5.06943 q^{52} -4.71276 q^{53} +8.95516 q^{55} +3.89426 q^{56} +3.79088 q^{58} +1.00000 q^{59} -3.91583 q^{61} +3.54047 q^{62} -10.8790 q^{64} -4.85089 q^{65} +13.1112 q^{67} +2.41071 q^{68} +21.8726 q^{70} +3.91855 q^{71} -3.15443 q^{73} +17.8917 q^{74} -11.0162 q^{76} +17.5269 q^{77} +6.51359 q^{79} -6.94321 q^{80} -2.32838 q^{82} +15.5265 q^{83} +2.30678 q^{85} +17.5244 q^{86} +3.34851 q^{88} +6.03654 q^{89} -9.49409 q^{91} +6.52547 q^{92} -21.2192 q^{94} -10.5413 q^{95} +13.4902 q^{97} +28.1075 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 34 q^{4} + 3 q^{5} + 19 q^{7} - 3 q^{8}+O(q^{10}) \) 24 * q - 2 * q^2 + 34 * q^4 + 3 * q^5 + 19 * q^7 - 3 * q^8	\( 24 q - 2 q^{2} + 34 q^{4} + 3 q^{5} + 19 q^{7} - 3 q^{8} + 10 q^{10} - 11 q^{11} + 21 q^{13} - 9 q^{14} + 50 q^{16} + 24 q^{17} + 13 q^{19} - q^{20} + 5 q^{22} - 18 q^{23} + 39 q^{25} + 7 q^{26} + 40 q^{28} + 10 q^{29} + 47 q^{31} - 18 q^{32} - 2 q^{34} - 5 q^{35} + 54 q^{37} - 5 q^{38} + 29 q^{40} + 16 q^{43} - 7 q^{44} - 8 q^{46} - 10 q^{47} + 55 q^{49} - 21 q^{50} + 68 q^{52} - 6 q^{53} + 15 q^{55} + 5 q^{56} + q^{58} + 24 q^{59} + 42 q^{61} + 7 q^{62} + 53 q^{64} + 9 q^{65} + 28 q^{67} + 34 q^{68} - 20 q^{70} - 54 q^{71} + 33 q^{73} + 17 q^{74} - 56 q^{76} + 23 q^{77} + 35 q^{79} - 3 q^{80} - 12 q^{82} + 17 q^{83} + 3 q^{85} - 36 q^{86} + 47 q^{88} - 15 q^{89} + 74 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{95} + 52 q^{97} + 48 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 34 * q^4 + 3 * q^5 + 19 * q^7 - 3 * q^8 + 10 * q^10 - 11 * q^11 + 21 * q^13 - 9 * q^14 + 50 * q^16 + 24 * q^17 + 13 * q^19 - q^20 + 5 * q^22 - 18 * q^23 + 39 * q^25 + 7 * q^26 + 40 * q^28 + 10 * q^29 + 47 * q^31 - 18 * q^32 - 2 * q^34 - 5 * q^35 + 54 * q^37 - 5 * q^38 + 29 * q^40 + 16 * q^43 - 7 * q^44 - 8 * q^46 - 10 * q^47 + 55 * q^49 - 21 * q^50 + 68 * q^52 - 6 * q^53 + 15 * q^55 + 5 * q^56 + q^58 + 24 * q^59 + 42 * q^61 + 7 * q^62 + 53 * q^64 + 9 * q^65 + 28 * q^67 + 34 * q^68 - 20 * q^70 - 54 * q^71 + 33 * q^73 + 17 * q^74 - 56 * q^76 + 23 * q^77 + 35 * q^79 - 3 * q^80 - 12 * q^82 + 17 * q^83 + 3 * q^85 - 36 * q^86 + 47 * q^88 - 15 * q^89 + 74 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^95 + 52 * q^97 + 48 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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