LMFDB - Embedded newform 2673.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2673,2,Mod(1,2673)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2673, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2673.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2673 = 3^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2673.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:	$21.3440124603$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.864654912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 14x^2 - 16*x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.480247$ of defining polynomial
Character	$\chi$	$=$	2673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.480247 q^{2} -1.76936 q^{4} -2.76936 q^{5} -3.14826 q^{7} +1.81022 q^{8} +O(q^{10})$	\(q-0.480247 q^{2} -1.76936 q^{4} -2.76936 q^{5} -3.14826 q^{7} +1.81022 q^{8} +1.32998 q^{10} +1.00000 q^{11} +0.0316969 q^{13} +1.51194 q^{14} +2.66937 q^{16} -5.47043 q^{17} -5.28266 q^{19} +4.90001 q^{20} -0.480247 q^{22} -7.06589 q^{23} +2.66937 q^{25} -0.0152223 q^{26} +5.57042 q^{28} +1.17390 q^{29} -10.4137 q^{31} -4.90241 q^{32} +2.62716 q^{34} +8.71869 q^{35} -2.92099 q^{37} +2.53698 q^{38} -5.01317 q^{40} -6.64434 q^{41} -5.43603 q^{43} -1.76936 q^{44} +3.39337 q^{46} +5.85779 q^{47} +2.91157 q^{49} -1.28196 q^{50} -0.0560833 q^{52} +6.81829 q^{53} -2.76936 q^{55} -5.69906 q^{56} -0.563763 q^{58} +3.02389 q^{59} -1.46437 q^{61} +5.00114 q^{62} -2.98438 q^{64} -0.0877802 q^{65} -2.43874 q^{67} +9.67918 q^{68} -4.18712 q^{70} +10.6933 q^{71} -12.4162 q^{73} +1.40279 q^{74} +9.34695 q^{76} -3.14826 q^{77} -3.36167 q^{79} -7.39246 q^{80} +3.19092 q^{82} +10.2047 q^{83} +15.1496 q^{85} +2.61063 q^{86} +1.81022 q^{88} +3.72986 q^{89} -0.0997902 q^{91} +12.5021 q^{92} -2.81319 q^{94} +14.6296 q^{95} -4.75995 q^{97} -1.39827 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.480247	−0.339586	−0.169793	−	0.985480i	$-0.554310\pi$
$2$	−0.480247	−0.339586	−0.169793	+	0.985480i	$0.554310\pi$
$3$	0	0
$3$	0	0
$4$	−1.76936	−0.884682
$5$	−2.76936	−1.23850	−0.619248	−	0.785195i	$-0.712563\pi$
$5$	−2.76936	−1.23850	−0.619248	+	0.785195i	$0.712563\pi$
$6$	0	0
$7$	−3.14826	−1.18993	−0.594966	−	0.803751i	$-0.702835\pi$
$7$	−3.14826	−1.18993	−0.594966	+	0.803751i	$0.702835\pi$
$8$	1.81022	0.640011
$9$	0	0
$10$	1.32998	0.420576
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.0316969	0.00879113	0.00439557	−	0.999990i	$-0.498601\pi$
$13$	0.0316969	0.00879113	0.00439557	+	0.999990i	$0.498601\pi$
$14$	1.51194	0.404084
$15$	0	0
$16$	2.66937	0.667343
$17$	−5.47043	−1.32677	−0.663387	−	0.748276i	$-0.730882\pi$
$17$	−5.47043	−1.32677	−0.663387	+	0.748276i	$0.730882\pi$
$18$	0	0
$19$	−5.28266	−1.21193	−0.605963	−	0.795493i	$-0.707212\pi$
$19$	−5.28266	−1.21193	−0.605963	+	0.795493i	$0.707212\pi$
$20$	4.90001	1.09568
$21$	0	0
$22$	−0.480247	−0.102389
$23$	−7.06589	−1.47334	−0.736670	−	0.676252i	$-0.763603\pi$
$23$	−7.06589	−1.47334	−0.736670	+	0.676252i	$0.763603\pi$
$24$	0	0
$25$	2.66937	0.533874
$26$	−0.0152223	−0.00298534
$27$	0	0
$28$	5.57042	1.05271
$29$	1.17390	0.217988	0.108994	−	0.994042i	$-0.465237\pi$
$29$	1.17390	0.217988	0.108994	+	0.994042i	$0.465237\pi$
$30$	0	0
$31$	−10.4137	−1.87036	−0.935178	−	0.354179i	$-0.884760\pi$
$31$	−10.4137	−1.87036	−0.935178	+	0.354179i	$0.884760\pi$
$32$	−4.90241	−0.866631
$33$	0	0
$34$	2.62716	0.450554
$35$	8.71869	1.47373
$36$	0	0
$37$	−2.92099	−0.480207	−0.240104	−	0.970747i	$-0.577181\pi$
$37$	−2.92099	−0.480207	−0.240104	+	0.970747i	$0.577181\pi$
$38$	2.53698	0.411553
$39$	0	0
$40$	−5.01317	−0.792651
$41$	−6.64434	−1.03767	−0.518835	−	0.854874i	$-0.673634\pi$
$41$	−6.64434	−1.03767	−0.518835	+	0.854874i	$0.673634\pi$
$42$	0	0
$43$	−5.43603	−0.828986	−0.414493	−	0.910052i	$-0.636041\pi$
$43$	−5.43603	−0.828986	−0.414493	+	0.910052i	$0.636041\pi$
$44$	−1.76936	−0.266742
$45$	0	0
$46$	3.39337	0.500325
$47$	5.85779	0.854447	0.427224	−	0.904146i	$-0.359492\pi$
$47$	5.85779	0.854447	0.427224	+	0.904146i	$0.359492\pi$
$48$	0	0
$49$	2.91157	0.415938
$50$	−1.28196	−0.181296
$51$	0	0
$52$	−0.0560833	−0.00777735
$53$	6.81829	0.936564	0.468282	−	0.883579i	$-0.344873\pi$
$53$	6.81829	0.936564	0.468282	+	0.883579i	$0.344873\pi$
$54$	0	0
$55$	−2.76936	−0.373421
$56$	−5.69906	−0.761570
$57$	0	0
$58$	−0.563763	−0.0740257
$59$	3.02389	0.393677	0.196838	−	0.980436i	$-0.436933\pi$
$59$	3.02389	0.393677	0.196838	+	0.980436i	$0.436933\pi$
$60$	0	0
$61$	−1.46437	−0.187494	−0.0937469	−	0.995596i	$-0.529884\pi$
$61$	−1.46437	−0.187494	−0.0937469	+	0.995596i	$0.529884\pi$
$62$	5.00114	0.635146
$63$	0	0
$64$	−2.98438	−0.373048
$65$	−0.0877802	−0.0108878
$66$	0	0
$67$	−2.43874	−0.297939	−0.148969	−	0.988842i	$-0.547596\pi$
$67$	−2.43874	−0.297939	−0.148969	+	0.988842i	$0.547596\pi$
$68$	9.67918	1.17377
$69$	0	0
$70$	−4.18712	−0.500457
$71$	10.6933	1.26906	0.634528	−	0.772900i	$-0.281194\pi$
$71$	10.6933	1.26906	0.634528	+	0.772900i	$0.281194\pi$
$72$	0	0
$73$	−12.4162	−1.45321	−0.726603	−	0.687058i	$-0.758902\pi$
$73$	−12.4162	−1.45321	−0.726603	+	0.687058i	$0.758902\pi$
$74$	1.40279	0.163072
$75$	0	0
$76$	9.34695	1.07217
$77$	−3.14826	−0.358778
$78$	0	0
$79$	−3.36167	−0.378218	−0.189109	−	0.981956i	$-0.560560\pi$
$79$	−3.36167	−0.378218	−0.189109	+	0.981956i	$0.560560\pi$
$80$	−7.39246	−0.826502
$81$	0	0
$82$	3.19092	0.352378
$83$	10.2047	1.12012	0.560058	−	0.828454i	$-0.310779\pi$
$83$	10.2047	1.12012	0.560058	+	0.828454i	$0.310779\pi$
$84$	0	0
$85$	15.1496	1.64321
$86$	2.61063	0.281512
$87$	0	0
$88$	1.81022	0.192971
$89$	3.72986	0.395364	0.197682	−	0.980266i	$-0.436659\pi$
$89$	3.72986	0.395364	0.197682	+	0.980266i	$0.436659\pi$
$90$	0	0
$91$	−0.0997902	−0.0104609
$92$	12.5021	1.30344
$93$	0	0
$94$	−2.81319	−0.290158
$95$	14.6296	1.50097
$96$	0	0
$97$	−4.75995	−0.483299	−0.241650	−	0.970364i	$-0.577688\pi$
$97$	−4.75995	−0.483299	−0.241650	+	0.970364i	$0.577688\pi$
$98$	−1.39827	−0.141247
$99$	0	0
$100$	−4.72309	−0.472309
$101$	−8.43969	−0.839781	−0.419890	−	0.907575i	$-0.637931\pi$
$101$	−8.43969	−0.839781	−0.419890	+	0.907575i	$0.637931\pi$
$102$	0	0
$103$	16.9869	1.67377	0.836883	−	0.547381i	$-0.184375\pi$
$103$	16.9869	1.67377	0.836883	+	0.547381i	$0.184375\pi$
$104$	0.0573785	0.00562642
$105$	0	0
$106$	−3.27446	−0.318044
$107$	18.6553	1.80348	0.901738	−	0.432284i	$-0.142292\pi$
$107$	18.6553	1.80348	0.901738	+	0.432284i	$0.142292\pi$
$108$	0	0
$109$	−2.11948	−0.203009	−0.101505	−	0.994835i	$-0.532366\pi$
$109$	−2.11948	−0.203009	−0.101505	+	0.994835i	$0.532366\pi$
$110$	1.32998	0.126808
$111$	0	0
$112$	−8.40389	−0.794093
$113$	−17.3648	−1.63354	−0.816770	−	0.576963i	$-0.804238\pi$
$113$	−17.3648	−1.63354	−0.816770	+	0.576963i	$0.804238\pi$
$114$	0	0
$115$	19.5680	1.82473
$116$	−2.07706	−0.192850
$117$	0	0
$118$	−1.45221	−0.133687
$119$	17.2224	1.57877
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.703261	0.0636702
$123$	0	0
$124$	18.4256	1.65467
$125$	6.45435	0.577295
$126$	0	0
$127$	5.19693	0.461153	0.230576	−	0.973054i	$-0.425939\pi$
$127$	5.19693	0.461153	0.230576	+	0.973054i	$0.425939\pi$
$128$	11.2380	0.993313
$129$	0	0
$130$	0.0421561	0.00369734
$131$	0.733416	0.0640789	0.0320394	−	0.999487i	$-0.489800\pi$
$131$	0.733416	0.0640789	0.0320394	+	0.999487i	$0.489800\pi$
$132$	0	0
$133$	16.6312	1.44211
$134$	1.17119	0.101176
$135$	0	0
$136$	−9.90271	−0.849150
$137$	5.54143	0.473437	0.236718	−	0.971578i	$-0.423928\pi$
$137$	5.54143	0.473437	0.236718	+	0.971578i	$0.423928\pi$
$138$	0	0
$139$	4.89705	0.415362	0.207681	−	0.978197i	$-0.433408\pi$
$139$	4.89705	0.415362	0.207681	+	0.978197i	$0.433408\pi$
$140$	−15.4265	−1.30378
$141$	0	0
$142$	−5.13540	−0.430953
$143$	0.0316969	0.00265063
$144$	0	0
$145$	−3.25096	−0.269978
$146$	5.96284	0.493488
$147$	0	0
$148$	5.16829	0.424831
$149$	19.1228	1.56660	0.783299	−	0.621645i	$-0.213535\pi$
$149$	19.1228	1.56660	0.783299	+	0.621645i	$0.213535\pi$
$150$	0	0
$151$	−17.0152	−1.38468	−0.692339	−	0.721572i	$-0.743420\pi$
$151$	−17.0152	−1.38468	−0.692339	+	0.721572i	$0.743420\pi$
$152$	−9.56280	−0.775646
$153$	0	0
$154$	1.51194	0.121836
$155$	28.8393	2.31643
$156$	0	0
$157$	−5.72335	−0.456773	−0.228386	−	0.973571i	$-0.573345\pi$
$157$	−5.72335	−0.456773	−0.228386	+	0.973571i	$0.573345\pi$
$158$	1.61443	0.128437
$159$	0	0
$160$	13.5765	1.07332
$161$	22.2453	1.75318
$162$	0	0
$163$	19.6734	1.54094	0.770471	−	0.637475i	$-0.220021\pi$
$163$	19.6734	1.54094	0.770471	+	0.637475i	$0.220021\pi$
$164$	11.7562	0.918008
$165$	0	0
$166$	−4.90079	−0.380375
$167$	−10.9382	−0.846420	−0.423210	−	0.906032i	$-0.639097\pi$
$167$	−10.9382	−0.846420	−0.423210	+	0.906032i	$0.639097\pi$
$168$	0	0
$169$	−12.9990	−0.999923
$170$	−7.27555	−0.558009
$171$	0	0
$172$	9.61831	0.733389
$173$	4.36211	0.331645	0.165823	−	0.986156i	$-0.446972\pi$
$173$	4.36211	0.331645	0.165823	+	0.986156i	$0.446972\pi$
$174$	0	0
$175$	−8.40389	−0.635274
$176$	2.66937	0.201211
$177$	0	0
$178$	−1.79125	−0.134260
$179$	−23.1555	−1.73072	−0.865361	−	0.501150i	$-0.832911\pi$
$179$	−23.1555	−1.73072	−0.865361	+	0.501150i	$0.832911\pi$
$180$	0	0
$181$	14.4068	1.07085	0.535424	−	0.844584i	$-0.320152\pi$
$181$	14.4068	1.07085	0.535424	+	0.844584i	$0.320152\pi$
$182$	0.0479239	0.00355236
$183$	0	0
$184$	−12.7908	−0.942954
$185$	8.08927	0.594735
$186$	0	0
$187$	−5.47043	−0.400038
$188$	−10.3646	−0.755914
$189$	0	0
$190$	−7.02582	−0.509707
$191$	−17.8669	−1.29280	−0.646402	−	0.762997i	$-0.723727\pi$
$191$	−17.8669	−1.29280	−0.646402	+	0.762997i	$0.723727\pi$
$192$	0	0
$193$	16.3284	1.17535	0.587673	−	0.809099i	$-0.300044\pi$
$193$	16.3284	1.17535	0.587673	+	0.809099i	$0.300044\pi$
$194$	2.28595	0.164122
$195$	0	0
$196$	−5.15162	−0.367973
$197$	−23.0639	−1.64323	−0.821617	−	0.570040i	$-0.806928\pi$
$197$	−23.0639	−1.64323	−0.821617	+	0.570040i	$0.806928\pi$
$198$	0	0
$199$	−22.6992	−1.60911	−0.804554	−	0.593880i	$-0.797595\pi$
$199$	−22.6992	−1.60911	−0.804554	+	0.593880i	$0.797595\pi$
$200$	4.83216	0.341685
$201$	0	0
$202$	4.05313	0.285178
$203$	−3.69576	−0.259391
$204$	0	0
$205$	18.4006	1.28515
$206$	−8.15789	−0.568387
$207$	0	0
$208$	0.0846108	0.00586670
$209$	−5.28266	−0.365409
$210$	0	0
$211$	−17.4125	−1.19873	−0.599364	−	0.800477i	$-0.704580\pi$
$211$	−17.4125	−1.19873	−0.599364	+	0.800477i	$0.704580\pi$
$212$	−12.0640	−0.828561
$213$	0	0
$214$	−8.95914	−0.612434
$215$	15.0543	1.02670
$216$	0	0
$217$	32.7851	2.22560
$218$	1.01787	0.0689390
$219$	0	0
$220$	4.90001	0.330359
$221$	−0.173396	−0.0116639
$222$	0	0
$223$	−2.81850	−0.188741	−0.0943703	−	0.995537i	$-0.530084\pi$
$223$	−2.81850	−0.188741	−0.0943703	+	0.995537i	$0.530084\pi$
$224$	15.4341	1.03123
$225$	0	0
$226$	8.33938	0.554727
$227$	−10.1060	−0.670759	−0.335379	−	0.942083i	$-0.608865\pi$
$227$	−10.1060	−0.670759	−0.335379	+	0.942083i	$0.608865\pi$
$228$	0	0
$229$	8.69847	0.574811	0.287405	−	0.957809i	$-0.407207\pi$
$229$	8.69847	0.574811	0.287405	+	0.957809i	$0.407207\pi$
$230$	−9.39748	−0.619651
$231$	0	0
$232$	2.12503	0.139515
$233$	12.3695	0.810352	0.405176	−	0.914239i	$-0.367210\pi$
$233$	12.3695	0.810352	0.405176	+	0.914239i	$0.367210\pi$
$234$	0	0
$235$	−16.2224	−1.05823
$236$	−5.35035	−0.348278
$237$	0	0
$238$	−8.27098	−0.536128
$239$	30.3432	1.96274	0.981371	−	0.192124i	$-0.0615377\pi$
$239$	30.3432	1.96274	0.981371	+	0.192124i	$0.0615377\pi$
$240$	0	0
$241$	−25.3909	−1.63557	−0.817787	−	0.575520i	$-0.804800\pi$
$241$	−25.3909	−1.63557	−0.817787	+	0.575520i	$0.804800\pi$
$242$	−0.480247	−0.0308714
$243$	0	0
$244$	2.59101	0.165872
$245$	−8.06319	−0.515139
$246$	0	0
$247$	−0.167444	−0.0106542
$248$	−18.8511	−1.19705
$249$	0	0
$250$	−3.09968	−0.196041
$251$	−30.6482	−1.93450	−0.967248	−	0.253833i	$-0.918309\pi$
$251$	−30.6482	−1.93450	−0.967248	+	0.253833i	$0.918309\pi$
$252$	0	0
$253$	−7.06589	−0.444229
$254$	−2.49581	−0.156601
$255$	0	0
$256$	0.571725	0.0357328
$257$	1.02233	0.0637711	0.0318855	−	0.999492i	$-0.489849\pi$
$257$	1.02233	0.0637711	0.0318855	+	0.999492i	$0.489849\pi$
$258$	0	0
$259$	9.19604	0.571414
$260$	0.155315	0.00963223
$261$	0	0
$262$	−0.352221	−0.0217603
$263$	27.0072	1.66534	0.832668	−	0.553773i	$-0.186812\pi$
$263$	27.0072	1.66534	0.832668	+	0.553773i	$0.186812\pi$
$264$	0	0
$265$	−18.8823	−1.15993
$266$	−7.98709	−0.489720
$267$	0	0
$268$	4.31501	0.263581
$269$	27.9060	1.70146	0.850729	−	0.525604i	$-0.176161\pi$
$269$	27.9060	1.70146	0.850729	+	0.525604i	$0.176161\pi$
$270$	0	0
$271$	−25.2139	−1.53164	−0.765818	−	0.643058i	$-0.777665\pi$
$271$	−25.2139	−1.53164	−0.765818	+	0.643058i	$0.777665\pi$
$272$	−14.6026	−0.885414
$273$	0	0
$274$	−2.66126	−0.160772
$275$	2.66937	0.160969
$276$	0	0
$277$	−12.2224	−0.734371	−0.367185	−	0.930148i	$-0.619679\pi$
$277$	−12.2224	−0.734371	−0.367185	+	0.930148i	$0.619679\pi$
$278$	−2.35179	−0.141051
$279$	0	0
$280$	15.7828	0.943201
$281$	0.949535	0.0566445	0.0283222	−	0.999599i	$-0.490984\pi$
$281$	0.949535	0.0566445	0.0283222	+	0.999599i	$0.490984\pi$
$282$	0	0
$283$	−2.54048	−0.151016	−0.0755079	−	0.997145i	$-0.524058\pi$
$283$	−2.54048	−0.151016	−0.0755079	+	0.997145i	$0.524058\pi$
$284$	−18.9203	−1.12271
$285$	0	0
$286$	−0.0152223	−0.000900115 0
$287$	20.9181	1.23476
$288$	0	0
$289$	12.9256	0.760331
$290$	1.56126	0.0916806
$291$	0	0
$292$	21.9688	1.28562
$293$	−4.83230	−0.282306	−0.141153	−	0.989988i	$-0.545081\pi$
$293$	−4.83230	−0.282306	−0.141153	+	0.989988i	$0.545081\pi$
$294$	0	0
$295$	−8.37424	−0.487567
$296$	−5.28764	−0.307338
$297$	0	0
$298$	−9.18365	−0.531994
$299$	−0.223967	−0.0129523
$300$	0	0
$301$	17.1140	0.986437
$302$	8.17150	0.470217
$303$	0	0
$304$	−14.1014	−0.808770
$305$	4.05538	0.232211
$306$	0	0
$307$	3.36653	0.192138	0.0960689	−	0.995375i	$-0.469373\pi$
$307$	3.36653	0.192138	0.0960689	+	0.995375i	$0.469373\pi$
$308$	5.57042	0.317404
$309$	0	0
$310$	−13.8500	−0.786626
$311$	−8.94952	−0.507480	−0.253740	−	0.967272i	$-0.581661\pi$
$311$	−8.94952	−0.507480	−0.253740	+	0.967272i	$0.581661\pi$
$312$	0	0
$313$	4.60598	0.260345	0.130173	−	0.991491i	$-0.458447\pi$
$313$	4.60598	0.260345	0.130173	+	0.991491i	$0.458447\pi$
$314$	2.74862	0.155114
$315$	0	0
$316$	5.94802	0.334602
$317$	−5.21637	−0.292980	−0.146490	−	0.989212i	$-0.546798\pi$
$317$	−5.21637	−0.292980	−0.146490	+	0.989212i	$0.546798\pi$
$318$	0	0
$319$	1.17390	0.0657260
$320$	8.26483	0.462018
$321$	0	0
$322$	−10.6832	−0.595353
$323$	28.8984	1.60795
$324$	0	0
$325$	0.0846108	0.00469336
$326$	−9.44810	−0.523282
$327$	0	0
$328$	−12.0277	−0.664121
$329$	−18.4419	−1.01673
$330$	0	0
$331$	14.0896	0.774433	0.387216	−	0.921989i	$-0.373437\pi$
$331$	14.0896	0.774433	0.387216	+	0.921989i	$0.373437\pi$
$332$	−18.0559	−0.990946
$333$	0	0
$334$	5.25301	0.287432
$335$	6.75374	0.368996
$336$	0	0
$337$	−24.6466	−1.34258	−0.671292	−	0.741193i	$-0.734260\pi$
$337$	−24.6466	−1.34258	−0.671292	+	0.741193i	$0.734260\pi$
$338$	6.24272	0.339559
$339$	0	0
$340$	−26.8052	−1.45371
$341$	−10.4137	−0.563933
$342$	0	0
$343$	12.8715	0.694994
$344$	−9.84043	−0.530560
$345$	0	0
$346$	−2.09489	−0.112622
$347$	16.2938	0.874698	0.437349	−	0.899292i	$-0.355917\pi$
$347$	16.2938	0.874698	0.437349	+	0.899292i	$0.355917\pi$
$348$	0	0
$349$	−5.75489	−0.308052	−0.154026	−	0.988067i	$-0.549224\pi$
$349$	−5.75489	−0.308052	−0.154026	+	0.988067i	$0.549224\pi$
$350$	4.03594	0.215730
$351$	0	0
$352$	−4.90241	−0.261299
$353$	−12.5239	−0.666579	−0.333290	−	0.942824i	$-0.608159\pi$
$353$	−12.5239	−0.666579	−0.333290	+	0.942824i	$0.608159\pi$
$354$	0	0
$355$	−29.6135	−1.57172
$356$	−6.59947	−0.349771
$357$	0	0
$358$	11.1203	0.587728
$359$	12.7086	0.670735	0.335368	−	0.942087i	$-0.391139\pi$
$359$	12.7086	0.670735	0.335368	+	0.942087i	$0.391139\pi$
$360$	0	0
$361$	8.90651	0.468764
$362$	−6.91881	−0.363644
$363$	0	0
$364$	0.176565	0.00925452
$365$	34.3850	1.79979
$366$	0	0
$367$	24.7112	1.28991	0.644956	−	0.764220i	$-0.276876\pi$
$367$	24.7112	1.28991	0.644956	+	0.764220i	$0.276876\pi$
$368$	−18.8615	−0.983223
$369$	0	0
$370$	−3.88485	−0.201964
$371$	−21.4658	−1.11445
$372$	0	0
$373$	5.42672	0.280985	0.140492	−	0.990082i	$-0.455131\pi$
$373$	5.42672	0.280985	0.140492	+	0.990082i	$0.455131\pi$
$374$	2.62716	0.135847
$375$	0	0
$376$	10.6039	0.546855
$377$	0.0372091	0.00191636
$378$	0	0
$379$	10.7285	0.551086	0.275543	−	0.961289i	$-0.411142\pi$
$379$	10.7285	0.551086	0.275543	+	0.961289i	$0.411142\pi$
$380$	−25.8851	−1.32788
$381$	0	0
$382$	8.58052	0.439018
$383$	−15.7118	−0.802835	−0.401417	−	0.915895i	$-0.631482\pi$
$383$	−15.7118	−0.802835	−0.401417	+	0.915895i	$0.631482\pi$
$384$	0	0
$385$	8.71869	0.444345
$386$	−7.84167	−0.399130
$387$	0	0
$388$	8.42207	0.427566
$389$	−28.6915	−1.45472	−0.727358	−	0.686258i	$-0.759252\pi$
$389$	−28.6915	−1.45472	−0.727358	+	0.686258i	$0.759252\pi$
$390$	0	0
$391$	38.6535	1.95479
$392$	5.27059	0.266205
$393$	0	0
$394$	11.0764	0.558019
$395$	9.30970	0.468422
$396$	0	0
$397$	27.9886	1.40471	0.702355	−	0.711827i	$-0.252132\pi$
$397$	27.9886	1.40471	0.702355	+	0.711827i	$0.252132\pi$
$398$	10.9012	0.546430
$399$	0	0
$400$	7.12555	0.356277
$401$	−11.0860	−0.553607	−0.276804	−	0.960926i	$-0.589275\pi$
$401$	−11.0860	−0.553607	−0.276804	+	0.960926i	$0.589275\pi$
$402$	0	0
$403$	−0.330082	−0.0164425
$404$	14.9329	0.742939
$405$	0	0
$406$	1.77488	0.0880856
$407$	−2.92099	−0.144788
$408$	0	0
$409$	4.73566	0.234163	0.117082	−	0.993122i	$-0.462646\pi$
$409$	4.73566	0.234163	0.117082	+	0.993122i	$0.462646\pi$
$410$	−8.83682	−0.436419
$411$	0	0
$412$	−30.0560	−1.48075
$413$	−9.52000	−0.468448
$414$	0	0
$415$	−28.2606	−1.38726
$416$	−0.155391	−0.00761867
$417$	0	0
$418$	2.53698	0.124088
$419$	−2.30679	−0.112694	−0.0563470	−	0.998411i	$-0.517945\pi$
$419$	−2.30679	−0.112694	−0.0563470	+	0.998411i	$0.517945\pi$
$420$	0	0
$421$	−9.78976	−0.477124	−0.238562	−	0.971127i	$-0.576676\pi$
$421$	−9.78976	−0.477124	−0.238562	+	0.971127i	$0.576676\pi$
$422$	8.36231	0.407071
$423$	0	0
$424$	12.3426	0.599411
$425$	−14.6026	−0.708331
$426$	0	0
$427$	4.61024	0.223105
$428$	−33.0080	−1.59550
$429$	0	0
$430$	−7.22979	−0.348652
$431$	−19.8453	−0.955916	−0.477958	−	0.878383i	$-0.658623\pi$
$431$	−19.8453	−0.955916	−0.477958	+	0.878383i	$0.658623\pi$
$432$	0	0
$433$	2.58364	0.124162	0.0620809	−	0.998071i	$-0.480226\pi$
$433$	2.58364	0.124162	0.0620809	+	0.998071i	$0.480226\pi$
$434$	−15.7449	−0.755781
$435$	0	0
$436$	3.75012	0.179598
$437$	37.3267	1.78558
$438$	0	0
$439$	13.1250	0.626420	0.313210	−	0.949684i	$-0.398596\pi$
$439$	13.1250	0.626420	0.313210	+	0.949684i	$0.398596\pi$
$440$	−5.01317	−0.238993
$441$	0	0
$442$	0.0832727	0.00396088
$443$	−13.6522	−0.648636	−0.324318	−	0.945948i	$-0.605135\pi$
$443$	−13.6522	−0.648636	−0.324318	+	0.945948i	$0.605135\pi$
$444$	0	0
$445$	−10.3293	−0.489657
$446$	1.35357	0.0640936
$447$	0	0
$448$	9.39562	0.443901
$449$	16.4178	0.774802	0.387401	−	0.921911i	$-0.373373\pi$
$449$	16.4178	0.774802	0.387401	+	0.921911i	$0.373373\pi$
$450$	0	0
$451$	−6.64434	−0.312870
$452$	30.7246	1.44516
$453$	0	0
$454$	4.85337	0.227780
$455$	0.276355	0.0129557
$456$	0	0
$457$	3.92594	0.183648	0.0918238	−	0.995775i	$-0.470730\pi$
$457$	3.92594	0.183648	0.0918238	+	0.995775i	$0.470730\pi$
$458$	−4.17741	−0.195198
$459$	0	0
$460$	−34.6229	−1.61430
$461$	−14.0884	−0.656163	−0.328082	−	0.944649i	$-0.606402\pi$
$461$	−14.0884	−0.656163	−0.328082	+	0.944649i	$0.606402\pi$
$462$	0	0
$463$	39.5825	1.83955	0.919777	−	0.392442i	$-0.128369\pi$
$463$	39.5825	1.83955	0.919777	+	0.392442i	$0.128369\pi$
$464$	3.13358	0.145473
$465$	0	0
$466$	−5.94040	−0.275184
$467$	−11.8206	−0.546994	−0.273497	−	0.961873i	$-0.588180\pi$
$467$	−11.8206	−0.546994	−0.273497	+	0.961873i	$0.588180\pi$
$468$	0	0
$469$	7.67778	0.354527
$470$	7.79073	0.359360
$471$	0	0
$472$	5.47391	0.251957
$473$	−5.43603	−0.249949
$474$	0	0
$475$	−14.1014	−0.647016
$476$	−30.4726	−1.39671
$477$	0	0
$478$	−14.5722	−0.666519
$479$	19.6955	0.899912	0.449956	−	0.893051i	$-0.351440\pi$
$479$	19.6955	0.899912	0.449956	+	0.893051i	$0.351440\pi$
$480$	0	0
$481$	−0.0925862	−0.00422157
$482$	12.1939	0.555418
$483$	0	0
$484$	−1.76936	−0.0804256
$485$	13.1820	0.598565
$486$	0	0
$487$	−41.8405	−1.89597	−0.947986	−	0.318312i	$-0.896884\pi$
$487$	−41.8405	−1.89597	−0.947986	+	0.318312i	$0.896884\pi$
$488$	−2.65085	−0.119998
$489$	0	0
$490$	3.87232	0.174934
$491$	−10.7080	−0.483244	−0.241622	−	0.970370i	$-0.577679\pi$
$491$	−10.7080	−0.483244	−0.241622	+	0.970370i	$0.577679\pi$
$492$	0	0
$493$	−6.42176	−0.289221
$494$	0.0804144	0.00361801
$495$	0	0
$496$	−27.7980	−1.24817
$497$	−33.6652	−1.51009
$498$	0	0
$499$	22.6270	1.01292	0.506461	−	0.862263i	$-0.330953\pi$
$499$	22.6270	1.01292	0.506461	+	0.862263i	$0.330953\pi$
$500$	−11.4201	−0.510722
$501$	0	0
$502$	14.7187	0.656927
$503$	−10.8123	−0.482095	−0.241047	−	0.970513i	$-0.577491\pi$
$503$	−10.8123	−0.482095	−0.241047	+	0.970513i	$0.577491\pi$
$504$	0	0
$505$	23.3726	1.04007
$506$	3.39337	0.150854
$507$	0	0
$508$	−9.19526	−0.407973
$509$	20.8877	0.925830	0.462915	−	0.886403i	$-0.346803\pi$
$509$	20.8877	0.925830	0.462915	+	0.886403i	$0.346803\pi$
$510$	0	0
$511$	39.0895	1.72922
$512$	−22.7507	−1.00545
$513$	0	0
$514$	−0.490970	−0.0216557
$515$	−47.0428	−2.07296
$516$	0	0
$517$	5.85779	0.257625
$518$	−4.41637	−0.194044
$519$	0	0
$520$	−0.158902	−0.00696831
$521$	−17.3227	−0.758921	−0.379461	−	0.925208i	$-0.623890\pi$
$521$	−17.3227	−0.758921	−0.379461	+	0.925208i	$0.623890\pi$
$522$	0	0
$523$	9.01522	0.394208	0.197104	−	0.980383i	$-0.436846\pi$
$523$	9.01522	0.394208	0.197104	+	0.980383i	$0.436846\pi$
$524$	−1.29768	−0.0566894
$525$	0	0
$526$	−12.9701	−0.565524
$527$	56.9674	2.48154
$528$	0	0
$529$	26.9268	1.17073
$530$	9.06817	0.393896
$531$	0	0
$532$	−29.4267	−1.27581
$533$	−0.210605	−0.00912230
$534$	0	0
$535$	−51.6633	−2.23360
$536$	−4.41466	−0.190684
$537$	0	0
$538$	−13.4018	−0.577791
$539$	2.91157	0.125410
$540$	0	0
$541$	36.7209	1.57876	0.789378	−	0.613907i	$-0.210403\pi$
$541$	36.7209	1.57876	0.789378	+	0.613907i	$0.210403\pi$
$542$	12.1089	0.520121
$543$	0	0
$544$	26.8183	1.14982
$545$	5.86960	0.251426
$546$	0	0
$547$	43.3238	1.85239	0.926195	−	0.377044i	$-0.123060\pi$
$547$	43.3238	1.85239	0.926195	+	0.377044i	$0.123060\pi$
$548$	−9.80481	−0.418841
$549$	0	0
$550$	−1.28196	−0.0546628
$551$	−6.20133	−0.264186
$552$	0	0
$553$	10.5834	0.450054
$554$	5.86975	0.249382
$555$	0	0
$556$	−8.66466	−0.367463
$557$	1.04347	0.0442133	0.0221066	−	0.999756i	$-0.492963\pi$
$557$	1.04347	0.0442133	0.0221066	+	0.999756i	$0.492963\pi$
$558$	0	0
$559$	−0.172305	−0.00728773
$560$	23.2734	0.983482
$561$	0	0
$562$	−0.456011	−0.0192357
$563$	31.5745	1.33071	0.665353	−	0.746529i	$-0.268281\pi$
$563$	31.5745	1.33071	0.665353	+	0.746529i	$0.268281\pi$
$564$	0	0
$565$	48.0894	2.02313
$566$	1.22006	0.0512828
$567$	0	0
$568$	19.3572	0.812210
$569$	−0.0268909	−0.00112732	−0.000563662	−	1.00000i	$-0.500179\pi$
$569$	−0.0268909	−0.00112732	−0.000563662		1.00000i	$0.500179\pi$
$570$	0	0
$571$	−27.1764	−1.13730	−0.568648	−	0.822581i	$-0.692533\pi$
$571$	−27.1764	−1.13730	−0.568648	+	0.822581i	$0.692533\pi$
$572$	−0.0560833	−0.00234496
$573$	0	0
$574$	−10.0459	−0.419306
$575$	−18.8615	−0.786579
$576$	0	0
$577$	−13.4957	−0.561834	−0.280917	−	0.959732i	$-0.590639\pi$
$577$	−13.4957	−0.561834	−0.280917	+	0.959732i	$0.590639\pi$
$578$	−6.20749	−0.258198
$579$	0	0
$580$	5.75214	0.238844
$581$	−32.1272	−1.33286
$582$	0	0
$583$	6.81829	0.282385
$584$	−22.4761	−0.930068
$585$	0	0
$586$	2.32069	0.0958670
$587$	19.8904	0.820964	0.410482	−	0.911869i	$-0.365360\pi$
$587$	19.8904	0.820964	0.410482	+	0.911869i	$0.365360\pi$
$588$	0	0
$589$	55.0120	2.26673
$590$	4.02170	0.165571
$591$	0	0
$592$	−7.79720	−0.320463
$593$	−31.5692	−1.29639	−0.648195	−	0.761474i	$-0.724476\pi$
$593$	−31.5692	−1.29639	−0.648195	+	0.761474i	$0.724476\pi$
$594$	0	0
$595$	−47.6950	−1.95530
$596$	−33.8351	−1.38594
$597$	0	0
$598$	0.107559	0.00439843
$599$	21.7910	0.890355	0.445178	−	0.895442i	$-0.353141\pi$
$599$	21.7910	0.890355	0.445178	+	0.895442i	$0.353141\pi$
$600$	0	0
$601$	−39.7207	−1.62024	−0.810120	−	0.586264i	$-0.800598\pi$
$601$	−39.7207	−1.62024	−0.810120	+	0.586264i	$0.800598\pi$
$602$	−8.21897	−0.334980
$603$	0	0
$604$	30.1061	1.22500
$605$	−2.76936	−0.112591
$606$	0	0
$607$	−32.7113	−1.32771	−0.663855	−	0.747862i	$-0.731081\pi$
$607$	−32.7113	−1.32771	−0.663855	+	0.747862i	$0.731081\pi$
$608$	25.8977	1.05029
$609$	0	0
$610$	−1.94758	−0.0788554
$611$	0.185674	0.00751156
$612$	0	0
$613$	3.48321	0.140685	0.0703427	−	0.997523i	$-0.477591\pi$
$613$	3.48321	0.140685	0.0703427	+	0.997523i	$0.477591\pi$
$614$	−1.61676	−0.0652472
$615$	0	0
$616$	−5.69906	−0.229622
$617$	−2.72835	−0.109839	−0.0549195	−	0.998491i	$-0.517490\pi$
$617$	−2.72835	−0.109839	−0.0549195	+	0.998491i	$0.517490\pi$
$618$	0	0
$619$	−14.7930	−0.594582	−0.297291	−	0.954787i	$-0.596083\pi$
$619$	−14.7930	−0.594582	−0.297291	+	0.954787i	$0.596083\pi$
$620$	−51.0272	−2.04930
$621$	0	0
$622$	4.29797	0.172333
$623$	−11.7426	−0.470456
$624$	0	0
$625$	−31.2213	−1.24885
$626$	−2.21201	−0.0884095
$627$	0	0
$628$	10.1267	0.404099
$629$	15.9791	0.637127
$630$	0	0
$631$	31.4299	1.25120	0.625602	−	0.780143i	$-0.284854\pi$
$631$	31.4299	1.25120	0.625602	+	0.780143i	$0.284854\pi$
$632$	−6.08538	−0.242064
$633$	0	0
$634$	2.50514	0.0994920
$635$	−14.3922	−0.571136
$636$	0	0
$637$	0.0922877	0.00365657
$638$	−0.563763	−0.0223196
$639$	0	0
$640$	−31.1222	−1.23021
$641$	−10.5348	−0.416101	−0.208050	−	0.978118i	$-0.566712\pi$
$641$	−10.5348	−0.416101	−0.208050	+	0.978118i	$0.566712\pi$
$642$	0	0
$643$	−1.80752	−0.0712816	−0.0356408	−	0.999365i	$-0.511347\pi$
$643$	−1.80752	−0.0712816	−0.0356408	+	0.999365i	$0.511347\pi$
$644$	−39.3600	−1.55100
$645$	0	0
$646$	−13.8784	−0.546038
$647$	27.2395	1.07090	0.535449	−	0.844568i	$-0.320143\pi$
$647$	27.2395	1.07090	0.535449	+	0.844568i	$0.320143\pi$
$648$	0	0
$649$	3.02389	0.118698
$650$	−0.0406340	−0.00159380
$651$	0	0
$652$	−34.8094	−1.36324
$653$	−9.02790	−0.353289	−0.176644	−	0.984275i	$-0.556524\pi$
$653$	−9.02790	−0.353289	−0.176644	+	0.984275i	$0.556524\pi$
$654$	0	0
$655$	−2.03110	−0.0793615
$656$	−17.7362	−0.692482
$657$	0	0
$658$	8.85665	0.345268
$659$	−28.4222	−1.10717	−0.553585	−	0.832793i	$-0.686741\pi$
$659$	−28.4222	−1.10717	−0.553585	+	0.832793i	$0.686741\pi$
$660$	0	0
$661$	33.6417	1.30851	0.654255	−	0.756274i	$-0.272982\pi$
$661$	33.6417	1.30851	0.654255	+	0.756274i	$0.272982\pi$
$662$	−6.76647	−0.262986
$663$	0	0
$664$	18.4729	0.716886
$665$	−46.0579	−1.78605
$666$	0	0
$667$	−8.29467	−0.321171
$668$	19.3536	0.748812
$669$	0	0
$670$	−3.24346	−0.125306
$671$	−1.46437	−0.0565315
$672$	0	0
$673$	−5.15293	−0.198631	−0.0993153	−	0.995056i	$-0.531665\pi$
$673$	−5.15293	−0.198631	−0.0993153	+	0.995056i	$0.531665\pi$
$674$	11.8364	0.455922
$675$	0	0
$676$	22.9999	0.884613
$677$	−48.9736	−1.88221	−0.941104	−	0.338116i	$-0.890210\pi$
$677$	−48.9736	−1.88221	−0.941104	+	0.338116i	$0.890210\pi$
$678$	0	0
$679$	14.9856	0.575093
$680$	27.4242	1.05167
$681$	0	0
$682$	5.00114	0.191504
$683$	−19.7008	−0.753830	−0.376915	−	0.926248i	$-0.623015\pi$
$683$	−19.7008	−0.753830	−0.376915	+	0.926248i	$0.623015\pi$
$684$	0	0
$685$	−15.3462	−0.586350
$686$	−6.18148	−0.236010
$687$	0	0
$688$	−14.5108	−0.553218
$689$	0.216118	0.00823346
$690$	0	0
$691$	−18.8984	−0.718931	−0.359465	−	0.933158i	$-0.617041\pi$
$691$	−18.8984	−0.718931	−0.359465	+	0.933158i	$0.617041\pi$
$692$	−7.71816	−0.293401
$693$	0	0
$694$	−7.82505	−0.297035
$695$	−13.5617	−0.514425
$696$	0	0
$697$	36.3474	1.37676
$698$	2.76377	0.104610
$699$	0	0
$700$	14.8695	0.562015
$701$	8.39322	0.317008	0.158504	−	0.987358i	$-0.449333\pi$
$701$	8.39322	0.317008	0.158504	+	0.987358i	$0.449333\pi$
$702$	0	0
$703$	15.4306	0.581975
$704$	−2.98438	−0.112478
$705$	0	0
$706$	6.01456	0.226361
$707$	26.5704	0.999282
$708$	0	0
$709$	−18.4494	−0.692882	−0.346441	−	0.938072i	$-0.612610\pi$
$709$	−18.4494	−0.692882	−0.346441	+	0.938072i	$0.612610\pi$
$710$	14.2218	0.533734
$711$	0	0
$712$	6.75188	0.253037
$713$	73.5821	2.75567
$714$	0	0
$715$	−0.0877802	−0.00328279
$716$	40.9704	1.53114
$717$	0	0
$718$	−6.10327	−0.227772
$719$	−28.4958	−1.06271	−0.531357	−	0.847148i	$-0.678318\pi$
$719$	−28.4958	−1.06271	−0.531357	+	0.847148i	$0.678318\pi$
$720$	0	0
$721$	−53.4792	−1.99167
$722$	−4.27732	−0.159185
$723$	0	0
$724$	−25.4908	−0.947359
$725$	3.13358	0.116378
$726$	0	0
$727$	−39.1447	−1.45180	−0.725899	−	0.687801i	$-0.758576\pi$
$727$	−39.1447	−1.45180	−0.725899	+	0.687801i	$0.758576\pi$
$728$	−0.180643	−0.00669506
$729$	0	0
$730$	−16.5133	−0.611183
$731$	29.7374	1.09988
$732$	0	0
$733$	−50.0775	−1.84966	−0.924828	−	0.380386i	$-0.875791\pi$
$733$	−50.0775	−1.84966	−0.924828	+	0.380386i	$0.875791\pi$
$734$	−11.8675	−0.438036
$735$	0	0
$736$	34.6399	1.27684
$737$	−2.43874	−0.0898320
$738$	0	0
$739$	10.2350	0.376502	0.188251	−	0.982121i	$-0.439718\pi$
$739$	10.2350	0.376502	0.188251	+	0.982121i	$0.439718\pi$
$740$	−14.3129	−0.526151
$741$	0	0
$742$	10.3089	0.378450
$743$	17.6206	0.646435	0.323218	−	0.946325i	$-0.395235\pi$
$743$	17.6206	0.646435	0.323218	+	0.946325i	$0.395235\pi$
$744$	0	0
$745$	−52.9579	−1.94023
$746$	−2.60616	−0.0954184
$747$	0	0
$748$	9.67918	0.353906
$749$	−58.7318	−2.14601
$750$	0	0
$751$	43.2266	1.57736	0.788681	−	0.614803i	$-0.210764\pi$
$751$	43.2266	1.57736	0.788681	+	0.614803i	$0.210764\pi$
$752$	15.6366	0.570209
$753$	0	0
$754$	−0.0178695	−0.000650770 0
$755$	47.1213	1.71492
$756$	0	0
$757$	21.4873	0.780969	0.390485	−	0.920610i	$-0.372308\pi$
$757$	21.4873	0.780969	0.390485	+	0.920610i	$0.372308\pi$
$758$	−5.15233	−0.187141
$759$	0	0
$760$	26.4829	0.960635
$761$	−23.3489	−0.846399	−0.423199	−	0.906037i	$-0.639093\pi$
$761$	−23.3489	−0.846399	−0.423199	+	0.906037i	$0.639093\pi$
$762$	0	0
$763$	6.67267	0.241567
$764$	31.6130	1.14372
$765$	0	0
$766$	7.54553	0.272631
$767$	0.0958478	0.00346086
$768$	0	0
$769$	−37.2012	−1.34151	−0.670754	−	0.741680i	$-0.734029\pi$
$769$	−37.2012	−1.34151	−0.670754	+	0.741680i	$0.734029\pi$
$770$	−4.18712	−0.150893
$771$	0	0
$772$	−28.8909	−1.03981
$773$	29.4954	1.06088	0.530438	−	0.847724i	$-0.322028\pi$
$773$	29.4954	1.06088	0.530438	+	0.847724i	$0.322028\pi$
$774$	0	0
$775$	−27.7980	−0.998535
$776$	−8.61657	−0.309317
$777$	0	0
$778$	13.7790	0.494001
$779$	35.0998	1.25758
$780$	0	0
$781$	10.6933	0.382635
$782$	−18.5632	−0.663819
$783$	0	0
$784$	7.77206	0.277574
$785$	15.8500	0.565712
$786$	0	0
$787$	−47.4563	−1.69163	−0.845817	−	0.533472i	$-0.820887\pi$
$787$	−47.4563	−1.69163	−0.845817	+	0.533472i	$0.820887\pi$
$788$	40.8084	1.45374
$789$	0	0
$790$	−4.47095	−0.159069
$791$	54.6689	1.94380
$792$	0	0
$793$	−0.0464161	−0.00164828
$794$	−13.4415	−0.477019
$795$	0	0
$796$	40.1632	1.42355
$797$	20.2750	0.718176	0.359088	−	0.933304i	$-0.383088\pi$
$797$	20.2750	0.718176	0.359088	+	0.933304i	$0.383088\pi$
$798$	0	0
$799$	−32.0447	−1.13366
$800$	−13.0863	−0.462672
$801$	0	0
$802$	5.32401	0.187997
$803$	−12.4162	−0.438158
$804$	0	0
$805$	−61.6053	−2.17130
$806$	0.158521	0.00558365
$807$	0	0
$808$	−15.2777	−0.537469
$809$	−21.4143	−0.752885	−0.376443	−	0.926440i	$-0.622853\pi$
$809$	−21.4143	−0.752885	−0.376443	+	0.926440i	$0.622853\pi$
$810$	0	0
$811$	15.9258	0.559231	0.279616	−	0.960112i	$-0.409793\pi$
$811$	15.9258	0.559231	0.279616	+	0.960112i	$0.409793\pi$
$812$	6.53914	0.229479
$813$	0	0
$814$	1.40279	0.0491679
$815$	−54.4829	−1.90845
$816$	0	0
$817$	28.7167	1.00467
$818$	−2.27428	−0.0795184
$819$	0	0
$820$	−32.5573	−1.13695
$821$	21.3210	0.744109	0.372055	−	0.928211i	$-0.378653\pi$
$821$	21.3210	0.744109	0.372055	+	0.928211i	$0.378653\pi$
$822$	0	0
$823$	−20.6697	−0.720501	−0.360251	−	0.932856i	$-0.617309\pi$
$823$	−20.6697	−0.720501	−0.360251	+	0.932856i	$0.617309\pi$
$824$	30.7501	1.07123
$825$	0	0
$826$	4.57195	0.159078
$827$	−12.3472	−0.429353	−0.214676	−	0.976685i	$-0.568870\pi$
$827$	−12.3472	−0.429353	−0.214676	+	0.976685i	$0.568870\pi$
$828$	0	0
$829$	−26.6081	−0.924138	−0.462069	−	0.886844i	$-0.652893\pi$
$829$	−26.6081	−0.924138	−0.462069	+	0.886844i	$0.652893\pi$
$830$	13.5721	0.471094
$831$	0	0
$832$	−0.0945956	−0.00327951
$833$	−15.9275	−0.551857
$834$	0	0
$835$	30.2917	1.04829
$836$	9.34695	0.323271
$837$	0	0
$838$	1.10783	0.0382693
$839$	−17.9614	−0.620097	−0.310049	−	0.950721i	$-0.600345\pi$
$839$	−17.9614	−0.620097	−0.310049	+	0.950721i	$0.600345\pi$
$840$	0	0
$841$	−27.6220	−0.952481
$842$	4.70150	0.162024
$843$	0	0
$844$	30.8091	1.06049
$845$	35.9989	1.23840
$846$	0	0
$847$	−3.14826	−0.108176
$848$	18.2005	0.625009
$849$	0	0
$850$	7.01286	0.240539
$851$	20.6394	0.707509
$852$	0	0
$853$	−33.3651	−1.14240	−0.571199	−	0.820812i	$-0.693522\pi$
$853$	−33.3651	−1.14240	−0.571199	+	0.820812i	$0.693522\pi$
$854$	−2.21405	−0.0757633
$855$	0	0
$856$	33.7703	1.15424
$857$	−11.6363	−0.397490	−0.198745	−	0.980051i	$-0.563686\pi$
$857$	−11.6363	−0.397490	−0.198745	+	0.980051i	$0.563686\pi$
$858$	0	0
$859$	−37.6989	−1.28627	−0.643134	−	0.765753i	$-0.722366\pi$
$859$	−37.6989	−1.28627	−0.643134	+	0.765753i	$0.722366\pi$
$860$	−26.6366	−0.908300
$861$	0	0
$862$	9.53065	0.324615
$863$	21.8806	0.744824	0.372412	−	0.928068i	$-0.378531\pi$
$863$	21.8806	0.744824	0.372412	+	0.928068i	$0.378531\pi$
$864$	0	0
$865$	−12.0803	−0.410742
$866$	−1.24078	−0.0421636
$867$	0	0
$868$	−58.0087	−1.96894
$869$	−3.36167	−0.114037
$870$	0	0
$871$	−0.0773003	−0.00261922
$872$	−3.83673	−0.129928
$873$	0	0
$874$	−17.9260	−0.606357
$875$	−20.3200	−0.686942
$876$	0	0
$877$	−20.5789	−0.694901	−0.347450	−	0.937698i	$-0.612952\pi$
$877$	−20.5789	−0.694901	−0.347450	+	0.937698i	$0.612952\pi$
$878$	−6.30322	−0.212723
$879$	0	0
$880$	−7.39246	−0.249200
$881$	57.8921	1.95043	0.975217	−	0.221250i	$-0.0710137\pi$
$881$	57.8921	1.95043	0.975217	+	0.221250i	$0.0710137\pi$
$882$	0	0
$883$	31.2182	1.05057	0.525287	−	0.850925i	$-0.323958\pi$
$883$	31.2182	1.05057	0.525287	+	0.850925i	$0.323958\pi$
$884$	0.306800	0.0103188
$885$	0	0
$886$	6.55642	0.220267
$887$	−18.7117	−0.628278	−0.314139	−	0.949377i	$-0.601716\pi$
$887$	−18.7117	−0.628278	−0.314139	+	0.949377i	$0.601716\pi$
$888$	0	0
$889$	−16.3613	−0.548741
$890$	4.96062	0.166281
$891$	0	0
$892$	4.98695	0.166975
$893$	−30.9447	−1.03553
$894$	0	0
$895$	64.1259	2.14349
$896$	−35.3804	−1.18197
$897$	0	0
$898$	−7.88457	−0.263112
$899$	−12.2247	−0.407716
$900$	0	0
$901$	−37.2990	−1.24261
$902$	3.19092	0.106246
$903$	0	0
$904$	−31.4341	−1.04548
$905$	−39.8976	−1.32624
$906$	0	0
$907$	−52.3884	−1.73953	−0.869764	−	0.493468i	$-0.835729\pi$
$907$	−52.3884	−1.73953	−0.869764	+	0.493468i	$0.835729\pi$
$908$	17.8812	0.593408
$909$	0	0
$910$	−0.132719	−0.00439958
$911$	−26.8137	−0.888378	−0.444189	−	0.895933i	$-0.646508\pi$
$911$	−26.8137	−0.888378	−0.444189	+	0.895933i	$0.646508\pi$
$912$	0	0
$913$	10.2047	0.337728
$914$	−1.88542	−0.0623641
$915$	0	0
$916$	−15.3907	−0.508525
$917$	−2.30899	−0.0762495
$918$	0	0
$919$	10.7913	0.355971	0.177986	−	0.984033i	$-0.443042\pi$
$919$	10.7913	0.355971	0.177986	+	0.984033i	$0.443042\pi$
$920$	35.4225	1.16785
$921$	0	0
$922$	6.76592	0.222824
$923$	0.338943	0.0111564
$924$	0	0
$925$	−7.79720	−0.256370
$926$	−19.0094	−0.624686
$927$	0	0
$928$	−5.75495	−0.188915
$929$	12.9482	0.424815	0.212408	−	0.977181i	$-0.431870\pi$
$929$	12.9482	0.424815	0.212408	+	0.977181i	$0.431870\pi$
$930$	0	0
$931$	−15.3808	−0.504086
$932$	−21.8861	−0.716903
$933$	0	0
$934$	5.67682	0.185751
$935$	15.1496	0.495445
$936$	0	0
$937$	5.28523	0.172661	0.0863304	−	0.996267i	$-0.472486\pi$
$937$	5.28523	0.172661	0.0863304	+	0.996267i	$0.472486\pi$
$938$	−3.68723	−0.120392
$939$	0	0
$940$	28.7032	0.936197
$941$	−47.5753	−1.55091	−0.775456	−	0.631402i	$-0.782480\pi$
$941$	−47.5753	−1.55091	−0.775456	+	0.631402i	$0.782480\pi$
$942$	0	0
$943$	46.9482	1.52884
$944$	8.07188	0.262717
$945$	0	0
$946$	2.61063	0.0848790
$947$	46.7724	1.51990	0.759950	−	0.649982i	$-0.225224\pi$
$947$	46.7724	1.51990	0.759950	+	0.649982i	$0.225224\pi$
$948$	0	0
$949$	−0.393555	−0.0127753
$950$	6.77214	0.219717
$951$	0	0
$952$	31.1763	1.01043
$953$	−54.5321	−1.76647	−0.883234	−	0.468933i	$-0.844639\pi$
$953$	−54.5321	−1.76647	−0.883234	+	0.468933i	$0.844639\pi$
$954$	0	0
$955$	49.4799	1.60113
$956$	−53.6882	−1.73640
$957$	0	0
$958$	−9.45872	−0.305597
$959$	−17.4459	−0.563358
$960$	0	0
$961$	77.4451	2.49823
$962$	0.0444642	0.00143358
$963$	0	0
$964$	44.9258	1.44696
$965$	−45.2193	−1.45566
$966$	0	0
$967$	−12.7407	−0.409715	−0.204857	−	0.978792i	$-0.565673\pi$
$967$	−12.7407	−0.409715	−0.204857	+	0.978792i	$0.565673\pi$
$968$	1.81022	0.0581828
$969$	0	0
$970$	−6.33062	−0.203264
$971$	1.14705	0.0368106	0.0184053	−	0.999831i	$-0.494141\pi$
$971$	1.14705	0.0368106	0.0184053	+	0.999831i	$0.494141\pi$
$972$	0	0
$973$	−15.4172	−0.494253
$974$	20.0937	0.643845
$975$	0	0
$976$	−3.90896	−0.125123
$977$	−18.9182	−0.605248	−0.302624	−	0.953110i	$-0.597863\pi$
$977$	−18.9182	−0.605248	−0.302624	+	0.953110i	$0.597863\pi$
$978$	0	0
$979$	3.72986	0.119207
$980$	14.2667	0.455734
$981$	0	0
$982$	5.14247	0.164103
$983$	34.7690	1.10896	0.554479	−	0.832197i	$-0.312917\pi$
$983$	34.7690	1.10896	0.554479	+	0.832197i	$0.312917\pi$
$984$	0	0
$985$	63.8723	2.03514
$986$	3.08403	0.0982155
$987$	0	0
$988$	0.296269	0.00942557
$989$	38.4104	1.22138
$990$	0	0
$991$	18.6437	0.592237	0.296119	−	0.955151i	$-0.404308\pi$
$991$	18.6437	0.592237	0.296119	+	0.955151i	$0.404308\pi$
$992$	51.0522	1.62091
$993$	0	0
$994$	16.1676	0.512805
$995$	62.8625	1.99287
$996$	0	0
$997$	2.07630	0.0657572	0.0328786	−	0.999459i	$-0.489533\pi$
$997$	2.07630	0.0657572	0.0328786	+	0.999459i	$0.489533\pi$
$998$	−10.8665	−0.343974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2673.2.a.p.1.3	yes	6
3.2	odd	2		2673.2.a.j.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2673.2.a.j.1.4	✓	6	3.2	odd	2
2673.2.a.p.1.3	yes	6	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:	\( N \)	\(=\)	\( 2673 = 3^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2673.a (trivial)

\(f(q)\)	\(=\)	\(q-0.480247 q^{2} -1.76936 q^{4} -2.76936 q^{5} -3.14826 q^{7} +1.81022 q^{8} +O(q^{10})\)	\(q-0.480247 q^{2} -1.76936 q^{4} -2.76936 q^{5} -3.14826 q^{7} +1.81022 q^{8} +1.32998 q^{10} +1.00000 q^{11} +0.0316969 q^{13} +1.51194 q^{14} +2.66937 q^{16} -5.47043 q^{17} -5.28266 q^{19} +4.90001 q^{20} -0.480247 q^{22} -7.06589 q^{23} +2.66937 q^{25} -0.0152223 q^{26} +5.57042 q^{28} +1.17390 q^{29} -10.4137 q^{31} -4.90241 q^{32} +2.62716 q^{34} +8.71869 q^{35} -2.92099 q^{37} +2.53698 q^{38} -5.01317 q^{40} -6.64434 q^{41} -5.43603 q^{43} -1.76936 q^{44} +3.39337 q^{46} +5.85779 q^{47} +2.91157 q^{49} -1.28196 q^{50} -0.0560833 q^{52} +6.81829 q^{53} -2.76936 q^{55} -5.69906 q^{56} -0.563763 q^{58} +3.02389 q^{59} -1.46437 q^{61} +5.00114 q^{62} -2.98438 q^{64} -0.0877802 q^{65} -2.43874 q^{67} +9.67918 q^{68} -4.18712 q^{70} +10.6933 q^{71} -12.4162 q^{73} +1.40279 q^{74} +9.34695 q^{76} -3.14826 q^{77} -3.36167 q^{79} -7.39246 q^{80} +3.19092 q^{82} +10.2047 q^{83} +15.1496 q^{85} +2.61063 q^{86} +1.81022 q^{88} +3.72986 q^{89} -0.0997902 q^{91} +12.5021 q^{92} -2.81319 q^{94} +14.6296 q^{95} -4.75995 q^{97} -1.39827 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Introduction

L-functions

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