LMFDB - Embedded newform 2151.4.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.976149 q^{2} -7.04713 q^{4} +2.51110 q^{5} +23.2375 q^{7} -14.6882 q^{8} +O(q^{10})$	\(q+0.976149 q^{2} -7.04713 q^{4} +2.51110 q^{5} +23.2375 q^{7} -14.6882 q^{8} +2.45121 q^{10} +4.23674 q^{11} -37.8137 q^{13} +22.6832 q^{14} +42.0392 q^{16} -93.4441 q^{17} +109.684 q^{19} -17.6961 q^{20} +4.13569 q^{22} +138.490 q^{23} -118.694 q^{25} -36.9118 q^{26} -163.758 q^{28} +36.2019 q^{29} -313.573 q^{31} +158.542 q^{32} -91.2153 q^{34} +58.3517 q^{35} +143.875 q^{37} +107.068 q^{38} -36.8836 q^{40} -216.572 q^{41} -166.582 q^{43} -29.8569 q^{44} +135.187 q^{46} +117.564 q^{47} +196.981 q^{49} -115.863 q^{50} +266.478 q^{52} +172.946 q^{53} +10.6389 q^{55} -341.318 q^{56} +35.3384 q^{58} -306.349 q^{59} +452.047 q^{61} -306.094 q^{62} -181.552 q^{64} -94.9540 q^{65} +300.560 q^{67} +658.513 q^{68} +56.9599 q^{70} +489.165 q^{71} -878.181 q^{73} +140.443 q^{74} -772.959 q^{76} +98.4511 q^{77} +314.910 q^{79} +105.565 q^{80} -211.407 q^{82} -437.897 q^{83} -234.647 q^{85} -162.609 q^{86} -62.2302 q^{88} +1458.92 q^{89} -878.695 q^{91} -975.956 q^{92} +114.760 q^{94} +275.428 q^{95} -1571.68 q^{97} +192.283 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	0.976149	0.345121	0.172560	−	0.984999i	$-0.444796\pi$
$2$	0.976149	0.345121	0.172560	+	0.984999i	$0.444796\pi$
$3$	0	0
$3$	0	0
$4$	−7.04713	−0.880892
$5$	2.51110	0.224600	0.112300	−	0.993674i	$-0.464178\pi$
$5$	2.51110	0.224600	0.112300	+	0.993674i	$0.464178\pi$
$6$	0	0
$7$	23.2375	1.25471	0.627353	−	0.778735i	$-0.284138\pi$
$7$	23.2375	1.25471	0.627353	+	0.778735i	$0.284138\pi$
$8$	−14.6882	−0.649135
$9$	0	0
$10$	2.45121	0.0775140
$11$	4.23674	0.116129	0.0580647	−	0.998313i	$-0.481507\pi$
$11$	4.23674	0.116129	0.0580647	+	0.998313i	$0.481507\pi$
$12$	0	0
$13$	−37.8137	−0.806741	−0.403370	−	0.915037i	$-0.632161\pi$
$13$	−37.8137	−0.806741	−0.403370	+	0.915037i	$0.632161\pi$
$14$	22.6832	0.433025
$15$	0	0
$16$	42.0392	0.656862
$17$	−93.4441	−1.33315	−0.666574	−	0.745439i	$-0.732240\pi$
$17$	−93.4441	−1.33315	−0.666574	+	0.745439i	$0.732240\pi$
$18$	0	0
$19$	109.684	1.32438	0.662192	−	0.749334i	$-0.269626\pi$
$19$	109.684	1.32438	0.662192	+	0.749334i	$0.269626\pi$
$20$	−17.6961	−0.197848
$21$	0	0
$22$	4.13569	0.0400787
$23$	138.490	1.25553	0.627763	−	0.778404i	$-0.283971\pi$
$23$	138.490	1.25553	0.627763	+	0.778404i	$0.283971\pi$
$24$	0	0
$25$	−118.694	−0.949555
$26$	−36.9118	−0.278423
$27$	0	0
$28$	−163.758	−1.10526
$29$	36.2019	0.231811	0.115905	−	0.993260i	$-0.463023\pi$
$29$	36.2019	0.231811	0.115905	+	0.993260i	$0.463023\pi$
$30$	0	0
$31$	−313.573	−1.81675	−0.908377	−	0.418151i	$-0.862678\pi$
$31$	−313.573	−1.81675	−0.908377	+	0.418151i	$0.862678\pi$
$32$	158.542	0.875831
$33$	0	0
$34$	−91.2153	−0.460097
$35$	58.3517	0.281807
$36$	0	0
$37$	143.875	0.639267	0.319634	−	0.947541i	$-0.396440\pi$
$37$	143.875	0.639267	0.319634	+	0.947541i	$0.396440\pi$
$38$	107.068	0.457072
$39$	0	0
$40$	−36.8836	−0.145795
$41$	−216.572	−0.824949	−0.412474	−	0.910969i	$-0.635335\pi$
$41$	−216.572	−0.824949	−0.412474	+	0.910969i	$0.635335\pi$
$42$	0	0
$43$	−166.582	−0.590779	−0.295389	−	0.955377i	$-0.595449\pi$
$43$	−166.582	−0.590779	−0.295389	+	0.955377i	$0.595449\pi$
$44$	−29.8569	−0.102298
$45$	0	0
$46$	135.187	0.433308
$47$	117.564	0.364860	0.182430	−	0.983219i	$-0.441604\pi$
$47$	117.564	0.364860	0.182430	+	0.983219i	$0.441604\pi$
$48$	0	0
$49$	196.981	0.574288
$50$	−115.863	−0.327711
$51$	0	0
$52$	266.478	0.710651
$53$	172.946	0.448225	0.224113	−	0.974563i	$-0.428052\pi$
$53$	172.946	0.448225	0.224113	+	0.974563i	$0.428052\pi$
$54$	0	0
$55$	10.6389	0.0260826
$56$	−341.318	−0.814473
$57$	0	0
$58$	35.3384	0.0800028
$59$	−306.349	−0.675987	−0.337993	−	0.941149i	$-0.609748\pi$
$59$	−306.349	−0.675987	−0.337993	+	0.941149i	$0.609748\pi$
$60$	0	0
$61$	452.047	0.948831	0.474416	−	0.880301i	$-0.342659\pi$
$61$	452.047	0.948831	0.474416	+	0.880301i	$0.342659\pi$
$62$	−306.094	−0.627000
$63$	0	0
$64$	−181.552	−0.354594
$65$	−94.9540	−0.181194
$66$	0	0
$67$	300.560	0.548048	0.274024	−	0.961723i	$-0.411645\pi$
$67$	300.560	0.548048	0.274024	+	0.961723i	$0.411645\pi$
$68$	658.513	1.17436
$69$	0	0
$70$	56.9599	0.0972573
$71$	489.165	0.817651	0.408825	−	0.912613i	$-0.365939\pi$
$71$	489.165	0.817651	0.408825	+	0.912613i	$0.365939\pi$
$72$	0	0
$73$	−878.181	−1.40799	−0.703995	−	0.710204i	$-0.748602\pi$
$73$	−878.181	−1.40799	−0.703995	+	0.710204i	$0.748602\pi$
$74$	140.443	0.220624
$75$	0	0
$76$	−772.959	−1.16664
$77$	98.4511	0.145708
$78$	0	0
$79$	314.910	0.448483	0.224241	−	0.974534i	$-0.428010\pi$
$79$	314.910	0.448483	0.224241	+	0.974534i	$0.428010\pi$
$80$	105.565	0.147531
$81$	0	0
$82$	−211.407	−0.284707
$83$	−437.897	−0.579102	−0.289551	−	0.957163i	$-0.593506\pi$
$83$	−437.897	−0.579102	−0.289551	+	0.957163i	$0.593506\pi$
$84$	0	0
$85$	−234.647	−0.299425
$86$	−162.609	−0.203890
$87$	0	0
$88$	−62.2302	−0.0753837
$89$	1458.92	1.73759	0.868795	−	0.495172i	$-0.164895\pi$
$89$	1458.92	1.73759	0.868795	+	0.495172i	$0.164895\pi$
$90$	0	0
$91$	−878.695	−1.01222
$92$	−975.956	−1.10598
$93$	0	0
$94$	114.760	0.125921
$95$	275.428	0.297456
$96$	0	0
$97$	−1571.68	−1.64515	−0.822575	−	0.568657i	$-0.807463\pi$
$97$	−1571.68	−1.64515	−0.822575	+	0.568657i	$0.807463\pi$
$98$	192.283	0.198199
$99$	0	0
$100$	836.455	0.836455
$101$	−27.9358	−0.0275220	−0.0137610	−	0.999905i	$-0.504380\pi$
$101$	−27.9358	−0.0275220	−0.0137610	+	0.999905i	$0.504380\pi$
$102$	0	0
$103$	253.262	0.242278	0.121139	−	0.992636i	$-0.461345\pi$
$103$	253.262	0.242278	0.121139	+	0.992636i	$0.461345\pi$
$104$	555.416	0.523683
$105$	0	0
$106$	168.821	0.154692
$107$	−1445.97	−1.30642	−0.653212	−	0.757175i	$-0.726579\pi$
$107$	−1445.97	−1.30642	−0.653212	+	0.757175i	$0.726579\pi$
$108$	0	0
$109$	−1507.62	−1.32480	−0.662402	−	0.749149i	$-0.730463\pi$
$109$	−1507.62	−1.32480	−0.662402	+	0.749149i	$0.730463\pi$
$110$	10.3851	0.00900166
$111$	0	0
$112$	976.885	0.824169
$113$	1639.11	1.36455	0.682275	−	0.731095i	$-0.260991\pi$
$113$	1639.11	1.36455	0.682275	+	0.731095i	$0.260991\pi$
$114$	0	0
$115$	347.762	0.281991
$116$	−255.119	−0.204200
$117$	0	0
$118$	−299.042	−0.233297
$119$	−2171.41	−1.67271
$120$	0	0
$121$	−1313.05	−0.986514
$122$	441.265	0.327461
$123$	0	0
$124$	2209.79	1.60036
$125$	−611.941	−0.437869
$126$	0	0
$127$	−1400.38	−0.978456	−0.489228	−	0.872156i	$-0.662721\pi$
$127$	−1400.38	−0.978456	−0.489228	+	0.872156i	$0.662721\pi$
$128$	−1445.56	−0.998209
$129$	0	0
$130$	−92.6892	−0.0625337
$131$	837.760	0.558744	0.279372	−	0.960183i	$-0.409874\pi$
$131$	837.760	0.558744	0.279372	+	0.960183i	$0.409874\pi$
$132$	0	0
$133$	2548.79	1.66171
$134$	293.391	0.189143
$135$	0	0
$136$	1372.53	0.865393
$137$	1577.17	0.983554	0.491777	−	0.870721i	$-0.336347\pi$
$137$	1577.17	0.983554	0.491777	+	0.870721i	$0.336347\pi$
$138$	0	0
$139$	−2297.75	−1.40211	−0.701054	−	0.713108i	$-0.747287\pi$
$139$	−2297.75	−1.40211	−0.701054	+	0.713108i	$0.747287\pi$
$140$	−411.212	−0.248241
$141$	0	0
$142$	477.498	0.282188
$143$	−160.207	−0.0936864
$144$	0	0
$145$	90.9065	0.0520647
$146$	−857.235	−0.485927
$147$	0	0
$148$	−1013.91	−0.563125
$149$	1136.78	0.625024	0.312512	−	0.949914i	$-0.398830\pi$
$149$	1136.78	0.625024	0.312512	+	0.949914i	$0.398830\pi$
$150$	0	0
$151$	2843.11	1.53225	0.766124	−	0.642693i	$-0.222183\pi$
$151$	2843.11	1.53225	0.766124	+	0.642693i	$0.222183\pi$
$152$	−1611.07	−0.859703
$153$	0	0
$154$	96.1029	0.0502870
$155$	−787.414	−0.408042
$156$	0	0
$157$	−1499.43	−0.762215	−0.381108	−	0.924531i	$-0.624457\pi$
$157$	−1499.43	−0.762215	−0.381108	+	0.924531i	$0.624457\pi$
$158$	307.399	0.154781
$159$	0	0
$160$	398.116	0.196711
$161$	3218.15	1.57532
$162$	0	0
$163$	693.374	0.333186	0.166593	−	0.986026i	$-0.446723\pi$
$163$	693.374	0.333186	0.166593	+	0.986026i	$0.446723\pi$
$164$	1526.21	0.726690
$165$	0	0
$166$	−427.453	−0.199860
$167$	−2451.60	−1.13599	−0.567995	−	0.823032i	$-0.692281\pi$
$167$	−2451.60	−1.13599	−0.567995	+	0.823032i	$0.692281\pi$
$168$	0	0
$169$	−767.125	−0.349169
$170$	−229.051	−0.103338
$171$	0	0
$172$	1173.92	0.520412
$173$	−43.8218	−0.0192584	−0.00962922	−	0.999954i	$-0.503065\pi$
$173$	−43.8218	−0.0192584	−0.00962922	+	0.999954i	$0.503065\pi$
$174$	0	0
$175$	−2758.16	−1.19141
$176$	178.109	0.0762810
$177$	0	0
$178$	1424.13	0.599678
$179$	445.439	0.185998	0.0929991	−	0.995666i	$-0.470355\pi$
$179$	445.439	0.185998	0.0929991	+	0.995666i	$0.470355\pi$
$180$	0	0
$181$	−2028.86	−0.833169	−0.416585	−	0.909097i	$-0.636773\pi$
$181$	−2028.86	−0.833169	−0.416585	+	0.909097i	$0.636773\pi$
$182$	−857.737	−0.349339
$183$	0	0
$184$	−2034.17	−0.815006
$185$	361.284	0.143579
$186$	0	0
$187$	−395.898	−0.154818
$188$	−828.487	−0.321402
$189$	0	0
$190$	268.859	0.102658
$191$	−2043.16	−0.774019	−0.387010	−	0.922076i	$-0.626492\pi$
$191$	−2043.16	−0.774019	−0.387010	+	0.922076i	$0.626492\pi$
$192$	0	0
$193$	3175.38	1.18430	0.592148	−	0.805830i	$-0.298280\pi$
$193$	3175.38	1.18430	0.592148	+	0.805830i	$0.298280\pi$
$194$	−1534.19	−0.567775
$195$	0	0
$196$	−1388.15	−0.505886
$197$	−3766.78	−1.36229	−0.681147	−	0.732146i	$-0.738519\pi$
$197$	−3766.78	−1.36229	−0.681147	+	0.732146i	$0.738519\pi$
$198$	0	0
$199$	−1845.97	−0.657575	−0.328787	−	0.944404i	$-0.606640\pi$
$199$	−1845.97	−0.657575	−0.328787	+	0.944404i	$0.606640\pi$
$200$	1743.41	0.616389
$201$	0	0
$202$	−27.2695	−0.00949840
$203$	841.240	0.290855
$204$	0	0
$205$	−543.834	−0.185283
$206$	247.222	0.0836153
$207$	0	0
$208$	−1589.66	−0.529917
$209$	464.703	0.153800
$210$	0	0
$211$	−724.344	−0.236331	−0.118166	−	0.992994i	$-0.537701\pi$
$211$	−724.344	−0.236331	−0.118166	+	0.992994i	$0.537701\pi$
$212$	−1218.77	−0.394838
$213$	0	0
$214$	−1411.48	−0.450874
$215$	−418.304	−0.132689
$216$	0	0
$217$	−7286.65	−2.27949
$218$	−1471.66	−0.457217
$219$	0	0
$220$	−74.9736	−0.0229760
$221$	3533.47	1.07551
$222$	0	0
$223$	−3808.13	−1.14355	−0.571774	−	0.820411i	$-0.693744\pi$
$223$	−3808.13	−1.14355	−0.571774	+	0.820411i	$0.693744\pi$
$224$	3684.13	1.09891
$225$	0	0
$226$	1600.01	0.470935
$227$	1500.22	0.438648	0.219324	−	0.975652i	$-0.429615\pi$
$227$	1500.22	0.438648	0.219324	+	0.975652i	$0.429615\pi$
$228$	0	0
$229$	−4847.05	−1.39870	−0.699349	−	0.714780i	$-0.746527\pi$
$229$	−4847.05	−1.39870	−0.699349	+	0.714780i	$0.746527\pi$
$230$	339.467	0.0973209
$231$	0	0
$232$	−531.742	−0.150477
$233$	−2485.18	−0.698753	−0.349376	−	0.936982i	$-0.613607\pi$
$233$	−2485.18	−0.698753	−0.349376	+	0.936982i	$0.613607\pi$
$234$	0	0
$235$	295.214	0.0819475
$236$	2158.88	0.595471
$237$	0	0
$238$	−2119.62	−0.577287
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−6360.18	−1.69998	−0.849991	−	0.526798i	$-0.823393\pi$
$241$	−6360.18	−1.69998	−0.849991	+	0.526798i	$0.823393\pi$
$242$	−1281.73	−0.340466
$243$	0	0
$244$	−3185.64	−0.835817
$245$	494.639	0.128985
$246$	0	0
$247$	−4147.56	−1.06843
$248$	4605.84	1.17932
$249$	0	0
$250$	−597.345	−0.151118
$251$	−5788.57	−1.45566	−0.727831	−	0.685756i	$-0.759472\pi$
$251$	−5788.57	−1.45566	−0.727831	+	0.685756i	$0.759472\pi$
$252$	0	0
$253$	586.745	0.145804
$254$	−1366.98	−0.337686
$255$	0	0
$256$	41.3362	0.0100918
$257$	2034.41	0.493786	0.246893	−	0.969043i	$-0.420590\pi$
$257$	2034.41	0.493786	0.246893	+	0.969043i	$0.420590\pi$
$258$	0	0
$259$	3343.29	0.802093
$260$	669.153	0.159612
$261$	0	0
$262$	817.778	0.192834
$263$	−7756.61	−1.81861	−0.909303	−	0.416135i	$-0.863384\pi$
$263$	−7756.61	−1.81861	−0.909303	+	0.416135i	$0.863384\pi$
$264$	0	0
$265$	434.285	0.100671
$266$	2487.99	0.573491
$267$	0	0
$268$	−2118.08	−0.482771
$269$	−6988.40	−1.58398	−0.791989	−	0.610535i	$-0.790954\pi$
$269$	−6988.40	−1.58398	−0.791989	+	0.610535i	$0.790954\pi$
$270$	0	0
$271$	−3053.46	−0.684444	−0.342222	−	0.939619i	$-0.611179\pi$
$271$	−3053.46	−0.684444	−0.342222	+	0.939619i	$0.611179\pi$
$272$	−3928.31	−0.875694
$273$	0	0
$274$	1539.55	0.339445
$275$	−502.877	−0.110271
$276$	0	0
$277$	6738.33	1.46161	0.730807	−	0.682584i	$-0.239144\pi$
$277$	6738.33	1.46161	0.730807	+	0.682584i	$0.239144\pi$
$278$	−2242.95	−0.483896
$279$	0	0
$280$	−857.083	−0.182930
$281$	4312.04	0.915425	0.457713	−	0.889100i	$-0.348669\pi$
$281$	4312.04	0.915425	0.457713	+	0.889100i	$0.348669\pi$
$282$	0	0
$283$	292.823	0.0615072	0.0307536	−	0.999527i	$-0.490209\pi$
$283$	292.823	0.0615072	0.0307536	+	0.999527i	$0.490209\pi$
$284$	−3447.21	−0.720262
$285$	0	0
$286$	−156.386	−0.0323331
$287$	−5032.59	−1.03507
$288$	0	0
$289$	3818.80	0.777284
$290$	88.7383	0.0179686
$291$	0	0
$292$	6188.66	1.24029
$293$	7478.64	1.49115	0.745575	−	0.666422i	$-0.232175\pi$
$293$	7478.64	1.49115	0.745575	+	0.666422i	$0.232175\pi$
$294$	0	0
$295$	−769.273	−0.151826
$296$	−2113.27	−0.414971
$297$	0	0
$298$	1109.67	0.215709
$299$	−5236.81	−1.01288
$300$	0	0
$301$	−3870.94	−0.741254
$302$	2775.30	0.528810
$303$	0	0
$304$	4611.03	0.869937
$305$	1135.14	0.213107
$306$	0	0
$307$	−6555.64	−1.21873	−0.609365	−	0.792890i	$-0.708575\pi$
$307$	−6555.64	−1.21873	−0.609365	+	0.792890i	$0.708575\pi$
$308$	−693.798	−0.128353
$309$	0	0
$310$	−768.633	−0.140824
$311$	2631.21	0.479751	0.239875	−	0.970804i	$-0.422893\pi$
$311$	2631.21	0.479751	0.239875	+	0.970804i	$0.422893\pi$
$312$	0	0
$313$	−201.749	−0.0364330	−0.0182165	−	0.999834i	$-0.505799\pi$
$313$	−201.749	−0.0364330	−0.0182165	+	0.999834i	$0.505799\pi$
$314$	−1463.67	−0.263056
$315$	0	0
$316$	−2219.21	−0.395065
$317$	−4240.58	−0.751340	−0.375670	−	0.926754i	$-0.622587\pi$
$317$	−4240.58	−0.751340	−0.375670	+	0.926754i	$0.622587\pi$
$318$	0	0
$319$	153.378	0.0269201
$320$	−455.896	−0.0796418
$321$	0	0
$322$	3141.40	0.543675
$323$	−10249.3	−1.76560
$324$	0	0
$325$	4488.27	0.766045
$326$	676.836	0.114989
$327$	0	0
$328$	3181.06	0.535503
$329$	2731.88	0.457792
$330$	0	0
$331$	−3852.01	−0.639655	−0.319828	−	0.947476i	$-0.603625\pi$
$331$	−3852.01	−0.639655	−0.319828	+	0.947476i	$0.603625\pi$
$332$	3085.92	0.510126
$333$	0	0
$334$	−2393.12	−0.392054
$335$	754.735	0.123091
$336$	0	0
$337$	4495.42	0.726651	0.363325	−	0.931662i	$-0.381641\pi$
$337$	4495.42	0.726651	0.363325	+	0.931662i	$0.381641\pi$
$338$	−748.828	−0.120506
$339$	0	0
$340$	1653.59	0.263761
$341$	−1328.53	−0.210979
$342$	0	0
$343$	−3393.12	−0.534143
$344$	2446.79	0.383495
$345$	0	0
$346$	−42.7766	−0.00664649
$347$	−8509.26	−1.31643	−0.658214	−	0.752831i	$-0.728688\pi$
$347$	−8509.26	−1.31643	−0.658214	+	0.752831i	$0.728688\pi$
$348$	0	0
$349$	702.296	0.107716	0.0538582	−	0.998549i	$-0.482848\pi$
$349$	702.296	0.107716	0.0538582	+	0.998549i	$0.482848\pi$
$350$	−2692.37	−0.411181
$351$	0	0
$352$	671.703	0.101710
$353$	−1894.19	−0.285602	−0.142801	−	0.989751i	$-0.545611\pi$
$353$	−1894.19	−0.285602	−0.142801	+	0.989751i	$0.545611\pi$
$354$	0	0
$355$	1228.34	0.183644
$356$	−10281.2	−1.53063
$357$	0	0
$358$	434.815	0.0641918
$359$	−6566.75	−0.965404	−0.482702	−	0.875785i	$-0.660345\pi$
$359$	−6566.75	−0.965404	−0.482702	+	0.875785i	$0.660345\pi$
$360$	0	0
$361$	5171.63	0.753991
$362$	−1980.46	−0.287544
$363$	0	0
$364$	6192.28	0.891659
$365$	−2205.20	−0.316234
$366$	0	0
$367$	−8279.24	−1.17758	−0.588791	−	0.808285i	$-0.700396\pi$
$367$	−8279.24	−1.17758	−0.588791	+	0.808285i	$0.700396\pi$
$368$	5821.99	0.824708
$369$	0	0
$370$	352.667	0.0495522
$371$	4018.83	0.562391
$372$	0	0
$373$	270.675	0.0375738	0.0187869	−	0.999824i	$-0.494020\pi$
$373$	270.675	0.0375738	0.0187869	+	0.999824i	$0.494020\pi$
$374$	−386.455	−0.0534308
$375$	0	0
$376$	−1726.80	−0.236843
$377$	−1368.93	−0.187011
$378$	0	0
$379$	−13682.3	−1.85439	−0.927193	−	0.374584i	$-0.877786\pi$
$379$	−13682.3	−1.85439	−0.927193	+	0.374584i	$0.877786\pi$
$380$	−1940.98	−0.262027
$381$	0	0
$382$	−1994.43	−0.267130
$383$	−11819.4	−1.57688	−0.788439	−	0.615112i	$-0.789111\pi$
$383$	−11819.4	−1.57688	−0.788439	+	0.615112i	$0.789111\pi$
$384$	0	0
$385$	247.221	0.0327261
$386$	3099.64	0.408725
$387$	0	0
$388$	11075.8	1.44920
$389$	−2622.88	−0.341864	−0.170932	−	0.985283i	$-0.554678\pi$
$389$	−2622.88	−0.341864	−0.170932	+	0.985283i	$0.554678\pi$
$390$	0	0
$391$	−12941.1	−1.67380
$392$	−2893.30	−0.372790
$393$	0	0
$394$	−3676.94	−0.470156
$395$	790.771	0.100729
$396$	0	0
$397$	−11115.5	−1.40522	−0.702610	−	0.711575i	$-0.747982\pi$
$397$	−11115.5	−1.40522	−0.702610	+	0.711575i	$0.747982\pi$
$398$	−1801.94	−0.226943
$399$	0	0
$400$	−4989.81	−0.623727
$401$	13368.8	1.66485	0.832427	−	0.554135i	$-0.186951\pi$
$401$	13368.8	1.66485	0.832427	+	0.554135i	$0.186951\pi$
$402$	0	0
$403$	11857.4	1.46565
$404$	196.868	0.0242439
$405$	0	0
$406$	821.176	0.100380
$407$	609.560	0.0742378
$408$	0	0
$409$	1848.87	0.223523	0.111761	−	0.993735i	$-0.464351\pi$
$409$	1848.87	0.223523	0.111761	+	0.993735i	$0.464351\pi$
$410$	−530.863	−0.0639451
$411$	0	0
$412$	−1784.77	−0.213421
$413$	−7118.78	−0.848165
$414$	0	0
$415$	−1099.60	−0.130066
$416$	−5995.07	−0.706569
$417$	0	0
$418$	453.619	0.0530796
$419$	5326.04	0.620988	0.310494	−	0.950575i	$-0.399506\pi$
$419$	5326.04	0.620988	0.310494	+	0.950575i	$0.399506\pi$
$420$	0	0
$421$	−8961.11	−1.03738	−0.518691	−	0.854962i	$-0.673580\pi$
$421$	−8961.11	−1.03738	−0.518691	+	0.854962i	$0.673580\pi$
$422$	−707.067	−0.0815628
$423$	0	0
$424$	−2540.27	−0.290959
$425$	11091.3	1.26590
$426$	0	0
$427$	10504.4	1.19050
$428$	10190.0	1.15082
$429$	0	0
$430$	−408.327	−0.0457936
$431$	12440.4	1.39033	0.695163	−	0.718852i	$-0.255332\pi$
$431$	12440.4	1.39033	0.695163	+	0.718852i	$0.255332\pi$
$432$	0	0
$433$	11326.0	1.25703	0.628516	−	0.777797i	$-0.283663\pi$
$433$	11326.0	1.25703	0.628516	+	0.777797i	$0.283663\pi$
$434$	−7112.86	−0.786700
$435$	0	0
$436$	10624.4	1.16701
$437$	15190.1	1.66280
$438$	0	0
$439$	−305.470	−0.0332102	−0.0166051	−	0.999862i	$-0.505286\pi$
$439$	−305.470	−0.0332102	−0.0166051	+	0.999862i	$0.505286\pi$
$440$	−156.266	−0.0169311
$441$	0	0
$442$	3449.19	0.371179
$443$	12806.0	1.37343	0.686716	−	0.726926i	$-0.259052\pi$
$443$	12806.0	1.37343	0.686716	+	0.726926i	$0.259052\pi$
$444$	0	0
$445$	3663.50	0.390262
$446$	−3717.30	−0.394662
$447$	0	0
$448$	−4218.82	−0.444912
$449$	13219.7	1.38947	0.694737	−	0.719264i	$-0.255521\pi$
$449$	13219.7	1.38947	0.694737	+	0.719264i	$0.255521\pi$
$450$	0	0
$451$	−917.559	−0.0958009
$452$	−11551.0	−1.20202
$453$	0	0
$454$	1464.44	0.151386
$455$	−2206.49	−0.227345
$456$	0	0
$457$	6693.36	0.685126	0.342563	−	0.939495i	$-0.388705\pi$
$457$	6693.36	0.685126	0.342563	+	0.939495i	$0.388705\pi$
$458$	−4731.44	−0.482720
$459$	0	0
$460$	−2450.72	−0.248403
$461$	−10114.0	−1.02181	−0.510905	−	0.859637i	$-0.670689\pi$
$461$	−10114.0	−1.02181	−0.510905	+	0.859637i	$0.670689\pi$
$462$	0	0
$463$	16596.1	1.66585	0.832924	−	0.553387i	$-0.186665\pi$
$463$	16596.1	1.66585	0.832924	+	0.553387i	$0.186665\pi$
$464$	1521.90	0.152268
$465$	0	0
$466$	−2425.90	−0.241154
$467$	−5486.93	−0.543694	−0.271847	−	0.962341i	$-0.587634\pi$
$467$	−5486.93	−0.543694	−0.271847	+	0.962341i	$0.587634\pi$
$468$	0	0
$469$	6984.25	0.687639
$470$	288.173	0.0282818
$471$	0	0
$472$	4499.72	0.438806
$473$	−705.764	−0.0686069
$474$	0	0
$475$	−13018.9	−1.25757
$476$	15302.2	1.47348
$477$	0	0
$478$	−233.300	−0.0223240
$479$	−8449.72	−0.806008	−0.403004	−	0.915198i	$-0.632034\pi$
$479$	−8449.72	−0.806008	−0.403004	+	0.915198i	$0.632034\pi$
$480$	0	0
$481$	−5440.44	−0.515723
$482$	−6208.49	−0.586699
$483$	0	0
$484$	9253.24	0.869012
$485$	−3946.64	−0.369500
$486$	0	0
$487$	15114.2	1.40634	0.703170	−	0.711022i	$-0.251767\pi$
$487$	15114.2	1.40634	0.703170	+	0.711022i	$0.251767\pi$
$488$	−6639.78	−0.615919
$489$	0	0
$490$	482.841	0.0445154
$491$	6023.76	0.553663	0.276832	−	0.960918i	$-0.410716\pi$
$491$	6023.76	0.553663	0.276832	+	0.960918i	$0.410716\pi$
$492$	0	0
$493$	−3382.85	−0.309038
$494$	−4048.64	−0.368739
$495$	0	0
$496$	−13182.4	−1.19336
$497$	11367.0	1.02591
$498$	0	0
$499$	−9394.43	−0.842791	−0.421395	−	0.906877i	$-0.638460\pi$
$499$	−9394.43	−0.842791	−0.421395	+	0.906877i	$0.638460\pi$
$500$	4312.43	0.385715
$501$	0	0
$502$	−5650.50	−0.502379
$503$	6233.17	0.552532	0.276266	−	0.961081i	$-0.410903\pi$
$503$	6233.17	0.552532	0.276266	+	0.961081i	$0.410903\pi$
$504$	0	0
$505$	−70.1497	−0.00618143
$506$	572.750	0.0503199
$507$	0	0
$508$	9868.69	0.861914
$509$	12428.9	1.08232	0.541159	−	0.840920i	$-0.317986\pi$
$509$	12428.9	1.08232	0.541159	+	0.840920i	$0.317986\pi$
$510$	0	0
$511$	−20406.7	−1.76662
$512$	11604.8	1.00169
$513$	0	0
$514$	1985.89	0.170416
$515$	635.967	0.0544157
$516$	0	0
$517$	498.086	0.0423710
$518$	3263.55	0.276819
$519$	0	0
$520$	1394.71	0.117619
$521$	−21485.4	−1.80670	−0.903350	−	0.428904i	$-0.858900\pi$
$521$	−21485.4	−1.80670	−0.903350	+	0.428904i	$0.858900\pi$
$522$	0	0
$523$	−12446.8	−1.04065	−0.520327	−	0.853967i	$-0.674190\pi$
$523$	−12446.8	−1.04065	−0.520327	+	0.853967i	$0.674190\pi$
$524$	−5903.81	−0.492193
$525$	0	0
$526$	−7571.61	−0.627638
$527$	29301.6	2.42200
$528$	0	0
$529$	7012.42	0.576347
$530$	423.926	0.0347437
$531$	0	0
$532$	−17961.6	−1.46379
$533$	8189.39	0.665520
$534$	0	0
$535$	−3630.98	−0.293422
$536$	−4414.69	−0.355757
$537$	0	0
$538$	−6821.71	−0.546663
$539$	834.556	0.0666918
$540$	0	0
$541$	14328.1	1.13866	0.569328	−	0.822110i	$-0.307203\pi$
$541$	14328.1	1.13866	0.569328	+	0.822110i	$0.307203\pi$
$542$	−2980.63	−0.236216
$543$	0	0
$544$	−14814.9	−1.16761
$545$	−3785.78	−0.297550
$546$	0	0
$547$	5027.35	0.392969	0.196484	−	0.980507i	$-0.437048\pi$
$547$	5027.35	0.392969	0.196484	+	0.980507i	$0.437048\pi$
$548$	−11114.5	−0.866405
$549$	0	0
$550$	−490.883	−0.0380569
$551$	3970.77	0.307007
$552$	0	0
$553$	7317.72	0.562714
$554$	6577.61	0.504433
$555$	0	0
$556$	16192.6	1.23510
$557$	−10522.6	−0.800461	−0.400230	−	0.916415i	$-0.631070\pi$
$557$	−10522.6	−0.800461	−0.400230	+	0.916415i	$0.631070\pi$
$558$	0	0
$559$	6299.07	0.476605
$560$	2453.06	0.185108
$561$	0	0
$562$	4209.19	0.315932
$563$	15124.0	1.13215	0.566076	−	0.824353i	$-0.308461\pi$
$563$	15124.0	1.13215	0.566076	+	0.824353i	$0.308461\pi$
$564$	0	0
$565$	4115.96	0.306478
$566$	285.839	0.0212274
$567$	0	0
$568$	−7184.97	−0.530765
$569$	25227.0	1.85865	0.929326	−	0.369260i	$-0.120389\pi$
$569$	25227.0	1.85865	0.929326	+	0.369260i	$0.120389\pi$
$570$	0	0
$571$	−16718.5	−1.22530	−0.612650	−	0.790354i	$-0.709896\pi$
$571$	−16718.5	−1.22530	−0.612650	+	0.790354i	$0.709896\pi$
$572$	1129.00	0.0825276
$573$	0	0
$574$	−4912.56	−0.357223
$575$	−16438.0	−1.19219
$576$	0	0
$577$	4804.27	0.346628	0.173314	−	0.984867i	$-0.444552\pi$
$577$	4804.27	0.346628	0.173314	+	0.984867i	$0.444552\pi$
$578$	3727.72	0.268257
$579$	0	0
$580$	−640.630	−0.0458633
$581$	−10175.6	−0.726603
$582$	0	0
$583$	732.726	0.0520522
$584$	12898.9	0.913976
$585$	0	0
$586$	7300.27	0.514627
$587$	3426.00	0.240896	0.120448	−	0.992720i	$-0.461567\pi$
$587$	3426.00	0.240896	0.120448	+	0.992720i	$0.461567\pi$
$588$	0	0
$589$	−34394.0	−2.40608
$590$	−750.924	−0.0523984
$591$	0	0
$592$	6048.38	0.419910
$593$	2088.23	0.144609	0.0723046	−	0.997383i	$-0.476965\pi$
$593$	2088.23	0.144609	0.0723046	+	0.997383i	$0.476965\pi$
$594$	0	0
$595$	−5452.62	−0.375690
$596$	−8011.04	−0.550579
$597$	0	0
$598$	−5111.90	−0.349567
$599$	14495.1	0.988739	0.494369	−	0.869252i	$-0.335399\pi$
$599$	14495.1	0.988739	0.494369	+	0.869252i	$0.335399\pi$
$600$	0	0
$601$	3752.69	0.254701	0.127350	−	0.991858i	$-0.459353\pi$
$601$	3752.69	0.254701	0.127350	+	0.991858i	$0.459353\pi$
$602$	−3778.62	−0.255822
$603$	0	0
$604$	−20035.8	−1.34974
$605$	−3297.20	−0.221571
$606$	0	0
$607$	11914.7	0.796712	0.398356	−	0.917231i	$-0.369581\pi$
$607$	11914.7	0.796712	0.398356	+	0.917231i	$0.369581\pi$
$608$	17389.6	1.15994
$609$	0	0
$610$	1108.06	0.0735477
$611$	−4445.52	−0.294348
$612$	0	0
$613$	4403.34	0.290129	0.145064	−	0.989422i	$-0.453661\pi$
$613$	4403.34	0.290129	0.145064	+	0.989422i	$0.453661\pi$
$614$	−6399.28	−0.420609
$615$	0	0
$616$	−1446.07	−0.0945844
$617$	1126.89	0.0735279	0.0367640	−	0.999324i	$-0.488295\pi$
$617$	1126.89	0.0735279	0.0367640	+	0.999324i	$0.488295\pi$
$618$	0	0
$619$	7478.09	0.485573	0.242787	−	0.970080i	$-0.421939\pi$
$619$	7478.09	0.485573	0.242787	+	0.970080i	$0.421939\pi$
$620$	5549.01	0.359441
$621$	0	0
$622$	2568.46	0.165572
$623$	33901.7	2.18016
$624$	0	0
$625$	13300.2	0.851210
$626$	−196.937	−0.0125738
$627$	0	0
$628$	10566.7	0.671429
$629$	−13444.3	−0.852238
$630$	0	0
$631$	295.066	0.0186155	0.00930775	−	0.999957i	$-0.497037\pi$
$631$	295.066	0.0186155	0.00930775	+	0.999957i	$0.497037\pi$
$632$	−4625.47	−0.291126
$633$	0	0
$634$	−4139.44	−0.259303
$635$	−3516.50	−0.219761
$636$	0	0
$637$	−7448.57	−0.463302
$638$	149.720	0.00929068
$639$	0	0
$640$	−3629.95	−0.224197
$641$	−10599.5	−0.653129	−0.326564	−	0.945175i	$-0.605891\pi$
$641$	−10599.5	−0.653129	−0.326564	+	0.945175i	$0.605891\pi$
$642$	0	0
$643$	586.645	0.0359798	0.0179899	−	0.999838i	$-0.494273\pi$
$643$	586.645	0.0359798	0.0179899	+	0.999838i	$0.494273\pi$
$644$	−22678.8	−1.38768
$645$	0	0
$646$	−10004.9	−0.609345
$647$	4354.14	0.264574	0.132287	−	0.991211i	$-0.457768\pi$
$647$	4354.14	0.264574	0.132287	+	0.991211i	$0.457768\pi$
$648$	0	0
$649$	−1297.92	−0.0785020
$650$	4381.22	0.264378
$651$	0	0
$652$	−4886.30	−0.293500
$653$	−8268.30	−0.495503	−0.247752	−	0.968824i	$-0.579692\pi$
$653$	−8268.30	−0.495503	−0.247752	+	0.968824i	$0.579692\pi$
$654$	0	0
$655$	2103.70	0.125494
$656$	−9104.51	−0.541877
$657$	0	0
$658$	2666.73	0.157994
$659$	−16681.6	−0.986073	−0.493036	−	0.870009i	$-0.664113\pi$
$659$	−16681.6	−0.986073	−0.493036	+	0.870009i	$0.664113\pi$
$660$	0	0
$661$	4800.66	0.282487	0.141244	−	0.989975i	$-0.454890\pi$
$661$	4800.66	0.282487	0.141244	+	0.989975i	$0.454890\pi$
$662$	−3760.14	−0.220758
$663$	0	0
$664$	6431.94	0.375915
$665$	6400.26	0.373220
$666$	0	0
$667$	5013.59	0.291045
$668$	17276.7	1.00068
$669$	0	0
$670$	736.734	0.0424814
$671$	1915.20	0.110187
$672$	0	0
$673$	−15573.7	−0.892008	−0.446004	−	0.895031i	$-0.647153\pi$
$673$	−15573.7	−0.892008	−0.446004	+	0.895031i	$0.647153\pi$
$674$	4388.20	0.250782
$675$	0	0
$676$	5406.03	0.307580
$677$	−277.024	−0.0157266	−0.00786329	−	0.999969i	$-0.502503\pi$
$677$	−277.024	−0.0157266	−0.00786329	+	0.999969i	$0.502503\pi$
$678$	0	0
$679$	−36521.8	−2.06418
$680$	3446.56	0.194367
$681$	0	0
$682$	−1296.84	−0.0728131
$683$	−9968.69	−0.558479	−0.279240	−	0.960221i	$-0.590082\pi$
$683$	−9968.69	−0.558479	−0.279240	+	0.960221i	$0.590082\pi$
$684$	0	0
$685$	3960.44	0.220906
$686$	−3312.19	−0.184344
$687$	0	0
$688$	−7002.96	−0.388060
$689$	−6539.72	−0.361602
$690$	0	0
$691$	2380.80	0.131071	0.0655355	−	0.997850i	$-0.479124\pi$
$691$	2380.80	0.131071	0.0655355	+	0.997850i	$0.479124\pi$
$692$	308.818	0.0169646
$693$	0	0
$694$	−8306.30	−0.454327
$695$	−5769.89	−0.314913
$696$	0	0
$697$	20237.4	1.09978
$698$	685.546	0.0371752
$699$	0	0
$700$	19437.1	1.04951
$701$	19847.7	1.06938	0.534692	−	0.845047i	$-0.320428\pi$
$701$	19847.7	1.06938	0.534692	+	0.845047i	$0.320428\pi$
$702$	0	0
$703$	15780.8	0.846635
$704$	−769.190	−0.0411789
$705$	0	0
$706$	−1849.01	−0.0985672
$707$	−649.159	−0.0345320
$708$	0	0
$709$	23792.5	1.26029	0.630146	−	0.776477i	$-0.282995\pi$
$709$	23792.5	1.26029	0.630146	+	0.776477i	$0.282995\pi$
$710$	1199.04	0.0633794
$711$	0	0
$712$	−21429.0	−1.12793
$713$	−43426.7	−2.28098
$714$	0	0
$715$	−402.295	−0.0210419
$716$	−3139.07	−0.163844
$717$	0	0
$718$	−6410.13	−0.333181
$719$	4682.16	0.242858	0.121429	−	0.992600i	$-0.461252\pi$
$719$	4682.16	0.242858	0.121429	+	0.992600i	$0.461252\pi$
$720$	0	0
$721$	5885.18	0.303988
$722$	5048.28	0.260218
$723$	0	0
$724$	14297.6	0.733932
$725$	−4296.96	−0.220117
$726$	0	0
$727$	18986.7	0.968608	0.484304	−	0.874900i	$-0.339073\pi$
$727$	18986.7	0.968608	0.484304	+	0.874900i	$0.339073\pi$
$728$	12906.5	0.657069
$729$	0	0
$730$	−2152.60	−0.109139
$731$	15566.1	0.787596
$732$	0	0
$733$	−5165.38	−0.260283	−0.130142	−	0.991495i	$-0.541543\pi$
$733$	−5165.38	−0.260283	−0.130142	+	0.991495i	$0.541543\pi$
$734$	−8081.77	−0.406408
$735$	0	0
$736$	21956.5	1.09963
$737$	1273.39	0.0636445
$738$	0	0
$739$	−3311.05	−0.164816	−0.0824079	−	0.996599i	$-0.526261\pi$
$739$	−3311.05	−0.164816	−0.0824079	+	0.996599i	$0.526261\pi$
$740$	−2546.02	−0.126478
$741$	0	0
$742$	3922.97	0.194093
$743$	−1827.40	−0.0902300	−0.0451150	−	0.998982i	$-0.514365\pi$
$743$	−1827.40	−0.0902300	−0.0451150	+	0.998982i	$0.514365\pi$
$744$	0	0
$745$	2854.57	0.140380
$746$	264.219	0.0129675
$747$	0	0
$748$	2789.95	0.136378
$749$	−33600.7	−1.63918
$750$	0	0
$751$	11503.1	0.558929	0.279464	−	0.960156i	$-0.409843\pi$
$751$	11503.1	0.558929	0.279464	+	0.960156i	$0.409843\pi$
$752$	4942.28	0.239663
$753$	0	0
$754$	−1336.28	−0.0645415
$755$	7139.35	0.344142
$756$	0	0
$757$	3435.51	0.164948	0.0824740	−	0.996593i	$-0.473718\pi$
$757$	3435.51	0.164948	0.0824740	+	0.996593i	$0.473718\pi$
$758$	−13356.0	−0.639987
$759$	0	0
$760$	−4045.55	−0.193089
$761$	−38724.0	−1.84461	−0.922303	−	0.386468i	$-0.873695\pi$
$761$	−38724.0	−1.84461	−0.922303	+	0.386468i	$0.873695\pi$
$762$	0	0
$763$	−35033.2	−1.66224
$764$	14398.4	0.681827
$765$	0	0
$766$	−11537.5	−0.544214
$767$	11584.2	0.545346
$768$	0	0
$769$	−14913.7	−0.699351	−0.349676	−	0.936871i	$-0.613708\pi$
$769$	−14913.7	−0.699351	−0.349676	+	0.936871i	$0.613708\pi$
$770$	241.324	0.0112944
$771$	0	0
$772$	−22377.3	−1.04324
$773$	−31444.2	−1.46309	−0.731545	−	0.681793i	$-0.761200\pi$
$773$	−31444.2	−1.46309	−0.731545	+	0.681793i	$0.761200\pi$
$774$	0	0
$775$	37219.4	1.72511
$776$	23085.2	1.06792
$777$	0	0
$778$	−2560.32	−0.117984
$779$	−23754.5	−1.09255
$780$	0	0
$781$	2072.46	0.0949534
$782$	−12632.4	−0.577664
$783$	0	0
$784$	8280.91	0.377228
$785$	−3765.23	−0.171193
$786$	0	0
$787$	−4795.01	−0.217184	−0.108592	−	0.994086i	$-0.534634\pi$
$787$	−4795.01	−0.217184	−0.108592	+	0.994086i	$0.534634\pi$
$788$	26545.0	1.20003
$789$	0	0
$790$	771.910	0.0347637
$791$	38088.7	1.71211
$792$	0	0
$793$	−17093.6	−0.765461
$794$	−10850.4	−0.484971
$795$	0	0
$796$	13008.8	0.579252
$797$	37451.4	1.66449	0.832244	−	0.554409i	$-0.187056\pi$
$797$	37451.4	1.66449	0.832244	+	0.554409i	$0.187056\pi$
$798$	0	0
$799$	−10985.6	−0.486413
$800$	−18818.1	−0.831650
$801$	0	0
$802$	13049.9	0.574575
$803$	−3720.62	−0.163509
$804$	0	0
$805$	8081.11	0.353816
$806$	11574.5	0.505826
$807$	0	0
$808$	410.328	0.0178655
$809$	29648.5	1.28849	0.644244	−	0.764820i	$-0.277172\pi$
$809$	29648.5	1.28849	0.644244	+	0.764820i	$0.277172\pi$
$810$	0	0
$811$	4515.07	0.195494	0.0977469	−	0.995211i	$-0.468836\pi$
$811$	4515.07	0.195494	0.0977469	+	0.995211i	$0.468836\pi$
$812$	−5928.33	−0.256211
$813$	0	0
$814$	595.021	0.0256210
$815$	1741.13	0.0748334
$816$	0	0
$817$	−18271.4	−0.782418
$818$	1804.77	0.0771424
$819$	0	0
$820$	3832.47	0.163214
$821$	9171.22	0.389863	0.194932	−	0.980817i	$-0.437552\pi$
$821$	9171.22	0.389863	0.194932	+	0.980817i	$0.437552\pi$
$822$	0	0
$823$	26386.7	1.11760	0.558798	−	0.829304i	$-0.311263\pi$
$823$	26386.7	1.11760	0.558798	+	0.829304i	$0.311263\pi$
$824$	−3719.98	−0.157271
$825$	0	0
$826$	−6948.98	−0.292719
$827$	41044.1	1.72581	0.862905	−	0.505367i	$-0.168643\pi$
$827$	41044.1	1.72581	0.862905	+	0.505367i	$0.168643\pi$
$828$	0	0
$829$	−35648.5	−1.49352	−0.746758	−	0.665096i	$-0.768391\pi$
$829$	−35648.5	−1.49352	−0.746758	+	0.665096i	$0.768391\pi$
$830$	−1073.38	−0.0448885
$831$	0	0
$832$	6865.16	0.286066
$833$	−18406.7	−0.765611
$834$	0	0
$835$	−6156.21	−0.255143
$836$	−3274.83	−0.135481
$837$	0	0
$838$	5199.00	0.214316
$839$	−37723.6	−1.55228	−0.776140	−	0.630560i	$-0.782825\pi$
$839$	−37723.6	−1.55228	−0.776140	+	0.630560i	$0.782825\pi$
$840$	0	0
$841$	−23078.4	−0.946264
$842$	−8747.38	−0.358022
$843$	0	0
$844$	5104.55	0.208182
$845$	−1926.33	−0.0784233
$846$	0	0
$847$	−30512.0	−1.23779
$848$	7270.50	0.294422
$849$	0	0
$850$	10826.7	0.436887
$851$	19925.2	0.802617
$852$	0	0
$853$	6360.23	0.255299	0.127650	−	0.991819i	$-0.459257\pi$
$853$	6360.23	0.255299	0.127650	+	0.991819i	$0.459257\pi$
$854$	10253.9	0.410868
$855$	0	0
$856$	21238.8	0.848045
$857$	36763.2	1.46535	0.732677	−	0.680576i	$-0.238271\pi$
$857$	36763.2	1.46535	0.732677	+	0.680576i	$0.238271\pi$
$858$	0	0
$859$	9073.94	0.360418	0.180209	−	0.983628i	$-0.442323\pi$
$859$	9073.94	0.360418	0.180209	+	0.983628i	$0.442323\pi$
$860$	2947.84	0.116884
$861$	0	0
$862$	12143.6	0.479831
$863$	−42127.7	−1.66170	−0.830848	−	0.556500i	$-0.812144\pi$
$863$	−42127.7	−1.66170	−0.830848	+	0.556500i	$0.812144\pi$
$864$	0	0
$865$	−110.041	−0.00432544
$866$	11055.9	0.433828
$867$	0	0
$868$	51350.0	2.00799
$869$	1334.19	0.0520821
$870$	0	0
$871$	−11365.3	−0.442132
$872$	22144.2	0.859976
$873$	0	0
$874$	14827.8	0.573866
$875$	−14220.0	−0.549397
$876$	0	0
$877$	−48611.9	−1.87173	−0.935865	−	0.352358i	$-0.885380\pi$
$877$	−48611.9	−1.87173	−0.935865	+	0.352358i	$0.885380\pi$
$878$	−298.184	−0.0114615
$879$	0	0
$880$	447.249	0.0171327
$881$	26493.5	1.01316	0.506578	−	0.862194i	$-0.330910\pi$
$881$	26493.5	1.01316	0.506578	+	0.862194i	$0.330910\pi$
$882$	0	0
$883$	−13915.9	−0.530361	−0.265180	−	0.964199i	$-0.585431\pi$
$883$	−13915.9	−0.530361	−0.265180	+	0.964199i	$0.585431\pi$
$884$	−24900.8	−0.947404
$885$	0	0
$886$	12500.5	0.474000
$887$	4037.63	0.152841	0.0764207	−	0.997076i	$-0.475651\pi$
$887$	4037.63	0.152841	0.0764207	+	0.997076i	$0.475651\pi$
$888$	0	0
$889$	−32541.4	−1.22768
$890$	3576.12	0.134687
$891$	0	0
$892$	26836.4	1.00734
$893$	12894.9	0.483215
$894$	0	0
$895$	1118.54	0.0417751
$896$	−33591.2	−1.25246
$897$	0	0
$898$	12904.3	0.479536
$899$	−11351.9	−0.421144
$900$	0	0
$901$	−16160.8	−0.597551
$902$	−895.674	−0.0330629
$903$	0	0
$904$	−24075.6	−0.885777
$905$	−5094.66	−0.187130
$906$	0	0
$907$	26661.3	0.976046	0.488023	−	0.872831i	$-0.337718\pi$
$907$	26661.3	0.976046	0.488023	+	0.872831i	$0.337718\pi$
$908$	−10572.2	−0.386401
$909$	0	0
$910$	−2153.86	−0.0784614
$911$	−22777.6	−0.828381	−0.414190	−	0.910190i	$-0.635935\pi$
$911$	−22777.6	−0.828381	−0.414190	+	0.910190i	$0.635935\pi$
$912$	0	0
$913$	−1855.26	−0.0672509
$914$	6533.72	0.236451
$915$	0	0
$916$	34157.8	1.23210
$917$	19467.4	0.701059
$918$	0	0
$919$	−408.187	−0.0146516	−0.00732581	−	0.999973i	$-0.502332\pi$
$919$	−408.187	−0.0146516	−0.00732581	+	0.999973i	$0.502332\pi$
$920$	−5108.01	−0.183050
$921$	0	0
$922$	−9872.73	−0.352647
$923$	−18497.1	−0.659632
$924$	0	0
$925$	−17077.1	−0.607020
$926$	16200.3	0.574919
$927$	0	0
$928$	5739.53	0.203027
$929$	−46017.5	−1.62517	−0.812586	−	0.582841i	$-0.801941\pi$
$929$	−46017.5	−1.62517	−0.812586	+	0.582841i	$0.801941\pi$
$930$	0	0
$931$	21605.7	0.760578
$932$	17513.4	0.615526
$933$	0	0
$934$	−5356.06	−0.187640
$935$	−994.140	−0.0347720
$936$	0	0
$937$	11682.9	0.407324	0.203662	−	0.979041i	$-0.434716\pi$
$937$	11682.9	0.407324	0.203662	+	0.979041i	$0.434716\pi$
$938$	6817.67	0.237318
$939$	0	0
$940$	−2080.41	−0.0721868
$941$	43127.8	1.49408	0.747038	−	0.664781i	$-0.231475\pi$
$941$	43127.8	1.49408	0.747038	+	0.664781i	$0.231475\pi$
$942$	0	0
$943$	−29993.0	−1.03574
$944$	−12878.6	−0.444030
$945$	0	0
$946$	−688.930	−0.0236776
$947$	−36008.0	−1.23559	−0.617795	−	0.786339i	$-0.711974\pi$
$947$	−36008.0	−1.23559	−0.617795	+	0.786339i	$0.711974\pi$
$948$	0	0
$949$	33207.3	1.13588
$950$	−12708.4	−0.434015
$951$	0	0
$952$	31894.1	1.08581
$953$	−25875.8	−0.879538	−0.439769	−	0.898111i	$-0.644940\pi$
$953$	−25875.8	−0.879538	−0.439769	+	0.898111i	$0.644940\pi$
$954$	0	0
$955$	−5130.57	−0.173844
$956$	1684.26	0.0569801
$957$	0	0
$958$	−8248.19	−0.278170
$959$	36649.5	1.23407
$960$	0	0
$961$	68537.1	2.30060
$962$	−5310.68	−0.177987
$963$	0	0
$964$	44821.1	1.49750
$965$	7973.70	0.265992
$966$	0	0
$967$	−2361.85	−0.0785440	−0.0392720	−	0.999229i	$-0.512504\pi$
$967$	−2361.85	−0.0785440	−0.0392720	+	0.999229i	$0.512504\pi$
$968$	19286.4	0.640380
$969$	0	0
$970$	−3852.50	−0.127522
$971$	35962.1	1.18855	0.594273	−	0.804264i	$-0.297440\pi$
$971$	35962.1	1.18855	0.594273	+	0.804264i	$0.297440\pi$
$972$	0	0
$973$	−53394.0	−1.75923
$974$	14753.7	0.485357
$975$	0	0
$976$	19003.7	0.623251
$977$	−4482.88	−0.146796	−0.0733982	−	0.997303i	$-0.523384\pi$
$977$	−4482.88	−0.146796	−0.0733982	+	0.997303i	$0.523384\pi$
$978$	0	0
$979$	6181.07	0.201785
$980$	−3485.78	−0.113622
$981$	0	0
$982$	5880.09	0.191081
$983$	36130.0	1.17230	0.586148	−	0.810204i	$-0.300644\pi$
$983$	36130.0	1.17230	0.586148	+	0.810204i	$0.300644\pi$
$984$	0	0
$985$	−9458.76	−0.305971
$986$	−3302.16	−0.106656
$987$	0	0
$988$	29228.4	0.941175
$989$	−23069.9	−0.741739
$990$	0	0
$991$	37030.9	1.18701	0.593503	−	0.804831i	$-0.297744\pi$
$991$	37030.9	1.18701	0.593503	+	0.804831i	$0.297744\pi$
$992$	−49714.6	−1.59117
$993$	0	0
$994$	11095.8	0.354063
$995$	−4635.42	−0.147691
$996$	0	0
$997$	−40087.4	−1.27340	−0.636701	−	0.771111i	$-0.719701\pi$
$997$	−40087.4	−1.27340	−0.636701	+	0.771111i	$0.719701\pi$
$998$	−9170.36	−0.290865
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.13		22
3.2	odd	2		239.4.a.a.1.10	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.10	✓	22	3.2	odd	2
2151.4.a.a.1.13		22	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.976149 q^{2} -7.04713 q^{4} +2.51110 q^{5} +23.2375 q^{7} -14.6882 q^{8} +O(q^{10})\)	\(q+0.976149 q^{2} -7.04713 q^{4} +2.51110 q^{5} +23.2375 q^{7} -14.6882 q^{8} +2.45121 q^{10} +4.23674 q^{11} -37.8137 q^{13} +22.6832 q^{14} +42.0392 q^{16} -93.4441 q^{17} +109.684 q^{19} -17.6961 q^{20} +4.13569 q^{22} +138.490 q^{23} -118.694 q^{25} -36.9118 q^{26} -163.758 q^{28} +36.2019 q^{29} -313.573 q^{31} +158.542 q^{32} -91.2153 q^{34} +58.3517 q^{35} +143.875 q^{37} +107.068 q^{38} -36.8836 q^{40} -216.572 q^{41} -166.582 q^{43} -29.8569 q^{44} +135.187 q^{46} +117.564 q^{47} +196.981 q^{49} -115.863 q^{50} +266.478 q^{52} +172.946 q^{53} +10.6389 q^{55} -341.318 q^{56} +35.3384 q^{58} -306.349 q^{59} +452.047 q^{61} -306.094 q^{62} -181.552 q^{64} -94.9540 q^{65} +300.560 q^{67} +658.513 q^{68} +56.9599 q^{70} +489.165 q^{71} -878.181 q^{73} +140.443 q^{74} -772.959 q^{76} +98.4511 q^{77} +314.910 q^{79} +105.565 q^{80} -211.407 q^{82} -437.897 q^{83} -234.647 q^{85} -162.609 q^{86} -62.2302 q^{88} +1458.92 q^{89} -878.695 q^{91} -975.956 q^{92} +114.760 q^{94} +275.428 q^{95} -1571.68 q^{97} +192.283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

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