LMFDB - Embedded newform 2151.4.a.b.1.9

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:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.61692 q^{2} -5.38557 q^{4} -9.90974 q^{5} -20.2386 q^{7} +21.6434 q^{8} +O(q^{10})$	\(q-1.61692 q^{2} -5.38557 q^{4} -9.90974 q^{5} -20.2386 q^{7} +21.6434 q^{8} +16.0233 q^{10} -0.449426 q^{11} +70.3224 q^{13} +32.7242 q^{14} +8.08893 q^{16} -134.632 q^{17} -66.4130 q^{19} +53.3696 q^{20} +0.726686 q^{22} -12.7265 q^{23} -26.7971 q^{25} -113.706 q^{26} +108.997 q^{28} +248.943 q^{29} -142.046 q^{31} -186.226 q^{32} +217.689 q^{34} +200.559 q^{35} +15.0524 q^{37} +107.384 q^{38} -214.480 q^{40} +121.579 q^{41} -68.8365 q^{43} +2.42042 q^{44} +20.5777 q^{46} -107.380 q^{47} +66.6017 q^{49} +43.3287 q^{50} -378.726 q^{52} +640.459 q^{53} +4.45370 q^{55} -438.032 q^{56} -402.520 q^{58} +477.849 q^{59} -79.9338 q^{61} +229.676 q^{62} +236.402 q^{64} -696.877 q^{65} +119.927 q^{67} +725.068 q^{68} -324.289 q^{70} +682.298 q^{71} +564.694 q^{73} -24.3385 q^{74} +357.672 q^{76} +9.09577 q^{77} +828.051 q^{79} -80.1591 q^{80} -196.584 q^{82} +1066.49 q^{83} +1334.16 q^{85} +111.303 q^{86} -9.72711 q^{88} +738.470 q^{89} -1423.23 q^{91} +68.5394 q^{92} +173.624 q^{94} +658.135 q^{95} +725.871 q^{97} -107.690 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * 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 $

$2$	−1.61692	−0.571668	−0.285834	−	0.958279i	$-0.592271\pi$
$2$	−1.61692	−0.571668	−0.285834	+	0.958279i	$0.592271\pi$
$3$	0	0
$3$	0	0
$4$	−5.38557	−0.673196
$5$	−9.90974	−0.886354	−0.443177	−	0.896434i	$-0.646149\pi$
$5$	−9.90974	−0.886354	−0.443177	+	0.896434i	$0.646149\pi$
$6$	0	0
$7$	−20.2386	−1.09278	−0.546391	−	0.837530i	$-0.683999\pi$
$7$	−20.2386	−1.09278	−0.546391	+	0.837530i	$0.683999\pi$
$8$	21.6434	0.956512
$9$	0	0
$10$	16.0233	0.506700
$11$	−0.449426	−0.0123188	−0.00615941	−	0.999981i	$-0.501961\pi$
$11$	−0.449426	−0.0123188	−0.00615941	+	0.999981i	$0.501961\pi$
$12$	0	0
$13$	70.3224	1.50030	0.750151	−	0.661267i	$-0.229981\pi$
$13$	70.3224	1.50030	0.750151	+	0.661267i	$0.229981\pi$
$14$	32.7242	0.624708
$15$	0	0
$16$	8.08893	0.126389
$17$	−134.632	−1.92076	−0.960382	−	0.278688i	$-0.910100\pi$
$17$	−134.632	−1.92076	−0.960382	+	0.278688i	$0.910100\pi$
$18$	0	0
$19$	−66.4130	−0.801904	−0.400952	−	0.916099i	$-0.631321\pi$
$19$	−66.4130	−0.801904	−0.400952	+	0.916099i	$0.631321\pi$
$20$	53.3696	0.596690
$21$	0	0
$22$	0.726686	0.00704227
$23$	−12.7265	−0.115376	−0.0576882	−	0.998335i	$-0.518373\pi$
$23$	−12.7265	−0.115376	−0.0576882	+	0.998335i	$0.518373\pi$
$24$	0	0
$25$	−26.7971	−0.214377
$26$	−113.706	−0.857674
$27$	0	0
$28$	108.997	0.735657
$29$	248.943	1.59405	0.797026	−	0.603945i	$-0.206405\pi$
$29$	248.943	1.59405	0.797026	+	0.603945i	$0.206405\pi$
$30$	0	0
$31$	−142.046	−0.822973	−0.411486	−	0.911416i	$-0.634990\pi$
$31$	−142.046	−0.822973	−0.411486	+	0.911416i	$0.634990\pi$
$32$	−186.226	−1.02876
$33$	0	0
$34$	217.689	1.09804
$35$	200.559	0.968592
$36$	0	0
$37$	15.0524	0.0668809	0.0334404	−	0.999441i	$-0.489354\pi$
$37$	15.0524	0.0668809	0.0334404	+	0.999441i	$0.489354\pi$
$38$	107.384	0.458423
$39$	0	0
$40$	−214.480	−0.847808
$41$	121.579	0.463110	0.231555	−	0.972822i	$-0.425619\pi$
$41$	121.579	0.463110	0.231555	+	0.972822i	$0.425619\pi$
$42$	0	0
$43$	−68.8365	−0.244127	−0.122064	−	0.992522i	$-0.538951\pi$
$43$	−68.8365	−0.244127	−0.122064	+	0.992522i	$0.538951\pi$
$44$	2.42042	0.00829299
$45$	0	0
$46$	20.5777	0.0659569
$47$	−107.380	−0.333254	−0.166627	−	0.986020i	$-0.553288\pi$
$47$	−107.380	−0.333254	−0.166627	+	0.986020i	$0.553288\pi$
$48$	0	0
$49$	66.6017	0.194174
$50$	43.3287	0.122552
$51$	0	0
$52$	−378.726	−1.01000
$53$	640.459	1.65988	0.829942	−	0.557850i	$-0.188374\pi$
$53$	640.459	1.65988	0.829942	+	0.557850i	$0.188374\pi$
$54$	0	0
$55$	4.45370	0.0109188
$56$	−438.032	−1.04526
$57$	0	0
$58$	−402.520	−0.911267
$59$	477.849	1.05442	0.527209	−	0.849736i	$-0.323238\pi$
$59$	477.849	1.05442	0.527209	+	0.849736i	$0.323238\pi$
$60$	0	0
$61$	−79.9338	−0.167778	−0.0838892	−	0.996475i	$-0.526734\pi$
$61$	−79.9338	−0.167778	−0.0838892	+	0.996475i	$0.526734\pi$
$62$	229.676	0.470467
$63$	0	0
$64$	236.402	0.461722
$65$	−696.877	−1.32980
$66$	0	0
$67$	119.927	0.218678	0.109339	−	0.994005i	$-0.465127\pi$
$67$	119.927	0.218678	0.109339	+	0.994005i	$0.465127\pi$
$68$	725.068	1.29305
$69$	0	0
$70$	−324.289	−0.553713
$71$	682.298	1.14048	0.570239	−	0.821479i	$-0.306851\pi$
$71$	682.298	1.14048	0.570239	+	0.821479i	$0.306851\pi$
$72$	0	0
$73$	564.694	0.905375	0.452688	−	0.891669i	$-0.350465\pi$
$73$	564.694	0.905375	0.452688	+	0.891669i	$0.350465\pi$
$74$	−24.3385	−0.0382336
$75$	0	0
$76$	357.672	0.539839
$77$	9.09577	0.0134618
$78$	0	0
$79$	828.051	1.17928	0.589640	−	0.807666i	$-0.299270\pi$
$79$	828.051	1.17928	0.589640	+	0.807666i	$0.299270\pi$
$80$	−80.1591	−0.112026
$81$	0	0
$82$	−196.584	−0.264745
$83$	1066.49	1.41040	0.705198	−	0.709011i	$-0.250858\pi$
$83$	1066.49	1.41040	0.705198	+	0.709011i	$0.250858\pi$
$84$	0	0
$85$	1334.16	1.70248
$86$	111.303	0.139560
$87$	0	0
$88$	−9.72711	−0.0117831
$89$	738.470	0.879524	0.439762	−	0.898114i	$-0.355063\pi$
$89$	738.470	0.879524	0.439762	+	0.898114i	$0.355063\pi$
$90$	0	0
$91$	−1423.23	−1.63950
$92$	68.5394	0.0776709
$93$	0	0
$94$	173.624	0.190510
$95$	658.135	0.710771
$96$	0	0
$97$	725.871	0.759804	0.379902	−	0.925027i	$-0.375958\pi$
$97$	725.871	0.759804	0.379902	+	0.925027i	$0.375958\pi$
$98$	−107.690	−0.111003
$99$	0	0
$100$	144.318	0.144318
$101$	200.823	0.197848	0.0989240	−	0.995095i	$-0.468460\pi$
$101$	200.823	0.197848	0.0989240	+	0.995095i	$0.468460\pi$
$102$	0	0
$103$	489.973	0.468723	0.234362	−	0.972150i	$-0.424700\pi$
$103$	489.973	0.468723	0.234362	+	0.972150i	$0.424700\pi$
$104$	1522.02	1.43506
$105$	0	0
$106$	−1035.57	−0.948901
$107$	611.410	0.552404	0.276202	−	0.961100i	$-0.410924\pi$
$107$	611.410	0.552404	0.276202	+	0.961100i	$0.410924\pi$
$108$	0	0
$109$	−1646.07	−1.44647	−0.723233	−	0.690604i	$-0.757345\pi$
$109$	−1646.07	−1.44647	−0.723233	+	0.690604i	$0.757345\pi$
$110$	−7.20127	−0.00624195
$111$	0	0
$112$	−163.709	−0.138116
$113$	−1678.05	−1.39697	−0.698486	−	0.715623i	$-0.746143\pi$
$113$	−1678.05	−1.39697	−0.698486	+	0.715623i	$0.746143\pi$
$114$	0	0
$115$	126.116	0.102264
$116$	−1340.70	−1.07311
$117$	0	0
$118$	−772.644	−0.602777
$119$	2724.76	2.09898
$120$	0	0
$121$	−1330.80	−0.999848
$122$	129.247	0.0959134
$123$	0	0
$124$	764.997	0.554022
$125$	1504.27	1.07637
$126$	0	0
$127$	−960.946	−0.671419	−0.335709	−	0.941966i	$-0.608976\pi$
$127$	−960.946	−0.671419	−0.335709	+	0.941966i	$0.608976\pi$
$128$	1107.57	0.764813
$129$	0	0
$130$	1126.79	0.760203
$131$	2036.42	1.35819	0.679094	−	0.734051i	$-0.262373\pi$
$131$	2036.42	1.35819	0.679094	+	0.734051i	$0.262373\pi$
$132$	0	0
$133$	1344.11	0.876307
$134$	−193.912	−0.125011
$135$	0	0
$136$	−2913.89	−1.83723
$137$	−1135.91	−0.708372	−0.354186	−	0.935175i	$-0.615242\pi$
$137$	−1135.91	−0.708372	−0.354186	+	0.935175i	$0.615242\pi$
$138$	0	0
$139$	−1650.00	−1.00684	−0.503421	−	0.864042i	$-0.667925\pi$
$139$	−1650.00	−1.00684	−0.503421	+	0.864042i	$0.667925\pi$
$140$	−1080.13	−0.652053
$141$	0	0
$142$	−1103.22	−0.651974
$143$	−31.6047	−0.0184820
$144$	0	0
$145$	−2466.96	−1.41289
$146$	−913.064	−0.517574
$147$	0	0
$148$	−81.0655	−0.0450240
$149$	−1438.31	−0.790813	−0.395406	−	0.918506i	$-0.629396\pi$
$149$	−1438.31	−0.790813	−0.395406	+	0.918506i	$0.629396\pi$
$150$	0	0
$151$	−3115.60	−1.67910	−0.839550	−	0.543283i	$-0.817181\pi$
$151$	−3115.60	−1.67910	−0.839550	+	0.543283i	$0.817181\pi$
$152$	−1437.40	−0.767031
$153$	0	0
$154$	−14.7071	−0.00769567
$155$	1407.64	0.729445
$156$	0	0
$157$	1501.21	0.763117	0.381558	−	0.924345i	$-0.375387\pi$
$157$	1501.21	0.763117	0.381558	+	0.924345i	$0.375387\pi$
$158$	−1338.89	−0.674156
$159$	0	0
$160$	1845.45	0.911850
$161$	257.567	0.126081
$162$	0	0
$163$	−2552.72	−1.22666	−0.613328	−	0.789829i	$-0.710169\pi$
$163$	−2552.72	−1.22666	−0.613328	+	0.789829i	$0.710169\pi$
$164$	−654.775	−0.311764
$165$	0	0
$166$	−1724.43	−0.806277
$167$	129.769	0.0601307	0.0300653	−	0.999548i	$-0.490428\pi$
$167$	129.769	0.0601307	0.0300653	+	0.999548i	$0.490428\pi$
$168$	0	0
$169$	2748.24	1.25091
$170$	−2157.24	−0.973250
$171$	0	0
$172$	370.724	0.164346
$173$	−1827.14	−0.802975	−0.401487	−	0.915865i	$-0.631507\pi$
$173$	−1827.14	−0.802975	−0.401487	+	0.915865i	$0.631507\pi$
$174$	0	0
$175$	542.336	0.234267
$176$	−3.63538	−0.00155697
$177$	0	0
$178$	−1194.05	−0.502795
$179$	761.132	0.317819	0.158910	−	0.987293i	$-0.449202\pi$
$179$	761.132	0.317819	0.158910	+	0.987293i	$0.449202\pi$
$180$	0	0
$181$	−3518.22	−1.44479	−0.722396	−	0.691480i	$-0.756959\pi$
$181$	−3518.22	−1.44479	−0.722396	+	0.691480i	$0.756959\pi$
$182$	2301.25	0.937251
$183$	0	0
$184$	−275.444	−0.110359
$185$	−149.165	−0.0592801
$186$	0	0
$187$	60.5070	0.0236616
$188$	578.301	0.224345
$189$	0	0
$190$	−1064.15	−0.406325
$191$	−2137.56	−0.809784	−0.404892	−	0.914365i	$-0.632691\pi$
$191$	−2137.56	−0.809784	−0.404892	+	0.914365i	$0.632691\pi$
$192$	0	0
$193$	1691.34	0.630803	0.315402	−	0.948958i	$-0.397861\pi$
$193$	1691.34	0.630803	0.315402	+	0.948958i	$0.397861\pi$
$194$	−1173.67	−0.434355
$195$	0	0
$196$	−358.688	−0.130717
$197$	−5252.94	−1.89978	−0.949889	−	0.312586i	$-0.898805\pi$
$197$	−5252.94	−1.89978	−0.949889	+	0.312586i	$0.898805\pi$
$198$	0	0
$199$	2859.08	1.01846	0.509232	−	0.860629i	$-0.329929\pi$
$199$	2859.08	1.01846	0.509232	+	0.860629i	$0.329929\pi$
$200$	−579.980	−0.205054
$201$	0	0
$202$	−324.715	−0.113103
$203$	−5038.26	−1.74195
$204$	0	0
$205$	−1204.82	−0.410480
$206$	−792.247	−0.267954
$207$	0	0
$208$	568.833	0.189622
$209$	29.8477	0.00987852
$210$	0	0
$211$	5553.87	1.81206	0.906029	−	0.423216i	$-0.139099\pi$
$211$	5553.87	1.81206	0.906029	+	0.423216i	$0.139099\pi$
$212$	−3449.24	−1.11743
$213$	0	0
$214$	−988.601	−0.315791
$215$	682.152	0.216383
$216$	0	0
$217$	2874.81	0.899330
$218$	2661.56	0.826898
$219$	0	0
$220$	−23.9857	−0.00735052
$221$	−9467.62	−2.88172
$222$	0	0
$223$	328.762	0.0987242	0.0493621	−	0.998781i	$-0.484281\pi$
$223$	328.762	0.0987242	0.0493621	+	0.998781i	$0.484281\pi$
$224$	3768.96	1.12422
$225$	0	0
$226$	2713.28	0.798604
$227$	2208.75	0.645814	0.322907	−	0.946431i	$-0.395340\pi$
$227$	2208.75	0.645814	0.322907	+	0.946431i	$0.395340\pi$
$228$	0	0
$229$	−5196.80	−1.49962	−0.749812	−	0.661651i	$-0.769856\pi$
$229$	−5196.80	−1.49962	−0.749812	+	0.661651i	$0.769856\pi$
$230$	−203.920	−0.0584612
$231$	0	0
$232$	5387.96	1.52473
$233$	4697.49	1.32079	0.660393	−	0.750920i	$-0.270390\pi$
$233$	4697.49	1.32079	0.660393	+	0.750920i	$0.270390\pi$
$234$	0	0
$235$	1064.10	0.295381
$236$	−2573.49	−0.709830
$237$	0	0
$238$	−4405.72	−1.19992
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	2648.39	0.707874	0.353937	−	0.935269i	$-0.384843\pi$
$241$	2648.39	0.707874	0.353937	+	0.935269i	$0.384843\pi$
$242$	2151.79	0.571581
$243$	0	0
$244$	430.489	0.112948
$245$	−660.005	−0.172107
$246$	0	0
$247$	−4670.32	−1.20310
$248$	−3074.35	−0.787183
$249$	0	0
$250$	−2432.28	−0.615324
$251$	1180.03	0.296745	0.148372	−	0.988932i	$-0.452597\pi$
$251$	1180.03	0.296745	0.148372	+	0.988932i	$0.452597\pi$
$252$	0	0
$253$	5.71962	0.00142130
$254$	1553.77	0.383828
$255$	0	0
$256$	−3682.06	−0.898941
$257$	7624.84	1.85068	0.925339	−	0.379141i	$-0.123780\pi$
$257$	7624.84	1.85068	0.925339	+	0.379141i	$0.123780\pi$
$258$	0	0
$259$	−304.639	−0.0730863
$260$	3753.08	0.895215
$261$	0	0
$262$	−3292.72	−0.776432
$263$	−1953.58	−0.458034	−0.229017	−	0.973422i	$-0.573551\pi$
$263$	−1953.58	−0.458034	−0.229017	+	0.973422i	$0.573551\pi$
$264$	0	0
$265$	−6346.78	−1.47124
$266$	−2173.31	−0.500956
$267$	0	0
$268$	−645.875	−0.147213
$269$	−2739.13	−0.620847	−0.310423	−	0.950598i	$-0.600471\pi$
$269$	−2739.13	−0.620847	−0.310423	+	0.950598i	$0.600471\pi$
$270$	0	0
$271$	−2632.08	−0.589991	−0.294996	−	0.955499i	$-0.595318\pi$
$271$	−2632.08	−0.589991	−0.294996	+	0.955499i	$0.595318\pi$
$272$	−1089.03	−0.242764
$273$	0	0
$274$	1836.67	0.404953
$275$	12.0433	0.00264087
$276$	0	0
$277$	−1268.68	−0.275189	−0.137594	−	0.990489i	$-0.543937\pi$
$277$	−1268.68	−0.275189	−0.137594	+	0.990489i	$0.543937\pi$
$278$	2667.91	0.575578
$279$	0	0
$280$	4340.79	0.926470
$281$	−2875.97	−0.610556	−0.305278	−	0.952263i	$-0.598749\pi$
$281$	−2875.97	−0.610556	−0.305278	+	0.952263i	$0.598749\pi$
$282$	0	0
$283$	986.458	0.207204	0.103602	−	0.994619i	$-0.466963\pi$
$283$	986.458	0.207204	0.103602	+	0.994619i	$0.466963\pi$
$284$	−3674.56	−0.767765
$285$	0	0
$286$	51.1023	0.0105655
$287$	−2460.60	−0.506079
$288$	0	0
$289$	13212.7	2.68933
$290$	3988.87	0.807706
$291$	0	0
$292$	−3041.20	−0.609495
$293$	6624.36	1.32082	0.660408	−	0.750907i	$-0.270383\pi$
$293$	6624.36	1.32082	0.660408	+	0.750907i	$0.270383\pi$
$294$	0	0
$295$	−4735.36	−0.934588
$296$	325.784	0.0639724
$297$	0	0
$298$	2325.64	0.452082
$299$	−894.957	−0.173099
$300$	0	0
$301$	1393.16	0.266778
$302$	5037.68	0.959887
$303$	0	0
$304$	−537.210	−0.101352
$305$	792.124	0.148711
$306$	0	0
$307$	7009.16	1.30304	0.651521	−	0.758631i	$-0.274131\pi$
$307$	7009.16	1.30304	0.651521	+	0.758631i	$0.274131\pi$
$308$	−48.9859	−0.00906243
$309$	0	0
$310$	−2276.03	−0.417000
$311$	−2947.22	−0.537368	−0.268684	−	0.963228i	$-0.586589\pi$
$311$	−2947.22	−0.537368	−0.268684	+	0.963228i	$0.586589\pi$
$312$	0	0
$313$	−9192.12	−1.65997	−0.829983	−	0.557789i	$-0.811650\pi$
$313$	−9192.12	−1.65997	−0.829983	+	0.557789i	$0.811650\pi$
$314$	−2427.33	−0.436249
$315$	0	0
$316$	−4459.53	−0.793886
$317$	−7206.12	−1.27677	−0.638385	−	0.769717i	$-0.720397\pi$
$317$	−7206.12	−1.27677	−0.638385	+	0.769717i	$0.720397\pi$
$318$	0	0
$319$	−111.881	−0.0196368
$320$	−2342.68	−0.409249
$321$	0	0
$322$	−416.464	−0.0720766
$323$	8941.29	1.54027
$324$	0	0
$325$	−1884.43	−0.321630
$326$	4127.55	0.701239
$327$	0	0
$328$	2631.39	0.442970
$329$	2173.22	0.364174
$330$	0	0
$331$	−8568.35	−1.42284	−0.711419	−	0.702768i	$-0.751947\pi$
$331$	−8568.35	−1.42284	−0.711419	+	0.702768i	$0.751947\pi$
$332$	−5743.67	−0.949473
$333$	0	0
$334$	−209.826	−0.0343748
$335$	−1188.44	−0.193826
$336$	0	0
$337$	−5871.71	−0.949117	−0.474559	−	0.880224i	$-0.657392\pi$
$337$	−5871.71	−0.949117	−0.474559	+	0.880224i	$0.657392\pi$
$338$	−4443.68	−0.715102
$339$	0	0
$340$	−7185.24	−1.14610
$341$	63.8390	0.0101381
$342$	0	0
$343$	5593.92	0.880593
$344$	−1489.86	−0.233511
$345$	0	0
$346$	2954.33	0.459034
$347$	−7234.33	−1.11919	−0.559595	−	0.828766i	$-0.689043\pi$
$347$	−7234.33	−1.11919	−0.559595	+	0.828766i	$0.689043\pi$
$348$	0	0
$349$	−4561.86	−0.699687	−0.349844	−	0.936808i	$-0.613765\pi$
$349$	−4561.86	−0.699687	−0.349844	+	0.936808i	$0.613765\pi$
$350$	−876.914	−0.133923
$351$	0	0
$352$	83.6950	0.0126732
$353$	−6847.59	−1.03247	−0.516233	−	0.856448i	$-0.672666\pi$
$353$	−6847.59	−1.03247	−0.516233	+	0.856448i	$0.672666\pi$
$354$	0	0
$355$	−6761.40	−1.01087
$356$	−3977.08	−0.592092
$357$	0	0
$358$	−1230.69	−0.181687
$359$	8304.02	1.22081	0.610403	−	0.792091i	$-0.291008\pi$
$359$	8304.02	1.22081	0.610403	+	0.792091i	$0.291008\pi$
$360$	0	0
$361$	−2448.32	−0.356949
$362$	5688.68	0.825940
$363$	0	0
$364$	7664.90	1.10371
$365$	−5595.97	−0.802483
$366$	0	0
$367$	3315.48	0.471572	0.235786	−	0.971805i	$-0.424234\pi$
$367$	3315.48	0.471572	0.235786	+	0.971805i	$0.424234\pi$
$368$	−102.944	−0.0145824
$369$	0	0
$370$	241.188	0.0338885
$371$	−12962.0	−1.81389
$372$	0	0
$373$	4349.37	0.603759	0.301879	−	0.953346i	$-0.402386\pi$
$373$	4349.37	0.603759	0.301879	+	0.953346i	$0.402386\pi$
$374$	−97.8350	−0.0135265
$375$	0	0
$376$	−2324.06	−0.318761
$377$	17506.2	2.39156
$378$	0	0
$379$	5391.39	0.730704	0.365352	−	0.930869i	$-0.380949\pi$
$379$	5391.39	0.730704	0.365352	+	0.930869i	$0.380949\pi$
$380$	−3544.43	−0.478489
$381$	0	0
$382$	3456.27	0.462927
$383$	892.011	0.119007	0.0595034	−	0.998228i	$-0.481048\pi$
$383$	892.011	0.119007	0.0595034	+	0.998228i	$0.481048\pi$
$384$	0	0
$385$	−90.1367	−0.0119319
$386$	−2734.75	−0.360610
$387$	0	0
$388$	−3909.23	−0.511497
$389$	4297.05	0.560075	0.280038	−	0.959989i	$-0.409653\pi$
$389$	4297.05	0.560075	0.280038	+	0.959989i	$0.409653\pi$
$390$	0	0
$391$	1713.39	0.221611
$392$	1441.49	0.185730
$393$	0	0
$394$	8493.58	1.08604
$395$	−8205.77	−1.04526
$396$	0	0
$397$	−11866.7	−1.50019	−0.750094	−	0.661331i	$-0.769992\pi$
$397$	−11866.7	−1.50019	−0.750094	+	0.661331i	$0.769992\pi$
$398$	−4622.90	−0.582223
$399$	0	0
$400$	−216.760	−0.0270949
$401$	−1884.96	−0.234739	−0.117370	−	0.993088i	$-0.537446\pi$
$401$	−1884.96	−0.234739	−0.117370	+	0.993088i	$0.537446\pi$
$402$	0	0
$403$	−9988.99	−1.23471
$404$	−1081.55	−0.133191
$405$	0	0
$406$	8146.46	0.995817
$407$	−6.76492	−0.000823894 0
$408$	0	0
$409$	−5035.01	−0.608717	−0.304358	−	0.952558i	$-0.598442\pi$
$409$	−5035.01	−0.608717	−0.304358	+	0.952558i	$0.598442\pi$
$410$	1948.10	0.234658
$411$	0	0
$412$	−2638.78	−0.315543
$413$	−9671.01	−1.15225
$414$	0	0
$415$	−10568.7	−1.25011
$416$	−13095.9	−1.54346
$417$	0	0
$418$	−48.2614	−0.00564723
$419$	−2814.12	−0.328112	−0.164056	−	0.986451i	$-0.552458\pi$
$419$	−2814.12	−0.328112	−0.164056	+	0.986451i	$0.552458\pi$
$420$	0	0
$421$	13017.0	1.50692	0.753458	−	0.657496i	$-0.228384\pi$
$421$	13017.0	1.50692	0.753458	+	0.657496i	$0.228384\pi$
$422$	−8980.16	−1.03589
$423$	0	0
$424$	13861.7	1.58770
$425$	3607.73	0.411767
$426$	0	0
$427$	1617.75	0.183345
$428$	−3292.79	−0.371876
$429$	0	0
$430$	−1102.98	−0.123699
$431$	−4295.90	−0.480108	−0.240054	−	0.970760i	$-0.577165\pi$
$431$	−4295.90	−0.480108	−0.240054	+	0.970760i	$0.577165\pi$
$432$	0	0
$433$	7628.16	0.846619	0.423310	−	0.905985i	$-0.360868\pi$
$433$	7628.16	0.846619	0.423310	+	0.905985i	$0.360868\pi$
$434$	−4648.33	−0.514118
$435$	0	0
$436$	8865.02	0.973755
$437$	845.204	0.0925208
$438$	0	0
$439$	−7640.41	−0.830653	−0.415327	−	0.909672i	$-0.636333\pi$
$439$	−7640.41	−0.830653	−0.415327	+	0.909672i	$0.636333\pi$
$440$	96.3931	0.0104440
$441$	0	0
$442$	15308.4	1.64739
$443$	7098.93	0.761355	0.380677	−	0.924708i	$-0.375691\pi$
$443$	7098.93	0.761355	0.380677	+	0.924708i	$0.375691\pi$
$444$	0	0
$445$	−7318.04	−0.779570
$446$	−531.581	−0.0564374
$447$	0	0
$448$	−4784.44	−0.504562
$449$	12141.4	1.27614	0.638071	−	0.769978i	$-0.279733\pi$
$449$	12141.4	1.27614	0.638071	+	0.769978i	$0.279733\pi$
$450$	0	0
$451$	−54.6410	−0.00570497
$452$	9037.27	0.940437
$453$	0	0
$454$	−3571.37	−0.369191
$455$	14103.8	1.45318
$456$	0	0
$457$	−5460.29	−0.558909	−0.279454	−	0.960159i	$-0.590154\pi$
$457$	−5460.29	−0.558909	−0.279454	+	0.960159i	$0.590154\pi$
$458$	8402.80	0.857286
$459$	0	0
$460$	−679.207	−0.0688439
$461$	3459.29	0.349491	0.174745	−	0.984614i	$-0.444090\pi$
$461$	3459.29	0.349491	0.174745	+	0.984614i	$0.444090\pi$
$462$	0	0
$463$	10738.3	1.07786	0.538930	−	0.842350i	$-0.318829\pi$
$463$	10738.3	1.07786	0.538930	+	0.842350i	$0.318829\pi$
$464$	2013.68	0.201471
$465$	0	0
$466$	−7595.47	−0.755050
$467$	−6381.82	−0.632367	−0.316184	−	0.948698i	$-0.602402\pi$
$467$	−6381.82	−0.632367	−0.316184	+	0.948698i	$0.602402\pi$
$468$	0	0
$469$	−2427.16	−0.238967
$470$	−1720.57	−0.168860
$471$	0	0
$472$	10342.3	1.00856
$473$	30.9369	0.00300736
$474$	0	0
$475$	1779.67	0.171910
$476$	−14674.4	−1.41302
$477$	0	0
$478$	386.444	0.0369781
$479$	−3093.68	−0.295102	−0.147551	−	0.989054i	$-0.547139\pi$
$479$	−3093.68	−0.295102	−0.147551	+	0.989054i	$0.547139\pi$
$480$	0	0
$481$	1058.52	0.100342
$482$	−4282.23	−0.404669
$483$	0	0
$484$	7167.11	0.673094
$485$	−7193.19	−0.673455
$486$	0	0
$487$	3698.42	0.344130	0.172065	−	0.985086i	$-0.444956\pi$
$487$	3698.42	0.344130	0.172065	+	0.985086i	$0.444956\pi$
$488$	−1730.04	−0.160482
$489$	0	0
$490$	1067.18	0.0983880
$491$	17407.3	1.59996	0.799980	−	0.600027i	$-0.204843\pi$
$491$	17407.3	1.59996	0.799980	+	0.600027i	$0.204843\pi$
$492$	0	0
$493$	−33515.6	−3.06180
$494$	7551.53	0.687772
$495$	0	0
$496$	−1149.00	−0.104015
$497$	−13808.8	−1.24629
$498$	0	0
$499$	13368.4	1.19930	0.599652	−	0.800261i	$-0.295305\pi$
$499$	13368.4	1.19930	0.599652	+	0.800261i	$0.295305\pi$
$500$	−8101.35	−0.724607
$501$	0	0
$502$	−1908.02	−0.169639
$503$	20018.2	1.77449	0.887244	−	0.461301i	$-0.152617\pi$
$503$	20018.2	1.77449	0.887244	+	0.461301i	$0.152617\pi$
$504$	0	0
$505$	−1990.10	−0.175363
$506$	−9.24816	−0.000812512 0
$507$	0	0
$508$	5175.24	0.451997
$509$	1413.40	0.123081	0.0615403	−	0.998105i	$-0.480399\pi$
$509$	1413.40	0.123081	0.0615403	+	0.998105i	$0.480399\pi$
$510$	0	0
$511$	−11428.6	−0.989378
$512$	−2906.95	−0.250918
$513$	0	0
$514$	−12328.8	−1.05797
$515$	−4855.51	−0.415455
$516$	0	0
$517$	48.2592	0.00410530
$518$	492.577	0.0417810
$519$	0	0
$520$	−15082.8	−1.27197
$521$	−7182.34	−0.603962	−0.301981	−	0.953314i	$-0.597648\pi$
$521$	−7182.34	−0.603962	−0.301981	+	0.953314i	$0.597648\pi$
$522$	0	0
$523$	−18112.4	−1.51434	−0.757170	−	0.653218i	$-0.773419\pi$
$523$	−18112.4	−1.51434	−0.757170	+	0.653218i	$0.773419\pi$
$524$	−10967.3	−0.914327
$525$	0	0
$526$	3158.78	0.261843
$527$	19123.8	1.58074
$528$	0	0
$529$	−12005.0	−0.986688
$530$	10262.2	0.841063
$531$	0	0
$532$	−7238.78	−0.589927
$533$	8549.76	0.694805
$534$	0	0
$535$	−6058.91	−0.489625
$536$	2595.63	0.209168
$537$	0	0
$538$	4428.96	0.354918
$539$	−29.9326	−0.00239200
$540$	0	0
$541$	3163.82	0.251429	0.125715	−	0.992066i	$-0.459878\pi$
$541$	3163.82	0.251429	0.125715	+	0.992066i	$0.459878\pi$
$542$	4255.87	0.337279
$543$	0	0
$544$	25072.0	1.97601
$545$	16312.1	1.28208
$546$	0	0
$547$	−21858.3	−1.70858	−0.854291	−	0.519796i	$-0.826008\pi$
$547$	−21858.3	−1.70858	−0.854291	+	0.519796i	$0.826008\pi$
$548$	6117.50	0.476874
$549$	0	0
$550$	−19.4731	−0.00150970
$551$	−16533.0	−1.27828
$552$	0	0
$553$	−16758.6	−1.28870
$554$	2051.35	0.157317
$555$	0	0
$556$	8886.17	0.677802
$557$	7138.08	0.542998	0.271499	−	0.962439i	$-0.412481\pi$
$557$	7138.08	0.542998	0.271499	+	0.962439i	$0.412481\pi$
$558$	0	0
$559$	−4840.75	−0.366264
$560$	1622.31	0.122420
$561$	0	0
$562$	4650.22	0.349035
$563$	−17864.1	−1.33727	−0.668635	−	0.743591i	$-0.733121\pi$
$563$	−17864.1	−1.33727	−0.668635	+	0.743591i	$0.733121\pi$
$564$	0	0
$565$	16629.1	1.23821
$566$	−1595.02	−0.118452
$567$	0	0
$568$	14767.2	1.09088
$569$	−23420.4	−1.72554	−0.862772	−	0.505593i	$-0.831274\pi$
$569$	−23420.4	−1.72554	−0.862772	+	0.505593i	$0.831274\pi$
$570$	0	0
$571$	−4627.77	−0.339170	−0.169585	−	0.985516i	$-0.554243\pi$
$571$	−4627.77	−0.339170	−0.169585	+	0.985516i	$0.554243\pi$
$572$	170.209	0.0124420
$573$	0	0
$574$	3978.59	0.289309
$575$	341.033	0.0247340
$576$	0	0
$577$	22441.1	1.61913	0.809563	−	0.587034i	$-0.199704\pi$
$577$	22441.1	1.61913	0.809563	+	0.587034i	$0.199704\pi$
$578$	−21363.9	−1.53740
$579$	0	0
$580$	13286.0	0.951155
$581$	−21584.4	−1.54126
$582$	0	0
$583$	−287.839	−0.0204478
$584$	12221.9	0.866002
$585$	0	0
$586$	−10711.1	−0.755068
$587$	−15701.2	−1.10402	−0.552009	−	0.833838i	$-0.686139\pi$
$587$	−15701.2	−1.10402	−0.552009	+	0.833838i	$0.686139\pi$
$588$	0	0
$589$	9433.68	0.659945
$590$	7656.70	0.534273
$591$	0	0
$592$	121.757	0.00845304
$593$	−18452.0	−1.27779	−0.638896	−	0.769293i	$-0.720609\pi$
$593$	−18452.0	−1.27779	−0.638896	+	0.769293i	$0.720609\pi$
$594$	0	0
$595$	−27001.7	−1.86044
$596$	7746.13	0.532372
$597$	0	0
$598$	1447.07	0.0989553
$599$	−12950.7	−0.883389	−0.441694	−	0.897166i	$-0.645622\pi$
$599$	−12950.7	−0.883389	−0.441694	+	0.897166i	$0.645622\pi$
$600$	0	0
$601$	1566.40	0.106314	0.0531571	−	0.998586i	$-0.483072\pi$
$601$	1566.40	0.106314	0.0531571	+	0.998586i	$0.483072\pi$
$602$	−2252.62	−0.152508
$603$	0	0
$604$	16779.3	1.13036
$605$	13187.9	0.886219
$606$	0	0
$607$	−11018.3	−0.736766	−0.368383	−	0.929674i	$-0.620089\pi$
$607$	−11018.3	−0.736766	−0.368383	+	0.929674i	$0.620089\pi$
$608$	12367.8	0.824971
$609$	0	0
$610$	−1280.80	−0.0850133
$611$	−7551.20	−0.499982
$612$	0	0
$613$	−2044.29	−0.134695	−0.0673474	−	0.997730i	$-0.521454\pi$
$613$	−2044.29	−0.134695	−0.0673474	+	0.997730i	$0.521454\pi$
$614$	−11333.2	−0.744906
$615$	0	0
$616$	196.863	0.0128764
$617$	−10312.8	−0.672897	−0.336448	−	0.941702i	$-0.609226\pi$
$617$	−10312.8	−0.672897	−0.336448	+	0.941702i	$0.609226\pi$
$618$	0	0
$619$	−12799.4	−0.831099	−0.415549	−	0.909571i	$-0.636411\pi$
$619$	−12799.4	−0.831099	−0.415549	+	0.909571i	$0.636411\pi$
$620$	−7580.92	−0.491060
$621$	0	0
$622$	4765.42	0.307196
$623$	−14945.6	−0.961129
$624$	0	0
$625$	−11557.3	−0.739666
$626$	14862.9	0.948948
$627$	0	0
$628$	−8084.86	−0.513728
$629$	−2026.52	−0.128462
$630$	0	0
$631$	−2807.16	−0.177102	−0.0885509	−	0.996072i	$-0.528224\pi$
$631$	−2807.16	−0.177102	−0.0885509	+	0.996072i	$0.528224\pi$
$632$	17921.8	1.12799
$633$	0	0
$634$	11651.7	0.729888
$635$	9522.73	0.595115
$636$	0	0
$637$	4683.59	0.291320
$638$	180.903	0.0112257
$639$	0	0
$640$	−10975.7	−0.677895
$641$	−4268.95	−0.263047	−0.131524	−	0.991313i	$-0.541987\pi$
$641$	−4268.95	−0.263047	−0.131524	+	0.991313i	$0.541987\pi$
$642$	0	0
$643$	−25641.5	−1.57263	−0.786317	−	0.617824i	$-0.788015\pi$
$643$	−25641.5	−1.57263	−0.786317	+	0.617824i	$0.788015\pi$
$644$	−1387.14	−0.0848774
$645$	0	0
$646$	−14457.4	−0.880521
$647$	16971.5	1.03125	0.515625	−	0.856814i	$-0.327560\pi$
$647$	16971.5	1.03125	0.515625	+	0.856814i	$0.327560\pi$
$648$	0	0
$649$	−214.758	−0.0129892
$650$	3046.98	0.183865
$651$	0	0
$652$	13747.9	0.825780
$653$	−6164.60	−0.369433	−0.184716	−	0.982792i	$-0.559137\pi$
$653$	−6164.60	−0.369433	−0.184716	+	0.982792i	$0.559137\pi$
$654$	0	0
$655$	−20180.4	−1.20384
$656$	983.447	0.0585323
$657$	0	0
$658$	−3513.92	−0.208187
$659$	−14722.3	−0.870255	−0.435128	−	0.900369i	$-0.643297\pi$
$659$	−14722.3	−0.870255	−0.435128	+	0.900369i	$0.643297\pi$
$660$	0	0
$661$	−10616.6	−0.624717	−0.312358	−	0.949964i	$-0.601119\pi$
$661$	−10616.6	−0.624717	−0.312358	+	0.949964i	$0.601119\pi$
$662$	13854.3	0.813390
$663$	0	0
$664$	23082.5	1.34906
$665$	−13319.8	−0.776718
$666$	0	0
$667$	−3168.17	−0.183916
$668$	−698.880	−0.0404798
$669$	0	0
$670$	1921.62	0.110804
$671$	35.9244	0.00206683
$672$	0	0
$673$	−12596.4	−0.721481	−0.360740	−	0.932666i	$-0.617476\pi$
$673$	−12596.4	−0.721481	−0.360740	+	0.932666i	$0.617476\pi$
$674$	9494.09	0.542579
$675$	0	0
$676$	−14800.8	−0.842105
$677$	−31525.3	−1.78968	−0.894841	−	0.446385i	$-0.852711\pi$
$677$	−31525.3	−1.78968	−0.894841	+	0.446385i	$0.852711\pi$
$678$	0	0
$679$	−14690.6	−0.830301
$680$	28875.9	1.62844
$681$	0	0
$682$	−103.223	−0.00579560
$683$	30069.3	1.68459	0.842293	−	0.539021i	$-0.181206\pi$
$683$	30069.3	1.68459	0.842293	+	0.539021i	$0.181206\pi$
$684$	0	0
$685$	11256.5	0.627869
$686$	−9044.92	−0.503406
$687$	0	0
$688$	−556.813	−0.0308551
$689$	45038.6	2.49033
$690$	0	0
$691$	−6662.00	−0.366765	−0.183382	−	0.983042i	$-0.558705\pi$
$691$	−6662.00	−0.366765	−0.183382	+	0.983042i	$0.558705\pi$
$692$	9840.17	0.540559
$693$	0	0
$694$	11697.3	0.639805
$695$	16351.0	0.892418
$696$	0	0
$697$	−16368.4	−0.889525
$698$	7376.16	0.399988
$699$	0	0
$700$	−2920.79	−0.157708
$701$	−6651.13	−0.358359	−0.179180	−	0.983816i	$-0.557344\pi$
$701$	−6651.13	−0.358359	−0.179180	+	0.983816i	$0.557344\pi$
$702$	0	0
$703$	−999.672	−0.0536321
$704$	−106.245	−0.00568787
$705$	0	0
$706$	11072.0	0.590227
$707$	−4064.38	−0.216205
$708$	0	0
$709$	5137.79	0.272149	0.136075	−	0.990699i	$-0.456551\pi$
$709$	5137.79	0.272149	0.136075	+	0.990699i	$0.456551\pi$
$710$	10932.6	0.577880
$711$	0	0
$712$	15983.0	0.841275
$713$	1807.74	0.0949516
$714$	0	0
$715$	313.195	0.0163816
$716$	−4099.13	−0.213955
$717$	0	0
$718$	−13426.9	−0.697895
$719$	4723.59	0.245007	0.122504	−	0.992468i	$-0.460908\pi$
$719$	4723.59	0.245007	0.122504	+	0.992468i	$0.460908\pi$
$720$	0	0
$721$	−9916.38	−0.512213
$722$	3958.73	0.204056
$723$	0	0
$724$	18947.6	0.972628
$725$	−6670.94	−0.341727
$726$	0	0
$727$	30223.2	1.54184	0.770919	−	0.636933i	$-0.219797\pi$
$727$	30223.2	1.54184	0.770919	+	0.636933i	$0.219797\pi$
$728$	−30803.5	−1.56821
$729$	0	0
$730$	9048.23	0.458753
$731$	9267.57	0.468911
$732$	0	0
$733$	−30170.5	−1.52029	−0.760146	−	0.649753i	$-0.774873\pi$
$733$	−30170.5	−1.52029	−0.760146	+	0.649753i	$0.774873\pi$
$734$	−5360.87	−0.269582
$735$	0	0
$736$	2370.01	0.118695
$737$	−53.8983	−0.00269385
$738$	0	0
$739$	24980.8	1.24348	0.621741	−	0.783223i	$-0.286426\pi$
$739$	24980.8	1.24348	0.621741	+	0.783223i	$0.286426\pi$
$740$	803.338	0.0399072
$741$	0	0
$742$	20958.5	1.03694
$743$	−4884.33	−0.241169	−0.120584	−	0.992703i	$-0.538477\pi$
$743$	−4884.33	−0.241169	−0.120584	+	0.992703i	$0.538477\pi$
$744$	0	0
$745$	14253.3	0.700940
$746$	−7032.59	−0.345149
$747$	0	0
$748$	−325.865	−0.0159289
$749$	−12374.1	−0.603657
$750$	0	0
$751$	12186.1	0.592113	0.296057	−	0.955170i	$-0.404328\pi$
$751$	12186.1	0.592113	0.296057	+	0.955170i	$0.404328\pi$
$752$	−868.586	−0.0421198
$753$	0	0
$754$	−28306.2	−1.36718
$755$	30874.8	1.48828
$756$	0	0
$757$	7001.81	0.336176	0.168088	−	0.985772i	$-0.446241\pi$
$757$	7001.81	0.336176	0.168088	+	0.985772i	$0.446241\pi$
$758$	−8717.44	−0.417720
$759$	0	0
$760$	14244.3	0.679861
$761$	8125.85	0.387072	0.193536	−	0.981093i	$-0.438004\pi$
$761$	8125.85	0.387072	0.193536	+	0.981093i	$0.438004\pi$
$762$	0	0
$763$	33314.2	1.58067
$764$	11512.0	0.545144
$765$	0	0
$766$	−1442.31	−0.0680323
$767$	33603.5	1.58195
$768$	0	0
$769$	−1853.98	−0.0869391	−0.0434695	−	0.999055i	$-0.513841\pi$
$769$	−1853.98	−0.0869391	−0.0434695	+	0.999055i	$0.513841\pi$
$770$	145.744	0.00682109
$771$	0	0
$772$	−9108.81	−0.424654
$773$	−6353.53	−0.295628	−0.147814	−	0.989015i	$-0.547224\pi$
$773$	−6353.53	−0.295628	−0.147814	+	0.989015i	$0.547224\pi$
$774$	0	0
$775$	3806.41	0.176426
$776$	15710.3	0.726762
$777$	0	0
$778$	−6947.99	−0.320177
$779$	−8074.45	−0.371370
$780$	0	0
$781$	−306.643	−0.0140493
$782$	−2770.41	−0.126688
$783$	0	0
$784$	538.736	0.0245416
$785$	−14876.6	−0.676392
$786$	0	0
$787$	29640.5	1.34253	0.671264	−	0.741218i	$-0.265752\pi$
$787$	29640.5	1.34253	0.671264	+	0.741218i	$0.265752\pi$
$788$	28290.1	1.27892
$789$	0	0
$790$	13268.1	0.597541
$791$	33961.5	1.52659
$792$	0	0
$793$	−5621.14	−0.251718
$794$	19187.6	0.857609
$795$	0	0
$796$	−15397.8	−0.685627
$797$	−34591.7	−1.53739	−0.768696	−	0.639615i	$-0.779094\pi$
$797$	−34591.7	−1.53739	−0.768696	+	0.639615i	$0.779094\pi$
$798$	0	0
$799$	14456.7	0.640102
$800$	4990.32	0.220543
$801$	0	0
$802$	3047.83	0.134193
$803$	−253.788	−0.0111532
$804$	0	0
$805$	−2552.42	−0.111753
$806$	16151.4	0.705842
$807$	0	0
$808$	4346.49	0.189244
$809$	41642.5	1.80973	0.904865	−	0.425698i	$-0.139971\pi$
$809$	41642.5	1.80973	0.904865	+	0.425698i	$0.139971\pi$
$810$	0	0
$811$	−24600.9	−1.06517	−0.532587	−	0.846375i	$-0.678780\pi$
$811$	−24600.9	−1.06517	−0.532587	+	0.846375i	$0.678780\pi$
$812$	27133.9	1.17268
$813$	0	0
$814$	10.9383	0.000470993 0
$815$	25296.8	1.08725
$816$	0	0
$817$	4571.64	0.195767
$818$	8141.21	0.347984
$819$	0	0
$820$	6488.65	0.276333
$821$	16662.2	0.708300	0.354150	−	0.935189i	$-0.384770\pi$
$821$	16662.2	0.708300	0.354150	+	0.935189i	$0.384770\pi$
$822$	0	0
$823$	−28936.7	−1.22560	−0.612801	−	0.790238i	$-0.709957\pi$
$823$	−28936.7	−1.22560	−0.612801	+	0.790238i	$0.709957\pi$
$824$	10604.7	0.448339
$825$	0	0
$826$	15637.2	0.658704
$827$	23609.9	0.992739	0.496370	−	0.868111i	$-0.334666\pi$
$827$	23609.9	0.992739	0.496370	+	0.868111i	$0.334666\pi$
$828$	0	0
$829$	−30659.9	−1.28451	−0.642257	−	0.766489i	$-0.722002\pi$
$829$	−30659.9	−1.28451	−0.642257	+	0.766489i	$0.722002\pi$
$830$	17088.7	0.714647
$831$	0	0
$832$	16624.3	0.692722
$833$	−8966.70	−0.372962
$834$	0	0
$835$	−1285.98	−0.0532971
$836$	−160.747	−0.00665018
$837$	0	0
$838$	4550.21	0.187571
$839$	−24528.5	−1.00932	−0.504659	−	0.863319i	$-0.668382\pi$
$839$	−24528.5	−1.00932	−0.504659	+	0.863319i	$0.668382\pi$
$840$	0	0
$841$	37583.5	1.54100
$842$	−21047.5	−0.861455
$843$	0	0
$844$	−29910.8	−1.21987
$845$	−27234.3	−1.10875
$846$	0	0
$847$	26933.5	1.09262
$848$	5180.63	0.209792
$849$	0	0
$850$	−5833.42	−0.235394
$851$	−191.564	−0.00771647
$852$	0	0
$853$	−10782.1	−0.432792	−0.216396	−	0.976306i	$-0.569430\pi$
$853$	−10782.1	−0.432792	−0.216396	+	0.976306i	$0.569430\pi$
$854$	−2615.77	−0.104813
$855$	0	0
$856$	13233.0	0.528381
$857$	41551.0	1.65619	0.828095	−	0.560587i	$-0.189425\pi$
$857$	41551.0	1.65619	0.828095	+	0.560587i	$0.189425\pi$
$858$	0	0
$859$	21325.0	0.847032	0.423516	−	0.905889i	$-0.360796\pi$
$859$	21325.0	0.847032	0.423516	+	0.905889i	$0.360796\pi$
$860$	−3673.78	−0.145668
$861$	0	0
$862$	6946.13	0.274462
$863$	−28229.7	−1.11350	−0.556750	−	0.830680i	$-0.687952\pi$
$863$	−28229.7	−1.11350	−0.556750	+	0.830680i	$0.687952\pi$
$864$	0	0
$865$	18106.4	0.711720
$866$	−12334.1	−0.483985
$867$	0	0
$868$	−15482.5	−0.605426
$869$	−372.148	−0.0145273
$870$	0	0
$871$	8433.55	0.328083
$872$	−35626.5	−1.38356
$873$	0	0
$874$	−1366.63	−0.0528911
$875$	−30444.3	−1.17624
$876$	0	0
$877$	−21778.1	−0.838535	−0.419267	−	0.907863i	$-0.637713\pi$
$877$	−21778.1	−0.838535	−0.419267	+	0.907863i	$0.637713\pi$
$878$	12353.9	0.474858
$879$	0	0
$880$	36.0256	0.00138003
$881$	30885.2	1.18110	0.590550	−	0.807001i	$-0.298911\pi$
$881$	30885.2	1.18110	0.590550	+	0.807001i	$0.298911\pi$
$882$	0	0
$883$	16773.2	0.639257	0.319629	−	0.947543i	$-0.396442\pi$
$883$	16773.2	0.639257	0.319629	+	0.947543i	$0.396442\pi$
$884$	50988.5	1.93997
$885$	0	0
$886$	−11478.4	−0.435242
$887$	−31237.2	−1.18246	−0.591229	−	0.806503i	$-0.701357\pi$
$887$	−31237.2	−1.18246	−0.591229	+	0.806503i	$0.701357\pi$
$888$	0	0
$889$	19448.2	0.733715
$890$	11832.7	0.445655
$891$	0	0
$892$	−1770.57	−0.0664608
$893$	7131.41	0.267238
$894$	0	0
$895$	−7542.62	−0.281700
$896$	−22415.6	−0.835775
$897$	0	0
$898$	−19631.6	−0.729528
$899$	−35361.2	−1.31186
$900$	0	0
$901$	−86226.1	−3.18824
$902$	88.3501	0.00326135
$903$	0	0
$904$	−36318.8	−1.33622
$905$	34864.6	1.28060
$906$	0	0
$907$	41722.6	1.52743	0.763713	−	0.645556i	$-0.223374\pi$
$907$	41722.6	1.52743	0.763713	+	0.645556i	$0.223374\pi$
$908$	−11895.4	−0.434759
$909$	0	0
$910$	−22804.7	−0.830736
$911$	−40589.4	−1.47617	−0.738083	−	0.674710i	$-0.764268\pi$
$911$	−40589.4	−1.47617	−0.738083	+	0.674710i	$0.764268\pi$
$912$	0	0
$913$	−479.310	−0.0173744
$914$	8828.84	0.319510
$915$	0	0
$916$	27987.7	1.00954
$917$	−41214.3	−1.48420
$918$	0	0
$919$	24510.8	0.879802	0.439901	−	0.898046i	$-0.355014\pi$
$919$	24510.8	0.879802	0.439901	+	0.898046i	$0.355014\pi$
$920$	2729.58	0.0978170
$921$	0	0
$922$	−5593.39	−0.199792
$923$	47980.8	1.71106
$924$	0	0
$925$	−403.359	−0.0143377
$926$	−17362.9	−0.616178
$927$	0	0
$928$	−46359.7	−1.63990
$929$	−5945.07	−0.209959	−0.104979	−	0.994474i	$-0.533478\pi$
$929$	−5945.07	−0.209959	−0.104979	+	0.994474i	$0.533478\pi$
$930$	0	0
$931$	−4423.22	−0.155709
$932$	−25298.7	−0.889148
$933$	0	0
$934$	10318.9	0.361504
$935$	−599.609	−0.0209725
$936$	0	0
$937$	14937.6	0.520801	0.260400	−	0.965501i	$-0.416145\pi$
$937$	14937.6	0.520801	0.260400	+	0.965501i	$0.416145\pi$
$938$	3924.52	0.136610
$939$	0	0
$940$	−5730.81	−0.198849
$941$	−33699.9	−1.16747	−0.583733	−	0.811946i	$-0.698409\pi$
$941$	−33699.9	−1.16747	−0.583733	+	0.811946i	$0.698409\pi$
$942$	0	0
$943$	−1547.28	−0.0534320
$944$	3865.29	0.133267
$945$	0	0
$946$	−50.0225	−0.00171921
$947$	−32419.7	−1.11246	−0.556230	−	0.831028i	$-0.687753\pi$
$947$	−32419.7	−1.11246	−0.556230	+	0.831028i	$0.687753\pi$
$948$	0	0
$949$	39710.6	1.35834
$950$	−2877.59	−0.0982751
$951$	0	0
$952$	58973.0	2.00770
$953$	−14462.5	−0.491592	−0.245796	−	0.969322i	$-0.579049\pi$
$953$	−14462.5	−0.491592	−0.245796	+	0.969322i	$0.579049\pi$
$954$	0	0
$955$	21182.7	0.717755
$956$	1287.15	0.0435454
$957$	0	0
$958$	5002.24	0.168700
$959$	22989.2	0.774097
$960$	0	0
$961$	−9614.03	−0.322716
$962$	−1711.54	−0.0573620
$963$	0	0
$964$	−14263.1	−0.476538
$965$	−16760.7	−0.559115
$966$	0	0
$967$	15624.7	0.519603	0.259802	−	0.965662i	$-0.416343\pi$
$967$	15624.7	0.519603	0.259802	+	0.965662i	$0.416343\pi$
$968$	−28803.0	−0.956367
$969$	0	0
$970$	11630.8	0.384993
$971$	−31379.4	−1.03709	−0.518543	−	0.855051i	$-0.673526\pi$
$971$	−31379.4	−1.03709	−0.518543	+	0.855051i	$0.673526\pi$
$972$	0	0
$973$	33393.7	1.10026
$974$	−5980.05	−0.196728
$975$	0	0
$976$	−646.579	−0.0212054
$977$	14535.6	0.475983	0.237991	−	0.971267i	$-0.423511\pi$
$977$	14535.6	0.475983	0.237991	+	0.971267i	$0.423511\pi$
$978$	0	0
$979$	−331.888	−0.0108347
$980$	3554.51	0.115862
$981$	0	0
$982$	−28146.2	−0.914645
$983$	28463.7	0.923550	0.461775	−	0.886997i	$-0.347213\pi$
$983$	28463.7	0.923550	0.461775	+	0.886997i	$0.347213\pi$
$984$	0	0
$985$	52055.3	1.68388
$986$	54192.0	1.75033
$987$	0	0
$988$	25152.3	0.809922
$989$	876.047	0.0281665
$990$	0	0
$991$	26846.8	0.860563	0.430281	−	0.902695i	$-0.358414\pi$
$991$	26846.8	0.860563	0.430281	+	0.902695i	$0.358414\pi$
$992$	26452.6	0.846645
$993$	0	0
$994$	22327.7	0.712466
$995$	−28332.7	−0.902720
$996$	0	0
$997$	784.416	0.0249174	0.0124587	−	0.999922i	$-0.496034\pi$
$997$	784.416	0.0249174	0.0124587	+	0.999922i	$0.496034\pi$
$998$	−21615.7	−0.685604
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.9		28
3.2	odd	2		717.4.a.b.1.20	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.20	✓	28	3.2	odd	2
2151.4.a.b.1.9		28	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)

\(f(q)\)	\(=\)	\(q-1.61692 q^{2} -5.38557 q^{4} -9.90974 q^{5} -20.2386 q^{7} +21.6434 q^{8} +O(q^{10})\)	\(q-1.61692 q^{2} -5.38557 q^{4} -9.90974 q^{5} -20.2386 q^{7} +21.6434 q^{8} +16.0233 q^{10} -0.449426 q^{11} +70.3224 q^{13} +32.7242 q^{14} +8.08893 q^{16} -134.632 q^{17} -66.4130 q^{19} +53.3696 q^{20} +0.726686 q^{22} -12.7265 q^{23} -26.7971 q^{25} -113.706 q^{26} +108.997 q^{28} +248.943 q^{29} -142.046 q^{31} -186.226 q^{32} +217.689 q^{34} +200.559 q^{35} +15.0524 q^{37} +107.384 q^{38} -214.480 q^{40} +121.579 q^{41} -68.8365 q^{43} +2.42042 q^{44} +20.5777 q^{46} -107.380 q^{47} +66.6017 q^{49} +43.3287 q^{50} -378.726 q^{52} +640.459 q^{53} +4.45370 q^{55} -438.032 q^{56} -402.520 q^{58} +477.849 q^{59} -79.9338 q^{61} +229.676 q^{62} +236.402 q^{64} -696.877 q^{65} +119.927 q^{67} +725.068 q^{68} -324.289 q^{70} +682.298 q^{71} +564.694 q^{73} -24.3385 q^{74} +357.672 q^{76} +9.09577 q^{77} +828.051 q^{79} -80.1591 q^{80} -196.584 q^{82} +1066.49 q^{83} +1334.16 q^{85} +111.303 q^{86} -9.72711 q^{88} +738.470 q^{89} -1423.23 q^{91} +68.5394 q^{92} +173.624 q^{94} +658.135 q^{95} +725.871 q^{97} -107.690 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

L-functions

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