LMFDB - Embedded newform 289.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [289,4,Mod(1,289)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("289.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(289, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.0515519917$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + \cdots + 1156 $ x^12 - 4x^11 - 58x^10 + 204x^9 + 1191x^8 - 3456x^7 - 10364x^6 + 21448x^5 + 38476x^4 - 32336x^3 - 57024x^2 - 15776*x + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.62703$ of defining polynomial
Character	$\chi$	$=$	289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.86166 q^{2} +5.06554 q^{3} +15.6358 q^{4} -18.5634 q^{5} -24.6269 q^{6} +14.0210 q^{7} -37.1225 q^{8} -1.34032 q^{9} +90.2489 q^{10} +19.4791 q^{11} +79.2036 q^{12} +29.9060 q^{13} -68.1655 q^{14} -94.0335 q^{15} +55.3912 q^{16} +6.51621 q^{18} -45.7882 q^{19} -290.253 q^{20} +71.0240 q^{21} -94.7006 q^{22} -89.3259 q^{23} -188.046 q^{24} +219.599 q^{25} -145.393 q^{26} -143.559 q^{27} +219.229 q^{28} +57.9077 q^{29} +457.159 q^{30} +161.949 q^{31} +27.6872 q^{32} +98.6719 q^{33} -260.277 q^{35} -20.9570 q^{36} -135.583 q^{37} +222.607 q^{38} +151.490 q^{39} +689.120 q^{40} -56.8918 q^{41} -345.295 q^{42} +52.1442 q^{43} +304.570 q^{44} +24.8809 q^{45} +434.272 q^{46} -482.699 q^{47} +280.586 q^{48} -146.411 q^{49} -1067.62 q^{50} +467.603 q^{52} -529.972 q^{53} +697.935 q^{54} -361.597 q^{55} -520.496 q^{56} -231.942 q^{57} -281.528 q^{58} -280.058 q^{59} -1470.29 q^{60} -586.656 q^{61} -787.341 q^{62} -18.7927 q^{63} -577.735 q^{64} -555.156 q^{65} -479.710 q^{66} +367.471 q^{67} -452.484 q^{69} +1265.38 q^{70} +60.5496 q^{71} +49.7563 q^{72} -368.086 q^{73} +659.161 q^{74} +1112.39 q^{75} -715.933 q^{76} +273.116 q^{77} -736.493 q^{78} +217.721 q^{79} -1028.25 q^{80} -691.015 q^{81} +276.589 q^{82} -594.154 q^{83} +1110.51 q^{84} -253.508 q^{86} +293.334 q^{87} -723.112 q^{88} -887.553 q^{89} -120.963 q^{90} +419.312 q^{91} -1396.68 q^{92} +820.358 q^{93} +2346.72 q^{94} +849.983 q^{95} +140.251 q^{96} -884.682 q^{97} +711.801 q^{98} -26.1083 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36}+ \cdots + 3464 q^{98}+O(q^{100}) $ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * q^98

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 $

$2$	−4.86166	−1.71886	−0.859429	−	0.511255i	$-0.829181\pi$
$2$	−4.86166	−1.71886	−0.859429	+	0.511255i	$0.829181\pi$
$3$	5.06554	0.974863	0.487432	−	0.873161i	$-0.337934\pi$
$3$	5.06554	0.974863	0.487432	+	0.873161i	$0.337934\pi$
$4$	15.6358	1.95447
$5$	−18.5634	−1.66036	−0.830179	−	0.557497i	$-0.811762\pi$
$5$	−18.5634	−1.66036	−0.830179	+	0.557497i	$0.811762\pi$
$6$	−24.6269	−1.67565
$7$	14.0210	0.757064	0.378532	−	0.925588i	$-0.376429\pi$
$7$	14.0210	0.757064	0.378532	+	0.925588i	$0.376429\pi$
$8$	−37.1225	−1.64060
$9$	−1.34032	−0.0496417
$10$	90.2489	2.85392
$11$	19.4791	0.533924	0.266962	−	0.963707i	$-0.413980\pi$
$11$	19.4791	0.533924	0.266962	+	0.963707i	$0.413980\pi$
$12$	79.2036	1.90534
$13$	29.9060	0.638033	0.319016	−	0.947749i	$-0.396648\pi$
$13$	29.9060	0.638033	0.319016	+	0.947749i	$0.396648\pi$
$14$	−68.1655	−1.30128
$15$	−94.0335	−1.61862
$16$	55.3912	0.865487
$17$	0	0
$17$	0	0
$18$	6.51621	0.0853269
$19$	−45.7882	−0.552870	−0.276435	−	0.961033i	$-0.589153\pi$
$19$	−45.7882	−0.552870	−0.276435	+	0.961033i	$0.589153\pi$
$20$	−290.253	−3.24512
$21$	71.0240	0.738034
$22$	−94.7006	−0.917738
$23$	−89.3259	−0.809814	−0.404907	−	0.914358i	$-0.632696\pi$
$23$	−89.3259	−0.809814	−0.404907	+	0.914358i	$0.632696\pi$
$24$	−188.046	−1.59936
$25$	219.599	1.75679
$26$	−145.393	−1.09669
$27$	−143.559	−1.02326
$28$	219.229	1.47966
$29$	57.9077	0.370800	0.185400	−	0.982663i	$-0.440642\pi$
$29$	57.9077	0.370800	0.185400	+	0.982663i	$0.440642\pi$
$30$	457.159	2.78218
$31$	161.949	0.938286	0.469143	−	0.883122i	$-0.344563\pi$
$31$	161.949	0.938286	0.469143	+	0.883122i	$0.344563\pi$
$32$	27.6872	0.152952
$33$	98.6719	0.520502
$34$	0	0
$35$	−260.277	−1.25700
$36$	−20.9570	−0.0970232
$37$	−135.583	−0.602426	−0.301213	−	0.953557i	$-0.597392\pi$
$37$	−135.583	−0.602426	−0.301213	+	0.953557i	$0.597392\pi$
$38$	222.607	0.950304
$39$	151.490	0.621994
$40$	689.120	2.72398
$41$	−56.8918	−0.216708	−0.108354	−	0.994112i	$-0.534558\pi$
$41$	−56.8918	−0.216708	−0.108354	+	0.994112i	$0.534558\pi$
$42$	−345.295	−1.26857
$43$	52.1442	0.184928	0.0924642	−	0.995716i	$-0.470526\pi$
$43$	52.1442	0.184928	0.0924642	+	0.995716i	$0.470526\pi$
$44$	304.570	1.04354
$45$	24.8809	0.0824229
$46$	434.272	1.39196
$47$	−482.699	−1.49806	−0.749030	−	0.662536i	$-0.769480\pi$
$47$	−482.699	−1.49806	−0.749030	+	0.662536i	$0.769480\pi$
$48$	280.586	0.843731
$49$	−146.411	−0.426855
$50$	−1067.62	−3.01967
$51$	0	0
$52$	467.603	1.24702
$53$	−529.972	−1.37353	−0.686766	−	0.726879i	$-0.740970\pi$
$53$	−529.972	−1.37353	−0.686766	+	0.726879i	$0.740970\pi$
$54$	697.935	1.75883
$55$	−361.597	−0.886504
$56$	−520.496	−1.24204
$57$	−231.942	−0.538972
$58$	−281.528	−0.637352
$59$	−280.058	−0.617973	−0.308986	−	0.951066i	$-0.599990\pi$
$59$	−280.058	−0.617973	−0.308986	+	0.951066i	$0.599990\pi$
$60$	−1470.29	−3.16355
$61$	−586.656	−1.23137	−0.615686	−	0.787992i	$-0.711121\pi$
$61$	−586.656	−1.23137	−0.615686	+	0.787992i	$0.711121\pi$
$62$	−787.341	−1.61278
$63$	−18.7927	−0.0375819
$64$	−577.735	−1.12839
$65$	−555.156	−1.05936
$66$	−479.710	−0.894669
$67$	367.471	0.670056	0.335028	−	0.942208i	$-0.391254\pi$
$67$	367.471	0.670056	0.335028	+	0.942208i	$0.391254\pi$
$68$	0	0
$69$	−452.484	−0.789458
$70$	1265.38	2.16060
$71$	60.5496	0.101210	0.0506051	−	0.998719i	$-0.483885\pi$
$71$	60.5496	0.101210	0.0506051	+	0.998719i	$0.483885\pi$
$72$	49.7563	0.0814421
$73$	−368.086	−0.590154	−0.295077	−	0.955473i	$-0.595345\pi$
$73$	−368.086	−0.590154	−0.295077	+	0.955473i	$0.595345\pi$
$74$	659.161	1.03548
$75$	1112.39	1.71263
$76$	−715.933	−1.08057
$77$	273.116	0.404214
$78$	−736.493	−1.06912
$79$	217.721	0.310070	0.155035	−	0.987909i	$-0.450451\pi$
$79$	217.721	0.310070	0.155035	+	0.987909i	$0.450451\pi$
$80$	−1028.25	−1.43702
$81$	−691.015	−0.947894
$82$	276.589	0.372490
$83$	−594.154	−0.785746	−0.392873	−	0.919593i	$-0.628519\pi$
$83$	−594.154	−0.785746	−0.392873	+	0.919593i	$0.628519\pi$
$84$	1110.51	1.44247
$85$	0	0
$86$	−253.508	−0.317866
$87$	293.334	0.361479
$88$	−723.112	−0.875955
$89$	−887.553	−1.05708	−0.528542	−	0.848907i	$-0.677261\pi$
$89$	−887.553	−1.05708	−0.528542	+	0.848907i	$0.677261\pi$
$90$	−120.963	−0.141673
$91$	419.312	0.483031
$92$	−1396.68	−1.58276
$93$	820.358	0.914701
$94$	2346.72	2.57495
$95$	849.983	0.917962
$96$	140.251	0.149107
$97$	−884.682	−0.926040	−0.463020	−	0.886348i	$-0.653234\pi$
$97$	−884.682	−0.926040	−0.463020	+	0.886348i	$0.653234\pi$
$98$	711.801	0.733702
$99$	−26.1083	−0.0265048
$100$	3433.60	3.43360
$101$	−1845.31	−1.81798	−0.908988	−	0.416822i	$-0.863144\pi$
$101$	−1845.31	−1.81798	−0.908988	+	0.416822i	$0.863144\pi$
$102$	0	0
$103$	−1823.07	−1.74400	−0.872002	−	0.489502i	$-0.837178\pi$
$103$	−1823.07	−1.74400	−0.872002	+	0.489502i	$0.837178\pi$
$104$	−1110.19	−1.04676
$105$	−1318.44	−1.22540
$106$	2576.54	2.36091
$107$	1855.35	1.67629	0.838146	−	0.545445i	$-0.183639\pi$
$107$	1855.35	1.67629	0.838146	+	0.545445i	$0.183639\pi$
$108$	−2244.66	−1.99993
$109$	1387.46	1.21922	0.609610	−	0.792702i	$-0.291326\pi$
$109$	1387.46	1.21922	0.609610	+	0.792702i	$0.291326\pi$
$110$	1757.96	1.52377
$111$	−686.803	−0.587283
$112$	776.640	0.655229
$113$	−613.115	−0.510416	−0.255208	−	0.966886i	$-0.582144\pi$
$113$	−613.115	−0.510416	−0.255208	+	0.966886i	$0.582144\pi$
$114$	1127.62	0.926417
$115$	1658.19	1.34458
$116$	905.432	0.724717
$117$	−40.0837	−0.0316730
$118$	1361.55	1.06221
$119$	0	0
$120$	3490.76	2.65551
$121$	−951.566	−0.714926
$122$	2852.13	2.11655
$123$	−288.188	−0.211260
$124$	2532.19	1.83385
$125$	−1756.07	−1.25654
$126$	91.3639	0.0645979
$127$	111.688	0.0780368	0.0390184	−	0.999238i	$-0.487577\pi$
$127$	111.688	0.0780368	0.0390184	+	0.999238i	$0.487577\pi$
$128$	2587.26	1.78659
$129$	264.139	0.180280
$130$	2698.98	1.82089
$131$	2132.04	1.42196	0.710981	−	0.703212i	$-0.248251\pi$
$131$	2132.04	1.42196	0.710981	+	0.703212i	$0.248251\pi$
$132$	1542.81	1.01731
$133$	−641.996	−0.418558
$134$	−1786.52	−1.15173
$135$	2664.94	1.69897
$136$	0	0
$137$	3087.68	1.92554	0.962769	−	0.270327i	$-0.0871317\pi$
$137$	3087.68	1.92554	0.962769	+	0.270327i	$0.0871317\pi$
$138$	2199.82	1.35697
$139$	−469.825	−0.286691	−0.143346	−	0.989673i	$-0.545786\pi$
$139$	−469.825	−0.286691	−0.143346	+	0.989673i	$0.545786\pi$
$140$	−4069.64	−2.45677
$141$	−2445.13	−1.46040
$142$	−294.372	−0.173966
$143$	582.540	0.340661
$144$	−74.2421	−0.0429642
$145$	−1074.96	−0.615660
$146$	1789.51	1.01439
$147$	−741.651	−0.416125
$148$	−2119.95	−1.17742
$149$	−2209.36	−1.21475	−0.607374	−	0.794416i	$-0.707777\pi$
$149$	−2209.36	−1.21475	−0.607374	+	0.794416i	$0.707777\pi$
$150$	−5408.05	−2.94377
$151$	−752.366	−0.405475	−0.202737	−	0.979233i	$-0.564984\pi$
$151$	−752.366	−0.405475	−0.202737	+	0.979233i	$0.564984\pi$
$152$	1699.77	0.907038
$153$	0	0
$154$	−1327.80	−0.694787
$155$	−3006.32	−1.55789
$156$	2368.66	1.21567
$157$	579.544	0.294603	0.147301	−	0.989092i	$-0.452941\pi$
$157$	579.544	0.294603	0.147301	+	0.989092i	$0.452941\pi$
$158$	−1058.49	−0.532966
$159$	−2684.59	−1.33901
$160$	−513.968	−0.253955
$161$	−1252.44	−0.613081
$162$	3359.48	1.62929
$163$	−1777.74	−0.854253	−0.427126	−	0.904192i	$-0.640474\pi$
$163$	−1777.74	−0.854253	−0.427126	+	0.904192i	$0.640474\pi$
$164$	−889.548	−0.423549
$165$	−1831.68	−0.864221
$166$	2888.58	1.35059
$167$	3134.84	1.45258	0.726291	−	0.687388i	$-0.241243\pi$
$167$	3134.84	1.45258	0.726291	+	0.687388i	$0.241243\pi$
$168$	−2636.59	−1.21082
$169$	−1302.63	−0.592914
$170$	0	0
$171$	61.3710	0.0274454
$172$	815.315	0.361437
$173$	1169.33	0.513886	0.256943	−	0.966427i	$-0.417285\pi$
$173$	1169.33	0.513886	0.256943	+	0.966427i	$0.417285\pi$
$174$	−1426.09	−0.621331
$175$	3079.00	1.33000
$176$	1078.97	0.462104
$177$	−1418.64	−0.602439
$178$	4314.99	1.81698
$179$	−1552.54	−0.648281	−0.324140	−	0.946009i	$-0.605075\pi$
$179$	−1552.54	−0.648281	−0.324140	+	0.946009i	$0.605075\pi$
$180$	389.033	0.161093
$181$	−1495.92	−0.614313	−0.307156	−	0.951659i	$-0.599377\pi$
$181$	−1495.92	−0.614313	−0.307156	+	0.951659i	$0.599377\pi$
$182$	−2038.55	−0.830262
$183$	−2971.73	−1.20042
$184$	3316.00	1.32858
$185$	2516.88	1.00024
$186$	−3988.30	−1.57224
$187$	0	0
$188$	−7547.37	−2.92792
$189$	−2012.84	−0.774671
$190$	−4132.33	−1.57785
$191$	3008.79	1.13983	0.569917	−	0.821702i	$-0.306975\pi$
$191$	3008.79	1.13983	0.569917	+	0.821702i	$0.306975\pi$
$192$	−2926.54	−1.10002
$193$	3893.56	1.45215	0.726074	−	0.687616i	$-0.241343\pi$
$193$	3893.56	1.45215	0.726074	+	0.687616i	$0.241343\pi$
$194$	4301.03	1.59173
$195$	−2812.16	−1.03273
$196$	−2289.25	−0.834275
$197$	946.676	0.342375	0.171187	−	0.985238i	$-0.445240\pi$
$197$	946.676	0.342375	0.171187	+	0.985238i	$0.445240\pi$
$198$	126.930	0.0455581
$199$	−1765.84	−0.629031	−0.314515	−	0.949252i	$-0.601842\pi$
$199$	−1765.84	−0.629031	−0.314515	+	0.949252i	$0.601842\pi$
$200$	−8152.07	−2.88219
$201$	1861.44	0.653213
$202$	8971.29	3.12484
$203$	811.925	0.280719
$204$	0	0
$205$	1056.10	0.359812
$206$	8863.15	2.99769
$207$	119.726	0.0402005
$208$	1656.53	0.552209
$209$	−891.910	−0.295190
$210$	6409.83	2.10629
$211$	−371.398	−0.121176	−0.0605879	−	0.998163i	$-0.519298\pi$
$211$	−371.398	−0.121176	−0.0605879	+	0.998163i	$0.519298\pi$
$212$	−8286.52	−2.68453
$213$	306.716	0.0986660
$214$	−9020.08	−2.88131
$215$	−967.973	−0.307047
$216$	5329.28	1.67876
$217$	2270.69	0.710342
$218$	−6745.38	−2.09566
$219$	−1864.56	−0.575319
$220$	−5653.85	−1.73265
$221$	0	0
$222$	3339.00	1.00946
$223$	2573.90	0.772919	0.386459	−	0.922306i	$-0.373698\pi$
$223$	2573.90	0.772919	0.386459	+	0.922306i	$0.373698\pi$
$224$	388.203	0.115794
$225$	−294.334	−0.0872100
$226$	2980.76	0.877332
$227$	5204.65	1.52178	0.760892	−	0.648879i	$-0.224762\pi$
$227$	5204.65	1.52178	0.760892	+	0.648879i	$0.224762\pi$
$228$	−3626.59	−1.05341
$229$	3198.73	0.923048	0.461524	−	0.887128i	$-0.347303\pi$
$229$	3198.73	0.923048	0.461524	+	0.887128i	$0.347303\pi$
$230$	−8061.56	−2.31115
$231$	1383.48	0.394053
$232$	−2149.68	−0.608334
$233$	−853.101	−0.239865	−0.119932	−	0.992782i	$-0.538268\pi$
$233$	−853.101	−0.239865	−0.119932	+	0.992782i	$0.538268\pi$
$234$	194.873	0.0544414
$235$	8960.51	2.48732
$236$	−4378.92	−1.20781
$237$	1102.87	0.302276
$238$	0	0
$239$	1892.67	0.512246	0.256123	−	0.966644i	$-0.417555\pi$
$239$	1892.67	0.512246	0.256123	+	0.966644i	$0.417555\pi$
$240$	−5208.62	−1.40090
$241$	−1849.42	−0.494322	−0.247161	−	0.968974i	$-0.579498\pi$
$241$	−1849.42	−0.494322	−0.247161	+	0.968974i	$0.579498\pi$
$242$	4626.19	1.22886
$243$	375.731	0.0991900
$244$	−9172.83	−2.40668
$245$	2717.88	0.708732
$246$	1401.07	0.363126
$247$	−1369.34	−0.352749
$248$	−6011.95	−1.53935
$249$	−3009.71	−0.765995
$250$	8537.43	2.15982
$251$	1026.39	0.258109	0.129055	−	0.991637i	$-0.458806\pi$
$251$	1026.39	0.258109	0.129055	+	0.991637i	$0.458806\pi$
$252$	−293.839	−0.0734527
$253$	−1739.98	−0.432379
$254$	−542.987	−0.134134
$255$	0	0
$256$	−7956.49	−1.94250
$257$	2615.84	0.634910	0.317455	−	0.948273i	$-0.397172\pi$
$257$	2615.84	0.634910	0.317455	+	0.948273i	$0.397172\pi$
$258$	−1284.15	−0.309875
$259$	−1901.02	−0.456075
$260$	−8680.29	−2.07049
$261$	−77.6151	−0.0184071
$262$	−10365.2	−2.44415
$263$	−4992.16	−1.17045	−0.585227	−	0.810869i	$-0.698995\pi$
$263$	−4992.16	−1.17045	−0.585227	+	0.810869i	$0.698995\pi$
$264$	−3662.95	−0.853936
$265$	9838.06	2.28056
$266$	3121.17	0.719441
$267$	−4495.94	−1.03051
$268$	5745.70	1.30961
$269$	−3410.52	−0.773023	−0.386512	−	0.922284i	$-0.626320\pi$
$269$	−3410.52	−0.773023	−0.386512	+	0.922284i	$0.626320\pi$
$270$	−12956.0	−2.92029
$271$	75.0911	0.0168319	0.00841597	−	0.999965i	$-0.497321\pi$
$271$	75.0911	0.0168319	0.00841597	+	0.999965i	$0.497321\pi$
$272$	0	0
$273$	2124.04	0.470889
$274$	−15011.3	−3.30972
$275$	4277.58	0.937992
$276$	−7074.93	−1.54297
$277$	−3441.88	−0.746580	−0.373290	−	0.927715i	$-0.621770\pi$
$277$	−3441.88	−0.746580	−0.373290	+	0.927715i	$0.621770\pi$
$278$	2284.13	0.492781
$279$	−217.064	−0.0465781
$280$	9662.16	2.06223
$281$	1673.05	0.355181	0.177591	−	0.984104i	$-0.443170\pi$
$281$	1673.05	0.355181	0.177591	+	0.984104i	$0.443170\pi$
$282$	11887.4	2.51023
$283$	6792.22	1.42670	0.713349	−	0.700809i	$-0.247178\pi$
$283$	6792.22	1.42670	0.713349	+	0.700809i	$0.247178\pi$
$284$	946.740	0.197812
$285$	4305.62	0.894887
$286$	−2832.11	−0.585547
$287$	−797.682	−0.164062
$288$	−37.1098	−0.00759277
$289$	0	0
$290$	5226.11	1.05823
$291$	−4481.39	−0.902762
$292$	−5755.31	−1.15344
$293$	−6436.90	−1.28344	−0.641720	−	0.766939i	$-0.721779\pi$
$293$	−6436.90	−1.28344	−0.641720	+	0.766939i	$0.721779\pi$
$294$	3605.66	0.715259
$295$	5198.81	1.02606
$296$	5033.20	0.988340
$297$	−2796.39	−0.546341
$298$	10741.2	2.08798
$299$	−2671.38	−0.516688
$300$	17393.0	3.34729
$301$	731.115	0.140003
$302$	3657.75	0.696953
$303$	−9347.51	−1.77228
$304$	−2536.26	−0.478501
$305$	10890.3	2.04452
$306$	0	0
$307$	5129.87	0.953672	0.476836	−	0.878992i	$-0.341784\pi$
$307$	5129.87	0.953672	0.476836	+	0.878992i	$0.341784\pi$
$308$	4270.38	0.790025
$309$	−9234.83	−1.70017
$310$	14615.7	2.67779
$311$	−3422.67	−0.624058	−0.312029	−	0.950073i	$-0.601009\pi$
$311$	−3422.67	−0.624058	−0.312029	+	0.950073i	$0.601009\pi$
$312$	−5623.69	−1.02044
$313$	−4283.53	−0.773544	−0.386772	−	0.922175i	$-0.626410\pi$
$313$	−4283.53	−0.773544	−0.386772	+	0.922175i	$0.626410\pi$
$314$	−2817.55	−0.506380
$315$	348.856	0.0623994
$316$	3404.23	0.606022
$317$	390.350	0.0691617	0.0345808	−	0.999402i	$-0.488990\pi$
$317$	390.350	0.0691617	0.0345808	+	0.999402i	$0.488990\pi$
$318$	13051.6	2.30156
$319$	1127.99	0.197979
$320$	10724.7	1.87353
$321$	9398.34	1.63416
$322$	6088.94	1.05380
$323$	0	0
$324$	−10804.5	−1.85263
$325$	6567.31	1.12089
$326$	8642.77	1.46834
$327$	7028.25	1.18857
$328$	2111.97	0.355531
$329$	−6767.93	−1.13413
$330$	8905.03	1.48547
$331$	3824.89	0.635151	0.317575	−	0.948233i	$-0.397131\pi$
$331$	3824.89	0.635151	0.317575	+	0.948233i	$0.397131\pi$
$332$	−9290.06	−1.53572
$333$	181.726	0.0299054
$334$	−15240.5	−2.49678
$335$	−6821.51	−1.11253
$336$	3934.10	0.638758
$337$	2376.90	0.384208	0.192104	−	0.981375i	$-0.438469\pi$
$337$	2376.90	0.384208	0.192104	+	0.981375i	$0.438469\pi$
$338$	6332.96	1.01914
$339$	−3105.76	−0.497586
$340$	0	0
$341$	3154.61	0.500973
$342$	−298.365	−0.0471747
$343$	−6862.04	−1.08022
$344$	−1935.73	−0.303394
$345$	8399.62	1.31078
$346$	−5684.87	−0.883296
$347$	−11923.2	−1.84459	−0.922293	−	0.386491i	$-0.873687\pi$
$347$	−11923.2	−1.84459	−0.922293	+	0.386491i	$0.873687\pi$
$348$	4586.50	0.706500
$349$	−7389.94	−1.13345	−0.566726	−	0.823906i	$-0.691790\pi$
$349$	−7389.94	−1.13345	−0.566726	+	0.823906i	$0.691790\pi$
$350$	−14969.1	−2.28608
$351$	−4293.27	−0.652871
$352$	539.321	0.0816645
$353$	2966.18	0.447235	0.223617	−	0.974677i	$-0.428213\pi$
$353$	2966.18	0.447235	0.223617	+	0.974677i	$0.428213\pi$
$354$	6896.96	1.03551
$355$	−1124.01	−0.168045
$356$	−13877.6	−2.06604
$357$	0	0
$358$	7547.93	1.11430
$359$	−6526.33	−0.959460	−0.479730	−	0.877416i	$-0.659265\pi$
$359$	−6526.33	−0.959460	−0.479730	+	0.877416i	$0.659265\pi$
$360$	−923.644	−0.135223
$361$	−4762.44	−0.694335
$362$	7272.64	1.05592
$363$	−4820.19	−0.696955
$364$	6556.27	0.944071
$365$	6832.92	0.979867
$366$	14447.6	2.06335
$367$	−5896.51	−0.838679	−0.419340	−	0.907829i	$-0.637738\pi$
$367$	−5896.51	−0.838679	−0.419340	+	0.907829i	$0.637738\pi$
$368$	−4947.86	−0.700884
$369$	76.2536	0.0107577
$370$	−12236.2	−1.71928
$371$	−7430.74	−1.03985
$372$	12826.9	1.78776
$373$	−9234.98	−1.28195	−0.640977	−	0.767560i	$-0.721471\pi$
$373$	−9234.98	−1.28195	−0.640977	+	0.767560i	$0.721471\pi$
$374$	0	0
$375$	−8895.45	−1.22496
$376$	17919.0	2.45772
$377$	1731.79	0.236582
$378$	9785.77	1.33155
$379$	−1449.66	−0.196475	−0.0982373	−	0.995163i	$-0.531320\pi$
$379$	−1449.66	−0.196475	−0.0982373	+	0.995163i	$0.531320\pi$
$380$	13290.1	1.79413
$381$	565.758	0.0760752
$382$	−14627.7	−1.95921
$383$	8159.07	1.08854	0.544268	−	0.838911i	$-0.316808\pi$
$383$	8159.07	1.08854	0.544268	+	0.838911i	$0.316808\pi$
$384$	13105.8	1.74168
$385$	−5069.96	−0.671140
$386$	−18929.2	−2.49604
$387$	−69.8902	−0.00918015
$388$	−13832.7	−1.80992
$389$	−1467.86	−0.191320	−0.0956598	−	0.995414i	$-0.530496\pi$
$389$	−1467.86	−0.191320	−0.0956598	+	0.995414i	$0.530496\pi$
$390$	13671.8	1.77512
$391$	0	0
$392$	5435.15	0.700298
$393$	10799.9	1.38622
$394$	−4602.42	−0.588494
$395$	−4041.63	−0.514827
$396$	−408.223	−0.0518030
$397$	6018.22	0.760821	0.380411	−	0.924818i	$-0.375783\pi$
$397$	6018.22	0.760821	0.380411	+	0.924818i	$0.375783\pi$
$398$	8584.92	1.08121
$399$	−3252.06	−0.408036
$400$	12163.8	1.52048
$401$	−5899.13	−0.734634	−0.367317	−	0.930096i	$-0.619724\pi$
$401$	−5899.13	−0.734634	−0.367317	+	0.930096i	$0.619724\pi$
$402$	−9049.70	−1.12278
$403$	4843.24	0.598657
$404$	−28852.9	−3.55318
$405$	12827.6	1.57384
$406$	−3947.31	−0.482516
$407$	−2641.04	−0.321649
$408$	0	0
$409$	−9261.09	−1.11964	−0.559818	−	0.828616i	$-0.689129\pi$
$409$	−9261.09	−1.11964	−0.559818	+	0.828616i	$0.689129\pi$
$410$	−5134.42	−0.618466
$411$	15640.8	1.87714
$412$	−28505.1	−3.40861
$413$	−3926.69	−0.467845
$414$	−582.066	−0.0690990
$415$	11029.5	1.30462
$416$	828.013	0.0975881
$417$	−2379.92	−0.279485
$418$	4336.17	0.507390
$419$	8538.07	0.995494	0.497747	−	0.867322i	$-0.334161\pi$
$419$	8538.07	0.995494	0.497747	+	0.867322i	$0.334161\pi$
$420$	−20614.9	−2.39501
$421$	−1617.31	−0.187228	−0.0936140	−	0.995609i	$-0.529842\pi$
$421$	−1617.31	−0.187228	−0.0936140	+	0.995609i	$0.529842\pi$
$422$	1805.61	0.208284
$423$	646.973	0.0743662
$424$	19673.9	2.25342
$425$	0	0
$426$	−1491.15	−0.169593
$427$	−8225.52	−0.932227
$428$	29009.8	3.27627
$429$	2950.88	0.332097
$430$	4705.96	0.527771
$431$	−2537.10	−0.283544	−0.141772	−	0.989899i	$-0.545280\pi$
$431$	−2537.10	−0.283544	−0.141772	+	0.989899i	$0.545280\pi$
$432$	−7951.90	−0.885616
$433$	1833.33	0.203474	0.101737	−	0.994811i	$-0.467560\pi$
$433$	1833.33	0.203474	0.101737	+	0.994811i	$0.467560\pi$
$434$	−11039.3	−1.22098
$435$	−5445.26	−0.600185
$436$	21694.1	2.38293
$437$	4090.07	0.447722
$438$	9064.84	0.988892
$439$	−10398.5	−1.13051	−0.565257	−	0.824915i	$-0.691223\pi$
$439$	−10398.5	−1.13051	−0.565257	+	0.824915i	$0.691223\pi$
$440$	13423.4	1.45440
$441$	196.238	0.0211898
$442$	0	0
$443$	−3984.14	−0.427296	−0.213648	−	0.976911i	$-0.568535\pi$
$443$	−3984.14	−0.427296	−0.213648	+	0.976911i	$0.568535\pi$
$444$	−10738.7	−1.14783
$445$	16476.0	1.75514
$446$	−12513.4	−1.32854
$447$	−11191.6	−1.18421
$448$	−8100.43	−0.854262
$449$	13184.1	1.38574	0.692868	−	0.721064i	$-0.256347\pi$
$449$	13184.1	1.38574	0.692868	+	0.721064i	$0.256347\pi$
$450$	1430.95	0.149902
$451$	−1108.20	−0.115705
$452$	−9586.52	−0.997593
$453$	−3811.14	−0.395282
$454$	−25303.3	−2.61573
$455$	−7783.85	−0.802005
$456$	8610.26	0.884238
$457$	−16241.9	−1.66250	−0.831252	−	0.555896i	$-0.812375\pi$
$457$	−16241.9	−1.66250	−0.831252	+	0.555896i	$0.812375\pi$
$458$	−15551.2	−1.58659
$459$	0	0
$460$	25927.1	2.62795
$461$	11054.7	1.11685	0.558425	−	0.829555i	$-0.311406\pi$
$461$	11054.7	1.11685	0.558425	+	0.829555i	$0.311406\pi$
$462$	−6726.02	−0.677322
$463$	14185.7	1.42390	0.711952	−	0.702228i	$-0.247811\pi$
$463$	14185.7	1.42390	0.711952	+	0.702228i	$0.247811\pi$
$464$	3207.58	0.320922
$465$	−15228.6	−1.51873
$466$	4147.49	0.412293
$467$	9689.39	0.960110	0.480055	−	0.877238i	$-0.340617\pi$
$467$	9689.39	0.960110	0.480055	+	0.877238i	$0.340617\pi$
$468$	−626.740	−0.0619040
$469$	5152.32	0.507275
$470$	−43563.0	−4.27534
$471$	2935.70	0.287197
$472$	10396.5	1.01385
$473$	1015.72	0.0987376
$474$	−5361.80	−0.519569
$475$	−10055.0	−0.971276
$476$	0	0
$477$	710.334	0.0681844
$478$	−9201.54	−0.880478
$479$	−13237.0	−1.26266	−0.631330	−	0.775514i	$-0.717491\pi$
$479$	−13237.0	−1.26266	−0.631330	+	0.775514i	$0.717491\pi$
$480$	−2603.52	−0.247571
$481$	−4054.75	−0.384367
$482$	8991.25	0.849669
$483$	−6344.28	−0.597670
$484$	−14878.5	−1.39730
$485$	16422.7	1.53756
$486$	−1826.68	−0.170493
$487$	19300.3	1.79585	0.897924	−	0.440150i	$-0.145075\pi$
$487$	19300.3	1.79585	0.897924	+	0.440150i	$0.145075\pi$
$488$	21778.2	2.02019
$489$	−9005.20	−0.832780
$490$	−13213.4	−1.21821
$491$	−72.2707	−0.00664263	−0.00332131	−	0.999994i	$-0.501057\pi$
$491$	−72.2707	−0.00664263	−0.00332131	+	0.999994i	$0.501057\pi$
$492$	−4506.04	−0.412902
$493$	0	0
$494$	6657.27	0.606325
$495$	484.658	0.0440075
$496$	8970.53	0.812074
$497$	848.967	0.0766225
$498$	14632.2	1.31664
$499$	14805.5	1.32823	0.664116	−	0.747630i	$-0.268808\pi$
$499$	14805.5	1.32823	0.664116	+	0.747630i	$0.268808\pi$
$500$	−27457.6	−2.45588
$501$	15879.6	1.41607
$502$	−4989.98	−0.443653
$503$	−6693.90	−0.593372	−0.296686	−	0.954975i	$-0.595882\pi$
$503$	−6693.90	−0.593372	−0.296686	+	0.954975i	$0.595882\pi$
$504$	697.633	0.0616569
$505$	34255.2	3.01849
$506$	8459.22	0.743198
$507$	−6598.54	−0.578011
$508$	1746.32	0.152521
$509$	4414.17	0.384390	0.192195	−	0.981357i	$-0.438439\pi$
$509$	4414.17	0.384390	0.192195	+	0.981357i	$0.438439\pi$
$510$	0	0
$511$	−5160.94	−0.446784
$512$	17983.7	1.55230
$513$	6573.30	0.565728
$514$	−12717.4	−1.09132
$515$	33842.3	2.89567
$516$	4130.01	0.352352
$517$	−9402.52	−0.799850
$518$	9242.10	0.783928
$519$	5923.27	0.500968
$520$	20608.8	1.73799
$521$	−1152.08	−0.0968781	−0.0484391	−	0.998826i	$-0.515425\pi$
$521$	−1152.08	−0.0968781	−0.0484391	+	0.998826i	$0.515425\pi$
$522$	377.339	0.0316392
$523$	8548.79	0.714747	0.357373	−	0.933962i	$-0.383672\pi$
$523$	8548.79	0.714747	0.357373	+	0.933962i	$0.383672\pi$
$524$	33336.0	2.77918
$525$	15596.8	1.29657
$526$	24270.2	2.01185
$527$	0	0
$528$	5465.55	0.450488
$529$	−4187.89	−0.344201
$530$	−47829.3	−3.91995
$531$	375.368	0.0306772
$532$	−10038.1	−0.818059
$533$	−1701.41	−0.138267
$534$	21857.7	1.77130
$535$	−34441.5	−2.78325
$536$	−13641.5	−1.09929
$537$	−7864.45	−0.631985
$538$	16580.8	1.32872
$539$	−2851.95	−0.227908
$540$	41668.4	3.32060
$541$	−1820.54	−0.144679	−0.0723393	−	0.997380i	$-0.523046\pi$
$541$	−1820.54	−0.144679	−0.0723393	+	0.997380i	$0.523046\pi$
$542$	−365.067	−0.0289317
$543$	−7577.62	−0.598871
$544$	0	0
$545$	−25756.0	−2.02434
$546$	−10326.4	−0.809392
$547$	82.7848	0.00647097	0.00323549	−	0.999995i	$-0.498970\pi$
$547$	82.7848	0.00647097	0.00323549	+	0.999995i	$0.498970\pi$
$548$	48278.3	3.76341
$549$	786.310	0.0611273
$550$	−20796.1	−1.61227
$551$	−2651.49	−0.205004
$552$	16797.3	1.29519
$553$	3052.67	0.234743
$554$	16733.3	1.28326
$555$	12749.4	0.975100
$556$	−7346.08	−0.560329
$557$	−12671.9	−0.963963	−0.481981	−	0.876181i	$-0.660083\pi$
$557$	−12671.9	−0.963963	−0.481981	+	0.876181i	$0.660083\pi$
$558$	1055.29	0.0800611
$559$	1559.42	0.117990
$560$	−14417.1	−1.08791
$561$	0	0
$562$	−8133.82	−0.610506
$563$	4708.47	0.352466	0.176233	−	0.984348i	$-0.443609\pi$
$563$	4708.47	0.352466	0.176233	+	0.984348i	$0.443609\pi$
$564$	−38231.5	−2.85432
$565$	11381.5	0.847473
$566$	−33021.5	−2.45229
$567$	−9688.73	−0.717616
$568$	−2247.76	−0.166045
$569$	8492.74	0.625719	0.312860	−	0.949799i	$-0.398713\pi$
$569$	8492.74	0.625719	0.312860	+	0.949799i	$0.398713\pi$
$570$	−20932.5	−1.53818
$571$	−1128.62	−0.0827166	−0.0413583	−	0.999144i	$-0.513169\pi$
$571$	−1128.62	−0.0827166	−0.0413583	+	0.999144i	$0.513169\pi$
$572$	9108.47	0.665811
$573$	15241.1	1.11118
$574$	3878.06	0.281998
$575$	−19615.9	−1.42267
$576$	774.353	0.0560151
$577$	18288.9	1.31955	0.659773	−	0.751465i	$-0.270652\pi$
$577$	18288.9	1.31955	0.659773	+	0.751465i	$0.270652\pi$
$578$	0	0
$579$	19723.0	1.41565
$580$	−16807.9	−1.20329
$581$	−8330.65	−0.594860
$582$	21787.0	1.55172
$583$	−10323.4	−0.733361
$584$	13664.3	0.968207
$585$	744.089	0.0525885
$586$	31294.1	2.20605
$587$	11310.2	0.795265	0.397633	−	0.917545i	$-0.369832\pi$
$587$	11310.2	0.795265	0.397633	+	0.917545i	$0.369832\pi$
$588$	−11596.3	−0.813304
$589$	−7415.34	−0.518750
$590$	−25274.9	−1.76364
$591$	4795.42	0.333769
$592$	−7510.12	−0.521392
$593$	5283.47	0.365879	0.182939	−	0.983124i	$-0.441439\pi$
$593$	5283.47	0.365879	0.182939	+	0.983124i	$0.441439\pi$
$594$	13595.1	0.939082
$595$	0	0
$596$	−34545.0	−2.37419
$597$	−8944.93	−0.613219
$598$	12987.3	0.888113
$599$	−22287.6	−1.52028	−0.760138	−	0.649762i	$-0.774869\pi$
$599$	−22287.6	−1.52028	−0.760138	+	0.649762i	$0.774869\pi$
$600$	−41294.6	−2.80974
$601$	−15046.8	−1.02125	−0.510625	−	0.859804i	$-0.670586\pi$
$601$	−15046.8	−1.02125	−0.510625	+	0.859804i	$0.670586\pi$
$602$	−3554.44	−0.240644
$603$	−492.531	−0.0332627
$604$	−11763.8	−0.792489
$605$	17664.3	1.18703
$606$	45444.4	3.04629
$607$	11288.7	0.754848	0.377424	−	0.926041i	$-0.376810\pi$
$607$	11288.7	0.754848	0.377424	+	0.926041i	$0.376810\pi$
$608$	−1267.75	−0.0845623
$609$	4112.84	0.273663
$610$	−52945.1	−3.51424
$611$	−14435.6	−0.955811
$612$	0	0
$613$	14753.0	0.972054	0.486027	−	0.873944i	$-0.338446\pi$
$613$	14753.0	0.972054	0.486027	+	0.873944i	$0.338446\pi$
$614$	−24939.7	−1.63923
$615$	5349.74	0.350768
$616$	−10138.8	−0.663154
$617$	6693.56	0.436747	0.218373	−	0.975865i	$-0.429925\pi$
$617$	6693.56	0.436747	0.218373	+	0.975865i	$0.429925\pi$
$618$	44896.6	2.92234
$619$	1285.36	0.0834617	0.0417308	−	0.999129i	$-0.486713\pi$
$619$	1285.36	0.0834617	0.0417308	+	0.999129i	$0.486713\pi$
$620$	−47006.1	−3.04485
$621$	12823.5	0.828648
$622$	16639.9	1.07267
$623$	−12444.4	−0.800280
$624$	8391.20	0.538328
$625$	5148.78	0.329522
$626$	20825.1	1.32961
$627$	−4518.01	−0.287770
$628$	9061.61	0.575792
$629$	0	0
$630$	−1696.02	−0.107256
$631$	27186.8	1.71520	0.857598	−	0.514320i	$-0.171956\pi$
$631$	27186.8	1.71520	0.857598	+	0.514320i	$0.171956\pi$
$632$	−8082.35	−0.508700
$633$	−1881.33	−0.118130
$634$	−1897.75	−0.118879
$635$	−2073.30	−0.129569
$636$	−41975.7	−2.61705
$637$	−4378.57	−0.272347
$638$	−5483.90	−0.340297
$639$	−81.1562	−0.00502424
$640$	−48028.2	−2.96638
$641$	12321.5	0.759236	0.379618	−	0.925143i	$-0.376055\pi$
$641$	12321.5	0.759236	0.379618	+	0.925143i	$0.376055\pi$
$642$	−45691.6	−2.80888
$643$	−25577.1	−1.56868	−0.784341	−	0.620330i	$-0.786999\pi$
$643$	−25577.1	−1.56868	−0.784341	+	0.620330i	$0.786999\pi$
$644$	−19582.9	−1.19825
$645$	−4903.30	−0.299329
$646$	0	0
$647$	−28275.5	−1.71812	−0.859062	−	0.511872i	$-0.828952\pi$
$647$	−28275.5	−1.71812	−0.859062	+	0.511872i	$0.828952\pi$
$648$	25652.2	1.55512
$649$	−5455.26	−0.329950
$650$	−31928.1	−1.92665
$651$	11502.3	0.692487
$652$	−27796.3	−1.66961
$653$	2354.19	0.141082	0.0705411	−	0.997509i	$-0.477527\pi$
$653$	2354.19	0.141082	0.0705411	+	0.997509i	$0.477527\pi$
$654$	−34169.0	−2.04299
$655$	−39577.8	−2.36097
$656$	−3151.31	−0.187558
$657$	493.355	0.0292962
$658$	32903.4	1.94940
$659$	21272.6	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$659$	21272.6	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$660$	−28639.8	−1.68909
$661$	−16621.3	−0.978051	−0.489026	−	0.872269i	$-0.662648\pi$
$661$	−16621.3	−0.978051	−0.489026	+	0.872269i	$0.662648\pi$
$662$	−18595.3	−1.09173
$663$	0	0
$664$	22056.5	1.28910
$665$	11917.6	0.694956
$666$	−883.489	−0.0514032
$667$	−5172.66	−0.300279
$668$	49015.6	2.83903
$669$	13038.2	0.753490
$670$	33163.9	1.91229
$671$	−11427.5	−0.657458
$672$	1966.46	0.112883
$673$	228.679	0.0130979	0.00654897	−	0.999979i	$-0.497915\pi$
$673$	228.679	0.0130979	0.00654897	+	0.999979i	$0.497915\pi$
$674$	−11555.7	−0.660399
$675$	−31525.4	−1.79765
$676$	−20367.7	−1.15883
$677$	9975.67	0.566316	0.283158	−	0.959073i	$-0.408618\pi$
$677$	9975.67	0.566316	0.283158	+	0.959073i	$0.408618\pi$
$678$	15099.1	0.855279
$679$	−12404.1	−0.701071
$680$	0	0
$681$	26364.4	1.48353
$682$	−15336.7	−0.861101
$683$	31110.8	1.74293	0.871464	−	0.490459i	$-0.163171\pi$
$683$	31110.8	1.74293	0.871464	+	0.490459i	$0.163171\pi$
$684$	959.583	0.0536412
$685$	−57317.8	−3.19708
$686$	33360.9	1.85674
$687$	16203.3	0.899846
$688$	2888.33	0.160053
$689$	−15849.3	−0.876358
$690$	−40836.1	−2.25305
$691$	−5108.91	−0.281262	−0.140631	−	0.990062i	$-0.544913\pi$
$691$	−5108.91	−0.281262	−0.140631	+	0.990062i	$0.544913\pi$
$692$	18283.3	1.00437
$693$	−366.064	−0.0200659
$694$	57966.6	3.17058
$695$	8721.54	0.476010
$696$	−10889.3	−0.593043
$697$	0	0
$698$	35927.4	1.94824
$699$	−4321.41	−0.233835
$700$	48142.5	2.59945
$701$	11204.5	0.603691	0.301846	−	0.953357i	$-0.402397\pi$
$701$	11204.5	0.603691	0.301846	+	0.953357i	$0.402397\pi$
$702$	20872.4	1.12219
$703$	6208.11	0.333063
$704$	−11253.7	−0.602473
$705$	45389.8	2.42479
$706$	−14420.6	−0.768733
$707$	−25873.2	−1.37632
$708$	−22181.6	−1.17745
$709$	−33950.0	−1.79833	−0.899167	−	0.437605i	$-0.855827\pi$
$709$	−33950.0	−1.79833	−0.899167	+	0.437605i	$0.855827\pi$
$710$	5464.54	0.288846
$711$	−291.817	−0.0153924
$712$	32948.2	1.73425
$713$	−14466.2	−0.759838
$714$	0	0
$715$	−10813.9	−0.565619
$716$	−24275.2	−1.26705
$717$	9587.41	0.499370
$718$	31728.8	1.64918
$719$	6231.07	0.323198	0.161599	−	0.986856i	$-0.448335\pi$
$719$	6231.07	0.323198	0.161599	+	0.986856i	$0.448335\pi$
$720$	1378.18	0.0713360
$721$	−25561.3	−1.32032
$722$	23153.4	1.19346
$723$	−9368.30	−0.481896
$724$	−23389.8	−1.20066
$725$	12716.5	0.651417
$726$	23434.2	1.19797
$727$	−5206.88	−0.265629	−0.132815	−	0.991141i	$-0.542402\pi$
$727$	−5206.88	−0.265629	−0.132815	+	0.991141i	$0.542402\pi$
$728$	−15565.9	−0.792461
$729$	20560.7	1.04459
$730$	−33219.4	−1.68425
$731$	0	0
$732$	−46465.3	−2.34618
$733$	3817.57	0.192367	0.0961835	−	0.995364i	$-0.469336\pi$
$733$	3817.57	0.192367	0.0961835	+	0.995364i	$0.469336\pi$
$734$	28666.8	1.44157
$735$	13767.5	0.690916
$736$	−2473.18	−0.123862
$737$	7158.00	0.357759
$738$	−370.719	−0.0184910
$739$	−9211.40	−0.458521	−0.229260	−	0.973365i	$-0.573631\pi$
$739$	−9211.40	−0.458521	−0.229260	+	0.973365i	$0.573631\pi$
$740$	39353.4	1.95495
$741$	−6936.44	−0.343882
$742$	36125.8	1.78736
$743$	−23138.9	−1.14251	−0.571255	−	0.820773i	$-0.693543\pi$
$743$	−23138.9	−1.14251	−0.571255	+	0.820773i	$0.693543\pi$
$744$	−30453.8	−1.50066
$745$	41013.1	2.01692
$746$	44897.4	2.20350
$747$	796.360	0.0390057
$748$	0	0
$749$	26013.9	1.26906
$750$	43246.7	2.10553
$751$	−31141.4	−1.51314	−0.756569	−	0.653914i	$-0.773126\pi$
$751$	−31141.4	−1.51314	−0.756569	+	0.653914i	$0.773126\pi$
$752$	−26737.2	−1.29655
$753$	5199.24	0.251621
$754$	−8419.36	−0.406651
$755$	13966.4	0.673233
$756$	−31472.4	−1.51407
$757$	30991.3	1.48798	0.743988	−	0.668193i	$-0.232932\pi$
$757$	30991.3	1.48798	0.743988	+	0.668193i	$0.232932\pi$
$758$	7047.74	0.337712
$759$	−8813.96	−0.421510
$760$	−31553.5	−1.50601
$761$	−14150.5	−0.674053	−0.337026	−	0.941495i	$-0.609421\pi$
$761$	−14150.5	−0.674053	−0.337026	+	0.941495i	$0.609421\pi$
$762$	−2750.52	−0.130762
$763$	19453.7	0.923027
$764$	47044.7	2.22777
$765$	0	0
$766$	−39666.7	−1.87104
$767$	−8375.39	−0.394287
$768$	−40303.9	−1.89367
$769$	−619.823	−0.0290655	−0.0145328	−	0.999894i	$-0.504626\pi$
$769$	−619.823	−0.0290655	−0.0145328	+	0.999894i	$0.504626\pi$
$770$	24648.4	1.15359
$771$	13250.7	0.618950
$772$	60878.8	2.83818
$773$	10196.4	0.474436	0.237218	−	0.971456i	$-0.423764\pi$
$773$	10196.4	0.474436	0.237218	+	0.971456i	$0.423764\pi$
$774$	339.783	0.0157794
$775$	35563.8	1.64837
$776$	32841.7	1.51926
$777$	−9629.67	−0.444611
$778$	7136.23	0.328851
$779$	2604.97	0.119811
$780$	−43970.3	−2.01845
$781$	1179.45	0.0540385
$782$	0	0
$783$	−8313.17	−0.379423
$784$	−8109.88	−0.369437
$785$	−10758.3	−0.489146
$786$	−52505.5	−2.38271
$787$	33206.7	1.50405	0.752027	−	0.659132i	$-0.229076\pi$
$787$	33206.7	1.50405	0.752027	+	0.659132i	$0.229076\pi$
$788$	14802.0	0.669162
$789$	−25288.0	−1.14103
$790$	19649.1	0.884914
$791$	−8596.49	−0.386417
$792$	969.205	0.0434839
$793$	−17544.5	−0.785655
$794$	−29258.6	−1.30774
$795$	49835.1	2.22323
$796$	−27610.3	−1.22942
$797$	−4351.64	−0.193404	−0.0967020	−	0.995313i	$-0.530829\pi$
$797$	−4351.64	−0.193404	−0.0967020	+	0.995313i	$0.530829\pi$
$798$	15810.4	0.701356
$799$	0	0
$800$	6080.08	0.268704
$801$	1189.61	0.0524754
$802$	28679.6	1.26273
$803$	−7169.98	−0.315097
$804$	29105.1	1.27669
$805$	23249.5	1.01793
$806$	−23546.2	−1.02901
$807$	−17276.1	−0.753592
$808$	68502.7	2.98257
$809$	13515.2	0.587355	0.293678	−	0.955905i	$-0.405121\pi$
$809$	13515.2	0.587355	0.293678	+	0.955905i	$0.405121\pi$
$810$	−62363.3	−2.70521
$811$	5758.39	0.249327	0.124664	−	0.992199i	$-0.460215\pi$
$811$	5758.39	0.249327	0.124664	+	0.992199i	$0.460215\pi$
$812$	12695.1	0.548657
$813$	380.377	0.0164088
$814$	12839.8	0.552870
$815$	33000.8	1.41837
$816$	0	0
$817$	−2387.59	−0.102241
$818$	45024.3	1.92450
$819$	−562.014	−0.0239785
$820$	16513.0	0.703243
$821$	−27956.3	−1.18841	−0.594203	−	0.804315i	$-0.702532\pi$
$821$	−27956.3	−1.18841	−0.594203	+	0.804315i	$0.702532\pi$
$822$	−76040.2	−3.22653
$823$	−38040.6	−1.61119	−0.805596	−	0.592465i	$-0.798155\pi$
$823$	−38040.6	−1.61119	−0.805596	+	0.592465i	$0.798155\pi$
$824$	67677.0	2.86121
$825$	21668.2	0.914414
$826$	19090.3	0.804159
$827$	14467.5	0.608324	0.304162	−	0.952620i	$-0.401624\pi$
$827$	14467.5	0.608324	0.304162	+	0.952620i	$0.401624\pi$
$828$	1872.00	0.0785708
$829$	−24519.4	−1.02726	−0.513628	−	0.858013i	$-0.671699\pi$
$829$	−24519.4	−1.02726	−0.513628	+	0.858013i	$0.671699\pi$
$830$	−53621.8	−2.24246
$831$	−17435.0	−0.727813
$832$	−17277.7	−0.719949
$833$	0	0
$834$	11570.4	0.480394
$835$	−58193.2	−2.41181
$836$	−13945.7	−0.576941
$837$	−23249.2	−0.960108
$838$	−41509.2	−1.71111
$839$	−38066.2	−1.56638	−0.783188	−	0.621785i	$-0.786408\pi$
$839$	−38066.2	−1.56638	−0.783188	+	0.621785i	$0.786408\pi$
$840$	48944.0	2.01039
$841$	−21035.7	−0.862508
$842$	7862.83	0.321818
$843$	8474.91	0.346253
$844$	−5807.09	−0.236835
$845$	24181.3	0.984451
$846$	−3145.36	−0.127825
$847$	−13341.9	−0.541244
$848$	−29355.7	−1.18877
$849$	34406.2	1.39083
$850$	0	0
$851$	12111.1	0.487853
$852$	4795.75	0.192840
$853$	−37413.0	−1.50175	−0.750877	−	0.660442i	$-0.770369\pi$
$853$	−37413.0	−1.50175	−0.750877	+	0.660442i	$0.770369\pi$
$854$	39989.7	1.60237
$855$	−1139.25	−0.0455691
$856$	−68875.3	−2.75013
$857$	29090.0	1.15951	0.579753	−	0.814793i	$-0.303149\pi$
$857$	29090.0	1.15951	0.579753	+	0.814793i	$0.303149\pi$
$858$	−14346.2	−0.570828
$859$	−43523.6	−1.72876	−0.864381	−	0.502837i	$-0.832290\pi$
$859$	−43523.6	−1.72876	−0.864381	+	0.502837i	$0.832290\pi$
$860$	−15135.0	−0.600115
$861$	−4040.69	−0.159938
$862$	12334.5	0.487373
$863$	15659.4	0.617672	0.308836	−	0.951115i	$-0.400061\pi$
$863$	15659.4	0.617672	0.308836	+	0.951115i	$0.400061\pi$
$864$	−3974.75	−0.156509
$865$	−21706.6	−0.853234
$866$	−8913.05	−0.349743
$867$	0	0
$868$	35503.9	1.38834
$869$	4241.00	0.165553
$870$	26473.0	1.03163
$871$	10989.6	0.427518
$872$	−51506.2	−2.00025
$873$	1185.76	0.0459702
$874$	−19884.5	−0.769570
$875$	−24621.9	−0.951283
$876$	−29153.8	−1.12445
$877$	−28891.8	−1.11244	−0.556219	−	0.831036i	$-0.687748\pi$
$877$	−28891.8	−1.11244	−0.556219	+	0.831036i	$0.687748\pi$
$878$	50554.2	1.94319
$879$	−32606.4	−1.25118
$880$	−20029.3	−0.767258
$881$	22094.9	0.844943	0.422471	−	0.906376i	$-0.361163\pi$
$881$	22094.9	0.844943	0.422471	+	0.906376i	$0.361163\pi$
$882$	−954.045	−0.0364222
$883$	40158.3	1.53050	0.765252	−	0.643731i	$-0.222615\pi$
$883$	40158.3	1.53050	0.765252	+	0.643731i	$0.222615\pi$
$884$	0	0
$885$	26334.8	1.00026
$886$	19369.6	0.734462
$887$	27801.7	1.05241	0.526207	−	0.850357i	$-0.323614\pi$
$887$	27801.7	1.05241	0.526207	+	0.850357i	$0.323614\pi$
$888$	25495.9	0.963497
$889$	1565.97	0.0590788
$890$	−80100.7	−3.01683
$891$	−13460.3	−0.506103
$892$	40244.9	1.51065
$893$	22101.9	0.828232
$894$	54409.7	2.03550
$895$	28820.4	1.07638
$896$	36276.0	1.35256
$897$	−13532.0	−0.503700
$898$	−64096.6	−2.38188
$899$	9378.09	0.347916
$900$	−4602.13	−0.170449
$901$	0	0
$902$	5387.70	0.198881
$903$	3703.49	0.136483
$904$	22760.4	0.837388
$905$	27769.3	1.01998
$906$	18528.5	0.679434
$907$	18413.3	0.674095	0.337048	−	0.941488i	$-0.390572\pi$
$907$	18413.3	0.674095	0.337048	+	0.941488i	$0.390572\pi$
$908$	81378.8	2.97428
$909$	2473.32	0.0902473
$910$	37842.4	1.37853
$911$	32271.8	1.17367	0.586834	−	0.809708i	$-0.300374\pi$
$911$	32271.8	1.17367	0.586834	+	0.809708i	$0.300374\pi$
$912$	−12847.5	−0.466473
$913$	−11573.6	−0.419528
$914$	78962.7	2.85761
$915$	55165.3	1.99313
$916$	50014.6	1.80407
$917$	29893.3	1.07652
$918$	0	0
$919$	19185.3	0.688643	0.344322	−	0.938852i	$-0.388109\pi$
$919$	19185.3	0.688643	0.344322	+	0.938852i	$0.388109\pi$
$920$	−61556.2	−2.20592
$921$	25985.6	0.929700
$922$	−53744.2	−1.91971
$923$	1810.80	0.0645754
$924$	21631.8	0.770166
$925$	−29773.9	−1.05834
$926$	−68966.3	−2.44749
$927$	2443.51	0.0865752
$928$	1603.30	0.0567144
$929$	−20823.7	−0.735419	−0.367709	−	0.929941i	$-0.619858\pi$
$929$	−20823.7	−0.735419	−0.367709	+	0.929941i	$0.619858\pi$
$930$	74036.4	2.61048
$931$	6703.89	0.235995
$932$	−13338.9	−0.468809
$933$	−17337.7	−0.608371
$934$	−47106.5	−1.65029
$935$	0	0
$936$	1488.01	0.0519627
$937$	21072.0	0.734677	0.367338	−	0.930087i	$-0.380269\pi$
$937$	21072.0	0.734677	0.367338	+	0.930087i	$0.380269\pi$
$938$	−25048.9	−0.871934
$939$	−21698.4	−0.754100
$940$	140105.	4.86139
$941$	11156.4	0.386491	0.193245	−	0.981150i	$-0.438099\pi$
$941$	11156.4	0.386491	0.193245	+	0.981150i	$0.438099\pi$
$942$	−14272.4	−0.493651
$943$	5081.91	0.175493
$944$	−15512.7	−0.534847
$945$	37365.2	1.28623
$946$	−4938.09	−0.169716
$947$	3045.57	0.104506	0.0522532	−	0.998634i	$-0.483360\pi$
$947$	3045.57	0.104506	0.0522532	+	0.998634i	$0.483360\pi$
$948$	17244.3	0.590789
$949$	−11008.0	−0.376537
$950$	48884.1	1.66949
$951$	1977.33	0.0674232
$952$	0	0
$953$	6037.53	0.205220	0.102610	−	0.994722i	$-0.467281\pi$
$953$	6037.53	0.205220	0.102610	+	0.994722i	$0.467281\pi$
$954$	−3453.41	−0.117199
$955$	−55853.2	−1.89253
$956$	29593.4	1.00117
$957$	5713.87	0.193002
$958$	64353.9	2.17033
$959$	43292.5	1.45775
$960$	54326.4	1.82644
$961$	−3563.58	−0.119619
$962$	19712.8	0.660673
$963$	−2486.77	−0.0832140
$964$	−28917.1	−0.966138
$965$	−72277.6	−2.41109
$966$	30843.8	1.02731
$967$	−11151.0	−0.370829	−0.185414	−	0.982660i	$-0.559363\pi$
$967$	−11151.0	−0.370829	−0.185414	+	0.982660i	$0.559363\pi$
$968$	35324.6	1.17291
$969$	0	0
$970$	−79841.6	−2.64284
$971$	29051.5	0.960152	0.480076	−	0.877227i	$-0.340609\pi$
$971$	29051.5	0.960152	0.480076	+	0.877227i	$0.340609\pi$
$972$	5874.85	0.193864
$973$	−6587.43	−0.217043
$974$	−93831.4	−3.08681
$975$	33267.0	1.09271
$976$	−32495.6	−1.06574
$977$	2430.15	0.0795778	0.0397889	−	0.999208i	$-0.487331\pi$
$977$	2430.15	0.0795778	0.0397889	+	0.999208i	$0.487331\pi$
$978$	43780.3	1.43143
$979$	−17288.7	−0.564402
$980$	42496.2	1.38520
$981$	−1859.65	−0.0605241
$982$	351.356	0.0114177
$983$	−2167.32	−0.0703222	−0.0351611	−	0.999382i	$-0.511194\pi$
$983$	−2167.32	−0.0703222	−0.0351611	+	0.999382i	$0.511194\pi$
$984$	10698.3	0.346594
$985$	−17573.5	−0.568465
$986$	0	0
$987$	−34283.2	−1.10562
$988$	−21410.7	−0.689437
$989$	−4657.83	−0.149758
$990$	−2356.24	−0.0756427
$991$	19533.1	0.626124	0.313062	−	0.949733i	$-0.398645\pi$
$991$	19533.1	0.626124	0.313062	+	0.949733i	$0.398645\pi$
$992$	4483.91	0.143512
$993$	19375.1	0.619185
$994$	−4127.39	−0.131703
$995$	32780.0	1.04442
$996$	−47059.2	−1.49712
$997$	−8892.06	−0.282462	−0.141231	−	0.989977i	$-0.545106\pi$
$997$	−8892.06	−0.282462	−0.141231	+	0.989977i	$0.545106\pi$
$998$	−71979.6	−2.28304
$999$	19464.2	0.616437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.a.g.1.2		12
17.4	even	4		289.4.b.e.288.11		12
17.11	odd	16		17.4.d.a.2.1	✓	12
17.13	even	4		289.4.b.e.288.12		12
17.14	odd	16		17.4.d.a.9.1	yes	12
17.16	even	2	inner	289.4.a.g.1.1		12
51.11	even	16		153.4.l.a.19.3		12
51.14	even	16		153.4.l.a.145.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.d.a.2.1	✓	12	17.11	odd	16
17.4.d.a.9.1	yes	12	17.14	odd	16
153.4.l.a.19.3		12	51.11	even	16
153.4.l.a.145.3		12	51.14	even	16
289.4.a.g.1.1		12	17.16	even	2	inner
289.4.a.g.1.2		12	1.1	even	1	trivial
289.4.b.e.288.11		12	17.4	even	4
289.4.b.e.288.12		12	17.13	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.a (trivial)

\(f(q)\)	\(=\)	\(q-4.86166 q^{2} +5.06554 q^{3} +15.6358 q^{4} -18.5634 q^{5} -24.6269 q^{6} +14.0210 q^{7} -37.1225 q^{8} -1.34032 q^{9} +90.2489 q^{10} +19.4791 q^{11} +79.2036 q^{12} +29.9060 q^{13} -68.1655 q^{14} -94.0335 q^{15} +55.3912 q^{16} +6.51621 q^{18} -45.7882 q^{19} -290.253 q^{20} +71.0240 q^{21} -94.7006 q^{22} -89.3259 q^{23} -188.046 q^{24} +219.599 q^{25} -145.393 q^{26} -143.559 q^{27} +219.229 q^{28} +57.9077 q^{29} +457.159 q^{30} +161.949 q^{31} +27.6872 q^{32} +98.6719 q^{33} -260.277 q^{35} -20.9570 q^{36} -135.583 q^{37} +222.607 q^{38} +151.490 q^{39} +689.120 q^{40} -56.8918 q^{41} -345.295 q^{42} +52.1442 q^{43} +304.570 q^{44} +24.8809 q^{45} +434.272 q^{46} -482.699 q^{47} +280.586 q^{48} -146.411 q^{49} -1067.62 q^{50} +467.603 q^{52} -529.972 q^{53} +697.935 q^{54} -361.597 q^{55} -520.496 q^{56} -231.942 q^{57} -281.528 q^{58} -280.058 q^{59} -1470.29 q^{60} -586.656 q^{61} -787.341 q^{62} -18.7927 q^{63} -577.735 q^{64} -555.156 q^{65} -479.710 q^{66} +367.471 q^{67} -452.484 q^{69} +1265.38 q^{70} +60.5496 q^{71} +49.7563 q^{72} -368.086 q^{73} +659.161 q^{74} +1112.39 q^{75} -715.933 q^{76} +273.116 q^{77} -736.493 q^{78} +217.721 q^{79} -1028.25 q^{80} -691.015 q^{81} +276.589 q^{82} -594.154 q^{83} +1110.51 q^{84} -253.508 q^{86} +293.334 q^{87} -723.112 q^{88} -887.553 q^{89} -120.963 q^{90} +419.312 q^{91} -1396.68 q^{92} +820.358 q^{93} +2346.72 q^{94} +849.983 q^{95} +140.251 q^{96} -884.682 q^{97} +711.801 q^{98} -26.1083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36}+ \cdots + 3464 q^{98}+O(q^{100}) \) 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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