LMFDB - Embedded newform 1521.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +O(q^{10})$	$q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +1.43845 q^{10} +64.7386 q^{11} -46.5616 q^{14} -50.4233 q^{16} +25.5464 q^{17} +107.970 q^{19} -0.807764 q^{20} +165.831 q^{22} -73.2614 q^{23} -124.685 q^{25} +26.1468 q^{28} -175.909 q^{29} +113.093 q^{31} +64.2547 q^{32} +65.4384 q^{34} -10.2074 q^{35} -114.808 q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} -187.663 q^{46} -31.9479 q^{47} -12.5937 q^{49} -319.386 q^{50} -2.84658 q^{53} +36.3542 q^{55} +439.469 q^{56} -450.600 q^{58} +71.6325 q^{59} -920.695 q^{61} +289.693 q^{62} +567.978 q^{64} +444.280 q^{67} -36.7471 q^{68} -26.1468 q^{70} -541.719 q^{71} -764.004 q^{73} -294.086 q^{74} -155.309 q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153 q^{80} -178.401 q^{82} +603.797 q^{83} +14.3457 q^{85} +1122.73 q^{86} -1565.19 q^{88} -1159.88 q^{89} +105.383 q^{92} -81.8362 q^{94} +60.6307 q^{95} -583.269 q^{97} -32.2595 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8	$ 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.56155	0.905646	0.452823	−	0.891601i	$-0.350417\pi$
$2$	2.56155	0.905646	0.452823	+	0.891601i	$0.350417\pi$
$3$	0	0
$3$	0	0
$4$	−1.43845	−0.179806
$5$	0.561553	0.0502268	0.0251134	−	0.999685i	$-0.492005\pi$
$5$	0.561553	0.0502268	0.0251134	+	0.999685i	$0.492005\pi$
$6$	0	0
$7$	−18.1771	−0.981470	−0.490735	−	0.871309i	$-0.663272\pi$
$7$	−18.1771	−0.981470	−0.490735	+	0.871309i	$0.663272\pi$
$8$	−24.1771	−1.06849
$9$	0	0
$10$	1.43845	0.0454877
$11$	64.7386	1.77449	0.887247	−	0.461295i	$-0.152615\pi$
$11$	64.7386	1.77449	0.887247	+	0.461295i	$0.152615\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−46.5616	−0.888864
$15$	0	0
$16$	−50.4233	−0.787864
$17$	25.5464	0.364465	0.182233	−	0.983255i	$-0.441668\pi$
$17$	25.5464	0.364465	0.182233	+	0.983255i	$0.441668\pi$
$18$	0	0
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	−0.807764	−0.00903108
$21$	0	0
$22$	165.831	1.60706
$23$	−73.2614	−0.664176	−0.332088	−	0.943248i	$-0.607753\pi$
$23$	−73.2614	−0.664176	−0.332088	+	0.943248i	$0.607753\pi$
$24$	0	0
$25$	−124.685	−0.997477
$26$	0	0
$27$	0	0
$28$	26.1468	0.176474
$29$	−175.909	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$29$	−175.909	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$30$	0	0
$31$	113.093	0.655228	0.327614	−	0.944812i	$-0.393755\pi$
$31$	113.093	0.655228	0.327614	+	0.944812i	$0.393755\pi$
$32$	64.2547	0.354961
$33$	0	0
$34$	65.4384	0.330077
$35$	−10.2074	−0.0492961
$36$	0	0
$37$	−114.808	−0.510116	−0.255058	−	0.966926i	$-0.582095\pi$
$37$	−114.808	−0.510116	−0.255058	+	0.966926i	$0.582095\pi$
$38$	276.570	1.18067
$39$	0	0
$40$	−13.5767	−0.0536666
$41$	−69.6458	−0.265289	−0.132645	−	0.991164i	$-0.542347\pi$
$41$	−69.6458	−0.265289	−0.132645	+	0.991164i	$0.542347\pi$
$42$	0	0
$43$	438.302	1.55443	0.777214	−	0.629236i	$-0.216632\pi$
$43$	438.302	1.55443	0.777214	+	0.629236i	$0.216632\pi$
$44$	−93.1231	−0.319064
$45$	0	0
$46$	−187.663	−0.601508
$47$	−31.9479	−0.0991506	−0.0495753	−	0.998770i	$-0.515787\pi$
$47$	−31.9479	−0.0991506	−0.0495753	+	0.998770i	$0.515787\pi$
$48$	0	0
$49$	−12.5937	−0.0367164
$50$	−319.386	−0.903361
$51$	0	0
$52$	0	0
$53$	−2.84658	−0.00737752	−0.00368876	−	0.999993i	$-0.501174\pi$
$53$	−2.84658	−0.00737752	−0.00368876	+	0.999993i	$0.501174\pi$
$54$	0	0
$55$	36.3542	0.0891272
$56$	439.469	1.04869
$57$	0	0
$58$	−450.600	−1.02012
$59$	71.6325	0.158064	0.0790319	−	0.996872i	$-0.474817\pi$
$59$	71.6325	0.158064	0.0790319	+	0.996872i	$0.474817\pi$
$60$	0	0
$61$	−920.695	−1.93251	−0.966253	−	0.257593i	$-0.917071\pi$
$61$	−920.695	−1.93251	−0.966253	+	0.257593i	$0.917071\pi$
$62$	289.693	0.593404
$63$	0	0
$64$	567.978	1.10933
$65$	0	0
$66$	0	0
$67$	444.280	0.810112	0.405056	−	0.914292i	$-0.367252\pi$
$67$	444.280	0.810112	0.405056	+	0.914292i	$0.367252\pi$
$68$	−36.7471	−0.0655330
$69$	0	0
$70$	−26.1468	−0.0446448
$71$	−541.719	−0.905496	−0.452748	−	0.891639i	$-0.649556\pi$
$71$	−541.719	−0.905496	−0.452748	+	0.891639i	$0.649556\pi$
$72$	0	0
$73$	−764.004	−1.22493	−0.612465	−	0.790498i	$-0.709822\pi$
$73$	−764.004	−1.22493	−0.612465	+	0.790498i	$0.709822\pi$
$74$	−294.086	−0.461984
$75$	0	0
$76$	−155.309	−0.234410
$77$	−1176.76	−1.74161
$78$	0	0
$79$	−421.538	−0.600338	−0.300169	−	0.953886i	$-0.597043\pi$
$79$	−421.538	−0.600338	−0.300169	+	0.953886i	$0.597043\pi$
$80$	−28.3153	−0.0395719
$81$	0	0
$82$	−178.401	−0.240258
$83$	603.797	0.798498	0.399249	−	0.916842i	$-0.369271\pi$
$83$	603.797	0.798498	0.399249	+	0.916842i	$0.369271\pi$
$84$	0	0
$85$	14.3457	0.0183059
$86$	1122.73	1.40776
$87$	0	0
$88$	−1565.19	−1.89602
$89$	−1159.88	−1.38143	−0.690715	−	0.723127i	$-0.742704\pi$
$89$	−1159.88	−1.38143	−0.690715	+	0.723127i	$0.742704\pi$
$90$	0	0
$91$	0	0
$92$	105.383	0.119423
$93$	0	0
$94$	−81.8362	−0.0897953
$95$	60.6307	0.0654798
$96$	0	0
$97$	−583.269	−0.610536	−0.305268	−	0.952267i	$-0.598746\pi$
$97$	−583.269	−0.610536	−0.305268	+	0.952267i	$0.598746\pi$
$98$	−32.2595	−0.0332521
$99$	0	0
$100$	179.352	0.179352
$101$	−921.740	−0.908085	−0.454043	−	0.890980i	$-0.650019\pi$
$101$	−921.740	−0.908085	−0.454043	+	0.890980i	$0.650019\pi$
$102$	0	0
$103$	−930.712	−0.890347	−0.445174	−	0.895444i	$-0.646858\pi$
$103$	−930.712	−0.890347	−0.445174	+	0.895444i	$0.646858\pi$
$104$	0	0
$105$	0	0
$106$	−7.29168	−0.00668142
$107$	−857.383	−0.774638	−0.387319	−	0.921946i	$-0.626599\pi$
$107$	−857.383	−0.774638	−0.387319	+	0.921946i	$0.626599\pi$
$108$	0	0
$109$	−671.853	−0.590384	−0.295192	−	0.955438i	$-0.595384\pi$
$109$	−671.853	−0.590384	−0.295192	+	0.955438i	$0.595384\pi$
$110$	93.1231	0.0807176
$111$	0	0
$112$	916.548	0.773265
$113$	−641.474	−0.534024	−0.267012	−	0.963693i	$-0.586036\pi$
$113$	−641.474	−0.534024	−0.267012	+	0.963693i	$0.586036\pi$
$114$	0	0
$115$	−41.1401	−0.0333594
$116$	253.036	0.202533
$117$	0	0
$118$	183.491	0.143150
$119$	−464.359	−0.357712
$120$	0	0
$121$	2860.09	2.14883
$122$	−2358.41	−1.75017
$123$	0	0
$124$	−162.678	−0.117814
$125$	−140.211	−0.100327
$126$	0	0
$127$	−553.174	−0.386506	−0.193253	−	0.981149i	$-0.561904\pi$
$127$	−553.174	−0.386506	−0.193253	+	0.981149i	$0.561904\pi$
$128$	940.868	0.649702
$129$	0	0
$130$	0	0
$131$	−2056.40	−1.37152	−0.685758	−	0.727830i	$-0.740529\pi$
$131$	−2056.40	−1.37152	−0.685758	+	0.727830i	$0.740529\pi$
$132$	0	0
$133$	−1962.57	−1.27952
$134$	1138.05	0.733674
$135$	0	0
$136$	−617.637	−0.389426
$137$	−1808.57	−1.12786	−0.563928	−	0.825824i	$-0.690710\pi$
$137$	−1808.57	−1.12786	−0.563928	+	0.825824i	$0.690710\pi$
$138$	0	0
$139$	1493.64	0.911428	0.455714	−	0.890126i	$-0.349384\pi$
$139$	1493.64	0.911428	0.455714	+	0.890126i	$0.349384\pi$
$140$	14.6828	0.00886373
$141$	0	0
$142$	−1387.64	−0.820058
$143$	0	0
$144$	0	0
$145$	−98.7822	−0.0565753
$146$	−1957.04	−1.10935
$147$	0	0
$148$	165.145	0.0917218
$149$	−2759.02	−1.51696	−0.758482	−	0.651694i	$-0.774059\pi$
$149$	−2759.02	−1.51696	−0.758482	+	0.651694i	$0.774059\pi$
$150$	0	0
$151$	976.355	0.526190	0.263095	−	0.964770i	$-0.415257\pi$
$151$	976.355	0.526190	0.263095	+	0.964770i	$0.415257\pi$
$152$	−2610.39	−1.39297
$153$	0	0
$154$	−3014.33	−1.57728
$155$	63.5076	0.0329100
$156$	0	0
$157$	−564.875	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$157$	−564.875	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$158$	−1079.79	−0.543694
$159$	0	0
$160$	36.0824	0.0178285
$161$	1331.68	0.651869
$162$	0	0
$163$	−1508.53	−0.724892	−0.362446	−	0.932005i	$-0.618058\pi$
$163$	−1508.53	−0.724892	−0.362446	+	0.932005i	$0.618058\pi$
$164$	100.182	0.0477005
$165$	0	0
$166$	1546.66	0.723157
$167$	592.521	0.274555	0.137277	−	0.990533i	$-0.456165\pi$
$167$	592.521	0.274555	0.137277	+	0.990533i	$0.456165\pi$
$168$	0	0
$169$	0	0
$170$	36.7471	0.0165787
$171$	0	0
$172$	−630.474	−0.279495
$173$	4495.57	1.97568	0.987838	−	0.155488i	$-0.0496952\pi$
$173$	4495.57	1.97568	0.987838	+	0.155488i	$0.0496952\pi$
$174$	0	0
$175$	2266.40	0.978994
$176$	−3264.34	−1.39806
$177$	0	0
$178$	−2971.10	−1.25109
$179$	154.285	0.0644235	0.0322117	−	0.999481i	$-0.489745\pi$
$179$	154.285	0.0644235	0.0322117	+	0.999481i	$0.489745\pi$
$180$	0	0
$181$	1071.35	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$181$	1071.35	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$182$	0	0
$183$	0	0
$184$	1771.25	0.709663
$185$	−64.4706	−0.0256215
$186$	0	0
$187$	1653.84	0.646742
$188$	45.9554	0.0178279
$189$	0	0
$190$	155.309	0.0593015
$191$	−677.203	−0.256548	−0.128274	−	0.991739i	$-0.540944\pi$
$191$	−677.203	−0.256548	−0.128274	+	0.991739i	$0.540944\pi$
$192$	0	0
$193$	−1321.68	−0.492936	−0.246468	−	0.969151i	$-0.579270\pi$
$193$	−1321.68	−0.492936	−0.246468	+	0.969151i	$0.579270\pi$
$194$	−1494.07	−0.552929
$195$	0	0
$196$	18.1154	0.00660183
$197$	1267.37	0.458356	0.229178	−	0.973385i	$-0.426396\pi$
$197$	1267.37	0.458356	0.229178	+	0.973385i	$0.426396\pi$
$198$	0	0
$199$	2396.24	0.853593	0.426796	−	0.904348i	$-0.359642\pi$
$199$	2396.24	0.853593	0.426796	+	0.904348i	$0.359642\pi$
$200$	3014.51	1.06579
$201$	0	0
$202$	−2361.09	−0.822403
$203$	3197.51	1.10552
$204$	0	0
$205$	−39.1098	−0.0133246
$206$	−2384.07	−0.806339
$207$	0	0
$208$	0	0
$209$	6989.81	2.31337
$210$	0	0
$211$	−91.5539	−0.0298712	−0.0149356	−	0.999888i	$-0.504754\pi$
$211$	−91.5539	−0.0298712	−0.0149356	+	0.999888i	$0.504754\pi$
$212$	4.09466	0.00132652
$213$	0	0
$214$	−2196.23	−0.701548
$215$	246.130	0.0780740
$216$	0	0
$217$	−2055.70	−0.643087
$218$	−1720.99	−0.534679
$219$	0	0
$220$	−52.2935	−0.0160256
$221$	0	0
$222$	0	0
$223$	−1235.42	−0.370985	−0.185493	−	0.982646i	$-0.559388\pi$
$223$	−1235.42	−0.370985	−0.185493	+	0.982646i	$0.559388\pi$
$224$	−1167.96	−0.348383
$225$	0	0
$226$	−1643.17	−0.483637
$227$	3301.66	0.965370	0.482685	−	0.875794i	$-0.339662\pi$
$227$	3301.66	0.965370	0.482685	+	0.875794i	$0.339662\pi$
$228$	0	0
$229$	−211.283	−0.0609694	−0.0304847	−	0.999535i	$-0.509705\pi$
$229$	−211.283	−0.0609694	−0.0304847	+	0.999535i	$0.509705\pi$
$230$	−105.383	−0.0302118
$231$	0	0
$232$	4252.97	1.20354
$233$	256.724	0.0721827	0.0360913	−	0.999348i	$-0.488509\pi$
$233$	256.724	0.0721827	0.0360913	+	0.999348i	$0.488509\pi$
$234$	0	0
$235$	−17.9404	−0.00498002
$236$	−103.040	−0.0284208
$237$	0	0
$238$	−1189.48	−0.323960
$239$	−3549.62	−0.960694	−0.480347	−	0.877078i	$-0.659489\pi$
$239$	−3549.62	−0.960694	−0.480347	+	0.877078i	$0.659489\pi$
$240$	0	0
$241$	5030.10	1.34447	0.672235	−	0.740338i	$-0.265335\pi$
$241$	5030.10	1.34447	0.672235	+	0.740338i	$0.265335\pi$
$242$	7326.27	1.94608
$243$	0	0
$244$	1324.37	0.347476
$245$	−7.07204	−0.00184415
$246$	0	0
$247$	0	0
$248$	−2734.25	−0.700102
$249$	0	0
$250$	−359.158	−0.0908606
$251$	718.784	0.180754	0.0903770	−	0.995908i	$-0.471193\pi$
$251$	718.784	0.180754	0.0903770	+	0.995908i	$0.471193\pi$
$252$	0	0
$253$	−4742.84	−1.17858
$254$	−1416.98	−0.350038
$255$	0	0
$256$	−2133.74	−0.520933
$257$	−1280.79	−0.310871	−0.155435	−	0.987846i	$-0.549678\pi$
$257$	−1280.79	−0.310871	−0.155435	+	0.987846i	$0.549678\pi$
$258$	0	0
$259$	2086.87	0.500663
$260$	0	0
$261$	0	0
$262$	−5267.58	−1.24211
$263$	−5225.55	−1.22517	−0.612587	−	0.790403i	$-0.709871\pi$
$263$	−5225.55	−1.22517	−0.612587	+	0.790403i	$0.709871\pi$
$264$	0	0
$265$	−1.59851	−0.000370549 0
$266$	−5027.24	−1.15880
$267$	0	0
$268$	−639.074	−0.145663
$269$	−6443.80	−1.46054	−0.730270	−	0.683158i	$-0.760606\pi$
$269$	−6443.80	−1.46054	−0.730270	+	0.683158i	$0.760606\pi$
$270$	0	0
$271$	3929.93	0.880909	0.440455	−	0.897775i	$-0.354817\pi$
$271$	3929.93	0.880909	0.440455	+	0.897775i	$0.354817\pi$
$272$	−1288.13	−0.287149
$273$	0	0
$274$	−4632.74	−1.02144
$275$	−8071.91	−1.77002
$276$	0	0
$277$	−5884.40	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$277$	−5884.40	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$278$	3826.03	0.825431
$279$	0	0
$280$	246.785	0.0526722
$281$	3529.79	0.749358	0.374679	−	0.927155i	$-0.377753\pi$
$281$	3529.79	0.749358	0.374679	+	0.927155i	$0.377753\pi$
$282$	0	0
$283$	−2611.00	−0.548438	−0.274219	−	0.961667i	$-0.588419\pi$
$283$	−2611.00	−0.548438	−0.274219	+	0.961667i	$0.588419\pi$
$284$	779.234	0.162813
$285$	0	0
$286$	0	0
$287$	1265.96	0.260373
$288$	0	0
$289$	−4260.38	−0.867165
$290$	−253.036	−0.0512372
$291$	0	0
$292$	1098.98	0.220250
$293$	−5491.03	−1.09484	−0.547422	−	0.836857i	$-0.684391\pi$
$293$	−5491.03	−1.09484	−0.547422	+	0.836857i	$0.684391\pi$
$294$	0	0
$295$	40.2255	0.00793904
$296$	2775.72	0.545052
$297$	0	0
$298$	−7067.37	−1.37383
$299$	0	0
$300$	0	0
$301$	−7967.05	−1.52563
$302$	2500.99	0.476542
$303$	0	0
$304$	−5444.19	−1.02712
$305$	−517.019	−0.0970637
$306$	0	0
$307$	7307.59	1.35852	0.679261	−	0.733897i	$-0.262300\pi$
$307$	7307.59	1.35852	0.679261	+	0.733897i	$0.262300\pi$
$308$	1692.71	0.313152
$309$	0	0
$310$	162.678	0.0298048
$311$	−7904.92	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$311$	−7904.92	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$312$	0	0
$313$	10002.4	1.80629	0.903145	−	0.429336i	$-0.141252\pi$
$313$	10002.4	1.80629	0.903145	+	0.429336i	$0.141252\pi$
$314$	−1446.96	−0.260053
$315$	0	0
$316$	606.360	0.107944
$317$	−6230.81	−1.10397	−0.551983	−	0.833856i	$-0.686129\pi$
$317$	−6230.81	−1.10397	−0.551983	+	0.833856i	$0.686129\pi$
$318$	0	0
$319$	−11388.1	−1.99878
$320$	318.950	0.0557182
$321$	0	0
$322$	3411.16	0.590362
$323$	2758.24	0.475147
$324$	0	0
$325$	0	0
$326$	−3864.19	−0.656495
$327$	0	0
$328$	1683.83	0.283458
$329$	580.719	0.0973134
$330$	0	0
$331$	4634.51	0.769594	0.384797	−	0.923001i	$-0.374271\pi$
$331$	4634.51	0.769594	0.384797	+	0.923001i	$0.374271\pi$
$332$	−868.531	−0.143575
$333$	0	0
$334$	1517.77	0.248649
$335$	249.487	0.0406893
$336$	0	0
$337$	3029.82	0.489747	0.244874	−	0.969555i	$-0.421254\pi$
$337$	3029.82	0.489747	0.244874	+	0.969555i	$0.421254\pi$
$338$	0	0
$339$	0	0
$340$	−20.6355	−0.00329151
$341$	7321.47	1.16270
$342$	0	0
$343$	6463.66	1.01751
$344$	−10596.9	−1.66089
$345$	0	0
$346$	11515.6	1.78926
$347$	−2841.60	−0.439611	−0.219805	−	0.975544i	$-0.570542\pi$
$347$	−2841.60	−0.439611	−0.219805	+	0.975544i	$0.570542\pi$
$348$	0	0
$349$	−7565.68	−1.16040	−0.580202	−	0.814472i	$-0.697027\pi$
$349$	−7565.68	−1.16040	−0.580202	+	0.814472i	$0.697027\pi$
$350$	5805.51	0.886622
$351$	0	0
$352$	4159.76	0.629875
$353$	−2339.44	−0.352736	−0.176368	−	0.984324i	$-0.556435\pi$
$353$	−2339.44	−0.352736	−0.176368	+	0.984324i	$0.556435\pi$
$354$	0	0
$355$	−304.204	−0.0454802
$356$	1668.43	0.248389
$357$	0	0
$358$	395.209	0.0583449
$359$	−2531.68	−0.372192	−0.186096	−	0.982532i	$-0.559583\pi$
$359$	−2531.68	−0.372192	−0.186096	+	0.982532i	$0.559583\pi$
$360$	0	0
$361$	4798.45	0.699585
$362$	2744.31	0.398447
$363$	0	0
$364$	0	0
$365$	−429.028	−0.0615243
$366$	0	0
$367$	6577.81	0.935583	0.467792	−	0.883839i	$-0.345050\pi$
$367$	6577.81	0.935583	0.467792	+	0.883839i	$0.345050\pi$
$368$	3694.08	0.523280
$369$	0	0
$370$	−165.145	−0.0232040
$371$	51.7426	0.00724081
$372$	0	0
$373$	2902.72	0.402942	0.201471	−	0.979495i	$-0.435428\pi$
$373$	2902.72	0.402942	0.201471	+	0.979495i	$0.435428\pi$
$374$	4236.40	0.585719
$375$	0	0
$376$	772.407	0.105941
$377$	0	0
$378$	0	0
$379$	1865.73	0.252866	0.126433	−	0.991975i	$-0.459647\pi$
$379$	1865.73	0.252866	0.126433	+	0.991975i	$0.459647\pi$
$380$	−87.2140	−0.0117736
$381$	0	0
$382$	−1734.69	−0.232342
$383$	10836.0	1.44567	0.722837	−	0.691019i	$-0.242838\pi$
$383$	10836.0	1.44567	0.722837	+	0.691019i	$0.242838\pi$
$384$	0	0
$385$	−660.813	−0.0874757
$386$	−3385.55	−0.446425
$387$	0	0
$388$	839.001	0.109778
$389$	9520.34	1.24088	0.620438	−	0.784256i	$-0.286955\pi$
$389$	9520.34	1.24088	0.620438	+	0.784256i	$0.286955\pi$
$390$	0	0
$391$	−1871.56	−0.242069
$392$	304.480	0.0392310
$393$	0	0
$394$	3246.43	0.415108
$395$	−236.716	−0.0301531
$396$	0	0
$397$	10108.8	1.27796	0.638978	−	0.769225i	$-0.279358\pi$
$397$	10108.8	1.27796	0.638978	+	0.769225i	$0.279358\pi$
$398$	6138.10	0.773053
$399$	0	0
$400$	6287.01	0.785876
$401$	2084.38	0.259573	0.129787	−	0.991542i	$-0.458571\pi$
$401$	2084.38	0.259573	0.129787	+	0.991542i	$0.458571\pi$
$402$	0	0
$403$	0	0
$404$	1325.88	0.163279
$405$	0	0
$406$	8190.60	1.00121
$407$	−7432.50	−0.905197
$408$	0	0
$409$	9716.53	1.17470	0.587349	−	0.809334i	$-0.300172\pi$
$409$	9716.53	1.17470	0.587349	+	0.809334i	$0.300172\pi$
$410$	−100.182	−0.0120674
$411$	0	0
$412$	1338.78	0.160090
$413$	−1302.07	−0.155135
$414$	0	0
$415$	339.064	0.0401060
$416$	0	0
$417$	0	0
$418$	17904.8	2.09510
$419$	−13381.9	−1.56026	−0.780129	−	0.625619i	$-0.784847\pi$
$419$	−13381.9	−1.56026	−0.780129	+	0.625619i	$0.784847\pi$
$420$	0	0
$421$	9463.37	1.09553	0.547763	−	0.836633i	$-0.315479\pi$
$421$	9463.37	1.09553	0.547763	+	0.836633i	$0.315479\pi$
$422$	−234.520	−0.0270527
$423$	0	0
$424$	68.8221	0.00788278
$425$	−3185.24	−0.363546
$426$	0	0
$427$	16735.5	1.89670
$428$	1233.30	0.139285
$429$	0	0
$430$	630.474	0.0707074
$431$	4852.28	0.542288	0.271144	−	0.962539i	$-0.412598\pi$
$431$	4852.28	0.542288	0.271144	+	0.962539i	$0.412598\pi$
$432$	0	0
$433$	−8208.00	−0.910973	−0.455486	−	0.890243i	$-0.650535\pi$
$433$	−8208.00	−0.910973	−0.455486	+	0.890243i	$0.650535\pi$
$434$	−5265.78	−0.582409
$435$	0	0
$436$	966.425	0.106155
$437$	−7910.01	−0.865874
$438$	0	0
$439$	−2993.80	−0.325481	−0.162741	−	0.986669i	$-0.552033\pi$
$439$	−2993.80	−0.325481	−0.162741	+	0.986669i	$0.552033\pi$
$440$	−878.938	−0.0952311
$441$	0	0
$442$	0	0
$443$	−9743.67	−1.04500	−0.522501	−	0.852639i	$-0.675001\pi$
$443$	−9743.67	−1.04500	−0.522501	+	0.852639i	$0.675001\pi$
$444$	0	0
$445$	−651.335	−0.0693848
$446$	−3164.59	−0.335981
$447$	0	0
$448$	−10324.2	−1.08878
$449$	−561.459	−0.0590131	−0.0295065	−	0.999565i	$-0.509394\pi$
$449$	−561.459	−0.0590131	−0.0295065	+	0.999565i	$0.509394\pi$
$450$	0	0
$451$	−4508.78	−0.470754
$452$	922.726	0.0960207
$453$	0	0
$454$	8457.38	0.874283
$455$	0	0
$456$	0	0
$457$	−13758.4	−1.40830	−0.704148	−	0.710054i	$-0.748671\pi$
$457$	−13758.4	−1.40830	−0.704148	+	0.710054i	$0.748671\pi$
$458$	−541.213	−0.0552166
$459$	0	0
$460$	59.1779	0.00599823
$461$	12009.2	1.21329	0.606644	−	0.794974i	$-0.292515\pi$
$461$	12009.2	1.21329	0.606644	+	0.794974i	$0.292515\pi$
$462$	0	0
$463$	−13635.7	−1.36870	−0.684348	−	0.729156i	$-0.739913\pi$
$463$	−13635.7	−1.36870	−0.684348	+	0.729156i	$0.739913\pi$
$464$	8869.91	0.887447
$465$	0	0
$466$	657.613	0.0653719
$467$	−8821.95	−0.874157	−0.437079	−	0.899423i	$-0.643987\pi$
$467$	−8821.95	−0.874157	−0.437079	+	0.899423i	$0.643987\pi$
$468$	0	0
$469$	−8075.72	−0.795100
$470$	−45.9554	−0.00451013
$471$	0	0
$472$	−1731.87	−0.168889
$473$	28375.1	2.75832
$474$	0	0
$475$	−13462.2	−1.30039
$476$	667.956	0.0643187
$477$	0	0
$478$	−9092.54	−0.870049
$479$	−14620.0	−1.39459	−0.697293	−	0.716786i	$-0.745612\pi$
$479$	−14620.0	−1.39459	−0.697293	+	0.716786i	$0.745612\pi$
$480$	0	0
$481$	0	0
$482$	12884.9	1.21761
$483$	0	0
$484$	−4114.09	−0.386372
$485$	−327.536	−0.0306653
$486$	0	0
$487$	9798.86	0.911763	0.455882	−	0.890040i	$-0.349324\pi$
$487$	9798.86	0.911763	0.455882	+	0.890040i	$0.349324\pi$
$488$	22259.7	2.06486
$489$	0	0
$490$	−18.1154	−0.00167014
$491$	10836.1	0.995977	0.497989	−	0.867184i	$-0.334072\pi$
$491$	10836.1	0.995977	0.497989	+	0.867184i	$0.334072\pi$
$492$	0	0
$493$	−4493.84	−0.410532
$494$	0	0
$495$	0	0
$496$	−5702.51	−0.516230
$497$	9846.86	0.888717
$498$	0	0
$499$	−2589.96	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$499$	−2589.96	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$500$	201.686	0.0180394
$501$	0	0
$502$	1841.20	0.163699
$503$	17067.5	1.51292	0.756462	−	0.654038i	$-0.226926\pi$
$503$	17067.5	1.51292	0.756462	+	0.654038i	$0.226926\pi$
$504$	0	0
$505$	−517.606	−0.0456102
$506$	−12149.0	−1.06737
$507$	0	0
$508$	795.712	0.0694961
$509$	−1012.89	−0.0882038	−0.0441019	−	0.999027i	$-0.514043\pi$
$509$	−1012.89	−0.0882038	−0.0441019	+	0.999027i	$0.514043\pi$
$510$	0	0
$511$	13887.4	1.20223
$512$	−12992.6	−1.12148
$513$	0	0
$514$	−3280.82	−0.281539
$515$	−522.644	−0.0447193
$516$	0	0
$517$	−2068.26	−0.175942
$518$	5345.63	0.453424
$519$	0	0
$520$	0	0
$521$	14367.7	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$521$	14367.7	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$522$	0	0
$523$	−16219.9	−1.35611	−0.678057	−	0.735010i	$-0.737178\pi$
$523$	−16219.9	−1.35611	−0.678057	+	0.735010i	$0.737178\pi$
$524$	2958.02	0.246607
$525$	0	0
$526$	−13385.5	−1.10957
$527$	2889.11	0.238808
$528$	0	0
$529$	−6799.77	−0.558870
$530$	−4.09466	−0.000335586 0
$531$	0	0
$532$	2823.06	0.230066
$533$	0	0
$534$	0	0
$535$	−481.466	−0.0389076
$536$	−10741.4	−0.865593
$537$	0	0
$538$	−16506.1	−1.32273
$539$	−815.301	−0.0651530
$540$	0	0
$541$	−17592.2	−1.39806	−0.699029	−	0.715094i	$-0.746384\pi$
$541$	−17592.2	−1.39806	−0.699029	+	0.715094i	$0.746384\pi$
$542$	10066.7	0.797792
$543$	0	0
$544$	1641.48	0.129371
$545$	−377.281	−0.0296531
$546$	0	0
$547$	10504.6	0.821103	0.410552	−	0.911837i	$-0.365336\pi$
$547$	10504.6	0.821103	0.410552	+	0.911837i	$0.365336\pi$
$548$	2601.53	0.202795
$549$	0	0
$550$	−20676.6	−1.60301
$551$	−18992.8	−1.46846
$552$	0	0
$553$	7662.33	0.589214
$554$	−15073.2	−1.15596
$555$	0	0
$556$	−2148.52	−0.163880
$557$	−507.558	−0.0386102	−0.0193051	−	0.999814i	$-0.506145\pi$
$557$	−507.558	−0.0386102	−0.0193051	+	0.999814i	$0.506145\pi$
$558$	0	0
$559$	0	0
$560$	514.690	0.0388386
$561$	0	0
$562$	9041.75	0.678653
$563$	3443.14	0.257746	0.128873	−	0.991661i	$-0.458864\pi$
$563$	3443.14	0.257746	0.128873	+	0.991661i	$0.458864\pi$
$564$	0	0
$565$	−360.221	−0.0268223
$566$	−6688.21	−0.496690
$567$	0	0
$568$	13097.2	0.967509
$569$	−23972.2	−1.76620	−0.883098	−	0.469189i	$-0.844546\pi$
$569$	−23972.2	−1.76620	−0.883098	+	0.469189i	$0.844546\pi$
$570$	0	0
$571$	−7458.32	−0.546622	−0.273311	−	0.961926i	$-0.588119\pi$
$571$	−7458.32	−0.546622	−0.273311	+	0.961926i	$0.588119\pi$
$572$	0	0
$573$	0	0
$574$	3242.82	0.235806
$575$	9134.57	0.662501
$576$	0	0
$577$	−5669.57	−0.409059	−0.204530	−	0.978860i	$-0.565566\pi$
$577$	−5669.57	−0.409059	−0.204530	+	0.978860i	$0.565566\pi$
$578$	−10913.2	−0.785344
$579$	0	0
$580$	142.093	0.0101726
$581$	−10975.3	−0.783702
$582$	0	0
$583$	−184.284	−0.0130914
$584$	18471.4	1.30882
$585$	0	0
$586$	−14065.6	−0.991541
$587$	1017.39	0.0715371	0.0357685	−	0.999360i	$-0.488612\pi$
$587$	1017.39	0.0715371	0.0357685	+	0.999360i	$0.488612\pi$
$588$	0	0
$589$	12210.6	0.854208
$590$	103.040	0.00718996
$591$	0	0
$592$	5788.99	0.401902
$593$	−10198.2	−0.706221	−0.353111	−	0.935582i	$-0.614876\pi$
$593$	−10198.2	−0.706221	−0.353111	+	0.935582i	$0.614876\pi$
$594$	0	0
$595$	−260.762	−0.0179667
$596$	3968.70	0.272759
$597$	0	0
$598$	0	0
$599$	−12516.3	−0.853763	−0.426881	−	0.904308i	$-0.640388\pi$
$599$	−12516.3	−0.853763	−0.426881	+	0.904308i	$0.640388\pi$
$600$	0	0
$601$	9627.46	0.653431	0.326716	−	0.945123i	$-0.394058\pi$
$601$	9627.46	0.653431	0.326716	+	0.945123i	$0.394058\pi$
$602$	−20408.0	−1.38168
$603$	0	0
$604$	−1404.44	−0.0946120
$605$	1606.09	0.107929
$606$	0	0
$607$	6667.20	0.445821	0.222910	−	0.974839i	$-0.428444\pi$
$607$	6667.20	0.445821	0.222910	+	0.974839i	$0.428444\pi$
$608$	6937.56	0.462755
$609$	0	0
$610$	−1324.37	−0.0879053
$611$	0	0
$612$	0	0
$613$	23085.4	1.52106	0.760530	−	0.649302i	$-0.224939\pi$
$613$	23085.4	1.52106	0.760530	+	0.649302i	$0.224939\pi$
$614$	18718.8	1.23034
$615$	0	0
$616$	28450.6	1.86089
$617$	3049.24	0.198959	0.0994796	−	0.995040i	$-0.468282\pi$
$617$	3049.24	0.198959	0.0994796	+	0.995040i	$0.468282\pi$
$618$	0	0
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	−91.3523	−0.00591741
$621$	0	0
$622$	−20248.9	−1.30531
$623$	21083.3	1.35583
$624$	0	0
$625$	15506.8	0.992438
$626$	25621.7	1.63586
$627$	0	0
$628$	812.543	0.0516305
$629$	−2932.92	−0.185920
$630$	0	0
$631$	23829.5	1.50339	0.751694	−	0.659512i	$-0.229237\pi$
$631$	23829.5	1.50339	0.751694	+	0.659512i	$0.229237\pi$
$632$	10191.6	0.641453
$633$	0	0
$634$	−15960.5	−0.999801
$635$	−310.637	−0.0194130
$636$	0	0
$637$	0	0
$638$	−29171.3	−1.81019
$639$	0	0
$640$	528.347	0.0326324
$641$	−13405.3	−0.826016	−0.413008	−	0.910727i	$-0.635522\pi$
$641$	−13405.3	−0.826016	−0.413008	+	0.910727i	$0.635522\pi$
$642$	0	0
$643$	−5251.51	−0.322083	−0.161042	−	0.986948i	$-0.551485\pi$
$643$	−5251.51	−0.322083	−0.161042	+	0.986948i	$0.551485\pi$
$644$	−1915.55	−0.117210
$645$	0	0
$646$	7065.37	0.430315
$647$	−21611.4	−1.31319	−0.656595	−	0.754244i	$-0.728004\pi$
$647$	−21611.4	−1.31319	−0.656595	+	0.754244i	$0.728004\pi$
$648$	0	0
$649$	4637.39	0.280483
$650$	0	0
$651$	0	0
$652$	2169.94	0.130340
$653$	21595.8	1.29420	0.647099	−	0.762406i	$-0.275982\pi$
$653$	21595.8	1.29420	0.647099	+	0.762406i	$0.275982\pi$
$654$	0	0
$655$	−1154.78	−0.0688869
$656$	3511.77	0.209012
$657$	0	0
$658$	1487.54	0.0881314
$659$	16642.6	0.983768	0.491884	−	0.870661i	$-0.336308\pi$
$659$	16642.6	0.983768	0.491884	+	0.870661i	$0.336308\pi$
$660$	0	0
$661$	−26981.1	−1.58766	−0.793831	−	0.608139i	$-0.791916\pi$
$661$	−26981.1	−1.58766	−0.793831	+	0.608139i	$0.791916\pi$
$662$	11871.5	0.696980
$663$	0	0
$664$	−14598.1	−0.853185
$665$	−1102.09	−0.0642664
$666$	0	0
$667$	12887.3	0.748126
$668$	−852.310	−0.0493665
$669$	0	0
$670$	639.074	0.0368501
$671$	−59604.5	−3.42922
$672$	0	0
$673$	11149.2	0.638591	0.319296	−	0.947655i	$-0.396554\pi$
$673$	11149.2	0.638591	0.319296	+	0.947655i	$0.396554\pi$
$674$	7761.04	0.443537
$675$	0	0
$676$	0	0
$677$	−3314.33	−0.188154	−0.0940769	−	0.995565i	$-0.529990\pi$
$677$	−3314.33	−0.188154	−0.0940769	+	0.995565i	$0.529990\pi$
$678$	0	0
$679$	10602.1	0.599223
$680$	−346.836	−0.0195596
$681$	0	0
$682$	18754.3	1.05299
$683$	24505.2	1.37287	0.686433	−	0.727193i	$-0.259176\pi$
$683$	24505.2	1.37287	0.686433	+	0.727193i	$0.259176\pi$
$684$	0	0
$685$	−1015.61	−0.0566486
$686$	16557.0	0.921500
$687$	0	0
$688$	−22100.6	−1.22468
$689$	0	0
$690$	0	0
$691$	21752.8	1.19756	0.598782	−	0.800912i	$-0.295652\pi$
$691$	21752.8	1.19756	0.598782	+	0.800912i	$0.295652\pi$
$692$	−6466.64	−0.355238
$693$	0	0
$694$	−7278.90	−0.398132
$695$	838.755	0.0457781
$696$	0	0
$697$	−1779.20	−0.0966887
$698$	−19379.9	−1.05092
$699$	0	0
$700$	−3260.10	−0.176029
$701$	−34250.9	−1.84542	−0.922709	−	0.385496i	$-0.874030\pi$
$701$	−34250.9	−1.84542	−0.922709	+	0.385496i	$0.874030\pi$
$702$	0	0
$703$	−12395.8	−0.665028
$704$	36770.1	1.96850
$705$	0	0
$706$	−5992.59	−0.319454
$707$	16754.6	0.891259
$708$	0	0
$709$	5527.11	0.292771	0.146386	−	0.989228i	$-0.453236\pi$
$709$	5527.11	0.292771	0.146386	+	0.989228i	$0.453236\pi$
$710$	−779.234	−0.0411889
$711$	0	0
$712$	28042.6	1.47604
$713$	−8285.33	−0.435187
$714$	0	0
$715$	0	0
$716$	−221.931	−0.0115837
$717$	0	0
$718$	−6485.02	−0.337074
$719$	3777.78	0.195949	0.0979745	−	0.995189i	$-0.468764\pi$
$719$	3777.78	0.195949	0.0979745	+	0.995189i	$0.468764\pi$
$720$	0	0
$721$	16917.6	0.873849
$722$	12291.5	0.633576
$723$	0	0
$724$	−1541.08	−0.0791072
$725$	21933.2	1.12355
$726$	0	0
$727$	19076.8	0.973204	0.486602	−	0.873624i	$-0.338236\pi$
$727$	19076.8	0.973204	0.486602	+	0.873624i	$0.338236\pi$
$728$	0	0
$729$	0	0
$730$	−1098.98	−0.0557192
$731$	11197.0	0.566535
$732$	0	0
$733$	−7997.30	−0.402984	−0.201492	−	0.979490i	$-0.564579\pi$
$733$	−7997.30	−0.402984	−0.201492	+	0.979490i	$0.564579\pi$
$734$	16849.4	0.847307
$735$	0	0
$736$	−4707.39	−0.235756
$737$	28762.1	1.43754
$738$	0	0
$739$	−28983.6	−1.44273	−0.721367	−	0.692553i	$-0.756486\pi$
$739$	−28983.6	−1.44273	−0.721367	+	0.692553i	$0.756486\pi$
$740$	92.7376	0.00460689
$741$	0	0
$742$	132.541	0.00655761
$743$	−19145.4	−0.945324	−0.472662	−	0.881244i	$-0.656707\pi$
$743$	−19145.4	−0.945324	−0.472662	+	0.881244i	$0.656707\pi$
$744$	0	0
$745$	−1549.33	−0.0761923
$746$	7435.47	0.364922
$747$	0	0
$748$	−2378.96	−0.116288
$749$	15584.7	0.760284
$750$	0	0
$751$	−25516.9	−1.23985	−0.619923	−	0.784663i	$-0.712836\pi$
$751$	−25516.9	−1.23985	−0.619923	+	0.784663i	$0.712836\pi$
$752$	1610.92	0.0781172
$753$	0	0
$754$	0	0
$755$	548.275	0.0264288
$756$	0	0
$757$	−17230.6	−0.827289	−0.413645	−	0.910438i	$-0.635744\pi$
$757$	−17230.6	−0.827289	−0.413645	+	0.910438i	$0.635744\pi$
$758$	4779.17	0.229007
$759$	0	0
$760$	−1465.87	−0.0699642
$761$	−2343.06	−0.111611	−0.0558053	−	0.998442i	$-0.517773\pi$
$761$	−2343.06	−0.111611	−0.0558053	+	0.998442i	$0.517773\pi$
$762$	0	0
$763$	12212.3	0.579444
$764$	974.121	0.0461289
$765$	0	0
$766$	27756.9	1.30927
$767$	0	0
$768$	0	0
$769$	7100.18	0.332950	0.166475	−	0.986046i	$-0.446761\pi$
$769$	7100.18	0.332950	0.166475	+	0.986046i	$0.446761\pi$
$770$	−1692.71	−0.0792219
$771$	0	0
$772$	1901.17	0.0886328
$773$	12270.4	0.570940	0.285470	−	0.958388i	$-0.407850\pi$
$773$	12270.4	0.570940	0.285470	+	0.958388i	$0.407850\pi$
$774$	0	0
$775$	−14100.9	−0.653575
$776$	14101.7	0.652349
$777$	0	0
$778$	24386.9	1.12379
$779$	−7519.64	−0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	−4794.11	−0.219229
$783$	0	0
$784$	635.017	0.0289275
$785$	−317.207	−0.0144224
$786$	0	0
$787$	−3425.04	−0.155133	−0.0775663	−	0.996987i	$-0.524715\pi$
$787$	−3425.04	−0.155133	−0.0775663	+	0.996987i	$0.524715\pi$
$788$	−1823.04	−0.0824151
$789$	0	0
$790$	−606.360	−0.0273080
$791$	11660.1	0.524129
$792$	0	0
$793$	0	0
$794$	25894.3	1.15737
$795$	0	0
$796$	−3446.87	−0.153481
$797$	11781.1	0.523600	0.261800	−	0.965122i	$-0.415684\pi$
$797$	11781.1	0.523600	0.261800	+	0.965122i	$0.415684\pi$
$798$	0	0
$799$	−816.154	−0.0361370
$800$	−8011.58	−0.354065
$801$	0	0
$802$	5339.24	0.235081
$803$	−49460.6	−2.17363
$804$	0	0
$805$	747.807	0.0327413
$806$	0	0
$807$	0	0
$808$	22285.0	0.970276
$809$	−18910.1	−0.821810	−0.410905	−	0.911678i	$-0.634787\pi$
$809$	−18910.1	−0.821810	−0.410905	+	0.911678i	$0.634787\pi$
$810$	0	0
$811$	−12803.3	−0.554359	−0.277180	−	0.960818i	$-0.589400\pi$
$811$	−12803.3	−0.554359	−0.277180	+	0.960818i	$0.589400\pi$
$812$	−4599.45	−0.198780
$813$	0	0
$814$	−19038.7	−0.819788
$815$	−847.121	−0.0364090
$816$	0	0
$817$	47323.3	2.02648
$818$	24889.4	1.06386
$819$	0	0
$820$	56.2574	0.00239585
$821$	19335.1	0.821923	0.410962	−	0.911653i	$-0.365193\pi$
$821$	19335.1	0.821923	0.410962	+	0.911653i	$0.365193\pi$
$822$	0	0
$823$	−2125.90	−0.0900417	−0.0450209	−	0.998986i	$-0.514335\pi$
$823$	−2125.90	−0.0900417	−0.0450209	+	0.998986i	$0.514335\pi$
$824$	22501.9	0.951324
$825$	0	0
$826$	−3335.32	−0.140497
$827$	−6989.24	−0.293881	−0.146941	−	0.989145i	$-0.546943\pi$
$827$	−6989.24	−0.293881	−0.146941	+	0.989145i	$0.546943\pi$
$828$	0	0
$829$	−32649.7	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$829$	−32649.7	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$830$	868.531	0.0363219
$831$	0	0
$832$	0	0
$833$	−321.724	−0.0133819
$834$	0	0
$835$	332.732	0.0137900
$836$	−10054.5	−0.415958
$837$	0	0
$838$	−34278.4	−1.41304
$839$	−4038.23	−0.166168	−0.0830841	−	0.996543i	$-0.526477\pi$
$839$	−4038.23	−0.166168	−0.0830841	+	0.996543i	$0.526477\pi$
$840$	0	0
$841$	6555.00	0.268769
$842$	24240.9	0.992159
$843$	0	0
$844$	131.695	0.00537102
$845$	0	0
$846$	0	0
$847$	−51988.1	−2.10901
$848$	143.534	0.00581248
$849$	0	0
$850$	−8159.17	−0.329244
$851$	8410.97	0.338807
$852$	0	0
$853$	−8114.12	−0.325700	−0.162850	−	0.986651i	$-0.552069\pi$
$853$	−8114.12	−0.325700	−0.162850	+	0.986651i	$0.552069\pi$
$854$	42869.0	1.71774
$855$	0	0
$856$	20729.0	0.827690
$857$	22298.1	0.888786	0.444393	−	0.895832i	$-0.353419\pi$
$857$	22298.1	0.888786	0.444393	+	0.895832i	$0.353419\pi$
$858$	0	0
$859$	33550.5	1.33263	0.666315	−	0.745670i	$-0.267870\pi$
$859$	33550.5	1.33263	0.666315	+	0.745670i	$0.267870\pi$
$860$	−354.045	−0.0140382
$861$	0	0
$862$	12429.4	0.491121
$863$	−14120.5	−0.556972	−0.278486	−	0.960440i	$-0.589833\pi$
$863$	−14120.5	−0.556972	−0.278486	+	0.960440i	$0.589833\pi$
$864$	0	0
$865$	2524.50	0.0992319
$866$	−21025.2	−0.825018
$867$	0	0
$868$	2957.01	0.115631
$869$	−27289.8	−1.06530
$870$	0	0
$871$	0	0
$872$	16243.4	0.630817
$873$	0	0
$874$	−20261.9	−0.784175
$875$	2548.63	0.0984679
$876$	0	0
$877$	1941.69	0.0747619	0.0373809	−	0.999301i	$-0.488099\pi$
$877$	1941.69	0.0747619	0.0373809	+	0.999301i	$0.488099\pi$
$878$	−7668.78	−0.294771
$879$	0	0
$880$	−1833.10	−0.0702201
$881$	790.231	0.0302197	0.0151099	−	0.999886i	$-0.495190\pi$
$881$	790.231	0.0302197	0.0151099	+	0.999886i	$0.495190\pi$
$882$	0	0
$883$	−36638.6	−1.39636	−0.698180	−	0.715922i	$-0.746007\pi$
$883$	−36638.6	−1.39636	−0.698180	+	0.715922i	$0.746007\pi$
$884$	0	0
$885$	0	0
$886$	−24958.9	−0.946401
$887$	40686.3	1.54015	0.770075	−	0.637954i	$-0.220219\pi$
$887$	40686.3	1.54015	0.770075	+	0.637954i	$0.220219\pi$
$888$	0	0
$889$	10055.1	0.379344
$890$	−1668.43	−0.0628381
$891$	0	0
$892$	1777.08	0.0667053
$893$	−3449.40	−0.129261
$894$	0	0
$895$	86.6392	0.00323579
$896$	−17102.2	−0.637663
$897$	0	0
$898$	−1438.21	−0.0534449
$899$	−19894.0	−0.738046
$900$	0	0
$901$	−72.7200	−0.00268885
$902$	−11549.5	−0.426336
$903$	0	0
$904$	15509.0	0.570598
$905$	601.618	0.0220977
$906$	0	0
$907$	−10464.4	−0.383093	−0.191547	−	0.981484i	$-0.561350\pi$
$907$	−10464.4	−0.383093	−0.191547	+	0.981484i	$0.561350\pi$
$908$	−4749.26	−0.173579
$909$	0	0
$910$	0	0
$911$	35611.5	1.29513	0.647563	−	0.762011i	$-0.275788\pi$
$911$	35611.5	1.29513	0.647563	+	0.762011i	$0.275788\pi$
$912$	0	0
$913$	39089.0	1.41693
$914$	−35242.9	−1.27542
$915$	0	0
$916$	303.920	0.0109627
$917$	37379.4	1.34610
$918$	0	0
$919$	1077.25	0.0386674	0.0193337	−	0.999813i	$-0.493846\pi$
$919$	1077.25	0.0386674	0.0193337	+	0.999813i	$0.493846\pi$
$920$	994.648	0.0356441
$921$	0	0
$922$	30762.3	1.09881
$923$	0	0
$924$	0	0
$925$	14314.8	0.508829
$926$	−34928.6	−1.23955
$927$	0	0
$928$	−11303.0	−0.399826
$929$	55733.8	1.96832	0.984159	−	0.177290i	$-0.0567330\pi$
$929$	55733.8	1.96832	0.984159	+	0.177290i	$0.0567330\pi$
$930$	0	0
$931$	−1359.74	−0.0478665
$932$	−369.284	−0.0129789
$933$	0	0
$934$	−22597.9	−0.791677
$935$	928.718	0.0324838
$936$	0	0
$937$	−3198.60	−0.111519	−0.0557596	−	0.998444i	$-0.517758\pi$
$937$	−3198.60	−0.111519	−0.0557596	+	0.998444i	$0.517758\pi$
$938$	−20686.4	−0.720079
$939$	0	0
$940$	25.8064	0.000895437 0
$941$	8823.35	0.305667	0.152834	−	0.988252i	$-0.451160\pi$
$941$	8823.35	0.305667	0.152834	+	0.988252i	$0.451160\pi$
$942$	0	0
$943$	5102.35	0.176199
$944$	−3611.95	−0.124533
$945$	0	0
$946$	72684.3	2.49806
$947$	28290.4	0.970766	0.485383	−	0.874301i	$-0.338680\pi$
$947$	28290.4	0.970766	0.485383	+	0.874301i	$0.338680\pi$
$948$	0	0
$949$	0	0
$950$	−34484.0	−1.17769
$951$	0	0
$952$	11226.8	0.382210
$953$	12399.0	0.421452	0.210726	−	0.977545i	$-0.432417\pi$
$953$	12399.0	0.421452	0.210726	+	0.977545i	$0.432417\pi$
$954$	0	0
$955$	−380.285	−0.0128856
$956$	5105.94	0.172739
$957$	0	0
$958$	−37450.0	−1.26300
$959$	32874.5	1.10696
$960$	0	0
$961$	−17001.0	−0.570676
$962$	0	0
$963$	0	0
$964$	−7235.53	−0.241744
$965$	−742.193	−0.0247586
$966$	0	0
$967$	26667.1	0.886820	0.443410	−	0.896319i	$-0.353769\pi$
$967$	26667.1	0.886820	0.443410	+	0.896319i	$0.353769\pi$
$968$	−69148.6	−2.29599
$969$	0	0
$970$	−839.001	−0.0277719
$971$	−49420.7	−1.63335	−0.816676	−	0.577096i	$-0.804186\pi$
$971$	−49420.7	−1.63335	−0.816676	+	0.577096i	$0.804186\pi$
$972$	0	0
$973$	−27149.9	−0.894539
$974$	25100.3	0.825735
$975$	0	0
$976$	46424.5	1.52255
$977$	778.759	0.0255012	0.0127506	−	0.999919i	$-0.495941\pi$
$977$	778.759	0.0255012	0.0127506	+	0.999919i	$0.495941\pi$
$978$	0	0
$979$	−75089.2	−2.45134
$980$	10.1728	0.000331589 0
$981$	0	0
$982$	27757.2	0.902003
$983$	5997.90	0.194612	0.0973059	−	0.995255i	$-0.468977\pi$
$983$	5997.90	0.194612	0.0973059	+	0.995255i	$0.468977\pi$
$984$	0	0
$985$	711.693	0.0230218
$986$	−11511.2	−0.371797
$987$	0	0
$988$	0	0
$989$	−32110.6	−1.03241
$990$	0	0
$991$	8974.94	0.287688	0.143844	−	0.989600i	$-0.454054\pi$
$991$	8974.94	0.287688	0.143844	+	0.989600i	$0.454054\pi$
$992$	7266.75	0.232580
$993$	0	0
$994$	25223.3	0.804863
$995$	1345.62	0.0428732
$996$	0	0
$997$	28530.2	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$997$	28530.2	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$998$	−6634.31	−0.210426
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.r.1.2		2
3.2	odd	2		169.4.a.g.1.1		2
13.12	even	2		117.4.a.d.1.1		2
39.2	even	12		169.4.e.f.147.1		8
39.5	even	4		169.4.b.f.168.4		4
39.8	even	4		169.4.b.f.168.1		4
39.11	even	12		169.4.e.f.147.4		8
39.17	odd	6		169.4.c.g.146.1		4
39.20	even	12		169.4.e.f.23.1		8
39.23	odd	6		169.4.c.g.22.1		4
39.29	odd	6		169.4.c.j.22.2		4
39.32	even	12		169.4.e.f.23.4		8
39.35	odd	6		169.4.c.j.146.2		4
39.38	odd	2		13.4.a.b.1.2	✓	2
52.51	odd	2		1872.4.a.bb.1.1		2
156.155	even	2		208.4.a.h.1.2		2
195.38	even	4		325.4.b.e.274.1		4
195.77	even	4		325.4.b.e.274.4		4
195.194	odd	2		325.4.a.f.1.1		2
273.272	even	2		637.4.a.b.1.2		2
312.77	odd	2		832.4.a.s.1.2		2
312.155	even	2		832.4.a.z.1.1		2
429.428	even	2		1573.4.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	39.38	odd	2
117.4.a.d.1.1		2	13.12	even	2
169.4.a.g.1.1		2	3.2	odd	2
169.4.b.f.168.1		4	39.8	even	4
169.4.b.f.168.4		4	39.5	even	4
169.4.c.g.22.1		4	39.23	odd	6
169.4.c.g.146.1		4	39.17	odd	6
169.4.c.j.22.2		4	39.29	odd	6
169.4.c.j.146.2		4	39.35	odd	6
169.4.e.f.23.1		8	39.20	even	12
169.4.e.f.23.4		8	39.32	even	12
169.4.e.f.147.1		8	39.2	even	12
169.4.e.f.147.4		8	39.11	even	12
208.4.a.h.1.2		2	156.155	even	2
325.4.a.f.1.1		2	195.194	odd	2
325.4.b.e.274.1		4	195.38	even	4
325.4.b.e.274.4		4	195.77	even	4
637.4.a.b.1.2		2	273.272	even	2
832.4.a.s.1.2		2	312.77	odd	2
832.4.a.z.1.1		2	312.155	even	2
1521.4.a.r.1.2		2	1.1	even	1	trivial
1573.4.a.b.1.1		2	429.428	even	2
1872.4.a.bb.1.1		2	52.51	odd	2

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +O(q^{10})\)	\(q+2.56155 q^{2} -1.43845 q^{4} +0.561553 q^{5} -18.1771 q^{7} -24.1771 q^{8} +1.43845 q^{10} +64.7386 q^{11} -46.5616 q^{14} -50.4233 q^{16} +25.5464 q^{17} +107.970 q^{19} -0.807764 q^{20} +165.831 q^{22} -73.2614 q^{23} -124.685 q^{25} +26.1468 q^{28} -175.909 q^{29} +113.093 q^{31} +64.2547 q^{32} +65.4384 q^{34} -10.2074 q^{35} -114.808 q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458 q^{41} +438.302 q^{43} -93.1231 q^{44} -187.663 q^{46} -31.9479 q^{47} -12.5937 q^{49} -319.386 q^{50} -2.84658 q^{53} +36.3542 q^{55} +439.469 q^{56} -450.600 q^{58} +71.6325 q^{59} -920.695 q^{61} +289.693 q^{62} +567.978 q^{64} +444.280 q^{67} -36.7471 q^{68} -26.1468 q^{70} -541.719 q^{71} -764.004 q^{73} -294.086 q^{74} -155.309 q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153 q^{80} -178.401 q^{82} +603.797 q^{83} +14.3457 q^{85} +1122.73 q^{86} -1565.19 q^{88} -1159.88 q^{89} +105.383 q^{92} -81.8362 q^{94} +60.6307 q^{95} -583.269 q^{97} -32.2595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8	\( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31} - 123 q^{32} + 135 q^{34} - 107 q^{35} - 209 q^{37} + 314 q^{38} - 89 q^{40} - 230 q^{41} + 287 q^{43} - 178 q^{44} + 4 q^{46} + 435 q^{47} + 383 q^{49} - 144 q^{50} + 118 q^{53} - 18 q^{55} + 1015 q^{56} - 794 q^{58} - 368 q^{59} - 1058 q^{61} + 332 q^{62} + 769 q^{64} - 68 q^{67} + 211 q^{68} + 125 q^{70} - 131 q^{71} - 456 q^{73} - 147 q^{74} - 22 q^{76} - 762 q^{77} - 1008 q^{79} - 69 q^{80} + 72 q^{82} + 1958 q^{83} + 173 q^{85} + 1359 q^{86} - 1242 q^{88} - 720 q^{89} + 788 q^{92} - 811 q^{94} + 146 q^{95} + 928 q^{97} - 650 q^{98}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * q^98

Introduction

L-functions

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