LMFDB - Embedded newform 1521.4.a.l.1.1

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			1.1
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)

$f(q)$	$=$	$q-4.56155 q^{2} +12.8078 q^{4} +2.80776 q^{5} -9.56155 q^{7} -21.9309 q^{8} -12.8078 q^{10} +39.4233 q^{11} +43.6155 q^{14} -2.42329 q^{16} -2.01515 q^{17} +60.1922 q^{19} +35.9612 q^{20} -179.831 q^{22} -4.46876 q^{23} -117.116 q^{25} -122.462 q^{28} -140.693 q^{29} -136.155 q^{31} +186.501 q^{32} +9.19224 q^{34} -26.8466 q^{35} +185.708 q^{37} -274.570 q^{38} -61.5767 q^{40} +310.231 q^{41} +427.471 q^{43} +504.924 q^{44} +20.3845 q^{46} -258.617 q^{47} -251.577 q^{49} +534.233 q^{50} -612.656 q^{53} +110.691 q^{55} +209.693 q^{56} +641.779 q^{58} -517.885 q^{59} -161.311 q^{61} +621.080 q^{62} -831.348 q^{64} +49.8987 q^{67} -25.8096 q^{68} +122.462 q^{70} +279.963 q^{71} -467.732 q^{73} -847.118 q^{74} +770.928 q^{76} -376.948 q^{77} +37.5379 q^{79} -6.80403 q^{80} -1415.14 q^{82} -76.1553 q^{83} -5.65808 q^{85} -1949.93 q^{86} -864.587 q^{88} +202.806 q^{89} -57.2348 q^{92} +1179.70 q^{94} +169.006 q^{95} +1174.37 q^{97} +1147.58 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8} - 5 q^{10} + 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} + 141 q^{19} + 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} - 80 q^{28} - 34 q^{29}+ \cdots + 1285 q^{98}+O(q^{100}) $ 2 * q - 5 * q^2 + 5 * q^4 - 15 * q^5 - 15 * q^7 - 15 * q^8 - 5 * q^10 + 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 + 141 * q^19 + 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 - 80 * q^28 - 34 * q^29 + 140 * q^31 + 105 * q^32 + 39 * q^34 + 70 * q^35 + 190 * q^37 - 310 * q^38 - 185 * q^40 + 538 * q^41 + 455 * q^43 + 680 * q^44 + 82 * q^46 + 60 * q^47 - 565 * q^49 + 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 + 595 * q^58 - 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 + 475 * q^67 + 505 * q^68 + 80 * q^70 + 127 * q^71 - 585 * q^73 - 849 * q^74 + 140 * q^76 - 255 * q^77 + 240 * q^79 - 1065 * q^80 - 1515 * q^82 + 260 * q^83 + 1205 * q^85 - 1962 * q^86 - 1020 * q^88 + 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 + 415 * q^97 + 1285 * 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 $

$2$	−4.56155	−1.61275	−0.806376	−	0.591403i	$-0.798574\pi$
$2$	−4.56155	−1.61275	−0.806376	+	0.591403i	$0.798574\pi$
$3$	0	0
$3$	0	0
$4$	12.8078	1.60097
$5$	2.80776	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	2.80776	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	−9.56155	−0.516275	−0.258138	−	0.966108i	$-0.583109\pi$
$7$	−9.56155	−0.516275	−0.258138	+	0.966108i	$0.583109\pi$
$8$	−21.9309	−0.969217
$9$	0	0
$10$	−12.8078	−0.405017
$11$	39.4233	1.08060	0.540299	−	0.841473i	$-0.318311\pi$
$11$	39.4233	1.08060	0.540299	+	0.841473i	$0.318311\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	43.6155	0.832624
$15$	0	0
$16$	−2.42329	−0.0378639
$17$	−2.01515	−0.0287498	−0.0143749	−	0.999897i	$-0.504576\pi$
$17$	−2.01515	−0.0287498	−0.0143749	+	0.999897i	$0.504576\pi$
$18$	0	0
$19$	60.1922	0.726792	0.363396	−	0.931635i	$-0.381617\pi$
$19$	60.1922	0.726792	0.363396	+	0.931635i	$0.381617\pi$
$20$	35.9612	0.402058
$21$	0	0
$22$	−179.831	−1.74274
$23$	−4.46876	−0.0405131	−0.0202565	−	0.999795i	$-0.506448\pi$
$23$	−4.46876	−0.0405131	−0.0202565	+	0.999795i	$0.506448\pi$
$24$	0	0
$25$	−117.116	−0.936932
$26$	0	0
$27$	0	0
$28$	−122.462	−0.826542
$29$	−140.693	−0.900899	−0.450449	−	0.892802i	$-0.648736\pi$
$29$	−140.693	−0.900899	−0.450449	+	0.892802i	$0.648736\pi$
$30$	0	0
$31$	−136.155	−0.788845	−0.394423	−	0.918929i	$-0.629055\pi$
$31$	−136.155	−0.788845	−0.394423	+	0.918929i	$0.629055\pi$
$32$	186.501	1.03028
$33$	0	0
$34$	9.19224	0.0463663
$35$	−26.8466	−0.129654
$36$	0	0
$37$	185.708	0.825142	0.412571	−	0.910925i	$-0.364631\pi$
$37$	185.708	0.825142	0.412571	+	0.910925i	$0.364631\pi$
$38$	−274.570	−1.17214
$39$	0	0
$40$	−61.5767	−0.243403
$41$	310.231	1.18171	0.590853	−	0.806779i	$-0.298791\pi$
$41$	310.231	1.18171	0.590853	+	0.806779i	$0.298791\pi$
$42$	0	0
$43$	427.471	1.51602	0.758008	−	0.652246i	$-0.226173\pi$
$43$	427.471	1.51602	0.758008	+	0.652246i	$0.226173\pi$
$44$	504.924	1.73000
$45$	0	0
$46$	20.3845	0.0653375
$47$	−258.617	−0.802622	−0.401311	−	0.915942i	$-0.631445\pi$
$47$	−258.617	−0.802622	−0.401311	+	0.915942i	$0.631445\pi$
$48$	0	0
$49$	−251.577	−0.733460
$50$	534.233	1.51104
$51$	0	0
$52$	0	0
$53$	−612.656	−1.58783	−0.793913	−	0.608031i	$-0.791960\pi$
$53$	−612.656	−1.58783	−0.793913	+	0.608031i	$0.791960\pi$
$54$	0	0
$55$	110.691	0.271375
$56$	209.693	0.500383
$57$	0	0
$58$	641.779	1.45293
$59$	−517.885	−1.14276	−0.571381	−	0.820685i	$-0.693592\pi$
$59$	−517.885	−1.14276	−0.571381	+	0.820685i	$0.693592\pi$
$60$	0	0
$61$	−161.311	−0.338585	−0.169293	−	0.985566i	$-0.554148\pi$
$61$	−161.311	−0.338585	−0.169293	+	0.985566i	$0.554148\pi$
$62$	621.080	1.27221
$63$	0	0
$64$	−831.348	−1.62373
$65$	0	0
$66$	0	0
$67$	49.8987	0.0909865	0.0454933	−	0.998965i	$-0.485514\pi$
$67$	49.8987	0.0909865	0.0454933	+	0.998965i	$0.485514\pi$
$68$	−25.8096	−0.0460276
$69$	0	0
$70$	122.462	0.209100
$71$	279.963	0.467965	0.233982	−	0.972241i	$-0.424824\pi$
$71$	279.963	0.467965	0.233982	+	0.972241i	$0.424824\pi$
$72$	0	0
$73$	−467.732	−0.749916	−0.374958	−	0.927042i	$-0.622343\pi$
$73$	−467.732	−0.749916	−0.374958	+	0.927042i	$0.622343\pi$
$74$	−847.118	−1.33075
$75$	0	0
$76$	770.928	1.16357
$77$	−376.948	−0.557886
$78$	0	0
$79$	37.5379	0.0534600	0.0267300	−	0.999643i	$-0.491491\pi$
$79$	37.5379	0.0534600	0.0267300	+	0.999643i	$0.491491\pi$
$80$	−6.80403	−0.00950892
$81$	0	0
$82$	−1415.14	−1.90580
$83$	−76.1553	−0.100712	−0.0503562	−	0.998731i	$-0.516036\pi$
$83$	−76.1553	−0.100712	−0.0503562	+	0.998731i	$0.516036\pi$
$84$	0	0
$85$	−5.65808	−0.00722006
$86$	−1949.93	−2.44496
$87$	0	0
$88$	−864.587	−1.04733
$89$	202.806	0.241544	0.120772	−	0.992680i	$-0.461463\pi$
$89$	202.806	0.241544	0.120772	+	0.992680i	$0.461463\pi$
$90$	0	0
$91$	0	0
$92$	−57.2348	−0.0648602
$93$	0	0
$94$	1179.70	1.29443
$95$	169.006	0.182522
$96$	0	0
$97$	1174.37	1.22927	0.614634	−	0.788812i	$-0.289304\pi$
$97$	1174.37	1.22927	0.614634	+	0.788812i	$0.289304\pi$
$98$	1147.58	1.18289
$99$	0	0
$100$	−1500.00	−1.50000
$101$	−970.697	−0.956316	−0.478158	−	0.878274i	$-0.658695\pi$
$101$	−970.697	−0.956316	−0.478158	+	0.878274i	$0.658695\pi$
$102$	0	0
$103$	−1899.70	−1.81731	−0.908654	−	0.417550i	$-0.862889\pi$
$103$	−1899.70	−1.81731	−0.908654	+	0.417550i	$0.862889\pi$
$104$	0	0
$105$	0	0
$106$	2794.66	2.56077
$107$	1906.49	1.72250	0.861251	−	0.508180i	$-0.169682\pi$
$107$	1906.49	1.72250	0.861251	+	0.508180i	$0.169682\pi$
$108$	0	0
$109$	896.004	0.787354	0.393677	−	0.919249i	$-0.371203\pi$
$109$	896.004	0.787354	0.393677	+	0.919249i	$0.371203\pi$
$110$	−504.924	−0.437660
$111$	0	0
$112$	23.1704	0.0195482
$113$	334.882	0.278788	0.139394	−	0.990237i	$-0.455484\pi$
$113$	334.882	0.278788	0.139394	+	0.990237i	$0.455484\pi$
$114$	0	0
$115$	−12.5472	−0.0101742
$116$	−1801.96	−1.44231
$117$	0	0
$118$	2362.36	1.84299
$119$	19.2680	0.0148428
$120$	0	0
$121$	223.196	0.167690
$122$	735.827	0.546054
$123$	0	0
$124$	−1743.84	−1.26292
$125$	−679.806	−0.486430
$126$	0	0
$127$	620.893	0.433822	0.216911	−	0.976191i	$-0.430402\pi$
$127$	620.893	0.433822	0.216911	+	0.976191i	$0.430402\pi$
$128$	2300.23	1.58839
$129$	0	0
$130$	0	0
$131$	1331.70	0.888180	0.444090	−	0.895982i	$-0.353527\pi$
$131$	1331.70	0.888180	0.444090	+	0.895982i	$0.353527\pi$
$132$	0	0
$133$	−575.531	−0.375225
$134$	−227.616	−0.146739
$135$	0	0
$136$	44.1941	0.0278648
$137$	622.015	0.387900	0.193950	−	0.981011i	$-0.437870\pi$
$137$	622.015	0.387900	0.193950	+	0.981011i	$0.437870\pi$
$138$	0	0
$139$	330.580	0.201723	0.100861	−	0.994900i	$-0.467840\pi$
$139$	330.580	0.201723	0.100861	+	0.994900i	$0.467840\pi$
$140$	−343.845	−0.207573
$141$	0	0
$142$	−1277.07	−0.754711
$143$	0	0
$144$	0	0
$145$	−395.033	−0.226246
$146$	2133.58	1.20943
$147$	0	0
$148$	2378.51	1.32103
$149$	−1810.54	−0.995470	−0.497735	−	0.867329i	$-0.665835\pi$
$149$	−1810.54	−0.995470	−0.497735	+	0.867329i	$0.665835\pi$
$150$	0	0
$151$	423.239	0.228097	0.114049	−	0.993475i	$-0.463618\pi$
$151$	423.239	0.228097	0.114049	+	0.993475i	$0.463618\pi$
$152$	−1320.07	−0.704419
$153$	0	0
$154$	1719.47	0.899732
$155$	−382.292	−0.198106
$156$	0	0
$157$	1322.17	0.672105	0.336052	−	0.941843i	$-0.390908\pi$
$157$	1322.17	0.672105	0.336052	+	0.941843i	$0.390908\pi$
$158$	−171.231	−0.0862178
$159$	0	0
$160$	523.651	0.258739
$161$	42.7283	0.0209159
$162$	0	0
$163$	3606.39	1.73297	0.866486	−	0.499201i	$-0.166373\pi$
$163$	3606.39	1.73297	0.866486	+	0.499201i	$0.166373\pi$
$164$	3973.37	1.89188
$165$	0	0
$166$	347.386	0.162424
$167$	−3415.43	−1.58260	−0.791300	−	0.611429i	$-0.790595\pi$
$167$	−3415.43	−1.58260	−0.791300	+	0.611429i	$0.790595\pi$
$168$	0	0
$169$	0	0
$170$	25.8096	0.0116442
$171$	0	0
$172$	5474.94	2.42710
$173$	−2342.23	−1.02934	−0.514671	−	0.857388i	$-0.672086\pi$
$173$	−2342.23	−1.02934	−0.514671	+	0.857388i	$0.672086\pi$
$174$	0	0
$175$	1119.82	0.483715
$176$	−95.5342	−0.0409157
$177$	0	0
$178$	−925.110	−0.389550
$179$	666.891	0.278468	0.139234	−	0.990260i	$-0.455536\pi$
$179$	666.891	0.278468	0.139234	+	0.990260i	$0.455536\pi$
$180$	0	0
$181$	−701.037	−0.287888	−0.143944	−	0.989586i	$-0.545978\pi$
$181$	−701.037	−0.287888	−0.143944	+	0.989586i	$0.545978\pi$
$182$	0	0
$183$	0	0
$184$	98.0037	0.0392659
$185$	521.425	0.207221
$186$	0	0
$187$	−79.4440	−0.0310670
$188$	−3312.31	−1.28497
$189$	0	0
$190$	−770.928	−0.294363
$191$	−1300.88	−0.492819	−0.246409	−	0.969166i	$-0.579251\pi$
$191$	−1300.88	−0.492819	−0.246409	+	0.969166i	$0.579251\pi$
$192$	0	0
$193$	519.333	0.193691	0.0968457	−	0.995299i	$-0.469125\pi$
$193$	519.333	0.193691	0.0968457	+	0.995299i	$0.469125\pi$
$194$	−5356.94	−1.98251
$195$	0	0
$196$	−3222.14	−1.17425
$197$	3121.05	1.12876	0.564379	−	0.825516i	$-0.309116\pi$
$197$	3121.05	1.12876	0.564379	+	0.825516i	$0.309116\pi$
$198$	0	0
$199$	1237.06	0.440667	0.220333	−	0.975425i	$-0.429285\pi$
$199$	1237.06	0.440667	0.220333	+	0.975425i	$0.429285\pi$
$200$	2568.47	0.908090
$201$	0	0
$202$	4427.89	1.54230
$203$	1345.25	0.465112
$204$	0	0
$205$	871.056	0.296767
$206$	8665.57	2.93087
$207$	0	0
$208$	0	0
$209$	2372.98	0.785369
$210$	0	0
$211$	−2531.67	−0.826006	−0.413003	−	0.910730i	$-0.635520\pi$
$211$	−2531.67	−0.826006	−0.413003	+	0.910730i	$0.635520\pi$
$212$	−7846.76	−2.54206
$213$	0	0
$214$	−8696.57	−2.77797
$215$	1200.24	0.380723
$216$	0	0
$217$	1301.86	0.407261
$218$	−4087.17	−1.26981
$219$	0	0
$220$	1417.71	0.434463
$221$	0	0
$222$	0	0
$223$	1195.53	0.359008	0.179504	−	0.983757i	$-0.442551\pi$
$223$	1195.53	0.359008	0.179504	+	0.983757i	$0.442551\pi$
$224$	−1783.24	−0.531909
$225$	0	0
$226$	−1527.58	−0.449617
$227$	−869.192	−0.254142	−0.127071	−	0.991894i	$-0.540558\pi$
$227$	−869.192	−0.254142	−0.127071	+	0.991894i	$0.540558\pi$
$228$	0	0
$229$	−4684.64	−1.35183	−0.675916	−	0.736978i	$-0.736252\pi$
$229$	−4684.64	−1.35183	−0.675916	+	0.736978i	$0.736252\pi$
$230$	57.2348	0.0164085
$231$	0	0
$232$	3085.52	0.873166
$233$	4868.99	1.36900	0.684502	−	0.729011i	$-0.260020\pi$
$233$	4868.99	1.36900	0.684502	+	0.729011i	$0.260020\pi$
$234$	0	0
$235$	−726.137	−0.201566
$236$	−6632.95	−1.82953
$237$	0	0
$238$	−87.8920	−0.0239378
$239$	4807.53	1.30114	0.650572	−	0.759444i	$-0.274529\pi$
$239$	4807.53	1.30114	0.650572	+	0.759444i	$0.274529\pi$
$240$	0	0
$241$	−5875.96	−1.57056	−0.785278	−	0.619143i	$-0.787480\pi$
$241$	−5875.96	−1.57056	−0.785278	+	0.619143i	$0.787480\pi$
$242$	−1018.12	−0.270443
$243$	0	0
$244$	−2066.03	−0.542065
$245$	−706.368	−0.184197
$246$	0	0
$247$	0	0
$248$	2986.00	0.764562
$249$	0	0
$250$	3100.97	0.784490
$251$	−5806.27	−1.46011	−0.730057	−	0.683386i	$-0.760506\pi$
$251$	−5806.27	−1.46011	−0.730057	+	0.683386i	$0.760506\pi$
$252$	0	0
$253$	−176.173	−0.0437783
$254$	−2832.24	−0.699647
$255$	0	0
$256$	−3841.83	−0.937947
$257$	1195.86	0.290256	0.145128	−	0.989413i	$-0.453641\pi$
$257$	1195.86	0.290256	0.145128	+	0.989413i	$0.453641\pi$
$258$	0	0
$259$	−1775.66	−0.426001
$260$	0	0
$261$	0	0
$262$	−6074.64	−1.43241
$263$	−234.184	−0.0549066	−0.0274533	−	0.999623i	$-0.508740\pi$
$263$	−234.184	−0.0549066	−0.0274533	+	0.999623i	$0.508740\pi$
$264$	0	0
$265$	−1720.19	−0.398757
$266$	2625.32	0.605145
$267$	0	0
$268$	639.091	0.145667
$269$	2668.27	0.604785	0.302393	−	0.953183i	$-0.402215\pi$
$269$	2668.27	0.604785	0.302393	+	0.953183i	$0.402215\pi$
$270$	0	0
$271$	−5701.28	−1.27796	−0.638982	−	0.769222i	$-0.720644\pi$
$271$	−5701.28	−1.27796	−0.638982	+	0.769222i	$0.720644\pi$
$272$	4.88331	0.00108858
$273$	0	0
$274$	−2837.35	−0.625587
$275$	−4617.12	−1.01245
$276$	0	0
$277$	−7152.49	−1.55145	−0.775725	−	0.631072i	$-0.782615\pi$
$277$	−7152.49	−1.55145	−0.775725	+	0.631072i	$0.782615\pi$
$278$	−1507.96	−0.325329
$279$	0	0
$280$	588.769	0.125663
$281$	−6132.87	−1.30198	−0.650990	−	0.759086i	$-0.725646\pi$
$281$	−6132.87	−1.30198	−0.650990	+	0.759086i	$0.725646\pi$
$282$	0	0
$283$	3377.15	0.709367	0.354683	−	0.934986i	$-0.384589\pi$
$283$	3377.15	0.709367	0.354683	+	0.934986i	$0.384589\pi$
$284$	3585.70	0.749198
$285$	0	0
$286$	0	0
$287$	−2966.29	−0.610086
$288$	0	0
$289$	−4908.94	−0.999173
$290$	1801.96	0.364879
$291$	0	0
$292$	−5990.60	−1.20059
$293$	−4704.77	−0.938073	−0.469037	−	0.883179i	$-0.655399\pi$
$293$	−4704.77	−0.938073	−0.469037	+	0.883179i	$0.655399\pi$
$294$	0	0
$295$	−1454.10	−0.286986
$296$	−4072.75	−0.799742
$297$	0	0
$298$	8258.86	1.60545
$299$	0	0
$300$	0	0
$301$	−4087.28	−0.782681
$302$	−1930.62	−0.367864
$303$	0	0
$304$	−145.863	−0.0275192
$305$	−452.922	−0.0850303
$306$	0	0
$307$	−5130.49	−0.953787	−0.476894	−	0.878961i	$-0.658237\pi$
$307$	−5130.49	−0.953787	−0.476894	+	0.878961i	$0.658237\pi$
$308$	−4827.86	−0.893159
$309$	0	0
$310$	1743.84	0.319496
$311$	−7948.94	−1.44933	−0.724667	−	0.689099i	$-0.758006\pi$
$311$	−7948.94	−1.44933	−0.724667	+	0.689099i	$0.758006\pi$
$312$	0	0
$313$	−8521.87	−1.53893	−0.769465	−	0.638689i	$-0.779477\pi$
$313$	−8521.87	−1.53893	−0.769465	+	0.638689i	$0.779477\pi$
$314$	−6031.14	−1.08394
$315$	0	0
$316$	480.776	0.0855879
$317$	−6662.46	−1.18044	−0.590222	−	0.807241i	$-0.700960\pi$
$317$	−6662.46	−1.18044	−0.590222	+	0.807241i	$0.700960\pi$
$318$	0	0
$319$	−5546.59	−0.973509
$320$	−2334.23	−0.407773
$321$	0	0
$322$	−194.907	−0.0337322
$323$	−121.297	−0.0208951
$324$	0	0
$325$	0	0
$326$	−16450.8	−2.79486
$327$	0	0
$328$	−6803.64	−1.14533
$329$	2472.78	0.414374
$330$	0	0
$331$	3911.77	0.649578	0.324789	−	0.945786i	$-0.394707\pi$
$331$	3911.77	0.649578	0.324789	+	0.945786i	$0.394707\pi$
$332$	−975.379	−0.161238
$333$	0	0
$334$	15579.7	2.55234
$335$	140.104	0.0228498
$336$	0	0
$337$	−627.211	−0.101384	−0.0506919	−	0.998714i	$-0.516143\pi$
$337$	−627.211	−0.101384	−0.0506919	+	0.998714i	$0.516143\pi$
$338$	0	0
$339$	0	0
$340$	−72.4674	−0.0115591
$341$	−5367.69	−0.852424
$342$	0	0
$343$	5685.08	0.894943
$344$	−9374.80	−1.46935
$345$	0	0
$346$	10684.2	1.66007
$347$	3823.02	0.591442	0.295721	−	0.955274i	$-0.404440\pi$
$347$	3823.02	0.591442	0.295721	+	0.955274i	$0.404440\pi$
$348$	0	0
$349$	−3410.67	−0.523120	−0.261560	−	0.965187i	$-0.584237\pi$
$349$	−3410.67	−0.523120	−0.261560	+	0.965187i	$0.584237\pi$
$350$	−5108.10	−0.780112
$351$	0	0
$352$	7352.48	1.11332
$353$	−5587.64	−0.842492	−0.421246	−	0.906946i	$-0.638407\pi$
$353$	−5587.64	−0.842492	−0.421246	+	0.906946i	$0.638407\pi$
$354$	0	0
$355$	786.070	0.117522
$356$	2597.49	0.386704
$357$	0	0
$358$	−3042.06	−0.449100
$359$	2230.14	0.327861	0.163931	−	0.986472i	$-0.447583\pi$
$359$	2230.14	0.327861	0.163931	+	0.986472i	$0.447583\pi$
$360$	0	0
$361$	−3235.89	−0.471774
$362$	3197.82	0.464292
$363$	0	0
$364$	0	0
$365$	−1313.28	−0.188330
$366$	0	0
$367$	−8699.14	−1.23731	−0.618653	−	0.785664i	$-0.712321\pi$
$367$	−8699.14	−1.23731	−0.618653	+	0.785664i	$0.712321\pi$
$368$	10.8291	0.00153398
$369$	0	0
$370$	−2378.51	−0.334197
$371$	5857.94	0.819756
$372$	0	0
$373$	10964.2	1.52199	0.760997	−	0.648755i	$-0.224710\pi$
$373$	10964.2	1.52199	0.760997	+	0.648755i	$0.224710\pi$
$374$	362.388	0.0501033
$375$	0	0
$376$	5671.70	0.777914
$377$	0	0
$378$	0	0
$379$	−13910.1	−1.88526	−0.942631	−	0.333835i	$-0.891657\pi$
$379$	−13910.1	−1.88526	−0.942631	+	0.333835i	$0.891657\pi$
$380$	2164.58	0.292213
$381$	0	0
$382$	5934.03	0.794794
$383$	−494.753	−0.0660071	−0.0330035	−	0.999455i	$-0.510507\pi$
$383$	−494.753	−0.0660071	−0.0330035	+	0.999455i	$0.510507\pi$
$384$	0	0
$385$	−1058.38	−0.140104
$386$	−2368.97	−0.312376
$387$	0	0
$388$	15041.0	1.96802
$389$	4140.47	0.539666	0.269833	−	0.962907i	$-0.413032\pi$
$389$	4140.47	0.539666	0.269833	+	0.962907i	$0.413032\pi$
$390$	0	0
$391$	9.00524	0.00116474
$392$	5517.30	0.710881
$393$	0	0
$394$	−14236.8	−1.82041
$395$	105.398	0.0134256
$396$	0	0
$397$	1881.79	0.237895	0.118948	−	0.992901i	$-0.462048\pi$
$397$	1881.79	0.237895	0.118948	+	0.992901i	$0.462048\pi$
$398$	−5642.90	−0.710686
$399$	0	0
$400$	283.807	0.0354759
$401$	421.765	0.0525236	0.0262618	−	0.999655i	$-0.491640\pi$
$401$	421.765	0.0525236	0.0262618	+	0.999655i	$0.491640\pi$
$402$	0	0
$403$	0	0
$404$	−12432.5	−1.53103
$405$	0	0
$406$	−6136.41	−0.750110
$407$	7321.23	0.891646
$408$	0	0
$409$	−2550.22	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$409$	−2550.22	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$410$	−3973.37	−0.478611
$411$	0	0
$412$	−24330.9	−2.90946
$413$	4951.79	0.589980
$414$	0	0
$415$	−213.826	−0.0252923
$416$	0	0
$417$	0	0
$418$	−10824.5	−1.26661
$419$	−12384.8	−1.44400	−0.722002	−	0.691891i	$-0.756778\pi$
$419$	−12384.8	−1.44400	−0.722002	+	0.691891i	$0.756778\pi$
$420$	0	0
$421$	−10463.0	−1.21124	−0.605622	−	0.795752i	$-0.707076\pi$
$421$	−10463.0	−1.21124	−0.605622	+	0.795752i	$0.707076\pi$
$422$	11548.3	1.33214
$423$	0	0
$424$	13436.1	1.53895
$425$	236.008	0.0269366
$426$	0	0
$427$	1542.38	0.174803
$428$	24417.9	2.75767
$429$	0	0
$430$	−5474.94	−0.614012
$431$	−3962.39	−0.442834	−0.221417	−	0.975179i	$-0.571068\pi$
$431$	−3962.39	−0.442834	−0.221417	+	0.975179i	$0.571068\pi$
$432$	0	0
$433$	−8394.14	−0.931632	−0.465816	−	0.884882i	$-0.654239\pi$
$433$	−8394.14	−0.931632	−0.465816	+	0.884882i	$0.654239\pi$
$434$	−5938.48	−0.656812
$435$	0	0
$436$	11475.8	1.26053
$437$	−268.984	−0.0294446
$438$	0	0
$439$	10174.5	1.10616	0.553079	−	0.833129i	$-0.313453\pi$
$439$	10174.5	1.10616	0.553079	+	0.833129i	$0.313453\pi$
$440$	−2427.56	−0.263021
$441$	0	0
$442$	0	0
$443$	5880.74	0.630705	0.315353	−	0.948975i	$-0.397877\pi$
$443$	5880.74	0.630705	0.315353	+	0.948975i	$0.397877\pi$
$444$	0	0
$445$	569.431	0.0606598
$446$	−5453.48	−0.578990
$447$	0	0
$448$	7948.97	0.838289
$449$	10664.9	1.12095	0.560475	−	0.828172i	$-0.310619\pi$
$449$	10664.9	1.12095	0.560475	+	0.828172i	$0.310619\pi$
$450$	0	0
$451$	12230.3	1.27695
$452$	4289.10	0.446332
$453$	0	0
$454$	3964.87	0.409869
$455$	0	0
$456$	0	0
$457$	14828.9	1.51787	0.758933	−	0.651169i	$-0.225721\pi$
$457$	14828.9	1.51787	0.758933	+	0.651169i	$0.225721\pi$
$458$	21369.2	2.18017
$459$	0	0
$460$	−160.702	−0.0162886
$461$	−9711.70	−0.981169	−0.490585	−	0.871394i	$-0.663217\pi$
$461$	−9711.70	−0.981169	−0.490585	+	0.871394i	$0.663217\pi$
$462$	0	0
$463$	−11353.5	−1.13962	−0.569809	−	0.821777i	$-0.692983\pi$
$463$	−11353.5	−1.13962	−0.569809	+	0.821777i	$0.692983\pi$
$464$	340.941	0.0341116
$465$	0	0
$466$	−22210.1	−2.20787
$467$	−6451.31	−0.639252	−0.319626	−	0.947544i	$-0.603557\pi$
$467$	−6451.31	−0.639252	−0.319626	+	0.947544i	$0.603557\pi$
$468$	0	0
$469$	−477.109	−0.0469741
$470$	3312.31	0.325076
$471$	0	0
$472$	11357.7	1.10758
$473$	16852.3	1.63820
$474$	0	0
$475$	−7049.50	−0.680954
$476$	246.780	0.0237629
$477$	0	0
$478$	−21929.8	−2.09842
$479$	−9566.46	−0.912531	−0.456266	−	0.889844i	$-0.650813\pi$
$479$	−9566.46	−0.912531	−0.456266	+	0.889844i	$0.650813\pi$
$480$	0	0
$481$	0	0
$482$	26803.5	2.53292
$483$	0	0
$484$	2858.64	0.268467
$485$	3297.35	0.308711
$486$	0	0
$487$	−4917.11	−0.457527	−0.228764	−	0.973482i	$-0.573468\pi$
$487$	−4917.11	−0.457527	−0.228764	+	0.973482i	$0.573468\pi$
$488$	3537.68	0.328162
$489$	0	0
$490$	3222.14	0.297064
$491$	−2950.82	−0.271220	−0.135610	−	0.990762i	$-0.543299\pi$
$491$	−2950.82	−0.271220	−0.135610	+	0.990762i	$0.543299\pi$
$492$	0	0
$493$	283.519	0.0259007
$494$	0	0
$495$	0	0
$496$	329.944	0.0298688
$497$	−2676.88	−0.241599
$498$	0	0
$499$	−13430.1	−1.20484	−0.602418	−	0.798180i	$-0.705796\pi$
$499$	−13430.1	−1.20484	−0.602418	+	0.798180i	$0.705796\pi$
$500$	−8706.79	−0.778759
$501$	0	0
$502$	26485.6	2.35480
$503$	−1320.29	−0.117035	−0.0585175	−	0.998286i	$-0.518637\pi$
$503$	−1320.29	−0.117035	−0.0585175	+	0.998286i	$0.518637\pi$
$504$	0	0
$505$	−2725.49	−0.240164
$506$	803.623	0.0706036
$507$	0	0
$508$	7952.25	0.694536
$509$	20916.4	1.82143	0.910713	−	0.413041i	$-0.135533\pi$
$509$	20916.4	1.82143	0.910713	+	0.413041i	$0.135533\pi$
$510$	0	0
$511$	4472.24	0.387163
$512$	−877.105	−0.0757089
$513$	0	0
$514$	−5454.98	−0.468110
$515$	−5333.90	−0.456388
$516$	0	0
$517$	−10195.5	−0.867311
$518$	8099.77	0.687033
$519$	0	0
$520$	0	0
$521$	10104.2	0.849661	0.424831	−	0.905273i	$-0.360334\pi$
$521$	10104.2	0.849661	0.424831	+	0.905273i	$0.360334\pi$
$522$	0	0
$523$	7131.22	0.596227	0.298113	−	0.954530i	$-0.403643\pi$
$523$	7131.22	0.596227	0.298113	+	0.954530i	$0.403643\pi$
$524$	17056.2	1.42195
$525$	0	0
$526$	1068.24	0.0885507
$527$	274.374	0.0226792
$528$	0	0
$529$	−12147.0	−0.998359
$530$	7846.76	0.643097
$531$	0	0
$532$	−7371.27	−0.600724
$533$	0	0
$534$	0	0
$535$	5352.98	0.432579
$536$	−1094.32	−0.0881856
$537$	0	0
$538$	−12171.5	−0.975369
$539$	−9917.98	−0.792575
$540$	0	0
$541$	−16831.7	−1.33762	−0.668809	−	0.743435i	$-0.733195\pi$
$541$	−16831.7	−1.33762	−0.668809	+	0.743435i	$0.733195\pi$
$542$	26006.7	2.06104
$543$	0	0
$544$	−375.828	−0.0296204
$545$	2515.77	0.197731
$546$	0	0
$547$	−9560.55	−0.747312	−0.373656	−	0.927567i	$-0.621896\pi$
$547$	−9560.55	−0.747312	−0.373656	+	0.927567i	$0.621896\pi$
$548$	7966.62	0.621017
$549$	0	0
$550$	21061.2	1.63282
$551$	−8468.64	−0.654766
$552$	0	0
$553$	−358.920	−0.0276001
$554$	32626.5	2.50210
$555$	0	0
$556$	4234.00	0.322952
$557$	−22827.9	−1.73653	−0.868267	−	0.496097i	$-0.834766\pi$
$557$	−22827.9	−1.73653	−0.868267	+	0.496097i	$0.834766\pi$
$558$	0	0
$559$	0	0
$560$	65.0571	0.00490922
$561$	0	0
$562$	27975.4	2.09977
$563$	−21629.7	−1.61916	−0.809578	−	0.587013i	$-0.800304\pi$
$563$	−21629.7	−1.61916	−0.809578	+	0.587013i	$0.800304\pi$
$564$	0	0
$565$	940.271	0.0700133
$566$	−15405.1	−1.14403
$567$	0	0
$568$	−6139.83	−0.453559
$569$	10589.9	0.780229	0.390114	−	0.920766i	$-0.372436\pi$
$569$	10589.9	0.780229	0.390114	+	0.920766i	$0.372436\pi$
$570$	0	0
$571$	−1757.27	−0.128791	−0.0643954	−	0.997924i	$-0.520512\pi$
$571$	−1757.27	−0.128791	−0.0643954	+	0.997924i	$0.520512\pi$
$572$	0	0
$573$	0	0
$574$	13530.9	0.983917
$575$	523.365	0.0379580
$576$	0	0
$577$	13580.6	0.979840	0.489920	−	0.871767i	$-0.337026\pi$
$577$	13580.6	0.979840	0.489920	+	0.871767i	$0.337026\pi$
$578$	22392.4	1.61142
$579$	0	0
$580$	−5059.49	−0.362214
$581$	728.163	0.0519953
$582$	0	0
$583$	−24152.9	−1.71580
$584$	10257.8	0.726831
$585$	0	0
$586$	21461.0	1.51288
$587$	−957.326	−0.0673136	−0.0336568	−	0.999433i	$-0.510715\pi$
$587$	−957.326	−0.0673136	−0.0336568	+	0.999433i	$0.510715\pi$
$588$	0	0
$589$	−8195.49	−0.573327
$590$	6632.95	0.462838
$591$	0	0
$592$	−450.026	−0.0312431
$593$	6729.49	0.466015	0.233007	−	0.972475i	$-0.425143\pi$
$593$	6729.49	0.466015	0.233007	+	0.972475i	$0.425143\pi$
$594$	0	0
$595$	54.1000	0.00372754
$596$	−23188.9	−1.59372
$597$	0	0
$598$	0	0
$599$	−2281.52	−0.155626	−0.0778132	−	0.996968i	$-0.524794\pi$
$599$	−2281.52	−0.155626	−0.0778132	+	0.996968i	$0.524794\pi$
$600$	0	0
$601$	6401.42	0.434475	0.217237	−	0.976119i	$-0.430295\pi$
$601$	6401.42	0.434475	0.217237	+	0.976119i	$0.430295\pi$
$602$	18644.4	1.26227
$603$	0	0
$604$	5420.74	0.365177
$605$	626.682	0.0421128
$606$	0	0
$607$	2779.24	0.185841	0.0929207	−	0.995674i	$-0.470380\pi$
$607$	2779.24	0.185841	0.0929207	+	0.995674i	$0.470380\pi$
$608$	11225.9	0.748800
$609$	0	0
$610$	2066.03	0.137133
$611$	0	0
$612$	0	0
$613$	−22620.8	−1.49045	−0.745226	−	0.666812i	$-0.767658\pi$
$613$	−22620.8	−1.49045	−0.745226	+	0.666812i	$0.767658\pi$
$614$	23403.0	1.53822
$615$	0	0
$616$	8266.80	0.540712
$617$	21974.0	1.43378	0.716889	−	0.697187i	$-0.245565\pi$
$617$	21974.0	1.43378	0.716889	+	0.697187i	$0.245565\pi$
$618$	0	0
$619$	−7145.19	−0.463957	−0.231979	−	0.972721i	$-0.574520\pi$
$619$	−7145.19	−0.463957	−0.231979	+	0.972721i	$0.574520\pi$
$620$	−4896.30	−0.317162
$621$	0	0
$622$	36259.5	2.33742
$623$	−1939.14	−0.124703
$624$	0	0
$625$	12730.8	0.814773
$626$	38873.0	2.48191
$627$	0	0
$628$	16934.0	1.07602
$629$	−374.231	−0.0237227
$630$	0	0
$631$	18883.2	1.19133	0.595666	−	0.803232i	$-0.296888\pi$
$631$	18883.2	1.19133	0.595666	+	0.803232i	$0.296888\pi$
$632$	−823.239	−0.0518144
$633$	0	0
$634$	30391.1	1.90376
$635$	1743.32	0.108947
$636$	0	0
$637$	0	0
$638$	25301.1	1.57003
$639$	0	0
$640$	6458.50	0.398898
$641$	3631.08	0.223743	0.111871	−	0.993723i	$-0.464316\pi$
$641$	3631.08	0.223743	0.111871	+	0.993723i	$0.464316\pi$
$642$	0	0
$643$	10772.0	0.660660	0.330330	−	0.943866i	$-0.392840\pi$
$643$	10772.0	0.660660	0.330330	+	0.943866i	$0.392840\pi$
$644$	547.253	0.0334857
$645$	0	0
$646$	553.301	0.0336987
$647$	−15148.3	−0.920464	−0.460232	−	0.887799i	$-0.652234\pi$
$647$	−15148.3	−0.920464	−0.460232	+	0.887799i	$0.652234\pi$
$648$	0	0
$649$	−20416.7	−1.23487
$650$	0	0
$651$	0	0
$652$	46189.8	2.77444
$653$	−7358.89	−0.441004	−0.220502	−	0.975387i	$-0.570770\pi$
$653$	−7358.89	−0.441004	−0.220502	+	0.975387i	$0.570770\pi$
$654$	0	0
$655$	3739.11	0.223052
$656$	−751.780	−0.0447441
$657$	0	0
$658$	−11279.7	−0.668282
$659$	−28333.3	−1.67482	−0.837411	−	0.546574i	$-0.815932\pi$
$659$	−28333.3	−1.67482	−0.837411	+	0.546574i	$0.815932\pi$
$660$	0	0
$661$	−1109.68	−0.0652975	−0.0326488	−	0.999467i	$-0.510394\pi$
$661$	−1109.68	−0.0652975	−0.0326488	+	0.999467i	$0.510394\pi$
$662$	−17843.8	−1.04761
$663$	0	0
$664$	1670.15	0.0976121
$665$	−1615.96	−0.0942317
$666$	0	0
$667$	628.724	0.0364982
$668$	−43744.1	−2.53369
$669$	0	0
$670$	−639.091	−0.0368511
$671$	−6359.39	−0.365874
$672$	0	0
$673$	20979.1	1.20161	0.600806	−	0.799395i	$-0.294846\pi$
$673$	20979.1	1.20161	0.600806	+	0.799395i	$0.294846\pi$
$674$	2861.06	0.163507
$675$	0	0
$676$	0	0
$677$	−30941.9	−1.75656	−0.878282	−	0.478142i	$-0.841310\pi$
$677$	−30941.9	−1.75656	−0.878282	+	0.478142i	$0.841310\pi$
$678$	0	0
$679$	−11228.8	−0.634641
$680$	124.087	0.00699780
$681$	0	0
$682$	24485.0	1.37475
$683$	5426.21	0.303995	0.151997	−	0.988381i	$-0.451429\pi$
$683$	5426.21	0.303995	0.151997	+	0.988381i	$0.451429\pi$
$684$	0	0
$685$	1746.47	0.0974150
$686$	−25932.8	−1.44332
$687$	0	0
$688$	−1035.89	−0.0574023
$689$	0	0
$690$	0	0
$691$	33792.7	1.86040	0.930199	−	0.367056i	$-0.119634\pi$
$691$	33792.7	1.86040	0.930199	+	0.367056i	$0.119634\pi$
$692$	−29998.7	−1.64795
$693$	0	0
$694$	−17438.9	−0.953849
$695$	928.192	0.0506595
$696$	0	0
$697$	−625.164	−0.0339738
$698$	15557.9	0.843662
$699$	0	0
$700$	14342.3	0.774413
$701$	−6905.96	−0.372089	−0.186045	−	0.982541i	$-0.559567\pi$
$701$	−6905.96	−0.372089	−0.186045	+	0.982541i	$0.559567\pi$
$702$	0	0
$703$	11178.2	0.599707
$704$	−32774.5	−1.75459
$705$	0	0
$706$	25488.3	1.35873
$707$	9281.37	0.493723
$708$	0	0
$709$	2007.13	0.106318	0.0531589	−	0.998586i	$-0.483071\pi$
$709$	2007.13	0.106318	0.0531589	+	0.998586i	$0.483071\pi$
$710$	−3585.70	−0.189534
$711$	0	0
$712$	−4447.71	−0.234108
$713$	608.445	0.0319585
$714$	0	0
$715$	0	0
$716$	8541.38	0.445819
$717$	0	0
$718$	−10172.9	−0.528759
$719$	12787.4	0.663270	0.331635	−	0.943408i	$-0.392400\pi$
$719$	12787.4	0.663270	0.331635	+	0.943408i	$0.392400\pi$
$720$	0	0
$721$	18164.1	0.938231
$722$	14760.7	0.760854
$723$	0	0
$724$	−8978.72	−0.460900
$725$	16477.5	0.844081
$726$	0	0
$727$	−6090.70	−0.310717	−0.155359	−	0.987858i	$-0.549653\pi$
$727$	−6090.70	−0.310717	−0.155359	+	0.987858i	$0.549653\pi$
$728$	0	0
$729$	0	0
$730$	5990.60	0.303729
$731$	−861.420	−0.0435852
$732$	0	0
$733$	38846.5	1.95747	0.978737	−	0.205117i	$-0.0657574\pi$
$733$	38846.5	1.95747	0.978737	+	0.205117i	$0.0657574\pi$
$734$	39681.6	1.99547
$735$	0	0
$736$	−833.427	−0.0417399
$737$	1967.17	0.0983198
$738$	0	0
$739$	−14457.5	−0.719661	−0.359830	−	0.933018i	$-0.617165\pi$
$739$	−14457.5	−0.719661	−0.359830	+	0.933018i	$0.617165\pi$
$740$	6678.29	0.331755
$741$	0	0
$742$	−26721.3	−1.32206
$743$	−1277.80	−0.0630929	−0.0315464	−	0.999502i	$-0.510043\pi$
$743$	−1277.80	−0.0630929	−0.0315464	+	0.999502i	$0.510043\pi$
$744$	0	0
$745$	−5083.56	−0.249996
$746$	−50013.7	−2.45460
$747$	0	0
$748$	−1017.50	−0.0497373
$749$	−18229.0	−0.889285
$750$	0	0
$751$	−13007.9	−0.632042	−0.316021	−	0.948752i	$-0.602347\pi$
$751$	−13007.9	−0.632042	−0.316021	+	0.948752i	$0.602347\pi$
$752$	626.706	0.0303904
$753$	0	0
$754$	0	0
$755$	1188.35	0.0572829
$756$	0	0
$757$	−10723.2	−0.514850	−0.257425	−	0.966298i	$-0.582874\pi$
$757$	−10723.2	−0.514850	−0.257425	+	0.966298i	$0.582874\pi$
$758$	63451.7	3.04046
$759$	0	0
$760$	−3706.44	−0.176904
$761$	13621.8	0.648870	0.324435	−	0.945908i	$-0.394826\pi$
$761$	13621.8	0.648870	0.324435	+	0.945908i	$0.394826\pi$
$762$	0	0
$763$	−8567.19	−0.406491
$764$	−16661.4	−0.788988
$765$	0	0
$766$	2256.84	0.106453
$767$	0	0
$768$	0	0
$769$	8495.15	0.398365	0.199183	−	0.979962i	$-0.436171\pi$
$769$	8495.15	0.398365	0.199183	+	0.979962i	$0.436171\pi$
$770$	4827.86	0.225953
$771$	0	0
$772$	6651.50	0.310094
$773$	34262.5	1.59423	0.797113	−	0.603830i	$-0.206359\pi$
$773$	34262.5	1.59423	0.797113	+	0.603830i	$0.206359\pi$
$774$	0	0
$775$	15946.0	0.739094
$776$	−25754.9	−1.19143
$777$	0	0
$778$	−18887.0	−0.870348
$779$	18673.5	0.858854
$780$	0	0
$781$	11037.1	0.505681
$782$	−41.0779	−0.00187844
$783$	0	0
$784$	609.644	0.0277717
$785$	3712.33	0.168788
$786$	0	0
$787$	12642.6	0.572629	0.286315	−	0.958136i	$-0.407570\pi$
$787$	12642.6	0.572629	0.286315	+	0.958136i	$0.407570\pi$
$788$	39973.6	1.80711
$789$	0	0
$790$	−480.776	−0.0216522
$791$	−3202.00	−0.143932
$792$	0	0
$793$	0	0
$794$	−8583.89	−0.383666
$795$	0	0
$796$	15843.9	0.705494
$797$	−19084.4	−0.848184	−0.424092	−	0.905619i	$-0.639407\pi$
$797$	−19084.4	−0.848184	−0.424092	+	0.905619i	$0.639407\pi$
$798$	0	0
$799$	521.154	0.0230752
$800$	−21842.3	−0.965304
$801$	0	0
$802$	−1923.90	−0.0847075
$803$	−18439.5	−0.810357
$804$	0	0
$805$	119.971	0.00525269
$806$	0	0
$807$	0	0
$808$	21288.2	0.926878
$809$	−11610.0	−0.504558	−0.252279	−	0.967655i	$-0.581180\pi$
$809$	−11610.0	−0.504558	−0.252279	+	0.967655i	$0.581180\pi$
$810$	0	0
$811$	−9613.36	−0.416240	−0.208120	−	0.978103i	$-0.566734\pi$
$811$	−9613.36	−0.416240	−0.208120	+	0.978103i	$0.566734\pi$
$812$	17229.6	0.744630
$813$	0	0
$814$	−33396.2	−1.43800
$815$	10125.9	0.435208
$816$	0	0
$817$	25730.4	1.10183
$818$	11632.9	0.497233
$819$	0	0
$820$	11156.3	0.475115
$821$	−26481.5	−1.12571	−0.562856	−	0.826555i	$-0.690297\pi$
$821$	−26481.5	−1.12571	−0.562856	+	0.826555i	$0.690297\pi$
$822$	0	0
$823$	13814.5	0.585107	0.292553	−	0.956249i	$-0.405495\pi$
$823$	13814.5	0.585107	0.292553	+	0.956249i	$0.405495\pi$
$824$	41662.0	1.76136
$825$	0	0
$826$	−22587.8	−0.951491
$827$	−44401.0	−1.86696	−0.933479	−	0.358633i	$-0.883243\pi$
$827$	−44401.0	−1.86696	−0.933479	+	0.358633i	$0.883243\pi$
$828$	0	0
$829$	24337.4	1.01963	0.509815	−	0.860284i	$-0.329714\pi$
$829$	24337.4	1.01963	0.509815	+	0.860284i	$0.329714\pi$
$830$	975.379	0.0407902
$831$	0	0
$832$	0	0
$833$	506.966	0.0210868
$834$	0	0
$835$	−9589.73	−0.397445
$836$	30392.5	1.25735
$837$	0	0
$838$	56494.0	2.32882
$839$	−24680.1	−1.01556	−0.507778	−	0.861488i	$-0.669533\pi$
$839$	−24680.1	−1.01556	−0.507778	+	0.861488i	$0.669533\pi$
$840$	0	0
$841$	−4594.43	−0.188381
$842$	47727.4	1.95344
$843$	0	0
$844$	−32425.0	−1.32241
$845$	0	0
$846$	0	0
$847$	−2134.10	−0.0865744
$848$	1484.65	0.0601214
$849$	0	0
$850$	−1076.56	−0.0434421
$851$	−829.885	−0.0334290
$852$	0	0
$853$	10151.7	0.407490	0.203745	−	0.979024i	$-0.434689\pi$
$853$	10151.7	0.407490	0.203745	+	0.979024i	$0.434689\pi$
$854$	−7035.65	−0.281914
$855$	0	0
$856$	−41811.1	−1.66948
$857$	−2028.92	−0.0808713	−0.0404357	−	0.999182i	$-0.512875\pi$
$857$	−2028.92	−0.0808713	−0.0404357	+	0.999182i	$0.512875\pi$
$858$	0	0
$859$	6655.76	0.264367	0.132184	−	0.991225i	$-0.457801\pi$
$859$	6655.76	0.264367	0.132184	+	0.991225i	$0.457801\pi$
$860$	15372.3	0.609526
$861$	0	0
$862$	18074.6	0.714182
$863$	45690.8	1.80224	0.901121	−	0.433568i	$-0.142746\pi$
$863$	45690.8	1.80224	0.901121	+	0.433568i	$0.142746\pi$
$864$	0	0
$865$	−6576.42	−0.258503
$866$	38290.3	1.50249
$867$	0	0
$868$	16673.9	0.652014
$869$	1479.87	0.0577688
$870$	0	0
$871$	0	0
$872$	−19650.1	−0.763117
$873$	0	0
$874$	1226.99	0.0474868
$875$	6500.00	0.251132
$876$	0	0
$877$	−30447.5	−1.17234	−0.586168	−	0.810189i	$-0.699364\pi$
$877$	−30447.5	−1.17234	−0.586168	+	0.810189i	$0.699364\pi$
$878$	−46411.6	−1.78396
$879$	0	0
$880$	−268.237	−0.0102753
$881$	−32542.0	−1.24446	−0.622230	−	0.782835i	$-0.713773\pi$
$881$	−32542.0	−1.24446	−0.622230	+	0.782835i	$0.713773\pi$
$882$	0	0
$883$	27641.9	1.05348	0.526741	−	0.850026i	$-0.323414\pi$
$883$	27641.9	1.05348	0.526741	+	0.850026i	$0.323414\pi$
$884$	0	0
$885$	0	0
$886$	−26825.3	−1.01717
$887$	−40099.9	−1.51795	−0.758976	−	0.651119i	$-0.774300\pi$
$887$	−40099.9	−1.51795	−0.758976	+	0.651119i	$0.774300\pi$
$888$	0	0
$889$	−5936.70	−0.223971
$890$	−2597.49	−0.0978293
$891$	0	0
$892$	15312.1	0.574761
$893$	−15566.8	−0.583339
$894$	0	0
$895$	1872.47	0.0699328
$896$	−21993.8	−0.820044
$897$	0	0
$898$	−48648.4	−1.80781
$899$	19156.1	0.710670
$900$	0	0
$901$	1234.60	0.0456497
$902$	−55789.3	−2.05940
$903$	0	0
$904$	−7344.26	−0.270206
$905$	−1968.35	−0.0722984
$906$	0	0
$907$	−36824.9	−1.34812	−0.674062	−	0.738674i	$-0.735452\pi$
$907$	−36824.9	−1.34812	−0.674062	+	0.738674i	$0.735452\pi$
$908$	−11132.4	−0.406874
$909$	0	0
$910$	0	0
$911$	−34520.5	−1.25545	−0.627725	−	0.778435i	$-0.716014\pi$
$911$	−34520.5	−1.25545	−0.627725	+	0.778435i	$0.716014\pi$
$912$	0	0
$913$	−3002.29	−0.108830
$914$	−67642.6	−2.44794
$915$	0	0
$916$	−59999.8	−2.16424
$917$	−12733.2	−0.458545
$918$	0	0
$919$	23522.8	0.844336	0.422168	−	0.906518i	$-0.361269\pi$
$919$	23522.8	0.844336	0.422168	+	0.906518i	$0.361269\pi$
$920$	275.171	0.00986101
$921$	0	0
$922$	44300.4	1.58238
$923$	0	0
$924$	0	0
$925$	−21749.5	−0.773102
$926$	51789.7	1.83792
$927$	0	0
$928$	−26239.4	−0.928180
$929$	−24563.2	−0.867482	−0.433741	−	0.901038i	$-0.642807\pi$
$929$	−24563.2	−0.867482	−0.433741	+	0.901038i	$0.642807\pi$
$930$	0	0
$931$	−15143.0	−0.533073
$932$	62360.9	2.19174
$933$	0	0
$934$	29428.0	1.03096
$935$	−223.060	−0.00780197
$936$	0	0
$937$	−12115.6	−0.422411	−0.211206	−	0.977442i	$-0.567739\pi$
$937$	−12115.6	−0.422411	−0.211206	+	0.977442i	$0.567739\pi$
$938$	2176.36	0.0757576
$939$	0	0
$940$	−9300.19	−0.322701
$941$	14898.3	0.516123	0.258062	−	0.966128i	$-0.416916\pi$
$941$	14898.3	0.516123	0.258062	+	0.966128i	$0.416916\pi$
$942$	0	0
$943$	−1386.35	−0.0478745
$944$	1254.99	0.0432695
$945$	0	0
$946$	−76872.7	−2.64201
$947$	7434.32	0.255103	0.127552	−	0.991832i	$-0.459288\pi$
$947$	7434.32	0.255103	0.127552	+	0.991832i	$0.459288\pi$
$948$	0	0
$949$	0	0
$950$	32156.7	1.09821
$951$	0	0
$952$	−422.564	−0.0143859
$953$	23528.6	0.799754	0.399877	−	0.916569i	$-0.369053\pi$
$953$	23528.6	0.799754	0.399877	+	0.916569i	$0.369053\pi$
$954$	0	0
$955$	−3652.56	−0.123764
$956$	61573.8	2.08309
$957$	0	0
$958$	43637.9	1.47169
$959$	−5947.43	−0.200263
$960$	0	0
$961$	−11252.7	−0.377723
$962$	0	0
$963$	0	0
$964$	−75257.9	−2.51441
$965$	1458.17	0.0486425
$966$	0	0
$967$	−23558.0	−0.783427	−0.391713	−	0.920087i	$-0.628118\pi$
$967$	−23558.0	−0.783427	−0.391713	+	0.920087i	$0.628118\pi$
$968$	−4894.88	−0.162528
$969$	0	0
$970$	−15041.0	−0.497875
$971$	262.338	0.00867026	0.00433513	−	0.999991i	$-0.498620\pi$
$971$	262.338	0.00867026	0.00433513	+	0.999991i	$0.498620\pi$
$972$	0	0
$973$	−3160.86	−0.104144
$974$	22429.7	0.737878
$975$	0	0
$976$	390.903	0.0128202
$977$	33144.4	1.08535	0.542673	−	0.839944i	$-0.317412\pi$
$977$	33144.4	1.08535	0.542673	+	0.839944i	$0.317412\pi$
$978$	0	0
$979$	7995.28	0.261011
$980$	−9047.00	−0.294894
$981$	0	0
$982$	13460.3	0.437410
$983$	−4866.80	−0.157911	−0.0789557	−	0.996878i	$-0.525159\pi$
$983$	−4866.80	−0.157911	−0.0789557	+	0.996878i	$0.525159\pi$
$984$	0	0
$985$	8763.17	0.283470
$986$	−1293.28	−0.0417714
$987$	0	0
$988$	0	0
$989$	−1910.26	−0.0614184
$990$	0	0
$991$	12533.9	0.401770	0.200885	−	0.979615i	$-0.435618\pi$
$991$	12533.9	0.401770	0.200885	+	0.979615i	$0.435618\pi$
$992$	−25393.1	−0.812733
$993$	0	0
$994$	12210.7	0.389639
$995$	3473.37	0.110666
$996$	0	0
$997$	−3560.92	−0.113115	−0.0565574	−	0.998399i	$-0.518012\pi$
$997$	−3560.92	−0.113115	−0.0565574	+	0.998399i	$0.518012\pi$
$998$	61262.1	1.94310
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.l.1.1		2
3.2	odd	2		169.4.a.j.1.2		2
13.4	even	6		117.4.g.d.55.1		4
13.10	even	6		117.4.g.d.100.1		4
13.12	even	2		1521.4.a.t.1.2		2
39.2	even	12		169.4.e.g.147.4		8
39.5	even	4		169.4.b.e.168.1		4
39.8	even	4		169.4.b.e.168.4		4
39.11	even	12		169.4.e.g.147.1		8
39.17	odd	6		13.4.c.b.3.2	✓	4
39.20	even	12		169.4.e.g.23.4		8
39.23	odd	6		13.4.c.b.9.2	yes	4
39.29	odd	6		169.4.c.f.22.1		4
39.32	even	12		169.4.e.g.23.1		8
39.35	odd	6		169.4.c.f.146.1		4
39.38	odd	2		169.4.a.f.1.1		2
156.23	even	6		208.4.i.e.113.2		4
156.95	even	6		208.4.i.e.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.2	✓	4	39.17	odd	6
13.4.c.b.9.2	yes	4	39.23	odd	6
117.4.g.d.55.1		4	13.4	even	6
117.4.g.d.100.1		4	13.10	even	6
169.4.a.f.1.1		2	39.38	odd	2
169.4.a.j.1.2		2	3.2	odd	2
169.4.b.e.168.1		4	39.5	even	4
169.4.b.e.168.4		4	39.8	even	4
169.4.c.f.22.1		4	39.29	odd	6
169.4.c.f.146.1		4	39.35	odd	6
169.4.e.g.23.1		8	39.32	even	12
169.4.e.g.23.4		8	39.20	even	12
169.4.e.g.147.1		8	39.11	even	12
169.4.e.g.147.4		8	39.2	even	12
208.4.i.e.81.2		4	156.95	even	6
208.4.i.e.113.2		4	156.23	even	6
1521.4.a.l.1.1		2	1.1	even	1	trivial
1521.4.a.t.1.2		2	13.12	even	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)

\(f(q)\)	\(=\)	\(q-4.56155 q^{2} +12.8078 q^{4} +2.80776 q^{5} -9.56155 q^{7} -21.9309 q^{8} -12.8078 q^{10} +39.4233 q^{11} +43.6155 q^{14} -2.42329 q^{16} -2.01515 q^{17} +60.1922 q^{19} +35.9612 q^{20} -179.831 q^{22} -4.46876 q^{23} -117.116 q^{25} -122.462 q^{28} -140.693 q^{29} -136.155 q^{31} +186.501 q^{32} +9.19224 q^{34} -26.8466 q^{35} +185.708 q^{37} -274.570 q^{38} -61.5767 q^{40} +310.231 q^{41} +427.471 q^{43} +504.924 q^{44} +20.3845 q^{46} -258.617 q^{47} -251.577 q^{49} +534.233 q^{50} -612.656 q^{53} +110.691 q^{55} +209.693 q^{56} +641.779 q^{58} -517.885 q^{59} -161.311 q^{61} +621.080 q^{62} -831.348 q^{64} +49.8987 q^{67} -25.8096 q^{68} +122.462 q^{70} +279.963 q^{71} -467.732 q^{73} -847.118 q^{74} +770.928 q^{76} -376.948 q^{77} +37.5379 q^{79} -6.80403 q^{80} -1415.14 q^{82} -76.1553 q^{83} -5.65808 q^{85} -1949.93 q^{86} -864.587 q^{88} +202.806 q^{89} -57.2348 q^{92} +1179.70 q^{94} +169.006 q^{95} +1174.37 q^{97} +1147.58 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} + 5 q^{4} - 15 q^{5} - 15 q^{7} - 15 q^{8} - 5 q^{10} + 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} + 141 q^{19} + 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} - 80 q^{28} - 34 q^{29}+ \cdots + 1285 q^{98}+O(q^{100}) \) 2 * q - 5 * q^2 + 5 * q^4 - 15 * q^5 - 15 * q^7 - 15 * q^8 - 5 * q^10 + 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 + 141 * q^19 + 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 - 80 * q^28 - 34 * q^29 + 140 * q^31 + 105 * q^32 + 39 * q^34 + 70 * q^35 + 190 * q^37 - 310 * q^38 - 185 * q^40 + 538 * q^41 + 455 * q^43 + 680 * q^44 + 82 * q^46 + 60 * q^47 - 565 * q^49 + 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 + 595 * q^58 - 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 + 475 * q^67 + 505 * q^68 + 80 * q^70 + 127 * q^71 - 585 * q^73 - 849 * q^74 + 140 * q^76 - 255 * q^77 + 240 * q^79 - 1065 * q^80 - 1515 * q^82 + 260 * q^83 + 1205 * q^85 - 1962 * q^86 - 1020 * q^88 + 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 + 415 * q^97 + 1285 * q^98

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