LMFDB - Embedded newform 387.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,4,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	387.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:	$22.8337391722$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.17112$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.17112 q^{2} +9.39827 q^{4} +7.54340 q^{5} +4.58222 q^{7} -5.83236 q^{8} +O(q^{10})$	\(q-4.17112 q^{2} +9.39827 q^{4} +7.54340 q^{5} +4.58222 q^{7} -5.83236 q^{8} -31.4645 q^{10} -26.9150 q^{11} -15.6529 q^{13} -19.1130 q^{14} -50.8587 q^{16} -27.2420 q^{17} +38.3104 q^{19} +70.8949 q^{20} +112.266 q^{22} -82.5575 q^{23} -68.0971 q^{25} +65.2903 q^{26} +43.0650 q^{28} +34.2852 q^{29} +119.055 q^{31} +258.797 q^{32} +113.630 q^{34} +34.5655 q^{35} +378.527 q^{37} -159.797 q^{38} -43.9958 q^{40} -385.478 q^{41} -43.0000 q^{43} -252.955 q^{44} +344.358 q^{46} -271.022 q^{47} -322.003 q^{49} +284.041 q^{50} -147.110 q^{52} +329.363 q^{53} -203.031 q^{55} -26.7252 q^{56} -143.008 q^{58} +173.956 q^{59} +54.5012 q^{61} -496.592 q^{62} -672.604 q^{64} -118.076 q^{65} -906.954 q^{67} -256.028 q^{68} -144.177 q^{70} +621.376 q^{71} -1025.87 q^{73} -1578.88 q^{74} +360.052 q^{76} -123.331 q^{77} -737.945 q^{79} -383.647 q^{80} +1607.88 q^{82} -558.465 q^{83} -205.498 q^{85} +179.358 q^{86} +156.978 q^{88} -1631.31 q^{89} -71.7252 q^{91} -775.898 q^{92} +1130.47 q^{94} +288.991 q^{95} -406.607 q^{97} +1343.12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * 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 $

$2$	−4.17112	−1.47471	−0.737357	−	0.675503i	$-0.763927\pi$
$2$	−4.17112	−1.47471	−0.737357	+	0.675503i	$0.763927\pi$
$3$	0	0
$3$	0	0
$4$	9.39827	1.17478
$5$	7.54340	0.674702	0.337351	−	0.941379i	$-0.390469\pi$
$5$	7.54340	0.674702	0.337351	+	0.941379i	$0.390469\pi$
$6$	0	0
$7$	4.58222	0.247417	0.123708	−	0.992319i	$-0.460521\pi$
$7$	4.58222	0.247417	0.123708	+	0.992319i	$0.460521\pi$
$8$	−5.83236	−0.257756
$9$	0	0
$10$	−31.4645	−0.994994
$11$	−26.9150	−0.737744	−0.368872	−	0.929480i	$-0.620256\pi$
$11$	−26.9150	−0.737744	−0.368872	+	0.929480i	$0.620256\pi$
$12$	0	0
$13$	−15.6529	−0.333949	−0.166975	−	0.985961i	$-0.553400\pi$
$13$	−15.6529	−0.333949	−0.166975	+	0.985961i	$0.553400\pi$
$14$	−19.1130	−0.364869
$15$	0	0
$16$	−50.8587	−0.794667
$17$	−27.2420	−0.388657	−0.194328	−	0.980937i	$-0.562253\pi$
$17$	−27.2420	−0.388657	−0.194328	+	0.980937i	$0.562253\pi$
$18$	0	0
$19$	38.3104	0.462580	0.231290	−	0.972885i	$-0.425705\pi$
$19$	38.3104	0.462580	0.231290	+	0.972885i	$0.425705\pi$
$20$	70.8949	0.792629
$21$	0	0
$22$	112.266	1.08796
$23$	−82.5575	−0.748453	−0.374227	−	0.927337i	$-0.622092\pi$
$23$	−82.5575	−0.748453	−0.374227	+	0.927337i	$0.622092\pi$
$24$	0	0
$25$	−68.0971	−0.544777
$26$	65.2903	0.492480
$27$	0	0
$28$	43.0650	0.290661
$29$	34.2852	0.219538	0.109769	−	0.993957i	$-0.464989\pi$
$29$	34.2852	0.219538	0.109769	+	0.993957i	$0.464989\pi$
$30$	0	0
$31$	119.055	0.689770	0.344885	−	0.938645i	$-0.387918\pi$
$31$	119.055	0.689770	0.344885	+	0.938645i	$0.387918\pi$
$32$	258.797	1.42966
$33$	0	0
$34$	113.630	0.573158
$35$	34.5655	0.166933
$36$	0	0
$37$	378.527	1.68188	0.840939	−	0.541129i	$-0.182003\pi$
$37$	378.527	1.68188	0.840939	+	0.541129i	$0.182003\pi$
$38$	−159.797	−0.682173
$39$	0	0
$40$	−43.9958	−0.173909
$41$	−385.478	−1.46833	−0.734166	−	0.678970i	$-0.762427\pi$
$41$	−385.478	−1.46833	−0.734166	+	0.678970i	$0.762427\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−252.955	−0.866689
$45$	0	0
$46$	344.358	1.10376
$47$	−271.022	−0.841120	−0.420560	−	0.907265i	$-0.638166\pi$
$47$	−271.022	−0.841120	−0.420560	+	0.907265i	$0.638166\pi$
$48$	0	0
$49$	−322.003	−0.938785
$50$	284.041	0.803390
$51$	0	0
$52$	−147.110	−0.392318
$53$	329.363	0.853612	0.426806	−	0.904343i	$-0.359639\pi$
$53$	329.363	0.853612	0.426806	+	0.904343i	$0.359639\pi$
$54$	0	0
$55$	−203.031	−0.497757
$56$	−26.7252	−0.0637732
$57$	0	0
$58$	−143.008	−0.323756
$59$	173.956	0.383849	0.191924	−	0.981410i	$-0.438527\pi$
$59$	173.956	0.383849	0.191924	+	0.981410i	$0.438527\pi$
$60$	0	0
$61$	54.5012	0.114396	0.0571981	−	0.998363i	$-0.481783\pi$
$61$	54.5012	0.114396	0.0571981	+	0.998363i	$0.481783\pi$
$62$	−496.592	−1.01721
$63$	0	0
$64$	−672.604	−1.31368
$65$	−118.076	−0.225316
$66$	0	0
$67$	−906.954	−1.65376	−0.826881	−	0.562377i	$-0.809887\pi$
$67$	−906.954	−1.65376	−0.826881	+	0.562377i	$0.809887\pi$
$68$	−256.028	−0.456588
$69$	0	0
$70$	−144.177	−0.246178
$71$	621.376	1.03864	0.519322	−	0.854579i	$-0.326184\pi$
$71$	621.376	1.03864	0.519322	+	0.854579i	$0.326184\pi$
$72$	0	0
$73$	−1025.87	−1.64477	−0.822387	−	0.568928i	$-0.807358\pi$
$73$	−1025.87	−1.64477	−0.822387	+	0.568928i	$0.807358\pi$
$74$	−1578.88	−2.48029
$75$	0	0
$76$	360.052	0.543431
$77$	−123.331	−0.182530
$78$	0	0
$79$	−737.945	−1.05095	−0.525476	−	0.850808i	$-0.676113\pi$
$79$	−737.945	−1.05095	−0.525476	+	0.850808i	$0.676113\pi$
$80$	−383.647	−0.536164
$81$	0	0
$82$	1607.88	2.16537
$83$	−558.465	−0.738548	−0.369274	−	0.929321i	$-0.620394\pi$
$83$	−558.465	−0.738548	−0.369274	+	0.929321i	$0.620394\pi$
$84$	0	0
$85$	−205.498	−0.262228
$86$	179.358	0.224892
$87$	0	0
$88$	156.978	0.190158
$89$	−1631.31	−1.94291	−0.971453	−	0.237231i	$-0.923760\pi$
$89$	−1631.31	−1.94291	−0.971453	+	0.237231i	$0.923760\pi$
$90$	0	0
$91$	−71.7252	−0.0826247
$92$	−775.898	−0.879271
$93$	0	0
$94$	1130.47	1.24041
$95$	288.991	0.312104
$96$	0	0
$97$	−406.607	−0.425616	−0.212808	−	0.977094i	$-0.568261\pi$
$97$	−406.607	−0.425616	−0.212808	+	0.977094i	$0.568261\pi$
$98$	1343.12	1.38444
$99$	0	0
$100$	−639.995	−0.639995
$101$	−1000.43	−0.985606	−0.492803	−	0.870141i	$-0.664028\pi$
$101$	−1000.43	−0.985606	−0.492803	+	0.870141i	$0.664028\pi$
$102$	0	0
$103$	−1659.81	−1.58783	−0.793913	−	0.608031i	$-0.791960\pi$
$103$	−1659.81	−1.58783	−0.793913	+	0.608031i	$0.791960\pi$
$104$	91.2935	0.0860775
$105$	0	0
$106$	−1373.81	−1.25883
$107$	151.590	0.136961	0.0684803	−	0.997652i	$-0.478185\pi$
$107$	151.590	0.136961	0.0684803	+	0.997652i	$0.478185\pi$
$108$	0	0
$109$	1092.76	0.960254	0.480127	−	0.877199i	$-0.340591\pi$
$109$	1092.76	0.960254	0.480127	+	0.877199i	$0.340591\pi$
$110$	846.866	0.734050
$111$	0	0
$112$	−233.046	−0.196614
$113$	970.442	0.807889	0.403945	−	0.914783i	$-0.367639\pi$
$113$	970.442	0.807889	0.403945	+	0.914783i	$0.367639\pi$
$114$	0	0
$115$	−622.764	−0.504983
$116$	322.222	0.257910
$117$	0	0
$118$	−725.590	−0.566068
$119$	−124.829	−0.0961602
$120$	0	0
$121$	−606.582	−0.455734
$122$	−227.331	−0.168702
$123$	0	0
$124$	1118.91	0.810331
$125$	−1456.61	−1.04226
$126$	0	0
$127$	2115.78	1.47831	0.739155	−	0.673535i	$-0.235225\pi$
$127$	2115.78	1.47831	0.739155	+	0.673535i	$0.235225\pi$
$128$	735.139	0.507638
$129$	0	0
$130$	492.511	0.332277
$131$	1695.44	1.13077	0.565386	−	0.824826i	$-0.308727\pi$
$131$	1695.44	1.13077	0.565386	+	0.824826i	$0.308727\pi$
$132$	0	0
$133$	175.547	0.114450
$134$	3783.02	2.43883
$135$	0	0
$136$	158.885	0.100179
$137$	−1613.51	−1.00622	−0.503108	−	0.864224i	$-0.667810\pi$
$137$	−1613.51	−1.00622	−0.503108	+	0.864224i	$0.667810\pi$
$138$	0	0
$139$	2072.45	1.26463	0.632313	−	0.774713i	$-0.282106\pi$
$139$	2072.45	1.26463	0.632313	+	0.774713i	$0.282106\pi$
$140$	324.856	0.196110
$141$	0	0
$142$	−2591.84	−1.53170
$143$	421.299	0.246369
$144$	0	0
$145$	258.627	0.148123
$146$	4279.01	2.42557
$147$	0	0
$148$	3557.50	1.97584
$149$	811.396	0.446122	0.223061	−	0.974805i	$-0.428395\pi$
$149$	811.396	0.446122	0.223061	+	0.974805i	$0.428395\pi$
$150$	0	0
$151$	−944.326	−0.508928	−0.254464	−	0.967082i	$-0.581899\pi$
$151$	−944.326	−0.508928	−0.254464	+	0.967082i	$0.581899\pi$
$152$	−223.440	−0.119233
$153$	0	0
$154$	514.427	0.269180
$155$	898.078	0.465390
$156$	0	0
$157$	2138.58	1.08712	0.543558	−	0.839372i	$-0.317077\pi$
$157$	2138.58	1.08712	0.543558	+	0.839372i	$0.317077\pi$
$158$	3078.06	1.54986
$159$	0	0
$160$	1952.21	0.964597
$161$	−378.297	−0.185180
$162$	0	0
$163$	−2184.57	−1.04975	−0.524873	−	0.851180i	$-0.675887\pi$
$163$	−2184.57	−1.04975	−0.524873	+	0.851180i	$0.675887\pi$
$164$	−3622.83	−1.72497
$165$	0	0
$166$	2329.43	1.08915
$167$	−3334.42	−1.54506	−0.772531	−	0.634978i	$-0.781009\pi$
$167$	−3334.42	−1.54506	−0.772531	+	0.634978i	$0.781009\pi$
$168$	0	0
$169$	−1951.99	−0.888478
$170$	857.156	0.386711
$171$	0	0
$172$	−404.126	−0.179153
$173$	−1169.82	−0.514105	−0.257052	−	0.966397i	$-0.582751\pi$
$173$	−1169.82	−0.514105	−0.257052	+	0.966397i	$0.582751\pi$
$174$	0	0
$175$	−312.036	−0.134787
$176$	1368.86	0.586260
$177$	0	0
$178$	6804.40	2.86523
$179$	−206.597	−0.0862669	−0.0431334	−	0.999069i	$-0.513734\pi$
$179$	−206.597	−0.0862669	−0.0431334	+	0.999069i	$0.513734\pi$
$180$	0	0
$181$	652.242	0.267849	0.133925	−	0.990992i	$-0.457242\pi$
$181$	652.242	0.267849	0.133925	+	0.990992i	$0.457242\pi$
$182$	299.175	0.121848
$183$	0	0
$184$	481.505	0.192919
$185$	2855.38	1.13477
$186$	0	0
$187$	733.220	0.286729
$188$	−2547.14	−0.988134
$189$	0	0
$190$	−1205.42	−0.460264
$191$	1128.21	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$191$	1128.21	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$192$	0	0
$193$	355.606	0.132627	0.0663136	−	0.997799i	$-0.478876\pi$
$193$	355.606	0.132627	0.0663136	+	0.997799i	$0.478876\pi$
$194$	1696.01	0.627662
$195$	0	0
$196$	−3026.27	−1.10287
$197$	−4347.00	−1.57214	−0.786068	−	0.618140i	$-0.787886\pi$
$197$	−4347.00	−1.57214	−0.786068	+	0.618140i	$0.787886\pi$
$198$	0	0
$199$	430.375	0.153309	0.0766545	−	0.997058i	$-0.475576\pi$
$199$	430.375	0.153309	0.0766545	+	0.997058i	$0.475576\pi$
$200$	397.167	0.140420
$201$	0	0
$202$	4172.91	1.45349
$203$	157.102	0.0543174
$204$	0	0
$205$	−2907.82	−0.990687
$206$	6923.28	2.34159
$207$	0	0
$208$	796.087	0.265378
$209$	−1031.13	−0.341265
$210$	0	0
$211$	103.702	0.0338347	0.0169174	−	0.999857i	$-0.494615\pi$
$211$	103.702	0.0338347	0.0169174	+	0.999857i	$0.494615\pi$
$212$	3095.44	1.00281
$213$	0	0
$214$	−632.302	−0.201978
$215$	−324.366	−0.102891
$216$	0	0
$217$	545.535	0.170661
$218$	−4558.05	−1.41610
$219$	0	0
$220$	−1908.14	−0.584757
$221$	426.418	0.129792
$222$	0	0
$223$	4447.94	1.33568	0.667839	−	0.744306i	$-0.267219\pi$
$223$	4447.94	1.33568	0.667839	+	0.744306i	$0.267219\pi$
$224$	1185.86	0.353723
$225$	0	0
$226$	−4047.83	−1.19141
$227$	−509.808	−0.149062	−0.0745311	−	0.997219i	$-0.523746\pi$
$227$	−509.808	−0.149062	−0.0745311	+	0.997219i	$0.523746\pi$
$228$	0	0
$229$	−4754.86	−1.37210	−0.686049	−	0.727556i	$-0.740656\pi$
$229$	−4754.86	−1.37210	−0.686049	+	0.727556i	$0.740656\pi$
$230$	2597.63	0.744706
$231$	0	0
$232$	−199.964	−0.0565873
$233$	−893.223	−0.251146	−0.125573	−	0.992084i	$-0.540077\pi$
$233$	−893.223	−0.251146	−0.125573	+	0.992084i	$0.540077\pi$
$234$	0	0
$235$	−2044.43	−0.567506
$236$	1634.88	0.450939
$237$	0	0
$238$	520.677	0.141809
$239$	3883.75	1.05112	0.525562	−	0.850755i	$-0.323855\pi$
$239$	3883.75	1.05112	0.525562	+	0.850755i	$0.323855\pi$
$240$	0	0
$241$	−3118.83	−0.833616	−0.416808	−	0.908995i	$-0.636851\pi$
$241$	−3118.83	−0.833616	−0.416808	+	0.908995i	$0.636851\pi$
$242$	2530.13	0.672078
$243$	0	0
$244$	512.217	0.134391
$245$	−2429.00	−0.633400
$246$	0	0
$247$	−599.670	−0.154478
$248$	−694.370	−0.177793
$249$	0	0
$250$	6075.70	1.53704
$251$	−5196.19	−1.30669	−0.653347	−	0.757058i	$-0.726636\pi$
$251$	−5196.19	−1.30669	−0.653347	+	0.757058i	$0.726636\pi$
$252$	0	0
$253$	2222.04	0.552167
$254$	−8825.19	−2.18009
$255$	0	0
$256$	2314.47	0.565057
$257$	2441.32	0.592550	0.296275	−	0.955103i	$-0.404256\pi$
$257$	2441.32	0.592550	0.296275	+	0.955103i	$0.404256\pi$
$258$	0	0
$259$	1734.50	0.416125
$260$	−1109.71	−0.264698
$261$	0	0
$262$	−7071.89	−1.66757
$263$	−5831.01	−1.36713	−0.683566	−	0.729889i	$-0.739572\pi$
$263$	−5831.01	−1.36713	−0.683566	+	0.729889i	$0.739572\pi$
$264$	0	0
$265$	2484.52	0.575934
$266$	−732.227	−0.168781
$267$	0	0
$268$	−8523.80	−1.94281
$269$	2605.41	0.590538	0.295269	−	0.955414i	$-0.404591\pi$
$269$	2605.41	0.590538	0.295269	+	0.955414i	$0.404591\pi$
$270$	0	0
$271$	1793.84	0.402097	0.201049	−	0.979581i	$-0.435565\pi$
$271$	1793.84	0.402097	0.201049	+	0.979581i	$0.435565\pi$
$272$	1385.49	0.308853
$273$	0	0
$274$	6730.16	1.48388
$275$	1832.83	0.401906
$276$	0	0
$277$	−825.071	−0.178967	−0.0894833	−	0.995988i	$-0.528522\pi$
$277$	−825.071	−0.178967	−0.0894833	+	0.995988i	$0.528522\pi$
$278$	−8644.45	−1.86496
$279$	0	0
$280$	−201.599	−0.0430279
$281$	5114.69	1.08582	0.542912	−	0.839789i	$-0.317322\pi$
$281$	5114.69	1.08582	0.542912	+	0.839789i	$0.317322\pi$
$282$	0	0
$283$	−3703.40	−0.777895	−0.388947	−	0.921260i	$-0.627161\pi$
$283$	−3703.40	−0.777895	−0.388947	+	0.921260i	$0.627161\pi$
$284$	5839.86	1.22018
$285$	0	0
$286$	−1757.29	−0.363324
$287$	−1766.35	−0.363290
$288$	0	0
$289$	−4170.87	−0.848946
$290$	−1078.77	−0.218439
$291$	0	0
$292$	−9641.37	−1.93225
$293$	−749.193	−0.149380	−0.0746899	−	0.997207i	$-0.523797\pi$
$293$	−749.193	−0.149380	−0.0746899	+	0.997207i	$0.523797\pi$
$294$	0	0
$295$	1312.22	0.258984
$296$	−2207.71	−0.433515
$297$	0	0
$298$	−3384.43	−0.657902
$299$	1292.27	0.249946
$300$	0	0
$301$	−197.036	−0.0377307
$302$	3938.90	0.750524
$303$	0	0
$304$	−1948.42	−0.367597
$305$	411.125	0.0771834
$306$	0	0
$307$	−4761.00	−0.885097	−0.442548	−	0.896745i	$-0.645925\pi$
$307$	−4761.00	−0.885097	−0.442548	+	0.896745i	$0.645925\pi$
$308$	−1159.09	−0.214433
$309$	0	0
$310$	−3745.99	−0.686317
$311$	5995.99	1.09325	0.546626	−	0.837377i	$-0.315912\pi$
$311$	5995.99	1.09325	0.546626	+	0.837377i	$0.315912\pi$
$312$	0	0
$313$	398.261	0.0719203	0.0359602	−	0.999353i	$-0.488551\pi$
$313$	398.261	0.0719203	0.0359602	+	0.999353i	$0.488551\pi$
$314$	−8920.27	−1.60318
$315$	0	0
$316$	−6935.40	−1.23464
$317$	−4657.19	−0.825154	−0.412577	−	0.910923i	$-0.635371\pi$
$317$	−4657.19	−0.825154	−0.412577	+	0.910923i	$0.635371\pi$
$318$	0	0
$319$	−922.786	−0.161963
$320$	−5073.72	−0.886342
$321$	0	0
$322$	1577.92	0.273088
$323$	−1043.65	−0.179785
$324$	0	0
$325$	1065.92	0.181928
$326$	9112.11	1.54808
$327$	0	0
$328$	2248.25	0.378472
$329$	−1241.88	−0.208107
$330$	0	0
$331$	10013.6	1.66282	0.831412	−	0.555656i	$-0.187533\pi$
$331$	10013.6	1.66282	0.831412	+	0.555656i	$0.187533\pi$
$332$	−5248.60	−0.867634
$333$	0	0
$334$	13908.3	2.27852
$335$	−6841.52	−1.11580
$336$	0	0
$337$	−3872.11	−0.625897	−0.312949	−	0.949770i	$-0.601317\pi$
$337$	−3872.11	−0.625897	−0.312949	+	0.949770i	$0.601317\pi$
$338$	8141.97	1.31025
$339$	0	0
$340$	−1931.32	−0.308061
$341$	−3204.36	−0.508874
$342$	0	0
$343$	−3047.19	−0.479688
$344$	250.791	0.0393075
$345$	0	0
$346$	4879.48	0.758158
$347$	−10290.8	−1.59204	−0.796022	−	0.605267i	$-0.793066\pi$
$347$	−10290.8	−1.59204	−0.796022	+	0.605267i	$0.793066\pi$
$348$	0	0
$349$	700.572	0.107452	0.0537260	−	0.998556i	$-0.482890\pi$
$349$	700.572	0.107452	0.0537260	+	0.998556i	$0.482890\pi$
$350$	1301.54	0.198772
$351$	0	0
$352$	−6965.52	−1.05473
$353$	−3608.33	−0.544057	−0.272028	−	0.962289i	$-0.587694\pi$
$353$	−3608.33	−0.544057	−0.272028	+	0.962289i	$0.587694\pi$
$354$	0	0
$355$	4687.29	0.700776
$356$	−15331.5	−2.28250
$357$	0	0
$358$	861.741	0.127219
$359$	12085.9	1.77680	0.888399	−	0.459072i	$-0.151818\pi$
$359$	12085.9	1.77680	0.888399	+	0.459072i	$0.151818\pi$
$360$	0	0
$361$	−5391.31	−0.786020
$362$	−2720.58	−0.395001
$363$	0	0
$364$	−674.093	−0.0970661
$365$	−7738.52	−1.10973
$366$	0	0
$367$	968.974	0.137820	0.0689101	−	0.997623i	$-0.478048\pi$
$367$	968.974	0.137820	0.0689101	+	0.997623i	$0.478048\pi$
$368$	4198.77	0.594771
$369$	0	0
$370$	−11910.2	−1.67346
$371$	1509.21	0.211198
$372$	0	0
$373$	13006.0	1.80543	0.902713	−	0.430243i	$-0.141572\pi$
$373$	13006.0	1.80543	0.902713	+	0.430243i	$0.141572\pi$
$374$	−3058.35	−0.422844
$375$	0	0
$376$	1580.70	0.216804
$377$	−536.664	−0.0733146
$378$	0	0
$379$	1901.08	0.257657	0.128828	−	0.991667i	$-0.458878\pi$
$379$	1901.08	0.257657	0.128828	+	0.991667i	$0.458878\pi$
$380$	2716.01	0.366654
$381$	0	0
$382$	−4705.89	−0.630299
$383$	9501.65	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$383$	9501.65	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$384$	0	0
$385$	−930.332	−0.123154
$386$	−1483.27	−0.195587
$387$	0	0
$388$	−3821.40	−0.500006
$389$	−4640.26	−0.604808	−0.302404	−	0.953180i	$-0.597789\pi$
$389$	−4640.26	−0.604808	−0.302404	+	0.953180i	$0.597789\pi$
$390$	0	0
$391$	2249.03	0.290891
$392$	1878.04	0.241978
$393$	0	0
$394$	18131.9	2.31845
$395$	−5566.61	−0.709080
$396$	0	0
$397$	4633.90	0.585815	0.292908	−	0.956141i	$-0.405377\pi$
$397$	4633.90	0.585815	0.292908	+	0.956141i	$0.405377\pi$
$398$	−1795.15	−0.226087
$399$	0	0
$400$	3463.33	0.432916
$401$	11801.3	1.46965	0.734823	−	0.678258i	$-0.237265\pi$
$401$	11801.3	1.46965	0.734823	+	0.678258i	$0.237265\pi$
$402$	0	0
$403$	−1863.56	−0.230348
$404$	−9402.29	−1.15787
$405$	0	0
$406$	−655.293	−0.0801026
$407$	−10188.1	−1.24080
$408$	0	0
$409$	5588.72	0.675658	0.337829	−	0.941207i	$-0.390307\pi$
$409$	5588.72	0.675658	0.337829	+	0.941207i	$0.390307\pi$
$410$	12128.9	1.46098
$411$	0	0
$412$	−15599.4	−1.86535
$413$	797.103	0.0949706
$414$	0	0
$415$	−4212.72	−0.498300
$416$	−4050.93	−0.477435
$417$	0	0
$418$	4300.95	0.503269
$419$	6908.94	0.805547	0.402773	−	0.915300i	$-0.368046\pi$
$419$	6908.94	0.805547	0.402773	+	0.915300i	$0.368046\pi$
$420$	0	0
$421$	16499.6	1.91007	0.955036	−	0.296491i	$-0.0958165\pi$
$421$	16499.6	1.91007	0.955036	+	0.296491i	$0.0958165\pi$
$422$	−432.553	−0.0498966
$423$	0	0
$424$	−1920.96	−0.220024
$425$	1855.10	0.211731
$426$	0	0
$427$	249.737	0.0283035
$428$	1424.69	0.160899
$429$	0	0
$430$	1352.97	0.151735
$431$	7101.67	0.793679	0.396839	−	0.917888i	$-0.370107\pi$
$431$	7101.67	0.793679	0.396839	+	0.917888i	$0.370107\pi$
$432$	0	0
$433$	−8935.72	−0.991739	−0.495870	−	0.868397i	$-0.665151\pi$
$433$	−8935.72	−0.991739	−0.495870	+	0.868397i	$0.665151\pi$
$434$	−2275.50	−0.251676
$435$	0	0
$436$	10270.1	1.12809
$437$	−3162.81	−0.346219
$438$	0	0
$439$	17789.3	1.93403	0.967015	−	0.254720i	$-0.0819833\pi$
$439$	17789.3	1.93403	0.967015	+	0.254720i	$0.0819833\pi$
$440$	1184.15	0.128300
$441$	0	0
$442$	−1778.64	−0.191406
$443$	−2744.62	−0.294359	−0.147180	−	0.989110i	$-0.547020\pi$
$443$	−2744.62	−0.294359	−0.147180	+	0.989110i	$0.547020\pi$
$444$	0	0
$445$	−12305.6	−1.31088
$446$	−18552.9	−1.96974
$447$	0	0
$448$	−3082.02	−0.325026
$449$	12051.9	1.26673	0.633366	−	0.773853i	$-0.281673\pi$
$449$	12051.9	1.26673	0.633366	+	0.773853i	$0.281673\pi$
$450$	0	0
$451$	10375.2	1.08325
$452$	9120.47	0.949095
$453$	0	0
$454$	2126.47	0.219824
$455$	−541.052	−0.0557471
$456$	0	0
$457$	130.425	0.0133502	0.00667509	−	0.999978i	$-0.497875\pi$
$457$	130.425	0.0133502	0.00667509	+	0.999978i	$0.497875\pi$
$458$	19833.1	2.02345
$459$	0	0
$460$	−5852.91	−0.593246
$461$	981.122	0.0991223	0.0495612	−	0.998771i	$-0.484218\pi$
$461$	981.122	0.0991223	0.0495612	+	0.998771i	$0.484218\pi$
$462$	0	0
$463$	6359.89	0.638378	0.319189	−	0.947691i	$-0.396589\pi$
$463$	6359.89	0.638378	0.319189	+	0.947691i	$0.396589\pi$
$464$	−1743.70	−0.174460
$465$	0	0
$466$	3725.75	0.370369
$467$	14159.9	1.40309	0.701546	−	0.712624i	$-0.252494\pi$
$467$	14159.9	1.40309	0.701546	+	0.712624i	$0.252494\pi$
$468$	0	0
$469$	−4155.86	−0.409168
$470$	8527.57	0.836909
$471$	0	0
$472$	−1014.57	−0.0989395
$473$	1157.35	0.112505
$474$	0	0
$475$	−2608.83	−0.252003
$476$	−1173.18	−0.112967
$477$	0	0
$478$	−16199.6	−1.55011
$479$	20583.3	1.96342	0.981709	−	0.190390i	$-0.0609751\pi$
$479$	20583.3	1.96342	0.981709	+	0.190390i	$0.0609751\pi$
$480$	0	0
$481$	−5925.06	−0.561662
$482$	13009.0	1.22935
$483$	0	0
$484$	−5700.82	−0.535389
$485$	−3067.20	−0.287164
$486$	0	0
$487$	−17672.3	−1.64437	−0.822186	−	0.569218i	$-0.807246\pi$
$487$	−17672.3	−1.64437	−0.822186	+	0.569218i	$0.807246\pi$
$488$	−317.871	−0.0294863
$489$	0	0
$490$	10131.7	0.934085
$491$	3484.50	0.320271	0.160136	−	0.987095i	$-0.448807\pi$
$491$	3484.50	0.320271	0.160136	+	0.987095i	$0.448807\pi$
$492$	0	0
$493$	−933.998	−0.0853249
$494$	2501.30	0.227811
$495$	0	0
$496$	−6054.97	−0.548137
$497$	2847.28	0.256978
$498$	0	0
$499$	1393.45	0.125009	0.0625046	−	0.998045i	$-0.480091\pi$
$499$	1393.45	0.125009	0.0625046	+	0.998045i	$0.480091\pi$
$500$	−13689.6	−1.22444
$501$	0	0
$502$	21673.9	1.92700
$503$	−3191.15	−0.282875	−0.141438	−	0.989947i	$-0.545172\pi$
$503$	−3191.15	−0.282875	−0.141438	+	0.989947i	$0.545172\pi$
$504$	0	0
$505$	−7546.63	−0.664991
$506$	−9268.39	−0.814289
$507$	0	0
$508$	19884.7	1.73670
$509$	−10831.8	−0.943247	−0.471624	−	0.881800i	$-0.656332\pi$
$509$	−10831.8	−0.943247	−0.471624	+	0.881800i	$0.656332\pi$
$510$	0	0
$511$	−4700.75	−0.406945
$512$	−15535.1	−1.34094
$513$	0	0
$514$	−10183.0	−0.873842
$515$	−12520.6	−1.07131
$516$	0	0
$517$	7294.56	0.620531
$518$	−7234.80	−0.613666
$519$	0	0
$520$	688.664	0.0580767
$521$	−14277.5	−1.20059	−0.600296	−	0.799778i	$-0.704951\pi$
$521$	−14277.5	−1.20059	−0.600296	+	0.799778i	$0.704951\pi$
$522$	0	0
$523$	4181.28	0.349588	0.174794	−	0.984605i	$-0.444074\pi$
$523$	4181.28	0.349588	0.174794	+	0.984605i	$0.444074\pi$
$524$	15934.2	1.32841
$525$	0	0
$526$	24321.9	2.01613
$527$	−3243.29	−0.268084
$528$	0	0
$529$	−5351.26	−0.439817
$530$	−10363.2	−0.849339
$531$	0	0
$532$	1649.84	0.134454
$533$	6033.87	0.490348
$534$	0	0
$535$	1143.51	0.0924077
$536$	5289.68	0.426267
$537$	0	0
$538$	−10867.5	−0.870875
$539$	8666.72	0.692583
$540$	0	0
$541$	5123.26	0.407146	0.203573	−	0.979060i	$-0.434745\pi$
$541$	5123.26	0.407146	0.203573	+	0.979060i	$0.434745\pi$
$542$	−7482.35	−0.592979
$543$	0	0
$544$	−7050.15	−0.555648
$545$	8243.14	0.647885
$546$	0	0
$547$	−15435.6	−1.20654	−0.603271	−	0.797537i	$-0.706136\pi$
$547$	−15435.6	−1.20654	−0.603271	+	0.797537i	$0.706136\pi$
$548$	−15164.2	−1.18209
$549$	0	0
$550$	−7644.98	−0.592696
$551$	1313.48	0.101554
$552$	0	0
$553$	−3381.43	−0.260023
$554$	3441.47	0.263925
$555$	0	0
$556$	19477.5	1.48566
$557$	978.643	0.0744460	0.0372230	−	0.999307i	$-0.488149\pi$
$557$	978.643	0.0744460	0.0372230	+	0.999307i	$0.488149\pi$
$558$	0	0
$559$	673.076	0.0509268
$560$	−1757.96	−0.132656
$561$	0	0
$562$	−21334.0	−1.60128
$563$	24362.4	1.82371	0.911857	−	0.410507i	$-0.134648\pi$
$563$	24362.4	1.82371	0.911857	+	0.410507i	$0.134648\pi$
$564$	0	0
$565$	7320.43	0.545085
$566$	15447.3	1.14717
$567$	0	0
$568$	−3624.09	−0.267717
$569$	12733.1	0.938135	0.469067	−	0.883162i	$-0.344590\pi$
$569$	12733.1	0.938135	0.469067	+	0.883162i	$0.344590\pi$
$570$	0	0
$571$	−6363.95	−0.466415	−0.233208	−	0.972427i	$-0.574922\pi$
$571$	−6363.95	−0.466415	−0.233208	+	0.972427i	$0.574922\pi$
$572$	3959.48	0.289430
$573$	0	0
$574$	7367.65	0.535749
$575$	5621.93	0.407740
$576$	0	0
$577$	−11981.3	−0.864449	−0.432225	−	0.901766i	$-0.642271\pi$
$577$	−11981.3	−0.864449	−0.432225	+	0.901766i	$0.642271\pi$
$578$	17397.2	1.25195
$579$	0	0
$580$	2430.65	0.174012
$581$	−2559.01	−0.182729
$582$	0	0
$583$	−8864.80	−0.629747
$584$	5983.22	0.423951
$585$	0	0
$586$	3124.97	0.220293
$587$	−17927.4	−1.26055	−0.630276	−	0.776371i	$-0.717058\pi$
$587$	−17927.4	−1.26055	−0.630276	+	0.776371i	$0.717058\pi$
$588$	0	0
$589$	4561.04	0.319074
$590$	−5473.42	−0.381927
$591$	0	0
$592$	−19251.4	−1.33653
$593$	−19927.7	−1.37999	−0.689993	−	0.723816i	$-0.742387\pi$
$593$	−19927.7	−1.37999	−0.689993	+	0.723816i	$0.742387\pi$
$594$	0	0
$595$	−941.636	−0.0648795
$596$	7625.72	0.524097
$597$	0	0
$598$	−5390.20	−0.368598
$599$	4155.72	0.283469	0.141735	−	0.989905i	$-0.454732\pi$
$599$	4155.72	0.283469	0.141735	+	0.989905i	$0.454732\pi$
$600$	0	0
$601$	−4104.20	−0.278559	−0.139279	−	0.990253i	$-0.544479\pi$
$601$	−4104.20	−0.278559	−0.139279	+	0.990253i	$0.544479\pi$
$602$	821.860	0.0556420
$603$	0	0
$604$	−8875.03	−0.597880
$605$	−4575.69	−0.307485
$606$	0	0
$607$	5870.16	0.392525	0.196262	−	0.980551i	$-0.437120\pi$
$607$	5870.16	0.392525	0.196262	+	0.980551i	$0.437120\pi$
$608$	9914.61	0.661333
$609$	0	0
$610$	−1714.85	−0.113823
$611$	4242.29	0.280891
$612$	0	0
$613$	29802.4	1.96363	0.981817	−	0.189830i	$-0.0607936\pi$
$613$	29802.4	1.96363	0.981817	+	0.189830i	$0.0607936\pi$
$614$	19858.7	1.30527
$615$	0	0
$616$	719.308	0.0470483
$617$	−27394.8	−1.78748	−0.893738	−	0.448588i	$-0.851927\pi$
$617$	−27394.8	−1.78748	−0.893738	+	0.448588i	$0.851927\pi$
$618$	0	0
$619$	−25129.8	−1.63174	−0.815872	−	0.578232i	$-0.803743\pi$
$619$	−25129.8	−1.63174	−0.815872	+	0.578232i	$0.803743\pi$
$620$	8440.38	0.546732
$621$	0	0
$622$	−25010.0	−1.61223
$623$	−7475.03	−0.480708
$624$	0	0
$625$	−2475.65	−0.158442
$626$	−1661.20	−0.106062
$627$	0	0
$628$	20098.9	1.27713
$629$	−10311.9	−0.653673
$630$	0	0
$631$	−1190.63	−0.0751160	−0.0375580	−	0.999294i	$-0.511958\pi$
$631$	−1190.63	−0.0751160	−0.0375580	+	0.999294i	$0.511958\pi$
$632$	4303.96	0.270890
$633$	0	0
$634$	19425.7	1.21687
$635$	15960.2	0.997420
$636$	0	0
$637$	5040.29	0.313507
$638$	3849.06	0.238849
$639$	0	0
$640$	5545.45	0.342505
$641$	−28828.0	−1.77635	−0.888174	−	0.459508i	$-0.848026\pi$
$641$	−28828.0	−1.77635	−0.888174	+	0.459508i	$0.848026\pi$
$642$	0	0
$643$	16853.2	1.03363	0.516816	−	0.856096i	$-0.327117\pi$
$643$	16853.2	1.03363	0.516816	+	0.856096i	$0.327117\pi$
$644$	−3555.34	−0.217546
$645$	0	0
$646$	4353.21	0.265131
$647$	20234.3	1.22951	0.614754	−	0.788719i	$-0.289255\pi$
$647$	20234.3	1.22951	0.614754	+	0.788719i	$0.289255\pi$
$648$	0	0
$649$	−4682.02	−0.283182
$650$	−4446.08	−0.268292
$651$	0	0
$652$	−20531.2	−1.23323
$653$	−24965.3	−1.49612	−0.748062	−	0.663629i	$-0.769015\pi$
$653$	−24965.3	−1.49612	−0.748062	+	0.663629i	$0.769015\pi$
$654$	0	0
$655$	12789.4	0.762935
$656$	19604.9	1.16683
$657$	0	0
$658$	5180.05	0.306899
$659$	−9033.00	−0.533954	−0.266977	−	0.963703i	$-0.586025\pi$
$659$	−9033.00	−0.533954	−0.266977	+	0.963703i	$0.586025\pi$
$660$	0	0
$661$	−19710.5	−1.15983	−0.579915	−	0.814677i	$-0.696914\pi$
$661$	−19710.5	−1.15983	−0.579915	+	0.814677i	$0.696914\pi$
$662$	−41767.8	−2.45219
$663$	0	0
$664$	3257.17	0.190365
$665$	1324.22	0.0772196
$666$	0	0
$667$	−2830.50	−0.164314
$668$	−31337.8	−1.81511
$669$	0	0
$670$	28536.8	1.64548
$671$	−1466.90	−0.0843951
$672$	0	0
$673$	19293.1	1.10504	0.552521	−	0.833499i	$-0.313666\pi$
$673$	19293.1	1.10504	0.552521	+	0.833499i	$0.313666\pi$
$674$	16151.1	0.923020
$675$	0	0
$676$	−18345.3	−1.04377
$677$	16029.1	0.909968	0.454984	−	0.890500i	$-0.349645\pi$
$677$	16029.1	0.909968	0.454984	+	0.890500i	$0.349645\pi$
$678$	0	0
$679$	−1863.16	−0.105304
$680$	1198.54	0.0675908
$681$	0	0
$682$	13365.8	0.750443
$683$	4293.48	0.240535	0.120267	−	0.992742i	$-0.461625\pi$
$683$	4293.48	0.240535	0.120267	+	0.992742i	$0.461625\pi$
$684$	0	0
$685$	−12171.4	−0.678896
$686$	12710.2	0.707403
$687$	0	0
$688$	2186.92	0.121186
$689$	−5155.49	−0.285063
$690$	0	0
$691$	−19631.7	−1.08079	−0.540393	−	0.841413i	$-0.681724\pi$
$691$	−19631.7	−1.08079	−0.540393	+	0.841413i	$0.681724\pi$
$692$	−10994.3	−0.603962
$693$	0	0
$694$	42924.2	2.34781
$695$	15633.3	0.853246
$696$	0	0
$697$	10501.2	0.570677
$698$	−2922.17	−0.158461
$699$	0	0
$700$	−2932.60	−0.158345
$701$	13399.7	0.721967	0.360983	−	0.932572i	$-0.382441\pi$
$701$	13399.7	0.721967	0.360983	+	0.932572i	$0.382441\pi$
$702$	0	0
$703$	14501.5	0.778003
$704$	18103.1	0.969158
$705$	0	0
$706$	15050.8	0.802329
$707$	−4584.18	−0.243855
$708$	0	0
$709$	−510.833	−0.0270589	−0.0135294	−	0.999908i	$-0.504307\pi$
$709$	−510.833	−0.0270589	−0.0135294	+	0.999908i	$0.504307\pi$
$710$	−19551.3	−1.03344
$711$	0	0
$712$	9514.40	0.500796
$713$	−9828.87	−0.516261
$714$	0	0
$715$	3178.03	0.166226
$716$	−1941.65	−0.101345
$717$	0	0
$718$	−50411.9	−2.62027
$719$	29685.4	1.53975	0.769875	−	0.638195i	$-0.220319\pi$
$719$	29685.4	1.53975	0.769875	+	0.638195i	$0.220319\pi$
$720$	0	0
$721$	−7605.63	−0.392855
$722$	22487.8	1.15916
$723$	0	0
$724$	6129.94	0.314665
$725$	−2334.72	−0.119599
$726$	0	0
$727$	−10575.9	−0.539529	−0.269765	−	0.962926i	$-0.586946\pi$
$727$	−10575.9	−0.539529	−0.269765	+	0.962926i	$0.586946\pi$
$728$	418.327	0.0212970
$729$	0	0
$730$	32278.3	1.63654
$731$	1171.41	0.0592696
$732$	0	0
$733$	−4428.93	−0.223173	−0.111587	−	0.993755i	$-0.535593\pi$
$733$	−4428.93	−0.223173	−0.111587	+	0.993755i	$0.535593\pi$
$734$	−4041.71	−0.203246
$735$	0	0
$736$	−21365.6	−1.07004
$737$	24410.7	1.22005
$738$	0	0
$739$	20131.6	1.00210	0.501052	−	0.865417i	$-0.332947\pi$
$739$	20131.6	1.00210	0.501052	+	0.865417i	$0.332947\pi$
$740$	26835.7	1.33311
$741$	0	0
$742$	−6295.11	−0.311457
$743$	7362.63	0.363538	0.181769	−	0.983341i	$-0.441818\pi$
$743$	7362.63	0.363538	0.181769	+	0.983341i	$0.441818\pi$
$744$	0	0
$745$	6120.69	0.300999
$746$	−54249.5	−2.66249
$747$	0	0
$748$	6891.00	0.336845
$749$	694.621	0.0338864
$750$	0	0
$751$	−7494.93	−0.364173	−0.182087	−	0.983283i	$-0.558285\pi$
$751$	−7494.93	−0.364173	−0.182087	+	0.983283i	$0.558285\pi$
$752$	13783.8	0.668410
$753$	0	0
$754$	2238.49	0.108118
$755$	−7123.43	−0.343375
$756$	0	0
$757$	26060.4	1.25123	0.625615	−	0.780132i	$-0.284848\pi$
$757$	26060.4	1.25123	0.625615	+	0.780132i	$0.284848\pi$
$758$	−7929.64	−0.379971
$759$	0	0
$760$	−1685.50	−0.0804467
$761$	23064.0	1.09865	0.549323	−	0.835610i	$-0.314886\pi$
$761$	23064.0	1.09865	0.549323	+	0.835610i	$0.314886\pi$
$762$	0	0
$763$	5007.28	0.237583
$764$	10603.2	0.502108
$765$	0	0
$766$	−39632.6	−1.86943
$767$	−2722.91	−0.128186
$768$	0	0
$769$	19455.7	0.912341	0.456171	−	0.889892i	$-0.349221\pi$
$769$	19455.7	0.912341	0.456171	+	0.889892i	$0.349221\pi$
$770$	3880.53	0.181616
$771$	0	0
$772$	3342.08	0.155808
$773$	14025.4	0.652599	0.326300	−	0.945266i	$-0.394198\pi$
$773$	14025.4	0.652599	0.326300	+	0.945266i	$0.394198\pi$
$774$	0	0
$775$	−8107.28	−0.375771
$776$	2371.48	0.109705
$777$	0	0
$778$	19355.1	0.891920
$779$	−14767.8	−0.679220
$780$	0	0
$781$	−16724.3	−0.766253
$782$	−9381.00	−0.428982
$783$	0	0
$784$	16376.7	0.746021
$785$	16132.1	0.733479
$786$	0	0
$787$	−41219.6	−1.86699	−0.933495	−	0.358589i	$-0.883258\pi$
$787$	−41219.6	−1.86699	−0.933495	+	0.358589i	$0.883258\pi$
$788$	−40854.3	−1.84692
$789$	0	0
$790$	23219.0	1.04569
$791$	4446.78	0.199885
$792$	0	0
$793$	−853.104	−0.0382025
$794$	−19328.6	−0.863910
$795$	0	0
$796$	4044.78	0.180105
$797$	−10289.5	−0.457305	−0.228653	−	0.973508i	$-0.573432\pi$
$797$	−10289.5	−0.457305	−0.228653	+	0.973508i	$0.573432\pi$
$798$	0	0
$799$	7383.19	0.326907
$800$	−17623.3	−0.778847
$801$	0	0
$802$	−49224.6	−2.16731
$803$	27611.2	1.21342
$804$	0	0
$805$	−2853.65	−0.124941
$806$	7773.12	0.339698
$807$	0	0
$808$	5834.85	0.254046
$809$	20063.0	0.871912	0.435956	−	0.899968i	$-0.356410\pi$
$809$	20063.0	0.871912	0.435956	+	0.899968i	$0.356410\pi$
$810$	0	0
$811$	26523.6	1.14842	0.574210	−	0.818708i	$-0.305309\pi$
$811$	26523.6	1.14842	0.574210	+	0.818708i	$0.305309\pi$
$812$	1476.49	0.0638112
$813$	0	0
$814$	42495.7	1.82982
$815$	−16479.1	−0.708267
$816$	0	0
$817$	−1647.35	−0.0705427
$818$	−23311.2	−0.996403
$819$	0	0
$820$	−27328.5	−1.16384
$821$	9204.45	0.391276	0.195638	−	0.980676i	$-0.437322\pi$
$821$	9204.45	0.391276	0.195638	+	0.980676i	$0.437322\pi$
$822$	0	0
$823$	15569.2	0.659427	0.329713	−	0.944081i	$-0.393048\pi$
$823$	15569.2	0.659427	0.329713	+	0.944081i	$0.393048\pi$
$824$	9680.62	0.409272
$825$	0	0
$826$	−3324.82	−0.140055
$827$	11786.2	0.495582	0.247791	−	0.968814i	$-0.420295\pi$
$827$	11786.2	0.495582	0.247791	+	0.968814i	$0.420295\pi$
$828$	0	0
$829$	−33661.5	−1.41027	−0.705133	−	0.709075i	$-0.749113\pi$
$829$	−33661.5	−1.41027	−0.705133	+	0.709075i	$0.749113\pi$
$830$	17571.8	0.734850
$831$	0	0
$832$	10528.2	0.438702
$833$	8772.02	0.364865
$834$	0	0
$835$	−25152.9	−1.04246
$836$	−9690.79	−0.400913
$837$	0	0
$838$	−28818.1	−1.18795
$839$	2686.31	0.110538	0.0552691	−	0.998471i	$-0.482398\pi$
$839$	2686.31	0.110538	0.0552691	+	0.998471i	$0.482398\pi$
$840$	0	0
$841$	−23213.5	−0.951803
$842$	−68821.8	−2.81681
$843$	0	0
$844$	974.618	0.0397485
$845$	−14724.6	−0.599458
$846$	0	0
$847$	−2779.49	−0.112756
$848$	−16750.9	−0.678337
$849$	0	0
$850$	−7737.86	−0.312243
$851$	−31250.3	−1.25881
$852$	0	0
$853$	7920.78	0.317940	0.158970	−	0.987283i	$-0.449183\pi$
$853$	7920.78	0.317940	0.158970	+	0.987283i	$0.449183\pi$
$854$	−1041.68	−0.0417396
$855$	0	0
$856$	−884.129	−0.0353025
$857$	−36678.1	−1.46196	−0.730980	−	0.682398i	$-0.760937\pi$
$857$	−36678.1	−1.46196	−0.730980	+	0.682398i	$0.760937\pi$
$858$	0	0
$859$	31205.9	1.23950	0.619750	−	0.784799i	$-0.287234\pi$
$859$	31205.9	1.23950	0.619750	+	0.784799i	$0.287234\pi$
$860$	−3048.48	−0.120875
$861$	0	0
$862$	−29622.0	−1.17045
$863$	−38340.4	−1.51231	−0.756155	−	0.654392i	$-0.772924\pi$
$863$	−38340.4	−1.51231	−0.756155	+	0.654392i	$0.772924\pi$
$864$	0	0
$865$	−8824.46	−0.346868
$866$	37272.0	1.46253
$867$	0	0
$868$	5127.09	0.200489
$869$	19861.8	0.775334
$870$	0	0
$871$	14196.5	0.552273
$872$	−6373.38	−0.247511
$873$	0	0
$874$	13192.5	0.510575
$875$	−6674.51	−0.257874
$876$	0	0
$877$	−38409.6	−1.47890	−0.739452	−	0.673210i	$-0.764915\pi$
$877$	−38409.6	−1.47890	−0.739452	+	0.673210i	$0.764915\pi$
$878$	−74201.5	−2.85214
$879$	0	0
$880$	10325.9	0.395551
$881$	−41203.4	−1.57568	−0.787842	−	0.615877i	$-0.788802\pi$
$881$	−41203.4	−1.57568	−0.787842	+	0.615877i	$0.788802\pi$
$882$	0	0
$883$	30194.6	1.15077	0.575384	−	0.817883i	$-0.304852\pi$
$883$	30194.6	1.15077	0.575384	+	0.817883i	$0.304852\pi$
$884$	4007.59	0.152477
$885$	0	0
$886$	11448.2	0.434096
$887$	13570.8	0.513712	0.256856	−	0.966450i	$-0.417313\pi$
$887$	13570.8	0.513712	0.256856	+	0.966450i	$0.417313\pi$
$888$	0	0
$889$	9694.99	0.365759
$890$	51328.3	1.93318
$891$	0	0
$892$	41803.0	1.56913
$893$	−10383.0	−0.389085
$894$	0	0
$895$	−1558.44	−0.0582045
$896$	3368.57	0.125598
$897$	0	0
$898$	−50269.8	−1.86807
$899$	4081.82	0.151431
$900$	0	0
$901$	−8972.51	−0.331762
$902$	−43276.1	−1.59749
$903$	0	0
$904$	−5659.96	−0.208239
$905$	4920.12	0.180719
$906$	0	0
$907$	21966.6	0.804176	0.402088	−	0.915601i	$-0.368285\pi$
$907$	21966.6	0.804176	0.402088	+	0.915601i	$0.368285\pi$
$908$	−4791.31	−0.175116
$909$	0	0
$910$	2256.79	0.0822110
$911$	−27091.0	−0.985252	−0.492626	−	0.870241i	$-0.663963\pi$
$911$	−27091.0	−0.985252	−0.492626	+	0.870241i	$0.663963\pi$
$912$	0	0
$913$	15031.1	0.544859
$914$	−544.020	−0.0196877
$915$	0	0
$916$	−44687.5	−1.61192
$917$	7768.88	0.279772
$918$	0	0
$919$	−24533.7	−0.880622	−0.440311	−	0.897845i	$-0.645132\pi$
$919$	−24533.7	−0.880622	−0.440311	+	0.897845i	$0.645132\pi$
$920$	3632.19	0.130163
$921$	0	0
$922$	−4092.38	−0.146177
$923$	−9726.35	−0.346855
$924$	0	0
$925$	−25776.6	−0.916248
$926$	−26527.9	−0.941426
$927$	0	0
$928$	8872.89	0.313865
$929$	−9220.40	−0.325631	−0.162816	−	0.986657i	$-0.552058\pi$
$929$	−9220.40	−0.325631	−0.162816	+	0.986657i	$0.552058\pi$
$930$	0	0
$931$	−12336.1	−0.434263
$932$	−8394.76	−0.295042
$933$	0	0
$934$	−59062.9	−2.06916
$935$	5530.97	0.193457
$936$	0	0
$937$	22526.6	0.785392	0.392696	−	0.919668i	$-0.371542\pi$
$937$	22526.6	0.785392	0.392696	+	0.919668i	$0.371542\pi$
$938$	17334.6	0.603407
$939$	0	0
$940$	−19214.1	−0.666697
$941$	776.782	0.0269100	0.0134550	−	0.999909i	$-0.495717\pi$
$941$	776.782	0.0269100	0.0134550	+	0.999909i	$0.495717\pi$
$942$	0	0
$943$	31824.1	1.09898
$944$	−8847.15	−0.305032
$945$	0	0
$946$	−4827.43	−0.165913
$947$	45677.6	1.56739	0.783697	−	0.621143i	$-0.213331\pi$
$947$	45677.6	1.56739	0.783697	+	0.621143i	$0.213331\pi$
$948$	0	0
$949$	16057.8	0.549271
$950$	10881.7	0.371632
$951$	0	0
$952$	728.048	0.0247859
$953$	41584.5	1.41349	0.706743	−	0.707470i	$-0.250164\pi$
$953$	41584.5	1.41349	0.706743	+	0.707470i	$0.250164\pi$
$954$	0	0
$955$	8510.52	0.288371
$956$	36500.5	1.23484
$957$	0	0
$958$	−85855.7	−2.89548
$959$	−7393.47	−0.248955
$960$	0	0
$961$	−15617.0	−0.524217
$962$	24714.2	0.828292
$963$	0	0
$964$	−29311.6	−0.979318
$965$	2682.48	0.0894839
$966$	0	0
$967$	5961.61	0.198255	0.0991274	−	0.995075i	$-0.468395\pi$
$967$	5961.61	0.198255	0.0991274	+	0.995075i	$0.468395\pi$
$968$	3537.80	0.117468
$969$	0	0
$970$	12793.7	0.423485
$971$	25246.9	0.834410	0.417205	−	0.908812i	$-0.363010\pi$
$971$	25246.9	0.834410	0.417205	+	0.908812i	$0.363010\pi$
$972$	0	0
$973$	9496.43	0.312890
$974$	73713.5	2.42498
$975$	0	0
$976$	−2771.86	−0.0909068
$977$	−44505.1	−1.45736	−0.728682	−	0.684852i	$-0.759867\pi$
$977$	−44505.1	−1.45736	−0.728682	+	0.684852i	$0.759867\pi$
$978$	0	0
$979$	43906.8	1.43337
$980$	−22828.4	−0.744109
$981$	0	0
$982$	−14534.3	−0.472308
$983$	−14970.7	−0.485750	−0.242875	−	0.970058i	$-0.578090\pi$
$983$	−14970.7	−0.485750	−0.242875	+	0.970058i	$0.578090\pi$
$984$	0	0
$985$	−32791.1	−1.06072
$986$	3895.82	0.125830
$987$	0	0
$988$	−5635.86	−0.181478
$989$	3549.97	0.114138
$990$	0	0
$991$	−10815.1	−0.346674	−0.173337	−	0.984863i	$-0.555455\pi$
$991$	−10815.1	−0.346674	−0.173337	+	0.984863i	$0.555455\pi$
$992$	30811.0	0.986139
$993$	0	0
$994$	−11876.4	−0.378969
$995$	3246.49	0.103438
$996$	0	0
$997$	−39556.1	−1.25652	−0.628262	−	0.778002i	$-0.716233\pi$
$997$	−39556.1	−1.25652	−0.628262	+	0.778002i	$0.716233\pi$
$998$	−5812.27	−0.184353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.4.a.h.1.2		6
3.2	odd	2		43.4.a.b.1.5	✓	6
12.11	even	2		688.4.a.i.1.3		6
15.14	odd	2		1075.4.a.b.1.2		6
21.20	even	2		2107.4.a.c.1.5		6
129.128	even	2		1849.4.a.c.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.5	✓	6	3.2	odd	2
387.4.a.h.1.2		6	1.1	even	1	trivial
688.4.a.i.1.3		6	12.11	even	2
1075.4.a.b.1.2		6	15.14	odd	2
1849.4.a.c.1.2		6	129.128	even	2
2107.4.a.c.1.5		6	21.20	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-4.17112 q^{2} +9.39827 q^{4} +7.54340 q^{5} +4.58222 q^{7} -5.83236 q^{8} +O(q^{10})\)	\(q-4.17112 q^{2} +9.39827 q^{4} +7.54340 q^{5} +4.58222 q^{7} -5.83236 q^{8} -31.4645 q^{10} -26.9150 q^{11} -15.6529 q^{13} -19.1130 q^{14} -50.8587 q^{16} -27.2420 q^{17} +38.3104 q^{19} +70.8949 q^{20} +112.266 q^{22} -82.5575 q^{23} -68.0971 q^{25} +65.2903 q^{26} +43.0650 q^{28} +34.2852 q^{29} +119.055 q^{31} +258.797 q^{32} +113.630 q^{34} +34.5655 q^{35} +378.527 q^{37} -159.797 q^{38} -43.9958 q^{40} -385.478 q^{41} -43.0000 q^{43} -252.955 q^{44} +344.358 q^{46} -271.022 q^{47} -322.003 q^{49} +284.041 q^{50} -147.110 q^{52} +329.363 q^{53} -203.031 q^{55} -26.7252 q^{56} -143.008 q^{58} +173.956 q^{59} +54.5012 q^{61} -496.592 q^{62} -672.604 q^{64} -118.076 q^{65} -906.954 q^{67} -256.028 q^{68} -144.177 q^{70} +621.376 q^{71} -1025.87 q^{73} -1578.88 q^{74} +360.052 q^{76} -123.331 q^{77} -737.945 q^{79} -383.647 q^{80} +1607.88 q^{82} -558.465 q^{83} -205.498 q^{85} +179.358 q^{86} +156.978 q^{88} -1631.31 q^{89} -71.7252 q^{91} -775.898 q^{92} +1130.47 q^{94} +288.991 q^{95} -406.607 q^{97} +1343.12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * q^98

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