LMFDB - Embedded newform 605.4.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,4,Mod(1,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, 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("605.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	605.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:	$35.6961555535$
Analytic rank:	$0$
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} - 3x^{5} - 33x^{4} + 67x^{3} + 256x^{2} - 236x + 44 $ x^6 - 3x^5 - 33x^4 + 67x^3 + 256x^2 - 236*x + 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.270958$ of defining polynomial
Character	$\chi$	$=$	605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.729042 q^{2} -0.737592 q^{3} -7.46850 q^{4} +5.00000 q^{5} -0.537736 q^{6} -19.3354 q^{7} -11.2772 q^{8} -26.4560 q^{9} +3.64521 q^{10} +5.50870 q^{12} -19.1434 q^{13} -14.0963 q^{14} -3.68796 q^{15} +51.5264 q^{16} +34.8156 q^{17} -19.2875 q^{18} -37.2672 q^{19} -37.3425 q^{20} +14.2616 q^{21} -114.071 q^{23} +8.31796 q^{24} +25.0000 q^{25} -13.9564 q^{26} +39.4287 q^{27} +144.406 q^{28} -26.9925 q^{29} -2.68868 q^{30} +45.6445 q^{31} +127.782 q^{32} +25.3821 q^{34} -96.6769 q^{35} +197.586 q^{36} +331.873 q^{37} -27.1694 q^{38} +14.1200 q^{39} -56.3860 q^{40} +241.369 q^{41} +10.3973 q^{42} +285.332 q^{43} -132.280 q^{45} -83.1628 q^{46} -72.3478 q^{47} -38.0055 q^{48} +30.8567 q^{49} +18.2261 q^{50} -25.6797 q^{51} +142.973 q^{52} -317.740 q^{53} +28.7452 q^{54} +218.049 q^{56} +27.4880 q^{57} -19.6787 q^{58} +755.784 q^{59} +27.5435 q^{60} -908.535 q^{61} +33.2768 q^{62} +511.536 q^{63} -319.053 q^{64} -95.7171 q^{65} +626.261 q^{67} -260.020 q^{68} +84.1380 q^{69} -70.4815 q^{70} +968.978 q^{71} +298.349 q^{72} -1027.71 q^{73} +241.950 q^{74} -18.4398 q^{75} +278.330 q^{76} +10.2941 q^{78} -459.166 q^{79} +257.632 q^{80} +685.229 q^{81} +175.968 q^{82} +637.420 q^{83} -106.513 q^{84} +174.078 q^{85} +208.019 q^{86} +19.9095 q^{87} +756.844 q^{89} -96.4376 q^{90} +370.145 q^{91} +851.941 q^{92} -33.6670 q^{93} -52.7446 q^{94} -186.336 q^{95} -94.2513 q^{96} +513.113 q^{97} +22.4958 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 12 q^{3} + 27 q^{4} + 30 q^{5} + 45 q^{6} - 10 q^{7} + 51 q^{8} + 104 q^{9} + 15 q^{10} + 137 q^{12} + 80 q^{13} + 7 q^{14} + 60 q^{15} + 155 q^{16} - 12 q^{17} + 14 q^{18} + 86 q^{19} + 135 q^{20}+ \cdots + 6850 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 + 12 * q^3 + 27 * q^4 + 30 * q^5 + 45 * q^6 - 10 * q^7 + 51 * q^8 + 104 * q^9 + 15 * q^10 + 137 * q^12 + 80 * q^13 + 7 * q^14 + 60 * q^15 + 155 * q^16 - 12 * q^17 + 14 * q^18 + 86 * q^19 + 135 * q^20 + 14 * q^21 + 406 * q^23 + 229 * q^24 + 150 * q^25 + 352 * q^26 + 342 * q^27 - 633 * q^28 - 372 * q^29 + 225 * q^30 - 122 * q^31 + 711 * q^32 - 56 * q^34 - 50 * q^35 + 410 * q^36 + 576 * q^37 + 652 * q^38 - 210 * q^39 + 255 * q^40 + 46 * q^41 - 1375 * q^42 + 10 * q^43 + 520 * q^45 - 1632 * q^46 + 568 * q^47 + 2813 * q^48 + 1012 * q^49 + 75 * q^50 - 908 * q^51 + 1036 * q^52 - 184 * q^53 + 2455 * q^54 - 1961 * q^56 - 2052 * q^57 - 186 * q^58 + 1598 * q^59 + 685 * q^60 + 1598 * q^61 - 450 * q^62 + 286 * q^63 - 653 * q^64 + 400 * q^65 + 528 * q^67 - 2104 * q^68 - 156 * q^69 + 35 * q^70 + 142 * q^71 + 3678 * q^72 - 268 * q^73 + 1272 * q^74 + 300 * q^75 + 3136 * q^76 + 900 * q^78 - 624 * q^79 + 775 * q^80 + 2538 * q^81 + 723 * q^82 - 2786 * q^83 - 8583 * q^84 - 60 * q^85 - 3997 * q^86 + 1224 * q^87 + 634 * q^89 + 70 * q^90 + 1690 * q^91 + 1164 * q^92 + 3304 * q^93 + 1681 * q^94 + 430 * q^95 + 5821 * q^96 + 648 * q^97 + 6850 * 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 $

$2$	0.729042	0.257755	0.128878	−	0.991660i	$-0.458863\pi$
$2$	0.729042	0.257755	0.128878	+	0.991660i	$0.458863\pi$
$3$	−0.737592	−0.141950	−0.0709748	−	0.997478i	$-0.522611\pi$
$3$	−0.737592	−0.141950	−0.0709748	+	0.997478i	$0.522611\pi$
$4$	−7.46850	−0.933562
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−0.537736	−0.0365883
$7$	−19.3354	−1.04401	−0.522006	−	0.852942i	$-0.674816\pi$
$7$	−19.3354	−1.04401	−0.522006	+	0.852942i	$0.674816\pi$
$8$	−11.2772	−0.498386
$9$	−26.4560	−0.979850
$10$	3.64521	0.115272
$11$	0	0
$11$	0	0
$12$	5.50870	0.132519
$13$	−19.1434	−0.408418	−0.204209	−	0.978927i	$-0.565462\pi$
$13$	−19.1434	−0.408418	−0.204209	+	0.978927i	$0.565462\pi$
$14$	−14.0963	−0.269100
$15$	−3.68796	−0.0634818
$16$	51.5264	0.805100
$17$	34.8156	0.496707	0.248354	−	0.968669i	$-0.420110\pi$
$17$	34.8156	0.496707	0.248354	+	0.968669i	$0.420110\pi$
$18$	−19.2875	−0.252562
$19$	−37.2672	−0.449984	−0.224992	−	0.974361i	$-0.572236\pi$
$19$	−37.2672	−0.449984	−0.224992	+	0.974361i	$0.572236\pi$
$20$	−37.3425	−0.417502
$21$	14.2616	0.148197
$22$	0	0
$23$	−114.071	−1.03415	−0.517076	−	0.855939i	$-0.672980\pi$
$23$	−114.071	−1.03415	−0.517076	+	0.855939i	$0.672980\pi$
$24$	8.31796	0.0707457
$25$	25.0000	0.200000
$26$	−13.9564	−0.105272
$27$	39.4287	0.281039
$28$	144.406	0.974650
$29$	−26.9925	−0.172841	−0.0864205	−	0.996259i	$-0.527543\pi$
$29$	−26.9925	−0.172841	−0.0864205	+	0.996259i	$0.527543\pi$
$30$	−2.68868	−0.0163628
$31$	45.6445	0.264452	0.132226	−	0.991220i	$-0.457788\pi$
$31$	45.6445	0.264452	0.132226	+	0.991220i	$0.457788\pi$
$32$	127.782	0.705905
$33$	0	0
$34$	25.3821	0.128029
$35$	−96.6769	−0.466896
$36$	197.586	0.914751
$37$	331.873	1.47458	0.737292	−	0.675574i	$-0.236104\pi$
$37$	331.873	1.47458	0.737292	+	0.675574i	$0.236104\pi$
$38$	−27.1694	−0.115986
$39$	14.1200	0.0579747
$40$	−56.3860	−0.222885
$41$	241.369	0.919401	0.459701	−	0.888074i	$-0.347957\pi$
$41$	241.369	0.919401	0.459701	+	0.888074i	$0.347957\pi$
$42$	10.3973	0.0381986
$43$	285.332	1.01192	0.505962	−	0.862556i	$-0.331137\pi$
$43$	285.332	1.01192	0.505962	+	0.862556i	$0.331137\pi$
$44$	0	0
$45$	−132.280	−0.438202
$46$	−83.1628	−0.266558
$47$	−72.3478	−0.224532	−0.112266	−	0.993678i	$-0.535811\pi$
$47$	−72.3478	−0.224532	−0.112266	+	0.993678i	$0.535811\pi$
$48$	−38.0055	−0.114284
$49$	30.8567	0.0899611
$50$	18.2261	0.0515511
$51$	−25.6797	−0.0705074
$52$	142.973	0.381283
$53$	−317.740	−0.823491	−0.411745	−	0.911299i	$-0.635081\pi$
$53$	−317.740	−0.823491	−0.411745	+	0.911299i	$0.635081\pi$
$54$	28.7452	0.0724393
$55$	0	0
$56$	218.049	0.520321
$57$	27.4880	0.0638750
$58$	−19.6787	−0.0445507
$59$	755.784	1.66771	0.833853	−	0.551986i	$-0.186130\pi$
$59$	755.784	1.66771	0.833853	+	0.551986i	$0.186130\pi$
$60$	27.5435	0.0592642
$61$	−908.535	−1.90698	−0.953492	−	0.301420i	$-0.902539\pi$
$61$	−908.535	−1.90698	−0.953492	+	0.301420i	$0.902539\pi$
$62$	33.2768	0.0681638
$63$	511.536	1.02298
$64$	−319.053	−0.623150
$65$	−95.7171	−0.182650
$66$	0	0
$67$	626.261	1.14194	0.570969	−	0.820971i	$-0.306567\pi$
$67$	626.261	1.14194	0.570969	+	0.820971i	$0.306567\pi$
$68$	−260.020	−0.463707
$69$	84.1380	0.146797
$70$	−70.4815	−0.120345
$71$	968.978	1.61967	0.809834	−	0.586659i	$-0.199557\pi$
$71$	968.978	1.61967	0.809834	+	0.586659i	$0.199557\pi$
$72$	298.349	0.488344
$73$	−1027.71	−1.64773	−0.823863	−	0.566788i	$-0.808186\pi$
$73$	−1027.71	−1.64773	−0.823863	+	0.566788i	$0.808186\pi$
$74$	241.950	0.380082
$75$	−18.4398	−0.0283899
$76$	278.330	0.420088
$77$	0	0
$78$	10.2941	0.0149433
$79$	−459.166	−0.653927	−0.326963	−	0.945037i	$-0.606025\pi$
$79$	−459.166	−0.653927	−0.326963	+	0.945037i	$0.606025\pi$
$80$	257.632	0.360052
$81$	685.229	0.939957
$82$	175.968	0.236981
$83$	637.420	0.842964	0.421482	−	0.906837i	$-0.361510\pi$
$83$	637.420	0.842964	0.421482	+	0.906837i	$0.361510\pi$
$84$	−106.513	−0.138351
$85$	174.078	0.222134
$86$	208.019	0.260829
$87$	19.9095	0.0245347
$88$	0	0
$89$	756.844	0.901408	0.450704	−	0.892673i	$-0.351173\pi$
$89$	756.844	0.901408	0.450704	+	0.892673i	$0.351173\pi$
$90$	−96.4376	−0.112949
$91$	370.145	0.426393
$92$	851.941	0.965445
$93$	−33.6670	−0.0375388
$94$	−52.7446	−0.0578744
$95$	−186.336	−0.201239
$96$	−94.2513	−0.100203
$97$	513.113	0.537100	0.268550	−	0.963266i	$-0.413455\pi$
$97$	513.113	0.537100	0.268550	+	0.963266i	$0.413455\pi$
$98$	22.4958	0.0231880
$99$	0	0
$100$	−186.712	−0.186712
$101$	−591.132	−0.582375	−0.291187	−	0.956666i	$-0.594050\pi$
$101$	−591.132	−0.582375	−0.291187	+	0.956666i	$0.594050\pi$
$102$	−18.7216	−0.0181737
$103$	613.367	0.586765	0.293383	−	0.955995i	$-0.405219\pi$
$103$	613.367	0.586765	0.293383	+	0.955995i	$0.405219\pi$
$104$	215.884	0.203550
$105$	71.3081	0.0662757
$106$	−231.646	−0.212259
$107$	−575.014	−0.519521	−0.259761	−	0.965673i	$-0.583644\pi$
$107$	−575.014	−0.519521	−0.259761	+	0.965673i	$0.583644\pi$
$108$	−294.473	−0.262367
$109$	2052.08	1.80325	0.901624	−	0.432521i	$-0.142376\pi$
$109$	2052.08	1.80325	0.901624	+	0.432521i	$0.142376\pi$
$110$	0	0
$111$	−244.787	−0.209317
$112$	−996.283	−0.840535
$113$	1091.59	0.908742	0.454371	−	0.890812i	$-0.349864\pi$
$113$	1091.59	0.908742	0.454371	+	0.890812i	$0.349864\pi$
$114$	20.0399	0.0164641
$115$	−570.356	−0.462487
$116$	201.594	0.161358
$117$	506.458	0.400188
$118$	550.999	0.429860
$119$	−673.173	−0.518569
$120$	41.5898	0.0316384
$121$	0	0
$122$	−662.360	−0.491535
$123$	−178.031	−0.130509
$124$	−340.896	−0.246882
$125$	125.000	0.0894427
$126$	372.931	0.263677
$127$	636.023	0.444393	0.222197	−	0.975002i	$-0.428677\pi$
$127$	636.023	0.444393	0.222197	+	0.975002i	$0.428677\pi$
$128$	−1254.86	−0.866525
$129$	−210.459	−0.143642
$130$	−69.7818	−0.0470790
$131$	2593.85	1.72997	0.864984	−	0.501799i	$-0.167328\pi$
$131$	2593.85	1.72997	0.864984	+	0.501799i	$0.167328\pi$
$132$	0	0
$133$	720.576	0.469788
$134$	456.571	0.294341
$135$	197.143	0.125684
$136$	−392.622	−0.247552
$137$	2168.42	1.35227	0.676133	−	0.736780i	$-0.263655\pi$
$137$	2168.42	1.35227	0.676133	+	0.736780i	$0.263655\pi$
$138$	61.3402	0.0378378
$139$	−1201.35	−0.733073	−0.366537	−	0.930404i	$-0.619457\pi$
$139$	−1201.35	−0.733073	−0.366537	+	0.930404i	$0.619457\pi$
$140$	722.031	0.435877
$141$	53.3632	0.0318723
$142$	706.426	0.417478
$143$	0	0
$144$	−1363.18	−0.788878
$145$	−134.963	−0.0772969
$146$	−749.242	−0.424711
$147$	−22.7596	−0.0127699
$148$	−2478.60	−1.37662
$149$	−1679.92	−0.923654	−0.461827	−	0.886970i	$-0.652806\pi$
$149$	−1679.92	−0.923654	−0.461827	+	0.886970i	$0.652806\pi$
$150$	−13.4434	−0.00731765
$151$	−2181.82	−1.17586	−0.587928	−	0.808913i	$-0.700056\pi$
$151$	−2181.82	−1.17586	−0.587928	+	0.808913i	$0.700056\pi$
$152$	420.270	0.224266
$153$	−921.080	−0.486699
$154$	0	0
$155$	228.223	0.118266
$156$	−105.455	−0.0541230
$157$	742.382	0.377379	0.188690	−	0.982037i	$-0.439576\pi$
$157$	742.382	0.377379	0.188690	+	0.982037i	$0.439576\pi$
$158$	−334.751	−0.168553
$159$	234.363	0.116894
$160$	638.912	0.315690
$161$	2205.61	1.07967
$162$	499.561	0.242279
$163$	−2136.53	−1.02666	−0.513330	−	0.858191i	$-0.671588\pi$
$163$	−2136.53	−1.02666	−0.513330	+	0.858191i	$0.671588\pi$
$164$	−1802.66	−0.858318
$165$	0	0
$166$	464.707	0.217278
$167$	−2509.44	−1.16279	−0.581395	−	0.813621i	$-0.697493\pi$
$167$	−2509.44	−1.16279	−0.581395	+	0.813621i	$0.697493\pi$
$168$	−160.831	−0.0738594
$169$	−1830.53	−0.833195
$170$	126.910	0.0572563
$171$	985.941	0.440917
$172$	−2131.00	−0.944695
$173$	511.948	0.224986	0.112493	−	0.993652i	$-0.464116\pi$
$173$	511.948	0.224986	0.112493	+	0.993652i	$0.464116\pi$
$174$	14.5149	0.00632395
$175$	−483.384	−0.208802
$176$	0	0
$177$	−557.460	−0.236730
$178$	551.772	0.232343
$179$	−2690.21	−1.12333	−0.561665	−	0.827365i	$-0.689839\pi$
$179$	−2690.21	−1.12333	−0.561665	+	0.827365i	$0.689839\pi$
$180$	987.931	0.409089
$181$	−745.715	−0.306235	−0.153118	−	0.988208i	$-0.548931\pi$
$181$	−745.715	−0.306235	−0.153118	+	0.988208i	$0.548931\pi$
$182$	269.852	0.109905
$183$	670.128	0.270695
$184$	1286.40	0.515407
$185$	1659.37	0.659454
$186$	−24.5447	−0.00967583
$187$	0	0
$188$	540.330	0.209615
$189$	−762.368	−0.293408
$190$	−135.847	−0.0518704
$191$	1522.08	0.576617	0.288308	−	0.957538i	$-0.406907\pi$
$191$	1522.08	0.576617	0.288308	+	0.957538i	$0.406907\pi$
$192$	235.331	0.0884558
$193$	−2697.01	−1.00588	−0.502941	−	0.864321i	$-0.667749\pi$
$193$	−2697.01	−1.00588	−0.502941	+	0.864321i	$0.667749\pi$
$194$	374.081	0.138441
$195$	70.6001	0.0259271
$196$	−230.453	−0.0839843
$197$	−5167.15	−1.86875	−0.934376	−	0.356288i	$-0.884042\pi$
$197$	−5167.15	−1.86875	−0.934376	+	0.356288i	$0.884042\pi$
$198$	0	0
$199$	2225.27	0.792690	0.396345	−	0.918102i	$-0.370278\pi$
$199$	2225.27	0.792690	0.396345	+	0.918102i	$0.370278\pi$
$200$	−281.930	−0.0996772
$201$	−461.925	−0.162098
$202$	−430.961	−0.150110
$203$	521.911	0.180448
$204$	191.789	0.0658231
$205$	1206.84	0.411169
$206$	447.170	0.151242
$207$	3017.86	1.01331
$208$	−986.392	−0.328817
$209$	0	0
$210$	51.9866	0.0170829
$211$	4611.47	1.50458	0.752291	−	0.658831i	$-0.228949\pi$
$211$	4611.47	1.50458	0.752291	+	0.658831i	$0.228949\pi$
$212$	2373.04	0.768780
$213$	−714.710	−0.229911
$214$	−419.210	−0.133909
$215$	1426.66	0.452547
$216$	−444.645	−0.140066
$217$	−882.554	−0.276091
$218$	1496.06	0.464797
$219$	758.028	0.233894
$220$	0	0
$221$	−666.490	−0.202864
$222$	−178.460	−0.0539525
$223$	−1217.14	−0.365496	−0.182748	−	0.983160i	$-0.558499\pi$
$223$	−1217.14	−0.365496	−0.182748	+	0.983160i	$0.558499\pi$
$224$	−2470.72	−0.736973
$225$	−661.399	−0.195970
$226$	795.813	0.234233
$227$	6354.43	1.85797	0.928983	−	0.370121i	$-0.120684\pi$
$227$	6354.43	1.85797	0.928983	+	0.370121i	$0.120684\pi$
$228$	−205.294	−0.0596313
$229$	−2602.50	−0.750996	−0.375498	−	0.926823i	$-0.622528\pi$
$229$	−2602.50	−0.750996	−0.375498	+	0.926823i	$0.622528\pi$
$230$	−415.814	−0.119209
$231$	0	0
$232$	304.400	0.0861416
$233$	4853.85	1.36475	0.682373	−	0.731004i	$-0.260948\pi$
$233$	4853.85	1.36475	0.682373	+	0.731004i	$0.260948\pi$
$234$	369.229	0.103151
$235$	−361.739	−0.100414
$236$	−5644.57	−1.55691
$237$	338.677	0.0928246
$238$	−490.772	−0.133664
$239$	−1973.44	−0.534105	−0.267052	−	0.963682i	$-0.586050\pi$
$239$	−1973.44	−0.534105	−0.267052	+	0.963682i	$0.586050\pi$
$240$	−190.027	−0.0511092
$241$	−1288.83	−0.344486	−0.172243	−	0.985055i	$-0.555101\pi$
$241$	−1288.83	−0.344486	−0.172243	+	0.985055i	$0.555101\pi$
$242$	0	0
$243$	−1569.99	−0.414465
$244$	6785.39	1.78029
$245$	154.283	0.0402318
$246$	−129.793	−0.0336393
$247$	713.423	0.183781
$248$	−514.742	−0.131799
$249$	−470.156	−0.119658
$250$	91.1303	0.0230543
$251$	2128.34	0.535219	0.267609	−	0.963527i	$-0.413766\pi$
$251$	2128.34	0.535219	0.267609	+	0.963527i	$0.413766\pi$
$252$	−3820.40	−0.955011
$253$	0	0
$254$	463.688	0.114545
$255$	−128.399	−0.0315319
$256$	1637.57	0.399798
$257$	−2985.66	−0.724672	−0.362336	−	0.932048i	$-0.618021\pi$
$257$	−2985.66	−0.724672	−0.362336	+	0.932048i	$0.618021\pi$
$258$	−153.433	−0.0370246
$259$	−6416.89	−1.53948
$260$	714.863	0.170515
$261$	714.114	0.169358
$262$	1891.03	0.445909
$263$	3974.84	0.931937	0.465968	−	0.884801i	$-0.345706\pi$
$263$	3974.84	0.931937	0.465968	+	0.884801i	$0.345706\pi$
$264$	0	0
$265$	−1588.70	−0.368276
$266$	525.330	0.121091
$267$	−558.242	−0.127955
$268$	−4677.23	−1.06607
$269$	−1490.18	−0.337761	−0.168880	−	0.985637i	$-0.554015\pi$
$269$	−1490.18	−0.337761	−0.168880	+	0.985637i	$0.554015\pi$
$270$	143.726	0.0323958
$271$	756.295	0.169526	0.0847632	−	0.996401i	$-0.472987\pi$
$271$	756.295	0.169526	0.0847632	+	0.996401i	$0.472987\pi$
$272$	1793.92	0.399899
$273$	−273.016	−0.0605263
$274$	1580.87	0.348554
$275$	0	0
$276$	−628.384	−0.137045
$277$	1097.85	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$277$	1097.85	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$278$	−875.835	−0.188954
$279$	−1207.57	−0.259123
$280$	1090.24	0.232695
$281$	−2880.06	−0.611422	−0.305711	−	0.952124i	$-0.598894\pi$
$281$	−2880.06	−0.611422	−0.305711	+	0.952124i	$0.598894\pi$
$282$	38.9040	0.00821525
$283$	−7875.47	−1.65423	−0.827117	−	0.562029i	$-0.810021\pi$
$283$	−7875.47	−1.65423	−0.827117	+	0.562029i	$0.810021\pi$
$284$	−7236.81	−1.51206
$285$	137.440	0.0285658
$286$	0	0
$287$	−4666.95	−0.959866
$288$	−3380.61	−0.691681
$289$	−3700.87	−0.753282
$290$	−98.3935	−0.0199237
$291$	−378.468	−0.0762412
$292$	7675.43	1.53826
$293$	5591.66	1.11491	0.557455	−	0.830207i	$-0.311778\pi$
$293$	5591.66	1.11491	0.557455	+	0.830207i	$0.311778\pi$
$294$	−16.5927	−0.00329152
$295$	3778.92	0.745821
$296$	−3742.60	−0.734913
$297$	0	0
$298$	−1224.73	−0.238077
$299$	2183.71	0.422366
$300$	137.718	0.0265038
$301$	−5517.01	−1.05646
$302$	−1590.64	−0.303083
$303$	436.014	0.0826679
$304$	−1920.25	−0.362282
$305$	−4542.67	−0.852829
$306$	−671.507	−0.125449
$307$	−9699.86	−1.80326	−0.901629	−	0.432510i	$-0.857628\pi$
$307$	−9699.86	−1.80326	−0.901629	+	0.432510i	$0.857628\pi$
$308$	0	0
$309$	−452.414	−0.0832911
$310$	166.384	0.0304838
$311$	−1105.67	−0.201597	−0.100798	−	0.994907i	$-0.532140\pi$
$311$	−1105.67	−0.201597	−0.100798	+	0.994907i	$0.532140\pi$
$312$	−159.234	−0.0288938
$313$	9151.92	1.65271	0.826353	−	0.563152i	$-0.190411\pi$
$313$	9151.92	1.65271	0.826353	+	0.563152i	$0.190411\pi$
$314$	541.228	0.0972715
$315$	2557.68	0.457489
$316$	3429.28	0.610481
$317$	8929.63	1.58214	0.791070	−	0.611726i	$-0.209525\pi$
$317$	8929.63	1.58214	0.791070	+	0.611726i	$0.209525\pi$
$318$	170.860	0.0301301
$319$	0	0
$320$	−1595.26	−0.278681
$321$	424.126	0.0737458
$322$	1607.98	0.278290
$323$	−1297.48	−0.223510
$324$	−5117.63	−0.877508
$325$	−478.586	−0.0816835
$326$	−1557.62	−0.264627
$327$	−1513.60	−0.255970
$328$	−2721.96	−0.458217
$329$	1398.87	0.234414
$330$	0	0
$331$	−2111.51	−0.350632	−0.175316	−	0.984512i	$-0.556095\pi$
$331$	−2111.51	−0.350632	−0.175316	+	0.984512i	$0.556095\pi$
$332$	−4760.57	−0.786959
$333$	−8780.03	−1.44487
$334$	−1829.49	−0.299716
$335$	3131.30	0.510691
$336$	734.850	0.119314
$337$	4170.40	0.674113	0.337057	−	0.941484i	$-0.390569\pi$
$337$	4170.40	0.674113	0.337057	+	0.941484i	$0.390569\pi$
$338$	−1334.53	−0.214761
$339$	−805.146	−0.128996
$340$	−1300.10	−0.207376
$341$	0	0
$342$	718.792	0.113649
$343$	6035.41	0.950092
$344$	−3217.75	−0.504329
$345$	420.690	0.0656498
$346$	373.232	0.0579915
$347$	11233.4	1.73788	0.868938	−	0.494921i	$-0.164803\pi$
$347$	11233.4	1.73788	0.868938	+	0.494921i	$0.164803\pi$
$348$	−148.694	−0.0229047
$349$	−7487.15	−1.14836	−0.574181	−	0.818729i	$-0.694679\pi$
$349$	−7487.15	−1.14836	−0.574181	+	0.818729i	$0.694679\pi$
$350$	−352.408	−0.0538199
$351$	−754.800	−0.114781
$352$	0	0
$353$	3775.45	0.569256	0.284628	−	0.958638i	$-0.408130\pi$
$353$	3775.45	0.569256	0.284628	+	0.958638i	$0.408130\pi$
$354$	−406.412	−0.0610185
$355$	4844.89	0.724338
$356$	−5652.49	−0.841521
$357$	496.527	0.0736106
$358$	−1961.28	−0.289544
$359$	−2014.08	−0.296097	−0.148049	−	0.988980i	$-0.547299\pi$
$359$	−2014.08	−0.296097	−0.148049	+	0.988980i	$0.547299\pi$
$360$	1491.74	0.218394
$361$	−5470.15	−0.797515
$362$	−543.658	−0.0789338
$363$	0	0
$364$	−2764.43	−0.398064
$365$	−5138.54	−0.736886
$366$	488.552	0.0697732
$367$	11609.9	1.65132	0.825658	−	0.564172i	$-0.190804\pi$
$367$	11609.9	1.65132	0.825658	+	0.564172i	$0.190804\pi$
$368$	−5877.68	−0.832596
$369$	−6385.64	−0.900876
$370$	1209.75	0.169978
$371$	6143.63	0.859734
$372$	251.442	0.0350448
$373$	5231.95	0.726273	0.363137	−	0.931736i	$-0.381706\pi$
$373$	5231.95	0.726273	0.363137	+	0.931736i	$0.381706\pi$
$374$	0	0
$375$	−92.1990	−0.0126964
$376$	815.880	0.111904
$377$	516.730	0.0705913
$378$	−555.799	−0.0756275
$379$	−2938.72	−0.398290	−0.199145	−	0.979970i	$-0.563816\pi$
$379$	−2938.72	−0.398290	−0.199145	+	0.979970i	$0.563816\pi$
$380$	1391.65	0.187869
$381$	−469.126	−0.0630814
$382$	1109.66	0.148626
$383$	22.9411	0.00306067	0.00153033	−	0.999999i	$-0.499513\pi$
$383$	22.9411	0.00306067	0.00153033	+	0.999999i	$0.499513\pi$
$384$	925.576	0.123003
$385$	0	0
$386$	−1966.24	−0.259271
$387$	−7548.74	−0.991535
$388$	−3832.18	−0.501417
$389$	−2898.85	−0.377834	−0.188917	−	0.981993i	$-0.560498\pi$
$389$	−2898.85	−0.377834	−0.188917	+	0.981993i	$0.560498\pi$
$390$	51.4705	0.00668285
$391$	−3971.46	−0.513671
$392$	−347.976	−0.0448354
$393$	−1913.20	−0.245568
$394$	−3767.07	−0.481681
$395$	−2295.83	−0.292445
$396$	0	0
$397$	2030.77	0.256730	0.128365	−	0.991727i	$-0.459027\pi$
$397$	2030.77	0.256730	0.128365	+	0.991727i	$0.459027\pi$
$398$	1622.32	0.204320
$399$	−531.491	−0.0666863
$400$	1288.16	0.161020
$401$	−5125.75	−0.638324	−0.319162	−	0.947700i	$-0.603401\pi$
$401$	−5125.75	−0.638324	−0.319162	+	0.947700i	$0.603401\pi$
$402$	−336.763	−0.0417816
$403$	−873.792	−0.108007
$404$	4414.87	0.543683
$405$	3426.14	0.420362
$406$	380.495	0.0465115
$407$	0	0
$408$	289.595	0.0351399
$409$	13227.0	1.59910	0.799549	−	0.600601i	$-0.205072\pi$
$409$	13227.0	1.59910	0.799549	+	0.600601i	$0.205072\pi$
$410$	879.840	0.105981
$411$	−1599.41	−0.191953
$412$	−4580.93	−0.547782
$413$	−14613.4	−1.74111
$414$	2200.15	0.261187
$415$	3187.10	0.376985
$416$	−2446.19	−0.288304
$417$	886.106	0.104059
$418$	0	0
$419$	14065.7	1.63998	0.819991	−	0.572376i	$-0.193978\pi$
$419$	14065.7	1.63998	0.819991	+	0.572376i	$0.193978\pi$
$420$	−532.564	−0.0618725
$421$	7149.16	0.827622	0.413811	−	0.910363i	$-0.364197\pi$
$421$	7149.16	0.827622	0.413811	+	0.910363i	$0.364197\pi$
$422$	3361.96	0.387814
$423$	1914.03	0.220008
$424$	3583.22	0.410416
$425$	870.390	0.0993415
$426$	−521.054	−0.0592609
$427$	17566.9	1.99091
$428$	4294.49	0.485005
$429$	0	0
$430$	1040.10	0.116646
$431$	−8002.27	−0.894329	−0.447164	−	0.894452i	$-0.647566\pi$
$431$	−8002.27	−0.894329	−0.447164	+	0.894452i	$0.647566\pi$
$432$	2031.62	0.226265
$433$	3651.31	0.405244	0.202622	−	0.979257i	$-0.435054\pi$
$433$	3651.31	0.405244	0.202622	+	0.979257i	$0.435054\pi$
$434$	−643.419	−0.0711638
$435$	99.5474	0.0109723
$436$	−15326.0	−1.68344
$437$	4251.12	0.465352
$438$	552.635	0.0602875
$439$	12493.6	1.35829	0.679144	−	0.734005i	$-0.262351\pi$
$439$	12493.6	1.35829	0.679144	+	0.734005i	$0.262351\pi$
$440$	0	0
$441$	−816.342	−0.0881484
$442$	−485.900	−0.0522893
$443$	5985.71	0.641964	0.320982	−	0.947085i	$-0.395987\pi$
$443$	5985.71	0.641964	0.320982	+	0.947085i	$0.395987\pi$
$444$	1828.19	0.195410
$445$	3784.22	0.403122
$446$	−887.346	−0.0942087
$447$	1239.10	0.131112
$448$	6169.00	0.650576
$449$	−12352.2	−1.29830	−0.649152	−	0.760659i	$-0.724876\pi$
$449$	−12352.2	−1.29830	−0.649152	+	0.760659i	$0.724876\pi$
$450$	−482.188	−0.0505123
$451$	0	0
$452$	−8152.52	−0.848368
$453$	1609.29	0.166912
$454$	4632.65	0.478901
$455$	1850.73	0.190689
$456$	−309.987	−0.0318344
$457$	15282.9	1.56434	0.782171	−	0.623064i	$-0.214112\pi$
$457$	15282.9	1.56434	0.782171	+	0.623064i	$0.214112\pi$
$458$	−1897.33	−0.193573
$459$	1372.73	0.139594
$460$	4259.70	0.431760
$461$	6227.38	0.629150	0.314575	−	0.949233i	$-0.398138\pi$
$461$	6227.38	0.629150	0.314575	+	0.949233i	$0.398138\pi$
$462$	0	0
$463$	14097.6	1.41506	0.707529	−	0.706684i	$-0.249810\pi$
$463$	14097.6	1.41506	0.707529	+	0.706684i	$0.249810\pi$
$464$	−1390.83	−0.139154
$465$	−168.335	−0.0167879
$466$	3538.66	0.351771
$467$	10616.8	1.05201	0.526004	−	0.850482i	$-0.323690\pi$
$467$	10616.8	1.05201	0.526004	+	0.850482i	$0.323690\pi$
$468$	−3782.48	−0.373601
$469$	−12109.0	−1.19220
$470$	−263.723	−0.0258822
$471$	−547.575	−0.0535688
$472$	−8523.12	−0.831162
$473$	0	0
$474$	246.910	0.0239260
$475$	−931.681	−0.0899968
$476$	5027.59	0.484116
$477$	8406.13	0.806898
$478$	−1438.72	−0.137668
$479$	−14627.4	−1.39529	−0.697645	−	0.716443i	$-0.745769\pi$
$479$	−14627.4	−1.39529	−0.697645	+	0.716443i	$0.745769\pi$
$480$	−471.256	−0.0448121
$481$	−6353.19	−0.602247
$482$	−939.614	−0.0887931
$483$	−1626.84	−0.153258
$484$	0	0
$485$	2565.56	0.240199
$486$	−1144.59	−0.106831
$487$	19666.3	1.82991	0.914954	−	0.403557i	$-0.132226\pi$
$487$	19666.3	1.82991	0.914954	+	0.403557i	$0.132226\pi$
$488$	10245.7	0.950414
$489$	1575.88	0.145734
$490$	112.479	0.0103700
$491$	8451.42	0.776796	0.388398	−	0.921492i	$-0.373029\pi$
$491$	8451.42	0.776796	0.388398	+	0.921492i	$0.373029\pi$
$492$	1329.63	0.121838
$493$	−939.762	−0.0858514
$494$	520.115	0.0473706
$495$	0	0
$496$	2351.90	0.212910
$497$	−18735.5	−1.69095
$498$	−342.764	−0.0308426
$499$	−734.629	−0.0659049	−0.0329524	−	0.999457i	$-0.510491\pi$
$499$	−734.629	−0.0659049	−0.0329524	+	0.999457i	$0.510491\pi$
$500$	−933.562	−0.0835003
$501$	1850.94	0.165058
$502$	1551.65	0.137956
$503$	14370.7	1.27387	0.636937	−	0.770916i	$-0.280201\pi$
$503$	14370.7	1.27387	0.636937	+	0.770916i	$0.280201\pi$
$504$	−5768.69	−0.509837
$505$	−2955.66	−0.260446
$506$	0	0
$507$	1350.18	0.118272
$508$	−4750.14	−0.414869
$509$	−18638.7	−1.62308	−0.811539	−	0.584298i	$-0.801370\pi$
$509$	−18638.7	−1.62308	−0.811539	+	0.584298i	$0.801370\pi$
$510$	−93.6080	−0.00812751
$511$	19871.1	1.72025
$512$	11232.8	0.969575
$513$	−1469.40	−0.126463
$514$	−2176.68	−0.186788
$515$	3066.83	0.262409
$516$	1571.81	0.134099
$517$	0	0
$518$	−4678.19	−0.396810
$519$	−377.608	−0.0319367
$520$	1079.42	0.0910302
$521$	10565.2	0.888422	0.444211	−	0.895922i	$-0.353484\pi$
$521$	10565.2	0.888422	0.444211	+	0.895922i	$0.353484\pi$
$522$	520.619	0.0436530
$523$	7469.83	0.624537	0.312268	−	0.949994i	$-0.398911\pi$
$523$	7469.83	0.624537	0.312268	+	0.949994i	$0.398911\pi$
$524$	−19372.2	−1.61503
$525$	356.540	0.0296394
$526$	2897.83	0.240212
$527$	1589.14	0.131355
$528$	0	0
$529$	845.251	0.0694708
$530$	−1158.23	−0.0949252
$531$	−19995.0	−1.63410
$532$	−5381.62	−0.438577
$533$	−4620.62	−0.375500
$534$	−406.982	−0.0329810
$535$	−2875.07	−0.232337
$536$	−7062.46	−0.569126
$537$	1984.28	0.159456
$538$	−1086.40	−0.0870597
$539$	0	0
$540$	−1472.36	−0.117334
$541$	−8518.88	−0.676997	−0.338498	−	0.940967i	$-0.609919\pi$
$541$	−8518.88	−0.676997	−0.338498	+	0.940967i	$0.609919\pi$
$542$	551.371	0.0436964
$543$	550.033	0.0434700
$544$	4448.83	0.350628
$545$	10260.4	0.806437
$546$	−199.040	−0.0156010
$547$	−3710.72	−0.290053	−0.145026	−	0.989428i	$-0.546327\pi$
$547$	−3710.72	−0.290053	−0.145026	+	0.989428i	$0.546327\pi$
$548$	−16194.8	−1.26242
$549$	24036.2	1.86856
$550$	0	0
$551$	1005.94	0.0777756
$552$	−948.840	−0.0731618
$553$	8878.15	0.682707
$554$	800.378	0.0613805
$555$	−1223.94	−0.0936093
$556$	8972.28	0.684369
$557$	8174.43	0.621834	0.310917	−	0.950437i	$-0.399364\pi$
$557$	8174.43	0.621834	0.310917	+	0.950437i	$0.399364\pi$
$558$	−880.369	−0.0667903
$559$	−5462.24	−0.413288
$560$	−4981.41	−0.375898
$561$	0	0
$562$	−2099.68	−0.157597
$563$	−6354.30	−0.475669	−0.237835	−	0.971306i	$-0.576438\pi$
$563$	−6354.30	−0.475669	−0.237835	+	0.971306i	$0.576438\pi$
$564$	−398.543	−0.0297547
$565$	5457.94	0.406402
$566$	−5741.56	−0.426388
$567$	−13249.2	−0.981326
$568$	−10927.3	−0.807220
$569$	11023.2	0.812158	0.406079	−	0.913838i	$-0.366896\pi$
$569$	11023.2	0.812158	0.406079	+	0.913838i	$0.366896\pi$
$570$	100.200	0.00736298
$571$	10506.5	0.770020	0.385010	−	0.922912i	$-0.374198\pi$
$571$	10506.5	0.770020	0.385010	+	0.922912i	$0.374198\pi$
$572$	0	0
$573$	−1122.67	−0.0818505
$574$	−3402.41	−0.247411
$575$	−2851.78	−0.206830
$576$	8440.84	0.610593
$577$	15541.6	1.12132	0.560662	−	0.828045i	$-0.310547\pi$
$577$	15541.6	1.12132	0.560662	+	0.828045i	$0.310547\pi$
$578$	−2698.09	−0.194162
$579$	1989.29	0.142784
$580$	1007.97	0.0721614
$581$	−12324.8	−0.880064
$582$	−275.919	−0.0196516
$583$	0	0
$584$	11589.7	0.821204
$585$	2532.29	0.178970
$586$	4076.56	0.287374
$587$	8327.99	0.585576	0.292788	−	0.956177i	$-0.405417\pi$
$587$	8327.99	0.585576	0.292788	+	0.956177i	$0.405417\pi$
$588$	169.980	0.0119215
$589$	−1701.05	−0.118999
$590$	2754.99	0.192239
$591$	3811.25	0.265269
$592$	17100.2	1.18719
$593$	20186.4	1.39790	0.698950	−	0.715170i	$-0.253651\pi$
$593$	20186.4	1.39790	0.698950	+	0.715170i	$0.253651\pi$
$594$	0	0
$595$	−3365.86	−0.231911
$596$	12546.5	0.862289
$597$	−1641.34	−0.112522
$598$	1592.02	0.108867
$599$	9706.44	0.662094	0.331047	−	0.943614i	$-0.392598\pi$
$599$	9706.44	0.662094	0.331047	+	0.943614i	$0.392598\pi$
$600$	207.949	0.0141491
$601$	−4522.84	−0.306972	−0.153486	−	0.988151i	$-0.549050\pi$
$601$	−4522.84	−0.306972	−0.153486	+	0.988151i	$0.549050\pi$
$602$	−4022.13	−0.272309
$603$	−16568.3	−1.11893
$604$	16294.9	1.09773
$605$	0	0
$606$	317.873	0.0213081
$607$	−13750.4	−0.919457	−0.459729	−	0.888059i	$-0.652053\pi$
$607$	−13750.4	−0.919457	−0.459729	+	0.888059i	$0.652053\pi$
$608$	−4762.10	−0.317646
$609$	−384.957	−0.0256145
$610$	−3311.80	−0.219821
$611$	1384.99	0.0917029
$612$	6879.09	0.454364
$613$	−17727.0	−1.16801	−0.584003	−	0.811751i	$-0.698514\pi$
$613$	−17727.0	−1.16801	−0.584003	+	0.811751i	$0.698514\pi$
$614$	−7071.61	−0.464799
$615$	−890.157	−0.0583652
$616$	0	0
$617$	−19390.1	−1.26518	−0.632590	−	0.774487i	$-0.718008\pi$
$617$	−19390.1	−1.26518	−0.632590	+	0.774487i	$0.718008\pi$
$618$	−329.829	−0.0214687
$619$	17350.9	1.12664	0.563321	−	0.826238i	$-0.309524\pi$
$619$	17350.9	1.12664	0.563321	+	0.826238i	$0.309524\pi$
$620$	−1704.48	−0.110409
$621$	−4497.68	−0.290637
$622$	−806.078	−0.0519627
$623$	−14633.9	−0.941081
$624$	727.555	0.0466755
$625$	625.000	0.0400000
$626$	6672.14	0.425994
$627$	0	0
$628$	−5544.48	−0.352307
$629$	11554.4	0.732437
$630$	1864.66	0.117920
$631$	8553.79	0.539653	0.269826	−	0.962909i	$-0.413034\pi$
$631$	8553.79	0.539653	0.269826	+	0.962909i	$0.413034\pi$
$632$	5178.10	0.325908
$633$	−3401.38	−0.213575
$634$	6510.08	0.407805
$635$	3180.12	0.198739
$636$	−1750.34	−0.109128
$637$	−590.702	−0.0367417
$638$	0	0
$639$	−25635.2	−1.58703
$640$	−6274.31	−0.387522
$641$	−17543.7	−1.08102	−0.540511	−	0.841337i	$-0.681769\pi$
$641$	−17543.7	−1.08102	−0.540511	+	0.841337i	$0.681769\pi$
$642$	309.206	0.0190084
$643$	−11059.1	−0.678271	−0.339135	−	0.940738i	$-0.610134\pi$
$643$	−11059.1	−0.678271	−0.339135	+	0.940738i	$0.610134\pi$
$644$	−16472.6	−1.00794
$645$	−1052.29	−0.0642388
$646$	−945.919	−0.0576110
$647$	10935.6	0.664486	0.332243	−	0.943194i	$-0.392195\pi$
$647$	10935.6	0.664486	0.332243	+	0.943194i	$0.392195\pi$
$648$	−7727.45	−0.468461
$649$	0	0
$650$	−348.909	−0.0210544
$651$	650.964	0.0391909
$652$	15956.6	0.958452
$653$	−12195.4	−0.730844	−0.365422	−	0.930842i	$-0.619075\pi$
$653$	−12195.4	−0.730844	−0.365422	+	0.930842i	$0.619075\pi$
$654$	−1103.48	−0.0659777
$655$	12969.3	0.773665
$656$	12436.9	0.740210
$657$	27189.0	1.61453
$658$	1019.84	0.0604216
$659$	−20533.8	−1.21378	−0.606891	−	0.794785i	$-0.707583\pi$
$659$	−20533.8	−1.21378	−0.606891	+	0.794785i	$0.707583\pi$
$660$	0	0
$661$	22534.4	1.32600	0.663001	−	0.748618i	$-0.269282\pi$
$661$	22534.4	1.32600	0.663001	+	0.748618i	$0.269282\pi$
$662$	−1539.38	−0.0903772
$663$	491.598	0.0287965
$664$	−7188.31	−0.420121
$665$	3602.88	0.210096
$666$	−6401.01	−0.372424
$667$	3079.07	0.178744
$668$	18741.7	1.08554
$669$	897.752	0.0518821
$670$	2282.85	0.131633
$671$	0	0
$672$	1822.38	0.104613
$673$	15795.7	0.904722	0.452361	−	0.891835i	$-0.350582\pi$
$673$	15795.7	0.904722	0.452361	+	0.891835i	$0.350582\pi$
$674$	3040.40	0.173756
$675$	985.717	0.0562078
$676$	13671.3	0.777839
$677$	3288.35	0.186679	0.0933394	−	0.995634i	$-0.470246\pi$
$677$	3288.35	0.186679	0.0933394	+	0.995634i	$0.470246\pi$
$678$	−586.985	−0.0332493
$679$	−9921.23	−0.560739
$680$	−1963.11	−0.110709
$681$	−4686.98	−0.263738
$682$	0	0
$683$	−14806.7	−0.829519	−0.414760	−	0.909931i	$-0.636134\pi$
$683$	−14806.7	−0.829519	−0.414760	+	0.909931i	$0.636134\pi$
$684$	−7363.49	−0.411623
$685$	10842.1	0.604751
$686$	4400.07	0.244891
$687$	1919.58	0.106604
$688$	14702.2	0.814701
$689$	6082.64	0.336328
$690$	306.701	0.0169216
$691$	21669.1	1.19295	0.596477	−	0.802630i	$-0.296567\pi$
$691$	21669.1	1.19295	0.596477	+	0.802630i	$0.296567\pi$
$692$	−3823.48	−0.210039
$693$	0	0
$694$	8189.66	0.447947
$695$	−6006.75	−0.327840
$696$	−224.523	−0.0122278
$697$	8403.40	0.456673
$698$	−5458.45	−0.295996
$699$	−3580.16	−0.193725
$700$	3610.15	0.194930
$701$	−8397.07	−0.452429	−0.226215	−	0.974078i	$-0.572635\pi$
$701$	−8397.07	−0.452429	−0.226215	+	0.974078i	$0.572635\pi$
$702$	−550.281	−0.0295855
$703$	−12368.0	−0.663539
$704$	0	0
$705$	266.816	0.0142537
$706$	2752.47	0.146729
$707$	11429.8	0.608006
$708$	4163.39	0.221002
$709$	−14939.9	−0.791370	−0.395685	−	0.918386i	$-0.629493\pi$
$709$	−14939.9	−0.791370	−0.395685	+	0.918386i	$0.629493\pi$
$710$	3532.13	0.186702
$711$	12147.7	0.640750
$712$	−8535.08	−0.449249
$713$	−5206.73	−0.273483
$714$	361.989	0.0189735
$715$	0	0
$716$	20091.8	1.04870
$717$	1455.59	0.0758160
$718$	−1468.35	−0.0763206
$719$	125.385	0.00650359	0.00325179	−	0.999995i	$-0.498965\pi$
$719$	125.385	0.00650359	0.00325179	+	0.999995i	$0.498965\pi$
$720$	−6815.91	−0.352797
$721$	−11859.7	−0.612590
$722$	−3987.97	−0.205564
$723$	950.633	0.0488996
$724$	5569.37	0.285890
$725$	−674.814	−0.0345682
$726$	0	0
$727$	15436.0	0.787470	0.393735	−	0.919224i	$-0.371183\pi$
$727$	15436.0	0.787470	0.393735	+	0.919224i	$0.371183\pi$
$728$	−4174.20	−0.212508
$729$	−17343.2	−0.881124
$730$	−3746.21	−0.189936
$731$	9934.02	0.502631
$732$	−5004.85	−0.252711
$733$	−25237.5	−1.27172	−0.635858	−	0.771806i	$-0.719354\pi$
$733$	−25237.5	−1.27172	−0.635858	+	0.771806i	$0.719354\pi$
$734$	8464.12	0.425635
$735$	−113.798	−0.00571089
$736$	−14576.3	−0.730013
$737$	0	0
$738$	−4655.40	−0.232206
$739$	24196.0	1.20442	0.602209	−	0.798339i	$-0.294288\pi$
$739$	24196.0	1.20442	0.602209	+	0.798339i	$0.294288\pi$
$740$	−12393.0	−0.615642
$741$	−526.215	−0.0260877
$742$	4478.97	0.221601
$743$	13967.8	0.689676	0.344838	−	0.938662i	$-0.387934\pi$
$743$	13967.8	0.689676	0.344838	+	0.938662i	$0.387934\pi$
$744$	379.669	0.0187088
$745$	−8399.61	−0.413071
$746$	3814.31	0.187201
$747$	−16863.6	−0.825978
$748$	0	0
$749$	11118.1	0.542386
$750$	−67.2170	−0.00327255
$751$	772.044	0.0375130	0.0187565	−	0.999824i	$-0.494029\pi$
$751$	772.044	0.0375130	0.0187565	+	0.999824i	$0.494029\pi$
$752$	−3727.83	−0.180771
$753$	−1569.85	−0.0759741
$754$	376.718	0.0181953
$755$	−10909.1	−0.525859
$756$	5693.74	0.273915
$757$	12428.9	0.596746	0.298373	−	0.954449i	$-0.403556\pi$
$757$	12428.9	0.596746	0.298373	+	0.954449i	$0.403556\pi$
$758$	−2142.45	−0.102661
$759$	0	0
$760$	2101.35	0.100295
$761$	16501.3	0.786035	0.393018	−	0.919531i	$-0.371431\pi$
$761$	16501.3	0.786035	0.393018	+	0.919531i	$0.371431\pi$
$762$	−342.012	−0.0162596
$763$	−39677.8	−1.88261
$764$	−11367.6	−0.538308
$765$	−4605.40	−0.217658
$766$	16.7250	0.000788904 0
$767$	−14468.3	−0.681121
$768$	−1207.86	−0.0567512
$769$	−5711.77	−0.267844	−0.133922	−	0.990992i	$-0.542757\pi$
$769$	−5711.77	−0.267844	−0.133922	+	0.990992i	$0.542757\pi$
$770$	0	0
$771$	2202.20	0.102867
$772$	20142.6	0.939053
$773$	−2620.00	−0.121908	−0.0609540	−	0.998141i	$-0.519414\pi$
$773$	−2620.00	−0.121908	−0.0609540	+	0.998141i	$0.519414\pi$
$774$	−5503.35	−0.255573
$775$	1141.11	0.0528903
$776$	−5786.47	−0.267683
$777$	4733.05	0.218529
$778$	−2113.38	−0.0973888
$779$	−8995.14	−0.413716
$780$	−527.277	−0.0242045
$781$	0	0
$782$	−2895.36	−0.132402
$783$	−1064.28	−0.0485751
$784$	1589.93	0.0724277
$785$	3711.91	0.168769
$786$	−1394.81	−0.0632966
$787$	5340.68	0.241899	0.120950	−	0.992659i	$-0.461406\pi$
$787$	5340.68	0.241899	0.120950	+	0.992659i	$0.461406\pi$
$788$	38590.9	1.74460
$789$	−2931.81	−0.132288
$790$	−1673.76	−0.0753793
$791$	−21106.2	−0.948738
$792$	0	0
$793$	17392.5	0.778846
$794$	1480.52	0.0661735
$795$	1171.81	0.0522767
$796$	−16619.4	−0.740026
$797$	−25516.3	−1.13405	−0.567023	−	0.823702i	$-0.691905\pi$
$797$	−25516.3	−1.13405	−0.567023	+	0.823702i	$0.691905\pi$
$798$	−387.479	−0.0171887
$799$	−2518.83	−0.111527
$800$	3194.56	0.141181
$801$	−20023.0	−0.883245
$802$	−3736.89	−0.164531
$803$	0	0
$804$	3449.88	0.151328
$805$	11028.1	0.482842
$806$	−637.032	−0.0278393
$807$	1099.14	0.0479450
$808$	6666.31	0.290248
$809$	−18588.3	−0.807826	−0.403913	−	0.914797i	$-0.632350\pi$
$809$	−18588.3	−0.807826	−0.403913	+	0.914797i	$0.632350\pi$
$810$	2497.80	0.108350
$811$	−10519.5	−0.455474	−0.227737	−	0.973723i	$-0.573133\pi$
$811$	−10519.5	−0.455474	−0.227737	+	0.973723i	$0.573133\pi$
$812$	−3897.89	−0.168460
$813$	−557.837	−0.0240642
$814$	0	0
$815$	−10682.6	−0.459137
$816$	−1323.18	−0.0567656
$817$	−10633.5	−0.455350
$818$	9643.02	0.412176
$819$	−9792.55	−0.417801
$820$	−9013.31	−0.383852
$821$	−7678.75	−0.326419	−0.163210	−	0.986591i	$-0.552185\pi$
$821$	−7678.75	−0.326419	−0.163210	+	0.986591i	$0.552185\pi$
$822$	−1166.03	−0.0494770
$823$	26903.4	1.13948	0.569740	−	0.821825i	$-0.307044\pi$
$823$	26903.4	1.13948	0.569740	+	0.821825i	$0.307044\pi$
$824$	−6917.05	−0.292436
$825$	0	0
$826$	−10653.8	−0.448779
$827$	−5583.42	−0.234770	−0.117385	−	0.993087i	$-0.537451\pi$
$827$	−5583.42	−0.234770	−0.117385	+	0.993087i	$0.537451\pi$
$828$	−22538.9	−0.945992
$829$	−136.924	−0.00573649	−0.00286825	−	0.999996i	$-0.500913\pi$
$829$	−136.924	−0.00573649	−0.00286825	+	0.999996i	$0.500913\pi$
$830$	2323.53	0.0971699
$831$	−809.764	−0.0338031
$832$	6107.76	0.254505
$833$	1074.29	0.0446843
$834$	646.009	0.0268219
$835$	−12547.2	−0.520016
$836$	0	0
$837$	1799.70	0.0743212
$838$	10254.5	0.422714
$839$	−15151.1	−0.623450	−0.311725	−	0.950172i	$-0.600907\pi$
$839$	−15151.1	−0.623450	−0.311725	+	0.950172i	$0.600907\pi$
$840$	−804.154	−0.0330309
$841$	−23660.4	−0.970126
$842$	5212.04	0.213324
$843$	2124.31	0.0867912
$844$	−34440.8	−1.40462
$845$	−9152.65	−0.372616
$846$	1395.41	0.0567083
$847$	0	0
$848$	−16372.0	−0.662993
$849$	5808.88	0.234818
$850$	634.552	0.0256058
$851$	−37857.2	−1.52495
$852$	5337.81	0.214636
$853$	10425.4	0.418475	0.209237	−	0.977865i	$-0.432902\pi$
$853$	10425.4	0.418475	0.209237	+	0.977865i	$0.432902\pi$
$854$	12807.0	0.513169
$855$	4929.70	0.197184
$856$	6484.55	0.258922
$857$	27375.9	1.09118	0.545592	−	0.838051i	$-0.316305\pi$
$857$	27375.9	1.09118	0.545592	+	0.838051i	$0.316305\pi$
$858$	0	0
$859$	−28410.3	−1.12846	−0.564230	−	0.825618i	$-0.690827\pi$
$859$	−28410.3	−1.12846	−0.564230	+	0.825618i	$0.690827\pi$
$860$	−10655.0	−0.422480
$861$	3442.31	0.136253
$862$	−5833.99	−0.230518
$863$	−27051.8	−1.06704	−0.533520	−	0.845788i	$-0.679131\pi$
$863$	−27051.8	−1.06704	−0.533520	+	0.845788i	$0.679131\pi$
$864$	5038.29	0.198387
$865$	2559.74	0.100617
$866$	2661.96	0.104454
$867$	2729.73	0.106928
$868$	6591.35	0.257748
$869$	0	0
$870$	72.5743	0.00282816
$871$	−11988.8	−0.466388
$872$	−23141.7	−0.898714
$873$	−13574.9	−0.526278
$874$	3099.25	0.119947
$875$	−2416.92	−0.0933793
$876$	−5661.33	−0.218355
$877$	7719.49	0.297228	0.148614	−	0.988895i	$-0.452519\pi$
$877$	7719.49	0.297228	0.148614	+	0.988895i	$0.452519\pi$
$878$	9108.38	0.350106
$879$	−4124.36	−0.158261
$880$	0	0
$881$	−47510.2	−1.81687	−0.908434	−	0.418029i	$-0.862721\pi$
$881$	−47510.2	−1.81687	−0.908434	+	0.418029i	$0.862721\pi$
$882$	−595.148	−0.0227207
$883$	31428.7	1.19780	0.598902	−	0.800822i	$-0.295604\pi$
$883$	31428.7	1.19780	0.598902	+	0.800822i	$0.295604\pi$
$884$	4977.68	0.189386
$885$	−2787.30	−0.105869
$886$	4363.84	0.165470
$887$	40887.6	1.54777	0.773884	−	0.633327i	$-0.218311\pi$
$887$	40887.6	1.54777	0.773884	+	0.633327i	$0.218311\pi$
$888$	2760.51	0.104321
$889$	−12297.7	−0.463952
$890$	2758.86	0.103907
$891$	0	0
$892$	9090.20	0.341214
$893$	2696.20	0.101036
$894$	903.353	0.0337949
$895$	−13451.1	−0.502368
$896$	24263.2	0.904663
$897$	−1610.69	−0.0599547
$898$	−9005.31	−0.334645
$899$	−1232.06	−0.0457081
$900$	4939.66	0.182950
$901$	−11062.3	−0.409034
$902$	0	0
$903$	4069.30	0.149964
$904$	−12310.0	−0.452905
$905$	−3728.58	−0.136953
$906$	1173.24	0.0430225
$907$	−33552.6	−1.22833	−0.614165	−	0.789177i	$-0.710507\pi$
$907$	−33552.6	−1.22833	−0.614165	+	0.789177i	$0.710507\pi$
$908$	−47458.1	−1.73453
$909$	15639.0	0.570640
$910$	1349.26	0.0491511
$911$	29572.6	1.07550	0.537751	−	0.843103i	$-0.319274\pi$
$911$	29572.6	1.07550	0.537751	+	0.843103i	$0.319274\pi$
$912$	1416.36	0.0514258
$913$	0	0
$914$	11141.9	0.403217
$915$	3350.64	0.121059
$916$	19436.8	0.701101
$917$	−50153.1	−1.80611
$918$	1000.78	0.0359811
$919$	48752.3	1.74994	0.874968	−	0.484181i	$-0.160882\pi$
$919$	48752.3	1.74994	0.874968	+	0.484181i	$0.160882\pi$
$920$	6432.02	0.230497
$921$	7154.53	0.255972
$922$	4540.02	0.162167
$923$	−18549.5	−0.661501
$924$	0	0
$925$	8296.83	0.294917
$926$	10277.8	0.364739
$927$	−16227.2	−0.574942
$928$	−3449.17	−0.122009
$929$	−5301.99	−0.187247	−0.0936235	−	0.995608i	$-0.529845\pi$
$929$	−5301.99	−0.187247	−0.0936235	+	0.995608i	$0.529845\pi$
$930$	−122.723	−0.00432716
$931$	−1149.94	−0.0404810
$932$	−36250.9	−1.27408
$933$	815.531	0.0286166
$934$	7740.12	0.271161
$935$	0	0
$936$	−5711.42	−0.199448
$937$	48059.5	1.67560	0.837800	−	0.545978i	$-0.183842\pi$
$937$	48059.5	1.67560	0.837800	+	0.545978i	$0.183842\pi$
$938$	−8827.96	−0.307295
$939$	−6750.38	−0.234601
$940$	2701.65	0.0937426
$941$	−13528.1	−0.468654	−0.234327	−	0.972158i	$-0.575289\pi$
$941$	−13528.1	−0.468654	−0.234327	+	0.972158i	$0.575289\pi$
$942$	−399.205	−0.0138076
$943$	−27533.2	−0.950801
$944$	38942.8	1.34267
$945$	−3811.84	−0.131216
$946$	0	0
$947$	−38482.1	−1.32049	−0.660243	−	0.751052i	$-0.729547\pi$
$947$	−38482.1	−1.32049	−0.660243	+	0.751052i	$0.729547\pi$
$948$	−2529.41	−0.0866576
$949$	19673.8	0.672961
$950$	−679.235	−0.0231971
$951$	−6586.42	−0.224584
$952$	7591.50	0.258447
$953$	20764.6	0.705806	0.352903	−	0.935660i	$-0.385195\pi$
$953$	20764.6	0.705806	0.352903	+	0.935660i	$0.385195\pi$
$954$	6128.42	0.207982
$955$	7610.40	0.257871
$956$	14738.6	0.498620
$957$	0	0
$958$	−10664.0	−0.359644
$959$	−41927.1	−1.41178
$960$	1176.65	0.0395586
$961$	−27707.6	−0.930065
$962$	−4631.75	−0.155232
$963$	15212.6	0.509053
$964$	9625.65	0.321599
$965$	−13485.1	−0.449844
$966$	−1186.04	−0.0395032
$967$	−16064.0	−0.534212	−0.267106	−	0.963667i	$-0.586067\pi$
$967$	−16064.0	−0.534212	−0.267106	+	0.963667i	$0.586067\pi$
$968$	0	0
$969$	957.012	0.0317272
$970$	1870.41	0.0619125
$971$	−15509.7	−0.512595	−0.256298	−	0.966598i	$-0.582503\pi$
$971$	−15509.7	−0.512595	−0.256298	+	0.966598i	$0.582503\pi$
$972$	11725.5	0.386929
$973$	23228.5	0.765337
$974$	14337.6	0.471669
$975$	353.001	0.0115949
$976$	−46813.6	−1.53531
$977$	−46382.5	−1.51884	−0.759420	−	0.650600i	$-0.774517\pi$
$977$	−46382.5	−1.51884	−0.759420	+	0.650600i	$0.774517\pi$
$978$	1148.89	0.0375637
$979$	0	0
$980$	−1152.26	−0.0375589
$981$	−54289.9	−1.76691
$982$	6161.44	0.200223
$983$	47620.6	1.54513	0.772564	−	0.634937i	$-0.218974\pi$
$983$	47620.6	1.54513	0.772564	+	0.634937i	$0.218974\pi$
$984$	2007.70	0.0650437
$985$	−25835.8	−0.835732
$986$	−685.126	−0.0221287
$987$	−1031.80	−0.0332750
$988$	−5328.19	−0.171571
$989$	−32548.2	−1.04648
$990$	0	0
$991$	−5116.46	−0.164006	−0.0820028	−	0.996632i	$-0.526132\pi$
$991$	−5116.46	−0.164006	−0.0820028	+	0.996632i	$0.526132\pi$
$992$	5832.57	0.186678
$993$	1557.43	0.0497720
$994$	−13659.0	−0.435852
$995$	11126.4	0.354502
$996$	3511.36	0.111708
$997$	−57836.2	−1.83720	−0.918601	−	0.395187i	$-0.870680\pi$
$997$	−57836.2	−1.83720	−0.918601	+	0.395187i	$0.870680\pi$
$998$	−535.576	−0.0169873
$999$	13085.3	0.414416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.4.a.n.1.4	yes	6
11.10	odd	2		605.4.a.m.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.4.a.m.1.3	✓	6	11.10	odd	2
605.4.a.n.1.4	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

\(f(q)\)	\(=\)	\(q+0.729042 q^{2} -0.737592 q^{3} -7.46850 q^{4} +5.00000 q^{5} -0.537736 q^{6} -19.3354 q^{7} -11.2772 q^{8} -26.4560 q^{9} +3.64521 q^{10} +5.50870 q^{12} -19.1434 q^{13} -14.0963 q^{14} -3.68796 q^{15} +51.5264 q^{16} +34.8156 q^{17} -19.2875 q^{18} -37.2672 q^{19} -37.3425 q^{20} +14.2616 q^{21} -114.071 q^{23} +8.31796 q^{24} +25.0000 q^{25} -13.9564 q^{26} +39.4287 q^{27} +144.406 q^{28} -26.9925 q^{29} -2.68868 q^{30} +45.6445 q^{31} +127.782 q^{32} +25.3821 q^{34} -96.6769 q^{35} +197.586 q^{36} +331.873 q^{37} -27.1694 q^{38} +14.1200 q^{39} -56.3860 q^{40} +241.369 q^{41} +10.3973 q^{42} +285.332 q^{43} -132.280 q^{45} -83.1628 q^{46} -72.3478 q^{47} -38.0055 q^{48} +30.8567 q^{49} +18.2261 q^{50} -25.6797 q^{51} +142.973 q^{52} -317.740 q^{53} +28.7452 q^{54} +218.049 q^{56} +27.4880 q^{57} -19.6787 q^{58} +755.784 q^{59} +27.5435 q^{60} -908.535 q^{61} +33.2768 q^{62} +511.536 q^{63} -319.053 q^{64} -95.7171 q^{65} +626.261 q^{67} -260.020 q^{68} +84.1380 q^{69} -70.4815 q^{70} +968.978 q^{71} +298.349 q^{72} -1027.71 q^{73} +241.950 q^{74} -18.4398 q^{75} +278.330 q^{76} +10.2941 q^{78} -459.166 q^{79} +257.632 q^{80} +685.229 q^{81} +175.968 q^{82} +637.420 q^{83} -106.513 q^{84} +174.078 q^{85} +208.019 q^{86} +19.9095 q^{87} +756.844 q^{89} -96.4376 q^{90} +370.145 q^{91} +851.941 q^{92} -33.6670 q^{93} -52.7446 q^{94} -186.336 q^{95} -94.2513 q^{96} +513.113 q^{97} +22.4958 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 12 q^{3} + 27 q^{4} + 30 q^{5} + 45 q^{6} - 10 q^{7} + 51 q^{8} + 104 q^{9} + 15 q^{10} + 137 q^{12} + 80 q^{13} + 7 q^{14} + 60 q^{15} + 155 q^{16} - 12 q^{17} + 14 q^{18} + 86 q^{19} + 135 q^{20}+ \cdots + 6850 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 + 12 * q^3 + 27 * q^4 + 30 * q^5 + 45 * q^6 - 10 * q^7 + 51 * q^8 + 104 * q^9 + 15 * q^10 + 137 * q^12 + 80 * q^13 + 7 * q^14 + 60 * q^15 + 155 * q^16 - 12 * q^17 + 14 * q^18 + 86 * q^19 + 135 * q^20 + 14 * q^21 + 406 * q^23 + 229 * q^24 + 150 * q^25 + 352 * q^26 + 342 * q^27 - 633 * q^28 - 372 * q^29 + 225 * q^30 - 122 * q^31 + 711 * q^32 - 56 * q^34 - 50 * q^35 + 410 * q^36 + 576 * q^37 + 652 * q^38 - 210 * q^39 + 255 * q^40 + 46 * q^41 - 1375 * q^42 + 10 * q^43 + 520 * q^45 - 1632 * q^46 + 568 * q^47 + 2813 * q^48 + 1012 * q^49 + 75 * q^50 - 908 * q^51 + 1036 * q^52 - 184 * q^53 + 2455 * q^54 - 1961 * q^56 - 2052 * q^57 - 186 * q^58 + 1598 * q^59 + 685 * q^60 + 1598 * q^61 - 450 * q^62 + 286 * q^63 - 653 * q^64 + 400 * q^65 + 528 * q^67 - 2104 * q^68 - 156 * q^69 + 35 * q^70 + 142 * q^71 + 3678 * q^72 - 268 * q^73 + 1272 * q^74 + 300 * q^75 + 3136 * q^76 + 900 * q^78 - 624 * q^79 + 775 * q^80 + 2538 * q^81 + 723 * q^82 - 2786 * q^83 - 8583 * q^84 - 60 * q^85 - 3997 * q^86 + 1224 * q^87 + 634 * q^89 + 70 * q^90 + 1690 * q^91 + 1164 * q^92 + 3304 * q^93 + 1681 * q^94 + 430 * q^95 + 5821 * q^96 + 648 * q^97 + 6850 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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