LMFDB - Embedded newform 2070.4.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,4,Mod(1,2070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2070.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8,0,16,-20,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 60x^{2} - 45x + 108 $ x^4 - 2x^3 - 60x^2 - 45*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.12571$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -24.6431 q^{7} -8.00000 q^{8} +10.0000 q^{10} +17.3867 q^{11} +4.00699 q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740 q^{17} +79.3172 q^{19} -20.0000 q^{20} -34.7734 q^{22} +23.0000 q^{23} +25.0000 q^{25} -8.01398 q^{26} -98.5723 q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000 q^{32} +96.5480 q^{34} +123.215 q^{35} -422.904 q^{37} -158.634 q^{38} +40.0000 q^{40} +170.251 q^{41} -228.920 q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087 q^{47} +264.282 q^{49} -50.0000 q^{50} +16.0280 q^{52} +260.354 q^{53} -86.9335 q^{55} +197.145 q^{56} -508.535 q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392 q^{62} +64.0000 q^{64} -20.0349 q^{65} -820.011 q^{67} -193.096 q^{68} -246.431 q^{70} -614.845 q^{71} +511.586 q^{73} +845.808 q^{74} +317.269 q^{76} -428.462 q^{77} +160.464 q^{79} -80.0000 q^{80} -340.502 q^{82} +32.5646 q^{83} +241.370 q^{85} +457.840 q^{86} -139.094 q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000 q^{92} +1160.17 q^{94} -396.586 q^{95} -249.798 q^{97} -528.563 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 8 * q^7 - 32 * q^8 + 40 * q^10 - 21 * q^11 + 70 * q^13 - 16 * q^14 + 64 * q^16 - 56 * q^17 + 173 * q^19 - 80 * q^20 + 42 * q^22 + 92 * q^23 + 100 * q^25 - 140 * q^26 + 32 * q^28 + 118 * q^29 + 17 * q^31 - 128 * q^32 + 112 * q^34 - 40 * q^35 - 343 * q^37 - 346 * q^38 + 160 * q^40 - 139 * q^41 - 50 * q^43 - 84 * q^44 - 184 * q^46 - 367 * q^47 - 124 * q^49 - 200 * q^50 + 280 * q^52 + 353 * q^53 + 105 * q^55 - 64 * q^56 - 236 * q^58 + 453 * q^59 - 327 * q^61 - 34 * q^62 + 256 * q^64 - 350 * q^65 - 455 * q^67 - 224 * q^68 + 80 * q^70 - 195 * q^71 - 633 * q^73 + 686 * q^74 + 692 * q^76 + 2 * q^77 - 1140 * q^79 - 320 * q^80 + 278 * q^82 + 1199 * q^83 + 280 * q^85 + 100 * q^86 + 168 * q^88 + 2170 * q^89 + 557 * q^91 + 368 * q^92 + 734 * q^94 - 865 * q^95 - 703 * q^97 + 248 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−24.6431	−1.33060	−0.665301	−	0.746575i	$-0.731697\pi$
$7$	−24.6431	−1.33060	−0.665301	+	0.746575i	$0.731697\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	17.3867	0.476572	0.238286	−	0.971195i	$-0.423414\pi$
$11$	17.3867	0.476572	0.238286	+	0.971195i	$0.423414\pi$
$12$	0	0
$13$	4.00699	0.0854876	0.0427438	−	0.999086i	$-0.486390\pi$
$13$	4.00699	0.0854876	0.0427438	+	0.999086i	$0.486390\pi$
$14$	49.2862	0.940877
$15$	0	0
$16$	16.0000	0.250000
$17$	−48.2740	−0.688716	−0.344358	−	0.938838i	$-0.611903\pi$
$17$	−48.2740	−0.688716	−0.344358	+	0.938838i	$0.611903\pi$
$18$	0	0
$19$	79.3172	0.957717	0.478859	−	0.877892i	$-0.341051\pi$
$19$	79.3172	0.957717	0.478859	+	0.877892i	$0.341051\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−34.7734	−0.336987
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−8.01398	−0.0604488
$27$	0	0
$28$	−98.5723	−0.665301
$29$	254.267	1.62815	0.814074	−	0.580761i	$-0.197245\pi$
$29$	254.267	1.62815	0.814074	+	0.580761i	$0.197245\pi$
$30$	0	0
$31$	−220.696	−1.27865	−0.639325	−	0.768937i	$-0.720786\pi$
$31$	−220.696	−1.27865	−0.639325	+	0.768937i	$0.720786\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	96.5480	0.486996
$35$	123.215	0.595063
$36$	0	0
$37$	−422.904	−1.87905	−0.939527	−	0.342476i	$-0.888735\pi$
$37$	−422.904	−1.87905	−0.939527	+	0.342476i	$0.888735\pi$
$38$	−158.634	−0.677208
$39$	0	0
$40$	40.0000	0.158114
$41$	170.251	0.648505	0.324252	−	0.945971i	$-0.394887\pi$
$41$	170.251	0.648505	0.324252	+	0.945971i	$0.394887\pi$
$42$	0	0
$43$	−228.920	−0.811860	−0.405930	−	0.913904i	$-0.633052\pi$
$43$	−228.920	−0.811860	−0.405930	+	0.913904i	$0.633052\pi$
$44$	69.5468	0.238286
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−580.087	−1.80031	−0.900154	−	0.435572i	$-0.856546\pi$
$47$	−580.087	−1.80031	−0.900154	+	0.435572i	$0.856546\pi$
$48$	0	0
$49$	264.282	0.770500
$50$	−50.0000	−0.141421
$51$	0	0
$52$	16.0280	0.0427438
$53$	260.354	0.674762	0.337381	−	0.941368i	$-0.390459\pi$
$53$	260.354	0.674762	0.337381	+	0.941368i	$0.390459\pi$
$54$	0	0
$55$	−86.9335	−0.213129
$56$	197.145	0.470439
$57$	0	0
$58$	−508.535	−1.15127
$59$	−353.130	−0.779215	−0.389607	−	0.920981i	$-0.627389\pi$
$59$	−353.130	−0.779215	−0.389607	+	0.920981i	$0.627389\pi$
$60$	0	0
$61$	−80.6108	−0.169199	−0.0845996	−	0.996415i	$-0.526961\pi$
$61$	−80.6108	−0.169199	−0.0845996	+	0.996415i	$0.526961\pi$
$62$	441.392	0.904142
$63$	0	0
$64$	64.0000	0.125000
$65$	−20.0349	−0.0382312
$66$	0	0
$67$	−820.011	−1.49523	−0.747614	−	0.664133i	$-0.768801\pi$
$67$	−820.011	−1.49523	−0.747614	+	0.664133i	$0.768801\pi$
$68$	−193.096	−0.344358
$69$	0	0
$70$	−246.431	−0.420773
$71$	−614.845	−1.02773	−0.513864	−	0.857872i	$-0.671786\pi$
$71$	−614.845	−1.02773	−0.513864	+	0.857872i	$0.671786\pi$
$72$	0	0
$73$	511.586	0.820228	0.410114	−	0.912034i	$-0.365489\pi$
$73$	511.586	0.820228	0.410114	+	0.912034i	$0.365489\pi$
$74$	845.808	1.32869
$75$	0	0
$76$	317.269	0.478859
$77$	−428.462	−0.634127
$78$	0	0
$79$	160.464	0.228526	0.114263	−	0.993451i	$-0.463549\pi$
$79$	160.464	0.228526	0.114263	+	0.993451i	$0.463549\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	−340.502	−0.458562
$83$	32.5646	0.0430654	0.0215327	−	0.999768i	$-0.493145\pi$
$83$	32.5646	0.0430654	0.0215327	+	0.999768i	$0.493145\pi$
$84$	0	0
$85$	241.370	0.308003
$86$	457.840	0.574072
$87$	0	0
$88$	−139.094	−0.168494
$89$	25.0375	0.0298199	0.0149100	−	0.999889i	$-0.495254\pi$
$89$	25.0375	0.0298199	0.0149100	+	0.999889i	$0.495254\pi$
$90$	0	0
$91$	−98.7445	−0.113750
$92$	92.0000	0.104257
$93$	0	0
$94$	1160.17	1.27301
$95$	−396.586	−0.428304
$96$	0	0
$97$	−249.798	−0.261476	−0.130738	−	0.991417i	$-0.541735\pi$
$97$	−249.798	−0.261476	−0.130738	+	0.991417i	$0.541735\pi$
$98$	−528.563	−0.544826
$99$	0	0
$100$	100.000	0.100000
$101$	−620.493	−0.611300	−0.305650	−	0.952144i	$-0.598874\pi$
$101$	−620.493	−0.611300	−0.305650	+	0.952144i	$0.598874\pi$
$102$	0	0
$103$	1473.24	1.40935	0.704673	−	0.709532i	$-0.251094\pi$
$103$	1473.24	1.40935	0.704673	+	0.709532i	$0.251094\pi$
$104$	−32.0559	−0.0302244
$105$	0	0
$106$	−520.708	−0.477129
$107$	940.141	0.849410	0.424705	−	0.905332i	$-0.360378\pi$
$107$	940.141	0.849410	0.424705	+	0.905332i	$0.360378\pi$
$108$	0	0
$109$	−636.264	−0.559111	−0.279555	−	0.960130i	$-0.590187\pi$
$109$	−636.264	−0.559111	−0.279555	+	0.960130i	$0.590187\pi$
$110$	173.867	0.150705
$111$	0	0
$112$	−394.289	−0.332650
$113$	−832.451	−0.693013	−0.346506	−	0.938048i	$-0.612632\pi$
$113$	−832.451	−0.693013	−0.346506	+	0.938048i	$0.612632\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	1017.07	0.814074
$117$	0	0
$118$	706.261	0.550988
$119$	1189.62	0.916406
$120$	0	0
$121$	−1028.70	−0.772879
$122$	161.222	0.119642
$123$	0	0
$124$	−882.783	−0.639325
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1614.60	1.12813	0.564067	−	0.825729i	$-0.309236\pi$
$127$	1614.60	1.12813	0.564067	+	0.825729i	$0.309236\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	40.0699	0.0270335
$131$	1974.56	1.31693	0.658464	−	0.752612i	$-0.271206\pi$
$131$	1974.56	1.31693	0.658464	+	0.752612i	$0.271206\pi$
$132$	0	0
$133$	−1954.62	−1.27434
$134$	1640.02	1.05729
$135$	0	0
$136$	386.192	0.243498
$137$	−753.805	−0.470087	−0.235044	−	0.971985i	$-0.575523\pi$
$137$	−753.805	−0.470087	−0.235044	+	0.971985i	$0.575523\pi$
$138$	0	0
$139$	1014.10	0.618810	0.309405	−	0.950930i	$-0.399870\pi$
$139$	1014.10	0.618810	0.309405	+	0.950930i	$0.399870\pi$
$140$	492.862	0.297532
$141$	0	0
$142$	1229.69	0.726714
$143$	69.6683	0.0407410
$144$	0	0
$145$	−1271.34	−0.728130
$146$	−1023.17	−0.579989
$147$	0	0
$148$	−1691.62	−0.939527
$149$	2771.80	1.52399	0.761995	−	0.647583i	$-0.224220\pi$
$149$	2771.80	1.52399	0.761995	+	0.647583i	$0.224220\pi$
$150$	0	0
$151$	3108.62	1.67534	0.837668	−	0.546180i	$-0.183918\pi$
$151$	3108.62	1.67534	0.837668	+	0.546180i	$0.183918\pi$
$152$	−634.538	−0.338604
$153$	0	0
$154$	856.924	0.448396
$155$	1103.48	0.571830
$156$	0	0
$157$	3712.87	1.88739	0.943693	−	0.330822i	$-0.107326\pi$
$157$	3712.87	1.88739	0.943693	+	0.330822i	$0.107326\pi$
$158$	−320.928	−0.161593
$159$	0	0
$160$	160.000	0.0790569
$161$	−566.791	−0.277450
$162$	0	0
$163$	915.791	0.440063	0.220032	−	0.975493i	$-0.429384\pi$
$163$	915.791	0.440063	0.220032	+	0.975493i	$0.429384\pi$
$164$	681.003	0.324252
$165$	0	0
$166$	−65.1292	−0.0304518
$167$	−1432.32	−0.663688	−0.331844	−	0.943334i	$-0.607671\pi$
$167$	−1432.32	−0.663688	−0.331844	+	0.943334i	$0.607671\pi$
$168$	0	0
$169$	−2180.94	−0.992692
$170$	−482.740	−0.217791
$171$	0	0
$172$	−915.680	−0.405930
$173$	3479.54	1.52916	0.764580	−	0.644529i	$-0.222946\pi$
$173$	3479.54	1.52916	0.764580	+	0.644529i	$0.222946\pi$
$174$	0	0
$175$	−616.077	−0.266120
$176$	278.187	0.119143
$177$	0	0
$178$	−50.0751	−0.0210859
$179$	−3642.71	−1.52105	−0.760527	−	0.649307i	$-0.775059\pi$
$179$	−3642.71	−1.52105	−0.760527	+	0.649307i	$0.775059\pi$
$180$	0	0
$181$	2409.15	0.989342	0.494671	−	0.869080i	$-0.335289\pi$
$181$	2409.15	0.989342	0.494671	+	0.869080i	$0.335289\pi$
$182$	197.489	0.0804333
$183$	0	0
$184$	−184.000	−0.0737210
$185$	2114.52	0.840338
$186$	0	0
$187$	−839.326	−0.328223
$188$	−2320.35	−0.900154
$189$	0	0
$190$	793.172	0.302857
$191$	189.608	0.0718299	0.0359150	−	0.999355i	$-0.488565\pi$
$191$	189.608	0.0718299	0.0359150	+	0.999355i	$0.488565\pi$
$192$	0	0
$193$	−1855.45	−0.692012	−0.346006	−	0.938232i	$-0.612462\pi$
$193$	−1855.45	−0.692012	−0.346006	+	0.938232i	$0.612462\pi$
$194$	499.597	0.184891
$195$	0	0
$196$	1057.13	0.385250
$197$	−2429.85	−0.878779	−0.439390	−	0.898297i	$-0.644805\pi$
$197$	−2429.85	−0.878779	−0.439390	+	0.898297i	$0.644805\pi$
$198$	0	0
$199$	4333.49	1.54368	0.771842	−	0.635815i	$-0.219336\pi$
$199$	4333.49	1.54368	0.771842	+	0.635815i	$0.219336\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	1240.99	0.432255
$203$	−6265.94	−2.16642
$204$	0	0
$205$	−851.254	−0.290020
$206$	−2946.48	−0.996558
$207$	0	0
$208$	64.1118	0.0213719
$209$	1379.07	0.456421
$210$	0	0
$211$	−816.788	−0.266493	−0.133247	−	0.991083i	$-0.542540\pi$
$211$	−816.788	−0.266493	−0.133247	+	0.991083i	$0.542540\pi$
$212$	1041.42	0.337381
$213$	0	0
$214$	−1880.28	−0.600624
$215$	1144.60	0.363075
$216$	0	0
$217$	5438.63	1.70137
$218$	1272.53	0.395351
$219$	0	0
$220$	−347.734	−0.106565
$221$	−193.433	−0.0588767
$222$	0	0
$223$	−4513.80	−1.35546	−0.677728	−	0.735313i	$-0.737035\pi$
$223$	−4513.80	−1.35546	−0.677728	+	0.735313i	$0.737035\pi$
$224$	788.579	0.235219
$225$	0	0
$226$	1664.90	0.490034
$227$	−2792.85	−0.816599	−0.408300	−	0.912848i	$-0.633878\pi$
$227$	−2792.85	−0.816599	−0.408300	+	0.912848i	$0.633878\pi$
$228$	0	0
$229$	1404.50	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$229$	1404.50	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	−2034.14	−0.575637
$233$	−1073.79	−0.301916	−0.150958	−	0.988540i	$-0.548236\pi$
$233$	−1073.79	−0.301916	−0.150958	+	0.988540i	$0.548236\pi$
$234$	0	0
$235$	2900.44	0.805122
$236$	−1412.52	−0.389607
$237$	0	0
$238$	−2379.24	−0.647997
$239$	2573.18	0.696423	0.348212	−	0.937416i	$-0.386789\pi$
$239$	2573.18	0.696423	0.348212	+	0.937416i	$0.386789\pi$
$240$	0	0
$241$	−696.127	−0.186064	−0.0930321	−	0.995663i	$-0.529656\pi$
$241$	−696.127	−0.186064	−0.0930321	+	0.995663i	$0.529656\pi$
$242$	2057.40	0.546508
$243$	0	0
$244$	−322.443	−0.0845996
$245$	−1321.41	−0.344578
$246$	0	0
$247$	317.823	0.0818729
$248$	1765.57	0.452071
$249$	0	0
$250$	250.000	0.0632456
$251$	−6467.81	−1.62647	−0.813236	−	0.581934i	$-0.802296\pi$
$251$	−6467.81	−1.62647	−0.813236	+	0.581934i	$0.802296\pi$
$252$	0	0
$253$	399.894	0.0993721
$254$	−3229.21	−0.797711
$255$	0	0
$256$	256.000	0.0625000
$257$	−5332.72	−1.29434	−0.647172	−	0.762344i	$-0.724048\pi$
$257$	−5332.72	−1.29434	−0.647172	+	0.762344i	$0.724048\pi$
$258$	0	0
$259$	10421.7	2.50027
$260$	−80.1398	−0.0191156
$261$	0	0
$262$	−3949.11	−0.931209
$263$	6872.85	1.61140	0.805699	−	0.592325i	$-0.201790\pi$
$263$	6872.85	1.61140	0.805699	+	0.592325i	$0.201790\pi$
$264$	0	0
$265$	−1301.77	−0.301763
$266$	3909.24	0.901094
$267$	0	0
$268$	−3280.04	−0.747614
$269$	1926.13	0.436572	0.218286	−	0.975885i	$-0.429953\pi$
$269$	1926.13	0.436572	0.218286	+	0.975885i	$0.429953\pi$
$270$	0	0
$271$	−3653.06	−0.818847	−0.409423	−	0.912344i	$-0.634270\pi$
$271$	−3653.06	−0.818847	−0.409423	+	0.912344i	$0.634270\pi$
$272$	−772.384	−0.172179
$273$	0	0
$274$	1507.61	0.332402
$275$	434.668	0.0953144
$276$	0	0
$277$	−1047.37	−0.227185	−0.113592	−	0.993527i	$-0.536236\pi$
$277$	−1047.37	−0.227185	−0.113592	+	0.993527i	$0.536236\pi$
$278$	−2028.19	−0.437565
$279$	0	0
$280$	−985.723	−0.210387
$281$	2758.90	0.585701	0.292851	−	0.956158i	$-0.405396\pi$
$281$	2758.90	0.585701	0.292851	+	0.956158i	$0.405396\pi$
$282$	0	0
$283$	3355.58	0.704837	0.352418	−	0.935843i	$-0.385359\pi$
$283$	3355.58	0.704837	0.352418	+	0.935843i	$0.385359\pi$
$284$	−2459.38	−0.513864
$285$	0	0
$286$	−139.337	−0.0288082
$287$	−4195.50	−0.862902
$288$	0	0
$289$	−2582.62	−0.525670
$290$	2542.67	0.514866
$291$	0	0
$292$	2046.35	0.410114
$293$	−419.625	−0.0836681	−0.0418341	−	0.999125i	$-0.513320\pi$
$293$	−419.625	−0.0836681	−0.0418341	+	0.999125i	$0.513320\pi$
$294$	0	0
$295$	1765.65	0.348475
$296$	3383.23	0.664346
$297$	0	0
$298$	−5543.59	−1.07762
$299$	92.1607	0.0178254
$300$	0	0
$301$	5641.30	1.08026
$302$	−6217.24	−1.18464
$303$	0	0
$304$	1269.08	0.239429
$305$	403.054	0.0756682
$306$	0	0
$307$	−4133.41	−0.768425	−0.384212	−	0.923245i	$-0.625527\pi$
$307$	−4133.41	−0.768425	−0.384212	+	0.923245i	$0.625527\pi$
$308$	−1713.85	−0.317064
$309$	0	0
$310$	−2206.96	−0.404345
$311$	6991.03	1.27468	0.637339	−	0.770584i	$-0.280035\pi$
$311$	6991.03	1.27468	0.637339	+	0.770584i	$0.280035\pi$
$312$	0	0
$313$	9380.53	1.69399	0.846996	−	0.531600i	$-0.178409\pi$
$313$	9380.53	1.69399	0.846996	+	0.531600i	$0.178409\pi$
$314$	−7425.75	−1.33458
$315$	0	0
$316$	641.855	0.114263
$317$	7995.35	1.41660	0.708302	−	0.705910i	$-0.249462\pi$
$317$	7995.35	1.41660	0.708302	+	0.705910i	$0.249462\pi$
$318$	0	0
$319$	4420.87	0.775929
$320$	−320.000	−0.0559017
$321$	0	0
$322$	1133.58	0.196186
$323$	−3828.96	−0.659595
$324$	0	0
$325$	100.175	0.0170975
$326$	−1831.58	−0.311172
$327$	0	0
$328$	−1362.01	−0.229281
$329$	14295.1	2.39549
$330$	0	0
$331$	−4798.86	−0.796886	−0.398443	−	0.917193i	$-0.630449\pi$
$331$	−4798.86	−0.796886	−0.398443	+	0.917193i	$0.630449\pi$
$332$	130.258	0.0215327
$333$	0	0
$334$	2864.63	0.469298
$335$	4100.06	0.668687
$336$	0	0
$337$	6895.51	1.11461	0.557303	−	0.830309i	$-0.311836\pi$
$337$	6895.51	1.11461	0.557303	+	0.830309i	$0.311836\pi$
$338$	4361.89	0.701939
$339$	0	0
$340$	965.480	0.154002
$341$	−3837.17	−0.609368
$342$	0	0
$343$	1939.86	0.305372
$344$	1831.36	0.287036
$345$	0	0
$346$	−6959.08	−1.08128
$347$	8744.17	1.35277	0.676386	−	0.736548i	$-0.263545\pi$
$347$	8744.17	1.35277	0.676386	+	0.736548i	$0.263545\pi$
$348$	0	0
$349$	−6387.08	−0.979635	−0.489818	−	0.871825i	$-0.662937\pi$
$349$	−6387.08	−0.979635	−0.489818	+	0.871825i	$0.662937\pi$
$350$	1232.15	0.188175
$351$	0	0
$352$	−556.375	−0.0842468
$353$	589.384	0.0888661	0.0444331	−	0.999012i	$-0.485852\pi$
$353$	589.384	0.0888661	0.0444331	+	0.999012i	$0.485852\pi$
$354$	0	0
$355$	3074.23	0.459614
$356$	100.150	0.0149100
$357$	0	0
$358$	7285.41	1.07555
$359$	−7214.13	−1.06058	−0.530289	−	0.847817i	$-0.677917\pi$
$359$	−7214.13	−1.06058	−0.530289	+	0.847817i	$0.677917\pi$
$360$	0	0
$361$	−567.774	−0.0827780
$362$	−4818.30	−0.699570
$363$	0	0
$364$	−394.978	−0.0568749
$365$	−2557.93	−0.366817
$366$	0	0
$367$	12356.4	1.75750	0.878748	−	0.477286i	$-0.158379\pi$
$367$	12356.4	1.75750	0.878748	+	0.477286i	$0.158379\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−4229.04	−0.594209
$371$	−6415.92	−0.897839
$372$	0	0
$373$	7145.98	0.991969	0.495985	−	0.868331i	$-0.334807\pi$
$373$	7145.98	0.991969	0.495985	+	0.868331i	$0.334807\pi$
$374$	1678.65	0.232088
$375$	0	0
$376$	4640.70	0.636505
$377$	1018.85	0.139186
$378$	0	0
$379$	2170.69	0.294197	0.147099	−	0.989122i	$-0.453007\pi$
$379$	2170.69	0.294197	0.147099	+	0.989122i	$0.453007\pi$
$380$	−1586.34	−0.214152
$381$	0	0
$382$	−379.215	−0.0507914
$383$	7967.21	1.06294	0.531469	−	0.847078i	$-0.321640\pi$
$383$	7967.21	1.06294	0.531469	+	0.847078i	$0.321640\pi$
$384$	0	0
$385$	2142.31	0.283590
$386$	3710.90	0.489326
$387$	0	0
$388$	−999.193	−0.130738
$389$	568.951	0.0741567	0.0370783	−	0.999312i	$-0.488195\pi$
$389$	568.951	0.0741567	0.0370783	+	0.999312i	$0.488195\pi$
$390$	0	0
$391$	−1110.30	−0.143607
$392$	−2114.25	−0.272413
$393$	0	0
$394$	4859.70	0.621391
$395$	−802.319	−0.102200
$396$	0	0
$397$	8564.88	1.08277	0.541384	−	0.840775i	$-0.317900\pi$
$397$	8564.88	1.08277	0.541384	+	0.840775i	$0.317900\pi$
$398$	−8666.98	−1.09155
$399$	0	0
$400$	400.000	0.0500000
$401$	12455.6	1.55113	0.775563	−	0.631270i	$-0.217466\pi$
$401$	12455.6	1.55113	0.775563	+	0.631270i	$0.217466\pi$
$402$	0	0
$403$	−884.325	−0.109309
$404$	−2481.97	−0.305650
$405$	0	0
$406$	12531.9	1.53189
$407$	−7352.91	−0.895504
$408$	0	0
$409$	11838.9	1.43129	0.715645	−	0.698465i	$-0.246133\pi$
$409$	11838.9	1.43129	0.715645	+	0.698465i	$0.246133\pi$
$410$	1702.51	0.205075
$411$	0	0
$412$	5892.96	0.704673
$413$	8702.22	1.03682
$414$	0	0
$415$	−162.823	−0.0192594
$416$	−128.224	−0.0151122
$417$	0	0
$418$	−2758.13	−0.322738
$419$	1531.77	0.178596	0.0892981	−	0.996005i	$-0.471538\pi$
$419$	1531.77	0.178596	0.0892981	+	0.996005i	$0.471538\pi$
$420$	0	0
$421$	−9985.06	−1.15592	−0.577959	−	0.816066i	$-0.696151\pi$
$421$	−9985.06	−1.15592	−0.577959	+	0.816066i	$0.696151\pi$
$422$	1633.58	0.188439
$423$	0	0
$424$	−2082.83	−0.238564
$425$	−1206.85	−0.137743
$426$	0	0
$427$	1986.50	0.225137
$428$	3760.56	0.424705
$429$	0	0
$430$	−2289.20	−0.256733
$431$	13786.7	1.54079	0.770396	−	0.637566i	$-0.220059\pi$
$431$	13786.7	1.54079	0.770396	+	0.637566i	$0.220059\pi$
$432$	0	0
$433$	−2621.92	−0.290996	−0.145498	−	0.989359i	$-0.546478\pi$
$433$	−2621.92	−0.290996	−0.145498	+	0.989359i	$0.546478\pi$
$434$	−10877.3	−1.20305
$435$	0	0
$436$	−2545.06	−0.279555
$437$	1824.30	0.199698
$438$	0	0
$439$	12062.7	1.31143	0.655717	−	0.755007i	$-0.272367\pi$
$439$	12062.7	1.31143	0.655717	+	0.755007i	$0.272367\pi$
$440$	695.468	0.0753526
$441$	0	0
$442$	386.867	0.0416321
$443$	−3659.26	−0.392453	−0.196227	−	0.980559i	$-0.562869\pi$
$443$	−3659.26	−0.392453	−0.196227	+	0.980559i	$0.562869\pi$
$444$	0	0
$445$	−125.188	−0.0133359
$446$	9027.60	0.958452
$447$	0	0
$448$	−1577.16	−0.166325
$449$	10529.6	1.10674	0.553368	−	0.832937i	$-0.313342\pi$
$449$	10529.6	1.10674	0.553368	+	0.832937i	$0.313342\pi$
$450$	0	0
$451$	2960.10	0.309059
$452$	−3329.81	−0.346506
$453$	0	0
$454$	5585.70	0.577423
$455$	493.723	0.0508705
$456$	0	0
$457$	6443.23	0.659522	0.329761	−	0.944064i	$-0.393032\pi$
$457$	6443.23	0.659522	0.329761	+	0.944064i	$0.393032\pi$
$458$	−2809.00	−0.286585
$459$	0	0
$460$	−460.000	−0.0466252
$461$	3263.86	0.329747	0.164873	−	0.986315i	$-0.447278\pi$
$461$	3263.86	0.329747	0.164873	+	0.986315i	$0.447278\pi$
$462$	0	0
$463$	9518.12	0.955388	0.477694	−	0.878526i	$-0.341473\pi$
$463$	9518.12	0.955388	0.477694	+	0.878526i	$0.341473\pi$
$464$	4068.28	0.407037
$465$	0	0
$466$	2147.59	0.213487
$467$	−19092.6	−1.89187	−0.945934	−	0.324360i	$-0.894851\pi$
$467$	−19092.6	−1.89187	−0.945934	+	0.324360i	$0.894851\pi$
$468$	0	0
$469$	20207.6	1.98955
$470$	−5800.87	−0.569307
$471$	0	0
$472$	2825.04	0.275494
$473$	−3980.17	−0.386910
$474$	0	0
$475$	1982.93	0.191543
$476$	4758.48	0.458203
$477$	0	0
$478$	−5146.36	−0.492446
$479$	6324.81	0.603315	0.301658	−	0.953416i	$-0.402460\pi$
$479$	6324.81	0.603315	0.301658	+	0.953416i	$0.402460\pi$
$480$	0	0
$481$	−1694.57	−0.160636
$482$	1392.25	0.131567
$483$	0	0
$484$	−4114.81	−0.386440
$485$	1248.99	0.116936
$486$	0	0
$487$	−7873.07	−0.732573	−0.366286	−	0.930502i	$-0.619371\pi$
$487$	−7873.07	−0.732573	−0.366286	+	0.930502i	$0.619371\pi$
$488$	644.886	0.0598209
$489$	0	0
$490$	2642.82	0.243654
$491$	3556.82	0.326918	0.163459	−	0.986550i	$-0.447735\pi$
$491$	3556.82	0.326918	0.163459	+	0.986550i	$0.447735\pi$
$492$	0	0
$493$	−12274.5	−1.12133
$494$	−635.646	−0.0578929
$495$	0	0
$496$	−3531.13	−0.319662
$497$	15151.7	1.36750
$498$	0	0
$499$	−1933.37	−0.173446	−0.0867231	−	0.996232i	$-0.527640\pi$
$499$	−1933.37	−0.173446	−0.0867231	+	0.996232i	$0.527640\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	12935.6	1.15009
$503$	−2114.36	−0.187425	−0.0937123	−	0.995599i	$-0.529873\pi$
$503$	−2114.36	−0.187425	−0.0937123	+	0.995599i	$0.529873\pi$
$504$	0	0
$505$	3102.46	0.273382
$506$	−799.789	−0.0702667
$507$	0	0
$508$	6458.42	0.564067
$509$	316.452	0.0275570	0.0137785	−	0.999905i	$-0.495614\pi$
$509$	316.452	0.0275570	0.0137785	+	0.999905i	$0.495614\pi$
$510$	0	0
$511$	−12607.1	−1.09140
$512$	−512.000	−0.0441942
$513$	0	0
$514$	10665.4	0.915239
$515$	−7366.20	−0.630279
$516$	0	0
$517$	−10085.8	−0.857976
$518$	−20843.3	−1.76796
$519$	0	0
$520$	160.280	0.0135168
$521$	309.041	0.0259872	0.0129936	−	0.999916i	$-0.495864\pi$
$521$	309.041	0.0259872	0.0129936	+	0.999916i	$0.495864\pi$
$522$	0	0
$523$	−5892.46	−0.492656	−0.246328	−	0.969186i	$-0.579224\pi$
$523$	−5892.46	−0.492656	−0.246328	+	0.969186i	$0.579224\pi$
$524$	7898.22	0.658464
$525$	0	0
$526$	−13745.7	−1.13943
$527$	10653.9	0.880626
$528$	0	0
$529$	529.000	0.0434783
$530$	2603.54	0.213378
$531$	0	0
$532$	−7818.49	−0.637170
$533$	682.193	0.0554391
$534$	0	0
$535$	−4700.71	−0.379868
$536$	6560.09	0.528643
$537$	0	0
$538$	−3852.25	−0.308703
$539$	4594.99	0.367199
$540$	0	0
$541$	1560.55	0.124017	0.0620085	−	0.998076i	$-0.480249\pi$
$541$	1560.55	0.124017	0.0620085	+	0.998076i	$0.480249\pi$
$542$	7306.12	0.579012
$543$	0	0
$544$	1544.77	0.121749
$545$	3181.32	0.250042
$546$	0	0
$547$	17756.3	1.38795	0.693973	−	0.720001i	$-0.255859\pi$
$547$	17756.3	1.38795	0.693973	+	0.720001i	$0.255859\pi$
$548$	−3015.22	−0.235044
$549$	0	0
$550$	−869.335	−0.0673974
$551$	20167.8	1.55930
$552$	0	0
$553$	−3954.32	−0.304078
$554$	2094.73	0.160644
$555$	0	0
$556$	4056.39	0.309405
$557$	−1212.77	−0.0922559	−0.0461280	−	0.998936i	$-0.514688\pi$
$557$	−1212.77	−0.0922559	−0.0461280	+	0.998936i	$0.514688\pi$
$558$	0	0
$559$	−917.280	−0.0694040
$560$	1971.45	0.148766
$561$	0	0
$562$	−5517.80	−0.414153
$563$	−12558.2	−0.940078	−0.470039	−	0.882646i	$-0.655760\pi$
$563$	−12558.2	−0.940078	−0.470039	+	0.882646i	$0.655760\pi$
$564$	0	0
$565$	4162.26	0.309925
$566$	−6711.17	−0.498395
$567$	0	0
$568$	4918.76	0.363357
$569$	−9776.72	−0.720319	−0.360159	−	0.932891i	$-0.617278\pi$
$569$	−9776.72	−0.720319	−0.360159	+	0.932891i	$0.617278\pi$
$570$	0	0
$571$	−18733.9	−1.37301	−0.686505	−	0.727125i	$-0.740856\pi$
$571$	−18733.9	−1.37301	−0.686505	+	0.727125i	$0.740856\pi$
$572$	278.673	0.0203705
$573$	0	0
$574$	8391.01	0.610164
$575$	575.000	0.0417029
$576$	0	0
$577$	5113.58	0.368944	0.184472	−	0.982838i	$-0.440942\pi$
$577$	5113.58	0.368944	0.184472	+	0.982838i	$0.440942\pi$
$578$	5165.24	0.371705
$579$	0	0
$580$	−5085.35	−0.364065
$581$	−802.492	−0.0573029
$582$	0	0
$583$	4526.70	0.321572
$584$	−4092.69	−0.289995
$585$	0	0
$586$	839.250	0.0591623
$587$	5379.95	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$587$	5379.95	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$588$	0	0
$589$	−17505.0	−1.22458
$590$	−3531.30	−0.246409
$591$	0	0
$592$	−6766.46	−0.469763
$593$	15060.3	1.04292	0.521461	−	0.853275i	$-0.325387\pi$
$593$	15060.3	1.04292	0.521461	+	0.853275i	$0.325387\pi$
$594$	0	0
$595$	−5948.10	−0.409829
$596$	11087.2	0.761995
$597$	0	0
$598$	−184.321	−0.0126045
$599$	3772.04	0.257298	0.128649	−	0.991690i	$-0.458936\pi$
$599$	3772.04	0.257298	0.128649	+	0.991690i	$0.458936\pi$
$600$	0	0
$601$	−14663.9	−0.995261	−0.497631	−	0.867389i	$-0.665796\pi$
$601$	−14663.9	−0.995261	−0.497631	+	0.867389i	$0.665796\pi$
$602$	−11282.6	−0.763861
$603$	0	0
$604$	12434.5	0.837668
$605$	5143.51	0.345642
$606$	0	0
$607$	−24514.5	−1.63923	−0.819614	−	0.572915i	$-0.805812\pi$
$607$	−24514.5	−1.63923	−0.819614	+	0.572915i	$0.805812\pi$
$608$	−2538.15	−0.169302
$609$	0	0
$610$	−806.108	−0.0535055
$611$	−2324.40	−0.153904
$612$	0	0
$613$	14451.2	0.952167	0.476084	−	0.879400i	$-0.342056\pi$
$613$	14451.2	0.952167	0.476084	+	0.879400i	$0.342056\pi$
$614$	8266.83	0.543358
$615$	0	0
$616$	3427.70	0.224198
$617$	−3292.19	−0.214811	−0.107406	−	0.994215i	$-0.534254\pi$
$617$	−3292.19	−0.214811	−0.107406	+	0.994215i	$0.534254\pi$
$618$	0	0
$619$	−12595.5	−0.817858	−0.408929	−	0.912566i	$-0.634098\pi$
$619$	−12595.5	−0.817858	−0.408929	+	0.912566i	$0.634098\pi$
$620$	4413.92	0.285915
$621$	0	0
$622$	−13982.1	−0.901333
$623$	−617.002	−0.0396785
$624$	0	0
$625$	625.000	0.0400000
$626$	−18761.1	−1.19783
$627$	0	0
$628$	14851.5	0.943693
$629$	20415.3	1.29413
$630$	0	0
$631$	19889.3	1.25480	0.627401	−	0.778697i	$-0.284119\pi$
$631$	19889.3	1.25480	0.627401	+	0.778697i	$0.284119\pi$
$632$	−1283.71	−0.0807963
$633$	0	0
$634$	−15990.7	−1.00169
$635$	−8073.02	−0.504517
$636$	0	0
$637$	1058.97	0.0658682
$638$	−8841.75	−0.548665
$639$	0	0
$640$	640.000	0.0395285
$641$	19276.0	1.18776	0.593880	−	0.804553i	$-0.297595\pi$
$641$	19276.0	1.18776	0.593880	+	0.804553i	$0.297595\pi$
$642$	0	0
$643$	−10219.2	−0.626758	−0.313379	−	0.949628i	$-0.601461\pi$
$643$	−10219.2	−0.626758	−0.313379	+	0.949628i	$0.601461\pi$
$644$	−2267.16	−0.138725
$645$	0	0
$646$	7657.93	0.466404
$647$	−20818.4	−1.26500	−0.632500	−	0.774560i	$-0.717971\pi$
$647$	−20818.4	−1.26500	−0.632500	+	0.774560i	$0.717971\pi$
$648$	0	0
$649$	−6139.77	−0.371352
$650$	−200.349	−0.0120898
$651$	0	0
$652$	3663.17	0.220032
$653$	15135.3	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$653$	15135.3	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$654$	0	0
$655$	−9872.78	−0.588949
$656$	2724.01	0.162126
$657$	0	0
$658$	−28590.3	−1.69387
$659$	13207.9	0.780737	0.390369	−	0.920659i	$-0.372348\pi$
$659$	13207.9	0.780737	0.390369	+	0.920659i	$0.372348\pi$
$660$	0	0
$661$	6671.34	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$661$	6671.34	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$662$	9597.72	0.563484
$663$	0	0
$664$	−260.517	−0.0152259
$665$	9773.11	0.569902
$666$	0	0
$667$	5848.15	0.339492
$668$	−5729.26	−0.331844
$669$	0	0
$670$	−8200.11	−0.472833
$671$	−1401.56	−0.0806355
$672$	0	0
$673$	−9534.41	−0.546099	−0.273049	−	0.962000i	$-0.588032\pi$
$673$	−9534.41	−0.546099	−0.273049	+	0.962000i	$0.588032\pi$
$674$	−13791.0	−0.788146
$675$	0	0
$676$	−8723.78	−0.496346
$677$	−17748.4	−1.00757	−0.503787	−	0.863828i	$-0.668060\pi$
$677$	−17748.4	−1.00757	−0.503787	+	0.863828i	$0.668060\pi$
$678$	0	0
$679$	6155.80	0.347920
$680$	−1930.96	−0.108896
$681$	0	0
$682$	7674.35	0.430888
$683$	−9583.57	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$683$	−9583.57	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$684$	0	0
$685$	3769.03	0.210229
$686$	−3879.73	−0.215931
$687$	0	0
$688$	−3662.72	−0.202965
$689$	1043.24	0.0576837
$690$	0	0
$691$	9410.74	0.518092	0.259046	−	0.965865i	$-0.416592\pi$
$691$	9410.74	0.518092	0.259046	+	0.965865i	$0.416592\pi$
$692$	13918.2	0.764580
$693$	0	0
$694$	−17488.3	−0.956554
$695$	−5070.48	−0.276740
$696$	0	0
$697$	−8218.69	−0.446636
$698$	12774.2	0.692707
$699$	0	0
$700$	−2464.31	−0.133060
$701$	19517.9	1.05161	0.525806	−	0.850605i	$-0.323764\pi$
$701$	19517.9	1.05161	0.525806	+	0.850605i	$0.323764\pi$
$702$	0	0
$703$	−33543.6	−1.79960
$704$	1112.75	0.0595715
$705$	0	0
$706$	−1178.77	−0.0628378
$707$	15290.9	0.813397
$708$	0	0
$709$	−28430.7	−1.50598	−0.752990	−	0.658033i	$-0.771389\pi$
$709$	−28430.7	−1.50598	−0.752990	+	0.658033i	$0.771389\pi$
$710$	−6148.45	−0.324996
$711$	0	0
$712$	−200.300	−0.0105429
$713$	−5076.00	−0.266617
$714$	0	0
$715$	−348.342	−0.0182199
$716$	−14570.8	−0.760527
$717$	0	0
$718$	14428.3	0.749942
$719$	−18716.0	−0.970779	−0.485390	−	0.874298i	$-0.661322\pi$
$719$	−18716.0	−0.970779	−0.485390	+	0.874298i	$0.661322\pi$
$720$	0	0
$721$	−36305.2	−1.87528
$722$	1135.55	0.0585329
$723$	0	0
$724$	9636.61	0.494671
$725$	6356.69	0.325630
$726$	0	0
$727$	18419.5	0.939670	0.469835	−	0.882754i	$-0.344313\pi$
$727$	18419.5	0.939670	0.469835	+	0.882754i	$0.344313\pi$
$728$	789.956	0.0402167
$729$	0	0
$730$	5115.86	0.259379
$731$	11050.9	0.559141
$732$	0	0
$733$	−21548.4	−1.08582	−0.542912	−	0.839790i	$-0.682678\pi$
$733$	−21548.4	−1.08582	−0.542912	+	0.839790i	$0.682678\pi$
$734$	−24712.9	−1.24274
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−14257.3	−0.712584
$738$	0	0
$739$	12066.4	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$739$	12066.4	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$740$	8458.08	0.420169
$741$	0	0
$742$	12831.8	0.634868
$743$	21951.0	1.08385	0.541926	−	0.840426i	$-0.317695\pi$
$743$	21951.0	1.08385	0.541926	+	0.840426i	$0.317695\pi$
$744$	0	0
$745$	−13859.0	−0.681549
$746$	−14292.0	−0.701428
$747$	0	0
$748$	−3357.31	−0.164111
$749$	−23168.0	−1.13023
$750$	0	0
$751$	−3112.43	−0.151230	−0.0756152	−	0.997137i	$-0.524092\pi$
$751$	−3112.43	−0.151230	−0.0756152	+	0.997137i	$0.524092\pi$
$752$	−9281.40	−0.450077
$753$	0	0
$754$	−2037.69	−0.0984197
$755$	−15543.1	−0.749233
$756$	0	0
$757$	−7684.64	−0.368960	−0.184480	−	0.982836i	$-0.559060\pi$
$757$	−7684.64	−0.368960	−0.184480	+	0.982836i	$0.559060\pi$
$758$	−4341.38	−0.208029
$759$	0	0
$760$	3172.69	0.151428
$761$	36484.6	1.73793	0.868965	−	0.494874i	$-0.164786\pi$
$761$	36484.6	1.73793	0.868965	+	0.494874i	$0.164786\pi$
$762$	0	0
$763$	15679.5	0.743954
$764$	758.430	0.0359150
$765$	0	0
$766$	−15934.4	−0.751611
$767$	−1414.99	−0.0666132
$768$	0	0
$769$	2004.39	0.0939925	0.0469962	−	0.998895i	$-0.485035\pi$
$769$	2004.39	0.0939925	0.0469962	+	0.998895i	$0.485035\pi$
$770$	−4284.62	−0.200529
$771$	0	0
$772$	−7421.81	−0.346006
$773$	−7716.17	−0.359031	−0.179516	−	0.983755i	$-0.557453\pi$
$773$	−7716.17	−0.359031	−0.179516	+	0.983755i	$0.557453\pi$
$774$	0	0
$775$	−5517.40	−0.255730
$776$	1998.39	0.0924457
$777$	0	0
$778$	−1137.90	−0.0524367
$779$	13503.8	0.621084
$780$	0	0
$781$	−10690.1	−0.489786
$782$	2220.61	0.101546
$783$	0	0
$784$	4228.51	0.192625
$785$	−18564.4	−0.844065
$786$	0	0
$787$	−57.0149	−0.00258241	−0.00129121	−	0.999999i	$-0.500411\pi$
$787$	−57.0149	−0.00258241	−0.00129121	+	0.999999i	$0.500411\pi$
$788$	−9719.39	−0.439390
$789$	0	0
$790$	1604.64	0.0722664
$791$	20514.2	0.922124
$792$	0	0
$793$	−323.006	−0.0144644
$794$	−17129.8	−0.765633
$795$	0	0
$796$	17334.0	0.771842
$797$	15184.7	0.674870	0.337435	−	0.941349i	$-0.390441\pi$
$797$	15184.7	0.674870	0.337435	+	0.941349i	$0.390441\pi$
$798$	0	0
$799$	28003.2	1.23990
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−24911.1	−1.09681
$803$	8894.80	0.390898
$804$	0	0
$805$	2833.95	0.124079
$806$	1768.65	0.0772929
$807$	0	0
$808$	4963.94	0.216127
$809$	30286.4	1.31621	0.658105	−	0.752926i	$-0.271358\pi$
$809$	30286.4	1.31621	0.658105	+	0.752926i	$0.271358\pi$
$810$	0	0
$811$	−43936.2	−1.90235	−0.951176	−	0.308650i	$-0.900123\pi$
$811$	−43936.2	−1.90235	−0.951176	+	0.308650i	$0.900123\pi$
$812$	−25063.7	−1.08321
$813$	0	0
$814$	14705.8	0.633217
$815$	−4578.96	−0.196802
$816$	0	0
$817$	−18157.3	−0.777532
$818$	−23677.9	−1.01207
$819$	0	0
$820$	−3405.02	−0.145010
$821$	5245.69	0.222991	0.111496	−	0.993765i	$-0.464436\pi$
$821$	5245.69	0.222991	0.111496	+	0.993765i	$0.464436\pi$
$822$	0	0
$823$	10678.0	0.452260	0.226130	−	0.974097i	$-0.427393\pi$
$823$	10678.0	0.452260	0.226130	+	0.974097i	$0.427393\pi$
$824$	−11785.9	−0.498279
$825$	0	0
$826$	−17404.4	−0.733145
$827$	−3393.69	−0.142697	−0.0713484	−	0.997451i	$-0.522730\pi$
$827$	−3393.69	−0.142697	−0.0713484	+	0.997451i	$0.522730\pi$
$828$	0	0
$829$	9601.74	0.402270	0.201135	−	0.979564i	$-0.435537\pi$
$829$	9601.74	0.402270	0.201135	+	0.979564i	$0.435537\pi$
$830$	325.646	0.0136185
$831$	0	0
$832$	256.447	0.0106859
$833$	−12757.9	−0.530656
$834$	0	0
$835$	7161.58	0.296810
$836$	5516.26	0.228210
$837$	0	0
$838$	−3063.54	−0.126287
$839$	−11992.8	−0.493489	−0.246744	−	0.969081i	$-0.579361\pi$
$839$	−11992.8	−0.493489	−0.246744	+	0.969081i	$0.579361\pi$
$840$	0	0
$841$	40263.0	1.65087
$842$	19970.1	0.817358
$843$	0	0
$844$	−3267.15	−0.133247
$845$	10904.7	0.443945
$846$	0	0
$847$	25350.4	1.02839
$848$	4165.66	0.168690
$849$	0	0
$850$	2413.70	0.0973991
$851$	−9726.79	−0.391810
$852$	0	0
$853$	−15441.5	−0.619821	−0.309911	−	0.950766i	$-0.600299\pi$
$853$	−15441.5	−0.619821	−0.309911	+	0.950766i	$0.600299\pi$
$854$	−3973.00	−0.159196
$855$	0	0
$856$	−7521.13	−0.300312
$857$	44572.4	1.77662	0.888310	−	0.459244i	$-0.151880\pi$
$857$	44572.4	1.77662	0.888310	+	0.459244i	$0.151880\pi$
$858$	0	0
$859$	2519.56	0.100077	0.0500386	−	0.998747i	$-0.484066\pi$
$859$	2519.56	0.100077	0.0500386	+	0.998747i	$0.484066\pi$
$860$	4578.40	0.181537
$861$	0	0
$862$	−27573.4	−1.08950
$863$	−28980.1	−1.14310	−0.571548	−	0.820568i	$-0.693657\pi$
$863$	−28980.1	−1.14310	−0.571548	+	0.820568i	$0.693657\pi$
$864$	0	0
$865$	−17397.7	−0.683861
$866$	5243.83	0.205765
$867$	0	0
$868$	21754.5	0.850687
$869$	2789.94	0.108909
$870$	0	0
$871$	−3285.77	−0.127823
$872$	5090.11	0.197675
$873$	0	0
$874$	−3648.59	−0.141208
$875$	3080.39	0.119013
$876$	0	0
$877$	−7955.28	−0.306307	−0.153153	−	0.988202i	$-0.548943\pi$
$877$	−7955.28	−0.306307	−0.153153	+	0.988202i	$0.548943\pi$
$878$	−24125.3	−0.927323
$879$	0	0
$880$	−1390.94	−0.0532823
$881$	−29722.0	−1.13662	−0.568309	−	0.822815i	$-0.692402\pi$
$881$	−29722.0	−1.13662	−0.568309	+	0.822815i	$0.692402\pi$
$882$	0	0
$883$	−19379.7	−0.738596	−0.369298	−	0.929311i	$-0.620402\pi$
$883$	−19379.7	−0.738596	−0.369298	+	0.929311i	$0.620402\pi$
$884$	−773.734	−0.0294383
$885$	0	0
$886$	7318.53	0.277507
$887$	−22901.7	−0.866925	−0.433463	−	0.901172i	$-0.642708\pi$
$887$	−22901.7	−0.866925	−0.433463	+	0.901172i	$0.642708\pi$
$888$	0	0
$889$	−39788.8	−1.50110
$890$	250.375	0.00942989
$891$	0	0
$892$	−18055.2	−0.677728
$893$	−46010.9	−1.72419
$894$	0	0
$895$	18213.5	0.680236
$896$	3154.31	0.117610
$897$	0	0
$898$	−21059.3	−0.782581
$899$	−56115.8	−2.08183
$900$	0	0
$901$	−12568.3	−0.464719
$902$	−5920.20	−0.218538
$903$	0	0
$904$	6659.61	0.245017
$905$	−12045.8	−0.442447
$906$	0	0
$907$	42542.2	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$907$	42542.2	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$908$	−11171.4	−0.408300
$909$	0	0
$910$	−987.445	−0.0359709
$911$	−220.864	−0.00803243	−0.00401622	−	0.999992i	$-0.501278\pi$
$911$	−220.864	−0.00803243	−0.00401622	+	0.999992i	$0.501278\pi$
$912$	0	0
$913$	566.191	0.0205237
$914$	−12886.5	−0.466352
$915$	0	0
$916$	5617.99	0.202646
$917$	−48659.1	−1.75231
$918$	0	0
$919$	27835.4	0.999135	0.499567	−	0.866275i	$-0.333492\pi$
$919$	27835.4	0.999135	0.499567	+	0.866275i	$0.333492\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−6527.72	−0.233166
$923$	−2463.68	−0.0878580
$924$	0	0
$925$	−10572.6	−0.375811
$926$	−19036.2	−0.675562
$927$	0	0
$928$	−8136.56	−0.287819
$929$	2172.71	0.0767323	0.0383661	−	0.999264i	$-0.487785\pi$
$929$	2172.71	0.0767323	0.0383661	+	0.999264i	$0.487785\pi$
$930$	0	0
$931$	20962.1	0.737921
$932$	−4295.17	−0.150958
$933$	0	0
$934$	38185.3	1.33775
$935$	4196.63	0.146786
$936$	0	0
$937$	54906.5	1.91432	0.957160	−	0.289560i	$-0.0935090\pi$
$937$	54906.5	1.91432	0.957160	+	0.289560i	$0.0935090\pi$
$938$	−40415.2	−1.40683
$939$	0	0
$940$	11601.7	0.402561
$941$	−5980.06	−0.207167	−0.103584	−	0.994621i	$-0.533031\pi$
$941$	−5980.06	−0.207167	−0.103584	+	0.994621i	$0.533031\pi$
$942$	0	0
$943$	3915.77	0.135223
$944$	−5650.09	−0.194804
$945$	0	0
$946$	7960.33	0.273586
$947$	20268.6	0.695504	0.347752	−	0.937587i	$-0.386945\pi$
$947$	20268.6	0.695504	0.347752	+	0.937587i	$0.386945\pi$
$948$	0	0
$949$	2049.92	0.0701193
$950$	−3965.86	−0.135442
$951$	0	0
$952$	−9516.97	−0.323999
$953$	21797.9	0.740925	0.370463	−	0.928847i	$-0.379199\pi$
$953$	21797.9	0.740925	0.370463	+	0.928847i	$0.379199\pi$
$954$	0	0
$955$	−948.038	−0.0321233
$956$	10292.7	0.348212
$957$	0	0
$958$	−12649.6	−0.426608
$959$	18576.1	0.625499
$960$	0	0
$961$	18915.6	0.634945
$962$	3389.14	0.113587
$963$	0	0
$964$	−2784.51	−0.0930321
$965$	9277.26	0.309477
$966$	0	0
$967$	48950.6	1.62786	0.813932	−	0.580960i	$-0.197323\pi$
$967$	48950.6	1.62786	0.813932	+	0.580960i	$0.197323\pi$
$968$	8229.62	0.273254
$969$	0	0
$970$	−2497.98	−0.0826860
$971$	26426.2	0.873386	0.436693	−	0.899611i	$-0.356150\pi$
$971$	26426.2	0.873386	0.436693	+	0.899611i	$0.356150\pi$
$972$	0	0
$973$	−24990.5	−0.823389
$974$	15746.1	0.518007
$975$	0	0
$976$	−1289.77	−0.0422998
$977$	−5770.09	−0.188947	−0.0944736	−	0.995527i	$-0.530117\pi$
$977$	−5770.09	−0.188947	−0.0944736	+	0.995527i	$0.530117\pi$
$978$	0	0
$979$	435.320	0.0142113
$980$	−5285.63	−0.172289
$981$	0	0
$982$	−7113.64	−0.231166
$983$	484.532	0.0157214	0.00786071	−	0.999969i	$-0.497498\pi$
$983$	484.532	0.0157214	0.00786071	+	0.999969i	$0.497498\pi$
$984$	0	0
$985$	12149.2	0.393002
$986$	24549.0	0.792901
$987$	0	0
$988$	1271.29	0.0409365
$989$	−5265.16	−0.169285
$990$	0	0
$991$	−26511.5	−0.849813	−0.424907	−	0.905237i	$-0.639693\pi$
$991$	−26511.5	−0.849813	−0.424907	+	0.905237i	$0.639693\pi$
$992$	7062.27	0.226035
$993$	0	0
$994$	−30303.4	−0.966966
$995$	−21667.4	−0.690356
$996$	0	0
$997$	−4628.20	−0.147018	−0.0735088	−	0.997295i	$-0.523420\pi$
$997$	−4628.20	−0.147018	−0.0735088	+	0.997295i	$0.523420\pi$
$998$	3866.74	0.122645
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bg.1.1		4
3.2	odd	2		230.4.a.j.1.4	✓	4
12.11	even	2		1840.4.a.k.1.1		4
15.2	even	4		1150.4.b.o.599.5		8
15.8	even	4		1150.4.b.o.599.4		8
15.14	odd	2		1150.4.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.4	✓	4	3.2	odd	2
1150.4.a.n.1.1		4	15.14	odd	2
1150.4.b.o.599.4		8	15.8	even	4
1150.4.b.o.599.5		8	15.2	even	4
1840.4.a.k.1.1		4	12.11	even	2
2070.4.a.bg.1.1		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -24.6431 q^{7} -8.00000 q^{8} +10.0000 q^{10} +17.3867 q^{11} +4.00699 q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740 q^{17} +79.3172 q^{19} -20.0000 q^{20} -34.7734 q^{22} +23.0000 q^{23} +25.0000 q^{25} -8.01398 q^{26} -98.5723 q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000 q^{32} +96.5480 q^{34} +123.215 q^{35} -422.904 q^{37} -158.634 q^{38} +40.0000 q^{40} +170.251 q^{41} -228.920 q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087 q^{47} +264.282 q^{49} -50.0000 q^{50} +16.0280 q^{52} +260.354 q^{53} -86.9335 q^{55} +197.145 q^{56} -508.535 q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392 q^{62} +64.0000 q^{64} -20.0349 q^{65} -820.011 q^{67} -193.096 q^{68} -246.431 q^{70} -614.845 q^{71} +511.586 q^{73} +845.808 q^{74} +317.269 q^{76} -428.462 q^{77} +160.464 q^{79} -80.0000 q^{80} -340.502 q^{82} +32.5646 q^{83} +241.370 q^{85} +457.840 q^{86} -139.094 q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000 q^{92} +1160.17 q^{94} -396.586 q^{95} -249.798 q^{97} -528.563 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 8 * q^7 - 32 * q^8 + 40 * q^10 - 21 * q^11 + 70 * q^13 - 16 * q^14 + 64 * q^16 - 56 * q^17 + 173 * q^19 - 80 * q^20 + 42 * q^22 + 92 * q^23 + 100 * q^25 - 140 * q^26 + 32 * q^28 + 118 * q^29 + 17 * q^31 - 128 * q^32 + 112 * q^34 - 40 * q^35 - 343 * q^37 - 346 * q^38 + 160 * q^40 - 139 * q^41 - 50 * q^43 - 84 * q^44 - 184 * q^46 - 367 * q^47 - 124 * q^49 - 200 * q^50 + 280 * q^52 + 353 * q^53 + 105 * q^55 - 64 * q^56 - 236 * q^58 + 453 * q^59 - 327 * q^61 - 34 * q^62 + 256 * q^64 - 350 * q^65 - 455 * q^67 - 224 * q^68 + 80 * q^70 - 195 * q^71 - 633 * q^73 + 686 * q^74 + 692 * q^76 + 2 * q^77 - 1140 * q^79 - 320 * q^80 + 278 * q^82 + 1199 * q^83 + 280 * q^85 + 100 * q^86 + 168 * q^88 + 2170 * q^89 + 557 * q^91 + 368 * q^92 + 734 * q^94 - 865 * q^95 - 703 * q^97 + 248 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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