LMFDB - Embedded newform 2070.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,4,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			1.2
Root			$1.01192$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +2.98517 q^{7} -8.00000 q^{8} +10.0000 q^{10} -68.6262 q^{11} -12.0028 q^{13} -5.97035 q^{14} +16.0000 q^{16} -106.208 q^{17} -48.4891 q^{19} -20.0000 q^{20} +137.252 q^{22} +23.0000 q^{23} +25.0000 q^{25} +24.0055 q^{26} +11.9407 q^{28} -135.226 q^{29} -230.782 q^{31} -32.0000 q^{32} +212.416 q^{34} -14.9259 q^{35} -107.956 q^{37} +96.9782 q^{38} +40.0000 q^{40} -394.637 q^{41} +136.063 q^{43} -274.505 q^{44} -46.0000 q^{46} +50.4512 q^{47} -334.089 q^{49} -50.0000 q^{50} -48.0110 q^{52} +414.707 q^{53} +343.131 q^{55} -23.8814 q^{56} +270.452 q^{58} +183.586 q^{59} -98.4751 q^{61} +461.565 q^{62} +64.0000 q^{64} +60.0138 q^{65} +136.925 q^{67} -424.831 q^{68} +29.8517 q^{70} +708.800 q^{71} -689.605 q^{73} +215.912 q^{74} -193.956 q^{76} -204.861 q^{77} +546.583 q^{79} -80.0000 q^{80} +789.275 q^{82} +20.2622 q^{83} +531.039 q^{85} -272.125 q^{86} +549.010 q^{88} +1087.31 q^{89} -35.8303 q^{91} +92.0000 q^{92} -100.902 q^{94} +242.445 q^{95} -1115.90 q^{97} +668.177 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.98517	0.161184	0.0805921	−	0.996747i	$-0.474319\pi$
$7$	2.98517	0.161184	0.0805921	+	0.996747i	$0.474319\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	−68.6262	−1.88105	−0.940527	−	0.339720i	$-0.889668\pi$
$11$	−68.6262	−1.88105	−0.940527	+	0.339720i	$0.889668\pi$
$12$	0	0
$13$	−12.0028	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$13$	−12.0028	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$14$	−5.97035	−0.113974
$15$	0	0
$16$	16.0000	0.250000
$17$	−106.208	−1.51525	−0.757623	−	0.652693i	$-0.773639\pi$
$17$	−106.208	−1.51525	−0.757623	+	0.652693i	$0.773639\pi$
$18$	0	0
$19$	−48.4891	−0.585482	−0.292741	−	0.956192i	$-0.594567\pi$
$19$	−48.4891	−0.585482	−0.292741	+	0.956192i	$0.594567\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	137.252	1.33011
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	24.0055	0.181072
$27$	0	0
$28$	11.9407	0.0805921
$29$	−135.226	−0.865890	−0.432945	−	0.901420i	$-0.642526\pi$
$29$	−135.226	−0.865890	−0.432945	+	0.901420i	$0.642526\pi$
$30$	0	0
$31$	−230.782	−1.33709	−0.668544	−	0.743673i	$-0.733082\pi$
$31$	−230.782	−1.33709	−0.668544	+	0.743673i	$0.733082\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	212.416	1.07144
$35$	−14.9259	−0.0720838
$36$	0	0
$37$	−107.956	−0.479672	−0.239836	−	0.970813i	$-0.577094\pi$
$37$	−107.956	−0.479672	−0.239836	+	0.970813i	$0.577094\pi$
$38$	96.9782	0.413998
$39$	0	0
$40$	40.0000	0.158114
$41$	−394.637	−1.50322	−0.751610	−	0.659608i	$-0.770722\pi$
$41$	−394.637	−1.50322	−0.751610	+	0.659608i	$0.770722\pi$
$42$	0	0
$43$	136.063	0.482543	0.241272	−	0.970458i	$-0.422436\pi$
$43$	136.063	0.482543	0.241272	+	0.970458i	$0.422436\pi$
$44$	−274.505	−0.940527
$45$	0	0
$46$	−46.0000	−0.147442
$47$	50.4512	0.156576	0.0782880	−	0.996931i	$-0.475055\pi$
$47$	50.4512	0.156576	0.0782880	+	0.996931i	$0.475055\pi$
$48$	0	0
$49$	−334.089	−0.974020
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−48.0110	−0.128037
$53$	414.707	1.07480	0.537400	−	0.843327i	$-0.319406\pi$
$53$	414.707	1.07480	0.537400	+	0.843327i	$0.319406\pi$
$54$	0	0
$55$	343.131	0.841232
$56$	−23.8814	−0.0569872
$57$	0	0
$58$	270.452	0.612277
$59$	183.586	0.405100	0.202550	−	0.979272i	$-0.435077\pi$
$59$	183.586	0.405100	0.202550	+	0.979272i	$0.435077\pi$
$60$	0	0
$61$	−98.4751	−0.206696	−0.103348	−	0.994645i	$-0.532956\pi$
$61$	−98.4751	−0.206696	−0.103348	+	0.994645i	$0.532956\pi$
$62$	461.565	0.945464
$63$	0	0
$64$	64.0000	0.125000
$65$	60.0138	0.114520
$66$	0	0
$67$	136.925	0.249672	0.124836	−	0.992177i	$-0.460160\pi$
$67$	136.925	0.249672	0.124836	+	0.992177i	$0.460160\pi$
$68$	−424.831	−0.757623
$69$	0	0
$70$	29.8517	0.0509709
$71$	708.800	1.18478	0.592388	−	0.805653i	$-0.298185\pi$
$71$	708.800	1.18478	0.592388	+	0.805653i	$0.298185\pi$
$72$	0	0
$73$	−689.605	−1.10565	−0.552823	−	0.833299i	$-0.686449\pi$
$73$	−689.605	−1.10565	−0.552823	+	0.833299i	$0.686449\pi$
$74$	215.912	0.339179
$75$	0	0
$76$	−193.956	−0.292741
$77$	−204.861	−0.303196
$78$	0	0
$79$	546.583	0.778423	0.389212	−	0.921148i	$-0.372748\pi$
$79$	546.583	0.778423	0.389212	+	0.921148i	$0.372748\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	789.275	1.06294
$83$	20.2622	0.0267959	0.0133980	−	0.999910i	$-0.495735\pi$
$83$	20.2622	0.0267959	0.0133980	+	0.999910i	$0.495735\pi$
$84$	0	0
$85$	531.039	0.677638
$86$	−272.125	−0.341210
$87$	0	0
$88$	549.010	0.665053
$89$	1087.31	1.29499	0.647495	−	0.762069i	$-0.275817\pi$
$89$	1087.31	1.29499	0.647495	+	0.762069i	$0.275817\pi$
$90$	0	0
$91$	−35.8303	−0.0412751
$92$	92.0000	0.104257
$93$	0	0
$94$	−100.902	−0.110716
$95$	242.445	0.261836
$96$	0	0
$97$	−1115.90	−1.16807	−0.584036	−	0.811728i	$-0.698527\pi$
$97$	−1115.90	−1.16807	−0.584036	+	0.811728i	$0.698527\pi$
$98$	668.177	0.688736
$99$	0	0
$100$	100.000	0.100000
$101$	134.565	0.132572	0.0662859	−	0.997801i	$-0.478885\pi$
$101$	134.565	0.132572	0.0662859	+	0.997801i	$0.478885\pi$
$102$	0	0
$103$	−925.953	−0.885795	−0.442898	−	0.896572i	$-0.646049\pi$
$103$	−925.953	−0.885795	−0.442898	+	0.896572i	$0.646049\pi$
$104$	96.0221	0.0905360
$105$	0	0
$106$	−829.415	−0.759999
$107$	−1871.13	−1.69055	−0.845273	−	0.534334i	$-0.820562\pi$
$107$	−1871.13	−1.69055	−0.845273	+	0.534334i	$0.820562\pi$
$108$	0	0
$109$	−978.242	−0.859620	−0.429810	−	0.902919i	$-0.641419\pi$
$109$	−978.242	−0.859620	−0.429810	+	0.902919i	$0.641419\pi$
$110$	−686.262	−0.594841
$111$	0	0
$112$	47.7628	0.0402961
$113$	1760.45	1.46557	0.732786	−	0.680459i	$-0.238220\pi$
$113$	1760.45	1.46557	0.732786	+	0.680459i	$0.238220\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	−540.903	−0.432945
$117$	0	0
$118$	−367.172	−0.286449
$119$	−317.049	−0.244234
$120$	0	0
$121$	3378.56	2.53836
$122$	196.950	0.146156
$123$	0	0
$124$	−923.129	−0.668544
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1260.87	−0.880980	−0.440490	−	0.897758i	$-0.645195\pi$
$127$	−1260.87	−0.880980	−0.440490	+	0.897758i	$0.645195\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−120.028	−0.0809778
$131$	−444.283	−0.296314	−0.148157	−	0.988964i	$-0.547334\pi$
$131$	−444.283	−0.296314	−0.148157	+	0.988964i	$0.547334\pi$
$132$	0	0
$133$	−144.748	−0.0943705
$134$	−273.849	−0.176545
$135$	0	0
$136$	849.662	0.535720
$137$	−429.072	−0.267577	−0.133789	−	0.991010i	$-0.542714\pi$
$137$	−429.072	−0.267577	−0.133789	+	0.991010i	$0.542714\pi$
$138$	0	0
$139$	−249.755	−0.152402	−0.0762012	−	0.997092i	$-0.524279\pi$
$139$	−249.755	−0.152402	−0.0762012	+	0.997092i	$0.524279\pi$
$140$	−59.7035	−0.0360419
$141$	0	0
$142$	−1417.60	−0.837763
$143$	823.704	0.481689
$144$	0	0
$145$	676.129	0.387238
$146$	1379.21	0.781809
$147$	0	0
$148$	−431.824	−0.239836
$149$	1564.18	0.860020	0.430010	−	0.902824i	$-0.358510\pi$
$149$	1564.18	0.860020	0.430010	+	0.902824i	$0.358510\pi$
$150$	0	0
$151$	−1860.04	−1.00244	−0.501219	−	0.865321i	$-0.667115\pi$
$151$	−1860.04	−1.00244	−0.501219	+	0.865321i	$0.667115\pi$
$152$	387.913	0.206999
$153$	0	0
$154$	409.722	0.214392
$155$	1153.91	0.597964
$156$	0	0
$157$	2834.52	1.44089	0.720445	−	0.693512i	$-0.243938\pi$
$157$	2834.52	1.44089	0.720445	+	0.693512i	$0.243938\pi$
$158$	−1093.17	−0.550428
$159$	0	0
$160$	160.000	0.0790569
$161$	68.6590	0.0336092
$162$	0	0
$163$	2114.51	1.01608	0.508041	−	0.861333i	$-0.330370\pi$
$163$	2114.51	1.01608	0.508041	+	0.861333i	$0.330370\pi$
$164$	−1578.55	−0.751610
$165$	0	0
$166$	−40.5244	−0.0189476
$167$	−2487.41	−1.15258	−0.576291	−	0.817244i	$-0.695501\pi$
$167$	−2487.41	−1.15258	−0.576291	+	0.817244i	$0.695501\pi$
$168$	0	0
$169$	−2052.93	−0.934426
$170$	−1062.08	−0.479163
$171$	0	0
$172$	544.251	0.241272
$173$	−3672.32	−1.61388	−0.806939	−	0.590634i	$-0.798878\pi$
$173$	−3672.32	−1.61388	−0.806939	+	0.590634i	$0.798878\pi$
$174$	0	0
$175$	74.6293	0.0322368
$176$	−1098.02	−0.470263
$177$	0	0
$178$	−2174.61	−0.915697
$179$	3224.63	1.34648	0.673240	−	0.739424i	$-0.264902\pi$
$179$	3224.63	1.34648	0.673240	+	0.739424i	$0.264902\pi$
$180$	0	0
$181$	−2224.23	−0.913401	−0.456700	−	0.889621i	$-0.650969\pi$
$181$	−2224.23	−0.913401	−0.456700	+	0.889621i	$0.650969\pi$
$182$	71.6606	0.0291859
$183$	0	0
$184$	−184.000	−0.0737210
$185$	539.780	0.214516
$186$	0	0
$187$	7288.64	2.85026
$188$	201.805	0.0782880
$189$	0	0
$190$	−484.891	−0.185146
$191$	−3273.53	−1.24013	−0.620063	−	0.784552i	$-0.712893\pi$
$191$	−3273.53	−1.24013	−0.620063	+	0.784552i	$0.712893\pi$
$192$	0	0
$193$	5126.86	1.91212	0.956062	−	0.293166i	$-0.0947089\pi$
$193$	5126.86	1.91212	0.956062	+	0.293166i	$0.0947089\pi$
$194$	2231.81	0.825951
$195$	0	0
$196$	−1336.35	−0.487010
$197$	−3262.57	−1.17994	−0.589971	−	0.807425i	$-0.700861\pi$
$197$	−3262.57	−1.17994	−0.589971	+	0.807425i	$0.700861\pi$
$198$	0	0
$199$	−168.282	−0.0599457	−0.0299728	−	0.999551i	$-0.509542\pi$
$199$	−168.282	−0.0599457	−0.0299728	+	0.999551i	$0.509542\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−269.131	−0.0937424
$203$	−403.673	−0.139568
$204$	0	0
$205$	1973.19	0.672260
$206$	1851.91	0.626352
$207$	0	0
$208$	−192.044	−0.0640186
$209$	3327.62	1.10132
$210$	0	0
$211$	4197.01	1.36936	0.684678	−	0.728846i	$-0.259943\pi$
$211$	4197.01	1.36936	0.684678	+	0.728846i	$0.259943\pi$
$212$	1658.83	0.537400
$213$	0	0
$214$	3742.25	1.19540
$215$	−680.313	−0.215800
$216$	0	0
$217$	−688.925	−0.215518
$218$	1956.48	0.607843
$219$	0	0
$220$	1372.52	0.420616
$221$	1274.79	0.388015
$222$	0	0
$223$	5192.66	1.55931	0.779656	−	0.626208i	$-0.215394\pi$
$223$	5192.66	1.55931	0.779656	+	0.626208i	$0.215394\pi$
$224$	−95.5256	−0.0284936
$225$	0	0
$226$	−3520.91	−1.03632
$227$	−4701.24	−1.37459	−0.687295	−	0.726378i	$-0.741202\pi$
$227$	−4701.24	−1.37459	−0.687295	+	0.726378i	$0.741202\pi$
$228$	0	0
$229$	−2125.17	−0.613255	−0.306628	−	0.951830i	$-0.599201\pi$
$229$	−2125.17	−0.613255	−0.306628	+	0.951830i	$0.599201\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	1081.81	0.306138
$233$	1646.67	0.462992	0.231496	−	0.972836i	$-0.425638\pi$
$233$	1646.67	0.462992	0.231496	+	0.972836i	$0.425638\pi$
$234$	0	0
$235$	−252.256	−0.0700229
$236$	734.345	0.202550
$237$	0	0
$238$	634.097	0.172699
$239$	−2877.88	−0.778888	−0.389444	−	0.921050i	$-0.627333\pi$
$239$	−2877.88	−0.778888	−0.389444	+	0.921050i	$0.627333\pi$
$240$	0	0
$241$	−4244.54	−1.13450	−0.567251	−	0.823545i	$-0.691993\pi$
$241$	−4244.54	−1.13450	−0.567251	+	0.823545i	$0.691993\pi$
$242$	−6757.12	−1.79489
$243$	0	0
$244$	−393.900	−0.103348
$245$	1670.44	0.435595
$246$	0	0
$247$	582.003	0.149927
$248$	1846.26	0.472732
$249$	0	0
$250$	250.000	0.0632456
$251$	−4401.99	−1.10698	−0.553488	−	0.832857i	$-0.686704\pi$
$251$	−4401.99	−1.10698	−0.553488	+	0.832857i	$0.686704\pi$
$252$	0	0
$253$	−1578.40	−0.392227
$254$	2521.75	0.622947
$255$	0	0
$256$	256.000	0.0625000
$257$	−1694.78	−0.411352	−0.205676	−	0.978620i	$-0.565939\pi$
$257$	−1694.78	−0.411352	−0.205676	+	0.978620i	$0.565939\pi$
$258$	0	0
$259$	−322.268	−0.0773156
$260$	240.055	0.0572600
$261$	0	0
$262$	888.566	0.209526
$263$	727.462	0.170560	0.0852799	−	0.996357i	$-0.472822\pi$
$263$	727.462	0.170560	0.0852799	+	0.996357i	$0.472822\pi$
$264$	0	0
$265$	−2073.54	−0.480666
$266$	289.497	0.0667300
$267$	0	0
$268$	547.698	0.124836
$269$	−94.9937	−0.0215311	−0.0107656	−	0.999942i	$-0.503427\pi$
$269$	−94.9937	−0.0215311	−0.0107656	+	0.999942i	$0.503427\pi$
$270$	0	0
$271$	5321.54	1.19284	0.596422	−	0.802671i	$-0.296589\pi$
$271$	5321.54	1.19284	0.596422	+	0.802671i	$0.296589\pi$
$272$	−1699.32	−0.378811
$273$	0	0
$274$	858.143	0.189206
$275$	−1715.66	−0.376211
$276$	0	0
$277$	2561.25	0.555562	0.277781	−	0.960644i	$-0.410401\pi$
$277$	2561.25	0.555562	0.277781	+	0.960644i	$0.410401\pi$
$278$	499.510	0.107765
$279$	0	0
$280$	119.407	0.0254855
$281$	8753.73	1.85838	0.929188	−	0.369606i	$-0.120507\pi$
$281$	8753.73	1.85838	0.929188	+	0.369606i	$0.120507\pi$
$282$	0	0
$283$	−2400.14	−0.504147	−0.252073	−	0.967708i	$-0.581112\pi$
$283$	−2400.14	−0.504147	−0.252073	+	0.967708i	$0.581112\pi$
$284$	2835.20	0.592388
$285$	0	0
$286$	−1647.41	−0.340606
$287$	−1178.06	−0.242295
$288$	0	0
$289$	6367.09	1.29597
$290$	−1352.26	−0.273818
$291$	0	0
$292$	−2758.42	−0.552823
$293$	−1045.56	−0.208472	−0.104236	−	0.994553i	$-0.533240\pi$
$293$	−1045.56	−0.208472	−0.104236	+	0.994553i	$0.533240\pi$
$294$	0	0
$295$	−917.931	−0.181166
$296$	863.649	0.169590
$297$	0	0
$298$	−3128.37	−0.608126
$299$	−276.063	−0.0533952
$300$	0	0
$301$	406.171	0.0777784
$302$	3720.08	0.708830
$303$	0	0
$304$	−775.826	−0.146371
$305$	492.376	0.0924372
$306$	0	0
$307$	−7905.84	−1.46974	−0.734870	−	0.678208i	$-0.762757\pi$
$307$	−7905.84	−1.46974	−0.734870	+	0.678208i	$0.762757\pi$
$308$	−819.445	−0.151598
$309$	0	0
$310$	−2307.82	−0.422824
$311$	9227.63	1.68248	0.841240	−	0.540662i	$-0.181826\pi$
$311$	9227.63	1.68248	0.841240	+	0.540662i	$0.181826\pi$
$312$	0	0
$313$	2712.57	0.489852	0.244926	−	0.969542i	$-0.421236\pi$
$313$	2712.57	0.489852	0.244926	+	0.969542i	$0.421236\pi$
$314$	−5669.05	−1.01886
$315$	0	0
$316$	2186.33	0.389212
$317$	3440.29	0.609546	0.304773	−	0.952425i	$-0.401419\pi$
$317$	3440.29	0.609546	0.304773	+	0.952425i	$0.401419\pi$
$318$	0	0
$319$	9280.04	1.62879
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−137.318	−0.0237653
$323$	5149.92	0.887149
$324$	0	0
$325$	−300.069	−0.0512149
$326$	−4229.03	−0.718479
$327$	0	0
$328$	3157.10	0.531468
$329$	150.606	0.0252376
$330$	0	0
$331$	3259.75	0.541305	0.270652	−	0.962677i	$-0.412761\pi$
$331$	3259.75	0.541305	0.270652	+	0.962677i	$0.412761\pi$
$332$	81.0487	0.0133980
$333$	0	0
$334$	4974.81	0.814999
$335$	−684.623	−0.111657
$336$	0	0
$337$	7163.38	1.15790	0.578952	−	0.815361i	$-0.303462\pi$
$337$	7163.38	1.15790	0.578952	+	0.815361i	$0.303462\pi$
$338$	4105.87	0.660739
$339$	0	0
$340$	2124.16	0.338819
$341$	15837.7	2.51513
$342$	0	0
$343$	−2021.23	−0.318181
$344$	−1088.50	−0.170605
$345$	0	0
$346$	7344.63	1.14118
$347$	1833.45	0.283645	0.141822	−	0.989892i	$-0.454704\pi$
$347$	1833.45	0.283645	0.141822	+	0.989892i	$0.454704\pi$
$348$	0	0
$349$	−10400.3	−1.59517	−0.797583	−	0.603209i	$-0.793888\pi$
$349$	−10400.3	−1.59517	−0.797583	+	0.603209i	$0.793888\pi$
$350$	−149.259	−0.0227949
$351$	0	0
$352$	2196.04	0.332526
$353$	3156.48	0.475928	0.237964	−	0.971274i	$-0.423520\pi$
$353$	3156.48	0.475928	0.237964	+	0.971274i	$0.423520\pi$
$354$	0	0
$355$	−3544.00	−0.529848
$356$	4349.22	0.647495
$357$	0	0
$358$	−6449.25	−0.952105
$359$	9324.16	1.37078	0.685390	−	0.728176i	$-0.259632\pi$
$359$	9324.16	1.37078	0.685390	+	0.728176i	$0.259632\pi$
$360$	0	0
$361$	−4507.81	−0.657211
$362$	4448.45	0.645872
$363$	0	0
$364$	−143.321	−0.0206376
$365$	3448.02	0.494460
$366$	0	0
$367$	3802.68	0.540867	0.270434	−	0.962739i	$-0.412833\pi$
$367$	3802.68	0.540867	0.270434	+	0.962739i	$0.412833\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−1079.56	−0.151686
$371$	1237.97	0.173241
$372$	0	0
$373$	−8928.03	−1.23935	−0.619673	−	0.784860i	$-0.712735\pi$
$373$	−8928.03	−1.23935	−0.619673	+	0.784860i	$0.712735\pi$
$374$	−14577.3	−2.01544
$375$	0	0
$376$	−403.610	−0.0553580
$377$	1623.08	0.221732
$378$	0	0
$379$	3798.13	0.514768	0.257384	−	0.966309i	$-0.417140\pi$
$379$	3798.13	0.514768	0.257384	+	0.966309i	$0.417140\pi$
$380$	969.782	0.130918
$381$	0	0
$382$	6547.05	0.876901
$383$	−1456.41	−0.194306	−0.0971530	−	0.995269i	$-0.530974\pi$
$383$	−1456.41	−0.194306	−0.0971530	+	0.995269i	$0.530974\pi$
$384$	0	0
$385$	1024.31	0.135593
$386$	−10253.7	−1.35208
$387$	0	0
$388$	−4463.62	−0.584036
$389$	8211.34	1.07026	0.535130	−	0.844769i	$-0.320262\pi$
$389$	8211.34	1.07026	0.535130	+	0.844769i	$0.320262\pi$
$390$	0	0
$391$	−2442.78	−0.315950
$392$	2672.71	0.344368
$393$	0	0
$394$	6525.14	0.834345
$395$	−2732.92	−0.348121
$396$	0	0
$397$	914.365	0.115594	0.0577968	−	0.998328i	$-0.481592\pi$
$397$	914.365	0.115594	0.0577968	+	0.998328i	$0.481592\pi$
$398$	336.564	0.0423880
$399$	0	0
$400$	400.000	0.0500000
$401$	−9414.97	−1.17247	−0.586236	−	0.810140i	$-0.699391\pi$
$401$	−9414.97	−1.17247	−0.586236	+	0.810140i	$0.699391\pi$
$402$	0	0
$403$	2770.02	0.342394
$404$	538.261	0.0662859
$405$	0	0
$406$	807.345	0.0986894
$407$	7408.62	0.902289
$408$	0	0
$409$	3299.41	0.398889	0.199444	−	0.979909i	$-0.436086\pi$
$409$	3299.41	0.398889	0.199444	+	0.979909i	$0.436086\pi$
$410$	−3946.37	−0.475360
$411$	0	0
$412$	−3703.81	−0.442898
$413$	548.037	0.0652957
$414$	0	0
$415$	−101.311	−0.0119835
$416$	384.088	0.0452680
$417$	0	0
$418$	−6655.25	−0.778753
$419$	3554.64	0.414452	0.207226	−	0.978293i	$-0.433556\pi$
$419$	3554.64	0.414452	0.207226	+	0.978293i	$0.433556\pi$
$420$	0	0
$421$	11655.5	1.34930	0.674649	−	0.738139i	$-0.264295\pi$
$421$	11655.5	1.34930	0.674649	+	0.738139i	$0.264295\pi$
$422$	−8394.02	−0.968281
$423$	0	0
$424$	−3317.66	−0.380000
$425$	−2655.19	−0.303049
$426$	0	0
$427$	−293.965	−0.0333161
$428$	−7484.50	−0.845273
$429$	0	0
$430$	1360.63	0.152594
$431$	−4727.38	−0.528330	−0.264165	−	0.964478i	$-0.585096\pi$
$431$	−4727.38	−0.528330	−0.264165	+	0.964478i	$0.585096\pi$
$432$	0	0
$433$	3936.24	0.436868	0.218434	−	0.975852i	$-0.429905\pi$
$433$	3936.24	0.436868	0.218434	+	0.975852i	$0.429905\pi$
$434$	1377.85	0.152394
$435$	0	0
$436$	−3912.97	−0.429810
$437$	−1115.25	−0.122081
$438$	0	0
$439$	6859.77	0.745784	0.372892	−	0.927875i	$-0.378366\pi$
$439$	6859.77	0.745784	0.372892	+	0.927875i	$0.378366\pi$
$440$	−2745.05	−0.297421
$441$	0	0
$442$	−2549.57	−0.274368
$443$	−16814.4	−1.80334	−0.901668	−	0.432430i	$-0.857656\pi$
$443$	−16814.4	−1.80334	−0.901668	+	0.432430i	$0.857656\pi$
$444$	0	0
$445$	−5436.53	−0.579137
$446$	−10385.3	−1.10260
$447$	0	0
$448$	191.051	0.0201480
$449$	−10673.8	−1.12189	−0.560943	−	0.827855i	$-0.689561\pi$
$449$	−10673.8	−1.12189	−0.560943	+	0.827855i	$0.689561\pi$
$450$	0	0
$451$	27082.5	2.82764
$452$	7041.82	0.732786
$453$	0	0
$454$	9402.47	0.971982
$455$	179.152	0.0184588
$456$	0	0
$457$	15493.0	1.58585	0.792926	−	0.609318i	$-0.208557\pi$
$457$	15493.0	1.58585	0.792926	+	0.609318i	$0.208557\pi$
$458$	4250.35	0.433637
$459$	0	0
$460$	−460.000	−0.0466252
$461$	10793.0	1.09041	0.545207	−	0.838302i	$-0.316451\pi$
$461$	10793.0	1.09041	0.545207	+	0.838302i	$0.316451\pi$
$462$	0	0
$463$	−6165.87	−0.618904	−0.309452	−	0.950915i	$-0.600146\pi$
$463$	−6165.87	−0.618904	−0.309452	+	0.950915i	$0.600146\pi$
$464$	−2163.61	−0.216473
$465$	0	0
$466$	−3293.34	−0.327385
$467$	10736.4	1.06386	0.531929	−	0.846789i	$-0.321467\pi$
$467$	10736.4	1.06386	0.531929	+	0.846789i	$0.321467\pi$
$468$	0	0
$469$	408.744	0.0402431
$470$	504.512	0.0495137
$471$	0	0
$472$	−1468.69	−0.143224
$473$	−9337.47	−0.907690
$474$	0	0
$475$	−1212.23	−0.117096
$476$	−1268.19	−0.122117
$477$	0	0
$478$	5755.75	0.550757
$479$	−6333.30	−0.604125	−0.302062	−	0.953288i	$-0.597675\pi$
$479$	−6333.30	−0.604125	−0.302062	+	0.953288i	$0.597675\pi$
$480$	0	0
$481$	1295.77	0.122832
$482$	8489.08	0.802213
$483$	0	0
$484$	13514.2	1.26918
$485$	5579.52	0.522377
$486$	0	0
$487$	−16833.3	−1.56630	−0.783149	−	0.621834i	$-0.786388\pi$
$487$	−16833.3	−1.56630	−0.783149	+	0.621834i	$0.786388\pi$
$488$	787.801	0.0730780
$489$	0	0
$490$	−3340.89	−0.308012
$491$	4.80428	0.000441577 0	0.000220788	−	1.00000i	$-0.499930\pi$
$491$	4.80428	0.000441577 0	0.000220788		1.00000i	$0.499930\pi$
$492$	0	0
$493$	14362.0	1.31204
$494$	−1164.01	−0.106014
$495$	0	0
$496$	−3692.52	−0.334272
$497$	2115.89	0.190967
$498$	0	0
$499$	21206.9	1.90251	0.951253	−	0.308412i	$-0.0997975\pi$
$499$	21206.9	1.90251	0.951253	+	0.308412i	$0.0997975\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	8803.98	0.782750
$503$	−14513.7	−1.28655	−0.643275	−	0.765635i	$-0.722425\pi$
$503$	−14513.7	−1.28655	−0.643275	+	0.765635i	$0.722425\pi$
$504$	0	0
$505$	−672.827	−0.0592879
$506$	3156.81	0.277346
$507$	0	0
$508$	−5043.49	−0.440490
$509$	2545.77	0.221689	0.110844	−	0.993838i	$-0.464645\pi$
$509$	2545.77	0.221689	0.110844	+	0.993838i	$0.464645\pi$
$510$	0	0
$511$	−2058.59	−0.178213
$512$	−512.000	−0.0441942
$513$	0	0
$514$	3389.56	0.290870
$515$	4629.77	0.396140
$516$	0	0
$517$	−3462.28	−0.294528
$518$	644.535	0.0546704
$519$	0	0
$520$	−480.110	−0.0404889
$521$	21343.1	1.79473	0.897367	−	0.441285i	$-0.145477\pi$
$521$	21343.1	1.79473	0.897367	+	0.441285i	$0.145477\pi$
$522$	0	0
$523$	−789.191	−0.0659827	−0.0329913	−	0.999456i	$-0.510503\pi$
$523$	−789.191	−0.0659827	−0.0329913	+	0.999456i	$0.510503\pi$
$524$	−1777.13	−0.148157
$525$	0	0
$526$	−1454.92	−0.120604
$527$	24510.9	2.02602
$528$	0	0
$529$	529.000	0.0434783
$530$	4147.07	0.339882
$531$	0	0
$532$	−578.994	−0.0471852
$533$	4736.74	0.384936
$534$	0	0
$535$	9355.63	0.756036
$536$	−1095.40	−0.0882723
$537$	0	0
$538$	189.987	0.0152248
$539$	22927.2	1.83218
$540$	0	0
$541$	13405.1	1.06531	0.532653	−	0.846334i	$-0.321195\pi$
$541$	13405.1	1.06531	0.532653	+	0.846334i	$0.321195\pi$
$542$	−10643.1	−0.843468
$543$	0	0
$544$	3398.65	0.267860
$545$	4891.21	0.384434
$546$	0	0
$547$	10088.8	0.788605	0.394303	−	0.918981i	$-0.370986\pi$
$547$	10088.8	0.788605	0.394303	+	0.918981i	$0.370986\pi$
$548$	−1716.29	−0.133789
$549$	0	0
$550$	3431.31	0.266021
$551$	6556.98	0.506963
$552$	0	0
$553$	1631.65	0.125470
$554$	−5122.50	−0.392842
$555$	0	0
$556$	−999.019	−0.0762012
$557$	11657.7	0.886809	0.443405	−	0.896322i	$-0.353770\pi$
$557$	11657.7	0.886809	0.443405	+	0.896322i	$0.353770\pi$
$558$	0	0
$559$	−1633.13	−0.123567
$560$	−238.814	−0.0180209
$561$	0	0
$562$	−17507.5	−1.31407
$563$	−4839.43	−0.362270	−0.181135	−	0.983458i	$-0.557977\pi$
$563$	−4839.43	−0.362270	−0.181135	+	0.983458i	$0.557977\pi$
$564$	0	0
$565$	−8802.27	−0.655424
$566$	4800.28	0.356486
$567$	0	0
$568$	−5670.40	−0.418882
$569$	646.680	0.0476454	0.0238227	−	0.999716i	$-0.492416\pi$
$569$	646.680	0.0476454	0.0238227	+	0.999716i	$0.492416\pi$
$570$	0	0
$571$	3263.66	0.239194	0.119597	−	0.992822i	$-0.461840\pi$
$571$	3263.66	0.239194	0.119597	+	0.992822i	$0.461840\pi$
$572$	3294.82	0.240845
$573$	0	0
$574$	2356.12	0.171329
$575$	575.000	0.0417029
$576$	0	0
$577$	−8980.46	−0.647940	−0.323970	−	0.946067i	$-0.605018\pi$
$577$	−8980.46	−0.647940	−0.323970	+	0.946067i	$0.605018\pi$
$578$	−12734.2	−0.916388
$579$	0	0
$580$	2704.52	0.193619
$581$	60.4861	0.00431908
$582$	0	0
$583$	−28459.8	−2.02176
$584$	5516.84	0.390905
$585$	0	0
$586$	2091.12	0.147412
$587$	6538.05	0.459718	0.229859	−	0.973224i	$-0.426174\pi$
$587$	6538.05	0.459718	0.229859	+	0.973224i	$0.426174\pi$
$588$	0	0
$589$	11190.4	0.782841
$590$	1835.86	0.128104
$591$	0	0
$592$	−1727.30	−0.119918
$593$	−7803.41	−0.540384	−0.270192	−	0.962806i	$-0.587087\pi$
$593$	−7803.41	−0.540384	−0.270192	+	0.962806i	$0.587087\pi$
$594$	0	0
$595$	1585.24	0.109225
$596$	6256.73	0.430010
$597$	0	0
$598$	552.127	0.0377561
$599$	27812.0	1.89711	0.948555	−	0.316613i	$-0.102546\pi$
$599$	27812.0	1.89711	0.948555	+	0.316613i	$0.102546\pi$
$600$	0	0
$601$	483.216	0.0327966	0.0163983	−	0.999866i	$-0.494780\pi$
$601$	483.216	0.0327966	0.0163983	+	0.999866i	$0.494780\pi$
$602$	−812.342	−0.0549976
$603$	0	0
$604$	−7440.17	−0.501219
$605$	−16892.8	−1.13519
$606$	0	0
$607$	−18104.0	−1.21058	−0.605289	−	0.796006i	$-0.706942\pi$
$607$	−18104.0	−1.21058	−0.605289	+	0.796006i	$0.706942\pi$
$608$	1551.65	0.103500
$609$	0	0
$610$	−984.751	−0.0653630
$611$	−605.554	−0.0400951
$612$	0	0
$613$	2191.19	0.144374	0.0721870	−	0.997391i	$-0.477002\pi$
$613$	2191.19	0.144374	0.0721870	+	0.997391i	$0.477002\pi$
$614$	15811.7	1.03926
$615$	0	0
$616$	1638.89	0.107196
$617$	8243.06	0.537849	0.268925	−	0.963161i	$-0.413332\pi$
$617$	8243.06	0.537849	0.268925	+	0.963161i	$0.413332\pi$
$618$	0	0
$619$	18298.0	1.18814	0.594069	−	0.804414i	$-0.297521\pi$
$619$	18298.0	1.18814	0.594069	+	0.804414i	$0.297521\pi$
$620$	4615.65	0.298982
$621$	0	0
$622$	−18455.3	−1.18969
$623$	3245.80	0.208732
$624$	0	0
$625$	625.000	0.0400000
$626$	−5425.15	−0.346378
$627$	0	0
$628$	11338.1	0.720445
$629$	11465.8	0.726821
$630$	0	0
$631$	−26472.0	−1.67010	−0.835051	−	0.550172i	$-0.814562\pi$
$631$	−26472.0	−1.67010	−0.835051	+	0.550172i	$0.814562\pi$
$632$	−4372.67	−0.275214
$633$	0	0
$634$	−6880.58	−0.431014
$635$	6304.37	0.393986
$636$	0	0
$637$	4009.99	0.249421
$638$	−18560.1	−1.15173
$639$	0	0
$640$	640.000	0.0395285
$641$	−7411.93	−0.456714	−0.228357	−	0.973577i	$-0.573335\pi$
$641$	−7411.93	−0.456714	−0.228357	+	0.973577i	$0.573335\pi$
$642$	0	0
$643$	8405.96	0.515550	0.257775	−	0.966205i	$-0.417011\pi$
$643$	8405.96	0.515550	0.257775	+	0.966205i	$0.417011\pi$
$644$	274.636	0.0168046
$645$	0	0
$646$	−10299.8	−0.627309
$647$	−27878.3	−1.69398	−0.846992	−	0.531605i	$-0.821589\pi$
$647$	−27878.3	−1.69398	−0.846992	+	0.531605i	$0.821589\pi$
$648$	0	0
$649$	−12598.8	−0.762014
$650$	600.138	0.0362144
$651$	0	0
$652$	8458.06	0.508041
$653$	18871.1	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$653$	18871.1	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$654$	0	0
$655$	2221.42	0.132516
$656$	−6314.20	−0.375805
$657$	0	0
$658$	−301.211	−0.0178457
$659$	12144.4	0.717872	0.358936	−	0.933362i	$-0.383140\pi$
$659$	12144.4	0.717872	0.358936	+	0.933362i	$0.383140\pi$
$660$	0	0
$661$	−32504.2	−1.91266	−0.956328	−	0.292295i	$-0.905581\pi$
$661$	−32504.2	−1.91266	−0.956328	+	0.292295i	$0.905581\pi$
$662$	−6519.49	−0.382760
$663$	0	0
$664$	−162.097	−0.00947380
$665$	723.742	0.0422038
$666$	0	0
$667$	−3110.19	−0.180551
$668$	−9949.62	−0.576291
$669$	0	0
$670$	1369.25	0.0789531
$671$	6757.98	0.388806
$672$	0	0
$673$	−21387.8	−1.22502	−0.612511	−	0.790462i	$-0.709840\pi$
$673$	−21387.8	−1.22502	−0.612511	+	0.790462i	$0.709840\pi$
$674$	−14326.8	−0.818762
$675$	0	0
$676$	−8211.74	−0.467213
$677$	−10494.6	−0.595776	−0.297888	−	0.954601i	$-0.596282\pi$
$677$	−10494.6	−0.595776	−0.297888	+	0.954601i	$0.596282\pi$
$678$	0	0
$679$	−3331.17	−0.188275
$680$	−4248.31	−0.239581
$681$	0	0
$682$	−31675.4	−1.77847
$683$	−24014.2	−1.34535	−0.672677	−	0.739937i	$-0.734855\pi$
$683$	−24014.2	−1.34535	−0.672677	+	0.739937i	$0.734855\pi$
$684$	0	0
$685$	2145.36	0.119664
$686$	4042.45	0.224988
$687$	0	0
$688$	2177.00	0.120636
$689$	−4977.63	−0.275229
$690$	0	0
$691$	−10825.8	−0.595995	−0.297997	−	0.954567i	$-0.596319\pi$
$691$	−10825.8	−0.595995	−0.297997	+	0.954567i	$0.596319\pi$
$692$	−14689.3	−0.806939
$693$	0	0
$694$	−3666.90	−0.200567
$695$	1248.77	0.0681564
$696$	0	0
$697$	41913.6	2.27775
$698$	20800.5	1.12795
$699$	0	0
$700$	298.517	0.0161184
$701$	171.021	0.00921451	0.00460726	−	0.999989i	$-0.498533\pi$
$701$	171.021	0.00921451	0.00460726	+	0.999989i	$0.498533\pi$
$702$	0	0
$703$	5234.69	0.280840
$704$	−4392.08	−0.235132
$705$	0	0
$706$	−6312.96	−0.336532
$707$	401.701	0.0213685
$708$	0	0
$709$	28732.7	1.52197	0.760987	−	0.648767i	$-0.224715\pi$
$709$	28732.7	1.52197	0.760987	+	0.648767i	$0.224715\pi$
$710$	7088.00	0.374659
$711$	0	0
$712$	−8698.45	−0.457848
$713$	−5307.99	−0.278802
$714$	0	0
$715$	−4118.52	−0.215418
$716$	12898.5	0.673240
$717$	0	0
$718$	−18648.3	−0.969288
$719$	−6976.22	−0.361849	−0.180924	−	0.983497i	$-0.557909\pi$
$719$	−6976.22	−0.361849	−0.180924	+	0.983497i	$0.557909\pi$
$720$	0	0
$721$	−2764.13	−0.142776
$722$	9015.61	0.464718
$723$	0	0
$724$	−8896.91	−0.456700
$725$	−3380.65	−0.173178
$726$	0	0
$727$	13114.6	0.669042	0.334521	−	0.942388i	$-0.391426\pi$
$727$	13114.6	0.669042	0.334521	+	0.942388i	$0.391426\pi$
$728$	286.643	0.0145930
$729$	0	0
$730$	−6896.05	−0.349636
$731$	−14450.9	−0.731172
$732$	0	0
$733$	−22681.7	−1.14293	−0.571466	−	0.820626i	$-0.693625\pi$
$733$	−22681.7	−1.14293	−0.571466	+	0.820626i	$0.693625\pi$
$734$	−7605.36	−0.382451
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−9396.62	−0.469646
$738$	0	0
$739$	24237.8	1.20650	0.603249	−	0.797553i	$-0.293873\pi$
$739$	24237.8	1.20650	0.603249	+	0.797553i	$0.293873\pi$
$740$	2159.12	0.107258
$741$	0	0
$742$	−2475.95	−0.122500
$743$	−27356.7	−1.35077	−0.675384	−	0.737466i	$-0.736022\pi$
$743$	−27356.7	−1.35077	−0.675384	+	0.737466i	$0.736022\pi$
$744$	0	0
$745$	−7820.92	−0.384612
$746$	17856.1	0.876350
$747$	0	0
$748$	29154.6	1.42513
$749$	−5585.63	−0.272489
$750$	0	0
$751$	22297.8	1.08343	0.541716	−	0.840561i	$-0.317775\pi$
$751$	22297.8	1.08343	0.541716	+	0.840561i	$0.317775\pi$
$752$	807.220	0.0391440
$753$	0	0
$754$	−3246.17	−0.156788
$755$	9300.21	0.448304
$756$	0	0
$757$	−16088.4	−0.772447	−0.386224	−	0.922405i	$-0.626221\pi$
$757$	−16088.4	−0.772447	−0.386224	+	0.922405i	$0.626221\pi$
$758$	−7596.26	−0.363996
$759$	0	0
$760$	−1939.56	−0.0925729
$761$	−22500.2	−1.07179	−0.535894	−	0.844285i	$-0.680025\pi$
$761$	−22500.2	−1.07179	−0.535894	+	0.844285i	$0.680025\pi$
$762$	0	0
$763$	−2920.22	−0.138557
$764$	−13094.1	−0.620063
$765$	0	0
$766$	2912.82	0.137395
$767$	−2203.54	−0.103736
$768$	0	0
$769$	−22677.7	−1.06343	−0.531716	−	0.846923i	$-0.678453\pi$
$769$	−22677.7	−1.06343	−0.531716	+	0.846923i	$0.678453\pi$
$770$	−2048.61	−0.0958790
$771$	0	0
$772$	20507.5	0.956062
$773$	−20465.1	−0.952235	−0.476118	−	0.879382i	$-0.657956\pi$
$773$	−20465.1	−0.952235	−0.476118	+	0.879382i	$0.657956\pi$
$774$	0	0
$775$	−5769.56	−0.267418
$776$	8927.24	0.412976
$777$	0	0
$778$	−16422.7	−0.756789
$779$	19135.6	0.880108
$780$	0	0
$781$	−48642.3	−2.22863
$782$	4885.56	0.223411
$783$	0	0
$784$	−5345.42	−0.243505
$785$	−14172.6	−0.644386
$786$	0	0
$787$	−28850.8	−1.30676	−0.653379	−	0.757031i	$-0.726649\pi$
$787$	−28850.8	−1.30676	−0.653379	+	0.757031i	$0.726649\pi$
$788$	−13050.3	−0.589971
$789$	0	0
$790$	5465.83	0.246159
$791$	5255.26	0.236227
$792$	0	0
$793$	1181.97	0.0529295
$794$	−1828.73	−0.0817370
$795$	0	0
$796$	−673.128	−0.0299728
$797$	39332.9	1.74811	0.874054	−	0.485828i	$-0.161482\pi$
$797$	39332.9	1.74811	0.874054	+	0.485828i	$0.161482\pi$
$798$	0	0
$799$	−5358.31	−0.237251
$800$	−800.000	−0.0353553
$801$	0	0
$802$	18829.9	0.829063
$803$	47325.0	2.07978
$804$	0	0
$805$	−343.295	−0.0150305
$806$	−5540.05	−0.242109
$807$	0	0
$808$	−1076.52	−0.0468712
$809$	−33946.5	−1.47527	−0.737637	−	0.675198i	$-0.764058\pi$
$809$	−33946.5	−1.47527	−0.737637	+	0.675198i	$0.764058\pi$
$810$	0	0
$811$	−20772.2	−0.899398	−0.449699	−	0.893180i	$-0.648469\pi$
$811$	−20772.2	−0.899398	−0.449699	+	0.893180i	$0.648469\pi$
$812$	−1614.69	−0.0697839
$813$	0	0
$814$	−14817.2	−0.638015
$815$	−10572.6	−0.454406
$816$	0	0
$817$	−6597.56	−0.282521
$818$	−6598.83	−0.282057
$819$	0	0
$820$	7892.75	0.336130
$821$	−6466.37	−0.274882	−0.137441	−	0.990510i	$-0.543888\pi$
$821$	−6466.37	−0.274882	−0.137441	+	0.990510i	$0.543888\pi$
$822$	0	0
$823$	−16503.7	−0.699008	−0.349504	−	0.936935i	$-0.613650\pi$
$823$	−16503.7	−0.699008	−0.349504	+	0.936935i	$0.613650\pi$
$824$	7407.63	0.313176
$825$	0	0
$826$	−1096.07	−0.0461710
$827$	−36540.8	−1.53645	−0.768227	−	0.640177i	$-0.778861\pi$
$827$	−36540.8	−1.53645	−0.768227	+	0.640177i	$0.778861\pi$
$828$	0	0
$829$	−24874.5	−1.04213	−0.521066	−	0.853516i	$-0.674465\pi$
$829$	−24874.5	−1.04213	−0.521066	+	0.853516i	$0.674465\pi$
$830$	202.622	0.00847362
$831$	0	0
$832$	−768.177	−0.0320093
$833$	35482.8	1.47588
$834$	0	0
$835$	12437.0	0.515450
$836$	13310.5	0.550662
$837$	0	0
$838$	−7109.28	−0.293062
$839$	−15814.3	−0.650741	−0.325370	−	0.945587i	$-0.605489\pi$
$839$	−15814.3	−0.650741	−0.325370	+	0.945587i	$0.605489\pi$
$840$	0	0
$841$	−6102.97	−0.250234
$842$	−23311.0	−0.954097
$843$	0	0
$844$	16788.0	0.684678
$845$	10264.7	0.417888
$846$	0	0
$847$	10085.6	0.409144
$848$	6635.32	0.268700
$849$	0	0
$850$	5310.39	0.214288
$851$	−2482.99	−0.100019
$852$	0	0
$853$	−31093.2	−1.24808	−0.624039	−	0.781394i	$-0.714509\pi$
$853$	−31093.2	−1.24808	−0.624039	+	0.781394i	$0.714509\pi$
$854$	587.931	0.0235580
$855$	0	0
$856$	14969.0	0.597699
$857$	−34212.4	−1.36368	−0.681840	−	0.731501i	$-0.738820\pi$
$857$	−34212.4	−1.36368	−0.681840	+	0.731501i	$0.738820\pi$
$858$	0	0
$859$	4821.57	0.191513	0.0957567	−	0.995405i	$-0.469473\pi$
$859$	4821.57	0.191513	0.0957567	+	0.995405i	$0.469473\pi$
$860$	−2721.25	−0.107900
$861$	0	0
$862$	9454.77	0.373585
$863$	−19145.6	−0.755185	−0.377593	−	0.925972i	$-0.623248\pi$
$863$	−19145.6	−0.755185	−0.377593	+	0.925972i	$0.623248\pi$
$864$	0	0
$865$	18361.6	0.721748
$866$	−7872.49	−0.308912
$867$	0	0
$868$	−2755.70	−0.107759
$869$	−37509.9	−1.46426
$870$	0	0
$871$	−1643.47	−0.0639345
$872$	7825.93	0.303921
$873$	0	0
$874$	2230.50	0.0863246
$875$	−373.147	−0.0144168
$876$	0	0
$877$	16890.8	0.650354	0.325177	−	0.945653i	$-0.394576\pi$
$877$	16890.8	0.650354	0.325177	+	0.945653i	$0.394576\pi$
$878$	−13719.5	−0.527349
$879$	0	0
$880$	5490.10	0.210308
$881$	27802.3	1.06321	0.531603	−	0.846994i	$-0.321590\pi$
$881$	27802.3	1.06321	0.531603	+	0.846994i	$0.321590\pi$
$882$	0	0
$883$	9794.15	0.373272	0.186636	−	0.982429i	$-0.440241\pi$
$883$	9794.15	0.373272	0.186636	+	0.982429i	$0.440241\pi$
$884$	5099.15	0.194008
$885$	0	0
$886$	33628.9	1.27515
$887$	41652.4	1.57672	0.788360	−	0.615214i	$-0.210930\pi$
$887$	41652.4	1.57672	0.788360	+	0.615214i	$0.210930\pi$
$888$	0	0
$889$	−3763.93	−0.142000
$890$	10873.1	0.409512
$891$	0	0
$892$	20770.7	0.779656
$893$	−2446.34	−0.0916724
$894$	0	0
$895$	−16123.1	−0.602164
$896$	−382.102	−0.0142468
$897$	0	0
$898$	21347.5	0.793293
$899$	31207.7	1.15777
$900$	0	0
$901$	−44045.2	−1.62859
$902$	−54164.9	−1.99944
$903$	0	0
$904$	−14083.6	−0.518158
$905$	11121.1	0.408485
$906$	0	0
$907$	5467.21	0.200150	0.100075	−	0.994980i	$-0.468092\pi$
$907$	5467.21	0.200150	0.100075	+	0.994980i	$0.468092\pi$
$908$	−18804.9	−0.687295
$909$	0	0
$910$	−358.303	−0.0130523
$911$	−33520.9	−1.21910	−0.609549	−	0.792749i	$-0.708649\pi$
$911$	−33520.9	−1.21910	−0.609549	+	0.792749i	$0.708649\pi$
$912$	0	0
$913$	−1390.52	−0.0504046
$914$	−30986.1	−1.12137
$915$	0	0
$916$	−8500.70	−0.306628
$917$	−1326.26	−0.0477612
$918$	0	0
$919$	21423.9	0.768996	0.384498	−	0.923126i	$-0.374375\pi$
$919$	21423.9	0.768996	0.384498	+	0.923126i	$0.374375\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−21586.0	−0.771039
$923$	−8507.56	−0.303391
$924$	0	0
$925$	−2698.90	−0.0959344
$926$	12331.7	0.437631
$927$	0	0
$928$	4327.23	0.153069
$929$	−25838.9	−0.912538	−0.456269	−	0.889842i	$-0.650814\pi$
$929$	−25838.9	−0.912538	−0.456269	+	0.889842i	$0.650814\pi$
$930$	0	0
$931$	16199.7	0.570271
$932$	6586.69	0.231496
$933$	0	0
$934$	−21472.8	−0.752261
$935$	−36443.2	−1.27467
$936$	0	0
$937$	48095.9	1.67687	0.838434	−	0.545003i	$-0.183472\pi$
$937$	48095.9	1.67687	0.838434	+	0.545003i	$0.183472\pi$
$938$	−817.487	−0.0284562
$939$	0	0
$940$	−1009.02	−0.0350114
$941$	32450.4	1.12418	0.562089	−	0.827077i	$-0.309998\pi$
$941$	32450.4	1.12418	0.562089	+	0.827077i	$0.309998\pi$
$942$	0	0
$943$	−9076.66	−0.313443
$944$	2937.38	0.101275
$945$	0	0
$946$	18674.9	0.641834
$947$	−39067.9	−1.34059	−0.670294	−	0.742096i	$-0.733832\pi$
$947$	−39067.9	−1.34059	−0.670294	+	0.742096i	$0.733832\pi$
$948$	0	0
$949$	8277.16	0.283127
$950$	2424.45	0.0827997
$951$	0	0
$952$	2536.39	0.0863496
$953$	−17756.2	−0.603546	−0.301773	−	0.953380i	$-0.597578\pi$
$953$	−17756.2	−0.603546	−0.301773	+	0.953380i	$0.597578\pi$
$954$	0	0
$955$	16367.6	0.554601
$956$	−11511.5	−0.389444
$957$	0	0
$958$	12666.6	0.427181
$959$	−1280.85	−0.0431292
$960$	0	0
$961$	23469.5	0.787805
$962$	−2591.54	−0.0868552
$963$	0	0
$964$	−16978.2	−0.567251
$965$	−25634.3	−0.855127
$966$	0	0
$967$	−10696.2	−0.355705	−0.177853	−	0.984057i	$-0.556915\pi$
$967$	−10696.2	−0.355705	−0.177853	+	0.984057i	$0.556915\pi$
$968$	−27028.5	−0.897446
$969$	0	0
$970$	−11159.0	−0.369377
$971$	53635.1	1.77264	0.886320	−	0.463074i	$-0.153254\pi$
$971$	53635.1	1.77264	0.886320	+	0.463074i	$0.153254\pi$
$972$	0	0
$973$	−745.561	−0.0245649
$974$	33666.5	1.10754
$975$	0	0
$976$	−1575.60	−0.0516740
$977$	21627.7	0.708220	0.354110	−	0.935204i	$-0.384784\pi$
$977$	21627.7	0.708220	0.354110	+	0.935204i	$0.384784\pi$
$978$	0	0
$979$	−74617.7	−2.43595
$980$	6681.77	0.217797
$981$	0	0
$982$	−9.60856	−0.000312242 0
$983$	−12031.4	−0.390378	−0.195189	−	0.980766i	$-0.562532\pi$
$983$	−12031.4	−0.390378	−0.195189	+	0.980766i	$0.562532\pi$
$984$	0	0
$985$	16312.8	0.527686
$986$	−28724.1	−0.927749
$987$	0	0
$988$	2328.01	0.0749635
$989$	3129.44	0.100617
$990$	0	0
$991$	−24573.0	−0.787675	−0.393838	−	0.919180i	$-0.628853\pi$
$991$	−24573.0	−0.787675	−0.393838	+	0.919180i	$0.628853\pi$
$992$	7385.03	0.236366
$993$	0	0
$994$	−4231.78	−0.135034
$995$	841.410	0.0268085
$996$	0	0
$997$	−14209.7	−0.451379	−0.225690	−	0.974199i	$-0.572464\pi$
$997$	−14209.7	−0.451379	−0.225690	+	0.974199i	$0.572464\pi$
$998$	−42413.8	−1.34527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bg.1.2		4
3.2	odd	2		230.4.a.j.1.2	✓	4
12.11	even	2		1840.4.a.k.1.3		4
15.2	even	4		1150.4.b.o.599.7		8
15.8	even	4		1150.4.b.o.599.2		8
15.14	odd	2		1150.4.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.2	✓	4	3.2	odd	2
1150.4.a.n.1.3		4	15.14	odd	2
1150.4.b.o.599.2		8	15.8	even	4
1150.4.b.o.599.7		8	15.2	even	4
1840.4.a.k.1.3		4	12.11	even	2
2070.4.a.bg.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +2.98517 q^{7} -8.00000 q^{8} +10.0000 q^{10} -68.6262 q^{11} -12.0028 q^{13} -5.97035 q^{14} +16.0000 q^{16} -106.208 q^{17} -48.4891 q^{19} -20.0000 q^{20} +137.252 q^{22} +23.0000 q^{23} +25.0000 q^{25} +24.0055 q^{26} +11.9407 q^{28} -135.226 q^{29} -230.782 q^{31} -32.0000 q^{32} +212.416 q^{34} -14.9259 q^{35} -107.956 q^{37} +96.9782 q^{38} +40.0000 q^{40} -394.637 q^{41} +136.063 q^{43} -274.505 q^{44} -46.0000 q^{46} +50.4512 q^{47} -334.089 q^{49} -50.0000 q^{50} -48.0110 q^{52} +414.707 q^{53} +343.131 q^{55} -23.8814 q^{56} +270.452 q^{58} +183.586 q^{59} -98.4751 q^{61} +461.565 q^{62} +64.0000 q^{64} +60.0138 q^{65} +136.925 q^{67} -424.831 q^{68} +29.8517 q^{70} +708.800 q^{71} -689.605 q^{73} +215.912 q^{74} -193.956 q^{76} -204.861 q^{77} +546.583 q^{79} -80.0000 q^{80} +789.275 q^{82} +20.2622 q^{83} +531.039 q^{85} -272.125 q^{86} +549.010 q^{88} +1087.31 q^{89} -35.8303 q^{91} +92.0000 q^{92} -100.902 q^{94} +242.445 q^{95} -1115.90 q^{97} +668.177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) 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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