LMFDB - Embedded newform 2070.4.a.bg.1.3

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			1.3
Root			$9.04090$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +5.03071 q^{7} -8.00000 q^{8} +10.0000 q^{10} +5.58219 q^{11} +62.7277 q^{13} -10.0614 q^{14} +16.0000 q^{16} +19.7435 q^{17} +158.545 q^{19} -20.0000 q^{20} -11.1644 q^{22} +23.0000 q^{23} +25.0000 q^{25} -125.455 q^{26} +20.1228 q^{28} +35.5033 q^{29} +282.041 q^{31} -32.0000 q^{32} -39.4870 q^{34} -25.1536 q^{35} -139.981 q^{37} -317.090 q^{38} +40.0000 q^{40} -227.680 q^{41} +436.962 q^{43} +22.3288 q^{44} -46.0000 q^{46} -90.2701 q^{47} -317.692 q^{49} -50.0000 q^{50} +250.911 q^{52} -330.183 q^{53} -27.9109 q^{55} -40.2457 q^{56} -71.0066 q^{58} +796.203 q^{59} -568.580 q^{61} -564.081 q^{62} +64.0000 q^{64} -313.638 q^{65} +85.1419 q^{67} +78.9740 q^{68} +50.3071 q^{70} +369.578 q^{71} -310.188 q^{73} +279.963 q^{74} +634.180 q^{76} +28.0824 q^{77} -1325.46 q^{79} -80.0000 q^{80} +455.360 q^{82} +158.806 q^{83} -98.7175 q^{85} -873.924 q^{86} -44.6575 q^{88} +1233.89 q^{89} +315.565 q^{91} +92.0000 q^{92} +180.540 q^{94} -792.725 q^{95} -106.389 q^{97} +635.384 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	5.03071	0.271633	0.135816	−	0.990734i	$-0.456634\pi$
$7$	5.03071	0.271633	0.135816	+	0.990734i	$0.456634\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	5.58219	0.153008	0.0765042	−	0.997069i	$-0.475624\pi$
$11$	5.58219	0.153008	0.0765042	+	0.997069i	$0.475624\pi$
$12$	0	0
$13$	62.7277	1.33827	0.669136	−	0.743140i	$-0.266664\pi$
$13$	62.7277	1.33827	0.669136	+	0.743140i	$0.266664\pi$
$14$	−10.0614	−0.192073
$15$	0	0
$16$	16.0000	0.250000
$17$	19.7435	0.281677	0.140838	−	0.990033i	$-0.455020\pi$
$17$	19.7435	0.281677	0.140838	+	0.990033i	$0.455020\pi$
$18$	0	0
$19$	158.545	1.91435	0.957177	−	0.289505i	$-0.0934907\pi$
$19$	158.545	1.91435	0.957177	+	0.289505i	$0.0934907\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−11.1644	−0.108193
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−125.455	−0.946301
$27$	0	0
$28$	20.1228	0.135816
$29$	35.5033	0.227338	0.113669	−	0.993519i	$-0.463740\pi$
$29$	35.5033	0.227338	0.113669	+	0.993519i	$0.463740\pi$
$30$	0	0
$31$	282.041	1.63406	0.817032	−	0.576592i	$-0.195618\pi$
$31$	282.041	1.63406	0.817032	+	0.576592i	$0.195618\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−39.4870	−0.199175
$35$	−25.1536	−0.121478
$36$	0	0
$37$	−139.981	−0.621967	−0.310984	−	0.950415i	$-0.600658\pi$
$37$	−139.981	−0.621967	−0.310984	+	0.950415i	$0.600658\pi$
$38$	−317.090	−1.35365
$39$	0	0
$40$	40.0000	0.158114
$41$	−227.680	−0.867260	−0.433630	−	0.901091i	$-0.642768\pi$
$41$	−227.680	−0.867260	−0.433630	+	0.901091i	$0.642768\pi$
$42$	0	0
$43$	436.962	1.54968	0.774838	−	0.632159i	$-0.217831\pi$
$43$	436.962	1.54968	0.774838	+	0.632159i	$0.217831\pi$
$44$	22.3288	0.0765042
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−90.2701	−0.280154	−0.140077	−	0.990141i	$-0.544735\pi$
$47$	−90.2701	−0.280154	−0.140077	+	0.990141i	$0.544735\pi$
$48$	0	0
$49$	−317.692	−0.926216
$50$	−50.0000	−0.141421
$51$	0	0
$52$	250.911	0.669136
$53$	−330.183	−0.855737	−0.427869	−	0.903841i	$-0.640735\pi$
$53$	−330.183	−0.855737	−0.427869	+	0.903841i	$0.640735\pi$
$54$	0	0
$55$	−27.9109	−0.0684275
$56$	−40.2457	−0.0960367
$57$	0	0
$58$	−71.0066	−0.160752
$59$	796.203	1.75689	0.878447	−	0.477839i	$-0.158580\pi$
$59$	796.203	1.75689	0.878447	+	0.477839i	$0.158580\pi$
$60$	0	0
$61$	−568.580	−1.19343	−0.596715	−	0.802453i	$-0.703528\pi$
$61$	−568.580	−1.19343	−0.596715	+	0.802453i	$0.703528\pi$
$62$	−564.081	−1.15546
$63$	0	0
$64$	64.0000	0.125000
$65$	−313.638	−0.598493
$66$	0	0
$67$	85.1419	0.155250	0.0776250	−	0.996983i	$-0.475266\pi$
$67$	85.1419	0.155250	0.0776250	+	0.996983i	$0.475266\pi$
$68$	78.9740	0.140838
$69$	0	0
$70$	50.3071	0.0858978
$71$	369.578	0.617758	0.308879	−	0.951101i	$-0.400046\pi$
$71$	369.578	0.617758	0.308879	+	0.951101i	$0.400046\pi$
$72$	0	0
$73$	−310.188	−0.497325	−0.248662	−	0.968590i	$-0.579991\pi$
$73$	−310.188	−0.497325	−0.248662	+	0.968590i	$0.579991\pi$
$74$	279.963	0.439797
$75$	0	0
$76$	634.180	0.957177
$77$	28.0824	0.0415621
$78$	0	0
$79$	−1325.46	−1.88766	−0.943832	−	0.330427i	$-0.892807\pi$
$79$	−1325.46	−1.88766	−0.943832	+	0.330427i	$0.892807\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	455.360	0.613246
$83$	158.806	0.210014	0.105007	−	0.994471i	$-0.466513\pi$
$83$	158.806	0.210014	0.105007	+	0.994471i	$0.466513\pi$
$84$	0	0
$85$	−98.7175	−0.125970
$86$	−873.924	−1.09579
$87$	0	0
$88$	−44.6575	−0.0540967
$89$	1233.89	1.46957	0.734787	−	0.678298i	$-0.237282\pi$
$89$	1233.89	1.46957	0.734787	+	0.678298i	$0.237282\pi$
$90$	0	0
$91$	315.565	0.363519
$92$	92.0000	0.104257
$93$	0	0
$94$	180.540	0.198099
$95$	−792.725	−0.856125
$96$	0	0
$97$	−106.389	−0.111363	−0.0556814	−	0.998449i	$-0.517733\pi$
$97$	−106.389	−0.111363	−0.0556814	+	0.998449i	$0.517733\pi$
$98$	635.384	0.654933
$99$	0	0
$100$	100.000	0.100000
$101$	−642.676	−0.633155	−0.316578	−	0.948567i	$-0.602534\pi$
$101$	−642.676	−0.633155	−0.316578	+	0.948567i	$0.602534\pi$
$102$	0	0
$103$	1621.99	1.55165	0.775823	−	0.630951i	$-0.217335\pi$
$103$	1621.99	1.55165	0.775823	+	0.630951i	$0.217335\pi$
$104$	−501.822	−0.473150
$105$	0	0
$106$	660.365	0.605098
$107$	−1490.86	−1.34698	−0.673490	−	0.739196i	$-0.735205\pi$
$107$	−1490.86	−1.34698	−0.673490	+	0.739196i	$0.735205\pi$
$108$	0	0
$109$	1204.00	1.05800	0.529001	−	0.848621i	$-0.322567\pi$
$109$	1204.00	1.05800	0.529001	+	0.848621i	$0.322567\pi$
$110$	55.8219	0.0483855
$111$	0	0
$112$	80.4914	0.0679082
$113$	−276.771	−0.230411	−0.115206	−	0.993342i	$-0.536753\pi$
$113$	−276.771	−0.230411	−0.115206	+	0.993342i	$0.536753\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	142.013	0.113669
$117$	0	0
$118$	−1592.41	−1.24231
$119$	99.3239	0.0765126
$120$	0	0
$121$	−1299.84	−0.976588
$122$	1137.16	0.843883
$123$	0	0
$124$	1128.16	0.817032
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1552.79	1.08494	0.542471	−	0.840075i	$-0.317489\pi$
$127$	1552.79	1.08494	0.542471	+	0.840075i	$0.317489\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	627.277	0.423199
$131$	313.873	0.209337	0.104669	−	0.994507i	$-0.466622\pi$
$131$	313.873	0.209337	0.104669	+	0.994507i	$0.466622\pi$
$132$	0	0
$133$	797.594	0.520001
$134$	−170.284	−0.109778
$135$	0	0
$136$	−157.948	−0.0995877
$137$	1066.09	0.664835	0.332417	−	0.943132i	$-0.392136\pi$
$137$	1066.09	0.664835	0.332417	+	0.943132i	$0.392136\pi$
$138$	0	0
$139$	−1594.79	−0.973152	−0.486576	−	0.873638i	$-0.661754\pi$
$139$	−1594.79	−0.973152	−0.486576	+	0.873638i	$0.661754\pi$
$140$	−100.614	−0.0607389
$141$	0	0
$142$	−739.155	−0.436821
$143$	350.158	0.204767
$144$	0	0
$145$	−177.517	−0.101669
$146$	620.375	0.351662
$147$	0	0
$148$	−559.925	−0.310984
$149$	1096.91	0.603102	0.301551	−	0.953450i	$-0.402496\pi$
$149$	1096.91	0.603102	0.301551	+	0.953450i	$0.402496\pi$
$150$	0	0
$151$	2734.41	1.47366	0.736832	−	0.676076i	$-0.236321\pi$
$151$	2734.41	1.47366	0.736832	+	0.676076i	$0.236321\pi$
$152$	−1268.36	−0.676826
$153$	0	0
$154$	−56.1647	−0.0293889
$155$	−1410.20	−0.730776
$156$	0	0
$157$	−1439.63	−0.731813	−0.365907	−	0.930652i	$-0.619241\pi$
$157$	−1439.63	−0.731813	−0.365907	+	0.930652i	$0.619241\pi$
$158$	2650.91	1.33478
$159$	0	0
$160$	160.000	0.0790569
$161$	115.706	0.0566394
$162$	0	0
$163$	−2995.82	−1.43957	−0.719787	−	0.694195i	$-0.755760\pi$
$163$	−2995.82	−1.43957	−0.719787	+	0.694195i	$0.755760\pi$
$164$	−910.721	−0.433630
$165$	0	0
$166$	−317.611	−0.148503
$167$	351.532	0.162888	0.0814442	−	0.996678i	$-0.474047\pi$
$167$	351.532	0.162888	0.0814442	+	0.996678i	$0.474047\pi$
$168$	0	0
$169$	1737.76	0.790971
$170$	197.435	0.0890740
$171$	0	0
$172$	1747.85	0.774838
$173$	3701.77	1.62682	0.813411	−	0.581690i	$-0.197608\pi$
$173$	3701.77	1.62682	0.813411	+	0.581690i	$0.197608\pi$
$174$	0	0
$175$	125.768	0.0543266
$176$	89.3150	0.0382521
$177$	0	0
$178$	−2467.78	−1.03915
$179$	−3584.09	−1.49658	−0.748290	−	0.663372i	$-0.769125\pi$
$179$	−3584.09	−1.49658	−0.748290	+	0.663372i	$0.769125\pi$
$180$	0	0
$181$	783.672	0.321822	0.160911	−	0.986969i	$-0.448557\pi$
$181$	783.672	0.321822	0.160911	+	0.986969i	$0.448557\pi$
$182$	−631.130	−0.257046
$183$	0	0
$184$	−184.000	−0.0737210
$185$	699.907	0.278152
$186$	0	0
$187$	110.212	0.0430989
$188$	−361.080	−0.140077
$189$	0	0
$190$	1585.45	0.605372
$191$	−1491.95	−0.565201	−0.282601	−	0.959238i	$-0.591197\pi$
$191$	−1491.95	−0.565201	−0.282601	+	0.959238i	$0.591197\pi$
$192$	0	0
$193$	3091.09	1.15286	0.576430	−	0.817147i	$-0.304446\pi$
$193$	3091.09	1.15286	0.576430	+	0.817147i	$0.304446\pi$
$194$	212.778	0.0787454
$195$	0	0
$196$	−1270.77	−0.463108
$197$	−2820.30	−1.01999	−0.509995	−	0.860178i	$-0.670353\pi$
$197$	−2820.30	−1.01999	−0.509995	+	0.860178i	$0.670353\pi$
$198$	0	0
$199$	1316.11	0.468825	0.234413	−	0.972137i	$-0.424683\pi$
$199$	1316.11	0.468825	0.234413	+	0.972137i	$0.424683\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	1285.35	0.447708
$203$	178.607	0.0617524
$204$	0	0
$205$	1138.40	0.387851
$206$	−3243.98	−1.09718
$207$	0	0
$208$	1003.64	0.334568
$209$	885.028	0.292912
$210$	0	0
$211$	−3399.19	−1.10905	−0.554525	−	0.832167i	$-0.687100\pi$
$211$	−3399.19	−1.10905	−0.554525	+	0.832167i	$0.687100\pi$
$212$	−1320.73	−0.427869
$213$	0	0
$214$	2981.72	0.952458
$215$	−2184.81	−0.693037
$216$	0	0
$217$	1418.86	0.443865
$218$	−2408.00	−0.748121
$219$	0	0
$220$	−111.644	−0.0342137
$221$	1238.46	0.376960
$222$	0	0
$223$	864.660	0.259650	0.129825	−	0.991537i	$-0.458558\pi$
$223$	864.660	0.259650	0.129825	+	0.991537i	$0.458558\pi$
$224$	−160.983	−0.0480184
$225$	0	0
$226$	553.543	0.162925
$227$	1979.40	0.578754	0.289377	−	0.957215i	$-0.406552\pi$
$227$	1979.40	0.578754	0.289377	+	0.957215i	$0.406552\pi$
$228$	0	0
$229$	−3113.53	−0.898462	−0.449231	−	0.893416i	$-0.648302\pi$
$229$	−3113.53	−0.898462	−0.449231	+	0.893416i	$0.648302\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	−284.026	−0.0803761
$233$	−6018.65	−1.69225	−0.846126	−	0.532983i	$-0.821071\pi$
$233$	−6018.65	−1.69225	−0.846126	+	0.532983i	$0.821071\pi$
$234$	0	0
$235$	451.350	0.125289
$236$	3184.81	0.878447
$237$	0	0
$238$	−198.648	−0.0541026
$239$	4224.88	1.14345	0.571725	−	0.820445i	$-0.306274\pi$
$239$	4224.88	1.14345	0.571725	+	0.820445i	$0.306274\pi$
$240$	0	0
$241$	1394.24	0.372658	0.186329	−	0.982487i	$-0.440341\pi$
$241$	1394.24	0.372658	0.186329	+	0.982487i	$0.440341\pi$
$242$	2599.68	0.690552
$243$	0	0
$244$	−2274.32	−0.596715
$245$	1588.46	0.414216
$246$	0	0
$247$	9945.16	2.56192
$248$	−2256.32	−0.577729
$249$	0	0
$250$	250.000	0.0632456
$251$	5968.83	1.50099	0.750497	−	0.660874i	$-0.229814\pi$
$251$	5968.83	1.50099	0.750497	+	0.660874i	$0.229814\pi$
$252$	0	0
$253$	128.390	0.0319045
$254$	−3105.57	−0.767169
$255$	0	0
$256$	256.000	0.0625000
$257$	1636.06	0.397100	0.198550	−	0.980091i	$-0.436377\pi$
$257$	1636.06	0.397100	0.198550	+	0.980091i	$0.436377\pi$
$258$	0	0
$259$	−704.206	−0.168947
$260$	−1254.55	−0.299247
$261$	0	0
$262$	−627.745	−0.148024
$263$	−2995.55	−0.702333	−0.351166	−	0.936313i	$-0.614215\pi$
$263$	−2995.55	−0.702333	−0.351166	+	0.936313i	$0.614215\pi$
$264$	0	0
$265$	1650.91	0.382697
$266$	−1595.19	−0.367696
$267$	0	0
$268$	340.568	0.0776250
$269$	5190.17	1.17640	0.588198	−	0.808717i	$-0.299838\pi$
$269$	5190.17	1.17640	0.588198	+	0.808717i	$0.299838\pi$
$270$	0	0
$271$	6720.43	1.50641	0.753205	−	0.657786i	$-0.228507\pi$
$271$	6720.43	1.50641	0.753205	+	0.657786i	$0.228507\pi$
$272$	315.896	0.0704192
$273$	0	0
$274$	−2132.18	−0.470109
$275$	139.555	0.0306017
$276$	0	0
$277$	−6114.74	−1.32635	−0.663176	−	0.748464i	$-0.730792\pi$
$277$	−6114.74	−1.32635	−0.663176	+	0.748464i	$0.730792\pi$
$278$	3189.58	0.688123
$279$	0	0
$280$	201.228	0.0429489
$281$	3549.07	0.753450	0.376725	−	0.926325i	$-0.377050\pi$
$281$	3549.07	0.753450	0.376725	+	0.926325i	$0.377050\pi$
$282$	0	0
$283$	3103.11	0.651806	0.325903	−	0.945403i	$-0.394332\pi$
$283$	3103.11	0.651806	0.325903	+	0.945403i	$0.394332\pi$
$284$	1478.31	0.308879
$285$	0	0
$286$	−700.316	−0.144792
$287$	−1145.39	−0.235576
$288$	0	0
$289$	−4523.19	−0.920658
$290$	355.033	0.0718905
$291$	0	0
$292$	−1240.75	−0.248662
$293$	−1218.51	−0.242957	−0.121478	−	0.992594i	$-0.538763\pi$
$293$	−1218.51	−0.242957	−0.121478	+	0.992594i	$0.538763\pi$
$294$	0	0
$295$	−3981.01	−0.785707
$296$	1119.85	0.219899
$297$	0	0
$298$	−2193.82	−0.426458
$299$	1442.74	0.279049
$300$	0	0
$301$	2198.23	0.420943
$302$	−5468.82	−1.04204
$303$	0	0
$304$	2536.72	0.478588
$305$	2842.90	0.533718
$306$	0	0
$307$	2564.45	0.476745	0.238373	−	0.971174i	$-0.423386\pi$
$307$	2564.45	0.476745	0.238373	+	0.971174i	$0.423386\pi$
$308$	112.329	0.0207811
$309$	0	0
$310$	2820.41	0.516736
$311$	92.6253	0.0168884	0.00844421	−	0.999964i	$-0.497312\pi$
$311$	92.6253	0.0168884	0.00844421	+	0.999964i	$0.497312\pi$
$312$	0	0
$313$	−4276.44	−0.772263	−0.386132	−	0.922444i	$-0.626189\pi$
$313$	−4276.44	−0.772263	−0.386132	+	0.922444i	$0.626189\pi$
$314$	2879.25	0.517470
$315$	0	0
$316$	−5301.82	−0.943832
$317$	2964.24	0.525199	0.262600	−	0.964905i	$-0.415420\pi$
$317$	2964.24	0.525199	0.262600	+	0.964905i	$0.415420\pi$
$318$	0	0
$319$	198.186	0.0347846
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−231.413	−0.0400501
$323$	3130.23	0.539229
$324$	0	0
$325$	1568.19	0.267654
$326$	5991.63	1.01793
$327$	0	0
$328$	1821.44	0.306623
$329$	−454.123	−0.0760991
$330$	0	0
$331$	−1348.81	−0.223979	−0.111989	−	0.993709i	$-0.535722\pi$
$331$	−1348.81	−0.223979	−0.111989	+	0.993709i	$0.535722\pi$
$332$	635.223	0.105007
$333$	0	0
$334$	−703.064	−0.115180
$335$	−425.710	−0.0694299
$336$	0	0
$337$	−5764.30	−0.931755	−0.465878	−	0.884849i	$-0.654261\pi$
$337$	−5764.30	−0.931755	−0.465878	+	0.884849i	$0.654261\pi$
$338$	−3475.53	−0.559301
$339$	0	0
$340$	−394.870	−0.0629848
$341$	1574.40	0.250026
$342$	0	0
$343$	−3323.75	−0.523223
$344$	−3495.70	−0.547894
$345$	0	0
$346$	−7403.53	−1.15034
$347$	−4066.16	−0.629057	−0.314528	−	0.949248i	$-0.601846\pi$
$347$	−4066.16	−0.629057	−0.314528	+	0.949248i	$0.601846\pi$
$348$	0	0
$349$	8407.82	1.28957	0.644786	−	0.764363i	$-0.276947\pi$
$349$	8407.82	1.28957	0.644786	+	0.764363i	$0.276947\pi$
$350$	−251.536	−0.0384147
$351$	0	0
$352$	−178.630	−0.0270483
$353$	−256.068	−0.0386095	−0.0193047	−	0.999814i	$-0.506145\pi$
$353$	−256.068	−0.0386095	−0.0193047	+	0.999814i	$0.506145\pi$
$354$	0	0
$355$	−1847.89	−0.276270
$356$	4935.56	0.734787
$357$	0	0
$358$	7168.19	1.05824
$359$	12052.5	1.77189	0.885943	−	0.463794i	$-0.153512\pi$
$359$	12052.5	1.77189	0.885943	+	0.463794i	$0.153512\pi$
$360$	0	0
$361$	18277.5	2.66475
$362$	−1567.34	−0.227563
$363$	0	0
$364$	1262.26	0.181759
$365$	1550.94	0.222410
$366$	0	0
$367$	−9245.40	−1.31500	−0.657502	−	0.753453i	$-0.728387\pi$
$367$	−9245.40	−1.31500	−0.657502	+	0.753453i	$0.728387\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−1399.81	−0.196683
$371$	−1661.05	−0.232446
$372$	0	0
$373$	−869.265	−0.120667	−0.0603335	−	0.998178i	$-0.519216\pi$
$373$	−869.265	−0.120667	−0.0603335	+	0.998178i	$0.519216\pi$
$374$	−220.424	−0.0304755
$375$	0	0
$376$	722.161	0.0990495
$377$	2227.04	0.304240
$378$	0	0
$379$	1278.76	0.173313	0.0866566	−	0.996238i	$-0.472382\pi$
$379$	1278.76	0.173313	0.0866566	+	0.996238i	$0.472382\pi$
$380$	−3170.90	−0.428062
$381$	0	0
$382$	2983.89	0.399658
$383$	5343.75	0.712932	0.356466	−	0.934308i	$-0.383982\pi$
$383$	5343.75	0.712932	0.356466	+	0.934308i	$0.383982\pi$
$384$	0	0
$385$	−140.412	−0.0185871
$386$	−6182.19	−0.815194
$387$	0	0
$388$	−425.557	−0.0556814
$389$	708.346	0.0923254	0.0461627	−	0.998934i	$-0.485301\pi$
$389$	708.346	0.0923254	0.0461627	+	0.998934i	$0.485301\pi$
$390$	0	0
$391$	454.101	0.0587336
$392$	2541.54	0.327467
$393$	0	0
$394$	5640.60	0.721241
$395$	6627.28	0.844189
$396$	0	0
$397$	−5924.01	−0.748911	−0.374455	−	0.927245i	$-0.622170\pi$
$397$	−5924.01	−0.748911	−0.374455	+	0.927245i	$0.622170\pi$
$398$	−2632.21	−0.331510
$399$	0	0
$400$	400.000	0.0500000
$401$	11393.6	1.41888	0.709441	−	0.704765i	$-0.248948\pi$
$401$	11393.6	1.41888	0.709441	+	0.704765i	$0.248948\pi$
$402$	0	0
$403$	17691.8	2.18682
$404$	−2570.71	−0.316578
$405$	0	0
$406$	−357.214	−0.0436656
$407$	−781.402	−0.0951663
$408$	0	0
$409$	13804.4	1.66891	0.834457	−	0.551073i	$-0.185782\pi$
$409$	13804.4	1.66891	0.834457	+	0.551073i	$0.185782\pi$
$410$	−2276.80	−0.274252
$411$	0	0
$412$	6487.96	0.775823
$413$	4005.47	0.477230
$414$	0	0
$415$	−794.028	−0.0939212
$416$	−2007.29	−0.236575
$417$	0	0
$418$	−1770.06	−0.207120
$419$	14795.8	1.72512	0.862558	−	0.505958i	$-0.168861\pi$
$419$	14795.8	1.72512	0.862558	+	0.505958i	$0.168861\pi$
$420$	0	0
$421$	5804.66	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$421$	5804.66	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$422$	6798.37	0.784217
$423$	0	0
$424$	2641.46	0.302549
$425$	493.588	0.0563353
$426$	0	0
$427$	−2860.36	−0.324175
$428$	−5963.44	−0.673490
$429$	0	0
$430$	4369.62	0.490051
$431$	−15268.2	−1.70636	−0.853181	−	0.521615i	$-0.825330\pi$
$431$	−15268.2	−1.70636	−0.853181	+	0.521615i	$0.825330\pi$
$432$	0	0
$433$	2250.62	0.249788	0.124894	−	0.992170i	$-0.460141\pi$
$433$	2250.62	0.249788	0.124894	+	0.992170i	$0.460141\pi$
$434$	−2837.73	−0.313860
$435$	0	0
$436$	4816.00	0.529001
$437$	3646.53	0.399170
$438$	0	0
$439$	10366.2	1.12700	0.563499	−	0.826117i	$-0.309455\pi$
$439$	10366.2	1.12700	0.563499	+	0.826117i	$0.309455\pi$
$440$	223.288	0.0241928
$441$	0	0
$442$	−2476.93	−0.266551
$443$	3881.11	0.416246	0.208123	−	0.978103i	$-0.433264\pi$
$443$	3881.11	0.416246	0.208123	+	0.978103i	$0.433264\pi$
$444$	0	0
$445$	−6169.45	−0.657214
$446$	−1729.32	−0.183600
$447$	0	0
$448$	321.965	0.0339541
$449$	10146.5	1.06647	0.533233	−	0.845968i	$-0.320977\pi$
$449$	10146.5	1.06647	0.533233	+	0.845968i	$0.320977\pi$
$450$	0	0
$451$	−1270.95	−0.132698
$452$	−1107.09	−0.115206
$453$	0	0
$454$	−3958.79	−0.409241
$455$	−1577.82	−0.162570
$456$	0	0
$457$	−4728.87	−0.484042	−0.242021	−	0.970271i	$-0.577810\pi$
$457$	−4728.87	−0.484042	−0.242021	+	0.970271i	$0.577810\pi$
$458$	6227.06	0.635308
$459$	0	0
$460$	−460.000	−0.0466252
$461$	−8380.69	−0.846698	−0.423349	−	0.905967i	$-0.639146\pi$
$461$	−8380.69	−0.846698	−0.423349	+	0.905967i	$0.639146\pi$
$462$	0	0
$463$	6845.17	0.687089	0.343545	−	0.939136i	$-0.388372\pi$
$463$	6845.17	0.687089	0.343545	+	0.939136i	$0.388372\pi$
$464$	568.053	0.0568345
$465$	0	0
$466$	12037.3	1.19660
$467$	14884.0	1.47484	0.737418	−	0.675436i	$-0.236045\pi$
$467$	14884.0	1.47484	0.737418	+	0.675436i	$0.236045\pi$
$468$	0	0
$469$	428.324	0.0421710
$470$	−902.701	−0.0885925
$471$	0	0
$472$	−6369.62	−0.621156
$473$	2439.20	0.237114
$474$	0	0
$475$	3963.62	0.382871
$476$	397.295	0.0382563
$477$	0	0
$478$	−8449.76	−0.808542
$479$	−6048.69	−0.576976	−0.288488	−	0.957483i	$-0.593153\pi$
$479$	−6048.69	−0.576976	−0.288488	+	0.957483i	$0.593153\pi$
$480$	0	0
$481$	−8780.71	−0.832361
$482$	−2788.47	−0.263509
$483$	0	0
$484$	−5199.36	−0.488294
$485$	531.946	0.0498029
$486$	0	0
$487$	3199.35	0.297693	0.148846	−	0.988860i	$-0.452444\pi$
$487$	3199.35	0.297693	0.148846	+	0.988860i	$0.452444\pi$
$488$	4548.64	0.421941
$489$	0	0
$490$	−3176.92	−0.292895
$491$	1614.41	0.148385	0.0741926	−	0.997244i	$-0.476362\pi$
$491$	1614.41	0.148385	0.0741926	+	0.997244i	$0.476362\pi$
$492$	0	0
$493$	700.960	0.0640358
$494$	−19890.3	−1.81155
$495$	0	0
$496$	4512.65	0.408516
$497$	1859.24	0.167803
$498$	0	0
$499$	−5392.04	−0.483729	−0.241864	−	0.970310i	$-0.577759\pi$
$499$	−5392.04	−0.483729	−0.241864	+	0.970310i	$0.577759\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−11937.7	−1.06136
$503$	−12625.4	−1.11917	−0.559583	−	0.828775i	$-0.689039\pi$
$503$	−12625.4	−1.11917	−0.559583	+	0.828775i	$0.689039\pi$
$504$	0	0
$505$	3213.38	0.283156
$506$	−256.781	−0.0225599
$507$	0	0
$508$	6211.15	0.542471
$509$	−7065.76	−0.615294	−0.307647	−	0.951501i	$-0.599541\pi$
$509$	−7065.76	−0.615294	−0.307647	+	0.951501i	$0.599541\pi$
$510$	0	0
$511$	−1560.46	−0.135090
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−3272.12	−0.280792
$515$	−8109.96	−0.693917
$516$	0	0
$517$	−503.905	−0.0428660
$518$	1408.41	0.119463
$519$	0	0
$520$	2509.11	0.211599
$521$	8399.20	0.706287	0.353144	−	0.935569i	$-0.385113\pi$
$521$	8399.20	0.706287	0.353144	+	0.935569i	$0.385113\pi$
$522$	0	0
$523$	−7002.06	−0.585428	−0.292714	−	0.956200i	$-0.594558\pi$
$523$	−7002.06	−0.585428	−0.292714	+	0.956200i	$0.594558\pi$
$524$	1255.49	0.104669
$525$	0	0
$526$	5991.10	0.496624
$527$	5568.47	0.460278
$528$	0	0
$529$	529.000	0.0434783
$530$	−3301.83	−0.270608
$531$	0	0
$532$	3190.37	0.260001
$533$	−14281.9	−1.16063
$534$	0	0
$535$	7454.30	0.602388
$536$	−681.136	−0.0548891
$537$	0	0
$538$	−10380.3	−0.831838
$539$	−1773.42	−0.141719
$540$	0	0
$541$	3338.13	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$541$	3338.13	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$542$	−13440.9	−1.06519
$543$	0	0
$544$	−631.792	−0.0497939
$545$	−6020.00	−0.473153
$546$	0	0
$547$	9609.09	0.751106	0.375553	−	0.926801i	$-0.377453\pi$
$547$	9609.09	0.751106	0.375553	+	0.926801i	$0.377453\pi$
$548$	4264.37	0.332417
$549$	0	0
$550$	−279.109	−0.0216387
$551$	5628.87	0.435205
$552$	0	0
$553$	−6667.98	−0.512751
$554$	12229.5	0.937872
$555$	0	0
$556$	−6379.15	−0.486576
$557$	11931.1	0.907608	0.453804	−	0.891102i	$-0.350067\pi$
$557$	11931.1	0.907608	0.453804	+	0.891102i	$0.350067\pi$
$558$	0	0
$559$	27409.6	2.07389
$560$	−402.457	−0.0303695
$561$	0	0
$562$	−7098.13	−0.532770
$563$	10082.4	0.754746	0.377373	−	0.926061i	$-0.376827\pi$
$563$	10082.4	0.754746	0.377373	+	0.926061i	$0.376827\pi$
$564$	0	0
$565$	1383.86	0.103043
$566$	−6206.23	−0.460896
$567$	0	0
$568$	−2956.62	−0.218410
$569$	18710.5	1.37853	0.689267	−	0.724507i	$-0.257933\pi$
$569$	18710.5	1.37853	0.689267	+	0.724507i	$0.257933\pi$
$570$	0	0
$571$	13510.1	0.990159	0.495079	−	0.868848i	$-0.335139\pi$
$571$	13510.1	0.990159	0.495079	+	0.868848i	$0.335139\pi$
$572$	1400.63	0.102383
$573$	0	0
$574$	2290.79	0.166578
$575$	575.000	0.0417029
$576$	0	0
$577$	11582.3	0.835662	0.417831	−	0.908525i	$-0.362790\pi$
$577$	11582.3	0.835662	0.417831	+	0.908525i	$0.362790\pi$
$578$	9046.39	0.651004
$579$	0	0
$580$	−710.066	−0.0508343
$581$	798.905	0.0570468
$582$	0	0
$583$	−1843.14	−0.130935
$584$	2481.50	0.175831
$585$	0	0
$586$	2437.03	0.171796
$587$	6019.34	0.423245	0.211623	−	0.977351i	$-0.432125\pi$
$587$	6019.34	0.423245	0.211623	+	0.977351i	$0.432125\pi$
$588$	0	0
$589$	44716.1	3.12818
$590$	7962.03	0.555579
$591$	0	0
$592$	−2239.70	−0.155492
$593$	−26299.3	−1.82122	−0.910608	−	0.413270i	$-0.864387\pi$
$593$	−26299.3	−1.82122	−0.910608	+	0.413270i	$0.864387\pi$
$594$	0	0
$595$	−496.619	−0.0342175
$596$	4387.63	0.301551
$597$	0	0
$598$	−2885.47	−0.197317
$599$	27368.0	1.86682	0.933409	−	0.358813i	$-0.116819\pi$
$599$	27368.0	1.86682	0.933409	+	0.358813i	$0.116819\pi$
$600$	0	0
$601$	−3472.33	−0.235673	−0.117836	−	0.993033i	$-0.537596\pi$
$601$	−3472.33	−0.235673	−0.117836	+	0.993033i	$0.537596\pi$
$602$	−4396.46	−0.297652
$603$	0	0
$604$	10937.6	0.736832
$605$	6499.20	0.436744
$606$	0	0
$607$	5061.10	0.338425	0.169212	−	0.985580i	$-0.445878\pi$
$607$	5061.10	0.338425	0.169212	+	0.985580i	$0.445878\pi$
$608$	−5073.44	−0.338413
$609$	0	0
$610$	−5685.80	−0.377396
$611$	−5662.44	−0.374922
$612$	0	0
$613$	−28272.6	−1.86283	−0.931417	−	0.363953i	$-0.881427\pi$
$613$	−28272.6	−1.86283	−0.931417	+	0.363953i	$0.881427\pi$
$614$	−5128.90	−0.337110
$615$	0	0
$616$	−224.659	−0.0146944
$617$	2222.47	0.145013	0.0725066	−	0.997368i	$-0.476900\pi$
$617$	2222.47	0.145013	0.0725066	+	0.997368i	$0.476900\pi$
$618$	0	0
$619$	−10498.2	−0.681679	−0.340840	−	0.940121i	$-0.610711\pi$
$619$	−10498.2	−0.681679	−0.340840	+	0.940121i	$0.610711\pi$
$620$	−5640.81	−0.365388
$621$	0	0
$622$	−185.251	−0.0119419
$623$	6207.34	0.399185
$624$	0	0
$625$	625.000	0.0400000
$626$	8552.87	0.546073
$627$	0	0
$628$	−5758.51	−0.365907
$629$	−2763.72	−0.175194
$630$	0	0
$631$	−11402.2	−0.719355	−0.359678	−	0.933077i	$-0.617113\pi$
$631$	−11402.2	−0.719355	−0.359678	+	0.933077i	$0.617113\pi$
$632$	10603.6	0.667390
$633$	0	0
$634$	−5928.47	−0.371372
$635$	−7763.93	−0.485200
$636$	0	0
$637$	−19928.1	−1.23953
$638$	−396.372	−0.0245964
$639$	0	0
$640$	640.000	0.0395285
$641$	−26128.3	−1.60999	−0.804996	−	0.593280i	$-0.797833\pi$
$641$	−26128.3	−1.60999	−0.804996	+	0.593280i	$0.797833\pi$
$642$	0	0
$643$	863.502	0.0529599	0.0264799	−	0.999649i	$-0.491570\pi$
$643$	863.502	0.0529599	0.0264799	+	0.999649i	$0.491570\pi$
$644$	462.825	0.0283197
$645$	0	0
$646$	−6260.47	−0.381292
$647$	5477.43	0.332828	0.166414	−	0.986056i	$-0.446781\pi$
$647$	5477.43	0.332828	0.166414	+	0.986056i	$0.446781\pi$
$648$	0	0
$649$	4444.55	0.268820
$650$	−3136.38	−0.189260
$651$	0	0
$652$	−11983.3	−0.719787
$653$	−19233.2	−1.15261	−0.576305	−	0.817235i	$-0.695506\pi$
$653$	−19233.2	−1.15261	−0.576305	+	0.817235i	$0.695506\pi$
$654$	0	0
$655$	−1569.36	−0.0936185
$656$	−3642.88	−0.216815
$657$	0	0
$658$	908.245	0.0538102
$659$	−9952.51	−0.588307	−0.294154	−	0.955758i	$-0.595038\pi$
$659$	−9952.51	−0.588307	−0.294154	+	0.955758i	$0.595038\pi$
$660$	0	0
$661$	3047.96	0.179352	0.0896762	−	0.995971i	$-0.471417\pi$
$661$	3047.96	0.179352	0.0896762	+	0.995971i	$0.471417\pi$
$662$	2697.61	0.158377
$663$	0	0
$664$	−1270.45	−0.0742513
$665$	−3987.97	−0.232552
$666$	0	0
$667$	816.576	0.0474032
$668$	1406.13	0.0814442
$669$	0	0
$670$	851.419	0.0490943
$671$	−3173.92	−0.182605
$672$	0	0
$673$	16245.9	0.930508	0.465254	−	0.885177i	$-0.345963\pi$
$673$	16245.9	0.930508	0.465254	+	0.885177i	$0.345963\pi$
$674$	11528.6	0.658850
$675$	0	0
$676$	6951.05	0.395486
$677$	30056.6	1.70631	0.853154	−	0.521659i	$-0.174687\pi$
$677$	30056.6	1.70631	0.853154	+	0.521659i	$0.174687\pi$
$678$	0	0
$679$	−535.213	−0.0302498
$680$	789.740	0.0445370
$681$	0	0
$682$	−3148.81	−0.176795
$683$	−10064.8	−0.563866	−0.281933	−	0.959434i	$-0.590976\pi$
$683$	−10064.8	−0.563866	−0.281933	+	0.959434i	$0.590976\pi$
$684$	0	0
$685$	−5330.46	−0.297323
$686$	6647.50	0.369975
$687$	0	0
$688$	6991.39	0.387419
$689$	−20711.6	−1.14521
$690$	0	0
$691$	10260.7	0.564886	0.282443	−	0.959284i	$-0.408855\pi$
$691$	10260.7	0.564886	0.282443	+	0.959284i	$0.408855\pi$
$692$	14807.1	0.813411
$693$	0	0
$694$	8132.31	0.444810
$695$	7973.94	0.435207
$696$	0	0
$697$	−4495.21	−0.244287
$698$	−16815.6	−0.911865
$699$	0	0
$700$	503.071	0.0271633
$701$	19815.6	1.06766	0.533828	−	0.845593i	$-0.320753\pi$
$701$	19815.6	1.06766	0.533828	+	0.845593i	$0.320753\pi$
$702$	0	0
$703$	−22193.3	−1.19067
$704$	357.260	0.0191261
$705$	0	0
$706$	512.136	0.0273010
$707$	−3233.12	−0.171986
$708$	0	0
$709$	−30533.5	−1.61736	−0.808681	−	0.588247i	$-0.799818\pi$
$709$	−30533.5	−1.61736	−0.808681	+	0.588247i	$0.799818\pi$
$710$	3695.78	0.195352
$711$	0	0
$712$	−9871.12	−0.519573
$713$	6486.93	0.340726
$714$	0	0
$715$	−1750.79	−0.0915745
$716$	−14336.4	−0.748290
$717$	0	0
$718$	−24105.0	−1.25291
$719$	−9125.45	−0.473327	−0.236663	−	0.971592i	$-0.576054\pi$
$719$	−9125.45	−0.473327	−0.236663	+	0.971592i	$0.576054\pi$
$720$	0	0
$721$	8159.77	0.421478
$722$	−36555.0	−1.88426
$723$	0	0
$724$	3134.69	0.160911
$725$	887.583	0.0454676
$726$	0	0
$727$	2635.53	0.134452	0.0672259	−	0.997738i	$-0.478585\pi$
$727$	2635.53	0.134452	0.0672259	+	0.997738i	$0.478585\pi$
$728$	−2524.52	−0.128523
$729$	0	0
$730$	−3101.88	−0.157268
$731$	8627.17	0.436508
$732$	0	0
$733$	10778.0	0.543102	0.271551	−	0.962424i	$-0.412463\pi$
$733$	10778.0	0.543102	0.271551	+	0.962424i	$0.412463\pi$
$734$	18490.8	0.929848
$735$	0	0
$736$	−736.000	−0.0368605
$737$	475.278	0.0237545
$738$	0	0
$739$	−1758.72	−0.0875449	−0.0437725	−	0.999042i	$-0.513938\pi$
$739$	−1758.72	−0.0875449	−0.0437725	+	0.999042i	$0.513938\pi$
$740$	2799.63	0.139076
$741$	0	0
$742$	3322.11	0.164364
$743$	−9737.97	−0.480823	−0.240412	−	0.970671i	$-0.577282\pi$
$743$	−9737.97	−0.480823	−0.240412	+	0.970671i	$0.577282\pi$
$744$	0	0
$745$	−5484.54	−0.269716
$746$	1738.53	0.0853245
$747$	0	0
$748$	440.848	0.0215495
$749$	−7500.08	−0.365884
$750$	0	0
$751$	−23795.7	−1.15621	−0.578107	−	0.815961i	$-0.696208\pi$
$751$	−23795.7	−1.15621	−0.578107	+	0.815961i	$0.696208\pi$
$752$	−1444.32	−0.0700385
$753$	0	0
$754$	−4454.08	−0.215130
$755$	−13672.1	−0.659043
$756$	0	0
$757$	23647.6	1.13538	0.567692	−	0.823241i	$-0.307837\pi$
$757$	23647.6	1.13538	0.567692	+	0.823241i	$0.307837\pi$
$758$	−2557.53	−0.122551
$759$	0	0
$760$	6341.80	0.302686
$761$	4606.09	0.219410	0.109705	−	0.993964i	$-0.465009\pi$
$761$	4606.09	0.219410	0.109705	+	0.993964i	$0.465009\pi$
$762$	0	0
$763$	6056.98	0.287388
$764$	−5967.79	−0.282601
$765$	0	0
$766$	−10687.5	−0.504119
$767$	49944.0	2.35120
$768$	0	0
$769$	17760.9	0.832867	0.416433	−	0.909166i	$-0.363280\pi$
$769$	17760.9	0.832867	0.416433	+	0.909166i	$0.363280\pi$
$770$	280.824	0.0131431
$771$	0	0
$772$	12364.4	0.576430
$773$	−7147.47	−0.332570	−0.166285	−	0.986078i	$-0.553177\pi$
$773$	−7147.47	−0.332570	−0.166285	+	0.986078i	$0.553177\pi$
$774$	0	0
$775$	7051.02	0.326813
$776$	851.114	0.0393727
$777$	0	0
$778$	−1416.69	−0.0652839
$779$	−36097.6	−1.66024
$780$	0	0
$781$	2063.05	0.0945222
$782$	−908.201	−0.0415310
$783$	0	0
$784$	−5083.07	−0.231554
$785$	7198.13	0.327277
$786$	0	0
$787$	−31719.1	−1.43668	−0.718338	−	0.695694i	$-0.755097\pi$
$787$	−31719.1	−1.43668	−0.718338	+	0.695694i	$0.755097\pi$
$788$	−11281.2	−0.509995
$789$	0	0
$790$	−13254.6	−0.596932
$791$	−1392.36	−0.0625872
$792$	0	0
$793$	−35665.7	−1.59713
$794$	11848.0	0.529560
$795$	0	0
$796$	5264.42	0.234413
$797$	−30828.3	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$797$	−30828.3	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$798$	0	0
$799$	−1782.25	−0.0789129
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−22787.3	−1.00330
$803$	−1731.53	−0.0760949
$804$	0	0
$805$	−578.532	−0.0253299
$806$	−35383.5	−1.54632
$807$	0	0
$808$	5141.41	0.223854
$809$	−36679.2	−1.59403	−0.797015	−	0.603959i	$-0.793589\pi$
$809$	−36679.2	−1.59403	−0.797015	+	0.603959i	$0.793589\pi$
$810$	0	0
$811$	−12210.0	−0.528672	−0.264336	−	0.964431i	$-0.585153\pi$
$811$	−12210.0	−0.528672	−0.264336	+	0.964431i	$0.585153\pi$
$812$	714.427	0.0308762
$813$	0	0
$814$	1562.80	0.0672927
$815$	14979.1	0.643797
$816$	0	0
$817$	69278.2	2.96663
$818$	−27608.9	−1.18010
$819$	0	0
$820$	4553.60	0.193925
$821$	9996.47	0.424944	0.212472	−	0.977167i	$-0.431849\pi$
$821$	9996.47	0.424944	0.212472	+	0.977167i	$0.431849\pi$
$822$	0	0
$823$	−42868.8	−1.81569	−0.907845	−	0.419306i	$-0.862273\pi$
$823$	−42868.8	−1.81569	−0.907845	+	0.419306i	$0.862273\pi$
$824$	−12975.9	−0.548590
$825$	0	0
$826$	−8010.93	−0.337453
$827$	15833.4	0.665759	0.332880	−	0.942969i	$-0.391980\pi$
$827$	15833.4	0.665759	0.332880	+	0.942969i	$0.391980\pi$
$828$	0	0
$829$	−17722.3	−0.742484	−0.371242	−	0.928536i	$-0.621068\pi$
$829$	−17722.3	−0.742484	−0.371242	+	0.928536i	$0.621068\pi$
$830$	1588.06	0.0664123
$831$	0	0
$832$	4014.57	0.167284
$833$	−6272.35	−0.260893
$834$	0	0
$835$	−1757.66	−0.0728459
$836$	3540.11	0.146456
$837$	0	0
$838$	−29591.7	−1.21984
$839$	40528.4	1.66769	0.833847	−	0.551996i	$-0.186134\pi$
$839$	40528.4	1.66769	0.833847	+	0.551996i	$0.186134\pi$
$840$	0	0
$841$	−23128.5	−0.948317
$842$	−11609.3	−0.475159
$843$	0	0
$844$	−13596.7	−0.554525
$845$	−8688.82	−0.353733
$846$	0	0
$847$	−6539.11	−0.265273
$848$	−5282.92	−0.213934
$849$	0	0
$850$	−987.175	−0.0398351
$851$	−3219.57	−0.129689
$852$	0	0
$853$	8116.97	0.325814	0.162907	−	0.986641i	$-0.447913\pi$
$853$	8116.97	0.325814	0.162907	+	0.986641i	$0.447913\pi$
$854$	5720.73	0.229226
$855$	0	0
$856$	11926.9	0.476229
$857$	−16285.5	−0.649126	−0.324563	−	0.945864i	$-0.605217\pi$
$857$	−16285.5	−0.649126	−0.324563	+	0.945864i	$0.605217\pi$
$858$	0	0
$859$	−26487.5	−1.05209	−0.526043	−	0.850458i	$-0.676325\pi$
$859$	−26487.5	−1.05209	−0.526043	+	0.850458i	$0.676325\pi$
$860$	−8739.24	−0.346518
$861$	0	0
$862$	30536.3	1.20658
$863$	−40396.2	−1.59340	−0.796699	−	0.604376i	$-0.793422\pi$
$863$	−40396.2	−1.59340	−0.796699	+	0.604376i	$0.793422\pi$
$864$	0	0
$865$	−18508.8	−0.727537
$866$	−4501.25	−0.176627
$867$	0	0
$868$	5675.46	0.221933
$869$	−7398.94	−0.288828
$870$	0	0
$871$	5340.76	0.207767
$872$	−9632.00	−0.374060
$873$	0	0
$874$	−7293.07	−0.282256
$875$	−628.839	−0.0242956
$876$	0	0
$877$	10962.9	0.422109	0.211054	−	0.977474i	$-0.432310\pi$
$877$	10962.9	0.422109	0.211054	+	0.977474i	$0.432310\pi$
$878$	−20732.4	−0.796908
$879$	0	0
$880$	−446.575	−0.0171069
$881$	−23906.5	−0.914222	−0.457111	−	0.889410i	$-0.651116\pi$
$881$	−23906.5	−0.914222	−0.457111	+	0.889410i	$0.651116\pi$
$882$	0	0
$883$	−30509.1	−1.16275	−0.581377	−	0.813634i	$-0.697486\pi$
$883$	−30509.1	−1.16275	−0.581377	+	0.813634i	$0.697486\pi$
$884$	4953.86	0.188480
$885$	0	0
$886$	−7762.22	−0.294331
$887$	40034.3	1.51547	0.757733	−	0.652565i	$-0.226307\pi$
$887$	40034.3	1.51547	0.757733	+	0.652565i	$0.226307\pi$
$888$	0	0
$889$	7811.62	0.294706
$890$	12338.9	0.464720
$891$	0	0
$892$	3458.64	0.129825
$893$	−14311.9	−0.536314
$894$	0	0
$895$	17920.5	0.669291
$896$	−643.931	−0.0240092
$897$	0	0
$898$	−20293.0	−0.754106
$899$	10013.4	0.371485
$900$	0	0
$901$	−6518.96	−0.241041
$902$	2541.91	0.0938318
$903$	0	0
$904$	2214.17	0.0814626
$905$	−3918.36	−0.143923
$906$	0	0
$907$	−2843.48	−0.104097	−0.0520486	−	0.998645i	$-0.516575\pi$
$907$	−2843.48	−0.104097	−0.0520486	+	0.998645i	$0.516575\pi$
$908$	7917.58	0.289377
$909$	0	0
$910$	3155.65	0.114955
$911$	−42268.6	−1.53723	−0.768617	−	0.639710i	$-0.779055\pi$
$911$	−42268.6	−1.53723	−0.768617	+	0.639710i	$0.779055\pi$
$912$	0	0
$913$	886.483	0.0321340
$914$	9457.74	0.342269
$915$	0	0
$916$	−12454.1	−0.449231
$917$	1579.00	0.0568629
$918$	0	0
$919$	−12699.6	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$919$	−12699.6	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	16761.4	0.598706
$923$	23182.8	0.826728
$924$	0	0
$925$	−3499.53	−0.124393
$926$	−13690.3	−0.485845
$927$	0	0
$928$	−1136.11	−0.0401880
$929$	24889.6	0.879011	0.439505	−	0.898240i	$-0.355154\pi$
$929$	24889.6	0.879011	0.439505	+	0.898240i	$0.355154\pi$
$930$	0	0
$931$	−50368.5	−1.77310
$932$	−24074.6	−0.846126
$933$	0	0
$934$	−29768.0	−1.04287
$935$	−551.060	−0.0192744
$936$	0	0
$937$	9585.48	0.334199	0.167099	−	0.985940i	$-0.446560\pi$
$937$	9585.48	0.334199	0.167099	+	0.985940i	$0.446560\pi$
$938$	−856.649	−0.0298194
$939$	0	0
$940$	1805.40	0.0626444
$941$	21146.9	0.732592	0.366296	−	0.930498i	$-0.380626\pi$
$941$	21146.9	0.732592	0.366296	+	0.930498i	$0.380626\pi$
$942$	0	0
$943$	−5236.65	−0.180836
$944$	12739.2	0.439224
$945$	0	0
$946$	−4878.41	−0.167665
$947$	−48747.2	−1.67272	−0.836362	−	0.548177i	$-0.815322\pi$
$947$	−48747.2	−1.67272	−0.836362	+	0.548177i	$0.815322\pi$
$948$	0	0
$949$	−19457.3	−0.665556
$950$	−7927.25	−0.270730
$951$	0	0
$952$	−794.591	−0.0270513
$953$	−24925.9	−0.847249	−0.423625	−	0.905838i	$-0.639242\pi$
$953$	−24925.9	−0.847249	−0.423625	+	0.905838i	$0.639242\pi$
$954$	0	0
$955$	7459.73	0.252766
$956$	16899.5	0.571725
$957$	0	0
$958$	12097.4	0.407984
$959$	5363.20	0.180591
$960$	0	0
$961$	49755.9	1.67017
$962$	17561.4	0.588568
$963$	0	0
$964$	5576.95	0.186329
$965$	−15455.5	−0.515574
$966$	0	0
$967$	−1881.92	−0.0625838	−0.0312919	−	0.999510i	$-0.509962\pi$
$967$	−1881.92	−0.0625838	−0.0312919	+	0.999510i	$0.509962\pi$
$968$	10398.7	0.345276
$969$	0	0
$970$	−1063.89	−0.0352160
$971$	14637.3	0.483763	0.241881	−	0.970306i	$-0.422236\pi$
$971$	14637.3	0.483763	0.241881	+	0.970306i	$0.422236\pi$
$972$	0	0
$973$	−8022.92	−0.264340
$974$	−6398.70	−0.210501
$975$	0	0
$976$	−9097.28	−0.298358
$977$	−42591.4	−1.39470	−0.697348	−	0.716733i	$-0.745637\pi$
$977$	−42591.4	−1.39470	−0.697348	+	0.716733i	$0.745637\pi$
$978$	0	0
$979$	6887.81	0.224857
$980$	6353.84	0.207108
$981$	0	0
$982$	−3228.81	−0.104924
$983$	48858.3	1.58529	0.792644	−	0.609684i	$-0.208704\pi$
$983$	48858.3	1.58529	0.792644	+	0.609684i	$0.208704\pi$
$984$	0	0
$985$	14101.5	0.456153
$986$	−1401.92	−0.0452801
$987$	0	0
$988$	39780.6	1.28096
$989$	10050.1	0.323130
$990$	0	0
$991$	−56522.1	−1.81179	−0.905895	−	0.423502i	$-0.860801\pi$
$991$	−56522.1	−1.81179	−0.905895	+	0.423502i	$0.860801\pi$
$992$	−9025.30	−0.288864
$993$	0	0
$994$	−3718.48	−0.118655
$995$	−6580.53	−0.209665
$996$	0	0
$997$	−20767.9	−0.659706	−0.329853	−	0.944032i	$-0.606999\pi$
$997$	−20767.9	−0.659706	−0.329853	+	0.944032i	$0.606999\pi$
$998$	10784.1	0.342048
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bg.1.3		4
3.2	odd	2		230.4.a.j.1.1	✓	4
12.11	even	2		1840.4.a.k.1.4		4
15.2	even	4		1150.4.b.o.599.8		8
15.8	even	4		1150.4.b.o.599.1		8
15.14	odd	2		1150.4.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.1	✓	4	3.2	odd	2
1150.4.a.n.1.4		4	15.14	odd	2
1150.4.b.o.599.1		8	15.8	even	4
1150.4.b.o.599.8		8	15.2	even	4
1840.4.a.k.1.4		4	12.11	even	2
2070.4.a.bg.1.3		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +5.03071 q^{7} -8.00000 q^{8} +10.0000 q^{10} +5.58219 q^{11} +62.7277 q^{13} -10.0614 q^{14} +16.0000 q^{16} +19.7435 q^{17} +158.545 q^{19} -20.0000 q^{20} -11.1644 q^{22} +23.0000 q^{23} +25.0000 q^{25} -125.455 q^{26} +20.1228 q^{28} +35.5033 q^{29} +282.041 q^{31} -32.0000 q^{32} -39.4870 q^{34} -25.1536 q^{35} -139.981 q^{37} -317.090 q^{38} +40.0000 q^{40} -227.680 q^{41} +436.962 q^{43} +22.3288 q^{44} -46.0000 q^{46} -90.2701 q^{47} -317.692 q^{49} -50.0000 q^{50} +250.911 q^{52} -330.183 q^{53} -27.9109 q^{55} -40.2457 q^{56} -71.0066 q^{58} +796.203 q^{59} -568.580 q^{61} -564.081 q^{62} +64.0000 q^{64} -313.638 q^{65} +85.1419 q^{67} +78.9740 q^{68} +50.3071 q^{70} +369.578 q^{71} -310.188 q^{73} +279.963 q^{74} +634.180 q^{76} +28.0824 q^{77} -1325.46 q^{79} -80.0000 q^{80} +455.360 q^{82} +158.806 q^{83} -98.7175 q^{85} -873.924 q^{86} -44.6575 q^{88} +1233.89 q^{89} +315.565 q^{91} +92.0000 q^{92} +180.540 q^{94} -792.725 q^{95} -106.389 q^{97} +635.384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) 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

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