LMFDB - Embedded newform 2142.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,4,Mod(1,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2142.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2142.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,4,0,8,-9,0,-14,16,0,-18,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$126.382091232$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{93}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 23 $ x^2 - x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.32183$ of defining polynomial
Character	$\chi$	$=$	2142.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} +0.321825 q^{5} -7.00000 q^{7} +8.00000 q^{8} +0.643651 q^{10} -36.5746 q^{11} +24.5746 q^{13} -14.0000 q^{14} +16.0000 q^{16} -17.0000 q^{17} +25.2873 q^{19} +1.28730 q^{20} -73.1492 q^{22} -34.6437 q^{23} -124.896 q^{25} +49.1492 q^{26} -28.0000 q^{28} +239.011 q^{29} +224.689 q^{31} +32.0000 q^{32} -34.0000 q^{34} -2.25278 q^{35} -314.298 q^{37} +50.5746 q^{38} +2.57460 q^{40} +194.747 q^{41} -367.206 q^{43} -146.298 q^{44} -69.2873 q^{46} +415.931 q^{47} +49.0000 q^{49} -249.793 q^{50} +98.2984 q^{52} +293.173 q^{53} -11.7706 q^{55} -56.0000 q^{56} +478.022 q^{58} -148.000 q^{59} +163.206 q^{61} +449.379 q^{62} +64.0000 q^{64} +7.90873 q^{65} +469.769 q^{67} -68.0000 q^{68} -4.50556 q^{70} -30.8508 q^{71} +511.758 q^{73} -628.597 q^{74} +101.149 q^{76} +256.022 q^{77} +391.102 q^{79} +5.14921 q^{80} +389.494 q^{82} +940.848 q^{83} -5.47103 q^{85} -734.412 q^{86} -292.597 q^{88} +642.229 q^{89} -172.022 q^{91} -138.575 q^{92} +831.862 q^{94} +8.13810 q^{95} -983.079 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} - 9 q^{5} - 14 q^{7} + 16 q^{8} - 18 q^{10} + 4 q^{11} - 28 q^{13} - 28 q^{14} + 32 q^{16} - 34 q^{17} + 12 q^{19} - 36 q^{20} + 8 q^{22} - 50 q^{23} - 163 q^{25} - 56 q^{26} - 56 q^{28}+ \cdots + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 - 9 * q^5 - 14 * q^7 + 16 * q^8 - 18 * q^10 + 4 * q^11 - 28 * q^13 - 28 * q^14 + 32 * q^16 - 34 * q^17 + 12 * q^19 - 36 * q^20 + 8 * q^22 - 50 * q^23 - 163 * q^25 - 56 * q^26 - 56 * q^28 + 208 * q^29 + 189 * q^31 + 64 * q^32 - 68 * q^34 + 63 * q^35 - 320 * q^37 + 24 * q^38 - 72 * q^40 + 457 * q^41 - 69 * q^43 + 16 * q^44 - 100 * q^46 + 774 * q^47 + 98 * q^49 - 326 * q^50 - 112 * q^52 + 731 * q^53 - 390 * q^55 - 112 * q^56 + 416 * q^58 - 296 * q^59 - 339 * q^61 + 378 * q^62 + 128 * q^64 + 498 * q^65 + 467 * q^67 - 136 * q^68 + 126 * q^70 - 216 * q^71 + 821 * q^73 - 640 * q^74 + 48 * q^76 - 28 * q^77 + 30 * q^79 - 144 * q^80 + 914 * q^82 + 358 * q^83 + 153 * q^85 - 138 * q^86 + 32 * q^88 + 918 * q^89 + 196 * q^91 - 200 * q^92 + 1548 * q^94 + 132 * q^95 - 915 * q^97 + 196 * 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
$5$	0.321825	0.0287849	0.0143925	−	0.999896i	$-0.495419\pi$
$5$	0.321825	0.0287849	0.0143925	+	0.999896i	$0.495419\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	0.643651	0.0203540
$11$	−36.5746	−1.00251	−0.501257	−	0.865298i	$-0.667129\pi$
$11$	−36.5746	−1.00251	−0.501257	+	0.865298i	$0.667129\pi$
$12$	0	0
$13$	24.5746	0.524290	0.262145	−	0.965029i	$-0.415570\pi$
$13$	24.5746	0.524290	0.262145	+	0.965029i	$0.415570\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	25.2873	0.305332	0.152666	−	0.988278i	$-0.451214\pi$
$19$	25.2873	0.305332	0.152666	+	0.988278i	$0.451214\pi$
$20$	1.28730	0.0143925
$21$	0	0
$22$	−73.1492	−0.708885
$23$	−34.6437	−0.314074	−0.157037	−	0.987593i	$-0.550194\pi$
$23$	−34.6437	−0.314074	−0.157037	+	0.987593i	$0.550194\pi$
$24$	0	0
$25$	−124.896	−0.999171
$26$	49.1492	0.370729
$27$	0	0
$28$	−28.0000	−0.188982
$29$	239.011	1.53046	0.765228	−	0.643759i	$-0.222626\pi$
$29$	239.011	1.53046	0.765228	+	0.643759i	$0.222626\pi$
$30$	0	0
$31$	224.689	1.30179	0.650893	−	0.759169i	$-0.274395\pi$
$31$	224.689	1.30179	0.650893	+	0.759169i	$0.274395\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−34.0000	−0.171499
$35$	−2.25278	−0.0108797
$36$	0	0
$37$	−314.298	−1.39650	−0.698248	−	0.715856i	$-0.746037\pi$
$37$	−314.298	−1.39650	−0.698248	+	0.715856i	$0.746037\pi$
$38$	50.5746	0.215902
$39$	0	0
$40$	2.57460	0.0101770
$41$	194.747	0.741815	0.370907	−	0.928670i	$-0.379047\pi$
$41$	194.747	0.741815	0.370907	+	0.928670i	$0.379047\pi$
$42$	0	0
$43$	−367.206	−1.30229	−0.651144	−	0.758954i	$-0.725711\pi$
$43$	−367.206	−1.30229	−0.651144	+	0.758954i	$0.725711\pi$
$44$	−146.298	−0.501257
$45$	0	0
$46$	−69.2873	−0.222084
$47$	415.931	1.29085	0.645423	−	0.763825i	$-0.276681\pi$
$47$	415.931	1.29085	0.645423	+	0.763825i	$0.276681\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−249.793	−0.706521
$51$	0	0
$52$	98.2984	0.262145
$53$	293.173	0.759818	0.379909	−	0.925024i	$-0.375955\pi$
$53$	293.173	0.759818	0.379909	+	0.925024i	$0.375955\pi$
$54$	0	0
$55$	−11.7706	−0.0288573
$56$	−56.0000	−0.133631
$57$	0	0
$58$	478.022	1.08220
$59$	−148.000	−0.326576	−0.163288	−	0.986578i	$-0.552210\pi$
$59$	−148.000	−0.326576	−0.163288	+	0.986578i	$0.552210\pi$
$60$	0	0
$61$	163.206	0.342564	0.171282	−	0.985222i	$-0.445209\pi$
$61$	163.206	0.342564	0.171282	+	0.985222i	$0.445209\pi$
$62$	449.379	0.920502
$63$	0	0
$64$	64.0000	0.125000
$65$	7.90873	0.0150917
$66$	0	0
$67$	469.769	0.856589	0.428295	−	0.903639i	$-0.359115\pi$
$67$	469.769	0.856589	0.428295	+	0.903639i	$0.359115\pi$
$68$	−68.0000	−0.121268
$69$	0	0
$70$	−4.50556	−0.00769310
$71$	−30.8508	−0.0515678	−0.0257839	−	0.999668i	$-0.508208\pi$
$71$	−30.8508	−0.0515678	−0.0257839	+	0.999668i	$0.508208\pi$
$72$	0	0
$73$	511.758	0.820504	0.410252	−	0.911972i	$-0.365441\pi$
$73$	511.758	0.820504	0.410252	+	0.911972i	$0.365441\pi$
$74$	−628.597	−0.987472
$75$	0	0
$76$	101.149	0.152666
$77$	256.022	0.378915
$78$	0	0
$79$	391.102	0.556993	0.278497	−	0.960437i	$-0.410164\pi$
$79$	391.102	0.556993	0.278497	+	0.960437i	$0.410164\pi$
$80$	5.14921	0.00719623
$81$	0	0
$82$	389.494	0.524542
$83$	940.848	1.24424	0.622118	−	0.782924i	$-0.286273\pi$
$83$	940.848	1.24424	0.622118	+	0.782924i	$0.286273\pi$
$84$	0	0
$85$	−5.47103	−0.00698137
$86$	−734.412	−0.920857
$87$	0	0
$88$	−292.597	−0.354442
$89$	642.229	0.764901	0.382450	−	0.923976i	$-0.375080\pi$
$89$	642.229	0.764901	0.382450	+	0.923976i	$0.375080\pi$
$90$	0	0
$91$	−172.022	−0.198163
$92$	−138.575	−0.157037
$93$	0	0
$94$	831.862	0.912766
$95$	8.13810	0.00878896
$96$	0	0
$97$	−983.079	−1.02904	−0.514518	−	0.857479i	$-0.672029\pi$
$97$	−983.079	−1.02904	−0.514518	+	0.857479i	$0.672029\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−499.586	−0.499586
$101$	−1254.34	−1.23576	−0.617880	−	0.786272i	$-0.712008\pi$
$101$	−1254.34	−1.23576	−0.617880	+	0.786272i	$0.712008\pi$
$102$	0	0
$103$	58.7817	0.0562324	0.0281162	−	0.999605i	$-0.491049\pi$
$103$	58.7817	0.0562324	0.0281162	+	0.999605i	$0.491049\pi$
$104$	196.597	0.185364
$105$	0	0
$106$	586.345	0.537272
$107$	1297.17	1.17198	0.585992	−	0.810317i	$-0.300705\pi$
$107$	1297.17	1.17198	0.585992	+	0.810317i	$0.300705\pi$
$108$	0	0
$109$	830.232	0.729558	0.364779	−	0.931094i	$-0.381145\pi$
$109$	830.232	0.729558	0.364779	+	0.931094i	$0.381145\pi$
$110$	−23.5413	−0.0204052
$111$	0	0
$112$	−112.000	−0.0944911
$113$	308.345	0.256696	0.128348	−	0.991729i	$-0.459033\pi$
$113$	308.345	0.256696	0.128348	+	0.991729i	$0.459033\pi$
$114$	0	0
$115$	−11.1492	−0.00904060
$116$	956.044	0.765228
$117$	0	0
$118$	−296.000	−0.230924
$119$	119.000	0.0916698
$120$	0	0
$121$	6.70159	0.00503500
$122$	326.412	0.242229
$123$	0	0
$124$	898.757	0.650893
$125$	−80.4230	−0.0575460
$126$	0	0
$127$	−1165.86	−0.814592	−0.407296	−	0.913296i	$-0.633528\pi$
$127$	−1165.86	−0.814592	−0.407296	+	0.913296i	$0.633528\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	15.8175	0.0106714
$131$	1202.80	0.802208	0.401104	−	0.916032i	$-0.368627\pi$
$131$	1202.80	0.802208	0.401104	+	0.916032i	$0.368627\pi$
$132$	0	0
$133$	−177.011	−0.115405
$134$	939.539	0.605700
$135$	0	0
$136$	−136.000	−0.0857493
$137$	2658.90	1.65814	0.829071	−	0.559144i	$-0.188870\pi$
$137$	2658.90	1.65814	0.829071	+	0.559144i	$0.188870\pi$
$138$	0	0
$139$	989.607	0.603866	0.301933	−	0.953329i	$-0.402368\pi$
$139$	989.607	0.603866	0.301933	+	0.953329i	$0.402368\pi$
$140$	−9.01111	−0.00543984
$141$	0	0
$142$	−61.7016	−0.0364640
$143$	−898.806	−0.525608
$144$	0	0
$145$	76.9198	0.0440541
$146$	1023.52	0.580184
$147$	0	0
$148$	−1257.19	−0.698248
$149$	3297.39	1.81297	0.906485	−	0.422238i	$-0.138755\pi$
$149$	3297.39	1.81297	0.906485	+	0.422238i	$0.138755\pi$
$150$	0	0
$151$	778.865	0.419756	0.209878	−	0.977728i	$-0.432693\pi$
$151$	778.865	0.419756	0.209878	+	0.977728i	$0.432693\pi$
$152$	202.298	0.107951
$153$	0	0
$154$	512.044	0.267933
$155$	72.3107	0.0374718
$156$	0	0
$157$	−439.492	−0.223409	−0.111705	−	0.993741i	$-0.535631\pi$
$157$	−439.492	−0.223409	−0.111705	+	0.993741i	$0.535631\pi$
$158$	782.205	0.393854
$159$	0	0
$160$	10.2984	0.00508851
$161$	242.506	0.118709
$162$	0	0
$163$	1936.59	0.930586	0.465293	−	0.885157i	$-0.345949\pi$
$163$	1936.59	0.930586	0.465293	+	0.885157i	$0.345949\pi$
$164$	778.989	0.370907
$165$	0	0
$166$	1881.70	0.879807
$167$	953.023	0.441600	0.220800	−	0.975319i	$-0.429133\pi$
$167$	953.023	0.441600	0.220800	+	0.975319i	$0.429133\pi$
$168$	0	0
$169$	−1593.09	−0.725120
$170$	−10.9421	−0.00493658
$171$	0	0
$172$	−1468.82	−0.651144
$173$	87.4179	0.0384177	0.0192088	−	0.999815i	$-0.493885\pi$
$173$	87.4179	0.0384177	0.0192088	+	0.999815i	$0.493885\pi$
$174$	0	0
$175$	874.275	0.377651
$176$	−585.194	−0.250629
$177$	0	0
$178$	1284.46	0.540867
$179$	−4657.04	−1.94460	−0.972299	−	0.233738i	$-0.924904\pi$
$179$	−4657.04	−1.94460	−0.972299	+	0.233738i	$0.924904\pi$
$180$	0	0
$181$	445.724	0.183041	0.0915204	−	0.995803i	$-0.470827\pi$
$181$	445.724	0.183041	0.0915204	+	0.995803i	$0.470827\pi$
$182$	−344.044	−0.140122
$183$	0	0
$184$	−277.149	−0.111042
$185$	−101.149	−0.0401980
$186$	0	0
$187$	621.768	0.243145
$188$	1663.72	0.645423
$189$	0	0
$190$	16.2762	0.00621473
$191$	3758.78	1.42396	0.711978	−	0.702202i	$-0.247800\pi$
$191$	3758.78	1.42396	0.711978	+	0.702202i	$0.247800\pi$
$192$	0	0
$193$	−1783.24	−0.665079	−0.332540	−	0.943089i	$-0.607906\pi$
$193$	−1783.24	−0.665079	−0.332540	+	0.943089i	$0.607906\pi$
$194$	−1966.16	−0.727639
$195$	0	0
$196$	196.000	0.0714286
$197$	−979.394	−0.354208	−0.177104	−	0.984192i	$-0.556673\pi$
$197$	−979.394	−0.354208	−0.177104	+	0.984192i	$0.556673\pi$
$198$	0	0
$199$	−3357.70	−1.19609	−0.598044	−	0.801464i	$-0.704055\pi$
$199$	−3357.70	−1.19609	−0.598044	+	0.801464i	$0.704055\pi$
$200$	−999.171	−0.353260
$201$	0	0
$202$	−2508.69	−0.873814
$203$	−1673.08	−0.578458
$204$	0	0
$205$	62.6746	0.0213531
$206$	117.563	0.0397623
$207$	0	0
$208$	393.194	0.131072
$209$	−924.873	−0.306100
$210$	0	0
$211$	2802.83	0.914478	0.457239	−	0.889344i	$-0.348838\pi$
$211$	2802.83	0.914478	0.457239	+	0.889344i	$0.348838\pi$
$212$	1172.69	0.379909
$213$	0	0
$214$	2594.34	0.828718
$215$	−118.176	−0.0374863
$216$	0	0
$217$	−1572.82	−0.492029
$218$	1660.46	0.515875
$219$	0	0
$220$	−47.0825	−0.0144287
$221$	−417.768	−0.127159
$222$	0	0
$223$	−569.149	−0.170911	−0.0854553	−	0.996342i	$-0.527234\pi$
$223$	−569.149	−0.170911	−0.0854553	+	0.996342i	$0.527234\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	616.690	0.181512
$227$	6224.90	1.82009	0.910047	−	0.414505i	$-0.136045\pi$
$227$	6224.90	1.82009	0.910047	+	0.414505i	$0.136045\pi$
$228$	0	0
$229$	−2936.16	−0.847278	−0.423639	−	0.905831i	$-0.639247\pi$
$229$	−2936.16	−0.847278	−0.423639	+	0.905831i	$0.639247\pi$
$230$	−22.2984	−0.00639267
$231$	0	0
$232$	1912.09	0.541098
$233$	2085.27	0.586311	0.293156	−	0.956065i	$-0.405295\pi$
$233$	2085.27	0.586311	0.293156	+	0.956065i	$0.405295\pi$
$234$	0	0
$235$	133.857	0.0371569
$236$	−592.000	−0.163288
$237$	0	0
$238$	238.000	0.0648204
$239$	4271.33	1.15602	0.578012	−	0.816028i	$-0.303829\pi$
$239$	4271.33	1.15602	0.578012	+	0.816028i	$0.303829\pi$
$240$	0	0
$241$	−144.317	−0.0385738	−0.0192869	−	0.999814i	$-0.506140\pi$
$241$	−144.317	−0.0385738	−0.0192869	+	0.999814i	$0.506140\pi$
$242$	13.4032	0.00356028
$243$	0	0
$244$	652.824	0.171282
$245$	15.7694	0.00411213
$246$	0	0
$247$	621.425	0.160082
$248$	1797.51	0.460251
$249$	0	0
$250$	−160.846	−0.0406912
$251$	2104.09	0.529119	0.264560	−	0.964369i	$-0.414773\pi$
$251$	2104.09	0.529119	0.264560	+	0.964369i	$0.414773\pi$
$252$	0	0
$253$	1267.08	0.314864
$254$	−2331.72	−0.576004
$255$	0	0
$256$	256.000	0.0625000
$257$	2675.34	0.649351	0.324676	−	0.945825i	$-0.394745\pi$
$257$	2675.34	0.649351	0.324676	+	0.945825i	$0.394745\pi$
$258$	0	0
$259$	2200.09	0.527826
$260$	31.6349	0.00754583
$261$	0	0
$262$	2405.60	0.567247
$263$	166.397	0.0390132	0.0195066	−	0.999810i	$-0.493790\pi$
$263$	166.397	0.0390132	0.0195066	+	0.999810i	$0.493790\pi$
$264$	0	0
$265$	94.3504	0.0218713
$266$	−354.022	−0.0816034
$267$	0	0
$268$	1879.08	0.428295
$269$	−8052.60	−1.82519	−0.912594	−	0.408866i	$-0.865924\pi$
$269$	−8052.60	−1.82519	−0.912594	+	0.408866i	$0.865924\pi$
$270$	0	0
$271$	−1858.91	−0.416681	−0.208340	−	0.978056i	$-0.566806\pi$
$271$	−1858.91	−0.416681	−0.208340	+	0.978056i	$0.566806\pi$
$272$	−272.000	−0.0606339
$273$	0	0
$274$	5317.81	1.17248
$275$	4568.04	1.00168
$276$	0	0
$277$	3211.52	0.696611	0.348306	−	0.937381i	$-0.386757\pi$
$277$	3211.52	0.696611	0.348306	+	0.937381i	$0.386757\pi$
$278$	1979.21	0.426998
$279$	0	0
$280$	−18.0222	−0.00384655
$281$	−863.925	−0.183407	−0.0917036	−	0.995786i	$-0.529231\pi$
$281$	−863.925	−0.183407	−0.0917036	+	0.995786i	$0.529231\pi$
$282$	0	0
$283$	8971.90	1.88454	0.942269	−	0.334858i	$-0.108688\pi$
$283$	8971.90	1.88454	0.942269	+	0.334858i	$0.108688\pi$
$284$	−123.403	−0.0257839
$285$	0	0
$286$	−1797.61	−0.371661
$287$	−1363.23	−0.280380
$288$	0	0
$289$	289.000	0.0588235
$290$	153.840	0.0311510
$291$	0	0
$292$	2047.03	0.410252
$293$	−1972.88	−0.393367	−0.196684	−	0.980467i	$-0.563017\pi$
$293$	−1972.88	−0.393367	−0.196684	+	0.980467i	$0.563017\pi$
$294$	0	0
$295$	−47.6302	−0.00940046
$296$	−2514.39	−0.493736
$297$	0	0
$298$	6594.78	1.28196
$299$	−851.354	−0.164666
$300$	0	0
$301$	2570.44	0.492219
$302$	1557.73	0.296812
$303$	0	0
$304$	404.597	0.0763330
$305$	52.5238	0.00986067
$306$	0	0
$307$	6751.99	1.25523	0.627616	−	0.778523i	$-0.284031\pi$
$307$	6751.99	1.25523	0.627616	+	0.778523i	$0.284031\pi$
$308$	1024.09	0.189457
$309$	0	0
$310$	144.621	0.0264966
$311$	−5372.74	−0.979615	−0.489807	−	0.871831i	$-0.662933\pi$
$311$	−5372.74	−0.979615	−0.489807	+	0.871831i	$0.662933\pi$
$312$	0	0
$313$	7540.67	1.36174	0.680868	−	0.732406i	$-0.261603\pi$
$313$	7540.67	1.36174	0.680868	+	0.732406i	$0.261603\pi$
$314$	−878.984	−0.157974
$315$	0	0
$316$	1564.41	0.278497
$317$	−3660.08	−0.648488	−0.324244	−	0.945973i	$-0.605110\pi$
$317$	−3660.08	−0.648488	−0.324244	+	0.945973i	$0.605110\pi$
$318$	0	0
$319$	−8741.74	−1.53430
$320$	20.5968	0.00359812
$321$	0	0
$322$	485.011	0.0839398
$323$	−429.884	−0.0740538
$324$	0	0
$325$	−3069.28	−0.523855
$326$	3873.18	0.658024
$327$	0	0
$328$	1557.98	0.262271
$329$	−2911.52	−0.487894
$330$	0	0
$331$	86.2655	0.0143250	0.00716251	−	0.999974i	$-0.497720\pi$
$331$	86.2655	0.0143250	0.00716251	+	0.999974i	$0.497720\pi$
$332$	3763.39	0.622118
$333$	0	0
$334$	1906.05	0.312258
$335$	151.184	0.0246569
$336$	0	0
$337$	−4028.19	−0.651126	−0.325563	−	0.945520i	$-0.605554\pi$
$337$	−4028.19	−0.651126	−0.325563	+	0.945520i	$0.605554\pi$
$338$	−3186.18	−0.512737
$339$	0	0
$340$	−21.8841	−0.00349069
$341$	−8217.92	−1.30506
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−2937.65	−0.460428
$345$	0	0
$346$	174.836	0.0271654
$347$	4054.26	0.627217	0.313608	−	0.949552i	$-0.398462\pi$
$347$	4054.26	0.627217	0.313608	+	0.949552i	$0.398462\pi$
$348$	0	0
$349$	−7112.95	−1.09097	−0.545484	−	0.838122i	$-0.683654\pi$
$349$	−7112.95	−1.09097	−0.545484	+	0.838122i	$0.683654\pi$
$350$	1748.55	0.267040
$351$	0	0
$352$	−1170.39	−0.177221
$353$	5701.18	0.859613	0.429806	−	0.902921i	$-0.358582\pi$
$353$	5701.18	0.859613	0.429806	+	0.902921i	$0.358582\pi$
$354$	0	0
$355$	−9.92857	−0.00148438
$356$	2568.92	0.382450
$357$	0	0
$358$	−9314.07	−1.37504
$359$	−2952.81	−0.434105	−0.217052	−	0.976160i	$-0.569644\pi$
$359$	−2952.81	−0.434105	−0.217052	+	0.976160i	$0.569644\pi$
$360$	0	0
$361$	−6219.55	−0.906772
$362$	891.448	0.129429
$363$	0	0
$364$	−688.089	−0.0990815
$365$	164.697	0.0236182
$366$	0	0
$367$	−3854.72	−0.548268	−0.274134	−	0.961691i	$-0.588391\pi$
$367$	−3854.72	−0.548268	−0.274134	+	0.961691i	$0.588391\pi$
$368$	−554.298	−0.0785185
$369$	0	0
$370$	−202.298	−0.0284243
$371$	−2052.21	−0.287184
$372$	0	0
$373$	676.560	0.0939167	0.0469584	−	0.998897i	$-0.485047\pi$
$373$	676.560	0.0939167	0.0469584	+	0.998897i	$0.485047\pi$
$374$	1243.54	0.171930
$375$	0	0
$376$	3327.45	0.456383
$377$	5873.60	0.802403
$378$	0	0
$379$	1642.91	0.222666	0.111333	−	0.993783i	$-0.464488\pi$
$379$	1642.91	0.222666	0.111333	+	0.993783i	$0.464488\pi$
$380$	32.5524	0.00439448
$381$	0	0
$382$	7517.56	1.00689
$383$	−4595.64	−0.613123	−0.306562	−	0.951851i	$-0.599179\pi$
$383$	−4595.64	−0.613123	−0.306562	+	0.951851i	$0.599179\pi$
$384$	0	0
$385$	82.3944	0.0109070
$386$	−3566.48	−0.470282
$387$	0	0
$388$	−3932.32	−0.514518
$389$	11416.3	1.48800	0.743999	−	0.668181i	$-0.232927\pi$
$389$	11416.3	1.48800	0.743999	+	0.668181i	$0.232927\pi$
$390$	0	0
$391$	588.942	0.0761741
$392$	392.000	0.0505076
$393$	0	0
$394$	−1958.79	−0.250463
$395$	125.867	0.0160330
$396$	0	0
$397$	12195.9	1.54180	0.770901	−	0.636954i	$-0.219806\pi$
$397$	12195.9	1.54180	0.770901	+	0.636954i	$0.219806\pi$
$398$	−6715.41	−0.845761
$399$	0	0
$400$	−1998.34	−0.249793
$401$	670.010	0.0834381	0.0417191	−	0.999129i	$-0.486717\pi$
$401$	670.010	0.0834381	0.0417191	+	0.999129i	$0.486717\pi$
$402$	0	0
$403$	5521.65	0.682514
$404$	−5017.37	−0.617880
$405$	0	0
$406$	−3346.16	−0.409032
$407$	11495.3	1.40001
$408$	0	0
$409$	12142.6	1.46800	0.734002	−	0.679147i	$-0.237650\pi$
$409$	12142.6	1.46800	0.734002	+	0.679147i	$0.237650\pi$
$410$	125.349	0.0150989
$411$	0	0
$412$	235.127	0.0281162
$413$	1036.00	0.123434
$414$	0	0
$415$	302.789	0.0358152
$416$	786.387	0.0926822
$417$	0	0
$418$	−1849.75	−0.216445
$419$	9804.10	1.14311	0.571553	−	0.820565i	$-0.306341\pi$
$419$	9804.10	1.14311	0.571553	+	0.820565i	$0.306341\pi$
$420$	0	0
$421$	−12913.2	−1.49490	−0.747449	−	0.664319i	$-0.768721\pi$
$421$	−12913.2	−1.49490	−0.747449	+	0.664319i	$0.768721\pi$
$422$	5605.66	0.646634
$423$	0	0
$424$	2345.38	0.268636
$425$	2123.24	0.242335
$426$	0	0
$427$	−1142.44	−0.129477
$428$	5188.69	0.585992
$429$	0	0
$430$	−236.352	−0.0265068
$431$	14434.6	1.61320	0.806599	−	0.591100i	$-0.201306\pi$
$431$	14434.6	1.61320	0.806599	+	0.591100i	$0.201306\pi$
$432$	0	0
$433$	−15643.1	−1.73617	−0.868084	−	0.496417i	$-0.834649\pi$
$433$	−15643.1	−1.73617	−0.868084	+	0.496417i	$0.834649\pi$
$434$	−3145.65	−0.347917
$435$	0	0
$436$	3320.93	0.364779
$437$	−876.044	−0.0958968
$438$	0	0
$439$	−5059.90	−0.550105	−0.275052	−	0.961429i	$-0.588695\pi$
$439$	−5059.90	−0.550105	−0.275052	+	0.961429i	$0.588695\pi$
$440$	−94.1651	−0.0102026
$441$	0	0
$442$	−835.537	−0.0899150
$443$	−16096.5	−1.72634	−0.863170	−	0.504914i	$-0.831524\pi$
$443$	−16096.5	−1.72634	−0.863170	+	0.504914i	$0.831524\pi$
$444$	0	0
$445$	206.686	0.0220176
$446$	−1138.30	−0.120852
$447$	0	0
$448$	−448.000	−0.0472456
$449$	7808.68	0.820744	0.410372	−	0.911918i	$-0.365399\pi$
$449$	7808.68	0.820744	0.410372	+	0.911918i	$0.365399\pi$
$450$	0	0
$451$	−7122.80	−0.743680
$452$	1233.38	0.128348
$453$	0	0
$454$	12449.8	1.28700
$455$	−55.3611	−0.00570411
$456$	0	0
$457$	11624.9	1.18991	0.594956	−	0.803758i	$-0.297169\pi$
$457$	11624.9	1.18991	0.594956	+	0.803758i	$0.297169\pi$
$458$	−5872.31	−0.599116
$459$	0	0
$460$	−44.5968	−0.00452030
$461$	−7703.19	−0.778250	−0.389125	−	0.921185i	$-0.627223\pi$
$461$	−7703.19	−0.778250	−0.389125	+	0.921185i	$0.627223\pi$
$462$	0	0
$463$	12414.3	1.24609	0.623045	−	0.782186i	$-0.285895\pi$
$463$	12414.3	1.24609	0.623045	+	0.782186i	$0.285895\pi$
$464$	3824.18	0.382614
$465$	0	0
$466$	4170.54	0.414585
$467$	5384.78	0.533571	0.266786	−	0.963756i	$-0.414038\pi$
$467$	5384.78	0.533571	0.266786	+	0.963756i	$0.414038\pi$
$468$	0	0
$469$	−3288.39	−0.323760
$470$	267.714	0.0262739
$471$	0	0
$472$	−1184.00	−0.115462
$473$	13430.4	1.30556
$474$	0	0
$475$	−3158.29	−0.305079
$476$	476.000	0.0458349
$477$	0	0
$478$	8542.67	0.817432
$479$	−3136.40	−0.299177	−0.149589	−	0.988748i	$-0.547795\pi$
$479$	−3136.40	−0.299177	−0.149589	+	0.988748i	$0.547795\pi$
$480$	0	0
$481$	−7723.76	−0.732169
$482$	−288.634	−0.0272758
$483$	0	0
$484$	26.8064	0.00251750
$485$	−316.380	−0.0296208
$486$	0	0
$487$	−7712.95	−0.717674	−0.358837	−	0.933400i	$-0.616827\pi$
$487$	−7712.95	−0.717674	−0.358837	+	0.933400i	$0.616827\pi$
$488$	1305.65	0.121115
$489$	0	0
$490$	31.5389	0.00290772
$491$	−13651.9	−1.25479	−0.627395	−	0.778701i	$-0.715879\pi$
$491$	−13651.9	−1.25479	−0.627395	+	0.778701i	$0.715879\pi$
$492$	0	0
$493$	−4063.19	−0.371190
$494$	1242.85	0.113195
$495$	0	0
$496$	3595.03	0.325447
$497$	215.956	0.0194908
$498$	0	0
$499$	4197.83	0.376595	0.188297	−	0.982112i	$-0.439703\pi$
$499$	4197.83	0.376595	0.188297	+	0.982112i	$0.439703\pi$
$500$	−321.692	−0.0287730
$501$	0	0
$502$	4208.18	0.374144
$503$	15598.7	1.38272	0.691362	−	0.722508i	$-0.257011\pi$
$503$	15598.7	1.38272	0.691362	+	0.722508i	$0.257011\pi$
$504$	0	0
$505$	−403.679	−0.0355713
$506$	2534.16	0.222642
$507$	0	0
$508$	−4663.43	−0.407296
$509$	−3968.13	−0.345549	−0.172774	−	0.984961i	$-0.555273\pi$
$509$	−3968.13	−0.345549	−0.172774	+	0.984961i	$0.555273\pi$
$510$	0	0
$511$	−3582.31	−0.310121
$512$	512.000	0.0441942
$513$	0	0
$514$	5350.68	0.459161
$515$	18.9175	0.00161865
$516$	0	0
$517$	−15212.5	−1.29409
$518$	4400.18	0.373229
$519$	0	0
$520$	63.2698	0.00533570
$521$	−19228.6	−1.61693	−0.808463	−	0.588547i	$-0.799700\pi$
$521$	−19228.6	−1.61693	−0.808463	+	0.588547i	$0.799700\pi$
$522$	0	0
$523$	−19149.0	−1.60101	−0.800506	−	0.599325i	$-0.795436\pi$
$523$	−19149.0	−1.60101	−0.800506	+	0.599325i	$0.795436\pi$
$524$	4811.21	0.401104
$525$	0	0
$526$	332.794	0.0275865
$527$	−3819.72	−0.315730
$528$	0	0
$529$	−10966.8	−0.901358
$530$	188.701	0.0154654
$531$	0	0
$532$	−708.044	−0.0577023
$533$	4785.84	0.388926
$534$	0	0
$535$	417.463	0.0337355
$536$	3758.16	0.302850
$537$	0	0
$538$	−16105.2	−1.29060
$539$	−1792.16	−0.143216
$540$	0	0
$541$	3615.70	0.287341	0.143670	−	0.989626i	$-0.454110\pi$
$541$	3615.70	0.287341	0.143670	+	0.989626i	$0.454110\pi$
$542$	−3717.81	−0.294638
$543$	0	0
$544$	−544.000	−0.0428746
$545$	267.190	0.0210003
$546$	0	0
$547$	9915.19	0.775033	0.387516	−	0.921863i	$-0.373333\pi$
$547$	9915.19	0.775033	0.387516	+	0.921863i	$0.373333\pi$
$548$	10635.6	0.829071
$549$	0	0
$550$	9136.07	0.708297
$551$	6043.95	0.467297
$552$	0	0
$553$	−2737.72	−0.210524
$554$	6423.03	0.492579
$555$	0	0
$556$	3958.43	0.301933
$557$	15373.4	1.16947	0.584733	−	0.811226i	$-0.301199\pi$
$557$	15373.4	1.16947	0.584733	+	0.811226i	$0.301199\pi$
$558$	0	0
$559$	−9023.94	−0.682776
$560$	−36.0444	−0.00271992
$561$	0	0
$562$	−1727.85	−0.129689
$563$	−8808.97	−0.659421	−0.329710	−	0.944082i	$-0.606951\pi$
$563$	−8808.97	−0.659421	−0.329710	+	0.944082i	$0.606951\pi$
$564$	0	0
$565$	99.2333	0.00738899
$566$	17943.8	1.33257
$567$	0	0
$568$	−246.806	−0.0182320
$569$	−2974.86	−0.219179	−0.109589	−	0.993977i	$-0.534954\pi$
$569$	−2974.86	−0.219179	−0.109589	+	0.993977i	$0.534954\pi$
$570$	0	0
$571$	1089.80	0.0798720	0.0399360	−	0.999202i	$-0.487285\pi$
$571$	1089.80	0.0798720	0.0399360	+	0.999202i	$0.487285\pi$
$572$	−3595.23	−0.262804
$573$	0	0
$574$	−2726.46	−0.198258
$575$	4326.87	0.313814
$576$	0	0
$577$	−7188.66	−0.518661	−0.259331	−	0.965789i	$-0.583502\pi$
$577$	−7188.66	−0.518661	−0.259331	+	0.965789i	$0.583502\pi$
$578$	578.000	0.0415945
$579$	0	0
$580$	307.679	0.0220271
$581$	−6585.94	−0.470277
$582$	0	0
$583$	−10722.7	−0.761728
$584$	4094.07	0.290092
$585$	0	0
$586$	−3945.75	−0.278153
$587$	−9458.69	−0.665080	−0.332540	−	0.943089i	$-0.607906\pi$
$587$	−9458.69	−0.665080	−0.332540	+	0.943089i	$0.607906\pi$
$588$	0	0
$589$	5681.79	0.397477
$590$	−95.2603	−0.00664713
$591$	0	0
$592$	−5028.77	−0.349124
$593$	−14197.7	−0.983190	−0.491595	−	0.870824i	$-0.663586\pi$
$593$	−14197.7	−0.983190	−0.491595	+	0.870824i	$0.663586\pi$
$594$	0	0
$595$	38.2972	0.00263871
$596$	13189.6	0.906485
$597$	0	0
$598$	−1702.71	−0.116436
$599$	−16757.3	−1.14305	−0.571524	−	0.820586i	$-0.693647\pi$
$599$	−16757.3	−1.14305	−0.571524	+	0.820586i	$0.693647\pi$
$600$	0	0
$601$	13320.1	0.904059	0.452030	−	0.892003i	$-0.350700\pi$
$601$	13320.1	0.904059	0.452030	+	0.892003i	$0.350700\pi$
$602$	5140.88	0.348051
$603$	0	0
$604$	3115.46	0.209878
$605$	2.15674	0.000144932 0
$606$	0	0
$607$	−5381.21	−0.359830	−0.179915	−	0.983682i	$-0.557582\pi$
$607$	−5381.21	−0.359830	−0.179915	+	0.983682i	$0.557582\pi$
$608$	809.194	0.0539756
$609$	0	0
$610$	105.048	0.00697255
$611$	10221.3	0.676778
$612$	0	0
$613$	−15421.0	−1.01606	−0.508031	−	0.861339i	$-0.669627\pi$
$613$	−15421.0	−1.01606	−0.508031	+	0.861339i	$0.669627\pi$
$614$	13504.0	0.887583
$615$	0	0
$616$	2048.18	0.133967
$617$	13277.1	0.866314	0.433157	−	0.901319i	$-0.357400\pi$
$617$	13277.1	0.866314	0.433157	+	0.901319i	$0.357400\pi$
$618$	0	0
$619$	−14770.3	−0.959078	−0.479539	−	0.877521i	$-0.659196\pi$
$619$	−14770.3	−0.959078	−0.479539	+	0.877521i	$0.659196\pi$
$620$	289.243	0.0187359
$621$	0	0
$622$	−10745.5	−0.692692
$623$	−4495.61	−0.289105
$624$	0	0
$625$	15586.2	0.997515
$626$	15081.3	0.962893
$627$	0	0
$628$	−1757.97	−0.111705
$629$	5343.07	0.338700
$630$	0	0
$631$	29388.0	1.85407	0.927036	−	0.374972i	$-0.122348\pi$
$631$	29388.0	1.85407	0.927036	+	0.374972i	$0.122348\pi$
$632$	3128.82	0.196927
$633$	0	0
$634$	−7320.16	−0.458550
$635$	−375.203	−0.0234480
$636$	0	0
$637$	1204.16	0.0748986
$638$	−17483.5	−1.08492
$639$	0	0
$640$	41.1936	0.00254425
$641$	1199.83	0.0739318	0.0369659	−	0.999317i	$-0.488231\pi$
$641$	1199.83	0.0739318	0.0369659	+	0.999317i	$0.488231\pi$
$642$	0	0
$643$	−14000.2	−0.858654	−0.429327	−	0.903149i	$-0.641249\pi$
$643$	−14000.2	−0.858654	−0.429327	+	0.903149i	$0.641249\pi$
$644$	970.022	0.0593544
$645$	0	0
$646$	−859.768	−0.0523640
$647$	−1398.98	−0.0850071	−0.0425036	−	0.999096i	$-0.513533\pi$
$647$	−1398.98	−0.0850071	−0.0425036	+	0.999096i	$0.513533\pi$
$648$	0	0
$649$	5413.04	0.327397
$650$	−6138.56	−0.370422
$651$	0	0
$652$	7746.37	0.465293
$653$	−13756.7	−0.824414	−0.412207	−	0.911090i	$-0.635242\pi$
$653$	−13756.7	−0.824414	−0.412207	+	0.911090i	$0.635242\pi$
$654$	0	0
$655$	387.092	0.0230915
$656$	3115.96	0.185454
$657$	0	0
$658$	−5823.03	−0.344993
$659$	16023.1	0.947150	0.473575	−	0.880754i	$-0.342963\pi$
$659$	16023.1	0.947150	0.473575	+	0.880754i	$0.342963\pi$
$660$	0	0
$661$	16622.1	0.978102	0.489051	−	0.872255i	$-0.337343\pi$
$661$	16622.1	0.978102	0.489051	+	0.872255i	$0.337343\pi$
$662$	172.531	0.0101293
$663$	0	0
$664$	7526.79	0.439904
$665$	−56.9667	−0.00332191
$666$	0	0
$667$	−8280.22	−0.480677
$668$	3812.09	0.220800
$669$	0	0
$670$	302.367	0.0174350
$671$	−5969.19	−0.343425
$672$	0	0
$673$	13474.9	0.771798	0.385899	−	0.922541i	$-0.373891\pi$
$673$	13474.9	0.771798	0.385899	+	0.922541i	$0.373891\pi$
$674$	−8056.37	−0.460415
$675$	0	0
$676$	−6372.36	−0.362560
$677$	−13117.9	−0.744700	−0.372350	−	0.928092i	$-0.621448\pi$
$677$	−13117.9	−0.744700	−0.372350	+	0.928092i	$0.621448\pi$
$678$	0	0
$679$	6881.55	0.388939
$680$	−43.7683	−0.00246829
$681$	0	0
$682$	−16435.8	−0.922817
$683$	−17228.5	−0.965200	−0.482600	−	0.875841i	$-0.660308\pi$
$683$	−17228.5	−0.965200	−0.482600	+	0.875841i	$0.660308\pi$
$684$	0	0
$685$	855.702	0.0477295
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−5875.30	−0.325572
$689$	7204.60	0.398365
$690$	0	0
$691$	7359.60	0.405170	0.202585	−	0.979265i	$-0.435066\pi$
$691$	7359.60	0.405170	0.202585	+	0.979265i	$0.435066\pi$
$692$	349.671	0.0192088
$693$	0	0
$694$	8108.52	0.443509
$695$	318.481	0.0173822
$696$	0	0
$697$	−3310.70	−0.179917
$698$	−14225.9	−0.771430
$699$	0	0
$700$	3497.10	0.188826
$701$	−34022.9	−1.83313	−0.916566	−	0.399882i	$-0.869051\pi$
$701$	−34022.9	−1.83313	−0.916566	+	0.399882i	$0.869051\pi$
$702$	0	0
$703$	−7947.76	−0.426395
$704$	−2340.77	−0.125314
$705$	0	0
$706$	11402.4	0.607838
$707$	8780.40	0.467073
$708$	0	0
$709$	24086.4	1.27586	0.637929	−	0.770095i	$-0.279791\pi$
$709$	24086.4	1.27586	0.637929	+	0.770095i	$0.279791\pi$
$710$	−19.8571	−0.00104961
$711$	0	0
$712$	5137.83	0.270433
$713$	−7784.06	−0.408857
$714$	0	0
$715$	−289.259	−0.0151296
$716$	−18628.1	−0.972299
$717$	0	0
$718$	−5905.63	−0.306958
$719$	11813.1	0.612732	0.306366	−	0.951914i	$-0.400887\pi$
$719$	11813.1	0.612732	0.306366	+	0.951914i	$0.400887\pi$
$720$	0	0
$721$	−411.472	−0.0212539
$722$	−12439.1	−0.641185
$723$	0	0
$724$	1782.90	0.0915204
$725$	−29851.6	−1.52919
$726$	0	0
$727$	24829.9	1.26670	0.633348	−	0.773867i	$-0.281680\pi$
$727$	24829.9	1.26670	0.633348	+	0.773867i	$0.281680\pi$
$728$	−1376.18	−0.0700612
$729$	0	0
$730$	329.394	0.0167006
$731$	6242.50	0.315851
$732$	0	0
$733$	−3605.67	−0.181690	−0.0908448	−	0.995865i	$-0.528957\pi$
$733$	−3605.67	−0.181690	−0.0908448	+	0.995865i	$0.528957\pi$
$734$	−7709.43	−0.387684
$735$	0	0
$736$	−1108.60	−0.0555210
$737$	−17181.6	−0.858743
$738$	0	0
$739$	−31007.3	−1.54347	−0.771734	−	0.635945i	$-0.780610\pi$
$739$	−31007.3	−1.54347	−0.771734	+	0.635945i	$0.780610\pi$
$740$	−404.597	−0.0200990
$741$	0	0
$742$	−4104.42	−0.203070
$743$	−35263.7	−1.74119	−0.870593	−	0.492004i	$-0.836265\pi$
$743$	−35263.7	−1.74119	−0.870593	+	0.492004i	$0.836265\pi$
$744$	0	0
$745$	1061.18	0.0521862
$746$	1353.12	0.0664092
$747$	0	0
$748$	2487.07	0.121573
$749$	−9080.20	−0.442968
$750$	0	0
$751$	−21219.7	−1.03105	−0.515526	−	0.856874i	$-0.672403\pi$
$751$	−21219.7	−1.03105	−0.515526	+	0.856874i	$0.672403\pi$
$752$	6654.90	0.322712
$753$	0	0
$754$	11747.2	0.567385
$755$	250.659	0.0120827
$756$	0	0
$757$	−21688.5	−1.04132	−0.520661	−	0.853764i	$-0.674314\pi$
$757$	−21688.5	−1.04132	−0.520661	+	0.853764i	$0.674314\pi$
$758$	3285.82	0.157449
$759$	0	0
$760$	65.1048	0.00310737
$761$	−5023.28	−0.239282	−0.119641	−	0.992817i	$-0.538174\pi$
$761$	−5023.28	−0.239282	−0.119641	+	0.992817i	$0.538174\pi$
$762$	0	0
$763$	−5811.62	−0.275747
$764$	15035.1	0.711978
$765$	0	0
$766$	−9191.28	−0.433543
$767$	−3637.04	−0.171220
$768$	0	0
$769$	−37085.8	−1.73908	−0.869538	−	0.493867i	$-0.835583\pi$
$769$	−37085.8	−1.73908	−0.869538	+	0.493867i	$0.835583\pi$
$770$	164.789	0.00771244
$771$	0	0
$772$	−7132.95	−0.332540
$773$	19327.3	0.899295	0.449647	−	0.893206i	$-0.351550\pi$
$773$	19327.3	0.899295	0.449647	+	0.893206i	$0.351550\pi$
$774$	0	0
$775$	−28062.9	−1.30071
$776$	−7864.63	−0.363819
$777$	0	0
$778$	22832.7	1.05217
$779$	4924.63	0.226500
$780$	0	0
$781$	1128.36	0.0516975
$782$	1177.88	0.0538632
$783$	0	0
$784$	784.000	0.0357143
$785$	−141.440	−0.00643083
$786$	0	0
$787$	−24247.1	−1.09824	−0.549120	−	0.835743i	$-0.685037\pi$
$787$	−24247.1	−1.09824	−0.549120	+	0.835743i	$0.685037\pi$
$788$	−3917.57	−0.177104
$789$	0	0
$790$	251.733	0.0113371
$791$	−2158.42	−0.0970221
$792$	0	0
$793$	4010.72	0.179603
$794$	24391.8	1.09022
$795$	0	0
$796$	−13430.8	−0.598044
$797$	−37105.7	−1.64913	−0.824563	−	0.565771i	$-0.808579\pi$
$797$	−37105.7	−1.64913	−0.824563	+	0.565771i	$0.808579\pi$
$798$	0	0
$799$	−7070.83	−0.313076
$800$	−3996.69	−0.176630
$801$	0	0
$802$	1340.02	0.0589996
$803$	−18717.4	−0.822567
$804$	0	0
$805$	78.0444	0.00341703
$806$	11043.3	0.482610
$807$	0	0
$808$	−10034.7	−0.436907
$809$	38611.6	1.67801	0.839006	−	0.544123i	$-0.183137\pi$
$809$	38611.6	1.67801	0.839006	+	0.544123i	$0.183137\pi$
$810$	0	0
$811$	25379.0	1.09886	0.549431	−	0.835539i	$-0.314844\pi$
$811$	25379.0	1.09886	0.549431	+	0.835539i	$0.314844\pi$
$812$	−6692.31	−0.289229
$813$	0	0
$814$	22990.7	0.989954
$815$	623.244	0.0267869
$816$	0	0
$817$	−9285.65	−0.397630
$818$	24285.2	1.03804
$819$	0	0
$820$	250.698	0.0106765
$821$	8770.80	0.372842	0.186421	−	0.982470i	$-0.440311\pi$
$821$	8770.80	0.372842	0.186421	+	0.982470i	$0.440311\pi$
$822$	0	0
$823$	19292.1	0.817107	0.408553	−	0.912734i	$-0.366033\pi$
$823$	19292.1	0.817107	0.408553	+	0.912734i	$0.366033\pi$
$824$	470.254	0.0198812
$825$	0	0
$826$	2072.00	0.0872810
$827$	−4800.71	−0.201859	−0.100929	−	0.994894i	$-0.532182\pi$
$827$	−4800.71	−0.201859	−0.100929	+	0.994894i	$0.532182\pi$
$828$	0	0
$829$	−24515.2	−1.02708	−0.513540	−	0.858066i	$-0.671666\pi$
$829$	−24515.2	−1.02708	−0.513540	+	0.858066i	$0.671666\pi$
$830$	605.578	0.0253252
$831$	0	0
$832$	1572.77	0.0655362
$833$	−833.000	−0.0346479
$834$	0	0
$835$	306.707	0.0127114
$836$	−3699.49	−0.153050
$837$	0	0
$838$	19608.2	0.808299
$839$	9288.33	0.382204	0.191102	−	0.981570i	$-0.438794\pi$
$839$	9288.33	0.382204	0.191102	+	0.981570i	$0.438794\pi$
$840$	0	0
$841$	32737.3	1.34230
$842$	−25826.4	−1.05705
$843$	0	0
$844$	11211.3	0.457239
$845$	−512.696	−0.0208725
$846$	0	0
$847$	−46.9111	−0.00190305
$848$	4690.76	0.189955
$849$	0	0
$850$	4246.48	0.171356
$851$	10888.4	0.438603
$852$	0	0
$853$	−25831.5	−1.03688	−0.518438	−	0.855115i	$-0.673486\pi$
$853$	−25831.5	−1.03688	−0.518438	+	0.855115i	$0.673486\pi$
$854$	−2284.88	−0.0915540
$855$	0	0
$856$	10377.4	0.414359
$857$	−18969.5	−0.756110	−0.378055	−	0.925783i	$-0.623407\pi$
$857$	−18969.5	−0.756110	−0.378055	+	0.925783i	$0.623407\pi$
$858$	0	0
$859$	−46123.5	−1.83203	−0.916015	−	0.401145i	$-0.868612\pi$
$859$	−46123.5	−1.83203	−0.916015	+	0.401145i	$0.868612\pi$
$860$	−472.705	−0.0187431
$861$	0	0
$862$	28869.1	1.14070
$863$	12453.4	0.491216	0.245608	−	0.969369i	$-0.421012\pi$
$863$	12453.4	0.491216	0.245608	+	0.969369i	$0.421012\pi$
$864$	0	0
$865$	28.1333	0.00110585
$866$	−31286.3	−1.22766
$867$	0	0
$868$	−6291.30	−0.246015
$869$	−14304.4	−0.558394
$870$	0	0
$871$	11544.4	0.449101
$872$	6641.85	0.257938
$873$	0	0
$874$	−1752.09	−0.0678093
$875$	562.961	0.0217504
$876$	0	0
$877$	51198.4	1.97132	0.985660	−	0.168745i	$-0.0539714\pi$
$877$	51198.4	1.97132	0.985660	+	0.168745i	$0.0539714\pi$
$878$	−10119.8	−0.388983
$879$	0	0
$880$	−188.330	−0.00721433
$881$	−32011.5	−1.22417	−0.612085	−	0.790792i	$-0.709669\pi$
$881$	−32011.5	−1.22417	−0.612085	+	0.790792i	$0.709669\pi$
$882$	0	0
$883$	−15601.3	−0.594595	−0.297297	−	0.954785i	$-0.596085\pi$
$883$	−15601.3	−0.594595	−0.297297	+	0.954785i	$0.596085\pi$
$884$	−1671.07	−0.0635795
$885$	0	0
$886$	−32193.0	−1.22071
$887$	4233.82	0.160268	0.0801341	−	0.996784i	$-0.474465\pi$
$887$	4233.82	0.160268	0.0801341	+	0.996784i	$0.474465\pi$
$888$	0	0
$889$	8161.01	0.307887
$890$	413.371	0.0155688
$891$	0	0
$892$	−2276.60	−0.0854553
$893$	10517.8	0.394136
$894$	0	0
$895$	−1498.75	−0.0559752
$896$	−896.000	−0.0334077
$897$	0	0
$898$	15617.4	0.580354
$899$	53703.2	1.99233
$900$	0	0
$901$	−4983.93	−0.184283
$902$	−14245.6	−0.525861
$903$	0	0
$904$	2466.76	0.0907558
$905$	143.445	0.00526882
$906$	0	0
$907$	11642.5	0.426221	0.213111	−	0.977028i	$-0.431640\pi$
$907$	11642.5	0.426221	0.213111	+	0.977028i	$0.431640\pi$
$908$	24899.6	0.910047
$909$	0	0
$910$	−110.722	−0.00403341
$911$	−39616.4	−1.44078	−0.720389	−	0.693570i	$-0.756037\pi$
$911$	−39616.4	−1.44078	−0.720389	+	0.693570i	$0.756037\pi$
$912$	0	0
$913$	−34411.2	−1.24736
$914$	23249.8	0.841396
$915$	0	0
$916$	−11744.6	−0.423639
$917$	−8419.61	−0.303206
$918$	0	0
$919$	29003.4	1.04106	0.520530	−	0.853843i	$-0.325734\pi$
$919$	29003.4	1.04106	0.520530	+	0.853843i	$0.325734\pi$
$920$	−89.1936	−0.00319633
$921$	0	0
$922$	−15406.4	−0.550306
$923$	−758.146	−0.0270365
$924$	0	0
$925$	39254.7	1.39534
$926$	24828.5	0.881119
$927$	0	0
$928$	7648.36	0.270549
$929$	21155.8	0.747145	0.373573	−	0.927601i	$-0.378133\pi$
$929$	21155.8	0.747145	0.373573	+	0.927601i	$0.378133\pi$
$930$	0	0
$931$	1239.08	0.0436188
$932$	8341.08	0.293156
$933$	0	0
$934$	10769.6	0.377292
$935$	200.101	0.00699893
$936$	0	0
$937$	4392.11	0.153131	0.0765656	−	0.997065i	$-0.475605\pi$
$937$	4392.11	0.153131	0.0765656	+	0.997065i	$0.475605\pi$
$938$	−6576.77	−0.228933
$939$	0	0
$940$	535.429	0.0185785
$941$	−46042.0	−1.59503	−0.797517	−	0.603296i	$-0.793854\pi$
$941$	−46042.0	−1.59503	−0.797517	+	0.603296i	$0.793854\pi$
$942$	0	0
$943$	−6746.75	−0.232985
$944$	−2368.00	−0.0816439
$945$	0	0
$946$	26860.8	0.923172
$947$	−31088.7	−1.06679	−0.533393	−	0.845867i	$-0.679083\pi$
$947$	−31088.7	−1.06679	−0.533393	+	0.845867i	$0.679083\pi$
$948$	0	0
$949$	12576.3	0.430182
$950$	−6316.59	−0.215723
$951$	0	0
$952$	952.000	0.0324102
$953$	−10751.0	−0.365433	−0.182717	−	0.983166i	$-0.558489\pi$
$953$	−10751.0	−0.365433	−0.182717	+	0.983166i	$0.558489\pi$
$954$	0	0
$955$	1209.67	0.0409885
$956$	17085.3	0.578012
$957$	0	0
$958$	−6272.81	−0.211550
$959$	−18612.3	−0.626719
$960$	0	0
$961$	20694.3	0.694649
$962$	−15447.5	−0.517721
$963$	0	0
$964$	−577.268	−0.0192869
$965$	−573.891	−0.0191443
$966$	0	0
$967$	−31774.9	−1.05668	−0.528342	−	0.849032i	$-0.677186\pi$
$967$	−31774.9	−1.05668	−0.528342	+	0.849032i	$0.677186\pi$
$968$	53.6127	0.00178014
$969$	0	0
$970$	−632.760	−0.0209450
$971$	34696.9	1.14673	0.573365	−	0.819300i	$-0.305638\pi$
$971$	34696.9	1.14673	0.573365	+	0.819300i	$0.305638\pi$
$972$	0	0
$973$	−6927.25	−0.228240
$974$	−15425.9	−0.507472
$975$	0	0
$976$	2611.30	0.0856409
$977$	5014.34	0.164200	0.0820998	−	0.996624i	$-0.473837\pi$
$977$	5014.34	0.164200	0.0820998	+	0.996624i	$0.473837\pi$
$978$	0	0
$979$	−23489.3	−0.766824
$980$	63.0778	0.00205607
$981$	0	0
$982$	−27303.8	−0.887270
$983$	−19781.9	−0.641857	−0.320929	−	0.947103i	$-0.603995\pi$
$983$	−19781.9	−0.641857	−0.320929	+	0.947103i	$0.603995\pi$
$984$	0	0
$985$	−315.194	−0.0101958
$986$	−8126.38	−0.262471
$987$	0	0
$988$	2485.70	0.0800412
$989$	12721.4	0.409015
$990$	0	0
$991$	16865.3	0.540610	0.270305	−	0.962775i	$-0.412876\pi$
$991$	16865.3	0.540610	0.270305	+	0.962775i	$0.412876\pi$
$992$	7190.06	0.230126
$993$	0	0
$994$	431.911	0.0137821
$995$	−1080.59	−0.0344293
$996$	0	0
$997$	51696.8	1.64218	0.821091	−	0.570798i	$-0.193366\pi$
$997$	51696.8	1.64218	0.821091	+	0.570798i	$0.193366\pi$
$998$	8395.67	0.266293
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2142.4.a.l.1.2		2
3.2	odd	2		238.4.a.a.1.1	✓	2
12.11	even	2		1904.4.a.b.1.2		2
21.20	even	2		1666.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.4.a.a.1.1	✓	2	3.2	odd	2
1666.4.a.c.1.2		2	21.20	even	2
1904.4.a.b.1.2		2	12.11	even	2
2142.4.a.l.1.2		2	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:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2142.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +0.321825 q^{5} -7.00000 q^{7} +8.00000 q^{8} +0.643651 q^{10} -36.5746 q^{11} +24.5746 q^{13} -14.0000 q^{14} +16.0000 q^{16} -17.0000 q^{17} +25.2873 q^{19} +1.28730 q^{20} -73.1492 q^{22} -34.6437 q^{23} -124.896 q^{25} +49.1492 q^{26} -28.0000 q^{28} +239.011 q^{29} +224.689 q^{31} +32.0000 q^{32} -34.0000 q^{34} -2.25278 q^{35} -314.298 q^{37} +50.5746 q^{38} +2.57460 q^{40} +194.747 q^{41} -367.206 q^{43} -146.298 q^{44} -69.2873 q^{46} +415.931 q^{47} +49.0000 q^{49} -249.793 q^{50} +98.2984 q^{52} +293.173 q^{53} -11.7706 q^{55} -56.0000 q^{56} +478.022 q^{58} -148.000 q^{59} +163.206 q^{61} +449.379 q^{62} +64.0000 q^{64} +7.90873 q^{65} +469.769 q^{67} -68.0000 q^{68} -4.50556 q^{70} -30.8508 q^{71} +511.758 q^{73} -628.597 q^{74} +101.149 q^{76} +256.022 q^{77} +391.102 q^{79} +5.14921 q^{80} +389.494 q^{82} +940.848 q^{83} -5.47103 q^{85} -734.412 q^{86} -292.597 q^{88} +642.229 q^{89} -172.022 q^{91} -138.575 q^{92} +831.862 q^{94} +8.13810 q^{95} -983.079 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 9 q^{5} - 14 q^{7} + 16 q^{8} - 18 q^{10} + 4 q^{11} - 28 q^{13} - 28 q^{14} + 32 q^{16} - 34 q^{17} + 12 q^{19} - 36 q^{20} + 8 q^{22} - 50 q^{23} - 163 q^{25} - 56 q^{26} - 56 q^{28}+ \cdots + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 - 9 * q^5 - 14 * q^7 + 16 * q^8 - 18 * q^10 + 4 * q^11 - 28 * q^13 - 28 * q^14 + 32 * q^16 - 34 * q^17 + 12 * q^19 - 36 * q^20 + 8 * q^22 - 50 * q^23 - 163 * q^25 - 56 * q^26 - 56 * q^28 + 208 * q^29 + 189 * q^31 + 64 * q^32 - 68 * q^34 + 63 * q^35 - 320 * q^37 + 24 * q^38 - 72 * q^40 + 457 * q^41 - 69 * q^43 + 16 * q^44 - 100 * q^46 + 774 * q^47 + 98 * q^49 - 326 * q^50 - 112 * q^52 + 731 * q^53 - 390 * q^55 - 112 * q^56 + 416 * q^58 - 296 * q^59 - 339 * q^61 + 378 * q^62 + 128 * q^64 + 498 * q^65 + 467 * q^67 - 136 * q^68 + 126 * q^70 - 216 * q^71 + 821 * q^73 - 640 * q^74 + 48 * q^76 - 28 * q^77 + 30 * q^79 - 144 * q^80 + 914 * q^82 + 358 * q^83 + 153 * q^85 - 138 * q^86 + 32 * q^88 + 918 * q^89 + 196 * q^91 - 200 * q^92 + 1548 * q^94 + 132 * q^95 - 915 * q^97 + 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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