LMFDB - Embedded newform 2178.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +1.61803 q^{5} +14.7984 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +1.61803 q^{5} +14.7984 q^{7} -8.00000 q^{8} -3.23607 q^{10} -50.9230 q^{13} -29.5967 q^{14} +16.0000 q^{16} +60.3131 q^{17} -26.4377 q^{19} +6.47214 q^{20} +29.1672 q^{23} -122.382 q^{25} +101.846 q^{26} +59.1935 q^{28} +150.267 q^{29} +29.3789 q^{31} -32.0000 q^{32} -120.626 q^{34} +23.9443 q^{35} -224.825 q^{37} +52.8754 q^{38} -12.9443 q^{40} -119.726 q^{41} +213.974 q^{43} -58.3344 q^{46} +175.021 q^{47} -124.008 q^{49} +244.764 q^{50} -203.692 q^{52} +63.1784 q^{53} -118.387 q^{56} -300.535 q^{58} +262.028 q^{59} -708.969 q^{61} -58.7577 q^{62} +64.0000 q^{64} -82.3951 q^{65} -151.692 q^{67} +241.252 q^{68} -47.8885 q^{70} -1055.55 q^{71} -31.5197 q^{73} +449.649 q^{74} -105.751 q^{76} +65.1157 q^{79} +25.8885 q^{80} +239.453 q^{82} +230.566 q^{83} +97.5886 q^{85} -427.947 q^{86} +618.952 q^{89} -753.577 q^{91} +116.669 q^{92} -350.043 q^{94} -42.7771 q^{95} -1099.99 q^{97} +248.016 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + q^{5} + 5 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + q^5 + 5 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + q^{5} + 5 q^{7} - 16 q^{8} - 2 q^{10} - 37 q^{13} - 10 q^{14} + 32 q^{16} + 181 q^{17} - 73 q^{19} + 4 q^{20} + 112 q^{23} - 247 q^{25} + 74 q^{26} + 20 q^{28} - 55 q^{29} - 261 q^{31} - 64 q^{32} - 362 q^{34} + 30 q^{35} - 273 q^{37} + 146 q^{38} - 8 q^{40} + 49 q^{41} + 580 q^{43} - 224 q^{46} + 397 q^{47} - 371 q^{49} + 494 q^{50} - 148 q^{52} + 625 q^{53} - 40 q^{56} + 110 q^{58} - 149 q^{59} - 405 q^{61} + 522 q^{62} + 128 q^{64} - 91 q^{65} - 44 q^{67} + 724 q^{68} - 60 q^{70} - 1125 q^{71} + 51 q^{73} + 546 q^{74} - 292 q^{76} + 1331 q^{79} + 16 q^{80} - 98 q^{82} - 601 q^{83} + 23 q^{85} - 1160 q^{86} - 864 q^{89} - 890 q^{91} + 448 q^{92} - 794 q^{94} - 14 q^{95} - 713 q^{97} + 742 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + q^5 + 5 * q^7 - 16 * q^8 - 2 * q^10 - 37 * q^13 - 10 * q^14 + 32 * q^16 + 181 * q^17 - 73 * q^19 + 4 * q^20 + 112 * q^23 - 247 * q^25 + 74 * q^26 + 20 * q^28 - 55 * q^29 - 261 * q^31 - 64 * q^32 - 362 * q^34 + 30 * q^35 - 273 * q^37 + 146 * q^38 - 8 * q^40 + 49 * q^41 + 580 * q^43 - 224 * q^46 + 397 * q^47 - 371 * q^49 + 494 * q^50 - 148 * q^52 + 625 * q^53 - 40 * q^56 + 110 * q^58 - 149 * q^59 - 405 * q^61 + 522 * q^62 + 128 * q^64 - 91 * q^65 - 44 * q^67 + 724 * q^68 - 60 * q^70 - 1125 * q^71 + 51 * q^73 + 546 * q^74 - 292 * q^76 + 1331 * q^79 + 16 * q^80 - 98 * q^82 - 601 * q^83 + 23 * q^85 - 1160 * q^86 - 864 * q^89 - 890 * q^91 + 448 * q^92 - 794 * q^94 - 14 * q^95 - 713 * q^97 + 742 * q^98

Display 100 coefficients 10 coefficients

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
$5$	1.61803	0.144721	0.0723607	−	0.997379i	$-0.476947\pi$
$5$	1.61803	0.144721	0.0723607	+	0.997379i	$0.476947\pi$
$6$	0	0
$7$	14.7984	0.799037	0.399519	−	0.916725i	$-0.369177\pi$
$7$	14.7984	0.799037	0.399519	+	0.916725i	$0.369177\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−3.23607	−0.102333
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−50.9230	−1.08642	−0.543211	−	0.839596i	$-0.682792\pi$
$13$	−50.9230	−1.08642	−0.543211	+	0.839596i	$0.682792\pi$
$14$	−29.5967	−0.565005
$15$	0	0
$16$	16.0000	0.250000
$17$	60.3131	0.860475	0.430237	−	0.902716i	$-0.358430\pi$
$17$	60.3131	0.860475	0.430237	+	0.902716i	$0.358430\pi$
$18$	0	0
$19$	−26.4377	−0.319222	−0.159611	−	0.987180i	$-0.551024\pi$
$19$	−26.4377	−0.319222	−0.159611	+	0.987180i	$0.551024\pi$
$20$	6.47214	0.0723607
$21$	0	0
$22$	0	0
$23$	29.1672	0.264425	0.132213	−	0.991221i	$-0.457792\pi$
$23$	29.1672	0.264425	0.132213	+	0.991221i	$0.457792\pi$
$24$	0	0
$25$	−122.382	−0.979056
$26$	101.846	0.768217
$27$	0	0
$28$	59.1935	0.399519
$29$	150.267	0.962205	0.481103	−	0.876664i	$-0.340236\pi$
$29$	150.267	0.962205	0.481103	+	0.876664i	$0.340236\pi$
$30$	0	0
$31$	29.3789	0.170213	0.0851064	−	0.996372i	$-0.472877\pi$
$31$	29.3789	0.170213	0.0851064	+	0.996372i	$0.472877\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−120.626	−0.608448
$35$	23.9443	0.115638
$36$	0	0
$37$	−224.825	−0.998945	−0.499472	−	0.866330i	$-0.666473\pi$
$37$	−224.825	−0.998945	−0.499472	+	0.866330i	$0.666473\pi$
$38$	52.8754	0.225724
$39$	0	0
$40$	−12.9443	−0.0511667
$41$	−119.726	−0.456052	−0.228026	−	0.973655i	$-0.573227\pi$
$41$	−119.726	−0.456052	−0.228026	+	0.973655i	$0.573227\pi$
$42$	0	0
$43$	213.974	0.758853	0.379427	−	0.925222i	$-0.376121\pi$
$43$	213.974	0.758853	0.379427	+	0.925222i	$0.376121\pi$
$44$	0	0
$45$	0	0
$46$	−58.3344	−0.186977
$47$	175.021	0.543180	0.271590	−	0.962413i	$-0.412450\pi$
$47$	175.021	0.543180	0.271590	+	0.962413i	$0.412450\pi$
$48$	0	0
$49$	−124.008	−0.361540
$50$	244.764	0.692297
$51$	0	0
$52$	−203.692	−0.543211
$53$	63.1784	0.163740	0.0818700	−	0.996643i	$-0.473911\pi$
$53$	63.1784	0.163740	0.0818700	+	0.996643i	$0.473911\pi$
$54$	0	0
$55$	0	0
$56$	−118.387	−0.282502
$57$	0	0
$58$	−300.535	−0.680382
$59$	262.028	0.578189	0.289095	−	0.957301i	$-0.406646\pi$
$59$	262.028	0.578189	0.289095	+	0.957301i	$0.406646\pi$
$60$	0	0
$61$	−708.969	−1.48810	−0.744051	−	0.668123i	$-0.767098\pi$
$61$	−708.969	−1.48810	−0.744051	+	0.668123i	$0.767098\pi$
$62$	−58.7577	−0.120359
$63$	0	0
$64$	64.0000	0.125000
$65$	−82.3951	−0.157229
$66$	0	0
$67$	−151.692	−0.276599	−0.138299	−	0.990390i	$-0.544164\pi$
$67$	−151.692	−0.276599	−0.138299	+	0.990390i	$0.544164\pi$
$68$	241.252	0.430237
$69$	0	0
$70$	−47.8885	−0.0817682
$71$	−1055.55	−1.76438	−0.882191	−	0.470892i	$-0.843932\pi$
$71$	−1055.55	−1.76438	−0.882191	+	0.470892i	$0.843932\pi$
$72$	0	0
$73$	−31.5197	−0.0505357	−0.0252678	−	0.999681i	$-0.508044\pi$
$73$	−31.5197	−0.0505357	−0.0252678	+	0.999681i	$0.508044\pi$
$74$	449.649	0.706361
$75$	0	0
$76$	−105.751	−0.159611
$77$	0	0
$78$	0	0
$79$	65.1157	0.0927354	0.0463677	−	0.998924i	$-0.485235\pi$
$79$	65.1157	0.0927354	0.0463677	+	0.998924i	$0.485235\pi$
$80$	25.8885	0.0361803
$81$	0	0
$82$	239.453	0.322477
$83$	230.566	0.304915	0.152457	−	0.988310i	$-0.451281\pi$
$83$	230.566	0.304915	0.152457	+	0.988310i	$0.451281\pi$
$84$	0	0
$85$	97.5886	0.124529
$86$	−427.947	−0.536590
$87$	0	0
$88$	0	0
$89$	618.952	0.737177	0.368589	−	0.929593i	$-0.379841\pi$
$89$	618.952	0.737177	0.368589	+	0.929593i	$0.379841\pi$
$90$	0	0
$91$	−753.577	−0.868092
$92$	116.669	0.132213
$93$	0	0
$94$	−350.043	−0.384087
$95$	−42.7771	−0.0461983
$96$	0	0
$97$	−1099.99	−1.15142	−0.575708	−	0.817655i	$-0.695273\pi$
$97$	−1099.99	−1.15142	−0.575708	+	0.817655i	$0.695273\pi$
$98$	248.016	0.255647
$99$	0	0
$100$	−489.528	−0.489528
$101$	−1349.28	−1.32929	−0.664643	−	0.747161i	$-0.731416\pi$
$101$	−1349.28	−1.32929	−0.664643	+	0.747161i	$0.731416\pi$
$102$	0	0
$103$	−1484.72	−1.42033	−0.710166	−	0.704035i	$-0.751380\pi$
$103$	−1484.72	−1.42033	−0.710166	+	0.704035i	$0.751380\pi$
$104$	407.384	0.384108
$105$	0	0
$106$	−126.357	−0.115782
$107$	1762.19	1.59212	0.796062	−	0.605215i	$-0.206913\pi$
$107$	1762.19	1.59212	0.796062	+	0.605215i	$0.206913\pi$
$108$	0	0
$109$	688.087	0.604649	0.302325	−	0.953205i	$-0.402237\pi$
$109$	688.087	0.604649	0.302325	+	0.953205i	$0.402237\pi$
$110$	0	0
$111$	0	0
$112$	236.774	0.199759
$113$	−303.195	−0.252409	−0.126205	−	0.992004i	$-0.540280\pi$
$113$	−303.195	−0.252409	−0.126205	+	0.992004i	$0.540280\pi$
$114$	0	0
$115$	47.1935	0.0382680
$116$	601.070	0.481103
$117$	0	0
$118$	−524.056	−0.408842
$119$	892.536	0.687551
$120$	0	0
$121$	0	0
$122$	1417.94	1.05225
$123$	0	0
$124$	117.515	0.0851064
$125$	−400.272	−0.286412
$126$	0	0
$127$	2265.63	1.58301	0.791505	−	0.611162i	$-0.209298\pi$
$127$	2265.63	1.58301	0.791505	+	0.611162i	$0.209298\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	164.790	0.111177
$131$	−1457.48	−0.972068	−0.486034	−	0.873940i	$-0.661557\pi$
$131$	−1457.48	−0.972068	−0.486034	+	0.873940i	$0.661557\pi$
$132$	0	0
$133$	−391.235	−0.255070
$134$	303.384	0.195585
$135$	0	0
$136$	−482.505	−0.304224
$137$	196.856	0.122763	0.0613817	−	0.998114i	$-0.480449\pi$
$137$	196.856	0.122763	0.0613817	+	0.998114i	$0.480449\pi$
$138$	0	0
$139$	−39.6537	−0.0241970	−0.0120985	−	0.999927i	$-0.503851\pi$
$139$	−39.6537	−0.0241970	−0.0120985	+	0.999927i	$0.503851\pi$
$140$	95.7771	0.0578189
$141$	0	0
$142$	2111.11	1.24761
$143$	0	0
$144$	0	0
$145$	243.138	0.139252
$146$	63.0395	0.0357341
$147$	0	0
$148$	−899.299	−0.499472
$149$	2451.19	1.34771	0.673856	−	0.738863i	$-0.264637\pi$
$149$	2451.19	1.34771	0.673856	+	0.738863i	$0.264637\pi$
$150$	0	0
$151$	2763.52	1.48935	0.744677	−	0.667425i	$-0.232604\pi$
$151$	2763.52	1.48935	0.744677	+	0.667425i	$0.232604\pi$
$152$	211.502	0.112862
$153$	0	0
$154$	0	0
$155$	47.5360	0.0246334
$156$	0	0
$157$	−3568.04	−1.81376	−0.906881	−	0.421386i	$-0.861544\pi$
$157$	−3568.04	−1.81376	−0.906881	+	0.421386i	$0.861544\pi$
$158$	−130.231	−0.0655738
$159$	0	0
$160$	−51.7771	−0.0255834
$161$	431.627	0.211285
$162$	0	0
$163$	−70.9741	−0.0341051	−0.0170525	−	0.999855i	$-0.505428\pi$
$163$	−70.9741	−0.0341051	−0.0170525	+	0.999855i	$0.505428\pi$
$164$	−478.906	−0.228026
$165$	0	0
$166$	−461.132	−0.215607
$167$	−2742.82	−1.27093	−0.635466	−	0.772129i	$-0.719192\pi$
$167$	−2742.82	−1.27093	−0.635466	+	0.772129i	$0.719192\pi$
$168$	0	0
$169$	396.150	0.180314
$170$	−195.177	−0.0880554
$171$	0	0
$172$	855.895	0.379427
$173$	−1333.66	−0.586107	−0.293053	−	0.956096i	$-0.594671\pi$
$173$	−1333.66	−0.586107	−0.293053	+	0.956096i	$0.594671\pi$
$174$	0	0
$175$	−1811.05	−0.782302
$176$	0	0
$177$	0	0
$178$	−1237.90	−0.521263
$179$	−3241.11	−1.35336	−0.676682	−	0.736276i	$-0.736583\pi$
$179$	−3241.11	−1.35336	−0.676682	+	0.736276i	$0.736583\pi$
$180$	0	0
$181$	2660.23	1.09245	0.546225	−	0.837638i	$-0.316064\pi$
$181$	2660.23	1.09245	0.546225	+	0.837638i	$0.316064\pi$
$182$	1507.15	0.613834
$183$	0	0
$184$	−233.337	−0.0934884
$185$	−363.774	−0.144569
$186$	0	0
$187$	0	0
$188$	700.085	0.271590
$189$	0	0
$190$	85.5542	0.0326671
$191$	998.627	0.378315	0.189157	−	0.981947i	$-0.439424\pi$
$191$	998.627	0.378315	0.189157	+	0.981947i	$0.439424\pi$
$192$	0	0
$193$	3841.74	1.43282	0.716411	−	0.697678i	$-0.245783\pi$
$193$	3841.74	1.43282	0.716411	+	0.697678i	$0.245783\pi$
$194$	2199.99	0.814174
$195$	0	0
$196$	−496.033	−0.180770
$197$	3386.25	1.22467	0.612335	−	0.790598i	$-0.290230\pi$
$197$	3386.25	1.22467	0.612335	+	0.790598i	$0.290230\pi$
$198$	0	0
$199$	−4895.30	−1.74381	−0.871906	−	0.489674i	$-0.837116\pi$
$199$	−4895.30	−1.74381	−0.871906	+	0.489674i	$0.837116\pi$
$200$	979.056	0.346148
$201$	0	0
$202$	2698.55	0.939947
$203$	2223.71	0.768838
$204$	0	0
$205$	−193.721	−0.0660004
$206$	2969.45	1.00433
$207$	0	0
$208$	−814.768	−0.271606
$209$	0	0
$210$	0	0
$211$	−1469.43	−0.479430	−0.239715	−	0.970843i	$-0.577054\pi$
$211$	−1469.43	−0.479430	−0.239715	+	0.970843i	$0.577054\pi$
$212$	252.714	0.0818700
$213$	0	0
$214$	−3524.38	−1.12580
$215$	346.217	0.109822
$216$	0	0
$217$	434.759	0.136006
$218$	−1376.17	−0.427551
$219$	0	0
$220$	0	0
$221$	−3071.32	−0.934839
$222$	0	0
$223$	1745.47	0.524149	0.262074	−	0.965048i	$-0.415593\pi$
$223$	1745.47	0.524149	0.262074	+	0.965048i	$0.415593\pi$
$224$	−473.548	−0.141251
$225$	0	0
$226$	606.391	0.178480
$227$	−2363.87	−0.691169	−0.345585	−	0.938388i	$-0.612319\pi$
$227$	−2363.87	−0.691169	−0.345585	+	0.938388i	$0.612319\pi$
$228$	0	0
$229$	−3093.75	−0.892754	−0.446377	−	0.894845i	$-0.647286\pi$
$229$	−3093.75	−0.892754	−0.446377	+	0.894845i	$0.647286\pi$
$230$	−94.3870	−0.0270595
$231$	0	0
$232$	−1202.14	−0.340191
$233$	−2700.06	−0.759170	−0.379585	−	0.925157i	$-0.623933\pi$
$233$	−2700.06	−0.759170	−0.379585	+	0.925157i	$0.623933\pi$
$234$	0	0
$235$	283.190	0.0786098
$236$	1048.11	0.289095
$237$	0	0
$238$	−1785.07	−0.486172
$239$	−357.923	−0.0968708	−0.0484354	−	0.998826i	$-0.515423\pi$
$239$	−357.923	−0.0968708	−0.0484354	+	0.998826i	$0.515423\pi$
$240$	0	0
$241$	4290.72	1.14685	0.573423	−	0.819260i	$-0.305615\pi$
$241$	4290.72	1.14685	0.573423	+	0.819260i	$0.305615\pi$
$242$	0	0
$243$	0	0
$244$	−2835.88	−0.744051
$245$	−200.649	−0.0523225
$246$	0	0
$247$	1346.29	0.346810
$248$	−235.031	−0.0601793
$249$	0	0
$250$	800.545	0.202524
$251$	5392.60	1.35609	0.678044	−	0.735022i	$-0.262828\pi$
$251$	5392.60	1.35609	0.678044	+	0.735022i	$0.262828\pi$
$252$	0	0
$253$	0	0
$254$	−4531.26	−1.11936
$255$	0	0
$256$	256.000	0.0625000
$257$	−4077.31	−0.989632	−0.494816	−	0.868998i	$-0.664764\pi$
$257$	−4077.31	−0.989632	−0.494816	+	0.868998i	$0.664764\pi$
$258$	0	0
$259$	−3327.04	−0.798194
$260$	−329.580	−0.0786143
$261$	0	0
$262$	2914.97	0.687356
$263$	3256.10	0.763421	0.381711	−	0.924282i	$-0.375335\pi$
$263$	3256.10	0.763421	0.381711	+	0.924282i	$0.375335\pi$
$264$	0	0
$265$	102.225	0.0236967
$266$	782.470	0.180362
$267$	0	0
$268$	−606.768	−0.138299
$269$	−2060.04	−0.466925	−0.233463	−	0.972366i	$-0.575006\pi$
$269$	−2060.04	−0.466925	−0.233463	+	0.972366i	$0.575006\pi$
$270$	0	0
$271$	−5349.01	−1.19900	−0.599500	−	0.800375i	$-0.704634\pi$
$271$	−5349.01	−1.19900	−0.599500	+	0.800375i	$0.704634\pi$
$272$	965.009	0.215119
$273$	0	0
$274$	−393.713	−0.0868068
$275$	0	0
$276$	0	0
$277$	−8404.77	−1.82308	−0.911541	−	0.411209i	$-0.865107\pi$
$277$	−8404.77	−1.82308	−0.911541	+	0.411209i	$0.865107\pi$
$278$	79.3073	0.0171098
$279$	0	0
$280$	−191.554	−0.0408841
$281$	−7035.59	−1.49362	−0.746812	−	0.665035i	$-0.768416\pi$
$281$	−7035.59	−1.49362	−0.746812	+	0.665035i	$0.768416\pi$
$282$	0	0
$283$	7553.62	1.58663	0.793314	−	0.608812i	$-0.208354\pi$
$283$	7553.62	1.58663	0.793314	+	0.608812i	$0.208354\pi$
$284$	−4222.21	−0.882191
$285$	0	0
$286$	0	0
$287$	−1771.76	−0.364402
$288$	0	0
$289$	−1275.33	−0.259583
$290$	−486.276	−0.0984658
$291$	0	0
$292$	−126.079	−0.0252678
$293$	−3263.90	−0.650782	−0.325391	−	0.945580i	$-0.605496\pi$
$293$	−3263.90	−0.650782	−0.325391	+	0.945580i	$0.605496\pi$
$294$	0	0
$295$	423.971	0.0836763
$296$	1798.60	0.353180
$297$	0	0
$298$	−4902.37	−0.952976
$299$	−1485.28	−0.287277
$300$	0	0
$301$	3166.46	0.606352
$302$	−5527.05	−1.05313
$303$	0	0
$304$	−423.003	−0.0798056
$305$	−1147.14	−0.215360
$306$	0	0
$307$	−5505.74	−1.02355	−0.511774	−	0.859120i	$-0.671012\pi$
$307$	−5505.74	−1.02355	−0.511774	+	0.859120i	$0.671012\pi$
$308$	0	0
$309$	0	0
$310$	−95.0720	−0.0174185
$311$	1210.19	0.220654	0.110327	−	0.993895i	$-0.464810\pi$
$311$	1210.19	0.220654	0.110327	+	0.993895i	$0.464810\pi$
$312$	0	0
$313$	−4903.30	−0.885467	−0.442733	−	0.896653i	$-0.645991\pi$
$313$	−4903.30	−0.885467	−0.442733	+	0.896653i	$0.645991\pi$
$314$	7136.08	1.28252
$315$	0	0
$316$	260.463	0.0463677
$317$	−7689.15	−1.36235	−0.681176	−	0.732120i	$-0.738531\pi$
$317$	−7689.15	−1.36235	−0.681176	+	0.732120i	$0.738531\pi$
$318$	0	0
$319$	0	0
$320$	103.554	0.0180902
$321$	0	0
$322$	−863.254	−0.149401
$323$	−1594.54	−0.274683
$324$	0	0
$325$	6232.06	1.06367
$326$	141.948	0.0241159
$327$	0	0
$328$	957.811	0.161239
$329$	2590.03	0.434021
$330$	0	0
$331$	3085.35	0.512344	0.256172	−	0.966631i	$-0.417539\pi$
$331$	3085.35	0.512344	0.256172	+	0.966631i	$0.417539\pi$
$332$	922.265	0.152457
$333$	0	0
$334$	5485.63	0.898684
$335$	−245.443	−0.0400298
$336$	0	0
$337$	−5776.57	−0.933738	−0.466869	−	0.884327i	$-0.654618\pi$
$337$	−5776.57	−0.933738	−0.466869	+	0.884327i	$0.654618\pi$
$338$	−792.301	−0.127501
$339$	0	0
$340$	390.354	0.0622645
$341$	0	0
$342$	0	0
$343$	−6910.96	−1.08792
$344$	−1711.79	−0.268295
$345$	0	0
$346$	2667.33	0.414440
$347$	−909.285	−0.140671	−0.0703357	−	0.997523i	$-0.522407\pi$
$347$	−909.285	−0.140671	−0.0703357	+	0.997523i	$0.522407\pi$
$348$	0	0
$349$	11434.2	1.75375	0.876874	−	0.480720i	$-0.159625\pi$
$349$	11434.2	1.75375	0.876874	+	0.480720i	$0.159625\pi$
$350$	3622.11	0.553171
$351$	0	0
$352$	0	0
$353$	−2773.55	−0.418190	−0.209095	−	0.977895i	$-0.567052\pi$
$353$	−2773.55	−0.418190	−0.209095	+	0.977895i	$0.567052\pi$
$354$	0	0
$355$	−1707.92	−0.255344
$356$	2475.81	0.368589
$357$	0	0
$358$	6482.22	0.956973
$359$	2522.86	0.370895	0.185448	−	0.982654i	$-0.440627\pi$
$359$	2522.86	0.370895	0.185448	+	0.982654i	$0.440627\pi$
$360$	0	0
$361$	−6160.05	−0.898097
$362$	−5320.47	−0.772479
$363$	0	0
$364$	−3014.31	−0.434046
$365$	−51.0000	−0.00731359
$366$	0	0
$367$	5254.27	0.747331	0.373666	−	0.927563i	$-0.378101\pi$
$367$	5254.27	0.747331	0.373666	+	0.927563i	$0.378101\pi$
$368$	466.675	0.0661063
$369$	0	0
$370$	727.548	0.102225
$371$	934.938	0.130834
$372$	0	0
$373$	316.370	0.0439169	0.0219585	−	0.999759i	$-0.493010\pi$
$373$	316.370	0.0439169	0.0219585	+	0.999759i	$0.493010\pi$
$374$	0	0
$375$	0	0
$376$	−1400.17	−0.192043
$377$	−7652.06	−1.04536
$378$	0	0
$379$	5257.04	0.712497	0.356248	−	0.934391i	$-0.384056\pi$
$379$	5257.04	0.712497	0.356248	+	0.934391i	$0.384056\pi$
$380$	−171.108	−0.0230991
$381$	0	0
$382$	−1997.25	−0.267509
$383$	1424.22	0.190011	0.0950057	−	0.995477i	$-0.469713\pi$
$383$	1424.22	0.190011	0.0950057	+	0.995477i	$0.469713\pi$
$384$	0	0
$385$	0	0
$386$	−7683.49	−1.01316
$387$	0	0
$388$	−4399.97	−0.575708
$389$	1970.12	0.256784	0.128392	−	0.991724i	$-0.459018\pi$
$389$	1970.12	0.256784	0.128392	+	0.991724i	$0.459018\pi$
$390$	0	0
$391$	1759.16	0.227531
$392$	992.065	0.127824
$393$	0	0
$394$	−6772.49	−0.865973
$395$	105.359	0.0134208
$396$	0	0
$397$	−4148.62	−0.524467	−0.262233	−	0.965004i	$-0.584459\pi$
$397$	−4148.62	−0.524467	−0.262233	+	0.965004i	$0.584459\pi$
$398$	9790.60	1.23306
$399$	0	0
$400$	−1958.11	−0.244764
$401$	7710.20	0.960171	0.480086	−	0.877222i	$-0.340606\pi$
$401$	7710.20	0.960171	0.480086	+	0.877222i	$0.340606\pi$
$402$	0	0
$403$	−1496.06	−0.184923
$404$	−5397.10	−0.664643
$405$	0	0
$406$	−4447.43	−0.543650
$407$	0	0
$408$	0	0
$409$	−3495.64	−0.422612	−0.211306	−	0.977420i	$-0.567772\pi$
$409$	−3495.64	−0.422612	−0.211306	+	0.977420i	$0.567772\pi$
$410$	387.443	0.0466693
$411$	0	0
$412$	−5938.89	−0.710166
$413$	3877.59	0.461995
$414$	0	0
$415$	373.064	0.0441277
$416$	1629.54	0.192054
$417$	0	0
$418$	0	0
$419$	−6493.78	−0.757140	−0.378570	−	0.925573i	$-0.623584\pi$
$419$	−6493.78	−0.757140	−0.378570	+	0.925573i	$0.623584\pi$
$420$	0	0
$421$	−9961.59	−1.15320	−0.576601	−	0.817026i	$-0.695621\pi$
$421$	−9961.59	−1.15320	−0.576601	+	0.817026i	$0.695621\pi$
$422$	2938.86	0.339008
$423$	0	0
$424$	−505.427	−0.0578909
$425$	−7381.23	−0.842453
$426$	0	0
$427$	−10491.6	−1.18905
$428$	7048.76	0.796062
$429$	0	0
$430$	−692.433	−0.0776561
$431$	−5496.82	−0.614321	−0.307161	−	0.951658i	$-0.599379\pi$
$431$	−5496.82	−0.614321	−0.307161	+	0.951658i	$0.599379\pi$
$432$	0	0
$433$	98.3315	0.0109134	0.00545671	−	0.999985i	$-0.498263\pi$
$433$	98.3315	0.0109134	0.00545671	+	0.999985i	$0.498263\pi$
$434$	−869.519	−0.0961710
$435$	0	0
$436$	2752.35	0.302325
$437$	−771.113	−0.0844104
$438$	0	0
$439$	−1951.20	−0.212131	−0.106065	−	0.994359i	$-0.533825\pi$
$439$	−1951.20	−0.212131	−0.106065	+	0.994359i	$0.533825\pi$
$440$	0	0
$441$	0	0
$442$	6142.64	0.661031
$443$	−9438.69	−1.01229	−0.506146	−	0.862448i	$-0.668930\pi$
$443$	−9438.69	−1.01229	−0.506146	+	0.862448i	$0.668930\pi$
$444$	0	0
$445$	1001.49	0.106685
$446$	−3490.94	−0.370629
$447$	0	0
$448$	947.096	0.0998796
$449$	−12878.8	−1.35365	−0.676825	−	0.736144i	$-0.736645\pi$
$449$	−12878.8	−1.35365	−0.676825	+	0.736144i	$0.736645\pi$
$450$	0	0
$451$	0	0
$452$	−1212.78	−0.126205
$453$	0	0
$454$	4727.74	0.488730
$455$	−1219.31	−0.125631
$456$	0	0
$457$	−7757.62	−0.794061	−0.397031	−	0.917805i	$-0.629959\pi$
$457$	−7757.62	−0.794061	−0.397031	+	0.917805i	$0.629959\pi$
$458$	6187.50	0.631272
$459$	0	0
$460$	188.774	0.0191340
$461$	−13529.3	−1.36686	−0.683432	−	0.730014i	$-0.739513\pi$
$461$	−13529.3	−1.36686	−0.683432	+	0.730014i	$0.739513\pi$
$462$	0	0
$463$	−14558.3	−1.46130	−0.730651	−	0.682752i	$-0.760783\pi$
$463$	−14558.3	−1.46130	−0.730651	+	0.682752i	$0.760783\pi$
$464$	2404.28	0.240551
$465$	0	0
$466$	5400.12	0.536815
$467$	4633.16	0.459094	0.229547	−	0.973298i	$-0.426275\pi$
$467$	4633.16	0.459094	0.229547	+	0.973298i	$0.426275\pi$
$468$	0	0
$469$	−2244.79	−0.221013
$470$	−566.381	−0.0555855
$471$	0	0
$472$	−2096.23	−0.204421
$473$	0	0
$474$	0	0
$475$	3235.50	0.312536
$476$	3570.14	0.343776
$477$	0	0
$478$	715.846	0.0684980
$479$	−19601.1	−1.86972	−0.934860	−	0.355018i	$-0.884475\pi$
$479$	−19601.1	−1.86972	−0.934860	+	0.355018i	$0.884475\pi$
$480$	0	0
$481$	11448.7	1.08528
$482$	−8581.45	−0.810942
$483$	0	0
$484$	0	0
$485$	−1779.83	−0.166634
$486$	0	0
$487$	−3675.21	−0.341971	−0.170985	−	0.985274i	$-0.554695\pi$
$487$	−3675.21	−0.341971	−0.170985	+	0.985274i	$0.554695\pi$
$488$	5671.76	0.526124
$489$	0	0
$490$	401.299	0.0369976
$491$	−20571.1	−1.89075	−0.945377	−	0.325980i	$-0.894306\pi$
$491$	−20571.1	−1.89075	−0.945377	+	0.325980i	$0.894306\pi$
$492$	0	0
$493$	9063.09	0.827953
$494$	−2692.57	−0.245232
$495$	0	0
$496$	470.062	0.0425532
$497$	−15620.5	−1.40981
$498$	0	0
$499$	968.806	0.0869133	0.0434566	−	0.999055i	$-0.486163\pi$
$499$	968.806	0.0869133	0.0434566	+	0.999055i	$0.486163\pi$
$500$	−1601.09	−0.143206
$501$	0	0
$502$	−10785.2	−0.958899
$503$	12912.5	1.14461	0.572307	−	0.820040i	$-0.306049\pi$
$503$	12912.5	1.14461	0.572307	+	0.820040i	$0.306049\pi$
$504$	0	0
$505$	−2183.17	−0.192376
$506$	0	0
$507$	0	0
$508$	9062.53	0.791505
$509$	2401.08	0.209088	0.104544	−	0.994520i	$-0.466662\pi$
$509$	2401.08	0.209088	0.104544	+	0.994520i	$0.466662\pi$
$510$	0	0
$511$	−466.441	−0.0403799
$512$	−512.000	−0.0441942
$513$	0	0
$514$	8154.61	0.699775
$515$	−2402.33	−0.205552
$516$	0	0
$517$	0	0
$518$	6654.08	0.564408
$519$	0	0
$520$	659.161	0.0555887
$521$	−14283.4	−1.20109	−0.600544	−	0.799591i	$-0.705049\pi$
$521$	−14283.4	−1.20109	−0.600544	+	0.799591i	$0.705049\pi$
$522$	0	0
$523$	−9266.32	−0.774737	−0.387369	−	0.921925i	$-0.626616\pi$
$523$	−9266.32	−0.774737	−0.387369	+	0.921925i	$0.626616\pi$
$524$	−5829.93	−0.486034
$525$	0	0
$526$	−6512.20	−0.539820
$527$	1771.93	0.146464
$528$	0	0
$529$	−11316.3	−0.930079
$530$	−204.450	−0.0167561
$531$	0	0
$532$	−1564.94	−0.127535
$533$	6096.82	0.495465
$534$	0	0
$535$	2851.28	0.230414
$536$	1213.54	0.0977924
$537$	0	0
$538$	4120.08	0.330166
$539$	0	0
$540$	0	0
$541$	5268.19	0.418664	0.209332	−	0.977845i	$-0.432871\pi$
$541$	5268.19	0.418664	0.209332	+	0.977845i	$0.432871\pi$
$542$	10698.0	0.847821
$543$	0	0
$544$	−1930.02	−0.152112
$545$	1113.35	0.0875056
$546$	0	0
$547$	−19377.9	−1.51470	−0.757348	−	0.653012i	$-0.773505\pi$
$547$	−19377.9	−1.51470	−0.757348	+	0.653012i	$0.773505\pi$
$548$	787.426	0.0613817
$549$	0	0
$550$	0	0
$551$	−3972.72	−0.307157
$552$	0	0
$553$	963.607	0.0740990
$554$	16809.5	1.28911
$555$	0	0
$556$	−158.615	−0.0120985
$557$	5309.74	0.403915	0.201958	−	0.979394i	$-0.435270\pi$
$557$	5309.74	0.403915	0.201958	+	0.979394i	$0.435270\pi$
$558$	0	0
$559$	−10896.2	−0.824435
$560$	383.108	0.0289094
$561$	0	0
$562$	14071.2	1.05615
$563$	4494.72	0.336465	0.168232	−	0.985747i	$-0.446194\pi$
$563$	4494.72	0.336465	0.168232	+	0.985747i	$0.446194\pi$
$564$	0	0
$565$	−490.580	−0.0365290
$566$	−15107.2	−1.12192
$567$	0	0
$568$	8444.42	0.623803
$569$	22234.4	1.63816	0.819082	−	0.573676i	$-0.194483\pi$
$569$	22234.4	1.63816	0.819082	+	0.573676i	$0.194483\pi$
$570$	0	0
$571$	2370.18	0.173711	0.0868554	−	0.996221i	$-0.472318\pi$
$571$	2370.18	0.173711	0.0868554	+	0.996221i	$0.472318\pi$
$572$	0	0
$573$	0	0
$574$	3543.51	0.257671
$575$	−3569.54	−0.258887
$576$	0	0
$577$	−13848.0	−0.999136	−0.499568	−	0.866275i	$-0.666508\pi$
$577$	−13848.0	−0.999136	−0.499568	+	0.866275i	$0.666508\pi$
$578$	2550.66	0.183553
$579$	0	0
$580$	972.551	0.0696258
$581$	3412.00	0.243638
$582$	0	0
$583$	0	0
$584$	252.158	0.0178671
$585$	0	0
$586$	6527.80	0.460172
$587$	19878.7	1.39776	0.698879	−	0.715240i	$-0.253683\pi$
$587$	19878.7	1.39776	0.698879	+	0.715240i	$0.253683\pi$
$588$	0	0
$589$	−776.709	−0.0543357
$590$	−847.941	−0.0591681
$591$	0	0
$592$	−3597.19	−0.249736
$593$	−23426.3	−1.62226	−0.811132	−	0.584863i	$-0.801148\pi$
$593$	−23426.3	−1.62226	−0.811132	+	0.584863i	$0.801148\pi$
$594$	0	0
$595$	1444.15	0.0995034
$596$	9804.74	0.673856
$597$	0	0
$598$	2970.56	0.203136
$599$	884.679	0.0603456	0.0301728	−	0.999545i	$-0.490394\pi$
$599$	884.679	0.0603456	0.0301728	+	0.999545i	$0.490394\pi$
$600$	0	0
$601$	−6902.37	−0.468475	−0.234238	−	0.972179i	$-0.575259\pi$
$601$	−6902.37	−0.468475	−0.234238	+	0.972179i	$0.575259\pi$
$602$	−6332.93	−0.428755
$603$	0	0
$604$	11054.1	0.744677
$605$	0	0
$606$	0	0
$607$	−18984.5	−1.26945	−0.634726	−	0.772738i	$-0.718887\pi$
$607$	−18984.5	−1.26945	−0.634726	+	0.772738i	$0.718887\pi$
$608$	846.006	0.0564311
$609$	0	0
$610$	2294.27	0.152283
$611$	−8912.61	−0.590124
$612$	0	0
$613$	−18129.4	−1.19452	−0.597261	−	0.802047i	$-0.703744\pi$
$613$	−18129.4	−1.19452	−0.597261	+	0.802047i	$0.703744\pi$
$614$	11011.5	0.723758
$615$	0	0
$616$	0	0
$617$	15257.7	0.995549	0.497774	−	0.867307i	$-0.334151\pi$
$617$	15257.7	0.995549	0.497774	+	0.867307i	$0.334151\pi$
$618$	0	0
$619$	−7610.30	−0.494158	−0.247079	−	0.968995i	$-0.579471\pi$
$619$	−7610.30	−0.494158	−0.247079	+	0.968995i	$0.579471\pi$
$620$	190.144	0.0123167
$621$	0	0
$622$	−2420.37	−0.156026
$623$	9159.48	0.589032
$624$	0	0
$625$	14650.1	0.937606
$626$	9806.61	0.626120
$627$	0	0
$628$	−14272.2	−0.906881
$629$	−13559.9	−0.859567
$630$	0	0
$631$	−26065.6	−1.64446	−0.822231	−	0.569154i	$-0.807271\pi$
$631$	−26065.6	−1.64446	−0.822231	+	0.569154i	$0.807271\pi$
$632$	−520.926	−0.0327869
$633$	0	0
$634$	15378.3	0.963329
$635$	3665.87	0.229095
$636$	0	0
$637$	6314.86	0.392785
$638$	0	0
$639$	0	0
$640$	−207.108	−0.0127917
$641$	22980.0	1.41600	0.708000	−	0.706212i	$-0.249598\pi$
$641$	22980.0	1.41600	0.708000	+	0.706212i	$0.249598\pi$
$642$	0	0
$643$	6551.27	0.401799	0.200900	−	0.979612i	$-0.435614\pi$
$643$	6551.27	0.401799	0.200900	+	0.979612i	$0.435614\pi$
$644$	1726.51	0.105643
$645$	0	0
$646$	3189.08	0.194230
$647$	6281.33	0.381676	0.190838	−	0.981622i	$-0.438879\pi$
$647$	6281.33	0.381676	0.190838	+	0.981622i	$0.438879\pi$
$648$	0	0
$649$	0	0
$650$	−12464.1	−0.752127
$651$	0	0
$652$	−283.897	−0.0170525
$653$	−1178.52	−0.0706266	−0.0353133	−	0.999376i	$-0.511243\pi$
$653$	−1178.52	−0.0706266	−0.0353133	+	0.999376i	$0.511243\pi$
$654$	0	0
$655$	−2358.26	−0.140679
$656$	−1915.62	−0.114013
$657$	0	0
$658$	−5180.06	−0.306899
$659$	15421.0	0.911561	0.455781	−	0.890092i	$-0.349360\pi$
$659$	15421.0	0.911561	0.455781	+	0.890092i	$0.349360\pi$
$660$	0	0
$661$	−9587.68	−0.564172	−0.282086	−	0.959389i	$-0.591026\pi$
$661$	−9587.68	−0.564172	−0.282086	+	0.959389i	$0.591026\pi$
$662$	−6170.69	−0.362282
$663$	0	0
$664$	−1844.53	−0.107804
$665$	−633.031	−0.0369141
$666$	0	0
$667$	4382.88	0.254431
$668$	−10971.3	−0.635466
$669$	0	0
$670$	490.885	0.0283053
$671$	0	0
$672$	0	0
$673$	16151.6	0.925108	0.462554	−	0.886591i	$-0.346933\pi$
$673$	16151.6	0.925108	0.462554	+	0.886591i	$0.346933\pi$
$674$	11553.1	0.660252
$675$	0	0
$676$	1584.60	0.0901571
$677$	8026.53	0.455664	0.227832	−	0.973700i	$-0.426836\pi$
$677$	8026.53	0.455664	0.227832	+	0.973700i	$0.426836\pi$
$678$	0	0
$679$	−16278.1	−0.920024
$680$	−780.709	−0.0440277
$681$	0	0
$682$	0	0
$683$	17912.7	1.00353	0.501765	−	0.865004i	$-0.332684\pi$
$683$	17912.7	1.00353	0.501765	+	0.865004i	$0.332684\pi$
$684$	0	0
$685$	318.520	0.0177665
$686$	13821.9	0.769276
$687$	0	0
$688$	3423.58	0.189713
$689$	−3217.23	−0.177891
$690$	0	0
$691$	27493.4	1.51360	0.756800	−	0.653646i	$-0.226762\pi$
$691$	27493.4	1.51360	0.756800	+	0.653646i	$0.226762\pi$
$692$	−5334.65	−0.293053
$693$	0	0
$694$	1818.57	0.0994698
$695$	−64.1610	−0.00350182
$696$	0	0
$697$	−7221.07	−0.392421
$698$	−22868.4	−1.24009
$699$	0	0
$700$	−7244.22	−0.391151
$701$	−2238.33	−0.120600	−0.0603000	−	0.998180i	$-0.519206\pi$
$701$	−2238.33	−0.120600	−0.0603000	+	0.998180i	$0.519206\pi$
$702$	0	0
$703$	5943.85	0.318885
$704$	0	0
$705$	0	0
$706$	5547.09	0.295705
$707$	−19967.1	−1.06215
$708$	0	0
$709$	24224.5	1.28317	0.641586	−	0.767051i	$-0.278277\pi$
$709$	24224.5	1.28317	0.641586	+	0.767051i	$0.278277\pi$
$710$	3415.84	0.180555
$711$	0	0
$712$	−4951.62	−0.260632
$713$	856.899	0.0450086
$714$	0	0
$715$	0	0
$716$	−12964.4	−0.676682
$717$	0	0
$718$	−5045.72	−0.262262
$719$	−9744.02	−0.505411	−0.252705	−	0.967543i	$-0.581320\pi$
$719$	−9744.02	−0.505411	−0.252705	+	0.967543i	$0.581320\pi$
$720$	0	0
$721$	−21971.5	−1.13490
$722$	12320.1	0.635051
$723$	0	0
$724$	10640.9	0.546225
$725$	−18390.0	−0.942053
$726$	0	0
$727$	14917.2	0.761002	0.380501	−	0.924781i	$-0.375752\pi$
$727$	14917.2	0.761002	0.380501	+	0.924781i	$0.375752\pi$
$728$	6028.62	0.306917
$729$	0	0
$730$	102.000	0.00517149
$731$	12905.4	0.652974
$732$	0	0
$733$	−7094.38	−0.357485	−0.178743	−	0.983896i	$-0.557203\pi$
$733$	−7094.38	−0.357485	−0.178743	+	0.983896i	$0.557203\pi$
$734$	−10508.5	−0.528443
$735$	0	0
$736$	−933.350	−0.0467442
$737$	0	0
$738$	0	0
$739$	−29689.9	−1.47789	−0.738944	−	0.673767i	$-0.764675\pi$
$739$	−29689.9	−1.47789	−0.738944	+	0.673767i	$0.764675\pi$
$740$	−1455.10	−0.0722843
$741$	0	0
$742$	−1869.88	−0.0925139
$743$	19302.6	0.953089	0.476545	−	0.879150i	$-0.341889\pi$
$743$	19302.6	0.953089	0.476545	+	0.879150i	$0.341889\pi$
$744$	0	0
$745$	3966.10	0.195043
$746$	−632.740	−0.0310539
$747$	0	0
$748$	0	0
$749$	26077.5	1.27217
$750$	0	0
$751$	12582.7	0.611384	0.305692	−	0.952130i	$-0.401112\pi$
$751$	12582.7	0.611384	0.305692	+	0.952130i	$0.401112\pi$
$752$	2800.34	0.135795
$753$	0	0
$754$	15304.1	0.739182
$755$	4471.48	0.215541
$756$	0	0
$757$	2728.73	0.131014	0.0655068	−	0.997852i	$-0.479134\pi$
$757$	2728.73	0.131014	0.0655068	+	0.997852i	$0.479134\pi$
$758$	−10514.1	−0.503811
$759$	0	0
$760$	342.217	0.0163336
$761$	21134.6	1.00674	0.503371	−	0.864070i	$-0.332093\pi$
$761$	21134.6	1.00674	0.503371	+	0.864070i	$0.332093\pi$
$762$	0	0
$763$	10182.6	0.483137
$764$	3994.51	0.189157
$765$	0	0
$766$	−2848.44	−0.134358
$767$	−13343.3	−0.628158
$768$	0	0
$769$	10493.4	0.492068	0.246034	−	0.969261i	$-0.420873\pi$
$769$	10493.4	0.492068	0.246034	+	0.969261i	$0.420873\pi$
$770$	0	0
$771$	0	0
$772$	15367.0	0.716411
$773$	14732.9	0.685516	0.342758	−	0.939424i	$-0.388639\pi$
$773$	14732.9	0.685516	0.342758	+	0.939424i	$0.388639\pi$
$774$	0	0
$775$	−3595.44	−0.166648
$776$	8799.94	0.407087
$777$	0	0
$778$	−3940.23	−0.181574
$779$	3165.29	0.145582
$780$	0	0
$781$	0	0
$782$	−3518.33	−0.160889
$783$	0	0
$784$	−1984.13	−0.0903849
$785$	−5773.21	−0.262490
$786$	0	0
$787$	−25252.0	−1.14376	−0.571879	−	0.820338i	$-0.693785\pi$
$787$	−25252.0	−1.14376	−0.571879	+	0.820338i	$0.693785\pi$
$788$	13545.0	0.612335
$789$	0	0
$790$	−210.719	−0.00948993
$791$	−4486.80	−0.201684
$792$	0	0
$793$	36102.8	1.61671
$794$	8297.24	0.370854
$795$	0	0
$796$	−19581.2	−0.871906
$797$	29313.5	1.30281	0.651403	−	0.758732i	$-0.274181\pi$
$797$	29313.5	1.30281	0.651403	+	0.758732i	$0.274181\pi$
$798$	0	0
$799$	10556.1	0.467393
$800$	3916.22	0.173074
$801$	0	0
$802$	−15420.4	−0.678944
$803$	0	0
$804$	0	0
$805$	698.387	0.0305775
$806$	2992.12	0.130760
$807$	0	0
$808$	10794.2	0.469974
$809$	−7235.09	−0.314428	−0.157214	−	0.987565i	$-0.550251\pi$
$809$	−7235.09	−0.314428	−0.157214	+	0.987565i	$0.550251\pi$
$810$	0	0
$811$	−18954.3	−0.820685	−0.410342	−	0.911932i	$-0.634591\pi$
$811$	−18954.3	−0.820685	−0.410342	+	0.911932i	$0.634591\pi$
$812$	8894.85	0.384419
$813$	0	0
$814$	0	0
$815$	−114.839	−0.00493573
$816$	0	0
$817$	−5656.97	−0.242243
$818$	6991.29	0.298832
$819$	0	0
$820$	−774.885	−0.0330002
$821$	−12893.1	−0.548077	−0.274039	−	0.961719i	$-0.588360\pi$
$821$	−12893.1	−0.548077	−0.274039	+	0.961719i	$0.588360\pi$
$822$	0	0
$823$	−17780.4	−0.753083	−0.376542	−	0.926400i	$-0.622887\pi$
$823$	−17780.4	−0.753083	−0.376542	+	0.926400i	$0.622887\pi$
$824$	11877.8	0.502163
$825$	0	0
$826$	−7755.18	−0.326680
$827$	−19993.0	−0.840659	−0.420329	−	0.907372i	$-0.638085\pi$
$827$	−19993.0	−0.840659	−0.420329	+	0.907372i	$0.638085\pi$
$828$	0	0
$829$	−3499.09	−0.146596	−0.0732982	−	0.997310i	$-0.523352\pi$
$829$	−3499.09	−0.146596	−0.0732982	+	0.997310i	$0.523352\pi$
$830$	−746.128	−0.0312030
$831$	0	0
$832$	−3259.07	−0.135803
$833$	−7479.31	−0.311096
$834$	0	0
$835$	−4437.97	−0.183931
$836$	0	0
$837$	0	0
$838$	12987.6	0.535379
$839$	−18827.6	−0.774734	−0.387367	−	0.921926i	$-0.626615\pi$
$839$	−18827.6	−0.774734	−0.387367	+	0.921926i	$0.626615\pi$
$840$	0	0
$841$	−1808.71	−0.0741608
$842$	19923.2	0.815438
$843$	0	0
$844$	−5877.72	−0.239715
$845$	640.985	0.0260953
$846$	0	0
$847$	0	0
$848$	1010.85	0.0409350
$849$	0	0
$850$	14762.5	0.595704
$851$	−6557.50	−0.264146
$852$	0	0
$853$	28919.7	1.16083	0.580417	−	0.814320i	$-0.302890\pi$
$853$	28919.7	1.16083	0.580417	+	0.814320i	$0.302890\pi$
$854$	20983.2	0.840784
$855$	0	0
$856$	−14097.5	−0.562901
$857$	13906.3	0.554294	0.277147	−	0.960827i	$-0.410611\pi$
$857$	13906.3	0.554294	0.277147	+	0.960827i	$0.410611\pi$
$858$	0	0
$859$	18829.8	0.747923	0.373962	−	0.927444i	$-0.377999\pi$
$859$	18829.8	0.747923	0.373962	+	0.927444i	$0.377999\pi$
$860$	1384.87	0.0549111
$861$	0	0
$862$	10993.6	0.434391
$863$	−26157.4	−1.03176	−0.515879	−	0.856661i	$-0.672535\pi$
$863$	−26157.4	−1.03176	−0.515879	+	0.856661i	$0.672535\pi$
$864$	0	0
$865$	−2157.91	−0.0848222
$866$	−196.663	−0.00771696
$867$	0	0
$868$	1739.04	0.0680032
$869$	0	0
$870$	0	0
$871$	7724.61	0.300503
$872$	−5504.69	−0.213776
$873$	0	0
$874$	1542.23	0.0596872
$875$	−5923.38	−0.228854
$876$	0	0
$877$	1415.24	0.0544917	0.0272458	−	0.999629i	$-0.491326\pi$
$877$	1415.24	0.0544917	0.0272458	+	0.999629i	$0.491326\pi$
$878$	3902.39	0.149999
$879$	0	0
$880$	0	0
$881$	5355.31	0.204796	0.102398	−	0.994744i	$-0.467349\pi$
$881$	5355.31	0.204796	0.102398	+	0.994744i	$0.467349\pi$
$882$	0	0
$883$	48865.1	1.86234	0.931168	−	0.364590i	$-0.118791\pi$
$883$	48865.1	1.86234	0.931168	+	0.364590i	$0.118791\pi$
$884$	−12285.3	−0.467420
$885$	0	0
$886$	18877.4	0.715799
$887$	−12614.5	−0.477511	−0.238755	−	0.971080i	$-0.576739\pi$
$887$	−12614.5	−0.477511	−0.238755	+	0.971080i	$0.576739\pi$
$888$	0	0
$889$	33527.7	1.26488
$890$	−2002.97	−0.0754379
$891$	0	0
$892$	6981.87	0.262074
$893$	−4627.16	−0.173395
$894$	0	0
$895$	−5244.23	−0.195861
$896$	−1894.19	−0.0706256
$897$	0	0
$898$	25757.6	0.957175
$899$	4414.69	0.163780
$900$	0	0
$901$	3810.49	0.140894
$902$	0	0
$903$	0	0
$904$	2425.56	0.0892401
$905$	4304.35	0.158101
$906$	0	0
$907$	5592.74	0.204745	0.102373	−	0.994746i	$-0.467357\pi$
$907$	5592.74	0.204745	0.102373	+	0.994746i	$0.467357\pi$
$908$	−9455.47	−0.345585
$909$	0	0
$910$	2438.63	0.0888349
$911$	22598.1	0.821854	0.410927	−	0.911668i	$-0.365205\pi$
$911$	22598.1	0.821854	0.410927	+	0.911668i	$0.365205\pi$
$912$	0	0
$913$	0	0
$914$	15515.2	0.561486
$915$	0	0
$916$	−12375.0	−0.446377
$917$	−21568.4	−0.776718
$918$	0	0
$919$	−6798.94	−0.244044	−0.122022	−	0.992527i	$-0.538938\pi$
$919$	−6798.94	−0.244044	−0.122022	+	0.992527i	$0.538938\pi$
$920$	−377.548	−0.0135298
$921$	0	0
$922$	27058.7	0.966519
$923$	53751.9	1.91686
$924$	0	0
$925$	27514.5	0.978022
$926$	29116.6	1.03330
$927$	0	0
$928$	−4808.56	−0.170095
$929$	15869.5	0.560453	0.280227	−	0.959934i	$-0.409590\pi$
$929$	15869.5	0.560453	0.280227	+	0.959934i	$0.409590\pi$
$930$	0	0
$931$	3278.49	0.115412
$932$	−10800.2	−0.379585
$933$	0	0
$934$	−9266.32	−0.324629
$935$	0	0
$936$	0	0
$937$	20320.0	0.708457	0.354229	−	0.935159i	$-0.384743\pi$
$937$	20320.0	0.708457	0.354229	+	0.935159i	$0.384743\pi$
$938$	4489.59	0.156280
$939$	0	0
$940$	1132.76	0.0393049
$941$	−2080.02	−0.0720583	−0.0360292	−	0.999351i	$-0.511471\pi$
$941$	−2080.02	−0.0720583	−0.0360292	+	0.999351i	$0.511471\pi$
$942$	0	0
$943$	−3492.08	−0.120592
$944$	4192.45	0.144547
$945$	0	0
$946$	0	0
$947$	21613.7	0.741658	0.370829	−	0.928701i	$-0.379074\pi$
$947$	21613.7	0.741658	0.370829	+	0.928701i	$0.379074\pi$
$948$	0	0
$949$	1605.08	0.0549031
$950$	−6470.99	−0.220997
$951$	0	0
$952$	−7140.28	−0.243086
$953$	15318.4	0.520685	0.260342	−	0.965516i	$-0.416165\pi$
$953$	15318.4	0.520685	0.260342	+	0.965516i	$0.416165\pi$
$954$	0	0
$955$	1615.81	0.0547502
$956$	−1431.69	−0.0484354
$957$	0	0
$958$	39202.1	1.32209
$959$	2913.16	0.0980925
$960$	0	0
$961$	−28927.9	−0.971028
$962$	−22897.5	−0.767406
$963$	0	0
$964$	17162.9	0.573423
$965$	6216.07	0.207360
$966$	0	0
$967$	−33356.0	−1.10926	−0.554631	−	0.832096i	$-0.687141\pi$
$967$	−33356.0	−1.10926	−0.554631	+	0.832096i	$0.687141\pi$
$968$	0	0
$969$	0	0
$970$	3559.65	0.117828
$971$	24491.3	0.809436	0.404718	−	0.914442i	$-0.367370\pi$
$971$	24491.3	0.809436	0.404718	+	0.914442i	$0.367370\pi$
$972$	0	0
$973$	−586.810	−0.0193343
$974$	7350.42	0.241810
$975$	0	0
$976$	−11343.5	−0.372026
$977$	−43967.8	−1.43977	−0.719885	−	0.694093i	$-0.755806\pi$
$977$	−43967.8	−1.43977	−0.719885	+	0.694093i	$0.755806\pi$
$978$	0	0
$979$	0	0
$980$	−802.597	−0.0261613
$981$	0	0
$982$	41142.2	1.33696
$983$	−34681.2	−1.12529	−0.562644	−	0.826699i	$-0.690216\pi$
$983$	−34681.2	−1.12529	−0.562644	+	0.826699i	$0.690216\pi$
$984$	0	0
$985$	5479.06	0.177236
$986$	−18126.2	−0.585452
$987$	0	0
$988$	5385.15	0.173405
$989$	6241.01	0.200660
$990$	0	0
$991$	42734.1	1.36982	0.684910	−	0.728627i	$-0.259841\pi$
$991$	42734.1	1.36982	0.684910	+	0.728627i	$0.259841\pi$
$992$	−940.124	−0.0300897
$993$	0	0
$994$	31240.9	0.996884
$995$	−7920.76	−0.252367
$996$	0	0
$997$	32108.6	1.01995	0.509975	−	0.860189i	$-0.329655\pi$
$997$	32108.6	1.01995	0.509975	+	0.860189i	$0.329655\pi$
$998$	−1937.61	−0.0614570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.z.1.2		2
3.2	odd	2		242.4.a.k.1.1		2
11.5	even	5		198.4.f.b.91.1		4
11.9	even	5		198.4.f.b.37.1		4
11.10	odd	2		2178.4.a.bi.1.2		2
12.11	even	2		1936.4.a.bb.1.2		2
33.2	even	10		242.4.c.j.81.1		4
33.5	odd	10		22.4.c.a.3.1	✓	4
33.8	even	10		242.4.c.m.9.1		4
33.14	odd	10		242.4.c.f.9.1		4
33.17	even	10		242.4.c.j.3.1		4
33.20	odd	10		22.4.c.a.15.1	yes	4
33.26	odd	10		242.4.c.f.27.1		4
33.29	even	10		242.4.c.m.27.1		4
33.32	even	2		242.4.a.h.1.1		2
132.71	even	10		176.4.m.a.113.1		4
132.119	even	10		176.4.m.a.81.1		4
132.131	odd	2		1936.4.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.a.3.1	✓	4	33.5	odd	10
22.4.c.a.15.1	yes	4	33.20	odd	10
176.4.m.a.81.1		4	132.119	even	10
176.4.m.a.113.1		4	132.71	even	10
198.4.f.b.37.1		4	11.9	even	5
198.4.f.b.91.1		4	11.5	even	5
242.4.a.h.1.1		2	33.32	even	2
242.4.a.k.1.1		2	3.2	odd	2
242.4.c.f.9.1		4	33.14	odd	10
242.4.c.f.27.1		4	33.26	odd	10
242.4.c.j.3.1		4	33.17	even	10
242.4.c.j.81.1		4	33.2	even	10
242.4.c.m.9.1		4	33.8	even	10
242.4.c.m.27.1		4	33.29	even	10
1936.4.a.bb.1.2		2	12.11	even	2
1936.4.a.bc.1.2		2	132.131	odd	2
2178.4.a.z.1.2		2	1.1	even	1	trivial
2178.4.a.bi.1.2		2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +1.61803 q^{5} +14.7984 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +1.61803 q^{5} +14.7984 q^{7} -8.00000 q^{8} -3.23607 q^{10} -50.9230 q^{13} -29.5967 q^{14} +16.0000 q^{16} +60.3131 q^{17} -26.4377 q^{19} +6.47214 q^{20} +29.1672 q^{23} -122.382 q^{25} +101.846 q^{26} +59.1935 q^{28} +150.267 q^{29} +29.3789 q^{31} -32.0000 q^{32} -120.626 q^{34} +23.9443 q^{35} -224.825 q^{37} +52.8754 q^{38} -12.9443 q^{40} -119.726 q^{41} +213.974 q^{43} -58.3344 q^{46} +175.021 q^{47} -124.008 q^{49} +244.764 q^{50} -203.692 q^{52} +63.1784 q^{53} -118.387 q^{56} -300.535 q^{58} +262.028 q^{59} -708.969 q^{61} -58.7577 q^{62} +64.0000 q^{64} -82.3951 q^{65} -151.692 q^{67} +241.252 q^{68} -47.8885 q^{70} -1055.55 q^{71} -31.5197 q^{73} +449.649 q^{74} -105.751 q^{76} +65.1157 q^{79} +25.8885 q^{80} +239.453 q^{82} +230.566 q^{83} +97.5886 q^{85} -427.947 q^{86} +618.952 q^{89} -753.577 q^{91} +116.669 q^{92} -350.043 q^{94} -42.7771 q^{95} -1099.99 q^{97} +248.016 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + q^{5} + 5 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + q^5 + 5 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + q^{5} + 5 q^{7} - 16 q^{8} - 2 q^{10} - 37 q^{13} - 10 q^{14} + 32 q^{16} + 181 q^{17} - 73 q^{19} + 4 q^{20} + 112 q^{23} - 247 q^{25} + 74 q^{26} + 20 q^{28} - 55 q^{29} - 261 q^{31} - 64 q^{32} - 362 q^{34} + 30 q^{35} - 273 q^{37} + 146 q^{38} - 8 q^{40} + 49 q^{41} + 580 q^{43} - 224 q^{46} + 397 q^{47} - 371 q^{49} + 494 q^{50} - 148 q^{52} + 625 q^{53} - 40 q^{56} + 110 q^{58} - 149 q^{59} - 405 q^{61} + 522 q^{62} + 128 q^{64} - 91 q^{65} - 44 q^{67} + 724 q^{68} - 60 q^{70} - 1125 q^{71} + 51 q^{73} + 546 q^{74} - 292 q^{76} + 1331 q^{79} + 16 q^{80} - 98 q^{82} - 601 q^{83} + 23 q^{85} - 1160 q^{86} - 864 q^{89} - 890 q^{91} + 448 q^{92} - 794 q^{94} - 14 q^{95} - 713 q^{97} + 742 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + q^5 + 5 * q^7 - 16 * q^8 - 2 * q^10 - 37 * q^13 - 10 * q^14 + 32 * q^16 + 181 * q^17 - 73 * q^19 + 4 * q^20 + 112 * q^23 - 247 * q^25 + 74 * q^26 + 20 * q^28 - 55 * q^29 - 261 * q^31 - 64 * q^32 - 362 * q^34 + 30 * q^35 - 273 * q^37 + 146 * q^38 - 8 * q^40 + 49 * q^41 + 580 * q^43 - 224 * q^46 + 397 * q^47 - 371 * q^49 + 494 * q^50 - 148 * q^52 + 625 * q^53 - 40 * q^56 + 110 * q^58 - 149 * q^59 - 405 * q^61 + 522 * q^62 + 128 * q^64 - 91 * q^65 - 44 * q^67 + 724 * q^68 - 60 * q^70 - 1125 * q^71 + 51 * q^73 + 546 * q^74 - 292 * q^76 + 1331 * q^79 + 16 * q^80 - 98 * q^82 - 601 * q^83 + 23 * q^85 - 1160 * q^86 - 864 * q^89 - 890 * q^91 + 448 * q^92 - 794 * q^94 - 14 * q^95 - 713 * q^97 + 742 * q^98

Introduction

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