LMFDB - Embedded newform 1856.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +O(q^{10})$	$q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +52.3485 q^{11} +5.40408 q^{13} -33.4495 q^{15} +17.1918 q^{17} -44.2020 q^{19} -95.1918 q^{21} -205.060 q^{23} -30.9694 q^{25} -145.227 q^{27} -29.0000 q^{29} +299.994 q^{31} +180.576 q^{33} +267.596 q^{35} -29.7980 q^{37} +18.6413 q^{39} -43.9592 q^{41} -64.8230 q^{43} +146.434 q^{45} -499.499 q^{47} +418.535 q^{49} +59.3031 q^{51} +351.627 q^{53} -507.620 q^{55} -152.474 q^{57} -522.372 q^{59} -484.606 q^{61} +416.727 q^{63} -52.4031 q^{65} +504.990 q^{67} -707.353 q^{69} +481.283 q^{71} +3.11019 q^{73} -106.829 q^{75} -1444.60 q^{77} -1043.27 q^{79} -93.2316 q^{81} +1007.08 q^{83} -166.708 q^{85} -100.035 q^{87} -295.637 q^{89} -149.131 q^{91} +1034.83 q^{93} +428.624 q^{95} +428.949 q^{97} -790.515 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 10 * q^5 - 16 * q^7 - 40 * q^9	$ 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + 90 q^{11} + 50 q^{13} - 62 q^{15} - 44 q^{17} - 108 q^{19} - 112 q^{21} - 28 q^{23} + 232 q^{25} - 70 q^{27} - 58 q^{29} + 66 q^{31} + 126 q^{33} + 496 q^{35} - 40 q^{37} - 46 q^{39} + 304 q^{41} + 130 q^{43} - 344 q^{45} - 514 q^{47} + 210 q^{49} + 148 q^{51} + 958 q^{53} + 234 q^{55} - 60 q^{57} + 180 q^{59} - 1028 q^{61} + 128 q^{63} + 826 q^{65} + 912 q^{67} - 964 q^{69} + 796 q^{71} - 856 q^{73} - 488 q^{75} - 1008 q^{77} - 318 q^{79} + 470 q^{81} + 1828 q^{83} - 1372 q^{85} - 58 q^{87} - 944 q^{89} + 368 q^{91} + 1374 q^{93} - 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 10 * q^5 - 16 * q^7 - 40 * q^9 + 90 * q^11 + 50 * q^13 - 62 * q^15 - 44 * q^17 - 108 * q^19 - 112 * q^21 - 28 * q^23 + 232 * q^25 - 70 * q^27 - 58 * q^29 + 66 * q^31 + 126 * q^33 + 496 * q^35 - 40 * q^37 - 46 * q^39 + 304 * q^41 + 130 * q^43 - 344 * q^45 - 514 * q^47 + 210 * q^49 + 148 * q^51 + 958 * q^53 + 234 * q^55 - 60 * q^57 + 180 * q^59 - 1028 * q^61 + 128 * q^63 + 826 * q^65 + 912 * q^67 - 964 * q^69 + 796 * q^71 - 856 * q^73 - 488 * q^75 - 1008 * q^77 - 318 * q^79 + 470 * q^81 + 1828 * q^83 - 1372 * q^85 - 58 * q^87 - 944 * q^89 + 368 * q^91 + 1374 * q^93 - 828 * q^95 + 368 * q^97 - 1728 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	3.44949	0.663855	0.331927	−	0.943305i	$-0.392301\pi$
$3$	3.44949	0.663855	0.331927	+	0.943305i	$0.392301\pi$
$4$	0	0
$5$	−9.69694	−0.867321	−0.433660	−	0.901076i	$-0.642778\pi$
$5$	−9.69694	−0.867321	−0.433660	+	0.901076i	$0.642778\pi$
$6$	0	0
$7$	−27.5959	−1.49004	−0.745020	−	0.667042i	$-0.767560\pi$
$7$	−27.5959	−1.49004	−0.745020	+	0.667042i	$0.767560\pi$
$8$	0	0
$9$	−15.1010	−0.559297
$10$	0	0
$11$	52.3485	1.43488	0.717439	−	0.696621i	$-0.245314\pi$
$11$	52.3485	1.43488	0.717439	+	0.696621i	$0.245314\pi$
$12$	0	0
$13$	5.40408	0.115294	0.0576470	−	0.998337i	$-0.481640\pi$
$13$	5.40408	0.115294	0.0576470	+	0.998337i	$0.481640\pi$
$14$	0	0
$15$	−33.4495	−0.575775
$16$	0	0
$17$	17.1918	0.245273	0.122636	−	0.992452i	$-0.460865\pi$
$17$	17.1918	0.245273	0.122636	+	0.992452i	$0.460865\pi$
$18$	0	0
$19$	−44.2020	−0.533718	−0.266859	−	0.963736i	$-0.585986\pi$
$19$	−44.2020	−0.533718	−0.266859	+	0.963736i	$0.585986\pi$
$20$	0	0
$21$	−95.1918	−0.989170
$22$	0	0
$23$	−205.060	−1.85904	−0.929522	−	0.368767i	$-0.879780\pi$
$23$	−205.060	−1.85904	−0.929522	+	0.368767i	$0.879780\pi$
$24$	0	0
$25$	−30.9694	−0.247755
$26$	0	0
$27$	−145.227	−1.03515
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	299.994	1.73808	0.869042	−	0.494739i	$-0.164736\pi$
$31$	299.994	1.73808	0.869042	+	0.494739i	$0.164736\pi$
$32$	0	0
$33$	180.576	0.952550
$34$	0	0
$35$	267.596	1.29234
$36$	0	0
$37$	−29.7980	−0.132399	−0.0661994	−	0.997806i	$-0.521087\pi$
$37$	−29.7980	−0.132399	−0.0661994	+	0.997806i	$0.521087\pi$
$38$	0	0
$39$	18.6413	0.0765385
$40$	0	0
$41$	−43.9592	−0.167446	−0.0837228	−	0.996489i	$-0.526681\pi$
$41$	−43.9592	−0.167446	−0.0837228	+	0.996489i	$0.526681\pi$
$42$	0	0
$43$	−64.8230	−0.229893	−0.114947	−	0.993372i	$-0.536670\pi$
$43$	−64.8230	−0.229893	−0.114947	+	0.993372i	$0.536670\pi$
$44$	0	0
$45$	146.434	0.485090
$46$	0	0
$47$	−499.499	−1.55020	−0.775101	−	0.631837i	$-0.782301\pi$
$47$	−499.499	−1.55020	−0.775101	+	0.631837i	$0.782301\pi$
$48$	0	0
$49$	418.535	1.22022
$50$	0	0
$51$	59.3031	0.162825
$52$	0	0
$53$	351.627	0.911314	0.455657	−	0.890156i	$-0.349404\pi$
$53$	351.627	0.911314	0.455657	+	0.890156i	$0.349404\pi$
$54$	0	0
$55$	−507.620	−1.24450
$56$	0	0
$57$	−152.474	−0.354311
$58$	0	0
$59$	−522.372	−1.15266	−0.576331	−	0.817216i	$-0.695516\pi$
$59$	−522.372	−1.15266	−0.576331	+	0.817216i	$0.695516\pi$
$60$	0	0
$61$	−484.606	−1.01717	−0.508586	−	0.861011i	$-0.669832\pi$
$61$	−484.606	−1.01717	−0.508586	+	0.861011i	$0.669832\pi$
$62$	0	0
$63$	416.727	0.833375
$64$	0	0
$65$	−52.4031	−0.0999969
$66$	0	0
$67$	504.990	0.920811	0.460405	−	0.887709i	$-0.347704\pi$
$67$	504.990	0.920811	0.460405	+	0.887709i	$0.347704\pi$
$68$	0	0
$69$	−707.353	−1.23413
$70$	0	0
$71$	481.283	0.804475	0.402238	−	0.915535i	$-0.368233\pi$
$71$	481.283	0.804475	0.402238	+	0.915535i	$0.368233\pi$
$72$	0	0
$73$	3.11019	0.00498659	0.00249329	−	0.999997i	$-0.499206\pi$
$73$	3.11019	0.00498659	0.00249329	+	0.999997i	$0.499206\pi$
$74$	0	0
$75$	−106.829	−0.164473
$76$	0	0
$77$	−1444.60	−2.13802
$78$	0	0
$79$	−1043.27	−1.48578	−0.742890	−	0.669414i	$-0.766545\pi$
$79$	−1043.27	−1.48578	−0.742890	+	0.669414i	$0.766545\pi$
$80$	0	0
$81$	−93.2316	−0.127890
$82$	0	0
$83$	1007.08	1.33182	0.665912	−	0.746030i	$-0.268042\pi$
$83$	1007.08	1.33182	0.665912	+	0.746030i	$0.268042\pi$
$84$	0	0
$85$	−166.708	−0.212730
$86$	0	0
$87$	−100.035	−0.123275
$88$	0	0
$89$	−295.637	−0.352106	−0.176053	−	0.984381i	$-0.556333\pi$
$89$	−295.637	−0.352106	−0.176053	+	0.984381i	$0.556333\pi$
$90$	0	0
$91$	−149.131	−0.171793
$92$	0	0
$93$	1034.83	1.15383
$94$	0	0
$95$	428.624	0.462905
$96$	0	0
$97$	428.949	0.449002	0.224501	−	0.974474i	$-0.427925\pi$
$97$	428.949	0.449002	0.224501	+	0.974474i	$0.427925\pi$
$98$	0	0
$99$	−790.515	−0.802523
$100$	0	0
$101$	1212.16	1.19420	0.597101	−	0.802166i	$-0.296319\pi$
$101$	1212.16	1.19420	0.597101	+	0.802166i	$0.296319\pi$
$102$	0	0
$103$	1032.07	0.987310	0.493655	−	0.869658i	$-0.335661\pi$
$103$	1032.07	0.987310	0.493655	+	0.869658i	$0.335661\pi$
$104$	0	0
$105$	923.069	0.857927
$106$	0	0
$107$	1176.06	1.06256	0.531280	−	0.847196i	$-0.321711\pi$
$107$	1176.06	1.06256	0.531280	+	0.847196i	$0.321711\pi$
$108$	0	0
$109$	2167.86	1.90498	0.952491	−	0.304568i	$-0.0985121\pi$
$109$	2167.86	1.90498	0.952491	+	0.304568i	$0.0985121\pi$
$110$	0	0
$111$	−102.788	−0.0878935
$112$	0	0
$113$	1623.65	1.35169	0.675843	−	0.737046i	$-0.263780\pi$
$113$	1623.65	1.35169	0.675843	+	0.737046i	$0.263780\pi$
$114$	0	0
$115$	1988.46	1.61239
$116$	0	0
$117$	−81.6072	−0.0644836
$118$	0	0
$119$	−474.424	−0.365466
$120$	0	0
$121$	1409.36	1.05887
$122$	0	0
$123$	−151.637	−0.111160
$124$	0	0
$125$	1512.43	1.08220
$126$	0	0
$127$	333.473	0.233000	0.116500	−	0.993191i	$-0.462833\pi$
$127$	333.473	0.233000	0.116500	+	0.993191i	$0.462833\pi$
$128$	0	0
$129$	−223.606	−0.152616
$130$	0	0
$131$	933.492	0.622592	0.311296	−	0.950313i	$-0.399237\pi$
$131$	933.492	0.622592	0.311296	+	0.950313i	$0.399237\pi$
$132$	0	0
$133$	1219.80	0.795261
$134$	0	0
$135$	1408.26	0.897804
$136$	0	0
$137$	−1090.32	−0.679944	−0.339972	−	0.940436i	$-0.610418\pi$
$137$	−1090.32	−0.679944	−0.339972	+	0.940436i	$0.610418\pi$
$138$	0	0
$139$	1924.60	1.17440	0.587202	−	0.809440i	$-0.300229\pi$
$139$	1924.60	1.17440	0.587202	+	0.809440i	$0.300229\pi$
$140$	0	0
$141$	−1723.02	−1.02911
$142$	0	0
$143$	282.895	0.165433
$144$	0	0
$145$	281.211	0.161057
$146$	0	0
$147$	1443.73	0.810047
$148$	0	0
$149$	1703.62	0.936687	0.468343	−	0.883546i	$-0.344851\pi$
$149$	1703.62	0.936687	0.468343	+	0.883546i	$0.344851\pi$
$150$	0	0
$151$	1082.81	0.583560	0.291780	−	0.956486i	$-0.405753\pi$
$151$	1082.81	0.583560	0.291780	+	0.956486i	$0.405753\pi$
$152$	0	0
$153$	−259.614	−0.137180
$154$	0	0
$155$	−2909.03	−1.50748
$156$	0	0
$157$	−2407.90	−1.22402	−0.612010	−	0.790850i	$-0.709639\pi$
$157$	−2407.90	−1.22402	−0.612010	+	0.790850i	$0.709639\pi$
$158$	0	0
$159$	1212.93	0.604980
$160$	0	0
$161$	5658.82	2.77005
$162$	0	0
$163$	−1061.59	−0.510123	−0.255061	−	0.966925i	$-0.582096\pi$
$163$	−1061.59	−0.510123	−0.255061	+	0.966925i	$0.582096\pi$
$164$	0	0
$165$	−1751.03	−0.826166
$166$	0	0
$167$	1315.63	0.609618	0.304809	−	0.952414i	$-0.401407\pi$
$167$	1315.63	0.609618	0.304809	+	0.952414i	$0.401407\pi$
$168$	0	0
$169$	−2167.80	−0.986707
$170$	0	0
$171$	667.496	0.298507
$172$	0	0
$173$	653.800	0.287327	0.143663	−	0.989627i	$-0.454112\pi$
$173$	653.800	0.287327	0.143663	+	0.989627i	$0.454112\pi$
$174$	0	0
$175$	854.629	0.369165
$176$	0	0
$177$	−1801.92	−0.765200
$178$	0	0
$179$	−2055.28	−0.858208	−0.429104	−	0.903255i	$-0.641171\pi$
$179$	−2055.28	−0.858208	−0.429104	+	0.903255i	$0.641171\pi$
$180$	0	0
$181$	−3398.36	−1.39557	−0.697785	−	0.716307i	$-0.745831\pi$
$181$	−3398.36	−1.39557	−0.697785	+	0.716307i	$0.745831\pi$
$182$	0	0
$183$	−1671.64	−0.675254
$184$	0	0
$185$	288.949	0.114832
$186$	0	0
$187$	899.966	0.351936
$188$	0	0
$189$	4007.67	1.54241
$190$	0	0
$191$	4853.92	1.83883	0.919417	−	0.393285i	$-0.128661\pi$
$191$	4853.92	1.83883	0.919417	+	0.393285i	$0.128661\pi$
$192$	0	0
$193$	−4877.24	−1.81902	−0.909512	−	0.415677i	$-0.863545\pi$
$193$	−4877.24	−1.81902	−0.909512	+	0.415677i	$0.863545\pi$
$194$	0	0
$195$	−180.764	−0.0663834
$196$	0	0
$197$	1031.42	0.373022	0.186511	−	0.982453i	$-0.440282\pi$
$197$	1031.42	0.373022	0.186511	+	0.982453i	$0.440282\pi$
$198$	0	0
$199$	−1167.70	−0.415960	−0.207980	−	0.978133i	$-0.566689\pi$
$199$	−1167.70	−0.415960	−0.207980	+	0.978133i	$0.566689\pi$
$200$	0	0
$201$	1741.96	0.611284
$202$	0	0
$203$	800.282	0.276693
$204$	0	0
$205$	426.269	0.145229
$206$	0	0
$207$	3096.62	1.03976
$208$	0	0
$209$	−2313.91	−0.765820
$210$	0	0
$211$	−794.341	−0.259169	−0.129585	−	0.991568i	$-0.541364\pi$
$211$	−794.341	−0.259169	−0.129585	+	0.991568i	$0.541364\pi$
$212$	0	0
$213$	1660.18	0.534055
$214$	0	0
$215$	628.584	0.199391
$216$	0	0
$217$	−8278.62	−2.58981
$218$	0	0
$219$	10.7286	0.00331037
$220$	0	0
$221$	92.9061	0.0282785
$222$	0	0
$223$	5136.08	1.54232	0.771161	−	0.636641i	$-0.219676\pi$
$223$	5136.08	1.54232	0.771161	+	0.636641i	$0.219676\pi$
$224$	0	0
$225$	467.669	0.138569
$226$	0	0
$227$	5032.84	1.47155	0.735774	−	0.677228i	$-0.236819\pi$
$227$	5032.84	1.47155	0.735774	+	0.677228i	$0.236819\pi$
$228$	0	0
$229$	6213.12	1.79290	0.896451	−	0.443142i	$-0.146136\pi$
$229$	6213.12	1.79290	0.896451	+	0.443142i	$0.146136\pi$
$230$	0	0
$231$	−4983.15	−1.41934
$232$	0	0
$233$	−3117.59	−0.876566	−0.438283	−	0.898837i	$-0.644413\pi$
$233$	−3117.59	−0.876566	−0.438283	+	0.898837i	$0.644413\pi$
$234$	0	0
$235$	4843.62	1.34452
$236$	0	0
$237$	−3598.73	−0.986342
$238$	0	0
$239$	7193.09	1.94679	0.973394	−	0.229136i	$-0.0735902\pi$
$239$	7193.09	1.94679	0.973394	+	0.229136i	$0.0735902\pi$
$240$	0	0
$241$	−837.347	−0.223810	−0.111905	−	0.993719i	$-0.535695\pi$
$241$	−837.347	−0.223810	−0.111905	+	0.993719i	$0.535695\pi$
$242$	0	0
$243$	3599.53	0.950246
$244$	0	0
$245$	−4058.51	−1.05832
$246$	0	0
$247$	−238.871	−0.0615345
$248$	0	0
$249$	3473.91	0.884138
$250$	0	0
$251$	−4306.56	−1.08298	−0.541489	−	0.840708i	$-0.682139\pi$
$251$	−4306.56	−1.08298	−0.541489	+	0.840708i	$0.682139\pi$
$252$	0	0
$253$	−10734.6	−2.66750
$254$	0	0
$255$	−575.058	−0.141222
$256$	0	0
$257$	401.238	0.0973873	0.0486936	−	0.998814i	$-0.484494\pi$
$257$	401.238	0.0973873	0.0486936	+	0.998814i	$0.484494\pi$
$258$	0	0
$259$	822.302	0.197279
$260$	0	0
$261$	437.930	0.103859
$262$	0	0
$263$	−2682.72	−0.628988	−0.314494	−	0.949260i	$-0.601835\pi$
$263$	−2682.72	−0.628988	−0.314494	+	0.949260i	$0.601835\pi$
$264$	0	0
$265$	−3409.70	−0.790401
$266$	0	0
$267$	−1019.80	−0.233747
$268$	0	0
$269$	4732.38	1.07263	0.536316	−	0.844017i	$-0.319816\pi$
$269$	4732.38	1.07263	0.536316	+	0.844017i	$0.319816\pi$
$270$	0	0
$271$	4529.66	1.01534	0.507670	−	0.861552i	$-0.330507\pi$
$271$	4529.66	1.01534	0.507670	+	0.861552i	$0.330507\pi$
$272$	0	0
$273$	−514.424	−0.114045
$274$	0	0
$275$	−1621.20	−0.355498
$276$	0	0
$277$	−7158.69	−1.55279	−0.776397	−	0.630245i	$-0.782955\pi$
$277$	−7158.69	−1.55279	−0.776397	+	0.630245i	$0.782955\pi$
$278$	0	0
$279$	−4530.22	−0.972105
$280$	0	0
$281$	−5769.75	−1.22489	−0.612446	−	0.790512i	$-0.709814\pi$
$281$	−5769.75	−1.22489	−0.612446	+	0.790512i	$0.709814\pi$
$282$	0	0
$283$	3815.25	0.801389	0.400694	−	0.916212i	$-0.368769\pi$
$283$	3815.25	0.801389	0.400694	+	0.916212i	$0.368769\pi$
$284$	0	0
$285$	1478.54	0.307301
$286$	0	0
$287$	1213.09	0.249501
$288$	0	0
$289$	−4617.44	−0.939841
$290$	0	0
$291$	1479.66	0.298072
$292$	0	0
$293$	−5416.66	−1.08002	−0.540008	−	0.841660i	$-0.681579\pi$
$293$	−5416.66	−1.08002	−0.540008	+	0.841660i	$0.681579\pi$
$294$	0	0
$295$	5065.41	0.999728
$296$	0	0
$297$	−7602.41	−1.48531
$298$	0	0
$299$	−1108.16	−0.214337
$300$	0	0
$301$	1788.85	0.342550
$302$	0	0
$303$	4181.33	0.792776
$304$	0	0
$305$	4699.20	0.882214
$306$	0	0
$307$	−4929.06	−0.916339	−0.458169	−	0.888865i	$-0.651495\pi$
$307$	−4929.06	−0.916339	−0.458169	+	0.888865i	$0.651495\pi$
$308$	0	0
$309$	3560.12	0.655430
$310$	0	0
$311$	1117.93	0.203832	0.101916	−	0.994793i	$-0.467503\pi$
$311$	1117.93	0.203832	0.101916	+	0.994793i	$0.467503\pi$
$312$	0	0
$313$	2032.21	0.366988	0.183494	−	0.983021i	$-0.441259\pi$
$313$	2032.21	0.366988	0.183494	+	0.983021i	$0.441259\pi$
$314$	0	0
$315$	−4040.97	−0.722803
$316$	0	0
$317$	−2051.75	−0.363526	−0.181763	−	0.983342i	$-0.558180\pi$
$317$	−2051.75	−0.363526	−0.181763	+	0.983342i	$0.558180\pi$
$318$	0	0
$319$	−1518.11	−0.266450
$320$	0	0
$321$	4056.80	0.705385
$322$	0	0
$323$	−759.914	−0.130906
$324$	0	0
$325$	−167.361	−0.0285647
$326$	0	0
$327$	7478.00	1.26463
$328$	0	0
$329$	13784.1	2.30986
$330$	0	0
$331$	9652.92	1.60294	0.801469	−	0.598036i	$-0.204052\pi$
$331$	9652.92	1.60294	0.801469	+	0.598036i	$0.204052\pi$
$332$	0	0
$333$	449.980	0.0740502
$334$	0	0
$335$	−4896.85	−0.798638
$336$	0	0
$337$	4277.57	0.691437	0.345719	−	0.938338i	$-0.387635\pi$
$337$	4277.57	0.691437	0.345719	+	0.938338i	$0.387635\pi$
$338$	0	0
$339$	5600.77	0.897322
$340$	0	0
$341$	15704.2	2.49394
$342$	0	0
$343$	−2084.45	−0.328133
$344$	0	0
$345$	6859.16	1.07039
$346$	0	0
$347$	9475.83	1.46596	0.732982	−	0.680248i	$-0.238128\pi$
$347$	9475.83	1.46596	0.732982	+	0.680248i	$0.238128\pi$
$348$	0	0
$349$	−2968.57	−0.455313	−0.227656	−	0.973742i	$-0.573106\pi$
$349$	−2968.57	−0.455313	−0.227656	+	0.973742i	$0.573106\pi$
$350$	0	0
$351$	−784.819	−0.119346
$352$	0	0
$353$	3969.16	0.598461	0.299231	−	0.954181i	$-0.403270\pi$
$353$	3969.16	0.598461	0.299231	+	0.954181i	$0.403270\pi$
$354$	0	0
$355$	−4666.97	−0.697738
$356$	0	0
$357$	−1636.52	−0.242616
$358$	0	0
$359$	−3503.03	−0.514994	−0.257497	−	0.966279i	$-0.582898\pi$
$359$	−3503.03	−0.514994	−0.257497	+	0.966279i	$0.582898\pi$
$360$	0	0
$361$	−4905.18	−0.715145
$362$	0	0
$363$	4861.58	0.702939
$364$	0	0
$365$	−30.1594	−0.00432497
$366$	0	0
$367$	−8253.53	−1.17393	−0.586963	−	0.809614i	$-0.699677\pi$
$367$	−8253.53	−1.17393	−0.586963	+	0.809614i	$0.699677\pi$
$368$	0	0
$369$	663.828	0.0936518
$370$	0	0
$371$	−9703.46	−1.35789
$372$	0	0
$373$	−1253.95	−0.174067	−0.0870335	−	0.996205i	$-0.527739\pi$
$373$	−1253.95	−0.174067	−0.0870335	+	0.996205i	$0.527739\pi$
$374$	0	0
$375$	5217.10	0.718426
$376$	0	0
$377$	−156.718	−0.0214096
$378$	0	0
$379$	6966.52	0.944185	0.472092	−	0.881549i	$-0.343499\pi$
$379$	6966.52	0.944185	0.472092	+	0.881549i	$0.343499\pi$
$380$	0	0
$381$	1150.31	0.154678
$382$	0	0
$383$	−9821.84	−1.31037	−0.655186	−	0.755467i	$-0.727410\pi$
$383$	−9821.84	−1.31037	−0.655186	+	0.755467i	$0.727410\pi$
$384$	0	0
$385$	14008.2	1.85435
$386$	0	0
$387$	978.893	0.128579
$388$	0	0
$389$	−3942.96	−0.513923	−0.256962	−	0.966422i	$-0.582721\pi$
$389$	−3942.96	−0.513923	−0.256962	+	0.966422i	$0.582721\pi$
$390$	0	0
$391$	−3525.36	−0.455972
$392$	0	0
$393$	3220.07	0.413311
$394$	0	0
$395$	10116.5	1.28865
$396$	0	0
$397$	−8725.50	−1.10307	−0.551537	−	0.834150i	$-0.685959\pi$
$397$	−8725.50	−1.10307	−0.551537	+	0.834150i	$0.685959\pi$
$398$	0	0
$399$	4207.67	0.527938
$400$	0	0
$401$	3986.86	0.496494	0.248247	−	0.968697i	$-0.420146\pi$
$401$	3986.86	0.496494	0.248247	+	0.968697i	$0.420146\pi$
$402$	0	0
$403$	1621.19	0.200391
$404$	0	0
$405$	904.061	0.110921
$406$	0	0
$407$	−1559.88	−0.189976
$408$	0	0
$409$	−704.847	−0.0852138	−0.0426069	−	0.999092i	$-0.513566\pi$
$409$	−704.847	−0.0852138	−0.0426069	+	0.999092i	$0.513566\pi$
$410$	0	0
$411$	−3761.05	−0.451384
$412$	0	0
$413$	14415.3	1.71751
$414$	0	0
$415$	−9765.60	−1.15512
$416$	0	0
$417$	6638.88	0.779634
$418$	0	0
$419$	−6705.33	−0.781807	−0.390903	−	0.920432i	$-0.627837\pi$
$419$	−6705.33	−0.781807	−0.390903	+	0.920432i	$0.627837\pi$
$420$	0	0
$421$	−8996.55	−1.04148	−0.520742	−	0.853714i	$-0.674345\pi$
$421$	−8996.55	−1.04148	−0.520742	+	0.853714i	$0.674345\pi$
$422$	0	0
$423$	7542.95	0.867023
$424$	0	0
$425$	−532.421	−0.0607675
$426$	0	0
$427$	13373.2	1.51563
$428$	0	0
$429$	975.845	0.109823
$430$	0	0
$431$	5383.16	0.601619	0.300810	−	0.953684i	$-0.402743\pi$
$431$	5383.16	0.601619	0.300810	+	0.953684i	$0.402743\pi$
$432$	0	0
$433$	−6901.64	−0.765985	−0.382993	−	0.923751i	$-0.625107\pi$
$433$	−6901.64	−0.765985	−0.382993	+	0.923751i	$0.625107\pi$
$434$	0	0
$435$	970.035	0.106919
$436$	0	0
$437$	9064.08	0.992205
$438$	0	0
$439$	12119.4	1.31760	0.658802	−	0.752316i	$-0.271063\pi$
$439$	12119.4	1.31760	0.658802	+	0.752316i	$0.271063\pi$
$440$	0	0
$441$	−6320.30	−0.682464
$442$	0	0
$443$	−5229.40	−0.560849	−0.280425	−	0.959876i	$-0.590475\pi$
$443$	−5229.40	−0.560849	−0.280425	+	0.959876i	$0.590475\pi$
$444$	0	0
$445$	2866.77	0.305389
$446$	0	0
$447$	5876.64	0.621824
$448$	0	0
$449$	−10220.3	−1.07422	−0.537112	−	0.843511i	$-0.680485\pi$
$449$	−10220.3	−1.07422	−0.537112	+	0.843511i	$0.680485\pi$
$450$	0	0
$451$	−2301.20	−0.240264
$452$	0	0
$453$	3735.13	0.387399
$454$	0	0
$455$	1446.11	0.148999
$456$	0	0
$457$	12030.6	1.23143	0.615717	−	0.787967i	$-0.288866\pi$
$457$	12030.6	1.23143	0.615717	+	0.787967i	$0.288866\pi$
$458$	0	0
$459$	−2496.72	−0.253893
$460$	0	0
$461$	−1063.24	−0.107418	−0.0537092	−	0.998557i	$-0.517104\pi$
$461$	−1063.24	−0.107418	−0.0537092	+	0.998557i	$0.517104\pi$
$462$	0	0
$463$	15498.8	1.55570	0.777851	−	0.628449i	$-0.216310\pi$
$463$	15498.8	1.55570	0.777851	+	0.628449i	$0.216310\pi$
$464$	0	0
$465$	−10034.7	−1.00074
$466$	0	0
$467$	−13215.6	−1.30952	−0.654759	−	0.755838i	$-0.727230\pi$
$467$	−13215.6	−1.30952	−0.654759	+	0.755838i	$0.727230\pi$
$468$	0	0
$469$	−13935.7	−1.37204
$470$	0	0
$471$	−8306.01	−0.812571
$472$	0	0
$473$	−3393.38	−0.329869
$474$	0	0
$475$	1368.91	0.132231
$476$	0	0
$477$	−5309.92	−0.509695
$478$	0	0
$479$	−19058.5	−1.81796	−0.908981	−	0.416837i	$-0.863139\pi$
$479$	−19058.5	−1.81796	−0.908981	+	0.416837i	$0.863139\pi$
$480$	0	0
$481$	−161.031	−0.0152648
$482$	0	0
$483$	19520.1	1.83891
$484$	0	0
$485$	−4159.49	−0.389428
$486$	0	0
$487$	−487.214	−0.0453343	−0.0226671	−	0.999743i	$-0.507216\pi$
$487$	−487.214	−0.0453343	−0.0226671	+	0.999743i	$0.507216\pi$
$488$	0	0
$489$	−3661.94	−0.338647
$490$	0	0
$491$	6462.89	0.594025	0.297013	−	0.954874i	$-0.404010\pi$
$491$	6462.89	0.594025	0.297013	+	0.954874i	$0.404010\pi$
$492$	0	0
$493$	−498.563	−0.0455460
$494$	0	0
$495$	7665.58	0.696045
$496$	0	0
$497$	−13281.4	−1.19870
$498$	0	0
$499$	7947.79	0.713010	0.356505	−	0.934293i	$-0.383968\pi$
$499$	7947.79	0.713010	0.356505	+	0.934293i	$0.383968\pi$
$500$	0	0
$501$	4538.24	0.404698
$502$	0	0
$503$	−2307.81	−0.204573	−0.102287	−	0.994755i	$-0.532616\pi$
$503$	−2307.81	−0.204573	−0.102287	+	0.994755i	$0.532616\pi$
$504$	0	0
$505$	−11754.2	−1.03576
$506$	0	0
$507$	−7477.79	−0.655030
$508$	0	0
$509$	16496.2	1.43651	0.718253	−	0.695782i	$-0.244942\pi$
$509$	16496.2	1.43651	0.718253	+	0.695782i	$0.244942\pi$
$510$	0	0
$511$	−85.8287	−0.00743021
$512$	0	0
$513$	6419.33	0.552476
$514$	0	0
$515$	−10007.9	−0.856314
$516$	0	0
$517$	−26148.0	−2.22435
$518$	0	0
$519$	2255.28	0.190743
$520$	0	0
$521$	−10386.5	−0.873398	−0.436699	−	0.899608i	$-0.643853\pi$
$521$	−10386.5	−0.873398	−0.436699	+	0.899608i	$0.643853\pi$
$522$	0	0
$523$	918.174	0.0767666	0.0383833	−	0.999263i	$-0.487779\pi$
$523$	918.174	0.0767666	0.0383833	+	0.999263i	$0.487779\pi$
$524$	0	0
$525$	2948.03	0.245072
$526$	0	0
$527$	5157.45	0.426304
$528$	0	0
$529$	29882.7	2.45604
$530$	0	0
$531$	7888.36	0.644681
$532$	0	0
$533$	−237.559	−0.0193055
$534$	0	0
$535$	−11404.2	−0.921580
$536$	0	0
$537$	−7089.68	−0.569725
$538$	0	0
$539$	21909.7	1.75086
$540$	0	0
$541$	12860.2	1.02200	0.511001	−	0.859580i	$-0.329275\pi$
$541$	12860.2	1.02200	0.511001	+	0.859580i	$0.329275\pi$
$542$	0	0
$543$	−11722.6	−0.926456
$544$	0	0
$545$	−21021.6	−1.65223
$546$	0	0
$547$	−4471.84	−0.349547	−0.174774	−	0.984609i	$-0.555919\pi$
$547$	−4471.84	−0.349547	−0.174774	+	0.984609i	$0.555919\pi$
$548$	0	0
$549$	7318.05	0.568901
$550$	0	0
$551$	1281.86	0.0991090
$552$	0	0
$553$	28789.9	2.21387
$554$	0	0
$555$	996.727	0.0762319
$556$	0	0
$557$	−2649.40	−0.201542	−0.100771	−	0.994910i	$-0.532131\pi$
$557$	−2649.40	−0.201542	−0.100771	+	0.994910i	$0.532131\pi$
$558$	0	0
$559$	−350.309	−0.0265053
$560$	0	0
$561$	3104.42	0.233634
$562$	0	0
$563$	−12615.8	−0.944395	−0.472198	−	0.881493i	$-0.656539\pi$
$563$	−12615.8	−0.944395	−0.472198	+	0.881493i	$0.656539\pi$
$564$	0	0
$565$	−15744.5	−1.17234
$566$	0	0
$567$	2572.81	0.190561
$568$	0	0
$569$	617.443	0.0454913	0.0227456	−	0.999741i	$-0.492759\pi$
$569$	617.443	0.0454913	0.0227456	+	0.999741i	$0.492759\pi$
$570$	0	0
$571$	−8005.50	−0.586725	−0.293362	−	0.956001i	$-0.594774\pi$
$571$	−8005.50	−0.586725	−0.293362	+	0.956001i	$0.594774\pi$
$572$	0	0
$573$	16743.5	1.22072
$574$	0	0
$575$	6350.59	0.460588
$576$	0	0
$577$	11761.1	0.848565	0.424282	−	0.905530i	$-0.360526\pi$
$577$	11761.1	0.848565	0.424282	+	0.905530i	$0.360526\pi$
$578$	0	0
$579$	−16824.0	−1.20757
$580$	0	0
$581$	−27791.3	−1.98447
$582$	0	0
$583$	18407.1	1.30762
$584$	0	0
$585$	791.340	0.0559280
$586$	0	0
$587$	1211.03	0.0851528	0.0425764	−	0.999093i	$-0.486443\pi$
$587$	1211.03	0.0851528	0.0425764	+	0.999093i	$0.486443\pi$
$588$	0	0
$589$	−13260.4	−0.927646
$590$	0	0
$591$	3557.86	0.247633
$592$	0	0
$593$	459.376	0.0318117	0.0159058	−	0.999873i	$-0.494937\pi$
$593$	459.376	0.0318117	0.0159058	+	0.999873i	$0.494937\pi$
$594$	0	0
$595$	4600.47	0.316976
$596$	0	0
$597$	−4027.97	−0.276137
$598$	0	0
$599$	16686.2	1.13820	0.569099	−	0.822269i	$-0.307292\pi$
$599$	16686.2	1.13820	0.569099	+	0.822269i	$0.307292\pi$
$600$	0	0
$601$	11625.9	0.789072	0.394536	−	0.918881i	$-0.370905\pi$
$601$	11625.9	0.789072	0.394536	+	0.918881i	$0.370905\pi$
$602$	0	0
$603$	−7625.86	−0.515007
$604$	0	0
$605$	−13666.5	−0.918384
$606$	0	0
$607$	−21556.9	−1.44146	−0.720731	−	0.693215i	$-0.756194\pi$
$607$	−21556.9	−1.44146	−0.720731	+	0.693215i	$0.756194\pi$
$608$	0	0
$609$	2760.56	0.183684
$610$	0	0
$611$	−2699.34	−0.178729
$612$	0	0
$613$	19273.2	1.26988	0.634940	−	0.772561i	$-0.281025\pi$
$613$	19273.2	1.26988	0.634940	+	0.772561i	$0.281025\pi$
$614$	0	0
$615$	1470.41	0.0964110
$616$	0	0
$617$	−3270.70	−0.213409	−0.106704	−	0.994291i	$-0.534030\pi$
$617$	−3270.70	−0.213409	−0.106704	+	0.994291i	$0.534030\pi$
$618$	0	0
$619$	−75.8068	−0.00492234	−0.00246117	−	0.999997i	$-0.500783\pi$
$619$	−75.8068	−0.00492234	−0.00246117	+	0.999997i	$0.500783\pi$
$620$	0	0
$621$	29780.3	1.92438
$622$	0	0
$623$	8158.37	0.524652
$624$	0	0
$625$	−10794.7	−0.690862
$626$	0	0
$627$	−7981.81	−0.508393
$628$	0	0
$629$	−512.282	−0.0324738
$630$	0	0
$631$	−4915.71	−0.310129	−0.155064	−	0.987904i	$-0.549559\pi$
$631$	−4915.71	−0.310129	−0.155064	+	0.987904i	$0.549559\pi$
$632$	0	0
$633$	−2740.07	−0.172051
$634$	0	0
$635$	−3233.67	−0.202086
$636$	0	0
$637$	2261.80	0.140684
$638$	0	0
$639$	−7267.86	−0.449941
$640$	0	0
$641$	3317.63	0.204428	0.102214	−	0.994762i	$-0.467407\pi$
$641$	3317.63	0.204428	0.102214	+	0.994762i	$0.467407\pi$
$642$	0	0
$643$	27222.7	1.66961	0.834805	−	0.550546i	$-0.185580\pi$
$643$	27222.7	1.66961	0.834805	+	0.550546i	$0.185580\pi$
$644$	0	0
$645$	2168.29	0.132367
$646$	0	0
$647$	24433.6	1.48468	0.742338	−	0.670025i	$-0.233717\pi$
$647$	24433.6	1.48468	0.742338	+	0.670025i	$0.233717\pi$
$648$	0	0
$649$	−27345.4	−1.65393
$650$	0	0
$651$	−28557.0	−1.71926
$652$	0	0
$653$	−5755.36	−0.344907	−0.172454	−	0.985018i	$-0.555170\pi$
$653$	−5755.36	−0.344907	−0.172454	+	0.985018i	$0.555170\pi$
$654$	0	0
$655$	−9052.01	−0.539987
$656$	0	0
$657$	−46.9671	−0.00278898
$658$	0	0
$659$	−965.300	−0.0570603	−0.0285301	−	0.999593i	$-0.509083\pi$
$659$	−965.300	−0.0570603	−0.0285301	+	0.999593i	$0.509083\pi$
$660$	0	0
$661$	22791.0	1.34110	0.670549	−	0.741866i	$-0.266059\pi$
$661$	22791.0	1.34110	0.670549	+	0.741866i	$0.266059\pi$
$662$	0	0
$663$	320.479	0.0187728
$664$	0	0
$665$	−11828.3	−0.689746
$666$	0	0
$667$	5946.75	0.345216
$668$	0	0
$669$	17716.9	1.02388
$670$	0	0
$671$	−25368.4	−1.45952
$672$	0	0
$673$	12412.1	0.710925	0.355463	−	0.934690i	$-0.384323\pi$
$673$	12412.1	0.710925	0.355463	+	0.934690i	$0.384323\pi$
$674$	0	0
$675$	4497.59	0.256463
$676$	0	0
$677$	−8998.03	−0.510816	−0.255408	−	0.966833i	$-0.582210\pi$
$677$	−8998.03	−0.510816	−0.255408	+	0.966833i	$0.582210\pi$
$678$	0	0
$679$	−11837.2	−0.669030
$680$	0	0
$681$	17360.7	0.976893
$682$	0	0
$683$	−13439.1	−0.752904	−0.376452	−	0.926436i	$-0.622856\pi$
$683$	−13439.1	−0.752904	−0.376452	+	0.926436i	$0.622856\pi$
$684$	0	0
$685$	10572.8	0.589730
$686$	0	0
$687$	21432.1	1.19023
$688$	0	0
$689$	1900.22	0.105069
$690$	0	0
$691$	16220.4	0.892983	0.446492	−	0.894788i	$-0.352673\pi$
$691$	16220.4	0.892983	0.446492	+	0.894788i	$0.352673\pi$
$692$	0	0
$693$	21815.0	1.19579
$694$	0	0
$695$	−18662.7	−1.01858
$696$	0	0
$697$	−755.739	−0.0410698
$698$	0	0
$699$	−10754.1	−0.581912
$700$	0	0
$701$	18919.6	1.01938	0.509689	−	0.860359i	$-0.329761\pi$
$701$	18919.6	1.01938	0.509689	+	0.860359i	$0.329761\pi$
$702$	0	0
$703$	1317.13	0.0706636
$704$	0	0
$705$	16708.0	0.892567
$706$	0	0
$707$	−33450.6	−1.77941
$708$	0	0
$709$	4781.99	0.253302	0.126651	−	0.991947i	$-0.459577\pi$
$709$	4781.99	0.253302	0.126651	+	0.991947i	$0.459577\pi$
$710$	0	0
$711$	15754.4	0.830992
$712$	0	0
$713$	−61516.9	−3.23117
$714$	0	0
$715$	−2743.22	−0.143483
$716$	0	0
$717$	24812.5	1.29238
$718$	0	0
$719$	13020.1	0.675336	0.337668	−	0.941265i	$-0.390362\pi$
$719$	13020.1	0.675336	0.337668	+	0.941265i	$0.390362\pi$
$720$	0	0
$721$	−28480.9	−1.47113
$722$	0	0
$723$	−2888.42	−0.148577
$724$	0	0
$725$	898.112	0.0460070
$726$	0	0
$727$	−8738.68	−0.445804	−0.222902	−	0.974841i	$-0.571553\pi$
$727$	−8738.68	−0.445804	−0.222902	+	0.974841i	$0.571553\pi$
$728$	0	0
$729$	14933.8	0.758715
$730$	0	0
$731$	−1114.43	−0.0563865
$732$	0	0
$733$	12775.2	0.643744	0.321872	−	0.946783i	$-0.395688\pi$
$733$	12775.2	0.643744	0.321872	+	0.946783i	$0.395688\pi$
$734$	0	0
$735$	−13999.8	−0.702571
$736$	0	0
$737$	26435.4	1.32125
$738$	0	0
$739$	−5397.68	−0.268683	−0.134342	−	0.990935i	$-0.542892\pi$
$739$	−5397.68	−0.268683	−0.134342	+	0.990935i	$0.542892\pi$
$740$	0	0
$741$	−823.985	−0.0408500
$742$	0	0
$743$	−37984.2	−1.87551	−0.937757	−	0.347293i	$-0.887101\pi$
$743$	−37984.2	−1.87551	−0.937757	+	0.347293i	$0.887101\pi$
$744$	0	0
$745$	−16519.9	−0.812408
$746$	0	0
$747$	−15207.9	−0.744886
$748$	0	0
$749$	−32454.4	−1.58326
$750$	0	0
$751$	26384.8	1.28202	0.641009	−	0.767534i	$-0.278516\pi$
$751$	26384.8	1.28202	0.641009	+	0.767534i	$0.278516\pi$
$752$	0	0
$753$	−14855.4	−0.718940
$754$	0	0
$755$	−10499.9	−0.506133
$756$	0	0
$757$	−6889.95	−0.330805	−0.165403	−	0.986226i	$-0.552892\pi$
$757$	−6889.95	−0.330805	−0.165403	+	0.986226i	$0.552892\pi$
$758$	0	0
$759$	−37028.8	−1.77083
$760$	0	0
$761$	31579.3	1.50427	0.752134	−	0.659010i	$-0.229025\pi$
$761$	31579.3	1.50427	0.752134	+	0.659010i	$0.229025\pi$
$762$	0	0
$763$	−59824.0	−2.83850
$764$	0	0
$765$	2517.46	0.118979
$766$	0	0
$767$	−2822.94	−0.132895
$768$	0	0
$769$	−12232.7	−0.573632	−0.286816	−	0.957986i	$-0.592597\pi$
$769$	−12232.7	−0.573632	−0.286816	+	0.957986i	$0.592597\pi$
$770$	0	0
$771$	1384.07	0.0646510
$772$	0	0
$773$	−9958.46	−0.463365	−0.231682	−	0.972792i	$-0.574423\pi$
$773$	−9958.46	−0.463365	−0.231682	+	0.972792i	$0.574423\pi$
$774$	0	0
$775$	−9290.64	−0.430619
$776$	0	0
$777$	2836.52	0.130965
$778$	0	0
$779$	1943.09	0.0893688
$780$	0	0
$781$	25194.4	1.15432
$782$	0	0
$783$	4211.58	0.192222
$784$	0	0
$785$	23349.2	1.06162
$786$	0	0
$787$	−103.911	−0.00470653	−0.00235326	−	0.999997i	$-0.500749\pi$
$787$	−103.911	−0.00470653	−0.00235326	+	0.999997i	$0.500749\pi$
$788$	0	0
$789$	−9254.02	−0.417556
$790$	0	0
$791$	−44806.2	−2.01406
$792$	0	0
$793$	−2618.85	−0.117274
$794$	0	0
$795$	−11761.7	−0.524711
$796$	0	0
$797$	−30624.4	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$797$	−30624.4	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$798$	0	0
$799$	−8587.31	−0.380222
$800$	0	0
$801$	4464.42	0.196932
$802$	0	0
$803$	162.814	0.00715514
$804$	0	0
$805$	−54873.3	−2.40252
$806$	0	0
$807$	16324.3	0.712072
$808$	0	0
$809$	−7737.50	−0.336262	−0.168131	−	0.985765i	$-0.553773\pi$
$809$	−7737.50	−0.336262	−0.168131	+	0.985765i	$0.553773\pi$
$810$	0	0
$811$	12974.1	0.561756	0.280878	−	0.959744i	$-0.409374\pi$
$811$	12974.1	0.561756	0.280878	+	0.959744i	$0.409374\pi$
$812$	0	0
$813$	15625.0	0.674038
$814$	0	0
$815$	10294.2	0.442440
$816$	0	0
$817$	2865.31	0.122698
$818$	0	0
$819$	2252.02	0.0960831
$820$	0	0
$821$	−3264.26	−0.138762	−0.0693809	−	0.997590i	$-0.522102\pi$
$821$	−3264.26	−0.138762	−0.0693809	+	0.997590i	$0.522102\pi$
$822$	0	0
$823$	23509.7	0.995745	0.497872	−	0.867250i	$-0.334115\pi$
$823$	23509.7	0.995745	0.497872	+	0.867250i	$0.334115\pi$
$824$	0	0
$825$	−5592.31	−0.235999
$826$	0	0
$827$	−27116.5	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$827$	−27116.5	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$828$	0	0
$829$	−15393.6	−0.644923	−0.322462	−	0.946583i	$-0.604510\pi$
$829$	−15393.6	−0.644923	−0.322462	+	0.946583i	$0.604510\pi$
$830$	0	0
$831$	−24693.8	−1.03083
$832$	0	0
$833$	7195.38	0.299286
$834$	0	0
$835$	−12757.5	−0.528734
$836$	0	0
$837$	−43567.3	−1.79917
$838$	0	0
$839$	−7906.51	−0.325343	−0.162672	−	0.986680i	$-0.552011\pi$
$839$	−7906.51	−0.325343	−0.162672	+	0.986680i	$0.552011\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−19902.7	−0.813150
$844$	0	0
$845$	21021.0	0.855791
$846$	0	0
$847$	−38892.6	−1.57777
$848$	0	0
$849$	13160.7	0.532005
$850$	0	0
$851$	6110.38	0.246135
$852$	0	0
$853$	11887.1	0.477146	0.238573	−	0.971125i	$-0.423320\pi$
$853$	11887.1	0.477146	0.238573	+	0.971125i	$0.423320\pi$
$854$	0	0
$855$	−6472.67	−0.258901
$856$	0	0
$857$	−655.328	−0.0261208	−0.0130604	−	0.999915i	$-0.504157\pi$
$857$	−655.328	−0.0261208	−0.0130604	+	0.999915i	$0.504157\pi$
$858$	0	0
$859$	48758.4	1.93669	0.968345	−	0.249617i	$-0.0803048\pi$
$859$	48758.4	1.93669	0.968345	+	0.249617i	$0.0803048\pi$
$860$	0	0
$861$	4184.55	0.165632
$862$	0	0
$863$	−10398.2	−0.410150	−0.205075	−	0.978746i	$-0.565744\pi$
$863$	−10398.2	−0.410150	−0.205075	+	0.978746i	$0.565744\pi$
$864$	0	0
$865$	−6339.86	−0.249204
$866$	0	0
$867$	−15927.8	−0.623918
$868$	0	0
$869$	−54613.4	−2.13191
$870$	0	0
$871$	2729.01	0.106164
$872$	0	0
$873$	−6477.57	−0.251125
$874$	0	0
$875$	−41736.8	−1.61253
$876$	0	0
$877$	31024.2	1.19454	0.597272	−	0.802039i	$-0.296251\pi$
$877$	31024.2	1.19454	0.597272	+	0.802039i	$0.296251\pi$
$878$	0	0
$879$	−18684.7	−0.716973
$880$	0	0
$881$	−1510.50	−0.0577641	−0.0288821	−	0.999583i	$-0.509195\pi$
$881$	−1510.50	−0.0577641	−0.0288821	+	0.999583i	$0.509195\pi$
$882$	0	0
$883$	32376.4	1.23392	0.616961	−	0.786993i	$-0.288363\pi$
$883$	32376.4	1.23392	0.616961	+	0.786993i	$0.288363\pi$
$884$	0	0
$885$	17473.1	0.663674
$886$	0	0
$887$	10474.6	0.396510	0.198255	−	0.980150i	$-0.436473\pi$
$887$	10474.6	0.396510	0.198255	+	0.980150i	$0.436473\pi$
$888$	0	0
$889$	−9202.51	−0.347179
$890$	0	0
$891$	−4880.53	−0.183506
$892$	0	0
$893$	22078.9	0.827371
$894$	0	0
$895$	19930.0	0.744341
$896$	0	0
$897$	−3822.59	−0.142288
$898$	0	0
$899$	−8699.84	−0.322754
$900$	0	0
$901$	6045.11	0.223520
$902$	0	0
$903$	6170.62	0.227403
$904$	0	0
$905$	32953.7	1.21041
$906$	0	0
$907$	34634.3	1.26793	0.633965	−	0.773362i	$-0.281426\pi$
$907$	34634.3	1.26793	0.633965	+	0.773362i	$0.281426\pi$
$908$	0	0
$909$	−18304.8	−0.667913
$910$	0	0
$911$	1186.14	0.0431377	0.0215688	−	0.999767i	$-0.493134\pi$
$911$	1186.14	0.0431377	0.0215688	+	0.999767i	$0.493134\pi$
$912$	0	0
$913$	52719.1	1.91101
$914$	0	0
$915$	16209.8	0.585662
$916$	0	0
$917$	−25760.6	−0.927687
$918$	0	0
$919$	−5685.58	−0.204080	−0.102040	−	0.994780i	$-0.532537\pi$
$919$	−5685.58	−0.204080	−0.102040	+	0.994780i	$0.532537\pi$
$920$	0	0
$921$	−17002.7	−0.608316
$922$	0	0
$923$	2600.89	0.0927512
$924$	0	0
$925$	922.824	0.0328025
$926$	0	0
$927$	−15585.3	−0.552200
$928$	0	0
$929$	−19143.7	−0.676087	−0.338044	−	0.941130i	$-0.609765\pi$
$929$	−19143.7	−0.676087	−0.338044	+	0.941130i	$0.609765\pi$
$930$	0	0
$931$	−18500.1	−0.651252
$932$	0	0
$933$	3856.28	0.135315
$934$	0	0
$935$	−8726.92	−0.305241
$936$	0	0
$937$	−53623.5	−1.86959	−0.934794	−	0.355191i	$-0.884416\pi$
$937$	−53623.5	−1.86959	−0.934794	+	0.355191i	$0.884416\pi$
$938$	0	0
$939$	7010.08	0.243626
$940$	0	0
$941$	7947.57	0.275328	0.137664	−	0.990479i	$-0.456041\pi$
$941$	7947.57	0.275328	0.137664	+	0.990479i	$0.456041\pi$
$942$	0	0
$943$	9014.28	0.311289
$944$	0	0
$945$	−38862.2	−1.33776
$946$	0	0
$947$	41047.8	1.40853	0.704263	−	0.709939i	$-0.251277\pi$
$947$	41047.8	1.40853	0.704263	+	0.709939i	$0.251277\pi$
$948$	0	0
$949$	16.8077	0.000574924 0
$950$	0	0
$951$	−7077.49	−0.241328
$952$	0	0
$953$	−12000.8	−0.407915	−0.203957	−	0.978980i	$-0.565380\pi$
$953$	−12000.8	−0.407915	−0.203957	+	0.978980i	$0.565380\pi$
$954$	0	0
$955$	−47068.1	−1.59486
$956$	0	0
$957$	−5236.69	−0.176884
$958$	0	0
$959$	30088.4	1.01314
$960$	0	0
$961$	60205.6	2.02093
$962$	0	0
$963$	−17759.7	−0.594287
$964$	0	0
$965$	47294.3	1.57768
$966$	0	0
$967$	−55391.0	−1.84204	−0.921021	−	0.389514i	$-0.872643\pi$
$967$	−55391.0	−1.84204	−0.921021	+	0.389514i	$0.872643\pi$
$968$	0	0
$969$	−2621.32	−0.0869028
$970$	0	0
$971$	23937.0	0.791117	0.395558	−	0.918441i	$-0.370551\pi$
$971$	23937.0	0.791117	0.395558	+	0.918441i	$0.370551\pi$
$972$	0	0
$973$	−53111.0	−1.74991
$974$	0	0
$975$	−577.310	−0.0189628
$976$	0	0
$977$	−43982.3	−1.44024	−0.720122	−	0.693847i	$-0.755914\pi$
$977$	−43982.3	−1.44024	−0.720122	+	0.693847i	$0.755914\pi$
$978$	0	0
$979$	−15476.1	−0.505229
$980$	0	0
$981$	−32736.8	−1.06545
$982$	0	0
$983$	2724.78	0.0884100	0.0442050	−	0.999022i	$-0.485925\pi$
$983$	2724.78	0.0884100	0.0442050	+	0.999022i	$0.485925\pi$
$984$	0	0
$985$	−10001.6	−0.323530
$986$	0	0
$987$	47548.3	1.53341
$988$	0	0
$989$	13292.6	0.427382
$990$	0	0
$991$	−23748.6	−0.761250	−0.380625	−	0.924729i	$-0.624291\pi$
$991$	−23748.6	−0.761250	−0.380625	+	0.924729i	$0.624291\pi$
$992$	0	0
$993$	33297.7	1.06412
$994$	0	0
$995$	11323.1	0.360771
$996$	0	0
$997$	−24718.0	−0.785182	−0.392591	−	0.919713i	$-0.628421\pi$
$997$	−24718.0	−0.785182	−0.392591	+	0.919713i	$0.628421\pi$
$998$	0	0
$999$	4327.47	0.137052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.l.1.2		2
4.3	odd	2		1856.4.a.i.1.1		2
8.3	odd	2		464.4.a.e.1.2		2
8.5	even	2		58.4.a.c.1.1	✓	2
24.5	odd	2		522.4.a.j.1.1		2
40.29	even	2		1450.4.a.g.1.2		2
232.173	even	2		1682.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.c.1.1	✓	2	8.5	even	2
464.4.a.e.1.2		2	8.3	odd	2
522.4.a.j.1.1		2	24.5	odd	2
1450.4.a.g.1.2		2	40.29	even	2
1682.4.a.c.1.2		2	232.173	even	2
1856.4.a.i.1.1		2	4.3	odd	2
1856.4.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +O(q^{10})\)	\(q+3.44949 q^{3} -9.69694 q^{5} -27.5959 q^{7} -15.1010 q^{9} +52.3485 q^{11} +5.40408 q^{13} -33.4495 q^{15} +17.1918 q^{17} -44.2020 q^{19} -95.1918 q^{21} -205.060 q^{23} -30.9694 q^{25} -145.227 q^{27} -29.0000 q^{29} +299.994 q^{31} +180.576 q^{33} +267.596 q^{35} -29.7980 q^{37} +18.6413 q^{39} -43.9592 q^{41} -64.8230 q^{43} +146.434 q^{45} -499.499 q^{47} +418.535 q^{49} +59.3031 q^{51} +351.627 q^{53} -507.620 q^{55} -152.474 q^{57} -522.372 q^{59} -484.606 q^{61} +416.727 q^{63} -52.4031 q^{65} +504.990 q^{67} -707.353 q^{69} +481.283 q^{71} +3.11019 q^{73} -106.829 q^{75} -1444.60 q^{77} -1043.27 q^{79} -93.2316 q^{81} +1007.08 q^{83} -166.708 q^{85} -100.035 q^{87} -295.637 q^{89} -149.131 q^{91} +1034.83 q^{93} +428.624 q^{95} +428.949 q^{97} -790.515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 10 * q^5 - 16 * q^7 - 40 * q^9	\( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + 90 q^{11} + 50 q^{13} - 62 q^{15} - 44 q^{17} - 108 q^{19} - 112 q^{21} - 28 q^{23} + 232 q^{25} - 70 q^{27} - 58 q^{29} + 66 q^{31} + 126 q^{33} + 496 q^{35} - 40 q^{37} - 46 q^{39} + 304 q^{41} + 130 q^{43} - 344 q^{45} - 514 q^{47} + 210 q^{49} + 148 q^{51} + 958 q^{53} + 234 q^{55} - 60 q^{57} + 180 q^{59} - 1028 q^{61} + 128 q^{63} + 826 q^{65} + 912 q^{67} - 964 q^{69} + 796 q^{71} - 856 q^{73} - 488 q^{75} - 1008 q^{77} - 318 q^{79} + 470 q^{81} + 1828 q^{83} - 1372 q^{85} - 58 q^{87} - 944 q^{89} + 368 q^{91} + 1374 q^{93} - 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 10 * q^5 - 16 * q^7 - 40 * q^9 + 90 * q^11 + 50 * q^13 - 62 * q^15 - 44 * q^17 - 108 * q^19 - 112 * q^21 - 28 * q^23 + 232 * q^25 - 70 * q^27 - 58 * q^29 + 66 * q^31 + 126 * q^33 + 496 * q^35 - 40 * q^37 - 46 * q^39 + 304 * q^41 + 130 * q^43 - 344 * q^45 - 514 * q^47 + 210 * q^49 + 148 * q^51 + 958 * q^53 + 234 * q^55 - 60 * q^57 + 180 * q^59 - 1028 * q^61 + 128 * q^63 + 826 * q^65 + 912 * q^67 - 964 * q^69 + 796 * q^71 - 856 * q^73 - 488 * q^75 - 1008 * q^77 - 318 * q^79 + 470 * q^81 + 1828 * q^83 - 1372 * q^85 - 58 * q^87 - 944 * q^89 + 368 * q^91 + 1374 * q^93 - 828 * q^95 + 368 * q^97 - 1728 * q^99

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