LMFDB - Embedded newform 1856.4.a.be.1.6

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 19x^{6} + 92x^{4} - 51x^{2} + 5 $ x^8 - 19x^6 + 92x^4 - 51*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.34163$ 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)

$f(q)$	$=$	$q+3.89793 q^{3} +15.3330 q^{5} -5.27577 q^{7} -11.8061 q^{9} +O(q^{10})$	$q+3.89793 q^{3} +15.3330 q^{5} -5.27577 q^{7} -11.8061 q^{9} +28.3477 q^{11} -61.6285 q^{13} +59.7668 q^{15} +39.9279 q^{17} -146.682 q^{19} -20.5646 q^{21} +43.8973 q^{23} +110.099 q^{25} -151.264 q^{27} -29.0000 q^{29} -169.034 q^{31} +110.497 q^{33} -80.8931 q^{35} -69.8939 q^{37} -240.224 q^{39} -229.327 q^{41} +150.534 q^{43} -181.023 q^{45} +237.886 q^{47} -315.166 q^{49} +155.636 q^{51} +218.454 q^{53} +434.654 q^{55} -571.757 q^{57} -115.370 q^{59} -211.782 q^{61} +62.2865 q^{63} -944.947 q^{65} -442.508 q^{67} +171.109 q^{69} -663.263 q^{71} -676.680 q^{73} +429.160 q^{75} -149.556 q^{77} +1319.35 q^{79} -270.849 q^{81} -1266.38 q^{83} +612.212 q^{85} -113.040 q^{87} -1356.44 q^{89} +325.138 q^{91} -658.884 q^{93} -2249.07 q^{95} +823.177 q^{97} -334.678 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) $ 8 * q + 20 * q^5 + 40 * q^9	$ 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) $ 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

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
$3$	3.89793	0.750157	0.375078	−	0.926993i	$-0.377616\pi$
$3$	3.89793	0.750157	0.375078	+	0.926993i	$0.377616\pi$
$4$	0	0
$5$	15.3330	1.37142	0.685710	−	0.727874i	$-0.259492\pi$
$5$	15.3330	1.37142	0.685710	+	0.727874i	$0.259492\pi$
$6$	0	0
$7$	−5.27577	−0.284865	−0.142432	−	0.989805i	$-0.545492\pi$
$7$	−5.27577	−0.284865	−0.142432	+	0.989805i	$0.545492\pi$
$8$	0	0
$9$	−11.8061	−0.437265
$10$	0	0
$11$	28.3477	0.777015	0.388507	−	0.921446i	$-0.372991\pi$
$11$	28.3477	0.777015	0.388507	+	0.921446i	$0.372991\pi$
$12$	0	0
$13$	−61.6285	−1.31482	−0.657411	−	0.753533i	$-0.728348\pi$
$13$	−61.6285	−1.31482	−0.657411	+	0.753533i	$0.728348\pi$
$14$	0	0
$15$	59.7668	1.02878
$16$	0	0
$17$	39.9279	0.569643	0.284822	−	0.958581i	$-0.408066\pi$
$17$	39.9279	0.569643	0.284822	+	0.958581i	$0.408066\pi$
$18$	0	0
$19$	−146.682	−1.77112	−0.885558	−	0.464528i	$-0.846224\pi$
$19$	−146.682	−1.77112	−0.885558	+	0.464528i	$0.846224\pi$
$20$	0	0
$21$	−20.5646	−0.213693
$22$	0	0
$23$	43.8973	0.397966	0.198983	−	0.980003i	$-0.436236\pi$
$23$	43.8973	0.397966	0.198983	+	0.980003i	$0.436236\pi$
$24$	0	0
$25$	110.099	0.880795
$26$	0	0
$27$	−151.264	−1.07817
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−169.034	−0.979337	−0.489669	−	0.871909i	$-0.662882\pi$
$31$	−169.034	−0.979337	−0.489669	+	0.871909i	$0.662882\pi$
$32$	0	0
$33$	110.497	0.582883
$34$	0	0
$35$	−80.8931	−0.390670
$36$	0	0
$37$	−69.8939	−0.310554	−0.155277	−	0.987871i	$-0.549627\pi$
$37$	−69.8939	−0.310554	−0.155277	+	0.987871i	$0.549627\pi$
$38$	0	0
$39$	−240.224	−0.986322
$40$	0	0
$41$	−229.327	−0.873533	−0.436766	−	0.899575i	$-0.643876\pi$
$41$	−229.327	−0.873533	−0.436766	+	0.899575i	$0.643876\pi$
$42$	0	0
$43$	150.534	0.533864	0.266932	−	0.963715i	$-0.413990\pi$
$43$	150.534	0.533864	0.266932	+	0.963715i	$0.413990\pi$
$44$	0	0
$45$	−181.023	−0.599674
$46$	0	0
$47$	237.886	0.738282	0.369141	−	0.929373i	$-0.379652\pi$
$47$	237.886	0.738282	0.369141	+	0.929373i	$0.379652\pi$
$48$	0	0
$49$	−315.166	−0.918852
$50$	0	0
$51$	155.636	0.427322
$52$	0	0
$53$	218.454	0.566169	0.283084	−	0.959095i	$-0.408642\pi$
$53$	218.454	0.566169	0.283084	+	0.959095i	$0.408642\pi$
$54$	0	0
$55$	434.654	1.06561
$56$	0	0
$57$	−571.757	−1.32862
$58$	0	0
$59$	−115.370	−0.254575	−0.127287	−	0.991866i	$-0.540627\pi$
$59$	−115.370	−0.254575	−0.127287	+	0.991866i	$0.540627\pi$
$60$	0	0
$61$	−211.782	−0.444524	−0.222262	−	0.974987i	$-0.571344\pi$
$61$	−211.782	−0.444524	−0.222262	+	0.974987i	$0.571344\pi$
$62$	0	0
$63$	62.2865	0.124561
$64$	0	0
$65$	−944.947	−1.80317
$66$	0	0
$67$	−442.508	−0.806880	−0.403440	−	0.915006i	$-0.632186\pi$
$67$	−442.508	−0.806880	−0.403440	+	0.915006i	$0.632186\pi$
$68$	0	0
$69$	171.109	0.298537
$70$	0	0
$71$	−663.263	−1.10866	−0.554330	−	0.832297i	$-0.687025\pi$
$71$	−663.263	−1.10866	−0.554330	+	0.832297i	$0.687025\pi$
$72$	0	0
$73$	−676.680	−1.08492	−0.542462	−	0.840081i	$-0.682508\pi$
$73$	−676.680	−1.08492	−0.542462	+	0.840081i	$0.682508\pi$
$74$	0	0
$75$	429.160	0.660735
$76$	0	0
$77$	−149.556	−0.221344
$78$	0	0
$79$	1319.35	1.87897	0.939486	−	0.342588i	$-0.111304\pi$
$79$	1319.35	1.87897	0.939486	+	0.342588i	$0.111304\pi$
$80$	0	0
$81$	−270.849	−0.371535
$82$	0	0
$83$	−1266.38	−1.67473	−0.837367	−	0.546641i	$-0.815906\pi$
$83$	−1266.38	−1.67473	−0.837367	+	0.546641i	$0.815906\pi$
$84$	0	0
$85$	612.212	0.781221
$86$	0	0
$87$	−113.040	−0.139301
$88$	0	0
$89$	−1356.44	−1.61553	−0.807763	−	0.589507i	$-0.799322\pi$
$89$	−1356.44	−1.61553	−0.807763	+	0.589507i	$0.799322\pi$
$90$	0	0
$91$	325.138	0.374546
$92$	0	0
$93$	−658.884	−0.734656
$94$	0	0
$95$	−2249.07	−2.42895
$96$	0	0
$97$	823.177	0.861659	0.430830	−	0.902433i	$-0.358221\pi$
$97$	823.177	0.861659	0.430830	+	0.902433i	$0.358221\pi$
$98$	0	0
$99$	−334.678	−0.339761
$100$	0	0
$101$	895.738	0.882468	0.441234	−	0.897392i	$-0.354541\pi$
$101$	895.738	0.882468	0.441234	+	0.897392i	$0.354541\pi$
$102$	0	0
$103$	774.903	0.741296	0.370648	−	0.928773i	$-0.379136\pi$
$103$	774.903	0.741296	0.370648	+	0.928773i	$0.379136\pi$
$104$	0	0
$105$	−315.316	−0.293063
$106$	0	0
$107$	868.181	0.784394	0.392197	−	0.919881i	$-0.371715\pi$
$107$	868.181	0.784394	0.392197	+	0.919881i	$0.371715\pi$
$108$	0	0
$109$	328.137	0.288347	0.144173	−	0.989552i	$-0.453948\pi$
$109$	328.137	0.288347	0.144173	+	0.989552i	$0.453948\pi$
$110$	0	0
$111$	−272.441	−0.232964
$112$	0	0
$113$	−1095.47	−0.911971	−0.455986	−	0.889987i	$-0.650713\pi$
$113$	−1095.47	−0.911971	−0.455986	+	0.889987i	$0.650713\pi$
$114$	0	0
$115$	673.076	0.545779
$116$	0	0
$117$	727.595	0.574925
$118$	0	0
$119$	−210.650	−0.162271
$120$	0	0
$121$	−527.406	−0.396248
$122$	0	0
$123$	−893.900	−0.655286
$124$	0	0
$125$	−228.470	−0.163480
$126$	0	0
$127$	−948.925	−0.663019	−0.331510	−	0.943452i	$-0.607558\pi$
$127$	−948.925	−0.663019	−0.331510	+	0.943452i	$0.607558\pi$
$128$	0	0
$129$	586.769	0.400482
$130$	0	0
$131$	45.1750	0.0301294	0.0150647	−	0.999887i	$-0.495205\pi$
$131$	45.1750	0.0301294	0.0150647	+	0.999887i	$0.495205\pi$
$132$	0	0
$133$	773.862	0.504529
$134$	0	0
$135$	−2319.32	−1.47863
$136$	0	0
$137$	−1006.30	−0.627550	−0.313775	−	0.949497i	$-0.601594\pi$
$137$	−1006.30	−0.627550	−0.313775	+	0.949497i	$0.601594\pi$
$138$	0	0
$139$	−177.789	−0.108488	−0.0542441	−	0.998528i	$-0.517275\pi$
$139$	−177.789	−0.108488	−0.0542441	+	0.998528i	$0.517275\pi$
$140$	0	0
$141$	927.263	0.553827
$142$	0	0
$143$	−1747.03	−1.02164
$144$	0	0
$145$	−444.656	−0.254666
$146$	0	0
$147$	−1228.50	−0.689283
$148$	0	0
$149$	2275.95	1.25136	0.625682	−	0.780078i	$-0.284821\pi$
$149$	2275.95	1.25136	0.625682	+	0.780078i	$0.284821\pi$
$150$	0	0
$151$	22.3187	0.0120283	0.00601414	−	0.999982i	$-0.498086\pi$
$151$	22.3187	0.0120283	0.00601414	+	0.999982i	$0.498086\pi$
$152$	0	0
$153$	−471.395	−0.249085
$154$	0	0
$155$	−2591.79	−1.34308
$156$	0	0
$157$	−2749.00	−1.39741	−0.698707	−	0.715408i	$-0.746241\pi$
$157$	−2749.00	−1.39741	−0.698707	+	0.715408i	$0.746241\pi$
$158$	0	0
$159$	851.517	0.424715
$160$	0	0
$161$	−231.592	−0.113367
$162$	0	0
$163$	1552.50	0.746018	0.373009	−	0.927828i	$-0.378326\pi$
$163$	1552.50	0.746018	0.373009	+	0.927828i	$0.378326\pi$
$164$	0	0
$165$	1694.25	0.799378
$166$	0	0
$167$	2322.26	1.07606	0.538029	−	0.842926i	$-0.319169\pi$
$167$	2322.26	1.07606	0.538029	+	0.842926i	$0.319169\pi$
$168$	0	0
$169$	1601.07	0.728755
$170$	0	0
$171$	1731.75	0.774447
$172$	0	0
$173$	−363.841	−0.159898	−0.0799490	−	0.996799i	$-0.525476\pi$
$173$	−363.841	−0.159898	−0.0799490	+	0.996799i	$0.525476\pi$
$174$	0	0
$175$	−580.859	−0.250908
$176$	0	0
$177$	−449.705	−0.190971
$178$	0	0
$179$	2532.15	1.05733	0.528665	−	0.848831i	$-0.322693\pi$
$179$	2532.15	1.05733	0.528665	+	0.848831i	$0.322693\pi$
$180$	0	0
$181$	1163.94	0.477982	0.238991	−	0.971022i	$-0.423183\pi$
$181$	1163.94	0.477982	0.238991	+	0.971022i	$0.423183\pi$
$182$	0	0
$183$	−825.512	−0.333462
$184$	0	0
$185$	−1071.68	−0.425900
$186$	0	0
$187$	1131.87	0.442621
$188$	0	0
$189$	798.032	0.307134
$190$	0	0
$191$	1677.77	0.635599	0.317799	−	0.948158i	$-0.397056\pi$
$191$	1677.77	0.635599	0.317799	+	0.948158i	$0.397056\pi$
$192$	0	0
$193$	2249.50	0.838978	0.419489	−	0.907760i	$-0.362209\pi$
$193$	2249.50	0.838978	0.419489	+	0.907760i	$0.362209\pi$
$194$	0	0
$195$	−3683.34	−1.35266
$196$	0	0
$197$	−2233.37	−0.807720	−0.403860	−	0.914821i	$-0.632332\pi$
$197$	−2233.37	−0.807720	−0.403860	+	0.914821i	$0.632332\pi$
$198$	0	0
$199$	2986.69	1.06392	0.531962	−	0.846768i	$-0.321455\pi$
$199$	2986.69	1.06392	0.531962	+	0.846768i	$0.321455\pi$
$200$	0	0
$201$	−1724.86	−0.605286
$202$	0	0
$203$	152.997	0.0528981
$204$	0	0
$205$	−3516.26	−1.19798
$206$	0	0
$207$	−518.259	−0.174017
$208$	0	0
$209$	−4158.11	−1.37618
$210$	0	0
$211$	5179.02	1.68975	0.844877	−	0.534960i	$-0.179673\pi$
$211$	5179.02	1.68975	0.844877	+	0.534960i	$0.179673\pi$
$212$	0	0
$213$	−2585.35	−0.831669
$214$	0	0
$215$	2308.13	0.732153
$216$	0	0
$217$	891.786	0.278979
$218$	0	0
$219$	−2637.65	−0.813863
$220$	0	0
$221$	−2460.70	−0.748979
$222$	0	0
$223$	−5527.03	−1.65972	−0.829859	−	0.557973i	$-0.811579\pi$
$223$	−5527.03	−1.65972	−0.829859	+	0.557973i	$0.811579\pi$
$224$	0	0
$225$	−1299.85	−0.385141
$226$	0	0
$227$	2788.28	0.815262	0.407631	−	0.913147i	$-0.366355\pi$
$227$	2788.28	0.815262	0.407631	+	0.913147i	$0.366355\pi$
$228$	0	0
$229$	2527.23	0.729274	0.364637	−	0.931150i	$-0.381193\pi$
$229$	2527.23	0.729274	0.364637	+	0.931150i	$0.381193\pi$
$230$	0	0
$231$	−582.959	−0.166043
$232$	0	0
$233$	−5606.98	−1.57650	−0.788252	−	0.615352i	$-0.789014\pi$
$233$	−5606.98	−1.57650	−0.788252	+	0.615352i	$0.789014\pi$
$234$	0	0
$235$	3647.49	1.01249
$236$	0	0
$237$	5142.74	1.40952
$238$	0	0
$239$	5924.09	1.60334	0.801668	−	0.597769i	$-0.203946\pi$
$239$	5924.09	1.60334	0.801668	+	0.597769i	$0.203946\pi$
$240$	0	0
$241$	−876.073	−0.234161	−0.117081	−	0.993122i	$-0.537354\pi$
$241$	−876.073	−0.234161	−0.117081	+	0.993122i	$0.537354\pi$
$242$	0	0
$243$	3028.37	0.799465
$244$	0	0
$245$	−4832.43	−1.26013
$246$	0	0
$247$	9039.81	2.32870
$248$	0	0
$249$	−4936.25	−1.25631
$250$	0	0
$251$	−5914.86	−1.48742	−0.743711	−	0.668502i	$-0.766936\pi$
$251$	−5914.86	−1.48742	−0.743711	+	0.668502i	$0.766936\pi$
$252$	0	0
$253$	1244.39	0.309226
$254$	0	0
$255$	2386.36	0.586038
$256$	0	0
$257$	−1589.22	−0.385730	−0.192865	−	0.981225i	$-0.561778\pi$
$257$	−1589.22	−0.385730	−0.192865	+	0.981225i	$0.561778\pi$
$258$	0	0
$259$	368.744	0.0884658
$260$	0	0
$261$	342.378	0.0811980
$262$	0	0
$263$	−7777.46	−1.82349	−0.911747	−	0.410752i	$-0.865266\pi$
$263$	−7777.46	−1.82349	−0.911747	+	0.410752i	$0.865266\pi$
$264$	0	0
$265$	3349.54	0.776456
$266$	0	0
$267$	−5287.29	−1.21190
$268$	0	0
$269$	925.953	0.209875	0.104937	−	0.994479i	$-0.466536\pi$
$269$	925.953	0.209875	0.104937	+	0.994479i	$0.466536\pi$
$270$	0	0
$271$	−2235.53	−0.501103	−0.250551	−	0.968103i	$-0.580612\pi$
$271$	−2235.53	−0.501103	−0.250551	+	0.968103i	$0.580612\pi$
$272$	0	0
$273$	1267.36	0.280968
$274$	0	0
$275$	3121.07	0.684391
$276$	0	0
$277$	−1810.59	−0.392736	−0.196368	−	0.980530i	$-0.562915\pi$
$277$	−1810.59	−0.392736	−0.196368	+	0.980530i	$0.562915\pi$
$278$	0	0
$279$	1995.64	0.428230
$280$	0	0
$281$	−7795.69	−1.65499	−0.827495	−	0.561474i	$-0.810235\pi$
$281$	−7795.69	−1.65499	−0.827495	+	0.561474i	$0.810235\pi$
$282$	0	0
$283$	−7693.60	−1.61603	−0.808016	−	0.589160i	$-0.799459\pi$
$283$	−7693.60	−1.61603	−0.808016	+	0.589160i	$0.799459\pi$
$284$	0	0
$285$	−8766.72	−1.82209
$286$	0	0
$287$	1209.88	0.248839
$288$	0	0
$289$	−3318.76	−0.675507
$290$	0	0
$291$	3208.69	0.646380
$292$	0	0
$293$	2093.31	0.417381	0.208691	−	0.977982i	$-0.433080\pi$
$293$	2093.31	0.417381	0.208691	+	0.977982i	$0.433080\pi$
$294$	0	0
$295$	−1768.97	−0.349129
$296$	0	0
$297$	−4287.98	−0.837757
$298$	0	0
$299$	−2705.33	−0.523255
$300$	0	0
$301$	−794.181	−0.152079
$302$	0	0
$303$	3491.52	0.661989
$304$	0	0
$305$	−3247.25	−0.609629
$306$	0	0
$307$	3700.08	0.687865	0.343933	−	0.938994i	$-0.388241\pi$
$307$	3700.08	0.687865	0.343933	+	0.938994i	$0.388241\pi$
$308$	0	0
$309$	3020.52	0.556088
$310$	0	0
$311$	758.661	0.138327	0.0691635	−	0.997605i	$-0.477967\pi$
$311$	758.661	0.138327	0.0691635	+	0.997605i	$0.477967\pi$
$312$	0	0
$313$	2441.99	0.440989	0.220494	−	0.975388i	$-0.429233\pi$
$313$	2441.99	0.440989	0.220494	+	0.975388i	$0.429233\pi$
$314$	0	0
$315$	955.036	0.170826
$316$	0	0
$317$	−5664.85	−1.00369	−0.501845	−	0.864958i	$-0.667345\pi$
$317$	−5664.85	−1.00369	−0.501845	+	0.864958i	$0.667345\pi$
$318$	0	0
$319$	−822.084	−0.144288
$320$	0	0
$321$	3384.11	0.588419
$322$	0	0
$323$	−5856.71	−1.00890
$324$	0	0
$325$	−6785.26	−1.15809
$326$	0	0
$327$	1279.05	0.216305
$328$	0	0
$329$	−1255.03	−0.210310
$330$	0	0
$331$	5046.15	0.837950	0.418975	−	0.907998i	$-0.362389\pi$
$331$	5046.15	0.837950	0.418975	+	0.907998i	$0.362389\pi$
$332$	0	0
$333$	825.178	0.135794
$334$	0	0
$335$	−6784.95	−1.10657
$336$	0	0
$337$	−2064.77	−0.333754	−0.166877	−	0.985978i	$-0.553368\pi$
$337$	−2064.77	−0.333754	−0.166877	+	0.985978i	$0.553368\pi$
$338$	0	0
$339$	−4270.05	−0.684121
$340$	0	0
$341$	−4791.74	−0.760959
$342$	0	0
$343$	3472.33	0.546613
$344$	0	0
$345$	2623.60	0.409420
$346$	0	0
$347$	10261.9	1.58757	0.793785	−	0.608199i	$-0.208108\pi$
$347$	10261.9	1.58757	0.793785	+	0.608199i	$0.208108\pi$
$348$	0	0
$349$	−9965.45	−1.52848	−0.764238	−	0.644934i	$-0.776885\pi$
$349$	−9965.45	−1.52848	−0.764238	+	0.644934i	$0.776885\pi$
$350$	0	0
$351$	9322.15	1.41761
$352$	0	0
$353$	−2978.90	−0.449153	−0.224576	−	0.974457i	$-0.572100\pi$
$353$	−2978.90	−0.449153	−0.224576	+	0.974457i	$0.572100\pi$
$354$	0	0
$355$	−10169.8	−1.52044
$356$	0	0
$357$	−821.100	−0.121729
$358$	0	0
$359$	8156.81	1.19916	0.599582	−	0.800313i	$-0.295333\pi$
$359$	8156.81	1.19916	0.599582	+	0.800313i	$0.295333\pi$
$360$	0	0
$361$	14656.7	2.13686
$362$	0	0
$363$	−2055.79	−0.297248
$364$	0	0
$365$	−10375.5	−1.48789
$366$	0	0
$367$	−6202.75	−0.882237	−0.441119	−	0.897449i	$-0.645418\pi$
$367$	−6202.75	−0.882237	−0.441119	+	0.897449i	$0.645418\pi$
$368$	0	0
$369$	2707.47	0.381965
$370$	0	0
$371$	−1152.51	−0.161282
$372$	0	0
$373$	−5940.53	−0.824635	−0.412317	−	0.911040i	$-0.635281\pi$
$373$	−5940.53	−0.824635	−0.412317	+	0.911040i	$0.635281\pi$
$374$	0	0
$375$	−890.559	−0.122635
$376$	0	0
$377$	1787.23	0.244156
$378$	0	0
$379$	3283.69	0.445045	0.222522	−	0.974928i	$-0.428571\pi$
$379$	3283.69	0.445045	0.222522	+	0.974928i	$0.428571\pi$
$380$	0	0
$381$	−3698.84	−0.497369
$382$	0	0
$383$	−6651.64	−0.887423	−0.443712	−	0.896170i	$-0.646339\pi$
$383$	−6651.64	−0.887423	−0.443712	+	0.896170i	$0.646339\pi$
$384$	0	0
$385$	−2293.14	−0.303556
$386$	0	0
$387$	−1777.22	−0.233440
$388$	0	0
$389$	−10607.5	−1.38257	−0.691285	−	0.722582i	$-0.742955\pi$
$389$	−10607.5	−1.38257	−0.691285	+	0.722582i	$0.742955\pi$
$390$	0	0
$391$	1752.73	0.226699
$392$	0	0
$393$	176.089	0.0226018
$394$	0	0
$395$	20229.6	2.57686
$396$	0	0
$397$	−4945.27	−0.625178	−0.312589	−	0.949888i	$-0.601196\pi$
$397$	−4945.27	−0.625178	−0.312589	+	0.949888i	$0.601196\pi$
$398$	0	0
$399$	3016.46	0.378476
$400$	0	0
$401$	14041.2	1.74859	0.874295	−	0.485396i	$-0.161324\pi$
$401$	14041.2	1.74859	0.874295	+	0.485396i	$0.161324\pi$
$402$	0	0
$403$	10417.3	1.28765
$404$	0	0
$405$	−4152.91	−0.509531
$406$	0	0
$407$	−1981.33	−0.241305
$408$	0	0
$409$	−11566.1	−1.39830	−0.699150	−	0.714975i	$-0.746438\pi$
$409$	−11566.1	−1.39830	−0.699150	+	0.714975i	$0.746438\pi$
$410$	0	0
$411$	−3922.50	−0.470761
$412$	0	0
$413$	608.666	0.0725194
$414$	0	0
$415$	−19417.3	−2.29677
$416$	0	0
$417$	−693.008	−0.0813831
$418$	0	0
$419$	10233.3	1.19314	0.596572	−	0.802560i	$-0.296529\pi$
$419$	10233.3	1.19314	0.596572	+	0.802560i	$0.296529\pi$
$420$	0	0
$421$	11582.5	1.34085	0.670423	−	0.741979i	$-0.266113\pi$
$421$	11582.5	1.34085	0.670423	+	0.741979i	$0.266113\pi$
$422$	0	0
$423$	−2808.52	−0.322825
$424$	0	0
$425$	4396.04	0.501739
$426$	0	0
$427$	1117.31	0.126629
$428$	0	0
$429$	−6809.79	−0.766387
$430$	0	0
$431$	1754.30	0.196060	0.0980299	−	0.995183i	$-0.468746\pi$
$431$	1754.30	0.196060	0.0980299	+	0.995183i	$0.468746\pi$
$432$	0	0
$433$	1060.29	0.117677	0.0588387	−	0.998268i	$-0.481260\pi$
$433$	1060.29	0.117677	0.0588387	+	0.998268i	$0.481260\pi$
$434$	0	0
$435$	−1733.24	−0.191040
$436$	0	0
$437$	−6438.96	−0.704845
$438$	0	0
$439$	−11608.1	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$439$	−11608.1	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$440$	0	0
$441$	3720.90	0.401782
$442$	0	0
$443$	−4090.97	−0.438754	−0.219377	−	0.975640i	$-0.570402\pi$
$443$	−4090.97	−0.438754	−0.219377	+	0.975640i	$0.570402\pi$
$444$	0	0
$445$	−20798.2	−2.21557
$446$	0	0
$447$	8871.50	0.938720
$448$	0	0
$449$	−16360.6	−1.71961	−0.859807	−	0.510620i	$-0.829416\pi$
$449$	−16360.6	−1.71961	−0.859807	+	0.510620i	$0.829416\pi$
$450$	0	0
$451$	−6500.90	−0.678748
$452$	0	0
$453$	86.9967	0.00902309
$454$	0	0
$455$	4985.32	0.513661
$456$	0	0
$457$	11693.7	1.19696	0.598479	−	0.801138i	$-0.295772\pi$
$457$	11693.7	1.19696	0.598479	+	0.801138i	$0.295772\pi$
$458$	0	0
$459$	−6039.64	−0.614175
$460$	0	0
$461$	14730.0	1.48817	0.744084	−	0.668086i	$-0.232886\pi$
$461$	14730.0	1.48817	0.744084	+	0.668086i	$0.232886\pi$
$462$	0	0
$463$	12020.1	1.20653	0.603263	−	0.797542i	$-0.293867\pi$
$463$	12020.1	1.20653	0.603263	+	0.797542i	$0.293867\pi$
$464$	0	0
$465$	−10102.6	−1.00752
$466$	0	0
$467$	−5973.63	−0.591920	−0.295960	−	0.955200i	$-0.595640\pi$
$467$	−5973.63	−0.591920	−0.295960	+	0.955200i	$0.595640\pi$
$468$	0	0
$469$	2334.57	0.229852
$470$	0	0
$471$	−10715.4	−1.04828
$472$	0	0
$473$	4267.29	0.414820
$474$	0	0
$475$	−16149.6	−1.55999
$476$	0	0
$477$	−2579.10	−0.247566
$478$	0	0
$479$	−17413.9	−1.66109	−0.830545	−	0.556951i	$-0.811971\pi$
$479$	−17413.9	−1.66109	−0.830545	+	0.556951i	$0.811971\pi$
$480$	0	0
$481$	4307.46	0.408322
$482$	0	0
$483$	−902.730	−0.0850427
$484$	0	0
$485$	12621.7	1.18170
$486$	0	0
$487$	3885.79	0.361565	0.180782	−	0.983523i	$-0.442137\pi$
$487$	3885.79	0.361565	0.180782	+	0.983523i	$0.442137\pi$
$488$	0	0
$489$	6051.53	0.559631
$490$	0	0
$491$	−14369.1	−1.32071	−0.660354	−	0.750954i	$-0.729594\pi$
$491$	−14369.1	−1.32071	−0.660354	+	0.750954i	$0.729594\pi$
$492$	0	0
$493$	−1157.91	−0.105780
$494$	0	0
$495$	−5131.60	−0.465956
$496$	0	0
$497$	3499.22	0.315818
$498$	0	0
$499$	466.364	0.0418383	0.0209191	−	0.999781i	$-0.493341\pi$
$499$	466.364	0.0418383	0.0209191	+	0.999781i	$0.493341\pi$
$500$	0	0
$501$	9052.00	0.807212
$502$	0	0
$503$	−3145.80	−0.278856	−0.139428	−	0.990232i	$-0.544526\pi$
$503$	−3145.80	−0.278856	−0.139428	+	0.990232i	$0.544526\pi$
$504$	0	0
$505$	13734.3	1.21024
$506$	0	0
$507$	6240.87	0.546680
$508$	0	0
$509$	1033.27	0.0899779	0.0449889	−	0.998987i	$-0.485675\pi$
$509$	1033.27	0.0899779	0.0449889	+	0.998987i	$0.485675\pi$
$510$	0	0
$511$	3570.01	0.309056
$512$	0	0
$513$	22187.7	1.90957
$514$	0	0
$515$	11881.6	1.01663
$516$	0	0
$517$	6743.53	0.573656
$518$	0	0
$519$	−1418.23	−0.119949
$520$	0	0
$521$	−11615.8	−0.976767	−0.488384	−	0.872629i	$-0.662413\pi$
$521$	−11615.8	−0.976767	−0.488384	+	0.872629i	$0.662413\pi$
$522$	0	0
$523$	−14249.8	−1.19140	−0.595700	−	0.803207i	$-0.703125\pi$
$523$	−14249.8	−1.19140	−0.595700	+	0.803207i	$0.703125\pi$
$524$	0	0
$525$	−2264.15	−0.188220
$526$	0	0
$527$	−6749.18	−0.557873
$528$	0	0
$529$	−10240.0	−0.841623
$530$	0	0
$531$	1362.08	0.111317
$532$	0	0
$533$	14133.1	1.14854
$534$	0	0
$535$	13311.8	1.07573
$536$	0	0
$537$	9870.15	0.793163
$538$	0	0
$539$	−8934.25	−0.713962
$540$	0	0
$541$	1221.09	0.0970404	0.0485202	−	0.998822i	$-0.484549\pi$
$541$	1221.09	0.0970404	0.0485202	+	0.998822i	$0.484549\pi$
$542$	0	0
$543$	4536.95	0.358562
$544$	0	0
$545$	5031.30	0.395445
$546$	0	0
$547$	23448.2	1.83285	0.916427	−	0.400201i	$-0.131060\pi$
$547$	23448.2	1.83285	0.916427	+	0.400201i	$0.131060\pi$
$548$	0	0
$549$	2500.33	0.194375
$550$	0	0
$551$	4253.79	0.328888
$552$	0	0
$553$	−6960.60	−0.535253
$554$	0	0
$555$	−4177.33	−0.319492
$556$	0	0
$557$	14174.7	1.07828	0.539140	−	0.842216i	$-0.318749\pi$
$557$	14174.7	1.07828	0.539140	+	0.842216i	$0.318749\pi$
$558$	0	0
$559$	−9277.16	−0.701936
$560$	0	0
$561$	4411.93	0.332035
$562$	0	0
$563$	−3348.85	−0.250688	−0.125344	−	0.992113i	$-0.540003\pi$
$563$	−3348.85	−0.250688	−0.125344	+	0.992113i	$0.540003\pi$
$564$	0	0
$565$	−16796.7	−1.25070
$566$	0	0
$567$	1428.94	0.105837
$568$	0	0
$569$	10533.7	0.776091	0.388045	−	0.921640i	$-0.373150\pi$
$569$	10533.7	0.776091	0.388045	+	0.921640i	$0.373150\pi$
$570$	0	0
$571$	19267.0	1.41208	0.706041	−	0.708171i	$-0.250480\pi$
$571$	19267.0	1.41208	0.706041	+	0.708171i	$0.250480\pi$
$572$	0	0
$573$	6539.84	0.476799
$574$	0	0
$575$	4833.07	0.350527
$576$	0	0
$577$	4747.42	0.342526	0.171263	−	0.985225i	$-0.445215\pi$
$577$	4747.42	0.342526	0.171263	+	0.985225i	$0.445215\pi$
$578$	0	0
$579$	8768.41	0.629365
$580$	0	0
$581$	6681.11	0.477073
$582$	0	0
$583$	6192.67	0.439921
$584$	0	0
$585$	11156.2	0.788464
$586$	0	0
$587$	7603.14	0.534608	0.267304	−	0.963612i	$-0.413867\pi$
$587$	7603.14	0.534608	0.267304	+	0.963612i	$0.413867\pi$
$588$	0	0
$589$	24794.3	1.73452
$590$	0	0
$591$	−8705.51	−0.605917
$592$	0	0
$593$	1892.76	0.131073	0.0655366	−	0.997850i	$-0.479124\pi$
$593$	1892.76	0.131073	0.0655366	+	0.997850i	$0.479124\pi$
$594$	0	0
$595$	−3229.89	−0.222542
$596$	0	0
$597$	11641.9	0.798110
$598$	0	0
$599$	18998.8	1.29594	0.647970	−	0.761665i	$-0.275618\pi$
$599$	18998.8	1.29594	0.647970	+	0.761665i	$0.275618\pi$
$600$	0	0
$601$	2034.70	0.138098	0.0690491	−	0.997613i	$-0.478003\pi$
$601$	2034.70	0.138098	0.0690491	+	0.997613i	$0.478003\pi$
$602$	0	0
$603$	5224.31	0.352820
$604$	0	0
$605$	−8086.69	−0.543423
$606$	0	0
$607$	1691.66	0.113117	0.0565587	−	0.998399i	$-0.481987\pi$
$607$	1691.66	0.113117	0.0565587	+	0.998399i	$0.481987\pi$
$608$	0	0
$609$	596.373	0.0396818
$610$	0	0
$611$	−14660.6	−0.970708
$612$	0	0
$613$	2051.86	0.135194	0.0675969	−	0.997713i	$-0.478467\pi$
$613$	2051.86	0.135194	0.0675969	+	0.997713i	$0.478467\pi$
$614$	0	0
$615$	−13706.1	−0.898674
$616$	0	0
$617$	−21839.0	−1.42497	−0.712483	−	0.701690i	$-0.752429\pi$
$617$	−21839.0	−1.42497	−0.712483	+	0.701690i	$0.752429\pi$
$618$	0	0
$619$	−5707.58	−0.370609	−0.185305	−	0.982681i	$-0.559327\pi$
$619$	−5707.58	−0.370609	−0.185305	+	0.982681i	$0.559327\pi$
$620$	0	0
$621$	−6640.07	−0.429077
$622$	0	0
$623$	7156.24	0.460207
$624$	0	0
$625$	−17265.5	−1.10499
$626$	0	0
$627$	−16208.0	−1.03235
$628$	0	0
$629$	−2790.72	−0.176905
$630$	0	0
$631$	−848.735	−0.0535461	−0.0267731	−	0.999642i	$-0.508523\pi$
$631$	−848.735	−0.0535461	−0.0267731	+	0.999642i	$0.508523\pi$
$632$	0	0
$633$	20187.4	1.26758
$634$	0	0
$635$	−14549.8	−0.909279
$636$	0	0
$637$	19423.2	1.20813
$638$	0	0
$639$	7830.59	0.484778
$640$	0	0
$641$	−25537.5	−1.57359	−0.786796	−	0.617213i	$-0.788262\pi$
$641$	−25537.5	−1.57359	−0.786796	+	0.617213i	$0.788262\pi$
$642$	0	0
$643$	4902.36	0.300669	0.150334	−	0.988635i	$-0.451965\pi$
$643$	4902.36	0.300669	0.150334	+	0.988635i	$0.451965\pi$
$644$	0	0
$645$	8996.91	0.549229
$646$	0	0
$647$	3920.19	0.238205	0.119103	−	0.992882i	$-0.461998\pi$
$647$	3920.19	0.238205	0.119103	+	0.992882i	$0.461998\pi$
$648$	0	0
$649$	−3270.48	−0.197808
$650$	0	0
$651$	3476.12	0.209278
$652$	0	0
$653$	3640.59	0.218174	0.109087	−	0.994032i	$-0.465207\pi$
$653$	3640.59	0.218174	0.109087	+	0.994032i	$0.465207\pi$
$654$	0	0
$655$	692.666	0.0413201
$656$	0	0
$657$	7988.99	0.474399
$658$	0	0
$659$	14826.8	0.876437	0.438219	−	0.898868i	$-0.355610\pi$
$659$	14826.8	0.876437	0.438219	+	0.898868i	$0.355610\pi$
$660$	0	0
$661$	−25302.1	−1.48886	−0.744432	−	0.667698i	$-0.767280\pi$
$661$	−25302.1	−1.48886	−0.744432	+	0.667698i	$0.767280\pi$
$662$	0	0
$663$	−9591.62	−0.561852
$664$	0	0
$665$	11865.6	0.691921
$666$	0	0
$667$	−1273.02	−0.0739005
$668$	0	0
$669$	−21544.0	−1.24505
$670$	0	0
$671$	−6003.55	−0.345402
$672$	0	0
$673$	−18983.4	−1.08730	−0.543652	−	0.839311i	$-0.682959\pi$
$673$	−18983.4	−1.08730	−0.543652	+	0.839311i	$0.682959\pi$
$674$	0	0
$675$	−16654.0	−0.949651
$676$	0	0
$677$	−16033.8	−0.910236	−0.455118	−	0.890431i	$-0.650403\pi$
$677$	−16033.8	−0.910236	−0.455118	+	0.890431i	$0.650403\pi$
$678$	0	0
$679$	−4342.89	−0.245456
$680$	0	0
$681$	10868.5	0.611574
$682$	0	0
$683$	1451.21	0.0813017	0.0406509	−	0.999173i	$-0.487057\pi$
$683$	1451.21	0.0813017	0.0406509	+	0.999173i	$0.487057\pi$
$684$	0	0
$685$	−15429.6	−0.860635
$686$	0	0
$687$	9850.95	0.547070
$688$	0	0
$689$	−13463.0	−0.744410
$690$	0	0
$691$	−8126.55	−0.447393	−0.223697	−	0.974659i	$-0.571812\pi$
$691$	−8126.55	−0.447393	−0.223697	+	0.974659i	$0.571812\pi$
$692$	0	0
$693$	1765.68	0.0967860
$694$	0	0
$695$	−2726.03	−0.148783
$696$	0	0
$697$	−9156.54	−0.497602
$698$	0	0
$699$	−21855.6	−1.18263
$700$	0	0
$701$	−1475.94	−0.0795229	−0.0397614	−	0.999209i	$-0.512660\pi$
$701$	−1475.94	−0.0795229	−0.0397614	+	0.999209i	$0.512660\pi$
$702$	0	0
$703$	10252.2	0.550027
$704$	0	0
$705$	14217.7	0.759530
$706$	0	0
$707$	−4725.71	−0.251384
$708$	0	0
$709$	228.875	0.0121235	0.00606177	−	0.999982i	$-0.498070\pi$
$709$	228.875	0.0121235	0.00606177	+	0.999982i	$0.498070\pi$
$710$	0	0
$711$	−15576.5	−0.821608
$712$	0	0
$713$	−7420.15	−0.389743
$714$	0	0
$715$	−26787.1	−1.40109
$716$	0	0
$717$	23091.7	1.20275
$718$	0	0
$719$	−15696.4	−0.814155	−0.407077	−	0.913394i	$-0.633452\pi$
$719$	−15696.4	−0.814155	−0.407077	+	0.913394i	$0.633452\pi$
$720$	0	0
$721$	−4088.21	−0.211169
$722$	0	0
$723$	−3414.87	−0.175658
$724$	0	0
$725$	−3192.88	−0.163560
$726$	0	0
$727$	−14329.5	−0.731020	−0.365510	−	0.930807i	$-0.619105\pi$
$727$	−14329.5	−0.731020	−0.365510	+	0.930807i	$0.619105\pi$
$728$	0	0
$729$	19117.3	0.971259
$730$	0	0
$731$	6010.49	0.304112
$732$	0	0
$733$	15912.1	0.801810	0.400905	−	0.916120i	$-0.368696\pi$
$733$	15912.1	0.801810	0.400905	+	0.916120i	$0.368696\pi$
$734$	0	0
$735$	−18836.5	−0.945297
$736$	0	0
$737$	−12544.1	−0.626958
$738$	0	0
$739$	−12354.6	−0.614983	−0.307492	−	0.951551i	$-0.599490\pi$
$739$	−12354.6	−0.614983	−0.307492	+	0.951551i	$0.599490\pi$
$740$	0	0
$741$	35236.5	1.74689
$742$	0	0
$743$	5080.77	0.250869	0.125434	−	0.992102i	$-0.459968\pi$
$743$	5080.77	0.250869	0.125434	+	0.992102i	$0.459968\pi$
$744$	0	0
$745$	34897.1	1.71615
$746$	0	0
$747$	14951.0	0.732302
$748$	0	0
$749$	−4580.32	−0.223446
$750$	0	0
$751$	34750.4	1.68850	0.844248	−	0.535952i	$-0.180047\pi$
$751$	34750.4	1.68850	0.844248	+	0.535952i	$0.180047\pi$
$752$	0	0
$753$	−23055.7	−1.11580
$754$	0	0
$755$	342.212	0.0164958
$756$	0	0
$757$	2602.27	0.124942	0.0624709	−	0.998047i	$-0.480102\pi$
$757$	2602.27	0.124942	0.0624709	+	0.998047i	$0.480102\pi$
$758$	0	0
$759$	4850.54	0.231968
$760$	0	0
$761$	26480.5	1.26139	0.630694	−	0.776031i	$-0.282770\pi$
$761$	26480.5	1.26139	0.630694	+	0.776031i	$0.282770\pi$
$762$	0	0
$763$	−1731.17	−0.0821398
$764$	0	0
$765$	−7227.87	−0.341600
$766$	0	0
$767$	7110.09	0.334720
$768$	0	0
$769$	−7464.68	−0.350043	−0.175022	−	0.984565i	$-0.556000\pi$
$769$	−7464.68	−0.350043	−0.175022	+	0.984565i	$0.556000\pi$
$770$	0	0
$771$	−6194.65	−0.289358
$772$	0	0
$773$	39751.4	1.84962	0.924812	−	0.380424i	$-0.124222\pi$
$773$	39751.4	1.84962	0.924812	+	0.380424i	$0.124222\pi$
$774$	0	0
$775$	−18610.6	−0.862596
$776$	0	0
$777$	1437.34	0.0663632
$778$	0	0
$779$	33638.2	1.54713
$780$	0	0
$781$	−18802.0	−0.861446
$782$	0	0
$783$	4386.64	0.200212
$784$	0	0
$785$	−42150.3	−1.91644
$786$	0	0
$787$	16132.1	0.730681	0.365341	−	0.930874i	$-0.380952\pi$
$787$	16132.1	0.730681	0.365341	+	0.930874i	$0.380952\pi$
$788$	0	0
$789$	−30316.0	−1.36791
$790$	0	0
$791$	5779.42	0.259788
$792$	0	0
$793$	13051.8	0.584469
$794$	0	0
$795$	13056.3	0.582463
$796$	0	0
$797$	30649.4	1.36218	0.681089	−	0.732201i	$-0.261507\pi$
$797$	30649.4	1.36218	0.681089	+	0.732201i	$0.261507\pi$
$798$	0	0
$799$	9498.28	0.420557
$800$	0	0
$801$	16014.3	0.706413
$802$	0	0
$803$	−19182.3	−0.843002
$804$	0	0
$805$	−3550.99	−0.155473
$806$	0	0
$807$	3609.30	0.157439
$808$	0	0
$809$	31988.2	1.39017	0.695083	−	0.718929i	$-0.255367\pi$
$809$	31988.2	1.39017	0.695083	+	0.718929i	$0.255367\pi$
$810$	0	0
$811$	39086.8	1.69238	0.846192	−	0.532877i	$-0.178889\pi$
$811$	39086.8	1.69238	0.846192	+	0.532877i	$0.178889\pi$
$812$	0	0
$813$	−8713.94	−0.375906
$814$	0	0
$815$	23804.4	1.02311
$816$	0	0
$817$	−22080.6	−0.945536
$818$	0	0
$819$	−3838.63	−0.163776
$820$	0	0
$821$	20056.3	0.852582	0.426291	−	0.904586i	$-0.359820\pi$
$821$	20056.3	0.852582	0.426291	+	0.904586i	$0.359820\pi$
$822$	0	0
$823$	−24931.0	−1.05594	−0.527970	−	0.849263i	$-0.677047\pi$
$823$	−24931.0	−1.05594	−0.527970	+	0.849263i	$0.677047\pi$
$824$	0	0
$825$	12165.7	0.513401
$826$	0	0
$827$	40104.5	1.68630	0.843150	−	0.537678i	$-0.180698\pi$
$827$	40104.5	1.68630	0.843150	+	0.537678i	$0.180698\pi$
$828$	0	0
$829$	35722.2	1.49660	0.748300	−	0.663360i	$-0.230870\pi$
$829$	35722.2	1.49660	0.748300	+	0.663360i	$0.230870\pi$
$830$	0	0
$831$	−7057.56	−0.294614
$832$	0	0
$833$	−12583.9	−0.523418
$834$	0	0
$835$	35607.1	1.47573
$836$	0	0
$837$	25568.7	1.05590
$838$	0	0
$839$	185.813	0.00764598	0.00382299	−	0.999993i	$-0.498783\pi$
$839$	185.813	0.00764598	0.00382299	+	0.999993i	$0.498783\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−30387.1	−1.24150
$844$	0	0
$845$	24549.2	0.999429
$846$	0	0
$847$	2782.47	0.112877
$848$	0	0
$849$	−29989.1	−1.21228
$850$	0	0
$851$	−3068.16	−0.123590
$852$	0	0
$853$	32958.5	1.32295	0.661475	−	0.749967i	$-0.269931\pi$
$853$	32958.5	1.32295	0.661475	+	0.749967i	$0.269931\pi$
$854$	0	0
$855$	26552.9	1.06209
$856$	0	0
$857$	−36681.9	−1.46211	−0.731056	−	0.682318i	$-0.760972\pi$
$857$	−36681.9	−1.46211	−0.731056	+	0.682318i	$0.760972\pi$
$858$	0	0
$859$	−25084.0	−0.996340	−0.498170	−	0.867079i	$-0.665994\pi$
$859$	−25084.0	−0.996340	−0.498170	+	0.867079i	$0.665994\pi$
$860$	0	0
$861$	4716.01	0.186668
$862$	0	0
$863$	−6552.57	−0.258461	−0.129231	−	0.991615i	$-0.541251\pi$
$863$	−6552.57	−0.258461	−0.129231	+	0.991615i	$0.541251\pi$
$864$	0	0
$865$	−5578.76	−0.219287
$866$	0	0
$867$	−12936.3	−0.506736
$868$	0	0
$869$	37400.6	1.45999
$870$	0	0
$871$	27271.1	1.06090
$872$	0	0
$873$	−9718.55	−0.376773
$874$	0	0
$875$	1205.35	0.0465696
$876$	0	0
$877$	39373.9	1.51603	0.758017	−	0.652234i	$-0.226168\pi$
$877$	39373.9	1.51603	0.758017	+	0.652234i	$0.226168\pi$
$878$	0	0
$879$	8159.59	0.313101
$880$	0	0
$881$	−8182.51	−0.312912	−0.156456	−	0.987685i	$-0.550007\pi$
$881$	−8182.51	−0.312912	−0.156456	+	0.987685i	$0.550007\pi$
$882$	0	0
$883$	7461.63	0.284376	0.142188	−	0.989840i	$-0.454586\pi$
$883$	7461.63	0.284376	0.142188	+	0.989840i	$0.454586\pi$
$884$	0	0
$885$	−6895.30	−0.261902
$886$	0	0
$887$	−29745.2	−1.12598	−0.562990	−	0.826463i	$-0.690349\pi$
$887$	−29745.2	−1.12598	−0.562990	+	0.826463i	$0.690349\pi$
$888$	0	0
$889$	5006.31	0.188871
$890$	0	0
$891$	−7677.95	−0.288688
$892$	0	0
$893$	−34893.7	−1.30758
$894$	0	0
$895$	38825.4	1.45004
$896$	0	0
$897$	−10545.2	−0.392523
$898$	0	0
$899$	4901.99	0.181858
$900$	0	0
$901$	8722.40	0.322514
$902$	0	0
$903$	−3095.66	−0.114083
$904$	0	0
$905$	17846.6	0.655515
$906$	0	0
$907$	34626.6	1.26765	0.633825	−	0.773476i	$-0.281484\pi$
$907$	34626.6	1.26765	0.633825	+	0.773476i	$0.281484\pi$
$908$	0	0
$909$	−10575.2	−0.385872
$910$	0	0
$911$	6172.29	0.224476	0.112238	−	0.993681i	$-0.464198\pi$
$911$	6172.29	0.224476	0.112238	+	0.993681i	$0.464198\pi$
$912$	0	0
$913$	−35898.9	−1.30129
$914$	0	0
$915$	−12657.5	−0.457317
$916$	0	0
$917$	−238.333	−0.00858281
$918$	0	0
$919$	15779.0	0.566376	0.283188	−	0.959064i	$-0.408608\pi$
$919$	15779.0	0.566376	0.283188	+	0.959064i	$0.408608\pi$
$920$	0	0
$921$	14422.6	0.516007
$922$	0	0
$923$	40875.9	1.45769
$924$	0	0
$925$	−7695.28	−0.273534
$926$	0	0
$927$	−9148.62	−0.324143
$928$	0	0
$929$	−9692.85	−0.342316	−0.171158	−	0.985244i	$-0.554751\pi$
$929$	−9692.85	−0.342316	−0.171158	+	0.985244i	$0.554751\pi$
$930$	0	0
$931$	46229.3	1.62739
$932$	0	0
$933$	2957.21	0.103767
$934$	0	0
$935$	17354.8	0.607020
$936$	0	0
$937$	17253.8	0.601554	0.300777	−	0.953694i	$-0.402754\pi$
$937$	17253.8	0.601554	0.300777	+	0.953694i	$0.402754\pi$
$938$	0	0
$939$	9518.70	0.330811
$940$	0	0
$941$	−6249.78	−0.216511	−0.108256	−	0.994123i	$-0.534526\pi$
$941$	−6249.78	−0.216511	−0.108256	+	0.994123i	$0.534526\pi$
$942$	0	0
$943$	−10066.8	−0.347637
$944$	0	0
$945$	12236.2	0.421210
$946$	0	0
$947$	38713.7	1.32843	0.664217	−	0.747540i	$-0.268765\pi$
$947$	38713.7	1.32843	0.664217	+	0.747540i	$0.268765\pi$
$948$	0	0
$949$	41702.8	1.42648
$950$	0	0
$951$	−22081.2	−0.752924
$952$	0	0
$953$	28936.5	0.983573	0.491787	−	0.870716i	$-0.336344\pi$
$953$	28936.5	0.983573	0.491787	+	0.870716i	$0.336344\pi$
$954$	0	0
$955$	25725.2	0.871673
$956$	0	0
$957$	−3204.43	−0.108239
$958$	0	0
$959$	5309.03	0.178767
$960$	0	0
$961$	−1218.42	−0.0408989
$962$	0	0
$963$	−10249.9	−0.342988
$964$	0	0
$965$	34491.5	1.15059
$966$	0	0
$967$	8697.74	0.289245	0.144623	−	0.989487i	$-0.453803\pi$
$967$	8697.74	0.289245	0.144623	+	0.989487i	$0.453803\pi$
$968$	0	0
$969$	−22829.1	−0.756837
$970$	0	0
$971$	−41062.0	−1.35710	−0.678549	−	0.734555i	$-0.737391\pi$
$971$	−41062.0	−1.35710	−0.678549	+	0.734555i	$0.737391\pi$
$972$	0	0
$973$	937.973	0.0309045
$974$	0	0
$975$	−26448.5	−0.868748
$976$	0	0
$977$	35397.4	1.15912	0.579561	−	0.814929i	$-0.303224\pi$
$977$	35397.4	1.15912	0.579561	+	0.814929i	$0.303224\pi$
$978$	0	0
$979$	−38451.9	−1.25529
$980$	0	0
$981$	−3874.03	−0.126084
$982$	0	0
$983$	−51453.4	−1.66949	−0.834746	−	0.550636i	$-0.814385\pi$
$983$	−51453.4	−1.66949	−0.834746	+	0.550636i	$0.814385\pi$
$984$	0	0
$985$	−34244.1	−1.10772
$986$	0	0
$987$	−4892.02	−0.157766
$988$	0	0
$989$	6608.03	0.212460
$990$	0	0
$991$	−43563.5	−1.39641	−0.698205	−	0.715898i	$-0.746017\pi$
$991$	−43563.5	−1.39641	−0.698205	+	0.715898i	$0.746017\pi$
$992$	0	0
$993$	19669.5	0.628594
$994$	0	0
$995$	45794.8	1.45909
$996$	0	0
$997$	45472.8	1.44447	0.722236	−	0.691647i	$-0.243114\pi$
$997$	45472.8	1.44447	0.722236	+	0.691647i	$0.243114\pi$
$998$	0	0
$999$	10572.4	0.334831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.be.1.6		8
4.3	odd	2	inner	1856.4.a.be.1.3		8
8.3	odd	2		928.4.a.c.1.6	yes	8
8.5	even	2		928.4.a.c.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.c.1.3	✓	8	8.5	even	2
928.4.a.c.1.6	yes	8	8.3	odd	2
1856.4.a.be.1.3		8	4.3	odd	2	inner
1856.4.a.be.1.6		8	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)

\(f(q)\)	\(=\)	\(q+3.89793 q^{3} +15.3330 q^{5} -5.27577 q^{7} -11.8061 q^{9} +O(q^{10})\)	\(q+3.89793 q^{3} +15.3330 q^{5} -5.27577 q^{7} -11.8061 q^{9} +28.3477 q^{11} -61.6285 q^{13} +59.7668 q^{15} +39.9279 q^{17} -146.682 q^{19} -20.5646 q^{21} +43.8973 q^{23} +110.099 q^{25} -151.264 q^{27} -29.0000 q^{29} -169.034 q^{31} +110.497 q^{33} -80.8931 q^{35} -69.8939 q^{37} -240.224 q^{39} -229.327 q^{41} +150.534 q^{43} -181.023 q^{45} +237.886 q^{47} -315.166 q^{49} +155.636 q^{51} +218.454 q^{53} +434.654 q^{55} -571.757 q^{57} -115.370 q^{59} -211.782 q^{61} +62.2865 q^{63} -944.947 q^{65} -442.508 q^{67} +171.109 q^{69} -663.263 q^{71} -676.680 q^{73} +429.160 q^{75} -149.556 q^{77} +1319.35 q^{79} -270.849 q^{81} -1266.38 q^{83} +612.212 q^{85} -113.040 q^{87} -1356.44 q^{89} +325.138 q^{91} -658.884 q^{93} -2249.07 q^{95} +823.177 q^{97} -334.678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) \) 8 * q + 20 * q^5 + 40 * q^9	\( 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) \) 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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