LMFDB - Embedded newform 1840.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 68x^{2} - 111x + 342 $ x^4 - 68x^2 - 111x + 342
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.57209$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.57209 q^{3} -5.00000 q^{5} +35.4229 q^{7} +30.3365 q^{9} +O(q^{10})$	$q+7.57209 q^{3} -5.00000 q^{5} +35.4229 q^{7} +30.3365 q^{9} +16.6298 q^{11} -79.9132 q^{13} -37.8604 q^{15} -46.8219 q^{17} +110.653 q^{19} +268.225 q^{21} +23.0000 q^{23} +25.0000 q^{25} +25.2641 q^{27} -0.836422 q^{29} +119.836 q^{31} +125.923 q^{33} -177.114 q^{35} +368.201 q^{37} -605.109 q^{39} -95.7927 q^{41} -331.961 q^{43} -151.682 q^{45} +535.037 q^{47} +911.782 q^{49} -354.539 q^{51} +409.345 q^{53} -83.1492 q^{55} +837.876 q^{57} +352.950 q^{59} -507.223 q^{61} +1074.61 q^{63} +399.566 q^{65} +820.056 q^{67} +174.158 q^{69} +733.770 q^{71} -91.4599 q^{73} +189.302 q^{75} +589.077 q^{77} -329.381 q^{79} -627.783 q^{81} +753.834 q^{83} +234.110 q^{85} -6.33346 q^{87} -1050.14 q^{89} -2830.76 q^{91} +907.407 q^{93} -553.267 q^{95} -271.928 q^{97} +504.491 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9	$ 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9 + 39 * q^11 - 20 * q^13 - 20 * q^15 - 23 * q^17 - 53 * q^19 + 300 * q^21 + 92 * q^23 + 100 * q^25 - 137 * q^27 + 161 * q^29 - 388 * q^31 + 87 * q^33 - 5 * q^35 + 466 * q^37 - 1047 * q^39 + 484 * q^41 - 894 * q^43 - 160 * q^45 + 265 * q^47 + 1643 * q^49 - 1825 * q^51 + 576 * q^53 - 195 * q^55 + 178 * q^57 + 94 * q^59 + 1153 * q^61 - 60 * q^63 + 100 * q^65 + 1472 * q^67 + 92 * q^69 - 200 * q^71 + 1147 * q^73 + 100 * q^75 - 2176 * q^77 + 908 * q^79 - 1056 * q^81 + 1048 * q^83 + 115 * q^85 + 2167 * q^87 - 1784 * q^89 - 2329 * q^91 + 1483 * q^93 + 265 * q^95 - 2047 * q^97 + 2665 * 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
$3$	7.57209	1.45725	0.728624	−	0.684914i	$-0.240160\pi$
$3$	7.57209	1.45725	0.728624	+	0.684914i	$0.240160\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	35.4229	1.91266	0.956328	−	0.292294i	$-0.0944187\pi$
$7$	35.4229	1.91266	0.956328	+	0.292294i	$0.0944187\pi$
$8$	0	0
$9$	30.3365	1.12357
$10$	0	0
$11$	16.6298	0.455826	0.227913	−	0.973682i	$-0.426810\pi$
$11$	16.6298	0.455826	0.227913	+	0.973682i	$0.426810\pi$
$12$	0	0
$13$	−79.9132	−1.70492	−0.852459	−	0.522795i	$-0.824889\pi$
$13$	−79.9132	−1.70492	−0.852459	+	0.522795i	$0.824889\pi$
$14$	0	0
$15$	−37.8604	−0.651701
$16$	0	0
$17$	−46.8219	−0.667999	−0.333999	−	0.942573i	$-0.608398\pi$
$17$	−46.8219	−0.667999	−0.333999	+	0.942573i	$0.608398\pi$
$18$	0	0
$19$	110.653	1.33608	0.668042	−	0.744123i	$-0.267132\pi$
$19$	110.653	1.33608	0.668042	+	0.744123i	$0.267132\pi$
$20$	0	0
$21$	268.225	2.78722
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	25.2641	0.180077
$28$	0	0
$29$	−0.836422	−0.00535585	−0.00267793	−	0.999996i	$-0.500852\pi$
$29$	−0.836422	−0.00535585	−0.00267793	+	0.999996i	$0.500852\pi$
$30$	0	0
$31$	119.836	0.694295	0.347147	−	0.937811i	$-0.387150\pi$
$31$	119.836	0.694295	0.347147	+	0.937811i	$0.387150\pi$
$32$	0	0
$33$	125.923	0.664252
$34$	0	0
$35$	−177.114	−0.855366
$36$	0	0
$37$	368.201	1.63600	0.817998	−	0.575221i	$-0.195084\pi$
$37$	368.201	1.63600	0.817998	+	0.575221i	$0.195084\pi$
$38$	0	0
$39$	−605.109	−2.48449
$40$	0	0
$41$	−95.7927	−0.364886	−0.182443	−	0.983216i	$-0.558400\pi$
$41$	−95.7927	−0.364886	−0.182443	+	0.983216i	$0.558400\pi$
$42$	0	0
$43$	−331.961	−1.17729	−0.588647	−	0.808390i	$-0.700339\pi$
$43$	−331.961	−1.17729	−0.588647	+	0.808390i	$0.700339\pi$
$44$	0	0
$45$	−151.682	−0.502477
$46$	0	0
$47$	535.037	1.66049	0.830246	−	0.557397i	$-0.188200\pi$
$47$	535.037	1.66049	0.830246	+	0.557397i	$0.188200\pi$
$48$	0	0
$49$	911.782	2.65826
$50$	0	0
$51$	−354.539	−0.973440
$52$	0	0
$53$	409.345	1.06090	0.530452	−	0.847715i	$-0.322022\pi$
$53$	409.345	1.06090	0.530452	+	0.847715i	$0.322022\pi$
$54$	0	0
$55$	−83.1492	−0.203852
$56$	0	0
$57$	837.876	1.94701
$58$	0	0
$59$	352.950	0.778817	0.389408	−	0.921065i	$-0.372679\pi$
$59$	352.950	0.778817	0.389408	+	0.921065i	$0.372679\pi$
$60$	0	0
$61$	−507.223	−1.06464	−0.532322	−	0.846542i	$-0.678680\pi$
$61$	−507.223	−1.06464	−0.532322	+	0.846542i	$0.678680\pi$
$62$	0	0
$63$	1074.61	2.14901
$64$	0	0
$65$	399.566	0.762462
$66$	0	0
$67$	820.056	1.49531	0.747655	−	0.664087i	$-0.231180\pi$
$67$	820.056	1.49531	0.747655	+	0.664087i	$0.231180\pi$
$68$	0	0
$69$	174.158	0.303857
$70$	0	0
$71$	733.770	1.22651	0.613257	−	0.789884i	$-0.289859\pi$
$71$	733.770	1.22651	0.613257	+	0.789884i	$0.289859\pi$
$72$	0	0
$73$	−91.4599	−0.146638	−0.0733190	−	0.997309i	$-0.523359\pi$
$73$	−91.4599	−0.146638	−0.0733190	+	0.997309i	$0.523359\pi$
$74$	0	0
$75$	189.302	0.291450
$76$	0	0
$77$	589.077	0.871838
$78$	0	0
$79$	−329.381	−0.469092	−0.234546	−	0.972105i	$-0.575360\pi$
$79$	−329.381	−0.469092	−0.234546	+	0.972105i	$0.575360\pi$
$80$	0	0
$81$	−627.783	−0.861156
$82$	0	0
$83$	753.834	0.996916	0.498458	−	0.866914i	$-0.333900\pi$
$83$	753.834	0.996916	0.498458	+	0.866914i	$0.333900\pi$
$84$	0	0
$85$	234.110	0.298738
$86$	0	0
$87$	−6.33346	−0.00780481
$88$	0	0
$89$	−1050.14	−1.25073	−0.625365	−	0.780332i	$-0.715050\pi$
$89$	−1050.14	−1.25073	−0.625365	+	0.780332i	$0.715050\pi$
$90$	0	0
$91$	−2830.76	−3.26092
$92$	0	0
$93$	907.407	1.01176
$94$	0	0
$95$	−553.267	−0.597515
$96$	0	0
$97$	−271.928	−0.284641	−0.142320	−	0.989821i	$-0.545456\pi$
$97$	−271.928	−0.284641	−0.142320	+	0.989821i	$0.545456\pi$
$98$	0	0
$99$	504.491	0.512154
$100$	0	0
$101$	1658.68	1.63410	0.817052	−	0.576563i	$-0.195607\pi$
$101$	1658.68	1.63410	0.817052	+	0.576563i	$0.195607\pi$
$102$	0	0
$103$	−1735.52	−1.66025	−0.830123	−	0.557580i	$-0.811730\pi$
$103$	−1735.52	−1.66025	−0.830123	+	0.557580i	$0.811730\pi$
$104$	0	0
$105$	−1341.13	−1.24648
$106$	0	0
$107$	1629.88	1.47258	0.736292	−	0.676664i	$-0.236575\pi$
$107$	1629.88	1.47258	0.736292	+	0.676664i	$0.236575\pi$
$108$	0	0
$109$	−432.151	−0.379748	−0.189874	−	0.981808i	$-0.560808\pi$
$109$	−432.151	−0.379748	−0.189874	+	0.981808i	$0.560808\pi$
$110$	0	0
$111$	2788.05	2.38405
$112$	0	0
$113$	−240.115	−0.199895	−0.0999476	−	0.994993i	$-0.531868\pi$
$113$	−240.115	−0.199895	−0.0999476	+	0.994993i	$0.531868\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−2424.28	−1.91560
$118$	0	0
$119$	−1658.57	−1.27765
$120$	0	0
$121$	−1054.45	−0.792223
$122$	0	0
$123$	−725.350	−0.531729
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	210.993	0.147422	0.0737110	−	0.997280i	$-0.476516\pi$
$127$	210.993	0.147422	0.0737110	+	0.997280i	$0.476516\pi$
$128$	0	0
$129$	−2513.64	−1.71561
$130$	0	0
$131$	746.178	0.497663	0.248832	−	0.968547i	$-0.419953\pi$
$131$	746.178	0.497663	0.248832	+	0.968547i	$0.419953\pi$
$132$	0	0
$133$	3919.66	2.55547
$134$	0	0
$135$	−126.320	−0.0805329
$136$	0	0
$137$	−2420.07	−1.50920	−0.754601	−	0.656184i	$-0.772170\pi$
$137$	−2420.07	−1.50920	−0.754601	+	0.656184i	$0.772170\pi$
$138$	0	0
$139$	−924.030	−0.563850	−0.281925	−	0.959436i	$-0.590973\pi$
$139$	−924.030	−0.563850	−0.281925	+	0.959436i	$0.590973\pi$
$140$	0	0
$141$	4051.34	2.41975
$142$	0	0
$143$	−1328.94	−0.777145
$144$	0	0
$145$	4.18211	0.00239521
$146$	0	0
$147$	6904.09	3.87374
$148$	0	0
$149$	430.614	0.236760	0.118380	−	0.992968i	$-0.462230\pi$
$149$	430.614	0.236760	0.118380	+	0.992968i	$0.462230\pi$
$150$	0	0
$151$	25.4118	0.0136953	0.00684763	−	0.999977i	$-0.497820\pi$
$151$	25.4118	0.0136953	0.00684763	+	0.999977i	$0.497820\pi$
$152$	0	0
$153$	−1420.41	−0.750546
$154$	0	0
$155$	−599.179	−0.310498
$156$	0	0
$157$	−1580.29	−0.803317	−0.401658	−	0.915790i	$-0.631566\pi$
$157$	−1580.29	−0.803317	−0.401658	+	0.915790i	$0.631566\pi$
$158$	0	0
$159$	3099.60	1.54600
$160$	0	0
$161$	814.727	0.398817
$162$	0	0
$163$	−458.739	−0.220437	−0.110218	−	0.993907i	$-0.535155\pi$
$163$	−458.739	−0.220437	−0.110218	+	0.993907i	$0.535155\pi$
$164$	0	0
$165$	−629.613	−0.297062
$166$	0	0
$167$	561.275	0.260076	0.130038	−	0.991509i	$-0.458490\pi$
$167$	561.275	0.260076	0.130038	+	0.991509i	$0.458490\pi$
$168$	0	0
$169$	4189.11	1.90674
$170$	0	0
$171$	3356.83	1.50119
$172$	0	0
$173$	1573.95	0.691705	0.345852	−	0.938289i	$-0.387590\pi$
$173$	1573.95	0.691705	0.345852	+	0.938289i	$0.387590\pi$
$174$	0	0
$175$	885.572	0.382531
$176$	0	0
$177$	2672.57	1.13493
$178$	0	0
$179$	−2215.59	−0.925145	−0.462572	−	0.886581i	$-0.653073\pi$
$179$	−2215.59	−0.925145	−0.462572	+	0.886581i	$0.653073\pi$
$180$	0	0
$181$	873.497	0.358710	0.179355	−	0.983784i	$-0.442599\pi$
$181$	873.497	0.358710	0.179355	+	0.983784i	$0.442599\pi$
$182$	0	0
$183$	−3840.74	−1.55145
$184$	0	0
$185$	−1841.00	−0.731640
$186$	0	0
$187$	−778.641	−0.304491
$188$	0	0
$189$	894.928	0.344425
$190$	0	0
$191$	2497.74	0.946230	0.473115	−	0.881001i	$-0.343130\pi$
$191$	2497.74	0.946230	0.473115	+	0.881001i	$0.343130\pi$
$192$	0	0
$193$	909.155	0.339080	0.169540	−	0.985523i	$-0.445772\pi$
$193$	909.155	0.339080	0.169540	+	0.985523i	$0.445772\pi$
$194$	0	0
$195$	3025.55	1.11110
$196$	0	0
$197$	608.627	0.220116	0.110058	−	0.993925i	$-0.464896\pi$
$197$	608.627	0.220116	0.110058	+	0.993925i	$0.464896\pi$
$198$	0	0
$199$	2304.98	0.821083	0.410542	−	0.911842i	$-0.365340\pi$
$199$	2304.98	0.821083	0.410542	+	0.911842i	$0.365340\pi$
$200$	0	0
$201$	6209.54	2.17904
$202$	0	0
$203$	−29.6285	−0.0102439
$204$	0	0
$205$	478.963	0.163182
$206$	0	0
$207$	697.739	0.234281
$208$	0	0
$209$	1840.15	0.609022
$210$	0	0
$211$	4373.37	1.42690	0.713448	−	0.700708i	$-0.247132\pi$
$211$	4373.37	1.42690	0.713448	+	0.700708i	$0.247132\pi$
$212$	0	0
$213$	5556.17	1.78733
$214$	0	0
$215$	1659.81	0.526502
$216$	0	0
$217$	4244.93	1.32795
$218$	0	0
$219$	−692.542	−0.213688
$220$	0	0
$221$	3741.69	1.13888
$222$	0	0
$223$	−5439.50	−1.63343	−0.816717	−	0.577038i	$-0.804208\pi$
$223$	−5439.50	−1.63343	−0.816717	+	0.577038i	$0.804208\pi$
$224$	0	0
$225$	758.412	0.224715
$226$	0	0
$227$	−216.498	−0.0633017	−0.0316508	−	0.999499i	$-0.510076\pi$
$227$	−216.498	−0.0633017	−0.0316508	+	0.999499i	$0.510076\pi$
$228$	0	0
$229$	−924.664	−0.266828	−0.133414	−	0.991060i	$-0.542594\pi$
$229$	−924.664	−0.266828	−0.133414	+	0.991060i	$0.542594\pi$
$230$	0	0
$231$	4460.54	1.27049
$232$	0	0
$233$	−2369.69	−0.666280	−0.333140	−	0.942877i	$-0.608108\pi$
$233$	−2369.69	−0.666280	−0.333140	+	0.942877i	$0.608108\pi$
$234$	0	0
$235$	−2675.18	−0.742595
$236$	0	0
$237$	−2494.10	−0.683584
$238$	0	0
$239$	2769.60	0.749583	0.374791	−	0.927109i	$-0.377714\pi$
$239$	2769.60	0.749583	0.374791	+	0.927109i	$0.377714\pi$
$240$	0	0
$241$	−5334.10	−1.42572	−0.712862	−	0.701305i	$-0.752601\pi$
$241$	−5334.10	−1.42572	−0.712862	+	0.701305i	$0.752601\pi$
$242$	0	0
$243$	−5435.76	−1.43500
$244$	0	0
$245$	−4558.91	−1.18881
$246$	0	0
$247$	−8842.66	−2.27791
$248$	0	0
$249$	5708.10	1.45275
$250$	0	0
$251$	−4677.13	−1.17617	−0.588084	−	0.808800i	$-0.700118\pi$
$251$	−4677.13	−1.17617	−0.588084	+	0.808800i	$0.700118\pi$
$252$	0	0
$253$	382.486	0.0950463
$254$	0	0
$255$	1772.70	0.435336
$256$	0	0
$257$	−2602.99	−0.631789	−0.315895	−	0.948794i	$-0.602305\pi$
$257$	−2602.99	−0.631789	−0.315895	+	0.948794i	$0.602305\pi$
$258$	0	0
$259$	13042.7	3.12910
$260$	0	0
$261$	−25.3741	−0.00601769
$262$	0	0
$263$	3411.30	0.799809	0.399904	−	0.916557i	$-0.369043\pi$
$263$	3411.30	0.799809	0.399904	+	0.916557i	$0.369043\pi$
$264$	0	0
$265$	−2046.73	−0.474451
$266$	0	0
$267$	−7951.77	−1.82262
$268$	0	0
$269$	4366.56	0.989718	0.494859	−	0.868973i	$-0.335220\pi$
$269$	4366.56	0.989718	0.494859	+	0.868973i	$0.335220\pi$
$270$	0	0
$271$	−5682.45	−1.27374	−0.636872	−	0.770970i	$-0.719772\pi$
$271$	−5682.45	−1.27374	−0.636872	+	0.770970i	$0.719772\pi$
$272$	0	0
$273$	−21434.7	−4.75197
$274$	0	0
$275$	415.746	0.0911652
$276$	0	0
$277$	−1884.62	−0.408794	−0.204397	−	0.978888i	$-0.565523\pi$
$277$	−1884.62	−0.408794	−0.204397	+	0.978888i	$0.565523\pi$
$278$	0	0
$279$	3635.39	0.780091
$280$	0	0
$281$	−2706.42	−0.574561	−0.287281	−	0.957846i	$-0.592751\pi$
$281$	−2706.42	−0.574561	−0.287281	+	0.957846i	$0.592751\pi$
$282$	0	0
$283$	5963.64	1.25266	0.626328	−	0.779560i	$-0.284557\pi$
$283$	5963.64	1.25266	0.626328	+	0.779560i	$0.284557\pi$
$284$	0	0
$285$	−4189.38	−0.870728
$286$	0	0
$287$	−3393.26	−0.697901
$288$	0	0
$289$	−2720.71	−0.553778
$290$	0	0
$291$	−2059.06	−0.414792
$292$	0	0
$293$	−4735.85	−0.944272	−0.472136	−	0.881526i	$-0.656517\pi$
$293$	−4735.85	−0.944272	−0.472136	+	0.881526i	$0.656517\pi$
$294$	0	0
$295$	−1764.75	−0.348297
$296$	0	0
$297$	420.138	0.0820837
$298$	0	0
$299$	−1838.00	−0.355500
$300$	0	0
$301$	−11759.0	−2.25176
$302$	0	0
$303$	12559.6	2.38130
$304$	0	0
$305$	2536.12	0.476123
$306$	0	0
$307$	769.241	0.143006	0.0715031	−	0.997440i	$-0.477220\pi$
$307$	769.241	0.143006	0.0715031	+	0.997440i	$0.477220\pi$
$308$	0	0
$309$	−13141.5	−2.41939
$310$	0	0
$311$	5592.71	1.01972	0.509861	−	0.860257i	$-0.329697\pi$
$311$	5592.71	1.01972	0.509861	+	0.860257i	$0.329697\pi$
$312$	0	0
$313$	9777.19	1.76562	0.882811	−	0.469729i	$-0.155648\pi$
$313$	9777.19	1.76562	0.882811	+	0.469729i	$0.155648\pi$
$314$	0	0
$315$	−5373.03	−0.961066
$316$	0	0
$317$	1868.51	0.331060	0.165530	−	0.986205i	$-0.447066\pi$
$317$	1868.51	0.331060	0.165530	+	0.986205i	$0.447066\pi$
$318$	0	0
$319$	−13.9096	−0.00244134
$320$	0	0
$321$	12341.6	2.14592
$322$	0	0
$323$	−5181.00	−0.892503
$324$	0	0
$325$	−1997.83	−0.340983
$326$	0	0
$327$	−3272.29	−0.553388
$328$	0	0
$329$	18952.6	3.17595
$330$	0	0
$331$	−6966.91	−1.15691	−0.578454	−	0.815715i	$-0.696344\pi$
$331$	−6966.91	−1.15691	−0.578454	+	0.815715i	$0.696344\pi$
$332$	0	0
$333$	11169.9	1.83816
$334$	0	0
$335$	−4100.28	−0.668723
$336$	0	0
$337$	−17.5487	−0.00283661	−0.00141830	−	0.999999i	$-0.500451\pi$
$337$	−17.5487	−0.00283661	−0.00141830	+	0.999999i	$0.500451\pi$
$338$	0	0
$339$	−1818.17	−0.291297
$340$	0	0
$341$	1992.85	0.316477
$342$	0	0
$343$	20147.9	3.17167
$344$	0	0
$345$	−870.790	−0.135889
$346$	0	0
$347$	9859.30	1.52529	0.762644	−	0.646818i	$-0.223901\pi$
$347$	9859.30	1.52529	0.762644	+	0.646818i	$0.223901\pi$
$348$	0	0
$349$	7200.25	1.10436	0.552178	−	0.833726i	$-0.313797\pi$
$349$	7200.25	1.10436	0.552178	+	0.833726i	$0.313797\pi$
$350$	0	0
$351$	−2018.93	−0.307016
$352$	0	0
$353$	−6054.58	−0.912897	−0.456449	−	0.889750i	$-0.650879\pi$
$353$	−6054.58	−0.912897	−0.456449	+	0.889750i	$0.650879\pi$
$354$	0	0
$355$	−3668.85	−0.548513
$356$	0	0
$357$	−12558.8	−1.86186
$358$	0	0
$359$	−2734.26	−0.401974	−0.200987	−	0.979594i	$-0.564415\pi$
$359$	−2734.26	−0.401974	−0.200987	+	0.979594i	$0.564415\pi$
$360$	0	0
$361$	5385.16	0.785123
$362$	0	0
$363$	−7984.37	−1.15447
$364$	0	0
$365$	457.300	0.0655785
$366$	0	0
$367$	2465.50	0.350676	0.175338	−	0.984508i	$-0.443898\pi$
$367$	2465.50	0.350676	0.175338	+	0.984508i	$0.443898\pi$
$368$	0	0
$369$	−2906.01	−0.409976
$370$	0	0
$371$	14500.2	2.02915
$372$	0	0
$373$	5704.45	0.791863	0.395932	−	0.918280i	$-0.370422\pi$
$373$	5704.45	0.791863	0.395932	+	0.918280i	$0.370422\pi$
$374$	0	0
$375$	−946.511	−0.130340
$376$	0	0
$377$	66.8411	0.00913128
$378$	0	0
$379$	−10231.9	−1.38675	−0.693376	−	0.720576i	$-0.743878\pi$
$379$	−10231.9	−1.38675	−0.693376	+	0.720576i	$0.743878\pi$
$380$	0	0
$381$	1597.66	0.214830
$382$	0	0
$383$	−6321.34	−0.843356	−0.421678	−	0.906746i	$-0.638559\pi$
$383$	−6321.34	−0.843356	−0.421678	+	0.906746i	$0.638559\pi$
$384$	0	0
$385$	−2945.38	−0.389898
$386$	0	0
$387$	−10070.5	−1.32278
$388$	0	0
$389$	−925.594	−0.120641	−0.0603207	−	0.998179i	$-0.519212\pi$
$389$	−925.594	−0.120641	−0.0603207	+	0.998179i	$0.519212\pi$
$390$	0	0
$391$	−1076.90	−0.139287
$392$	0	0
$393$	5650.13	0.725219
$394$	0	0
$395$	1646.91	0.209784
$396$	0	0
$397$	−2706.75	−0.342187	−0.171093	−	0.985255i	$-0.554730\pi$
$397$	−2706.75	−0.342187	−0.171093	+	0.985255i	$0.554730\pi$
$398$	0	0
$399$	29680.0	3.72396
$400$	0	0
$401$	−14095.3	−1.75532	−0.877661	−	0.479281i	$-0.840897\pi$
$401$	−14095.3	−1.75532	−0.877661	+	0.479281i	$0.840897\pi$
$402$	0	0
$403$	−9576.45	−1.18372
$404$	0	0
$405$	3138.92	0.385121
$406$	0	0
$407$	6123.12	0.745729
$408$	0	0
$409$	−10846.0	−1.31125	−0.655625	−	0.755087i	$-0.727595\pi$
$409$	−10846.0	−1.31125	−0.655625	+	0.755087i	$0.727595\pi$
$410$	0	0
$411$	−18325.0	−2.19928
$412$	0	0
$413$	12502.5	1.48961
$414$	0	0
$415$	−3769.17	−0.445834
$416$	0	0
$417$	−6996.84	−0.821670
$418$	0	0
$419$	5626.55	0.656026	0.328013	−	0.944673i	$-0.393621\pi$
$419$	5626.55	0.656026	0.328013	+	0.944673i	$0.393621\pi$
$420$	0	0
$421$	−7109.09	−0.822983	−0.411491	−	0.911414i	$-0.634992\pi$
$421$	−7109.09	−0.822983	−0.411491	+	0.911414i	$0.634992\pi$
$422$	0	0
$423$	16231.1	1.86568
$424$	0	0
$425$	−1170.55	−0.133600
$426$	0	0
$427$	−17967.3	−2.03630
$428$	0	0
$429$	−10062.9	−1.13249
$430$	0	0
$431$	−4464.14	−0.498909	−0.249455	−	0.968387i	$-0.580251\pi$
$431$	−4464.14	−0.498909	−0.249455	+	0.968387i	$0.580251\pi$
$432$	0	0
$433$	11009.4	1.22189	0.610947	−	0.791672i	$-0.290789\pi$
$433$	11009.4	1.22189	0.610947	+	0.791672i	$0.290789\pi$
$434$	0	0
$435$	31.6673	0.00349042
$436$	0	0
$437$	2545.03	0.278593
$438$	0	0
$439$	3052.70	0.331885	0.165943	−	0.986135i	$-0.446933\pi$
$439$	3052.70	0.331885	0.165943	+	0.986135i	$0.446933\pi$
$440$	0	0
$441$	27660.2	2.98675
$442$	0	0
$443$	4465.82	0.478956	0.239478	−	0.970902i	$-0.423024\pi$
$443$	4465.82	0.478956	0.239478	+	0.970902i	$0.423024\pi$
$444$	0	0
$445$	5250.72	0.559343
$446$	0	0
$447$	3260.64	0.345018
$448$	0	0
$449$	9040.14	0.950179	0.475090	−	0.879937i	$-0.342416\pi$
$449$	9040.14	0.950179	0.475090	+	0.879937i	$0.342416\pi$
$450$	0	0
$451$	−1593.02	−0.166324
$452$	0	0
$453$	192.420	0.0199574
$454$	0	0
$455$	14153.8	1.45833
$456$	0	0
$457$	7756.88	0.793986	0.396993	−	0.917822i	$-0.370054\pi$
$457$	7756.88	0.793986	0.396993	+	0.917822i	$0.370054\pi$
$458$	0	0
$459$	−1182.91	−0.120291
$460$	0	0
$461$	7064.19	0.713692	0.356846	−	0.934163i	$-0.383852\pi$
$461$	7064.19	0.713692	0.356846	+	0.934163i	$0.383852\pi$
$462$	0	0
$463$	−12599.7	−1.26470	−0.632350	−	0.774682i	$-0.717910\pi$
$463$	−12599.7	−1.26470	−0.632350	+	0.774682i	$0.717910\pi$
$464$	0	0
$465$	−4537.03	−0.452473
$466$	0	0
$467$	−14746.0	−1.46117	−0.730583	−	0.682824i	$-0.760751\pi$
$467$	−14746.0	−1.46117	−0.730583	+	0.682824i	$0.760751\pi$
$468$	0	0
$469$	29048.8	2.86002
$470$	0	0
$471$	−11966.1	−1.17063
$472$	0	0
$473$	−5520.46	−0.536641
$474$	0	0
$475$	2766.33	0.267217
$476$	0	0
$477$	12418.1	1.19200
$478$	0	0
$479$	17411.2	1.66084	0.830418	−	0.557141i	$-0.188102\pi$
$479$	17411.2	1.66084	0.830418	+	0.557141i	$0.188102\pi$
$480$	0	0
$481$	−29424.1	−2.78924
$482$	0	0
$483$	6169.18	0.581175
$484$	0	0
$485$	1359.64	0.127295
$486$	0	0
$487$	−1320.34	−0.122855	−0.0614276	−	0.998112i	$-0.519565\pi$
$487$	−1320.34	−0.122855	−0.0614276	+	0.998112i	$0.519565\pi$
$488$	0	0
$489$	−3473.61	−0.321231
$490$	0	0
$491$	−17115.8	−1.57317	−0.786583	−	0.617485i	$-0.788152\pi$
$491$	−17115.8	−1.57317	−0.786583	+	0.617485i	$0.788152\pi$
$492$	0	0
$493$	39.1629	0.00357770
$494$	0	0
$495$	−2522.45	−0.229042
$496$	0	0
$497$	25992.3	2.34590
$498$	0	0
$499$	−17540.6	−1.57360	−0.786798	−	0.617210i	$-0.788263\pi$
$499$	−17540.6	−1.57360	−0.786798	+	0.617210i	$0.788263\pi$
$500$	0	0
$501$	4250.02	0.378996
$502$	0	0
$503$	4934.98	0.437455	0.218727	−	0.975786i	$-0.429809\pi$
$503$	4934.98	0.437455	0.218727	+	0.975786i	$0.429809\pi$
$504$	0	0
$505$	−8293.39	−0.730794
$506$	0	0
$507$	31720.3	2.77860
$508$	0	0
$509$	11927.3	1.03865	0.519323	−	0.854578i	$-0.326184\pi$
$509$	11927.3	1.03865	0.519323	+	0.854578i	$0.326184\pi$
$510$	0	0
$511$	−3239.78	−0.280468
$512$	0	0
$513$	2795.56	0.240598
$514$	0	0
$515$	8677.58	0.742485
$516$	0	0
$517$	8897.57	0.756895
$518$	0	0
$519$	11918.1	1.00799
$520$	0	0
$521$	−9592.76	−0.806653	−0.403327	−	0.915056i	$-0.632146\pi$
$521$	−9592.76	−0.806653	−0.403327	+	0.915056i	$0.632146\pi$
$522$	0	0
$523$	−6525.20	−0.545558	−0.272779	−	0.962077i	$-0.587943\pi$
$523$	−6525.20	−0.545558	−0.272779	+	0.962077i	$0.587943\pi$
$524$	0	0
$525$	6705.63	0.557443
$526$	0	0
$527$	−5610.94	−0.463788
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	10707.3	0.875058
$532$	0	0
$533$	7655.10	0.622100
$534$	0	0
$535$	−8149.40	−0.658560
$536$	0	0
$537$	−16776.6	−1.34817
$538$	0	0
$539$	15162.8	1.21170
$540$	0	0
$541$	5634.05	0.447739	0.223870	−	0.974619i	$-0.428131\pi$
$541$	5634.05	0.447739	0.223870	+	0.974619i	$0.428131\pi$
$542$	0	0
$543$	6614.19	0.522729
$544$	0	0
$545$	2160.76	0.169829
$546$	0	0
$547$	−6093.08	−0.476273	−0.238136	−	0.971232i	$-0.576537\pi$
$547$	−6093.08	−0.476273	−0.238136	+	0.971232i	$0.576537\pi$
$548$	0	0
$549$	−15387.4	−1.19620
$550$	0	0
$551$	−92.5529	−0.00715587
$552$	0	0
$553$	−11667.6	−0.897212
$554$	0	0
$555$	−13940.2	−1.06618
$556$	0	0
$557$	21461.9	1.63262	0.816311	−	0.577612i	$-0.196015\pi$
$557$	21461.9	1.63262	0.816311	+	0.577612i	$0.196015\pi$
$558$	0	0
$559$	26528.1	2.00719
$560$	0	0
$561$	−5895.93	−0.443719
$562$	0	0
$563$	11038.5	0.826316	0.413158	−	0.910659i	$-0.364426\pi$
$563$	11038.5	0.826316	0.413158	+	0.910659i	$0.364426\pi$
$564$	0	0
$565$	1200.58	0.0893959
$566$	0	0
$567$	−22237.9	−1.64710
$568$	0	0
$569$	−19210.3	−1.41535	−0.707677	−	0.706536i	$-0.750257\pi$
$569$	−19210.3	−1.41535	−0.707677	+	0.706536i	$0.750257\pi$
$570$	0	0
$571$	11967.0	0.877066	0.438533	−	0.898715i	$-0.355498\pi$
$571$	11967.0	0.877066	0.438533	+	0.898715i	$0.355498\pi$
$572$	0	0
$573$	18913.1	1.37889
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−14982.3	−1.08097	−0.540487	−	0.841352i	$-0.681760\pi$
$577$	−14982.3	−1.08097	−0.540487	+	0.841352i	$0.681760\pi$
$578$	0	0
$579$	6884.20	0.494124
$580$	0	0
$581$	26703.0	1.90676
$582$	0	0
$583$	6807.35	0.483587
$584$	0	0
$585$	12121.4	0.856682
$586$	0	0
$587$	−24002.9	−1.68774	−0.843871	−	0.536547i	$-0.819729\pi$
$587$	−24002.9	−1.68774	−0.843871	+	0.536547i	$0.819729\pi$
$588$	0	0
$589$	13260.2	0.927637
$590$	0	0
$591$	4608.58	0.320764
$592$	0	0
$593$	5124.67	0.354882	0.177441	−	0.984131i	$-0.443218\pi$
$593$	5124.67	0.354882	0.177441	+	0.984131i	$0.443218\pi$
$594$	0	0
$595$	8292.84	0.571384
$596$	0	0
$597$	17453.5	1.19652
$598$	0	0
$599$	23776.8	1.62186	0.810928	−	0.585146i	$-0.198963\pi$
$599$	23776.8	1.62186	0.810928	+	0.585146i	$0.198963\pi$
$600$	0	0
$601$	−25435.3	−1.72633	−0.863166	−	0.504920i	$-0.831522\pi$
$601$	−25435.3	−1.72633	−0.863166	+	0.504920i	$0.831522\pi$
$602$	0	0
$603$	24877.6	1.68009
$604$	0	0
$605$	5272.24	0.354293
$606$	0	0
$607$	7445.94	0.497894	0.248947	−	0.968517i	$-0.419916\pi$
$607$	7445.94	0.497894	0.248947	+	0.968517i	$0.419916\pi$
$608$	0	0
$609$	−224.349	−0.0149279
$610$	0	0
$611$	−42756.5	−2.83100
$612$	0	0
$613$	−12874.4	−0.848275	−0.424138	−	0.905598i	$-0.639423\pi$
$613$	−12874.4	−0.848275	−0.424138	+	0.905598i	$0.639423\pi$
$614$	0	0
$615$	3626.75	0.237796
$616$	0	0
$617$	−19247.2	−1.25585	−0.627927	−	0.778272i	$-0.716096\pi$
$617$	−19247.2	−1.25585	−0.627927	+	0.778272i	$0.716096\pi$
$618$	0	0
$619$	14496.4	0.941293	0.470647	−	0.882322i	$-0.344021\pi$
$619$	14496.4	0.941293	0.470647	+	0.882322i	$0.344021\pi$
$620$	0	0
$621$	581.074	0.0375486
$622$	0	0
$623$	−37199.1	−2.39222
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	13933.7	0.887496
$628$	0	0
$629$	−17239.9	−1.09284
$630$	0	0
$631$	18030.9	1.13756	0.568779	−	0.822490i	$-0.307416\pi$
$631$	18030.9	1.13756	0.568779	+	0.822490i	$0.307416\pi$
$632$	0	0
$633$	33115.5	2.07934
$634$	0	0
$635$	−1054.96	−0.0659291
$636$	0	0
$637$	−72863.4	−4.53211
$638$	0	0
$639$	22260.0	1.37808
$640$	0	0
$641$	11776.4	0.725646	0.362823	−	0.931858i	$-0.381813\pi$
$641$	11776.4	0.725646	0.362823	+	0.931858i	$0.381813\pi$
$642$	0	0
$643$	−20207.4	−1.23935	−0.619676	−	0.784858i	$-0.712736\pi$
$643$	−20207.4	−1.23935	−0.619676	+	0.784858i	$0.712736\pi$
$644$	0	0
$645$	12568.2	0.767244
$646$	0	0
$647$	−21452.8	−1.30355	−0.651776	−	0.758412i	$-0.725976\pi$
$647$	−21452.8	−1.30355	−0.651776	+	0.758412i	$0.725976\pi$
$648$	0	0
$649$	5869.50	0.355005
$650$	0	0
$651$	32143.0	1.93515
$652$	0	0
$653$	−15095.2	−0.904624	−0.452312	−	0.891860i	$-0.649401\pi$
$653$	−15095.2	−0.904624	−0.452312	+	0.891860i	$0.649401\pi$
$654$	0	0
$655$	−3730.89	−0.222562
$656$	0	0
$657$	−2774.57	−0.164759
$658$	0	0
$659$	−6964.68	−0.411693	−0.205846	−	0.978584i	$-0.565995\pi$
$659$	−6964.68	−0.411693	−0.205846	+	0.978584i	$0.565995\pi$
$660$	0	0
$661$	−17380.4	−1.02272	−0.511361	−	0.859366i	$-0.670858\pi$
$661$	−17380.4	−1.02272	−0.511361	+	0.859366i	$0.670858\pi$
$662$	0	0
$663$	28332.4	1.65964
$664$	0	0
$665$	−19598.3	−1.14284
$666$	0	0
$667$	−19.2377	−0.00111677
$668$	0	0
$669$	−41188.3	−2.38032
$670$	0	0
$671$	−8435.04	−0.485292
$672$	0	0
$673$	12383.2	0.709271	0.354635	−	0.935005i	$-0.384605\pi$
$673$	12383.2	0.709271	0.354635	+	0.935005i	$0.384605\pi$
$674$	0	0
$675$	631.602	0.0360154
$676$	0	0
$677$	−8375.74	−0.475489	−0.237744	−	0.971328i	$-0.576408\pi$
$677$	−8375.74	−0.475489	−0.237744	+	0.971328i	$0.576408\pi$
$678$	0	0
$679$	−9632.49	−0.544420
$680$	0	0
$681$	−1639.34	−0.0922463
$682$	0	0
$683$	4434.66	0.248444	0.124222	−	0.992254i	$-0.460356\pi$
$683$	4434.66	0.248444	0.124222	+	0.992254i	$0.460356\pi$
$684$	0	0
$685$	12100.4	0.674936
$686$	0	0
$687$	−7001.64	−0.388834
$688$	0	0
$689$	−32712.1	−1.80875
$690$	0	0
$691$	−11032.8	−0.607389	−0.303694	−	0.952770i	$-0.598220\pi$
$691$	−11032.8	−0.607389	−0.303694	+	0.952770i	$0.598220\pi$
$692$	0	0
$693$	17870.5	0.979574
$694$	0	0
$695$	4620.15	0.252162
$696$	0	0
$697$	4485.20	0.243743
$698$	0	0
$699$	−17943.5	−0.970936
$700$	0	0
$701$	3708.40	0.199806	0.0999032	−	0.994997i	$-0.468147\pi$
$701$	3708.40	0.199806	0.0999032	+	0.994997i	$0.468147\pi$
$702$	0	0
$703$	40742.6	2.18583
$704$	0	0
$705$	−20256.7	−1.08215
$706$	0	0
$707$	58755.2	3.12548
$708$	0	0
$709$	−6315.04	−0.334508	−0.167254	−	0.985914i	$-0.553490\pi$
$709$	−6315.04	−0.334508	−0.167254	+	0.985914i	$0.553490\pi$
$710$	0	0
$711$	−9992.26	−0.527059
$712$	0	0
$713$	2756.22	0.144770
$714$	0	0
$715$	6644.71	0.347550
$716$	0	0
$717$	20971.6	1.09233
$718$	0	0
$719$	−24736.9	−1.28307	−0.641536	−	0.767093i	$-0.721703\pi$
$719$	−24736.9	−1.28307	−0.641536	+	0.767093i	$0.721703\pi$
$720$	0	0
$721$	−61477.0	−3.17548
$722$	0	0
$723$	−40390.2	−2.07763
$724$	0	0
$725$	−20.9106	−0.00107117
$726$	0	0
$727$	−33283.6	−1.69797	−0.848983	−	0.528420i	$-0.822785\pi$
$727$	−33283.6	−1.69797	−0.848983	+	0.528420i	$0.822785\pi$
$728$	0	0
$729$	−24209.9	−1.22999
$730$	0	0
$731$	15543.1	0.786430
$732$	0	0
$733$	−33636.5	−1.69494	−0.847471	−	0.530841i	$-0.821876\pi$
$733$	−33636.5	−1.69494	−0.847471	+	0.530841i	$0.821876\pi$
$734$	0	0
$735$	−34520.4	−1.73239
$736$	0	0
$737$	13637.4	0.681601
$738$	0	0
$739$	−36575.5	−1.82064	−0.910319	−	0.413907i	$-0.864164\pi$
$739$	−36575.5	−1.82064	−0.910319	+	0.413907i	$0.864164\pi$
$740$	0	0
$741$	−66957.3	−3.31949
$742$	0	0
$743$	17704.7	0.874190	0.437095	−	0.899415i	$-0.356007\pi$
$743$	17704.7	0.874190	0.437095	+	0.899415i	$0.356007\pi$
$744$	0	0
$745$	−2153.07	−0.105882
$746$	0	0
$747$	22868.7	1.12011
$748$	0	0
$749$	57735.1	2.81655
$750$	0	0
$751$	−8858.37	−0.430421	−0.215211	−	0.976568i	$-0.569044\pi$
$751$	−8858.37	−0.430421	−0.215211	+	0.976568i	$0.569044\pi$
$752$	0	0
$753$	−35415.7	−1.71397
$754$	0	0
$755$	−127.059	−0.00612470
$756$	0	0
$757$	7899.26	0.379265	0.189633	−	0.981855i	$-0.439270\pi$
$757$	7899.26	0.379265	0.189633	+	0.981855i	$0.439270\pi$
$758$	0	0
$759$	2896.22	0.138506
$760$	0	0
$761$	23437.4	1.11643	0.558216	−	0.829696i	$-0.311486\pi$
$761$	23437.4	1.11643	0.558216	+	0.829696i	$0.311486\pi$
$762$	0	0
$763$	−15308.1	−0.726329
$764$	0	0
$765$	7102.06	0.335654
$766$	0	0
$767$	−28205.4	−1.32782
$768$	0	0
$769$	31447.7	1.47468	0.737342	−	0.675519i	$-0.236081\pi$
$769$	31447.7	1.47468	0.737342	+	0.675519i	$0.236081\pi$
$770$	0	0
$771$	−19710.0	−0.920674
$772$	0	0
$773$	2397.42	0.111551	0.0557756	−	0.998443i	$-0.482237\pi$
$773$	2397.42	0.111551	0.0557756	+	0.998443i	$0.482237\pi$
$774$	0	0
$775$	2995.89	0.138859
$776$	0	0
$777$	98760.8	4.55987
$778$	0	0
$779$	−10599.8	−0.487518
$780$	0	0
$781$	12202.5	0.559077
$782$	0	0
$783$	−21.1314	−0.000964465 0
$784$	0	0
$785$	7901.44	0.359254
$786$	0	0
$787$	19407.0	0.879014	0.439507	−	0.898239i	$-0.355153\pi$
$787$	19407.0	0.879014	0.439507	+	0.898239i	$0.355153\pi$
$788$	0	0
$789$	25830.7	1.16552
$790$	0	0
$791$	−8505.58	−0.382331
$792$	0	0
$793$	40533.8	1.81513
$794$	0	0
$795$	−15498.0	−0.691393
$796$	0	0
$797$	−28631.1	−1.27248	−0.636240	−	0.771491i	$-0.719511\pi$
$797$	−28631.1	−1.27248	−0.636240	+	0.771491i	$0.719511\pi$
$798$	0	0
$799$	−25051.4	−1.10921
$800$	0	0
$801$	−31857.6	−1.40529
$802$	0	0
$803$	−1520.96	−0.0668414
$804$	0	0
$805$	−4073.63	−0.178356
$806$	0	0
$807$	33064.0	1.44227
$808$	0	0
$809$	−9191.25	−0.399440	−0.199720	−	0.979853i	$-0.564003\pi$
$809$	−9191.25	−0.399440	−0.199720	+	0.979853i	$0.564003\pi$
$810$	0	0
$811$	567.477	0.0245706	0.0122853	−	0.999925i	$-0.496089\pi$
$811$	567.477	0.0245706	0.0122853	+	0.999925i	$0.496089\pi$
$812$	0	0
$813$	−43028.0	−1.85616
$814$	0	0
$815$	2293.69	0.0985823
$816$	0	0
$817$	−36732.6	−1.57296
$818$	0	0
$819$	−85875.2	−3.66388
$820$	0	0
$821$	−6057.35	−0.257494	−0.128747	−	0.991677i	$-0.541096\pi$
$821$	−6057.35	−0.257494	−0.128747	+	0.991677i	$0.541096\pi$
$822$	0	0
$823$	21327.0	0.903294	0.451647	−	0.892197i	$-0.350837\pi$
$823$	21327.0	0.903294	0.451647	+	0.892197i	$0.350837\pi$
$824$	0	0
$825$	3148.06	0.132850
$826$	0	0
$827$	27553.8	1.15857	0.579286	−	0.815124i	$-0.303332\pi$
$827$	27553.8	1.15857	0.579286	+	0.815124i	$0.303332\pi$
$828$	0	0
$829$	11181.6	0.468461	0.234231	−	0.972181i	$-0.424743\pi$
$829$	11181.6	0.468461	0.234231	+	0.972181i	$0.424743\pi$
$830$	0	0
$831$	−14270.5	−0.595715
$832$	0	0
$833$	−42691.4	−1.77571
$834$	0	0
$835$	−2806.37	−0.116310
$836$	0	0
$837$	3027.54	0.125026
$838$	0	0
$839$	−9811.83	−0.403745	−0.201872	−	0.979412i	$-0.564703\pi$
$839$	−9811.83	−0.403745	−0.201872	+	0.979412i	$0.564703\pi$
$840$	0	0
$841$	−24388.3	−0.999971
$842$	0	0
$843$	−20493.3	−0.837279
$844$	0	0
$845$	−20945.6	−0.852721
$846$	0	0
$847$	−37351.6	−1.51525
$848$	0	0
$849$	45157.2	1.82543
$850$	0	0
$851$	8468.62	0.341129
$852$	0	0
$853$	−24030.8	−0.964594	−0.482297	−	0.876008i	$-0.660197\pi$
$853$	−24030.8	−0.964594	−0.482297	+	0.876008i	$0.660197\pi$
$854$	0	0
$855$	−16784.2	−0.671352
$856$	0	0
$857$	−16934.8	−0.675008	−0.337504	−	0.941324i	$-0.609583\pi$
$857$	−16934.8	−0.675008	−0.337504	+	0.941324i	$0.609583\pi$
$858$	0	0
$859$	31057.0	1.23359	0.616793	−	0.787125i	$-0.288432\pi$
$859$	31057.0	1.23359	0.616793	+	0.787125i	$0.288432\pi$
$860$	0	0
$861$	−25694.0	−1.01701
$862$	0	0
$863$	16157.8	0.637332	0.318666	−	0.947867i	$-0.396765\pi$
$863$	16157.8	0.637332	0.318666	+	0.947867i	$0.396765\pi$
$864$	0	0
$865$	−7869.73	−0.309340
$866$	0	0
$867$	−20601.4	−0.806992
$868$	0	0
$869$	−5477.55	−0.213824
$870$	0	0
$871$	−65533.3	−2.54938
$872$	0	0
$873$	−8249.35	−0.319815
$874$	0	0
$875$	−4427.86	−0.171073
$876$	0	0
$877$	−32473.9	−1.25036	−0.625180	−	0.780480i	$-0.714975\pi$
$877$	−32473.9	−1.25036	−0.625180	+	0.780480i	$0.714975\pi$
$878$	0	0
$879$	−35860.3	−1.37604
$880$	0	0
$881$	−15703.2	−0.600517	−0.300258	−	0.953858i	$-0.597073\pi$
$881$	−15703.2	−0.600517	−0.300258	+	0.953858i	$0.597073\pi$
$882$	0	0
$883$	14330.4	0.546155	0.273078	−	0.961992i	$-0.411958\pi$
$883$	14330.4	0.546155	0.273078	+	0.961992i	$0.411958\pi$
$884$	0	0
$885$	−13362.8	−0.507556
$886$	0	0
$887$	−23238.2	−0.879664	−0.439832	−	0.898080i	$-0.644962\pi$
$887$	−23238.2	−0.879664	−0.439832	+	0.898080i	$0.644962\pi$
$888$	0	0
$889$	7473.98	0.281968
$890$	0	0
$891$	−10439.9	−0.392537
$892$	0	0
$893$	59203.6	2.21856
$894$	0	0
$895$	11077.9	0.413737
$896$	0	0
$897$	−13917.5	−0.518052
$898$	0	0
$899$	−100.233	−0.00371854
$900$	0	0
$901$	−19166.3	−0.708683
$902$	0	0
$903$	−89040.4	−3.28137
$904$	0	0
$905$	−4367.48	−0.160420
$906$	0	0
$907$	3667.46	0.134263	0.0671313	−	0.997744i	$-0.478615\pi$
$907$	3667.46	0.134263	0.0671313	+	0.997744i	$0.478615\pi$
$908$	0	0
$909$	50318.4	1.83604
$910$	0	0
$911$	−21457.9	−0.780385	−0.390192	−	0.920733i	$-0.627591\pi$
$911$	−21457.9	−0.780385	−0.390192	+	0.920733i	$0.627591\pi$
$912$	0	0
$913$	12536.1	0.454420
$914$	0	0
$915$	19203.7	0.693830
$916$	0	0
$917$	26431.8	0.951859
$918$	0	0
$919$	11109.3	0.398762	0.199381	−	0.979922i	$-0.436107\pi$
$919$	11109.3	0.398762	0.199381	+	0.979922i	$0.436107\pi$
$920$	0	0
$921$	5824.76	0.208396
$922$	0	0
$923$	−58637.9	−2.09110
$924$	0	0
$925$	9205.02	0.327199
$926$	0	0
$927$	−52649.4	−1.86541
$928$	0	0
$929$	−20804.3	−0.734732	−0.367366	−	0.930076i	$-0.619740\pi$
$929$	−20804.3	−0.734732	−0.367366	+	0.930076i	$0.619740\pi$
$930$	0	0
$931$	100892.	3.55166
$932$	0	0
$933$	42348.5	1.48599
$934$	0	0
$935$	3893.20	0.136173
$936$	0	0
$937$	−35550.5	−1.23947	−0.619735	−	0.784811i	$-0.712760\pi$
$937$	−35550.5	−1.23947	−0.619735	+	0.784811i	$0.712760\pi$
$938$	0	0
$939$	74033.7	2.57295
$940$	0	0
$941$	49674.4	1.72087	0.860436	−	0.509558i	$-0.170191\pi$
$941$	49674.4	1.72087	0.860436	+	0.509558i	$0.170191\pi$
$942$	0	0
$943$	−2203.23	−0.0760839
$944$	0	0
$945$	−4474.64	−0.154032
$946$	0	0
$947$	−33466.2	−1.14837	−0.574184	−	0.818726i	$-0.694681\pi$
$947$	−33466.2	−1.14837	−0.574184	+	0.818726i	$0.694681\pi$
$948$	0	0
$949$	7308.85	0.250006
$950$	0	0
$951$	14148.5	0.482437
$952$	0	0
$953$	−17298.4	−0.587984	−0.293992	−	0.955808i	$-0.594984\pi$
$953$	−17298.4	−0.587984	−0.293992	+	0.955808i	$0.594984\pi$
$954$	0	0
$955$	−12488.7	−0.423167
$956$	0	0
$957$	−105.324	−0.00355763
$958$	0	0
$959$	−85726.0	−2.88659
$960$	0	0
$961$	−15430.4	−0.517955
$962$	0	0
$963$	49444.8	1.65456
$964$	0	0
$965$	−4545.78	−0.151641
$966$	0	0
$967$	14869.7	0.494495	0.247248	−	0.968952i	$-0.420474\pi$
$967$	14869.7	0.494495	0.247248	+	0.968952i	$0.420474\pi$
$968$	0	0
$969$	−39231.0	−1.30060
$970$	0	0
$971$	−26634.9	−0.880283	−0.440142	−	0.897928i	$-0.645072\pi$
$971$	−26634.9	−0.880283	−0.440142	+	0.897928i	$0.645072\pi$
$972$	0	0
$973$	−32731.8	−1.07845
$974$	0	0
$975$	−15127.7	−0.496898
$976$	0	0
$977$	40803.6	1.33615	0.668077	−	0.744092i	$-0.267118\pi$
$977$	40803.6	1.33615	0.668077	+	0.744092i	$0.267118\pi$
$978$	0	0
$979$	−17463.7	−0.570115
$980$	0	0
$981$	−13109.9	−0.426675
$982$	0	0
$983$	18615.8	0.604021	0.302011	−	0.953305i	$-0.402342\pi$
$983$	18615.8	0.604021	0.302011	+	0.953305i	$0.402342\pi$
$984$	0	0
$985$	−3043.14	−0.0984389
$986$	0	0
$987$	143510.	4.62815
$988$	0	0
$989$	−7635.11	−0.245483
$990$	0	0
$991$	−46185.1	−1.48044	−0.740220	−	0.672364i	$-0.765279\pi$
$991$	−46185.1	−1.48044	−0.740220	+	0.672364i	$0.765279\pi$
$992$	0	0
$993$	−52754.1	−1.68590
$994$	0	0
$995$	−11524.9	−0.367200
$996$	0	0
$997$	16544.3	0.525541	0.262771	−	0.964858i	$-0.415364\pi$
$997$	16544.3	0.525541	0.262771	+	0.964858i	$0.415364\pi$
$998$	0	0
$999$	9302.26	0.294605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.m.1.4		4
4.3	odd	2		230.4.a.h.1.1	✓	4
12.11	even	2		2070.4.a.bj.1.1		4
20.3	even	4		1150.4.b.n.599.5		8
20.7	even	4		1150.4.b.n.599.4		8
20.19	odd	2		1150.4.a.p.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.1	✓	4	4.3	odd	2
1150.4.a.p.1.4		4	20.19	odd	2
1150.4.b.n.599.4		8	20.7	even	4
1150.4.b.n.599.5		8	20.3	even	4
1840.4.a.m.1.4		4	1.1	even	1	trivial
2070.4.a.bj.1.1		4	12.11	even	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+7.57209 q^{3} -5.00000 q^{5} +35.4229 q^{7} +30.3365 q^{9} +O(q^{10})\)	\(q+7.57209 q^{3} -5.00000 q^{5} +35.4229 q^{7} +30.3365 q^{9} +16.6298 q^{11} -79.9132 q^{13} -37.8604 q^{15} -46.8219 q^{17} +110.653 q^{19} +268.225 q^{21} +23.0000 q^{23} +25.0000 q^{25} +25.2641 q^{27} -0.836422 q^{29} +119.836 q^{31} +125.923 q^{33} -177.114 q^{35} +368.201 q^{37} -605.109 q^{39} -95.7927 q^{41} -331.961 q^{43} -151.682 q^{45} +535.037 q^{47} +911.782 q^{49} -354.539 q^{51} +409.345 q^{53} -83.1492 q^{55} +837.876 q^{57} +352.950 q^{59} -507.223 q^{61} +1074.61 q^{63} +399.566 q^{65} +820.056 q^{67} +174.158 q^{69} +733.770 q^{71} -91.4599 q^{73} +189.302 q^{75} +589.077 q^{77} -329.381 q^{79} -627.783 q^{81} +753.834 q^{83} +234.110 q^{85} -6.33346 q^{87} -1050.14 q^{89} -2830.76 q^{91} +907.407 q^{93} -553.267 q^{95} -271.928 q^{97} +504.491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9	\( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 20 * q^5 + q^7 + 32 * q^9 + 39 * q^11 - 20 * q^13 - 20 * q^15 - 23 * q^17 - 53 * q^19 + 300 * q^21 + 92 * q^23 + 100 * q^25 - 137 * q^27 + 161 * q^29 - 388 * q^31 + 87 * q^33 - 5 * q^35 + 466 * q^37 - 1047 * q^39 + 484 * q^41 - 894 * q^43 - 160 * q^45 + 265 * q^47 + 1643 * q^49 - 1825 * q^51 + 576 * q^53 - 195 * q^55 + 178 * q^57 + 94 * q^59 + 1153 * q^61 - 60 * q^63 + 100 * q^65 + 1472 * q^67 + 92 * q^69 - 200 * q^71 + 1147 * q^73 + 100 * q^75 - 2176 * q^77 + 908 * q^79 - 1056 * q^81 + 1048 * q^83 + 115 * q^85 + 2167 * q^87 - 1784 * q^89 - 2329 * q^91 + 1483 * q^93 + 265 * q^95 - 2047 * q^97 + 2665 * q^99

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