LMFDB - Embedded newform 1840.4.a.o.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-7.88744$ 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)

$f(q)$	$=$	$q+8.88744 q^{3} -5.00000 q^{5} +15.9808 q^{7} +51.9866 q^{9} +O(q^{10})$	$q+8.88744 q^{3} -5.00000 q^{5} +15.9808 q^{7} +51.9866 q^{9} -9.63486 q^{11} +42.2114 q^{13} -44.4372 q^{15} +40.5247 q^{17} -18.3222 q^{19} +142.029 q^{21} +23.0000 q^{23} +25.0000 q^{25} +222.067 q^{27} +31.6437 q^{29} -63.3450 q^{31} -85.6293 q^{33} -79.9042 q^{35} -30.9604 q^{37} +375.151 q^{39} +203.566 q^{41} +170.854 q^{43} -259.933 q^{45} +505.645 q^{47} -87.6127 q^{49} +360.161 q^{51} -224.035 q^{53} +48.1743 q^{55} -162.838 q^{57} -263.713 q^{59} +501.998 q^{61} +830.789 q^{63} -211.057 q^{65} +184.030 q^{67} +204.411 q^{69} -109.717 q^{71} -679.979 q^{73} +222.186 q^{75} -153.973 q^{77} +331.528 q^{79} +569.968 q^{81} +1305.31 q^{83} -202.624 q^{85} +281.231 q^{87} +122.494 q^{89} +674.574 q^{91} -562.975 q^{93} +91.6111 q^{95} -220.692 q^{97} -500.884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 25 * q^5 - 8 * q^7 + 30 * q^9	$ 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9} - 7 q^{11} + 5 q^{13} - 15 q^{15} - 24 q^{17} + 13 q^{19} + 115 q^{23} + 125 q^{25} + 204 q^{27} - 253 q^{29} + 98 q^{31} - 473 q^{33} + 40 q^{35} - 435 q^{37} + 410 q^{39} - 774 q^{41} + 498 q^{43} - 150 q^{45} + 572 q^{47} - 683 q^{49} + 657 q^{51} - 665 q^{53} + 35 q^{55} - 932 q^{57} + 763 q^{59} + 337 q^{61} + 1527 q^{63} - 25 q^{65} + 305 q^{67} + 69 q^{69} + 1504 q^{71} - 1304 q^{73} + 75 q^{75} + 182 q^{77} - 626 q^{79} - 959 q^{81} + 1703 q^{83} + 120 q^{85} + 1354 q^{87} + 646 q^{89} + 767 q^{91} - 452 q^{93} - 65 q^{95} - 233 q^{97} - 111 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 25 * q^5 - 8 * q^7 + 30 * q^9 - 7 * q^11 + 5 * q^13 - 15 * q^15 - 24 * q^17 + 13 * q^19 + 115 * q^23 + 125 * q^25 + 204 * q^27 - 253 * q^29 + 98 * q^31 - 473 * q^33 + 40 * q^35 - 435 * q^37 + 410 * q^39 - 774 * q^41 + 498 * q^43 - 150 * q^45 + 572 * q^47 - 683 * q^49 + 657 * q^51 - 665 * q^53 + 35 * q^55 - 932 * q^57 + 763 * q^59 + 337 * q^61 + 1527 * q^63 - 25 * q^65 + 305 * q^67 + 69 * q^69 + 1504 * q^71 - 1304 * q^73 + 75 * q^75 + 182 * q^77 - 626 * q^79 - 959 * q^81 + 1703 * q^83 + 120 * q^85 + 1354 * q^87 + 646 * q^89 + 767 * q^91 - 452 * q^93 - 65 * q^95 - 233 * q^97 - 111 * 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
$3$	8.88744	1.71039	0.855194	−	0.518307i	$-0.173438\pi$
$3$	8.88744	1.71039	0.855194	+	0.518307i	$0.173438\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	15.9808	0.862884	0.431442	−	0.902141i	$-0.358005\pi$
$7$	15.9808	0.862884	0.431442	+	0.902141i	$0.358005\pi$
$8$	0	0
$9$	51.9866	1.92543
$10$	0	0
$11$	−9.63486	−0.264093	−0.132046	−	0.991244i	$-0.542155\pi$
$11$	−9.63486	−0.264093	−0.132046	+	0.991244i	$0.542155\pi$
$12$	0	0
$13$	42.2114	0.900565	0.450282	−	0.892886i	$-0.351323\pi$
$13$	42.2114	0.900565	0.450282	+	0.892886i	$0.351323\pi$
$14$	0	0
$15$	−44.4372	−0.764909
$16$	0	0
$17$	40.5247	0.578158	0.289079	−	0.957305i	$-0.406651\pi$
$17$	40.5247	0.578158	0.289079	+	0.957305i	$0.406651\pi$
$18$	0	0
$19$	−18.3222	−0.221232	−0.110616	−	0.993863i	$-0.535282\pi$
$19$	−18.3222	−0.221232	−0.110616	+	0.993863i	$0.535282\pi$
$20$	0	0
$21$	142.029	1.47587
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	222.067	1.58284
$28$	0	0
$29$	31.6437	0.202624	0.101312	−	0.994855i	$-0.467696\pi$
$29$	31.6437	0.202624	0.101312	+	0.994855i	$0.467696\pi$
$30$	0	0
$31$	−63.3450	−0.367003	−0.183502	−	0.983019i	$-0.558743\pi$
$31$	−63.3450	−0.367003	−0.183502	+	0.983019i	$0.558743\pi$
$32$	0	0
$33$	−85.6293	−0.451701
$34$	0	0
$35$	−79.9042	−0.385894
$36$	0	0
$37$	−30.9604	−0.137564	−0.0687819	−	0.997632i	$-0.521911\pi$
$37$	−30.9604	−0.137564	−0.0687819	+	0.997632i	$0.521911\pi$
$38$	0	0
$39$	375.151	1.54032
$40$	0	0
$41$	203.566	0.775407	0.387704	−	0.921784i	$-0.373268\pi$
$41$	203.566	0.775407	0.387704	+	0.921784i	$0.373268\pi$
$42$	0	0
$43$	170.854	0.605929	0.302964	−	0.953002i	$-0.402024\pi$
$43$	170.854	0.605929	0.302964	+	0.953002i	$0.402024\pi$
$44$	0	0
$45$	−259.933	−0.861078
$46$	0	0
$47$	505.645	1.56927	0.784637	−	0.619955i	$-0.212849\pi$
$47$	505.645	1.56927	0.784637	+	0.619955i	$0.212849\pi$
$48$	0	0
$49$	−87.6127	−0.255431
$50$	0	0
$51$	360.161	0.988875
$52$	0	0
$53$	−224.035	−0.580632	−0.290316	−	0.956931i	$-0.593760\pi$
$53$	−224.035	−0.580632	−0.290316	+	0.956931i	$0.593760\pi$
$54$	0	0
$55$	48.1743	0.118106
$56$	0	0
$57$	−162.838	−0.378393
$58$	0	0
$59$	−263.713	−0.581906	−0.290953	−	0.956737i	$-0.593972\pi$
$59$	−263.713	−0.581906	−0.290953	+	0.956737i	$0.593972\pi$
$60$	0	0
$61$	501.998	1.05368	0.526838	−	0.849966i	$-0.323377\pi$
$61$	501.998	1.05368	0.526838	+	0.849966i	$0.323377\pi$
$62$	0	0
$63$	830.789	1.66142
$64$	0	0
$65$	−211.057	−0.402745
$66$	0	0
$67$	184.030	0.335565	0.167783	−	0.985824i	$-0.446339\pi$
$67$	184.030	0.335565	0.167783	+	0.985824i	$0.446339\pi$
$68$	0	0
$69$	204.411	0.356641
$70$	0	0
$71$	−109.717	−0.183395	−0.0916976	−	0.995787i	$-0.529229\pi$
$71$	−109.717	−0.183395	−0.0916976	+	0.995787i	$0.529229\pi$
$72$	0	0
$73$	−679.979	−1.09021	−0.545107	−	0.838367i	$-0.683511\pi$
$73$	−679.979	−1.09021	−0.545107	+	0.838367i	$0.683511\pi$
$74$	0	0
$75$	222.186	0.342078
$76$	0	0
$77$	−153.973	−0.227882
$78$	0	0
$79$	331.528	0.472150	0.236075	−	0.971735i	$-0.424139\pi$
$79$	331.528	0.472150	0.236075	+	0.971735i	$0.424139\pi$
$80$	0	0
$81$	569.968	0.781849
$82$	0	0
$83$	1305.31	1.72622	0.863108	−	0.505019i	$-0.168515\pi$
$83$	1305.31	1.72622	0.863108	+	0.505019i	$0.168515\pi$
$84$	0	0
$85$	−202.624	−0.258560
$86$	0	0
$87$	281.231	0.346565
$88$	0	0
$89$	122.494	0.145891	0.0729456	−	0.997336i	$-0.476760\pi$
$89$	122.494	0.145891	0.0729456	+	0.997336i	$0.476760\pi$
$90$	0	0
$91$	674.574	0.777083
$92$	0	0
$93$	−562.975	−0.627718
$94$	0	0
$95$	91.6111	0.0989379
$96$	0	0
$97$	−220.692	−0.231009	−0.115504	−	0.993307i	$-0.536848\pi$
$97$	−220.692	−0.231009	−0.115504	+	0.993307i	$0.536848\pi$
$98$	0	0
$99$	−500.884	−0.508492
$100$	0	0
$101$	−258.949	−0.255113	−0.127557	−	0.991831i	$-0.540713\pi$
$101$	−258.949	−0.255113	−0.127557	+	0.991831i	$0.540713\pi$
$102$	0	0
$103$	1028.64	0.984032	0.492016	−	0.870586i	$-0.336260\pi$
$103$	1028.64	0.984032	0.492016	+	0.870586i	$0.336260\pi$
$104$	0	0
$105$	−710.144	−0.660028
$106$	0	0
$107$	−1855.11	−1.67608	−0.838038	−	0.545612i	$-0.816297\pi$
$107$	−1855.11	−1.67608	−0.838038	+	0.545612i	$0.816297\pi$
$108$	0	0
$109$	628.684	0.552450	0.276225	−	0.961093i	$-0.410917\pi$
$109$	628.684	0.552450	0.276225	+	0.961093i	$0.410917\pi$
$110$	0	0
$111$	−275.159	−0.235288
$112$	0	0
$113$	−816.227	−0.679506	−0.339753	−	0.940515i	$-0.610344\pi$
$113$	−816.227	−0.679506	−0.339753	+	0.940515i	$0.610344\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	2194.43	1.73397
$118$	0	0
$119$	647.619	0.498884
$120$	0	0
$121$	−1238.17	−0.930255
$122$	0	0
$123$	1809.18	1.32625
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2264.62	−1.58230	−0.791150	−	0.611622i	$-0.790518\pi$
$127$	−2264.62	−1.58230	−0.791150	+	0.611622i	$0.790518\pi$
$128$	0	0
$129$	1518.45	1.03637
$130$	0	0
$131$	1411.42	0.941347	0.470674	−	0.882307i	$-0.344011\pi$
$131$	1411.42	0.941347	0.470674	+	0.882307i	$0.344011\pi$
$132$	0	0
$133$	−292.805	−0.190898
$134$	0	0
$135$	−1110.33	−0.707869
$136$	0	0
$137$	835.349	0.520939	0.260470	−	0.965482i	$-0.416123\pi$
$137$	835.349	0.520939	0.260470	+	0.965482i	$0.416123\pi$
$138$	0	0
$139$	−277.196	−0.169147	−0.0845735	−	0.996417i	$-0.526953\pi$
$139$	−277.196	−0.169147	−0.0845735	+	0.996417i	$0.526953\pi$
$140$	0	0
$141$	4493.89	2.68407
$142$	0	0
$143$	−406.701	−0.237833
$144$	0	0
$145$	−158.218	−0.0906161
$146$	0	0
$147$	−778.653	−0.436886
$148$	0	0
$149$	1137.79	0.625582	0.312791	−	0.949822i	$-0.398736\pi$
$149$	1137.79	0.625582	0.312791	+	0.949822i	$0.398736\pi$
$150$	0	0
$151$	−414.752	−0.223524	−0.111762	−	0.993735i	$-0.535649\pi$
$151$	−414.752	−0.223524	−0.111762	+	0.993735i	$0.535649\pi$
$152$	0	0
$153$	2106.74	1.11320
$154$	0	0
$155$	316.725	0.164129
$156$	0	0
$157$	3545.22	1.80216	0.901080	−	0.433653i	$-0.142776\pi$
$157$	3545.22	1.80216	0.901080	+	0.433653i	$0.142776\pi$
$158$	0	0
$159$	−1991.09	−0.993107
$160$	0	0
$161$	367.559	0.179924
$162$	0	0
$163$	1942.88	0.933609	0.466805	−	0.884360i	$-0.345405\pi$
$163$	1942.88	0.933609	0.466805	+	0.884360i	$0.345405\pi$
$164$	0	0
$165$	428.146	0.202007
$166$	0	0
$167$	2696.57	1.24950	0.624750	−	0.780825i	$-0.285201\pi$
$167$	2696.57	1.24950	0.624750	+	0.780825i	$0.285201\pi$
$168$	0	0
$169$	−415.196	−0.188983
$170$	0	0
$171$	−952.510	−0.425967
$172$	0	0
$173$	3656.58	1.60697	0.803483	−	0.595328i	$-0.202978\pi$
$173$	3656.58	1.60697	0.803483	+	0.595328i	$0.202978\pi$
$174$	0	0
$175$	399.521	0.172577
$176$	0	0
$177$	−2343.73	−0.995285
$178$	0	0
$179$	1984.62	0.828701	0.414351	−	0.910117i	$-0.364009\pi$
$179$	1984.62	0.828701	0.414351	+	0.910117i	$0.364009\pi$
$180$	0	0
$181$	2426.26	0.996365	0.498183	−	0.867072i	$-0.334001\pi$
$181$	2426.26	0.996365	0.498183	+	0.867072i	$0.334001\pi$
$182$	0	0
$183$	4461.48	1.80220
$184$	0	0
$185$	154.802	0.0615204
$186$	0	0
$187$	−390.450	−0.152687
$188$	0	0
$189$	3548.82	1.36581
$190$	0	0
$191$	4350.91	1.64828	0.824139	−	0.566387i	$-0.191659\pi$
$191$	4350.91	1.64828	0.824139	+	0.566387i	$0.191659\pi$
$192$	0	0
$193$	−2425.67	−0.904680	−0.452340	−	0.891846i	$-0.649411\pi$
$193$	−2425.67	−0.904680	−0.452340	+	0.891846i	$0.649411\pi$
$194$	0	0
$195$	−1875.76	−0.688850
$196$	0	0
$197$	−4818.67	−1.74272	−0.871360	−	0.490644i	$-0.836761\pi$
$197$	−4818.67	−1.74272	−0.871360	+	0.490644i	$0.836761\pi$
$198$	0	0
$199$	−1679.82	−0.598390	−0.299195	−	0.954192i	$-0.596718\pi$
$199$	−1679.82	−0.598390	−0.299195	+	0.954192i	$0.596718\pi$
$200$	0	0
$201$	1635.56	0.573947
$202$	0	0
$203$	505.693	0.174841
$204$	0	0
$205$	−1017.83	−0.346773
$206$	0	0
$207$	1195.69	0.401480
$208$	0	0
$209$	176.532	0.0584258
$210$	0	0
$211$	583.138	0.190260	0.0951301	−	0.995465i	$-0.469673\pi$
$211$	583.138	0.190260	0.0951301	+	0.995465i	$0.469673\pi$
$212$	0	0
$213$	−975.107	−0.313677
$214$	0	0
$215$	−854.268	−0.270980
$216$	0	0
$217$	−1012.31	−0.316681
$218$	0	0
$219$	−6043.28	−1.86469
$220$	0	0
$221$	1710.61	0.520669
$222$	0	0
$223$	3083.05	0.925814	0.462907	−	0.886407i	$-0.346806\pi$
$223$	3083.05	0.925814	0.462907	+	0.886407i	$0.346806\pi$
$224$	0	0
$225$	1299.66	0.385086
$226$	0	0
$227$	−682.335	−0.199507	−0.0997537	−	0.995012i	$-0.531805\pi$
$227$	−682.335	−0.199507	−0.0997537	+	0.995012i	$0.531805\pi$
$228$	0	0
$229$	−4400.81	−1.26993	−0.634964	−	0.772541i	$-0.718985\pi$
$229$	−4400.81	−1.26993	−0.634964	+	0.772541i	$0.718985\pi$
$230$	0	0
$231$	−1368.43	−0.389766
$232$	0	0
$233$	−3391.37	−0.953545	−0.476772	−	0.879027i	$-0.658193\pi$
$233$	−3391.37	−0.953545	−0.476772	+	0.879027i	$0.658193\pi$
$234$	0	0
$235$	−2528.22	−0.701801
$236$	0	0
$237$	2946.44	0.807560
$238$	0	0
$239$	750.785	0.203198	0.101599	−	0.994825i	$-0.467604\pi$
$239$	750.785	0.203198	0.101599	+	0.994825i	$0.467604\pi$
$240$	0	0
$241$	−349.695	−0.0934682	−0.0467341	−	0.998907i	$-0.514881\pi$
$241$	−349.695	−0.0934682	−0.0467341	+	0.998907i	$0.514881\pi$
$242$	0	0
$243$	−930.249	−0.245578
$244$	0	0
$245$	438.064	0.114232
$246$	0	0
$247$	−773.407	−0.199234
$248$	0	0
$249$	11600.8	2.95250
$250$	0	0
$251$	2597.01	0.653074	0.326537	−	0.945184i	$-0.394118\pi$
$251$	2597.01	0.653074	0.326537	+	0.945184i	$0.394118\pi$
$252$	0	0
$253$	−221.602	−0.0550672
$254$	0	0
$255$	−1800.81	−0.442238
$256$	0	0
$257$	−221.689	−0.0538077	−0.0269039	−	0.999638i	$-0.508565\pi$
$257$	−221.689	−0.0538077	−0.0269039	+	0.999638i	$0.508565\pi$
$258$	0	0
$259$	−494.773	−0.118702
$260$	0	0
$261$	1645.05	0.390138
$262$	0	0
$263$	2779.66	0.651716	0.325858	−	0.945419i	$-0.394347\pi$
$263$	2779.66	0.651716	0.325858	+	0.945419i	$0.394347\pi$
$264$	0	0
$265$	1120.17	0.259667
$266$	0	0
$267$	1088.66	0.249531
$268$	0	0
$269$	−4836.46	−1.09622	−0.548112	−	0.836405i	$-0.684653\pi$
$269$	−4836.46	−1.09622	−0.548112	+	0.836405i	$0.684653\pi$
$270$	0	0
$271$	−2166.54	−0.485637	−0.242819	−	0.970072i	$-0.578072\pi$
$271$	−2166.54	−0.485637	−0.242819	+	0.970072i	$0.578072\pi$
$272$	0	0
$273$	5995.24	1.32911
$274$	0	0
$275$	−240.872	−0.0528186
$276$	0	0
$277$	−5271.29	−1.14340	−0.571699	−	0.820463i	$-0.693716\pi$
$277$	−5271.29	−1.14340	−0.571699	+	0.820463i	$0.693716\pi$
$278$	0	0
$279$	−3293.09	−0.706639
$280$	0	0
$281$	−4213.36	−0.894476	−0.447238	−	0.894415i	$-0.647592\pi$
$281$	−4213.36	−0.894476	−0.447238	+	0.894415i	$0.647592\pi$
$282$	0	0
$283$	−6015.10	−1.26346	−0.631732	−	0.775187i	$-0.717656\pi$
$283$	−6015.10	−1.26346	−0.631732	+	0.775187i	$0.717656\pi$
$284$	0	0
$285$	814.189	0.169222
$286$	0	0
$287$	3253.16	0.669087
$288$	0	0
$289$	−3270.75	−0.665733
$290$	0	0
$291$	−1961.39	−0.395115
$292$	0	0
$293$	1705.88	0.340131	0.170065	−	0.985433i	$-0.445602\pi$
$293$	1705.88	0.340131	0.170065	+	0.985433i	$0.445602\pi$
$294$	0	0
$295$	1318.56	0.260236
$296$	0	0
$297$	−2139.58	−0.418018
$298$	0	0
$299$	970.863	0.187781
$300$	0	0
$301$	2730.38	0.522846
$302$	0	0
$303$	−2301.40	−0.436342
$304$	0	0
$305$	−2509.99	−0.471218
$306$	0	0
$307$	−7171.22	−1.33317	−0.666585	−	0.745429i	$-0.732245\pi$
$307$	−7171.22	−1.33317	−0.666585	+	0.745429i	$0.732245\pi$
$308$	0	0
$309$	9142.01	1.68308
$310$	0	0
$311$	6909.11	1.25974	0.629871	−	0.776700i	$-0.283108\pi$
$311$	6909.11	1.25974	0.629871	+	0.776700i	$0.283108\pi$
$312$	0	0
$313$	−2726.24	−0.492320	−0.246160	−	0.969229i	$-0.579169\pi$
$313$	−2726.24	−0.492320	−0.246160	+	0.969229i	$0.579169\pi$
$314$	0	0
$315$	−4153.95	−0.743011
$316$	0	0
$317$	−10048.4	−1.78036	−0.890181	−	0.455607i	$-0.849422\pi$
$317$	−10048.4	−1.78036	−0.890181	+	0.455607i	$0.849422\pi$
$318$	0	0
$319$	−304.883	−0.0535115
$320$	0	0
$321$	−16487.2	−2.86674
$322$	0	0
$323$	−742.503	−0.127907
$324$	0	0
$325$	1055.29	0.180113
$326$	0	0
$327$	5587.39	0.944904
$328$	0	0
$329$	8080.63	1.35410
$330$	0	0
$331$	2907.79	0.482859	0.241430	−	0.970418i	$-0.422384\pi$
$331$	2907.79	0.482859	0.241430	+	0.970418i	$0.422384\pi$
$332$	0	0
$333$	−1609.53	−0.264869
$334$	0	0
$335$	−920.151	−0.150069
$336$	0	0
$337$	1439.69	0.232715	0.116358	−	0.993207i	$-0.462878\pi$
$337$	1439.69	0.232715	0.116358	+	0.993207i	$0.462878\pi$
$338$	0	0
$339$	−7254.17	−1.16222
$340$	0	0
$341$	610.321	0.0969229
$342$	0	0
$343$	−6881.55	−1.08329
$344$	0	0
$345$	−1022.06	−0.159495
$346$	0	0
$347$	−5642.54	−0.872933	−0.436466	−	0.899721i	$-0.643770\pi$
$347$	−5642.54	−0.872933	−0.436466	+	0.899721i	$0.643770\pi$
$348$	0	0
$349$	8114.51	1.24458	0.622292	−	0.782785i	$-0.286202\pi$
$349$	8114.51	1.24458	0.622292	+	0.782785i	$0.286202\pi$
$350$	0	0
$351$	9373.76	1.42545
$352$	0	0
$353$	2256.43	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$353$	2256.43	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$354$	0	0
$355$	548.587	0.0820168
$356$	0	0
$357$	5755.68	0.853285
$358$	0	0
$359$	12942.1	1.90266	0.951332	−	0.308169i	$-0.0997162\pi$
$359$	12942.1	1.90266	0.951332	+	0.308169i	$0.0997162\pi$
$360$	0	0
$361$	−6523.30	−0.951056
$362$	0	0
$363$	−11004.2	−1.59110
$364$	0	0
$365$	3399.90	0.487558
$366$	0	0
$367$	−3183.88	−0.452854	−0.226427	−	0.974028i	$-0.572704\pi$
$367$	−3183.88	−0.452854	−0.226427	+	0.974028i	$0.572704\pi$
$368$	0	0
$369$	10582.7	1.49299
$370$	0	0
$371$	−3580.26	−0.501019
$372$	0	0
$373$	−9005.13	−1.25005	−0.625024	−	0.780606i	$-0.714911\pi$
$373$	−9005.13	−1.25005	−0.625024	+	0.780606i	$0.714911\pi$
$374$	0	0
$375$	−1110.93	−0.152982
$376$	0	0
$377$	1335.73	0.182476
$378$	0	0
$379$	8772.11	1.18890	0.594450	−	0.804133i	$-0.297370\pi$
$379$	8772.11	1.18890	0.594450	+	0.804133i	$0.297370\pi$
$380$	0	0
$381$	−20126.6	−2.70635
$382$	0	0
$383$	746.557	0.0996013	0.0498006	−	0.998759i	$-0.484141\pi$
$383$	746.557	0.0996013	0.0498006	+	0.998759i	$0.484141\pi$
$384$	0	0
$385$	769.866	0.101912
$386$	0	0
$387$	8882.10	1.16667
$388$	0	0
$389$	6657.68	0.867757	0.433879	−	0.900971i	$-0.357145\pi$
$389$	6657.68	0.867757	0.433879	+	0.900971i	$0.357145\pi$
$390$	0	0
$391$	932.069	0.120554
$392$	0	0
$393$	12543.9	1.61007
$394$	0	0
$395$	−1657.64	−0.211152
$396$	0	0
$397$	3734.41	0.472102	0.236051	−	0.971741i	$-0.424147\pi$
$397$	3734.41	0.472102	0.236051	+	0.971741i	$0.424147\pi$
$398$	0	0
$399$	−2602.28	−0.326509
$400$	0	0
$401$	−11824.3	−1.47251	−0.736253	−	0.676706i	$-0.763407\pi$
$401$	−11824.3	−1.47251	−0.736253	+	0.676706i	$0.763407\pi$
$402$	0	0
$403$	−2673.88	−0.330510
$404$	0	0
$405$	−2849.84	−0.349654
$406$	0	0
$407$	298.299	0.0363296
$408$	0	0
$409$	9697.52	1.17240	0.586200	−	0.810166i	$-0.300623\pi$
$409$	9697.52	1.17240	0.586200	+	0.810166i	$0.300623\pi$
$410$	0	0
$411$	7424.11	0.891008
$412$	0	0
$413$	−4214.35	−0.502118
$414$	0	0
$415$	−6526.53	−0.771987
$416$	0	0
$417$	−2463.56	−0.289307
$418$	0	0
$419$	9358.00	1.09109	0.545547	−	0.838080i	$-0.316322\pi$
$419$	9358.00	1.09109	0.545547	+	0.838080i	$0.316322\pi$
$420$	0	0
$421$	1372.14	0.158846	0.0794228	−	0.996841i	$-0.474692\pi$
$421$	1372.14	0.158846	0.0794228	+	0.996841i	$0.474692\pi$
$422$	0	0
$423$	26286.8	3.02153
$424$	0	0
$425$	1013.12	0.115632
$426$	0	0
$427$	8022.35	0.909201
$428$	0	0
$429$	−3614.53	−0.406786
$430$	0	0
$431$	−1101.08	−0.123056	−0.0615279	−	0.998105i	$-0.519597\pi$
$431$	−1101.08	−0.123056	−0.0615279	+	0.998105i	$0.519597\pi$
$432$	0	0
$433$	−7940.12	−0.881242	−0.440621	−	0.897693i	$-0.645242\pi$
$433$	−7940.12	−0.881242	−0.440621	+	0.897693i	$0.645242\pi$
$434$	0	0
$435$	−1406.16	−0.154989
$436$	0	0
$437$	−421.411	−0.0461301
$438$	0	0
$439$	9759.66	1.06106	0.530528	−	0.847668i	$-0.321994\pi$
$439$	9759.66	1.06106	0.530528	+	0.847668i	$0.321994\pi$
$440$	0	0
$441$	−4554.69	−0.491814
$442$	0	0
$443$	13742.3	1.47385	0.736927	−	0.675972i	$-0.236276\pi$
$443$	13742.3	1.47385	0.736927	+	0.675972i	$0.236276\pi$
$444$	0	0
$445$	−612.469	−0.0652446
$446$	0	0
$447$	10112.1	1.06999
$448$	0	0
$449$	930.841	0.0978377	0.0489188	−	0.998803i	$-0.484422\pi$
$449$	930.841	0.0978377	0.0489188	+	0.998803i	$0.484422\pi$
$450$	0	0
$451$	−1961.33	−0.204779
$452$	0	0
$453$	−3686.09	−0.382312
$454$	0	0
$455$	−3372.87	−0.347522
$456$	0	0
$457$	−11416.7	−1.16861	−0.584303	−	0.811536i	$-0.698632\pi$
$457$	−11416.7	−1.16861	−0.584303	+	0.811536i	$0.698632\pi$
$458$	0	0
$459$	8999.20	0.915134
$460$	0	0
$461$	−1852.77	−0.187184	−0.0935922	−	0.995611i	$-0.529835\pi$
$461$	−1852.77	−0.187184	−0.0935922	+	0.995611i	$0.529835\pi$
$462$	0	0
$463$	17102.8	1.71671	0.858353	−	0.513060i	$-0.171488\pi$
$463$	17102.8	1.71671	0.858353	+	0.513060i	$0.171488\pi$
$464$	0	0
$465$	2814.87	0.280724
$466$	0	0
$467$	1986.47	0.196837	0.0984184	−	0.995145i	$-0.468622\pi$
$467$	1986.47	0.196837	0.0984184	+	0.995145i	$0.468622\pi$
$468$	0	0
$469$	2940.96	0.289554
$470$	0	0
$471$	31507.9	3.08239
$472$	0	0
$473$	−1646.15	−0.160021
$474$	0	0
$475$	−458.056	−0.0442464
$476$	0	0
$477$	−11646.8	−1.11797
$478$	0	0
$479$	333.523	0.0318143	0.0159072	−	0.999873i	$-0.494936\pi$
$479$	333.523	0.0318143	0.0159072	+	0.999873i	$0.494936\pi$
$480$	0	0
$481$	−1306.88	−0.123885
$482$	0	0
$483$	3266.66	0.307740
$484$	0	0
$485$	1103.46	0.103310
$486$	0	0
$487$	−19465.1	−1.81118	−0.905592	−	0.424150i	$-0.860573\pi$
$487$	−19465.1	−1.81118	−0.905592	+	0.424150i	$0.860573\pi$
$488$	0	0
$489$	17267.3	1.59683
$490$	0	0
$491$	−18882.3	−1.73553	−0.867764	−	0.496976i	$-0.834444\pi$
$491$	−18882.3	−1.73553	−0.867764	+	0.496976i	$0.834444\pi$
$492$	0	0
$493$	1282.35	0.117149
$494$	0	0
$495$	2504.42	0.227405
$496$	0	0
$497$	−1753.38	−0.158249
$498$	0	0
$499$	−6415.92	−0.575583	−0.287792	−	0.957693i	$-0.592921\pi$
$499$	−6415.92	−0.575583	−0.287792	+	0.957693i	$0.592921\pi$
$500$	0	0
$501$	23965.6	2.13713
$502$	0	0
$503$	−3760.79	−0.333371	−0.166685	−	0.986010i	$-0.553306\pi$
$503$	−3760.79	−0.333371	−0.166685	+	0.986010i	$0.553306\pi$
$504$	0	0
$505$	1294.75	0.114090
$506$	0	0
$507$	−3690.03	−0.323235
$508$	0	0
$509$	−7206.18	−0.627521	−0.313760	−	0.949502i	$-0.601589\pi$
$509$	−7206.18	−0.627521	−0.313760	+	0.949502i	$0.601589\pi$
$510$	0	0
$511$	−10866.6	−0.940728
$512$	0	0
$513$	−4068.76	−0.350176
$514$	0	0
$515$	−5143.22	−0.440073
$516$	0	0
$517$	−4871.82	−0.414434
$518$	0	0
$519$	32497.7	2.74853
$520$	0	0
$521$	−1295.94	−0.108976	−0.0544879	−	0.998514i	$-0.517353\pi$
$521$	−1295.94	−0.108976	−0.0544879	+	0.998514i	$0.517353\pi$
$522$	0	0
$523$	−19924.3	−1.66583	−0.832913	−	0.553404i	$-0.813329\pi$
$523$	−19924.3	−1.66583	−0.832913	+	0.553404i	$0.813329\pi$
$524$	0	0
$525$	3550.72	0.295174
$526$	0	0
$527$	−2567.04	−0.212186
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−13709.5	−1.12042
$532$	0	0
$533$	8592.81	0.698304
$534$	0	0
$535$	9275.54	0.749564
$536$	0	0
$537$	17638.2	1.41740
$538$	0	0
$539$	844.137	0.0674574
$540$	0	0
$541$	10543.1	0.837866	0.418933	−	0.908017i	$-0.362404\pi$
$541$	10543.1	0.837866	0.418933	+	0.908017i	$0.362404\pi$
$542$	0	0
$543$	21563.2	1.70417
$544$	0	0
$545$	−3143.42	−0.247063
$546$	0	0
$547$	−7341.41	−0.573850	−0.286925	−	0.957953i	$-0.592633\pi$
$547$	−7341.41	−0.573850	−0.286925	+	0.957953i	$0.592633\pi$
$548$	0	0
$549$	26097.2	2.02878
$550$	0	0
$551$	−579.783	−0.0448268
$552$	0	0
$553$	5298.10	0.407411
$554$	0	0
$555$	1375.79	0.105224
$556$	0	0
$557$	−3231.59	−0.245829	−0.122915	−	0.992417i	$-0.539224\pi$
$557$	−3231.59	−0.245829	−0.122915	+	0.992417i	$0.539224\pi$
$558$	0	0
$559$	7211.97	0.545678
$560$	0	0
$561$	−3470.10	−0.261155
$562$	0	0
$563$	22615.7	1.69296	0.846481	−	0.532419i	$-0.178717\pi$
$563$	22615.7	1.69296	0.846481	+	0.532419i	$0.178717\pi$
$564$	0	0
$565$	4081.14	0.303884
$566$	0	0
$567$	9108.57	0.674645
$568$	0	0
$569$	11686.8	0.861046	0.430523	−	0.902580i	$-0.358329\pi$
$569$	11686.8	0.861046	0.430523	+	0.902580i	$0.358329\pi$
$570$	0	0
$571$	−19619.6	−1.43793	−0.718963	−	0.695048i	$-0.755383\pi$
$571$	−19619.6	−1.43793	−0.718963	+	0.695048i	$0.755383\pi$
$572$	0	0
$573$	38668.5	2.81920
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	11451.6	0.826230	0.413115	−	0.910679i	$-0.364441\pi$
$577$	11451.6	0.826230	0.413115	+	0.910679i	$0.364441\pi$
$578$	0	0
$579$	−21558.0	−1.54735
$580$	0	0
$581$	20859.9	1.48952
$582$	0	0
$583$	2158.54	0.153341
$584$	0	0
$585$	−10972.1	−0.775457
$586$	0	0
$587$	−14547.0	−1.02286	−0.511430	−	0.859325i	$-0.670884\pi$
$587$	−14547.0	−1.02286	−0.511430	+	0.859325i	$0.670884\pi$
$588$	0	0
$589$	1160.62	0.0811928
$590$	0	0
$591$	−42825.6	−2.98073
$592$	0	0
$593$	−202.333	−0.0140115	−0.00700575	−	0.999975i	$-0.502230\pi$
$593$	−202.333	−0.0140115	−0.00700575	+	0.999975i	$0.502230\pi$
$594$	0	0
$595$	−3238.10	−0.223108
$596$	0	0
$597$	−14929.3	−1.02348
$598$	0	0
$599$	4251.93	0.290032	0.145016	−	0.989429i	$-0.453677\pi$
$599$	4251.93	0.290032	0.145016	+	0.989429i	$0.453677\pi$
$600$	0	0
$601$	4098.39	0.278165	0.139082	−	0.990281i	$-0.455585\pi$
$601$	4098.39	0.278165	0.139082	+	0.990281i	$0.455585\pi$
$602$	0	0
$603$	9567.11	0.646107
$604$	0	0
$605$	6190.85	0.416023
$606$	0	0
$607$	−11481.8	−0.767765	−0.383882	−	0.923382i	$-0.625413\pi$
$607$	−11481.8	−0.767765	−0.383882	+	0.923382i	$0.625413\pi$
$608$	0	0
$609$	4494.32	0.299046
$610$	0	0
$611$	21344.0	1.41323
$612$	0	0
$613$	−26833.5	−1.76802	−0.884008	−	0.467471i	$-0.845165\pi$
$613$	−26833.5	−1.76802	−0.884008	+	0.467471i	$0.845165\pi$
$614$	0	0
$615$	−9045.91	−0.593116
$616$	0	0
$617$	9603.82	0.626637	0.313319	−	0.949648i	$-0.398559\pi$
$617$	9603.82	0.626637	0.313319	+	0.949648i	$0.398559\pi$
$618$	0	0
$619$	−6805.10	−0.441874	−0.220937	−	0.975288i	$-0.570912\pi$
$619$	−6805.10	−0.441874	−0.220937	+	0.975288i	$0.570912\pi$
$620$	0	0
$621$	5107.54	0.330046
$622$	0	0
$623$	1957.55	0.125887
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1568.92	0.0999308
$628$	0	0
$629$	−1254.66	−0.0795336
$630$	0	0
$631$	−13205.5	−0.833126	−0.416563	−	0.909107i	$-0.636765\pi$
$631$	−13205.5	−0.833126	−0.416563	+	0.909107i	$0.636765\pi$
$632$	0	0
$633$	5182.61	0.325419
$634$	0	0
$635$	11323.1	0.707626
$636$	0	0
$637$	−3698.26	−0.230032
$638$	0	0
$639$	−5703.83	−0.353114
$640$	0	0
$641$	−25910.3	−1.59656	−0.798280	−	0.602286i	$-0.794257\pi$
$641$	−25910.3	−1.59656	−0.798280	+	0.602286i	$0.794257\pi$
$642$	0	0
$643$	−25201.9	−1.54567	−0.772834	−	0.634608i	$-0.781162\pi$
$643$	−25201.9	−1.54567	−0.772834	+	0.634608i	$0.781162\pi$
$644$	0	0
$645$	−7592.26	−0.463480
$646$	0	0
$647$	−20569.2	−1.24986	−0.624929	−	0.780681i	$-0.714872\pi$
$647$	−20569.2	−1.24986	−0.624929	+	0.780681i	$0.714872\pi$
$648$	0	0
$649$	2540.83	0.153677
$650$	0	0
$651$	−8996.81	−0.541648
$652$	0	0
$653$	−15604.2	−0.935126	−0.467563	−	0.883960i	$-0.654868\pi$
$653$	−15604.2	−0.935126	−0.467563	+	0.883960i	$0.654868\pi$
$654$	0	0
$655$	−7057.11	−0.420983
$656$	0	0
$657$	−35349.8	−2.09913
$658$	0	0
$659$	19937.8	1.17855	0.589277	−	0.807931i	$-0.299413\pi$
$659$	19937.8	1.17855	0.589277	+	0.807931i	$0.299413\pi$
$660$	0	0
$661$	22533.1	1.32592	0.662961	−	0.748654i	$-0.269300\pi$
$661$	22533.1	1.32592	0.662961	+	0.748654i	$0.269300\pi$
$662$	0	0
$663$	15202.9	0.890546
$664$	0	0
$665$	1464.02	0.0853720
$666$	0	0
$667$	727.805	0.0422500
$668$	0	0
$669$	27400.5	1.58350
$670$	0	0
$671$	−4836.68	−0.278268
$672$	0	0
$673$	−32433.5	−1.85768	−0.928842	−	0.370477i	$-0.879194\pi$
$673$	−32433.5	−1.85768	−0.928842	+	0.370477i	$0.879194\pi$
$674$	0	0
$675$	5551.67	0.316569
$676$	0	0
$677$	3491.22	0.198196	0.0990979	−	0.995078i	$-0.468404\pi$
$677$	3491.22	0.198196	0.0990979	+	0.995078i	$0.468404\pi$
$678$	0	0
$679$	−3526.84	−0.199334
$680$	0	0
$681$	−6064.21	−0.341235
$682$	0	0
$683$	−21137.5	−1.18419	−0.592096	−	0.805868i	$-0.701699\pi$
$683$	−21137.5	−1.18419	−0.592096	+	0.805868i	$0.701699\pi$
$684$	0	0
$685$	−4176.74	−0.232971
$686$	0	0
$687$	−39111.9	−2.17207
$688$	0	0
$689$	−9456.82	−0.522897
$690$	0	0
$691$	16463.1	0.906347	0.453174	−	0.891422i	$-0.350292\pi$
$691$	16463.1	0.906347	0.453174	+	0.891422i	$0.350292\pi$
$692$	0	0
$693$	−8004.54	−0.438770
$694$	0	0
$695$	1385.98	0.0756448
$696$	0	0
$697$	8249.46	0.448308
$698$	0	0
$699$	−30140.6	−1.63093
$700$	0	0
$701$	−13393.8	−0.721648	−0.360824	−	0.932634i	$-0.617505\pi$
$701$	−13393.8	−0.721648	−0.360824	+	0.932634i	$0.617505\pi$
$702$	0	0
$703$	567.264	0.0304335
$704$	0	0
$705$	−22469.4	−1.20035
$706$	0	0
$707$	−4138.23	−0.220133
$708$	0	0
$709$	17609.5	0.932779	0.466390	−	0.884579i	$-0.345555\pi$
$709$	17609.5	0.932779	0.466390	+	0.884579i	$0.345555\pi$
$710$	0	0
$711$	17235.0	0.909091
$712$	0	0
$713$	−1456.94	−0.0765255
$714$	0	0
$715$	2033.51	0.106362
$716$	0	0
$717$	6672.56	0.347547
$718$	0	0
$719$	−8610.73	−0.446629	−0.223314	−	0.974746i	$-0.571688\pi$
$719$	−8610.73	−0.446629	−0.223314	+	0.974746i	$0.571688\pi$
$720$	0	0
$721$	16438.6	0.849106
$722$	0	0
$723$	−3107.89	−0.159867
$724$	0	0
$725$	791.092	0.0405247
$726$	0	0
$727$	−9153.66	−0.466974	−0.233487	−	0.972360i	$-0.575014\pi$
$727$	−9153.66	−0.466974	−0.233487	+	0.972360i	$0.575014\pi$
$728$	0	0
$729$	−23656.7	−1.20188
$730$	0	0
$731$	6923.80	0.350323
$732$	0	0
$733$	35745.4	1.80121	0.900604	−	0.434642i	$-0.143125\pi$
$733$	35745.4	1.80121	0.900604	+	0.434642i	$0.143125\pi$
$734$	0	0
$735$	3893.26	0.195381
$736$	0	0
$737$	−1773.11	−0.0886204
$738$	0	0
$739$	−16842.9	−0.838396	−0.419198	−	0.907895i	$-0.637689\pi$
$739$	−16842.9	−0.838396	−0.419198	+	0.907895i	$0.637689\pi$
$740$	0	0
$741$	−6873.61	−0.340767
$742$	0	0
$743$	−6577.22	−0.324758	−0.162379	−	0.986729i	$-0.551917\pi$
$743$	−6577.22	−0.324758	−0.162379	+	0.986729i	$0.551917\pi$
$744$	0	0
$745$	−5688.97	−0.279769
$746$	0	0
$747$	67858.4	3.32371
$748$	0	0
$749$	−29646.2	−1.44626
$750$	0	0
$751$	−32188.7	−1.56403	−0.782013	−	0.623263i	$-0.785807\pi$
$751$	−32188.7	−1.56403	−0.782013	+	0.623263i	$0.785807\pi$
$752$	0	0
$753$	23080.8	1.11701
$754$	0	0
$755$	2073.76	0.0999628
$756$	0	0
$757$	16659.1	0.799848	0.399924	−	0.916548i	$-0.369037\pi$
$757$	16659.1	0.799848	0.399924	+	0.916548i	$0.369037\pi$
$758$	0	0
$759$	−1969.47	−0.0941862
$760$	0	0
$761$	−30186.6	−1.43793	−0.718965	−	0.695046i	$-0.755384\pi$
$761$	−30186.6	−1.43793	−0.718965	+	0.695046i	$0.755384\pi$
$762$	0	0
$763$	10046.9	0.476700
$764$	0	0
$765$	−10533.7	−0.497839
$766$	0	0
$767$	−11131.7	−0.524044
$768$	0	0
$769$	−27171.2	−1.27415	−0.637073	−	0.770803i	$-0.719855\pi$
$769$	−27171.2	−1.27415	−0.637073	+	0.770803i	$0.719855\pi$
$770$	0	0
$771$	−1970.25	−0.0920321
$772$	0	0
$773$	−36528.6	−1.69967	−0.849833	−	0.527052i	$-0.823297\pi$
$773$	−36528.6	−1.69967	−0.849833	+	0.527052i	$0.823297\pi$
$774$	0	0
$775$	−1583.63	−0.0734006
$776$	0	0
$777$	−4397.27	−0.203026
$778$	0	0
$779$	−3729.79	−0.171545
$780$	0	0
$781$	1057.11	0.0484333
$782$	0	0
$783$	7027.02	0.320722
$784$	0	0
$785$	−17726.1	−0.805951
$786$	0	0
$787$	3314.84	0.150141	0.0750706	−	0.997178i	$-0.476082\pi$
$787$	3314.84	0.150141	0.0750706	+	0.997178i	$0.476082\pi$
$788$	0	0
$789$	24704.1	1.11469
$790$	0	0
$791$	−13044.0	−0.586335
$792$	0	0
$793$	21190.0	0.948903
$794$	0	0
$795$	9955.47	0.444131
$796$	0	0
$797$	526.572	0.0234029	0.0117015	−	0.999932i	$-0.496275\pi$
$797$	526.572	0.0234029	0.0117015	+	0.999932i	$0.496275\pi$
$798$	0	0
$799$	20491.1	0.907289
$800$	0	0
$801$	6368.04	0.280903
$802$	0	0
$803$	6551.51	0.287917
$804$	0	0
$805$	−1837.80	−0.0804644
$806$	0	0
$807$	−42983.7	−1.87497
$808$	0	0
$809$	−23931.8	−1.04005	−0.520023	−	0.854152i	$-0.674077\pi$
$809$	−23931.8	−1.04005	−0.520023	+	0.854152i	$0.674077\pi$
$810$	0	0
$811$	−10319.3	−0.446807	−0.223404	−	0.974726i	$-0.571717\pi$
$811$	−10319.3	−0.446807	−0.223404	+	0.974726i	$0.571717\pi$
$812$	0	0
$813$	−19255.0	−0.830628
$814$	0	0
$815$	−9714.41	−0.417523
$816$	0	0
$817$	−3130.42	−0.134051
$818$	0	0
$819$	35068.8	1.49622
$820$	0	0
$821$	−22460.0	−0.954760	−0.477380	−	0.878697i	$-0.658414\pi$
$821$	−22460.0	−0.954760	−0.477380	+	0.878697i	$0.658414\pi$
$822$	0	0
$823$	10598.1	0.448880	0.224440	−	0.974488i	$-0.427945\pi$
$823$	10598.1	0.448880	0.224440	+	0.974488i	$0.427945\pi$
$824$	0	0
$825$	−2140.73	−0.0903403
$826$	0	0
$827$	28688.4	1.20628	0.603139	−	0.797636i	$-0.293916\pi$
$827$	28688.4	1.20628	0.603139	+	0.797636i	$0.293916\pi$
$828$	0	0
$829$	15013.9	0.629018	0.314509	−	0.949255i	$-0.398160\pi$
$829$	15013.9	0.629018	0.314509	+	0.949255i	$0.398160\pi$
$830$	0	0
$831$	−46848.3	−1.95566
$832$	0	0
$833$	−3550.48	−0.147679
$834$	0	0
$835$	−13482.8	−0.558793
$836$	0	0
$837$	−14066.8	−0.580909
$838$	0	0
$839$	−18368.8	−0.755853	−0.377927	−	0.925836i	$-0.623363\pi$
$839$	−18368.8	−0.755853	−0.377927	+	0.925836i	$0.623363\pi$
$840$	0	0
$841$	−23387.7	−0.958944
$842$	0	0
$843$	−37445.9	−1.52990
$844$	0	0
$845$	2075.98	0.0845159
$846$	0	0
$847$	−19787.0	−0.802702
$848$	0	0
$849$	−53458.8	−2.16102
$850$	0	0
$851$	−712.089	−0.0286840
$852$	0	0
$853$	27876.5	1.11896	0.559480	−	0.828844i	$-0.311001\pi$
$853$	27876.5	1.11896	0.559480	+	0.828844i	$0.311001\pi$
$854$	0	0
$855$	4762.55	0.190498
$856$	0	0
$857$	41457.7	1.65247	0.826237	−	0.563323i	$-0.190477\pi$
$857$	41457.7	1.65247	0.826237	+	0.563323i	$0.190477\pi$
$858$	0	0
$859$	10862.0	0.431440	0.215720	−	0.976455i	$-0.430790\pi$
$859$	10862.0	0.431440	0.215720	+	0.976455i	$0.430790\pi$
$860$	0	0
$861$	28912.2	1.14440
$862$	0	0
$863$	5007.70	0.197525	0.0987624	−	0.995111i	$-0.468512\pi$
$863$	5007.70	0.197525	0.0987624	+	0.995111i	$0.468512\pi$
$864$	0	0
$865$	−18282.9	−0.718657
$866$	0	0
$867$	−29068.6	−1.13866
$868$	0	0
$869$	−3194.23	−0.124691
$870$	0	0
$871$	7768.18	0.302198
$872$	0	0
$873$	−11473.0	−0.444791
$874$	0	0
$875$	−1997.61	−0.0771787
$876$	0	0
$877$	−11303.7	−0.435231	−0.217616	−	0.976035i	$-0.569828\pi$
$877$	−11303.7	−0.435231	−0.217616	+	0.976035i	$0.569828\pi$
$878$	0	0
$879$	15160.9	0.581756
$880$	0	0
$881$	16913.5	0.646801	0.323401	−	0.946262i	$-0.395174\pi$
$881$	16913.5	0.646801	0.323401	+	0.946262i	$0.395174\pi$
$882$	0	0
$883$	24529.2	0.934853	0.467426	−	0.884032i	$-0.345181\pi$
$883$	24529.2	0.934853	0.467426	+	0.884032i	$0.345181\pi$
$884$	0	0
$885$	11718.6	0.445105
$886$	0	0
$887$	−8626.24	−0.326540	−0.163270	−	0.986581i	$-0.552204\pi$
$887$	−8626.24	−0.326540	−0.163270	+	0.986581i	$0.552204\pi$
$888$	0	0
$889$	−36190.5	−1.36534
$890$	0	0
$891$	−5491.56	−0.206481
$892$	0	0
$893$	−9264.54	−0.347174
$894$	0	0
$895$	−9923.10	−0.370606
$896$	0	0
$897$	8628.48	0.321178
$898$	0	0
$899$	−2004.47	−0.0743635
$900$	0	0
$901$	−9078.94	−0.335697
$902$	0	0
$903$	24266.1	0.894270
$904$	0	0
$905$	−12131.3	−0.445588
$906$	0	0
$907$	−20552.6	−0.752412	−0.376206	−	0.926536i	$-0.622771\pi$
$907$	−20552.6	−0.752412	−0.376206	+	0.926536i	$0.622771\pi$
$908$	0	0
$909$	−13461.9	−0.491202
$910$	0	0
$911$	−20249.3	−0.736431	−0.368215	−	0.929741i	$-0.620031\pi$
$911$	−20249.3	−0.736431	−0.368215	+	0.929741i	$0.620031\pi$
$912$	0	0
$913$	−12576.4	−0.455881
$914$	0	0
$915$	−22307.4	−0.805966
$916$	0	0
$917$	22555.7	0.812274
$918$	0	0
$919$	−17245.1	−0.619003	−0.309501	−	0.950899i	$-0.600162\pi$
$919$	−17245.1	−0.619003	−0.309501	+	0.950899i	$0.600162\pi$
$920$	0	0
$921$	−63733.8	−2.28024
$922$	0	0
$923$	−4631.33	−0.165159
$924$	0	0
$925$	−774.010	−0.0275128
$926$	0	0
$927$	53475.7	1.89468
$928$	0	0
$929$	−18564.9	−0.655647	−0.327823	−	0.944739i	$-0.606315\pi$
$929$	−18564.9	−0.655647	−0.327823	+	0.944739i	$0.606315\pi$
$930$	0	0
$931$	1605.26	0.0565094
$932$	0	0
$933$	61404.3	2.15465
$934$	0	0
$935$	1952.25	0.0682839
$936$	0	0
$937$	10059.7	0.350733	0.175367	−	0.984503i	$-0.443889\pi$
$937$	10059.7	0.350733	0.175367	+	0.984503i	$0.443889\pi$
$938$	0	0
$939$	−24229.3	−0.842059
$940$	0	0
$941$	−11428.2	−0.395906	−0.197953	−	0.980211i	$-0.563429\pi$
$941$	−11428.2	−0.395906	−0.197953	+	0.980211i	$0.563429\pi$
$942$	0	0
$943$	4682.02	0.161684
$944$	0	0
$945$	−17744.1	−0.610809
$946$	0	0
$947$	33503.1	1.14964	0.574818	−	0.818281i	$-0.305073\pi$
$947$	33503.1	1.14964	0.574818	+	0.818281i	$0.305073\pi$
$948$	0	0
$949$	−28702.9	−0.981808
$950$	0	0
$951$	−89304.6	−3.04511
$952$	0	0
$953$	18496.0	0.628694	0.314347	−	0.949308i	$-0.398214\pi$
$953$	18496.0	0.628694	0.314347	+	0.949308i	$0.398214\pi$
$954$	0	0
$955$	−21754.6	−0.737132
$956$	0	0
$957$	−2709.63	−0.0915254
$958$	0	0
$959$	13349.6	0.449510
$960$	0	0
$961$	−25778.4	−0.865309
$962$	0	0
$963$	−96440.8	−3.22717
$964$	0	0
$965$	12128.3	0.404585
$966$	0	0
$967$	−2658.65	−0.0884140	−0.0442070	−	0.999022i	$-0.514076\pi$
$967$	−2658.65	−0.0884140	−0.0442070	+	0.999022i	$0.514076\pi$
$968$	0	0
$969$	−6598.95	−0.218771
$970$	0	0
$971$	46521.0	1.53752	0.768760	−	0.639538i	$-0.220874\pi$
$971$	46521.0	1.53752	0.768760	+	0.639538i	$0.220874\pi$
$972$	0	0
$973$	−4429.82	−0.145954
$974$	0	0
$975$	9378.79	0.308063
$976$	0	0
$977$	−5874.33	−0.192361	−0.0961803	−	0.995364i	$-0.530663\pi$
$977$	−5874.33	−0.192361	−0.0961803	+	0.995364i	$0.530663\pi$
$978$	0	0
$979$	−1180.21	−0.0385288
$980$	0	0
$981$	32683.1	1.06370
$982$	0	0
$983$	−9002.56	−0.292103	−0.146051	−	0.989277i	$-0.546657\pi$
$983$	−9002.56	−0.292103	−0.146051	+	0.989277i	$0.546657\pi$
$984$	0	0
$985$	24093.3	0.779368
$986$	0	0
$987$	71816.1	2.31604
$988$	0	0
$989$	3929.63	0.126345
$990$	0	0
$991$	−26421.4	−0.846926	−0.423463	−	0.905913i	$-0.639186\pi$
$991$	−26421.4	−0.846926	−0.423463	+	0.905913i	$0.639186\pi$
$992$	0	0
$993$	25842.8	0.825877
$994$	0	0
$995$	8399.12	0.267608
$996$	0	0
$997$	−47185.5	−1.49888	−0.749438	−	0.662074i	$-0.769676\pi$
$997$	−47185.5	−1.49888	−0.749438	+	0.662074i	$0.769676\pi$
$998$	0	0
$999$	−6875.28	−0.217742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.o.1.5		5
4.3	odd	2		460.4.a.b.1.1	✓	5
20.3	even	4		2300.4.c.d.1749.1		10
20.7	even	4		2300.4.c.d.1749.10		10
20.19	odd	2		2300.4.a.c.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.1	✓	5	4.3	odd	2
1840.4.a.o.1.5		5	1.1	even	1	trivial
2300.4.a.c.1.5		5	20.19	odd	2
2300.4.c.d.1749.1		10	20.3	even	4
2300.4.c.d.1749.10		10	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q+8.88744 q^{3} -5.00000 q^{5} +15.9808 q^{7} +51.9866 q^{9} +O(q^{10})\)	\(q+8.88744 q^{3} -5.00000 q^{5} +15.9808 q^{7} +51.9866 q^{9} -9.63486 q^{11} +42.2114 q^{13} -44.4372 q^{15} +40.5247 q^{17} -18.3222 q^{19} +142.029 q^{21} +23.0000 q^{23} +25.0000 q^{25} +222.067 q^{27} +31.6437 q^{29} -63.3450 q^{31} -85.6293 q^{33} -79.9042 q^{35} -30.9604 q^{37} +375.151 q^{39} +203.566 q^{41} +170.854 q^{43} -259.933 q^{45} +505.645 q^{47} -87.6127 q^{49} +360.161 q^{51} -224.035 q^{53} +48.1743 q^{55} -162.838 q^{57} -263.713 q^{59} +501.998 q^{61} +830.789 q^{63} -211.057 q^{65} +184.030 q^{67} +204.411 q^{69} -109.717 q^{71} -679.979 q^{73} +222.186 q^{75} -153.973 q^{77} +331.528 q^{79} +569.968 q^{81} +1305.31 q^{83} -202.624 q^{85} +281.231 q^{87} +122.494 q^{89} +674.574 q^{91} -562.975 q^{93} +91.6111 q^{95} -220.692 q^{97} -500.884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 25 * q^5 - 8 * q^7 + 30 * q^9	\( 5 q + 3 q^{3} - 25 q^{5} - 8 q^{7} + 30 q^{9} - 7 q^{11} + 5 q^{13} - 15 q^{15} - 24 q^{17} + 13 q^{19} + 115 q^{23} + 125 q^{25} + 204 q^{27} - 253 q^{29} + 98 q^{31} - 473 q^{33} + 40 q^{35} - 435 q^{37} + 410 q^{39} - 774 q^{41} + 498 q^{43} - 150 q^{45} + 572 q^{47} - 683 q^{49} + 657 q^{51} - 665 q^{53} + 35 q^{55} - 932 q^{57} + 763 q^{59} + 337 q^{61} + 1527 q^{63} - 25 q^{65} + 305 q^{67} + 69 q^{69} + 1504 q^{71} - 1304 q^{73} + 75 q^{75} + 182 q^{77} - 626 q^{79} - 959 q^{81} + 1703 q^{83} + 120 q^{85} + 1354 q^{87} + 646 q^{89} + 767 q^{91} - 452 q^{93} - 65 q^{95} - 233 q^{97} - 111 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 25 * q^5 - 8 * q^7 + 30 * q^9 - 7 * q^11 + 5 * q^13 - 15 * q^15 - 24 * q^17 + 13 * q^19 + 115 * q^23 + 125 * q^25 + 204 * q^27 - 253 * q^29 + 98 * q^31 - 473 * q^33 + 40 * q^35 - 435 * q^37 + 410 * q^39 - 774 * q^41 + 498 * q^43 - 150 * q^45 + 572 * q^47 - 683 * q^49 + 657 * q^51 - 665 * q^53 + 35 * q^55 - 932 * q^57 + 763 * q^59 + 337 * q^61 + 1527 * q^63 - 25 * q^65 + 305 * q^67 + 69 * q^69 + 1504 * q^71 - 1304 * q^73 + 75 * q^75 + 182 * q^77 - 626 * q^79 - 959 * q^81 + 1703 * q^83 + 120 * q^85 + 1354 * q^87 + 646 * q^89 + 767 * q^91 - 452 * q^93 - 65 * q^95 - 233 * q^97 - 111 * q^99

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