LMFDB - Embedded newform 1840.4.a.m.1.1

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			1.1
Root			$8.73081$ 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)

$f(q)$	$=$	$q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +O(q^{10})$	$q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +32.1352 q^{11} +40.0811 q^{13} +38.6540 q^{15} +126.165 q^{17} -0.232742 q^{19} +182.155 q^{21} +23.0000 q^{23} +25.0000 q^{25} -44.5711 q^{27} -137.226 q^{29} -112.866 q^{31} -248.431 q^{33} +117.811 q^{35} +45.7057 q^{37} -309.859 q^{39} -135.385 q^{41} -543.528 q^{43} -163.827 q^{45} -26.4344 q^{47} +212.180 q^{49} -975.359 q^{51} +43.6958 q^{53} -160.676 q^{55} +1.79928 q^{57} -202.248 q^{59} +150.279 q^{61} -772.026 q^{63} -200.405 q^{65} +420.722 q^{67} -177.809 q^{69} -667.381 q^{71} +602.960 q^{73} -193.270 q^{75} -757.178 q^{77} +1378.88 q^{79} -540.095 q^{81} +485.178 q^{83} -630.826 q^{85} +1060.86 q^{87} -1127.71 q^{89} -944.400 q^{91} +872.545 q^{93} +1.16371 q^{95} -1486.24 q^{97} +1052.92 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.73081	−1.48779	−0.743897	−	0.668294i	$-0.767025\pi$
$3$	−7.73081	−1.48779	−0.743897	+	0.668294i	$0.767025\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−23.5622	−1.27224	−0.636121	−	0.771589i	$-0.719462\pi$
$7$	−23.5622	−1.27224	−0.636121	+	0.771589i	$0.719462\pi$
$8$	0	0
$9$	32.7654	1.21353
$10$	0	0
$11$	32.1352	0.880830	0.440415	−	0.897794i	$-0.354831\pi$
$11$	32.1352	0.880830	0.440415	+	0.897794i	$0.354831\pi$
$12$	0	0
$13$	40.0811	0.855115	0.427557	−	0.903988i	$-0.359374\pi$
$13$	40.0811	0.855115	0.427557	+	0.903988i	$0.359374\pi$
$14$	0	0
$15$	38.6540	0.665362
$16$	0	0
$17$	126.165	1.79997	0.899987	−	0.435916i	$-0.143576\pi$
$17$	126.165	1.79997	0.899987	+	0.435916i	$0.143576\pi$
$18$	0	0
$19$	−0.232742	−0.00281024	−0.00140512	−	0.999999i	$-0.500447\pi$
$19$	−0.232742	−0.00281024	−0.00140512	+	0.999999i	$0.500447\pi$
$20$	0	0
$21$	182.155	1.89283
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−44.5711	−0.317693
$28$	0	0
$29$	−137.226	−0.878695	−0.439347	−	0.898317i	$-0.644790\pi$
$29$	−137.226	−0.878695	−0.439347	+	0.898317i	$0.644790\pi$
$30$	0	0
$31$	−112.866	−0.653914	−0.326957	−	0.945039i	$-0.606023\pi$
$31$	−112.866	−0.653914	−0.326957	+	0.945039i	$0.606023\pi$
$32$	0	0
$33$	−248.431	−1.31049
$34$	0	0
$35$	117.811	0.568964
$36$	0	0
$37$	45.7057	0.203080	0.101540	−	0.994831i	$-0.467623\pi$
$37$	45.7057	0.203080	0.101540	+	0.994831i	$0.467623\pi$
$38$	0	0
$39$	−309.859	−1.27223
$40$	0	0
$41$	−135.385	−0.515696	−0.257848	−	0.966186i	$-0.583013\pi$
$41$	−135.385	−0.515696	−0.257848	+	0.966186i	$0.583013\pi$
$42$	0	0
$43$	−543.528	−1.92761	−0.963805	−	0.266608i	$-0.914097\pi$
$43$	−543.528	−1.92761	−0.963805	+	0.266608i	$0.914097\pi$
$44$	0	0
$45$	−163.827	−0.542708
$46$	0	0
$47$	−26.4344	−0.0820394	−0.0410197	−	0.999158i	$-0.513061\pi$
$47$	−26.4344	−0.0820394	−0.0410197	+	0.999158i	$0.513061\pi$
$48$	0	0
$49$	212.180	0.618599
$50$	0	0
$51$	−975.359	−2.67799
$52$	0	0
$53$	43.6958	0.113247	0.0566235	−	0.998396i	$-0.481967\pi$
$53$	43.6958	0.113247	0.0566235	+	0.998396i	$0.481967\pi$
$54$	0	0
$55$	−160.676	−0.393919
$56$	0	0
$57$	1.79928	0.00418106
$58$	0	0
$59$	−202.248	−0.446278	−0.223139	−	0.974787i	$-0.571630\pi$
$59$	−202.248	−0.446278	−0.223139	+	0.974787i	$0.571630\pi$
$60$	0	0
$61$	150.279	0.315430	0.157715	−	0.987485i	$-0.449587\pi$
$61$	150.279	0.315430	0.157715	+	0.987485i	$0.449587\pi$
$62$	0	0
$63$	−772.026	−1.54391
$64$	0	0
$65$	−200.405	−0.382419
$66$	0	0
$67$	420.722	0.767155	0.383577	−	0.923509i	$-0.374692\pi$
$67$	420.722	0.767155	0.383577	+	0.923509i	$0.374692\pi$
$68$	0	0
$69$	−177.809	−0.310227
$70$	0	0
$71$	−667.381	−1.11554	−0.557771	−	0.829995i	$-0.688343\pi$
$71$	−667.381	−1.11554	−0.557771	+	0.829995i	$0.688343\pi$
$72$	0	0
$73$	602.960	0.966728	0.483364	−	0.875420i	$-0.339415\pi$
$73$	602.960	0.966728	0.483364	+	0.875420i	$0.339415\pi$
$74$	0	0
$75$	−193.270	−0.297559
$76$	0	0
$77$	−757.178	−1.12063
$78$	0	0
$79$	1378.88	1.96374	0.981872	−	0.189545i	$-0.0607013\pi$
$79$	1378.88	1.96374	0.981872	+	0.189545i	$0.0607013\pi$
$80$	0	0
$81$	−540.095	−0.740871
$82$	0	0
$83$	485.178	0.641629	0.320815	−	0.947142i	$-0.396043\pi$
$83$	485.178	0.641629	0.320815	+	0.947142i	$0.396043\pi$
$84$	0	0
$85$	−630.826	−0.804973
$86$	0	0
$87$	1060.86	1.30732
$88$	0	0
$89$	−1127.71	−1.34311	−0.671556	−	0.740954i	$-0.734374\pi$
$89$	−1127.71	−1.34311	−0.671556	+	0.740954i	$0.734374\pi$
$90$	0	0
$91$	−944.400	−1.08791
$92$	0	0
$93$	872.545	0.972890
$94$	0	0
$95$	1.16371	0.00125678
$96$	0	0
$97$	−1486.24	−1.55572	−0.777858	−	0.628441i	$-0.783694\pi$
$97$	−1486.24	−1.55572	−0.777858	+	0.628441i	$0.783694\pi$
$98$	0	0
$99$	1052.92	1.06892
$100$	0	0
$101$	1888.86	1.86087	0.930437	−	0.366451i	$-0.119427\pi$
$101$	1888.86	1.86087	0.930437	+	0.366451i	$0.119427\pi$
$102$	0	0
$103$	1497.17	1.43224	0.716118	−	0.697979i	$-0.245917\pi$
$103$	1497.17	1.43224	0.716118	+	0.697979i	$0.245917\pi$
$104$	0	0
$105$	−910.776	−0.846501
$106$	0	0
$107$	585.150	0.528678	0.264339	−	0.964430i	$-0.414846\pi$
$107$	585.150	0.528678	0.264339	+	0.964430i	$0.414846\pi$
$108$	0	0
$109$	−139.166	−0.122290	−0.0611452	−	0.998129i	$-0.519475\pi$
$109$	−139.166	−0.122290	−0.0611452	+	0.998129i	$0.519475\pi$
$110$	0	0
$111$	−353.342	−0.302142
$112$	0	0
$113$	−1293.18	−1.07656	−0.538282	−	0.842765i	$-0.680927\pi$
$113$	−1293.18	−1.07656	−0.538282	+	0.842765i	$0.680927\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	1313.27	1.03771
$118$	0	0
$119$	−2972.74	−2.29000
$120$	0	0
$121$	−298.328	−0.224138
$122$	0	0
$123$	1046.63	0.767250
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1298.43	0.907221	0.453610	−	0.891200i	$-0.350136\pi$
$127$	1298.43	0.907221	0.453610	+	0.891200i	$0.350136\pi$
$128$	0	0
$129$	4201.91	2.86789
$130$	0	0
$131$	525.247	0.350313	0.175157	−	0.984541i	$-0.443957\pi$
$131$	525.247	0.350313	0.175157	+	0.984541i	$0.443957\pi$
$132$	0	0
$133$	5.48392	0.00357531
$134$	0	0
$135$	222.855	0.142077
$136$	0	0
$137$	−2428.12	−1.51422	−0.757109	−	0.653288i	$-0.773389\pi$
$137$	−2428.12	−1.51422	−0.757109	+	0.653288i	$0.773389\pi$
$138$	0	0
$139$	−2456.46	−1.49895	−0.749476	−	0.662031i	$-0.769695\pi$
$139$	−2456.46	−1.49895	−0.749476	+	0.662031i	$0.769695\pi$
$140$	0	0
$141$	204.359	0.122058
$142$	0	0
$143$	1288.01	0.753211
$144$	0	0
$145$	686.128	0.392964
$146$	0	0
$147$	−1640.32	−0.920349
$148$	0	0
$149$	−2405.39	−1.32253	−0.661266	−	0.750151i	$-0.729981\pi$
$149$	−2405.39	−1.32253	−0.661266	+	0.750151i	$0.729981\pi$
$150$	0	0
$151$	649.276	0.349916	0.174958	−	0.984576i	$-0.444021\pi$
$151$	649.276	0.349916	0.174958	+	0.984576i	$0.444021\pi$
$152$	0	0
$153$	4133.85	2.18433
$154$	0	0
$155$	564.330	0.292439
$156$	0	0
$157$	3665.04	1.86307	0.931536	−	0.363650i	$-0.118470\pi$
$157$	3665.04	1.86307	0.931536	+	0.363650i	$0.118470\pi$
$158$	0	0
$159$	−337.804	−0.168488
$160$	0	0
$161$	−541.932	−0.265281
$162$	0	0
$163$	1055.27	0.507088	0.253544	−	0.967324i	$-0.418404\pi$
$163$	1055.27	0.507088	0.253544	+	0.967324i	$0.418404\pi$
$164$	0	0
$165$	1242.16	0.586071
$166$	0	0
$167$	731.805	0.339095	0.169547	−	0.985522i	$-0.445769\pi$
$167$	731.805	0.339095	0.169547	+	0.985522i	$0.445769\pi$
$168$	0	0
$169$	−590.507	−0.268779
$170$	0	0
$171$	−7.62587	−0.00341032
$172$	0	0
$173$	−521.773	−0.229304	−0.114652	−	0.993406i	$-0.536575\pi$
$173$	−521.773	−0.229304	−0.114652	+	0.993406i	$0.536575\pi$
$174$	0	0
$175$	−589.056	−0.254448
$176$	0	0
$177$	1563.54	0.663970
$178$	0	0
$179$	1183.07	0.494005	0.247002	−	0.969015i	$-0.420554\pi$
$179$	1183.07	0.494005	0.247002	+	0.969015i	$0.420554\pi$
$180$	0	0
$181$	1723.92	0.707946	0.353973	−	0.935256i	$-0.384830\pi$
$181$	1723.92	0.707946	0.353973	+	0.935256i	$0.384830\pi$
$182$	0	0
$183$	−1161.78	−0.469295
$184$	0	0
$185$	−228.529	−0.0908204
$186$	0	0
$187$	4054.35	1.58547
$188$	0	0
$189$	1050.20	0.404182
$190$	0	0
$191$	2798.12	1.06002	0.530012	−	0.847990i	$-0.322187\pi$
$191$	2798.12	1.06002	0.530012	+	0.847990i	$0.322187\pi$
$192$	0	0
$193$	−497.686	−0.185618	−0.0928089	−	0.995684i	$-0.529585\pi$
$193$	−497.686	−0.185618	−0.0928089	+	0.995684i	$0.529585\pi$
$194$	0	0
$195$	1549.30	0.568961
$196$	0	0
$197$	296.489	0.107228	0.0536141	−	0.998562i	$-0.482926\pi$
$197$	296.489	0.107228	0.0536141	+	0.998562i	$0.482926\pi$
$198$	0	0
$199$	−2347.43	−0.836207	−0.418103	−	0.908399i	$-0.637305\pi$
$199$	−2347.43	−0.836207	−0.418103	+	0.908399i	$0.637305\pi$
$200$	0	0
$201$	−3252.52	−1.14137
$202$	0	0
$203$	3233.34	1.11791
$204$	0	0
$205$	676.923	0.230626
$206$	0	0
$207$	753.604	0.253039
$208$	0	0
$209$	−7.47920	−0.00247535
$210$	0	0
$211$	−2838.61	−0.926153	−0.463076	−	0.886318i	$-0.653254\pi$
$211$	−2838.61	−0.926153	−0.463076	+	0.886318i	$0.653254\pi$
$212$	0	0
$213$	5159.39	1.65970
$214$	0	0
$215$	2717.64	0.862053
$216$	0	0
$217$	2659.38	0.831937
$218$	0	0
$219$	−4661.37	−1.43829
$220$	0	0
$221$	5056.84	1.53918
$222$	0	0
$223$	2124.68	0.638024	0.319012	−	0.947751i	$-0.396649\pi$
$223$	2124.68	0.638024	0.319012	+	0.947751i	$0.396649\pi$
$224$	0	0
$225$	819.135	0.242707
$226$	0	0
$227$	−1551.97	−0.453780	−0.226890	−	0.973920i	$-0.572856\pi$
$227$	−1551.97	−0.453780	−0.226890	+	0.973920i	$0.572856\pi$
$228$	0	0
$229$	−158.531	−0.0457469	−0.0228735	−	0.999738i	$-0.507281\pi$
$229$	−158.531	−0.0457469	−0.0228735	+	0.999738i	$0.507281\pi$
$230$	0	0
$231$	5853.60	1.66727
$232$	0	0
$233$	−728.999	−0.204971	−0.102486	−	0.994734i	$-0.532680\pi$
$233$	−728.999	−0.204971	−0.102486	+	0.994734i	$0.532680\pi$
$234$	0	0
$235$	132.172	0.0366891
$236$	0	0
$237$	−10659.8	−2.92165
$238$	0	0
$239$	6201.60	1.67844	0.839222	−	0.543789i	$-0.183011\pi$
$239$	6201.60	1.67844	0.839222	+	0.543789i	$0.183011\pi$
$240$	0	0
$241$	−7223.04	−1.93061	−0.965305	−	0.261126i	$-0.915906\pi$
$241$	−7223.04	−1.93061	−0.965305	+	0.261126i	$0.915906\pi$
$242$	0	0
$243$	5378.79	1.41996
$244$	0	0
$245$	−1060.90	−0.276646
$246$	0	0
$247$	−9.32853	−0.00240308
$248$	0	0
$249$	−3750.82	−0.954612
$250$	0	0
$251$	−5042.78	−1.26812	−0.634059	−	0.773285i	$-0.718612\pi$
$251$	−5042.78	−1.26812	−0.634059	+	0.773285i	$0.718612\pi$
$252$	0	0
$253$	739.110	0.183666
$254$	0	0
$255$	4876.80	1.19763
$256$	0	0
$257$	2981.15	0.723577	0.361788	−	0.932260i	$-0.382166\pi$
$257$	2981.15	0.723577	0.361788	+	0.932260i	$0.382166\pi$
$258$	0	0
$259$	−1076.93	−0.258367
$260$	0	0
$261$	−4496.25	−1.06632
$262$	0	0
$263$	−7242.86	−1.69815	−0.849076	−	0.528271i	$-0.822841\pi$
$263$	−7242.86	−1.69815	−0.849076	+	0.528271i	$0.822841\pi$
$264$	0	0
$265$	−218.479	−0.0506456
$266$	0	0
$267$	8718.10	1.99827
$268$	0	0
$269$	−2567.58	−0.581962	−0.290981	−	0.956729i	$-0.593982\pi$
$269$	−2567.58	−0.581962	−0.290981	+	0.956729i	$0.593982\pi$
$270$	0	0
$271$	−8066.27	−1.80808	−0.904042	−	0.427443i	$-0.859414\pi$
$271$	−8066.27	−1.80808	−0.904042	+	0.427443i	$0.859414\pi$
$272$	0	0
$273$	7300.98	1.61859
$274$	0	0
$275$	803.380	0.176166
$276$	0	0
$277$	8991.54	1.95036	0.975179	−	0.221418i	$-0.0710684\pi$
$277$	8991.54	1.95036	0.975179	+	0.221418i	$0.0710684\pi$
$278$	0	0
$279$	−3698.10	−0.793546
$280$	0	0
$281$	968.130	0.205529	0.102765	−	0.994706i	$-0.467231\pi$
$281$	968.130	0.205529	0.102765	+	0.994706i	$0.467231\pi$
$282$	0	0
$283$	2252.65	0.473167	0.236583	−	0.971611i	$-0.423972\pi$
$283$	2252.65	0.473167	0.236583	+	0.971611i	$0.423972\pi$
$284$	0	0
$285$	−8.99640	−0.00186983
$286$	0	0
$287$	3189.97	0.656090
$288$	0	0
$289$	11004.7	2.23991
$290$	0	0
$291$	11489.8	2.31458
$292$	0	0
$293$	2734.35	0.545196	0.272598	−	0.962128i	$-0.412117\pi$
$293$	2734.35	0.545196	0.272598	+	0.962128i	$0.412117\pi$
$294$	0	0
$295$	1011.24	0.199582
$296$	0	0
$297$	−1432.30	−0.279834
$298$	0	0
$299$	921.865	0.178304
$300$	0	0
$301$	12806.7	2.45239
$302$	0	0
$303$	−14602.4	−2.76860
$304$	0	0
$305$	−751.394	−0.141065
$306$	0	0
$307$	7977.09	1.48299	0.741493	−	0.670961i	$-0.234118\pi$
$307$	7977.09	1.48299	0.741493	+	0.670961i	$0.234118\pi$
$308$	0	0
$309$	−11574.3	−2.13087
$310$	0	0
$311$	−7729.43	−1.40931	−0.704655	−	0.709550i	$-0.748898\pi$
$311$	−7729.43	−1.40931	−0.704655	+	0.709550i	$0.748898\pi$
$312$	0	0
$313$	5506.25	0.994350	0.497175	−	0.867650i	$-0.334371\pi$
$313$	5506.25	0.994350	0.497175	+	0.867650i	$0.334371\pi$
$314$	0	0
$315$	3860.13	0.690456
$316$	0	0
$317$	5231.37	0.926887	0.463443	−	0.886126i	$-0.346614\pi$
$317$	5231.37	0.926887	0.463443	+	0.886126i	$0.346614\pi$
$318$	0	0
$319$	−4409.77	−0.773981
$320$	0	0
$321$	−4523.68	−0.786565
$322$	0	0
$323$	−29.3639	−0.00505836
$324$	0	0
$325$	1002.03	0.171023
$326$	0	0
$327$	1075.86	0.181943
$328$	0	0
$329$	622.854	0.104374
$330$	0	0
$331$	3551.61	0.589771	0.294885	−	0.955533i	$-0.404719\pi$
$331$	3551.61	0.589771	0.294885	+	0.955533i	$0.404719\pi$
$332$	0	0
$333$	1497.57	0.246445
$334$	0	0
$335$	−2103.61	−0.343082
$336$	0	0
$337$	−7002.99	−1.13198	−0.565990	−	0.824412i	$-0.691506\pi$
$337$	−7002.99	−1.13198	−0.565990	+	0.824412i	$0.691506\pi$
$338$	0	0
$339$	9997.29	1.60171
$340$	0	0
$341$	−3626.97	−0.575987
$342$	0	0
$343$	3082.42	0.485234
$344$	0	0
$345$	889.043	0.138738
$346$	0	0
$347$	10268.2	1.58854	0.794272	−	0.607562i	$-0.207852\pi$
$347$	10268.2	1.58854	0.794272	+	0.607562i	$0.207852\pi$
$348$	0	0
$349$	4515.58	0.692588	0.346294	−	0.938126i	$-0.387440\pi$
$349$	4515.58	0.692588	0.346294	+	0.938126i	$0.387440\pi$
$350$	0	0
$351$	−1786.46	−0.271664
$352$	0	0
$353$	7838.63	1.18189	0.590947	−	0.806711i	$-0.298754\pi$
$353$	7838.63	1.18189	0.590947	+	0.806711i	$0.298754\pi$
$354$	0	0
$355$	3336.90	0.498886
$356$	0	0
$357$	22981.7	3.40705
$358$	0	0
$359$	12464.8	1.83250	0.916250	−	0.400606i	$-0.131201\pi$
$359$	12464.8	1.83250	0.916250	+	0.400606i	$0.131201\pi$
$360$	0	0
$361$	−6858.95	−0.999992
$362$	0	0
$363$	2306.32	0.333472
$364$	0	0
$365$	−3014.80	−0.432334
$366$	0	0
$367$	−3936.14	−0.559850	−0.279925	−	0.960022i	$-0.590310\pi$
$367$	−3936.14	−0.559850	−0.279925	+	0.960022i	$0.590310\pi$
$368$	0	0
$369$	−4435.93	−0.625814
$370$	0	0
$371$	−1029.57	−0.144077
$372$	0	0
$373$	1920.72	0.266625	0.133313	−	0.991074i	$-0.457439\pi$
$373$	1920.72	0.266625	0.133313	+	0.991074i	$0.457439\pi$
$374$	0	0
$375$	966.351	0.133072
$376$	0	0
$377$	−5500.15	−0.751385
$378$	0	0
$379$	13074.5	1.77201	0.886004	−	0.463678i	$-0.153470\pi$
$379$	13074.5	1.77201	0.886004	+	0.463678i	$0.153470\pi$
$380$	0	0
$381$	−10037.9	−1.34976
$382$	0	0
$383$	−8838.22	−1.17914	−0.589571	−	0.807716i	$-0.700703\pi$
$383$	−8838.22	−1.17914	−0.589571	+	0.807716i	$0.700703\pi$
$384$	0	0
$385$	3785.89	0.501160
$386$	0	0
$387$	−17808.9	−2.33922
$388$	0	0
$389$	−12687.3	−1.65365	−0.826827	−	0.562456i	$-0.809856\pi$
$389$	−12687.3	−1.65365	−0.826827	+	0.562456i	$0.809856\pi$
$390$	0	0
$391$	2901.80	0.375321
$392$	0	0
$393$	−4060.58	−0.521194
$394$	0	0
$395$	−6894.38	−0.878213
$396$	0	0
$397$	8060.49	1.01900	0.509501	−	0.860470i	$-0.329830\pi$
$397$	8060.49	1.01900	0.509501	+	0.860470i	$0.329830\pi$
$398$	0	0
$399$	−42.3951	−0.00531932
$400$	0	0
$401$	12325.1	1.53488	0.767440	−	0.641121i	$-0.221530\pi$
$401$	12325.1	1.53488	0.767440	+	0.641121i	$0.221530\pi$
$402$	0	0
$403$	−4523.79	−0.559171
$404$	0	0
$405$	2700.47	0.331328
$406$	0	0
$407$	1468.76	0.178879
$408$	0	0
$409$	−10806.7	−1.30650	−0.653250	−	0.757143i	$-0.726595\pi$
$409$	−10806.7	−1.30650	−0.653250	+	0.757143i	$0.726595\pi$
$410$	0	0
$411$	18771.3	2.25285
$412$	0	0
$413$	4765.41	0.567774
$414$	0	0
$415$	−2425.89	−0.286945
$416$	0	0
$417$	18990.4	2.23013
$418$	0	0
$419$	1823.74	0.212639	0.106319	−	0.994332i	$-0.466093\pi$
$419$	1823.74	0.212639	0.106319	+	0.994332i	$0.466093\pi$
$420$	0	0
$421$	5995.43	0.694060	0.347030	−	0.937854i	$-0.387190\pi$
$421$	5995.43	0.694060	0.347030	+	0.937854i	$0.387190\pi$
$422$	0	0
$423$	−866.133	−0.0995575
$424$	0	0
$425$	3154.13	0.359995
$426$	0	0
$427$	−3540.91	−0.401303
$428$	0	0
$429$	−9957.39	−1.12062
$430$	0	0
$431$	−3887.80	−0.434499	−0.217249	−	0.976116i	$-0.569708\pi$
$431$	−3887.80	−0.434499	−0.217249	+	0.976116i	$0.569708\pi$
$432$	0	0
$433$	−4422.93	−0.490884	−0.245442	−	0.969411i	$-0.578933\pi$
$433$	−4422.93	−0.490884	−0.245442	+	0.969411i	$0.578933\pi$
$434$	0	0
$435$	−5304.32	−0.584650
$436$	0	0
$437$	−5.35306	−0.000585976 0
$438$	0	0
$439$	1748.08	0.190048	0.0950242	−	0.995475i	$-0.469707\pi$
$439$	1748.08	0.190048	0.0950242	+	0.995475i	$0.469707\pi$
$440$	0	0
$441$	6952.14	0.750690
$442$	0	0
$443$	−3371.31	−0.361571	−0.180785	−	0.983523i	$-0.557864\pi$
$443$	−3371.31	−0.361571	−0.180785	+	0.983523i	$0.557864\pi$
$444$	0	0
$445$	5638.55	0.600658
$446$	0	0
$447$	18595.6	1.96766
$448$	0	0
$449$	13314.7	1.39947	0.699734	−	0.714404i	$-0.253302\pi$
$449$	13314.7	1.39947	0.699734	+	0.714404i	$0.253302\pi$
$450$	0	0
$451$	−4350.61	−0.454240
$452$	0	0
$453$	−5019.43	−0.520603
$454$	0	0
$455$	4722.00	0.486529
$456$	0	0
$457$	3767.20	0.385607	0.192803	−	0.981237i	$-0.438242\pi$
$457$	3767.20	0.385607	0.192803	+	0.981237i	$0.438242\pi$
$458$	0	0
$459$	−5623.32	−0.571839
$460$	0	0
$461$	9674.06	0.977366	0.488683	−	0.872461i	$-0.337477\pi$
$461$	9674.06	0.977366	0.488683	+	0.872461i	$0.337477\pi$
$462$	0	0
$463$	−2977.73	−0.298892	−0.149446	−	0.988770i	$-0.547749\pi$
$463$	−2977.73	−0.298892	−0.149446	+	0.988770i	$0.547749\pi$
$464$	0	0
$465$	−4362.73	−0.435089
$466$	0	0
$467$	10701.0	1.06035	0.530176	−	0.847888i	$-0.322126\pi$
$467$	10701.0	1.06035	0.530176	+	0.847888i	$0.322126\pi$
$468$	0	0
$469$	−9913.16	−0.976006
$470$	0	0
$471$	−28333.7	−2.77187
$472$	0	0
$473$	−17466.4	−1.69790
$474$	0	0
$475$	−5.81854	−0.000562048 0
$476$	0	0
$477$	1431.71	0.137429
$478$	0	0
$479$	−2337.15	−0.222937	−0.111469	−	0.993768i	$-0.535555\pi$
$479$	−2337.15	−0.222937	−0.111469	+	0.993768i	$0.535555\pi$
$480$	0	0
$481$	1831.94	0.173657
$482$	0	0
$483$	4189.57	0.394683
$484$	0	0
$485$	7431.18	0.695737
$486$	0	0
$487$	7183.85	0.668442	0.334221	−	0.942495i	$-0.391527\pi$
$487$	7183.85	0.668442	0.334221	+	0.942495i	$0.391527\pi$
$488$	0	0
$489$	−8158.12	−0.754443
$490$	0	0
$491$	12084.2	1.11070	0.555350	−	0.831617i	$-0.312584\pi$
$491$	12084.2	1.11070	0.555350	+	0.831617i	$0.312584\pi$
$492$	0	0
$493$	−17313.1	−1.58163
$494$	0	0
$495$	−5264.61	−0.478034
$496$	0	0
$497$	15725.0	1.41924
$498$	0	0
$499$	−7145.78	−0.641060	−0.320530	−	0.947238i	$-0.603861\pi$
$499$	−7145.78	−0.641060	−0.320530	+	0.947238i	$0.603861\pi$
$500$	0	0
$501$	−5657.45	−0.504503
$502$	0	0
$503$	−20436.2	−1.81154	−0.905770	−	0.423771i	$-0.860706\pi$
$503$	−20436.2	−1.81154	−0.905770	+	0.423771i	$0.860706\pi$
$504$	0	0
$505$	−9444.29	−0.832208
$506$	0	0
$507$	4565.10	0.399888
$508$	0	0
$509$	19721.7	1.71738	0.858690	−	0.512495i	$-0.171279\pi$
$509$	19721.7	1.71738	0.858690	+	0.512495i	$0.171279\pi$
$510$	0	0
$511$	−14207.1	−1.22991
$512$	0	0
$513$	10.3735	0.000892794 0
$514$	0	0
$515$	−7485.84	−0.640516
$516$	0	0
$517$	−849.475	−0.0722628
$518$	0	0
$519$	4033.73	0.341158
$520$	0	0
$521$	3483.23	0.292904	0.146452	−	0.989218i	$-0.453215\pi$
$521$	3483.23	0.292904	0.146452	+	0.989218i	$0.453215\pi$
$522$	0	0
$523$	−15689.0	−1.31172	−0.655862	−	0.754881i	$-0.727695\pi$
$523$	−15689.0	−1.31172	−0.655862	+	0.754881i	$0.727695\pi$
$524$	0	0
$525$	4553.88	0.378567
$526$	0	0
$527$	−14239.8	−1.17703
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−6626.72	−0.541573
$532$	0	0
$533$	−5426.36	−0.440979
$534$	0	0
$535$	−2925.75	−0.236432
$536$	0	0
$537$	−9146.09	−0.734977
$538$	0	0
$539$	6818.43	0.544881
$540$	0	0
$541$	7620.73	0.605621	0.302810	−	0.953051i	$-0.402075\pi$
$541$	7620.73	0.605621	0.302810	+	0.953051i	$0.402075\pi$
$542$	0	0
$543$	−13327.3	−1.05328
$544$	0	0
$545$	695.828	0.0546899
$546$	0	0
$547$	−5925.62	−0.463183	−0.231592	−	0.972813i	$-0.574393\pi$
$547$	−5925.62	−0.463183	−0.231592	+	0.972813i	$0.574393\pi$
$548$	0	0
$549$	4923.94	0.382784
$550$	0	0
$551$	31.9381	0.00246934
$552$	0	0
$553$	−32489.4	−2.49836
$554$	0	0
$555$	1766.71	0.135122
$556$	0	0
$557$	−5456.16	−0.415054	−0.207527	−	0.978229i	$-0.566541\pi$
$557$	−5456.16	−0.415054	−0.207527	+	0.978229i	$0.566541\pi$
$558$	0	0
$559$	−21785.2	−1.64833
$560$	0	0
$561$	−31343.4	−2.35886
$562$	0	0
$563$	9194.29	0.688265	0.344132	−	0.938921i	$-0.388173\pi$
$563$	9194.29	0.688265	0.344132	+	0.938921i	$0.388173\pi$
$564$	0	0
$565$	6465.88	0.481454
$566$	0	0
$567$	12725.8	0.942567
$568$	0	0
$569$	338.831	0.0249640	0.0124820	−	0.999922i	$-0.496027\pi$
$569$	338.831	0.0249640	0.0124820	+	0.999922i	$0.496027\pi$
$570$	0	0
$571$	1725.34	0.126451	0.0632254	−	0.997999i	$-0.479861\pi$
$571$	1725.34	0.126451	0.0632254	+	0.997999i	$0.479861\pi$
$572$	0	0
$573$	−21631.7	−1.57710
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	23300.0	1.68109	0.840547	−	0.541738i	$-0.182234\pi$
$577$	23300.0	1.68109	0.840547	+	0.541738i	$0.182234\pi$
$578$	0	0
$579$	3847.52	0.276161
$580$	0	0
$581$	−11431.9	−0.816307
$582$	0	0
$583$	1404.18	0.0997513
$584$	0	0
$585$	−6566.36	−0.464078
$586$	0	0
$587$	−15653.3	−1.10065	−0.550325	−	0.834950i	$-0.685496\pi$
$587$	−15653.3	−1.10065	−0.550325	+	0.834950i	$0.685496\pi$
$588$	0	0
$589$	26.2686	0.00183766
$590$	0	0
$591$	−2292.10	−0.159533
$592$	0	0
$593$	−2658.08	−0.184072	−0.0920358	−	0.995756i	$-0.529337\pi$
$593$	−2658.08	−0.184072	−0.0920358	+	0.995756i	$0.529337\pi$
$594$	0	0
$595$	14863.7	1.02412
$596$	0	0
$597$	18147.6	1.24410
$598$	0	0
$599$	−19417.6	−1.32451	−0.662256	−	0.749278i	$-0.730401\pi$
$599$	−19417.6	−1.32451	−0.662256	+	0.749278i	$0.730401\pi$
$600$	0	0
$601$	−18469.0	−1.25352	−0.626760	−	0.779213i	$-0.715619\pi$
$601$	−18469.0	−1.25352	−0.626760	+	0.779213i	$0.715619\pi$
$602$	0	0
$603$	13785.1	0.930968
$604$	0	0
$605$	1491.64	0.100238
$606$	0	0
$607$	−3968.56	−0.265369	−0.132684	−	0.991158i	$-0.542360\pi$
$607$	−3968.56	−0.265369	−0.132684	+	0.991158i	$0.542360\pi$
$608$	0	0
$609$	−24996.4	−1.66322
$610$	0	0
$611$	−1059.52	−0.0701531
$612$	0	0
$613$	−11478.7	−0.756311	−0.378156	−	0.925742i	$-0.623442\pi$
$613$	−11478.7	−0.756311	−0.378156	+	0.925742i	$0.623442\pi$
$614$	0	0
$615$	−5233.16	−0.343124
$616$	0	0
$617$	−15691.1	−1.02382	−0.511911	−	0.859039i	$-0.671062\pi$
$617$	−15691.1	−1.02382	−0.511911	+	0.859039i	$0.671062\pi$
$618$	0	0
$619$	18249.4	1.18499	0.592494	−	0.805575i	$-0.298144\pi$
$619$	18249.4	1.18499	0.592494	+	0.805575i	$0.298144\pi$
$620$	0	0
$621$	−1025.14	−0.0662436
$622$	0	0
$623$	26571.4	1.70876
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	57.8203	0.00368281
$628$	0	0
$629$	5766.48	0.365540
$630$	0	0
$631$	23528.8	1.48442	0.742208	−	0.670169i	$-0.233778\pi$
$631$	23528.8	1.48442	0.742208	+	0.670169i	$0.233778\pi$
$632$	0	0
$633$	21944.8	1.37793
$634$	0	0
$635$	−6492.15	−0.405722
$636$	0	0
$637$	8504.38	0.528973
$638$	0	0
$639$	−21867.0	−1.35375
$640$	0	0
$641$	3748.79	0.230996	0.115498	−	0.993308i	$-0.463154\pi$
$641$	3748.79	0.230996	0.115498	+	0.993308i	$0.463154\pi$
$642$	0	0
$643$	−15924.3	−0.976660	−0.488330	−	0.872659i	$-0.662394\pi$
$643$	−15924.3	−0.976660	−0.488330	+	0.872659i	$0.662394\pi$
$644$	0	0
$645$	−21009.5	−1.28256
$646$	0	0
$647$	2854.11	0.173426	0.0867129	−	0.996233i	$-0.472364\pi$
$647$	2854.11	0.173426	0.0867129	+	0.996233i	$0.472364\pi$
$648$	0	0
$649$	−6499.27	−0.393095
$650$	0	0
$651$	−20559.1	−1.23775
$652$	0	0
$653$	7925.97	0.474988	0.237494	−	0.971389i	$-0.423674\pi$
$653$	7925.97	0.474988	0.237494	+	0.971389i	$0.423674\pi$
$654$	0	0
$655$	−2626.24	−0.156665
$656$	0	0
$657$	19756.2	1.17316
$658$	0	0
$659$	5799.43	0.342813	0.171406	−	0.985200i	$-0.445169\pi$
$659$	5799.43	0.342813	0.171406	+	0.985200i	$0.445169\pi$
$660$	0	0
$661$	22324.8	1.31367	0.656833	−	0.754036i	$-0.271896\pi$
$661$	22324.8	1.31367	0.656833	+	0.754036i	$0.271896\pi$
$662$	0	0
$663$	−39093.4	−2.28999
$664$	0	0
$665$	−27.4196	−0.00159893
$666$	0	0
$667$	−3156.19	−0.183221
$668$	0	0
$669$	−16425.5	−0.949249
$670$	0	0
$671$	4829.24	0.277840
$672$	0	0
$673$	−23253.5	−1.33188	−0.665940	−	0.746005i	$-0.731969\pi$
$673$	−23253.5	−1.33188	−0.665940	+	0.746005i	$0.731969\pi$
$674$	0	0
$675$	−1114.28	−0.0635386
$676$	0	0
$677$	−19003.7	−1.07884	−0.539419	−	0.842038i	$-0.681356\pi$
$677$	−19003.7	−1.07884	−0.539419	+	0.842038i	$0.681356\pi$
$678$	0	0
$679$	35019.1	1.97925
$680$	0	0
$681$	11998.0	0.675131
$682$	0	0
$683$	11222.5	0.628722	0.314361	−	0.949304i	$-0.398210\pi$
$683$	11222.5	0.628722	0.314361	+	0.949304i	$0.398210\pi$
$684$	0	0
$685$	12140.6	0.677179
$686$	0	0
$687$	1225.58	0.0680620
$688$	0	0
$689$	1751.38	0.0968391
$690$	0	0
$691$	−4297.68	−0.236601	−0.118301	−	0.992978i	$-0.537745\pi$
$691$	−4297.68	−0.236601	−0.118301	+	0.992978i	$0.537745\pi$
$692$	0	0
$693$	−24809.2	−1.35992
$694$	0	0
$695$	12282.3	0.670352
$696$	0	0
$697$	−17080.8	−0.928240
$698$	0	0
$699$	5635.75	0.304955
$700$	0	0
$701$	25769.6	1.38845	0.694225	−	0.719758i	$-0.255747\pi$
$701$	25769.6	1.38845	0.694225	+	0.719758i	$0.255747\pi$
$702$	0	0
$703$	−10.6376	−0.000570705 0
$704$	0	0
$705$	−1021.80	−0.0545859
$706$	0	0
$707$	−44505.7	−2.36748
$708$	0	0
$709$	−4900.01	−0.259554	−0.129777	−	0.991543i	$-0.541426\pi$
$709$	−4900.01	−0.259554	−0.129777	+	0.991543i	$0.541426\pi$
$710$	0	0
$711$	45179.4	2.38307
$712$	0	0
$713$	−2595.92	−0.136350
$714$	0	0
$715$	−6440.07	−0.336846
$716$	0	0
$717$	−47943.4	−2.49718
$718$	0	0
$719$	−7631.85	−0.395855	−0.197928	−	0.980217i	$-0.563421\pi$
$719$	−7631.85	−0.395855	−0.197928	+	0.980217i	$0.563421\pi$
$720$	0	0
$721$	−35276.6	−1.82215
$722$	0	0
$723$	55839.9	2.87235
$724$	0	0
$725$	−3430.64	−0.175739
$726$	0	0
$727$	36985.6	1.88682	0.943410	−	0.331628i	$-0.107598\pi$
$727$	36985.6	1.88682	0.943410	+	0.331628i	$0.107598\pi$
$728$	0	0
$729$	−26999.8	−1.37173
$730$	0	0
$731$	−68574.3	−3.46965
$732$	0	0
$733$	−22227.3	−1.12003	−0.560015	−	0.828482i	$-0.689205\pi$
$733$	−22227.3	−1.12003	−0.560015	+	0.828482i	$0.689205\pi$
$734$	0	0
$735$	8201.60	0.411592
$736$	0	0
$737$	13520.0	0.675733
$738$	0	0
$739$	17899.1	0.890974	0.445487	−	0.895288i	$-0.353030\pi$
$739$	17899.1	0.890974	0.445487	+	0.895288i	$0.353030\pi$
$740$	0	0
$741$	72.1171	0.00357529
$742$	0	0
$743$	29843.4	1.47355	0.736776	−	0.676137i	$-0.236347\pi$
$743$	29843.4	1.47355	0.736776	+	0.676137i	$0.236347\pi$
$744$	0	0
$745$	12027.0	0.591455
$746$	0	0
$747$	15897.0	0.778638
$748$	0	0
$749$	−13787.4	−0.672606
$750$	0	0
$751$	9399.65	0.456722	0.228361	−	0.973577i	$-0.426663\pi$
$751$	9399.65	0.456722	0.228361	+	0.973577i	$0.426663\pi$
$752$	0	0
$753$	38984.8	1.88670
$754$	0	0
$755$	−3246.38	−0.156487
$756$	0	0
$757$	29871.6	1.43422	0.717109	−	0.696961i	$-0.245465\pi$
$757$	29871.6	1.43422	0.717109	+	0.696961i	$0.245465\pi$
$758$	0	0
$759$	−5713.92	−0.273257
$760$	0	0
$761$	−273.955	−0.0130497	−0.00652487	−	0.999979i	$-0.502077\pi$
$761$	−273.955	−0.0130497	−0.00652487	+	0.999979i	$0.502077\pi$
$762$	0	0
$763$	3279.05	0.155583
$764$	0	0
$765$	−20669.3	−0.976861
$766$	0	0
$767$	−8106.31	−0.381619
$768$	0	0
$769$	−13273.8	−0.622450	−0.311225	−	0.950336i	$-0.600739\pi$
$769$	−13273.8	−0.622450	−0.311225	+	0.950336i	$0.600739\pi$
$770$	0	0
$771$	−23046.7	−1.07653
$772$	0	0
$773$	34161.1	1.58951	0.794753	−	0.606933i	$-0.207600\pi$
$773$	34161.1	1.58951	0.794753	+	0.606933i	$0.207600\pi$
$774$	0	0
$775$	−2821.65	−0.130783
$776$	0	0
$777$	8325.54	0.384398
$778$	0	0
$779$	31.5096	0.00144923
$780$	0	0
$781$	−21446.4	−0.982603
$782$	0	0
$783$	6116.29	0.279155
$784$	0	0
$785$	−18325.2	−0.833191
$786$	0	0
$787$	38489.6	1.74334	0.871669	−	0.490095i	$-0.163038\pi$
$787$	38489.6	1.74334	0.871669	+	0.490095i	$0.163038\pi$
$788$	0	0
$789$	55993.2	2.52650
$790$	0	0
$791$	30470.1	1.36965
$792$	0	0
$793$	6023.33	0.269729
$794$	0	0
$795$	1689.02	0.0753502
$796$	0	0
$797$	17176.8	0.763405	0.381702	−	0.924285i	$-0.375338\pi$
$797$	17176.8	0.763405	0.381702	+	0.924285i	$0.375338\pi$
$798$	0	0
$799$	−3335.10	−0.147669
$800$	0	0
$801$	−36949.8	−1.62991
$802$	0	0
$803$	19376.2	0.851523
$804$	0	0
$805$	2709.66	0.118637
$806$	0	0
$807$	19849.4	0.865841
$808$	0	0
$809$	8226.61	0.357518	0.178759	−	0.983893i	$-0.442792\pi$
$809$	8226.61	0.357518	0.178759	+	0.983893i	$0.442792\pi$
$810$	0	0
$811$	−27867.9	−1.20662	−0.603312	−	0.797505i	$-0.706153\pi$
$811$	−27867.9	−1.20662	−0.603312	+	0.797505i	$0.706153\pi$
$812$	0	0
$813$	62358.8	2.69006
$814$	0	0
$815$	−5276.37	−0.226777
$816$	0	0
$817$	126.502	0.00541705
$818$	0	0
$819$	−30943.6	−1.32022
$820$	0	0
$821$	−17637.5	−0.749761	−0.374880	−	0.927073i	$-0.622316\pi$
$821$	−17637.5	−0.749761	−0.374880	+	0.927073i	$0.622316\pi$
$822$	0	0
$823$	−28048.6	−1.18799	−0.593993	−	0.804470i	$-0.702449\pi$
$823$	−28048.6	−1.18799	−0.593993	+	0.804470i	$0.702449\pi$
$824$	0	0
$825$	−6210.78	−0.262099
$826$	0	0
$827$	32592.9	1.37046	0.685228	−	0.728329i	$-0.259703\pi$
$827$	32592.9	1.37046	0.685228	+	0.728329i	$0.259703\pi$
$828$	0	0
$829$	18815.9	0.788303	0.394152	−	0.919045i	$-0.371038\pi$
$829$	18815.9	0.788303	0.394152	+	0.919045i	$0.371038\pi$
$830$	0	0
$831$	−69511.9	−2.90173
$832$	0	0
$833$	26769.7	1.11346
$834$	0	0
$835$	−3659.03	−0.151648
$836$	0	0
$837$	5030.56	0.207744
$838$	0	0
$839$	−7612.72	−0.313254	−0.156627	−	0.987658i	$-0.550062\pi$
$839$	−7612.72	−0.313254	−0.156627	+	0.987658i	$0.550062\pi$
$840$	0	0
$841$	−5558.14	−0.227896
$842$	0	0
$843$	−7484.43	−0.305786
$844$	0	0
$845$	2952.54	0.120202
$846$	0	0
$847$	7029.28	0.285158
$848$	0	0
$849$	−17414.8	−0.703975
$850$	0	0
$851$	1051.23	0.0423452
$852$	0	0
$853$	−31421.4	−1.26125	−0.630627	−	0.776086i	$-0.717202\pi$
$853$	−31421.4	−1.26125	−0.630627	+	0.776086i	$0.717202\pi$
$854$	0	0
$855$	38.1293	0.00152514
$856$	0	0
$857$	20909.5	0.833435	0.416718	−	0.909036i	$-0.363180\pi$
$857$	20909.5	0.833435	0.416718	+	0.909036i	$0.363180\pi$
$858$	0	0
$859$	−23304.5	−0.925655	−0.462828	−	0.886448i	$-0.653165\pi$
$859$	−23304.5	−0.925655	−0.462828	+	0.886448i	$0.653165\pi$
$860$	0	0
$861$	−24661.0	−0.976127
$862$	0	0
$863$	19146.3	0.755211	0.377606	−	0.925966i	$-0.376747\pi$
$863$	19146.3	0.755211	0.377606	+	0.925966i	$0.376747\pi$
$864$	0	0
$865$	2608.86	0.102548
$866$	0	0
$867$	−85075.0	−3.33252
$868$	0	0
$869$	44310.5	1.72972
$870$	0	0
$871$	16863.0	0.656005
$872$	0	0
$873$	−48697.1	−1.88791
$874$	0	0
$875$	2945.28	0.113793
$876$	0	0
$877$	31389.0	1.20859	0.604293	−	0.796762i	$-0.293456\pi$
$877$	31389.0	1.20859	0.604293	+	0.796762i	$0.293456\pi$
$878$	0	0
$879$	−21138.7	−0.811139
$880$	0	0
$881$	44496.3	1.70161	0.850806	−	0.525480i	$-0.176114\pi$
$881$	44496.3	1.70161	0.850806	+	0.525480i	$0.176114\pi$
$882$	0	0
$883$	−8906.65	−0.339448	−0.169724	−	0.985492i	$-0.554288\pi$
$883$	−8906.65	−0.339448	−0.169724	+	0.985492i	$0.554288\pi$
$884$	0	0
$885$	−7817.69	−0.296936
$886$	0	0
$887$	36584.7	1.38489	0.692444	−	0.721472i	$-0.256534\pi$
$887$	36584.7	1.38489	0.692444	+	0.721472i	$0.256534\pi$
$888$	0	0
$889$	−30593.9	−1.15420
$890$	0	0
$891$	−17356.1	−0.652581
$892$	0	0
$893$	6.15238	0.000230551 0
$894$	0	0
$895$	−5915.35	−0.220926
$896$	0	0
$897$	−7126.76	−0.265279
$898$	0	0
$899$	15488.1	0.574591
$900$	0	0
$901$	5512.90	0.203842
$902$	0	0
$903$	−99006.4	−3.64865
$904$	0	0
$905$	−8619.62	−0.316603
$906$	0	0
$907$	21346.1	0.781463	0.390731	−	0.920505i	$-0.372222\pi$
$907$	21346.1	0.781463	0.390731	+	0.920505i	$0.372222\pi$
$908$	0	0
$909$	61889.1	2.25823
$910$	0	0
$911$	10208.0	0.371246	0.185623	−	0.982621i	$-0.440570\pi$
$911$	10208.0	0.371246	0.185623	+	0.982621i	$0.440570\pi$
$912$	0	0
$913$	15591.3	0.565166
$914$	0	0
$915$	5808.88	0.209875
$916$	0	0
$917$	−12376.0	−0.445683
$918$	0	0
$919$	−1758.00	−0.0631025	−0.0315513	−	0.999502i	$-0.510045\pi$
$919$	−1758.00	−0.0631025	−0.0315513	+	0.999502i	$0.510045\pi$
$920$	0	0
$921$	−61669.3	−2.20638
$922$	0	0
$923$	−26749.3	−0.953917
$924$	0	0
$925$	1142.64	0.0406161
$926$	0	0
$927$	49055.3	1.73807
$928$	0	0
$929$	−845.008	−0.0298426	−0.0149213	−	0.999889i	$-0.504750\pi$
$929$	−845.008	−0.0298426	−0.0149213	+	0.999889i	$0.504750\pi$
$930$	0	0
$931$	−49.3830	−0.00173841
$932$	0	0
$933$	59754.7	2.09676
$934$	0	0
$935$	−20271.7	−0.709045
$936$	0	0
$937$	3851.61	0.134287	0.0671433	−	0.997743i	$-0.478612\pi$
$937$	3851.61	0.134287	0.0671433	+	0.997743i	$0.478612\pi$
$938$	0	0
$939$	−42567.7	−1.47939
$940$	0	0
$941$	−4379.36	−0.151714	−0.0758571	−	0.997119i	$-0.524169\pi$
$941$	−4379.36	−0.151714	−0.0758571	+	0.997119i	$0.524169\pi$
$942$	0	0
$943$	−3113.85	−0.107530
$944$	0	0
$945$	−5250.98	−0.180756
$946$	0	0
$947$	−29746.9	−1.02074	−0.510371	−	0.859954i	$-0.670492\pi$
$947$	−29746.9	−1.02074	−0.510371	+	0.859954i	$0.670492\pi$
$948$	0	0
$949$	24167.3	0.826663
$950$	0	0
$951$	−40442.7	−1.37902
$952$	0	0
$953$	4306.61	0.146385	0.0731924	−	0.997318i	$-0.476681\pi$
$953$	4306.61	0.146385	0.0731924	+	0.997318i	$0.476681\pi$
$954$	0	0
$955$	−13990.6	−0.474057
$956$	0	0
$957$	34091.1	1.15152
$958$	0	0
$959$	57211.9	1.92645
$960$	0	0
$961$	−17052.3	−0.572397
$962$	0	0
$963$	19172.7	0.641568
$964$	0	0
$965$	2488.43	0.0830108
$966$	0	0
$967$	12233.4	0.406824	0.203412	−	0.979093i	$-0.434797\pi$
$967$	12233.4	0.406824	0.203412	+	0.979093i	$0.434797\pi$
$968$	0	0
$969$	227.007	0.00752581
$970$	0	0
$971$	−48207.7	−1.59326	−0.796631	−	0.604465i	$-0.793387\pi$
$971$	−48207.7	−1.59326	−0.796631	+	0.604465i	$0.793387\pi$
$972$	0	0
$973$	57879.8	1.90703
$974$	0	0
$975$	−7746.48	−0.254447
$976$	0	0
$977$	28662.9	0.938595	0.469298	−	0.883040i	$-0.344507\pi$
$977$	28662.9	0.938595	0.469298	+	0.883040i	$0.344507\pi$
$978$	0	0
$979$	−36239.2	−1.18305
$980$	0	0
$981$	−4559.81	−0.148403
$982$	0	0
$983$	20944.1	0.679567	0.339783	−	0.940504i	$-0.389646\pi$
$983$	20944.1	0.679567	0.339783	+	0.940504i	$0.389646\pi$
$984$	0	0
$985$	−1482.44	−0.0479539
$986$	0	0
$987$	−4815.16	−0.155287
$988$	0	0
$989$	−12501.1	−0.401934
$990$	0	0
$991$	36440.2	1.16808	0.584038	−	0.811727i	$-0.301472\pi$
$991$	36440.2	1.16808	0.584038	+	0.811727i	$0.301472\pi$
$992$	0	0
$993$	−27456.8	−0.877458
$994$	0	0
$995$	11737.2	0.373963
$996$	0	0
$997$	−11945.0	−0.379439	−0.189720	−	0.981838i	$-0.560758\pi$
$997$	−11945.0	−0.379439	−0.189720	+	0.981838i	$0.560758\pi$
$998$	0	0
$999$	−2037.15	−0.0645172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.m.1.1		4
4.3	odd	2		230.4.a.h.1.4	✓	4
12.11	even	2		2070.4.a.bj.1.3		4
20.3	even	4		1150.4.b.n.599.8		8
20.7	even	4		1150.4.b.n.599.1		8
20.19	odd	2		1150.4.a.p.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.4	✓	4	4.3	odd	2
1150.4.a.p.1.1		4	20.19	odd	2
1150.4.b.n.599.1		8	20.7	even	4
1150.4.b.n.599.8		8	20.3	even	4
1840.4.a.m.1.1		4	1.1	even	1	trivial
2070.4.a.bj.1.3		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)

\(f(q)\)	\(=\)	\(q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +O(q^{10})\)	\(q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +32.1352 q^{11} +40.0811 q^{13} +38.6540 q^{15} +126.165 q^{17} -0.232742 q^{19} +182.155 q^{21} +23.0000 q^{23} +25.0000 q^{25} -44.5711 q^{27} -137.226 q^{29} -112.866 q^{31} -248.431 q^{33} +117.811 q^{35} +45.7057 q^{37} -309.859 q^{39} -135.385 q^{41} -543.528 q^{43} -163.827 q^{45} -26.4344 q^{47} +212.180 q^{49} -975.359 q^{51} +43.6958 q^{53} -160.676 q^{55} +1.79928 q^{57} -202.248 q^{59} +150.279 q^{61} -772.026 q^{63} -200.405 q^{65} +420.722 q^{67} -177.809 q^{69} -667.381 q^{71} +602.960 q^{73} -193.270 q^{75} -757.178 q^{77} +1378.88 q^{79} -540.095 q^{81} +485.178 q^{83} -630.826 q^{85} +1060.86 q^{87} -1127.71 q^{89} -944.400 q^{91} +872.545 q^{93} +1.16371 q^{95} -1486.24 q^{97} +1052.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more