LMFDB - Embedded newform 1840.4.a.t.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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 118x^{4} + 155x^{3} + 3095x^{2} - 6472x + 2800 $ x^6 - x^5 - 118x^4 + 155x^3 + 3095x^2 - 6472x + 2800
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.1
Root			$-8.72903$ 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)

$f(q)$	$=$	$q-8.72903 q^{3} +5.00000 q^{5} +5.16219 q^{7} +49.1960 q^{9} +O(q^{10})$	$q-8.72903 q^{3} +5.00000 q^{5} +5.16219 q^{7} +49.1960 q^{9} -38.0098 q^{11} +39.8463 q^{13} -43.6452 q^{15} -52.1600 q^{17} +35.3134 q^{19} -45.0609 q^{21} +23.0000 q^{23} +25.0000 q^{25} -193.749 q^{27} +154.377 q^{29} -275.161 q^{31} +331.789 q^{33} +25.8110 q^{35} -315.469 q^{37} -347.820 q^{39} +195.373 q^{41} -158.622 q^{43} +245.980 q^{45} +358.521 q^{47} -316.352 q^{49} +455.306 q^{51} +50.4171 q^{53} -190.049 q^{55} -308.251 q^{57} -408.817 q^{59} +218.668 q^{61} +253.959 q^{63} +199.231 q^{65} -228.419 q^{67} -200.768 q^{69} +850.928 q^{71} +675.795 q^{73} -218.226 q^{75} -196.214 q^{77} +933.178 q^{79} +362.952 q^{81} +663.096 q^{83} -260.800 q^{85} -1347.56 q^{87} -243.459 q^{89} +205.694 q^{91} +2401.88 q^{93} +176.567 q^{95} +585.232 q^{97} -1869.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + 30 q^{5} - 24 q^{7} + 75 q^{9}+O(q^{10}) $ 6 * q + q^3 + 30 * q^5 - 24 * q^7 + 75 * q^9	$ 6 q + q^{3} + 30 q^{5} - 24 q^{7} + 75 q^{9} - 117 q^{11} - 59 q^{13} + 5 q^{15} + 88 q^{17} - 105 q^{19} + 44 q^{21} + 138 q^{23} + 150 q^{25} - 164 q^{27} - 71 q^{29} - 396 q^{31} - 85 q^{33} - 120 q^{35} + 57 q^{37} - 718 q^{39} + 692 q^{41} - 778 q^{43} + 375 q^{45} - 248 q^{47} + 958 q^{49} - 545 q^{51} - 49 q^{53} - 585 q^{55} + 396 q^{57} - 1539 q^{59} + 461 q^{61} - 921 q^{63} - 295 q^{65} - 2815 q^{67} + 23 q^{69} - 378 q^{71} + 518 q^{73} + 25 q^{75} - 286 q^{77} - 1694 q^{79} + 498 q^{81} - 1757 q^{83} + 440 q^{85} - 2918 q^{87} + 688 q^{89} - 2363 q^{91} + 36 q^{93} - 525 q^{95} + 1029 q^{97} - 4077 q^{99}+O(q^{100}) $ 6 * q + q^3 + 30 * q^5 - 24 * q^7 + 75 * q^9 - 117 * q^11 - 59 * q^13 + 5 * q^15 + 88 * q^17 - 105 * q^19 + 44 * q^21 + 138 * q^23 + 150 * q^25 - 164 * q^27 - 71 * q^29 - 396 * q^31 - 85 * q^33 - 120 * q^35 + 57 * q^37 - 718 * q^39 + 692 * q^41 - 778 * q^43 + 375 * q^45 - 248 * q^47 + 958 * q^49 - 545 * q^51 - 49 * q^53 - 585 * q^55 + 396 * q^57 - 1539 * q^59 + 461 * q^61 - 921 * q^63 - 295 * q^65 - 2815 * q^67 + 23 * q^69 - 378 * q^71 + 518 * q^73 + 25 * q^75 - 286 * q^77 - 1694 * q^79 + 498 * q^81 - 1757 * q^83 + 440 * q^85 - 2918 * q^87 + 688 * q^89 - 2363 * q^91 + 36 * q^93 - 525 * q^95 + 1029 * q^97 - 4077 * 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
$3$	−8.72903	−1.67990	−0.839951	−	0.542662i	$-0.817417\pi$
$3$	−8.72903	−1.67990	−0.839951	+	0.542662i	$0.817417\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	5.16219	0.278732	0.139366	−	0.990241i	$-0.455494\pi$
$7$	5.16219	0.278732	0.139366	+	0.990241i	$0.455494\pi$
$8$	0	0
$9$	49.1960	1.82207
$10$	0	0
$11$	−38.0098	−1.04185	−0.520927	−	0.853601i	$-0.674414\pi$
$11$	−38.0098	−1.04185	−0.520927	+	0.853601i	$0.674414\pi$
$12$	0	0
$13$	39.8463	0.850106	0.425053	−	0.905169i	$-0.360255\pi$
$13$	39.8463	0.850106	0.425053	+	0.905169i	$0.360255\pi$
$14$	0	0
$15$	−43.6452	−0.751275
$16$	0	0
$17$	−52.1600	−0.744157	−0.372078	−	0.928201i	$-0.621355\pi$
$17$	−52.1600	−0.744157	−0.372078	+	0.928201i	$0.621355\pi$
$18$	0	0
$19$	35.3134	0.426392	0.213196	−	0.977009i	$-0.431613\pi$
$19$	35.3134	0.426392	0.213196	+	0.977009i	$0.431613\pi$
$20$	0	0
$21$	−45.0609	−0.468243
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−193.749	−1.38100
$28$	0	0
$29$	154.377	0.988518	0.494259	−	0.869315i	$-0.335439\pi$
$29$	154.377	0.988518	0.494259	+	0.869315i	$0.335439\pi$
$30$	0	0
$31$	−275.161	−1.59420	−0.797101	−	0.603845i	$-0.793634\pi$
$31$	−275.161	−1.59420	−0.797101	+	0.603845i	$0.793634\pi$
$32$	0	0
$33$	331.789	1.75021
$34$	0	0
$35$	25.8110	0.124653
$36$	0	0
$37$	−315.469	−1.40170	−0.700849	−	0.713310i	$-0.747195\pi$
$37$	−315.469	−1.40170	−0.700849	+	0.713310i	$0.747195\pi$
$38$	0	0
$39$	−347.820	−1.42809
$40$	0	0
$41$	195.373	0.744200	0.372100	−	0.928193i	$-0.378638\pi$
$41$	195.373	0.744200	0.372100	+	0.928193i	$0.378638\pi$
$42$	0	0
$43$	−158.622	−0.562550	−0.281275	−	0.959627i	$-0.590757\pi$
$43$	−158.622	−0.562550	−0.281275	+	0.959627i	$0.590757\pi$
$44$	0	0
$45$	245.980	0.814856
$46$	0	0
$47$	358.521	1.11267	0.556337	−	0.830957i	$-0.312206\pi$
$47$	358.521	1.11267	0.556337	+	0.830957i	$0.312206\pi$
$48$	0	0
$49$	−316.352	−0.922308
$50$	0	0
$51$	455.306	1.25011
$52$	0	0
$53$	50.4171	0.130666	0.0653332	−	0.997864i	$-0.479189\pi$
$53$	50.4171	0.130666	0.0653332	+	0.997864i	$0.479189\pi$
$54$	0	0
$55$	−190.049	−0.465931
$56$	0	0
$57$	−308.251	−0.716296
$58$	0	0
$59$	−408.817	−0.902092	−0.451046	−	0.892501i	$-0.648949\pi$
$59$	−408.817	−0.902092	−0.451046	+	0.892501i	$0.648949\pi$
$60$	0	0
$61$	218.668	0.458977	0.229488	−	0.973311i	$-0.426295\pi$
$61$	218.668	0.458977	0.229488	+	0.973311i	$0.426295\pi$
$62$	0	0
$63$	253.959	0.507870
$64$	0	0
$65$	199.231	0.380179
$66$	0	0
$67$	−228.419	−0.416505	−0.208253	−	0.978075i	$-0.566778\pi$
$67$	−228.419	−0.416505	−0.208253	+	0.978075i	$0.566778\pi$
$68$	0	0
$69$	−200.768	−0.350284
$70$	0	0
$71$	850.928	1.42235	0.711173	−	0.703017i	$-0.248164\pi$
$71$	850.928	1.42235	0.711173	+	0.703017i	$0.248164\pi$
$72$	0	0
$73$	675.795	1.08350	0.541752	−	0.840538i	$-0.317761\pi$
$73$	675.795	1.08350	0.541752	+	0.840538i	$0.317761\pi$
$74$	0	0
$75$	−218.226	−0.335981
$76$	0	0
$77$	−196.214	−0.290398
$78$	0	0
$79$	933.178	1.32900	0.664498	−	0.747290i	$-0.268645\pi$
$79$	933.178	1.32900	0.664498	+	0.747290i	$0.268645\pi$
$80$	0	0
$81$	362.952	0.497877
$82$	0	0
$83$	663.096	0.876919	0.438459	−	0.898751i	$-0.355524\pi$
$83$	663.096	0.876919	0.438459	+	0.898751i	$0.355524\pi$
$84$	0	0
$85$	−260.800	−0.332797
$86$	0	0
$87$	−1347.56	−1.66061
$88$	0	0
$89$	−243.459	−0.289962	−0.144981	−	0.989434i	$-0.546312\pi$
$89$	−243.459	−0.289962	−0.144981	+	0.989434i	$0.546312\pi$
$90$	0	0
$91$	205.694	0.236952
$92$	0	0
$93$	2401.88	2.67811
$94$	0	0
$95$	176.567	0.190688
$96$	0	0
$97$	585.232	0.612591	0.306296	−	0.951936i	$-0.400910\pi$
$97$	585.232	0.612591	0.306296	+	0.951936i	$0.400910\pi$
$98$	0	0
$99$	−1869.93	−1.89833
$100$	0	0
$101$	−443.740	−0.437166	−0.218583	−	0.975818i	$-0.570143\pi$
$101$	−443.740	−0.437166	−0.218583	+	0.975818i	$0.570143\pi$
$102$	0	0
$103$	−213.096	−0.203854	−0.101927	−	0.994792i	$-0.532501\pi$
$103$	−213.096	−0.203854	−0.101927	+	0.994792i	$0.532501\pi$
$104$	0	0
$105$	−225.305	−0.209405
$106$	0	0
$107$	685.864	0.619672	0.309836	−	0.950790i	$-0.399726\pi$
$107$	685.864	0.619672	0.309836	+	0.950790i	$0.399726\pi$
$108$	0	0
$109$	275.012	0.241664	0.120832	−	0.992673i	$-0.461444\pi$
$109$	275.012	0.241664	0.120832	+	0.992673i	$0.461444\pi$
$110$	0	0
$111$	2753.74	2.35472
$112$	0	0
$113$	1423.71	1.18523	0.592617	−	0.805485i	$-0.298095\pi$
$113$	1423.71	1.18523	0.592617	+	0.805485i	$0.298095\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	1960.28	1.54895
$118$	0	0
$119$	−269.260	−0.207420
$120$	0	0
$121$	113.748	0.0854605
$122$	0	0
$123$	−1705.42	−1.25018
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1360.84	0.950826	0.475413	−	0.879763i	$-0.342299\pi$
$127$	1360.84	0.950826	0.475413	+	0.879763i	$0.342299\pi$
$128$	0	0
$129$	1384.62	0.945029
$130$	0	0
$131$	−465.737	−0.310623	−0.155312	−	0.987866i	$-0.549638\pi$
$131$	−465.737	−0.310623	−0.155312	+	0.987866i	$0.549638\pi$
$132$	0	0
$133$	182.294	0.118849
$134$	0	0
$135$	−968.746	−0.617603
$136$	0	0
$137$	−1532.93	−0.955963	−0.477982	−	0.878370i	$-0.658631\pi$
$137$	−1532.93	−0.955963	−0.477982	+	0.878370i	$0.658631\pi$
$138$	0	0
$139$	−2391.08	−1.45905	−0.729527	−	0.683952i	$-0.760260\pi$
$139$	−2391.08	−1.45905	−0.729527	+	0.683952i	$0.760260\pi$
$140$	0	0
$141$	−3129.54	−1.86918
$142$	0	0
$143$	−1514.55	−0.885686
$144$	0	0
$145$	771.883	0.442079
$146$	0	0
$147$	2761.44	1.54939
$148$	0	0
$149$	−1474.01	−0.810442	−0.405221	−	0.914219i	$-0.632805\pi$
$149$	−1474.01	−0.810442	−0.405221	+	0.914219i	$0.632805\pi$
$150$	0	0
$151$	2188.82	1.17962	0.589812	−	0.807541i	$-0.299202\pi$
$151$	2188.82	1.17962	0.589812	+	0.807541i	$0.299202\pi$
$152$	0	0
$153$	−2566.06	−1.35591
$154$	0	0
$155$	−1375.80	−0.712949
$156$	0	0
$157$	579.541	0.294601	0.147301	−	0.989092i	$-0.452942\pi$
$157$	579.541	0.294601	0.147301	+	0.989092i	$0.452942\pi$
$158$	0	0
$159$	−440.092	−0.219507
$160$	0	0
$161$	118.730	0.0581197
$162$	0	0
$163$	517.107	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$163$	517.107	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$164$	0	0
$165$	1658.95	0.782719
$166$	0	0
$167$	−1503.58	−0.696708	−0.348354	−	0.937363i	$-0.613259\pi$
$167$	−1503.58	−0.696708	−0.348354	+	0.937363i	$0.613259\pi$
$168$	0	0
$169$	−609.272	−0.277320
$170$	0	0
$171$	1737.28	0.776917
$172$	0	0
$173$	−3366.76	−1.47960	−0.739798	−	0.672829i	$-0.765079\pi$
$173$	−3366.76	−1.47960	−0.739798	+	0.672829i	$0.765079\pi$
$174$	0	0
$175$	129.055	0.0557464
$176$	0	0
$177$	3568.58	1.51543
$178$	0	0
$179$	−2849.55	−1.18986	−0.594931	−	0.803777i	$-0.702820\pi$
$179$	−2849.55	−1.18986	−0.594931	+	0.803777i	$0.702820\pi$
$180$	0	0
$181$	−2659.06	−1.09197	−0.545984	−	0.837796i	$-0.683844\pi$
$181$	−2659.06	−1.09197	−0.545984	+	0.837796i	$0.683844\pi$
$182$	0	0
$183$	−1908.76	−0.771036
$184$	0	0
$185$	−1577.35	−0.626858
$186$	0	0
$187$	1982.59	0.775303
$188$	0	0
$189$	−1000.17	−0.384930
$190$	0	0
$191$	−850.736	−0.322289	−0.161144	−	0.986931i	$-0.551518\pi$
$191$	−850.736	−0.322289	−0.161144	+	0.986931i	$0.551518\pi$
$192$	0	0
$193$	−854.757	−0.318792	−0.159396	−	0.987215i	$-0.550955\pi$
$193$	−854.757	−0.318792	−0.159396	+	0.987215i	$0.550955\pi$
$194$	0	0
$195$	−1739.10	−0.638663
$196$	0	0
$197$	−4662.14	−1.68611	−0.843055	−	0.537827i	$-0.819245\pi$
$197$	−4662.14	−1.68611	−0.843055	+	0.537827i	$0.819245\pi$
$198$	0	0
$199$	4706.99	1.67673	0.838366	−	0.545108i	$-0.183511\pi$
$199$	4706.99	1.67673	0.838366	+	0.545108i	$0.183511\pi$
$200$	0	0
$201$	1993.88	0.699688
$202$	0	0
$203$	796.922	0.275532
$204$	0	0
$205$	976.867	0.332816
$206$	0	0
$207$	1131.51	0.379928
$208$	0	0
$209$	−1342.26	−0.444238
$210$	0	0
$211$	−4307.70	−1.40547	−0.702735	−	0.711452i	$-0.748038\pi$
$211$	−4307.70	−1.40547	−0.702735	+	0.711452i	$0.748038\pi$
$212$	0	0
$213$	−7427.78	−2.38940
$214$	0	0
$215$	−793.110	−0.251580
$216$	0	0
$217$	−1420.43	−0.444356
$218$	0	0
$219$	−5899.03	−1.82018
$220$	0	0
$221$	−2078.38	−0.632612
$222$	0	0
$223$	6129.19	1.84054	0.920271	−	0.391283i	$-0.127968\pi$
$223$	6129.19	1.84054	0.920271	+	0.391283i	$0.127968\pi$
$224$	0	0
$225$	1229.90	0.364415
$226$	0	0
$227$	−380.409	−0.111228	−0.0556138	−	0.998452i	$-0.517712\pi$
$227$	−380.409	−0.111228	−0.0556138	+	0.998452i	$0.517712\pi$
$228$	0	0
$229$	−2280.21	−0.657994	−0.328997	−	0.944331i	$-0.606711\pi$
$229$	−2280.21	−0.657994	−0.328997	+	0.944331i	$0.606711\pi$
$230$	0	0
$231$	1712.76	0.487841
$232$	0	0
$233$	−2162.63	−0.608062	−0.304031	−	0.952662i	$-0.598333\pi$
$233$	−2162.63	−0.608062	−0.304031	+	0.952662i	$0.598333\pi$
$234$	0	0
$235$	1792.60	0.497603
$236$	0	0
$237$	−8145.73	−2.23258
$238$	0	0
$239$	−3111.07	−0.842000	−0.421000	−	0.907061i	$-0.638321\pi$
$239$	−3111.07	−0.842000	−0.421000	+	0.907061i	$0.638321\pi$
$240$	0	0
$241$	−5330.64	−1.42480	−0.712400	−	0.701774i	$-0.752392\pi$
$241$	−5330.64	−1.42480	−0.712400	+	0.701774i	$0.752392\pi$
$242$	0	0
$243$	2063.01	0.544618
$244$	0	0
$245$	−1581.76	−0.412469
$246$	0	0
$247$	1407.11	0.362478
$248$	0	0
$249$	−5788.19	−1.47314
$250$	0	0
$251$	−6350.09	−1.59687	−0.798434	−	0.602082i	$-0.794338\pi$
$251$	−6350.09	−1.59687	−0.798434	+	0.602082i	$0.794338\pi$
$252$	0	0
$253$	−874.226	−0.217242
$254$	0	0
$255$	2276.53	0.559067
$256$	0	0
$257$	−3024.25	−0.734038	−0.367019	−	0.930213i	$-0.619622\pi$
$257$	−3024.25	−0.734038	−0.367019	+	0.930213i	$0.619622\pi$
$258$	0	0
$259$	−1628.51	−0.390698
$260$	0	0
$261$	7594.71	1.80115
$262$	0	0
$263$	3721.96	0.872646	0.436323	−	0.899790i	$-0.356281\pi$
$263$	3721.96	0.872646	0.436323	+	0.899790i	$0.356281\pi$
$264$	0	0
$265$	252.085	0.0584358
$266$	0	0
$267$	2125.16	0.487107
$268$	0	0
$269$	2767.39	0.627253	0.313626	−	0.949546i	$-0.398456\pi$
$269$	2767.39	0.627253	0.313626	+	0.949546i	$0.398456\pi$
$270$	0	0
$271$	8043.73	1.80303	0.901516	−	0.432746i	$-0.142455\pi$
$271$	8043.73	1.80303	0.901516	+	0.432746i	$0.142455\pi$
$272$	0	0
$273$	−1795.51	−0.398056
$274$	0	0
$275$	−950.246	−0.208371
$276$	0	0
$277$	4637.94	1.00602	0.503009	−	0.864281i	$-0.332226\pi$
$277$	4637.94	1.00602	0.503009	+	0.864281i	$0.332226\pi$
$278$	0	0
$279$	−13536.8	−2.90475
$280$	0	0
$281$	−3052.82	−0.648100	−0.324050	−	0.946040i	$-0.605045\pi$
$281$	−3052.82	−0.648100	−0.324050	+	0.946040i	$0.605045\pi$
$282$	0	0
$283$	−3693.93	−0.775906	−0.387953	−	0.921679i	$-0.626818\pi$
$283$	−3693.93	−0.775906	−0.387953	+	0.921679i	$0.626818\pi$
$284$	0	0
$285$	−1541.26	−0.320338
$286$	0	0
$287$	1008.56	0.207432
$288$	0	0
$289$	−2192.33	−0.446231
$290$	0	0
$291$	−5108.51	−1.02909
$292$	0	0
$293$	3468.45	0.691566	0.345783	−	0.938315i	$-0.387613\pi$
$293$	3468.45	0.691566	0.345783	+	0.938315i	$0.387613\pi$
$294$	0	0
$295$	−2044.08	−0.403428
$296$	0	0
$297$	7364.38	1.43880
$298$	0	0
$299$	916.465	0.177259
$300$	0	0
$301$	−818.837	−0.156801
$302$	0	0
$303$	3873.42	0.734396
$304$	0	0
$305$	1093.34	0.205261
$306$	0	0
$307$	−4702.66	−0.874251	−0.437125	−	0.899401i	$-0.644003\pi$
$307$	−4702.66	−0.874251	−0.437125	+	0.899401i	$0.644003\pi$
$308$	0	0
$309$	1860.12	0.342455
$310$	0	0
$311$	−8242.68	−1.50289	−0.751446	−	0.659794i	$-0.770643\pi$
$311$	−8242.68	−1.50289	−0.751446	+	0.659794i	$0.770643\pi$
$312$	0	0
$313$	−9561.64	−1.72670	−0.863348	−	0.504610i	$-0.831636\pi$
$313$	−9561.64	−1.72670	−0.863348	+	0.504610i	$0.831636\pi$
$314$	0	0
$315$	1269.80	0.227126
$316$	0	0
$317$	8811.57	1.56122	0.780611	−	0.625018i	$-0.214908\pi$
$317$	8811.57	1.56122	0.780611	+	0.625018i	$0.214908\pi$
$318$	0	0
$319$	−5867.83	−1.02989
$320$	0	0
$321$	−5986.92	−1.04099
$322$	0	0
$323$	−1841.95	−0.317302
$324$	0	0
$325$	996.157	0.170021
$326$	0	0
$327$	−2400.59	−0.405972
$328$	0	0
$329$	1850.75	0.310138
$330$	0	0
$331$	−7709.72	−1.28026	−0.640128	−	0.768268i	$-0.721119\pi$
$331$	−7709.72	−1.28026	−0.640128	+	0.768268i	$0.721119\pi$
$332$	0	0
$333$	−15519.8	−2.55400
$334$	0	0
$335$	−1142.10	−0.186267
$336$	0	0
$337$	3874.65	0.626308	0.313154	−	0.949702i	$-0.398614\pi$
$337$	3874.65	0.626308	0.313154	+	0.949702i	$0.398614\pi$
$338$	0	0
$339$	−12427.6	−1.99108
$340$	0	0
$341$	10458.8	1.66093
$342$	0	0
$343$	−3403.70	−0.535809
$344$	0	0
$345$	−1003.84	−0.156652
$346$	0	0
$347$	−2660.21	−0.411549	−0.205775	−	0.978599i	$-0.565971\pi$
$347$	−2660.21	−0.411549	−0.205775	+	0.978599i	$0.565971\pi$
$348$	0	0
$349$	5871.11	0.900496	0.450248	−	0.892904i	$-0.351336\pi$
$349$	5871.11	0.900496	0.450248	+	0.892904i	$0.351336\pi$
$350$	0	0
$351$	−7720.19	−1.17400
$352$	0	0
$353$	−821.405	−0.123850	−0.0619249	−	0.998081i	$-0.519724\pi$
$353$	−821.405	−0.123850	−0.0619249	+	0.998081i	$0.519724\pi$
$354$	0	0
$355$	4254.64	0.636093
$356$	0	0
$357$	2350.38	0.348446
$358$	0	0
$359$	−1539.67	−0.226352	−0.113176	−	0.993575i	$-0.536102\pi$
$359$	−1539.67	−0.226352	−0.113176	+	0.993575i	$0.536102\pi$
$360$	0	0
$361$	−5611.97	−0.818190
$362$	0	0
$363$	−992.910	−0.143565
$364$	0	0
$365$	3378.97	0.484558
$366$	0	0
$367$	8566.05	1.21838	0.609188	−	0.793026i	$-0.291495\pi$
$367$	8566.05	1.21838	0.609188	+	0.793026i	$0.291495\pi$
$368$	0	0
$369$	9611.59	1.35599
$370$	0	0
$371$	260.262	0.0364209
$372$	0	0
$373$	−6795.52	−0.943322	−0.471661	−	0.881780i	$-0.656345\pi$
$373$	−6795.52	−0.943322	−0.471661	+	0.881780i	$0.656345\pi$
$374$	0	0
$375$	−1091.13	−0.150255
$376$	0	0
$377$	6151.34	0.840345
$378$	0	0
$379$	3495.70	0.473778	0.236889	−	0.971537i	$-0.423872\pi$
$379$	3495.70	0.473778	0.236889	+	0.971537i	$0.423872\pi$
$380$	0	0
$381$	−11878.8	−1.59729
$382$	0	0
$383$	−1622.75	−0.216498	−0.108249	−	0.994124i	$-0.534524\pi$
$383$	−1622.75	−0.216498	−0.108249	+	0.994124i	$0.534524\pi$
$384$	0	0
$385$	−981.070	−0.129870
$386$	0	0
$387$	−7803.57	−1.02501
$388$	0	0
$389$	−9773.83	−1.27391	−0.636957	−	0.770899i	$-0.719807\pi$
$389$	−9773.83	−1.27391	−0.636957	+	0.770899i	$0.719807\pi$
$390$	0	0
$391$	−1199.68	−0.155167
$392$	0	0
$393$	4065.43	0.521816
$394$	0	0
$395$	4665.89	0.594345
$396$	0	0
$397$	4147.13	0.524279	0.262139	−	0.965030i	$-0.415572\pi$
$397$	4147.13	0.524279	0.262139	+	0.965030i	$0.415572\pi$
$398$	0	0
$399$	−1591.25	−0.199655
$400$	0	0
$401$	−5786.11	−0.720559	−0.360280	−	0.932844i	$-0.617319\pi$
$401$	−5786.11	−0.720559	−0.360280	+	0.932844i	$0.617319\pi$
$402$	0	0
$403$	−10964.1	−1.35524
$404$	0	0
$405$	1814.76	0.222657
$406$	0	0
$407$	11990.9	1.46036
$408$	0	0
$409$	−756.583	−0.0914685	−0.0457342	−	0.998954i	$-0.514563\pi$
$409$	−756.583	−0.0914685	−0.0457342	+	0.998954i	$0.514563\pi$
$410$	0	0
$411$	13381.0	1.60593
$412$	0	0
$413$	−2110.39	−0.251442
$414$	0	0
$415$	3315.48	0.392170
$416$	0	0
$417$	20871.8	2.45107
$418$	0	0
$419$	−8412.65	−0.980870	−0.490435	−	0.871478i	$-0.663162\pi$
$419$	−8412.65	−0.980870	−0.490435	+	0.871478i	$0.663162\pi$
$420$	0	0
$421$	4671.31	0.540774	0.270387	−	0.962752i	$-0.412848\pi$
$421$	4671.31	0.540774	0.270387	+	0.962752i	$0.412848\pi$
$422$	0	0
$423$	17637.8	2.02737
$424$	0	0
$425$	−1304.00	−0.148831
$426$	0	0
$427$	1128.81	0.127932
$428$	0	0
$429$	13220.6	1.48787
$430$	0	0
$431$	−6924.66	−0.773896	−0.386948	−	0.922102i	$-0.626471\pi$
$431$	−6924.66	−0.773896	−0.386948	+	0.922102i	$0.626471\pi$
$432$	0	0
$433$	−13654.8	−1.51549	−0.757746	−	0.652550i	$-0.773699\pi$
$433$	−13654.8	−1.51549	−0.757746	+	0.652550i	$0.773699\pi$
$434$	0	0
$435$	−6737.79	−0.742649
$436$	0	0
$437$	812.207	0.0889088
$438$	0	0
$439$	−14879.9	−1.61772	−0.808859	−	0.588003i	$-0.799914\pi$
$439$	−14879.9	−1.61772	−0.808859	+	0.588003i	$0.799914\pi$
$440$	0	0
$441$	−15563.2	−1.68051
$442$	0	0
$443$	5404.18	0.579594	0.289797	−	0.957088i	$-0.406412\pi$
$443$	5404.18	0.579594	0.289797	+	0.957088i	$0.406412\pi$
$444$	0	0
$445$	−1217.29	−0.129675
$446$	0	0
$447$	12866.7	1.36146
$448$	0	0
$449$	10215.5	1.07371	0.536857	−	0.843673i	$-0.319612\pi$
$449$	10215.5	1.07371	0.536857	+	0.843673i	$0.319612\pi$
$450$	0	0
$451$	−7426.11	−0.775348
$452$	0	0
$453$	−19106.2	−1.98165
$454$	0	0
$455$	1028.47	0.105968
$456$	0	0
$457$	660.128	0.0675700	0.0337850	−	0.999429i	$-0.489244\pi$
$457$	660.128	0.0675700	0.0337850	+	0.999429i	$0.489244\pi$
$458$	0	0
$459$	10106.0	1.02768
$460$	0	0
$461$	3308.55	0.334261	0.167131	−	0.985935i	$-0.446550\pi$
$461$	3308.55	0.334261	0.167131	+	0.985935i	$0.446550\pi$
$462$	0	0
$463$	3413.57	0.342640	0.171320	−	0.985215i	$-0.445197\pi$
$463$	3413.57	0.342640	0.171320	+	0.985215i	$0.445197\pi$
$464$	0	0
$465$	12009.4	1.19769
$466$	0	0
$467$	11026.0	1.09256	0.546278	−	0.837604i	$-0.316044\pi$
$467$	11026.0	1.09256	0.546278	+	0.837604i	$0.316044\pi$
$468$	0	0
$469$	−1179.14	−0.116093
$470$	0	0
$471$	−5058.83	−0.494901
$472$	0	0
$473$	6029.20	0.586095
$474$	0	0
$475$	882.834	0.0852783
$476$	0	0
$477$	2480.32	0.238084
$478$	0	0
$479$	−13542.0	−1.29175	−0.645876	−	0.763443i	$-0.723508\pi$
$479$	−13542.0	−1.29175	−0.645876	+	0.763443i	$0.723508\pi$
$480$	0	0
$481$	−12570.3	−1.19159
$482$	0	0
$483$	−1036.40	−0.0976354
$484$	0	0
$485$	2926.16	0.273959
$486$	0	0
$487$	−1440.00	−0.133989	−0.0669944	−	0.997753i	$-0.521341\pi$
$487$	−1440.00	−0.133989	−0.0669944	+	0.997753i	$0.521341\pi$
$488$	0	0
$489$	−4513.85	−0.417430
$490$	0	0
$491$	−14147.1	−1.30031	−0.650153	−	0.759804i	$-0.725295\pi$
$491$	−14147.1	−1.30031	−0.650153	+	0.759804i	$0.725295\pi$
$492$	0	0
$493$	−8052.29	−0.735612
$494$	0	0
$495$	−9349.65	−0.848961
$496$	0	0
$497$	4392.65	0.396454
$498$	0	0
$499$	3711.47	0.332962	0.166481	−	0.986045i	$-0.446760\pi$
$499$	3711.47	0.332962	0.166481	+	0.986045i	$0.446760\pi$
$500$	0	0
$501$	13124.8	1.17040
$502$	0	0
$503$	−21745.2	−1.92758	−0.963789	−	0.266665i	$-0.914078\pi$
$503$	−21745.2	−1.92758	−0.963789	+	0.266665i	$0.914078\pi$
$504$	0	0
$505$	−2218.70	−0.195506
$506$	0	0
$507$	5318.36	0.465871
$508$	0	0
$509$	2069.49	0.180213	0.0901064	−	0.995932i	$-0.471279\pi$
$509$	2069.49	0.180213	0.0901064	+	0.995932i	$0.471279\pi$
$510$	0	0
$511$	3488.58	0.302007
$512$	0	0
$513$	−6841.94	−0.588848
$514$	0	0
$515$	−1065.48	−0.0911662
$516$	0	0
$517$	−13627.3	−1.15924
$518$	0	0
$519$	29388.6	2.48558
$520$	0	0
$521$	−4167.35	−0.350432	−0.175216	−	0.984530i	$-0.556062\pi$
$521$	−4167.35	−0.350432	−0.175216	+	0.984530i	$0.556062\pi$
$522$	0	0
$523$	11965.4	1.00040	0.500200	−	0.865910i	$-0.333260\pi$
$523$	11965.4	1.00040	0.500200	+	0.865910i	$0.333260\pi$
$524$	0	0
$525$	−1126.52	−0.0936486
$526$	0	0
$527$	14352.4	1.18634
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−20112.1	−1.64368
$532$	0	0
$533$	7784.91	0.632649
$534$	0	0
$535$	3429.32	0.277126
$536$	0	0
$537$	24873.8	1.99885
$538$	0	0
$539$	12024.5	0.960911
$540$	0	0
$541$	13853.3	1.10092	0.550462	−	0.834860i	$-0.314452\pi$
$541$	13853.3	1.10092	0.550462	+	0.834860i	$0.314452\pi$
$542$	0	0
$543$	23211.0	1.83440
$544$	0	0
$545$	1375.06	0.108075
$546$	0	0
$547$	13072.6	1.02183	0.510917	−	0.859630i	$-0.329306\pi$
$547$	13072.6	1.02183	0.510917	+	0.859630i	$0.329306\pi$
$548$	0	0
$549$	10757.6	0.836289
$550$	0	0
$551$	5451.56	0.421496
$552$	0	0
$553$	4817.24	0.370434
$554$	0	0
$555$	13768.7	1.05306
$556$	0	0
$557$	5378.57	0.409152	0.204576	−	0.978851i	$-0.434418\pi$
$557$	5378.57	0.409152	0.204576	+	0.978851i	$0.434418\pi$
$558$	0	0
$559$	−6320.50	−0.478227
$560$	0	0
$561$	−17306.1	−1.30243
$562$	0	0
$563$	−19049.9	−1.42603	−0.713016	−	0.701148i	$-0.752671\pi$
$563$	−19049.9	−1.42603	−0.713016	+	0.701148i	$0.752671\pi$
$564$	0	0
$565$	7118.55	0.530052
$566$	0	0
$567$	1873.63	0.138774
$568$	0	0
$569$	18403.2	1.35589	0.677945	−	0.735112i	$-0.262871\pi$
$569$	18403.2	1.35589	0.677945	+	0.735112i	$0.262871\pi$
$570$	0	0
$571$	−15486.9	−1.13504	−0.567518	−	0.823361i	$-0.692096\pi$
$571$	−15486.9	−1.13504	−0.567518	+	0.823361i	$0.692096\pi$
$572$	0	0
$573$	7426.10	0.541414
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−17193.6	−1.24052	−0.620260	−	0.784397i	$-0.712973\pi$
$577$	−17193.6	−1.24052	−0.620260	+	0.784397i	$0.712973\pi$
$578$	0	0
$579$	7461.20	0.535539
$580$	0	0
$581$	3423.03	0.244425
$582$	0	0
$583$	−1916.34	−0.136135
$584$	0	0
$585$	9801.39	0.692714
$586$	0	0
$587$	−26790.1	−1.88372	−0.941861	−	0.336003i	$-0.890925\pi$
$587$	−26790.1	−1.88372	−0.941861	+	0.336003i	$0.890925\pi$
$588$	0	0
$589$	−9716.84	−0.679755
$590$	0	0
$591$	40696.0	2.83250
$592$	0	0
$593$	23314.3	1.61451	0.807254	−	0.590204i	$-0.200953\pi$
$593$	23314.3	1.61451	0.807254	+	0.590204i	$0.200953\pi$
$594$	0	0
$595$	−1346.30	−0.0927612
$596$	0	0
$597$	−41087.4	−2.81675
$598$	0	0
$599$	−4311.34	−0.294085	−0.147042	−	0.989130i	$-0.546975\pi$
$599$	−4311.34	−0.294085	−0.147042	+	0.989130i	$0.546975\pi$
$600$	0	0
$601$	28384.5	1.92650	0.963251	−	0.268602i	$-0.0865617\pi$
$601$	28384.5	1.92650	0.963251	+	0.268602i	$0.0865617\pi$
$602$	0	0
$603$	−11237.3	−0.758903
$604$	0	0
$605$	568.740	0.0382191
$606$	0	0
$607$	3132.56	0.209467	0.104734	−	0.994500i	$-0.466601\pi$
$607$	3132.56	0.209467	0.104734	+	0.994500i	$0.466601\pi$
$608$	0	0
$609$	−6956.35	−0.462867
$610$	0	0
$611$	14285.7	0.945890
$612$	0	0
$613$	24702.7	1.62762	0.813811	−	0.581129i	$-0.197389\pi$
$613$	24702.7	1.62762	0.813811	+	0.581129i	$0.197389\pi$
$614$	0	0
$615$	−8527.10	−0.559099
$616$	0	0
$617$	21754.9	1.41948	0.709739	−	0.704464i	$-0.248813\pi$
$617$	21754.9	1.41948	0.709739	+	0.704464i	$0.248813\pi$
$618$	0	0
$619$	−9291.07	−0.603295	−0.301647	−	0.953420i	$-0.597537\pi$
$619$	−9291.07	−0.603295	−0.301647	+	0.953420i	$0.597537\pi$
$620$	0	0
$621$	−4456.23	−0.287959
$622$	0	0
$623$	−1256.78	−0.0808216
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	11716.6	0.746277
$628$	0	0
$629$	16454.9	1.04308
$630$	0	0
$631$	13411.8	0.846143	0.423072	−	0.906096i	$-0.360952\pi$
$631$	13411.8	0.846143	0.423072	+	0.906096i	$0.360952\pi$
$632$	0	0
$633$	37602.0	2.36105
$634$	0	0
$635$	6804.19	0.425222
$636$	0	0
$637$	−12605.4	−0.784060
$638$	0	0
$639$	41862.2	2.59162
$640$	0	0
$641$	−28688.3	−1.76774	−0.883868	−	0.467736i	$-0.845070\pi$
$641$	−28688.3	−1.76774	−0.883868	+	0.467736i	$0.845070\pi$
$642$	0	0
$643$	−223.993	−0.0137378	−0.00686891	−	0.999976i	$-0.502186\pi$
$643$	−223.993	−0.0137378	−0.00686891	+	0.999976i	$0.502186\pi$
$644$	0	0
$645$	6923.08	0.422630
$646$	0	0
$647$	19074.2	1.15901	0.579507	−	0.814967i	$-0.303245\pi$
$647$	19074.2	1.15901	0.579507	+	0.814967i	$0.303245\pi$
$648$	0	0
$649$	15539.1	0.939849
$650$	0	0
$651$	12399.0	0.746474
$652$	0	0
$653$	−5587.94	−0.334874	−0.167437	−	0.985883i	$-0.553549\pi$
$653$	−5587.94	−0.334874	−0.167437	+	0.985883i	$0.553549\pi$
$654$	0	0
$655$	−2328.68	−0.138915
$656$	0	0
$657$	33246.4	1.97422
$658$	0	0
$659$	−17983.8	−1.06305	−0.531526	−	0.847042i	$-0.678381\pi$
$659$	−17983.8	−1.06305	−0.531526	+	0.847042i	$0.678381\pi$
$660$	0	0
$661$	−9991.31	−0.587923	−0.293961	−	0.955817i	$-0.594974\pi$
$661$	−9991.31	−0.587923	−0.293961	+	0.955817i	$0.594974\pi$
$662$	0	0
$663$	18142.3	1.06273
$664$	0	0
$665$	911.472	0.0531509
$666$	0	0
$667$	3550.66	0.206120
$668$	0	0
$669$	−53501.9	−3.09193
$670$	0	0
$671$	−8311.54	−0.478187
$672$	0	0
$673$	−4963.63	−0.284300	−0.142150	−	0.989845i	$-0.545401\pi$
$673$	−4963.63	−0.284300	−0.142150	+	0.989845i	$0.545401\pi$
$674$	0	0
$675$	−4843.73	−0.276201
$676$	0	0
$677$	−5947.62	−0.337645	−0.168822	−	0.985646i	$-0.553996\pi$
$677$	−5947.62	−0.337645	−0.168822	+	0.985646i	$0.553996\pi$
$678$	0	0
$679$	3021.08	0.170749
$680$	0	0
$681$	3320.61	0.186851
$682$	0	0
$683$	−7882.10	−0.441582	−0.220791	−	0.975321i	$-0.570864\pi$
$683$	−7882.10	−0.441582	−0.220791	+	0.975321i	$0.570864\pi$
$684$	0	0
$685$	−7664.65	−0.427520
$686$	0	0
$687$	19904.0	1.10537
$688$	0	0
$689$	2008.93	0.111080
$690$	0	0
$691$	−5702.63	−0.313948	−0.156974	−	0.987603i	$-0.550174\pi$
$691$	−5702.63	−0.313948	−0.156974	+	0.987603i	$0.550174\pi$
$692$	0	0
$693$	−9652.94	−0.529127
$694$	0	0
$695$	−11955.4	−0.652509
$696$	0	0
$697$	−10190.7	−0.553802
$698$	0	0
$699$	18877.6	1.02148
$700$	0	0
$701$	−29520.0	−1.59052	−0.795261	−	0.606268i	$-0.792666\pi$
$701$	−29520.0	−1.59052	−0.795261	+	0.606268i	$0.792666\pi$
$702$	0	0
$703$	−11140.3	−0.597672
$704$	0	0
$705$	−15647.7	−0.835924
$706$	0	0
$707$	−2290.67	−0.121852
$708$	0	0
$709$	−25193.5	−1.33450	−0.667251	−	0.744833i	$-0.732529\pi$
$709$	−25193.5	−1.33450	−0.667251	+	0.744833i	$0.732529\pi$
$710$	0	0
$711$	45908.6	2.42153
$712$	0	0
$713$	−6328.69	−0.332414
$714$	0	0
$715$	−7572.76	−0.396091
$716$	0	0
$717$	27156.6	1.41448
$718$	0	0
$719$	−22488.0	−1.16642	−0.583212	−	0.812320i	$-0.698204\pi$
$719$	−22488.0	−1.16642	−0.583212	+	0.812320i	$0.698204\pi$
$720$	0	0
$721$	−1100.04	−0.0568206
$722$	0	0
$723$	46531.3	2.39352
$724$	0	0
$725$	3859.42	0.197704
$726$	0	0
$727$	13224.7	0.674657	0.337328	−	0.941387i	$-0.390477\pi$
$727$	13224.7	0.674657	0.337328	+	0.941387i	$0.390477\pi$
$728$	0	0
$729$	−27807.8	−1.41278
$730$	0	0
$731$	8273.73	0.418625
$732$	0	0
$733$	9962.01	0.501985	0.250993	−	0.967989i	$-0.419243\pi$
$733$	9962.01	0.501985	0.250993	+	0.967989i	$0.419243\pi$
$734$	0	0
$735$	13807.2	0.692908
$736$	0	0
$737$	8682.18	0.433938
$738$	0	0
$739$	−5841.02	−0.290751	−0.145376	−	0.989377i	$-0.546439\pi$
$739$	−5841.02	−0.290751	−0.145376	+	0.989377i	$0.546439\pi$
$740$	0	0
$741$	−12282.7	−0.608928
$742$	0	0
$743$	−32537.0	−1.60655	−0.803274	−	0.595609i	$-0.796911\pi$
$743$	−32537.0	−1.60655	−0.803274	+	0.595609i	$0.796911\pi$
$744$	0	0
$745$	−7370.07	−0.362441
$746$	0	0
$747$	32621.7	1.59781
$748$	0	0
$749$	3540.56	0.172723
$750$	0	0
$751$	1296.30	0.0629860	0.0314930	−	0.999504i	$-0.489974\pi$
$751$	1296.30	0.0629860	0.0314930	+	0.999504i	$0.489974\pi$
$752$	0	0
$753$	55430.1	2.68258
$754$	0	0
$755$	10944.1	0.527544
$756$	0	0
$757$	−10560.1	−0.507017	−0.253509	−	0.967333i	$-0.581585\pi$
$757$	−10560.1	−0.507017	−0.253509	+	0.967333i	$0.581585\pi$
$758$	0	0
$759$	7631.15	0.364945
$760$	0	0
$761$	10648.0	0.507213	0.253606	−	0.967307i	$-0.418383\pi$
$761$	10648.0	0.507213	0.253606	+	0.967307i	$0.418383\pi$
$762$	0	0
$763$	1419.66	0.0673594
$764$	0	0
$765$	−12830.3	−0.606380
$766$	0	0
$767$	−16289.8	−0.766874
$768$	0	0
$769$	34957.9	1.63929	0.819645	−	0.572871i	$-0.194171\pi$
$769$	34957.9	1.63929	0.819645	+	0.572871i	$0.194171\pi$
$770$	0	0
$771$	26398.8	1.23311
$772$	0	0
$773$	−34404.7	−1.60084	−0.800422	−	0.599437i	$-0.795391\pi$
$773$	−34404.7	−1.60084	−0.800422	+	0.599437i	$0.795391\pi$
$774$	0	0
$775$	−6879.01	−0.318841
$776$	0	0
$777$	14215.3	0.656335
$778$	0	0
$779$	6899.29	0.317321
$780$	0	0
$781$	−32343.6	−1.48188
$782$	0	0
$783$	−29910.4	−1.36515
$784$	0	0
$785$	2897.70	0.131750
$786$	0	0
$787$	28711.0	1.30043	0.650213	−	0.759752i	$-0.274680\pi$
$787$	28711.0	1.30043	0.650213	+	0.759752i	$0.274680\pi$
$788$	0	0
$789$	−32489.1	−1.46596
$790$	0	0
$791$	7349.46	0.330363
$792$	0	0
$793$	8713.11	0.390179
$794$	0	0
$795$	−2200.46	−0.0981664
$796$	0	0
$797$	−13725.0	−0.609995	−0.304998	−	0.952353i	$-0.598656\pi$
$797$	−13725.0	−0.609995	−0.304998	+	0.952353i	$0.598656\pi$
$798$	0	0
$799$	−18700.5	−0.828003
$800$	0	0
$801$	−11977.2	−0.528331
$802$	0	0
$803$	−25686.9	−1.12885
$804$	0	0
$805$	593.652	0.0259919
$806$	0	0
$807$	−24156.7	−1.05372
$808$	0	0
$809$	2335.57	0.101501	0.0507504	−	0.998711i	$-0.483839\pi$
$809$	2335.57	0.101501	0.0507504	+	0.998711i	$0.483839\pi$
$810$	0	0
$811$	−26955.3	−1.16711	−0.583556	−	0.812073i	$-0.698339\pi$
$811$	−26955.3	−1.16711	−0.583556	+	0.812073i	$0.698339\pi$
$812$	0	0
$813$	−70214.0	−3.02892
$814$	0	0
$815$	2585.54	0.111126
$816$	0	0
$817$	−5601.48	−0.239866
$818$	0	0
$819$	10119.3	0.431743
$820$	0	0
$821$	7822.07	0.332512	0.166256	−	0.986083i	$-0.446832\pi$
$821$	7822.07	0.332512	0.166256	+	0.986083i	$0.446832\pi$
$822$	0	0
$823$	−45298.9	−1.91861	−0.959307	−	0.282366i	$-0.908881\pi$
$823$	−45298.9	−1.91861	−0.959307	+	0.282366i	$0.908881\pi$
$824$	0	0
$825$	8294.73	0.350043
$826$	0	0
$827$	−35367.7	−1.48713	−0.743563	−	0.668665i	$-0.766866\pi$
$827$	−35367.7	−1.48713	−0.743563	+	0.668665i	$0.766866\pi$
$828$	0	0
$829$	−4234.85	−0.177422	−0.0887108	−	0.996057i	$-0.528275\pi$
$829$	−4234.85	−0.177422	−0.0887108	+	0.996057i	$0.528275\pi$
$830$	0	0
$831$	−40484.7	−1.69001
$832$	0	0
$833$	16500.9	0.686342
$834$	0	0
$835$	−7517.88	−0.311577
$836$	0	0
$837$	53312.2	2.20160
$838$	0	0
$839$	−23205.3	−0.954869	−0.477435	−	0.878667i	$-0.658433\pi$
$839$	−23205.3	−0.954869	−0.477435	+	0.878667i	$0.658433\pi$
$840$	0	0
$841$	−556.847	−0.0228319
$842$	0	0
$843$	26648.2	1.08875
$844$	0	0
$845$	−3046.36	−0.124021
$846$	0	0
$847$	587.189	0.0238206
$848$	0	0
$849$	32244.5	1.30345
$850$	0	0
$851$	−7255.79	−0.292274
$852$	0	0
$853$	14766.9	0.592742	0.296371	−	0.955073i	$-0.404224\pi$
$853$	14766.9	0.592742	0.296371	+	0.955073i	$0.404224\pi$
$854$	0	0
$855$	8686.38	0.347448
$856$	0	0
$857$	8867.95	0.353469	0.176735	−	0.984259i	$-0.443447\pi$
$857$	8867.95	0.353469	0.176735	+	0.984259i	$0.443447\pi$
$858$	0	0
$859$	2456.28	0.0975638	0.0487819	−	0.998809i	$-0.484466\pi$
$859$	2456.28	0.0975638	0.0487819	+	0.998809i	$0.484466\pi$
$860$	0	0
$861$	−8803.71	−0.348466
$862$	0	0
$863$	5523.03	0.217852	0.108926	−	0.994050i	$-0.465259\pi$
$863$	5523.03	0.217852	0.108926	+	0.994050i	$0.465259\pi$
$864$	0	0
$865$	−16833.8	−0.661696
$866$	0	0
$867$	19136.9	0.749624
$868$	0	0
$869$	−35469.9	−1.38462
$870$	0	0
$871$	−9101.66	−0.354074
$872$	0	0
$873$	28791.1	1.11619
$874$	0	0
$875$	645.274	0.0249306
$876$	0	0
$877$	−31016.4	−1.19424	−0.597120	−	0.802152i	$-0.703688\pi$
$877$	−31016.4	−1.19424	−0.597120	+	0.802152i	$0.703688\pi$
$878$	0	0
$879$	−30276.2	−1.16176
$880$	0	0
$881$	−1126.17	−0.0430667	−0.0215334	−	0.999768i	$-0.506855\pi$
$881$	−1126.17	−0.0430667	−0.0215334	+	0.999768i	$0.506855\pi$
$882$	0	0
$883$	−18161.5	−0.692168	−0.346084	−	0.938204i	$-0.612489\pi$
$883$	−18161.5	−0.692168	−0.346084	+	0.938204i	$0.612489\pi$
$884$	0	0
$885$	17842.9	0.677719
$886$	0	0
$887$	2413.56	0.0913634	0.0456817	−	0.998956i	$-0.485454\pi$
$887$	2413.56	0.0913634	0.0456817	+	0.998956i	$0.485454\pi$
$888$	0	0
$889$	7024.91	0.265026
$890$	0	0
$891$	−13795.8	−0.518715
$892$	0	0
$893$	12660.6	0.474435
$894$	0	0
$895$	−14247.7	−0.532122
$896$	0	0
$897$	−7999.85	−0.297778
$898$	0	0
$899$	−42478.4	−1.57590
$900$	0	0
$901$	−2629.76	−0.0972362
$902$	0	0
$903$	7147.66	0.263410
$904$	0	0
$905$	−13295.3	−0.488343
$906$	0	0
$907$	−28064.7	−1.02742	−0.513711	−	0.857963i	$-0.671730\pi$
$907$	−28064.7	−1.02742	−0.513711	+	0.857963i	$0.671730\pi$
$908$	0	0
$909$	−21830.2	−0.796548
$910$	0	0
$911$	−47846.5	−1.74009	−0.870047	−	0.492969i	$-0.835912\pi$
$911$	−47846.5	−1.74009	−0.870047	+	0.492969i	$0.835912\pi$
$912$	0	0
$913$	−25204.2	−0.913622
$914$	0	0
$915$	−9543.80	−0.344818
$916$	0	0
$917$	−2404.22	−0.0865806
$918$	0	0
$919$	35931.0	1.28972	0.644861	−	0.764300i	$-0.276915\pi$
$919$	35931.0	1.28972	0.644861	+	0.764300i	$0.276915\pi$
$920$	0	0
$921$	41049.7	1.46866
$922$	0	0
$923$	33906.3	1.20914
$924$	0	0
$925$	−7886.73	−0.280340
$926$	0	0
$927$	−10483.4	−0.371437
$928$	0	0
$929$	−27780.1	−0.981093	−0.490546	−	0.871415i	$-0.663203\pi$
$929$	−27780.1	−0.981093	−0.490546	+	0.871415i	$0.663203\pi$
$930$	0	0
$931$	−11171.4	−0.393265
$932$	0	0
$933$	71950.6	2.52471
$934$	0	0
$935$	9912.97	0.346726
$936$	0	0
$937$	18978.7	0.661694	0.330847	−	0.943684i	$-0.392666\pi$
$937$	18978.7	0.661694	0.330847	+	0.943684i	$0.392666\pi$
$938$	0	0
$939$	83463.8	2.90068
$940$	0	0
$941$	53850.9	1.86556	0.932778	−	0.360450i	$-0.117377\pi$
$941$	53850.9	1.86556	0.932778	+	0.360450i	$0.117377\pi$
$942$	0	0
$943$	4493.59	0.155176
$944$	0	0
$945$	−5000.85	−0.172146
$946$	0	0
$947$	34403.0	1.18051	0.590257	−	0.807215i	$-0.299026\pi$
$947$	34403.0	1.18051	0.590257	+	0.807215i	$0.299026\pi$
$948$	0	0
$949$	26927.9	0.921093
$950$	0	0
$951$	−76916.5	−2.62270
$952$	0	0
$953$	−32692.9	−1.11126	−0.555628	−	0.831431i	$-0.687522\pi$
$953$	−32692.9	−1.11126	−0.555628	+	0.831431i	$0.687522\pi$
$954$	0	0
$955$	−4253.68	−0.144132
$956$	0	0
$957$	51220.5	1.73012
$958$	0	0
$959$	−7913.27	−0.266458
$960$	0	0
$961$	45922.3	1.54148
$962$	0	0
$963$	33741.7	1.12909
$964$	0	0
$965$	−4273.79	−0.142568
$966$	0	0
$967$	26256.8	0.873177	0.436588	−	0.899661i	$-0.356187\pi$
$967$	26256.8	0.873177	0.436588	+	0.899661i	$0.356187\pi$
$968$	0	0
$969$	16078.4	0.533037
$970$	0	0
$971$	−46759.7	−1.54541	−0.772704	−	0.634767i	$-0.781096\pi$
$971$	−46759.7	−1.54541	−0.772704	+	0.634767i	$0.781096\pi$
$972$	0	0
$973$	−12343.2	−0.406685
$974$	0	0
$975$	−8695.49	−0.285619
$976$	0	0
$977$	−33207.1	−1.08740	−0.543700	−	0.839279i	$-0.682977\pi$
$977$	−33207.1	−1.08740	−0.543700	+	0.839279i	$0.682977\pi$
$978$	0	0
$979$	9253.83	0.302098
$980$	0	0
$981$	13529.5	0.440329
$982$	0	0
$983$	10045.9	0.325956	0.162978	−	0.986630i	$-0.447890\pi$
$983$	10045.9	0.325956	0.162978	+	0.986630i	$0.447890\pi$
$984$	0	0
$985$	−23310.7	−0.754051
$986$	0	0
$987$	−16155.3	−0.521001
$988$	0	0
$989$	−3648.31	−0.117300
$990$	0	0
$991$	17797.6	0.570495	0.285248	−	0.958454i	$-0.407924\pi$
$991$	17797.6	0.570495	0.285248	+	0.958454i	$0.407924\pi$
$992$	0	0
$993$	67298.4	2.15070
$994$	0	0
$995$	23534.9	0.749857
$996$	0	0
$997$	28689.0	0.911324	0.455662	−	0.890153i	$-0.349403\pi$
$997$	28689.0	0.911324	0.455662	+	0.890153i	$0.349403\pi$
$998$	0	0
$999$	61121.9	1.93575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.t.1.1		6
4.3	odd	2		460.4.a.c.1.6	✓	6
20.3	even	4		2300.4.c.f.1749.11		12
20.7	even	4		2300.4.c.f.1749.2		12
20.19	odd	2		2300.4.a.f.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.c.1.6	✓	6	4.3	odd	2
1840.4.a.t.1.1		6	1.1	even	1	trivial
2300.4.a.f.1.1		6	20.19	odd	2
2300.4.c.f.1749.2		12	20.7	even	4
2300.4.c.f.1749.11		12	20.3	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)

\(f(q)\)	\(=\)	\(q-8.72903 q^{3} +5.00000 q^{5} +5.16219 q^{7} +49.1960 q^{9} +O(q^{10})\)	\(q-8.72903 q^{3} +5.00000 q^{5} +5.16219 q^{7} +49.1960 q^{9} -38.0098 q^{11} +39.8463 q^{13} -43.6452 q^{15} -52.1600 q^{17} +35.3134 q^{19} -45.0609 q^{21} +23.0000 q^{23} +25.0000 q^{25} -193.749 q^{27} +154.377 q^{29} -275.161 q^{31} +331.789 q^{33} +25.8110 q^{35} -315.469 q^{37} -347.820 q^{39} +195.373 q^{41} -158.622 q^{43} +245.980 q^{45} +358.521 q^{47} -316.352 q^{49} +455.306 q^{51} +50.4171 q^{53} -190.049 q^{55} -308.251 q^{57} -408.817 q^{59} +218.668 q^{61} +253.959 q^{63} +199.231 q^{65} -228.419 q^{67} -200.768 q^{69} +850.928 q^{71} +675.795 q^{73} -218.226 q^{75} -196.214 q^{77} +933.178 q^{79} +362.952 q^{81} +663.096 q^{83} -260.800 q^{85} -1347.56 q^{87} -243.459 q^{89} +205.694 q^{91} +2401.88 q^{93} +176.567 q^{95} +585.232 q^{97} -1869.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + 30 q^{5} - 24 q^{7} + 75 q^{9}+O(q^{10}) \) 6 * q + q^3 + 30 * q^5 - 24 * q^7 + 75 * q^9	\( 6 q + q^{3} + 30 q^{5} - 24 q^{7} + 75 q^{9} - 117 q^{11} - 59 q^{13} + 5 q^{15} + 88 q^{17} - 105 q^{19} + 44 q^{21} + 138 q^{23} + 150 q^{25} - 164 q^{27} - 71 q^{29} - 396 q^{31} - 85 q^{33} - 120 q^{35} + 57 q^{37} - 718 q^{39} + 692 q^{41} - 778 q^{43} + 375 q^{45} - 248 q^{47} + 958 q^{49} - 545 q^{51} - 49 q^{53} - 585 q^{55} + 396 q^{57} - 1539 q^{59} + 461 q^{61} - 921 q^{63} - 295 q^{65} - 2815 q^{67} + 23 q^{69} - 378 q^{71} + 518 q^{73} + 25 q^{75} - 286 q^{77} - 1694 q^{79} + 498 q^{81} - 1757 q^{83} + 440 q^{85} - 2918 q^{87} + 688 q^{89} - 2363 q^{91} + 36 q^{93} - 525 q^{95} + 1029 q^{97} - 4077 q^{99}+O(q^{100}) \) 6 * q + q^3 + 30 * q^5 - 24 * q^7 + 75 * q^9 - 117 * q^11 - 59 * q^13 + 5 * q^15 + 88 * q^17 - 105 * q^19 + 44 * q^21 + 138 * q^23 + 150 * q^25 - 164 * q^27 - 71 * q^29 - 396 * q^31 - 85 * q^33 - 120 * q^35 + 57 * q^37 - 718 * q^39 + 692 * q^41 - 778 * q^43 + 375 * q^45 - 248 * q^47 + 958 * q^49 - 545 * q^51 - 49 * q^53 - 585 * q^55 + 396 * q^57 - 1539 * q^59 + 461 * q^61 - 921 * q^63 - 295 * q^65 - 2815 * q^67 + 23 * q^69 - 378 * q^71 + 518 * q^73 + 25 * q^75 - 286 * q^77 - 1694 * q^79 + 498 * q^81 - 1757 * q^83 + 440 * q^85 - 2918 * q^87 + 688 * q^89 - 2363 * q^91 + 36 * q^93 - 525 * q^95 + 1029 * q^97 - 4077 * q^99

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