LMFDB - Embedded newform 1856.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.507544971$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 70x + 238 $ x^3 - x^2 - 70*x + 238
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.29111$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.29111 q^{3} +5.83968 q^{5} -20.2616 q^{7} +12.5781 q^{9} +O(q^{10})$	$q+6.29111 q^{3} +5.83968 q^{5} -20.2616 q^{7} +12.5781 q^{9} +59.1940 q^{11} -21.2575 q^{13} +36.7381 q^{15} -93.2318 q^{17} +69.4851 q^{19} -127.468 q^{21} -34.2444 q^{23} -90.8981 q^{25} -90.7298 q^{27} -29.0000 q^{29} -155.135 q^{31} +372.396 q^{33} -118.321 q^{35} -117.443 q^{37} -133.733 q^{39} -325.822 q^{41} -358.494 q^{43} +73.4521 q^{45} +287.430 q^{47} +67.5320 q^{49} -586.531 q^{51} +496.598 q^{53} +345.674 q^{55} +437.138 q^{57} -474.799 q^{59} -136.910 q^{61} -254.852 q^{63} -124.137 q^{65} -42.5994 q^{67} -215.436 q^{69} -266.035 q^{71} +735.992 q^{73} -571.850 q^{75} -1199.36 q^{77} +1180.09 q^{79} -910.400 q^{81} +962.664 q^{83} -544.444 q^{85} -182.442 q^{87} +709.823 q^{89} +430.710 q^{91} -975.971 q^{93} +405.771 q^{95} -413.230 q^{97} +744.547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9}+O(q^{10}) $ 3 * q - 10 * q^3 + 20 * q^5 - 8 * q^7 + 93 * q^9	$ 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9} + 86 q^{11} - 124 q^{13} - 54 q^{15} + 14 q^{17} + 88 q^{19} - 280 q^{21} - 68 q^{23} + 111 q^{25} - 334 q^{27} - 87 q^{29} - 326 q^{31} + 110 q^{33} - 784 q^{35} - 166 q^{37} + 682 q^{39} + 34 q^{41} - 946 q^{43} + 242 q^{45} - 234 q^{47} + 1067 q^{49} - 1428 q^{51} + 1144 q^{53} - 94 q^{55} + 244 q^{57} + 488 q^{59} - 450 q^{61} + 1096 q^{63} - 1154 q^{65} - 52 q^{67} + 404 q^{69} - 1196 q^{71} + 2434 q^{73} - 1868 q^{75} + 312 q^{77} - 742 q^{79} - 849 q^{81} + 464 q^{83} + 672 q^{85} + 290 q^{87} + 1986 q^{89} + 448 q^{91} + 358 q^{93} - 68 q^{95} - 406 q^{97} + 2540 q^{99}+O(q^{100}) $ 3 * q - 10 * q^3 + 20 * q^5 - 8 * q^7 + 93 * q^9 + 86 * q^11 - 124 * q^13 - 54 * q^15 + 14 * q^17 + 88 * q^19 - 280 * q^21 - 68 * q^23 + 111 * q^25 - 334 * q^27 - 87 * q^29 - 326 * q^31 + 110 * q^33 - 784 * q^35 - 166 * q^37 + 682 * q^39 + 34 * q^41 - 946 * q^43 + 242 * q^45 - 234 * q^47 + 1067 * q^49 - 1428 * q^51 + 1144 * q^53 - 94 * q^55 + 244 * q^57 + 488 * q^59 - 450 * q^61 + 1096 * q^63 - 1154 * q^65 - 52 * q^67 + 404 * q^69 - 1196 * q^71 + 2434 * q^73 - 1868 * q^75 + 312 * q^77 - 742 * q^79 - 849 * q^81 + 464 * q^83 + 672 * q^85 + 290 * q^87 + 1986 * q^89 + 448 * q^91 + 358 * q^93 - 68 * q^95 - 406 * q^97 + 2540 * 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
$3$	6.29111	1.21073	0.605363	−	0.795950i	$-0.293028\pi$
$3$	6.29111	1.21073	0.605363	+	0.795950i	$0.293028\pi$
$4$	0	0
$5$	5.83968	0.522317	0.261159	−	0.965296i	$-0.415895\pi$
$5$	5.83968	0.522317	0.261159	+	0.965296i	$0.415895\pi$
$6$	0	0
$7$	−20.2616	−1.09402	−0.547012	−	0.837125i	$-0.684235\pi$
$7$	−20.2616	−1.09402	−0.547012	+	0.837125i	$0.684235\pi$
$8$	0	0
$9$	12.5781	0.465855
$10$	0	0
$11$	59.1940	1.62251	0.811257	−	0.584690i	$-0.198784\pi$
$11$	59.1940	1.62251	0.811257	+	0.584690i	$0.198784\pi$
$12$	0	0
$13$	−21.2575	−0.453520	−0.226760	−	0.973951i	$-0.572813\pi$
$13$	−21.2575	−0.453520	−0.226760	+	0.973951i	$0.572813\pi$
$14$	0	0
$15$	36.7381	0.632382
$16$	0	0
$17$	−93.2318	−1.33012	−0.665059	−	0.746790i	$-0.731594\pi$
$17$	−93.2318	−1.33012	−0.665059	+	0.746790i	$0.731594\pi$
$18$	0	0
$19$	69.4851	0.838998	0.419499	−	0.907756i	$-0.362206\pi$
$19$	69.4851	0.838998	0.419499	+	0.907756i	$0.362206\pi$
$20$	0	0
$21$	−127.468	−1.32456
$22$	0	0
$23$	−34.2444	−0.310455	−0.155227	−	0.987879i	$-0.549611\pi$
$23$	−34.2444	−0.310455	−0.155227	+	0.987879i	$0.549611\pi$
$24$	0	0
$25$	−90.8981	−0.727185
$26$	0	0
$27$	−90.7298	−0.646702
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−155.135	−0.898808	−0.449404	−	0.893329i	$-0.648364\pi$
$31$	−155.135	−0.898808	−0.449404	+	0.893329i	$0.648364\pi$
$32$	0	0
$33$	372.396	1.96442
$34$	0	0
$35$	−118.321	−0.571427
$36$	0	0
$37$	−117.443	−0.521825	−0.260913	−	0.965362i	$-0.584023\pi$
$37$	−117.443	−0.521825	−0.260913	+	0.965362i	$0.584023\pi$
$38$	0	0
$39$	−133.733	−0.549088
$40$	0	0
$41$	−325.822	−1.24109	−0.620546	−	0.784170i	$-0.713089\pi$
$41$	−325.822	−1.24109	−0.620546	+	0.784170i	$0.713089\pi$
$42$	0	0
$43$	−358.494	−1.27139	−0.635695	−	0.771940i	$-0.719286\pi$
$43$	−358.494	−1.27139	−0.635695	+	0.771940i	$0.719286\pi$
$44$	0	0
$45$	73.4521	0.243324
$46$	0	0
$47$	287.430	0.892041	0.446020	−	0.895023i	$-0.352841\pi$
$47$	287.430	0.892041	0.446020	+	0.895023i	$0.352841\pi$
$48$	0	0
$49$	67.5320	0.196886
$50$	0	0
$51$	−586.531	−1.61041
$52$	0	0
$53$	496.598	1.28704	0.643519	−	0.765430i	$-0.277474\pi$
$53$	496.598	1.28704	0.643519	+	0.765430i	$0.277474\pi$
$54$	0	0
$55$	345.674	0.847467
$56$	0	0
$57$	437.138	1.01580
$58$	0	0
$59$	−474.799	−1.04769	−0.523844	−	0.851814i	$-0.675503\pi$
$59$	−474.799	−1.04769	−0.523844	+	0.851814i	$0.675503\pi$
$60$	0	0
$61$	−136.910	−0.287370	−0.143685	−	0.989623i	$-0.545895\pi$
$61$	−136.910	−0.287370	−0.143685	+	0.989623i	$0.545895\pi$
$62$	0	0
$63$	−254.852	−0.509656
$64$	0	0
$65$	−124.137	−0.236881
$66$	0	0
$67$	−42.5994	−0.0776767	−0.0388384	−	0.999246i	$-0.512366\pi$
$67$	−42.5994	−0.0776767	−0.0388384	+	0.999246i	$0.512366\pi$
$68$	0	0
$69$	−215.436	−0.375875
$70$	0	0
$71$	−266.035	−0.444684	−0.222342	−	0.974969i	$-0.571370\pi$
$71$	−266.035	−0.444684	−0.222342	+	0.974969i	$0.571370\pi$
$72$	0	0
$73$	735.992	1.18002	0.590009	−	0.807397i	$-0.299124\pi$
$73$	735.992	1.18002	0.590009	+	0.807397i	$0.299124\pi$
$74$	0	0
$75$	−571.850	−0.880421
$76$	0	0
$77$	−1199.36	−1.77507
$78$	0	0
$79$	1180.09	1.68064	0.840322	−	0.542088i	$-0.182366\pi$
$79$	1180.09	1.68064	0.840322	+	0.542088i	$0.182366\pi$
$80$	0	0
$81$	−910.400	−1.24883
$82$	0	0
$83$	962.664	1.27308	0.636542	−	0.771242i	$-0.280364\pi$
$83$	962.664	1.27308	0.636542	+	0.771242i	$0.280364\pi$
$84$	0	0
$85$	−544.444	−0.694744
$86$	0	0
$87$	−182.442	−0.224826
$88$	0	0
$89$	709.823	0.845405	0.422703	−	0.906268i	$-0.361081\pi$
$89$	709.823	0.845405	0.422703	+	0.906268i	$0.361081\pi$
$90$	0	0
$91$	430.710	0.496161
$92$	0	0
$93$	−975.971	−1.08821
$94$	0	0
$95$	405.771	0.438223
$96$	0	0
$97$	−413.230	−0.432548	−0.216274	−	0.976333i	$-0.569390\pi$
$97$	−413.230	−0.432548	−0.216274	+	0.976333i	$0.569390\pi$
$98$	0	0
$99$	744.547	0.755857
$100$	0	0
$101$	−245.292	−0.241658	−0.120829	−	0.992673i	$-0.538555\pi$
$101$	−245.292	−0.241658	−0.120829	+	0.992673i	$0.538555\pi$
$102$	0	0
$103$	−1085.57	−1.03849	−0.519247	−	0.854624i	$-0.673787\pi$
$103$	−1085.57	−1.03849	−0.519247	+	0.854624i	$0.673787\pi$
$104$	0	0
$105$	−744.372	−0.691841
$106$	0	0
$107$	−775.162	−0.700353	−0.350176	−	0.936684i	$-0.613878\pi$
$107$	−775.162	−0.700353	−0.350176	+	0.936684i	$0.613878\pi$
$108$	0	0
$109$	−494.737	−0.434745	−0.217372	−	0.976089i	$-0.569749\pi$
$109$	−494.737	−0.434745	−0.217372	+	0.976089i	$0.569749\pi$
$110$	0	0
$111$	−738.848	−0.631787
$112$	0	0
$113$	−1030.47	−0.857861	−0.428931	−	0.903337i	$-0.641110\pi$
$113$	−1030.47	−0.857861	−0.428931	+	0.903337i	$0.641110\pi$
$114$	0	0
$115$	−199.977	−0.162156
$116$	0	0
$117$	−267.378	−0.211275
$118$	0	0
$119$	1889.02	1.45518
$120$	0	0
$121$	2172.93	1.63255
$122$	0	0
$123$	−2049.78	−1.50262
$124$	0	0
$125$	−1260.78	−0.902138
$126$	0	0
$127$	−2743.10	−1.91662	−0.958309	−	0.285734i	$-0.907763\pi$
$127$	−2743.10	−1.91662	−0.958309	+	0.285734i	$0.907763\pi$
$128$	0	0
$129$	−2255.32	−1.53930
$130$	0	0
$131$	−1082.55	−0.722004	−0.361002	−	0.932565i	$-0.617565\pi$
$131$	−1082.55	−0.722004	−0.361002	+	0.932565i	$0.617565\pi$
$132$	0	0
$133$	−1407.88	−0.917884
$134$	0	0
$135$	−529.833	−0.337784
$136$	0	0
$137$	−2826.17	−1.76245	−0.881225	−	0.472696i	$-0.843281\pi$
$137$	−2826.17	−1.76245	−0.881225	+	0.472696i	$0.843281\pi$
$138$	0	0
$139$	1151.71	0.702782	0.351391	−	0.936229i	$-0.385709\pi$
$139$	1151.71	0.702782	0.351391	+	0.936229i	$0.385709\pi$
$140$	0	0
$141$	1808.25	1.08002
$142$	0	0
$143$	−1258.31	−0.735842
$144$	0	0
$145$	−169.351	−0.0969919
$146$	0	0
$147$	424.852	0.238375
$148$	0	0
$149$	−1318.06	−0.724698	−0.362349	−	0.932042i	$-0.618025\pi$
$149$	−1318.06	−0.724698	−0.362349	+	0.932042i	$0.618025\pi$
$150$	0	0
$151$	−2955.00	−1.59255	−0.796274	−	0.604936i	$-0.793199\pi$
$151$	−2955.00	−1.59255	−0.796274	+	0.604936i	$0.793199\pi$
$152$	0	0
$153$	−1172.68	−0.619643
$154$	0	0
$155$	−905.939	−0.469463
$156$	0	0
$157$	2675.18	1.35989	0.679945	−	0.733263i	$-0.262004\pi$
$157$	2675.18	1.35989	0.679945	+	0.733263i	$0.262004\pi$
$158$	0	0
$159$	3124.16	1.55825
$160$	0	0
$161$	693.847	0.339645
$162$	0	0
$163$	3678.35	1.76755	0.883774	−	0.467915i	$-0.154994\pi$
$163$	3678.35	1.76755	0.883774	+	0.467915i	$0.154994\pi$
$164$	0	0
$165$	2174.67	1.02605
$166$	0	0
$167$	−1625.06	−0.753002	−0.376501	−	0.926416i	$-0.622873\pi$
$167$	−1625.06	−0.753002	−0.376501	+	0.926416i	$0.622873\pi$
$168$	0	0
$169$	−1745.12	−0.794320
$170$	0	0
$171$	873.990	0.390852
$172$	0	0
$173$	2326.16	1.02228	0.511142	−	0.859497i	$-0.329223\pi$
$173$	2326.16	1.02228	0.511142	+	0.859497i	$0.329223\pi$
$174$	0	0
$175$	1841.74	0.795557
$176$	0	0
$177$	−2987.02	−1.26846
$178$	0	0
$179$	−2006.91	−0.838009	−0.419004	−	0.907984i	$-0.637621\pi$
$179$	−2006.91	−0.838009	−0.419004	+	0.907984i	$0.637621\pi$
$180$	0	0
$181$	−2477.72	−1.01750	−0.508751	−	0.860914i	$-0.669892\pi$
$181$	−2477.72	−1.01750	−0.508751	+	0.860914i	$0.669892\pi$
$182$	0	0
$183$	−861.319	−0.347926
$184$	0	0
$185$	−685.831	−0.272558
$186$	0	0
$187$	−5518.76	−2.15814
$188$	0	0
$189$	1838.33	0.707507
$190$	0	0
$191$	217.988	0.0825815	0.0412907	−	0.999147i	$-0.486853\pi$
$191$	217.988	0.0825815	0.0412907	+	0.999147i	$0.486853\pi$
$192$	0	0
$193$	4652.02	1.73502	0.867512	−	0.497415i	$-0.165718\pi$
$193$	4652.02	1.73502	0.867512	+	0.497415i	$0.165718\pi$
$194$	0	0
$195$	−780.959	−0.286798
$196$	0	0
$197$	1080.72	0.390852	0.195426	−	0.980718i	$-0.437391\pi$
$197$	1080.72	0.390852	0.195426	+	0.980718i	$0.437391\pi$
$198$	0	0
$199$	1289.66	0.459404	0.229702	−	0.973261i	$-0.426225\pi$
$199$	1289.66	0.459404	0.229702	+	0.973261i	$0.426225\pi$
$200$	0	0
$201$	−267.997	−0.0940452
$202$	0	0
$203$	587.586	0.203155
$204$	0	0
$205$	−1902.69	−0.648244
$206$	0	0
$207$	−430.730	−0.144627
$208$	0	0
$209$	4113.10	1.36129
$210$	0	0
$211$	−4593.00	−1.49856	−0.749278	−	0.662255i	$-0.769599\pi$
$211$	−4593.00	−1.49856	−0.749278	+	0.662255i	$0.769599\pi$
$212$	0	0
$213$	−1673.66	−0.538390
$214$	0	0
$215$	−2093.49	−0.664069
$216$	0	0
$217$	3143.28	0.983317
$218$	0	0
$219$	4630.21	1.42868
$220$	0	0
$221$	1981.87	0.603235
$222$	0	0
$223$	−4151.48	−1.24665	−0.623326	−	0.781962i	$-0.714219\pi$
$223$	−4151.48	−1.24665	−0.623326	+	0.781962i	$0.714219\pi$
$224$	0	0
$225$	−1143.32	−0.338763
$226$	0	0
$227$	−2329.86	−0.681225	−0.340613	−	0.940204i	$-0.610634\pi$
$227$	−2329.86	−0.681225	−0.340613	+	0.940204i	$0.610634\pi$
$228$	0	0
$229$	2176.28	0.628002	0.314001	−	0.949423i	$-0.398330\pi$
$229$	2176.28	0.628002	0.314001	+	0.949423i	$0.398330\pi$
$230$	0	0
$231$	−7545.33	−2.14912
$232$	0	0
$233$	−386.468	−0.108662	−0.0543312	−	0.998523i	$-0.517303\pi$
$233$	−386.468	−0.108662	−0.0543312	+	0.998523i	$0.517303\pi$
$234$	0	0
$235$	1678.50	0.465928
$236$	0	0
$237$	7424.10	2.03480
$238$	0	0
$239$	−2749.41	−0.744120	−0.372060	−	0.928209i	$-0.621348\pi$
$239$	−2749.41	−0.744120	−0.372060	+	0.928209i	$0.621348\pi$
$240$	0	0
$241$	3343.87	0.893767	0.446883	−	0.894592i	$-0.352534\pi$
$241$	3343.87	0.893767	0.446883	+	0.894592i	$0.352534\pi$
$242$	0	0
$243$	−3277.72	−0.865292
$244$	0	0
$245$	394.366	0.102837
$246$	0	0
$247$	−1477.08	−0.380502
$248$	0	0
$249$	6056.22	1.54136
$250$	0	0
$251$	6408.70	1.61161	0.805803	−	0.592183i	$-0.201734\pi$
$251$	6408.70	1.61161	0.805803	+	0.592183i	$0.201734\pi$
$252$	0	0
$253$	−2027.06	−0.503717
$254$	0	0
$255$	−3425.16	−0.841144
$256$	0	0
$257$	−398.382	−0.0966941	−0.0483470	−	0.998831i	$-0.515395\pi$
$257$	−398.382	−0.0966941	−0.0483470	+	0.998831i	$0.515395\pi$
$258$	0	0
$259$	2379.59	0.570889
$260$	0	0
$261$	−364.765	−0.0865072
$262$	0	0
$263$	266.162	0.0624039	0.0312020	−	0.999513i	$-0.490066\pi$
$263$	266.162	0.0624039	0.0312020	+	0.999513i	$0.490066\pi$
$264$	0	0
$265$	2899.98	0.672242
$266$	0	0
$267$	4465.58	1.02355
$268$	0	0
$269$	−487.757	−0.110554	−0.0552771	−	0.998471i	$-0.517604\pi$
$269$	−487.757	−0.110554	−0.0552771	+	0.998471i	$0.517604\pi$
$270$	0	0
$271$	7689.29	1.72358	0.861792	−	0.507262i	$-0.169342\pi$
$271$	7689.29	1.72358	0.861792	+	0.507262i	$0.169342\pi$
$272$	0	0
$273$	2709.64	0.600715
$274$	0	0
$275$	−5380.62	−1.17987
$276$	0	0
$277$	5650.25	1.22560	0.612799	−	0.790239i	$-0.290044\pi$
$277$	5650.25	1.22560	0.612799	+	0.790239i	$0.290044\pi$
$278$	0	0
$279$	−1951.30	−0.418715
$280$	0	0
$281$	−5440.01	−1.15489	−0.577444	−	0.816430i	$-0.695950\pi$
$281$	−5440.01	−1.15489	−0.577444	+	0.816430i	$0.695950\pi$
$282$	0	0
$283$	−7067.13	−1.48444	−0.742221	−	0.670155i	$-0.766227\pi$
$283$	−7067.13	−1.48444	−0.742221	+	0.670155i	$0.766227\pi$
$284$	0	0
$285$	2552.75	0.530568
$286$	0	0
$287$	6601.66	1.35778
$288$	0	0
$289$	3779.16	0.769216
$290$	0	0
$291$	−2599.68	−0.523696
$292$	0	0
$293$	4723.82	0.941872	0.470936	−	0.882167i	$-0.343916\pi$
$293$	4723.82	0.941872	0.470936	+	0.882167i	$0.343916\pi$
$294$	0	0
$295$	−2772.68	−0.547226
$296$	0	0
$297$	−5370.66	−1.04928
$298$	0	0
$299$	727.950	0.140797
$300$	0	0
$301$	7263.65	1.39093
$302$	0	0
$303$	−1543.16	−0.292582
$304$	0	0
$305$	−799.514	−0.150098
$306$	0	0
$307$	−1235.75	−0.229732	−0.114866	−	0.993381i	$-0.536644\pi$
$307$	−1235.75	−0.229732	−0.114866	+	0.993381i	$0.536644\pi$
$308$	0	0
$309$	−6829.47	−1.25733
$310$	0	0
$311$	−9884.18	−1.80219	−0.901094	−	0.433625i	$-0.857234\pi$
$311$	−9884.18	−1.80219	−0.901094	+	0.433625i	$0.857234\pi$
$312$	0	0
$313$	8993.99	1.62419	0.812093	−	0.583528i	$-0.198328\pi$
$313$	8993.99	1.62419	0.812093	+	0.583528i	$0.198328\pi$
$314$	0	0
$315$	−1488.26	−0.266202
$316$	0	0
$317$	−4948.34	−0.876740	−0.438370	−	0.898795i	$-0.644444\pi$
$317$	−4948.34	−0.876740	−0.438370	+	0.898795i	$0.644444\pi$
$318$	0	0
$319$	−1716.63	−0.301293
$320$	0	0
$321$	−4876.63	−0.847935
$322$	0	0
$323$	−6478.22	−1.11597
$324$	0	0
$325$	1932.26	0.329793
$326$	0	0
$327$	−3112.44	−0.526356
$328$	0	0
$329$	−5823.78	−0.975913
$330$	0	0
$331$	−5661.27	−0.940094	−0.470047	−	0.882641i	$-0.655763\pi$
$331$	−5661.27	−0.940094	−0.470047	+	0.882641i	$0.655763\pi$
$332$	0	0
$333$	−1477.21	−0.243095
$334$	0	0
$335$	−248.767	−0.0405719
$336$	0	0
$337$	5778.58	0.934063	0.467032	−	0.884241i	$-0.345323\pi$
$337$	5778.58	0.934063	0.467032	+	0.884241i	$0.345323\pi$
$338$	0	0
$339$	−6482.79	−1.03863
$340$	0	0
$341$	−9183.05	−1.45833
$342$	0	0
$343$	5581.42	0.878625
$344$	0	0
$345$	−1258.08	−0.196326
$346$	0	0
$347$	712.832	0.110279	0.0551395	−	0.998479i	$-0.482440\pi$
$347$	712.832	0.110279	0.0551395	+	0.998479i	$0.482440\pi$
$348$	0	0
$349$	−5142.57	−0.788754	−0.394377	−	0.918949i	$-0.629040\pi$
$349$	−5142.57	−0.788754	−0.394377	+	0.918949i	$0.629040\pi$
$350$	0	0
$351$	1928.69	0.293292
$352$	0	0
$353$	−7641.32	−1.15214	−0.576071	−	0.817399i	$-0.695415\pi$
$353$	−7641.32	−1.15214	−0.576071	+	0.817399i	$0.695415\pi$
$354$	0	0
$355$	−1553.56	−0.232266
$356$	0	0
$357$	11884.1	1.76182
$358$	0	0
$359$	−11682.6	−1.71751	−0.858754	−	0.512388i	$-0.828761\pi$
$359$	−11682.6	−1.71751	−0.858754	+	0.512388i	$0.828761\pi$
$360$	0	0
$361$	−2030.82	−0.296082
$362$	0	0
$363$	13670.1	1.97657
$364$	0	0
$365$	4297.96	0.616344
$366$	0	0
$367$	10555.2	1.50130	0.750651	−	0.660699i	$-0.229740\pi$
$367$	10555.2	1.50130	0.750651	+	0.660699i	$0.229740\pi$
$368$	0	0
$369$	−4098.21	−0.578169
$370$	0	0
$371$	−10061.9	−1.40805
$372$	0	0
$373$	9492.41	1.31769	0.658845	−	0.752279i	$-0.271045\pi$
$373$	9492.41	1.31769	0.658845	+	0.752279i	$0.271045\pi$
$374$	0	0
$375$	−7931.69	−1.09224
$376$	0	0
$377$	616.466	0.0842165
$378$	0	0
$379$	−5301.89	−0.718575	−0.359288	−	0.933227i	$-0.616980\pi$
$379$	−5301.89	−0.718575	−0.359288	+	0.933227i	$0.616980\pi$
$380$	0	0
$381$	−17257.1	−2.32050
$382$	0	0
$383$	4247.46	0.566671	0.283335	−	0.959021i	$-0.408559\pi$
$383$	4247.46	0.566671	0.283335	+	0.959021i	$0.408559\pi$
$384$	0	0
$385$	−7003.91	−0.927148
$386$	0	0
$387$	−4509.17	−0.592284
$388$	0	0
$389$	−11931.5	−1.55515	−0.777573	−	0.628792i	$-0.783550\pi$
$389$	−11931.5	−1.55515	−0.777573	+	0.628792i	$0.783550\pi$
$390$	0	0
$391$	3192.67	0.412942
$392$	0	0
$393$	−6810.42	−0.874149
$394$	0	0
$395$	6891.37	0.877829
$396$	0	0
$397$	5377.48	0.679818	0.339909	−	0.940458i	$-0.389604\pi$
$397$	5377.48	0.679818	0.339909	+	0.940458i	$0.389604\pi$
$398$	0	0
$399$	−8857.12	−1.11130
$400$	0	0
$401$	5777.76	0.719520	0.359760	−	0.933045i	$-0.382859\pi$
$401$	5777.76	0.719520	0.359760	+	0.933045i	$0.382859\pi$
$402$	0	0
$403$	3297.77	0.407627
$404$	0	0
$405$	−5316.45	−0.652287
$406$	0	0
$407$	−6951.93	−0.846669
$408$	0	0
$409$	−1908.37	−0.230716	−0.115358	−	0.993324i	$-0.536802\pi$
$409$	−1908.37	−0.230716	−0.115358	+	0.993324i	$0.536802\pi$
$410$	0	0
$411$	−17779.7	−2.13384
$412$	0	0
$413$	9620.19	1.14620
$414$	0	0
$415$	5621.65	0.664954
$416$	0	0
$417$	7245.53	0.850875
$418$	0	0
$419$	1684.27	0.196377	0.0981885	−	0.995168i	$-0.468695\pi$
$419$	1684.27	0.196377	0.0981885	+	0.995168i	$0.468695\pi$
$420$	0	0
$421$	−13168.2	−1.52441	−0.762207	−	0.647333i	$-0.775884\pi$
$421$	−13168.2	−1.52441	−0.762207	+	0.647333i	$0.775884\pi$
$422$	0	0
$423$	3615.32	0.415562
$424$	0	0
$425$	8474.59	0.967242
$426$	0	0
$427$	2774.02	0.314390
$428$	0	0
$429$	−7916.19	−0.890903
$430$	0	0
$431$	15674.0	1.75172	0.875859	−	0.482567i	$-0.160296\pi$
$431$	15674.0	1.75172	0.875859	+	0.482567i	$0.160296\pi$
$432$	0	0
$433$	14592.2	1.61953	0.809763	−	0.586756i	$-0.199595\pi$
$433$	14592.2	1.61953	0.809763	+	0.586756i	$0.199595\pi$
$434$	0	0
$435$	−1065.40	−0.117430
$436$	0	0
$437$	−2379.48	−0.260471
$438$	0	0
$439$	1442.96	0.156876	0.0784381	−	0.996919i	$-0.475007\pi$
$439$	1442.96	0.156876	0.0784381	+	0.996919i	$0.475007\pi$
$440$	0	0
$441$	849.424	0.0917206
$442$	0	0
$443$	11318.5	1.21391	0.606953	−	0.794738i	$-0.292392\pi$
$443$	11318.5	1.21391	0.606953	+	0.794738i	$0.292392\pi$
$444$	0	0
$445$	4145.14	0.441570
$446$	0	0
$447$	−8292.09	−0.877410
$448$	0	0
$449$	14063.9	1.47821	0.739103	−	0.673592i	$-0.235250\pi$
$449$	14063.9	1.47821	0.739103	+	0.673592i	$0.235250\pi$
$450$	0	0
$451$	−19286.7	−2.01369
$452$	0	0
$453$	−18590.3	−1.92814
$454$	0	0
$455$	2515.21	0.259153
$456$	0	0
$457$	3706.09	0.379351	0.189676	−	0.981847i	$-0.439256\pi$
$457$	3706.09	0.379351	0.189676	+	0.981847i	$0.439256\pi$
$458$	0	0
$459$	8458.90	0.860191
$460$	0	0
$461$	−18085.8	−1.82720	−0.913601	−	0.406613i	$-0.866710\pi$
$461$	−18085.8	−1.82720	−0.913601	+	0.406613i	$0.866710\pi$
$462$	0	0
$463$	−8928.44	−0.896198	−0.448099	−	0.893984i	$-0.647899\pi$
$463$	−8928.44	−0.896198	−0.448099	+	0.893984i	$0.647899\pi$
$464$	0	0
$465$	−5699.36	−0.568391
$466$	0	0
$467$	−6181.73	−0.612540	−0.306270	−	0.951945i	$-0.599081\pi$
$467$	−6181.73	−0.612540	−0.306270	+	0.951945i	$0.599081\pi$
$468$	0	0
$469$	863.131	0.0849801
$470$	0	0
$471$	16829.9	1.64645
$472$	0	0
$473$	−21220.7	−2.06285
$474$	0	0
$475$	−6316.06	−0.610107
$476$	0	0
$477$	6246.26	0.599574
$478$	0	0
$479$	−18038.3	−1.72065	−0.860327	−	0.509743i	$-0.829740\pi$
$479$	−18038.3	−1.72065	−0.860327	+	0.509743i	$0.829740\pi$
$480$	0	0
$481$	2496.54	0.236658
$482$	0	0
$483$	4365.07	0.411216
$484$	0	0
$485$	−2413.13	−0.225927
$486$	0	0
$487$	5188.78	0.482805	0.241403	−	0.970425i	$-0.422393\pi$
$487$	5188.78	0.482805	0.241403	+	0.970425i	$0.422393\pi$
$488$	0	0
$489$	23140.9	2.14001
$490$	0	0
$491$	−2954.05	−0.271516	−0.135758	−	0.990742i	$-0.543347\pi$
$491$	−2954.05	−0.271516	−0.135758	+	0.990742i	$0.543347\pi$
$492$	0	0
$493$	2703.72	0.246997
$494$	0	0
$495$	4347.92	0.394797
$496$	0	0
$497$	5390.29	0.486494
$498$	0	0
$499$	13333.1	1.19613	0.598067	−	0.801446i	$-0.295936\pi$
$499$	13333.1	1.19613	0.598067	+	0.801446i	$0.295936\pi$
$500$	0	0
$501$	−10223.5	−0.911678
$502$	0	0
$503$	−3740.58	−0.331579	−0.165789	−	0.986161i	$-0.553017\pi$
$503$	−3740.58	−0.331579	−0.165789	+	0.986161i	$0.553017\pi$
$504$	0	0
$505$	−1432.43	−0.126222
$506$	0	0
$507$	−10978.7	−0.961703
$508$	0	0
$509$	11433.2	0.995613	0.497807	−	0.867288i	$-0.334139\pi$
$509$	11433.2	0.995613	0.497807	+	0.867288i	$0.334139\pi$
$510$	0	0
$511$	−14912.4	−1.29097
$512$	0	0
$513$	−6304.37	−0.542582
$514$	0	0
$515$	−6339.41	−0.542423
$516$	0	0
$517$	17014.1	1.44735
$518$	0	0
$519$	14634.2	1.23770
$520$	0	0
$521$	14855.8	1.24922	0.624611	−	0.780936i	$-0.285257\pi$
$521$	14855.8	1.24922	0.624611	+	0.780936i	$0.285257\pi$
$522$	0	0
$523$	13970.3	1.16803	0.584013	−	0.811744i	$-0.301482\pi$
$523$	13970.3	1.16803	0.584013	+	0.811744i	$0.301482\pi$
$524$	0	0
$525$	11586.6	0.963201
$526$	0	0
$527$	14463.5	1.19552
$528$	0	0
$529$	−10994.3	−0.903618
$530$	0	0
$531$	−5972.07	−0.488071
$532$	0	0
$533$	6926.14	0.562860
$534$	0	0
$535$	−4526.70	−0.365806
$536$	0	0
$537$	−12625.7	−1.01460
$538$	0	0
$539$	3997.49	0.319451
$540$	0	0
$541$	4989.89	0.396548	0.198274	−	0.980147i	$-0.436466\pi$
$541$	4989.89	0.396548	0.198274	+	0.980147i	$0.436466\pi$
$542$	0	0
$543$	−15587.6	−1.23191
$544$	0	0
$545$	−2889.11	−0.227075
$546$	0	0
$547$	−21396.6	−1.67249	−0.836246	−	0.548355i	$-0.815254\pi$
$547$	−21396.6	−1.67249	−0.836246	+	0.548355i	$0.815254\pi$
$548$	0	0
$549$	−1722.07	−0.133873
$550$	0	0
$551$	−2015.07	−0.155798
$552$	0	0
$553$	−23910.6	−1.83866
$554$	0	0
$555$	−4314.64	−0.329993
$556$	0	0
$557$	21947.8	1.66958	0.834790	−	0.550568i	$-0.185589\pi$
$557$	21947.8	1.66958	0.834790	+	0.550568i	$0.185589\pi$
$558$	0	0
$559$	7620.66	0.576601
$560$	0	0
$561$	−34719.1	−2.61291
$562$	0	0
$563$	23157.0	1.73349	0.866743	−	0.498755i	$-0.166209\pi$
$563$	23157.0	1.73349	0.866743	+	0.498755i	$0.166209\pi$
$564$	0	0
$565$	−6017.61	−0.448076
$566$	0	0
$567$	18446.2	1.36625
$568$	0	0
$569$	−520.110	−0.0383201	−0.0191601	−	0.999816i	$-0.506099\pi$
$569$	−520.110	−0.0383201	−0.0191601	+	0.999816i	$0.506099\pi$
$570$	0	0
$571$	15243.1	1.11717	0.558587	−	0.829446i	$-0.311344\pi$
$571$	15243.1	1.11717	0.558587	+	0.829446i	$0.311344\pi$
$572$	0	0
$573$	1371.39	0.0999834
$574$	0	0
$575$	3112.75	0.225758
$576$	0	0
$577$	4498.02	0.324532	0.162266	−	0.986747i	$-0.448120\pi$
$577$	4498.02	0.324532	0.162266	+	0.986747i	$0.448120\pi$
$578$	0	0
$579$	29266.4	2.10064
$580$	0	0
$581$	−19505.1	−1.39278
$582$	0	0
$583$	29395.6	2.08824
$584$	0	0
$585$	−1561.40	−0.110352
$586$	0	0
$587$	11145.6	0.783691	0.391846	−	0.920031i	$-0.371837\pi$
$587$	11145.6	0.783691	0.391846	+	0.920031i	$0.371837\pi$
$588$	0	0
$589$	−10779.6	−0.754099
$590$	0	0
$591$	6798.91	0.473214
$592$	0	0
$593$	14167.8	0.981117	0.490558	−	0.871408i	$-0.336793\pi$
$593$	14167.8	0.981117	0.490558	+	0.871408i	$0.336793\pi$
$594$	0	0
$595$	11031.3	0.760066
$596$	0	0
$597$	8113.37	0.556211
$598$	0	0
$599$	−26385.6	−1.79981	−0.899906	−	0.436083i	$-0.856365\pi$
$599$	−26385.6	−1.79981	−0.899906	+	0.436083i	$0.856365\pi$
$600$	0	0
$601$	5090.67	0.345512	0.172756	−	0.984965i	$-0.444733\pi$
$601$	5090.67	0.345512	0.172756	+	0.984965i	$0.444733\pi$
$602$	0	0
$603$	−535.819	−0.0361861
$604$	0	0
$605$	12689.2	0.852710
$606$	0	0
$607$	15255.1	1.02008	0.510039	−	0.860151i	$-0.329631\pi$
$607$	15255.1	1.02008	0.510039	+	0.860151i	$0.329631\pi$
$608$	0	0
$609$	3696.57	0.245965
$610$	0	0
$611$	−6110.02	−0.404558
$612$	0	0
$613$	−3159.88	−0.208200	−0.104100	−	0.994567i	$-0.533196\pi$
$613$	−3159.88	−0.208200	−0.104100	+	0.994567i	$0.533196\pi$
$614$	0	0
$615$	−11970.1	−0.784845
$616$	0	0
$617$	−23789.5	−1.55223	−0.776117	−	0.630588i	$-0.782814\pi$
$617$	−23789.5	−1.55223	−0.776117	+	0.630588i	$0.782814\pi$
$618$	0	0
$619$	−4068.12	−0.264155	−0.132077	−	0.991239i	$-0.542165\pi$
$619$	−4068.12	−0.264155	−0.132077	+	0.991239i	$0.542165\pi$
$620$	0	0
$621$	3106.99	0.200772
$622$	0	0
$623$	−14382.1	−0.924893
$624$	0	0
$625$	3999.73	0.255983
$626$	0	0
$627$	25876.0	1.64814
$628$	0	0
$629$	10949.4	0.694090
$630$	0	0
$631$	−12496.5	−0.788396	−0.394198	−	0.919026i	$-0.628978\pi$
$631$	−12496.5	−0.788396	−0.394198	+	0.919026i	$0.628978\pi$
$632$	0	0
$633$	−28895.1	−1.81434
$634$	0	0
$635$	−16018.8	−1.00108
$636$	0	0
$637$	−1435.56	−0.0892919
$638$	0	0
$639$	−3346.21	−0.207158
$640$	0	0
$641$	24314.1	1.49821	0.749104	−	0.662453i	$-0.230485\pi$
$641$	24314.1	1.49821	0.749104	+	0.662453i	$0.230485\pi$
$642$	0	0
$643$	−14538.7	−0.891683	−0.445841	−	0.895112i	$-0.647095\pi$
$643$	−14538.7	−0.891683	−0.445841	+	0.895112i	$0.647095\pi$
$644$	0	0
$645$	−13170.4	−0.804005
$646$	0	0
$647$	−11215.5	−0.681492	−0.340746	−	0.940155i	$-0.610680\pi$
$647$	−11215.5	−0.681492	−0.340746	+	0.940155i	$0.610680\pi$
$648$	0	0
$649$	−28105.3	−1.69989
$650$	0	0
$651$	19774.7	1.19053
$652$	0	0
$653$	6036.71	0.361768	0.180884	−	0.983504i	$-0.442104\pi$
$653$	6036.71	0.361768	0.180884	+	0.983504i	$0.442104\pi$
$654$	0	0
$655$	−6321.73	−0.377115
$656$	0	0
$657$	9257.37	0.549718
$658$	0	0
$659$	254.000	0.0150143	0.00750716	−	0.999972i	$-0.497610\pi$
$659$	254.000	0.0150143	0.00750716	+	0.999972i	$0.497610\pi$
$660$	0	0
$661$	−18788.8	−1.10560	−0.552798	−	0.833315i	$-0.686440\pi$
$661$	−18788.8	−1.10560	−0.552798	+	0.833315i	$0.686440\pi$
$662$	0	0
$663$	12468.2	0.730352
$664$	0	0
$665$	−8221.56	−0.479426
$666$	0	0
$667$	993.089	0.0576500
$668$	0	0
$669$	−26117.4	−1.50935
$670$	0	0
$671$	−8104.27	−0.466262
$672$	0	0
$673$	−11489.8	−0.658098	−0.329049	−	0.944313i	$-0.606728\pi$
$673$	−11489.8	−0.658098	−0.329049	+	0.944313i	$0.606728\pi$
$674$	0	0
$675$	8247.17	0.470272
$676$	0	0
$677$	10076.7	0.572050	0.286025	−	0.958222i	$-0.407666\pi$
$677$	10076.7	0.572050	0.286025	+	0.958222i	$0.407666\pi$
$678$	0	0
$679$	8372.69	0.473217
$680$	0	0
$681$	−14657.4	−0.824777
$682$	0	0
$683$	−4084.29	−0.228816	−0.114408	−	0.993434i	$-0.536497\pi$
$683$	−4084.29	−0.228816	−0.114408	+	0.993434i	$0.536497\pi$
$684$	0	0
$685$	−16503.9	−0.920558
$686$	0	0
$687$	13691.2	0.760338
$688$	0	0
$689$	−10556.4	−0.583698
$690$	0	0
$691$	13053.9	0.718659	0.359330	−	0.933211i	$-0.383005\pi$
$691$	13053.9	0.718659	0.359330	+	0.933211i	$0.383005\pi$
$692$	0	0
$693$	−15085.7	−0.826925
$694$	0	0
$695$	6725.61	0.367075
$696$	0	0
$697$	30376.9	1.65080
$698$	0	0
$699$	−2431.31	−0.131560
$700$	0	0
$701$	33902.8	1.82666	0.913331	−	0.407217i	$-0.133501\pi$
$701$	33902.8	1.82666	0.913331	+	0.407217i	$0.133501\pi$
$702$	0	0
$703$	−8160.55	−0.437811
$704$	0	0
$705$	10559.6	0.564111
$706$	0	0
$707$	4970.01	0.264380
$708$	0	0
$709$	33368.4	1.76753	0.883765	−	0.467932i	$-0.155001\pi$
$709$	33368.4	1.76753	0.883765	+	0.467932i	$0.155001\pi$
$710$	0	0
$711$	14843.3	0.782937
$712$	0	0
$713$	5312.51	0.279039
$714$	0	0
$715$	−7348.15	−0.384343
$716$	0	0
$717$	−17296.9	−0.900925
$718$	0	0
$719$	3046.49	0.158018	0.0790089	−	0.996874i	$-0.474824\pi$
$719$	3046.49	0.158018	0.0790089	+	0.996874i	$0.474824\pi$
$720$	0	0
$721$	21995.5	1.13614
$722$	0	0
$723$	21036.7	1.08211
$724$	0	0
$725$	2636.05	0.135035
$726$	0	0
$727$	−12440.5	−0.634653	−0.317326	−	0.948316i	$-0.602785\pi$
$727$	−12440.5	−0.634653	−0.317326	+	0.948316i	$0.602785\pi$
$728$	0	0
$729$	3960.28	0.201203
$730$	0	0
$731$	33423.0	1.69110
$732$	0	0
$733$	−22658.6	−1.14177	−0.570884	−	0.821031i	$-0.693400\pi$
$733$	−22658.6	−1.14177	−0.570884	+	0.821031i	$0.693400\pi$
$734$	0	0
$735$	2481.00	0.124507
$736$	0	0
$737$	−2521.63	−0.126032
$738$	0	0
$739$	−34256.2	−1.70519	−0.852595	−	0.522573i	$-0.824972\pi$
$739$	−34256.2	−1.70519	−0.852595	+	0.522573i	$0.824972\pi$
$740$	0	0
$741$	−9292.45	−0.460684
$742$	0	0
$743$	−9839.65	−0.485843	−0.242922	−	0.970046i	$-0.578106\pi$
$743$	−9839.65	−0.485843	−0.242922	+	0.970046i	$0.578106\pi$
$744$	0	0
$745$	−7697.08	−0.378522
$746$	0	0
$747$	12108.5	0.593073
$748$	0	0
$749$	15706.0	0.766202
$750$	0	0
$751$	11982.5	0.582222	0.291111	−	0.956689i	$-0.405975\pi$
$751$	11982.5	0.582222	0.291111	+	0.956689i	$0.405975\pi$
$752$	0	0
$753$	40317.8	1.95121
$754$	0	0
$755$	−17256.3	−0.831816
$756$	0	0
$757$	2208.86	0.106053	0.0530267	−	0.998593i	$-0.483113\pi$
$757$	2208.86	0.106053	0.0530267	+	0.998593i	$0.483113\pi$
$758$	0	0
$759$	−12752.5	−0.609863
$760$	0	0
$761$	−25957.1	−1.23646	−0.618230	−	0.785997i	$-0.712150\pi$
$761$	−25957.1	−1.23646	−0.618230	+	0.785997i	$0.712150\pi$
$762$	0	0
$763$	10024.2	0.475621
$764$	0	0
$765$	−6848.07	−0.323650
$766$	0	0
$767$	10093.0	0.475148
$768$	0	0
$769$	7432.15	0.348518	0.174259	−	0.984700i	$-0.444247\pi$
$769$	7432.15	0.348518	0.174259	+	0.984700i	$0.444247\pi$
$770$	0	0
$771$	−2506.26	−0.117070
$772$	0	0
$773$	−16360.6	−0.761254	−0.380627	−	0.924729i	$-0.624292\pi$
$773$	−16360.6	−0.761254	−0.380627	+	0.924729i	$0.624292\pi$
$774$	0	0
$775$	14101.5	0.653600
$776$	0	0
$777$	14970.2	0.691190
$778$	0	0
$779$	−22639.7	−1.04127
$780$	0	0
$781$	−15747.7	−0.721505
$782$	0	0
$783$	2631.17	0.120090
$784$	0	0
$785$	15622.2	0.710294
$786$	0	0
$787$	6195.13	0.280601	0.140300	−	0.990109i	$-0.455193\pi$
$787$	6195.13	0.280601	0.140300	+	0.990109i	$0.455193\pi$
$788$	0	0
$789$	1674.45	0.0755540
$790$	0	0
$791$	20878.9	0.938520
$792$	0	0
$793$	2910.37	0.130328
$794$	0	0
$795$	18244.1	0.813901
$796$	0	0
$797$	−692.544	−0.0307794	−0.0153897	−	0.999882i	$-0.504899\pi$
$797$	−692.544	−0.0307794	−0.0153897	+	0.999882i	$0.504899\pi$
$798$	0	0
$799$	−26797.6	−1.18652
$800$	0	0
$801$	8928.22	0.393837
$802$	0	0
$803$	43566.3	1.91460
$804$	0	0
$805$	4051.85	0.177402
$806$	0	0
$807$	−3068.53	−0.133851
$808$	0	0
$809$	−16062.0	−0.698036	−0.349018	−	0.937116i	$-0.613485\pi$
$809$	−16062.0	−0.698036	−0.349018	+	0.937116i	$0.613485\pi$
$810$	0	0
$811$	−21373.9	−0.925447	−0.462723	−	0.886503i	$-0.653128\pi$
$811$	−21373.9	−0.925447	−0.462723	+	0.886503i	$0.653128\pi$
$812$	0	0
$813$	48374.2	2.08679
$814$	0	0
$815$	21480.4	0.923220
$816$	0	0
$817$	−24910.0	−1.06669
$818$	0	0
$819$	5417.51	0.231139
$820$	0	0
$821$	22311.7	0.948457	0.474229	−	0.880402i	$-0.342727\pi$
$821$	22311.7	0.948457	0.474229	+	0.880402i	$0.342727\pi$
$822$	0	0
$823$	11459.0	0.485341	0.242670	−	0.970109i	$-0.421977\pi$
$823$	11459.0	0.485341	0.242670	+	0.970109i	$0.421977\pi$
$824$	0	0
$825$	−33850.1	−1.42850
$826$	0	0
$827$	11856.5	0.498540	0.249270	−	0.968434i	$-0.419809\pi$
$827$	11856.5	0.498540	0.249270	+	0.968434i	$0.419809\pi$
$828$	0	0
$829$	−18446.9	−0.772844	−0.386422	−	0.922322i	$-0.626289\pi$
$829$	−18446.9	−0.772844	−0.386422	+	0.922322i	$0.626289\pi$
$830$	0	0
$831$	35546.3	1.48386
$832$	0	0
$833$	−6296.13	−0.261882
$834$	0	0
$835$	−9489.86	−0.393306
$836$	0	0
$837$	14075.4	0.581261
$838$	0	0
$839$	−14510.3	−0.597080	−0.298540	−	0.954397i	$-0.596500\pi$
$839$	−14510.3	−0.597080	−0.298540	+	0.954397i	$0.596500\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−34223.7	−1.39825
$844$	0	0
$845$	−10190.9	−0.414887
$846$	0	0
$847$	−44026.9	−1.78605
$848$	0	0
$849$	−44460.1	−1.79725
$850$	0	0
$851$	4021.78	0.162003
$852$	0	0
$853$	−16175.5	−0.649284	−0.324642	−	0.945837i	$-0.605244\pi$
$853$	−16175.5	−0.649284	−0.324642	+	0.945837i	$0.605244\pi$
$854$	0	0
$855$	5103.82	0.204149
$856$	0	0
$857$	18482.5	0.736696	0.368348	−	0.929688i	$-0.379923\pi$
$857$	18482.5	0.736696	0.368348	+	0.929688i	$0.379923\pi$
$858$	0	0
$859$	−37632.8	−1.49478	−0.747389	−	0.664387i	$-0.768693\pi$
$859$	−37632.8	−1.49478	−0.747389	+	0.664387i	$0.768693\pi$
$860$	0	0
$861$	41531.8	1.64390
$862$	0	0
$863$	6431.80	0.253698	0.126849	−	0.991922i	$-0.459514\pi$
$863$	6431.80	0.253698	0.126849	+	0.991922i	$0.459514\pi$
$864$	0	0
$865$	13584.1	0.533956
$866$	0	0
$867$	23775.1	0.931310
$868$	0	0
$869$	69854.4	2.72687
$870$	0	0
$871$	905.554	0.0352279
$872$	0	0
$873$	−5197.64	−0.201505
$874$	0	0
$875$	25545.3	0.986960
$876$	0	0
$877$	41357.6	1.59241	0.796206	−	0.605025i	$-0.206837\pi$
$877$	41357.6	1.59241	0.796206	+	0.605025i	$0.206837\pi$
$878$	0	0
$879$	29718.1	1.14035
$880$	0	0
$881$	43426.8	1.66071	0.830355	−	0.557234i	$-0.188137\pi$
$881$	43426.8	1.66071	0.830355	+	0.557234i	$0.188137\pi$
$882$	0	0
$883$	−21255.0	−0.810066	−0.405033	−	0.914302i	$-0.632740\pi$
$883$	−21255.0	−0.810066	−0.405033	+	0.914302i	$0.632740\pi$
$884$	0	0
$885$	−17443.2	−0.662540
$886$	0	0
$887$	29267.1	1.10789	0.553943	−	0.832555i	$-0.313123\pi$
$887$	29267.1	1.10789	0.553943	+	0.832555i	$0.313123\pi$
$888$	0	0
$889$	55579.5	2.09682
$890$	0	0
$891$	−53890.2	−2.02625
$892$	0	0
$893$	19972.1	0.748421
$894$	0	0
$895$	−11719.7	−0.437706
$896$	0	0
$897$	4579.61	0.170467
$898$	0	0
$899$	4498.91	0.166904
$900$	0	0
$901$	−46298.7	−1.71191
$902$	0	0
$903$	45696.4	1.68403
$904$	0	0
$905$	−14469.1	−0.531458
$906$	0	0
$907$	19327.7	0.707571	0.353785	−	0.935327i	$-0.384894\pi$
$907$	19327.7	0.707571	0.353785	+	0.935327i	$0.384894\pi$
$908$	0	0
$909$	−3085.31	−0.112578
$910$	0	0
$911$	24325.2	0.884665	0.442333	−	0.896851i	$-0.354151\pi$
$911$	24325.2	0.884665	0.442333	+	0.896851i	$0.354151\pi$
$912$	0	0
$913$	56983.9	2.06560
$914$	0	0
$915$	−5029.83	−0.181728
$916$	0	0
$917$	21934.1	0.789889
$918$	0	0
$919$	8954.43	0.321414	0.160707	−	0.987002i	$-0.448623\pi$
$919$	8954.43	0.321414	0.160707	+	0.987002i	$0.448623\pi$
$920$	0	0
$921$	−7774.22	−0.278143
$922$	0	0
$923$	5655.23	0.201673
$924$	0	0
$925$	10675.4	0.379464
$926$	0	0
$927$	−13654.5	−0.483788
$928$	0	0
$929$	−5986.57	−0.211424	−0.105712	−	0.994397i	$-0.533712\pi$
$929$	−5986.57	−0.211424	−0.105712	+	0.994397i	$0.533712\pi$
$930$	0	0
$931$	4692.47	0.165187
$932$	0	0
$933$	−62182.5	−2.18195
$934$	0	0
$935$	−32227.8	−1.12723
$936$	0	0
$937$	47478.9	1.65535	0.827677	−	0.561205i	$-0.189662\pi$
$937$	47478.9	1.65535	0.827677	+	0.561205i	$0.189662\pi$
$938$	0	0
$939$	56582.2	1.96644
$940$	0	0
$941$	−3.21123	−0.000111246 0	−5.56232e−5	−	1.00000i	$-0.500018\pi$
$941$	−3.21123	−0.000111246 0	−5.56232e−5		1.00000i	$0.500018\pi$
$942$	0	0
$943$	11157.6	0.385303
$944$	0	0
$945$	10735.3	0.369543
$946$	0	0
$947$	27172.0	0.932389	0.466195	−	0.884682i	$-0.345625\pi$
$947$	27172.0	0.932389	0.466195	+	0.884682i	$0.345625\pi$
$948$	0	0
$949$	−15645.3	−0.535162
$950$	0	0
$951$	−31130.6	−1.06149
$952$	0	0
$953$	21321.3	0.724726	0.362363	−	0.932037i	$-0.381970\pi$
$953$	21321.3	0.724726	0.362363	+	0.932037i	$0.381970\pi$
$954$	0	0
$955$	1272.98	0.0431337
$956$	0	0
$957$	−10799.5	−0.364783
$958$	0	0
$959$	57262.6	1.92816
$960$	0	0
$961$	−5724.15	−0.192144
$962$	0	0
$963$	−9750.06	−0.326263
$964$	0	0
$965$	27166.3	0.906233
$966$	0	0
$967$	1250.96	0.0416009	0.0208005	−	0.999784i	$-0.493379\pi$
$967$	1250.96	0.0416009	0.0208005	+	0.999784i	$0.493379\pi$
$968$	0	0
$969$	−40755.2	−1.35113
$970$	0	0
$971$	−25.7091	−0.000849685 0	−0.000424843	−	1.00000i	$-0.500135\pi$
$971$	−25.7091	−0.000849685 0	−0.000424843		1.00000i	$0.500135\pi$
$972$	0	0
$973$	−23335.4	−0.768859
$974$	0	0
$975$	12156.1	0.399288
$976$	0	0
$977$	8959.93	0.293402	0.146701	−	0.989181i	$-0.453135\pi$
$977$	8959.93	0.293402	0.146701	+	0.989181i	$0.453135\pi$
$978$	0	0
$979$	42017.2	1.37168
$980$	0	0
$981$	−6222.84	−0.202528
$982$	0	0
$983$	20484.4	0.664650	0.332325	−	0.943165i	$-0.392167\pi$
$983$	20484.4	0.664650	0.332325	+	0.943165i	$0.392167\pi$
$984$	0	0
$985$	6311.04	0.204149
$986$	0	0
$987$	−36638.1	−1.18156
$988$	0	0
$989$	12276.4	0.394709
$990$	0	0
$991$	−1572.84	−0.0504167	−0.0252083	−	0.999682i	$-0.508025\pi$
$991$	−1572.84	−0.0504167	−0.0252083	+	0.999682i	$0.508025\pi$
$992$	0	0
$993$	−35615.7	−1.13820
$994$	0	0
$995$	7531.18	0.239954
$996$	0	0
$997$	596.888	0.0189605	0.00948026	−	0.999955i	$-0.496982\pi$
$997$	596.888	0.0189605	0.00948026	+	0.999955i	$0.496982\pi$
$998$	0	0
$999$	10655.6	0.337466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.o.1.3		3
4.3	odd	2		1856.4.a.v.1.1		3
8.3	odd	2		116.4.a.c.1.3	✓	3
8.5	even	2		464.4.a.j.1.1		3
24.11	even	2		1044.4.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.4.a.c.1.3	✓	3	8.3	odd	2
464.4.a.j.1.1		3	8.5	even	2
1044.4.a.f.1.2		3	24.11	even	2
1856.4.a.o.1.3		3	1.1	even	1	trivial
1856.4.a.v.1.1		3	4.3	odd	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+6.29111 q^{3} +5.83968 q^{5} -20.2616 q^{7} +12.5781 q^{9} +O(q^{10})\)	\(q+6.29111 q^{3} +5.83968 q^{5} -20.2616 q^{7} +12.5781 q^{9} +59.1940 q^{11} -21.2575 q^{13} +36.7381 q^{15} -93.2318 q^{17} +69.4851 q^{19} -127.468 q^{21} -34.2444 q^{23} -90.8981 q^{25} -90.7298 q^{27} -29.0000 q^{29} -155.135 q^{31} +372.396 q^{33} -118.321 q^{35} -117.443 q^{37} -133.733 q^{39} -325.822 q^{41} -358.494 q^{43} +73.4521 q^{45} +287.430 q^{47} +67.5320 q^{49} -586.531 q^{51} +496.598 q^{53} +345.674 q^{55} +437.138 q^{57} -474.799 q^{59} -136.910 q^{61} -254.852 q^{63} -124.137 q^{65} -42.5994 q^{67} -215.436 q^{69} -266.035 q^{71} +735.992 q^{73} -571.850 q^{75} -1199.36 q^{77} +1180.09 q^{79} -910.400 q^{81} +962.664 q^{83} -544.444 q^{85} -182.442 q^{87} +709.823 q^{89} +430.710 q^{91} -975.971 q^{93} +405.771 q^{95} -413.230 q^{97} +744.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9}+O(q^{10}) \) 3 * q - 10 * q^3 + 20 * q^5 - 8 * q^7 + 93 * q^9	\( 3 q - 10 q^{3} + 20 q^{5} - 8 q^{7} + 93 q^{9} + 86 q^{11} - 124 q^{13} - 54 q^{15} + 14 q^{17} + 88 q^{19} - 280 q^{21} - 68 q^{23} + 111 q^{25} - 334 q^{27} - 87 q^{29} - 326 q^{31} + 110 q^{33} - 784 q^{35} - 166 q^{37} + 682 q^{39} + 34 q^{41} - 946 q^{43} + 242 q^{45} - 234 q^{47} + 1067 q^{49} - 1428 q^{51} + 1144 q^{53} - 94 q^{55} + 244 q^{57} + 488 q^{59} - 450 q^{61} + 1096 q^{63} - 1154 q^{65} - 52 q^{67} + 404 q^{69} - 1196 q^{71} + 2434 q^{73} - 1868 q^{75} + 312 q^{77} - 742 q^{79} - 849 q^{81} + 464 q^{83} + 672 q^{85} + 290 q^{87} + 1986 q^{89} + 448 q^{91} + 358 q^{93} - 68 q^{95} - 406 q^{97} + 2540 q^{99}+O(q^{100}) \) 3 * q - 10 * q^3 + 20 * q^5 - 8 * q^7 + 93 * q^9 + 86 * q^11 - 124 * q^13 - 54 * q^15 + 14 * q^17 + 88 * q^19 - 280 * q^21 - 68 * q^23 + 111 * q^25 - 334 * q^27 - 87 * q^29 - 326 * q^31 + 110 * q^33 - 784 * q^35 - 166 * q^37 + 682 * q^39 + 34 * q^41 - 946 * q^43 + 242 * q^45 - 234 * q^47 + 1067 * q^49 - 1428 * q^51 + 1144 * q^53 - 94 * q^55 + 244 * q^57 + 488 * q^59 - 450 * q^61 + 1096 * q^63 - 1154 * q^65 - 52 * q^67 + 404 * q^69 - 1196 * q^71 + 2434 * q^73 - 1868 * q^75 + 312 * q^77 - 742 * q^79 - 849 * q^81 + 464 * q^83 + 672 * q^85 + 290 * q^87 + 1986 * q^89 + 448 * q^91 + 358 * q^93 - 68 * q^95 - 406 * q^97 + 2540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more