LMFDB - Embedded newform 729.6.a.e.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,6,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	729.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:	$116.919804644$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.88095 q^{2} -23.7001 q^{4} -16.0197 q^{5} +243.284 q^{7} -160.469 q^{8} +O(q^{10})$	\(q+2.88095 q^{2} -23.7001 q^{4} -16.0197 q^{5} +243.284 q^{7} -160.469 q^{8} -46.1520 q^{10} -571.492 q^{11} -491.745 q^{13} +700.890 q^{14} +296.099 q^{16} +870.541 q^{17} +67.4900 q^{19} +379.669 q^{20} -1646.44 q^{22} +4374.47 q^{23} -2868.37 q^{25} -1416.69 q^{26} -5765.86 q^{28} -4119.77 q^{29} -1719.15 q^{31} +5988.07 q^{32} +2507.99 q^{34} -3897.34 q^{35} -6229.10 q^{37} +194.435 q^{38} +2570.67 q^{40} +7963.69 q^{41} -13898.1 q^{43} +13544.4 q^{44} +12602.6 q^{46} +909.526 q^{47} +42380.2 q^{49} -8263.63 q^{50} +11654.4 q^{52} +2259.64 q^{53} +9155.14 q^{55} -39039.7 q^{56} -11868.9 q^{58} +17004.7 q^{59} -10404.9 q^{61} -4952.79 q^{62} +7776.16 q^{64} +7877.61 q^{65} +2493.04 q^{67} -20631.9 q^{68} -11228.1 q^{70} +24371.2 q^{71} +26187.6 q^{73} -17945.7 q^{74} -1599.52 q^{76} -139035. q^{77} +32925.2 q^{79} -4743.43 q^{80} +22943.0 q^{82} -28173.2 q^{83} -13945.8 q^{85} -40039.7 q^{86} +91707.0 q^{88} +10984.5 q^{89} -119634. q^{91} -103675. q^{92} +2620.30 q^{94} -1081.17 q^{95} -114098. q^{97} +122095. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8}+O(q^{10}) $ 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8	$ 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28} + 13008 q^{29} + 24273 q^{32} + 35868 q^{35} + 3 q^{37} + 51801 q^{38} + 96 q^{40} + 55833 q^{41} + 110757 q^{44} + 3 q^{46} + 90129 q^{47} + 57624 q^{49} + 145362 q^{50} + 3072 q^{52} + 103203 q^{53} - 6 q^{55} + 227154 q^{56} - 192 q^{58} + 176856 q^{59} - 31851 q^{61} + 246066 q^{62} + 86019 q^{64} + 167160 q^{65} - 801 q^{67} + 374589 q^{68} + 9375 q^{70} + 279531 q^{71} + 27012 q^{73} + 413970 q^{74} + 96 q^{76} + 185190 q^{77} + 462057 q^{80} - 6 q^{82} + 295536 q^{83} + 319803 q^{86} + 3072 q^{88} + 154827 q^{89} + 91002 q^{91} + 330558 q^{92} + 96 q^{94} + 353244 q^{95} + 463410 q^{98}+O(q^{100}) $ 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * q^98

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 $

$2$	2.88095	0.509285	0.254643	−	0.967035i	$-0.418042\pi$
$2$	2.88095	0.509285	0.254643	+	0.967035i	$0.418042\pi$
$3$	0	0
$3$	0	0
$4$	−23.7001	−0.740629
$5$	−16.0197	−0.286569	−0.143285	−	0.989682i	$-0.545766\pi$
$5$	−16.0197	−0.286569	−0.143285	+	0.989682i	$0.545766\pi$
$6$	0	0
$7$	243.284	1.87659	0.938294	−	0.345839i	$-0.112406\pi$
$7$	243.284	1.87659	0.938294	+	0.345839i	$0.112406\pi$
$8$	−160.469	−0.886476
$9$	0	0
$10$	−46.1520	−0.145946
$11$	−571.492	−1.42406	−0.712031	−	0.702149i	$-0.752224\pi$
$11$	−571.492	−1.42406	−0.712031	+	0.702149i	$0.752224\pi$
$12$	0	0
$13$	−491.745	−0.807015	−0.403507	−	0.914976i	$-0.632209\pi$
$13$	−491.745	−0.807015	−0.403507	+	0.914976i	$0.632209\pi$
$14$	700.890	0.955718
$15$	0	0
$16$	296.099	0.289159
$17$	870.541	0.730578	0.365289	−	0.930894i	$-0.380970\pi$
$17$	870.541	0.730578	0.365289	+	0.930894i	$0.380970\pi$
$18$	0	0
$19$	67.4900	0.0428899	0.0214450	−	0.999770i	$-0.493173\pi$
$19$	67.4900	0.0428899	0.0214450	+	0.999770i	$0.493173\pi$
$20$	379.669	0.212242
$21$	0	0
$22$	−1646.44	−0.725253
$23$	4374.47	1.72427	0.862136	−	0.506678i	$-0.169127\pi$
$23$	4374.47	1.72427	0.862136	+	0.506678i	$0.169127\pi$
$24$	0	0
$25$	−2868.37	−0.917878
$26$	−1416.69	−0.411001
$27$	0	0
$28$	−5765.86	−1.38985
$29$	−4119.77	−0.909658	−0.454829	−	0.890579i	$-0.650300\pi$
$29$	−4119.77	−0.909658	−0.454829	+	0.890579i	$0.650300\pi$
$30$	0	0
$31$	−1719.15	−0.321299	−0.160650	−	0.987012i	$-0.551359\pi$
$31$	−1719.15	−0.321299	−0.160650	+	0.987012i	$0.551359\pi$
$32$	5988.07	1.03374
$33$	0	0
$34$	2507.99	0.372073
$35$	−3897.34	−0.537773
$36$	0	0
$37$	−6229.10	−0.748034	−0.374017	−	0.927422i	$-0.622020\pi$
$37$	−6229.10	−0.748034	−0.374017	+	0.927422i	$0.622020\pi$
$38$	194.435	0.0218432
$39$	0	0
$40$	2570.67	0.254037
$41$	7963.69	0.739868	0.369934	−	0.929058i	$-0.379380\pi$
$41$	7963.69	0.739868	0.369934	+	0.929058i	$0.379380\pi$
$42$	0	0
$43$	−13898.1	−1.14626	−0.573131	−	0.819464i	$-0.694271\pi$
$43$	−13898.1	−1.14626	−0.573131	+	0.819464i	$0.694271\pi$
$44$	13544.4	1.05470
$45$	0	0
$46$	12602.6	0.878146
$47$	909.526	0.0600579	0.0300290	−	0.999549i	$-0.490440\pi$
$47$	909.526	0.0600579	0.0300290	+	0.999549i	$0.490440\pi$
$48$	0	0
$49$	42380.2	2.52158
$50$	−8263.63	−0.467462
$51$	0	0
$52$	11654.4	0.597698
$53$	2259.64	0.110497	0.0552484	−	0.998473i	$-0.482405\pi$
$53$	2259.64	0.110497	0.0552484	+	0.998473i	$0.482405\pi$
$54$	0	0
$55$	9155.14	0.408092
$56$	−39039.7	−1.66355
$57$	0	0
$58$	−11868.9	−0.463275
$59$	17004.7	0.635974	0.317987	−	0.948095i	$-0.396993\pi$
$59$	17004.7	0.635974	0.317987	+	0.948095i	$0.396993\pi$
$60$	0	0
$61$	−10404.9	−0.358026	−0.179013	−	0.983847i	$-0.557290\pi$
$61$	−10404.9	−0.358026	−0.179013	+	0.983847i	$0.557290\pi$
$62$	−4952.79	−0.163633
$63$	0	0
$64$	7776.16	0.237310
$65$	7877.61	0.231266
$66$	0	0
$67$	2493.04	0.0678489	0.0339245	−	0.999424i	$-0.489199\pi$
$67$	2493.04	0.0678489	0.0339245	+	0.999424i	$0.489199\pi$
$68$	−20631.9	−0.541087
$69$	0	0
$70$	−11228.1	−0.273880
$71$	24371.2	0.573762	0.286881	−	0.957966i	$-0.407382\pi$
$71$	24371.2	0.573762	0.286881	+	0.957966i	$0.407382\pi$
$72$	0	0
$73$	26187.6	0.575159	0.287580	−	0.957757i	$-0.407149\pi$
$73$	26187.6	0.575159	0.287580	+	0.957757i	$0.407149\pi$
$74$	−17945.7	−0.380962
$75$	0	0
$76$	−1599.52	−0.0317655
$77$	−139035.	−2.67238
$78$	0	0
$79$	32925.2	0.593554	0.296777	−	0.954947i	$-0.404088\pi$
$79$	32925.2	0.593554	0.296777	+	0.954947i	$0.404088\pi$
$80$	−4743.43	−0.0828642
$81$	0	0
$82$	22943.0	0.376804
$83$	−28173.2	−0.448891	−0.224445	−	0.974487i	$-0.572057\pi$
$83$	−28173.2	−0.448891	−0.224445	+	0.974487i	$0.572057\pi$
$84$	0	0
$85$	−13945.8	−0.209361
$86$	−40039.7	−0.583774
$87$	0	0
$88$	91707.0	1.26240
$89$	10984.5	0.146996	0.0734982	−	0.997295i	$-0.476584\pi$
$89$	10984.5	0.146996	0.0734982	+	0.997295i	$0.476584\pi$
$90$	0	0
$91$	−119634.	−1.51443
$92$	−103675.	−1.27704
$93$	0	0
$94$	2620.30	0.0305866
$95$	−1081.17	−0.0122909
$96$	0	0
$97$	−114098.	−1.23126	−0.615630	−	0.788035i	$-0.711098\pi$
$97$	−114098.	−1.23126	−0.615630	+	0.788035i	$0.711098\pi$
$98$	122095.	1.28420
$99$	0	0
$100$	67980.7	0.679807
$101$	201213.	1.96269	0.981345	−	0.192253i	$-0.0615793\pi$
$101$	201213.	1.96269	0.981345	+	0.192253i	$0.0615793\pi$
$102$	0	0
$103$	69627.0	0.646672	0.323336	−	0.946284i	$-0.395196\pi$
$103$	69627.0	0.646672	0.323336	+	0.946284i	$0.395196\pi$
$104$	78910.0	0.715399
$105$	0	0
$106$	6509.92	0.0562744
$107$	3025.81	0.0255495	0.0127748	−	0.999918i	$-0.495934\pi$
$107$	3025.81	0.0255495	0.0127748	+	0.999918i	$0.495934\pi$
$108$	0	0
$109$	−74086.5	−0.597273	−0.298637	−	0.954367i	$-0.596532\pi$
$109$	−74086.5	−0.597273	−0.298637	+	0.954367i	$0.596532\pi$
$110$	26375.5	0.207835
$111$	0	0
$112$	72036.3	0.542633
$113$	142339.	1.04864	0.524321	−	0.851521i	$-0.324319\pi$
$113$	142339.	1.04864	0.524321	+	0.851521i	$0.324319\pi$
$114$	0	0
$115$	−70077.7	−0.494123
$116$	97639.1	0.673719
$117$	0	0
$118$	48989.8	0.323892
$119$	211789.	1.37099
$120$	0	0
$121$	165552.	1.02795
$122$	−29976.1	−0.182337
$123$	0	0
$124$	40744.0	0.237963
$125$	96012.1	0.549605
$126$	0	0
$127$	−14357.7	−0.0789909	−0.0394954	−	0.999220i	$-0.512575\pi$
$127$	−14357.7	−0.0789909	−0.0394954	+	0.999220i	$0.512575\pi$
$128$	−169215.	−0.912883
$129$	0	0
$130$	22695.0	0.117780
$131$	189280.	0.963667	0.481834	−	0.876263i	$-0.339971\pi$
$131$	189280.	0.963667	0.481834	+	0.876263i	$0.339971\pi$
$132$	0	0
$133$	16419.2	0.0804867
$134$	7182.34	0.0345544
$135$	0	0
$136$	−139695.	−0.647641
$137$	201213.	0.915914	0.457957	−	0.888974i	$-0.348581\pi$
$137$	201213.	0.915914	0.457957	+	0.888974i	$0.348581\pi$
$138$	0	0
$139$	162685.	0.714187	0.357093	−	0.934069i	$-0.383768\pi$
$139$	162685.	0.714187	0.357093	+	0.934069i	$0.383768\pi$
$140$	92367.5	0.398290
$141$	0	0
$142$	70212.3	0.292208
$143$	281028.	1.14924
$144$	0	0
$145$	65997.6	0.260680
$146$	75445.1	0.292920
$147$	0	0
$148$	147630.	0.554015
$149$	381580.	1.40805	0.704027	−	0.710173i	$-0.251383\pi$
$149$	381580.	1.40805	0.704027	+	0.710173i	$0.251383\pi$
$150$	0	0
$151$	411812.	1.46979	0.734897	−	0.678179i	$-0.237230\pi$
$151$	411812.	1.46979	0.734897	+	0.678179i	$0.237230\pi$
$152$	−10830.1	−0.0380209
$153$	0	0
$154$	−400553.	−1.36100
$155$	27540.3	0.0920745
$156$	0	0
$157$	281537.	0.911562	0.455781	−	0.890092i	$-0.349360\pi$
$157$	281537.	0.911562	0.455781	+	0.890092i	$0.349360\pi$
$158$	94855.8	0.302288
$159$	0	0
$160$	−95927.1	−0.296239
$161$	1.06424e6	3.23575
$162$	0	0
$163$	450153.	1.32706	0.663532	−	0.748148i	$-0.269057\pi$
$163$	450153.	1.32706	0.663532	+	0.748148i	$0.269057\pi$
$164$	−188740.	−0.547968
$165$	0	0
$166$	−81165.6	−0.228613
$167$	−456289.	−1.26605	−0.633023	−	0.774133i	$-0.718186\pi$
$167$	−456289.	−1.26605	−0.633023	+	0.774133i	$0.718186\pi$
$168$	0	0
$169$	−129480.	−0.348727
$170$	−40177.2	−0.106625
$171$	0	0
$172$	329386.	0.848955
$173$	−209439.	−0.532037	−0.266019	−	0.963968i	$-0.585708\pi$
$173$	−209439.	−0.532037	−0.266019	+	0.963968i	$0.585708\pi$
$174$	0	0
$175$	−697829.	−1.72248
$176$	−169218.	−0.411781
$177$	0	0
$178$	31645.9	0.0748631
$179$	37655.8	0.0878414	0.0439207	−	0.999035i	$-0.486015\pi$
$179$	37655.8	0.0878414	0.0439207	+	0.999035i	$0.486015\pi$
$180$	0	0
$181$	68951.2	0.156439	0.0782196	−	0.996936i	$-0.475076\pi$
$181$	68951.2	0.156439	0.0782196	+	0.996936i	$0.475076\pi$
$182$	−344659.	−0.771279
$183$	0	0
$184$	−701968.	−1.52853
$185$	99788.4	0.214364
$186$	0	0
$187$	−497507.	−1.04039
$188$	−21555.9	−0.0444806
$189$	0	0
$190$	−3114.80	−0.00625959
$191$	787299.	1.56155	0.780776	−	0.624811i	$-0.214824\pi$
$191$	787299.	1.56155	0.780776	+	0.624811i	$0.214824\pi$
$192$	0	0
$193$	379435.	0.733237	0.366618	−	0.930371i	$-0.380515\pi$
$193$	379435.	0.733237	0.366618	+	0.930371i	$0.380515\pi$
$194$	−328712.	−0.627063
$195$	0	0
$196$	−1.00442e6	−1.86756
$197$	791502.	1.45307	0.726535	−	0.687130i	$-0.241130\pi$
$197$	791502.	1.45307	0.726535	+	0.687130i	$0.241130\pi$
$198$	0	0
$199$	302422.	0.541353	0.270676	−	0.962670i	$-0.412753\pi$
$199$	302422.	0.541353	0.270676	+	0.962670i	$0.412753\pi$
$200$	460285.	0.813677
$201$	0	0
$202$	579684.	0.999569
$203$	−1.00228e6	−1.70705
$204$	0	0
$205$	−127576.	−0.212024
$206$	200592.	0.329341
$207$	0	0
$208$	−145605.	−0.233356
$209$	−38570.0	−0.0610779
$210$	0	0
$211$	−804886.	−1.24460	−0.622298	−	0.782781i	$-0.713801\pi$
$211$	−804886.	−1.24460	−0.622298	+	0.782781i	$0.713801\pi$
$212$	−53553.8	−0.0818372
$213$	0	0
$214$	8717.22	0.0130120
$215$	222643.	0.328484
$216$	0	0
$217$	−418242.	−0.602946
$218$	−213440.	−0.304182
$219$	0	0
$220$	−216978.	−0.302245
$221$	−428084.	−0.589588
$222$	0	0
$223$	1.31339e6	1.76861	0.884303	−	0.466913i	$-0.154634\pi$
$223$	1.31339e6	1.76861	0.884303	+	0.466913i	$0.154634\pi$
$224$	1.45680e6	1.93991
$225$	0	0
$226$	410071.	0.534058
$227$	−1.28035e6	−1.64917	−0.824586	−	0.565737i	$-0.808592\pi$
$227$	−1.28035e6	−1.64917	−0.824586	+	0.565737i	$0.808592\pi$
$228$	0	0
$229$	433685.	0.546494	0.273247	−	0.961944i	$-0.411902\pi$
$229$	433685.	0.546494	0.273247	+	0.961944i	$0.411902\pi$
$230$	−201891.	−0.251650
$231$	0	0
$232$	661097.	0.806390
$233$	1.29710e6	1.56525	0.782624	−	0.622495i	$-0.213881\pi$
$233$	1.29710e6	1.56525	0.782624	+	0.622495i	$0.213881\pi$
$234$	0	0
$235$	−14570.3	−0.0172108
$236$	−403014.	−0.471020
$237$	0	0
$238$	610154.	0.698227
$239$	−187774.	−0.212637	−0.106319	−	0.994332i	$-0.533906\pi$
$239$	−187774.	−0.212637	−0.106319	+	0.994332i	$0.533906\pi$
$240$	0	0
$241$	721347.	0.800021	0.400010	−	0.916511i	$-0.369006\pi$
$241$	721347.	0.800021	0.400010	+	0.916511i	$0.369006\pi$
$242$	476948.	0.523520
$243$	0	0
$244$	246598.	0.265165
$245$	−678919.	−0.722608
$246$	0	0
$247$	−33187.8	−0.0346128
$248$	275871.	0.284824
$249$	0	0
$250$	276606.	0.279906
$251$	−331259.	−0.331881	−0.165941	−	0.986136i	$-0.553066\pi$
$251$	−331259.	−0.331881	−0.165941	+	0.986136i	$0.553066\pi$
$252$	0	0
$253$	−2.49997e6	−2.45547
$254$	−41364.0	−0.0402289
$255$	0	0
$256$	−736339.	−0.702227
$257$	1.75004e6	1.65278	0.826391	−	0.563096i	$-0.190390\pi$
$257$	1.75004e6	1.65278	0.826391	+	0.563096i	$0.190390\pi$
$258$	0	0
$259$	−1.51544e6	−1.40375
$260$	−186700.	−0.171282
$261$	0	0
$262$	545307.	0.490781
$263$	268807.	0.239635	0.119818	−	0.992796i	$-0.461769\pi$
$263$	268807.	0.239635	0.119818	+	0.992796i	$0.461769\pi$
$264$	0	0
$265$	−36198.8	−0.0316650
$266$	47303.1	0.0409907
$267$	0	0
$268$	−59085.4	−0.0502509
$269$	856225.	0.721452	0.360726	−	0.932672i	$-0.382529\pi$
$269$	856225.	0.721452	0.360726	+	0.932672i	$0.382529\pi$
$270$	0	0
$271$	−766331.	−0.633860	−0.316930	−	0.948449i	$-0.602652\pi$
$271$	−766331.	−0.633860	−0.316930	+	0.948449i	$0.602652\pi$
$272$	257766.	0.211254
$273$	0	0
$274$	579685.	0.466462
$275$	1.63925e6	1.30711
$276$	0	0
$277$	−944155.	−0.739339	−0.369670	−	0.929163i	$-0.620529\pi$
$277$	−944155.	−0.739339	−0.369670	+	0.929163i	$0.620529\pi$
$278$	468689.	0.363725
$279$	0	0
$280$	625404.	0.476723
$281$	−863619.	−0.652464	−0.326232	−	0.945290i	$-0.605779\pi$
$281$	−863619.	−0.652464	−0.326232	+	0.945290i	$0.605779\pi$
$282$	0	0
$283$	−778763.	−0.578015	−0.289008	−	0.957327i	$-0.593325\pi$
$283$	−778763.	−0.578015	−0.289008	+	0.957327i	$0.593325\pi$
$284$	−577601.	−0.424944
$285$	0	0
$286$	809629.	0.585290
$287$	1.93744e6	1.38843
$288$	0	0
$289$	−662016.	−0.466255
$290$	190136.	0.132761
$291$	0	0
$292$	−620649.	−0.425979
$293$	−229461.	−0.156149	−0.0780745	−	0.996948i	$-0.524877\pi$
$293$	−229461.	−0.156149	−0.0780745	+	0.996948i	$0.524877\pi$
$294$	0	0
$295$	−272411.	−0.182251
$296$	999580.	0.663114
$297$	0	0
$298$	1.09931e6	0.717101
$299$	−2.15112e6	−1.39151
$300$	0	0
$301$	−3.38119e6	−2.15106
$302$	1.18641e6	0.748544
$303$	0	0
$304$	19983.7	0.0124020
$305$	166684.	0.102599
$306$	0	0
$307$	1.68495e6	1.02033	0.510167	−	0.860075i	$-0.329584\pi$
$307$	1.68495e6	1.02033	0.510167	+	0.860075i	$0.329584\pi$
$308$	3.29515e6	1.97924
$309$	0	0
$310$	79342.2	0.0468922
$311$	1.92662e6	1.12952	0.564761	−	0.825254i	$-0.308968\pi$
$311$	1.92662e6	1.12952	0.564761	+	0.825254i	$0.308968\pi$
$312$	0	0
$313$	−1.09859e6	−0.633835	−0.316918	−	0.948453i	$-0.602648\pi$
$313$	−1.09859e6	−0.633835	−0.316918	+	0.948453i	$0.602648\pi$
$314$	811094.	0.464245
$315$	0	0
$316$	−780330.	−0.439603
$317$	595900.	0.333062	0.166531	−	0.986036i	$-0.446743\pi$
$317$	595900.	0.333062	0.166531	+	0.986036i	$0.446743\pi$
$318$	0	0
$319$	2.35442e6	1.29541
$320$	−124572.	−0.0680057
$321$	0	0
$322$	3.06602e6	1.64792
$323$	58752.8	0.0313345
$324$	0	0
$325$	1.41051e6	0.740741
$326$	1.29687e6	0.675854
$327$	0	0
$328$	−1.27793e6	−0.655876
$329$	221273.	0.112704
$330$	0	0
$331$	1.44753e6	0.726202	0.363101	−	0.931750i	$-0.381718\pi$
$331$	1.44753e6	0.726202	0.363101	+	0.931750i	$0.381718\pi$
$332$	667708.	0.332461
$333$	0	0
$334$	−1.31455e6	−0.644778
$335$	−39937.9	−0.0194434
$336$	0	0
$337$	−3.71843e6	−1.78355	−0.891775	−	0.452480i	$-0.850539\pi$
$337$	−3.71843e6	−1.78355	−0.891775	+	0.452480i	$0.850539\pi$
$338$	−373026.	−0.177602
$339$	0	0
$340$	330518.	0.155059
$341$	982481.	0.457549
$342$	0	0
$343$	6.22156e6	2.85538
$344$	2.23022e6	1.01613
$345$	0	0
$346$	−603383.	−0.270959
$347$	−1.31123e6	−0.584597	−0.292298	−	0.956327i	$-0.594420\pi$
$347$	−1.31123e6	−0.584597	−0.292298	+	0.956327i	$0.594420\pi$
$348$	0	0
$349$	−3.17705e6	−1.39624	−0.698120	−	0.715981i	$-0.745980\pi$
$349$	−3.17705e6	−1.39624	−0.698120	+	0.715981i	$0.745980\pi$
$350$	−2.01041e6	−0.877233
$351$	0	0
$352$	−3.42213e6	−1.47211
$353$	−1.35303e6	−0.577922	−0.288961	−	0.957341i	$-0.593310\pi$
$353$	−1.35303e6	−0.577922	−0.288961	+	0.957341i	$0.593310\pi$
$354$	0	0
$355$	−390420.	−0.164422
$356$	−260335.	−0.108870
$357$	0	0
$358$	108484.	0.0447363
$359$	−3.99325e6	−1.63527	−0.817636	−	0.575735i	$-0.804716\pi$
$359$	−3.99325e6	−1.63527	−0.817636	+	0.575735i	$0.804716\pi$
$360$	0	0
$361$	−2.47154e6	−0.998160
$362$	198645.	0.0796722
$363$	0	0
$364$	2.83533e6	1.12163
$365$	−419518.	−0.164823
$366$	0	0
$367$	−3.25068e6	−1.25982	−0.629910	−	0.776668i	$-0.716908\pi$
$367$	−3.25068e6	−1.25982	−0.629910	+	0.776668i	$0.716908\pi$
$368$	1.29528e6	0.498589
$369$	0	0
$370$	287486.	0.109172
$371$	549735.	0.207357
$372$	0	0
$373$	2.38266e6	0.886726	0.443363	−	0.896342i	$-0.353785\pi$
$373$	2.38266e6	0.886726	0.443363	+	0.896342i	$0.353785\pi$
$374$	−1.43329e6	−0.529854
$375$	0	0
$376$	−145951.	−0.0532399
$377$	2.02588e6	0.734107
$378$	0	0
$379$	411391.	0.147115	0.0735575	−	0.997291i	$-0.476565\pi$
$379$	411391.	0.147115	0.0735575	+	0.997291i	$0.476565\pi$
$380$	25623.9	0.00910302
$381$	0	0
$382$	2.26817e6	0.795275
$383$	−3.85622e6	−1.34328	−0.671638	−	0.740880i	$-0.734409\pi$
$383$	−3.85622e6	−1.34328	−0.671638	+	0.740880i	$0.734409\pi$
$384$	0	0
$385$	2.22730e6	0.765821
$386$	1.09313e6	0.373427
$387$	0	0
$388$	2.70414e6	0.911907
$389$	1.43691e6	0.481454	0.240727	−	0.970593i	$-0.422614\pi$
$389$	1.43691e6	0.481454	0.240727	+	0.970593i	$0.422614\pi$
$390$	0	0
$391$	3.80815e6	1.25972
$392$	−6.80073e6	−2.23532
$393$	0	0
$394$	2.28028e6	0.740027
$395$	−527452.	−0.170094
$396$	0	0
$397$	−4.38023e6	−1.39483	−0.697414	−	0.716669i	$-0.745666\pi$
$397$	−4.38023e6	−1.39483	−0.697414	+	0.716669i	$0.745666\pi$
$398$	871263.	0.275703
$399$	0	0
$400$	−849322.	−0.265413
$401$	1.03147e6	0.320328	0.160164	−	0.987090i	$-0.448798\pi$
$401$	1.03147e6	0.320328	0.160164	+	0.987090i	$0.448798\pi$
$402$	0	0
$403$	845383.	0.259293
$404$	−4.76876e6	−1.45363
$405$	0	0
$406$	−2.88751e6	−0.869377
$407$	3.55988e6	1.06525
$408$	0	0
$409$	3.00281e6	0.887603	0.443802	−	0.896125i	$-0.353629\pi$
$409$	3.00281e6	0.887603	0.443802	+	0.896125i	$0.353629\pi$
$410$	−367540.	−0.107980
$411$	0	0
$412$	−1.65017e6	−0.478944
$413$	4.13698e6	1.19346
$414$	0	0
$415$	451326.	0.128638
$416$	−2.94460e6	−0.834244
$417$	0	0
$418$	−111118.	−0.0311061
$419$	678561.	0.188823	0.0944113	−	0.995533i	$-0.469903\pi$
$419$	678561.	0.188823	0.0944113	+	0.995533i	$0.469903\pi$
$420$	0	0
$421$	839853.	0.230939	0.115470	−	0.993311i	$-0.463163\pi$
$421$	839853.	0.230939	0.115470	+	0.993311i	$0.463163\pi$
$422$	−2.31884e6	−0.633854
$423$	0	0
$424$	−362603.	−0.0979529
$425$	−2.49703e6	−0.670582
$426$	0	0
$427$	−2.53136e6	−0.671868
$428$	−71712.1	−0.0189227
$429$	0	0
$430$	641425.	0.167292
$431$	−105184.	−0.0272745	−0.0136372	−	0.999907i	$-0.504341\pi$
$431$	−105184.	−0.0272745	−0.0136372	+	0.999907i	$0.504341\pi$
$432$	0	0
$433$	5.79597e6	1.48561	0.742807	−	0.669505i	$-0.233494\pi$
$433$	5.79597e6	1.48561	0.742807	+	0.669505i	$0.233494\pi$
$434$	−1.20493e6	−0.307071
$435$	0	0
$436$	1.75586e6	0.442358
$437$	295233.	0.0739539
$438$	0	0
$439$	−6.07162e6	−1.50364	−0.751819	−	0.659370i	$-0.770823\pi$
$439$	−6.07162e6	−1.50364	−0.751819	+	0.659370i	$0.770823\pi$
$440$	−1.46912e6	−0.361764
$441$	0	0
$442$	−1.23329e6	−0.300268
$443$	−3.52341e6	−0.853010	−0.426505	−	0.904485i	$-0.640255\pi$
$443$	−3.52341e6	−0.853010	−0.426505	+	0.904485i	$0.640255\pi$
$444$	0	0
$445$	−175969.	−0.0421247
$446$	3.78381e6	0.900725
$447$	0	0
$448$	1.89182e6	0.445332
$449$	−6.93998e6	−1.62458	−0.812292	−	0.583250i	$-0.801781\pi$
$449$	−6.93998e6	−1.62458	−0.812292	+	0.583250i	$0.801781\pi$
$450$	0	0
$451$	−4.55118e6	−1.05362
$452$	−3.37345e6	−0.776654
$453$	0	0
$454$	−3.68864e6	−0.839898
$455$	1.91650e6	0.433990
$456$	0	0
$457$	−2.74592e6	−0.615030	−0.307515	−	0.951543i	$-0.599498\pi$
$457$	−2.74592e6	−0.615030	−0.307515	+	0.951543i	$0.599498\pi$
$458$	1.24942e6	0.278321
$459$	0	0
$460$	1.66085e6	0.365962
$461$	3.22834e6	0.707501	0.353751	−	0.935340i	$-0.384906\pi$
$461$	3.22834e6	0.707501	0.353751	+	0.935340i	$0.384906\pi$
$462$	0	0
$463$	2.68538e6	0.582174	0.291087	−	0.956697i	$-0.405983\pi$
$463$	2.68538e6	0.582174	0.291087	+	0.956697i	$0.405983\pi$
$464$	−1.21986e6	−0.263036
$465$	0	0
$466$	3.73688e6	0.797158
$467$	−8.45458e6	−1.79391	−0.896953	−	0.442126i	$-0.854225\pi$
$467$	−8.45458e6	−1.79391	−0.896953	+	0.442126i	$0.854225\pi$
$468$	0	0
$469$	606518.	0.127324
$470$	−41976.5	−0.00876519
$471$	0	0
$472$	−2.72874e6	−0.563776
$473$	7.94265e6	1.63235
$474$	0	0
$475$	−193586.	−0.0393677
$476$	−5.01942e6	−1.01540
$477$	0	0
$478$	−540967.	−0.108293
$479$	1.47771e6	0.294274	0.147137	−	0.989116i	$-0.452994\pi$
$479$	1.47771e6	0.294274	0.147137	+	0.989116i	$0.452994\pi$
$480$	0	0
$481$	3.06313e6	0.603674
$482$	2.07816e6	0.407439
$483$	0	0
$484$	−3.92361e6	−0.761329
$485$	1.82782e6	0.352842
$486$	0	0
$487$	−4.05953e6	−0.775627	−0.387814	−	0.921738i	$-0.626770\pi$
$487$	−4.05953e6	−0.775627	−0.387814	+	0.921738i	$0.626770\pi$
$488$	1.66967e6	0.317382
$489$	0	0
$490$	−1.95593e6	−0.368014
$491$	7.87400e6	1.47398	0.736990	−	0.675903i	$-0.236246\pi$
$491$	7.87400e6	1.47398	0.736990	+	0.675903i	$0.236246\pi$
$492$	0	0
$493$	−3.58643e6	−0.664577
$494$	−95612.6	−0.0176278
$495$	0	0
$496$	−509039.	−0.0929066
$497$	5.92913e6	1.07671
$498$	0	0
$499$	2.85276e6	0.512877	0.256439	−	0.966561i	$-0.417451\pi$
$499$	2.85276e6	0.512877	0.256439	+	0.966561i	$0.417451\pi$
$500$	−2.27550e6	−0.407053
$501$	0	0
$502$	−954340.	−0.169022
$503$	−8.31648e6	−1.46561	−0.732807	−	0.680436i	$-0.761790\pi$
$503$	−8.31648e6	−1.46561	−0.732807	+	0.680436i	$0.761790\pi$
$504$	0	0
$505$	−3.22337e6	−0.562447
$506$	−7.20231e6	−1.25053
$507$	0	0
$508$	340280.	0.0585029
$509$	5.62172e6	0.961778	0.480889	−	0.876781i	$-0.340314\pi$
$509$	5.62172e6	0.961778	0.480889	+	0.876781i	$0.340314\pi$
$510$	0	0
$511$	6.37102e6	1.07934
$512$	3.29354e6	0.555249
$513$	0	0
$514$	5.04179e6	0.841738
$515$	−1.11540e6	−0.185317
$516$	0	0
$517$	−519787.	−0.0855261
$518$	−4.36592e6	−0.714909
$519$	0	0
$520$	−1.26412e6	−0.205012
$521$	6.67470e6	1.07730	0.538651	−	0.842529i	$-0.318934\pi$
$521$	6.67470e6	1.07730	0.538651	+	0.842529i	$0.318934\pi$
$522$	0	0
$523$	6.35577e6	1.01605	0.508023	−	0.861343i	$-0.330376\pi$
$523$	6.35577e6	1.01605	0.508023	+	0.861343i	$0.330376\pi$
$524$	−4.48596e6	−0.713719
$525$	0	0
$526$	774420.	0.122043
$527$	−1.49659e6	−0.234734
$528$	0	0
$529$	1.26996e7	1.97311
$530$	−104287.	−0.0161265
$531$	0	0
$532$	−389138.	−0.0596107
$533$	−3.91610e6	−0.597085
$534$	0	0
$535$	−48472.7	−0.00732171
$536$	−400057.	−0.0601465
$537$	0	0
$538$	2.46674e6	0.367425
$539$	−2.42200e7	−3.59089
$540$	0	0
$541$	−703405.	−0.103327	−0.0516633	−	0.998665i	$-0.516452\pi$
$541$	−703405.	−0.103327	−0.0516633	+	0.998665i	$0.516452\pi$
$542$	−2.20776e6	−0.322815
$543$	0	0
$544$	5.21286e6	0.755229
$545$	1.18685e6	0.171160
$546$	0	0
$547$	491046.	0.0701704	0.0350852	−	0.999384i	$-0.488830\pi$
$547$	491046.	0.0701704	0.0350852	+	0.999384i	$0.488830\pi$
$548$	−4.76877e6	−0.678352
$549$	0	0
$550$	4.72260e6	0.665694
$551$	−278043.	−0.0390152
$552$	0	0
$553$	8.01017e6	1.11386
$554$	−2.72006e6	−0.376534
$555$	0	0
$556$	−3.85566e6	−0.528947
$557$	195151.	0.0266522	0.0133261	−	0.999911i	$-0.495758\pi$
$557$	195151.	0.0266522	0.0133261	+	0.999911i	$0.495758\pi$
$558$	0	0
$559$	6.83431e6	0.925050
$560$	−1.15400e6	−0.155502
$561$	0	0
$562$	−2.48805e6	−0.332290
$563$	5.73710e6	0.762818	0.381409	−	0.924406i	$-0.375439\pi$
$563$	5.73710e6	0.762818	0.381409	+	0.924406i	$0.375439\pi$
$564$	0	0
$565$	−2.28023e6	−0.300509
$566$	−2.24358e6	−0.294375
$567$	0	0
$568$	−3.91083e6	−0.508626
$569$	474236.	0.0614065	0.0307032	−	0.999529i	$-0.490225\pi$
$569$	474236.	0.0614065	0.0307032	+	0.999529i	$0.490225\pi$
$570$	0	0
$571$	4.15432e6	0.533224	0.266612	−	0.963804i	$-0.414096\pi$
$571$	4.15432e6	0.533224	0.266612	+	0.963804i	$0.414096\pi$
$572$	−6.66040e6	−0.851159
$573$	0	0
$574$	5.58167e6	0.707106
$575$	−1.25476e7	−1.58267
$576$	0	0
$577$	3.24254e6	0.405458	0.202729	−	0.979235i	$-0.435019\pi$
$577$	3.24254e6	0.405458	0.202729	+	0.979235i	$0.435019\pi$
$578$	−1.90723e6	−0.237457
$579$	0	0
$580$	−1.56415e6	−0.193067
$581$	−6.85409e6	−0.842383
$582$	0	0
$583$	−1.29137e6	−0.157354
$584$	−4.20230e6	−0.509865
$585$	0	0
$586$	−661065.	−0.0795244
$587$	8.49157e6	1.01717	0.508584	−	0.861012i	$-0.330169\pi$
$587$	8.49157e6	1.01717	0.508584	+	0.861012i	$0.330169\pi$
$588$	0	0
$589$	−116025.	−0.0137805
$590$	−784802.	−0.0928176
$591$	0	0
$592$	−1.84443e6	−0.216301
$593$	1.09793e7	1.28215	0.641073	−	0.767480i	$-0.278490\pi$
$593$	1.09793e7	1.28215	0.641073	+	0.767480i	$0.278490\pi$
$594$	0	0
$595$	−3.39280e6	−0.392885
$596$	−9.04348e6	−1.04285
$597$	0	0
$598$	−6.19728e6	−0.708677
$599$	−1.73611e6	−0.197702	−0.0988509	−	0.995102i	$-0.531517\pi$
$599$	−1.73611e6	−0.197702	−0.0988509	+	0.995102i	$0.531517\pi$
$600$	0	0
$601$	9.21484e6	1.04064	0.520321	−	0.853971i	$-0.325812\pi$
$601$	9.21484e6	1.04064	0.520321	+	0.853971i	$0.325812\pi$
$602$	−9.74103e6	−1.09550
$603$	0	0
$604$	−9.75999e6	−1.08857
$605$	−2.65210e6	−0.294579
$606$	0	0
$607$	−7.61252e6	−0.838604	−0.419302	−	0.907847i	$-0.637725\pi$
$607$	−7.61252e6	−0.838604	−0.419302	+	0.907847i	$0.637725\pi$
$608$	404134.	0.0443371
$609$	0	0
$610$	480209.	0.0522523
$611$	−447255.	−0.0484676
$612$	0	0
$613$	1.13037e7	1.21498	0.607489	−	0.794328i	$-0.292177\pi$
$613$	1.13037e7	1.21498	0.607489	+	0.794328i	$0.292177\pi$
$614$	4.85427e6	0.519641
$615$	0	0
$616$	2.23109e7	2.36900
$617$	−7.51811e6	−0.795052	−0.397526	−	0.917591i	$-0.630131\pi$
$617$	−7.51811e6	−0.795052	−0.397526	+	0.917591i	$0.630131\pi$
$618$	0	0
$619$	−1.31127e7	−1.37552	−0.687758	−	0.725940i	$-0.741405\pi$
$619$	−1.31127e7	−1.37552	−0.687758	+	0.725940i	$0.741405\pi$
$620$	−652708.	−0.0681930
$621$	0	0
$622$	5.55050e6	0.575249
$623$	2.67237e6	0.275852
$624$	0	0
$625$	7.42557e6	0.760378
$626$	−3.16500e6	−0.322803
$627$	0	0
$628$	−6.67246e6	−0.675129
$629$	−5.42269e6	−0.546497
$630$	0	0
$631$	−1.11507e7	−1.11489	−0.557443	−	0.830216i	$-0.688217\pi$
$631$	−1.11507e7	−1.11489	−0.557443	+	0.830216i	$0.688217\pi$
$632$	−5.28348e6	−0.526172
$633$	0	0
$634$	1.71676e6	0.169624
$635$	230007.	0.0226364
$636$	0	0
$637$	−2.08403e7	−2.03495
$638$	6.78296e6	0.659732
$639$	0	0
$640$	2.71078e6	0.261604
$641$	5.68407e6	0.546404	0.273202	−	0.961957i	$-0.411917\pi$
$641$	5.68407e6	0.546404	0.273202	+	0.961957i	$0.411917\pi$
$642$	0	0
$643$	−2.06111e7	−1.96595	−0.982976	−	0.183736i	$-0.941181\pi$
$643$	−2.06111e7	−1.96595	−0.982976	+	0.183736i	$0.941181\pi$
$644$	−2.52226e7	−2.39649
$645$	0	0
$646$	169264.	0.0159582
$647$	−1.27533e7	−1.19774	−0.598870	−	0.800846i	$-0.704383\pi$
$647$	−1.27533e7	−1.19774	−0.598870	+	0.800846i	$0.704383\pi$
$648$	0	0
$649$	−9.71806e6	−0.905666
$650$	4.06360e6	0.377248
$651$	0	0
$652$	−1.06687e7	−0.982861
$653$	−6.22126e6	−0.570946	−0.285473	−	0.958387i	$-0.592151\pi$
$653$	−6.22126e6	−0.570946	−0.285473	+	0.958387i	$0.592151\pi$
$654$	0	0
$655$	−3.03222e6	−0.276157
$656$	2.35804e6	0.213940
$657$	0	0
$658$	637478.	0.0573985
$659$	1.78339e7	1.59968	0.799840	−	0.600214i	$-0.204918\pi$
$659$	1.78339e7	1.59968	0.799840	+	0.600214i	$0.204918\pi$
$660$	0	0
$661$	−8.38243e6	−0.746219	−0.373110	−	0.927787i	$-0.621708\pi$
$661$	−8.38243e6	−0.746219	−0.373110	+	0.927787i	$0.621708\pi$
$662$	4.17026e6	0.369844
$663$	0	0
$664$	4.52093e6	0.397931
$665$	−263032.	−0.0230650
$666$	0	0
$667$	−1.80218e7	−1.56850
$668$	1.08141e7	0.937669
$669$	0	0
$670$	−115059.	−0.00990225
$671$	5.94634e6	0.509851
$672$	0	0
$673$	1.51575e7	1.29000	0.644999	−	0.764183i	$-0.276858\pi$
$673$	1.51575e7	1.29000	0.644999	+	0.764183i	$0.276858\pi$
$674$	−1.07126e7	−0.908335
$675$	0	0
$676$	3.06869e6	0.258277
$677$	4.08012e6	0.342138	0.171069	−	0.985259i	$-0.445278\pi$
$677$	4.08012e6	0.342138	0.171069	+	0.985259i	$0.445278\pi$
$678$	0	0
$679$	−2.77583e7	−2.31057
$680$	2.23788e6	0.185594
$681$	0	0
$682$	2.83048e6	0.233023
$683$	6.27269e6	0.514519	0.257260	−	0.966342i	$-0.417180\pi$
$683$	6.27269e6	0.514519	0.257260	+	0.966342i	$0.417180\pi$
$684$	0	0
$685$	−3.22338e6	−0.262473
$686$	1.79240e7	1.45420
$687$	0	0
$688$	−4.11521e6	−0.331452
$689$	−1.11117e6	−0.0891726
$690$	0	0
$691$	6.23905e6	0.497076	0.248538	−	0.968622i	$-0.420050\pi$
$691$	6.23905e6	0.497076	0.248538	+	0.968622i	$0.420050\pi$
$692$	4.96373e6	0.394042
$693$	0	0
$694$	−3.77760e6	−0.297727
$695$	−2.60617e6	−0.204664
$696$	0	0
$697$	6.93271e6	0.540532
$698$	−9.15292e6	−0.711084
$699$	0	0
$700$	1.65386e7	1.27572
$701$	−6.79050e6	−0.521924	−0.260962	−	0.965349i	$-0.584040\pi$
$701$	−6.79050e6	−0.521924	−0.260962	+	0.965349i	$0.584040\pi$
$702$	0	0
$703$	−420402.	−0.0320831
$704$	−4.44401e6	−0.337943
$705$	0	0
$706$	−3.89800e6	−0.294327
$707$	4.89519e7	3.68316
$708$	0	0
$709$	−1.21897e7	−0.910705	−0.455353	−	0.890311i	$-0.650487\pi$
$709$	−1.21897e7	−0.910705	−0.455353	+	0.890311i	$0.650487\pi$
$710$	−1.12478e6	−0.0837379
$711$	0	0
$712$	−1.76268e6	−0.130309
$713$	−7.52036e6	−0.554007
$714$	0	0
$715$	−4.50199e6	−0.329337
$716$	−892446.	−0.0650578
$717$	0	0
$718$	−1.15044e7	−0.832820
$719$	−1.60413e7	−1.15722	−0.578611	−	0.815604i	$-0.696405\pi$
$719$	−1.60413e7	−1.15722	−0.578611	+	0.815604i	$0.696405\pi$
$720$	0	0
$721$	1.69391e7	1.21354
$722$	−7.12040e6	−0.508348
$723$	0	0
$724$	−1.63415e6	−0.115863
$725$	1.18170e7	0.834955
$726$	0	0
$727$	4.05141e6	0.284296	0.142148	−	0.989845i	$-0.454599\pi$
$727$	4.05141e6	0.284296	0.142148	+	0.989845i	$0.454599\pi$
$728$	1.91976e7	1.34251
$729$	0	0
$730$	−1.20861e6	−0.0839419
$731$	−1.20989e7	−0.837435
$732$	0	0
$733$	4.65706e6	0.320149	0.160074	−	0.987105i	$-0.448827\pi$
$733$	4.65706e6	0.320149	0.160074	+	0.987105i	$0.448827\pi$
$734$	−9.36504e6	−0.641608
$735$	0	0
$736$	2.61946e7	1.78245
$737$	−1.42476e6	−0.0966210
$738$	0	0
$739$	1.60416e6	0.108053	0.0540265	−	0.998540i	$-0.482794\pi$
$739$	1.60416e6	0.108053	0.0540265	+	0.998540i	$0.482794\pi$
$740$	−2.36500e6	−0.158764
$741$	0	0
$742$	1.58376e6	0.105604
$743$	1.14508e7	0.760964	0.380482	−	0.924788i	$-0.375758\pi$
$743$	1.14508e7	0.760964	0.380482	+	0.924788i	$0.375758\pi$
$744$	0	0
$745$	−6.11280e6	−0.403505
$746$	6.86432e6	0.451597
$747$	0	0
$748$	1.17910e7	0.770541
$749$	736133.	0.0479459
$750$	0	0
$751$	6.30532e6	0.407950	0.203975	−	0.978976i	$-0.434614\pi$
$751$	6.30532e6	0.407950	0.203975	+	0.978976i	$0.434614\pi$
$752$	269310.	0.0173663
$753$	0	0
$754$	5.83645e6	0.373870
$755$	−6.59711e6	−0.421198
$756$	0	0
$757$	1.50138e7	0.952248	0.476124	−	0.879378i	$-0.342041\pi$
$757$	1.50138e7	0.952248	0.476124	+	0.879378i	$0.342041\pi$
$758$	1.18520e6	0.0749235
$759$	0	0
$760$	173495.	0.0108956
$761$	1.09639e7	0.686286	0.343143	−	0.939283i	$-0.388509\pi$
$761$	1.09639e7	0.686286	0.343143	+	0.939283i	$0.388509\pi$
$762$	0	0
$763$	−1.80241e7	−1.12084
$764$	−1.86591e7	−1.15653
$765$	0	0
$766$	−1.11096e7	−0.684110
$767$	−8.36198e6	−0.513240
$768$	0	0
$769$	1.91484e6	0.116766	0.0583830	−	0.998294i	$-0.481406\pi$
$769$	1.91484e6	0.116766	0.0583830	+	0.998294i	$0.481406\pi$
$770$	6.41675e6	0.390021
$771$	0	0
$772$	−8.99266e6	−0.543056
$773$	2.64741e7	1.59358	0.796788	−	0.604259i	$-0.206531\pi$
$773$	2.64741e7	1.59358	0.796788	+	0.604259i	$0.206531\pi$
$774$	0	0
$775$	4.93115e6	0.294913
$776$	1.83093e7	1.09148
$777$	0	0
$778$	4.13966e6	0.245197
$779$	537469.	0.0317329
$780$	0	0
$781$	−1.39280e7	−0.817071
$782$	1.09711e7	0.641554
$783$	0	0
$784$	1.25487e7	0.729139
$785$	−4.51014e6	−0.261226
$786$	0	0
$787$	1.22581e7	0.705480	0.352740	−	0.935721i	$-0.385250\pi$
$787$	1.22581e7	0.705480	0.352740	+	0.935721i	$0.385250\pi$
$788$	−1.87587e7	−1.07618
$789$	0	0
$790$	−1.51956e6	−0.0866266
$791$	3.46288e7	1.96787
$792$	0	0
$793$	5.11658e6	0.288932
$794$	−1.26192e7	−0.710365
$795$	0	0
$796$	−7.16743e6	−0.400941
$797$	1.97321e7	1.10034	0.550170	−	0.835052i	$-0.314563\pi$
$797$	1.97321e7	1.10034	0.550170	+	0.835052i	$0.314563\pi$
$798$	0	0
$799$	791779.	0.0438770
$800$	−1.71760e7	−0.948848
$801$	0	0
$802$	2.97161e6	0.163138
$803$	−1.49660e7	−0.819062
$804$	0	0
$805$	−1.70488e7	−0.927266
$806$	2.43551e6	0.132054
$807$	0	0
$808$	−3.22885e7	−1.73988
$809$	1.64424e6	0.0883270	0.0441635	−	0.999024i	$-0.485938\pi$
$809$	1.64424e6	0.0883270	0.0441635	+	0.999024i	$0.485938\pi$
$810$	0	0
$811$	−2.05261e7	−1.09586	−0.547930	−	0.836524i	$-0.684584\pi$
$811$	−2.05261e7	−1.09586	−0.547930	+	0.836524i	$0.684584\pi$
$812$	2.37540e7	1.26429
$813$	0	0
$814$	1.02559e7	0.542514
$815$	−7.21133e6	−0.380296
$816$	0	0
$817$	−937982.	−0.0491631
$818$	8.65094e6	0.452043
$819$	0	0
$820$	3.02357e6	0.157031
$821$	−1.00137e7	−0.518485	−0.259243	−	0.965812i	$-0.583473\pi$
$821$	−1.00137e7	−0.518485	−0.259243	+	0.965812i	$0.583473\pi$
$822$	0	0
$823$	2.64559e7	1.36152	0.680758	−	0.732509i	$-0.261651\pi$
$823$	2.64559e7	1.36152	0.680758	+	0.732509i	$0.261651\pi$
$824$	−1.11730e7	−0.573260
$825$	0	0
$826$	1.19184e7	0.607812
$827$	2.10541e7	1.07047	0.535233	−	0.844704i	$-0.320224\pi$
$827$	2.10541e7	1.07047	0.535233	+	0.844704i	$0.320224\pi$
$828$	0	0
$829$	−2.27856e7	−1.15153	−0.575763	−	0.817616i	$-0.695295\pi$
$829$	−2.27856e7	−1.15153	−0.575763	+	0.817616i	$0.695295\pi$
$830$	1.30025e6	0.0655136
$831$	0	0
$832$	−3.82389e6	−0.191512
$833$	3.68937e7	1.84221
$834$	0	0
$835$	7.30963e6	0.362810
$836$	914113.	0.0452360
$837$	0	0
$838$	1.95490e6	0.0961645
$839$	−1.81390e7	−0.889630	−0.444815	−	0.895622i	$-0.646731\pi$
$839$	−1.81390e7	−0.889630	−0.444815	+	0.895622i	$0.646731\pi$
$840$	0	0
$841$	−3.53863e6	−0.172522
$842$	2.41958e6	0.117614
$843$	0	0
$844$	1.90759e7	0.921783
$845$	2.07423e6	0.0999346
$846$	0	0
$847$	4.02763e7	1.92904
$848$	669078.	0.0319512
$849$	0	0
$850$	−7.19383e6	−0.341517
$851$	−2.72490e7	−1.28981
$852$	0	0
$853$	3.48951e7	1.64207	0.821036	−	0.570877i	$-0.193397\pi$
$853$	3.48951e7	1.64207	0.821036	+	0.570877i	$0.193397\pi$
$854$	−7.29272e6	−0.342172
$855$	0	0
$856$	−485550.	−0.0226490
$857$	−3.10576e7	−1.44450	−0.722248	−	0.691634i	$-0.756891\pi$
$857$	−3.10576e7	−1.44450	−0.722248	+	0.691634i	$0.756891\pi$
$858$	0	0
$859$	−2.46968e7	−1.14198	−0.570989	−	0.820958i	$-0.693440\pi$
$859$	−2.46968e7	−1.14198	−0.570989	+	0.820958i	$0.693440\pi$
$860$	−5.27668e6	−0.243284
$861$	0	0
$862$	−303030.	−0.0138905
$863$	2.68237e7	1.22600	0.613002	−	0.790081i	$-0.289962\pi$
$863$	2.68237e7	1.22600	0.613002	+	0.790081i	$0.289962\pi$
$864$	0	0
$865$	3.35515e6	0.152466
$866$	1.66979e7	0.756601
$867$	0	0
$868$	9.91238e6	0.446559
$869$	−1.88165e7	−0.845257
$870$	0	0
$871$	−1.22594e6	−0.0547551
$872$	1.18886e7	0.529469
$873$	0	0
$874$	850551.	0.0376636
$875$	2.33582e7	1.03138
$876$	0	0
$877$	−730308.	−0.0320632	−0.0160316	−	0.999871i	$-0.505103\pi$
$877$	−730308.	−0.0320632	−0.0160316	+	0.999871i	$0.505103\pi$
$878$	−1.74920e7	−0.765780
$879$	0	0
$880$	2.71083e6	0.118004
$881$	−9.00874e6	−0.391043	−0.195521	−	0.980699i	$-0.562640\pi$
$881$	−9.00874e6	−0.391043	−0.195521	+	0.980699i	$0.562640\pi$
$882$	0	0
$883$	−2.47086e7	−1.06646	−0.533232	−	0.845969i	$-0.679023\pi$
$883$	−2.47086e7	−1.06646	−0.533232	+	0.845969i	$0.679023\pi$
$884$	1.01456e7	0.436665
$885$	0	0
$886$	−1.01508e7	−0.434425
$887$	−6.26605e6	−0.267414	−0.133707	−	0.991021i	$-0.542688\pi$
$887$	−6.26605e6	−0.267414	−0.133707	+	0.991021i	$0.542688\pi$
$888$	0	0
$889$	−3.49301e6	−0.148233
$890$	−506959.	−0.0214535
$891$	0	0
$892$	−3.11275e7	−1.30988
$893$	61383.9	0.00257588
$894$	0	0
$895$	−603235.	−0.0251726
$896$	−4.11674e7	−1.71310
$897$	0	0
$898$	−1.99938e7	−0.827377
$899$	7.08250e6	0.292272
$900$	0	0
$901$	1.96711e6	0.0807267
$902$	−1.31117e7	−0.536592
$903$	0	0
$904$	−2.28410e7	−0.929596
$905$	−1.10458e6	−0.0448307
$906$	0	0
$907$	4.15955e7	1.67891	0.839456	−	0.543428i	$-0.182874\pi$
$907$	4.15955e7	1.67891	0.839456	+	0.543428i	$0.182874\pi$
$908$	3.03446e7	1.22142
$909$	0	0
$910$	5.52134e6	0.221025
$911$	−2.66624e7	−1.06440	−0.532198	−	0.846620i	$-0.678634\pi$
$911$	−2.66624e7	−1.06440	−0.532198	+	0.846620i	$0.678634\pi$
$912$	0	0
$913$	1.61008e7	0.639248
$914$	−7.91085e6	−0.313226
$915$	0	0
$916$	−1.02784e7	−0.404749
$917$	4.60489e7	1.80841
$918$	0	0
$919$	−5.07075e6	−0.198054	−0.0990270	−	0.995085i	$-0.531573\pi$
$919$	−5.07075e6	−0.198054	−0.0990270	+	0.995085i	$0.531573\pi$
$920$	1.12453e7	0.438029
$921$	0	0
$922$	9.30070e6	0.360320
$923$	−1.19844e7	−0.463034
$924$	0	0
$925$	1.78674e7	0.686604
$926$	7.73644e6	0.296493
$927$	0	0
$928$	−2.46695e7	−0.940351
$929$	−2.42377e7	−0.921408	−0.460704	−	0.887554i	$-0.652403\pi$
$929$	−2.42377e7	−0.921408	−0.460704	+	0.887554i	$0.652403\pi$
$930$	0	0
$931$	2.86024e6	0.108150
$932$	−3.07414e7	−1.15927
$933$	0	0
$934$	−2.43572e7	−0.913610
$935$	7.96993e6	0.298143
$936$	0	0
$937$	−5.83540e6	−0.217131	−0.108565	−	0.994089i	$-0.534626\pi$
$937$	−5.83540e6	−0.217131	−0.108565	+	0.994089i	$0.534626\pi$
$938$	1.74735e6	0.0648445
$939$	0	0
$940$	345319.	0.0127468
$941$	9.47091e6	0.348673	0.174336	−	0.984686i	$-0.444222\pi$
$941$	9.47091e6	0.348673	0.174336	+	0.984686i	$0.444222\pi$
$942$	0	0
$943$	3.48369e7	1.27573
$944$	5.03508e6	0.183898
$945$	0	0
$946$	2.28824e7	0.831330
$947$	−2.48506e7	−0.900455	−0.450227	−	0.892914i	$-0.648657\pi$
$947$	−2.48506e7	−0.900455	−0.450227	+	0.892914i	$0.648657\pi$
$948$	0	0
$949$	−1.28776e7	−0.464162
$950$	−557712.	−0.0200494
$951$	0	0
$952$	−3.39856e7	−1.21535
$953$	3.83092e7	1.36638	0.683188	−	0.730242i	$-0.260593\pi$
$953$	3.83092e7	1.36638	0.683188	+	0.730242i	$0.260593\pi$
$954$	0	0
$955$	−1.26123e7	−0.447493
$956$	4.45026e6	0.157485
$957$	0	0
$958$	4.25722e6	0.149869
$959$	4.89520e7	1.71879
$960$	0	0
$961$	−2.56737e7	−0.896767
$962$	8.82473e6	0.307442
$963$	0	0
$964$	−1.70960e7	−0.592518
$965$	−6.07844e6	−0.210123
$966$	0	0
$967$	2.45508e7	0.844306	0.422153	−	0.906525i	$-0.361274\pi$
$967$	2.45508e7	0.844306	0.422153	+	0.906525i	$0.361274\pi$
$968$	−2.65661e7	−0.911253
$969$	0	0
$970$	5.26587e6	0.179697
$971$	3.28194e7	1.11707	0.558537	−	0.829479i	$-0.311363\pi$
$971$	3.28194e7	1.11707	0.558537	+	0.829479i	$0.311363\pi$
$972$	0	0
$973$	3.95788e7	1.34023
$974$	−1.16953e7	−0.395015
$975$	0	0
$976$	−3.08089e6	−0.103527
$977$	2.06278e7	0.691381	0.345690	−	0.938349i	$-0.387645\pi$
$977$	2.06278e7	0.691381	0.345690	+	0.938349i	$0.387645\pi$
$978$	0	0
$979$	−6.27758e6	−0.209332
$980$	1.60905e7	0.535184
$981$	0	0
$982$	2.26846e7	0.750677
$983$	5.09115e7	1.68047	0.840237	−	0.542219i	$-0.182416\pi$
$983$	5.09115e7	1.68047	0.840237	+	0.542219i	$0.182416\pi$
$984$	0	0
$985$	−1.26796e7	−0.416405
$986$	−1.03323e7	−0.338459
$987$	0	0
$988$	786556.	0.0256352
$989$	−6.07967e7	−1.97647
$990$	0	0
$991$	−1.56610e7	−0.506565	−0.253283	−	0.967392i	$-0.581510\pi$
$991$	−1.56610e7	−0.506565	−0.253283	+	0.967392i	$0.581510\pi$
$992$	−1.02944e7	−0.332140
$993$	0	0
$994$	1.70815e7	0.548354
$995$	−4.84471e6	−0.155135
$996$	0	0
$997$	−5.47132e7	−1.74323	−0.871613	−	0.490194i	$-0.836926\pi$
$997$	−5.47132e7	−1.74323	−0.871613	+	0.490194i	$0.836926\pi$
$998$	8.21865e6	0.261201
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.e.1.24		42
3.2	odd	2		729.6.a.c.1.19		42
27.5	odd	18		27.6.e.a.25.7	yes	84
27.11	odd	18		27.6.e.a.13.7	✓	84
27.16	even	9		81.6.e.a.10.8		84
27.22	even	9		81.6.e.a.73.8		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.13.7	✓	84	27.11	odd	18
27.6.e.a.25.7	yes	84	27.5	odd	18
81.6.e.a.10.8		84	27.16	even	9
81.6.e.a.73.8		84	27.22	even	9
729.6.a.c.1.19		42	3.2	odd	2
729.6.a.e.1.24		42	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+2.88095 q^{2} -23.7001 q^{4} -16.0197 q^{5} +243.284 q^{7} -160.469 q^{8} +O(q^{10})\)	\(q+2.88095 q^{2} -23.7001 q^{4} -16.0197 q^{5} +243.284 q^{7} -160.469 q^{8} -46.1520 q^{10} -571.492 q^{11} -491.745 q^{13} +700.890 q^{14} +296.099 q^{16} +870.541 q^{17} +67.4900 q^{19} +379.669 q^{20} -1646.44 q^{22} +4374.47 q^{23} -2868.37 q^{25} -1416.69 q^{26} -5765.86 q^{28} -4119.77 q^{29} -1719.15 q^{31} +5988.07 q^{32} +2507.99 q^{34} -3897.34 q^{35} -6229.10 q^{37} +194.435 q^{38} +2570.67 q^{40} +7963.69 q^{41} -13898.1 q^{43} +13544.4 q^{44} +12602.6 q^{46} +909.526 q^{47} +42380.2 q^{49} -8263.63 q^{50} +11654.4 q^{52} +2259.64 q^{53} +9155.14 q^{55} -39039.7 q^{56} -11868.9 q^{58} +17004.7 q^{59} -10404.9 q^{61} -4952.79 q^{62} +7776.16 q^{64} +7877.61 q^{65} +2493.04 q^{67} -20631.9 q^{68} -11228.1 q^{70} +24371.2 q^{71} +26187.6 q^{73} -17945.7 q^{74} -1599.52 q^{76} -139035. q^{77} +32925.2 q^{79} -4743.43 q^{80} +22943.0 q^{82} -28173.2 q^{83} -13945.8 q^{85} -40039.7 q^{86} +91707.0 q^{88} +10984.5 q^{89} -119634. q^{91} -103675. q^{92} +2620.30 q^{94} -1081.17 q^{95} -114098. q^{97} +122095. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8}+O(q^{10}) \) 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8	\( 42 q + 12 q^{2} + 624 q^{4} + 150 q^{5} + 573 q^{8} + 3 q^{10} + 1452 q^{11} + 2256 q^{14} + 8448 q^{16} + 3465 q^{17} + 3 q^{19} + 4128 q^{20} + 96 q^{22} + 5019 q^{23} + 18750 q^{25} + 3903 q^{26} - 6 q^{28} + 13008 q^{29} + 24273 q^{32} + 35868 q^{35} + 3 q^{37} + 51801 q^{38} + 96 q^{40} + 55833 q^{41} + 110757 q^{44} + 3 q^{46} + 90129 q^{47} + 57624 q^{49} + 145362 q^{50} + 3072 q^{52} + 103203 q^{53} - 6 q^{55} + 227154 q^{56} - 192 q^{58} + 176856 q^{59} - 31851 q^{61} + 246066 q^{62} + 86019 q^{64} + 167160 q^{65} - 801 q^{67} + 374589 q^{68} + 9375 q^{70} + 279531 q^{71} + 27012 q^{73} + 413970 q^{74} + 96 q^{76} + 185190 q^{77} + 462057 q^{80} - 6 q^{82} + 295536 q^{83} + 319803 q^{86} + 3072 q^{88} + 154827 q^{89} + 91002 q^{91} + 330558 q^{92} + 96 q^{94} + 353244 q^{95} + 463410 q^{98}+O(q^{100}) \) 42 * q + 12 * q^2 + 624 * q^4 + 150 * q^5 + 573 * q^8 + 3 * q^10 + 1452 * q^11 + 2256 * q^14 + 8448 * q^16 + 3465 * q^17 + 3 * q^19 + 4128 * q^20 + 96 * q^22 + 5019 * q^23 + 18750 * q^25 + 3903 * q^26 - 6 * q^28 + 13008 * q^29 + 24273 * q^32 + 35868 * q^35 + 3 * q^37 + 51801 * q^38 + 96 * q^40 + 55833 * q^41 + 110757 * q^44 + 3 * q^46 + 90129 * q^47 + 57624 * q^49 + 145362 * q^50 + 3072 * q^52 + 103203 * q^53 - 6 * q^55 + 227154 * q^56 - 192 * q^58 + 176856 * q^59 - 31851 * q^61 + 246066 * q^62 + 86019 * q^64 + 167160 * q^65 - 801 * q^67 + 374589 * q^68 + 9375 * q^70 + 279531 * q^71 + 27012 * q^73 + 413970 * q^74 + 96 * q^76 + 185190 * q^77 + 462057 * q^80 - 6 * q^82 + 295536 * q^83 + 319803 * q^86 + 3072 * q^88 + 154827 * q^89 + 91002 * q^91 + 330558 * q^92 + 96 * q^94 + 353244 * q^95 + 463410 * q^98

Introduction

L-functions

Modular forms

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Database

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