LMFDB - Embedded newform 810.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(1,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("810.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.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:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8657424.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 125x^{2} + 126x + 3726 $ x^4 - 2x^3 - 125x^2 + 126*x + 3726
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.37910$ of defining polynomial
Character	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.5296 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.5296 q^{7} +8.00000 q^{8} -10.0000 q^{10} -47.3429 q^{11} +19.0491 q^{13} -73.0592 q^{14} +16.0000 q^{16} +113.461 q^{17} -21.3538 q^{19} -20.0000 q^{20} -94.6859 q^{22} +41.0199 q^{23} +25.0000 q^{25} +38.0982 q^{26} -146.118 q^{28} +16.1660 q^{29} +204.374 q^{31} +32.0000 q^{32} +226.922 q^{34} +182.648 q^{35} +1.34030 q^{37} -42.7077 q^{38} -40.0000 q^{40} +385.648 q^{41} +155.072 q^{43} -189.372 q^{44} +82.0398 q^{46} +310.889 q^{47} +991.413 q^{49} +50.0000 q^{50} +76.1964 q^{52} -397.930 q^{53} +236.715 q^{55} -292.237 q^{56} +32.3321 q^{58} -453.805 q^{59} +803.829 q^{61} +408.747 q^{62} +64.0000 q^{64} -95.2454 q^{65} -733.819 q^{67} +453.844 q^{68} +365.296 q^{70} -121.560 q^{71} -284.679 q^{73} +2.68060 q^{74} -85.4153 q^{76} +1729.42 q^{77} -629.730 q^{79} -80.0000 q^{80} +771.297 q^{82} -1388.26 q^{83} -567.305 q^{85} +310.144 q^{86} -378.744 q^{88} +1215.74 q^{89} -695.856 q^{91} +164.080 q^{92} +621.779 q^{94} +106.769 q^{95} +1296.08 q^{97} +1982.83 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8} - 40 q^{10} - 36 q^{11} + 32 q^{13} + 4 q^{14} + 64 q^{16} + 90 q^{17} + 164 q^{19} - 80 q^{20} - 72 q^{22} + 42 q^{23} + 100 q^{25} + 64 q^{26} + 8 q^{28} + 36 q^{29} + 446 q^{31} + 128 q^{32} + 180 q^{34} - 10 q^{35} + 686 q^{37} + 328 q^{38} - 160 q^{40} + 108 q^{41} + 470 q^{43} - 144 q^{44} + 84 q^{46} + 804 q^{47} + 1338 q^{49} + 200 q^{50} + 128 q^{52} + 168 q^{53} + 180 q^{55} + 16 q^{56} + 72 q^{58} + 768 q^{59} + 596 q^{61} + 892 q^{62} + 256 q^{64} - 160 q^{65} - 424 q^{67} + 360 q^{68} - 20 q^{70} + 348 q^{71} - 94 q^{73} + 1372 q^{74} + 656 q^{76} + 630 q^{77} + 1088 q^{79} - 320 q^{80} + 216 q^{82} + 522 q^{83} - 450 q^{85} + 940 q^{86} - 288 q^{88} + 1128 q^{89} + 2842 q^{91} + 168 q^{92} + 1608 q^{94} - 820 q^{95} + 2486 q^{97} + 2676 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8 - 40 * q^10 - 36 * q^11 + 32 * q^13 + 4 * q^14 + 64 * q^16 + 90 * q^17 + 164 * q^19 - 80 * q^20 - 72 * q^22 + 42 * q^23 + 100 * q^25 + 64 * q^26 + 8 * q^28 + 36 * q^29 + 446 * q^31 + 128 * q^32 + 180 * q^34 - 10 * q^35 + 686 * q^37 + 328 * q^38 - 160 * q^40 + 108 * q^41 + 470 * q^43 - 144 * q^44 + 84 * q^46 + 804 * q^47 + 1338 * q^49 + 200 * q^50 + 128 * q^52 + 168 * q^53 + 180 * q^55 + 16 * q^56 + 72 * q^58 + 768 * q^59 + 596 * q^61 + 892 * q^62 + 256 * q^64 - 160 * q^65 - 424 * q^67 + 360 * q^68 - 20 * q^70 + 348 * q^71 - 94 * q^73 + 1372 * q^74 + 656 * q^76 + 630 * q^77 + 1088 * q^79 - 320 * q^80 + 216 * q^82 + 522 * q^83 - 450 * q^85 + 940 * q^86 - 288 * q^88 + 1128 * q^89 + 2842 * q^91 + 168 * q^92 + 1608 * q^94 - 820 * q^95 + 2486 * q^97 + 2676 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−36.5296	−1.97241	−0.986207	−	0.165518i	$-0.947070\pi$
$7$	−36.5296	−1.97241	−0.986207	+	0.165518i	$0.947070\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−47.3429	−1.29768	−0.648838	−	0.760927i	$-0.724745\pi$
$11$	−47.3429	−1.29768	−0.648838	+	0.760927i	$0.724745\pi$
$12$	0	0
$13$	19.0491	0.406405	0.203203	−	0.979137i	$-0.434865\pi$
$13$	19.0491	0.406405	0.203203	+	0.979137i	$0.434865\pi$
$14$	−73.0592	−1.39471
$15$	0	0
$16$	16.0000	0.250000
$17$	113.461	1.61873	0.809363	−	0.587309i	$-0.199813\pi$
$17$	113.461	1.61873	0.809363	+	0.587309i	$0.199813\pi$
$18$	0	0
$19$	−21.3538	−0.257837	−0.128919	−	0.991655i	$-0.541151\pi$
$19$	−21.3538	−0.257837	−0.128919	+	0.991655i	$0.541151\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−94.6859	−0.917595
$23$	41.0199	0.371880	0.185940	−	0.982561i	$-0.440467\pi$
$23$	41.0199	0.371880	0.185940	+	0.982561i	$0.440467\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	38.0982	0.287372
$27$	0	0
$28$	−146.118	−0.986207
$29$	16.1660	0.103516	0.0517579	−	0.998660i	$-0.483518\pi$
$29$	16.1660	0.103516	0.0517579	+	0.998660i	$0.483518\pi$
$30$	0	0
$31$	204.374	1.18408	0.592042	−	0.805907i	$-0.298322\pi$
$31$	204.374	1.18408	0.592042	+	0.805907i	$0.298322\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	226.922	1.14461
$35$	182.648	0.882090
$36$	0	0
$37$	1.34030	0.00595524	0.00297762	−	0.999996i	$-0.499052\pi$
$37$	1.34030	0.00595524	0.00297762	+	0.999996i	$0.499052\pi$
$38$	−42.7077	−0.182318
$39$	0	0
$40$	−40.0000	−0.158114
$41$	385.648	1.46898	0.734490	−	0.678620i	$-0.237422\pi$
$41$	385.648	1.46898	0.734490	+	0.678620i	$0.237422\pi$
$42$	0	0
$43$	155.072	0.549960	0.274980	−	0.961450i	$-0.411329\pi$
$43$	155.072	0.549960	0.274980	+	0.961450i	$0.411329\pi$
$44$	−189.372	−0.648838
$45$	0	0
$46$	82.0398	0.262959
$47$	310.889	0.964848	0.482424	−	0.875938i	$-0.339756\pi$
$47$	310.889	0.964848	0.482424	+	0.875938i	$0.339756\pi$
$48$	0	0
$49$	991.413	2.89042
$50$	50.0000	0.141421
$51$	0	0
$52$	76.1964	0.203203
$53$	−397.930	−1.03132	−0.515659	−	0.856794i	$-0.672453\pi$
$53$	−397.930	−1.03132	−0.515659	+	0.856794i	$0.672453\pi$
$54$	0	0
$55$	236.715	0.580338
$56$	−292.237	−0.697354
$57$	0	0
$58$	32.3321	0.0731967
$59$	−453.805	−1.00136	−0.500681	−	0.865632i	$-0.666917\pi$
$59$	−453.805	−1.00136	−0.500681	+	0.865632i	$0.666917\pi$
$60$	0	0
$61$	803.829	1.68721	0.843605	−	0.536965i	$-0.180429\pi$
$61$	803.829	1.68721	0.843605	+	0.536965i	$0.180429\pi$
$62$	408.747	0.837274
$63$	0	0
$64$	64.0000	0.125000
$65$	−95.2454	−0.181750
$66$	0	0
$67$	−733.819	−1.33806	−0.669032	−	0.743234i	$-0.733291\pi$
$67$	−733.819	−1.33806	−0.669032	+	0.743234i	$0.733291\pi$
$68$	453.844	0.809363
$69$	0	0
$70$	365.296	0.623732
$71$	−121.560	−0.203190	−0.101595	−	0.994826i	$-0.532395\pi$
$71$	−121.560	−0.203190	−0.101595	+	0.994826i	$0.532395\pi$
$72$	0	0
$73$	−284.679	−0.456427	−0.228213	−	0.973611i	$-0.573288\pi$
$73$	−284.679	−0.456427	−0.228213	+	0.973611i	$0.573288\pi$
$74$	2.68060	0.00421099
$75$	0	0
$76$	−85.4153	−0.128919
$77$	1729.42	2.55955
$78$	0	0
$79$	−629.730	−0.896838	−0.448419	−	0.893824i	$-0.648013\pi$
$79$	−629.730	−0.896838	−0.448419	+	0.893824i	$0.648013\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	771.297	1.03873
$83$	−1388.26	−1.83592	−0.917959	−	0.396674i	$-0.870164\pi$
$83$	−1388.26	−1.83592	−0.917959	+	0.396674i	$0.870164\pi$
$84$	0	0
$85$	−567.305	−0.723916
$86$	310.144	0.388880
$87$	0	0
$88$	−378.744	−0.458798
$89$	1215.74	1.44795	0.723976	−	0.689825i	$-0.242312\pi$
$89$	1215.74	1.44795	0.723976	+	0.689825i	$0.242312\pi$
$90$	0	0
$91$	−695.856	−0.801599
$92$	164.080	0.185940
$93$	0	0
$94$	621.779	0.682251
$95$	106.769	0.115308
$96$	0	0
$97$	1296.08	1.35667	0.678336	−	0.734752i	$-0.262701\pi$
$97$	1296.08	1.35667	0.678336	+	0.734752i	$0.262701\pi$
$98$	1982.83	2.04383
$99$	0	0
$100$	100.000	0.100000
$101$	−612.771	−0.603693	−0.301847	−	0.953357i	$-0.597603\pi$
$101$	−612.771	−0.603693	−0.301847	+	0.953357i	$0.597603\pi$
$102$	0	0
$103$	1684.22	1.61117	0.805586	−	0.592479i	$-0.201851\pi$
$103$	1684.22	1.61117	0.805586	+	0.592479i	$0.201851\pi$
$104$	152.393	0.143686
$105$	0	0
$106$	−795.860	−0.729252
$107$	238.432	0.215421	0.107711	−	0.994182i	$-0.465648\pi$
$107$	238.432	0.215421	0.107711	+	0.994182i	$0.465648\pi$
$108$	0	0
$109$	−875.428	−0.769273	−0.384637	−	0.923068i	$-0.625673\pi$
$109$	−875.428	−0.769273	−0.384637	+	0.923068i	$0.625673\pi$
$110$	473.429	0.410361
$111$	0	0
$112$	−584.474	−0.493103
$113$	127.116	0.105824	0.0529119	−	0.998599i	$-0.483150\pi$
$113$	127.116	0.105824	0.0529119	+	0.998599i	$0.483150\pi$
$114$	0	0
$115$	−205.099	−0.166310
$116$	64.6641	0.0517579
$117$	0	0
$118$	−907.610	−0.708070
$119$	−4144.69	−3.19280
$120$	0	0
$121$	910.354	0.683962
$122$	1607.66	1.19304
$123$	0	0
$124$	817.495	0.592042
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1886.53	1.31813	0.659064	−	0.752087i	$-0.270952\pi$
$127$	1886.53	1.31813	0.659064	+	0.752087i	$0.270952\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−190.491	−0.128517
$131$	−1613.16	−1.07589	−0.537947	−	0.842979i	$-0.680800\pi$
$131$	−1613.16	−1.07589	−0.537947	+	0.842979i	$0.680800\pi$
$132$	0	0
$133$	780.047	0.508561
$134$	−1467.64	−0.946154
$135$	0	0
$136$	907.688	0.572306
$137$	−1545.67	−0.963907	−0.481953	−	0.876197i	$-0.660073\pi$
$137$	−1545.67	−0.963907	−0.481953	+	0.876197i	$0.660073\pi$
$138$	0	0
$139$	707.199	0.431538	0.215769	−	0.976444i	$-0.430774\pi$
$139$	707.199	0.431538	0.215769	+	0.976444i	$0.430774\pi$
$140$	730.592	0.441045
$141$	0	0
$142$	−243.119	−0.143677
$143$	−901.840	−0.527382
$144$	0	0
$145$	−80.8301	−0.0462936
$146$	−569.358	−0.322743
$147$	0	0
$148$	5.36119	0.00297762
$149$	−637.410	−0.350461	−0.175230	−	0.984527i	$-0.556067\pi$
$149$	−637.410	−0.350461	−0.175230	+	0.984527i	$0.556067\pi$
$150$	0	0
$151$	2538.49	1.36807	0.684037	−	0.729447i	$-0.260223\pi$
$151$	2538.49	1.36807	0.684037	+	0.729447i	$0.260223\pi$
$152$	−170.831	−0.0911592
$153$	0	0
$154$	3458.84	1.80988
$155$	−1021.87	−0.529538
$156$	0	0
$157$	2987.54	1.51867	0.759337	−	0.650698i	$-0.225523\pi$
$157$	2987.54	1.51867	0.759337	+	0.650698i	$0.225523\pi$
$158$	−1259.46	−0.634160
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−1498.44	−0.733501
$162$	0	0
$163$	3504.01	1.68377	0.841886	−	0.539655i	$-0.181445\pi$
$163$	3504.01	1.68377	0.841886	+	0.539655i	$0.181445\pi$
$164$	1542.59	0.734490
$165$	0	0
$166$	−2776.52	−1.29819
$167$	1827.59	0.846847	0.423423	−	0.905932i	$-0.360828\pi$
$167$	1827.59	0.846847	0.423423	+	0.905932i	$0.360828\pi$
$168$	0	0
$169$	−1834.13	−0.834835
$170$	−1134.61	−0.511886
$171$	0	0
$172$	620.288	0.274980
$173$	1718.62	0.755285	0.377642	−	0.925951i	$-0.376735\pi$
$173$	1718.62	0.755285	0.377642	+	0.925951i	$0.376735\pi$
$174$	0	0
$175$	−913.240	−0.394483
$176$	−757.487	−0.324419
$177$	0	0
$178$	2431.47	1.02386
$179$	−2.07651	−0.000867073 0	−0.000433536	−	1.00000i	$-0.500138\pi$
$179$	−2.07651	−0.000867073 0	−0.000433536		1.00000i	$0.500138\pi$
$180$	0	0
$181$	72.6855	0.0298490	0.0149245	−	0.999889i	$-0.495249\pi$
$181$	72.6855	0.0298490	0.0149245	+	0.999889i	$0.495249\pi$
$182$	−1391.71	−0.566816
$183$	0	0
$184$	328.159	0.131479
$185$	−6.70149	−0.00266326
$186$	0	0
$187$	−5371.58	−2.10058
$188$	1243.56	0.482424
$189$	0	0
$190$	213.538	0.0815352
$191$	218.407	0.0827400	0.0413700	−	0.999144i	$-0.486828\pi$
$191$	218.407	0.0827400	0.0413700	+	0.999144i	$0.486828\pi$
$192$	0	0
$193$	4479.82	1.67080	0.835400	−	0.549642i	$-0.185236\pi$
$193$	4479.82	1.67080	0.835400	+	0.549642i	$0.185236\pi$
$194$	2592.16	0.959312
$195$	0	0
$196$	3965.65	1.44521
$197$	−615.095	−0.222455	−0.111228	−	0.993795i	$-0.535478\pi$
$197$	−615.095	−0.222455	−0.111228	+	0.993795i	$0.535478\pi$
$198$	0	0
$199$	3797.68	1.35282	0.676408	−	0.736527i	$-0.263535\pi$
$199$	3797.68	1.35282	0.676408	+	0.736527i	$0.263535\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−1225.54	−0.426876
$203$	−590.539	−0.204176
$204$	0	0
$205$	−1928.24	−0.656948
$206$	3368.43	1.13927
$207$	0	0
$208$	304.785	0.101601
$209$	1010.95	0.334589
$210$	0	0
$211$	3166.56	1.03315	0.516575	−	0.856242i	$-0.327207\pi$
$211$	3166.56	1.03315	0.516575	+	0.856242i	$0.327207\pi$
$212$	−1591.72	−0.515659
$213$	0	0
$214$	476.863	0.152326
$215$	−775.360	−0.245949
$216$	0	0
$217$	−7465.69	−2.33550
$218$	−1750.86	−0.543958
$219$	0	0
$220$	946.859	0.290169
$221$	2161.33	0.657858
$222$	0	0
$223$	2551.23	0.766111	0.383055	−	0.923725i	$-0.374872\pi$
$223$	2551.23	0.766111	0.383055	+	0.923725i	$0.374872\pi$
$224$	−1168.95	−0.348677
$225$	0	0
$226$	254.232	0.0748287
$227$	−3608.05	−1.05495	−0.527477	−	0.849569i	$-0.676862\pi$
$227$	−3608.05	−1.05495	−0.527477	+	0.849569i	$0.676862\pi$
$228$	0	0
$229$	−5190.98	−1.49795	−0.748973	−	0.662600i	$-0.769453\pi$
$229$	−5190.98	−1.49795	−0.748973	+	0.662600i	$0.769453\pi$
$230$	−410.199	−0.117599
$231$	0	0
$232$	129.328	0.0365983
$233$	2386.67	0.671057	0.335528	−	0.942030i	$-0.391085\pi$
$233$	2386.67	0.671057	0.335528	+	0.942030i	$0.391085\pi$
$234$	0	0
$235$	−1554.45	−0.431493
$236$	−1815.22	−0.500681
$237$	0	0
$238$	−8289.37	−2.25765
$239$	−6757.75	−1.82897	−0.914483	−	0.404625i	$-0.867402\pi$
$239$	−6757.75	−1.82897	−0.914483	+	0.404625i	$0.867402\pi$
$240$	0	0
$241$	−4495.74	−1.20164	−0.600822	−	0.799383i	$-0.705160\pi$
$241$	−4495.74	−1.20164	−0.600822	+	0.799383i	$0.705160\pi$
$242$	1820.71	0.483634
$243$	0	0
$244$	3215.32	0.843605
$245$	−4957.06	−1.29263
$246$	0	0
$247$	−406.771	−0.104786
$248$	1634.99	0.418637
$249$	0	0
$250$	−250.000	−0.0632456
$251$	2572.47	0.646904	0.323452	−	0.946245i	$-0.395157\pi$
$251$	2572.47	0.646904	0.323452	+	0.946245i	$0.395157\pi$
$252$	0	0
$253$	−1942.00	−0.482580
$254$	3773.06	0.932058
$255$	0	0
$256$	256.000	0.0625000
$257$	−3944.18	−0.957321	−0.478660	−	0.878000i	$-0.658877\pi$
$257$	−3944.18	−0.957321	−0.478660	+	0.878000i	$0.658877\pi$
$258$	0	0
$259$	−48.9606	−0.0117462
$260$	−380.982	−0.0908749
$261$	0	0
$262$	−3226.31	−0.760772
$263$	7197.96	1.68762	0.843812	−	0.536638i	$-0.180306\pi$
$263$	7197.96	1.68762	0.843812	+	0.536638i	$0.180306\pi$
$264$	0	0
$265$	1989.65	0.461220
$266$	1560.09	0.359607
$267$	0	0
$268$	−2935.28	−0.669032
$269$	4125.03	0.934972	0.467486	−	0.884000i	$-0.345160\pi$
$269$	4125.03	0.934972	0.467486	+	0.884000i	$0.345160\pi$
$270$	0	0
$271$	−1822.17	−0.408446	−0.204223	−	0.978924i	$-0.565467\pi$
$271$	−1822.17	−0.408446	−0.204223	+	0.978924i	$0.565467\pi$
$272$	1815.38	0.404681
$273$	0	0
$274$	−3091.33	−0.681585
$275$	−1183.57	−0.259535
$276$	0	0
$277$	−5492.33	−1.19134	−0.595672	−	0.803228i	$-0.703114\pi$
$277$	−5492.33	−1.19134	−0.595672	+	0.803228i	$0.703114\pi$
$278$	1414.40	0.305144
$279$	0	0
$280$	1461.18	0.311866
$281$	−1159.60	−0.246178	−0.123089	−	0.992396i	$-0.539280\pi$
$281$	−1159.60	−0.246178	−0.123089	+	0.992396i	$0.539280\pi$
$282$	0	0
$283$	1837.07	0.385876	0.192938	−	0.981211i	$-0.438198\pi$
$283$	1837.07	0.385876	0.192938	+	0.981211i	$0.438198\pi$
$284$	−486.238	−0.101595
$285$	0	0
$286$	−1803.68	−0.372915
$287$	−14087.6	−2.89743
$288$	0	0
$289$	7960.40	1.62027
$290$	−161.660	−0.0327345
$291$	0	0
$292$	−1138.72	−0.228213
$293$	7137.21	1.42307	0.711536	−	0.702649i	$-0.248000\pi$
$293$	7137.21	1.42307	0.711536	+	0.702649i	$0.248000\pi$
$294$	0	0
$295$	2269.03	0.447823
$296$	10.7224	0.00210549
$297$	0	0
$298$	−1274.82	−0.247813
$299$	781.392	0.151134
$300$	0	0
$301$	−5664.72	−1.08475
$302$	5076.97	0.967374
$303$	0	0
$304$	−341.661	−0.0644593
$305$	−4019.15	−0.754543
$306$	0	0
$307$	−7987.37	−1.48490	−0.742448	−	0.669904i	$-0.766335\pi$
$307$	−7987.37	−1.48490	−0.742448	+	0.669904i	$0.766335\pi$
$308$	6917.68	1.27978
$309$	0	0
$310$	−2043.74	−0.374440
$311$	−1651.26	−0.301075	−0.150538	−	0.988604i	$-0.548100\pi$
$311$	−1651.26	−0.301075	−0.150538	+	0.988604i	$0.548100\pi$
$312$	0	0
$313$	6879.45	1.24233	0.621165	−	0.783680i	$-0.286660\pi$
$313$	6879.45	1.24233	0.621165	+	0.783680i	$0.286660\pi$
$314$	5975.08	1.07386
$315$	0	0
$316$	−2518.92	−0.448419
$317$	4939.57	0.875186	0.437593	−	0.899173i	$-0.355831\pi$
$317$	4939.57	0.875186	0.437593	+	0.899173i	$0.355831\pi$
$318$	0	0
$319$	−765.347	−0.134330
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−2996.88	−0.518664
$323$	−2422.83	−0.417367
$324$	0	0
$325$	476.227	0.0812810
$326$	7008.01	1.19061
$327$	0	0
$328$	3085.19	0.519363
$329$	−11356.7	−1.90308
$330$	0	0
$331$	4849.20	0.805245	0.402623	−	0.915366i	$-0.368099\pi$
$331$	4849.20	0.805245	0.402623	+	0.915366i	$0.368099\pi$
$332$	−5553.04	−0.917959
$333$	0	0
$334$	3655.19	0.598811
$335$	3669.10	0.598400
$336$	0	0
$337$	11022.4	1.78169	0.890844	−	0.454309i	$-0.150114\pi$
$337$	11022.4	1.78169	0.890844	+	0.454309i	$0.150114\pi$
$338$	−3668.26	−0.590317
$339$	0	0
$340$	−2269.22	−0.361958
$341$	−9675.65	−1.53656
$342$	0	0
$343$	−23686.3	−3.72868
$344$	1240.58	0.194440
$345$	0	0
$346$	3437.24	0.534067
$347$	3605.38	0.557772	0.278886	−	0.960324i	$-0.410035\pi$
$347$	3605.38	0.557772	0.278886	+	0.960324i	$0.410035\pi$
$348$	0	0
$349$	−1269.90	−0.194775	−0.0973874	−	0.995247i	$-0.531049\pi$
$349$	−1269.90	−0.194775	−0.0973874	+	0.995247i	$0.531049\pi$
$350$	−1826.48	−0.278941
$351$	0	0
$352$	−1514.97	−0.229399
$353$	−2646.23	−0.398993	−0.199497	−	0.979899i	$-0.563931\pi$
$353$	−2646.23	−0.398993	−0.199497	+	0.979899i	$0.563931\pi$
$354$	0	0
$355$	607.798	0.0908692
$356$	4862.95	0.723976
$357$	0	0
$358$	−4.15303	−0.000613113 0
$359$	−2963.94	−0.435741	−0.217870	−	0.975978i	$-0.569911\pi$
$359$	−2963.94	−0.435741	−0.217870	+	0.975978i	$0.569911\pi$
$360$	0	0
$361$	−6403.01	−0.933520
$362$	145.371	0.0211064
$363$	0	0
$364$	−2783.42	−0.400800
$365$	1423.40	0.204120
$366$	0	0
$367$	7307.47	1.03936	0.519682	−	0.854360i	$-0.326050\pi$
$367$	7307.47	1.03936	0.519682	+	0.854360i	$0.326050\pi$
$368$	656.318	0.0929700
$369$	0	0
$370$	−13.4030	−0.00188321
$371$	14536.2	2.03419
$372$	0	0
$373$	−3636.78	−0.504840	−0.252420	−	0.967618i	$-0.581227\pi$
$373$	−3636.78	−0.504840	−0.252420	+	0.967618i	$0.581227\pi$
$374$	−10743.2	−1.48534
$375$	0	0
$376$	2487.11	0.341125
$377$	307.948	0.0420693
$378$	0	0
$379$	148.874	0.0201772	0.0100886	−	0.999949i	$-0.496789\pi$
$379$	148.874	0.0201772	0.0100886	+	0.999949i	$0.496789\pi$
$380$	427.077	0.0576541
$381$	0	0
$382$	436.813	0.0585060
$383$	−2137.40	−0.285159	−0.142579	−	0.989783i	$-0.545540\pi$
$383$	−2137.40	−0.285159	−0.142579	+	0.989783i	$0.545540\pi$
$384$	0	0
$385$	−8647.10	−1.14467
$386$	8959.64	1.18143
$387$	0	0
$388$	5184.33	0.678336
$389$	5511.81	0.718406	0.359203	−	0.933260i	$-0.383049\pi$
$389$	5511.81	0.718406	0.359203	+	0.933260i	$0.383049\pi$
$390$	0	0
$391$	4654.16	0.601972
$392$	7931.30	1.02192
$393$	0	0
$394$	−1230.19	−0.157300
$395$	3148.65	0.401078
$396$	0	0
$397$	−140.629	−0.0177783	−0.00888914	−	0.999960i	$-0.502830\pi$
$397$	−140.629	−0.0177783	−0.00888914	+	0.999960i	$0.502830\pi$
$398$	7595.37	0.956586
$399$	0	0
$400$	400.000	0.0500000
$401$	−6118.20	−0.761916	−0.380958	−	0.924592i	$-0.624406\pi$
$401$	−6118.20	−0.761916	−0.380958	+	0.924592i	$0.624406\pi$
$402$	0	0
$403$	3893.13	0.481218
$404$	−2451.08	−0.301847
$405$	0	0
$406$	−1181.08	−0.144374
$407$	−63.4537	−0.00772797
$408$	0	0
$409$	5653.42	0.683481	0.341741	−	0.939794i	$-0.388984\pi$
$409$	5653.42	0.683481	0.341741	+	0.939794i	$0.388984\pi$
$410$	−3856.48	−0.464532
$411$	0	0
$412$	6736.86	0.805586
$413$	16577.3	1.97510
$414$	0	0
$415$	6941.30	0.821048
$416$	609.571	0.0718430
$417$	0	0
$418$	2021.91	0.236590
$419$	−7970.71	−0.929343	−0.464671	−	0.885483i	$-0.653828\pi$
$419$	−7970.71	−0.929343	−0.464671	+	0.885483i	$0.653828\pi$
$420$	0	0
$421$	−15557.4	−1.80100	−0.900500	−	0.434855i	$-0.856800\pi$
$421$	−15557.4	−1.80100	−0.900500	+	0.434855i	$0.856800\pi$
$422$	6333.12	0.730548
$423$	0	0
$424$	−3183.44	−0.364626
$425$	2836.53	0.323745
$426$	0	0
$427$	−29363.6	−3.32787
$428$	953.727	0.107711
$429$	0	0
$430$	−1550.72	−0.173913
$431$	12353.4	1.38061	0.690304	−	0.723520i	$-0.257477\pi$
$431$	12353.4	1.38061	0.690304	+	0.723520i	$0.257477\pi$
$432$	0	0
$433$	−2730.36	−0.303032	−0.151516	−	0.988455i	$-0.548415\pi$
$433$	−2730.36	−0.303032	−0.151516	+	0.988455i	$0.548415\pi$
$434$	−14931.4	−1.65145
$435$	0	0
$436$	−3501.71	−0.384637
$437$	−875.932	−0.0958844
$438$	0	0
$439$	−12257.3	−1.33260	−0.666298	−	0.745686i	$-0.732122\pi$
$439$	−12257.3	−1.33260	−0.666298	+	0.745686i	$0.732122\pi$
$440$	1893.72	0.205181
$441$	0	0
$442$	4322.66	0.465176
$443$	−382.919	−0.0410678	−0.0205339	−	0.999789i	$-0.506537\pi$
$443$	−382.919	−0.0410678	−0.0205339	+	0.999789i	$0.506537\pi$
$444$	0	0
$445$	−6078.68	−0.647544
$446$	5102.45	0.541722
$447$	0	0
$448$	−2337.90	−0.246552
$449$	−13682.2	−1.43809	−0.719045	−	0.694963i	$-0.755421\pi$
$449$	−13682.2	−1.43809	−0.719045	+	0.694963i	$0.755421\pi$
$450$	0	0
$451$	−18257.7	−1.90626
$452$	508.465	0.0529119
$453$	0	0
$454$	−7216.09	−0.745965
$455$	3479.28	0.358486
$456$	0	0
$457$	−5975.64	−0.611660	−0.305830	−	0.952086i	$-0.598934\pi$
$457$	−5975.64	−0.611660	−0.305830	+	0.952086i	$0.598934\pi$
$458$	−10382.0	−1.05921
$459$	0	0
$460$	−820.398	−0.0831549
$461$	10808.6	1.09199	0.545995	−	0.837788i	$-0.316152\pi$
$461$	10808.6	1.09199	0.545995	+	0.837788i	$0.316152\pi$
$462$	0	0
$463$	7145.79	0.717264	0.358632	−	0.933479i	$-0.383243\pi$
$463$	7145.79	0.717264	0.358632	+	0.933479i	$0.383243\pi$
$464$	258.656	0.0258789
$465$	0	0
$466$	4773.35	0.474509
$467$	11434.1	1.13299	0.566497	−	0.824064i	$-0.308298\pi$
$467$	11434.1	1.13299	0.566497	+	0.824064i	$0.308298\pi$
$468$	0	0
$469$	26806.1	2.63922
$470$	−3108.89	−0.305112
$471$	0	0
$472$	−3630.44	−0.354035
$473$	−7341.57	−0.713669
$474$	0	0
$475$	−533.846	−0.0515674
$476$	−16578.7	−1.59640
$477$	0	0
$478$	−13515.5	−1.29327
$479$	−11229.6	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$479$	−11229.6	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$480$	0	0
$481$	25.5315	0.00242024
$482$	−8991.48	−0.849691
$483$	0	0
$484$	3641.42	0.341981
$485$	−6480.41	−0.606722
$486$	0	0
$487$	637.597	0.0593271	0.0296635	−	0.999560i	$-0.490556\pi$
$487$	637.597	0.0593271	0.0296635	+	0.999560i	$0.490556\pi$
$488$	6430.63	0.596519
$489$	0	0
$490$	−9914.13	−0.914030
$491$	−16678.8	−1.53300	−0.766501	−	0.642243i	$-0.778004\pi$
$491$	−16678.8	−1.53300	−0.766501	+	0.642243i	$0.778004\pi$
$492$	0	0
$493$	1834.21	0.167564
$494$	−813.542	−0.0740951
$495$	0	0
$496$	3269.98	0.296021
$497$	4440.52	0.400774
$498$	0	0
$499$	10754.0	0.964762	0.482381	−	0.875962i	$-0.339772\pi$
$499$	10754.0	0.964762	0.482381	+	0.875962i	$0.339772\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	5144.94	0.457430
$503$	20264.5	1.79632	0.898160	−	0.439669i	$-0.144904\pi$
$503$	20264.5	1.79632	0.898160	+	0.439669i	$0.144904\pi$
$504$	0	0
$505$	3063.86	0.269980
$506$	−3884.00	−0.341235
$507$	0	0
$508$	7546.11	0.659064
$509$	−2391.64	−0.208267	−0.104133	−	0.994563i	$-0.533207\pi$
$509$	−2391.64	−0.208267	−0.104133	+	0.994563i	$0.533207\pi$
$510$	0	0
$511$	10399.2	0.900263
$512$	512.000	0.0441942
$513$	0	0
$514$	−7888.37	−0.676928
$515$	−8421.08	−0.720538
$516$	0	0
$517$	−14718.4	−1.25206
$518$	−97.9212	−0.00830581
$519$	0	0
$520$	−761.964	−0.0642583
$521$	5389.68	0.453217	0.226609	−	0.973986i	$-0.427236\pi$
$521$	5389.68	0.453217	0.226609	+	0.973986i	$0.427236\pi$
$522$	0	0
$523$	−4302.99	−0.359764	−0.179882	−	0.983688i	$-0.557572\pi$
$523$	−4302.99	−0.359764	−0.179882	+	0.983688i	$0.557572\pi$
$524$	−6452.62	−0.537947
$525$	0	0
$526$	14395.9	1.19333
$527$	23188.4	1.91671
$528$	0	0
$529$	−10484.4	−0.861705
$530$	3979.30	0.326131
$531$	0	0
$532$	3120.19	0.254281
$533$	7346.25	0.597001
$534$	0	0
$535$	−1192.16	−0.0963393
$536$	−5870.55	−0.473077
$537$	0	0
$538$	8250.06	0.661125
$539$	−46936.4	−3.75082
$540$	0	0
$541$	12892.9	1.02460	0.512301	−	0.858806i	$-0.328793\pi$
$541$	12892.9	1.02460	0.512301	+	0.858806i	$0.328793\pi$
$542$	−3644.33	−0.288815
$543$	0	0
$544$	3630.75	0.286153
$545$	4377.14	0.344030
$546$	0	0
$547$	6270.76	0.490161	0.245081	−	0.969503i	$-0.421186\pi$
$547$	6270.76	0.490161	0.245081	+	0.969503i	$0.421186\pi$
$548$	−6182.67	−0.481953
$549$	0	0
$550$	−2367.15	−0.183519
$551$	−345.207	−0.0266902
$552$	0	0
$553$	23003.8	1.76893
$554$	−10984.7	−0.842408
$555$	0	0
$556$	2828.80	0.215769
$557$	4969.24	0.378013	0.189007	−	0.981976i	$-0.439473\pi$
$557$	4969.24	0.378013	0.189007	+	0.981976i	$0.439473\pi$
$558$	0	0
$559$	2953.98	0.223506
$560$	2922.37	0.220523
$561$	0	0
$562$	−2319.21	−0.174074
$563$	14321.6	1.07209	0.536043	−	0.844190i	$-0.319918\pi$
$563$	14321.6	1.07209	0.536043	+	0.844190i	$0.319918\pi$
$564$	0	0
$565$	−635.581	−0.0473258
$566$	3674.15	0.272855
$567$	0	0
$568$	−972.477	−0.0718384
$569$	52.2575	0.00385017	0.00192509	−	0.999998i	$-0.499387\pi$
$569$	52.2575	0.00385017	0.00192509	+	0.999998i	$0.499387\pi$
$570$	0	0
$571$	−699.598	−0.0512737	−0.0256368	−	0.999671i	$-0.508161\pi$
$571$	−699.598	−0.0512737	−0.0256368	+	0.999671i	$0.508161\pi$
$572$	−3607.36	−0.263691
$573$	0	0
$574$	−28175.2	−2.04880
$575$	1025.50	0.0743760
$576$	0	0
$577$	10230.3	0.738114	0.369057	−	0.929407i	$-0.379681\pi$
$577$	10230.3	0.738114	0.369057	+	0.929407i	$0.379681\pi$
$578$	15920.8	1.14571
$579$	0	0
$580$	−323.321	−0.0231468
$581$	50712.6	3.62119
$582$	0	0
$583$	18839.2	1.33832
$584$	−2277.43	−0.161371
$585$	0	0
$586$	14274.4	1.00626
$587$	7359.17	0.517454	0.258727	−	0.965950i	$-0.416697\pi$
$587$	7359.17	0.517454	0.258727	+	0.965950i	$0.416697\pi$
$588$	0	0
$589$	−4364.16	−0.305301
$590$	4538.05	0.316659
$591$	0	0
$592$	21.4448	0.00148881
$593$	345.832	0.0239488	0.0119744	−	0.999928i	$-0.496188\pi$
$593$	345.832	0.0239488	0.0119744	+	0.999928i	$0.496188\pi$
$594$	0	0
$595$	20723.4	1.42786
$596$	−2549.64	−0.175230
$597$	0	0
$598$	1562.78	0.106868
$599$	5882.29	0.401242	0.200621	−	0.979669i	$-0.435704\pi$
$599$	5882.29	0.401242	0.200621	+	0.979669i	$0.435704\pi$
$600$	0	0
$601$	−5928.44	−0.402373	−0.201186	−	0.979553i	$-0.564480\pi$
$601$	−5928.44	−0.402373	−0.201186	+	0.979553i	$0.564480\pi$
$602$	−11329.4	−0.767033
$603$	0	0
$604$	10153.9	0.684037
$605$	−4551.77	−0.305877
$606$	0	0
$607$	−10983.7	−0.734456	−0.367228	−	0.930131i	$-0.619693\pi$
$607$	−10983.7	−0.734456	−0.367228	+	0.930131i	$0.619693\pi$
$608$	−683.323	−0.0455796
$609$	0	0
$610$	−8038.29	−0.533542
$611$	5922.16	0.392119
$612$	0	0
$613$	−1147.81	−0.0756273	−0.0378136	−	0.999285i	$-0.512039\pi$
$613$	−1147.81	−0.0756273	−0.0378136	+	0.999285i	$0.512039\pi$
$614$	−15974.7	−1.04998
$615$	0	0
$616$	13835.4	0.904939
$617$	−28192.9	−1.83955	−0.919775	−	0.392446i	$-0.871629\pi$
$617$	−28192.9	−1.83955	−0.919775	+	0.392446i	$0.871629\pi$
$618$	0	0
$619$	−9675.37	−0.628249	−0.314124	−	0.949382i	$-0.601711\pi$
$619$	−9675.37	−0.628249	−0.314124	+	0.949382i	$0.601711\pi$
$620$	−4087.47	−0.264769
$621$	0	0
$622$	−3302.52	−0.212892
$623$	−44410.4	−2.85596
$624$	0	0
$625$	625.000	0.0400000
$626$	13758.9	0.878460
$627$	0	0
$628$	11950.2	0.759337
$629$	152.072	0.00963989
$630$	0	0
$631$	6947.43	0.438308	0.219154	−	0.975690i	$-0.429670\pi$
$631$	6947.43	0.438308	0.219154	+	0.975690i	$0.429670\pi$
$632$	−5037.84	−0.317080
$633$	0	0
$634$	9879.14	0.618850
$635$	−9432.64	−0.589485
$636$	0	0
$637$	18885.5	1.17468
$638$	−1530.69	−0.0949855
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−14075.8	−0.867331	−0.433666	−	0.901074i	$-0.642780\pi$
$641$	−14075.8	−0.867331	−0.433666	+	0.901074i	$0.642780\pi$
$642$	0	0
$643$	−15047.4	−0.922880	−0.461440	−	0.887171i	$-0.652667\pi$
$643$	−15047.4	−0.922880	−0.461440	+	0.887171i	$0.652667\pi$
$644$	−5993.76	−0.366751
$645$	0	0
$646$	−4845.65	−0.295123
$647$	−8961.35	−0.544524	−0.272262	−	0.962223i	$-0.587772\pi$
$647$	−8961.35	−0.544524	−0.272262	+	0.962223i	$0.587772\pi$
$648$	0	0
$649$	21484.5	1.29944
$650$	952.454	0.0574744
$651$	0	0
$652$	14016.0	0.841886
$653$	14026.6	0.840590	0.420295	−	0.907388i	$-0.361927\pi$
$653$	14026.6	0.840590	0.420295	+	0.907388i	$0.361927\pi$
$654$	0	0
$655$	8065.78	0.481154
$656$	6170.37	0.367245
$657$	0	0
$658$	−22713.3	−1.34568
$659$	−29440.9	−1.74030	−0.870148	−	0.492790i	$-0.835977\pi$
$659$	−29440.9	−1.74030	−0.870148	+	0.492790i	$0.835977\pi$
$660$	0	0
$661$	3351.62	0.197221	0.0986104	−	0.995126i	$-0.468560\pi$
$661$	3351.62	0.197221	0.0986104	+	0.995126i	$0.468560\pi$
$662$	9698.40	0.569394
$663$	0	0
$664$	−11106.1	−0.649095
$665$	−3900.24	−0.227436
$666$	0	0
$667$	663.129	0.0384954
$668$	7310.37	0.423423
$669$	0	0
$670$	7338.19	0.423133
$671$	−38055.6	−2.18945
$672$	0	0
$673$	−8921.46	−0.510991	−0.255496	−	0.966810i	$-0.582239\pi$
$673$	−8921.46	−0.510991	−0.255496	+	0.966810i	$0.582239\pi$
$674$	22044.8	1.25984
$675$	0	0
$676$	−7336.53	−0.417417
$677$	10983.8	0.623549	0.311774	−	0.950156i	$-0.399077\pi$
$677$	10983.8	0.623549	0.311774	+	0.950156i	$0.399077\pi$
$678$	0	0
$679$	−47345.4	−2.67592
$680$	−4538.44	−0.255943
$681$	0	0
$682$	−19351.3	−1.08651
$683$	4172.20	0.233740	0.116870	−	0.993147i	$-0.462714\pi$
$683$	4172.20	0.233740	0.116870	+	0.993147i	$0.462714\pi$
$684$	0	0
$685$	7728.33	0.431072
$686$	−47372.5	−2.63658
$687$	0	0
$688$	2481.15	0.137490
$689$	−7580.20	−0.419133
$690$	0	0
$691$	24998.7	1.37626	0.688129	−	0.725588i	$-0.258432\pi$
$691$	24998.7	1.37626	0.688129	+	0.725588i	$0.258432\pi$
$692$	6874.48	0.377642
$693$	0	0
$694$	7210.76	0.394404
$695$	−3535.99	−0.192990
$696$	0	0
$697$	43756.1	2.37787
$698$	−2539.81	−0.137727
$699$	0	0
$700$	−3652.96	−0.197241
$701$	22656.5	1.22072	0.610361	−	0.792123i	$-0.291024\pi$
$701$	22656.5	1.22072	0.610361	+	0.792123i	$0.291024\pi$
$702$	0	0
$703$	−28.6205	−0.00153548
$704$	−3029.95	−0.162209
$705$	0	0
$706$	−5292.46	−0.282131
$707$	22384.3	1.19073
$708$	0	0
$709$	10612.4	0.562138	0.281069	−	0.959688i	$-0.409311\pi$
$709$	10612.4	0.562138	0.281069	+	0.959688i	$0.409311\pi$
$710$	1215.60	0.0642542
$711$	0	0
$712$	9725.89	0.511929
$713$	8383.39	0.440337
$714$	0	0
$715$	4509.20	0.235852
$716$	−8.30606	−0.000433536 0
$717$	0	0
$718$	−5927.89	−0.308115
$719$	15401.9	0.798878	0.399439	−	0.916760i	$-0.369205\pi$
$719$	15401.9	0.798878	0.399439	+	0.916760i	$0.369205\pi$
$720$	0	0
$721$	−61523.7	−3.17790
$722$	−12806.0	−0.660098
$723$	0	0
$724$	290.742	0.0149245
$725$	404.151	0.0207031
$726$	0	0
$727$	−6859.46	−0.349936	−0.174968	−	0.984574i	$-0.555982\pi$
$727$	−6859.46	−0.349936	−0.174968	+	0.984574i	$0.555982\pi$
$728$	−5566.85	−0.283408
$729$	0	0
$730$	2846.79	0.144335
$731$	17594.6	0.890234
$732$	0	0
$733$	−15629.9	−0.787592	−0.393796	−	0.919198i	$-0.628838\pi$
$733$	−15629.9	−0.787592	−0.393796	+	0.919198i	$0.628838\pi$
$734$	14614.9	0.734942
$735$	0	0
$736$	1312.64	0.0657397
$737$	34741.2	1.73637
$738$	0	0
$739$	27457.7	1.36678	0.683388	−	0.730056i	$-0.260506\pi$
$739$	27457.7	1.36678	0.683388	+	0.730056i	$0.260506\pi$
$740$	−26.8060	−0.00133163
$741$	0	0
$742$	29072.4	1.43839
$743$	24470.0	1.20823	0.604116	−	0.796896i	$-0.293526\pi$
$743$	24470.0	1.20823	0.604116	+	0.796896i	$0.293526\pi$
$744$	0	0
$745$	3187.05	0.156731
$746$	−7273.56	−0.356976
$747$	0	0
$748$	−21486.3	−1.05029
$749$	−8709.82	−0.424900
$750$	0	0
$751$	−12690.8	−0.616635	−0.308318	−	0.951283i	$-0.599766\pi$
$751$	−12690.8	−0.616635	−0.308318	+	0.951283i	$0.599766\pi$
$752$	4974.23	0.241212
$753$	0	0
$754$	615.896	0.0297475
$755$	−12692.4	−0.611821
$756$	0	0
$757$	5447.66	0.261557	0.130779	−	0.991412i	$-0.458252\pi$
$757$	5447.66	0.261557	0.130779	+	0.991412i	$0.458252\pi$
$758$	297.749	0.0142675
$759$	0	0
$760$	854.153	0.0407676
$761$	34556.6	1.64609	0.823045	−	0.567976i	$-0.192273\pi$
$761$	34556.6	1.64609	0.823045	+	0.567976i	$0.192273\pi$
$762$	0	0
$763$	31979.0	1.51733
$764$	873.626	0.0413700
$765$	0	0
$766$	−4274.79	−0.201638
$767$	−8644.57	−0.406959
$768$	0	0
$769$	−4001.90	−0.187662	−0.0938312	−	0.995588i	$-0.529911\pi$
$769$	−4001.90	−0.187662	−0.0938312	+	0.995588i	$0.529911\pi$
$770$	−17294.2	−0.809402
$771$	0	0
$772$	17919.3	0.835400
$773$	39635.0	1.84421	0.922104	−	0.386942i	$-0.126469\pi$
$773$	39635.0	1.84421	0.922104	+	0.386942i	$0.126469\pi$
$774$	0	0
$775$	5109.34	0.236817
$776$	10368.7	0.479656
$777$	0	0
$778$	11023.6	0.507990
$779$	−8235.07	−0.378757
$780$	0	0
$781$	5754.99	0.263674
$782$	9308.32	0.425658
$783$	0	0
$784$	15862.6	0.722604
$785$	−14937.7	−0.679172
$786$	0	0
$787$	12805.4	0.580005	0.290003	−	0.957026i	$-0.406344\pi$
$787$	12805.4	0.580005	0.290003	+	0.957026i	$0.406344\pi$
$788$	−2460.38	−0.111228
$789$	0	0
$790$	6297.30	0.283605
$791$	−4643.51	−0.208728
$792$	0	0
$793$	15312.2	0.685690
$794$	−281.259	−0.0125711
$795$	0	0
$796$	15190.7	0.676408
$797$	−23750.2	−1.05555	−0.527775	−	0.849384i	$-0.676974\pi$
$797$	−23750.2	−1.05555	−0.527775	+	0.849384i	$0.676974\pi$
$798$	0	0
$799$	35273.8	1.56182
$800$	800.000	0.0353553
$801$	0	0
$802$	−12236.4	−0.538756
$803$	13477.5	0.592294
$804$	0	0
$805$	7492.20	0.328032
$806$	7786.27	0.340272
$807$	0	0
$808$	−4902.17	−0.213438
$809$	22179.3	0.963886	0.481943	−	0.876203i	$-0.339931\pi$
$809$	22179.3	0.963886	0.481943	+	0.876203i	$0.339931\pi$
$810$	0	0
$811$	44054.4	1.90747	0.953735	−	0.300649i	$-0.0972032\pi$
$811$	44054.4	1.90747	0.953735	+	0.300649i	$0.0972032\pi$
$812$	−2362.15	−0.102088
$813$	0	0
$814$	−126.907	−0.00546450
$815$	−17520.0	−0.753006
$816$	0	0
$817$	−3311.38	−0.141800
$818$	11306.8	0.483294
$819$	0	0
$820$	−7712.97	−0.328474
$821$	46741.3	1.98695	0.993474	−	0.114059i	$-0.0363852\pi$
$821$	46741.3	1.98695	0.993474	+	0.114059i	$0.0363852\pi$
$822$	0	0
$823$	16252.8	0.688381	0.344191	−	0.938900i	$-0.388153\pi$
$823$	16252.8	0.688381	0.344191	+	0.938900i	$0.388153\pi$
$824$	13473.7	0.569635
$825$	0	0
$826$	33154.6	1.39661
$827$	−35839.5	−1.50697	−0.753483	−	0.657467i	$-0.771628\pi$
$827$	−35839.5	−1.50697	−0.753483	+	0.657467i	$0.771628\pi$
$828$	0	0
$829$	42719.3	1.78975	0.894875	−	0.446316i	$-0.147264\pi$
$829$	42719.3	1.78975	0.894875	+	0.446316i	$0.147264\pi$
$830$	13882.6	0.580569
$831$	0	0
$832$	1219.14	0.0508006
$833$	112487.	4.67879
$834$	0	0
$835$	−9137.97	−0.378721
$836$	4043.81	0.167294
$837$	0	0
$838$	−15941.4	−0.657145
$839$	9452.74	0.388969	0.194484	−	0.980906i	$-0.437697\pi$
$839$	9452.74	0.388969	0.194484	+	0.980906i	$0.437697\pi$
$840$	0	0
$841$	−24127.7	−0.989284
$842$	−31114.8	−1.27350
$843$	0	0
$844$	12666.2	0.516575
$845$	9170.66	0.373350
$846$	0	0
$847$	−33254.9	−1.34906
$848$	−6366.88	−0.257830
$849$	0	0
$850$	5673.05	0.228922
$851$	54.9789	0.00221463
$852$	0	0
$853$	32551.5	1.30662	0.653308	−	0.757093i	$-0.273381\pi$
$853$	32551.5	1.30662	0.653308	+	0.757093i	$0.273381\pi$
$854$	−58727.1	−2.35316
$855$	0	0
$856$	1907.45	0.0761629
$857$	−23045.5	−0.918575	−0.459288	−	0.888288i	$-0.651895\pi$
$857$	−23045.5	−0.918575	−0.459288	+	0.888288i	$0.651895\pi$
$858$	0	0
$859$	−24979.2	−0.992176	−0.496088	−	0.868272i	$-0.665231\pi$
$859$	−24979.2	−0.992176	−0.496088	+	0.868272i	$0.665231\pi$
$860$	−3101.44	−0.122975
$861$	0	0
$862$	24706.8	0.976237
$863$	−31583.2	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$863$	−31583.2	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$864$	0	0
$865$	−8593.10	−0.337774
$866$	−5460.72	−0.214276
$867$	0	0
$868$	−29862.8	−1.16775
$869$	29813.3	1.16380
$870$	0	0
$871$	−13978.6	−0.543796
$872$	−7003.43	−0.271979
$873$	0	0
$874$	−1751.86	−0.0678005
$875$	4566.20	0.176418
$876$	0	0
$877$	46002.9	1.77128	0.885638	−	0.464377i	$-0.153722\pi$
$877$	46002.9	1.77128	0.885638	+	0.464377i	$0.153722\pi$
$878$	−24514.6	−0.942287
$879$	0	0
$880$	3787.44	0.145085
$881$	−25034.4	−0.957357	−0.478678	−	0.877990i	$-0.658884\pi$
$881$	−25034.4	−0.957357	−0.478678	+	0.877990i	$0.658884\pi$
$882$	0	0
$883$	−4286.38	−0.163361	−0.0816807	−	0.996659i	$-0.526029\pi$
$883$	−4286.38	−0.163361	−0.0816807	+	0.996659i	$0.526029\pi$
$884$	8645.32	0.328929
$885$	0	0
$886$	−765.837	−0.0290393
$887$	−25228.5	−0.955007	−0.477504	−	0.878630i	$-0.658458\pi$
$887$	−25228.5	−0.955007	−0.477504	+	0.878630i	$0.658458\pi$
$888$	0	0
$889$	−68914.2	−2.59989
$890$	−12157.4	−0.457883
$891$	0	0
$892$	10204.9	0.383055
$893$	−6638.68	−0.248774
$894$	0	0
$895$	10.3826	0.000387767 0
$896$	−4675.79	−0.174338
$897$	0	0
$898$	−27364.4	−1.01688
$899$	3303.91	0.122571
$900$	0	0
$901$	−45149.5	−1.66942
$902$	−36515.5	−1.34793
$903$	0	0
$904$	1016.93	0.0374144
$905$	−363.428	−0.0133489
$906$	0	0
$907$	15127.2	0.553794	0.276897	−	0.960900i	$-0.410694\pi$
$907$	15127.2	0.553794	0.276897	+	0.960900i	$0.410694\pi$
$908$	−14432.2	−0.527477
$909$	0	0
$910$	6958.56	0.253488
$911$	−8359.71	−0.304028	−0.152014	−	0.988378i	$-0.548576\pi$
$911$	−8359.71	−0.304028	−0.152014	+	0.988378i	$0.548576\pi$
$912$	0	0
$913$	65724.3	2.38243
$914$	−11951.3	−0.432509
$915$	0	0
$916$	−20763.9	−0.748973
$917$	58928.0	2.12211
$918$	0	0
$919$	14411.8	0.517304	0.258652	−	0.965971i	$-0.416722\pi$
$919$	14411.8	0.517304	0.258652	+	0.965971i	$0.416722\pi$
$920$	−1640.80	−0.0587994
$921$	0	0
$922$	21617.2	0.772154
$923$	−2315.60	−0.0825773
$924$	0	0
$925$	33.5075	0.00119105
$926$	14291.6	0.507182
$927$	0	0
$928$	517.313	0.0182992
$929$	28619.1	1.01072	0.505362	−	0.862908i	$-0.331359\pi$
$929$	28619.1	1.01072	0.505362	+	0.862908i	$0.331359\pi$
$930$	0	0
$931$	−21170.5	−0.745256
$932$	9546.70	0.335528
$933$	0	0
$934$	22868.3	0.801148
$935$	26857.9	0.939408
$936$	0	0
$937$	22287.3	0.777050	0.388525	−	0.921438i	$-0.372985\pi$
$937$	22287.3	0.777050	0.388525	+	0.921438i	$0.372985\pi$
$938$	53612.3	1.86621
$939$	0	0
$940$	−6217.79	−0.215747
$941$	−18919.0	−0.655411	−0.327706	−	0.944780i	$-0.606275\pi$
$941$	−18919.0	−0.655411	−0.327706	+	0.944780i	$0.606275\pi$
$942$	0	0
$943$	15819.3	0.546284
$944$	−7260.88	−0.250341
$945$	0	0
$946$	−14683.1	−0.504641
$947$	11526.1	0.395511	0.197756	−	0.980251i	$-0.436635\pi$
$947$	11526.1	0.395511	0.197756	+	0.980251i	$0.436635\pi$
$948$	0	0
$949$	−5422.88	−0.185494
$950$	−1067.69	−0.0364637
$951$	0	0
$952$	−33157.5	−1.12882
$953$	18715.8	0.636165	0.318083	−	0.948063i	$-0.396961\pi$
$953$	18715.8	0.636165	0.318083	+	0.948063i	$0.396961\pi$
$954$	0	0
$955$	−1092.03	−0.0370025
$956$	−27031.0	−0.914483
$957$	0	0
$958$	−22459.3	−0.757439
$959$	56462.6	1.90122
$960$	0	0
$961$	11977.6	0.402055
$962$	51.0629	0.00171137
$963$	0	0
$964$	−17983.0	−0.600822
$965$	−22399.1	−0.747205
$966$	0	0
$967$	5297.79	0.176179	0.0880896	−	0.996113i	$-0.471924\pi$
$967$	5297.79	0.176179	0.0880896	+	0.996113i	$0.471924\pi$
$968$	7282.83	0.241817
$969$	0	0
$970$	−12960.8	−0.429017
$971$	−24832.1	−0.820699	−0.410350	−	0.911928i	$-0.634593\pi$
$971$	−24832.1	−0.820699	−0.410350	+	0.911928i	$0.634593\pi$
$972$	0	0
$973$	−25833.7	−0.851172
$974$	1275.19	0.0419506
$975$	0	0
$976$	12861.3	0.421802
$977$	−14030.4	−0.459440	−0.229720	−	0.973257i	$-0.573781\pi$
$977$	−14030.4	−0.459440	−0.229720	+	0.973257i	$0.573781\pi$
$978$	0	0
$979$	−57556.5	−1.87897
$980$	−19828.3	−0.646317
$981$	0	0
$982$	−33357.6	−1.08400
$983$	5432.26	0.176259	0.0881293	−	0.996109i	$-0.471911\pi$
$983$	5432.26	0.176259	0.0881293	+	0.996109i	$0.471911\pi$
$984$	0	0
$985$	3075.47	0.0994850
$986$	3668.43	0.118485
$987$	0	0
$988$	−1627.08	−0.0523932
$989$	6361.04	0.204519
$990$	0	0
$991$	−10376.6	−0.332617	−0.166308	−	0.986074i	$-0.553185\pi$
$991$	−10376.6	−0.332617	−0.166308	+	0.986074i	$0.553185\pi$
$992$	6539.96	0.209318
$993$	0	0
$994$	8881.05	0.283390
$995$	−18988.4	−0.604998
$996$	0	0
$997$	−47582.0	−1.51147	−0.755736	−	0.654877i	$-0.772721\pi$
$997$	−47582.0	−1.51147	−0.755736	+	0.654877i	$0.772721\pi$
$998$	21508.0	0.682190
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.a.u.1.1	yes	4
3.2	odd	2		810.4.a.t.1.1	✓	4
9.2	odd	6		810.4.e.bh.271.4		8
9.4	even	3		810.4.e.bg.541.4		8
9.5	odd	6		810.4.e.bh.541.4		8
9.7	even	3		810.4.e.bg.271.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.4.a.t.1.1	✓	4	3.2	odd	2
810.4.a.u.1.1	yes	4	1.1	even	1	trivial
810.4.e.bg.271.4		8	9.7	even	3
810.4.e.bg.541.4		8	9.4	even	3
810.4.e.bh.271.4		8	9.2	odd	6
810.4.e.bh.541.4		8	9.5	odd	6

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.5296 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.5296 q^{7} +8.00000 q^{8} -10.0000 q^{10} -47.3429 q^{11} +19.0491 q^{13} -73.0592 q^{14} +16.0000 q^{16} +113.461 q^{17} -21.3538 q^{19} -20.0000 q^{20} -94.6859 q^{22} +41.0199 q^{23} +25.0000 q^{25} +38.0982 q^{26} -146.118 q^{28} +16.1660 q^{29} +204.374 q^{31} +32.0000 q^{32} +226.922 q^{34} +182.648 q^{35} +1.34030 q^{37} -42.7077 q^{38} -40.0000 q^{40} +385.648 q^{41} +155.072 q^{43} -189.372 q^{44} +82.0398 q^{46} +310.889 q^{47} +991.413 q^{49} +50.0000 q^{50} +76.1964 q^{52} -397.930 q^{53} +236.715 q^{55} -292.237 q^{56} +32.3321 q^{58} -453.805 q^{59} +803.829 q^{61} +408.747 q^{62} +64.0000 q^{64} -95.2454 q^{65} -733.819 q^{67} +453.844 q^{68} +365.296 q^{70} -121.560 q^{71} -284.679 q^{73} +2.68060 q^{74} -85.4153 q^{76} +1729.42 q^{77} -629.730 q^{79} -80.0000 q^{80} +771.297 q^{82} -1388.26 q^{83} -567.305 q^{85} +310.144 q^{86} -378.744 q^{88} +1215.74 q^{89} -695.856 q^{91} +164.080 q^{92} +621.779 q^{94} +106.769 q^{95} +1296.08 q^{97} +1982.83 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} + 2 q^{7} + 32 q^{8} - 40 q^{10} - 36 q^{11} + 32 q^{13} + 4 q^{14} + 64 q^{16} + 90 q^{17} + 164 q^{19} - 80 q^{20} - 72 q^{22} + 42 q^{23} + 100 q^{25} + 64 q^{26} + 8 q^{28} + 36 q^{29} + 446 q^{31} + 128 q^{32} + 180 q^{34} - 10 q^{35} + 686 q^{37} + 328 q^{38} - 160 q^{40} + 108 q^{41} + 470 q^{43} - 144 q^{44} + 84 q^{46} + 804 q^{47} + 1338 q^{49} + 200 q^{50} + 128 q^{52} + 168 q^{53} + 180 q^{55} + 16 q^{56} + 72 q^{58} + 768 q^{59} + 596 q^{61} + 892 q^{62} + 256 q^{64} - 160 q^{65} - 424 q^{67} + 360 q^{68} - 20 q^{70} + 348 q^{71} - 94 q^{73} + 1372 q^{74} + 656 q^{76} + 630 q^{77} + 1088 q^{79} - 320 q^{80} + 216 q^{82} + 522 q^{83} - 450 q^{85} + 940 q^{86} - 288 q^{88} + 1128 q^{89} + 2842 q^{91} + 168 q^{92} + 1608 q^{94} - 820 q^{95} + 2486 q^{97} + 2676 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 + 2 * q^7 + 32 * q^8 - 40 * q^10 - 36 * q^11 + 32 * q^13 + 4 * q^14 + 64 * q^16 + 90 * q^17 + 164 * q^19 - 80 * q^20 - 72 * q^22 + 42 * q^23 + 100 * q^25 + 64 * q^26 + 8 * q^28 + 36 * q^29 + 446 * q^31 + 128 * q^32 + 180 * q^34 - 10 * q^35 + 686 * q^37 + 328 * q^38 - 160 * q^40 + 108 * q^41 + 470 * q^43 - 144 * q^44 + 84 * q^46 + 804 * q^47 + 1338 * q^49 + 200 * q^50 + 128 * q^52 + 168 * q^53 + 180 * q^55 + 16 * q^56 + 72 * q^58 + 768 * q^59 + 596 * q^61 + 892 * q^62 + 256 * q^64 - 160 * q^65 - 424 * q^67 + 360 * q^68 - 20 * q^70 + 348 * q^71 - 94 * q^73 + 1372 * q^74 + 656 * q^76 + 630 * q^77 + 1088 * q^79 - 320 * q^80 + 216 * q^82 + 522 * q^83 - 450 * q^85 + 940 * q^86 - 288 * q^88 + 1128 * q^89 + 2842 * q^91 + 168 * q^92 + 1608 * q^94 - 820 * q^95 + 2486 * q^97 + 2676 * q^98

Introduction

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