LMFDB - Embedded newform 405.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} +O(q^{10})$	\(q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} -3.66025 q^{10} -37.4974 q^{11} -38.9282 q^{13} -5.07180 q^{14} +51.4256 q^{16} +80.9948 q^{17} +112.779 q^{19} +37.3205 q^{20} -27.4500 q^{22} -13.4256 q^{23} +25.0000 q^{25} -28.4974 q^{26} +51.7128 q^{28} +43.4308 q^{29} -149.785 q^{31} +128.210 q^{32} +59.2923 q^{34} +34.6410 q^{35} -218.344 q^{37} +82.5603 q^{38} +56.6025 q^{40} +372.138 q^{41} +460.200 q^{43} +279.885 q^{44} -9.82824 q^{46} +214.287 q^{47} -295.000 q^{49} +18.3013 q^{50} +290.564 q^{52} +445.205 q^{53} +187.487 q^{55} +78.4308 q^{56} +31.7935 q^{58} +401.497 q^{59} +1.44624 q^{61} -109.650 q^{62} -317.549 q^{64} +194.641 q^{65} -816.067 q^{67} -604.554 q^{68} +25.3590 q^{70} +147.518 q^{71} +432.651 q^{73} -159.839 q^{74} -841.797 q^{76} +259.790 q^{77} +384.133 q^{79} -257.128 q^{80} +272.424 q^{82} +1261.23 q^{83} -404.974 q^{85} +336.890 q^{86} +424.490 q^{88} +513.000 q^{89} +269.703 q^{91} +100.210 q^{92} +156.869 q^{94} -563.897 q^{95} -1096.10 q^{97} -215.955 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 8 * q^4 - 10 * q^5 + 12 * q^8	$ 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + 10 q^{10} + 22 q^{11} - 64 q^{13} - 24 q^{14} - 8 q^{16} - 32 q^{17} - 10 q^{19} + 40 q^{20} - 190 q^{22} + 84 q^{23} + 50 q^{25} + 40 q^{26} + 48 q^{28} + 170 q^{29} - 258 q^{31} + 104 q^{32} + 368 q^{34} + 76 q^{37} + 418 q^{38} - 60 q^{40} + 578 q^{41} + 380 q^{43} + 248 q^{44} - 276 q^{46} + 484 q^{47} - 590 q^{49} - 50 q^{50} + 304 q^{52} + 544 q^{53} - 110 q^{55} + 240 q^{56} - 314 q^{58} + 706 q^{59} + 668 q^{61} + 186 q^{62} + 224 q^{64} + 320 q^{65} - 1452 q^{67} - 544 q^{68} + 120 q^{70} + 974 q^{71} + 1184 q^{73} - 964 q^{74} - 776 q^{76} + 672 q^{77} + 408 q^{79} + 40 q^{80} - 290 q^{82} + 444 q^{83} + 160 q^{85} + 556 q^{86} + 1812 q^{88} + 1026 q^{89} + 96 q^{91} + 48 q^{92} - 580 q^{94} + 50 q^{95} - 668 q^{97} + 590 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 8 * q^4 - 10 * q^5 + 12 * q^8 + 10 * q^10 + 22 * q^11 - 64 * q^13 - 24 * q^14 - 8 * q^16 - 32 * q^17 - 10 * q^19 + 40 * q^20 - 190 * q^22 + 84 * q^23 + 50 * q^25 + 40 * q^26 + 48 * q^28 + 170 * q^29 - 258 * q^31 + 104 * q^32 + 368 * q^34 + 76 * q^37 + 418 * q^38 - 60 * q^40 + 578 * q^41 + 380 * q^43 + 248 * q^44 - 276 * q^46 + 484 * q^47 - 590 * q^49 - 50 * q^50 + 304 * q^52 + 544 * q^53 - 110 * q^55 + 240 * q^56 - 314 * q^58 + 706 * q^59 + 668 * q^61 + 186 * q^62 + 224 * q^64 + 320 * q^65 - 1452 * q^67 - 544 * q^68 + 120 * q^70 + 974 * q^71 + 1184 * q^73 - 964 * q^74 - 776 * q^76 + 672 * q^77 + 408 * q^79 + 40 * q^80 - 290 * q^82 + 444 * q^83 + 160 * q^85 + 556 * q^86 + 1812 * q^88 + 1026 * q^89 + 96 * q^91 + 48 * q^92 - 580 * q^94 + 50 * q^95 - 668 * q^97 + 590 * 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 $

$2$	0.732051	0.258819	0.129410	−	0.991591i	$-0.458692\pi$
$2$	0.732051	0.258819	0.129410	+	0.991591i	$0.458692\pi$
$3$	0	0
$3$	0	0
$4$	−7.46410	−0.933013
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−6.92820	−0.374088	−0.187044	−	0.982352i	$-0.559891\pi$
$7$	−6.92820	−0.374088	−0.187044	+	0.982352i	$0.559891\pi$
$8$	−11.3205	−0.500301
$9$	0	0
$10$	−3.66025	−0.115747
$11$	−37.4974	−1.02781	−0.513904	−	0.857847i	$-0.671801\pi$
$11$	−37.4974	−1.02781	−0.513904	+	0.857847i	$0.671801\pi$
$12$	0	0
$13$	−38.9282	−0.830519	−0.415259	−	0.909703i	$-0.636309\pi$
$13$	−38.9282	−0.830519	−0.415259	+	0.909703i	$0.636309\pi$
$14$	−5.07180	−0.0968211
$15$	0	0
$16$	51.4256	0.803525
$17$	80.9948	1.15554	0.577769	−	0.816201i	$-0.303924\pi$
$17$	80.9948	1.15554	0.577769	+	0.816201i	$0.303924\pi$
$18$	0	0
$19$	112.779	1.36176	0.680878	−	0.732396i	$-0.261598\pi$
$19$	112.779	1.36176	0.680878	+	0.732396i	$0.261598\pi$
$20$	37.3205	0.417256
$21$	0	0
$22$	−27.4500	−0.266017
$23$	−13.4256	−0.121715	−0.0608573	−	0.998146i	$-0.519383\pi$
$23$	−13.4256	−0.121715	−0.0608573	+	0.998146i	$0.519383\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−28.4974	−0.214954
$27$	0	0
$28$	51.7128	0.349029
$29$	43.4308	0.278100	0.139050	−	0.990285i	$-0.455595\pi$
$29$	43.4308	0.278100	0.139050	+	0.990285i	$0.455595\pi$
$30$	0	0
$31$	−149.785	−0.867810	−0.433905	−	0.900959i	$-0.642865\pi$
$31$	−149.785	−0.867810	−0.433905	+	0.900959i	$0.642865\pi$
$32$	128.210	0.708268
$33$	0	0
$34$	59.2923	0.299075
$35$	34.6410	0.167297
$36$	0	0
$37$	−218.344	−0.970147	−0.485074	−	0.874473i	$-0.661207\pi$
$37$	−218.344	−0.970147	−0.485074	+	0.874473i	$0.661207\pi$
$38$	82.5603	0.352449
$39$	0	0
$40$	56.6025	0.223741
$41$	372.138	1.41752	0.708759	−	0.705450i	$-0.249255\pi$
$41$	372.138	1.41752	0.708759	+	0.705450i	$0.249255\pi$
$42$	0	0
$43$	460.200	1.63209	0.816045	−	0.577989i	$-0.196162\pi$
$43$	460.200	1.63209	0.816045	+	0.577989i	$0.196162\pi$
$44$	279.885	0.958959
$45$	0	0
$46$	−9.82824	−0.0315021
$47$	214.287	0.665043	0.332521	−	0.943096i	$-0.392101\pi$
$47$	214.287	0.665043	0.332521	+	0.943096i	$0.392101\pi$
$48$	0	0
$49$	−295.000	−0.860058
$50$	18.3013	0.0517638
$51$	0	0
$52$	290.564	0.774884
$53$	445.205	1.15384	0.576921	−	0.816800i	$-0.304254\pi$
$53$	445.205	1.15384	0.576921	+	0.816800i	$0.304254\pi$
$54$	0	0
$55$	187.487	0.459650
$56$	78.4308	0.187156
$57$	0	0
$58$	31.7935	0.0719775
$59$	401.497	0.885941	0.442970	−	0.896536i	$-0.353925\pi$
$59$	401.497	0.885941	0.442970	+	0.896536i	$0.353925\pi$
$60$	0	0
$61$	1.44624	0.00303562	0.00151781	−	0.999999i	$-0.499517\pi$
$61$	1.44624	0.00303562	0.00151781	+	0.999999i	$0.499517\pi$
$62$	−109.650	−0.224606
$63$	0	0
$64$	−317.549	−0.620212
$65$	194.641	0.371419
$66$	0	0
$67$	−816.067	−1.48804	−0.744018	−	0.668160i	$-0.767082\pi$
$67$	−816.067	−1.48804	−0.744018	+	0.668160i	$0.767082\pi$
$68$	−604.554	−1.07813
$69$	0	0
$70$	25.3590	0.0432997
$71$	147.518	0.246580	0.123290	−	0.992371i	$-0.460655\pi$
$71$	147.518	0.246580	0.123290	+	0.992371i	$0.460655\pi$
$72$	0	0
$73$	432.651	0.693671	0.346836	−	0.937926i	$-0.387256\pi$
$73$	432.651	0.693671	0.346836	+	0.937926i	$0.387256\pi$
$74$	−159.839	−0.251093
$75$	0	0
$76$	−841.797	−1.27054
$77$	259.790	0.384491
$78$	0	0
$79$	384.133	0.547068	0.273534	−	0.961862i	$-0.411807\pi$
$79$	384.133	0.547068	0.273534	+	0.961862i	$0.411807\pi$
$80$	−257.128	−0.359347
$81$	0	0
$82$	272.424	0.366881
$83$	1261.23	1.66793	0.833964	−	0.551819i	$-0.186066\pi$
$83$	1261.23	1.66793	0.833964	+	0.551819i	$0.186066\pi$
$84$	0	0
$85$	−404.974	−0.516772
$86$	336.890	0.422416
$87$	0	0
$88$	424.490	0.514213
$89$	513.000	0.610988	0.305494	−	0.952194i	$-0.401178\pi$
$89$	513.000	0.610988	0.305494	+	0.952194i	$0.401178\pi$
$90$	0	0
$91$	269.703	0.310687
$92$	100.210	0.113561
$93$	0	0
$94$	156.869	0.172126
$95$	−563.897	−0.608996
$96$	0	0
$97$	−1096.10	−1.14734	−0.573672	−	0.819085i	$-0.694482\pi$
$97$	−1096.10	−1.14734	−0.573672	+	0.819085i	$0.694482\pi$
$98$	−215.955	−0.222599
$99$	0	0
$100$	−186.603	−0.186603
$101$	−1194.09	−1.17640	−0.588198	−	0.808717i	$-0.700163\pi$
$101$	−1194.09	−1.17640	−0.588198	+	0.808717i	$0.700163\pi$
$102$	0	0
$103$	181.503	0.173631	0.0868154	−	0.996224i	$-0.472331\pi$
$103$	181.503	0.173631	0.0868154	+	0.996224i	$0.472331\pi$
$104$	440.687	0.415509
$105$	0	0
$106$	325.913	0.298636
$107$	−611.667	−0.552636	−0.276318	−	0.961066i	$-0.589114\pi$
$107$	−611.667	−0.552636	−0.276318	+	0.961066i	$0.589114\pi$
$108$	0	0
$109$	550.467	0.483717	0.241859	−	0.970312i	$-0.422243\pi$
$109$	550.467	0.483717	0.241859	+	0.970312i	$0.422243\pi$
$110$	137.250	0.118966
$111$	0	0
$112$	−356.287	−0.300589
$113$	475.902	0.396187	0.198094	−	0.980183i	$-0.436525\pi$
$113$	475.902	0.396187	0.198094	+	0.980183i	$0.436525\pi$
$114$	0	0
$115$	67.1281	0.0544324
$116$	−324.172	−0.259471
$117$	0	0
$118$	293.917	0.229298
$119$	−561.149	−0.432272
$120$	0	0
$121$	75.0567	0.0563912
$122$	1.05872	0.000785676 0
$123$	0	0
$124$	1118.01	0.809678
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1645.91	−1.15001	−0.575003	−	0.818152i	$-0.694999\pi$
$127$	−1645.91	−1.15001	−0.575003	+	0.818152i	$0.694999\pi$
$128$	−1258.14	−0.868791
$129$	0	0
$130$	142.487	0.0961304
$131$	−386.707	−0.257914	−0.128957	−	0.991650i	$-0.541163\pi$
$131$	−386.707	−0.257914	−0.128957	+	0.991650i	$0.541163\pi$
$132$	0	0
$133$	−781.359	−0.509417
$134$	−597.402	−0.385132
$135$	0	0
$136$	−916.903	−0.578116
$137$	2973.22	1.85416	0.927078	−	0.374869i	$-0.122312\pi$
$137$	2973.22	1.85416	0.927078	+	0.374869i	$0.122312\pi$
$138$	0	0
$139$	2347.35	1.43237	0.716187	−	0.697909i	$-0.245886\pi$
$139$	2347.35	1.43237	0.716187	+	0.697909i	$0.245886\pi$
$140$	−258.564	−0.156090
$141$	0	0
$142$	107.991	0.0638196
$143$	1459.71	0.853614
$144$	0	0
$145$	−217.154	−0.124370
$146$	316.723	0.179535
$147$	0	0
$148$	1629.74	0.905160
$149$	−1402.09	−0.770898	−0.385449	−	0.922729i	$-0.625953\pi$
$149$	−1402.09	−0.770898	−0.385449	+	0.922729i	$0.625953\pi$
$150$	0	0
$151$	−2941.26	−1.58514	−0.792571	−	0.609780i	$-0.791258\pi$
$151$	−2941.26	−1.58514	−0.792571	+	0.609780i	$0.791258\pi$
$152$	−1276.72	−0.681288
$153$	0	0
$154$	190.179	0.0995135
$155$	748.923	0.388096
$156$	0	0
$157$	2551.98	1.29726	0.648631	−	0.761103i	$-0.275342\pi$
$157$	2551.98	1.29726	0.648631	+	0.761103i	$0.275342\pi$
$158$	281.205	0.141592
$159$	0	0
$160$	−641.051	−0.316747
$161$	93.0155	0.0455320
$162$	0	0
$163$	351.559	0.168934	0.0844669	−	0.996426i	$-0.473081\pi$
$163$	351.559	0.168934	0.0844669	+	0.996426i	$0.473081\pi$
$164$	−2777.68	−1.32256
$165$	0	0
$166$	923.285	0.431692
$167$	20.9893	0.00972576	0.00486288	−	0.999988i	$-0.498452\pi$
$167$	20.9893	0.00972576	0.00486288	+	0.999988i	$0.498452\pi$
$168$	0	0
$169$	−681.595	−0.310239
$170$	−296.462	−0.133750
$171$	0	0
$172$	−3434.98	−1.52276
$173$	−4524.61	−1.98844	−0.994219	−	0.107374i	$-0.965756\pi$
$173$	−4524.61	−1.98844	−0.994219	+	0.107374i	$0.965756\pi$
$174$	0	0
$175$	−173.205	−0.0748176
$176$	−1928.33	−0.825871
$177$	0	0
$178$	375.542	0.158135
$179$	3626.79	1.51441	0.757204	−	0.653179i	$-0.226565\pi$
$179$	3626.79	1.51441	0.757204	+	0.653179i	$0.226565\pi$
$180$	0	0
$181$	3551.41	1.45842	0.729210	−	0.684289i	$-0.239888\pi$
$181$	3551.41	1.45842	0.729210	+	0.684289i	$0.239888\pi$
$182$	197.436	0.0804117
$183$	0	0
$184$	151.985	0.0608939
$185$	1091.72	0.433863
$186$	0	0
$187$	−3037.10	−1.18767
$188$	−1599.46	−0.620493
$189$	0	0
$190$	−412.801	−0.157620
$191$	−279.651	−0.105941	−0.0529707	−	0.998596i	$-0.516869\pi$
$191$	−279.651	−0.105941	−0.0529707	+	0.998596i	$0.516869\pi$
$192$	0	0
$193$	266.159	0.0992672	0.0496336	−	0.998767i	$-0.484195\pi$
$193$	266.159	0.0992672	0.0496336	+	0.998767i	$0.484195\pi$
$194$	−802.403	−0.296954
$195$	0	0
$196$	2201.91	0.802445
$197$	−4731.12	−1.71106	−0.855528	−	0.517756i	$-0.826768\pi$
$197$	−4731.12	−1.71106	−0.855528	+	0.517756i	$0.826768\pi$
$198$	0	0
$199$	2879.69	1.02581	0.512904	−	0.858446i	$-0.328570\pi$
$199$	2879.69	1.02581	0.512904	+	0.858446i	$0.328570\pi$
$200$	−283.013	−0.100060
$201$	0	0
$202$	−874.132	−0.304474
$203$	−300.897	−0.104034
$204$	0	0
$205$	−1860.69	−0.633934
$206$	132.869	0.0449390
$207$	0	0
$208$	−2001.91	−0.667343
$209$	−4228.94	−1.39963
$210$	0	0
$211$	2728.21	0.890130	0.445065	−	0.895498i	$-0.353181\pi$
$211$	2728.21	0.890130	0.445065	+	0.895498i	$0.353181\pi$
$212$	−3323.06	−1.07655
$213$	0	0
$214$	−447.771	−0.143033
$215$	−2301.00	−0.729892
$216$	0	0
$217$	1037.74	0.324637
$218$	402.970	0.125195
$219$	0	0
$220$	−1399.42	−0.428859
$221$	−3152.98	−0.959695
$222$	0	0
$223$	−3674.85	−1.10352	−0.551762	−	0.834002i	$-0.686044\pi$
$223$	−3674.85	−1.10352	−0.551762	+	0.834002i	$0.686044\pi$
$224$	−888.267	−0.264954
$225$	0	0
$226$	348.385	0.102541
$227$	5773.91	1.68823	0.844115	−	0.536163i	$-0.180127\pi$
$227$	5773.91	1.68823	0.844115	+	0.536163i	$0.180127\pi$
$228$	0	0
$229$	4818.28	1.39040	0.695198	−	0.718818i	$-0.255317\pi$
$229$	4818.28	1.39040	0.695198	+	0.718818i	$0.255317\pi$
$230$	49.1412	0.0140882
$231$	0	0
$232$	−491.659	−0.139133
$233$	91.0828	0.0256096	0.0128048	−	0.999918i	$-0.495924\pi$
$233$	91.0828	0.0256096	0.0128048	+	0.999918i	$0.495924\pi$
$234$	0	0
$235$	−1071.44	−0.297416
$236$	−2996.82	−0.826594
$237$	0	0
$238$	−410.789	−0.111880
$239$	5486.84	1.48500	0.742498	−	0.669848i	$-0.233641\pi$
$239$	5486.84	1.48500	0.742498	+	0.669848i	$0.233641\pi$
$240$	0	0
$241$	3017.19	0.806450	0.403225	−	0.915101i	$-0.367889\pi$
$241$	3017.19	0.806450	0.403225	+	0.915101i	$0.367889\pi$
$242$	54.9453	0.0145951
$243$	0	0
$244$	−10.7949	−0.00283227
$245$	1475.00	0.384630
$246$	0	0
$247$	−4390.30	−1.13096
$248$	1695.64	0.434166
$249$	0	0
$250$	−91.5064	−0.0231495
$251$	−2740.98	−0.689280	−0.344640	−	0.938735i	$-0.611999\pi$
$251$	−2740.98	−0.689280	−0.344640	+	0.938735i	$0.611999\pi$
$252$	0	0
$253$	503.426	0.125099
$254$	−1204.89	−0.297643
$255$	0	0
$256$	1619.36	0.395352
$257$	−2521.40	−0.611987	−0.305993	−	0.952034i	$-0.598989\pi$
$257$	−2521.40	−0.611987	−0.305993	+	0.952034i	$0.598989\pi$
$258$	0	0
$259$	1512.73	0.362920
$260$	−1452.82	−0.346539
$261$	0	0
$262$	−283.089	−0.0667531
$263$	−4487.72	−1.05218	−0.526092	−	0.850428i	$-0.676343\pi$
$263$	−4487.72	−1.05218	−0.526092	+	0.850428i	$0.676343\pi$
$264$	0	0
$265$	−2226.03	−0.516014
$266$	−571.994	−0.131847
$267$	0	0
$268$	6091.20	1.38836
$269$	8531.47	1.93373	0.966864	−	0.255291i	$-0.0821712\pi$
$269$	8531.47	1.93373	0.966864	+	0.255291i	$0.0821712\pi$
$270$	0	0
$271$	−123.097	−0.0275927	−0.0137964	−	0.999905i	$-0.504392\pi$
$271$	−123.097	−0.0275927	−0.0137964	+	0.999905i	$0.504392\pi$
$272$	4165.21	0.928504
$273$	0	0
$274$	2176.55	0.479891
$275$	−937.436	−0.205562
$276$	0	0
$277$	−6557.58	−1.42241	−0.711204	−	0.702986i	$-0.751850\pi$
$277$	−6557.58	−1.42241	−0.711204	+	0.702986i	$0.751850\pi$
$278$	1718.38	0.370726
$279$	0	0
$280$	−392.154	−0.0836989
$281$	7084.04	1.50391	0.751954	−	0.659215i	$-0.229111\pi$
$281$	7084.04	1.50391	0.751954	+	0.659215i	$0.229111\pi$
$282$	0	0
$283$	2072.93	0.435417	0.217709	−	0.976014i	$-0.430142\pi$
$283$	2072.93	0.435417	0.217709	+	0.976014i	$0.430142\pi$
$284$	−1101.09	−0.230062
$285$	0	0
$286$	1068.58	0.220932
$287$	−2578.25	−0.530276
$288$	0	0
$289$	1647.16	0.335267
$290$	−158.968	−0.0321893
$291$	0	0
$292$	−3229.35	−0.647204
$293$	7128.40	1.42132	0.710658	−	0.703537i	$-0.248397\pi$
$293$	7128.40	1.42132	0.710658	+	0.703537i	$0.248397\pi$
$294$	0	0
$295$	−2007.49	−0.396205
$296$	2471.76	0.485365
$297$	0	0
$298$	−1026.40	−0.199523
$299$	522.635	0.101086
$300$	0	0
$301$	−3188.36	−0.610545
$302$	−2153.15	−0.410265
$303$	0	0
$304$	5799.75	1.09421
$305$	−7.23122	−0.00135757
$306$	0	0
$307$	−6334.70	−1.17766	−0.588828	−	0.808259i	$-0.700410\pi$
$307$	−6334.70	−1.17766	−0.588828	+	0.808259i	$0.700410\pi$
$308$	−1939.10	−0.358735
$309$	0	0
$310$	548.250	0.100447
$311$	7393.90	1.34813	0.674067	−	0.738671i	$-0.264546\pi$
$311$	7393.90	1.34813	0.674067	+	0.738671i	$0.264546\pi$
$312$	0	0
$313$	−933.055	−0.168496	−0.0842482	−	0.996445i	$-0.526849\pi$
$313$	−933.055	−0.168496	−0.0842482	+	0.996445i	$0.526849\pi$
$314$	1868.18	0.335756
$315$	0	0
$316$	−2867.21	−0.510421
$317$	8734.68	1.54760	0.773799	−	0.633432i	$-0.218354\pi$
$317$	8734.68	1.54760	0.773799	+	0.633432i	$0.218354\pi$
$318$	0	0
$319$	−1628.54	−0.285833
$320$	1587.74	0.277367
$321$	0	0
$322$	68.0920	0.0117845
$323$	9134.55	1.57356
$324$	0	0
$325$	−973.205	−0.166104
$326$	257.359	0.0437233
$327$	0	0
$328$	−4212.80	−0.709185
$329$	−1484.63	−0.248784
$330$	0	0
$331$	−1290.53	−0.214302	−0.107151	−	0.994243i	$-0.534173\pi$
$331$	−1290.53	−0.214302	−0.107151	+	0.994243i	$0.534173\pi$
$332$	−9413.95	−1.55620
$333$	0	0
$334$	15.3652	0.00251721
$335$	4080.33	0.665470
$336$	0	0
$337$	−10001.3	−1.61664	−0.808318	−	0.588746i	$-0.799622\pi$
$337$	−10001.3	−1.61664	−0.808318	+	0.588746i	$0.799622\pi$
$338$	−498.962	−0.0802958
$339$	0	0
$340$	3022.77	0.482155
$341$	5616.54	0.891943
$342$	0	0
$343$	4420.19	0.695825
$344$	−5209.70	−0.816535
$345$	0	0
$346$	−3312.24	−0.514645
$347$	6468.37	1.00069	0.500346	−	0.865825i	$-0.333206\pi$
$347$	6468.37	1.00069	0.500346	+	0.865825i	$0.333206\pi$
$348$	0	0
$349$	11169.8	1.71319	0.856597	−	0.515985i	$-0.172574\pi$
$349$	11169.8	1.71319	0.856597	+	0.515985i	$0.172574\pi$
$350$	−126.795	−0.0193642
$351$	0	0
$352$	−4807.55	−0.727964
$353$	1816.77	0.273930	0.136965	−	0.990576i	$-0.456265\pi$
$353$	1816.77	0.273930	0.136965	+	0.990576i	$0.456265\pi$
$354$	0	0
$355$	−737.590	−0.110274
$356$	−3829.08	−0.570059
$357$	0	0
$358$	2654.99	0.391957
$359$	918.073	0.134969	0.0674847	−	0.997720i	$-0.478503\pi$
$359$	918.073	0.134969	0.0674847	+	0.997720i	$0.478503\pi$
$360$	0	0
$361$	5860.21	0.854382
$362$	2599.81	0.377467
$363$	0	0
$364$	−2013.09	−0.289875
$365$	−2163.26	−0.310219
$366$	0	0
$367$	7354.27	1.04602	0.523011	−	0.852326i	$-0.324809\pi$
$367$	7354.27	1.04602	0.523011	+	0.852326i	$0.324809\pi$
$368$	−690.421	−0.0978008
$369$	0	0
$370$	799.193	0.112292
$371$	−3084.47	−0.431638
$372$	0	0
$373$	−602.607	−0.0836509	−0.0418255	−	0.999125i	$-0.513317\pi$
$373$	−602.607	−0.0836509	−0.0418255	+	0.999125i	$0.513317\pi$
$374$	−2223.31	−0.307392
$375$	0	0
$376$	−2425.84	−0.332721
$377$	−1690.68	−0.230967
$378$	0	0
$379$	13319.0	1.80515	0.902575	−	0.430533i	$-0.141674\pi$
$379$	13319.0	1.80515	0.902575	+	0.430533i	$0.141674\pi$
$380$	4208.99	0.568201
$381$	0	0
$382$	−204.718	−0.0274197
$383$	−8537.75	−1.13906	−0.569528	−	0.821972i	$-0.692874\pi$
$383$	−8537.75	−1.13906	−0.569528	+	0.821972i	$0.692874\pi$
$384$	0	0
$385$	−1298.95	−0.171950
$386$	194.842	0.0256922
$387$	0	0
$388$	8181.42	1.07049
$389$	−12640.0	−1.64749	−0.823744	−	0.566962i	$-0.808119\pi$
$389$	−12640.0	−1.64749	−0.823744	+	0.566962i	$0.808119\pi$
$390$	0	0
$391$	−1087.41	−0.140646
$392$	3339.55	0.430288
$393$	0	0
$394$	−3463.42	−0.442854
$395$	−1920.67	−0.244656
$396$	0	0
$397$	5473.94	0.692013	0.346006	−	0.938232i	$-0.387538\pi$
$397$	5473.94	0.692013	0.346006	+	0.938232i	$0.387538\pi$
$398$	2108.08	0.265499
$399$	0	0
$400$	1285.64	0.160705
$401$	13045.6	1.62461	0.812305	−	0.583233i	$-0.198212\pi$
$401$	13045.6	1.62461	0.812305	+	0.583233i	$0.198212\pi$
$402$	0	0
$403$	5830.85	0.720732
$404$	8912.79	1.09759
$405$	0	0
$406$	−220.272	−0.0269259
$407$	8187.32	0.997126
$408$	0	0
$409$	−7182.72	−0.868368	−0.434184	−	0.900824i	$-0.642963\pi$
$409$	−7182.72	−0.868368	−0.434184	+	0.900824i	$0.642963\pi$
$410$	−1362.12	−0.164074
$411$	0	0
$412$	−1354.75	−0.162000
$413$	−2781.66	−0.331420
$414$	0	0
$415$	−6306.15	−0.745920
$416$	−4990.99	−0.588230
$417$	0	0
$418$	−3095.80	−0.362250
$419$	8720.40	1.01675	0.508376	−	0.861135i	$-0.330246\pi$
$419$	8720.40	1.01675	0.508376	+	0.861135i	$0.330246\pi$
$420$	0	0
$421$	6553.03	0.758611	0.379305	−	0.925272i	$-0.376163\pi$
$421$	6553.03	0.758611	0.379305	+	0.925272i	$0.376163\pi$
$422$	1997.18	0.230383
$423$	0	0
$424$	−5039.95	−0.577268
$425$	2024.87	0.231107
$426$	0	0
$427$	−10.0199	−0.00113559
$428$	4565.54	0.515616
$429$	0	0
$430$	−1684.45	−0.188910
$431$	5217.70	0.583127	0.291564	−	0.956551i	$-0.405824\pi$
$431$	5217.70	0.583127	0.291564	+	0.956551i	$0.405824\pi$
$432$	0	0
$433$	3377.56	0.374862	0.187431	−	0.982278i	$-0.439984\pi$
$433$	3377.56	0.374862	0.187431	+	0.982278i	$0.439984\pi$
$434$	759.677	0.0840223
$435$	0	0
$436$	−4108.74	−0.451314
$437$	−1514.13	−0.165746
$438$	0	0
$439$	2138.92	0.232540	0.116270	−	0.993218i	$-0.462906\pi$
$439$	2138.92	0.232540	0.116270	+	0.993218i	$0.462906\pi$
$440$	−2122.45	−0.229963
$441$	0	0
$442$	−2308.14	−0.248387
$443$	8512.62	0.912973	0.456486	−	0.889730i	$-0.349108\pi$
$443$	8512.62	0.912973	0.456486	+	0.889730i	$0.349108\pi$
$444$	0	0
$445$	−2565.00	−0.273242
$446$	−2690.17	−0.285613
$447$	0	0
$448$	2200.04	0.232014
$449$	−4542.67	−0.477465	−0.238733	−	0.971085i	$-0.576732\pi$
$449$	−4542.67	−0.477465	−0.238733	+	0.971085i	$0.576732\pi$
$450$	0	0
$451$	−13954.2	−1.45694
$452$	−3552.18	−0.369648
$453$	0	0
$454$	4226.80	0.436946
$455$	−1348.51	−0.138943
$456$	0	0
$457$	−17942.2	−1.83654	−0.918272	−	0.395951i	$-0.870415\pi$
$457$	−17942.2	−1.83654	−0.918272	+	0.395951i	$0.870415\pi$
$458$	3527.22	0.359861
$459$	0	0
$460$	−501.051	−0.0507862
$461$	−5180.89	−0.523423	−0.261711	−	0.965146i	$-0.584287\pi$
$461$	−5180.89	−0.523423	−0.261711	+	0.965146i	$0.584287\pi$
$462$	0	0
$463$	14192.9	1.42462	0.712310	−	0.701864i	$-0.247649\pi$
$463$	14192.9	1.42462	0.712310	+	0.701864i	$0.247649\pi$
$464$	2233.46	0.223460
$465$	0	0
$466$	66.6773	0.00662825
$467$	−11886.3	−1.17780	−0.588901	−	0.808205i	$-0.700439\pi$
$467$	−11886.3	−1.17780	−0.588901	+	0.808205i	$0.700439\pi$
$468$	0	0
$469$	5653.88	0.556656
$470$	−784.346	−0.0769769
$471$	0	0
$472$	−4545.15	−0.443237
$473$	−17256.3	−1.67748
$474$	0	0
$475$	2819.49	0.272351
$476$	4188.47	0.403316
$477$	0	0
$478$	4016.65	0.384345
$479$	−4319.46	−0.412027	−0.206014	−	0.978549i	$-0.566049\pi$
$479$	−4319.46	−0.412027	−0.206014	+	0.978549i	$0.566049\pi$
$480$	0	0
$481$	8499.72	0.805725
$482$	2208.74	0.208725
$483$	0	0
$484$	−560.231	−0.0526137
$485$	5480.51	0.513108
$486$	0	0
$487$	−6436.94	−0.598944	−0.299472	−	0.954105i	$-0.596810\pi$
$487$	−6436.94	−0.598944	−0.299472	+	0.954105i	$0.596810\pi$
$488$	−16.3722	−0.00151872
$489$	0	0
$490$	1079.77	0.0995495
$491$	−2897.73	−0.266340	−0.133170	−	0.991093i	$-0.542516\pi$
$491$	−2897.73	−0.266340	−0.133170	+	0.991093i	$0.542516\pi$
$492$	0	0
$493$	3517.67	0.321355
$494$	−3213.92	−0.292715
$495$	0	0
$496$	−7702.77	−0.697307
$497$	−1022.03	−0.0922425
$498$	0	0
$499$	15054.6	1.35057	0.675286	−	0.737556i	$-0.264020\pi$
$499$	15054.6	1.35057	0.675286	+	0.737556i	$0.264020\pi$
$500$	933.013	0.0834512
$501$	0	0
$502$	−2006.54	−0.178399
$503$	15582.5	1.38129	0.690646	−	0.723193i	$-0.257326\pi$
$503$	15582.5	1.38129	0.690646	+	0.723193i	$0.257326\pi$
$504$	0	0
$505$	5970.43	0.526101
$506$	368.534	0.0323781
$507$	0	0
$508$	12285.2	1.07297
$509$	4071.01	0.354508	0.177254	−	0.984165i	$-0.443279\pi$
$509$	4071.01	0.354508	0.177254	+	0.984165i	$0.443279\pi$
$510$	0	0
$511$	−2997.50	−0.259494
$512$	11250.6	0.971116
$513$	0	0
$514$	−1845.79	−0.158394
$515$	−907.513	−0.0776501
$516$	0	0
$517$	−8035.22	−0.683537
$518$	1107.39	0.0939307
$519$	0	0
$520$	−2203.44	−0.185821
$521$	−18597.9	−1.56389	−0.781947	−	0.623345i	$-0.785773\pi$
$521$	−18597.9	−1.56389	−0.781947	+	0.623345i	$0.785773\pi$
$522$	0	0
$523$	−12073.5	−1.00944	−0.504722	−	0.863282i	$-0.668405\pi$
$523$	−12073.5	−1.00944	−0.504722	+	0.863282i	$0.668405\pi$
$524$	2886.42	0.240637
$525$	0	0
$526$	−3285.24	−0.272325
$527$	−12131.8	−1.00279
$528$	0	0
$529$	−11986.8	−0.985186
$530$	−1629.56	−0.133554
$531$	0	0
$532$	5832.14	0.475292
$533$	−14486.7	−1.17728
$534$	0	0
$535$	3058.33	0.247146
$536$	9238.29	0.744465
$537$	0	0
$538$	6245.47	0.500486
$539$	11061.7	0.883976
$540$	0	0
$541$	1802.04	0.143208	0.0716041	−	0.997433i	$-0.477188\pi$
$541$	1802.04	0.143208	0.0716041	+	0.997433i	$0.477188\pi$
$542$	−90.1134	−0.00714152
$543$	0	0
$544$	10384.4	0.818430
$545$	−2752.33	−0.216325
$546$	0	0
$547$	6589.73	0.515094	0.257547	−	0.966266i	$-0.417086\pi$
$547$	6589.73	0.515094	0.257547	+	0.966266i	$0.417086\pi$
$548$	−22192.4	−1.72995
$549$	0	0
$550$	−686.250	−0.0532033
$551$	4898.10	0.378704
$552$	0	0
$553$	−2661.35	−0.204651
$554$	−4800.48	−0.368146
$555$	0	0
$556$	−17520.9	−1.33642
$557$	−21128.6	−1.60727	−0.803633	−	0.595125i	$-0.797103\pi$
$557$	−21128.6	−1.60727	−0.803633	+	0.595125i	$0.797103\pi$
$558$	0	0
$559$	−17914.8	−1.35548
$560$	1781.44	0.134428
$561$	0	0
$562$	5185.88	0.389240
$563$	110.427	0.00826636	0.00413318	−	0.999991i	$-0.498684\pi$
$563$	110.427	0.00826636	0.00413318	+	0.999991i	$0.498684\pi$
$564$	0	0
$565$	−2379.51	−0.177180
$566$	1517.49	0.112694
$567$	0	0
$568$	−1669.98	−0.123364
$569$	6930.08	0.510587	0.255294	−	0.966864i	$-0.417828\pi$
$569$	6930.08	0.510587	0.255294	+	0.966864i	$0.417828\pi$
$570$	0	0
$571$	21264.6	1.55849	0.779244	−	0.626721i	$-0.215603\pi$
$571$	21264.6	1.55849	0.779244	+	0.626721i	$0.215603\pi$
$572$	−10895.4	−0.796433
$573$	0	0
$574$	−1887.41	−0.137246
$575$	−335.641	−0.0243429
$576$	0	0
$577$	2927.39	0.211211	0.105606	−	0.994408i	$-0.466322\pi$
$577$	2927.39	0.211211	0.105606	+	0.994408i	$0.466322\pi$
$578$	1205.81	0.0867734
$579$	0	0
$580$	1620.86	0.116039
$581$	−8738.06	−0.623952
$582$	0	0
$583$	−16694.0	−1.18593
$584$	−4897.83	−0.347044
$585$	0	0
$586$	5218.35	0.367864
$587$	−21750.5	−1.52937	−0.764683	−	0.644407i	$-0.777104\pi$
$587$	−21750.5	−1.52937	−0.764683	+	0.644407i	$0.777104\pi$
$588$	0	0
$589$	−16892.6	−1.18175
$590$	−1469.58	−0.102545
$591$	0	0
$592$	−11228.5	−0.779538
$593$	−5635.88	−0.390283	−0.195142	−	0.980775i	$-0.562517\pi$
$593$	−5635.88	−0.390283	−0.195142	+	0.980775i	$0.562517\pi$
$594$	0	0
$595$	2805.74	0.193318
$596$	10465.4	0.719258
$597$	0	0
$598$	382.596	0.0261630
$599$	−18324.4	−1.24994	−0.624970	−	0.780648i	$-0.714889\pi$
$599$	−18324.4	−1.24994	−0.624970	+	0.780648i	$0.714889\pi$
$600$	0	0
$601$	−2503.90	−0.169944	−0.0849719	−	0.996383i	$-0.527080\pi$
$601$	−2503.90	−0.169944	−0.0849719	+	0.996383i	$0.527080\pi$
$602$	−2334.04	−0.158021
$603$	0	0
$604$	21953.9	1.47896
$605$	−375.284	−0.0252189
$606$	0	0
$607$	17466.6	1.16796	0.583978	−	0.811770i	$-0.301495\pi$
$607$	17466.6	1.16796	0.583978	+	0.811770i	$0.301495\pi$
$608$	14459.5	0.964489
$609$	0	0
$610$	−5.29362	−0.000351365 0
$611$	−8341.82	−0.552330
$612$	0	0
$613$	−2244.45	−0.147883	−0.0739417	−	0.997263i	$-0.523558\pi$
$613$	−2244.45	−0.147883	−0.0739417	+	0.997263i	$0.523558\pi$
$614$	−4637.32	−0.304800
$615$	0	0
$616$	−2940.95	−0.192361
$617$	5809.84	0.379085	0.189542	−	0.981873i	$-0.439300\pi$
$617$	5809.84	0.379085	0.189542	+	0.981873i	$0.439300\pi$
$618$	0	0
$619$	−1543.48	−0.100223	−0.0501114	−	0.998744i	$-0.515958\pi$
$619$	−1543.48	−0.100223	−0.0501114	+	0.998744i	$0.515958\pi$
$620$	−5590.04	−0.362099
$621$	0	0
$622$	5412.71	0.348922
$623$	−3554.17	−0.228563
$624$	0	0
$625$	625.000	0.0400000
$626$	−683.044	−0.0436101
$627$	0	0
$628$	−19048.2	−1.21036
$629$	−17684.7	−1.12104
$630$	0	0
$631$	7992.93	0.504269	0.252134	−	0.967692i	$-0.418868\pi$
$631$	7992.93	0.504269	0.252134	+	0.967692i	$0.418868\pi$
$632$	−4348.58	−0.273698
$633$	0	0
$634$	6394.23	0.400548
$635$	8229.54	0.514298
$636$	0	0
$637$	11483.8	0.714294
$638$	−1192.18	−0.0739791
$639$	0	0
$640$	6290.72	0.388535
$641$	−26129.8	−1.61009	−0.805044	−	0.593215i	$-0.797858\pi$
$641$	−26129.8	−1.61009	−0.805044	+	0.593215i	$0.797858\pi$
$642$	0	0
$643$	−28772.1	−1.76464	−0.882319	−	0.470653i	$-0.844018\pi$
$643$	−28772.1	−1.76464	−0.882319	+	0.470653i	$0.844018\pi$
$644$	−694.277	−0.0424819
$645$	0	0
$646$	6686.96	0.407268
$647$	7156.38	0.434847	0.217424	−	0.976077i	$-0.430235\pi$
$647$	7156.38	0.434847	0.217424	+	0.976077i	$0.430235\pi$
$648$	0	0
$649$	−15055.1	−0.910578
$650$	−712.436	−0.0429908
$651$	0	0
$652$	−2624.07	−0.157617
$653$	−15434.9	−0.924981	−0.462491	−	0.886624i	$-0.653044\pi$
$653$	−15434.9	−0.924981	−0.462491	+	0.886624i	$0.653044\pi$
$654$	0	0
$655$	1933.54	0.115343
$656$	19137.5	1.13901
$657$	0	0
$658$	−1086.82	−0.0643901
$659$	6462.64	0.382016	0.191008	−	0.981588i	$-0.438824\pi$
$659$	6462.64	0.382016	0.191008	+	0.981588i	$0.438824\pi$
$660$	0	0
$661$	1747.60	0.102835	0.0514174	−	0.998677i	$-0.483626\pi$
$661$	1747.60	0.102835	0.0514174	+	0.998677i	$0.483626\pi$
$662$	−944.735	−0.0554655
$663$	0	0
$664$	−14277.8	−0.834465
$665$	3906.79	0.227818
$666$	0	0
$667$	−583.085	−0.0338488
$668$	−156.666	−0.00907426
$669$	0	0
$670$	2987.01	0.172236
$671$	−54.2305	−0.00312004
$672$	0	0
$673$	−5478.38	−0.313783	−0.156892	−	0.987616i	$-0.550147\pi$
$673$	−5478.38	−0.313783	−0.156892	+	0.987616i	$0.550147\pi$
$674$	−7321.47	−0.418416
$675$	0	0
$676$	5087.49	0.289457
$677$	4018.53	0.228131	0.114066	−	0.993473i	$-0.463613\pi$
$677$	4018.53	0.228131	0.114066	+	0.993473i	$0.463613\pi$
$678$	0	0
$679$	7594.02	0.429207
$680$	4584.51	0.258541
$681$	0	0
$682$	4111.59	0.230852
$683$	−9894.95	−0.554348	−0.277174	−	0.960820i	$-0.589398\pi$
$683$	−9894.95	−0.554348	−0.277174	+	0.960820i	$0.589398\pi$
$684$	0	0
$685$	−14866.1	−0.829204
$686$	3235.81	0.180093
$687$	0	0
$688$	23666.1	1.31143
$689$	−17331.0	−0.958287
$690$	0	0
$691$	21847.9	1.20280	0.601400	−	0.798948i	$-0.294610\pi$
$691$	21847.9	1.20280	0.601400	+	0.798948i	$0.294610\pi$
$692$	33772.1	1.85524
$693$	0	0
$694$	4735.18	0.258998
$695$	−11736.8	−0.640577
$696$	0	0
$697$	30141.3	1.63800
$698$	8176.85	0.443407
$699$	0	0
$700$	1292.82	0.0698057
$701$	−15506.5	−0.835480	−0.417740	−	0.908567i	$-0.637178\pi$
$701$	−15506.5	−0.835480	−0.417740	+	0.908567i	$0.637178\pi$
$702$	0	0
$703$	−24624.7	−1.32110
$704$	11907.3	0.637460
$705$	0	0
$706$	1329.97	0.0708982
$707$	8272.88	0.440076
$708$	0	0
$709$	6399.70	0.338993	0.169496	−	0.985531i	$-0.445786\pi$
$709$	6399.70	0.338993	0.169496	+	0.985531i	$0.445786\pi$
$710$	−539.954	−0.0285410
$711$	0	0
$712$	−5807.42	−0.305677
$713$	2010.95	0.105625
$714$	0	0
$715$	−7298.54	−0.381748
$716$	−27070.7	−1.41296
$717$	0	0
$718$	672.076	0.0349327
$719$	−1771.66	−0.0918941	−0.0459470	−	0.998944i	$-0.514631\pi$
$719$	−1771.66	−0.0918941	−0.0459470	+	0.998944i	$0.514631\pi$
$720$	0	0
$721$	−1257.49	−0.0649532
$722$	4289.97	0.221130
$723$	0	0
$724$	−26508.1	−1.36073
$725$	1085.77	0.0556200
$726$	0	0
$727$	9575.06	0.488472	0.244236	−	0.969716i	$-0.421463\pi$
$727$	9575.06	0.488472	0.244236	+	0.969716i	$0.421463\pi$
$728$	−3053.17	−0.155437
$729$	0	0
$730$	−1583.61	−0.0802907
$731$	37273.8	1.88594
$732$	0	0
$733$	−19790.5	−0.997243	−0.498622	−	0.866820i	$-0.666160\pi$
$733$	−19790.5	−0.997243	−0.498622	+	0.866820i	$0.666160\pi$
$734$	5383.70	0.270730
$735$	0	0
$736$	−1721.30	−0.0862066
$737$	30600.4	1.52942
$738$	0	0
$739$	−13106.5	−0.652407	−0.326203	−	0.945300i	$-0.605769\pi$
$739$	−13106.5	−0.652407	−0.326203	+	0.945300i	$0.605769\pi$
$740$	−8148.69	−0.404800
$741$	0	0
$742$	−2257.99	−0.111716
$743$	17068.9	0.842795	0.421398	−	0.906876i	$-0.361540\pi$
$743$	17068.9	0.842795	0.421398	+	0.906876i	$0.361540\pi$
$744$	0	0
$745$	7010.46	0.344756
$746$	−441.139	−0.0216505
$747$	0	0
$748$	22669.2	1.10811
$749$	4237.75	0.206734
$750$	0	0
$751$	4112.93	0.199844	0.0999220	−	0.994995i	$-0.468141\pi$
$751$	4112.93	0.199844	0.0999220	+	0.994995i	$0.468141\pi$
$752$	11019.9	0.534379
$753$	0	0
$754$	−1237.67	−0.0597787
$755$	14706.3	0.708897
$756$	0	0
$757$	17493.6	0.839915	0.419958	−	0.907544i	$-0.362045\pi$
$757$	17493.6	0.839915	0.419958	+	0.907544i	$0.362045\pi$
$758$	9750.19	0.467207
$759$	0	0
$760$	6383.60	0.304681
$761$	−5212.04	−0.248274	−0.124137	−	0.992265i	$-0.539616\pi$
$761$	−5212.04	−0.248274	−0.124137	+	0.992265i	$0.539616\pi$
$762$	0	0
$763$	−3813.75	−0.180953
$764$	2087.34	0.0988447
$765$	0	0
$766$	−6250.07	−0.294809
$767$	−15629.6	−0.735790
$768$	0	0
$769$	−23324.7	−1.09377	−0.546885	−	0.837208i	$-0.684186\pi$
$769$	−23324.7	−1.09377	−0.546885	+	0.837208i	$0.684186\pi$
$770$	−950.897	−0.0445038
$771$	0	0
$772$	−1986.64	−0.0926176
$773$	20455.0	0.951765	0.475883	−	0.879509i	$-0.342129\pi$
$773$	20455.0	0.951765	0.475883	+	0.879509i	$0.342129\pi$
$774$	0	0
$775$	−3744.62	−0.173562
$776$	12408.4	0.574017
$777$	0	0
$778$	−9253.11	−0.426401
$779$	41969.6	1.93032
$780$	0	0
$781$	−5531.55	−0.253437
$782$	−796.037	−0.0364018
$783$	0	0
$784$	−15170.6	−0.691079
$785$	−12759.9	−0.580153
$786$	0	0
$787$	9110.97	0.412670	0.206335	−	0.978481i	$-0.433846\pi$
$787$	9110.97	0.412670	0.206335	+	0.978481i	$0.433846\pi$
$788$	35313.5	1.59644
$789$	0	0
$790$	−1406.03	−0.0633217
$791$	−3297.15	−0.148209
$792$	0	0
$793$	−56.2997	−0.00252114
$794$	4007.20	0.179106
$795$	0	0
$796$	−21494.3	−0.957093
$797$	−7925.09	−0.352222	−0.176111	−	0.984370i	$-0.556352\pi$
$797$	−7925.09	−0.352222	−0.176111	+	0.984370i	$0.556352\pi$
$798$	0	0
$799$	17356.2	0.768481
$800$	3205.26	0.141654
$801$	0	0
$802$	9550.08	0.420480
$803$	−16223.3	−0.712962
$804$	0	0
$805$	−465.077	−0.0203625
$806$	4268.48	0.186539
$807$	0	0
$808$	13517.7	0.588552
$809$	−33705.8	−1.46481	−0.732406	−	0.680868i	$-0.761603\pi$
$809$	−33705.8	−1.46481	−0.732406	+	0.680868i	$0.761603\pi$
$810$	0	0
$811$	10424.5	0.451360	0.225680	−	0.974201i	$-0.427540\pi$
$811$	10424.5	0.451360	0.225680	+	0.974201i	$0.427540\pi$
$812$	2245.93	0.0970648
$813$	0	0
$814$	5993.53	0.258075
$815$	−1757.79	−0.0755495
$816$	0	0
$817$	51901.1	2.22251
$818$	−5258.11	−0.224750
$819$	0	0
$820$	13888.4	0.591468
$821$	−9467.66	−0.402465	−0.201232	−	0.979544i	$-0.564495\pi$
$821$	−9467.66	−0.402465	−0.201232	+	0.979544i	$0.564495\pi$
$822$	0	0
$823$	1160.55	0.0491545	0.0245772	−	0.999698i	$-0.492176\pi$
$823$	1160.55	0.0491545	0.0245772	+	0.999698i	$0.492176\pi$
$824$	−2054.70	−0.0868676
$825$	0	0
$826$	−2036.31	−0.0857777
$827$	26499.9	1.11426	0.557128	−	0.830426i	$-0.311903\pi$
$827$	26499.9	1.11426	0.557128	+	0.830426i	$0.311903\pi$
$828$	0	0
$829$	653.861	0.0273939	0.0136969	−	0.999906i	$-0.495640\pi$
$829$	653.861	0.0273939	0.0136969	+	0.999906i	$0.495640\pi$
$830$	−4616.42	−0.193058
$831$	0	0
$832$	12361.6	0.515098
$833$	−23893.5	−0.993830
$834$	0	0
$835$	−104.947	−0.00434949
$836$	31565.2	1.30587
$837$	0	0
$838$	6383.77	0.263155
$839$	41269.7	1.69820	0.849099	−	0.528234i	$-0.177146\pi$
$839$	41269.7	1.69820	0.849099	+	0.528234i	$0.177146\pi$
$840$	0	0
$841$	−22502.8	−0.922661
$842$	4797.15	0.196343
$843$	0	0
$844$	−20363.6	−0.830502
$845$	3407.97	0.138743
$846$	0	0
$847$	−520.008	−0.0210953
$848$	22894.9	0.927141
$849$	0	0
$850$	1482.31	0.0598150
$851$	2931.40	0.118081
$852$	0	0
$853$	5069.39	0.203485	0.101743	−	0.994811i	$-0.467558\pi$
$853$	5069.39	0.203485	0.101743	+	0.994811i	$0.467558\pi$
$854$	−7.33506	−0.000293912 0
$855$	0	0
$856$	6924.38	0.276484
$857$	−43639.0	−1.73941	−0.869707	−	0.493568i	$-0.835693\pi$
$857$	−43639.0	−1.73941	−0.869707	+	0.493568i	$0.835693\pi$
$858$	0	0
$859$	−12898.1	−0.512315	−0.256157	−	0.966635i	$-0.582457\pi$
$859$	−12898.1	−0.512315	−0.256157	+	0.966635i	$0.582457\pi$
$860$	17174.9	0.680999
$861$	0	0
$862$	3819.62	0.150924
$863$	8962.98	0.353538	0.176769	−	0.984252i	$-0.443435\pi$
$863$	8962.98	0.353538	0.176769	+	0.984252i	$0.443435\pi$
$864$	0	0
$865$	22623.0	0.889256
$866$	2472.54	0.0970214
$867$	0	0
$868$	−7745.78	−0.302891
$869$	−14404.0	−0.562281
$870$	0	0
$871$	31768.0	1.23584
$872$	−6231.56	−0.242004
$873$	0	0
$874$	−1108.42	−0.0428982
$875$	866.025	0.0334594
$876$	0	0
$877$	−37720.1	−1.45236	−0.726179	−	0.687506i	$-0.758705\pi$
$877$	−37720.1	−1.45236	−0.726179	+	0.687506i	$0.758705\pi$
$878$	1565.80	0.0601859
$879$	0	0
$880$	9641.64	0.369341
$881$	−7961.44	−0.304458	−0.152229	−	0.988345i	$-0.548645\pi$
$881$	−7961.44	−0.304458	−0.152229	+	0.988345i	$0.548645\pi$
$882$	0	0
$883$	−37426.4	−1.42639	−0.713193	−	0.700968i	$-0.752752\pi$
$883$	−37426.4	−1.42639	−0.713193	+	0.700968i	$0.752752\pi$
$884$	23534.2	0.895408
$885$	0	0
$886$	6231.67	0.236295
$887$	26853.3	1.01651	0.508256	−	0.861206i	$-0.330290\pi$
$887$	26853.3	1.01651	0.508256	+	0.861206i	$0.330290\pi$
$888$	0	0
$889$	11403.2	0.430203
$890$	−1877.71	−0.0707202
$891$	0	0
$892$	27429.4	1.02960
$893$	24167.2	0.905626
$894$	0	0
$895$	−18133.9	−0.677264
$896$	8716.67	0.325004
$897$	0	0
$898$	−3325.46	−0.123577
$899$	−6505.26	−0.241338
$900$	0	0
$901$	36059.3	1.33331
$902$	−10215.2	−0.377083
$903$	0	0
$904$	−5387.46	−0.198213
$905$	−17757.0	−0.652226
$906$	0	0
$907$	27150.9	0.993969	0.496985	−	0.867759i	$-0.334441\pi$
$907$	27150.9	0.993969	0.496985	+	0.867759i	$0.334441\pi$
$908$	−43097.1	−1.57514
$909$	0	0
$910$	−987.180	−0.0359612
$911$	15300.5	0.556452	0.278226	−	0.960516i	$-0.410254\pi$
$911$	15300.5	0.556452	0.278226	+	0.960516i	$0.410254\pi$
$912$	0	0
$913$	−47292.9	−1.71431
$914$	−13134.6	−0.475332
$915$	0	0
$916$	−35964.1	−1.29726
$917$	2679.19	0.0964826
$918$	0	0
$919$	−26325.6	−0.944943	−0.472471	−	0.881346i	$-0.656638\pi$
$919$	−26325.6	−0.944943	−0.472471	+	0.881346i	$0.656638\pi$
$920$	−759.925	−0.0272326
$921$	0	0
$922$	−3792.67	−0.135472
$923$	−5742.61	−0.204789
$924$	0	0
$925$	−5458.59	−0.194029
$926$	10389.9	0.368719
$927$	0	0
$928$	5568.27	0.196969
$929$	35507.1	1.25398	0.626991	−	0.779027i	$-0.284286\pi$
$929$	35507.1	1.25398	0.626991	+	0.779027i	$0.284286\pi$
$930$	0	0
$931$	−33269.9	−1.17119
$932$	−679.852	−0.0238941
$933$	0	0
$934$	−8701.39	−0.304838
$935$	15185.5	0.531143
$936$	0	0
$937$	35105.4	1.22395	0.611975	−	0.790877i	$-0.290375\pi$
$937$	35105.4	1.22395	0.611975	+	0.790877i	$0.290375\pi$
$938$	4138.92	0.144073
$939$	0	0
$940$	7997.31	0.277493
$941$	3574.47	0.123830	0.0619151	−	0.998081i	$-0.480279\pi$
$941$	3574.47	0.123830	0.0619151	+	0.998081i	$0.480279\pi$
$942$	0	0
$943$	−4996.19	−0.172533
$944$	20647.3	0.711876
$945$	0	0
$946$	−12632.5	−0.434163
$947$	−33058.5	−1.13438	−0.567189	−	0.823588i	$-0.691969\pi$
$947$	−33058.5	−1.13438	−0.567189	+	0.823588i	$0.691969\pi$
$948$	0	0
$949$	−16842.3	−0.576107
$950$	2064.01	0.0704897
$951$	0	0
$952$	6352.49	0.216266
$953$	−1355.24	−0.0460656	−0.0230328	−	0.999735i	$-0.507332\pi$
$953$	−1355.24	−0.0460656	−0.0230328	+	0.999735i	$0.507332\pi$
$954$	0	0
$955$	1398.25	0.0473784
$956$	−40954.3	−1.38552
$957$	0	0
$958$	−3162.06	−0.106641
$959$	−20599.1	−0.693617
$960$	0	0
$961$	−7355.57	−0.246906
$962$	6222.23	0.208537
$963$	0	0
$964$	−22520.6	−0.752428
$965$	−1330.80	−0.0443937
$966$	0	0
$967$	56118.0	1.86622	0.933110	−	0.359592i	$-0.117084\pi$
$967$	56118.0	1.86622	0.933110	+	0.359592i	$0.117084\pi$
$968$	−849.680	−0.0282126
$969$	0	0
$970$	4012.01	0.132802
$971$	−32718.5	−1.08135	−0.540673	−	0.841233i	$-0.681830\pi$
$971$	−32718.5	−1.08135	−0.540673	+	0.841233i	$0.681830\pi$
$972$	0	0
$973$	−16262.9	−0.535834
$974$	−4712.17	−0.155018
$975$	0	0
$976$	74.3741	0.00243920
$977$	8850.88	0.289831	0.144915	−	0.989444i	$-0.453709\pi$
$977$	8850.88	0.289831	0.144915	+	0.989444i	$0.453709\pi$
$978$	0	0
$979$	−19236.2	−0.627978
$980$	−11009.5	−0.358864
$981$	0	0
$982$	−2121.29	−0.0689338
$983$	34107.9	1.10669	0.553343	−	0.832953i	$-0.313352\pi$
$983$	34107.9	1.10669	0.553343	+	0.832953i	$0.313352\pi$
$984$	0	0
$985$	23655.6	0.765208
$986$	2575.11	0.0831727
$987$	0	0
$988$	32769.7	1.05520
$989$	−6178.47	−0.198649
$990$	0	0
$991$	−52154.2	−1.67178	−0.835890	−	0.548897i	$-0.815048\pi$
$991$	−52154.2	−1.67178	−0.835890	+	0.548897i	$0.815048\pi$
$992$	−19203.9	−0.614642
$993$	0	0
$994$	−748.182	−0.0238741
$995$	−14398.5	−0.458756
$996$	0	0
$997$	2111.10	0.0670604	0.0335302	−	0.999438i	$-0.489325\pi$
$997$	2111.10	0.0670604	0.0335302	+	0.999438i	$0.489325\pi$
$998$	11020.7	0.349554
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.c.1.2	✓	2
3.2	odd	2		405.4.a.f.1.1	yes	2
5.4	even	2		2025.4.a.m.1.1		2
9.2	odd	6		405.4.e.o.271.2		4
9.4	even	3		405.4.e.p.136.1		4
9.5	odd	6		405.4.e.o.136.2		4
9.7	even	3		405.4.e.p.271.1		4
15.14	odd	2		2025.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.c.1.2	✓	2	1.1	even	1	trivial
405.4.a.f.1.1	yes	2	3.2	odd	2
405.4.e.o.136.2		4	9.5	odd	6
405.4.e.o.271.2		4	9.2	odd	6
405.4.e.p.136.1		4	9.4	even	3
405.4.e.p.271.1		4	9.7	even	3
2025.4.a.i.1.2		2	15.14	odd	2
2025.4.a.m.1.1		2	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} -3.66025 q^{10} -37.4974 q^{11} -38.9282 q^{13} -5.07180 q^{14} +51.4256 q^{16} +80.9948 q^{17} +112.779 q^{19} +37.3205 q^{20} -27.4500 q^{22} -13.4256 q^{23} +25.0000 q^{25} -28.4974 q^{26} +51.7128 q^{28} +43.4308 q^{29} -149.785 q^{31} +128.210 q^{32} +59.2923 q^{34} +34.6410 q^{35} -218.344 q^{37} +82.5603 q^{38} +56.6025 q^{40} +372.138 q^{41} +460.200 q^{43} +279.885 q^{44} -9.82824 q^{46} +214.287 q^{47} -295.000 q^{49} +18.3013 q^{50} +290.564 q^{52} +445.205 q^{53} +187.487 q^{55} +78.4308 q^{56} +31.7935 q^{58} +401.497 q^{59} +1.44624 q^{61} -109.650 q^{62} -317.549 q^{64} +194.641 q^{65} -816.067 q^{67} -604.554 q^{68} +25.3590 q^{70} +147.518 q^{71} +432.651 q^{73} -159.839 q^{74} -841.797 q^{76} +259.790 q^{77} +384.133 q^{79} -257.128 q^{80} +272.424 q^{82} +1261.23 q^{83} -404.974 q^{85} +336.890 q^{86} +424.490 q^{88} +513.000 q^{89} +269.703 q^{91} +100.210 q^{92} +156.869 q^{94} -563.897 q^{95} -1096.10 q^{97} -215.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 8 * q^4 - 10 * q^5 + 12 * q^8	\( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + 10 q^{10} + 22 q^{11} - 64 q^{13} - 24 q^{14} - 8 q^{16} - 32 q^{17} - 10 q^{19} + 40 q^{20} - 190 q^{22} + 84 q^{23} + 50 q^{25} + 40 q^{26} + 48 q^{28} + 170 q^{29} - 258 q^{31} + 104 q^{32} + 368 q^{34} + 76 q^{37} + 418 q^{38} - 60 q^{40} + 578 q^{41} + 380 q^{43} + 248 q^{44} - 276 q^{46} + 484 q^{47} - 590 q^{49} - 50 q^{50} + 304 q^{52} + 544 q^{53} - 110 q^{55} + 240 q^{56} - 314 q^{58} + 706 q^{59} + 668 q^{61} + 186 q^{62} + 224 q^{64} + 320 q^{65} - 1452 q^{67} - 544 q^{68} + 120 q^{70} + 974 q^{71} + 1184 q^{73} - 964 q^{74} - 776 q^{76} + 672 q^{77} + 408 q^{79} + 40 q^{80} - 290 q^{82} + 444 q^{83} + 160 q^{85} + 556 q^{86} + 1812 q^{88} + 1026 q^{89} + 96 q^{91} + 48 q^{92} - 580 q^{94} + 50 q^{95} - 668 q^{97} + 590 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 8 * q^4 - 10 * q^5 + 12 * q^8 + 10 * q^10 + 22 * q^11 - 64 * q^13 - 24 * q^14 - 8 * q^16 - 32 * q^17 - 10 * q^19 + 40 * q^20 - 190 * q^22 + 84 * q^23 + 50 * q^25 + 40 * q^26 + 48 * q^28 + 170 * q^29 - 258 * q^31 + 104 * q^32 + 368 * q^34 + 76 * q^37 + 418 * q^38 - 60 * q^40 + 578 * q^41 + 380 * q^43 + 248 * q^44 - 276 * q^46 + 484 * q^47 - 590 * q^49 - 50 * q^50 + 304 * q^52 + 544 * q^53 - 110 * q^55 + 240 * q^56 - 314 * q^58 + 706 * q^59 + 668 * q^61 + 186 * q^62 + 224 * q^64 + 320 * q^65 - 1452 * q^67 - 544 * q^68 + 120 * q^70 + 974 * q^71 + 1184 * q^73 - 964 * q^74 - 776 * q^76 + 672 * q^77 + 408 * q^79 + 40 * q^80 - 290 * q^82 + 444 * q^83 + 160 * q^85 + 556 * q^86 + 1812 * q^88 + 1026 * q^89 + 96 * q^91 + 48 * q^92 - 580 * q^94 + 50 * q^95 - 668 * q^97 + 590 * q^98

Introduction

L-functions

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