LMFDB - Embedded newform 405.5.d.a.404.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,5,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$41.8648350490$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.25
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.09451 q^{2} -14.8020 q^{4} +(24.2165 + 6.20970i) q^{5} -19.5383i q^{7} -33.7133 q^{8} +(26.5053 + 6.79660i) q^{10} -115.938i q^{11} +183.284i q^{13} -21.3849i q^{14} +199.933 q^{16} +89.0232 q^{17} -52.2295 q^{19} +(-358.454 - 91.9161i) q^{20} -126.896i q^{22} -464.004 q^{23} +(547.879 + 300.754i) q^{25} +200.607i q^{26} +289.206i q^{28} -1449.62i q^{29} -1535.88 q^{31} +758.242 q^{32} +97.4372 q^{34} +(121.327 - 473.149i) q^{35} -641.462i q^{37} -57.1659 q^{38} +(-816.418 - 209.349i) q^{40} +1380.22i q^{41} -2050.85i q^{43} +1716.12i q^{44} -507.859 q^{46} +1178.26 q^{47} +2019.26 q^{49} +(599.662 + 329.180i) q^{50} -2712.97i q^{52} -64.4267 q^{53} +(719.940 - 2807.62i) q^{55} +658.700i q^{56} -1586.63i q^{58} -3276.80i q^{59} -1858.82 q^{61} -1681.05 q^{62} -2369.02 q^{64} +(-1138.14 + 4438.49i) q^{65} -8380.10i q^{67} -1317.72 q^{68} +(132.794 - 517.869i) q^{70} -4564.71i q^{71} -8077.75i q^{73} -702.089i q^{74} +773.103 q^{76} -2265.23 q^{77} -8657.39 q^{79} +(4841.68 + 1241.52i) q^{80} +1510.67i q^{82} -3340.11 q^{83} +(2155.83 + 552.807i) q^{85} -2244.69i q^{86} +3908.65i q^{88} -10168.3i q^{89} +3581.05 q^{91} +6868.21 q^{92} +1289.62 q^{94} +(-1264.82 - 324.329i) q^{95} -13704.3i q^{97} +2210.10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) $ 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.09451			0.273629			0.136814	−	0.990597i	$-0.456314\pi$
$2$	1.09451			0.273629			0.136814	+	0.990597i	$0.456314\pi$
$3$		0			0
$3$		0			0
$4$	−14.8020			−0.925127
$5$	24.2165	+	6.20970i	0.968661	+	0.248388i
$5$	24.2165	+	6.20970i	0.968661	+	0.248388i
$6$		0			0
$7$		−	19.5383i		−	0.398741i	−0.979924	−	0.199370i	$-0.936110\pi$
$7$		−	19.5383i		−	0.398741i	0.979924	−	0.199370i	$-0.0638897\pi$
$8$	−33.7133			−0.526770
$9$		0			0
$10$	26.5053	+	6.79660i	0.265053	+	0.0679660i
$11$		−	115.938i		−	0.958166i	−0.877770	−	0.479083i	$-0.840969\pi$
$11$		−	115.938i		−	0.958166i	0.877770	−	0.479083i	$-0.159031\pi$
$12$		0			0
$13$			183.284i			1.08452i	0.840211	+	0.542259i	$0.182431\pi$
$13$			183.284i			1.08452i	−0.840211	+	0.542259i	$0.817569\pi$
$14$		−	21.3849i		−	0.109107i
$15$		0			0
$16$	199.933			0.780988
$17$	89.0232			0.308039			0.154019	−	0.988068i	$-0.450778\pi$
$17$	89.0232			0.308039			0.154019	+	0.988068i	$0.450778\pi$
$18$		0			0
$19$	−52.2295			−0.144680			−0.0723400	−	0.997380i	$-0.523047\pi$
$19$	−52.2295			−0.144680			−0.0723400	+	0.997380i	$0.523047\pi$
$20$	−358.454	−	91.9161i	−0.896135	−	0.229790i
$21$		0			0
$22$		−	126.896i		−	0.262182i
$23$	−464.004			−0.877134			−0.438567	−	0.898698i	$-0.644514\pi$
$23$	−464.004			−0.877134			−0.438567	+	0.898698i	$0.644514\pi$
$24$		0			0
$25$	547.879	+	300.754i	0.876607	+	0.481207i
$26$			200.607i			0.296755i
$27$		0			0
$28$			289.206i			0.368886i
$29$		−	1449.62i		−	1.72369i	−0.507174	−	0.861844i	$-0.669310\pi$
$29$		−	1449.62i		−	1.72369i	0.507174	−	0.861844i	$-0.330690\pi$
$30$		0			0
$31$	−1535.88			−1.59821			−0.799107	−	0.601189i	$-0.794694\pi$
$31$	−1535.88			−1.59821			−0.799107	+	0.601189i	$0.794694\pi$
$32$	758.242			0.740471
$33$		0			0
$34$	97.4372			0.0842882
$35$	121.327	−	473.149i	0.0990423	−	0.386244i
$36$		0			0
$37$		−	641.462i		−	0.468562i	−0.972169	−	0.234281i	$-0.924726\pi$
$37$		−	641.462i		−	0.468562i	0.972169	−	0.234281i	$-0.0752736\pi$
$38$	−57.1659			−0.0395886
$39$		0			0
$40$	−816.418	−	209.349i	−0.510261	−	0.130843i
$41$			1380.22i			0.821072i	0.911844	+	0.410536i	$0.134658\pi$
$41$			1380.22i			0.821072i	−0.911844	+	0.410536i	$0.865342\pi$
$42$		0			0
$43$		−	2050.85i		−	1.10917i	−0.832128	−	0.554584i	$-0.812877\pi$
$43$		−	2050.85i		−	1.10917i	0.832128	−	0.554584i	$-0.187123\pi$
$44$			1716.12i			0.886426i
$45$		0			0
$46$	−507.859			−0.240009
$47$	1178.26			0.533390			0.266695	−	0.963781i	$-0.414068\pi$
$47$	1178.26			0.533390			0.266695	+	0.963781i	$0.414068\pi$
$48$		0			0
$49$	2019.26			0.841006
$50$	599.662	+	329.180i	0.239865	+	0.131672i
$51$		0			0
$52$		−	2712.97i		−	1.00332i
$53$	−64.4267			−0.0229358			−0.0114679	−	0.999934i	$-0.503650\pi$
$53$	−64.4267			−0.0229358			−0.0114679	+	0.999934i	$0.503650\pi$
$54$		0			0
$55$	719.940	−	2807.62i	0.237997	−	0.928138i
$56$			658.700i			0.210045i
$57$		0			0
$58$		−	1586.63i		−	0.471650i
$59$		−	3276.80i		−	0.941338i	−0.882310	−	0.470669i	$-0.844013\pi$
$59$		−	3276.80i		−	0.941338i	0.882310	−	0.470669i	$-0.155987\pi$
$60$		0			0
$61$	−1858.82			−0.499548			−0.249774	−	0.968304i	$-0.580356\pi$
$61$	−1858.82			−0.499548			−0.249774	+	0.968304i	$0.580356\pi$
$62$	−1681.05			−0.437317
$63$		0			0
$64$	−2369.02			−0.578374
$65$	−1138.14	+	4438.49i	−0.269381	+	1.05053i
$66$		0			0
$67$		−	8380.10i		−	1.86681i	−0.358827	−	0.933404i	$-0.616823\pi$
$67$		−	8380.10i		−	1.86681i	0.358827	−	0.933404i	$-0.383177\pi$
$68$	−1317.72			−0.284975
$69$		0			0
$70$	132.794	−	517.869i	0.0271008	−	0.105687i
$71$		−	4564.71i		−	0.905517i	−0.891633	−	0.452759i	$-0.850440\pi$
$71$		−	4564.71i		−	0.905517i	0.891633	−	0.452759i	$-0.149560\pi$
$72$		0			0
$73$		−	8077.75i		−	1.51581i	−0.652365	−	0.757905i	$-0.726223\pi$
$73$		−	8077.75i		−	1.51581i	0.652365	−	0.757905i	$-0.273777\pi$
$74$		−	702.089i		−	0.128212i
$75$		0			0
$76$	773.103			0.133847
$77$	−2265.23			−0.382060
$78$		0			0
$79$	−8657.39			−1.38718			−0.693590	−	0.720370i	$-0.743972\pi$
$79$	−8657.39			−1.38718			−0.693590	+	0.720370i	$0.743972\pi$
$80$	4841.68	+	1241.52i	0.756512	+	0.193988i
$81$		0			0
$82$			1510.67i			0.224669i
$83$	−3340.11			−0.484847			−0.242424	−	0.970171i	$-0.577942\pi$
$83$	−3340.11			−0.484847			−0.242424	+	0.970171i	$0.577942\pi$
$84$		0			0
$85$	2155.83	+	552.807i	0.298385	+	0.0765131i
$86$		−	2244.69i		−	0.303500i
$87$		0			0
$88$			3908.65i			0.504733i
$89$		−	10168.3i		−	1.28371i	−0.766825	−	0.641856i	$-0.778165\pi$
$89$		−	10168.3i		−	1.28371i	0.766825	−	0.641856i	$-0.221835\pi$
$90$		0			0
$91$	3581.05			0.432441
$92$	6868.21			0.811461
$93$		0			0
$94$	1289.62			0.145951
$95$	−1264.82	−	324.329i	−0.140146	−	0.0359367i
$96$		0			0
$97$		−	13704.3i		−	1.45651i	−0.685306	−	0.728255i	$-0.740332\pi$
$97$		−	13704.3i		−	1.45651i	0.685306	−	0.728255i	$-0.259668\pi$
$98$	2210.10			0.230123
$99$		0			0
$100$	−8109.73	−	4451.78i	−0.810973	−	0.445178i
$101$			7098.89i			0.695901i	0.937513	+	0.347951i	$0.113122\pi$
$101$			7098.89i			0.695901i	−0.937513	+	0.347951i	$0.886878\pi$
$102$		0			0
$103$		−	4686.40i		−	0.441738i	−0.975303	−	0.220869i	$-0.929111\pi$
$103$		−	4686.40i		−	0.441738i	0.975303	−	0.220869i	$-0.0708894\pi$
$104$		−	6179.09i		−	0.571292i
$105$		0			0
$106$	−70.5160			−0.00627590
$107$	−20235.1			−1.76741			−0.883705	−	0.468044i	$-0.844959\pi$
$107$	−20235.1			−1.76741			−0.883705	+	0.468044i	$0.844959\pi$
$108$		0			0
$109$	2264.65			0.190611			0.0953053	−	0.995448i	$-0.469617\pi$
$109$	2264.65			0.190611			0.0953053	+	0.995448i	$0.469617\pi$
$110$	787.985	−	3072.98i	0.0651227	−	0.253965i
$111$		0			0
$112$		−	3906.35i		−	0.311412i
$113$	10010.5			0.783970			0.391985	−	0.919972i	$-0.371789\pi$
$113$	10010.5			0.783970			0.391985	+	0.919972i	$0.371789\pi$
$114$		0			0
$115$	−11236.6	−	2881.32i	−0.849646	−	0.217870i
$116$			21457.4i			1.59463i
$117$		0			0
$118$		−	3586.50i		−	0.257577i
$119$		−	1739.36i		−	0.122828i
$120$		0			0
$121$	1199.36			0.0819177
$122$	−2034.50			−0.136691
$123$		0			0
$124$	22734.2			1.47855
$125$	11400.1	+	10685.4i	0.729609	+	0.683865i
$126$		0			0
$127$			19581.7i			1.21406i	0.794677	+	0.607032i	$0.207640\pi$
$127$			19581.7i			1.21406i	−0.794677	+	0.607032i	$0.792360\pi$
$128$	−14724.8			−0.898730
$129$		0			0
$130$	−1245.71	+	4857.99i	−0.0737104	+	0.287455i
$131$			28479.5i			1.65955i	0.558102	+	0.829773i	$0.311530\pi$
$131$			28479.5i			1.65955i	−0.558102	+	0.829773i	$0.688470\pi$
$132$		0			0
$133$			1020.47i			0.0576898i
$134$		−	9172.14i		−	0.510812i
$135$		0			0
$136$	−3001.26			−0.162266
$137$	26236.5			1.39786			0.698931	−	0.715189i	$-0.253659\pi$
$137$	26236.5			1.39786			0.698931	+	0.715189i	$0.253659\pi$
$138$		0			0
$139$	−33308.1			−1.72393			−0.861967	−	0.506964i	$-0.830768\pi$
$139$	−33308.1			−1.72393			−0.861967	+	0.506964i	$0.830768\pi$
$140$	−1795.88	+	7003.57i	−0.0916267	+	0.357325i
$141$		0			0
$142$		−	4996.14i		−	0.247775i
$143$	21249.5			1.03915
$144$		0			0
$145$	9001.71	−	35104.8i	0.428143	−	1.66967i
$146$		−	8841.22i		−	0.414769i
$147$		0			0
$148$			9494.94i			0.433480i
$149$			5263.88i			0.237101i	0.992948	+	0.118551i	$0.0378248\pi$
$149$			5263.88i			0.237101i	−0.992948	+	0.118551i	$0.962175\pi$
$150$		0			0
$151$	11273.8			0.494444			0.247222	−	0.968959i	$-0.420482\pi$
$151$	11273.8			0.494444			0.247222	+	0.968959i	$0.420482\pi$
$152$	1760.83			0.0762131
$153$		0			0
$154$	−2479.33			−0.104542
$155$	−37193.8	−	9537.37i	−1.54813	−	0.396977i
$156$		0			0
$157$		−	20139.1i		−	0.817034i	−0.912751	−	0.408517i	$-0.866046\pi$
$157$		−	20139.1i		−	0.817034i	0.912751	−	0.408517i	$-0.133954\pi$
$158$	−9475.64			−0.379572
$159$		0			0
$160$	18362.0	+	4708.45i	0.717265	+	0.183924i
$161$			9065.85i			0.349749i
$162$		0			0
$163$		−	15388.7i		−	0.579197i	−0.957148	−	0.289598i	$-0.906478\pi$
$163$		−	15388.7i		−	0.579197i	0.957148	−	0.289598i	$-0.0935218\pi$
$164$		−	20430.1i		−	0.759596i
$165$		0			0
$166$	−3655.80			−0.132668
$167$	17047.9			0.611278			0.305639	−	0.952148i	$-0.401130\pi$
$167$	17047.9			0.611278			0.305639	+	0.952148i	$0.401130\pi$
$168$		0			0
$169$	−5031.86			−0.176180
$170$	2359.59	+	605.055i	0.0816467	+	0.0209362i
$171$		0			0
$172$			30356.8i			1.02612i
$173$	19408.7			0.648492			0.324246	−	0.945973i	$-0.394889\pi$
$173$	19408.7			0.648492			0.324246	+	0.945973i	$0.394889\pi$
$174$		0			0
$175$	5876.23	−	10704.6i	0.191877	−	0.349539i
$176$		−	23179.8i		−	0.748316i
$177$		0			0
$178$		−	11129.3i		−	0.351260i
$179$		−	6659.35i		−	0.207838i	−0.994586	−	0.103919i	$-0.966862\pi$
$179$		−	6659.35i		−	0.207838i	0.994586	−	0.103919i	$-0.0331383\pi$
$180$		0			0
$181$	−15739.8			−0.480443			−0.240221	−	0.970718i	$-0.577220\pi$
$181$	−15739.8			−0.480443			−0.240221	+	0.970718i	$0.577220\pi$
$182$	3919.51			0.118328
$183$		0			0
$184$	15643.1			0.462048
$185$	3983.28	−	15534.0i	0.116385	−	0.453878i
$186$		0			0
$187$		−	10321.2i		−	0.295152i
$188$	−17440.6			−0.493454
$189$		0			0
$190$	−1384.36	−	354.983i	−0.0383479	−	0.00983332i
$191$			29252.1i			0.801844i	0.916112	+	0.400922i	$0.131310\pi$
$191$			29252.1i			0.801844i	−0.916112	+	0.400922i	$0.868690\pi$
$192$		0			0
$193$			42991.8i			1.15417i	0.816683	+	0.577086i	$0.195810\pi$
$193$			42991.8i			1.15417i	−0.816683	+	0.577086i	$0.804190\pi$
$194$		−	14999.6i		−	0.398543i
$195$		0			0
$196$	−29889.1			−0.778038
$197$	−48322.8			−1.24514			−0.622571	−	0.782563i	$-0.713912\pi$
$197$	−48322.8			−1.24514			−0.622571	+	0.782563i	$0.713912\pi$
$198$		0			0
$199$	10246.4			0.258740			0.129370	−	0.991596i	$-0.458705\pi$
$199$	10246.4			0.258740			0.129370	+	0.991596i	$0.458705\pi$
$200$	−18470.8	−	10139.4i	−0.461770	−	0.253485i
$201$		0			0
$202$			7769.84i			0.190419i
$203$	−28323.1			−0.687304
$204$		0			0
$205$	−8570.76	+	33424.2i	−0.203944	+	0.795340i
$206$		−	5129.34i		−	0.120872i
$207$		0			0
$208$			36644.4i			0.846996i
$209$			6055.39i			0.138627i
$210$		0			0
$211$	11134.0			0.250084			0.125042	−	0.992151i	$-0.460093\pi$
$211$	11134.0			0.250084			0.125042	+	0.992151i	$0.460093\pi$
$212$	953.647			0.0212186
$213$		0			0
$214$	−22147.6			−0.483614
$215$	12735.2	−	49664.5i	0.275504	−	1.07441i
$216$		0			0
$217$			30008.5i			0.637273i
$218$	2478.69			0.0521565
$219$		0			0
$220$	−10656.6	+	41558.5i	−0.220177	+	0.858646i
$221$			16316.5i			0.334074i
$222$		0			0
$223$			48339.8i			0.972065i	0.873941	+	0.486032i	$0.161556\pi$
$223$			48339.8i			0.972065i	−0.873941	+	0.486032i	$0.838444\pi$
$224$		−	14814.7i		−	0.295256i
$225$		0			0
$226$	10956.7			0.214517
$227$	19661.4			0.381560			0.190780	−	0.981633i	$-0.438898\pi$
$227$	19661.4			0.381560			0.190780	+	0.981633i	$0.438898\pi$
$228$		0			0
$229$	82115.8			1.56587			0.782935	−	0.622103i	$-0.213722\pi$
$229$	82115.8			1.56587			0.782935	+	0.622103i	$0.213722\pi$
$230$	−12298.6	−	3153.65i	−0.232487	−	0.0596153i
$231$		0			0
$232$			48871.5i			0.907987i
$233$	36907.6			0.679837			0.339918	−	0.940455i	$-0.389601\pi$
$233$	36907.6			0.679837			0.339918	+	0.940455i	$0.389601\pi$
$234$		0			0
$235$	28533.3	+	7316.63i	0.516674	+	0.132488i
$236$			48503.3i			0.870858i
$237$		0			0
$238$		−	1903.76i		−	0.0336091i
$239$		−	26719.8i		−	0.467775i	−0.972264	−	0.233888i	$-0.924855\pi$
$239$		−	26719.8i		−	0.467775i	0.972264	−	0.233888i	$-0.0751448\pi$
$240$		0			0
$241$	15275.6			0.263006			0.131503	−	0.991316i	$-0.458020\pi$
$241$	15275.6			0.263006			0.131503	+	0.991316i	$0.458020\pi$
$242$	1312.71			0.0224150
$243$		0			0
$244$	27514.3			0.462145
$245$	48899.3	+	12539.0i	0.814649	+	0.208896i
$246$		0			0
$247$		−	9572.81i		−	0.156908i
$248$	51779.7			0.841891
$249$		0			0
$250$	12477.6	+	11695.3i	0.199642	+	0.187125i
$251$			17504.6i			0.277847i	0.990303	+	0.138923i	$0.0443642\pi$
$251$			17504.6i			0.277847i	−0.990303	+	0.138923i	$0.955636\pi$
$252$		0			0
$253$			53795.8i			0.840441i
$254$			21432.4i			0.332203i
$255$		0			0
$256$	21787.8			0.332456
$257$	−57020.7			−0.863309			−0.431654	−	0.902039i	$-0.642070\pi$
$257$	−57020.7			−0.863309			−0.431654	+	0.902039i	$0.642070\pi$
$258$		0			0
$259$	−12533.1			−0.186835
$260$	16846.7	−	65698.7i	0.249212	−	0.971874i
$261$		0			0
$262$			31171.2i			0.454099i
$263$	−59434.2			−0.859261			−0.429630	−	0.903005i	$-0.641356\pi$
$263$	−59434.2			−0.859261			−0.429630	+	0.903005i	$0.641356\pi$
$264$		0			0
$265$	−1560.19	−	400.070i	−0.0222170	−	0.00569698i
$266$			1116.92i			0.0157856i
$267$		0			0
$268$			124043.i			1.72704i
$269$			107707.i			1.48847i	0.667919	+	0.744234i	$0.267185\pi$
$269$			107707.i			1.48847i	−0.667919	+	0.744234i	$0.732815\pi$
$270$		0			0
$271$	−11971.8			−0.163012			−0.0815059	−	0.996673i	$-0.525973\pi$
$271$	−11971.8			−0.163012			−0.0815059	+	0.996673i	$0.525973\pi$
$272$	17798.7			0.240575
$273$		0			0
$274$	28716.2			0.382495
$275$	34868.9	−	63520.1i	0.461076	−	0.839935i
$276$		0			0
$277$		−	26795.2i		−	0.349219i	−0.984638	−	0.174609i	$-0.944134\pi$
$277$		−	26795.2i		−	0.349219i	0.984638	−	0.174609i	$-0.0558662\pi$
$278$	−36456.2			−0.471718
$279$		0			0
$280$	−4090.32	+	15951.4i	−0.0521725	+	0.203462i
$281$			43545.0i			0.551474i	0.961233	+	0.275737i	$0.0889219\pi$
$281$			43545.0i			0.551474i	−0.961233	+	0.275737i	$0.911078\pi$
$282$		0			0
$283$			62179.6i			0.776381i	0.921579	+	0.388190i	$0.126900\pi$
$283$			62179.6i			0.776381i	−0.921579	+	0.388190i	$0.873100\pi$
$284$			67567.0i			0.837719i
$285$		0			0
$286$	23257.9			0.284341
$287$	26967.2			0.327395
$288$		0			0
$289$	−75595.9			−0.905112
$290$	9852.50	−	38422.7i	0.117152	−	0.456869i
$291$		0			0
$292$			119567.i			1.40232i
$293$	17216.0			0.200539			0.100269	−	0.994960i	$-0.468030\pi$
$293$	17216.0			0.200539			0.100269	+	0.994960i	$0.468030\pi$
$294$		0			0
$295$	20347.9	−	79352.7i	0.233817	−	0.911838i
$296$			21625.8i			0.246825i
$297$		0			0
$298$			5761.40i			0.0648777i
$299$		−	85044.3i		−	0.951268i
$300$		0			0
$301$	−40070.1			−0.442270
$302$	12339.3			0.135294
$303$		0			0
$304$	−10442.4			−0.112993
$305$	−45014.1	−	11542.7i	−0.483892	−	0.124082i
$306$		0			0
$307$		−	107387.i		−	1.13940i	−0.821854	−	0.569698i	$-0.807060\pi$
$307$		−	107387.i		−	1.13940i	0.821854	−	0.569698i	$-0.192940\pi$
$308$	33530.0			0.353454
$309$		0			0
$310$	−40709.1	−	10438.8i	−0.423612	−	0.108624i
$311$			136394.i			1.41018i	0.709117	+	0.705091i	$0.249094\pi$
$311$			136394.i			1.41018i	−0.709117	+	0.705091i	$0.750906\pi$
$312$		0			0
$313$		−	94570.2i		−	0.965307i	−0.875811	−	0.482654i	$-0.839673\pi$
$313$		−	94570.2i		−	0.965307i	0.875811	−	0.482654i	$-0.160327\pi$
$314$		−	22042.5i		−	0.223564i
$315$		0			0
$316$	128147.			1.28332
$317$	−141514.			−1.40826			−0.704129	−	0.710072i	$-0.748662\pi$
$317$	−141514.			−1.40826			−0.704129	+	0.710072i	$0.748662\pi$
$318$		0			0
$319$	−168066.			−1.65158
$320$	−57369.4	−	14710.9i	−0.560248	−	0.143661i
$321$		0			0
$322$			9922.70i			0.0957014i
$323$	−4649.64			−0.0445670
$324$		0			0
$325$	−55123.3	+	100417.i	−0.521878	+	0.950696i
$326$		−	16843.1i		−	0.158485i
$327$		0			0
$328$		−	46531.8i		−	0.432516i
$329$		−	23021.2i		−	0.212684i
$330$		0			0
$331$	82751.2			0.755298			0.377649	−	0.925949i	$-0.376733\pi$
$331$	82751.2			0.755298			0.377649	+	0.925949i	$0.376733\pi$
$332$	49440.5			0.448545
$333$		0			0
$334$	18659.2			0.167263
$335$	52037.9	−	202937.i	0.463692	−	1.80830i
$336$		0			0
$337$		−	83200.1i		−	0.732595i	−0.930498	−	0.366297i	$-0.880625\pi$
$337$		−	83200.1i		−	0.732595i	0.930498	−	0.366297i	$-0.119375\pi$
$338$	−5507.45			−0.0482078
$339$		0			0
$340$	−31910.7	−	8182.67i	−0.276044	−	0.0707843i
$341$			178067.i			1.53135i
$342$		0			0
$343$		−	86364.2i		−	0.734084i
$344$			69141.0i			0.584277i
$345$		0			0
$346$	21243.1			0.177446
$347$	103628.			0.860635			0.430317	−	0.902678i	$-0.358402\pi$
$347$	103628.			0.860635			0.430317	+	0.902678i	$0.358402\pi$
$348$		0			0
$349$	150713.			1.23737			0.618685	−	0.785639i	$-0.287666\pi$
$349$	150713.			1.23737			0.618685	+	0.785639i	$0.287666\pi$
$350$	6431.61	−	11716.4i	0.0525030	−	0.0956438i
$351$		0			0
$352$		−	87909.1i		−	0.709494i
$353$	22137.7			0.177657			0.0888286	−	0.996047i	$-0.471688\pi$
$353$	22137.7			0.177657			0.0888286	+	0.996047i	$0.471688\pi$
$354$		0			0
$355$	28345.5	−	110541.i	0.224919	−	0.877139i
$356$			150511.i			1.18760i
$357$		0			0
$358$		−	7288.75i		−	0.0568705i
$359$		−	90926.7i		−	0.705509i	−0.935716	−	0.352755i	$-0.885245\pi$
$359$		−	90926.7i		−	0.705509i	0.935716	−	0.352755i	$-0.114755\pi$
$360$		0			0
$361$	−127593.			−0.979068
$362$	−17227.4			−0.131463
$363$		0			0
$364$	−53006.8			−0.400063
$365$	50160.4	−	195615.i	0.376509	−	1.46831i
$366$		0			0
$367$			19749.6i			0.146631i	0.997309	+	0.0733156i	$0.0233580\pi$
$367$			19749.6i			0.146631i	−0.997309	+	0.0733156i	$0.976642\pi$
$368$	−92769.7			−0.685032
$369$		0			0
$370$	4359.76	−	17002.2i	0.0318463	−	0.124194i
$371$			1258.79i			0.00914544i
$372$		0			0
$373$		−	109344.i		−	0.785920i	−0.919555	−	0.392960i	$-0.871451\pi$
$373$		−	109344.i		−	0.785920i	0.919555	−	0.392960i	$-0.128549\pi$
$374$		−	11296.7i		−	0.0807621i
$375$		0			0
$376$	−39723.0			−0.280974
$377$	265692.			1.86937
$378$		0			0
$379$	75560.0			0.526034			0.263017	−	0.964791i	$-0.415283\pi$
$379$	75560.0			0.526034			0.263017	+	0.964791i	$0.415283\pi$
$380$	18721.9	+	4800.73i	0.129653	+	0.0332461i
$381$		0			0
$382$			32016.8i			0.219407i
$383$	−216249.			−1.47420			−0.737100	−	0.675783i	$-0.763806\pi$
$383$	−216249.			−1.47420			−0.737100	+	0.675783i	$0.763806\pi$
$384$		0			0
$385$	−54856.0	−	14066.4i	−0.370086	−	0.0948990i
$386$			47055.1i			0.315815i
$387$		0			0
$388$			202852.i			1.34746i
$389$			56554.2i			0.373737i	0.982385	+	0.186868i	$0.0598338\pi$
$389$			56554.2i			0.373737i	−0.982385	+	0.186868i	$0.940166\pi$
$390$		0			0
$391$	−41307.1			−0.270191
$392$	−68075.7			−0.443017
$393$		0			0
$394$	−52890.0			−0.340707
$395$	−209652.	−	53759.8i	−1.34371	−	0.344559i
$396$		0			0
$397$		−	254481.i		−	1.61464i	−0.590117	−	0.807318i	$-0.700918\pi$
$397$		−	254481.i		−	1.61464i	0.590117	−	0.807318i	$-0.299082\pi$
$398$	11214.8			0.0707986
$399$		0			0
$400$	109539.	+	60130.7i	0.684620	+	0.375817i
$401$		−	56231.5i		−	0.349696i	−0.984595	−	0.174848i	$-0.944057\pi$
$401$		−	56231.5i		−	0.349696i	0.984595	−	0.174848i	$-0.0559434\pi$
$402$		0			0
$403$		−	281502.i		−	1.73329i
$404$		−	105078.i		−	0.643797i
$405$		0			0
$406$	−31000.1			−0.188066
$407$	−74369.9			−0.448961
$408$		0			0
$409$	102314.			0.611632			0.305816	−	0.952091i	$-0.401071\pi$
$409$	102314.			0.611632			0.305816	+	0.952091i	$0.401071\pi$
$410$	−9380.82	+	36583.2i	−0.0558050	+	0.217628i
$411$		0			0
$412$			69368.3i			0.408664i
$413$	−64023.0			−0.375350
$414$		0			0
$415$	−80885.9	−	20741.1i	−0.469652	−	0.120430i
$416$			138973.i			0.803054i
$417$		0			0
$418$			6627.71i			0.0379324i
$419$			58412.2i			0.332717i	0.986065	+	0.166359i	$0.0532009\pi$
$419$			58412.2i			0.332717i	−0.986065	+	0.166359i	$0.946799\pi$
$420$		0			0
$421$	−235037.			−1.32609			−0.663043	−	0.748582i	$-0.730735\pi$
$421$	−235037.			−1.32609			−0.663043	+	0.748582i	$0.730735\pi$
$422$	12186.3			0.0684301
$423$		0			0
$424$	2172.04			0.0120819
$425$	48774.0	+	26774.1i	0.270029	+	0.148230i
$426$		0			0
$427$			36318.1i			0.199190i
$428$	299520.			1.63508
$429$		0			0
$430$	13938.8	−	54358.5i	0.0753858	−	0.293989i
$431$			43601.5i			0.234718i	0.993090	+	0.117359i	$0.0374429\pi$
$431$			43601.5i			0.234718i	−0.993090	+	0.117359i	$0.962557\pi$
$432$		0			0
$433$		−	102452.i		−	0.546445i	−0.961951	−	0.273222i	$-0.911910\pi$
$433$		−	102452.i		−	0.546445i	0.961951	−	0.273222i	$-0.0880895\pi$
$434$			32844.8i			0.174376i
$435$		0			0
$436$	−33521.4			−0.176339
$437$	24234.7			0.126904
$438$		0			0
$439$	−102464.			−0.531671			−0.265835	−	0.964018i	$-0.585648\pi$
$439$	−102464.			−0.531671			−0.265835	+	0.964018i	$0.585648\pi$
$440$	−24271.5	+	94654.0i	−0.125370	+	0.488915i
$441$		0			0
$442$			17858.6i			0.0914121i
$443$	247629.			1.26181			0.630906	−	0.775859i	$-0.282683\pi$
$443$	247629.			1.26181			0.630906	+	0.775859i	$0.282683\pi$
$444$		0			0
$445$	63141.9	−	246240.i	0.318858	−	1.24348i
$446$			52908.6i			0.265985i
$447$		0			0
$448$			46286.6i			0.230621i
$449$		−	79287.5i		−	0.393289i	−0.980475	−	0.196645i	$-0.936995\pi$
$449$		−	79287.5i		−	0.393289i	0.980475	−	0.196645i	$-0.0630045\pi$
$450$		0			0
$451$	160020.			0.786723
$452$	−148176.			−0.725272
$453$		0			0
$454$	21519.7			0.104406
$455$	86720.5	+	22237.2i	0.418889	+	0.107413i
$456$		0			0
$457$			117509.i			0.562651i	0.959612	+	0.281325i	$0.0907740\pi$
$457$			117509.i			0.562651i	−0.959612	+	0.281325i	$0.909226\pi$
$458$	89876.9			0.428467
$459$		0			0
$460$	166324.	+	42649.5i	0.786030	+	0.201557i
$461$			367089.i			1.72731i	0.504087	+	0.863653i	$0.331829\pi$
$461$			367089.i			1.72731i	−0.504087	+	0.863653i	$0.668171\pi$
$462$		0			0
$463$		−	92371.9i		−	0.430901i	−0.976515	−	0.215451i	$-0.930878\pi$
$463$		−	92371.9i		−	0.430901i	0.976515	−	0.215451i	$-0.0691220\pi$
$464$		−	289827.i		−	1.34618i
$465$		0			0
$466$	40396.0			0.186023
$467$	−82073.0			−0.376328			−0.188164	−	0.982138i	$-0.560254\pi$
$467$	−82073.0			−0.376328			−0.188164	+	0.982138i	$0.560254\pi$
$468$		0			0
$469$	−163733.			−0.744372
$470$	31230.1	+	8008.16i	0.141377	+	0.0362524i
$471$		0			0
$472$			110472.i			0.495869i
$473$	−237772.			−1.06277
$474$		0			0
$475$	−28615.5	−	15708.2i	−0.126828	−	0.0696210i
$476$			25746.1i			0.113631i
$477$		0			0
$478$		−	29245.2i		−	0.127997i
$479$		−	90721.3i		−	0.395401i	−0.980262	−	0.197701i	$-0.936653\pi$
$479$		−	90721.3i		−	0.395401i	0.980262	−	0.197701i	$-0.0633474\pi$
$480$		0			0
$481$	117569.			0.508164
$482$	16719.4			0.0719660
$483$		0			0
$484$	−17752.9			−0.0757843
$485$	85099.6	−	331871.i	0.361779	−	1.41086i
$486$		0			0
$487$		−	240214.i		−	1.01284i	−0.862287	−	0.506421i	$-0.830968\pi$
$487$		−	240214.i		−	1.01284i	0.862287	−	0.506421i	$-0.169032\pi$
$488$	62666.8			0.263147
$489$		0			0
$490$	53521.0	+	13724.1i	0.222911	+	0.0571598i
$491$			228899.i			0.949469i	0.880129	+	0.474734i	$0.157456\pi$
$491$			228899.i			0.949469i	−0.880129	+	0.474734i	$0.842544\pi$
$492$		0			0
$493$		−	129050.i		−	0.530963i
$494$		−	10477.6i		−	0.0429345i
$495$		0			0
$496$	−307074.			−1.24819
$497$	−89186.6			−0.361066
$498$		0			0
$499$	10464.5			0.0420260			0.0210130	−	0.999779i	$-0.493311\pi$
$499$	10464.5			0.0420260			0.0210130	+	0.999779i	$0.493311\pi$
$500$	−168745.	−	158166.i	−0.674981	−	0.632662i
$501$		0			0
$502$			19159.1i			0.0760269i
$503$	87312.8			0.345097			0.172549	−	0.985001i	$-0.444800\pi$
$503$	87312.8			0.345097			0.172549	+	0.985001i	$0.444800\pi$
$504$		0			0
$505$	−44081.9	+	171910.i	−0.172853	+	0.674092i
$506$			58880.2i			0.229969i
$507$		0			0
$508$		−	289848.i		−	1.12316i
$509$		−	244061.i		−	0.942026i	−0.882126	−	0.471013i	$-0.843888\pi$
$509$		−	244061.i		−	0.942026i	0.882126	−	0.471013i	$-0.156112\pi$
$510$		0			0
$511$	−157825.			−0.604415
$512$	259444.			0.989700
$513$		0			0
$514$	−62409.9			−0.236226
$515$	29101.1	−	113488.i	0.109722	−	0.427895i
$516$		0			0
$517$		−	136605.i		−	0.511076i
$518$	−13717.6			−0.0511233
$519$		0			0
$520$	38370.3	−	149636.i	0.141902	−	0.553388i
$521$			342391.i			1.26138i	0.776034	+	0.630692i	$0.217229\pi$
$521$			342391.i			1.26138i	−0.776034	+	0.630692i	$0.782771\pi$
$522$		0			0
$523$		−	313862.i		−	1.14746i	−0.819046	−	0.573728i	$-0.805497\pi$
$523$		−	313862.i		−	1.14746i	0.819046	−	0.573728i	$-0.194503\pi$
$524$		−	421554.i		−	1.53529i
$525$		0			0
$526$	−65051.6			−0.235118
$527$	−136729.			−0.492312
$528$		0			0
$529$	−64541.2			−0.230635
$530$	−1707.65	−	437.883i	−0.00607922	−	0.00155886i
$531$		0			0
$532$		−	15105.1i		−	0.0533704i
$533$	−252972.			−0.890467
$534$		0			0
$535$	−490023.	−	125654.i	−1.71202	−	0.439003i
$536$			282521.i			0.983378i
$537$		0			0
$538$			117887.i			0.407288i
$539$		−	234109.i		−	0.805823i
$540$		0			0
$541$	81438.3			0.278249			0.139125	−	0.990275i	$-0.455571\pi$
$541$	81438.3			0.278249			0.139125	+	0.990275i	$0.455571\pi$
$542$	−13103.3			−0.0446047
$543$		0			0
$544$	67501.1			0.228094
$545$	54841.8	+	14062.8i	0.184637	+	0.0473454i
$546$		0			0
$547$		−	399520.i		−	1.33525i	−0.744496	−	0.667627i	$-0.767310\pi$
$547$		−	399520.i		−	1.33525i	0.744496	−	0.667627i	$-0.232690\pi$
$548$	−388353.			−1.29320
$549$		0			0
$550$	38164.5	−	69523.7i	0.126164	−	0.229830i
$551$			75713.0i			0.249383i
$552$		0			0
$553$			169151.i			0.553125i
$554$		−	29327.7i		−	0.0955562i
$555$		0			0
$556$	493028.			1.59486
$557$	−464065.			−1.49578			−0.747892	−	0.663821i	$-0.768934\pi$
$557$	−464065.			−1.49578			−0.747892	+	0.663821i	$0.768934\pi$
$558$		0			0
$559$	375888.			1.20291
$560$	24257.2	−	94598.1i	0.0773508	−	0.301652i
$561$		0			0
$562$			47660.6i			0.150899i
$563$	564722.			1.78163			0.890816	−	0.454365i	$-0.150134\pi$
$563$	564722.			1.78163			0.890816	+	0.454365i	$0.150134\pi$
$564$		0			0
$565$	242420.	+	62162.2i	0.759401	+	0.194729i
$566$			68056.4i			0.212440i
$567$		0			0
$568$			153891.i			0.476999i
$569$		−	551230.i		−	1.70258i	−0.524692	−	0.851292i	$-0.675819\pi$
$569$		−	551230.i		−	1.70258i	0.524692	−	0.851292i	$-0.324181\pi$
$570$		0			0
$571$	−215672.			−0.661488			−0.330744	−	0.943721i	$-0.607300\pi$
$571$	−215672.			−0.661488			−0.330744	+	0.943721i	$0.607300\pi$
$572$	−314537.			−0.961345
$573$		0			0
$574$	29516.0			0.0895846
$575$	−254218.	−	139551.i	−0.768902	−	0.422083i
$576$		0			0
$577$			302487.i			0.908562i	0.890858	+	0.454281i	$0.150104\pi$
$577$			302487.i			0.908562i	−0.890858	+	0.454281i	$0.849896\pi$
$578$	−82740.8			−0.247665
$579$		0			0
$580$	−133244.	+	519622.i	−0.396087	+	1.54466i
$581$			65260.1i			0.193328i
$582$		0			0
$583$			7469.51i			0.0219763i
$584$			272327.i			0.798483i
$585$		0			0
$586$	18843.2			0.0548731
$587$	135754.			0.393982			0.196991	−	0.980405i	$-0.436883\pi$
$587$	135754.			0.393982			0.196991	+	0.980405i	$0.436883\pi$
$588$		0			0
$589$	80218.4			0.231230
$590$	22271.1	−	86852.6i	0.0639790	−	0.249505i
$591$		0			0
$592$		−	128249.i		−	0.365942i
$593$	612873.			1.74285			0.871427	−	0.490525i	$-0.163195\pi$
$593$	612873.			1.74285			0.871427	+	0.490525i	$0.163195\pi$
$594$		0			0
$595$	10800.9	−	42121.3i	0.0305089	−	0.118978i
$596$		−	77916.2i		−	0.219349i
$597$		0			0
$598$		−	93082.3i		−	0.260294i
$599$			176343.i			0.491479i	0.969336	+	0.245740i	$0.0790308\pi$
$599$			176343.i			0.491479i	−0.969336	+	0.245740i	$0.920969\pi$
$600$		0			0
$601$	110103.			0.304825			0.152412	−	0.988317i	$-0.451296\pi$
$601$	110103.			0.304825			0.152412	+	0.988317i	$0.451296\pi$
$602$	−43857.4			−0.121018
$603$		0			0
$604$	−166875.			−0.457423
$605$	29044.2	+	7447.64i	0.0793505	+	0.0203474i
$606$		0			0
$607$		−	219306.i		−	0.595213i	−0.954689	−	0.297607i	$-0.903812\pi$
$607$		−	219306.i		−	0.595213i	0.954689	−	0.297607i	$-0.0961884\pi$
$608$	−39602.6			−0.107131
$609$		0			0
$610$	−49268.6	−	12633.6i	−0.132407	−	0.0339523i
$611$			215955.i			0.578471i
$612$		0			0
$613$			75985.1i			0.202212i	0.994876	+	0.101106i	$0.0322382\pi$
$613$			75985.1i			0.202212i	−0.994876	+	0.101106i	$0.967762\pi$
$614$		−	117537.i		−	0.311771i
$615$		0			0
$616$	76368.4			0.201258
$617$	484132.			1.27173			0.635863	−	0.771802i	$-0.280644\pi$
$617$	484132.			1.27173			0.635863	+	0.771802i	$0.280644\pi$
$618$		0			0
$619$	46103.0			0.120323			0.0601614	−	0.998189i	$-0.480838\pi$
$619$	46103.0			0.120323			0.0601614	+	0.998189i	$0.480838\pi$
$620$	550543.	+	141173.i	1.43221	+	0.367254i
$621$		0			0
$622$			149285.i			0.385866i
$623$	−198671.			−0.511868
$624$		0			0
$625$	209719.	+	329554.i	0.536880	+	0.843659i
$626$		−	103508.i		−	0.264136i
$627$		0			0
$628$			298099.i			0.755861i
$629$		−	57105.0i		−	0.144335i
$630$		0			0
$631$	240096.			0.603013			0.301507	−	0.953464i	$-0.402510\pi$
$631$	240096.			0.603013			0.301507	+	0.953464i	$0.402510\pi$
$632$	291869.			0.730725
$633$		0			0
$634$	−154890.			−0.385340
$635$	−121596.	+	474199.i	−0.301559	+	1.17602i
$636$		0			0
$637$			370096.i			0.912086i
$638$	−183951.			−0.451919
$639$		0			0
$640$	−356583.	−	91436.5i	−0.870565	−	0.223234i
$641$			555022.i			1.35081i	0.737447	+	0.675405i	$0.236031\pi$
$641$			555022.i			1.35081i	−0.737447	+	0.675405i	$0.763969\pi$
$642$		0			0
$643$			646721.i			1.56421i	0.623147	+	0.782105i	$0.285854\pi$
$643$			646721.i			1.56421i	−0.623147	+	0.782105i	$0.714146\pi$
$644$		−	134193.i		−	0.323562i
$645$		0			0
$646$	−5089.09			−0.0121948
$647$	347688.			0.830579			0.415289	−	0.909689i	$-0.363680\pi$
$647$	347688.			0.830579			0.415289	+	0.909689i	$0.363680\pi$
$648$		0			0
$649$	−379906.			−0.901959
$650$	−60333.3	+	109908.i	−0.142801	+	0.260138i
$651$		0			0
$652$			227784.i			0.535831i
$653$	−609079.			−1.42839			−0.714196	−	0.699946i	$-0.753207\pi$
$653$	−609079.			−1.42839			−0.714196	+	0.699946i	$0.753207\pi$
$654$		0			0
$655$	−176849.	+	689673.i	−0.412211	+	1.60754i
$656$			275952.i			0.641247i
$657$		0			0
$658$		−	25197.0i		−	0.0581965i
$659$			623594.i			1.43592i	0.696083	+	0.717961i	$0.254924\pi$
$659$			623594.i			1.43592i	−0.696083	+	0.717961i	$0.745076\pi$
$660$		0			0
$661$	369284.			0.845197			0.422598	−	0.906317i	$-0.361118\pi$
$661$	369284.			0.845197			0.422598	+	0.906317i	$0.361118\pi$
$662$	90572.4			0.206671
$663$		0			0
$664$	112606.			0.255403
$665$	−6336.84	+	24712.3i	−0.0143294	+	0.0558818i
$666$		0			0
$667$			672630.i			1.51191i
$668$	−252344.			−0.565510
$669$		0			0
$670$	56956.2	−	222117.i	0.126880	−	0.494804i
$671$			215508.i			0.478650i
$672$		0			0
$673$			150223.i			0.331670i	0.986154	+	0.165835i	$0.0530319\pi$
$673$			150223.i			0.331670i	−0.986154	+	0.165835i	$0.946968\pi$
$674$		−	91063.7i		−	0.200459i
$675$		0			0
$676$	74481.8			0.162988
$677$	580295.			1.26611			0.633055	−	0.774107i	$-0.281801\pi$
$677$	580295.			1.26611			0.633055	+	0.774107i	$0.281801\pi$
$678$		0			0
$679$	−267759.			−0.580770
$680$	−72680.1	−	18636.9i	−0.157180	−	0.0403048i
$681$		0			0
$682$			194897.i			0.419022i
$683$	11404.6			0.0244477			0.0122238	−	0.999925i	$-0.496109\pi$
$683$	11404.6			0.0244477			0.0122238	+	0.999925i	$0.496109\pi$
$684$		0			0
$685$	635356.	+	162921.i	1.35405	+	0.347212i
$686$		−	94526.9i		−	0.200866i
$687$		0			0
$688$		−	410033.i		−	0.866247i
$689$		−	11808.4i		−	0.0248743i
$690$		0			0
$691$	−100807.			−0.211123			−0.105562	−	0.994413i	$-0.533664\pi$
$691$	−100807.			−0.211123			−0.105562	+	0.994413i	$0.533664\pi$
$692$	−287289.			−0.599938
$693$		0			0
$694$	113423.			0.235494
$695$	−806607.	−	206833.i	−1.66991	−	0.428204i
$696$		0			0
$697$			122872.i			0.252922i
$698$	164957.			0.338580
$699$		0			0
$700$	−86980.1	+	158450.i	−0.177510	+	0.323368i
$701$		−	220659.i		−	0.449040i	−0.974469	−	0.224520i	$-0.927919\pi$
$701$		−	220659.i		−	0.449040i	0.974469	−	0.224520i	$-0.0720814\pi$
$702$		0			0
$703$			33503.2i			0.0677916i
$704$			274660.i			0.554178i
$705$		0			0
$706$	24230.0			0.0486121
$707$	138700.			0.277484
$708$		0			0
$709$	−129462.			−0.257543			−0.128772	−	0.991674i	$-0.541103\pi$
$709$	−129462.			−0.257543			−0.128772	+	0.991674i	$0.541103\pi$
$710$	31024.5	−	120989.i	0.0615444	−	0.240010i
$711$		0			0
$712$			342806.i			0.676221i
$713$	712656.			1.40185
$714$		0			0
$715$	514590.	+	131953.i	1.00658	+	0.258112i
$716$			98571.9i			0.192277i
$717$		0			0
$718$		−	99520.6i		−	0.193048i
$719$		−	354459.i		−	0.685659i	−0.939398	−	0.342829i	$-0.888615\pi$
$719$		−	354459.i		−	0.685659i	0.939398	−	0.342829i	$-0.111385\pi$
$720$		0			0
$721$	−91564.3			−0.176139
$722$	−139652.			−0.267901
$723$		0			0
$724$	232981.			0.444471
$725$	435980.	−	794218.i	0.829451	−	1.51100i
$726$		0			0
$727$			199949.i			0.378312i	0.981947	+	0.189156i	$0.0605753\pi$
$727$			199949.i			0.378312i	−0.981947	+	0.189156i	$0.939425\pi$
$728$	−120729.			−0.227797
$729$		0			0
$730$	54901.3	−	214103.i	0.103024	−	0.401770i
$731$		−	182573.i		−	0.341667i
$732$		0			0
$733$			862301.i			1.60491i	0.596712	+	0.802455i	$0.296473\pi$
$733$			862301.i			1.60491i	−0.596712	+	0.802455i	$0.703527\pi$
$734$			21616.2i			0.0401225i
$735$		0			0
$736$	−351827.			−0.649492
$737$	−971573.			−1.78871
$738$		0			0
$739$	684605.			1.25358			0.626789	−	0.779189i	$-0.284369\pi$
$739$	684605.			1.25358			0.626789	+	0.779189i	$0.284369\pi$
$740$	−58960.7	+	229934.i	−0.107671	+	0.419895i
$741$		0			0
$742$			1377.76i			0.00250246i
$743$	−801856.			−1.45251			−0.726254	−	0.687426i	$-0.758740\pi$
$743$	−801856.			−1.45251			−0.726254	+	0.687426i	$0.758740\pi$
$744$		0			0
$745$	−32687.1	+	127473.i	−0.0588931	+	0.229671i
$746$		−	119679.i		−	0.215050i
$747$		0			0
$748$			152775.i			0.273053i
$749$			395359.i			0.704738i
$750$		0			0
$751$	509079.			0.902620			0.451310	−	0.892367i	$-0.350957\pi$
$751$	509079.			0.902620			0.451310	+	0.892367i	$0.350957\pi$
$752$	235573.			0.416571
$753$		0			0
$754$	290803.			0.511513
$755$	273012.	+	70006.9i	0.478948	+	0.122814i
$756$		0			0
$757$			766769.i			1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$757$			766769.i			1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$758$	82701.5			0.143938
$759$		0			0
$760$	42641.1	+	10934.2i	0.0738246	+	0.0189304i
$761$		−	327796.i		−	0.566023i	−0.959117	−	0.283011i	$-0.908667\pi$
$761$		−	327796.i		−	0.566023i	0.959117	−	0.283011i	$-0.0913335\pi$
$762$		0			0
$763$		−	44247.3i		−	0.0760042i
$764$		−	432990.i		−	0.741808i
$765$		0			0
$766$	−236688.			−0.403384
$767$	600583.			1.02090
$768$		0			0
$769$	394875.			0.667740			0.333870	−	0.942619i	$-0.391645\pi$
$769$	394875.			0.667740			0.333870	+	0.942619i	$0.391645\pi$
$770$	−60040.7	−	15395.9i	−0.101266	−	0.0259671i
$771$		0			0
$772$		−	636366.i		−	1.06776i
$773$	121076.			0.202628			0.101314	−	0.994855i	$-0.467695\pi$
$773$	121076.			0.202628			0.101314	+	0.994855i	$0.467695\pi$
$774$		0			0
$775$	−841479.	−	461924.i	−1.40101	−	0.769072i
$776$			462017.i			0.767246i
$777$		0			0
$778$			61899.4i			0.102265i
$779$		−	72088.3i		−	0.118793i
$780$		0			0
$781$	−529224.			−0.867636
$782$	−45211.3			−0.0739321
$783$		0			0
$784$	403716.			0.656816
$785$	125058.	−	487698.i	0.202941	−	0.791429i
$786$		0			0
$787$		−	604224.i		−	0.975548i	−0.872970	−	0.487774i	$-0.837809\pi$
$787$		−	604224.i		−	0.975548i	0.872970	−	0.487774i	$-0.162191\pi$
$788$	715275.			1.15192
$789$		0			0
$790$	−229467.	−	58840.8i	−0.367677	−	0.0942811i
$791$		−	195588.i		−	0.312601i
$792$		0			0
$793$		−	340691.i		−	0.541769i
$794$		−	278533.i		−	0.441810i
$795$		0			0
$796$	−151667.			−0.239367
$797$	−889702.			−1.40064			−0.700322	−	0.713827i	$-0.746960\pi$
$797$	−889702.			−1.40064			−0.700322	+	0.713827i	$0.746960\pi$
$798$		0			0
$799$	104892.			0.164305
$800$	415425.	+	228045.i	0.649102	+	0.356320i
$801$		0			0
$802$		−	61546.2i		−	0.0956868i
$803$	−936519.			−1.45240
$804$		0			0
$805$	−56296.1	+	219543.i	−0.0868734	+	0.338788i
$806$		−	308108.i		−	0.474278i
$807$		0			0
$808$		−	239327.i		−	0.366580i
$809$			351052.i			0.536382i	0.963366	+	0.268191i	$0.0864258\pi$
$809$			351052.i			0.536382i	−0.963366	+	0.268191i	$0.913574\pi$
$810$		0			0
$811$	−316054.			−0.480529			−0.240264	−	0.970707i	$-0.577234\pi$
$811$	−316054.			−0.480529			−0.240264	+	0.970707i	$0.577234\pi$
$812$	419240.			0.635844
$813$		0			0
$814$	−81398.9			−0.122848
$815$	95559.0	−	372660.i	0.143865	−	0.561045i
$816$		0			0
$817$			107115.i			0.160475i
$818$	111985.			0.167360
$819$		0			0
$820$	126865.	−	494746.i	0.188674	−	0.735791i
$821$		−	985939.i		−	1.46273i	−0.681987	−	0.731364i	$-0.738884\pi$
$821$		−	985939.i		−	1.46273i	0.681987	−	0.731364i	$-0.261116\pi$
$822$		0			0
$823$		−	632153.i		−	0.933303i	−0.884441	−	0.466651i	$-0.845460\pi$
$823$		−	632153.i		−	0.933303i	0.884441	−	0.466651i	$-0.154540\pi$
$824$			157994.i			0.232695i
$825$		0			0
$826$	−70074.1			−0.102706
$827$	−309027.			−0.451840			−0.225920	−	0.974146i	$-0.572539\pi$
$827$	−309027.			−0.451840			−0.225920	+	0.974146i	$0.572539\pi$
$828$		0			0
$829$	−832364.			−1.21117			−0.605584	−	0.795782i	$-0.707060\pi$
$829$	−832364.			−1.21117			−0.605584	+	0.795782i	$0.707060\pi$
$830$	−88530.8	−	22701.4i	−0.128510	−	0.0329531i
$831$		0			0
$832$		−	434202.i		−	0.627257i
$833$	179761.			0.259062
$834$		0			0
$835$	412841.	+	105862.i	0.592121	+	0.151834i
$836$		−	89632.1i		−	0.128248i
$837$		0			0
$838$			63933.0i			0.0910410i
$839$			829659.i			1.17863i	0.807905	+	0.589313i	$0.200601\pi$
$839$			829659.i			1.17863i	−0.807905	+	0.589313i	$0.799399\pi$
$840$		0			0
$841$	−1.39412e6			−1.97110
$842$	−257251.			−0.362855
$843$		0			0
$844$	−164806.			−0.231359
$845$	−121854.	−	31246.3i	−0.170658	−	0.0437608i
$846$		0			0
$847$		−	23433.4i		−	0.0326639i
$848$	−12881.0			−0.0179126
$849$		0			0
$850$	53383.8	+	29304.7i	0.0738876	+	0.0405601i
$851$			297641.i			0.410992i
$852$		0			0
$853$			701609.i			0.964266i	0.876098	+	0.482133i	$0.160138\pi$
$853$			701609.i			0.964266i	−0.876098	+	0.482133i	$0.839862\pi$
$854$			39750.7i			0.0545041i
$855$		0			0
$856$	682191.			0.931019
$857$	−1.08522e6			−1.47760			−0.738801	−	0.673923i	$-0.764608\pi$
$857$	−1.08522e6			−1.47760			−0.738801	+	0.673923i	$0.764608\pi$
$858$		0			0
$859$	−884781.			−1.19908			−0.599542	−	0.800343i	$-0.704651\pi$
$859$	−884781.			−1.19908			−0.599542	+	0.800343i	$0.704651\pi$
$860$	−188506.	+	735136.i	−0.254876	+	0.993964i
$861$		0			0
$862$			47722.5i			0.0642256i
$863$	848540.			1.13933			0.569667	−	0.821876i	$-0.307072\pi$
$863$	848540.			1.13933			0.569667	+	0.821876i	$0.307072\pi$
$864$		0			0
$865$	470011.	+	120522.i	0.628169	+	0.161077i
$866$		−	112136.i		−	0.149523i
$867$		0			0
$868$		−	444188.i		−	0.589558i
$869$			1.00372e6i			1.32915i
$870$		0			0
$871$	1.53593e6			2.02459
$872$	−76348.6			−0.100408
$873$		0			0
$874$	26525.2			0.0347245
$875$	208774.	−	222739.i	0.272685	−	0.290925i
$876$		0			0
$877$			513895.i			0.668152i	0.942546	+	0.334076i	$0.108424\pi$
$877$			513895.i			0.668152i	−0.942546	+	0.334076i	$0.891576\pi$
$878$	−112148.			−0.145480
$879$		0			0
$880$	143940.	−	561335.i	0.185873	−	0.724865i
$881$		−	222212.i		−	0.286297i	−0.989701	−	0.143148i	$-0.954277\pi$
$881$		−	222212.i		−	0.286297i	0.989701	−	0.143148i	$-0.0457226\pi$
$882$		0			0
$883$		−	248366.i		−	0.318545i	−0.987235	−	0.159272i	$-0.949085\pi$
$883$		−	248366.i		−	0.318545i	0.987235	−	0.159272i	$-0.0509148\pi$
$884$		−	241517.i		−	0.309061i
$885$		0			0
$886$	271034.			0.345268
$887$	864526.			1.09883			0.549415	−	0.835549i	$-0.314850\pi$
$887$	864526.			1.09883			0.549415	+	0.835549i	$0.314850\pi$
$888$		0			0
$889$	382592.			0.484097
$890$	69109.7	−	269514.i	0.0872488	−	0.340252i
$891$		0			0
$892$		−	715528.i		−	0.899284i
$893$	−61539.9			−0.0771709
$894$		0			0
$895$	41352.5	−	161266.i	0.0516245	−	0.201325i
$896$			287697.i			0.358360i
$897$		0			0
$898$		−	86781.3i		−	0.107615i
$899$			2.22645e6i			2.75482i
$900$		0			0
$901$	−5735.47			−0.00706512
$902$	175145.			0.215270
$903$		0			0
$904$	−337487.			−0.412972
$905$	−381163.	−	97739.3i	−0.465386	−	0.119336i
$906$		0			0
$907$		−	1.07355e6i		−	1.30499i	−0.757794	−	0.652494i	$-0.773723\pi$
$907$		−	1.07355e6i		−	1.30499i	0.757794	−	0.652494i	$-0.226277\pi$
$908$	−291029.			−0.352992
$909$		0			0
$910$	94916.8	+	24338.9i	0.114620	+	0.0293913i
$911$		−	1.00771e6i		−	1.21422i	−0.794618	−	0.607110i	$-0.792329\pi$
$911$		−	1.00771e6i		−	1.21422i	0.794618	−	0.607110i	$-0.207671\pi$
$912$		0			0
$913$			387246.i			0.464564i
$914$			128615.i			0.153957i
$915$		0			0
$916$	−1.21548e6			−1.44863
$917$	556440.			0.661728
$918$		0			0
$919$	−576271.			−0.682332			−0.341166	−	0.940003i	$-0.610822\pi$
$919$	−576271.			−0.682332			−0.341166	+	0.940003i	$0.610822\pi$
$920$	378821.	+	97138.9i	0.447568	+	0.114767i
$921$		0			0
$922$			401784.i			0.472640i
$923$	836637.			0.982050
$924$		0			0
$925$	192922.	−	351444.i	0.225475	−	0.410745i
$926$		−	101102.i		−	0.117907i
$927$		0			0
$928$		−	1.09916e6i		−	1.27634i
$929$		−	1.02738e6i		−	1.19041i	−0.803572	−	0.595207i	$-0.797070\pi$
$929$		−	1.02738e6i		−	1.19041i	0.803572	−	0.595207i	$-0.202930\pi$
$930$		0			0
$931$	−105465.			−0.121677
$932$	−546308.			−0.628935
$933$		0			0
$934$	−89830.1			−0.102974
$935$	64091.4	−	249943.i	0.0733122	−	0.285902i
$936$		0			0
$937$			358312.i			0.408115i	0.978959	+	0.204057i	$0.0654129\pi$
$937$			358312.i			0.408115i	−0.978959	+	0.204057i	$0.934587\pi$
$938$	−179208.			−0.203682
$939$		0			0
$940$	−422351.	−	108301.i	−0.477989	−	0.122568i
$941$		−	732982.i		−	0.827778i	−0.910327	−	0.413889i	$-0.864170\pi$
$941$		−	732982.i		−	0.827778i	0.910327	−	0.413889i	$-0.135830\pi$
$942$		0			0
$943$		−	640429.i		−	0.720191i
$944$		−	655140.i		−	0.735174i
$945$		0			0
$946$	−260245.			−0.290804
$947$	722514.			0.805650			0.402825	−	0.915277i	$-0.368028\pi$
$947$	722514.			0.805650			0.402825	+	0.915277i	$0.368028\pi$
$948$		0			0
$949$	1.48052e6			1.64392
$950$	−31320.0	−	17192.9i	−0.0347036	−	0.0190503i
$951$		0			0
$952$			58639.6i			0.0647019i
$953$	−686812.			−0.756227			−0.378113	−	0.925759i	$-0.623427\pi$
$953$	−686812.			−0.756227			−0.378113	+	0.925759i	$0.623427\pi$
$954$		0			0
$955$	−181646.	+	708383.i	−0.199168	+	0.776715i
$956$			395507.i			0.432752i
$957$		0			0
$958$		−	99295.8i		−	0.108193i
$959$		−	512616.i		−	0.557384i
$960$		0			0
$961$	1.43542e6			1.55429
$962$	128681.			0.139048
$963$		0			0
$964$	−226111.			−0.243314
$965$	−266966.	+	1.04111e6i	−0.286682	+	1.11800i
$966$		0			0
$967$		−	251363.i		−	0.268811i	−0.990926	−	0.134406i	$-0.957087\pi$
$967$		−	251363.i		−	0.268811i	0.990926	−	0.134406i	$-0.0429125\pi$
$968$	−40434.3			−0.0431518
$969$		0			0
$970$	93142.7	−	363237.i	0.0989932	−	0.386053i
$971$		−	299319.i		−	0.317465i	−0.987322	−	0.158733i	$-0.949259\pi$
$971$		−	299319.i		−	0.317465i	0.987322	−	0.158733i	$-0.0507408\pi$
$972$		0			0
$973$			650784.i			0.687403i
$974$		−	262918.i		−	0.277142i
$975$		0			0
$976$	−371639.			−0.390141
$977$	704266.			0.737815			0.368907	−	0.929466i	$-0.379732\pi$
$977$	704266.			0.737815			0.368907	+	0.929466i	$0.379732\pi$
$978$		0			0
$979$	−1.17889e6			−1.23001
$980$	−723810.	−	185602.i	−0.753654	−	0.193255i
$981$		0			0
$982$			250533.i			0.259802i
$983$	687286.			0.711263			0.355632	−	0.934626i	$-0.384266\pi$
$983$	687286.			0.711263			0.355632	+	0.934626i	$0.384266\pi$
$984$		0			0
$985$	−1.17021e6	−	300070.i	−1.20612	−	0.309278i
$986$		−	141247.i		−	0.145287i
$987$		0			0
$988$			141697.i			0.145160i
$989$			951604.i			0.972890i
$990$		0			0
$991$	315848.			0.321611			0.160806	−	0.986986i	$-0.448591\pi$
$991$	315848.			0.321611			0.160806	+	0.986986i	$0.448591\pi$
$992$	−1.16457e6			−1.18343
$993$		0			0
$994$	−97616.1			−0.0987981
$995$	248131.	+	63626.7i	0.250631	+	0.0642678i
$996$		0			0
$997$			529342.i			0.532532i	0.963900	+	0.266266i	$0.0857899\pi$
$997$			529342.i			0.532532i	−0.963900	+	0.266266i	$0.914210\pi$
$998$	11453.6			0.0114995
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.25		44
3.2	odd	2	inner	405.5.d.a.404.19		44
5.4	even	2	inner	405.5.d.a.404.20		44
9.2	odd	6		135.5.h.a.44.13		44
9.4	even	3		135.5.h.a.89.10		44
9.5	odd	6		45.5.h.a.29.13	yes	44
9.7	even	3		45.5.h.a.14.10	✓	44
15.14	odd	2	inner	405.5.d.a.404.26		44
45.4	even	6		135.5.h.a.89.13		44
45.14	odd	6		45.5.h.a.29.10	yes	44
45.29	odd	6		135.5.h.a.44.10		44
45.34	even	6		45.5.h.a.14.13	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.10	✓	44	9.7	even	3
45.5.h.a.14.13	yes	44	45.34	even	6
45.5.h.a.29.10	yes	44	45.14	odd	6
45.5.h.a.29.13	yes	44	9.5	odd	6
135.5.h.a.44.10		44	45.29	odd	6
135.5.h.a.44.13		44	9.2	odd	6
135.5.h.a.89.10		44	9.4	even	3
135.5.h.a.89.13		44	45.4	even	6
405.5.d.a.404.19		44	3.2	odd	2	inner
405.5.d.a.404.20		44	5.4	even	2	inner
405.5.d.a.404.25		44	1.1	even	1	trivial
405.5.d.a.404.26		44	15.14	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.09451 q^{2} -14.8020 q^{4} +(24.2165 + 6.20970i) q^{5} -19.5383i q^{7} -33.7133 q^{8} +(26.5053 + 6.79660i) q^{10} -115.938i q^{11} +183.284i q^{13} -21.3849i q^{14} +199.933 q^{16} +89.0232 q^{17} -52.2295 q^{19} +(-358.454 - 91.9161i) q^{20} -126.896i q^{22} -464.004 q^{23} +(547.879 + 300.754i) q^{25} +200.607i q^{26} +289.206i q^{28} -1449.62i q^{29} -1535.88 q^{31} +758.242 q^{32} +97.4372 q^{34} +(121.327 - 473.149i) q^{35} -641.462i q^{37} -57.1659 q^{38} +(-816.418 - 209.349i) q^{40} +1380.22i q^{41} -2050.85i q^{43} +1716.12i q^{44} -507.859 q^{46} +1178.26 q^{47} +2019.26 q^{49} +(599.662 + 329.180i) q^{50} -2712.97i q^{52} -64.4267 q^{53} +(719.940 - 2807.62i) q^{55} +658.700i q^{56} -1586.63i q^{58} -3276.80i q^{59} -1858.82 q^{61} -1681.05 q^{62} -2369.02 q^{64} +(-1138.14 + 4438.49i) q^{65} -8380.10i q^{67} -1317.72 q^{68} +(132.794 - 517.869i) q^{70} -4564.71i q^{71} -8077.75i q^{73} -702.089i q^{74} +773.103 q^{76} -2265.23 q^{77} -8657.39 q^{79} +(4841.68 + 1241.52i) q^{80} +1510.67i q^{82} -3340.11 q^{83} +(2155.83 + 552.807i) q^{85} -2244.69i q^{86} +3908.65i q^{88} -10168.3i q^{89} +3581.05 q^{91} +6868.21 q^{92} +1289.62 q^{94} +(-1264.82 - 324.329i) q^{95} -13704.3i q^{97} +2210.10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

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