LMFDB - Embedded newform 405.5.d.a.404.3

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			404.3
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7.28232 q^{2} +37.0321 q^{4} +(21.3751 - 12.9656i) q^{5} -44.4060i q^{7} -153.163 q^{8} +(-155.660 + 94.4195i) q^{10} +164.451i q^{11} -162.158i q^{13} +323.378i q^{14} +522.865 q^{16} -79.6141 q^{17} +493.552 q^{19} +(791.564 - 480.144i) q^{20} -1197.58i q^{22} +198.547 q^{23} +(288.787 - 554.281i) q^{25} +1180.89i q^{26} -1644.45i q^{28} +106.209i q^{29} +1034.85 q^{31} -1357.06 q^{32} +579.775 q^{34} +(-575.750 - 949.181i) q^{35} -1042.31i q^{37} -3594.20 q^{38} +(-3273.86 + 1985.84i) q^{40} -44.6454i q^{41} +3448.84i q^{43} +6089.97i q^{44} -1445.88 q^{46} +836.017 q^{47} +429.108 q^{49} +(-2103.04 + 4036.45i) q^{50} -6005.05i q^{52} +307.853 q^{53} +(2132.20 + 3515.15i) q^{55} +6801.34i q^{56} -773.449i q^{58} +4467.53i q^{59} -1752.57 q^{61} -7536.11 q^{62} +1516.74 q^{64} +(-2102.47 - 3466.14i) q^{65} -4072.59i q^{67} -2948.28 q^{68} +(4192.79 + 6912.24i) q^{70} -5455.14i q^{71} +486.298i q^{73} +7590.40i q^{74} +18277.3 q^{76} +7302.60 q^{77} -7265.20 q^{79} +(11176.3 - 6779.25i) q^{80} +325.122i q^{82} +3510.00 q^{83} +(-1701.76 + 1032.24i) q^{85} -25115.6i q^{86} -25187.7i q^{88} -5210.89i q^{89} -7200.78 q^{91} +7352.63 q^{92} -6088.14 q^{94} +(10549.7 - 6399.19i) q^{95} -1743.98i q^{97} -3124.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−7.28232			−1.82058			−0.910290	−	0.413972i	$-0.864141\pi$
$2$	−7.28232			−1.82058			−0.910290	+	0.413972i	$0.864141\pi$
$3$		0			0
$3$		0			0
$4$	37.0321			2.31451
$5$	21.3751	−	12.9656i	0.855003	−	0.518624i
$5$	21.3751	−	12.9656i	0.855003	−	0.518624i
$6$		0			0
$7$		−	44.4060i		−	0.906245i	−0.891448	−	0.453122i	$-0.850310\pi$
$7$		−	44.4060i		−	0.906245i	0.891448	−	0.453122i	$-0.149690\pi$
$8$	−153.163			−2.39317
$9$		0			0
$10$	−155.660	+	94.4195i	−1.55660	+	0.944195i
$11$			164.451i			1.35910i	0.733630	+	0.679549i	$0.237824\pi$
$11$			164.451i			1.35910i	−0.733630	+	0.679549i	$0.762176\pi$
$12$		0			0
$13$		−	162.158i		−	0.959514i	−0.877401	−	0.479757i	$-0.840725\pi$
$13$		−	162.158i		−	0.959514i	0.877401	−	0.479757i	$-0.159275\pi$
$14$			323.378i			1.64989i
$15$		0			0
$16$	522.865			2.04244
$17$	−79.6141			−0.275481			−0.137741	−	0.990468i	$-0.543984\pi$
$17$	−79.6141			−0.275481			−0.137741	+	0.990468i	$0.543984\pi$
$18$		0			0
$19$	493.552			1.36718			0.683590	−	0.729866i	$-0.260418\pi$
$19$	493.552			1.36718			0.683590	+	0.729866i	$0.260418\pi$
$20$	791.564	−	480.144i	1.97891	−	1.20036i
$21$		0			0
$22$		−	1197.58i		−	2.47435i
$23$	198.547			0.375326			0.187663	−	0.982234i	$-0.439909\pi$
$23$	198.547			0.375326			0.187663	+	0.982234i	$0.439909\pi$
$24$		0			0
$25$	288.787	−	554.281i	0.462059	−	0.886849i
$26$			1180.89i			1.74687i
$27$		0			0
$28$		−	1644.45i		−	2.09751i
$29$			106.209i			0.126289i	0.998004	+	0.0631446i	$0.0201129\pi$
$29$			106.209i			0.126289i	−0.998004	+	0.0631446i	$0.979887\pi$
$30$		0			0
$31$	1034.85			1.07685			0.538424	−	0.842674i	$-0.319020\pi$
$31$	1034.85			1.07685			0.538424	+	0.842674i	$0.319020\pi$
$32$	−1357.06			−1.32526
$33$		0			0
$34$	579.775			0.501536
$35$	−575.750	−	949.181i	−0.470000	−	0.774842i
$36$		0			0
$37$		−	1042.31i		−	0.761363i	−0.924706	−	0.380681i	$-0.875689\pi$
$37$		−	1042.31i		−	0.761363i	0.924706	−	0.380681i	$-0.124311\pi$
$38$	−3594.20			−2.48906
$39$		0			0
$40$	−3273.86	+	1985.84i	−2.04616	+	1.24115i
$41$		−	44.6454i		−	0.0265588i	−0.999912	−	0.0132794i	$-0.995773\pi$
$41$		−	44.6454i		−	0.0265588i	0.999912	−	0.0132794i	$-0.00422709\pi$
$42$		0			0
$43$			3448.84i			1.86525i	0.360850	+	0.932624i	$0.382487\pi$
$43$			3448.84i			1.86525i	−0.360850	+	0.932624i	$0.617513\pi$
$44$			6089.97i			3.14564i
$45$		0			0
$46$	−1445.88			−0.683310
$47$	836.017			0.378460			0.189230	−	0.981933i	$-0.439401\pi$
$47$	836.017			0.378460			0.189230	+	0.981933i	$0.439401\pi$
$48$		0			0
$49$	429.108			0.178721
$50$	−2103.04	+	4036.45i	−0.841215	+	1.61458i
$51$		0			0
$52$		−	6005.05i		−	2.22080i
$53$	307.853			0.109595			0.0547975	−	0.998497i	$-0.482549\pi$
$53$	307.853			0.109595			0.0547975	+	0.998497i	$0.482549\pi$
$54$		0			0
$55$	2132.20	+	3515.15i	0.704860	+	1.16203i
$56$			6801.34i			2.16879i
$57$		0			0
$58$		−	773.449i		−	0.229919i
$59$			4467.53i			1.28341i	0.766953	+	0.641703i	$0.221772\pi$
$59$			4467.53i			1.28341i	−0.766953	+	0.641703i	$0.778228\pi$
$60$		0			0
$61$	−1752.57			−0.470995			−0.235497	−	0.971875i	$-0.575672\pi$
$61$	−1752.57			−0.470995			−0.235497	+	0.971875i	$0.575672\pi$
$62$	−7536.11			−1.96049
$63$		0			0
$64$	1516.74			0.370297
$65$	−2102.47	−	3466.14i	−0.497627	−	0.820387i
$66$		0			0
$67$		−	4072.59i		−	0.907237i	−0.891196	−	0.453619i	$-0.850133\pi$
$67$		−	4072.59i		−	0.907237i	0.891196	−	0.453619i	$-0.149867\pi$
$68$	−2948.28			−0.637604
$69$		0			0
$70$	4192.79	+	6912.24i	0.855672	+	1.41066i
$71$		−	5455.14i		−	1.08215i	−0.840973	−	0.541077i	$-0.818017\pi$
$71$		−	5455.14i		−	1.08215i	0.840973	−	0.541077i	$-0.181983\pi$
$72$		0			0
$73$			486.298i			0.0912549i	0.998959	+	0.0456275i	$0.0145287\pi$
$73$			486.298i			0.0912549i	−0.998959	+	0.0456275i	$0.985471\pi$
$74$			7590.40i			1.38612i
$75$		0			0
$76$	18277.3			3.16435
$77$	7302.60			1.23168
$78$		0			0
$79$	−7265.20			−1.16411			−0.582054	−	0.813150i	$-0.697751\pi$
$79$	−7265.20			−1.16411			−0.582054	+	0.813150i	$0.697751\pi$
$80$	11176.3	−	6779.25i	1.74629	−	1.05926i
$81$		0			0
$82$			325.122i			0.0483524i
$83$	3510.00			0.509508			0.254754	−	0.967006i	$-0.418006\pi$
$83$	3510.00			0.509508			0.254754	+	0.967006i	$0.418006\pi$
$84$		0			0
$85$	−1701.76	+	1032.24i	−0.235537	+	0.142871i
$86$		−	25115.6i		−	3.39583i
$87$		0			0
$88$		−	25187.7i		−	3.25255i
$89$		−	5210.89i		−	0.657857i	−0.944355	−	0.328929i	$-0.893312\pi$
$89$		−	5210.89i		−	0.657857i	0.944355	−	0.328929i	$-0.106688\pi$
$90$		0			0
$91$	−7200.78			−0.869555
$92$	7352.63			0.868695
$93$		0			0
$94$	−6088.14			−0.689016
$95$	10549.7	−	6399.19i	1.16894	−	0.709052i
$96$		0			0
$97$		−	1743.98i		−	0.185352i	−0.995696	−	0.0926761i	$-0.970458\pi$
$97$		−	1743.98i		−	0.185352i	0.995696	−	0.0926761i	$-0.0295421\pi$
$98$	−3124.90			−0.325375
$99$		0			0
$100$	10694.4	−	20526.2i	1.06944	−	2.05262i
$101$		−	6603.41i		−	0.647329i	−0.946172	−	0.323665i	$-0.895085\pi$
$101$		−	6603.41i		−	0.647329i	0.946172	−	0.323665i	$-0.104915\pi$
$102$		0			0
$103$		−	7034.38i		−	0.663058i	−0.943445	−	0.331529i	$-0.892436\pi$
$103$		−	7034.38i		−	0.663058i	0.943445	−	0.331529i	$-0.107564\pi$
$104$			24836.5i			2.29628i
$105$		0			0
$106$	−2241.88			−0.199527
$107$	17346.9			1.51515			0.757574	−	0.652749i	$-0.226384\pi$
$107$	17346.9			1.51515			0.757574	+	0.652749i	$0.226384\pi$
$108$		0			0
$109$	17571.4			1.47895			0.739475	−	0.673184i	$-0.235074\pi$
$109$	17571.4			1.47895			0.739475	+	0.673184i	$0.235074\pi$
$110$	−15527.4	−	25598.4i	−1.28325	−	2.11557i
$111$		0			0
$112$		−	23218.3i		−	1.85095i
$113$	−15639.3			−1.22479			−0.612395	−	0.790552i	$-0.709794\pi$
$113$	−15639.3			−1.22479			−0.612395	+	0.790552i	$0.709794\pi$
$114$		0			0
$115$	4243.96	−	2574.28i	0.320904	−	0.194653i
$116$			3933.15i			0.292297i
$117$		0			0
$118$		−	32534.0i		−	2.33654i
$119$			3535.34i			0.249653i
$120$		0			0
$121$	−12403.1			−0.847147
$122$	12762.8			0.857484
$123$		0			0
$124$	38322.7			2.49237
$125$	−1013.74	−	15592.1i	−0.0648793	−	0.997893i
$126$		0			0
$127$			17452.3i			1.08205i	0.841007	+	0.541024i	$0.181963\pi$
$127$			17452.3i			1.08205i	−0.841007	+	0.541024i	$0.818037\pi$
$128$	10667.7			0.651103
$129$		0			0
$130$	15310.9	+	25241.5i	0.905969	+	1.49358i
$131$			2379.49i			0.138657i	0.997594	+	0.0693285i	$0.0220856\pi$
$131$			2379.49i			0.138657i	−0.997594	+	0.0693285i	$0.977914\pi$
$132$		0			0
$133$		−	21916.7i		−	1.23900i
$134$			29657.9i			1.65170i
$135$		0			0
$136$	12193.9			0.659273
$137$	5702.00			0.303799			0.151900	−	0.988396i	$-0.451461\pi$
$137$	5702.00			0.303799			0.151900	+	0.988396i	$0.451461\pi$
$138$		0			0
$139$	14738.5			0.762823			0.381412	−	0.924405i	$-0.375438\pi$
$139$	14738.5			0.762823			0.381412	+	0.924405i	$0.375438\pi$
$140$	−21321.2	−	35150.2i	−1.08782	−	1.79338i
$141$		0			0
$142$			39726.1i			1.97015i
$143$	26667.0			1.30407
$144$		0			0
$145$	1377.06	+	2270.23i	0.0654965	+	0.107978i
$146$		−	3541.37i		−	0.166137i
$147$		0			0
$148$		−	38598.8i		−	1.76218i
$149$		−	18357.0i		−	0.826854i	−0.910537	−	0.413427i	$-0.864332\pi$
$149$		−	18357.0i		−	0.826854i	0.910537	−	0.413427i	$-0.135668\pi$
$150$		0			0
$151$	1143.51			0.0501515			0.0250758	−	0.999686i	$-0.492017\pi$
$151$	1143.51			0.0501515			0.0250758	+	0.999686i	$0.492017\pi$
$152$	−75593.7			−3.27189
$153$		0			0
$154$	−53179.9			−2.24236
$155$	22120.0	−	13417.4i	0.920707	−	0.558478i
$156$		0			0
$157$			31703.1i			1.28618i	0.765791	+	0.643090i	$0.222348\pi$
$157$			31703.1i			1.28618i	−0.765791	+	0.643090i	$0.777652\pi$
$158$	52907.5			2.11935
$159$		0			0
$160$	−29007.3	+	17595.1i	−1.13310	+	0.687310i
$161$		−	8816.69i		−	0.340137i
$162$		0			0
$163$		−	37271.3i		−	1.40281i	−0.712762	−	0.701406i	$-0.752556\pi$
$163$		−	37271.3i		−	1.40281i	0.712762	−	0.701406i	$-0.247444\pi$
$164$		−	1653.31i		−	0.0614706i
$165$		0			0
$166$	−25560.9			−0.927599
$167$	−25246.6			−0.905253			−0.452627	−	0.891700i	$-0.649513\pi$
$167$	−25246.6			−0.905253			−0.452627	+	0.891700i	$0.649513\pi$
$168$		0			0
$169$	2265.82			0.0793327
$170$	12392.7	−	7517.13i	0.428814	−	0.260108i
$171$		0			0
$172$			127718.i			4.31713i
$173$	−8008.46			−0.267582			−0.133791	−	0.991010i	$-0.542715\pi$
$173$	−8008.46			−0.267582			−0.133791	+	0.991010i	$0.542715\pi$
$174$		0			0
$175$	−24613.4	−	12823.9i	−0.803702	−	0.418739i
$176$			85985.6i			2.77588i
$177$		0			0
$178$			37947.3i			1.19768i
$179$		−	41994.7i		−	1.31065i	−0.755345	−	0.655327i	$-0.772531\pi$
$179$		−	41994.7i		−	1.31065i	0.755345	−	0.655327i	$-0.227469\pi$
$180$		0			0
$181$	−4820.58			−0.147144			−0.0735719	−	0.997290i	$-0.523440\pi$
$181$	−4820.58			−0.147144			−0.0735719	+	0.997290i	$0.523440\pi$
$182$	52438.4			1.58309
$183$		0			0
$184$	−30410.0			−0.898217
$185$	−13514.1	−	22279.4i	−0.394861	−	0.650967i
$186$		0			0
$187$		−	13092.6i		−	0.374406i
$188$	30959.5			0.875948
$189$		0			0
$190$	−76826.3	+	46601.0i	−2.12815	+	1.29089i
$191$		−	66372.4i		−	1.81937i	−0.415301	−	0.909684i	$-0.636324\pi$
$191$		−	66372.4i		−	1.81937i	0.415301	−	0.909684i	$-0.363676\pi$
$192$		0			0
$193$		−	1127.60i		−	0.0302721i	−0.999885	−	0.0151360i	$-0.995182\pi$
$193$		−	1127.60i		−	0.0302721i	0.999885	−	0.0151360i	$-0.00481813\pi$
$194$			12700.2i			0.337448i
$195$		0			0
$196$	15890.8			0.413650
$197$	−44846.7			−1.15557			−0.577787	−	0.816188i	$-0.696084\pi$
$197$	−44846.7			−1.15557			−0.577787	+	0.816188i	$0.696084\pi$
$198$		0			0
$199$	57005.6			1.43950			0.719749	−	0.694234i	$-0.244257\pi$
$199$	57005.6			1.43950			0.719749	+	0.694234i	$0.244257\pi$
$200$	−44231.4	+	84895.1i	−1.10578	+	2.12238i
$201$		0			0
$202$			48088.1i			1.17851i
$203$	4716.32			0.114449
$204$		0			0
$205$	−578.853	−	954.297i	−0.0137740	−	0.0227078i
$206$			51226.6i			1.20715i
$207$		0			0
$208$		−	84786.7i		−	1.95975i
$209$			81165.0i			1.85813i
$210$		0			0
$211$	−30364.6			−0.682028			−0.341014	−	0.940058i	$-0.610770\pi$
$211$	−30364.6			−0.682028			−0.341014	+	0.940058i	$0.610770\pi$
$212$	11400.4			0.253659
$213$		0			0
$214$	−126326.			−2.75845
$215$	44716.3	+	73719.2i	0.967362	+	1.59479i
$216$		0			0
$217$		−	45953.5i		−	0.975887i
$218$	−127961.			−2.69255
$219$		0			0
$220$	78960.0	+	130173.i	1.63141	+	2.68953i
$221$			12910.1i			0.264328i
$222$		0			0
$223$		−	47054.4i		−	0.946216i	−0.881004	−	0.473108i	$-0.843132\pi$
$223$		−	47054.4i		−	0.946216i	0.881004	−	0.473108i	$-0.156868\pi$
$224$			60261.8i			1.20101i
$225$		0			0
$226$	113891.			2.22983
$227$	−43194.0			−0.838247			−0.419123	−	0.907929i	$-0.637663\pi$
$227$	−43194.0			−0.838247			−0.419123	+	0.907929i	$0.637663\pi$
$228$		0			0
$229$	86075.2			1.64137			0.820686	−	0.571380i	$-0.193592\pi$
$229$	86075.2			1.64137			0.820686	+	0.571380i	$0.193592\pi$
$230$	−30905.9	+	18746.7i	−0.584232	+	0.354381i
$231$		0			0
$232$		−	16267.3i		−	0.302231i
$233$	−14612.3			−0.269158			−0.134579	−	0.990903i	$-0.542968\pi$
$233$	−14612.3			−0.269158			−0.134579	+	0.990903i	$0.542968\pi$
$234$		0			0
$235$	17869.9	−	10839.5i	0.323584	−	0.196278i
$236$			165442.i			2.97045i
$237$		0			0
$238$		−	25745.5i		−	0.454514i
$239$		−	53349.5i		−	0.933973i	−0.884264	−	0.466987i	$-0.845340\pi$
$239$		−	53349.5i		−	0.933973i	0.884264	−	0.466987i	$-0.154660\pi$
$240$		0			0
$241$	77412.8			1.33284			0.666421	−	0.745576i	$-0.267825\pi$
$241$	77412.8			1.33284			0.666421	+	0.745576i	$0.267825\pi$
$242$	90323.1			1.54230
$243$		0			0
$244$	−64901.5			−1.09012
$245$	9172.21	−	5563.64i	0.152807	−	0.0926887i
$246$		0			0
$247$		−	80033.3i		−	1.31183i
$248$	−158500.			−2.57707
$249$		0			0
$250$	7382.37	+	113546.i	0.118118	+	1.81674i
$251$		−	36216.5i		−	0.574855i	−0.957802	−	0.287428i	$-0.907200\pi$
$251$		−	36216.5i		−	0.574855i	0.957802	−	0.287428i	$-0.0928001\pi$
$252$		0			0
$253$			32651.3i			0.510104i
$254$		−	127094.i		−	1.96995i
$255$		0			0
$256$	−101953.			−1.55568
$257$	27669.3			0.418921			0.209461	−	0.977817i	$-0.432829\pi$
$257$	27669.3			0.418921			0.209461	+	0.977817i	$0.432829\pi$
$258$		0			0
$259$	−46284.6			−0.689981
$260$	−77859.1	−	128358.i	−1.15176	−	1.89879i
$261$		0			0
$262$		−	17328.2i		−	0.252436i
$263$	102579.			1.48301			0.741507	−	0.670946i	$-0.234112\pi$
$263$	102579.			1.48301			0.741507	+	0.670946i	$0.234112\pi$
$264$		0			0
$265$	6580.37	−	3991.49i	0.0937041	−	0.0568386i
$266$			159604.i			2.25570i
$267$		0			0
$268$		−	150817.i		−	2.09981i
$269$		−	36102.4i		−	0.498921i	−0.968385	−	0.249461i	$-0.919747\pi$
$269$		−	36102.4i		−	0.498921i	0.968385	−	0.249461i	$-0.0802533\pi$
$270$		0			0
$271$	−62347.3			−0.848944			−0.424472	−	0.905441i	$-0.639540\pi$
$271$	−62347.3			−0.848944			−0.424472	+	0.905441i	$0.639540\pi$
$272$	−41627.4			−0.562654
$273$		0			0
$274$	−41523.8			−0.553090
$275$	91151.9	+	47491.2i	1.20531	+	0.627983i
$276$		0			0
$277$			81719.9i			1.06505i	0.846416	+	0.532523i	$0.178756\pi$
$277$			81719.9i			1.06505i	−0.846416	+	0.532523i	$0.821244\pi$
$278$	−107330.			−1.38878
$279$		0			0
$280$	88183.4	+	145379.i	1.12479	+	1.85433i
$281$		−	56352.4i		−	0.713674i	−0.934167	−	0.356837i	$-0.883855\pi$
$281$		−	56352.4i		−	0.713674i	0.934167	−	0.356837i	$-0.116145\pi$
$282$		0			0
$283$		−	45852.1i		−	0.572514i	−0.958153	−	0.286257i	$-0.907589\pi$
$283$		−	45852.1i		−	0.572514i	0.958153	−	0.286257i	$-0.0924111\pi$
$284$		−	202015.i		−	2.50466i
$285$		0			0
$286$	−194198.			−2.37417
$287$	−1982.52			−0.0240688
$288$		0			0
$289$	−77182.6			−0.924110
$290$	−10028.2	−	16532.5i	−0.119242	−	0.196582i
$291$		0			0
$292$			18008.6i			0.211210i
$293$	123204.			1.43512			0.717560	−	0.696496i	$-0.245259\pi$
$293$	123204.			1.43512			0.717560	+	0.696496i	$0.245259\pi$
$294$		0			0
$295$	57924.2	+	95493.8i	0.665604	+	1.09732i
$296$			159642.i			1.82207i
$297$		0			0
$298$			133681.i			1.50535i
$299$		−	32196.0i		−	0.360130i
$300$		0			0
$301$	153149.			1.69037
$302$	−8327.37			−0.0913048
$303$		0			0
$304$	258061.			2.79238
$305$	−37461.3	+	22723.1i	−0.402702	+	0.244269i
$306$		0			0
$307$			15411.3i			0.163517i	0.996652	+	0.0817587i	$0.0260537\pi$
$307$			15411.3i			0.163517i	−0.996652	+	0.0817587i	$0.973946\pi$
$308$	270431.			2.85072
$309$		0			0
$310$	−161085.	+	97710.1i	−1.67622	+	1.01675i
$311$			16751.2i			0.173191i	0.996244	+	0.0865957i	$0.0275988\pi$
$311$			16751.2i			0.173191i	−0.996244	+	0.0865957i	$0.972401\pi$
$312$		0			0
$313$		−	90525.5i		−	0.924022i	−0.886874	−	0.462011i	$-0.847128\pi$
$313$		−	90525.5i		−	0.924022i	0.886874	−	0.462011i	$-0.152872\pi$
$314$		−	230872.i		−	2.34159i
$315$		0			0
$316$	−269046.			−2.69434
$317$	−17921.2			−0.178340			−0.0891699	−	0.996016i	$-0.528421\pi$
$317$	−17921.2			−0.178340			−0.0891699	+	0.996016i	$0.528421\pi$
$318$		0			0
$319$	−17466.2			−0.171639
$320$	32420.4	−	19665.4i	0.316605	−	0.192045i
$321$		0			0
$322$			64205.9i			0.619246i
$323$	−39293.7			−0.376633
$324$		0			0
$325$	−89881.0	−	46829.1i	−0.850944	−	0.443352i
$326$			271422.i			2.55393i
$327$		0			0
$328$			6838.00i			0.0635597i
$329$		−	37124.2i		−	0.342977i
$330$		0			0
$331$	95796.7			0.874369			0.437184	−	0.899372i	$-0.355976\pi$
$331$	95796.7			0.874369			0.437184	+	0.899372i	$0.355976\pi$
$332$	129983.			1.17926
$333$		0			0
$334$	183854.			1.64809
$335$	−52803.5	−	87051.8i	−0.470515	−	0.775690i
$336$		0			0
$337$		−	2224.38i		−	0.0195862i	−0.999952	−	0.00979309i	$-0.996883\pi$
$337$		−	2224.38i		−	0.0195862i	0.999952	−	0.00979309i	$-0.00311729\pi$
$338$	−16500.4			−0.144431
$339$		0			0
$340$	−63019.7	+	38226.2i	−0.545153	+	0.330676i
$341$			170182.i			1.46354i
$342$		0			0
$343$		−	125674.i		−	1.06821i
$344$		−	528234.i		−	4.46385i
$345$		0			0
$346$	58320.2			0.487154
$347$	−4956.62			−0.0411648			−0.0205824	−	0.999788i	$-0.506552\pi$
$347$	−4956.62			−0.0411648			−0.0205824	+	0.999788i	$0.506552\pi$
$348$		0			0
$349$	−205224.			−1.68491			−0.842456	−	0.538765i	$-0.818891\pi$
$349$	−205224.			−1.68491			−0.842456	+	0.538765i	$0.818891\pi$
$350$	179242.	+	93387.5i	1.46320	+	0.762347i
$351$		0			0
$352$		−	223170.i		−	1.80116i
$353$	−154011.			−1.23595			−0.617977	−	0.786196i	$-0.712047\pi$
$353$	−154011.			−1.23595			−0.617977	+	0.786196i	$0.712047\pi$
$354$		0			0
$355$	−70729.1	−	116604.i	−0.561231	−	0.925245i
$356$		−	192970.i		−	1.52262i
$357$		0			0
$358$			305818.i			2.38615i
$359$		−	175796.i		−	1.36401i	−0.731345	−	0.682007i	$-0.761107\pi$
$359$		−	175796.i		−	1.36401i	0.731345	−	0.682007i	$-0.238893\pi$
$360$		0			0
$361$	113273.			0.869181
$362$	35105.0			0.267887
$363$		0			0
$364$	−266660.			−2.01259
$365$	6305.14	+	10394.6i	0.0473270	+	0.0780232i
$366$		0			0
$367$			9134.69i			0.0678206i	0.999425	+	0.0339103i	$0.0107961\pi$
$367$			9134.69i			0.0678206i	−0.999425	+	0.0339103i	$0.989204\pi$
$368$	103813.			0.766581
$369$		0			0
$370$	98414.0	+	162245.i	0.718875	+	1.18514i
$371$		−	13670.5i		−	0.0993200i
$372$		0			0
$373$			10430.1i			0.0749673i	0.999297	+	0.0374837i	$0.0119342\pi$
$373$			10430.1i			0.0749673i	−0.999297	+	0.0374837i	$0.988066\pi$
$374$			95344.5i			0.681636i
$375$		0			0
$376$	−128047.			−0.905717
$377$	17222.7			0.121176
$378$		0			0
$379$	103658.			0.721646			0.360823	−	0.932634i	$-0.382496\pi$
$379$	103658.			0.721646			0.360823	+	0.932634i	$0.382496\pi$
$380$	390678.	−	236976.i	2.70553	−	1.64111i
$381$		0			0
$382$			483345.i			3.31230i
$383$	−140256.			−0.956145			−0.478072	−	0.878320i	$-0.658664\pi$
$383$	−140256.			−0.956145			−0.478072	+	0.878320i	$0.658664\pi$
$384$		0			0
$385$	156094.	−	94682.6i	1.05309	−	0.638776i
$386$			8211.57i			0.0551127i
$387$		0			0
$388$		−	64583.3i		−	0.428999i
$389$			136993.i			0.905317i	0.891684	+	0.452658i	$0.149524\pi$
$389$			136993.i			0.905317i	−0.891684	+	0.452658i	$0.850476\pi$
$390$		0			0
$391$	−15807.2			−0.103395
$392$	−65723.3			−0.427708
$393$		0			0
$394$	326588.			2.10381
$395$	−155294.	+	94197.6i	−0.995316	+	0.603734i
$396$		0			0
$397$			37281.0i			0.236541i	0.992981	+	0.118270i	$0.0377350\pi$
$397$			37281.0i			0.236541i	−0.992981	+	0.118270i	$0.962265\pi$
$398$	−415133.			−2.62072
$399$		0			0
$400$	150997.	−	289814.i	0.943728	−	1.81134i
$401$		−	260586.i		−	1.62055i	−0.586050	−	0.810275i	$-0.699318\pi$
$401$		−	260586.i		−	1.62055i	0.586050	−	0.810275i	$-0.300682\pi$
$402$		0			0
$403$		−	167809.i		−	1.03325i
$404$		−	244538.i		−	1.49825i
$405$		0			0
$406$	−34345.8			−0.208363
$407$	171408.			1.03477
$408$		0			0
$409$	180520.			1.07914			0.539572	−	0.841940i	$-0.318586\pi$
$409$	180520.			1.07914			0.539572	+	0.841940i	$0.318586\pi$
$410$	4215.39	+	6949.50i	0.0250767	+	0.0413414i
$411$		0			0
$412$		−	260498.i		−	1.53465i
$413$	198385.			1.16308
$414$		0			0
$415$	75026.4	−	45509.2i	0.435630	−	0.264243i
$416$			220059.i			1.27160i
$417$		0			0
$418$		−	591070.i		−	3.38288i
$419$			56367.4i			0.321070i	0.987030	+	0.160535i	$0.0513219\pi$
$419$			56367.4i			0.321070i	−0.987030	+	0.160535i	$0.948678\pi$
$420$		0			0
$421$	53682.2			0.302877			0.151438	−	0.988467i	$-0.451609\pi$
$421$	53682.2			0.302877			0.151438	+	0.988467i	$0.451609\pi$
$422$	221124.			1.24169
$423$		0			0
$424$	−47151.5			−0.262279
$425$	−22991.5	+	44128.6i	−0.127289	+	0.244310i
$426$		0			0
$427$			77824.7i			0.426837i
$428$	642394.			3.50682
$429$		0			0
$430$	−325638.	−	536847.i	−1.76116	−	2.90344i
$431$			35602.6i			0.191658i	0.995398	+	0.0958290i	$0.0305502\pi$
$431$			35602.6i			0.191658i	−0.995398	+	0.0958290i	$0.969450\pi$
$432$		0			0
$433$		−	303764.i		−	1.62017i	−0.586314	−	0.810084i	$-0.699421\pi$
$433$		−	303764.i		−	1.62017i	0.586314	−	0.810084i	$-0.300579\pi$
$434$			334648.i			1.77668i
$435$		0			0
$436$	650707.			3.42304
$437$	97993.4			0.513138
$438$		0			0
$439$	104573.			0.542612			0.271306	−	0.962493i	$-0.412544\pi$
$439$	104573.			0.542612			0.271306	+	0.962493i	$0.412544\pi$
$440$	−326574.	−	538389.i	−1.68685	−	2.78094i
$441$		0			0
$442$		−	94015.1i		−	0.481230i
$443$	215655.			1.09889			0.549443	−	0.835531i	$-0.314840\pi$
$443$	215655.			1.09889			0.549443	+	0.835531i	$0.314840\pi$
$444$		0			0
$445$	−67562.3	−	111383.i	−0.341180	−	0.562470i
$446$			342665.i			1.72266i
$447$		0			0
$448$		−	67352.2i		−	0.335580i
$449$			356746.i			1.76957i	0.466004	+	0.884783i	$0.345693\pi$
$449$			356746.i			1.76957i	−0.466004	+	0.884783i	$0.654307\pi$
$450$		0			0
$451$	7341.97			0.0360960
$452$	−579158.			−2.83479
$453$		0			0
$454$	314552.			1.52609
$455$	−153917.	+	93362.4i	−0.743471	+	0.450972i
$456$		0			0
$457$			44999.8i			0.215466i	0.994180	+	0.107733i	$0.0343591\pi$
$457$			44999.8i			0.215466i	−0.994180	+	0.107733i	$0.965641\pi$
$458$	−626827.			−2.98825
$459$		0			0
$460$	157163.	−	95331.2i	0.742736	−	0.450526i
$461$			334697.i			1.57489i	0.616387	+	0.787443i	$0.288596\pi$
$461$			334697.i			1.57489i	−0.616387	+	0.787443i	$0.711404\pi$
$462$		0			0
$463$			208550.i			0.972854i	0.873721	+	0.486427i	$0.161700\pi$
$463$			208550.i			0.972854i	−0.873721	+	0.486427i	$0.838300\pi$
$464$			55533.1i			0.257938i
$465$		0			0
$466$	106412.			0.490024
$467$	−294193.			−1.34896			−0.674479	−	0.738294i	$-0.735632\pi$
$467$	−294193.			−1.34896			−0.674479	+	0.738294i	$0.735632\pi$
$468$		0			0
$469$	−180847.			−0.822179
$470$	−130134.	+	78936.4i	−0.589110	+	0.357340i
$471$		0			0
$472$		−	684260.i		−	3.07140i
$473$	−567165.			−2.53505
$474$		0			0
$475$	142531.	−	273566.i	0.631718	−	1.21248i
$476$			130921.i			0.577825i
$477$		0			0
$478$			388508.i			1.70037i
$479$			209628.i			0.913648i	0.889557	+	0.456824i	$0.151013\pi$
$479$			209628.i			0.913648i	−0.889557	+	0.456824i	$0.848987\pi$
$480$		0			0
$481$	−169018.			−0.730538
$482$	−563744.			−2.42654
$483$		0			0
$484$	−459312.			−1.96073
$485$	−22611.7	−	37277.7i	−0.0961281	−	0.158477i
$486$		0			0
$487$			121334.i			0.511593i	0.966731	+	0.255796i	$0.0823376\pi$
$487$			121334.i			0.511593i	−0.966731	+	0.255796i	$0.917662\pi$
$488$	268429.			1.12717
$489$		0			0
$490$	−66795.0	+	40516.2i	−0.278196	+	0.168747i
$491$			433367.i			1.79760i	0.438361	+	0.898799i	$0.355559\pi$
$491$			433367.i			1.79760i	−0.438361	+	0.898799i	$0.644441\pi$
$492$		0			0
$493$		−	8455.75i		−	0.0347903i
$494$			582828.i			2.38829i
$495$		0			0
$496$	541087.			2.19940
$497$	−242241.			−0.980697
$498$		0			0
$499$	−403937.			−1.62223			−0.811115	−	0.584886i	$-0.801139\pi$
$499$	−403937.			−1.62223			−0.811115	+	0.584886i	$0.801139\pi$
$500$	−37540.9	−	577408.i	−0.150164	−	2.30963i
$501$		0			0
$502$			263740.i			1.04657i
$503$	381137.			1.50642			0.753208	−	0.657782i	$-0.228505\pi$
$503$	381137.			1.50642			0.753208	+	0.657782i	$0.228505\pi$
$504$		0			0
$505$	−85617.1	−	141148.i	−0.335720	−	0.553468i
$506$		−	237777.i		−	0.928685i
$507$		0			0
$508$			646298.i			2.50441i
$509$			342697.i			1.32274i	0.750060	+	0.661370i	$0.230025\pi$
$509$			342697.i			1.32274i	−0.750060	+	0.661370i	$0.769975\pi$
$510$		0			0
$511$	21594.5			0.0826993
$512$	571773.			2.18114
$513$		0			0
$514$	−201497.			−0.762679
$515$	−91204.9	−	150360.i	−0.343878	−	0.566916i
$516$		0			0
$517$			137484.i			0.514364i
$518$	337059.			1.25617
$519$		0			0
$520$	322020.	+	530883.i	1.19090	+	1.96332i
$521$		−	85.0427i		−	0.000313301i	−1.00000		0.000156650i	$-0.999950\pi$
$521$		−	85.0427i		−	0.000313301i	1.00000		0.000156650i	$-4.98634e-5\pi$
$522$		0			0
$523$			251271.i			0.918626i	0.888275	+	0.459313i	$0.151904\pi$
$523$			251271.i			0.918626i	−0.888275	+	0.459313i	$0.848096\pi$
$524$			88117.6i			0.320923i
$525$		0			0
$526$	−747009.			−2.69994
$527$	−82388.7			−0.296651
$528$		0			0
$529$	−240420.			−0.859131
$530$	−47920.3	+	29067.3i	−0.170596	+	0.103479i
$531$		0			0
$532$		−	811621.i		−	2.86768i
$533$	−7239.60			−0.0254836
$534$		0			0
$535$	370792.	−	224913.i	1.29546	−	0.785792i
$536$			623768.i			2.17117i
$537$		0			0
$538$			262909.i			0.908325i
$539$			70567.2i			0.242899i
$540$		0			0
$541$	142630.			0.487321			0.243661	−	0.969861i	$-0.421652\pi$
$541$	142630.			0.487321			0.243661	+	0.969861i	$0.421652\pi$
$542$	454033.			1.54557
$543$		0			0
$544$	108041.			0.365084
$545$	375590.	−	227824.i	1.26451	−	0.767019i
$546$		0			0
$547$			31256.4i			0.104464i	0.998635	+	0.0522318i	$0.0166335\pi$
$547$			31256.4i			0.104464i	−0.998635	+	0.0522318i	$0.983367\pi$
$548$	211157.			0.703145
$549$		0			0
$550$	−663797.	−	345846.i	−2.19437	−	1.14329i
$551$			52419.8i			0.172660i
$552$		0			0
$553$			322618.i			1.05497i
$554$		−	595110.i		−	1.93900i
$555$		0			0
$556$	545798.			1.76556
$557$	129530.			0.417503			0.208752	−	0.977969i	$-0.433060\pi$
$557$	129530.			0.417503			0.208752	+	0.977969i	$0.433060\pi$
$558$		0			0
$559$	559257.			1.78973
$560$	−301039.	−	496293.i	−0.959947	−	1.58257i
$561$		0			0
$562$			410376.i			1.29930i
$563$	−249165.			−0.786088			−0.393044	−	0.919520i	$-0.628578\pi$
$563$	−249165.			−0.786088			−0.393044	+	0.919520i	$0.628578\pi$
$564$		0			0
$565$	−334292.	+	202773.i	−1.04720	+	0.635205i
$566$			333909.i			1.04231i
$567$		0			0
$568$			835524.i			2.58978i
$569$			63614.8i			0.196487i	0.995162	+	0.0982434i	$0.0313224\pi$
$569$			63614.8i			0.196487i	−0.995162	+	0.0982434i	$0.968678\pi$
$570$		0			0
$571$	−90423.1			−0.277337			−0.138668	−	0.990339i	$-0.544282\pi$
$571$	−90423.1			−0.277337			−0.138668	+	0.990339i	$0.544282\pi$
$572$	987536.			3.01829
$573$		0			0
$574$	14437.3			0.0438191
$575$	57337.9	−	110051.i	0.173423	−	0.332857i
$576$		0			0
$577$			328301.i			0.986100i	0.870001	+	0.493050i	$0.164118\pi$
$577$			328301.i			0.986100i	−0.870001	+	0.493050i	$0.835882\pi$
$578$	562068.			1.68242
$579$		0			0
$580$	50995.7	+	84071.4i	0.151592	+	0.249915i
$581$		−	155865.i		−	0.461739i
$582$		0			0
$583$			50626.6i			0.148950i
$584$		−	74482.6i		−	0.218388i
$585$		0			0
$586$	−897208.			−2.61275
$587$	−333132.			−0.966807			−0.483403	−	0.875398i	$-0.660600\pi$
$587$	−333132.			−0.966807			−0.483403	+	0.875398i	$0.660600\pi$
$588$		0			0
$589$	510752.			1.47224
$590$	−421823.	−	695416.i	−1.21179	−	1.99775i
$591$		0			0
$592$		−	544985.i		−	1.55504i
$593$	−596831.			−1.69724			−0.848618	−	0.529006i	$-0.822565\pi$
$593$	−596831.			−1.69724			−0.848618	+	0.529006i	$0.822565\pi$
$594$		0			0
$595$	45837.8	+	75568.2i	0.129476	+	0.213454i
$596$		−	679798.i		−	1.91376i
$597$		0			0
$598$			234462.i			0.655646i
$599$			417722.i			1.16422i	0.813112	+	0.582108i	$0.197772\pi$
$599$			417722.i			1.16422i	−0.813112	+	0.582108i	$0.802228\pi$
$600$		0			0
$601$	−525842.			−1.45581			−0.727907	−	0.685675i	$-0.759507\pi$
$601$	−525842.			−1.45581			−0.727907	+	0.685675i	$0.759507\pi$
$602$	−1.11528e6			−3.07745
$603$		0			0
$604$	42346.4			0.116076
$605$	−265117.	+	160813.i	−0.724313	+	0.439350i
$606$		0			0
$607$		−	325005.i		−	0.882089i	−0.897485	−	0.441044i	$-0.854608\pi$
$607$		−	325005.i		−	0.882089i	0.897485	−	0.441044i	$-0.145392\pi$
$608$	−669782.			−1.81187
$609$		0			0
$610$	272805.	−	165477.i	0.733151	−	0.444711i
$611$		−	135567.i		−	0.363137i
$612$		0			0
$613$			180164.i			0.479453i	0.970840	+	0.239727i	$0.0770578\pi$
$613$			180164.i			0.479453i	−0.970840	+	0.239727i	$0.922942\pi$
$614$		−	112230.i		−	0.297696i
$615$		0			0
$616$	−1.11849e6			−2.94760
$617$	210750.			0.553602			0.276801	−	0.960927i	$-0.410726\pi$
$617$	210750.			0.553602			0.276801	+	0.960927i	$0.410726\pi$
$618$		0			0
$619$	733785.			1.91508			0.957541	−	0.288297i	$-0.0930892\pi$
$619$	733785.			1.91508			0.957541	+	0.288297i	$0.0930892\pi$
$620$	819151.	−	496877.i	2.13098	−	1.29260i
$621$		0			0
$622$		−	121988.i		−	0.315309i
$623$	−231395.			−0.596180
$624$		0			0
$625$	−223829.	−	320138.i	−0.573003	−	0.819553i
$626$			659236.i			1.68226i
$627$		0			0
$628$			1.17403e6i			2.97687i
$629$			82982.2i			0.209741i
$630$		0			0
$631$	502459.			1.26195			0.630975	−	0.775803i	$-0.282655\pi$
$631$	502459.			1.26195			0.630975	+	0.775803i	$0.282655\pi$
$632$	1.11276e6			2.78590
$633$		0			0
$634$	130508.			0.324682
$635$	226280.	+	373045.i	0.561176	+	0.925154i
$636$		0			0
$637$		−	69583.3i		−	0.171485i
$638$	127194.			0.312483
$639$		0			0
$640$	228022.	−	138313.i	0.556695	−	0.337677i
$641$		−	408773.i		−	0.994869i	−0.867502	−	0.497434i	$-0.834276\pi$
$641$		−	408773.i		−	0.994869i	0.867502	−	0.497434i	$-0.165724\pi$
$642$		0			0
$643$		−	671727.i		−	1.62469i	−0.583176	−	0.812346i	$-0.698190\pi$
$643$		−	671727.i		−	1.62469i	0.583176	−	0.812346i	$-0.301810\pi$
$644$		−	326501.i		−	0.787250i
$645$		0			0
$646$	286149.			0.685689
$647$	−120699.			−0.288334			−0.144167	−	0.989553i	$-0.546050\pi$
$647$	−120699.			−0.288334			−0.144167	+	0.989553i	$0.546050\pi$
$648$		0			0
$649$	−734690.			−1.74427
$650$	654542.	+	341024.i	1.54921	+	0.807158i
$651$		0			0
$652$		−	1.38024e6i		−	3.24682i
$653$	−62540.7			−0.146668			−0.0733342	−	0.997307i	$-0.523364\pi$
$653$	−62540.7			−0.146668			−0.0733342	+	0.997307i	$0.523364\pi$
$654$		0			0
$655$	30851.5	+	50861.8i	0.0719108	+	0.118552i
$656$		−	23343.5i		−	0.0542448i
$657$		0			0
$658$			270350.i			0.624417i
$659$			112155.i			0.258254i	0.991628	+	0.129127i	$0.0412174\pi$
$659$			112155.i			0.258254i	−0.991628	+	0.129127i	$0.958783\pi$
$660$		0			0
$661$	158493.			0.362750			0.181375	−	0.983414i	$-0.441945\pi$
$661$	158493.			0.362750			0.181375	+	0.983414i	$0.441945\pi$
$662$	−697622.			−1.59186
$663$		0			0
$664$	−537601.			−1.21934
$665$	−284163.	−	468470.i	−0.642575	−	1.05935i
$666$		0			0
$667$			21087.5i			0.0473996i
$668$	−934936.			−2.09522
$669$		0			0
$670$	384532.	+	633939.i	0.856609	+	1.41221i
$671$		−	288212.i		−	0.640128i
$672$		0			0
$673$		−	260162.i		−	0.574399i	−0.957871	−	0.287199i	$-0.907276\pi$
$673$		−	260162.i		−	0.574399i	0.957871	−	0.287199i	$-0.0927243\pi$
$674$			16198.7i			0.0356582i
$675$		0			0
$676$	83908.2			0.183616
$677$	155603.			0.339501			0.169751	−	0.985487i	$-0.445704\pi$
$677$	155603.			0.339501			0.169751	+	0.985487i	$0.445704\pi$
$678$		0			0
$679$	−77443.1			−0.167975
$680$	260646.	−	158101.i	0.563680	−	0.341914i
$681$		0			0
$682$		−	1.23932e6i		−	2.66449i
$683$	296453.			0.635499			0.317749	−	0.948175i	$-0.397073\pi$
$683$	296453.			0.635499			0.317749	+	0.948175i	$0.397073\pi$
$684$		0			0
$685$	121881.	−	73929.9i	0.259749	−	0.157557i
$686$			915196.i			1.94476i
$687$		0			0
$688$			1.80328e6i			3.80966i
$689$		−	49920.7i		−	0.105158i
$690$		0			0
$691$	−658162.			−1.37840			−0.689202	−	0.724569i	$-0.742039\pi$
$691$	−658162.			−1.37840			−0.689202	+	0.724569i	$0.742039\pi$
$692$	−296571.			−0.619321
$693$		0			0
$694$	36095.7			0.0749438
$695$	315037.	−	191093.i	0.652216	−	0.395618i
$696$		0			0
$697$			3554.40i			0.00731646i
$698$	1.49451e6			3.06752
$699$		0			0
$700$	−911486.	−	474895.i	−1.86018	−	0.969174i
$701$		−	915321.i		−	1.86268i	−0.364155	−	0.931338i	$-0.618642\pi$
$701$		−	915321.i		−	1.86268i	0.364155	−	0.931338i	$-0.381358\pi$
$702$		0			0
$703$		−	514432.i		−	1.04092i
$704$			249429.i			0.503270i
$705$		0			0
$706$	1.12156e6			2.25015
$707$	−293231.			−0.586639
$708$		0			0
$709$	258768.			0.514775			0.257388	−	0.966308i	$-0.417138\pi$
$709$	258768.			0.514775			0.257388	+	0.966308i	$0.417138\pi$
$710$	515072.	+	849147.i	1.02177	+	1.68448i
$711$		0			0
$712$			798114.i			1.57436i
$713$	205467.			0.404168
$714$		0			0
$715$	570009.	−	345753.i	1.11499	−	0.676323i
$716$		−	1.55515e6i		−	3.03352i
$717$		0			0
$718$			1.28020e6i			2.48330i
$719$			140839.i			0.272437i	0.990679	+	0.136218i	$0.0434949\pi$
$719$			140839.i			0.272437i	−0.990679	+	0.136218i	$0.956505\pi$
$720$		0			0
$721$	−312369.			−0.600893
$722$	−824887.			−1.58241
$723$		0			0
$724$	−178516.			−0.340566
$725$	58869.7	+	30671.8i	0.111999	+	0.0583530i
$726$		0			0
$727$			972686.i			1.84037i	0.391489	+	0.920183i	$0.371960\pi$
$727$			972686.i			1.84037i	−0.391489	+	0.920183i	$0.628040\pi$
$728$	1.10289e6			2.08099
$729$		0			0
$730$	−45916.0	−	75697.1i	−0.0861625	−	0.142047i
$731$		−	274577.i		−	0.513841i
$732$		0			0
$733$			846888.i			1.57622i	0.615531	+	0.788112i	$0.288941\pi$
$733$			846888.i			1.57622i	−0.615531	+	0.788112i	$0.711059\pi$
$734$		−	66521.7i		−	0.123473i
$735$		0			0
$736$	−269442.			−0.497404
$737$	669740.			1.23302
$738$		0			0
$739$	−30292.9			−0.0554692			−0.0277346	−	0.999615i	$-0.508829\pi$
$739$	−30292.9			−0.0554692			−0.0277346	+	0.999615i	$0.508829\pi$
$740$	−500456.	−	825052.i	−0.913909	−	1.50667i
$741$		0			0
$742$			99552.9i			0.180820i
$743$	601215.			1.08906			0.544530	−	0.838741i	$-0.316708\pi$
$743$	601215.			1.08906			0.544530	+	0.838741i	$0.316708\pi$
$744$		0			0
$745$	−238009.	−	392382.i	−0.428826	−	0.706962i
$746$		−	75955.5i		−	0.136484i
$747$		0			0
$748$		−	484847.i		−	0.866566i
$749$		−	770308.i		−	1.37310i
$750$		0			0
$751$	−324728.			−0.575758			−0.287879	−	0.957667i	$-0.592950\pi$
$751$	−324728.			−0.575758			−0.287879	+	0.957667i	$0.592950\pi$
$752$	437124.			0.772981
$753$		0			0
$754$	−125421.			−0.220611
$755$	24442.5	−	14826.2i	0.0428797	−	0.0260098i
$756$		0			0
$757$			176604.i			0.308183i	0.988057	+	0.154092i	$0.0492451\pi$
$757$			176604.i			0.308183i	−0.988057	+	0.154092i	$0.950755\pi$
$758$	−754870.			−1.31381
$759$		0			0
$760$	−1.61582e6	+	980118.i	−2.79747	+	1.69688i
$761$		−	219754.i		−	0.379462i	−0.981836	−	0.189731i	$-0.939238\pi$
$761$		−	219754.i		−	0.379462i	0.981836	−	0.189731i	$-0.0607616\pi$
$762$		0			0
$763$		−	780276.i		−	1.34029i
$764$		−	2.45791e6i		−	4.21094i
$765$		0			0
$766$	1.02139e6			1.74074
$767$	724446.			1.23145
$768$		0			0
$769$	−985249.			−1.66607			−0.833035	−	0.553220i	$-0.813399\pi$
$769$	−985249.			−1.66607			−0.833035	+	0.553220i	$0.813399\pi$
$770$	−1.13672e6	+	689508.i	−1.91723	+	1.16294i
$771$		0			0
$772$		−	41757.6i		−	0.0700649i
$773$	142411.			0.238333			0.119167	−	0.992874i	$-0.461978\pi$
$773$	142411.			0.238333			0.119167	+	0.992874i	$0.461978\pi$
$774$		0			0
$775$	298851.	−	573598.i	0.497567	−	0.955001i
$776$			267113.i			0.443579i
$777$		0			0
$778$		−	997629.i		−	1.64820i
$779$		−	22034.8i		−	0.0363107i
$780$		0			0
$781$	897102.			1.47075
$782$	115113.			0.188239
$783$		0			0
$784$	224366.			0.365026
$785$	411049.	+	677655.i	0.667043	+	1.09969i
$786$		0			0
$787$			847294.i			1.36800i	0.729484	+	0.683998i	$0.239760\pi$
$787$			847294.i			1.36800i	−0.729484	+	0.683998i	$0.760240\pi$
$788$	−1.66077e6			−2.67459
$789$		0			0
$790$	1.13090e6	−	685977.i	1.81205	−	1.09915i
$791$			694481.i			1.10996i
$792$		0			0
$793$			284193.i			0.451926i
$794$		−	271492.i		−	0.430641i
$795$		0			0
$796$	2.11104e6			3.33173
$797$	1.08930e6			1.71487			0.857433	−	0.514596i	$-0.172058\pi$
$797$	1.08930e6			1.71487			0.857433	+	0.514596i	$0.172058\pi$
$798$		0			0
$799$	−66558.8			−0.104259
$800$	−391902.	+	752195.i	−0.612348	+	1.17530i
$801$		0			0
$802$			1.89767e6i			2.95034i
$803$	−79972.0			−0.124024
$804$		0			0
$805$	−114314.	−	188457.i	−0.176403	−	0.290818i
$806$			1.22204e6i			1.88111i
$807$		0			0
$808$			1.01140e6i			1.54917i
$809$			525045.i			0.802231i	0.916027	+	0.401116i	$0.131377\pi$
$809$			525045.i			0.802231i	−0.916027	+	0.401116i	$0.868623\pi$
$810$		0			0
$811$	−891294.			−1.35512			−0.677562	−	0.735466i	$-0.736963\pi$
$811$	−891294.			−1.35512			−0.677562	+	0.735466i	$0.736963\pi$
$812$	174656.			0.264893
$813$		0			0
$814$	−1.24825e6			−1.88387
$815$	−483245.	−	796677.i	−0.727532	−	1.19941i
$816$		0			0
$817$			1.70218e6i			2.55013i
$818$	−1.31461e6			−1.96467
$819$		0			0
$820$	−21436.2	−	35339.7i	−0.0318801	−	0.0525575i
$821$			1.15283e6i			1.71032i	0.518365	+	0.855159i	$0.326541\pi$
$821$			1.15283e6i			1.71032i	−0.518365	+	0.855159i	$0.673459\pi$
$822$		0			0
$823$			260008.i			0.383873i	0.981407	+	0.191937i	$0.0614768\pi$
$823$			260008.i			0.383873i	−0.981407	+	0.191937i	$0.938523\pi$
$824$			1.07740e6i			1.58681i
$825$		0			0
$826$	−1.44470e6			−2.11748
$827$	−103528.			−0.151373			−0.0756863	−	0.997132i	$-0.524115\pi$
$827$	−103528.			−0.151373			−0.0756863	+	0.997132i	$0.524115\pi$
$828$		0			0
$829$	456869.			0.664787			0.332394	−	0.943141i	$-0.392144\pi$
$829$	456869.			0.664787			0.332394	+	0.943141i	$0.392144\pi$
$830$	−546366.	+	331412.i	−0.793099	+	0.481075i
$831$		0			0
$832$		−	245951.i		−	0.355306i
$833$	−34163.1			−0.0492342
$834$		0			0
$835$	−539648.	+	327337.i	−0.773994	+	0.469486i
$836$			3.00571e6i			4.30066i
$837$		0			0
$838$		−	410485.i		−	0.584533i
$839$		−	780649.i		−	1.10900i	−0.832183	−	0.554501i	$-0.812909\pi$
$839$		−	780649.i		−	1.10900i	0.832183	−	0.554501i	$-0.187091\pi$
$840$		0			0
$841$	696001.			0.984051
$842$	−390931.			−0.551411
$843$		0			0
$844$	−1.12446e6			−1.57856
$845$	48432.1	−	29377.7i	0.0678296	−	0.0411438i
$846$		0			0
$847$			550771.i			0.767722i
$848$	160965.			0.223841
$849$		0			0
$850$	167431.	−	321358.i	0.231739	−	0.444786i
$851$		−	206947.i		−	0.285759i
$852$		0			0
$853$		−	547049.i		−	0.751845i	−0.926651	−	0.375922i	$-0.877326\pi$
$853$		−	547049.i		−	0.751845i	0.926651	−	0.375922i	$-0.122674\pi$
$854$		−	566744.i		−	0.777090i
$855$		0			0
$856$	−2.65690e6			−3.62600
$857$	607168.			0.826699			0.413349	−	0.910572i	$-0.364359\pi$
$857$	607168.			0.826699			0.413349	+	0.910572i	$0.364359\pi$
$858$		0			0
$859$	−362565.			−0.491360			−0.245680	−	0.969351i	$-0.579011\pi$
$859$	−362565.			−0.491360			−0.245680	+	0.969351i	$0.579011\pi$
$860$	1.65594e6	+	2.72998e6i	2.23897	+	3.69116i
$861$		0			0
$862$		−	259269.i		−	0.348929i
$863$	428729.			0.575653			0.287827	−	0.957683i	$-0.407067\pi$
$863$	428729.			0.575653			0.287827	+	0.957683i	$0.407067\pi$
$864$		0			0
$865$	−171181.	+	103834.i	−0.228783	+	0.138774i
$866$			2.21210e6i			2.94964i
$867$		0			0
$868$		−	1.70176e6i		−	2.25870i
$869$		−	1.19477e6i		−	1.58214i
$870$		0			0
$871$	−660402.			−0.870507
$872$	−2.69128e6			−3.53937
$873$		0			0
$874$	−713619.			−0.934208
$875$	−692382.	+	45016.1i	−0.904335	+	0.0587965i
$876$		0			0
$877$			1.52002e6i			1.97629i	0.153530	+	0.988144i	$0.450936\pi$
$877$			1.52002e6i			1.97629i	−0.153530	+	0.988144i	$0.549064\pi$
$878$	−761532.			−0.987869
$879$		0			0
$880$	1.11485e6	+	1.83795e6i	1.43964	+	2.37338i
$881$		−	605464.i		−	0.780076i	−0.920799	−	0.390038i	$-0.872462\pi$
$881$		−	605464.i		−	0.780076i	0.920799	−	0.390038i	$-0.127538\pi$
$882$		0			0
$883$			632279.i			0.810938i	0.914109	+	0.405469i	$0.132892\pi$
$883$			632279.i			0.810938i	−0.914109	+	0.405469i	$0.867108\pi$
$884$			478087.i			0.611790i
$885$		0			0
$886$	−1.57047e6			−2.00061
$887$	−501531.			−0.637456			−0.318728	−	0.947846i	$-0.603256\pi$
$887$	−501531.			−0.637456			−0.318728	+	0.947846i	$0.603256\pi$
$888$		0			0
$889$	774989.			0.980600
$890$	492010.	+	811127.i	0.621146	+	1.02402i
$891$		0			0
$892$		−	1.74252e6i		−	2.19002i
$893$	412618.			0.517422
$894$		0			0
$895$	−544486.	−	897639.i	−0.679736	−	1.12061i
$896$		−	473709.i		−	0.590059i
$897$		0			0
$898$		−	2.59794e6i		−	3.22163i
$899$			109911.i			0.135994i
$900$		0			0
$901$	−24509.4			−0.0301914
$902$	−53466.5			−0.0657157
$903$		0			0
$904$	2.39536e6			2.93113
$905$	−103040.	+	62501.6i	−0.125808	+	0.0763122i
$906$		0			0
$907$		−	714256.i		−	0.868239i	−0.900855	−	0.434120i	$-0.857059\pi$
$907$		−	714256.i		−	0.868239i	0.900855	−	0.434120i	$-0.142941\pi$
$908$	−1.59957e6			−1.94013
$909$		0			0
$910$	1.12087e6	−	679894.i	1.35355	−	0.821029i
$911$		−	439887.i		−	0.530035i	−0.964244	−	0.265017i	$-0.914622\pi$
$911$		−	439887.i		−	0.530035i	0.964244	−	0.265017i	$-0.0853777\pi$
$912$		0			0
$913$			577222.i			0.692471i
$914$		−	327703.i		−	0.392272i
$915$		0			0
$916$	3.18755e6			3.79897
$917$	105664.			0.125657
$918$		0			0
$919$	−541978.			−0.641728			−0.320864	−	0.947125i	$-0.603973\pi$
$919$	−541978.			−0.641728			−0.320864	+	0.947125i	$0.603973\pi$
$920$	−650016.	+	394284.i	−0.767978	+	0.465837i
$921$		0			0
$922$		−	2.43737e6i		−	2.86721i
$923$	−884594.			−1.03834
$924$		0			0
$925$	−577730.	−	301004.i	−0.675214	−	0.351795i
$926$		−	1.51872e6i		−	1.77116i
$927$		0			0
$928$		−	144133.i		−	0.167366i
$929$		−	1.66062e6i		−	1.92416i	−0.272776	−	0.962078i	$-0.587942\pi$
$929$		−	1.66062e6i		−	1.92416i	0.272776	−	0.962078i	$-0.412058\pi$
$930$		0			0
$931$	211787.			0.244343
$932$	−541126.			−0.622970
$933$		0			0
$934$	2.14241e6			2.45589
$935$	−169753.	−	279855.i	−0.194176	−	0.320118i
$936$		0			0
$937$			612690.i			0.697850i	0.937151	+	0.348925i	$0.113453\pi$
$937$			612690.i			0.697850i	−0.937151	+	0.348925i	$0.886547\pi$
$938$	1.31699e6			1.49684
$939$		0			0
$940$	661761.	−	401408.i	0.748938	−	0.454287i
$941$		−	123312.i		−	0.139260i	−0.997573	−	0.0696299i	$-0.977818\pi$
$941$		−	123312.i		−	0.139260i	0.997573	−	0.0696299i	$-0.0221818\pi$
$942$		0			0
$943$		−	8864.21i		−	0.00996820i
$944$			2.33592e6i			2.62128i
$945$		0			0
$946$	4.13028e6			4.61527
$947$	−963764.			−1.07466			−0.537329	−	0.843372i	$-0.680567\pi$
$947$	−963764.			−1.07466			−0.537329	+	0.843372i	$0.680567\pi$
$948$		0			0
$949$	78857.0			0.0875604
$950$	−1.03796e6	+	1.99220e6i	−1.15009	+	2.20742i
$951$		0			0
$952$		−	541483.i		−	0.597462i
$953$	43511.5			0.0479092			0.0239546	−	0.999713i	$-0.492374\pi$
$953$	43511.5			0.0479092			0.0239546	+	0.999713i	$0.492374\pi$
$954$		0			0
$955$	−860557.	−	1.41871e6i	−0.943567	−	1.55556i
$956$		−	1.97565e6i		−	2.16169i
$957$		0			0
$958$		−	1.52658e6i		−	1.66337i
$959$		−	253203.i		−	0.275316i
$960$		0			0
$961$	147394.			0.159600
$962$	1.23084e6			1.33000
$963$		0			0
$964$	2.86676e6			3.08487
$965$	−14620.1	−	24102.6i	−0.0156998	−	0.0258827i
$966$		0			0
$967$			391299.i			0.418462i	0.977866	+	0.209231i	$0.0670961\pi$
$967$			391299.i			0.418462i	−0.977866	+	0.209231i	$0.932904\pi$
$968$	1.89969e6			2.02736
$969$		0			0
$970$	164666.	+	271468.i	0.175009	+	0.288519i
$971$			30706.0i			0.0325676i	0.999867	+	0.0162838i	$0.00518352\pi$
$971$			30706.0i			0.0325676i	−0.999867	+	0.0162838i	$0.994816\pi$
$972$		0			0
$973$		−	654478.i		−	0.691304i
$974$		−	883592.i		−	0.931395i
$975$		0			0
$976$	−916358.			−0.961979
$977$	633745.			0.663935			0.331968	−	0.943291i	$-0.392288\pi$
$977$	633745.			0.663935			0.331968	+	0.943291i	$0.392288\pi$
$978$		0			0
$979$	856935.			0.894093
$980$	339667.	−	206033.i	0.353672	−	0.214529i
$981$		0			0
$982$		−	3.15591e6i		−	3.27267i
$983$	324701.			0.336029			0.168014	−	0.985785i	$-0.446265\pi$
$983$	324701.			0.336029			0.168014	+	0.985785i	$0.446265\pi$
$984$		0			0
$985$	−958601.	+	581464.i	−0.988019	+	0.599308i
$986$			61577.4i			0.0633385i
$987$		0			0
$988$		−	2.96381e6i		−	3.03624i
$989$			684758.i			0.700075i
$990$		0			0
$991$	−655544.			−0.667505			−0.333753	−	0.942661i	$-0.608315\pi$
$991$	−655544.			−0.667505			−0.333753	+	0.942661i	$0.608315\pi$
$992$	−1.40436e6			−1.42710
$993$		0			0
$994$	1.76407e6			1.78544
$995$	1.21850e6	−	739111.i	1.23077	−	0.746558i
$996$		0			0
$997$			927873.i			0.933465i	0.884398	+	0.466733i	$0.154569\pi$
$997$			927873.i			0.933465i	−0.884398	+	0.466733i	$0.845431\pi$
$998$	2.94160e6			2.95340
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.3		44
3.2	odd	2	inner	405.5.d.a.404.41		44
5.4	even	2	inner	405.5.d.a.404.42		44
9.2	odd	6		135.5.h.a.44.2		44
9.4	even	3		135.5.h.a.89.21		44
9.5	odd	6		45.5.h.a.29.2	yes	44
9.7	even	3		45.5.h.a.14.21	yes	44
15.14	odd	2	inner	405.5.d.a.404.4		44
45.4	even	6		135.5.h.a.89.2		44
45.14	odd	6		45.5.h.a.29.21	yes	44
45.29	odd	6		135.5.h.a.44.21		44
45.34	even	6		45.5.h.a.14.2	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.2	✓	44	45.34	even	6
45.5.h.a.14.21	yes	44	9.7	even	3
45.5.h.a.29.2	yes	44	9.5	odd	6
45.5.h.a.29.21	yes	44	45.14	odd	6
135.5.h.a.44.2		44	9.2	odd	6
135.5.h.a.44.21		44	45.29	odd	6
135.5.h.a.89.2		44	45.4	even	6
135.5.h.a.89.21		44	9.4	even	3
405.5.d.a.404.3		44	1.1	even	1	trivial
405.5.d.a.404.4		44	15.14	odd	2	inner
405.5.d.a.404.41		44	3.2	odd	2	inner
405.5.d.a.404.42		44	5.4	even	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)

\(f(q)\)	\(=\)	\(q-7.28232 q^{2} +37.0321 q^{4} +(21.3751 - 12.9656i) q^{5} -44.4060i q^{7} -153.163 q^{8} +(-155.660 + 94.4195i) q^{10} +164.451i q^{11} -162.158i q^{13} +323.378i q^{14} +522.865 q^{16} -79.6141 q^{17} +493.552 q^{19} +(791.564 - 480.144i) q^{20} -1197.58i q^{22} +198.547 q^{23} +(288.787 - 554.281i) q^{25} +1180.89i q^{26} -1644.45i q^{28} +106.209i q^{29} +1034.85 q^{31} -1357.06 q^{32} +579.775 q^{34} +(-575.750 - 949.181i) q^{35} -1042.31i q^{37} -3594.20 q^{38} +(-3273.86 + 1985.84i) q^{40} -44.6454i q^{41} +3448.84i q^{43} +6089.97i q^{44} -1445.88 q^{46} +836.017 q^{47} +429.108 q^{49} +(-2103.04 + 4036.45i) q^{50} -6005.05i q^{52} +307.853 q^{53} +(2132.20 + 3515.15i) q^{55} +6801.34i q^{56} -773.449i q^{58} +4467.53i q^{59} -1752.57 q^{61} -7536.11 q^{62} +1516.74 q^{64} +(-2102.47 - 3466.14i) q^{65} -4072.59i q^{67} -2948.28 q^{68} +(4192.79 + 6912.24i) q^{70} -5455.14i q^{71} +486.298i q^{73} +7590.40i q^{74} +18277.3 q^{76} +7302.60 q^{77} -7265.20 q^{79} +(11176.3 - 6779.25i) q^{80} +325.122i q^{82} +3510.00 q^{83} +(-1701.76 + 1032.24i) q^{85} -25115.6i q^{86} -25187.7i q^{88} -5210.89i q^{89} -7200.78 q^{91} +7352.63 q^{92} -6088.14 q^{94} +(10549.7 - 6399.19i) q^{95} -1743.98i q^{97} -3124.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more