LMFDB - Embedded newform 1458.3.b.c.1457.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.17
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +7.93206i q^{5} +0.453662 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +7.93206i q^{5} +0.453662 q^{7} +2.82843i q^{8} +11.2176 q^{10} -14.1039i q^{11} -12.8008 q^{13} -0.641574i q^{14} +4.00000 q^{16} +32.9763i q^{17} -0.405584 q^{19} -15.8641i q^{20} -19.9459 q^{22} -14.4803i q^{23} -37.9176 q^{25} +18.1031i q^{26} -0.907323 q^{28} +26.2212i q^{29} +24.9862 q^{31} -5.65685i q^{32} +46.6355 q^{34} +3.59847i q^{35} -7.68953 q^{37} +0.573582i q^{38} -22.4353 q^{40} -24.7833i q^{41} +17.9487 q^{43} +28.2077i q^{44} -20.4782 q^{46} -47.4703i q^{47} -48.7942 q^{49} +53.6236i q^{50} +25.6016 q^{52} +0.261457i q^{53} +111.873 q^{55} +1.28315i q^{56} +37.0824 q^{58} +55.1024i q^{59} -104.672 q^{61} -35.3359i q^{62} -8.00000 q^{64} -101.537i q^{65} -64.6084 q^{67} -65.9526i q^{68} +5.08901 q^{70} -109.255i q^{71} -62.9410 q^{73} +10.8746i q^{74} +0.811168 q^{76} -6.39839i q^{77} -19.2590 q^{79} +31.7283i q^{80} -35.0489 q^{82} -57.1293i q^{83} -261.570 q^{85} -25.3832i q^{86} +39.8918 q^{88} -103.681i q^{89} -5.80724 q^{91} +28.9606i q^{92} -67.1331 q^{94} -3.21712i q^{95} +56.2132 q^{97} +69.0054i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4}+O(q^{10}) $ 36 * q - 72 * q^4	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	7.93206i	1.58641i	0.608953	+	0.793206i	$0.291590\pi$
$5$	7.93206i	1.58641i	−0.608953	+	0.793206i	$0.708410\pi$
$6$	0	0
$7$	0.453662	0.0648088	0.0324044	−	0.999475i	$-0.489684\pi$
$7$	0.453662	0.0648088	0.0324044	+	0.999475i	$0.489684\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	11.2176	1.12176
$11$	− 14.1039i	− 1.28217i	−0.767470	−	0.641085i	$-0.778485\pi$
$11$	− 14.1039i	− 1.28217i	0.767470	−	0.641085i	$-0.221515\pi$
$12$	0	0
$13$	−12.8008	−0.984678	−0.492339	−	0.870404i	$-0.663858\pi$
$13$	−12.8008	−0.984678	−0.492339	+	0.870404i	$0.663858\pi$
$14$	− 0.641574i	− 0.0458267i
$15$	0	0
$16$	4.00000	0.250000
$17$	32.9763i	1.93978i	0.243538	+	0.969891i	$0.421692\pi$
$17$	32.9763i	1.93978i	−0.243538	+	0.969891i	$0.578308\pi$
$18$	0	0
$19$	−0.405584	−0.0213465	−0.0106733	−	0.999943i	$-0.503397\pi$
$19$	−0.405584	−0.0213465	−0.0106733	+	0.999943i	$0.503397\pi$
$20$	− 15.8641i	− 0.793206i
$21$	0	0
$22$	−19.9459	−0.906631
$23$	− 14.4803i	− 0.629578i	−0.949162	−	0.314789i	$-0.898066\pi$
$23$	− 14.4803i	− 0.629578i	0.949162	−	0.314789i	$-0.101934\pi$
$24$	0	0
$25$	−37.9176	−1.51671
$26$	18.1031i	0.696272i
$27$	0	0
$28$	−0.907323	−0.0324044
$29$	26.2212i	0.904179i	0.891972	+	0.452090i	$0.149321\pi$
$29$	26.2212i	0.904179i	−0.891972	+	0.452090i	$0.850679\pi$
$30$	0	0
$31$	24.9862	0.806008	0.403004	−	0.915198i	$-0.367966\pi$
$31$	24.9862	0.806008	0.403004	+	0.915198i	$0.367966\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	46.6355	1.37163
$35$	3.59847i	0.102814i
$36$	0	0
$37$	−7.68953	−0.207825	−0.103913	−	0.994586i	$-0.533136\pi$
$37$	−7.68953	−0.207825	−0.103913	+	0.994586i	$0.533136\pi$
$38$	0.573582i	0.0150943i
$39$	0	0
$40$	−22.4353	−0.560882
$41$	− 24.7833i	− 0.604472i	−0.953233	−	0.302236i	$-0.902267\pi$
$41$	− 24.7833i	− 0.604472i	0.953233	−	0.302236i	$-0.0977330\pi$
$42$	0	0
$43$	17.9487	0.417411	0.208705	−	0.977979i	$-0.433075\pi$
$43$	17.9487	0.417411	0.208705	+	0.977979i	$0.433075\pi$
$44$	28.2077i	0.641085i
$45$	0	0
$46$	−20.4782	−0.445179
$47$	− 47.4703i	− 1.01001i	−0.863118	−	0.505003i	$-0.831491\pi$
$47$	− 47.4703i	− 1.01001i	0.863118	−	0.505003i	$-0.168509\pi$
$48$	0	0
$49$	−48.7942	−0.995800
$50$	53.6236i	1.07247i
$51$	0	0
$52$	25.6016	0.492339
$53$	0.261457i	0.00493315i	0.999997	+	0.00246657i	$0.000785136\pi$
$53$	0.261457i	0.00493315i	−0.999997	+	0.00246657i	$0.999215\pi$
$54$	0	0
$55$	111.873	2.03405
$56$	1.28315i	0.0229134i
$57$	0	0
$58$	37.0824	0.639351
$59$	55.1024i	0.933939i	0.884274	+	0.466969i	$0.154654\pi$
$59$	55.1024i	0.933939i	−0.884274	+	0.466969i	$0.845346\pi$
$60$	0	0
$61$	−104.672	−1.71593	−0.857967	−	0.513705i	$-0.828273\pi$
$61$	−104.672	−1.71593	−0.857967	+	0.513705i	$0.828273\pi$
$62$	− 35.3359i	− 0.569933i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 101.537i	− 1.56211i
$66$	0	0
$67$	−64.6084	−0.964305	−0.482152	−	0.876087i	$-0.660145\pi$
$67$	−64.6084	−0.964305	−0.482152	+	0.876087i	$0.660145\pi$
$68$	− 65.9526i	− 0.969891i
$69$	0	0
$70$	5.08901	0.0727001
$71$	− 109.255i	− 1.53881i	−0.638761	−	0.769405i	$-0.720553\pi$
$71$	− 109.255i	− 1.53881i	0.638761	−	0.769405i	$-0.279447\pi$
$72$	0	0
$73$	−62.9410	−0.862205	−0.431103	−	0.902303i	$-0.641875\pi$
$73$	−62.9410	−0.862205	−0.431103	+	0.902303i	$0.641875\pi$
$74$	10.8746i	0.146955i
$75$	0	0
$76$	0.811168	0.0106733
$77$	− 6.39839i	− 0.0830959i
$78$	0	0
$79$	−19.2590	−0.243785	−0.121892	−	0.992543i	$-0.538896\pi$
$79$	−19.2590	−0.243785	−0.121892	+	0.992543i	$0.538896\pi$
$80$	31.7283i	0.396603i
$81$	0	0
$82$	−35.0489	−0.427426
$83$	− 57.1293i	− 0.688305i	−0.938914	−	0.344153i	$-0.888166\pi$
$83$	− 57.1293i	− 0.688305i	0.938914	−	0.344153i	$-0.111834\pi$
$84$	0	0
$85$	−261.570	−3.07730
$86$	− 25.3832i	− 0.295154i
$87$	0	0
$88$	39.8918	0.453316
$89$	− 103.681i	− 1.16495i	−0.812848	−	0.582476i	$-0.802084\pi$
$89$	− 103.681i	− 1.16495i	0.812848	−	0.582476i	$-0.197916\pi$
$90$	0	0
$91$	−5.80724	−0.0638158
$92$	28.9606i	0.314789i
$93$	0	0
$94$	−67.1331	−0.714182
$95$	− 3.21712i	− 0.0338644i
$96$	0	0
$97$	56.2132	0.579517	0.289759	−	0.957100i	$-0.406425\pi$
$97$	56.2132	0.579517	0.289759	+	0.957100i	$0.406425\pi$
$98$	69.0054i	0.704137i
$99$	0	0
$100$	75.8353	0.758353
$101$	− 44.9754i	− 0.445301i	−0.974898	−	0.222651i	$-0.928529\pi$
$101$	− 44.9754i	− 0.445301i	0.974898	−	0.222651i	$-0.0714709\pi$
$102$	0	0
$103$	−91.1393	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$103$	−91.1393	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$104$	− 36.2062i	− 0.348136i
$105$	0	0
$106$	0.369756	0.00348826
$107$	− 206.049i	− 1.92569i	−0.270046	−	0.962847i	$-0.587039\pi$
$107$	− 206.049i	− 1.92569i	0.270046	−	0.962847i	$-0.412961\pi$
$108$	0	0
$109$	−140.161	−1.28588	−0.642942	−	0.765915i	$-0.722286\pi$
$109$	−140.161	−1.28588	−0.642942	+	0.765915i	$0.722286\pi$
$110$	− 158.212i	− 1.43829i
$111$	0	0
$112$	1.81465	0.0162022
$113$	− 72.9125i	− 0.645243i	−0.946528	−	0.322622i	$-0.895436\pi$
$113$	− 72.9125i	− 0.645243i	0.946528	−	0.322622i	$-0.104564\pi$
$114$	0	0
$115$	114.859	0.998771
$116$	− 52.4424i	− 0.452090i
$117$	0	0
$118$	77.9265	0.660394
$119$	14.9601i	0.125715i
$120$	0	0
$121$	−77.9192	−0.643960
$122$	148.029i	1.21335i
$123$	0	0
$124$	−49.9725	−0.403004
$125$	− 102.464i	− 0.819708i
$126$	0	0
$127$	10.9869	0.0865113	0.0432557	−	0.999064i	$-0.486227\pi$
$127$	10.9869	0.0865113	0.0432557	+	0.999064i	$0.486227\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−143.595	−1.10458
$131$	− 37.4800i	− 0.286107i	−0.989715	−	0.143053i	$-0.954308\pi$
$131$	− 37.4800i	− 0.286107i	0.989715	−	0.143053i	$-0.0456921\pi$
$132$	0	0
$133$	−0.183998	−0.00138344
$134$	91.3701i	0.681866i
$135$	0	0
$136$	−93.2711	−0.685817
$137$	42.3296i	0.308975i	0.987995	+	0.154488i	$0.0493727\pi$
$137$	42.3296i	0.308975i	−0.987995	+	0.154488i	$0.950627\pi$
$138$	0	0
$139$	−162.947	−1.17228	−0.586142	−	0.810209i	$-0.699354\pi$
$139$	−162.947	−1.17228	−0.586142	+	0.810209i	$0.699354\pi$
$140$	− 7.19695i	− 0.0514068i
$141$	0	0
$142$	−154.511	−1.08810
$143$	180.541i	1.26252i
$144$	0	0
$145$	−207.988	−1.43440
$146$	89.0120i	0.609671i
$147$	0	0
$148$	15.3791	0.103913
$149$	179.560i	1.20510i	0.798081	+	0.602550i	$0.205849\pi$
$149$	179.560i	1.20510i	−0.798081	+	0.602550i	$0.794151\pi$
$150$	0	0
$151$	−160.931	−1.06577	−0.532883	−	0.846189i	$-0.678891\pi$
$151$	−160.931	−1.06577	−0.532883	+	0.846189i	$0.678891\pi$
$152$	− 1.14716i	− 0.00754713i
$153$	0	0
$154$	−9.04868	−0.0587577
$155$	198.192i	1.27866i
$156$	0	0
$157$	77.2951	0.492325	0.246163	−	0.969229i	$-0.420830\pi$
$157$	77.2951	0.492325	0.246163	+	0.969229i	$0.420830\pi$
$158$	27.2363i	0.172382i
$159$	0	0
$160$	44.8705	0.280441
$161$	− 6.56915i	− 0.0408022i
$162$	0	0
$163$	−2.14092	−0.0131345	−0.00656723	−	0.999978i	$-0.502090\pi$
$163$	−2.14092	−0.0131345	−0.00656723	+	0.999978i	$0.502090\pi$
$164$	49.5667i	0.302236i
$165$	0	0
$166$	−80.7931	−0.486705
$167$	89.7592i	0.537480i	0.963213	+	0.268740i	$0.0866072\pi$
$167$	89.7592i	0.537480i	−0.963213	+	0.268740i	$0.913393\pi$
$168$	0	0
$169$	−5.13926	−0.0304098
$170$	369.916i	2.17598i
$171$	0	0
$172$	−35.8973	−0.208705
$173$	182.352i	1.05406i	0.849847	+	0.527030i	$0.176694\pi$
$173$	182.352i	1.05406i	−0.849847	+	0.527030i	$0.823306\pi$
$174$	0	0
$175$	−17.2018	−0.0982959
$176$	− 56.4155i	− 0.320543i
$177$	0	0
$178$	−146.627	−0.823746
$179$	− 120.126i	− 0.671093i	−0.942024	−	0.335546i	$-0.891079\pi$
$179$	− 120.126i	− 0.671093i	0.942024	−	0.335546i	$-0.108921\pi$
$180$	0	0
$181$	159.986	0.883898	0.441949	−	0.897040i	$-0.354287\pi$
$181$	159.986	0.883898	0.441949	+	0.897040i	$0.354287\pi$
$182$	8.21267i	0.0451246i
$183$	0	0
$184$	40.9565	0.222589
$185$	− 60.9939i	− 0.329697i
$186$	0	0
$187$	465.094	2.48713
$188$	94.9406i	0.505003i
$189$	0	0
$190$	−4.54969	−0.0239457
$191$	127.695i	0.668558i	0.942474	+	0.334279i	$0.108493\pi$
$191$	127.695i	0.668558i	−0.942474	+	0.334279i	$0.891507\pi$
$192$	0	0
$193$	99.7012	0.516587	0.258293	−	0.966067i	$-0.416840\pi$
$193$	99.7012	0.516587	0.258293	+	0.966067i	$0.416840\pi$
$194$	− 79.4974i	− 0.409781i
$195$	0	0
$196$	97.5884	0.497900
$197$	38.0900i	0.193350i	0.995316	+	0.0966751i	$0.0308208\pi$
$197$	38.0900i	0.193350i	−0.995316	+	0.0966751i	$0.969179\pi$
$198$	0	0
$199$	385.435	1.93686	0.968430	−	0.249284i	$-0.0801954\pi$
$199$	385.435	1.93686	0.968430	+	0.249284i	$0.0801954\pi$
$200$	− 107.247i	− 0.536236i
$201$	0	0
$202$	−63.6049	−0.314876
$203$	11.8956i	0.0585988i
$204$	0	0
$205$	196.583	0.958942
$206$	128.890i	0.625682i
$207$	0	0
$208$	−51.2032	−0.246169
$209$	5.72030i	0.0273699i
$210$	0	0
$211$	−341.198	−1.61705	−0.808527	−	0.588459i	$-0.799735\pi$
$211$	−341.198	−1.61705	−0.808527	+	0.588459i	$0.799735\pi$
$212$	− 0.522914i	− 0.00246657i
$213$	0	0
$214$	−291.398	−1.36167
$215$	142.370i	0.662186i
$216$	0	0
$217$	11.3353	0.0522364
$218$	198.218i	0.909257i
$219$	0	0
$220$	−223.746	−1.01703
$221$	− 422.123i	− 1.91006i
$222$	0	0
$223$	−58.4755	−0.262222	−0.131111	−	0.991368i	$-0.541854\pi$
$223$	−58.4755	−0.262222	−0.131111	+	0.991368i	$0.541854\pi$
$224$	− 2.56630i	− 0.0114567i
$225$	0	0
$226$	−103.114	−0.456256
$227$	− 41.9987i	− 0.185016i	−0.995712	−	0.0925081i	$-0.970512\pi$
$227$	− 41.9987i	− 0.185016i	0.995712	−	0.0925081i	$-0.0294884\pi$
$228$	0	0
$229$	89.0101	0.388690	0.194345	−	0.980933i	$-0.437742\pi$
$229$	89.0101	0.388690	0.194345	+	0.980933i	$0.437742\pi$
$230$	− 162.435i	− 0.706238i
$231$	0	0
$232$	−74.1648	−0.319676
$233$	140.437i	0.602733i	0.953508	+	0.301367i	$0.0974428\pi$
$233$	140.437i	0.602733i	−0.953508	+	0.301367i	$0.902557\pi$
$234$	0	0
$235$	376.537	1.60229
$236$	− 110.205i	− 0.466969i
$237$	0	0
$238$	21.1568	0.0888939
$239$	29.0536i	0.121563i	0.998151	+	0.0607815i	$0.0193593\pi$
$239$	29.0536i	0.121563i	−0.998151	+	0.0607815i	$0.980641\pi$
$240$	0	0
$241$	−148.449	−0.615971	−0.307985	−	0.951391i	$-0.599655\pi$
$241$	−148.449	−0.615971	−0.307985	+	0.951391i	$0.599655\pi$
$242$	110.194i	0.455349i
$243$	0	0
$244$	209.344	0.857967
$245$	− 387.039i	− 1.57975i
$246$	0	0
$247$	5.19180	0.0210194
$248$	70.6717i	0.284967i
$249$	0	0
$250$	−144.905	−0.579621
$251$	− 312.054i	− 1.24324i	−0.783318	−	0.621621i	$-0.786474\pi$
$251$	− 312.054i	− 1.24324i	0.783318	−	0.621621i	$-0.213526\pi$
$252$	0	0
$253$	−204.228	−0.807226
$254$	− 15.5379i	− 0.0611728i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 52.6698i	− 0.204941i	−0.994736	−	0.102470i	$-0.967325\pi$
$257$	− 52.6698i	− 0.204941i	0.994736	−	0.102470i	$-0.0326747\pi$
$258$	0	0
$259$	−3.48845	−0.0134689
$260$	203.074i	0.781053i
$261$	0	0
$262$	−53.0047	−0.202308
$263$	− 16.0349i	− 0.0609691i	−0.999535	−	0.0304846i	$-0.990295\pi$
$263$	− 16.0349i	− 0.0609691i	0.999535	−	0.0304846i	$-0.00970504\pi$
$264$	0	0
$265$	−2.07389	−0.00782601
$266$	0.260212i	0 0.000978241i
$267$	0	0
$268$	129.217	0.482152
$269$	− 233.865i	− 0.869388i	−0.900578	−	0.434694i	$-0.856857\pi$
$269$	− 233.865i	− 0.869388i	0.900578	−	0.434694i	$-0.143143\pi$
$270$	0	0
$271$	−229.874	−0.848243	−0.424122	−	0.905605i	$-0.639417\pi$
$271$	−229.874	−0.848243	−0.424122	+	0.905605i	$0.639417\pi$
$272$	131.905i	0.484946i
$273$	0	0
$274$	59.8631	0.218479
$275$	534.786i	1.94467i
$276$	0	0
$277$	263.042	0.949610	0.474805	−	0.880091i	$-0.342519\pi$
$277$	263.042	0.949610	0.474805	+	0.880091i	$0.342519\pi$
$278$	230.442i	0.828929i
$279$	0	0
$280$	−10.1780	−0.0363501
$281$	492.576i	1.75294i	0.481458	+	0.876469i	$0.340107\pi$
$281$	492.576i	1.75294i	−0.481458	+	0.876469i	$0.659893\pi$
$282$	0	0
$283$	−489.966	−1.73133	−0.865665	−	0.500624i	$-0.833104\pi$
$283$	−489.966	−1.73133	−0.865665	+	0.500624i	$0.833104\pi$
$284$	218.511i	0.769405i
$285$	0	0
$286$	255.324	0.892740
$287$	− 11.2432i	− 0.0391751i
$288$	0	0
$289$	−798.437	−2.76276
$290$	294.140i	1.01428i
$291$	0	0
$292$	125.882	0.431103
$293$	152.267i	0.519683i	0.965651	+	0.259842i	$0.0836704\pi$
$293$	152.267i	0.519683i	−0.965651	+	0.259842i	$0.916330\pi$
$294$	0	0
$295$	−437.076	−1.48161
$296$	− 21.7493i	− 0.0734773i
$297$	0	0
$298$	253.936	0.852135
$299$	185.360i	0.619932i
$300$	0	0
$301$	8.14262	0.0270519
$302$	227.590i	0.753611i
$303$	0	0
$304$	−1.62234	−0.00533663
$305$	− 830.265i	− 2.72218i
$306$	0	0
$307$	305.403	0.994798	0.497399	−	0.867522i	$-0.334289\pi$
$307$	305.403	0.994798	0.497399	+	0.867522i	$0.334289\pi$
$308$	12.7968i	0.0415480i
$309$	0	0
$310$	280.286	0.904150
$311$	− 353.846i	− 1.13777i	−0.822417	−	0.568885i	$-0.807375\pi$
$311$	− 353.846i	− 1.13777i	0.822417	−	0.568885i	$-0.192625\pi$
$312$	0	0
$313$	116.156	0.371106	0.185553	−	0.982634i	$-0.440592\pi$
$313$	116.156	0.371106	0.185553	+	0.982634i	$0.440592\pi$
$314$	− 109.312i	− 0.348127i
$315$	0	0
$316$	38.5180	0.121892
$317$	227.211i	0.716755i	0.933577	+	0.358378i	$0.116670\pi$
$317$	227.211i	0.716755i	−0.933577	+	0.358378i	$0.883330\pi$
$318$	0	0
$319$	369.820	1.15931
$320$	− 63.4565i	− 0.198302i
$321$	0	0
$322$	−9.29019	−0.0288515
$323$	− 13.3747i	− 0.0414076i
$324$	0	0
$325$	485.376	1.49347
$326$	3.02771i	0.00928747i
$327$	0	0
$328$	70.0979	0.213713
$329$	− 21.5354i	− 0.0654573i
$330$	0	0
$331$	−94.2026	−0.284600	−0.142300	−	0.989824i	$-0.545450\pi$
$331$	−94.2026	−0.284600	−0.142300	+	0.989824i	$0.545450\pi$
$332$	114.259i	0.344153i
$333$	0	0
$334$	126.939	0.380056
$335$	− 512.478i	− 1.52979i
$336$	0	0
$337$	72.4556	0.215002	0.107501	−	0.994205i	$-0.465715\pi$
$337$	72.4556	0.215002	0.107501	+	0.994205i	$0.465715\pi$
$338$	7.26802i	0.0215030i
$339$	0	0
$340$	523.140	1.53865
$341$	− 352.403i	− 1.03344i
$342$	0	0
$343$	−44.3655	−0.129345
$344$	50.7665i	0.147577i
$345$	0	0
$346$	257.885	0.745333
$347$	413.366i	1.19126i	0.803261	+	0.595628i	$0.203097\pi$
$347$	413.366i	1.19126i	−0.803261	+	0.595628i	$0.796903\pi$
$348$	0	0
$349$	−70.3384	−0.201543	−0.100771	−	0.994910i	$-0.532131\pi$
$349$	−70.3384	−0.201543	−0.100771	+	0.994910i	$0.532131\pi$
$350$	24.3270i	0.0695057i
$351$	0	0
$352$	−79.7835	−0.226658
$353$	496.127i	1.40546i	0.711458	+	0.702729i	$0.248035\pi$
$353$	496.127i	1.40546i	−0.711458	+	0.702729i	$0.751965\pi$
$354$	0	0
$355$	866.622	2.44119
$356$	207.362i	0.582476i
$357$	0	0
$358$	−169.883	−0.474534
$359$	− 290.630i	− 0.809554i	−0.914415	−	0.404777i	$-0.867349\pi$
$359$	− 290.630i	− 0.809554i	0.914415	−	0.404777i	$-0.132651\pi$
$360$	0	0
$361$	−360.836	−0.999544
$362$	− 226.254i	− 0.625010i
$363$	0	0
$364$	11.6145	0.0319079
$365$	− 499.252i	− 1.36781i
$366$	0	0
$367$	−570.671	−1.55496	−0.777481	−	0.628906i	$-0.783503\pi$
$367$	−570.671	−1.55496	−0.777481	+	0.628906i	$0.783503\pi$
$368$	− 57.9212i	− 0.157395i
$369$	0	0
$370$	−86.2583	−0.233131
$371$	0.118613i	0 0.000319711i
$372$	0	0
$373$	−19.9479	−0.0534797	−0.0267398	−	0.999642i	$-0.508513\pi$
$373$	−19.9479	−0.0534797	−0.0267398	+	0.999642i	$0.508513\pi$
$374$	− 657.742i	− 1.75867i
$375$	0	0
$376$	134.266	0.357091
$377$	− 335.653i	− 0.890325i
$378$	0	0
$379$	−37.0647	−0.0977961	−0.0488981	−	0.998804i	$-0.515571\pi$
$379$	−37.0647	−0.0977961	−0.0488981	+	0.998804i	$0.515571\pi$
$380$	6.43423i	0.0169322i
$381$	0	0
$382$	180.587	0.472742
$383$	− 406.659i	− 1.06177i	−0.847443	−	0.530886i	$-0.821859\pi$
$383$	− 406.659i	− 1.06177i	0.847443	−	0.530886i	$-0.178141\pi$
$384$	0	0
$385$	50.7524	0.131824
$386$	− 140.999i	− 0.365282i
$387$	0	0
$388$	−112.426	−0.289759
$389$	155.756i	0.400400i	0.979755	+	0.200200i	$0.0641592\pi$
$389$	155.756i	0.400400i	−0.979755	+	0.200200i	$0.935841\pi$
$390$	0	0
$391$	477.507	1.22124
$392$	− 138.011i	− 0.352068i
$393$	0	0
$394$	53.8674	0.136719
$395$	− 152.764i	− 0.386743i
$396$	0	0
$397$	407.439	1.02629	0.513147	−	0.858300i	$-0.328479\pi$
$397$	407.439	1.02629	0.513147	+	0.858300i	$0.328479\pi$
$398$	− 545.088i	− 1.36957i
$399$	0	0
$400$	−151.671	−0.379176
$401$	58.0631i	0.144796i	0.997376	+	0.0723979i	$0.0230652\pi$
$401$	58.0631i	0.144796i	−0.997376	+	0.0723979i	$0.976935\pi$
$402$	0	0
$403$	−319.844	−0.793658
$404$	89.9509i	0.222651i
$405$	0	0
$406$	16.8229	0.0414356
$407$	108.452i	0.266467i
$408$	0	0
$409$	289.436	0.707666	0.353833	−	0.935309i	$-0.384878\pi$
$409$	289.436	0.707666	0.353833	+	0.935309i	$0.384878\pi$
$410$	− 278.010i	− 0.678074i
$411$	0	0
$412$	182.279	0.442424
$413$	24.9978i	0.0605274i
$414$	0	0
$415$	453.154	1.09194
$416$	72.4123i	0.174068i
$417$	0	0
$418$	8.08973	0.0193534
$419$	672.165i	1.60421i	0.597182	+	0.802106i	$0.296287\pi$
$419$	672.165i	1.60421i	−0.597182	+	0.802106i	$0.703713\pi$
$420$	0	0
$421$	−224.643	−0.533594	−0.266797	−	0.963753i	$-0.585965\pi$
$421$	−224.643	−0.533594	−0.266797	+	0.963753i	$0.585965\pi$
$422$	482.527i	1.14343i
$423$	0	0
$424$	−0.739512	−0.00174413
$425$	− 1250.38i	− 2.94208i
$426$	0	0
$427$	−47.4857	−0.111208
$428$	412.099i	0.962847i
$429$	0	0
$430$	201.342	0.468236
$431$	− 189.283i	− 0.439171i	−0.975593	−	0.219585i	$-0.929530\pi$
$431$	− 189.283i	− 0.439171i	0.975593	−	0.219585i	$-0.0704705\pi$
$432$	0	0
$433$	−364.876	−0.842669	−0.421334	−	0.906905i	$-0.638438\pi$
$433$	−364.876	−0.842669	−0.421334	+	0.906905i	$0.638438\pi$
$434$	− 16.0305i	− 0.0369367i
$435$	0	0
$436$	280.323	0.642942
$437$	5.87297i	0.0134393i
$438$	0	0
$439$	424.089	0.966034	0.483017	−	0.875611i	$-0.339541\pi$
$439$	424.089	0.966034	0.483017	+	0.875611i	$0.339541\pi$
$440$	316.424i	0.719146i
$441$	0	0
$442$	−596.973	−1.35062
$443$	− 179.779i	− 0.405822i	−0.979197	−	0.202911i	$-0.934960\pi$
$443$	− 179.779i	− 0.405822i	0.979197	−	0.202911i	$-0.0650401\pi$
$444$	0	0
$445$	822.403	1.84810
$446$	82.6969i	0.185419i
$447$	0	0
$448$	−3.62929	−0.00810110
$449$	644.477i	1.43536i	0.696373	+	0.717681i	$0.254796\pi$
$449$	644.477i	1.43536i	−0.696373	+	0.717681i	$0.745204\pi$
$450$	0	0
$451$	−349.541	−0.775035
$452$	145.825i	0.322622i
$453$	0	0
$454$	−59.3951	−0.130826
$455$	− 46.0634i	− 0.101238i
$456$	0	0
$457$	259.655	0.568173	0.284086	−	0.958799i	$-0.408310\pi$
$457$	259.655	0.568173	0.284086	+	0.958799i	$0.408310\pi$
$458$	− 125.879i	− 0.274846i
$459$	0	0
$460$	−229.717	−0.499385
$461$	− 486.957i	− 1.05631i	−0.849149	−	0.528153i	$-0.822885\pi$
$461$	− 486.957i	− 1.05631i	0.849149	−	0.528153i	$-0.177115\pi$
$462$	0	0
$463$	599.520	1.29486	0.647430	−	0.762125i	$-0.275844\pi$
$463$	599.520	1.29486	0.647430	+	0.762125i	$0.275844\pi$
$464$	104.885i	0.226045i
$465$	0	0
$466$	198.608	0.426197
$467$	636.857i	1.36372i	0.731483	+	0.681860i	$0.238828\pi$
$467$	636.857i	1.36372i	−0.731483	+	0.681860i	$0.761172\pi$
$468$	0	0
$469$	−29.3104	−0.0624954
$470$	− 532.504i	− 1.13299i
$471$	0	0
$472$	−155.853	−0.330197
$473$	− 253.146i	− 0.535192i
$474$	0	0
$475$	15.3788	0.0323764
$476$	− 29.9202i	− 0.0628575i
$477$	0	0
$478$	41.0880	0.0859581
$479$	399.155i	0.833309i	0.909065	+	0.416655i	$0.136798\pi$
$479$	399.155i	0.833309i	−0.909065	+	0.416655i	$0.863202\pi$
$480$	0	0
$481$	98.4322	0.204641
$482$	209.939i	0.435557i
$483$	0	0
$484$	155.838	0.321980
$485$	445.886i	0.919354i
$486$	0	0
$487$	−379.619	−0.779505	−0.389753	−	0.920920i	$-0.627440\pi$
$487$	−379.619	−0.779505	−0.389753	+	0.920920i	$0.627440\pi$
$488$	− 296.057i	− 0.606674i
$489$	0	0
$490$	−547.355	−1.11705
$491$	598.486i	1.21891i	0.792820	+	0.609456i	$0.208612\pi$
$491$	598.486i	1.21891i	−0.792820	+	0.609456i	$0.791388\pi$
$492$	0	0
$493$	−864.678	−1.75391
$494$	− 7.34232i	− 0.0148630i
$495$	0	0
$496$	99.9449	0.201502
$497$	− 49.5650i	− 0.0997284i
$498$	0	0
$499$	−46.3120	−0.0928097	−0.0464049	−	0.998923i	$-0.514776\pi$
$499$	−46.3120	−0.0928097	−0.0464049	+	0.998923i	$0.514776\pi$
$500$	204.927i	0.409854i
$501$	0	0
$502$	−441.311	−0.879105
$503$	− 841.766i	− 1.67349i	−0.547592	−	0.836745i	$-0.684456\pi$
$503$	− 841.766i	− 1.67349i	0.547592	−	0.836745i	$-0.315544\pi$
$504$	0	0
$505$	356.748	0.706432
$506$	288.822i	0.570795i
$507$	0	0
$508$	−21.9739	−0.0432557
$509$	− 510.795i	− 1.00353i	−0.865005	−	0.501763i	$-0.832685\pi$
$509$	− 510.795i	− 1.00353i	0.865005	−	0.501763i	$-0.167315\pi$
$510$	0	0
$511$	−28.5539	−0.0558785
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−74.4863	−0.144915
$515$	− 722.923i	− 1.40373i
$516$	0	0
$517$	−669.515	−1.29500
$518$	4.93341i	0.00952395i
$519$	0	0
$520$	287.190	0.552288
$521$	573.687i	1.10113i	0.834793	+	0.550563i	$0.185587\pi$
$521$	573.687i	1.10113i	−0.834793	+	0.550563i	$0.814413\pi$
$522$	0	0
$523$	−514.806	−0.984333	−0.492166	−	0.870501i	$-0.663795\pi$
$523$	−514.806	−0.984333	−0.492166	+	0.870501i	$0.663795\pi$
$524$	74.9600i	0.143053i
$525$	0	0
$526$	−22.6767	−0.0431117
$527$	823.954i	1.56348i
$528$	0	0
$529$	319.321	0.603631
$530$	2.93293i	0.00553382i
$531$	0	0
$532$	0.367996	0.000691721 0
$533$	317.247i	0.595210i
$534$	0	0
$535$	1634.40	3.05495
$536$	− 182.740i	− 0.340933i
$537$	0	0
$538$	−330.736	−0.614750
$539$	688.187i	1.27678i
$540$	0	0
$541$	104.119	0.192457	0.0962284	−	0.995359i	$-0.469322\pi$
$541$	104.119	0.192457	0.0962284	+	0.995359i	$0.469322\pi$
$542$	325.091i	0.599799i
$543$	0	0
$544$	186.542	0.342908
$545$	− 1111.77i	− 2.03994i
$546$	0	0
$547$	−199.944	−0.365528	−0.182764	−	0.983157i	$-0.558504\pi$
$547$	−199.944	−0.365528	−0.182764	+	0.983157i	$0.558504\pi$
$548$	− 84.6593i	− 0.154488i
$549$	0	0
$550$	756.301	1.37509
$551$	− 10.6349i	− 0.0193011i
$552$	0	0
$553$	−8.73706	−0.0157994
$554$	− 371.997i	− 0.671475i
$555$	0	0
$556$	325.895	0.586142
$557$	333.759i	0.599208i	0.954064	+	0.299604i	$0.0968546\pi$
$557$	333.759i	0.599208i	−0.954064	+	0.299604i	$0.903145\pi$
$558$	0	0
$559$	−229.757	−0.411015
$560$	14.3939i	0.0257034i
$561$	0	0
$562$	696.607	1.23951
$563$	− 1004.44i	− 1.78408i	−0.451958	−	0.892039i	$-0.649274\pi$
$563$	− 1004.44i	− 1.78408i	0.451958	−	0.892039i	$-0.350726\pi$
$564$	0	0
$565$	578.346	1.02362
$566$	692.917i	1.22423i
$567$	0	0
$568$	309.021	0.544051
$569$	− 686.999i	− 1.20738i	−0.797219	−	0.603690i	$-0.793697\pi$
$569$	− 686.999i	− 1.20738i	0.797219	−	0.603690i	$-0.206303\pi$
$570$	0	0
$571$	−387.458	−0.678560	−0.339280	−	0.940685i	$-0.610183\pi$
$571$	−387.458	−0.678560	−0.339280	+	0.940685i	$0.610183\pi$
$572$	− 361.082i	− 0.631262i
$573$	0	0
$574$	−15.9004	−0.0277010
$575$	549.059i	0.954885i
$576$	0	0
$577$	559.264	0.969262	0.484631	−	0.874719i	$-0.338954\pi$
$577$	559.264	0.969262	0.484631	+	0.874719i	$0.338954\pi$
$578$	1129.16i	1.95356i
$579$	0	0
$580$	415.976	0.717201
$581$	− 25.9174i	− 0.0446082i
$582$	0	0
$583$	3.68755	0.00632514
$584$	− 178.024i	− 0.304836i
$585$	0	0
$586$	215.338	0.367472
$587$	506.831i	0.863426i	0.902011	+	0.431713i	$0.142091\pi$
$587$	506.831i	0.863426i	−0.902011	+	0.431713i	$0.857909\pi$
$588$	0	0
$589$	−10.1340	−0.0172055
$590$	618.118i	1.04766i
$591$	0	0
$592$	−30.7581	−0.0519563
$593$	− 598.516i	− 1.00930i	−0.863324	−	0.504651i	$-0.831621\pi$
$593$	− 598.516i	− 1.00930i	0.863324	−	0.504651i	$-0.168379\pi$
$594$	0	0
$595$	−118.664	−0.199436
$596$	− 359.120i	− 0.602550i
$597$	0	0
$598$	262.138	0.438358
$599$	772.726i	1.29003i	0.764171	+	0.645014i	$0.223148\pi$
$599$	772.726i	1.29003i	−0.764171	+	0.645014i	$0.776852\pi$
$600$	0	0
$601$	−130.251	−0.216724	−0.108362	−	0.994112i	$-0.534561\pi$
$601$	−130.251	−0.216724	−0.108362	+	0.994112i	$0.534561\pi$
$602$	− 11.5154i	− 0.0191286i
$603$	0	0
$604$	321.861	0.532883
$605$	− 618.060i	− 1.02159i
$606$	0	0
$607$	633.411	1.04351	0.521756	−	0.853095i	$-0.325277\pi$
$607$	633.411	1.04351	0.521756	+	0.853095i	$0.325277\pi$
$608$	2.29433i	0.00377357i
$609$	0	0
$610$	−1174.17	−1.92487
$611$	607.658i	0.994530i
$612$	0	0
$613$	776.793	1.26720	0.633599	−	0.773661i	$-0.281577\pi$
$613$	776.793	1.26720	0.633599	+	0.773661i	$0.281577\pi$
$614$	− 431.905i	− 0.703428i
$615$	0	0
$616$	18.0974	0.0293788
$617$	366.108i	0.593368i	0.954976	+	0.296684i	$0.0958808\pi$
$617$	366.108i	0.593368i	−0.954976	+	0.296684i	$0.904119\pi$
$618$	0	0
$619$	446.196	0.720833	0.360417	−	0.932791i	$-0.382635\pi$
$619$	446.196	0.720833	0.360417	+	0.932791i	$0.382635\pi$
$620$	− 396.385i	− 0.639330i
$621$	0	0
$622$	−500.414	−0.804524
$623$	− 47.0360i	− 0.0754992i
$624$	0	0
$625$	−135.194	−0.216310
$626$	− 164.270i	− 0.262412i
$627$	0	0
$628$	−154.590	−0.246163
$629$	− 253.572i	− 0.403136i
$630$	0	0
$631$	−889.598	−1.40982	−0.704911	−	0.709295i	$-0.749013\pi$
$631$	−889.598	−1.40982	−0.704911	+	0.709295i	$0.749013\pi$
$632$	− 54.4726i	− 0.0861909i
$633$	0	0
$634$	321.325	0.506822
$635$	87.1491i	0.137243i
$636$	0	0
$637$	624.605	0.980542
$638$	− 523.005i	− 0.819757i
$639$	0	0
$640$	−89.7411	−0.140220
$641$	467.862i	0.729893i	0.931028	+	0.364947i	$0.118913\pi$
$641$	467.862i	0.729893i	−0.931028	+	0.364947i	$0.881087\pi$
$642$	0	0
$643$	−390.946	−0.608003	−0.304001	−	0.952672i	$-0.598323\pi$
$643$	−390.946	−0.608003	−0.304001	+	0.952672i	$0.598323\pi$
$644$	13.1383i	0.0204011i
$645$	0	0
$646$	−18.9146	−0.0292796
$647$	− 343.874i	− 0.531490i	−0.964043	−	0.265745i	$-0.914382\pi$
$647$	− 343.874i	− 0.531490i	0.964043	−	0.265745i	$-0.0856180\pi$
$648$	0	0
$649$	777.157	1.19747
$650$	− 686.426i	− 1.05604i
$651$	0	0
$652$	4.28184	0.00656723
$653$	118.140i	0.180919i	0.995900	+	0.0904595i	$0.0288336\pi$
$653$	118.140i	0.180919i	−0.995900	+	0.0904595i	$0.971166\pi$
$654$	0	0
$655$	297.294	0.453884
$656$	− 99.1333i	− 0.151118i
$657$	0	0
$658$	−30.4557	−0.0462853
$659$	690.725i	1.04814i	0.851675	+	0.524070i	$0.175587\pi$
$659$	690.725i	1.04814i	−0.851675	+	0.524070i	$0.824413\pi$
$660$	0	0
$661$	1001.46	1.51506	0.757532	−	0.652798i	$-0.226405\pi$
$661$	1001.46	1.51506	0.757532	+	0.652798i	$0.226405\pi$
$662$	133.223i	0.201243i
$663$	0	0
$664$	161.586	0.243353
$665$	− 1.45948i	− 0.00219471i
$666$	0	0
$667$	379.691	0.569251
$668$	− 179.518i	− 0.268740i
$669$	0	0
$670$	−724.753	−1.08172
$671$	1476.28i	2.20012i
$672$	0	0
$673$	−159.339	−0.236759	−0.118380	−	0.992968i	$-0.537770\pi$
$673$	−159.339	−0.236759	−0.118380	+	0.992968i	$0.537770\pi$
$674$	− 102.468i	− 0.152029i
$675$	0	0
$676$	10.2785	0.0152049
$677$	1022.70i	1.51064i	0.655358	+	0.755318i	$0.272518\pi$
$677$	1022.70i	1.51064i	−0.655358	+	0.755318i	$0.727482\pi$
$678$	0	0
$679$	25.5018	0.0375578
$680$	− 739.832i	− 1.08799i
$681$	0	0
$682$	−498.373	−0.730752
$683$	− 434.662i	− 0.636401i	−0.948023	−	0.318200i	$-0.896922\pi$
$683$	− 434.662i	− 0.636401i	0.948023	−	0.318200i	$-0.103078\pi$
$684$	0	0
$685$	−335.761	−0.490163
$686$	62.7423i	0.0914610i
$687$	0	0
$688$	71.7947	0.104353
$689$	− 3.34686i	− 0.00485756i
$690$	0	0
$691$	613.032	0.887167	0.443584	−	0.896233i	$-0.353707\pi$
$691$	613.032	0.887167	0.443584	+	0.896233i	$0.353707\pi$
$692$	− 364.705i	− 0.527030i
$693$	0	0
$694$	584.587	0.842345
$695$	− 1292.51i	− 1.85972i
$696$	0	0
$697$	817.263	1.17254
$698$	99.4735i	0.142512i
$699$	0	0
$700$	34.4036	0.0491479
$701$	644.120i	0.918859i	0.888214	+	0.459430i	$0.151946\pi$
$701$	644.120i	0.918859i	−0.888214	+	0.459430i	$0.848054\pi$
$702$	0	0
$703$	3.11875	0.00443634
$704$	112.831i	0.160271i
$705$	0	0
$706$	701.629	0.993809
$707$	− 20.4036i	− 0.0288594i
$708$	0	0
$709$	313.146	0.441673	0.220837	−	0.975311i	$-0.429121\pi$
$709$	313.146	0.441673	0.220837	+	0.975311i	$0.429121\pi$
$710$	− 1225.59i	− 1.72618i
$711$	0	0
$712$	293.254	0.411873
$713$	− 361.808i	− 0.507445i
$714$	0	0
$715$	−1432.06	−2.00288
$716$	240.251i	0.335546i
$717$	0	0
$718$	−411.013	−0.572441
$719$	− 1310.13i	− 1.82216i	−0.412232	−	0.911079i	$-0.635251\pi$
$719$	− 1310.13i	− 1.82216i	0.412232	−	0.911079i	$-0.364749\pi$
$720$	0	0
$721$	−41.3464	−0.0573459
$722$	510.298i	0.706785i
$723$	0	0
$724$	−319.971	−0.441949
$725$	− 994.246i	− 1.37137i
$726$	0	0
$727$	1260.51	1.73386	0.866928	−	0.498434i	$-0.166091\pi$
$727$	1260.51	1.73386	0.866928	+	0.498434i	$0.166091\pi$
$728$	− 16.4253i	− 0.0225623i
$729$	0	0
$730$	−706.049	−0.967190
$731$	591.881i	0.809686i
$732$	0	0
$733$	211.903	0.289090	0.144545	−	0.989498i	$-0.453828\pi$
$733$	211.903	0.289090	0.144545	+	0.989498i	$0.453828\pi$
$734$	807.051i	1.09952i
$735$	0	0
$736$	−81.9129	−0.111295
$737$	911.229i	1.23640i
$738$	0	0
$739$	−52.4229	−0.0709377	−0.0354688	−	0.999371i	$-0.511292\pi$
$739$	−52.4229	−0.0709377	−0.0354688	+	0.999371i	$0.511292\pi$
$740$	121.988i	0.164848i
$741$	0	0
$742$	0.167744	0.000226070 0
$743$	− 19.8274i	− 0.0266856i	−0.999911	−	0.0133428i	$-0.995753\pi$
$743$	− 19.8274i	− 0.0266856i	0.999911	−	0.0133428i	$-0.00424726\pi$
$744$	0	0
$745$	−1424.28	−1.91179
$746$	28.2106i	0.0378158i
$747$	0	0
$748$	−930.187	−1.24357
$749$	− 93.4767i	− 0.124802i
$750$	0	0
$751$	−1341.81	−1.78670	−0.893351	−	0.449360i	$-0.851652\pi$
$751$	−1341.81	−1.78670	−0.893351	+	0.449360i	$0.851652\pi$
$752$	− 189.881i	− 0.252501i
$753$	0	0
$754$	−474.684	−0.629555
$755$	− 1276.51i	− 1.69075i
$756$	0	0
$757$	1369.68	1.80935	0.904675	−	0.426102i	$-0.140114\pi$
$757$	1369.68	1.80935	0.904675	+	0.426102i	$0.140114\pi$
$758$	52.4175i	0.0691523i
$759$	0	0
$760$	9.09938	0.0119729
$761$	− 55.4905i	− 0.0729179i	−0.999335	−	0.0364590i	$-0.988392\pi$
$761$	− 55.4905i	− 0.0729179i	0.999335	−	0.0364590i	$-0.0116078\pi$
$762$	0	0
$763$	−63.5858	−0.0833366
$764$	− 255.389i	− 0.334279i
$765$	0	0
$766$	−575.103	−0.750787
$767$	− 705.355i	− 0.919628i
$768$	0	0
$769$	−1121.57	−1.45847	−0.729236	−	0.684262i	$-0.760124\pi$
$769$	−1121.57	−1.45847	−0.729236	+	0.684262i	$0.760124\pi$
$770$	− 71.7747i	− 0.0932139i
$771$	0	0
$772$	−199.402	−0.258293
$773$	− 146.615i	− 0.189670i	−0.995493	−	0.0948349i	$-0.969768\pi$
$773$	− 146.615i	− 0.189670i	0.995493	−	0.0948349i	$-0.0302323\pi$
$774$	0	0
$775$	−947.419	−1.22248
$776$	158.995i	0.204890i
$777$	0	0
$778$	220.272	0.283126
$779$	10.0517i	0.0129034i
$780$	0	0
$781$	−1540.93	−1.97302
$782$	− 675.296i	− 0.863550i
$783$	0	0
$784$	−195.177	−0.248950
$785$	613.110i	0.781031i
$786$	0	0
$787$	−1438.60	−1.82795	−0.913975	−	0.405771i	$-0.867003\pi$
$787$	−1438.60	−1.82795	−0.913975	+	0.405771i	$0.867003\pi$
$788$	− 76.1799i	− 0.0966751i
$789$	0	0
$790$	−216.040	−0.273469
$791$	− 33.0776i	− 0.0418174i
$792$	0	0
$793$	1339.89	1.68964
$794$	− 576.206i	− 0.725700i
$795$	0	0
$796$	−770.871	−0.968430
$797$	− 1348.77i	− 1.69231i	−0.532935	−	0.846156i	$-0.678911\pi$
$797$	− 1348.77i	− 1.69231i	0.532935	−	0.846156i	$-0.321089\pi$
$798$	0	0
$799$	1565.39	1.95919
$800$	214.495i	0.268118i
$801$	0	0
$802$	82.1137	0.102386
$803$	887.711i	1.10549i
$804$	0	0
$805$	52.1070	0.0647291
$806$	452.328i	0.561201i
$807$	0	0
$808$	127.210	0.157438
$809$	596.082i	0.736813i	0.929665	+	0.368407i	$0.120097\pi$
$809$	596.082i	0.736813i	−0.929665	+	0.368407i	$0.879903\pi$
$810$	0	0
$811$	−304.801	−0.375833	−0.187917	−	0.982185i	$-0.560173\pi$
$811$	−304.801	−0.375833	−0.187917	+	0.982185i	$0.560173\pi$
$812$	− 23.7911i	− 0.0292994i
$813$	0	0
$814$	153.375	0.188421
$815$	− 16.9819i	− 0.0208367i
$816$	0	0
$817$	−7.27969	−0.00891027
$818$	− 409.324i	− 0.500396i
$819$	0	0
$820$	−393.166	−0.479471
$821$	662.573i	0.807032i	0.914973	+	0.403516i	$0.132212\pi$
$821$	662.573i	0.807032i	−0.914973	+	0.403516i	$0.867788\pi$
$822$	0	0
$823$	1093.11	1.32820	0.664102	−	0.747642i	$-0.268814\pi$
$823$	1093.11	1.32820	0.664102	+	0.747642i	$0.268814\pi$
$824$	− 257.781i	− 0.312841i
$825$	0	0
$826$	35.3523	0.0427994
$827$	1011.97i	1.22367i	0.790985	+	0.611835i	$0.209568\pi$
$827$	1011.97i	1.22367i	−0.790985	+	0.611835i	$0.790432\pi$
$828$	0	0
$829$	−964.811	−1.16383	−0.581913	−	0.813251i	$-0.697695\pi$
$829$	−964.811	−1.16383	−0.581913	+	0.813251i	$0.697695\pi$
$830$	− 640.856i	− 0.772116i
$831$	0	0
$832$	102.406	0.123085
$833$	− 1609.05i	− 1.93164i
$834$	0	0
$835$	−711.975	−0.852665
$836$	− 11.4406i	− 0.0136849i
$837$	0	0
$838$	950.584	1.13435
$839$	998.533i	1.19015i	0.803671	+	0.595073i	$0.202877\pi$
$839$	998.533i	1.19015i	−0.803671	+	0.595073i	$0.797123\pi$
$840$	0	0
$841$	153.449	0.182460
$842$	317.693i	0.377308i
$843$	0	0
$844$	682.397	0.808527
$845$	− 40.7650i	− 0.0482426i
$846$	0	0
$847$	−35.3489	−0.0417343
$848$	1.04583i	0.00123329i
$849$	0	0
$850$	−1768.31	−2.08036
$851$	111.347i	0.130842i
$852$	0	0
$853$	1123.05	1.31659	0.658296	−	0.752759i	$-0.271277\pi$
$853$	1123.05	1.31659	0.658296	+	0.752759i	$0.271277\pi$
$854$	67.1549i	0.0786357i
$855$	0	0
$856$	582.796	0.680836
$857$	391.017i	0.456262i	0.973630	+	0.228131i	$0.0732615\pi$
$857$	391.017i	0.456262i	−0.973630	+	0.228131i	$0.926738\pi$
$858$	0	0
$859$	−1450.02	−1.68804	−0.844019	−	0.536314i	$-0.819816\pi$
$859$	−1450.02	−1.68804	−0.844019	+	0.536314i	$0.819816\pi$
$860$	− 284.740i	− 0.331093i
$861$	0	0
$862$	−267.686	−0.310541
$863$	953.368i	1.10471i	0.833608	+	0.552357i	$0.186271\pi$
$863$	953.368i	1.10471i	−0.833608	+	0.552357i	$0.813729\pi$
$864$	0	0
$865$	−1446.43	−1.67217
$866$	516.012i	0.595857i
$867$	0	0
$868$	−22.6706	−0.0261182
$869$	271.626i	0.312573i
$870$	0	0
$871$	827.040	0.949529
$872$	− 396.436i	− 0.454629i
$873$	0	0
$874$	8.30564	0.00950302
$875$	− 46.4838i	− 0.0531243i
$876$	0	0
$877$	−1239.07	−1.41285	−0.706423	−	0.707790i	$-0.749692\pi$
$877$	−1239.07	−1.41285	−0.706423	+	0.707790i	$0.749692\pi$
$878$	− 599.752i	− 0.683089i
$879$	0	0
$880$	447.491	0.508513
$881$	− 49.7743i	− 0.0564974i	−0.999601	−	0.0282487i	$-0.991007\pi$
$881$	− 49.7743i	− 0.0564974i	0.999601	−	0.0282487i	$-0.00899304\pi$
$882$	0	0
$883$	−429.923	−0.486889	−0.243444	−	0.969915i	$-0.578277\pi$
$883$	−429.923	−0.486889	−0.243444	+	0.969915i	$0.578277\pi$
$884$	844.247i	0.955030i
$885$	0	0
$886$	−254.246	−0.286959
$887$	620.652i	0.699721i	0.936802	+	0.349860i	$0.113771\pi$
$887$	620.652i	0.699721i	−0.936802	+	0.349860i	$0.886229\pi$
$888$	0	0
$889$	4.98435	0.00560670
$890$	− 1163.05i	− 1.30680i
$891$	0	0
$892$	116.951	0.131111
$893$	19.2532i	0.0215601i
$894$	0	0
$895$	952.844	1.06463
$896$	5.13260i	0.00572834i
$897$	0	0
$898$	911.428	1.01495
$899$	655.169i	0.728775i
$900$	0	0
$901$	−8.62188	−0.00956924
$902$	494.326i	0.548033i
$903$	0	0
$904$	206.228	0.228128
$905$	1269.02i	1.40223i
$906$	0	0
$907$	−1727.90	−1.90507	−0.952536	−	0.304425i	$-0.901536\pi$
$907$	−1727.90	−1.90507	−0.952536	+	0.304425i	$0.901536\pi$
$908$	83.9973i	0.0925081i
$909$	0	0
$910$	−65.1434	−0.0715862
$911$	− 643.276i	− 0.706121i	−0.935600	−	0.353061i	$-0.885141\pi$
$911$	− 643.276i	− 0.706121i	0.935600	−	0.353061i	$-0.114859\pi$
$912$	0	0
$913$	−805.745	−0.882525
$914$	− 367.208i	− 0.401759i
$915$	0	0
$916$	−178.020	−0.194345
$917$	− 17.0032i	− 0.0185422i
$918$	0	0
$919$	436.696	0.475186	0.237593	−	0.971365i	$-0.423642\pi$
$919$	436.696	0.475186	0.237593	+	0.971365i	$0.423642\pi$
$920$	324.869i	0.353119i
$921$	0	0
$922$	−688.662	−0.746921
$923$	1398.56i	1.51523i
$924$	0	0
$925$	291.569	0.315210
$926$	− 847.850i	− 0.915604i
$927$	0	0
$928$	148.330	0.159838
$929$	1570.67i	1.69071i	0.534202	+	0.845357i	$0.320612\pi$
$929$	1570.67i	1.69071i	−0.534202	+	0.845357i	$0.679388\pi$
$930$	0	0
$931$	19.7901	0.0212569
$932$	− 280.874i	− 0.301367i
$933$	0	0
$934$	900.652	0.964295
$935$	3689.15i	3.94562i
$936$	0	0
$937$	984.535	1.05073	0.525366	−	0.850877i	$-0.323929\pi$
$937$	984.535	1.05073	0.525366	+	0.850877i	$0.323929\pi$
$938$	41.4511i	0.0441909i
$939$	0	0
$940$	−753.075	−0.801143
$941$	205.664i	0.218559i	0.994011	+	0.109280i	$0.0348544\pi$
$941$	205.664i	0.218559i	−0.994011	+	0.109280i	$0.965146\pi$
$942$	0	0
$943$	−358.870	−0.380562
$944$	220.410i	0.233485i
$945$	0	0
$946$	−358.002	−0.378438
$947$	624.378i	0.659323i	0.944099	+	0.329661i	$0.106935\pi$
$947$	624.378i	0.659323i	−0.944099	+	0.329661i	$0.893065\pi$
$948$	0	0
$949$	805.696	0.848994
$950$	− 21.7489i	− 0.0228936i
$951$	0	0
$952$	−42.3135	−0.0444470
$953$	− 685.103i	− 0.718891i	−0.933166	−	0.359445i	$-0.882966\pi$
$953$	− 685.103i	− 0.718891i	0.933166	−	0.359445i	$-0.117034\pi$
$954$	0	0
$955$	−1012.88	−1.06061
$956$	− 58.1071i	− 0.0607815i
$957$	0	0
$958$	564.490	0.589239
$959$	19.2033i	0.0200243i
$960$	0	0
$961$	−336.688	−0.350352
$962$	− 139.204i	− 0.144703i
$963$	0	0
$964$	296.898	0.307985
$965$	790.836i	0.819519i
$966$	0	0
$967$	707.828	0.731983	0.365992	−	0.930618i	$-0.380730\pi$
$967$	707.828	0.731983	0.365992	+	0.930618i	$0.380730\pi$
$968$	− 220.389i	− 0.227674i
$969$	0	0
$970$	630.579	0.650081
$971$	1276.82i	1.31496i	0.753473	+	0.657479i	$0.228377\pi$
$971$	1276.82i	1.31496i	−0.753473	+	0.657479i	$0.771623\pi$
$972$	0	0
$973$	−73.9230	−0.0759743
$974$	536.863i	0.551194i
$975$	0	0
$976$	−418.688	−0.428984
$977$	− 1614.68i	− 1.65269i	−0.563166	−	0.826344i	$-0.690417\pi$
$977$	− 1614.68i	− 1.65269i	0.563166	−	0.826344i	$-0.309583\pi$
$978$	0	0
$979$	−1462.30	−1.49367
$980$	774.077i	0.789875i
$981$	0	0
$982$	846.387	0.861901
$983$	901.667i	0.917260i	0.888627	+	0.458630i	$0.151660\pi$
$983$	901.667i	0.917260i	−0.888627	+	0.458630i	$0.848340\pi$
$984$	0	0
$985$	−302.132	−0.306733
$986$	1222.84i	1.24020i
$987$	0	0
$988$	−10.3836	−0.0105097
$989$	− 259.902i	− 0.262793i
$990$	0	0
$991$	−885.858	−0.893903	−0.446952	−	0.894558i	$-0.647490\pi$
$991$	−885.858	−0.893903	−0.446952	+	0.894558i	$0.647490\pi$
$992$	− 141.343i	− 0.142483i
$993$	0	0
$994$	−70.0955	−0.0705186
$995$	3057.30i	3.07266i
$996$	0	0
$997$	−1256.06	−1.25984	−0.629922	−	0.776658i	$-0.716913\pi$
$997$	−1256.06	−1.25984	−0.629922	+	0.776658i	$0.716913\pi$
$998$	65.4951i	0.0656264i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.17		36
3.2	odd	2	inner	1458.3.b.c.1457.20		36
27.4	even	9		54.3.f.a.11.4	yes	36
27.7	even	9		162.3.f.a.125.1		36
27.20	odd	18		54.3.f.a.5.4	✓	36
27.23	odd	18		162.3.f.a.35.1		36
108.31	odd	18		432.3.bc.c.65.6		36
108.47	even	18		432.3.bc.c.113.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.5.4	✓	36	27.20	odd	18
54.3.f.a.11.4	yes	36	27.4	even	9
162.3.f.a.35.1		36	27.23	odd	18
162.3.f.a.125.1		36	27.7	even	9
432.3.bc.c.65.6		36	108.31	odd	18
432.3.bc.c.113.6		36	108.47	even	18
1458.3.b.c.1457.17		36	1.1	even	1	trivial
1458.3.b.c.1457.20		36	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +7.93206i q^{5} +0.453662 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +7.93206i q^{5} +0.453662 q^{7} +2.82843i q^{8} +11.2176 q^{10} -14.1039i q^{11} -12.8008 q^{13} -0.641574i q^{14} +4.00000 q^{16} +32.9763i q^{17} -0.405584 q^{19} -15.8641i q^{20} -19.9459 q^{22} -14.4803i q^{23} -37.9176 q^{25} +18.1031i q^{26} -0.907323 q^{28} +26.2212i q^{29} +24.9862 q^{31} -5.65685i q^{32} +46.6355 q^{34} +3.59847i q^{35} -7.68953 q^{37} +0.573582i q^{38} -22.4353 q^{40} -24.7833i q^{41} +17.9487 q^{43} +28.2077i q^{44} -20.4782 q^{46} -47.4703i q^{47} -48.7942 q^{49} +53.6236i q^{50} +25.6016 q^{52} +0.261457i q^{53} +111.873 q^{55} +1.28315i q^{56} +37.0824 q^{58} +55.1024i q^{59} -104.672 q^{61} -35.3359i q^{62} -8.00000 q^{64} -101.537i q^{65} -64.6084 q^{67} -65.9526i q^{68} +5.08901 q^{70} -109.255i q^{71} -62.9410 q^{73} +10.8746i q^{74} +0.811168 q^{76} -6.39839i q^{77} -19.2590 q^{79} +31.7283i q^{80} -35.0489 q^{82} -57.1293i q^{83} -261.570 q^{85} -25.3832i q^{86} +39.8918 q^{88} -103.681i q^{89} -5.80724 q^{91} +28.9606i q^{92} -67.1331 q^{94} -3.21712i q^{95} +56.2132 q^{97} +69.0054i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4}+O(q^{10}) \) 36 * q - 72 * q^4	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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