LMFDB - Embedded newform 5445.2.a.cb.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 $ x^8 - 4x^7 - 3x^6 + 22x^5 - 3x^4 - 32x^3 + 9x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.48523$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.485227 q^{2} -1.76455 q^{4} +1.00000 q^{5} +1.61947 q^{7} -1.82666 q^{8} +O(q^{10})$	$q+0.485227 q^{2} -1.76455 q^{4} +1.00000 q^{5} +1.61947 q^{7} -1.82666 q^{8} +0.485227 q^{10} -4.95793 q^{13} +0.785812 q^{14} +2.64276 q^{16} -3.13866 q^{17} -1.09825 q^{19} -1.76455 q^{20} +6.75778 q^{23} +1.00000 q^{25} -2.40572 q^{26} -2.85765 q^{28} +3.55099 q^{29} +8.53688 q^{31} +4.93567 q^{32} -1.52296 q^{34} +1.61947 q^{35} -5.35976 q^{37} -0.532901 q^{38} -1.82666 q^{40} -5.32036 q^{41} -3.41085 q^{43} +3.27906 q^{46} -6.60156 q^{47} -4.37731 q^{49} +0.485227 q^{50} +8.74854 q^{52} -9.71622 q^{53} -2.95823 q^{56} +1.72304 q^{58} -7.18545 q^{59} +6.63497 q^{61} +4.14232 q^{62} -2.89061 q^{64} -4.95793 q^{65} +2.36043 q^{67} +5.53834 q^{68} +0.785812 q^{70} +16.6559 q^{71} -8.49548 q^{73} -2.60070 q^{74} +1.93792 q^{76} +1.99189 q^{79} +2.64276 q^{80} -2.58158 q^{82} +9.54929 q^{83} -3.13866 q^{85} -1.65504 q^{86} +2.65371 q^{89} -8.02924 q^{91} -11.9245 q^{92} -3.20326 q^{94} -1.09825 q^{95} -14.5239 q^{97} -2.12399 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 6 * q^4 + 8 * q^5 - 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} - 4 q^{10} - 6 q^{13} + 14 q^{14} + 14 q^{16} - 8 q^{17} + 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} - 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} - 20 q^{38} - 12 q^{40} - 22 q^{41} - 14 q^{43} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} + 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} - 2 q^{71} - 16 q^{73} - 24 q^{74} + 22 q^{76} + 16 q^{79} + 14 q^{80} + 10 q^{82} - 46 q^{83} - 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} + 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 6 * q^4 + 8 * q^5 - 8 * q^7 - 12 * q^8 - 4 * q^10 - 6 * q^13 + 14 * q^14 + 14 * q^16 - 8 * q^17 + 2 * q^19 + 6 * q^20 + 4 * q^23 + 8 * q^25 - 2 * q^26 - 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 - 20 * q^38 - 12 * q^40 - 22 * q^41 - 14 * q^43 + 2 * q^46 + 10 * q^47 - 4 * q^50 + 10 * q^52 - 18 * q^53 + 34 * q^56 + 12 * q^58 + 2 * q^59 + 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 - 2 * q^71 - 16 * q^73 - 24 * q^74 + 22 * q^76 + 16 * q^79 + 14 * q^80 + 10 * q^82 - 46 * q^83 - 8 * q^85 - 28 * q^86 + 38 * q^89 + 8 * q^91 - 24 * q^92 + 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.485227	0.343107	0.171554	−	0.985175i	$-0.445121\pi$
$2$	0.485227	0.343107	0.171554	+	0.985175i	$0.445121\pi$
$3$	0	0
$3$	0	0
$4$	−1.76455	−0.882277
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.61947	0.612103	0.306052	−	0.952015i	$-0.400992\pi$
$7$	1.61947	0.612103	0.306052	+	0.952015i	$0.400992\pi$
$8$	−1.82666	−0.645823
$9$	0	0
$10$	0.485227	0.153442
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.95793	−1.37508	−0.687542	−	0.726145i	$-0.741310\pi$
$13$	−4.95793	−1.37508	−0.687542	+	0.726145i	$0.741310\pi$
$14$	0.785812	0.210017
$15$	0	0
$16$	2.64276	0.660691
$17$	−3.13866	−0.761238	−0.380619	−	0.924732i	$-0.624289\pi$
$17$	−3.13866	−0.761238	−0.380619	+	0.924732i	$0.624289\pi$
$18$	0	0
$19$	−1.09825	−0.251956	−0.125978	−	0.992033i	$-0.540207\pi$
$19$	−1.09825	−0.251956	−0.125978	+	0.992033i	$0.540207\pi$
$20$	−1.76455	−0.394566
$21$	0	0
$22$	0	0
$23$	6.75778	1.40909	0.704547	−	0.709657i	$-0.251150\pi$
$23$	6.75778	1.40909	0.704547	+	0.709657i	$0.251150\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.40572	−0.471801
$27$	0	0
$28$	−2.85765	−0.540045
$29$	3.55099	0.659402	0.329701	−	0.944085i	$-0.393052\pi$
$29$	3.55099	0.659402	0.329701	+	0.944085i	$0.393052\pi$
$30$	0	0
$31$	8.53688	1.53327	0.766634	−	0.642084i	$-0.221930\pi$
$31$	8.53688	1.53327	0.766634	+	0.642084i	$0.221930\pi$
$32$	4.93567	0.872511
$33$	0	0
$34$	−1.52296	−0.261186
$35$	1.61947	0.273741
$36$	0	0
$37$	−5.35976	−0.881139	−0.440570	−	0.897718i	$-0.645223\pi$
$37$	−5.35976	−0.881139	−0.440570	+	0.897718i	$0.645223\pi$
$38$	−0.532901	−0.0864479
$39$	0	0
$40$	−1.82666	−0.288821
$41$	−5.32036	−0.830901	−0.415451	−	0.909616i	$-0.636376\pi$
$41$	−5.32036	−0.830901	−0.415451	+	0.909616i	$0.636376\pi$
$42$	0	0
$43$	−3.41085	−0.520150	−0.260075	−	0.965588i	$-0.583747\pi$
$43$	−3.41085	−0.520150	−0.260075	+	0.965588i	$0.583747\pi$
$44$	0	0
$45$	0	0
$46$	3.27906	0.483471
$47$	−6.60156	−0.962937	−0.481469	−	0.876463i	$-0.659896\pi$
$47$	−6.60156	−0.962937	−0.481469	+	0.876463i	$0.659896\pi$
$48$	0	0
$49$	−4.37731	−0.625330
$50$	0.485227	0.0686215
$51$	0	0
$52$	8.74854	1.21320
$53$	−9.71622	−1.33463	−0.667313	−	0.744778i	$-0.732556\pi$
$53$	−9.71622	−1.33463	−0.667313	+	0.744778i	$0.732556\pi$
$54$	0	0
$55$	0	0
$56$	−2.95823	−0.395310
$57$	0	0
$58$	1.72304	0.226246
$59$	−7.18545	−0.935466	−0.467733	−	0.883870i	$-0.654929\pi$
$59$	−7.18545	−0.935466	−0.467733	+	0.883870i	$0.654929\pi$
$60$	0	0
$61$	6.63497	0.849521	0.424760	−	0.905306i	$-0.360358\pi$
$61$	6.63497	0.849521	0.424760	+	0.905306i	$0.360358\pi$
$62$	4.14232	0.526076
$63$	0	0
$64$	−2.89061	−0.361326
$65$	−4.95793	−0.614956
$66$	0	0
$67$	2.36043	0.288373	0.144186	−	0.989551i	$-0.453944\pi$
$67$	2.36043	0.288373	0.144186	+	0.989551i	$0.453944\pi$
$68$	5.53834	0.671623
$69$	0	0
$70$	0.785812	0.0939225
$71$	16.6559	1.97670	0.988348	−	0.152212i	$-0.0486397\pi$
$71$	16.6559	1.97670	0.988348	+	0.152212i	$0.0486397\pi$
$72$	0	0
$73$	−8.49548	−0.994320	−0.497160	−	0.867659i	$-0.665624\pi$
$73$	−8.49548	−0.994320	−0.497160	+	0.867659i	$0.665624\pi$
$74$	−2.60070	−0.302325
$75$	0	0
$76$	1.93792	0.222295
$77$	0	0
$78$	0	0
$79$	1.99189	0.224105	0.112053	−	0.993702i	$-0.464258\pi$
$79$	1.99189	0.224105	0.112053	+	0.993702i	$0.464258\pi$
$80$	2.64276	0.295470
$81$	0	0
$82$	−2.58158	−0.285088
$83$	9.54929	1.04817	0.524085	−	0.851666i	$-0.324407\pi$
$83$	9.54929	1.04817	0.524085	+	0.851666i	$0.324407\pi$
$84$	0	0
$85$	−3.13866	−0.340436
$86$	−1.65504	−0.178467
$87$	0	0
$88$	0	0
$89$	2.65371	0.281293	0.140647	−	0.990060i	$-0.455082\pi$
$89$	2.65371	0.281293	0.140647	+	0.990060i	$0.455082\pi$
$90$	0	0
$91$	−8.02924	−0.841693
$92$	−11.9245	−1.24321
$93$	0	0
$94$	−3.20326	−0.330391
$95$	−1.09825	−0.112678
$96$	0	0
$97$	−14.5239	−1.47468	−0.737338	−	0.675524i	$-0.763918\pi$
$97$	−14.5239	−1.47468	−0.737338	+	0.675524i	$0.763918\pi$
$98$	−2.12399	−0.214555
$99$	0	0
$100$	−1.76455	−0.176455
$101$	−17.6927	−1.76049	−0.880247	−	0.474517i	$-0.842623\pi$
$101$	−17.6927	−1.76049	−0.880247	+	0.474517i	$0.842623\pi$
$102$	0	0
$103$	−4.93631	−0.486389	−0.243195	−	0.969978i	$-0.578195\pi$
$103$	−4.93631	−0.486389	−0.243195	+	0.969978i	$0.578195\pi$
$104$	9.05648	0.888061
$105$	0	0
$106$	−4.71457	−0.457920
$107$	−12.7475	−1.23235	−0.616173	−	0.787611i	$-0.711318\pi$
$107$	−12.7475	−1.23235	−0.616173	+	0.787611i	$0.711318\pi$
$108$	0	0
$109$	−15.5953	−1.49376	−0.746879	−	0.664960i	$-0.768449\pi$
$109$	−15.5953	−1.49376	−0.746879	+	0.664960i	$0.768449\pi$
$110$	0	0
$111$	0	0
$112$	4.27988	0.404411
$113$	−5.63373	−0.529977	−0.264988	−	0.964252i	$-0.585368\pi$
$113$	−5.63373	−0.529977	−0.264988	+	0.964252i	$0.585368\pi$
$114$	0	0
$115$	6.75778	0.630166
$116$	−6.26592	−0.581776
$117$	0	0
$118$	−3.48658	−0.320965
$119$	−5.08298	−0.465956
$120$	0	0
$121$	0	0
$122$	3.21947	0.291477
$123$	0	0
$124$	−15.0638	−1.35277
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.36289	−0.653351	−0.326676	−	0.945137i	$-0.605928\pi$
$127$	−7.36289	−0.653351	−0.326676	+	0.945137i	$0.605928\pi$
$128$	−11.2739	−0.996485
$129$	0	0
$130$	−2.40572	−0.210996
$131$	6.74024	0.588897	0.294449	−	0.955667i	$-0.404864\pi$
$131$	6.74024	0.588897	0.294449	+	0.955667i	$0.404864\pi$
$132$	0	0
$133$	−1.77859	−0.154223
$134$	1.14535	0.0989428
$135$	0	0
$136$	5.73328	0.491625
$137$	6.32003	0.539957	0.269978	−	0.962866i	$-0.412983\pi$
$137$	6.32003	0.539957	0.269978	+	0.962866i	$0.412983\pi$
$138$	0	0
$139$	14.0838	1.19457	0.597285	−	0.802029i	$-0.296246\pi$
$139$	14.0838	1.19457	0.597285	+	0.802029i	$0.296246\pi$
$140$	−2.85765	−0.241515
$141$	0	0
$142$	8.08191	0.678219
$143$	0	0
$144$	0	0
$145$	3.55099	0.294894
$146$	−4.12224	−0.341159
$147$	0	0
$148$	9.45759	0.777409
$149$	−17.3475	−1.42116	−0.710582	−	0.703614i	$-0.751568\pi$
$149$	−17.3475	−1.42116	−0.710582	+	0.703614i	$0.751568\pi$
$150$	0	0
$151$	9.01455	0.733593	0.366797	−	0.930301i	$-0.380454\pi$
$151$	9.01455	0.733593	0.366797	+	0.930301i	$0.380454\pi$
$152$	2.00613	0.162719
$153$	0	0
$154$	0	0
$155$	8.53688	0.685698
$156$	0	0
$157$	11.9364	0.952628	0.476314	−	0.879275i	$-0.341972\pi$
$157$	11.9364	0.952628	0.476314	+	0.879275i	$0.341972\pi$
$158$	0.966519	0.0768921
$159$	0	0
$160$	4.93567	0.390199
$161$	10.9440	0.862511
$162$	0	0
$163$	−2.32743	−0.182298	−0.0911491	−	0.995837i	$-0.529054\pi$
$163$	−2.32743	−0.182298	−0.0911491	+	0.995837i	$0.529054\pi$
$164$	9.38807	0.733085
$165$	0	0
$166$	4.63357	0.359635
$167$	−9.82442	−0.760237	−0.380118	−	0.924938i	$-0.624117\pi$
$167$	−9.82442	−0.760237	−0.380118	+	0.924938i	$0.624117\pi$
$168$	0	0
$169$	11.5811	0.890854
$170$	−1.52296	−0.116806
$171$	0	0
$172$	6.01864	0.458917
$173$	−23.2400	−1.76690	−0.883452	−	0.468521i	$-0.844787\pi$
$173$	−23.2400	−1.76690	−0.883452	+	0.468521i	$0.844787\pi$
$174$	0	0
$175$	1.61947	0.122421
$176$	0	0
$177$	0	0
$178$	1.28765	0.0965137
$179$	−0.376550	−0.0281447	−0.0140724	−	0.999901i	$-0.504480\pi$
$179$	−0.376550	−0.0281447	−0.0140724	+	0.999901i	$0.504480\pi$
$180$	0	0
$181$	−7.58158	−0.563534	−0.281767	−	0.959483i	$-0.590921\pi$
$181$	−7.58158	−0.563534	−0.281767	+	0.959483i	$0.590921\pi$
$182$	−3.89600	−0.288791
$183$	0	0
$184$	−12.3442	−0.910026
$185$	−5.35976	−0.394057
$186$	0	0
$187$	0	0
$188$	11.6488	0.849578
$189$	0	0
$190$	−0.532901	−0.0386607
$191$	2.16401	0.156583	0.0782913	−	0.996931i	$-0.475054\pi$
$191$	2.16401	0.156583	0.0782913	+	0.996931i	$0.475054\pi$
$192$	0	0
$193$	18.5881	1.33800	0.668999	−	0.743263i	$-0.266723\pi$
$193$	18.5881	1.33800	0.668999	+	0.743263i	$0.266723\pi$
$194$	−7.04738	−0.505972
$195$	0	0
$196$	7.72400	0.551714
$197$	8.70640	0.620306	0.310153	−	0.950687i	$-0.399620\pi$
$197$	8.70640	0.620306	0.310153	+	0.950687i	$0.399620\pi$
$198$	0	0
$199$	−18.9231	−1.34142	−0.670712	−	0.741718i	$-0.734011\pi$
$199$	−18.9231	−1.34142	−0.670712	+	0.741718i	$0.734011\pi$
$200$	−1.82666	−0.129165
$201$	0	0
$202$	−8.58499	−0.604038
$203$	5.75073	0.403622
$204$	0	0
$205$	−5.32036	−0.371590
$206$	−2.39523	−0.166884
$207$	0	0
$208$	−13.1026	−0.908505
$209$	0	0
$210$	0	0
$211$	−17.3522	−1.19457	−0.597287	−	0.802027i	$-0.703755\pi$
$211$	−17.3522	−1.19457	−0.597287	+	0.802027i	$0.703755\pi$
$212$	17.1448	1.17751
$213$	0	0
$214$	−6.18542	−0.422827
$215$	−3.41085	−0.232618
$216$	0	0
$217$	13.8252	0.938519
$218$	−7.56726	−0.512519
$219$	0	0
$220$	0	0
$221$	15.5613	1.04677
$222$	0	0
$223$	7.19082	0.481533	0.240767	−	0.970583i	$-0.422601\pi$
$223$	7.19082	0.481533	0.240767	+	0.970583i	$0.422601\pi$
$224$	7.99318	0.534067
$225$	0	0
$226$	−2.73364	−0.181839
$227$	−27.3012	−1.81205	−0.906023	−	0.423229i	$-0.860897\pi$
$227$	−27.3012	−1.81205	−0.906023	+	0.423229i	$0.860897\pi$
$228$	0	0
$229$	12.3763	0.817848	0.408924	−	0.912568i	$-0.365904\pi$
$229$	12.3763	0.817848	0.408924	+	0.912568i	$0.365904\pi$
$230$	3.27906	0.216215
$231$	0	0
$232$	−6.48646	−0.425857
$233$	−14.5447	−0.952857	−0.476429	−	0.879213i	$-0.658069\pi$
$233$	−14.5447	−0.952857	−0.476429	+	0.879213i	$0.658069\pi$
$234$	0	0
$235$	−6.60156	−0.430639
$236$	12.6791	0.825341
$237$	0	0
$238$	−2.46640	−0.159873
$239$	−28.9267	−1.87112	−0.935558	−	0.353173i	$-0.885103\pi$
$239$	−28.9267	−1.87112	−0.935558	+	0.353173i	$0.885103\pi$
$240$	0	0
$241$	9.40610	0.605900	0.302950	−	0.953006i	$-0.402028\pi$
$241$	9.40610	0.605900	0.302950	+	0.953006i	$0.402028\pi$
$242$	0	0
$243$	0	0
$244$	−11.7078	−0.749513
$245$	−4.37731	−0.279656
$246$	0	0
$247$	5.44505	0.346460
$248$	−15.5940	−0.990220
$249$	0	0
$250$	0.485227	0.0306885
$251$	0.0325382	0.00205379	0.00102690	−	0.999999i	$-0.499673\pi$
$251$	0.0325382	0.00205379	0.00102690	+	0.999999i	$0.499673\pi$
$252$	0	0
$253$	0	0
$254$	−3.57267	−0.224170
$255$	0	0
$256$	0.310793	0.0194246
$257$	6.61680	0.412745	0.206372	−	0.978474i	$-0.433834\pi$
$257$	6.61680	0.412745	0.206372	+	0.978474i	$0.433834\pi$
$258$	0	0
$259$	−8.67999	−0.539348
$260$	8.74854	0.542562
$261$	0	0
$262$	3.27055	0.202055
$263$	−5.19057	−0.320064	−0.160032	−	0.987112i	$-0.551160\pi$
$263$	−5.19057	−0.320064	−0.160032	+	0.987112i	$0.551160\pi$
$264$	0	0
$265$	−9.71622	−0.596863
$266$	−0.863018	−0.0529150
$267$	0	0
$268$	−4.16511	−0.254425
$269$	−15.8774	−0.968060	−0.484030	−	0.875052i	$-0.660827\pi$
$269$	−15.8774	−0.968060	−0.484030	+	0.875052i	$0.660827\pi$
$270$	0	0
$271$	26.2337	1.59358	0.796791	−	0.604255i	$-0.206529\pi$
$271$	26.2337	1.59358	0.796791	+	0.604255i	$0.206529\pi$
$272$	−8.29474	−0.502943
$273$	0	0
$274$	3.06665	0.185263
$275$	0	0
$276$	0	0
$277$	6.51114	0.391216	0.195608	−	0.980682i	$-0.437332\pi$
$277$	6.51114	0.391216	0.195608	+	0.980682i	$0.437332\pi$
$278$	6.83383	0.409866
$279$	0	0
$280$	−2.95823	−0.176788
$281$	−11.2188	−0.669258	−0.334629	−	0.942350i	$-0.608611\pi$
$281$	−11.2188	−0.669258	−0.334629	+	0.942350i	$0.608611\pi$
$282$	0	0
$283$	−8.27653	−0.491989	−0.245994	−	0.969271i	$-0.579114\pi$
$283$	−8.27653	−0.491989	−0.245994	+	0.969271i	$0.579114\pi$
$284$	−29.3903	−1.74399
$285$	0	0
$286$	0	0
$287$	−8.61619	−0.508597
$288$	0	0
$289$	−7.14879	−0.420517
$290$	1.72304	0.101180
$291$	0	0
$292$	14.9907	0.877266
$293$	0.418459	0.0244466	0.0122233	−	0.999925i	$-0.496109\pi$
$293$	0.418459	0.0244466	0.0122233	+	0.999925i	$0.496109\pi$
$294$	0	0
$295$	−7.18545	−0.418353
$296$	9.79048	0.569060
$297$	0	0
$298$	−8.41749	−0.487612
$299$	−33.5046	−1.93762
$300$	0	0
$301$	−5.52378	−0.318386
$302$	4.37410	0.251701
$303$	0	0
$304$	−2.90241	−0.166465
$305$	6.63497	0.379917
$306$	0	0
$307$	32.7941	1.87166	0.935828	−	0.352457i	$-0.114654\pi$
$307$	32.7941	1.87166	0.935828	+	0.352457i	$0.114654\pi$
$308$	0	0
$309$	0	0
$310$	4.14232	0.235268
$311$	−0.509602	−0.0288969	−0.0144484	−	0.999896i	$-0.504599\pi$
$311$	−0.509602	−0.0288969	−0.0144484	+	0.999896i	$0.504599\pi$
$312$	0	0
$313$	2.83268	0.160113	0.0800563	−	0.996790i	$-0.474490\pi$
$313$	2.83268	0.160113	0.0800563	+	0.996790i	$0.474490\pi$
$314$	5.79186	0.326854
$315$	0	0
$316$	−3.51480	−0.197723
$317$	−26.9913	−1.51598	−0.757992	−	0.652264i	$-0.773819\pi$
$317$	−26.9913	−1.51598	−0.757992	+	0.652264i	$0.773819\pi$
$318$	0	0
$319$	0	0
$320$	−2.89061	−0.161590
$321$	0	0
$322$	5.31034	0.295934
$323$	3.44704	0.191798
$324$	0	0
$325$	−4.95793	−0.275017
$326$	−1.12933	−0.0625479
$327$	0	0
$328$	9.71852	0.536615
$329$	−10.6911	−0.589417
$330$	0	0
$331$	21.5474	1.18435	0.592175	−	0.805809i	$-0.298269\pi$
$331$	21.5474	1.18435	0.592175	+	0.805809i	$0.298269\pi$
$332$	−16.8502	−0.924777
$333$	0	0
$334$	−4.76708	−0.260843
$335$	2.36043	0.128964
$336$	0	0
$337$	−10.9766	−0.597932	−0.298966	−	0.954264i	$-0.596642\pi$
$337$	−10.9766	−0.597932	−0.298966	+	0.954264i	$0.596642\pi$
$338$	5.61946	0.305658
$339$	0	0
$340$	5.53834	0.300359
$341$	0	0
$342$	0	0
$343$	−18.4252	−0.994870
$344$	6.23048	0.335925
$345$	0	0
$346$	−11.2767	−0.606238
$347$	−16.7192	−0.897532	−0.448766	−	0.893649i	$-0.648136\pi$
$347$	−16.7192	−0.897532	−0.448766	+	0.893649i	$0.648136\pi$
$348$	0	0
$349$	−4.59837	−0.246145	−0.123072	−	0.992398i	$-0.539275\pi$
$349$	−4.59837	−0.246145	−0.123072	+	0.992398i	$0.539275\pi$
$350$	0.785812	0.0420034
$351$	0	0
$352$	0	0
$353$	−14.4628	−0.769775	−0.384887	−	0.922964i	$-0.625760\pi$
$353$	−14.4628	−0.769775	−0.384887	+	0.922964i	$0.625760\pi$
$354$	0	0
$355$	16.6559	0.884005
$356$	−4.68262	−0.248179
$357$	0	0
$358$	−0.182712	−0.00965665
$359$	8.56634	0.452114	0.226057	−	0.974114i	$-0.427416\pi$
$359$	8.56634	0.452114	0.226057	+	0.974114i	$0.427416\pi$
$360$	0	0
$361$	−17.7938	−0.936518
$362$	−3.67879	−0.193353
$363$	0	0
$364$	14.1680	0.742607
$365$	−8.49548	−0.444674
$366$	0	0
$367$	1.14927	0.0599913	0.0299956	−	0.999550i	$-0.490451\pi$
$367$	1.14927	0.0599913	0.0299956	+	0.999550i	$0.490451\pi$
$368$	17.8592	0.930975
$369$	0	0
$370$	−2.60070	−0.135204
$371$	−15.7352	−0.816929
$372$	0	0
$373$	−32.1348	−1.66388	−0.831939	−	0.554867i	$-0.812769\pi$
$373$	−32.1348	−1.66388	−0.831939	+	0.554867i	$0.812769\pi$
$374$	0	0
$375$	0	0
$376$	12.0588	0.621887
$377$	−17.6056	−0.906733
$378$	0	0
$379$	14.7386	0.757073	0.378537	−	0.925586i	$-0.376427\pi$
$379$	14.7386	0.757073	0.378537	+	0.925586i	$0.376427\pi$
$380$	1.93792	0.0994133
$381$	0	0
$382$	1.05004	0.0537246
$383$	30.0887	1.53746	0.768732	−	0.639572i	$-0.220888\pi$
$383$	30.0887	1.53746	0.768732	+	0.639572i	$0.220888\pi$
$384$	0	0
$385$	0	0
$386$	9.01943	0.459077
$387$	0	0
$388$	25.6282	1.30107
$389$	−34.6757	−1.75813	−0.879064	−	0.476704i	$-0.841831\pi$
$389$	−34.6757	−1.75813	−0.879064	+	0.476704i	$0.841831\pi$
$390$	0	0
$391$	−21.2104	−1.07266
$392$	7.99587	0.403852
$393$	0	0
$394$	4.22458	0.212831
$395$	1.99189	0.100223
$396$	0	0
$397$	36.4341	1.82858	0.914288	−	0.405064i	$-0.132751\pi$
$397$	36.4341	1.82858	0.914288	+	0.405064i	$0.132751\pi$
$398$	−9.18200	−0.460252
$399$	0	0
$400$	2.64276	0.132138
$401$	−21.5435	−1.07583	−0.537917	−	0.842998i	$-0.680788\pi$
$401$	−21.5435	−1.07583	−0.537917	+	0.842998i	$0.680788\pi$
$402$	0	0
$403$	−42.3253	−2.10837
$404$	31.2198	1.55324
$405$	0	0
$406$	2.79041	0.138486
$407$	0	0
$408$	0	0
$409$	5.77447	0.285529	0.142765	−	0.989757i	$-0.454401\pi$
$409$	5.77447	0.285529	0.142765	+	0.989757i	$0.454401\pi$
$410$	−2.58158	−0.127495
$411$	0	0
$412$	8.71039	0.429130
$413$	−11.6366	−0.572602
$414$	0	0
$415$	9.54929	0.468756
$416$	−24.4707	−1.19978
$417$	0	0
$418$	0	0
$419$	13.4745	0.658273	0.329137	−	0.944282i	$-0.393242\pi$
$419$	13.4745	0.658273	0.329137	+	0.944282i	$0.393242\pi$
$420$	0	0
$421$	−24.2366	−1.18122	−0.590610	−	0.806957i	$-0.701113\pi$
$421$	−24.2366	−1.18122	−0.590610	+	0.806957i	$0.701113\pi$
$422$	−8.41976	−0.409867
$423$	0	0
$424$	17.7483	0.861932
$425$	−3.13866	−0.152248
$426$	0	0
$427$	10.7452	0.519994
$428$	22.4936	1.08727
$429$	0	0
$430$	−1.65504	−0.0798130
$431$	−0.775904	−0.0373740	−0.0186870	−	0.999825i	$-0.505949\pi$
$431$	−0.775904	−0.0373740	−0.0186870	+	0.999825i	$0.505949\pi$
$432$	0	0
$433$	34.5838	1.66199	0.830995	−	0.556280i	$-0.187772\pi$
$433$	34.5838	1.66199	0.830995	+	0.556280i	$0.187772\pi$
$434$	6.70838	0.322013
$435$	0	0
$436$	27.5187	1.31791
$437$	−7.42173	−0.355030
$438$	0	0
$439$	−7.24100	−0.345594	−0.172797	−	0.984957i	$-0.555280\pi$
$439$	−7.24100	−0.345594	−0.172797	+	0.984957i	$0.555280\pi$
$440$	0	0
$441$	0	0
$442$	7.55076	0.359153
$443$	5.41215	0.257139	0.128570	−	0.991701i	$-0.458961\pi$
$443$	5.41215	0.257139	0.128570	+	0.991701i	$0.458961\pi$
$444$	0	0
$445$	2.65371	0.125798
$446$	3.48918	0.165218
$447$	0	0
$448$	−4.68126	−0.221169
$449$	−12.8909	−0.608359	−0.304180	−	0.952615i	$-0.598382\pi$
$449$	−12.8909	−0.608359	−0.304180	+	0.952615i	$0.598382\pi$
$450$	0	0
$451$	0	0
$452$	9.94102	0.467586
$453$	0	0
$454$	−13.2473	−0.621726
$455$	−8.02924	−0.376416
$456$	0	0
$457$	−1.70545	−0.0797777	−0.0398888	−	0.999204i	$-0.512700\pi$
$457$	−1.70545	−0.0797777	−0.0398888	+	0.999204i	$0.512700\pi$
$458$	6.00531	0.280610
$459$	0	0
$460$	−11.9245	−0.555981
$461$	−29.9226	−1.39364	−0.696818	−	0.717248i	$-0.745401\pi$
$461$	−29.9226	−1.39364	−0.696818	+	0.717248i	$0.745401\pi$
$462$	0	0
$463$	−33.9765	−1.57902	−0.789511	−	0.613736i	$-0.789666\pi$
$463$	−33.9765	−1.57902	−0.789511	+	0.613736i	$0.789666\pi$
$464$	9.38442	0.435661
$465$	0	0
$466$	−7.05750	−0.326932
$467$	0.169922	0.00786304	0.00393152	−	0.999992i	$-0.498749\pi$
$467$	0.169922	0.00786304	0.00393152	+	0.999992i	$0.498749\pi$
$468$	0	0
$469$	3.82265	0.176514
$470$	−3.20326	−0.147755
$471$	0	0
$472$	13.1254	0.604146
$473$	0	0
$474$	0	0
$475$	−1.09825	−0.0503912
$476$	8.96920	0.411103
$477$	0	0
$478$	−14.0360	−0.641994
$479$	−14.6180	−0.667912	−0.333956	−	0.942589i	$-0.608384\pi$
$479$	−14.6180	−0.667912	−0.333956	+	0.942589i	$0.608384\pi$
$480$	0	0
$481$	26.5733	1.21164
$482$	4.56410	0.207889
$483$	0	0
$484$	0	0
$485$	−14.5239	−0.659495
$486$	0	0
$487$	−23.9520	−1.08537	−0.542684	−	0.839937i	$-0.682592\pi$
$487$	−23.9520	−1.08537	−0.542684	+	0.839937i	$0.682592\pi$
$488$	−12.1199	−0.548640
$489$	0	0
$490$	−2.12399	−0.0959520
$491$	−31.7288	−1.43190	−0.715950	−	0.698151i	$-0.754006\pi$
$491$	−31.7288	−1.43190	−0.715950	+	0.698151i	$0.754006\pi$
$492$	0	0
$493$	−11.1454	−0.501962
$494$	2.64209	0.118873
$495$	0	0
$496$	22.5609	1.01302
$497$	26.9738	1.20994
$498$	0	0
$499$	33.1017	1.48183	0.740917	−	0.671596i	$-0.234391\pi$
$499$	33.1017	1.48183	0.740917	+	0.671596i	$0.234391\pi$
$500$	−1.76455	−0.0789133
$501$	0	0
$502$	0.0157884	0.000704672 0
$503$	−18.1421	−0.808917	−0.404459	−	0.914556i	$-0.632540\pi$
$503$	−18.1421	−0.808917	−0.404459	+	0.914556i	$0.632540\pi$
$504$	0	0
$505$	−17.6927	−0.787316
$506$	0	0
$507$	0	0
$508$	12.9922	0.576437
$509$	−16.6736	−0.739044	−0.369522	−	0.929222i	$-0.620478\pi$
$509$	−16.6736	−0.739044	−0.369522	+	0.929222i	$0.620478\pi$
$510$	0	0
$511$	−13.7582	−0.608627
$512$	22.6987	1.00315
$513$	0	0
$514$	3.21065	0.141616
$515$	−4.93631	−0.217520
$516$	0	0
$517$	0	0
$518$	−4.21176	−0.185054
$519$	0	0
$520$	9.05648	0.397153
$521$	20.5225	0.899108	0.449554	−	0.893253i	$-0.351583\pi$
$521$	20.5225	0.899108	0.449554	+	0.893253i	$0.351583\pi$
$522$	0	0
$523$	−0.404664	−0.0176947	−0.00884736	−	0.999961i	$-0.502816\pi$
$523$	−0.404664	−0.0176947	−0.00884736	+	0.999961i	$0.502816\pi$
$524$	−11.8935	−0.519571
$525$	0	0
$526$	−2.51860	−0.109816
$527$	−26.7944	−1.16718
$528$	0	0
$529$	22.6676	0.985546
$530$	−4.71457	−0.204788
$531$	0	0
$532$	3.13841	0.136067
$533$	26.3780	1.14256
$534$	0	0
$535$	−12.7475	−0.551122
$536$	−4.31171	−0.186238
$537$	0	0
$538$	−7.70412	−0.332148
$539$	0	0
$540$	0	0
$541$	17.7037	0.761142	0.380571	−	0.924752i	$-0.375728\pi$
$541$	17.7037	0.761142	0.380571	+	0.924752i	$0.375728\pi$
$542$	12.7293	0.546769
$543$	0	0
$544$	−15.4914	−0.664188
$545$	−15.5953	−0.668029
$546$	0	0
$547$	−5.79243	−0.247666	−0.123833	−	0.992303i	$-0.539519\pi$
$547$	−5.79243	−0.247666	−0.123833	+	0.992303i	$0.539519\pi$
$548$	−11.1520	−0.476392
$549$	0	0
$550$	0	0
$551$	−3.89987	−0.166140
$552$	0	0
$553$	3.22581	0.137175
$554$	3.15938	0.134229
$555$	0	0
$556$	−24.8516	−1.05394
$557$	41.1689	1.74438	0.872191	−	0.489166i	$-0.162699\pi$
$557$	41.1689	1.74438	0.872191	+	0.489166i	$0.162699\pi$
$558$	0	0
$559$	16.9108	0.715250
$560$	4.27988	0.180858
$561$	0	0
$562$	−5.44367	−0.229627
$563$	−12.0373	−0.507314	−0.253657	−	0.967294i	$-0.581633\pi$
$563$	−12.0373	−0.507314	−0.253657	+	0.967294i	$0.581633\pi$
$564$	0	0
$565$	−5.63373	−0.237013
$566$	−4.01600	−0.168805
$567$	0	0
$568$	−30.4248	−1.27660
$569$	−3.69378	−0.154851	−0.0774256	−	0.996998i	$-0.524670\pi$
$569$	−3.69378	−0.154851	−0.0774256	+	0.996998i	$0.524670\pi$
$570$	0	0
$571$	5.17733	0.216665	0.108332	−	0.994115i	$-0.465449\pi$
$571$	5.17733	0.216665	0.108332	+	0.994115i	$0.465449\pi$
$572$	0	0
$573$	0	0
$574$	−4.18081	−0.174503
$575$	6.75778	0.281819
$576$	0	0
$577$	0.984135	0.0409701	0.0204850	−	0.999790i	$-0.493479\pi$
$577$	0.984135	0.0409701	0.0204850	+	0.999790i	$0.493479\pi$
$578$	−3.46879	−0.144282
$579$	0	0
$580$	−6.26592	−0.260178
$581$	15.4648	0.641589
$582$	0	0
$583$	0	0
$584$	15.5184	0.642155
$585$	0	0
$586$	0.203048	0.00838782
$587$	31.6510	1.30638	0.653188	−	0.757196i	$-0.273431\pi$
$587$	31.6510	1.30638	0.653188	+	0.757196i	$0.273431\pi$
$588$	0	0
$589$	−9.37563	−0.386316
$590$	−3.48658	−0.143540
$591$	0	0
$592$	−14.1646	−0.582160
$593$	38.9330	1.59879	0.799394	−	0.600808i	$-0.205154\pi$
$593$	38.9330	1.59879	0.799394	+	0.600808i	$0.205154\pi$
$594$	0	0
$595$	−5.08298	−0.208382
$596$	30.6107	1.25386
$597$	0	0
$598$	−16.2573	−0.664812
$599$	1.69868	0.0694063	0.0347031	−	0.999398i	$-0.488951\pi$
$599$	1.69868	0.0694063	0.0347031	+	0.999398i	$0.488951\pi$
$600$	0	0
$601$	41.0817	1.67576	0.837879	−	0.545856i	$-0.183795\pi$
$601$	41.0817	1.67576	0.837879	+	0.545856i	$0.183795\pi$
$602$	−2.68029	−0.109240
$603$	0	0
$604$	−15.9067	−0.647233
$605$	0	0
$606$	0	0
$607$	21.8428	0.886573	0.443287	−	0.896380i	$-0.353812\pi$
$607$	21.8428	0.886573	0.443287	+	0.896380i	$0.353812\pi$
$608$	−5.42060	−0.219834
$609$	0	0
$610$	3.21947	0.130352
$611$	32.7301	1.32412
$612$	0	0
$613$	−45.7809	−1.84907	−0.924536	−	0.381096i	$-0.875547\pi$
$613$	−45.7809	−1.84907	−0.924536	+	0.381096i	$0.875547\pi$
$614$	15.9126	0.642179
$615$	0	0
$616$	0	0
$617$	37.0763	1.49263	0.746317	−	0.665590i	$-0.231820\pi$
$617$	37.0763	1.49263	0.746317	+	0.665590i	$0.231820\pi$
$618$	0	0
$619$	16.2548	0.653334	0.326667	−	0.945139i	$-0.394074\pi$
$619$	16.2548	0.653334	0.326667	+	0.945139i	$0.394074\pi$
$620$	−15.0638	−0.604976
$621$	0	0
$622$	−0.247273	−0.00991474
$623$	4.29762	0.172180
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.37449	0.0549358
$627$	0	0
$628$	−21.0624	−0.840482
$629$	16.8225	0.670756
$630$	0	0
$631$	−9.08331	−0.361601	−0.180800	−	0.983520i	$-0.557869\pi$
$631$	−9.08331	−0.361601	−0.180800	+	0.983520i	$0.557869\pi$
$632$	−3.63851	−0.144732
$633$	0	0
$634$	−13.0969	−0.520145
$635$	−7.36289	−0.292187
$636$	0	0
$637$	21.7024	0.859880
$638$	0	0
$639$	0	0
$640$	−11.2739	−0.445641
$641$	7.45471	0.294443	0.147222	−	0.989104i	$-0.452967\pi$
$641$	7.45471	0.294443	0.147222	+	0.989104i	$0.452967\pi$
$642$	0	0
$643$	44.1591	1.74147	0.870733	−	0.491756i	$-0.163645\pi$
$643$	44.1591	1.74147	0.870733	+	0.491756i	$0.163645\pi$
$644$	−19.3114	−0.760974
$645$	0	0
$646$	1.67260	0.0658074
$647$	11.4388	0.449704	0.224852	−	0.974393i	$-0.427810\pi$
$647$	11.4388	0.449704	0.224852	+	0.974393i	$0.427810\pi$
$648$	0	0
$649$	0	0
$650$	−2.40572	−0.0943602
$651$	0	0
$652$	4.10687	0.160838
$653$	−1.97887	−0.0774393	−0.0387196	−	0.999250i	$-0.512328\pi$
$653$	−1.97887	−0.0774393	−0.0387196	+	0.999250i	$0.512328\pi$
$654$	0	0
$655$	6.74024	0.263363
$656$	−14.0605	−0.548969
$657$	0	0
$658$	−5.18759	−0.202233
$659$	−18.3062	−0.713107	−0.356553	−	0.934275i	$-0.616048\pi$
$659$	−18.3062	−0.713107	−0.356553	+	0.934275i	$0.616048\pi$
$660$	0	0
$661$	−1.16189	−0.0451922	−0.0225961	−	0.999745i	$-0.507193\pi$
$661$	−1.16189	−0.0451922	−0.0225961	+	0.999745i	$0.507193\pi$
$662$	10.4554	0.406359
$663$	0	0
$664$	−17.4433	−0.676933
$665$	−1.77859	−0.0689706
$666$	0	0
$667$	23.9968	0.929160
$668$	17.3357	0.670740
$669$	0	0
$670$	1.14535	0.0442485
$671$	0	0
$672$	0	0
$673$	−1.31368	−0.0506385	−0.0253192	−	0.999679i	$-0.508060\pi$
$673$	−1.31368	−0.0506385	−0.0253192	+	0.999679i	$0.508060\pi$
$674$	−5.32613	−0.205155
$675$	0	0
$676$	−20.4355	−0.785980
$677$	19.2668	0.740484	0.370242	−	0.928935i	$-0.379275\pi$
$677$	19.2668	0.740484	0.370242	+	0.928935i	$0.379275\pi$
$678$	0	0
$679$	−23.5210	−0.902654
$680$	5.73328	0.219861
$681$	0	0
$682$	0	0
$683$	−7.15293	−0.273699	−0.136850	−	0.990592i	$-0.543698\pi$
$683$	−7.15293	−0.273699	−0.136850	+	0.990592i	$0.543698\pi$
$684$	0	0
$685$	6.32003	0.241476
$686$	−8.94043	−0.341347
$687$	0	0
$688$	−9.01407	−0.343658
$689$	48.1724	1.83522
$690$	0	0
$691$	20.8320	0.792485	0.396242	−	0.918146i	$-0.370314\pi$
$691$	20.8320	0.792485	0.396242	+	0.918146i	$0.370314\pi$
$692$	41.0083	1.55890
$693$	0	0
$694$	−8.11259	−0.307950
$695$	14.0838	0.534228
$696$	0	0
$697$	16.6988	0.632513
$698$	−2.23125	−0.0844541
$699$	0	0
$700$	−2.85765	−0.108009
$701$	−18.7296	−0.707408	−0.353704	−	0.935357i	$-0.615078\pi$
$701$	−18.7296	−0.707408	−0.353704	+	0.935357i	$0.615078\pi$
$702$	0	0
$703$	5.88636	0.222008
$704$	0	0
$705$	0	0
$706$	−7.01772	−0.264115
$707$	−28.6529	−1.07760
$708$	0	0
$709$	−18.0328	−0.677236	−0.338618	−	0.940924i	$-0.609959\pi$
$709$	−18.0328	−0.677236	−0.338618	+	0.940924i	$0.609959\pi$
$710$	8.08191	0.303309
$711$	0	0
$712$	−4.84744	−0.181666
$713$	57.6903	2.16052
$714$	0	0
$715$	0	0
$716$	0.664444	0.0248314
$717$	0	0
$718$	4.15662	0.155124
$719$	39.4673	1.47188	0.735942	−	0.677045i	$-0.236740\pi$
$719$	39.4673	1.47188	0.735942	+	0.677045i	$0.236740\pi$
$720$	0	0
$721$	−7.99423	−0.297721
$722$	−8.63406	−0.321326
$723$	0	0
$724$	13.3781	0.497193
$725$	3.55099	0.131880
$726$	0	0
$727$	12.4002	0.459899	0.229950	−	0.973203i	$-0.426144\pi$
$727$	12.4002	0.459899	0.229950	+	0.973203i	$0.426144\pi$
$728$	14.6667	0.543585
$729$	0	0
$730$	−4.12224	−0.152571
$731$	10.7055	0.395958
$732$	0	0
$733$	−10.9465	−0.404320	−0.202160	−	0.979353i	$-0.564796\pi$
$733$	−10.9465	−0.404320	−0.202160	+	0.979353i	$0.564796\pi$
$734$	0.557656	0.0205834
$735$	0	0
$736$	33.3541	1.22945
$737$	0	0
$738$	0	0
$739$	−14.7047	−0.540920	−0.270460	−	0.962731i	$-0.587176\pi$
$739$	−14.7047	−0.540920	−0.270460	+	0.962731i	$0.587176\pi$
$740$	9.45759	0.347668
$741$	0	0
$742$	−7.63512	−0.280294
$743$	28.8578	1.05869	0.529345	−	0.848407i	$-0.322438\pi$
$743$	28.8578	1.05869	0.529345	+	0.848407i	$0.322438\pi$
$744$	0	0
$745$	−17.3475	−0.635564
$746$	−15.5927	−0.570889
$747$	0	0
$748$	0	0
$749$	−20.6442	−0.754323
$750$	0	0
$751$	−25.7450	−0.939447	−0.469723	−	0.882814i	$-0.655646\pi$
$751$	−25.7450	−0.939447	−0.469723	+	0.882814i	$0.655646\pi$
$752$	−17.4464	−0.636204
$753$	0	0
$754$	−8.54270	−0.311107
$755$	9.01455	0.328073
$756$	0	0
$757$	39.6545	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$757$	39.6545	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$758$	7.15159	0.259757
$759$	0	0
$760$	2.00613	0.0727701
$761$	31.1718	1.12998	0.564988	−	0.825099i	$-0.308881\pi$
$761$	31.1718	1.12998	0.564988	+	0.825099i	$0.308881\pi$
$762$	0	0
$763$	−25.2562	−0.914334
$764$	−3.81852	−0.138149
$765$	0	0
$766$	14.5999	0.527515
$767$	35.6250	1.28634
$768$	0	0
$769$	38.5273	1.38933	0.694666	−	0.719333i	$-0.255552\pi$
$769$	38.5273	1.38933	0.694666	+	0.719333i	$0.255552\pi$
$770$	0	0
$771$	0	0
$772$	−32.7997	−1.18049
$773$	−42.8510	−1.54124	−0.770621	−	0.637293i	$-0.780054\pi$
$773$	−42.8510	−1.54124	−0.770621	+	0.637293i	$0.780054\pi$
$774$	0	0
$775$	8.53688	0.306654
$776$	26.5302	0.952380
$777$	0	0
$778$	−16.8256	−0.603226
$779$	5.84309	0.209350
$780$	0	0
$781$	0	0
$782$	−10.2919	−0.368036
$783$	0	0
$784$	−11.5682	−0.413149
$785$	11.9364	0.426028
$786$	0	0
$787$	−27.0485	−0.964176	−0.482088	−	0.876123i	$-0.660121\pi$
$787$	−27.0485	−0.964176	−0.482088	+	0.876123i	$0.660121\pi$
$788$	−15.3629	−0.547282
$789$	0	0
$790$	0.966519	0.0343872
$791$	−9.12367	−0.324400
$792$	0	0
$793$	−32.8957	−1.16816
$794$	17.6788	0.627398
$795$	0	0
$796$	33.3909	1.18351
$797$	10.3959	0.368241	0.184121	−	0.982904i	$-0.441056\pi$
$797$	10.3959	0.368241	0.184121	+	0.982904i	$0.441056\pi$
$798$	0	0
$799$	20.7201	0.733024
$800$	4.93567	0.174502
$801$	0	0
$802$	−10.4535	−0.369126
$803$	0	0
$804$	0	0
$805$	10.9440	0.385727
$806$	−20.5374	−0.723398
$807$	0	0
$808$	32.3187	1.13697
$809$	39.9399	1.40421	0.702106	−	0.712073i	$-0.252243\pi$
$809$	39.9399	1.40421	0.702106	+	0.712073i	$0.252243\pi$
$810$	0	0
$811$	15.5894	0.547417	0.273709	−	0.961813i	$-0.411750\pi$
$811$	15.5894	0.547417	0.273709	+	0.961813i	$0.411750\pi$
$812$	−10.1475	−0.356107
$813$	0	0
$814$	0	0
$815$	−2.32743	−0.0815263
$816$	0	0
$817$	3.74597	0.131055
$818$	2.80193	0.0979671
$819$	0	0
$820$	9.38807	0.327846
$821$	−2.46398	−0.0859934	−0.0429967	−	0.999075i	$-0.513690\pi$
$821$	−2.46398	−0.0859934	−0.0429967	+	0.999075i	$0.513690\pi$
$822$	0	0
$823$	−55.1708	−1.92313	−0.961566	−	0.274573i	$-0.911464\pi$
$823$	−55.1708	−1.92313	−0.961566	+	0.274573i	$0.911464\pi$
$824$	9.01698	0.314122
$825$	0	0
$826$	−5.64642	−0.196464
$827$	−30.7906	−1.07069	−0.535346	−	0.844633i	$-0.679819\pi$
$827$	−30.7906	−1.07069	−0.535346	+	0.844633i	$0.679819\pi$
$828$	0	0
$829$	−25.8676	−0.898418	−0.449209	−	0.893427i	$-0.648294\pi$
$829$	−25.8676	−0.898418	−0.449209	+	0.893427i	$0.648294\pi$
$830$	4.63357	0.160834
$831$	0	0
$832$	14.3314	0.496853
$833$	13.7389	0.476025
$834$	0	0
$835$	−9.82442	−0.339988
$836$	0	0
$837$	0	0
$838$	6.53820	0.225858
$839$	52.4369	1.81032	0.905161	−	0.425070i	$-0.139750\pi$
$839$	52.4369	1.81032	0.905161	+	0.425070i	$0.139750\pi$
$840$	0	0
$841$	−16.3905	−0.565189
$842$	−11.7603	−0.405285
$843$	0	0
$844$	30.6189	1.05395
$845$	11.5811	0.398402
$846$	0	0
$847$	0	0
$848$	−25.6777	−0.881775
$849$	0	0
$850$	−1.52296	−0.0522373
$851$	−36.2201	−1.24161
$852$	0	0
$853$	29.8759	1.02293	0.511465	−	0.859304i	$-0.329103\pi$
$853$	29.8759	1.02293	0.511465	+	0.859304i	$0.329103\pi$
$854$	5.21384	0.178414
$855$	0	0
$856$	23.2854	0.795877
$857$	−50.9131	−1.73916	−0.869579	−	0.493794i	$-0.835610\pi$
$857$	−50.9131	−1.73916	−0.869579	+	0.493794i	$0.835610\pi$
$858$	0	0
$859$	30.2338	1.03156	0.515782	−	0.856720i	$-0.327501\pi$
$859$	30.2338	1.03156	0.515782	+	0.856720i	$0.327501\pi$
$860$	6.01864	0.205234
$861$	0	0
$862$	−0.376490	−0.0128233
$863$	−18.8078	−0.640225	−0.320112	−	0.947380i	$-0.603721\pi$
$863$	−18.8078	−0.640225	−0.320112	+	0.947380i	$0.603721\pi$
$864$	0	0
$865$	−23.2400	−0.790184
$866$	16.7810	0.570241
$867$	0	0
$868$	−24.3954	−0.828034
$869$	0	0
$870$	0	0
$871$	−11.7029	−0.396536
$872$	28.4874	0.964703
$873$	0	0
$874$	−3.60122	−0.121813
$875$	1.61947	0.0547482
$876$	0	0
$877$	45.2915	1.52939	0.764693	−	0.644395i	$-0.222891\pi$
$877$	45.2915	1.52939	0.764693	+	0.644395i	$0.222891\pi$
$878$	−3.51353	−0.118576
$879$	0	0
$880$	0	0
$881$	−24.6233	−0.829581	−0.414791	−	0.909917i	$-0.636145\pi$
$881$	−24.6233	−0.829581	−0.414791	+	0.909917i	$0.636145\pi$
$882$	0	0
$883$	55.9629	1.88330	0.941651	−	0.336591i	$-0.109274\pi$
$883$	55.9629	1.88330	0.941651	+	0.336591i	$0.109274\pi$
$884$	−27.4587	−0.923537
$885$	0	0
$886$	2.62612	0.0882263
$887$	−23.5352	−0.790234	−0.395117	−	0.918631i	$-0.629296\pi$
$887$	−23.5352	−0.790234	−0.395117	+	0.918631i	$0.629296\pi$
$888$	0	0
$889$	−11.9240	−0.399918
$890$	1.28765	0.0431623
$891$	0	0
$892$	−12.6886	−0.424846
$893$	7.25017	0.242618
$894$	0	0
$895$	−0.376550	−0.0125867
$896$	−18.2578	−0.609951
$897$	0	0
$898$	−6.25501	−0.208732
$899$	30.3144	1.01104
$900$	0	0
$901$	30.4959	1.01597
$902$	0	0
$903$	0	0
$904$	10.2909	0.342271
$905$	−7.58158	−0.252020
$906$	0	0
$907$	−45.6222	−1.51486	−0.757430	−	0.652916i	$-0.773545\pi$
$907$	−45.6222	−1.51486	−0.757430	+	0.652916i	$0.773545\pi$
$908$	48.1745	1.59873
$909$	0	0
$910$	−3.89600	−0.129151
$911$	46.8905	1.55355	0.776776	−	0.629778i	$-0.216854\pi$
$911$	46.8905	1.55355	0.776776	+	0.629778i	$0.216854\pi$
$912$	0	0
$913$	0	0
$914$	−0.827531	−0.0273723
$915$	0	0
$916$	−21.8386	−0.721569
$917$	10.9156	0.360466
$918$	0	0
$919$	33.8040	1.11509	0.557545	−	0.830146i	$-0.311743\pi$
$919$	33.8040	1.11509	0.557545	+	0.830146i	$0.311743\pi$
$920$	−12.3442	−0.406976
$921$	0	0
$922$	−14.5193	−0.478167
$923$	−82.5790	−2.71812
$924$	0	0
$925$	−5.35976	−0.176228
$926$	−16.4863	−0.541774
$927$	0	0
$928$	17.5265	0.575336
$929$	−15.5132	−0.508971	−0.254485	−	0.967077i	$-0.581906\pi$
$929$	−15.5132	−0.508971	−0.254485	+	0.967077i	$0.581906\pi$
$930$	0	0
$931$	4.80738	0.157555
$932$	25.6650	0.840684
$933$	0	0
$934$	0.0824506	0.00269787
$935$	0	0
$936$	0	0
$937$	4.34747	0.142026	0.0710128	−	0.997475i	$-0.477377\pi$
$937$	4.34747	0.142026	0.0710128	+	0.997475i	$0.477377\pi$
$938$	1.85486	0.0605632
$939$	0	0
$940$	11.6488	0.379943
$941$	−18.7131	−0.610030	−0.305015	−	0.952348i	$-0.598662\pi$
$941$	−18.7131	−0.610030	−0.305015	+	0.952348i	$0.598662\pi$
$942$	0	0
$943$	−35.9538	−1.17082
$944$	−18.9894	−0.618054
$945$	0	0
$946$	0	0
$947$	−18.6269	−0.605294	−0.302647	−	0.953103i	$-0.597870\pi$
$947$	−18.6269	−0.605294	−0.302647	+	0.953103i	$0.597870\pi$
$948$	0	0
$949$	42.1200	1.36727
$950$	−0.532901	−0.0172896
$951$	0	0
$952$	9.28490	0.300925
$953$	8.74307	0.283216	0.141608	−	0.989923i	$-0.454773\pi$
$953$	8.74307	0.283216	0.141608	+	0.989923i	$0.454773\pi$
$954$	0	0
$955$	2.16401	0.0700259
$956$	51.0428	1.65084
$957$	0	0
$958$	−7.09304	−0.229166
$959$	10.2351	0.330509
$960$	0	0
$961$	41.8783	1.35091
$962$	12.8941	0.415722
$963$	0	0
$964$	−16.5976	−0.534572
$965$	18.5881	0.598371
$966$	0	0
$967$	−47.6843	−1.53342	−0.766712	−	0.641992i	$-0.778108\pi$
$967$	−47.6843	−1.53342	−0.766712	+	0.641992i	$0.778108\pi$
$968$	0	0
$969$	0	0
$970$	−7.04738	−0.226278
$971$	−32.8229	−1.05334	−0.526669	−	0.850070i	$-0.676559\pi$
$971$	−32.8229	−1.05334	−0.526669	+	0.850070i	$0.676559\pi$
$972$	0	0
$973$	22.8083	0.731200
$974$	−11.6222	−0.372398
$975$	0	0
$976$	17.5346	0.561270
$977$	46.1269	1.47573	0.737864	−	0.674949i	$-0.235834\pi$
$977$	46.1269	1.47573	0.737864	+	0.674949i	$0.235834\pi$
$978$	0	0
$979$	0	0
$980$	7.72400	0.246734
$981$	0	0
$982$	−15.3957	−0.491296
$983$	16.1706	0.515761	0.257881	−	0.966177i	$-0.416976\pi$
$983$	16.1706	0.515761	0.257881	+	0.966177i	$0.416976\pi$
$984$	0	0
$985$	8.70640	0.277409
$986$	−5.40803	−0.172227
$987$	0	0
$988$	−9.60809	−0.305674
$989$	−23.0498	−0.732940
$990$	0	0
$991$	60.7308	1.92918	0.964589	−	0.263758i	$-0.0849622\pi$
$991$	60.7308	1.92918	0.964589	+	0.263758i	$0.0849622\pi$
$992$	42.1352	1.33779
$993$	0	0
$994$	13.0884	0.415140
$995$	−18.9231	−0.599903
$996$	0	0
$997$	46.9915	1.48824	0.744118	−	0.668048i	$-0.232870\pi$
$997$	46.9915	1.48824	0.744118	+	0.668048i	$0.232870\pi$
$998$	16.0618	0.508428
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.cb.1.6		8
3.2	odd	2		5445.2.a.cc.1.3		8
11.2	odd	10		495.2.n.g.136.3	yes	16
11.6	odd	10		495.2.n.g.91.3	✓	16
11.10	odd	2		5445.2.a.cd.1.3		8
33.2	even	10		495.2.n.h.136.2	yes	16
33.17	even	10		495.2.n.h.91.2	yes	16
33.32	even	2		5445.2.a.ca.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.91.3	✓	16	11.6	odd	10
495.2.n.g.136.3	yes	16	11.2	odd	10
495.2.n.h.91.2	yes	16	33.17	even	10
495.2.n.h.136.2	yes	16	33.2	even	10
5445.2.a.ca.1.6		8	33.32	even	2
5445.2.a.cb.1.6		8	1.1	even	1	trivial
5445.2.a.cc.1.3		8	3.2	odd	2
5445.2.a.cd.1.3		8	11.10	odd	2

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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.485227 q^{2} -1.76455 q^{4} +1.00000 q^{5} +1.61947 q^{7} -1.82666 q^{8} +O(q^{10})\)	\(q+0.485227 q^{2} -1.76455 q^{4} +1.00000 q^{5} +1.61947 q^{7} -1.82666 q^{8} +0.485227 q^{10} -4.95793 q^{13} +0.785812 q^{14} +2.64276 q^{16} -3.13866 q^{17} -1.09825 q^{19} -1.76455 q^{20} +6.75778 q^{23} +1.00000 q^{25} -2.40572 q^{26} -2.85765 q^{28} +3.55099 q^{29} +8.53688 q^{31} +4.93567 q^{32} -1.52296 q^{34} +1.61947 q^{35} -5.35976 q^{37} -0.532901 q^{38} -1.82666 q^{40} -5.32036 q^{41} -3.41085 q^{43} +3.27906 q^{46} -6.60156 q^{47} -4.37731 q^{49} +0.485227 q^{50} +8.74854 q^{52} -9.71622 q^{53} -2.95823 q^{56} +1.72304 q^{58} -7.18545 q^{59} +6.63497 q^{61} +4.14232 q^{62} -2.89061 q^{64} -4.95793 q^{65} +2.36043 q^{67} +5.53834 q^{68} +0.785812 q^{70} +16.6559 q^{71} -8.49548 q^{73} -2.60070 q^{74} +1.93792 q^{76} +1.99189 q^{79} +2.64276 q^{80} -2.58158 q^{82} +9.54929 q^{83} -3.13866 q^{85} -1.65504 q^{86} +2.65371 q^{89} -8.02924 q^{91} -11.9245 q^{92} -3.20326 q^{94} -1.09825 q^{95} -14.5239 q^{97} -2.12399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 6 * q^4 + 8 * q^5 - 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} - 4 q^{10} - 6 q^{13} + 14 q^{14} + 14 q^{16} - 8 q^{17} + 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} - 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} - 20 q^{38} - 12 q^{40} - 22 q^{41} - 14 q^{43} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} + 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} - 2 q^{71} - 16 q^{73} - 24 q^{74} + 22 q^{76} + 16 q^{79} + 14 q^{80} + 10 q^{82} - 46 q^{83} - 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} + 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 6 * q^4 + 8 * q^5 - 8 * q^7 - 12 * q^8 - 4 * q^10 - 6 * q^13 + 14 * q^14 + 14 * q^16 - 8 * q^17 + 2 * q^19 + 6 * q^20 + 4 * q^23 + 8 * q^25 - 2 * q^26 - 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 - 20 * q^38 - 12 * q^40 - 22 * q^41 - 14 * q^43 + 2 * q^46 + 10 * q^47 - 4 * q^50 + 10 * q^52 - 18 * q^53 + 34 * q^56 + 12 * q^58 + 2 * q^59 + 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 - 2 * q^71 - 16 * q^73 - 24 * q^74 + 22 * q^76 + 16 * q^79 + 14 * q^80 + 10 * q^82 - 46 * q^83 - 8 * q^85 - 28 * q^86 + 38 * q^89 + 8 * q^91 - 24 * q^92 + 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * q^98

Introduction

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