LMFDB - Embedded newform 8752.2.a.s.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, 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("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.72204$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +O(q^{10})$	$q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +1.84194 q^{11} -0.700934 q^{13} +0.830099 q^{15} +0.793975 q^{17} -3.65845 q^{19} -1.83505 q^{21} +3.65159 q^{23} -4.77939 q^{25} -5.08377 q^{27} +5.52848 q^{29} -6.13605 q^{31} +3.25534 q^{33} -0.487683 q^{35} -2.73114 q^{37} -1.23879 q^{39} -9.69220 q^{41} +4.25340 q^{43} +0.0580027 q^{45} +5.21084 q^{47} -5.92191 q^{49} +1.40322 q^{51} -7.01107 q^{53} +0.865138 q^{55} -6.46572 q^{57} -6.98691 q^{59} -3.87686 q^{61} -0.128223 q^{63} -0.329220 q^{65} +5.10753 q^{67} +6.45360 q^{69} -1.33192 q^{71} -7.83377 q^{73} -8.44681 q^{75} -1.91251 q^{77} -4.36166 q^{79} -9.35523 q^{81} +13.3348 q^{83} +0.372921 q^{85} +9.77071 q^{87} -0.208861 q^{89} +0.727788 q^{91} -10.8445 q^{93} -1.71833 q^{95} +3.08809 q^{97} +0.227465 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	$ 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) $ 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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	0
$2$	0	0
$3$	1.76734	1.02037	0.510187	−	0.860063i	$-0.329576\pi$
$3$	1.76734	1.02037	0.510187	+	0.860063i	$0.329576\pi$
$4$	0	0
$5$	0.469688	0.210051	0.105025	−	0.994470i	$-0.466508\pi$
$5$	0.469688	0.210051	0.105025	+	0.994470i	$0.466508\pi$
$6$	0	0
$7$	−1.03831	−0.392445	−0.196222	−	0.980559i	$-0.562867\pi$
$7$	−1.03831	−0.392445	−0.196222	+	0.980559i	$0.562867\pi$
$8$	0	0
$9$	0.123492	0.0411640
$10$	0	0
$11$	1.84194	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$11$	1.84194	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$12$	0	0
$13$	−0.700934	−0.194404	−0.0972021	−	0.995265i	$-0.530989\pi$
$13$	−0.700934	−0.194404	−0.0972021	+	0.995265i	$0.530989\pi$
$14$	0	0
$15$	0.830099	0.214331
$16$	0	0
$17$	0.793975	0.192567	0.0962837	−	0.995354i	$-0.469304\pi$
$17$	0.793975	0.192567	0.0962837	+	0.995354i	$0.469304\pi$
$18$	0	0
$19$	−3.65845	−0.839306	−0.419653	−	0.907685i	$-0.637848\pi$
$19$	−3.65845	−0.839306	−0.419653	+	0.907685i	$0.637848\pi$
$20$	0	0
$21$	−1.83505	−0.400441
$22$	0	0
$23$	3.65159	0.761409	0.380704	−	0.924697i	$-0.375682\pi$
$23$	3.65159	0.761409	0.380704	+	0.924697i	$0.375682\pi$
$24$	0	0
$25$	−4.77939	−0.955879
$26$	0	0
$27$	−5.08377	−0.978372
$28$	0	0
$29$	5.52848	1.02661	0.513307	−	0.858205i	$-0.328420\pi$
$29$	5.52848	1.02661	0.513307	+	0.858205i	$0.328420\pi$
$30$	0	0
$31$	−6.13605	−1.10207	−0.551034	−	0.834483i	$-0.685766\pi$
$31$	−6.13605	−1.10207	−0.551034	+	0.834483i	$0.685766\pi$
$32$	0	0
$33$	3.25534	0.566681
$34$	0	0
$35$	−0.487683	−0.0824334
$36$	0	0
$37$	−2.73114	−0.448997	−0.224498	−	0.974474i	$-0.572074\pi$
$37$	−2.73114	−0.448997	−0.224498	+	0.974474i	$0.572074\pi$
$38$	0	0
$39$	−1.23879	−0.198365
$40$	0	0
$41$	−9.69220	−1.51367	−0.756833	−	0.653608i	$-0.773255\pi$
$41$	−9.69220	−1.51367	−0.756833	+	0.653608i	$0.773255\pi$
$42$	0	0
$43$	4.25340	0.648638	0.324319	−	0.945948i	$-0.394865\pi$
$43$	4.25340	0.648638	0.324319	+	0.945948i	$0.394865\pi$
$44$	0	0
$45$	0.0580027	0.00864653
$46$	0	0
$47$	5.21084	0.760078	0.380039	−	0.924970i	$-0.375910\pi$
$47$	5.21084	0.760078	0.380039	+	0.924970i	$0.375910\pi$
$48$	0	0
$49$	−5.92191	−0.845987
$50$	0	0
$51$	1.40322	0.196491
$52$	0	0
$53$	−7.01107	−0.963044	−0.481522	−	0.876434i	$-0.659916\pi$
$53$	−7.01107	−0.963044	−0.481522	+	0.876434i	$0.659916\pi$
$54$	0	0
$55$	0.865138	0.116655
$56$	0	0
$57$	−6.46572	−0.856406
$58$	0	0
$59$	−6.98691	−0.909619	−0.454809	−	0.890589i	$-0.650293\pi$
$59$	−6.98691	−0.909619	−0.454809	+	0.890589i	$0.650293\pi$
$60$	0	0
$61$	−3.87686	−0.496381	−0.248191	−	0.968711i	$-0.579836\pi$
$61$	−3.87686	−0.496381	−0.248191	+	0.968711i	$0.579836\pi$
$62$	0	0
$63$	−0.128223	−0.0161546
$64$	0	0
$65$	−0.329220	−0.0408348
$66$	0	0
$67$	5.10753	0.623984	0.311992	−	0.950085i	$-0.399004\pi$
$67$	5.10753	0.623984	0.311992	+	0.950085i	$0.399004\pi$
$68$	0	0
$69$	6.45360	0.776922
$70$	0	0
$71$	−1.33192	−0.158070	−0.0790350	−	0.996872i	$-0.525184\pi$
$71$	−1.33192	−0.158070	−0.0790350	+	0.996872i	$0.525184\pi$
$72$	0	0
$73$	−7.83377	−0.916874	−0.458437	−	0.888727i	$-0.651591\pi$
$73$	−7.83377	−0.916874	−0.458437	+	0.888727i	$0.651591\pi$
$74$	0	0
$75$	−8.44681	−0.975354
$76$	0	0
$77$	−1.91251	−0.217950
$78$	0	0
$79$	−4.36166	−0.490726	−0.245363	−	0.969431i	$-0.578907\pi$
$79$	−4.36166	−0.490726	−0.245363	+	0.969431i	$0.578907\pi$
$80$	0	0
$81$	−9.35523	−1.03947
$82$	0	0
$83$	13.3348	1.46368	0.731842	−	0.681475i	$-0.238661\pi$
$83$	13.3348	1.46368	0.731842	+	0.681475i	$0.238661\pi$
$84$	0	0
$85$	0.372921	0.0404489
$86$	0	0
$87$	9.77071	1.04753
$88$	0	0
$89$	−0.208861	−0.0221393	−0.0110696	−	0.999939i	$-0.503524\pi$
$89$	−0.208861	−0.0221393	−0.0110696	+	0.999939i	$0.503524\pi$
$90$	0	0
$91$	0.727788	0.0762929
$92$	0	0
$93$	−10.8445	−1.12452
$94$	0	0
$95$	−1.71833	−0.176297
$96$	0	0
$97$	3.08809	0.313549	0.156774	−	0.987634i	$-0.449890\pi$
$97$	3.08809	0.313549	0.156774	+	0.987634i	$0.449890\pi$
$98$	0	0
$99$	0.227465	0.0228611
$100$	0	0
$101$	−8.00261	−0.796290	−0.398145	−	0.917323i	$-0.630346\pi$
$101$	−8.00261	−0.796290	−0.398145	+	0.917323i	$0.630346\pi$
$102$	0	0
$103$	3.26470	0.321681	0.160840	−	0.986980i	$-0.448580\pi$
$103$	3.26470	0.321681	0.160840	+	0.986980i	$0.448580\pi$
$104$	0	0
$105$	−0.861901	−0.0841129
$106$	0	0
$107$	18.4884	1.78735	0.893673	−	0.448719i	$-0.148120\pi$
$107$	18.4884	1.78735	0.893673	+	0.448719i	$0.148120\pi$
$108$	0	0
$109$	−18.6769	−1.78893	−0.894463	−	0.447141i	$-0.852442\pi$
$109$	−18.6769	−1.78893	−0.894463	+	0.447141i	$0.852442\pi$
$110$	0	0
$111$	−4.82686	−0.458145
$112$	0	0
$113$	7.85874	0.739288	0.369644	−	0.929173i	$-0.379480\pi$
$113$	7.85874	0.739288	0.369644	+	0.929173i	$0.379480\pi$
$114$	0	0
$115$	1.71511	0.159935
$116$	0	0
$117$	−0.0865597	−0.00800245
$118$	0	0
$119$	−0.824394	−0.0755720
$120$	0	0
$121$	−7.60726	−0.691569
$122$	0	0
$123$	−17.1294	−1.54451
$124$	0	0
$125$	−4.59326	−0.410834
$126$	0	0
$127$	−9.84052	−0.873205	−0.436602	−	0.899655i	$-0.643818\pi$
$127$	−9.84052	−0.873205	−0.436602	+	0.899655i	$0.643818\pi$
$128$	0	0
$129$	7.51721	0.661854
$130$	0	0
$131$	−8.02091	−0.700791	−0.350395	−	0.936602i	$-0.613953\pi$
$131$	−8.02091	−0.700791	−0.350395	+	0.936602i	$0.613953\pi$
$132$	0	0
$133$	3.79861	0.329381
$134$	0	0
$135$	−2.38779	−0.205508
$136$	0	0
$137$	16.8434	1.43903	0.719516	−	0.694476i	$-0.244364\pi$
$137$	16.8434	1.43903	0.719516	+	0.694476i	$0.244364\pi$
$138$	0	0
$139$	−11.2506	−0.954260	−0.477130	−	0.878833i	$-0.658323\pi$
$139$	−11.2506	−0.954260	−0.477130	+	0.878833i	$0.658323\pi$
$140$	0	0
$141$	9.20932	0.775565
$142$	0	0
$143$	−1.29108	−0.107965
$144$	0	0
$145$	2.59666	0.215641
$146$	0	0
$147$	−10.4660	−0.863224
$148$	0	0
$149$	−18.8662	−1.54558	−0.772788	−	0.634664i	$-0.781138\pi$
$149$	−18.8662	−1.54558	−0.772788	+	0.634664i	$0.781138\pi$
$150$	0	0
$151$	6.40292	0.521062	0.260531	−	0.965465i	$-0.416102\pi$
$151$	6.40292	0.521062	0.260531	+	0.965465i	$0.416102\pi$
$152$	0	0
$153$	0.0980495	0.00792684
$154$	0	0
$155$	−2.88203	−0.231490
$156$	0	0
$157$	−14.9254	−1.19118	−0.595589	−	0.803290i	$-0.703081\pi$
$157$	−14.9254	−1.19118	−0.595589	+	0.803290i	$0.703081\pi$
$158$	0	0
$159$	−12.3909	−0.982665
$160$	0	0
$161$	−3.79148	−0.298811
$162$	0	0
$163$	20.3834	1.59655	0.798274	−	0.602294i	$-0.205747\pi$
$163$	20.3834	1.59655	0.798274	+	0.602294i	$0.205747\pi$
$164$	0	0
$165$	1.52899	0.119032
$166$	0	0
$167$	−15.8853	−1.22924	−0.614622	−	0.788822i	$-0.710691\pi$
$167$	−15.8853	−1.22924	−0.614622	+	0.788822i	$0.710691\pi$
$168$	0	0
$169$	−12.5087	−0.962207
$170$	0	0
$171$	−0.451789	−0.0345492
$172$	0	0
$173$	16.4110	1.24771	0.623854	−	0.781541i	$-0.285566\pi$
$173$	16.4110	1.24771	0.623854	+	0.781541i	$0.285566\pi$
$174$	0	0
$175$	4.96250	0.375130
$176$	0	0
$177$	−12.3483	−0.928151
$178$	0	0
$179$	20.4269	1.52678	0.763390	−	0.645938i	$-0.223534\pi$
$179$	20.4269	1.52678	0.763390	+	0.645938i	$0.223534\pi$
$180$	0	0
$181$	−10.3683	−0.770666	−0.385333	−	0.922778i	$-0.625913\pi$
$181$	−10.3683	−0.770666	−0.385333	+	0.922778i	$0.625913\pi$
$182$	0	0
$183$	−6.85173	−0.506495
$184$	0	0
$185$	−1.28279	−0.0943122
$186$	0	0
$187$	1.46246	0.106945
$188$	0	0
$189$	5.27853	0.383957
$190$	0	0
$191$	7.95483	0.575591	0.287796	−	0.957692i	$-0.407078\pi$
$191$	7.95483	0.575591	0.287796	+	0.957692i	$0.407078\pi$
$192$	0	0
$193$	23.9566	1.72443	0.862217	−	0.506539i	$-0.169075\pi$
$193$	23.9566	1.72443	0.862217	+	0.506539i	$0.169075\pi$
$194$	0	0
$195$	−0.581845	−0.0416668
$196$	0	0
$197$	5.15127	0.367013	0.183506	−	0.983019i	$-0.441255\pi$
$197$	5.15127	0.367013	0.183506	+	0.983019i	$0.441255\pi$
$198$	0	0
$199$	−1.54311	−0.109388	−0.0546940	−	0.998503i	$-0.517418\pi$
$199$	−1.54311	−0.109388	−0.0546940	+	0.998503i	$0.517418\pi$
$200$	0	0
$201$	9.02674	0.636697
$202$	0	0
$203$	−5.74028	−0.402889
$204$	0	0
$205$	−4.55231	−0.317947
$206$	0	0
$207$	0.450942	0.0313426
$208$	0	0
$209$	−6.73864	−0.466122
$210$	0	0
$211$	−8.81178	−0.606628	−0.303314	−	0.952891i	$-0.598093\pi$
$211$	−8.81178	−0.606628	−0.303314	+	0.952891i	$0.598093\pi$
$212$	0	0
$213$	−2.35396	−0.161291
$214$	0	0
$215$	1.99777	0.136247
$216$	0	0
$217$	6.37113	0.432501
$218$	0	0
$219$	−13.8449	−0.935555
$220$	0	0
$221$	−0.556524	−0.0374359
$222$	0	0
$223$	−24.2050	−1.62089	−0.810445	−	0.585815i	$-0.800774\pi$
$223$	−24.2050	−1.62089	−0.810445	+	0.585815i	$0.800774\pi$
$224$	0	0
$225$	−0.590216	−0.0393478
$226$	0	0
$227$	3.55162	0.235729	0.117865	−	0.993030i	$-0.462395\pi$
$227$	3.55162	0.235729	0.117865	+	0.993030i	$0.462395\pi$
$228$	0	0
$229$	−18.8164	−1.24342	−0.621711	−	0.783247i	$-0.713562\pi$
$229$	−18.8164	−1.24342	−0.621711	+	0.783247i	$0.713562\pi$
$230$	0	0
$231$	−3.38005	−0.222391
$232$	0	0
$233$	12.1897	0.798575	0.399288	−	0.916826i	$-0.369258\pi$
$233$	12.1897	0.798575	0.399288	+	0.916826i	$0.369258\pi$
$234$	0	0
$235$	2.44747	0.159655
$236$	0	0
$237$	−7.70855	−0.500724
$238$	0	0
$239$	−30.6423	−1.98208	−0.991042	−	0.133553i	$-0.957361\pi$
$239$	−30.6423	−1.98208	−0.991042	+	0.133553i	$0.957361\pi$
$240$	0	0
$241$	−7.26294	−0.467847	−0.233924	−	0.972255i	$-0.575157\pi$
$241$	−7.26294	−0.467847	−0.233924	+	0.972255i	$0.575157\pi$
$242$	0	0
$243$	−1.28256	−0.0822763
$244$	0	0
$245$	−2.78145	−0.177700
$246$	0	0
$247$	2.56433	0.163164
$248$	0	0
$249$	23.5671	1.49350
$250$	0	0
$251$	25.2813	1.59574	0.797871	−	0.602828i	$-0.205959\pi$
$251$	25.2813	1.59574	0.797871	+	0.602828i	$0.205959\pi$
$252$	0	0
$253$	6.72601	0.422860
$254$	0	0
$255$	0.659078	0.0412731
$256$	0	0
$257$	−26.5017	−1.65313	−0.826564	−	0.562842i	$-0.809708\pi$
$257$	−26.5017	−1.65313	−0.826564	+	0.562842i	$0.809708\pi$
$258$	0	0
$259$	2.83578	0.176207
$260$	0	0
$261$	0.682723	0.0422595
$262$	0	0
$263$	−12.5983	−0.776845	−0.388422	−	0.921481i	$-0.626980\pi$
$263$	−12.5983	−0.776845	−0.388422	+	0.921481i	$0.626980\pi$
$264$	0	0
$265$	−3.29301	−0.202288
$266$	0	0
$267$	−0.369129	−0.0225903
$268$	0	0
$269$	1.45914	0.0889652	0.0444826	−	0.999010i	$-0.485836\pi$
$269$	1.45914	0.0889652	0.0444826	+	0.999010i	$0.485836\pi$
$270$	0	0
$271$	−5.16205	−0.313572	−0.156786	−	0.987633i	$-0.550113\pi$
$271$	−5.16205	−0.313572	−0.156786	+	0.987633i	$0.550113\pi$
$272$	0	0
$273$	1.28625	0.0778473
$274$	0	0
$275$	−8.80336	−0.530862
$276$	0	0
$277$	14.0110	0.841837	0.420918	−	0.907098i	$-0.361708\pi$
$277$	14.0110	0.841837	0.420918	+	0.907098i	$0.361708\pi$
$278$	0	0
$279$	−0.757753	−0.0453655
$280$	0	0
$281$	16.3617	0.976059	0.488030	−	0.872827i	$-0.337716\pi$
$281$	16.3617	0.976059	0.488030	+	0.872827i	$0.337716\pi$
$282$	0	0
$283$	−18.5182	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$283$	−18.5182	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$284$	0	0
$285$	−3.03687	−0.179889
$286$	0	0
$287$	10.0635	0.594031
$288$	0	0
$289$	−16.3696	−0.962918
$290$	0	0
$291$	5.45771	0.319937
$292$	0	0
$293$	30.5239	1.78323	0.891613	−	0.452798i	$-0.149574\pi$
$293$	30.5239	1.78323	0.891613	+	0.452798i	$0.149574\pi$
$294$	0	0
$295$	−3.28167	−0.191066
$296$	0	0
$297$	−9.36400	−0.543354
$298$	0	0
$299$	−2.55952	−0.148021
$300$	0	0
$301$	−4.41636	−0.254555
$302$	0	0
$303$	−14.1433	−0.812514
$304$	0	0
$305$	−1.82092	−0.104265
$306$	0	0
$307$	−3.36035	−0.191785	−0.0958926	−	0.995392i	$-0.530571\pi$
$307$	−3.36035	−0.191785	−0.0958926	+	0.995392i	$0.530571\pi$
$308$	0	0
$309$	5.76984	0.328235
$310$	0	0
$311$	−26.0631	−1.47791	−0.738953	−	0.673757i	$-0.764679\pi$
$311$	−26.0631	−1.47791	−0.738953	+	0.673757i	$0.764679\pi$
$312$	0	0
$313$	−28.4960	−1.61069	−0.805345	−	0.592807i	$-0.798020\pi$
$313$	−28.4960	−1.61069	−0.805345	+	0.592807i	$0.798020\pi$
$314$	0	0
$315$	−0.0602249	−0.00339329
$316$	0	0
$317$	−8.26530	−0.464225	−0.232113	−	0.972689i	$-0.574564\pi$
$317$	−8.26530	−0.464225	−0.232113	+	0.972689i	$0.574564\pi$
$318$	0	0
$319$	10.1831	0.570146
$320$	0	0
$321$	32.6754	1.82376
$322$	0	0
$323$	−2.90472	−0.161623
$324$	0	0
$325$	3.35004	0.185827
$326$	0	0
$327$	−33.0085	−1.82537
$328$	0	0
$329$	−5.41047	−0.298289
$330$	0	0
$331$	−12.5162	−0.687951	−0.343976	−	0.938979i	$-0.611774\pi$
$331$	−12.5162	−0.687951	−0.343976	+	0.938979i	$0.611774\pi$
$332$	0	0
$333$	−0.337274	−0.0184825
$334$	0	0
$335$	2.39895	0.131068
$336$	0	0
$337$	−20.0461	−1.09198	−0.545990	−	0.837792i	$-0.683846\pi$
$337$	−20.0461	−1.09198	−0.545990	+	0.837792i	$0.683846\pi$
$338$	0	0
$339$	13.8891	0.754351
$340$	0	0
$341$	−11.3022	−0.612051
$342$	0	0
$343$	13.4170	0.724448
$344$	0	0
$345$	3.03118	0.163193
$346$	0	0
$347$	−0.748255	−0.0401684	−0.0200842	−	0.999798i	$-0.506393\pi$
$347$	−0.748255	−0.0401684	−0.0200842	+	0.999798i	$0.506393\pi$
$348$	0	0
$349$	4.16941	0.223183	0.111592	−	0.993754i	$-0.464405\pi$
$349$	4.16941	0.223183	0.111592	+	0.993754i	$0.464405\pi$
$350$	0	0
$351$	3.56339	0.190200
$352$	0	0
$353$	28.9002	1.53820	0.769100	−	0.639128i	$-0.220705\pi$
$353$	28.9002	1.53820	0.769100	+	0.639128i	$0.220705\pi$
$354$	0	0
$355$	−0.625588	−0.0332028
$356$	0	0
$357$	−1.45698	−0.0771118
$358$	0	0
$359$	−22.5238	−1.18876	−0.594380	−	0.804185i	$-0.702602\pi$
$359$	−22.5238	−1.18876	−0.594380	+	0.804185i	$0.702602\pi$
$360$	0	0
$361$	−5.61576	−0.295566
$362$	0	0
$363$	−13.4446	−0.705659
$364$	0	0
$365$	−3.67943	−0.192590
$366$	0	0
$367$	3.07935	0.160741	0.0803705	−	0.996765i	$-0.474390\pi$
$367$	3.07935	0.160741	0.0803705	+	0.996765i	$0.474390\pi$
$368$	0	0
$369$	−1.19691	−0.0623085
$370$	0	0
$371$	7.27967	0.377942
$372$	0	0
$373$	−23.8756	−1.23623	−0.618115	−	0.786088i	$-0.712103\pi$
$373$	−23.8756	−1.23623	−0.618115	+	0.786088i	$0.712103\pi$
$374$	0	0
$375$	−8.11786	−0.419205
$376$	0	0
$377$	−3.87510	−0.199578
$378$	0	0
$379$	−3.60479	−0.185166	−0.0925828	−	0.995705i	$-0.529512\pi$
$379$	−3.60479	−0.185166	−0.0925828	+	0.995705i	$0.529512\pi$
$380$	0	0
$381$	−17.3915	−0.890996
$382$	0	0
$383$	3.63594	0.185788	0.0928939	−	0.995676i	$-0.470388\pi$
$383$	3.63594	0.185788	0.0928939	+	0.995676i	$0.470388\pi$
$384$	0	0
$385$	−0.898282	−0.0457807
$386$	0	0
$387$	0.525261	0.0267005
$388$	0	0
$389$	21.6735	1.09889	0.549445	−	0.835530i	$-0.314839\pi$
$389$	21.6735	1.09889	0.549445	+	0.835530i	$0.314839\pi$
$390$	0	0
$391$	2.89927	0.146622
$392$	0	0
$393$	−14.1757	−0.715069
$394$	0	0
$395$	−2.04862	−0.103077
$396$	0	0
$397$	6.59119	0.330802	0.165401	−	0.986226i	$-0.447108\pi$
$397$	6.59119	0.330802	0.165401	+	0.986226i	$0.447108\pi$
$398$	0	0
$399$	6.71343	0.336092
$400$	0	0
$401$	−4.35340	−0.217398	−0.108699	−	0.994075i	$-0.534668\pi$
$401$	−4.35340	−0.217398	−0.108699	+	0.994075i	$0.534668\pi$
$402$	0	0
$403$	4.30097	0.214246
$404$	0	0
$405$	−4.39404	−0.218342
$406$	0	0
$407$	−5.03060	−0.249358
$408$	0	0
$409$	31.8412	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$409$	31.8412	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$410$	0	0
$411$	29.7681	1.46835
$412$	0	0
$413$	7.25459	0.356975
$414$	0	0
$415$	6.26319	0.307448
$416$	0	0
$417$	−19.8836	−0.973703
$418$	0	0
$419$	29.8225	1.45692	0.728462	−	0.685087i	$-0.240236\pi$
$419$	29.8225	1.45692	0.728462	+	0.685087i	$0.240236\pi$
$420$	0	0
$421$	−7.93092	−0.386530	−0.193265	−	0.981147i	$-0.561908\pi$
$421$	−7.93092	−0.386530	−0.193265	+	0.981147i	$0.561908\pi$
$422$	0	0
$423$	0.643496	0.0312879
$424$	0	0
$425$	−3.79472	−0.184071
$426$	0	0
$427$	4.02539	0.194802
$428$	0	0
$429$	−2.28178	−0.110165
$430$	0	0
$431$	24.9537	1.20198	0.600988	−	0.799258i	$-0.294774\pi$
$431$	24.9537	1.20198	0.600988	+	0.799258i	$0.294774\pi$
$432$	0	0
$433$	23.1505	1.11254	0.556272	−	0.831000i	$-0.312231\pi$
$433$	23.1505	1.11254	0.556272	+	0.831000i	$0.312231\pi$
$434$	0	0
$435$	4.58918	0.220035
$436$	0	0
$437$	−13.3591	−0.639054
$438$	0	0
$439$	−18.9654	−0.905167	−0.452584	−	0.891722i	$-0.649498\pi$
$439$	−18.9654	−0.905167	−0.452584	+	0.891722i	$0.649498\pi$
$440$	0	0
$441$	−0.731308	−0.0348242
$442$	0	0
$443$	−21.6203	−1.02721	−0.513606	−	0.858026i	$-0.671691\pi$
$443$	−21.6203	−1.02721	−0.513606	+	0.858026i	$0.671691\pi$
$444$	0	0
$445$	−0.0980997	−0.00465037
$446$	0	0
$447$	−33.3429	−1.57707
$448$	0	0
$449$	15.1299	0.714023	0.357011	−	0.934100i	$-0.383796\pi$
$449$	15.1299	0.714023	0.357011	+	0.934100i	$0.383796\pi$
$450$	0	0
$451$	−17.8524	−0.840639
$452$	0	0
$453$	11.3161	0.531678
$454$	0	0
$455$	0.341833	0.0160254
$456$	0	0
$457$	−7.87882	−0.368556	−0.184278	−	0.982874i	$-0.558995\pi$
$457$	−7.87882	−0.368556	−0.184278	+	0.982874i	$0.558995\pi$
$458$	0	0
$459$	−4.03639	−0.188402
$460$	0	0
$461$	−33.2729	−1.54967	−0.774836	−	0.632162i	$-0.782168\pi$
$461$	−33.2729	−1.54967	−0.774836	+	0.632162i	$0.782168\pi$
$462$	0	0
$463$	−31.9493	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$463$	−31.9493	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$464$	0	0
$465$	−5.09353	−0.236207
$466$	0	0
$467$	−6.03218	−0.279136	−0.139568	−	0.990213i	$-0.544571\pi$
$467$	−6.03218	−0.279136	−0.139568	+	0.990213i	$0.544571\pi$
$468$	0	0
$469$	−5.30320	−0.244879
$470$	0	0
$471$	−26.3783	−1.21545
$472$	0	0
$473$	7.83451	0.360231
$474$	0	0
$475$	17.4852	0.802274
$476$	0	0
$477$	−0.865810	−0.0396427
$478$	0	0
$479$	−18.1368	−0.828693	−0.414347	−	0.910119i	$-0.635990\pi$
$479$	−18.1368	−0.828693	−0.414347	+	0.910119i	$0.635990\pi$
$480$	0	0
$481$	1.91435	0.0872869
$482$	0	0
$483$	−6.70084	−0.304899
$484$	0	0
$485$	1.45044	0.0658612
$486$	0	0
$487$	23.3591	1.05850	0.529251	−	0.848465i	$-0.322473\pi$
$487$	23.3591	1.05850	0.529251	+	0.848465i	$0.322473\pi$
$488$	0	0
$489$	36.0243	1.62908
$490$	0	0
$491$	4.28889	0.193555	0.0967775	−	0.995306i	$-0.469146\pi$
$491$	4.28889	0.193555	0.0967775	+	0.995306i	$0.469146\pi$
$492$	0	0
$493$	4.38948	0.197692
$494$	0	0
$495$	0.106838	0.00480199
$496$	0	0
$497$	1.38295	0.0620338
$498$	0	0
$499$	12.0079	0.537549	0.268775	−	0.963203i	$-0.413381\pi$
$499$	12.0079	0.537549	0.268775	+	0.963203i	$0.413381\pi$
$500$	0	0
$501$	−28.0748	−1.25429
$502$	0	0
$503$	22.0435	0.982872	0.491436	−	0.870914i	$-0.336472\pi$
$503$	22.0435	0.982872	0.491436	+	0.870914i	$0.336472\pi$
$504$	0	0
$505$	−3.75873	−0.167261
$506$	0	0
$507$	−22.1071	−0.981811
$508$	0	0
$509$	−11.9642	−0.530302	−0.265151	−	0.964207i	$-0.585422\pi$
$509$	−11.9642	−0.530302	−0.265151	+	0.964207i	$0.585422\pi$
$510$	0	0
$511$	8.13390	0.359822
$512$	0	0
$513$	18.5987	0.821153
$514$	0	0
$515$	1.53339	0.0675693
$516$	0	0
$517$	9.59805	0.422122
$518$	0	0
$519$	29.0039	1.27313
$520$	0	0
$521$	−5.47217	−0.239740	−0.119870	−	0.992790i	$-0.538248\pi$
$521$	−5.47217	−0.239740	−0.119870	+	0.992790i	$0.538248\pi$
$522$	0	0
$523$	31.1363	1.36149	0.680747	−	0.732519i	$-0.261655\pi$
$523$	31.1363	1.36149	0.680747	+	0.732519i	$0.261655\pi$
$524$	0	0
$525$	8.77042	0.382773
$526$	0	0
$527$	−4.87187	−0.212222
$528$	0	0
$529$	−9.66591	−0.420257
$530$	0	0
$531$	−0.862827	−0.0374435
$532$	0	0
$533$	6.79359	0.294263
$534$	0	0
$535$	8.68380	0.375434
$536$	0	0
$537$	36.1013	1.55789
$538$	0	0
$539$	−10.9078	−0.469832
$540$	0	0
$541$	−15.0578	−0.647387	−0.323694	−	0.946162i	$-0.604925\pi$
$541$	−15.0578	−0.647387	−0.323694	+	0.946162i	$0.604925\pi$
$542$	0	0
$543$	−18.3242	−0.786368
$544$	0	0
$545$	−8.77234	−0.375766
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	−0.478761	−0.0204330
$550$	0	0
$551$	−20.2257	−0.861642
$552$	0	0
$553$	4.52877	0.192583
$554$	0	0
$555$	−2.26712	−0.0962338
$556$	0	0
$557$	23.5488	0.997796	0.498898	−	0.866661i	$-0.333738\pi$
$557$	23.5488	0.997796	0.498898	+	0.866661i	$0.333738\pi$
$558$	0	0
$559$	−2.98136	−0.126098
$560$	0	0
$561$	2.58466	0.109124
$562$	0	0
$563$	29.0653	1.22496	0.612478	−	0.790488i	$-0.290173\pi$
$563$	29.0653	1.22496	0.612478	+	0.790488i	$0.290173\pi$
$564$	0	0
$565$	3.69116	0.155288
$566$	0	0
$567$	9.71364	0.407934
$568$	0	0
$569$	−0.227367	−0.00953174	−0.00476587	−	0.999989i	$-0.501517\pi$
$569$	−0.227367	−0.00953174	−0.00476587	+	0.999989i	$0.501517\pi$
$570$	0	0
$571$	−0.874566	−0.0365994	−0.0182997	−	0.999833i	$-0.505825\pi$
$571$	−0.874566	−0.0365994	−0.0182997	+	0.999833i	$0.505825\pi$
$572$	0	0
$573$	14.0589	0.587319
$574$	0	0
$575$	−17.4524	−0.727814
$576$	0	0
$577$	−29.2997	−1.21976	−0.609882	−	0.792492i	$-0.708783\pi$
$577$	−29.2997	−1.21976	−0.609882	+	0.792492i	$0.708783\pi$
$578$	0	0
$579$	42.3395	1.75957
$580$	0	0
$581$	−13.8457	−0.574415
$582$	0	0
$583$	−12.9140	−0.534842
$584$	0	0
$585$	−0.0406561	−0.00168092
$586$	0	0
$587$	10.9789	0.453149	0.226574	−	0.973994i	$-0.427247\pi$
$587$	10.9789	0.453149	0.226574	+	0.973994i	$0.427247\pi$
$588$	0	0
$589$	22.4484	0.924971
$590$	0	0
$591$	9.10404	0.374490
$592$	0	0
$593$	−42.7926	−1.75728	−0.878641	−	0.477483i	$-0.841549\pi$
$593$	−42.7926	−1.75728	−0.878641	+	0.477483i	$0.841549\pi$
$594$	0	0
$595$	−0.387208	−0.0158740
$596$	0	0
$597$	−2.72720	−0.111617
$598$	0	0
$599$	−0.841737	−0.0343925	−0.0171962	−	0.999852i	$-0.505474\pi$
$599$	−0.841737	−0.0343925	−0.0171962	+	0.999852i	$0.505474\pi$
$600$	0	0
$601$	−19.7479	−0.805536	−0.402768	−	0.915302i	$-0.631952\pi$
$601$	−19.7479	−0.805536	−0.402768	+	0.915302i	$0.631952\pi$
$602$	0	0
$603$	0.630738	0.0256857
$604$	0	0
$605$	−3.57304	−0.145265
$606$	0	0
$607$	0.928802	0.0376989	0.0188495	−	0.999822i	$-0.494000\pi$
$607$	0.928802	0.0376989	0.0188495	+	0.999822i	$0.494000\pi$
$608$	0	0
$609$	−10.1450	−0.411098
$610$	0	0
$611$	−3.65245	−0.147762
$612$	0	0
$613$	20.1162	0.812485	0.406242	−	0.913765i	$-0.366839\pi$
$613$	20.1162	0.812485	0.406242	+	0.913765i	$0.366839\pi$
$614$	0	0
$615$	−8.04548	−0.324425
$616$	0	0
$617$	−13.2513	−0.533479	−0.266739	−	0.963769i	$-0.585946\pi$
$617$	−13.2513	−0.533479	−0.266739	+	0.963769i	$0.585946\pi$
$618$	0	0
$619$	29.2870	1.17714	0.588572	−	0.808445i	$-0.299690\pi$
$619$	29.2870	1.17714	0.588572	+	0.808445i	$0.299690\pi$
$620$	0	0
$621$	−18.5638	−0.744941
$622$	0	0
$623$	0.216863	0.00868844
$624$	0	0
$625$	21.7396	0.869583
$626$	0	0
$627$	−11.9095	−0.475619
$628$	0	0
$629$	−2.16846	−0.0864621
$630$	0	0
$631$	46.5766	1.85418	0.927092	−	0.374834i	$-0.122300\pi$
$631$	46.5766	1.85418	0.927092	+	0.374834i	$0.122300\pi$
$632$	0	0
$633$	−15.5734	−0.618987
$634$	0	0
$635$	−4.62197	−0.183417
$636$	0	0
$637$	4.15087	0.164463
$638$	0	0
$639$	−0.164482	−0.00650679
$640$	0	0
$641$	−5.11165	−0.201898	−0.100949	−	0.994892i	$-0.532188\pi$
$641$	−5.11165	−0.201898	−0.100949	+	0.994892i	$0.532188\pi$
$642$	0	0
$643$	4.32994	0.170756	0.0853781	−	0.996349i	$-0.472790\pi$
$643$	4.32994	0.170756	0.0853781	+	0.996349i	$0.472790\pi$
$644$	0	0
$645$	3.53074	0.139023
$646$	0	0
$647$	12.0401	0.473343	0.236672	−	0.971590i	$-0.423943\pi$
$647$	12.0401	0.473343	0.236672	+	0.971590i	$0.423943\pi$
$648$	0	0
$649$	−12.8695	−0.505171
$650$	0	0
$651$	11.2600	0.441313
$652$	0	0
$653$	39.4366	1.54327	0.771636	−	0.636064i	$-0.219439\pi$
$653$	39.4366	1.54327	0.771636	+	0.636064i	$0.219439\pi$
$654$	0	0
$655$	−3.76733	−0.147202
$656$	0	0
$657$	−0.967408	−0.0377422
$658$	0	0
$659$	−43.8625	−1.70864	−0.854320	−	0.519748i	$-0.826026\pi$
$659$	−43.8625	−1.70864	−0.854320	+	0.519748i	$0.826026\pi$
$660$	0	0
$661$	−5.18941	−0.201845	−0.100922	−	0.994894i	$-0.532179\pi$
$661$	−5.18941	−0.201845	−0.100922	+	0.994894i	$0.532179\pi$
$662$	0	0
$663$	−0.983568	−0.0381986
$664$	0	0
$665$	1.78416	0.0691868
$666$	0	0
$667$	20.1877	0.781672
$668$	0	0
$669$	−42.7785	−1.65391
$670$	0	0
$671$	−7.14094	−0.275673
$672$	0	0
$673$	37.7606	1.45557	0.727783	−	0.685808i	$-0.240551\pi$
$673$	37.7606	1.45557	0.727783	+	0.685808i	$0.240551\pi$
$674$	0	0
$675$	24.2973	0.935205
$676$	0	0
$677$	−36.6368	−1.40806	−0.704032	−	0.710168i	$-0.748619\pi$
$677$	−36.6368	−1.40806	−0.704032	+	0.710168i	$0.748619\pi$
$678$	0	0
$679$	−3.20640	−0.123050
$680$	0	0
$681$	6.27692	0.240532
$682$	0	0
$683$	−31.5096	−1.20568	−0.602841	−	0.797861i	$-0.705965\pi$
$683$	−31.5096	−1.20568	−0.602841	+	0.797861i	$0.705965\pi$
$684$	0	0
$685$	7.91116	0.302270
$686$	0	0
$687$	−33.2549	−1.26876
$688$	0	0
$689$	4.91430	0.187220
$690$	0	0
$691$	−18.9407	−0.720540	−0.360270	−	0.932848i	$-0.617315\pi$
$691$	−18.9407	−0.720540	−0.360270	+	0.932848i	$0.617315\pi$
$692$	0	0
$693$	−0.236179	−0.00897171
$694$	0	0
$695$	−5.28426	−0.200443
$696$	0	0
$697$	−7.69537	−0.291483
$698$	0	0
$699$	21.5434	0.814846
$700$	0	0
$701$	42.8730	1.61929	0.809646	−	0.586919i	$-0.199659\pi$
$701$	42.8730	1.61929	0.809646	+	0.586919i	$0.199659\pi$
$702$	0	0
$703$	9.99174	0.376846
$704$	0	0
$705$	4.32551	0.162908
$706$	0	0
$707$	8.30921	0.312500
$708$	0	0
$709$	35.4811	1.33252	0.666260	−	0.745719i	$-0.267894\pi$
$709$	35.4811	1.33252	0.666260	+	0.745719i	$0.267894\pi$
$710$	0	0
$711$	−0.538630	−0.0202002
$712$	0	0
$713$	−22.4063	−0.839124
$714$	0	0
$715$	−0.606404	−0.0226782
$716$	0	0
$717$	−54.1553	−2.02247
$718$	0	0
$719$	24.4790	0.912913	0.456456	−	0.889746i	$-0.349118\pi$
$719$	24.4790	0.912913	0.456456	+	0.889746i	$0.349118\pi$
$720$	0	0
$721$	−3.38978	−0.126242
$722$	0	0
$723$	−12.8361	−0.477379
$724$	0	0
$725$	−26.4228	−0.981317
$726$	0	0
$727$	28.7796	1.06737	0.533687	−	0.845682i	$-0.320806\pi$
$727$	28.7796	1.06737	0.533687	+	0.845682i	$0.320806\pi$
$728$	0	0
$729$	25.7990	0.955517
$730$	0	0
$731$	3.37710	0.124906
$732$	0	0
$733$	23.0494	0.851350	0.425675	−	0.904876i	$-0.360037\pi$
$733$	23.0494	0.851350	0.425675	+	0.904876i	$0.360037\pi$
$734$	0	0
$735$	−4.91577	−0.181321
$736$	0	0
$737$	9.40776	0.346539
$738$	0	0
$739$	9.02906	0.332139	0.166070	−	0.986114i	$-0.446892\pi$
$739$	9.02906	0.332139	0.166070	+	0.986114i	$0.446892\pi$
$740$	0	0
$741$	4.53205	0.166489
$742$	0	0
$743$	47.5028	1.74271	0.871355	−	0.490653i	$-0.163242\pi$
$743$	47.5028	1.74271	0.871355	+	0.490653i	$0.163242\pi$
$744$	0	0
$745$	−8.86121	−0.324650
$746$	0	0
$747$	1.64674	0.0602510
$748$	0	0
$749$	−19.1968	−0.701434
$750$	0	0
$751$	20.5388	0.749473	0.374737	−	0.927131i	$-0.377733\pi$
$751$	20.5388	0.749473	0.374737	+	0.927131i	$0.377733\pi$
$752$	0	0
$753$	44.6807	1.62825
$754$	0	0
$755$	3.00737	0.109450
$756$	0	0
$757$	36.3077	1.31962	0.659812	−	0.751430i	$-0.270636\pi$
$757$	36.3077	1.31962	0.659812	+	0.751430i	$0.270636\pi$
$758$	0	0
$759$	11.8871	0.431476
$760$	0	0
$761$	23.0597	0.835915	0.417957	−	0.908467i	$-0.362746\pi$
$761$	23.0597	0.835915	0.417957	+	0.908467i	$0.362746\pi$
$762$	0	0
$763$	19.3925	0.702055
$764$	0	0
$765$	0.0460527	0.00166504
$766$	0	0
$767$	4.89737	0.176834
$768$	0	0
$769$	4.40403	0.158813	0.0794066	−	0.996842i	$-0.474697\pi$
$769$	4.40403	0.158813	0.0794066	+	0.996842i	$0.474697\pi$
$770$	0	0
$771$	−46.8375	−1.68681
$772$	0	0
$773$	−25.6259	−0.921699	−0.460849	−	0.887478i	$-0.652455\pi$
$773$	−25.6259	−0.921699	−0.460849	+	0.887478i	$0.652455\pi$
$774$	0	0
$775$	29.3266	1.05344
$776$	0	0
$777$	5.01178	0.179797
$778$	0	0
$779$	35.4584	1.27043
$780$	0	0
$781$	−2.45332	−0.0877867
$782$	0	0
$783$	−28.1055	−1.00441
$784$	0	0
$785$	−7.01029	−0.250208
$786$	0	0
$787$	−46.4038	−1.65412	−0.827059	−	0.562115i	$-0.809988\pi$
$787$	−46.4038	−1.65412	−0.827059	+	0.562115i	$0.809988\pi$
$788$	0	0
$789$	−22.2655	−0.792673
$790$	0	0
$791$	−8.15982	−0.290130
$792$	0	0
$793$	2.71742	0.0964985
$794$	0	0
$795$	−5.81988	−0.206410
$796$	0	0
$797$	−33.6692	−1.19262	−0.596312	−	0.802753i	$-0.703368\pi$
$797$	−33.6692	−1.19262	−0.596312	+	0.802753i	$0.703368\pi$
$798$	0	0
$799$	4.13727	0.146366
$800$	0	0
$801$	−0.0257927	−0.000911340 0
$802$	0	0
$803$	−14.4293	−0.509200
$804$	0	0
$805$	−1.78082	−0.0627655
$806$	0	0
$807$	2.57879	0.0907778
$808$	0	0
$809$	32.2939	1.13539	0.567696	−	0.823238i	$-0.307835\pi$
$809$	32.2939	1.13539	0.567696	+	0.823238i	$0.307835\pi$
$810$	0	0
$811$	−27.4278	−0.963119	−0.481559	−	0.876413i	$-0.659929\pi$
$811$	−27.4278	−0.963119	−0.481559	+	0.876413i	$0.659929\pi$
$812$	0	0
$813$	−9.12310	−0.319961
$814$	0	0
$815$	9.57382	0.335356
$816$	0	0
$817$	−15.5609	−0.544405
$818$	0	0
$819$	0.0898759	0.00314052
$820$	0	0
$821$	−17.5646	−0.613008	−0.306504	−	0.951869i	$-0.599159\pi$
$821$	−17.5646	−0.613008	−0.306504	+	0.951869i	$0.599159\pi$
$822$	0	0
$823$	7.94705	0.277017	0.138508	−	0.990361i	$-0.455769\pi$
$823$	7.94705	0.277017	0.138508	+	0.990361i	$0.455769\pi$
$824$	0	0
$825$	−15.5585	−0.541678
$826$	0	0
$827$	5.16367	0.179558	0.0897792	−	0.995962i	$-0.471384\pi$
$827$	5.16367	0.179558	0.0897792	+	0.995962i	$0.471384\pi$
$828$	0	0
$829$	−20.0606	−0.696734	−0.348367	−	0.937358i	$-0.613264\pi$
$829$	−20.0606	−0.696734	−0.348367	+	0.937358i	$0.613264\pi$
$830$	0	0
$831$	24.7621	0.858989
$832$	0	0
$833$	−4.70185	−0.162909
$834$	0	0
$835$	−7.46115	−0.258204
$836$	0	0
$837$	31.1943	1.07823
$838$	0	0
$839$	−39.3019	−1.35685	−0.678426	−	0.734669i	$-0.737338\pi$
$839$	−39.3019	−1.35685	−0.678426	+	0.734669i	$0.737338\pi$
$840$	0	0
$841$	1.56409	0.0539343
$842$	0	0
$843$	28.9168	0.995946
$844$	0	0
$845$	−5.87518	−0.202112
$846$	0	0
$847$	7.89870	0.271403
$848$	0	0
$849$	−32.7280	−1.12322
$850$	0	0
$851$	−9.97300	−0.341870
$852$	0	0
$853$	41.2623	1.41280	0.706398	−	0.707815i	$-0.250319\pi$
$853$	41.2623	1.41280	0.706398	+	0.707815i	$0.250319\pi$
$854$	0	0
$855$	−0.212200	−0.00725708
$856$	0	0
$857$	4.93880	0.168706	0.0843532	−	0.996436i	$-0.473118\pi$
$857$	4.93880	0.168706	0.0843532	+	0.996436i	$0.473118\pi$
$858$	0	0
$859$	38.1044	1.30011	0.650053	−	0.759889i	$-0.274747\pi$
$859$	38.1044	1.30011	0.650053	+	0.759889i	$0.274747\pi$
$860$	0	0
$861$	17.7857	0.606134
$862$	0	0
$863$	1.71273	0.0583019	0.0291510	−	0.999575i	$-0.490720\pi$
$863$	1.71273	0.0583019	0.0291510	+	0.999575i	$0.490720\pi$
$864$	0	0
$865$	7.70808	0.262082
$866$	0	0
$867$	−28.9307	−0.982537
$868$	0	0
$869$	−8.03393	−0.272532
$870$	0	0
$871$	−3.58004	−0.121305
$872$	0	0
$873$	0.381355	0.0129069
$874$	0	0
$875$	4.76924	0.161230
$876$	0	0
$877$	19.2114	0.648724	0.324362	−	0.945933i	$-0.394850\pi$
$877$	19.2114	0.648724	0.324362	+	0.945933i	$0.394850\pi$
$878$	0	0
$879$	53.9462	1.81956
$880$	0	0
$881$	0.0458679	0.00154533	0.000772664	−	1.00000i	$-0.499754\pi$
$881$	0.0458679	0.00154533	0.000772664		1.00000i	$0.499754\pi$
$882$	0	0
$883$	−42.1763	−1.41935	−0.709674	−	0.704531i	$-0.751158\pi$
$883$	−42.1763	−1.41935	−0.709674	+	0.704531i	$0.751158\pi$
$884$	0	0
$885$	−5.79983	−0.194959
$886$	0	0
$887$	−36.1371	−1.21337	−0.606683	−	0.794944i	$-0.707500\pi$
$887$	−36.1371	−1.21337	−0.606683	+	0.794944i	$0.707500\pi$
$888$	0	0
$889$	10.2175	0.342685
$890$	0	0
$891$	−17.2318	−0.577286
$892$	0	0
$893$	−19.0636	−0.637938
$894$	0	0
$895$	9.59428	0.320701
$896$	0	0
$897$	−4.52355	−0.151037
$898$	0	0
$899$	−33.9230	−1.13140
$900$	0	0
$901$	−5.56661	−0.185451
$902$	0	0
$903$	−7.80520	−0.259741
$904$	0	0
$905$	−4.86985	−0.161879
$906$	0	0
$907$	21.2700	0.706259	0.353129	−	0.935574i	$-0.385118\pi$
$907$	21.2700	0.706259	0.353129	+	0.935574i	$0.385118\pi$
$908$	0	0
$909$	−0.988258	−0.0327785
$910$	0	0
$911$	10.2529	0.339694	0.169847	−	0.985470i	$-0.445673\pi$
$911$	10.2529	0.339694	0.169847	+	0.985470i	$0.445673\pi$
$912$	0	0
$913$	24.5619	0.812880
$914$	0	0
$915$	−3.21818	−0.106390
$916$	0	0
$917$	8.32821	0.275022
$918$	0	0
$919$	39.3681	1.29864	0.649318	−	0.760517i	$-0.275055\pi$
$919$	39.3681	1.29864	0.649318	+	0.760517i	$0.275055\pi$
$920$	0	0
$921$	−5.93888	−0.195693
$922$	0	0
$923$	0.933589	0.0307295
$924$	0	0
$925$	13.0532	0.429187
$926$	0	0
$927$	0.403164	0.0132416
$928$	0	0
$929$	48.6956	1.59765	0.798826	−	0.601562i	$-0.205455\pi$
$929$	48.6956	1.59765	0.798826	+	0.601562i	$0.205455\pi$
$930$	0	0
$931$	21.6650	0.710042
$932$	0	0
$933$	−46.0625	−1.50802
$934$	0	0
$935$	0.686898	0.0224640
$936$	0	0
$937$	−12.4806	−0.407723	−0.203862	−	0.979000i	$-0.565349\pi$
$937$	−12.4806	−0.407723	−0.203862	+	0.979000i	$0.565349\pi$
$938$	0	0
$939$	−50.3621	−1.64351
$940$	0	0
$941$	−44.4531	−1.44913	−0.724565	−	0.689207i	$-0.757959\pi$
$941$	−44.4531	−1.44913	−0.724565	+	0.689207i	$0.757959\pi$
$942$	0	0
$943$	−35.3919	−1.15252
$944$	0	0
$945$	2.47927	0.0806505
$946$	0	0
$947$	7.49160	0.243444	0.121722	−	0.992564i	$-0.461158\pi$
$947$	7.49160	0.243444	0.121722	+	0.992564i	$0.461158\pi$
$948$	0	0
$949$	5.49096	0.178244
$950$	0	0
$951$	−14.6076	−0.473684
$952$	0	0
$953$	−14.6341	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$953$	−14.6341	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$954$	0	0
$955$	3.73629	0.120903
$956$	0	0
$957$	17.9971	0.581762
$958$	0	0
$959$	−17.4887	−0.564741
$960$	0	0
$961$	6.65114	0.214553
$962$	0	0
$963$	2.28317	0.0735743
$964$	0	0
$965$	11.2521	0.362219
$966$	0	0
$967$	−1.24602	−0.0400692	−0.0200346	−	0.999799i	$-0.506378\pi$
$967$	−1.24602	−0.0400692	−0.0200346	+	0.999799i	$0.506378\pi$
$968$	0	0
$969$	−5.13362	−0.164916
$970$	0	0
$971$	−15.4693	−0.496434	−0.248217	−	0.968704i	$-0.579845\pi$
$971$	−15.4693	−0.496434	−0.248217	+	0.968704i	$0.579845\pi$
$972$	0	0
$973$	11.6816	0.374494
$974$	0	0
$975$	5.92066	0.189613
$976$	0	0
$977$	29.2177	0.934757	0.467379	−	0.884057i	$-0.345199\pi$
$977$	29.2177	0.934757	0.467379	+	0.884057i	$0.345199\pi$
$978$	0	0
$979$	−0.384710	−0.0122954
$980$	0	0
$981$	−2.30645	−0.0736393
$982$	0	0
$983$	2.39930	0.0765259	0.0382629	−	0.999268i	$-0.487818\pi$
$983$	2.39930	0.0765259	0.0382629	+	0.999268i	$0.487818\pi$
$984$	0	0
$985$	2.41949	0.0770913
$986$	0	0
$987$	−9.56214	−0.304366
$988$	0	0
$989$	15.5317	0.493878
$990$	0	0
$991$	−51.4773	−1.63523	−0.817614	−	0.575766i	$-0.804704\pi$
$991$	−51.4773	−1.63523	−0.817614	+	0.575766i	$0.804704\pi$
$992$	0	0
$993$	−22.1203	−0.701968
$994$	0	0
$995$	−0.724779	−0.0229771
$996$	0	0
$997$	−41.2171	−1.30536	−0.652680	−	0.757634i	$-0.726355\pi$
$997$	−41.2171	−1.30536	−0.652680	+	0.757634i	$0.726355\pi$
$998$	0	0
$999$	13.8845	0.439286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.13		18
4.3	odd	2		547.2.a.b.1.1	✓	18
12.11	even	2		4923.2.a.l.1.18		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.1	✓	18	4.3	odd	2
4923.2.a.l.1.18		18	12.11	even	2
8752.2.a.s.1.13		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +O(q^{10})\)	\(q+1.76734 q^{3} +0.469688 q^{5} -1.03831 q^{7} +0.123492 q^{9} +1.84194 q^{11} -0.700934 q^{13} +0.830099 q^{15} +0.793975 q^{17} -3.65845 q^{19} -1.83505 q^{21} +3.65159 q^{23} -4.77939 q^{25} -5.08377 q^{27} +5.52848 q^{29} -6.13605 q^{31} +3.25534 q^{33} -0.487683 q^{35} -2.73114 q^{37} -1.23879 q^{39} -9.69220 q^{41} +4.25340 q^{43} +0.0580027 q^{45} +5.21084 q^{47} -5.92191 q^{49} +1.40322 q^{51} -7.01107 q^{53} +0.865138 q^{55} -6.46572 q^{57} -6.98691 q^{59} -3.87686 q^{61} -0.128223 q^{63} -0.329220 q^{65} +5.10753 q^{67} +6.45360 q^{69} -1.33192 q^{71} -7.83377 q^{73} -8.44681 q^{75} -1.91251 q^{77} -4.36166 q^{79} -9.35523 q^{81} +13.3348 q^{83} +0.372921 q^{85} +9.77071 q^{87} -0.208861 q^{89} +0.727788 q^{91} -10.8445 q^{93} -1.71833 q^{95} +3.08809 q^{97} +0.227465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 18 * q + 10 * q^3 - 27 * q^5 + 11 * q^7 + 14 * q^9 - 2 * q^11 - 25 * q^13 - 9 * q^15 - 30 * q^17 - 4 * q^19 - 16 * q^21 + 26 * q^23 + 31 * q^25 + 37 * q^27 - 18 * q^29 + 5 * q^31 - 10 * q^33 + 9 * q^35 - 18 * q^37 - 7 * q^39 - 17 * q^41 - 8 * q^43 - 44 * q^45 + 52 * q^47 + 29 * q^49 - 19 * q^51 - 60 * q^53 - 11 * q^55 + 4 * q^57 + 8 * q^59 - 26 * q^61 + q^63 - 6 * q^65 - 12 * q^67 - 38 * q^69 + q^71 - 2 * q^73 + 17 * q^75 - 73 * q^77 - 18 * q^79 + 18 * q^81 + 43 * q^83 + 51 * q^85 - 3 * q^87 - 28 * q^89 + q^91 - 60 * q^93 + 18 * q^95 - 34 * q^97 - 15 * q^99

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