LMFDB - Embedded newform 8752.2.a.s.1.9

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			1.9
Root			$-1.15793$ 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)

$f(q)$	$=$	$q+0.220625 q^{3} -0.421419 q^{5} +0.645304 q^{7} -2.95132 q^{9} +O(q^{10})$	$q+0.220625 q^{3} -0.421419 q^{5} +0.645304 q^{7} -2.95132 q^{9} +0.640037 q^{11} -4.07156 q^{13} -0.0929756 q^{15} +6.47629 q^{17} +4.15152 q^{19} +0.142370 q^{21} +7.48578 q^{23} -4.82241 q^{25} -1.31301 q^{27} -0.298162 q^{29} -10.2445 q^{31} +0.141208 q^{33} -0.271944 q^{35} -6.30257 q^{37} -0.898287 q^{39} +3.79643 q^{41} -0.987983 q^{43} +1.24375 q^{45} -0.803927 q^{47} -6.58358 q^{49} +1.42883 q^{51} -0.867236 q^{53} -0.269724 q^{55} +0.915929 q^{57} -0.841036 q^{59} -5.88765 q^{61} -1.90450 q^{63} +1.71583 q^{65} -3.61378 q^{67} +1.65155 q^{69} +4.80121 q^{71} +9.79568 q^{73} -1.06394 q^{75} +0.413018 q^{77} -5.23123 q^{79} +8.56429 q^{81} +17.0050 q^{83} -2.72923 q^{85} -0.0657820 q^{87} -9.06607 q^{89} -2.62739 q^{91} -2.26019 q^{93} -1.74953 q^{95} +5.13040 q^{97} -1.88896 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.220625	0.127378	0.0636889	−	0.997970i	$-0.479713\pi$
$3$	0.220625	0.127378	0.0636889	+	0.997970i	$0.479713\pi$
$4$	0	0
$5$	−0.421419	−0.188464	−0.0942322	−	0.995550i	$-0.530040\pi$
$5$	−0.421419	−0.188464	−0.0942322	+	0.995550i	$0.530040\pi$
$6$	0	0
$7$	0.645304	0.243902	0.121951	−	0.992536i	$-0.461085\pi$
$7$	0.645304	0.243902	0.121951	+	0.992536i	$0.461085\pi$
$8$	0	0
$9$	−2.95132	−0.983775
$10$	0	0
$11$	0.640037	0.192978	0.0964892	−	0.995334i	$-0.469239\pi$
$11$	0.640037	0.192978	0.0964892	+	0.995334i	$0.469239\pi$
$12$	0	0
$13$	−4.07156	−1.12925	−0.564623	−	0.825349i	$-0.690979\pi$
$13$	−4.07156	−1.12925	−0.564623	+	0.825349i	$0.690979\pi$
$14$	0	0
$15$	−0.0929756	−0.0240062
$16$	0	0
$17$	6.47629	1.57073	0.785365	−	0.619033i	$-0.212475\pi$
$17$	6.47629	1.57073	0.785365	+	0.619033i	$0.212475\pi$
$18$	0	0
$19$	4.15152	0.952425	0.476212	−	0.879330i	$-0.342009\pi$
$19$	4.15152	0.952425	0.476212	+	0.879330i	$0.342009\pi$
$20$	0	0
$21$	0.142370	0.0310677
$22$	0	0
$23$	7.48578	1.56089	0.780447	−	0.625222i	$-0.214992\pi$
$23$	7.48578	1.56089	0.780447	+	0.625222i	$0.214992\pi$
$24$	0	0
$25$	−4.82241	−0.964481
$26$	0	0
$27$	−1.31301	−0.252689
$28$	0	0
$29$	−0.298162	−0.0553673	−0.0276837	−	0.999617i	$-0.508813\pi$
$29$	−0.298162	−0.0553673	−0.0276837	+	0.999617i	$0.508813\pi$
$30$	0	0
$31$	−10.2445	−1.83996	−0.919981	−	0.391962i	$-0.871797\pi$
$31$	−10.2445	−1.83996	−0.919981	+	0.391962i	$0.871797\pi$
$32$	0	0
$33$	0.141208	0.0245812
$34$	0	0
$35$	−0.271944	−0.0459668
$36$	0	0
$37$	−6.30257	−1.03614	−0.518068	−	0.855339i	$-0.673349\pi$
$37$	−6.30257	−1.03614	−0.518068	+	0.855339i	$0.673349\pi$
$38$	0	0
$39$	−0.898287	−0.143841
$40$	0	0
$41$	3.79643	0.592903	0.296452	−	0.955048i	$-0.404197\pi$
$41$	3.79643	0.592903	0.296452	+	0.955048i	$0.404197\pi$
$42$	0	0
$43$	−0.987983	−0.150666	−0.0753330	−	0.997158i	$-0.524002\pi$
$43$	−0.987983	−0.150666	−0.0753330	+	0.997158i	$0.524002\pi$
$44$	0	0
$45$	1.24375	0.185407
$46$	0	0
$47$	−0.803927	−0.117265	−0.0586324	−	0.998280i	$-0.518674\pi$
$47$	−0.803927	−0.117265	−0.0586324	+	0.998280i	$0.518674\pi$
$48$	0	0
$49$	−6.58358	−0.940512
$50$	0	0
$51$	1.42883	0.200076
$52$	0	0
$53$	−0.867236	−0.119124	−0.0595620	−	0.998225i	$-0.518970\pi$
$53$	−0.867236	−0.119124	−0.0595620	+	0.998225i	$0.518970\pi$
$54$	0	0
$55$	−0.269724	−0.0363696
$56$	0	0
$57$	0.915929	0.121318
$58$	0	0
$59$	−0.841036	−0.109494	−0.0547468	−	0.998500i	$-0.517435\pi$
$59$	−0.841036	−0.109494	−0.0547468	+	0.998500i	$0.517435\pi$
$60$	0	0
$61$	−5.88765	−0.753836	−0.376918	−	0.926247i	$-0.623016\pi$
$61$	−5.88765	−0.753836	−0.376918	+	0.926247i	$0.623016\pi$
$62$	0	0
$63$	−1.90450	−0.239945
$64$	0	0
$65$	1.71583	0.212823
$66$	0	0
$67$	−3.61378	−0.441493	−0.220747	−	0.975331i	$-0.570849\pi$
$67$	−3.61378	−0.441493	−0.220747	+	0.975331i	$0.570849\pi$
$68$	0	0
$69$	1.65155	0.198823
$70$	0	0
$71$	4.80121	0.569799	0.284900	−	0.958557i	$-0.408040\pi$
$71$	4.80121	0.569799	0.284900	+	0.958557i	$0.408040\pi$
$72$	0	0
$73$	9.79568	1.14650	0.573249	−	0.819382i	$-0.305683\pi$
$73$	9.79568	1.14650	0.573249	+	0.819382i	$0.305683\pi$
$74$	0	0
$75$	−1.06394	−0.122854
$76$	0	0
$77$	0.413018	0.0470678
$78$	0	0
$79$	−5.23123	−0.588559	−0.294280	−	0.955719i	$-0.595080\pi$
$79$	−5.23123	−0.588559	−0.294280	+	0.955719i	$0.595080\pi$
$80$	0	0
$81$	8.56429	0.951588
$82$	0	0
$83$	17.0050	1.86654	0.933272	−	0.359170i	$-0.116940\pi$
$83$	17.0050	1.86654	0.933272	+	0.359170i	$0.116940\pi$
$84$	0	0
$85$	−2.72923	−0.296027
$86$	0	0
$87$	−0.0657820	−0.00705257
$88$	0	0
$89$	−9.06607	−0.961001	−0.480501	−	0.876994i	$-0.659545\pi$
$89$	−9.06607	−0.961001	−0.480501	+	0.876994i	$0.659545\pi$
$90$	0	0
$91$	−2.62739	−0.275425
$92$	0	0
$93$	−2.26019	−0.234371
$94$	0	0
$95$	−1.74953	−0.179498
$96$	0	0
$97$	5.13040	0.520913	0.260457	−	0.965486i	$-0.416127\pi$
$97$	5.13040	0.520913	0.260457	+	0.965486i	$0.416127\pi$
$98$	0	0
$99$	−1.88896	−0.189847
$100$	0	0
$101$	−0.429975	−0.0427841	−0.0213921	−	0.999771i	$-0.506810\pi$
$101$	−0.429975	−0.0427841	−0.0213921	+	0.999771i	$0.506810\pi$
$102$	0	0
$103$	−6.28129	−0.618914	−0.309457	−	0.950913i	$-0.600147\pi$
$103$	−6.28129	−0.618914	−0.309457	+	0.950913i	$0.600147\pi$
$104$	0	0
$105$	−0.0599975	−0.00585516
$106$	0	0
$107$	−8.04485	−0.777725	−0.388863	−	0.921296i	$-0.627132\pi$
$107$	−8.04485	−0.777725	−0.388863	+	0.921296i	$0.627132\pi$
$108$	0	0
$109$	−1.54427	−0.147914	−0.0739571	−	0.997261i	$-0.523563\pi$
$109$	−1.54427	−0.147914	−0.0739571	+	0.997261i	$0.523563\pi$
$110$	0	0
$111$	−1.39050	−0.131981
$112$	0	0
$113$	12.8195	1.20596	0.602979	−	0.797757i	$-0.293980\pi$
$113$	12.8195	1.20596	0.602979	+	0.797757i	$0.293980\pi$
$114$	0	0
$115$	−3.15465	−0.294173
$116$	0	0
$117$	12.0165	1.11092
$118$	0	0
$119$	4.17917	0.383104
$120$	0	0
$121$	−10.5904	−0.962759
$122$	0	0
$123$	0.837588	0.0755228
$124$	0	0
$125$	4.13935	0.370235
$126$	0	0
$127$	20.4407	1.81381	0.906907	−	0.421330i	$-0.138437\pi$
$127$	20.4407	1.81381	0.906907	+	0.421330i	$0.138437\pi$
$128$	0	0
$129$	−0.217974	−0.0191915
$130$	0	0
$131$	−0.581501	−0.0508060	−0.0254030	−	0.999677i	$-0.508087\pi$
$131$	−0.581501	−0.0508060	−0.0254030	+	0.999677i	$0.508087\pi$
$132$	0	0
$133$	2.67899	0.232298
$134$	0	0
$135$	0.553328	0.0476229
$136$	0	0
$137$	−12.2771	−1.04890	−0.524452	−	0.851440i	$-0.675730\pi$
$137$	−12.2771	−1.04890	−0.524452	+	0.851440i	$0.675730\pi$
$138$	0	0
$139$	1.35565	0.114985	0.0574923	−	0.998346i	$-0.481690\pi$
$139$	1.35565	0.114985	0.0574923	+	0.998346i	$0.481690\pi$
$140$	0	0
$141$	−0.177366	−0.0149369
$142$	0	0
$143$	−2.60595	−0.217920
$144$	0	0
$145$	0.125651	0.0104348
$146$	0	0
$147$	−1.45250	−0.119800
$148$	0	0
$149$	3.61357	0.296035	0.148017	−	0.988985i	$-0.452711\pi$
$149$	3.61357	0.296035	0.148017	+	0.988985i	$0.452711\pi$
$150$	0	0
$151$	4.04873	0.329481	0.164741	−	0.986337i	$-0.447321\pi$
$151$	4.04873	0.329481	0.164741	+	0.986337i	$0.447321\pi$
$152$	0	0
$153$	−19.1136	−1.54524
$154$	0	0
$155$	4.31722	0.346768
$156$	0	0
$157$	−16.8338	−1.34348	−0.671742	−	0.740786i	$-0.734453\pi$
$157$	−16.8338	−1.34348	−0.671742	+	0.740786i	$0.734453\pi$
$158$	0	0
$159$	−0.191334	−0.0151738
$160$	0	0
$161$	4.83061	0.380705
$162$	0	0
$163$	−23.5478	−1.84441	−0.922203	−	0.386706i	$-0.873613\pi$
$163$	−23.5478	−1.84441	−0.922203	+	0.386706i	$0.873613\pi$
$164$	0	0
$165$	−0.0595078	−0.00463268
$166$	0	0
$167$	11.2055	0.867108	0.433554	−	0.901128i	$-0.357259\pi$
$167$	11.2055	0.867108	0.433554	+	0.901128i	$0.357259\pi$
$168$	0	0
$169$	3.57757	0.275198
$170$	0	0
$171$	−12.2525	−0.936972
$172$	0	0
$173$	−6.09402	−0.463320	−0.231660	−	0.972797i	$-0.574416\pi$
$173$	−6.09402	−0.463320	−0.231660	+	0.972797i	$0.574416\pi$
$174$	0	0
$175$	−3.11192	−0.235239
$176$	0	0
$177$	−0.185553	−0.0139470
$178$	0	0
$179$	11.8489	0.885629	0.442814	−	0.896613i	$-0.353980\pi$
$179$	11.8489	0.885629	0.442814	+	0.896613i	$0.353980\pi$
$180$	0	0
$181$	−11.3326	−0.842346	−0.421173	−	0.906980i	$-0.638381\pi$
$181$	−11.3326	−0.842346	−0.421173	+	0.906980i	$0.638381\pi$
$182$	0	0
$183$	−1.29896	−0.0960220
$184$	0	0
$185$	2.65602	0.195275
$186$	0	0
$187$	4.14506	0.303117
$188$	0	0
$189$	−0.847291	−0.0616313
$190$	0	0
$191$	−6.48714	−0.469393	−0.234697	−	0.972069i	$-0.575410\pi$
$191$	−6.48714	−0.469393	−0.234697	+	0.972069i	$0.575410\pi$
$192$	0	0
$193$	−11.6701	−0.840033	−0.420017	−	0.907516i	$-0.637976\pi$
$193$	−11.6701	−0.840033	−0.420017	+	0.907516i	$0.637976\pi$
$194$	0	0
$195$	0.378555	0.0271089
$196$	0	0
$197$	−19.8657	−1.41537	−0.707685	−	0.706528i	$-0.750260\pi$
$197$	−19.8657	−1.41537	−0.707685	+	0.706528i	$0.750260\pi$
$198$	0	0
$199$	−21.3669	−1.51466	−0.757331	−	0.653031i	$-0.773497\pi$
$199$	−21.3669	−1.51466	−0.757331	+	0.653031i	$0.773497\pi$
$200$	0	0
$201$	−0.797289	−0.0562364
$202$	0	0
$203$	−0.192405	−0.0135042
$204$	0	0
$205$	−1.59989	−0.111741
$206$	0	0
$207$	−22.0930	−1.53557
$208$	0	0
$209$	2.65713	0.183797
$210$	0	0
$211$	−17.2121	−1.18493	−0.592465	−	0.805596i	$-0.701845\pi$
$211$	−17.2121	−1.18493	−0.592465	+	0.805596i	$0.701845\pi$
$212$	0	0
$213$	1.05927	0.0725798
$214$	0	0
$215$	0.416355	0.0283952
$216$	0	0
$217$	−6.61080	−0.448771
$218$	0	0
$219$	2.16117	0.146038
$220$	0	0
$221$	−26.3686	−1.77374
$222$	0	0
$223$	19.3469	1.29557	0.647783	−	0.761825i	$-0.275696\pi$
$223$	19.3469	1.29557	0.647783	+	0.761825i	$0.275696\pi$
$224$	0	0
$225$	14.2325	0.948832
$226$	0	0
$227$	25.0121	1.66011	0.830057	−	0.557679i	$-0.188308\pi$
$227$	25.0121	1.66011	0.830057	+	0.557679i	$0.188308\pi$
$228$	0	0
$229$	−14.3939	−0.951178	−0.475589	−	0.879668i	$-0.657765\pi$
$229$	−14.3939	−0.951178	−0.475589	+	0.879668i	$0.657765\pi$
$230$	0	0
$231$	0.0911222	0.00599540
$232$	0	0
$233$	−19.0426	−1.24752	−0.623762	−	0.781614i	$-0.714397\pi$
$233$	−19.0426	−1.24752	−0.623762	+	0.781614i	$0.714397\pi$
$234$	0	0
$235$	0.338790	0.0221002
$236$	0	0
$237$	−1.15414	−0.0749694
$238$	0	0
$239$	−11.6405	−0.752963	−0.376481	−	0.926424i	$-0.622866\pi$
$239$	−11.6405	−0.752963	−0.376481	+	0.926424i	$0.622866\pi$
$240$	0	0
$241$	−19.3895	−1.24899	−0.624494	−	0.781029i	$-0.714695\pi$
$241$	−19.3895	−1.24899	−0.624494	+	0.781029i	$0.714695\pi$
$242$	0	0
$243$	5.82853	0.373900
$244$	0	0
$245$	2.77445	0.177253
$246$	0	0
$247$	−16.9032	−1.07552
$248$	0	0
$249$	3.75173	0.237756
$250$	0	0
$251$	2.07993	0.131284	0.0656419	−	0.997843i	$-0.479090\pi$
$251$	2.07993	0.131284	0.0656419	+	0.997843i	$0.479090\pi$
$252$	0	0
$253$	4.79118	0.301219
$254$	0	0
$255$	−0.602137	−0.0377073
$256$	0	0
$257$	23.6693	1.47645	0.738225	−	0.674555i	$-0.235664\pi$
$257$	23.6693	1.47645	0.738225	+	0.674555i	$0.235664\pi$
$258$	0	0
$259$	−4.06707	−0.252716
$260$	0	0
$261$	0.879974	0.0544690
$262$	0	0
$263$	−13.5122	−0.833198	−0.416599	−	0.909090i	$-0.636778\pi$
$263$	−13.5122	−0.833198	−0.416599	+	0.909090i	$0.636778\pi$
$264$	0	0
$265$	0.365470	0.0224506
$266$	0	0
$267$	−2.00020	−0.122410
$268$	0	0
$269$	−13.0912	−0.798186	−0.399093	−	0.916910i	$-0.630675\pi$
$269$	−13.0912	−0.798186	−0.399093	+	0.916910i	$0.630675\pi$
$270$	0	0
$271$	−22.6734	−1.37731	−0.688656	−	0.725089i	$-0.741799\pi$
$271$	−22.6734	−1.37731	−0.688656	+	0.725089i	$0.741799\pi$
$272$	0	0
$273$	−0.579668	−0.0350831
$274$	0	0
$275$	−3.08652	−0.186124
$276$	0	0
$277$	11.2897	0.678335	0.339167	−	0.940726i	$-0.389855\pi$
$277$	11.2897	0.678335	0.339167	+	0.940726i	$0.389855\pi$
$278$	0	0
$279$	30.2348	1.81011
$280$	0	0
$281$	−31.5464	−1.88190	−0.940951	−	0.338543i	$-0.890066\pi$
$281$	−31.5464	−1.88190	−0.940951	+	0.338543i	$0.890066\pi$
$282$	0	0
$283$	−12.2749	−0.729666	−0.364833	−	0.931073i	$-0.618874\pi$
$283$	−12.2749	−0.729666	−0.364833	+	0.931073i	$0.618874\pi$
$284$	0	0
$285$	−0.385990	−0.0228641
$286$	0	0
$287$	2.44985	0.144610
$288$	0	0
$289$	24.9423	1.46719
$290$	0	0
$291$	1.13189	0.0663528
$292$	0	0
$293$	−17.1994	−1.00480	−0.502400	−	0.864635i	$-0.667550\pi$
$293$	−17.1994	−1.00480	−0.502400	+	0.864635i	$0.667550\pi$
$294$	0	0
$295$	0.354429	0.0206356
$296$	0	0
$297$	−0.840375	−0.0487635
$298$	0	0
$299$	−30.4788	−1.76263
$300$	0	0
$301$	−0.637549	−0.0367477
$302$	0	0
$303$	−0.0948632	−0.00544975
$304$	0	0
$305$	2.48117	0.142071
$306$	0	0
$307$	−31.1189	−1.77605	−0.888025	−	0.459795i	$-0.847923\pi$
$307$	−31.1189	−1.77605	−0.888025	+	0.459795i	$0.847923\pi$
$308$	0	0
$309$	−1.38581	−0.0788360
$310$	0	0
$311$	−3.57300	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$311$	−3.57300	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$312$	0	0
$313$	25.7114	1.45330	0.726648	−	0.687010i	$-0.241077\pi$
$313$	25.7114	1.45330	0.726648	+	0.687010i	$0.241077\pi$
$314$	0	0
$315$	0.802594	0.0452210
$316$	0	0
$317$	−18.3410	−1.03013	−0.515067	−	0.857150i	$-0.672233\pi$
$317$	−18.3410	−1.03013	−0.515067	+	0.857150i	$0.672233\pi$
$318$	0	0
$319$	−0.190835	−0.0106847
$320$	0	0
$321$	−1.77489	−0.0990650
$322$	0	0
$323$	26.8865	1.49600
$324$	0	0
$325$	19.6347	1.08914
$326$	0	0
$327$	−0.340704	−0.0188410
$328$	0	0
$329$	−0.518777	−0.0286011
$330$	0	0
$331$	6.93883	0.381393	0.190696	−	0.981649i	$-0.438925\pi$
$331$	6.93883	0.381393	0.190696	+	0.981649i	$0.438925\pi$
$332$	0	0
$333$	18.6009	1.01932
$334$	0	0
$335$	1.52292	0.0832057
$336$	0	0
$337$	−17.6976	−0.964047	−0.482024	−	0.876158i	$-0.660098\pi$
$337$	−17.6976	−0.964047	−0.482024	+	0.876158i	$0.660098\pi$
$338$	0	0
$339$	2.82830	0.153612
$340$	0	0
$341$	−6.55685	−0.355073
$342$	0	0
$343$	−8.76554	−0.473295
$344$	0	0
$345$	−0.695995	−0.0374711
$346$	0	0
$347$	−8.98600	−0.482393	−0.241197	−	0.970476i	$-0.577540\pi$
$347$	−8.98600	−0.482393	−0.241197	+	0.970476i	$0.577540\pi$
$348$	0	0
$349$	−29.3335	−1.57018	−0.785092	−	0.619379i	$-0.787385\pi$
$349$	−29.3335	−1.57018	−0.785092	+	0.619379i	$0.787385\pi$
$350$	0	0
$351$	5.34600	0.285348
$352$	0	0
$353$	−31.5641	−1.67999	−0.839995	−	0.542594i	$-0.817442\pi$
$353$	−31.5641	−1.67999	−0.839995	+	0.542594i	$0.817442\pi$
$354$	0	0
$355$	−2.02332	−0.107387
$356$	0	0
$357$	0.922030	0.0487990
$358$	0	0
$359$	27.6250	1.45799	0.728997	−	0.684517i	$-0.239987\pi$
$359$	27.6250	1.45799	0.728997	+	0.684517i	$0.239987\pi$
$360$	0	0
$361$	−1.76485	−0.0928870
$362$	0	0
$363$	−2.33650	−0.122634
$364$	0	0
$365$	−4.12809	−0.216074
$366$	0	0
$367$	−8.43787	−0.440453	−0.220227	−	0.975449i	$-0.570680\pi$
$367$	−8.43787	−0.440453	−0.220227	+	0.975449i	$0.570680\pi$
$368$	0	0
$369$	−11.2045	−0.583284
$370$	0	0
$371$	−0.559631	−0.0290546
$372$	0	0
$373$	10.8965	0.564201	0.282101	−	0.959385i	$-0.408969\pi$
$373$	10.8965	0.564201	0.282101	+	0.959385i	$0.408969\pi$
$374$	0	0
$375$	0.913244	0.0471597
$376$	0	0
$377$	1.21398	0.0625234
$378$	0	0
$379$	−20.0041	−1.02754	−0.513770	−	0.857928i	$-0.671752\pi$
$379$	−20.0041	−1.02754	−0.513770	+	0.857928i	$0.671752\pi$
$380$	0	0
$381$	4.50972	0.231040
$382$	0	0
$383$	−2.92786	−0.149607	−0.0748033	−	0.997198i	$-0.523833\pi$
$383$	−2.92786	−0.149607	−0.0748033	+	0.997198i	$0.523833\pi$
$384$	0	0
$385$	−0.174054	−0.00887061
$386$	0	0
$387$	2.91586	0.148221
$388$	0	0
$389$	−8.39847	−0.425819	−0.212910	−	0.977072i	$-0.568294\pi$
$389$	−8.39847	−0.425819	−0.212910	+	0.977072i	$0.568294\pi$
$390$	0	0
$391$	48.4801	2.45174
$392$	0	0
$393$	−0.128294	−0.00647156
$394$	0	0
$395$	2.20454	0.110922
$396$	0	0
$397$	32.4143	1.62683	0.813413	−	0.581686i	$-0.197607\pi$
$397$	32.4143	1.62683	0.813413	+	0.581686i	$0.197607\pi$
$398$	0	0
$399$	0.591053	0.0295897
$400$	0	0
$401$	5.01237	0.250306	0.125153	−	0.992137i	$-0.460058\pi$
$401$	5.01237	0.250306	0.125153	+	0.992137i	$0.460058\pi$
$402$	0	0
$403$	41.7110	2.07777
$404$	0	0
$405$	−3.60916	−0.179340
$406$	0	0
$407$	−4.03388	−0.199952
$408$	0	0
$409$	25.6745	1.26952	0.634760	−	0.772709i	$-0.281099\pi$
$409$	25.6745	1.26952	0.634760	+	0.772709i	$0.281099\pi$
$410$	0	0
$411$	−2.70863	−0.133607
$412$	0	0
$413$	−0.542724	−0.0267057
$414$	0	0
$415$	−7.16625	−0.351777
$416$	0	0
$417$	0.299090	0.0146465
$418$	0	0
$419$	−2.84015	−0.138751	−0.0693753	−	0.997591i	$-0.522101\pi$
$419$	−2.84015	−0.138751	−0.0693753	+	0.997591i	$0.522101\pi$
$420$	0	0
$421$	32.1637	1.56757	0.783783	−	0.621035i	$-0.213288\pi$
$421$	32.1637	1.56757	0.783783	+	0.621035i	$0.213288\pi$
$422$	0	0
$423$	2.37265	0.115362
$424$	0	0
$425$	−31.2313	−1.51494
$426$	0	0
$427$	−3.79932	−0.183862
$428$	0	0
$429$	−0.574937	−0.0277582
$430$	0	0
$431$	−7.34330	−0.353714	−0.176857	−	0.984237i	$-0.556593\pi$
$431$	−7.34330	−0.353714	−0.176857	+	0.984237i	$0.556593\pi$
$432$	0	0
$433$	−10.5912	−0.508983	−0.254491	−	0.967075i	$-0.581908\pi$
$433$	−10.5912	−0.508983	−0.254491	+	0.967075i	$0.581908\pi$
$434$	0	0
$435$	0.0277218	0.00132916
$436$	0	0
$437$	31.0774	1.48663
$438$	0	0
$439$	10.6116	0.506464	0.253232	−	0.967406i	$-0.418506\pi$
$439$	10.6116	0.506464	0.253232	+	0.967406i	$0.418506\pi$
$440$	0	0
$441$	19.4303	0.925252
$442$	0	0
$443$	31.0558	1.47551	0.737753	−	0.675071i	$-0.235887\pi$
$443$	31.0558	1.47551	0.737753	+	0.675071i	$0.235887\pi$
$444$	0	0
$445$	3.82062	0.181115
$446$	0	0
$447$	0.797243	0.0377083
$448$	0	0
$449$	27.9951	1.32117	0.660586	−	0.750751i	$-0.270308\pi$
$449$	27.9951	1.32117	0.660586	+	0.750751i	$0.270308\pi$
$450$	0	0
$451$	2.42986	0.114418
$452$	0	0
$453$	0.893251	0.0419686
$454$	0	0
$455$	1.10723	0.0519079
$456$	0	0
$457$	26.8641	1.25665	0.628326	−	0.777950i	$-0.283741\pi$
$457$	26.8641	1.25665	0.628326	+	0.777950i	$0.283741\pi$
$458$	0	0
$459$	−8.50343	−0.396906
$460$	0	0
$461$	30.6776	1.42880	0.714399	−	0.699738i	$-0.246700\pi$
$461$	30.6776	1.42880	0.714399	+	0.699738i	$0.246700\pi$
$462$	0	0
$463$	−15.2132	−0.707015	−0.353508	−	0.935432i	$-0.615011\pi$
$463$	−15.2132	−0.707015	−0.353508	+	0.935432i	$0.615011\pi$
$464$	0	0
$465$	0.952487	0.0441705
$466$	0	0
$467$	17.9194	0.829209	0.414604	−	0.910002i	$-0.363920\pi$
$467$	17.9194	0.829209	0.414604	+	0.910002i	$0.363920\pi$
$468$	0	0
$469$	−2.33198	−0.107681
$470$	0	0
$471$	−3.71395	−0.171130
$472$	0	0
$473$	−0.632346	−0.0290753
$474$	0	0
$475$	−20.0203	−0.918596
$476$	0	0
$477$	2.55949	0.117191
$478$	0	0
$479$	−34.8507	−1.59237	−0.796186	−	0.605052i	$-0.793152\pi$
$479$	−34.8507	−1.59237	−0.796186	+	0.605052i	$0.793152\pi$
$480$	0	0
$481$	25.6613	1.17005
$482$	0	0
$483$	1.06575	0.0484934
$484$	0	0
$485$	−2.16205	−0.0981737
$486$	0	0
$487$	−3.51576	−0.159314	−0.0796571	−	0.996822i	$-0.525383\pi$
$487$	−3.51576	−0.159314	−0.0796571	+	0.996822i	$0.525383\pi$
$488$	0	0
$489$	−5.19523	−0.234936
$490$	0	0
$491$	−17.8253	−0.804446	−0.402223	−	0.915542i	$-0.631762\pi$
$491$	−17.8253	−0.804446	−0.402223	+	0.915542i	$0.631762\pi$
$492$	0	0
$493$	−1.93098	−0.0869672
$494$	0	0
$495$	0.796043	0.0357795
$496$	0	0
$497$	3.09824	0.138975
$498$	0	0
$499$	−32.2263	−1.44265	−0.721323	−	0.692599i	$-0.756466\pi$
$499$	−32.2263	−1.44265	−0.721323	+	0.692599i	$0.756466\pi$
$500$	0	0
$501$	2.47221	0.110450
$502$	0	0
$503$	−11.2441	−0.501350	−0.250675	−	0.968071i	$-0.580652\pi$
$503$	−11.2441	−0.501350	−0.250675	+	0.968071i	$0.580652\pi$
$504$	0	0
$505$	0.181200	0.00806329
$506$	0	0
$507$	0.789301	0.0350541
$508$	0	0
$509$	−27.6441	−1.22530	−0.612651	−	0.790353i	$-0.709897\pi$
$509$	−27.6441	−1.22530	−0.612651	+	0.790353i	$0.709897\pi$
$510$	0	0
$511$	6.32119	0.279633
$512$	0	0
$513$	−5.45099	−0.240667
$514$	0	0
$515$	2.64706	0.116643
$516$	0	0
$517$	−0.514543	−0.0226296
$518$	0	0
$519$	−1.34449	−0.0590167
$520$	0	0
$521$	−24.0447	−1.05342	−0.526709	−	0.850046i	$-0.676574\pi$
$521$	−24.0447	−1.05342	−0.526709	+	0.850046i	$0.676574\pi$
$522$	0	0
$523$	6.79138	0.296966	0.148483	−	0.988915i	$-0.452561\pi$
$523$	6.79138	0.296966	0.148483	+	0.988915i	$0.452561\pi$
$524$	0	0
$525$	−0.686566	−0.0299642
$526$	0	0
$527$	−66.3462	−2.89009
$528$	0	0
$529$	33.0370	1.43639
$530$	0	0
$531$	2.48217	0.107717
$532$	0	0
$533$	−15.4574	−0.669534
$534$	0	0
$535$	3.39026	0.146574
$536$	0	0
$537$	2.61416	0.112809
$538$	0	0
$539$	−4.21374	−0.181499
$540$	0	0
$541$	−3.22089	−0.138477	−0.0692385	−	0.997600i	$-0.522057\pi$
$541$	−3.22089	−0.138477	−0.0692385	+	0.997600i	$0.522057\pi$
$542$	0	0
$543$	−2.50025	−0.107296
$544$	0	0
$545$	0.650785	0.0278766
$546$	0	0
$547$	1.00000	0.0427569
$547$	1.00000	0.0427569
$548$	0	0
$549$	17.3764	0.741605
$550$	0	0
$551$	−1.23783	−0.0527332
$552$	0	0
$553$	−3.37573	−0.143551
$554$	0	0
$555$	0.585985	0.0248737
$556$	0	0
$557$	2.24964	0.0953202	0.0476601	−	0.998864i	$-0.484824\pi$
$557$	2.24964	0.0953202	0.0476601	+	0.998864i	$0.484824\pi$
$558$	0	0
$559$	4.02263	0.170139
$560$	0	0
$561$	0.914504	0.0386104
$562$	0	0
$563$	−13.2181	−0.557077	−0.278539	−	0.960425i	$-0.589850\pi$
$563$	−13.2181	−0.557077	−0.278539	+	0.960425i	$0.589850\pi$
$564$	0	0
$565$	−5.40238	−0.227280
$566$	0	0
$567$	5.52657	0.232094
$568$	0	0
$569$	9.91716	0.415749	0.207875	−	0.978156i	$-0.433345\pi$
$569$	9.91716	0.415749	0.207875	+	0.978156i	$0.433345\pi$
$570$	0	0
$571$	−37.4262	−1.56624	−0.783119	−	0.621872i	$-0.786373\pi$
$571$	−37.4262	−1.56624	−0.783119	+	0.621872i	$0.786373\pi$
$572$	0	0
$573$	−1.43123	−0.0597903
$574$	0	0
$575$	−36.0995	−1.50545
$576$	0	0
$577$	−3.61492	−0.150491	−0.0752455	−	0.997165i	$-0.523974\pi$
$577$	−3.61492	−0.150491	−0.0752455	+	0.997165i	$0.523974\pi$
$578$	0	0
$579$	−2.57472	−0.107002
$580$	0	0
$581$	10.9734	0.455254
$582$	0	0
$583$	−0.555063	−0.0229884
$584$	0	0
$585$	−5.06398	−0.209370
$586$	0	0
$587$	22.0003	0.908048	0.454024	−	0.890989i	$-0.349988\pi$
$587$	22.0003	0.908048	0.454024	+	0.890989i	$0.349988\pi$
$588$	0	0
$589$	−42.5302	−1.75243
$590$	0	0
$591$	−4.38286	−0.180287
$592$	0	0
$593$	−23.0882	−0.948117	−0.474059	−	0.880493i	$-0.657212\pi$
$593$	−23.0882	−0.948117	−0.474059	+	0.880493i	$0.657212\pi$
$594$	0	0
$595$	−1.76118	−0.0722015
$596$	0	0
$597$	−4.71408	−0.192934
$598$	0	0
$599$	−0.624248	−0.0255061	−0.0127530	−	0.999919i	$-0.504060\pi$
$599$	−0.624248	−0.0255061	−0.0127530	+	0.999919i	$0.504060\pi$
$600$	0	0
$601$	26.0763	1.06368	0.531838	−	0.846846i	$-0.321501\pi$
$601$	26.0763	1.06368	0.531838	+	0.846846i	$0.321501\pi$
$602$	0	0
$603$	10.6654	0.434330
$604$	0	0
$605$	4.46298	0.181446
$606$	0	0
$607$	4.24424	0.172268	0.0861341	−	0.996284i	$-0.472549\pi$
$607$	4.24424	0.172268	0.0861341	+	0.996284i	$0.472549\pi$
$608$	0	0
$609$	−0.0424494	−0.00172014
$610$	0	0
$611$	3.27323	0.132421
$612$	0	0
$613$	−43.4402	−1.75453	−0.877267	−	0.480003i	$-0.840636\pi$
$613$	−43.4402	−1.75453	−0.877267	+	0.480003i	$0.840636\pi$
$614$	0	0
$615$	−0.352976	−0.0142334
$616$	0	0
$617$	31.9260	1.28529	0.642646	−	0.766163i	$-0.277836\pi$
$617$	31.9260	1.28529	0.642646	+	0.766163i	$0.277836\pi$
$618$	0	0
$619$	−41.8091	−1.68045	−0.840225	−	0.542237i	$-0.817577\pi$
$619$	−41.8091	−1.68045	−0.840225	+	0.542237i	$0.817577\pi$
$620$	0	0
$621$	−9.82891	−0.394421
$622$	0	0
$623$	−5.85037	−0.234390
$624$	0	0
$625$	22.3676	0.894705
$626$	0	0
$627$	0.586229	0.0234117
$628$	0	0
$629$	−40.8172	−1.62749
$630$	0	0
$631$	−10.6280	−0.423094	−0.211547	−	0.977368i	$-0.567850\pi$
$631$	−10.6280	−0.423094	−0.211547	+	0.977368i	$0.567850\pi$
$632$	0	0
$633$	−3.79742	−0.150934
$634$	0	0
$635$	−8.61409	−0.341840
$636$	0	0
$637$	26.8054	1.06207
$638$	0	0
$639$	−14.1699	−0.560554
$640$	0	0
$641$	3.46304	0.136782	0.0683909	−	0.997659i	$-0.478213\pi$
$641$	3.46304	0.136782	0.0683909	+	0.997659i	$0.478213\pi$
$642$	0	0
$643$	−12.0057	−0.473459	−0.236730	−	0.971576i	$-0.576076\pi$
$643$	−12.0057	−0.473459	−0.236730	+	0.971576i	$0.576076\pi$
$644$	0	0
$645$	0.0918583	0.00361692
$646$	0	0
$647$	17.6257	0.692937	0.346468	−	0.938062i	$-0.387381\pi$
$647$	17.6257	0.692937	0.346468	+	0.938062i	$0.387381\pi$
$648$	0	0
$649$	−0.538294	−0.0211299
$650$	0	0
$651$	−1.45851	−0.0571634
$652$	0	0
$653$	47.1062	1.84341	0.921704	−	0.387895i	$-0.126798\pi$
$653$	47.1062	1.84341	0.921704	+	0.387895i	$0.126798\pi$
$654$	0	0
$655$	0.245056	0.00957512
$656$	0	0
$657$	−28.9102	−1.12790
$658$	0	0
$659$	−11.8304	−0.460848	−0.230424	−	0.973090i	$-0.574011\pi$
$659$	−11.8304	−0.460848	−0.230424	+	0.973090i	$0.574011\pi$
$660$	0	0
$661$	11.7551	0.457219	0.228610	−	0.973518i	$-0.426582\pi$
$661$	11.7551	0.457219	0.228610	+	0.973518i	$0.426582\pi$
$662$	0	0
$663$	−5.81756	−0.225935
$664$	0	0
$665$	−1.12898	−0.0437800
$666$	0	0
$667$	−2.23198	−0.0864226
$668$	0	0
$669$	4.26842	0.165027
$670$	0	0
$671$	−3.76831	−0.145474
$672$	0	0
$673$	−24.5574	−0.946618	−0.473309	−	0.880897i	$-0.656941\pi$
$673$	−24.5574	−0.946618	−0.473309	+	0.880897i	$0.656941\pi$
$674$	0	0
$675$	6.33187	0.243714
$676$	0	0
$677$	−39.3171	−1.51108	−0.755540	−	0.655103i	$-0.772625\pi$
$677$	−39.3171	−1.51108	−0.755540	+	0.655103i	$0.772625\pi$
$678$	0	0
$679$	3.31067	0.127052
$680$	0	0
$681$	5.51830	0.211462
$682$	0	0
$683$	6.62542	0.253515	0.126757	−	0.991934i	$-0.459543\pi$
$683$	6.62542	0.253515	0.126757	+	0.991934i	$0.459543\pi$
$684$	0	0
$685$	5.17381	0.197681
$686$	0	0
$687$	−3.17566	−0.121159
$688$	0	0
$689$	3.53100	0.134520
$690$	0	0
$691$	21.9346	0.834430	0.417215	−	0.908808i	$-0.363006\pi$
$691$	21.9346	0.834430	0.417215	+	0.908808i	$0.363006\pi$
$692$	0	0
$693$	−1.21895	−0.0463041
$694$	0	0
$695$	−0.571297	−0.0216705
$696$	0	0
$697$	24.5868	0.931291
$698$	0	0
$699$	−4.20128	−0.158907
$700$	0	0
$701$	−18.0547	−0.681917	−0.340958	−	0.940078i	$-0.610752\pi$
$701$	−18.0547	−0.681917	−0.340958	+	0.940078i	$0.610752\pi$
$702$	0	0
$703$	−26.1653	−0.986842
$704$	0	0
$705$	0.0747456	0.00281508
$706$	0	0
$707$	−0.277465	−0.0104351
$708$	0	0
$709$	14.5503	0.546448	0.273224	−	0.961950i	$-0.411910\pi$
$709$	14.5503	0.546448	0.273224	+	0.961950i	$0.411910\pi$
$710$	0	0
$711$	15.4390	0.579010
$712$	0	0
$713$	−76.6880	−2.87199
$714$	0	0
$715$	1.09820	0.0410702
$716$	0	0
$717$	−2.56819	−0.0959107
$718$	0	0
$719$	−2.03755	−0.0759879	−0.0379939	−	0.999278i	$-0.512097\pi$
$719$	−2.03755	−0.0759879	−0.0379939	+	0.999278i	$0.512097\pi$
$720$	0	0
$721$	−4.05334	−0.150954
$722$	0	0
$723$	−4.27781	−0.159093
$724$	0	0
$725$	1.43786	0.0534008
$726$	0	0
$727$	36.1230	1.33973	0.669865	−	0.742483i	$-0.266352\pi$
$727$	36.1230	1.33973	0.669865	+	0.742483i	$0.266352\pi$
$728$	0	0
$729$	−24.4070	−0.903961
$730$	0	0
$731$	−6.39846	−0.236656
$732$	0	0
$733$	−19.5342	−0.721512	−0.360756	−	0.932660i	$-0.617481\pi$
$733$	−19.5342	−0.721512	−0.360756	+	0.932660i	$0.617481\pi$
$734$	0	0
$735$	0.612112	0.0225781
$736$	0	0
$737$	−2.31295	−0.0851986
$738$	0	0
$739$	18.9474	0.696990	0.348495	−	0.937311i	$-0.386693\pi$
$739$	18.9474	0.696990	0.348495	+	0.937311i	$0.386693\pi$
$740$	0	0
$741$	−3.72926	−0.136998
$742$	0	0
$743$	−19.7628	−0.725028	−0.362514	−	0.931978i	$-0.618082\pi$
$743$	−19.7628	−0.725028	−0.362514	+	0.931978i	$0.618082\pi$
$744$	0	0
$745$	−1.52283	−0.0557920
$746$	0	0
$747$	−50.1874	−1.83626
$748$	0	0
$749$	−5.19138	−0.189689
$750$	0	0
$751$	−28.5750	−1.04272	−0.521359	−	0.853337i	$-0.674575\pi$
$751$	−28.5750	−1.04272	−0.521359	+	0.853337i	$0.674575\pi$
$752$	0	0
$753$	0.458884	0.0167227
$754$	0	0
$755$	−1.70621	−0.0620955
$756$	0	0
$757$	−27.6434	−1.00472	−0.502358	−	0.864660i	$-0.667534\pi$
$757$	−27.6434	−1.00472	−0.502358	+	0.864660i	$0.667534\pi$
$758$	0	0
$759$	1.05705	0.0383686
$760$	0	0
$761$	20.4285	0.740531	0.370266	−	0.928926i	$-0.379267\pi$
$761$	20.4285	0.740531	0.370266	+	0.928926i	$0.379267\pi$
$762$	0	0
$763$	−0.996524	−0.0360766
$764$	0	0
$765$	8.05485	0.291224
$766$	0	0
$767$	3.42432	0.123645
$768$	0	0
$769$	14.6134	0.526974	0.263487	−	0.964663i	$-0.415127\pi$
$769$	14.6134	0.526974	0.263487	+	0.964663i	$0.415127\pi$
$770$	0	0
$771$	5.22203	0.188067
$772$	0	0
$773$	−8.68704	−0.312451	−0.156225	−	0.987721i	$-0.549933\pi$
$773$	−8.68704	−0.312451	−0.156225	+	0.987721i	$0.549933\pi$
$774$	0	0
$775$	49.4030	1.77461
$776$	0	0
$777$	−0.897297	−0.0321904
$778$	0	0
$779$	15.7610	0.564696
$780$	0	0
$781$	3.07296	0.109959
$782$	0	0
$783$	0.391490	0.0139907
$784$	0	0
$785$	7.09409	0.253199
$786$	0	0
$787$	50.1106	1.78625	0.893125	−	0.449808i	$-0.148507\pi$
$787$	50.1106	1.78625	0.893125	+	0.449808i	$0.148507\pi$
$788$	0	0
$789$	−2.98113	−0.106131
$790$	0	0
$791$	8.27247	0.294135
$792$	0	0
$793$	23.9719	0.851267
$794$	0	0
$795$	0.0806318	0.00285971
$796$	0	0
$797$	−23.7864	−0.842557	−0.421279	−	0.906931i	$-0.638419\pi$
$797$	−23.7864	−0.842557	−0.421279	+	0.906931i	$0.638419\pi$
$798$	0	0
$799$	−5.20646	−0.184191
$800$	0	0
$801$	26.7569	0.945409
$802$	0	0
$803$	6.26960	0.221249
$804$	0	0
$805$	−2.03571	−0.0717494
$806$	0	0
$807$	−2.88825	−0.101671
$808$	0	0
$809$	37.8000	1.32898	0.664489	−	0.747298i	$-0.268649\pi$
$809$	37.8000	1.32898	0.664489	+	0.747298i	$0.268649\pi$
$810$	0	0
$811$	16.7465	0.588049	0.294024	−	0.955798i	$-0.405005\pi$
$811$	16.7465	0.588049	0.294024	+	0.955798i	$0.405005\pi$
$812$	0	0
$813$	−5.00232	−0.175439
$814$	0	0
$815$	9.92350	0.347605
$816$	0	0
$817$	−4.10163	−0.143498
$818$	0	0
$819$	7.75428	0.270957
$820$	0	0
$821$	−6.67989	−0.233130	−0.116565	−	0.993183i	$-0.537188\pi$
$821$	−6.67989	−0.233130	−0.116565	+	0.993183i	$0.537188\pi$
$822$	0	0
$823$	42.3883	1.47756	0.738782	−	0.673944i	$-0.235401\pi$
$823$	42.3883	1.47756	0.738782	+	0.673944i	$0.235401\pi$
$824$	0	0
$825$	−0.680963	−0.0237081
$826$	0	0
$827$	−36.9785	−1.28587	−0.642934	−	0.765922i	$-0.722283\pi$
$827$	−36.9785	−1.28587	−0.642934	+	0.765922i	$0.722283\pi$
$828$	0	0
$829$	−10.0901	−0.350442	−0.175221	−	0.984529i	$-0.556064\pi$
$829$	−10.0901	−0.350442	−0.175221	+	0.984529i	$0.556064\pi$
$830$	0	0
$831$	2.49080	0.0864048
$832$	0	0
$833$	−42.6372	−1.47729
$834$	0	0
$835$	−4.72222	−0.163419
$836$	0	0
$837$	13.4511	0.464938
$838$	0	0
$839$	−1.93149	−0.0666825	−0.0333412	−	0.999444i	$-0.510615\pi$
$839$	−1.93149	−0.0666825	−0.0333412	+	0.999444i	$0.510615\pi$
$840$	0	0
$841$	−28.9111	−0.996934
$842$	0	0
$843$	−6.95993	−0.239713
$844$	0	0
$845$	−1.50766	−0.0518650
$846$	0	0
$847$	−6.83400	−0.234819
$848$	0	0
$849$	−2.70815	−0.0929433
$850$	0	0
$851$	−47.1797	−1.61730
$852$	0	0
$853$	−33.0856	−1.13283	−0.566414	−	0.824121i	$-0.691670\pi$
$853$	−33.0856	−1.13283	−0.566414	+	0.824121i	$0.691670\pi$
$854$	0	0
$855$	5.16344	0.176586
$856$	0	0
$857$	2.23398	0.0763113	0.0381557	−	0.999272i	$-0.487852\pi$
$857$	2.23398	0.0763113	0.0381557	+	0.999272i	$0.487852\pi$
$858$	0	0
$859$	53.8019	1.83570	0.917848	−	0.396932i	$-0.129925\pi$
$859$	53.8019	1.83570	0.917848	+	0.396932i	$0.129925\pi$
$860$	0	0
$861$	0.540499	0.0184202
$862$	0	0
$863$	−22.7366	−0.773962	−0.386981	−	0.922088i	$-0.626482\pi$
$863$	−22.7366	−0.773962	−0.386981	+	0.922088i	$0.626482\pi$
$864$	0	0
$865$	2.56814	0.0873193
$866$	0	0
$867$	5.50289	0.186888
$868$	0	0
$869$	−3.34818	−0.113579
$870$	0	0
$871$	14.7137	0.498554
$872$	0	0
$873$	−15.1415	−0.512462
$874$	0	0
$875$	2.67114	0.0903010
$876$	0	0
$877$	−19.3780	−0.654349	−0.327175	−	0.944964i	$-0.606097\pi$
$877$	−19.3780	−0.654349	−0.327175	+	0.944964i	$0.606097\pi$
$878$	0	0
$879$	−3.79462	−0.127989
$880$	0	0
$881$	−33.8051	−1.13892	−0.569461	−	0.822018i	$-0.692848\pi$
$881$	−33.8051	−1.13892	−0.569461	+	0.822018i	$0.692848\pi$
$882$	0	0
$883$	0.646989	0.0217729	0.0108864	−	0.999941i	$-0.496535\pi$
$883$	0.646989	0.0217729	0.0108864	+	0.999941i	$0.496535\pi$
$884$	0	0
$885$	0.0781958	0.00262852
$886$	0	0
$887$	52.9615	1.77827	0.889136	−	0.457643i	$-0.151306\pi$
$887$	52.9615	1.77827	0.889136	+	0.457643i	$0.151306\pi$
$888$	0	0
$889$	13.1904	0.442393
$890$	0	0
$891$	5.48146	0.183636
$892$	0	0
$893$	−3.33752	−0.111686
$894$	0	0
$895$	−4.99336	−0.166910
$896$	0	0
$897$	−6.72438	−0.224521
$898$	0	0
$899$	3.05452	0.101874
$900$	0	0
$901$	−5.61647	−0.187112
$902$	0	0
$903$	−0.140659	−0.00468085
$904$	0	0
$905$	4.77578	0.158752
$906$	0	0
$907$	17.7548	0.589539	0.294770	−	0.955568i	$-0.404757\pi$
$907$	17.7548	0.589539	0.294770	+	0.955568i	$0.404757\pi$
$908$	0	0
$909$	1.26900	0.0420899
$910$	0	0
$911$	49.6346	1.64447	0.822233	−	0.569150i	$-0.192728\pi$
$911$	49.6346	1.64447	0.822233	+	0.569150i	$0.192728\pi$
$912$	0	0
$913$	10.8839	0.360203
$914$	0	0
$915$	0.547408	0.0180967
$916$	0	0
$917$	−0.375245	−0.0123917
$918$	0	0
$919$	−34.5625	−1.14011	−0.570055	−	0.821606i	$-0.693078\pi$
$919$	−34.5625	−1.14011	−0.570055	+	0.821606i	$0.693078\pi$
$920$	0	0
$921$	−6.86561	−0.226229
$922$	0	0
$923$	−19.5484	−0.643444
$924$	0	0
$925$	30.3935	0.999334
$926$	0	0
$927$	18.5381	0.608872
$928$	0	0
$929$	37.0060	1.21413	0.607063	−	0.794654i	$-0.292348\pi$
$929$	37.0060	1.21413	0.607063	+	0.794654i	$0.292348\pi$
$930$	0	0
$931$	−27.3319	−0.895767
$932$	0	0
$933$	−0.788293	−0.0258075
$934$	0	0
$935$	−1.74681	−0.0571268
$936$	0	0
$937$	13.3304	0.435487	0.217743	−	0.976006i	$-0.430130\pi$
$937$	13.3304	0.435487	0.217743	+	0.976006i	$0.430130\pi$
$938$	0	0
$939$	5.67258	0.185118
$940$	0	0
$941$	−2.02721	−0.0660850	−0.0330425	−	0.999454i	$-0.510520\pi$
$941$	−2.02721	−0.0660850	−0.0330425	+	0.999454i	$0.510520\pi$
$942$	0	0
$943$	28.4193	0.925459
$944$	0	0
$945$	0.357065	0.0116153
$946$	0	0
$947$	40.3563	1.31140	0.655701	−	0.755020i	$-0.272373\pi$
$947$	40.3563	1.31140	0.655701	+	0.755020i	$0.272373\pi$
$948$	0	0
$949$	−39.8836	−1.29468
$950$	0	0
$951$	−4.04648	−0.131216
$952$	0	0
$953$	22.7099	0.735646	0.367823	−	0.929896i	$-0.380103\pi$
$953$	22.7099	0.735646	0.367823	+	0.929896i	$0.380103\pi$
$954$	0	0
$955$	2.73381	0.0884639
$956$	0	0
$957$	−0.0421029	−0.00136099
$958$	0	0
$959$	−7.92246	−0.255830
$960$	0	0
$961$	73.9494	2.38546
$962$	0	0
$963$	23.7430	0.765107
$964$	0	0
$965$	4.91801	0.158316
$966$	0	0
$967$	14.9197	0.479785	0.239892	−	0.970799i	$-0.422888\pi$
$967$	14.9197	0.479785	0.239892	+	0.970799i	$0.422888\pi$
$968$	0	0
$969$	5.93182	0.190558
$970$	0	0
$971$	−49.3577	−1.58396	−0.791982	−	0.610544i	$-0.790951\pi$
$971$	−49.3577	−1.58396	−0.791982	+	0.610544i	$0.790951\pi$
$972$	0	0
$973$	0.874806	0.0280450
$974$	0	0
$975$	4.33190	0.138732
$976$	0	0
$977$	4.29541	0.137422	0.0687112	−	0.997637i	$-0.478111\pi$
$977$	4.29541	0.137422	0.0687112	+	0.997637i	$0.478111\pi$
$978$	0	0
$979$	−5.80262	−0.185453
$980$	0	0
$981$	4.55764	0.145514
$982$	0	0
$983$	44.2697	1.41198	0.705992	−	0.708220i	$-0.250501\pi$
$983$	44.2697	1.41198	0.705992	+	0.708220i	$0.250501\pi$
$984$	0	0
$985$	8.37177	0.266747
$986$	0	0
$987$	−0.114455	−0.00364315
$988$	0	0
$989$	−7.39583	−0.235174
$990$	0	0
$991$	4.60210	0.146190	0.0730952	−	0.997325i	$-0.476712\pi$
$991$	4.60210	0.146190	0.0730952	+	0.997325i	$0.476712\pi$
$992$	0	0
$993$	1.53088	0.0485810
$994$	0	0
$995$	9.00444	0.285460
$996$	0	0
$997$	−4.30801	−0.136436	−0.0682180	−	0.997670i	$-0.521731\pi$
$997$	−4.30801	−0.136436	−0.0682180	+	0.997670i	$0.521731\pi$
$998$	0	0
$999$	8.27534	0.261820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.s.1.9		18
4.3	odd	2		547.2.a.b.1.14	✓	18
12.11	even	2		4923.2.a.l.1.5		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.14	✓	18	4.3	odd	2
4923.2.a.l.1.5		18	12.11	even	2
8752.2.a.s.1.9		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)

\(f(q)\)	\(=\)	\(q+0.220625 q^{3} -0.421419 q^{5} +0.645304 q^{7} -2.95132 q^{9} +O(q^{10})\)	\(q+0.220625 q^{3} -0.421419 q^{5} +0.645304 q^{7} -2.95132 q^{9} +0.640037 q^{11} -4.07156 q^{13} -0.0929756 q^{15} +6.47629 q^{17} +4.15152 q^{19} +0.142370 q^{21} +7.48578 q^{23} -4.82241 q^{25} -1.31301 q^{27} -0.298162 q^{29} -10.2445 q^{31} +0.141208 q^{33} -0.271944 q^{35} -6.30257 q^{37} -0.898287 q^{39} +3.79643 q^{41} -0.987983 q^{43} +1.24375 q^{45} -0.803927 q^{47} -6.58358 q^{49} +1.42883 q^{51} -0.867236 q^{53} -0.269724 q^{55} +0.915929 q^{57} -0.841036 q^{59} -5.88765 q^{61} -1.90450 q^{63} +1.71583 q^{65} -3.61378 q^{67} +1.65155 q^{69} +4.80121 q^{71} +9.79568 q^{73} -1.06394 q^{75} +0.413018 q^{77} -5.23123 q^{79} +8.56429 q^{81} +17.0050 q^{83} -2.72923 q^{85} -0.0657820 q^{87} -9.06607 q^{89} -2.62739 q^{91} -2.26019 q^{93} -1.74953 q^{95} +5.13040 q^{97} -1.88896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) 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

Introduction

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