LMFDB - Embedded newform 8512.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.10463409.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 11x^{3} + 17x - 4 $ x^5 - 11x^3 + 17x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.05218$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05218 q^{3} -0.712910 q^{5} +1.00000 q^{7} +6.31583 q^{9} +O(q^{10})$	$q-3.05218 q^{3} -0.712910 q^{5} +1.00000 q^{7} +6.31583 q^{9} -0.888842 q^{11} +0.163342 q^{13} +2.17593 q^{15} -4.19208 q^{17} +1.00000 q^{19} -3.05218 q^{21} +2.16334 q^{23} -4.49176 q^{25} -10.1205 q^{27} -9.06833 q^{29} +6.19208 q^{31} +2.71291 q^{33} -0.712910 q^{35} -9.39146 q^{37} -0.498549 q^{39} +5.53814 q^{41} +7.80230 q^{43} -4.50262 q^{45} +1.71408 q^{47} +1.00000 q^{49} +12.7950 q^{51} +7.33197 q^{53} +0.633664 q^{55} -3.05218 q^{57} +11.0971 q^{59} -4.96447 q^{61} +6.31583 q^{63} -0.116448 q^{65} +7.01376 q^{67} -6.60292 q^{69} -9.06884 q^{71} +11.2346 q^{73} +13.7097 q^{75} -0.888842 q^{77} -9.55480 q^{79} +11.9422 q^{81} -1.92437 q^{83} +2.98857 q^{85} +27.6782 q^{87} -3.98178 q^{89} +0.163342 q^{91} -18.8994 q^{93} -0.712910 q^{95} -14.1703 q^{97} -5.61377 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 3 * q^5 + 5 * q^7 + 7 * q^9	$ 5 q - 3 q^{5} + 5 q^{7} + 7 q^{9} + 6 q^{11} - 4 q^{13} + q^{15} + 9 q^{17} + 5 q^{19} + 6 q^{23} + 12 q^{25} - 10 q^{29} + q^{31} + 13 q^{33} - 3 q^{35} - 17 q^{37} - 9 q^{39} + 24 q^{41} + 15 q^{43} - 3 q^{45} - 5 q^{47} + 5 q^{49} + 10 q^{51} - 8 q^{53} + 19 q^{55} - 5 q^{59} - 9 q^{61} + 7 q^{63} + 24 q^{65} + 17 q^{67} - 9 q^{69} + q^{71} - 17 q^{73} + 12 q^{75} + 6 q^{77} - 13 q^{79} + 21 q^{81} - 15 q^{83} + 7 q^{85} + 20 q^{87} + 29 q^{89} - 4 q^{91} - 10 q^{93} - 3 q^{95} - 7 q^{97} - 19 q^{99}+O(q^{100}) $ 5 * q - 3 * q^5 + 5 * q^7 + 7 * q^9 + 6 * q^11 - 4 * q^13 + q^15 + 9 * q^17 + 5 * q^19 + 6 * q^23 + 12 * q^25 - 10 * q^29 + q^31 + 13 * q^33 - 3 * q^35 - 17 * q^37 - 9 * q^39 + 24 * q^41 + 15 * q^43 - 3 * q^45 - 5 * q^47 + 5 * q^49 + 10 * q^51 - 8 * q^53 + 19 * q^55 - 5 * q^59 - 9 * q^61 + 7 * q^63 + 24 * q^65 + 17 * q^67 - 9 * q^69 + q^71 - 17 * q^73 + 12 * q^75 + 6 * q^77 - 13 * q^79 + 21 * q^81 - 15 * q^83 + 7 * q^85 + 20 * q^87 + 29 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 7 * q^97 - 19 * 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
$3$	−3.05218	−1.76218	−0.881090	−	0.472949i	$-0.843189\pi$
$3$	−3.05218	−1.76218	−0.881090	+	0.472949i	$0.843189\pi$
$4$	0	0
$5$	−0.712910	−0.318823	−0.159412	−	0.987212i	$-0.550960\pi$
$5$	−0.712910	−0.318823	−0.159412	+	0.987212i	$0.550960\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.31583	2.10528
$10$	0	0
$11$	−0.888842	−0.267996	−0.133998	−	0.990982i	$-0.542782\pi$
$11$	−0.888842	−0.267996	−0.133998	+	0.990982i	$0.542782\pi$
$12$	0	0
$13$	0.163342	0.0453029	0.0226514	−	0.999743i	$-0.492789\pi$
$13$	0.163342	0.0453029	0.0226514	+	0.999743i	$0.492789\pi$
$14$	0	0
$15$	2.17593	0.561823
$16$	0	0
$17$	−4.19208	−1.01673	−0.508364	−	0.861142i	$-0.669750\pi$
$17$	−4.19208	−1.01673	−0.508364	+	0.861142i	$0.669750\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.05218	−0.666041
$22$	0	0
$23$	2.16334	0.451088	0.225544	−	0.974233i	$-0.427584\pi$
$23$	2.16334	0.451088	0.225544	+	0.974233i	$0.427584\pi$
$24$	0	0
$25$	−4.49176	−0.898352
$26$	0	0
$27$	−10.1205	−1.94769
$28$	0	0
$29$	−9.06833	−1.68395	−0.841973	−	0.539519i	$-0.818606\pi$
$29$	−9.06833	−1.68395	−0.841973	+	0.539519i	$0.818606\pi$
$30$	0	0
$31$	6.19208	1.11213	0.556065	−	0.831139i	$-0.312311\pi$
$31$	6.19208	1.11213	0.556065	+	0.831139i	$0.312311\pi$
$32$	0	0
$33$	2.71291	0.472257
$34$	0	0
$35$	−0.712910	−0.120504
$36$	0	0
$37$	−9.39146	−1.54395	−0.771973	−	0.635655i	$-0.780730\pi$
$37$	−9.39146	−1.54395	−0.771973	+	0.635655i	$0.780730\pi$
$38$	0	0
$39$	−0.498549	−0.0798318
$40$	0	0
$41$	5.53814	0.864913	0.432456	−	0.901655i	$-0.357647\pi$
$41$	5.53814	0.864913	0.432456	+	0.901655i	$0.357647\pi$
$42$	0	0
$43$	7.80230	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$43$	7.80230	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$44$	0	0
$45$	−4.50262	−0.671210
$46$	0	0
$47$	1.71408	0.250024	0.125012	−	0.992155i	$-0.460103\pi$
$47$	1.71408	0.250024	0.125012	+	0.992155i	$0.460103\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.7950	1.79166
$52$	0	0
$53$	7.33197	1.00712	0.503562	−	0.863959i	$-0.332023\pi$
$53$	7.33197	1.00712	0.503562	+	0.863959i	$0.332023\pi$
$54$	0	0
$55$	0.633664	0.0854433
$56$	0	0
$57$	−3.05218	−0.404272
$58$	0	0
$59$	11.0971	1.44472	0.722358	−	0.691520i	$-0.243058\pi$
$59$	11.0971	1.44472	0.722358	+	0.691520i	$0.243058\pi$
$60$	0	0
$61$	−4.96447	−0.635636	−0.317818	−	0.948152i	$-0.602950\pi$
$61$	−4.96447	−0.635636	−0.317818	+	0.948152i	$0.602950\pi$
$62$	0	0
$63$	6.31583	0.795719
$64$	0	0
$65$	−0.116448	−0.0144436
$66$	0	0
$67$	7.01376	0.856867	0.428433	−	0.903573i	$-0.359066\pi$
$67$	7.01376	0.856867	0.428433	+	0.903573i	$0.359066\pi$
$68$	0	0
$69$	−6.60292	−0.794898
$70$	0	0
$71$	−9.06884	−1.07627	−0.538137	−	0.842858i	$-0.680872\pi$
$71$	−9.06884	−1.07627	−0.538137	+	0.842858i	$0.680872\pi$
$72$	0	0
$73$	11.2346	1.31491	0.657454	−	0.753495i	$-0.271634\pi$
$73$	11.2346	1.31491	0.657454	+	0.753495i	$0.271634\pi$
$74$	0	0
$75$	13.7097	1.58306
$76$	0	0
$77$	−0.888842	−0.101293
$78$	0	0
$79$	−9.55480	−1.07500	−0.537499	−	0.843264i	$-0.680631\pi$
$79$	−9.55480	−1.07500	−0.537499	+	0.843264i	$0.680631\pi$
$80$	0	0
$81$	11.9422	1.32691
$82$	0	0
$83$	−1.92437	−0.211227	−0.105613	−	0.994407i	$-0.533681\pi$
$83$	−1.92437	−0.211227	−0.105613	+	0.994407i	$0.533681\pi$
$84$	0	0
$85$	2.98857	0.324156
$86$	0	0
$87$	27.6782	2.96742
$88$	0	0
$89$	−3.98178	−0.422068	−0.211034	−	0.977479i	$-0.567683\pi$
$89$	−3.98178	−0.422068	−0.211034	+	0.977479i	$0.567683\pi$
$90$	0	0
$91$	0.163342	0.0171229
$92$	0	0
$93$	−18.8994	−1.95977
$94$	0	0
$95$	−0.712910	−0.0731430
$96$	0	0
$97$	−14.1703	−1.43878	−0.719388	−	0.694608i	$-0.755578\pi$
$97$	−14.1703	−1.43878	−0.719388	+	0.694608i	$0.755578\pi$
$98$	0	0
$99$	−5.61377	−0.564205
$100$	0	0
$101$	−1.91585	−0.190634	−0.0953169	−	0.995447i	$-0.530386\pi$
$101$	−1.91585	−0.190634	−0.0953169	+	0.995447i	$0.530386\pi$
$102$	0	0
$103$	−10.0454	−0.989802	−0.494901	−	0.868949i	$-0.664796\pi$
$103$	−10.0454	−0.989802	−0.494901	+	0.868949i	$0.664796\pi$
$104$	0	0
$105$	2.17593	0.212349
$106$	0	0
$107$	−2.98502	−0.288573	−0.144286	−	0.989536i	$-0.546089\pi$
$107$	−2.98502	−0.288573	−0.144286	+	0.989536i	$0.546089\pi$
$108$	0	0
$109$	−10.3903	−0.995210	−0.497605	−	0.867404i	$-0.665787\pi$
$109$	−10.3903	−0.995210	−0.497605	+	0.867404i	$0.665787\pi$
$110$	0	0
$111$	28.6645	2.72071
$112$	0	0
$113$	3.45456	0.324977	0.162489	−	0.986710i	$-0.448048\pi$
$113$	3.45456	0.324977	0.162489	+	0.986710i	$0.448048\pi$
$114$	0	0
$115$	−1.54227	−0.143817
$116$	0	0
$117$	1.03164	0.0953750
$118$	0	0
$119$	−4.19208	−0.384287
$120$	0	0
$121$	−10.2100	−0.928178
$122$	0	0
$123$	−16.9034	−1.52413
$124$	0	0
$125$	6.76677	0.605238
$126$	0	0
$127$	−13.1601	−1.16777	−0.583885	−	0.811836i	$-0.698468\pi$
$127$	−13.1601	−1.16777	−0.583885	+	0.811836i	$0.698468\pi$
$128$	0	0
$129$	−23.8140	−2.09671
$130$	0	0
$131$	7.44248	0.650252	0.325126	−	0.945671i	$-0.394593\pi$
$131$	7.44248	0.650252	0.325126	+	0.945671i	$0.394593\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	7.21502	0.620970
$136$	0	0
$137$	15.7877	1.34883	0.674417	−	0.738351i	$-0.264395\pi$
$137$	15.7877	1.34883	0.674417	+	0.738351i	$0.264395\pi$
$138$	0	0
$139$	−3.58916	−0.304429	−0.152214	−	0.988348i	$-0.548640\pi$
$139$	−3.58916	−0.304429	−0.152214	+	0.988348i	$0.548640\pi$
$140$	0	0
$141$	−5.23167	−0.440586
$142$	0	0
$143$	−0.145185	−0.0121410
$144$	0	0
$145$	6.46490	0.536881
$146$	0	0
$147$	−3.05218	−0.251740
$148$	0	0
$149$	−7.08939	−0.580785	−0.290393	−	0.956908i	$-0.593786\pi$
$149$	−7.08939	−0.580785	−0.290393	+	0.956908i	$0.593786\pi$
$150$	0	0
$151$	−14.5439	−1.18357	−0.591785	−	0.806096i	$-0.701576\pi$
$151$	−14.5439	−1.18357	−0.591785	+	0.806096i	$0.701576\pi$
$152$	0	0
$153$	−26.4764	−2.14049
$154$	0	0
$155$	−4.41439	−0.354573
$156$	0	0
$157$	−0.0372025	−0.00296908	−0.00148454	−	0.999999i	$-0.500473\pi$
$157$	−0.0372025	−0.00296908	−0.00148454	+	0.999999i	$0.500473\pi$
$158$	0	0
$159$	−22.3785	−1.77473
$160$	0	0
$161$	2.16334	0.170495
$162$	0	0
$163$	−13.6838	−1.07180	−0.535900	−	0.844281i	$-0.680028\pi$
$163$	−13.6838	−1.07180	−0.535900	+	0.844281i	$0.680028\pi$
$164$	0	0
$165$	−1.93406	−0.150566
$166$	0	0
$167$	19.1193	1.47950	0.739750	−	0.672882i	$-0.234944\pi$
$167$	19.1193	1.47950	0.739750	+	0.672882i	$0.234944\pi$
$168$	0	0
$169$	−12.9733	−0.997948
$170$	0	0
$171$	6.31583	0.482983
$172$	0	0
$173$	−5.25273	−0.399358	−0.199679	−	0.979861i	$-0.563990\pi$
$173$	−5.25273	−0.399358	−0.199679	+	0.979861i	$0.563990\pi$
$174$	0	0
$175$	−4.49176	−0.339545
$176$	0	0
$177$	−33.8703	−2.54585
$178$	0	0
$179$	4.41439	0.329947	0.164974	−	0.986298i	$-0.447246\pi$
$179$	4.41439	0.329947	0.164974	+	0.986298i	$0.447246\pi$
$180$	0	0
$181$	5.94458	0.441857	0.220929	−	0.975290i	$-0.429091\pi$
$181$	5.94458	0.441857	0.220929	+	0.975290i	$0.429091\pi$
$182$	0	0
$183$	15.1525	1.12010
$184$	0	0
$185$	6.69526	0.492246
$186$	0	0
$187$	3.72610	0.272479
$188$	0	0
$189$	−10.1205	−0.736159
$190$	0	0
$191$	−4.06103	−0.293846	−0.146923	−	0.989148i	$-0.546937\pi$
$191$	−4.06103	−0.293846	−0.146923	+	0.989148i	$0.546937\pi$
$192$	0	0
$193$	−1.44248	−0.103832	−0.0519159	−	0.998651i	$-0.516533\pi$
$193$	−1.44248	−0.103832	−0.0519159	+	0.998651i	$0.516533\pi$
$194$	0	0
$195$	0.355421	0.0254522
$196$	0	0
$197$	19.3114	1.37588	0.687941	−	0.725766i	$-0.258515\pi$
$197$	19.3114	1.37588	0.687941	+	0.725766i	$0.258515\pi$
$198$	0	0
$199$	9.32040	0.660706	0.330353	−	0.943857i	$-0.392832\pi$
$199$	9.32040	0.660706	0.330353	+	0.943857i	$0.392832\pi$
$200$	0	0
$201$	−21.4073	−1.50995
$202$	0	0
$203$	−9.06833	−0.636472
$204$	0	0
$205$	−3.94820	−0.275754
$206$	0	0
$207$	13.6633	0.949665
$208$	0	0
$209$	−0.888842	−0.0614825
$210$	0	0
$211$	14.2844	0.983376	0.491688	−	0.870771i	$-0.336380\pi$
$211$	14.2844	0.983376	0.491688	+	0.870771i	$0.336380\pi$
$212$	0	0
$213$	27.6798	1.89659
$214$	0	0
$215$	−5.56233	−0.379348
$216$	0	0
$217$	6.19208	0.420346
$218$	0	0
$219$	−34.2900	−2.31710
$220$	0	0
$221$	−0.684742	−0.0460607
$222$	0	0
$223$	−7.16857	−0.480043	−0.240022	−	0.970768i	$-0.577155\pi$
$223$	−7.16857	−0.480043	−0.240022	+	0.970768i	$0.577155\pi$
$224$	0	0
$225$	−28.3692	−1.89128
$226$	0	0
$227$	−3.98567	−0.264538	−0.132269	−	0.991214i	$-0.542226\pi$
$227$	−3.98567	−0.264538	−0.132269	+	0.991214i	$0.542226\pi$
$228$	0	0
$229$	15.6375	1.03335	0.516676	−	0.856181i	$-0.327169\pi$
$229$	15.6375	1.03335	0.516676	+	0.856181i	$0.327169\pi$
$230$	0	0
$231$	2.71291	0.178496
$232$	0	0
$233$	11.9515	0.782972	0.391486	−	0.920184i	$-0.371961\pi$
$233$	11.9515	0.782972	0.391486	+	0.920184i	$0.371961\pi$
$234$	0	0
$235$	−1.22198	−0.0797133
$236$	0	0
$237$	29.1630	1.89434
$238$	0	0
$239$	−11.7210	−0.758171	−0.379086	−	0.925362i	$-0.623761\pi$
$239$	−11.7210	−0.758171	−0.379086	+	0.925362i	$0.623761\pi$
$240$	0	0
$241$	30.2187	1.94656	0.973280	−	0.229620i	$-0.0737484\pi$
$241$	30.2187	1.94656	0.973280	+	0.229620i	$0.0737484\pi$
$242$	0	0
$243$	−6.08822	−0.390560
$244$	0	0
$245$	−0.712910	−0.0455461
$246$	0	0
$247$	0.163342	0.0103932
$248$	0	0
$249$	5.87353	0.372220
$250$	0	0
$251$	−4.05542	−0.255976	−0.127988	−	0.991776i	$-0.540852\pi$
$251$	−4.05542	−0.255976	−0.127988	+	0.991776i	$0.540852\pi$
$252$	0	0
$253$	−1.92287	−0.120890
$254$	0	0
$255$	−9.12168	−0.571222
$256$	0	0
$257$	−12.4627	−0.777401	−0.388701	−	0.921364i	$-0.627076\pi$
$257$	−12.4627	−0.777401	−0.388701	+	0.921364i	$0.627076\pi$
$258$	0	0
$259$	−9.39146	−0.583557
$260$	0	0
$261$	−57.2740	−3.54517
$262$	0	0
$263$	31.4158	1.93718	0.968590	−	0.248661i	$-0.0799906\pi$
$263$	31.4158	1.93718	0.968590	+	0.248661i	$0.0799906\pi$
$264$	0	0
$265$	−5.22704	−0.321094
$266$	0	0
$267$	12.1531	0.743760
$268$	0	0
$269$	20.4598	1.24746	0.623728	−	0.781642i	$-0.285617\pi$
$269$	20.4598	1.24746	0.623728	+	0.781642i	$0.285617\pi$
$270$	0	0
$271$	−23.1268	−1.40485	−0.702427	−	0.711755i	$-0.747900\pi$
$271$	−23.1268	−1.40485	−0.702427	+	0.711755i	$0.747900\pi$
$272$	0	0
$273$	−0.498549	−0.0301736
$274$	0	0
$275$	3.99247	0.240755
$276$	0	0
$277$	−0.0135796	−0.000815920 0	−0.000407960	−	1.00000i	$-0.500130\pi$
$277$	−0.0135796	−0.000815920 0	−0.000407960		1.00000i	$0.500130\pi$
$278$	0	0
$279$	39.1081	2.34134
$280$	0	0
$281$	−20.7927	−1.24039	−0.620193	−	0.784449i	$-0.712946\pi$
$281$	−20.7927	−1.24039	−0.620193	+	0.784449i	$0.712946\pi$
$282$	0	0
$283$	1.27100	0.0755532	0.0377766	−	0.999286i	$-0.487972\pi$
$283$	1.27100	0.0755532	0.0377766	+	0.999286i	$0.487972\pi$
$284$	0	0
$285$	2.17593	0.128891
$286$	0	0
$287$	5.53814	0.326906
$288$	0	0
$289$	0.573525	0.0337368
$290$	0	0
$291$	43.2504	2.53538
$292$	0	0
$293$	−1.13628	−0.0663822	−0.0331911	−	0.999449i	$-0.510567\pi$
$293$	−1.13628	−0.0663822	−0.0331911	+	0.999449i	$0.510567\pi$
$294$	0	0
$295$	−7.91121	−0.460608
$296$	0	0
$297$	8.99554	0.521974
$298$	0	0
$299$	0.353364	0.0204356
$300$	0	0
$301$	7.80230	0.449717
$302$	0	0
$303$	5.84751	0.335931
$304$	0	0
$305$	3.53922	0.202655
$306$	0	0
$307$	31.6289	1.80516	0.902579	−	0.430524i	$-0.141671\pi$
$307$	31.6289	1.80516	0.902579	+	0.430524i	$0.141671\pi$
$308$	0	0
$309$	30.6604	1.74421
$310$	0	0
$311$	−9.24071	−0.523992	−0.261996	−	0.965069i	$-0.584381\pi$
$311$	−9.24071	−0.523992	−0.261996	+	0.965069i	$0.584381\pi$
$312$	0	0
$313$	27.4469	1.55139	0.775694	−	0.631109i	$-0.217400\pi$
$313$	27.4469	1.55139	0.775694	+	0.631109i	$0.217400\pi$
$314$	0	0
$315$	−4.50262	−0.253694
$316$	0	0
$317$	8.98702	0.504761	0.252381	−	0.967628i	$-0.418787\pi$
$317$	8.98702	0.504761	0.252381	+	0.967628i	$0.418787\pi$
$318$	0	0
$319$	8.06031	0.451291
$320$	0	0
$321$	9.11083	0.508517
$322$	0	0
$323$	−4.19208	−0.233254
$324$	0	0
$325$	−0.733692	−0.0406979
$326$	0	0
$327$	31.7131	1.75374
$328$	0	0
$329$	1.71408	0.0945000
$330$	0	0
$331$	−7.13834	−0.392358	−0.196179	−	0.980568i	$-0.562853\pi$
$331$	−7.13834	−0.392358	−0.196179	+	0.980568i	$0.562853\pi$
$332$	0	0
$333$	−59.3148	−3.25043
$334$	0	0
$335$	−5.00018	−0.273189
$336$	0	0
$337$	33.8589	1.84441	0.922207	−	0.386697i	$-0.126384\pi$
$337$	33.8589	1.84441	0.922207	+	0.386697i	$0.126384\pi$
$338$	0	0
$339$	−10.5439	−0.572668
$340$	0	0
$341$	−5.50378	−0.298046
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.70729	0.253432
$346$	0	0
$347$	4.37480	0.234852	0.117426	−	0.993082i	$-0.462536\pi$
$347$	4.37480	0.234852	0.117426	+	0.993082i	$0.462536\pi$
$348$	0	0
$349$	−30.2181	−1.61754	−0.808769	−	0.588127i	$-0.799866\pi$
$349$	−30.2181	−1.61754	−0.808769	+	0.588127i	$0.799866\pi$
$350$	0	0
$351$	−1.65310	−0.0882361
$352$	0	0
$353$	−17.2225	−0.916661	−0.458330	−	0.888782i	$-0.651552\pi$
$353$	−17.2225	−0.916661	−0.458330	+	0.888782i	$0.651552\pi$
$354$	0	0
$355$	6.46527	0.343141
$356$	0	0
$357$	12.7950	0.677183
$358$	0	0
$359$	24.3298	1.28408	0.642038	−	0.766673i	$-0.278089\pi$
$359$	24.3298	1.28408	0.642038	+	0.766673i	$0.278089\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	31.1627	1.63562
$364$	0	0
$365$	−8.00924	−0.419223
$366$	0	0
$367$	−16.5222	−0.862450	−0.431225	−	0.902244i	$-0.641919\pi$
$367$	−16.5222	−0.862450	−0.431225	+	0.902244i	$0.641919\pi$
$368$	0	0
$369$	34.9780	1.82088
$370$	0	0
$371$	7.33197	0.380657
$372$	0	0
$373$	−34.2404	−1.77290	−0.886450	−	0.462825i	$-0.846836\pi$
$373$	−34.2404	−1.77290	−0.886450	+	0.462825i	$0.846836\pi$
$374$	0	0
$375$	−20.6534	−1.06654
$376$	0	0
$377$	−1.48124	−0.0762876
$378$	0	0
$379$	17.3668	0.892075	0.446037	−	0.895014i	$-0.352835\pi$
$379$	17.3668	0.892075	0.446037	+	0.895014i	$0.352835\pi$
$380$	0	0
$381$	40.1671	2.05782
$382$	0	0
$383$	24.2483	1.23903	0.619516	−	0.784984i	$-0.287329\pi$
$383$	24.2483	1.23903	0.619516	+	0.784984i	$0.287329\pi$
$384$	0	0
$385$	0.633664	0.0322945
$386$	0	0
$387$	49.2780	2.50494
$388$	0	0
$389$	−20.8777	−1.05854	−0.529272	−	0.848452i	$-0.677535\pi$
$389$	−20.8777	−1.05854	−0.529272	+	0.848452i	$0.677535\pi$
$390$	0	0
$391$	−9.06890	−0.458634
$392$	0	0
$393$	−22.7158	−1.14586
$394$	0	0
$395$	6.81171	0.342734
$396$	0	0
$397$	1.68506	0.0845710	0.0422855	−	0.999106i	$-0.486536\pi$
$397$	1.68506	0.0845710	0.0422855	+	0.999106i	$0.486536\pi$
$398$	0	0
$399$	−3.05218	−0.152800
$400$	0	0
$401$	25.1287	1.25487	0.627434	−	0.778670i	$-0.284105\pi$
$401$	25.1287	1.25487	0.627434	+	0.778670i	$0.284105\pi$
$402$	0	0
$403$	1.01143	0.0503827
$404$	0	0
$405$	−8.51371	−0.423049
$406$	0	0
$407$	8.34752	0.413771
$408$	0	0
$409$	−33.1226	−1.63781	−0.818903	−	0.573932i	$-0.805417\pi$
$409$	−33.1226	−1.63781	−0.818903	+	0.573932i	$0.805417\pi$
$410$	0	0
$411$	−48.1870	−2.37689
$412$	0	0
$413$	11.0971	0.546051
$414$	0	0
$415$	1.37190	0.0673440
$416$	0	0
$417$	10.9548	0.536458
$418$	0	0
$419$	21.1469	1.03309	0.516546	−	0.856260i	$-0.327218\pi$
$419$	21.1469	1.03309	0.516546	+	0.856260i	$0.327218\pi$
$420$	0	0
$421$	17.3029	0.843292	0.421646	−	0.906761i	$-0.361453\pi$
$421$	17.3029	0.843292	0.421646	+	0.906761i	$0.361453\pi$
$422$	0	0
$423$	10.8258	0.526368
$424$	0	0
$425$	18.8298	0.913380
$426$	0	0
$427$	−4.96447	−0.240248
$428$	0	0
$429$	0.443132	0.0213946
$430$	0	0
$431$	8.35548	0.402469	0.201235	−	0.979543i	$-0.435505\pi$
$431$	8.35548	0.402469	0.201235	+	0.979543i	$0.435505\pi$
$432$	0	0
$433$	2.18795	0.105146	0.0525732	−	0.998617i	$-0.483258\pi$
$433$	2.18795	0.105146	0.0525732	+	0.998617i	$0.483258\pi$
$434$	0	0
$435$	−19.7321	−0.946080
$436$	0	0
$437$	2.16334	0.103487
$438$	0	0
$439$	−9.50734	−0.453760	−0.226880	−	0.973923i	$-0.572853\pi$
$439$	−9.50734	−0.453760	−0.226880	+	0.973923i	$0.572853\pi$
$440$	0	0
$441$	6.31583	0.300754
$442$	0	0
$443$	−28.4937	−1.35378	−0.676888	−	0.736086i	$-0.736672\pi$
$443$	−28.4937	−1.35378	−0.676888	+	0.736086i	$0.736672\pi$
$444$	0	0
$445$	2.83865	0.134565
$446$	0	0
$447$	21.6381	1.02345
$448$	0	0
$449$	−35.4685	−1.67386	−0.836931	−	0.547309i	$-0.815652\pi$
$449$	−35.4685	−1.67386	−0.836931	+	0.547309i	$0.815652\pi$
$450$	0	0
$451$	−4.92253	−0.231793
$452$	0	0
$453$	44.3908	2.08566
$454$	0	0
$455$	−0.116448	−0.00545917
$456$	0	0
$457$	−36.7055	−1.71701	−0.858504	−	0.512806i	$-0.828606\pi$
$457$	−36.7055	−1.71701	−0.858504	+	0.512806i	$0.828606\pi$
$458$	0	0
$459$	42.4260	1.98028
$460$	0	0
$461$	−8.16178	−0.380132	−0.190066	−	0.981771i	$-0.560870\pi$
$461$	−8.16178	−0.380132	−0.190066	+	0.981771i	$0.560870\pi$
$462$	0	0
$463$	−21.0402	−0.977819	−0.488910	−	0.872334i	$-0.662605\pi$
$463$	−21.0402	−0.977819	−0.488910	+	0.872334i	$0.662605\pi$
$464$	0	0
$465$	13.4735	0.624821
$466$	0	0
$467$	34.2973	1.58709	0.793544	−	0.608512i	$-0.208233\pi$
$467$	34.2973	1.58709	0.793544	+	0.608512i	$0.208233\pi$
$468$	0	0
$469$	7.01376	0.323865
$470$	0	0
$471$	0.113549	0.00523205
$472$	0	0
$473$	−6.93501	−0.318872
$474$	0	0
$475$	−4.49176	−0.206096
$476$	0	0
$477$	46.3075	2.12027
$478$	0	0
$479$	41.2805	1.88616	0.943078	−	0.332572i	$-0.107916\pi$
$479$	41.2805	1.88616	0.943078	+	0.332572i	$0.107916\pi$
$480$	0	0
$481$	−1.53402	−0.0699452
$482$	0	0
$483$	−6.60292	−0.300443
$484$	0	0
$485$	10.1022	0.458715
$486$	0	0
$487$	13.8969	0.629729	0.314865	−	0.949137i	$-0.398041\pi$
$487$	13.8969	0.629729	0.314865	+	0.949137i	$0.398041\pi$
$488$	0	0
$489$	41.7656	1.88871
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	38.0152	1.71212
$494$	0	0
$495$	4.00211	0.179882
$496$	0	0
$497$	−9.06884	−0.406793
$498$	0	0
$499$	15.7121	0.703369	0.351684	−	0.936119i	$-0.385609\pi$
$499$	15.7121	0.703369	0.351684	+	0.936119i	$0.385609\pi$
$500$	0	0
$501$	−58.3558	−2.60714
$502$	0	0
$503$	18.8304	0.839605	0.419803	−	0.907615i	$-0.362099\pi$
$503$	18.8304	0.839605	0.419803	+	0.907615i	$0.362099\pi$
$504$	0	0
$505$	1.36583	0.0607784
$506$	0	0
$507$	39.5970	1.75856
$508$	0	0
$509$	8.25563	0.365924	0.182962	−	0.983120i	$-0.441431\pi$
$509$	8.25563	0.365924	0.182962	+	0.983120i	$0.441431\pi$
$510$	0	0
$511$	11.2346	0.496988
$512$	0	0
$513$	−10.1205	−0.446832
$514$	0	0
$515$	7.16146	0.315572
$516$	0	0
$517$	−1.52354	−0.0670053
$518$	0	0
$519$	16.0323	0.703740
$520$	0	0
$521$	35.3304	1.54785	0.773927	−	0.633275i	$-0.218290\pi$
$521$	35.3304	1.54785	0.773927	+	0.633275i	$0.218290\pi$
$522$	0	0
$523$	−21.0541	−0.920631	−0.460316	−	0.887755i	$-0.652264\pi$
$523$	−21.0541	−0.920631	−0.460316	+	0.887755i	$0.652264\pi$
$524$	0	0
$525$	13.7097	0.598339
$526$	0	0
$527$	−25.9577	−1.13073
$528$	0	0
$529$	−18.3200	−0.796520
$530$	0	0
$531$	70.0872	3.04152
$532$	0	0
$533$	0.904610	0.0391830
$534$	0	0
$535$	2.12805	0.0920036
$536$	0	0
$537$	−13.4735	−0.581426
$538$	0	0
$539$	−0.888842	−0.0382851
$540$	0	0
$541$	29.3748	1.26292	0.631460	−	0.775408i	$-0.282456\pi$
$541$	29.3748	1.26292	0.631460	+	0.775408i	$0.282456\pi$
$542$	0	0
$543$	−18.1440	−0.778632
$544$	0	0
$545$	7.40734	0.317296
$546$	0	0
$547$	−1.03397	−0.0442093	−0.0221047	−	0.999756i	$-0.507037\pi$
$547$	−1.03397	−0.0442093	−0.0221047	+	0.999756i	$0.507037\pi$
$548$	0	0
$549$	−31.3548	−1.33819
$550$	0	0
$551$	−9.06833	−0.386324
$552$	0	0
$553$	−9.55480	−0.406311
$554$	0	0
$555$	−20.4352	−0.867425
$556$	0	0
$557$	21.9527	0.930166	0.465083	−	0.885267i	$-0.346024\pi$
$557$	21.9527	0.930166	0.465083	+	0.885267i	$0.346024\pi$
$558$	0	0
$559$	1.27444	0.0539031
$560$	0	0
$561$	−11.3727	−0.480157
$562$	0	0
$563$	−5.44009	−0.229272	−0.114636	−	0.993408i	$-0.536570\pi$
$563$	−5.44009	−0.229272	−0.114636	+	0.993408i	$0.536570\pi$
$564$	0	0
$565$	−2.46279	−0.103610
$566$	0	0
$567$	11.9422	0.501525
$568$	0	0
$569$	−21.2764	−0.891953	−0.445977	−	0.895045i	$-0.647144\pi$
$569$	−21.2764	−0.891953	−0.445977	+	0.895045i	$0.647144\pi$
$570$	0	0
$571$	45.9311	1.92216	0.961078	−	0.276279i	$-0.0891013\pi$
$571$	45.9311	1.92216	0.961078	+	0.276279i	$0.0891013\pi$
$572$	0	0
$573$	12.3950	0.517809
$574$	0	0
$575$	−9.71721	−0.405236
$576$	0	0
$577$	39.8710	1.65985	0.829926	−	0.557874i	$-0.188383\pi$
$577$	39.8710	1.65985	0.829926	+	0.557874i	$0.188383\pi$
$578$	0	0
$579$	4.40271	0.182970
$580$	0	0
$581$	−1.92437	−0.0798363
$582$	0	0
$583$	−6.51697	−0.269905
$584$	0	0
$585$	−0.735465	−0.0304078
$586$	0	0
$587$	10.1295	0.418091	0.209046	−	0.977906i	$-0.432964\pi$
$587$	10.1295	0.418091	0.209046	+	0.977906i	$0.432964\pi$
$588$	0	0
$589$	6.19208	0.255140
$590$	0	0
$591$	−58.9420	−2.42455
$592$	0	0
$593$	38.8114	1.59379	0.796896	−	0.604116i	$-0.206474\pi$
$593$	38.8114	1.59379	0.796896	+	0.604116i	$0.206474\pi$
$594$	0	0
$595$	2.98857	0.122520
$596$	0	0
$597$	−28.4476	−1.16428
$598$	0	0
$599$	38.3342	1.56629	0.783147	−	0.621837i	$-0.213613\pi$
$599$	38.3342	1.56629	0.783147	+	0.621837i	$0.213613\pi$
$600$	0	0
$601$	−19.7197	−0.804385	−0.402193	−	0.915555i	$-0.631752\pi$
$601$	−19.7197	−0.804385	−0.402193	+	0.915555i	$0.631752\pi$
$602$	0	0
$603$	44.2977	1.80394
$604$	0	0
$605$	7.27878	0.295925
$606$	0	0
$607$	11.1353	0.451966	0.225983	−	0.974131i	$-0.427441\pi$
$607$	11.1353	0.451966	0.225983	+	0.974131i	$0.427441\pi$
$608$	0	0
$609$	27.6782	1.12158
$610$	0	0
$611$	0.279980	0.0113268
$612$	0	0
$613$	25.0204	1.01056	0.505282	−	0.862954i	$-0.331388\pi$
$613$	25.0204	1.01056	0.505282	+	0.862954i	$0.331388\pi$
$614$	0	0
$615$	12.0506	0.485928
$616$	0	0
$617$	−16.9000	−0.680369	−0.340185	−	0.940359i	$-0.610490\pi$
$617$	−16.9000	−0.680369	−0.340185	+	0.940359i	$0.610490\pi$
$618$	0	0
$619$	−0.0539180	−0.00216715	−0.00108357	−	0.999999i	$-0.500345\pi$
$619$	−0.0539180	−0.00216715	−0.00108357	+	0.999999i	$0.500345\pi$
$620$	0	0
$621$	−21.8941	−0.878581
$622$	0	0
$623$	−3.98178	−0.159527
$624$	0	0
$625$	17.6347	0.705388
$626$	0	0
$627$	2.71291	0.108343
$628$	0	0
$629$	39.3697	1.56977
$630$	0	0
$631$	−23.6466	−0.941358	−0.470679	−	0.882305i	$-0.655991\pi$
$631$	−23.6466	−0.941358	−0.470679	+	0.882305i	$0.655991\pi$
$632$	0	0
$633$	−43.5985	−1.73289
$634$	0	0
$635$	9.38197	0.372312
$636$	0	0
$637$	0.163342	0.00647184
$638$	0	0
$639$	−57.2772	−2.26585
$640$	0	0
$641$	4.78911	0.189158	0.0945792	−	0.995517i	$-0.469849\pi$
$641$	4.78911	0.189158	0.0945792	+	0.995517i	$0.469849\pi$
$642$	0	0
$643$	18.3562	0.723897	0.361948	−	0.932198i	$-0.382112\pi$
$643$	18.3562	0.723897	0.361948	+	0.932198i	$0.382112\pi$
$644$	0	0
$645$	16.9773	0.668479
$646$	0	0
$647$	23.1705	0.910924	0.455462	−	0.890255i	$-0.349474\pi$
$647$	23.1705	0.910924	0.455462	+	0.890255i	$0.349474\pi$
$648$	0	0
$649$	−9.86354	−0.387178
$650$	0	0
$651$	−18.8994	−0.740724
$652$	0	0
$653$	45.1666	1.76751	0.883753	−	0.467954i	$-0.155009\pi$
$653$	45.1666	1.76751	0.883753	+	0.467954i	$0.155009\pi$
$654$	0	0
$655$	−5.30582	−0.207315
$656$	0	0
$657$	70.9556	2.76824
$658$	0	0
$659$	−45.7844	−1.78351	−0.891754	−	0.452521i	$-0.850525\pi$
$659$	−45.7844	−1.78351	−0.891754	+	0.452521i	$0.850525\pi$
$660$	0	0
$661$	−16.9067	−0.657593	−0.328796	−	0.944401i	$-0.606643\pi$
$661$	−16.9067	−0.657593	−0.328796	+	0.944401i	$0.606643\pi$
$662$	0	0
$663$	2.08996	0.0811672
$664$	0	0
$665$	−0.712910	−0.0276455
$666$	0	0
$667$	−19.6179	−0.759608
$668$	0	0
$669$	21.8798	0.845922
$670$	0	0
$671$	4.41263	0.170348
$672$	0	0
$673$	43.5623	1.67920	0.839601	−	0.543203i	$-0.182789\pi$
$673$	43.5623	1.67920	0.839601	+	0.543203i	$0.182789\pi$
$674$	0	0
$675$	45.4589	1.74971
$676$	0	0
$677$	−12.1936	−0.468637	−0.234319	−	0.972160i	$-0.575286\pi$
$677$	−12.1936	−0.468637	−0.234319	+	0.972160i	$0.575286\pi$
$678$	0	0
$679$	−14.1703	−0.543806
$680$	0	0
$681$	12.1650	0.466164
$682$	0	0
$683$	31.6523	1.21114	0.605571	−	0.795791i	$-0.292945\pi$
$683$	31.6523	1.21114	0.605571	+	0.795791i	$0.292945\pi$
$684$	0	0
$685$	−11.2552	−0.430039
$686$	0	0
$687$	−47.7284	−1.82095
$688$	0	0
$689$	1.19762	0.0456256
$690$	0	0
$691$	−3.66207	−0.139312	−0.0696558	−	0.997571i	$-0.522190\pi$
$691$	−3.66207	−0.139312	−0.0696558	+	0.997571i	$0.522190\pi$
$692$	0	0
$693$	−5.61377	−0.213250
$694$	0	0
$695$	2.55875	0.0970589
$696$	0	0
$697$	−23.2163	−0.879381
$698$	0	0
$699$	−36.4783	−1.37974
$700$	0	0
$701$	−28.1085	−1.06164	−0.530821	−	0.847484i	$-0.678116\pi$
$701$	−28.1085	−1.06164	−0.530821	+	0.847484i	$0.678116\pi$
$702$	0	0
$703$	−9.39146	−0.354206
$704$	0	0
$705$	3.72971	0.140469
$706$	0	0
$707$	−1.91585	−0.0720528
$708$	0	0
$709$	2.33745	0.0877849	0.0438925	−	0.999036i	$-0.486024\pi$
$709$	2.33745	0.0877849	0.0438925	+	0.999036i	$0.486024\pi$
$710$	0	0
$711$	−60.3465	−2.26317
$712$	0	0
$713$	13.3956	0.501669
$714$	0	0
$715$	0.103504	0.00387083
$716$	0	0
$717$	35.7748	1.33603
$718$	0	0
$719$	9.29251	0.346552	0.173276	−	0.984873i	$-0.444565\pi$
$719$	9.29251	0.346552	0.173276	+	0.984873i	$0.444565\pi$
$720$	0	0
$721$	−10.0454	−0.374110
$722$	0	0
$723$	−92.2332	−3.43019
$724$	0	0
$725$	40.7328	1.51278
$726$	0	0
$727$	28.6037	1.06085	0.530426	−	0.847731i	$-0.322032\pi$
$727$	28.6037	1.06085	0.530426	+	0.847731i	$0.322032\pi$
$728$	0	0
$729$	−17.2442	−0.638674
$730$	0	0
$731$	−32.7078	−1.20974
$732$	0	0
$733$	34.1671	1.26199	0.630995	−	0.775787i	$-0.282647\pi$
$733$	34.1671	1.26199	0.630995	+	0.775787i	$0.282647\pi$
$734$	0	0
$735$	2.17593	0.0802605
$736$	0	0
$737$	−6.23412	−0.229637
$738$	0	0
$739$	33.5169	1.23294	0.616469	−	0.787379i	$-0.288563\pi$
$739$	33.5169	1.23294	0.616469	+	0.787379i	$0.288563\pi$
$740$	0	0
$741$	−0.498549	−0.0183147
$742$	0	0
$743$	40.0506	1.46931	0.734657	−	0.678438i	$-0.237343\pi$
$743$	40.0506	1.46931	0.734657	+	0.678438i	$0.237343\pi$
$744$	0	0
$745$	5.05409	0.185168
$746$	0	0
$747$	−12.1540	−0.444691
$748$	0	0
$749$	−2.98502	−0.109070
$750$	0	0
$751$	24.7183	0.901984	0.450992	−	0.892528i	$-0.351070\pi$
$751$	24.7183	0.901984	0.450992	+	0.892528i	$0.351070\pi$
$752$	0	0
$753$	12.3779	0.451075
$754$	0	0
$755$	10.3685	0.377349
$756$	0	0
$757$	16.7061	0.607192	0.303596	−	0.952801i	$-0.401813\pi$
$757$	16.7061	0.607192	0.303596	+	0.952801i	$0.401813\pi$
$758$	0	0
$759$	5.86895	0.213029
$760$	0	0
$761$	45.5691	1.65188	0.825940	−	0.563759i	$-0.190645\pi$
$761$	45.5691	1.65188	0.825940	+	0.563759i	$0.190645\pi$
$762$	0	0
$763$	−10.3903	−0.376154
$764$	0	0
$765$	18.8753	0.682439
$766$	0	0
$767$	1.81262	0.0654497
$768$	0	0
$769$	43.4115	1.56546	0.782729	−	0.622363i	$-0.213827\pi$
$769$	43.4115	1.56546	0.782729	+	0.622363i	$0.213827\pi$
$770$	0	0
$771$	38.0384	1.36992
$772$	0	0
$773$	−49.6944	−1.78738	−0.893691	−	0.448683i	$-0.851893\pi$
$773$	−49.6944	−1.78738	−0.893691	+	0.448683i	$0.851893\pi$
$774$	0	0
$775$	−27.8133	−0.999084
$776$	0	0
$777$	28.6645	1.02833
$778$	0	0
$779$	5.53814	0.198425
$780$	0	0
$781$	8.06077	0.288437
$782$	0	0
$783$	91.7762	3.27981
$784$	0	0
$785$	0.0265220	0.000946612 0
$786$	0	0
$787$	23.4392	0.835517	0.417758	−	0.908558i	$-0.362816\pi$
$787$	23.4392	0.835517	0.417758	+	0.908558i	$0.362816\pi$
$788$	0	0
$789$	−95.8868	−3.41366
$790$	0	0
$791$	3.45456	0.122830
$792$	0	0
$793$	−0.810906	−0.0287961
$794$	0	0
$795$	15.9539	0.565826
$796$	0	0
$797$	−32.2050	−1.14076	−0.570380	−	0.821381i	$-0.693204\pi$
$797$	−32.2050	−1.14076	−0.570380	+	0.821381i	$0.693204\pi$
$798$	0	0
$799$	−7.18554	−0.254206
$800$	0	0
$801$	−25.1483	−0.888570
$802$	0	0
$803$	−9.98576	−0.352390
$804$	0	0
$805$	−1.54227	−0.0543578
$806$	0	0
$807$	−62.4470	−2.19824
$808$	0	0
$809$	12.7920	0.449742	0.224871	−	0.974389i	$-0.427804\pi$
$809$	12.7920	0.449742	0.224871	+	0.974389i	$0.427804\pi$
$810$	0	0
$811$	0.457400	0.0160615	0.00803074	−	0.999968i	$-0.497444\pi$
$811$	0.457400	0.0160615	0.00803074	+	0.999968i	$0.497444\pi$
$812$	0	0
$813$	70.5873	2.47561
$814$	0	0
$815$	9.75534	0.341715
$816$	0	0
$817$	7.80230	0.272968
$818$	0	0
$819$	1.03164	0.0360484
$820$	0	0
$821$	30.2376	1.05530	0.527649	−	0.849462i	$-0.323074\pi$
$821$	30.2376	1.05530	0.527649	+	0.849462i	$0.323074\pi$
$822$	0	0
$823$	−22.2135	−0.774315	−0.387157	−	0.922014i	$-0.626543\pi$
$823$	−22.2135	−0.774315	−0.387157	+	0.922014i	$0.626543\pi$
$824$	0	0
$825$	−12.1857	−0.424253
$826$	0	0
$827$	−38.7895	−1.34884	−0.674422	−	0.738346i	$-0.735607\pi$
$827$	−38.7895	−1.34884	−0.674422	+	0.738346i	$0.735607\pi$
$828$	0	0
$829$	32.2972	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$829$	32.2972	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$830$	0	0
$831$	0.0414475	0.00143780
$832$	0	0
$833$	−4.19208	−0.145247
$834$	0	0
$835$	−13.6304	−0.471699
$836$	0	0
$837$	−62.6670	−2.16609
$838$	0	0
$839$	−46.8210	−1.61644	−0.808220	−	0.588880i	$-0.799569\pi$
$839$	−46.8210	−1.61644	−0.808220	+	0.588880i	$0.799569\pi$
$840$	0	0
$841$	53.2346	1.83568
$842$	0	0
$843$	63.4630	2.18578
$844$	0	0
$845$	9.24881	0.318169
$846$	0	0
$847$	−10.2100	−0.350818
$848$	0	0
$849$	−3.87933	−0.133138
$850$	0	0
$851$	−20.3169	−0.696456
$852$	0	0
$853$	39.9173	1.36674	0.683372	−	0.730071i	$-0.260513\pi$
$853$	39.9173	1.36674	0.683372	+	0.730071i	$0.260513\pi$
$854$	0	0
$855$	−4.50262	−0.153986
$856$	0	0
$857$	15.0689	0.514744	0.257372	−	0.966312i	$-0.417143\pi$
$857$	15.0689	0.514744	0.257372	+	0.966312i	$0.417143\pi$
$858$	0	0
$859$	−34.5715	−1.17956	−0.589781	−	0.807563i	$-0.700786\pi$
$859$	−34.5715	−1.17956	−0.589781	+	0.807563i	$0.700786\pi$
$860$	0	0
$861$	−16.9034	−0.576067
$862$	0	0
$863$	36.8101	1.25303	0.626515	−	0.779410i	$-0.284481\pi$
$863$	36.8101	1.25303	0.626515	+	0.779410i	$0.284481\pi$
$864$	0	0
$865$	3.74472	0.127324
$866$	0	0
$867$	−1.75051	−0.0594503
$868$	0	0
$869$	8.49271	0.288095
$870$	0	0
$871$	1.14564	0.0388185
$872$	0	0
$873$	−89.4972	−3.02902
$874$	0	0
$875$	6.76677	0.228759
$876$	0	0
$877$	−22.8416	−0.771305	−0.385652	−	0.922644i	$-0.626024\pi$
$877$	−22.8416	−0.771305	−0.385652	+	0.922644i	$0.626024\pi$
$878$	0	0
$879$	3.46814	0.116977
$880$	0	0
$881$	−7.62686	−0.256956	−0.128478	−	0.991712i	$-0.541009\pi$
$881$	−7.62686	−0.256956	−0.128478	+	0.991712i	$0.541009\pi$
$882$	0	0
$883$	26.2414	0.883092	0.441546	−	0.897239i	$-0.354430\pi$
$883$	26.2414	0.883092	0.441546	+	0.897239i	$0.354430\pi$
$884$	0	0
$885$	24.1465	0.811675
$886$	0	0
$887$	−34.6260	−1.16263	−0.581314	−	0.813679i	$-0.697461\pi$
$887$	−34.6260	−1.16263	−0.581314	+	0.813679i	$0.697461\pi$
$888$	0	0
$889$	−13.1601	−0.441376
$890$	0	0
$891$	−10.6147	−0.355607
$892$	0	0
$893$	1.71408	0.0573593
$894$	0	0
$895$	−3.14707	−0.105195
$896$	0	0
$897$	−1.07853	−0.0360111
$898$	0	0
$899$	−56.1518	−1.87277
$900$	0	0
$901$	−30.7362	−1.02397
$902$	0	0
$903$	−23.8140	−0.792482
$904$	0	0
$905$	−4.23795	−0.140874
$906$	0	0
$907$	−7.05063	−0.234112	−0.117056	−	0.993125i	$-0.537346\pi$
$907$	−7.05063	−0.234112	−0.117056	+	0.993125i	$0.537346\pi$
$908$	0	0
$909$	−12.1001	−0.401337
$910$	0	0
$911$	−32.9215	−1.09074	−0.545368	−	0.838197i	$-0.683610\pi$
$911$	−32.9215	−1.09074	−0.545368	+	0.838197i	$0.683610\pi$
$912$	0	0
$913$	1.71046	0.0566080
$914$	0	0
$915$	−10.8024	−0.357115
$916$	0	0
$917$	7.44248	0.245772
$918$	0	0
$919$	49.2083	1.62323	0.811617	−	0.584191i	$-0.198588\pi$
$919$	49.2083	1.62323	0.811617	+	0.584191i	$0.198588\pi$
$920$	0	0
$921$	−96.5373	−3.18101
$922$	0	0
$923$	−1.48132	−0.0487583
$924$	0	0
$925$	42.1842	1.38701
$926$	0	0
$927$	−63.4450	−2.08381
$928$	0	0
$929$	33.2081	1.08952	0.544761	−	0.838591i	$-0.316620\pi$
$929$	33.2081	1.08952	0.544761	+	0.838591i	$0.316620\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	28.2043	0.923369
$934$	0	0
$935$	−2.65637	−0.0868726
$936$	0	0
$937$	−35.2525	−1.15165	−0.575824	−	0.817574i	$-0.695319\pi$
$937$	−35.2525	−1.15165	−0.575824	+	0.817574i	$0.695319\pi$
$938$	0	0
$939$	−83.7729	−2.73382
$940$	0	0
$941$	−48.0258	−1.56560	−0.782799	−	0.622275i	$-0.786209\pi$
$941$	−48.0258	−1.56560	−0.782799	+	0.622275i	$0.786209\pi$
$942$	0	0
$943$	11.9809	0.390152
$944$	0	0
$945$	7.21502	0.234704
$946$	0	0
$947$	36.5688	1.18833	0.594163	−	0.804345i	$-0.297483\pi$
$947$	36.5688	1.18833	0.594163	+	0.804345i	$0.297483\pi$
$948$	0	0
$949$	1.83508	0.0595691
$950$	0	0
$951$	−27.4300	−0.889480
$952$	0	0
$953$	38.2681	1.23963	0.619813	−	0.784749i	$-0.287208\pi$
$953$	38.2681	1.23963	0.619813	+	0.784749i	$0.287208\pi$
$954$	0	0
$955$	2.89515	0.0936848
$956$	0	0
$957$	−24.6016	−0.795256
$958$	0	0
$959$	15.7877	0.509811
$960$	0	0
$961$	7.34184	0.236834
$962$	0	0
$963$	−18.8529	−0.607525
$964$	0	0
$965$	1.02836	0.0331040
$966$	0	0
$967$	13.0736	0.420417	0.210209	−	0.977657i	$-0.432586\pi$
$967$	13.0736	0.420417	0.210209	+	0.977657i	$0.432586\pi$
$968$	0	0
$969$	12.7950	0.411034
$970$	0	0
$971$	−36.3540	−1.16665	−0.583327	−	0.812237i	$-0.698250\pi$
$971$	−36.3540	−1.16665	−0.583327	+	0.812237i	$0.698250\pi$
$972$	0	0
$973$	−3.58916	−0.115063
$974$	0	0
$975$	2.23936	0.0717170
$976$	0	0
$977$	−34.6068	−1.10717	−0.553585	−	0.832793i	$-0.686741\pi$
$977$	−34.6068	−1.10717	−0.553585	+	0.832793i	$0.686741\pi$
$978$	0	0
$979$	3.53918	0.113113
$980$	0	0
$981$	−65.6233	−2.09519
$982$	0	0
$983$	11.2142	0.357676	0.178838	−	0.983879i	$-0.442766\pi$
$983$	11.2142	0.357676	0.178838	+	0.983879i	$0.442766\pi$
$984$	0	0
$985$	−13.7673	−0.438663
$986$	0	0
$987$	−5.23167	−0.166526
$988$	0	0
$989$	16.8790	0.536722
$990$	0	0
$991$	−49.8880	−1.58474	−0.792372	−	0.610038i	$-0.791154\pi$
$991$	−49.8880	−1.58474	−0.792372	+	0.610038i	$0.791154\pi$
$992$	0	0
$993$	21.7875	0.691406
$994$	0	0
$995$	−6.64461	−0.210648
$996$	0	0
$997$	−19.1301	−0.605857	−0.302928	−	0.953013i	$-0.597964\pi$
$997$	−19.1301	−0.605857	−0.302928	+	0.953013i	$0.597964\pi$
$998$	0	0
$999$	95.0464	3.00713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.by.1.1		5
4.3	odd	2		8512.2.a.bx.1.5		5
8.3	odd	2		1064.2.a.i.1.1	✓	5
8.5	even	2		2128.2.a.v.1.5		5
24.11	even	2		9576.2.a.cm.1.3		5
56.27	even	2		7448.2.a.bl.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.i.1.1	✓	5	8.3	odd	2
2128.2.a.v.1.5		5	8.5	even	2
7448.2.a.bl.1.5		5	56.27	even	2
8512.2.a.bx.1.5		5	4.3	odd	2
8512.2.a.by.1.1		5	1.1	even	1	trivial
9576.2.a.cm.1.3		5	24.11	even	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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05218 q^{3} -0.712910 q^{5} +1.00000 q^{7} +6.31583 q^{9} +O(q^{10})\)	\(q-3.05218 q^{3} -0.712910 q^{5} +1.00000 q^{7} +6.31583 q^{9} -0.888842 q^{11} +0.163342 q^{13} +2.17593 q^{15} -4.19208 q^{17} +1.00000 q^{19} -3.05218 q^{21} +2.16334 q^{23} -4.49176 q^{25} -10.1205 q^{27} -9.06833 q^{29} +6.19208 q^{31} +2.71291 q^{33} -0.712910 q^{35} -9.39146 q^{37} -0.498549 q^{39} +5.53814 q^{41} +7.80230 q^{43} -4.50262 q^{45} +1.71408 q^{47} +1.00000 q^{49} +12.7950 q^{51} +7.33197 q^{53} +0.633664 q^{55} -3.05218 q^{57} +11.0971 q^{59} -4.96447 q^{61} +6.31583 q^{63} -0.116448 q^{65} +7.01376 q^{67} -6.60292 q^{69} -9.06884 q^{71} +11.2346 q^{73} +13.7097 q^{75} -0.888842 q^{77} -9.55480 q^{79} +11.9422 q^{81} -1.92437 q^{83} +2.98857 q^{85} +27.6782 q^{87} -3.98178 q^{89} +0.163342 q^{91} -18.8994 q^{93} -0.712910 q^{95} -14.1703 q^{97} -5.61377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 3 * q^5 + 5 * q^7 + 7 * q^9	\( 5 q - 3 q^{5} + 5 q^{7} + 7 q^{9} + 6 q^{11} - 4 q^{13} + q^{15} + 9 q^{17} + 5 q^{19} + 6 q^{23} + 12 q^{25} - 10 q^{29} + q^{31} + 13 q^{33} - 3 q^{35} - 17 q^{37} - 9 q^{39} + 24 q^{41} + 15 q^{43} - 3 q^{45} - 5 q^{47} + 5 q^{49} + 10 q^{51} - 8 q^{53} + 19 q^{55} - 5 q^{59} - 9 q^{61} + 7 q^{63} + 24 q^{65} + 17 q^{67} - 9 q^{69} + q^{71} - 17 q^{73} + 12 q^{75} + 6 q^{77} - 13 q^{79} + 21 q^{81} - 15 q^{83} + 7 q^{85} + 20 q^{87} + 29 q^{89} - 4 q^{91} - 10 q^{93} - 3 q^{95} - 7 q^{97} - 19 q^{99}+O(q^{100}) \) 5 * q - 3 * q^5 + 5 * q^7 + 7 * q^9 + 6 * q^11 - 4 * q^13 + q^15 + 9 * q^17 + 5 * q^19 + 6 * q^23 + 12 * q^25 - 10 * q^29 + q^31 + 13 * q^33 - 3 * q^35 - 17 * q^37 - 9 * q^39 + 24 * q^41 + 15 * q^43 - 3 * q^45 - 5 * q^47 + 5 * q^49 + 10 * q^51 - 8 * q^53 + 19 * q^55 - 5 * q^59 - 9 * q^61 + 7 * q^63 + 24 * q^65 + 17 * q^67 - 9 * q^69 + q^71 - 17 * q^73 + 12 * q^75 + 6 * q^77 - 13 * q^79 + 21 * q^81 - 15 * q^83 + 7 * q^85 + 20 * q^87 + 29 * q^89 - 4 * q^91 - 10 * q^93 - 3 * q^95 - 7 * q^97 - 19 * q^99

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