LMFDB - Embedded newform 8624.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} -1.00000 q^{11} +3.23607 q^{13} +1.52786 q^{15} -2.47214 q^{17} -7.23607 q^{19} -4.00000 q^{23} -3.47214 q^{25} -5.52786 q^{27} +4.47214 q^{29} +2.00000 q^{31} -1.23607 q^{33} +6.94427 q^{37} +4.00000 q^{39} +2.47214 q^{41} +10.4721 q^{43} -1.81966 q^{45} -2.00000 q^{47} -3.05573 q^{51} +8.47214 q^{53} -1.23607 q^{55} -8.94427 q^{57} +2.76393 q^{59} +0.763932 q^{61} +4.00000 q^{65} -11.4164 q^{67} -4.94427 q^{69} -6.47214 q^{71} -12.9443 q^{73} -4.29180 q^{75} -2.41641 q^{81} -12.1803 q^{83} -3.05573 q^{85} +5.52786 q^{87} -10.0000 q^{89} +2.47214 q^{93} -8.94427 q^{95} -12.4721 q^{97} +1.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 12 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^23 + 2 * q^25 - 20 * q^27 + 4 * q^31 + 2 * q^33 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 - 26 * q^45 - 4 * q^47 - 24 * q^51 + 8 * q^53 + 2 * q^55 + 10 * q^59 + 6 * q^61 + 8 * q^65 + 4 * q^67 + 8 * q^69 - 4 * q^71 - 8 * q^73 - 22 * q^75 + 22 * q^81 - 2 * q^83 - 24 * q^85 + 20 * q^87 - 20 * q^89 - 4 * q^93 - 16 * q^97 - 6 * 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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	0	0
$15$	1.52786	0.394493
$16$	0	0
$17$	−2.47214	−0.599581	−0.299791	−	0.954005i	$-0.596917\pi$
$17$	−2.47214	−0.599581	−0.299791	+	0.954005i	$0.596917\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	−1.23607	−0.215172
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	10.4721	1.59699	0.798493	−	0.602004i	$-0.205631\pi$
$43$	10.4721	1.59699	0.798493	+	0.602004i	$0.205631\pi$
$44$	0	0
$45$	−1.81966	−0.271259
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.05573	−0.427888
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	−1.23607	−0.166671
$56$	0	0
$57$	−8.94427	−1.18470
$58$	0	0
$59$	2.76393	0.359833	0.179917	−	0.983682i	$-0.442417\pi$
$59$	2.76393	0.359833	0.179917	+	0.983682i	$0.442417\pi$
$60$	0	0
$61$	0.763932	0.0978115	0.0489057	−	0.998803i	$-0.484427\pi$
$61$	0.763932	0.0978115	0.0489057	+	0.998803i	$0.484427\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−11.4164	−1.39474	−0.697368	−	0.716713i	$-0.745646\pi$
$67$	−11.4164	−1.39474	−0.697368	+	0.716713i	$0.745646\pi$
$68$	0	0
$69$	−4.94427	−0.595220
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	−12.9443	−1.51501	−0.757506	−	0.652828i	$-0.773582\pi$
$73$	−12.9443	−1.51501	−0.757506	+	0.652828i	$0.773582\pi$
$74$	0	0
$75$	−4.29180	−0.495574
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−12.1803	−1.33697	−0.668483	−	0.743727i	$-0.733056\pi$
$83$	−12.1803	−1.33697	−0.668483	+	0.743727i	$0.733056\pi$
$84$	0	0
$85$	−3.05573	−0.331440
$86$	0	0
$87$	5.52786	0.592649
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.47214	0.256349
$94$	0	0
$95$	−8.94427	−0.917663
$96$	0	0
$97$	−12.4721	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$98$	0	0
$99$	1.47214	0.147955
$100$	0	0
$101$	−8.18034	−0.813974	−0.406987	−	0.913434i	$-0.633421\pi$
$101$	−8.18034	−0.813974	−0.406987	+	0.913434i	$0.633421\pi$
$102$	0	0
$103$	−14.9443	−1.47250	−0.736251	−	0.676708i	$-0.763406\pi$
$103$	−14.9443	−1.47250	−0.736251	+	0.676708i	$0.763406\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	8.58359	0.814719
$112$	0	0
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	0	0
$115$	−4.94427	−0.461056
$116$	0	0
$117$	−4.76393	−0.440426
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.05573	0.275526
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	12.9443	1.13968
$130$	0	0
$131$	4.76393	0.416227	0.208113	−	0.978105i	$-0.433268\pi$
$131$	4.76393	0.416227	0.208113	+	0.978105i	$0.433268\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.83282	−0.588075
$136$	0	0
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	0	0
$139$	−21.7082	−1.84127	−0.920633	−	0.390429i	$-0.872327\pi$
$139$	−21.7082	−1.84127	−0.920633	+	0.390429i	$0.872327\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	0	0
$143$	−3.23607	−0.270614
$144$	0	0
$145$	5.52786	0.459064
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.3607	−1.83186	−0.915929	−	0.401340i	$-0.868545\pi$
$149$	−22.3607	−1.83186	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	3.63932	0.294222
$154$	0	0
$155$	2.47214	0.198567
$156$	0	0
$157$	12.6525	1.00978	0.504889	−	0.863184i	$-0.331534\pi$
$157$	12.6525	1.00978	0.504889	+	0.863184i	$0.331534\pi$
$158$	0	0
$159$	10.4721	0.830494
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.4164	1.52081	0.760405	−	0.649449i	$-0.225000\pi$
$163$	19.4164	1.52081	0.760405	+	0.649449i	$0.225000\pi$
$164$	0	0
$165$	−1.52786	−0.118944
$166$	0	0
$167$	11.4164	0.883428	0.441714	−	0.897156i	$-0.354371\pi$
$167$	11.4164	0.883428	0.441714	+	0.897156i	$0.354371\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	10.6525	0.814615
$172$	0	0
$173$	3.23607	0.246034	0.123017	−	0.992405i	$-0.460743\pi$
$173$	3.23607	0.246034	0.123017	+	0.992405i	$0.460743\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.41641	0.256793
$178$	0	0
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	−9.23607	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$181$	−9.23607	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$182$	0	0
$183$	0.944272	0.0698026
$184$	0	0
$185$	8.58359	0.631078
$186$	0	0
$187$	2.47214	0.180780
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.47214	0.178877	0.0894387	−	0.995992i	$-0.471493\pi$
$191$	2.47214	0.178877	0.0894387	+	0.995992i	$0.471493\pi$
$192$	0	0
$193$	−14.9443	−1.07571	−0.537856	−	0.843037i	$-0.680766\pi$
$193$	−14.9443	−1.07571	−0.537856	+	0.843037i	$0.680766\pi$
$194$	0	0
$195$	4.94427	0.354067
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	18.9443	1.34292	0.671462	−	0.741039i	$-0.265667\pi$
$199$	18.9443	1.34292	0.671462	+	0.741039i	$0.265667\pi$
$200$	0	0
$201$	−14.1115	−0.995345
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.05573	0.213421
$206$	0	0
$207$	5.88854	0.409282
$208$	0	0
$209$	7.23607	0.500529
$210$	0	0
$211$	13.5279	0.931297	0.465648	−	0.884970i	$-0.345821\pi$
$211$	13.5279	0.931297	0.465648	+	0.884970i	$0.345821\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	12.9443	0.882792
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−0.472136	−0.0316166	−0.0158083	−	0.999875i	$-0.505032\pi$
$223$	−0.472136	−0.0316166	−0.0158083	+	0.999875i	$0.505032\pi$
$224$	0	0
$225$	5.11146	0.340764
$226$	0	0
$227$	−19.2361	−1.27674	−0.638371	−	0.769729i	$-0.720392\pi$
$227$	−19.2361	−1.27674	−0.638371	+	0.769729i	$0.720392\pi$
$228$	0	0
$229$	−17.2361	−1.13899	−0.569496	−	0.821994i	$-0.692861\pi$
$229$	−17.2361	−1.13899	−0.569496	+	0.821994i	$0.692861\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	−2.47214	−0.161264
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−15.4164	−0.993058	−0.496529	−	0.868020i	$-0.665392\pi$
$241$	−15.4164	−0.993058	−0.496529	+	0.868020i	$0.665392\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−23.4164	−1.48995
$248$	0	0
$249$	−15.0557	−0.954118
$250$	0	0
$251$	29.2361	1.84536	0.922682	−	0.385562i	$-0.125992\pi$
$251$	29.2361	1.84536	0.922682	+	0.385562i	$0.125992\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	−3.77709	−0.236530
$256$	0	0
$257$	−6.94427	−0.433172	−0.216586	−	0.976264i	$-0.569492\pi$
$257$	−6.94427	−0.433172	−0.216586	+	0.976264i	$0.569492\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.58359	−0.407514
$262$	0	0
$263$	4.94427	0.304877	0.152438	−	0.988313i	$-0.451287\pi$
$263$	4.94427	0.304877	0.152438	+	0.988313i	$0.451287\pi$
$264$	0	0
$265$	10.4721	0.643298
$266$	0	0
$267$	−12.3607	−0.756461
$268$	0	0
$269$	22.7639	1.38794	0.693971	−	0.720003i	$-0.255860\pi$
$269$	22.7639	1.38794	0.693971	+	0.720003i	$0.255860\pi$
$270$	0	0
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.47214	0.209378
$276$	0	0
$277$	3.52786	0.211969	0.105984	−	0.994368i	$-0.466201\pi$
$277$	3.52786	0.211969	0.105984	+	0.994368i	$0.466201\pi$
$278$	0	0
$279$	−2.94427	−0.176269
$280$	0	0
$281$	28.8328	1.72002	0.860011	−	0.510276i	$-0.170457\pi$
$281$	28.8328	1.72002	0.860011	+	0.510276i	$0.170457\pi$
$282$	0	0
$283$	14.6525	0.870999	0.435500	−	0.900189i	$-0.356572\pi$
$283$	14.6525	0.870999	0.435500	+	0.900189i	$0.356572\pi$
$284$	0	0
$285$	−11.0557	−0.654885
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−15.4164	−0.903726
$292$	0	0
$293$	26.6525	1.55705	0.778527	−	0.627611i	$-0.215967\pi$
$293$	26.6525	1.55705	0.778527	+	0.627611i	$0.215967\pi$
$294$	0	0
$295$	3.41641	0.198911
$296$	0	0
$297$	5.52786	0.320759
$298$	0	0
$299$	−12.9443	−0.748587
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.1115	−0.580888
$304$	0	0
$305$	0.944272	0.0540689
$306$	0	0
$307$	−26.0689	−1.48783	−0.743915	−	0.668274i	$-0.767033\pi$
$307$	−26.0689	−1.48783	−0.743915	+	0.668274i	$0.767033\pi$
$308$	0	0
$309$	−18.4721	−1.05084
$310$	0	0
$311$	−21.4164	−1.21441	−0.607207	−	0.794544i	$-0.707710\pi$
$311$	−21.4164	−1.21441	−0.607207	+	0.794544i	$0.707710\pi$
$312$	0	0
$313$	−19.5279	−1.10378	−0.551890	−	0.833917i	$-0.686093\pi$
$313$	−19.5279	−1.10378	−0.551890	+	0.833917i	$0.686093\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.9443	−1.73800	−0.869002	−	0.494809i	$-0.835238\pi$
$317$	−30.9443	−1.73800	−0.869002	+	0.494809i	$0.835238\pi$
$318$	0	0
$319$	−4.47214	−0.250392
$320$	0	0
$321$	−3.05573	−0.170554
$322$	0	0
$323$	17.8885	0.995345
$324$	0	0
$325$	−11.2361	−0.623265
$326$	0	0
$327$	−12.3607	−0.683547
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.9443	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$331$	16.9443	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$332$	0	0
$333$	−10.2229	−0.560212
$334$	0	0
$335$	−14.1115	−0.770991
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−0.583592	−0.0316964
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.11146	−0.329030
$346$	0	0
$347$	−2.47214	−0.132711	−0.0663556	−	0.997796i	$-0.521137\pi$
$347$	−2.47214	−0.132711	−0.0663556	+	0.997796i	$0.521137\pi$
$348$	0	0
$349$	−21.7082	−1.16201	−0.581007	−	0.813899i	$-0.697341\pi$
$349$	−21.7082	−1.16201	−0.581007	+	0.813899i	$0.697341\pi$
$350$	0	0
$351$	−17.8885	−0.954820
$352$	0	0
$353$	17.0557	0.907785	0.453892	−	0.891056i	$-0.350035\pi$
$353$	17.0557	0.907785	0.453892	+	0.891056i	$0.350035\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.8328	−1.41618	−0.708091	−	0.706121i	$-0.750443\pi$
$359$	−26.8328	−1.41618	−0.708091	+	0.706121i	$0.750443\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	1.23607	0.0648767
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	−5.41641	−0.282734	−0.141367	−	0.989957i	$-0.545150\pi$
$367$	−5.41641	−0.282734	−0.141367	+	0.989957i	$0.545150\pi$
$368$	0	0
$369$	−3.63932	−0.189455
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−12.9443	−0.668439
$376$	0	0
$377$	14.4721	0.745353
$378$	0	0
$379$	−14.4721	−0.743384	−0.371692	−	0.928356i	$-0.621222\pi$
$379$	−14.4721	−0.743384	−0.371692	+	0.928356i	$0.621222\pi$
$380$	0	0
$381$	14.8328	0.759908
$382$	0	0
$383$	−23.8885	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$383$	−23.8885	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.4164	−0.783660
$388$	0	0
$389$	33.4164	1.69428	0.847140	−	0.531370i	$-0.178323\pi$
$389$	33.4164	1.69428	0.847140	+	0.531370i	$0.178323\pi$
$390$	0	0
$391$	9.88854	0.500085
$392$	0	0
$393$	5.88854	0.297038
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.7082	1.18988	0.594940	−	0.803770i	$-0.297176\pi$
$397$	23.7082	1.18988	0.594940	+	0.803770i	$0.297176\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.3607	0.717138	0.358569	−	0.933503i	$-0.383265\pi$
$401$	14.3607	0.717138	0.358569	+	0.933503i	$0.383265\pi$
$402$	0	0
$403$	6.47214	0.322400
$404$	0	0
$405$	−2.98684	−0.148417
$406$	0	0
$407$	−6.94427	−0.344215
$408$	0	0
$409$	−3.41641	−0.168930	−0.0844652	−	0.996426i	$-0.526918\pi$
$409$	−3.41641	−0.168930	−0.0844652	+	0.996426i	$0.526918\pi$
$410$	0	0
$411$	−24.5836	−1.21262
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−15.0557	−0.739057
$416$	0	0
$417$	−26.8328	−1.31401
$418$	0	0
$419$	17.2361	0.842037	0.421019	−	0.907052i	$-0.361673\pi$
$419$	17.2361	0.842037	0.421019	+	0.907052i	$0.361673\pi$
$420$	0	0
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	0	0
$423$	2.94427	0.143155
$424$	0	0
$425$	8.58359	0.416365
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−23.0557	−1.11056	−0.555278	−	0.831665i	$-0.687388\pi$
$431$	−23.0557	−1.11056	−0.555278	+	0.831665i	$0.687388\pi$
$432$	0	0
$433$	−28.4721	−1.36828	−0.684142	−	0.729349i	$-0.739823\pi$
$433$	−28.4721	−1.36828	−0.684142	+	0.729349i	$0.739823\pi$
$434$	0	0
$435$	6.83282	0.327608
$436$	0	0
$437$	28.9443	1.38459
$438$	0	0
$439$	8.94427	0.426887	0.213443	−	0.976955i	$-0.431532\pi$
$439$	8.94427	0.426887	0.213443	+	0.976955i	$0.431532\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.9443	1.18514	0.592569	−	0.805520i	$-0.298114\pi$
$443$	24.9443	1.18514	0.592569	+	0.805520i	$0.298114\pi$
$444$	0	0
$445$	−12.3607	−0.585952
$446$	0	0
$447$	−27.6393	−1.30729
$448$	0	0
$449$	18.9443	0.894035	0.447018	−	0.894525i	$-0.352486\pi$
$449$	18.9443	0.894035	0.447018	+	0.894525i	$0.352486\pi$
$450$	0	0
$451$	−2.47214	−0.116408
$452$	0	0
$453$	−14.8328	−0.696906
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.9443	1.26040	0.630200	−	0.776433i	$-0.282973\pi$
$457$	26.9443	1.26040	0.630200	+	0.776433i	$0.282973\pi$
$458$	0	0
$459$	13.6656	0.637857
$460$	0	0
$461$	−24.7639	−1.15337	−0.576686	−	0.816966i	$-0.695654\pi$
$461$	−24.7639	−1.15337	−0.576686	+	0.816966i	$0.695654\pi$
$462$	0	0
$463$	30.4721	1.41616	0.708080	−	0.706132i	$-0.249562\pi$
$463$	30.4721	1.41616	0.708080	+	0.706132i	$0.249562\pi$
$464$	0	0
$465$	3.05573	0.141706
$466$	0	0
$467$	−27.1246	−1.25518	−0.627589	−	0.778545i	$-0.715958\pi$
$467$	−27.1246	−1.25518	−0.627589	+	0.778545i	$0.715958\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	15.6393	0.720622
$472$	0	0
$473$	−10.4721	−0.481509
$474$	0	0
$475$	25.1246	1.15280
$476$	0	0
$477$	−12.4721	−0.571060
$478$	0	0
$479$	12.3607	0.564774	0.282387	−	0.959301i	$-0.408874\pi$
$479$	12.3607	0.564774	0.282387	+	0.959301i	$0.408874\pi$
$480$	0	0
$481$	22.4721	1.02464
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.4164	−0.700023
$486$	0	0
$487$	−16.9443	−0.767818	−0.383909	−	0.923371i	$-0.625422\pi$
$487$	−16.9443	−0.767818	−0.383909	+	0.923371i	$0.625422\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	16.9443	0.764684	0.382342	−	0.924021i	$-0.375118\pi$
$491$	16.9443	0.764684	0.382342	+	0.924021i	$0.375118\pi$
$492$	0	0
$493$	−11.0557	−0.497925
$494$	0	0
$495$	1.81966	0.0817876
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.3607	1.44866	0.724331	−	0.689452i	$-0.242149\pi$
$499$	32.3607	1.44866	0.724331	+	0.689452i	$0.242149\pi$
$500$	0	0
$501$	14.1115	0.630453
$502$	0	0
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−10.1115	−0.449954
$506$	0	0
$507$	−3.12461	−0.138769
$508$	0	0
$509$	−24.0689	−1.06683	−0.533417	−	0.845852i	$-0.679092\pi$
$509$	−24.0689	−1.06683	−0.533417	+	0.845852i	$0.679092\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	−18.4721	−0.813980
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	10.3607	0.453910	0.226955	−	0.973905i	$-0.427123\pi$
$521$	10.3607	0.453910	0.226955	+	0.973905i	$0.427123\pi$
$522$	0	0
$523$	−14.2918	−0.624937	−0.312468	−	0.949928i	$-0.601156\pi$
$523$	−14.2918	−0.624937	−0.312468	+	0.949928i	$0.601156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.94427	−0.215376
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.06888	−0.176575
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−3.05573	−0.132111
$536$	0	0
$537$	11.0557	0.477090
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−26.9443	−1.15842	−0.579212	−	0.815177i	$-0.696640\pi$
$541$	−26.9443	−1.15842	−0.579212	+	0.815177i	$0.696640\pi$
$542$	0	0
$543$	−11.4164	−0.489925
$544$	0	0
$545$	−12.3607	−0.529473
$546$	0	0
$547$	0.944272	0.0403742	0.0201871	−	0.999796i	$-0.493574\pi$
$547$	0.944272	0.0403742	0.0201871	+	0.999796i	$0.493574\pi$
$548$	0	0
$549$	−1.12461	−0.0479973
$550$	0	0
$551$	−32.3607	−1.37861
$552$	0	0
$553$	0	0
$554$	0	0
$555$	10.6099	0.450365
$556$	0	0
$557$	24.8328	1.05220	0.526100	−	0.850423i	$-0.323654\pi$
$557$	24.8328	1.05220	0.526100	+	0.850423i	$0.323654\pi$
$558$	0	0
$559$	33.8885	1.43333
$560$	0	0
$561$	3.05573	0.129013
$562$	0	0
$563$	31.2361	1.31644	0.658222	−	0.752824i	$-0.271309\pi$
$563$	31.2361	1.31644	0.658222	+	0.752824i	$0.271309\pi$
$564$	0	0
$565$	−0.583592	−0.0245519
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.8328	−1.54411	−0.772056	−	0.635555i	$-0.780772\pi$
$569$	−36.8328	−1.54411	−0.772056	+	0.635555i	$0.780772\pi$
$570$	0	0
$571$	10.1115	0.423151	0.211576	−	0.977362i	$-0.432141\pi$
$571$	10.1115	0.423151	0.211576	+	0.977362i	$0.432141\pi$
$572$	0	0
$573$	3.05573	0.127655
$574$	0	0
$575$	13.8885	0.579192
$576$	0	0
$577$	−26.9443	−1.12170	−0.560852	−	0.827916i	$-0.689526\pi$
$577$	−26.9443	−1.12170	−0.560852	+	0.827916i	$0.689526\pi$
$578$	0	0
$579$	−18.4721	−0.767676
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.47214	−0.350880
$584$	0	0
$585$	−5.88854	−0.243461
$586$	0	0
$587$	−5.81966	−0.240203	−0.120102	−	0.992762i	$-0.538322\pi$
$587$	−5.81966	−0.240203	−0.120102	+	0.992762i	$0.538322\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	0	0
$591$	22.2492	0.915211
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.4164	0.958370
$598$	0	0
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	34.8328	1.42086	0.710430	−	0.703768i	$-0.248500\pi$
$601$	34.8328	1.42086	0.710430	+	0.703768i	$0.248500\pi$
$602$	0	0
$603$	16.8065	0.684414
$604$	0	0
$605$	1.23607	0.0502533
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.47214	−0.261835
$612$	0	0
$613$	28.4721	1.14998	0.574989	−	0.818161i	$-0.305006\pi$
$613$	28.4721	1.14998	0.574989	+	0.818161i	$0.305006\pi$
$614$	0	0
$615$	3.77709	0.152307
$616$	0	0
$617$	21.4164	0.862192	0.431096	−	0.902306i	$-0.358127\pi$
$617$	21.4164	0.862192	0.431096	+	0.902306i	$0.358127\pi$
$618$	0	0
$619$	18.5410	0.745227	0.372613	−	0.927987i	$-0.378462\pi$
$619$	18.5410	0.745227	0.372613	+	0.927987i	$0.378462\pi$
$620$	0	0
$621$	22.1115	0.887302
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	8.94427	0.357200
$628$	0	0
$629$	−17.1672	−0.684500
$630$	0	0
$631$	31.4164	1.25067	0.625334	−	0.780357i	$-0.284963\pi$
$631$	31.4164	1.25067	0.625334	+	0.780357i	$0.284963\pi$
$632$	0	0
$633$	16.7214	0.664614
$634$	0	0
$635$	14.8328	0.588622
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.52786	0.376916
$640$	0	0
$641$	27.5279	1.08729	0.543643	−	0.839317i	$-0.317045\pi$
$641$	27.5279	1.08729	0.543643	+	0.839317i	$0.317045\pi$
$642$	0	0
$643$	−18.7639	−0.739977	−0.369989	−	0.929036i	$-0.620638\pi$
$643$	−18.7639	−0.739977	−0.369989	+	0.929036i	$0.620638\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	−28.8328	−1.13353	−0.566767	−	0.823878i	$-0.691806\pi$
$647$	−28.8328	−1.13353	−0.566767	+	0.823878i	$0.691806\pi$
$648$	0	0
$649$	−2.76393	−0.108494
$650$	0	0
$651$	0	0
$652$	0	0
$653$	46.3607	1.81423	0.907117	−	0.420879i	$-0.138278\pi$
$653$	46.3607	1.81423	0.907117	+	0.420879i	$0.138278\pi$
$654$	0	0
$655$	5.88854	0.230084
$656$	0	0
$657$	19.0557	0.743435
$658$	0	0
$659$	−16.5836	−0.646005	−0.323003	−	0.946398i	$-0.604692\pi$
$659$	−16.5836	−0.646005	−0.323003	+	0.946398i	$0.604692\pi$
$660$	0	0
$661$	3.12461	0.121533	0.0607667	−	0.998152i	$-0.480645\pi$
$661$	3.12461	0.121533	0.0607667	+	0.998152i	$0.480645\pi$
$662$	0	0
$663$	−9.88854	−0.384039
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.8885	−0.692647
$668$	0	0
$669$	−0.583592	−0.0225630
$670$	0	0
$671$	−0.763932	−0.0294913
$672$	0	0
$673$	−3.88854	−0.149892	−0.0749462	−	0.997188i	$-0.523878\pi$
$673$	−3.88854	−0.149892	−0.0749462	+	0.997188i	$0.523878\pi$
$674$	0	0
$675$	19.1935	0.738758
$676$	0	0
$677$	26.0689	1.00191	0.500954	−	0.865474i	$-0.332982\pi$
$677$	26.0689	1.00191	0.500954	+	0.865474i	$0.332982\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−23.7771	−0.911140
$682$	0	0
$683$	−32.9443	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$684$	0	0
$685$	−24.5836	−0.939291
$686$	0	0
$687$	−21.3050	−0.812835
$688$	0	0
$689$	27.4164	1.04448
$690$	0	0
$691$	12.6525	0.481323	0.240661	−	0.970609i	$-0.422636\pi$
$691$	12.6525	0.481323	0.240661	+	0.970609i	$0.422636\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−26.8328	−1.01783
$696$	0	0
$697$	−6.11146	−0.231488
$698$	0	0
$699$	−18.4721	−0.698680
$700$	0	0
$701$	−42.7214	−1.61356	−0.806782	−	0.590850i	$-0.798793\pi$
$701$	−42.7214	−1.61356	−0.806782	+	0.590850i	$0.798793\pi$
$702$	0	0
$703$	−50.2492	−1.89519
$704$	0	0
$705$	−3.05573	−0.115085
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.47214	0.167955	0.0839773	−	0.996468i	$-0.473238\pi$
$709$	4.47214	0.167955	0.0839773	+	0.996468i	$0.473238\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−24.7214	−0.923236
$718$	0	0
$719$	16.8328	0.627758	0.313879	−	0.949463i	$-0.398371\pi$
$719$	16.8328	0.627758	0.313879	+	0.949463i	$0.398371\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19.0557	−0.708690
$724$	0	0
$725$	−15.5279	−0.576690
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−25.8885	−0.957522
$732$	0	0
$733$	−49.1246	−1.81446	−0.907229	−	0.420636i	$-0.861807\pi$
$733$	−49.1246	−1.81446	−0.907229	+	0.420636i	$0.861807\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.4164	0.420529
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−28.9443	−1.06329
$742$	0	0
$743$	−21.8885	−0.803013	−0.401506	−	0.915856i	$-0.631513\pi$
$743$	−21.8885	−0.803013	−0.401506	+	0.915856i	$0.631513\pi$
$744$	0	0
$745$	−27.6393	−1.01263
$746$	0	0
$747$	17.9311	0.656065
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.9443	0.618305	0.309153	−	0.951012i	$-0.399955\pi$
$751$	16.9443	0.618305	0.309153	+	0.951012i	$0.399955\pi$
$752$	0	0
$753$	36.1378	1.31693
$754$	0	0
$755$	−14.8328	−0.539821
$756$	0	0
$757$	−23.3050	−0.847033	−0.423516	−	0.905888i	$-0.639204\pi$
$757$	−23.3050	−0.847033	−0.423516	+	0.905888i	$0.639204\pi$
$758$	0	0
$759$	4.94427	0.179466
$760$	0	0
$761$	11.4164	0.413844	0.206922	−	0.978357i	$-0.433655\pi$
$761$	11.4164	0.413844	0.206922	+	0.978357i	$0.433655\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.49845	0.162642
$766$	0	0
$767$	8.94427	0.322959
$768$	0	0
$769$	43.4164	1.56564	0.782818	−	0.622251i	$-0.213782\pi$
$769$	43.4164	1.56564	0.782818	+	0.622251i	$0.213782\pi$
$770$	0	0
$771$	−8.58359	−0.309131
$772$	0	0
$773$	−15.7082	−0.564985	−0.282492	−	0.959270i	$-0.591161\pi$
$773$	−15.7082	−0.564985	−0.282492	+	0.959270i	$0.591161\pi$
$774$	0	0
$775$	−6.94427	−0.249446
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.8885	−0.640924
$780$	0	0
$781$	6.47214	0.231591
$782$	0	0
$783$	−24.7214	−0.883469
$784$	0	0
$785$	15.6393	0.558191
$786$	0	0
$787$	−28.1803	−1.00452	−0.502260	−	0.864716i	$-0.667498\pi$
$787$	−28.1803	−1.00452	−0.502260	+	0.864716i	$0.667498\pi$
$788$	0	0
$789$	6.11146	0.217574
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.47214	0.0877881
$794$	0	0
$795$	12.9443	0.459086
$796$	0	0
$797$	41.5967	1.47343	0.736716	−	0.676202i	$-0.236375\pi$
$797$	41.5967	1.47343	0.736716	+	0.676202i	$0.236375\pi$
$798$	0	0
$799$	4.94427	0.174916
$800$	0	0
$801$	14.7214	0.520154
$802$	0	0
$803$	12.9443	0.456793
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.1378	0.990496
$808$	0	0
$809$	−21.0557	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$809$	−21.0557	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$810$	0	0
$811$	4.76393	0.167284	0.0836421	−	0.996496i	$-0.473345\pi$
$811$	4.76393	0.167284	0.0836421	+	0.996496i	$0.473345\pi$
$812$	0	0
$813$	1.16718	0.0409349
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	−75.7771	−2.65110
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.41641	−0.0494330	−0.0247165	−	0.999695i	$-0.507868\pi$
$821$	−1.41641	−0.0494330	−0.0247165	+	0.999695i	$0.507868\pi$
$822$	0	0
$823$	46.2492	1.61215	0.806073	−	0.591816i	$-0.201589\pi$
$823$	46.2492	1.61215	0.806073	+	0.591816i	$0.201589\pi$
$824$	0	0
$825$	4.29180	0.149421
$826$	0	0
$827$	−16.9443	−0.589210	−0.294605	−	0.955619i	$-0.595188\pi$
$827$	−16.9443	−0.589210	−0.294605	+	0.955619i	$0.595188\pi$
$828$	0	0
$829$	−11.7082	−0.406643	−0.203321	−	0.979112i	$-0.565174\pi$
$829$	−11.7082	−0.406643	−0.203321	+	0.979112i	$0.565174\pi$
$830$	0	0
$831$	4.36068	0.151270
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14.1115	0.488347
$836$	0	0
$837$	−11.0557	−0.382142
$838$	0	0
$839$	16.8328	0.581133	0.290567	−	0.956855i	$-0.406156\pi$
$839$	16.8328	0.581133	0.290567	+	0.956855i	$0.406156\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	35.6393	1.22748
$844$	0	0
$845$	−3.12461	−0.107490
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18.1115	0.621584
$850$	0	0
$851$	−27.7771	−0.952186
$852$	0	0
$853$	−32.5410	−1.11418	−0.557092	−	0.830451i	$-0.688083\pi$
$853$	−32.5410	−1.11418	−0.557092	+	0.830451i	$0.688083\pi$
$854$	0	0
$855$	13.1672	0.450308
$856$	0	0
$857$	46.4721	1.58746	0.793729	−	0.608272i	$-0.208137\pi$
$857$	46.4721	1.58746	0.793729	+	0.608272i	$0.208137\pi$
$858$	0	0
$859$	15.1246	0.516045	0.258023	−	0.966139i	$-0.416929\pi$
$859$	15.1246	0.516045	0.258023	+	0.966139i	$0.416929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.583592	−0.0198657	−0.00993285	−	0.999951i	$-0.503162\pi$
$863$	−0.583592	−0.0198657	−0.00993285	+	0.999951i	$0.503162\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	−13.4590	−0.457091
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−36.9443	−1.25181
$872$	0	0
$873$	18.3607	0.621415
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.05573	0.305790	0.152895	−	0.988242i	$-0.451140\pi$
$877$	9.05573	0.305790	0.152895	+	0.988242i	$0.451140\pi$
$878$	0	0
$879$	32.9443	1.11118
$880$	0	0
$881$	−28.8328	−0.971402	−0.485701	−	0.874125i	$-0.661436\pi$
$881$	−28.8328	−0.971402	−0.485701	+	0.874125i	$0.661436\pi$
$882$	0	0
$883$	2.83282	0.0953318	0.0476659	−	0.998863i	$-0.484822\pi$
$883$	2.83282	0.0953318	0.0476659	+	0.998863i	$0.484822\pi$
$884$	0	0
$885$	4.22291	0.141952
$886$	0	0
$887$	−44.3607	−1.48949	−0.744743	−	0.667351i	$-0.767428\pi$
$887$	−44.3607	−1.48949	−0.744743	+	0.667351i	$0.767428\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.41641	0.0809527
$892$	0	0
$893$	14.4721	0.484292
$894$	0	0
$895$	11.0557	0.369552
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	8.94427	0.298308
$900$	0	0
$901$	−20.9443	−0.697755
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.4164	−0.379494
$906$	0	0
$907$	24.3607	0.808883	0.404442	−	0.914564i	$-0.367466\pi$
$907$	24.3607	0.808883	0.404442	+	0.914564i	$0.367466\pi$
$908$	0	0
$909$	12.0426	0.399427
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	12.1803	0.403110
$914$	0	0
$915$	1.16718	0.0385859
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.1115	0.729390	0.364695	−	0.931127i	$-0.381173\pi$
$919$	22.1115	0.729390	0.364695	+	0.931127i	$0.381173\pi$
$920$	0	0
$921$	−32.2229	−1.06178
$922$	0	0
$923$	−20.9443	−0.689389
$924$	0	0
$925$	−24.1115	−0.792780
$926$	0	0
$927$	22.0000	0.722575
$928$	0	0
$929$	−40.2492	−1.32053	−0.660267	−	0.751031i	$-0.729557\pi$
$929$	−40.2492	−1.32053	−0.660267	+	0.751031i	$0.729557\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−26.4721	−0.866659
$934$	0	0
$935$	3.05573	0.0999330
$936$	0	0
$937$	3.05573	0.0998263	0.0499131	−	0.998754i	$-0.484106\pi$
$937$	3.05573	0.0998263	0.0499131	+	0.998754i	$0.484106\pi$
$938$	0	0
$939$	−24.1378	−0.787706
$940$	0	0
$941$	11.8197	0.385310	0.192655	−	0.981267i	$-0.438290\pi$
$941$	11.8197	0.385310	0.192655	+	0.981267i	$0.438290\pi$
$942$	0	0
$943$	−9.88854	−0.322015
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.9443	−0.550615	−0.275307	−	0.961356i	$-0.588780\pi$
$947$	−16.9443	−0.550615	−0.275307	+	0.961356i	$0.588780\pi$
$948$	0	0
$949$	−41.8885	−1.35976
$950$	0	0
$951$	−38.2492	−1.24032
$952$	0	0
$953$	22.9443	0.743238	0.371619	−	0.928385i	$-0.378803\pi$
$953$	22.9443	0.743238	0.371619	+	0.928385i	$0.378803\pi$
$954$	0	0
$955$	3.05573	0.0988810
$956$	0	0
$957$	−5.52786	−0.178690
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	3.63932	0.117275
$964$	0	0
$965$	−18.4721	−0.594639
$966$	0	0
$967$	−45.8885	−1.47568	−0.737838	−	0.674978i	$-0.764153\pi$
$967$	−45.8885	−1.47568	−0.737838	+	0.674978i	$0.764153\pi$
$968$	0	0
$969$	22.1115	0.710322
$970$	0	0
$971$	50.5410	1.62194	0.810969	−	0.585089i	$-0.198940\pi$
$971$	50.5410	1.62194	0.810969	+	0.585089i	$0.198940\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−13.8885	−0.444789
$976$	0	0
$977$	−28.8328	−0.922443	−0.461222	−	0.887285i	$-0.652589\pi$
$977$	−28.8328	−0.922443	−0.461222	+	0.887285i	$0.652589\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	14.7214	0.470017
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	22.2492	0.708919
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.8885	−1.33198
$990$	0	0
$991$	0.360680	0.0114574	0.00572869	−	0.999984i	$-0.498176\pi$
$991$	0.360680	0.0114574	0.00572869	+	0.999984i	$0.498176\pi$
$992$	0	0
$993$	20.9443	0.664646
$994$	0	0
$995$	23.4164	0.742350
$996$	0	0
$997$	−24.1803	−0.765799	−0.382900	−	0.923790i	$-0.625074\pi$
$997$	−24.1803	−0.765799	−0.382900	+	0.923790i	$0.625074\pi$
$998$	0	0
$999$	−38.3870	−1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bf.1.2		2
4.3	odd	2		1078.2.a.w.1.1		2
7.6	odd	2		1232.2.a.p.1.1		2
12.11	even	2		9702.2.a.cu.1.1		2
28.3	even	6		1078.2.e.q.177.1		4
28.11	odd	6		1078.2.e.n.177.2		4
28.19	even	6		1078.2.e.q.67.1		4
28.23	odd	6		1078.2.e.n.67.2		4
28.27	even	2		154.2.a.d.1.2	✓	2
56.13	odd	2		4928.2.a.bk.1.2		2
56.27	even	2		4928.2.a.bt.1.1		2
84.83	odd	2		1386.2.a.m.1.2		2
140.27	odd	4		3850.2.c.q.1849.3		4
140.83	odd	4		3850.2.c.q.1849.2		4
140.139	even	2		3850.2.a.bj.1.1		2
308.307	odd	2		1694.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.d.1.2	✓	2	28.27	even	2
1078.2.a.w.1.1		2	4.3	odd	2
1078.2.e.n.67.2		4	28.23	odd	6
1078.2.e.n.177.2		4	28.11	odd	6
1078.2.e.q.67.1		4	28.19	even	6
1078.2.e.q.177.1		4	28.3	even	6
1232.2.a.p.1.1		2	7.6	odd	2
1386.2.a.m.1.2		2	84.83	odd	2
1694.2.a.l.1.2		2	308.307	odd	2
3850.2.a.bj.1.1		2	140.139	even	2
3850.2.c.q.1849.2		4	140.83	odd	4
3850.2.c.q.1849.3		4	140.27	odd	4
4928.2.a.bk.1.2		2	56.13	odd	2
4928.2.a.bt.1.1		2	56.27	even	2
8624.2.a.bf.1.2		2	1.1	even	1	trivial
9702.2.a.cu.1.1		2	12.11	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} -1.00000 q^{11} +3.23607 q^{13} +1.52786 q^{15} -2.47214 q^{17} -7.23607 q^{19} -4.00000 q^{23} -3.47214 q^{25} -5.52786 q^{27} +4.47214 q^{29} +2.00000 q^{31} -1.23607 q^{33} +6.94427 q^{37} +4.00000 q^{39} +2.47214 q^{41} +10.4721 q^{43} -1.81966 q^{45} -2.00000 q^{47} -3.05573 q^{51} +8.47214 q^{53} -1.23607 q^{55} -8.94427 q^{57} +2.76393 q^{59} +0.763932 q^{61} +4.00000 q^{65} -11.4164 q^{67} -4.94427 q^{69} -6.47214 q^{71} -12.9443 q^{73} -4.29180 q^{75} -2.41641 q^{81} -12.1803 q^{83} -3.05573 q^{85} +5.52786 q^{87} -10.0000 q^{89} +2.47214 q^{93} -8.94427 q^{95} -12.4721 q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 12 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^23 + 2 * q^25 - 20 * q^27 + 4 * q^31 + 2 * q^33 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 - 26 * q^45 - 4 * q^47 - 24 * q^51 + 8 * q^53 + 2 * q^55 + 10 * q^59 + 6 * q^61 + 8 * q^65 + 4 * q^67 + 8 * q^69 - 4 * q^71 - 8 * q^73 - 22 * q^75 + 22 * q^81 - 2 * q^83 - 24 * q^85 + 20 * q^87 - 20 * q^89 - 4 * q^93 - 16 * q^97 - 6 * q^99

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