LMFDB - Embedded newform 8512.2.a.e.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:	$1$
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:	$ 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			$1.61803$ 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)

$f(q)$	$=$	$q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +O(q^{10})$	$q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +5.85410 q^{11} -5.23607 q^{13} +9.47214 q^{15} -2.47214 q^{17} +1.00000 q^{19} -2.61803 q^{21} -5.70820 q^{23} +8.09017 q^{25} -2.23607 q^{27} +0.854102 q^{29} +4.47214 q^{31} -15.3262 q^{33} -3.61803 q^{35} +0.0901699 q^{37} +13.7082 q^{39} -6.09017 q^{41} -6.85410 q^{43} -13.9443 q^{45} -11.8541 q^{47} +1.00000 q^{49} +6.47214 q^{51} +3.38197 q^{53} -21.1803 q^{55} -2.61803 q^{57} -4.09017 q^{59} +7.85410 q^{61} +3.85410 q^{63} +18.9443 q^{65} +14.9443 q^{67} +14.9443 q^{69} +13.5623 q^{71} +8.76393 q^{73} -21.1803 q^{75} +5.85410 q^{77} -6.85410 q^{79} -5.70820 q^{81} -4.00000 q^{83} +8.94427 q^{85} -2.23607 q^{87} +1.09017 q^{89} -5.23607 q^{91} -11.7082 q^{93} -3.61803 q^{95} -3.56231 q^{97} +22.5623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9	$ 2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9} + 5 q^{11} - 6 q^{13} + 10 q^{15} + 4 q^{17} + 2 q^{19} - 3 q^{21} + 2 q^{23} + 5 q^{25} - 5 q^{29} - 15 q^{33} - 5 q^{35} - 11 q^{37} + 14 q^{39} - q^{41} - 7 q^{43} - 10 q^{45} - 17 q^{47} + 2 q^{49} + 4 q^{51} + 9 q^{53} - 20 q^{55} - 3 q^{57} + 3 q^{59} + 9 q^{61} + q^{63} + 20 q^{65} + 12 q^{67} + 12 q^{69} + 7 q^{71} + 22 q^{73} - 20 q^{75} + 5 q^{77} - 7 q^{79} + 2 q^{81} - 8 q^{83} - 9 q^{89} - 6 q^{91} - 10 q^{93} - 5 q^{95} + 13 q^{97} + 25 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9 + 5 * q^11 - 6 * q^13 + 10 * q^15 + 4 * q^17 + 2 * q^19 - 3 * q^21 + 2 * q^23 + 5 * q^25 - 5 * q^29 - 15 * q^33 - 5 * q^35 - 11 * q^37 + 14 * q^39 - q^41 - 7 * q^43 - 10 * q^45 - 17 * q^47 + 2 * q^49 + 4 * q^51 + 9 * q^53 - 20 * q^55 - 3 * q^57 + 3 * q^59 + 9 * q^61 + q^63 + 20 * q^65 + 12 * q^67 + 12 * q^69 + 7 * q^71 + 22 * q^73 - 20 * q^75 + 5 * q^77 - 7 * q^79 + 2 * q^81 - 8 * q^83 - 9 * q^89 - 6 * q^91 - 10 * q^93 - 5 * q^95 + 13 * q^97 + 25 * 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	5.85410	1.76508	0.882539	−	0.470239i	$-0.155832\pi$
$11$	5.85410	1.76508	0.882539	+	0.470239i	$0.155832\pi$
$12$	0	0
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	0	0
$15$	9.47214	2.44569
$16$	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
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.61803	−0.571302
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	0.854102	0.158603	0.0793014	−	0.996851i	$-0.474731\pi$
$29$	0.854102	0.158603	0.0793014	+	0.996851i	$0.474731\pi$
$30$	0	0
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	0	0
$33$	−15.3262	−2.66796
$34$	0	0
$35$	−3.61803	−0.611559
$36$	0	0
$37$	0.0901699	0.0148238	0.00741192	−	0.999973i	$-0.497641\pi$
$37$	0.0901699	0.0148238	0.00741192	+	0.999973i	$0.497641\pi$
$38$	0	0
$39$	13.7082	2.19507
$40$	0	0
$41$	−6.09017	−0.951125	−0.475562	−	0.879682i	$-0.657755\pi$
$41$	−6.09017	−0.951125	−0.475562	+	0.879682i	$0.657755\pi$
$42$	0	0
$43$	−6.85410	−1.04524	−0.522620	−	0.852566i	$-0.675045\pi$
$43$	−6.85410	−1.04524	−0.522620	+	0.852566i	$0.675045\pi$
$44$	0	0
$45$	−13.9443	−2.07869
$46$	0	0
$47$	−11.8541	−1.72910	−0.864549	−	0.502548i	$-0.832396\pi$
$47$	−11.8541	−1.72910	−0.864549	+	0.502548i	$0.832396\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.47214	0.906280
$52$	0	0
$53$	3.38197	0.464549	0.232274	−	0.972650i	$-0.425383\pi$
$53$	3.38197	0.464549	0.232274	+	0.972650i	$0.425383\pi$
$54$	0	0
$55$	−21.1803	−2.85596
$56$	0	0
$57$	−2.61803	−0.346767
$58$	0	0
$59$	−4.09017	−0.532495	−0.266247	−	0.963905i	$-0.585784\pi$
$59$	−4.09017	−0.532495	−0.266247	+	0.963905i	$0.585784\pi$
$60$	0	0
$61$	7.85410	1.00561	0.502807	−	0.864398i	$-0.332301\pi$
$61$	7.85410	1.00561	0.502807	+	0.864398i	$0.332301\pi$
$62$	0	0
$63$	3.85410	0.485571
$64$	0	0
$65$	18.9443	2.34975
$66$	0	0
$67$	14.9443	1.82573	0.912867	−	0.408258i	$-0.133864\pi$
$67$	14.9443	1.82573	0.912867	+	0.408258i	$0.133864\pi$
$68$	0	0
$69$	14.9443	1.79908
$70$	0	0
$71$	13.5623	1.60955	0.804775	−	0.593580i	$-0.202286\pi$
$71$	13.5623	1.60955	0.804775	+	0.593580i	$0.202286\pi$
$72$	0	0
$73$	8.76393	1.02574	0.512870	−	0.858466i	$-0.328582\pi$
$73$	8.76393	1.02574	0.512870	+	0.858466i	$0.328582\pi$
$74$	0	0
$75$	−21.1803	−2.44569
$76$	0	0
$77$	5.85410	0.667137
$78$	0	0
$79$	−6.85410	−0.771147	−0.385573	−	0.922677i	$-0.625996\pi$
$79$	−6.85410	−0.771147	−0.385573	+	0.922677i	$0.625996\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	0	0
$87$	−2.23607	−0.239732
$88$	0	0
$89$	1.09017	0.115558	0.0577789	−	0.998329i	$-0.481598\pi$
$89$	1.09017	0.115558	0.0577789	+	0.998329i	$0.481598\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	0	0
$93$	−11.7082	−1.21408
$94$	0	0
$95$	−3.61803	−0.371202
$96$	0	0
$97$	−3.56231	−0.361697	−0.180849	−	0.983511i	$-0.557884\pi$
$97$	−3.56231	−0.361697	−0.180849	+	0.983511i	$0.557884\pi$
$98$	0	0
$99$	22.5623	2.26760
$100$	0	0
$101$	10.9443	1.08900	0.544498	−	0.838762i	$-0.316720\pi$
$101$	10.9443	1.08900	0.544498	+	0.838762i	$0.316720\pi$
$102$	0	0
$103$	13.7082	1.35071	0.675355	−	0.737493i	$-0.263991\pi$
$103$	13.7082	1.35071	0.675355	+	0.737493i	$0.263991\pi$
$104$	0	0
$105$	9.47214	0.924386
$106$	0	0
$107$	−12.1803	−1.17752	−0.588759	−	0.808309i	$-0.700383\pi$
$107$	−12.1803	−1.17752	−0.588759	+	0.808309i	$0.700383\pi$
$108$	0	0
$109$	−9.56231	−0.915903	−0.457951	−	0.888977i	$-0.651417\pi$
$109$	−9.56231	−0.915903	−0.457951	+	0.888977i	$0.651417\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	0	0
$113$	16.9443	1.59398	0.796992	−	0.603991i	$-0.206424\pi$
$113$	16.9443	1.59398	0.796992	+	0.603991i	$0.206424\pi$
$114$	0	0
$115$	20.6525	1.92585
$116$	0	0
$117$	−20.1803	−1.86567
$118$	0	0
$119$	−2.47214	−0.226620
$120$	0	0
$121$	23.2705	2.11550
$122$	0	0
$123$	15.9443	1.43765
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	2.14590	0.190418	0.0952088	−	0.995457i	$-0.469648\pi$
$127$	2.14590	0.190418	0.0952088	+	0.995457i	$0.469648\pi$
$128$	0	0
$129$	17.9443	1.57991
$130$	0	0
$131$	22.1803	1.93791	0.968953	−	0.247246i	$-0.0795257\pi$
$131$	22.1803	1.93791	0.968953	+	0.247246i	$0.0795257\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	8.09017	0.696291
$136$	0	0
$137$	18.5623	1.58588	0.792942	−	0.609297i	$-0.208548\pi$
$137$	18.5623	1.58588	0.792942	+	0.609297i	$0.208548\pi$
$138$	0	0
$139$	2.47214	0.209684	0.104842	−	0.994489i	$-0.466566\pi$
$139$	2.47214	0.209684	0.104842	+	0.994489i	$0.466566\pi$
$140$	0	0
$141$	31.0344	2.61357
$142$	0	0
$143$	−30.6525	−2.56329
$144$	0	0
$145$	−3.09017	−0.256625
$146$	0	0
$147$	−2.61803	−0.215932
$148$	0	0
$149$	8.94427	0.732743	0.366372	−	0.930469i	$-0.380600\pi$
$149$	8.94427	0.732743	0.366372	+	0.930469i	$0.380600\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−9.52786	−0.770282
$154$	0	0
$155$	−16.1803	−1.29964
$156$	0	0
$157$	−18.6180	−1.48588	−0.742940	−	0.669358i	$-0.766569\pi$
$157$	−18.6180	−1.48588	−0.742940	+	0.669358i	$0.766569\pi$
$158$	0	0
$159$	−8.85410	−0.702176
$160$	0	0
$161$	−5.70820	−0.449869
$162$	0	0
$163$	−1.56231	−0.122369	−0.0611846	−	0.998126i	$-0.519488\pi$
$163$	−1.56231	−0.122369	−0.0611846	+	0.998126i	$0.519488\pi$
$164$	0	0
$165$	55.4508	4.31684
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	3.85410	0.294731
$172$	0	0
$173$	−0.180340	−0.0137110	−0.00685549	−	0.999977i	$-0.502182\pi$
$173$	−0.180340	−0.0137110	−0.00685549	+	0.999977i	$0.502182\pi$
$174$	0	0
$175$	8.09017	0.611559
$176$	0	0
$177$	10.7082	0.804878
$178$	0	0
$179$	−21.1246	−1.57893	−0.789464	−	0.613797i	$-0.789641\pi$
$179$	−21.1246	−1.57893	−0.789464	+	0.613797i	$0.789641\pi$
$180$	0	0
$181$	7.23607	0.537853	0.268926	−	0.963161i	$-0.413331\pi$
$181$	7.23607	0.537853	0.268926	+	0.963161i	$0.413331\pi$
$182$	0	0
$183$	−20.5623	−1.52001
$184$	0	0
$185$	−0.326238	−0.0239855
$186$	0	0
$187$	−14.4721	−1.05831
$188$	0	0
$189$	−2.23607	−0.162650
$190$	0	0
$191$	1.70820	0.123601	0.0618006	−	0.998089i	$-0.480316\pi$
$191$	1.70820	0.123601	0.0618006	+	0.998089i	$0.480316\pi$
$192$	0	0
$193$	7.23607	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$193$	7.23607	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$194$	0	0
$195$	−49.5967	−3.55170
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	1.03444	0.0733296	0.0366648	−	0.999328i	$-0.488327\pi$
$199$	1.03444	0.0733296	0.0366648	+	0.999328i	$0.488327\pi$
$200$	0	0
$201$	−39.1246	−2.75964
$202$	0	0
$203$	0.854102	0.0599462
$204$	0	0
$205$	22.0344	1.53895
$206$	0	0
$207$	−22.0000	−1.52911
$208$	0	0
$209$	5.85410	0.404937
$210$	0	0
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	0	0
$213$	−35.5066	−2.43287
$214$	0	0
$215$	24.7984	1.69124
$216$	0	0
$217$	4.47214	0.303588
$218$	0	0
$219$	−22.9443	−1.55043
$220$	0	0
$221$	12.9443	0.870726
$222$	0	0
$223$	2.18034	0.146006	0.0730032	−	0.997332i	$-0.476742\pi$
$223$	2.18034	0.146006	0.0730032	+	0.997332i	$0.476742\pi$
$224$	0	0
$225$	31.1803	2.07869
$226$	0	0
$227$	12.9443	0.859142	0.429571	−	0.903033i	$-0.358665\pi$
$227$	12.9443	0.859142	0.429571	+	0.903033i	$0.358665\pi$
$228$	0	0
$229$	9.32624	0.616295	0.308148	−	0.951339i	$-0.400291\pi$
$229$	9.32624	0.616295	0.308148	+	0.951339i	$0.400291\pi$
$230$	0	0
$231$	−15.3262	−1.00839
$232$	0	0
$233$	−10.1459	−0.664680	−0.332340	−	0.943160i	$-0.607838\pi$
$233$	−10.1459	−0.664680	−0.332340	+	0.943160i	$0.607838\pi$
$234$	0	0
$235$	42.8885	2.79774
$236$	0	0
$237$	17.9443	1.16561
$238$	0	0
$239$	−8.29180	−0.536352	−0.268176	−	0.963370i	$-0.586421\pi$
$239$	−8.29180	−0.536352	−0.268176	+	0.963370i	$0.586421\pi$
$240$	0	0
$241$	4.56231	0.293884	0.146942	−	0.989145i	$-0.453057\pi$
$241$	4.56231	0.293884	0.146942	+	0.989145i	$0.453057\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	−3.61803	−0.231148
$246$	0	0
$247$	−5.23607	−0.333163
$248$	0	0
$249$	10.4721	0.663645
$250$	0	0
$251$	−29.2361	−1.84536	−0.922682	−	0.385562i	$-0.874008\pi$
$251$	−29.2361	−1.84536	−0.922682	+	0.385562i	$0.874008\pi$
$252$	0	0
$253$	−33.4164	−2.10087
$254$	0	0
$255$	−23.4164	−1.46639
$256$	0	0
$257$	−2.67376	−0.166785	−0.0833923	−	0.996517i	$-0.526575\pi$
$257$	−2.67376	−0.166785	−0.0833923	+	0.996517i	$0.526575\pi$
$258$	0	0
$259$	0.0901699	0.00560289
$260$	0	0
$261$	3.29180	0.203757
$262$	0	0
$263$	−25.2361	−1.55612	−0.778061	−	0.628188i	$-0.783797\pi$
$263$	−25.2361	−1.55612	−0.778061	+	0.628188i	$0.783797\pi$
$264$	0	0
$265$	−12.2361	−0.751656
$266$	0	0
$267$	−2.85410	−0.174668
$268$	0	0
$269$	29.4164	1.79355	0.896775	−	0.442487i	$-0.145904\pi$
$269$	29.4164	1.79355	0.896775	+	0.442487i	$0.145904\pi$
$270$	0	0
$271$	16.8541	1.02381	0.511907	−	0.859041i	$-0.328939\pi$
$271$	16.8541	1.02381	0.511907	+	0.859041i	$0.328939\pi$
$272$	0	0
$273$	13.7082	0.829658
$274$	0	0
$275$	47.3607	2.85596
$276$	0	0
$277$	18.2918	1.09905	0.549524	−	0.835478i	$-0.314809\pi$
$277$	18.2918	1.09905	0.549524	+	0.835478i	$0.314809\pi$
$278$	0	0
$279$	17.2361	1.03190
$280$	0	0
$281$	−17.8885	−1.06714	−0.533571	−	0.845756i	$-0.679150\pi$
$281$	−17.8885	−1.06714	−0.533571	+	0.845756i	$0.679150\pi$
$282$	0	0
$283$	−5.52786	−0.328597	−0.164299	−	0.986411i	$-0.552536\pi$
$283$	−5.52786	−0.328597	−0.164299	+	0.986411i	$0.552536\pi$
$284$	0	0
$285$	9.47214	0.561081
$286$	0	0
$287$	−6.09017	−0.359491
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	9.32624	0.546714
$292$	0	0
$293$	−5.23607	−0.305894	−0.152947	−	0.988234i	$-0.548876\pi$
$293$	−5.23607	−0.305894	−0.152947	+	0.988234i	$0.548876\pi$
$294$	0	0
$295$	14.7984	0.861595
$296$	0	0
$297$	−13.0902	−0.759569
$298$	0	0
$299$	29.8885	1.72850
$300$	0	0
$301$	−6.85410	−0.395064
$302$	0	0
$303$	−28.6525	−1.64604
$304$	0	0
$305$	−28.4164	−1.62712
$306$	0	0
$307$	−3.90983	−0.223146	−0.111573	−	0.993756i	$-0.535589\pi$
$307$	−3.90983	−0.223146	−0.111573	+	0.993756i	$0.535589\pi$
$308$	0	0
$309$	−35.8885	−2.04163
$310$	0	0
$311$	25.4508	1.44319	0.721593	−	0.692318i	$-0.243410\pi$
$311$	25.4508	1.44319	0.721593	+	0.692318i	$0.243410\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	−13.9443	−0.785671
$316$	0	0
$317$	7.67376	0.431001	0.215501	−	0.976504i	$-0.430862\pi$
$317$	7.67376	0.431001	0.215501	+	0.976504i	$0.430862\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	31.8885	1.77984
$322$	0	0
$323$	−2.47214	−0.137553
$324$	0	0
$325$	−42.3607	−2.34975
$326$	0	0
$327$	25.0344	1.38441
$328$	0	0
$329$	−11.8541	−0.653538
$330$	0	0
$331$	27.3050	1.50082	0.750408	−	0.660975i	$-0.229857\pi$
$331$	27.3050	1.50082	0.750408	+	0.660975i	$0.229857\pi$
$332$	0	0
$333$	0.347524	0.0190442
$334$	0	0
$335$	−54.0689	−2.95410
$336$	0	0
$337$	−9.23607	−0.503121	−0.251560	−	0.967842i	$-0.580944\pi$
$337$	−9.23607	−0.503121	−0.251560	+	0.967842i	$0.580944\pi$
$338$	0	0
$339$	−44.3607	−2.40934
$340$	0	0
$341$	26.1803	1.41774
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−54.0689	−2.91097
$346$	0	0
$347$	2.47214	0.132711	0.0663556	−	0.997796i	$-0.478863\pi$
$347$	2.47214	0.132711	0.0663556	+	0.997796i	$0.478863\pi$
$348$	0	0
$349$	15.8885	0.850494	0.425247	−	0.905077i	$-0.360187\pi$
$349$	15.8885	0.850494	0.425247	+	0.905077i	$0.360187\pi$
$350$	0	0
$351$	11.7082	0.624938
$352$	0	0
$353$	−24.3607	−1.29659	−0.648294	−	0.761390i	$-0.724517\pi$
$353$	−24.3607	−1.29659	−0.648294	+	0.761390i	$0.724517\pi$
$354$	0	0
$355$	−49.0689	−2.60431
$356$	0	0
$357$	6.47214	0.342542
$358$	0	0
$359$	−12.7639	−0.673655	−0.336827	−	0.941566i	$-0.609354\pi$
$359$	−12.7639	−0.673655	−0.336827	+	0.941566i	$0.609354\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−60.9230	−3.19763
$364$	0	0
$365$	−31.7082	−1.65968
$366$	0	0
$367$	5.67376	0.296168	0.148084	−	0.988975i	$-0.452689\pi$
$367$	5.67376	0.296168	0.148084	+	0.988975i	$0.452689\pi$
$368$	0	0
$369$	−23.4721	−1.22191
$370$	0	0
$371$	3.38197	0.175583
$372$	0	0
$373$	−9.03444	−0.467786	−0.233893	−	0.972262i	$-0.575146\pi$
$373$	−9.03444	−0.467786	−0.233893	+	0.972262i	$0.575146\pi$
$374$	0	0
$375$	29.2705	1.51152
$376$	0	0
$377$	−4.47214	−0.230327
$378$	0	0
$379$	−35.1246	−1.80423	−0.902115	−	0.431496i	$-0.857986\pi$
$379$	−35.1246	−1.80423	−0.902115	+	0.431496i	$0.857986\pi$
$380$	0	0
$381$	−5.61803	−0.287821
$382$	0	0
$383$	26.0689	1.33206	0.666029	−	0.745926i	$-0.267993\pi$
$383$	26.0689	1.33206	0.666029	+	0.745926i	$0.267993\pi$
$384$	0	0
$385$	−21.1803	−1.07945
$386$	0	0
$387$	−26.4164	−1.34282
$388$	0	0
$389$	−30.3607	−1.53935	−0.769674	−	0.638437i	$-0.779581\pi$
$389$	−30.3607	−1.53935	−0.769674	+	0.638437i	$0.779581\pi$
$390$	0	0
$391$	14.1115	0.713647
$392$	0	0
$393$	−58.0689	−2.92919
$394$	0	0
$395$	24.7984	1.24774
$396$	0	0
$397$	−36.5066	−1.83221	−0.916106	−	0.400935i	$-0.868685\pi$
$397$	−36.5066	−1.83221	−0.916106	+	0.400935i	$0.868685\pi$
$398$	0	0
$399$	−2.61803	−0.131066
$400$	0	0
$401$	9.88854	0.493810	0.246905	−	0.969040i	$-0.420586\pi$
$401$	9.88854	0.493810	0.246905	+	0.969040i	$0.420586\pi$
$402$	0	0
$403$	−23.4164	−1.16645
$404$	0	0
$405$	20.6525	1.02623
$406$	0	0
$407$	0.527864	0.0261652
$408$	0	0
$409$	−9.09017	−0.449480	−0.224740	−	0.974419i	$-0.572153\pi$
$409$	−9.09017	−0.449480	−0.224740	+	0.974419i	$0.572153\pi$
$410$	0	0
$411$	−48.5967	−2.39710
$412$	0	0
$413$	−4.09017	−0.201264
$414$	0	0
$415$	14.4721	0.710409
$416$	0	0
$417$	−6.47214	−0.316942
$418$	0	0
$419$	−28.4721	−1.39095	−0.695477	−	0.718548i	$-0.744807\pi$
$419$	−28.4721	−1.39095	−0.695477	+	0.718548i	$0.744807\pi$
$420$	0	0
$421$	20.4721	0.997751	0.498875	−	0.866674i	$-0.333747\pi$
$421$	20.4721	0.997751	0.498875	+	0.866674i	$0.333747\pi$
$422$	0	0
$423$	−45.6869	−2.22137
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	7.85410	0.380087
$428$	0	0
$429$	80.2492	3.87447
$430$	0	0
$431$	−23.0902	−1.11221	−0.556107	−	0.831111i	$-0.687706\pi$
$431$	−23.0902	−1.11221	−0.556107	+	0.831111i	$0.687706\pi$
$432$	0	0
$433$	−18.2705	−0.878025	−0.439012	−	0.898481i	$-0.644672\pi$
$433$	−18.2705	−0.878025	−0.439012	+	0.898481i	$0.644672\pi$
$434$	0	0
$435$	8.09017	0.387894
$436$	0	0
$437$	−5.70820	−0.273060
$438$	0	0
$439$	−23.3050	−1.11228	−0.556142	−	0.831087i	$-0.687719\pi$
$439$	−23.3050	−1.11228	−0.556142	+	0.831087i	$0.687719\pi$
$440$	0	0
$441$	3.85410	0.183529
$442$	0	0
$443$	6.27051	0.297921	0.148960	−	0.988843i	$-0.452407\pi$
$443$	6.27051	0.297921	0.148960	+	0.988843i	$0.452407\pi$
$444$	0	0
$445$	−3.94427	−0.186976
$446$	0	0
$447$	−23.4164	−1.10756
$448$	0	0
$449$	26.1803	1.23553	0.617763	−	0.786364i	$-0.288039\pi$
$449$	26.1803	1.23553	0.617763	+	0.786364i	$0.288039\pi$
$450$	0	0
$451$	−35.6525	−1.67881
$452$	0	0
$453$	52.3607	2.46012
$454$	0	0
$455$	18.9443	0.888121
$456$	0	0
$457$	29.2705	1.36922	0.684608	−	0.728911i	$-0.259973\pi$
$457$	29.2705	1.36922	0.684608	+	0.728911i	$0.259973\pi$
$458$	0	0
$459$	5.52786	0.258019
$460$	0	0
$461$	−26.0344	−1.21254	−0.606272	−	0.795257i	$-0.707336\pi$
$461$	−26.0344	−1.21254	−0.606272	+	0.795257i	$0.707336\pi$
$462$	0	0
$463$	5.23607	0.243341	0.121670	−	0.992571i	$-0.461175\pi$
$463$	5.23607	0.243341	0.121670	+	0.992571i	$0.461175\pi$
$464$	0	0
$465$	42.3607	1.96443
$466$	0	0
$467$	4.18034	0.193443	0.0967215	−	0.995311i	$-0.469164\pi$
$467$	4.18034	0.193443	0.0967215	+	0.995311i	$0.469164\pi$
$468$	0	0
$469$	14.9443	0.690062
$470$	0	0
$471$	48.7426	2.24594
$472$	0	0
$473$	−40.1246	−1.84493
$474$	0	0
$475$	8.09017	0.371202
$476$	0	0
$477$	13.0344	0.596806
$478$	0	0
$479$	−37.3951	−1.70863	−0.854313	−	0.519758i	$-0.826022\pi$
$479$	−37.3951	−1.70863	−0.854313	+	0.519758i	$0.826022\pi$
$480$	0	0
$481$	−0.472136	−0.0215275
$482$	0	0
$483$	14.9443	0.679988
$484$	0	0
$485$	12.8885	0.585239
$486$	0	0
$487$	17.1459	0.776955	0.388477	−	0.921458i	$-0.373001\pi$
$487$	17.1459	0.776955	0.388477	+	0.921458i	$0.373001\pi$
$488$	0	0
$489$	4.09017	0.184964
$490$	0	0
$491$	5.88854	0.265746	0.132873	−	0.991133i	$-0.457580\pi$
$491$	5.88854	0.265746	0.132873	+	0.991133i	$0.457580\pi$
$492$	0	0
$493$	−2.11146	−0.0950952
$494$	0	0
$495$	−81.6312	−3.66905
$496$	0	0
$497$	13.5623	0.608353
$498$	0	0
$499$	−31.0344	−1.38929	−0.694646	−	0.719352i	$-0.744439\pi$
$499$	−31.0344	−1.38929	−0.694646	+	0.719352i	$0.744439\pi$
$500$	0	0
$501$	5.23607	0.233930
$502$	0	0
$503$	−7.14590	−0.318620	−0.159310	−	0.987229i	$-0.550927\pi$
$503$	−7.14590	−0.318620	−0.159310	+	0.987229i	$0.550927\pi$
$504$	0	0
$505$	−39.5967	−1.76203
$506$	0	0
$507$	−37.7426	−1.67621
$508$	0	0
$509$	−4.76393	−0.211158	−0.105579	−	0.994411i	$-0.533670\pi$
$509$	−4.76393	−0.211158	−0.105579	+	0.994411i	$0.533670\pi$
$510$	0	0
$511$	8.76393	0.387694
$512$	0	0
$513$	−2.23607	−0.0987248
$514$	0	0
$515$	−49.5967	−2.18549
$516$	0	0
$517$	−69.3951	−3.05199
$518$	0	0
$519$	0.472136	0.0207245
$520$	0	0
$521$	23.3050	1.02101	0.510504	−	0.859875i	$-0.329459\pi$
$521$	23.3050	1.02101	0.510504	+	0.859875i	$0.329459\pi$
$522$	0	0
$523$	−33.8885	−1.48184	−0.740921	−	0.671592i	$-0.765611\pi$
$523$	−33.8885	−1.48184	−0.740921	+	0.671592i	$0.765611\pi$
$524$	0	0
$525$	−21.1803	−0.924386
$526$	0	0
$527$	−11.0557	−0.481595
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	−15.7639	−0.684096
$532$	0	0
$533$	31.8885	1.38125
$534$	0	0
$535$	44.0689	1.90526
$536$	0	0
$537$	55.3050	2.38658
$538$	0	0
$539$	5.85410	0.252154
$540$	0	0
$541$	−13.7082	−0.589362	−0.294681	−	0.955596i	$-0.595213\pi$
$541$	−13.7082	−0.589362	−0.294681	+	0.955596i	$0.595213\pi$
$542$	0	0
$543$	−18.9443	−0.812977
$544$	0	0
$545$	34.5967	1.48196
$546$	0	0
$547$	25.7082	1.09920	0.549602	−	0.835427i	$-0.314779\pi$
$547$	25.7082	1.09920	0.549602	+	0.835427i	$0.314779\pi$
$548$	0	0
$549$	30.2705	1.29191
$550$	0	0
$551$	0.854102	0.0363860
$552$	0	0
$553$	−6.85410	−0.291466
$554$	0	0
$555$	0.854102	0.0362546
$556$	0	0
$557$	−1.81966	−0.0771015	−0.0385507	−	0.999257i	$-0.512274\pi$
$557$	−1.81966	−0.0771015	−0.0385507	+	0.999257i	$0.512274\pi$
$558$	0	0
$559$	35.8885	1.51792
$560$	0	0
$561$	37.8885	1.59966
$562$	0	0
$563$	−20.7984	−0.876547	−0.438273	−	0.898842i	$-0.644410\pi$
$563$	−20.7984	−0.876547	−0.438273	+	0.898842i	$0.644410\pi$
$564$	0	0
$565$	−61.3050	−2.57912
$566$	0	0
$567$	−5.70820	−0.239722
$568$	0	0
$569$	13.7082	0.574678	0.287339	−	0.957829i	$-0.407229\pi$
$569$	13.7082	0.574678	0.287339	+	0.957829i	$0.407229\pi$
$570$	0	0
$571$	−34.2148	−1.43184	−0.715922	−	0.698180i	$-0.753993\pi$
$571$	−34.2148	−1.43184	−0.715922	+	0.698180i	$0.753993\pi$
$572$	0	0
$573$	−4.47214	−0.186826
$574$	0	0
$575$	−46.1803	−1.92585
$576$	0	0
$577$	−18.9443	−0.788660	−0.394330	−	0.918969i	$-0.629023\pi$
$577$	−18.9443	−0.788660	−0.394330	+	0.918969i	$0.629023\pi$
$578$	0	0
$579$	−18.9443	−0.787297
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	19.7984	0.819965
$584$	0	0
$585$	73.0132	3.01872
$586$	0	0
$587$	−21.1246	−0.871906	−0.435953	−	0.899969i	$-0.643589\pi$
$587$	−21.1246	−0.871906	−0.435953	+	0.899969i	$0.643589\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	47.1246	1.93845
$592$	0	0
$593$	−13.1246	−0.538963	−0.269482	−	0.963006i	$-0.586852\pi$
$593$	−13.1246	−0.538963	−0.269482	+	0.963006i	$0.586852\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	−2.70820	−0.110839
$598$	0	0
$599$	−42.7984	−1.74869	−0.874347	−	0.485301i	$-0.838710\pi$
$599$	−42.7984	−1.74869	−0.874347	+	0.485301i	$0.838710\pi$
$600$	0	0
$601$	7.88854	0.321780	0.160890	−	0.986972i	$-0.448563\pi$
$601$	7.88854	0.321780	0.160890	+	0.986972i	$0.448563\pi$
$602$	0	0
$603$	57.5967	2.34552
$604$	0	0
$605$	−84.1935	−3.42295
$606$	0	0
$607$	9.23607	0.374880	0.187440	−	0.982276i	$-0.439981\pi$
$607$	9.23607	0.374880	0.187440	+	0.982276i	$0.439981\pi$
$608$	0	0
$609$	−2.23607	−0.0906100
$610$	0	0
$611$	62.0689	2.51104
$612$	0	0
$613$	−16.9443	−0.684373	−0.342186	−	0.939632i	$-0.611167\pi$
$613$	−16.9443	−0.684373	−0.342186	+	0.939632i	$0.611167\pi$
$614$	0	0
$615$	−57.6869	−2.32616
$616$	0	0
$617$	31.9098	1.28464	0.642321	−	0.766436i	$-0.277972\pi$
$617$	31.9098	1.28464	0.642321	+	0.766436i	$0.277972\pi$
$618$	0	0
$619$	−18.2918	−0.735209	−0.367605	−	0.929982i	$-0.619822\pi$
$619$	−18.2918	−0.735209	−0.367605	+	0.929982i	$0.619822\pi$
$620$	0	0
$621$	12.7639	0.512199
$622$	0	0
$623$	1.09017	0.0436767
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−15.3262	−0.612071
$628$	0	0
$629$	−0.222912	−0.00888810
$630$	0	0
$631$	40.3607	1.60673	0.803367	−	0.595485i	$-0.203040\pi$
$631$	40.3607	1.60673	0.803367	+	0.595485i	$0.203040\pi$
$632$	0	0
$633$	−36.3607	−1.44521
$634$	0	0
$635$	−7.76393	−0.308102
$636$	0	0
$637$	−5.23607	−0.207461
$638$	0	0
$639$	52.2705	2.06779
$640$	0	0
$641$	−22.6525	−0.894719	−0.447360	−	0.894354i	$-0.647636\pi$
$641$	−22.6525	−0.894719	−0.447360	+	0.894354i	$0.647636\pi$
$642$	0	0
$643$	−37.5279	−1.47995	−0.739977	−	0.672632i	$-0.765164\pi$
$643$	−37.5279	−1.47995	−0.739977	+	0.672632i	$0.765164\pi$
$644$	0	0
$645$	−64.9230	−2.55634
$646$	0	0
$647$	−20.2705	−0.796916	−0.398458	−	0.917187i	$-0.630455\pi$
$647$	−20.2705	−0.796916	−0.398458	+	0.917187i	$0.630455\pi$
$648$	0	0
$649$	−23.9443	−0.939895
$650$	0	0
$651$	−11.7082	−0.458881
$652$	0	0
$653$	32.6525	1.27779	0.638895	−	0.769294i	$-0.279392\pi$
$653$	32.6525	1.27779	0.638895	+	0.769294i	$0.279392\pi$
$654$	0	0
$655$	−80.2492	−3.13560
$656$	0	0
$657$	33.7771	1.31777
$658$	0	0
$659$	8.94427	0.348419	0.174210	−	0.984709i	$-0.444263\pi$
$659$	8.94427	0.348419	0.174210	+	0.984709i	$0.444263\pi$
$660$	0	0
$661$	16.8328	0.654721	0.327360	−	0.944900i	$-0.393841\pi$
$661$	16.8328	0.654721	0.327360	+	0.944900i	$0.393841\pi$
$662$	0	0
$663$	−33.8885	−1.31612
$664$	0	0
$665$	−3.61803	−0.140301
$666$	0	0
$667$	−4.87539	−0.188776
$668$	0	0
$669$	−5.70820	−0.220692
$670$	0	0
$671$	45.9787	1.77499
$672$	0	0
$673$	−25.2361	−0.972779	−0.486389	−	0.873742i	$-0.661686\pi$
$673$	−25.2361	−0.972779	−0.486389	+	0.873742i	$0.661686\pi$
$674$	0	0
$675$	−18.0902	−0.696291
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−3.56231	−0.136709
$680$	0	0
$681$	−33.8885	−1.29861
$682$	0	0
$683$	−23.8885	−0.914070	−0.457035	−	0.889449i	$-0.651089\pi$
$683$	−23.8885	−0.914070	−0.457035	+	0.889449i	$0.651089\pi$
$684$	0	0
$685$	−67.1591	−2.56602
$686$	0	0
$687$	−24.4164	−0.931544
$688$	0	0
$689$	−17.7082	−0.674629
$690$	0	0
$691$	38.6525	1.47041	0.735205	−	0.677845i	$-0.237086\pi$
$691$	38.6525	1.47041	0.735205	+	0.677845i	$0.237086\pi$
$692$	0	0
$693$	22.5623	0.857071
$694$	0	0
$695$	−8.94427	−0.339276
$696$	0	0
$697$	15.0557	0.570276
$698$	0	0
$699$	26.5623	1.00468
$700$	0	0
$701$	−39.2361	−1.48193	−0.740963	−	0.671546i	$-0.765631\pi$
$701$	−39.2361	−1.48193	−0.740963	+	0.671546i	$0.765631\pi$
$702$	0	0
$703$	0.0901699	0.00340082
$704$	0	0
$705$	−112.284	−4.22885
$706$	0	0
$707$	10.9443	0.411602
$708$	0	0
$709$	−18.1803	−0.682777	−0.341388	−	0.939922i	$-0.610897\pi$
$709$	−18.1803	−0.682777	−0.341388	+	0.939922i	$0.610897\pi$
$710$	0	0
$711$	−26.4164	−0.990693
$712$	0	0
$713$	−25.5279	−0.956026
$714$	0	0
$715$	110.902	4.14749
$716$	0	0
$717$	21.7082	0.810708
$718$	0	0
$719$	25.8885	0.965480	0.482740	−	0.875764i	$-0.339642\pi$
$719$	25.8885	0.965480	0.482740	+	0.875764i	$0.339642\pi$
$720$	0	0
$721$	13.7082	0.510520
$722$	0	0
$723$	−11.9443	−0.444212
$724$	0	0
$725$	6.90983	0.256625
$726$	0	0
$727$	15.5623	0.577174	0.288587	−	0.957454i	$-0.406815\pi$
$727$	15.5623	0.577174	0.288587	+	0.957454i	$0.406815\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	16.9443	0.626707
$732$	0	0
$733$	27.4377	1.01343	0.506717	−	0.862112i	$-0.330859\pi$
$733$	27.4377	1.01343	0.506717	+	0.862112i	$0.330859\pi$
$734$	0	0
$735$	9.47214	0.349385
$736$	0	0
$737$	87.4853	3.22256
$738$	0	0
$739$	32.5623	1.19782	0.598912	−	0.800815i	$-0.295600\pi$
$739$	32.5623	1.19782	0.598912	+	0.800815i	$0.295600\pi$
$740$	0	0
$741$	13.7082	0.503583
$742$	0	0
$743$	3.79837	0.139349	0.0696744	−	0.997570i	$-0.477804\pi$
$743$	3.79837	0.139349	0.0696744	+	0.997570i	$0.477804\pi$
$744$	0	0
$745$	−32.3607	−1.18560
$746$	0	0
$747$	−15.4164	−0.564057
$748$	0	0
$749$	−12.1803	−0.445060
$750$	0	0
$751$	13.9098	0.507577	0.253788	−	0.967260i	$-0.418323\pi$
$751$	13.9098	0.507577	0.253788	+	0.967260i	$0.418323\pi$
$752$	0	0
$753$	76.5410	2.78931
$754$	0	0
$755$	72.3607	2.63347
$756$	0	0
$757$	−21.5279	−0.782444	−0.391222	−	0.920296i	$-0.627947\pi$
$757$	−21.5279	−0.782444	−0.391222	+	0.920296i	$0.627947\pi$
$758$	0	0
$759$	87.4853	3.17551
$760$	0	0
$761$	−10.9443	−0.396730	−0.198365	−	0.980128i	$-0.563563\pi$
$761$	−10.9443	−0.396730	−0.198365	+	0.980128i	$0.563563\pi$
$762$	0	0
$763$	−9.56231	−0.346179
$764$	0	0
$765$	34.4721	1.24634
$766$	0	0
$767$	21.4164	0.773302
$768$	0	0
$769$	30.6525	1.10536	0.552678	−	0.833395i	$-0.313606\pi$
$769$	30.6525	1.10536	0.552678	+	0.833395i	$0.313606\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	0	0
$773$	37.7082	1.35627	0.678135	−	0.734937i	$-0.262789\pi$
$773$	37.7082	1.35627	0.678135	+	0.734937i	$0.262789\pi$
$774$	0	0
$775$	36.1803	1.29964
$776$	0	0
$777$	−0.236068	−0.00846889
$778$	0	0
$779$	−6.09017	−0.218203
$780$	0	0
$781$	79.3951	2.84098
$782$	0	0
$783$	−1.90983	−0.0682518
$784$	0	0
$785$	67.3607	2.40421
$786$	0	0
$787$	28.6738	1.02211	0.511055	−	0.859548i	$-0.329255\pi$
$787$	28.6738	1.02211	0.511055	+	0.859548i	$0.329255\pi$
$788$	0	0
$789$	66.0689	2.35211
$790$	0	0
$791$	16.9443	0.602469
$792$	0	0
$793$	−41.1246	−1.46038
$794$	0	0
$795$	32.0344	1.13614
$796$	0	0
$797$	−49.7082	−1.76075	−0.880377	−	0.474274i	$-0.842711\pi$
$797$	−49.7082	−1.76075	−0.880377	+	0.474274i	$0.842711\pi$
$798$	0	0
$799$	29.3050	1.03673
$800$	0	0
$801$	4.20163	0.148457
$802$	0	0
$803$	51.3050	1.81051
$804$	0	0
$805$	20.6525	0.727904
$806$	0	0
$807$	−77.0132	−2.71099
$808$	0	0
$809$	9.03444	0.317634	0.158817	−	0.987308i	$-0.449232\pi$
$809$	9.03444	0.317634	0.158817	+	0.987308i	$0.449232\pi$
$810$	0	0
$811$	5.45085	0.191405	0.0957026	−	0.995410i	$-0.469490\pi$
$811$	5.45085	0.191405	0.0957026	+	0.995410i	$0.469490\pi$
$812$	0	0
$813$	−44.1246	−1.54752
$814$	0	0
$815$	5.65248	0.197998
$816$	0	0
$817$	−6.85410	−0.239795
$818$	0	0
$819$	−20.1803	−0.705158
$820$	0	0
$821$	−33.4164	−1.16624	−0.583120	−	0.812386i	$-0.698168\pi$
$821$	−33.4164	−1.16624	−0.583120	+	0.812386i	$0.698168\pi$
$822$	0	0
$823$	−29.4164	−1.02539	−0.512696	−	0.858570i	$-0.671353\pi$
$823$	−29.4164	−1.02539	−0.512696	+	0.858570i	$0.671353\pi$
$824$	0	0
$825$	−123.992	−4.31684
$826$	0	0
$827$	−18.0689	−0.628317	−0.314158	−	0.949371i	$-0.601722\pi$
$827$	−18.0689	−0.628317	−0.314158	+	0.949371i	$0.601722\pi$
$828$	0	0
$829$	31.4164	1.09114	0.545568	−	0.838066i	$-0.316314\pi$
$829$	31.4164	1.09114	0.545568	+	0.838066i	$0.316314\pi$
$830$	0	0
$831$	−47.8885	−1.66124
$832$	0	0
$833$	−2.47214	−0.0856544
$834$	0	0
$835$	7.23607	0.250414
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−21.0557	−0.726924	−0.363462	−	0.931609i	$-0.618405\pi$
$839$	−21.0557	−0.726924	−0.363462	+	0.931609i	$0.618405\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	46.8328	1.61301
$844$	0	0
$845$	−52.1591	−1.79433
$846$	0	0
$847$	23.2705	0.799584
$848$	0	0
$849$	14.4721	0.496682
$850$	0	0
$851$	−0.514708	−0.0176440
$852$	0	0
$853$	−1.90983	−0.0653913	−0.0326957	−	0.999465i	$-0.510409\pi$
$853$	−1.90983	−0.0653913	−0.0326957	+	0.999465i	$0.510409\pi$
$854$	0	0
$855$	−13.9443	−0.476884
$856$	0	0
$857$	50.0000	1.70797	0.853984	−	0.520300i	$-0.174180\pi$
$857$	50.0000	1.70797	0.853984	+	0.520300i	$0.174180\pi$
$858$	0	0
$859$	7.59675	0.259198	0.129599	−	0.991567i	$-0.458631\pi$
$859$	7.59675	0.259198	0.129599	+	0.991567i	$0.458631\pi$
$860$	0	0
$861$	15.9443	0.543379
$862$	0	0
$863$	6.27051	0.213451	0.106725	−	0.994289i	$-0.465963\pi$
$863$	6.27051	0.213451	0.106725	+	0.994289i	$0.465963\pi$
$864$	0	0
$865$	0.652476	0.0221848
$866$	0	0
$867$	28.5066	0.968134
$868$	0	0
$869$	−40.1246	−1.36113
$870$	0	0
$871$	−78.2492	−2.65137
$872$	0	0
$873$	−13.7295	−0.464673
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	34.4377	1.16288	0.581439	−	0.813590i	$-0.302490\pi$
$877$	34.4377	1.16288	0.581439	+	0.813590i	$0.302490\pi$
$878$	0	0
$879$	13.7082	0.462366
$880$	0	0
$881$	17.5967	0.592849	0.296425	−	0.955056i	$-0.404206\pi$
$881$	17.5967	0.592849	0.296425	+	0.955056i	$0.404206\pi$
$882$	0	0
$883$	19.9656	0.671895	0.335947	−	0.941881i	$-0.390944\pi$
$883$	19.9656	0.671895	0.335947	+	0.941881i	$0.390944\pi$
$884$	0	0
$885$	−38.7426	−1.30232
$886$	0	0
$887$	−17.4164	−0.584786	−0.292393	−	0.956298i	$-0.594451\pi$
$887$	−17.4164	−0.584786	−0.292393	+	0.956298i	$0.594451\pi$
$888$	0	0
$889$	2.14590	0.0719711
$890$	0	0
$891$	−33.4164	−1.11949
$892$	0	0
$893$	−11.8541	−0.396682
$894$	0	0
$895$	76.4296	2.55476
$896$	0	0
$897$	−78.2492	−2.61267
$898$	0	0
$899$	3.81966	0.127393
$900$	0	0
$901$	−8.36068	−0.278535
$902$	0	0
$903$	17.9443	0.597148
$904$	0	0
$905$	−26.1803	−0.870264
$906$	0	0
$907$	−29.0132	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$907$	−29.0132	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$908$	0	0
$909$	42.1803	1.39903
$910$	0	0
$911$	−43.9787	−1.45708	−0.728540	−	0.685003i	$-0.759801\pi$
$911$	−43.9787	−1.45708	−0.728540	+	0.685003i	$0.759801\pi$
$912$	0	0
$913$	−23.4164	−0.774970
$914$	0	0
$915$	74.3951	2.45943
$916$	0	0
$917$	22.1803	0.732459
$918$	0	0
$919$	39.5967	1.30618	0.653088	−	0.757282i	$-0.273473\pi$
$919$	39.5967	1.30618	0.653088	+	0.757282i	$0.273473\pi$
$920$	0	0
$921$	10.2361	0.337290
$922$	0	0
$923$	−71.0132	−2.33743
$924$	0	0
$925$	0.729490	0.0239855
$926$	0	0
$927$	52.8328	1.73526
$928$	0	0
$929$	−2.76393	−0.0906817	−0.0453408	−	0.998972i	$-0.514437\pi$
$929$	−2.76393	−0.0906817	−0.0453408	+	0.998972i	$0.514437\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−66.6312	−2.18141
$934$	0	0
$935$	52.3607	1.71238
$936$	0	0
$937$	−25.8885	−0.845742	−0.422871	−	0.906190i	$-0.638978\pi$
$937$	−25.8885	−0.845742	−0.422871	+	0.906190i	$0.638978\pi$
$938$	0	0
$939$	20.9443	0.683490
$940$	0	0
$941$	58.5410	1.90838	0.954191	−	0.299197i	$-0.0967188\pi$
$941$	58.5410	1.90838	0.954191	+	0.299197i	$0.0967188\pi$
$942$	0	0
$943$	34.7639	1.13207
$944$	0	0
$945$	8.09017	0.263173
$946$	0	0
$947$	−12.6869	−0.412269	−0.206135	−	0.978524i	$-0.566089\pi$
$947$	−12.6869	−0.412269	−0.206135	+	0.978524i	$0.566089\pi$
$948$	0	0
$949$	−45.8885	−1.48961
$950$	0	0
$951$	−20.0902	−0.651468
$952$	0	0
$953$	−36.4721	−1.18145	−0.590724	−	0.806874i	$-0.701158\pi$
$953$	−36.4721	−1.18145	−0.590724	+	0.806874i	$0.701158\pi$
$954$	0	0
$955$	−6.18034	−0.199991
$956$	0	0
$957$	−13.0902	−0.423145
$958$	0	0
$959$	18.5623	0.599408
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	−46.9443	−1.51276
$964$	0	0
$965$	−26.1803	−0.842775
$966$	0	0
$967$	−15.4164	−0.495758	−0.247879	−	0.968791i	$-0.579734\pi$
$967$	−15.4164	−0.495758	−0.247879	+	0.968791i	$0.579734\pi$
$968$	0	0
$969$	6.47214	0.207915
$970$	0	0
$971$	4.43769	0.142412	0.0712062	−	0.997462i	$-0.477315\pi$
$971$	4.43769	0.142412	0.0712062	+	0.997462i	$0.477315\pi$
$972$	0	0
$973$	2.47214	0.0792530
$974$	0	0
$975$	110.902	3.55170
$976$	0	0
$977$	−2.06888	−0.0661895	−0.0330947	−	0.999452i	$-0.510536\pi$
$977$	−2.06888	−0.0661895	−0.0330947	+	0.999452i	$0.510536\pi$
$978$	0	0
$979$	6.38197	0.203969
$980$	0	0
$981$	−36.8541	−1.17666
$982$	0	0
$983$	−31.0557	−0.990524	−0.495262	−	0.868744i	$-0.664928\pi$
$983$	−31.0557	−0.990524	−0.495262	+	0.868744i	$0.664928\pi$
$984$	0	0
$985$	65.1246	2.07504
$986$	0	0
$987$	31.0344	0.987837
$988$	0	0
$989$	39.1246	1.24409
$990$	0	0
$991$	23.6738	0.752022	0.376011	−	0.926615i	$-0.377296\pi$
$991$	23.6738	0.752022	0.376011	+	0.926615i	$0.377296\pi$
$992$	0	0
$993$	−71.4853	−2.26852
$994$	0	0
$995$	−3.74265	−0.118650
$996$	0	0
$997$	14.4508	0.457663	0.228832	−	0.973466i	$-0.426510\pi$
$997$	14.4508	0.457663	0.228832	+	0.973466i	$0.426510\pi$
$998$	0	0
$999$	−0.201626	−0.00637917

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.e.1.1		2
4.3	odd	2		8512.2.a.ba.1.2		2
8.3	odd	2		2128.2.a.d.1.1		2
8.5	even	2		1064.2.a.e.1.2	✓	2
24.5	odd	2		9576.2.a.bc.1.1		2
56.13	odd	2		7448.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.e.1.2	✓	2	8.5	even	2
2128.2.a.d.1.1		2	8.3	odd	2
7448.2.a.y.1.1		2	56.13	odd	2
8512.2.a.e.1.1		2	1.1	even	1	trivial
8512.2.a.ba.1.2		2	4.3	odd	2
9576.2.a.bc.1.1		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +O(q^{10})\)	\(q-2.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} +3.85410 q^{9} +5.85410 q^{11} -5.23607 q^{13} +9.47214 q^{15} -2.47214 q^{17} +1.00000 q^{19} -2.61803 q^{21} -5.70820 q^{23} +8.09017 q^{25} -2.23607 q^{27} +0.854102 q^{29} +4.47214 q^{31} -15.3262 q^{33} -3.61803 q^{35} +0.0901699 q^{37} +13.7082 q^{39} -6.09017 q^{41} -6.85410 q^{43} -13.9443 q^{45} -11.8541 q^{47} +1.00000 q^{49} +6.47214 q^{51} +3.38197 q^{53} -21.1803 q^{55} -2.61803 q^{57} -4.09017 q^{59} +7.85410 q^{61} +3.85410 q^{63} +18.9443 q^{65} +14.9443 q^{67} +14.9443 q^{69} +13.5623 q^{71} +8.76393 q^{73} -21.1803 q^{75} +5.85410 q^{77} -6.85410 q^{79} -5.70820 q^{81} -4.00000 q^{83} +8.94427 q^{85} -2.23607 q^{87} +1.09017 q^{89} -5.23607 q^{91} -11.7082 q^{93} -3.61803 q^{95} -3.56231 q^{97} +22.5623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9	\( 2 q - 3 q^{3} - 5 q^{5} + 2 q^{7} + q^{9} + 5 q^{11} - 6 q^{13} + 10 q^{15} + 4 q^{17} + 2 q^{19} - 3 q^{21} + 2 q^{23} + 5 q^{25} - 5 q^{29} - 15 q^{33} - 5 q^{35} - 11 q^{37} + 14 q^{39} - q^{41} - 7 q^{43} - 10 q^{45} - 17 q^{47} + 2 q^{49} + 4 q^{51} + 9 q^{53} - 20 q^{55} - 3 q^{57} + 3 q^{59} + 9 q^{61} + q^{63} + 20 q^{65} + 12 q^{67} + 12 q^{69} + 7 q^{71} + 22 q^{73} - 20 q^{75} + 5 q^{77} - 7 q^{79} + 2 q^{81} - 8 q^{83} - 9 q^{89} - 6 q^{91} - 10 q^{93} - 5 q^{95} + 13 q^{97} + 25 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 5 * q^5 + 2 * q^7 + q^9 + 5 * q^11 - 6 * q^13 + 10 * q^15 + 4 * q^17 + 2 * q^19 - 3 * q^21 + 2 * q^23 + 5 * q^25 - 5 * q^29 - 15 * q^33 - 5 * q^35 - 11 * q^37 + 14 * q^39 - q^41 - 7 * q^43 - 10 * q^45 - 17 * q^47 + 2 * q^49 + 4 * q^51 + 9 * q^53 - 20 * q^55 - 3 * q^57 + 3 * q^59 + 9 * q^61 + q^63 + 20 * q^65 + 12 * q^67 + 12 * q^69 + 7 * q^71 + 22 * q^73 - 20 * q^75 + 5 * q^77 - 7 * q^79 + 2 * q^81 - 8 * q^83 - 9 * q^89 - 6 * q^91 - 10 * q^93 - 5 * q^95 + 13 * q^97 + 25 * q^99

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