LMFDB - Embedded newform 8752.2.a.v.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$69.8850718490$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.649055 q^{3} -2.10046 q^{5} -0.493203 q^{7} -2.57873 q^{9} +O(q^{10})$	$q-0.649055 q^{3} -2.10046 q^{5} -0.493203 q^{7} -2.57873 q^{9} -5.42323 q^{11} +1.33548 q^{13} +1.36332 q^{15} +3.05264 q^{17} -5.61982 q^{19} +0.320116 q^{21} -3.92639 q^{23} -0.588056 q^{25} +3.62090 q^{27} -2.34163 q^{29} -5.46702 q^{31} +3.51997 q^{33} +1.03596 q^{35} +2.01724 q^{37} -0.866801 q^{39} -1.47915 q^{41} -1.71874 q^{43} +5.41652 q^{45} -10.7954 q^{47} -6.75675 q^{49} -1.98133 q^{51} -0.235550 q^{53} +11.3913 q^{55} +3.64757 q^{57} +10.5854 q^{59} -6.86905 q^{61} +1.27184 q^{63} -2.80513 q^{65} -5.66862 q^{67} +2.54844 q^{69} -13.6646 q^{71} -11.5461 q^{73} +0.381681 q^{75} +2.67475 q^{77} -13.3971 q^{79} +5.38602 q^{81} -15.9517 q^{83} -6.41195 q^{85} +1.51985 q^{87} +10.7028 q^{89} -0.658664 q^{91} +3.54840 q^{93} +11.8042 q^{95} -8.23233 q^{97} +13.9850 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9}+O(q^{10}) $ 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9	$ 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9} - 10 q^{11} + 19 q^{13} - 5 q^{15} + 40 q^{17} - 8 q^{21} - 26 q^{23} + 36 q^{25} - 11 q^{27} + 30 q^{29} + 5 q^{31} + 10 q^{33} - 11 q^{35} + 26 q^{37} + 17 q^{39} + 9 q^{41} + 10 q^{43} + 64 q^{45} - 28 q^{47} + 20 q^{49} + 9 q^{51} + 80 q^{53} + q^{55} - 8 q^{57} + 2 q^{59} + 22 q^{61} + 9 q^{63} + 30 q^{65} + 16 q^{67} + 22 q^{69} + q^{71} + 2 q^{73} + 31 q^{75} + 67 q^{77} + 34 q^{79} - 11 q^{81} - 15 q^{83} + 15 q^{85} + 29 q^{87} + 38 q^{89} + 41 q^{91} - 4 q^{93} + 46 q^{95} - 2 q^{97} + 41 q^{99}+O(q^{100}) $ 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9 - 10 * q^11 + 19 * q^13 - 5 * q^15 + 40 * q^17 - 8 * q^21 - 26 * q^23 + 36 * q^25 - 11 * q^27 + 30 * q^29 + 5 * q^31 + 10 * q^33 - 11 * q^35 + 26 * q^37 + 17 * q^39 + 9 * q^41 + 10 * q^43 + 64 * q^45 - 28 * q^47 + 20 * q^49 + 9 * q^51 + 80 * q^53 + q^55 - 8 * q^57 + 2 * q^59 + 22 * q^61 + 9 * q^63 + 30 * q^65 + 16 * q^67 + 22 * q^69 + q^71 + 2 * q^73 + 31 * q^75 + 67 * q^77 + 34 * q^79 - 11 * q^81 - 15 * q^83 + 15 * q^85 + 29 * q^87 + 38 * q^89 + 41 * q^91 - 4 * q^93 + 46 * q^95 - 2 * q^97 + 41 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.649055	−0.374732	−0.187366	−	0.982290i	$-0.559995\pi$
$3$	−0.649055	−0.374732	−0.187366	+	0.982290i	$0.559995\pi$
$4$	0	0
$5$	−2.10046	−0.939356	−0.469678	−	0.882838i	$-0.655630\pi$
$5$	−2.10046	−0.939356	−0.469678	+	0.882838i	$0.655630\pi$
$6$	0	0
$7$	−0.493203	−0.186413	−0.0932067	−	0.995647i	$-0.529712\pi$
$7$	−0.493203	−0.186413	−0.0932067	+	0.995647i	$0.529712\pi$
$8$	0	0
$9$	−2.57873	−0.859576
$10$	0	0
$11$	−5.42323	−1.63517	−0.817583	−	0.575811i	$-0.804686\pi$
$11$	−5.42323	−1.63517	−0.817583	+	0.575811i	$0.804686\pi$
$12$	0	0
$13$	1.33548	0.370396	0.185198	−	0.982701i	$-0.440707\pi$
$13$	1.33548	0.370396	0.185198	+	0.982701i	$0.440707\pi$
$14$	0	0
$15$	1.36332	0.352007
$16$	0	0
$17$	3.05264	0.740373	0.370187	−	0.928957i	$-0.379294\pi$
$17$	3.05264	0.740373	0.370187	+	0.928957i	$0.379294\pi$
$18$	0	0
$19$	−5.61982	−1.28928	−0.644638	−	0.764488i	$-0.722992\pi$
$19$	−5.61982	−1.28928	−0.644638	+	0.764488i	$0.722992\pi$
$20$	0	0
$21$	0.320116	0.0698551
$22$	0	0
$23$	−3.92639	−0.818709	−0.409354	−	0.912375i	$-0.634246\pi$
$23$	−3.92639	−0.818709	−0.409354	+	0.912375i	$0.634246\pi$
$24$	0	0
$25$	−0.588056	−0.117611
$26$	0	0
$27$	3.62090	0.696843
$28$	0	0
$29$	−2.34163	−0.434830	−0.217415	−	0.976079i	$-0.569763\pi$
$29$	−2.34163	−0.434830	−0.217415	+	0.976079i	$0.569763\pi$
$30$	0	0
$31$	−5.46702	−0.981907	−0.490953	−	0.871186i	$-0.663351\pi$
$31$	−5.46702	−0.981907	−0.490953	+	0.871186i	$0.663351\pi$
$32$	0	0
$33$	3.51997	0.612749
$34$	0	0
$35$	1.03596	0.175108
$36$	0	0
$37$	2.01724	0.331632	0.165816	−	0.986157i	$-0.446974\pi$
$37$	2.01724	0.331632	0.165816	+	0.986157i	$0.446974\pi$
$38$	0	0
$39$	−0.866801	−0.138799
$40$	0	0
$41$	−1.47915	−0.231005	−0.115502	−	0.993307i	$-0.536848\pi$
$41$	−1.47915	−0.231005	−0.115502	+	0.993307i	$0.536848\pi$
$42$	0	0
$43$	−1.71874	−0.262106	−0.131053	−	0.991375i	$-0.541836\pi$
$43$	−1.71874	−0.262106	−0.131053	+	0.991375i	$0.541836\pi$
$44$	0	0
$45$	5.41652	0.807447
$46$	0	0
$47$	−10.7954	−1.57467	−0.787334	−	0.616526i	$-0.788539\pi$
$47$	−10.7954	−1.57467	−0.787334	+	0.616526i	$0.788539\pi$
$48$	0	0
$49$	−6.75675	−0.965250
$50$	0	0
$51$	−1.98133	−0.277442
$52$	0	0
$53$	−0.235550	−0.0323552	−0.0161776	−	0.999869i	$-0.505150\pi$
$53$	−0.235550	−0.0323552	−0.0161776	+	0.999869i	$0.505150\pi$
$54$	0	0
$55$	11.3913	1.53600
$56$	0	0
$57$	3.64757	0.483133
$58$	0	0
$59$	10.5854	1.37810	0.689052	−	0.724712i	$-0.258027\pi$
$59$	10.5854	1.37810	0.689052	+	0.724712i	$0.258027\pi$
$60$	0	0
$61$	−6.86905	−0.879491	−0.439746	−	0.898122i	$-0.644931\pi$
$61$	−6.86905	−0.879491	−0.439746	+	0.898122i	$0.644931\pi$
$62$	0	0
$63$	1.27184	0.160236
$64$	0	0
$65$	−2.80513	−0.347933
$66$	0	0
$67$	−5.66862	−0.692532	−0.346266	−	0.938136i	$-0.612550\pi$
$67$	−5.66862	−0.692532	−0.346266	+	0.938136i	$0.612550\pi$
$68$	0	0
$69$	2.54844	0.306797
$70$	0	0
$71$	−13.6646	−1.62169	−0.810843	−	0.585264i	$-0.800991\pi$
$71$	−13.6646	−1.62169	−0.810843	+	0.585264i	$0.800991\pi$
$72$	0	0
$73$	−11.5461	−1.35137	−0.675685	−	0.737191i	$-0.736152\pi$
$73$	−11.5461	−1.35137	−0.675685	+	0.737191i	$0.736152\pi$
$74$	0	0
$75$	0.381681	0.0440727
$76$	0	0
$77$	2.67475	0.304817
$78$	0	0
$79$	−13.3971	−1.50730	−0.753648	−	0.657279i	$-0.771708\pi$
$79$	−13.3971	−1.50730	−0.753648	+	0.657279i	$0.771708\pi$
$80$	0	0
$81$	5.38602	0.598446
$82$	0	0
$83$	−15.9517	−1.75092	−0.875462	−	0.483288i	$-0.839442\pi$
$83$	−15.9517	−1.75092	−0.875462	+	0.483288i	$0.839442\pi$
$84$	0	0
$85$	−6.41195	−0.695474
$86$	0	0
$87$	1.51985	0.162945
$88$	0	0
$89$	10.7028	1.13449	0.567246	−	0.823549i	$-0.308009\pi$
$89$	10.7028	1.13449	0.567246	+	0.823549i	$0.308009\pi$
$90$	0	0
$91$	−0.658664	−0.0690467
$92$	0	0
$93$	3.54840	0.367952
$94$	0	0
$95$	11.8042	1.21109
$96$	0	0
$97$	−8.23233	−0.835867	−0.417933	−	0.908478i	$-0.637245\pi$
$97$	−8.23233	−0.835867	−0.417933	+	0.908478i	$0.637245\pi$
$98$	0	0
$99$	13.9850	1.40555
$100$	0	0
$101$	2.05291	0.204272	0.102136	−	0.994770i	$-0.467432\pi$
$101$	2.05291	0.204272	0.102136	+	0.994770i	$0.467432\pi$
$102$	0	0
$103$	6.59981	0.650298	0.325149	−	0.945663i	$-0.394585\pi$
$103$	6.59981	0.650298	0.325149	+	0.945663i	$0.394585\pi$
$104$	0	0
$105$	−0.672392	−0.0656187
$106$	0	0
$107$	7.71776	0.746104	0.373052	−	0.927811i	$-0.378311\pi$
$107$	7.71776	0.746104	0.373052	+	0.927811i	$0.378311\pi$
$108$	0	0
$109$	8.85070	0.847744	0.423872	−	0.905722i	$-0.360671\pi$
$109$	8.85070	0.847744	0.423872	+	0.905722i	$0.360671\pi$
$110$	0	0
$111$	−1.30930	−0.124273
$112$	0	0
$113$	−1.61098	−0.151548	−0.0757742	−	0.997125i	$-0.524143\pi$
$113$	−1.61098	−0.151548	−0.0757742	+	0.997125i	$0.524143\pi$
$114$	0	0
$115$	8.24724	0.769059
$116$	0	0
$117$	−3.44384	−0.318383
$118$	0	0
$119$	−1.50557	−0.138015
$120$	0	0
$121$	18.4114	1.67377
$122$	0	0
$123$	0.960051	0.0865649
$124$	0	0
$125$	11.7375	1.04983
$126$	0	0
$127$	18.4435	1.63659	0.818297	−	0.574795i	$-0.194918\pi$
$127$	18.4435	1.63659	0.818297	+	0.574795i	$0.194918\pi$
$128$	0	0
$129$	1.11556	0.0982195
$130$	0	0
$131$	−0.0700687	−0.00612193	−0.00306097	−	0.999995i	$-0.500974\pi$
$131$	−0.0700687	−0.00612193	−0.00306097	+	0.999995i	$0.500974\pi$
$132$	0	0
$133$	2.77171	0.240338
$134$	0	0
$135$	−7.60557	−0.654583
$136$	0	0
$137$	−16.1390	−1.37885	−0.689425	−	0.724357i	$-0.742137\pi$
$137$	−16.1390	−1.37885	−0.689425	+	0.724357i	$0.742137\pi$
$138$	0	0
$139$	−8.39117	−0.711729	−0.355865	−	0.934537i	$-0.615814\pi$
$139$	−8.39117	−0.711729	−0.355865	+	0.934537i	$0.615814\pi$
$140$	0	0
$141$	7.00680	0.590079
$142$	0	0
$143$	−7.24262	−0.605658
$144$	0	0
$145$	4.91851	0.408460
$146$	0	0
$147$	4.38550	0.361710
$148$	0	0
$149$	−5.26038	−0.430947	−0.215474	−	0.976510i	$-0.569130\pi$
$149$	−5.26038	−0.430947	−0.215474	+	0.976510i	$0.569130\pi$
$150$	0	0
$151$	−17.1490	−1.39556	−0.697782	−	0.716310i	$-0.745830\pi$
$151$	−17.1490	−1.39556	−0.697782	+	0.716310i	$0.745830\pi$
$152$	0	0
$153$	−7.87192	−0.636407
$154$	0	0
$155$	11.4833	0.922359
$156$	0	0
$157$	−7.75151	−0.618638	−0.309319	−	0.950958i	$-0.600101\pi$
$157$	−7.75151	−0.618638	−0.309319	+	0.950958i	$0.600101\pi$
$158$	0	0
$159$	0.152885	0.0121245
$160$	0	0
$161$	1.93651	0.152618
$162$	0	0
$163$	15.3781	1.20450	0.602252	−	0.798306i	$-0.294270\pi$
$163$	15.3781	1.20450	0.602252	+	0.798306i	$0.294270\pi$
$164$	0	0
$165$	−7.39358	−0.575589
$166$	0	0
$167$	−20.8947	−1.61688	−0.808439	−	0.588580i	$-0.799687\pi$
$167$	−20.8947	−1.61688	−0.808439	+	0.588580i	$0.799687\pi$
$168$	0	0
$169$	−11.2165	−0.862807
$170$	0	0
$171$	14.4920	1.10823
$172$	0	0
$173$	13.1899	1.00281	0.501403	−	0.865214i	$-0.332817\pi$
$173$	13.1899	1.00281	0.501403	+	0.865214i	$0.332817\pi$
$174$	0	0
$175$	0.290031	0.0219243
$176$	0	0
$177$	−6.87052	−0.516420
$178$	0	0
$179$	22.3612	1.67136	0.835678	−	0.549220i	$-0.185075\pi$
$179$	22.3612	1.67136	0.835678	+	0.549220i	$0.185075\pi$
$180$	0	0
$181$	−14.6153	−1.08635	−0.543174	−	0.839620i	$-0.682778\pi$
$181$	−14.6153	−1.08635	−0.543174	+	0.839620i	$0.682778\pi$
$182$	0	0
$183$	4.45839	0.329574
$184$	0	0
$185$	−4.23713	−0.311520
$186$	0	0
$187$	−16.5552	−1.21063
$188$	0	0
$189$	−1.78584	−0.129901
$190$	0	0
$191$	0.977419	0.0707236	0.0353618	−	0.999375i	$-0.488742\pi$
$191$	0.977419	0.0707236	0.0353618	+	0.999375i	$0.488742\pi$
$192$	0	0
$193$	−13.1179	−0.944251	−0.472125	−	0.881531i	$-0.656513\pi$
$193$	−13.1179	−0.944251	−0.472125	+	0.881531i	$0.656513\pi$
$194$	0	0
$195$	1.82068	0.130382
$196$	0	0
$197$	7.34184	0.523084	0.261542	−	0.965192i	$-0.415769\pi$
$197$	7.34184	0.523084	0.261542	+	0.965192i	$0.415769\pi$
$198$	0	0
$199$	20.4950	1.45285	0.726427	−	0.687244i	$-0.241180\pi$
$199$	20.4950	1.45285	0.726427	+	0.687244i	$0.241180\pi$
$200$	0	0
$201$	3.67924	0.259514
$202$	0	0
$203$	1.15490	0.0810581
$204$	0	0
$205$	3.10690	0.216996
$206$	0	0
$207$	10.1251	0.703742
$208$	0	0
$209$	30.4776	2.10818
$210$	0	0
$211$	4.00330	0.275599	0.137799	−	0.990460i	$-0.455997\pi$
$211$	4.00330	0.275599	0.137799	+	0.990460i	$0.455997\pi$
$212$	0	0
$213$	8.86906	0.607698
$214$	0	0
$215$	3.61016	0.246211
$216$	0	0
$217$	2.69635	0.183040
$218$	0	0
$219$	7.49406	0.506402
$220$	0	0
$221$	4.07674	0.274231
$222$	0	0
$223$	−5.31598	−0.355984	−0.177992	−	0.984032i	$-0.556960\pi$
$223$	−5.31598	−0.355984	−0.177992	+	0.984032i	$0.556960\pi$
$224$	0	0
$225$	1.51644	0.101096
$226$	0	0
$227$	−10.4150	−0.691268	−0.345634	−	0.938369i	$-0.612336\pi$
$227$	−10.4150	−0.691268	−0.345634	+	0.938369i	$0.612336\pi$
$228$	0	0
$229$	10.8225	0.715172	0.357586	−	0.933880i	$-0.383600\pi$
$229$	10.8225	0.715172	0.357586	+	0.933880i	$0.383600\pi$
$230$	0	0
$231$	−1.73606	−0.114225
$232$	0	0
$233$	14.4689	0.947891	0.473946	−	0.880554i	$-0.342829\pi$
$233$	14.4689	0.947891	0.473946	+	0.880554i	$0.342829\pi$
$234$	0	0
$235$	22.6753	1.47917
$236$	0	0
$237$	8.69548	0.564832
$238$	0	0
$239$	−14.6986	−0.950776	−0.475388	−	0.879776i	$-0.657692\pi$
$239$	−14.6986	−0.950776	−0.475388	+	0.879776i	$0.657692\pi$
$240$	0	0
$241$	−11.8481	−0.763204	−0.381602	−	0.924327i	$-0.624627\pi$
$241$	−11.8481	−0.763204	−0.381602	+	0.924327i	$0.624627\pi$
$242$	0	0
$243$	−14.3585	−0.921100
$244$	0	0
$245$	14.1923	0.906713
$246$	0	0
$247$	−7.50516	−0.477542
$248$	0	0
$249$	10.3535	0.656127
$250$	0	0
$251$	29.7811	1.87976	0.939882	−	0.341500i	$-0.110935\pi$
$251$	29.7811	1.87976	0.939882	+	0.341500i	$0.110935\pi$
$252$	0	0
$253$	21.2937	1.33872
$254$	0	0
$255$	4.16171	0.260616
$256$	0	0
$257$	17.8819	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$257$	17.8819	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$258$	0	0
$259$	−0.994908	−0.0618206
$260$	0	0
$261$	6.03843	0.373770
$262$	0	0
$263$	−12.4323	−0.766610	−0.383305	−	0.923622i	$-0.625214\pi$
$263$	−12.4323	−0.766610	−0.383305	+	0.923622i	$0.625214\pi$
$264$	0	0
$265$	0.494763	0.0303931
$266$	0	0
$267$	−6.94669	−0.425130
$268$	0	0
$269$	4.42443	0.269762	0.134881	−	0.990862i	$-0.456935\pi$
$269$	4.42443	0.269762	0.134881	+	0.990862i	$0.456935\pi$
$270$	0	0
$271$	−1.86417	−0.113240	−0.0566201	−	0.998396i	$-0.518032\pi$
$271$	−1.86417	−0.113240	−0.0566201	+	0.998396i	$0.518032\pi$
$272$	0	0
$273$	0.427509	0.0258740
$274$	0	0
$275$	3.18916	0.192314
$276$	0	0
$277$	13.6838	0.822181	0.411091	−	0.911594i	$-0.365148\pi$
$277$	13.6838	0.822181	0.411091	+	0.911594i	$0.365148\pi$
$278$	0	0
$279$	14.0980	0.844023
$280$	0	0
$281$	3.51406	0.209631	0.104815	−	0.994492i	$-0.466575\pi$
$281$	3.51406	0.209631	0.104815	+	0.994492i	$0.466575\pi$
$282$	0	0
$283$	19.3392	1.14959	0.574797	−	0.818296i	$-0.305081\pi$
$283$	19.3392	1.14959	0.574797	+	0.818296i	$0.305081\pi$
$284$	0	0
$285$	−7.66159	−0.453833
$286$	0	0
$287$	0.729523	0.0430624
$288$	0	0
$289$	−7.68140	−0.451847
$290$	0	0
$291$	5.34324	0.313226
$292$	0	0
$293$	−15.6698	−0.915439	−0.457719	−	0.889097i	$-0.651334\pi$
$293$	−15.6698	−0.915439	−0.457719	+	0.889097i	$0.651334\pi$
$294$	0	0
$295$	−22.2343	−1.29453
$296$	0	0
$297$	−19.6370	−1.13945
$298$	0	0
$299$	−5.24362	−0.303246
$300$	0	0
$301$	0.847690	0.0488600
$302$	0	0
$303$	−1.33245	−0.0765472
$304$	0	0
$305$	14.4282	0.826155
$306$	0	0
$307$	−27.8779	−1.59107	−0.795536	−	0.605906i	$-0.792811\pi$
$307$	−27.8779	−1.59107	−0.795536	+	0.605906i	$0.792811\pi$
$308$	0	0
$309$	−4.28364	−0.243688
$310$	0	0
$311$	18.7487	1.06314	0.531570	−	0.847014i	$-0.321602\pi$
$311$	18.7487	1.06314	0.531570	+	0.847014i	$0.321602\pi$
$312$	0	0
$313$	20.5219	1.15997	0.579984	−	0.814628i	$-0.303059\pi$
$313$	20.5219	1.15997	0.579984	+	0.814628i	$0.303059\pi$
$314$	0	0
$315$	−2.67145	−0.150519
$316$	0	0
$317$	8.81980	0.495369	0.247685	−	0.968841i	$-0.420330\pi$
$317$	8.81980	0.495369	0.247685	+	0.968841i	$0.420330\pi$
$318$	0	0
$319$	12.6992	0.711019
$320$	0	0
$321$	−5.00925	−0.279589
$322$	0	0
$323$	−17.1553	−0.954545
$324$	0	0
$325$	−0.785338	−0.0435627
$326$	0	0
$327$	−5.74459	−0.317677
$328$	0	0
$329$	5.32432	0.293539
$330$	0	0
$331$	−7.96208	−0.437636	−0.218818	−	0.975766i	$-0.570220\pi$
$331$	−7.96208	−0.437636	−0.218818	+	0.975766i	$0.570220\pi$
$332$	0	0
$333$	−5.20191	−0.285063
$334$	0	0
$335$	11.9067	0.650533
$336$	0	0
$337$	−23.1994	−1.26375	−0.631876	−	0.775070i	$-0.717715\pi$
$337$	−23.1994	−1.26375	−0.631876	+	0.775070i	$0.717715\pi$
$338$	0	0
$339$	1.04562	0.0567901
$340$	0	0
$341$	29.6489	1.60558
$342$	0	0
$343$	6.78488	0.366349
$344$	0	0
$345$	−5.35291	−0.288191
$346$	0	0
$347$	−13.3799	−0.718271	−0.359135	−	0.933285i	$-0.616928\pi$
$347$	−13.3799	−0.718271	−0.359135	+	0.933285i	$0.616928\pi$
$348$	0	0
$349$	−4.33553	−0.232076	−0.116038	−	0.993245i	$-0.537019\pi$
$349$	−4.33553	−0.232076	−0.116038	+	0.993245i	$0.537019\pi$
$350$	0	0
$351$	4.83565	0.258108
$352$	0	0
$353$	16.8608	0.897411	0.448705	−	0.893680i	$-0.351885\pi$
$353$	16.8608	0.897411	0.448705	+	0.893680i	$0.351885\pi$
$354$	0	0
$355$	28.7019	1.52334
$356$	0	0
$357$	0.977198	0.0517188
$358$	0	0
$359$	−10.5231	−0.555387	−0.277694	−	0.960670i	$-0.589570\pi$
$359$	−10.5231	−0.555387	−0.277694	+	0.960670i	$0.589570\pi$
$360$	0	0
$361$	12.5824	0.662230
$362$	0	0
$363$	−11.9500	−0.627214
$364$	0	0
$365$	24.2522	1.26942
$366$	0	0
$367$	22.6463	1.18213	0.591065	−	0.806624i	$-0.298708\pi$
$367$	22.6463	1.18213	0.591065	+	0.806624i	$0.298708\pi$
$368$	0	0
$369$	3.81433	0.198566
$370$	0	0
$371$	0.116174	0.00603145
$372$	0	0
$373$	−24.6880	−1.27830	−0.639148	−	0.769084i	$-0.720713\pi$
$373$	−24.6880	−1.27830	−0.639148	+	0.769084i	$0.720713\pi$
$374$	0	0
$375$	−7.61829	−0.393407
$376$	0	0
$377$	−3.12721	−0.161059
$378$	0	0
$379$	7.56088	0.388376	0.194188	−	0.980964i	$-0.437793\pi$
$379$	7.56088	0.388376	0.194188	+	0.980964i	$0.437793\pi$
$380$	0	0
$381$	−11.9708	−0.613285
$382$	0	0
$383$	12.9946	0.663992	0.331996	−	0.943281i	$-0.392278\pi$
$383$	12.9946	0.663992	0.331996	+	0.943281i	$0.392278\pi$
$384$	0	0
$385$	−5.61822	−0.286331
$386$	0	0
$387$	4.43217	0.225300
$388$	0	0
$389$	7.49062	0.379789	0.189895	−	0.981804i	$-0.439185\pi$
$389$	7.49062	0.379789	0.189895	+	0.981804i	$0.439185\pi$
$390$	0	0
$391$	−11.9858	−0.606150
$392$	0	0
$393$	0.0454785	0.00229409
$394$	0	0
$395$	28.1402	1.41589
$396$	0	0
$397$	−4.41646	−0.221656	−0.110828	−	0.993840i	$-0.535350\pi$
$397$	−4.41646	−0.221656	−0.110828	+	0.993840i	$0.535350\pi$
$398$	0	0
$399$	−1.79899	−0.0900624
$400$	0	0
$401$	−1.63103	−0.0814499	−0.0407250	−	0.999170i	$-0.512967\pi$
$401$	−1.63103	−0.0814499	−0.0407250	+	0.999170i	$0.512967\pi$
$402$	0	0
$403$	−7.30111	−0.363694
$404$	0	0
$405$	−11.3131	−0.562154
$406$	0	0
$407$	−10.9399	−0.542273
$408$	0	0
$409$	−17.1384	−0.847438	−0.423719	−	0.905794i	$-0.639276\pi$
$409$	−17.1384	−0.847438	−0.423719	+	0.905794i	$0.639276\pi$
$410$	0	0
$411$	10.4751	0.516699
$412$	0	0
$413$	−5.22076	−0.256897
$414$	0	0
$415$	33.5059	1.64474
$416$	0	0
$417$	5.44633	0.266708
$418$	0	0
$419$	−12.0759	−0.589944	−0.294972	−	0.955506i	$-0.595310\pi$
$419$	−12.0759	−0.589944	−0.294972	+	0.955506i	$0.595310\pi$
$420$	0	0
$421$	15.8866	0.774265	0.387132	−	0.922024i	$-0.373466\pi$
$421$	15.8866	0.774265	0.387132	+	0.922024i	$0.373466\pi$
$422$	0	0
$423$	27.8384	1.35355
$424$	0	0
$425$	−1.79512	−0.0870762
$426$	0	0
$427$	3.38784	0.163949
$428$	0	0
$429$	4.70086	0.226960
$430$	0	0
$431$	40.0820	1.93068	0.965341	−	0.260993i	$-0.0840499\pi$
$431$	40.0820	1.93068	0.965341	+	0.260993i	$0.0840499\pi$
$432$	0	0
$433$	−27.3917	−1.31636	−0.658181	−	0.752860i	$-0.728674\pi$
$433$	−27.3917	−1.31636	−0.658181	+	0.752860i	$0.728674\pi$
$434$	0	0
$435$	−3.19238	−0.153063
$436$	0	0
$437$	22.0656	1.05554
$438$	0	0
$439$	22.9705	1.09632	0.548162	−	0.836372i	$-0.315328\pi$
$439$	22.9705	1.09632	0.548162	+	0.836372i	$0.315328\pi$
$440$	0	0
$441$	17.4238	0.829706
$442$	0	0
$443$	29.4665	1.39999	0.699997	−	0.714145i	$-0.253184\pi$
$443$	29.4665	1.39999	0.699997	+	0.714145i	$0.253184\pi$
$444$	0	0
$445$	−22.4808	−1.06569
$446$	0	0
$447$	3.41428	0.161490
$448$	0	0
$449$	11.7695	0.555438	0.277719	−	0.960662i	$-0.410422\pi$
$449$	11.7695	0.555438	0.277719	+	0.960662i	$0.410422\pi$
$450$	0	0
$451$	8.02178	0.377731
$452$	0	0
$453$	11.1306	0.522963
$454$	0	0
$455$	1.38350	0.0648594
$456$	0	0
$457$	−19.3291	−0.904178	−0.452089	−	0.891973i	$-0.649321\pi$
$457$	−19.3291	−0.904178	−0.452089	+	0.891973i	$0.649321\pi$
$458$	0	0
$459$	11.0533	0.515924
$460$	0	0
$461$	−13.0842	−0.609394	−0.304697	−	0.952449i	$-0.598555\pi$
$461$	−13.0842	−0.609394	−0.304697	+	0.952449i	$0.598555\pi$
$462$	0	0
$463$	−31.8576	−1.48055	−0.740275	−	0.672304i	$-0.765305\pi$
$463$	−31.8576	−1.48055	−0.740275	+	0.672304i	$0.765305\pi$
$464$	0	0
$465$	−7.45328	−0.345638
$466$	0	0
$467$	−27.4348	−1.26953	−0.634766	−	0.772704i	$-0.718904\pi$
$467$	−27.4348	−1.26953	−0.634766	+	0.772704i	$0.718904\pi$
$468$	0	0
$469$	2.79578	0.129097
$470$	0	0
$471$	5.03116	0.231824
$472$	0	0
$473$	9.32114	0.428586
$474$	0	0
$475$	3.30477	0.151633
$476$	0	0
$477$	0.607419	0.0278118
$478$	0	0
$479$	−0.530850	−0.0242552	−0.0121276	−	0.999926i	$-0.503860\pi$
$479$	−0.530850	−0.0242552	−0.0121276	+	0.999926i	$0.503860\pi$
$480$	0	0
$481$	2.69398	0.122835
$482$	0	0
$483$	−1.25690	−0.0571910
$484$	0	0
$485$	17.2917	0.785176
$486$	0	0
$487$	29.1736	1.32198	0.660990	−	0.750395i	$-0.270137\pi$
$487$	29.1736	1.32198	0.660990	+	0.750395i	$0.270137\pi$
$488$	0	0
$489$	−9.98121	−0.451366
$490$	0	0
$491$	1.33706	0.0603408	0.0301704	−	0.999545i	$-0.490395\pi$
$491$	1.33706	0.0603408	0.0301704	+	0.999545i	$0.490395\pi$
$492$	0	0
$493$	−7.14815	−0.321937
$494$	0	0
$495$	−29.3750	−1.32031
$496$	0	0
$497$	6.73941	0.302304
$498$	0	0
$499$	10.4066	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$499$	10.4066	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$500$	0	0
$501$	13.5618	0.605896
$502$	0	0
$503$	−9.76116	−0.435229	−0.217614	−	0.976035i	$-0.569827\pi$
$503$	−9.76116	−0.435229	−0.217614	+	0.976035i	$0.569827\pi$
$504$	0	0
$505$	−4.31205	−0.191884
$506$	0	0
$507$	7.28012	0.323321
$508$	0	0
$509$	−26.9932	−1.19645	−0.598227	−	0.801327i	$-0.704128\pi$
$509$	−26.9932	−1.19645	−0.598227	+	0.801327i	$0.704128\pi$
$510$	0	0
$511$	5.69458	0.251913
$512$	0	0
$513$	−20.3488	−0.898422
$514$	0	0
$515$	−13.8626	−0.610861
$516$	0	0
$517$	58.5459	2.57484
$518$	0	0
$519$	−8.56094	−0.375784
$520$	0	0
$521$	−12.5942	−0.551764	−0.275882	−	0.961192i	$-0.588970\pi$
$521$	−12.5942	−0.551764	−0.275882	+	0.961192i	$0.588970\pi$
$522$	0	0
$523$	−17.5745	−0.768481	−0.384240	−	0.923233i	$-0.625537\pi$
$523$	−17.5745	−0.768481	−0.384240	+	0.923233i	$0.625537\pi$
$524$	0	0
$525$	−0.188246	−0.00821574
$526$	0	0
$527$	−16.6888	−0.726977
$528$	0	0
$529$	−7.58346	−0.329716
$530$	0	0
$531$	−27.2969	−1.18458
$532$	0	0
$533$	−1.97538	−0.0855632
$534$	0	0
$535$	−16.2109	−0.700857
$536$	0	0
$537$	−14.5137	−0.626311
$538$	0	0
$539$	36.6434	1.57834
$540$	0	0
$541$	−36.0347	−1.54925	−0.774625	−	0.632420i	$-0.782062\pi$
$541$	−36.0347	−1.54925	−0.774625	+	0.632420i	$0.782062\pi$
$542$	0	0
$543$	9.48615	0.407090
$544$	0	0
$545$	−18.5906	−0.796333
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	17.7134	0.755989
$550$	0	0
$551$	13.1596	0.560616
$552$	0	0
$553$	6.60751	0.280980
$554$	0	0
$555$	2.75013	0.116737
$556$	0	0
$557$	−9.76135	−0.413602	−0.206801	−	0.978383i	$-0.566305\pi$
$557$	−9.76135	−0.413602	−0.206801	+	0.978383i	$0.566305\pi$
$558$	0	0
$559$	−2.29535	−0.0970830
$560$	0	0
$561$	10.7452	0.453663
$562$	0	0
$563$	−23.6853	−0.998218	−0.499109	−	0.866539i	$-0.666339\pi$
$563$	−23.6853	−0.998218	−0.499109	+	0.866539i	$0.666339\pi$
$564$	0	0
$565$	3.38381	0.142358
$566$	0	0
$567$	−2.65640	−0.111558
$568$	0	0
$569$	−8.39071	−0.351757	−0.175878	−	0.984412i	$-0.556277\pi$
$569$	−8.39071	−0.351757	−0.175878	+	0.984412i	$0.556277\pi$
$570$	0	0
$571$	14.7123	0.615690	0.307845	−	0.951437i	$-0.400392\pi$
$571$	14.7123	0.615690	0.307845	+	0.951437i	$0.400392\pi$
$572$	0	0
$573$	−0.634399	−0.0265024
$574$	0	0
$575$	2.30894	0.0962893
$576$	0	0
$577$	−20.6874	−0.861229	−0.430615	−	0.902536i	$-0.641703\pi$
$577$	−20.6874	−0.861229	−0.430615	+	0.902536i	$0.641703\pi$
$578$	0	0
$579$	8.51427	0.353841
$580$	0	0
$581$	7.86742	0.326395
$582$	0	0
$583$	1.27744	0.0529062
$584$	0	0
$585$	7.23366	0.299075
$586$	0	0
$587$	−18.3978	−0.759360	−0.379680	−	0.925118i	$-0.623966\pi$
$587$	−18.3978	−0.759360	−0.379680	+	0.925118i	$0.623966\pi$
$588$	0	0
$589$	30.7237	1.26595
$590$	0	0
$591$	−4.76526	−0.196016
$592$	0	0
$593$	−4.27476	−0.175543	−0.0877716	−	0.996141i	$-0.527975\pi$
$593$	−4.27476	−0.175543	−0.0877716	+	0.996141i	$0.527975\pi$
$594$	0	0
$595$	3.16240	0.129646
$596$	0	0
$597$	−13.3024	−0.544431
$598$	0	0
$599$	23.3231	0.952957	0.476478	−	0.879186i	$-0.341913\pi$
$599$	23.3231	0.952957	0.476478	+	0.879186i	$0.341913\pi$
$600$	0	0
$601$	−1.86001	−0.0758716	−0.0379358	−	0.999280i	$-0.512078\pi$
$601$	−1.86001	−0.0758716	−0.0379358	+	0.999280i	$0.512078\pi$
$602$	0	0
$603$	14.6178	0.595283
$604$	0	0
$605$	−38.6725	−1.57226
$606$	0	0
$607$	−2.00439	−0.0813557	−0.0406778	−	0.999172i	$-0.512952\pi$
$607$	−2.00439	−0.0813557	−0.0406778	+	0.999172i	$0.512952\pi$
$608$	0	0
$609$	−0.749594	−0.0303751
$610$	0	0
$611$	−14.4170	−0.583251
$612$	0	0
$613$	37.7597	1.52510	0.762549	−	0.646930i	$-0.223948\pi$
$613$	37.7597	1.52510	0.762549	+	0.646930i	$0.223948\pi$
$614$	0	0
$615$	−2.01655	−0.0813152
$616$	0	0
$617$	−45.6774	−1.83890	−0.919451	−	0.393205i	$-0.871366\pi$
$617$	−45.6774	−1.83890	−0.919451	+	0.393205i	$0.871366\pi$
$618$	0	0
$619$	−12.7954	−0.514292	−0.257146	−	0.966373i	$-0.582782\pi$
$619$	−12.7954	−0.514292	−0.257146	+	0.966373i	$0.582782\pi$
$620$	0	0
$621$	−14.2171	−0.570511
$622$	0	0
$623$	−5.27864	−0.211484
$624$	0	0
$625$	−21.7139	−0.868556
$626$	0	0
$627$	−19.7816	−0.790002
$628$	0	0
$629$	6.15789	0.245531
$630$	0	0
$631$	26.6777	1.06202	0.531011	−	0.847365i	$-0.321812\pi$
$631$	26.6777	1.06202	0.531011	+	0.847365i	$0.321812\pi$
$632$	0	0
$633$	−2.59836	−0.103276
$634$	0	0
$635$	−38.7399	−1.53734
$636$	0	0
$637$	−9.02351	−0.357525
$638$	0	0
$639$	35.2372	1.39396
$640$	0	0
$641$	−5.40348	−0.213425	−0.106712	−	0.994290i	$-0.534032\pi$
$641$	−5.40348	−0.213425	−0.106712	+	0.994290i	$0.534032\pi$
$642$	0	0
$643$	−45.8847	−1.80952	−0.904758	−	0.425926i	$-0.859948\pi$
$643$	−45.8847	−1.80952	−0.904758	+	0.425926i	$0.859948\pi$
$644$	0	0
$645$	−2.34319	−0.0922630
$646$	0	0
$647$	39.9658	1.57122	0.785608	−	0.618724i	$-0.212350\pi$
$647$	39.9658	1.57122	0.785608	+	0.618724i	$0.212350\pi$
$648$	0	0
$649$	−57.4071	−2.25343
$650$	0	0
$651$	−1.75008	−0.0685911
$652$	0	0
$653$	−19.0410	−0.745134	−0.372567	−	0.928005i	$-0.621522\pi$
$653$	−19.0410	−0.745134	−0.372567	+	0.928005i	$0.621522\pi$
$654$	0	0
$655$	0.147177	0.00575067
$656$	0	0
$657$	29.7743	1.16160
$658$	0	0
$659$	45.0530	1.75502	0.877509	−	0.479561i	$-0.159204\pi$
$659$	45.0530	1.75502	0.877509	+	0.479561i	$0.159204\pi$
$660$	0	0
$661$	25.6704	0.998464	0.499232	−	0.866468i	$-0.333616\pi$
$661$	25.6704	0.998464	0.499232	+	0.866468i	$0.333616\pi$
$662$	0	0
$663$	−2.64603	−0.102763
$664$	0	0
$665$	−5.82188	−0.225763
$666$	0	0
$667$	9.19416	0.355999
$668$	0	0
$669$	3.45036	0.133399
$670$	0	0
$671$	37.2524	1.43811
$672$	0	0
$673$	44.8000	1.72691	0.863456	−	0.504425i	$-0.168295\pi$
$673$	44.8000	1.72691	0.863456	+	0.504425i	$0.168295\pi$
$674$	0	0
$675$	−2.12929	−0.0819565
$676$	0	0
$677$	16.9021	0.649601	0.324801	−	0.945783i	$-0.394703\pi$
$677$	16.9021	0.649601	0.324801	+	0.945783i	$0.394703\pi$
$678$	0	0
$679$	4.06021	0.155817
$680$	0	0
$681$	6.75991	0.259040
$682$	0	0
$683$	25.8450	0.988933	0.494467	−	0.869197i	$-0.335363\pi$
$683$	25.8450	0.988933	0.494467	+	0.869197i	$0.335363\pi$
$684$	0	0
$685$	33.8994	1.29523
$686$	0	0
$687$	−7.02441	−0.267998
$688$	0	0
$689$	−0.314572	−0.0119842
$690$	0	0
$691$	−46.1110	−1.75415	−0.877073	−	0.480356i	$-0.840507\pi$
$691$	−46.1110	−1.75415	−0.877073	+	0.480356i	$0.840507\pi$
$692$	0	0
$693$	−6.89746	−0.262013
$694$	0	0
$695$	17.6253	0.668567
$696$	0	0
$697$	−4.51531	−0.171030
$698$	0	0
$699$	−9.39114	−0.355205
$700$	0	0
$701$	33.3947	1.26130	0.630649	−	0.776068i	$-0.282789\pi$
$701$	33.3947	1.26130	0.630649	+	0.776068i	$0.282789\pi$
$702$	0	0
$703$	−11.3365	−0.427565
$704$	0	0
$705$	−14.7175	−0.554294
$706$	0	0
$707$	−1.01250	−0.0380790
$708$	0	0
$709$	−12.1775	−0.457336	−0.228668	−	0.973504i	$-0.573437\pi$
$709$	−12.1775	−0.457336	−0.228668	+	0.973504i	$0.573437\pi$
$710$	0	0
$711$	34.5476	1.29563
$712$	0	0
$713$	21.4657	0.803896
$714$	0	0
$715$	15.2129	0.568929
$716$	0	0
$717$	9.54023	0.356286
$718$	0	0
$719$	31.0757	1.15893	0.579463	−	0.814998i	$-0.303262\pi$
$719$	31.0757	1.15893	0.579463	+	0.814998i	$0.303262\pi$
$720$	0	0
$721$	−3.25505	−0.121224
$722$	0	0
$723$	7.69007	0.285997
$724$	0	0
$725$	1.37701	0.0511409
$726$	0	0
$727$	−29.0355	−1.07687	−0.538433	−	0.842668i	$-0.680984\pi$
$727$	−29.0355	−1.07687	−0.538433	+	0.842668i	$0.680984\pi$
$728$	0	0
$729$	−6.83858	−0.253281
$730$	0	0
$731$	−5.24670	−0.194056
$732$	0	0
$733$	18.9678	0.700590	0.350295	−	0.936639i	$-0.386081\pi$
$733$	18.9678	0.700590	0.350295	+	0.936639i	$0.386081\pi$
$734$	0	0
$735$	−9.21159	−0.339774
$736$	0	0
$737$	30.7422	1.13240
$738$	0	0
$739$	−23.4947	−0.864266	−0.432133	−	0.901810i	$-0.642239\pi$
$739$	−23.4947	−0.864266	−0.432133	+	0.901810i	$0.642239\pi$
$740$	0	0
$741$	4.87126	0.178950
$742$	0	0
$743$	11.2292	0.411961	0.205980	−	0.978556i	$-0.433962\pi$
$743$	11.2292	0.411961	0.205980	+	0.978556i	$0.433962\pi$
$744$	0	0
$745$	11.0492	0.404813
$746$	0	0
$747$	41.1350	1.50505
$748$	0	0
$749$	−3.80642	−0.139084
$750$	0	0
$751$	47.5855	1.73642	0.868210	−	0.496197i	$-0.165270\pi$
$751$	47.5855	1.73642	0.868210	+	0.496197i	$0.165270\pi$
$752$	0	0
$753$	−19.3295	−0.704408
$754$	0	0
$755$	36.0208	1.31093
$756$	0	0
$757$	19.4143	0.705626	0.352813	−	0.935694i	$-0.385225\pi$
$757$	19.4143	0.705626	0.352813	+	0.935694i	$0.385225\pi$
$758$	0	0
$759$	−13.8208	−0.501663
$760$	0	0
$761$	25.9744	0.941571	0.470785	−	0.882248i	$-0.343971\pi$
$761$	25.9744	0.941571	0.470785	+	0.882248i	$0.343971\pi$
$762$	0	0
$763$	−4.36520	−0.158031
$764$	0	0
$765$	16.5347	0.597812
$766$	0	0
$767$	14.1366	0.510444
$768$	0	0
$769$	−36.9822	−1.33361	−0.666806	−	0.745231i	$-0.732339\pi$
$769$	−36.9822	−1.33361	−0.666806	+	0.745231i	$0.732339\pi$
$770$	0	0
$771$	−11.6064	−0.417993
$772$	0	0
$773$	15.4954	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$773$	15.4954	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$774$	0	0
$775$	3.21492	0.115483
$776$	0	0
$777$	0.645750	0.0231662
$778$	0	0
$779$	8.31257	0.297829
$780$	0	0
$781$	74.1061	2.65172
$782$	0	0
$783$	−8.47882	−0.303008
$784$	0	0
$785$	16.2818	0.581121
$786$	0	0
$787$	−22.1934	−0.791110	−0.395555	−	0.918442i	$-0.629448\pi$
$787$	−22.1934	−0.791110	−0.395555	+	0.918442i	$0.629448\pi$
$788$	0	0
$789$	8.06927	0.287274
$790$	0	0
$791$	0.794542	0.0282507
$792$	0	0
$793$	−9.17348	−0.325760
$794$	0	0
$795$	−0.321129	−0.0113893
$796$	0	0
$797$	−33.1295	−1.17351	−0.586754	−	0.809765i	$-0.699595\pi$
$797$	−33.1295	−1.17351	−0.586754	+	0.809765i	$0.699595\pi$
$798$	0	0
$799$	−32.9544	−1.16584
$800$	0	0
$801$	−27.5995	−0.975182
$802$	0	0
$803$	62.6172	2.20971
$804$	0	0
$805$	−4.06756	−0.143363
$806$	0	0
$807$	−2.87170	−0.101089
$808$	0	0
$809$	36.4939	1.28306	0.641528	−	0.767100i	$-0.278301\pi$
$809$	36.4939	1.28306	0.641528	+	0.767100i	$0.278301\pi$
$810$	0	0
$811$	27.2943	0.958433	0.479216	−	0.877697i	$-0.340921\pi$
$811$	27.2943	0.958433	0.479216	+	0.877697i	$0.340921\pi$
$812$	0	0
$813$	1.20995	0.0424347
$814$	0	0
$815$	−32.3011	−1.13146
$816$	0	0
$817$	9.65903	0.337927
$818$	0	0
$819$	1.69851	0.0593509
$820$	0	0
$821$	−32.5051	−1.13444	−0.567218	−	0.823568i	$-0.691980\pi$
$821$	−32.5051	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$822$	0	0
$823$	−8.50744	−0.296551	−0.148275	−	0.988946i	$-0.547372\pi$
$823$	−8.50744	−0.296551	−0.148275	+	0.988946i	$0.547372\pi$
$824$	0	0
$825$	−2.06994	−0.0720661
$826$	0	0
$827$	−31.7519	−1.10412	−0.552061	−	0.833804i	$-0.686159\pi$
$827$	−31.7519	−1.10412	−0.552061	+	0.833804i	$0.686159\pi$
$828$	0	0
$829$	−21.1405	−0.734239	−0.367120	−	0.930174i	$-0.619656\pi$
$829$	−21.1405	−0.734239	−0.367120	+	0.930174i	$0.619656\pi$
$830$	0	0
$831$	−8.88156	−0.308098
$832$	0	0
$833$	−20.6259	−0.714645
$834$	0	0
$835$	43.8885	1.51882
$836$	0	0
$837$	−19.7956	−0.684235
$838$	0	0
$839$	−36.9463	−1.27553	−0.637764	−	0.770232i	$-0.720140\pi$
$839$	−36.9463	−1.27553	−0.637764	+	0.770232i	$0.720140\pi$
$840$	0	0
$841$	−23.5168	−0.810923
$842$	0	0
$843$	−2.28082	−0.0785555
$844$	0	0
$845$	23.5598	0.810482
$846$	0	0
$847$	−9.08057	−0.312012
$848$	0	0
$849$	−12.5522	−0.430790
$850$	0	0
$851$	−7.92046	−0.271510
$852$	0	0
$853$	−32.2923	−1.10567	−0.552834	−	0.833291i	$-0.686454\pi$
$853$	−32.2923	−1.10567	−0.552834	+	0.833291i	$0.686454\pi$
$854$	0	0
$855$	−30.4399	−1.04102
$856$	0	0
$857$	−50.1115	−1.71178	−0.855889	−	0.517160i	$-0.826989\pi$
$857$	−50.1115	−1.71178	−0.855889	+	0.517160i	$0.826989\pi$
$858$	0	0
$859$	2.56245	0.0874298	0.0437149	−	0.999044i	$-0.486081\pi$
$859$	2.56245	0.0874298	0.0437149	+	0.999044i	$0.486081\pi$
$860$	0	0
$861$	−0.473500	−0.0161369
$862$	0	0
$863$	−34.5586	−1.17639	−0.588194	−	0.808720i	$-0.700161\pi$
$863$	−34.5586	−1.17639	−0.588194	+	0.808720i	$0.700161\pi$
$864$	0	0
$865$	−27.7048	−0.941991
$866$	0	0
$867$	4.98565	0.169322
$868$	0	0
$869$	72.6557	2.46468
$870$	0	0
$871$	−7.57033	−0.256511
$872$	0	0
$873$	21.2289	0.718491
$874$	0	0
$875$	−5.78898	−0.195703
$876$	0	0
$877$	42.8424	1.44669	0.723343	−	0.690489i	$-0.242604\pi$
$877$	42.8424	1.44669	0.723343	+	0.690489i	$0.242604\pi$
$878$	0	0
$879$	10.1706	0.343044
$880$	0	0
$881$	19.9855	0.673330	0.336665	−	0.941625i	$-0.390701\pi$
$881$	19.9855	0.673330	0.336665	+	0.941625i	$0.390701\pi$
$882$	0	0
$883$	−49.7566	−1.67444	−0.837221	−	0.546865i	$-0.815821\pi$
$883$	−49.7566	−1.67444	−0.837221	+	0.546865i	$0.815821\pi$
$884$	0	0
$885$	14.4313	0.485102
$886$	0	0
$887$	5.04201	0.169294	0.0846470	−	0.996411i	$-0.473024\pi$
$887$	5.04201	0.169294	0.0846470	+	0.996411i	$0.473024\pi$
$888$	0	0
$889$	−9.09639	−0.305083
$890$	0	0
$891$	−29.2096	−0.978559
$892$	0	0
$893$	60.6681	2.03018
$894$	0	0
$895$	−46.9689	−1.57000
$896$	0	0
$897$	3.40340	0.113636
$898$	0	0
$899$	12.8018	0.426963
$900$	0	0
$901$	−0.719048	−0.0239550
$902$	0	0
$903$	−0.550197	−0.0183094
$904$	0	0
$905$	30.6989	1.02047
$906$	0	0
$907$	−24.4048	−0.810347	−0.405174	−	0.914240i	$-0.632789\pi$
$907$	−24.4048	−0.810347	−0.405174	+	0.914240i	$0.632789\pi$
$908$	0	0
$909$	−5.29388	−0.175587
$910$	0	0
$911$	10.2748	0.340419	0.170209	−	0.985408i	$-0.445556\pi$
$911$	10.2748	0.340419	0.170209	+	0.985408i	$0.445556\pi$
$912$	0	0
$913$	86.5096	2.86305
$914$	0	0
$915$	−9.36468	−0.309587
$916$	0	0
$917$	0.0345581	0.00114121
$918$	0	0
$919$	−39.7302	−1.31058	−0.655289	−	0.755378i	$-0.727453\pi$
$919$	−39.7302	−1.31058	−0.655289	+	0.755378i	$0.727453\pi$
$920$	0	0
$921$	18.0943	0.596226
$922$	0	0
$923$	−18.2488	−0.600666
$924$	0	0
$925$	−1.18625	−0.0390036
$926$	0	0
$927$	−17.0191	−0.558981
$928$	0	0
$929$	28.4071	0.932007	0.466003	−	0.884783i	$-0.345693\pi$
$929$	28.4071	0.932007	0.466003	+	0.884783i	$0.345693\pi$
$930$	0	0
$931$	37.9717	1.24447
$932$	0	0
$933$	−12.1689	−0.398393
$934$	0	0
$935$	34.7735	1.13721
$936$	0	0
$937$	−11.2515	−0.367572	−0.183786	−	0.982966i	$-0.558835\pi$
$937$	−11.2515	−0.367572	−0.183786	+	0.982966i	$0.558835\pi$
$938$	0	0
$939$	−13.3199	−0.434678
$940$	0	0
$941$	39.5265	1.28853	0.644264	−	0.764803i	$-0.277164\pi$
$941$	39.5265	1.28853	0.644264	+	0.764803i	$0.277164\pi$
$942$	0	0
$943$	5.80773	0.189126
$944$	0	0
$945$	3.75109	0.122023
$946$	0	0
$947$	22.4627	0.729938	0.364969	−	0.931020i	$-0.381080\pi$
$947$	22.4627	0.729938	0.364969	+	0.931020i	$0.381080\pi$
$948$	0	0
$949$	−15.4196	−0.500542
$950$	0	0
$951$	−5.72454	−0.185631
$952$	0	0
$953$	−55.0446	−1.78307	−0.891534	−	0.452953i	$-0.850370\pi$
$953$	−55.0446	−1.78307	−0.891534	+	0.452953i	$0.850370\pi$
$954$	0	0
$955$	−2.05303	−0.0664346
$956$	0	0
$957$	−8.24248	−0.266442
$958$	0	0
$959$	7.95982	0.257036
$960$	0	0
$961$	−1.11164	−0.0358594
$962$	0	0
$963$	−19.9020	−0.641333
$964$	0	0
$965$	27.5538	0.886987
$966$	0	0
$967$	−28.6510	−0.921354	−0.460677	−	0.887568i	$-0.652393\pi$
$967$	−28.6510	−0.921354	−0.460677	+	0.887568i	$0.652393\pi$
$968$	0	0
$969$	11.1347	0.357699
$970$	0	0
$971$	14.0405	0.450583	0.225291	−	0.974291i	$-0.427667\pi$
$971$	14.0405	0.450583	0.225291	+	0.974291i	$0.427667\pi$
$972$	0	0
$973$	4.13855	0.132676
$974$	0	0
$975$	0.509727	0.0163243
$976$	0	0
$977$	−7.80427	−0.249681	−0.124840	−	0.992177i	$-0.539842\pi$
$977$	−7.80427	−0.249681	−0.124840	+	0.992177i	$0.539842\pi$
$978$	0	0
$979$	−58.0436	−1.85508
$980$	0	0
$981$	−22.8236	−0.728700
$982$	0	0
$983$	43.0140	1.37193	0.685967	−	0.727633i	$-0.259379\pi$
$983$	43.0140	1.37193	0.685967	+	0.727633i	$0.259379\pi$
$984$	0	0
$985$	−15.4213	−0.491362
$986$	0	0
$987$	−3.45578	−0.109999
$988$	0	0
$989$	6.74846	0.214588
$990$	0	0
$991$	15.8637	0.503926	0.251963	−	0.967737i	$-0.418924\pi$
$991$	15.8637	0.503926	0.251963	+	0.967737i	$0.418924\pi$
$992$	0	0
$993$	5.16783	0.163996
$994$	0	0
$995$	−43.0490	−1.36475
$996$	0	0
$997$	−23.2483	−0.736280	−0.368140	−	0.929770i	$-0.620005\pi$
$997$	−23.2483	−0.736280	−0.368140	+	0.929770i	$0.620005\pi$
$998$	0	0
$999$	7.30422	0.231095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.v.1.14		25
4.3	odd	2		547.2.a.c.1.11	✓	25
12.11	even	2		4923.2.a.n.1.15		25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.c.1.11	✓	25	4.3	odd	2
4923.2.a.n.1.15		25	12.11	even	2
8752.2.a.v.1.14		25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.649055 q^{3} -2.10046 q^{5} -0.493203 q^{7} -2.57873 q^{9} +O(q^{10})\)	\(q-0.649055 q^{3} -2.10046 q^{5} -0.493203 q^{7} -2.57873 q^{9} -5.42323 q^{11} +1.33548 q^{13} +1.36332 q^{15} +3.05264 q^{17} -5.61982 q^{19} +0.320116 q^{21} -3.92639 q^{23} -0.588056 q^{25} +3.62090 q^{27} -2.34163 q^{29} -5.46702 q^{31} +3.51997 q^{33} +1.03596 q^{35} +2.01724 q^{37} -0.866801 q^{39} -1.47915 q^{41} -1.71874 q^{43} +5.41652 q^{45} -10.7954 q^{47} -6.75675 q^{49} -1.98133 q^{51} -0.235550 q^{53} +11.3913 q^{55} +3.64757 q^{57} +10.5854 q^{59} -6.86905 q^{61} +1.27184 q^{63} -2.80513 q^{65} -5.66862 q^{67} +2.54844 q^{69} -13.6646 q^{71} -11.5461 q^{73} +0.381681 q^{75} +2.67475 q^{77} -13.3971 q^{79} +5.38602 q^{81} -15.9517 q^{83} -6.41195 q^{85} +1.51985 q^{87} +10.7028 q^{89} -0.658664 q^{91} +3.54840 q^{93} +11.8042 q^{95} -8.23233 q^{97} +13.9850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9}+O(q^{10}) \) 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9	\( 25 q - 8 q^{3} + 29 q^{5} - 5 q^{7} + 29 q^{9} - 10 q^{11} + 19 q^{13} - 5 q^{15} + 40 q^{17} - 8 q^{21} - 26 q^{23} + 36 q^{25} - 11 q^{27} + 30 q^{29} + 5 q^{31} + 10 q^{33} - 11 q^{35} + 26 q^{37} + 17 q^{39} + 9 q^{41} + 10 q^{43} + 64 q^{45} - 28 q^{47} + 20 q^{49} + 9 q^{51} + 80 q^{53} + q^{55} - 8 q^{57} + 2 q^{59} + 22 q^{61} + 9 q^{63} + 30 q^{65} + 16 q^{67} + 22 q^{69} + q^{71} + 2 q^{73} + 31 q^{75} + 67 q^{77} + 34 q^{79} - 11 q^{81} - 15 q^{83} + 15 q^{85} + 29 q^{87} + 38 q^{89} + 41 q^{91} - 4 q^{93} + 46 q^{95} - 2 q^{97} + 41 q^{99}+O(q^{100}) \) 25 * q - 8 * q^3 + 29 * q^5 - 5 * q^7 + 29 * q^9 - 10 * q^11 + 19 * q^13 - 5 * q^15 + 40 * q^17 - 8 * q^21 - 26 * q^23 + 36 * q^25 - 11 * q^27 + 30 * q^29 + 5 * q^31 + 10 * q^33 - 11 * q^35 + 26 * q^37 + 17 * q^39 + 9 * q^41 + 10 * q^43 + 64 * q^45 - 28 * q^47 + 20 * q^49 + 9 * q^51 + 80 * q^53 + q^55 - 8 * q^57 + 2 * q^59 + 22 * q^61 + 9 * q^63 + 30 * q^65 + 16 * q^67 + 22 * q^69 + q^71 + 2 * q^73 + 31 * q^75 + 67 * q^77 + 34 * q^79 - 11 * q^81 - 15 * q^83 + 15 * q^85 + 29 * q^87 + 38 * q^89 + 41 * q^91 - 4 * q^93 + 46 * q^95 - 2 * q^97 + 41 * q^99

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