LMFDB - Embedded newform 8128.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8128,2,Mod(1,8128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8128, 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("8128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8128 = 2^{6} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8128.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:	$64.9024067629$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	8128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34730 q^{3} +0.120615 q^{5} -0.879385 q^{7} -1.18479 q^{9} +O(q^{10})$	$q-1.34730 q^{3} +0.120615 q^{5} -0.879385 q^{7} -1.18479 q^{9} -2.71688 q^{11} +5.10607 q^{13} -0.162504 q^{15} -4.46791 q^{17} +2.87939 q^{19} +1.18479 q^{21} +5.22668 q^{23} -4.98545 q^{25} +5.63816 q^{27} -5.94356 q^{29} -1.42602 q^{31} +3.66044 q^{33} -0.106067 q^{35} +10.5817 q^{37} -6.87939 q^{39} -5.38919 q^{41} +8.27631 q^{43} -0.142903 q^{45} +6.43376 q^{47} -6.22668 q^{49} +6.01960 q^{51} -12.6236 q^{53} -0.327696 q^{55} -3.87939 q^{57} +2.71688 q^{59} +1.98040 q^{61} +1.04189 q^{63} +0.615867 q^{65} -2.87939 q^{67} -7.04189 q^{69} -9.49794 q^{71} +11.0915 q^{73} +6.71688 q^{75} +2.38919 q^{77} +7.72462 q^{79} -4.04189 q^{81} +11.4561 q^{83} -0.538896 q^{85} +8.00774 q^{87} -2.64496 q^{89} -4.49020 q^{91} +1.92127 q^{93} +0.347296 q^{95} +1.68004 q^{97} +3.21894 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^5 + 3 * q^7	$ 3 q - 3 q^{3} + 6 q^{5} + 3 q^{7} + 3 q^{13} - 3 q^{15} - 18 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} - 3 q^{29} - 12 q^{31} - 12 q^{33} + 12 q^{35} - 15 q^{39} - 12 q^{41} - 9 q^{43} + 3 q^{47} - 12 q^{49} + 21 q^{51} - 3 q^{53} + 3 q^{55} - 6 q^{57} + 3 q^{61} - 9 q^{65} - 3 q^{67} - 18 q^{69} - 3 q^{71} + 3 q^{73} + 12 q^{75} + 3 q^{77} - 9 q^{79} - 9 q^{81} + 12 q^{83} - 39 q^{85} - 33 q^{89} - 12 q^{91} - 3 q^{93} - 15 q^{97} + 27 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^5 + 3 * q^7 + 3 * q^13 - 3 * q^15 - 18 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 - 3 * q^29 - 12 * q^31 - 12 * q^33 + 12 * q^35 - 15 * q^39 - 12 * q^41 - 9 * q^43 + 3 * q^47 - 12 * q^49 + 21 * q^51 - 3 * q^53 + 3 * q^55 - 6 * q^57 + 3 * q^61 - 9 * q^65 - 3 * q^67 - 18 * q^69 - 3 * q^71 + 3 * q^73 + 12 * q^75 + 3 * q^77 - 9 * q^79 - 9 * q^81 + 12 * q^83 - 39 * q^85 - 33 * q^89 - 12 * q^91 - 3 * q^93 - 15 * q^97 + 27 * 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
$3$	−1.34730	−0.777862	−0.388931	−	0.921267i	$-0.627156\pi$
$3$	−1.34730	−0.777862	−0.388931	+	0.921267i	$0.627156\pi$
$4$	0	0
$5$	0.120615	0.0539406	0.0269703	−	0.999636i	$-0.491414\pi$
$5$	0.120615	0.0539406	0.0269703	+	0.999636i	$0.491414\pi$
$6$	0	0
$7$	−0.879385	−0.332376	−0.166188	−	0.986094i	$-0.553146\pi$
$7$	−0.879385	−0.332376	−0.166188	+	0.986094i	$0.553146\pi$
$8$	0	0
$9$	−1.18479	−0.394931
$10$	0	0
$11$	−2.71688	−0.819171	−0.409585	−	0.912272i	$-0.634327\pi$
$11$	−2.71688	−0.819171	−0.409585	+	0.912272i	$0.634327\pi$
$12$	0	0
$13$	5.10607	1.41617	0.708084	−	0.706128i	$-0.249560\pi$
$13$	5.10607	1.41617	0.708084	+	0.706128i	$0.249560\pi$
$14$	0	0
$15$	−0.162504	−0.0419583
$16$	0	0
$17$	−4.46791	−1.08363	−0.541814	−	0.840499i	$-0.682262\pi$
$17$	−4.46791	−1.08363	−0.541814	+	0.840499i	$0.682262\pi$
$18$	0	0
$19$	2.87939	0.660576	0.330288	−	0.943880i	$-0.392854\pi$
$19$	2.87939	0.660576	0.330288	+	0.943880i	$0.392854\pi$
$20$	0	0
$21$	1.18479	0.258543
$22$	0	0
$23$	5.22668	1.08984	0.544919	−	0.838489i	$-0.316560\pi$
$23$	5.22668	1.08984	0.544919	+	0.838489i	$0.316560\pi$
$24$	0	0
$25$	−4.98545	−0.997090
$26$	0	0
$27$	5.63816	1.08506
$28$	0	0
$29$	−5.94356	−1.10369	−0.551846	−	0.833946i	$-0.686076\pi$
$29$	−5.94356	−1.10369	−0.551846	+	0.833946i	$0.686076\pi$
$30$	0	0
$31$	−1.42602	−0.256121	−0.128061	−	0.991766i	$-0.540875\pi$
$31$	−1.42602	−0.256121	−0.128061	+	0.991766i	$0.540875\pi$
$32$	0	0
$33$	3.66044	0.637202
$34$	0	0
$35$	−0.106067	−0.0179286
$36$	0	0
$37$	10.5817	1.73962	0.869812	−	0.493383i	$-0.164240\pi$
$37$	10.5817	1.73962	0.869812	+	0.493383i	$0.164240\pi$
$38$	0	0
$39$	−6.87939	−1.10158
$40$	0	0
$41$	−5.38919	−0.841649	−0.420825	−	0.907142i	$-0.638259\pi$
$41$	−5.38919	−0.841649	−0.420825	+	0.907142i	$0.638259\pi$
$42$	0	0
$43$	8.27631	1.26213	0.631063	−	0.775732i	$-0.282619\pi$
$43$	8.27631	1.26213	0.631063	+	0.775732i	$0.282619\pi$
$44$	0	0
$45$	−0.142903	−0.0213028
$46$	0	0
$47$	6.43376	0.938461	0.469230	−	0.883076i	$-0.344531\pi$
$47$	6.43376	0.938461	0.469230	+	0.883076i	$0.344531\pi$
$48$	0	0
$49$	−6.22668	−0.889526
$50$	0	0
$51$	6.01960	0.842913
$52$	0	0
$53$	−12.6236	−1.73399	−0.866993	−	0.498320i	$-0.833950\pi$
$53$	−12.6236	−1.73399	−0.866993	+	0.498320i	$0.833950\pi$
$54$	0	0
$55$	−0.327696	−0.0441865
$56$	0	0
$57$	−3.87939	−0.513837
$58$	0	0
$59$	2.71688	0.353708	0.176854	−	0.984237i	$-0.443408\pi$
$59$	2.71688	0.353708	0.176854	+	0.984237i	$0.443408\pi$
$60$	0	0
$61$	1.98040	0.253564	0.126782	−	0.991931i	$-0.459535\pi$
$61$	1.98040	0.253564	0.126782	+	0.991931i	$0.459535\pi$
$62$	0	0
$63$	1.04189	0.131266
$64$	0	0
$65$	0.615867	0.0763889
$66$	0	0
$67$	−2.87939	−0.351773	−0.175886	−	0.984410i	$-0.556279\pi$
$67$	−2.87939	−0.351773	−0.175886	+	0.984410i	$0.556279\pi$
$68$	0	0
$69$	−7.04189	−0.847744
$70$	0	0
$71$	−9.49794	−1.12720	−0.563599	−	0.826048i	$-0.690584\pi$
$71$	−9.49794	−1.12720	−0.563599	+	0.826048i	$0.690584\pi$
$72$	0	0
$73$	11.0915	1.29816	0.649082	−	0.760718i	$-0.275153\pi$
$73$	11.0915	1.29816	0.649082	+	0.760718i	$0.275153\pi$
$74$	0	0
$75$	6.71688	0.775599
$76$	0	0
$77$	2.38919	0.272273
$78$	0	0
$79$	7.72462	0.869088	0.434544	−	0.900651i	$-0.356910\pi$
$79$	7.72462	0.869088	0.434544	+	0.900651i	$0.356910\pi$
$80$	0	0
$81$	−4.04189	−0.449099
$82$	0	0
$83$	11.4561	1.25747	0.628733	−	0.777622i	$-0.283574\pi$
$83$	11.4561	1.25747	0.628733	+	0.777622i	$0.283574\pi$
$84$	0	0
$85$	−0.538896	−0.0584515
$86$	0	0
$87$	8.00774	0.858520
$88$	0	0
$89$	−2.64496	−0.280366	−0.140183	−	0.990126i	$-0.544769\pi$
$89$	−2.64496	−0.280366	−0.140183	+	0.990126i	$0.544769\pi$
$90$	0	0
$91$	−4.49020	−0.470701
$92$	0	0
$93$	1.92127	0.199227
$94$	0	0
$95$	0.347296	0.0356319
$96$	0	0
$97$	1.68004	0.170583	0.0852914	−	0.996356i	$-0.472818\pi$
$97$	1.68004	0.170583	0.0852914	+	0.996356i	$0.472818\pi$
$98$	0	0
$99$	3.21894	0.323516
$100$	0	0
$101$	16.7246	1.66416	0.832081	−	0.554654i	$-0.187149\pi$
$101$	16.7246	1.66416	0.832081	+	0.554654i	$0.187149\pi$
$102$	0	0
$103$	−0.162504	−0.0160120	−0.00800599	−	0.999968i	$-0.502548\pi$
$103$	−0.162504	−0.0160120	−0.00800599	+	0.999968i	$0.502548\pi$
$104$	0	0
$105$	0.142903	0.0139460
$106$	0	0
$107$	14.4757	1.39941	0.699707	−	0.714430i	$-0.253314\pi$
$107$	14.4757	1.39941	0.699707	+	0.714430i	$0.253314\pi$
$108$	0	0
$109$	−9.36959	−0.897443	−0.448722	−	0.893672i	$-0.648121\pi$
$109$	−9.36959	−0.897443	−0.448722	+	0.893672i	$0.648121\pi$
$110$	0	0
$111$	−14.2567	−1.35319
$112$	0	0
$113$	−8.90167	−0.837399	−0.418700	−	0.908125i	$-0.637514\pi$
$113$	−8.90167	−0.837399	−0.418700	+	0.908125i	$0.637514\pi$
$114$	0	0
$115$	0.630415	0.0587865
$116$	0	0
$117$	−6.04963	−0.559288
$118$	0	0
$119$	3.92902	0.360172
$120$	0	0
$121$	−3.61856	−0.328960
$122$	0	0
$123$	7.26083	0.654687
$124$	0	0
$125$	−1.20439	−0.107724
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	−11.1506	−0.981759
$130$	0	0
$131$	−22.0009	−1.92223	−0.961115	−	0.276148i	$-0.910942\pi$
$131$	−22.0009	−1.92223	−0.961115	+	0.276148i	$0.910942\pi$
$132$	0	0
$133$	−2.53209	−0.219560
$134$	0	0
$135$	0.680045	0.0585289
$136$	0	0
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	0	0
$139$	−18.0351	−1.52972	−0.764858	−	0.644199i	$-0.777191\pi$
$139$	−18.0351	−1.52972	−0.764858	+	0.644199i	$0.777191\pi$
$140$	0	0
$141$	−8.66819	−0.729993
$142$	0	0
$143$	−13.8726	−1.16008
$144$	0	0
$145$	−0.716881	−0.0595338
$146$	0	0
$147$	8.38919	0.691928
$148$	0	0
$149$	−23.8452	−1.95348	−0.976739	−	0.214432i	$-0.931210\pi$
$149$	−23.8452	−1.95348	−0.976739	+	0.214432i	$0.931210\pi$
$150$	0	0
$151$	7.53714	0.613364	0.306682	−	0.951812i	$-0.400781\pi$
$151$	7.53714	0.613364	0.306682	+	0.951812i	$0.400781\pi$
$152$	0	0
$153$	5.29355	0.427958
$154$	0	0
$155$	−0.171999	−0.0138153
$156$	0	0
$157$	18.0915	1.44386	0.721930	−	0.691966i	$-0.243255\pi$
$157$	18.0915	1.44386	0.721930	+	0.691966i	$0.243255\pi$
$158$	0	0
$159$	17.0077	1.34880
$160$	0	0
$161$	−4.59627	−0.362237
$162$	0	0
$163$	−1.63041	−0.127704	−0.0638520	−	0.997959i	$-0.520339\pi$
$163$	−1.63041	−0.127704	−0.0638520	+	0.997959i	$0.520339\pi$
$164$	0	0
$165$	0.441504	0.0343710
$166$	0	0
$167$	−3.45336	−0.267229	−0.133615	−	0.991033i	$-0.542658\pi$
$167$	−3.45336	−0.267229	−0.133615	+	0.991033i	$0.542658\pi$
$168$	0	0
$169$	13.0719	1.00553
$170$	0	0
$171$	−3.41147	−0.260882
$172$	0	0
$173$	9.23173	0.701876	0.350938	−	0.936399i	$-0.385863\pi$
$173$	9.23173	0.701876	0.350938	+	0.936399i	$0.385863\pi$
$174$	0	0
$175$	4.38413	0.331409
$176$	0	0
$177$	−3.66044	−0.275136
$178$	0	0
$179$	−9.80840	−0.733114	−0.366557	−	0.930396i	$-0.619464\pi$
$179$	−9.80840	−0.733114	−0.366557	+	0.930396i	$0.619464\pi$
$180$	0	0
$181$	7.98545	0.593554	0.296777	−	0.954947i	$-0.404088\pi$
$181$	7.98545	0.593554	0.296777	+	0.954947i	$0.404088\pi$
$182$	0	0
$183$	−2.66819	−0.197238
$184$	0	0
$185$	1.27631	0.0938363
$186$	0	0
$187$	12.1388	0.887676
$188$	0	0
$189$	−4.95811	−0.360650
$190$	0	0
$191$	4.20439	0.304219	0.152110	−	0.988364i	$-0.451393\pi$
$191$	4.20439	0.304219	0.152110	+	0.988364i	$0.451393\pi$
$192$	0	0
$193$	11.1257	0.800843	0.400422	−	0.916331i	$-0.368864\pi$
$193$	11.1257	0.800843	0.400422	+	0.916331i	$0.368864\pi$
$194$	0	0
$195$	−0.829755	−0.0594200
$196$	0	0
$197$	−5.44831	−0.388176	−0.194088	−	0.980984i	$-0.562175\pi$
$197$	−5.44831	−0.388176	−0.194088	+	0.980984i	$0.562175\pi$
$198$	0	0
$199$	−6.27631	−0.444916	−0.222458	−	0.974942i	$-0.571408\pi$
$199$	−6.27631	−0.444916	−0.222458	+	0.974942i	$0.571408\pi$
$200$	0	0
$201$	3.87939	0.273631
$202$	0	0
$203$	5.22668	0.366841
$204$	0	0
$205$	−0.650015	−0.0453990
$206$	0	0
$207$	−6.19253	−0.430411
$208$	0	0
$209$	−7.82295	−0.541125
$210$	0	0
$211$	5.09152	0.350515	0.175257	−	0.984523i	$-0.443924\pi$
$211$	5.09152	0.350515	0.175257	+	0.984523i	$0.443924\pi$
$212$	0	0
$213$	12.7965	0.876805
$214$	0	0
$215$	0.998245	0.0680798
$216$	0	0
$217$	1.25402	0.0851286
$218$	0	0
$219$	−14.9436	−1.00979
$220$	0	0
$221$	−22.8135	−1.53460
$222$	0	0
$223$	−16.1530	−1.08169	−0.540843	−	0.841124i	$-0.681895\pi$
$223$	−16.1530	−1.08169	−0.540843	+	0.841124i	$0.681895\pi$
$224$	0	0
$225$	5.90673	0.393782
$226$	0	0
$227$	−9.34730	−0.620402	−0.310201	−	0.950671i	$-0.600396\pi$
$227$	−9.34730	−0.620402	−0.310201	+	0.950671i	$0.600396\pi$
$228$	0	0
$229$	10.5321	0.695980	0.347990	−	0.937498i	$-0.386864\pi$
$229$	10.5321	0.695980	0.347990	+	0.937498i	$0.386864\pi$
$230$	0	0
$231$	−3.21894	−0.211791
$232$	0	0
$233$	−10.5544	−0.691440	−0.345720	−	0.938338i	$-0.612365\pi$
$233$	−10.5544	−0.691440	−0.345720	+	0.938338i	$0.612365\pi$
$234$	0	0
$235$	0.776007	0.0506211
$236$	0	0
$237$	−10.4074	−0.676030
$238$	0	0
$239$	−15.4466	−0.999155	−0.499577	−	0.866269i	$-0.666511\pi$
$239$	−15.4466	−0.999155	−0.499577	+	0.866269i	$0.666511\pi$
$240$	0	0
$241$	8.05232	0.518695	0.259348	−	0.965784i	$-0.416492\pi$
$241$	8.05232	0.518695	0.259348	+	0.965784i	$0.416492\pi$
$242$	0	0
$243$	−11.4688	−0.735727
$244$	0	0
$245$	−0.751030	−0.0479815
$246$	0	0
$247$	14.7023	0.935487
$248$	0	0
$249$	−15.4347	−0.978134
$250$	0	0
$251$	−5.52528	−0.348753	−0.174376	−	0.984679i	$-0.555791\pi$
$251$	−5.52528	−0.348753	−0.174376	+	0.984679i	$0.555791\pi$
$252$	0	0
$253$	−14.2003	−0.892764
$254$	0	0
$255$	0.726053	0.0454672
$256$	0	0
$257$	−0.608126	−0.0379339	−0.0189669	−	0.999820i	$-0.506038\pi$
$257$	−0.608126	−0.0379339	−0.0189669	+	0.999820i	$0.506038\pi$
$258$	0	0
$259$	−9.30541	−0.578210
$260$	0	0
$261$	7.04189	0.435882
$262$	0	0
$263$	8.95636	0.552273	0.276136	−	0.961118i	$-0.410946\pi$
$263$	8.95636	0.552273	0.276136	+	0.961118i	$0.410946\pi$
$264$	0	0
$265$	−1.52259	−0.0935322
$266$	0	0
$267$	3.56355	0.218086
$268$	0	0
$269$	−9.49794	−0.579100	−0.289550	−	0.957163i	$-0.593506\pi$
$269$	−9.49794	−0.579100	−0.289550	+	0.957163i	$0.593506\pi$
$270$	0	0
$271$	−10.5749	−0.642380	−0.321190	−	0.947015i	$-0.604083\pi$
$271$	−10.5749	−0.642380	−0.321190	+	0.947015i	$0.604083\pi$
$272$	0	0
$273$	6.04963	0.366140
$274$	0	0
$275$	13.5449	0.816787
$276$	0	0
$277$	−23.8161	−1.43097	−0.715487	−	0.698626i	$-0.753795\pi$
$277$	−23.8161	−1.43097	−0.715487	+	0.698626i	$0.753795\pi$
$278$	0	0
$279$	1.68954	0.101150
$280$	0	0
$281$	−17.1215	−1.02139	−0.510693	−	0.859763i	$-0.670611\pi$
$281$	−17.1215	−1.02139	−0.510693	+	0.859763i	$0.670611\pi$
$282$	0	0
$283$	24.6236	1.46372	0.731861	−	0.681454i	$-0.238652\pi$
$283$	24.6236	1.46372	0.731861	+	0.681454i	$0.238652\pi$
$284$	0	0
$285$	−0.467911	−0.0277167
$286$	0	0
$287$	4.73917	0.279744
$288$	0	0
$289$	2.96223	0.174249
$290$	0	0
$291$	−2.26352	−0.132690
$292$	0	0
$293$	13.2739	0.775472	0.387736	−	0.921770i	$-0.373257\pi$
$293$	13.2739	0.775472	0.387736	+	0.921770i	$0.373257\pi$
$294$	0	0
$295$	0.327696	0.0190792
$296$	0	0
$297$	−15.3182	−0.888852
$298$	0	0
$299$	26.6878	1.54339
$300$	0	0
$301$	−7.27807	−0.419501
$302$	0	0
$303$	−22.5330	−1.29449
$304$	0	0
$305$	0.238865	0.0136774
$306$	0	0
$307$	−26.3182	−1.50206	−0.751030	−	0.660269i	$-0.770442\pi$
$307$	−26.3182	−1.50206	−0.751030	+	0.660269i	$0.770442\pi$
$308$	0	0
$309$	0.218941	0.0124551
$310$	0	0
$311$	4.31315	0.244576	0.122288	−	0.992495i	$-0.460977\pi$
$311$	4.31315	0.244576	0.122288	+	0.992495i	$0.460977\pi$
$312$	0	0
$313$	−34.6732	−1.95985	−0.979924	−	0.199373i	$-0.936109\pi$
$313$	−34.6732	−1.95985	−0.979924	+	0.199373i	$0.936109\pi$
$314$	0	0
$315$	0.125667	0.00708054
$316$	0	0
$317$	21.2371	1.19279	0.596397	−	0.802689i	$-0.296598\pi$
$317$	21.2371	1.19279	0.596397	+	0.802689i	$0.296598\pi$
$318$	0	0
$319$	16.1480	0.904112
$320$	0	0
$321$	−19.5030	−1.08855
$322$	0	0
$323$	−12.8648	−0.715819
$324$	0	0
$325$	−25.4561	−1.41205
$326$	0	0
$327$	12.6236	0.698087
$328$	0	0
$329$	−5.65776	−0.311922
$330$	0	0
$331$	22.8598	1.25649	0.628244	−	0.778017i	$-0.283774\pi$
$331$	22.8598	1.25649	0.628244	+	0.778017i	$0.283774\pi$
$332$	0	0
$333$	−12.5371	−0.687031
$334$	0	0
$335$	−0.347296	−0.0189748
$336$	0	0
$337$	7.49020	0.408017	0.204009	−	0.978969i	$-0.434603\pi$
$337$	7.49020	0.408017	0.204009	+	0.978969i	$0.434603\pi$
$338$	0	0
$339$	11.9932	0.651381
$340$	0	0
$341$	3.87433	0.209807
$342$	0	0
$343$	11.6313	0.628034
$344$	0	0
$345$	−0.849356	−0.0457278
$346$	0	0
$347$	−22.9581	−1.23246	−0.616228	−	0.787568i	$-0.711340\pi$
$347$	−22.9581	−1.23246	−0.616228	+	0.787568i	$0.711340\pi$
$348$	0	0
$349$	−23.2131	−1.24257	−0.621284	−	0.783586i	$-0.713388\pi$
$349$	−23.2131	−1.24257	−0.621284	+	0.783586i	$0.713388\pi$
$350$	0	0
$351$	28.7888	1.53663
$352$	0	0
$353$	−22.1807	−1.18056	−0.590279	−	0.807199i	$-0.700982\pi$
$353$	−22.1807	−1.18056	−0.590279	+	0.807199i	$0.700982\pi$
$354$	0	0
$355$	−1.14559	−0.0608017
$356$	0	0
$357$	−5.29355	−0.280164
$358$	0	0
$359$	2.83750	0.149757	0.0748787	−	0.997193i	$-0.476143\pi$
$359$	2.83750	0.149757	0.0748787	+	0.997193i	$0.476143\pi$
$360$	0	0
$361$	−10.7091	−0.563639
$362$	0	0
$363$	4.87527	0.255885
$364$	0	0
$365$	1.33780	0.0700237
$366$	0	0
$367$	38.1147	1.98957	0.994787	−	0.101978i	$-0.0325171\pi$
$367$	38.1147	1.98957	0.994787	+	0.101978i	$0.0325171\pi$
$368$	0	0
$369$	6.38507	0.332393
$370$	0	0
$371$	11.1010	0.576336
$372$	0	0
$373$	−35.2841	−1.82694	−0.913469	−	0.406907i	$-0.866607\pi$
$373$	−35.2841	−1.82694	−0.913469	+	0.406907i	$0.866607\pi$
$374$	0	0
$375$	1.62267	0.0837945
$376$	0	0
$377$	−30.3482	−1.56301
$378$	0	0
$379$	−19.2422	−0.988404	−0.494202	−	0.869347i	$-0.664540\pi$
$379$	−19.2422	−0.988404	−0.494202	+	0.869347i	$0.664540\pi$
$380$	0	0
$381$	−1.34730	−0.0690241
$382$	0	0
$383$	30.7811	1.57284	0.786419	−	0.617693i	$-0.211933\pi$
$383$	30.7811	1.57284	0.786419	+	0.617693i	$0.211933\pi$
$384$	0	0
$385$	0.288171	0.0146866
$386$	0	0
$387$	−9.80571	−0.498452
$388$	0	0
$389$	−8.57491	−0.434765	−0.217383	−	0.976086i	$-0.569752\pi$
$389$	−8.57491	−0.434765	−0.217383	+	0.976086i	$0.569752\pi$
$390$	0	0
$391$	−23.3523	−1.18098
$392$	0	0
$393$	29.6418	1.49523
$394$	0	0
$395$	0.931703	0.0468791
$396$	0	0
$397$	12.8307	0.643954	0.321977	−	0.946748i	$-0.395653\pi$
$397$	12.8307	0.643954	0.321977	+	0.946748i	$0.395653\pi$
$398$	0	0
$399$	3.41147	0.170787
$400$	0	0
$401$	8.71688	0.435300	0.217650	−	0.976027i	$-0.430161\pi$
$401$	8.71688	0.435300	0.217650	+	0.976027i	$0.430161\pi$
$402$	0	0
$403$	−7.28136	−0.362711
$404$	0	0
$405$	−0.487511	−0.0242246
$406$	0	0
$407$	−28.7493	−1.42505
$408$	0	0
$409$	−22.1438	−1.09494	−0.547471	−	0.836825i	$-0.684409\pi$
$409$	−22.1438	−1.09494	−0.547471	+	0.836825i	$0.684409\pi$
$410$	0	0
$411$	14.8203	0.731030
$412$	0	0
$413$	−2.38919	−0.117564
$414$	0	0
$415$	1.38177	0.0678284
$416$	0	0
$417$	24.2986	1.18991
$418$	0	0
$419$	−9.52704	−0.465426	−0.232713	−	0.972545i	$-0.574760\pi$
$419$	−9.52704	−0.465426	−0.232713	+	0.972545i	$0.574760\pi$
$420$	0	0
$421$	−25.6759	−1.25137	−0.625684	−	0.780077i	$-0.715180\pi$
$421$	−25.6759	−1.25137	−0.625684	+	0.780077i	$0.715180\pi$
$422$	0	0
$423$	−7.62267	−0.370627
$424$	0	0
$425$	22.2746	1.08047
$426$	0	0
$427$	−1.74153	−0.0842787
$428$	0	0
$429$	18.6905	0.902385
$430$	0	0
$431$	5.68779	0.273971	0.136985	−	0.990573i	$-0.456259\pi$
$431$	5.68779	0.273971	0.136985	+	0.990573i	$0.456259\pi$
$432$	0	0
$433$	5.65095	0.271567	0.135784	−	0.990739i	$-0.456645\pi$
$433$	5.65095	0.271567	0.135784	+	0.990739i	$0.456645\pi$
$434$	0	0
$435$	0.965852	0.0463090
$436$	0	0
$437$	15.0496	0.719921
$438$	0	0
$439$	−25.3874	−1.21168	−0.605838	−	0.795588i	$-0.707162\pi$
$439$	−25.3874	−1.21168	−0.605838	+	0.795588i	$0.707162\pi$
$440$	0	0
$441$	7.37733	0.351301
$442$	0	0
$443$	−9.18210	−0.436255	−0.218127	−	0.975920i	$-0.569995\pi$
$443$	−9.18210	−0.436255	−0.218127	+	0.975920i	$0.569995\pi$
$444$	0	0
$445$	−0.319022	−0.0151231
$446$	0	0
$447$	32.1266	1.51954
$448$	0	0
$449$	−12.5175	−0.590739	−0.295370	−	0.955383i	$-0.595443\pi$
$449$	−12.5175	−0.590739	−0.295370	+	0.955383i	$0.595443\pi$
$450$	0	0
$451$	14.6418	0.689454
$452$	0	0
$453$	−10.1548	−0.477112
$454$	0	0
$455$	−0.541584	−0.0253899
$456$	0	0
$457$	13.2439	0.619524	0.309762	−	0.950814i	$-0.399751\pi$
$457$	13.2439	0.619524	0.309762	+	0.950814i	$0.399751\pi$
$458$	0	0
$459$	−25.1908	−1.17580
$460$	0	0
$461$	−3.85441	−0.179518	−0.0897588	−	0.995964i	$-0.528610\pi$
$461$	−3.85441	−0.179518	−0.0897588	+	0.995964i	$0.528610\pi$
$462$	0	0
$463$	−6.10876	−0.283898	−0.141949	−	0.989874i	$-0.545337\pi$
$463$	−6.10876	−0.283898	−0.141949	+	0.989874i	$0.545337\pi$
$464$	0	0
$465$	0.231734	0.0107464
$466$	0	0
$467$	33.6705	1.55809	0.779044	−	0.626970i	$-0.215705\pi$
$467$	33.6705	1.55809	0.779044	+	0.626970i	$0.215705\pi$
$468$	0	0
$469$	2.53209	0.116921
$470$	0	0
$471$	−24.3746	−1.12312
$472$	0	0
$473$	−22.4858	−1.03390
$474$	0	0
$475$	−14.3550	−0.658654
$476$	0	0
$477$	14.9564	0.684805
$478$	0	0
$479$	16.1712	0.738880	0.369440	−	0.929255i	$-0.379550\pi$
$479$	16.1712	0.738880	0.369440	+	0.929255i	$0.379550\pi$
$480$	0	0
$481$	54.0310	2.46360
$482$	0	0
$483$	6.19253	0.281770
$484$	0	0
$485$	0.202638	0.00920133
$486$	0	0
$487$	−40.2695	−1.82479	−0.912393	−	0.409316i	$-0.865767\pi$
$487$	−40.2695	−1.82479	−0.912393	+	0.409316i	$0.865767\pi$
$488$	0	0
$489$	2.19665	0.0993360
$490$	0	0
$491$	−19.6682	−0.887613	−0.443806	−	0.896123i	$-0.646372\pi$
$491$	−19.6682	−0.887613	−0.443806	+	0.896123i	$0.646372\pi$
$492$	0	0
$493$	26.5553	1.19599
$494$	0	0
$495$	0.388252	0.0174506
$496$	0	0
$497$	8.35235	0.374654
$498$	0	0
$499$	−27.4175	−1.22737	−0.613687	−	0.789549i	$-0.710314\pi$
$499$	−27.4175	−1.22737	−0.613687	+	0.789549i	$0.710314\pi$
$500$	0	0
$501$	4.65270	0.207867
$502$	0	0
$503$	−40.8607	−1.82189	−0.910945	−	0.412528i	$-0.864646\pi$
$503$	−40.8607	−1.82189	−0.910945	+	0.412528i	$0.864646\pi$
$504$	0	0
$505$	2.01724	0.0897658
$506$	0	0
$507$	−17.6117	−0.782165
$508$	0	0
$509$	−26.9172	−1.19308	−0.596541	−	0.802583i	$-0.703459\pi$
$509$	−26.9172	−1.19308	−0.596541	+	0.802583i	$0.703459\pi$
$510$	0	0
$511$	−9.75372	−0.431479
$512$	0	0
$513$	16.2344	0.716767
$514$	0	0
$515$	−0.0196004	−0.000863695 0
$516$	0	0
$517$	−17.4798	−0.768759
$518$	0	0
$519$	−12.4379	−0.545962
$520$	0	0
$521$	2.32501	0.101860	0.0509302	−	0.998702i	$-0.483781\pi$
$521$	2.32501	0.101860	0.0509302	+	0.998702i	$0.483781\pi$
$522$	0	0
$523$	−2.46616	−0.107837	−0.0539187	−	0.998545i	$-0.517171\pi$
$523$	−2.46616	−0.107837	−0.0539187	+	0.998545i	$0.517171\pi$
$524$	0	0
$525$	−5.90673	−0.257791
$526$	0	0
$527$	6.37134	0.277540
$528$	0	0
$529$	4.31820	0.187748
$530$	0	0
$531$	−3.21894	−0.139690
$532$	0	0
$533$	−27.5175	−1.19192
$534$	0	0
$535$	1.74598	0.0754852
$536$	0	0
$537$	13.2148	0.570262
$538$	0	0
$539$	16.9172	0.728673
$540$	0	0
$541$	−9.33956	−0.401539	−0.200769	−	0.979639i	$-0.564344\pi$
$541$	−9.33956	−0.401539	−0.200769	+	0.979639i	$0.564344\pi$
$542$	0	0
$543$	−10.7588	−0.461703
$544$	0	0
$545$	−1.13011	−0.0484086
$546$	0	0
$547$	−18.3669	−0.785312	−0.392656	−	0.919685i	$-0.628444\pi$
$547$	−18.3669	−0.785312	−0.392656	+	0.919685i	$0.628444\pi$
$548$	0	0
$549$	−2.34636	−0.100140
$550$	0	0
$551$	−17.1138	−0.729073
$552$	0	0
$553$	−6.79292	−0.288864
$554$	0	0
$555$	−1.71957	−0.0729917
$556$	0	0
$557$	13.5936	0.575978	0.287989	−	0.957634i	$-0.407013\pi$
$557$	13.5936	0.575978	0.287989	+	0.957634i	$0.407013\pi$
$558$	0	0
$559$	42.2594	1.78738
$560$	0	0
$561$	−16.3545	−0.690489
$562$	0	0
$563$	−31.0428	−1.30830	−0.654149	−	0.756365i	$-0.726973\pi$
$563$	−31.0428	−1.30830	−0.654149	+	0.756365i	$0.726973\pi$
$564$	0	0
$565$	−1.07367	−0.0451698
$566$	0	0
$567$	3.55438	0.149270
$568$	0	0
$569$	20.9436	0.878000	0.439000	−	0.898487i	$-0.355333\pi$
$569$	20.9436	0.878000	0.439000	+	0.898487i	$0.355333\pi$
$570$	0	0
$571$	9.05737	0.379039	0.189520	−	0.981877i	$-0.439307\pi$
$571$	9.05737	0.379039	0.189520	+	0.981877i	$0.439307\pi$
$572$	0	0
$573$	−5.66456	−0.236641
$574$	0	0
$575$	−26.0574	−1.08667
$576$	0	0
$577$	3.36783	0.140205	0.0701023	−	0.997540i	$-0.477667\pi$
$577$	3.36783	0.140205	0.0701023	+	0.997540i	$0.477667\pi$
$578$	0	0
$579$	−14.9896	−0.622945
$580$	0	0
$581$	−10.0743	−0.417952
$582$	0	0
$583$	34.2968	1.42043
$584$	0	0
$585$	−0.729675	−0.0301683
$586$	0	0
$587$	−20.2831	−0.837174	−0.418587	−	0.908177i	$-0.637474\pi$
$587$	−20.2831	−0.837174	−0.418587	+	0.908177i	$0.637474\pi$
$588$	0	0
$589$	−4.10607	−0.169188
$590$	0	0
$591$	7.34049	0.301947
$592$	0	0
$593$	5.12061	0.210278	0.105139	−	0.994458i	$-0.466471\pi$
$593$	5.12061	0.210278	0.105139	+	0.994458i	$0.466471\pi$
$594$	0	0
$595$	0.473897	0.0194279
$596$	0	0
$597$	8.45605	0.346083
$598$	0	0
$599$	−6.12155	−0.250120	−0.125060	−	0.992149i	$-0.539912\pi$
$599$	−6.12155	−0.250120	−0.125060	+	0.992149i	$0.539912\pi$
$600$	0	0
$601$	−29.4766	−1.20238	−0.601188	−	0.799108i	$-0.705306\pi$
$601$	−29.4766	−1.20238	−0.601188	+	0.799108i	$0.705306\pi$
$602$	0	0
$603$	3.41147	0.138926
$604$	0	0
$605$	−0.436451	−0.0177443
$606$	0	0
$607$	14.2591	0.578758	0.289379	−	0.957215i	$-0.406551\pi$
$607$	14.2591	0.578758	0.289379	+	0.957215i	$0.406551\pi$
$608$	0	0
$609$	−7.04189	−0.285352
$610$	0	0
$611$	32.8512	1.32902
$612$	0	0
$613$	−15.5449	−0.627852	−0.313926	−	0.949447i	$-0.601644\pi$
$613$	−15.5449	−0.627852	−0.313926	+	0.949447i	$0.601644\pi$
$614$	0	0
$615$	0.875763	0.0353142
$616$	0	0
$617$	−15.5439	−0.625776	−0.312888	−	0.949790i	$-0.601296\pi$
$617$	−15.5439	−0.625776	−0.312888	+	0.949790i	$0.601296\pi$
$618$	0	0
$619$	45.4739	1.82775	0.913875	−	0.405995i	$-0.133075\pi$
$619$	45.4739	1.82775	0.913875	+	0.405995i	$0.133075\pi$
$620$	0	0
$621$	29.4688	1.18254
$622$	0	0
$623$	2.32594	0.0931869
$624$	0	0
$625$	24.7820	0.991280
$626$	0	0
$627$	10.5398	0.420920
$628$	0	0
$629$	−47.2782	−1.88510
$630$	0	0
$631$	−14.9513	−0.595202	−0.297601	−	0.954690i	$-0.596187\pi$
$631$	−14.9513	−0.595202	−0.297601	+	0.954690i	$0.596187\pi$
$632$	0	0
$633$	−6.85978	−0.272652
$634$	0	0
$635$	0.120615	0.00478645
$636$	0	0
$637$	−31.7939	−1.25972
$638$	0	0
$639$	11.2531	0.445165
$640$	0	0
$641$	1.98040	0.0782211	0.0391105	−	0.999235i	$-0.487548\pi$
$641$	1.98040	0.0782211	0.0391105	+	0.999235i	$0.487548\pi$
$642$	0	0
$643$	34.1147	1.34535	0.672677	−	0.739936i	$-0.265144\pi$
$643$	34.1147	1.34535	0.672677	+	0.739936i	$0.265144\pi$
$644$	0	0
$645$	−1.34493	−0.0529567
$646$	0	0
$647$	−14.7570	−0.580158	−0.290079	−	0.957003i	$-0.593682\pi$
$647$	−14.7570	−0.580158	−0.290079	+	0.957003i	$0.593682\pi$
$648$	0	0
$649$	−7.38144	−0.289747
$650$	0	0
$651$	−1.68954	−0.0662183
$652$	0	0
$653$	−29.9986	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$653$	−29.9986	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$654$	0	0
$655$	−2.65364	−0.103686
$656$	0	0
$657$	−13.1411	−0.512685
$658$	0	0
$659$	3.29860	0.128495	0.0642476	−	0.997934i	$-0.479535\pi$
$659$	3.29860	0.128495	0.0642476	+	0.997934i	$0.479535\pi$
$660$	0	0
$661$	28.0128	1.08957	0.544786	−	0.838575i	$-0.316611\pi$
$661$	28.0128	1.08957	0.544786	+	0.838575i	$0.316611\pi$
$662$	0	0
$663$	30.7365	1.19371
$664$	0	0
$665$	−0.305407	−0.0118432
$666$	0	0
$667$	−31.0651	−1.20285
$668$	0	0
$669$	21.7629	0.841402
$670$	0	0
$671$	−5.38051	−0.207712
$672$	0	0
$673$	−22.8512	−0.880850	−0.440425	−	0.897789i	$-0.645172\pi$
$673$	−22.8512	−0.880850	−0.440425	+	0.897789i	$0.645172\pi$
$674$	0	0
$675$	−28.1088	−1.08191
$676$	0	0
$677$	12.7469	0.489904	0.244952	−	0.969535i	$-0.421228\pi$
$677$	12.7469	0.489904	0.244952	+	0.969535i	$0.421228\pi$
$678$	0	0
$679$	−1.47741	−0.0566977
$680$	0	0
$681$	12.5936	0.482587
$682$	0	0
$683$	20.3013	0.776807	0.388404	−	0.921489i	$-0.373027\pi$
$683$	20.3013	0.776807	0.388404	+	0.921489i	$0.373027\pi$
$684$	0	0
$685$	−1.32676	−0.0506930
$686$	0	0
$687$	−14.1898	−0.541376
$688$	0	0
$689$	−64.4570	−2.45562
$690$	0	0
$691$	−37.4543	−1.42483	−0.712414	−	0.701759i	$-0.752398\pi$
$691$	−37.4543	−1.42483	−0.712414	+	0.701759i	$0.752398\pi$
$692$	0	0
$693$	−2.83069	−0.107529
$694$	0	0
$695$	−2.17530	−0.0825137
$696$	0	0
$697$	24.0784	0.912034
$698$	0	0
$699$	14.2199	0.537845
$700$	0	0
$701$	22.5422	0.851407	0.425703	−	0.904863i	$-0.360027\pi$
$701$	22.5422	0.851407	0.425703	+	0.904863i	$0.360027\pi$
$702$	0	0
$703$	30.4688	1.14915
$704$	0	0
$705$	−1.04551	−0.0393762
$706$	0	0
$707$	−14.7074	−0.553128
$708$	0	0
$709$	−2.90848	−0.109230	−0.0546151	−	0.998507i	$-0.517393\pi$
$709$	−2.90848	−0.109230	−0.0546151	+	0.998507i	$0.517393\pi$
$710$	0	0
$711$	−9.15207	−0.343230
$712$	0	0
$713$	−7.45336	−0.279131
$714$	0	0
$715$	−1.67324	−0.0625755
$716$	0	0
$717$	20.8111	0.777204
$718$	0	0
$719$	4.05880	0.151368	0.0756839	−	0.997132i	$-0.475886\pi$
$719$	4.05880	0.151368	0.0756839	+	0.997132i	$0.475886\pi$
$720$	0	0
$721$	0.142903	0.00532200
$722$	0	0
$723$	−10.8489	−0.403473
$724$	0	0
$725$	29.6313	1.10048
$726$	0	0
$727$	−21.6637	−0.803464	−0.401732	−	0.915757i	$-0.631592\pi$
$727$	−21.6637	−0.803464	−0.401732	+	0.915757i	$0.631592\pi$
$728$	0	0
$729$	27.5776	1.02139
$730$	0	0
$731$	−36.9778	−1.36767
$732$	0	0
$733$	23.1993	0.856887	0.428444	−	0.903569i	$-0.359062\pi$
$733$	23.1993	0.856887	0.428444	+	0.903569i	$0.359062\pi$
$734$	0	0
$735$	1.01186	0.0373230
$736$	0	0
$737$	7.82295	0.288162
$738$	0	0
$739$	−48.5844	−1.78721	−0.893603	−	0.448858i	$-0.851831\pi$
$739$	−48.5844	−1.78721	−0.893603	+	0.448858i	$0.851831\pi$
$740$	0	0
$741$	−19.8084	−0.727680
$742$	0	0
$743$	16.7033	0.612783	0.306392	−	0.951906i	$-0.400878\pi$
$743$	16.7033	0.612783	0.306392	+	0.951906i	$0.400878\pi$
$744$	0	0
$745$	−2.87609	−0.105372
$746$	0	0
$747$	−13.5730	−0.496612
$748$	0	0
$749$	−12.7297	−0.465132
$750$	0	0
$751$	22.1634	0.808755	0.404378	−	0.914592i	$-0.367488\pi$
$751$	22.1634	0.808755	0.404378	+	0.914592i	$0.367488\pi$
$752$	0	0
$753$	7.44419	0.271281
$754$	0	0
$755$	0.909090	0.0330852
$756$	0	0
$757$	−24.1625	−0.878201	−0.439101	−	0.898438i	$-0.644703\pi$
$757$	−24.1625	−0.878201	−0.439101	+	0.898438i	$0.644703\pi$
$758$	0	0
$759$	19.1320	0.694447
$760$	0	0
$761$	−38.8675	−1.40895	−0.704473	−	0.709730i	$-0.748817\pi$
$761$	−38.8675	−1.40895	−0.704473	+	0.709730i	$0.748817\pi$
$762$	0	0
$763$	8.23947	0.298289
$764$	0	0
$765$	0.638480	0.0230843
$766$	0	0
$767$	13.8726	0.500910
$768$	0	0
$769$	6.27126	0.226147	0.113074	−	0.993587i	$-0.463930\pi$
$769$	6.27126	0.226147	0.113074	+	0.993587i	$0.463930\pi$
$770$	0	0
$771$	0.819326	0.0295073
$772$	0	0
$773$	−2.46522	−0.0886679	−0.0443340	−	0.999017i	$-0.514117\pi$
$773$	−2.46522	−0.0886679	−0.0443340	+	0.999017i	$0.514117\pi$
$774$	0	0
$775$	7.10936	0.255376
$776$	0	0
$777$	12.5371	0.449767
$778$	0	0
$779$	−15.5175	−0.555974
$780$	0	0
$781$	25.8048	0.923368
$782$	0	0
$783$	−33.5107	−1.19758
$784$	0	0
$785$	2.18210	0.0778826
$786$	0	0
$787$	26.3422	0.939000	0.469500	−	0.882933i	$-0.344434\pi$
$787$	26.3422	0.939000	0.469500	+	0.882933i	$0.344434\pi$
$788$	0	0
$789$	−12.0669	−0.429592
$790$	0	0
$791$	7.82800	0.278332
$792$	0	0
$793$	10.1121	0.359090
$794$	0	0
$795$	2.05138	0.0727551
$796$	0	0
$797$	47.1498	1.67013	0.835066	−	0.550149i	$-0.185429\pi$
$797$	47.1498	1.67013	0.835066	+	0.550149i	$0.185429\pi$
$798$	0	0
$799$	−28.7455	−1.01694
$800$	0	0
$801$	3.13373	0.110725
$802$	0	0
$803$	−30.1343	−1.06342
$804$	0	0
$805$	−0.554378	−0.0195392
$806$	0	0
$807$	12.7965	0.450460
$808$	0	0
$809$	−14.3824	−0.505657	−0.252829	−	0.967511i	$-0.581361\pi$
$809$	−14.3824	−0.505657	−0.252829	+	0.967511i	$0.581361\pi$
$810$	0	0
$811$	−35.8120	−1.25753	−0.628765	−	0.777595i	$-0.716439\pi$
$811$	−35.8120	−1.25753	−0.628765	+	0.777595i	$0.716439\pi$
$812$	0	0
$813$	14.2475	0.499683
$814$	0	0
$815$	−0.196652	−0.00688842
$816$	0	0
$817$	23.8307	0.833730
$818$	0	0
$819$	5.31996	0.185894
$820$	0	0
$821$	−14.0969	−0.491985	−0.245993	−	0.969272i	$-0.579114\pi$
$821$	−14.0969	−0.491985	−0.245993	+	0.969272i	$0.579114\pi$
$822$	0	0
$823$	−11.0888	−0.386532	−0.193266	−	0.981146i	$-0.561908\pi$
$823$	−11.0888	−0.386532	−0.193266	+	0.981146i	$0.561908\pi$
$824$	0	0
$825$	−18.2490	−0.635348
$826$	0	0
$827$	32.3429	1.12467	0.562336	−	0.826909i	$-0.309903\pi$
$827$	32.3429	1.12467	0.562336	+	0.826909i	$0.309903\pi$
$828$	0	0
$829$	12.2463	0.425331	0.212665	−	0.977125i	$-0.431786\pi$
$829$	12.2463	0.425331	0.212665	+	0.977125i	$0.431786\pi$
$830$	0	0
$831$	32.0874	1.11310
$832$	0	0
$833$	27.8203	0.963915
$834$	0	0
$835$	−0.416527	−0.0144145
$836$	0	0
$837$	−8.04013	−0.277908
$838$	0	0
$839$	22.0283	0.760501	0.380250	−	0.924884i	$-0.375838\pi$
$839$	22.0283	0.760501	0.380250	+	0.924884i	$0.375838\pi$
$840$	0	0
$841$	6.32594	0.218136
$842$	0	0
$843$	23.0678	0.794497
$844$	0	0
$845$	1.57667	0.0542390
$846$	0	0
$847$	3.18210	0.109338
$848$	0	0
$849$	−33.1753	−1.13857
$850$	0	0
$851$	55.3073	1.89591
$852$	0	0
$853$	−26.4935	−0.907120	−0.453560	−	0.891226i	$-0.649846\pi$
$853$	−26.4935	−0.907120	−0.453560	+	0.891226i	$0.649846\pi$
$854$	0	0
$855$	−0.411474	−0.0140721
$856$	0	0
$857$	−40.8452	−1.39525	−0.697623	−	0.716465i	$-0.745759\pi$
$857$	−40.8452	−1.39525	−0.697623	+	0.716465i	$0.745759\pi$
$858$	0	0
$859$	41.7279	1.42374	0.711869	−	0.702312i	$-0.247849\pi$
$859$	41.7279	1.42374	0.711869	+	0.702312i	$0.247849\pi$
$860$	0	0
$861$	−6.38507	−0.217602
$862$	0	0
$863$	−30.2814	−1.03079	−0.515395	−	0.856953i	$-0.672355\pi$
$863$	−30.2814	−1.03079	−0.515395	+	0.856953i	$0.672355\pi$
$864$	0	0
$865$	1.11348	0.0378596
$866$	0	0
$867$	−3.99100	−0.135542
$868$	0	0
$869$	−20.9869	−0.711931
$870$	0	0
$871$	−14.7023	−0.498170
$872$	0	0
$873$	−1.99050	−0.0673684
$874$	0	0
$875$	1.05913	0.0358050
$876$	0	0
$877$	4.08553	0.137959	0.0689793	−	0.997618i	$-0.478026\pi$
$877$	4.08553	0.137959	0.0689793	+	0.997618i	$0.478026\pi$
$878$	0	0
$879$	−17.8839	−0.603210
$880$	0	0
$881$	21.5871	0.727288	0.363644	−	0.931538i	$-0.381533\pi$
$881$	21.5871	0.727288	0.363644	+	0.931538i	$0.381533\pi$
$882$	0	0
$883$	−48.9136	−1.64608	−0.823038	−	0.567987i	$-0.807722\pi$
$883$	−48.9136	−1.64608	−0.823038	+	0.567987i	$0.807722\pi$
$884$	0	0
$885$	−0.441504	−0.0148410
$886$	0	0
$887$	−31.3688	−1.05326	−0.526630	−	0.850095i	$-0.676545\pi$
$887$	−31.3688	−1.05326	−0.526630	+	0.850095i	$0.676545\pi$
$888$	0	0
$889$	−0.879385	−0.0294936
$890$	0	0
$891$	10.9813	0.367889
$892$	0	0
$893$	18.5253	0.619925
$894$	0	0
$895$	−1.18304	−0.0395446
$896$	0	0
$897$	−35.9564	−1.20055
$898$	0	0
$899$	8.47565	0.282679
$900$	0	0
$901$	56.4012	1.87899
$902$	0	0
$903$	9.80571	0.326314
$904$	0	0
$905$	0.963163	0.0320166
$906$	0	0
$907$	51.0607	1.69544	0.847721	−	0.530443i	$-0.177974\pi$
$907$	51.0607	1.69544	0.847721	+	0.530443i	$0.177974\pi$
$908$	0	0
$909$	−19.8152	−0.657229
$910$	0	0
$911$	−56.2191	−1.86262	−0.931310	−	0.364227i	$-0.881333\pi$
$911$	−56.2191	−1.86262	−0.931310	+	0.364227i	$0.881333\pi$
$912$	0	0
$913$	−31.1247	−1.03008
$914$	0	0
$915$	−0.321823	−0.0106391
$916$	0	0
$917$	19.3473	0.638904
$918$	0	0
$919$	−15.1756	−0.500598	−0.250299	−	0.968169i	$-0.580529\pi$
$919$	−15.1756	−0.500598	−0.250299	+	0.968169i	$0.580529\pi$
$920$	0	0
$921$	35.4584	1.16839
$922$	0	0
$923$	−48.4971	−1.59630
$924$	0	0
$925$	−52.7547	−1.73456
$926$	0	0
$927$	0.192533	0.00632362
$928$	0	0
$929$	26.8084	0.879555	0.439778	−	0.898107i	$-0.355057\pi$
$929$	26.8084	0.879555	0.439778	+	0.898107i	$0.355057\pi$
$930$	0	0
$931$	−17.9290	−0.587600
$932$	0	0
$933$	−5.81109	−0.190247
$934$	0	0
$935$	1.46412	0.0478817
$936$	0	0
$937$	−33.0273	−1.07896	−0.539478	−	0.842000i	$-0.681378\pi$
$937$	−33.0273	−1.07896	−0.539478	+	0.842000i	$0.681378\pi$
$938$	0	0
$939$	46.7151	1.52449
$940$	0	0
$941$	−16.3628	−0.533411	−0.266706	−	0.963778i	$-0.585935\pi$
$941$	−16.3628	−0.533411	−0.266706	+	0.963778i	$0.585935\pi$
$942$	0	0
$943$	−28.1676	−0.917262
$944$	0	0
$945$	−0.598021	−0.0194536
$946$	0	0
$947$	53.7766	1.74751	0.873753	−	0.486371i	$-0.161680\pi$
$947$	53.7766	1.74751	0.873753	+	0.486371i	$0.161680\pi$
$948$	0	0
$949$	56.6340	1.83842
$950$	0	0
$951$	−28.6127	−0.927830
$952$	0	0
$953$	33.7897	1.09456	0.547278	−	0.836951i	$-0.315664\pi$
$953$	33.7897	1.09456	0.547278	+	0.836951i	$0.315664\pi$
$954$	0	0
$955$	0.507112	0.0164098
$956$	0	0
$957$	−21.7561	−0.703274
$958$	0	0
$959$	9.67324	0.312365
$960$	0	0
$961$	−28.9665	−0.934402
$962$	0	0
$963$	−17.1506	−0.552672
$964$	0	0
$965$	1.34192	0.0431979
$966$	0	0
$967$	−24.2713	−0.780511	−0.390256	−	0.920707i	$-0.627613\pi$
$967$	−24.2713	−0.780511	−0.390256	+	0.920707i	$0.627613\pi$
$968$	0	0
$969$	17.3327	0.556808
$970$	0	0
$971$	1.10513	0.0354654	0.0177327	−	0.999843i	$-0.494355\pi$
$971$	1.10513	0.0354654	0.0177327	+	0.999843i	$0.494355\pi$
$972$	0	0
$973$	15.8598	0.508441
$974$	0	0
$975$	34.2968	1.09838
$976$	0	0
$977$	−49.4867	−1.58322	−0.791610	−	0.611027i	$-0.790757\pi$
$977$	−49.4867	−1.58322	−0.791610	+	0.611027i	$0.790757\pi$
$978$	0	0
$979$	7.18605	0.229667
$980$	0	0
$981$	11.1010	0.354428
$982$	0	0
$983$	−29.0259	−0.925783	−0.462891	−	0.886415i	$-0.653188\pi$
$983$	−29.0259	−0.925783	−0.462891	+	0.886415i	$0.653188\pi$
$984$	0	0
$985$	−0.657147	−0.0209384
$986$	0	0
$987$	7.62267	0.242632
$988$	0	0
$989$	43.2576	1.37551
$990$	0	0
$991$	−14.8298	−0.471083	−0.235541	−	0.971864i	$-0.575686\pi$
$991$	−14.8298	−0.471083	−0.235541	+	0.971864i	$0.575686\pi$
$992$	0	0
$993$	−30.7989	−0.977374
$994$	0	0
$995$	−0.757016	−0.0239990
$996$	0	0
$997$	41.9941	1.32997	0.664984	−	0.746858i	$-0.268438\pi$
$997$	41.9941	1.32997	0.664984	+	0.746858i	$0.268438\pi$
$998$	0	0
$999$	59.6614	1.88760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8128.2.a.w.1.2		3
4.3	odd	2		8128.2.a.bd.1.2		3
8.3	odd	2		127.2.a.a.1.1	✓	3
8.5	even	2		2032.2.a.k.1.2		3
24.11	even	2		1143.2.a.e.1.3		3
40.19	odd	2		3175.2.a.h.1.3		3
56.27	even	2		6223.2.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.2.a.a.1.1	✓	3	8.3	odd	2
1143.2.a.e.1.3		3	24.11	even	2
2032.2.a.k.1.2		3	8.5	even	2
3175.2.a.h.1.3		3	40.19	odd	2
6223.2.a.e.1.1		3	56.27	even	2
8128.2.a.w.1.2		3	1.1	even	1	trivial
8128.2.a.bd.1.2		3	4.3	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:	\( N \)	\(=\)	\( 8128 = 2^{6} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8128.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34730 q^{3} +0.120615 q^{5} -0.879385 q^{7} -1.18479 q^{9} +O(q^{10})\)	\(q-1.34730 q^{3} +0.120615 q^{5} -0.879385 q^{7} -1.18479 q^{9} -2.71688 q^{11} +5.10607 q^{13} -0.162504 q^{15} -4.46791 q^{17} +2.87939 q^{19} +1.18479 q^{21} +5.22668 q^{23} -4.98545 q^{25} +5.63816 q^{27} -5.94356 q^{29} -1.42602 q^{31} +3.66044 q^{33} -0.106067 q^{35} +10.5817 q^{37} -6.87939 q^{39} -5.38919 q^{41} +8.27631 q^{43} -0.142903 q^{45} +6.43376 q^{47} -6.22668 q^{49} +6.01960 q^{51} -12.6236 q^{53} -0.327696 q^{55} -3.87939 q^{57} +2.71688 q^{59} +1.98040 q^{61} +1.04189 q^{63} +0.615867 q^{65} -2.87939 q^{67} -7.04189 q^{69} -9.49794 q^{71} +11.0915 q^{73} +6.71688 q^{75} +2.38919 q^{77} +7.72462 q^{79} -4.04189 q^{81} +11.4561 q^{83} -0.538896 q^{85} +8.00774 q^{87} -2.64496 q^{89} -4.49020 q^{91} +1.92127 q^{93} +0.347296 q^{95} +1.68004 q^{97} +3.21894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^5 + 3 * q^7	\( 3 q - 3 q^{3} + 6 q^{5} + 3 q^{7} + 3 q^{13} - 3 q^{15} - 18 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} - 3 q^{29} - 12 q^{31} - 12 q^{33} + 12 q^{35} - 15 q^{39} - 12 q^{41} - 9 q^{43} + 3 q^{47} - 12 q^{49} + 21 q^{51} - 3 q^{53} + 3 q^{55} - 6 q^{57} + 3 q^{61} - 9 q^{65} - 3 q^{67} - 18 q^{69} - 3 q^{71} + 3 q^{73} + 12 q^{75} + 3 q^{77} - 9 q^{79} - 9 q^{81} + 12 q^{83} - 39 q^{85} - 33 q^{89} - 12 q^{91} - 3 q^{93} - 15 q^{97} + 27 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^5 + 3 * q^7 + 3 * q^13 - 3 * q^15 - 18 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 - 3 * q^29 - 12 * q^31 - 12 * q^33 + 12 * q^35 - 15 * q^39 - 12 * q^41 - 9 * q^43 + 3 * q^47 - 12 * q^49 + 21 * q^51 - 3 * q^53 + 3 * q^55 - 6 * q^57 + 3 * q^61 - 9 * q^65 - 3 * q^67 - 18 * q^69 - 3 * q^71 + 3 * q^73 + 12 * q^75 + 3 * q^77 - 9 * q^79 - 9 * q^81 + 12 * q^83 - 39 * q^85 - 33 * q^89 - 12 * q^91 - 3 * q^93 - 15 * q^97 + 27 * q^99

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