LMFDB - Embedded newform 8024.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74430 q^{3} +1.05481 q^{5} +1.84260 q^{7} +0.0425790 q^{9} +O(q^{10})$	$q-1.74430 q^{3} +1.05481 q^{5} +1.84260 q^{7} +0.0425790 q^{9} +6.02375 q^{11} -0.598711 q^{13} -1.83991 q^{15} -1.00000 q^{17} +2.57381 q^{19} -3.21405 q^{21} -5.74361 q^{23} -3.88737 q^{25} +5.15863 q^{27} -2.33430 q^{29} -4.57189 q^{31} -10.5072 q^{33} +1.94360 q^{35} +8.60753 q^{37} +1.04433 q^{39} -10.8058 q^{41} -4.58088 q^{43} +0.0449129 q^{45} +2.41769 q^{47} -3.60481 q^{49} +1.74430 q^{51} -2.12083 q^{53} +6.35394 q^{55} -4.48949 q^{57} -1.00000 q^{59} -0.502075 q^{61} +0.0784562 q^{63} -0.631529 q^{65} -8.85713 q^{67} +10.0186 q^{69} -7.47697 q^{71} +2.03259 q^{73} +6.78073 q^{75} +11.0994 q^{77} -15.5536 q^{79} -9.12592 q^{81} -15.7037 q^{83} -1.05481 q^{85} +4.07171 q^{87} -2.18348 q^{89} -1.10319 q^{91} +7.97475 q^{93} +2.71489 q^{95} -11.0393 q^{97} +0.256485 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	−1.74430	−1.00707	−0.503536	−	0.863974i	$-0.667968\pi$
$3$	−1.74430	−1.00707	−0.503536	+	0.863974i	$0.667968\pi$
$4$	0	0
$5$	1.05481	0.471727	0.235864	−	0.971786i	$-0.424208\pi$
$5$	1.05481	0.471727	0.235864	+	0.971786i	$0.424208\pi$
$6$	0	0
$7$	1.84260	0.696439	0.348219	−	0.937413i	$-0.386786\pi$
$7$	1.84260	0.696439	0.348219	+	0.937413i	$0.386786\pi$
$8$	0	0
$9$	0.0425790	0.0141930
$10$	0	0
$11$	6.02375	1.81623	0.908115	−	0.418721i	$-0.137521\pi$
$11$	6.02375	1.81623	0.908115	+	0.418721i	$0.137521\pi$
$12$	0	0
$13$	−0.598711	−0.166053	−0.0830263	−	0.996547i	$-0.526459\pi$
$13$	−0.598711	−0.166053	−0.0830263	+	0.996547i	$0.526459\pi$
$14$	0	0
$15$	−1.83991	−0.475063
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.57381	0.590472	0.295236	−	0.955424i	$-0.404602\pi$
$19$	2.57381	0.590472	0.295236	+	0.955424i	$0.404602\pi$
$20$	0	0
$21$	−3.21405	−0.701364
$22$	0	0
$23$	−5.74361	−1.19763	−0.598813	−	0.800889i	$-0.704361\pi$
$23$	−5.74361	−1.19763	−0.598813	+	0.800889i	$0.704361\pi$
$24$	0	0
$25$	−3.88737	−0.777474
$26$	0	0
$27$	5.15863	0.992778
$28$	0	0
$29$	−2.33430	−0.433468	−0.216734	−	0.976231i	$-0.569540\pi$
$29$	−2.33430	−0.433468	−0.216734	+	0.976231i	$0.569540\pi$
$30$	0	0
$31$	−4.57189	−0.821136	−0.410568	−	0.911830i	$-0.634670\pi$
$31$	−4.57189	−0.821136	−0.410568	+	0.911830i	$0.634670\pi$
$32$	0	0
$33$	−10.5072	−1.82907
$34$	0	0
$35$	1.94360	0.328529
$36$	0	0
$37$	8.60753	1.41507	0.707535	−	0.706678i	$-0.249807\pi$
$37$	8.60753	1.41507	0.707535	+	0.706678i	$0.249807\pi$
$38$	0	0
$39$	1.04433	0.167227
$40$	0	0
$41$	−10.8058	−1.68758	−0.843789	−	0.536675i	$-0.819680\pi$
$41$	−10.8058	−1.68758	−0.843789	+	0.536675i	$0.819680\pi$
$42$	0	0
$43$	−4.58088	−0.698577	−0.349289	−	0.937015i	$-0.613577\pi$
$43$	−4.58088	−0.698577	−0.349289	+	0.937015i	$0.613577\pi$
$44$	0	0
$45$	0.0449129	0.00669522
$46$	0	0
$47$	2.41769	0.352656	0.176328	−	0.984331i	$-0.443578\pi$
$47$	2.41769	0.352656	0.176328	+	0.984331i	$0.443578\pi$
$48$	0	0
$49$	−3.60481	−0.514973
$50$	0	0
$51$	1.74430	0.244251
$52$	0	0
$53$	−2.12083	−0.291319	−0.145659	−	0.989335i	$-0.546530\pi$
$53$	−2.12083	−0.291319	−0.145659	+	0.989335i	$0.546530\pi$
$54$	0	0
$55$	6.35394	0.856765
$56$	0	0
$57$	−4.48949	−0.594647
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−0.502075	−0.0642841	−0.0321421	−	0.999483i	$-0.510233\pi$
$61$	−0.502075	−0.0642841	−0.0321421	+	0.999483i	$0.510233\pi$
$62$	0	0
$63$	0.0784562	0.00988455
$64$	0	0
$65$	−0.631529	−0.0783315
$66$	0	0
$67$	−8.85713	−1.08207	−0.541035	−	0.841000i	$-0.681967\pi$
$67$	−8.85713	−1.08207	−0.541035	+	0.841000i	$0.681967\pi$
$68$	0	0
$69$	10.0186	1.20609
$70$	0	0
$71$	−7.47697	−0.887353	−0.443677	−	0.896187i	$-0.646326\pi$
$71$	−7.47697	−0.887353	−0.443677	+	0.896187i	$0.646326\pi$
$72$	0	0
$73$	2.03259	0.237897	0.118948	−	0.992900i	$-0.462048\pi$
$73$	2.03259	0.237897	0.118948	+	0.992900i	$0.462048\pi$
$74$	0	0
$75$	6.78073	0.782971
$76$	0	0
$77$	11.0994	1.26489
$78$	0	0
$79$	−15.5536	−1.74991	−0.874957	−	0.484200i	$-0.839111\pi$
$79$	−15.5536	−1.74991	−0.874957	+	0.484200i	$0.839111\pi$
$80$	0	0
$81$	−9.12592	−1.01399
$82$	0	0
$83$	−15.7037	−1.72371	−0.861853	−	0.507159i	$-0.830696\pi$
$83$	−15.7037	−1.72371	−0.861853	+	0.507159i	$0.830696\pi$
$84$	0	0
$85$	−1.05481	−0.114411
$86$	0	0
$87$	4.07171	0.436534
$88$	0	0
$89$	−2.18348	−0.231448	−0.115724	−	0.993281i	$-0.536919\pi$
$89$	−2.18348	−0.231448	−0.115724	+	0.993281i	$0.536919\pi$
$90$	0	0
$91$	−1.10319	−0.115645
$92$	0	0
$93$	7.97475	0.826943
$94$	0	0
$95$	2.71489	0.278541
$96$	0	0
$97$	−11.0393	−1.12087	−0.560435	−	0.828198i	$-0.689366\pi$
$97$	−11.0393	−1.12087	−0.560435	+	0.828198i	$0.689366\pi$
$98$	0	0
$99$	0.256485	0.0257777
$100$	0	0
$101$	10.8829	1.08289	0.541443	−	0.840737i	$-0.317878\pi$
$101$	10.8829	1.08289	0.541443	+	0.840737i	$0.317878\pi$
$102$	0	0
$103$	−13.1289	−1.29363	−0.646813	−	0.762649i	$-0.723898\pi$
$103$	−13.1289	−1.29363	−0.646813	+	0.762649i	$0.723898\pi$
$104$	0	0
$105$	−3.39023	−0.330852
$106$	0	0
$107$	−5.55060	−0.536596	−0.268298	−	0.963336i	$-0.586461\pi$
$107$	−5.55060	−0.536596	−0.268298	+	0.963336i	$0.586461\pi$
$108$	0	0
$109$	−2.34236	−0.224357	−0.112179	−	0.993688i	$-0.535783\pi$
$109$	−2.34236	−0.224357	−0.112179	+	0.993688i	$0.535783\pi$
$110$	0	0
$111$	−15.0141	−1.42508
$112$	0	0
$113$	14.4344	1.35787	0.678936	−	0.734197i	$-0.262441\pi$
$113$	14.4344	1.35787	0.678936	+	0.734197i	$0.262441\pi$
$114$	0	0
$115$	−6.05844	−0.564953
$116$	0	0
$117$	−0.0254925	−0.00235678
$118$	0	0
$119$	−1.84260	−0.168911
$120$	0	0
$121$	25.2856	2.29869
$122$	0	0
$123$	18.8485	1.69951
$124$	0	0
$125$	−9.37452	−0.838482
$126$	0	0
$127$	10.9128	0.968358	0.484179	−	0.874969i	$-0.339118\pi$
$127$	10.9128	0.968358	0.484179	+	0.874969i	$0.339118\pi$
$128$	0	0
$129$	7.99042	0.703517
$130$	0	0
$131$	5.98620	0.523017	0.261508	−	0.965201i	$-0.415780\pi$
$131$	5.98620	0.523017	0.261508	+	0.965201i	$0.415780\pi$
$132$	0	0
$133$	4.74251	0.411227
$134$	0	0
$135$	5.44139	0.468320
$136$	0	0
$137$	20.6815	1.76694	0.883469	−	0.468490i	$-0.155202\pi$
$137$	20.6815	1.76694	0.883469	+	0.468490i	$0.155202\pi$
$138$	0	0
$139$	18.0761	1.53319	0.766597	−	0.642128i	$-0.221948\pi$
$139$	18.0761	1.53319	0.766597	+	0.642128i	$0.221948\pi$
$140$	0	0
$141$	−4.21717	−0.355150
$142$	0	0
$143$	−3.60649	−0.301590
$144$	0	0
$145$	−2.46225	−0.204479
$146$	0	0
$147$	6.28786	0.518614
$148$	0	0
$149$	19.8053	1.62252	0.811259	−	0.584687i	$-0.198783\pi$
$149$	19.8053	1.62252	0.811259	+	0.584687i	$0.198783\pi$
$150$	0	0
$151$	−4.89709	−0.398519	−0.199260	−	0.979947i	$-0.563854\pi$
$151$	−4.89709	−0.398519	−0.199260	+	0.979947i	$0.563854\pi$
$152$	0	0
$153$	−0.0425790	−0.00344231
$154$	0	0
$155$	−4.82250	−0.387352
$156$	0	0
$157$	−8.42032	−0.672014	−0.336007	−	0.941859i	$-0.609077\pi$
$157$	−8.42032	−0.672014	−0.336007	+	0.941859i	$0.609077\pi$
$158$	0	0
$159$	3.69937	0.293379
$160$	0	0
$161$	−10.5832	−0.834073
$162$	0	0
$163$	14.1317	1.10688	0.553442	−	0.832888i	$-0.313314\pi$
$163$	14.1317	1.10688	0.553442	+	0.832888i	$0.313314\pi$
$164$	0	0
$165$	−11.0832	−0.862824
$166$	0	0
$167$	−15.2263	−1.17824	−0.589122	−	0.808044i	$-0.700526\pi$
$167$	−15.2263	−1.17824	−0.589122	+	0.808044i	$0.700526\pi$
$168$	0	0
$169$	−12.6415	−0.972427
$170$	0	0
$171$	0.109590	0.00838056
$172$	0	0
$173$	−6.45291	−0.490605	−0.245303	−	0.969447i	$-0.578887\pi$
$173$	−6.45291	−0.490605	−0.245303	+	0.969447i	$0.578887\pi$
$174$	0	0
$175$	−7.16288	−0.541463
$176$	0	0
$177$	1.74430	0.131110
$178$	0	0
$179$	−5.99581	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$179$	−5.99581	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$180$	0	0
$181$	−11.2116	−0.833352	−0.416676	−	0.909055i	$-0.636805\pi$
$181$	−11.2116	−0.833352	−0.416676	+	0.909055i	$0.636805\pi$
$182$	0	0
$183$	0.875769	0.0647387
$184$	0	0
$185$	9.07935	0.667527
$186$	0	0
$187$	−6.02375	−0.440501
$188$	0	0
$189$	9.50531	0.691409
$190$	0	0
$191$	1.74364	0.126165	0.0630826	−	0.998008i	$-0.479907\pi$
$191$	1.74364	0.126165	0.0630826	+	0.998008i	$0.479907\pi$
$192$	0	0
$193$	1.07360	0.0772794	0.0386397	−	0.999253i	$-0.487698\pi$
$193$	1.07360	0.0772794	0.0386397	+	0.999253i	$0.487698\pi$
$194$	0	0
$195$	1.10158	0.0788854
$196$	0	0
$197$	−4.59941	−0.327695	−0.163847	−	0.986486i	$-0.552390\pi$
$197$	−4.59941	−0.327695	−0.163847	+	0.986486i	$0.552390\pi$
$198$	0	0
$199$	11.0181	0.781050	0.390525	−	0.920592i	$-0.372293\pi$
$199$	11.0181	0.781050	0.390525	+	0.920592i	$0.372293\pi$
$200$	0	0
$201$	15.4495	1.08972
$202$	0	0
$203$	−4.30119	−0.301884
$204$	0	0
$205$	−11.3981	−0.796076
$206$	0	0
$207$	−0.244557	−0.0169979
$208$	0	0
$209$	15.5040	1.07243
$210$	0	0
$211$	−23.5634	−1.62217	−0.811085	−	0.584928i	$-0.801123\pi$
$211$	−23.5634	−1.62217	−0.811085	+	0.584928i	$0.801123\pi$
$212$	0	0
$213$	13.0421	0.893628
$214$	0	0
$215$	−4.83197	−0.329538
$216$	0	0
$217$	−8.42419	−0.571871
$218$	0	0
$219$	−3.54544	−0.239579
$220$	0	0
$221$	0.598711	0.0402737
$222$	0	0
$223$	−0.620274	−0.0415366	−0.0207683	−	0.999784i	$-0.506611\pi$
$223$	−0.620274	−0.0415366	−0.0207683	+	0.999784i	$0.506611\pi$
$224$	0	0
$225$	−0.165520	−0.0110347
$226$	0	0
$227$	6.35233	0.421619	0.210810	−	0.977527i	$-0.432390\pi$
$227$	6.35233	0.421619	0.210810	+	0.977527i	$0.432390\pi$
$228$	0	0
$229$	28.4085	1.87729	0.938644	−	0.344887i	$-0.112083\pi$
$229$	28.4085	1.87729	0.938644	+	0.344887i	$0.112083\pi$
$230$	0	0
$231$	−19.3607	−1.27384
$232$	0	0
$233$	9.88718	0.647730	0.323865	−	0.946103i	$-0.395018\pi$
$233$	9.88718	0.647730	0.323865	+	0.946103i	$0.395018\pi$
$234$	0	0
$235$	2.55021	0.166358
$236$	0	0
$237$	27.1301	1.76229
$238$	0	0
$239$	5.23987	0.338939	0.169469	−	0.985535i	$-0.445795\pi$
$239$	5.23987	0.338939	0.169469	+	0.985535i	$0.445795\pi$
$240$	0	0
$241$	20.5495	1.32371	0.661855	−	0.749632i	$-0.269769\pi$
$241$	20.5495	1.32371	0.661855	+	0.749632i	$0.269769\pi$
$242$	0	0
$243$	0.442460	0.0283838
$244$	0	0
$245$	−3.80240	−0.242927
$246$	0	0
$247$	−1.54097	−0.0980493
$248$	0	0
$249$	27.3920	1.73589
$250$	0	0
$251$	−7.03512	−0.444053	−0.222026	−	0.975041i	$-0.571267\pi$
$251$	−7.03512	−0.444053	−0.222026	+	0.975041i	$0.571267\pi$
$252$	0	0
$253$	−34.5981	−2.17516
$254$	0	0
$255$	1.83991	0.115220
$256$	0	0
$257$	−7.69867	−0.480230	−0.240115	−	0.970744i	$-0.577185\pi$
$257$	−7.69867	−0.480230	−0.240115	+	0.970744i	$0.577185\pi$
$258$	0	0
$259$	15.8603	0.985510
$260$	0	0
$261$	−0.0993921	−0.00615221
$262$	0	0
$263$	−10.2284	−0.630708	−0.315354	−	0.948974i	$-0.602123\pi$
$263$	−10.2284	−0.630708	−0.315354	+	0.948974i	$0.602123\pi$
$264$	0	0
$265$	−2.23709	−0.137423
$266$	0	0
$267$	3.80864	0.233085
$268$	0	0
$269$	−23.3933	−1.42631	−0.713156	−	0.701006i	$-0.752735\pi$
$269$	−23.3933	−1.42631	−0.713156	+	0.701006i	$0.752735\pi$
$270$	0	0
$271$	−9.34130	−0.567444	−0.283722	−	0.958907i	$-0.591569\pi$
$271$	−9.34130	−0.567444	−0.283722	+	0.958907i	$0.591569\pi$
$272$	0	0
$273$	1.92429	0.116463
$274$	0	0
$275$	−23.4165	−1.41207
$276$	0	0
$277$	−16.3186	−0.980492	−0.490246	−	0.871584i	$-0.663093\pi$
$277$	−16.3186	−0.980492	−0.490246	+	0.871584i	$0.663093\pi$
$278$	0	0
$279$	−0.194667	−0.0116544
$280$	0	0
$281$	−13.0799	−0.780279	−0.390140	−	0.920756i	$-0.627573\pi$
$281$	−13.0799	−0.780279	−0.390140	+	0.920756i	$0.627573\pi$
$282$	0	0
$283$	−18.3941	−1.09342	−0.546708	−	0.837323i	$-0.684119\pi$
$283$	−18.3941	−1.09342	−0.546708	+	0.837323i	$0.684119\pi$
$284$	0	0
$285$	−4.73557	−0.280511
$286$	0	0
$287$	−19.9108	−1.17529
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	19.2558	1.12880
$292$	0	0
$293$	−0.946606	−0.0553013	−0.0276507	−	0.999618i	$-0.508803\pi$
$293$	−0.946606	−0.0553013	−0.0276507	+	0.999618i	$0.508803\pi$
$294$	0	0
$295$	−1.05481	−0.0614136
$296$	0	0
$297$	31.0743	1.80311
$298$	0	0
$299$	3.43876	0.198869
$300$	0	0
$301$	−8.44074	−0.486516
$302$	0	0
$303$	−18.9830	−1.09054
$304$	0	0
$305$	−0.529596	−0.0303246
$306$	0	0
$307$	30.6189	1.74751	0.873757	−	0.486364i	$-0.161677\pi$
$307$	30.6189	1.74751	0.873757	+	0.486364i	$0.161677\pi$
$308$	0	0
$309$	22.9007	1.30277
$310$	0	0
$311$	−31.2056	−1.76951	−0.884754	−	0.466059i	$-0.845673\pi$
$311$	−31.2056	−1.76951	−0.884754	+	0.466059i	$0.845673\pi$
$312$	0	0
$313$	−20.2398	−1.14402	−0.572011	−	0.820246i	$-0.693836\pi$
$313$	−20.2398	−1.14402	−0.572011	+	0.820246i	$0.693836\pi$
$314$	0	0
$315$	0.0827567	0.00466281
$316$	0	0
$317$	−9.78770	−0.549732	−0.274866	−	0.961483i	$-0.588634\pi$
$317$	−9.78770	−0.549732	−0.274866	+	0.961483i	$0.588634\pi$
$318$	0	0
$319$	−14.0612	−0.787278
$320$	0	0
$321$	9.68190	0.540391
$322$	0	0
$323$	−2.57381	−0.143210
$324$	0	0
$325$	2.32741	0.129101
$326$	0	0
$327$	4.08577	0.225944
$328$	0	0
$329$	4.45485	0.245604
$330$	0	0
$331$	−13.4945	−0.741726	−0.370863	−	0.928688i	$-0.620938\pi$
$331$	−13.4945	−0.741726	−0.370863	+	0.928688i	$0.620938\pi$
$332$	0	0
$333$	0.366500	0.0200841
$334$	0	0
$335$	−9.34262	−0.510442
$336$	0	0
$337$	21.2821	1.15931	0.579656	−	0.814861i	$-0.303187\pi$
$337$	21.2821	1.15931	0.579656	+	0.814861i	$0.303187\pi$
$338$	0	0
$339$	−25.1779	−1.36747
$340$	0	0
$341$	−27.5400	−1.49137
$342$	0	0
$343$	−19.5405	−1.05509
$344$	0	0
$345$	10.5677	0.568948
$346$	0	0
$347$	9.52748	0.511462	0.255731	−	0.966748i	$-0.417684\pi$
$347$	9.52748	0.511462	0.255731	+	0.966748i	$0.417684\pi$
$348$	0	0
$349$	5.71491	0.305912	0.152956	−	0.988233i	$-0.451121\pi$
$349$	5.71491	0.305912	0.152956	+	0.988233i	$0.451121\pi$
$350$	0	0
$351$	−3.08853	−0.164853
$352$	0	0
$353$	15.7744	0.839585	0.419792	−	0.907620i	$-0.362103\pi$
$353$	15.7744	0.839585	0.419792	+	0.907620i	$0.362103\pi$
$354$	0	0
$355$	−7.88681	−0.418589
$356$	0	0
$357$	3.21405	0.170106
$358$	0	0
$359$	2.42723	0.128104	0.0640521	−	0.997947i	$-0.479598\pi$
$359$	2.42723	0.128104	0.0640521	+	0.997947i	$0.479598\pi$
$360$	0	0
$361$	−12.3755	−0.651343
$362$	0	0
$363$	−44.1057	−2.31495
$364$	0	0
$365$	2.14400	0.112222
$366$	0	0
$367$	−2.31877	−0.121039	−0.0605194	−	0.998167i	$-0.519276\pi$
$367$	−2.31877	−0.121039	−0.0605194	+	0.998167i	$0.519276\pi$
$368$	0	0
$369$	−0.460099	−0.0239518
$370$	0	0
$371$	−3.90786	−0.202886
$372$	0	0
$373$	15.4912	0.802102	0.401051	−	0.916056i	$-0.368645\pi$
$373$	15.4912	0.802102	0.401051	+	0.916056i	$0.368645\pi$
$374$	0	0
$375$	16.3520	0.844412
$376$	0	0
$377$	1.39757	0.0719785
$378$	0	0
$379$	−10.6790	−0.548543	−0.274272	−	0.961652i	$-0.588437\pi$
$379$	−10.6790	−0.548543	−0.274272	+	0.961652i	$0.588437\pi$
$380$	0	0
$381$	−19.0353	−0.975206
$382$	0	0
$383$	1.85770	0.0949243	0.0474621	−	0.998873i	$-0.484887\pi$
$383$	1.85770	0.0949243	0.0474621	+	0.998873i	$0.484887\pi$
$384$	0	0
$385$	11.7078	0.596685
$386$	0	0
$387$	−0.195049	−0.00991490
$388$	0	0
$389$	23.4064	1.18675	0.593375	−	0.804926i	$-0.297795\pi$
$389$	23.4064	1.18675	0.593375	+	0.804926i	$0.297795\pi$
$390$	0	0
$391$	5.74361	0.290467
$392$	0	0
$393$	−10.4417	−0.526715
$394$	0	0
$395$	−16.4061	−0.825482
$396$	0	0
$397$	26.5294	1.33147	0.665737	−	0.746187i	$-0.268117\pi$
$397$	26.5294	1.33147	0.665737	+	0.746187i	$0.268117\pi$
$398$	0	0
$399$	−8.27235	−0.414135
$400$	0	0
$401$	−9.33532	−0.466184	−0.233092	−	0.972455i	$-0.574884\pi$
$401$	−9.33532	−0.466184	−0.233092	+	0.972455i	$0.574884\pi$
$402$	0	0
$403$	2.73724	0.136352
$404$	0	0
$405$	−9.62615	−0.478327
$406$	0	0
$407$	51.8497	2.57009
$408$	0	0
$409$	−7.33101	−0.362495	−0.181248	−	0.983438i	$-0.558014\pi$
$409$	−7.33101	−0.362495	−0.181248	+	0.983438i	$0.558014\pi$
$410$	0	0
$411$	−36.0747	−1.77943
$412$	0	0
$413$	−1.84260	−0.0906686
$414$	0	0
$415$	−16.5645	−0.813119
$416$	0	0
$417$	−31.5301	−1.54404
$418$	0	0
$419$	−14.1976	−0.693598	−0.346799	−	0.937939i	$-0.612731\pi$
$419$	−14.1976	−0.693598	−0.346799	+	0.937939i	$0.612731\pi$
$420$	0	0
$421$	−19.6703	−0.958672	−0.479336	−	0.877632i	$-0.659122\pi$
$421$	−19.6703	−0.958672	−0.479336	+	0.877632i	$0.659122\pi$
$422$	0	0
$423$	0.102943	0.00500525
$424$	0	0
$425$	3.88737	0.188565
$426$	0	0
$427$	−0.925126	−0.0447700
$428$	0	0
$429$	6.29079	0.303722
$430$	0	0
$431$	−18.7827	−0.904731	−0.452366	−	0.891833i	$-0.649420\pi$
$431$	−18.7827	−0.904731	−0.452366	+	0.891833i	$0.649420\pi$
$432$	0	0
$433$	19.5745	0.940692	0.470346	−	0.882482i	$-0.344129\pi$
$433$	19.5745	0.940692	0.470346	+	0.882482i	$0.344129\pi$
$434$	0	0
$435$	4.29490	0.205925
$436$	0	0
$437$	−14.7829	−0.707164
$438$	0	0
$439$	15.9031	0.759011	0.379506	−	0.925189i	$-0.376094\pi$
$439$	15.9031	0.759011	0.379506	+	0.925189i	$0.376094\pi$
$440$	0	0
$441$	−0.153489	−0.00730900
$442$	0	0
$443$	9.65861	0.458894	0.229447	−	0.973321i	$-0.426308\pi$
$443$	9.65861	0.458894	0.229447	+	0.973321i	$0.426308\pi$
$444$	0	0
$445$	−2.30316	−0.109180
$446$	0	0
$447$	−34.5464	−1.63399
$448$	0	0
$449$	−27.9027	−1.31681	−0.658406	−	0.752663i	$-0.728769\pi$
$449$	−27.9027	−1.31681	−0.658406	+	0.752663i	$0.728769\pi$
$450$	0	0
$451$	−65.0913	−3.06503
$452$	0	0
$453$	8.54198	0.401337
$454$	0	0
$455$	−1.16366	−0.0545531
$456$	0	0
$457$	−11.7190	−0.548191	−0.274095	−	0.961703i	$-0.588378\pi$
$457$	−11.7190	−0.548191	−0.274095	+	0.961703i	$0.588378\pi$
$458$	0	0
$459$	−5.15863	−0.240784
$460$	0	0
$461$	−3.60889	−0.168083	−0.0840413	−	0.996462i	$-0.526783\pi$
$461$	−3.60889	−0.168083	−0.0840413	+	0.996462i	$0.526783\pi$
$462$	0	0
$463$	40.8473	1.89833	0.949167	−	0.314774i	$-0.101929\pi$
$463$	40.8473	1.89833	0.949167	+	0.314774i	$0.101929\pi$
$464$	0	0
$465$	8.41188	0.390091
$466$	0	0
$467$	−24.6197	−1.13926	−0.569631	−	0.821901i	$-0.692914\pi$
$467$	−24.6197	−1.13926	−0.569631	+	0.821901i	$0.692914\pi$
$468$	0	0
$469$	−16.3202	−0.753596
$470$	0	0
$471$	14.6875	0.676766
$472$	0	0
$473$	−27.5941	−1.26878
$474$	0	0
$475$	−10.0053	−0.459076
$476$	0	0
$477$	−0.0903030	−0.00413469
$478$	0	0
$479$	32.9638	1.50616	0.753078	−	0.657932i	$-0.228568\pi$
$479$	32.9638	1.50616	0.753078	+	0.657932i	$0.228568\pi$
$480$	0	0
$481$	−5.15343	−0.234976
$482$	0	0
$483$	18.4603	0.839971
$484$	0	0
$485$	−11.6444	−0.528745
$486$	0	0
$487$	19.4456	0.881166	0.440583	−	0.897712i	$-0.354772\pi$
$487$	19.4456	0.881166	0.440583	+	0.897712i	$0.354772\pi$
$488$	0	0
$489$	−24.6500	−1.11471
$490$	0	0
$491$	−24.4257	−1.10231	−0.551157	−	0.834401i	$-0.685814\pi$
$491$	−24.4257	−1.10231	−0.551157	+	0.834401i	$0.685814\pi$
$492$	0	0
$493$	2.33430	0.105132
$494$	0	0
$495$	0.270544	0.0121601
$496$	0	0
$497$	−13.7771	−0.617987
$498$	0	0
$499$	−35.8808	−1.60624	−0.803122	−	0.595814i	$-0.796829\pi$
$499$	−35.8808	−1.60624	−0.803122	+	0.595814i	$0.796829\pi$
$500$	0	0
$501$	26.5592	1.18658
$502$	0	0
$503$	−33.1367	−1.47749	−0.738746	−	0.673984i	$-0.764582\pi$
$503$	−33.1367	−1.47749	−0.738746	+	0.673984i	$0.764582\pi$
$504$	0	0
$505$	11.4794	0.510827
$506$	0	0
$507$	22.0506	0.979303
$508$	0	0
$509$	−0.321190	−0.0142365	−0.00711825	−	0.999975i	$-0.502266\pi$
$509$	−0.321190	−0.0142365	−0.00711825	+	0.999975i	$0.502266\pi$
$510$	0	0
$511$	3.74526	0.165680
$512$	0	0
$513$	13.2773	0.586207
$514$	0	0
$515$	−13.8485	−0.610238
$516$	0	0
$517$	14.5636	0.640505
$518$	0	0
$519$	11.2558	0.494075
$520$	0	0
$521$	41.9003	1.83569	0.917843	−	0.396943i	$-0.129929\pi$
$521$	41.9003	1.83569	0.917843	+	0.396943i	$0.129929\pi$
$522$	0	0
$523$	28.7697	1.25801	0.629005	−	0.777402i	$-0.283463\pi$
$523$	28.7697	1.25801	0.629005	+	0.777402i	$0.283463\pi$
$524$	0	0
$525$	12.4942	0.545292
$526$	0	0
$527$	4.57189	0.199155
$528$	0	0
$529$	9.98907	0.434308
$530$	0	0
$531$	−0.0425790	−0.00184777
$532$	0	0
$533$	6.46953	0.280227
$534$	0	0
$535$	−5.85485	−0.253127
$536$	0	0
$537$	10.4585	0.451317
$538$	0	0
$539$	−21.7145	−0.935309
$540$	0	0
$541$	11.0728	0.476055	0.238028	−	0.971258i	$-0.423499\pi$
$541$	11.0728	0.476055	0.238028	+	0.971258i	$0.423499\pi$
$542$	0	0
$543$	19.5564	0.839245
$544$	0	0
$545$	−2.47075	−0.105835
$546$	0	0
$547$	25.8147	1.10376	0.551878	−	0.833925i	$-0.313911\pi$
$547$	25.8147	1.10376	0.551878	+	0.833925i	$0.313911\pi$
$548$	0	0
$549$	−0.0213778	−0.000912384 0
$550$	0	0
$551$	−6.00803	−0.255951
$552$	0	0
$553$	−28.6591	−1.21871
$554$	0	0
$555$	−15.8371	−0.672247
$556$	0	0
$557$	−35.0864	−1.48666	−0.743330	−	0.668925i	$-0.766755\pi$
$557$	−35.0864	−1.48666	−0.743330	+	0.668925i	$0.766755\pi$
$558$	0	0
$559$	2.74262	0.116001
$560$	0	0
$561$	10.5072	0.443615
$562$	0	0
$563$	−37.3591	−1.57450	−0.787250	−	0.616634i	$-0.788496\pi$
$563$	−37.3591	−1.57450	−0.787250	+	0.616634i	$0.788496\pi$
$564$	0	0
$565$	15.2256	0.640545
$566$	0	0
$567$	−16.8155	−0.706183
$568$	0	0
$569$	−31.4081	−1.31669	−0.658347	−	0.752715i	$-0.728744\pi$
$569$	−31.4081	−1.31669	−0.658347	+	0.752715i	$0.728744\pi$
$570$	0	0
$571$	−15.3355	−0.641769	−0.320884	−	0.947118i	$-0.603980\pi$
$571$	−15.3355	−0.641769	−0.320884	+	0.947118i	$0.603980\pi$
$572$	0	0
$573$	−3.04143	−0.127057
$574$	0	0
$575$	22.3275	0.931122
$576$	0	0
$577$	0.919402	0.0382752	0.0191376	−	0.999817i	$-0.493908\pi$
$577$	0.919402	0.0382752	0.0191376	+	0.999817i	$0.493908\pi$
$578$	0	0
$579$	−1.87268	−0.0778259
$580$	0	0
$581$	−28.9357	−1.20046
$582$	0	0
$583$	−12.7754	−0.529102
$584$	0	0
$585$	−0.0268898	−0.00111176
$586$	0	0
$587$	−5.59692	−0.231010	−0.115505	−	0.993307i	$-0.536849\pi$
$587$	−5.59692	−0.231010	−0.115505	+	0.993307i	$0.536849\pi$
$588$	0	0
$589$	−11.7672	−0.484858
$590$	0	0
$591$	8.02275	0.330012
$592$	0	0
$593$	−8.00852	−0.328871	−0.164435	−	0.986388i	$-0.552580\pi$
$593$	−8.00852	−0.328871	−0.164435	+	0.986388i	$0.552580\pi$
$594$	0	0
$595$	−1.94360	−0.0796800
$596$	0	0
$597$	−19.2188	−0.786574
$598$	0	0
$599$	−1.85097	−0.0756284	−0.0378142	−	0.999285i	$-0.512040\pi$
$599$	−1.85097	−0.0756284	−0.0378142	+	0.999285i	$0.512040\pi$
$600$	0	0
$601$	−13.3221	−0.543420	−0.271710	−	0.962379i	$-0.587589\pi$
$601$	−13.3221	−0.543420	−0.271710	+	0.962379i	$0.587589\pi$
$602$	0	0
$603$	−0.377127	−0.0153578
$604$	0	0
$605$	26.6716	1.08436
$606$	0	0
$607$	−30.0742	−1.22067	−0.610336	−	0.792142i	$-0.708966\pi$
$607$	−30.0742	−1.22067	−0.610336	+	0.792142i	$0.708966\pi$
$608$	0	0
$609$	7.50256	0.304019
$610$	0	0
$611$	−1.44750	−0.0585595
$612$	0	0
$613$	−30.9975	−1.25198	−0.625989	−	0.779832i	$-0.715305\pi$
$613$	−30.9975	−1.25198	−0.625989	+	0.779832i	$0.715305\pi$
$614$	0	0
$615$	19.8817	0.801706
$616$	0	0
$617$	38.3757	1.54495	0.772473	−	0.635048i	$-0.219020\pi$
$617$	38.3757	1.54495	0.772473	+	0.635048i	$0.219020\pi$
$618$	0	0
$619$	−42.0343	−1.68950	−0.844750	−	0.535160i	$-0.820251\pi$
$619$	−42.0343	−1.68950	−0.844750	+	0.535160i	$0.820251\pi$
$620$	0	0
$621$	−29.6291	−1.18898
$622$	0	0
$623$	−4.02329	−0.161190
$624$	0	0
$625$	9.54847	0.381939
$626$	0	0
$627$	−27.0436	−1.08002
$628$	0	0
$629$	−8.60753	−0.343205
$630$	0	0
$631$	−14.5397	−0.578815	−0.289408	−	0.957206i	$-0.593458\pi$
$631$	−14.5397	−0.578815	−0.289408	+	0.957206i	$0.593458\pi$
$632$	0	0
$633$	41.1016	1.63364
$634$	0	0
$635$	11.5110	0.456801
$636$	0	0
$637$	2.15824	0.0855126
$638$	0	0
$639$	−0.318362	−0.0125942
$640$	0	0
$641$	−22.7997	−0.900533	−0.450267	−	0.892894i	$-0.648671\pi$
$641$	−22.7997	−0.900533	−0.450267	+	0.892894i	$0.648671\pi$
$642$	0	0
$643$	8.76457	0.345641	0.172820	−	0.984953i	$-0.444712\pi$
$643$	8.76457	0.345641	0.172820	+	0.984953i	$0.444712\pi$
$644$	0	0
$645$	8.42840	0.331868
$646$	0	0
$647$	22.2589	0.875086	0.437543	−	0.899197i	$-0.355849\pi$
$647$	22.2589	0.875086	0.437543	+	0.899197i	$0.355849\pi$
$648$	0	0
$649$	−6.02375	−0.236453
$650$	0	0
$651$	14.6943	0.575915
$652$	0	0
$653$	22.8514	0.894246	0.447123	−	0.894472i	$-0.352449\pi$
$653$	22.8514	0.894246	0.447123	+	0.894472i	$0.352449\pi$
$654$	0	0
$655$	6.31433	0.246721
$656$	0	0
$657$	0.0865455	0.00337646
$658$	0	0
$659$	25.4683	0.992105	0.496052	−	0.868293i	$-0.334782\pi$
$659$	25.4683	0.992105	0.496052	+	0.868293i	$0.334782\pi$
$660$	0	0
$661$	10.4114	0.404956	0.202478	−	0.979287i	$-0.435100\pi$
$661$	10.4114	0.404956	0.202478	+	0.979287i	$0.435100\pi$
$662$	0	0
$663$	−1.04433	−0.0405585
$664$	0	0
$665$	5.00246	0.193987
$666$	0	0
$667$	13.4073	0.519133
$668$	0	0
$669$	1.08194	0.0418303
$670$	0	0
$671$	−3.02438	−0.116755
$672$	0	0
$673$	−6.49351	−0.250306	−0.125153	−	0.992137i	$-0.539942\pi$
$673$	−6.49351	−0.250306	−0.125153	+	0.992137i	$0.539942\pi$
$674$	0	0
$675$	−20.0535	−0.771859
$676$	0	0
$677$	−23.7743	−0.913719	−0.456859	−	0.889539i	$-0.651026\pi$
$677$	−23.7743	−0.913719	−0.456859	+	0.889539i	$0.651026\pi$
$678$	0	0
$679$	−20.3411	−0.780618
$680$	0	0
$681$	−11.0804	−0.424600
$682$	0	0
$683$	−22.8021	−0.872500	−0.436250	−	0.899826i	$-0.643694\pi$
$683$	−22.8021	−0.872500	−0.436250	+	0.899826i	$0.643694\pi$
$684$	0	0
$685$	21.8151	0.833513
$686$	0	0
$687$	−49.5530	−1.89056
$688$	0	0
$689$	1.26977	0.0483743
$690$	0	0
$691$	8.45066	0.321478	0.160739	−	0.986997i	$-0.448612\pi$
$691$	8.45066	0.321478	0.160739	+	0.986997i	$0.448612\pi$
$692$	0	0
$693$	0.472601	0.0179526
$694$	0	0
$695$	19.0669	0.723250
$696$	0	0
$697$	10.8058	0.409298
$698$	0	0
$699$	−17.2462	−0.652311
$700$	0	0
$701$	27.4604	1.03717	0.518583	−	0.855027i	$-0.326460\pi$
$701$	27.4604	1.03717	0.518583	+	0.855027i	$0.326460\pi$
$702$	0	0
$703$	22.1541	0.835559
$704$	0	0
$705$	−4.44833	−0.167534
$706$	0	0
$707$	20.0528	0.754164
$708$	0	0
$709$	18.0106	0.676403	0.338201	−	0.941074i	$-0.390181\pi$
$709$	18.0106	0.676403	0.338201	+	0.941074i	$0.390181\pi$
$710$	0	0
$711$	−0.662256	−0.0248365
$712$	0	0
$713$	26.2592	0.983414
$714$	0	0
$715$	−3.80417	−0.142268
$716$	0	0
$717$	−9.13990	−0.341336
$718$	0	0
$719$	3.79330	0.141466	0.0707332	−	0.997495i	$-0.477466\pi$
$719$	3.79330	0.141466	0.0707332	+	0.997495i	$0.477466\pi$
$720$	0	0
$721$	−24.1913	−0.900931
$722$	0	0
$723$	−35.8445	−1.33307
$724$	0	0
$725$	9.07428	0.337010
$726$	0	0
$727$	−25.3067	−0.938575	−0.469288	−	0.883045i	$-0.655489\pi$
$727$	−25.3067	−0.938575	−0.469288	+	0.883045i	$0.655489\pi$
$728$	0	0
$729$	26.6060	0.985407
$730$	0	0
$731$	4.58088	0.169430
$732$	0	0
$733$	−41.6292	−1.53761	−0.768804	−	0.639484i	$-0.779148\pi$
$733$	−41.6292	−1.53761	−0.768804	+	0.639484i	$0.779148\pi$
$734$	0	0
$735$	6.63253	0.244644
$736$	0	0
$737$	−53.3532	−1.96529
$738$	0	0
$739$	−23.5029	−0.864569	−0.432285	−	0.901737i	$-0.642292\pi$
$739$	−23.5029	−0.864569	−0.432285	+	0.901737i	$0.642292\pi$
$740$	0	0
$741$	2.68791	0.0987427
$742$	0	0
$743$	34.6695	1.27190	0.635951	−	0.771730i	$-0.280608\pi$
$743$	34.6695	1.27190	0.635951	+	0.771730i	$0.280608\pi$
$744$	0	0
$745$	20.8910	0.765385
$746$	0	0
$747$	−0.668648	−0.0244645
$748$	0	0
$749$	−10.2276	−0.373707
$750$	0	0
$751$	−27.1286	−0.989936	−0.494968	−	0.868911i	$-0.664820\pi$
$751$	−27.1286	−0.989936	−0.494968	+	0.868911i	$0.664820\pi$
$752$	0	0
$753$	12.2713	0.447193
$754$	0	0
$755$	−5.16552	−0.187992
$756$	0	0
$757$	−24.7890	−0.900972	−0.450486	−	0.892784i	$-0.648749\pi$
$757$	−24.7890	−0.900972	−0.450486	+	0.892784i	$0.648749\pi$
$758$	0	0
$759$	60.3494	2.19055
$760$	0	0
$761$	41.7373	1.51298	0.756488	−	0.654008i	$-0.226914\pi$
$761$	41.7373	1.51298	0.756488	+	0.654008i	$0.226914\pi$
$762$	0	0
$763$	−4.31604	−0.156251
$764$	0	0
$765$	−0.0449129	−0.00162383
$766$	0	0
$767$	0.598711	0.0216182
$768$	0	0
$769$	13.2623	0.478251	0.239125	−	0.970989i	$-0.423139\pi$
$769$	13.2623	0.478251	0.239125	+	0.970989i	$0.423139\pi$
$770$	0	0
$771$	13.4288	0.483626
$772$	0	0
$773$	23.0304	0.828345	0.414173	−	0.910198i	$-0.364071\pi$
$773$	23.0304	0.828345	0.414173	+	0.910198i	$0.364071\pi$
$774$	0	0
$775$	17.7726	0.638412
$776$	0	0
$777$	−27.6651	−0.992479
$778$	0	0
$779$	−27.8119	−0.996467
$780$	0	0
$781$	−45.0394	−1.61164
$782$	0	0
$783$	−12.0418	−0.430338
$784$	0	0
$785$	−8.88187	−0.317007
$786$	0	0
$787$	9.77237	0.348347	0.174174	−	0.984715i	$-0.444275\pi$
$787$	9.77237	0.348347	0.174174	+	0.984715i	$0.444275\pi$
$788$	0	0
$789$	17.8413	0.635168
$790$	0	0
$791$	26.5969	0.945675
$792$	0	0
$793$	0.300598	0.0106745
$794$	0	0
$795$	3.90215	0.138395
$796$	0	0
$797$	−24.8939	−0.881787	−0.440894	−	0.897559i	$-0.645338\pi$
$797$	−24.8939	−0.881787	−0.440894	+	0.897559i	$0.645338\pi$
$798$	0	0
$799$	−2.41769	−0.0855317
$800$	0	0
$801$	−0.0929703	−0.00328494
$802$	0	0
$803$	12.2438	0.432075
$804$	0	0
$805$	−11.1633	−0.393455
$806$	0	0
$807$	40.8048	1.43640
$808$	0	0
$809$	−45.1403	−1.58705	−0.793525	−	0.608538i	$-0.791756\pi$
$809$	−45.1403	−1.58705	−0.793525	+	0.608538i	$0.791756\pi$
$810$	0	0
$811$	−9.85589	−0.346087	−0.173043	−	0.984914i	$-0.555360\pi$
$811$	−9.85589	−0.346087	−0.173043	+	0.984914i	$0.555360\pi$
$812$	0	0
$813$	16.2940	0.571456
$814$	0	0
$815$	14.9064	0.522147
$816$	0	0
$817$	−11.7903	−0.412490
$818$	0	0
$819$	−0.0469726	−0.00164136
$820$	0	0
$821$	42.5113	1.48366	0.741828	−	0.670590i	$-0.233959\pi$
$821$	42.5113	1.48366	0.741828	+	0.670590i	$0.233959\pi$
$822$	0	0
$823$	−53.6620	−1.87054	−0.935270	−	0.353934i	$-0.884844\pi$
$823$	−53.6620	−1.87054	−0.935270	+	0.353934i	$0.884844\pi$
$824$	0	0
$825$	40.8455	1.42206
$826$	0	0
$827$	11.3397	0.394320	0.197160	−	0.980371i	$-0.436828\pi$
$827$	11.3397	0.394320	0.197160	+	0.980371i	$0.436828\pi$
$828$	0	0
$829$	−19.8919	−0.690874	−0.345437	−	0.938442i	$-0.612269\pi$
$829$	−19.8919	−0.690874	−0.345437	+	0.938442i	$0.612269\pi$
$830$	0	0
$831$	28.4646	0.987425
$832$	0	0
$833$	3.60481	0.124899
$834$	0	0
$835$	−16.0609	−0.555810
$836$	0	0
$837$	−23.5847	−0.815206
$838$	0	0
$839$	40.3093	1.39163	0.695816	−	0.718220i	$-0.255043\pi$
$839$	40.3093	1.39163	0.695816	+	0.718220i	$0.255043\pi$
$840$	0	0
$841$	−23.5510	−0.812105
$842$	0	0
$843$	22.8152	0.785797
$844$	0	0
$845$	−13.3345	−0.458720
$846$	0	0
$847$	46.5914	1.60090
$848$	0	0
$849$	32.0848	1.10115
$850$	0	0
$851$	−49.4383	−1.69472
$852$	0	0
$853$	21.4704	0.735133	0.367566	−	0.929997i	$-0.380191\pi$
$853$	21.4704	0.735133	0.367566	+	0.929997i	$0.380191\pi$
$854$	0	0
$855$	0.115597	0.00395334
$856$	0	0
$857$	−2.04572	−0.0698804	−0.0349402	−	0.999389i	$-0.511124\pi$
$857$	−2.04572	−0.0698804	−0.0349402	+	0.999389i	$0.511124\pi$
$858$	0	0
$859$	−8.24249	−0.281230	−0.140615	−	0.990064i	$-0.544908\pi$
$859$	−8.24249	−0.281230	−0.140615	+	0.990064i	$0.544908\pi$
$860$	0	0
$861$	34.7303	1.18361
$862$	0	0
$863$	43.8494	1.49265	0.746326	−	0.665581i	$-0.231816\pi$
$863$	43.8494	1.49265	0.746326	+	0.665581i	$0.231816\pi$
$864$	0	0
$865$	−6.80662	−0.231432
$866$	0	0
$867$	−1.74430	−0.0592395
$868$	0	0
$869$	−93.6910	−3.17825
$870$	0	0
$871$	5.30286	0.179681
$872$	0	0
$873$	−0.470042	−0.0159085
$874$	0	0
$875$	−17.2735	−0.583952
$876$	0	0
$877$	−22.6505	−0.764853	−0.382426	−	0.923986i	$-0.624911\pi$
$877$	−22.6505	−0.764853	−0.382426	+	0.923986i	$0.624911\pi$
$878$	0	0
$879$	1.65116	0.0556924
$880$	0	0
$881$	−23.9661	−0.807437	−0.403718	−	0.914883i	$-0.632282\pi$
$881$	−23.9661	−0.807437	−0.403718	+	0.914883i	$0.632282\pi$
$882$	0	0
$883$	22.8823	0.770052	0.385026	−	0.922906i	$-0.374192\pi$
$883$	22.8823	0.770052	0.385026	+	0.922906i	$0.374192\pi$
$884$	0	0
$885$	1.83991	0.0618479
$886$	0	0
$887$	−4.76460	−0.159980	−0.0799898	−	0.996796i	$-0.525489\pi$
$887$	−4.76460	−0.159980	−0.0799898	+	0.996796i	$0.525489\pi$
$888$	0	0
$889$	20.1080	0.674402
$890$	0	0
$891$	−54.9723	−1.84164
$892$	0	0
$893$	6.22266	0.208233
$894$	0	0
$895$	−6.32447	−0.211404
$896$	0	0
$897$	−5.99823	−0.200275
$898$	0	0
$899$	10.6722	0.355937
$900$	0	0
$901$	2.12083	0.0706552
$902$	0	0
$903$	14.7232	0.489957
$904$	0	0
$905$	−11.8262	−0.393115
$906$	0	0
$907$	53.7124	1.78349	0.891746	−	0.452536i	$-0.149481\pi$
$907$	53.7124	1.78349	0.891746	+	0.452536i	$0.149481\pi$
$908$	0	0
$909$	0.463382	0.0153694
$910$	0	0
$911$	−19.9907	−0.662320	−0.331160	−	0.943575i	$-0.607440\pi$
$911$	−19.9907	−0.662320	−0.331160	+	0.943575i	$0.607440\pi$
$912$	0	0
$913$	−94.5952	−3.13065
$914$	0	0
$915$	0.923774	0.0305390
$916$	0	0
$917$	11.0302	0.364249
$918$	0	0
$919$	−13.8769	−0.457757	−0.228878	−	0.973455i	$-0.573506\pi$
$919$	−13.8769	−0.457757	−0.228878	+	0.973455i	$0.573506\pi$
$920$	0	0
$921$	−53.4085	−1.75987
$922$	0	0
$923$	4.47655	0.147347
$924$	0	0
$925$	−33.4607	−1.10018
$926$	0	0
$927$	−0.559014	−0.0183604
$928$	0	0
$929$	−24.9854	−0.819744	−0.409872	−	0.912143i	$-0.634427\pi$
$929$	−24.9854	−0.819744	−0.409872	+	0.912143i	$0.634427\pi$
$930$	0	0
$931$	−9.27808	−0.304077
$932$	0	0
$933$	54.4319	1.78202
$934$	0	0
$935$	−6.35394	−0.207796
$936$	0	0
$937$	10.8038	0.352944	0.176472	−	0.984306i	$-0.443532\pi$
$937$	10.8038	0.352944	0.176472	+	0.984306i	$0.443532\pi$
$938$	0	0
$939$	35.3043	1.15211
$940$	0	0
$941$	−4.64358	−0.151376	−0.0756882	−	0.997132i	$-0.524115\pi$
$941$	−4.64358	−0.151376	−0.0756882	+	0.997132i	$0.524115\pi$
$942$	0	0
$943$	62.0641	2.02109
$944$	0	0
$945$	10.0263	0.326157
$946$	0	0
$947$	25.9173	0.842199	0.421099	−	0.907014i	$-0.361644\pi$
$947$	25.9173	0.842199	0.421099	+	0.907014i	$0.361644\pi$
$948$	0	0
$949$	−1.21693	−0.0395033
$950$	0	0
$951$	17.0727	0.553619
$952$	0	0
$953$	9.22969	0.298979	0.149490	−	0.988763i	$-0.452237\pi$
$953$	9.22969	0.298979	0.149490	+	0.988763i	$0.452237\pi$
$954$	0	0
$955$	1.83921	0.0595156
$956$	0	0
$957$	24.5270	0.792846
$958$	0	0
$959$	38.1078	1.23056
$960$	0	0
$961$	−10.0978	−0.325735
$962$	0	0
$963$	−0.236339	−0.00761591
$964$	0	0
$965$	1.13245	0.0364548
$966$	0	0
$967$	−54.1981	−1.74289	−0.871447	−	0.490490i	$-0.836818\pi$
$967$	−54.1981	−1.74289	−0.871447	+	0.490490i	$0.836818\pi$
$968$	0	0
$969$	4.48949	0.144223
$970$	0	0
$971$	−29.0364	−0.931822	−0.465911	−	0.884832i	$-0.654273\pi$
$971$	−29.0364	−0.931822	−0.465911	+	0.884832i	$0.654273\pi$
$972$	0	0
$973$	33.3071	1.06778
$974$	0	0
$975$	−4.05970	−0.130014
$976$	0	0
$977$	43.6200	1.39553	0.697763	−	0.716329i	$-0.254179\pi$
$977$	43.6200	1.39553	0.697763	+	0.716329i	$0.254179\pi$
$978$	0	0
$979$	−13.1527	−0.420363
$980$	0	0
$981$	−0.0997352	−0.00318430
$982$	0	0
$983$	−16.4486	−0.524629	−0.262314	−	0.964983i	$-0.584486\pi$
$983$	−16.4486	−0.524629	−0.262314	+	0.964983i	$0.584486\pi$
$984$	0	0
$985$	−4.85153	−0.154582
$986$	0	0
$987$	−7.77058	−0.247340
$988$	0	0
$989$	26.3108	0.836634
$990$	0	0
$991$	−20.1702	−0.640728	−0.320364	−	0.947295i	$-0.603805\pi$
$991$	−20.1702	−0.640728	−0.320364	+	0.947295i	$0.603805\pi$
$992$	0	0
$993$	23.5385	0.746971
$994$	0	0
$995$	11.6220	0.368443
$996$	0	0
$997$	42.1453	1.33476	0.667378	−	0.744719i	$-0.267417\pi$
$997$	42.1453	1.33476	0.667378	+	0.744719i	$0.267417\pi$
$998$	0	0
$999$	44.4031	1.40485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.6	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.6	✓	23	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74430 q^{3} +1.05481 q^{5} +1.84260 q^{7} +0.0425790 q^{9} +O(q^{10})\)	\(q-1.74430 q^{3} +1.05481 q^{5} +1.84260 q^{7} +0.0425790 q^{9} +6.02375 q^{11} -0.598711 q^{13} -1.83991 q^{15} -1.00000 q^{17} +2.57381 q^{19} -3.21405 q^{21} -5.74361 q^{23} -3.88737 q^{25} +5.15863 q^{27} -2.33430 q^{29} -4.57189 q^{31} -10.5072 q^{33} +1.94360 q^{35} +8.60753 q^{37} +1.04433 q^{39} -10.8058 q^{41} -4.58088 q^{43} +0.0449129 q^{45} +2.41769 q^{47} -3.60481 q^{49} +1.74430 q^{51} -2.12083 q^{53} +6.35394 q^{55} -4.48949 q^{57} -1.00000 q^{59} -0.502075 q^{61} +0.0784562 q^{63} -0.631529 q^{65} -8.85713 q^{67} +10.0186 q^{69} -7.47697 q^{71} +2.03259 q^{73} +6.78073 q^{75} +11.0994 q^{77} -15.5536 q^{79} -9.12592 q^{81} -15.7037 q^{83} -1.05481 q^{85} +4.07171 q^{87} -2.18348 q^{89} -1.10319 q^{91} +7.97475 q^{93} +2.71489 q^{95} -11.0393 q^{97} +0.256485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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