LMFDB - Embedded newform 8032.2.a.i.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70694 q^{3} +1.80279 q^{5} -0.510098 q^{7} -0.0863705 q^{9} +O(q^{10})$	$q-1.70694 q^{3} +1.80279 q^{5} -0.510098 q^{7} -0.0863705 q^{9} +2.94398 q^{11} -2.13332 q^{13} -3.07725 q^{15} -5.07079 q^{17} -3.44304 q^{19} +0.870704 q^{21} -5.81872 q^{23} -1.74994 q^{25} +5.26824 q^{27} -3.06392 q^{29} +0.190243 q^{31} -5.02519 q^{33} -0.919601 q^{35} -2.93430 q^{37} +3.64144 q^{39} -5.04756 q^{41} +4.54833 q^{43} -0.155708 q^{45} +3.80418 q^{47} -6.73980 q^{49} +8.65551 q^{51} +12.5215 q^{53} +5.30739 q^{55} +5.87705 q^{57} +7.35634 q^{59} -6.72949 q^{61} +0.0440574 q^{63} -3.84593 q^{65} -1.34776 q^{67} +9.93218 q^{69} +5.95576 q^{71} +12.5647 q^{73} +2.98703 q^{75} -1.50172 q^{77} +16.9929 q^{79} -8.73343 q^{81} -10.9781 q^{83} -9.14159 q^{85} +5.22991 q^{87} +11.2470 q^{89} +1.08820 q^{91} -0.324733 q^{93} -6.20709 q^{95} +8.33626 q^{97} -0.254273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * 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
$3$	−1.70694	−0.985500	−0.492750	−	0.870171i	$-0.664008\pi$
$3$	−1.70694	−0.985500	−0.492750	+	0.870171i	$0.664008\pi$
$4$	0	0
$5$	1.80279	0.806234	0.403117	−	0.915149i	$-0.367927\pi$
$5$	1.80279	0.806234	0.403117	+	0.915149i	$0.367927\pi$
$6$	0	0
$7$	−0.510098	−0.192799	−0.0963994	−	0.995343i	$-0.530733\pi$
$7$	−0.510098	−0.192799	−0.0963994	+	0.995343i	$0.530733\pi$
$8$	0	0
$9$	−0.0863705	−0.0287902
$10$	0	0
$11$	2.94398	0.887644	0.443822	−	0.896115i	$-0.353622\pi$
$11$	2.94398	0.887644	0.443822	+	0.896115i	$0.353622\pi$
$12$	0	0
$13$	−2.13332	−0.591676	−0.295838	−	0.955238i	$-0.595599\pi$
$13$	−2.13332	−0.591676	−0.295838	+	0.955238i	$0.595599\pi$
$14$	0	0
$15$	−3.07725	−0.794543
$16$	0	0
$17$	−5.07079	−1.22985	−0.614924	−	0.788587i	$-0.710813\pi$
$17$	−5.07079	−1.22985	−0.614924	+	0.788587i	$0.710813\pi$
$18$	0	0
$19$	−3.44304	−0.789888	−0.394944	−	0.918705i	$-0.629236\pi$
$19$	−3.44304	−0.789888	−0.394944	+	0.918705i	$0.629236\pi$
$20$	0	0
$21$	0.870704	0.190003
$22$	0	0
$23$	−5.81872	−1.21329	−0.606644	−	0.794974i	$-0.707485\pi$
$23$	−5.81872	−1.21329	−0.606644	+	0.794974i	$0.707485\pi$
$24$	0	0
$25$	−1.74994	−0.349988
$26$	0	0
$27$	5.26824	1.01387
$28$	0	0
$29$	−3.06392	−0.568955	−0.284477	−	0.958683i	$-0.591820\pi$
$29$	−3.06392	−0.568955	−0.284477	+	0.958683i	$0.591820\pi$
$30$	0	0
$31$	0.190243	0.0341687	0.0170844	−	0.999854i	$-0.494562\pi$
$31$	0.190243	0.0341687	0.0170844	+	0.999854i	$0.494562\pi$
$32$	0	0
$33$	−5.02519	−0.874773
$34$	0	0
$35$	−0.919601	−0.155441
$36$	0	0
$37$	−2.93430	−0.482396	−0.241198	−	0.970476i	$-0.577540\pi$
$37$	−2.93430	−0.482396	−0.241198	+	0.970476i	$0.577540\pi$
$38$	0	0
$39$	3.64144	0.583097
$40$	0	0
$41$	−5.04756	−0.788297	−0.394148	−	0.919047i	$-0.628960\pi$
$41$	−5.04756	−0.788297	−0.394148	+	0.919047i	$0.628960\pi$
$42$	0	0
$43$	4.54833	0.693613	0.346807	−	0.937937i	$-0.387266\pi$
$43$	4.54833	0.693613	0.346807	+	0.937937i	$0.387266\pi$
$44$	0	0
$45$	−0.155708	−0.0232116
$46$	0	0
$47$	3.80418	0.554897	0.277449	−	0.960741i	$-0.410511\pi$
$47$	3.80418	0.554897	0.277449	+	0.960741i	$0.410511\pi$
$48$	0	0
$49$	−6.73980	−0.962829
$50$	0	0
$51$	8.65551	1.21201
$52$	0	0
$53$	12.5215	1.71996	0.859980	−	0.510328i	$-0.170476\pi$
$53$	12.5215	1.71996	0.859980	+	0.510328i	$0.170476\pi$
$54$	0	0
$55$	5.30739	0.715648
$56$	0	0
$57$	5.87705	0.778434
$58$	0	0
$59$	7.35634	0.957713	0.478857	−	0.877893i	$-0.341051\pi$
$59$	7.35634	0.957713	0.478857	+	0.877893i	$0.341051\pi$
$60$	0	0
$61$	−6.72949	−0.861622	−0.430811	−	0.902442i	$-0.641772\pi$
$61$	−6.72949	−0.861622	−0.430811	+	0.902442i	$0.641772\pi$
$62$	0	0
$63$	0.0440574	0.00555071
$64$	0	0
$65$	−3.84593	−0.477029
$66$	0	0
$67$	−1.34776	−0.164655	−0.0823274	−	0.996605i	$-0.526235\pi$
$67$	−1.34776	−0.164655	−0.0823274	+	0.996605i	$0.526235\pi$
$68$	0	0
$69$	9.93218	1.19569
$70$	0	0
$71$	5.95576	0.706819	0.353409	−	0.935469i	$-0.385022\pi$
$71$	5.95576	0.706819	0.353409	+	0.935469i	$0.385022\pi$
$72$	0	0
$73$	12.5647	1.47059	0.735293	−	0.677750i	$-0.237045\pi$
$73$	12.5647	1.47059	0.735293	+	0.677750i	$0.237045\pi$
$74$	0	0
$75$	2.98703	0.344913
$76$	0	0
$77$	−1.50172	−0.171137
$78$	0	0
$79$	16.9929	1.91185	0.955923	−	0.293616i	$-0.0948588\pi$
$79$	16.9929	1.91185	0.955923	+	0.293616i	$0.0948588\pi$
$80$	0	0
$81$	−8.73343	−0.970381
$82$	0	0
$83$	−10.9781	−1.20500	−0.602502	−	0.798117i	$-0.705830\pi$
$83$	−10.9781	−1.20500	−0.602502	+	0.798117i	$0.705830\pi$
$84$	0	0
$85$	−9.14159	−0.991544
$86$	0	0
$87$	5.22991	0.560705
$88$	0	0
$89$	11.2470	1.19218	0.596092	−	0.802916i	$-0.296719\pi$
$89$	11.2470	1.19218	0.596092	+	0.802916i	$0.296719\pi$
$90$	0	0
$91$	1.08820	0.114075
$92$	0	0
$93$	−0.324733	−0.0336733
$94$	0	0
$95$	−6.20709	−0.636834
$96$	0	0
$97$	8.33626	0.846419	0.423209	−	0.906032i	$-0.360903\pi$
$97$	8.33626	0.846419	0.423209	+	0.906032i	$0.360903\pi$
$98$	0	0
$99$	−0.254273	−0.0255554
$100$	0	0
$101$	−12.7316	−1.26684	−0.633419	−	0.773809i	$-0.718349\pi$
$101$	−12.7316	−1.26684	−0.633419	+	0.773809i	$0.718349\pi$
$102$	0	0
$103$	1.49095	0.146907	0.0734537	−	0.997299i	$-0.476598\pi$
$103$	1.49095	0.146907	0.0734537	+	0.997299i	$0.476598\pi$
$104$	0	0
$105$	1.56970	0.153187
$106$	0	0
$107$	5.19086	0.501819	0.250910	−	0.968011i	$-0.419270\pi$
$107$	5.19086	0.501819	0.250910	+	0.968011i	$0.419270\pi$
$108$	0	0
$109$	2.56164	0.245360	0.122680	−	0.992446i	$-0.460851\pi$
$109$	2.56164	0.245360	0.122680	+	0.992446i	$0.460851\pi$
$110$	0	0
$111$	5.00866	0.475401
$112$	0	0
$113$	9.19627	0.865112	0.432556	−	0.901607i	$-0.357612\pi$
$113$	9.19627	0.865112	0.432556	+	0.901607i	$0.357612\pi$
$114$	0	0
$115$	−10.4899	−0.978193
$116$	0	0
$117$	0.184256	0.0170345
$118$	0	0
$119$	2.58660	0.237113
$120$	0	0
$121$	−2.33298	−0.212089
$122$	0	0
$123$	8.61587	0.776866
$124$	0	0
$125$	−12.1687	−1.08841
$126$	0	0
$127$	8.76003	0.777327	0.388663	−	0.921380i	$-0.372937\pi$
$127$	8.76003	0.777327	0.388663	+	0.921380i	$0.372937\pi$
$128$	0	0
$129$	−7.76370	−0.683556
$130$	0	0
$131$	0.841996	0.0735655	0.0367828	−	0.999323i	$-0.488289\pi$
$131$	0.841996	0.0735655	0.0367828	+	0.999323i	$0.488289\pi$
$132$	0	0
$133$	1.75629	0.152289
$134$	0	0
$135$	9.49754	0.817418
$136$	0	0
$137$	−9.80482	−0.837682	−0.418841	−	0.908060i	$-0.637564\pi$
$137$	−9.80482	−0.837682	−0.418841	+	0.908060i	$0.637564\pi$
$138$	0	0
$139$	13.2429	1.12325	0.561625	−	0.827392i	$-0.310176\pi$
$139$	13.2429	1.12325	0.561625	+	0.827392i	$0.310176\pi$
$140$	0	0
$141$	−6.49349	−0.546851
$142$	0	0
$143$	−6.28045	−0.525198
$144$	0	0
$145$	−5.52361	−0.458711
$146$	0	0
$147$	11.5044	0.948867
$148$	0	0
$149$	−6.79975	−0.557057	−0.278529	−	0.960428i	$-0.589847\pi$
$149$	−6.79975	−0.557057	−0.278529	+	0.960428i	$0.589847\pi$
$150$	0	0
$151$	−14.4526	−1.17614	−0.588068	−	0.808812i	$-0.700111\pi$
$151$	−14.4526	−1.17614	−0.588068	+	0.808812i	$0.700111\pi$
$152$	0	0
$153$	0.437967	0.0354075
$154$	0	0
$155$	0.342969	0.0275480
$156$	0	0
$157$	−4.31186	−0.344124	−0.172062	−	0.985086i	$-0.555043\pi$
$157$	−4.31186	−0.344124	−0.172062	+	0.985086i	$0.555043\pi$
$158$	0	0
$159$	−21.3734	−1.69502
$160$	0	0
$161$	2.96812	0.233920
$162$	0	0
$163$	−7.51400	−0.588542	−0.294271	−	0.955722i	$-0.595077\pi$
$163$	−7.51400	−0.588542	−0.294271	+	0.955722i	$0.595077\pi$
$164$	0	0
$165$	−9.05937	−0.705271
$166$	0	0
$167$	9.41684	0.728697	0.364349	−	0.931263i	$-0.381292\pi$
$167$	9.41684	0.728697	0.364349	+	0.931263i	$0.381292\pi$
$168$	0	0
$169$	−8.44895	−0.649919
$170$	0	0
$171$	0.297377	0.0227410
$172$	0	0
$173$	−0.862438	−0.0655699	−0.0327850	−	0.999462i	$-0.510438\pi$
$173$	−0.862438	−0.0655699	−0.0327850	+	0.999462i	$0.510438\pi$
$174$	0	0
$175$	0.892639	0.0674772
$176$	0	0
$177$	−12.5568	−0.943826
$178$	0	0
$179$	25.3400	1.89400	0.947000	−	0.321234i	$-0.104098\pi$
$179$	25.3400	1.89400	0.947000	+	0.321234i	$0.104098\pi$
$180$	0	0
$181$	7.97761	0.592971	0.296485	−	0.955037i	$-0.404185\pi$
$181$	7.97761	0.592971	0.296485	+	0.955037i	$0.404185\pi$
$182$	0	0
$183$	11.4868	0.849129
$184$	0	0
$185$	−5.28993	−0.388924
$186$	0	0
$187$	−14.9283	−1.09167
$188$	0	0
$189$	−2.68732	−0.195473
$190$	0	0
$191$	10.4265	0.754432	0.377216	−	0.926125i	$-0.376881\pi$
$191$	10.4265	0.754432	0.377216	+	0.926125i	$0.376881\pi$
$192$	0	0
$193$	24.6499	1.77434	0.887169	−	0.461445i	$-0.152669\pi$
$193$	24.6499	1.77434	0.887169	+	0.461445i	$0.152669\pi$
$194$	0	0
$195$	6.56476	0.470112
$196$	0	0
$197$	4.23749	0.301909	0.150954	−	0.988541i	$-0.451765\pi$
$197$	4.23749	0.301909	0.150954	+	0.988541i	$0.451765\pi$
$198$	0	0
$199$	25.0011	1.77228	0.886140	−	0.463418i	$-0.153377\pi$
$199$	25.0011	1.77228	0.886140	+	0.463418i	$0.153377\pi$
$200$	0	0
$201$	2.30053	0.162267
$202$	0	0
$203$	1.56290	0.109694
$204$	0	0
$205$	−9.09971	−0.635551
$206$	0	0
$207$	0.502566	0.0349307
$208$	0	0
$209$	−10.1362	−0.701139
$210$	0	0
$211$	0.749241	0.0515799	0.0257899	−	0.999667i	$-0.491790\pi$
$211$	0.749241	0.0515799	0.0257899	+	0.999667i	$0.491790\pi$
$212$	0	0
$213$	−10.1661	−0.696570
$214$	0	0
$215$	8.19969	0.559214
$216$	0	0
$217$	−0.0970428	−0.00658769
$218$	0	0
$219$	−21.4471	−1.44926
$220$	0	0
$221$	10.8176	0.727672
$222$	0	0
$223$	2.29984	0.154009	0.0770043	−	0.997031i	$-0.475464\pi$
$223$	2.29984	0.154009	0.0770043	+	0.997031i	$0.475464\pi$
$224$	0	0
$225$	0.151143	0.0100762
$226$	0	0
$227$	11.0444	0.733045	0.366522	−	0.930409i	$-0.380548\pi$
$227$	11.0444	0.733045	0.366522	+	0.930409i	$0.380548\pi$
$228$	0	0
$229$	−4.79281	−0.316718	−0.158359	−	0.987382i	$-0.550620\pi$
$229$	−4.79281	−0.316718	−0.158359	+	0.987382i	$0.550620\pi$
$230$	0	0
$231$	2.56334	0.168655
$232$	0	0
$233$	3.15821	0.206901	0.103451	−	0.994635i	$-0.467012\pi$
$233$	3.15821	0.206901	0.103451	+	0.994635i	$0.467012\pi$
$234$	0	0
$235$	6.85815	0.447377
$236$	0	0
$237$	−29.0057	−1.88412
$238$	0	0
$239$	−11.9311	−0.771762	−0.385881	−	0.922549i	$-0.626102\pi$
$239$	−11.9311	−0.771762	−0.385881	+	0.922549i	$0.626102\pi$
$240$	0	0
$241$	5.12934	0.330410	0.165205	−	0.986259i	$-0.447171\pi$
$241$	5.12934	0.330410	0.165205	+	0.986259i	$0.447171\pi$
$242$	0	0
$243$	−0.897307	−0.0575623
$244$	0	0
$245$	−12.1505	−0.776265
$246$	0	0
$247$	7.34511	0.467358
$248$	0	0
$249$	18.7389	1.18753
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−17.1302	−1.07697
$254$	0	0
$255$	15.6041	0.977167
$256$	0	0
$257$	−25.4413	−1.58698	−0.793491	−	0.608582i	$-0.791739\pi$
$257$	−25.4413	−1.58698	−0.793491	+	0.608582i	$0.791739\pi$
$258$	0	0
$259$	1.49678	0.0930054
$260$	0	0
$261$	0.264632	0.0163803
$262$	0	0
$263$	−27.9556	−1.72382	−0.861909	−	0.507063i	$-0.830731\pi$
$263$	−27.9556	−1.72382	−0.861909	+	0.507063i	$0.830731\pi$
$264$	0	0
$265$	22.5737	1.38669
$266$	0	0
$267$	−19.1980	−1.17490
$268$	0	0
$269$	−28.4442	−1.73427	−0.867137	−	0.498070i	$-0.834042\pi$
$269$	−28.4442	−1.73427	−0.867137	+	0.498070i	$0.834042\pi$
$270$	0	0
$271$	0.0409792	0.00248931	0.00124466	−	0.999999i	$-0.499604\pi$
$271$	0.0409792	0.00248931	0.00124466	+	0.999999i	$0.499604\pi$
$272$	0	0
$273$	−1.85749	−0.112420
$274$	0	0
$275$	−5.15178	−0.310664
$276$	0	0
$277$	25.3982	1.52603	0.763014	−	0.646382i	$-0.223719\pi$
$277$	25.3982	1.52603	0.763014	+	0.646382i	$0.223719\pi$
$278$	0	0
$279$	−0.0164314	−0.000983723 0
$280$	0	0
$281$	11.8749	0.708396	0.354198	−	0.935170i	$-0.384754\pi$
$281$	11.8749	0.708396	0.354198	+	0.935170i	$0.384754\pi$
$282$	0	0
$283$	−12.1057	−0.719609	−0.359805	−	0.933028i	$-0.617157\pi$
$283$	−12.1057	−0.719609	−0.359805	+	0.933028i	$0.617157\pi$
$284$	0	0
$285$	10.5951	0.627600
$286$	0	0
$287$	2.57475	0.151983
$288$	0	0
$289$	8.71292	0.512525
$290$	0	0
$291$	−14.2295	−0.834145
$292$	0	0
$293$	16.2159	0.947344	0.473672	−	0.880701i	$-0.342928\pi$
$293$	16.2159	0.947344	0.473672	+	0.880701i	$0.342928\pi$
$294$	0	0
$295$	13.2620	0.772141
$296$	0	0
$297$	15.5096	0.899957
$298$	0	0
$299$	12.4132	0.717873
$300$	0	0
$301$	−2.32009	−0.133728
$302$	0	0
$303$	21.7320	1.24847
$304$	0	0
$305$	−12.1319	−0.694669
$306$	0	0
$307$	−17.8907	−1.02107	−0.510537	−	0.859856i	$-0.670553\pi$
$307$	−17.8907	−1.02107	−0.510537	+	0.859856i	$0.670553\pi$
$308$	0	0
$309$	−2.54495	−0.144777
$310$	0	0
$311$	7.99807	0.453529	0.226765	−	0.973950i	$-0.427185\pi$
$311$	7.99807	0.453529	0.226765	+	0.973950i	$0.427185\pi$
$312$	0	0
$313$	1.23243	0.0696609	0.0348305	−	0.999393i	$-0.488911\pi$
$313$	1.23243	0.0696609	0.0348305	+	0.999393i	$0.488911\pi$
$314$	0	0
$315$	0.0794264	0.00447517
$316$	0	0
$317$	21.5936	1.21282	0.606408	−	0.795154i	$-0.292610\pi$
$317$	21.5936	1.21282	0.606408	+	0.795154i	$0.292610\pi$
$318$	0	0
$319$	−9.02011	−0.505029
$320$	0	0
$321$	−8.86046	−0.494543
$322$	0	0
$323$	17.4589	0.971442
$324$	0	0
$325$	3.73318	0.207079
$326$	0	0
$327$	−4.37255	−0.241803
$328$	0	0
$329$	−1.94051	−0.106984
$330$	0	0
$331$	−14.5465	−0.799545	−0.399773	−	0.916614i	$-0.630911\pi$
$331$	−14.5465	−0.799545	−0.399773	+	0.916614i	$0.630911\pi$
$332$	0	0
$333$	0.253437	0.0138883
$334$	0	0
$335$	−2.42973	−0.132750
$336$	0	0
$337$	3.75730	0.204673	0.102337	−	0.994750i	$-0.467368\pi$
$337$	3.75730	0.204673	0.102337	+	0.994750i	$0.467368\pi$
$338$	0	0
$339$	−15.6974	−0.852568
$340$	0	0
$341$	0.560073	0.0303296
$342$	0	0
$343$	7.00864	0.378431
$344$	0	0
$345$	17.9057	0.964009
$346$	0	0
$347$	0.977278	0.0524631	0.0262315	−	0.999656i	$-0.491649\pi$
$347$	0.977278	0.0524631	0.0262315	+	0.999656i	$0.491649\pi$
$348$	0	0
$349$	−14.5104	−0.776724	−0.388362	−	0.921507i	$-0.626959\pi$
$349$	−14.5104	−0.776724	−0.388362	+	0.921507i	$0.626959\pi$
$350$	0	0
$351$	−11.2388	−0.599884
$352$	0	0
$353$	17.7728	0.945948	0.472974	−	0.881076i	$-0.343180\pi$
$353$	17.7728	0.945948	0.472974	+	0.881076i	$0.343180\pi$
$354$	0	0
$355$	10.7370	0.569861
$356$	0	0
$357$	−4.41516	−0.233675
$358$	0	0
$359$	−13.6438	−0.720092	−0.360046	−	0.932935i	$-0.617239\pi$
$359$	−13.6438	−0.720092	−0.360046	+	0.932935i	$0.617239\pi$
$360$	0	0
$361$	−7.14547	−0.376077
$362$	0	0
$363$	3.98224	0.209014
$364$	0	0
$365$	22.6515	1.18564
$366$	0	0
$367$	−12.7521	−0.665653	−0.332826	−	0.942988i	$-0.608002\pi$
$367$	−12.7521	−0.665653	−0.332826	+	0.942988i	$0.608002\pi$
$368$	0	0
$369$	0.435961	0.0226952
$370$	0	0
$371$	−6.38719	−0.331606
$372$	0	0
$373$	8.69065	0.449985	0.224992	−	0.974361i	$-0.427764\pi$
$373$	8.69065	0.449985	0.224992	+	0.974361i	$0.427764\pi$
$374$	0	0
$375$	20.7713	1.07262
$376$	0	0
$377$	6.53631	0.336637
$378$	0	0
$379$	10.7407	0.551715	0.275858	−	0.961199i	$-0.411038\pi$
$379$	10.7407	0.551715	0.275858	+	0.961199i	$0.411038\pi$
$380$	0	0
$381$	−14.9528	−0.766055
$382$	0	0
$383$	24.7638	1.26537	0.632686	−	0.774408i	$-0.281952\pi$
$383$	24.7638	1.26537	0.632686	+	0.774408i	$0.281952\pi$
$384$	0	0
$385$	−2.70729	−0.137976
$386$	0	0
$387$	−0.392841	−0.0199692
$388$	0	0
$389$	−5.99654	−0.304037	−0.152018	−	0.988378i	$-0.548577\pi$
$389$	−5.99654	−0.304037	−0.152018	+	0.988378i	$0.548577\pi$
$390$	0	0
$391$	29.5055	1.49216
$392$	0	0
$393$	−1.43723	−0.0724988
$394$	0	0
$395$	30.6346	1.54139
$396$	0	0
$397$	−0.934048	−0.0468785	−0.0234393	−	0.999725i	$-0.507462\pi$
$397$	−0.934048	−0.0468785	−0.0234393	+	0.999725i	$0.507462\pi$
$398$	0	0
$399$	−2.99787	−0.150081
$400$	0	0
$401$	20.0509	1.00129	0.500646	−	0.865652i	$-0.333096\pi$
$401$	20.0509	1.00129	0.500646	+	0.865652i	$0.333096\pi$
$402$	0	0
$403$	−0.405850	−0.0202168
$404$	0	0
$405$	−15.7446	−0.782354
$406$	0	0
$407$	−8.63852	−0.428196
$408$	0	0
$409$	9.30135	0.459922	0.229961	−	0.973200i	$-0.426140\pi$
$409$	9.30135	0.459922	0.229961	+	0.973200i	$0.426140\pi$
$410$	0	0
$411$	16.7362	0.825535
$412$	0	0
$413$	−3.75245	−0.184646
$414$	0	0
$415$	−19.7913	−0.971515
$416$	0	0
$417$	−22.6048	−1.10696
$418$	0	0
$419$	−26.5745	−1.29825	−0.649124	−	0.760683i	$-0.724864\pi$
$419$	−26.5745	−1.29825	−0.649124	+	0.760683i	$0.724864\pi$
$420$	0	0
$421$	2.40663	0.117292	0.0586459	−	0.998279i	$-0.481322\pi$
$421$	2.40663	0.117292	0.0586459	+	0.998279i	$0.481322\pi$
$422$	0	0
$423$	−0.328569	−0.0159756
$424$	0	0
$425$	8.87357	0.430431
$426$	0	0
$427$	3.43270	0.166120
$428$	0	0
$429$	10.7203	0.517582
$430$	0	0
$431$	26.0054	1.25263	0.626317	−	0.779568i	$-0.284561\pi$
$431$	26.0054	1.25263	0.626317	+	0.779568i	$0.284561\pi$
$432$	0	0
$433$	26.5326	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$433$	26.5326	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$434$	0	0
$435$	9.42844	0.452059
$436$	0	0
$437$	20.0341	0.958361
$438$	0	0
$439$	−32.3029	−1.54173	−0.770866	−	0.636997i	$-0.780176\pi$
$439$	−32.3029	−1.54173	−0.770866	+	0.636997i	$0.780176\pi$
$440$	0	0
$441$	0.582120	0.0277200
$442$	0	0
$443$	6.73045	0.319773	0.159887	−	0.987135i	$-0.448887\pi$
$443$	6.73045	0.319773	0.159887	+	0.987135i	$0.448887\pi$
$444$	0	0
$445$	20.2761	0.961179
$446$	0	0
$447$	11.6067	0.548980
$448$	0	0
$449$	3.60685	0.170218	0.0851089	−	0.996372i	$-0.472876\pi$
$449$	3.60685	0.170218	0.0851089	+	0.996372i	$0.472876\pi$
$450$	0	0
$451$	−14.8599	−0.699727
$452$	0	0
$453$	24.6697	1.15908
$454$	0	0
$455$	1.96180	0.0919707
$456$	0	0
$457$	8.77713	0.410577	0.205289	−	0.978701i	$-0.434187\pi$
$457$	8.77713	0.410577	0.205289	+	0.978701i	$0.434187\pi$
$458$	0	0
$459$	−26.7141	−1.24691
$460$	0	0
$461$	28.1218	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$461$	28.1218	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$462$	0	0
$463$	36.1532	1.68018	0.840091	−	0.542445i	$-0.182501\pi$
$463$	36.1532	1.68018	0.840091	+	0.542445i	$0.182501\pi$
$464$	0	0
$465$	−0.585427	−0.0271485
$466$	0	0
$467$	24.3958	1.12890	0.564451	−	0.825467i	$-0.309088\pi$
$467$	24.3958	1.12890	0.564451	+	0.825467i	$0.309088\pi$
$468$	0	0
$469$	0.687488	0.0317453
$470$	0	0
$471$	7.36007	0.339134
$472$	0	0
$473$	13.3902	0.615681
$474$	0	0
$475$	6.02511	0.276451
$476$	0	0
$477$	−1.08149	−0.0495179
$478$	0	0
$479$	−3.95601	−0.180755	−0.0903773	−	0.995908i	$-0.528807\pi$
$479$	−3.95601	−0.180755	−0.0903773	+	0.995908i	$0.528807\pi$
$480$	0	0
$481$	6.25980	0.285422
$482$	0	0
$483$	−5.06639	−0.230529
$484$	0	0
$485$	15.0285	0.682411
$486$	0	0
$487$	38.7591	1.75634	0.878171	−	0.478347i	$-0.158764\pi$
$487$	38.7591	1.75634	0.878171	+	0.478347i	$0.158764\pi$
$488$	0	0
$489$	12.8259	0.580008
$490$	0	0
$491$	−29.5316	−1.33274	−0.666371	−	0.745621i	$-0.732153\pi$
$491$	−29.5316	−1.33274	−0.666371	+	0.745621i	$0.732153\pi$
$492$	0	0
$493$	15.5365	0.699728
$494$	0	0
$495$	−0.458402	−0.0206036
$496$	0	0
$497$	−3.03802	−0.136274
$498$	0	0
$499$	2.36668	0.105947	0.0529735	−	0.998596i	$-0.483130\pi$
$499$	2.36668	0.105947	0.0529735	+	0.998596i	$0.483130\pi$
$500$	0	0
$501$	−16.0739	−0.718131
$502$	0	0
$503$	28.4340	1.26781	0.633906	−	0.773410i	$-0.281451\pi$
$503$	28.4340	1.26781	0.633906	+	0.773410i	$0.281451\pi$
$504$	0	0
$505$	−22.9524	−1.02137
$506$	0	0
$507$	14.4218	0.640495
$508$	0	0
$509$	−30.8221	−1.36617	−0.683083	−	0.730341i	$-0.739361\pi$
$509$	−30.8221	−1.36617	−0.683083	+	0.730341i	$0.739361\pi$
$510$	0	0
$511$	−6.40922	−0.283527
$512$	0	0
$513$	−18.1388	−0.800846
$514$	0	0
$515$	2.68787	0.118442
$516$	0	0
$517$	11.1994	0.492551
$518$	0	0
$519$	1.47213	0.0646192
$520$	0	0
$521$	−14.8347	−0.649919	−0.324959	−	0.945728i	$-0.605351\pi$
$521$	−14.8347	−0.649919	−0.324959	+	0.945728i	$0.605351\pi$
$522$	0	0
$523$	19.7825	0.865028	0.432514	−	0.901627i	$-0.357627\pi$
$523$	19.7825	0.865028	0.432514	+	0.901627i	$0.357627\pi$
$524$	0	0
$525$	−1.52368	−0.0664988
$526$	0	0
$527$	−0.964685	−0.0420223
$528$	0	0
$529$	10.8575	0.472066
$530$	0	0
$531$	−0.635371	−0.0275727
$532$	0	0
$533$	10.7681	0.466417
$534$	0	0
$535$	9.35804	0.404584
$536$	0	0
$537$	−43.2537	−1.86654
$538$	0	0
$539$	−19.8418	−0.854649
$540$	0	0
$541$	3.56598	0.153313	0.0766566	−	0.997058i	$-0.475575\pi$
$541$	3.56598	0.153313	0.0766566	+	0.997058i	$0.475575\pi$
$542$	0	0
$543$	−13.6173	−0.584373
$544$	0	0
$545$	4.61810	0.197818
$546$	0	0
$547$	15.0952	0.645425	0.322712	−	0.946497i	$-0.395405\pi$
$547$	15.0952	0.645425	0.322712	+	0.946497i	$0.395405\pi$
$548$	0	0
$549$	0.581229	0.0248063
$550$	0	0
$551$	10.5492	0.449411
$552$	0	0
$553$	−8.66802	−0.368602
$554$	0	0
$555$	9.02958	0.383284
$556$	0	0
$557$	−20.4176	−0.865120	−0.432560	−	0.901605i	$-0.642390\pi$
$557$	−20.4176	−0.865120	−0.432560	+	0.901605i	$0.642390\pi$
$558$	0	0
$559$	−9.70303	−0.410394
$560$	0	0
$561$	25.4817	1.07584
$562$	0	0
$563$	21.5646	0.908840	0.454420	−	0.890787i	$-0.349846\pi$
$563$	21.5646	0.908840	0.454420	+	0.890787i	$0.349846\pi$
$564$	0	0
$565$	16.5790	0.697482
$566$	0	0
$567$	4.45490	0.187088
$568$	0	0
$569$	4.13101	0.173181	0.0865904	−	0.996244i	$-0.472403\pi$
$569$	4.13101	0.173181	0.0865904	+	0.996244i	$0.472403\pi$
$570$	0	0
$571$	36.9429	1.54601	0.773005	−	0.634400i	$-0.218753\pi$
$571$	36.9429	1.54601	0.773005	+	0.634400i	$0.218753\pi$
$572$	0	0
$573$	−17.7973	−0.743492
$574$	0	0
$575$	10.1824	0.424635
$576$	0	0
$577$	31.9151	1.32864	0.664322	−	0.747447i	$-0.268720\pi$
$577$	31.9151	1.32864	0.664322	+	0.747447i	$0.268720\pi$
$578$	0	0
$579$	−42.0758	−1.74861
$580$	0	0
$581$	5.59991	0.232323
$582$	0	0
$583$	36.8630	1.52671
$584$	0	0
$585$	0.332175	0.0137338
$586$	0	0
$587$	−36.2591	−1.49657	−0.748286	−	0.663376i	$-0.769123\pi$
$587$	−36.2591	−1.49657	−0.748286	+	0.663376i	$0.769123\pi$
$588$	0	0
$589$	−0.655016	−0.0269895
$590$	0	0
$591$	−7.23313	−0.297531
$592$	0	0
$593$	48.0786	1.97435	0.987176	−	0.159634i	$-0.0510314\pi$
$593$	48.0786	1.97435	0.987176	+	0.159634i	$0.0510314\pi$
$594$	0	0
$595$	4.66310	0.191169
$596$	0	0
$597$	−42.6752	−1.74658
$598$	0	0
$599$	11.3398	0.463333	0.231666	−	0.972795i	$-0.425582\pi$
$599$	11.3398	0.463333	0.231666	+	0.972795i	$0.425582\pi$
$600$	0	0
$601$	45.7525	1.86628	0.933142	−	0.359507i	$-0.117055\pi$
$601$	45.7525	1.86628	0.933142	+	0.359507i	$0.117055\pi$
$602$	0	0
$603$	0.116406	0.00474044
$604$	0	0
$605$	−4.20588	−0.170993
$606$	0	0
$607$	−7.92631	−0.321719	−0.160860	−	0.986977i	$-0.551427\pi$
$607$	−7.92631	−0.321719	−0.160860	+	0.986977i	$0.551427\pi$
$608$	0	0
$609$	−2.66776	−0.108103
$610$	0	0
$611$	−8.11554	−0.328319
$612$	0	0
$613$	−38.2769	−1.54599	−0.772994	−	0.634414i	$-0.781242\pi$
$613$	−38.2769	−1.54599	−0.772994	+	0.634414i	$0.781242\pi$
$614$	0	0
$615$	15.5326	0.626336
$616$	0	0
$617$	−44.6532	−1.79767	−0.898834	−	0.438289i	$-0.855585\pi$
$617$	−44.6532	−1.79767	−0.898834	+	0.438289i	$0.855585\pi$
$618$	0	0
$619$	−24.9899	−1.00443	−0.502214	−	0.864743i	$-0.667481\pi$
$619$	−24.9899	−1.00443	−0.502214	+	0.864743i	$0.667481\pi$
$620$	0	0
$621$	−30.6544	−1.23012
$622$	0	0
$623$	−5.73709	−0.229852
$624$	0	0
$625$	−13.1880	−0.527521
$626$	0	0
$627$	17.3019	0.690972
$628$	0	0
$629$	14.8792	0.593273
$630$	0	0
$631$	44.0995	1.75557	0.877787	−	0.479051i	$-0.159019\pi$
$631$	44.0995	1.75557	0.877787	+	0.479051i	$0.159019\pi$
$632$	0	0
$633$	−1.27891	−0.0508319
$634$	0	0
$635$	15.7925	0.626707
$636$	0	0
$637$	14.3781	0.569683
$638$	0	0
$639$	−0.514402	−0.0203494
$640$	0	0
$641$	−29.9919	−1.18461	−0.592305	−	0.805714i	$-0.701782\pi$
$641$	−29.9919	−1.18461	−0.592305	+	0.805714i	$0.701782\pi$
$642$	0	0
$643$	−9.88201	−0.389708	−0.194854	−	0.980832i	$-0.562423\pi$
$643$	−9.88201	−0.389708	−0.194854	+	0.980832i	$0.562423\pi$
$644$	0	0
$645$	−13.9963	−0.551105
$646$	0	0
$647$	−8.89154	−0.349562	−0.174781	−	0.984607i	$-0.555922\pi$
$647$	−8.89154	−0.349562	−0.174781	+	0.984607i	$0.555922\pi$
$648$	0	0
$649$	21.6569	0.850108
$650$	0	0
$651$	0.165646	0.00649217
$652$	0	0
$653$	18.9096	0.739988	0.369994	−	0.929034i	$-0.379360\pi$
$653$	18.9096	0.739988	0.369994	+	0.929034i	$0.379360\pi$
$654$	0	0
$655$	1.51794	0.0593110
$656$	0	0
$657$	−1.08522	−0.0423384
$658$	0	0
$659$	−36.5532	−1.42391	−0.711955	−	0.702225i	$-0.752190\pi$
$659$	−36.5532	−1.42391	−0.711955	+	0.702225i	$0.752190\pi$
$660$	0	0
$661$	42.1217	1.63835	0.819173	−	0.573547i	$-0.194433\pi$
$661$	42.1217	1.63835	0.819173	+	0.573547i	$0.194433\pi$
$662$	0	0
$663$	−18.4650	−0.717120
$664$	0	0
$665$	3.16622	0.122781
$666$	0	0
$667$	17.8281	0.690306
$668$	0	0
$669$	−3.92567	−0.151775
$670$	0	0
$671$	−19.8115	−0.764813
$672$	0	0
$673$	−31.5283	−1.21533	−0.607663	−	0.794195i	$-0.707893\pi$
$673$	−31.5283	−1.21533	−0.607663	+	0.794195i	$0.707893\pi$
$674$	0	0
$675$	−9.21908	−0.354843
$676$	0	0
$677$	−41.1644	−1.58208	−0.791039	−	0.611766i	$-0.790460\pi$
$677$	−41.1644	−1.58208	−0.791039	+	0.611766i	$0.790460\pi$
$678$	0	0
$679$	−4.25231	−0.163189
$680$	0	0
$681$	−18.8521	−0.722415
$682$	0	0
$683$	−4.40323	−0.168485	−0.0842424	−	0.996445i	$-0.526847\pi$
$683$	−4.40323	−0.168485	−0.0842424	+	0.996445i	$0.526847\pi$
$684$	0	0
$685$	−17.6761	−0.675367
$686$	0	0
$687$	8.18103	0.312126
$688$	0	0
$689$	−26.7124	−1.01766
$690$	0	0
$691$	−29.7836	−1.13302	−0.566511	−	0.824054i	$-0.691707\pi$
$691$	−29.7836	−1.13302	−0.566511	+	0.824054i	$0.691707\pi$
$692$	0	0
$693$	0.129704	0.00492705
$694$	0	0
$695$	23.8742	0.905601
$696$	0	0
$697$	25.5951	0.969485
$698$	0	0
$699$	−5.39086	−0.203901
$700$	0	0
$701$	15.5973	0.589102	0.294551	−	0.955636i	$-0.404830\pi$
$701$	15.5973	0.589102	0.294551	+	0.955636i	$0.404830\pi$
$702$	0	0
$703$	10.1029	0.381039
$704$	0	0
$705$	−11.7064	−0.440890
$706$	0	0
$707$	6.49434	0.244245
$708$	0	0
$709$	−5.23409	−0.196570	−0.0982851	−	0.995158i	$-0.531336\pi$
$709$	−5.23409	−0.196570	−0.0982851	+	0.995158i	$0.531336\pi$
$710$	0	0
$711$	−1.46768	−0.0550424
$712$	0	0
$713$	−1.10697	−0.0414565
$714$	0	0
$715$	−11.3224	−0.423432
$716$	0	0
$717$	20.3657	0.760571
$718$	0	0
$719$	36.7349	1.36998	0.684990	−	0.728552i	$-0.259806\pi$
$719$	36.7349	1.36998	0.684990	+	0.728552i	$0.259806\pi$
$720$	0	0
$721$	−0.760529	−0.0283236
$722$	0	0
$723$	−8.75546	−0.325619
$724$	0	0
$725$	5.36166	0.199127
$726$	0	0
$727$	1.63921	0.0607948	0.0303974	−	0.999538i	$-0.490323\pi$
$727$	1.63921	0.0607948	0.0303974	+	0.999538i	$0.490323\pi$
$728$	0	0
$729$	27.7319	1.02711
$730$	0	0
$731$	−23.0636	−0.853038
$732$	0	0
$733$	15.0686	0.556571	0.278285	−	0.960498i	$-0.410234\pi$
$733$	15.0686	0.556571	0.278285	+	0.960498i	$0.410234\pi$
$734$	0	0
$735$	20.7401	0.765009
$736$	0	0
$737$	−3.96777	−0.146155
$738$	0	0
$739$	1.55364	0.0571515	0.0285757	−	0.999592i	$-0.490903\pi$
$739$	1.55364	0.0571515	0.0285757	+	0.999592i	$0.490903\pi$
$740$	0	0
$741$	−12.5376	−0.460581
$742$	0	0
$743$	13.0824	0.479945	0.239973	−	0.970780i	$-0.422862\pi$
$743$	13.0824	0.479945	0.239973	+	0.970780i	$0.422862\pi$
$744$	0	0
$745$	−12.2585	−0.449118
$746$	0	0
$747$	0.948185	0.0346923
$748$	0	0
$749$	−2.64785	−0.0967502
$750$	0	0
$751$	43.4637	1.58601	0.793006	−	0.609214i	$-0.208515\pi$
$751$	43.4637	1.58601	0.793006	+	0.609214i	$0.208515\pi$
$752$	0	0
$753$	−1.70694	−0.0622042
$754$	0	0
$755$	−26.0550	−0.948240
$756$	0	0
$757$	16.2327	0.589989	0.294995	−	0.955499i	$-0.404682\pi$
$757$	16.2327	0.589989	0.294995	+	0.955499i	$0.404682\pi$
$758$	0	0
$759$	29.2402	1.06135
$760$	0	0
$761$	11.5507	0.418714	0.209357	−	0.977839i	$-0.432863\pi$
$761$	11.5507	0.418714	0.209357	+	0.977839i	$0.432863\pi$
$762$	0	0
$763$	−1.30669	−0.0473052
$764$	0	0
$765$	0.789563	0.0285467
$766$	0	0
$767$	−15.6934	−0.566656
$768$	0	0
$769$	−30.0273	−1.08281	−0.541407	−	0.840761i	$-0.682108\pi$
$769$	−30.0273	−1.08281	−0.541407	+	0.840761i	$0.682108\pi$
$770$	0	0
$771$	43.4266	1.56397
$772$	0	0
$773$	21.5093	0.773635	0.386818	−	0.922156i	$-0.373574\pi$
$773$	21.5093	0.773635	0.386818	+	0.922156i	$0.373574\pi$
$774$	0	0
$775$	−0.332914	−0.0119586
$776$	0	0
$777$	−2.55491	−0.0916568
$778$	0	0
$779$	17.3790	0.622666
$780$	0	0
$781$	17.5336	0.627403
$782$	0	0
$783$	−16.1414	−0.576848
$784$	0	0
$785$	−7.77339	−0.277444
$786$	0	0
$787$	−25.4188	−0.906083	−0.453042	−	0.891489i	$-0.649661\pi$
$787$	−25.4188	−0.906083	−0.453042	+	0.891489i	$0.649661\pi$
$788$	0	0
$789$	47.7185	1.69882
$790$	0	0
$791$	−4.69100	−0.166793
$792$	0	0
$793$	14.3561	0.509802
$794$	0	0
$795$	−38.5318	−1.36658
$796$	0	0
$797$	35.4408	1.25538	0.627689	−	0.778465i	$-0.284001\pi$
$797$	35.4408	1.25538	0.627689	+	0.778465i	$0.284001\pi$
$798$	0	0
$799$	−19.2902	−0.682439
$800$	0	0
$801$	−0.971413	−0.0343232
$802$	0	0
$803$	36.9902	1.30536
$804$	0	0
$805$	5.35090	0.188594
$806$	0	0
$807$	48.5524	1.70913
$808$	0	0
$809$	−46.7921	−1.64512	−0.822561	−	0.568677i	$-0.807456\pi$
$809$	−46.7921	−1.64512	−0.822561	+	0.568677i	$0.807456\pi$
$810$	0	0
$811$	−33.5831	−1.17926	−0.589631	−	0.807672i	$-0.700727\pi$
$811$	−33.5831	−1.17926	−0.589631	+	0.807672i	$0.700727\pi$
$812$	0	0
$813$	−0.0699489	−0.00245321
$814$	0	0
$815$	−13.5462	−0.474502
$816$	0	0
$817$	−15.6601	−0.547877
$818$	0	0
$819$	−0.0939885	−0.00328423
$820$	0	0
$821$	37.0349	1.29253	0.646264	−	0.763114i	$-0.276330\pi$
$821$	37.0349	1.29253	0.646264	+	0.763114i	$0.276330\pi$
$822$	0	0
$823$	42.2697	1.47343	0.736714	−	0.676204i	$-0.236376\pi$
$823$	42.2697	1.47343	0.736714	+	0.676204i	$0.236376\pi$
$824$	0	0
$825$	8.79376	0.306159
$826$	0	0
$827$	−27.4850	−0.955747	−0.477873	−	0.878429i	$-0.658592\pi$
$827$	−27.4850	−0.955747	−0.477873	+	0.878429i	$0.658592\pi$
$828$	0	0
$829$	−23.3934	−0.812485	−0.406243	−	0.913765i	$-0.633161\pi$
$829$	−23.3934	−0.812485	−0.406243	+	0.913765i	$0.633161\pi$
$830$	0	0
$831$	−43.3530	−1.50390
$832$	0	0
$833$	34.1761	1.18413
$834$	0	0
$835$	16.9766	0.587500
$836$	0	0
$837$	1.00225	0.0346427
$838$	0	0
$839$	46.0748	1.59068	0.795339	−	0.606165i	$-0.207293\pi$
$839$	46.0748	1.59068	0.795339	+	0.606165i	$0.207293\pi$
$840$	0	0
$841$	−19.6124	−0.676290
$842$	0	0
$843$	−20.2697	−0.698124
$844$	0	0
$845$	−15.2317	−0.523987
$846$	0	0
$847$	1.19005	0.0408905
$848$	0	0
$849$	20.6637	0.709175
$850$	0	0
$851$	17.0739	0.585285
$852$	0	0
$853$	42.8485	1.46710	0.733552	−	0.679634i	$-0.237861\pi$
$853$	42.8485	1.46710	0.733552	+	0.679634i	$0.237861\pi$
$854$	0	0
$855$	0.536110	0.0183346
$856$	0	0
$857$	−54.0657	−1.84685	−0.923424	−	0.383780i	$-0.874622\pi$
$857$	−54.0657	−1.84685	−0.923424	+	0.383780i	$0.874622\pi$
$858$	0	0
$859$	−40.0762	−1.36738	−0.683691	−	0.729772i	$-0.739626\pi$
$859$	−40.0762	−1.36738	−0.683691	+	0.729772i	$0.739626\pi$
$860$	0	0
$861$	−4.39493	−0.149779
$862$	0	0
$863$	−6.16434	−0.209837	−0.104918	−	0.994481i	$-0.533458\pi$
$863$	−6.16434	−0.209837	−0.104918	+	0.994481i	$0.533458\pi$
$864$	0	0
$865$	−1.55480	−0.0528647
$866$	0	0
$867$	−14.8724	−0.505093
$868$	0	0
$869$	50.0267	1.69704
$870$	0	0
$871$	2.87520	0.0974223
$872$	0	0
$873$	−0.720007	−0.0243685
$874$	0	0
$875$	6.20725	0.209843
$876$	0	0
$877$	43.2953	1.46198	0.730990	−	0.682389i	$-0.239059\pi$
$877$	43.2953	1.46198	0.730990	+	0.682389i	$0.239059\pi$
$878$	0	0
$879$	−27.6795	−0.933607
$880$	0	0
$881$	−2.68142	−0.0903394	−0.0451697	−	0.998979i	$-0.514383\pi$
$881$	−2.68142	−0.0903394	−0.0451697	+	0.998979i	$0.514383\pi$
$882$	0	0
$883$	−10.1886	−0.342873	−0.171437	−	0.985195i	$-0.554841\pi$
$883$	−10.1886	−0.342873	−0.171437	+	0.985195i	$0.554841\pi$
$884$	0	0
$885$	−22.6373	−0.760944
$886$	0	0
$887$	−5.34303	−0.179402	−0.0897008	−	0.995969i	$-0.528591\pi$
$887$	−5.34303	−0.179402	−0.0897008	+	0.995969i	$0.528591\pi$
$888$	0	0
$889$	−4.46847	−0.149868
$890$	0	0
$891$	−25.7110	−0.861352
$892$	0	0
$893$	−13.0980	−0.438306
$894$	0	0
$895$	45.6828	1.52701
$896$	0	0
$897$	−21.1885	−0.707464
$898$	0	0
$899$	−0.582890	−0.0194405
$900$	0	0
$901$	−63.4939	−2.11529
$902$	0	0
$903$	3.96025	0.131789
$904$	0	0
$905$	14.3820	0.478073
$906$	0	0
$907$	−16.4584	−0.546492	−0.273246	−	0.961944i	$-0.588097\pi$
$907$	−16.4584	−0.546492	−0.273246	+	0.961944i	$0.588097\pi$
$908$	0	0
$909$	1.09963	0.0364725
$910$	0	0
$911$	37.5492	1.24406	0.622030	−	0.782993i	$-0.286308\pi$
$911$	37.5492	1.24406	0.622030	+	0.782993i	$0.286308\pi$
$912$	0	0
$913$	−32.3194	−1.06961
$914$	0	0
$915$	20.7083	0.684596
$916$	0	0
$917$	−0.429500	−0.0141834
$918$	0	0
$919$	−0.307556	−0.0101453	−0.00507267	−	0.999987i	$-0.501615\pi$
$919$	−0.307556	−0.0101453	−0.00507267	+	0.999987i	$0.501615\pi$
$920$	0	0
$921$	30.5382	1.00627
$922$	0	0
$923$	−12.7055	−0.418208
$924$	0	0
$925$	5.13484	0.168833
$926$	0	0
$927$	−0.128774	−0.00422949
$928$	0	0
$929$	27.4555	0.900787	0.450393	−	0.892830i	$-0.351284\pi$
$929$	27.4555	0.900787	0.450393	+	0.892830i	$0.351284\pi$
$930$	0	0
$931$	23.2054	0.760527
$932$	0	0
$933$	−13.6522	−0.446953
$934$	0	0
$935$	−26.9127	−0.880138
$936$	0	0
$937$	14.3128	0.467577	0.233789	−	0.972287i	$-0.424888\pi$
$937$	14.3128	0.467577	0.233789	+	0.972287i	$0.424888\pi$
$938$	0	0
$939$	−2.10367	−0.0686508
$940$	0	0
$941$	10.8257	0.352908	0.176454	−	0.984309i	$-0.443537\pi$
$941$	10.8257	0.352908	0.176454	+	0.984309i	$0.443537\pi$
$942$	0	0
$943$	29.3704	0.956431
$944$	0	0
$945$	−4.84467	−0.157597
$946$	0	0
$947$	43.8729	1.42568	0.712838	−	0.701329i	$-0.247409\pi$
$947$	43.8729	1.42568	0.712838	+	0.701329i	$0.247409\pi$
$948$	0	0
$949$	−26.8045	−0.870111
$950$	0	0
$951$	−36.8589	−1.19523
$952$	0	0
$953$	−14.2915	−0.462946	−0.231473	−	0.972841i	$-0.574355\pi$
$953$	−14.2915	−0.462946	−0.231473	+	0.972841i	$0.574355\pi$
$954$	0	0
$955$	18.7967	0.608248
$956$	0	0
$957$	15.3967	0.497706
$958$	0	0
$959$	5.00142	0.161504
$960$	0	0
$961$	−30.9638	−0.998832
$962$	0	0
$963$	−0.448337	−0.0144475
$964$	0	0
$965$	44.4386	1.43053
$966$	0	0
$967$	−9.38554	−0.301818	−0.150909	−	0.988548i	$-0.548220\pi$
$967$	−9.38554	−0.301818	−0.150909	+	0.988548i	$0.548220\pi$
$968$	0	0
$969$	−29.8013	−0.957355
$970$	0	0
$971$	−40.0932	−1.28665	−0.643327	−	0.765592i	$-0.722446\pi$
$971$	−40.0932	−1.28665	−0.643327	+	0.765592i	$0.722446\pi$
$972$	0	0
$973$	−6.75518	−0.216561
$974$	0	0
$975$	−6.37229	−0.204077
$976$	0	0
$977$	−9.17214	−0.293443	−0.146721	−	0.989178i	$-0.546872\pi$
$977$	−9.17214	−0.293443	−0.146721	+	0.989178i	$0.546872\pi$
$978$	0	0
$979$	33.1111	1.05823
$980$	0	0
$981$	−0.221250	−0.00706396
$982$	0	0
$983$	−7.57981	−0.241758	−0.120879	−	0.992667i	$-0.538571\pi$
$983$	−7.57981	−0.241758	−0.120879	+	0.992667i	$0.538571\pi$
$984$	0	0
$985$	7.63932	0.243409
$986$	0	0
$987$	3.31232	0.105432
$988$	0	0
$989$	−26.4654	−0.841552
$990$	0	0
$991$	−31.0770	−0.987192	−0.493596	−	0.869691i	$-0.664318\pi$
$991$	−31.0770	−0.987192	−0.493596	+	0.869691i	$0.664318\pi$
$992$	0	0
$993$	24.8299	0.787952
$994$	0	0
$995$	45.0718	1.42887
$996$	0	0
$997$	10.6893	0.338532	0.169266	−	0.985570i	$-0.445860\pi$
$997$	10.6893	0.338532	0.169266	+	0.985570i	$0.445860\pi$
$998$	0	0
$999$	−15.4586	−0.489088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.8	yes	30
4.3	odd	2		8032.2.a.h.1.23	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.23	✓	30	4.3	odd	2
8032.2.a.i.1.8	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70694 q^{3} +1.80279 q^{5} -0.510098 q^{7} -0.0863705 q^{9} +O(q^{10})\)	\(q-1.70694 q^{3} +1.80279 q^{5} -0.510098 q^{7} -0.0863705 q^{9} +2.94398 q^{11} -2.13332 q^{13} -3.07725 q^{15} -5.07079 q^{17} -3.44304 q^{19} +0.870704 q^{21} -5.81872 q^{23} -1.74994 q^{25} +5.26824 q^{27} -3.06392 q^{29} +0.190243 q^{31} -5.02519 q^{33} -0.919601 q^{35} -2.93430 q^{37} +3.64144 q^{39} -5.04756 q^{41} +4.54833 q^{43} -0.155708 q^{45} +3.80418 q^{47} -6.73980 q^{49} +8.65551 q^{51} +12.5215 q^{53} +5.30739 q^{55} +5.87705 q^{57} +7.35634 q^{59} -6.72949 q^{61} +0.0440574 q^{63} -3.84593 q^{65} -1.34776 q^{67} +9.93218 q^{69} +5.95576 q^{71} +12.5647 q^{73} +2.98703 q^{75} -1.50172 q^{77} +16.9929 q^{79} -8.73343 q^{81} -10.9781 q^{83} -9.14159 q^{85} +5.22991 q^{87} +11.2470 q^{89} +1.08820 q^{91} -0.324733 q^{93} -6.20709 q^{95} +8.33626 q^{97} -0.254273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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