LMFDB - Embedded newform 8048.2.a.x.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.885832 q^{3} -0.282055 q^{5} +2.95093 q^{7} -2.21530 q^{9} +O(q^{10})$	$q-0.885832 q^{3} -0.282055 q^{5} +2.95093 q^{7} -2.21530 q^{9} +5.51650 q^{11} +6.01407 q^{13} +0.249853 q^{15} -3.87741 q^{17} -2.94287 q^{19} -2.61403 q^{21} -7.16204 q^{23} -4.92044 q^{25} +4.61988 q^{27} +8.17785 q^{29} -8.64136 q^{31} -4.88669 q^{33} -0.832326 q^{35} -7.02457 q^{37} -5.32745 q^{39} -4.63817 q^{41} +1.94637 q^{43} +0.624837 q^{45} +6.63886 q^{47} +1.70800 q^{49} +3.43474 q^{51} -5.95148 q^{53} -1.55596 q^{55} +2.60689 q^{57} -12.0529 q^{59} +1.90056 q^{61} -6.53721 q^{63} -1.69630 q^{65} -8.14767 q^{67} +6.34436 q^{69} -15.4828 q^{71} -7.81736 q^{73} +4.35869 q^{75} +16.2788 q^{77} -1.70982 q^{79} +2.55347 q^{81} -7.62297 q^{83} +1.09364 q^{85} -7.24420 q^{87} +0.476640 q^{89} +17.7471 q^{91} +7.65479 q^{93} +0.830052 q^{95} -2.06521 q^{97} -12.2207 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.885832	−0.511435	−0.255718	−	0.966752i	$-0.582312\pi$
$3$	−0.885832	−0.511435	−0.255718	+	0.966752i	$0.582312\pi$
$4$	0	0
$5$	−0.282055	−0.126139	−0.0630695	−	0.998009i	$-0.520089\pi$
$5$	−0.282055	−0.126139	−0.0630695	+	0.998009i	$0.520089\pi$
$6$	0	0
$7$	2.95093	1.11535	0.557674	−	0.830060i	$-0.311694\pi$
$7$	2.95093	1.11535	0.557674	+	0.830060i	$0.311694\pi$
$8$	0	0
$9$	−2.21530	−0.738434
$10$	0	0
$11$	5.51650	1.66329	0.831643	−	0.555311i	$-0.187401\pi$
$11$	5.51650	1.66329	0.831643	+	0.555311i	$0.187401\pi$
$12$	0	0
$13$	6.01407	1.66800	0.834001	−	0.551762i	$-0.186044\pi$
$13$	6.01407	1.66800	0.834001	+	0.551762i	$0.186044\pi$
$14$	0	0
$15$	0.249853	0.0645119
$16$	0	0
$17$	−3.87741	−0.940411	−0.470206	−	0.882557i	$-0.655820\pi$
$17$	−3.87741	−0.940411	−0.470206	+	0.882557i	$0.655820\pi$
$18$	0	0
$19$	−2.94287	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$19$	−2.94287	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$20$	0	0
$21$	−2.61403	−0.570428
$22$	0	0
$23$	−7.16204	−1.49339	−0.746694	−	0.665168i	$-0.768360\pi$
$23$	−7.16204	−1.49339	−0.746694	+	0.665168i	$0.768360\pi$
$24$	0	0
$25$	−4.92044	−0.984089
$26$	0	0
$27$	4.61988	0.889096
$28$	0	0
$29$	8.17785	1.51859	0.759294	−	0.650747i	$-0.225544\pi$
$29$	8.17785	1.51859	0.759294	+	0.650747i	$0.225544\pi$
$30$	0	0
$31$	−8.64136	−1.55203	−0.776017	−	0.630712i	$-0.782763\pi$
$31$	−8.64136	−1.55203	−0.776017	+	0.630712i	$0.782763\pi$
$32$	0	0
$33$	−4.88669	−0.850663
$34$	0	0
$35$	−0.832326	−0.140689
$36$	0	0
$37$	−7.02457	−1.15483	−0.577416	−	0.816450i	$-0.695939\pi$
$37$	−7.02457	−1.15483	−0.577416	+	0.816450i	$0.695939\pi$
$38$	0	0
$39$	−5.32745	−0.853075
$40$	0	0
$41$	−4.63817	−0.724361	−0.362180	−	0.932108i	$-0.617968\pi$
$41$	−4.63817	−0.724361	−0.362180	+	0.932108i	$0.617968\pi$
$42$	0	0
$43$	1.94637	0.296819	0.148410	−	0.988926i	$-0.452585\pi$
$43$	1.94637	0.296819	0.148410	+	0.988926i	$0.452585\pi$
$44$	0	0
$45$	0.624837	0.0931453
$46$	0	0
$47$	6.63886	0.968378	0.484189	−	0.874964i	$-0.339115\pi$
$47$	6.63886	0.968378	0.484189	+	0.874964i	$0.339115\pi$
$48$	0	0
$49$	1.70800	0.244000
$50$	0	0
$51$	3.43474	0.480959
$52$	0	0
$53$	−5.95148	−0.817499	−0.408750	−	0.912647i	$-0.634035\pi$
$53$	−5.95148	−0.817499	−0.408750	+	0.912647i	$0.634035\pi$
$54$	0	0
$55$	−1.55596	−0.209805
$56$	0	0
$57$	2.60689	0.345291
$58$	0	0
$59$	−12.0529	−1.56915	−0.784574	−	0.620035i	$-0.787118\pi$
$59$	−12.0529	−1.56915	−0.784574	+	0.620035i	$0.787118\pi$
$60$	0	0
$61$	1.90056	0.243341	0.121671	−	0.992571i	$-0.461175\pi$
$61$	1.90056	0.243341	0.121671	+	0.992571i	$0.461175\pi$
$62$	0	0
$63$	−6.53721	−0.823611
$64$	0	0
$65$	−1.69630	−0.210400
$66$	0	0
$67$	−8.14767	−0.995396	−0.497698	−	0.867350i	$-0.665821\pi$
$67$	−8.14767	−0.995396	−0.497698	+	0.867350i	$0.665821\pi$
$68$	0	0
$69$	6.34436	0.763771
$70$	0	0
$71$	−15.4828	−1.83748	−0.918738	−	0.394867i	$-0.870791\pi$
$71$	−15.4828	−1.83748	−0.918738	+	0.394867i	$0.870791\pi$
$72$	0	0
$73$	−7.81736	−0.914952	−0.457476	−	0.889222i	$-0.651246\pi$
$73$	−7.81736	−0.914952	−0.457476	+	0.889222i	$0.651246\pi$
$74$	0	0
$75$	4.35869	0.503298
$76$	0	0
$77$	16.2788	1.85514
$78$	0	0
$79$	−1.70982	−0.192370	−0.0961851	−	0.995363i	$-0.530664\pi$
$79$	−1.70982	−0.192370	−0.0961851	+	0.995363i	$0.530664\pi$
$80$	0	0
$81$	2.55347	0.283719
$82$	0	0
$83$	−7.62297	−0.836730	−0.418365	−	0.908279i	$-0.637397\pi$
$83$	−7.62297	−0.836730	−0.418365	+	0.908279i	$0.637397\pi$
$84$	0	0
$85$	1.09364	0.118622
$86$	0	0
$87$	−7.24420	−0.776660
$88$	0	0
$89$	0.476640	0.0505237	0.0252619	−	0.999681i	$-0.491958\pi$
$89$	0.476640	0.0505237	0.0252619	+	0.999681i	$0.491958\pi$
$90$	0	0
$91$	17.7471	1.86040
$92$	0	0
$93$	7.65479	0.793765
$94$	0	0
$95$	0.830052	0.0851615
$96$	0	0
$97$	−2.06521	−0.209690	−0.104845	−	0.994489i	$-0.533435\pi$
$97$	−2.06521	−0.209690	−0.104845	+	0.994489i	$0.533435\pi$
$98$	0	0
$99$	−12.2207	−1.22823
$100$	0	0
$101$	17.6251	1.75377	0.876884	−	0.480702i	$-0.159618\pi$
$101$	17.6251	1.75377	0.876884	+	0.480702i	$0.159618\pi$
$102$	0	0
$103$	−6.35126	−0.625809	−0.312904	−	0.949785i	$-0.601302\pi$
$103$	−6.35126	−0.625809	−0.312904	+	0.949785i	$0.601302\pi$
$104$	0	0
$105$	0.737301	0.0719532
$106$	0	0
$107$	−19.0273	−1.83944	−0.919720	−	0.392574i	$-0.871585\pi$
$107$	−19.0273	−1.83944	−0.919720	+	0.392574i	$0.871585\pi$
$108$	0	0
$109$	−5.72659	−0.548508	−0.274254	−	0.961657i	$-0.588431\pi$
$109$	−5.72659	−0.548508	−0.274254	+	0.961657i	$0.588431\pi$
$110$	0	0
$111$	6.22259	0.590622
$112$	0	0
$113$	11.0908	1.04334	0.521669	−	0.853148i	$-0.325310\pi$
$113$	11.0908	1.04334	0.521669	+	0.853148i	$0.325310\pi$
$114$	0	0
$115$	2.02009	0.188374
$116$	0	0
$117$	−13.3230	−1.23171
$118$	0	0
$119$	−11.4420	−1.04889
$120$	0	0
$121$	19.4317	1.76652
$122$	0	0
$123$	4.10864	0.370464
$124$	0	0
$125$	2.79811	0.250271
$126$	0	0
$127$	10.9820	0.974492	0.487246	−	0.873265i	$-0.338002\pi$
$127$	10.9820	0.974492	0.487246	+	0.873265i	$0.338002\pi$
$128$	0	0
$129$	−1.72416	−0.151804
$130$	0	0
$131$	−1.15872	−0.101238	−0.0506191	−	0.998718i	$-0.516119\pi$
$131$	−1.15872	−0.101238	−0.0506191	+	0.998718i	$0.516119\pi$
$132$	0	0
$133$	−8.68421	−0.753016
$134$	0	0
$135$	−1.30306	−0.112150
$136$	0	0
$137$	−5.35904	−0.457854	−0.228927	−	0.973444i	$-0.573522\pi$
$137$	−5.35904	−0.457854	−0.228927	+	0.973444i	$0.573522\pi$
$138$	0	0
$139$	−11.2043	−0.950336	−0.475168	−	0.879895i	$-0.657613\pi$
$139$	−11.2043	−0.950336	−0.475168	+	0.879895i	$0.657613\pi$
$140$	0	0
$141$	−5.88092	−0.495262
$142$	0	0
$143$	33.1766	2.77437
$144$	0	0
$145$	−2.30661	−0.191553
$146$	0	0
$147$	−1.51300	−0.124790
$148$	0	0
$149$	19.4390	1.59251	0.796254	−	0.604962i	$-0.206812\pi$
$149$	19.4390	1.59251	0.796254	+	0.604962i	$0.206812\pi$
$150$	0	0
$151$	7.78762	0.633747	0.316874	−	0.948468i	$-0.397367\pi$
$151$	7.78762	0.633747	0.316874	+	0.948468i	$0.397367\pi$
$152$	0	0
$153$	8.58964	0.694432
$154$	0	0
$155$	2.43734	0.195772
$156$	0	0
$157$	16.4637	1.31395	0.656975	−	0.753913i	$-0.271836\pi$
$157$	16.4637	1.31395	0.656975	+	0.753913i	$0.271836\pi$
$158$	0	0
$159$	5.27201	0.418098
$160$	0	0
$161$	−21.1347	−1.66565
$162$	0	0
$163$	−0.657264	−0.0514809	−0.0257404	−	0.999669i	$-0.508194\pi$
$163$	−0.657264	−0.0514809	−0.0257404	+	0.999669i	$0.508194\pi$
$164$	0	0
$165$	1.37832	0.107302
$166$	0	0
$167$	−12.9318	−1.00069	−0.500345	−	0.865826i	$-0.666793\pi$
$167$	−12.9318	−1.00069	−0.500345	+	0.865826i	$0.666793\pi$
$168$	0	0
$169$	23.1690	1.78223
$170$	0	0
$171$	6.51935	0.498547
$172$	0	0
$173$	16.6627	1.26684	0.633419	−	0.773809i	$-0.281651\pi$
$173$	16.6627	1.26684	0.633419	+	0.773809i	$0.281651\pi$
$174$	0	0
$175$	−14.5199	−1.09760
$176$	0	0
$177$	10.6768	0.802518
$178$	0	0
$179$	−19.3607	−1.44708	−0.723542	−	0.690281i	$-0.757487\pi$
$179$	−19.3607	−1.44708	−0.723542	+	0.690281i	$0.757487\pi$
$180$	0	0
$181$	−20.6750	−1.53676	−0.768379	−	0.639995i	$-0.778936\pi$
$181$	−20.6750	−1.53676	−0.768379	+	0.639995i	$0.778936\pi$
$182$	0	0
$183$	−1.68357	−0.124453
$184$	0	0
$185$	1.98132	0.145669
$186$	0	0
$187$	−21.3897	−1.56417
$188$	0	0
$189$	13.6330	0.991651
$190$	0	0
$191$	0.226364	0.0163792	0.00818958	−	0.999966i	$-0.497393\pi$
$191$	0.226364	0.0163792	0.00818958	+	0.999966i	$0.497393\pi$
$192$	0	0
$193$	4.06511	0.292613	0.146306	−	0.989239i	$-0.453261\pi$
$193$	4.06511	0.292613	0.146306	+	0.989239i	$0.453261\pi$
$194$	0	0
$195$	1.50264	0.107606
$196$	0	0
$197$	−2.54470	−0.181302	−0.0906510	−	0.995883i	$-0.528895\pi$
$197$	−2.54470	−0.181302	−0.0906510	+	0.995883i	$0.528895\pi$
$198$	0	0
$199$	5.37857	0.381277	0.190638	−	0.981660i	$-0.438944\pi$
$199$	5.37857	0.381277	0.190638	+	0.981660i	$0.438944\pi$
$200$	0	0
$201$	7.21746	0.509081
$202$	0	0
$203$	24.1323	1.69375
$204$	0	0
$205$	1.30822	0.0913701
$206$	0	0
$207$	15.8661	1.10277
$208$	0	0
$209$	−16.2343	−1.12295
$210$	0	0
$211$	22.2416	1.53118	0.765588	−	0.643331i	$-0.222448\pi$
$211$	22.2416	1.53118	0.765588	+	0.643331i	$0.222448\pi$
$212$	0	0
$213$	13.7152	0.939750
$214$	0	0
$215$	−0.548985	−0.0374405
$216$	0	0
$217$	−25.5001	−1.73106
$218$	0	0
$219$	6.92486	0.467939
$220$	0	0
$221$	−23.3190	−1.56861
$222$	0	0
$223$	4.05544	0.271572	0.135786	−	0.990738i	$-0.456644\pi$
$223$	4.05544	0.271572	0.135786	+	0.990738i	$0.456644\pi$
$224$	0	0
$225$	10.9003	0.726685
$226$	0	0
$227$	1.70236	0.112989	0.0564947	−	0.998403i	$-0.482008\pi$
$227$	1.70236	0.112989	0.0564947	+	0.998403i	$0.482008\pi$
$228$	0	0
$229$	26.2327	1.73351	0.866754	−	0.498735i	$-0.166202\pi$
$229$	26.2327	1.73351	0.866754	+	0.498735i	$0.166202\pi$
$230$	0	0
$231$	−14.4203	−0.948785
$232$	0	0
$233$	21.8557	1.43181	0.715906	−	0.698197i	$-0.246014\pi$
$233$	21.8557	1.43181	0.715906	+	0.698197i	$0.246014\pi$
$234$	0	0
$235$	−1.87253	−0.122150
$236$	0	0
$237$	1.51462	0.0983849
$238$	0	0
$239$	2.12013	0.137140	0.0685700	−	0.997646i	$-0.478156\pi$
$239$	2.12013	0.137140	0.0685700	+	0.997646i	$0.478156\pi$
$240$	0	0
$241$	−16.0051	−1.03098	−0.515490	−	0.856896i	$-0.672390\pi$
$241$	−16.0051	−1.03098	−0.515490	+	0.856896i	$0.672390\pi$
$242$	0	0
$243$	−16.1216	−1.03420
$244$	0	0
$245$	−0.481750	−0.0307779
$246$	0	0
$247$	−17.6986	−1.12614
$248$	0	0
$249$	6.75267	0.427933
$250$	0	0
$251$	4.72870	0.298473	0.149236	−	0.988802i	$-0.452318\pi$
$251$	4.72870	0.298473	0.149236	+	0.988802i	$0.452318\pi$
$252$	0	0
$253$	−39.5093	−2.48393
$254$	0	0
$255$	−0.968785	−0.0606677
$256$	0	0
$257$	−25.0582	−1.56309	−0.781545	−	0.623849i	$-0.785568\pi$
$257$	−25.0582	−1.56309	−0.781545	+	0.623849i	$0.785568\pi$
$258$	0	0
$259$	−20.7290	−1.28804
$260$	0	0
$261$	−18.1164	−1.12138
$262$	0	0
$263$	24.3320	1.50037	0.750186	−	0.661226i	$-0.229964\pi$
$263$	24.3320	1.50037	0.750186	+	0.661226i	$0.229964\pi$
$264$	0	0
$265$	1.67865	0.103118
$266$	0	0
$267$	−0.422223	−0.0258396
$268$	0	0
$269$	−15.7297	−0.959058	−0.479529	−	0.877526i	$-0.659192\pi$
$269$	−15.7297	−0.959058	−0.479529	+	0.877526i	$0.659192\pi$
$270$	0	0
$271$	9.13594	0.554969	0.277484	−	0.960730i	$-0.410499\pi$
$271$	9.13594	0.554969	0.277484	+	0.960730i	$0.410499\pi$
$272$	0	0
$273$	−15.7210	−0.951475
$274$	0	0
$275$	−27.1436	−1.63682
$276$	0	0
$277$	−30.7685	−1.84870	−0.924351	−	0.381544i	$-0.875393\pi$
$277$	−30.7685	−1.84870	−0.924351	+	0.381544i	$0.875393\pi$
$278$	0	0
$279$	19.1432	1.14607
$280$	0	0
$281$	−4.40287	−0.262653	−0.131327	−	0.991339i	$-0.541924\pi$
$281$	−4.40287	−0.262653	−0.131327	+	0.991339i	$0.541924\pi$
$282$	0	0
$283$	7.53953	0.448179	0.224089	−	0.974569i	$-0.428059\pi$
$283$	7.53953	0.448179	0.224089	+	0.974569i	$0.428059\pi$
$284$	0	0
$285$	−0.735286	−0.0435546
$286$	0	0
$287$	−13.6869	−0.807914
$288$	0	0
$289$	−1.96566	−0.115627
$290$	0	0
$291$	1.82942	0.107243
$292$	0	0
$293$	27.3310	1.59669	0.798346	−	0.602199i	$-0.205709\pi$
$293$	27.3310	1.59669	0.798346	+	0.602199i	$0.205709\pi$
$294$	0	0
$295$	3.39957	0.197931
$296$	0	0
$297$	25.4855	1.47882
$298$	0	0
$299$	−43.0730	−2.49098
$300$	0	0
$301$	5.74362	0.331057
$302$	0	0
$303$	−15.6129	−0.896939
$304$	0	0
$305$	−0.536062	−0.0306948
$306$	0	0
$307$	−8.51561	−0.486011	−0.243006	−	0.970025i	$-0.578133\pi$
$307$	−8.51561	−0.486011	−0.243006	+	0.970025i	$0.578133\pi$
$308$	0	0
$309$	5.62615	0.320061
$310$	0	0
$311$	−12.9703	−0.735478	−0.367739	−	0.929929i	$-0.619868\pi$
$311$	−12.9703	−0.735478	−0.367739	+	0.929929i	$0.619868\pi$
$312$	0	0
$313$	−20.4708	−1.15708	−0.578538	−	0.815656i	$-0.696376\pi$
$313$	−20.4708	−1.15708	−0.578538	+	0.815656i	$0.696376\pi$
$314$	0	0
$315$	1.84385	0.103889
$316$	0	0
$317$	−25.7333	−1.44533	−0.722663	−	0.691200i	$-0.757082\pi$
$317$	−25.7333	−1.44533	−0.722663	+	0.691200i	$0.757082\pi$
$318$	0	0
$319$	45.1131	2.52585
$320$	0	0
$321$	16.8550	0.940755
$322$	0	0
$323$	11.4107	0.634910
$324$	0	0
$325$	−29.5919	−1.64146
$326$	0	0
$327$	5.07279	0.280526
$328$	0	0
$329$	19.5908	1.08008
$330$	0	0
$331$	−18.3426	−1.00820	−0.504099	−	0.863646i	$-0.668175\pi$
$331$	−18.3426	−1.00820	−0.504099	+	0.863646i	$0.668175\pi$
$332$	0	0
$333$	15.5616	0.852768
$334$	0	0
$335$	2.29809	0.125558
$336$	0	0
$337$	−11.3668	−0.619189	−0.309595	−	0.950869i	$-0.600193\pi$
$337$	−11.3668	−0.619189	−0.309595	+	0.950869i	$0.600193\pi$
$338$	0	0
$339$	−9.82461	−0.533600
$340$	0	0
$341$	−47.6700	−2.58148
$342$	0	0
$343$	−15.6163	−0.843203
$344$	0	0
$345$	−1.78946	−0.0963413
$346$	0	0
$347$	23.8119	1.27829	0.639144	−	0.769087i	$-0.279289\pi$
$347$	23.8119	1.27829	0.639144	+	0.769087i	$0.279289\pi$
$348$	0	0
$349$	17.9131	0.958866	0.479433	−	0.877579i	$-0.340842\pi$
$349$	17.9131	0.958866	0.479433	+	0.877579i	$0.340842\pi$
$350$	0	0
$351$	27.7843	1.48302
$352$	0	0
$353$	26.5459	1.41289	0.706447	−	0.707766i	$-0.250297\pi$
$353$	26.5459	1.41289	0.706447	+	0.707766i	$0.250297\pi$
$354$	0	0
$355$	4.36702	0.231777
$356$	0	0
$357$	10.1357	0.536437
$358$	0	0
$359$	17.3779	0.917172	0.458586	−	0.888650i	$-0.348356\pi$
$359$	17.3779	0.917172	0.458586	+	0.888650i	$0.348356\pi$
$360$	0	0
$361$	−10.3395	−0.544185
$362$	0	0
$363$	−17.2132	−0.903461
$364$	0	0
$365$	2.20493	0.115411
$366$	0	0
$367$	−27.8490	−1.45371	−0.726853	−	0.686794i	$-0.759018\pi$
$367$	−27.8490	−1.45371	−0.726853	+	0.686794i	$0.759018\pi$
$368$	0	0
$369$	10.2750	0.534893
$370$	0	0
$371$	−17.5624	−0.911796
$372$	0	0
$373$	−35.1463	−1.81981	−0.909904	−	0.414820i	$-0.863845\pi$
$373$	−35.1463	−1.81981	−0.909904	+	0.414820i	$0.863845\pi$
$374$	0	0
$375$	−2.47866	−0.127997
$376$	0	0
$377$	49.1822	2.53301
$378$	0	0
$379$	−35.8962	−1.84387	−0.921933	−	0.387350i	$-0.873390\pi$
$379$	−35.8962	−1.84387	−0.921933	+	0.387350i	$0.873390\pi$
$380$	0	0
$381$	−9.72818	−0.498390
$382$	0	0
$383$	−26.5210	−1.35516	−0.677580	−	0.735449i	$-0.736971\pi$
$383$	−26.5210	−1.35516	−0.677580	+	0.735449i	$0.736971\pi$
$384$	0	0
$385$	−4.59152	−0.234006
$386$	0	0
$387$	−4.31181	−0.219181
$388$	0	0
$389$	−21.1473	−1.07221	−0.536106	−	0.844150i	$-0.680105\pi$
$389$	−21.1473	−1.07221	−0.536106	+	0.844150i	$0.680105\pi$
$390$	0	0
$391$	27.7702	1.40440
$392$	0	0
$393$	1.02643	0.0517768
$394$	0	0
$395$	0.482264	0.0242654
$396$	0	0
$397$	5.04736	0.253320	0.126660	−	0.991946i	$-0.459574\pi$
$397$	5.04736	0.253320	0.126660	+	0.991946i	$0.459574\pi$
$398$	0	0
$399$	7.69275	0.385119
$400$	0	0
$401$	−17.4767	−0.872745	−0.436372	−	0.899766i	$-0.643737\pi$
$401$	−17.4767	−0.872745	−0.436372	+	0.899766i	$0.643737\pi$
$402$	0	0
$403$	−51.9697	−2.58880
$404$	0	0
$405$	−0.720219	−0.0357880
$406$	0	0
$407$	−38.7510	−1.92082
$408$	0	0
$409$	−22.7225	−1.12356	−0.561778	−	0.827288i	$-0.689883\pi$
$409$	−22.7225	−1.12356	−0.561778	+	0.827288i	$0.689883\pi$
$410$	0	0
$411$	4.74721	0.234163
$412$	0	0
$413$	−35.5672	−1.75015
$414$	0	0
$415$	2.15010	0.105544
$416$	0	0
$417$	9.92512	0.486035
$418$	0	0
$419$	−0.130565	−0.00637852	−0.00318926	−	0.999995i	$-0.501015\pi$
$419$	−0.130565	−0.00637852	−0.00318926	+	0.999995i	$0.501015\pi$
$420$	0	0
$421$	−20.1879	−0.983896	−0.491948	−	0.870625i	$-0.663715\pi$
$421$	−20.1879	−0.983896	−0.491948	+	0.870625i	$0.663715\pi$
$422$	0	0
$423$	−14.7071	−0.715083
$424$	0	0
$425$	19.0786	0.925448
$426$	0	0
$427$	5.60842	0.271410
$428$	0	0
$429$	−29.3889	−1.41891
$430$	0	0
$431$	7.83612	0.377452	0.188726	−	0.982030i	$-0.439564\pi$
$431$	7.83612	0.377452	0.188726	+	0.982030i	$0.439564\pi$
$432$	0	0
$433$	15.6947	0.754238	0.377119	−	0.926165i	$-0.376915\pi$
$433$	15.6947	0.754238	0.377119	+	0.926165i	$0.376915\pi$
$434$	0	0
$435$	2.04326	0.0979670
$436$	0	0
$437$	21.0769	1.00825
$438$	0	0
$439$	2.85755	0.136383	0.0681917	−	0.997672i	$-0.478277\pi$
$439$	2.85755	0.136383	0.0681917	+	0.997672i	$0.478277\pi$
$440$	0	0
$441$	−3.78373	−0.180178
$442$	0	0
$443$	41.7607	1.98411	0.992056	−	0.125798i	$-0.0401490\pi$
$443$	41.7607	1.98411	0.992056	+	0.125798i	$0.0401490\pi$
$444$	0	0
$445$	−0.134439	−0.00637301
$446$	0	0
$447$	−17.2197	−0.814465
$448$	0	0
$449$	19.1688	0.904632	0.452316	−	0.891858i	$-0.350598\pi$
$449$	19.1688	0.904632	0.452316	+	0.891858i	$0.350598\pi$
$450$	0	0
$451$	−25.5865	−1.20482
$452$	0	0
$453$	−6.89852	−0.324121
$454$	0	0
$455$	−5.00566	−0.234669
$456$	0	0
$457$	20.1469	0.942433	0.471216	−	0.882018i	$-0.343815\pi$
$457$	20.1469	0.942433	0.471216	+	0.882018i	$0.343815\pi$
$458$	0	0
$459$	−17.9132	−0.836116
$460$	0	0
$461$	11.5057	0.535873	0.267936	−	0.963437i	$-0.413658\pi$
$461$	11.5057	0.535873	0.267936	+	0.963437i	$0.413658\pi$
$462$	0	0
$463$	−8.61504	−0.400375	−0.200187	−	0.979758i	$-0.564155\pi$
$463$	−8.61504	−0.400375	−0.200187	+	0.979758i	$0.564155\pi$
$464$	0	0
$465$	−2.15907	−0.100125
$466$	0	0
$467$	11.5000	0.532158	0.266079	−	0.963951i	$-0.414272\pi$
$467$	11.5000	0.532158	0.266079	+	0.963951i	$0.414272\pi$
$468$	0	0
$469$	−24.0432	−1.11021
$470$	0	0
$471$	−14.5841	−0.672000
$472$	0	0
$473$	10.7372	0.493695
$474$	0	0
$475$	14.4802	0.664399
$476$	0	0
$477$	13.1843	0.603669
$478$	0	0
$479$	−23.7147	−1.08355	−0.541776	−	0.840523i	$-0.682248\pi$
$479$	−23.7147	−1.08355	−0.541776	+	0.840523i	$0.682248\pi$
$480$	0	0
$481$	−42.2463	−1.92626
$482$	0	0
$483$	18.7218	0.851870
$484$	0	0
$485$	0.582502	0.0264500
$486$	0	0
$487$	−20.3401	−0.921696	−0.460848	−	0.887479i	$-0.652455\pi$
$487$	−20.3401	−0.921696	−0.460848	+	0.887479i	$0.652455\pi$
$488$	0	0
$489$	0.582225	0.0263291
$490$	0	0
$491$	−16.8349	−0.759746	−0.379873	−	0.925039i	$-0.624032\pi$
$491$	−16.8349	−0.759746	−0.379873	+	0.925039i	$0.624032\pi$
$492$	0	0
$493$	−31.7089	−1.42810
$494$	0	0
$495$	3.44691	0.154927
$496$	0	0
$497$	−45.6888	−2.04942
$498$	0	0
$499$	−14.0566	−0.629259	−0.314630	−	0.949215i	$-0.601880\pi$
$499$	−14.0566	−0.629259	−0.314630	+	0.949215i	$0.601880\pi$
$500$	0	0
$501$	11.4554	0.511788
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.97126	−0.221218
$506$	0	0
$507$	−20.5239	−0.911497
$508$	0	0
$509$	−17.4083	−0.771608	−0.385804	−	0.922581i	$-0.626076\pi$
$509$	−17.4083	−0.771608	−0.385804	+	0.922581i	$0.626076\pi$
$510$	0	0
$511$	−23.0685	−1.02049
$512$	0	0
$513$	−13.5957	−0.600265
$514$	0	0
$515$	1.79141	0.0789388
$516$	0	0
$517$	36.6233	1.61069
$518$	0	0
$519$	−14.7603	−0.647906
$520$	0	0
$521$	−12.5802	−0.551147	−0.275573	−	0.961280i	$-0.588868\pi$
$521$	−12.5802	−0.551147	−0.275573	+	0.961280i	$0.588868\pi$
$522$	0	0
$523$	8.11843	0.354994	0.177497	−	0.984121i	$-0.443200\pi$
$523$	8.11843	0.354994	0.177497	+	0.984121i	$0.443200\pi$
$524$	0	0
$525$	12.8622	0.561352
$526$	0	0
$527$	33.5061	1.45955
$528$	0	0
$529$	28.2948	1.23021
$530$	0	0
$531$	26.7007	1.15871
$532$	0	0
$533$	−27.8943	−1.20824
$534$	0	0
$535$	5.36675	0.232025
$536$	0	0
$537$	17.1503	0.740089
$538$	0	0
$539$	9.42217	0.405842
$540$	0	0
$541$	23.3391	1.00343	0.501713	−	0.865034i	$-0.332703\pi$
$541$	23.3391	1.00343	0.501713	+	0.865034i	$0.332703\pi$
$542$	0	0
$543$	18.3145	0.785952
$544$	0	0
$545$	1.61521	0.0691881
$546$	0	0
$547$	−6.83841	−0.292389	−0.146195	−	0.989256i	$-0.546703\pi$
$547$	−6.83841	−0.292389	−0.146195	+	0.989256i	$0.546703\pi$
$548$	0	0
$549$	−4.21031	−0.179692
$550$	0	0
$551$	−24.0664	−1.02526
$552$	0	0
$553$	−5.04557	−0.214560
$554$	0	0
$555$	−1.75511	−0.0745004
$556$	0	0
$557$	9.71959	0.411832	0.205916	−	0.978570i	$-0.433983\pi$
$557$	9.71959	0.411832	0.205916	+	0.978570i	$0.433983\pi$
$558$	0	0
$559$	11.7056	0.495095
$560$	0	0
$561$	18.9477	0.799973
$562$	0	0
$563$	−5.60044	−0.236031	−0.118015	−	0.993012i	$-0.537653\pi$
$563$	−5.60044	−0.236031	−0.118015	+	0.993012i	$0.537653\pi$
$564$	0	0
$565$	−3.12823	−0.131605
$566$	0	0
$567$	7.53512	0.316445
$568$	0	0
$569$	−2.87470	−0.120514	−0.0602568	−	0.998183i	$-0.519192\pi$
$569$	−2.87470	−0.120514	−0.0602568	+	0.998183i	$0.519192\pi$
$570$	0	0
$571$	14.5838	0.610312	0.305156	−	0.952302i	$-0.401291\pi$
$571$	14.5838	0.610312	0.305156	+	0.952302i	$0.401291\pi$
$572$	0	0
$573$	−0.200521	−0.00837688
$574$	0	0
$575$	35.2404	1.46963
$576$	0	0
$577$	−6.97898	−0.290539	−0.145269	−	0.989392i	$-0.546405\pi$
$577$	−6.97898	−0.290539	−0.145269	+	0.989392i	$0.546405\pi$
$578$	0	0
$579$	−3.60100	−0.149653
$580$	0	0
$581$	−22.4949	−0.933245
$582$	0	0
$583$	−32.8313	−1.35973
$584$	0	0
$585$	3.75782	0.155367
$586$	0	0
$587$	36.5461	1.50842	0.754210	−	0.656634i	$-0.228020\pi$
$587$	36.5461	1.50842	0.754210	+	0.656634i	$0.228020\pi$
$588$	0	0
$589$	25.4304	1.04784
$590$	0	0
$591$	2.25417	0.0927243
$592$	0	0
$593$	−6.80544	−0.279466	−0.139733	−	0.990189i	$-0.544624\pi$
$593$	−6.80544	−0.279466	−0.139733	+	0.990189i	$0.544624\pi$
$594$	0	0
$595$	3.22727	0.132305
$596$	0	0
$597$	−4.76451	−0.194998
$598$	0	0
$599$	28.5833	1.16788	0.583942	−	0.811796i	$-0.301510\pi$
$599$	28.5833	1.16788	0.583942	+	0.811796i	$0.301510\pi$
$600$	0	0
$601$	26.2943	1.07257	0.536283	−	0.844038i	$-0.319828\pi$
$601$	26.2943	1.07257	0.536283	+	0.844038i	$0.319828\pi$
$602$	0	0
$603$	18.0496	0.735035
$604$	0	0
$605$	−5.48082	−0.222827
$606$	0	0
$607$	−18.1016	−0.734722	−0.367361	−	0.930078i	$-0.619739\pi$
$607$	−18.1016	−0.734722	−0.367361	+	0.930078i	$0.619739\pi$
$608$	0	0
$609$	−21.3771	−0.866245
$610$	0	0
$611$	39.9266	1.61526
$612$	0	0
$613$	17.2313	0.695964	0.347982	−	0.937501i	$-0.386867\pi$
$613$	17.2313	0.695964	0.347982	+	0.937501i	$0.386867\pi$
$614$	0	0
$615$	−1.15886	−0.0467299
$616$	0	0
$617$	24.9655	1.00507	0.502537	−	0.864556i	$-0.332400\pi$
$617$	24.9655	1.00507	0.502537	+	0.864556i	$0.332400\pi$
$618$	0	0
$619$	−15.2755	−0.613972	−0.306986	−	0.951714i	$-0.599321\pi$
$619$	−15.2755	−0.613972	−0.306986	+	0.951714i	$0.599321\pi$
$620$	0	0
$621$	−33.0878	−1.32777
$622$	0	0
$623$	1.40653	0.0563515
$624$	0	0
$625$	23.8130	0.952520
$626$	0	0
$627$	14.3809	0.574317
$628$	0	0
$629$	27.2372	1.08602
$630$	0	0
$631$	27.0717	1.07771	0.538854	−	0.842399i	$-0.318857\pi$
$631$	27.0717	1.07771	0.538854	+	0.842399i	$0.318857\pi$
$632$	0	0
$633$	−19.7023	−0.783097
$634$	0	0
$635$	−3.09752	−0.122921
$636$	0	0
$637$	10.2720	0.406993
$638$	0	0
$639$	34.2992	1.35685
$640$	0	0
$641$	8.95425	0.353672	0.176836	−	0.984240i	$-0.443414\pi$
$641$	8.95425	0.353672	0.176836	+	0.984240i	$0.443414\pi$
$642$	0	0
$643$	9.02662	0.355975	0.177988	−	0.984033i	$-0.443041\pi$
$643$	9.02662	0.355975	0.177988	+	0.984033i	$0.443041\pi$
$644$	0	0
$645$	0.486308	0.0191484
$646$	0	0
$647$	5.33520	0.209748	0.104874	−	0.994485i	$-0.466556\pi$
$647$	5.33520	0.209748	0.104874	+	0.994485i	$0.466556\pi$
$648$	0	0
$649$	−66.4895	−2.60994
$650$	0	0
$651$	22.5888	0.885324
$652$	0	0
$653$	36.3096	1.42091	0.710453	−	0.703744i	$-0.248490\pi$
$653$	36.3096	1.42091	0.710453	+	0.703744i	$0.248490\pi$
$654$	0	0
$655$	0.326824	0.0127701
$656$	0	0
$657$	17.3178	0.675632
$658$	0	0
$659$	33.3149	1.29776	0.648882	−	0.760889i	$-0.275237\pi$
$659$	33.3149	1.29776	0.648882	+	0.760889i	$0.275237\pi$
$660$	0	0
$661$	4.74741	0.184653	0.0923264	−	0.995729i	$-0.470570\pi$
$661$	4.74741	0.184653	0.0923264	+	0.995729i	$0.470570\pi$
$662$	0	0
$663$	20.6567	0.802242
$664$	0	0
$665$	2.44943	0.0949847
$666$	0	0
$667$	−58.5701	−2.26784
$668$	0	0
$669$	−3.59244	−0.138892
$670$	0	0
$671$	10.4844	0.404746
$672$	0	0
$673$	13.3928	0.516254	0.258127	−	0.966111i	$-0.416895\pi$
$673$	13.3928	0.516254	0.258127	+	0.966111i	$0.416895\pi$
$674$	0	0
$675$	−22.7319	−0.874950
$676$	0	0
$677$	20.9997	0.807084	0.403542	−	0.914961i	$-0.367779\pi$
$677$	20.9997	0.807084	0.403542	+	0.914961i	$0.367779\pi$
$678$	0	0
$679$	−6.09428	−0.233877
$680$	0	0
$681$	−1.50800	−0.0577868
$682$	0	0
$683$	1.68394	0.0644340	0.0322170	−	0.999481i	$-0.489743\pi$
$683$	1.68394	0.0644340	0.0322170	+	0.999481i	$0.489743\pi$
$684$	0	0
$685$	1.51155	0.0577532
$686$	0	0
$687$	−23.2378	−0.886577
$688$	0	0
$689$	−35.7926	−1.36359
$690$	0	0
$691$	−26.8304	−1.02067	−0.510337	−	0.859974i	$-0.670480\pi$
$691$	−26.8304	−1.02067	−0.510337	+	0.859974i	$0.670480\pi$
$692$	0	0
$693$	−36.0625	−1.36990
$694$	0	0
$695$	3.16023	0.119874
$696$	0	0
$697$	17.9841	0.681197
$698$	0	0
$699$	−19.3604	−0.732279
$700$	0	0
$701$	25.0000	0.944238	0.472119	−	0.881535i	$-0.343489\pi$
$701$	25.0000	0.944238	0.472119	+	0.881535i	$0.343489\pi$
$702$	0	0
$703$	20.6724	0.779675
$704$	0	0
$705$	1.65874	0.0624719
$706$	0	0
$707$	52.0106	1.95606
$708$	0	0
$709$	−42.7399	−1.60513	−0.802566	−	0.596564i	$-0.796532\pi$
$709$	−42.7399	−1.60513	−0.802566	+	0.596564i	$0.796532\pi$
$710$	0	0
$711$	3.78777	0.142053
$712$	0	0
$713$	61.8897	2.31779
$714$	0	0
$715$	−9.35763	−0.349955
$716$	0	0
$717$	−1.87808	−0.0701383
$718$	0	0
$719$	−43.4014	−1.61860	−0.809300	−	0.587395i	$-0.800153\pi$
$719$	−43.4014	−1.61860	−0.809300	+	0.587395i	$0.800153\pi$
$720$	0	0
$721$	−18.7421	−0.697994
$722$	0	0
$723$	14.1778	0.527280
$724$	0	0
$725$	−40.2387	−1.49443
$726$	0	0
$727$	−19.5905	−0.726571	−0.363285	−	0.931678i	$-0.618345\pi$
$727$	−19.5905	−0.726571	−0.363285	+	0.931678i	$0.618345\pi$
$728$	0	0
$729$	6.62060	0.245208
$730$	0	0
$731$	−7.54690	−0.279132
$732$	0	0
$733$	31.7357	1.17219	0.586093	−	0.810244i	$-0.300665\pi$
$733$	31.7357	1.17219	0.586093	+	0.810244i	$0.300665\pi$
$734$	0	0
$735$	0.426750	0.0157409
$736$	0	0
$737$	−44.9466	−1.65563
$738$	0	0
$739$	37.3100	1.37247	0.686235	−	0.727380i	$-0.259262\pi$
$739$	37.3100	1.37247	0.686235	+	0.727380i	$0.259262\pi$
$740$	0	0
$741$	15.6780	0.575946
$742$	0	0
$743$	−27.0619	−0.992807	−0.496403	−	0.868092i	$-0.665346\pi$
$743$	−27.0619	−0.992807	−0.496403	+	0.868092i	$0.665346\pi$
$744$	0	0
$745$	−5.48288	−0.200877
$746$	0	0
$747$	16.8872	0.617870
$748$	0	0
$749$	−56.1483	−2.05162
$750$	0	0
$751$	−4.67853	−0.170722	−0.0853609	−	0.996350i	$-0.527204\pi$
$751$	−4.67853	−0.170722	−0.0853609	+	0.996350i	$0.527204\pi$
$752$	0	0
$753$	−4.18883	−0.152649
$754$	0	0
$755$	−2.19654	−0.0799402
$756$	0	0
$757$	−29.4509	−1.07041	−0.535205	−	0.844722i	$-0.679766\pi$
$757$	−29.4509	−1.07041	−0.535205	+	0.844722i	$0.679766\pi$
$758$	0	0
$759$	34.9986	1.27037
$760$	0	0
$761$	18.0423	0.654033	0.327017	−	0.945019i	$-0.393957\pi$
$761$	18.0423	0.654033	0.327017	+	0.945019i	$0.393957\pi$
$762$	0	0
$763$	−16.8988	−0.611777
$764$	0	0
$765$	−2.42275	−0.0875949
$766$	0	0
$767$	−72.4867	−2.61734
$768$	0	0
$769$	0.962749	0.0347176	0.0173588	−	0.999849i	$-0.494474\pi$
$769$	0.962749	0.0347176	0.0173588	+	0.999849i	$0.494474\pi$
$770$	0	0
$771$	22.1974	0.799419
$772$	0	0
$773$	−0.442527	−0.0159166	−0.00795829	−	0.999968i	$-0.502533\pi$
$773$	−0.442527	−0.0159166	−0.00795829	+	0.999968i	$0.502533\pi$
$774$	0	0
$775$	42.5193	1.52734
$776$	0	0
$777$	18.3624	0.658749
$778$	0	0
$779$	13.6495	0.489046
$780$	0	0
$781$	−85.4111	−3.05625
$782$	0	0
$783$	37.7807	1.35017
$784$	0	0
$785$	−4.64368	−0.165740
$786$	0	0
$787$	15.9060	0.566989	0.283495	−	0.958974i	$-0.408506\pi$
$787$	15.9060	0.566989	0.283495	+	0.958974i	$0.408506\pi$
$788$	0	0
$789$	−21.5540	−0.767344
$790$	0	0
$791$	32.7283	1.16368
$792$	0	0
$793$	11.4301	0.405894
$794$	0	0
$795$	−1.48700	−0.0527384
$796$	0	0
$797$	3.31088	0.117277	0.0586387	−	0.998279i	$-0.481324\pi$
$797$	3.31088	0.117277	0.0586387	+	0.998279i	$0.481324\pi$
$798$	0	0
$799$	−25.7416	−0.910673
$800$	0	0
$801$	−1.05590	−0.0373084
$802$	0	0
$803$	−43.1244	−1.52183
$804$	0	0
$805$	5.96115	0.210103
$806$	0	0
$807$	13.9339	0.490496
$808$	0	0
$809$	−28.2824	−0.994358	−0.497179	−	0.867648i	$-0.665631\pi$
$809$	−28.2824	−0.994358	−0.497179	+	0.867648i	$0.665631\pi$
$810$	0	0
$811$	−8.29369	−0.291231	−0.145615	−	0.989341i	$-0.546516\pi$
$811$	−8.29369	−0.291231	−0.145615	+	0.989341i	$0.546516\pi$
$812$	0	0
$813$	−8.09291	−0.283831
$814$	0	0
$815$	0.185385	0.00649374
$816$	0	0
$817$	−5.72793	−0.200395
$818$	0	0
$819$	−39.3152	−1.37378
$820$	0	0
$821$	−11.6816	−0.407691	−0.203845	−	0.979003i	$-0.565344\pi$
$821$	−11.6816	−0.407691	−0.203845	+	0.979003i	$0.565344\pi$
$822$	0	0
$823$	−24.7339	−0.862170	−0.431085	−	0.902311i	$-0.641869\pi$
$823$	−24.7339	−0.862170	−0.431085	+	0.902311i	$0.641869\pi$
$824$	0	0
$825$	24.0447	0.837128
$826$	0	0
$827$	−55.1154	−1.91655	−0.958276	−	0.285845i	$-0.907726\pi$
$827$	−55.1154	−1.91655	−0.958276	+	0.285845i	$0.907726\pi$
$828$	0	0
$829$	−3.88122	−0.134800	−0.0674001	−	0.997726i	$-0.521470\pi$
$829$	−3.88122	−0.134800	−0.0674001	+	0.997726i	$0.521470\pi$
$830$	0	0
$831$	27.2557	0.945491
$832$	0	0
$833$	−6.62262	−0.229460
$834$	0	0
$835$	3.64747	0.126226
$836$	0	0
$837$	−39.9220	−1.37991
$838$	0	0
$839$	49.7899	1.71894	0.859469	−	0.511187i	$-0.170794\pi$
$839$	49.7899	1.71894	0.859469	+	0.511187i	$0.170794\pi$
$840$	0	0
$841$	37.8772	1.30611
$842$	0	0
$843$	3.90020	0.134330
$844$	0	0
$845$	−6.53495	−0.224809
$846$	0	0
$847$	57.3417	1.97028
$848$	0	0
$849$	−6.67876	−0.229214
$850$	0	0
$851$	50.3103	1.72461
$852$	0	0
$853$	−13.4433	−0.460290	−0.230145	−	0.973156i	$-0.573920\pi$
$853$	−13.4433	−0.460290	−0.230145	+	0.973156i	$0.573920\pi$
$854$	0	0
$855$	−1.83882	−0.0628862
$856$	0	0
$857$	−1.40311	−0.0479293	−0.0239646	−	0.999713i	$-0.507629\pi$
$857$	−1.40311	−0.0479293	−0.0239646	+	0.999713i	$0.507629\pi$
$858$	0	0
$859$	−50.4578	−1.72160	−0.860800	−	0.508944i	$-0.830036\pi$
$859$	−50.4578	−1.72160	−0.860800	+	0.508944i	$0.830036\pi$
$860$	0	0
$861$	12.1243	0.413196
$862$	0	0
$863$	−18.9585	−0.645353	−0.322677	−	0.946509i	$-0.604583\pi$
$863$	−18.9585	−0.645353	−0.322677	+	0.946509i	$0.604583\pi$
$864$	0	0
$865$	−4.69979	−0.159798
$866$	0	0
$867$	1.74124	0.0591356
$868$	0	0
$869$	−9.43223	−0.319967
$870$	0	0
$871$	−49.0007	−1.66032
$872$	0	0
$873$	4.57505	0.154842
$874$	0	0
$875$	8.25704	0.279139
$876$	0	0
$877$	−37.0638	−1.25156	−0.625778	−	0.780001i	$-0.715218\pi$
$877$	−37.0638	−1.25156	−0.625778	+	0.780001i	$0.715218\pi$
$878$	0	0
$879$	−24.2106	−0.816605
$880$	0	0
$881$	−16.7981	−0.565941	−0.282971	−	0.959129i	$-0.591320\pi$
$881$	−16.7981	−0.565941	−0.282971	+	0.959129i	$0.591320\pi$
$882$	0	0
$883$	−21.3748	−0.719320	−0.359660	−	0.933083i	$-0.617107\pi$
$883$	−21.3748	−0.719320	−0.359660	+	0.933083i	$0.617107\pi$
$884$	0	0
$885$	−3.01145	−0.101229
$886$	0	0
$887$	−24.4406	−0.820637	−0.410318	−	0.911942i	$-0.634582\pi$
$887$	−24.4406	−0.820637	−0.410318	+	0.911942i	$0.634582\pi$
$888$	0	0
$889$	32.4070	1.08690
$890$	0	0
$891$	14.0862	0.471906
$892$	0	0
$893$	−19.5373	−0.653791
$894$	0	0
$895$	5.46077	0.182534
$896$	0	0
$897$	38.1554	1.27397
$898$	0	0
$899$	−70.6678	−2.35690
$900$	0	0
$901$	23.0764	0.768785
$902$	0	0
$903$	−5.08788	−0.169314
$904$	0	0
$905$	5.83148	0.193845
$906$	0	0
$907$	−31.8297	−1.05689	−0.528444	−	0.848968i	$-0.677224\pi$
$907$	−31.8297	−1.05689	−0.528444	+	0.848968i	$0.677224\pi$
$908$	0	0
$909$	−39.0450	−1.29504
$910$	0	0
$911$	−48.0090	−1.59061	−0.795305	−	0.606210i	$-0.792689\pi$
$911$	−48.0090	−1.59061	−0.795305	+	0.606210i	$0.792689\pi$
$912$	0	0
$913$	−42.0521	−1.39172
$914$	0	0
$915$	0.474861	0.0156984
$916$	0	0
$917$	−3.41932	−0.112916
$918$	0	0
$919$	−4.93119	−0.162665	−0.0813325	−	0.996687i	$-0.525918\pi$
$919$	−4.93119	−0.162665	−0.0813325	+	0.996687i	$0.525918\pi$
$920$	0	0
$921$	7.54340	0.248563
$922$	0	0
$923$	−93.1149	−3.06492
$924$	0	0
$925$	34.5640	1.13646
$926$	0	0
$927$	14.0700	0.462118
$928$	0	0
$929$	32.8804	1.07877	0.539385	−	0.842059i	$-0.318657\pi$
$929$	32.8804	1.07877	0.539385	+	0.842059i	$0.318657\pi$
$930$	0	0
$931$	−5.02642	−0.164734
$932$	0	0
$933$	11.4895	0.376150
$934$	0	0
$935$	6.03309	0.197303
$936$	0	0
$937$	−3.71111	−0.121237	−0.0606183	−	0.998161i	$-0.519307\pi$
$937$	−3.71111	−0.121237	−0.0606183	+	0.998161i	$0.519307\pi$
$938$	0	0
$939$	18.1336	0.591769
$940$	0	0
$941$	37.3293	1.21690	0.608450	−	0.793592i	$-0.291791\pi$
$941$	37.3293	1.21690	0.608450	+	0.793592i	$0.291791\pi$
$942$	0	0
$943$	33.2188	1.08175
$944$	0	0
$945$	−3.84524	−0.125086
$946$	0	0
$947$	35.3052	1.14726	0.573632	−	0.819113i	$-0.305534\pi$
$947$	35.3052	1.14726	0.573632	+	0.819113i	$0.305534\pi$
$948$	0	0
$949$	−47.0141	−1.52614
$950$	0	0
$951$	22.7954	0.739191
$952$	0	0
$953$	−23.5473	−0.762773	−0.381386	−	0.924416i	$-0.624553\pi$
$953$	−23.5473	−0.762773	−0.381386	+	0.924416i	$0.624553\pi$
$954$	0	0
$955$	−0.0638473	−0.00206605
$956$	0	0
$957$	−39.9626	−1.29181
$958$	0	0
$959$	−15.8142	−0.510666
$960$	0	0
$961$	43.6731	1.40881
$962$	0	0
$963$	42.1513	1.35831
$964$	0	0
$965$	−1.14659	−0.0369099
$966$	0	0
$967$	5.18902	0.166868	0.0834339	−	0.996513i	$-0.473411\pi$
$967$	5.18902	0.166868	0.0834339	+	0.996513i	$0.473411\pi$
$968$	0	0
$969$	−10.1080	−0.324715
$970$	0	0
$971$	42.5528	1.36558	0.682792	−	0.730613i	$-0.260766\pi$
$971$	42.5528	1.36558	0.682792	+	0.730613i	$0.260766\pi$
$972$	0	0
$973$	−33.0631	−1.05995
$974$	0	0
$975$	26.2134	0.839502
$976$	0	0
$977$	−53.1560	−1.70061	−0.850306	−	0.526289i	$-0.823583\pi$
$977$	−53.1560	−1.70061	−0.850306	+	0.526289i	$0.823583\pi$
$978$	0	0
$979$	2.62938	0.0840354
$980$	0	0
$981$	12.6861	0.405037
$982$	0	0
$983$	−15.6194	−0.498182	−0.249091	−	0.968480i	$-0.580132\pi$
$983$	−15.6194	−0.498182	−0.249091	+	0.968480i	$0.580132\pi$
$984$	0	0
$985$	0.717745	0.0228692
$986$	0	0
$987$	−17.3542	−0.552390
$988$	0	0
$989$	−13.9400	−0.443266
$990$	0	0
$991$	−17.3582	−0.551401	−0.275700	−	0.961244i	$-0.588910\pi$
$991$	−17.3582	−0.551401	−0.275700	+	0.961244i	$0.588910\pi$
$992$	0	0
$993$	16.2484	0.515628
$994$	0	0
$995$	−1.51705	−0.0480939
$996$	0	0
$997$	−18.8781	−0.597876	−0.298938	−	0.954273i	$-0.596632\pi$
$997$	−18.8781	−0.597876	−0.298938	+	0.954273i	$0.596632\pi$
$998$	0	0
$999$	−32.4527	−1.02676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.15		33
4.3	odd	2		4024.2.a.g.1.19	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.19	✓	33	4.3	odd	2
8048.2.a.x.1.15		33	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.885832 q^{3} -0.282055 q^{5} +2.95093 q^{7} -2.21530 q^{9} +O(q^{10})\)	\(q-0.885832 q^{3} -0.282055 q^{5} +2.95093 q^{7} -2.21530 q^{9} +5.51650 q^{11} +6.01407 q^{13} +0.249853 q^{15} -3.87741 q^{17} -2.94287 q^{19} -2.61403 q^{21} -7.16204 q^{23} -4.92044 q^{25} +4.61988 q^{27} +8.17785 q^{29} -8.64136 q^{31} -4.88669 q^{33} -0.832326 q^{35} -7.02457 q^{37} -5.32745 q^{39} -4.63817 q^{41} +1.94637 q^{43} +0.624837 q^{45} +6.63886 q^{47} +1.70800 q^{49} +3.43474 q^{51} -5.95148 q^{53} -1.55596 q^{55} +2.60689 q^{57} -12.0529 q^{59} +1.90056 q^{61} -6.53721 q^{63} -1.69630 q^{65} -8.14767 q^{67} +6.34436 q^{69} -15.4828 q^{71} -7.81736 q^{73} +4.35869 q^{75} +16.2788 q^{77} -1.70982 q^{79} +2.55347 q^{81} -7.62297 q^{83} +1.09364 q^{85} -7.24420 q^{87} +0.476640 q^{89} +17.7471 q^{91} +7.65479 q^{93} +0.830052 q^{95} -2.06521 q^{97} -12.2207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

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