LMFDB - Embedded newform 8048.2.a.s.1.5

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.44429 q^{3} +3.11768 q^{5} +1.05875 q^{7} +2.97454 q^{9} +O(q^{10})$	$q-2.44429 q^{3} +3.11768 q^{5} +1.05875 q^{7} +2.97454 q^{9} -0.276389 q^{11} +3.03669 q^{13} -7.62051 q^{15} -1.15214 q^{17} -6.80765 q^{19} -2.58790 q^{21} +8.16231 q^{23} +4.71994 q^{25} +0.0622392 q^{27} -0.470339 q^{29} -9.46734 q^{31} +0.675573 q^{33} +3.30086 q^{35} +0.964244 q^{37} -7.42254 q^{39} -9.66997 q^{41} -4.32259 q^{43} +9.27366 q^{45} +3.94684 q^{47} -5.87904 q^{49} +2.81616 q^{51} -0.187817 q^{53} -0.861691 q^{55} +16.6398 q^{57} -3.94030 q^{59} +1.07192 q^{61} +3.14930 q^{63} +9.46743 q^{65} +1.00059 q^{67} -19.9510 q^{69} -15.8917 q^{71} +0.0981513 q^{73} -11.5369 q^{75} -0.292627 q^{77} -1.16259 q^{79} -9.07574 q^{81} +1.14804 q^{83} -3.59200 q^{85} +1.14964 q^{87} -7.54676 q^{89} +3.21511 q^{91} +23.1409 q^{93} -21.2241 q^{95} +7.76129 q^{97} -0.822128 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * 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
$3$	−2.44429	−1.41121	−0.705605	−	0.708606i	$-0.749325\pi$
$3$	−2.44429	−1.41121	−0.705605	+	0.708606i	$0.749325\pi$
$4$	0	0
$5$	3.11768	1.39427	0.697135	−	0.716940i	$-0.254458\pi$
$5$	3.11768	1.39427	0.697135	+	0.716940i	$0.254458\pi$
$6$	0	0
$7$	1.05875	0.400171	0.200086	−	0.979778i	$-0.435878\pi$
$7$	1.05875	0.400171	0.200086	+	0.979778i	$0.435878\pi$
$8$	0	0
$9$	2.97454	0.991512
$10$	0	0
$11$	−0.276389	−0.0833343	−0.0416671	−	0.999132i	$-0.513267\pi$
$11$	−0.276389	−0.0833343	−0.0416671	+	0.999132i	$0.513267\pi$
$12$	0	0
$13$	3.03669	0.842226	0.421113	−	0.907008i	$-0.361640\pi$
$13$	3.03669	0.842226	0.421113	+	0.907008i	$0.361640\pi$
$14$	0	0
$15$	−7.62051	−1.96761
$16$	0	0
$17$	−1.15214	−0.279434	−0.139717	−	0.990191i	$-0.544619\pi$
$17$	−1.15214	−0.279434	−0.139717	+	0.990191i	$0.544619\pi$
$18$	0	0
$19$	−6.80765	−1.56178	−0.780891	−	0.624667i	$-0.785235\pi$
$19$	−6.80765	−1.56178	−0.780891	+	0.624667i	$0.785235\pi$
$20$	0	0
$21$	−2.58790	−0.564725
$22$	0	0
$23$	8.16231	1.70196	0.850980	−	0.525198i	$-0.176009\pi$
$23$	8.16231	1.70196	0.850980	+	0.525198i	$0.176009\pi$
$24$	0	0
$25$	4.71994	0.943988
$26$	0	0
$27$	0.0622392	0.0119779
$28$	0	0
$29$	−0.470339	−0.0873398	−0.0436699	−	0.999046i	$-0.513905\pi$
$29$	−0.470339	−0.0873398	−0.0436699	+	0.999046i	$0.513905\pi$
$30$	0	0
$31$	−9.46734	−1.70038	−0.850192	−	0.526473i	$-0.823514\pi$
$31$	−9.46734	−1.70038	−0.850192	+	0.526473i	$0.823514\pi$
$32$	0	0
$33$	0.675573	0.117602
$34$	0	0
$35$	3.30086	0.557947
$36$	0	0
$37$	0.964244	0.158521	0.0792604	−	0.996854i	$-0.474744\pi$
$37$	0.964244	0.158521	0.0792604	+	0.996854i	$0.474744\pi$
$38$	0	0
$39$	−7.42254	−1.18856
$40$	0	0
$41$	−9.66997	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$41$	−9.66997	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$42$	0	0
$43$	−4.32259	−0.659189	−0.329595	−	0.944123i	$-0.606912\pi$
$43$	−4.32259	−0.659189	−0.329595	+	0.944123i	$0.606912\pi$
$44$	0	0
$45$	9.27366	1.38244
$46$	0	0
$47$	3.94684	0.575705	0.287853	−	0.957675i	$-0.407059\pi$
$47$	3.94684	0.575705	0.287853	+	0.957675i	$0.407059\pi$
$48$	0	0
$49$	−5.87904	−0.839863
$50$	0	0
$51$	2.81616	0.394341
$52$	0	0
$53$	−0.187817	−0.0257987	−0.0128993	−	0.999917i	$-0.504106\pi$
$53$	−0.187817	−0.0257987	−0.0128993	+	0.999917i	$0.504106\pi$
$54$	0	0
$55$	−0.861691	−0.116190
$56$	0	0
$57$	16.6398	2.20400
$58$	0	0
$59$	−3.94030	−0.512983	−0.256492	−	0.966546i	$-0.582567\pi$
$59$	−3.94030	−0.512983	−0.256492	+	0.966546i	$0.582567\pi$
$60$	0	0
$61$	1.07192	0.137245	0.0686225	−	0.997643i	$-0.478140\pi$
$61$	1.07192	0.137245	0.0686225	+	0.997643i	$0.478140\pi$
$62$	0	0
$63$	3.14930	0.396775
$64$	0	0
$65$	9.46743	1.17429
$66$	0	0
$67$	1.00059	0.122241	0.0611207	−	0.998130i	$-0.480533\pi$
$67$	1.00059	0.122241	0.0611207	+	0.998130i	$0.480533\pi$
$68$	0	0
$69$	−19.9510	−2.40182
$70$	0	0
$71$	−15.8917	−1.88600	−0.942998	−	0.332798i	$-0.892007\pi$
$71$	−15.8917	−1.88600	−0.942998	+	0.332798i	$0.892007\pi$
$72$	0	0
$73$	0.0981513	0.0114877	0.00574387	−	0.999984i	$-0.498172\pi$
$73$	0.0981513	0.0114877	0.00574387	+	0.999984i	$0.498172\pi$
$74$	0	0
$75$	−11.5369	−1.33216
$76$	0	0
$77$	−0.292627	−0.0333480
$78$	0	0
$79$	−1.16259	−0.130801	−0.0654006	−	0.997859i	$-0.520833\pi$
$79$	−1.16259	−0.130801	−0.0654006	+	0.997859i	$0.520833\pi$
$80$	0	0
$81$	−9.07574	−1.00842
$82$	0	0
$83$	1.14804	0.126014	0.0630071	−	0.998013i	$-0.479931\pi$
$83$	1.14804	0.126014	0.0630071	+	0.998013i	$0.479931\pi$
$84$	0	0
$85$	−3.59200	−0.389607
$86$	0	0
$87$	1.14964	0.123255
$88$	0	0
$89$	−7.54676	−0.799955	−0.399977	−	0.916525i	$-0.630982\pi$
$89$	−7.54676	−0.799955	−0.399977	+	0.916525i	$0.630982\pi$
$90$	0	0
$91$	3.21511	0.337035
$92$	0	0
$93$	23.1409	2.39960
$94$	0	0
$95$	−21.2241	−2.17755
$96$	0	0
$97$	7.76129	0.788039	0.394020	−	0.919102i	$-0.371084\pi$
$97$	7.76129	0.788039	0.394020	+	0.919102i	$0.371084\pi$
$98$	0	0
$99$	−0.822128	−0.0826270
$100$	0	0
$101$	−17.0717	−1.69870	−0.849350	−	0.527829i	$-0.823006\pi$
$101$	−17.0717	−1.69870	−0.849350	+	0.527829i	$0.823006\pi$
$102$	0	0
$103$	7.12347	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$103$	7.12347	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$104$	0	0
$105$	−8.06824	−0.787380
$106$	0	0
$107$	−15.3454	−1.48349	−0.741746	−	0.670681i	$-0.766002\pi$
$107$	−15.3454	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$108$	0	0
$109$	−9.15312	−0.876710	−0.438355	−	0.898802i	$-0.644439\pi$
$109$	−9.15312	−0.876710	−0.438355	+	0.898802i	$0.644439\pi$
$110$	0	0
$111$	−2.35689	−0.223706
$112$	0	0
$113$	0.445965	0.0419529	0.0209764	−	0.999780i	$-0.493323\pi$
$113$	0.445965	0.0419529	0.0209764	+	0.999780i	$0.493323\pi$
$114$	0	0
$115$	25.4475	2.37299
$116$	0	0
$117$	9.03274	0.835077
$118$	0	0
$119$	−1.21983	−0.111822
$120$	0	0
$121$	−10.9236	−0.993055
$122$	0	0
$123$	23.6362	2.13120
$124$	0	0
$125$	−0.873137	−0.0780957
$126$	0	0
$127$	0.0608505	0.00539961	0.00269980	−	0.999996i	$-0.499141\pi$
$127$	0.0608505	0.00539961	0.00269980	+	0.999996i	$0.499141\pi$
$128$	0	0
$129$	10.5657	0.930254
$130$	0	0
$131$	7.88758	0.689141	0.344571	−	0.938760i	$-0.388024\pi$
$131$	7.88758	0.689141	0.344571	+	0.938760i	$0.388024\pi$
$132$	0	0
$133$	−7.20762	−0.624980
$134$	0	0
$135$	0.194042	0.0167005
$136$	0	0
$137$	−2.03603	−0.173949	−0.0869747	−	0.996211i	$-0.527720\pi$
$137$	−2.03603	−0.173949	−0.0869747	+	0.996211i	$0.527720\pi$
$138$	0	0
$139$	−0.0305856	−0.00259423	−0.00129712	−	0.999999i	$-0.500413\pi$
$139$	−0.0305856	−0.00259423	−0.00129712	+	0.999999i	$0.500413\pi$
$140$	0	0
$141$	−9.64720	−0.812441
$142$	0	0
$143$	−0.839306	−0.0701863
$144$	0	0
$145$	−1.46637	−0.121775
$146$	0	0
$147$	14.3701	1.18522
$148$	0	0
$149$	3.54160	0.290140	0.145070	−	0.989421i	$-0.453659\pi$
$149$	3.54160	0.290140	0.145070	+	0.989421i	$0.453659\pi$
$150$	0	0
$151$	24.4151	1.98687	0.993437	−	0.114382i	$-0.0364887\pi$
$151$	24.4151	1.98687	0.993437	+	0.114382i	$0.0364887\pi$
$152$	0	0
$153$	−3.42708	−0.277063
$154$	0	0
$155$	−29.5161	−2.37079
$156$	0	0
$157$	20.0853	1.60298	0.801489	−	0.598009i	$-0.204041\pi$
$157$	20.0853	1.60298	0.801489	+	0.598009i	$0.204041\pi$
$158$	0	0
$159$	0.459079	0.0364073
$160$	0	0
$161$	8.64188	0.681075
$162$	0	0
$163$	0.766902	0.0600684	0.0300342	−	0.999549i	$-0.490438\pi$
$163$	0.766902	0.0600684	0.0300342	+	0.999549i	$0.490438\pi$
$164$	0	0
$165$	2.10622	0.163969
$166$	0	0
$167$	−6.82968	−0.528496	−0.264248	−	0.964455i	$-0.585124\pi$
$167$	−6.82968	−0.528496	−0.264248	+	0.964455i	$0.585124\pi$
$168$	0	0
$169$	−3.77852	−0.290655
$170$	0	0
$171$	−20.2496	−1.54853
$172$	0	0
$173$	−11.1682	−0.849105	−0.424552	−	0.905403i	$-0.639569\pi$
$173$	−11.1682	−0.849105	−0.424552	+	0.905403i	$0.639569\pi$
$174$	0	0
$175$	4.99725	0.377757
$176$	0	0
$177$	9.63122	0.723927
$178$	0	0
$179$	18.0526	1.34932	0.674658	−	0.738130i	$-0.264291\pi$
$179$	18.0526	1.34932	0.674658	+	0.738130i	$0.264291\pi$
$180$	0	0
$181$	−4.98959	−0.370873	−0.185437	−	0.982656i	$-0.559370\pi$
$181$	−4.98959	−0.370873	−0.185437	+	0.982656i	$0.559370\pi$
$182$	0	0
$183$	−2.62008	−0.193682
$184$	0	0
$185$	3.00621	0.221021
$186$	0	0
$187$	0.318438	0.0232865
$188$	0	0
$189$	0.0658959	0.00479322
$190$	0	0
$191$	−7.22146	−0.522527	−0.261263	−	0.965268i	$-0.584139\pi$
$191$	−7.22146	−0.522527	−0.261263	+	0.965268i	$0.584139\pi$
$192$	0	0
$193$	−1.24449	−0.0895801	−0.0447900	−	0.998996i	$-0.514262\pi$
$193$	−1.24449	−0.0895801	−0.0447900	+	0.998996i	$0.514262\pi$
$194$	0	0
$195$	−23.1411	−1.65717
$196$	0	0
$197$	17.2185	1.22677	0.613385	−	0.789784i	$-0.289807\pi$
$197$	17.2185	1.22677	0.613385	+	0.789784i	$0.289807\pi$
$198$	0	0
$199$	−12.9911	−0.920913	−0.460456	−	0.887682i	$-0.652314\pi$
$199$	−12.9911	−0.920913	−0.460456	+	0.887682i	$0.652314\pi$
$200$	0	0
$201$	−2.44573	−0.172508
$202$	0	0
$203$	−0.497973	−0.0349509
$204$	0	0
$205$	−30.1479	−2.10562
$206$	0	0
$207$	24.2791	1.68751
$208$	0	0
$209$	1.88156	0.130150
$210$	0	0
$211$	5.68148	0.391129	0.195564	−	0.980691i	$-0.437346\pi$
$211$	5.68148	0.391129	0.195564	+	0.980691i	$0.437346\pi$
$212$	0	0
$213$	38.8438	2.66154
$214$	0	0
$215$	−13.4765	−0.919088
$216$	0	0
$217$	−10.0236	−0.680445
$218$	0	0
$219$	−0.239910	−0.0162116
$220$	0	0
$221$	−3.49868	−0.235347
$222$	0	0
$223$	−19.6897	−1.31852	−0.659260	−	0.751915i	$-0.729130\pi$
$223$	−19.6897	−1.31852	−0.659260	+	0.751915i	$0.729130\pi$
$224$	0	0
$225$	14.0396	0.935976
$226$	0	0
$227$	−4.52998	−0.300665	−0.150333	−	0.988635i	$-0.548034\pi$
$227$	−4.52998	−0.300665	−0.150333	+	0.988635i	$0.548034\pi$
$228$	0	0
$229$	21.6206	1.42873	0.714364	−	0.699774i	$-0.246716\pi$
$229$	21.6206	1.42873	0.714364	+	0.699774i	$0.246716\pi$
$230$	0	0
$231$	0.715265	0.0470610
$232$	0	0
$233$	−28.1191	−1.84214	−0.921071	−	0.389395i	$-0.872684\pi$
$233$	−28.1191	−1.84214	−0.921071	+	0.389395i	$0.872684\pi$
$234$	0	0
$235$	12.3050	0.802688
$236$	0	0
$237$	2.84169	0.184588
$238$	0	0
$239$	5.12585	0.331563	0.165782	−	0.986162i	$-0.446985\pi$
$239$	5.12585	0.331563	0.165782	+	0.986162i	$0.446985\pi$
$240$	0	0
$241$	16.2913	1.04942	0.524708	−	0.851283i	$-0.324175\pi$
$241$	16.2913	1.04942	0.524708	+	0.851283i	$0.324175\pi$
$242$	0	0
$243$	21.9970	1.41111
$244$	0	0
$245$	−18.3290	−1.17100
$246$	0	0
$247$	−20.6727	−1.31537
$248$	0	0
$249$	−2.80615	−0.177832
$250$	0	0
$251$	20.9946	1.32517	0.662584	−	0.748988i	$-0.269460\pi$
$251$	20.9946	1.32517	0.662584	+	0.748988i	$0.269460\pi$
$252$	0	0
$253$	−2.25597	−0.141832
$254$	0	0
$255$	8.77988	0.549817
$256$	0	0
$257$	20.7791	1.29617	0.648083	−	0.761570i	$-0.275571\pi$
$257$	20.7791	1.29617	0.648083	+	0.761570i	$0.275571\pi$
$258$	0	0
$259$	1.02090	0.0634354
$260$	0	0
$261$	−1.39904	−0.0865985
$262$	0	0
$263$	−2.32523	−0.143380	−0.0716899	−	0.997427i	$-0.522839\pi$
$263$	−2.32523	−0.143380	−0.0716899	+	0.997427i	$0.522839\pi$
$264$	0	0
$265$	−0.585554	−0.0359703
$266$	0	0
$267$	18.4464	1.12890
$268$	0	0
$269$	−16.3294	−0.995623	−0.497812	−	0.867285i	$-0.665863\pi$
$269$	−16.3294	−0.995623	−0.497812	+	0.867285i	$0.665863\pi$
$270$	0	0
$271$	−1.62614	−0.0987808	−0.0493904	−	0.998780i	$-0.515728\pi$
$271$	−1.62614	−0.0987808	−0.0493904	+	0.998780i	$0.515728\pi$
$272$	0	0
$273$	−7.85864	−0.475626
$274$	0	0
$275$	−1.30454	−0.0786666
$276$	0	0
$277$	−0.620729	−0.0372960	−0.0186480	−	0.999826i	$-0.505936\pi$
$277$	−0.620729	−0.0372960	−0.0186480	+	0.999826i	$0.505936\pi$
$278$	0	0
$279$	−28.1609	−1.68595
$280$	0	0
$281$	0.745909	0.0444972	0.0222486	−	0.999752i	$-0.492917\pi$
$281$	0.745909	0.0444972	0.0222486	+	0.999752i	$0.492917\pi$
$282$	0	0
$283$	25.3853	1.50900	0.754499	−	0.656302i	$-0.227880\pi$
$283$	25.3853	1.50900	0.754499	+	0.656302i	$0.227880\pi$
$284$	0	0
$285$	51.8777	3.07297
$286$	0	0
$287$	−10.2381	−0.604337
$288$	0	0
$289$	−15.6726	−0.921916
$290$	0	0
$291$	−18.9708	−1.11209
$292$	0	0
$293$	3.24794	0.189747	0.0948733	−	0.995489i	$-0.469755\pi$
$293$	3.24794	0.189747	0.0948733	+	0.995489i	$0.469755\pi$
$294$	0	0
$295$	−12.2846	−0.715237
$296$	0	0
$297$	−0.0172022	−0.000998172 0
$298$	0	0
$299$	24.7864	1.43343
$300$	0	0
$301$	−4.57656	−0.263789
$302$	0	0
$303$	41.7282	2.39722
$304$	0	0
$305$	3.34190	0.191357
$306$	0	0
$307$	12.6566	0.722351	0.361175	−	0.932498i	$-0.382376\pi$
$307$	12.6566	0.722351	0.361175	+	0.932498i	$0.382376\pi$
$308$	0	0
$309$	−17.4118	−0.990523
$310$	0	0
$311$	6.14123	0.348237	0.174119	−	0.984725i	$-0.444292\pi$
$311$	6.14123	0.348237	0.174119	+	0.984725i	$0.444292\pi$
$312$	0	0
$313$	9.41733	0.532299	0.266149	−	0.963932i	$-0.414249\pi$
$313$	9.41733	0.532299	0.266149	+	0.963932i	$0.414249\pi$
$314$	0	0
$315$	9.81852	0.553211
$316$	0	0
$317$	−9.29907	−0.522288	−0.261144	−	0.965300i	$-0.584100\pi$
$317$	−9.29907	−0.522288	−0.261144	+	0.965300i	$0.584100\pi$
$318$	0	0
$319$	0.129996	0.00727840
$320$	0	0
$321$	37.5084	2.09352
$322$	0	0
$323$	7.84335	0.436416
$324$	0	0
$325$	14.3330	0.795051
$326$	0	0
$327$	22.3729	1.23722
$328$	0	0
$329$	4.17873	0.230381
$330$	0	0
$331$	−31.9722	−1.75735	−0.878675	−	0.477420i	$-0.841572\pi$
$331$	−31.9722	−1.75735	−0.878675	+	0.477420i	$0.841572\pi$
$332$	0	0
$333$	2.86818	0.157175
$334$	0	0
$335$	3.11952	0.170438
$336$	0	0
$337$	4.53520	0.247048	0.123524	−	0.992342i	$-0.460580\pi$
$337$	4.53520	0.247048	0.123524	+	0.992342i	$0.460580\pi$
$338$	0	0
$339$	−1.09007	−0.0592043
$340$	0	0
$341$	2.61666	0.141700
$342$	0	0
$343$	−13.6357	−0.736260
$344$	0	0
$345$	−62.2010	−3.34879
$346$	0	0
$347$	11.8273	0.634922	0.317461	−	0.948271i	$-0.397170\pi$
$347$	11.8273	0.634922	0.317461	+	0.948271i	$0.397170\pi$
$348$	0	0
$349$	−1.22924	−0.0657997	−0.0328999	−	0.999459i	$-0.510474\pi$
$349$	−1.22924	−0.0657997	−0.0328999	+	0.999459i	$0.510474\pi$
$350$	0	0
$351$	0.189001	0.0100881
$352$	0	0
$353$	30.5968	1.62851	0.814253	−	0.580510i	$-0.197147\pi$
$353$	30.5968	1.62851	0.814253	+	0.580510i	$0.197147\pi$
$354$	0	0
$355$	−49.5452	−2.62959
$356$	0	0
$357$	2.98161	0.157804
$358$	0	0
$359$	−31.5497	−1.66513	−0.832564	−	0.553929i	$-0.813128\pi$
$359$	−31.5497	−1.66513	−0.832564	+	0.553929i	$0.813128\pi$
$360$	0	0
$361$	27.3441	1.43916
$362$	0	0
$363$	26.7004	1.40141
$364$	0	0
$365$	0.306004	0.0160170
$366$	0	0
$367$	−19.9703	−1.04244	−0.521220	−	0.853423i	$-0.674523\pi$
$367$	−19.9703	−1.04244	−0.521220	+	0.853423i	$0.674523\pi$
$368$	0	0
$369$	−28.7637	−1.49738
$370$	0	0
$371$	−0.198852	−0.0103239
$372$	0	0
$373$	−21.1843	−1.09688	−0.548441	−	0.836190i	$-0.684778\pi$
$373$	−21.1843	−1.09688	−0.548441	+	0.836190i	$0.684778\pi$
$374$	0	0
$375$	2.13420	0.110209
$376$	0	0
$377$	−1.42827	−0.0735598
$378$	0	0
$379$	−5.40698	−0.277738	−0.138869	−	0.990311i	$-0.544347\pi$
$379$	−5.40698	−0.277738	−0.138869	+	0.990311i	$0.544347\pi$
$380$	0	0
$381$	−0.148736	−0.00761998
$382$	0	0
$383$	33.3652	1.70488	0.852442	−	0.522822i	$-0.175121\pi$
$383$	33.3652	1.70488	0.852442	+	0.522822i	$0.175121\pi$
$384$	0	0
$385$	−0.912319	−0.0464961
$386$	0	0
$387$	−12.8577	−0.653594
$388$	0	0
$389$	−12.7076	−0.644303	−0.322151	−	0.946688i	$-0.604406\pi$
$389$	−12.7076	−0.644303	−0.322151	+	0.946688i	$0.604406\pi$
$390$	0	0
$391$	−9.40411	−0.475586
$392$	0	0
$393$	−19.2795	−0.972523
$394$	0	0
$395$	−3.62457	−0.182372
$396$	0	0
$397$	3.84092	0.192770	0.0963850	−	0.995344i	$-0.469272\pi$
$397$	3.84092	0.192770	0.0963850	+	0.995344i	$0.469272\pi$
$398$	0	0
$399$	17.6175	0.881978
$400$	0	0
$401$	12.1362	0.606055	0.303027	−	0.952982i	$-0.402003\pi$
$401$	12.1362	0.606055	0.303027	+	0.952982i	$0.402003\pi$
$402$	0	0
$403$	−28.7494	−1.43211
$404$	0	0
$405$	−28.2953	−1.40600
$406$	0	0
$407$	−0.266506	−0.0132102
$408$	0	0
$409$	−6.04417	−0.298865	−0.149432	−	0.988772i	$-0.547745\pi$
$409$	−6.04417	−0.298865	−0.149432	+	0.988772i	$0.547745\pi$
$410$	0	0
$411$	4.97663	0.245479
$412$	0	0
$413$	−4.17181	−0.205281
$414$	0	0
$415$	3.57924	0.175698
$416$	0	0
$417$	0.0747599	0.00366101
$418$	0	0
$419$	22.5850	1.10335	0.551676	−	0.834059i	$-0.313989\pi$
$419$	22.5850	1.10335	0.551676	+	0.834059i	$0.313989\pi$
$420$	0	0
$421$	−29.9232	−1.45837	−0.729185	−	0.684317i	$-0.760100\pi$
$421$	−29.9232	−1.45837	−0.729185	+	0.684317i	$0.760100\pi$
$422$	0	0
$423$	11.7400	0.570819
$424$	0	0
$425$	−5.43802	−0.263783
$426$	0	0
$427$	1.13490	0.0549215
$428$	0	0
$429$	2.05150	0.0990476
$430$	0	0
$431$	−25.8754	−1.24637	−0.623187	−	0.782073i	$-0.714163\pi$
$431$	−25.8754	−1.24637	−0.623187	+	0.782073i	$0.714163\pi$
$432$	0	0
$433$	−39.4966	−1.89808	−0.949042	−	0.315149i	$-0.897946\pi$
$433$	−39.4966	−1.89808	−0.949042	+	0.315149i	$0.897946\pi$
$434$	0	0
$435$	3.58422	0.171850
$436$	0	0
$437$	−55.5662	−2.65809
$438$	0	0
$439$	20.1240	0.960465	0.480232	−	0.877141i	$-0.340552\pi$
$439$	20.1240	0.960465	0.480232	+	0.877141i	$0.340552\pi$
$440$	0	0
$441$	−17.4874	−0.832734
$442$	0	0
$443$	−31.8393	−1.51273	−0.756365	−	0.654150i	$-0.773027\pi$
$443$	−31.8393	−1.51273	−0.756365	+	0.654150i	$0.773027\pi$
$444$	0	0
$445$	−23.5284	−1.11535
$446$	0	0
$447$	−8.65670	−0.409448
$448$	0	0
$449$	−34.5842	−1.63213	−0.816065	−	0.577960i	$-0.803849\pi$
$449$	−34.5842	−1.63213	−0.816065	+	0.577960i	$0.803849\pi$
$450$	0	0
$451$	2.67267	0.125851
$452$	0	0
$453$	−59.6775	−2.80390
$454$	0	0
$455$	10.0237	0.469917
$456$	0	0
$457$	−13.8410	−0.647455	−0.323728	−	0.946150i	$-0.604936\pi$
$457$	−13.8410	−0.647455	−0.323728	+	0.946150i	$0.604936\pi$
$458$	0	0
$459$	−0.0717081	−0.00334705
$460$	0	0
$461$	−11.2664	−0.524729	−0.262365	−	0.964969i	$-0.584502\pi$
$461$	−11.2664	−0.524729	−0.262365	+	0.964969i	$0.584502\pi$
$462$	0	0
$463$	−35.2585	−1.63860	−0.819301	−	0.573363i	$-0.805638\pi$
$463$	−35.2585	−1.63860	−0.819301	+	0.573363i	$0.805638\pi$
$464$	0	0
$465$	72.1459	3.34569
$466$	0	0
$467$	−28.0194	−1.29658	−0.648292	−	0.761392i	$-0.724516\pi$
$467$	−28.0194	−1.29658	−0.648292	+	0.761392i	$0.724516\pi$
$468$	0	0
$469$	1.05938	0.0489175
$470$	0	0
$471$	−49.0942	−2.26214
$472$	0	0
$473$	1.19472	0.0549331
$474$	0	0
$475$	−32.1317	−1.47430
$476$	0	0
$477$	−0.558669	−0.0255797
$478$	0	0
$479$	−2.12618	−0.0971476	−0.0485738	−	0.998820i	$-0.515468\pi$
$479$	−2.12618	−0.0971476	−0.0485738	+	0.998820i	$0.515468\pi$
$480$	0	0
$481$	2.92811	0.133510
$482$	0	0
$483$	−21.1232	−0.961140
$484$	0	0
$485$	24.1972	1.09874
$486$	0	0
$487$	−14.4187	−0.653374	−0.326687	−	0.945133i	$-0.605932\pi$
$487$	−14.4187	−0.653374	−0.326687	+	0.945133i	$0.605932\pi$
$488$	0	0
$489$	−1.87453	−0.0847691
$490$	0	0
$491$	27.8549	1.25707	0.628537	−	0.777780i	$-0.283654\pi$
$491$	27.8549	1.25707	0.628537	+	0.777780i	$0.283654\pi$
$492$	0	0
$493$	0.541896	0.0244058
$494$	0	0
$495$	−2.56313	−0.115204
$496$	0	0
$497$	−16.8254	−0.754722
$498$	0	0
$499$	31.5072	1.41046	0.705228	−	0.708980i	$-0.250845\pi$
$499$	31.5072	1.41046	0.705228	+	0.708980i	$0.250845\pi$
$500$	0	0
$501$	16.6937	0.745819
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−53.2242	−2.36845
$506$	0	0
$507$	9.23579	0.410176
$508$	0	0
$509$	−9.18764	−0.407235	−0.203618	−	0.979050i	$-0.565270\pi$
$509$	−9.18764	−0.407235	−0.203618	+	0.979050i	$0.565270\pi$
$510$	0	0
$511$	0.103918	0.00459706
$512$	0	0
$513$	−0.423702	−0.0187069
$514$	0	0
$515$	22.2087	0.978633
$516$	0	0
$517$	−1.09086	−0.0479760
$518$	0	0
$519$	27.2984	1.19826
$520$	0	0
$521$	−4.85097	−0.212525	−0.106262	−	0.994338i	$-0.533888\pi$
$521$	−4.85097	−0.212525	−0.106262	+	0.994338i	$0.533888\pi$
$522$	0	0
$523$	−1.06963	−0.0467717	−0.0233859	−	0.999727i	$-0.507445\pi$
$523$	−1.06963	−0.0467717	−0.0233859	+	0.999727i	$0.507445\pi$
$524$	0	0
$525$	−12.2147	−0.533094
$526$	0	0
$527$	10.9077	0.475146
$528$	0	0
$529$	43.6233	1.89667
$530$	0	0
$531$	−11.7206	−0.508629
$532$	0	0
$533$	−29.3647	−1.27193
$534$	0	0
$535$	−47.8419	−2.06839
$536$	0	0
$537$	−44.1258	−1.90417
$538$	0	0
$539$	1.62490	0.0699894
$540$	0	0
$541$	−4.43382	−0.190625	−0.0953124	−	0.995447i	$-0.530385\pi$
$541$	−4.43382	−0.190625	−0.0953124	+	0.995447i	$0.530385\pi$
$542$	0	0
$543$	12.1960	0.523380
$544$	0	0
$545$	−28.5365	−1.22237
$546$	0	0
$547$	11.9757	0.512046	0.256023	−	0.966671i	$-0.417588\pi$
$547$	11.9757	0.512046	0.256023	+	0.966671i	$0.417588\pi$
$548$	0	0
$549$	3.18846	0.136080
$550$	0	0
$551$	3.20190	0.136406
$552$	0	0
$553$	−1.23089	−0.0523429
$554$	0	0
$555$	−7.34803	−0.311906
$556$	0	0
$557$	−9.62893	−0.407991	−0.203995	−	0.978972i	$-0.565393\pi$
$557$	−9.62893	−0.407991	−0.203995	+	0.978972i	$0.565393\pi$
$558$	0	0
$559$	−13.1264	−0.555186
$560$	0	0
$561$	−0.778353	−0.0328621
$562$	0	0
$563$	24.6699	1.03971	0.519857	−	0.854253i	$-0.325985\pi$
$563$	24.6699	1.03971	0.519857	+	0.854253i	$0.325985\pi$
$564$	0	0
$565$	1.39038	0.0584936
$566$	0	0
$567$	−9.60897	−0.403539
$568$	0	0
$569$	−35.5110	−1.48870	−0.744348	−	0.667792i	$-0.767240\pi$
$569$	−35.5110	−1.48870	−0.744348	+	0.667792i	$0.767240\pi$
$570$	0	0
$571$	39.6208	1.65808	0.829040	−	0.559190i	$-0.188888\pi$
$571$	39.6208	1.65808	0.829040	+	0.559190i	$0.188888\pi$
$572$	0	0
$573$	17.6513	0.737395
$574$	0	0
$575$	38.5256	1.60663
$576$	0	0
$577$	22.9848	0.956868	0.478434	−	0.878123i	$-0.341205\pi$
$577$	22.9848	0.956868	0.478434	+	0.878123i	$0.341205\pi$
$578$	0	0
$579$	3.04188	0.126416
$580$	0	0
$581$	1.21550	0.0504273
$582$	0	0
$583$	0.0519105	0.00214991
$584$	0	0
$585$	28.1612	1.16432
$586$	0	0
$587$	−21.3179	−0.879886	−0.439943	−	0.898026i	$-0.645001\pi$
$587$	−21.3179	−0.879886	−0.439943	+	0.898026i	$0.645001\pi$
$588$	0	0
$589$	64.4503	2.65563
$590$	0	0
$591$	−42.0871	−1.73123
$592$	0	0
$593$	25.3210	1.03981	0.519905	−	0.854224i	$-0.325967\pi$
$593$	25.3210	1.03981	0.519905	+	0.854224i	$0.325967\pi$
$594$	0	0
$595$	−3.80304	−0.155910
$596$	0	0
$597$	31.7539	1.29960
$598$	0	0
$599$	−31.4467	−1.28488	−0.642440	−	0.766336i	$-0.722078\pi$
$599$	−31.4467	−1.28488	−0.642440	+	0.766336i	$0.722078\pi$
$600$	0	0
$601$	−25.0254	−1.02081	−0.510403	−	0.859935i	$-0.670504\pi$
$601$	−25.0254	−1.02081	−0.510403	+	0.859935i	$0.670504\pi$
$602$	0	0
$603$	2.97629	0.121204
$604$	0	0
$605$	−34.0563	−1.38459
$606$	0	0
$607$	−17.7298	−0.719632	−0.359816	−	0.933023i	$-0.617161\pi$
$607$	−17.7298	−0.719632	−0.359816	+	0.933023i	$0.617161\pi$
$608$	0	0
$609$	1.21719	0.0493230
$610$	0	0
$611$	11.9853	0.484874
$612$	0	0
$613$	11.9308	0.481880	0.240940	−	0.970540i	$-0.422544\pi$
$613$	11.9308	0.481880	0.240940	+	0.970540i	$0.422544\pi$
$614$	0	0
$615$	73.6900	2.97147
$616$	0	0
$617$	−27.1907	−1.09466	−0.547329	−	0.836918i	$-0.684355\pi$
$617$	−27.1907	−1.09466	−0.547329	+	0.836918i	$0.684355\pi$
$618$	0	0
$619$	−20.4989	−0.823922	−0.411961	−	0.911202i	$-0.635156\pi$
$619$	−20.4989	−0.823922	−0.411961	+	0.911202i	$0.635156\pi$
$620$	0	0
$621$	0.508016	0.0203860
$622$	0	0
$623$	−7.99016	−0.320119
$624$	0	0
$625$	−26.3219	−1.05287
$626$	0	0
$627$	−4.59906	−0.183669
$628$	0	0
$629$	−1.11094	−0.0442962
$630$	0	0
$631$	−19.5611	−0.778717	−0.389359	−	0.921086i	$-0.627303\pi$
$631$	−19.5611	−0.778717	−0.389359	+	0.921086i	$0.627303\pi$
$632$	0	0
$633$	−13.8872	−0.551965
$634$	0	0
$635$	0.189712	0.00752851
$636$	0	0
$637$	−17.8528	−0.707354
$638$	0	0
$639$	−47.2704	−1.86999
$640$	0	0
$641$	8.68502	0.343038	0.171519	−	0.985181i	$-0.445133\pi$
$641$	8.68502	0.343038	0.171519	+	0.985181i	$0.445133\pi$
$642$	0	0
$643$	−11.4019	−0.449647	−0.224823	−	0.974400i	$-0.572181\pi$
$643$	−11.4019	−0.449647	−0.224823	+	0.974400i	$0.572181\pi$
$644$	0	0
$645$	32.9404	1.29703
$646$	0	0
$647$	−8.81736	−0.346646	−0.173323	−	0.984865i	$-0.555451\pi$
$647$	−8.81736	−0.346646	−0.173323	+	0.984865i	$0.555451\pi$
$648$	0	0
$649$	1.08905	0.0427491
$650$	0	0
$651$	24.5005	0.960250
$652$	0	0
$653$	−30.6511	−1.19947	−0.599735	−	0.800199i	$-0.704727\pi$
$653$	−30.6511	−1.19947	−0.599735	+	0.800199i	$0.704727\pi$
$654$	0	0
$655$	24.5910	0.960849
$656$	0	0
$657$	0.291955	0.0113902
$658$	0	0
$659$	43.2181	1.68354	0.841770	−	0.539837i	$-0.181514\pi$
$659$	43.2181	1.68354	0.841770	+	0.539837i	$0.181514\pi$
$660$	0	0
$661$	−27.2617	−1.06036	−0.530179	−	0.847886i	$-0.677875\pi$
$661$	−27.2617	−1.06036	−0.530179	+	0.847886i	$0.677875\pi$
$662$	0	0
$663$	8.55179	0.332124
$664$	0	0
$665$	−22.4711	−0.871391
$666$	0	0
$667$	−3.83906	−0.148649
$668$	0	0
$669$	48.1272	1.86071
$670$	0	0
$671$	−0.296266	−0.0114372
$672$	0	0
$673$	−22.0821	−0.851201	−0.425601	−	0.904911i	$-0.639937\pi$
$673$	−22.0821	−0.851201	−0.425601	+	0.904911i	$0.639937\pi$
$674$	0	0
$675$	0.293765	0.0113070
$676$	0	0
$677$	−4.43187	−0.170331	−0.0851653	−	0.996367i	$-0.527142\pi$
$677$	−4.43187	−0.170331	−0.0851653	+	0.996367i	$0.527142\pi$
$678$	0	0
$679$	8.21729	0.315351
$680$	0	0
$681$	11.0726	0.424302
$682$	0	0
$683$	−35.9289	−1.37478	−0.687391	−	0.726287i	$-0.741244\pi$
$683$	−35.9289	−1.37478	−0.687391	+	0.726287i	$0.741244\pi$
$684$	0	0
$685$	−6.34768	−0.242532
$686$	0	0
$687$	−52.8469	−2.01623
$688$	0	0
$689$	−0.570343	−0.0217283
$690$	0	0
$691$	−50.1691	−1.90852	−0.954261	−	0.298975i	$-0.903355\pi$
$691$	−50.1691	−1.90852	−0.954261	+	0.298975i	$0.903355\pi$
$692$	0	0
$693$	−0.870431	−0.0330649
$694$	0	0
$695$	−0.0953561	−0.00361706
$696$	0	0
$697$	11.1411	0.422001
$698$	0	0
$699$	68.7311	2.59965
$700$	0	0
$701$	42.7688	1.61536	0.807679	−	0.589623i	$-0.200724\pi$
$701$	42.7688	1.61536	0.807679	+	0.589623i	$0.200724\pi$
$702$	0	0
$703$	−6.56424	−0.247575
$704$	0	0
$705$	−30.0769	−1.13276
$706$	0	0
$707$	−18.0748	−0.679771
$708$	0	0
$709$	30.6970	1.15285	0.576425	−	0.817150i	$-0.304447\pi$
$709$	30.6970	1.15285	0.576425	+	0.817150i	$0.304447\pi$
$710$	0	0
$711$	−3.45816	−0.129691
$712$	0	0
$713$	−77.2753	−2.89398
$714$	0	0
$715$	−2.61669	−0.0978586
$716$	0	0
$717$	−12.5290	−0.467906
$718$	0	0
$719$	−40.8172	−1.52223	−0.761113	−	0.648620i	$-0.775347\pi$
$719$	−40.8172	−1.52223	−0.761113	+	0.648620i	$0.775347\pi$
$720$	0	0
$721$	7.54200	0.280879
$722$	0	0
$723$	−39.8206	−1.48094
$724$	0	0
$725$	−2.21997	−0.0824477
$726$	0	0
$727$	−19.0601	−0.706899	−0.353450	−	0.935454i	$-0.614991\pi$
$727$	−19.0601	−0.706899	−0.353450	+	0.935454i	$0.614991\pi$
$728$	0	0
$729$	−26.5397	−0.982953
$730$	0	0
$731$	4.98022	0.184200
$732$	0	0
$733$	18.3940	0.679397	0.339699	−	0.940534i	$-0.389675\pi$
$733$	18.3940	0.679397	0.339699	+	0.940534i	$0.389675\pi$
$734$	0	0
$735$	44.8013	1.65252
$736$	0	0
$737$	−0.276552	−0.0101869
$738$	0	0
$739$	14.0525	0.516929	0.258464	−	0.966021i	$-0.416784\pi$
$739$	14.0525	0.516929	0.258464	+	0.966021i	$0.416784\pi$
$740$	0	0
$741$	50.5300	1.85627
$742$	0	0
$743$	−1.57408	−0.0577473	−0.0288737	−	0.999583i	$-0.509192\pi$
$743$	−1.57408	−0.0577473	−0.0288737	+	0.999583i	$0.509192\pi$
$744$	0	0
$745$	11.0416	0.404533
$746$	0	0
$747$	3.41490	0.124945
$748$	0	0
$749$	−16.2469	−0.593650
$750$	0	0
$751$	24.6073	0.897932	0.448966	−	0.893549i	$-0.351792\pi$
$751$	24.6073	0.897932	0.448966	+	0.893549i	$0.351792\pi$
$752$	0	0
$753$	−51.3168	−1.87009
$754$	0	0
$755$	76.1186	2.77024
$756$	0	0
$757$	37.3779	1.35852	0.679262	−	0.733896i	$-0.262300\pi$
$757$	37.3779	1.35852	0.679262	+	0.733896i	$0.262300\pi$
$758$	0	0
$759$	5.51424	0.200154
$760$	0	0
$761$	32.8067	1.18924	0.594622	−	0.804006i	$-0.297302\pi$
$761$	32.8067	1.18924	0.594622	+	0.804006i	$0.297302\pi$
$762$	0	0
$763$	−9.69090	−0.350834
$764$	0	0
$765$	−10.6845	−0.386300
$766$	0	0
$767$	−11.9655	−0.432048
$768$	0	0
$769$	−22.0785	−0.796172	−0.398086	−	0.917348i	$-0.630325\pi$
$769$	−22.0785	−0.796172	−0.398086	+	0.917348i	$0.630325\pi$
$770$	0	0
$771$	−50.7901	−1.82916
$772$	0	0
$773$	41.2993	1.48543	0.742717	−	0.669605i	$-0.233537\pi$
$773$	41.2993	1.48543	0.742717	+	0.669605i	$0.233537\pi$
$774$	0	0
$775$	−44.6853	−1.60514
$776$	0	0
$777$	−2.49536	−0.0895207
$778$	0	0
$779$	65.8297	2.35859
$780$	0	0
$781$	4.39228	0.157168
$782$	0	0
$783$	−0.0292735	−0.00104615
$784$	0	0
$785$	62.6195	2.23498
$786$	0	0
$787$	43.5177	1.55124	0.775620	−	0.631200i	$-0.217437\pi$
$787$	43.5177	1.55124	0.775620	+	0.631200i	$0.217437\pi$
$788$	0	0
$789$	5.68353	0.202339
$790$	0	0
$791$	0.472167	0.0167883
$792$	0	0
$793$	3.25508	0.115591
$794$	0	0
$795$	1.43126	0.0507617
$796$	0	0
$797$	4.27533	0.151440	0.0757199	−	0.997129i	$-0.475875\pi$
$797$	4.27533	0.151440	0.0757199	+	0.997129i	$0.475875\pi$
$798$	0	0
$799$	−4.54730	−0.160872
$800$	0	0
$801$	−22.4481	−0.793165
$802$	0	0
$803$	−0.0271279	−0.000957322 0
$804$	0	0
$805$	26.9426	0.949603
$806$	0	0
$807$	39.9138	1.40503
$808$	0	0
$809$	0.813101	0.0285871	0.0142936	−	0.999898i	$-0.495450\pi$
$809$	0.813101	0.0285871	0.0142936	+	0.999898i	$0.495450\pi$
$810$	0	0
$811$	−40.7366	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$811$	−40.7366	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$812$	0	0
$813$	3.97475	0.139400
$814$	0	0
$815$	2.39096	0.0837516
$816$	0	0
$817$	29.4267	1.02951
$818$	0	0
$819$	9.56345	0.334174
$820$	0	0
$821$	−11.7607	−0.410452	−0.205226	−	0.978715i	$-0.565793\pi$
$821$	−11.7607	−0.410452	−0.205226	+	0.978715i	$0.565793\pi$
$822$	0	0
$823$	47.7921	1.66593	0.832963	−	0.553329i	$-0.186643\pi$
$823$	47.7921	1.66593	0.832963	+	0.553329i	$0.186643\pi$
$824$	0	0
$825$	3.18866	0.111015
$826$	0	0
$827$	−0.128571	−0.00447085	−0.00223543	−	0.999998i	$-0.500712\pi$
$827$	−0.128571	−0.00447085	−0.00223543	+	0.999998i	$0.500712\pi$
$828$	0	0
$829$	1.62250	0.0563516	0.0281758	−	0.999603i	$-0.491030\pi$
$829$	1.62250	0.0563516	0.0281758	+	0.999603i	$0.491030\pi$
$830$	0	0
$831$	1.51724	0.0526325
$832$	0	0
$833$	6.77347	0.234687
$834$	0	0
$835$	−21.2928	−0.736866
$836$	0	0
$837$	−0.589239	−0.0203671
$838$	0	0
$839$	−48.5816	−1.67722	−0.838612	−	0.544729i	$-0.816633\pi$
$839$	−48.5816	−1.67722	−0.838612	+	0.544729i	$0.816633\pi$
$840$	0	0
$841$	−28.7788	−0.992372
$842$	0	0
$843$	−1.82322	−0.0627949
$844$	0	0
$845$	−11.7802	−0.405252
$846$	0	0
$847$	−11.5654	−0.397392
$848$	0	0
$849$	−62.0489	−2.12951
$850$	0	0
$851$	7.87046	0.269796
$852$	0	0
$853$	18.9867	0.650092	0.325046	−	0.945698i	$-0.394620\pi$
$853$	18.9867	0.650092	0.325046	+	0.945698i	$0.394620\pi$
$854$	0	0
$855$	−63.1318	−2.15906
$856$	0	0
$857$	7.82102	0.267161	0.133581	−	0.991038i	$-0.457353\pi$
$857$	7.82102	0.267161	0.133581	+	0.991038i	$0.457353\pi$
$858$	0	0
$859$	40.8407	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$859$	40.8407	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$860$	0	0
$861$	25.0249	0.852845
$862$	0	0
$863$	52.2611	1.77899	0.889494	−	0.456947i	$-0.151057\pi$
$863$	52.2611	1.77899	0.889494	+	0.456947i	$0.151057\pi$
$864$	0	0
$865$	−34.8190	−1.18388
$866$	0	0
$867$	38.3083	1.30102
$868$	0	0
$869$	0.321325	0.0109002
$870$	0	0
$871$	3.03848	0.102955
$872$	0	0
$873$	23.0862	0.781351
$874$	0	0
$875$	−0.924437	−0.0312517
$876$	0	0
$877$	−20.4381	−0.690145	−0.345072	−	0.938576i	$-0.612146\pi$
$877$	−20.4381	−0.690145	−0.345072	+	0.938576i	$0.612146\pi$
$878$	0	0
$879$	−7.93890	−0.267772
$880$	0	0
$881$	40.9096	1.37828	0.689140	−	0.724628i	$-0.257988\pi$
$881$	40.9096	1.37828	0.689140	+	0.724628i	$0.257988\pi$
$882$	0	0
$883$	−31.0930	−1.04636	−0.523181	−	0.852221i	$-0.675255\pi$
$883$	−31.0930	−1.04636	−0.523181	+	0.852221i	$0.675255\pi$
$884$	0	0
$885$	30.0271	1.00935
$886$	0	0
$887$	−20.1819	−0.677641	−0.338820	−	0.940851i	$-0.610028\pi$
$887$	−20.1819	−0.677641	−0.338820	+	0.940851i	$0.610028\pi$
$888$	0	0
$889$	0.0644256	0.00216077
$890$	0	0
$891$	2.50843	0.0840356
$892$	0	0
$893$	−26.8687	−0.899126
$894$	0	0
$895$	56.2823	1.88131
$896$	0	0
$897$	−60.5851	−2.02288
$898$	0	0
$899$	4.45286	0.148511
$900$	0	0
$901$	0.216391	0.00720904
$902$	0	0
$903$	11.1864	0.372261
$904$	0	0
$905$	−15.5559	−0.517097
$906$	0	0
$907$	0.517074	0.0171692	0.00858458	−	0.999963i	$-0.497267\pi$
$907$	0.517074	0.0171692	0.00858458	+	0.999963i	$0.497267\pi$
$908$	0	0
$909$	−50.7805	−1.68428
$910$	0	0
$911$	−27.4013	−0.907845	−0.453923	−	0.891041i	$-0.649976\pi$
$911$	−27.4013	−0.907845	−0.453923	+	0.891041i	$0.649976\pi$
$912$	0	0
$913$	−0.317306	−0.0105013
$914$	0	0
$915$	−8.16856	−0.270044
$916$	0	0
$917$	8.35101	0.275775
$918$	0	0
$919$	−17.9762	−0.592982	−0.296491	−	0.955036i	$-0.595816\pi$
$919$	−17.9762	−0.592982	−0.296491	+	0.955036i	$0.595816\pi$
$920$	0	0
$921$	−30.9364	−1.01939
$922$	0	0
$923$	−48.2581	−1.58844
$924$	0	0
$925$	4.55117	0.149642
$926$	0	0
$927$	21.1890	0.695939
$928$	0	0
$929$	−8.97943	−0.294606	−0.147303	−	0.989091i	$-0.547059\pi$
$929$	−8.97943	−0.294606	−0.147303	+	0.989091i	$0.547059\pi$
$930$	0	0
$931$	40.0224	1.31168
$932$	0	0
$933$	−15.0109	−0.491436
$934$	0	0
$935$	0.992787	0.0324676
$936$	0	0
$937$	−19.4867	−0.636603	−0.318302	−	0.947989i	$-0.603112\pi$
$937$	−19.4867	−0.636603	−0.318302	+	0.947989i	$0.603112\pi$
$938$	0	0
$939$	−23.0186	−0.751185
$940$	0	0
$941$	39.0622	1.27339	0.636696	−	0.771115i	$-0.280301\pi$
$941$	39.0622	1.27339	0.636696	+	0.771115i	$0.280301\pi$
$942$	0	0
$943$	−78.9293	−2.57029
$944$	0	0
$945$	0.205443	0.00668305
$946$	0	0
$947$	19.3629	0.629209	0.314604	−	0.949223i	$-0.398128\pi$
$947$	19.3629	0.629209	0.314604	+	0.949223i	$0.398128\pi$
$948$	0	0
$949$	0.298055	0.00967527
$950$	0	0
$951$	22.7296	0.737057
$952$	0	0
$953$	−15.8870	−0.514630	−0.257315	−	0.966328i	$-0.582838\pi$
$953$	−15.8870	−0.514630	−0.257315	+	0.966328i	$0.582838\pi$
$954$	0	0
$955$	−22.5142	−0.728543
$956$	0	0
$957$	−0.317748	−0.0102713
$958$	0	0
$959$	−2.15565	−0.0696095
$960$	0	0
$961$	58.6304	1.89130
$962$	0	0
$963$	−45.6453	−1.47090
$964$	0	0
$965$	−3.87991	−0.124899
$966$	0	0
$967$	−50.7896	−1.63328	−0.816642	−	0.577145i	$-0.804167\pi$
$967$	−50.7896	−1.63328	−0.816642	+	0.577145i	$0.804167\pi$
$968$	0	0
$969$	−19.1714	−0.615874
$970$	0	0
$971$	−10.7112	−0.343737	−0.171869	−	0.985120i	$-0.554980\pi$
$971$	−10.7112	−0.343737	−0.171869	+	0.985120i	$0.554980\pi$
$972$	0	0
$973$	−0.0323826	−0.00103814
$974$	0	0
$975$	−35.0339	−1.12198
$976$	0	0
$977$	20.3973	0.652567	0.326284	−	0.945272i	$-0.394204\pi$
$977$	20.3973	0.652567	0.326284	+	0.945272i	$0.394204\pi$
$978$	0	0
$979$	2.08584	0.0666637
$980$	0	0
$981$	−27.2263	−0.869269
$982$	0	0
$983$	−6.98986	−0.222942	−0.111471	−	0.993768i	$-0.535556\pi$
$983$	−6.98986	−0.222942	−0.111471	+	0.993768i	$0.535556\pi$
$984$	0	0
$985$	53.6819	1.71045
$986$	0	0
$987$	−10.2140	−0.325115
$988$	0	0
$989$	−35.2824	−1.12191
$990$	0	0
$991$	−28.9923	−0.920972	−0.460486	−	0.887667i	$-0.652325\pi$
$991$	−28.9923	−0.920972	−0.460486	+	0.887667i	$0.652325\pi$
$992$	0	0
$993$	78.1492	2.47999
$994$	0	0
$995$	−40.5020	−1.28400
$996$	0	0
$997$	−0.137773	−0.00436332	−0.00218166	−	0.999998i	$-0.500694\pi$
$997$	−0.137773	−0.00436332	−0.00218166	+	0.999998i	$0.500694\pi$
$998$	0	0
$999$	0.0600138	0.00189875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.s.1.5		21
4.3	odd	2		2012.2.a.b.1.17	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.17	✓	21	4.3	odd	2
8048.2.a.s.1.5		21	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)

\(f(q)\)	\(=\)	\(q-2.44429 q^{3} +3.11768 q^{5} +1.05875 q^{7} +2.97454 q^{9} +O(q^{10})\)	\(q-2.44429 q^{3} +3.11768 q^{5} +1.05875 q^{7} +2.97454 q^{9} -0.276389 q^{11} +3.03669 q^{13} -7.62051 q^{15} -1.15214 q^{17} -6.80765 q^{19} -2.58790 q^{21} +8.16231 q^{23} +4.71994 q^{25} +0.0622392 q^{27} -0.470339 q^{29} -9.46734 q^{31} +0.675573 q^{33} +3.30086 q^{35} +0.964244 q^{37} -7.42254 q^{39} -9.66997 q^{41} -4.32259 q^{43} +9.27366 q^{45} +3.94684 q^{47} -5.87904 q^{49} +2.81616 q^{51} -0.187817 q^{53} -0.861691 q^{55} +16.6398 q^{57} -3.94030 q^{59} +1.07192 q^{61} +3.14930 q^{63} +9.46743 q^{65} +1.00059 q^{67} -19.9510 q^{69} -15.8917 q^{71} +0.0981513 q^{73} -11.5369 q^{75} -0.292627 q^{77} -1.16259 q^{79} -9.07574 q^{81} +1.14804 q^{83} -3.59200 q^{85} +1.14964 q^{87} -7.54676 q^{89} +3.21511 q^{91} +23.1409 q^{93} -21.2241 q^{95} +7.76129 q^{97} -0.822128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

Introduction

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Modular forms

Varieties

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