LMFDB - Embedded newform 8048.2.a.x.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.05277 q^{3} -1.97996 q^{5} -4.35506 q^{7} -1.89168 q^{9} +O(q^{10})$	$q-1.05277 q^{3} -1.97996 q^{5} -4.35506 q^{7} -1.89168 q^{9} +3.68201 q^{11} -3.40459 q^{13} +2.08444 q^{15} +0.913284 q^{17} +2.61464 q^{19} +4.58486 q^{21} -3.15857 q^{23} -1.07975 q^{25} +5.14980 q^{27} +9.69811 q^{29} -7.70856 q^{31} -3.87630 q^{33} +8.62285 q^{35} +0.206922 q^{37} +3.58424 q^{39} +8.61588 q^{41} +5.42378 q^{43} +3.74546 q^{45} -8.71763 q^{47} +11.9665 q^{49} -0.961476 q^{51} -2.86324 q^{53} -7.29023 q^{55} -2.75261 q^{57} -11.4512 q^{59} +11.9633 q^{61} +8.23837 q^{63} +6.74096 q^{65} +9.74197 q^{67} +3.32524 q^{69} +1.93633 q^{71} +7.38898 q^{73} +1.13672 q^{75} -16.0353 q^{77} -0.551502 q^{79} +0.253493 q^{81} -5.03932 q^{83} -1.80827 q^{85} -10.2099 q^{87} +15.7210 q^{89} +14.8272 q^{91} +8.11532 q^{93} -5.17689 q^{95} -0.889467 q^{97} -6.96518 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.05277	−0.607816	−0.303908	−	0.952701i	$-0.598292\pi$
$3$	−1.05277	−0.607816	−0.303908	+	0.952701i	$0.598292\pi$
$4$	0	0
$5$	−1.97996	−0.885466	−0.442733	−	0.896653i	$-0.645991\pi$
$5$	−1.97996	−0.885466	−0.442733	+	0.896653i	$0.645991\pi$
$6$	0	0
$7$	−4.35506	−1.64606	−0.823028	−	0.568000i	$-0.807717\pi$
$7$	−4.35506	−1.64606	−0.823028	+	0.568000i	$0.807717\pi$
$8$	0	0
$9$	−1.89168	−0.630560
$10$	0	0
$11$	3.68201	1.11017	0.555083	−	0.831795i	$-0.312686\pi$
$11$	3.68201	1.11017	0.555083	+	0.831795i	$0.312686\pi$
$12$	0	0
$13$	−3.40459	−0.944263	−0.472131	−	0.881528i	$-0.656515\pi$
$13$	−3.40459	−0.944263	−0.472131	+	0.881528i	$0.656515\pi$
$14$	0	0
$15$	2.08444	0.538200
$16$	0	0
$17$	0.913284	0.221504	0.110752	−	0.993848i	$-0.464674\pi$
$17$	0.913284	0.221504	0.110752	+	0.993848i	$0.464674\pi$
$18$	0	0
$19$	2.61464	0.599840	0.299920	−	0.953964i	$-0.403040\pi$
$19$	2.61464	0.599840	0.299920	+	0.953964i	$0.403040\pi$
$20$	0	0
$21$	4.58486	1.00050
$22$	0	0
$23$	−3.15857	−0.658607	−0.329303	−	0.944224i	$-0.606814\pi$
$23$	−3.15857	−0.658607	−0.329303	+	0.944224i	$0.606814\pi$
$24$	0	0
$25$	−1.07975	−0.215950
$26$	0	0
$27$	5.14980	0.991080
$28$	0	0
$29$	9.69811	1.80089	0.900447	−	0.434967i	$-0.143240\pi$
$29$	9.69811	1.80089	0.900447	+	0.434967i	$0.143240\pi$
$30$	0	0
$31$	−7.70856	−1.38450	−0.692249	−	0.721659i	$-0.743380\pi$
$31$	−7.70856	−1.38450	−0.692249	+	0.721659i	$0.743380\pi$
$32$	0	0
$33$	−3.87630	−0.674777
$34$	0	0
$35$	8.62285	1.45753
$36$	0	0
$37$	0.206922	0.0340178	0.0170089	−	0.999855i	$-0.494586\pi$
$37$	0.206922	0.0340178	0.0170089	+	0.999855i	$0.494586\pi$
$38$	0	0
$39$	3.58424	0.573938
$40$	0	0
$41$	8.61588	1.34557	0.672787	−	0.739836i	$-0.265097\pi$
$41$	8.61588	1.34557	0.672787	+	0.739836i	$0.265097\pi$
$42$	0	0
$43$	5.42378	0.827118	0.413559	−	0.910477i	$-0.364285\pi$
$43$	5.42378	0.827118	0.413559	+	0.910477i	$0.364285\pi$
$44$	0	0
$45$	3.74546	0.558340
$46$	0	0
$47$	−8.71763	−1.27160	−0.635799	−	0.771855i	$-0.719329\pi$
$47$	−8.71763	−1.27160	−0.635799	+	0.771855i	$0.719329\pi$
$48$	0	0
$49$	11.9665	1.70950
$50$	0	0
$51$	−0.961476	−0.134634
$52$	0	0
$53$	−2.86324	−0.393296	−0.196648	−	0.980474i	$-0.563006\pi$
$53$	−2.86324	−0.393296	−0.196648	+	0.980474i	$0.563006\pi$
$54$	0	0
$55$	−7.29023	−0.983015
$56$	0	0
$57$	−2.75261	−0.364592
$58$	0	0
$59$	−11.4512	−1.49082	−0.745412	−	0.666604i	$-0.767747\pi$
$59$	−11.4512	−1.49082	−0.745412	+	0.666604i	$0.767747\pi$
$60$	0	0
$61$	11.9633	1.53175	0.765873	−	0.642992i	$-0.222307\pi$
$61$	11.9633	1.53175	0.765873	+	0.642992i	$0.222307\pi$
$62$	0	0
$63$	8.23837	1.03794
$64$	0	0
$65$	6.74096	0.836113
$66$	0	0
$67$	9.74197	1.19017	0.595086	−	0.803662i	$-0.297118\pi$
$67$	9.74197	1.19017	0.595086	+	0.803662i	$0.297118\pi$
$68$	0	0
$69$	3.32524	0.400312
$70$	0	0
$71$	1.93633	0.229800	0.114900	−	0.993377i	$-0.463345\pi$
$71$	1.93633	0.229800	0.114900	+	0.993377i	$0.463345\pi$
$72$	0	0
$73$	7.38898	0.864815	0.432407	−	0.901678i	$-0.357664\pi$
$73$	7.38898	0.864815	0.432407	+	0.901678i	$0.357664\pi$
$74$	0	0
$75$	1.13672	0.131258
$76$	0	0
$77$	−16.0353	−1.82740
$78$	0	0
$79$	−0.551502	−0.0620489	−0.0310244	−	0.999519i	$-0.509877\pi$
$79$	−0.551502	−0.0620489	−0.0310244	+	0.999519i	$0.509877\pi$
$80$	0	0
$81$	0.253493	0.0281659
$82$	0	0
$83$	−5.03932	−0.553137	−0.276568	−	0.960994i	$-0.589197\pi$
$83$	−5.03932	−0.553137	−0.276568	+	0.960994i	$0.589197\pi$
$84$	0	0
$85$	−1.80827	−0.196134
$86$	0	0
$87$	−10.2099	−1.09461
$88$	0	0
$89$	15.7210	1.66642	0.833209	−	0.552958i	$-0.186501\pi$
$89$	15.7210	1.66642	0.833209	+	0.552958i	$0.186501\pi$
$90$	0	0
$91$	14.8272	1.55431
$92$	0	0
$93$	8.11532	0.841520
$94$	0	0
$95$	−5.17689	−0.531138
$96$	0	0
$97$	−0.889467	−0.0903117	−0.0451559	−	0.998980i	$-0.514378\pi$
$97$	−0.889467	−0.0903117	−0.0451559	+	0.998980i	$0.514378\pi$
$98$	0	0
$99$	−6.96518	−0.700026
$100$	0	0
$101$	12.8762	1.28123	0.640614	−	0.767863i	$-0.278680\pi$
$101$	12.8762	1.28123	0.640614	+	0.767863i	$0.278680\pi$
$102$	0	0
$103$	−2.50928	−0.247247	−0.123623	−	0.992329i	$-0.539452\pi$
$103$	−2.50928	−0.247247	−0.123623	+	0.992329i	$0.539452\pi$
$104$	0	0
$105$	−9.07786	−0.885908
$106$	0	0
$107$	−8.87141	−0.857632	−0.428816	−	0.903392i	$-0.641069\pi$
$107$	−8.87141	−0.857632	−0.428816	+	0.903392i	$0.641069\pi$
$108$	0	0
$109$	2.98186	0.285611	0.142805	−	0.989751i	$-0.454388\pi$
$109$	2.98186	0.285611	0.142805	+	0.989751i	$0.454388\pi$
$110$	0	0
$111$	−0.217841	−0.0206765
$112$	0	0
$113$	2.12257	0.199674	0.0998372	−	0.995004i	$-0.468168\pi$
$113$	2.12257	0.199674	0.0998372	+	0.995004i	$0.468168\pi$
$114$	0	0
$115$	6.25384	0.583174
$116$	0	0
$117$	6.44039	0.595414
$118$	0	0
$119$	−3.97740	−0.364608
$120$	0	0
$121$	2.55716	0.232469
$122$	0	0
$123$	−9.07052	−0.817861
$124$	0	0
$125$	12.0377	1.07668
$126$	0	0
$127$	5.68457	0.504424	0.252212	−	0.967672i	$-0.418842\pi$
$127$	5.68457	0.504424	0.252212	+	0.967672i	$0.418842\pi$
$128$	0	0
$129$	−5.70998	−0.502736
$130$	0	0
$131$	−3.01810	−0.263693	−0.131846	−	0.991270i	$-0.542091\pi$
$131$	−3.01810	−0.263693	−0.131846	+	0.991270i	$0.542091\pi$
$132$	0	0
$133$	−11.3869	−0.987370
$134$	0	0
$135$	−10.1964	−0.877568
$136$	0	0
$137$	−21.8429	−1.86616	−0.933082	−	0.359663i	$-0.882892\pi$
$137$	−21.8429	−1.86616	−0.933082	+	0.359663i	$0.882892\pi$
$138$	0	0
$139$	14.3347	1.21585	0.607927	−	0.793993i	$-0.292001\pi$
$139$	14.3347	1.21585	0.607927	+	0.793993i	$0.292001\pi$
$140$	0	0
$141$	9.17764	0.772897
$142$	0	0
$143$	−12.5357	−1.04829
$144$	0	0
$145$	−19.2019	−1.59463
$146$	0	0
$147$	−12.5980	−1.03906
$148$	0	0
$149$	−21.8638	−1.79115	−0.895576	−	0.444908i	$-0.853236\pi$
$149$	−21.8638	−1.79115	−0.895576	+	0.444908i	$0.853236\pi$
$150$	0	0
$151$	−11.0819	−0.901832	−0.450916	−	0.892566i	$-0.648903\pi$
$151$	−11.0819	−0.901832	−0.450916	+	0.892566i	$0.648903\pi$
$152$	0	0
$153$	−1.72764	−0.139672
$154$	0	0
$155$	15.2627	1.22593
$156$	0	0
$157$	−4.42402	−0.353075	−0.176538	−	0.984294i	$-0.556490\pi$
$157$	−4.42402	−0.353075	−0.176538	+	0.984294i	$0.556490\pi$
$158$	0	0
$159$	3.01432	0.239051
$160$	0	0
$161$	13.7557	1.08410
$162$	0	0
$163$	13.9699	1.09420	0.547102	−	0.837066i	$-0.315731\pi$
$163$	13.9699	1.09420	0.547102	+	0.837066i	$0.315731\pi$
$164$	0	0
$165$	7.67492	0.597492
$166$	0	0
$167$	20.9194	1.61879	0.809397	−	0.587262i	$-0.199794\pi$
$167$	20.9194	1.61879	0.809397	+	0.587262i	$0.199794\pi$
$168$	0	0
$169$	−1.40878	−0.108368
$170$	0	0
$171$	−4.94606	−0.378235
$172$	0	0
$173$	−24.2827	−1.84618	−0.923091	−	0.384582i	$-0.874346\pi$
$173$	−24.2827	−1.84618	−0.923091	+	0.384582i	$0.874346\pi$
$174$	0	0
$175$	4.70236	0.355465
$176$	0	0
$177$	12.0555	0.906146
$178$	0	0
$179$	4.27827	0.319773	0.159886	−	0.987135i	$-0.448887\pi$
$179$	4.27827	0.319773	0.159886	+	0.987135i	$0.448887\pi$
$180$	0	0
$181$	23.5742	1.75226	0.876129	−	0.482076i	$-0.160117\pi$
$181$	23.5742	1.75226	0.876129	+	0.482076i	$0.160117\pi$
$182$	0	0
$183$	−12.5946	−0.931019
$184$	0	0
$185$	−0.409698	−0.0301216
$186$	0	0
$187$	3.36272	0.245906
$188$	0	0
$189$	−22.4277	−1.63137
$190$	0	0
$191$	7.82832	0.566437	0.283219	−	0.959055i	$-0.408598\pi$
$191$	7.82832	0.566437	0.283219	+	0.959055i	$0.408598\pi$
$192$	0	0
$193$	8.99381	0.647389	0.323694	−	0.946162i	$-0.395075\pi$
$193$	8.99381	0.647389	0.323694	+	0.946162i	$0.395075\pi$
$194$	0	0
$195$	−7.09666	−0.508203
$196$	0	0
$197$	−15.2216	−1.08449	−0.542246	−	0.840220i	$-0.682426\pi$
$197$	−15.2216	−1.08449	−0.542246	+	0.840220i	$0.682426\pi$
$198$	0	0
$199$	−2.86938	−0.203405	−0.101703	−	0.994815i	$-0.532429\pi$
$199$	−2.86938	−0.203405	−0.101703	+	0.994815i	$0.532429\pi$
$200$	0	0
$201$	−10.2560	−0.723405
$202$	0	0
$203$	−42.2358	−2.96437
$204$	0	0
$205$	−17.0591	−1.19146
$206$	0	0
$207$	5.97500	0.415291
$208$	0	0
$209$	9.62712	0.665922
$210$	0	0
$211$	−21.3156	−1.46743	−0.733714	−	0.679459i	$-0.762215\pi$
$211$	−21.3156	−1.46743	−0.733714	+	0.679459i	$0.762215\pi$
$212$	0	0
$213$	−2.03851	−0.139676
$214$	0	0
$215$	−10.7389	−0.732385
$216$	0	0
$217$	33.5712	2.27896
$218$	0	0
$219$	−7.77888	−0.525648
$220$	0	0
$221$	−3.10936	−0.209158
$222$	0	0
$223$	−1.14971	−0.0769905	−0.0384952	−	0.999259i	$-0.512256\pi$
$223$	−1.14971	−0.0769905	−0.0384952	+	0.999259i	$0.512256\pi$
$224$	0	0
$225$	2.04254	0.136169
$226$	0	0
$227$	−25.8190	−1.71367	−0.856833	−	0.515594i	$-0.827571\pi$
$227$	−25.8190	−1.71367	−0.856833	+	0.515594i	$0.827571\pi$
$228$	0	0
$229$	−13.3048	−0.879203	−0.439601	−	0.898193i	$-0.644880\pi$
$229$	−13.3048	−0.879203	−0.439601	+	0.898193i	$0.644880\pi$
$230$	0	0
$231$	16.8815	1.11072
$232$	0	0
$233$	1.91624	0.125537	0.0627684	−	0.998028i	$-0.480007\pi$
$233$	1.91624	0.125537	0.0627684	+	0.998028i	$0.480007\pi$
$234$	0	0
$235$	17.2606	1.12596
$236$	0	0
$237$	0.580604	0.0377143
$238$	0	0
$239$	−13.5241	−0.874804	−0.437402	−	0.899266i	$-0.644101\pi$
$239$	−13.5241	−0.874804	−0.437402	+	0.899266i	$0.644101\pi$
$240$	0	0
$241$	10.9682	0.706525	0.353262	−	0.935524i	$-0.385072\pi$
$241$	10.9682	0.706525	0.353262	+	0.935524i	$0.385072\pi$
$242$	0	0
$243$	−15.7163	−1.00820
$244$	0	0
$245$	−23.6933	−1.51371
$246$	0	0
$247$	−8.90177	−0.566406
$248$	0	0
$249$	5.30523	0.336205
$250$	0	0
$251$	−15.3993	−0.971998	−0.485999	−	0.873959i	$-0.661544\pi$
$251$	−15.3993	−0.971998	−0.485999	+	0.873959i	$0.661544\pi$
$252$	0	0
$253$	−11.6299	−0.731163
$254$	0	0
$255$	1.90369	0.119213
$256$	0	0
$257$	22.7850	1.42129	0.710646	−	0.703550i	$-0.248403\pi$
$257$	22.7850	1.42129	0.710646	+	0.703550i	$0.248403\pi$
$258$	0	0
$259$	−0.901157	−0.0559952
$260$	0	0
$261$	−18.3457	−1.13557
$262$	0	0
$263$	−18.6623	−1.15077	−0.575383	−	0.817884i	$-0.695147\pi$
$263$	−18.6623	−1.15077	−0.575383	+	0.817884i	$0.695147\pi$
$264$	0	0
$265$	5.66910	0.348250
$266$	0	0
$267$	−16.5505	−1.01288
$268$	0	0
$269$	26.9824	1.64515	0.822573	−	0.568660i	$-0.192538\pi$
$269$	26.9824	1.64515	0.822573	+	0.568660i	$0.192538\pi$
$270$	0	0
$271$	−3.58380	−0.217700	−0.108850	−	0.994058i	$-0.534717\pi$
$271$	−3.58380	−0.217700	−0.108850	+	0.994058i	$0.534717\pi$
$272$	0	0
$273$	−15.6096	−0.944734
$274$	0	0
$275$	−3.97564	−0.239740
$276$	0	0
$277$	−21.8115	−1.31053	−0.655263	−	0.755401i	$-0.727442\pi$
$277$	−21.8115	−1.31053	−0.655263	+	0.755401i	$0.727442\pi$
$278$	0	0
$279$	14.5821	0.873009
$280$	0	0
$281$	17.3217	1.03333	0.516664	−	0.856188i	$-0.327173\pi$
$281$	17.3217	1.03333	0.516664	+	0.856188i	$0.327173\pi$
$282$	0	0
$283$	14.9307	0.887535	0.443768	−	0.896142i	$-0.353642\pi$
$283$	14.9307	0.887535	0.443768	+	0.896142i	$0.353642\pi$
$284$	0	0
$285$	5.45006	0.322834
$286$	0	0
$287$	−37.5226	−2.21489
$288$	0	0
$289$	−16.1659	−0.950936
$290$	0	0
$291$	0.936402	0.0548929
$292$	0	0
$293$	3.36735	0.196723	0.0983613	−	0.995151i	$-0.468640\pi$
$293$	3.36735	0.196723	0.0983613	+	0.995151i	$0.468640\pi$
$294$	0	0
$295$	22.6730	1.32007
$296$	0	0
$297$	18.9616	1.10026
$298$	0	0
$299$	10.7536	0.621898
$300$	0	0
$301$	−23.6209	−1.36148
$302$	0	0
$303$	−13.5556	−0.778751
$304$	0	0
$305$	−23.6869	−1.35631
$306$	0	0
$307$	13.7459	0.784518	0.392259	−	0.919855i	$-0.371694\pi$
$307$	13.7459	0.784518	0.392259	+	0.919855i	$0.371694\pi$
$308$	0	0
$309$	2.64169	0.150281
$310$	0	0
$311$	−27.9073	−1.58248	−0.791238	−	0.611508i	$-0.790563\pi$
$311$	−27.9073	−1.58248	−0.791238	+	0.611508i	$0.790563\pi$
$312$	0	0
$313$	−28.0834	−1.58737	−0.793684	−	0.608330i	$-0.791840\pi$
$313$	−28.0834	−1.58737	−0.793684	+	0.608330i	$0.791840\pi$
$314$	0	0
$315$	−16.3117	−0.919059
$316$	0	0
$317$	28.5432	1.60315	0.801573	−	0.597897i	$-0.203997\pi$
$317$	28.5432	1.60315	0.801573	+	0.597897i	$0.203997\pi$
$318$	0	0
$319$	35.7085	1.99929
$320$	0	0
$321$	9.33954	0.521282
$322$	0	0
$323$	2.38791	0.132867
$324$	0	0
$325$	3.67610	0.203913
$326$	0	0
$327$	−3.13921	−0.173599
$328$	0	0
$329$	37.9658	2.09312
$330$	0	0
$331$	−27.3785	−1.50486	−0.752428	−	0.658674i	$-0.771118\pi$
$331$	−27.3785	−1.50486	−0.752428	+	0.658674i	$0.771118\pi$
$332$	0	0
$333$	−0.391430	−0.0214502
$334$	0	0
$335$	−19.2887	−1.05386
$336$	0	0
$337$	−5.33938	−0.290854	−0.145427	−	0.989369i	$-0.546456\pi$
$337$	−5.33938	−0.290854	−0.145427	+	0.989369i	$0.546456\pi$
$338$	0	0
$339$	−2.23457	−0.121365
$340$	0	0
$341$	−28.3830	−1.53702
$342$	0	0
$343$	−21.6295	−1.16788
$344$	0	0
$345$	−6.58385	−0.354462
$346$	0	0
$347$	−28.7963	−1.54587	−0.772933	−	0.634488i	$-0.781211\pi$
$347$	−28.7963	−1.54587	−0.772933	+	0.634488i	$0.781211\pi$
$348$	0	0
$349$	−36.3942	−1.94814	−0.974069	−	0.226253i	$-0.927352\pi$
$349$	−36.3942	−1.94814	−0.974069	+	0.226253i	$0.927352\pi$
$350$	0	0
$351$	−17.5330	−0.935840
$352$	0	0
$353$	−29.3860	−1.56406	−0.782029	−	0.623242i	$-0.785815\pi$
$353$	−29.3860	−1.56406	−0.782029	+	0.623242i	$0.785815\pi$
$354$	0	0
$355$	−3.83386	−0.203480
$356$	0	0
$357$	4.18728	0.221615
$358$	0	0
$359$	−1.20141	−0.0634081	−0.0317041	−	0.999497i	$-0.510093\pi$
$359$	−1.20141	−0.0634081	−0.0317041	+	0.999497i	$0.510093\pi$
$360$	0	0
$361$	−12.1637	−0.640192
$362$	0	0
$363$	−2.69210	−0.141299
$364$	0	0
$365$	−14.6299	−0.765764
$366$	0	0
$367$	−4.66844	−0.243691	−0.121845	−	0.992549i	$-0.538881\pi$
$367$	−4.66844	−0.243691	−0.121845	+	0.992549i	$0.538881\pi$
$368$	0	0
$369$	−16.2985	−0.848465
$370$	0	0
$371$	12.4696	0.647387
$372$	0	0
$373$	27.1376	1.40513	0.702567	−	0.711618i	$-0.252037\pi$
$373$	27.1376	1.40513	0.702567	+	0.711618i	$0.252037\pi$
$374$	0	0
$375$	−12.6729	−0.654424
$376$	0	0
$377$	−33.0181	−1.70052
$378$	0	0
$379$	6.49478	0.333614	0.166807	−	0.985990i	$-0.446654\pi$
$379$	6.49478	0.333614	0.166807	+	0.985990i	$0.446654\pi$
$380$	0	0
$381$	−5.98453	−0.306597
$382$	0	0
$383$	19.7309	1.00820	0.504101	−	0.863645i	$-0.331824\pi$
$383$	19.7309	1.00820	0.504101	+	0.863645i	$0.331824\pi$
$384$	0	0
$385$	31.7494	1.61810
$386$	0	0
$387$	−10.2601	−0.521548
$388$	0	0
$389$	−25.1184	−1.27356	−0.636778	−	0.771048i	$-0.719733\pi$
$389$	−25.1184	−1.27356	−0.636778	+	0.771048i	$0.719733\pi$
$390$	0	0
$391$	−2.88467	−0.145884
$392$	0	0
$393$	3.17736	0.160277
$394$	0	0
$395$	1.09195	0.0549422
$396$	0	0
$397$	17.7938	0.893044	0.446522	−	0.894773i	$-0.352662\pi$
$397$	17.7938	0.893044	0.446522	+	0.894773i	$0.352662\pi$
$398$	0	0
$399$	11.9878	0.600139
$400$	0	0
$401$	23.8512	1.19107	0.595535	−	0.803329i	$-0.296940\pi$
$401$	23.8512	1.19107	0.595535	+	0.803329i	$0.296940\pi$
$402$	0	0
$403$	26.2445	1.30733
$404$	0	0
$405$	−0.501908	−0.0249400
$406$	0	0
$407$	0.761888	0.0377654
$408$	0	0
$409$	24.6359	1.21817	0.609083	−	0.793106i	$-0.291537\pi$
$409$	24.6359	1.21817	0.609083	+	0.793106i	$0.291537\pi$
$410$	0	0
$411$	22.9955	1.13428
$412$	0	0
$413$	49.8708	2.45398
$414$	0	0
$415$	9.97766	0.489784
$416$	0	0
$417$	−15.0911	−0.739016
$418$	0	0
$419$	18.9579	0.926152	0.463076	−	0.886319i	$-0.346746\pi$
$419$	18.9579	0.926152	0.463076	+	0.886319i	$0.346746\pi$
$420$	0	0
$421$	−31.9110	−1.55525	−0.777624	−	0.628729i	$-0.783575\pi$
$421$	−31.9110	−1.55525	−0.777624	+	0.628729i	$0.783575\pi$
$422$	0	0
$423$	16.4910	0.801818
$424$	0	0
$425$	−0.986116	−0.0478337
$426$	0	0
$427$	−52.1009	−2.52134
$428$	0	0
$429$	13.1972	0.637166
$430$	0	0
$431$	9.62592	0.463664	0.231832	−	0.972756i	$-0.425528\pi$
$431$	9.62592	0.463664	0.231832	+	0.972756i	$0.425528\pi$
$432$	0	0
$433$	16.0909	0.773280	0.386640	−	0.922231i	$-0.373636\pi$
$433$	16.0909	0.773280	0.386640	+	0.922231i	$0.373636\pi$
$434$	0	0
$435$	20.2151	0.969241
$436$	0	0
$437$	−8.25852	−0.395058
$438$	0	0
$439$	−19.7116	−0.940784	−0.470392	−	0.882458i	$-0.655887\pi$
$439$	−19.7116	−0.940784	−0.470392	+	0.882458i	$0.655887\pi$
$440$	0	0
$441$	−22.6368	−1.07794
$442$	0	0
$443$	−3.60983	−0.171508	−0.0857541	−	0.996316i	$-0.527330\pi$
$443$	−3.60983	−0.171508	−0.0857541	+	0.996316i	$0.527330\pi$
$444$	0	0
$445$	−31.1269	−1.47556
$446$	0	0
$447$	23.0175	1.08869
$448$	0	0
$449$	25.6044	1.20835	0.604174	−	0.796853i	$-0.293503\pi$
$449$	25.6044	1.20835	0.604174	+	0.796853i	$0.293503\pi$
$450$	0	0
$451$	31.7237	1.49381
$452$	0	0
$453$	11.6667	0.548148
$454$	0	0
$455$	−29.3572	−1.37629
$456$	0	0
$457$	−26.4625	−1.23786	−0.618932	−	0.785444i	$-0.712435\pi$
$457$	−26.4625	−1.23786	−0.618932	+	0.785444i	$0.712435\pi$
$458$	0	0
$459$	4.70323	0.219528
$460$	0	0
$461$	19.9008	0.926871	0.463436	−	0.886131i	$-0.346617\pi$
$461$	19.9008	0.926871	0.463436	+	0.886131i	$0.346617\pi$
$462$	0	0
$463$	6.11723	0.284292	0.142146	−	0.989846i	$-0.454600\pi$
$463$	6.11723	0.284292	0.142146	+	0.989846i	$0.454600\pi$
$464$	0	0
$465$	−16.0680	−0.745137
$466$	0	0
$467$	−33.1853	−1.53563	−0.767817	−	0.640669i	$-0.778657\pi$
$467$	−33.1853	−1.53563	−0.767817	+	0.640669i	$0.778657\pi$
$468$	0	0
$469$	−42.4268	−1.95909
$470$	0	0
$471$	4.65747	0.214605
$472$	0	0
$473$	19.9704	0.918239
$474$	0	0
$475$	−2.82315	−0.129535
$476$	0	0
$477$	5.41633	0.247997
$478$	0	0
$479$	38.5769	1.76262	0.881311	−	0.472536i	$-0.156661\pi$
$479$	38.5769	1.76262	0.881311	+	0.472536i	$0.156661\pi$
$480$	0	0
$481$	−0.704484	−0.0321217
$482$	0	0
$483$	−14.4816	−0.658935
$484$	0	0
$485$	1.76111	0.0799680
$486$	0	0
$487$	9.85394	0.446525	0.223262	−	0.974758i	$-0.428329\pi$
$487$	9.85394	0.446525	0.223262	+	0.974758i	$0.428329\pi$
$488$	0	0
$489$	−14.7070	−0.665074
$490$	0	0
$491$	−4.75897	−0.214769	−0.107385	−	0.994218i	$-0.534248\pi$
$491$	−4.75897	−0.214769	−0.107385	+	0.994218i	$0.534248\pi$
$492$	0	0
$493$	8.85713	0.398905
$494$	0	0
$495$	13.7908	0.619850
$496$	0	0
$497$	−8.43283	−0.378264
$498$	0	0
$499$	6.60733	0.295785	0.147892	−	0.989003i	$-0.452751\pi$
$499$	6.60733	0.295785	0.147892	+	0.989003i	$0.452751\pi$
$500$	0	0
$501$	−22.0233	−0.983928
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−25.4944	−1.13448
$506$	0	0
$507$	1.48312	0.0658676
$508$	0	0
$509$	21.5716	0.956143	0.478071	−	0.878321i	$-0.341336\pi$
$509$	21.5716	0.956143	0.478071	+	0.878321i	$0.341336\pi$
$510$	0	0
$511$	−32.1794	−1.42353
$512$	0	0
$513$	13.4649	0.594489
$514$	0	0
$515$	4.96829	0.218929
$516$	0	0
$517$	−32.0984	−1.41168
$518$	0	0
$519$	25.5641	1.12214
$520$	0	0
$521$	−8.99183	−0.393939	−0.196970	−	0.980410i	$-0.563110\pi$
$521$	−8.99183	−0.393939	−0.196970	+	0.980410i	$0.563110\pi$
$522$	0	0
$523$	16.3341	0.714240	0.357120	−	0.934059i	$-0.383759\pi$
$523$	16.3341	0.714240	0.357120	+	0.934059i	$0.383759\pi$
$524$	0	0
$525$	−4.95049	−0.216057
$526$	0	0
$527$	−7.04010	−0.306672
$528$	0	0
$529$	−13.0235	−0.566237
$530$	0	0
$531$	21.6621	0.940054
$532$	0	0
$533$	−29.3335	−1.27058
$534$	0	0
$535$	17.5651	0.759404
$536$	0	0
$537$	−4.50402	−0.194363
$538$	0	0
$539$	44.0608	1.89783
$540$	0	0
$541$	20.2963	0.872607	0.436304	−	0.899800i	$-0.356287\pi$
$541$	20.2963	0.872607	0.436304	+	0.899800i	$0.356287\pi$
$542$	0	0
$543$	−24.8182	−1.06505
$544$	0	0
$545$	−5.90397	−0.252898
$546$	0	0
$547$	−34.5263	−1.47624	−0.738118	−	0.674671i	$-0.764285\pi$
$547$	−34.5263	−1.47624	−0.738118	+	0.674671i	$0.764285\pi$
$548$	0	0
$549$	−22.6308	−0.965857
$550$	0	0
$551$	25.3571	1.08025
$552$	0	0
$553$	2.40182	0.102136
$554$	0	0
$555$	0.431317	0.0183084
$556$	0	0
$557$	−8.41718	−0.356647	−0.178324	−	0.983972i	$-0.557067\pi$
$557$	−8.41718	−0.356647	−0.178324	+	0.983972i	$0.557067\pi$
$558$	0	0
$559$	−18.4657	−0.781017
$560$	0	0
$561$	−3.54016	−0.149466
$562$	0	0
$563$	32.4921	1.36938	0.684690	−	0.728834i	$-0.259938\pi$
$563$	32.4921	1.36938	0.684690	+	0.728834i	$0.259938\pi$
$564$	0	0
$565$	−4.20261	−0.176805
$566$	0	0
$567$	−1.10398	−0.0463627
$568$	0	0
$569$	−11.5526	−0.484310	−0.242155	−	0.970238i	$-0.577854\pi$
$569$	−11.5526	−0.484310	−0.242155	+	0.970238i	$0.577854\pi$
$570$	0	0
$571$	−2.36859	−0.0991225	−0.0495612	−	0.998771i	$-0.515782\pi$
$571$	−2.36859	−0.0991225	−0.0495612	+	0.998771i	$0.515782\pi$
$572$	0	0
$573$	−8.24140	−0.344290
$574$	0	0
$575$	3.41046	0.142226
$576$	0	0
$577$	3.77203	0.157032	0.0785158	−	0.996913i	$-0.474982\pi$
$577$	3.77203	0.157032	0.0785158	+	0.996913i	$0.474982\pi$
$578$	0	0
$579$	−9.46839	−0.393493
$580$	0	0
$581$	21.9465	0.910494
$582$	0	0
$583$	−10.5425	−0.436624
$584$	0	0
$585$	−12.7517	−0.527219
$586$	0	0
$587$	40.9189	1.68891	0.844453	−	0.535630i	$-0.179926\pi$
$587$	40.9189	1.68891	0.844453	+	0.535630i	$0.179926\pi$
$588$	0	0
$589$	−20.1551	−0.830477
$590$	0	0
$591$	16.0248	0.659171
$592$	0	0
$593$	0.874959	0.0359303	0.0179651	−	0.999839i	$-0.494281\pi$
$593$	0.874959	0.0359303	0.0179651	+	0.999839i	$0.494281\pi$
$594$	0	0
$595$	7.87511	0.322848
$596$	0	0
$597$	3.02080	0.123633
$598$	0	0
$599$	−34.7297	−1.41902	−0.709509	−	0.704696i	$-0.751083\pi$
$599$	−34.7297	−1.41902	−0.709509	+	0.704696i	$0.751083\pi$
$600$	0	0
$601$	21.2268	0.865860	0.432930	−	0.901428i	$-0.357480\pi$
$601$	21.2268	0.865860	0.432930	+	0.901428i	$0.357480\pi$
$602$	0	0
$603$	−18.4287	−0.750475
$604$	0	0
$605$	−5.06309	−0.205844
$606$	0	0
$607$	−25.5508	−1.03707	−0.518537	−	0.855055i	$-0.673523\pi$
$607$	−25.5508	−1.03707	−0.518537	+	0.855055i	$0.673523\pi$
$608$	0	0
$609$	44.4645	1.80179
$610$	0	0
$611$	29.6799	1.20072
$612$	0	0
$613$	27.6830	1.11811	0.559053	−	0.829132i	$-0.311165\pi$
$613$	27.6830	1.11811	0.559053	+	0.829132i	$0.311165\pi$
$614$	0	0
$615$	17.9593	0.724188
$616$	0	0
$617$	3.32890	0.134016	0.0670082	−	0.997752i	$-0.478655\pi$
$617$	3.32890	0.134016	0.0670082	+	0.997752i	$0.478655\pi$
$618$	0	0
$619$	−11.3192	−0.454956	−0.227478	−	0.973783i	$-0.573048\pi$
$619$	−11.3192	−0.454956	−0.227478	+	0.973783i	$0.573048\pi$
$620$	0	0
$621$	−16.2660	−0.652732
$622$	0	0
$623$	−68.4657	−2.74302
$624$	0	0
$625$	−18.4354	−0.737416
$626$	0	0
$627$	−10.1351	−0.404758
$628$	0	0
$629$	0.188979	0.00753507
$630$	0	0
$631$	10.7231	0.426880	0.213440	−	0.976956i	$-0.431533\pi$
$631$	10.7231	0.426880	0.213440	+	0.976956i	$0.431533\pi$
$632$	0	0
$633$	22.4404	0.891925
$634$	0	0
$635$	−11.2552	−0.446651
$636$	0	0
$637$	−40.7411	−1.61422
$638$	0	0
$639$	−3.66292	−0.144903
$640$	0	0
$641$	−20.9439	−0.827236	−0.413618	−	0.910451i	$-0.635735\pi$
$641$	−20.9439	−0.827236	−0.413618	+	0.910451i	$0.635735\pi$
$642$	0	0
$643$	−23.7360	−0.936058	−0.468029	−	0.883713i	$-0.655036\pi$
$643$	−23.7360	−0.936058	−0.468029	+	0.883713i	$0.655036\pi$
$644$	0	0
$645$	11.3055	0.445155
$646$	0	0
$647$	23.4441	0.921682	0.460841	−	0.887483i	$-0.347548\pi$
$647$	23.4441	0.921682	0.460841	+	0.887483i	$0.347548\pi$
$648$	0	0
$649$	−42.1635	−1.65506
$650$	0	0
$651$	−35.3427	−1.38519
$652$	0	0
$653$	42.5562	1.66535	0.832676	−	0.553760i	$-0.186808\pi$
$653$	42.5562	1.66535	0.832676	+	0.553760i	$0.186808\pi$
$654$	0	0
$655$	5.97573	0.233491
$656$	0	0
$657$	−13.9776	−0.545318
$658$	0	0
$659$	−15.8733	−0.618335	−0.309168	−	0.951008i	$-0.600050\pi$
$659$	−15.8733	−0.618335	−0.309168	+	0.951008i	$0.600050\pi$
$660$	0	0
$661$	−43.9621	−1.70993	−0.854963	−	0.518689i	$-0.826420\pi$
$661$	−43.9621	−1.70993	−0.854963	+	0.518689i	$0.826420\pi$
$662$	0	0
$663$	3.27343	0.127129
$664$	0	0
$665$	22.5456	0.874283
$666$	0	0
$667$	−30.6321	−1.18608
$668$	0	0
$669$	1.21038	0.0467960
$670$	0	0
$671$	44.0490	1.70049
$672$	0	0
$673$	−30.1398	−1.16181	−0.580903	−	0.813973i	$-0.697300\pi$
$673$	−30.1398	−1.16181	−0.580903	+	0.813973i	$0.697300\pi$
$674$	0	0
$675$	−5.56049	−0.214023
$676$	0	0
$677$	−19.0426	−0.731867	−0.365933	−	0.930641i	$-0.619250\pi$
$677$	−19.0426	−0.731867	−0.365933	+	0.930641i	$0.619250\pi$
$678$	0	0
$679$	3.87368	0.148658
$680$	0	0
$681$	27.1814	1.04159
$682$	0	0
$683$	0.289286	0.0110692	0.00553462	−	0.999985i	$-0.498238\pi$
$683$	0.289286	0.0110692	0.00553462	+	0.999985i	$0.498238\pi$
$684$	0	0
$685$	43.2481	1.65243
$686$	0	0
$687$	14.0068	0.534393
$688$	0	0
$689$	9.74814	0.371375
$690$	0	0
$691$	41.9662	1.59647	0.798235	−	0.602346i	$-0.205767\pi$
$691$	41.9662	1.59647	0.798235	+	0.602346i	$0.205767\pi$
$692$	0	0
$693$	30.3337	1.15228
$694$	0	0
$695$	−28.3822	−1.07660
$696$	0	0
$697$	7.86874	0.298050
$698$	0	0
$699$	−2.01735	−0.0763033
$700$	0	0
$701$	−4.99274	−0.188573	−0.0942866	−	0.995545i	$-0.530057\pi$
$701$	−4.99274	−0.188573	−0.0942866	+	0.995545i	$0.530057\pi$
$702$	0	0
$703$	0.541027	0.0204052
$704$	0	0
$705$	−18.1714	−0.684374
$706$	0	0
$707$	−56.0765	−2.10897
$708$	0	0
$709$	−4.74294	−0.178125	−0.0890625	−	0.996026i	$-0.528387\pi$
$709$	−4.74294	−0.178125	−0.0890625	+	0.996026i	$0.528387\pi$
$710$	0	0
$711$	1.04327	0.0391255
$712$	0	0
$713$	24.3480	0.911840
$714$	0	0
$715$	24.8202	0.928224
$716$	0	0
$717$	14.2378	0.531720
$718$	0	0
$719$	16.2036	0.604294	0.302147	−	0.953261i	$-0.402297\pi$
$719$	16.2036	0.604294	0.302147	+	0.953261i	$0.402297\pi$
$720$	0	0
$721$	10.9281	0.406983
$722$	0	0
$723$	−11.5470	−0.429437
$724$	0	0
$725$	−10.4715	−0.388902
$726$	0	0
$727$	33.4020	1.23881	0.619405	−	0.785072i	$-0.287374\pi$
$727$	33.4020	1.23881	0.619405	+	0.785072i	$0.287374\pi$
$728$	0	0
$729$	15.7851	0.584634
$730$	0	0
$731$	4.95345	0.183210
$732$	0	0
$733$	33.6148	1.24159	0.620795	−	0.783973i	$-0.286810\pi$
$733$	33.6148	1.24159	0.620795	+	0.783973i	$0.286810\pi$
$734$	0	0
$735$	24.9435	0.920055
$736$	0	0
$737$	35.8700	1.32129
$738$	0	0
$739$	−20.5752	−0.756872	−0.378436	−	0.925627i	$-0.623538\pi$
$739$	−20.5752	−0.756872	−0.378436	+	0.925627i	$0.623538\pi$
$740$	0	0
$741$	9.37150	0.344271
$742$	0	0
$743$	−2.54970	−0.0935396	−0.0467698	−	0.998906i	$-0.514893\pi$
$743$	−2.54970	−0.0935396	−0.0467698	+	0.998906i	$0.514893\pi$
$744$	0	0
$745$	43.2895	1.58601
$746$	0	0
$747$	9.53277	0.348786
$748$	0	0
$749$	38.6355	1.41171
$750$	0	0
$751$	48.3128	1.76296	0.881480	−	0.472221i	$-0.156548\pi$
$751$	48.3128	1.76296	0.881480	+	0.472221i	$0.156548\pi$
$752$	0	0
$753$	16.2119	0.590796
$754$	0	0
$755$	21.9417	0.798542
$756$	0	0
$757$	15.1383	0.550209	0.275105	−	0.961414i	$-0.411287\pi$
$757$	15.1383	0.550209	0.275105	+	0.961414i	$0.411287\pi$
$758$	0	0
$759$	12.2435	0.444412
$760$	0	0
$761$	−27.8681	−1.01022	−0.505110	−	0.863055i	$-0.668548\pi$
$761$	−27.8681	−1.01022	−0.505110	+	0.863055i	$0.668548\pi$
$762$	0	0
$763$	−12.9862	−0.470131
$764$	0	0
$765$	3.42067	0.123674
$766$	0	0
$767$	38.9867	1.40773
$768$	0	0
$769$	−10.9273	−0.394049	−0.197024	−	0.980399i	$-0.563128\pi$
$769$	−10.9273	−0.394049	−0.197024	+	0.980399i	$0.563128\pi$
$770$	0	0
$771$	−23.9874	−0.863884
$772$	0	0
$773$	−49.3445	−1.77480	−0.887399	−	0.461001i	$-0.847490\pi$
$773$	−49.3445	−1.77480	−0.887399	+	0.461001i	$0.847490\pi$
$774$	0	0
$775$	8.32330	0.298982
$776$	0	0
$777$	0.948709	0.0340348
$778$	0	0
$779$	22.5274	0.807128
$780$	0	0
$781$	7.12958	0.255116
$782$	0	0
$783$	49.9433	1.78483
$784$	0	0
$785$	8.75940	0.312636
$786$	0	0
$787$	17.6570	0.629404	0.314702	−	0.949191i	$-0.398095\pi$
$787$	17.6570	0.629404	0.314702	+	0.949191i	$0.398095\pi$
$788$	0	0
$789$	19.6471	0.699454
$790$	0	0
$791$	−9.24390	−0.328675
$792$	0	0
$793$	−40.7301	−1.44637
$794$	0	0
$795$	−5.96825	−0.211672
$796$	0	0
$797$	21.0277	0.744838	0.372419	−	0.928065i	$-0.378528\pi$
$797$	21.0277	0.744838	0.372419	+	0.928065i	$0.378528\pi$
$798$	0	0
$799$	−7.96168	−0.281664
$800$	0	0
$801$	−29.7390	−1.05078
$802$	0	0
$803$	27.2063	0.960088
$804$	0	0
$805$	−27.2358	−0.959937
$806$	0	0
$807$	−28.4062	−0.999945
$808$	0	0
$809$	−44.8072	−1.57534	−0.787669	−	0.616098i	$-0.788712\pi$
$809$	−44.8072	−1.57534	−0.787669	+	0.616098i	$0.788712\pi$
$810$	0	0
$811$	−7.30649	−0.256566	−0.128283	−	0.991738i	$-0.540947\pi$
$811$	−7.30649	−0.256566	−0.128283	+	0.991738i	$0.540947\pi$
$812$	0	0
$813$	3.77291	0.132322
$814$	0	0
$815$	−27.6598	−0.968880
$816$	0	0
$817$	14.1812	0.496138
$818$	0	0
$819$	−28.0483	−0.980086
$820$	0	0
$821$	19.6515	0.685841	0.342921	−	0.939364i	$-0.388584\pi$
$821$	19.6515	0.685841	0.342921	+	0.939364i	$0.388584\pi$
$822$	0	0
$823$	−3.88253	−0.135336	−0.0676682	−	0.997708i	$-0.521556\pi$
$823$	−3.88253	−0.135336	−0.0676682	+	0.997708i	$0.521556\pi$
$824$	0	0
$825$	4.18542	0.145718
$826$	0	0
$827$	−2.07211	−0.0720543	−0.0360271	−	0.999351i	$-0.511470\pi$
$827$	−2.07211	−0.0720543	−0.0360271	+	0.999351i	$0.511470\pi$
$828$	0	0
$829$	7.54915	0.262193	0.131096	−	0.991370i	$-0.458150\pi$
$829$	7.54915	0.262193	0.131096	+	0.991370i	$0.458150\pi$
$830$	0	0
$831$	22.9624	0.796558
$832$	0	0
$833$	10.9288	0.378661
$834$	0	0
$835$	−41.4197	−1.43339
$836$	0	0
$837$	−39.6976	−1.37215
$838$	0	0
$839$	57.6888	1.99164	0.995820	−	0.0913423i	$-0.0291158\pi$
$839$	57.6888	1.99164	0.995820	+	0.0913423i	$0.0291158\pi$
$840$	0	0
$841$	65.0533	2.24322
$842$	0	0
$843$	−18.2358	−0.628074
$844$	0	0
$845$	2.78933	0.0959560
$846$	0	0
$847$	−11.1366	−0.382658
$848$	0	0
$849$	−15.7185	−0.539458
$850$	0	0
$851$	−0.653577	−0.0224043
$852$	0	0
$853$	−48.8703	−1.67329	−0.836644	−	0.547748i	$-0.815485\pi$
$853$	−48.8703	−1.67329	−0.836644	+	0.547748i	$0.815485\pi$
$854$	0	0
$855$	9.79302	0.334914
$856$	0	0
$857$	−26.0704	−0.890548	−0.445274	−	0.895394i	$-0.646894\pi$
$857$	−26.0704	−0.890548	−0.445274	+	0.895394i	$0.646894\pi$
$858$	0	0
$859$	−23.7263	−0.809530	−0.404765	−	0.914421i	$-0.632647\pi$
$859$	−23.7263	−0.809530	−0.404765	+	0.914421i	$0.632647\pi$
$860$	0	0
$861$	39.5026	1.34625
$862$	0	0
$863$	18.9724	0.645828	0.322914	−	0.946428i	$-0.395338\pi$
$863$	18.9724	0.645828	0.322914	+	0.946428i	$0.395338\pi$
$864$	0	0
$865$	48.0789	1.63473
$866$	0	0
$867$	17.0190	0.577994
$868$	0	0
$869$	−2.03063	−0.0688846
$870$	0	0
$871$	−33.1674	−1.12383
$872$	0	0
$873$	1.68259	0.0569470
$874$	0	0
$875$	−52.4247	−1.77228
$876$	0	0
$877$	23.4497	0.791840	0.395920	−	0.918285i	$-0.370426\pi$
$877$	23.4497	0.791840	0.395920	+	0.918285i	$0.370426\pi$
$878$	0	0
$879$	−3.54504	−0.119571
$880$	0	0
$881$	−4.66510	−0.157171	−0.0785856	−	0.996907i	$-0.525040\pi$
$881$	−4.66510	−0.157171	−0.0785856	+	0.996907i	$0.525040\pi$
$882$	0	0
$883$	11.4860	0.386535	0.193268	−	0.981146i	$-0.438091\pi$
$883$	11.4860	0.386535	0.193268	+	0.981146i	$0.438091\pi$
$884$	0	0
$885$	−23.8694	−0.802362
$886$	0	0
$887$	15.1480	0.508621	0.254311	−	0.967123i	$-0.418151\pi$
$887$	15.1480	0.508621	0.254311	+	0.967123i	$0.418151\pi$
$888$	0	0
$889$	−24.7566	−0.830311
$890$	0	0
$891$	0.933364	0.0312689
$892$	0	0
$893$	−22.7935	−0.762754
$894$	0	0
$895$	−8.47081	−0.283148
$896$	0	0
$897$	−11.3211	−0.377999
$898$	0	0
$899$	−74.7584	−2.49333
$900$	0	0
$901$	−2.61495	−0.0871166
$902$	0	0
$903$	24.8673	0.827531
$904$	0	0
$905$	−46.6761	−1.55157
$906$	0	0
$907$	−48.0715	−1.59619	−0.798094	−	0.602533i	$-0.794158\pi$
$907$	−48.0715	−1.59619	−0.798094	+	0.602533i	$0.794158\pi$
$908$	0	0
$909$	−24.3576	−0.807891
$910$	0	0
$911$	−43.6332	−1.44563	−0.722817	−	0.691040i	$-0.757153\pi$
$911$	−43.6332	−1.44563	−0.722817	+	0.691040i	$0.757153\pi$
$912$	0	0
$913$	−18.5548	−0.614074
$914$	0	0
$915$	24.9368	0.824386
$916$	0	0
$917$	13.1440	0.434053
$918$	0	0
$919$	−24.3325	−0.802654	−0.401327	−	0.915935i	$-0.631451\pi$
$919$	−24.3325	−0.802654	−0.401327	+	0.915935i	$0.631451\pi$
$920$	0	0
$921$	−14.4712	−0.476842
$922$	0	0
$923$	−6.59241	−0.216992
$924$	0	0
$925$	−0.223424	−0.00734612
$926$	0	0
$927$	4.74676	0.155904
$928$	0	0
$929$	51.3363	1.68429	0.842145	−	0.539252i	$-0.181293\pi$
$929$	51.3363	1.68429	0.842145	+	0.539252i	$0.181293\pi$
$930$	0	0
$931$	31.2881	1.02543
$932$	0	0
$933$	29.3799	0.961854
$934$	0	0
$935$	−6.65805	−0.217742
$936$	0	0
$937$	−27.9591	−0.913383	−0.456692	−	0.889625i	$-0.650966\pi$
$937$	−27.9591	−0.913383	−0.456692	+	0.889625i	$0.650966\pi$
$938$	0	0
$939$	29.5653	0.964828
$940$	0	0
$941$	−6.64175	−0.216515	−0.108257	−	0.994123i	$-0.534527\pi$
$941$	−6.64175	−0.216515	−0.108257	+	0.994123i	$0.534527\pi$
$942$	0	0
$943$	−27.2138	−0.886204
$944$	0	0
$945$	44.4060	1.44453
$946$	0	0
$947$	−35.9576	−1.16846	−0.584232	−	0.811587i	$-0.698604\pi$
$947$	−35.9576	−1.16846	−0.584232	+	0.811587i	$0.698604\pi$
$948$	0	0
$949$	−25.1564	−0.816612
$950$	0	0
$951$	−30.0494	−0.974417
$952$	0	0
$953$	−2.13451	−0.0691436	−0.0345718	−	0.999402i	$-0.511007\pi$
$953$	−2.13451	−0.0691436	−0.0345718	+	0.999402i	$0.511007\pi$
$954$	0	0
$955$	−15.4998	−0.501561
$956$	0	0
$957$	−37.5927	−1.21520
$958$	0	0
$959$	95.1270	3.07181
$960$	0	0
$961$	28.4219	0.916835
$962$	0	0
$963$	16.7819	0.540788
$964$	0	0
$965$	−17.8074	−0.573241
$966$	0	0
$967$	−22.2062	−0.714104	−0.357052	−	0.934084i	$-0.616218\pi$
$967$	−22.2062	−0.714104	−0.357052	+	0.934084i	$0.616218\pi$
$968$	0	0
$969$	−2.51391	−0.0807586
$970$	0	0
$971$	−15.0609	−0.483328	−0.241664	−	0.970360i	$-0.577693\pi$
$971$	−15.0609	−0.483328	−0.241664	+	0.970360i	$0.577693\pi$
$972$	0	0
$973$	−62.4285	−2.00137
$974$	0	0
$975$	−3.87008	−0.123942
$976$	0	0
$977$	19.2226	0.614984	0.307492	−	0.951551i	$-0.400510\pi$
$977$	19.2226	0.614984	0.307492	+	0.951551i	$0.400510\pi$
$978$	0	0
$979$	57.8847	1.85000
$980$	0	0
$981$	−5.64073	−0.180095
$982$	0	0
$983$	−20.4859	−0.653400	−0.326700	−	0.945128i	$-0.605937\pi$
$983$	−20.4859	−0.653400	−0.326700	+	0.945128i	$0.605937\pi$
$984$	0	0
$985$	30.1381	0.960281
$986$	0	0
$987$	−39.9691	−1.27223
$988$	0	0
$989$	−17.1314	−0.544746
$990$	0	0
$991$	−45.0476	−1.43099	−0.715493	−	0.698620i	$-0.753798\pi$
$991$	−45.0476	−1.43099	−0.715493	+	0.698620i	$0.753798\pi$
$992$	0	0
$993$	28.8232	0.914676
$994$	0	0
$995$	5.68128	0.180109
$996$	0	0
$997$	−23.2108	−0.735094	−0.367547	−	0.930005i	$-0.619802\pi$
$997$	−23.2108	−0.735094	−0.367547	+	0.930005i	$0.619802\pi$
$998$	0	0
$999$	1.06561	0.0337143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.13		33
4.3	odd	2		4024.2.a.g.1.21	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.21	✓	33	4.3	odd	2
8048.2.a.x.1.13		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)

\(f(q)\)	\(=\)	\(q-1.05277 q^{3} -1.97996 q^{5} -4.35506 q^{7} -1.89168 q^{9} +O(q^{10})\)	\(q-1.05277 q^{3} -1.97996 q^{5} -4.35506 q^{7} -1.89168 q^{9} +3.68201 q^{11} -3.40459 q^{13} +2.08444 q^{15} +0.913284 q^{17} +2.61464 q^{19} +4.58486 q^{21} -3.15857 q^{23} -1.07975 q^{25} +5.14980 q^{27} +9.69811 q^{29} -7.70856 q^{31} -3.87630 q^{33} +8.62285 q^{35} +0.206922 q^{37} +3.58424 q^{39} +8.61588 q^{41} +5.42378 q^{43} +3.74546 q^{45} -8.71763 q^{47} +11.9665 q^{49} -0.961476 q^{51} -2.86324 q^{53} -7.29023 q^{55} -2.75261 q^{57} -11.4512 q^{59} +11.9633 q^{61} +8.23837 q^{63} +6.74096 q^{65} +9.74197 q^{67} +3.32524 q^{69} +1.93633 q^{71} +7.38898 q^{73} +1.13672 q^{75} -16.0353 q^{77} -0.551502 q^{79} +0.253493 q^{81} -5.03932 q^{83} -1.80827 q^{85} -10.2099 q^{87} +15.7210 q^{89} +14.8272 q^{91} +8.11532 q^{93} -5.17689 q^{95} -0.889467 q^{97} -6.96518 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 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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