LMFDB - Embedded newform 8048.2.a.t.1.8

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:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-0.301406 q^{3} +3.72608 q^{5} +3.06802 q^{7} -2.90915 q^{9} +O(q^{10})$	$q-0.301406 q^{3} +3.72608 q^{5} +3.06802 q^{7} -2.90915 q^{9} -1.17363 q^{11} -4.64937 q^{13} -1.12306 q^{15} -1.45150 q^{17} +3.85726 q^{19} -0.924717 q^{21} +5.40909 q^{23} +8.88368 q^{25} +1.78105 q^{27} +5.63184 q^{29} +4.49372 q^{31} +0.353740 q^{33} +11.4317 q^{35} -7.33783 q^{37} +1.40135 q^{39} -11.3803 q^{41} -1.68116 q^{43} -10.8397 q^{45} +0.152534 q^{47} +2.41272 q^{49} +0.437491 q^{51} +6.85851 q^{53} -4.37306 q^{55} -1.16260 q^{57} +10.8204 q^{59} +3.25765 q^{61} -8.92533 q^{63} -17.3239 q^{65} -8.34459 q^{67} -1.63033 q^{69} +15.5490 q^{71} +16.3257 q^{73} -2.67759 q^{75} -3.60073 q^{77} -11.8536 q^{79} +8.19064 q^{81} -1.33359 q^{83} -5.40841 q^{85} -1.69747 q^{87} +6.98475 q^{89} -14.2644 q^{91} -1.35443 q^{93} +14.3725 q^{95} +7.18127 q^{97} +3.41428 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	−0.301406	−0.174017	−0.0870083	−	0.996208i	$-0.527731\pi$
$3$	−0.301406	−0.174017	−0.0870083	+	0.996208i	$0.527731\pi$
$4$	0	0
$5$	3.72608	1.66635	0.833177	−	0.553006i	$-0.186519\pi$
$5$	3.72608	1.66635	0.833177	+	0.553006i	$0.186519\pi$
$6$	0	0
$7$	3.06802	1.15960	0.579800	−	0.814759i	$-0.303131\pi$
$7$	3.06802	1.15960	0.579800	+	0.814759i	$0.303131\pi$
$8$	0	0
$9$	−2.90915	−0.969718
$10$	0	0
$11$	−1.17363	−0.353864	−0.176932	−	0.984223i	$-0.556617\pi$
$11$	−1.17363	−0.353864	−0.176932	+	0.984223i	$0.556617\pi$
$12$	0	0
$13$	−4.64937	−1.28950	−0.644752	−	0.764392i	$-0.723039\pi$
$13$	−4.64937	−1.28950	−0.644752	+	0.764392i	$0.723039\pi$
$14$	0	0
$15$	−1.12306	−0.289973
$16$	0	0
$17$	−1.45150	−0.352041	−0.176020	−	0.984387i	$-0.556322\pi$
$17$	−1.45150	−0.352041	−0.176020	+	0.984387i	$0.556322\pi$
$18$	0	0
$19$	3.85726	0.884916	0.442458	−	0.896789i	$-0.354106\pi$
$19$	3.85726	0.884916	0.442458	+	0.896789i	$0.354106\pi$
$20$	0	0
$21$	−0.924717	−0.201790
$22$	0	0
$23$	5.40909	1.12787	0.563936	−	0.825818i	$-0.309286\pi$
$23$	5.40909	1.12787	0.563936	+	0.825818i	$0.309286\pi$
$24$	0	0
$25$	8.88368	1.77674
$26$	0	0
$27$	1.78105	0.342764
$28$	0	0
$29$	5.63184	1.04581	0.522903	−	0.852392i	$-0.324849\pi$
$29$	5.63184	1.04581	0.522903	+	0.852392i	$0.324849\pi$
$30$	0	0
$31$	4.49372	0.807096	0.403548	−	0.914958i	$-0.367777\pi$
$31$	4.49372	0.807096	0.403548	+	0.914958i	$0.367777\pi$
$32$	0	0
$33$	0.353740	0.0615783
$34$	0	0
$35$	11.4317	1.93231
$36$	0	0
$37$	−7.33783	−1.20633	−0.603166	−	0.797616i	$-0.706094\pi$
$37$	−7.33783	−1.20633	−0.603166	+	0.797616i	$0.706094\pi$
$38$	0	0
$39$	1.40135	0.224395
$40$	0	0
$41$	−11.3803	−1.77730	−0.888648	−	0.458589i	$-0.848355\pi$
$41$	−11.3803	−1.77730	−0.888648	+	0.458589i	$0.848355\pi$
$42$	0	0
$43$	−1.68116	−0.256374	−0.128187	−	0.991750i	$-0.540916\pi$
$43$	−1.68116	−0.256374	−0.128187	+	0.991750i	$0.540916\pi$
$44$	0	0
$45$	−10.8397	−1.61589
$46$	0	0
$47$	0.152534	0.0222494	0.0111247	−	0.999938i	$-0.496459\pi$
$47$	0.152534	0.0222494	0.0111247	+	0.999938i	$0.496459\pi$
$48$	0	0
$49$	2.41272	0.344674
$50$	0	0
$51$	0.437491	0.0612609
$52$	0	0
$53$	6.85851	0.942088	0.471044	−	0.882110i	$-0.343877\pi$
$53$	6.85851	0.942088	0.471044	+	0.882110i	$0.343877\pi$
$54$	0	0
$55$	−4.37306	−0.589663
$56$	0	0
$57$	−1.16260	−0.153990
$58$	0	0
$59$	10.8204	1.40870	0.704348	−	0.709855i	$-0.251240\pi$
$59$	10.8204	1.40870	0.704348	+	0.709855i	$0.251240\pi$
$60$	0	0
$61$	3.25765	0.417099	0.208550	−	0.978012i	$-0.433126\pi$
$61$	3.25765	0.417099	0.208550	+	0.978012i	$0.433126\pi$
$62$	0	0
$63$	−8.92533	−1.12449
$64$	0	0
$65$	−17.3239	−2.14877
$66$	0	0
$67$	−8.34459	−1.01945	−0.509727	−	0.860336i	$-0.670254\pi$
$67$	−8.34459	−1.01945	−0.509727	+	0.860336i	$0.670254\pi$
$68$	0	0
$69$	−1.63033	−0.196269
$70$	0	0
$71$	15.5490	1.84533	0.922665	−	0.385601i	$-0.126006\pi$
$71$	15.5490	1.84533	0.922665	+	0.385601i	$0.126006\pi$
$72$	0	0
$73$	16.3257	1.91078	0.955392	−	0.295341i	$-0.0954332\pi$
$73$	16.3257	1.91078	0.955392	+	0.295341i	$0.0954332\pi$
$74$	0	0
$75$	−2.67759	−0.309182
$76$	0	0
$77$	−3.60073	−0.410341
$78$	0	0
$79$	−11.8536	−1.33363	−0.666814	−	0.745224i	$-0.732343\pi$
$79$	−11.8536	−1.33363	−0.666814	+	0.745224i	$0.732343\pi$
$80$	0	0
$81$	8.19064	0.910072
$82$	0	0
$83$	−1.33359	−0.146381	−0.0731903	−	0.997318i	$-0.523318\pi$
$83$	−1.33359	−0.146381	−0.0731903	+	0.997318i	$0.523318\pi$
$84$	0	0
$85$	−5.40841	−0.586624
$86$	0	0
$87$	−1.69747	−0.181988
$88$	0	0
$89$	6.98475	0.740382	0.370191	−	0.928956i	$-0.379292\pi$
$89$	6.98475	0.740382	0.370191	+	0.928956i	$0.379292\pi$
$90$	0	0
$91$	−14.2644	−1.49531
$92$	0	0
$93$	−1.35443	−0.140448
$94$	0	0
$95$	14.3725	1.47458
$96$	0	0
$97$	7.18127	0.729147	0.364574	−	0.931175i	$-0.381215\pi$
$97$	7.18127	0.729147	0.364574	+	0.931175i	$0.381215\pi$
$98$	0	0
$99$	3.41428	0.343149
$100$	0	0
$101$	14.9931	1.49186	0.745932	−	0.666022i	$-0.232004\pi$
$101$	14.9931	1.49186	0.745932	+	0.666022i	$0.232004\pi$
$102$	0	0
$103$	17.3974	1.71422	0.857110	−	0.515133i	$-0.172258\pi$
$103$	17.3974	1.71422	0.857110	+	0.515133i	$0.172258\pi$
$104$	0	0
$105$	−3.44557	−0.336253
$106$	0	0
$107$	7.06215	0.682724	0.341362	−	0.939932i	$-0.389112\pi$
$107$	7.06215	0.682724	0.341362	+	0.939932i	$0.389112\pi$
$108$	0	0
$109$	−4.91081	−0.470370	−0.235185	−	0.971951i	$-0.575570\pi$
$109$	−4.91081	−0.470370	−0.235185	+	0.971951i	$0.575570\pi$
$110$	0	0
$111$	2.21166	0.209922
$112$	0	0
$113$	−18.9925	−1.78666	−0.893332	−	0.449398i	$-0.851639\pi$
$113$	−18.9925	−1.78666	−0.893332	+	0.449398i	$0.851639\pi$
$114$	0	0
$115$	20.1547	1.87944
$116$	0	0
$117$	13.5257	1.25046
$118$	0	0
$119$	−4.45323	−0.408227
$120$	0	0
$121$	−9.62258	−0.874780
$122$	0	0
$123$	3.43007	0.309279
$124$	0	0
$125$	14.4709	1.29432
$126$	0	0
$127$	10.7997	0.958315	0.479157	−	0.877729i	$-0.340942\pi$
$127$	10.7997	0.958315	0.479157	+	0.877729i	$0.340942\pi$
$128$	0	0
$129$	0.506710	0.0446133
$130$	0	0
$131$	−16.1541	−1.41139	−0.705696	−	0.708515i	$-0.749366\pi$
$131$	−16.1541	−1.41139	−0.705696	+	0.708515i	$0.749366\pi$
$132$	0	0
$133$	11.8341	1.02615
$134$	0	0
$135$	6.63635	0.571166
$136$	0	0
$137$	3.16222	0.270167	0.135084	−	0.990834i	$-0.456870\pi$
$137$	3.16222	0.270167	0.135084	+	0.990834i	$0.456870\pi$
$138$	0	0
$139$	−4.31329	−0.365848	−0.182924	−	0.983127i	$-0.558556\pi$
$139$	−4.31329	−0.365848	−0.182924	+	0.983127i	$0.558556\pi$
$140$	0	0
$141$	−0.0459748	−0.00387177
$142$	0	0
$143$	5.45667	0.456309
$144$	0	0
$145$	20.9847	1.74268
$146$	0	0
$147$	−0.727207	−0.0599790
$148$	0	0
$149$	15.8440	1.29799	0.648994	−	0.760793i	$-0.275190\pi$
$149$	15.8440	1.29799	0.648994	+	0.760793i	$0.275190\pi$
$150$	0	0
$151$	4.42965	0.360479	0.180240	−	0.983623i	$-0.442313\pi$
$151$	4.42965	0.360479	0.180240	+	0.983623i	$0.442313\pi$
$152$	0	0
$153$	4.22264	0.341380
$154$	0	0
$155$	16.7440	1.34491
$156$	0	0
$157$	4.02597	0.321308	0.160654	−	0.987011i	$-0.448640\pi$
$157$	4.02597	0.321308	0.160654	+	0.987011i	$0.448640\pi$
$158$	0	0
$159$	−2.06719	−0.163939
$160$	0	0
$161$	16.5952	1.30788
$162$	0	0
$163$	1.62470	0.127256	0.0636281	−	0.997974i	$-0.479733\pi$
$163$	1.62470	0.127256	0.0636281	+	0.997974i	$0.479733\pi$
$164$	0	0
$165$	1.31806	0.102611
$166$	0	0
$167$	17.7196	1.37118	0.685591	−	0.727987i	$-0.259544\pi$
$167$	17.7196	1.37118	0.685591	+	0.727987i	$0.259544\pi$
$168$	0	0
$169$	8.61668	0.662822
$170$	0	0
$171$	−11.2214	−0.858120
$172$	0	0
$173$	−5.23597	−0.398083	−0.199042	−	0.979991i	$-0.563783\pi$
$173$	−5.23597	−0.398083	−0.199042	+	0.979991i	$0.563783\pi$
$174$	0	0
$175$	27.2553	2.06031
$176$	0	0
$177$	−3.26133	−0.245136
$178$	0	0
$179$	−16.0376	−1.19871	−0.599355	−	0.800484i	$-0.704576\pi$
$179$	−16.0376	−1.19871	−0.599355	+	0.800484i	$0.704576\pi$
$180$	0	0
$181$	4.62033	0.343426	0.171713	−	0.985147i	$-0.445070\pi$
$181$	4.62033	0.343426	0.171713	+	0.985147i	$0.445070\pi$
$182$	0	0
$183$	−0.981873	−0.0725822
$184$	0	0
$185$	−27.3413	−2.01018
$186$	0	0
$187$	1.70353	0.124575
$188$	0	0
$189$	5.46430	0.397469
$190$	0	0
$191$	20.9325	1.51462	0.757310	−	0.653056i	$-0.226513\pi$
$191$	20.9325	1.51462	0.757310	+	0.653056i	$0.226513\pi$
$192$	0	0
$193$	−0.0646429	−0.00465310	−0.00232655	−	0.999997i	$-0.500741\pi$
$193$	−0.0646429	−0.00465310	−0.00232655	+	0.999997i	$0.500741\pi$
$194$	0	0
$195$	5.22154	0.373922
$196$	0	0
$197$	0.466359	0.0332267	0.0166134	−	0.999862i	$-0.494712\pi$
$197$	0.466359	0.0332267	0.0166134	+	0.999862i	$0.494712\pi$
$198$	0	0
$199$	−10.1551	−0.719879	−0.359940	−	0.932976i	$-0.617203\pi$
$199$	−10.1551	−0.719879	−0.359940	+	0.932976i	$0.617203\pi$
$200$	0	0
$201$	2.51511	0.177402
$202$	0	0
$203$	17.2786	1.21272
$204$	0	0
$205$	−42.4037	−2.96161
$206$	0	0
$207$	−15.7359	−1.09372
$208$	0	0
$209$	−4.52702	−0.313140
$210$	0	0
$211$	17.2361	1.18658	0.593292	−	0.804987i	$-0.297828\pi$
$211$	17.2361	1.18658	0.593292	+	0.804987i	$0.297828\pi$
$212$	0	0
$213$	−4.68657	−0.321118
$214$	0	0
$215$	−6.26412	−0.427210
$216$	0	0
$217$	13.7868	0.935910
$218$	0	0
$219$	−4.92067	−0.332508
$220$	0	0
$221$	6.74857	0.453958
$222$	0	0
$223$	23.1432	1.54979	0.774893	−	0.632093i	$-0.217804\pi$
$223$	23.1432	1.54979	0.774893	+	0.632093i	$0.217804\pi$
$224$	0	0
$225$	−25.8440	−1.72293
$226$	0	0
$227$	15.1287	1.00413	0.502064	−	0.864830i	$-0.332574\pi$
$227$	15.1287	1.00413	0.502064	+	0.864830i	$0.332574\pi$
$228$	0	0
$229$	23.7020	1.56628	0.783138	−	0.621848i	$-0.213618\pi$
$229$	23.7020	1.56628	0.783138	+	0.621848i	$0.213618\pi$
$230$	0	0
$231$	1.08528	0.0714062
$232$	0	0
$233$	−18.7419	−1.22782	−0.613911	−	0.789375i	$-0.710405\pi$
$233$	−18.7419	−1.22782	−0.613911	+	0.789375i	$0.710405\pi$
$234$	0	0
$235$	0.568356	0.0370754
$236$	0	0
$237$	3.57273	0.232074
$238$	0	0
$239$	−12.7737	−0.826264	−0.413132	−	0.910671i	$-0.635565\pi$
$239$	−12.7737	−0.826264	−0.413132	+	0.910671i	$0.635565\pi$
$240$	0	0
$241$	−2.74355	−0.176728	−0.0883639	−	0.996088i	$-0.528164\pi$
$241$	−2.74355	−0.176728	−0.0883639	+	0.996088i	$0.528164\pi$
$242$	0	0
$243$	−7.81187	−0.501131
$244$	0	0
$245$	8.98999	0.574349
$246$	0	0
$247$	−17.9339	−1.14110
$248$	0	0
$249$	0.401952	0.0254727
$250$	0	0
$251$	−2.43708	−0.153827	−0.0769136	−	0.997038i	$-0.524507\pi$
$251$	−2.43708	−0.153827	−0.0769136	+	0.997038i	$0.524507\pi$
$252$	0	0
$253$	−6.34829	−0.399114
$254$	0	0
$255$	1.63013	0.102082
$256$	0	0
$257$	−20.3908	−1.27194	−0.635971	−	0.771713i	$-0.719400\pi$
$257$	−20.3908	−1.27194	−0.635971	+	0.771713i	$0.719400\pi$
$258$	0	0
$259$	−22.5126	−1.39886
$260$	0	0
$261$	−16.3839	−1.01414
$262$	0	0
$263$	21.2691	1.31151	0.655754	−	0.754974i	$-0.272351\pi$
$263$	21.2691	1.31151	0.655754	+	0.754974i	$0.272351\pi$
$264$	0	0
$265$	25.5553	1.56985
$266$	0	0
$267$	−2.10524	−0.128839
$268$	0	0
$269$	−22.1856	−1.35268	−0.676341	−	0.736588i	$-0.736436\pi$
$269$	−22.1856	−1.35268	−0.676341	+	0.736588i	$0.736436\pi$
$270$	0	0
$271$	−13.1494	−0.798769	−0.399384	−	0.916784i	$-0.630776\pi$
$271$	−13.1494	−0.798769	−0.399384	+	0.916784i	$0.630776\pi$
$272$	0	0
$273$	4.29936	0.260209
$274$	0	0
$275$	−10.4262	−0.628723
$276$	0	0
$277$	3.33426	0.200336	0.100168	−	0.994971i	$-0.468062\pi$
$277$	3.33426	0.200336	0.100168	+	0.994971i	$0.468062\pi$
$278$	0	0
$279$	−13.0729	−0.782656
$280$	0	0
$281$	−15.0793	−0.899553	−0.449777	−	0.893141i	$-0.648496\pi$
$281$	−15.0793	−0.899553	−0.449777	+	0.893141i	$0.648496\pi$
$282$	0	0
$283$	4.52764	0.269140	0.134570	−	0.990904i	$-0.457035\pi$
$283$	4.52764	0.269140	0.134570	+	0.990904i	$0.457035\pi$
$284$	0	0
$285$	−4.33194	−0.256602
$286$	0	0
$287$	−34.9148	−2.06096
$288$	0	0
$289$	−14.8931	−0.876067
$290$	0	0
$291$	−2.16447	−0.126884
$292$	0	0
$293$	13.2843	0.776077	0.388038	−	0.921643i	$-0.373153\pi$
$293$	13.2843	0.776077	0.388038	+	0.921643i	$0.373153\pi$
$294$	0	0
$295$	40.3177	2.34739
$296$	0	0
$297$	−2.09031	−0.121292
$298$	0	0
$299$	−25.1489	−1.45440
$300$	0	0
$301$	−5.15781	−0.297291
$302$	0	0
$303$	−4.51899	−0.259609
$304$	0	0
$305$	12.1383	0.695035
$306$	0	0
$307$	−26.6964	−1.52365	−0.761823	−	0.647786i	$-0.775695\pi$
$307$	−26.6964	−1.52365	−0.761823	+	0.647786i	$0.775695\pi$
$308$	0	0
$309$	−5.24369	−0.298303
$310$	0	0
$311$	−5.85153	−0.331810	−0.165905	−	0.986142i	$-0.553054\pi$
$311$	−5.85153	−0.331810	−0.165905	+	0.986142i	$0.553054\pi$
$312$	0	0
$313$	−12.8900	−0.728587	−0.364293	−	0.931284i	$-0.618689\pi$
$313$	−12.8900	−0.728587	−0.364293	+	0.931284i	$0.618689\pi$
$314$	0	0
$315$	−33.2565	−1.87379
$316$	0	0
$317$	21.0687	1.18334	0.591669	−	0.806181i	$-0.298469\pi$
$317$	21.0687	1.18334	0.591669	+	0.806181i	$0.298469\pi$
$318$	0	0
$319$	−6.60972	−0.370073
$320$	0	0
$321$	−2.12857	−0.118805
$322$	0	0
$323$	−5.59882	−0.311527
$324$	0	0
$325$	−41.3036	−2.29111
$326$	0	0
$327$	1.48015	0.0818523
$328$	0	0
$329$	0.467978	0.0258005
$330$	0	0
$331$	−21.2904	−1.17023	−0.585113	−	0.810951i	$-0.698950\pi$
$331$	−21.2904	−1.17023	−0.585113	+	0.810951i	$0.698950\pi$
$332$	0	0
$333$	21.3469	1.16980
$334$	0	0
$335$	−31.0926	−1.69877
$336$	0	0
$337$	−21.6961	−1.18186	−0.590930	−	0.806723i	$-0.701239\pi$
$337$	−21.6961	−1.18186	−0.590930	+	0.806723i	$0.701239\pi$
$338$	0	0
$339$	5.72445	0.310909
$340$	0	0
$341$	−5.27399	−0.285603
$342$	0	0
$343$	−14.0738	−0.759916
$344$	0	0
$345$	−6.07474	−0.327053
$346$	0	0
$347$	−13.7539	−0.738347	−0.369173	−	0.929361i	$-0.620359\pi$
$347$	−13.7539	−0.738347	−0.369173	+	0.929361i	$0.620359\pi$
$348$	0	0
$349$	2.39400	0.128148	0.0640741	−	0.997945i	$-0.479591\pi$
$349$	2.39400	0.128148	0.0640741	+	0.997945i	$0.479591\pi$
$350$	0	0
$351$	−8.28078	−0.441995
$352$	0	0
$353$	18.8324	1.00235	0.501173	−	0.865347i	$-0.332902\pi$
$353$	18.8324	1.00235	0.501173	+	0.865347i	$0.332902\pi$
$354$	0	0
$355$	57.9370	3.07498
$356$	0	0
$357$	1.34223	0.0710382
$358$	0	0
$359$	−14.4127	−0.760673	−0.380336	−	0.924848i	$-0.624192\pi$
$359$	−14.4127	−0.760673	−0.380336	+	0.924848i	$0.624192\pi$
$360$	0	0
$361$	−4.12153	−0.216923
$362$	0	0
$363$	2.90030	0.152226
$364$	0	0
$365$	60.8311	3.18404
$366$	0	0
$367$	11.2947	0.589579	0.294790	−	0.955562i	$-0.404750\pi$
$367$	11.2947	0.589579	0.294790	+	0.955562i	$0.404750\pi$
$368$	0	0
$369$	33.1069	1.72348
$370$	0	0
$371$	21.0420	1.09245
$372$	0	0
$373$	2.25807	0.116918	0.0584592	−	0.998290i	$-0.481381\pi$
$373$	2.25807	0.116918	0.0584592	+	0.998290i	$0.481381\pi$
$374$	0	0
$375$	−4.36162	−0.225233
$376$	0	0
$377$	−26.1845	−1.34857
$378$	0	0
$379$	14.2568	0.732320	0.366160	−	0.930552i	$-0.380672\pi$
$379$	14.2568	0.732320	0.366160	+	0.930552i	$0.380672\pi$
$380$	0	0
$381$	−3.25508	−0.166763
$382$	0	0
$383$	5.62065	0.287202	0.143601	−	0.989636i	$-0.454132\pi$
$383$	5.62065	0.287202	0.143601	+	0.989636i	$0.454132\pi$
$384$	0	0
$385$	−13.4166	−0.683774
$386$	0	0
$387$	4.89074	0.248610
$388$	0	0
$389$	23.6700	1.20012	0.600058	−	0.799956i	$-0.295144\pi$
$389$	23.6700	1.20012	0.600058	+	0.799956i	$0.295144\pi$
$390$	0	0
$391$	−7.85130	−0.397057
$392$	0	0
$393$	4.86894	0.245606
$394$	0	0
$395$	−44.1673	−2.22230
$396$	0	0
$397$	0.156130	0.00783594	0.00391797	−	0.999992i	$-0.498753\pi$
$397$	0.156130	0.00783594	0.00391797	+	0.999992i	$0.498753\pi$
$398$	0	0
$399$	−3.56688	−0.178567
$400$	0	0
$401$	−4.82148	−0.240773	−0.120387	−	0.992727i	$-0.538413\pi$
$401$	−4.82148	−0.240773	−0.120387	+	0.992727i	$0.538413\pi$
$402$	0	0
$403$	−20.8930	−1.04075
$404$	0	0
$405$	30.5190	1.51650
$406$	0	0
$407$	8.61193	0.426878
$408$	0	0
$409$	20.1441	0.996059	0.498030	−	0.867160i	$-0.334057\pi$
$409$	20.1441	0.996059	0.498030	+	0.867160i	$0.334057\pi$
$410$	0	0
$411$	−0.953112	−0.0470136
$412$	0	0
$413$	33.1971	1.63352
$414$	0	0
$415$	−4.96907	−0.243922
$416$	0	0
$417$	1.30005	0.0636637
$418$	0	0
$419$	18.8326	0.920035	0.460017	−	0.887910i	$-0.347843\pi$
$419$	18.8326	0.920035	0.460017	+	0.887910i	$0.347843\pi$
$420$	0	0
$421$	18.7069	0.911716	0.455858	−	0.890052i	$-0.349332\pi$
$421$	18.7069	0.911716	0.455858	+	0.890052i	$0.349332\pi$
$422$	0	0
$423$	−0.443746	−0.0215757
$424$	0	0
$425$	−12.8947	−0.625483
$426$	0	0
$427$	9.99451	0.483668
$428$	0	0
$429$	−1.64467	−0.0794054
$430$	0	0
$431$	40.1512	1.93402	0.967008	−	0.254747i	$-0.0819923\pi$
$431$	40.1512	1.93402	0.967008	+	0.254747i	$0.0819923\pi$
$432$	0	0
$433$	−4.92321	−0.236594	−0.118297	−	0.992978i	$-0.537744\pi$
$433$	−4.92321	−0.236594	−0.118297	+	0.992978i	$0.537744\pi$
$434$	0	0
$435$	−6.32491	−0.303256
$436$	0	0
$437$	20.8643	0.998073
$438$	0	0
$439$	−8.33871	−0.397985	−0.198992	−	0.980001i	$-0.563767\pi$
$439$	−8.33871	−0.397985	−0.198992	+	0.980001i	$0.563767\pi$
$440$	0	0
$441$	−7.01897	−0.334237
$442$	0	0
$443$	−36.3769	−1.72832	−0.864158	−	0.503220i	$-0.832149\pi$
$443$	−36.3769	−1.72832	−0.864158	+	0.503220i	$0.832149\pi$
$444$	0	0
$445$	26.0258	1.23374
$446$	0	0
$447$	−4.77546	−0.225872
$448$	0	0
$449$	26.8393	1.26663	0.633313	−	0.773895i	$-0.281694\pi$
$449$	26.8393	1.26663	0.633313	+	0.773895i	$0.281694\pi$
$450$	0	0
$451$	13.3563	0.628922
$452$	0	0
$453$	−1.33512	−0.0627294
$454$	0	0
$455$	−53.1501	−2.49172
$456$	0	0
$457$	10.4784	0.490159	0.245080	−	0.969503i	$-0.421186\pi$
$457$	10.4784	0.490159	0.245080	+	0.969503i	$0.421186\pi$
$458$	0	0
$459$	−2.58520	−0.120667
$460$	0	0
$461$	−20.6985	−0.964024	−0.482012	−	0.876165i	$-0.660094\pi$
$461$	−20.6985	−0.964024	−0.482012	+	0.876165i	$0.660094\pi$
$462$	0	0
$463$	−20.9521	−0.973726	−0.486863	−	0.873478i	$-0.661859\pi$
$463$	−20.9521	−0.973726	−0.486863	+	0.873478i	$0.661859\pi$
$464$	0	0
$465$	−5.04673	−0.234037
$466$	0	0
$467$	12.0636	0.558235	0.279118	−	0.960257i	$-0.409958\pi$
$467$	12.0636	0.558235	0.279118	+	0.960257i	$0.409958\pi$
$468$	0	0
$469$	−25.6013	−1.18216
$470$	0	0
$471$	−1.21345	−0.0559129
$472$	0	0
$473$	1.97306	0.0907215
$474$	0	0
$475$	34.2667	1.57226
$476$	0	0
$477$	−19.9525	−0.913560
$478$	0	0
$479$	−24.2636	−1.10863	−0.554316	−	0.832306i	$-0.687020\pi$
$479$	−24.2636	−1.10863	−0.554316	+	0.832306i	$0.687020\pi$
$480$	0	0
$481$	34.1163	1.55557
$482$	0	0
$483$	−5.00188	−0.227593
$484$	0	0
$485$	26.7580	1.21502
$486$	0	0
$487$	22.2435	1.00795	0.503974	−	0.863719i	$-0.331871\pi$
$487$	22.2435	1.00795	0.503974	+	0.863719i	$0.331871\pi$
$488$	0	0
$489$	−0.489693	−0.0221447
$490$	0	0
$491$	−10.1889	−0.459819	−0.229909	−	0.973212i	$-0.573843\pi$
$491$	−10.1889	−0.459819	−0.229909	+	0.973212i	$0.573843\pi$
$492$	0	0
$493$	−8.17462	−0.368166
$494$	0	0
$495$	12.7219	0.571807
$496$	0	0
$497$	47.7047	2.13985
$498$	0	0
$499$	3.72453	0.166733	0.0833665	−	0.996519i	$-0.473433\pi$
$499$	3.72453	0.166733	0.0833665	+	0.996519i	$0.473433\pi$
$500$	0	0
$501$	−5.34078	−0.238608
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	55.8653	2.48597
$506$	0	0
$507$	−2.59712	−0.115342
$508$	0	0
$509$	−29.5337	−1.30906	−0.654528	−	0.756038i	$-0.727133\pi$
$509$	−29.5337	−1.30906	−0.654528	+	0.756038i	$0.727133\pi$
$510$	0	0
$511$	50.0876	2.21575
$512$	0	0
$513$	6.86999	0.303317
$514$	0	0
$515$	64.8243	2.85650
$516$	0	0
$517$	−0.179020	−0.00787328
$518$	0	0
$519$	1.57815	0.0692731
$520$	0	0
$521$	−35.4590	−1.55349	−0.776744	−	0.629816i	$-0.783130\pi$
$521$	−35.4590	−1.55349	−0.776744	+	0.629816i	$0.783130\pi$
$522$	0	0
$523$	−34.2287	−1.49672	−0.748358	−	0.663295i	$-0.769158\pi$
$523$	−34.2287	−1.49672	−0.748358	+	0.663295i	$0.769158\pi$
$524$	0	0
$525$	−8.21489	−0.358527
$526$	0	0
$527$	−6.52264	−0.284131
$528$	0	0
$529$	6.25824	0.272098
$530$	0	0
$531$	−31.4782	−1.36604
$532$	0	0
$533$	52.9111	2.29183
$534$	0	0
$535$	26.3142	1.13766
$536$	0	0
$537$	4.83384	0.208595
$538$	0	0
$539$	−2.83165	−0.121968
$540$	0	0
$541$	−35.8437	−1.54104	−0.770520	−	0.637416i	$-0.780003\pi$
$541$	−35.8437	−1.54104	−0.770520	+	0.637416i	$0.780003\pi$
$542$	0	0
$543$	−1.39259	−0.0597619
$544$	0	0
$545$	−18.2981	−0.783803
$546$	0	0
$547$	3.53567	0.151174	0.0755871	−	0.997139i	$-0.475917\pi$
$547$	3.53567	0.151174	0.0755871	+	0.997139i	$0.475917\pi$
$548$	0	0
$549$	−9.47700	−0.404469
$550$	0	0
$551$	21.7235	0.925451
$552$	0	0
$553$	−36.3669	−1.54648
$554$	0	0
$555$	8.24084	0.349804
$556$	0	0
$557$	2.87431	0.121788	0.0608942	−	0.998144i	$-0.480605\pi$
$557$	2.87431	0.121788	0.0608942	+	0.998144i	$0.480605\pi$
$558$	0	0
$559$	7.81632	0.330595
$560$	0	0
$561$	−0.513454	−0.0216780
$562$	0	0
$563$	−11.6026	−0.488990	−0.244495	−	0.969651i	$-0.578622\pi$
$563$	−11.6026	−0.488990	−0.244495	+	0.969651i	$0.578622\pi$
$564$	0	0
$565$	−70.7676	−2.97721
$566$	0	0
$567$	25.1290	1.05532
$568$	0	0
$569$	31.9509	1.33945	0.669725	−	0.742609i	$-0.266412\pi$
$569$	31.9509	1.33945	0.669725	+	0.742609i	$0.266412\pi$
$570$	0	0
$571$	−39.1988	−1.64042	−0.820209	−	0.572063i	$-0.806143\pi$
$571$	−39.1988	−1.64042	−0.820209	+	0.572063i	$0.806143\pi$
$572$	0	0
$573$	−6.30916	−0.263569
$574$	0	0
$575$	48.0526	2.00393
$576$	0	0
$577$	−22.3956	−0.932341	−0.466171	−	0.884695i	$-0.654367\pi$
$577$	−22.3956	−0.932341	−0.466171	+	0.884695i	$0.654367\pi$
$578$	0	0
$579$	0.0194837	0.000809716 0
$580$	0	0
$581$	−4.09148	−0.169743
$582$	0	0
$583$	−8.04938	−0.333371
$584$	0	0
$585$	50.3980	2.08370
$586$	0	0
$587$	−12.4270	−0.512916	−0.256458	−	0.966555i	$-0.582555\pi$
$587$	−12.4270	−0.512916	−0.256458	+	0.966555i	$0.582555\pi$
$588$	0	0
$589$	17.3335	0.714213
$590$	0	0
$591$	−0.140563	−0.00578200
$592$	0	0
$593$	9.80507	0.402646	0.201323	−	0.979525i	$-0.435476\pi$
$593$	9.80507	0.402646	0.201323	+	0.979525i	$0.435476\pi$
$594$	0	0
$595$	−16.5931	−0.680250
$596$	0	0
$597$	3.06082	0.125271
$598$	0	0
$599$	9.90435	0.404681	0.202340	−	0.979315i	$-0.435145\pi$
$599$	9.90435	0.404681	0.202340	+	0.979315i	$0.435145\pi$
$600$	0	0
$601$	41.6079	1.69722	0.848610	−	0.529018i	$-0.177440\pi$
$601$	41.6079	1.69722	0.848610	+	0.529018i	$0.177440\pi$
$602$	0	0
$603$	24.2757	0.988583
$604$	0	0
$605$	−35.8545	−1.45769
$606$	0	0
$607$	37.4240	1.51899	0.759497	−	0.650511i	$-0.225445\pi$
$607$	37.4240	1.51899	0.759497	+	0.650511i	$0.225445\pi$
$608$	0	0
$609$	−5.20786	−0.211033
$610$	0	0
$611$	−0.709190	−0.0286907
$612$	0	0
$613$	−33.4155	−1.34964	−0.674819	−	0.737983i	$-0.735778\pi$
$613$	−33.4155	−1.34964	−0.674819	+	0.737983i	$0.735778\pi$
$614$	0	0
$615$	12.7807	0.515369
$616$	0	0
$617$	22.1297	0.890909	0.445454	−	0.895305i	$-0.353042\pi$
$617$	22.1297	0.890909	0.445454	+	0.895305i	$0.353042\pi$
$618$	0	0
$619$	26.0337	1.04638	0.523191	−	0.852215i	$-0.324741\pi$
$619$	26.0337	1.04638	0.523191	+	0.852215i	$0.324741\pi$
$620$	0	0
$621$	9.63387	0.386594
$622$	0	0
$623$	21.4293	0.858548
$624$	0	0
$625$	9.50140	0.380056
$626$	0	0
$627$	1.36447	0.0544916
$628$	0	0
$629$	10.6509	0.424678
$630$	0	0
$631$	15.5675	0.619732	0.309866	−	0.950780i	$-0.399716\pi$
$631$	15.5675	0.619732	0.309866	+	0.950780i	$0.399716\pi$
$632$	0	0
$633$	−5.19507	−0.206485
$634$	0	0
$635$	40.2404	1.59689
$636$	0	0
$637$	−11.2176	−0.444459
$638$	0	0
$639$	−45.2345	−1.78945
$640$	0	0
$641$	32.0898	1.26747	0.633736	−	0.773550i	$-0.281521\pi$
$641$	32.0898	1.26747	0.633736	+	0.773550i	$0.281521\pi$
$642$	0	0
$643$	4.78556	0.188724	0.0943620	−	0.995538i	$-0.469919\pi$
$643$	4.78556	0.188724	0.0943620	+	0.995538i	$0.469919\pi$
$644$	0	0
$645$	1.88804	0.0743416
$646$	0	0
$647$	−15.8439	−0.622887	−0.311444	−	0.950265i	$-0.600812\pi$
$647$	−15.8439	−0.622887	−0.311444	+	0.950265i	$0.600812\pi$
$648$	0	0
$649$	−12.6992	−0.498487
$650$	0	0
$651$	−4.15542	−0.162864
$652$	0	0
$653$	−47.6996	−1.86663	−0.933315	−	0.359058i	$-0.883098\pi$
$653$	−47.6996	−1.86663	−0.933315	+	0.359058i	$0.883098\pi$
$654$	0	0
$655$	−60.1916	−2.35188
$656$	0	0
$657$	−47.4941	−1.85292
$658$	0	0
$659$	−16.6129	−0.647146	−0.323573	−	0.946203i	$-0.604884\pi$
$659$	−16.6129	−0.647146	−0.323573	+	0.946203i	$0.604884\pi$
$660$	0	0
$661$	38.9128	1.51353	0.756766	−	0.653685i	$-0.226778\pi$
$661$	38.9128	1.51353	0.756766	+	0.653685i	$0.226778\pi$
$662$	0	0
$663$	−2.03406	−0.0789962
$664$	0	0
$665$	44.0950	1.70993
$666$	0	0
$667$	30.4631	1.17954
$668$	0	0
$669$	−6.97550	−0.269689
$670$	0	0
$671$	−3.82329	−0.147596
$672$	0	0
$673$	−38.3230	−1.47724	−0.738622	−	0.674119i	$-0.764523\pi$
$673$	−38.3230	−1.47724	−0.738622	+	0.674119i	$0.764523\pi$
$674$	0	0
$675$	15.8223	0.609001
$676$	0	0
$677$	24.7614	0.951659	0.475829	−	0.879538i	$-0.342148\pi$
$677$	24.7614	0.951659	0.475829	+	0.879538i	$0.342148\pi$
$678$	0	0
$679$	22.0322	0.845520
$680$	0	0
$681$	−4.55988	−0.174735
$682$	0	0
$683$	−19.6227	−0.750843	−0.375422	−	0.926854i	$-0.622502\pi$
$683$	−19.6227	−0.750843	−0.375422	+	0.926854i	$0.622502\pi$
$684$	0	0
$685$	11.7827	0.450194
$686$	0	0
$687$	−7.14393	−0.272558
$688$	0	0
$689$	−31.8878	−1.21483
$690$	0	0
$691$	35.1053	1.33547	0.667734	−	0.744400i	$-0.267264\pi$
$691$	35.1053	1.33547	0.667734	+	0.744400i	$0.267264\pi$
$692$	0	0
$693$	10.4751	0.397915
$694$	0	0
$695$	−16.0717	−0.609633
$696$	0	0
$697$	16.5184	0.625681
$698$	0	0
$699$	5.64891	0.213662
$700$	0	0
$701$	−12.6272	−0.476923	−0.238461	−	0.971152i	$-0.576643\pi$
$701$	−12.6272	−0.476923	−0.238461	+	0.971152i	$0.576643\pi$
$702$	0	0
$703$	−28.3039	−1.06750
$704$	0	0
$705$	−0.171306	−0.00645175
$706$	0	0
$707$	45.9989	1.72997
$708$	0	0
$709$	−28.3950	−1.06640	−0.533198	−	0.845990i	$-0.679010\pi$
$709$	−28.3950	−1.06640	−0.533198	+	0.845990i	$0.679010\pi$
$710$	0	0
$711$	34.4838	1.29324
$712$	0	0
$713$	24.3069	0.910302
$714$	0	0
$715$	20.3320	0.760373
$716$	0	0
$717$	3.85007	0.143784
$718$	0	0
$719$	−30.2026	−1.12637	−0.563183	−	0.826332i	$-0.690423\pi$
$719$	−30.2026	−1.12637	−0.563183	+	0.826332i	$0.690423\pi$
$720$	0	0
$721$	53.3756	1.98781
$722$	0	0
$723$	0.826922	0.0307536
$724$	0	0
$725$	50.0315	1.85812
$726$	0	0
$727$	12.8672	0.477219	0.238609	−	0.971116i	$-0.423308\pi$
$727$	12.8672	0.477219	0.238609	+	0.971116i	$0.423308\pi$
$728$	0	0
$729$	−22.2174	−0.822866
$730$	0	0
$731$	2.44020	0.0902540
$732$	0	0
$733$	−10.8470	−0.400644	−0.200322	−	0.979730i	$-0.564199\pi$
$733$	−10.8470	−0.400644	−0.200322	+	0.979730i	$0.564199\pi$
$734$	0	0
$735$	−2.70963	−0.0999463
$736$	0	0
$737$	9.79350	0.360748
$738$	0	0
$739$	8.61885	0.317049	0.158525	−	0.987355i	$-0.449326\pi$
$739$	8.61885	0.317049	0.158525	+	0.987355i	$0.449326\pi$
$740$	0	0
$741$	5.40537	0.198571
$742$	0	0
$743$	4.10452	0.150580	0.0752902	−	0.997162i	$-0.476012\pi$
$743$	4.10452	0.150580	0.0752902	+	0.997162i	$0.476012\pi$
$744$	0	0
$745$	59.0359	2.16291
$746$	0	0
$747$	3.87962	0.141948
$748$	0	0
$749$	21.6668	0.791687
$750$	0	0
$751$	43.9781	1.60478	0.802391	−	0.596799i	$-0.203561\pi$
$751$	43.9781	1.60478	0.802391	+	0.596799i	$0.203561\pi$
$752$	0	0
$753$	0.734550	0.0267685
$754$	0	0
$755$	16.5052	0.600686
$756$	0	0
$757$	−4.83983	−0.175907	−0.0879533	−	0.996125i	$-0.528033\pi$
$757$	−4.83983	−0.175907	−0.0879533	+	0.996125i	$0.528033\pi$
$758$	0	0
$759$	1.91341	0.0694525
$760$	0	0
$761$	20.3984	0.739443	0.369721	−	0.929143i	$-0.379453\pi$
$761$	20.3984	0.739443	0.369721	+	0.929143i	$0.379453\pi$
$762$	0	0
$763$	−15.0664	−0.545442
$764$	0	0
$765$	15.7339	0.568860
$766$	0	0
$767$	−50.3081	−1.81652
$768$	0	0
$769$	26.8238	0.967292	0.483646	−	0.875264i	$-0.339312\pi$
$769$	26.8238	0.967292	0.483646	+	0.875264i	$0.339312\pi$
$770$	0	0
$771$	6.14590	0.221339
$772$	0	0
$773$	−24.0637	−0.865510	−0.432755	−	0.901511i	$-0.642459\pi$
$773$	−24.0637	−0.865510	−0.432755	+	0.901511i	$0.642459\pi$
$774$	0	0
$775$	39.9208	1.43400
$776$	0	0
$777$	6.78542	0.243425
$778$	0	0
$779$	−43.8966	−1.57276
$780$	0	0
$781$	−18.2489	−0.652997
$782$	0	0
$783$	10.0306	0.358465
$784$	0	0
$785$	15.0011	0.535412
$786$	0	0
$787$	19.9260	0.710285	0.355143	−	0.934812i	$-0.384432\pi$
$787$	19.9260	0.710285	0.355143	+	0.934812i	$0.384432\pi$
$788$	0	0
$789$	−6.41063	−0.228224
$790$	0	0
$791$	−58.2693	−2.07182
$792$	0	0
$793$	−15.1460	−0.537851
$794$	0	0
$795$	−7.70253	−0.273181
$796$	0	0
$797$	15.3504	0.543738	0.271869	−	0.962334i	$-0.412358\pi$
$797$	15.3504	0.543738	0.271869	+	0.962334i	$0.412358\pi$
$798$	0	0
$799$	−0.221404	−0.00783271
$800$	0	0
$801$	−20.3197	−0.717962
$802$	0	0
$803$	−19.1605	−0.676158
$804$	0	0
$805$	61.8349	2.17940
$806$	0	0
$807$	6.68688	0.235389
$808$	0	0
$809$	−19.7103	−0.692976	−0.346488	−	0.938054i	$-0.612626\pi$
$809$	−19.7103	−0.692976	−0.346488	+	0.938054i	$0.612626\pi$
$810$	0	0
$811$	−45.7177	−1.60537	−0.802683	−	0.596406i	$-0.796595\pi$
$811$	−45.7177	−1.60537	−0.802683	+	0.596406i	$0.796595\pi$
$812$	0	0
$813$	3.96330	0.138999
$814$	0	0
$815$	6.05376	0.212054
$816$	0	0
$817$	−6.48466	−0.226869
$818$	0	0
$819$	41.4972	1.45003
$820$	0	0
$821$	10.1844	0.355440	0.177720	−	0.984081i	$-0.443128\pi$
$821$	10.1844	0.355440	0.177720	+	0.984081i	$0.443128\pi$
$822$	0	0
$823$	23.0653	0.804006	0.402003	−	0.915638i	$-0.368314\pi$
$823$	23.0653	0.804006	0.402003	+	0.915638i	$0.368314\pi$
$824$	0	0
$825$	3.14252	0.109408
$826$	0	0
$827$	36.4008	1.26578	0.632889	−	0.774242i	$-0.281869\pi$
$827$	36.4008	1.26578	0.632889	+	0.774242i	$0.281869\pi$
$828$	0	0
$829$	50.9621	1.76999	0.884993	−	0.465604i	$-0.154163\pi$
$829$	50.9621	1.76999	0.884993	+	0.465604i	$0.154163\pi$
$830$	0	0
$831$	−1.00496	−0.0348618
$832$	0	0
$833$	−3.50206	−0.121339
$834$	0	0
$835$	66.0246	2.28487
$836$	0	0
$837$	8.00356	0.276643
$838$	0	0
$839$	32.2271	1.11260	0.556302	−	0.830980i	$-0.312220\pi$
$839$	32.2271	1.11260	0.556302	+	0.830980i	$0.312220\pi$
$840$	0	0
$841$	2.71762	0.0937112
$842$	0	0
$843$	4.54498	0.156537
$844$	0	0
$845$	32.1065	1.10450
$846$	0	0
$847$	−29.5222	−1.01440
$848$	0	0
$849$	−1.36466	−0.0468349
$850$	0	0
$851$	−39.6910	−1.36059
$852$	0	0
$853$	−45.0180	−1.54139	−0.770694	−	0.637206i	$-0.780090\pi$
$853$	−45.0180	−1.54139	−0.770694	+	0.637206i	$0.780090\pi$
$854$	0	0
$855$	−41.8117	−1.42993
$856$	0	0
$857$	−27.2873	−0.932115	−0.466057	−	0.884754i	$-0.654326\pi$
$857$	−27.2873	−0.932115	−0.466057	+	0.884754i	$0.654326\pi$
$858$	0	0
$859$	−45.4292	−1.55003	−0.775013	−	0.631946i	$-0.782257\pi$
$859$	−45.4292	−1.55003	−0.775013	+	0.631946i	$0.782257\pi$
$860$	0	0
$861$	10.5235	0.358641
$862$	0	0
$863$	46.2979	1.57600	0.788000	−	0.615675i	$-0.211117\pi$
$863$	46.2979	1.57600	0.788000	+	0.615675i	$0.211117\pi$
$864$	0	0
$865$	−19.5096	−0.663348
$866$	0	0
$867$	4.48888	0.152450
$868$	0	0
$869$	13.9117	0.471923
$870$	0	0
$871$	38.7971	1.31459
$872$	0	0
$873$	−20.8914	−0.707067
$874$	0	0
$875$	44.3970	1.50089
$876$	0	0
$877$	−20.2644	−0.684279	−0.342140	−	0.939649i	$-0.611152\pi$
$877$	−20.2644	−0.684279	−0.342140	+	0.939649i	$0.611152\pi$
$878$	0	0
$879$	−4.00396	−0.135050
$880$	0	0
$881$	−35.4246	−1.19348	−0.596742	−	0.802433i	$-0.703539\pi$
$881$	−35.4246	−1.19348	−0.596742	+	0.802433i	$0.703539\pi$
$882$	0	0
$883$	8.30116	0.279356	0.139678	−	0.990197i	$-0.455393\pi$
$883$	8.30116	0.279356	0.139678	+	0.990197i	$0.455393\pi$
$884$	0	0
$885$	−12.1520	−0.408484
$886$	0	0
$887$	27.4612	0.922055	0.461028	−	0.887386i	$-0.347481\pi$
$887$	27.4612	0.922055	0.461028	+	0.887386i	$0.347481\pi$
$888$	0	0
$889$	33.1335	1.11126
$890$	0	0
$891$	−9.61282	−0.322042
$892$	0	0
$893$	0.588365	0.0196889
$894$	0	0
$895$	−59.7576	−1.99747
$896$	0	0
$897$	7.58002	0.253089
$898$	0	0
$899$	25.3079	0.844067
$900$	0	0
$901$	−9.95512	−0.331653
$902$	0	0
$903$	1.55459	0.0517336
$904$	0	0
$905$	17.2157	0.572270
$906$	0	0
$907$	18.2682	0.606585	0.303292	−	0.952898i	$-0.401914\pi$
$907$	18.2682	0.606585	0.303292	+	0.952898i	$0.401914\pi$
$908$	0	0
$909$	−43.6171	−1.44669
$910$	0	0
$911$	−30.5548	−1.01233	−0.506163	−	0.862438i	$-0.668936\pi$
$911$	−30.5548	−1.01233	−0.506163	+	0.862438i	$0.668936\pi$
$912$	0	0
$913$	1.56515	0.0517989
$914$	0	0
$915$	−3.65854	−0.120948
$916$	0	0
$917$	−49.5611	−1.63665
$918$	0	0
$919$	−4.50873	−0.148729	−0.0743646	−	0.997231i	$-0.523693\pi$
$919$	−4.50873	−0.148729	−0.0743646	+	0.997231i	$0.523693\pi$
$920$	0	0
$921$	8.04645	0.265140
$922$	0	0
$923$	−72.2933	−2.37956
$924$	0	0
$925$	−65.1869	−2.14333
$926$	0	0
$927$	−50.6118	−1.66231
$928$	0	0
$929$	57.1290	1.87434	0.937171	−	0.348869i	$-0.113434\pi$
$929$	57.1290	1.87434	0.937171	+	0.348869i	$0.113434\pi$
$930$	0	0
$931$	9.30649	0.305008
$932$	0	0
$933$	1.76368	0.0577404
$934$	0	0
$935$	6.34750	0.207585
$936$	0	0
$937$	−22.3179	−0.729095	−0.364548	−	0.931185i	$-0.618776\pi$
$937$	−22.3179	−0.729095	−0.364548	+	0.931185i	$0.618776\pi$
$938$	0	0
$939$	3.88513	0.126786
$940$	0	0
$941$	−28.0794	−0.915361	−0.457680	−	0.889117i	$-0.651320\pi$
$941$	−28.0794	−0.915361	−0.457680	+	0.889117i	$0.651320\pi$
$942$	0	0
$943$	−61.5568	−2.00457
$944$	0	0
$945$	20.3604	0.662324
$946$	0	0
$947$	1.45613	0.0473177	0.0236589	−	0.999720i	$-0.492468\pi$
$947$	1.45613	0.0473177	0.0236589	+	0.999720i	$0.492468\pi$
$948$	0	0
$949$	−75.9045	−2.46396
$950$	0	0
$951$	−6.35024	−0.205921
$952$	0	0
$953$	33.8519	1.09657	0.548285	−	0.836292i	$-0.315281\pi$
$953$	33.8519	1.09657	0.548285	+	0.836292i	$0.315281\pi$
$954$	0	0
$955$	77.9961	2.52389
$956$	0	0
$957$	1.99221	0.0643989
$958$	0	0
$959$	9.70175	0.313286
$960$	0	0
$961$	−10.8065	−0.348595
$962$	0	0
$963$	−20.5449	−0.662050
$964$	0	0
$965$	−0.240865	−0.00775371
$966$	0	0
$967$	0.0534134	0.00171766	0.000858830	−	1.00000i	$-0.499727\pi$
$967$	0.0534134	0.00171766	0.000858830		1.00000i	$0.499727\pi$
$968$	0	0
$969$	1.68752	0.0542108
$970$	0	0
$971$	19.3689	0.621577	0.310788	−	0.950479i	$-0.399407\pi$
$971$	19.3689	0.621577	0.310788	+	0.950479i	$0.399407\pi$
$972$	0	0
$973$	−13.2332	−0.424238
$974$	0	0
$975$	12.4491	0.398691
$976$	0	0
$977$	−39.5302	−1.26468	−0.632341	−	0.774690i	$-0.717906\pi$
$977$	−39.5302	−1.26468	−0.632341	+	0.774690i	$0.717906\pi$
$978$	0	0
$979$	−8.19755	−0.261995
$980$	0	0
$981$	14.2863	0.456127
$982$	0	0
$983$	−45.8057	−1.46098	−0.730488	−	0.682926i	$-0.760707\pi$
$983$	−45.8057	−1.46098	−0.730488	+	0.682926i	$0.760707\pi$
$984$	0	0
$985$	1.73769	0.0553675
$986$	0	0
$987$	−0.141051	−0.00448971
$988$	0	0
$989$	−9.09352	−0.289157
$990$	0	0
$991$	27.4464	0.871865	0.435932	−	0.899979i	$-0.356419\pi$
$991$	27.4464	0.871865	0.435932	+	0.899979i	$0.356419\pi$
$992$	0	0
$993$	6.41705	0.203639
$994$	0	0
$995$	−37.8389	−1.19957
$996$	0	0
$997$	−43.2907	−1.37103	−0.685515	−	0.728058i	$-0.740423\pi$
$997$	−43.2907	−1.37103	−0.685515	+	0.728058i	$0.740423\pi$
$998$	0	0
$999$	−13.0691	−0.413487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.8		21
4.3	odd	2		2012.2.a.a.1.14	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.14	✓	21	4.3	odd	2
8048.2.a.t.1.8		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)

\(f(q)\)	\(=\)	\(q-0.301406 q^{3} +3.72608 q^{5} +3.06802 q^{7} -2.90915 q^{9} +O(q^{10})\)	\(q-0.301406 q^{3} +3.72608 q^{5} +3.06802 q^{7} -2.90915 q^{9} -1.17363 q^{11} -4.64937 q^{13} -1.12306 q^{15} -1.45150 q^{17} +3.85726 q^{19} -0.924717 q^{21} +5.40909 q^{23} +8.88368 q^{25} +1.78105 q^{27} +5.63184 q^{29} +4.49372 q^{31} +0.353740 q^{33} +11.4317 q^{35} -7.33783 q^{37} +1.40135 q^{39} -11.3803 q^{41} -1.68116 q^{43} -10.8397 q^{45} +0.152534 q^{47} +2.41272 q^{49} +0.437491 q^{51} +6.85851 q^{53} -4.37306 q^{55} -1.16260 q^{57} +10.8204 q^{59} +3.25765 q^{61} -8.92533 q^{63} -17.3239 q^{65} -8.34459 q^{67} -1.63033 q^{69} +15.5490 q^{71} +16.3257 q^{73} -2.67759 q^{75} -3.60073 q^{77} -11.8536 q^{79} +8.19064 q^{81} -1.33359 q^{83} -5.40841 q^{85} -1.69747 q^{87} +6.98475 q^{89} -14.2644 q^{91} -1.35443 q^{93} +14.3725 q^{95} +7.18127 q^{97} +3.41428 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

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