LMFDB - Embedded newform 8048.2.a.v.1.11

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:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-0.823496 q^{3} -2.12271 q^{5} -4.15456 q^{7} -2.32185 q^{9} +O(q^{10})$	$q-0.823496 q^{3} -2.12271 q^{5} -4.15456 q^{7} -2.32185 q^{9} -0.347254 q^{11} +3.83770 q^{13} +1.74804 q^{15} -6.72831 q^{17} +6.76295 q^{19} +3.42126 q^{21} -0.816344 q^{23} -0.494108 q^{25} +4.38253 q^{27} +1.71507 q^{29} -2.32959 q^{31} +0.285962 q^{33} +8.81891 q^{35} +5.48996 q^{37} -3.16033 q^{39} -1.70345 q^{41} +5.51038 q^{43} +4.92862 q^{45} +6.07376 q^{47} +10.2603 q^{49} +5.54074 q^{51} +3.81786 q^{53} +0.737119 q^{55} -5.56926 q^{57} +8.10272 q^{59} -15.2931 q^{61} +9.64627 q^{63} -8.14632 q^{65} +7.22250 q^{67} +0.672256 q^{69} +8.48015 q^{71} -4.46685 q^{73} +0.406896 q^{75} +1.44269 q^{77} -11.7094 q^{79} +3.35657 q^{81} +1.00287 q^{83} +14.2822 q^{85} -1.41235 q^{87} -8.30368 q^{89} -15.9439 q^{91} +1.91841 q^{93} -14.3558 q^{95} +15.0445 q^{97} +0.806273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−0.823496	−0.475446	−0.237723	−	0.971333i	$-0.576401\pi$
$3$	−0.823496	−0.475446	−0.237723	+	0.971333i	$0.576401\pi$
$4$	0	0
$5$	−2.12271	−0.949304	−0.474652	−	0.880174i	$-0.657426\pi$
$5$	−2.12271	−0.949304	−0.474652	+	0.880174i	$0.657426\pi$
$6$	0	0
$7$	−4.15456	−1.57027	−0.785137	−	0.619322i	$-0.787408\pi$
$7$	−4.15456	−1.57027	−0.785137	+	0.619322i	$0.787408\pi$
$8$	0	0
$9$	−2.32185	−0.773951
$10$	0	0
$11$	−0.347254	−0.104701	−0.0523505	−	0.998629i	$-0.516671\pi$
$11$	−0.347254	−0.104701	−0.0523505	+	0.998629i	$0.516671\pi$
$12$	0	0
$13$	3.83770	1.06439	0.532193	−	0.846623i	$-0.321368\pi$
$13$	3.83770	1.06439	0.532193	+	0.846623i	$0.321368\pi$
$14$	0	0
$15$	1.74804	0.451343
$16$	0	0
$17$	−6.72831	−1.63185	−0.815927	−	0.578154i	$-0.803773\pi$
$17$	−6.72831	−1.63185	−0.815927	+	0.578154i	$0.803773\pi$
$18$	0	0
$19$	6.76295	1.55153	0.775763	−	0.631024i	$-0.217365\pi$
$19$	6.76295	1.55153	0.775763	+	0.631024i	$0.217365\pi$
$20$	0	0
$21$	3.42126	0.746580
$22$	0	0
$23$	−0.816344	−0.170219	−0.0851097	−	0.996372i	$-0.527124\pi$
$23$	−0.816344	−0.170219	−0.0851097	+	0.996372i	$0.527124\pi$
$24$	0	0
$25$	−0.494108	−0.0988217
$26$	0	0
$27$	4.38253	0.843418
$28$	0	0
$29$	1.71507	0.318480	0.159240	−	0.987240i	$-0.449096\pi$
$29$	1.71507	0.318480	0.159240	+	0.987240i	$0.449096\pi$
$30$	0	0
$31$	−2.32959	−0.418406	−0.209203	−	0.977872i	$-0.567087\pi$
$31$	−2.32959	−0.418406	−0.209203	+	0.977872i	$0.567087\pi$
$32$	0	0
$33$	0.285962	0.0497797
$34$	0	0
$35$	8.81891	1.49067
$36$	0	0
$37$	5.48996	0.902544	0.451272	−	0.892386i	$-0.350970\pi$
$37$	5.48996	0.902544	0.451272	+	0.892386i	$0.350970\pi$
$38$	0	0
$39$	−3.16033	−0.506058
$40$	0	0
$41$	−1.70345	−0.266035	−0.133017	−	0.991114i	$-0.542467\pi$
$41$	−1.70345	−0.266035	−0.133017	+	0.991114i	$0.542467\pi$
$42$	0	0
$43$	5.51038	0.840325	0.420163	−	0.907449i	$-0.361973\pi$
$43$	5.51038	0.840325	0.420163	+	0.907449i	$0.361973\pi$
$44$	0	0
$45$	4.92862	0.734715
$46$	0	0
$47$	6.07376	0.885949	0.442974	−	0.896534i	$-0.353923\pi$
$47$	6.07376	0.885949	0.442974	+	0.896534i	$0.353923\pi$
$48$	0	0
$49$	10.2603	1.46576
$50$	0	0
$51$	5.54074	0.775858
$52$	0	0
$53$	3.81786	0.524424	0.262212	−	0.965010i	$-0.415548\pi$
$53$	3.81786	0.524424	0.262212	+	0.965010i	$0.415548\pi$
$54$	0	0
$55$	0.737119	0.0993931
$56$	0	0
$57$	−5.56926	−0.737667
$58$	0	0
$59$	8.10272	1.05488	0.527442	−	0.849591i	$-0.323151\pi$
$59$	8.10272	1.05488	0.527442	+	0.849591i	$0.323151\pi$
$60$	0	0
$61$	−15.2931	−1.95808	−0.979039	−	0.203674i	$-0.934712\pi$
$61$	−15.2931	−1.95808	−0.979039	+	0.203674i	$0.934712\pi$
$62$	0	0
$63$	9.64627	1.21532
$64$	0	0
$65$	−8.14632	−1.01043
$66$	0	0
$67$	7.22250	0.882368	0.441184	−	0.897417i	$-0.354559\pi$
$67$	7.22250	0.882368	0.441184	+	0.897417i	$0.354559\pi$
$68$	0	0
$69$	0.672256	0.0809301
$70$	0	0
$71$	8.48015	1.00641	0.503204	−	0.864167i	$-0.332154\pi$
$71$	8.48015	1.00641	0.503204	+	0.864167i	$0.332154\pi$
$72$	0	0
$73$	−4.46685	−0.522806	−0.261403	−	0.965230i	$-0.584185\pi$
$73$	−4.46685	−0.522806	−0.261403	+	0.965230i	$0.584185\pi$
$74$	0	0
$75$	0.406896	0.0469843
$76$	0	0
$77$	1.44269	0.164409
$78$	0	0
$79$	−11.7094	−1.31741	−0.658703	−	0.752403i	$-0.728895\pi$
$79$	−11.7094	−1.31741	−0.658703	+	0.752403i	$0.728895\pi$
$80$	0	0
$81$	3.35657	0.372952
$82$	0	0
$83$	1.00287	0.110079	0.0550394	−	0.998484i	$-0.482472\pi$
$83$	1.00287	0.110079	0.0550394	+	0.998484i	$0.482472\pi$
$84$	0	0
$85$	14.2822	1.54913
$86$	0	0
$87$	−1.41235	−0.151420
$88$	0	0
$89$	−8.30368	−0.880188	−0.440094	−	0.897952i	$-0.645055\pi$
$89$	−8.30368	−0.880188	−0.440094	+	0.897952i	$0.645055\pi$
$90$	0	0
$91$	−15.9439	−1.67138
$92$	0	0
$93$	1.91841	0.198929
$94$	0	0
$95$	−14.3558	−1.47287
$96$	0	0
$97$	15.0445	1.52753	0.763767	−	0.645492i	$-0.223348\pi$
$97$	15.0445	1.52753	0.763767	+	0.645492i	$0.223348\pi$
$98$	0	0
$99$	0.806273	0.0810335
$100$	0	0
$101$	13.6568	1.35890	0.679452	−	0.733720i	$-0.262218\pi$
$101$	13.6568	1.35890	0.679452	+	0.733720i	$0.262218\pi$
$102$	0	0
$103$	−11.5941	−1.14240	−0.571202	−	0.820809i	$-0.693523\pi$
$103$	−11.5941	−1.14240	−0.571202	+	0.820809i	$0.693523\pi$
$104$	0	0
$105$	−7.26234	−0.708732
$106$	0	0
$107$	0.824365	0.0796943	0.0398472	−	0.999206i	$-0.487313\pi$
$107$	0.824365	0.0796943	0.0398472	+	0.999206i	$0.487313\pi$
$108$	0	0
$109$	−7.67512	−0.735144	−0.367572	−	0.929995i	$-0.619811\pi$
$109$	−7.67512	−0.735144	−0.367572	+	0.929995i	$0.619811\pi$
$110$	0	0
$111$	−4.52096	−0.429111
$112$	0	0
$113$	1.59341	0.149895	0.0749475	−	0.997187i	$-0.476121\pi$
$113$	1.59341	0.149895	0.0749475	+	0.997187i	$0.476121\pi$
$114$	0	0
$115$	1.73286	0.161590
$116$	0	0
$117$	−8.91058	−0.823783
$118$	0	0
$119$	27.9531	2.56246
$120$	0	0
$121$	−10.8794	−0.989038
$122$	0	0
$123$	1.40279	0.126485
$124$	0	0
$125$	11.6624	1.04312
$126$	0	0
$127$	15.7093	1.39397	0.696987	−	0.717083i	$-0.254523\pi$
$127$	15.7093	1.39397	0.696987	+	0.717083i	$0.254523\pi$
$128$	0	0
$129$	−4.53778	−0.399529
$130$	0	0
$131$	19.7398	1.72468	0.862339	−	0.506332i	$-0.168999\pi$
$131$	19.7398	1.72468	0.862339	+	0.506332i	$0.168999\pi$
$132$	0	0
$133$	−28.0971	−2.43632
$134$	0	0
$135$	−9.30283	−0.800660
$136$	0	0
$137$	−0.120001	−0.0102524	−0.00512618	−	0.999987i	$-0.501632\pi$
$137$	−0.120001	−0.0102524	−0.00512618	+	0.999987i	$0.501632\pi$
$138$	0	0
$139$	−2.83196	−0.240204	−0.120102	−	0.992762i	$-0.538322\pi$
$139$	−2.83196	−0.240204	−0.120102	+	0.992762i	$0.538322\pi$
$140$	0	0
$141$	−5.00172	−0.421220
$142$	0	0
$143$	−1.33266	−0.111442
$144$	0	0
$145$	−3.64059	−0.302335
$146$	0	0
$147$	−8.44934	−0.696890
$148$	0	0
$149$	−8.31380	−0.681093	−0.340546	−	0.940228i	$-0.610612\pi$
$149$	−8.31380	−0.681093	−0.340546	+	0.940228i	$0.610612\pi$
$150$	0	0
$151$	−5.69855	−0.463742	−0.231871	−	0.972747i	$-0.574485\pi$
$151$	−5.69855	−0.463742	−0.231871	+	0.972747i	$0.574485\pi$
$152$	0	0
$153$	15.6222	1.26298
$154$	0	0
$155$	4.94503	0.397195
$156$	0	0
$157$	−2.49199	−0.198883	−0.0994413	−	0.995043i	$-0.531706\pi$
$157$	−2.49199	−0.198883	−0.0994413	+	0.995043i	$0.531706\pi$
$158$	0	0
$159$	−3.14400	−0.249335
$160$	0	0
$161$	3.39155	0.267291
$162$	0	0
$163$	−8.83664	−0.692139	−0.346070	−	0.938209i	$-0.612484\pi$
$163$	−8.83664	−0.692139	−0.346070	+	0.938209i	$0.612484\pi$
$164$	0	0
$165$	−0.607015	−0.0472560
$166$	0	0
$167$	−0.457582	−0.0354088	−0.0177044	−	0.999843i	$-0.505636\pi$
$167$	−0.457582	−0.0354088	−0.0177044	+	0.999843i	$0.505636\pi$
$168$	0	0
$169$	1.72794	0.132918
$170$	0	0
$171$	−15.7026	−1.20081
$172$	0	0
$173$	16.9072	1.28543	0.642717	−	0.766104i	$-0.277807\pi$
$173$	16.9072	1.28543	0.642717	+	0.766104i	$0.277807\pi$
$174$	0	0
$175$	2.05280	0.155177
$176$	0	0
$177$	−6.67256	−0.501540
$178$	0	0
$179$	3.81648	0.285257	0.142628	−	0.989776i	$-0.454445\pi$
$179$	3.81648	0.285257	0.142628	+	0.989776i	$0.454445\pi$
$180$	0	0
$181$	14.4118	1.07122	0.535611	−	0.844465i	$-0.320081\pi$
$181$	14.4118	1.07122	0.535611	+	0.844465i	$0.320081\pi$
$182$	0	0
$183$	12.5938	0.930960
$184$	0	0
$185$	−11.6536	−0.856789
$186$	0	0
$187$	2.33643	0.170857
$188$	0	0
$189$	−18.2074	−1.32440
$190$	0	0
$191$	18.5386	1.34140	0.670701	−	0.741728i	$-0.265993\pi$
$191$	18.5386	1.34140	0.670701	+	0.741728i	$0.265993\pi$
$192$	0	0
$193$	−24.7549	−1.78190	−0.890949	−	0.454103i	$-0.849960\pi$
$193$	−24.7549	−1.78190	−0.890949	+	0.454103i	$0.849960\pi$
$194$	0	0
$195$	6.70846	0.480403
$196$	0	0
$197$	−21.0824	−1.50206	−0.751030	−	0.660268i	$-0.770443\pi$
$197$	−21.0824	−1.50206	−0.751030	+	0.660268i	$0.770443\pi$
$198$	0	0
$199$	−24.5709	−1.74179	−0.870893	−	0.491472i	$-0.836459\pi$
$199$	−24.5709	−1.74179	−0.870893	+	0.491472i	$0.836459\pi$
$200$	0	0
$201$	−5.94770	−0.419518
$202$	0	0
$203$	−7.12535	−0.500102
$204$	0	0
$205$	3.61594	0.252548
$206$	0	0
$207$	1.89543	0.131742
$208$	0	0
$209$	−2.34846	−0.162447
$210$	0	0
$211$	4.17275	0.287264	0.143632	−	0.989631i	$-0.454122\pi$
$211$	4.17275	0.287264	0.143632	+	0.989631i	$0.454122\pi$
$212$	0	0
$213$	−6.98337	−0.478493
$214$	0	0
$215$	−11.6969	−0.797724
$216$	0	0
$217$	9.67840	0.657013
$218$	0	0
$219$	3.67844	0.248566
$220$	0	0
$221$	−25.8212	−1.73692
$222$	0	0
$223$	−12.7125	−0.851293	−0.425646	−	0.904890i	$-0.639953\pi$
$223$	−12.7125	−0.851293	−0.425646	+	0.904890i	$0.639953\pi$
$224$	0	0
$225$	1.14725	0.0764832
$226$	0	0
$227$	1.68166	0.111616	0.0558079	−	0.998442i	$-0.482227\pi$
$227$	1.68166	0.111616	0.0558079	+	0.998442i	$0.482227\pi$
$228$	0	0
$229$	−20.5569	−1.35844	−0.679218	−	0.733937i	$-0.737681\pi$
$229$	−20.5569	−1.35844	−0.679218	+	0.733937i	$0.737681\pi$
$230$	0	0
$231$	−1.18805	−0.0781677
$232$	0	0
$233$	−21.2247	−1.39047	−0.695237	−	0.718780i	$-0.744701\pi$
$233$	−21.2247	−1.39047	−0.695237	+	0.718780i	$0.744701\pi$
$234$	0	0
$235$	−12.8928	−0.841035
$236$	0	0
$237$	9.64262	0.626355
$238$	0	0
$239$	25.0093	1.61772	0.808859	−	0.588003i	$-0.200086\pi$
$239$	25.0093	1.61772	0.808859	+	0.588003i	$0.200086\pi$
$240$	0	0
$241$	0.373029	0.0240289	0.0120145	−	0.999928i	$-0.496176\pi$
$241$	0.373029	0.0240289	0.0120145	+	0.999928i	$0.496176\pi$
$242$	0	0
$243$	−15.9117	−1.02074
$244$	0	0
$245$	−21.7797	−1.39145
$246$	0	0
$247$	25.9542	1.65142
$248$	0	0
$249$	−0.825856	−0.0523365
$250$	0	0
$251$	3.81990	0.241110	0.120555	−	0.992707i	$-0.461533\pi$
$251$	3.81990	0.241110	0.120555	+	0.992707i	$0.461533\pi$
$252$	0	0
$253$	0.283479	0.0178222
$254$	0	0
$255$	−11.7614	−0.736526
$256$	0	0
$257$	−21.7329	−1.35566	−0.677831	−	0.735218i	$-0.737080\pi$
$257$	−21.7329	−1.35566	−0.677831	+	0.735218i	$0.737080\pi$
$258$	0	0
$259$	−22.8084	−1.41724
$260$	0	0
$261$	−3.98214	−0.246488
$262$	0	0
$263$	3.00331	0.185192	0.0925961	−	0.995704i	$-0.470483\pi$
$263$	3.00331	0.185192	0.0925961	+	0.995704i	$0.470483\pi$
$264$	0	0
$265$	−8.10421	−0.497838
$266$	0	0
$267$	6.83805	0.418482
$268$	0	0
$269$	7.97821	0.486440	0.243220	−	0.969971i	$-0.421796\pi$
$269$	7.97821	0.486440	0.243220	+	0.969971i	$0.421796\pi$
$270$	0	0
$271$	−19.9565	−1.21227	−0.606137	−	0.795360i	$-0.707282\pi$
$271$	−19.9565	−1.21227	−0.606137	+	0.795360i	$0.707282\pi$
$272$	0	0
$273$	13.1298	0.794650
$274$	0	0
$275$	0.171581	0.0103467
$276$	0	0
$277$	−25.8065	−1.55056	−0.775281	−	0.631616i	$-0.782392\pi$
$277$	−25.8065	−1.55056	−0.775281	+	0.631616i	$0.782392\pi$
$278$	0	0
$279$	5.40896	0.323826
$280$	0	0
$281$	−2.93470	−0.175070	−0.0875348	−	0.996161i	$-0.527899\pi$
$281$	−2.93470	−0.175070	−0.0875348	+	0.996161i	$0.527899\pi$
$282$	0	0
$283$	21.3900	1.27151	0.635753	−	0.771893i	$-0.280690\pi$
$283$	21.3900	1.27151	0.635753	+	0.771893i	$0.280690\pi$
$284$	0	0
$285$	11.8219	0.700270
$286$	0	0
$287$	7.07710	0.417748
$288$	0	0
$289$	28.2702	1.66295
$290$	0	0
$291$	−12.3891	−0.726260
$292$	0	0
$293$	−25.3928	−1.48346	−0.741731	−	0.670697i	$-0.765995\pi$
$293$	−25.3928	−1.48346	−0.741731	+	0.670697i	$0.765995\pi$
$294$	0	0
$295$	−17.1997	−1.00141
$296$	0	0
$297$	−1.52185	−0.0883067
$298$	0	0
$299$	−3.13288	−0.181179
$300$	0	0
$301$	−22.8932	−1.31954
$302$	0	0
$303$	−11.2463	−0.646085
$304$	0	0
$305$	32.4627	1.85881
$306$	0	0
$307$	−10.6838	−0.609757	−0.304879	−	0.952391i	$-0.598616\pi$
$307$	−10.6838	−0.609757	−0.304879	+	0.952391i	$0.598616\pi$
$308$	0	0
$309$	9.54773	0.543151
$310$	0	0
$311$	−19.3607	−1.09785	−0.548923	−	0.835873i	$-0.684962\pi$
$311$	−19.3607	−1.09785	−0.548923	+	0.835873i	$0.684962\pi$
$312$	0	0
$313$	7.47721	0.422637	0.211318	−	0.977417i	$-0.432224\pi$
$313$	7.47721	0.422637	0.211318	+	0.977417i	$0.432224\pi$
$314$	0	0
$315$	−20.4762	−1.15370
$316$	0	0
$317$	−27.9433	−1.56945	−0.784726	−	0.619843i	$-0.787196\pi$
$317$	−27.9433	−1.56945	−0.784726	+	0.619843i	$0.787196\pi$
$318$	0	0
$319$	−0.595565	−0.0333452
$320$	0	0
$321$	−0.678861	−0.0378903
$322$	0	0
$323$	−45.5032	−2.53187
$324$	0	0
$325$	−1.89624	−0.105184
$326$	0	0
$327$	6.32044	0.349521
$328$	0	0
$329$	−25.2338	−1.39118
$330$	0	0
$331$	8.06821	0.443469	0.221735	−	0.975107i	$-0.428828\pi$
$331$	8.06821	0.443469	0.221735	+	0.975107i	$0.428828\pi$
$332$	0	0
$333$	−12.7469	−0.698525
$334$	0	0
$335$	−15.3313	−0.837636
$336$	0	0
$337$	10.1703	0.554013	0.277007	−	0.960868i	$-0.410658\pi$
$337$	10.1703	0.554013	0.277007	+	0.960868i	$0.410658\pi$
$338$	0	0
$339$	−1.31216	−0.0712669
$340$	0	0
$341$	0.808959	0.0438076
$342$	0	0
$343$	−13.5452	−0.731374
$344$	0	0
$345$	−1.42700	−0.0768273
$346$	0	0
$347$	34.9374	1.87554	0.937768	−	0.347263i	$-0.112889\pi$
$347$	34.9374	1.87554	0.937768	+	0.347263i	$0.112889\pi$
$348$	0	0
$349$	1.37287	0.0734880	0.0367440	−	0.999325i	$-0.488301\pi$
$349$	1.37287	0.0734880	0.0367440	+	0.999325i	$0.488301\pi$
$350$	0	0
$351$	16.8188	0.897722
$352$	0	0
$353$	31.0396	1.65207	0.826035	−	0.563619i	$-0.190591\pi$
$353$	31.0396	1.65207	0.826035	+	0.563619i	$0.190591\pi$
$354$	0	0
$355$	−18.0009	−0.955388
$356$	0	0
$357$	−23.0193	−1.21831
$358$	0	0
$359$	11.8881	0.627431	0.313715	−	0.949517i	$-0.398426\pi$
$359$	11.8881	0.627431	0.313715	+	0.949517i	$0.398426\pi$
$360$	0	0
$361$	26.7375	1.40724
$362$	0	0
$363$	8.95916	0.470234
$364$	0	0
$365$	9.48183	0.496302
$366$	0	0
$367$	−9.16528	−0.478424	−0.239212	−	0.970967i	$-0.576889\pi$
$367$	−9.16528	−0.478424	−0.239212	+	0.970967i	$0.576889\pi$
$368$	0	0
$369$	3.95517	0.205898
$370$	0	0
$371$	−15.8615	−0.823489
$372$	0	0
$373$	−14.7199	−0.762166	−0.381083	−	0.924541i	$-0.624449\pi$
$373$	−14.7199	−0.762166	−0.381083	+	0.924541i	$0.624449\pi$
$374$	0	0
$375$	−9.60393	−0.495945
$376$	0	0
$377$	6.58192	0.338986
$378$	0	0
$379$	−7.82369	−0.401876	−0.200938	−	0.979604i	$-0.564399\pi$
$379$	−7.82369	−0.401876	−0.200938	+	0.979604i	$0.564399\pi$
$380$	0	0
$381$	−12.9365	−0.662759
$382$	0	0
$383$	−4.55131	−0.232561	−0.116280	−	0.993216i	$-0.537097\pi$
$383$	−4.55131	−0.232561	−0.116280	+	0.993216i	$0.537097\pi$
$384$	0	0
$385$	−3.06240	−0.156075
$386$	0	0
$387$	−12.7943	−0.650371
$388$	0	0
$389$	28.8205	1.46126	0.730628	−	0.682776i	$-0.239228\pi$
$389$	28.8205	1.46126	0.730628	+	0.682776i	$0.239228\pi$
$390$	0	0
$391$	5.49261	0.277773
$392$	0	0
$393$	−16.2557	−0.819991
$394$	0	0
$395$	24.8556	1.25062
$396$	0	0
$397$	−10.9789	−0.551013	−0.275506	−	0.961299i	$-0.588846\pi$
$397$	−10.9789	−0.551013	−0.275506	+	0.961299i	$0.588846\pi$
$398$	0	0
$399$	23.1378	1.15834
$400$	0	0
$401$	8.07798	0.403395	0.201697	−	0.979448i	$-0.435354\pi$
$401$	8.07798	0.403395	0.201697	+	0.979448i	$0.435354\pi$
$402$	0	0
$403$	−8.94026	−0.445346
$404$	0	0
$405$	−7.12502	−0.354045
$406$	0	0
$407$	−1.90641	−0.0944973
$408$	0	0
$409$	12.3101	0.608695	0.304348	−	0.952561i	$-0.401562\pi$
$409$	12.3101	0.608695	0.304348	+	0.952561i	$0.401562\pi$
$410$	0	0
$411$	0.0988202	0.00487444
$412$	0	0
$413$	−33.6632	−1.65646
$414$	0	0
$415$	−2.12879	−0.104498
$416$	0	0
$417$	2.33211	0.114204
$418$	0	0
$419$	−4.56527	−0.223028	−0.111514	−	0.993763i	$-0.535570\pi$
$419$	−4.56527	−0.223028	−0.111514	+	0.993763i	$0.535570\pi$
$420$	0	0
$421$	−6.55210	−0.319330	−0.159665	−	0.987171i	$-0.551041\pi$
$421$	−6.55210	−0.319330	−0.159665	+	0.987171i	$0.551041\pi$
$422$	0	0
$423$	−14.1024	−0.685681
$424$	0	0
$425$	3.32451	0.161263
$426$	0	0
$427$	63.5359	3.07472
$428$	0	0
$429$	1.09744	0.0529848
$430$	0	0
$431$	28.2823	1.36231	0.681155	−	0.732140i	$-0.261478\pi$
$431$	28.2823	1.36231	0.681155	+	0.732140i	$0.261478\pi$
$432$	0	0
$433$	4.34310	0.208716	0.104358	−	0.994540i	$-0.466721\pi$
$433$	4.34310	0.208716	0.104358	+	0.994540i	$0.466721\pi$
$434$	0	0
$435$	2.99801	0.143744
$436$	0	0
$437$	−5.52089	−0.264100
$438$	0	0
$439$	−35.3827	−1.68872	−0.844362	−	0.535773i	$-0.820020\pi$
$439$	−35.3827	−1.68872	−0.844362	+	0.535773i	$0.820020\pi$
$440$	0	0
$441$	−23.8230	−1.13443
$442$	0	0
$443$	−34.2931	−1.62932	−0.814658	−	0.579942i	$-0.803075\pi$
$443$	−34.2931	−1.62932	−0.814658	+	0.579942i	$0.803075\pi$
$444$	0	0
$445$	17.6263	0.835566
$446$	0	0
$447$	6.84638	0.323823
$448$	0	0
$449$	31.0109	1.46350	0.731748	−	0.681575i	$-0.238705\pi$
$449$	31.0109	1.46350	0.731748	+	0.681575i	$0.238705\pi$
$450$	0	0
$451$	0.591531	0.0278541
$452$	0	0
$453$	4.69273	0.220484
$454$	0	0
$455$	33.8443	1.58665
$456$	0	0
$457$	19.3168	0.903600	0.451800	−	0.892119i	$-0.350782\pi$
$457$	19.3168	0.903600	0.451800	+	0.892119i	$0.350782\pi$
$458$	0	0
$459$	−29.4870	−1.37634
$460$	0	0
$461$	39.8254	1.85485	0.927427	−	0.374004i	$-0.122015\pi$
$461$	39.8254	1.85485	0.927427	+	0.374004i	$0.122015\pi$
$462$	0	0
$463$	−41.4088	−1.92443	−0.962216	−	0.272287i	$-0.912220\pi$
$463$	−41.4088	−1.92443	−0.962216	+	0.272287i	$0.912220\pi$
$464$	0	0
$465$	−4.07222	−0.188845
$466$	0	0
$467$	28.5289	1.32016	0.660080	−	0.751195i	$-0.270522\pi$
$467$	28.5289	1.32016	0.660080	+	0.751195i	$0.270522\pi$
$468$	0	0
$469$	−30.0063	−1.38556
$470$	0	0
$471$	2.05215	0.0945579
$472$	0	0
$473$	−1.91350	−0.0879829
$474$	0	0
$475$	−3.34163	−0.153324
$476$	0	0
$477$	−8.86452	−0.405879
$478$	0	0
$479$	−26.6159	−1.21611	−0.608056	−	0.793894i	$-0.708050\pi$
$479$	−26.6159	−1.21611	−0.608056	+	0.793894i	$0.708050\pi$
$480$	0	0
$481$	21.0688	0.960656
$482$	0	0
$483$	−2.79292	−0.127082
$484$	0	0
$485$	−31.9350	−1.45009
$486$	0	0
$487$	−37.0051	−1.67686	−0.838430	−	0.545010i	$-0.816526\pi$
$487$	−37.0051	−1.67686	−0.838430	+	0.545010i	$0.816526\pi$
$488$	0	0
$489$	7.27694	0.329075
$490$	0	0
$491$	−17.0755	−0.770605	−0.385302	−	0.922790i	$-0.625903\pi$
$491$	−17.0755	−0.770605	−0.385302	+	0.922790i	$0.625903\pi$
$492$	0	0
$493$	−11.5395	−0.519714
$494$	0	0
$495$	−1.71148	−0.0769255
$496$	0	0
$497$	−35.2313	−1.58034
$498$	0	0
$499$	−29.3453	−1.31367	−0.656837	−	0.754033i	$-0.728106\pi$
$499$	−29.3453	−1.31367	−0.656837	+	0.754033i	$0.728106\pi$
$500$	0	0
$501$	0.376817	0.0168350
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−28.9894	−1.29001
$506$	0	0
$507$	−1.42295	−0.0631955
$508$	0	0
$509$	−10.3858	−0.460343	−0.230171	−	0.973150i	$-0.573929\pi$
$509$	−10.3858	−0.460343	−0.230171	+	0.973150i	$0.573929\pi$
$510$	0	0
$511$	18.5578	0.820949
$512$	0	0
$513$	29.6388	1.30859
$514$	0	0
$515$	24.6110	1.08449
$516$	0	0
$517$	−2.10914	−0.0927598
$518$	0	0
$519$	−13.9230	−0.611154
$520$	0	0
$521$	−26.5151	−1.16165	−0.580823	−	0.814030i	$-0.697269\pi$
$521$	−26.5151	−1.16165	−0.580823	+	0.814030i	$0.697269\pi$
$522$	0	0
$523$	37.8472	1.65494	0.827472	−	0.561507i	$-0.189778\pi$
$523$	37.8472	1.65494	0.827472	+	0.561507i	$0.189778\pi$
$524$	0	0
$525$	−1.69047	−0.0737783
$526$	0	0
$527$	15.6742	0.682778
$528$	0	0
$529$	−22.3336	−0.971025
$530$	0	0
$531$	−18.8133	−0.816429
$532$	0	0
$533$	−6.53735	−0.283164
$534$	0	0
$535$	−1.74989	−0.0756542
$536$	0	0
$537$	−3.14285	−0.135624
$538$	0	0
$539$	−3.56294	−0.153467
$540$	0	0
$541$	44.7987	1.92605	0.963024	−	0.269417i	$-0.0868308\pi$
$541$	44.7987	1.92605	0.963024	+	0.269417i	$0.0868308\pi$
$542$	0	0
$543$	−11.8681	−0.509308
$544$	0	0
$545$	16.2921	0.697875
$546$	0	0
$547$	18.3039	0.782618	0.391309	−	0.920259i	$-0.372022\pi$
$547$	18.3039	0.782618	0.391309	+	0.920259i	$0.372022\pi$
$548$	0	0
$549$	35.5083	1.51546
$550$	0	0
$551$	11.5989	0.494131
$552$	0	0
$553$	48.6472	2.06869
$554$	0	0
$555$	9.59669	0.407357
$556$	0	0
$557$	−27.3695	−1.15968	−0.579841	−	0.814730i	$-0.696885\pi$
$557$	−27.3695	−1.15968	−0.579841	+	0.814730i	$0.696885\pi$
$558$	0	0
$559$	21.1472	0.894431
$560$	0	0
$561$	−1.92404	−0.0812332
$562$	0	0
$563$	3.85582	0.162504	0.0812518	−	0.996694i	$-0.474108\pi$
$563$	3.85582	0.162504	0.0812518	+	0.996694i	$0.474108\pi$
$564$	0	0
$565$	−3.38234	−0.142296
$566$	0	0
$567$	−13.9451	−0.585637
$568$	0	0
$569$	7.22012	0.302683	0.151342	−	0.988482i	$-0.451641\pi$
$569$	7.22012	0.302683	0.151342	+	0.988482i	$0.451641\pi$
$570$	0	0
$571$	2.96941	0.124266	0.0621330	−	0.998068i	$-0.480210\pi$
$571$	2.96941	0.124266	0.0621330	+	0.998068i	$0.480210\pi$
$572$	0	0
$573$	−15.2664	−0.637764
$574$	0	0
$575$	0.403362	0.0168214
$576$	0	0
$577$	−18.8412	−0.784370	−0.392185	−	0.919886i	$-0.628281\pi$
$577$	−18.8412	−0.784370	−0.392185	+	0.919886i	$0.628281\pi$
$578$	0	0
$579$	20.3856	0.847196
$580$	0	0
$581$	−4.16646	−0.172854
$582$	0	0
$583$	−1.32577	−0.0549077
$584$	0	0
$585$	18.9146	0.782021
$586$	0	0
$587$	8.88738	0.366821	0.183411	−	0.983036i	$-0.441286\pi$
$587$	8.88738	0.366821	0.183411	+	0.983036i	$0.441286\pi$
$588$	0	0
$589$	−15.7549	−0.649169
$590$	0	0
$591$	17.3613	0.714148
$592$	0	0
$593$	9.62411	0.395215	0.197607	−	0.980281i	$-0.436683\pi$
$593$	9.62411	0.395215	0.197607	+	0.980281i	$0.436683\pi$
$594$	0	0
$595$	−59.3364	−2.43255
$596$	0	0
$597$	20.2341	0.828125
$598$	0	0
$599$	−27.0832	−1.10659	−0.553294	−	0.832986i	$-0.686629\pi$
$599$	−27.0832	−1.10659	−0.553294	+	0.832986i	$0.686629\pi$
$600$	0	0
$601$	24.6676	1.00621	0.503107	−	0.864224i	$-0.332190\pi$
$601$	24.6676	1.00621	0.503107	+	0.864224i	$0.332190\pi$
$602$	0	0
$603$	−16.7696	−0.682910
$604$	0	0
$605$	23.0938	0.938898
$606$	0	0
$607$	6.40644	0.260030	0.130015	−	0.991512i	$-0.458498\pi$
$607$	6.40644	0.260030	0.130015	+	0.991512i	$0.458498\pi$
$608$	0	0
$609$	5.86770	0.237771
$610$	0	0
$611$	23.3093	0.942992
$612$	0	0
$613$	0.516185	0.0208485	0.0104243	−	0.999946i	$-0.496682\pi$
$613$	0.516185	0.0208485	0.0104243	+	0.999946i	$0.496682\pi$
$614$	0	0
$615$	−2.97771	−0.120073
$616$	0	0
$617$	18.1430	0.730409	0.365204	−	0.930927i	$-0.380999\pi$
$617$	18.1430	0.730409	0.365204	+	0.930927i	$0.380999\pi$
$618$	0	0
$619$	−0.354113	−0.0142330	−0.00711650	−	0.999975i	$-0.502265\pi$
$619$	−0.354113	−0.0142330	−0.00711650	+	0.999975i	$0.502265\pi$
$620$	0	0
$621$	−3.57765	−0.143566
$622$	0	0
$623$	34.4981	1.38214
$624$	0	0
$625$	−22.2853	−0.891413
$626$	0	0
$627$	1.93395	0.0772345
$628$	0	0
$629$	−36.9382	−1.47282
$630$	0	0
$631$	42.9039	1.70798	0.853989	−	0.520291i	$-0.174176\pi$
$631$	42.9039	1.70798	0.853989	+	0.520291i	$0.174176\pi$
$632$	0	0
$633$	−3.43625	−0.136579
$634$	0	0
$635$	−33.3463	−1.32331
$636$	0	0
$637$	39.3761	1.56014
$638$	0	0
$639$	−19.6897	−0.778912
$640$	0	0
$641$	29.2037	1.15348	0.576739	−	0.816928i	$-0.304325\pi$
$641$	29.2037	1.15348	0.576739	+	0.816928i	$0.304325\pi$
$642$	0	0
$643$	−25.9413	−1.02303	−0.511513	−	0.859275i	$-0.670915\pi$
$643$	−25.9413	−1.02303	−0.511513	+	0.859275i	$0.670915\pi$
$644$	0	0
$645$	9.63238	0.379274
$646$	0	0
$647$	−38.2575	−1.50406	−0.752029	−	0.659130i	$-0.770925\pi$
$647$	−38.2575	−1.50406	−0.752029	+	0.659130i	$0.770925\pi$
$648$	0	0
$649$	−2.81370	−0.110447
$650$	0	0
$651$	−7.97012	−0.312374
$652$	0	0
$653$	−15.4300	−0.603824	−0.301912	−	0.953336i	$-0.597625\pi$
$653$	−15.4300	−0.603824	−0.301912	+	0.953336i	$0.597625\pi$
$654$	0	0
$655$	−41.9019	−1.63724
$656$	0	0
$657$	10.3714	0.404626
$658$	0	0
$659$	42.7609	1.66573	0.832864	−	0.553478i	$-0.186700\pi$
$659$	42.7609	1.66573	0.832864	+	0.553478i	$0.186700\pi$
$660$	0	0
$661$	22.3138	0.867906	0.433953	−	0.900936i	$-0.357118\pi$
$661$	22.3138	0.867906	0.433953	+	0.900936i	$0.357118\pi$
$662$	0	0
$663$	21.2637	0.825813
$664$	0	0
$665$	59.6418	2.31281
$666$	0	0
$667$	−1.40009	−0.0542115
$668$	0	0
$669$	10.4687	0.404743
$670$	0	0
$671$	5.31058	0.205013
$672$	0	0
$673$	−26.9712	−1.03966	−0.519832	−	0.854268i	$-0.674006\pi$
$673$	−26.9712	−1.03966	−0.519832	+	0.854268i	$0.674006\pi$
$674$	0	0
$675$	−2.16544	−0.0833479
$676$	0	0
$677$	−4.23350	−0.162706	−0.0813532	−	0.996685i	$-0.525924\pi$
$677$	−4.23350	−0.162706	−0.0813532	+	0.996685i	$0.525924\pi$
$678$	0	0
$679$	−62.5031	−2.39865
$680$	0	0
$681$	−1.38484	−0.0530673
$682$	0	0
$683$	−25.7643	−0.985842	−0.492921	−	0.870074i	$-0.664071\pi$
$683$	−25.7643	−0.985842	−0.492921	+	0.870074i	$0.664071\pi$
$684$	0	0
$685$	0.254727	0.00973261
$686$	0	0
$687$	16.9285	0.645863
$688$	0	0
$689$	14.6518	0.558190
$690$	0	0
$691$	6.35519	0.241763	0.120881	−	0.992667i	$-0.461428\pi$
$691$	6.35519	0.241763	0.120881	+	0.992667i	$0.461428\pi$
$692$	0	0
$693$	−3.34971	−0.127245
$694$	0	0
$695$	6.01144	0.228027
$696$	0	0
$697$	11.4614	0.434130
$698$	0	0
$699$	17.4784	0.661095
$700$	0	0
$701$	−4.91157	−0.185507	−0.0927537	−	0.995689i	$-0.529567\pi$
$701$	−4.91157	−0.185507	−0.0927537	+	0.995689i	$0.529567\pi$
$702$	0	0
$703$	37.1283	1.40032
$704$	0	0
$705$	10.6172	0.399866
$706$	0	0
$707$	−56.7380	−2.13385
$708$	0	0
$709$	47.9150	1.79948	0.899742	−	0.436421i	$-0.143754\pi$
$709$	47.9150	1.79948	0.899742	+	0.436421i	$0.143754\pi$
$710$	0	0
$711$	27.1874	1.01961
$712$	0	0
$713$	1.90174	0.0712209
$714$	0	0
$715$	2.82884	0.105793
$716$	0	0
$717$	−20.5951	−0.769137
$718$	0	0
$719$	0.628068	0.0234230	0.0117115	−	0.999931i	$-0.496272\pi$
$719$	0.628068	0.0234230	0.0117115	+	0.999931i	$0.496272\pi$
$720$	0	0
$721$	48.1685	1.79389
$722$	0	0
$723$	−0.307188	−0.0114244
$724$	0	0
$725$	−0.847430	−0.0314728
$726$	0	0
$727$	7.77808	0.288473	0.144236	−	0.989543i	$-0.453927\pi$
$727$	7.77808	0.288473	0.144236	+	0.989543i	$0.453927\pi$
$728$	0	0
$729$	3.03352	0.112352
$730$	0	0
$731$	−37.0755	−1.37129
$732$	0	0
$733$	−24.2134	−0.894341	−0.447170	−	0.894449i	$-0.647568\pi$
$733$	−24.2134	−0.894341	−0.447170	+	0.894449i	$0.647568\pi$
$734$	0	0
$735$	17.9355	0.661561
$736$	0	0
$737$	−2.50804	−0.0923849
$738$	0	0
$739$	20.5972	0.757679	0.378840	−	0.925462i	$-0.376323\pi$
$739$	20.5972	0.757679	0.378840	+	0.925462i	$0.376323\pi$
$740$	0	0
$741$	−21.3732	−0.785163
$742$	0	0
$743$	−50.6514	−1.85822	−0.929109	−	0.369805i	$-0.879424\pi$
$743$	−50.6514	−1.85822	−0.929109	+	0.369805i	$0.879424\pi$
$744$	0	0
$745$	17.6478	0.646564
$746$	0	0
$747$	−2.32851	−0.0851957
$748$	0	0
$749$	−3.42487	−0.125142
$750$	0	0
$751$	9.54226	0.348202	0.174101	−	0.984728i	$-0.444298\pi$
$751$	9.54226	0.348202	0.174101	+	0.984728i	$0.444298\pi$
$752$	0	0
$753$	−3.14568	−0.114635
$754$	0	0
$755$	12.0964	0.440232
$756$	0	0
$757$	−25.0891	−0.911877	−0.455939	−	0.890011i	$-0.650696\pi$
$757$	−25.0891	−0.911877	−0.455939	+	0.890011i	$0.650696\pi$
$758$	0	0
$759$	−0.233444	−0.00847347
$760$	0	0
$761$	−16.1210	−0.584387	−0.292193	−	0.956359i	$-0.594385\pi$
$761$	−16.1210	−0.584387	−0.292193	+	0.956359i	$0.594385\pi$
$762$	0	0
$763$	31.8867	1.15438
$764$	0	0
$765$	−33.1613	−1.19895
$766$	0	0
$767$	31.0958	1.12280
$768$	0	0
$769$	−3.70242	−0.133513	−0.0667563	−	0.997769i	$-0.521265\pi$
$769$	−3.70242	−0.133513	−0.0667563	+	0.997769i	$0.521265\pi$
$770$	0	0
$771$	17.8970	0.644543
$772$	0	0
$773$	−15.6202	−0.561820	−0.280910	−	0.959734i	$-0.590636\pi$
$773$	−15.6202	−0.561820	−0.280910	+	0.959734i	$0.590636\pi$
$774$	0	0
$775$	1.15107	0.0413476
$776$	0	0
$777$	18.7826	0.673822
$778$	0	0
$779$	−11.5204	−0.412760
$780$	0	0
$781$	−2.94477	−0.105372
$782$	0	0
$783$	7.51634	0.268612
$784$	0	0
$785$	5.28977	0.188800
$786$	0	0
$787$	−22.9804	−0.819162	−0.409581	−	0.912274i	$-0.634325\pi$
$787$	−22.9804	−0.819162	−0.409581	+	0.912274i	$0.634325\pi$
$788$	0	0
$789$	−2.47322	−0.0880488
$790$	0	0
$791$	−6.61989	−0.235376
$792$	0	0
$793$	−58.6902	−2.08415
$794$	0	0
$795$	6.67379	0.236695
$796$	0	0
$797$	−29.0389	−1.02861	−0.514304	−	0.857608i	$-0.671950\pi$
$797$	−29.0389	−1.02861	−0.514304	+	0.857608i	$0.671950\pi$
$798$	0	0
$799$	−40.8661	−1.44574
$800$	0	0
$801$	19.2799	0.681223
$802$	0	0
$803$	1.55113	0.0547383
$804$	0	0
$805$	−7.19926	−0.253741
$806$	0	0
$807$	−6.57002	−0.231276
$808$	0	0
$809$	54.4110	1.91299	0.956495	−	0.291749i	$-0.0942373\pi$
$809$	54.4110	1.91299	0.956495	+	0.291749i	$0.0942373\pi$
$810$	0	0
$811$	−2.15103	−0.0755329	−0.0377665	−	0.999287i	$-0.512024\pi$
$811$	−2.15103	−0.0755329	−0.0377665	+	0.999287i	$0.512024\pi$
$812$	0	0
$813$	16.4341	0.576370
$814$	0	0
$815$	18.7576	0.657051
$816$	0	0
$817$	37.2664	1.30379
$818$	0	0
$819$	37.0195	1.29357
$820$	0	0
$821$	1.22999	0.0429270	0.0214635	−	0.999770i	$-0.493167\pi$
$821$	1.22999	0.0429270	0.0214635	+	0.999770i	$0.493167\pi$
$822$	0	0
$823$	−6.62689	−0.230999	−0.115499	−	0.993308i	$-0.536847\pi$
$823$	−6.62689	−0.230999	−0.115499	+	0.993308i	$0.536847\pi$
$824$	0	0
$825$	−0.141296	−0.00491931
$826$	0	0
$827$	−44.2468	−1.53861	−0.769306	−	0.638880i	$-0.779398\pi$
$827$	−44.2468	−1.53861	−0.769306	+	0.638880i	$0.779398\pi$
$828$	0	0
$829$	53.0928	1.84399	0.921995	−	0.387202i	$-0.126558\pi$
$829$	53.0928	1.84399	0.921995	+	0.387202i	$0.126558\pi$
$830$	0	0
$831$	21.2516	0.737208
$832$	0	0
$833$	−69.0347	−2.39191
$834$	0	0
$835$	0.971314	0.0336137
$836$	0	0
$837$	−10.2095	−0.352891
$838$	0	0
$839$	−6.79163	−0.234473	−0.117237	−	0.993104i	$-0.537404\pi$
$839$	−6.79163	−0.234473	−0.117237	+	0.993104i	$0.537404\pi$
$840$	0	0
$841$	−26.0585	−0.898570
$842$	0	0
$843$	2.41671	0.0832361
$844$	0	0
$845$	−3.66791	−0.126180
$846$	0	0
$847$	45.1991	1.55306
$848$	0	0
$849$	−17.6146	−0.604532
$850$	0	0
$851$	−4.48170	−0.153631
$852$	0	0
$853$	24.2136	0.829057	0.414528	−	0.910036i	$-0.363947\pi$
$853$	24.2136	0.829057	0.414528	+	0.910036i	$0.363947\pi$
$854$	0	0
$855$	33.3320	1.13993
$856$	0	0
$857$	−12.5641	−0.429183	−0.214591	−	0.976704i	$-0.568842\pi$
$857$	−12.5641	−0.429183	−0.214591	+	0.976704i	$0.568842\pi$
$858$	0	0
$859$	25.0971	0.856301	0.428151	−	0.903707i	$-0.359165\pi$
$859$	25.0971	0.856301	0.428151	+	0.903707i	$0.359165\pi$
$860$	0	0
$861$	−5.82796	−0.198616
$862$	0	0
$863$	−9.28070	−0.315919	−0.157959	−	0.987446i	$-0.550491\pi$
$863$	−9.28070	−0.315919	−0.157959	+	0.987446i	$0.550491\pi$
$864$	0	0
$865$	−35.8891	−1.22027
$866$	0	0
$867$	−23.2804	−0.790643
$868$	0	0
$869$	4.06612	0.137934
$870$	0	0
$871$	27.7178	0.939181
$872$	0	0
$873$	−34.9311	−1.18224
$874$	0	0
$875$	−48.4521	−1.63798
$876$	0	0
$877$	−41.0652	−1.38667	−0.693336	−	0.720614i	$-0.743860\pi$
$877$	−41.0652	−1.38667	−0.693336	+	0.720614i	$0.743860\pi$
$878$	0	0
$879$	20.9109	0.705306
$880$	0	0
$881$	51.4251	1.73256	0.866278	−	0.499563i	$-0.166506\pi$
$881$	51.4251	1.73256	0.866278	+	0.499563i	$0.166506\pi$
$882$	0	0
$883$	−0.524245	−0.0176422	−0.00882112	−	0.999961i	$-0.502808\pi$
$883$	−0.524245	−0.0176422	−0.00882112	+	0.999961i	$0.502808\pi$
$884$	0	0
$885$	14.1639	0.476114
$886$	0	0
$887$	−35.0210	−1.17589	−0.587945	−	0.808901i	$-0.700063\pi$
$887$	−35.0210	−1.17589	−0.587945	+	0.808901i	$0.700063\pi$
$888$	0	0
$889$	−65.2651	−2.18892
$890$	0	0
$891$	−1.16558	−0.0390485
$892$	0	0
$893$	41.0765	1.37457
$894$	0	0
$895$	−8.10127	−0.270796
$896$	0	0
$897$	2.57992	0.0861409
$898$	0	0
$899$	−3.99540	−0.133254
$900$	0	0
$901$	−25.6878	−0.855784
$902$	0	0
$903$	18.8524	0.627370
$904$	0	0
$905$	−30.5921	−1.01692
$906$	0	0
$907$	23.0731	0.766128	0.383064	−	0.923722i	$-0.374869\pi$
$907$	23.0731	0.766128	0.383064	+	0.923722i	$0.374869\pi$
$908$	0	0
$909$	−31.7091	−1.05172
$910$	0	0
$911$	−29.7313	−0.985040	−0.492520	−	0.870301i	$-0.663924\pi$
$911$	−29.7313	−0.985040	−0.492520	+	0.870301i	$0.663924\pi$
$912$	0	0
$913$	−0.348249	−0.0115254
$914$	0	0
$915$	−26.7329	−0.883764
$916$	0	0
$917$	−82.0103	−2.70822
$918$	0	0
$919$	19.8975	0.656357	0.328179	−	0.944616i	$-0.393565\pi$
$919$	19.8975	0.656357	0.328179	+	0.944616i	$0.393565\pi$
$920$	0	0
$921$	8.79808	0.289907
$922$	0	0
$923$	32.5443	1.07121
$924$	0	0
$925$	−2.71264	−0.0891909
$926$	0	0
$927$	26.9199	0.884165
$928$	0	0
$929$	−25.4312	−0.834372	−0.417186	−	0.908821i	$-0.636984\pi$
$929$	−25.4312	−0.834372	−0.417186	+	0.908821i	$0.636984\pi$
$930$	0	0
$931$	69.3901	2.27417
$932$	0	0
$933$	15.9435	0.521967
$934$	0	0
$935$	−4.95957	−0.162195
$936$	0	0
$937$	57.2627	1.87069	0.935345	−	0.353737i	$-0.115089\pi$
$937$	57.2627	1.87069	0.935345	+	0.353737i	$0.115089\pi$
$938$	0	0
$939$	−6.15745	−0.200941
$940$	0	0
$941$	41.1114	1.34019	0.670096	−	0.742274i	$-0.266253\pi$
$941$	41.1114	1.34019	0.670096	+	0.742274i	$0.266253\pi$
$942$	0	0
$943$	1.39060	0.0452843
$944$	0	0
$945$	38.6491	1.25726
$946$	0	0
$947$	−45.6839	−1.48453	−0.742263	−	0.670109i	$-0.766247\pi$
$947$	−45.6839	−1.48453	−0.742263	+	0.670109i	$0.766247\pi$
$948$	0	0
$949$	−17.1424	−0.556467
$950$	0	0
$951$	23.0112	0.746189
$952$	0	0
$953$	−49.4998	−1.60346	−0.801728	−	0.597689i	$-0.796086\pi$
$953$	−49.4998	−1.60346	−0.801728	+	0.597689i	$0.796086\pi$
$954$	0	0
$955$	−39.3519	−1.27340
$956$	0	0
$957$	0.490445	0.0158538
$958$	0	0
$959$	0.498550	0.0160990
$960$	0	0
$961$	−25.5730	−0.824936
$962$	0	0
$963$	−1.91405	−0.0616795
$964$	0	0
$965$	52.5475	1.69156
$966$	0	0
$967$	−3.74494	−0.120429	−0.0602146	−	0.998185i	$-0.519178\pi$
$967$	−3.74494	−0.120429	−0.0602146	+	0.998185i	$0.519178\pi$
$968$	0	0
$969$	37.4717	1.20377
$970$	0	0
$971$	−41.5625	−1.33380	−0.666901	−	0.745146i	$-0.732380\pi$
$971$	−41.5625	−1.33380	−0.666901	+	0.745146i	$0.732380\pi$
$972$	0	0
$973$	11.7656	0.377186
$974$	0	0
$975$	1.56155	0.0500095
$976$	0	0
$977$	31.4357	1.00572	0.502859	−	0.864368i	$-0.332281\pi$
$977$	31.4357	1.00572	0.502859	+	0.864368i	$0.332281\pi$
$978$	0	0
$979$	2.88349	0.0921566
$980$	0	0
$981$	17.8205	0.568965
$982$	0	0
$983$	35.3372	1.12708	0.563540	−	0.826089i	$-0.309439\pi$
$983$	35.3372	1.12708	0.563540	+	0.826089i	$0.309439\pi$
$984$	0	0
$985$	44.7518	1.42591
$986$	0	0
$987$	20.7799	0.661432
$988$	0	0
$989$	−4.49836	−0.143040
$990$	0	0
$991$	−17.2922	−0.549304	−0.274652	−	0.961544i	$-0.588563\pi$
$991$	−17.2922	−0.549304	−0.274652	+	0.961544i	$0.588563\pi$
$992$	0	0
$993$	−6.64414	−0.210846
$994$	0	0
$995$	52.1569	1.65349
$996$	0	0
$997$	10.2403	0.324312	0.162156	−	0.986765i	$-0.448155\pi$
$997$	10.2403	0.324312	0.162156	+	0.986765i	$0.448155\pi$
$998$	0	0
$999$	24.0599	0.761222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.11		28
4.3	odd	2		4024.2.a.d.1.18	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.18	✓	28	4.3	odd	2
8048.2.a.v.1.11		28	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)

\(f(q)\)	\(=\)	\(q-0.823496 q^{3} -2.12271 q^{5} -4.15456 q^{7} -2.32185 q^{9} +O(q^{10})\)	\(q-0.823496 q^{3} -2.12271 q^{5} -4.15456 q^{7} -2.32185 q^{9} -0.347254 q^{11} +3.83770 q^{13} +1.74804 q^{15} -6.72831 q^{17} +6.76295 q^{19} +3.42126 q^{21} -0.816344 q^{23} -0.494108 q^{25} +4.38253 q^{27} +1.71507 q^{29} -2.32959 q^{31} +0.285962 q^{33} +8.81891 q^{35} +5.48996 q^{37} -3.16033 q^{39} -1.70345 q^{41} +5.51038 q^{43} +4.92862 q^{45} +6.07376 q^{47} +10.2603 q^{49} +5.54074 q^{51} +3.81786 q^{53} +0.737119 q^{55} -5.56926 q^{57} +8.10272 q^{59} -15.2931 q^{61} +9.64627 q^{63} -8.14632 q^{65} +7.22250 q^{67} +0.672256 q^{69} +8.48015 q^{71} -4.46685 q^{73} +0.406896 q^{75} +1.44269 q^{77} -11.7094 q^{79} +3.35657 q^{81} +1.00287 q^{83} +14.2822 q^{85} -1.41235 q^{87} -8.30368 q^{89} -15.9439 q^{91} +1.91841 q^{93} -14.3558 q^{95} +15.0445 q^{97} +0.806273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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