LMFDB - Embedded newform 8048.2.a.v.1.27

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			1.27
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.92644 q^{3} +0.124991 q^{5} -2.19824 q^{7} +5.56407 q^{9} +O(q^{10})$	$q+2.92644 q^{3} +0.124991 q^{5} -2.19824 q^{7} +5.56407 q^{9} -0.684798 q^{11} -5.23188 q^{13} +0.365779 q^{15} +3.19997 q^{17} +1.04142 q^{19} -6.43301 q^{21} -6.34708 q^{23} -4.98438 q^{25} +7.50361 q^{27} -6.90634 q^{29} +1.53143 q^{31} -2.00402 q^{33} -0.274759 q^{35} -1.84431 q^{37} -15.3108 q^{39} +10.2063 q^{41} +6.72891 q^{43} +0.695458 q^{45} +2.89294 q^{47} -2.16776 q^{49} +9.36454 q^{51} -6.45664 q^{53} -0.0855935 q^{55} +3.04766 q^{57} -3.81560 q^{59} +9.92318 q^{61} -12.2311 q^{63} -0.653937 q^{65} -10.4558 q^{67} -18.5744 q^{69} +2.63957 q^{71} -16.4537 q^{73} -14.5865 q^{75} +1.50535 q^{77} +0.990261 q^{79} +5.26667 q^{81} -12.7822 q^{83} +0.399967 q^{85} -20.2110 q^{87} +1.30655 q^{89} +11.5009 q^{91} +4.48164 q^{93} +0.130168 q^{95} +12.0738 q^{97} -3.81026 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.92644	1.68958	0.844791	−	0.535096i	$-0.179724\pi$
$3$	2.92644	1.68958	0.844791	+	0.535096i	$0.179724\pi$
$4$	0	0
$5$	0.124991	0.0558976	0.0279488	−	0.999609i	$-0.491102\pi$
$5$	0.124991	0.0558976	0.0279488	+	0.999609i	$0.491102\pi$
$6$	0	0
$7$	−2.19824	−0.830855	−0.415427	−	0.909626i	$-0.636368\pi$
$7$	−2.19824	−0.830855	−0.415427	+	0.909626i	$0.636368\pi$
$8$	0	0
$9$	5.56407	1.85469
$10$	0	0
$11$	−0.684798	−0.206474	−0.103237	−	0.994657i	$-0.532920\pi$
$11$	−0.684798	−0.206474	−0.103237	+	0.994657i	$0.532920\pi$
$12$	0	0
$13$	−5.23188	−1.45106	−0.725531	−	0.688190i	$-0.758406\pi$
$13$	−5.23188	−1.45106	−0.725531	+	0.688190i	$0.758406\pi$
$14$	0	0
$15$	0.365779	0.0944437
$16$	0	0
$17$	3.19997	0.776107	0.388054	−	0.921637i	$-0.373148\pi$
$17$	3.19997	0.776107	0.388054	+	0.921637i	$0.373148\pi$
$18$	0	0
$19$	1.04142	0.238919	0.119459	−	0.992839i	$-0.461884\pi$
$19$	1.04142	0.238919	0.119459	+	0.992839i	$0.461884\pi$
$20$	0	0
$21$	−6.43301	−1.40380
$22$	0	0
$23$	−6.34708	−1.32346	−0.661729	−	0.749743i	$-0.730177\pi$
$23$	−6.34708	−1.32346	−0.661729	+	0.749743i	$0.730177\pi$
$24$	0	0
$25$	−4.98438	−0.996875
$26$	0	0
$27$	7.50361	1.44407
$28$	0	0
$29$	−6.90634	−1.28247	−0.641237	−	0.767343i	$-0.721579\pi$
$29$	−6.90634	−1.28247	−0.641237	+	0.767343i	$0.721579\pi$
$30$	0	0
$31$	1.53143	0.275053	0.137527	−	0.990498i	$-0.456085\pi$
$31$	1.53143	0.275053	0.137527	+	0.990498i	$0.456085\pi$
$32$	0	0
$33$	−2.00402	−0.348856
$34$	0	0
$35$	−0.274759	−0.0464428
$36$	0	0
$37$	−1.84431	−0.303202	−0.151601	−	0.988442i	$-0.548443\pi$
$37$	−1.84431	−0.303202	−0.151601	+	0.988442i	$0.548443\pi$
$38$	0	0
$39$	−15.3108	−2.45169
$40$	0	0
$41$	10.2063	1.59395	0.796975	−	0.604013i	$-0.206432\pi$
$41$	10.2063	1.59395	0.796975	+	0.604013i	$0.206432\pi$
$42$	0	0
$43$	6.72891	1.02615	0.513074	−	0.858344i	$-0.328506\pi$
$43$	6.72891	1.02615	0.513074	+	0.858344i	$0.328506\pi$
$44$	0	0
$45$	0.695458	0.103673
$46$	0	0
$47$	2.89294	0.421978	0.210989	−	0.977488i	$-0.432332\pi$
$47$	2.89294	0.421978	0.210989	+	0.977488i	$0.432332\pi$
$48$	0	0
$49$	−2.16776	−0.309680
$50$	0	0
$51$	9.36454	1.31130
$52$	0	0
$53$	−6.45664	−0.886888	−0.443444	−	0.896302i	$-0.646244\pi$
$53$	−6.45664	−0.886888	−0.443444	+	0.896302i	$0.646244\pi$
$54$	0	0
$55$	−0.0855935	−0.0115414
$56$	0	0
$57$	3.04766	0.403673
$58$	0	0
$59$	−3.81560	−0.496748	−0.248374	−	0.968664i	$-0.579896\pi$
$59$	−3.81560	−0.496748	−0.248374	+	0.968664i	$0.579896\pi$
$60$	0	0
$61$	9.92318	1.27053	0.635267	−	0.772293i	$-0.280890\pi$
$61$	9.92318	1.27053	0.635267	+	0.772293i	$0.280890\pi$
$62$	0	0
$63$	−12.2311	−1.54098
$64$	0	0
$65$	−0.653937	−0.0811109
$66$	0	0
$67$	−10.4558	−1.27738	−0.638692	−	0.769462i	$-0.720524\pi$
$67$	−10.4558	−1.27738	−0.638692	+	0.769462i	$0.720524\pi$
$68$	0	0
$69$	−18.5744	−2.23609
$70$	0	0
$71$	2.63957	0.313260	0.156630	−	0.987657i	$-0.449937\pi$
$71$	2.63957	0.313260	0.156630	+	0.987657i	$0.449937\pi$
$72$	0	0
$73$	−16.4537	−1.92575	−0.962877	−	0.269939i	$-0.912996\pi$
$73$	−16.4537	−1.92575	−0.962877	+	0.269939i	$0.912996\pi$
$74$	0	0
$75$	−14.5865	−1.68430
$76$	0	0
$77$	1.50535	0.171550
$78$	0	0
$79$	0.990261	0.111413	0.0557065	−	0.998447i	$-0.482259\pi$
$79$	0.990261	0.111413	0.0557065	+	0.998447i	$0.482259\pi$
$80$	0	0
$81$	5.26667	0.585186
$82$	0	0
$83$	−12.7822	−1.40302	−0.701512	−	0.712657i	$-0.747491\pi$
$83$	−12.7822	−1.40302	−0.701512	+	0.712657i	$0.747491\pi$
$84$	0	0
$85$	0.399967	0.0433825
$86$	0	0
$87$	−20.2110	−2.16685
$88$	0	0
$89$	1.30655	0.138494	0.0692469	−	0.997600i	$-0.477940\pi$
$89$	1.30655	0.138494	0.0692469	+	0.997600i	$0.477940\pi$
$90$	0	0
$91$	11.5009	1.20562
$92$	0	0
$93$	4.48164	0.464725
$94$	0	0
$95$	0.130168	0.0133550
$96$	0	0
$97$	12.0738	1.22591	0.612953	−	0.790119i	$-0.289981\pi$
$97$	12.0738	1.22591	0.612953	+	0.790119i	$0.289981\pi$
$98$	0	0
$99$	−3.81026	−0.382946
$100$	0	0
$101$	−15.3447	−1.52686	−0.763429	−	0.645892i	$-0.776486\pi$
$101$	−15.3447	−1.52686	−0.763429	+	0.645892i	$0.776486\pi$
$102$	0	0
$103$	−14.9224	−1.47034	−0.735172	−	0.677880i	$-0.762899\pi$
$103$	−14.9224	−1.47034	−0.735172	+	0.677880i	$0.762899\pi$
$104$	0	0
$105$	−0.804068	−0.0784690
$106$	0	0
$107$	−17.8895	−1.72944	−0.864720	−	0.502255i	$-0.832504\pi$
$107$	−17.8895	−1.72944	−0.864720	+	0.502255i	$0.832504\pi$
$108$	0	0
$109$	−4.13401	−0.395967	−0.197983	−	0.980205i	$-0.563439\pi$
$109$	−4.13401	−0.395967	−0.197983	+	0.980205i	$0.563439\pi$
$110$	0	0
$111$	−5.39726	−0.512285
$112$	0	0
$113$	16.5479	1.55670	0.778350	−	0.627831i	$-0.216057\pi$
$113$	16.5479	1.55670	0.778350	+	0.627831i	$0.216057\pi$
$114$	0	0
$115$	−0.793327	−0.0739781
$116$	0	0
$117$	−29.1105	−2.69127
$118$	0	0
$119$	−7.03429	−0.644832
$120$	0	0
$121$	−10.5311	−0.957368
$122$	0	0
$123$	29.8680	2.69311
$124$	0	0
$125$	−1.24796	−0.111621
$126$	0	0
$127$	−1.42164	−0.126150	−0.0630752	−	0.998009i	$-0.520091\pi$
$127$	−1.42164	−0.126150	−0.0630752	+	0.998009i	$0.520091\pi$
$128$	0	0
$129$	19.6918	1.73376
$130$	0	0
$131$	8.96875	0.783603	0.391802	−	0.920050i	$-0.371852\pi$
$131$	8.96875	0.783603	0.391802	+	0.920050i	$0.371852\pi$
$132$	0	0
$133$	−2.28929	−0.198507
$134$	0	0
$135$	0.937882	0.0807201
$136$	0	0
$137$	−15.5701	−1.33024	−0.665120	−	0.746736i	$-0.731620\pi$
$137$	−15.5701	−1.33024	−0.665120	+	0.746736i	$0.731620\pi$
$138$	0	0
$139$	12.9396	1.09753	0.548763	−	0.835978i	$-0.315099\pi$
$139$	12.9396	1.09753	0.548763	+	0.835978i	$0.315099\pi$
$140$	0	0
$141$	8.46601	0.712967
$142$	0	0
$143$	3.58278	0.299607
$144$	0	0
$145$	−0.863229	−0.0716873
$146$	0	0
$147$	−6.34383	−0.523231
$148$	0	0
$149$	−23.0629	−1.88939	−0.944695	−	0.327951i	$-0.893642\pi$
$149$	−23.0629	−1.88939	−0.944695	+	0.327951i	$0.893642\pi$
$150$	0	0
$151$	−0.534439	−0.0434921	−0.0217460	−	0.999764i	$-0.506923\pi$
$151$	−0.534439	−0.0434921	−0.0217460	+	0.999764i	$0.506923\pi$
$152$	0	0
$153$	17.8049	1.43944
$154$	0	0
$155$	0.191415	0.0153748
$156$	0	0
$157$	−22.6311	−1.80616	−0.903080	−	0.429472i	$-0.858700\pi$
$157$	−22.6311	−1.80616	−0.903080	+	0.429472i	$0.858700\pi$
$158$	0	0
$159$	−18.8950	−1.49847
$160$	0	0
$161$	13.9524	1.09960
$162$	0	0
$163$	−2.19723	−0.172100	−0.0860501	−	0.996291i	$-0.527425\pi$
$163$	−2.19723	−0.172100	−0.0860501	+	0.996291i	$0.527425\pi$
$164$	0	0
$165$	−0.250485	−0.0195002
$166$	0	0
$167$	7.38930	0.571801	0.285901	−	0.958259i	$-0.407707\pi$
$167$	7.38930	0.571801	0.285901	+	0.958259i	$0.407707\pi$
$168$	0	0
$169$	14.3725	1.10558
$170$	0	0
$171$	5.79455	0.443120
$172$	0	0
$173$	−10.8931	−0.828185	−0.414093	−	0.910235i	$-0.635901\pi$
$173$	−10.8931	−0.828185	−0.414093	+	0.910235i	$0.635901\pi$
$174$	0	0
$175$	10.9568	0.828259
$176$	0	0
$177$	−11.1661	−0.839298
$178$	0	0
$179$	14.2624	1.06602	0.533012	−	0.846108i	$-0.321060\pi$
$179$	14.2624	1.06602	0.533012	+	0.846108i	$0.321060\pi$
$180$	0	0
$181$	17.3329	1.28835	0.644173	−	0.764880i	$-0.277202\pi$
$181$	17.3329	1.28835	0.644173	+	0.764880i	$0.277202\pi$
$182$	0	0
$183$	29.0396	2.14667
$184$	0	0
$185$	−0.230521	−0.0169483
$186$	0	0
$187$	−2.19133	−0.160246
$188$	0	0
$189$	−16.4947	−1.19981
$190$	0	0
$191$	5.90903	0.427563	0.213781	−	0.976882i	$-0.431422\pi$
$191$	5.90903	0.427563	0.213781	+	0.976882i	$0.431422\pi$
$192$	0	0
$193$	−18.2410	−1.31301	−0.656507	−	0.754320i	$-0.727967\pi$
$193$	−18.2410	−1.31301	−0.656507	+	0.754320i	$0.727967\pi$
$194$	0	0
$195$	−1.91371	−0.137044
$196$	0	0
$197$	13.7541	0.979937	0.489968	−	0.871740i	$-0.337008\pi$
$197$	13.7541	0.979937	0.489968	+	0.871740i	$0.337008\pi$
$198$	0	0
$199$	15.1280	1.07240	0.536198	−	0.844092i	$-0.319860\pi$
$199$	15.1280	1.07240	0.536198	+	0.844092i	$0.319860\pi$
$200$	0	0
$201$	−30.5984	−2.15825
$202$	0	0
$203$	15.1818	1.06555
$204$	0	0
$205$	1.27569	0.0890980
$206$	0	0
$207$	−35.3156	−2.45460
$208$	0	0
$209$	−0.713164	−0.0493306
$210$	0	0
$211$	0.0288291	0.00198467	0.000992337	−	1.00000i	$-0.499684\pi$
$211$	0.0288291	0.00198467	0.000992337		1.00000i	$0.499684\pi$
$212$	0	0
$213$	7.72456	0.529278
$214$	0	0
$215$	0.841052	0.0573593
$216$	0	0
$217$	−3.36644	−0.228529
$218$	0	0
$219$	−48.1507	−3.25372
$220$	0	0
$221$	−16.7419	−1.12618
$222$	0	0
$223$	19.1423	1.28186	0.640931	−	0.767599i	$-0.278549\pi$
$223$	19.1423	1.28186	0.640931	+	0.767599i	$0.278549\pi$
$224$	0	0
$225$	−27.7334	−1.84890
$226$	0	0
$227$	−20.2540	−1.34430	−0.672152	−	0.740413i	$-0.734630\pi$
$227$	−20.2540	−1.34430	−0.672152	+	0.740413i	$0.734630\pi$
$228$	0	0
$229$	17.6183	1.16425	0.582125	−	0.813100i	$-0.302222\pi$
$229$	17.6183	1.16425	0.582125	+	0.813100i	$0.302222\pi$
$230$	0	0
$231$	4.40531	0.289848
$232$	0	0
$233$	0.493520	0.0323316	0.0161658	−	0.999869i	$-0.494854\pi$
$233$	0.493520	0.0323316	0.0161658	+	0.999869i	$0.494854\pi$
$234$	0	0
$235$	0.361591	0.0235876
$236$	0	0
$237$	2.89794	0.188242
$238$	0	0
$239$	26.6049	1.72093	0.860463	−	0.509514i	$-0.170175\pi$
$239$	26.6049	1.72093	0.860463	+	0.509514i	$0.170175\pi$
$240$	0	0
$241$	3.16066	0.203596	0.101798	−	0.994805i	$-0.467540\pi$
$241$	3.16066	0.203596	0.101798	+	0.994805i	$0.467540\pi$
$242$	0	0
$243$	−7.09821	−0.455350
$244$	0	0
$245$	−0.270951	−0.0173104
$246$	0	0
$247$	−5.44859	−0.346686
$248$	0	0
$249$	−37.4063	−2.37053
$250$	0	0
$251$	−9.66267	−0.609903	−0.304951	−	0.952368i	$-0.598640\pi$
$251$	−9.66267	−0.609903	−0.304951	+	0.952368i	$0.598640\pi$
$252$	0	0
$253$	4.34647	0.273260
$254$	0	0
$255$	1.17048	0.0732984
$256$	0	0
$257$	−12.8046	−0.798727	−0.399364	−	0.916793i	$-0.630769\pi$
$257$	−12.8046	−0.798727	−0.399364	+	0.916793i	$0.630769\pi$
$258$	0	0
$259$	4.05422	0.251917
$260$	0	0
$261$	−38.4274	−2.37859
$262$	0	0
$263$	23.2755	1.43523	0.717613	−	0.696442i	$-0.245235\pi$
$263$	23.2755	1.43523	0.717613	+	0.696442i	$0.245235\pi$
$264$	0	0
$265$	−0.807022	−0.0495749
$266$	0	0
$267$	3.82354	0.233997
$268$	0	0
$269$	−6.70921	−0.409068	−0.204534	−	0.978859i	$-0.565568\pi$
$269$	−6.70921	−0.409068	−0.204534	+	0.978859i	$0.565568\pi$
$270$	0	0
$271$	−6.11292	−0.371334	−0.185667	−	0.982613i	$-0.559445\pi$
$271$	−6.11292	−0.371334	−0.185667	+	0.982613i	$0.559445\pi$
$272$	0	0
$273$	33.6567	2.03700
$274$	0	0
$275$	3.41329	0.205829
$276$	0	0
$277$	−11.6950	−0.702683	−0.351341	−	0.936247i	$-0.614274\pi$
$277$	−11.6950	−0.702683	−0.351341	+	0.936247i	$0.614274\pi$
$278$	0	0
$279$	8.52099	0.510138
$280$	0	0
$281$	−22.5841	−1.34725	−0.673627	−	0.739071i	$-0.735265\pi$
$281$	−22.5841	−1.34725	−0.673627	+	0.739071i	$0.735265\pi$
$282$	0	0
$283$	22.4502	1.33453	0.667264	−	0.744821i	$-0.267465\pi$
$283$	22.4502	1.33453	0.667264	+	0.744821i	$0.267465\pi$
$284$	0	0
$285$	0.380930	0.0225644
$286$	0	0
$287$	−22.4358	−1.32434
$288$	0	0
$289$	−6.76018	−0.397658
$290$	0	0
$291$	35.3332	2.07127
$292$	0	0
$293$	−24.8303	−1.45060	−0.725301	−	0.688432i	$-0.758300\pi$
$293$	−24.8303	−1.45060	−0.725301	+	0.688432i	$0.758300\pi$
$294$	0	0
$295$	−0.476915	−0.0277671
$296$	0	0
$297$	−5.13846	−0.298163
$298$	0	0
$299$	33.2071	1.92042
$300$	0	0
$301$	−14.7917	−0.852581
$302$	0	0
$303$	−44.9055	−2.57975
$304$	0	0
$305$	1.24031	0.0710198
$306$	0	0
$307$	−6.08657	−0.347379	−0.173690	−	0.984800i	$-0.555569\pi$
$307$	−6.08657	−0.347379	−0.173690	+	0.984800i	$0.555569\pi$
$308$	0	0
$309$	−43.6695	−2.48427
$310$	0	0
$311$	23.1084	1.31036	0.655178	−	0.755475i	$-0.272594\pi$
$311$	23.1084	1.31036	0.655178	+	0.755475i	$0.272594\pi$
$312$	0	0
$313$	13.6387	0.770902	0.385451	−	0.922728i	$-0.374046\pi$
$313$	13.6387	0.770902	0.385451	+	0.922728i	$0.374046\pi$
$314$	0	0
$315$	−1.52878	−0.0861370
$316$	0	0
$317$	−3.22419	−0.181089	−0.0905444	−	0.995892i	$-0.528861\pi$
$317$	−3.22419	−0.181089	−0.0905444	+	0.995892i	$0.528861\pi$
$318$	0	0
$319$	4.72945	0.264798
$320$	0	0
$321$	−52.3525	−2.92203
$322$	0	0
$323$	3.33252	0.185427
$324$	0	0
$325$	26.0776	1.44653
$326$	0	0
$327$	−12.0980	−0.669018
$328$	0	0
$329$	−6.35935	−0.350602
$330$	0	0
$331$	29.7151	1.63329	0.816644	−	0.577142i	$-0.195832\pi$
$331$	29.7151	1.63329	0.816644	+	0.577142i	$0.195832\pi$
$332$	0	0
$333$	−10.2618	−0.562346
$334$	0	0
$335$	−1.30689	−0.0714028
$336$	0	0
$337$	22.3421	1.21705	0.608526	−	0.793534i	$-0.291761\pi$
$337$	22.3421	1.21705	0.608526	+	0.793534i	$0.291761\pi$
$338$	0	0
$339$	48.4266	2.63017
$340$	0	0
$341$	−1.04872	−0.0567914
$342$	0	0
$343$	20.1529	1.08815
$344$	0	0
$345$	−2.32163	−0.124992
$346$	0	0
$347$	28.0064	1.50346	0.751730	−	0.659471i	$-0.229220\pi$
$347$	28.0064	1.50346	0.751730	+	0.659471i	$0.229220\pi$
$348$	0	0
$349$	16.2555	0.870138	0.435069	−	0.900397i	$-0.356724\pi$
$349$	16.2555	0.870138	0.435069	+	0.900397i	$0.356724\pi$
$350$	0	0
$351$	−39.2579	−2.09543
$352$	0	0
$353$	−14.7318	−0.784095	−0.392048	−	0.919945i	$-0.628233\pi$
$353$	−14.7318	−0.784095	−0.392048	+	0.919945i	$0.628233\pi$
$354$	0	0
$355$	0.329922	0.0175105
$356$	0	0
$357$	−20.5855	−1.08950
$358$	0	0
$359$	22.8293	1.20488	0.602441	−	0.798163i	$-0.294195\pi$
$359$	22.8293	1.20488	0.602441	+	0.798163i	$0.294195\pi$
$360$	0	0
$361$	−17.9154	−0.942918
$362$	0	0
$363$	−30.8185	−1.61755
$364$	0	0
$365$	−2.05656	−0.107645
$366$	0	0
$367$	−21.1688	−1.10500	−0.552501	−	0.833512i	$-0.686327\pi$
$367$	−21.1688	−1.10500	−0.552501	+	0.833512i	$0.686327\pi$
$368$	0	0
$369$	56.7883	2.95628
$370$	0	0
$371$	14.1932	0.736875
$372$	0	0
$373$	15.8669	0.821558	0.410779	−	0.911735i	$-0.365257\pi$
$373$	15.8669	0.821558	0.410779	+	0.911735i	$0.365257\pi$
$374$	0	0
$375$	−3.65207	−0.188592
$376$	0	0
$377$	36.1331	1.86095
$378$	0	0
$379$	−27.6503	−1.42030	−0.710149	−	0.704051i	$-0.751373\pi$
$379$	−27.6503	−1.42030	−0.710149	+	0.704051i	$0.751373\pi$
$380$	0	0
$381$	−4.16036	−0.213141
$382$	0	0
$383$	2.53070	0.129313	0.0646563	−	0.997908i	$-0.479405\pi$
$383$	2.53070	0.129313	0.0646563	+	0.997908i	$0.479405\pi$
$384$	0	0
$385$	0.188155	0.00958925
$386$	0	0
$387$	37.4401	1.90319
$388$	0	0
$389$	17.7301	0.898950	0.449475	−	0.893293i	$-0.351611\pi$
$389$	17.7301	0.898950	0.449475	+	0.893293i	$0.351611\pi$
$390$	0	0
$391$	−20.3105	−1.02715
$392$	0	0
$393$	26.2465	1.32396
$394$	0	0
$395$	0.123774	0.00622773
$396$	0	0
$397$	5.30065	0.266032	0.133016	−	0.991114i	$-0.457534\pi$
$397$	5.30065	0.266032	0.133016	+	0.991114i	$0.457534\pi$
$398$	0	0
$399$	−6.69948	−0.335394
$400$	0	0
$401$	8.10636	0.404812	0.202406	−	0.979302i	$-0.435124\pi$
$401$	8.10636	0.404812	0.202406	+	0.979302i	$0.435124\pi$
$402$	0	0
$403$	−8.01225	−0.399119
$404$	0	0
$405$	0.658286	0.0327105
$406$	0	0
$407$	1.26298	0.0626035
$408$	0	0
$409$	−18.8134	−0.930263	−0.465132	−	0.885241i	$-0.653993\pi$
$409$	−18.8134	−0.930263	−0.465132	+	0.885241i	$0.653993\pi$
$410$	0	0
$411$	−45.5649	−2.24755
$412$	0	0
$413$	8.38758	0.412726
$414$	0	0
$415$	−1.59765	−0.0784257
$416$	0	0
$417$	37.8671	1.85436
$418$	0	0
$419$	−20.7589	−1.01414	−0.507069	−	0.861906i	$-0.669271\pi$
$419$	−20.7589	−1.01414	−0.507069	+	0.861906i	$0.669271\pi$
$420$	0	0
$421$	25.4315	1.23945	0.619727	−	0.784817i	$-0.287243\pi$
$421$	25.4315	1.23945	0.619727	+	0.784817i	$0.287243\pi$
$422$	0	0
$423$	16.0965	0.782639
$424$	0	0
$425$	−15.9499	−0.773682
$426$	0	0
$427$	−21.8135	−1.05563
$428$	0	0
$429$	10.4848	0.506211
$430$	0	0
$431$	−20.4276	−0.983963	−0.491982	−	0.870606i	$-0.663727\pi$
$431$	−20.4276	−0.983963	−0.491982	+	0.870606i	$0.663727\pi$
$432$	0	0
$433$	−22.3315	−1.07318	−0.536592	−	0.843842i	$-0.680289\pi$
$433$	−22.3315	−1.07318	−0.536592	+	0.843842i	$0.680289\pi$
$434$	0	0
$435$	−2.52619	−0.121122
$436$	0	0
$437$	−6.60999	−0.316199
$438$	0	0
$439$	−10.7157	−0.511432	−0.255716	−	0.966752i	$-0.582311\pi$
$439$	−10.7157	−0.511432	−0.255716	+	0.966752i	$0.582311\pi$
$440$	0	0
$441$	−12.0616	−0.574361
$442$	0	0
$443$	−25.4030	−1.20693	−0.603466	−	0.797388i	$-0.706214\pi$
$443$	−25.4030	−1.20693	−0.603466	+	0.797388i	$0.706214\pi$
$444$	0	0
$445$	0.163307	0.00774148
$446$	0	0
$447$	−67.4924	−3.19228
$448$	0	0
$449$	−32.5238	−1.53490	−0.767448	−	0.641112i	$-0.778474\pi$
$449$	−32.5238	−1.53490	−0.767448	+	0.641112i	$0.778474\pi$
$450$	0	0
$451$	−6.98922	−0.329110
$452$	0	0
$453$	−1.56401	−0.0734834
$454$	0	0
$455$	1.43751	0.0673914
$456$	0	0
$457$	−7.75980	−0.362988	−0.181494	−	0.983392i	$-0.558093\pi$
$457$	−7.75980	−0.362988	−0.181494	+	0.983392i	$0.558093\pi$
$458$	0	0
$459$	24.0113	1.12075
$460$	0	0
$461$	20.1297	0.937532	0.468766	−	0.883322i	$-0.344699\pi$
$461$	20.1297	0.937532	0.468766	+	0.883322i	$0.344699\pi$
$462$	0	0
$463$	8.91696	0.414406	0.207203	−	0.978298i	$-0.433564\pi$
$463$	8.91696	0.414406	0.207203	+	0.978298i	$0.433564\pi$
$464$	0	0
$465$	0.560165	0.0259770
$466$	0	0
$467$	21.8796	1.01247	0.506233	−	0.862397i	$-0.331038\pi$
$467$	21.8796	1.01247	0.506233	+	0.862397i	$0.331038\pi$
$468$	0	0
$469$	22.9844	1.06132
$470$	0	0
$471$	−66.2287	−3.05166
$472$	0	0
$473$	−4.60794	−0.211873
$474$	0	0
$475$	−5.19084	−0.238172
$476$	0	0
$477$	−35.9252	−1.64490
$478$	0	0
$479$	−22.1664	−1.01281	−0.506404	−	0.862297i	$-0.669025\pi$
$479$	−22.1664	−1.01281	−0.506404	+	0.862297i	$0.669025\pi$
$480$	0	0
$481$	9.64918	0.439965
$482$	0	0
$483$	40.8308	1.85787
$484$	0	0
$485$	1.50911	0.0685252
$486$	0	0
$487$	1.47231	0.0667167	0.0333584	−	0.999443i	$-0.489380\pi$
$487$	1.47231	0.0667167	0.0333584	+	0.999443i	$0.489380\pi$
$488$	0	0
$489$	−6.43007	−0.290778
$490$	0	0
$491$	32.0435	1.44610	0.723051	−	0.690795i	$-0.242739\pi$
$491$	32.0435	1.44610	0.723051	+	0.690795i	$0.242739\pi$
$492$	0	0
$493$	−22.1001	−0.995338
$494$	0	0
$495$	−0.476248	−0.0214058
$496$	0	0
$497$	−5.80240	−0.260273
$498$	0	0
$499$	6.46266	0.289308	0.144654	−	0.989482i	$-0.453793\pi$
$499$	6.46266	0.289308	0.144654	+	0.989482i	$0.453793\pi$
$500$	0	0
$501$	21.6244	0.966105
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−1.91795	−0.0853477
$506$	0	0
$507$	42.0604	1.86797
$508$	0	0
$509$	−21.3454	−0.946119	−0.473059	−	0.881031i	$-0.656850\pi$
$509$	−21.3454	−0.946119	−0.473059	+	0.881031i	$0.656850\pi$
$510$	0	0
$511$	36.1690	1.60002
$512$	0	0
$513$	7.81443	0.345015
$514$	0	0
$515$	−1.86516	−0.0821888
$516$	0	0
$517$	−1.98108	−0.0871277
$518$	0	0
$519$	−31.8780	−1.39929
$520$	0	0
$521$	1.25630	0.0550395	0.0275198	−	0.999621i	$-0.491239\pi$
$521$	1.25630	0.0550395	0.0275198	+	0.999621i	$0.491239\pi$
$522$	0	0
$523$	11.5794	0.506334	0.253167	−	0.967423i	$-0.418528\pi$
$523$	11.5794	0.506334	0.253167	+	0.967423i	$0.418528\pi$
$524$	0	0
$525$	32.0646	1.39941
$526$	0	0
$527$	4.90054	0.213471
$528$	0	0
$529$	17.2854	0.751540
$530$	0	0
$531$	−21.2303	−0.921315
$532$	0	0
$533$	−53.3979	−2.31292
$534$	0	0
$535$	−2.23602	−0.0966716
$536$	0	0
$537$	41.7382	1.80114
$538$	0	0
$539$	1.48448	0.0639411
$540$	0	0
$541$	13.2144	0.568132	0.284066	−	0.958805i	$-0.408316\pi$
$541$	13.2144	0.568132	0.284066	+	0.958805i	$0.408316\pi$
$542$	0	0
$543$	50.7238	2.17677
$544$	0	0
$545$	−0.516714	−0.0221336
$546$	0	0
$547$	23.8882	1.02139	0.510693	−	0.859763i	$-0.329389\pi$
$547$	23.8882	1.02139	0.510693	+	0.859763i	$0.329389\pi$
$548$	0	0
$549$	55.2133	2.35645
$550$	0	0
$551$	−7.19242	−0.306407
$552$	0	0
$553$	−2.17683	−0.0925681
$554$	0	0
$555$	−0.674608	−0.0286355
$556$	0	0
$557$	31.1120	1.31826	0.659130	−	0.752029i	$-0.270925\pi$
$557$	31.1120	1.31826	0.659130	+	0.752029i	$0.270925\pi$
$558$	0	0
$559$	−35.2048	−1.48900
$560$	0	0
$561$	−6.41282	−0.270749
$562$	0	0
$563$	12.1112	0.510424	0.255212	−	0.966885i	$-0.417855\pi$
$563$	12.1112	0.510424	0.255212	+	0.966885i	$0.417855\pi$
$564$	0	0
$565$	2.06834	0.0870158
$566$	0	0
$567$	−11.5774	−0.486204
$568$	0	0
$569$	−4.88936	−0.204972	−0.102486	−	0.994734i	$-0.532680\pi$
$569$	−4.88936	−0.204972	−0.102486	+	0.994734i	$0.532680\pi$
$570$	0	0
$571$	16.7597	0.701371	0.350685	−	0.936493i	$-0.385949\pi$
$571$	16.7597	0.701371	0.350685	+	0.936493i	$0.385949\pi$
$572$	0	0
$573$	17.2925	0.722403
$574$	0	0
$575$	31.6362	1.31932
$576$	0	0
$577$	29.2363	1.21712	0.608561	−	0.793507i	$-0.291747\pi$
$577$	29.2363	1.21712	0.608561	+	0.793507i	$0.291747\pi$
$578$	0	0
$579$	−53.3812	−2.21845
$580$	0	0
$581$	28.0982	1.16571
$582$	0	0
$583$	4.42150	0.183120
$584$	0	0
$585$	−3.63855	−0.150436
$586$	0	0
$587$	−13.2629	−0.547418	−0.273709	−	0.961812i	$-0.588251\pi$
$587$	−13.2629	−0.547418	−0.273709	+	0.961812i	$0.588251\pi$
$588$	0	0
$589$	1.59487	0.0657153
$590$	0	0
$591$	40.2505	1.65568
$592$	0	0
$593$	15.9463	0.654834	0.327417	−	0.944880i	$-0.393822\pi$
$593$	15.9463	0.654834	0.327417	+	0.944880i	$0.393822\pi$
$594$	0	0
$595$	−0.879222	−0.0360446
$596$	0	0
$597$	44.2713	1.81190
$598$	0	0
$599$	−42.3185	−1.72909	−0.864544	−	0.502557i	$-0.832393\pi$
$599$	−42.3185	−1.72909	−0.864544	+	0.502557i	$0.832393\pi$
$600$	0	0
$601$	4.15062	0.169307	0.0846537	−	0.996410i	$-0.473022\pi$
$601$	4.15062	0.169307	0.0846537	+	0.996410i	$0.473022\pi$
$602$	0	0
$603$	−58.1771	−2.36915
$604$	0	0
$605$	−1.31629	−0.0535146
$606$	0	0
$607$	−36.4247	−1.47843	−0.739217	−	0.673467i	$-0.764804\pi$
$607$	−36.4247	−1.47843	−0.739217	+	0.673467i	$0.764804\pi$
$608$	0	0
$609$	44.4285	1.80034
$610$	0	0
$611$	−15.1355	−0.612316
$612$	0	0
$613$	−21.3761	−0.863372	−0.431686	−	0.902024i	$-0.642081\pi$
$613$	−21.3761	−0.863372	−0.431686	+	0.902024i	$0.642081\pi$
$614$	0	0
$615$	3.73323	0.150538
$616$	0	0
$617$	28.7707	1.15826	0.579131	−	0.815234i	$-0.303392\pi$
$617$	28.7707	1.15826	0.579131	+	0.815234i	$0.303392\pi$
$618$	0	0
$619$	−13.4867	−0.542076	−0.271038	−	0.962569i	$-0.587367\pi$
$619$	−13.4867	−0.542076	−0.271038	+	0.962569i	$0.587367\pi$
$620$	0	0
$621$	−47.6260	−1.91117
$622$	0	0
$623$	−2.87210	−0.115068
$624$	0	0
$625$	24.7659	0.990636
$626$	0	0
$627$	−2.08703	−0.0833481
$628$	0	0
$629$	−5.90173	−0.235317
$630$	0	0
$631$	32.1866	1.28133	0.640665	−	0.767821i	$-0.278659\pi$
$631$	32.1866	1.28133	0.640665	+	0.767821i	$0.278659\pi$
$632$	0	0
$633$	0.0843666	0.00335327
$634$	0	0
$635$	−0.177692	−0.00705150
$636$	0	0
$637$	11.3415	0.449365
$638$	0	0
$639$	14.6868	0.580999
$640$	0	0
$641$	−43.8329	−1.73129	−0.865647	−	0.500655i	$-0.833092\pi$
$641$	−43.8329	−1.73129	−0.865647	+	0.500655i	$0.833092\pi$
$642$	0	0
$643$	−35.1898	−1.38775	−0.693875	−	0.720096i	$-0.744098\pi$
$643$	−35.1898	−1.38775	−0.693875	+	0.720096i	$0.744098\pi$
$644$	0	0
$645$	2.46129	0.0969132
$646$	0	0
$647$	−0.759380	−0.0298543	−0.0149271	−	0.999889i	$-0.504752\pi$
$647$	−0.759380	−0.0298543	−0.0149271	+	0.999889i	$0.504752\pi$
$648$	0	0
$649$	2.61291	0.102566
$650$	0	0
$651$	−9.85171	−0.386119
$652$	0	0
$653$	−19.8143	−0.775395	−0.387698	−	0.921787i	$-0.626730\pi$
$653$	−19.8143	−0.775395	−0.387698	+	0.921787i	$0.626730\pi$
$654$	0	0
$655$	1.12101	0.0438016
$656$	0	0
$657$	−91.5493	−3.57168
$658$	0	0
$659$	48.9815	1.90805	0.954024	−	0.299729i	$-0.0968964\pi$
$659$	48.9815	1.90805	0.954024	+	0.299729i	$0.0968964\pi$
$660$	0	0
$661$	−33.7707	−1.31353	−0.656763	−	0.754097i	$-0.728075\pi$
$661$	−33.7707	−1.31353	−0.656763	+	0.754097i	$0.728075\pi$
$662$	0	0
$663$	−48.9941	−1.90277
$664$	0	0
$665$	−0.286141	−0.0110961
$666$	0	0
$667$	43.8351	1.69730
$668$	0	0
$669$	56.0188	2.16581
$670$	0	0
$671$	−6.79538	−0.262333
$672$	0	0
$673$	13.3901	0.516149	0.258075	−	0.966125i	$-0.416912\pi$
$673$	13.3901	0.516149	0.258075	+	0.966125i	$0.416912\pi$
$674$	0	0
$675$	−37.4008	−1.43956
$676$	0	0
$677$	−7.81952	−0.300529	−0.150264	−	0.988646i	$-0.548012\pi$
$677$	−7.81952	−0.300529	−0.150264	+	0.988646i	$0.548012\pi$
$678$	0	0
$679$	−26.5410	−1.01855
$680$	0	0
$681$	−59.2722	−2.27131
$682$	0	0
$683$	3.29209	0.125968	0.0629841	−	0.998015i	$-0.479938\pi$
$683$	3.29209	0.125968	0.0629841	+	0.998015i	$0.479938\pi$
$684$	0	0
$685$	−1.94612	−0.0743573
$686$	0	0
$687$	51.5589	1.96710
$688$	0	0
$689$	33.7804	1.28693
$690$	0	0
$691$	48.7786	1.85563	0.927813	−	0.373045i	$-0.121686\pi$
$691$	48.7786	1.85563	0.927813	+	0.373045i	$0.121686\pi$
$692$	0	0
$693$	8.37586	0.318173
$694$	0	0
$695$	1.61734	0.0613490
$696$	0	0
$697$	32.6597	1.23708
$698$	0	0
$699$	1.44426	0.0546269
$700$	0	0
$701$	−18.1750	−0.686462	−0.343231	−	0.939251i	$-0.611521\pi$
$701$	−18.1750	−0.686462	−0.343231	+	0.939251i	$0.611521\pi$
$702$	0	0
$703$	−1.92070	−0.0724406
$704$	0	0
$705$	1.05817	0.0398532
$706$	0	0
$707$	33.7313	1.26860
$708$	0	0
$709$	−17.9486	−0.674073	−0.337036	−	0.941492i	$-0.609425\pi$
$709$	−17.9486	−0.674073	−0.337036	+	0.941492i	$0.609425\pi$
$710$	0	0
$711$	5.50988	0.206637
$712$	0	0
$713$	−9.72011	−0.364021
$714$	0	0
$715$	0.447815	0.0167473
$716$	0	0
$717$	77.8576	2.90765
$718$	0	0
$719$	12.2202	0.455736	0.227868	−	0.973692i	$-0.426825\pi$
$719$	12.2202	0.455736	0.227868	+	0.973692i	$0.426825\pi$
$720$	0	0
$721$	32.8029	1.22164
$722$	0	0
$723$	9.24950	0.343993
$724$	0	0
$725$	34.4238	1.27847
$726$	0	0
$727$	−12.1125	−0.449228	−0.224614	−	0.974448i	$-0.572112\pi$
$727$	−12.1125	−0.449228	−0.224614	+	0.974448i	$0.572112\pi$
$728$	0	0
$729$	−36.5725	−1.35454
$730$	0	0
$731$	21.5323	0.796402
$732$	0	0
$733$	−8.94868	−0.330527	−0.165264	−	0.986249i	$-0.552847\pi$
$733$	−8.94868	−0.330527	−0.165264	+	0.986249i	$0.552847\pi$
$734$	0	0
$735$	−0.792922	−0.0292474
$736$	0	0
$737$	7.16014	0.263747
$738$	0	0
$739$	−20.4727	−0.753100	−0.376550	−	0.926396i	$-0.622890\pi$
$739$	−20.4727	−0.753100	−0.376550	+	0.926396i	$0.622890\pi$
$740$	0	0
$741$	−15.9450	−0.585754
$742$	0	0
$743$	3.47345	0.127429	0.0637143	−	0.997968i	$-0.479705\pi$
$743$	3.47345	0.127429	0.0637143	+	0.997968i	$0.479705\pi$
$744$	0	0
$745$	−2.88266	−0.105612
$746$	0	0
$747$	−71.1208	−2.60218
$748$	0	0
$749$	39.3252	1.43691
$750$	0	0
$751$	17.9585	0.655316	0.327658	−	0.944796i	$-0.393741\pi$
$751$	17.9585	0.655316	0.327658	+	0.944796i	$0.393741\pi$
$752$	0	0
$753$	−28.2773	−1.03048
$754$	0	0
$755$	−0.0668000	−0.00243110
$756$	0	0
$757$	27.5553	1.00152	0.500758	−	0.865588i	$-0.333055\pi$
$757$	27.5553	1.00152	0.500758	+	0.865588i	$0.333055\pi$
$758$	0	0
$759$	12.7197	0.461696
$760$	0	0
$761$	−15.2016	−0.551059	−0.275529	−	0.961293i	$-0.588853\pi$
$761$	−15.2016	−0.551059	−0.275529	+	0.961293i	$0.588853\pi$
$762$	0	0
$763$	9.08753	0.328991
$764$	0	0
$765$	2.22545	0.0804612
$766$	0	0
$767$	19.9627	0.720813
$768$	0	0
$769$	39.3884	1.42038	0.710190	−	0.704010i	$-0.248609\pi$
$769$	39.3884	1.42038	0.710190	+	0.704010i	$0.248609\pi$
$770$	0	0
$771$	−37.4719	−1.34952
$772$	0	0
$773$	2.28513	0.0821903	0.0410952	−	0.999155i	$-0.486915\pi$
$773$	2.28513	0.0821903	0.0410952	+	0.999155i	$0.486915\pi$
$774$	0	0
$775$	−7.63323	−0.274194
$776$	0	0
$777$	11.8644	0.425634
$778$	0	0
$779$	10.6290	0.380824
$780$	0	0
$781$	−1.80757	−0.0646801
$782$	0	0
$783$	−51.8225	−1.85198
$784$	0	0
$785$	−2.82868	−0.100960
$786$	0	0
$787$	−51.1008	−1.82155	−0.910773	−	0.412908i	$-0.864513\pi$
$787$	−51.1008	−1.82155	−0.910773	+	0.412908i	$0.864513\pi$
$788$	0	0
$789$	68.1143	2.42493
$790$	0	0
$791$	−36.3763	−1.29339
$792$	0	0
$793$	−51.9169	−1.84362
$794$	0	0
$795$	−2.36170	−0.0837610
$796$	0	0
$797$	−21.7081	−0.768941	−0.384470	−	0.923137i	$-0.625616\pi$
$797$	−21.7081	−0.768941	−0.384470	+	0.923137i	$0.625616\pi$
$798$	0	0
$799$	9.25731	0.327500
$800$	0	0
$801$	7.26973	0.256863
$802$	0	0
$803$	11.2674	0.397619
$804$	0	0
$805$	1.74392	0.0614651
$806$	0	0
$807$	−19.6341	−0.691154
$808$	0	0
$809$	25.3624	0.891694	0.445847	−	0.895109i	$-0.352903\pi$
$809$	25.3624	0.891694	0.445847	+	0.895109i	$0.352903\pi$
$810$	0	0
$811$	20.0122	0.702722	0.351361	−	0.936240i	$-0.385719\pi$
$811$	20.0122	0.702722	0.351361	+	0.936240i	$0.385719\pi$
$812$	0	0
$813$	−17.8891	−0.627399
$814$	0	0
$815$	−0.274634	−0.00962000
$816$	0	0
$817$	7.00764	0.245166
$818$	0	0
$819$	63.9918	2.23605
$820$	0	0
$821$	10.4190	0.363627	0.181813	−	0.983333i	$-0.441803\pi$
$821$	10.4190	0.363627	0.181813	+	0.983333i	$0.441803\pi$
$822$	0	0
$823$	−13.6657	−0.476355	−0.238177	−	0.971222i	$-0.576550\pi$
$823$	−13.6657	−0.476355	−0.238177	+	0.971222i	$0.576550\pi$
$824$	0	0
$825$	9.98880	0.347766
$826$	0	0
$827$	44.3157	1.54101	0.770504	−	0.637435i	$-0.220005\pi$
$827$	44.3157	1.54101	0.770504	+	0.637435i	$0.220005\pi$
$828$	0	0
$829$	−22.0998	−0.767559	−0.383780	−	0.923425i	$-0.625378\pi$
$829$	−22.0998	−0.767559	−0.383780	+	0.923425i	$0.625378\pi$
$830$	0	0
$831$	−34.2247	−1.18724
$832$	0	0
$833$	−6.93678	−0.240345
$834$	0	0
$835$	0.923595	0.0319623
$836$	0	0
$837$	11.4913	0.397196
$838$	0	0
$839$	23.8595	0.823722	0.411861	−	0.911247i	$-0.364879\pi$
$839$	23.8595	0.823722	0.411861	+	0.911247i	$0.364879\pi$
$840$	0	0
$841$	18.6975	0.644742
$842$	0	0
$843$	−66.0911	−2.27630
$844$	0	0
$845$	1.79643	0.0617992
$846$	0	0
$847$	23.1497	0.795434
$848$	0	0
$849$	65.6994	2.25480
$850$	0	0
$851$	11.7060	0.401275
$852$	0	0
$853$	−30.4220	−1.04163	−0.520815	−	0.853670i	$-0.674372\pi$
$853$	−30.4220	−1.04163	−0.520815	+	0.853670i	$0.674372\pi$
$854$	0	0
$855$	0.724266	0.0247694
$856$	0	0
$857$	4.61527	0.157654	0.0788272	−	0.996888i	$-0.474882\pi$
$857$	4.61527	0.157654	0.0788272	+	0.996888i	$0.474882\pi$
$858$	0	0
$859$	8.44274	0.288063	0.144031	−	0.989573i	$-0.453993\pi$
$859$	8.44274	0.288063	0.144031	+	0.989573i	$0.453993\pi$
$860$	0	0
$861$	−65.6570	−2.23758
$862$	0	0
$863$	14.4390	0.491509	0.245755	−	0.969332i	$-0.420964\pi$
$863$	14.4390	0.491509	0.245755	+	0.969332i	$0.420964\pi$
$864$	0	0
$865$	−1.36154	−0.0462936
$866$	0	0
$867$	−19.7833	−0.671875
$868$	0	0
$869$	−0.678129	−0.0230039
$870$	0	0
$871$	54.7037	1.85356
$872$	0	0
$873$	67.1793	2.27368
$874$	0	0
$875$	2.74330	0.0927405
$876$	0	0
$877$	24.3153	0.821069	0.410535	−	0.911845i	$-0.365342\pi$
$877$	24.3153	0.821069	0.410535	+	0.911845i	$0.365342\pi$
$878$	0	0
$879$	−72.6645	−2.45091
$880$	0	0
$881$	−27.7217	−0.933966	−0.466983	−	0.884266i	$-0.654659\pi$
$881$	−27.7217	−0.933966	−0.466983	+	0.884266i	$0.654659\pi$
$882$	0	0
$883$	22.8793	0.769948	0.384974	−	0.922927i	$-0.374210\pi$
$883$	22.8793	0.769948	0.384974	+	0.922927i	$0.374210\pi$
$884$	0	0
$885$	−1.39566	−0.0469147
$886$	0	0
$887$	−9.37976	−0.314942	−0.157471	−	0.987524i	$-0.550334\pi$
$887$	−9.37976	−0.314942	−0.157471	+	0.987524i	$0.550334\pi$
$888$	0	0
$889$	3.12510	0.104813
$890$	0	0
$891$	−3.60661	−0.120826
$892$	0	0
$893$	3.01277	0.100818
$894$	0	0
$895$	1.78267	0.0595882
$896$	0	0
$897$	97.1788	3.24471
$898$	0	0
$899$	−10.5766	−0.352749
$900$	0	0
$901$	−20.6611	−0.688320
$902$	0	0
$903$	−43.2871	−1.44051
$904$	0	0
$905$	2.16646	0.0720155
$906$	0	0
$907$	−19.4578	−0.646087	−0.323044	−	0.946384i	$-0.604706\pi$
$907$	−19.4578	−0.646087	−0.323044	+	0.946384i	$0.604706\pi$
$908$	0	0
$909$	−85.3791	−2.83185
$910$	0	0
$911$	25.2593	0.836878	0.418439	−	0.908245i	$-0.362577\pi$
$911$	25.2593	0.836878	0.418439	+	0.908245i	$0.362577\pi$
$912$	0	0
$913$	8.75320	0.289689
$914$	0	0
$915$	3.62969	0.119994
$916$	0	0
$917$	−19.7154	−0.651061
$918$	0	0
$919$	20.9774	0.691982	0.345991	−	0.938238i	$-0.387543\pi$
$919$	20.9774	0.691982	0.345991	+	0.938238i	$0.387543\pi$
$920$	0	0
$921$	−17.8120	−0.586926
$922$	0	0
$923$	−13.8099	−0.454559
$924$	0	0
$925$	9.19272	0.302255
$926$	0	0
$927$	−83.0291	−2.72703
$928$	0	0
$929$	60.5873	1.98780	0.993902	−	0.110263i	$-0.0351695\pi$
$929$	60.5873	1.98780	0.993902	+	0.110263i	$0.0351695\pi$
$930$	0	0
$931$	−2.25756	−0.0739884
$932$	0	0
$933$	67.6254	2.21395
$934$	0	0
$935$	−0.273897	−0.00895738
$936$	0	0
$937$	−27.5445	−0.899839	−0.449920	−	0.893069i	$-0.648547\pi$
$937$	−27.5445	−0.899839	−0.449920	+	0.893069i	$0.648547\pi$
$938$	0	0
$939$	39.9128	1.30250
$940$	0	0
$941$	1.17112	0.0381774	0.0190887	−	0.999818i	$-0.493924\pi$
$941$	1.17112	0.0381774	0.0190887	+	0.999818i	$0.493924\pi$
$942$	0	0
$943$	−64.7799	−2.10952
$944$	0	0
$945$	−2.06169	−0.0670667
$946$	0	0
$947$	12.8030	0.416040	0.208020	−	0.978125i	$-0.433298\pi$
$947$	12.8030	0.416040	0.208020	+	0.978125i	$0.433298\pi$
$948$	0	0
$949$	86.0835	2.79439
$950$	0	0
$951$	−9.43542	−0.305964
$952$	0	0
$953$	13.5203	0.437966	0.218983	−	0.975729i	$-0.429726\pi$
$953$	13.5203	0.437966	0.218983	+	0.975729i	$0.429726\pi$
$954$	0	0
$955$	0.738575	0.0238997
$956$	0	0
$957$	13.8405	0.447399
$958$	0	0
$959$	34.2267	1.10524
$960$	0	0
$961$	−28.6547	−0.924346
$962$	0	0
$963$	−99.5382	−3.20757
$964$	0	0
$965$	−2.27996	−0.0733944
$966$	0	0
$967$	62.1877	1.99982	0.999911	−	0.0133649i	$-0.00425429\pi$
$967$	62.1877	1.99982	0.999911	+	0.0133649i	$0.00425429\pi$
$968$	0	0
$969$	9.75244	0.313294
$970$	0	0
$971$	−19.5870	−0.628578	−0.314289	−	0.949327i	$-0.601766\pi$
$971$	−19.5870	−0.628578	−0.314289	+	0.949327i	$0.601766\pi$
$972$	0	0
$973$	−28.4443	−0.911884
$974$	0	0
$975$	76.3147	2.44403
$976$	0	0
$977$	−24.8114	−0.793787	−0.396893	−	0.917865i	$-0.629912\pi$
$977$	−24.8114	−0.793787	−0.396893	+	0.917865i	$0.629912\pi$
$978$	0	0
$979$	−0.894722	−0.0285954
$980$	0	0
$981$	−23.0019	−0.734395
$982$	0	0
$983$	−9.40956	−0.300118	−0.150059	−	0.988677i	$-0.547946\pi$
$983$	−9.40956	−0.300118	−0.150059	+	0.988677i	$0.547946\pi$
$984$	0	0
$985$	1.71913	0.0547761
$986$	0	0
$987$	−18.6103	−0.592372
$988$	0	0
$989$	−42.7089	−1.35806
$990$	0	0
$991$	16.3958	0.520829	0.260414	−	0.965497i	$-0.416141\pi$
$991$	16.3958	0.520829	0.260414	+	0.965497i	$0.416141\pi$
$992$	0	0
$993$	86.9594	2.75957
$994$	0	0
$995$	1.89086	0.0599444
$996$	0	0
$997$	−10.9983	−0.348320	−0.174160	−	0.984717i	$-0.555721\pi$
$997$	−10.9983	−0.348320	−0.174160	+	0.984717i	$0.555721\pi$
$998$	0	0
$999$	−13.8389	−0.437845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.27		28
4.3	odd	2		4024.2.a.d.1.2	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.2	✓	28	4.3	odd	2
8048.2.a.v.1.27		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)

\(f(q)\)	\(=\)	\(q+2.92644 q^{3} +0.124991 q^{5} -2.19824 q^{7} +5.56407 q^{9} +O(q^{10})\)	\(q+2.92644 q^{3} +0.124991 q^{5} -2.19824 q^{7} +5.56407 q^{9} -0.684798 q^{11} -5.23188 q^{13} +0.365779 q^{15} +3.19997 q^{17} +1.04142 q^{19} -6.43301 q^{21} -6.34708 q^{23} -4.98438 q^{25} +7.50361 q^{27} -6.90634 q^{29} +1.53143 q^{31} -2.00402 q^{33} -0.274759 q^{35} -1.84431 q^{37} -15.3108 q^{39} +10.2063 q^{41} +6.72891 q^{43} +0.695458 q^{45} +2.89294 q^{47} -2.16776 q^{49} +9.36454 q^{51} -6.45664 q^{53} -0.0855935 q^{55} +3.04766 q^{57} -3.81560 q^{59} +9.92318 q^{61} -12.2311 q^{63} -0.653937 q^{65} -10.4558 q^{67} -18.5744 q^{69} +2.63957 q^{71} -16.4537 q^{73} -14.5865 q^{75} +1.50535 q^{77} +0.990261 q^{79} +5.26667 q^{81} -12.7822 q^{83} +0.399967 q^{85} -20.2110 q^{87} +1.30655 q^{89} +11.5009 q^{91} +4.48164 q^{93} +0.130168 q^{95} +12.0738 q^{97} -3.81026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 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

Introduction

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