LMFDB - Embedded newform 5100.2.a.bd.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5100,2,Mod(1,5100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5100.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5100.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,0,0,-2,0,5,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.7237050309$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.4383968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 13x^{3} + 5x^{2} + 14x + 2 $ x^5 - x^4 - 13x^3 + 5x^2 + 14*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1020)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.805414$ of defining polynomial
Character	$\chi$	$=$	5100.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.94396 q^{7} +1.00000 q^{9} +2.22048 q^{11} +2.92695 q^{13} -1.00000 q^{17} +7.45305 q^{19} +2.94396 q^{21} +8.15951 q^{23} +1.00000 q^{27} +6.55478 q^{29} -7.49874 q^{31} +2.22048 q^{33} +2.74048 q^{37} +2.92695 q^{39} -6.88909 q^{41} -9.78125 q^{43} -8.16561 q^{47} +1.66687 q^{49} -1.00000 q^{51} +8.94396 q^{53} +7.45305 q^{57} +2.18569 q^{59} -14.7082 q^{61} +2.94396 q^{63} -2.18569 q^{67} +8.15951 q^{69} +5.85390 q^{71} -7.81604 q^{73} +6.53700 q^{77} -12.9975 q^{79} +1.00000 q^{81} -11.5383 q^{83} +6.55478 q^{87} +8.82029 q^{89} +8.61681 q^{91} -7.49874 q^{93} +3.01818 q^{97} +2.22048 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 2 q^{7} + 5 q^{9} + 6 q^{11} - 5 q^{17} + 12 q^{19} - 2 q^{21} + 6 q^{23} + 5 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{33} + 14 q^{41} - 4 q^{43} - 4 q^{47} + 15 q^{49} - 5 q^{51} + 28 q^{53}+ \cdots + 6 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 2 * q^7 + 5 * q^9 + 6 * q^11 - 5 * q^17 + 12 * q^19 - 2 * q^21 + 6 * q^23 + 5 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^33 + 14 * q^41 - 4 * q^43 - 4 * q^47 + 15 * q^49 - 5 * q^51 + 28 * q^53 + 12 * q^57 + 24 * q^59 - 14 * q^61 - 2 * q^63 - 24 * q^67 + 6 * q^69 + 24 * q^73 + 22 * q^77 + 22 * q^79 + 5 * q^81 + 2 * q^83 + 6 * q^87 + 18 * q^89 + 20 * q^91 + 6 * q^93 - 2 * q^97 + 6 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.94396	1.11271	0.556355	−	0.830945i	$-0.312199\pi$
$7$	2.94396	1.11271	0.556355	+	0.830945i	$0.312199\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.22048	0.669501	0.334750	−	0.942307i	$-0.391348\pi$
$11$	2.22048	0.669501	0.334750	+	0.942307i	$0.391348\pi$
$12$	0	0
$13$	2.92695	0.811790	0.405895	−	0.913920i	$-0.366960\pi$
$13$	2.92695	0.811790	0.405895	+	0.913920i	$0.366960\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	7.45305	1.70985	0.854923	−	0.518755i	$-0.173604\pi$
$19$	7.45305	1.70985	0.854923	+	0.518755i	$0.173604\pi$
$20$	0	0
$21$	2.94396	0.642424
$22$	0	0
$23$	8.15951	1.70138	0.850688	−	0.525671i	$-0.176186\pi$
$23$	8.15951	1.70138	0.850688	+	0.525671i	$0.176186\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.55478	1.21719	0.608596	−	0.793480i	$-0.291733\pi$
$29$	6.55478	1.21719	0.608596	+	0.793480i	$0.291733\pi$
$30$	0	0
$31$	−7.49874	−1.34681	−0.673407	−	0.739272i	$-0.735170\pi$
$31$	−7.49874	−1.34681	−0.673407	+	0.739272i	$0.735170\pi$
$32$	0	0
$33$	2.22048	0.386536
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.74048	0.450532	0.225266	−	0.974297i	$-0.427675\pi$
$37$	2.74048	0.450532	0.225266	+	0.974297i	$0.427675\pi$
$38$	0	0
$39$	2.92695	0.468687
$40$	0	0
$41$	−6.88909	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$41$	−6.88909	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$42$	0	0
$43$	−9.78125	−1.49163	−0.745813	−	0.666155i	$-0.767939\pi$
$43$	−9.78125	−1.49163	−0.745813	+	0.666155i	$0.767939\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.16561	−1.19108	−0.595539	−	0.803327i	$-0.703061\pi$
$47$	−8.16561	−1.19108	−0.595539	+	0.803327i	$0.703061\pi$
$48$	0	0
$49$	1.66687	0.238125
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	8.94396	1.22855	0.614273	−	0.789093i	$-0.289449\pi$
$53$	8.94396	1.22855	0.614273	+	0.789093i	$0.289449\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.45305	0.987180
$58$	0	0
$59$	2.18569	0.284553	0.142277	−	0.989827i	$-0.454558\pi$
$59$	2.18569	0.284553	0.142277	+	0.989827i	$0.454558\pi$
$60$	0	0
$61$	−14.7082	−1.88319	−0.941596	−	0.336745i	$-0.890674\pi$
$61$	−14.7082	−1.88319	−0.941596	+	0.336745i	$0.890674\pi$
$62$	0	0
$63$	2.94396	0.370904
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.18569	−0.267025	−0.133513	−	0.991047i	$-0.542626\pi$
$67$	−2.18569	−0.267025	−0.133513	+	0.991047i	$0.542626\pi$
$68$	0	0
$69$	8.15951	0.982290
$70$	0	0
$71$	5.85390	0.694730	0.347365	−	0.937730i	$-0.387076\pi$
$71$	5.85390	0.694730	0.347365	+	0.937730i	$0.387076\pi$
$72$	0	0
$73$	−7.81604	−0.914798	−0.457399	−	0.889262i	$-0.651219\pi$
$73$	−7.81604	−0.914798	−0.457399	+	0.889262i	$0.651219\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.53700	0.744960
$78$	0	0
$79$	−12.9975	−1.46233	−0.731165	−	0.682200i	$-0.761023\pi$
$79$	−12.9975	−1.46233	−0.731165	+	0.682200i	$0.761023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.5383	−1.26650	−0.633248	−	0.773949i	$-0.718279\pi$
$83$	−11.5383	−1.26650	−0.633248	+	0.773949i	$0.718279\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.55478	0.702747
$88$	0	0
$89$	8.82029	0.934948	0.467474	−	0.884007i	$-0.345164\pi$
$89$	8.82029	0.934948	0.467474	+	0.884007i	$0.345164\pi$
$90$	0	0
$91$	8.61681	0.903287
$92$	0	0
$93$	−7.49874	−0.777583
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.01818	0.306450	0.153225	−	0.988191i	$-0.451034\pi$
$97$	3.01818	0.306450	0.153225	+	0.988191i	$0.451034\pi$
$98$	0	0
$99$	2.22048	0.223167
$100$	0	0
$101$	6.09139	0.606116	0.303058	−	0.952972i	$-0.401992\pi$
$101$	6.09139	0.606116	0.303058	+	0.952972i	$0.401992\pi$
$102$	0	0
$103$	9.78125	0.963775	0.481887	−	0.876233i	$-0.339951\pi$
$103$	9.78125	0.963775	0.481887	+	0.876233i	$0.339951\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.78617	−0.269349	−0.134675	−	0.990890i	$-0.542999\pi$
$107$	−2.78617	−0.269349	−0.134675	+	0.990890i	$0.542999\pi$
$108$	0	0
$109$	12.9061	1.23618	0.618090	−	0.786108i	$-0.287907\pi$
$109$	12.9061	1.23618	0.618090	+	0.786108i	$0.287907\pi$
$110$	0	0
$111$	2.74048	0.260115
$112$	0	0
$113$	19.2691	1.81268	0.906341	−	0.422546i	$-0.138864\pi$
$113$	19.2691	1.81268	0.906341	+	0.422546i	$0.138864\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.92695	0.270597
$118$	0	0
$119$	−2.94396	−0.269872
$120$	0	0
$121$	−6.06946	−0.551769
$122$	0	0
$123$	−6.88909	−0.621168
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.0705	−1.42603	−0.713014	−	0.701149i	$-0.752671\pi$
$127$	−16.0705	−1.42603	−0.713014	+	0.701149i	$0.752671\pi$
$128$	0	0
$129$	−9.78125	−0.861191
$130$	0	0
$131$	15.5900	1.36210	0.681051	−	0.732236i	$-0.261523\pi$
$131$	15.5900	1.36210	0.681051	+	0.732236i	$0.261523\pi$
$132$	0	0
$133$	21.9414	1.90256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.4091	−0.974744	−0.487372	−	0.873195i	$-0.662044\pi$
$137$	−11.4091	−0.974744	−0.487372	+	0.873195i	$0.662044\pi$
$138$	0	0
$139$	12.2949	1.04284	0.521418	−	0.853301i	$-0.325403\pi$
$139$	12.2949	1.04284	0.521418	+	0.853301i	$0.325403\pi$
$140$	0	0
$141$	−8.16561	−0.687669
$142$	0	0
$143$	6.49924	0.543494
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.66687	0.137481
$148$	0	0
$149$	11.0918	0.908675	0.454337	−	0.890830i	$-0.349876\pi$
$149$	11.0918	0.908675	0.454337	+	0.890830i	$0.349876\pi$
$150$	0	0
$151$	14.7824	1.20298	0.601488	−	0.798882i	$-0.294575\pi$
$151$	14.7824	1.20298	0.601488	+	0.798882i	$0.294575\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.55361	−0.443226	−0.221613	−	0.975135i	$-0.571132\pi$
$157$	−5.55361	−0.443226	−0.221613	+	0.975135i	$0.571132\pi$
$158$	0	0
$159$	8.94396	0.709302
$160$	0	0
$161$	24.0212	1.89314
$162$	0	0
$163$	−16.4945	−1.29195	−0.645974	−	0.763359i	$-0.723549\pi$
$163$	−16.4945	−1.29195	−0.645974	+	0.763359i	$0.723549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.12590	−0.396654	−0.198327	−	0.980136i	$-0.563551\pi$
$167$	−5.12590	−0.396654	−0.198327	+	0.980136i	$0.563551\pi$
$168$	0	0
$169$	−4.43297	−0.340997
$170$	0	0
$171$	7.45305	0.569949
$172$	0	0
$173$	−9.75256	−0.741473	−0.370737	−	0.928738i	$-0.620895\pi$
$173$	−9.75256	−0.741473	−0.370737	+	0.928738i	$0.620895\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.18569	0.164287
$178$	0	0
$179$	−4.31903	−0.322819	−0.161410	−	0.986888i	$-0.551604\pi$
$179$	−4.31903	−0.322819	−0.161410	+	0.986888i	$0.551604\pi$
$180$	0	0
$181$	−0.219705	−0.0163306	−0.00816529	−	0.999967i	$-0.502599\pi$
$181$	−0.219705	−0.0163306	−0.00816529	+	0.999967i	$0.502599\pi$
$182$	0	0
$183$	−14.7082	−1.08726
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.22048	−0.162378
$188$	0	0
$189$	2.94396	0.214141
$190$	0	0
$191$	−22.2927	−1.61305	−0.806523	−	0.591203i	$-0.798653\pi$
$191$	−22.2927	−1.61305	−0.806523	+	0.591203i	$0.798653\pi$
$192$	0	0
$193$	−24.5225	−1.76517	−0.882584	−	0.470154i	$-0.844198\pi$
$193$	−24.5225	−1.76517	−0.882584	+	0.470154i	$0.844198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.6923	−0.833039	−0.416520	−	0.909127i	$-0.636750\pi$
$197$	−11.6923	−0.833039	−0.416520	+	0.909127i	$0.636750\pi$
$198$	0	0
$199$	−2.79400	−0.198062	−0.0990308	−	0.995084i	$-0.531574\pi$
$199$	−2.79400	−0.198062	−0.0990308	+	0.995084i	$0.531574\pi$
$200$	0	0
$201$	−2.18569	−0.154167
$202$	0	0
$203$	19.2970	1.35438
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.15951	0.567125
$208$	0	0
$209$	16.5494	1.14474
$210$	0	0
$211$	2.75653	0.189767	0.0948837	−	0.995488i	$-0.469752\pi$
$211$	2.75653	0.189767	0.0948837	+	0.995488i	$0.469752\pi$
$212$	0	0
$213$	5.85390	0.401103
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−22.0760	−1.49861
$218$	0	0
$219$	−7.81604	−0.528159
$220$	0	0
$221$	−2.92695	−0.196888
$222$	0	0
$223$	3.76929	0.252410	0.126205	−	0.992004i	$-0.459720\pi$
$223$	3.76929	0.252410	0.126205	+	0.992004i	$0.459720\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.4726	1.29244	0.646220	−	0.763152i	$-0.276349\pi$
$227$	19.4726	1.29244	0.646220	+	0.763152i	$0.276349\pi$
$228$	0	0
$229$	7.62799	0.504072	0.252036	−	0.967718i	$-0.418900\pi$
$229$	7.62799	0.504072	0.252036	+	0.967718i	$0.418900\pi$
$230$	0	0
$231$	6.53700	0.430103
$232$	0	0
$233$	−9.78812	−0.641241	−0.320621	−	0.947208i	$-0.603891\pi$
$233$	−9.78812	−0.641241	−0.320621	+	0.947208i	$0.603891\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.9975	−0.844277
$238$	0	0
$239$	−1.78029	−0.115158	−0.0575788	−	0.998341i	$-0.518338\pi$
$239$	−1.78029	−0.115158	−0.0575788	+	0.998341i	$0.518338\pi$
$240$	0	0
$241$	12.5961	0.811387	0.405693	−	0.914009i	$-0.367030\pi$
$241$	12.5961	0.811387	0.405693	+	0.914009i	$0.367030\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.8147	1.38804
$248$	0	0
$249$	−11.5383	−0.731212
$250$	0	0
$251$	21.4987	1.35699	0.678494	−	0.734606i	$-0.262633\pi$
$251$	21.4987	1.35699	0.678494	+	0.734606i	$0.262633\pi$
$252$	0	0
$253$	18.1181	1.13907
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.5505	1.71856	0.859278	−	0.511510i	$-0.170914\pi$
$257$	27.5505	1.71856	0.859278	+	0.511510i	$0.170914\pi$
$258$	0	0
$259$	8.06785	0.501312
$260$	0	0
$261$	6.55478	0.405731
$262$	0	0
$263$	−10.1415	−0.625349	−0.312674	−	0.949860i	$-0.601225\pi$
$263$	−10.1415	−0.625349	−0.312674	+	0.949860i	$0.601225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.82029	0.539793
$268$	0	0
$269$	−10.0303	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$269$	−10.0303	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$270$	0	0
$271$	22.4148	1.36160	0.680801	−	0.732469i	$-0.261632\pi$
$271$	22.4148	1.36160	0.680801	+	0.732469i	$0.261632\pi$
$272$	0	0
$273$	8.61681	0.521513
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.9975	0.780943	0.390471	−	0.920615i	$-0.372312\pi$
$277$	12.9975	0.780943	0.390471	+	0.920615i	$0.372312\pi$
$278$	0	0
$279$	−7.49874	−0.448938
$280$	0	0
$281$	−22.8514	−1.36320	−0.681599	−	0.731725i	$-0.738715\pi$
$281$	−22.8514	−1.36320	−0.681599	+	0.731725i	$0.738715\pi$
$282$	0	0
$283$	−13.7820	−0.819256	−0.409628	−	0.912253i	$-0.634342\pi$
$283$	−13.7820	−0.819256	−0.409628	+	0.912253i	$0.634342\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.2812	−1.19716
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.01818	0.176929
$292$	0	0
$293$	1.03188	0.0602832	0.0301416	−	0.999546i	$-0.490404\pi$
$293$	1.03188	0.0602832	0.0301416	+	0.999546i	$0.490404\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.22048	0.128845
$298$	0	0
$299$	23.8825	1.38116
$300$	0	0
$301$	−28.7956	−1.65975
$302$	0	0
$303$	6.09139	0.349941
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.0756	−1.20285	−0.601423	−	0.798931i	$-0.705399\pi$
$307$	−21.0756	−1.20285	−0.601423	+	0.798931i	$0.705399\pi$
$308$	0	0
$309$	9.78125	0.556436
$310$	0	0
$311$	25.1113	1.42393	0.711966	−	0.702214i	$-0.247805\pi$
$311$	25.1113	1.42393	0.711966	+	0.702214i	$0.247805\pi$
$312$	0	0
$313$	−15.2728	−0.863271	−0.431636	−	0.902048i	$-0.642063\pi$
$313$	−15.2728	−0.863271	−0.431636	+	0.902048i	$0.642063\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.29291	0.409611	0.204805	−	0.978803i	$-0.434344\pi$
$317$	7.29291	0.409611	0.204805	+	0.978803i	$0.434344\pi$
$318$	0	0
$319$	14.5548	0.814911
$320$	0	0
$321$	−2.78617	−0.155509
$322$	0	0
$323$	−7.45305	−0.414699
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.9061	0.713709
$328$	0	0
$329$	−24.0392	−1.32532
$330$	0	0
$331$	24.5504	1.34941	0.674706	−	0.738086i	$-0.264270\pi$
$331$	24.5504	1.34941	0.674706	+	0.738086i	$0.264270\pi$
$332$	0	0
$333$	2.74048	0.150177
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.35689	0.346282	0.173141	−	0.984897i	$-0.444608\pi$
$337$	6.35689	0.346282	0.173141	+	0.984897i	$0.444608\pi$
$338$	0	0
$339$	19.2691	1.04655
$340$	0	0
$341$	−16.6508	−0.901693
$342$	0	0
$343$	−15.7005	−0.847747
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.0371	−0.807232	−0.403616	−	0.914928i	$-0.632247\pi$
$347$	−15.0371	−0.807232	−0.403616	+	0.914928i	$0.632247\pi$
$348$	0	0
$349$	34.9730	1.87206	0.936032	−	0.351916i	$-0.114470\pi$
$349$	34.9730	1.87206	0.936032	+	0.351916i	$0.114470\pi$
$350$	0	0
$351$	2.92695	0.156229
$352$	0	0
$353$	13.7039	0.729387	0.364694	−	0.931128i	$-0.381174\pi$
$353$	13.7039	0.729387	0.364694	+	0.931128i	$0.381174\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.94396	−0.155811
$358$	0	0
$359$	−13.1274	−0.692835	−0.346418	−	0.938080i	$-0.612602\pi$
$359$	−13.1274	−0.692835	−0.346418	+	0.938080i	$0.612602\pi$
$360$	0	0
$361$	36.5479	1.92357
$362$	0	0
$363$	−6.06946	−0.318564
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.7635	1.08384	0.541922	−	0.840429i	$-0.317697\pi$
$367$	20.7635	1.08384	0.541922	+	0.840429i	$0.317697\pi$
$368$	0	0
$369$	−6.88909	−0.358632
$370$	0	0
$371$	26.3306	1.36702
$372$	0	0
$373$	−17.8879	−0.926201	−0.463100	−	0.886306i	$-0.653263\pi$
$373$	−17.8879	−0.926201	−0.463100	+	0.886306i	$0.653263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.1855	0.988105
$378$	0	0
$379$	6.52015	0.334918	0.167459	−	0.985879i	$-0.446444\pi$
$379$	6.52015	0.334918	0.167459	+	0.985879i	$0.446444\pi$
$380$	0	0
$381$	−16.0705	−0.823318
$382$	0	0
$383$	−8.43497	−0.431007	−0.215503	−	0.976503i	$-0.569139\pi$
$383$	−8.43497	−0.431007	−0.215503	+	0.976503i	$0.569139\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.78125	−0.497209
$388$	0	0
$389$	−20.9635	−1.06289	−0.531445	−	0.847093i	$-0.678351\pi$
$389$	−20.9635	−1.06289	−0.531445	+	0.847093i	$0.678351\pi$
$390$	0	0
$391$	−8.15951	−0.412644
$392$	0	0
$393$	15.5900	0.786410
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−24.6066	−1.23497	−0.617484	−	0.786583i	$-0.711848\pi$
$397$	−24.6066	−1.23497	−0.617484	+	0.786583i	$0.711848\pi$
$398$	0	0
$399$	21.9414	1.09845
$400$	0	0
$401$	−16.5153	−0.824737	−0.412369	−	0.911017i	$-0.635298\pi$
$401$	−16.5153	−0.824737	−0.412369	+	0.911017i	$0.635298\pi$
$402$	0	0
$403$	−21.9484	−1.09333
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.08518	0.301631
$408$	0	0
$409$	10.9340	0.540652	0.270326	−	0.962769i	$-0.412869\pi$
$409$	10.9340	0.540652	0.270326	+	0.962769i	$0.412869\pi$
$410$	0	0
$411$	−11.4091	−0.562769
$412$	0	0
$413$	6.43459	0.316625
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.2949	0.602082
$418$	0	0
$419$	21.5631	1.05343	0.526714	−	0.850043i	$-0.323424\pi$
$419$	21.5631	1.05343	0.526714	+	0.850043i	$0.323424\pi$
$420$	0	0
$421$	−12.5868	−0.613442	−0.306721	−	0.951799i	$-0.599232\pi$
$421$	−12.5868	−0.613442	−0.306721	+	0.951799i	$0.599232\pi$
$422$	0	0
$423$	−8.16561	−0.397026
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−43.3003	−2.09545
$428$	0	0
$429$	6.49924	0.313786
$430$	0	0
$431$	−9.88964	−0.476367	−0.238184	−	0.971220i	$-0.576552\pi$
$431$	−9.88964	−0.476367	−0.238184	+	0.971220i	$0.576552\pi$
$432$	0	0
$433$	11.1185	0.534319	0.267159	−	0.963652i	$-0.413915\pi$
$433$	11.1185	0.534319	0.267159	+	0.963652i	$0.413915\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	60.8132	2.90909
$438$	0	0
$439$	−7.87168	−0.375695	−0.187847	−	0.982198i	$-0.560151\pi$
$439$	−7.87168	−0.375695	−0.187847	+	0.982198i	$0.560151\pi$
$440$	0	0
$441$	1.66687	0.0793749
$442$	0	0
$443$	24.7964	1.17811	0.589055	−	0.808093i	$-0.299500\pi$
$443$	24.7964	1.17811	0.589055	+	0.808093i	$0.299500\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.0918	0.524623
$448$	0	0
$449$	0.451817	0.0213226	0.0106613	−	0.999943i	$-0.496606\pi$
$449$	0.451817	0.0213226	0.0106613	+	0.999943i	$0.496606\pi$
$450$	0	0
$451$	−15.2971	−0.720312
$452$	0	0
$453$	14.7824	0.694539
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.2642	−1.50925	−0.754627	−	0.656154i	$-0.772182\pi$
$457$	−32.2642	−1.50925	−0.754627	+	0.656154i	$0.772182\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−6.01778	−0.280276	−0.140138	−	0.990132i	$-0.544755\pi$
$461$	−6.01778	−0.280276	−0.140138	+	0.990132i	$0.544755\pi$
$462$	0	0
$463$	−7.48331	−0.347779	−0.173889	−	0.984765i	$-0.555634\pi$
$463$	−7.48331	−0.347779	−0.173889	+	0.984765i	$0.555634\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.18742	0.378869	0.189434	−	0.981893i	$-0.439335\pi$
$467$	8.18742	0.378869	0.189434	+	0.981893i	$0.439335\pi$
$468$	0	0
$469$	−6.43459	−0.297122
$470$	0	0
$471$	−5.55361	−0.255897
$472$	0	0
$473$	−21.7191	−0.998645
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.94396	0.409516
$478$	0	0
$479$	37.1891	1.69922	0.849608	−	0.527415i	$-0.176839\pi$
$479$	37.1891	1.69922	0.849608	+	0.527415i	$0.176839\pi$
$480$	0	0
$481$	8.02124	0.365737
$482$	0	0
$483$	24.0212	1.09300
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−37.6780	−1.70736	−0.853678	−	0.520802i	$-0.825633\pi$
$487$	−37.6780	−1.70736	−0.853678	+	0.520802i	$0.825633\pi$
$488$	0	0
$489$	−16.4945	−0.745907
$490$	0	0
$491$	−37.9828	−1.71414	−0.857069	−	0.515202i	$-0.827717\pi$
$491$	−37.9828	−1.71414	−0.857069	+	0.515202i	$0.827717\pi$
$492$	0	0
$493$	−6.55478	−0.295213
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.2336	0.773033
$498$	0	0
$499$	−6.63997	−0.297246	−0.148623	−	0.988894i	$-0.547484\pi$
$499$	−6.63997	−0.297246	−0.148623	+	0.988894i	$0.547484\pi$
$500$	0	0
$501$	−5.12590	−0.229008
$502$	0	0
$503$	−14.9224	−0.665358	−0.332679	−	0.943040i	$-0.607953\pi$
$503$	−14.9224	−0.665358	−0.332679	+	0.943040i	$0.607953\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.43297	−0.196875
$508$	0	0
$509$	5.68041	0.251780	0.125890	−	0.992044i	$-0.459821\pi$
$509$	5.68041	0.251780	0.125890	+	0.992044i	$0.459821\pi$
$510$	0	0
$511$	−23.0101	−1.01791
$512$	0	0
$513$	7.45305	0.329060
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−18.1316	−0.797427
$518$	0	0
$519$	−9.75256	−0.428090
$520$	0	0
$521$	−10.0303	−0.439436	−0.219718	−	0.975563i	$-0.570514\pi$
$521$	−10.0303	−0.439436	−0.219718	+	0.975563i	$0.570514\pi$
$522$	0	0
$523$	35.1154	1.53549	0.767744	−	0.640756i	$-0.221379\pi$
$523$	35.1154	1.53549	0.767744	+	0.640756i	$0.221379\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.49874	0.326650
$528$	0	0
$529$	43.5777	1.89468
$530$	0	0
$531$	2.18569	0.0948510
$532$	0	0
$533$	−20.1640	−0.873400
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.31903	−0.186380
$538$	0	0
$539$	3.70126	0.159425
$540$	0	0
$541$	−6.77276	−0.291184	−0.145592	−	0.989345i	$-0.546509\pi$
$541$	−6.77276	−0.291184	−0.145592	+	0.989345i	$0.546509\pi$
$542$	0	0
$543$	−0.219705	−0.00942847
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.0154	−1.66818	−0.834090	−	0.551629i	$-0.814007\pi$
$547$	−39.0154	−1.66818	−0.834090	+	0.551629i	$0.814007\pi$
$548$	0	0
$549$	−14.7082	−0.627731
$550$	0	0
$551$	48.8531	2.08121
$552$	0	0
$553$	−38.2640	−1.62715
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.89389	−0.207361	−0.103680	−	0.994611i	$-0.533062\pi$
$557$	−4.89389	−0.207361	−0.103680	+	0.994611i	$0.533062\pi$
$558$	0	0
$559$	−28.6292	−1.21089
$560$	0	0
$561$	−2.22048	−0.0937488
$562$	0	0
$563$	29.2863	1.23427	0.617136	−	0.786856i	$-0.288293\pi$
$563$	29.2863	1.23427	0.617136	+	0.786856i	$0.288293\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.94396	0.123635
$568$	0	0
$569$	27.7551	1.16356	0.581778	−	0.813348i	$-0.302357\pi$
$569$	27.7551	1.16356	0.581778	+	0.813348i	$0.302357\pi$
$570$	0	0
$571$	32.9491	1.37888	0.689439	−	0.724343i	$-0.257857\pi$
$571$	32.9491	1.37888	0.689439	+	0.724343i	$0.257857\pi$
$572$	0	0
$573$	−22.2927	−0.931293
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.3658	−1.30578	−0.652888	−	0.757455i	$-0.726443\pi$
$577$	−31.3658	−1.30578	−0.652888	+	0.757455i	$0.726443\pi$
$578$	0	0
$579$	−24.5225	−1.01912
$580$	0	0
$581$	−33.9683	−1.40924
$582$	0	0
$583$	19.8599	0.822513
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.7149	−1.39156	−0.695782	−	0.718253i	$-0.744942\pi$
$587$	−33.7149	−1.39156	−0.695782	+	0.718253i	$0.744942\pi$
$588$	0	0
$589$	−55.8885	−2.30284
$590$	0	0
$591$	−11.6923	−0.480955
$592$	0	0
$593$	−5.28289	−0.216942	−0.108471	−	0.994100i	$-0.534595\pi$
$593$	−5.28289	−0.216942	−0.108471	+	0.994100i	$0.534595\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.79400	−0.114351
$598$	0	0
$599$	−11.8098	−0.482537	−0.241268	−	0.970458i	$-0.577563\pi$
$599$	−11.8098	−0.482537	−0.241268	+	0.970458i	$0.577563\pi$
$600$	0	0
$601$	−7.80211	−0.318255	−0.159127	−	0.987258i	$-0.550868\pi$
$601$	−7.80211	−0.318255	−0.159127	+	0.987258i	$0.550868\pi$
$602$	0	0
$603$	−2.18569	−0.0890083
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.19576	0.210889	0.105445	−	0.994425i	$-0.466373\pi$
$607$	5.19576	0.210889	0.105445	+	0.994425i	$0.466373\pi$
$608$	0	0
$609$	19.2970	0.781954
$610$	0	0
$611$	−23.9003	−0.966904
$612$	0	0
$613$	9.52430	0.384683	0.192341	−	0.981328i	$-0.438392\pi$
$613$	9.52430	0.384683	0.192341	+	0.981328i	$0.438392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.5985	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$617$	34.5985	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$618$	0	0
$619$	−12.5225	−0.503322	−0.251661	−	0.967815i	$-0.580977\pi$
$619$	−12.5225	−0.503322	−0.251661	+	0.967815i	$0.580977\pi$
$620$	0	0
$621$	8.15951	0.327430
$622$	0	0
$623$	25.9665	1.04033
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.5494	0.660918
$628$	0	0
$629$	−2.74048	−0.109270
$630$	0	0
$631$	14.2986	0.569219	0.284609	−	0.958644i	$-0.408136\pi$
$631$	14.2986	0.569219	0.284609	+	0.958644i	$0.408136\pi$
$632$	0	0
$633$	2.75653	0.109562
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.87886	0.193307
$638$	0	0
$639$	5.85390	0.231577
$640$	0	0
$641$	6.33683	0.250290	0.125145	−	0.992138i	$-0.460060\pi$
$641$	6.33683	0.250290	0.125145	+	0.992138i	$0.460060\pi$
$642$	0	0
$643$	13.5652	0.534961	0.267480	−	0.963563i	$-0.413809\pi$
$643$	13.5652	0.534961	0.267480	+	0.963563i	$0.413809\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.03764	0.0407939	0.0203969	−	0.999792i	$-0.493507\pi$
$647$	1.03764	0.0407939	0.0203969	+	0.999792i	$0.493507\pi$
$648$	0	0
$649$	4.85330	0.190509
$650$	0	0
$651$	−22.0760	−0.865225
$652$	0	0
$653$	18.0163	0.705034	0.352517	−	0.935805i	$-0.385326\pi$
$653$	18.0163	0.705034	0.352517	+	0.935805i	$0.385326\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.81604	−0.304933
$658$	0	0
$659$	12.8514	0.500619	0.250309	−	0.968166i	$-0.419468\pi$
$659$	12.8514	0.500619	0.250309	+	0.968166i	$0.419468\pi$
$660$	0	0
$661$	−33.5247	−1.30396	−0.651981	−	0.758236i	$-0.726062\pi$
$661$	−33.5247	−1.30396	−0.651981	+	0.758236i	$0.726062\pi$
$662$	0	0
$663$	−2.92695	−0.113673
$664$	0	0
$665$	0	0
$666$	0	0
$667$	53.4838	2.07090
$668$	0	0
$669$	3.76929	0.145729
$670$	0	0
$671$	−32.6593	−1.26080
$672$	0	0
$673$	43.1130	1.66189	0.830943	−	0.556358i	$-0.187802\pi$
$673$	43.1130	1.66189	0.830943	+	0.556358i	$0.187802\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.254132	−0.00976710	−0.00488355	−	0.999988i	$-0.501554\pi$
$677$	−0.254132	−0.00976710	−0.00488355	+	0.999988i	$0.501554\pi$
$678$	0	0
$679$	8.88539	0.340990
$680$	0	0
$681$	19.4726	0.746190
$682$	0	0
$683$	−31.2210	−1.19464	−0.597319	−	0.802004i	$-0.703767\pi$
$683$	−31.2210	−1.19464	−0.597319	+	0.802004i	$0.703767\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.62799	0.291026
$688$	0	0
$689$	26.1785	0.997322
$690$	0	0
$691$	−1.31305	−0.0499506	−0.0249753	−	0.999688i	$-0.507951\pi$
$691$	−1.31305	−0.0499506	−0.0249753	+	0.999688i	$0.507951\pi$
$692$	0	0
$693$	6.53700	0.248320
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.88909	0.260943
$698$	0	0
$699$	−9.78812	−0.370221
$700$	0	0
$701$	−39.0578	−1.47519	−0.737596	−	0.675242i	$-0.764039\pi$
$701$	−39.0578	−1.47519	−0.737596	+	0.675242i	$0.764039\pi$
$702$	0	0
$703$	20.4249	0.770340
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.9328	0.674431
$708$	0	0
$709$	3.72235	0.139796	0.0698979	−	0.997554i	$-0.477733\pi$
$709$	3.72235	0.139796	0.0698979	+	0.997554i	$0.477733\pi$
$710$	0	0
$711$	−12.9975	−0.487444
$712$	0	0
$713$	−61.1861	−2.29144
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.78029	−0.0664863
$718$	0	0
$719$	43.2995	1.61480	0.807399	−	0.590005i	$-0.200874\pi$
$719$	43.2995	1.61480	0.807399	+	0.590005i	$0.200874\pi$
$720$	0	0
$721$	28.7956	1.07240
$722$	0	0
$723$	12.5961	0.468454
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.62706	0.134520	0.0672601	−	0.997735i	$-0.478574\pi$
$727$	3.62706	0.134520	0.0672601	+	0.997735i	$0.478574\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.78125	0.361773
$732$	0	0
$733$	−24.3743	−0.900285	−0.450142	−	0.892957i	$-0.648627\pi$
$733$	−24.3743	−0.900285	−0.450142	+	0.892957i	$0.648627\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.85330	−0.178773
$738$	0	0
$739$	3.76879	0.138637	0.0693186	−	0.997595i	$-0.477918\pi$
$739$	3.76879	0.138637	0.0693186	+	0.997595i	$0.477918\pi$
$740$	0	0
$741$	21.8147	0.801383
$742$	0	0
$743$	−45.5234	−1.67009	−0.835047	−	0.550179i	$-0.814559\pi$
$743$	−45.5234	−1.67009	−0.835047	+	0.550179i	$0.814559\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.5383	−0.422166
$748$	0	0
$749$	−8.20237	−0.299708
$750$	0	0
$751$	−5.23474	−0.191018	−0.0955092	−	0.995429i	$-0.530448\pi$
$751$	−5.23474	−0.191018	−0.0955092	+	0.995429i	$0.530448\pi$
$752$	0	0
$753$	21.4987	0.783458
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.7836	−1.44596	−0.722980	−	0.690869i	$-0.757228\pi$
$757$	−39.7836	−1.44596	−0.722980	+	0.690869i	$0.757228\pi$
$758$	0	0
$759$	18.1181	0.657644
$760$	0	0
$761$	27.4012	0.993293	0.496646	−	0.867953i	$-0.334565\pi$
$761$	27.4012	0.993293	0.496646	+	0.867953i	$0.334565\pi$
$762$	0	0
$763$	37.9950	1.37551
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.39742	0.230997
$768$	0	0
$769$	−28.7265	−1.03590	−0.517952	−	0.855410i	$-0.673305\pi$
$769$	−28.7265	−1.03590	−0.517952	+	0.855410i	$0.673305\pi$
$770$	0	0
$771$	27.5505	0.992208
$772$	0	0
$773$	9.08155	0.326641	0.163320	−	0.986573i	$-0.447780\pi$
$773$	9.08155	0.326641	0.163320	+	0.986573i	$0.447780\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.06785	0.289432
$778$	0	0
$779$	−51.3447	−1.83961
$780$	0	0
$781$	12.9985	0.465122
$782$	0	0
$783$	6.55478	0.234249
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.6783	0.950978	0.475489	−	0.879722i	$-0.342271\pi$
$787$	26.6783	0.950978	0.475489	+	0.879722i	$0.342271\pi$
$788$	0	0
$789$	−10.1415	−0.361045
$790$	0	0
$791$	56.7273	2.01699
$792$	0	0
$793$	−43.0502	−1.52876
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.8270	−0.560621	−0.280310	−	0.959909i	$-0.590437\pi$
$797$	−15.8270	−0.560621	−0.280310	+	0.959909i	$0.590437\pi$
$798$	0	0
$799$	8.16561	0.288879
$800$	0	0
$801$	8.82029	0.311649
$802$	0	0
$803$	−17.3554	−0.612458
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.0303	−0.353084
$808$	0	0
$809$	−22.0429	−0.774987	−0.387494	−	0.921872i	$-0.626659\pi$
$809$	−22.0429	−0.774987	−0.387494	+	0.921872i	$0.626659\pi$
$810$	0	0
$811$	−41.3144	−1.45074	−0.725372	−	0.688357i	$-0.758332\pi$
$811$	−41.3144	−1.45074	−0.725372	+	0.688357i	$0.758332\pi$
$812$	0	0
$813$	22.4148	0.786121
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−72.9001	−2.55045
$818$	0	0
$819$	8.61681	0.301096
$820$	0	0
$821$	−0.00462923	−0.000161561 0	−8.07806e−5	−	1.00000i	$-0.500026\pi$
$821$	−0.00462923	−0.000161561 0	−8.07806e−5		1.00000i	$0.500026\pi$
$822$	0	0
$823$	35.0154	1.22056	0.610281	−	0.792185i	$-0.291057\pi$
$823$	35.0154	1.22056	0.610281	+	0.792185i	$0.291057\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.1206	0.734436	0.367218	−	0.930135i	$-0.380310\pi$
$827$	21.1206	0.734436	0.367218	+	0.930135i	$0.380310\pi$
$828$	0	0
$829$	−22.3747	−0.777104	−0.388552	−	0.921427i	$-0.627025\pi$
$829$	−22.3747	−0.777104	−0.388552	+	0.921427i	$0.627025\pi$
$830$	0	0
$831$	12.9975	0.450878
$832$	0	0
$833$	−1.66687	−0.0577537
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.49874	−0.259194
$838$	0	0
$839$	11.2265	0.387581	0.193790	−	0.981043i	$-0.437922\pi$
$839$	11.2265	0.387581	0.193790	+	0.981043i	$0.437922\pi$
$840$	0	0
$841$	13.9652	0.481559
$842$	0	0
$843$	−22.8514	−0.787043
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.8682	−0.613959
$848$	0	0
$849$	−13.7820	−0.472998
$850$	0	0
$851$	22.3610	0.766524
$852$	0	0
$853$	−27.9221	−0.956035	−0.478018	−	0.878350i	$-0.658645\pi$
$853$	−27.9221	−0.956035	−0.478018	+	0.878350i	$0.658645\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.5855	−1.31806	−0.659028	−	0.752119i	$-0.729032\pi$
$857$	−38.5855	−1.31806	−0.659028	+	0.752119i	$0.729032\pi$
$858$	0	0
$859$	−30.1869	−1.02996	−0.514981	−	0.857202i	$-0.672201\pi$
$859$	−30.1869	−1.02996	−0.514981	+	0.857202i	$0.672201\pi$
$860$	0	0
$861$	−20.2812	−0.691180
$862$	0	0
$863$	−2.10589	−0.0716852	−0.0358426	−	0.999357i	$-0.511411\pi$
$863$	−2.10589	−0.0716852	−0.0358426	+	0.999357i	$0.511411\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−28.8607	−0.979031
$870$	0	0
$871$	−6.39742	−0.216768
$872$	0	0
$873$	3.01818	0.102150
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.0171621	−0.000579523 0	−0.000289762	−	1.00000i	$-0.500092\pi$
$877$	−0.0171621	−0.000579523 0	−0.000289762		1.00000i	$0.500092\pi$
$878$	0	0
$879$	1.03188	0.0348045
$880$	0	0
$881$	−17.6280	−0.593902	−0.296951	−	0.954893i	$-0.595970\pi$
$881$	−17.6280	−0.593902	−0.296951	+	0.954893i	$0.595970\pi$
$882$	0	0
$883$	35.7475	1.20300	0.601499	−	0.798874i	$-0.294570\pi$
$883$	35.7475	1.20300	0.601499	+	0.798874i	$0.294570\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.3002	−1.05096	−0.525479	−	0.850807i	$-0.676114\pi$
$887$	−31.3002	−1.05096	−0.525479	+	0.850807i	$0.676114\pi$
$888$	0	0
$889$	−47.3109	−1.58676
$890$	0	0
$891$	2.22048	0.0743890
$892$	0	0
$893$	−60.8587	−2.03656
$894$	0	0
$895$	0	0
$896$	0	0
$897$	23.8825	0.797413
$898$	0	0
$899$	−49.1526	−1.63933
$900$	0	0
$901$	−8.94396	−0.297966
$902$	0	0
$903$	−28.7956	−0.958256
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.6563	1.28356	0.641780	−	0.766889i	$-0.278196\pi$
$907$	38.6563	1.28356	0.641780	+	0.766889i	$0.278196\pi$
$908$	0	0
$909$	6.09139	0.202039
$910$	0	0
$911$	−7.51866	−0.249104	−0.124552	−	0.992213i	$-0.539749\pi$
$911$	−7.51866	−0.249104	−0.124552	+	0.992213i	$0.539749\pi$
$912$	0	0
$913$	−25.6207	−0.847920
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45.8962	1.51563
$918$	0	0
$919$	8.24705	0.272045	0.136023	−	0.990706i	$-0.456568\pi$
$919$	8.24705	0.272045	0.136023	+	0.990706i	$0.456568\pi$
$920$	0	0
$921$	−21.0756	−0.694463
$922$	0	0
$923$	17.1341	0.563975
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.78125	0.321258
$928$	0	0
$929$	−38.5663	−1.26532	−0.632660	−	0.774430i	$-0.718037\pi$
$929$	−38.5663	−1.26532	−0.632660	+	0.774430i	$0.718037\pi$
$930$	0	0
$931$	12.4233	0.407157
$932$	0	0
$933$	25.1113	0.822107
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.8518	0.713866	0.356933	−	0.934130i	$-0.383822\pi$
$937$	21.8518	0.713866	0.356933	+	0.934130i	$0.383822\pi$
$938$	0	0
$939$	−15.2728	−0.498410
$940$	0	0
$941$	−8.65899	−0.282275	−0.141138	−	0.989990i	$-0.545076\pi$
$941$	−8.65899	−0.282275	−0.141138	+	0.989990i	$0.545076\pi$
$942$	0	0
$943$	−56.2116	−1.83050
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.28370	0.204193	0.102096	−	0.994775i	$-0.467445\pi$
$947$	6.28370	0.204193	0.102096	+	0.994775i	$0.467445\pi$
$948$	0	0
$949$	−22.8771	−0.742624
$950$	0	0
$951$	7.29291	0.236489
$952$	0	0
$953$	44.7341	1.44908	0.724540	−	0.689233i	$-0.242052\pi$
$953$	44.7341	1.44908	0.724540	+	0.689233i	$0.242052\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.5548	0.470489
$958$	0	0
$959$	−33.5878	−1.08461
$960$	0	0
$961$	25.2311	0.813906
$962$	0	0
$963$	−2.78617	−0.0897832
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.14972	0.133446	0.0667230	−	0.997772i	$-0.478746\pi$
$967$	4.14972	0.133446	0.0667230	+	0.997772i	$0.478746\pi$
$968$	0	0
$969$	−7.45305	−0.239426
$970$	0	0
$971$	12.6864	0.407126	0.203563	−	0.979062i	$-0.434748\pi$
$971$	12.6864	0.407126	0.203563	+	0.979062i	$0.434748\pi$
$972$	0	0
$973$	36.1955	1.16038
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.2144	0.646717	0.323359	−	0.946277i	$-0.395188\pi$
$977$	20.2144	0.646717	0.323359	+	0.946277i	$0.395188\pi$
$978$	0	0
$979$	19.5853	0.625949
$980$	0	0
$981$	12.9061	0.412060
$982$	0	0
$983$	39.9959	1.27567	0.637835	−	0.770173i	$-0.279830\pi$
$983$	39.9959	1.27567	0.637835	+	0.770173i	$0.279830\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−24.0392	−0.765176
$988$	0	0
$989$	−79.8102	−2.53782
$990$	0	0
$991$	21.7724	0.691622	0.345811	−	0.938304i	$-0.387604\pi$
$991$	21.7724	0.691622	0.345811	+	0.938304i	$0.387604\pi$
$992$	0	0
$993$	24.5504	0.779084
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.4553	−1.24956	−0.624781	−	0.780800i	$-0.714812\pi$
$997$	−39.4553	−1.24956	−0.624781	+	0.780800i	$0.714812\pi$
$998$	0	0
$999$	2.74048	0.0867049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5100.2.a.bd.1.5		5
5.2	odd	4		1020.2.g.d.409.1	✓	10
5.3	odd	4		1020.2.g.d.409.6	yes	10
5.4	even	2		5100.2.a.bc.1.1		5
15.2	even	4		3060.2.g.g.2449.9		10
15.8	even	4		3060.2.g.g.2449.10		10
20.3	even	4		4080.2.m.r.2449.1		10
20.7	even	4		4080.2.m.r.2449.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.g.d.409.1	✓	10	5.2	odd	4
1020.2.g.d.409.6	yes	10	5.3	odd	4
3060.2.g.g.2449.9		10	15.2	even	4
3060.2.g.g.2449.10		10	15.8	even	4
4080.2.m.r.2449.1		10	20.3	even	4
4080.2.m.r.2449.6		10	20.7	even	4
5100.2.a.bc.1.1		5	5.4	even	2
5100.2.a.bd.1.5		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5100.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.94396 q^{7} +1.00000 q^{9} +2.22048 q^{11} +2.92695 q^{13} -1.00000 q^{17} +7.45305 q^{19} +2.94396 q^{21} +8.15951 q^{23} +1.00000 q^{27} +6.55478 q^{29} -7.49874 q^{31} +2.22048 q^{33} +2.74048 q^{37} +2.92695 q^{39} -6.88909 q^{41} -9.78125 q^{43} -8.16561 q^{47} +1.66687 q^{49} -1.00000 q^{51} +8.94396 q^{53} +7.45305 q^{57} +2.18569 q^{59} -14.7082 q^{61} +2.94396 q^{63} -2.18569 q^{67} +8.15951 q^{69} +5.85390 q^{71} -7.81604 q^{73} +6.53700 q^{77} -12.9975 q^{79} +1.00000 q^{81} -11.5383 q^{83} +6.55478 q^{87} +8.82029 q^{89} +8.61681 q^{91} -7.49874 q^{93} +3.01818 q^{97} +2.22048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 2 q^{7} + 5 q^{9} + 6 q^{11} - 5 q^{17} + 12 q^{19} - 2 q^{21} + 6 q^{23} + 5 q^{27} + 6 q^{29} + 6 q^{31} + 6 q^{33} + 14 q^{41} - 4 q^{43} - 4 q^{47} + 15 q^{49} - 5 q^{51} + 28 q^{53}+ \cdots + 6 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 2 * q^7 + 5 * q^9 + 6 * q^11 - 5 * q^17 + 12 * q^19 - 2 * q^21 + 6 * q^23 + 5 * q^27 + 6 * q^29 + 6 * q^31 + 6 * q^33 + 14 * q^41 - 4 * q^43 - 4 * q^47 + 15 * q^49 - 5 * q^51 + 28 * q^53 + 12 * q^57 + 24 * q^59 - 14 * q^61 - 2 * q^63 - 24 * q^67 + 6 * q^69 + 24 * q^73 + 22 * q^77 + 22 * q^79 + 5 * q^81 + 2 * q^83 + 6 * q^87 + 18 * q^89 + 20 * q^91 + 6 * q^93 - 2 * q^97 + 6 * q^99

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