LMFDB - Embedded newform 2793.2.a.be.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2793,2,Mod(1,2793)] 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("2793.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2793.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,-5,7,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.3022172845$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1240016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 2x^{2} + 16x + 6 $ x^5 - x^4 - 8x^3 + 2x^2 + 16*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.77304$ of defining polynomial
Character	$\chi$	$=$	2793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.91670 q^{2} -1.00000 q^{3} +1.67374 q^{4} -2.54204 q^{5} -1.91670 q^{6} -0.625344 q^{8} +1.00000 q^{9} -4.87234 q^{10} -5.54608 q^{11} -1.67374 q^{12} +1.83340 q^{13} +2.54204 q^{15} -4.54608 q^{16} +3.88952 q^{17} +1.91670 q^{18} -1.00000 q^{19} -4.25472 q^{20} -10.6302 q^{22} +5.54608 q^{23} +0.625344 q^{24} +1.46199 q^{25} +3.51408 q^{26} -1.00000 q^{27} +8.08812 q^{29} +4.87234 q^{30} +3.54608 q^{31} -7.46278 q^{32} +5.54608 q^{33} +7.45505 q^{34} +1.67374 q^{36} +5.73661 q^{37} -1.91670 q^{38} -1.83340 q^{39} +1.58965 q^{40} +2.48592 q^{41} +3.80947 q^{43} -9.28268 q^{44} -2.54204 q^{45} +10.6302 q^{46} +2.99597 q^{47} +4.54608 q^{48} +2.80219 q^{50} -3.88952 q^{51} +3.06863 q^{52} -3.31529 q^{53} -1.91670 q^{54} +14.0984 q^{55} +1.00000 q^{57} +15.5025 q^{58} +11.0922 q^{59} +4.25472 q^{60} -8.43157 q^{61} +6.79676 q^{62} -5.21175 q^{64} -4.66058 q^{65} +10.6302 q^{66} -6.14424 q^{67} +6.51005 q^{68} -5.54608 q^{69} +8.18491 q^{71} -0.625344 q^{72} -8.31932 q^{73} +10.9954 q^{74} -1.46199 q^{75} -1.67374 q^{76} -3.51408 q^{78} -4.79676 q^{79} +11.5563 q^{80} +1.00000 q^{81} +4.76477 q^{82} +15.3234 q^{83} -9.88734 q^{85} +7.30160 q^{86} -8.08812 q^{87} +3.46820 q^{88} +12.6541 q^{89} -4.87234 q^{90} +9.28268 q^{92} -3.54608 q^{93} +5.74237 q^{94} +2.54204 q^{95} +7.46278 q^{96} -15.1241 q^{97} -5.54608 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} - 5 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9} - 2 q^{11} - 7 q^{12} - 8 q^{13} - 2 q^{15} + 3 q^{16} + 2 q^{17} + q^{18} - 5 q^{19} + 2 q^{20} + 2 q^{22} + 2 q^{23} - 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 5 * q + q^2 - 5 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9 - 2 * q^11 - 7 * q^12 - 8 * q^13 - 2 * q^15 + 3 * q^16 + 2 * q^17 + q^18 - 5 * q^19 + 2 * q^20 + 2 * q^22 + 2 * q^23 - 3 * q^24 + 11 * q^25 + 32 * q^26 - 5 * q^27 - 8 * q^31 - 3 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^36 + 2 * q^37 - q^38 + 8 * q^39 + 36 * q^40 - 2 * q^41 + 20 * q^43 + 6 * q^44 + 2 * q^45 - 2 * q^46 + 26 * q^47 - 3 * q^48 - q^50 - 2 * q^51 - 4 * q^52 + 4 * q^53 - q^54 + 4 * q^55 + 5 * q^57 - 2 * q^58 + 4 * q^59 - 2 * q^60 - 10 * q^61 - 4 * q^62 - 21 * q^64 - 4 * q^65 - 2 * q^66 + 10 * q^67 + 58 * q^68 - 2 * q^69 + 10 * q^71 + 3 * q^72 - 10 * q^73 - 6 * q^74 - 11 * q^75 - 7 * q^76 - 32 * q^78 + 14 * q^79 + 6 * q^80 + 5 * q^81 + 26 * q^82 + 34 * q^83 - 36 * q^85 + 8 * q^86 + 6 * q^88 - 10 * q^89 - 6 * q^92 + 8 * q^93 + 8 * q^94 - 2 * q^95 + 3 * q^96 + 16 * q^97 - 2 * 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 $

$2$	1.91670	1.35531	0.677656	−	0.735379i	$-0.262996\pi$
$2$	1.91670	1.35531	0.677656	+	0.735379i	$0.262996\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.67374	0.836870
$5$	−2.54204	−1.13684	−0.568418	−	0.822740i	$-0.692444\pi$
$5$	−2.54204	−1.13684	−0.568418	+	0.822740i	$0.692444\pi$
$6$	−1.91670	−0.782490
$7$	0	0
$7$	0	0
$8$	−0.625344	−0.221092
$9$	1.00000	0.333333
$10$	−4.87234	−1.54077
$11$	−5.54608	−1.67220	−0.836102	−	0.548574i	$-0.815171\pi$
$11$	−5.54608	−1.67220	−0.836102	+	0.548574i	$0.815171\pi$
$12$	−1.67374	−0.483167
$13$	1.83340	0.508494	0.254247	−	0.967139i	$-0.418172\pi$
$13$	1.83340	0.508494	0.254247	+	0.967139i	$0.418172\pi$
$14$	0	0
$15$	2.54204	0.656353
$16$	−4.54608	−1.13652
$17$	3.88952	0.943348	0.471674	−	0.881773i	$-0.343650\pi$
$17$	3.88952	0.943348	0.471674	+	0.881773i	$0.343650\pi$
$18$	1.91670	0.451771
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.25472	−0.951384
$21$	0	0
$22$	−10.6302	−2.26636
$23$	5.54608	1.15644	0.578218	−	0.815882i	$-0.303748\pi$
$23$	5.54608	1.15644	0.578218	+	0.815882i	$0.303748\pi$
$24$	0.625344	0.127648
$25$	1.46199	0.292397
$26$	3.51408	0.689167
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.08812	1.50193	0.750963	−	0.660344i	$-0.229590\pi$
$29$	8.08812	1.50193	0.750963	+	0.660344i	$0.229590\pi$
$30$	4.87234	0.889563
$31$	3.54608	0.636894	0.318447	−	0.947941i	$-0.396839\pi$
$31$	3.54608	0.636894	0.318447	+	0.947941i	$0.396839\pi$
$32$	−7.46278	−1.31924
$33$	5.54608	0.965448
$34$	7.45505	1.27853
$35$	0	0
$36$	1.67374	0.278957
$37$	5.73661	0.943093	0.471546	−	0.881841i	$-0.343696\pi$
$37$	5.73661	0.943093	0.471546	+	0.881841i	$0.343696\pi$
$38$	−1.91670	−0.310930
$39$	−1.83340	−0.293579
$40$	1.58965	0.251346
$41$	2.48592	0.388236	0.194118	−	0.980978i	$-0.437816\pi$
$41$	2.48592	0.388236	0.194118	+	0.980978i	$0.437816\pi$
$42$	0	0
$43$	3.80947	0.580938	0.290469	−	0.956884i	$-0.406189\pi$
$43$	3.80947	0.580938	0.290469	+	0.956884i	$0.406189\pi$
$44$	−9.28268	−1.39942
$45$	−2.54204	−0.378946
$46$	10.6302	1.56733
$47$	2.99597	0.437007	0.218503	−	0.975836i	$-0.429883\pi$
$47$	2.99597	0.437007	0.218503	+	0.975836i	$0.429883\pi$
$48$	4.54608	0.656169
$49$	0	0
$50$	2.80219	0.396290
$51$	−3.88952	−0.544642
$52$	3.06863	0.425543
$53$	−3.31529	−0.455390	−0.227695	−	0.973732i	$-0.573119\pi$
$53$	−3.31529	−0.455390	−0.227695	+	0.973732i	$0.573119\pi$
$54$	−1.91670	−0.260830
$55$	14.0984	1.90102
$56$	0	0
$57$	1.00000	0.132453
$58$	15.5025	2.03558
$59$	11.0922	1.44408	0.722038	−	0.691854i	$-0.243206\pi$
$59$	11.0922	1.44408	0.722038	+	0.691854i	$0.243206\pi$
$60$	4.25472	0.549282
$61$	−8.43157	−1.07955	−0.539776	−	0.841809i	$-0.681491\pi$
$61$	−8.43157	−1.07955	−0.539776	+	0.841809i	$0.681491\pi$
$62$	6.79676	0.863190
$63$	0	0
$64$	−5.21175	−0.651469
$65$	−4.66058	−0.578074
$66$	10.6302	1.30848
$67$	−6.14424	−0.750639	−0.375319	−	0.926896i	$-0.622467\pi$
$67$	−6.14424	−0.750639	−0.375319	+	0.926896i	$0.622467\pi$
$68$	6.51005	0.789459
$69$	−5.54608	−0.667669
$70$	0	0
$71$	8.18491	0.971370	0.485685	−	0.874134i	$-0.338570\pi$
$71$	8.18491	0.971370	0.485685	+	0.874134i	$0.338570\pi$
$72$	−0.625344	−0.0736975
$73$	−8.31932	−0.973703	−0.486851	−	0.873485i	$-0.661855\pi$
$73$	−8.31932	−0.973703	−0.486851	+	0.873485i	$0.661855\pi$
$74$	10.9954	1.27818
$75$	−1.46199	−0.168816
$76$	−1.67374	−0.191991
$77$	0	0
$78$	−3.51408	−0.397891
$79$	−4.79676	−0.539678	−0.269839	−	0.962905i	$-0.586970\pi$
$79$	−4.79676	−0.539678	−0.269839	+	0.962905i	$0.586970\pi$
$80$	11.5563	1.29204
$81$	1.00000	0.111111
$82$	4.76477	0.526180
$83$	15.3234	1.68196	0.840978	−	0.541069i	$-0.181980\pi$
$83$	15.3234	1.68196	0.840978	+	0.541069i	$0.181980\pi$
$84$	0	0
$85$	−9.88734	−1.07243
$86$	7.30160	0.787352
$87$	−8.08812	−0.867137
$88$	3.46820	0.369712
$89$	12.6541	1.34133	0.670666	−	0.741760i	$-0.266008\pi$
$89$	12.6541	1.34133	0.670666	+	0.741760i	$0.266008\pi$
$90$	−4.87234	−0.513589
$91$	0	0
$92$	9.28268	0.967787
$93$	−3.54608	−0.367711
$94$	5.74237	0.592281
$95$	2.54204	0.260808
$96$	7.46278	0.761666
$97$	−15.1241	−1.53562	−0.767812	−	0.640675i	$-0.778655\pi$
$97$	−15.1241	−1.53562	−0.767812	+	0.640675i	$0.778655\pi$
$98$	0	0
$99$	−5.54608	−0.557402
$100$	2.44699	0.244699
$101$	−14.9817	−1.49073	−0.745366	−	0.666655i	$-0.767725\pi$
$101$	−14.9817	−1.49073	−0.745366	+	0.666655i	$0.767725\pi$
$102$	−7.45505	−0.738160
$103$	−6.36176	−0.626843	−0.313421	−	0.949614i	$-0.601475\pi$
$103$	−6.36176	−0.626843	−0.313421	+	0.949614i	$0.601475\pi$
$104$	−1.14651	−0.112424
$105$	0	0
$106$	−6.35442	−0.617196
$107$	17.9296	1.73332	0.866659	−	0.498901i	$-0.166263\pi$
$107$	17.9296	1.73332	0.866659	+	0.498901i	$0.166263\pi$
$108$	−1.67374	−0.161056
$109$	−4.00806	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$109$	−4.00806	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$110$	27.0223	2.57648
$111$	−5.73661	−0.544495
$112$	0	0
$113$	1.96781	0.185116	0.0925581	−	0.995707i	$-0.470496\pi$
$113$	1.96781	0.185116	0.0925581	+	0.995707i	$0.470496\pi$
$114$	1.91670	0.179515
$115$	−14.0984	−1.31468
$116$	13.5374	1.25692
$117$	1.83340	0.169498
$118$	21.2603	1.95717
$119$	0	0
$120$	−1.58965	−0.145115
$121$	19.7590	1.79627
$122$	−16.1608	−1.46313
$123$	−2.48592	−0.224148
$124$	5.93521	0.532997
$125$	8.99378	0.804428
$126$	0	0
$127$	7.44928	0.661017	0.330509	−	0.943803i	$-0.392780\pi$
$127$	7.44928	0.661017	0.330509	+	0.943803i	$0.392780\pi$
$128$	4.93619	0.436301
$129$	−3.80947	−0.335405
$130$	−8.93294	−0.783471
$131$	15.4356	1.34861	0.674307	−	0.738451i	$-0.264442\pi$
$131$	15.4356	1.34861	0.674307	+	0.738451i	$0.264442\pi$
$132$	9.28268	0.807954
$133$	0	0
$134$	−11.7767	−1.01735
$135$	2.54204	0.218784
$136$	−2.43229	−0.208567
$137$	−15.4539	−1.32032	−0.660158	−	0.751127i	$-0.729511\pi$
$137$	−15.4539	−1.32032	−0.660158	+	0.751127i	$0.729511\pi$
$138$	−10.6302	−0.904900
$139$	10.3193	0.875273	0.437637	−	0.899152i	$-0.355816\pi$
$139$	10.3193	0.875273	0.437637	+	0.899152i	$0.355816\pi$
$140$	0	0
$141$	−2.99597	−0.252306
$142$	15.6880	1.31651
$143$	−10.1682	−0.850306
$144$	−4.54608	−0.378840
$145$	−20.5604	−1.70744
$146$	−15.9456	−1.31967
$147$	0	0
$148$	9.60159	0.789246
$149$	12.0081	0.983739	0.491869	−	0.870669i	$-0.336314\pi$
$149$	12.0081	0.983739	0.491869	+	0.870669i	$0.336314\pi$
$150$	−2.80219	−0.228798
$151$	19.0463	1.54996	0.774982	−	0.631983i	$-0.217759\pi$
$151$	19.0463	1.54996	0.774982	+	0.631983i	$0.217759\pi$
$152$	0.625344	0.0507221
$153$	3.88952	0.314449
$154$	0	0
$155$	−9.01428	−0.724044
$156$	−3.06863	−0.245687
$157$	−5.93019	−0.473281	−0.236640	−	0.971597i	$-0.576046\pi$
$157$	−5.93019	−0.473281	−0.236640	+	0.971597i	$0.576046\pi$
$158$	−9.19396	−0.731432
$159$	3.31529	0.262920
$160$	18.9707	1.49977
$161$	0	0
$162$	1.91670	0.150590
$163$	−2.31084	−0.180999	−0.0904995	−	0.995896i	$-0.528846\pi$
$163$	−2.31084	−0.180999	−0.0904995	+	0.995896i	$0.528846\pi$
$164$	4.16078	0.324903
$165$	−14.0984	−1.09756
$166$	29.3703	2.27958
$167$	9.41729	0.728732	0.364366	−	0.931256i	$-0.381286\pi$
$167$	9.41729	0.728732	0.364366	+	0.931256i	$0.381286\pi$
$168$	0	0
$169$	−9.63864	−0.741434
$170$	−18.9511	−1.45348
$171$	−1.00000	−0.0764719
$172$	6.37605	0.486170
$173$	13.5781	1.03232	0.516161	−	0.856492i	$-0.327361\pi$
$173$	13.5781	1.03232	0.516161	+	0.856492i	$0.327361\pi$
$174$	−15.5025	−1.17524
$175$	0	0
$176$	25.2129	1.90049
$177$	−11.0922	−0.833737
$178$	24.2541	1.81792
$179$	1.55976	0.116582	0.0582910	−	0.998300i	$-0.481435\pi$
$179$	1.55976	0.116582	0.0582910	+	0.998300i	$0.481435\pi$
$180$	−4.25472	−0.317128
$181$	−18.5334	−1.37757	−0.688787	−	0.724963i	$-0.741857\pi$
$181$	−18.5334	−1.37757	−0.688787	+	0.724963i	$0.741857\pi$
$182$	0	0
$183$	8.43157	0.623279
$184$	−3.46820	−0.255679
$185$	−14.5827	−1.07214
$186$	−6.79676	−0.498363
$187$	−21.5716	−1.57747
$188$	5.01447	0.365718
$189$	0	0
$190$	4.87234	0.353476
$191$	−23.3811	−1.69179	−0.845897	−	0.533347i	$-0.820934\pi$
$191$	−23.3811	−1.69179	−0.845897	+	0.533347i	$0.820934\pi$
$192$	5.21175	0.376126
$193$	16.8288	1.21136	0.605680	−	0.795708i	$-0.292901\pi$
$193$	16.8288	1.21136	0.605680	+	0.795708i	$0.292901\pi$
$194$	−28.9885	−2.08125
$195$	4.66058	0.333751
$196$	0	0
$197$	−23.1207	−1.64728	−0.823641	−	0.567111i	$-0.808061\pi$
$197$	−23.1207	−1.64728	−0.823641	+	0.567111i	$0.808061\pi$
$198$	−10.6302	−0.755453
$199$	−12.1762	−0.863151	−0.431575	−	0.902077i	$-0.642042\pi$
$199$	−12.1762	−0.863151	−0.431575	+	0.902077i	$0.642042\pi$
$200$	−0.914245	−0.0646469
$201$	6.14424	0.433381
$202$	−28.7154	−2.02041
$203$	0	0
$204$	−6.51005	−0.455794
$205$	−6.31932	−0.441361
$206$	−12.1936	−0.849567
$207$	5.54608	0.385479
$208$	−8.33478	−0.577913
$209$	5.54608	0.383630
$210$	0	0
$211$	−6.95597	−0.478869	−0.239434	−	0.970913i	$-0.576962\pi$
$211$	−6.95597	−0.478869	−0.239434	+	0.970913i	$0.576962\pi$
$212$	−5.54893	−0.381102
$213$	−8.18491	−0.560821
$214$	34.3656	2.34919
$215$	−9.68383	−0.660432
$216$	0.625344	0.0425493
$217$	0	0
$218$	−7.68225	−0.520308
$219$	8.31932	0.562168
$220$	23.5970	1.59091
$221$	7.13105	0.479686
$222$	−10.9954	−0.737960
$223$	28.4092	1.90242	0.951211	−	0.308542i	$-0.0998411\pi$
$223$	28.4092	1.90242	0.951211	+	0.308542i	$0.0998411\pi$
$224$	0	0
$225$	1.46199	0.0974658
$226$	3.77170	0.250890
$227$	−3.31311	−0.219899	−0.109949	−	0.993937i	$-0.535069\pi$
$227$	−3.31311	−0.219899	−0.109949	+	0.993937i	$0.535069\pi$
$228$	1.67374	0.110846
$229$	−1.88776	−0.124746	−0.0623732	−	0.998053i	$-0.519867\pi$
$229$	−1.88776	−0.124746	−0.0623732	+	0.998053i	$0.519867\pi$
$230$	−27.0223	−1.78180
$231$	0	0
$232$	−5.05786	−0.332064
$233$	23.7227	1.55413	0.777064	−	0.629422i	$-0.216708\pi$
$233$	23.7227	1.55413	0.777064	+	0.629422i	$0.216708\pi$
$234$	3.51408	0.229722
$235$	−7.61588	−0.496805
$236$	18.5654	1.20850
$237$	4.79676	0.311583
$238$	0	0
$239$	21.9159	1.41762	0.708811	−	0.705399i	$-0.249232\pi$
$239$	21.9159	1.41762	0.708811	+	0.705399i	$0.249232\pi$
$240$	−11.5563	−0.745957
$241$	17.2445	1.11081	0.555406	−	0.831579i	$-0.312563\pi$
$241$	17.2445	1.11081	0.555406	+	0.831579i	$0.312563\pi$
$242$	37.8720	2.43450
$243$	−1.00000	−0.0641500
$244$	−14.1122	−0.903444
$245$	0	0
$246$	−4.76477	−0.303790
$247$	−1.83340	−0.116656
$248$	−2.21752	−0.140812
$249$	−15.3234	−0.971078
$250$	17.2384	1.09025
$251$	3.72636	0.235206	0.117603	−	0.993061i	$-0.462479\pi$
$251$	3.72636	0.235206	0.117603	+	0.993061i	$0.462479\pi$
$252$	0	0
$253$	−30.7590	−1.93380
$254$	14.2780	0.895884
$255$	9.88734	0.619169
$256$	19.8847	1.24279
$257$	−4.42999	−0.276335	−0.138168	−	0.990409i	$-0.544121\pi$
$257$	−4.42999	−0.276335	−0.138168	+	0.990409i	$0.544121\pi$
$258$	−7.30160	−0.454578
$259$	0	0
$260$	−7.80060	−0.483773
$261$	8.08812	0.500642
$262$	29.5854	1.82779
$263$	2.23297	0.137691	0.0688454	−	0.997627i	$-0.478068\pi$
$263$	2.23297	0.137691	0.0688454	+	0.997627i	$0.478068\pi$
$264$	−3.46820	−0.213453
$265$	8.42761	0.517704
$266$	0	0
$267$	−12.6541	−0.774418
$268$	−10.2839	−0.628187
$269$	−4.87505	−0.297237	−0.148619	−	0.988895i	$-0.547483\pi$
$269$	−4.87505	−0.297237	−0.148619	+	0.988895i	$0.547483\pi$
$270$	4.87234	0.296521
$271$	16.5519	1.00545	0.502727	−	0.864445i	$-0.332330\pi$
$271$	16.5519	1.00545	0.502727	+	0.864445i	$0.332330\pi$
$272$	−17.6821	−1.07213
$273$	0	0
$274$	−29.6205	−1.78944
$275$	−8.10829	−0.488948
$276$	−9.28268	−0.558752
$277$	28.0841	1.68741	0.843704	−	0.536808i	$-0.180370\pi$
$277$	28.0841	1.68741	0.843704	+	0.536808i	$0.180370\pi$
$278$	19.7790	1.18627
$279$	3.54608	0.212298
$280$	0	0
$281$	11.2674	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$281$	11.2674	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$282$	−5.74237	−0.341953
$283$	−15.7952	−0.938925	−0.469463	−	0.882952i	$-0.655552\pi$
$283$	−15.7952	−0.938925	−0.469463	+	0.882952i	$0.655552\pi$
$284$	13.6994	0.812910
$285$	−2.54204	−0.150578
$286$	−19.4893	−1.15243
$287$	0	0
$288$	−7.46278	−0.439748
$289$	−1.87161	−0.110095
$290$	−39.4080	−2.31412
$291$	15.1241	0.886593
$292$	−13.9244	−0.814862
$293$	−26.4573	−1.54565	−0.772827	−	0.634617i	$-0.781158\pi$
$293$	−26.4573	−1.54565	−0.772827	+	0.634617i	$0.781158\pi$
$294$	0	0
$295$	−28.1967	−1.64168
$296$	−3.58735	−0.208511
$297$	5.54608	0.321816
$298$	23.0159	1.33327
$299$	10.1682	0.588041
$300$	−2.44699	−0.141277
$301$	0	0
$302$	36.5060	2.10068
$303$	14.9817	0.860675
$304$	4.54608	0.260735
$305$	21.4334	1.22727
$306$	7.45505	0.426177
$307$	−11.7875	−0.672750	−0.336375	−	0.941728i	$-0.609201\pi$
$307$	−11.7875	−0.672750	−0.336375	+	0.941728i	$0.609201\pi$
$308$	0	0
$309$	6.36176	0.361908
$310$	−17.2777	−0.981306
$311$	−2.39938	−0.136056	−0.0680281	−	0.997683i	$-0.521671\pi$
$311$	−2.39938	−0.136056	−0.0680281	+	0.997683i	$0.521671\pi$
$312$	1.14651	0.0649081
$313$	−6.66058	−0.376478	−0.188239	−	0.982123i	$-0.560278\pi$
$313$	−6.66058	−0.376478	−0.188239	+	0.982123i	$0.560278\pi$
$314$	−11.3664	−0.641443
$315$	0	0
$316$	−8.02853	−0.451640
$317$	−18.3376	−1.02994	−0.514972	−	0.857207i	$-0.672198\pi$
$317$	−18.3376	−1.02994	−0.514972	+	0.857207i	$0.672198\pi$
$318$	6.35442	0.356338
$319$	−44.8573	−2.51153
$320$	13.2485	0.740614
$321$	−17.9296	−1.00073
$322$	0	0
$323$	−3.88952	−0.216419
$324$	1.67374	0.0929855
$325$	2.68041	0.148682
$326$	−4.42919	−0.245310
$327$	4.00806	0.221646
$328$	−1.55456	−0.0858360
$329$	0	0
$330$	−27.0223	−1.48753
$331$	−3.09099	−0.169896	−0.0849482	−	0.996385i	$-0.527072\pi$
$331$	−3.09099	−0.169896	−0.0849482	+	0.996385i	$0.527072\pi$
$332$	25.6473	1.40758
$333$	5.73661	0.314364
$334$	18.0501	0.987658
$335$	15.6189	0.853353
$336$	0	0
$337$	16.8233	0.916425	0.458213	−	0.888843i	$-0.348490\pi$
$337$	16.8233	0.916425	0.458213	+	0.888843i	$0.348490\pi$
$338$	−18.4744	−1.00487
$339$	−1.96781	−0.106877
$340$	−16.5488	−0.897486
$341$	−19.6668	−1.06502
$342$	−1.91670	−0.103643
$343$	0	0
$344$	−2.38223	−0.128441
$345$	14.0984	0.759031
$346$	26.0251	1.39912
$347$	−26.8877	−1.44341	−0.721705	−	0.692201i	$-0.756641\pi$
$347$	−26.8877	−1.44341	−0.721705	+	0.692201i	$0.756641\pi$
$348$	−13.5374	−0.725681
$349$	4.80367	0.257134	0.128567	−	0.991701i	$-0.458962\pi$
$349$	4.80367	0.257134	0.128567	+	0.991701i	$0.458962\pi$
$350$	0	0
$351$	−1.83340	−0.0978597
$352$	41.3891	2.20605
$353$	12.5775	0.669432	0.334716	−	0.942319i	$-0.391360\pi$
$353$	12.5775	0.669432	0.334716	+	0.942319i	$0.391360\pi$
$354$	−21.2603	−1.12997
$355$	−20.8064	−1.10429
$356$	21.1797	1.12252
$357$	0	0
$358$	2.98960	0.158005
$359$	−31.3332	−1.65370	−0.826851	−	0.562421i	$-0.809870\pi$
$359$	−31.3332	−1.65370	−0.826851	+	0.562421i	$0.809870\pi$
$360$	1.58965	0.0837820
$361$	1.00000	0.0526316
$362$	−35.5229	−1.86704
$363$	−19.7590	−1.03708
$364$	0	0
$365$	21.1481	1.10694
$366$	16.1608	0.844738
$367$	21.0841	1.10058	0.550290	−	0.834973i	$-0.314517\pi$
$367$	21.0841	1.10058	0.550290	+	0.834973i	$0.314517\pi$
$368$	−25.2129	−1.31431
$369$	2.48592	0.129412
$370$	−27.9507	−1.45309
$371$	0	0
$372$	−5.93521	−0.307726
$373$	−23.6097	−1.22246	−0.611231	−	0.791452i	$-0.709325\pi$
$373$	−23.6097	−1.22246	−0.611231	+	0.791452i	$0.709325\pi$
$374$	−41.3463	−2.13796
$375$	−8.99378	−0.464437
$376$	−1.87351	−0.0966189
$377$	14.8288	0.763720
$378$	0	0
$379$	7.13803	0.366656	0.183328	−	0.983052i	$-0.441313\pi$
$379$	7.13803	0.366656	0.183328	+	0.983052i	$0.441313\pi$
$380$	4.25472	0.218262
$381$	−7.44928	−0.381638
$382$	−44.8145	−2.29291
$383$	9.41729	0.481201	0.240600	−	0.970624i	$-0.422656\pi$
$383$	9.41729	0.481201	0.240600	+	0.970624i	$0.422656\pi$
$384$	−4.93619	−0.251899
$385$	0	0
$386$	32.2557	1.64177
$387$	3.80947	0.193646
$388$	−25.3139	−1.28512
$389$	−28.0721	−1.42331	−0.711655	−	0.702529i	$-0.752054\pi$
$389$	−28.0721	−1.42331	−0.711655	+	0.702529i	$0.752054\pi$
$390$	8.93294	0.452337
$391$	21.5716	1.09092
$392$	0	0
$393$	−15.4356	−0.778623
$394$	−44.3155	−2.23258
$395$	12.1936	0.613526
$396$	−9.28268	−0.466472
$397$	6.59078	0.330782	0.165391	−	0.986228i	$-0.447111\pi$
$397$	6.59078	0.330782	0.165391	+	0.986228i	$0.447111\pi$
$398$	−23.3382	−1.16984
$399$	0	0
$400$	−6.64630	−0.332315
$401$	34.1865	1.70719	0.853596	−	0.520936i	$-0.174417\pi$
$401$	34.1865	1.70719	0.853596	+	0.520936i	$0.174417\pi$
$402$	11.7767	0.587367
$403$	6.50138	0.323857
$404$	−25.0754	−1.24755
$405$	−2.54204	−0.126315
$406$	0	0
$407$	−31.8157	−1.57704
$408$	2.43229	0.120416
$409$	33.4555	1.65427	0.827134	−	0.562005i	$-0.189970\pi$
$409$	33.4555	1.65427	0.827134	+	0.562005i	$0.189970\pi$
$410$	−12.1122	−0.598181
$411$	15.4539	0.762285
$412$	−10.6479	−0.524586
$413$	0	0
$414$	10.6302	0.522444
$415$	−38.9526	−1.91211
$416$	−13.6823	−0.670828
$417$	−10.3193	−0.505339
$418$	10.6302	0.519938
$419$	30.4418	1.48718	0.743590	−	0.668636i	$-0.233121\pi$
$419$	30.4418	1.48718	0.743590	+	0.668636i	$0.233121\pi$
$420$	0	0
$421$	−17.4678	−0.851328	−0.425664	−	0.904881i	$-0.639959\pi$
$421$	−17.4678	−0.851328	−0.425664	+	0.904881i	$0.639959\pi$
$422$	−13.3325	−0.649017
$423$	2.99597	0.145669
$424$	2.07320	0.100683
$425$	5.68643	0.275833
$426$	−15.6880	−0.760087
$427$	0	0
$428$	30.0094	1.45056
$429$	10.1682	0.490924
$430$	−18.5610	−0.895091
$431$	24.8969	1.19924	0.599621	−	0.800284i	$-0.295318\pi$
$431$	24.8969	1.19924	0.599621	+	0.800284i	$0.295318\pi$
$432$	4.54608	0.218723
$433$	11.3177	0.543895	0.271948	−	0.962312i	$-0.412332\pi$
$433$	11.3177	0.543895	0.271948	+	0.962312i	$0.412332\pi$
$434$	0	0
$435$	20.5604	0.985794
$436$	−6.70845	−0.321277
$437$	−5.54608	−0.265305
$438$	15.9456	0.761912
$439$	−12.7424	−0.608162	−0.304081	−	0.952646i	$-0.598349\pi$
$439$	−12.7424	−0.608162	−0.304081	+	0.952646i	$0.598349\pi$
$440$	−8.81633	−0.420302
$441$	0	0
$442$	13.6681	0.650125
$443$	−22.6140	−1.07443	−0.537213	−	0.843447i	$-0.680523\pi$
$443$	−22.6140	−1.07443	−0.537213	+	0.843447i	$0.680523\pi$
$444$	−9.60159	−0.455671
$445$	−32.1673	−1.52487
$446$	54.4519	2.57837
$447$	−12.0081	−0.567962
$448$	0	0
$449$	5.06578	0.239069	0.119534	−	0.992830i	$-0.461860\pi$
$449$	5.06578	0.239069	0.119534	+	0.992830i	$0.461860\pi$
$450$	2.80219	0.132097
$451$	−13.7871	−0.649210
$452$	3.29360	0.154918
$453$	−19.0463	−0.894872
$454$	−6.35023	−0.298031
$455$	0	0
$456$	−0.625344	−0.0292844
$457$	−7.79559	−0.364662	−0.182331	−	0.983237i	$-0.558364\pi$
$457$	−7.79559	−0.364662	−0.182331	+	0.983237i	$0.558364\pi$
$458$	−3.61826	−0.169070
$459$	−3.88952	−0.181547
$460$	−23.5970	−1.10022
$461$	−6.89680	−0.321216	−0.160608	−	0.987018i	$-0.551345\pi$
$461$	−6.89680	−0.321216	−0.160608	+	0.987018i	$0.551345\pi$
$462$	0	0
$463$	10.6950	0.497037	0.248518	−	0.968627i	$-0.420056\pi$
$463$	10.6950	0.497037	0.248518	+	0.968627i	$0.420056\pi$
$464$	−36.7692	−1.70697
$465$	9.01428	0.418027
$466$	45.4694	2.10633
$467$	−11.2146	−0.518952	−0.259476	−	0.965750i	$-0.583550\pi$
$467$	−11.2146	−0.518952	−0.259476	+	0.965750i	$0.583550\pi$
$468$	3.06863	0.141848
$469$	0	0
$470$	−14.5974	−0.673326
$471$	5.93019	0.273249
$472$	−6.93641	−0.319274
$473$	−21.1276	−0.971447
$474$	9.19396	0.422292
$475$	−1.46199	−0.0670806
$476$	0	0
$477$	−3.31529	−0.151797
$478$	42.0062	1.92132
$479$	35.4607	1.62024	0.810120	−	0.586264i	$-0.199402\pi$
$479$	35.4607	1.62024	0.810120	+	0.586264i	$0.199402\pi$
$480$	−18.9707	−0.865890
$481$	10.5175	0.479557
$482$	33.0524	1.50550
$483$	0	0
$484$	33.0713	1.50324
$485$	38.4462	1.74575
$486$	−1.91670	−0.0869433
$487$	8.58734	0.389129	0.194565	−	0.980890i	$-0.437671\pi$
$487$	8.58734	0.389129	0.194565	+	0.980890i	$0.437671\pi$
$488$	5.27263	0.238681
$489$	2.31084	0.104500
$490$	0	0
$491$	−33.7384	−1.52259	−0.761297	−	0.648403i	$-0.775437\pi$
$491$	−33.7384	−1.52259	−0.761297	+	0.648403i	$0.775437\pi$
$492$	−4.16078	−0.187583
$493$	31.4589	1.41684
$494$	−3.51408	−0.158106
$495$	14.0984	0.633674
$496$	−16.1207	−0.723842
$497$	0	0
$498$	−29.3703	−1.31611
$499$	−10.9364	−0.489581	−0.244790	−	0.969576i	$-0.578719\pi$
$499$	−10.9364	−0.489581	−0.244790	+	0.969576i	$0.578719\pi$
$500$	15.0532	0.673202
$501$	−9.41729	−0.420733
$502$	7.14232	0.318777
$503$	−22.9454	−1.02309	−0.511543	−	0.859258i	$-0.670926\pi$
$503$	−22.9454	−1.02309	−0.511543	+	0.859258i	$0.670926\pi$
$504$	0	0
$505$	38.0841	1.69472
$506$	−58.9557	−2.62090
$507$	9.63864	0.428067
$508$	12.4682	0.553185
$509$	−7.57001	−0.335535	−0.167767	−	0.985827i	$-0.553656\pi$
$509$	−7.57001	−0.335535	−0.167767	+	0.985827i	$0.553656\pi$
$510$	18.9511	0.839167
$511$	0	0
$512$	28.2406	1.24807
$513$	1.00000	0.0441511
$514$	−8.49096	−0.374520
$515$	16.1719	0.712618
$516$	−6.37605	−0.280690
$517$	−16.6159	−0.730765
$518$	0	0
$519$	−13.5781	−0.596011
$520$	2.91447	0.127808
$521$	0.827184	0.0362396	0.0181198	−	0.999836i	$-0.494232\pi$
$521$	0.827184	0.0362396	0.0181198	+	0.999836i	$0.494232\pi$
$522$	15.5025	0.678526
$523$	17.2414	0.753915	0.376958	−	0.926230i	$-0.376970\pi$
$523$	17.2414	0.753915	0.376958	+	0.926230i	$0.376970\pi$
$524$	25.8352	1.12861
$525$	0	0
$526$	4.27993	0.186614
$527$	13.7925	0.600812
$528$	−25.2129	−1.09725
$529$	7.75895	0.337346
$530$	16.1532	0.701651
$531$	11.0922	0.481358
$532$	0	0
$533$	4.55769	0.197415
$534$	−24.2541	−1.04958
$535$	−45.5778	−1.97050
$536$	3.84226	0.165961
$537$	−1.55976	−0.0673087
$538$	−9.34401	−0.402849
$539$	0	0
$540$	4.25472	0.183094
$541$	34.1280	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$541$	34.1280	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$542$	31.7250	1.36270
$543$	18.5334	0.795343
$544$	−29.0266	−1.24451
$545$	10.1887	0.436435
$546$	0	0
$547$	30.0101	1.28314	0.641569	−	0.767066i	$-0.278284\pi$
$547$	30.0101	1.28314	0.641569	+	0.767066i	$0.278284\pi$
$548$	−25.8658	−1.10493
$549$	−8.43157	−0.359850
$550$	−15.5412	−0.662677
$551$	−8.08812	−0.344565
$552$	3.46820	0.147617
$553$	0	0
$554$	53.8287	2.28696
$555$	14.5827	0.619002
$556$	17.2719	0.732490
$557$	−34.9066	−1.47904	−0.739521	−	0.673134i	$-0.764948\pi$
$557$	−34.9066	−1.47904	−0.739521	+	0.673134i	$0.764948\pi$
$558$	6.79676	0.287730
$559$	6.98428	0.295403
$560$	0	0
$561$	21.5716	0.910753
$562$	21.5963	0.910984
$563$	−20.8358	−0.878123	−0.439061	−	0.898457i	$-0.644689\pi$
$563$	−20.8358	−0.878123	−0.439061	+	0.898457i	$0.644689\pi$
$564$	−5.01447	−0.211147
$565$	−5.00226	−0.210447
$566$	−30.2746	−1.27254
$567$	0	0
$568$	−5.11838	−0.214763
$569$	32.3180	1.35484	0.677421	−	0.735595i	$-0.263097\pi$
$569$	32.3180	1.35484	0.677421	+	0.735595i	$0.263097\pi$
$570$	−4.87234	−0.204080
$571$	20.4451	0.855599	0.427800	−	0.903874i	$-0.359289\pi$
$571$	20.4451	0.855599	0.427800	+	0.903874i	$0.359289\pi$
$572$	−17.0189	−0.711595
$573$	23.3811	0.976757
$574$	0	0
$575$	8.10829	0.338139
$576$	−5.21175	−0.217156
$577$	−20.1462	−0.838699	−0.419349	−	0.907825i	$-0.637742\pi$
$577$	−20.1462	−0.838699	−0.419349	+	0.907825i	$0.637742\pi$
$578$	−3.58732	−0.149213
$579$	−16.8288	−0.699379
$580$	−34.4127	−1.42891
$581$	0	0
$582$	28.9885	1.20161
$583$	18.3869	0.761506
$584$	5.20244	0.215278
$585$	−4.66058	−0.192691
$586$	−50.7108	−2.09484
$587$	9.60605	0.396484	0.198242	−	0.980153i	$-0.436477\pi$
$587$	9.60605	0.396484	0.198242	+	0.980153i	$0.436477\pi$
$588$	0	0
$589$	−3.54608	−0.146113
$590$	−54.0447	−2.22498
$591$	23.1207	0.951059
$592$	−26.0791	−1.07184
$593$	−36.4249	−1.49579	−0.747895	−	0.663817i	$-0.768936\pi$
$593$	−36.4249	−1.49579	−0.747895	+	0.663817i	$0.768936\pi$
$594$	10.6302	0.436161
$595$	0	0
$596$	20.0984	0.823261
$597$	12.1762	0.498340
$598$	19.4893	0.796979
$599$	12.6358	0.516284	0.258142	−	0.966107i	$-0.416890\pi$
$599$	12.6358	0.516284	0.258142	+	0.966107i	$0.416890\pi$
$600$	0.914245	0.0373239
$601$	2.95144	0.120392	0.0601960	−	0.998187i	$-0.480827\pi$
$601$	2.95144	0.120392	0.0601960	+	0.998187i	$0.480827\pi$
$602$	0	0
$603$	−6.14424	−0.250213
$604$	31.8785	1.29712
$605$	−50.2281	−2.04206
$606$	28.7154	1.16648
$607$	3.09215	0.125507	0.0627533	−	0.998029i	$-0.480012\pi$
$607$	3.09215	0.125507	0.0627533	+	0.998029i	$0.480012\pi$
$608$	7.46278	0.302656
$609$	0	0
$610$	41.0814	1.66334
$611$	5.49281	0.222215
$612$	6.51005	0.263153
$613$	−22.9195	−0.925709	−0.462855	−	0.886434i	$-0.653175\pi$
$613$	−22.9195	−0.925709	−0.462855	+	0.886434i	$0.653175\pi$
$614$	−22.5932	−0.911785
$615$	6.31932	0.254820
$616$	0	0
$617$	−13.8913	−0.559242	−0.279621	−	0.960110i	$-0.590209\pi$
$617$	−13.8913	−0.559242	−0.279621	+	0.960110i	$0.590209\pi$
$618$	12.1936	0.490498
$619$	42.8963	1.72415	0.862074	−	0.506783i	$-0.169165\pi$
$619$	42.8963	1.72415	0.862074	+	0.506783i	$0.169165\pi$
$620$	−15.0876	−0.605931
$621$	−5.54608	−0.222556
$622$	−4.59889	−0.184399
$623$	0	0
$624$	8.33478	0.333658
$625$	−30.1725	−1.20690
$626$	−12.7663	−0.510246
$627$	−5.54608	−0.221489
$628$	−9.92559	−0.396074
$629$	22.3127	0.889664
$630$	0	0
$631$	39.0613	1.55501	0.777504	−	0.628878i	$-0.216486\pi$
$631$	39.0613	1.55501	0.777504	+	0.628878i	$0.216486\pi$
$632$	2.99963	0.119319
$633$	6.95597	0.276475
$634$	−35.1477	−1.39590
$635$	−18.9364	−0.751468
$636$	5.54893	0.220029
$637$	0	0
$638$	−85.9780	−3.40390
$639$	8.18491	0.323790
$640$	−12.5480	−0.496003
$641$	49.8109	1.96741	0.983705	−	0.179789i	$-0.0575416\pi$
$641$	49.8109	1.96741	0.983705	+	0.179789i	$0.0575416\pi$
$642$	−34.3656	−1.35630
$643$	−19.9187	−0.785515	−0.392758	−	0.919642i	$-0.628479\pi$
$643$	−19.9187	−0.785515	−0.392758	+	0.919642i	$0.628479\pi$
$644$	0	0
$645$	9.68383	0.381300
$646$	−7.45505	−0.293315
$647$	38.5332	1.51490	0.757448	−	0.652896i	$-0.226446\pi$
$647$	38.5332	1.51490	0.757448	+	0.652896i	$0.226446\pi$
$648$	−0.625344	−0.0245658
$649$	−61.5179	−2.41479
$650$	5.13754	0.201511
$651$	0	0
$652$	−3.86775	−0.151473
$653$	−1.47322	−0.0576515	−0.0288257	−	0.999584i	$-0.509177\pi$
$653$	−1.47322	−0.0576515	−0.0288257	+	0.999584i	$0.509177\pi$
$654$	7.68225	0.300400
$655$	−39.2380	−1.53315
$656$	−11.3012	−0.441237
$657$	−8.31932	−0.324568
$658$	0	0
$659$	35.4063	1.37923	0.689616	−	0.724175i	$-0.257779\pi$
$659$	35.4063	1.37923	0.689616	+	0.724175i	$0.257779\pi$
$660$	−23.5970	−0.918512
$661$	−12.2390	−0.476043	−0.238022	−	0.971260i	$-0.576499\pi$
$661$	−12.2390	−0.476043	−0.238022	+	0.971260i	$0.576499\pi$
$662$	−5.92451	−0.230262
$663$	−7.13105	−0.276947
$664$	−9.58236	−0.371868
$665$	0	0
$666$	10.9954	0.426062
$667$	44.8573	1.73688
$668$	15.7621	0.609853
$669$	−28.4092	−1.09836
$670$	29.9368	1.15656
$671$	46.7621	1.80523
$672$	0	0
$673$	−39.6565	−1.52864	−0.764322	−	0.644835i	$-0.776926\pi$
$673$	−39.6565	−1.52864	−0.764322	+	0.644835i	$0.776926\pi$
$674$	32.2453	1.24204
$675$	−1.46199	−0.0562719
$676$	−16.1326	−0.620484
$677$	12.7354	0.489463	0.244731	−	0.969591i	$-0.421300\pi$
$677$	12.7354	0.489463	0.244731	+	0.969591i	$0.421300\pi$
$678$	−3.77170	−0.144851
$679$	0	0
$680$	6.18299	0.237107
$681$	3.31311	0.126958
$682$	−37.6954	−1.44343
$683$	−10.2019	−0.390366	−0.195183	−	0.980767i	$-0.562530\pi$
$683$	−10.2019	−0.390366	−0.195183	+	0.980767i	$0.562530\pi$
$684$	−1.67374	−0.0639970
$685$	39.2845	1.50098
$686$	0	0
$687$	1.88776	0.0720224
$688$	−17.3181	−0.660247
$689$	−6.07825	−0.231563
$690$	27.0223	1.02872
$691$	16.3435	0.621736	0.310868	−	0.950453i	$-0.399380\pi$
$691$	16.3435	0.621736	0.310868	+	0.950453i	$0.399380\pi$
$692$	22.7262	0.863919
$693$	0	0
$694$	−51.5357	−1.95627
$695$	−26.2322	−0.995043
$696$	5.05786	0.191718
$697$	9.66905	0.366241
$698$	9.20719	0.348497
$699$	−23.7227	−0.897276
$700$	0	0
$701$	19.9807	0.754660	0.377330	−	0.926079i	$-0.376842\pi$
$701$	19.9807	0.754660	0.377330	+	0.926079i	$0.376842\pi$
$702$	−3.51408	−0.132630
$703$	−5.73661	−0.216360
$704$	28.9048	1.08939
$705$	7.61588	0.286831
$706$	24.1073	0.907289
$707$	0	0
$708$	−18.5654	−0.697729
$709$	−2.38955	−0.0897414	−0.0448707	−	0.998993i	$-0.514288\pi$
$709$	−2.38955	−0.0897414	−0.0448707	+	0.998993i	$0.514288\pi$
$710$	−39.8796	−1.49666
$711$	−4.79676	−0.179893
$712$	−7.91316	−0.296558
$713$	19.6668	0.736527
$714$	0	0
$715$	25.8479	0.966659
$716$	2.61063	0.0975640
$717$	−21.9159	−0.818464
$718$	−60.0563	−2.24128
$719$	−48.8139	−1.82045	−0.910225	−	0.414113i	$-0.864092\pi$
$719$	−48.8139	−1.82045	−0.910225	+	0.414113i	$0.864092\pi$
$720$	11.5563	0.430679
$721$	0	0
$722$	1.91670	0.0713322
$723$	−17.2445	−0.641328
$724$	−31.0200	−1.15285
$725$	11.8247	0.439159
$726$	−37.8720	−1.40556
$727$	27.4034	1.01634	0.508168	−	0.861258i	$-0.330323\pi$
$727$	27.4034	1.01634	0.508168	+	0.861258i	$0.330323\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	40.5345	1.50025
$731$	14.8170	0.548027
$732$	14.1122	0.521604
$733$	−16.5229	−0.610288	−0.305144	−	0.952306i	$-0.598705\pi$
$733$	−16.5229	−0.610288	−0.305144	+	0.952306i	$0.598705\pi$
$734$	40.4119	1.49163
$735$	0	0
$736$	−41.3891	−1.52562
$737$	34.0764	1.25522
$738$	4.76477	0.175393
$739$	−22.6771	−0.834192	−0.417096	−	0.908863i	$-0.636952\pi$
$739$	−22.6771	−0.834192	−0.417096	+	0.908863i	$0.636952\pi$
$740$	−24.4077	−0.897243
$741$	1.83340	0.0673516
$742$	0	0
$743$	15.2185	0.558313	0.279156	−	0.960246i	$-0.409945\pi$
$743$	15.2185	0.558313	0.279156	+	0.960246i	$0.409945\pi$
$744$	2.21752	0.0812981
$745$	−30.5250	−1.11835
$746$	−45.2526	−1.65682
$747$	15.3234	0.560652
$748$	−36.1052	−1.32014
$749$	0	0
$750$	−17.2384	−0.629457
$751$	−12.3674	−0.451294	−0.225647	−	0.974209i	$-0.572450\pi$
$751$	−12.3674	−0.451294	−0.225647	+	0.974209i	$0.572450\pi$
$752$	−13.6199	−0.496667
$753$	−3.72636	−0.135796
$754$	28.4223	1.03508
$755$	−48.4165	−1.76206
$756$	0	0
$757$	−14.9271	−0.542536	−0.271268	−	0.962504i	$-0.587443\pi$
$757$	−14.9271	−0.542536	−0.271268	+	0.962504i	$0.587443\pi$
$758$	13.6815	0.496933
$759$	30.7590	1.11648
$760$	−1.58965	−0.0576627
$761$	−15.2563	−0.553041	−0.276520	−	0.961008i	$-0.589181\pi$
$761$	−15.2563	−0.553041	−0.276520	+	0.961008i	$0.589181\pi$
$762$	−14.2780	−0.517239
$763$	0	0
$764$	−39.1338	−1.41581
$765$	−9.88734	−0.357477
$766$	18.0501	0.652177
$767$	20.3364	0.734303
$768$	−19.8847	−0.717527
$769$	10.7563	0.387883	0.193941	−	0.981013i	$-0.437873\pi$
$769$	10.7563	0.387883	0.193941	+	0.981013i	$0.437873\pi$
$770$	0	0
$771$	4.42999	0.159542
$772$	28.1670	1.01375
$773$	13.8026	0.496444	0.248222	−	0.968703i	$-0.420154\pi$
$773$	13.8026	0.496444	0.248222	+	0.968703i	$0.420154\pi$
$774$	7.30160	0.262451
$775$	5.18432	0.186226
$776$	9.45779	0.339515
$777$	0	0
$778$	−53.8057	−1.92903
$779$	−2.48592	−0.0890674
$780$	7.80060	0.279306
$781$	−45.3941	−1.62433
$782$	41.3463	1.47854
$783$	−8.08812	−0.289046
$784$	0	0
$785$	15.0748	0.538043
$786$	−29.5854	−1.05528
$787$	19.6101	0.699023	0.349512	−	0.936932i	$-0.386347\pi$
$787$	19.6101	0.699023	0.349512	+	0.936932i	$0.386347\pi$
$788$	−38.6980	−1.37856
$789$	−2.23297	−0.0794958
$790$	23.3714	0.831519
$791$	0	0
$792$	3.46820	0.123237
$793$	−15.4584	−0.548945
$794$	12.6325	0.448312
$795$	−8.42761	−0.298897
$796$	−20.3798	−0.722345
$797$	25.2766	0.895344	0.447672	−	0.894198i	$-0.352253\pi$
$797$	25.2766	0.895344	0.447672	+	0.894198i	$0.352253\pi$
$798$	0	0
$799$	11.6529	0.412250
$800$	−10.9105	−0.385744
$801$	12.6541	0.447111
$802$	65.5252	2.31378
$803$	46.1396	1.62823
$804$	10.2839	0.362684
$805$	0	0
$806$	12.4612	0.438927
$807$	4.87505	0.171610
$808$	9.36870	0.329590
$809$	36.3718	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$809$	36.3718	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$810$	−4.87234	−0.171196
$811$	−38.3409	−1.34633	−0.673165	−	0.739492i	$-0.735066\pi$
$811$	−38.3409	−1.34633	−0.673165	+	0.739492i	$0.735066\pi$
$812$	0	0
$813$	−16.5519	−0.580500
$814$	−60.9811	−2.13739
$815$	5.87426	0.205766
$816$	17.6821	0.618996
$817$	−3.80947	−0.133276
$818$	64.1241	2.24205
$819$	0	0
$820$	−10.5769	−0.369361
$821$	7.02009	0.245003	0.122501	−	0.992468i	$-0.460908\pi$
$821$	7.02009	0.245003	0.122501	+	0.992468i	$0.460908\pi$
$822$	29.6205	1.03313
$823$	−18.4767	−0.644056	−0.322028	−	0.946730i	$-0.604365\pi$
$823$	−18.4767	−0.644056	−0.322028	+	0.946730i	$0.604365\pi$
$824$	3.97829	0.138590
$825$	8.10829	0.282294
$826$	0	0
$827$	−30.2879	−1.05321	−0.526606	−	0.850109i	$-0.676536\pi$
$827$	−30.2879	−1.05321	−0.526606	+	0.850109i	$0.676536\pi$
$828$	9.28268	0.322596
$829$	−5.07259	−0.176178	−0.0880891	−	0.996113i	$-0.528076\pi$
$829$	−5.07259	−0.176178	−0.0880891	+	0.996113i	$0.528076\pi$
$830$	−74.6605	−2.59150
$831$	−28.0841	−0.974226
$832$	−9.55523	−0.331268
$833$	0	0
$834$	−19.7790	−0.684892
$835$	−23.9392	−0.828449
$836$	9.28268	0.321048
$837$	−3.54608	−0.122570
$838$	58.3478	2.01559
$839$	−5.10458	−0.176230	−0.0881149	−	0.996110i	$-0.528084\pi$
$839$	−5.10458	−0.176230	−0.0881149	+	0.996110i	$0.528084\pi$
$840$	0	0
$841$	36.4177	1.25578
$842$	−33.4805	−1.15381
$843$	−11.2674	−0.388071
$844$	−11.6425	−0.400751
$845$	24.5019	0.842889
$846$	5.74237	0.197427
$847$	0	0
$848$	15.0716	0.517559
$849$	15.7952	0.542089
$850$	10.8992	0.373839
$851$	31.8157	1.09063
$852$	−13.6994	−0.469334
$853$	38.4821	1.31760	0.658800	−	0.752318i	$-0.271064\pi$
$853$	38.4821	1.31760	0.658800	+	0.752318i	$0.271064\pi$
$854$	0	0
$855$	2.54204	0.0869361
$856$	−11.2122	−0.383224
$857$	47.8034	1.63293	0.816466	−	0.577394i	$-0.195930\pi$
$857$	47.8034	1.63293	0.816466	+	0.577394i	$0.195930\pi$
$858$	19.4893	0.665355
$859$	10.5654	0.360486	0.180243	−	0.983622i	$-0.442312\pi$
$859$	10.5654	0.360486	0.180243	+	0.983622i	$0.442312\pi$
$860$	−16.2082	−0.552695
$861$	0	0
$862$	47.7199	1.62535
$863$	1.30828	0.0445344	0.0222672	−	0.999752i	$-0.492912\pi$
$863$	1.30828	0.0445344	0.0222672	+	0.999752i	$0.492912\pi$
$864$	7.46278	0.253889
$865$	−34.5161	−1.17358
$866$	21.6927	0.737148
$867$	1.87161	0.0635633
$868$	0	0
$869$	26.6032	0.902452
$870$	39.4080	1.33606
$871$	−11.2649	−0.381695
$872$	2.50642	0.0848780
$873$	−15.1241	−0.511875
$874$	−10.6302	−0.359571
$875$	0	0
$876$	13.9244	0.470461
$877$	47.8543	1.61592	0.807962	−	0.589235i	$-0.200571\pi$
$877$	47.8543	1.61592	0.807962	+	0.589235i	$0.200571\pi$
$878$	−24.4234	−0.824249
$879$	26.4573	0.892384
$880$	−64.0922	−2.16055
$881$	−45.8583	−1.54501	−0.772503	−	0.635011i	$-0.780995\pi$
$881$	−45.8583	−1.54501	−0.772503	+	0.635011i	$0.780995\pi$
$882$	0	0
$883$	28.6217	0.963196	0.481598	−	0.876392i	$-0.340056\pi$
$883$	28.6217	0.963196	0.481598	+	0.876392i	$0.340056\pi$
$884$	11.9355	0.401435
$885$	28.1967	0.947823
$886$	−43.3443	−1.45618
$887$	58.2209	1.95487	0.977434	−	0.211243i	$-0.0677511\pi$
$887$	58.2209	1.95487	0.977434	+	0.211243i	$0.0677511\pi$
$888$	3.58735	0.120384
$889$	0	0
$890$	−61.6550	−2.06668
$891$	−5.54608	−0.185801
$892$	47.5496	1.59208
$893$	−2.99597	−0.100256
$894$	−23.0159	−0.769765
$895$	−3.96498	−0.132535
$896$	0	0
$897$	−10.1682	−0.339506
$898$	9.70958	0.324013
$899$	28.6811	0.956568
$900$	2.44699	0.0815662
$901$	−12.8949	−0.429591
$902$	−26.4258	−0.879881
$903$	0	0
$904$	−1.23056	−0.0409278
$905$	47.1126	1.56608
$906$	−36.5060	−1.21283
$907$	−34.9591	−1.16080	−0.580399	−	0.814332i	$-0.697104\pi$
$907$	−34.9591	−1.16080	−0.580399	+	0.814332i	$0.697104\pi$
$908$	−5.54527	−0.184026
$909$	−14.9817	−0.496911
$910$	0	0
$911$	14.7994	0.490325	0.245162	−	0.969482i	$-0.421159\pi$
$911$	14.7994	0.490325	0.245162	+	0.969482i	$0.421159\pi$
$912$	−4.54608	−0.150536
$913$	−84.9845	−2.81258
$914$	−14.9418	−0.494231
$915$	−21.4334	−0.708567
$916$	−3.15961	−0.104396
$917$	0	0
$918$	−7.45505	−0.246053
$919$	0.762136	0.0251405	0.0125703	−	0.999921i	$-0.495999\pi$
$919$	0.762136	0.0251405	0.0125703	+	0.999921i	$0.495999\pi$
$920$	8.81633	0.290666
$921$	11.7875	0.388412
$922$	−13.2191	−0.435348
$923$	15.0062	0.493936
$924$	0	0
$925$	8.38685	0.275758
$926$	20.4990	0.673640
$927$	−6.36176	−0.208948
$928$	−60.3598	−1.98141
$929$	−26.5680	−0.871667	−0.435833	−	0.900027i	$-0.643546\pi$
$929$	−26.5680	−0.871667	−0.435833	+	0.900027i	$0.643546\pi$
$930$	17.2777	0.566557
$931$	0	0
$932$	39.7057	1.30060
$933$	2.39938	0.0785521
$934$	−21.4951	−0.703341
$935$	54.8359	1.79333
$936$	−1.14651	−0.0374747
$937$	28.7590	0.939514	0.469757	−	0.882796i	$-0.344342\pi$
$937$	28.7590	0.939514	0.469757	+	0.882796i	$0.344342\pi$
$938$	0	0
$939$	6.66058	0.217360
$940$	−12.7470	−0.415761
$941$	34.1732	1.11402	0.557008	−	0.830507i	$-0.311949\pi$
$941$	34.1732	1.11402	0.557008	+	0.830507i	$0.311949\pi$
$942$	11.3664	0.370337
$943$	13.7871	0.448970
$944$	−50.4258	−1.64122
$945$	0	0
$946$	−40.4952	−1.31661
$947$	−40.5626	−1.31811	−0.659054	−	0.752096i	$-0.729043\pi$
$947$	−40.5626	−1.31811	−0.659054	+	0.752096i	$0.729043\pi$
$948$	8.02853	0.260755
$949$	−15.2526	−0.495122
$950$	−2.80219	−0.0909151
$951$	18.3376	0.594638
$952$	0	0
$953$	−38.6589	−1.25229	−0.626143	−	0.779709i	$-0.715367\pi$
$953$	−38.6589	−1.25229	−0.626143	+	0.779709i	$0.715367\pi$
$954$	−6.35442	−0.205732
$955$	59.4357	1.92329
$956$	36.6815	1.18636
$957$	44.8573	1.45003
$958$	67.9675	2.19593
$959$	0	0
$960$	−13.2485	−0.427594
$961$	−18.4254	−0.594366
$962$	20.1589	0.649949
$963$	17.9296	0.577773
$964$	28.8627	0.929606
$965$	−42.7794	−1.37712
$966$	0	0
$967$	6.83153	0.219687	0.109844	−	0.993949i	$-0.464965\pi$
$967$	6.83153	0.219687	0.109844	+	0.993949i	$0.464965\pi$
$968$	−12.3561	−0.397141
$969$	3.88952	0.124949
$970$	73.6899	2.36604
$971$	−57.7215	−1.85237	−0.926186	−	0.377068i	$-0.876932\pi$
$971$	−57.7215	−1.85237	−0.926186	+	0.377068i	$0.876932\pi$
$972$	−1.67374	−0.0536852
$973$	0	0
$974$	16.4593	0.527392
$975$	−2.68041	−0.0858418
$976$	38.3305	1.22693
$977$	38.7294	1.23906	0.619532	−	0.784972i	$-0.287323\pi$
$977$	38.7294	1.23906	0.619532	+	0.784972i	$0.287323\pi$
$978$	4.42919	0.141630
$979$	−70.1806	−2.24298
$980$	0	0
$981$	−4.00806	−0.127968
$982$	−64.6665	−2.06359
$983$	−44.7814	−1.42831	−0.714153	−	0.699990i	$-0.753188\pi$
$983$	−44.7814	−1.42831	−0.714153	+	0.699990i	$0.753188\pi$
$984$	1.55456	0.0495574
$985$	58.7739	1.87269
$986$	60.2973	1.92026
$987$	0	0
$988$	−3.06863	−0.0976263
$989$	21.1276	0.671818
$990$	27.0223	0.858826
$991$	22.1647	0.704086	0.352043	−	0.935984i	$-0.385487\pi$
$991$	22.1647	0.704086	0.352043	+	0.935984i	$0.385487\pi$
$992$	−26.4636	−0.840219
$993$	3.09099	0.0980897
$994$	0	0
$995$	30.9525	0.981261
$996$	−25.6473	−0.812666
$997$	−5.54303	−0.175549	−0.0877747	−	0.996140i	$-0.527976\pi$
$997$	−5.54303	−0.175549	−0.0877747	+	0.996140i	$0.527976\pi$
$998$	−20.9618	−0.663535
$999$	−5.73661	−0.181498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.2.a.be.1.4		5
3.2	odd	2		8379.2.a.ce.1.2		5
7.6	odd	2		399.2.a.f.1.4	✓	5
21.20	even	2		1197.2.a.p.1.2		5
28.27	even	2		6384.2.a.cc.1.5		5
35.34	odd	2		9975.2.a.bq.1.2		5
133.132	even	2		7581.2.a.x.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.f.1.4	✓	5	7.6	odd	2
1197.2.a.p.1.2		5	21.20	even	2
2793.2.a.be.1.4		5	1.1	even	1	trivial
6384.2.a.cc.1.5		5	28.27	even	2
7581.2.a.x.1.2		5	133.132	even	2
8379.2.a.ce.1.2		5	3.2	odd	2
9975.2.a.bq.1.2		5	35.34	odd	2

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:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2793.a (trivial)

\(f(q)\)	\(=\)	\(q+1.91670 q^{2} -1.00000 q^{3} +1.67374 q^{4} -2.54204 q^{5} -1.91670 q^{6} -0.625344 q^{8} +1.00000 q^{9} -4.87234 q^{10} -5.54608 q^{11} -1.67374 q^{12} +1.83340 q^{13} +2.54204 q^{15} -4.54608 q^{16} +3.88952 q^{17} +1.91670 q^{18} -1.00000 q^{19} -4.25472 q^{20} -10.6302 q^{22} +5.54608 q^{23} +0.625344 q^{24} +1.46199 q^{25} +3.51408 q^{26} -1.00000 q^{27} +8.08812 q^{29} +4.87234 q^{30} +3.54608 q^{31} -7.46278 q^{32} +5.54608 q^{33} +7.45505 q^{34} +1.67374 q^{36} +5.73661 q^{37} -1.91670 q^{38} -1.83340 q^{39} +1.58965 q^{40} +2.48592 q^{41} +3.80947 q^{43} -9.28268 q^{44} -2.54204 q^{45} +10.6302 q^{46} +2.99597 q^{47} +4.54608 q^{48} +2.80219 q^{50} -3.88952 q^{51} +3.06863 q^{52} -3.31529 q^{53} -1.91670 q^{54} +14.0984 q^{55} +1.00000 q^{57} +15.5025 q^{58} +11.0922 q^{59} +4.25472 q^{60} -8.43157 q^{61} +6.79676 q^{62} -5.21175 q^{64} -4.66058 q^{65} +10.6302 q^{66} -6.14424 q^{67} +6.51005 q^{68} -5.54608 q^{69} +8.18491 q^{71} -0.625344 q^{72} -8.31932 q^{73} +10.9954 q^{74} -1.46199 q^{75} -1.67374 q^{76} -3.51408 q^{78} -4.79676 q^{79} +11.5563 q^{80} +1.00000 q^{81} +4.76477 q^{82} +15.3234 q^{83} -9.88734 q^{85} +7.30160 q^{86} -8.08812 q^{87} +3.46820 q^{88} +12.6541 q^{89} -4.87234 q^{90} +9.28268 q^{92} -3.54608 q^{93} +5.74237 q^{94} +2.54204 q^{95} +7.46278 q^{96} -15.1241 q^{97} -5.54608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} - 5 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + 5 q^{9} - 2 q^{11} - 7 q^{12} - 8 q^{13} - 2 q^{15} + 3 q^{16} + 2 q^{17} + q^{18} - 5 q^{19} + 2 q^{20} + 2 q^{22} + 2 q^{23} - 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 5 * q + q^2 - 5 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 3 * q^8 + 5 * q^9 - 2 * q^11 - 7 * q^12 - 8 * q^13 - 2 * q^15 + 3 * q^16 + 2 * q^17 + q^18 - 5 * q^19 + 2 * q^20 + 2 * q^22 + 2 * q^23 - 3 * q^24 + 11 * q^25 + 32 * q^26 - 5 * q^27 - 8 * q^31 - 3 * q^32 + 2 * q^33 + 8 * q^34 + 7 * q^36 + 2 * q^37 - q^38 + 8 * q^39 + 36 * q^40 - 2 * q^41 + 20 * q^43 + 6 * q^44 + 2 * q^45 - 2 * q^46 + 26 * q^47 - 3 * q^48 - q^50 - 2 * q^51 - 4 * q^52 + 4 * q^53 - q^54 + 4 * q^55 + 5 * q^57 - 2 * q^58 + 4 * q^59 - 2 * q^60 - 10 * q^61 - 4 * q^62 - 21 * q^64 - 4 * q^65 - 2 * q^66 + 10 * q^67 + 58 * q^68 - 2 * q^69 + 10 * q^71 + 3 * q^72 - 10 * q^73 - 6 * q^74 - 11 * q^75 - 7 * q^76 - 32 * q^78 + 14 * q^79 + 6 * q^80 + 5 * q^81 + 26 * q^82 + 34 * q^83 - 36 * q^85 + 8 * q^86 + 6 * q^88 - 10 * q^89 - 6 * q^92 + 8 * q^93 + 8 * q^94 - 2 * q^95 + 3 * q^96 + 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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