LMFDB - Embedded newform 2175.2.a.bc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-2,-8,12,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 $ x^8 - 2x^7 - 12x^6 + 23x^5 + 36x^4 - 62x^3 - 15x^2 + 14*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.510732$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.510732 q^{2} -1.00000 q^{3} -1.73915 q^{4} +0.510732 q^{6} +4.82343 q^{7} +1.90970 q^{8} +1.00000 q^{9} +4.88439 q^{11} +1.73915 q^{12} +4.59669 q^{13} -2.46348 q^{14} +2.50296 q^{16} +6.50987 q^{17} -0.510732 q^{18} +3.09205 q^{19} -4.82343 q^{21} -2.49461 q^{22} -5.62549 q^{23} -1.90970 q^{24} -2.34767 q^{26} -1.00000 q^{27} -8.38869 q^{28} +1.00000 q^{29} +9.24375 q^{31} -5.09775 q^{32} -4.88439 q^{33} -3.32480 q^{34} -1.73915 q^{36} -11.1261 q^{37} -1.57921 q^{38} -4.59669 q^{39} -2.84537 q^{41} +2.46348 q^{42} -4.58611 q^{43} -8.49470 q^{44} +2.87311 q^{46} +3.62713 q^{47} -2.50296 q^{48} +16.2655 q^{49} -6.50987 q^{51} -7.99434 q^{52} -0.967845 q^{53} +0.510732 q^{54} +9.21132 q^{56} -3.09205 q^{57} -0.510732 q^{58} -0.298882 q^{59} -0.786908 q^{61} -4.72108 q^{62} +4.82343 q^{63} -2.40234 q^{64} +2.49461 q^{66} +4.86742 q^{67} -11.3217 q^{68} +5.62549 q^{69} +0.741689 q^{71} +1.90970 q^{72} +5.52981 q^{73} +5.68246 q^{74} -5.37755 q^{76} +23.5595 q^{77} +2.34767 q^{78} -2.96278 q^{79} +1.00000 q^{81} +1.45322 q^{82} -13.6633 q^{83} +8.38869 q^{84} +2.34227 q^{86} -1.00000 q^{87} +9.32773 q^{88} +3.67835 q^{89} +22.1718 q^{91} +9.78358 q^{92} -9.24375 q^{93} -1.85249 q^{94} +5.09775 q^{96} -2.87658 q^{97} -8.30730 q^{98} +4.88439 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 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 $

$2$	−0.510732	−0.361142	−0.180571	−	0.983562i	$-0.557795\pi$
$2$	−0.510732	−0.361142	−0.180571	+	0.983562i	$0.557795\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.73915	−0.869577
$5$	0	0
$5$	0	0
$6$	0.510732	0.208505
$7$	4.82343	1.82309	0.911543	−	0.411205i	$-0.134892\pi$
$7$	4.82343	1.82309	0.911543	+	0.411205i	$0.134892\pi$
$8$	1.90970	0.675182
$9$	1.00000	0.333333
$10$	0	0
$11$	4.88439	1.47270	0.736349	−	0.676602i	$-0.236548\pi$
$11$	4.88439	1.47270	0.736349	+	0.676602i	$0.236548\pi$
$12$	1.73915	0.502050
$13$	4.59669	1.27489	0.637446	−	0.770495i	$-0.279991\pi$
$13$	4.59669	1.27489	0.637446	+	0.770495i	$0.279991\pi$
$14$	−2.46348	−0.658392
$15$	0	0
$16$	2.50296	0.625740
$17$	6.50987	1.57888	0.789438	−	0.613830i	$-0.210372\pi$
$17$	6.50987	1.57888	0.789438	+	0.613830i	$0.210372\pi$
$18$	−0.510732	−0.120381
$19$	3.09205	0.709364	0.354682	−	0.934987i	$-0.384589\pi$
$19$	3.09205	0.709364	0.354682	+	0.934987i	$0.384589\pi$
$20$	0	0
$21$	−4.82343	−1.05256
$22$	−2.49461	−0.531853
$23$	−5.62549	−1.17299	−0.586497	−	0.809951i	$-0.699494\pi$
$23$	−5.62549	−1.17299	−0.586497	+	0.809951i	$0.699494\pi$
$24$	−1.90970	−0.389817
$25$	0	0
$26$	−2.34767	−0.460416
$27$	−1.00000	−0.192450
$28$	−8.38869	−1.58531
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.24375	1.66023	0.830113	−	0.557595i	$-0.188276\pi$
$31$	9.24375	1.66023	0.830113	+	0.557595i	$0.188276\pi$
$32$	−5.09775	−0.901163
$33$	−4.88439	−0.850262
$34$	−3.32480	−0.570198
$35$	0	0
$36$	−1.73915	−0.289859
$37$	−11.1261	−1.82912	−0.914561	−	0.404448i	$-0.867464\pi$
$37$	−11.1261	−1.82912	−0.914561	+	0.404448i	$0.867464\pi$
$38$	−1.57921	−0.256181
$39$	−4.59669	−0.736059
$40$	0	0
$41$	−2.84537	−0.444372	−0.222186	−	0.975004i	$-0.571319\pi$
$41$	−2.84537	−0.444372	−0.222186	+	0.975004i	$0.571319\pi$
$42$	2.46348	0.380123
$43$	−4.58611	−0.699375	−0.349688	−	0.936866i	$-0.613712\pi$
$43$	−4.58611	−0.699375	−0.349688	+	0.936866i	$0.613712\pi$
$44$	−8.49470	−1.28062
$45$	0	0
$46$	2.87311	0.423617
$47$	3.62713	0.529071	0.264536	−	0.964376i	$-0.414781\pi$
$47$	3.62713	0.529071	0.264536	+	0.964376i	$0.414781\pi$
$48$	−2.50296	−0.361271
$49$	16.2655	2.32364
$50$	0	0
$51$	−6.50987	−0.911564
$52$	−7.99434	−1.10862
$53$	−0.967845	−0.132944	−0.0664719	−	0.997788i	$-0.521174\pi$
$53$	−0.967845	−0.132944	−0.0664719	+	0.997788i	$0.521174\pi$
$54$	0.510732	0.0695018
$55$	0	0
$56$	9.21132	1.23091
$57$	−3.09205	−0.409552
$58$	−0.510732	−0.0670623
$59$	−0.298882	−0.0389112	−0.0194556	−	0.999811i	$-0.506193\pi$
$59$	−0.298882	−0.0389112	−0.0194556	+	0.999811i	$0.506193\pi$
$60$	0	0
$61$	−0.786908	−0.100753	−0.0503766	−	0.998730i	$-0.516042\pi$
$61$	−0.786908	−0.100753	−0.0503766	+	0.998730i	$0.516042\pi$
$62$	−4.72108	−0.599577
$63$	4.82343	0.607695
$64$	−2.40234	−0.300293
$65$	0	0
$66$	2.49461	0.307065
$67$	4.86742	0.594650	0.297325	−	0.954776i	$-0.403906\pi$
$67$	4.86742	0.594650	0.297325	+	0.954776i	$0.403906\pi$
$68$	−11.3217	−1.37295
$69$	5.62549	0.677229
$70$	0	0
$71$	0.741689	0.0880223	0.0440112	−	0.999031i	$-0.485986\pi$
$71$	0.741689	0.0880223	0.0440112	+	0.999031i	$0.485986\pi$
$72$	1.90970	0.225061
$73$	5.52981	0.647215	0.323608	−	0.946191i	$-0.395104\pi$
$73$	5.52981	0.647215	0.323608	+	0.946191i	$0.395104\pi$
$74$	5.68246	0.660572
$75$	0	0
$76$	−5.37755	−0.616847
$77$	23.5595	2.68485
$78$	2.34767	0.265822
$79$	−2.96278	−0.333339	−0.166669	−	0.986013i	$-0.553301\pi$
$79$	−2.96278	−0.333339	−0.166669	+	0.986013i	$0.553301\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.45322	0.160481
$83$	−13.6633	−1.49975	−0.749874	−	0.661581i	$-0.769886\pi$
$83$	−13.6633	−1.49975	−0.749874	+	0.661581i	$0.769886\pi$
$84$	8.38869	0.915281
$85$	0	0
$86$	2.34227	0.252574
$87$	−1.00000	−0.107211
$88$	9.32773	0.994339
$89$	3.67835	0.389905	0.194952	−	0.980813i	$-0.437545\pi$
$89$	3.67835	0.389905	0.194952	+	0.980813i	$0.437545\pi$
$90$	0	0
$91$	22.1718	2.32424
$92$	9.78358	1.02001
$93$	−9.24375	−0.958532
$94$	−1.85249	−0.191070
$95$	0	0
$96$	5.09775	0.520287
$97$	−2.87658	−0.292072	−0.146036	−	0.989279i	$-0.546652\pi$
$97$	−2.87658	−0.292072	−0.146036	+	0.989279i	$0.546652\pi$
$98$	−8.30730	−0.839164
$99$	4.88439	0.490899
$100$	0	0
$101$	−14.6780	−1.46051	−0.730255	−	0.683174i	$-0.760599\pi$
$101$	−14.6780	−1.46051	−0.730255	+	0.683174i	$0.760599\pi$
$102$	3.32480	0.329204
$103$	−2.48154	−0.244514	−0.122257	−	0.992498i	$-0.539013\pi$
$103$	−2.48154	−0.244514	−0.122257	+	0.992498i	$0.539013\pi$
$104$	8.77831	0.860784
$105$	0	0
$106$	0.494309	0.0480115
$107$	−4.48423	−0.433507	−0.216753	−	0.976226i	$-0.569547\pi$
$107$	−4.48423	−0.433507	−0.216753	+	0.976226i	$0.569547\pi$
$108$	1.73915	0.167350
$109$	3.28140	0.314301	0.157151	−	0.987575i	$-0.449769\pi$
$109$	3.28140	0.314301	0.157151	+	0.987575i	$0.449769\pi$
$110$	0	0
$111$	11.1261	1.05604
$112$	12.0729	1.14078
$113$	−17.9820	−1.69160	−0.845800	−	0.533500i	$-0.820877\pi$
$113$	−17.9820	−1.69160	−0.845800	+	0.533500i	$0.820877\pi$
$114$	1.57921	0.147906
$115$	0	0
$116$	−1.73915	−0.161476
$117$	4.59669	0.424964
$118$	0.152649	0.0140524
$119$	31.3999	2.87843
$120$	0	0
$121$	12.8572	1.16884
$122$	0.401899	0.0363862
$123$	2.84537	0.256558
$124$	−16.0763	−1.44369
$125$	0	0
$126$	−2.46348	−0.219464
$127$	−13.5099	−1.19881	−0.599405	−	0.800446i	$-0.704596\pi$
$127$	−13.5099	−1.19881	−0.599405	+	0.800446i	$0.704596\pi$
$128$	11.4224	1.00961
$129$	4.58611	0.403784
$130$	0	0
$131$	12.1450	1.06111	0.530556	−	0.847650i	$-0.321983\pi$
$131$	12.1450	1.06111	0.530556	+	0.847650i	$0.321983\pi$
$132$	8.49470	0.739368
$133$	14.9143	1.29323
$134$	−2.48594	−0.214753
$135$	0	0
$136$	12.4319	1.06603
$137$	−9.97466	−0.852193	−0.426096	−	0.904678i	$-0.640112\pi$
$137$	−9.97466	−0.852193	−0.426096	+	0.904678i	$0.640112\pi$
$138$	−2.87311	−0.244576
$139$	2.82971	0.240013	0.120007	−	0.992773i	$-0.461708\pi$
$139$	2.82971	0.240013	0.120007	+	0.992773i	$0.461708\pi$
$140$	0	0
$141$	−3.62713	−0.305459
$142$	−0.378804	−0.0317885
$143$	22.4520	1.87753
$144$	2.50296	0.208580
$145$	0	0
$146$	−2.82425	−0.233736
$147$	−16.2655	−1.34155
$148$	19.3500	1.59056
$149$	−22.5396	−1.84652	−0.923259	−	0.384178i	$-0.874485\pi$
$149$	−22.5396	−1.84652	−0.923259	+	0.384178i	$0.874485\pi$
$150$	0	0
$151$	−14.2205	−1.15725	−0.578626	−	0.815593i	$-0.696411\pi$
$151$	−14.2205	−1.15725	−0.578626	+	0.815593i	$0.696411\pi$
$152$	5.90490	0.478950
$153$	6.50987	0.526292
$154$	−12.0326	−0.969613
$155$	0	0
$156$	7.99434	0.640059
$157$	10.7712	0.859632	0.429816	−	0.902917i	$-0.358579\pi$
$157$	10.7712	0.859632	0.429816	+	0.902917i	$0.358579\pi$
$158$	1.51319	0.120383
$159$	0.967845	0.0767551
$160$	0	0
$161$	−27.1341	−2.13847
$162$	−0.510732	−0.0401269
$163$	−10.6908	−0.837365	−0.418682	−	0.908133i	$-0.637508\pi$
$163$	−10.6908	−0.837365	−0.418682	+	0.908133i	$0.637508\pi$
$164$	4.94853	0.386415
$165$	0	0
$166$	6.97830	0.541621
$167$	7.90643	0.611818	0.305909	−	0.952061i	$-0.401040\pi$
$167$	7.90643	0.611818	0.305909	+	0.952061i	$0.401040\pi$
$168$	−9.21132	−0.710669
$169$	8.12952	0.625347
$170$	0	0
$171$	3.09205	0.236455
$172$	7.97595	0.608160
$173$	3.76760	0.286445	0.143223	−	0.989690i	$-0.454253\pi$
$173$	3.76760	0.286445	0.143223	+	0.989690i	$0.454253\pi$
$174$	0.510732	0.0387185
$175$	0	0
$176$	12.2254	0.921526
$177$	0.298882	0.0224654
$178$	−1.87865	−0.140811
$179$	10.3282	0.771964	0.385982	−	0.922506i	$-0.373863\pi$
$179$	10.3282	0.771964	0.385982	+	0.922506i	$0.373863\pi$
$180$	0	0
$181$	14.4533	1.07431	0.537154	−	0.843484i	$-0.319500\pi$
$181$	14.4533	1.07431	0.537154	+	0.843484i	$0.319500\pi$
$182$	−11.3238	−0.839378
$183$	0.786908	0.0581699
$184$	−10.7430	−0.791985
$185$	0	0
$186$	4.72108	0.346166
$187$	31.7967	2.32521
$188$	−6.30813	−0.460068
$189$	−4.82343	−0.350853
$190$	0	0
$191$	8.81550	0.637867	0.318934	−	0.947777i	$-0.396675\pi$
$191$	8.81550	0.637867	0.318934	+	0.947777i	$0.396675\pi$
$192$	2.40234	0.173374
$193$	17.2543	1.24199	0.620996	−	0.783814i	$-0.286728\pi$
$193$	17.2543	1.24199	0.620996	+	0.783814i	$0.286728\pi$
$194$	1.46916	0.105479
$195$	0	0
$196$	−28.2882	−2.02058
$197$	−22.5719	−1.60818	−0.804090	−	0.594508i	$-0.797347\pi$
$197$	−22.5719	−1.60818	−0.804090	+	0.594508i	$0.797347\pi$
$198$	−2.49461	−0.177284
$199$	−13.7975	−0.978078	−0.489039	−	0.872262i	$-0.662652\pi$
$199$	−13.7975	−0.978078	−0.489039	+	0.872262i	$0.662652\pi$
$200$	0	0
$201$	−4.86742	−0.343321
$202$	7.49649	0.527451
$203$	4.82343	0.338538
$204$	11.3217	0.792675
$205$	0	0
$206$	1.26740	0.0883041
$207$	−5.62549	−0.390998
$208$	11.5053	0.797751
$209$	15.1028	1.04468
$210$	0	0
$211$	16.4705	1.13387	0.566937	−	0.823761i	$-0.308128\pi$
$211$	16.4705	1.13387	0.566937	+	0.823761i	$0.308128\pi$
$212$	1.68323	0.115605
$213$	−0.741689	−0.0508197
$214$	2.29024	0.156557
$215$	0	0
$216$	−1.90970	−0.129939
$217$	44.5866	3.02674
$218$	−1.67592	−0.113507
$219$	−5.52981	−0.373670
$220$	0	0
$221$	29.9238	2.01289
$222$	−5.68246	−0.381382
$223$	−4.58903	−0.307304	−0.153652	−	0.988125i	$-0.549104\pi$
$223$	−4.58903	−0.307304	−0.153652	+	0.988125i	$0.549104\pi$
$224$	−24.5886	−1.64290
$225$	0	0
$226$	9.18396	0.610908
$227$	−25.5254	−1.69418	−0.847089	−	0.531452i	$-0.821647\pi$
$227$	−25.5254	−1.69418	−0.847089	+	0.531452i	$0.821647\pi$
$228$	5.37755	0.356137
$229$	−8.04982	−0.531947	−0.265974	−	0.963980i	$-0.585693\pi$
$229$	−8.04982	−0.531947	−0.265974	+	0.963980i	$0.585693\pi$
$230$	0	0
$231$	−23.5595	−1.55010
$232$	1.90970	0.125378
$233$	3.75903	0.246262	0.123131	−	0.992390i	$-0.460706\pi$
$233$	3.75903	0.246262	0.123131	+	0.992390i	$0.460706\pi$
$234$	−2.34767	−0.153472
$235$	0	0
$236$	0.519802	0.0338362
$237$	2.96278	0.192453
$238$	−16.0369	−1.03952
$239$	−3.53192	−0.228461	−0.114231	−	0.993454i	$-0.536440\pi$
$239$	−3.53192	−0.228461	−0.114231	+	0.993454i	$0.536440\pi$
$240$	0	0
$241$	3.05553	0.196824	0.0984118	−	0.995146i	$-0.468624\pi$
$241$	3.05553	0.196824	0.0984118	+	0.995146i	$0.468624\pi$
$242$	−6.56659	−0.422117
$243$	−1.00000	−0.0641500
$244$	1.36855	0.0876126
$245$	0	0
$246$	−1.45322	−0.0926538
$247$	14.2132	0.904363
$248$	17.6528	1.12096
$249$	13.6633	0.865880
$250$	0	0
$251$	−23.3283	−1.47247	−0.736233	−	0.676728i	$-0.763397\pi$
$251$	−23.3283	−1.47247	−0.736233	+	0.676728i	$0.763397\pi$
$252$	−8.38869	−0.528437
$253$	−27.4770	−1.72747
$254$	6.89994	0.432941
$255$	0	0
$256$	−1.02912	−0.0643202
$257$	28.2268	1.76074	0.880369	−	0.474289i	$-0.157295\pi$
$257$	28.2268	1.76074	0.880369	+	0.474289i	$0.157295\pi$
$258$	−2.34227	−0.145823
$259$	−53.6660	−3.33465
$260$	0	0
$261$	1.00000	0.0618984
$262$	−6.20283	−0.383212
$263$	−27.6688	−1.70613	−0.853067	−	0.521802i	$-0.825260\pi$
$263$	−27.6688	−1.70613	−0.853067	+	0.521802i	$0.825260\pi$
$264$	−9.32773	−0.574082
$265$	0	0
$266$	−7.61719	−0.467040
$267$	−3.67835	−0.225112
$268$	−8.46519	−0.517094
$269$	0.839052	0.0511579	0.0255790	−	0.999673i	$-0.491857\pi$
$269$	0.839052	0.0511579	0.0255790	+	0.999673i	$0.491857\pi$
$270$	0	0
$271$	1.27423	0.0774037	0.0387018	−	0.999251i	$-0.487678\pi$
$271$	1.27423	0.0774037	0.0387018	+	0.999251i	$0.487678\pi$
$272$	16.2940	0.987966
$273$	−22.1718	−1.34190
$274$	5.09437	0.307762
$275$	0	0
$276$	−9.78358	−0.588902
$277$	−26.7387	−1.60657	−0.803287	−	0.595592i	$-0.796917\pi$
$277$	−26.7387	−1.60657	−0.803287	+	0.595592i	$0.796917\pi$
$278$	−1.44522	−0.0866788
$279$	9.24375	0.553409
$280$	0	0
$281$	−16.3446	−0.975038	−0.487519	−	0.873112i	$-0.662098\pi$
$281$	−16.3446	−0.975038	−0.487519	+	0.873112i	$0.662098\pi$
$282$	1.85249	0.110314
$283$	23.7207	1.41005	0.705025	−	0.709183i	$-0.250936\pi$
$283$	23.7207	1.41005	0.705025	+	0.709183i	$0.250936\pi$
$284$	−1.28991	−0.0765422
$285$	0	0
$286$	−11.4669	−0.678054
$287$	−13.7244	−0.810127
$288$	−5.09775	−0.300388
$289$	25.3784	1.49285
$290$	0	0
$291$	2.87658	0.168628
$292$	−9.61718	−0.562803
$293$	−8.28535	−0.484035	−0.242018	−	0.970272i	$-0.577809\pi$
$293$	−8.28535	−0.484035	−0.242018	+	0.970272i	$0.577809\pi$
$294$	8.30730	0.484491
$295$	0	0
$296$	−21.2476	−1.23499
$297$	−4.88439	−0.283421
$298$	11.5117	0.666855
$299$	−25.8586	−1.49544
$300$	0	0
$301$	−22.1208	−1.27502
$302$	7.26288	0.417932
$303$	14.6780	0.843226
$304$	7.73927	0.443878
$305$	0	0
$306$	−3.32480	−0.190066
$307$	−16.3024	−0.930425	−0.465213	−	0.885199i	$-0.654022\pi$
$307$	−16.3024	−0.930425	−0.465213	+	0.885199i	$0.654022\pi$
$308$	−40.9736	−2.33469
$309$	2.48154	0.141170
$310$	0	0
$311$	−7.72085	−0.437809	−0.218905	−	0.975746i	$-0.570248\pi$
$311$	−7.72085	−0.437809	−0.218905	+	0.975746i	$0.570248\pi$
$312$	−8.77831	−0.496974
$313$	−6.22393	−0.351797	−0.175899	−	0.984408i	$-0.556283\pi$
$313$	−6.22393	−0.351797	−0.175899	+	0.984408i	$0.556283\pi$
$314$	−5.50117	−0.310449
$315$	0	0
$316$	5.15273	0.289864
$317$	−31.0672	−1.74491	−0.872454	−	0.488696i	$-0.837473\pi$
$317$	−31.0672	−1.74491	−0.872454	+	0.488696i	$0.837473\pi$
$318$	−0.494309	−0.0277195
$319$	4.88439	0.273473
$320$	0	0
$321$	4.48423	0.250285
$322$	13.8583	0.772291
$323$	20.1288	1.12000
$324$	−1.73915	−0.0966196
$325$	0	0
$326$	5.46011	0.302407
$327$	−3.28140	−0.181462
$328$	−5.43381	−0.300032
$329$	17.4952	0.964542
$330$	0	0
$331$	34.8731	1.91680	0.958399	−	0.285431i	$-0.0921366\pi$
$331$	34.8731	1.91680	0.958399	+	0.285431i	$0.0921366\pi$
$332$	23.7627	1.30415
$333$	−11.1261	−0.609707
$334$	−4.03806	−0.220953
$335$	0	0
$336$	−12.0729	−0.658628
$337$	−16.5875	−0.903579	−0.451790	−	0.892124i	$-0.649214\pi$
$337$	−16.5875	−0.903579	−0.451790	+	0.892124i	$0.649214\pi$
$338$	−4.15200	−0.225839
$339$	17.9820	0.976646
$340$	0	0
$341$	45.1500	2.44501
$342$	−1.57921	−0.0853937
$343$	44.6914	2.41311
$344$	−8.75811	−0.472206
$345$	0	0
$346$	−1.92423	−0.103447
$347$	0.413605	0.0222035	0.0111017	−	0.999938i	$-0.496466\pi$
$347$	0.413605	0.0222035	0.0111017	+	0.999938i	$0.496466\pi$
$348$	1.73915	0.0932284
$349$	35.5590	1.90343	0.951715	−	0.306982i	$-0.0993190\pi$
$349$	35.5590	1.90343	0.951715	+	0.306982i	$0.0993190\pi$
$350$	0	0
$351$	−4.59669	−0.245353
$352$	−24.8994	−1.32714
$353$	3.08030	0.163948	0.0819740	−	0.996634i	$-0.473878\pi$
$353$	3.08030	0.163948	0.0819740	+	0.996634i	$0.473878\pi$
$354$	−0.152649	−0.00811318
$355$	0	0
$356$	−6.39722	−0.339052
$357$	−31.3999	−1.66186
$358$	−5.27492	−0.278788
$359$	−14.3219	−0.755879	−0.377940	−	0.925830i	$-0.623367\pi$
$359$	−14.3219	−0.755879	−0.377940	+	0.925830i	$0.623367\pi$
$360$	0	0
$361$	−9.43924	−0.496802
$362$	−7.38177	−0.387977
$363$	−12.8572	−0.674829
$364$	−38.5601	−2.02110
$365$	0	0
$366$	−0.401899	−0.0210076
$367$	17.3009	0.903102	0.451551	−	0.892245i	$-0.350871\pi$
$367$	17.3009	0.903102	0.451551	+	0.892245i	$0.350871\pi$
$368$	−14.0804	−0.733990
$369$	−2.84537	−0.148124
$370$	0	0
$371$	−4.66833	−0.242368
$372$	16.0763	0.833517
$373$	31.8314	1.64817	0.824083	−	0.566469i	$-0.191691\pi$
$373$	31.8314	1.64817	0.824083	+	0.566469i	$0.191691\pi$
$374$	−16.2396	−0.839729
$375$	0	0
$376$	6.92674	0.357219
$377$	4.59669	0.236741
$378$	2.46348	0.126708
$379$	−18.1855	−0.934127	−0.467064	−	0.884224i	$-0.654688\pi$
$379$	−18.1855	−0.934127	−0.467064	+	0.884224i	$0.654688\pi$
$380$	0	0
$381$	13.5099	0.692134
$382$	−4.50236	−0.230361
$383$	32.3026	1.65059	0.825293	−	0.564704i	$-0.191010\pi$
$383$	32.3026	1.65059	0.825293	+	0.564704i	$0.191010\pi$
$384$	−11.4224	−0.582899
$385$	0	0
$386$	−8.81232	−0.448535
$387$	−4.58611	−0.233125
$388$	5.00281	0.253979
$389$	−26.5696	−1.34713	−0.673567	−	0.739126i	$-0.735239\pi$
$389$	−26.5696	−1.34713	−0.673567	+	0.739126i	$0.735239\pi$
$390$	0	0
$391$	−36.6212	−1.85201
$392$	31.0622	1.56888
$393$	−12.1450	−0.612634
$394$	11.5282	0.580781
$395$	0	0
$396$	−8.49470	−0.426875
$397$	8.87832	0.445590	0.222795	−	0.974865i	$-0.428482\pi$
$397$	8.87832	0.445590	0.222795	+	0.974865i	$0.428482\pi$
$398$	7.04681	0.353225
$399$	−14.9143	−0.746648
$400$	0	0
$401$	25.6502	1.28091	0.640455	−	0.767995i	$-0.278746\pi$
$401$	25.6502	1.28091	0.640455	+	0.767995i	$0.278746\pi$
$402$	2.48594	0.123988
$403$	42.4906	2.11661
$404$	25.5272	1.27003
$405$	0	0
$406$	−2.46348	−0.122260
$407$	−54.3442	−2.69374
$408$	−12.4319	−0.615472
$409$	9.92575	0.490797	0.245398	−	0.969422i	$-0.421081\pi$
$409$	9.92575	0.490797	0.245398	+	0.969422i	$0.421081\pi$
$410$	0	0
$411$	9.97466	0.492014
$412$	4.31579	0.212624
$413$	−1.44164	−0.0709384
$414$	2.87311	0.141206
$415$	0	0
$416$	−23.4327	−1.14888
$417$	−2.82971	−0.138572
$418$	−7.71345	−0.377277
$419$	8.12620	0.396991	0.198495	−	0.980102i	$-0.436394\pi$
$419$	8.12620	0.396991	0.198495	+	0.980102i	$0.436394\pi$
$420$	0	0
$421$	−8.37826	−0.408332	−0.204166	−	0.978936i	$-0.565448\pi$
$421$	−8.37826	−0.408332	−0.204166	+	0.978936i	$0.565448\pi$
$422$	−8.41198	−0.409489
$423$	3.62713	0.176357
$424$	−1.84830	−0.0897612
$425$	0	0
$426$	0.378804	0.0183531
$427$	−3.79559	−0.183682
$428$	7.79876	0.376967
$429$	−22.4520	−1.08399
$430$	0	0
$431$	−32.3877	−1.56006	−0.780030	−	0.625742i	$-0.784796\pi$
$431$	−32.3877	−1.56006	−0.780030	+	0.625742i	$0.784796\pi$
$432$	−2.50296	−0.120424
$433$	9.89328	0.475441	0.237720	−	0.971334i	$-0.423600\pi$
$433$	9.89328	0.475441	0.237720	+	0.971334i	$0.423600\pi$
$434$	−22.7718	−1.09308
$435$	0	0
$436$	−5.70686	−0.273309
$437$	−17.3943	−0.832081
$438$	2.82425	0.134948
$439$	5.84726	0.279075	0.139537	−	0.990217i	$-0.455438\pi$
$439$	5.84726	0.279075	0.139537	+	0.990217i	$0.455438\pi$
$440$	0	0
$441$	16.2655	0.774547
$442$	−15.2830	−0.726940
$443$	3.14176	0.149270	0.0746348	−	0.997211i	$-0.476221\pi$
$443$	3.14176	0.149270	0.0746348	+	0.997211i	$0.476221\pi$
$444$	−19.3500	−0.918311
$445$	0	0
$446$	2.34376	0.110980
$447$	22.5396	1.06609
$448$	−11.5875	−0.547459
$449$	27.8643	1.31500	0.657500	−	0.753455i	$-0.271614\pi$
$449$	27.8643	1.31500	0.657500	+	0.753455i	$0.271614\pi$
$450$	0	0
$451$	−13.8979	−0.654425
$452$	31.2734	1.47098
$453$	14.2205	0.668139
$454$	13.0366	0.611838
$455$	0	0
$456$	−5.90490	−0.276522
$457$	−31.8928	−1.49188	−0.745940	−	0.666013i	$-0.768000\pi$
$457$	−31.8928	−1.49188	−0.745940	+	0.666013i	$0.768000\pi$
$458$	4.11130	0.192108
$459$	−6.50987	−0.303855
$460$	0	0
$461$	5.64660	0.262989	0.131494	−	0.991317i	$-0.458022\pi$
$461$	5.64660	0.262989	0.131494	+	0.991317i	$0.458022\pi$
$462$	12.0326	0.559806
$463$	32.7602	1.52250	0.761248	−	0.648460i	$-0.224587\pi$
$463$	32.7602	1.52250	0.761248	+	0.648460i	$0.224587\pi$
$464$	2.50296	0.116197
$465$	0	0
$466$	−1.91985	−0.0889355
$467$	−37.4129	−1.73126	−0.865632	−	0.500680i	$-0.833083\pi$
$467$	−37.4129	−1.73126	−0.865632	+	0.500680i	$0.833083\pi$
$468$	−7.99434	−0.369539
$469$	23.4777	1.08410
$470$	0	0
$471$	−10.7712	−0.496308
$472$	−0.570777	−0.0262721
$473$	−22.4003	−1.02997
$474$	−1.51319	−0.0695029
$475$	0	0
$476$	−54.6093	−2.50301
$477$	−0.967845	−0.0443146
$478$	1.80387	0.0825069
$479$	−13.0770	−0.597503	−0.298751	−	0.954331i	$-0.596570\pi$
$479$	−13.0770	−0.597503	−0.298751	+	0.954331i	$0.596570\pi$
$480$	0	0
$481$	−51.1432	−2.33193
$482$	−1.56055	−0.0710813
$483$	27.1341	1.23465
$484$	−22.3607	−1.01639
$485$	0	0
$486$	0.510732	0.0231673
$487$	19.4492	0.881328	0.440664	−	0.897672i	$-0.354743\pi$
$487$	19.4492	0.881328	0.440664	+	0.897672i	$0.354743\pi$
$488$	−1.50276	−0.0680268
$489$	10.6908	0.483453
$490$	0	0
$491$	15.8221	0.714041	0.357020	−	0.934097i	$-0.383793\pi$
$491$	15.8221	0.714041	0.357020	+	0.934097i	$0.383793\pi$
$492$	−4.94853	−0.223097
$493$	6.50987	0.293190
$494$	−7.25912	−0.326603
$495$	0	0
$496$	23.1367	1.03887
$497$	3.57749	0.160472
$498$	−6.97830	−0.312705
$499$	4.88592	0.218724	0.109362	−	0.994002i	$-0.465119\pi$
$499$	4.88592	0.218724	0.109362	+	0.994002i	$0.465119\pi$
$500$	0	0
$501$	−7.90643	−0.353233
$502$	11.9145	0.531769
$503$	−16.9719	−0.756742	−0.378371	−	0.925654i	$-0.623516\pi$
$503$	−16.9719	−0.756742	−0.378371	+	0.925654i	$0.623516\pi$
$504$	9.21132	0.410305
$505$	0	0
$506$	14.0334	0.623860
$507$	−8.12952	−0.361045
$508$	23.4958	1.04246
$509$	−7.53223	−0.333860	−0.166930	−	0.985969i	$-0.553385\pi$
$509$	−7.53223	−0.333860	−0.166930	+	0.985969i	$0.553385\pi$
$510$	0	0
$511$	26.6726	1.17993
$512$	−22.3193	−0.986383
$513$	−3.09205	−0.136517
$514$	−14.4163	−0.635876
$515$	0	0
$516$	−7.97595	−0.351122
$517$	17.7163	0.779162
$518$	27.4089	1.20428
$519$	−3.76760	−0.165379
$520$	0	0
$521$	−0.748655	−0.0327992	−0.0163996	−	0.999866i	$-0.505220\pi$
$521$	−0.748655	−0.0327992	−0.0163996	+	0.999866i	$0.505220\pi$
$522$	−0.510732	−0.0223541
$523$	18.3909	0.804178	0.402089	−	0.915601i	$-0.368284\pi$
$523$	18.3909	0.804178	0.402089	+	0.915601i	$0.368284\pi$
$524$	−21.1220	−0.922719
$525$	0	0
$526$	14.1313	0.616156
$527$	60.1756	2.62129
$528$	−12.2254	−0.532043
$529$	8.64609	0.375917
$530$	0	0
$531$	−0.298882	−0.0129704
$532$	−25.9382	−1.12456
$533$	−13.0793	−0.566525
$534$	1.87865	0.0812972
$535$	0	0
$536$	9.29533	0.401497
$537$	−10.3282	−0.445693
$538$	−0.428531	−0.0184753
$539$	79.4469	3.42202
$540$	0	0
$541$	33.4160	1.43667	0.718334	−	0.695698i	$-0.244905\pi$
$541$	33.4160	1.43667	0.718334	+	0.695698i	$0.244905\pi$
$542$	−0.650787	−0.0279537
$543$	−14.4533	−0.620251
$544$	−33.1857	−1.42282
$545$	0	0
$546$	11.3238	0.484615
$547$	3.00020	0.128279	0.0641396	−	0.997941i	$-0.479570\pi$
$547$	3.00020	0.128279	0.0641396	+	0.997941i	$0.479570\pi$
$548$	17.3475	0.741047
$549$	−0.786908	−0.0335844
$550$	0	0
$551$	3.09205	0.131726
$552$	10.7430	0.457253
$553$	−14.2908	−0.607705
$554$	13.6563	0.580201
$555$	0	0
$556$	−4.92131	−0.208710
$557$	4.11112	0.174194	0.0870969	−	0.996200i	$-0.472241\pi$
$557$	4.11112	0.174194	0.0870969	+	0.996200i	$0.472241\pi$
$558$	−4.72108	−0.199859
$559$	−21.0809	−0.891627
$560$	0	0
$561$	−31.7967	−1.34246
$562$	8.34771	0.352127
$563$	−11.5127	−0.485204	−0.242602	−	0.970126i	$-0.578001\pi$
$563$	−11.5127	−0.485204	−0.242602	+	0.970126i	$0.578001\pi$
$564$	6.30813	0.265620
$565$	0	0
$566$	−12.1149	−0.509228
$567$	4.82343	0.202565
$568$	1.41641	0.0594311
$569$	32.9470	1.38121	0.690605	−	0.723232i	$-0.257344\pi$
$569$	32.9470	1.38121	0.690605	+	0.723232i	$0.257344\pi$
$570$	0	0
$571$	−22.3213	−0.934117	−0.467059	−	0.884226i	$-0.654686\pi$
$571$	−22.3213	−0.934117	−0.467059	+	0.884226i	$0.654686\pi$
$572$	−39.0474	−1.63266
$573$	−8.81550	−0.368273
$574$	7.00950	0.292571
$575$	0	0
$576$	−2.40234	−0.100098
$577$	−39.8197	−1.65772	−0.828859	−	0.559458i	$-0.811009\pi$
$577$	−39.8197	−1.65772	−0.828859	+	0.559458i	$0.811009\pi$
$578$	−12.9616	−0.539130
$579$	−17.2543	−0.717064
$580$	0	0
$581$	−65.9042	−2.73417
$582$	−1.46916	−0.0608986
$583$	−4.72733	−0.195786
$584$	10.5603	0.436988
$585$	0	0
$586$	4.23159	0.174805
$587$	−28.7487	−1.18659	−0.593293	−	0.804987i	$-0.702172\pi$
$587$	−28.7487	−1.18659	−0.593293	+	0.804987i	$0.702172\pi$
$588$	28.2882	1.16658
$589$	28.5821	1.17771
$590$	0	0
$591$	22.5719	0.928483
$592$	−27.8482	−1.14455
$593$	27.5400	1.13093	0.565467	−	0.824771i	$-0.308696\pi$
$593$	27.5400	1.13093	0.565467	+	0.824771i	$0.308696\pi$
$594$	2.49461	0.102355
$595$	0	0
$596$	39.1999	1.60569
$597$	13.7975	0.564693
$598$	13.2068	0.540066
$599$	−4.11339	−0.168069	−0.0840343	−	0.996463i	$-0.526781\pi$
$599$	−4.11339	−0.168069	−0.0840343	+	0.996463i	$0.526781\pi$
$600$	0	0
$601$	−0.656799	−0.0267914	−0.0133957	−	0.999910i	$-0.504264\pi$
$601$	−0.656799	−0.0267914	−0.0133957	+	0.999910i	$0.504264\pi$
$602$	11.2978	0.460463
$603$	4.86742	0.198217
$604$	24.7317	1.00632
$605$	0	0
$606$	−7.49649	−0.304524
$607$	−18.4210	−0.747684	−0.373842	−	0.927492i	$-0.621960\pi$
$607$	−18.4210	−0.747684	−0.373842	+	0.927492i	$0.621960\pi$
$608$	−15.7625	−0.639253
$609$	−4.82343	−0.195455
$610$	0	0
$611$	16.6728	0.674508
$612$	−11.3217	−0.457651
$613$	21.3356	0.861737	0.430869	−	0.902415i	$-0.358207\pi$
$613$	21.3356	0.861737	0.430869	+	0.902415i	$0.358207\pi$
$614$	8.32613	0.336015
$615$	0	0
$616$	44.9917	1.81277
$617$	−23.2463	−0.935861	−0.467931	−	0.883765i	$-0.655000\pi$
$617$	−23.2463	−0.935861	−0.467931	+	0.883765i	$0.655000\pi$
$618$	−1.26740	−0.0509824
$619$	−4.11413	−0.165361	−0.0826804	−	0.996576i	$-0.526348\pi$
$619$	−4.11413	−0.165361	−0.0826804	+	0.996576i	$0.526348\pi$
$620$	0	0
$621$	5.62549	0.225743
$622$	3.94328	0.158111
$623$	17.7423	0.710830
$624$	−11.5053	−0.460582
$625$	0	0
$626$	3.17876	0.127049
$627$	−15.1028	−0.603146
$628$	−18.7327	−0.747515
$629$	−72.4296	−2.88796
$630$	0	0
$631$	−45.7221	−1.82017	−0.910083	−	0.414426i	$-0.863982\pi$
$631$	−45.7221	−1.82017	−0.910083	+	0.414426i	$0.863982\pi$
$632$	−5.65803	−0.225065
$633$	−16.4705	−0.654642
$634$	15.8670	0.630159
$635$	0	0
$636$	−1.68323	−0.0667444
$637$	74.7673	2.96239
$638$	−2.49461	−0.0987626
$639$	0.741689	0.0293408
$640$	0	0
$641$	28.3443	1.11953	0.559767	−	0.828650i	$-0.310891\pi$
$641$	28.3443	1.11953	0.559767	+	0.828650i	$0.310891\pi$
$642$	−2.29024	−0.0903884
$643$	−17.4545	−0.688337	−0.344168	−	0.938908i	$-0.611839\pi$
$643$	−17.4545	−0.688337	−0.344168	+	0.938908i	$0.611839\pi$
$644$	47.1904	1.85956
$645$	0	0
$646$	−10.2804	−0.404478
$647$	17.3874	0.683569	0.341784	−	0.939778i	$-0.388969\pi$
$647$	17.3874	0.683569	0.341784	+	0.939778i	$0.388969\pi$
$648$	1.90970	0.0750202
$649$	−1.45986	−0.0573044
$650$	0	0
$651$	−44.5866	−1.74749
$652$	18.5929	0.728153
$653$	−30.2052	−1.18202	−0.591010	−	0.806664i	$-0.701271\pi$
$653$	−30.2052	−1.18202	−0.591010	+	0.806664i	$0.701271\pi$
$654$	1.67592	0.0655335
$655$	0	0
$656$	−7.12184	−0.278061
$657$	5.52981	0.215738
$658$	−8.93535	−0.348336
$659$	26.3659	1.02707	0.513535	−	0.858069i	$-0.328336\pi$
$659$	26.3659	1.02707	0.513535	+	0.858069i	$0.328336\pi$
$660$	0	0
$661$	15.8187	0.615274	0.307637	−	0.951504i	$-0.400462\pi$
$661$	15.8187	0.615274	0.307637	+	0.951504i	$0.400462\pi$
$662$	−17.8108	−0.692236
$663$	−29.9238	−1.16215
$664$	−26.0929	−1.01260
$665$	0	0
$666$	5.68246	0.220191
$667$	−5.62549	−0.217820
$668$	−13.7505	−0.532023
$669$	4.58903	0.177422
$670$	0	0
$671$	−3.84356	−0.148379
$672$	24.5886	0.948527
$673$	43.1070	1.66165	0.830827	−	0.556530i	$-0.187868\pi$
$673$	43.1070	1.66165	0.830827	+	0.556530i	$0.187868\pi$
$674$	8.47177	0.326320
$675$	0	0
$676$	−14.1385	−0.543788
$677$	40.6978	1.56415	0.782073	−	0.623187i	$-0.214163\pi$
$677$	40.6978	1.56415	0.782073	+	0.623187i	$0.214163\pi$
$678$	−9.18396	−0.352708
$679$	−13.8750	−0.532472
$680$	0	0
$681$	25.5254	0.978134
$682$	−23.0596	−0.882996
$683$	−21.0217	−0.804372	−0.402186	−	0.915558i	$-0.631750\pi$
$683$	−21.0217	−0.804372	−0.402186	+	0.915558i	$0.631750\pi$
$684$	−5.37755	−0.205616
$685$	0	0
$686$	−22.8253	−0.871475
$687$	8.04982	0.307120
$688$	−11.4789	−0.437627
$689$	−4.44888	−0.169489
$690$	0	0
$691$	6.98613	0.265765	0.132882	−	0.991132i	$-0.457577\pi$
$691$	6.98613	0.265765	0.132882	+	0.991132i	$0.457577\pi$
$692$	−6.55244	−0.249086
$693$	23.5595	0.894951
$694$	−0.211241	−0.00801860
$695$	0	0
$696$	−1.90970	−0.0723871
$697$	−18.5230	−0.701608
$698$	−18.1611	−0.687408
$699$	−3.75903	−0.142179
$700$	0	0
$701$	1.40572	0.0530934	0.0265467	−	0.999648i	$-0.491549\pi$
$701$	1.40572	0.0530934	0.0265467	+	0.999648i	$0.491549\pi$
$702$	2.34767	0.0886072
$703$	−34.4025	−1.29751
$704$	−11.7340	−0.442240
$705$	0	0
$706$	−1.57321	−0.0592085
$707$	−70.7981	−2.66264
$708$	−0.519802	−0.0195354
$709$	1.27110	0.0477372	0.0238686	−	0.999715i	$-0.492402\pi$
$709$	1.27110	0.0477372	0.0238686	+	0.999715i	$0.492402\pi$
$710$	0	0
$711$	−2.96278	−0.111113
$712$	7.02457	0.263257
$713$	−52.0006	−1.94744
$714$	16.0369	0.600167
$715$	0	0
$716$	−17.9623	−0.671282
$717$	3.53192	0.131902
$718$	7.31463	0.272979
$719$	27.8439	1.03840	0.519201	−	0.854652i	$-0.326230\pi$
$719$	27.8439	1.03840	0.519201	+	0.854652i	$0.326230\pi$
$720$	0	0
$721$	−11.9696	−0.445770
$722$	4.82092	0.179416
$723$	−3.05553	−0.113636
$724$	−25.1365	−0.934192
$725$	0	0
$726$	6.56659	0.243709
$727$	−8.97787	−0.332971	−0.166485	−	0.986044i	$-0.553242\pi$
$727$	−8.97787	−0.332971	−0.166485	+	0.986044i	$0.553242\pi$
$728$	42.3416	1.56928
$729$	1.00000	0.0370370
$730$	0	0
$731$	−29.8550	−1.10423
$732$	−1.36855	−0.0505832
$733$	−34.7466	−1.28339	−0.641697	−	0.766958i	$-0.721769\pi$
$733$	−34.7466	−1.28339	−0.641697	+	0.766958i	$0.721769\pi$
$734$	−8.83614	−0.326148
$735$	0	0
$736$	28.6773	1.05706
$737$	23.7744	0.875740
$738$	1.45322	0.0534937
$739$	−3.32518	−0.122319	−0.0611594	−	0.998128i	$-0.519480\pi$
$739$	−3.32518	−0.122319	−0.0611594	+	0.998128i	$0.519480\pi$
$740$	0	0
$741$	−14.2132	−0.522134
$742$	2.38426	0.0875291
$743$	−1.06314	−0.0390029	−0.0195015	−	0.999810i	$-0.506208\pi$
$743$	−1.06314	−0.0390029	−0.0195015	+	0.999810i	$0.506208\pi$
$744$	−17.6528	−0.647184
$745$	0	0
$746$	−16.2573	−0.595222
$747$	−13.6633	−0.499916
$748$	−55.2994	−2.02195
$749$	−21.6294	−0.790320
$750$	0	0
$751$	−38.7756	−1.41494	−0.707472	−	0.706742i	$-0.750164\pi$
$751$	−38.7756	−1.41494	−0.707472	+	0.706742i	$0.750164\pi$
$752$	9.07856	0.331061
$753$	23.3283	0.850129
$754$	−2.34767	−0.0854972
$755$	0	0
$756$	8.38869	0.305094
$757$	40.9695	1.48906	0.744531	−	0.667588i	$-0.232673\pi$
$757$	40.9695	1.48906	0.744531	+	0.667588i	$0.232673\pi$
$758$	9.28792	0.337352
$759$	27.4770	0.997354
$760$	0	0
$761$	13.8561	0.502283	0.251142	−	0.967950i	$-0.419194\pi$
$761$	13.8561	0.502283	0.251142	+	0.967950i	$0.419194\pi$
$762$	−6.89994	−0.249958
$763$	15.8276	0.572998
$764$	−15.3315	−0.554675
$765$	0	0
$766$	−16.4980	−0.596096
$767$	−1.37387	−0.0496075
$768$	1.02912	0.0371353
$769$	−6.73919	−0.243021	−0.121511	−	0.992590i	$-0.538774\pi$
$769$	−6.73919	−0.243021	−0.121511	+	0.992590i	$0.538774\pi$
$770$	0	0
$771$	−28.2268	−1.01656
$772$	−30.0079	−1.08001
$773$	7.45730	0.268220	0.134110	−	0.990966i	$-0.457182\pi$
$773$	7.45730	0.268220	0.134110	+	0.990966i	$0.457182\pi$
$774$	2.34227	0.0841912
$775$	0	0
$776$	−5.49341	−0.197202
$777$	53.6660	1.92526
$778$	13.5700	0.486506
$779$	−8.79801	−0.315221
$780$	0	0
$781$	3.62270	0.129630
$782$	18.7036	0.668839
$783$	−1.00000	−0.0357371
$784$	40.7119	1.45400
$785$	0	0
$786$	6.20283	0.221248
$787$	−7.81608	−0.278613	−0.139307	−	0.990249i	$-0.544487\pi$
$787$	−7.81608	−0.278613	−0.139307	+	0.990249i	$0.544487\pi$
$788$	39.2559	1.39844
$789$	27.6688	0.985037
$790$	0	0
$791$	−86.7347	−3.08393
$792$	9.32773	0.331446
$793$	−3.61717	−0.128449
$794$	−4.53444	−0.160921
$795$	0	0
$796$	23.9959	0.850514
$797$	21.8313	0.773305	0.386653	−	0.922225i	$-0.373631\pi$
$797$	21.8313	0.773305	0.386653	+	0.922225i	$0.373631\pi$
$798$	7.61719	0.269646
$799$	23.6121	0.835338
$800$	0	0
$801$	3.67835	0.129968
$802$	−13.1004	−0.462590
$803$	27.0097	0.953152
$804$	8.46519	0.298544
$805$	0	0
$806$	−21.7013	−0.764396
$807$	−0.839052	−0.0295360
$808$	−28.0305	−0.986111
$809$	5.71715	0.201004	0.100502	−	0.994937i	$-0.467955\pi$
$809$	5.71715	0.201004	0.100502	+	0.994937i	$0.467955\pi$
$810$	0	0
$811$	−44.8661	−1.57546	−0.787732	−	0.616019i	$-0.788745\pi$
$811$	−44.8661	−1.57546	−0.787732	+	0.616019i	$0.788745\pi$
$812$	−8.38869	−0.294385
$813$	−1.27423	−0.0446890
$814$	27.7553	0.972823
$815$	0	0
$816$	−16.2940	−0.570402
$817$	−14.1805	−0.496112
$818$	−5.06939	−0.177247
$819$	22.1718	0.774745
$820$	0	0
$821$	37.3699	1.30422	0.652110	−	0.758125i	$-0.273884\pi$
$821$	37.3699	1.30422	0.652110	+	0.758125i	$0.273884\pi$
$822$	−5.09437	−0.177687
$823$	−31.6982	−1.10493	−0.552464	−	0.833537i	$-0.686312\pi$
$823$	−31.6982	−1.10493	−0.552464	+	0.833537i	$0.686312\pi$
$824$	−4.73901	−0.165091
$825$	0	0
$826$	0.736290	0.0256188
$827$	−14.4089	−0.501046	−0.250523	−	0.968111i	$-0.580603\pi$
$827$	−14.4089	−0.501046	−0.250523	+	0.968111i	$0.580603\pi$
$828$	9.78358	0.340003
$829$	−20.6622	−0.717630	−0.358815	−	0.933409i	$-0.616819\pi$
$829$	−20.6622	−0.717630	−0.358815	+	0.933409i	$0.616819\pi$
$830$	0	0
$831$	26.7387	0.927556
$832$	−11.0428	−0.382840
$833$	105.886	3.66874
$834$	1.44522	0.0500440
$835$	0	0
$836$	−26.2660	−0.908429
$837$	−9.24375	−0.319511
$838$	−4.15031	−0.143370
$839$	1.64970	0.0569541	0.0284771	−	0.999594i	$-0.490934\pi$
$839$	1.64970	0.0569541	0.0284771	+	0.999594i	$0.490934\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	4.27904	0.147466
$843$	16.3446	0.562938
$844$	−28.6447	−0.985990
$845$	0	0
$846$	−1.85249	−0.0636899
$847$	62.0159	2.13089
$848$	−2.42248	−0.0831882
$849$	−23.7207	−0.814092
$850$	0	0
$851$	62.5898	2.14555
$852$	1.28991	0.0441916
$853$	−23.4066	−0.801428	−0.400714	−	0.916203i	$-0.631238\pi$
$853$	−23.4066	−0.801428	−0.400714	+	0.916203i	$0.631238\pi$
$854$	1.93853	0.0663351
$855$	0	0
$856$	−8.56355	−0.292696
$857$	18.7479	0.640415	0.320207	−	0.947347i	$-0.396247\pi$
$857$	18.7479	0.640415	0.320207	+	0.947347i	$0.396247\pi$
$858$	11.4669	0.391475
$859$	−29.1326	−0.993990	−0.496995	−	0.867753i	$-0.665563\pi$
$859$	−29.1326	−0.993990	−0.496995	+	0.867753i	$0.665563\pi$
$860$	0	0
$861$	13.7244	0.467727
$862$	16.5414	0.563403
$863$	25.5033	0.868142	0.434071	−	0.900879i	$-0.357077\pi$
$863$	25.5033	0.868142	0.434071	+	0.900879i	$0.357077\pi$
$864$	5.09775	0.173429
$865$	0	0
$866$	−5.05281	−0.171701
$867$	−25.3784	−0.861897
$868$	−77.5429	−2.63198
$869$	−14.4714	−0.490908
$870$	0	0
$871$	22.3740	0.758114
$872$	6.26651	0.212211
$873$	−2.87658	−0.0973573
$874$	8.88380	0.300499
$875$	0	0
$876$	9.61718	0.324935
$877$	24.3134	0.821005	0.410502	−	0.911860i	$-0.365353\pi$
$877$	24.3134	0.821005	0.410502	+	0.911860i	$0.365353\pi$
$878$	−2.98638	−0.100786
$879$	8.28535	0.279458
$880$	0	0
$881$	51.3387	1.72965	0.864823	−	0.502077i	$-0.167431\pi$
$881$	51.3387	1.72965	0.864823	+	0.502077i	$0.167431\pi$
$882$	−8.30730	−0.279721
$883$	17.1764	0.578033	0.289016	−	0.957324i	$-0.406672\pi$
$883$	17.1764	0.578033	0.289016	+	0.957324i	$0.406672\pi$
$884$	−52.0421	−1.75037
$885$	0	0
$886$	−1.60460	−0.0539075
$887$	38.2392	1.28395	0.641974	−	0.766727i	$-0.278116\pi$
$887$	38.2392	1.28395	0.641974	+	0.766727i	$0.278116\pi$
$888$	21.2476	0.713022
$889$	−65.1641	−2.18553
$890$	0	0
$891$	4.88439	0.163633
$892$	7.98103	0.267225
$893$	11.2153	0.375304
$894$	−11.5117	−0.385009
$895$	0	0
$896$	55.0954	1.84061
$897$	25.8586	0.863393
$898$	−14.2312	−0.474901
$899$	9.24375	0.308296
$900$	0	0
$901$	−6.30055	−0.209902
$902$	7.09808	0.236340
$903$	22.1208	0.736134
$904$	−34.3402	−1.14214
$905$	0	0
$906$	−7.26288	−0.241293
$907$	9.37007	0.311128	0.155564	−	0.987826i	$-0.450281\pi$
$907$	9.37007	0.311128	0.155564	+	0.987826i	$0.450281\pi$
$908$	44.3925	1.47322
$909$	−14.6780	−0.486837
$910$	0	0
$911$	−10.5220	−0.348611	−0.174305	−	0.984692i	$-0.555768\pi$
$911$	−10.5220	−0.348611	−0.174305	+	0.984692i	$0.555768\pi$
$912$	−7.73927	−0.256273
$913$	−66.7371	−2.20867
$914$	16.2886	0.538780
$915$	0	0
$916$	13.9999	0.462569
$917$	58.5805	1.93450
$918$	3.32480	0.109735
$919$	−47.9043	−1.58022	−0.790109	−	0.612967i	$-0.789976\pi$
$919$	−47.9043	−1.58022	−0.790109	+	0.612967i	$0.789976\pi$
$920$	0	0
$921$	16.3024	0.537181
$922$	−2.88390	−0.0949762
$923$	3.40931	0.112219
$924$	40.9736	1.34793
$925$	0	0
$926$	−16.7317	−0.549837
$927$	−2.48154	−0.0815046
$928$	−5.09775	−0.167342
$929$	32.0568	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$929$	32.0568	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$930$	0	0
$931$	50.2937	1.64831
$932$	−6.53752	−0.214144
$933$	7.72085	0.252769
$934$	19.1080	0.625232
$935$	0	0
$936$	8.77831	0.286928
$937$	23.8183	0.778109	0.389054	−	0.921215i	$-0.372802\pi$
$937$	23.8183	0.778109	0.389054	+	0.921215i	$0.372802\pi$
$938$	−11.9908	−0.391513
$939$	6.22393	0.203110
$940$	0	0
$941$	45.7836	1.49250	0.746251	−	0.665664i	$-0.231852\pi$
$941$	45.7836	1.49250	0.746251	+	0.665664i	$0.231852\pi$
$942$	5.50117	0.179238
$943$	16.0066	0.521246
$944$	−0.748091	−0.0243483
$945$	0	0
$946$	11.4406	0.371965
$947$	−4.69130	−0.152447	−0.0762233	−	0.997091i	$-0.524286\pi$
$947$	−4.69130	−0.152447	−0.0762233	+	0.997091i	$0.524286\pi$
$948$	−5.15273	−0.167353
$949$	25.4188	0.825129
$950$	0	0
$951$	31.0672	1.00742
$952$	59.9645	1.94346
$953$	56.4381	1.82821	0.914104	−	0.405479i	$-0.132895\pi$
$953$	56.4381	1.82821	0.914104	+	0.405479i	$0.132895\pi$
$954$	0.494309	0.0160038
$955$	0	0
$956$	6.14256	0.198665
$957$	−4.88439	−0.157890
$958$	6.67883	0.215783
$959$	−48.1121	−1.55362
$960$	0	0
$961$	54.4469	1.75635
$962$	26.1205	0.842158
$963$	−4.48423	−0.144502
$964$	−5.31403	−0.171153
$965$	0	0
$966$	−13.8583	−0.445882
$967$	−42.2611	−1.35902	−0.679512	−	0.733664i	$-0.737808\pi$
$967$	−42.2611	−1.35902	−0.679512	+	0.733664i	$0.737808\pi$
$968$	24.5535	0.789179
$969$	−20.1288	−0.646631
$970$	0	0
$971$	25.6613	0.823510	0.411755	−	0.911295i	$-0.364916\pi$
$971$	25.6613	0.823510	0.411755	+	0.911295i	$0.364916\pi$
$972$	1.73915	0.0557834
$973$	13.6489	0.437565
$974$	−9.93333	−0.318284
$975$	0	0
$976$	−1.96960	−0.0630453
$977$	7.35242	0.235225	0.117612	−	0.993060i	$-0.462476\pi$
$977$	7.35242	0.235225	0.117612	+	0.993060i	$0.462476\pi$
$978$	−5.46011	−0.174595
$979$	17.9665	0.574212
$980$	0	0
$981$	3.28140	0.104767
$982$	−8.08084	−0.257870
$983$	3.23911	0.103312	0.0516558	−	0.998665i	$-0.483550\pi$
$983$	3.23911	0.103312	0.0516558	+	0.998665i	$0.483550\pi$
$984$	5.43381	0.173223
$985$	0	0
$986$	−3.32480	−0.105883
$987$	−17.4952	−0.556879
$988$	−24.7189	−0.786413
$989$	25.7991	0.820364
$990$	0	0
$991$	3.19230	0.101407	0.0507034	−	0.998714i	$-0.483854\pi$
$991$	3.19230	0.101407	0.0507034	+	0.998714i	$0.483854\pi$
$992$	−47.1223	−1.49614
$993$	−34.8731	−1.10666
$994$	−1.82714	−0.0579532
$995$	0	0
$996$	−23.7627	−0.752949
$997$	−49.0253	−1.55265	−0.776323	−	0.630335i	$-0.782917\pi$
$997$	−49.0253	−1.55265	−0.776323	+	0.630335i	$0.782917\pi$
$998$	−2.49539	−0.0789903
$999$	11.1261	0.352015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.bc.1.4	✓	8
3.2	odd	2		6525.2.a.bz.1.5		8
5.2	odd	4		2175.2.c.p.349.6		16
5.3	odd	4		2175.2.c.p.349.11		16
5.4	even	2		2175.2.a.bd.1.5	yes	8
15.14	odd	2		6525.2.a.by.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.4	✓	8	1.1	even	1	trivial
2175.2.a.bd.1.5	yes	8	5.4	even	2
2175.2.c.p.349.6		16	5.2	odd	4
2175.2.c.p.349.11		16	5.3	odd	4
6525.2.a.by.1.4		8	15.14	odd	2
6525.2.a.bz.1.5		8	3.2	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-0.510732 q^{2} -1.00000 q^{3} -1.73915 q^{4} +0.510732 q^{6} +4.82343 q^{7} +1.90970 q^{8} +1.00000 q^{9} +4.88439 q^{11} +1.73915 q^{12} +4.59669 q^{13} -2.46348 q^{14} +2.50296 q^{16} +6.50987 q^{17} -0.510732 q^{18} +3.09205 q^{19} -4.82343 q^{21} -2.49461 q^{22} -5.62549 q^{23} -1.90970 q^{24} -2.34767 q^{26} -1.00000 q^{27} -8.38869 q^{28} +1.00000 q^{29} +9.24375 q^{31} -5.09775 q^{32} -4.88439 q^{33} -3.32480 q^{34} -1.73915 q^{36} -11.1261 q^{37} -1.57921 q^{38} -4.59669 q^{39} -2.84537 q^{41} +2.46348 q^{42} -4.58611 q^{43} -8.49470 q^{44} +2.87311 q^{46} +3.62713 q^{47} -2.50296 q^{48} +16.2655 q^{49} -6.50987 q^{51} -7.99434 q^{52} -0.967845 q^{53} +0.510732 q^{54} +9.21132 q^{56} -3.09205 q^{57} -0.510732 q^{58} -0.298882 q^{59} -0.786908 q^{61} -4.72108 q^{62} +4.82343 q^{63} -2.40234 q^{64} +2.49461 q^{66} +4.86742 q^{67} -11.3217 q^{68} +5.62549 q^{69} +0.741689 q^{71} +1.90970 q^{72} +5.52981 q^{73} +5.68246 q^{74} -5.37755 q^{76} +23.5595 q^{77} +2.34767 q^{78} -2.96278 q^{79} +1.00000 q^{81} +1.45322 q^{82} -13.6633 q^{83} +8.38869 q^{84} +2.34227 q^{86} -1.00000 q^{87} +9.32773 q^{88} +3.67835 q^{89} +22.1718 q^{91} +9.78358 q^{92} -9.24375 q^{93} -1.85249 q^{94} +5.09775 q^{96} -2.87658 q^{97} -8.30730 q^{98} +4.88439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 12 q^{4} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} + 6 q^{11} - 12 q^{12} + 6 q^{13} + 9 q^{14} + 32 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{21} + 3 q^{22} - 14 q^{23} + 3 q^{24} + 18 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 12 * q^4 + 2 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 + 6 * q^11 - 12 * q^12 + 6 * q^13 + 9 * q^14 + 32 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^21 + 3 * q^22 - 14 * q^23 + 3 * q^24 + 18 * q^26 - 8 * q^27 + 14 * q^28 + 8 * q^29 + 8 * q^31 + 2 * q^32 - 6 * q^33 - 13 * q^34 + 12 * q^36 + 4 * q^37 - 26 * q^38 - 6 * q^39 + 2 * q^41 - 9 * q^42 + 2 * q^43 - 15 * q^44 + 24 * q^46 - 12 * q^47 - 32 * q^48 + 38 * q^49 + 12 * q^51 + 49 * q^52 - 4 * q^53 + 2 * q^54 + 58 * q^56 - 2 * q^58 + 18 * q^59 + 12 * q^61 + 4 * q^62 + 2 * q^63 + 21 * q^64 - 3 * q^66 + 26 * q^67 - 81 * q^68 + 14 * q^69 + 24 * q^71 - 3 * q^72 - 14 * q^73 - 22 * q^74 + 26 * q^77 - 18 * q^78 + 10 * q^79 + 8 * q^81 + 48 * q^82 - 40 * q^83 - 14 * q^84 + 8 * q^86 - 8 * q^87 - 10 * q^88 + 34 * q^89 + 26 * q^91 + 18 * q^92 - 8 * q^93 - 43 * q^94 - 2 * q^96 + 30 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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