LMFDB - Embedded newform 275.4.b.f.199.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,4,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 80x^{8} + 2296x^{6} + 27417x^{4} + 110472x^{2} + 21904 $ x^10 + 80x^8 + 2296x^6 + 27417x^4 + 110472x^2 + 21904
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.8
Root			$4.72095i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.f.199.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.72095i q^{2} +8.29546i q^{3} -14.2874 q^{4} -39.1625 q^{6} +5.48389i q^{7} -29.6825i q^{8} -41.8147 q^{9} -11.0000 q^{11} -118.521i q^{12} -84.9102i q^{13} -25.8892 q^{14} +25.8306 q^{16} +119.626i q^{17} -197.405i q^{18} +54.4593 q^{19} -45.4914 q^{21} -51.9305i q^{22} -83.8353i q^{23} +246.230 q^{24} +400.857 q^{26} -122.895i q^{27} -78.3505i q^{28} -296.062 q^{29} -18.7281 q^{31} -115.515i q^{32} -91.2501i q^{33} -564.750 q^{34} +597.424 q^{36} +188.075i q^{37} +257.100i q^{38} +704.369 q^{39} -209.594 q^{41} -214.763i q^{42} +183.046i q^{43} +157.161 q^{44} +395.782 q^{46} +142.669i q^{47} +214.277i q^{48} +312.927 q^{49} -992.355 q^{51} +1213.15i q^{52} +610.881i q^{53} +580.182 q^{54} +162.776 q^{56} +451.766i q^{57} -1397.70i q^{58} -657.261 q^{59} +171.698 q^{61} -88.4144i q^{62} -229.307i q^{63} +751.986 q^{64} +430.788 q^{66} +116.641i q^{67} -1709.15i q^{68} +695.452 q^{69} -595.308 q^{71} +1241.17i q^{72} -629.849i q^{73} -887.895 q^{74} -778.082 q^{76} -60.3228i q^{77} +3325.29i q^{78} +935.886 q^{79} -109.526 q^{81} -989.485i q^{82} -81.2520i q^{83} +649.954 q^{84} -864.154 q^{86} -2455.98i q^{87} +326.508i q^{88} +245.174 q^{89} +465.638 q^{91} +1197.79i q^{92} -155.358i q^{93} -673.535 q^{94} +958.251 q^{96} -395.976i q^{97} +1477.31i q^{98} +459.962 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) $ 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.72095i	1.66911i	0.550925	+	0.834555i	$0.314275\pi$
$2$	4.72095i	1.66911i	−0.550925	+	0.834555i	$0.685725\pi$
$3$	8.29546i	1.59646i	0.602351	+	0.798231i	$0.294231\pi$
$3$	8.29546i	1.59646i	−0.602351	+	0.798231i	$0.705769\pi$
$4$	−14.2874	−1.78593
$5$	0	0
$5$	0	0
$6$	−39.1625	−2.66467
$7$	5.48389i	0.296102i	0.988980	+	0.148051i	$0.0473000\pi$
$7$	5.48389i	0.296102i	−0.988980	+	0.148051i	$0.952700\pi$
$8$	− 29.6825i	− 1.31179i
$9$	−41.8147	−1.54869
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 118.521i	− 2.85116i
$13$	− 84.9102i	− 1.81153i	−0.423784	−	0.905763i	$-0.639299\pi$
$13$	− 84.9102i	− 1.81153i	0.423784	−	0.905763i	$-0.360701\pi$
$14$	−25.8892	−0.494227
$15$	0	0
$16$	25.8306	0.403603
$17$	119.626i	1.70668i	0.521352	+	0.853342i	$0.325428\pi$
$17$	119.626i	1.70668i	−0.521352	+	0.853342i	$0.674572\pi$
$18$	− 197.405i	− 2.58494i
$19$	54.4593	0.657570	0.328785	−	0.944405i	$-0.393361\pi$
$19$	54.4593	0.657570	0.328785	+	0.944405i	$0.393361\pi$
$20$	0	0
$21$	−45.4914	−0.472716
$22$	− 51.9305i	− 0.503255i
$23$	− 83.8353i	− 0.760037i	−0.924979	−	0.380019i	$-0.875918\pi$
$23$	− 83.8353i	− 0.760037i	0.924979	−	0.380019i	$-0.124082\pi$
$24$	246.230	2.09423
$25$	0	0
$26$	400.857	3.02363
$27$	− 122.895i	− 0.875969i
$28$	− 78.3505i	− 0.528816i
$29$	−296.062	−1.89577	−0.947887	−	0.318608i	$-0.896785\pi$
$29$	−296.062	−1.89577	−0.947887	+	0.318608i	$0.896785\pi$
$30$	0	0
$31$	−18.7281	−0.108505	−0.0542526	−	0.998527i	$-0.517278\pi$
$31$	−18.7281	−0.108505	−0.0542526	+	0.998527i	$0.517278\pi$
$32$	− 115.515i	− 0.638137i
$33$	− 91.2501i	− 0.481352i
$34$	−564.750	−2.84864
$35$	0	0
$36$	597.424	2.76585
$37$	188.075i	0.835660i	0.908525	+	0.417830i	$0.137209\pi$
$37$	188.075i	0.835660i	−0.908525	+	0.417830i	$0.862791\pi$
$38$	257.100i	1.09756i
$39$	704.369	2.89203
$40$	0	0
$41$	−209.594	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$41$	−209.594	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$42$	− 214.763i	− 0.789015i
$43$	183.046i	0.649170i	0.945856	+	0.324585i	$0.105225\pi$
$43$	183.046i	0.649170i	−0.945856	+	0.324585i	$0.894775\pi$
$44$	157.161	0.538477
$45$	0	0
$46$	395.782	1.26859
$47$	142.669i	0.442775i	0.975186	+	0.221388i	$0.0710586\pi$
$47$	142.669i	0.442775i	−0.975186	+	0.221388i	$0.928941\pi$
$48$	214.277i	0.644338i
$49$	312.927	0.912323
$50$	0	0
$51$	−992.355	−2.72466
$52$	1213.15i	3.23525i
$53$	610.881i	1.58323i	0.611022	+	0.791613i	$0.290759\pi$
$53$	610.881i	1.58323i	−0.611022	+	0.791613i	$0.709241\pi$
$54$	580.182	1.46209
$55$	0	0
$56$	162.776	0.388425
$57$	451.766i	1.04979i
$58$	− 1397.70i	− 3.16425i
$59$	−657.261	−1.45031	−0.725154	−	0.688587i	$-0.758231\pi$
$59$	−657.261	−1.45031	−0.725154	+	0.688587i	$0.758231\pi$
$60$	0	0
$61$	171.698	0.360388	0.180194	−	0.983631i	$-0.442327\pi$
$61$	171.698	0.360388	0.180194	+	0.983631i	$0.442327\pi$
$62$	− 88.4144i	− 0.181107i
$63$	− 229.307i	− 0.458572i
$64$	751.986	1.46872
$65$	0	0
$66$	430.788	0.803428
$67$	116.641i	0.212687i	0.994329	+	0.106343i	$0.0339143\pi$
$67$	116.641i	0.212687i	−0.994329	+	0.106343i	$0.966086\pi$
$68$	− 1709.15i	− 3.04801i
$69$	695.452	1.21337
$70$	0	0
$71$	−595.308	−0.995071	−0.497536	−	0.867444i	$-0.665762\pi$
$71$	−595.308	−0.995071	−0.497536	+	0.867444i	$0.665762\pi$
$72$	1241.17i	2.03157i
$73$	− 629.849i	− 1.00984i	−0.863166	−	0.504920i	$-0.831522\pi$
$73$	− 629.849i	− 1.00984i	0.863166	−	0.504920i	$-0.168478\pi$
$74$	−887.895	−1.39481
$75$	0	0
$76$	−778.082	−1.17437
$77$	− 60.3228i	− 0.0892782i
$78$	3325.29i	4.82712i
$79$	935.886	1.33285	0.666426	−	0.745571i	$-0.267823\pi$
$79$	935.886	1.33285	0.666426	+	0.745571i	$0.267823\pi$
$80$	0	0
$81$	−109.526	−0.150241
$82$	− 989.485i	− 1.33257i
$83$	− 81.2520i	− 0.107453i	−0.998556	−	0.0537263i	$-0.982890\pi$
$83$	− 81.2520i	− 0.107453i	0.998556	−	0.0537263i	$-0.0171098\pi$
$84$	649.954	0.844236
$85$	0	0
$86$	−864.154	−1.08354
$87$	− 2455.98i	− 3.02653i
$88$	326.508i	0.395521i
$89$	245.174	0.292004	0.146002	−	0.989284i	$-0.453359\pi$
$89$	245.174	0.292004	0.146002	+	0.989284i	$0.453359\pi$
$90$	0	0
$91$	465.638	0.536397
$92$	1197.79i	1.35737i
$93$	− 155.358i	− 0.173225i
$94$	−673.535	−0.739040
$95$	0	0
$96$	958.251	1.01876
$97$	− 395.976i	− 0.414487i	−0.978289	−	0.207244i	$-0.933551\pi$
$97$	− 395.976i	− 0.414487i	0.978289	−	0.207244i	$-0.0664493\pi$
$98$	1477.31i	1.52277i
$99$	459.962	0.466949
$100$	0	0
$101$	−311.696	−0.307078	−0.153539	−	0.988143i	$-0.549067\pi$
$101$	−311.696	−0.307078	−0.153539	+	0.988143i	$0.549067\pi$
$102$	− 4684.86i	− 4.54775i
$103$	− 948.676i	− 0.907532i	−0.891121	−	0.453766i	$-0.850080\pi$
$103$	− 948.676i	− 0.907532i	0.891121	−	0.453766i	$-0.149920\pi$
$104$	−2520.35	−2.37635
$105$	0	0
$106$	−2883.94	−2.64258
$107$	− 1841.53i	− 1.66380i	−0.554923	−	0.831902i	$-0.687252\pi$
$107$	− 1841.53i	− 1.66380i	0.554923	−	0.831902i	$-0.312748\pi$
$108$	1755.85i	1.56442i
$109$	−1239.66	−1.08934	−0.544669	−	0.838651i	$-0.683345\pi$
$109$	−1239.66	−1.08934	−0.544669	+	0.838651i	$0.683345\pi$
$110$	0	0
$111$	−1560.17	−1.33410
$112$	141.652i	0.119508i
$113$	584.293i	0.486422i	0.969973	+	0.243211i	$0.0782007\pi$
$113$	584.293i	0.486422i	−0.969973	+	0.243211i	$0.921799\pi$
$114$	−2132.76	−1.75221
$115$	0	0
$116$	4229.96	3.38571
$117$	3550.50i	2.80550i
$118$	− 3102.90i	− 2.42072i
$119$	−656.017	−0.505353
$120$	0	0
$121$	121.000	0.0909091
$122$	810.577i	0.601526i
$123$	− 1738.68i	− 1.27457i
$124$	267.576	0.193782
$125$	0	0
$126$	1082.55	0.765406
$127$	1872.57i	1.30837i	0.756333	+	0.654187i	$0.226989\pi$
$127$	1872.57i	1.30837i	−0.756333	+	0.654187i	$0.773011\pi$
$128$	2625.97i	1.81332i
$129$	−1518.46	−1.03638
$130$	0	0
$131$	1602.54	1.06881	0.534406	−	0.845228i	$-0.320535\pi$
$131$	1602.54	1.06881	0.534406	+	0.845228i	$0.320535\pi$
$132$	1303.73i	0.859658i
$133$	298.649i	0.194708i
$134$	−550.659	−0.354998
$135$	0	0
$136$	3550.81	2.23882
$137$	475.189i	0.296337i	0.988962	+	0.148168i	$0.0473378\pi$
$137$	475.189i	0.296337i	−0.988962	+	0.148168i	$0.952662\pi$
$138$	3283.20i	2.02525i
$139$	−831.271	−0.507248	−0.253624	−	0.967303i	$-0.581623\pi$
$139$	−831.271	−0.507248	−0.253624	+	0.967303i	$0.581623\pi$
$140$	0	0
$141$	−1183.51	−0.706874
$142$	− 2810.42i	− 1.66088i
$143$	934.012i	0.546196i
$144$	−1080.10	−0.625058
$145$	0	0
$146$	2973.49	1.68553
$147$	2595.87i	1.45649i
$148$	− 2687.11i	− 1.49243i
$149$	2274.77	1.25072	0.625358	−	0.780338i	$-0.284953\pi$
$149$	2274.77	1.25072	0.625358	+	0.780338i	$0.284953\pi$
$150$	0	0
$151$	−2469.23	−1.33075	−0.665374	−	0.746510i	$-0.731728\pi$
$151$	−2469.23	−1.33075	−0.665374	+	0.746510i	$0.731728\pi$
$152$	− 1616.49i	− 0.862597i
$153$	− 5002.14i	− 2.64313i
$154$	284.781	0.149015
$155$	0	0
$156$	−10063.6	−5.16496
$157$	241.092i	0.122556i	0.998121	+	0.0612778i	$0.0195176\pi$
$157$	241.092i	0.122556i	−0.998121	+	0.0612778i	$0.980482\pi$
$158$	4418.27i	2.22468i
$159$	−5067.55	−2.52756
$160$	0	0
$161$	459.743	0.225049
$162$	− 517.066i	− 0.250769i
$163$	− 1325.45i	− 0.636915i	−0.947937	−	0.318458i	$-0.896835\pi$
$163$	− 1325.45i	− 0.636915i	0.947937	−	0.318458i	$-0.103165\pi$
$164$	2994.56	1.42583
$165$	0	0
$166$	383.587	0.179350
$167$	1154.76i	0.535077i	0.963547	+	0.267538i	$0.0862103\pi$
$167$	1154.76i	0.535077i	−0.963547	+	0.267538i	$0.913790\pi$
$168$	1350.30i	0.620107i
$169$	−5012.73	−2.28163
$170$	0	0
$171$	−2277.20	−1.01837
$172$	− 2615.26i	− 1.15937i
$173$	2281.20i	1.00252i	0.865296	+	0.501260i	$0.167130\pi$
$173$	2281.20i	1.00252i	−0.865296	+	0.501260i	$0.832870\pi$
$174$	11594.5	5.05161
$175$	0	0
$176$	−284.137	−0.121691
$177$	− 5452.29i	− 2.31536i
$178$	1157.45i	0.487386i
$179$	−2909.82	−1.21503	−0.607515	−	0.794308i	$-0.707834\pi$
$179$	−2909.82	−1.21503	−0.607515	+	0.794308i	$0.707834\pi$
$180$	0	0
$181$	1898.75	0.779739	0.389870	−	0.920870i	$-0.372520\pi$
$181$	1898.75	0.779739	0.389870	+	0.920870i	$0.372520\pi$
$182$	2198.25i	0.895305i
$183$	1424.31i	0.575346i
$184$	−2488.44	−0.997013
$185$	0	0
$186$	733.439	0.289131
$187$	− 1315.89i	− 0.514585i
$188$	− 2038.37i	− 0.790764i
$189$	673.943	0.259377
$190$	0	0
$191$	−1228.35	−0.465341	−0.232670	−	0.972556i	$-0.574746\pi$
$191$	−1228.35	−0.465341	−0.232670	+	0.972556i	$0.574746\pi$
$192$	6238.08i	2.34476i
$193$	4564.93i	1.70254i	0.524724	+	0.851272i	$0.324168\pi$
$193$	4564.93i	1.70254i	−0.524724	+	0.851272i	$0.675832\pi$
$194$	1869.38	0.691824
$195$	0	0
$196$	−4470.91	−1.62934
$197$	4619.27i	1.67061i	0.549790	+	0.835303i	$0.314708\pi$
$197$	4619.27i	1.67061i	−0.549790	+	0.835303i	$0.685292\pi$
$198$	2171.46i	0.779388i
$199$	1329.67	0.473658	0.236829	−	0.971551i	$-0.423892\pi$
$199$	1329.67	0.473658	0.236829	+	0.971551i	$0.423892\pi$
$200$	0	0
$201$	−967.595	−0.339547
$202$	− 1471.50i	− 0.512547i
$203$	− 1623.57i	− 0.561343i
$204$	14178.2	4.86603
$205$	0	0
$206$	4478.66	1.51477
$207$	3505.55i	1.17707i
$208$	− 2193.28i	− 0.731138i
$209$	−599.053	−0.198265
$210$	0	0
$211$	457.678	0.149326	0.0746632	−	0.997209i	$-0.476212\pi$
$211$	457.678	0.149326	0.0746632	+	0.997209i	$0.476212\pi$
$212$	− 8727.91i	− 2.82752i
$213$	− 4938.36i	− 1.58859i
$214$	8693.76	2.77707
$215$	0	0
$216$	−3647.84	−1.14909
$217$	− 102.703i	− 0.0321287i
$218$	− 5852.38i	− 1.81822i
$219$	5224.89	1.61217
$220$	0	0
$221$	10157.5	3.09170
$222$	− 7365.51i	− 2.22676i
$223$	1662.19i	0.499140i	0.968357	+	0.249570i	$0.0802893\pi$
$223$	1662.19i	0.499140i	−0.968357	+	0.249570i	$0.919711\pi$
$224$	633.472	0.188954
$225$	0	0
$226$	−2758.42	−0.811891
$227$	519.278i	0.151831i	0.997114	+	0.0759156i	$0.0241880\pi$
$227$	519.278i	0.151831i	−0.997114	+	0.0759156i	$0.975812\pi$
$228$	− 6454.56i	− 1.87484i
$229$	988.947	0.285378	0.142689	−	0.989768i	$-0.454425\pi$
$229$	988.947	0.285378	0.142689	+	0.989768i	$0.454425\pi$
$230$	0	0
$231$	500.406	0.142529
$232$	8787.88i	2.48687i
$233$	5335.96i	1.50030i	0.661267	+	0.750151i	$0.270019\pi$
$233$	5335.96i	1.50030i	−0.661267	+	0.750151i	$0.729981\pi$
$234$	−16761.7	−4.68268
$235$	0	0
$236$	9390.55	2.59014
$237$	7763.61i	2.12785i
$238$	− 3097.03i	− 0.843489i
$239$	4104.99	1.11100	0.555502	−	0.831516i	$-0.312526\pi$
$239$	4104.99	1.11100	0.555502	+	0.831516i	$0.312526\pi$
$240$	0	0
$241$	−168.748	−0.0451037	−0.0225518	−	0.999746i	$-0.507179\pi$
$241$	−168.748	−0.0451037	−0.0225518	+	0.999746i	$0.507179\pi$
$242$	571.235i	0.151737i
$243$	− 4226.74i	− 1.11582i
$244$	−2453.11	−0.643625
$245$	0	0
$246$	8208.24	2.12739
$247$	− 4624.15i	− 1.19121i
$248$	555.897i	0.142337i
$249$	674.023	0.171544
$250$	0	0
$251$	−7749.22	−1.94871	−0.974355	−	0.225017i	$-0.927756\pi$
$251$	−7749.22	−1.94871	−0.974355	+	0.225017i	$0.927756\pi$
$252$	3276.21i	0.818975i
$253$	922.188i	0.229160i
$254$	−8840.31	−2.18382
$255$	0	0
$256$	−6381.20	−1.55791
$257$	− 267.993i	− 0.0650465i	−0.999471	−	0.0325232i	$-0.989646\pi$
$257$	− 267.993i	− 0.0650465i	0.999471	−	0.0325232i	$-0.0103543\pi$
$258$	− 7168.56i	− 1.72982i
$259$	−1031.39	−0.247441
$260$	0	0
$261$	12379.8	2.93597
$262$	7565.50i	1.78396i
$263$	1194.86i	0.280144i	0.990141	+	0.140072i	$0.0447335\pi$
$263$	1194.86i	0.280144i	−0.990141	+	0.140072i	$0.955267\pi$
$264$	−2708.53	−0.631435
$265$	0	0
$266$	−1409.91	−0.324989
$267$	2033.83i	0.466173i
$268$	− 1666.50i	− 0.379843i
$269$	−1185.18	−0.268631	−0.134316	−	0.990939i	$-0.542884\pi$
$269$	−1185.18	−0.268631	−0.134316	+	0.990939i	$0.542884\pi$
$270$	0	0
$271$	−3252.40	−0.729037	−0.364519	−	0.931196i	$-0.618766\pi$
$271$	−3252.40	−0.729037	−0.364519	+	0.931196i	$0.618766\pi$
$272$	3090.02i	0.688823i
$273$	3862.68i	0.856338i
$274$	−2243.34	−0.494618
$275$	0	0
$276$	−9936.21	−2.16699
$277$	− 2904.35i	− 0.629984i	−0.949094	−	0.314992i	$-0.897998\pi$
$277$	− 2904.35i	− 0.629984i	0.949094	−	0.314992i	$-0.102002\pi$
$278$	− 3924.39i	− 0.846653i
$279$	783.110	0.168041
$280$	0	0
$281$	1592.03	0.337981	0.168991	−	0.985618i	$-0.445949\pi$
$281$	1592.03	0.337981	0.168991	+	0.985618i	$0.445949\pi$
$282$	− 5587.28i	− 1.17985i
$283$	4138.44i	0.869275i	0.900605	+	0.434638i	$0.143124\pi$
$283$	4138.44i	0.869275i	−0.900605	+	0.434638i	$0.856876\pi$
$284$	8505.40	1.77712
$285$	0	0
$286$	−4409.43	−0.911660
$287$	− 1149.39i	− 0.236399i
$288$	4830.23i	0.988278i
$289$	−9397.44	−1.91277
$290$	0	0
$291$	3284.80	0.661713
$292$	8998.91i	1.80350i
$293$	1348.05i	0.268784i	0.990928	+	0.134392i	$0.0429082\pi$
$293$	1348.05i	0.268784i	−0.990928	+	0.134392i	$0.957092\pi$
$294$	−12255.0	−2.43104
$295$	0	0
$296$	5582.55	1.09621
$297$	1351.85i	0.264115i
$298$	10739.1i	2.08758i
$299$	−7118.47	−1.37683
$300$	0	0
$301$	−1003.81	−0.192221
$302$	− 11657.1i	− 2.22116i
$303$	− 2585.66i	− 0.490239i
$304$	1406.72	0.265397
$305$	0	0
$306$	23614.9	4.41167
$307$	1741.35i	0.323727i	0.986813	+	0.161863i	$0.0517504\pi$
$307$	1741.35i	0.323727i	−0.986813	+	0.161863i	$0.948250\pi$
$308$	861.856i	0.159444i
$309$	7869.71	1.44884
$310$	0	0
$311$	−2842.37	−0.518252	−0.259126	−	0.965844i	$-0.583434\pi$
$311$	−2842.37	−0.518252	−0.259126	+	0.965844i	$0.583434\pi$
$312$	− 20907.5i	− 3.79376i
$313$	− 7279.32i	− 1.31454i	−0.753654	−	0.657271i	$-0.771711\pi$
$313$	− 7279.32i	− 1.31454i	0.753654	−	0.657271i	$-0.228289\pi$
$314$	−1138.18	−0.204559
$315$	0	0
$316$	−13371.4	−2.38037
$317$	7390.30i	1.30940i	0.755888	+	0.654701i	$0.227205\pi$
$317$	7390.30i	1.30940i	−0.755888	+	0.654701i	$0.772795\pi$
$318$	− 23923.6i	− 4.21878i
$319$	3256.69	0.571597
$320$	0	0
$321$	15276.3	2.65620
$322$	2170.43i	0.375631i
$323$	6514.77i	1.12226i
$324$	1564.84	0.268319
$325$	0	0
$326$	6257.38	1.06308
$327$	− 10283.6i	− 1.73909i
$328$	6221.29i	1.04730i
$329$	−782.382	−0.131107
$330$	0	0
$331$	−4830.91	−0.802209	−0.401104	−	0.916032i	$-0.631374\pi$
$331$	−4830.91	−0.802209	−0.401104	+	0.916032i	$0.631374\pi$
$332$	1160.88i	0.191902i
$333$	− 7864.33i	− 1.29418i
$334$	−5451.56	−0.893102
$335$	0	0
$336$	−1175.07	−0.190790
$337$	− 2556.55i	− 0.413247i	−0.978421	−	0.206623i	$-0.933753\pi$
$337$	− 2556.55i	− 0.413247i	0.978421	−	0.206623i	$-0.0662475\pi$
$338$	− 23664.9i	− 3.80828i
$339$	−4846.98	−0.776555
$340$	0	0
$341$	206.009	0.0327156
$342$	− 10750.6i	− 1.69978i
$343$	3597.03i	0.566243i
$344$	5433.28	0.851578
$345$	0	0
$346$	−10769.4	−1.67332
$347$	− 4704.78i	− 0.727856i	−0.931427	−	0.363928i	$-0.881435\pi$
$347$	− 4704.78i	− 0.727856i	0.931427	−	0.363928i	$-0.118565\pi$
$348$	35089.5i	5.40516i
$349$	10288.9	1.57809	0.789043	−	0.614338i	$-0.210577\pi$
$349$	10288.9	1.57809	0.789043	+	0.614338i	$0.210577\pi$
$350$	0	0
$351$	−10435.0	−1.58684
$352$	1270.67i	0.192405i
$353$	6832.15i	1.03014i	0.857149	+	0.515069i	$0.172234\pi$
$353$	6832.15i	1.03014i	−0.857149	+	0.515069i	$0.827766\pi$
$354$	25740.0	3.86459
$355$	0	0
$356$	−3502.89	−0.521497
$357$	− 5441.97i	− 0.806777i
$358$	− 13737.1i	− 2.02802i
$359$	11254.2	1.65452	0.827259	−	0.561821i	$-0.189899\pi$
$359$	11254.2	1.65452	0.827259	+	0.561821i	$0.189899\pi$
$360$	0	0
$361$	−3893.18	−0.567602
$362$	8963.90i	1.30147i
$363$	1003.75i	0.145133i
$364$	−6652.76	−0.957965
$365$	0	0
$366$	−6724.11	−0.960314
$367$	6565.52i	0.933835i	0.884301	+	0.466917i	$0.154635\pi$
$367$	6565.52i	0.933835i	−0.884301	+	0.466917i	$0.845365\pi$
$368$	− 2165.52i	− 0.306754i
$369$	8764.13	1.23643
$370$	0	0
$371$	−3350.01	−0.468797
$372$	2219.66i	0.309366i
$373$	2568.81i	0.356590i	0.983977	+	0.178295i	$0.0570582\pi$
$373$	2568.81i	0.356590i	−0.983977	+	0.178295i	$0.942942\pi$
$374$	6212.25	0.858898
$375$	0	0
$376$	4234.78	0.580830
$377$	25138.7i	3.43424i
$378$	3181.65i	0.432928i
$379$	−9975.58	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$379$	−9975.58	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$380$	0	0
$381$	−15533.8	−2.08877
$382$	− 5798.97i	− 0.776704i
$383$	5769.21i	0.769695i	0.922980	+	0.384847i	$0.125746\pi$
$383$	5769.21i	0.769695i	−0.922980	+	0.384847i	$0.874254\pi$
$384$	−21783.7	−2.89490
$385$	0	0
$386$	−21550.8	−2.84173
$387$	− 7654.04i	− 1.00537i
$388$	5657.47i	0.740243i
$389$	3040.21	0.396259	0.198129	−	0.980176i	$-0.436513\pi$
$389$	3040.21	0.396259	0.198129	+	0.980176i	$0.436513\pi$
$390$	0	0
$391$	10028.9	1.29714
$392$	− 9288.46i	− 1.19678i
$393$	13293.8i	1.70632i
$394$	−21807.4	−2.78842
$395$	0	0
$396$	−6571.66	−0.833936
$397$	7428.51i	0.939109i	0.882903	+	0.469555i	$0.155586\pi$
$397$	7428.51i	0.939109i	−0.882903	+	0.469555i	$0.844414\pi$
$398$	6277.32i	0.790587i
$399$	−2477.43	−0.310844
$400$	0	0
$401$	−4832.93	−0.601858	−0.300929	−	0.953647i	$-0.597297\pi$
$401$	−4832.93	−0.601858	−0.300929	+	0.953647i	$0.597297\pi$
$402$	− 4567.97i	− 0.566741i
$403$	1590.20i	0.196560i
$404$	4453.32	0.548418
$405$	0	0
$406$	7664.82	0.936942
$407$	− 2068.83i	− 0.251961i
$408$	29455.6i	3.57419i
$409$	−2479.52	−0.299766	−0.149883	−	0.988704i	$-0.547890\pi$
$409$	−2479.52	−0.299766	−0.149883	+	0.988704i	$0.547890\pi$
$410$	0	0
$411$	−3941.91	−0.473091
$412$	13554.1i	1.62078i
$413$	− 3604.35i	− 0.429439i
$414$	−16549.5	−1.96465
$415$	0	0
$416$	−9808.40	−1.15600
$417$	− 6895.78i	− 0.809803i
$418$	− 2828.10i	− 0.330926i
$419$	−470.068	−0.0548074	−0.0274037	−	0.999624i	$-0.508724\pi$
$419$	−470.068	−0.0548074	−0.0274037	+	0.999624i	$0.508724\pi$
$420$	0	0
$421$	425.224	0.0492260	0.0246130	−	0.999697i	$-0.492165\pi$
$421$	425.224	0.0492260	0.0246130	+	0.999697i	$0.492165\pi$
$422$	2160.68i	0.249242i
$423$	− 5965.68i	− 0.685724i
$424$	18132.5	2.07687
$425$	0	0
$426$	23313.8	2.65154
$427$	941.572i	0.106712i
$428$	26310.6i	2.97143i
$429$	−7748.06	−0.871981
$430$	0	0
$431$	9509.32	1.06276	0.531378	−	0.847135i	$-0.321674\pi$
$431$	9509.32	1.06276	0.531378	+	0.847135i	$0.321674\pi$
$432$	− 3174.46i	− 0.353544i
$433$	− 17742.8i	− 1.96920i	−0.174811	−	0.984602i	$-0.555932\pi$
$433$	− 17742.8i	− 1.96920i	0.174811	−	0.984602i	$-0.444068\pi$
$434$	484.855	0.0536262
$435$	0	0
$436$	17711.5	1.94548
$437$	− 4565.61i	− 0.499778i
$438$	24666.5i	2.69089i
$439$	2146.35	0.233348	0.116674	−	0.993170i	$-0.462777\pi$
$439$	2146.35	0.233348	0.116674	+	0.993170i	$0.462777\pi$
$440$	0	0
$441$	−13085.0	−1.41291
$442$	47953.0i	5.16039i
$443$	− 1174.80i	− 0.125996i	−0.998014	−	0.0629980i	$-0.979934\pi$
$443$	− 1174.80i	− 0.125996i	0.998014	−	0.0629980i	$-0.0200662\pi$
$444$	22290.8	2.38260
$445$	0	0
$446$	−7847.11	−0.833119
$447$	18870.3i	1.99672i
$448$	4123.81i	0.434892i
$449$	1293.65	0.135971	0.0679857	−	0.997686i	$-0.478343\pi$
$449$	1293.65	0.135971	0.0679857	+	0.997686i	$0.478343\pi$
$450$	0	0
$451$	2305.54	0.240717
$452$	− 8348.03i	− 0.868713i
$453$	− 20483.4i	− 2.12449i
$454$	−2451.49	−0.253423
$455$	0	0
$456$	13409.5	1.37710
$457$	− 4489.28i	− 0.459518i	−0.973248	−	0.229759i	$-0.926206\pi$
$457$	− 4489.28i	− 0.459518i	0.973248	−	0.229759i	$-0.0737937\pi$
$458$	4668.77i	0.476326i
$459$	14701.5	1.49500
$460$	0	0
$461$	−3695.39	−0.373344	−0.186672	−	0.982422i	$-0.559770\pi$
$461$	−3695.39	−0.373344	−0.186672	+	0.982422i	$0.559770\pi$
$462$	2362.39i	0.237897i
$463$	13467.7i	1.35183i	0.736980	+	0.675915i	$0.236251\pi$
$463$	13467.7i	1.35183i	−0.736980	+	0.675915i	$0.763749\pi$
$464$	−7647.47	−0.765140
$465$	0	0
$466$	−25190.8	−2.50417
$467$	− 19019.3i	− 1.88460i	−0.334769	−	0.942300i	$-0.608658\pi$
$467$	− 19019.3i	− 1.88460i	0.334769	−	0.942300i	$-0.391342\pi$
$468$	− 50727.4i	− 5.01041i
$469$	−639.649	−0.0629771
$470$	0	0
$471$	−1999.97	−0.195656
$472$	19509.2i	1.90251i
$473$	− 2013.51i	− 0.195732i
$474$	−36651.6	−3.55161
$475$	0	0
$476$	9372.78	0.902522
$477$	− 25543.8i	− 2.45193i
$478$	19379.5i	1.85439i
$479$	8105.77	0.773199	0.386599	−	0.922248i	$-0.373650\pi$
$479$	8105.77	0.773199	0.386599	+	0.922248i	$0.373650\pi$
$480$	0	0
$481$	15969.5	1.51382
$482$	− 796.650i	− 0.0752830i
$483$	3813.78i	0.359282i
$484$	−1728.78	−0.162357
$485$	0	0
$486$	19954.2	1.86243
$487$	− 13979.7i	− 1.30078i	−0.759601	−	0.650390i	$-0.774606\pi$
$487$	− 13979.7i	− 1.30078i	0.759601	−	0.650390i	$-0.225394\pi$
$488$	− 5096.42i	− 0.472755i
$489$	10995.2	1.01681
$490$	0	0
$491$	8699.28	0.799578	0.399789	−	0.916607i	$-0.369083\pi$
$491$	8699.28	0.799578	0.399789	+	0.916607i	$0.369083\pi$
$492$	24841.3i	2.27628i
$493$	− 35416.8i	− 3.23549i
$494$	21830.4	1.98825
$495$	0	0
$496$	−483.758	−0.0437931
$497$	− 3264.60i	− 0.294643i
$498$	3182.03i	0.286326i
$499$	−18008.7	−1.61559	−0.807795	−	0.589463i	$-0.799339\pi$
$499$	−18008.7	−1.61559	−0.807795	+	0.589463i	$0.799339\pi$
$500$	0	0
$501$	−9579.25	−0.854231
$502$	− 36583.7i	− 3.25261i
$503$	− 6633.20i	− 0.587991i	−0.955807	−	0.293996i	$-0.905015\pi$
$503$	− 6633.20i	− 0.587991i	0.955807	−	0.293996i	$-0.0949851\pi$
$504$	−6806.42	−0.601552
$505$	0	0
$506$	−4353.61	−0.382493
$507$	− 41583.0i	− 3.64253i
$508$	− 26754.1i	− 2.33666i
$509$	13765.3	1.19869	0.599347	−	0.800489i	$-0.295427\pi$
$509$	13765.3	1.19869	0.599347	+	0.800489i	$0.295427\pi$
$510$	0	0
$511$	3454.02	0.299016
$512$	− 9117.57i	− 0.786999i
$513$	− 6692.79i	− 0.576011i
$514$	1265.18	0.108570
$515$	0	0
$516$	21694.8	1.85089
$517$	− 1569.36i	− 0.133502i
$518$	− 4869.12i	− 0.413006i
$519$	−18923.6	−1.60049
$520$	0	0
$521$	−4547.84	−0.382427	−0.191214	−	0.981548i	$-0.561242\pi$
$521$	−4547.84	−0.382427	−0.191214	+	0.981548i	$0.561242\pi$
$522$	58444.3i	4.90046i
$523$	20913.6i	1.74855i	0.485434	+	0.874273i	$0.338662\pi$
$523$	20913.6i	1.74855i	−0.485434	+	0.874273i	$0.661338\pi$
$524$	−22896.1	−1.90882
$525$	0	0
$526$	−5640.86	−0.467591
$527$	− 2240.37i	− 0.185184i
$528$	− 2357.05i	− 0.194275i
$529$	5138.65	0.422343
$530$	0	0
$531$	27483.2	2.24608
$532$	− 4266.92i	− 0.347734i
$533$	17796.7i	1.44627i
$534$	−9601.61	−0.778094
$535$	0	0
$536$	3462.21	0.279002
$537$	− 24138.3i	− 1.93975i
$538$	− 5595.19i	− 0.448375i
$539$	−3442.20	−0.275076
$540$	0	0
$541$	6198.74	0.492615	0.246308	−	0.969192i	$-0.420783\pi$
$541$	6198.74	0.492615	0.246308	+	0.969192i	$0.420783\pi$
$542$	− 15354.4i	− 1.21684i
$543$	15751.0i	1.24482i
$544$	13818.6	1.08910
$545$	0	0
$546$	−18235.5	−1.42932
$547$	11376.1i	0.889228i	0.895722	+	0.444614i	$0.146659\pi$
$547$	11376.1i	0.889228i	−0.895722	+	0.444614i	$0.853341\pi$
$548$	− 6789.22i	− 0.529235i
$549$	−7179.50	−0.558130
$550$	0	0
$551$	−16123.4	−1.24660
$552$	− 20642.8i	− 1.59169i
$553$	5132.29i	0.394661i
$554$	13711.3	1.05151
$555$	0	0
$556$	11876.7	0.905907
$557$	− 16917.9i	− 1.28696i	−0.765465	−	0.643478i	$-0.777491\pi$
$557$	− 16917.9i	− 1.28696i	0.765465	−	0.643478i	$-0.222509\pi$
$558$	3697.03i	0.280480i
$559$	15542.5	1.17599
$560$	0	0
$561$	10915.9	0.821515
$562$	7515.92i	0.564128i
$563$	8104.37i	0.606676i	0.952883	+	0.303338i	$0.0981011\pi$
$563$	8104.37i	0.606676i	−0.952883	+	0.303338i	$0.901899\pi$
$564$	16909.2	1.26242
$565$	0	0
$566$	−19537.4	−1.45092
$567$	− 600.627i	− 0.0444867i
$568$	17670.2i	1.30533i
$569$	10154.5	0.748150	0.374075	−	0.927398i	$-0.377960\pi$
$569$	10154.5	0.748150	0.374075	+	0.927398i	$0.377960\pi$
$570$	0	0
$571$	2508.21	0.183827	0.0919135	−	0.995767i	$-0.470702\pi$
$571$	2508.21	0.183827	0.0919135	+	0.995767i	$0.470702\pi$
$572$	− 13344.6i	− 0.975465i
$573$	− 10189.7i	− 0.742899i
$574$	5426.23	0.394576
$575$	0	0
$576$	−31444.1	−2.27460
$577$	13325.7i	0.961447i	0.876872	+	0.480724i	$0.159626\pi$
$577$	13325.7i	0.961447i	−0.876872	+	0.480724i	$0.840374\pi$
$578$	− 44364.9i	− 3.19262i
$579$	−37868.2	−2.71805
$580$	0	0
$581$	445.577	0.0318170
$582$	15507.4i	1.10447i
$583$	− 6719.70i	− 0.477361i
$584$	−18695.5	−1.32470
$585$	0	0
$586$	−6364.07	−0.448630
$587$	16860.6i	1.18554i	0.805373	+	0.592769i	$0.201965\pi$
$587$	16860.6i	1.18554i	−0.805373	+	0.592769i	$0.798035\pi$
$588$	− 37088.3i	− 2.60118i
$589$	−1019.92	−0.0713498
$590$	0	0
$591$	−38319.0	−2.66706
$592$	4858.10i	0.337275i
$593$	− 20931.0i	− 1.44946i	−0.689031	−	0.724732i	$-0.741964\pi$
$593$	− 20931.0i	− 1.44946i	0.689031	−	0.724732i	$-0.258036\pi$
$594$	−6382.00	−0.440836
$595$	0	0
$596$	−32500.6	−2.23369
$597$	11030.3i	0.756178i
$598$	− 33605.9i	− 2.29808i
$599$	−2001.46	−0.136523	−0.0682616	−	0.997667i	$-0.521745\pi$
$599$	−2001.46	−0.136523	−0.0682616	+	0.997667i	$0.521745\pi$
$600$	0	0
$601$	24221.9	1.64398	0.821990	−	0.569502i	$-0.192864\pi$
$601$	24221.9	1.64398	0.821990	+	0.569502i	$0.192864\pi$
$602$	− 4738.92i	− 0.320837i
$603$	− 4877.33i	− 0.329387i
$604$	35278.9	2.37662
$605$	0	0
$606$	12206.8	0.818262
$607$	10737.9i	0.718017i	0.933334	+	0.359009i	$0.116885\pi$
$607$	10737.9i	0.718017i	−0.933334	+	0.359009i	$0.883115\pi$
$608$	− 6290.88i	− 0.419620i
$609$	13468.3	0.896163
$610$	0	0
$611$	12114.1	0.802099
$612$	71467.6i	4.72043i
$613$	− 13890.6i	− 0.915233i	−0.889150	−	0.457616i	$-0.848703\pi$
$613$	− 13890.6i	− 0.915233i	0.889150	−	0.457616i	$-0.151297\pi$
$614$	−8220.84	−0.540335
$615$	0	0
$616$	−1790.53	−0.117115
$617$	− 3195.60i	− 0.208509i	−0.994551	−	0.104254i	$-0.966754\pi$
$617$	− 3195.60i	− 0.208509i	0.994551	−	0.104254i	$-0.0332456\pi$
$618$	37152.5i	2.41827i
$619$	15177.6	0.985521	0.492761	−	0.870165i	$-0.335988\pi$
$619$	15177.6	0.985521	0.492761	+	0.870165i	$0.335988\pi$
$620$	0	0
$621$	−10302.9	−0.665770
$622$	− 13418.7i	− 0.865019i
$623$	1344.50i	0.0864630i
$624$	18194.3	1.16723
$625$	0	0
$626$	34365.4	2.19411
$627$	− 4969.42i	− 0.316522i
$628$	− 3444.58i	− 0.218875i
$629$	−22498.8	−1.42621
$630$	0	0
$631$	20594.6	1.29930	0.649649	−	0.760234i	$-0.274916\pi$
$631$	20594.6	1.29930	0.649649	+	0.760234i	$0.274916\pi$
$632$	− 27779.4i	− 1.74843i
$633$	3796.65i	0.238394i
$634$	−34889.2	−2.18553
$635$	0	0
$636$	72402.0	4.51404
$637$	− 26570.7i	− 1.65270i
$638$	15374.7i	0.954058i
$639$	24892.6	1.54106
$640$	0	0
$641$	−21402.0	−1.31876	−0.659381	−	0.751809i	$-0.729182\pi$
$641$	−21402.0	−1.31876	−0.659381	+	0.751809i	$0.729182\pi$
$642$	72118.7i	4.43349i
$643$	− 21970.6i	− 1.34749i	−0.738963	−	0.673746i	$-0.764684\pi$
$643$	− 21970.6i	− 1.34749i	0.738963	−	0.673746i	$-0.235316\pi$
$644$	−6568.54	−0.401920
$645$	0	0
$646$	−30755.9	−1.87318
$647$	− 7164.99i	− 0.435370i	−0.976019	−	0.217685i	$-0.930149\pi$
$647$	− 7164.99i	− 0.435370i	0.976019	−	0.217685i	$-0.0698506\pi$
$648$	3251.00i	0.197085i
$649$	7229.87	0.437284
$650$	0	0
$651$	851.967	0.0512922
$652$	18937.2i	1.13748i
$653$	6995.87i	0.419249i	0.977782	+	0.209624i	$0.0672241\pi$
$653$	6995.87i	0.419249i	−0.977782	+	0.209624i	$0.932776\pi$
$654$	48548.2	2.90273
$655$	0	0
$656$	−5413.95	−0.322224
$657$	26337.0i	1.56393i
$658$	− 3693.59i	− 0.218831i
$659$	2079.61	0.122929	0.0614644	−	0.998109i	$-0.480423\pi$
$659$	2079.61	0.122929	0.0614644	+	0.998109i	$0.480423\pi$
$660$	0	0
$661$	−15601.7	−0.918055	−0.459028	−	0.888422i	$-0.651802\pi$
$661$	−15601.7	−0.918055	−0.459028	+	0.888422i	$0.651802\pi$
$662$	− 22806.5i	− 1.33897i
$663$	84261.0i	4.93579i
$664$	−2411.76	−0.140956
$665$	0	0
$666$	37127.1	2.16013
$667$	24820.5i	1.44086i
$668$	− 16498.5i	− 0.955607i
$669$	−13788.6	−0.796859
$670$	0	0
$671$	−1888.68	−0.108661
$672$	5254.94i	0.301658i
$673$	13140.4i	0.752636i	0.926490	+	0.376318i	$0.122810\pi$
$673$	13140.4i	0.752636i	−0.926490	+	0.376318i	$0.877190\pi$
$674$	12069.4	0.689754
$675$	0	0
$676$	71618.9	4.07481
$677$	15337.7i	0.870716i	0.900257	+	0.435358i	$0.143378\pi$
$677$	15337.7i	0.870716i	−0.900257	+	0.435358i	$0.856622\pi$
$678$	− 22882.4i	− 1.29615i
$679$	2171.49	0.122731
$680$	0	0
$681$	−4307.65	−0.242393
$682$	972.559i	0.0546059i
$683$	− 21513.6i	− 1.20526i	−0.798019	−	0.602632i	$-0.794119\pi$
$683$	− 21513.6i	− 1.20526i	0.798019	−	0.602632i	$-0.205881\pi$
$684$	32535.3	1.81874
$685$	0	0
$686$	−16981.4	−0.945122
$687$	8203.78i	0.455595i
$688$	4728.20i	0.262007i
$689$	51870.0	2.86806
$690$	0	0
$691$	19044.3	1.04845	0.524226	−	0.851579i	$-0.324355\pi$
$691$	19044.3	1.04845	0.524226	+	0.851579i	$0.324355\pi$
$692$	− 32592.4i	− 1.79043i
$693$	2522.38i	0.138265i
$694$	22211.1	1.21487
$695$	0	0
$696$	−72899.6	−3.97019
$697$	− 25073.0i	− 1.36256i
$698$	48573.4i	2.63400i
$699$	−44264.3	−2.39518
$700$	0	0
$701$	15942.4	0.858968	0.429484	−	0.903074i	$-0.358695\pi$
$701$	15942.4	0.858968	0.429484	+	0.903074i	$0.358695\pi$
$702$	− 49263.4i	− 2.64861i
$703$	10242.5i	0.549505i
$704$	−8271.85	−0.442837
$705$	0	0
$706$	−32254.3	−1.71941
$707$	− 1709.30i	− 0.0909265i
$708$	77899.0i	4.13506i
$709$	−444.064	−0.0235221	−0.0117611	−	0.999931i	$-0.503744\pi$
$709$	−444.064	−0.0235221	−0.0117611	+	0.999931i	$0.503744\pi$
$710$	0	0
$711$	−39133.8	−2.06418
$712$	− 7277.37i	− 0.383049i
$713$	1570.07i	0.0824681i
$714$	25691.3	1.34660
$715$	0	0
$716$	41573.8	2.16995
$717$	34052.8i	1.77368i
$718$	53130.4i	2.76157i
$719$	−13715.0	−0.711379	−0.355689	−	0.934604i	$-0.615754\pi$
$719$	−13715.0	−0.711379	−0.355689	+	0.934604i	$0.615754\pi$
$720$	0	0
$721$	5202.43	0.268722
$722$	− 18379.5i	− 0.947389i
$723$	− 1399.84i	− 0.0720064i
$724$	−27128.2	−1.39256
$725$	0	0
$726$	−4738.66	−0.242243
$727$	18642.9i	0.951067i	0.879698	+	0.475534i	$0.157745\pi$
$727$	18642.9i	0.951067i	−0.879698	+	0.475534i	$0.842255\pi$
$728$	− 13821.3i	− 0.703643i
$729$	32105.5	1.63113
$730$	0	0
$731$	−21897.2	−1.10793
$732$	− 20349.7i	− 1.02752i
$733$	14418.4i	0.726541i	0.931684	+	0.363270i	$0.118340\pi$
$733$	14418.4i	0.726541i	−0.931684	+	0.363270i	$0.881660\pi$
$734$	−30995.5	−1.55867
$735$	0	0
$736$	−9684.24	−0.485008
$737$	− 1283.06i	− 0.0641275i
$738$	41375.1i	2.06374i
$739$	−22187.3	−1.10443	−0.552215	−	0.833702i	$-0.686217\pi$
$739$	−22187.3	−1.10443	−0.552215	+	0.833702i	$0.686217\pi$
$740$	0	0
$741$	38359.5	1.90171
$742$	− 15815.2i	− 0.782473i
$743$	− 3770.05i	− 0.186150i	−0.995659	−	0.0930752i	$-0.970330\pi$
$743$	− 3770.05i	− 0.186150i	0.995659	−	0.0930752i	$-0.0296697\pi$
$744$	−4611.42	−0.227235
$745$	0	0
$746$	−12127.2	−0.595188
$747$	3397.53i	0.166411i
$748$	18800.6i	0.919009i
$749$	10098.7	0.492656
$750$	0	0
$751$	28939.6	1.40615	0.703076	−	0.711115i	$-0.251810\pi$
$751$	28939.6	1.40615	0.703076	+	0.711115i	$0.251810\pi$
$752$	3685.23i	0.178706i
$753$	− 64283.3i	− 3.11104i
$754$	−118679.	−5.73213
$755$	0	0
$756$	−9628.90	−0.463227
$757$	31242.6i	1.50004i	0.661415	+	0.750020i	$0.269956\pi$
$757$	31242.6i	1.50004i	−0.661415	+	0.750020i	$0.730044\pi$
$758$	− 47094.2i	− 2.25665i
$759$	−7649.98	−0.365845
$760$	0	0
$761$	−3138.07	−0.149481	−0.0747404	−	0.997203i	$-0.523813\pi$
$761$	−3138.07	−0.149481	−0.0747404	+	0.997203i	$0.523813\pi$
$762$	− 73334.5i	− 3.48639i
$763$	− 6798.16i	− 0.322555i
$764$	17549.9	0.831064
$765$	0	0
$766$	−27236.2	−1.28470
$767$	55808.2i	2.62727i
$768$	− 52935.0i	− 2.48714i
$769$	1548.14	0.0725974	0.0362987	−	0.999341i	$-0.488443\pi$
$769$	1548.14	0.0725974	0.0362987	+	0.999341i	$0.488443\pi$
$770$	0	0
$771$	2223.13	0.103844
$772$	− 65221.0i	− 3.04062i
$773$	22990.0i	1.06972i	0.844941	+	0.534860i	$0.179635\pi$
$773$	22990.0i	1.06972i	−0.844941	+	0.534860i	$0.820365\pi$
$774$	36134.4	1.67807
$775$	0	0
$776$	−11753.6	−0.543722
$777$	− 8555.82i	− 0.395030i
$778$	14352.7i	0.661399i
$779$	−11414.4	−0.524984
$780$	0	0
$781$	6548.39	0.300025
$782$	47346.0i	2.16507i
$783$	36384.6i	1.66064i
$784$	8083.09	0.368217
$785$	0	0
$786$	−62759.4	−2.84803
$787$	− 1913.05i	− 0.0866493i	−0.999061	−	0.0433246i	$-0.986205\pi$
$787$	− 1913.05i	− 0.0866493i	0.999061	−	0.0433246i	$-0.0137950\pi$
$788$	− 65997.4i	− 2.98358i
$789$	−9911.88	−0.447240
$790$	0	0
$791$	−3204.20	−0.144031
$792$	− 13652.8i	− 0.612541i
$793$	− 14578.9i	− 0.652852i
$794$	−35069.7	−1.56748
$795$	0	0
$796$	−18997.6	−0.845918
$797$	− 32560.8i	− 1.44713i	−0.690256	−	0.723565i	$-0.742502\pi$
$797$	− 32560.8i	− 1.44713i	0.690256	−	0.723565i	$-0.257498\pi$
$798$	− 11695.8i	− 0.518833i
$799$	−17067.0	−0.755678
$800$	0	0
$801$	−10251.9	−0.452225
$802$	− 22816.0i	− 1.00457i
$803$	6928.34i	0.304478i
$804$	13824.4	0.606405
$805$	0	0
$806$	−7507.28	−0.328080
$807$	− 9831.63i	− 0.428860i
$808$	9251.91i	0.402823i
$809$	26822.2	1.16566	0.582831	−	0.812594i	$-0.301945\pi$
$809$	26822.2	1.16566	0.582831	+	0.812594i	$0.301945\pi$
$810$	0	0
$811$	−30337.2	−1.31354	−0.656771	−	0.754090i	$-0.728078\pi$
$811$	−30337.2	−1.31354	−0.656771	+	0.754090i	$0.728078\pi$
$812$	23196.7i	1.00252i
$813$	− 26980.2i	− 1.16388i
$814$	9766.85	0.420550
$815$	0	0
$816$	−25633.1	−1.09968
$817$	9968.59i	0.426875i
$818$	− 11705.7i	− 0.500342i
$819$	−19470.5	−0.830715
$820$	0	0
$821$	−19389.4	−0.824231	−0.412115	−	0.911132i	$-0.635210\pi$
$821$	−19389.4	−0.824231	−0.412115	+	0.911132i	$0.635210\pi$
$822$	− 18609.6i	− 0.789640i
$823$	− 19189.1i	− 0.812744i	−0.913708	−	0.406372i	$-0.866794\pi$
$823$	− 19189.1i	− 0.812744i	0.913708	−	0.406372i	$-0.133206\pi$
$824$	−28159.1	−1.19050
$825$	0	0
$826$	17016.0	0.716781
$827$	33109.8i	1.39219i	0.717949	+	0.696095i	$0.245081\pi$
$827$	33109.8i	1.39219i	−0.717949	+	0.696095i	$0.754919\pi$
$828$	− 50085.2i	− 2.10215i
$829$	23075.7	0.966771	0.483385	−	0.875408i	$-0.339407\pi$
$829$	23075.7	0.966771	0.483385	+	0.875408i	$0.339407\pi$
$830$	0	0
$831$	24092.9	1.00575
$832$	− 63851.3i	− 2.66063i
$833$	37434.3i	1.55705i
$834$	32554.7	1.35165
$835$	0	0
$836$	8558.91	0.354086
$837$	2301.59i	0.0950473i
$838$	− 2219.17i	− 0.0914796i
$839$	35936.4	1.47874	0.739369	−	0.673300i	$-0.235124\pi$
$839$	35936.4	1.47874	0.739369	+	0.673300i	$0.235124\pi$
$840$	0	0
$841$	63264.0	2.59396
$842$	2007.46i	0.0821636i
$843$	13206.7i	0.539575i
$844$	−6539.03	−0.266686
$845$	0	0
$846$	28163.7	1.14455
$847$	663.551i	0.0269184i
$848$	15779.4i	0.638996i
$849$	−34330.3	−1.38777
$850$	0	0
$851$	15767.4	0.635133
$852$	70556.3i	2.83711i
$853$	2359.85i	0.0947242i	0.998878	+	0.0473621i	$0.0150815\pi$
$853$	2359.85i	0.0947242i	−0.998878	+	0.0473621i	$0.984919\pi$
$854$	−4445.12	−0.178113
$855$	0	0
$856$	−54661.1	−2.18257
$857$	32425.6i	1.29246i	0.763142	+	0.646230i	$0.223655\pi$
$857$	32425.6i	1.29246i	−0.763142	+	0.646230i	$0.776345\pi$
$858$	− 36578.2i	− 1.45543i
$859$	−48098.2	−1.91046	−0.955232	−	0.295857i	$-0.904395\pi$
$859$	−48098.2	−1.91046	−0.955232	+	0.295857i	$0.904395\pi$
$860$	0	0
$861$	9534.74	0.377402
$862$	44893.1i	1.77386i
$863$	23607.2i	0.931167i	0.885004	+	0.465584i	$0.154156\pi$
$863$	23607.2i	0.931167i	−0.885004	+	0.465584i	$0.845844\pi$
$864$	−14196.2	−0.558988
$865$	0	0
$866$	83763.0	3.28682
$867$	− 77956.1i	− 3.05367i
$868$	1467.36i	0.0573794i
$869$	−10294.7	−0.401870
$870$	0	0
$871$	9904.04	0.385288
$872$	36796.2i	1.42899i
$873$	16557.6i	0.641914i
$874$	21554.0	0.834184
$875$	0	0
$876$	−74650.1	−2.87922
$877$	30843.7i	1.18759i	0.804616	+	0.593795i	$0.202371\pi$
$877$	30843.7i	1.18759i	−0.804616	+	0.593795i	$0.797629\pi$
$878$	10132.8i	0.389483i
$879$	−11182.7	−0.429104
$880$	0	0
$881$	−15477.7	−0.591891	−0.295946	−	0.955205i	$-0.595635\pi$
$881$	−15477.7	−0.591891	−0.295946	+	0.955205i	$0.595635\pi$
$882$	− 61773.5i	− 2.35830i
$883$	− 27661.4i	− 1.05423i	−0.849795	−	0.527113i	$-0.823274\pi$
$883$	− 27661.4i	− 1.05423i	0.849795	−	0.527113i	$-0.176726\pi$
$884$	−145124.	−5.52155
$885$	0	0
$886$	5546.16	0.210301
$887$	− 17217.1i	− 0.651740i	−0.945415	−	0.325870i	$-0.894343\pi$
$887$	− 17217.1i	− 0.651740i	0.945415	−	0.325870i	$-0.105657\pi$
$888$	46309.9i	1.75007i
$889$	−10269.0	−0.387413
$890$	0	0
$891$	1204.78	0.0452994
$892$	− 23748.3i	− 0.891427i
$893$	7769.67i	0.291156i
$894$	−89085.9	−3.33275
$895$	0	0
$896$	−14400.5	−0.536929
$897$	− 59051.0i	− 2.19805i
$898$	6107.27i	0.226951i
$899$	5544.68	0.205701
$900$	0	0
$901$	−73077.5	−2.70207
$902$	10884.3i	0.401784i
$903$	− 8327.04i	− 0.306873i
$904$	17343.3	0.638086
$905$	0	0
$906$	96701.2	3.54601
$907$	− 20648.2i	− 0.755910i	−0.925824	−	0.377955i	$-0.876627\pi$
$907$	− 20648.2i	− 0.755910i	0.925824	−	0.377955i	$-0.123373\pi$
$908$	− 7419.13i	− 0.271159i
$909$	13033.5	0.475570
$910$	0	0
$911$	14434.1	0.524944	0.262472	−	0.964940i	$-0.415462\pi$
$911$	14434.1	0.524944	0.262472	+	0.964940i	$0.415462\pi$
$912$	11669.4i	0.423697i
$913$	893.772i	0.0323982i
$914$	21193.7	0.766985
$915$	0	0
$916$	−14129.5	−0.509663
$917$	8788.14i	0.316478i
$918$	69405.0i	2.49532i
$919$	53819.1	1.93180	0.965902	−	0.258907i	$-0.0833624\pi$
$919$	53819.1	1.93180	0.965902	+	0.258907i	$0.0833624\pi$
$920$	0	0
$921$	−14445.3	−0.516818
$922$	− 17445.8i	− 0.623152i
$923$	50547.7i	1.80260i
$924$	−7149.49	−0.254547
$925$	0	0
$926$	−63580.4	−2.25635
$927$	39668.6i	1.40549i
$928$	34199.7i	1.20976i
$929$	6549.01	0.231288	0.115644	−	0.993291i	$-0.463107\pi$
$929$	6549.01	0.231288	0.115644	+	0.993291i	$0.463107\pi$
$930$	0	0
$931$	17041.8	0.599917
$932$	− 76237.0i	− 2.67943i
$933$	− 23578.8i	− 0.827370i
$934$	89789.2	3.14560
$935$	0	0
$936$	105388.	3.68024
$937$	14481.1i	0.504885i	0.967612	+	0.252443i	$0.0812339\pi$
$937$	14481.1i	0.504885i	−0.967612	+	0.252443i	$0.918766\pi$
$938$	− 3019.75i	− 0.105116i
$939$	60385.4	2.09862
$940$	0	0
$941$	27311.0	0.946134	0.473067	−	0.881026i	$-0.343147\pi$
$941$	27311.0	0.946134	0.473067	+	0.881026i	$0.343147\pi$
$942$	− 9441.77i	− 0.326570i
$943$	17571.4i	0.606790i
$944$	−16977.5	−0.585349
$945$	0	0
$946$	9505.69	0.326698
$947$	4437.34i	0.152264i	0.997098	+	0.0761322i	$0.0242571\pi$
$947$	4437.34i	0.152264i	−0.997098	+	0.0761322i	$0.975743\pi$
$948$	− 110922.i	− 3.80018i
$949$	−53480.6	−1.82935
$950$	0	0
$951$	−61305.9	−2.09041
$952$	19472.2i	0.662919i
$953$	− 41059.5i	− 1.39564i	−0.716272	−	0.697821i	$-0.754153\pi$
$953$	− 41059.5i	− 1.39564i	0.716272	−	0.697821i	$-0.245847\pi$
$954$	120591.	4.09254
$955$	0	0
$956$	−58649.6	−1.98417
$957$	27015.7i	0.912534i
$958$	38267.0i	1.29055i
$959$	−2605.88	−0.0877460
$960$	0	0
$961$	−29440.3	−0.988227
$962$	75391.3i	2.52673i
$963$	77002.9i	2.57672i
$964$	2410.96	0.0805518
$965$	0	0
$966$	−18004.7	−0.599681
$967$	− 23044.5i	− 0.766351i	−0.923676	−	0.383175i	$-0.874831\pi$
$967$	− 23044.5i	− 0.766351i	0.923676	−	0.383175i	$-0.125169\pi$
$968$	− 3591.59i	− 0.119254i
$969$	−54043.0	−1.79165
$970$	0	0
$971$	42514.9	1.40512	0.702559	−	0.711626i	$-0.252041\pi$
$971$	42514.9	1.40512	0.702559	+	0.711626i	$0.252041\pi$
$972$	60389.1i	1.99278i
$973$	− 4558.60i	− 0.150197i
$974$	65997.4	2.17114
$975$	0	0
$976$	4435.06	0.145454
$977$	30875.5i	1.01105i	0.862812	+	0.505525i	$0.168701\pi$
$977$	30875.5i	1.01105i	−0.862812	+	0.505525i	$0.831299\pi$
$978$	51907.9i	1.69717i
$979$	−2696.91	−0.0880425
$980$	0	0
$981$	51836.0	1.68705
$982$	41068.9i	1.33458i
$983$	50025.0i	1.62314i	0.584254	+	0.811571i	$0.301387\pi$
$983$	50025.0i	1.62314i	−0.584254	+	0.811571i	$0.698613\pi$
$984$	−51608.5	−1.67197
$985$	0	0
$986$	167201.	5.40038
$987$	− 6490.22i	− 0.209307i
$988$	66067.1i	2.12740i
$989$	15345.7	0.493394
$990$	0	0
$991$	−38794.2	−1.24353	−0.621765	−	0.783204i	$-0.713584\pi$
$991$	−38794.2	−1.24353	−0.621765	+	0.783204i	$0.713584\pi$
$992$	2163.38i	0.0692412i
$993$	− 40074.7i	− 1.28070i
$994$	15412.0	0.491791
$995$	0	0
$996$	−9630.04	−0.306365
$997$	− 22826.4i	− 0.725094i	−0.931966	−	0.362547i	$-0.881907\pi$
$997$	− 22826.4i	− 0.725094i	0.931966	−	0.362547i	$-0.118093\pi$
$998$	− 85018.2i	− 2.69660i
$999$	23113.6	0.732013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.b.f.199.8		10
5.2	odd	4		275.4.a.g.1.2	✓	5
5.3	odd	4		275.4.a.h.1.4	yes	5
5.4	even	2	inner	275.4.b.f.199.3		10
15.2	even	4		2475.4.a.bl.1.4		5
15.8	even	4		2475.4.a.bh.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.g.1.2	✓	5	5.2	odd	4
275.4.a.h.1.4	yes	5	5.3	odd	4
275.4.b.f.199.3		10	5.4	even	2	inner
275.4.b.f.199.8		10	1.1	even	1	trivial
2475.4.a.bh.1.2		5	15.8	even	4
2475.4.a.bl.1.4		5	15.2	even	4

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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.72095i q^{2} +8.29546i q^{3} -14.2874 q^{4} -39.1625 q^{6} +5.48389i q^{7} -29.6825i q^{8} -41.8147 q^{9} -11.0000 q^{11} -118.521i q^{12} -84.9102i q^{13} -25.8892 q^{14} +25.8306 q^{16} +119.626i q^{17} -197.405i q^{18} +54.4593 q^{19} -45.4914 q^{21} -51.9305i q^{22} -83.8353i q^{23} +246.230 q^{24} +400.857 q^{26} -122.895i q^{27} -78.3505i q^{28} -296.062 q^{29} -18.7281 q^{31} -115.515i q^{32} -91.2501i q^{33} -564.750 q^{34} +597.424 q^{36} +188.075i q^{37} +257.100i q^{38} +704.369 q^{39} -209.594 q^{41} -214.763i q^{42} +183.046i q^{43} +157.161 q^{44} +395.782 q^{46} +142.669i q^{47} +214.277i q^{48} +312.927 q^{49} -992.355 q^{51} +1213.15i q^{52} +610.881i q^{53} +580.182 q^{54} +162.776 q^{56} +451.766i q^{57} -1397.70i q^{58} -657.261 q^{59} +171.698 q^{61} -88.4144i q^{62} -229.307i q^{63} +751.986 q^{64} +430.788 q^{66} +116.641i q^{67} -1709.15i q^{68} +695.452 q^{69} -595.308 q^{71} +1241.17i q^{72} -629.849i q^{73} -887.895 q^{74} -778.082 q^{76} -60.3228i q^{77} +3325.29i q^{78} +935.886 q^{79} -109.526 q^{81} -989.485i q^{82} -81.2520i q^{83} +649.954 q^{84} -864.154 q^{86} -2455.98i q^{87} +326.508i q^{88} +245.174 q^{89} +465.638 q^{91} +1197.79i q^{92} -155.358i q^{93} -673.535 q^{94} +958.251 q^{96} -395.976i q^{97} +1477.31i q^{98} +459.962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) \) 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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