LMFDB - Embedded newform 1127.4.a.m.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1127,4,Mod(1,1127)] 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("1127.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1127.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4951525765$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	1127.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.93780 q^{2} -3.29133 q^{3} +0.630669 q^{4} +19.2434 q^{5} +9.66926 q^{6} +21.6496 q^{8} -16.1672 q^{9} -56.5334 q^{10} -50.0729 q^{11} -2.07574 q^{12} +35.7919 q^{13} -63.3364 q^{15} -68.6476 q^{16} -114.078 q^{17} +47.4959 q^{18} -137.489 q^{19} +12.1362 q^{20} +147.104 q^{22} +23.0000 q^{23} -71.2560 q^{24} +245.310 q^{25} -105.150 q^{26} +142.077 q^{27} +182.527 q^{29} +186.070 q^{30} +39.8629 q^{31} +28.4760 q^{32} +164.806 q^{33} +335.138 q^{34} -10.1961 q^{36} +177.627 q^{37} +403.916 q^{38} -117.803 q^{39} +416.613 q^{40} -10.3871 q^{41} -258.445 q^{43} -31.5794 q^{44} -311.112 q^{45} -67.5694 q^{46} -434.979 q^{47} +225.942 q^{48} -720.671 q^{50} +375.467 q^{51} +22.5729 q^{52} -209.452 q^{53} -417.395 q^{54} -963.574 q^{55} +452.522 q^{57} -536.227 q^{58} +71.1467 q^{59} -39.9443 q^{60} +594.367 q^{61} -117.109 q^{62} +465.524 q^{64} +688.760 q^{65} -484.167 q^{66} +256.024 q^{67} -71.9453 q^{68} -75.7005 q^{69} +341.011 q^{71} -350.013 q^{72} +397.070 q^{73} -521.832 q^{74} -807.394 q^{75} -86.7102 q^{76} +346.082 q^{78} -560.541 q^{79} -1321.02 q^{80} -31.1091 q^{81} +30.5151 q^{82} -285.389 q^{83} -2195.25 q^{85} +759.261 q^{86} -600.755 q^{87} -1084.06 q^{88} +958.424 q^{89} +913.984 q^{90} +14.5054 q^{92} -131.202 q^{93} +1277.88 q^{94} -2645.76 q^{95} -93.7238 q^{96} +325.599 q^{97} +809.536 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 18 q^{3} + 88 q^{4} + 20 q^{5} + 36 q^{6} + 21 q^{8} + 126 q^{9} + 200 q^{10} + 20 q^{11} + 161 q^{12} + 196 q^{13} + 20 q^{15} + 324 q^{16} + 242 q^{17} + 85 q^{18} + 128 q^{19} + 46 q^{20} + 14 q^{22}+ \cdots + 3570 q^{99}+O(q^{100}) $ 22 * q + 18 * q^3 + 88 * q^4 + 20 * q^5 + 36 * q^6 + 21 * q^8 + 126 * q^9 + 200 * q^10 + 20 * q^11 + 161 * q^12 + 196 * q^13 + 20 * q^15 + 324 * q^16 + 242 * q^17 + 85 * q^18 + 128 * q^19 + 46 * q^20 + 14 * q^22 + 506 * q^23 + 66 * q^24 + 668 * q^25 + 340 * q^26 + 378 * q^27 - 190 * q^29 + 493 * q^30 + 508 * q^31 - 487 * q^32 + 1240 * q^33 + 416 * q^34 - 470 * q^36 - 102 * q^37 + 423 * q^38 - 520 * q^39 + 2048 * q^40 + 920 * q^41 - 728 * q^43 + 1132 * q^44 + 570 * q^45 + 1384 * q^47 + 1323 * q^48 - 975 * q^50 + 1688 * q^51 + 1892 * q^52 - 480 * q^53 + 1695 * q^54 + 1254 * q^55 - 302 * q^57 + 791 * q^58 + 650 * q^59 - 2490 * q^60 + 960 * q^61 + 2554 * q^62 + 3791 * q^64 + 800 * q^65 - 1221 * q^66 - 2136 * q^67 + 2290 * q^68 + 414 * q^69 - 136 * q^71 + 4114 * q^72 + 2562 * q^73 - 1928 * q^74 + 1820 * q^75 + 4645 * q^76 + 2605 * q^78 + 790 * q^79 - 2248 * q^80 - 506 * q^81 + 3838 * q^82 + 2026 * q^83 + 1392 * q^85 + 3419 * q^86 + 3340 * q^87 - 5101 * q^88 + 2618 * q^89 + 6780 * q^90 + 2024 * q^92 + 206 * q^93 + 3516 * q^94 - 4410 * q^95 + 3818 * q^96 + 5158 * q^97 + 3570 * 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 $

$2$	−2.93780	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$2$	−2.93780	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$3$	−3.29133	−0.633416	−0.316708	−	0.948523i	$-0.602578\pi$
$3$	−3.29133	−0.633416	−0.316708	+	0.948523i	$0.602578\pi$
$4$	0.630669	0.0788337
$5$	19.2434	1.72118	0.860592	−	0.509294i	$-0.170094\pi$
$5$	19.2434	1.72118	0.860592	+	0.509294i	$0.170094\pi$
$6$	9.66926	0.657910
$7$	0	0
$7$	0	0
$8$	21.6496	0.956787
$9$	−16.1672	−0.598784
$10$	−56.5334	−1.78774
$11$	−50.0729	−1.37250	−0.686252	−	0.727364i	$-0.740745\pi$
$11$	−50.0729	−1.37250	−0.686252	+	0.727364i	$0.740745\pi$
$12$	−2.07574	−0.0499345
$13$	35.7919	0.763607	0.381804	−	0.924243i	$-0.375303\pi$
$13$	35.7919	0.763607	0.381804	+	0.924243i	$0.375303\pi$
$14$	0	0
$15$	−63.3364	−1.09023
$16$	−68.6476	−1.07262
$17$	−114.078	−1.62752	−0.813762	−	0.581198i	$-0.802584\pi$
$17$	−114.078	−1.62752	−0.813762	+	0.581198i	$0.802584\pi$
$18$	47.4959	0.621939
$19$	−137.489	−1.66011	−0.830057	−	0.557678i	$-0.811692\pi$
$19$	−137.489	−1.66011	−0.830057	+	0.557678i	$0.811692\pi$
$20$	12.1362	0.135687
$21$	0	0
$22$	147.104	1.42558
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−71.2560	−0.606044
$25$	245.310	1.96248
$26$	−105.150	−0.793136
$27$	142.077	1.01270
$28$	0	0
$29$	182.527	1.16877	0.584385	−	0.811476i	$-0.301336\pi$
$29$	182.527	1.16877	0.584385	+	0.811476i	$0.301336\pi$
$30$	186.070	1.13238
$31$	39.8629	0.230954	0.115477	−	0.993310i	$-0.463160\pi$
$31$	39.8629	0.230954	0.115477	+	0.993310i	$0.463160\pi$
$32$	28.4760	0.157309
$33$	164.806	0.869366
$34$	335.138	1.69046
$35$	0	0
$36$	−10.1961	−0.0472043
$37$	177.627	0.789235	0.394617	−	0.918846i	$-0.370877\pi$
$37$	177.627	0.789235	0.394617	+	0.918846i	$0.370877\pi$
$38$	403.916	1.72431
$39$	−117.803	−0.483681
$40$	416.613	1.64681
$41$	−10.3871	−0.0395655	−0.0197827	−	0.999804i	$-0.506297\pi$
$41$	−10.3871	−0.0395655	−0.0197827	+	0.999804i	$0.506297\pi$
$42$	0	0
$43$	−258.445	−0.916571	−0.458285	−	0.888805i	$-0.651536\pi$
$43$	−258.445	−0.916571	−0.458285	+	0.888805i	$0.651536\pi$
$44$	−31.5794	−0.108199
$45$	−311.112	−1.03062
$46$	−67.5694	−0.216577
$47$	−434.979	−1.34996	−0.674981	−	0.737835i	$-0.735848\pi$
$47$	−434.979	−1.34996	−0.674981	+	0.737835i	$0.735848\pi$
$48$	225.942	0.679414
$49$	0	0
$50$	−720.671	−2.03836
$51$	375.467	1.03090
$52$	22.5729	0.0601980
$53$	−209.452	−0.542839	−0.271419	−	0.962461i	$-0.587493\pi$
$53$	−209.452	−0.542839	−0.271419	+	0.962461i	$0.587493\pi$
$54$	−417.395	−1.05186
$55$	−963.574	−2.36233
$56$	0	0
$57$	452.522	1.05154
$58$	−536.227	−1.21397
$59$	71.1467	0.156992	0.0784958	−	0.996914i	$-0.474988\pi$
$59$	71.1467	0.156992	0.0784958	+	0.996914i	$0.474988\pi$
$60$	−39.9443	−0.0859465
$61$	594.367	1.24756	0.623778	−	0.781601i	$-0.285597\pi$
$61$	594.367	1.24756	0.623778	+	0.781601i	$0.285597\pi$
$62$	−117.109	−0.239885
$63$	0	0
$64$	465.524	0.909227
$65$	688.760	1.31431
$66$	−484.167	−0.902983
$67$	256.024	0.466841	0.233420	−	0.972376i	$-0.425008\pi$
$67$	256.024	0.466841	0.233420	+	0.972376i	$0.425008\pi$
$68$	−71.9453	−0.128304
$69$	−75.7005	−0.132076
$70$	0	0
$71$	341.011	0.570007	0.285004	−	0.958526i	$-0.408005\pi$
$71$	341.011	0.570007	0.285004	+	0.958526i	$0.408005\pi$
$72$	−350.013	−0.572909
$73$	397.070	0.636623	0.318312	−	0.947986i	$-0.396884\pi$
$73$	397.070	0.636623	0.318312	+	0.947986i	$0.396884\pi$
$74$	−521.832	−0.819754
$75$	−807.394	−1.24306
$76$	−86.7102	−0.130873
$77$	0	0
$78$	346.082	0.502385
$79$	−560.541	−0.798301	−0.399151	−	0.916885i	$-0.630695\pi$
$79$	−560.541	−0.798301	−0.399151	+	0.916885i	$0.630695\pi$
$80$	−1321.02	−1.84618
$81$	−31.1091	−0.0426736
$82$	30.5151	0.0410954
$83$	−285.389	−0.377416	−0.188708	−	0.982033i	$-0.560430\pi$
$83$	−285.389	−0.377416	−0.188708	+	0.982033i	$0.560430\pi$
$84$	0	0
$85$	−2195.25	−2.80127
$86$	759.261	0.952014
$87$	−600.755	−0.740318
$88$	−1084.06	−1.31319
$89$	958.424	1.14149	0.570746	−	0.821127i	$-0.306654\pi$
$89$	958.424	1.14149	0.570746	+	0.821127i	$0.306654\pi$
$90$	913.984	1.07047
$91$	0	0
$92$	14.5054	0.0164380
$93$	−131.202	−0.146290
$94$	1277.88	1.40216
$95$	−2645.76	−2.85736
$96$	−93.7238	−0.0996421
$97$	325.599	0.340821	0.170410	−	0.985373i	$-0.445491\pi$
$97$	325.599	0.340821	0.170410	+	0.985373i	$0.445491\pi$
$98$	0	0
$99$	809.536	0.821833
$100$	154.709	0.154709
$101$	−604.882	−0.595920	−0.297960	−	0.954578i	$-0.596306\pi$
$101$	−604.882	−0.595920	−0.297960	+	0.954578i	$0.596306\pi$
$102$	−1103.05	−1.07076
$103$	−1176.39	−1.12537	−0.562684	−	0.826672i	$-0.690231\pi$
$103$	−1176.39	−1.12537	−0.562684	+	0.826672i	$0.690231\pi$
$104$	774.882	0.730610
$105$	0	0
$106$	615.328	0.563830
$107$	2003.72	1.81035	0.905173	−	0.425044i	$-0.139741\pi$
$107$	2003.72	1.81035	0.905173	+	0.425044i	$0.139741\pi$
$108$	89.6038	0.0798345
$109$	−585.876	−0.514833	−0.257416	−	0.966301i	$-0.582871\pi$
$109$	−585.876	−0.514833	−0.257416	+	0.966301i	$0.582871\pi$
$110$	2830.79	2.45368
$111$	−584.628	−0.499914
$112$	0	0
$113$	−391.508	−0.325929	−0.162964	−	0.986632i	$-0.552106\pi$
$113$	−391.508	−0.325929	−0.162964	+	0.986632i	$0.552106\pi$
$114$	−1329.42	−1.09221
$115$	442.599	0.358892
$116$	115.114	0.0921385
$117$	−578.654	−0.457236
$118$	−209.015	−0.163062
$119$	0	0
$120$	−1371.21	−1.04311
$121$	1176.29	0.883765
$122$	−1746.13	−1.29580
$123$	34.1872	0.0250614
$124$	25.1403	0.0182070
$125$	2315.17	1.65660
$126$	0	0
$127$	−868.409	−0.606762	−0.303381	−	0.952869i	$-0.598116\pi$
$127$	−868.409	−0.606762	−0.303381	+	0.952869i	$0.598116\pi$
$128$	−1595.42	−1.10169
$129$	850.628	0.580571
$130$	−2023.44	−1.36513
$131$	2586.18	1.72485	0.862427	−	0.506182i	$-0.168944\pi$
$131$	2586.18	1.72485	0.862427	+	0.506182i	$0.168944\pi$
$132$	103.938	0.0685353
$133$	0	0
$134$	−752.148	−0.484893
$135$	2734.05	1.74304
$136$	−2469.74	−1.55719
$137$	105.139	0.0655665	0.0327833	−	0.999462i	$-0.489563\pi$
$137$	105.139	0.0655665	0.0327833	+	0.999462i	$0.489563\pi$
$138$	222.393	0.137184
$139$	−218.658	−0.133427	−0.0667133	−	0.997772i	$-0.521251\pi$
$139$	−218.658	−0.133427	−0.0667133	+	0.997772i	$0.521251\pi$
$140$	0	0
$141$	1431.66	0.855088
$142$	−1001.82	−0.592049
$143$	−1792.20	−1.04805
$144$	1109.84	0.642267
$145$	3512.44	2.01167
$146$	−1166.51	−0.661241
$147$	0	0
$148$	112.024	0.0622183
$149$	1832.68	1.00764	0.503822	−	0.863807i	$-0.331927\pi$
$149$	1832.68	1.00764	0.503822	+	0.863807i	$0.331927\pi$
$150$	2371.96	1.29113
$151$	912.028	0.491522	0.245761	−	0.969331i	$-0.420962\pi$
$151$	912.028	0.491522	0.245761	+	0.969331i	$0.420962\pi$
$152$	−2976.59	−1.58838
$153$	1844.31	0.974536
$154$	0	0
$155$	767.099	0.397515
$156$	−74.2947	−0.0381304
$157$	1467.44	0.745950	0.372975	−	0.927841i	$-0.378338\pi$
$157$	1467.44	0.745950	0.372975	+	0.927841i	$0.378338\pi$
$158$	1646.76	0.829171
$159$	689.375	0.343843
$160$	547.976	0.270758
$161$	0	0
$162$	91.3923	0.0443238
$163$	−1259.62	−0.605284	−0.302642	−	0.953104i	$-0.597869\pi$
$163$	−1259.62	−0.605284	−0.302642	+	0.953104i	$0.597869\pi$
$164$	−6.55079	−0.00311909
$165$	3171.44	1.49634
$166$	838.417	0.392011
$167$	1324.26	0.613621	0.306810	−	0.951771i	$-0.400738\pi$
$167$	1324.26	0.613621	0.306810	+	0.951771i	$0.400738\pi$
$168$	0	0
$169$	−915.937	−0.416904
$170$	6449.20	2.90959
$171$	2222.81	0.994050
$172$	−162.994	−0.0722566
$173$	−965.076	−0.424123	−0.212062	−	0.977256i	$-0.568018\pi$
$173$	−965.076	−0.424123	−0.212062	+	0.977256i	$0.568018\pi$
$174$	1764.90	0.768946
$175$	0	0
$176$	3437.38	1.47217
$177$	−234.167	−0.0994410
$178$	−2815.66	−1.18563
$179$	2489.30	1.03944	0.519719	−	0.854338i	$-0.326037\pi$
$179$	2489.30	1.03944	0.519719	+	0.854338i	$0.326037\pi$
$180$	−196.209	−0.0812474
$181$	−3035.25	−1.24645	−0.623227	−	0.782041i	$-0.714179\pi$
$181$	−3035.25	−1.24645	−0.623227	+	0.782041i	$0.714179\pi$
$182$	0	0
$183$	−1956.26	−0.790222
$184$	497.941	0.199504
$185$	3418.15	1.35842
$186$	385.445	0.151947
$187$	5712.20	2.23378
$188$	−274.328	−0.106423
$189$	0	0
$190$	7772.72	2.96786
$191$	−1501.58	−0.568852	−0.284426	−	0.958698i	$-0.591803\pi$
$191$	−1501.58	−0.568852	−0.284426	+	0.958698i	$0.591803\pi$
$192$	−1532.19	−0.575919
$193$	−1039.47	−0.387684	−0.193842	−	0.981033i	$-0.562095\pi$
$193$	−1039.47	−0.387684	−0.193842	+	0.981033i	$0.562095\pi$
$194$	−956.546	−0.354000
$195$	−2266.93	−0.832505
$196$	0	0
$197$	4871.01	1.76165	0.880825	−	0.473441i	$-0.156988\pi$
$197$	4871.01	1.76165	0.880825	+	0.473441i	$0.156988\pi$
$198$	−2378.26	−0.853613
$199$	5038.67	1.79489	0.897443	−	0.441131i	$-0.145423\pi$
$199$	5038.67	1.79489	0.897443	+	0.441131i	$0.145423\pi$
$200$	5310.86	1.87767
$201$	−842.659	−0.295704
$202$	1777.02	0.618964
$203$	0	0
$204$	236.796	0.0812696
$205$	−199.883	−0.0680995
$206$	3455.99	1.16889
$207$	−371.845	−0.124855
$208$	−2457.03	−0.819060
$209$	6884.48	2.27851
$210$	0	0
$211$	1105.56	0.360711	0.180356	−	0.983601i	$-0.442275\pi$
$211$	1105.56	0.360711	0.180356	+	0.983601i	$0.442275\pi$
$212$	−132.095	−0.0427940
$213$	−1122.38	−0.361052
$214$	−5886.53	−1.88035
$215$	−4973.38	−1.57759
$216$	3075.92	0.968934
$217$	0	0
$218$	1721.19	0.534741
$219$	−1306.89	−0.403247
$220$	−607.696	−0.186231
$221$	−4083.06	−1.24279
$222$	1717.52	0.519245
$223$	1988.19	0.597035	0.298518	−	0.954404i	$-0.403508\pi$
$223$	1988.19	0.597035	0.298518	+	0.954404i	$0.403508\pi$
$224$	0	0
$225$	−3965.96	−1.17510
$226$	1150.17	0.338532
$227$	−905.777	−0.264839	−0.132420	−	0.991194i	$-0.542275\pi$
$227$	−905.777	−0.264839	−0.132420	+	0.991194i	$0.542275\pi$
$228$	285.392	0.0828970
$229$	−1373.72	−0.396410	−0.198205	−	0.980161i	$-0.563511\pi$
$229$	−1373.72	−0.396410	−0.198205	+	0.980161i	$0.563511\pi$
$230$	−1300.27	−0.372770
$231$	0	0
$232$	3951.63	1.11826
$233$	−636.385	−0.178931	−0.0894656	−	0.995990i	$-0.528516\pi$
$233$	−636.385	−0.178931	−0.0894656	+	0.995990i	$0.528516\pi$
$234$	1699.97	0.474917
$235$	−8370.49	−2.32354
$236$	44.8700	0.0123762
$237$	1844.92	0.505657
$238$	0	0
$239$	−6820.45	−1.84594	−0.922968	−	0.384878i	$-0.874244\pi$
$239$	−6820.45	−1.84594	−0.922968	+	0.384878i	$0.874244\pi$
$240$	4347.89	1.16940
$241$	5804.83	1.55154	0.775772	−	0.631013i	$-0.217361\pi$
$241$	5804.83	1.55154	0.775772	+	0.631013i	$0.217361\pi$
$242$	−3455.71	−0.917939
$243$	−3733.70	−0.985665
$244$	374.849	0.0983494
$245$	0	0
$246$	−100.435	−0.0260305
$247$	−4921.00	−1.26768
$248$	863.016	0.220974
$249$	939.310	0.239062
$250$	−6801.51	−1.72066
$251$	2339.95	0.588433	0.294216	−	0.955739i	$-0.404941\pi$
$251$	2339.95	0.588433	0.294216	+	0.955739i	$0.404941\pi$
$252$	0	0
$253$	−1151.68	−0.286187
$254$	2551.21	0.630225
$255$	7225.27	1.77437
$256$	962.846	0.235070
$257$	1767.81	0.429079	0.214539	−	0.976715i	$-0.431175\pi$
$257$	1767.81	0.429079	0.214539	+	0.976715i	$0.431175\pi$
$258$	−2498.98	−0.603021
$259$	0	0
$260$	434.380	0.103612
$261$	−2950.94	−0.699841
$262$	−7597.69	−1.79155
$263$	6089.12	1.42765	0.713824	−	0.700325i	$-0.246962\pi$
$263$	6089.12	1.42765	0.713824	+	0.700325i	$0.246962\pi$
$264$	3567.99	0.831798
$265$	−4030.58	−0.934326
$266$	0	0
$267$	−3154.49	−0.723039
$268$	161.467	0.0368028
$269$	4370.73	0.990662	0.495331	−	0.868704i	$-0.335047\pi$
$269$	4370.73	0.990662	0.495331	+	0.868704i	$0.335047\pi$
$270$	−8032.10	−1.81044
$271$	6874.28	1.54090	0.770448	−	0.637503i	$-0.220033\pi$
$271$	6874.28	1.54090	0.770448	+	0.637503i	$0.220033\pi$
$272$	7831.16	1.74571
$273$	0	0
$274$	−308.877	−0.0681019
$275$	−12283.4	−2.69351
$276$	−47.7420	−0.0104121
$277$	7858.50	1.70459	0.852295	−	0.523062i	$-0.175210\pi$
$277$	7858.50	1.70459	0.852295	+	0.523062i	$0.175210\pi$
$278$	642.373	0.138586
$279$	−644.470	−0.138292
$280$	0	0
$281$	−4197.59	−0.891128	−0.445564	−	0.895250i	$-0.646997\pi$
$281$	−4197.59	−0.891128	−0.445564	+	0.895250i	$0.646997\pi$
$282$	−4205.93	−0.888154
$283$	−1656.03	−0.347848	−0.173924	−	0.984759i	$-0.555645\pi$
$283$	−1656.03	−0.347848	−0.173924	+	0.984759i	$0.555645\pi$
$284$	215.065	0.0449357
$285$	8708.07	1.80990
$286$	5265.14	1.08858
$287$	0	0
$288$	−460.376	−0.0941942
$289$	8100.73	1.64884
$290$	−10318.8	−2.08946
$291$	−1071.65	−0.215881
$292$	250.420	0.0501873
$293$	−4194.47	−0.836327	−0.418163	−	0.908372i	$-0.637326\pi$
$293$	−4194.47	−0.836327	−0.418163	+	0.908372i	$0.637326\pi$
$294$	0	0
$295$	1369.11	0.270212
$296$	3845.55	0.755130
$297$	−7114.21	−1.38993
$298$	−5384.05	−1.04661
$299$	823.215	0.159223
$300$	−509.199	−0.0979953
$301$	0	0
$302$	−2679.35	−0.510528
$303$	1990.86	0.377466
$304$	9438.30	1.78067
$305$	11437.7	2.14727
$306$	−5418.23	−1.01222
$307$	3773.47	0.701510	0.350755	−	0.936467i	$-0.385925\pi$
$307$	3773.47	0.701510	0.350755	+	0.936467i	$0.385925\pi$
$308$	0	0
$309$	3871.88	0.712827
$310$	−2253.58	−0.412887
$311$	5313.38	0.968791	0.484395	−	0.874849i	$-0.339040\pi$
$311$	5313.38	0.968791	0.484395	+	0.874849i	$0.339040\pi$
$312$	−2550.39	−0.462780
$313$	6019.17	1.08698	0.543488	−	0.839417i	$-0.317103\pi$
$313$	6019.17	1.08698	0.543488	+	0.839417i	$0.317103\pi$
$314$	−4311.03	−0.774795
$315$	0	0
$316$	−353.516	−0.0629330
$317$	−1729.55	−0.306440	−0.153220	−	0.988192i	$-0.548964\pi$
$317$	−1729.55	−0.306440	−0.153220	+	0.988192i	$0.548964\pi$
$318$	−2025.25	−0.357139
$319$	−9139.63	−1.60414
$320$	8958.28	1.56495
$321$	−6594.90	−1.14670
$322$	0	0
$323$	15684.5	2.70188
$324$	−19.6195	−0.00336412
$325$	8780.11	1.49856
$326$	3700.52	0.628690
$327$	1928.31	0.326103
$328$	−224.876	−0.0378557
$329$	0	0
$330$	−9317.04	−1.55420
$331$	−9507.72	−1.57883	−0.789414	−	0.613862i	$-0.789615\pi$
$331$	−9507.72	−1.57883	−0.789414	+	0.613862i	$0.789615\pi$
$332$	−179.986	−0.0297531
$333$	−2871.72	−0.472581
$334$	−3890.42	−0.637349
$335$	4926.78	0.803519
$336$	0	0
$337$	−1620.67	−0.261970	−0.130985	−	0.991384i	$-0.541814\pi$
$337$	−1620.67	−0.261970	−0.130985	+	0.991384i	$0.541814\pi$
$338$	2690.84	0.433025
$339$	1288.58	0.206449
$340$	−1384.47	−0.220834
$341$	−1996.05	−0.316986
$342$	−6530.17	−1.03249
$343$	0	0
$344$	−5595.24	−0.876963
$345$	−1456.74	−0.227328
$346$	2835.20	0.440524
$347$	4832.63	0.747635	0.373817	−	0.927502i	$-0.378049\pi$
$347$	4832.63	0.747635	0.373817	+	0.927502i	$0.378049\pi$
$348$	−378.878	−0.0583620
$349$	11217.7	1.72054	0.860269	−	0.509840i	$-0.170296\pi$
$349$	11217.7	1.72054	0.860269	+	0.509840i	$0.170296\pi$
$350$	0	0
$351$	5085.22	0.773302
$352$	−1425.87	−0.215907
$353$	−10532.5	−1.58807	−0.794036	−	0.607871i	$-0.792024\pi$
$353$	−10532.5	−1.58807	−0.794036	+	0.607871i	$0.792024\pi$
$354$	687.936	0.103286
$355$	6562.21	0.981088
$356$	604.449	0.0899880
$357$	0	0
$358$	−7313.07	−1.07963
$359$	4113.88	0.604798	0.302399	−	0.953181i	$-0.402212\pi$
$359$	4113.88	0.604798	0.302399	+	0.953181i	$0.402212\pi$
$360$	−6735.45	−0.986082
$361$	12044.3	1.75598
$362$	8916.95	1.29465
$363$	−3871.56	−0.559791
$364$	0	0
$365$	7640.98	1.09575
$366$	5747.09	0.820779
$367$	2208.81	0.314166	0.157083	−	0.987585i	$-0.449791\pi$
$367$	2208.81	0.314166	0.157083	+	0.987585i	$0.449791\pi$
$368$	−1578.90	−0.223657
$369$	167.929	0.0236912
$370$	−10041.8	−1.41095
$371$	0	0
$372$	−82.7449	−0.0115326
$373$	−753.446	−0.104590	−0.0522948	−	0.998632i	$-0.516654\pi$
$373$	−753.446	−0.104590	−0.0522948	+	0.998632i	$0.516654\pi$
$374$	−16781.3	−2.32016
$375$	−7619.98	−1.04932
$376$	−9417.14	−1.29163
$377$	6532.98	0.892482
$378$	0	0
$379$	373.749	0.0506548	0.0253274	−	0.999679i	$-0.491937\pi$
$379$	373.749	0.0506548	0.0253274	+	0.999679i	$0.491937\pi$
$380$	−1668.60	−0.225256
$381$	2858.22	0.384333
$382$	4411.35	0.590849
$383$	−9325.20	−1.24411	−0.622057	−	0.782972i	$-0.713703\pi$
$383$	−9325.20	−1.24411	−0.622057	+	0.782972i	$0.713703\pi$
$384$	5251.06	0.697831
$385$	0	0
$386$	3053.77	0.402675
$387$	4178.33	0.548828
$388$	205.346	0.0268681
$389$	−12888.7	−1.67990	−0.839952	−	0.542661i	$-0.817417\pi$
$389$	−12888.7	−1.67990	−0.839952	+	0.542661i	$0.817417\pi$
$390$	6659.80	0.864697
$391$	−2623.79	−0.339362
$392$	0	0
$393$	−8511.97	−1.09255
$394$	−14310.1	−1.82977
$395$	−10786.7	−1.37402
$396$	510.550	0.0647881
$397$	−321.656	−0.0406636	−0.0203318	−	0.999793i	$-0.506472\pi$
$397$	−321.656	−0.0406636	−0.0203318	+	0.999793i	$0.506472\pi$
$398$	−14802.6	−1.86429
$399$	0	0
$400$	−16839.9	−2.10499
$401$	−13843.9	−1.72402	−0.862009	−	0.506893i	$-0.830794\pi$
$401$	−13843.9	−1.72402	−0.862009	+	0.506893i	$0.830794\pi$
$402$	2475.56	0.307139
$403$	1426.77	0.176358
$404$	−381.480	−0.0469786
$405$	−598.646	−0.0734492
$406$	0	0
$407$	−8894.29	−1.08323
$408$	8128.72	0.986352
$409$	−13475.9	−1.62919	−0.814595	−	0.580031i	$-0.803041\pi$
$409$	−13475.9	−1.62919	−0.814595	+	0.580031i	$0.803041\pi$
$410$	587.215	0.0707329
$411$	−346.046	−0.0415309
$412$	−741.912	−0.0887169
$413$	0	0
$414$	1092.41	0.129683
$415$	−5491.87	−0.649603
$416$	1019.21	0.120122
$417$	719.674	0.0845146
$418$	−20225.2	−2.36662
$419$	4710.03	0.549165	0.274582	−	0.961564i	$-0.411460\pi$
$419$	4710.03	0.549165	0.274582	+	0.961564i	$0.411460\pi$
$420$	0	0
$421$	17209.4	1.99225	0.996124	−	0.0879655i	$-0.0280365\pi$
$421$	17209.4	1.99225	0.996124	+	0.0879655i	$0.0280365\pi$
$422$	−3247.92	−0.374660
$423$	7032.38	0.808336
$424$	−4534.56	−0.519381
$425$	−27984.4	−3.19398
$426$	3297.32	0.375013
$427$	0	0
$428$	1263.68	0.142716
$429$	5898.73	0.663854
$430$	14610.8	1.63859
$431$	12400.8	1.38591	0.692953	−	0.720982i	$-0.256309\pi$
$431$	12400.8	1.38591	0.692953	+	0.720982i	$0.256309\pi$
$432$	−9753.26	−1.08624
$433$	17265.4	1.91622	0.958111	−	0.286399i	$-0.0924582\pi$
$433$	17265.4	1.91622	0.958111	+	0.286399i	$0.0924582\pi$
$434$	0	0
$435$	−11560.6	−1.27422
$436$	−369.494	−0.0405862
$437$	−3162.25	−0.346158
$438$	3839.37	0.418841
$439$	−5740.65	−0.624115	−0.312057	−	0.950063i	$-0.601018\pi$
$439$	−5740.65	−0.624115	−0.312057	+	0.950063i	$0.601018\pi$
$440$	−20861.0	−2.26025
$441$	0	0
$442$	11995.2	1.29085
$443$	12144.9	1.30254	0.651268	−	0.758848i	$-0.274237\pi$
$443$	12144.9	1.30254	0.651268	+	0.758848i	$0.274237\pi$
$444$	−368.707	−0.0394100
$445$	18443.4	1.96472
$446$	−5840.90	−0.620122
$447$	−6031.95	−0.638258
$448$	0	0
$449$	−176.722	−0.0185747	−0.00928733	−	0.999957i	$-0.502956\pi$
$449$	−176.722	−0.0185747	−0.00928733	+	0.999957i	$0.502956\pi$
$450$	11651.2	1.22054
$451$	520.109	0.0543038
$452$	−246.912	−0.0256942
$453$	−3001.78	−0.311338
$454$	2660.99	0.275080
$455$	0	0
$456$	9796.93	1.00610
$457$	−263.872	−0.0270097	−0.0135048	−	0.999909i	$-0.504299\pi$
$457$	−263.872	−0.0270097	−0.0135048	+	0.999909i	$0.504299\pi$
$458$	4035.72	0.411739
$459$	−16207.9	−1.64819
$460$	279.134	0.0282928
$461$	−2205.17	−0.222787	−0.111394	−	0.993776i	$-0.535531\pi$
$461$	−2205.17	−0.222787	−0.111394	+	0.993776i	$0.535531\pi$
$462$	0	0
$463$	−14654.4	−1.47095	−0.735474	−	0.677553i	$-0.763040\pi$
$463$	−14654.4	−1.47095	−0.735474	+	0.677553i	$0.763040\pi$
$464$	−12530.0	−1.25365
$465$	−2524.77	−0.251792
$466$	1869.57	0.185850
$467$	−364.352	−0.0361032	−0.0180516	−	0.999837i	$-0.505746\pi$
$467$	−364.352	−0.0361032	−0.0180516	+	0.999837i	$0.505746\pi$
$468$	−364.939	−0.0360456
$469$	0	0
$470$	24590.8	2.41338
$471$	−4829.81	−0.472497
$472$	1540.30	0.150208
$473$	12941.1	1.25800
$474$	−5420.02	−0.525210
$475$	−33727.4	−3.25794
$476$	0	0
$477$	3386.25	0.325043
$478$	20037.1	1.91732
$479$	−8804.97	−0.839894	−0.419947	−	0.907549i	$-0.637951\pi$
$479$	−8804.97	−0.839894	−0.419947	+	0.907549i	$0.637951\pi$
$480$	−1803.57	−0.171503
$481$	6357.61	0.602666
$482$	−17053.4	−1.61154
$483$	0	0
$484$	741.851	0.0696704
$485$	6265.65	0.586615
$486$	10968.9	1.02378
$487$	5284.47	0.491709	0.245854	−	0.969307i	$-0.420931\pi$
$487$	5284.47	0.491709	0.245854	+	0.969307i	$0.420931\pi$
$488$	12867.8	1.19365
$489$	4145.83	0.383397
$490$	0	0
$491$	−9709.26	−0.892409	−0.446204	−	0.894931i	$-0.647225\pi$
$491$	−9709.26	−0.892409	−0.446204	+	0.894931i	$0.647225\pi$
$492$	21.5608	0.00197568
$493$	−20822.2	−1.90220
$494$	14456.9	1.31670
$495$	15578.3	1.41453
$496$	−2736.49	−0.247726
$497$	0	0
$498$	−2759.50	−0.248306
$499$	4450.25	0.399240	0.199620	−	0.979873i	$-0.436029\pi$
$499$	4450.25	0.399240	0.199620	+	0.979873i	$0.436029\pi$
$500$	1460.11	0.130596
$501$	−4358.59	−0.388677
$502$	−6874.32	−0.611187
$503$	−1315.14	−0.116579	−0.0582896	−	0.998300i	$-0.518565\pi$
$503$	−1315.14	−0.116579	−0.0582896	+	0.998300i	$0.518565\pi$
$504$	0	0
$505$	−11640.0	−1.02569
$506$	3383.39	0.297253
$507$	3014.65	0.264073
$508$	−547.679	−0.0478333
$509$	10670.6	0.929207	0.464603	−	0.885519i	$-0.346197\pi$
$509$	10670.6	0.929207	0.464603	+	0.885519i	$0.346197\pi$
$510$	−21226.4	−1.84298
$511$	0	0
$512$	9934.75	0.857535
$513$	−19534.1	−1.68119
$514$	−5193.48	−0.445671
$515$	−22637.7	−1.93697
$516$	536.465	0.0457685
$517$	21780.7	1.85283
$518$	0	0
$519$	3176.38	0.268647
$520$	14911.4	1.25751
$521$	11134.9	0.936333	0.468166	−	0.883640i	$-0.344915\pi$
$521$	11134.9	0.936333	0.468166	+	0.883640i	$0.344915\pi$
$522$	8669.27	0.726903
$523$	6797.83	0.568352	0.284176	−	0.958772i	$-0.408280\pi$
$523$	6797.83	0.568352	0.284176	+	0.958772i	$0.408280\pi$
$524$	1631.03	0.135977
$525$	0	0
$526$	−17888.6	−1.48285
$527$	−4547.47	−0.375884
$528$	−11313.5	−0.932498
$529$	529.000	0.0434783
$530$	11841.0	0.970455
$531$	−1150.24	−0.0940041
$532$	0	0
$533$	−371.773	−0.0302125
$534$	9267.25	0.750999
$535$	38558.4	3.11594
$536$	5542.83	0.446667
$537$	−8193.11	−0.658396
$538$	−12840.3	−1.02897
$539$	0	0
$540$	1724.28	0.137410
$541$	20092.9	1.59679	0.798395	−	0.602134i	$-0.205683\pi$
$541$	20092.9	1.59679	0.798395	+	0.602134i	$0.205683\pi$
$542$	−20195.3	−1.60048
$543$	9989.99	0.789524
$544$	−3248.48	−0.256024
$545$	−11274.3	−0.886122
$546$	0	0
$547$	8570.64	0.669934	0.334967	−	0.942230i	$-0.391275\pi$
$547$	8570.64	0.669934	0.334967	+	0.942230i	$0.391275\pi$
$548$	66.3078	0.00516885
$549$	−9609.24	−0.747017
$550$	36086.0	2.79766
$551$	−25095.4	−1.94029
$552$	−1638.89	−0.126369
$553$	0	0
$554$	−23086.7	−1.77050
$555$	−11250.3	−0.860444
$556$	−137.901	−0.0105185
$557$	16242.7	1.23560	0.617798	−	0.786337i	$-0.288025\pi$
$557$	16242.7	1.23560	0.617798	+	0.786337i	$0.288025\pi$
$558$	1893.32	0.143639
$559$	−9250.26	−0.699900
$560$	0	0
$561$	−18800.7	−1.41491
$562$	12331.7	0.925588
$563$	18479.9	1.38337	0.691684	−	0.722201i	$-0.256869\pi$
$563$	18479.9	1.38337	0.691684	+	0.722201i	$0.256869\pi$
$564$	902.903	0.0674097
$565$	−7533.95	−0.560984
$566$	4865.09	0.361299
$567$	0	0
$568$	7382.75	0.545375
$569$	15586.9	1.14840	0.574198	−	0.818717i	$-0.305314\pi$
$569$	15586.9	1.14840	0.574198	+	0.818717i	$0.305314\pi$
$570$	−25582.6	−1.87989
$571$	516.215	0.0378335	0.0189167	−	0.999821i	$-0.493978\pi$
$571$	516.215	0.0378335	0.0189167	+	0.999821i	$0.493978\pi$
$572$	−1130.29	−0.0826219
$573$	4942.20	0.360320
$574$	0	0
$575$	5642.12	0.409205
$576$	−7526.21	−0.544430
$577$	16474.2	1.18861	0.594306	−	0.804239i	$-0.297427\pi$
$577$	16474.2	1.18861	0.594306	+	0.804239i	$0.297427\pi$
$578$	−23798.3	−1.71259
$579$	3421.25	0.245565
$580$	2215.19	0.158587
$581$	0	0
$582$	3148.31	0.224229
$583$	10487.9	0.745048
$584$	8596.41	0.609113
$585$	−11135.3	−0.786988
$586$	12322.5	0.868667
$587$	−2484.61	−0.174703	−0.0873517	−	0.996178i	$-0.527840\pi$
$587$	−2484.61	−0.174703	−0.0873517	+	0.996178i	$0.527840\pi$
$588$	0	0
$589$	−5480.72	−0.383411
$590$	−4022.16	−0.280660
$591$	−16032.1	−1.11586
$592$	−12193.7	−0.846548
$593$	−5566.23	−0.385460	−0.192730	−	0.981252i	$-0.561734\pi$
$593$	−5566.23	−0.385460	−0.192730	+	0.981252i	$0.561734\pi$
$594$	20900.1	1.44368
$595$	0	0
$596$	1155.82	0.0794363
$597$	−16583.9	−1.13691
$598$	−2418.44	−0.165380
$599$	−6457.01	−0.440444	−0.220222	−	0.975450i	$-0.570678\pi$
$599$	−6457.01	−0.440444	−0.220222	+	0.975450i	$0.570678\pi$
$600$	−17479.8	−1.18935
$601$	21689.8	1.47212	0.736061	−	0.676915i	$-0.236683\pi$
$601$	21689.8	1.47212	0.736061	+	0.676915i	$0.236683\pi$
$602$	0	0
$603$	−4139.19	−0.279537
$604$	575.188	0.0387484
$605$	22635.9	1.52112
$606$	−5848.76	−0.392062
$607$	14277.1	0.954680	0.477340	−	0.878719i	$-0.341601\pi$
$607$	14277.1	0.954680	0.477340	+	0.878719i	$0.341601\pi$
$608$	−3915.14	−0.261151
$609$	0	0
$610$	−33601.6	−2.23031
$611$	−15568.8	−1.03084
$612$	1163.15	0.0768262
$613$	10965.4	0.722496	0.361248	−	0.932470i	$-0.382351\pi$
$613$	10965.4	0.722496	0.361248	+	0.932470i	$0.382351\pi$
$614$	−11085.7	−0.728637
$615$	657.879	0.0431353
$616$	0	0
$617$	3291.68	0.214778	0.107389	−	0.994217i	$-0.465751\pi$
$617$	3291.68	0.214778	0.107389	+	0.994217i	$0.465751\pi$
$618$	−11374.8	−0.740391
$619$	−22424.5	−1.45608	−0.728042	−	0.685533i	$-0.759569\pi$
$619$	−22424.5	−1.45608	−0.728042	+	0.685533i	$0.759569\pi$
$620$	483.786	0.0313376
$621$	3267.78	0.211162
$622$	−15609.6	−1.00625
$623$	0	0
$624$	8086.89	0.518806
$625$	13888.1	0.888839
$626$	−17683.1	−1.12901
$627$	−22659.1	−1.44325
$628$	925.467	0.0588060
$629$	−20263.3	−1.28450
$630$	0	0
$631$	−14811.1	−0.934423	−0.467211	−	0.884146i	$-0.654741\pi$
$631$	−14811.1	−0.934423	−0.467211	+	0.884146i	$0.654741\pi$
$632$	−12135.5	−0.763804
$633$	−3638.77	−0.228480
$634$	5081.08	0.318290
$635$	−16711.2	−1.04435
$636$	434.768	0.0271064
$637$	0	0
$638$	26850.4	1.66617
$639$	−5513.18	−0.341311
$640$	−30701.4	−1.89622
$641$	−20724.2	−1.27700	−0.638501	−	0.769621i	$-0.720445\pi$
$641$	−20724.2	−1.27700	−0.638501	+	0.769621i	$0.720445\pi$
$642$	19374.5	1.19104
$643$	8785.36	0.538819	0.269409	−	0.963026i	$-0.413171\pi$
$643$	8785.36	0.538819	0.269409	+	0.963026i	$0.413171\pi$
$644$	0	0
$645$	16369.0	0.999270
$646$	−46077.8	−2.80636
$647$	7715.71	0.468834	0.234417	−	0.972136i	$-0.424682\pi$
$647$	7715.71	0.468834	0.234417	+	0.972136i	$0.424682\pi$
$648$	−673.500	−0.0408296
$649$	−3562.52	−0.215472
$650$	−25794.2	−1.55651
$651$	0	0
$652$	−794.405	−0.0477167
$653$	−9535.26	−0.571430	−0.285715	−	0.958315i	$-0.592231\pi$
$653$	−9535.26	−0.571430	−0.285715	+	0.958315i	$0.592231\pi$
$654$	−5664.99	−0.338714
$655$	49767.0	2.96879
$656$	713.046	0.0424387
$657$	−6419.49	−0.381200
$658$	0	0
$659$	18803.7	1.11151	0.555756	−	0.831345i	$-0.312429\pi$
$659$	18803.7	1.11151	0.555756	+	0.831345i	$0.312429\pi$
$660$	2000.13	0.117962
$661$	−22332.3	−1.31411	−0.657055	−	0.753843i	$-0.728198\pi$
$661$	−22332.3	−1.31411	−0.657055	+	0.753843i	$0.728198\pi$
$662$	27931.8	1.63988
$663$	13438.7	0.787203
$664$	−6178.57	−0.361107
$665$	0	0
$666$	8436.55	0.490855
$667$	4198.11	0.243706
$668$	835.173	0.0483740
$669$	−6543.78	−0.378172
$670$	−14473.9	−0.834591
$671$	−29761.7	−1.71227
$672$	0	0
$673$	12526.2	0.717460	0.358730	−	0.933441i	$-0.383210\pi$
$673$	12526.2	0.717460	0.358730	+	0.933441i	$0.383210\pi$
$674$	4761.22	0.272100
$675$	34852.9	1.98739
$676$	−577.653	−0.0328660
$677$	−6314.26	−0.358459	−0.179229	−	0.983807i	$-0.557360\pi$
$677$	−6314.26	−0.358459	−0.179229	+	0.983807i	$0.557360\pi$
$678$	−3785.59	−0.214432
$679$	0	0
$680$	−47526.3	−2.68022
$681$	2981.21	0.167754
$682$	5863.99	0.329243
$683$	−27524.1	−1.54199	−0.770995	−	0.636841i	$-0.780241\pi$
$683$	−27524.1	−1.54199	−0.770995	+	0.636841i	$0.780241\pi$
$684$	1401.86	0.0783646
$685$	2023.23	0.112852
$686$	0	0
$687$	4521.36	0.251093
$688$	17741.7	0.983131
$689$	−7496.70	−0.414516
$690$	4279.60	0.236118
$691$	24378.6	1.34212	0.671060	−	0.741403i	$-0.265839\pi$
$691$	24378.6	1.34212	0.671060	+	0.741403i	$0.265839\pi$
$692$	−608.643	−0.0334352
$693$	0	0
$694$	−14197.3	−0.776545
$695$	−4207.73	−0.229652
$696$	−13006.1	−0.708327
$697$	1184.93	0.0643938
$698$	−32955.3	−1.78707
$699$	2094.55	0.113338
$700$	0	0
$701$	9933.10	0.535190	0.267595	−	0.963532i	$-0.413771\pi$
$701$	9933.10	0.535190	0.267595	+	0.963532i	$0.413771\pi$
$702$	−14939.4	−0.803205
$703$	−24421.8	−1.31022
$704$	−23310.1	−1.24792
$705$	27550.0	1.47176
$706$	30942.4	1.64948
$707$	0	0
$708$	−147.682	−0.00783930
$709$	18595.5	0.985006	0.492503	−	0.870311i	$-0.336082\pi$
$709$	18595.5	0.985006	0.492503	+	0.870311i	$0.336082\pi$
$710$	−19278.5	−1.01903
$711$	9062.36	0.478010
$712$	20749.5	1.09216
$713$	916.846	0.0481573
$714$	0	0
$715$	−34488.2	−1.80389
$716$	1569.93	0.0819426
$717$	22448.3	1.16924
$718$	−12085.8	−0.628185
$719$	12236.0	0.634668	0.317334	−	0.948314i	$-0.397212\pi$
$719$	12236.0	0.634668	0.317334	+	0.948314i	$0.397212\pi$
$720$	21357.1	1.10546
$721$	0	0
$722$	−35383.7	−1.82388
$723$	−19105.6	−0.982773
$724$	−1914.24	−0.0982625
$725$	44775.5	2.29369
$726$	11373.9	0.581438
$727$	3562.48	0.181740	0.0908700	−	0.995863i	$-0.471035\pi$
$727$	3562.48	0.181740	0.0908700	+	0.995863i	$0.471035\pi$
$728$	0	0
$729$	13128.8	0.667010
$730$	−22447.7	−1.13812
$731$	29482.9	1.49174
$732$	−1233.75	−0.0622961
$733$	9839.38	0.495806	0.247903	−	0.968785i	$-0.420259\pi$
$733$	9839.38	0.495806	0.247903	+	0.968785i	$0.420259\pi$
$734$	−6489.05	−0.326315
$735$	0	0
$736$	654.948	0.0328012
$737$	−12819.9	−0.640740
$738$	−493.342	−0.0246073
$739$	−8927.22	−0.444375	−0.222187	−	0.975004i	$-0.571320\pi$
$739$	−8927.22	−0.444375	−0.222187	+	0.975004i	$0.571320\pi$
$740$	2155.72	0.107089
$741$	16196.6	0.802967
$742$	0	0
$743$	−21791.5	−1.07598	−0.537990	−	0.842951i	$-0.680816\pi$
$743$	−21791.5	−1.07598	−0.537990	+	0.842951i	$0.680816\pi$
$744$	−2840.47	−0.139969
$745$	35267.1	1.73434
$746$	2213.47	0.108634
$747$	4613.94	0.225991
$748$	3602.51	0.176097
$749$	0	0
$750$	22386.0	1.08989
$751$	−37433.0	−1.81884	−0.909421	−	0.415876i	$-0.863475\pi$
$751$	−37433.0	−1.81884	−0.909421	+	0.415876i	$0.863475\pi$
$752$	29860.3	1.44800
$753$	−7701.55	−0.372723
$754$	−19192.6	−0.926994
$755$	17550.5	0.845999
$756$	0	0
$757$	32140.5	1.54315	0.771576	−	0.636137i	$-0.219469\pi$
$757$	32140.5	1.54315	0.771576	+	0.636137i	$0.219469\pi$
$758$	−1098.00	−0.0526136
$759$	3790.54	0.181275
$760$	−57279.8	−2.73389
$761$	2408.34	0.114721	0.0573603	−	0.998354i	$-0.481732\pi$
$761$	2408.34	0.114721	0.0573603	+	0.998354i	$0.481732\pi$
$762$	−8396.87	−0.399195
$763$	0	0
$764$	−947.002	−0.0448447
$765$	35490.9	1.67736
$766$	27395.6	1.29222
$767$	2546.48	0.119880
$768$	−3169.04	−0.148897
$769$	−8562.35	−0.401517	−0.200758	−	0.979641i	$-0.564341\pi$
$769$	−8562.35	−0.401517	−0.200758	+	0.979641i	$0.564341\pi$
$770$	0	0
$771$	−5818.45	−0.271785
$772$	−655.564	−0.0305625
$773$	3198.92	0.148845	0.0744226	−	0.997227i	$-0.476289\pi$
$773$	3198.92	0.148845	0.0744226	+	0.997227i	$0.476289\pi$
$774$	−12275.1	−0.570051
$775$	9778.75	0.453243
$776$	7049.10	0.326093
$777$	0	0
$778$	37864.4	1.74486
$779$	1428.11	0.0656833
$780$	−1429.68	−0.0656294
$781$	−17075.4	−0.782337
$782$	7708.16	0.352485
$783$	25932.9	1.18361
$784$	0	0
$785$	28238.5	1.28392
$786$	25006.5	1.13480
$787$	4162.29	0.188525	0.0942627	−	0.995547i	$-0.469951\pi$
$787$	4162.29	0.188525	0.0942627	+	0.995547i	$0.469951\pi$
$788$	3072.00	0.138877
$789$	−20041.3	−0.904295
$790$	31689.3	1.42716
$791$	0	0
$792$	17526.2	0.786319
$793$	21273.6	0.952643
$794$	944.960	0.0422360
$795$	13265.9	0.591817
$796$	3177.74	0.141497
$797$	−25456.6	−1.13139	−0.565696	−	0.824614i	$-0.691393\pi$
$797$	−25456.6	−1.13139	−0.565696	+	0.824614i	$0.691393\pi$
$798$	0	0
$799$	49621.5	2.19710
$800$	6985.44	0.308716
$801$	−15495.0	−0.683507
$802$	40670.6	1.79068
$803$	−19882.4	−0.873767
$804$	−531.439	−0.0233115
$805$	0	0
$806$	−4191.56	−0.183178
$807$	−14385.5	−0.627501
$808$	−13095.5	−0.570169
$809$	−17884.8	−0.777252	−0.388626	−	0.921396i	$-0.627050\pi$
$809$	−17884.8	−0.777252	−0.388626	+	0.921396i	$0.627050\pi$
$810$	1758.70	0.0762894
$811$	33047.0	1.43087	0.715435	−	0.698679i	$-0.246229\pi$
$811$	33047.0	1.43087	0.715435	+	0.698679i	$0.246229\pi$
$812$	0	0
$813$	−22625.5	−0.976028
$814$	26129.6	1.12511
$815$	−24239.5	−1.04181
$816$	−25774.9	−1.10576
$817$	35533.4	1.52161
$818$	39589.4	1.69219
$819$	0	0
$820$	−126.060	−0.00536853
$821$	−1766.15	−0.0750780	−0.0375390	−	0.999295i	$-0.511952\pi$
$821$	−1766.15	−0.0750780	−0.0375390	+	0.999295i	$0.511952\pi$
$822$	1016.61	0.0431369
$823$	18488.3	0.783062	0.391531	−	0.920165i	$-0.371946\pi$
$823$	18488.3	0.783062	0.391531	+	0.920165i	$0.371946\pi$
$824$	−25468.4	−1.07674
$825$	40428.5	1.70611
$826$	0	0
$827$	6221.29	0.261590	0.130795	−	0.991409i	$-0.458247\pi$
$827$	6221.29	0.261590	0.130795	+	0.991409i	$0.458247\pi$
$828$	−234.511	−0.00984278
$829$	8360.35	0.350262	0.175131	−	0.984545i	$-0.443965\pi$
$829$	8360.35	0.350262	0.175131	+	0.984545i	$0.443965\pi$
$830$	16134.0	0.674723
$831$	−25864.9	−1.07971
$832$	16662.0	0.694292
$833$	0	0
$834$	−2114.26	−0.0877827
$835$	25483.4	1.05615
$836$	4341.83	0.179624
$837$	5663.61	0.233886
$838$	−13837.1	−0.570400
$839$	−7505.00	−0.308822	−0.154411	−	0.988007i	$-0.549348\pi$
$839$	−7505.00	−0.308822	−0.154411	+	0.988007i	$0.549348\pi$
$840$	0	0
$841$	8926.98	0.366025
$842$	−50557.8	−2.06929
$843$	13815.6	0.564455
$844$	697.244	0.0284362
$845$	−17625.8	−0.717568
$846$	−20659.7	−0.839594
$847$	0	0
$848$	14378.4	0.582259
$849$	5450.54	0.220332
$850$	82212.5	3.31749
$851$	4085.42	0.164567
$852$	−707.849	−0.0284630
$853$	−4208.20	−0.168917	−0.0844583	−	0.996427i	$-0.526916\pi$
$853$	−4208.20	−0.168917	−0.0844583	+	0.996427i	$0.526916\pi$
$854$	0	0
$855$	42774.5	1.71094
$856$	43379.8	1.73211
$857$	−28478.9	−1.13515	−0.567573	−	0.823323i	$-0.692118\pi$
$857$	−28478.9	−1.13515	−0.567573	+	0.823323i	$0.692118\pi$
$858$	−17329.3	−0.689525
$859$	4323.06	0.171712	0.0858561	−	0.996308i	$-0.472637\pi$
$859$	4323.06	0.171712	0.0858561	+	0.996308i	$0.472637\pi$
$860$	−3136.55	−0.124367
$861$	0	0
$862$	−36431.1	−1.43950
$863$	23135.4	0.912558	0.456279	−	0.889837i	$-0.349182\pi$
$863$	23135.4	0.912558	0.456279	+	0.889837i	$0.349182\pi$
$864$	4045.79	0.159306
$865$	−18571.4	−0.729995
$866$	−50722.4	−1.99032
$867$	−26662.1	−1.04440
$868$	0	0
$869$	28067.9	1.09567
$870$	33962.7	1.32350
$871$	9163.60	0.356483
$872$	−12684.0	−0.492585
$873$	−5264.02	−0.204078
$874$	9290.06	0.359544
$875$	0	0
$876$	−824.213	−0.0317895
$877$	−10974.5	−0.422557	−0.211279	−	0.977426i	$-0.567763\pi$
$877$	−10974.5	−0.422557	−0.211279	+	0.977426i	$0.567763\pi$
$878$	16864.9	0.648249
$879$	13805.4	0.529743
$880$	66147.0	2.53388
$881$	−37410.7	−1.43064	−0.715322	−	0.698795i	$-0.753720\pi$
$881$	−37410.7	−1.43064	−0.715322	+	0.698795i	$0.753720\pi$
$882$	0	0
$883$	5633.68	0.214710	0.107355	−	0.994221i	$-0.465762\pi$
$883$	5633.68	0.214710	0.107355	+	0.994221i	$0.465762\pi$
$884$	−2575.06	−0.0979737
$885$	−4506.17	−0.171156
$886$	−35679.4	−1.35290
$887$	−16676.6	−0.631280	−0.315640	−	0.948879i	$-0.602219\pi$
$887$	−16676.6	−0.631280	−0.315640	+	0.948879i	$0.602219\pi$
$888$	−12657.0	−0.478311
$889$	0	0
$890$	−54182.9	−2.04069
$891$	1557.72	0.0585697
$892$	1253.89	0.0470665
$893$	59805.0	2.24109
$894$	17720.7	0.662939
$895$	47902.7	1.78906
$896$	0	0
$897$	−2709.47	−0.100855
$898$	519.173	0.0192929
$899$	7276.04	0.269933
$900$	−2501.21	−0.0926374
$901$	23893.8	0.883483
$902$	−1527.98	−0.0564036
$903$	0	0
$904$	−8475.99	−0.311844
$905$	−58408.5	−2.14538
$906$	8818.63	0.323377
$907$	−32588.8	−1.19305	−0.596523	−	0.802596i	$-0.703451\pi$
$907$	−32588.8	−1.19305	−0.596523	+	0.802596i	$0.703451\pi$
$908$	−571.246	−0.0208783
$909$	9779.22	0.356828
$910$	0	0
$911$	−36129.9	−1.31398	−0.656990	−	0.753899i	$-0.728171\pi$
$911$	−36129.9	−1.31398	−0.656990	+	0.753899i	$0.728171\pi$
$912$	−31064.5	−1.12791
$913$	14290.3	0.518005
$914$	775.204	0.0280541
$915$	−37645.1	−1.36012
$916$	−866.363	−0.0312505
$917$	0	0
$918$	47615.4	1.71192
$919$	−42821.3	−1.53704	−0.768522	−	0.639824i	$-0.779007\pi$
$919$	−42821.3	−1.53704	−0.768522	+	0.639824i	$0.779007\pi$
$920$	9582.10	0.343383
$921$	−12419.7	−0.444348
$922$	6478.34	0.231402
$923$	12205.4	0.435262
$924$	0	0
$925$	43573.6	1.54885
$926$	43051.8	1.52783
$927$	19018.9	0.673853
$928$	5197.63	0.183858
$929$	−42222.5	−1.49115	−0.745574	−	0.666423i	$-0.767825\pi$
$929$	−42222.5	−1.49115	−0.745574	+	0.666423i	$0.767825\pi$
$930$	7417.28	0.261529
$931$	0	0
$932$	−401.349	−0.0141058
$933$	−17488.1	−0.613648
$934$	1070.39	0.0374993
$935$	109922.	3.84475
$936$	−12527.6	−0.437477
$937$	30715.6	1.07090	0.535452	−	0.844566i	$-0.320141\pi$
$937$	30715.6	1.07090	0.535452	+	0.844566i	$0.320141\pi$
$938$	0	0
$939$	−19811.0	−0.688508
$940$	−5279.01	−0.183173
$941$	11106.6	0.384766	0.192383	−	0.981320i	$-0.438378\pi$
$941$	11106.6	0.384766	0.192383	+	0.981320i	$0.438378\pi$
$942$	14189.0	0.490768
$943$	−238.902	−0.00824997
$944$	−4884.05	−0.168392
$945$	0	0
$946$	−38018.4	−1.30664
$947$	−46397.3	−1.59209	−0.796045	−	0.605237i	$-0.793078\pi$
$947$	−46397.3	−1.59209	−0.796045	+	0.605237i	$0.793078\pi$
$948$	1163.54	0.0398628
$949$	14211.9	0.486130
$950$	99084.4	3.38392
$951$	5692.53	0.194104
$952$	0	0
$953$	17041.2	0.579242	0.289621	−	0.957141i	$-0.406471\pi$
$953$	17041.2	0.579242	0.289621	+	0.957141i	$0.406471\pi$
$954$	−9948.12	−0.337612
$955$	−28895.6	−0.979099
$956$	−4301.45	−0.145522
$957$	30081.5	1.01609
$958$	25867.2	0.872372
$959$	0	0
$960$	−29484.6	−0.991263
$961$	−28202.0	−0.946660
$962$	−18677.4	−0.625970
$963$	−32394.5	−1.08401
$964$	3660.93	0.122314
$965$	−20003.0	−0.667276
$966$	0	0
$967$	−21441.0	−0.713026	−0.356513	−	0.934290i	$-0.616034\pi$
$967$	−21441.0	−0.713026	−0.356513	+	0.934290i	$0.616034\pi$
$968$	25466.3	0.845575
$969$	−51622.7	−1.71141
$970$	−18407.2	−0.609299
$971$	−31927.2	−1.05519	−0.527597	−	0.849495i	$-0.676907\pi$
$971$	−31927.2	−1.05519	−0.527597	+	0.849495i	$0.676907\pi$
$972$	−2354.73	−0.0777036
$973$	0	0
$974$	−15524.7	−0.510723
$975$	−28898.2	−0.949213
$976$	−40801.9	−1.33815
$977$	−51368.2	−1.68210	−0.841051	−	0.540956i	$-0.818062\pi$
$977$	−51368.2	−1.68210	−0.841051	+	0.540956i	$0.818062\pi$
$978$	−12179.6	−0.398222
$979$	−47991.0	−1.56670
$980$	0	0
$981$	9471.96	0.308274
$982$	28523.9	0.926917
$983$	38166.8	1.23839	0.619193	−	0.785239i	$-0.287460\pi$
$983$	38166.8	1.23839	0.619193	+	0.785239i	$0.287460\pi$
$984$	740.139	0.0239784
$985$	93735.0	3.03213
$986$	61171.5	1.97576
$987$	0	0
$988$	−3103.53	−0.0999355
$989$	−5944.24	−0.191118
$990$	−45765.8	−1.46923
$991$	−5430.45	−0.174071	−0.0870353	−	0.996205i	$-0.527739\pi$
$991$	−5430.45	−0.174071	−0.0870353	+	0.996205i	$0.527739\pi$
$992$	1135.14	0.0363312
$993$	31293.0	1.00005
$994$	0	0
$995$	96961.4	3.08933
$996$	592.394	0.0188461
$997$	8739.64	0.277620	0.138810	−	0.990319i	$-0.455672\pi$
$997$	8739.64	0.277620	0.138810	+	0.990319i	$0.455672\pi$
$998$	−13074.0	−0.414678
$999$	25236.7	0.799254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.4.a.m.1.6		22
7.3	odd	6		161.4.e.b.93.17	✓	44
7.5	odd	6		161.4.e.b.116.17	yes	44
7.6	odd	2		1127.4.a.j.1.6		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.e.b.93.17	✓	44	7.3	odd	6
161.4.e.b.116.17	yes	44	7.5	odd	6
1127.4.a.j.1.6		22	7.6	odd	2
1127.4.a.m.1.6		22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93780 q^{2} -3.29133 q^{3} +0.630669 q^{4} +19.2434 q^{5} +9.66926 q^{6} +21.6496 q^{8} -16.1672 q^{9} -56.5334 q^{10} -50.0729 q^{11} -2.07574 q^{12} +35.7919 q^{13} -63.3364 q^{15} -68.6476 q^{16} -114.078 q^{17} +47.4959 q^{18} -137.489 q^{19} +12.1362 q^{20} +147.104 q^{22} +23.0000 q^{23} -71.2560 q^{24} +245.310 q^{25} -105.150 q^{26} +142.077 q^{27} +182.527 q^{29} +186.070 q^{30} +39.8629 q^{31} +28.4760 q^{32} +164.806 q^{33} +335.138 q^{34} -10.1961 q^{36} +177.627 q^{37} +403.916 q^{38} -117.803 q^{39} +416.613 q^{40} -10.3871 q^{41} -258.445 q^{43} -31.5794 q^{44} -311.112 q^{45} -67.5694 q^{46} -434.979 q^{47} +225.942 q^{48} -720.671 q^{50} +375.467 q^{51} +22.5729 q^{52} -209.452 q^{53} -417.395 q^{54} -963.574 q^{55} +452.522 q^{57} -536.227 q^{58} +71.1467 q^{59} -39.9443 q^{60} +594.367 q^{61} -117.109 q^{62} +465.524 q^{64} +688.760 q^{65} -484.167 q^{66} +256.024 q^{67} -71.9453 q^{68} -75.7005 q^{69} +341.011 q^{71} -350.013 q^{72} +397.070 q^{73} -521.832 q^{74} -807.394 q^{75} -86.7102 q^{76} +346.082 q^{78} -560.541 q^{79} -1321.02 q^{80} -31.1091 q^{81} +30.5151 q^{82} -285.389 q^{83} -2195.25 q^{85} +759.261 q^{86} -600.755 q^{87} -1084.06 q^{88} +958.424 q^{89} +913.984 q^{90} +14.5054 q^{92} -131.202 q^{93} +1277.88 q^{94} -2645.76 q^{95} -93.7238 q^{96} +325.599 q^{97} +809.536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 18 q^{3} + 88 q^{4} + 20 q^{5} + 36 q^{6} + 21 q^{8} + 126 q^{9} + 200 q^{10} + 20 q^{11} + 161 q^{12} + 196 q^{13} + 20 q^{15} + 324 q^{16} + 242 q^{17} + 85 q^{18} + 128 q^{19} + 46 q^{20} + 14 q^{22}+ \cdots + 3570 q^{99}+O(q^{100}) \) 22 * q + 18 * q^3 + 88 * q^4 + 20 * q^5 + 36 * q^6 + 21 * q^8 + 126 * q^9 + 200 * q^10 + 20 * q^11 + 161 * q^12 + 196 * q^13 + 20 * q^15 + 324 * q^16 + 242 * q^17 + 85 * q^18 + 128 * q^19 + 46 * q^20 + 14 * q^22 + 506 * q^23 + 66 * q^24 + 668 * q^25 + 340 * q^26 + 378 * q^27 - 190 * q^29 + 493 * q^30 + 508 * q^31 - 487 * q^32 + 1240 * q^33 + 416 * q^34 - 470 * q^36 - 102 * q^37 + 423 * q^38 - 520 * q^39 + 2048 * q^40 + 920 * q^41 - 728 * q^43 + 1132 * q^44 + 570 * q^45 + 1384 * q^47 + 1323 * q^48 - 975 * q^50 + 1688 * q^51 + 1892 * q^52 - 480 * q^53 + 1695 * q^54 + 1254 * q^55 - 302 * q^57 + 791 * q^58 + 650 * q^59 - 2490 * q^60 + 960 * q^61 + 2554 * q^62 + 3791 * q^64 + 800 * q^65 - 1221 * q^66 - 2136 * q^67 + 2290 * q^68 + 414 * q^69 - 136 * q^71 + 4114 * q^72 + 2562 * q^73 - 1928 * q^74 + 1820 * q^75 + 4645 * q^76 + 2605 * q^78 + 790 * q^79 - 2248 * q^80 - 506 * q^81 + 3838 * q^82 + 2026 * q^83 + 1392 * q^85 + 3419 * q^86 + 3340 * q^87 - 5101 * q^88 + 2618 * q^89 + 6780 * q^90 + 2024 * q^92 + 206 * q^93 + 3516 * q^94 - 4410 * q^95 + 3818 * q^96 + 5158 * q^97 + 3570 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more