LMFDB - Embedded newform 1519.4.a.h.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1519,4,Mod(1,1519)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1519, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1519.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1519.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23,5,-6,91,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.6239012987$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	1519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.06861 q^{2} -5.09781 q^{3} -6.85807 q^{4} +1.69042 q^{5} -5.44757 q^{6} -15.8775 q^{8} -1.01232 q^{9} +1.80640 q^{10} -11.3948 q^{11} +34.9612 q^{12} +63.5400 q^{13} -8.61743 q^{15} +37.8978 q^{16} -10.9774 q^{17} -1.08178 q^{18} -113.179 q^{19} -11.5930 q^{20} -12.1766 q^{22} -39.8450 q^{23} +80.9404 q^{24} -122.142 q^{25} +67.8994 q^{26} +142.802 q^{27} +278.148 q^{29} -9.20866 q^{30} -31.0000 q^{31} +167.518 q^{32} +58.0884 q^{33} -11.7305 q^{34} +6.94258 q^{36} -68.4809 q^{37} -120.944 q^{38} -323.915 q^{39} -26.8396 q^{40} +172.827 q^{41} +152.093 q^{43} +78.1462 q^{44} -1.71125 q^{45} -42.5787 q^{46} +440.422 q^{47} -193.196 q^{48} -130.523 q^{50} +55.9607 q^{51} -435.762 q^{52} +707.544 q^{53} +152.599 q^{54} -19.2619 q^{55} +576.966 q^{57} +297.232 q^{58} +9.06489 q^{59} +59.0990 q^{60} -616.443 q^{61} -33.1269 q^{62} -124.171 q^{64} +107.409 q^{65} +62.0738 q^{66} -29.5775 q^{67} +75.2838 q^{68} +203.122 q^{69} +779.280 q^{71} +16.0731 q^{72} -700.002 q^{73} -73.1793 q^{74} +622.659 q^{75} +776.191 q^{76} -346.138 q^{78} +988.243 q^{79} +64.0631 q^{80} -700.643 q^{81} +184.684 q^{82} -142.373 q^{83} -18.5564 q^{85} +162.528 q^{86} -1417.95 q^{87} +180.920 q^{88} -287.113 q^{89} -1.82865 q^{90} +273.260 q^{92} +158.032 q^{93} +470.639 q^{94} -191.320 q^{95} -853.974 q^{96} -1764.37 q^{97} +11.5352 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) $ 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Coefficient data

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

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

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

$2$	1.06861	0.377810	0.188905	−	0.981995i	$-0.439506\pi$
$2$	1.06861	0.377810	0.188905	+	0.981995i	$0.439506\pi$
$3$	−5.09781	−0.981074	−0.490537	−	0.871420i	$-0.663199\pi$
$3$	−5.09781	−0.981074	−0.490537	+	0.871420i	$0.663199\pi$
$4$	−6.85807	−0.857259
$5$	1.69042	0.151196	0.0755978	−	0.997138i	$-0.475914\pi$
$5$	1.69042	0.151196	0.0755978	+	0.997138i	$0.475914\pi$
$6$	−5.44757	−0.370660
$7$	0	0
$7$	0	0
$8$	−15.8775	−0.701692
$9$	−1.01232	−0.0374934
$10$	1.80640	0.0571232
$11$	−11.3948	−0.312332	−0.156166	−	0.987731i	$-0.549913\pi$
$11$	−11.3948	−0.312332	−0.156166	+	0.987731i	$0.549913\pi$
$12$	34.9612	0.841035
$13$	63.5400	1.35560	0.677801	−	0.735246i	$-0.262933\pi$
$13$	63.5400	1.35560	0.677801	+	0.735246i	$0.262933\pi$
$14$	0	0
$15$	−8.61743	−0.148334
$16$	37.8978	0.592153
$17$	−10.9774	−0.156612	−0.0783061	−	0.996929i	$-0.524951\pi$
$17$	−10.9774	−0.156612	−0.0783061	+	0.996929i	$0.524951\pi$
$18$	−1.08178	−0.0141654
$19$	−113.179	−1.36658	−0.683292	−	0.730145i	$-0.739452\pi$
$19$	−113.179	−1.36658	−0.683292	+	0.730145i	$0.739452\pi$
$20$	−11.5930	−0.129614
$21$	0	0
$22$	−12.1766	−0.118002
$23$	−39.8450	−0.361228	−0.180614	−	0.983554i	$-0.557809\pi$
$23$	−39.8450	−0.361228	−0.180614	+	0.983554i	$0.557809\pi$
$24$	80.9404	0.688412
$25$	−122.142	−0.977140
$26$	67.8994	0.512160
$27$	142.802	1.01786
$28$	0	0
$29$	278.148	1.78106	0.890532	−	0.454921i	$-0.150332\pi$
$29$	278.148	1.78106	0.890532	+	0.454921i	$0.150332\pi$
$30$	−9.20866	−0.0560421
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	167.518	0.925413
$33$	58.0884	0.306421
$34$	−11.7305	−0.0591697
$35$	0	0
$36$	6.94258	0.0321416
$37$	−68.4809	−0.304275	−0.152138	−	0.988359i	$-0.548616\pi$
$37$	−68.4809	−0.304275	−0.152138	+	0.988359i	$0.548616\pi$
$38$	−120.944	−0.516309
$39$	−323.915	−1.32995
$40$	−26.8396	−0.106093
$41$	172.827	0.658316	0.329158	−	0.944275i	$-0.393235\pi$
$41$	172.827	0.658316	0.329158	+	0.944275i	$0.393235\pi$
$42$	0	0
$43$	152.093	0.539393	0.269697	−	0.962945i	$-0.413077\pi$
$43$	152.093	0.539393	0.269697	+	0.962945i	$0.413077\pi$
$44$	78.1462	0.267750
$45$	−1.71125	−0.00566883
$46$	−42.5787	−0.136476
$47$	440.422	1.36685	0.683427	−	0.730018i	$-0.260489\pi$
$47$	440.422	1.36685	0.683427	+	0.730018i	$0.260489\pi$
$48$	−193.196	−0.580946
$49$	0	0
$50$	−130.523	−0.369174
$51$	55.9607	0.153648
$52$	−435.762	−1.16210
$53$	707.544	1.83375	0.916874	−	0.399177i	$-0.130704\pi$
$53$	707.544	1.83375	0.916874	+	0.399177i	$0.130704\pi$
$54$	152.599	0.384557
$55$	−19.2619	−0.0472232
$56$	0	0
$57$	576.966	1.34072
$58$	297.232	0.672904
$59$	9.06489	0.0200025	0.0100013	−	0.999950i	$-0.496816\pi$
$59$	9.06489	0.0200025	0.0100013	+	0.999950i	$0.496816\pi$
$60$	59.0990	0.127161
$61$	−616.443	−1.29389	−0.646947	−	0.762535i	$-0.723954\pi$
$61$	−616.443	−1.29389	−0.646947	+	0.762535i	$0.723954\pi$
$62$	−33.1269	−0.0678567
$63$	0	0
$64$	−124.171	−0.242522
$65$	107.409	0.204961
$66$	62.0738	0.115769
$67$	−29.5775	−0.0539323	−0.0269662	−	0.999636i	$-0.508585\pi$
$67$	−29.5775	−0.0539323	−0.0269662	+	0.999636i	$0.508585\pi$
$68$	75.2838	0.134257
$69$	203.122	0.354392
$70$	0	0
$71$	779.280	1.30258	0.651292	−	0.758827i	$-0.274227\pi$
$71$	779.280	1.30258	0.651292	+	0.758827i	$0.274227\pi$
$72$	16.0731	0.0263088
$73$	−700.002	−1.12232	−0.561158	−	0.827709i	$-0.689644\pi$
$73$	−700.002	−1.12232	−0.561158	+	0.827709i	$0.689644\pi$
$74$	−73.1793	−0.114958
$75$	622.659	0.958647
$76$	776.191	1.17152
$77$	0	0
$78$	−346.138	−0.502467
$79$	988.243	1.40742	0.703709	−	0.710488i	$-0.251526\pi$
$79$	988.243	1.40742	0.703709	+	0.710488i	$0.251526\pi$
$80$	64.0631	0.0895309
$81$	−700.643	−0.961101
$82$	184.684	0.248719
$83$	−142.373	−0.188282	−0.0941412	−	0.995559i	$-0.530011\pi$
$83$	−142.373	−0.188282	−0.0941412	+	0.995559i	$0.530011\pi$
$84$	0	0
$85$	−18.5564	−0.0236791
$86$	162.528	0.203788
$87$	−1417.95	−1.74736
$88$	180.920	0.219161
$89$	−287.113	−0.341954	−0.170977	−	0.985275i	$-0.554692\pi$
$89$	−287.113	−0.341954	−0.170977	+	0.985275i	$0.554692\pi$
$90$	−1.82865	−0.00214174
$91$	0	0
$92$	273.260	0.309666
$93$	158.032	0.176206
$94$	470.639	0.516412
$95$	−191.320	−0.206621
$96$	−853.974	−0.907899
$97$	−1764.37	−1.84685	−0.923427	−	0.383775i	$-0.874624\pi$
$97$	−1764.37	−1.84685	−0.923427	+	0.383775i	$0.874624\pi$
$98$	0	0
$99$	11.5352	0.0117104
$100$	837.662	0.837662
$101$	−972.472	−0.958066	−0.479033	−	0.877797i	$-0.659012\pi$
$101$	−972.472	−0.958066	−0.479033	+	0.877797i	$0.659012\pi$
$102$	59.8001	0.0580499
$103$	389.777	0.372872	0.186436	−	0.982467i	$-0.440306\pi$
$103$	389.777	0.372872	0.186436	+	0.982467i	$0.440306\pi$
$104$	−1008.85	−0.951215
$105$	0	0
$106$	756.088	0.692809
$107$	−1229.35	−1.11071	−0.555354	−	0.831614i	$-0.687417\pi$
$107$	−1229.35	−1.11071	−0.555354	+	0.831614i	$0.687417\pi$
$108$	−979.344	−0.872568
$109$	−1905.00	−1.67400	−0.837002	−	0.547200i	$-0.815694\pi$
$109$	−1905.00	−1.67400	−0.837002	+	0.547200i	$0.815694\pi$
$110$	−20.5835	−0.0178414
$111$	349.103	0.298517
$112$	0	0
$113$	−892.356	−0.742883	−0.371441	−	0.928456i	$-0.621136\pi$
$113$	−892.356	−0.742883	−0.371441	+	0.928456i	$0.621136\pi$
$114$	616.551	0.506538
$115$	−67.3547	−0.0546161
$116$	−1907.56	−1.52683
$117$	−64.3229	−0.0508261
$118$	9.68683	0.00755716
$119$	0	0
$120$	136.823	0.104085
$121$	−1201.16	−0.902449
$122$	−658.737	−0.488846
$123$	−881.037	−0.645857
$124$	212.600	0.153968
$125$	−417.774	−0.298935
$126$	0	0
$127$	492.761	0.344295	0.172147	−	0.985071i	$-0.444929\pi$
$127$	492.761	0.344295	0.172147	+	0.985071i	$0.444929\pi$
$128$	−1472.83	−1.01704
$129$	−775.339	−0.529185
$130$	114.778	0.0774364
$131$	1025.67	0.684071	0.342036	−	0.939687i	$-0.388884\pi$
$131$	1025.67	0.684071	0.342036	+	0.939687i	$0.388884\pi$
$132$	−398.375	−0.262682
$133$	0	0
$134$	−31.6068	−0.0203762
$135$	241.394	0.153896
$136$	174.293	0.109894
$137$	1875.23	1.16943	0.584715	−	0.811238i	$-0.301206\pi$
$137$	1875.23	1.16943	0.584715	+	0.811238i	$0.301206\pi$
$138$	217.058	0.133893
$139$	796.182	0.485837	0.242918	−	0.970047i	$-0.421895\pi$
$139$	796.182	0.485837	0.242918	+	0.970047i	$0.421895\pi$
$140$	0	0
$141$	−2245.19	−1.34099
$142$	832.746	0.492130
$143$	−724.024	−0.423398
$144$	−38.3647	−0.0222018
$145$	470.187	0.269289
$146$	−748.028	−0.424022
$147$	0	0
$148$	469.647	0.260843
$149$	744.054	0.409096	0.204548	−	0.978857i	$-0.434428\pi$
$149$	744.054	0.409096	0.204548	+	0.978857i	$0.434428\pi$
$150$	665.379	0.362187
$151$	3699.06	1.99354	0.996772	−	0.0802822i	$-0.0255822\pi$
$151$	3699.06	1.99354	0.996772	+	0.0802822i	$0.0255822\pi$
$152$	1797.00	0.958920
$153$	11.1126	0.00587192
$154$	0	0
$155$	−52.4029	−0.0271555
$156$	2221.43	1.14011
$157$	573.120	0.291337	0.145669	−	0.989333i	$-0.453467\pi$
$157$	573.120	0.291337	0.145669	+	0.989333i	$0.453467\pi$
$158$	1056.05	0.531737
$159$	−3606.92	−1.79904
$160$	283.175	0.139918
$161$	0	0
$162$	−748.713	−0.363114
$163$	−1222.94	−0.587655	−0.293827	−	0.955858i	$-0.594929\pi$
$163$	−1222.94	−0.587655	−0.293827	+	0.955858i	$0.594929\pi$
$164$	−1185.26	−0.564348
$165$	98.1936	0.0463295
$166$	−152.141	−0.0711350
$167$	−1957.71	−0.907139	−0.453570	−	0.891221i	$-0.649850\pi$
$167$	−1957.71	−0.907139	−0.453570	+	0.891221i	$0.649850\pi$
$168$	0	0
$169$	1840.33	0.837656
$170$	−19.8295	−0.00894620
$171$	114.574	0.0512378
$172$	−1043.06	−0.462400
$173$	−2739.84	−1.20408	−0.602041	−	0.798465i	$-0.705646\pi$
$173$	−2739.84	−1.20408	−0.602041	+	0.798465i	$0.705646\pi$
$174$	−1515.23	−0.660169
$175$	0	0
$176$	−431.837	−0.184948
$177$	−46.2111	−0.0196240
$178$	−306.812	−0.129194
$179$	4520.34	1.88752	0.943760	−	0.330632i	$-0.107262\pi$
$179$	4520.34	1.88752	0.943760	+	0.330632i	$0.107262\pi$
$180$	11.7359	0.00485966
$181$	−4122.94	−1.69312	−0.846562	−	0.532290i	$-0.821332\pi$
$181$	−4122.94	−1.69312	−0.846562	+	0.532290i	$0.821332\pi$
$182$	0	0
$183$	3142.51	1.26941
$184$	632.638	0.253471
$185$	−115.761	−0.0460051
$186$	168.875	0.0665725
$187$	125.085	0.0489150
$188$	−3020.45	−1.17175
$189$	0	0
$190$	−204.446	−0.0780637
$191$	−2179.90	−0.825820	−0.412910	−	0.910772i	$-0.635488\pi$
$191$	−2179.90	−0.825820	−0.412910	+	0.910772i	$0.635488\pi$
$192$	633.002	0.237932
$193$	−1806.85	−0.673884	−0.336942	−	0.941525i	$-0.609393\pi$
$193$	−1806.85	−0.673884	−0.336942	+	0.941525i	$0.609393\pi$
$194$	−1885.42	−0.697760
$195$	−547.551	−0.201082
$196$	0	0
$197$	−3637.58	−1.31557	−0.657784	−	0.753207i	$-0.728506\pi$
$197$	−3637.58	−1.31557	−0.657784	+	0.753207i	$0.728506\pi$
$198$	12.3266	0.00442431
$199$	−5283.43	−1.88207	−0.941036	−	0.338306i	$-0.890146\pi$
$199$	−5283.43	−1.88207	−0.941036	+	0.338306i	$0.890146\pi$
$200$	1939.31	0.685651
$201$	150.780	0.0529116
$202$	−1039.19	−0.361967
$203$	0	0
$204$	−383.782	−0.131716
$205$	292.149	0.0995345
$206$	416.519	0.140875
$207$	40.3359	0.0135437
$208$	2408.02	0.802723
$209$	1289.65	0.426828
$210$	0	0
$211$	−2103.57	−0.686329	−0.343165	−	0.939275i	$-0.611499\pi$
$211$	−2103.57	−0.686329	−0.343165	+	0.939275i	$0.611499\pi$
$212$	−4852.39	−1.57200
$213$	−3972.62	−1.27793
$214$	−1313.69	−0.419637
$215$	257.100	0.0815538
$216$	−2267.33	−0.714223
$217$	0	0
$218$	−2035.71	−0.632456
$219$	3568.48	1.10107
$220$	132.100	0.0404825
$221$	−697.503	−0.212304
$222$	373.054	0.112783
$223$	4045.91	1.21495	0.607476	−	0.794338i	$-0.292182\pi$
$223$	4045.91	1.21495	0.607476	+	0.794338i	$0.292182\pi$
$224$	0	0
$225$	123.647	0.0366363
$226$	−953.580	−0.280669
$227$	−2816.86	−0.823619	−0.411810	−	0.911270i	$-0.635103\pi$
$227$	−2816.86	−0.823619	−0.411810	+	0.911270i	$0.635103\pi$
$228$	−3956.88	−1.14934
$229$	5525.98	1.59462	0.797308	−	0.603572i	$-0.206256\pi$
$229$	5525.98	1.59462	0.797308	+	0.603572i	$0.206256\pi$
$230$	−71.9758	−0.0206345
$231$	0	0
$232$	−4416.29	−1.24976
$233$	−1720.37	−0.483713	−0.241857	−	0.970312i	$-0.577756\pi$
$233$	−1720.37	−0.483713	−0.241857	+	0.970312i	$0.577756\pi$
$234$	−68.7360	−0.0192026
$235$	744.497	0.206662
$236$	−62.1677	−0.0171473
$237$	−5037.88	−1.38078
$238$	0	0
$239$	1643.46	0.444798	0.222399	−	0.974956i	$-0.428611\pi$
$239$	1643.46	0.444798	0.222399	+	0.974956i	$0.428611\pi$
$240$	−326.581	−0.0878364
$241$	−4056.22	−1.08417	−0.542083	−	0.840325i	$-0.682364\pi$
$241$	−4056.22	−1.08417	−0.542083	+	0.840325i	$0.682364\pi$
$242$	−1283.57	−0.340954
$243$	−283.898	−0.0749467
$244$	4227.61	1.10920
$245$	0	0
$246$	−941.484	−0.244012
$247$	−7191.40	−1.85254
$248$	492.202	0.126028
$249$	725.789	0.184719
$250$	−446.437	−0.112941
$251$	−6943.08	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$251$	−6943.08	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$252$	0	0
$253$	454.025	0.112823
$254$	526.569	0.130078
$255$	94.5969	0.0232309
$256$	−580.511	−0.141726
$257$	5267.21	1.27844	0.639221	−	0.769023i	$-0.279257\pi$
$257$	5267.21	1.27844	0.639221	+	0.769023i	$0.279257\pi$
$258$	−828.535	−0.199931
$259$	0	0
$260$	−736.620	−0.175705
$261$	−281.576	−0.0667781
$262$	1096.04	0.258449
$263$	−2630.67	−0.616784	−0.308392	−	0.951259i	$-0.599791\pi$
$263$	−2630.67	−0.616784	−0.308392	+	0.951259i	$0.599791\pi$
$264$	−922.297	−0.215013
$265$	1196.04	0.277254
$266$	0	0
$267$	1463.65	0.335482
$268$	202.845	0.0462340
$269$	5708.71	1.29393	0.646963	−	0.762522i	$-0.276039\pi$
$269$	5708.71	1.29393	0.646963	+	0.762522i	$0.276039\pi$
$270$	257.956	0.0581433
$271$	2226.99	0.499188	0.249594	−	0.968351i	$-0.419703\pi$
$271$	2226.99	0.499188	0.249594	+	0.968351i	$0.419703\pi$
$272$	−416.019	−0.0927384
$273$	0	0
$274$	2003.89	0.441823
$275$	1391.79	0.305192
$276$	−1393.03	−0.303806
$277$	4948.83	1.07345	0.536726	−	0.843757i	$-0.319661\pi$
$277$	4948.83	1.07345	0.536726	+	0.843757i	$0.319661\pi$
$278$	850.807	0.183554
$279$	31.3820	0.00673401
$280$	0	0
$281$	−2871.94	−0.609699	−0.304850	−	0.952401i	$-0.598606\pi$
$281$	−2871.94	−0.609699	−0.304850	+	0.952401i	$0.598606\pi$
$282$	−2399.23	−0.506638
$283$	−6200.78	−1.30247	−0.651233	−	0.758878i	$-0.725748\pi$
$283$	−6200.78	−1.30247	−0.651233	+	0.758878i	$0.725748\pi$
$284$	−5344.36	−1.11665
$285$	975.314	0.202711
$286$	−773.698	−0.159964
$287$	0	0
$288$	−169.582	−0.0346969
$289$	−4792.50	−0.975473
$290$	502.446	0.101740
$291$	8994.43	1.81190
$292$	4800.66	0.962115
$293$	−4001.17	−0.797784	−0.398892	−	0.916998i	$-0.630605\pi$
$293$	−4001.17	−0.797784	−0.398892	+	0.916998i	$0.630605\pi$
$294$	0	0
$295$	15.3235	0.00302429
$296$	1087.30	0.213508
$297$	−1627.19	−0.317910
$298$	795.103	0.154561
$299$	−2531.75	−0.489682
$300$	−4270.24	−0.821809
$301$	0	0
$302$	3952.85	0.753182
$303$	4957.48	0.939933
$304$	−4289.24	−0.809226
$305$	−1042.05	−0.195631
$306$	11.8751	0.00221847
$307$	−2011.05	−0.373866	−0.186933	−	0.982373i	$-0.559855\pi$
$307$	−2011.05	−0.373866	−0.186933	+	0.982373i	$0.559855\pi$
$308$	0	0
$309$	−1987.01	−0.365815
$310$	−55.9983	−0.0102596
$311$	−4938.73	−0.900481	−0.450240	−	0.892907i	$-0.648662\pi$
$311$	−4938.73	−0.900481	−0.450240	+	0.892907i	$0.648662\pi$
$312$	5142.95	0.933212
$313$	3367.19	0.608066	0.304033	−	0.952661i	$-0.401667\pi$
$313$	3367.19	0.608066	0.304033	+	0.952661i	$0.401667\pi$
$314$	612.441	0.110070
$315$	0	0
$316$	−6777.45	−1.20652
$317$	−3010.50	−0.533397	−0.266698	−	0.963780i	$-0.585933\pi$
$317$	−3010.50	−0.533397	−0.266698	+	0.963780i	$0.585933\pi$
$318$	−3854.39	−0.679697
$319$	−3169.44	−0.556283
$320$	−209.901	−0.0366683
$321$	6266.99	1.08969
$322$	0	0
$323$	1242.41	0.214024
$324$	4805.06	0.823913
$325$	−7760.93	−1.32461
$326$	−1306.84	−0.222022
$327$	9711.35	1.64232
$328$	−2744.05	−0.461935
$329$	0	0
$330$	104.931	0.0175038
$331$	9031.19	1.49970	0.749848	−	0.661610i	$-0.230127\pi$
$331$	9031.19	1.49970	0.749848	+	0.661610i	$0.230127\pi$
$332$	976.403	0.161407
$333$	69.3247	0.0114083
$334$	−2092.03	−0.342727
$335$	−49.9983	−0.00815433
$336$	0	0
$337$	1908.11	0.308432	0.154216	−	0.988037i	$-0.450715\pi$
$337$	1908.11	0.308432	0.154216	+	0.988037i	$0.450715\pi$
$338$	1966.59	0.316475
$339$	4549.06	0.728823
$340$	127.261	0.0202991
$341$	353.238	0.0560965
$342$	122.435	0.0193582
$343$	0	0
$344$	−2414.85	−0.378488
$345$	343.361	0.0535825
$346$	−2927.82	−0.454915
$347$	−7424.64	−1.14863	−0.574316	−	0.818634i	$-0.694732\pi$
$347$	−7424.64	−1.14863	−0.574316	+	0.818634i	$0.694732\pi$
$348$	9724.39	1.49794
$349$	−4224.58	−0.647956	−0.323978	−	0.946065i	$-0.605020\pi$
$349$	−4224.58	−0.647956	−0.323978	+	0.946065i	$0.605020\pi$
$350$	0	0
$351$	9073.61	1.37981
$352$	−1908.83	−0.289036
$353$	728.325	0.109815	0.0549077	−	0.998491i	$-0.482514\pi$
$353$	728.325	0.109815	0.0549077	+	0.998491i	$0.482514\pi$
$354$	−49.3816	−0.00741413
$355$	1317.31	0.196945
$356$	1969.04	0.293143
$357$	0	0
$358$	4830.47	0.713124
$359$	−1968.81	−0.289443	−0.144721	−	0.989472i	$-0.546229\pi$
$359$	−1968.81	−0.289443	−0.144721	+	0.989472i	$0.546229\pi$
$360$	27.1703	0.00397777
$361$	5950.53	0.867550
$362$	−4405.81	−0.639680
$363$	6123.28	0.885369
$364$	0	0
$365$	−1183.30	−0.169689
$366$	3358.12	0.479595
$367$	9924.61	1.41161	0.705805	−	0.708407i	$-0.250586\pi$
$367$	9924.61	1.41161	0.705805	+	0.708407i	$0.250586\pi$
$368$	−1510.04	−0.213902
$369$	−174.956	−0.0246825
$370$	−123.704	−0.0173812
$371$	0	0
$372$	−1083.80	−0.151054
$373$	−7273.60	−1.00968	−0.504842	−	0.863211i	$-0.668449\pi$
$373$	−7273.60	−1.00968	−0.504842	+	0.863211i	$0.668449\pi$
$374$	133.667	0.0184806
$375$	2129.73	0.293277
$376$	−6992.79	−0.959111
$377$	17673.5	2.41441
$378$	0	0
$379$	6423.61	0.870603	0.435302	−	0.900285i	$-0.356642\pi$
$379$	6423.61	0.870603	0.435302	+	0.900285i	$0.356642\pi$
$380$	1312.09	0.177128
$381$	−2512.00	−0.337779
$382$	−2329.46	−0.312003
$383$	9574.31	1.27735	0.638674	−	0.769477i	$-0.279483\pi$
$383$	9574.31	1.27735	0.638674	+	0.769477i	$0.279483\pi$
$384$	7508.22	0.997792
$385$	0	0
$386$	−1930.81	−0.254601
$387$	−153.967	−0.0202237
$388$	12100.2	1.58323
$389$	4410.16	0.574817	0.287408	−	0.957808i	$-0.407206\pi$
$389$	4410.16	0.574817	0.287408	+	0.957808i	$0.407206\pi$
$390$	−585.118	−0.0759708
$391$	437.394	0.0565728
$392$	0	0
$393$	−5228.68	−0.671124
$394$	−3887.15	−0.497035
$395$	1670.54	0.212795
$396$	−79.1091	−0.0100388
$397$	−11630.5	−1.47033	−0.735163	−	0.677890i	$-0.762895\pi$
$397$	−11630.5	−1.47033	−0.735163	+	0.677890i	$0.762895\pi$
$398$	−5645.92	−0.711066
$399$	0	0
$400$	−4628.93	−0.578616
$401$	5975.71	0.744172	0.372086	−	0.928198i	$-0.378643\pi$
$401$	5975.71	0.744172	0.372086	+	0.928198i	$0.378643\pi$
$402$	161.125	0.0199906
$403$	−1969.74	−0.243473
$404$	6669.29	0.821311
$405$	−1184.38	−0.145314
$406$	0	0
$407$	780.324	0.0950350
$408$	−888.514	−0.107814
$409$	−12641.1	−1.52827	−0.764133	−	0.645058i	$-0.776833\pi$
$409$	−12641.1	−1.52827	−0.764133	+	0.645058i	$0.776833\pi$
$410$	312.193	0.0376052
$411$	−9559.59	−1.14730
$412$	−2673.12	−0.319648
$413$	0	0
$414$	43.1033	0.00511694
$415$	−240.669	−0.0284675
$416$	10644.1	1.25449
$417$	−4058.79	−0.476642
$418$	1378.13	0.161260
$419$	−16626.8	−1.93859	−0.969296	−	0.245896i	$-0.920918\pi$
$419$	−16626.8	−1.93859	−0.969296	+	0.245896i	$0.920918\pi$
$420$	0	0
$421$	7015.57	0.812157	0.406078	−	0.913838i	$-0.366896\pi$
$421$	7015.57	0.812157	0.406078	+	0.913838i	$0.366896\pi$
$422$	−2247.89	−0.259302
$423$	−445.849	−0.0512480
$424$	−11234.0	−1.28673
$425$	1340.81	0.153032
$426$	−4245.18	−0.482816
$427$	0	0
$428$	8430.97	0.952164
$429$	3690.94	0.415385
$430$	274.739	0.0308119
$431$	−9018.86	−1.00794	−0.503971	−	0.863720i	$-0.668128\pi$
$431$	−9018.86	−1.00794	−0.503971	+	0.863720i	$0.668128\pi$
$432$	5411.86	0.602728
$433$	−1858.88	−0.206310	−0.103155	−	0.994665i	$-0.532894\pi$
$433$	−1858.88	−0.206310	−0.103155	+	0.994665i	$0.532894\pi$
$434$	0	0
$435$	−2396.92	−0.264192
$436$	13064.7	1.43505
$437$	4509.62	0.493649
$438$	3813.31	0.415997
$439$	6259.52	0.680525	0.340263	−	0.940330i	$-0.389484\pi$
$439$	6259.52	0.680525	0.340263	+	0.940330i	$0.389484\pi$
$440$	305.831	0.0331361
$441$	0	0
$442$	−745.358	−0.0802106
$443$	11456.1	1.22866	0.614331	−	0.789048i	$-0.289426\pi$
$443$	11456.1	1.22866	0.614331	+	0.789048i	$0.289426\pi$
$444$	−2394.17	−0.255906
$445$	−485.341	−0.0517019
$446$	4323.50	0.459022
$447$	−3793.05	−0.401353
$448$	0	0
$449$	13324.0	1.40045	0.700223	−	0.713924i	$-0.253084\pi$
$449$	13324.0	1.40045	0.700223	+	0.713924i	$0.253084\pi$
$450$	132.131	0.0138416
$451$	−1969.32	−0.205613
$452$	6119.84	0.636843
$453$	−18857.1	−1.95581
$454$	−3010.12	−0.311172
$455$	0	0
$456$	−9160.76	−0.940772
$457$	−4740.07	−0.485189	−0.242594	−	0.970128i	$-0.577998\pi$
$457$	−4740.07	−0.485189	−0.242594	+	0.970128i	$0.577998\pi$
$458$	5905.12	0.602463
$459$	−1567.59	−0.159409
$460$	461.923	0.0468202
$461$	6297.43	0.636226	0.318113	−	0.948053i	$-0.396951\pi$
$461$	6297.43	0.636226	0.318113	+	0.948053i	$0.396951\pi$
$462$	0	0
$463$	−4769.45	−0.478737	−0.239368	−	0.970929i	$-0.576940\pi$
$463$	−4769.45	−0.478737	−0.239368	+	0.970929i	$0.576940\pi$
$464$	10541.2	1.05466
$465$	267.140	0.0266416
$466$	−1838.40	−0.182752
$467$	−523.918	−0.0519144	−0.0259572	−	0.999663i	$-0.508263\pi$
$467$	−523.918	−0.0519144	−0.0259572	+	0.999663i	$0.508263\pi$
$468$	441.131	0.0435711
$469$	0	0
$470$	795.577	0.0780792
$471$	−2921.66	−0.285823
$472$	−143.928	−0.0140356
$473$	−1733.06	−0.168470
$474$	−5383.52	−0.521674
$475$	13824.0	1.33534
$476$	0	0
$477$	−716.262	−0.0687534
$478$	1756.22	0.168049
$479$	10734.2	1.02392	0.511960	−	0.859009i	$-0.328919\pi$
$479$	10734.2	1.02392	0.511960	+	0.859009i	$0.328919\pi$
$480$	−1443.57	−0.137270
$481$	−4351.28	−0.412476
$482$	−4334.51	−0.409609
$483$	0	0
$484$	8237.64	0.773633
$485$	−2982.52	−0.279236
$486$	−303.376	−0.0283157
$487$	−1537.61	−0.143071	−0.0715357	−	0.997438i	$-0.522790\pi$
$487$	−1537.61	−0.143071	−0.0715357	+	0.997438i	$0.522790\pi$
$488$	9787.56	0.907914
$489$	6234.30	0.576533
$490$	0	0
$491$	−6701.98	−0.616000	−0.308000	−	0.951386i	$-0.599660\pi$
$491$	−6701.98	−0.616000	−0.308000	+	0.951386i	$0.599660\pi$
$492$	6042.22	0.553667
$493$	−3053.34	−0.278936
$494$	−7684.80	−0.699910
$495$	19.4993	0.00177056
$496$	−1174.83	−0.106354
$497$	0	0
$498$	775.585	0.0697888
$499$	9529.20	0.854881	0.427440	−	0.904044i	$-0.359415\pi$
$499$	9529.20	0.854881	0.427440	+	0.904044i	$0.359415\pi$
$500$	2865.13	0.256265
$501$	9980.04	0.889971
$502$	−7419.44	−0.659653
$503$	5089.01	0.451108	0.225554	−	0.974231i	$-0.427581\pi$
$503$	5089.01	0.451108	0.225554	+	0.974231i	$0.427581\pi$
$504$	0	0
$505$	−1643.88	−0.144855
$506$	485.175	0.0426258
$507$	−9381.65	−0.821803
$508$	−3379.39	−0.295150
$509$	20343.6	1.77154	0.885769	−	0.464127i	$-0.153632\pi$
$509$	20343.6	1.77154	0.885769	+	0.464127i	$0.153632\pi$
$510$	101.087	0.00877689
$511$	0	0
$512$	11162.3	0.963495
$513$	−16162.2	−1.39099
$514$	5628.59	0.483009
$515$	658.886	0.0563766
$516$	5317.33	0.453648
$517$	−5018.51	−0.426913
$518$	0	0
$519$	13967.2	1.18129
$520$	−1705.39	−0.143819
$521$	3059.55	0.257277	0.128638	−	0.991692i	$-0.458939\pi$
$521$	3059.55	0.257277	0.128638	+	0.991692i	$0.458939\pi$
$522$	−300.894	−0.0252295
$523$	−7266.06	−0.607500	−0.303750	−	0.952752i	$-0.598239\pi$
$523$	−7266.06	−0.607500	−0.303750	+	0.952752i	$0.598239\pi$
$524$	−7034.13	−0.586426
$525$	0	0
$526$	−2811.16	−0.233027
$527$	340.299	0.0281284
$528$	2201.42	0.181448
$529$	−10579.4	−0.869514
$530$	1278.10	0.104750
$531$	−9.17659	−0.000749962 0
$532$	0	0
$533$	10981.4	0.892415
$534$	1564.07	0.126749
$535$	−2078.11	−0.167934
$536$	469.616	0.0378439
$537$	−23043.8	−1.85180
$538$	6100.37	0.488858
$539$	0	0
$540$	−1655.50	−0.131928
$541$	−5752.04	−0.457116	−0.228558	−	0.973530i	$-0.573401\pi$
$541$	−5752.04	−0.457116	−0.228558	+	0.973530i	$0.573401\pi$
$542$	2379.78	0.188598
$543$	21018.0	1.66108
$544$	−1838.91	−0.144931
$545$	−3220.25	−0.253102
$546$	0	0
$547$	−10824.1	−0.846075	−0.423037	−	0.906112i	$-0.639036\pi$
$547$	−10824.1	−0.846075	−0.423037	+	0.906112i	$0.639036\pi$
$548$	−12860.5	−1.00251
$549$	624.039	0.0485124
$550$	1487.28	0.115305
$551$	−31480.6	−2.43397
$552$	−3225.07	−0.248674
$553$	0	0
$554$	5288.36	0.405561
$555$	590.129	0.0451344
$556$	−5460.28	−0.416488
$557$	−1717.55	−0.130656	−0.0653278	−	0.997864i	$-0.520809\pi$
$557$	−1717.55	−0.130656	−0.0653278	+	0.997864i	$0.520809\pi$
$558$	33.5351	0.00254418
$559$	9663.96	0.731202
$560$	0	0
$561$	−637.659	−0.0479893
$562$	−3068.98	−0.230351
$563$	7436.84	0.556706	0.278353	−	0.960479i	$-0.410212\pi$
$563$	7436.84	0.556706	0.278353	+	0.960479i	$0.410212\pi$
$564$	15397.7	1.14957
$565$	−1508.45	−0.112321
$566$	−6626.21	−0.492085
$567$	0	0
$568$	−12373.0	−0.914013
$569$	−10663.6	−0.785658	−0.392829	−	0.919611i	$-0.628504\pi$
$569$	−10663.6	−0.785658	−0.392829	+	0.919611i	$0.628504\pi$
$570$	1042.23	0.0765863
$571$	3526.22	0.258438	0.129219	−	0.991616i	$-0.458753\pi$
$571$	3526.22	0.258438	0.129219	+	0.991616i	$0.458753\pi$
$572$	4965.41	0.362962
$573$	11112.7	0.810191
$574$	0	0
$575$	4866.77	0.352971
$576$	125.701	0.00909298
$577$	8391.90	0.605476	0.302738	−	0.953074i	$-0.402099\pi$
$577$	8391.90	0.605476	0.302738	+	0.953074i	$0.402099\pi$
$578$	−5121.31	−0.368544
$579$	9210.96	0.661131
$580$	−3224.58	−0.230850
$581$	0	0
$582$	9611.53	0.684555
$583$	−8062.30	−0.572738
$584$	11114.3	0.787519
$585$	−108.733	−0.00768468
$586$	−4275.68	−0.301411
$587$	6802.34	0.478301	0.239150	−	0.970983i	$-0.423131\pi$
$587$	6802.34	0.478301	0.239150	+	0.970983i	$0.423131\pi$
$588$	0	0
$589$	3508.55	0.245446
$590$	16.3748	0.00114261
$591$	18543.7	1.29067
$592$	−2595.27	−0.180178
$593$	−13527.4	−0.936771	−0.468385	−	0.883524i	$-0.655164\pi$
$593$	−13527.4	−0.936771	−0.468385	+	0.883524i	$0.655164\pi$
$594$	−1738.83	−0.120110
$595$	0	0
$596$	−5102.78	−0.350701
$597$	26933.9	1.84645
$598$	−2705.45	−0.185007
$599$	10633.4	0.725323	0.362662	−	0.931921i	$-0.381868\pi$
$599$	10633.4	0.725323	0.362662	+	0.931921i	$0.381868\pi$
$600$	−9886.26	−0.672675
$601$	14943.1	1.01421	0.507105	−	0.861884i	$-0.330716\pi$
$601$	14943.1	1.01421	0.507105	+	0.861884i	$0.330716\pi$
$602$	0	0
$603$	29.9419	0.00202211
$604$	−25368.4	−1.70898
$605$	−2030.46	−0.136446
$606$	5297.61	0.355117
$607$	2596.39	0.173615	0.0868076	−	0.996225i	$-0.472333\pi$
$607$	2596.39	0.173615	0.0868076	+	0.996225i	$0.472333\pi$
$608$	−18959.5	−1.26465
$609$	0	0
$610$	−1113.54	−0.0739114
$611$	27984.4	1.85291
$612$	−76.2114	−0.00503376
$613$	−6716.68	−0.442552	−0.221276	−	0.975211i	$-0.571022\pi$
$613$	−6716.68	−0.442552	−0.221276	+	0.975211i	$0.571022\pi$
$614$	−2149.03	−0.141250
$615$	−1489.32	−0.0976507
$616$	0	0
$617$	10466.6	0.682934	0.341467	−	0.939894i	$-0.389076\pi$
$617$	10466.6	0.682934	0.341467	+	0.939894i	$0.389076\pi$
$618$	−2123.34	−0.138209
$619$	−4068.02	−0.264148	−0.132074	−	0.991240i	$-0.542164\pi$
$619$	−4068.02	−0.264148	−0.132074	+	0.991240i	$0.542164\pi$
$620$	359.383	0.0232793
$621$	−5689.92	−0.367679
$622$	−5277.57	−0.340211
$623$	0	0
$624$	−12275.7	−0.787531
$625$	14561.6	0.931942
$626$	3598.21	0.229734
$627$	−6574.40	−0.418750
$628$	−3930.50	−0.249751
$629$	751.742	0.0476533
$630$	0	0
$631$	4589.57	0.289553	0.144776	−	0.989464i	$-0.453754\pi$
$631$	4589.57	0.289553	0.144776	+	0.989464i	$0.453754\pi$
$632$	−15690.8	−0.987574
$633$	10723.6	0.673340
$634$	−3217.05	−0.201523
$635$	832.972	0.0520559
$636$	24736.6	1.54225
$637$	0	0
$638$	−3386.89	−0.210170
$639$	−788.882	−0.0488383
$640$	−2489.70	−0.153772
$641$	12887.9	0.794137	0.397069	−	0.917789i	$-0.370027\pi$
$641$	12887.9	0.794137	0.397069	+	0.917789i	$0.370027\pi$
$642$	6696.96	0.411695
$643$	12135.2	0.744267	0.372134	−	0.928179i	$-0.378626\pi$
$643$	12135.2	0.744267	0.372134	+	0.928179i	$0.378626\pi$
$644$	0	0
$645$	−1310.65	−0.0800103
$646$	1327.65	0.0808604
$647$	−21159.6	−1.28573	−0.642865	−	0.765979i	$-0.722255\pi$
$647$	−21159.6	−1.28573	−0.642865	+	0.765979i	$0.722255\pi$
$648$	11124.4	0.674397
$649$	−103.292	−0.00624743
$650$	−8293.40	−0.500452
$651$	0	0
$652$	8386.99	0.503773
$653$	−25356.0	−1.51954	−0.759769	−	0.650193i	$-0.774688\pi$
$653$	−25356.0	−1.51954	−0.759769	+	0.650193i	$0.774688\pi$
$654$	10377.6	0.620486
$655$	1733.81	0.103428
$656$	6549.74	0.389824
$657$	708.627	0.0420794
$658$	0	0
$659$	−19623.2	−1.15996	−0.579978	−	0.814632i	$-0.696939\pi$
$659$	−19623.2	−1.15996	−0.579978	+	0.814632i	$0.696939\pi$
$660$	−673.419	−0.0397164
$661$	−8106.33	−0.477004	−0.238502	−	0.971142i	$-0.576656\pi$
$661$	−8106.33	−0.477004	−0.238502	+	0.971142i	$0.576656\pi$
$662$	9650.81	0.566601
$663$	3555.74	0.208286
$664$	2260.52	0.132116
$665$	0	0
$666$	74.0810	0.00431018
$667$	−11082.8	−0.643371
$668$	13426.1	0.777654
$669$	−20625.3	−1.19196
$670$	−53.4286	−0.00308079
$671$	7024.23	0.404124
$672$	0	0
$673$	−22638.5	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$673$	−22638.5	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$674$	2039.03	0.116529
$675$	−17442.1	−0.994590
$676$	−12621.1	−0.718088
$677$	−18076.3	−1.02619	−0.513095	−	0.858332i	$-0.671501\pi$
$677$	−18076.3	−1.02619	−0.513095	+	0.858332i	$0.671501\pi$
$678$	4861.17	0.275357
$679$	0	0
$680$	294.628	0.0166154
$681$	14359.8	0.808032
$682$	377.473	0.0211938
$683$	−1272.88	−0.0713111	−0.0356556	−	0.999364i	$-0.511352\pi$
$683$	−1272.88	−0.0713111	−0.0356556	+	0.999364i	$0.511352\pi$
$684$	−785.755	−0.0439241
$685$	3169.93	0.176813
$686$	0	0
$687$	−28170.4	−1.56444
$688$	5763.97	0.319403
$689$	44957.3	2.48583
$690$	366.919	0.0202440
$691$	−15182.5	−0.835846	−0.417923	−	0.908482i	$-0.637242\pi$
$691$	−15182.5	−0.835846	−0.417923	+	0.908482i	$0.637242\pi$
$692$	18790.0	1.03221
$693$	0	0
$694$	−7934.03	−0.433965
$695$	1345.88	0.0734563
$696$	22513.4	1.22611
$697$	−1897.18	−0.103100
$698$	−4514.43	−0.244804
$699$	8770.12	0.474558
$700$	0	0
$701$	−23767.1	−1.28056	−0.640279	−	0.768142i	$-0.721181\pi$
$701$	−23767.1	−1.28056	−0.640279	+	0.768142i	$0.721181\pi$
$702$	9696.14	0.521307
$703$	7750.61	0.415818
$704$	1414.90	0.0757474
$705$	−3795.31	−0.202751
$706$	778.295	0.0414894
$707$	0	0
$708$	316.919	0.0168228
$709$	155.285	0.00822546	0.00411273	−	0.999992i	$-0.498691\pi$
$709$	155.285	0.00822546	0.00411273	+	0.999992i	$0.498691\pi$
$710$	1407.69	0.0744078
$711$	−1000.42	−0.0527689
$712$	4558.63	0.239946
$713$	1235.19	0.0648785
$714$	0	0
$715$	−1223.90	−0.0640159
$716$	−31000.8	−1.61809
$717$	−8378.07	−0.436380
$718$	−2103.89	−0.109354
$719$	−33058.8	−1.71472	−0.857362	−	0.514714i	$-0.827898\pi$
$719$	−33058.8	−1.71472	−0.857362	+	0.514714i	$0.827898\pi$
$720$	−64.8524	−0.00335682
$721$	0	0
$722$	6358.79	0.327769
$723$	20677.8	1.06365
$724$	28275.4	1.45145
$725$	−33973.7	−1.74035
$726$	6543.40	0.334502
$727$	−1738.18	−0.0886735	−0.0443368	−	0.999017i	$-0.514117\pi$
$727$	−1738.18	−0.0886735	−0.0443368	+	0.999017i	$0.514117\pi$
$728$	0	0
$729$	20364.6	1.03463
$730$	−1264.48	−0.0641103
$731$	−1669.58	−0.0844756
$732$	−21551.6	−1.08821
$733$	−735.878	−0.0370809	−0.0185404	−	0.999828i	$-0.505902\pi$
$733$	−735.878	−0.0370809	−0.0185404	+	0.999828i	$0.505902\pi$
$734$	10605.5	0.533321
$735$	0	0
$736$	−6674.74	−0.334286
$737$	337.029	0.0168448
$738$	−186.960	−0.00932531
$739$	26114.2	1.29990	0.649949	−	0.759977i	$-0.274790\pi$
$739$	26114.2	1.29990	0.649949	+	0.759977i	$0.274790\pi$
$740$	793.900	0.0394383
$741$	36660.4	1.81748
$742$	0	0
$743$	−28095.4	−1.38724	−0.693621	−	0.720340i	$-0.743986\pi$
$743$	−28095.4	−1.38724	−0.693621	+	0.720340i	$0.743986\pi$
$744$	−2509.15	−0.123642
$745$	1257.76	0.0618535
$746$	−7772.63	−0.381469
$747$	144.127	0.00705935
$748$	−857.841	−0.0419329
$749$	0	0
$750$	2275.85	0.110803
$751$	−2510.16	−0.121967	−0.0609833	−	0.998139i	$-0.519424\pi$
$751$	−2510.16	−0.121967	−0.0609833	+	0.998139i	$0.519424\pi$
$752$	16691.0	0.809387
$753$	35394.5	1.71295
$754$	18886.1	0.912190
$755$	6252.95	0.301415
$756$	0	0
$757$	8171.67	0.392344	0.196172	−	0.980569i	$-0.437149\pi$
$757$	8171.67	0.392344	0.196172	+	0.980569i	$0.437149\pi$
$758$	6864.33	0.328923
$759$	−2314.53	−0.110688
$760$	3037.68	0.144984
$761$	−7806.65	−0.371867	−0.185934	−	0.982562i	$-0.559531\pi$
$761$	−7806.65	−0.371867	−0.185934	+	0.982562i	$0.559531\pi$
$762$	−2684.35	−0.127616
$763$	0	0
$764$	14949.9	0.707942
$765$	18.7850	0.000887809 0
$766$	10231.2	0.482595
$767$	575.983	0.0271154
$768$	2959.34	0.139044
$769$	20857.7	0.978084	0.489042	−	0.872260i	$-0.337346\pi$
$769$	20857.7	0.978084	0.489042	+	0.872260i	$0.337346\pi$
$770$	0	0
$771$	−26851.3	−1.25425
$772$	12391.5	0.577694
$773$	−18249.3	−0.849136	−0.424568	−	0.905396i	$-0.639574\pi$
$773$	−18249.3	−0.849136	−0.424568	+	0.905396i	$0.639574\pi$
$774$	−164.530	−0.00764071
$775$	3786.42	0.175500
$776$	28013.8	1.29592
$777$	0	0
$778$	4712.73	0.217172
$779$	−19560.4	−0.899644
$780$	3755.15	0.172379
$781$	−8879.72	−0.406839
$782$	467.403	0.0213738
$783$	39720.0	1.81287
$784$	0	0
$785$	968.812	0.0440489
$786$	−5587.41	−0.253558
$787$	−19636.9	−0.889429	−0.444714	−	0.895672i	$-0.646695\pi$
$787$	−19636.9	−0.889429	−0.444714	+	0.895672i	$0.646695\pi$
$788$	24946.8	1.12778
$789$	13410.7	0.605110
$790$	1785.16	0.0803963
$791$	0	0
$792$	−183.149	−0.00821708
$793$	−39168.8	−1.75400
$794$	−12428.5	−0.555505
$795$	−6097.21	−0.272007
$796$	36234.1	1.61342
$797$	−36758.6	−1.63370	−0.816848	−	0.576852i	$-0.804281\pi$
$797$	−36758.6	−1.63370	−0.816848	+	0.576852i	$0.804281\pi$
$798$	0	0
$799$	−4834.69	−0.214066
$800$	−20461.0	−0.904258
$801$	290.651	0.0128210
$802$	6385.70	0.281156
$803$	7976.36	0.350535
$804$	−1034.06	−0.0453590
$805$	0	0
$806$	−2104.88	−0.0919867
$807$	−29101.9	−1.26944
$808$	15440.4	0.672267
$809$	−8895.84	−0.386602	−0.193301	−	0.981140i	$-0.561919\pi$
$809$	−8895.84	−0.386602	−0.193301	+	0.981140i	$0.561919\pi$
$810$	−1265.64	−0.0549012
$811$	9955.82	0.431068	0.215534	−	0.976496i	$-0.430851\pi$
$811$	9955.82	0.431068	0.215534	+	0.976496i	$0.430851\pi$
$812$	0	0
$813$	−11352.8	−0.489740
$814$	833.862	0.0359052
$815$	−2067.27	−0.0888508
$816$	2120.79	0.0909833
$817$	−17213.7	−0.737126
$818$	−13508.4	−0.577395
$819$	0	0
$820$	−2003.58	−0.0853269
$821$	−1460.65	−0.0620913	−0.0310457	−	0.999518i	$-0.509884\pi$
$821$	−1460.65	−0.0620913	−0.0310457	+	0.999518i	$0.509884\pi$
$822$	−10215.5	−0.433461
$823$	−33331.3	−1.41173	−0.705866	−	0.708345i	$-0.749442\pi$
$823$	−33331.3	−1.41173	−0.705866	+	0.708345i	$0.749442\pi$
$824$	−6188.67	−0.261642
$825$	−7095.06	−0.299416
$826$	0	0
$827$	20877.9	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$827$	20877.9	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$828$	−276.627	−0.0116104
$829$	−19886.9	−0.833171	−0.416586	−	0.909096i	$-0.636773\pi$
$829$	−19886.9	−0.833171	−0.416586	+	0.909096i	$0.636773\pi$
$830$	−257.182	−0.0107553
$831$	−25228.2	−1.05314
$832$	−7889.84	−0.328763
$833$	0	0
$834$	−4337.26	−0.180080
$835$	−3309.35	−0.137155
$836$	−8844.52	−0.365902
$837$	−4426.85	−0.182813
$838$	−17767.5	−0.732420
$839$	33409.9	1.37478	0.687388	−	0.726290i	$-0.258757\pi$
$839$	33409.9	1.37478	0.687388	+	0.726290i	$0.258757\pi$
$840$	0	0
$841$	52977.5	2.17219
$842$	7496.90	0.306841
$843$	14640.6	0.598160
$844$	14426.4	0.588362
$845$	3110.93	0.126650
$846$	−476.438	−0.0193620
$847$	0	0
$848$	26814.3	1.08586
$849$	31610.4	1.27782
$850$	1432.80	0.0578171
$851$	2728.62	0.109913
$852$	27244.5	1.09552
$853$	−15726.3	−0.631251	−0.315626	−	0.948884i	$-0.602214\pi$
$853$	−15726.3	−0.631251	−0.315626	+	0.948884i	$0.602214\pi$
$854$	0	0
$855$	193.677	0.00774693
$856$	19519.0	0.779374
$857$	24847.0	0.990383	0.495192	−	0.868784i	$-0.335098\pi$
$857$	24847.0	0.990383	0.495192	+	0.868784i	$0.335098\pi$
$858$	3944.17	0.156937
$859$	−26765.4	−1.06312	−0.531562	−	0.847019i	$-0.678395\pi$
$859$	−26765.4	−1.06312	−0.531562	+	0.847019i	$0.678395\pi$
$860$	−1763.21	−0.0699128
$861$	0	0
$862$	−9637.64	−0.380811
$863$	−3229.39	−0.127381	−0.0636905	−	0.997970i	$-0.520287\pi$
$863$	−3229.39	−0.127381	−0.0636905	+	0.997970i	$0.520287\pi$
$864$	23921.8	0.941939
$865$	−4631.47	−0.182052
$866$	−1986.42	−0.0779461
$867$	24431.2	0.957011
$868$	0	0
$869$	−11260.8	−0.439582
$870$	−2561.37	−0.0998146
$871$	−1879.35	−0.0731108
$872$	30246.7	1.17463
$873$	1786.11	0.0692448
$874$	4819.02	0.186506
$875$	0	0
$876$	−24472.9	−0.943906
$877$	6097.69	0.234783	0.117391	−	0.993086i	$-0.462547\pi$
$877$	6097.69	0.234783	0.117391	+	0.993086i	$0.462547\pi$
$878$	6688.98	0.257110
$879$	20397.2	0.782685
$880$	−729.984	−0.0279634
$881$	−29714.8	−1.13634	−0.568171	−	0.822910i	$-0.692349\pi$
$881$	−29714.8	−1.13634	−0.568171	+	0.822910i	$0.692349\pi$
$882$	0	0
$883$	−7649.70	−0.291543	−0.145772	−	0.989318i	$-0.546566\pi$
$883$	−7649.70	−0.291543	−0.145772	+	0.989318i	$0.546566\pi$
$884$	4783.53	0.181999
$885$	−78.1161	−0.00296705
$886$	12242.1	0.464201
$887$	28257.5	1.06967	0.534833	−	0.844958i	$-0.320374\pi$
$887$	28257.5	1.06967	0.534833	+	0.844958i	$0.320374\pi$
$888$	−5542.87	−0.209467
$889$	0	0
$890$	−518.640	−0.0195335
$891$	7983.66	0.300183
$892$	−27747.2	−1.04153
$893$	−49846.6	−1.86792
$894$	−4053.28	−0.151635
$895$	7641.26	0.285384
$896$	0	0
$897$	12906.4	0.480414
$898$	14238.2	0.529103
$899$	−8622.60	−0.319889
$900$	−847.984	−0.0314068
$901$	−7766.99	−0.287187
$902$	−2104.43	−0.0776829
$903$	0	0
$904$	14168.4	0.521275
$905$	−6969.49	−0.255993
$906$	−20150.9	−0.738927
$907$	40378.4	1.47822	0.739108	−	0.673587i	$-0.235247\pi$
$907$	40378.4	1.47822	0.739108	+	0.673587i	$0.235247\pi$
$908$	19318.2	0.706055
$909$	984.455	0.0359211
$910$	0	0
$911$	−14278.6	−0.519289	−0.259644	−	0.965704i	$-0.583605\pi$
$911$	−14278.6	−0.519289	−0.259644	+	0.965704i	$0.583605\pi$
$912$	21865.7	0.793911
$913$	1622.31	0.0588066
$914$	−5065.28	−0.183309
$915$	5312.16	0.191928
$916$	−37897.6	−1.36700
$917$	0	0
$918$	−1675.14	−0.0602264
$919$	−28188.0	−1.01179	−0.505895	−	0.862595i	$-0.668838\pi$
$919$	−28188.0	−1.01179	−0.505895	+	0.862595i	$0.668838\pi$
$920$	1069.42	0.0383237
$921$	10252.0	0.366790
$922$	6729.49	0.240373
$923$	49515.4	1.76579
$924$	0	0
$925$	8364.43	0.297320
$926$	−5096.68	−0.180872
$927$	−394.579	−0.0139802
$928$	46594.8	1.64822
$929$	−29553.5	−1.04372	−0.521862	−	0.853030i	$-0.674763\pi$
$929$	−29553.5	−1.04372	−0.521862	+	0.853030i	$0.674763\pi$
$930$	285.469	0.0100655
$931$	0	0
$932$	11798.4	0.414668
$933$	25176.7	0.883438
$934$	−559.863	−0.0196138
$935$	211.446	0.00739574
$936$	1021.29	0.0356643
$937$	10120.3	0.352845	0.176422	−	0.984315i	$-0.443548\pi$
$937$	10120.3	0.352845	0.176422	+	0.984315i	$0.443548\pi$
$938$	0	0
$939$	−17165.3	−0.596558
$940$	−5105.82	−0.177163
$941$	19374.2	0.671182	0.335591	−	0.942008i	$-0.391064\pi$
$941$	19374.2	0.671182	0.335591	+	0.942008i	$0.391064\pi$
$942$	−3122.11	−0.107987
$943$	−6886.27	−0.237803
$944$	343.539	0.0118445
$945$	0	0
$946$	−1851.96	−0.0636496
$947$	−3809.98	−0.130737	−0.0653685	−	0.997861i	$-0.520822\pi$
$947$	−3809.98	−0.130737	−0.0653685	+	0.997861i	$0.520822\pi$
$948$	34550.1	1.18369
$949$	−44478.1	−1.52141
$950$	14772.4	0.504507
$951$	15347.0	0.523302
$952$	0	0
$953$	−26227.8	−0.891501	−0.445750	−	0.895157i	$-0.647063\pi$
$953$	−26227.8	−0.891501	−0.445750	+	0.895157i	$0.647063\pi$
$954$	−765.404	−0.0259758
$955$	−3684.93	−0.124860
$956$	−11271.0	−0.381308
$957$	16157.2	0.545755
$958$	11470.7	0.386848
$959$	0	0
$960$	1070.04	0.0359743
$961$	961.000	0.0322581
$962$	−4649.81	−0.155838
$963$	1244.50	0.0416442
$964$	27817.9	0.929412
$965$	−3054.33	−0.101888
$966$	0	0
$967$	18620.7	0.619236	0.309618	−	0.950861i	$-0.399799\pi$
$967$	18620.7	0.619236	0.309618	+	0.950861i	$0.399799\pi$
$968$	19071.4	0.633241
$969$	−6333.58	−0.209973
$970$	−3187.15	−0.105498
$971$	−15203.3	−0.502468	−0.251234	−	0.967926i	$-0.580836\pi$
$971$	−15203.3	−0.502468	−0.251234	+	0.967926i	$0.580836\pi$
$972$	1946.99	0.0642488
$973$	0	0
$974$	−1643.10	−0.0540538
$975$	39563.8	1.29954
$976$	−23361.8	−0.766183
$977$	456.756	0.0149569	0.00747846	−	0.999972i	$-0.497620\pi$
$977$	456.756	0.0149569	0.00747846	+	0.999972i	$0.497620\pi$
$978$	6662.03	0.217820
$979$	3271.59	0.106803
$980$	0	0
$981$	1928.48	0.0627641
$982$	−7161.79	−0.232731
$983$	−17077.0	−0.554093	−0.277046	−	0.960857i	$-0.589356\pi$
$983$	−17077.0	−0.554093	−0.277046	+	0.960857i	$0.589356\pi$
$984$	13988.6	0.453193
$985$	−6149.03	−0.198908
$986$	−3262.83	−0.105385
$987$	0	0
$988$	49319.2	1.58811
$989$	−6060.13	−0.194844
$990$	20.8371	0.000668935 0
$991$	−20515.4	−0.657612	−0.328806	−	0.944397i	$-0.606646\pi$
$991$	−20515.4	−0.657612	−0.328806	+	0.944397i	$0.606646\pi$
$992$	−5193.05	−0.166209
$993$	−46039.3	−1.47131
$994$	0	0
$995$	−8931.20	−0.284561
$996$	−4977.52	−0.158352
$997$	17822.4	0.566139	0.283070	−	0.959099i	$-0.408647\pi$
$997$	17822.4	0.566139	0.283070	+	0.959099i	$0.408647\pi$
$998$	10183.0	0.322983
$999$	−9779.18	−0.309709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.13	✓	23
7.6	odd	2		1519.4.a.i.1.13	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.13	✓	23	1.1	even	1	trivial
1519.4.a.i.1.13	yes	23	7.6	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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q+1.06861 q^{2} -5.09781 q^{3} -6.85807 q^{4} +1.69042 q^{5} -5.44757 q^{6} -15.8775 q^{8} -1.01232 q^{9} +1.80640 q^{10} -11.3948 q^{11} +34.9612 q^{12} +63.5400 q^{13} -8.61743 q^{15} +37.8978 q^{16} -10.9774 q^{17} -1.08178 q^{18} -113.179 q^{19} -11.5930 q^{20} -12.1766 q^{22} -39.8450 q^{23} +80.9404 q^{24} -122.142 q^{25} +67.8994 q^{26} +142.802 q^{27} +278.148 q^{29} -9.20866 q^{30} -31.0000 q^{31} +167.518 q^{32} +58.0884 q^{33} -11.7305 q^{34} +6.94258 q^{36} -68.4809 q^{37} -120.944 q^{38} -323.915 q^{39} -26.8396 q^{40} +172.827 q^{41} +152.093 q^{43} +78.1462 q^{44} -1.71125 q^{45} -42.5787 q^{46} +440.422 q^{47} -193.196 q^{48} -130.523 q^{50} +55.9607 q^{51} -435.762 q^{52} +707.544 q^{53} +152.599 q^{54} -19.2619 q^{55} +576.966 q^{57} +297.232 q^{58} +9.06489 q^{59} +59.0990 q^{60} -616.443 q^{61} -33.1269 q^{62} -124.171 q^{64} +107.409 q^{65} +62.0738 q^{66} -29.5775 q^{67} +75.2838 q^{68} +203.122 q^{69} +779.280 q^{71} +16.0731 q^{72} -700.002 q^{73} -73.1793 q^{74} +622.659 q^{75} +776.191 q^{76} -346.138 q^{78} +988.243 q^{79} +64.0631 q^{80} -700.643 q^{81} +184.684 q^{82} -142.373 q^{83} -18.5564 q^{85} +162.528 q^{86} -1417.95 q^{87} +180.920 q^{88} -287.113 q^{89} -1.82865 q^{90} +273.260 q^{92} +158.032 q^{93} +470.639 q^{94} -191.320 q^{95} -853.974 q^{96} -1764.37 q^{97} +11.5352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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