LMFDB - Embedded newform 1519.4.a.h.1.18

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			1.18
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)

$f(q)$	$=$	\(q+3.44360 q^{2} +1.20269 q^{3} +3.85839 q^{4} +17.2945 q^{5} +4.14160 q^{6} -14.2621 q^{8} -25.5535 q^{9} +59.5555 q^{10} -36.6522 q^{11} +4.64046 q^{12} +9.94450 q^{13} +20.8000 q^{15} -79.9799 q^{16} -53.2459 q^{17} -87.9961 q^{18} +74.3481 q^{19} +66.7290 q^{20} -126.216 q^{22} -82.5557 q^{23} -17.1529 q^{24} +174.101 q^{25} +34.2449 q^{26} -63.2058 q^{27} -26.6505 q^{29} +71.6270 q^{30} -31.0000 q^{31} -161.322 q^{32} -44.0814 q^{33} -183.358 q^{34} -98.5954 q^{36} -439.883 q^{37} +256.025 q^{38} +11.9602 q^{39} -246.656 q^{40} +388.103 q^{41} -86.3249 q^{43} -141.418 q^{44} -441.937 q^{45} -284.289 q^{46} -475.285 q^{47} -96.1914 q^{48} +599.535 q^{50} -64.0385 q^{51} +38.3697 q^{52} +108.315 q^{53} -217.655 q^{54} -633.884 q^{55} +89.4180 q^{57} -91.7736 q^{58} -185.027 q^{59} +80.2545 q^{60} -789.250 q^{61} -106.752 q^{62} +84.3094 q^{64} +171.986 q^{65} -151.799 q^{66} -778.075 q^{67} -205.443 q^{68} -99.2892 q^{69} +692.485 q^{71} +364.446 q^{72} -174.777 q^{73} -1514.78 q^{74} +209.390 q^{75} +286.864 q^{76} +41.1861 q^{78} +325.641 q^{79} -1383.22 q^{80} +613.928 q^{81} +1336.47 q^{82} +880.935 q^{83} -920.863 q^{85} -297.269 q^{86} -32.0523 q^{87} +522.737 q^{88} -1196.89 q^{89} -1521.85 q^{90} -318.532 q^{92} -37.2835 q^{93} -1636.69 q^{94} +1285.82 q^{95} -194.021 q^{96} +663.731 q^{97} +936.594 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.44360	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$2$	3.44360	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$3$	1.20269	0.231458	0.115729	−	0.993281i	$-0.463080\pi$
$3$	1.20269	0.231458	0.115729	+	0.993281i	$0.463080\pi$
$4$	3.85839	0.482298
$5$	17.2945	1.54687	0.773435	−	0.633875i	$-0.218537\pi$
$5$	17.2945	1.54687	0.773435	+	0.633875i	$0.218537\pi$
$6$	4.14160	0.281800
$7$	0	0
$7$	0	0
$8$	−14.2621	−0.630300
$9$	−25.5535	−0.946427
$10$	59.5555	1.88331
$11$	−36.6522	−1.00464	−0.502321	−	0.864681i	$-0.667521\pi$
$11$	−36.6522	−1.00464	−0.502321	+	0.864681i	$0.667521\pi$
$12$	4.64046	0.111632
$13$	9.94450	0.212162	0.106081	−	0.994357i	$-0.466170\pi$
$13$	9.94450	0.212162	0.106081	+	0.994357i	$0.466170\pi$
$14$	0	0
$15$	20.8000	0.358036
$16$	−79.9799	−1.24969
$17$	−53.2459	−0.759648	−0.379824	−	0.925059i	$-0.624016\pi$
$17$	−53.2459	−0.759648	−0.379824	+	0.925059i	$0.624016\pi$
$18$	−87.9961	−1.15227
$19$	74.3481	0.897718	0.448859	−	0.893603i	$-0.351831\pi$
$19$	74.3481	0.897718	0.448859	+	0.893603i	$0.351831\pi$
$20$	66.7290	0.746053
$21$	0	0
$22$	−126.216	−1.22315
$23$	−82.5557	−0.748437	−0.374218	−	0.927341i	$-0.622089\pi$
$23$	−82.5557	−0.748437	−0.374218	+	0.927341i	$0.622089\pi$
$24$	−17.1529	−0.145888
$25$	174.101	1.39281
$26$	34.2449	0.258307
$27$	−63.2058	−0.450517
$28$	0	0
$29$	−26.6505	−0.170651	−0.0853253	−	0.996353i	$-0.527193\pi$
$29$	−26.6505	−0.170651	−0.0853253	+	0.996353i	$0.527193\pi$
$30$	71.6270	0.435908
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−161.322	−0.891189
$33$	−44.0814	−0.232533
$34$	−183.358	−0.924869
$35$	0	0
$36$	−98.5954	−0.456460
$37$	−439.883	−1.95450	−0.977248	−	0.212099i	$-0.931970\pi$
$37$	−439.883	−1.95450	−0.977248	+	0.212099i	$0.931970\pi$
$38$	256.025	1.09297
$39$	11.9602	0.0491067
$40$	−246.656	−0.974993
$41$	388.103	1.47833	0.739165	−	0.673524i	$-0.235220\pi$
$41$	388.103	1.47833	0.739165	+	0.673524i	$0.235220\pi$
$42$	0	0
$43$	−86.3249	−0.306149	−0.153075	−	0.988215i	$-0.548918\pi$
$43$	−86.3249	−0.306149	−0.153075	+	0.988215i	$0.548918\pi$
$44$	−141.418	−0.484537
$45$	−441.937	−1.46400
$46$	−284.289	−0.911219
$47$	−475.285	−1.47505	−0.737526	−	0.675319i	$-0.764006\pi$
$47$	−475.285	−1.47505	−0.737526	+	0.675319i	$0.764006\pi$
$48$	−96.1914	−0.289251
$49$	0	0
$50$	599.535	1.69574
$51$	−64.0385	−0.175827
$52$	38.3697	0.102325
$53$	108.315	0.280720	0.140360	−	0.990101i	$-0.455174\pi$
$53$	108.315	0.280720	0.140360	+	0.990101i	$0.455174\pi$
$54$	−217.655	−0.548503
$55$	−633.884	−1.55405
$56$	0	0
$57$	89.4180	0.207784
$58$	−91.7736	−0.207767
$59$	−185.027	−0.408278	−0.204139	−	0.978942i	$-0.565439\pi$
$59$	−185.027	−0.408278	−0.204139	+	0.978942i	$0.565439\pi$
$60$	80.2545	0.172680
$61$	−789.250	−1.65661	−0.828304	−	0.560279i	$-0.810694\pi$
$61$	−789.250	−1.65661	−0.828304	+	0.560279i	$0.810694\pi$
$62$	−106.752	−0.218669
$63$	0	0
$64$	84.3094	0.164667
$65$	171.986	0.328187
$66$	−151.799	−0.283108
$67$	−778.075	−1.41876	−0.709381	−	0.704825i	$-0.751025\pi$
$67$	−778.075	−1.41876	−0.709381	+	0.704825i	$0.751025\pi$
$68$	−205.443	−0.366377
$69$	−99.2892	−0.173232
$70$	0	0
$71$	692.485	1.15751	0.578753	−	0.815503i	$-0.303540\pi$
$71$	692.485	1.15751	0.578753	+	0.815503i	$0.303540\pi$
$72$	364.446	0.596533
$73$	−174.777	−0.280221	−0.140110	−	0.990136i	$-0.544746\pi$
$73$	−174.777	−0.280221	−0.140110	+	0.990136i	$0.544746\pi$
$74$	−1514.78	−2.37959
$75$	209.390	0.322378
$76$	286.864	0.432968
$77$	0	0
$78$	41.1861	0.0597873
$79$	325.641	0.463766	0.231883	−	0.972744i	$-0.425511\pi$
$79$	325.641	0.463766	0.231883	+	0.972744i	$0.425511\pi$
$80$	−1383.22	−1.93310
$81$	613.928	0.842151
$82$	1336.47	1.79986
$83$	880.935	1.16500	0.582501	−	0.812830i	$-0.302074\pi$
$83$	880.935	1.16500	0.582501	+	0.812830i	$0.302074\pi$
$84$	0	0
$85$	−920.863	−1.17508
$86$	−297.269	−0.372736
$87$	−32.0523	−0.0394985
$88$	522.737	0.633226
$89$	−1196.89	−1.42550	−0.712752	−	0.701416i	$-0.752552\pi$
$89$	−1196.89	−1.42550	−0.712752	+	0.701416i	$0.752552\pi$
$90$	−1521.85	−1.78242
$91$	0	0
$92$	−318.532	−0.360970
$93$	−37.2835	−0.0415712
$94$	−1636.69	−1.79587
$95$	1285.82	1.38865
$96$	−194.021	−0.206273
$97$	663.731	0.694760	0.347380	−	0.937724i	$-0.387071\pi$
$97$	663.731	0.694760	0.347380	+	0.937724i	$0.387071\pi$
$98$	0	0
$99$	936.594	0.950821
$100$	671.749	0.671749
$101$	−610.336	−0.601294	−0.300647	−	0.953735i	$-0.597203\pi$
$101$	−610.336	−0.601294	−0.300647	+	0.953735i	$0.597203\pi$
$102$	−220.523	−0.214069
$103$	574.420	0.549508	0.274754	−	0.961515i	$-0.411404\pi$
$103$	574.420	0.549508	0.274754	+	0.961515i	$0.411404\pi$
$104$	−141.829	−0.133726
$105$	0	0
$106$	372.993	0.341776
$107$	−1212.88	−1.09583	−0.547913	−	0.836536i	$-0.684577\pi$
$107$	−1212.88	−1.09583	−0.547913	+	0.836536i	$0.684577\pi$
$108$	−243.872	−0.217284
$109$	2140.31	1.88078	0.940389	−	0.340101i	$-0.110461\pi$
$109$	2140.31	1.88078	0.940389	+	0.340101i	$0.110461\pi$
$110$	−2182.84	−1.89205
$111$	−529.045	−0.452385
$112$	0	0
$113$	−1061.77	−0.883916	−0.441958	−	0.897036i	$-0.645716\pi$
$113$	−1061.77	−0.883916	−0.441958	+	0.897036i	$0.645716\pi$
$114$	307.920	0.252977
$115$	−1427.76	−1.15774
$116$	−102.828	−0.0823045
$117$	−254.117	−0.200796
$118$	−637.157	−0.497077
$119$	0	0
$120$	−296.651	−0.225670
$121$	12.3870	0.00930655
$122$	−2717.86	−2.01691
$123$	466.769	0.342172
$124$	−119.610	−0.0866233
$125$	849.182	0.607626
$126$	0	0
$127$	−628.574	−0.439189	−0.219594	−	0.975591i	$-0.570473\pi$
$127$	−628.574	−0.439189	−0.219594	+	0.975591i	$0.570473\pi$
$128$	1580.91	1.09167
$129$	−103.822	−0.0708609
$130$	592.250	0.399567
$131$	896.367	0.597832	0.298916	−	0.954279i	$-0.403375\pi$
$131$	896.367	0.597832	0.298916	+	0.954279i	$0.403375\pi$
$132$	−170.083	−0.112150
$133$	0	0
$134$	−2679.38	−1.72734
$135$	−1093.12	−0.696892
$136$	759.396	0.478807
$137$	−1213.83	−0.756967	−0.378484	−	0.925608i	$-0.623554\pi$
$137$	−1213.83	−0.756967	−0.378484	+	0.925608i	$0.623554\pi$
$138$	−341.912	−0.210909
$139$	1043.46	0.636730	0.318365	−	0.947968i	$-0.396866\pi$
$139$	1043.46	0.636730	0.318365	+	0.947968i	$0.396866\pi$
$140$	0	0
$141$	−571.622	−0.341413
$142$	2384.64	1.40926
$143$	−364.488	−0.213147
$144$	2043.77	1.18274
$145$	−460.908	−0.263974
$146$	−601.862	−0.341168
$147$	0	0
$148$	−1697.24	−0.942650
$149$	−2045.14	−1.12446	−0.562229	−	0.826982i	$-0.690056\pi$
$149$	−2045.14	−1.12446	−0.562229	+	0.826982i	$0.690056\pi$
$150$	721.057	0.392494
$151$	685.779	0.369589	0.184794	−	0.982777i	$-0.440838\pi$
$151$	685.779	0.369589	0.184794	+	0.982777i	$0.440838\pi$
$152$	−1060.36	−0.565832
$153$	1360.62	0.718952
$154$	0	0
$155$	−536.131	−0.277826
$156$	46.1470	0.0236841
$157$	1223.12	0.621758	0.310879	−	0.950450i	$-0.399377\pi$
$157$	1223.12	0.621758	0.310879	+	0.950450i	$0.399377\pi$
$158$	1121.38	0.564633
$159$	130.269	0.0649751
$160$	−2790.00	−1.37855
$161$	0	0
$162$	2114.12	1.02532
$163$	2828.71	1.35927	0.679636	−	0.733549i	$-0.262138\pi$
$163$	2828.71	1.35927	0.679636	+	0.733549i	$0.262138\pi$
$164$	1497.45	0.712996
$165$	−762.368	−0.359699
$166$	3033.59	1.41839
$167$	1525.36	0.706801	0.353401	−	0.935472i	$-0.385025\pi$
$167$	1525.36	0.706801	0.353401	+	0.935472i	$0.385025\pi$
$168$	0	0
$169$	−2098.11	−0.954987
$170$	−3171.08	−1.43065
$171$	−1899.86	−0.849624
$172$	−333.075	−0.147655
$173$	2371.34	1.04214	0.521068	−	0.853515i	$-0.325534\pi$
$173$	2371.34	1.04214	0.521068	+	0.853515i	$0.325534\pi$
$174$	−110.375	−0.0480893
$175$	0	0
$176$	2931.44	1.25549
$177$	−222.530	−0.0944994
$178$	−4121.61	−1.73555
$179$	−1616.89	−0.675150	−0.337575	−	0.941299i	$-0.609607\pi$
$179$	−1616.89	−0.675150	−0.337575	+	0.941299i	$0.609607\pi$
$180$	−1705.16	−0.706085
$181$	2311.94	0.949420	0.474710	−	0.880142i	$-0.342553\pi$
$181$	2311.94	0.949420	0.474710	+	0.880142i	$0.342553\pi$
$182$	0	0
$183$	−949.226	−0.383436
$184$	1177.41	0.471740
$185$	−7607.58	−3.02335
$186$	−128.389	−0.0506128
$187$	1951.58	0.763175
$188$	−1833.83	−0.711415
$189$	0	0
$190$	4427.84	1.69068
$191$	−705.215	−0.267160	−0.133580	−	0.991038i	$-0.542647\pi$
$191$	−705.215	−0.267160	−0.133580	+	0.991038i	$0.542647\pi$
$192$	101.398	0.0381135
$193$	1395.98	0.520646	0.260323	−	0.965522i	$-0.416171\pi$
$193$	1395.98	0.520646	0.260323	+	0.965522i	$0.416171\pi$
$194$	2285.63	0.845868
$195$	206.846	0.0759618
$196$	0	0
$197$	3745.35	1.35454	0.677272	−	0.735733i	$-0.263162\pi$
$197$	3745.35	1.35454	0.677272	+	0.735733i	$0.263162\pi$
$198$	3225.26	1.15762
$199$	−2443.91	−0.870574	−0.435287	−	0.900292i	$-0.643353\pi$
$199$	−2443.91	−0.870574	−0.435287	+	0.900292i	$0.643353\pi$
$200$	−2483.04	−0.877888
$201$	−935.786	−0.328384
$202$	−2101.75	−0.732074
$203$	0	0
$204$	−247.085	−0.0848011
$205$	6712.07	2.28679
$206$	1978.07	0.669024
$207$	2109.59	0.708341
$208$	−795.361	−0.265136
$209$	−2725.03	−0.901885
$210$	0	0
$211$	−3053.47	−0.996252	−0.498126	−	0.867105i	$-0.665978\pi$
$211$	−3053.47	−0.996252	−0.498126	+	0.867105i	$0.665978\pi$
$212$	417.920	0.135391
$213$	832.848	0.267914
$214$	−4176.67	−1.33416
$215$	−1492.95	−0.473574
$216$	901.445	0.283961
$217$	0	0
$218$	7370.38	2.28984
$219$	−210.203	−0.0648594
$220$	−2445.77	−0.749517
$221$	−529.504	−0.161169
$222$	−1821.82	−0.550777
$223$	−5229.07	−1.57024	−0.785122	−	0.619342i	$-0.787400\pi$
$223$	−5229.07	−1.57024	−0.785122	+	0.619342i	$0.787400\pi$
$224$	0	0
$225$	−4448.90	−1.31819
$226$	−3656.30	−1.07616
$227$	5787.06	1.69207	0.846037	−	0.533125i	$-0.178982\pi$
$227$	5787.06	1.69207	0.846037	+	0.533125i	$0.178982\pi$
$228$	345.009	0.100214
$229$	813.703	0.234808	0.117404	−	0.993084i	$-0.462543\pi$
$229$	813.703	0.234808	0.117404	+	0.993084i	$0.462543\pi$
$230$	−4916.64	−1.40954
$231$	0	0
$232$	380.091	0.107561
$233$	−2348.65	−0.660365	−0.330183	−	0.943917i	$-0.607110\pi$
$233$	−2348.65	−0.660365	−0.330183	+	0.943917i	$0.607110\pi$
$234$	−875.078	−0.244468
$235$	−8219.83	−2.28171
$236$	−713.904	−0.196912
$237$	391.647	0.107343
$238$	0	0
$239$	−928.497	−0.251295	−0.125647	−	0.992075i	$-0.540101\pi$
$239$	−928.497	−0.251295	−0.125647	+	0.992075i	$0.540101\pi$
$240$	−1663.59	−0.447433
$241$	−6720.53	−1.79630	−0.898148	−	0.439693i	$-0.855087\pi$
$241$	−6720.53	−1.79630	−0.898148	+	0.439693i	$0.855087\pi$
$242$	42.6560	0.0113307
$243$	2444.92	0.645440
$244$	−3045.23	−0.798979
$245$	0	0
$246$	1607.37	0.416593
$247$	739.355	0.190462
$248$	442.124	0.113205
$249$	1059.50	0.269650
$250$	2924.25	0.739782
$251$	−5518.02	−1.38763	−0.693814	−	0.720154i	$-0.744071\pi$
$251$	−5518.02	−1.38763	−0.693814	+	0.720154i	$0.744071\pi$
$252$	0	0
$253$	3025.85	0.751912
$254$	−2164.56	−0.534711
$255$	−1107.52	−0.271982
$256$	4769.54	1.16444
$257$	−1339.15	−0.325035	−0.162518	−	0.986706i	$-0.551961\pi$
$257$	−1339.15	−0.325035	−0.162518	+	0.986706i	$0.551961\pi$
$258$	−357.523	−0.0862729
$259$	0	0
$260$	663.587	0.158284
$261$	681.013	0.161508
$262$	3086.73	0.727858
$263$	1863.57	0.436929	0.218465	−	0.975845i	$-0.429895\pi$
$263$	1863.57	0.436929	0.218465	+	0.975845i	$0.429895\pi$
$264$	628.692	0.146566
$265$	1873.25	0.434238
$266$	0	0
$267$	−1439.49	−0.329945
$268$	−3002.11	−0.684266
$269$	−3911.21	−0.886509	−0.443254	−	0.896396i	$-0.646176\pi$
$269$	−3911.21	−0.886509	−0.443254	+	0.896396i	$0.646176\pi$
$270$	−3764.25	−0.848463
$271$	3975.48	0.891119	0.445560	−	0.895252i	$-0.353005\pi$
$271$	3975.48	0.891119	0.445560	+	0.895252i	$0.353005\pi$
$272$	4258.60	0.949322
$273$	0	0
$274$	−4179.95	−0.921605
$275$	−6381.20	−1.39928
$276$	−383.096	−0.0835495
$277$	−1805.31	−0.391592	−0.195796	−	0.980645i	$-0.562729\pi$
$277$	−1805.31	−0.391592	−0.195796	+	0.980645i	$0.562729\pi$
$278$	3593.27	0.775216
$279$	792.159	0.169983
$280$	0	0
$281$	−474.789	−0.100795	−0.0503977	−	0.998729i	$-0.516049\pi$
$281$	−474.789	−0.100795	−0.0503977	+	0.998729i	$0.516049\pi$
$282$	−1968.44	−0.415669
$283$	−2824.17	−0.593214	−0.296607	−	0.955000i	$-0.595855\pi$
$283$	−2824.17	−0.593214	−0.296607	+	0.955000i	$0.595855\pi$
$284$	2671.88	0.558263
$285$	1546.44	0.321416
$286$	−1255.15	−0.259506
$287$	0	0
$288$	4122.36	0.843445
$289$	−2077.88	−0.422934
$290$	−1587.18	−0.321388
$291$	798.265	0.160808
$292$	−674.357	−0.135150
$293$	−3036.28	−0.605396	−0.302698	−	0.953086i	$-0.597887\pi$
$293$	−3036.28	−0.605396	−0.302698	+	0.953086i	$0.597887\pi$
$294$	0	0
$295$	−3199.95	−0.631553
$296$	6273.64	1.23192
$297$	2316.63	0.452609
$298$	−7042.64	−1.36902
$299$	−820.975	−0.158790
$300$	807.909	0.155482
$301$	0	0
$302$	2361.55	0.449973
$303$	−734.047	−0.139175
$304$	−5946.36	−1.12187
$305$	−13649.7	−2.56256
$306$	4685.43	0.875321
$307$	5357.71	0.996029	0.498014	−	0.867169i	$-0.334063\pi$
$307$	5357.71	0.996029	0.498014	+	0.867169i	$0.334063\pi$
$308$	0	0
$309$	690.851	0.127188
$310$	−1846.22	−0.338252
$311$	−2498.80	−0.455608	−0.227804	−	0.973707i	$-0.573154\pi$
$311$	−2498.80	−0.455608	−0.227804	+	0.973707i	$0.573154\pi$
$312$	−170.577	−0.0309520
$313$	4948.73	0.893671	0.446835	−	0.894616i	$-0.352551\pi$
$313$	4948.73	0.893671	0.446835	+	0.894616i	$0.352551\pi$
$314$	4211.95	0.756988
$315$	0	0
$316$	1256.45	0.223673
$317$	−1670.35	−0.295951	−0.147975	−	0.988991i	$-0.547276\pi$
$317$	−1670.35	−0.295951	−0.147975	+	0.988991i	$0.547276\pi$
$318$	448.596	0.0791069
$319$	976.799	0.171443
$320$	1458.09	0.254718
$321$	−1458.72	−0.253638
$322$	0	0
$323$	−3958.73	−0.681950
$324$	2368.77	0.406168
$325$	1731.35	0.295501
$326$	9740.94	1.65491
$327$	2574.14	0.435322
$328$	−5535.15	−0.931792
$329$	0	0
$330$	−2625.29	−0.437932
$331$	−8848.24	−1.46931	−0.734657	−	0.678439i	$-0.762657\pi$
$331$	−8848.24	−1.46931	−0.734657	+	0.678439i	$0.762657\pi$
$332$	3398.99	0.561879
$333$	11240.6	1.84979
$334$	5252.73	0.860528
$335$	−13456.5	−2.19464
$336$	0	0
$337$	−4016.50	−0.649236	−0.324618	−	0.945845i	$-0.605236\pi$
$337$	−4016.50	−0.649236	−0.324618	+	0.945845i	$0.605236\pi$
$338$	−7225.04	−1.16269
$339$	−1276.98	−0.204590
$340$	−3553.04	−0.566738
$341$	1136.22	0.180439
$342$	−6542.35	−1.03441
$343$	0	0
$344$	1231.17	0.192966
$345$	−1717.16	−0.267968
$346$	8165.94	1.26880
$347$	5542.72	0.857490	0.428745	−	0.903426i	$-0.358956\pi$
$347$	5542.72	0.857490	0.428745	+	0.903426i	$0.358956\pi$
$348$	−123.670	−0.0190501
$349$	2828.15	0.433775	0.216887	−	0.976197i	$-0.430410\pi$
$349$	2828.15	0.433775	0.216887	+	0.976197i	$0.430410\pi$
$350$	0	0
$351$	−628.550	−0.0955827
$352$	5912.83	0.895326
$353$	5206.37	0.785005	0.392503	−	0.919751i	$-0.371609\pi$
$353$	5206.37	0.785005	0.392503	+	0.919751i	$0.371609\pi$
$354$	−766.305	−0.115053
$355$	11976.2	1.79051
$356$	−4618.06	−0.687518
$357$	0	0
$358$	−5567.92	−0.821993
$359$	1371.96	0.201697	0.100848	−	0.994902i	$-0.467844\pi$
$359$	1371.96	0.201697	0.100848	+	0.994902i	$0.467844\pi$
$360$	6302.93	0.922760
$361$	−1331.35	−0.194103
$362$	7961.40	1.15592
$363$	14.8978	0.00215408
$364$	0	0
$365$	−3022.69	−0.433465
$366$	−3268.75	−0.466832
$367$	2205.23	0.313657	0.156829	−	0.987626i	$-0.449873\pi$
$367$	2205.23	0.313657	0.156829	+	0.987626i	$0.449873\pi$
$368$	6602.80	0.935312
$369$	−9917.40	−1.39913
$370$	−26197.5	−3.68092
$371$	0	0
$372$	−143.854	−0.0200497
$373$	−2003.59	−0.278128	−0.139064	−	0.990283i	$-0.544409\pi$
$373$	−2003.59	−0.278128	−0.139064	+	0.990283i	$0.544409\pi$
$374$	6720.46	0.929163
$375$	1021.31	0.140640
$376$	6778.54	0.929725
$377$	−265.026	−0.0362056
$378$	0	0
$379$	13621.7	1.84617	0.923087	−	0.384591i	$-0.125658\pi$
$379$	13621.7	1.84617	0.923087	+	0.384591i	$0.125658\pi$
$380$	4961.18	0.669745
$381$	−755.982	−0.101654
$382$	−2428.48	−0.325266
$383$	−5198.10	−0.693501	−0.346750	−	0.937957i	$-0.612715\pi$
$383$	−5198.10	−0.693501	−0.346750	+	0.937957i	$0.612715\pi$
$384$	1901.35	0.252676
$385$	0	0
$386$	4807.19	0.633885
$387$	2205.91	0.289748
$388$	2560.93	0.335081
$389$	−921.571	−0.120117	−0.0600585	−	0.998195i	$-0.519129\pi$
$389$	−921.571	−0.120117	−0.0600585	+	0.998195i	$0.519129\pi$
$390$	712.295	0.0924832
$391$	4395.75	0.568549
$392$	0	0
$393$	1078.05	0.138373
$394$	12897.5	1.64915
$395$	5631.81	0.717386
$396$	3613.74	0.458579
$397$	9403.30	1.18876	0.594381	−	0.804184i	$-0.297397\pi$
$397$	9403.30	1.18876	0.594381	+	0.804184i	$0.297397\pi$
$398$	−8415.86	−1.05992
$399$	0	0
$400$	−13924.6	−1.74058
$401$	13146.7	1.63719	0.818596	−	0.574370i	$-0.194753\pi$
$401$	13146.7	1.63719	0.818596	+	0.574370i	$0.194753\pi$
$402$	−3222.47	−0.399807
$403$	−308.279	−0.0381054
$404$	−2354.91	−0.290003
$405$	10617.6	1.30270
$406$	0	0
$407$	16122.7	1.96357
$408$	913.321	0.110824
$409$	14944.6	1.80675	0.903377	−	0.428847i	$-0.141080\pi$
$409$	14944.6	1.80675	0.903377	+	0.428847i	$0.141080\pi$
$410$	23113.7	2.78415
$411$	−1459.87	−0.175206
$412$	2216.33	0.265026
$413$	0	0
$414$	7264.58	0.862403
$415$	15235.4	1.80211
$416$	−1604.27	−0.189077
$417$	1254.97	0.147377
$418$	−9383.90	−1.09804
$419$	8876.22	1.03492	0.517460	−	0.855707i	$-0.326877\pi$
$419$	8876.22	1.03492	0.517460	+	0.855707i	$0.326877\pi$
$420$	0	0
$421$	−15472.4	−1.79116	−0.895582	−	0.444896i	$-0.853240\pi$
$421$	−15472.4	−1.79116	−0.895582	+	0.444896i	$0.853240\pi$
$422$	−10514.9	−1.21293
$423$	12145.2	1.39603
$424$	−1544.79	−0.176938
$425$	−9270.17	−1.05805
$426$	2868.00	0.326185
$427$	0	0
$428$	−4679.75	−0.528515
$429$	−438.368	−0.0493347
$430$	−5141.12	−0.576574
$431$	−3752.07	−0.419329	−0.209664	−	0.977773i	$-0.567237\pi$
$431$	−3752.07	−0.419329	−0.209664	+	0.977773i	$0.567237\pi$
$432$	5055.20	0.563005
$433$	−15501.3	−1.72043	−0.860216	−	0.509930i	$-0.829671\pi$
$433$	−15501.3	−1.72043	−0.860216	+	0.509930i	$0.829671\pi$
$434$	0	0
$435$	−554.331	−0.0610991
$436$	8258.15	0.907096
$437$	−6137.86	−0.671885
$438$	−723.856	−0.0789661
$439$	14289.9	1.55357	0.776787	−	0.629763i	$-0.216848\pi$
$439$	14289.9	1.55357	0.776787	+	0.629763i	$0.216848\pi$
$440$	9040.49	0.979520
$441$	0	0
$442$	−1823.40	−0.196222
$443$	3125.58	0.335216	0.167608	−	0.985854i	$-0.446396\pi$
$443$	3125.58	0.335216	0.167608	+	0.985854i	$0.446396\pi$
$444$	−2041.26	−0.218184
$445$	−20699.6	−2.20507
$446$	−18006.8	−1.91177
$447$	−2459.67	−0.260265
$448$	0	0
$449$	5399.95	0.567571	0.283786	−	0.958888i	$-0.408410\pi$
$449$	5399.95	0.567571	0.283786	+	0.958888i	$0.408410\pi$
$450$	−15320.2	−1.60489
$451$	−14224.9	−1.48519
$452$	−4096.70	−0.426311
$453$	824.782	0.0855444
$454$	19928.3	2.06009
$455$	0	0
$456$	−1275.29	−0.130967
$457$	11505.3	1.17767	0.588835	−	0.808253i	$-0.299587\pi$
$457$	11505.3	1.17767	0.588835	+	0.808253i	$0.299587\pi$
$458$	2802.07	0.285878
$459$	3365.45	0.342235
$460$	−5508.86	−0.558374
$461$	5524.07	0.558094	0.279047	−	0.960277i	$-0.409981\pi$
$461$	5524.07	0.558094	0.279047	+	0.960277i	$0.409981\pi$
$462$	0	0
$463$	6929.37	0.695541	0.347770	−	0.937580i	$-0.386939\pi$
$463$	6929.37	0.695541	0.347770	+	0.937580i	$0.386939\pi$
$464$	2131.50	0.213260
$465$	−644.801	−0.0643052
$466$	−8087.81	−0.803992
$467$	−14849.9	−1.47146	−0.735728	−	0.677277i	$-0.763160\pi$
$467$	−14849.9	−1.47146	−0.735728	+	0.677277i	$0.763160\pi$
$468$	−980.481	−0.0968435
$469$	0	0
$470$	−28305.8	−2.77798
$471$	1471.04	0.143911
$472$	2638.86	0.257338
$473$	3164.00	0.307571
$474$	1348.67	0.130689
$475$	12944.1	1.25035
$476$	0	0
$477$	−2767.82	−0.265681
$478$	−3197.37	−0.305951
$479$	−17074.0	−1.62866	−0.814332	−	0.580399i	$-0.802896\pi$
$479$	−17074.0	−1.62866	−0.814332	+	0.580399i	$0.802896\pi$
$480$	−3355.51	−0.319078
$481$	−4374.42	−0.414670
$482$	−23142.8	−2.18698
$483$	0	0
$484$	47.7939	0.00448853
$485$	11478.9	1.07470
$486$	8419.34	0.785821
$487$	15073.3	1.40254	0.701268	−	0.712898i	$-0.252618\pi$
$487$	15073.3	1.40254	0.701268	+	0.712898i	$0.252618\pi$
$488$	11256.3	1.04416
$489$	3402.07	0.314615
$490$	0	0
$491$	15255.1	1.40214	0.701070	−	0.713092i	$-0.252706\pi$
$491$	15255.1	1.40214	0.701070	+	0.713092i	$0.252706\pi$
$492$	1800.98	0.165029
$493$	1419.03	0.129634
$494$	2546.04	0.231886
$495$	16198.0	1.47080
$496$	2479.38	0.224450
$497$	0	0
$498$	3648.48	0.328298
$499$	−15652.0	−1.40417	−0.702084	−	0.712094i	$-0.747747\pi$
$499$	−15652.0	−1.40417	−0.702084	+	0.712094i	$0.747747\pi$
$500$	3276.47	0.293057
$501$	1834.54	0.163595
$502$	−19001.9	−1.68943
$503$	14397.9	1.27629	0.638144	−	0.769917i	$-0.279703\pi$
$503$	14397.9	1.27629	0.638144	+	0.769917i	$0.279703\pi$
$504$	0	0
$505$	−10555.5	−0.930124
$506$	10419.8	0.915450
$507$	−2523.38	−0.221040
$508$	−2425.28	−0.211820
$509$	15549.2	1.35404	0.677021	−	0.735963i	$-0.263270\pi$
$509$	15549.2	1.35404	0.677021	+	0.735963i	$0.263270\pi$
$510$	−3813.84	−0.331137
$511$	0	0
$512$	3777.13	0.326029
$513$	−4699.23	−0.404437
$514$	−4611.51	−0.395729
$515$	9934.33	0.850017
$516$	−400.587	−0.0341761
$517$	17420.3	1.48190
$518$	0	0
$519$	2851.99	0.241211
$520$	−2452.87	−0.206857
$521$	−11793.5	−0.991716	−0.495858	−	0.868404i	$-0.665146\pi$
$521$	−11793.5	−0.991716	−0.495858	+	0.868404i	$0.665146\pi$
$522$	2345.14	0.196636
$523$	−11416.3	−0.954496	−0.477248	−	0.878769i	$-0.658366\pi$
$523$	−11416.3	−0.954496	−0.477248	+	0.878769i	$0.658366\pi$
$524$	3458.53	0.288333
$525$	0	0
$526$	6417.38	0.531960
$527$	1650.62	0.136437
$528$	3525.63	0.290593
$529$	−5351.56	−0.439842
$530$	6450.74	0.528683
$531$	4728.08	0.386405
$532$	0	0
$533$	3859.49	0.313646
$534$	−4957.03	−0.401707
$535$	−20976.2	−1.69510
$536$	11097.0	0.894246
$537$	−1944.62	−0.156269
$538$	−13468.7	−1.07932
$539$	0	0
$540$	−4217.66	−0.336110
$541$	−9599.17	−0.762848	−0.381424	−	0.924400i	$-0.624566\pi$
$541$	−9599.17	−0.762848	−0.381424	+	0.924400i	$0.624566\pi$
$542$	13690.0	1.08493
$543$	2780.55	0.219751
$544$	8589.76	0.676990
$545$	37015.7	2.90932
$546$	0	0
$547$	21296.5	1.66467	0.832333	−	0.554275i	$-0.187004\pi$
$547$	21296.5	1.66467	0.832333	+	0.554275i	$0.187004\pi$
$548$	−4683.42	−0.365084
$549$	20168.1	1.56786
$550$	−21974.3	−1.70361
$551$	−1981.41	−0.153196
$552$	1416.07	0.109188
$553$	0	0
$554$	−6216.78	−0.476761
$555$	−9149.59	−0.699781
$556$	4026.09	0.307094
$557$	−9480.36	−0.721177	−0.360589	−	0.932725i	$-0.617424\pi$
$557$	−9480.36	−0.721177	−0.360589	+	0.932725i	$0.617424\pi$
$558$	2727.88	0.206954
$559$	−858.458	−0.0649533
$560$	0	0
$561$	2347.15	0.176643
$562$	−1634.98	−0.122718
$563$	−16015.8	−1.19891	−0.599456	−	0.800408i	$-0.704616\pi$
$563$	−16015.8	−1.19891	−0.599456	+	0.800408i	$0.704616\pi$
$564$	−2205.54	−0.164663
$565$	−18362.8	−1.36730
$566$	−9725.32	−0.722236
$567$	0	0
$568$	−9876.27	−0.729576
$569$	13298.7	0.979811	0.489905	−	0.871776i	$-0.337031\pi$
$569$	13298.7	0.979811	0.489905	+	0.871776i	$0.337031\pi$
$570$	5325.34	0.391322
$571$	−1791.78	−0.131320	−0.0656600	−	0.997842i	$-0.520915\pi$
$571$	−1791.78	−0.131320	−0.0656600	+	0.997842i	$0.520915\pi$
$572$	−1406.34	−0.102800
$573$	−848.157	−0.0618364
$574$	0	0
$575$	−14373.0	−1.04243
$576$	−2154.40	−0.155845
$577$	9068.34	0.654280	0.327140	−	0.944976i	$-0.393915\pi$
$577$	9068.34	0.654280	0.327140	+	0.944976i	$0.393915\pi$
$578$	−7155.38	−0.514921
$579$	1678.93	0.120508
$580$	−1778.36	−0.127314
$581$	0	0
$582$	2748.91	0.195783
$583$	−3969.98	−0.282023
$584$	2492.68	0.176623
$585$	−4394.84	−0.310605
$586$	−10455.7	−0.737068
$587$	−1574.45	−0.110706	−0.0553532	−	0.998467i	$-0.517628\pi$
$587$	−1574.45	−0.110706	−0.0553532	+	0.998467i	$0.517628\pi$
$588$	0	0
$589$	−2304.79	−0.161235
$590$	−11019.3	−0.768914
$591$	4504.51	0.313521
$592$	35181.8	2.44251
$593$	−13871.3	−0.960580	−0.480290	−	0.877110i	$-0.659469\pi$
$593$	−13871.3	−0.960580	−0.480290	+	0.877110i	$0.659469\pi$
$594$	7977.56	0.551049
$595$	0	0
$596$	−7890.93	−0.542324
$597$	−2939.28	−0.201502
$598$	−2827.11	−0.193326
$599$	1448.02	0.0987719	0.0493860	−	0.998780i	$-0.484274\pi$
$599$	1448.02	0.0987719	0.0493860	+	0.998780i	$0.484274\pi$
$600$	−2986.34	−0.203195
$601$	20515.5	1.39242	0.696211	−	0.717837i	$-0.254868\pi$
$601$	20515.5	1.39242	0.696211	+	0.717837i	$0.254868\pi$
$602$	0	0
$603$	19882.6	1.34275
$604$	2646.00	0.178252
$605$	214.228	0.0143960
$606$	−2527.77	−0.169445
$607$	−2676.39	−0.178964	−0.0894821	−	0.995988i	$-0.528521\pi$
$607$	−2676.39	−0.178964	−0.0894821	+	0.995988i	$0.528521\pi$
$608$	−11994.0	−0.800036
$609$	0	0
$610$	−47004.2	−3.11991
$611$	−4726.47	−0.312950
$612$	5249.80	0.346749
$613$	−3187.50	−0.210019	−0.105010	−	0.994471i	$-0.533487\pi$
$613$	−3187.50	−0.210019	−0.105010	+	0.994471i	$0.533487\pi$
$614$	18449.8	1.21266
$615$	8072.56	0.529296
$616$	0	0
$617$	−10116.8	−0.660106	−0.330053	−	0.943962i	$-0.607067\pi$
$617$	−10116.8	−0.660106	−0.330053	+	0.943962i	$0.607067\pi$
$618$	2379.02	0.154851
$619$	−28167.8	−1.82901	−0.914506	−	0.404572i	$-0.867421\pi$
$619$	−28167.8	−1.82901	−0.914506	+	0.404572i	$0.867421\pi$
$620$	−2068.60	−0.133995
$621$	5218.00	0.337184
$622$	−8604.87	−0.554701
$623$	0	0
$624$	−956.575	−0.0613680
$625$	−7076.43	−0.452891
$626$	17041.5	1.08804
$627$	−3277.37	−0.208749
$628$	4719.29	0.299873
$629$	23422.0	1.48473
$630$	0	0
$631$	676.117	0.0426558	0.0213279	−	0.999773i	$-0.493211\pi$
$631$	676.117	0.0426558	0.0213279	+	0.999773i	$0.493211\pi$
$632$	−4644.32	−0.292312
$633$	−3672.38	−0.230591
$634$	−5752.03	−0.360319
$635$	−10870.9	−0.679368
$636$	502.630	0.0313374
$637$	0	0
$638$	3363.71	0.208731
$639$	−17695.4	−1.09549
$640$	27341.1	1.68867
$641$	−7421.14	−0.457281	−0.228641	−	0.973511i	$-0.573428\pi$
$641$	−7421.14	−0.457281	−0.228641	+	0.973511i	$0.573428\pi$
$642$	−5023.25	−0.308803
$643$	−6581.13	−0.403630	−0.201815	−	0.979424i	$-0.564684\pi$
$643$	−6581.13	−0.403630	−0.201815	+	0.979424i	$0.564684\pi$
$644$	0	0
$645$	−1795.56	−0.109613
$646$	−13632.3	−0.830271
$647$	20489.2	1.24500	0.622499	−	0.782621i	$-0.286117\pi$
$647$	20489.2	1.24500	0.622499	+	0.782621i	$0.286117\pi$
$648$	−8755.88	−0.530808
$649$	6781.64	0.410173
$650$	5962.07	0.359772
$651$	0	0
$652$	10914.2	0.655575
$653$	−563.311	−0.0337582	−0.0168791	−	0.999858i	$-0.505373\pi$
$653$	−563.311	−0.0337582	−0.0168791	+	0.999858i	$0.505373\pi$
$654$	8864.31	0.530003
$655$	15502.3	0.924768
$656$	−31040.5	−1.84745
$657$	4466.17	0.265208
$658$	0	0
$659$	−13681.1	−0.808709	−0.404354	−	0.914602i	$-0.632504\pi$
$659$	−13681.1	−0.808709	−0.404354	+	0.914602i	$0.632504\pi$
$660$	−2941.51	−0.173482
$661$	−3239.39	−0.190617	−0.0953084	−	0.995448i	$-0.530384\pi$
$661$	−3239.39	−0.190617	−0.0953084	+	0.995448i	$0.530384\pi$
$662$	−30469.8	−1.78889
$663$	−636.831	−0.0373038
$664$	−12564.0	−0.734301
$665$	0	0
$666$	38708.0	2.25211
$667$	2200.15	0.127721
$668$	5885.42	0.340889
$669$	−6288.96	−0.363446
$670$	−46338.7	−2.67197
$671$	28927.8	1.66430
$672$	0	0
$673$	17976.8	1.02965	0.514824	−	0.857296i	$-0.327857\pi$
$673$	17976.8	1.02965	0.514824	+	0.857296i	$0.327857\pi$
$674$	−13831.2	−0.790443
$675$	−11004.2	−0.627484
$676$	−8095.31	−0.460589
$677$	−30498.8	−1.73141	−0.865705	−	0.500554i	$-0.833130\pi$
$677$	−30498.8	−1.73141	−0.865705	+	0.500554i	$0.833130\pi$
$678$	−4397.41	−0.249088
$679$	0	0
$680$	13133.4	0.740652
$681$	6960.06	0.391645
$682$	3912.69	0.219684
$683$	29319.9	1.64260	0.821299	−	0.570499i	$-0.193250\pi$
$683$	29319.9	1.64260	0.821299	+	0.570499i	$0.193250\pi$
$684$	−7330.38	−0.409772
$685$	−20992.6	−1.17093
$686$	0	0
$687$	978.636	0.0543483
$688$	6904.26	0.382591
$689$	1077.14	0.0595582
$690$	−5913.22	−0.326250
$691$	8500.99	0.468007	0.234003	−	0.972236i	$-0.424817\pi$
$691$	8500.99	0.468007	0.234003	+	0.972236i	$0.424817\pi$
$692$	9149.53	0.502620
$693$	0	0
$694$	19086.9	1.04399
$695$	18046.2	0.984939
$696$	457.133	0.0248959
$697$	−20664.9	−1.12301
$698$	9739.02	0.528120
$699$	−2824.70	−0.152847
$700$	0	0
$701$	27949.8	1.50592	0.752960	−	0.658066i	$-0.228625\pi$
$701$	27949.8	1.50592	0.752960	+	0.658066i	$0.228625\pi$
$702$	−2164.47	−0.116372
$703$	−32704.5	−1.75459
$704$	−3090.13	−0.165431
$705$	−9885.94	−0.528122
$706$	17928.6	0.955741
$707$	0	0
$708$	−858.607	−0.0455769
$709$	−16677.3	−0.883396	−0.441698	−	0.897164i	$-0.645624\pi$
$709$	−16677.3	−0.883396	−0.441698	+	0.897164i	$0.645624\pi$
$710$	41241.3	2.17994
$711$	−8321.28	−0.438920
$712$	17070.1	0.898496
$713$	2559.23	0.134423
$714$	0	0
$715$	−6303.66	−0.329711
$716$	−6238.58	−0.325624
$717$	−1116.70	−0.0581643
$718$	4724.47	0.245565
$719$	−23329.1	−1.21005	−0.605026	−	0.796206i	$-0.706837\pi$
$719$	−23329.1	−1.21005	−0.605026	+	0.796206i	$0.706837\pi$
$720$	35346.1	1.82954
$721$	0	0
$722$	−4584.65	−0.236320
$723$	−8082.74	−0.415768
$724$	8920.35	0.457904
$725$	−4639.88	−0.237684
$726$	51.3020	0.00262259
$727$	−650.985	−0.0332100	−0.0166050	−	0.999862i	$-0.505286\pi$
$727$	−650.985	−0.0332100	−0.0166050	+	0.999862i	$0.505286\pi$
$728$	0	0
$729$	−13635.6	−0.692758
$730$	−10408.9	−0.527742
$731$	4596.45	0.232566
$732$	−3662.48	−0.184930
$733$	−15603.9	−0.786281	−0.393140	−	0.919478i	$-0.628611\pi$
$733$	−15603.9	−0.786281	−0.393140	+	0.919478i	$0.628611\pi$
$734$	7593.95	0.381877
$735$	0	0
$736$	13318.1	0.666999
$737$	28518.2	1.42535
$738$	−34151.6	−1.70344
$739$	13018.2	0.648015	0.324008	−	0.946054i	$-0.394970\pi$
$739$	13018.2	0.648015	0.324008	+	0.946054i	$0.394970\pi$
$740$	−29353.0	−1.45816
$741$	889.218	0.0440840
$742$	0	0
$743$	3851.56	0.190175	0.0950876	−	0.995469i	$-0.469687\pi$
$743$	3851.56	0.190175	0.0950876	+	0.995469i	$0.469687\pi$
$744$	531.740	0.0262023
$745$	−35369.7	−1.73939
$746$	−6899.55	−0.338620
$747$	−22511.0	−1.10259
$748$	7529.95	0.368078
$749$	0	0
$750$	3516.97	0.171229
$751$	−10149.4	−0.493150	−0.246575	−	0.969124i	$-0.579305\pi$
$751$	−10149.4	−0.493150	−0.246575	+	0.969124i	$0.579305\pi$
$752$	38013.2	1.84335
$753$	−6636.49	−0.321178
$754$	−912.642	−0.0440802
$755$	11860.2	0.571706
$756$	0	0
$757$	−11079.2	−0.531943	−0.265972	−	0.963981i	$-0.585693\pi$
$757$	−11079.2	−0.531943	−0.265972	+	0.963981i	$0.585693\pi$
$758$	46907.7	2.24771
$759$	3639.17	0.174036
$760$	−18338.4	−0.875268
$761$	−8719.95	−0.415372	−0.207686	−	0.978196i	$-0.566593\pi$
$761$	−8719.95	−0.415372	−0.207686	+	0.978196i	$0.566593\pi$
$762$	−2603.30	−0.123763
$763$	0	0
$764$	−2720.99	−0.128851
$765$	23531.3	1.11213
$766$	−17900.2	−0.844335
$767$	−1840.00	−0.0866211
$768$	5736.29	0.269519
$769$	−28723.7	−1.34695	−0.673475	−	0.739210i	$-0.735199\pi$
$769$	−28723.7	−1.34695	−0.673475	+	0.739210i	$0.735199\pi$
$770$	0	0
$771$	−1610.59	−0.0752322
$772$	5386.22	0.251107
$773$	−37831.3	−1.76028	−0.880140	−	0.474714i	$-0.842551\pi$
$773$	−37831.3	−1.76028	−0.880140	+	0.474714i	$0.842551\pi$
$774$	7596.26	0.352767
$775$	−5397.14	−0.250156
$776$	−9466.18	−0.437907
$777$	0	0
$778$	−3173.52	−0.146242
$779$	28854.7	1.32712
$780$	798.091	0.0366362
$781$	−25381.1	−1.16288
$782$	15137.2	0.692206
$783$	1684.46	0.0768810
$784$	0	0
$785$	21153.4	0.961779
$786$	3712.39	0.168469
$787$	36246.1	1.64172	0.820861	−	0.571129i	$-0.193494\pi$
$787$	36246.1	1.64172	0.820861	+	0.571129i	$0.193494\pi$
$788$	14451.0	0.653294
$789$	2241.30	0.101131
$790$	19393.7	0.873415
$791$	0	0
$792$	−13357.8	−0.599303
$793$	−7848.69	−0.351469
$794$	32381.2	1.44731
$795$	2252.95	0.100508
$796$	−9429.55	−0.419876
$797$	3544.15	0.157516	0.0787580	−	0.996894i	$-0.474905\pi$
$797$	3544.15	0.157516	0.0787580	+	0.996894i	$0.474905\pi$
$798$	0	0
$799$	25307.0	1.12052
$800$	−28086.4	−1.24126
$801$	30584.7	1.34914
$802$	45271.9	1.99328
$803$	6405.97	0.281521
$804$	−3610.62	−0.158379
$805$	0	0
$806$	−1061.59	−0.0463933
$807$	−4703.99	−0.205190
$808$	8704.65	0.378996
$809$	23499.3	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$809$	23499.3	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$810$	36562.8	1.58603
$811$	31217.4	1.35165	0.675826	−	0.737061i	$-0.263787\pi$
$811$	31217.4	1.35165	0.675826	+	0.737061i	$0.263787\pi$
$812$	0	0
$813$	4781.29	0.206257
$814$	55520.2	2.39064
$815$	48921.2	2.10262
$816$	5121.79	0.219729
$817$	−6418.10	−0.274836
$818$	51463.2	2.19972
$819$	0	0
$820$	25897.7	1.10291
$821$	−17438.2	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$821$	−17438.2	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$822$	−5027.19	−0.213313
$823$	−29386.5	−1.24465	−0.622326	−	0.782758i	$-0.713812\pi$
$823$	−29386.5	−1.24465	−0.622326	+	0.782758i	$0.713812\pi$
$824$	−8192.42	−0.346355
$825$	−7674.63	−0.323874
$826$	0	0
$827$	7745.71	0.325689	0.162845	−	0.986652i	$-0.447933\pi$
$827$	7745.71	0.325689	0.162845	+	0.986652i	$0.447933\pi$
$828$	8139.61	0.341632
$829$	−18385.9	−0.770289	−0.385144	−	0.922856i	$-0.625848\pi$
$829$	−18385.9	−0.770289	−0.385144	+	0.922856i	$0.625848\pi$
$830$	52464.5	2.19406
$831$	−2171.24	−0.0906372
$832$	838.415	0.0349361
$833$	0	0
$834$	4321.61	0.179430
$835$	26380.4	1.09333
$836$	−10514.2	−0.434978
$837$	1959.38	0.0809152
$838$	30566.2	1.26001
$839$	−17331.9	−0.713188	−0.356594	−	0.934259i	$-0.616062\pi$
$839$	−17331.9	−0.713188	−0.356594	+	0.934259i	$0.616062\pi$
$840$	0	0
$841$	−23678.8	−0.970878
$842$	−53280.9	−2.18074
$843$	−571.026	−0.0233300
$844$	−11781.4	−0.480491
$845$	−36285.8	−1.47724
$846$	41823.2	1.69966
$847$	0	0
$848$	−8663.00	−0.350812
$849$	−3396.61	−0.137304
$850$	−31922.8	−1.28817
$851$	36314.9	1.46282
$852$	3213.45	0.129215
$853$	5121.84	0.205590	0.102795	−	0.994703i	$-0.467221\pi$
$853$	5121.84	0.205590	0.102795	+	0.994703i	$0.467221\pi$
$854$	0	0
$855$	−32857.2	−1.31426
$856$	17298.1	0.690699
$857$	19629.6	0.782422	0.391211	−	0.920301i	$-0.372056\pi$
$857$	19629.6	0.782422	0.391211	+	0.920301i	$0.372056\pi$
$858$	−1509.56	−0.0600648
$859$	42818.6	1.70076	0.850379	−	0.526170i	$-0.176372\pi$
$859$	42818.6	1.70076	0.850379	+	0.526170i	$0.176372\pi$
$860$	−5760.38	−0.228404
$861$	0	0
$862$	−12920.6	−0.510531
$863$	32304.6	1.27423	0.637116	−	0.770768i	$-0.280127\pi$
$863$	32304.6	1.27423	0.637116	+	0.770768i	$0.280127\pi$
$864$	10196.5	0.401496
$865$	41011.2	1.61205
$866$	−53380.4	−2.09462
$867$	−2499.05	−0.0978918
$868$	0	0
$869$	−11935.5	−0.465919
$870$	−1908.89	−0.0743880
$871$	−7737.57	−0.301008
$872$	−30525.3	−1.18545
$873$	−16960.7	−0.657539
$874$	−21136.3	−0.818018
$875$	0	0
$876$	−811.045	−0.0312816
$877$	−27583.1	−1.06205	−0.531023	−	0.847357i	$-0.678192\pi$
$877$	−27583.1	−1.06205	−0.531023	+	0.847357i	$0.678192\pi$
$878$	49208.7	1.89147
$879$	−3651.71	−0.140124
$880$	50698.0	1.94208
$881$	−42842.4	−1.63836	−0.819182	−	0.573534i	$-0.805572\pi$
$881$	−42842.4	−1.63836	−0.819182	+	0.573534i	$0.805572\pi$
$882$	0	0
$883$	−1814.31	−0.0691466	−0.0345733	−	0.999402i	$-0.511007\pi$
$883$	−1814.31	−0.0691466	−0.0345733	+	0.999402i	$0.511007\pi$
$884$	−2043.03	−0.0777313
$885$	−3848.56	−0.146178
$886$	10763.2	0.408124
$887$	−35662.9	−1.34999	−0.674996	−	0.737821i	$-0.735855\pi$
$887$	−35662.9	−1.34999	−0.674996	+	0.737821i	$0.735855\pi$
$888$	7545.27	0.285138
$889$	0	0
$890$	−71281.3	−2.68467
$891$	−22501.8	−0.846061
$892$	−20175.8	−0.757325
$893$	−35336.5	−1.32418
$894$	−8470.14	−0.316872
$895$	−27963.3	−1.04437
$896$	0	0
$897$	−987.381	−0.0367533
$898$	18595.3	0.691016
$899$	826.164	0.0306497
$900$	−17165.6	−0.635762
$901$	−5767.31	−0.213249
$902$	−48984.7	−1.80822
$903$	0	0
$904$	15143.0	0.557133
$905$	39983.9	1.46863
$906$	2840.22	0.104150
$907$	−23655.3	−0.865998	−0.432999	−	0.901394i	$-0.642545\pi$
$907$	−23655.3	−0.865998	−0.432999	+	0.901394i	$0.642545\pi$
$908$	22328.7	0.816084
$909$	15596.2	0.569081
$910$	0	0
$911$	20104.4	0.731162	0.365581	−	0.930779i	$-0.380870\pi$
$911$	20104.4	0.731162	0.365581	+	0.930779i	$0.380870\pi$
$912$	−7151.65	−0.259665
$913$	−32288.3	−1.17041
$914$	39619.7	1.43381
$915$	−16416.4	−0.593126
$916$	3139.58	0.113247
$917$	0	0
$918$	11589.3	0.416669
$919$	−10431.2	−0.374420	−0.187210	−	0.982320i	$-0.559945\pi$
$919$	−10431.2	−0.374420	−0.187210	+	0.982320i	$0.559945\pi$
$920$	20362.8	0.729721
$921$	6443.69	0.230539
$922$	19022.7	0.679478
$923$	6886.42	0.245579
$924$	0	0
$925$	−76584.2	−2.72224
$926$	23862.0	0.846818
$927$	−14678.5	−0.520069
$928$	4299.32	0.152082
$929$	13881.5	0.490246	0.245123	−	0.969492i	$-0.421172\pi$
$929$	13881.5	0.490246	0.245123	+	0.969492i	$0.421172\pi$
$930$	−2220.44	−0.0782914
$931$	0	0
$932$	−9061.99	−0.318493
$933$	−3005.29	−0.105454
$934$	−51137.0	−1.79149
$935$	33751.7	1.18053
$936$	3624.23	0.126562
$937$	−32190.6	−1.12233	−0.561164	−	0.827704i	$-0.689646\pi$
$937$	−32190.6	−1.12233	−0.561164	+	0.827704i	$0.689646\pi$
$938$	0	0
$939$	5951.81	0.206848
$940$	−31715.3	−1.10047
$941$	−20400.5	−0.706736	−0.353368	−	0.935484i	$-0.614964\pi$
$941$	−20400.5	−0.706736	−0.353368	+	0.935484i	$0.614964\pi$
$942$	5065.69	0.175211
$943$	−32040.1	−1.10644
$944$	14798.4	0.510220
$945$	0	0
$946$	10895.6	0.374466
$947$	−47368.4	−1.62541	−0.812706	−	0.582673i	$-0.802007\pi$
$947$	−47368.4	−1.62541	−0.812706	+	0.582673i	$0.802007\pi$
$948$	1511.12	0.0517711
$949$	−1738.07	−0.0594522
$950$	44574.3	1.52230
$951$	−2008.92	−0.0685003
$952$	0	0
$953$	−25855.3	−0.878840	−0.439420	−	0.898282i	$-0.644816\pi$
$953$	−25855.3	−0.878840	−0.439420	+	0.898282i	$0.644816\pi$
$954$	−9531.28	−0.323466
$955$	−12196.4	−0.413262
$956$	−3582.50	−0.121199
$957$	1174.79	0.0396819
$958$	−58796.0	−1.98289
$959$	0	0
$960$	1753.64	0.0589567
$961$	961.000	0.0322581
$962$	−15063.8	−0.504859
$963$	30993.3	1.03712
$964$	−25930.4	−0.866350
$965$	24142.8	0.805372
$966$	0	0
$967$	−40511.0	−1.34720	−0.673602	−	0.739094i	$-0.735254\pi$
$967$	−40511.0	−1.34720	−0.673602	+	0.739094i	$0.735254\pi$
$968$	−176.665	−0.00586592
$969$	−4761.14	−0.157843
$970$	39528.8	1.30845
$971$	−18403.8	−0.608247	−0.304123	−	0.952633i	$-0.598364\pi$
$971$	−18403.8	−0.608247	−0.304123	+	0.952633i	$0.598364\pi$
$972$	9433.46	0.311295
$973$	0	0
$974$	51906.3	1.70758
$975$	2082.28	0.0683963
$976$	63124.1	2.07024
$977$	−35081.3	−1.14877	−0.574387	−	0.818584i	$-0.694759\pi$
$977$	−35081.3	−1.14877	−0.574387	+	0.818584i	$0.694759\pi$
$978$	11715.4	0.383043
$979$	43868.6	1.43212
$980$	0	0
$981$	−54692.5	−1.78002
$982$	52532.3	1.70710
$983$	−258.803	−0.00839728	−0.00419864	−	0.999991i	$-0.501336\pi$
$983$	−258.803	−0.00839728	−0.00419864	+	0.999991i	$0.501336\pi$
$984$	−6657.09	−0.215671
$985$	64774.1	2.09531
$986$	4886.56	0.157829
$987$	0	0
$988$	2852.72	0.0918593
$989$	7126.61	0.229134
$990$	55779.3	1.79069
$991$	28178.5	0.903247	0.451624	−	0.892209i	$-0.350845\pi$
$991$	28178.5	0.903247	0.451624	+	0.892209i	$0.350845\pi$
$992$	5001.00	0.160062
$993$	−10641.7	−0.340085
$994$	0	0
$995$	−42266.3	−1.34667
$996$	4087.94	0.130052
$997$	−22666.7	−0.720023	−0.360012	−	0.932948i	$-0.617227\pi$
$997$	−22666.7	−0.720023	−0.360012	+	0.932948i	$0.617227\pi$
$998$	−53899.2	−1.70957
$999$	27803.2	0.880534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.18	✓	23
7.6	odd	2		1519.4.a.i.1.18	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.18	✓	23	1.1	even	1	trivial
1519.4.a.i.1.18	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)

\(f(q)\)	\(=\)	\(q+3.44360 q^{2} +1.20269 q^{3} +3.85839 q^{4} +17.2945 q^{5} +4.14160 q^{6} -14.2621 q^{8} -25.5535 q^{9} +59.5555 q^{10} -36.6522 q^{11} +4.64046 q^{12} +9.94450 q^{13} +20.8000 q^{15} -79.9799 q^{16} -53.2459 q^{17} -87.9961 q^{18} +74.3481 q^{19} +66.7290 q^{20} -126.216 q^{22} -82.5557 q^{23} -17.1529 q^{24} +174.101 q^{25} +34.2449 q^{26} -63.2058 q^{27} -26.6505 q^{29} +71.6270 q^{30} -31.0000 q^{31} -161.322 q^{32} -44.0814 q^{33} -183.358 q^{34} -98.5954 q^{36} -439.883 q^{37} +256.025 q^{38} +11.9602 q^{39} -246.656 q^{40} +388.103 q^{41} -86.3249 q^{43} -141.418 q^{44} -441.937 q^{45} -284.289 q^{46} -475.285 q^{47} -96.1914 q^{48} +599.535 q^{50} -64.0385 q^{51} +38.3697 q^{52} +108.315 q^{53} -217.655 q^{54} -633.884 q^{55} +89.4180 q^{57} -91.7736 q^{58} -185.027 q^{59} +80.2545 q^{60} -789.250 q^{61} -106.752 q^{62} +84.3094 q^{64} +171.986 q^{65} -151.799 q^{66} -778.075 q^{67} -205.443 q^{68} -99.2892 q^{69} +692.485 q^{71} +364.446 q^{72} -174.777 q^{73} -1514.78 q^{74} +209.390 q^{75} +286.864 q^{76} +41.1861 q^{78} +325.641 q^{79} -1383.22 q^{80} +613.928 q^{81} +1336.47 q^{82} +880.935 q^{83} -920.863 q^{85} -297.269 q^{86} -32.0523 q^{87} +522.737 q^{88} -1196.89 q^{89} -1521.85 q^{90} -318.532 q^{92} -37.2835 q^{93} -1636.69 q^{94} +1285.82 q^{95} -194.021 q^{96} +663.731 q^{97} +936.594 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 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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