LMFDB - Embedded newform 1922.4.a.x.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1922,4,Mod(1,1922)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1922, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1922.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1922 = 2 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1922.a (trivial)

Newform invariants

comment:select newform

sage:traces = [32,-64,0] 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:	$113.401671031$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	1922.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -7.26205 q^{3} +4.00000 q^{4} +2.99959 q^{5} +14.5241 q^{6} +27.7754 q^{7} -8.00000 q^{8} +25.7374 q^{9} -5.99919 q^{10} -66.3689 q^{11} -29.0482 q^{12} -24.4628 q^{13} -55.5508 q^{14} -21.7832 q^{15} +16.0000 q^{16} -101.174 q^{17} -51.4747 q^{18} -43.1229 q^{19} +11.9984 q^{20} -201.707 q^{21} +132.738 q^{22} -157.557 q^{23} +58.0964 q^{24} -116.002 q^{25} +48.9257 q^{26} +9.16927 q^{27} +111.102 q^{28} -224.537 q^{29} +43.5664 q^{30} -32.0000 q^{32} +481.975 q^{33} +202.348 q^{34} +83.3150 q^{35} +102.949 q^{36} +57.8032 q^{37} +86.2458 q^{38} +177.650 q^{39} -23.9968 q^{40} -209.659 q^{41} +403.413 q^{42} -197.681 q^{43} -265.476 q^{44} +77.2017 q^{45} +315.114 q^{46} -55.7187 q^{47} -116.193 q^{48} +428.474 q^{49} +232.005 q^{50} +734.730 q^{51} -97.8514 q^{52} +386.731 q^{53} -18.3385 q^{54} -199.080 q^{55} -222.203 q^{56} +313.161 q^{57} +449.074 q^{58} +56.5098 q^{59} -87.1328 q^{60} +160.569 q^{61} +714.866 q^{63} +64.0000 q^{64} -73.3786 q^{65} -963.949 q^{66} -521.100 q^{67} -404.696 q^{68} +1144.19 q^{69} -166.630 q^{70} -407.358 q^{71} -205.899 q^{72} +404.228 q^{73} -115.606 q^{74} +842.415 q^{75} -172.492 q^{76} -1843.43 q^{77} -355.301 q^{78} +1096.41 q^{79} +47.9935 q^{80} -761.497 q^{81} +419.319 q^{82} -1327.61 q^{83} -806.826 q^{84} -303.481 q^{85} +395.362 q^{86} +1630.60 q^{87} +530.951 q^{88} +1589.32 q^{89} -154.403 q^{90} -679.466 q^{91} -630.228 q^{92} +111.437 q^{94} -129.351 q^{95} +232.386 q^{96} -703.502 q^{97} -856.948 q^{98} -1708.16 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) $ 32 * q - 64 * q^2 + 128 * q^4 + 112 * q^7 - 256 * q^8 + 288 * q^9 - 224 * q^14 + 512 * q^16 - 576 * q^18 + 304 * q^19 + 1200 * q^25 + 448 * q^28 - 1024 * q^32 - 272 * q^33 + 1152 * q^36 - 608 * q^38 + 1616 * q^39 + 368 * q^41 + 1360 * q^45 + 784 * q^47 + 3088 * q^49 - 2400 * q^50 - 464 * q^51 - 896 * q^56 - 848 * q^59 + 6992 * q^63 + 2048 * q^64 + 544 * q^66 + 3216 * q^67 - 1664 * q^69 + 2688 * q^71 - 2304 * q^72 + 1216 * q^76 - 3232 * q^78 + 3584 * q^81 - 736 * q^82 + 4960 * q^87 - 2720 * q^90 - 1568 * q^94 - 6816 * q^95 + 7504 * q^97 - 6176 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.26205	−1.39758	−0.698791	−	0.715326i	$-0.746278\pi$
$3$	−7.26205	−1.39758	−0.698791	+	0.715326i	$0.746278\pi$
$4$	4.00000	0.500000
$5$	2.99959	0.268292	0.134146	−	0.990962i	$-0.457171\pi$
$5$	2.99959	0.268292	0.134146	+	0.990962i	$0.457171\pi$
$6$	14.5241	0.988240
$7$	27.7754	1.49973	0.749866	−	0.661590i	$-0.230118\pi$
$7$	27.7754	1.49973	0.749866	+	0.661590i	$0.230118\pi$
$8$	−8.00000	−0.353553
$9$	25.7374	0.953236
$10$	−5.99919	−0.189711
$11$	−66.3689	−1.81918	−0.909590	−	0.415506i	$-0.863604\pi$
$11$	−66.3689	−1.81918	−0.909590	+	0.415506i	$0.863604\pi$
$12$	−29.0482	−0.698791
$13$	−24.4628	−0.521906	−0.260953	−	0.965352i	$-0.584037\pi$
$13$	−24.4628	−0.521906	−0.260953	+	0.965352i	$0.584037\pi$
$14$	−55.5508	−1.06047
$15$	−21.7832	−0.374960
$16$	16.0000	0.250000
$17$	−101.174	−1.44343	−0.721714	−	0.692191i	$-0.756646\pi$
$17$	−101.174	−1.44343	−0.721714	+	0.692191i	$0.756646\pi$
$18$	−51.4747	−0.674040
$19$	−43.1229	−0.520688	−0.260344	−	0.965516i	$-0.583836\pi$
$19$	−43.1229	−0.520688	−0.260344	+	0.965516i	$0.583836\pi$
$20$	11.9984	0.134146
$21$	−201.707	−2.09600
$22$	132.738	1.28635
$23$	−157.557	−1.42839	−0.714194	−	0.699948i	$-0.753206\pi$
$23$	−157.557	−1.42839	−0.714194	+	0.699948i	$0.753206\pi$
$24$	58.0964	0.494120
$25$	−116.002	−0.928019
$26$	48.9257	0.369043
$27$	9.16927	0.0653566
$28$	111.102	0.749866
$29$	−224.537	−1.43777	−0.718887	−	0.695127i	$-0.755348\pi$
$29$	−224.537	−1.43777	−0.718887	+	0.695127i	$0.755348\pi$
$30$	43.5664	0.265137
$31$	0	0
$31$	0	0
$32$	−32.0000	−0.176777
$33$	481.975	2.54245
$34$	202.348	1.02066
$35$	83.3150	0.402366
$36$	102.949	0.476618
$37$	57.8032	0.256832	0.128416	−	0.991720i	$-0.459011\pi$
$37$	57.8032	0.256832	0.128416	+	0.991720i	$0.459011\pi$
$38$	86.2458	0.368182
$39$	177.650	0.729406
$40$	−23.9968	−0.0948555
$41$	−209.659	−0.798617	−0.399308	−	0.916817i	$-0.630750\pi$
$41$	−209.659	−0.798617	−0.399308	+	0.916817i	$0.630750\pi$
$42$	403.413	1.48209
$43$	−197.681	−0.701072	−0.350536	−	0.936549i	$-0.614001\pi$
$43$	−197.681	−0.701072	−0.350536	+	0.936549i	$0.614001\pi$
$44$	−265.476	−0.909590
$45$	77.2017	0.255745
$46$	315.114	1.01002
$47$	−55.7187	−0.172923	−0.0864617	−	0.996255i	$-0.527556\pi$
$47$	−55.7187	−0.172923	−0.0864617	+	0.996255i	$0.527556\pi$
$48$	−116.193	−0.349396
$49$	428.474	1.24920
$50$	232.005	0.656209
$51$	734.730	2.01731
$52$	−97.8514	−0.260953
$53$	386.731	1.00229	0.501147	−	0.865362i	$-0.332911\pi$
$53$	386.731	1.00229	0.501147	+	0.865362i	$0.332911\pi$
$54$	−18.3385	−0.0462141
$55$	−199.080	−0.488071
$56$	−222.203	−0.530235
$57$	313.161	0.727704
$58$	449.074	1.01666
$59$	56.5098	0.124694	0.0623471	−	0.998055i	$-0.480141\pi$
$59$	56.5098	0.124694	0.0623471	+	0.998055i	$0.480141\pi$
$60$	−87.1328	−0.187480
$61$	160.569	0.337028	0.168514	−	0.985699i	$-0.446103\pi$
$61$	160.569	0.337028	0.168514	+	0.985699i	$0.446103\pi$
$62$	0	0
$63$	714.866	1.42960
$64$	64.0000	0.125000
$65$	−73.3786	−0.140023
$66$	−963.949	−1.79779
$67$	−521.100	−0.950187	−0.475093	−	0.879935i	$-0.657586\pi$
$67$	−521.100	−0.950187	−0.475093	+	0.879935i	$0.657586\pi$
$68$	−404.696	−0.721714
$69$	1144.19	1.99629
$70$	−166.630	−0.284516
$71$	−407.358	−0.680908	−0.340454	−	0.940261i	$-0.610581\pi$
$71$	−407.358	−0.680908	−0.340454	+	0.940261i	$0.610581\pi$
$72$	−205.899	−0.337020
$73$	404.228	0.648100	0.324050	−	0.946040i	$-0.394955\pi$
$73$	404.228	0.648100	0.324050	+	0.946040i	$0.394955\pi$
$74$	−115.606	−0.181608
$75$	842.415	1.29698
$76$	−172.492	−0.260344
$77$	−1843.43	−2.72828
$78$	−355.301	−0.515768
$79$	1096.41	1.56146	0.780731	−	0.624868i	$-0.214847\pi$
$79$	1096.41	1.56146	0.780731	+	0.624868i	$0.214847\pi$
$80$	47.9935	0.0670730
$81$	−761.497	−1.04458
$82$	419.319	0.564707
$83$	−1327.61	−1.75571	−0.877857	−	0.478922i	$-0.841028\pi$
$83$	−1327.61	−1.75571	−0.877857	+	0.478922i	$0.841028\pi$
$84$	−806.826	−1.04800
$85$	−303.481	−0.387260
$86$	395.362	0.495733
$87$	1630.60	2.00941
$88$	530.951	0.643177
$89$	1589.32	1.89289	0.946447	−	0.322858i	$-0.104644\pi$
$89$	1589.32	1.89289	0.946447	+	0.322858i	$0.104644\pi$
$90$	−154.403	−0.180839
$91$	−679.466	−0.782718
$92$	−630.228	−0.714194
$93$	0	0
$94$	111.437	0.122275
$95$	−129.351	−0.139696
$96$	232.386	0.247060
$97$	−703.502	−0.736390	−0.368195	−	0.929749i	$-0.620024\pi$
$97$	−703.502	−0.736390	−0.368195	+	0.929749i	$0.620024\pi$
$98$	−856.948	−0.883315
$99$	−1708.16	−1.73411
$100$	−464.010	−0.464010
$101$	−560.989	−0.552678	−0.276339	−	0.961060i	$-0.589121\pi$
$101$	−560.989	−0.552678	−0.276339	+	0.961060i	$0.589121\pi$
$102$	−1469.46	−1.42645
$103$	−1370.52	−1.31108	−0.655542	−	0.755159i	$-0.727560\pi$
$103$	−1370.52	−1.31108	−0.655542	+	0.755159i	$0.727560\pi$
$104$	195.703	0.184521
$105$	−605.038	−0.562339
$106$	−773.463	−0.708730
$107$	1464.29	1.32297	0.661485	−	0.749958i	$-0.269926\pi$
$107$	1464.29	1.32297	0.661485	+	0.749958i	$0.269926\pi$
$108$	36.6771	0.0326783
$109$	686.476	0.603234	0.301617	−	0.953429i	$-0.402474\pi$
$109$	686.476	0.603234	0.301617	+	0.953429i	$0.402474\pi$
$110$	398.160	0.345119
$111$	−419.770	−0.358944
$112$	444.407	0.374933
$113$	−2034.00	−1.69330	−0.846650	−	0.532151i	$-0.821384\pi$
$113$	−2034.00	−1.69330	−0.846650	+	0.532151i	$0.821384\pi$
$114$	−626.321	−0.514565
$115$	−472.607	−0.383225
$116$	−898.147	−0.718887
$117$	−629.609	−0.497499
$118$	−113.020	−0.0881721
$119$	−2810.15	−2.16476
$120$	174.266	0.132568
$121$	3073.84	2.30942
$122$	−321.137	−0.238315
$123$	1522.56	1.11613
$124$	0	0
$125$	−722.910	−0.517272
$126$	−1429.73	−1.01088
$127$	2232.93	1.56016	0.780079	−	0.625681i	$-0.215179\pi$
$127$	2232.93	1.56016	0.780079	+	0.625681i	$0.215179\pi$
$128$	−128.000	−0.0883883
$129$	1435.57	0.979806
$130$	146.757	0.0990112
$131$	−1042.78	−0.695484	−0.347742	−	0.937590i	$-0.613051\pi$
$131$	−1042.78	−0.695484	−0.347742	+	0.937590i	$0.613051\pi$
$132$	1927.90	1.27123
$133$	−1197.76	−0.780893
$134$	1042.20	0.671883
$135$	27.5041	0.0175346
$136$	809.392	0.510329
$137$	−355.363	−0.221611	−0.110805	−	0.993842i	$-0.535343\pi$
$137$	−355.363	−0.221611	−0.110805	+	0.993842i	$0.535343\pi$
$138$	−2288.37	−1.41159
$139$	−2674.58	−1.63205	−0.816025	−	0.578017i	$-0.803827\pi$
$139$	−2674.58	−1.63205	−0.816025	+	0.578017i	$0.803827\pi$
$140$	333.260	0.201183
$141$	404.632	0.241675
$142$	814.716	0.481475
$143$	1623.57	0.949440
$144$	411.798	0.238309
$145$	−673.519	−0.385743
$146$	−808.456	−0.458276
$147$	−3111.60	−1.74585
$148$	231.213	0.128416
$149$	−1064.43	−0.585244	−0.292622	−	0.956228i	$-0.594528\pi$
$149$	−1064.43	−0.585244	−0.292622	+	0.956228i	$0.594528\pi$
$150$	−1684.83	−0.917106
$151$	−1356.25	−0.730930	−0.365465	−	0.930825i	$-0.619090\pi$
$151$	−1356.25	−0.730930	−0.365465	+	0.930825i	$0.619090\pi$
$152$	344.983	0.184091
$153$	−2603.95	−1.37593
$154$	3686.85	1.92919
$155$	0	0
$156$	710.602	0.364703
$157$	1613.65	0.820275	0.410138	−	0.912024i	$-0.365481\pi$
$157$	1613.65	0.820275	0.410138	+	0.912024i	$0.365481\pi$
$158$	−2192.81	−1.10412
$159$	−2808.46	−1.40079
$160$	−95.9870	−0.0474277
$161$	−4376.21	−2.14220
$162$	1522.99	0.738628
$163$	−1598.87	−0.768301	−0.384151	−	0.923270i	$-0.625506\pi$
$163$	−1598.87	−0.768301	−0.384151	+	0.923270i	$0.625506\pi$
$164$	−838.637	−0.399308
$165$	1445.73	0.682120
$166$	2655.22	1.24148
$167$	2651.12	1.22844	0.614220	−	0.789135i	$-0.289471\pi$
$167$	2651.12	1.22844	0.614220	+	0.789135i	$0.289471\pi$
$168$	1613.65	0.741047
$169$	−1598.57	−0.727615
$170$	606.962	0.273834
$171$	−1109.87	−0.496339
$172$	−790.725	−0.350536
$173$	−2594.68	−1.14029	−0.570144	−	0.821545i	$-0.693113\pi$
$173$	−2594.68	−1.14029	−0.570144	+	0.821545i	$0.693113\pi$
$174$	−3261.20	−1.42087
$175$	−3222.02	−1.39178
$176$	−1061.90	−0.454795
$177$	−410.377	−0.174270
$178$	−3178.64	−1.33848
$179$	2047.48	0.854950	0.427475	−	0.904027i	$-0.359403\pi$
$179$	2047.48	0.854950	0.427475	+	0.904027i	$0.359403\pi$
$180$	308.807	0.127873
$181$	924.632	0.379709	0.189855	−	0.981812i	$-0.439198\pi$
$181$	924.632	0.379709	0.189855	+	0.981812i	$0.439198\pi$
$182$	1358.93	0.553465
$183$	−1166.06	−0.471024
$184$	1260.46	0.505011
$185$	173.386	0.0689060
$186$	0	0
$187$	6714.81	2.62586
$188$	−222.875	−0.0864617
$189$	254.680	0.0980173
$190$	258.702	0.0987803
$191$	1597.00	0.604999	0.302499	−	0.953150i	$-0.402179\pi$
$191$	1597.00	0.604999	0.302499	+	0.953150i	$0.402179\pi$
$192$	−464.771	−0.174698
$193$	3367.54	1.25596	0.627981	−	0.778228i	$-0.283881\pi$
$193$	3367.54	1.25596	0.627981	+	0.778228i	$0.283881\pi$
$194$	1407.00	0.520706
$195$	532.879	0.195694
$196$	1713.90	0.624598
$197$	−2036.90	−0.736666	−0.368333	−	0.929694i	$-0.620071\pi$
$197$	−2036.90	−0.736666	−0.368333	+	0.929694i	$0.620071\pi$
$198$	3416.32	1.22620
$199$	2650.71	0.944241	0.472121	−	0.881534i	$-0.343489\pi$
$199$	2650.71	0.944241	0.472121	+	0.881534i	$0.343489\pi$
$200$	928.019	0.328104
$201$	3784.25	1.32796
$202$	1121.98	0.390803
$203$	−6236.61	−2.15628
$204$	2938.92	1.00866
$205$	−628.893	−0.214262
$206$	2741.05	0.927076
$207$	−4055.10	−1.36159
$208$	−391.406	−0.130476
$209$	2862.02	0.947226
$210$	1210.08	0.397634
$211$	428.479	0.139800	0.0698998	−	0.997554i	$-0.477732\pi$
$211$	428.479	0.139800	0.0698998	+	0.997554i	$0.477732\pi$
$212$	1546.93	0.501147
$213$	2958.25	0.951625
$214$	−2928.57	−0.935482
$215$	−592.963	−0.188092
$216$	−73.3542	−0.0231070
$217$	0	0
$218$	−1372.95	−0.426551
$219$	−2935.53	−0.905774
$220$	−796.319	−0.244036
$221$	2475.00	0.753334
$222$	839.539	0.253812
$223$	4299.54	1.29111	0.645557	−	0.763712i	$-0.276625\pi$
$223$	4299.54	1.29111	0.645557	+	0.763712i	$0.276625\pi$
$224$	−888.814	−0.265118
$225$	−2985.60	−0.884622
$226$	4068.00	1.19734
$227$	2670.90	0.780943	0.390471	−	0.920615i	$-0.372312\pi$
$227$	2670.90	0.780943	0.390471	+	0.920615i	$0.372312\pi$
$228$	1252.64	0.363852
$229$	501.400	0.144688	0.0723438	−	0.997380i	$-0.476952\pi$
$229$	501.400	0.144688	0.0723438	+	0.997380i	$0.476952\pi$
$230$	945.214	0.270981
$231$	13387.0	3.81300
$232$	1796.29	0.508330
$233$	15.0863	0.00424179	0.00212090	−	0.999998i	$-0.499325\pi$
$233$	15.0863	0.00424179	0.00212090	+	0.999998i	$0.499325\pi$
$234$	1259.22	0.351785
$235$	−167.133	−0.0463940
$236$	226.039	0.0623471
$237$	−7962.16	−2.18227
$238$	5620.30	1.53071
$239$	3057.75	0.827571	0.413785	−	0.910374i	$-0.364206\pi$
$239$	3057.75	0.827571	0.413785	+	0.910374i	$0.364206\pi$
$240$	−348.531	−0.0937400
$241$	−4446.46	−1.18847	−0.594236	−	0.804290i	$-0.702546\pi$
$241$	−4446.46	−1.18847	−0.594236	+	0.804290i	$0.702546\pi$
$242$	−6147.67	−1.63301
$243$	5282.46	1.39453
$244$	642.274	0.168514
$245$	1285.25	0.335149
$246$	−3045.11	−0.789225
$247$	1054.91	0.271750
$248$	0	0
$249$	9641.18	2.45376
$250$	1445.82	0.365766
$251$	1701.29	0.427826	0.213913	−	0.976853i	$-0.431379\pi$
$251$	1701.29	0.427826	0.213913	+	0.976853i	$0.431379\pi$
$252$	2859.47	0.714799
$253$	10456.9	2.59850
$254$	−4465.85	−1.10320
$255$	2203.89	0.541228
$256$	256.000	0.0625000
$257$	−2072.58	−0.503050	−0.251525	−	0.967851i	$-0.580932\pi$
$257$	−2072.58	−0.503050	−0.251525	+	0.967851i	$0.580932\pi$
$258$	−2871.14	−0.692827
$259$	1605.51	0.385179
$260$	−293.514	−0.0700115
$261$	−5778.99	−1.37054
$262$	2085.57	0.491781
$263$	7708.31	1.80728	0.903640	−	0.428293i	$-0.140885\pi$
$263$	7708.31	1.80728	0.903640	+	0.428293i	$0.140885\pi$
$264$	−3855.80	−0.898893
$265$	1160.04	0.268908
$266$	2395.51	0.552174
$267$	−11541.7	−2.64548
$268$	−2084.40	−0.475093
$269$	−2033.51	−0.460912	−0.230456	−	0.973083i	$-0.574022\pi$
$269$	−2033.51	−0.460912	−0.230456	+	0.973083i	$0.574022\pi$
$270$	−55.0082	−0.0123989
$271$	4714.76	1.05683	0.528416	−	0.848986i	$-0.322786\pi$
$271$	4714.76	1.05683	0.528416	+	0.848986i	$0.322786\pi$
$272$	−1618.78	−0.360857
$273$	4934.32	1.09391
$274$	710.726	0.156703
$275$	7698.96	1.68824
$276$	4576.75	0.998145
$277$	−5348.72	−1.16019	−0.580097	−	0.814548i	$-0.696985\pi$
$277$	−5348.72	−1.16019	−0.580097	+	0.814548i	$0.696985\pi$
$278$	5349.16	1.15403
$279$	0	0
$280$	−666.520	−0.142258
$281$	−4822.38	−1.02377	−0.511884	−	0.859054i	$-0.671052\pi$
$281$	−4822.38	−1.02377	−0.511884	+	0.859054i	$0.671052\pi$
$282$	−809.263	−0.170890
$283$	−2897.52	−0.608621	−0.304311	−	0.952573i	$-0.598426\pi$
$283$	−2897.52	−0.608621	−0.304311	+	0.952573i	$0.598426\pi$
$284$	−1629.43	−0.340454
$285$	939.355	0.195237
$286$	−3247.15	−0.671356
$287$	−5823.38	−1.19771
$288$	−823.596	−0.168510
$289$	5323.17	1.08349
$290$	1347.04	0.272762
$291$	5108.87	1.02917
$292$	1616.91	0.324050
$293$	−69.2462	−0.0138068	−0.00690342	−	0.999976i	$-0.502197\pi$
$293$	−69.2462	−0.0138068	−0.00690342	+	0.999976i	$0.502197\pi$
$294$	6223.20	1.23451
$295$	169.507	0.0334544
$296$	−462.426	−0.0908039
$297$	−608.555	−0.118895
$298$	2128.86	0.413830
$299$	3854.29	0.745483
$300$	3369.66	0.648492
$301$	−5490.68	−1.05142
$302$	2712.51	0.516845
$303$	4073.93	0.772413
$304$	−689.967	−0.130172
$305$	481.641	0.0904218
$306$	5207.90	0.972928
$307$	2756.13	0.512379	0.256190	−	0.966627i	$-0.417533\pi$
$307$	2756.13	0.512379	0.256190	+	0.966627i	$0.417533\pi$
$308$	−7373.70	−1.36414
$309$	9952.81	1.83235
$310$	0	0
$311$	−4520.29	−0.824187	−0.412094	−	0.911142i	$-0.635202\pi$
$311$	−4520.29	−0.824187	−0.412094	+	0.911142i	$0.635202\pi$
$312$	−1421.20	−0.257884
$313$	−9570.98	−1.72838	−0.864192	−	0.503163i	$-0.832170\pi$
$313$	−9570.98	−1.72838	−0.864192	+	0.503163i	$0.832170\pi$
$314$	−3227.30	−0.580022
$315$	2144.31	0.383550
$316$	4385.63	0.780731
$317$	117.492	0.0208171	0.0104086	−	0.999946i	$-0.496687\pi$
$317$	117.492	0.0208171	0.0104086	+	0.999946i	$0.496687\pi$
$318$	5616.92	0.990508
$319$	14902.3	2.61557
$320$	191.974	0.0335365
$321$	−10633.7	−1.84896
$322$	8752.43	1.51476
$323$	4362.92	0.751576
$324$	−3045.99	−0.522289
$325$	2837.75	0.484339
$326$	3197.74	0.543271
$327$	−4985.23	−0.843069
$328$	1677.27	0.282354
$329$	−1547.61	−0.259339
$330$	−2891.46	−0.482332
$331$	−7547.63	−1.25334	−0.626669	−	0.779285i	$-0.715582\pi$
$331$	−7547.63	−1.25334	−0.626669	+	0.779285i	$0.715582\pi$
$332$	−5310.45	−0.877857
$333$	1487.70	0.244822
$334$	−5302.23	−0.868638
$335$	−1563.09	−0.254927
$336$	−3227.30	−0.524000
$337$	−5590.11	−0.903598	−0.451799	−	0.892120i	$-0.649218\pi$
$337$	−5590.11	−0.903598	−0.451799	+	0.892120i	$0.649218\pi$
$338$	3197.14	0.514501
$339$	14771.0	2.36653
$340$	−1213.92	−0.193630
$341$	0	0
$342$	2219.74	0.350964
$343$	2374.08	0.373727
$344$	1581.45	0.247866
$345$	3432.10	0.535588
$346$	5189.36	0.806305
$347$	6904.37	1.06814	0.534072	−	0.845439i	$-0.320661\pi$
$347$	6904.37	1.06814	0.534072	+	0.845439i	$0.320661\pi$
$348$	6522.39	1.00470
$349$	8332.65	1.27804	0.639021	−	0.769189i	$-0.279340\pi$
$349$	8332.65	1.27804	0.639021	+	0.769189i	$0.279340\pi$
$350$	6444.03	0.984137
$351$	−224.306	−0.0341100
$352$	2123.81	0.321589
$353$	4537.42	0.684144	0.342072	−	0.939674i	$-0.388871\pi$
$353$	4537.42	0.684144	0.342072	+	0.939674i	$0.388871\pi$
$354$	820.755	0.123228
$355$	−1221.91	−0.182682
$356$	6357.28	0.946447
$357$	20407.4	3.02542
$358$	−4094.97	−0.604541
$359$	3519.88	0.517471	0.258736	−	0.965948i	$-0.416694\pi$
$359$	3519.88	0.517471	0.258736	+	0.965948i	$0.416694\pi$
$360$	−617.613	−0.0904197
$361$	−4999.41	−0.728884
$362$	−1849.26	−0.268495
$363$	−22322.3	−3.22760
$364$	−2717.86	−0.391359
$365$	1212.52	0.173880
$366$	2332.11	0.333064
$367$	−12612.6	−1.79393	−0.896964	−	0.442103i	$-0.854232\pi$
$367$	−12612.6	−1.79393	−0.896964	+	0.442103i	$0.854232\pi$
$368$	−2520.91	−0.357097
$369$	−5396.08	−0.761270
$370$	−346.772	−0.0487239
$371$	10741.6	1.50317
$372$	0	0
$373$	1996.68	0.277169	0.138585	−	0.990351i	$-0.455745\pi$
$373$	1996.68	0.277169	0.138585	+	0.990351i	$0.455745\pi$
$374$	−13429.6	−1.85676
$375$	5249.80	0.722930
$376$	445.749	0.0611377
$377$	5492.81	0.750382
$378$	−509.361	−0.0693087
$379$	595.707	0.0807372	0.0403686	−	0.999185i	$-0.487147\pi$
$379$	595.707	0.0807372	0.0403686	+	0.999185i	$0.487147\pi$
$380$	−517.405	−0.0698482
$381$	−16215.6	−2.18045
$382$	−3194.00	−0.427799
$383$	−9811.32	−1.30897	−0.654484	−	0.756076i	$-0.727114\pi$
$383$	−9811.32	−1.30897	−0.654484	+	0.756076i	$0.727114\pi$
$384$	929.542	0.123530
$385$	−5529.53	−0.731976
$386$	−6735.08	−0.888100
$387$	−5087.79	−0.668287
$388$	−2814.01	−0.368195
$389$	−9410.62	−1.22657	−0.613287	−	0.789860i	$-0.710153\pi$
$389$	−9410.62	−1.22657	−0.613287	+	0.789860i	$0.710153\pi$
$390$	−1065.76	−0.138376
$391$	15940.7	2.06178
$392$	−3427.79	−0.441657
$393$	7572.74	0.971996
$394$	4073.80	0.520901
$395$	3288.78	0.418927
$396$	−6832.65	−0.867054
$397$	3847.61	0.486414	0.243207	−	0.969974i	$-0.421801\pi$
$397$	3847.61	0.486414	0.243207	+	0.969974i	$0.421801\pi$
$398$	−5301.42	−0.667679
$399$	8698.17	1.09136
$400$	−1856.04	−0.232005
$401$	−1522.21	−0.189565	−0.0947827	−	0.995498i	$-0.530216\pi$
$401$	−1522.21	−0.189565	−0.0947827	+	0.995498i	$0.530216\pi$
$402$	−7568.51	−0.939012
$403$	0	0
$404$	−2243.96	−0.276339
$405$	−2284.18	−0.280252
$406$	12473.2	1.52472
$407$	−3836.34	−0.467224
$408$	−5877.84	−0.713227
$409$	−8174.62	−0.988286	−0.494143	−	0.869381i	$-0.664518\pi$
$409$	−8174.62	−0.988286	−0.494143	+	0.869381i	$0.664518\pi$
$410$	1257.79	0.151506
$411$	2580.66	0.309719
$412$	−5482.09	−0.655542
$413$	1569.58	0.187008
$414$	8110.21	0.962790
$415$	−3982.30	−0.471044
$416$	782.811	0.0922607
$417$	19422.9	2.28092
$418$	−5724.04	−0.669790
$419$	−8671.32	−1.01103	−0.505515	−	0.862818i	$-0.668698\pi$
$419$	−8671.32	−1.01103	−0.505515	+	0.862818i	$0.668698\pi$
$420$	−2420.15	−0.281170
$421$	1211.88	0.140293	0.0701467	−	0.997537i	$-0.477653\pi$
$421$	1211.88	0.140293	0.0701467	+	0.997537i	$0.477653\pi$
$422$	−856.959	−0.0988533
$423$	−1434.05	−0.164837
$424$	−3093.85	−0.354365
$425$	11736.4	1.33953
$426$	−5916.51	−0.672901
$427$	4459.86	0.505451
$428$	5857.14	0.661485
$429$	−11790.5	−1.32692
$430$	1185.93	0.133001
$431$	15543.7	1.73716	0.868579	−	0.495551i	$-0.165034\pi$
$431$	15543.7	1.73716	0.868579	+	0.495551i	$0.165034\pi$
$432$	146.708	0.0163391
$433$	4281.07	0.475138	0.237569	−	0.971371i	$-0.423649\pi$
$433$	4281.07	0.475138	0.237569	+	0.971371i	$0.423649\pi$
$434$	0	0
$435$	4891.13	0.539108
$436$	2745.91	0.301617
$437$	6794.32	0.743744
$438$	5871.05	0.640479
$439$	6616.42	0.719327	0.359664	−	0.933082i	$-0.382891\pi$
$439$	6616.42	0.719327	0.359664	+	0.933082i	$0.382891\pi$
$440$	1592.64	0.172559
$441$	11027.8	1.19078
$442$	−4950.01	−0.532687
$443$	−2299.95	−0.246668	−0.123334	−	0.992365i	$-0.539359\pi$
$443$	−2299.95	−0.246668	−0.123334	+	0.992365i	$0.539359\pi$
$444$	−1679.08	−0.179472
$445$	4767.32	0.507848
$446$	−8599.08	−0.912956
$447$	7729.93	0.817927
$448$	1777.63	0.187466
$449$	−978.282	−0.102824	−0.0514120	−	0.998678i	$-0.516372\pi$
$449$	−978.282	−0.102824	−0.0514120	+	0.998678i	$0.516372\pi$
$450$	5971.20	0.625522
$451$	13914.9	1.45283
$452$	−8136.01	−0.846650
$453$	9849.18	1.02153
$454$	−5341.80	−0.552210
$455$	−2038.12	−0.209997
$456$	−2505.29	−0.257282
$457$	427.099	0.0437174	0.0218587	−	0.999761i	$-0.493042\pi$
$457$	427.099	0.0437174	0.0218587	+	0.999761i	$0.493042\pi$
$458$	−1002.80	−0.102310
$459$	−927.692	−0.0943375
$460$	−1890.43	−0.191612
$461$	−8967.28	−0.905960	−0.452980	−	0.891521i	$-0.649639\pi$
$461$	−8967.28	−0.905960	−0.452980	+	0.891521i	$0.649639\pi$
$462$	−26774.1	−2.69620
$463$	−3002.65	−0.301393	−0.150697	−	0.988580i	$-0.548152\pi$
$463$	−3002.65	−0.301393	−0.150697	+	0.988580i	$0.548152\pi$
$464$	−3592.59	−0.359444
$465$	0	0
$466$	−30.1726	−0.00299940
$467$	3088.95	0.306081	0.153040	−	0.988220i	$-0.451094\pi$
$467$	3088.95	0.306081	0.153040	+	0.988220i	$0.451094\pi$
$468$	−2518.44	−0.248750
$469$	−14473.8	−1.42503
$470$	334.267	0.0328055
$471$	−11718.4	−1.14640
$472$	−452.079	−0.0440860
$473$	13119.9	1.27538
$474$	15924.3	1.54310
$475$	5002.36	0.483209
$476$	−11240.6	−1.08238
$477$	9953.45	0.955424
$478$	−6115.50	−0.585181
$479$	12382.2	1.18112	0.590559	−	0.806994i	$-0.298907\pi$
$479$	12382.2	1.18112	0.590559	+	0.806994i	$0.298907\pi$
$480$	697.063	0.0662842
$481$	−1414.03	−0.134042
$482$	8892.93	0.840377
$483$	31780.3	2.99390
$484$	12295.3	1.15471
$485$	−2110.22	−0.197567
$486$	−10564.9	−0.986079
$487$	18783.9	1.74780	0.873900	−	0.486105i	$-0.161583\pi$
$487$	18783.9	1.74780	0.873900	+	0.486105i	$0.161583\pi$
$488$	−1284.55	−0.119157
$489$	11611.1	1.07376
$490$	−2570.50	−0.236986
$491$	−12311.2	−1.13156	−0.565782	−	0.824555i	$-0.691426\pi$
$491$	−12311.2	−1.13156	−0.565782	+	0.824555i	$0.691426\pi$
$492$	6090.23	0.558066
$493$	22717.3	2.07532
$494$	−2109.82	−0.192156
$495$	−5123.79	−0.465247
$496$	0	0
$497$	−11314.5	−1.02118
$498$	−19282.4	−1.73507
$499$	−12099.7	−1.08548	−0.542742	−	0.839900i	$-0.682614\pi$
$499$	−12099.7	−1.08548	−0.542742	+	0.839900i	$0.682614\pi$
$500$	−2891.64	−0.258636
$501$	−19252.5	−1.71685
$502$	−3402.57	−0.302518
$503$	13826.6	1.22564	0.612822	−	0.790221i	$-0.290034\pi$
$503$	13826.6	1.22564	0.612822	+	0.790221i	$0.290034\pi$
$504$	−5718.93	−0.505439
$505$	−1682.74	−0.148279
$506$	−20913.8	−1.83741
$507$	11608.9	1.01690
$508$	8931.70	0.780079
$509$	−16677.5	−1.45229	−0.726146	−	0.687541i	$-0.758690\pi$
$509$	−16677.5	−1.45229	−0.726146	+	0.687541i	$0.758690\pi$
$510$	−4407.79	−0.382706
$511$	11227.6	0.971977
$512$	−512.000	−0.0441942
$513$	−395.406	−0.0340304
$514$	4145.15	0.355710
$515$	−4111.01	−0.351753
$516$	5742.28	0.489903
$517$	3697.99	0.314579
$518$	−3211.02	−0.272363
$519$	18842.7	1.59365
$520$	587.029	0.0495056
$521$	11591.6	0.974738	0.487369	−	0.873196i	$-0.337957\pi$
$521$	11591.6	0.974738	0.487369	+	0.873196i	$0.337957\pi$
$522$	11558.0	0.969117
$523$	−11916.0	−0.996272	−0.498136	−	0.867099i	$-0.665982\pi$
$523$	−11916.0	−0.996272	−0.498136	+	0.867099i	$0.665982\pi$
$524$	−4171.13	−0.347742
$525$	23398.4	1.94513
$526$	−15416.6	−1.27794
$527$	0	0
$528$	7711.59	0.635614
$529$	12657.2	1.04029
$530$	−2320.07	−0.190146
$531$	1454.41	0.118863
$532$	−4791.03	−0.390446
$533$	5128.86	0.416802
$534$	23083.5	1.87063
$535$	4392.26	0.354942
$536$	4168.80	0.335942
$537$	−14868.9	−1.19486
$538$	4067.02	0.325914
$539$	−28437.4	−2.27251
$540$	110.016	0.00876732
$541$	−7724.63	−0.613878	−0.306939	−	0.951729i	$-0.599305\pi$
$541$	−7724.63	−0.613878	−0.306939	+	0.951729i	$0.599305\pi$
$542$	−9429.52	−0.747293
$543$	−6714.72	−0.530675
$544$	3237.57	0.255165
$545$	2059.15	0.161843
$546$	−9868.63	−0.773514
$547$	9613.26	0.751432	0.375716	−	0.926735i	$-0.377397\pi$
$547$	9613.26	0.751432	0.375716	+	0.926735i	$0.377397\pi$
$548$	−1421.45	−0.110805
$549$	4132.61	0.321267
$550$	−15397.9	−1.19376
$551$	9682.68	0.748632
$552$	−9153.50	−0.705795
$553$	30453.2	2.34177
$554$	10697.4	0.820380
$555$	−1259.14	−0.0963017
$556$	−10698.3	−0.816025
$557$	1862.26	0.141664	0.0708319	−	0.997488i	$-0.477435\pi$
$557$	1862.26	0.141664	0.0708319	+	0.997488i	$0.477435\pi$
$558$	0	0
$559$	4835.84	0.365893
$560$	1333.04	0.100591
$561$	−48763.3	−3.66985
$562$	9644.76	0.723914
$563$	−19241.5	−1.44037	−0.720187	−	0.693780i	$-0.755944\pi$
$563$	−19241.5	−1.44037	−0.720187	+	0.693780i	$0.755944\pi$
$564$	1618.53	0.120837
$565$	−6101.18	−0.454298
$566$	5795.05	0.430360
$567$	−21150.9	−1.56659
$568$	3258.86	0.240737
$569$	17092.3	1.25931	0.629655	−	0.776875i	$-0.283196\pi$
$569$	17092.3	1.25931	0.629655	+	0.776875i	$0.283196\pi$
$570$	−1878.71	−0.138054
$571$	−16223.0	−1.18899	−0.594494	−	0.804100i	$-0.702648\pi$
$571$	−16223.0	−1.18899	−0.594494	+	0.804100i	$0.702648\pi$
$572$	6494.29	0.474720
$573$	−11597.5	−0.845535
$574$	11646.8	0.846910
$575$	18277.0	1.32557
$576$	1647.19	0.119154
$577$	4128.91	0.297901	0.148951	−	0.988845i	$-0.452410\pi$
$577$	4128.91	0.297901	0.148951	+	0.988845i	$0.452410\pi$
$578$	−10646.3	−0.766141
$579$	−24455.2	−1.75531
$580$	−2694.08	−0.192872
$581$	−36875.0	−2.63310
$582$	−10217.7	−0.727730
$583$	−25666.9	−1.82336
$584$	−3233.83	−0.229138
$585$	−1888.57	−0.133475
$586$	138.492	0.00976292
$587$	9701.50	0.682153	0.341076	−	0.940036i	$-0.389209\pi$
$587$	9701.50	0.682153	0.341076	+	0.940036i	$0.389209\pi$
$588$	−12446.4	−0.872927
$589$	0	0
$590$	−339.013	−0.0236558
$591$	14792.1	1.02955
$592$	924.851	0.0642080
$593$	−9888.10	−0.684748	−0.342374	−	0.939564i	$-0.611231\pi$
$593$	−9888.10	−0.684748	−0.342374	+	0.939564i	$0.611231\pi$
$594$	1217.11	0.0840717
$595$	−8429.31	−0.580786
$596$	−4257.71	−0.292622
$597$	−19249.6	−1.31965
$598$	−7708.59	−0.527136
$599$	3291.60	0.224526	0.112263	−	0.993679i	$-0.464190\pi$
$599$	3291.60	0.224526	0.112263	+	0.993679i	$0.464190\pi$
$600$	−6739.32	−0.458553
$601$	−5313.08	−0.360607	−0.180304	−	0.983611i	$-0.557708\pi$
$601$	−5313.08	−0.360607	−0.180304	+	0.983611i	$0.557708\pi$
$602$	10981.4	0.743466
$603$	−13411.7	−0.905752
$604$	−5425.02	−0.365465
$605$	9220.26	0.619598
$606$	−8147.86	−0.546179
$607$	−8296.83	−0.554791	−0.277395	−	0.960756i	$-0.589471\pi$
$607$	−8296.83	−0.554791	−0.277395	+	0.960756i	$0.589471\pi$
$608$	1379.93	0.0920455
$609$	45290.5	3.01357
$610$	−963.281	−0.0639379
$611$	1363.04	0.0902497
$612$	−10415.8	−0.687964
$613$	−18295.9	−1.20549	−0.602745	−	0.797934i	$-0.705926\pi$
$613$	−18295.9	−1.20549	−0.602745	+	0.797934i	$0.705926\pi$
$614$	−5512.25	−0.362307
$615$	4567.05	0.299449
$616$	14747.4	0.964594
$617$	18054.1	1.17801	0.589003	−	0.808131i	$-0.299521\pi$
$617$	18054.1	1.17801	0.589003	+	0.808131i	$0.299521\pi$
$618$	−19905.6	−1.29567
$619$	12744.5	0.827533	0.413767	−	0.910383i	$-0.364213\pi$
$619$	12744.5	0.827533	0.413767	+	0.910383i	$0.364213\pi$
$620$	0	0
$621$	−1444.68	−0.0933545
$622$	9040.58	0.582788
$623$	44144.1	2.83883
$624$	2842.41	0.182351
$625$	12331.9	0.789240
$626$	19142.0	1.22215
$627$	−20784.1	−1.32383
$628$	6454.60	0.410138
$629$	−5848.18	−0.370719
$630$	−4288.62	−0.271211
$631$	7305.73	0.460913	0.230457	−	0.973083i	$-0.425978\pi$
$631$	7305.73	0.460913	0.230457	+	0.973083i	$0.425978\pi$
$632$	−8771.26	−0.552060
$633$	−3111.64	−0.195382
$634$	−234.984	−0.0147199
$635$	6697.87	0.418578
$636$	−11233.8	−0.700395
$637$	−10481.7	−0.651962
$638$	−29804.5	−1.84949
$639$	−10484.3	−0.649066
$640$	−383.948	−0.0237139
$641$	6609.83	0.407290	0.203645	−	0.979045i	$-0.434721\pi$
$641$	6609.83	0.407290	0.203645	+	0.979045i	$0.434721\pi$
$642$	21267.4	1.30741
$643$	25720.6	1.57748	0.788740	−	0.614726i	$-0.210734\pi$
$643$	25720.6	1.57748	0.788740	+	0.614726i	$0.210734\pi$
$644$	−17504.9	−1.07110
$645$	4306.13	0.262874
$646$	−8725.83	−0.531445
$647$	−10699.0	−0.650111	−0.325055	−	0.945695i	$-0.605383\pi$
$647$	−10699.0	−0.650111	−0.325055	+	0.945695i	$0.605383\pi$
$648$	6091.97	0.369314
$649$	−3750.50	−0.226841
$650$	−5675.50	−0.342479
$651$	0	0
$652$	−6395.48	−0.384151
$653$	32452.1	1.94479	0.972395	−	0.233343i	$-0.0749664\pi$
$653$	32452.1	1.94479	0.972395	+	0.233343i	$0.0749664\pi$
$654$	9970.45	0.596140
$655$	−3127.93	−0.186593
$656$	−3354.55	−0.199654
$657$	10403.8	0.617793
$658$	3095.22	0.183380
$659$	10849.9	0.641353	0.320676	−	0.947189i	$-0.396090\pi$
$659$	10849.9	0.641353	0.320676	+	0.947189i	$0.396090\pi$
$660$	5782.91	0.341060
$661$	−16809.2	−0.989113	−0.494557	−	0.869145i	$-0.664670\pi$
$661$	−16809.2	−0.989113	−0.494557	+	0.869145i	$0.664670\pi$
$662$	15095.3	0.886244
$663$	−17973.6	−1.05285
$664$	10620.9	0.620739
$665$	−3592.79	−0.209507
$666$	−2975.40	−0.173115
$667$	35377.4	2.05370
$668$	10604.5	0.614220
$669$	−31223.5	−1.80444
$670$	3126.18	0.180261
$671$	−10656.8	−0.613115
$672$	6454.61	0.370524
$673$	9275.00	0.531241	0.265620	−	0.964078i	$-0.414423\pi$
$673$	9275.00	0.531241	0.265620	+	0.964078i	$0.414423\pi$
$674$	11180.2	0.638940
$675$	−1063.66	−0.0606522
$676$	−6394.28	−0.363807
$677$	−7049.47	−0.400196	−0.200098	−	0.979776i	$-0.564126\pi$
$677$	−7049.47	−0.400196	−0.200098	+	0.979776i	$0.564126\pi$
$678$	−29542.1	−1.67339
$679$	−19540.1	−1.10439
$680$	2427.85	0.136917
$681$	−19396.2	−1.09143
$682$	0	0
$683$	2417.75	0.135450	0.0677251	−	0.997704i	$-0.478426\pi$
$683$	2417.75	0.135450	0.0677251	+	0.997704i	$0.478426\pi$
$684$	−4439.48	−0.248169
$685$	−1065.94	−0.0594564
$686$	−4748.16	−0.264265
$687$	−3641.20	−0.202213
$688$	−3162.90	−0.175268
$689$	−9460.55	−0.523103
$690$	−6864.19	−0.378718
$691$	−6258.42	−0.344546	−0.172273	−	0.985049i	$-0.555111\pi$
$691$	−6258.42	−0.344546	−0.172273	+	0.985049i	$0.555111\pi$
$692$	−10378.7	−0.570144
$693$	−47444.9	−2.60070
$694$	−13808.7	−0.755292
$695$	−8022.65	−0.437865
$696$	−13044.8	−0.710433
$697$	21212.1	1.15275
$698$	−16665.3	−0.903712
$699$	−109.558	−0.00592825
$700$	−12888.1	−0.695890
$701$	515.948	0.0277990	0.0138995	−	0.999903i	$-0.495576\pi$
$701$	515.948	0.0277990	0.0138995	+	0.999903i	$0.495576\pi$
$702$	448.613	0.0241194
$703$	−2492.64	−0.133729
$704$	−4247.61	−0.227398
$705$	1213.73	0.0648394
$706$	−9074.85	−0.483763
$707$	−15581.7	−0.828869
$708$	−1641.51	−0.0871351
$709$	−3084.27	−0.163374	−0.0816870	−	0.996658i	$-0.526031\pi$
$709$	−3084.27	−0.163374	−0.0816870	+	0.996658i	$0.526031\pi$
$710$	2443.82	0.129176
$711$	28218.6	1.48844
$712$	−12714.6	−0.669239
$713$	0	0
$714$	−40814.9	−2.13930
$715$	4870.06	0.254727
$716$	8189.94	0.427475
$717$	−22205.5	−1.15660
$718$	−7039.76	−0.365907
$719$	−5604.23	−0.290685	−0.145342	−	0.989381i	$-0.546428\pi$
$719$	−5604.23	−0.290685	−0.145342	+	0.989381i	$0.546428\pi$
$720$	1235.23	0.0639364
$721$	−38066.9	−1.96627
$722$	9998.83	0.515399
$723$	32290.4	1.66099
$724$	3698.53	0.189855
$725$	26046.8	1.33428
$726$	44644.7	2.28226
$727$	−16591.2	−0.846402	−0.423201	−	0.906036i	$-0.639094\pi$
$727$	−16591.2	−0.846402	−0.423201	+	0.906036i	$0.639094\pi$
$728$	5435.73	0.276733
$729$	−17801.1	−0.904387
$730$	−2425.04	−0.122952
$731$	20000.2	1.01195
$732$	−4664.23	−0.235512
$733$	26698.6	1.34534	0.672670	−	0.739943i	$-0.265147\pi$
$733$	26698.6	1.34534	0.672670	+	0.739943i	$0.265147\pi$
$734$	25225.2	1.26850
$735$	−9333.54	−0.468398
$736$	5041.83	0.252506
$737$	34584.9	1.72856
$738$	10792.2	0.538299
$739$	−3054.04	−0.152022	−0.0760112	−	0.997107i	$-0.524218\pi$
$739$	−3054.04	−0.152022	−0.0760112	+	0.997107i	$0.524218\pi$
$740$	693.545	0.0344530
$741$	−7660.80	−0.379793
$742$	−21483.3	−1.06290
$743$	−1359.08	−0.0671060	−0.0335530	−	0.999437i	$-0.510682\pi$
$743$	−1359.08	−0.0671060	−0.0335530	+	0.999437i	$0.510682\pi$
$744$	0	0
$745$	−3192.85	−0.157016
$746$	−3993.36	−0.195988
$747$	−34169.2	−1.67361
$748$	26859.2	1.31293
$749$	40671.2	1.98410
$750$	−10499.6	−0.511189
$751$	−28992.8	−1.40874	−0.704370	−	0.709834i	$-0.748770\pi$
$751$	−28992.8	−1.40874	−0.704370	+	0.709834i	$0.748770\pi$
$752$	−891.499	−0.0432309
$753$	−12354.8	−0.597921
$754$	−10985.6	−0.530600
$755$	−4068.21	−0.196102
$756$	1018.72	0.0490087
$757$	33394.5	1.60336	0.801680	−	0.597754i	$-0.203940\pi$
$757$	33394.5	1.60336	0.801680	+	0.597754i	$0.203940\pi$
$758$	−1191.41	−0.0570898
$759$	−75938.5	−3.63161
$760$	1034.81	0.0493901
$761$	5289.49	0.251963	0.125982	−	0.992033i	$-0.459792\pi$
$761$	5289.49	0.251963	0.125982	+	0.992033i	$0.459792\pi$
$762$	32431.2	1.54181
$763$	19067.2	0.904689
$764$	6387.99	0.302499
$765$	−7810.80	−0.369150
$766$	19622.6	0.925580
$767$	−1382.39	−0.0650786
$768$	−1859.08	−0.0873489
$769$	29830.7	1.39886	0.699429	−	0.714702i	$-0.253438\pi$
$769$	29830.7	1.39886	0.699429	+	0.714702i	$0.253438\pi$
$770$	11059.1	0.517585
$771$	15051.2	0.703053
$772$	13470.2	0.627981
$773$	18777.3	0.873705	0.436853	−	0.899533i	$-0.356093\pi$
$773$	18777.3	0.873705	0.436853	+	0.899533i	$0.356093\pi$
$774$	10175.6	0.472550
$775$	0	0
$776$	5628.02	0.260353
$777$	−11659.3	−0.538320
$778$	18821.2	0.867319
$779$	9041.12	0.415830
$780$	2131.52	0.0978468
$781$	27035.9	1.23870
$782$	−31881.3	−1.45790
$783$	−2058.84	−0.0939680
$784$	6855.59	0.312299
$785$	4840.29	0.220073
$786$	−15145.5	−0.687305
$787$	−17683.0	−0.800930	−0.400465	−	0.916312i	$-0.631151\pi$
$787$	−17683.0	−0.800930	−0.400465	+	0.916312i	$0.631151\pi$
$788$	−8147.60	−0.368333
$789$	−55978.1	−2.52582
$790$	−6577.55	−0.296226
$791$	−56495.3	−2.53950
$792$	13665.3	0.613100
$793$	−3927.96	−0.175897
$794$	−7695.23	−0.343946
$795$	−8424.25	−0.375820
$796$	10602.8	0.472121
$797$	−27842.5	−1.23743	−0.618714	−	0.785616i	$-0.712346\pi$
$797$	−27842.5	−1.23743	−0.618714	+	0.785616i	$0.712346\pi$
$798$	−17396.3	−0.771709
$799$	5637.28	0.249603
$800$	3712.08	0.164052
$801$	40904.9	1.80438
$802$	3044.43	0.134043
$803$	−26828.2	−1.17901
$804$	15137.0	0.663982
$805$	−13126.9	−0.574734
$806$	0	0
$807$	14767.4	0.644162
$808$	4487.91	0.195401
$809$	18385.8	0.799023	0.399511	−	0.916728i	$-0.369180\pi$
$809$	18385.8	0.799023	0.399511	+	0.916728i	$0.369180\pi$
$810$	4568.36	0.198168
$811$	−43648.0	−1.88988	−0.944938	−	0.327249i	$-0.893878\pi$
$811$	−43648.0	−1.88988	−0.944938	+	0.327249i	$0.893878\pi$
$812$	−24946.4	−1.07814
$813$	−34238.8	−1.47701
$814$	7672.67	0.330377
$815$	−4795.96	−0.206129
$816$	11755.7	0.504328
$817$	8524.59	0.365040
$818$	16349.2	0.698824
$819$	−17487.7	−0.746115
$820$	−2515.57	−0.107131
$821$	−20987.7	−0.892174	−0.446087	−	0.894990i	$-0.647183\pi$
$821$	−20987.7	−0.892174	−0.446087	+	0.894990i	$0.647183\pi$
$822$	−5161.33	−0.219005
$823$	1502.09	0.0636204	0.0318102	−	0.999494i	$-0.489873\pi$
$823$	1502.09	0.0636204	0.0318102	+	0.999494i	$0.489873\pi$
$824$	10964.2	0.463538
$825$	−55910.2	−2.35945
$826$	−3139.17	−0.132234
$827$	35311.1	1.48475	0.742373	−	0.669986i	$-0.233700\pi$
$827$	35311.1	1.48475	0.742373	+	0.669986i	$0.233700\pi$
$828$	−16220.4	−0.680795
$829$	10254.1	0.429603	0.214802	−	0.976658i	$-0.431089\pi$
$829$	10254.1	0.429603	0.214802	+	0.976658i	$0.431089\pi$
$830$	7964.59	0.333078
$831$	38842.7	1.62147
$832$	−1565.62	−0.0652382
$833$	−43350.4	−1.80313
$834$	−38845.9	−1.61286
$835$	7952.27	0.329581
$836$	11448.1	0.473613
$837$	0	0
$838$	17342.6	0.714906
$839$	−3721.24	−0.153124	−0.0765622	−	0.997065i	$-0.524394\pi$
$839$	−3721.24	−0.153124	−0.0765622	+	0.997065i	$0.524394\pi$
$840$	4840.30	0.198817
$841$	26027.8	1.06719
$842$	−2423.77	−0.0992025
$843$	35020.4	1.43080
$844$	1713.92	0.0698998
$845$	−4795.06	−0.195213
$846$	2868.10	0.116557
$847$	85377.1	3.46351
$848$	6187.70	0.250574
$849$	21042.0	0.850599
$850$	−23472.9	−0.947191
$851$	−9107.30	−0.366856
$852$	11833.0	0.475813
$853$	−26083.3	−1.04698	−0.523491	−	0.852031i	$-0.675371\pi$
$853$	−26083.3	−1.04698	−0.523491	+	0.852031i	$0.675371\pi$
$854$	−8919.72	−0.357408
$855$	−3329.16	−0.133164
$856$	−11714.3	−0.467741
$857$	−19018.4	−0.758059	−0.379029	−	0.925385i	$-0.623742\pi$
$857$	−19018.4	−0.758059	−0.379029	+	0.925385i	$0.623742\pi$
$858$	23580.9	0.938275
$859$	23095.2	0.917342	0.458671	−	0.888606i	$-0.348326\pi$
$859$	23095.2	0.917342	0.458671	+	0.888606i	$0.348326\pi$
$860$	−2371.85	−0.0940460
$861$	42289.6	1.67390
$862$	−31087.5	−1.22836
$863$	25483.1	1.00516	0.502580	−	0.864530i	$-0.332384\pi$
$863$	25483.1	1.00516	0.502580	+	0.864530i	$0.332384\pi$
$864$	−293.417	−0.0115535
$865$	−7782.98	−0.305930
$866$	−8562.13	−0.335974
$867$	−38657.1	−1.51426
$868$	0	0
$869$	−72767.4	−2.84058
$870$	−9782.26	−0.381207
$871$	12747.6	0.495908
$872$	−5491.81	−0.213275
$873$	−18106.3	−0.701953
$874$	−13588.6	−0.525907
$875$	−20079.1	−0.775769
$876$	−11742.1	−0.452887
$877$	−32841.6	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$877$	−32841.6	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$878$	−13232.8	−0.508641
$879$	502.869	0.0192962
$880$	−3185.28	−0.122018
$881$	−28768.9	−1.10017	−0.550084	−	0.835109i	$-0.685404\pi$
$881$	−28768.9	−1.10017	−0.550084	+	0.835109i	$0.685404\pi$
$882$	−22055.6	−0.842007
$883$	22367.2	0.852454	0.426227	−	0.904616i	$-0.359842\pi$
$883$	22367.2	0.852454	0.426227	+	0.904616i	$0.359842\pi$
$884$	9900.01	0.376667
$885$	−1230.97	−0.0467553
$886$	4599.90	0.174421
$887$	25158.7	0.952364	0.476182	−	0.879347i	$-0.342020\pi$
$887$	25158.7	0.952364	0.476182	+	0.879347i	$0.342020\pi$
$888$	3358.16	0.126906
$889$	62020.5	2.33982
$890$	−9534.63	−0.359103
$891$	50539.7	1.90027
$892$	17198.2	0.645557
$893$	2402.75	0.0900392
$894$	−15459.9	−0.578362
$895$	6141.62	0.229376
$896$	−3555.25	−0.132559
$897$	−27990.1	−1.04187
$898$	1956.56	0.0727076
$899$	0	0
$900$	−11942.4	−0.442311
$901$	−39127.1	−1.44674
$902$	−27829.7	−1.02730
$903$	39873.6	1.46945
$904$	16272.0	0.598672
$905$	2773.52	0.101873
$906$	−19698.4	−0.722334
$907$	11928.8	0.436701	0.218351	−	0.975870i	$-0.429932\pi$
$907$	11928.8	0.436701	0.218351	+	0.975870i	$0.429932\pi$
$908$	10683.6	0.390471
$909$	−14438.4	−0.526833
$910$	4076.24	0.148490
$911$	−14709.7	−0.534966	−0.267483	−	0.963563i	$-0.586192\pi$
$911$	−14709.7	−0.534966	−0.267483	+	0.963563i	$0.586192\pi$
$912$	5010.57	0.181926
$913$	88112.2	3.19396
$914$	−854.197	−0.0309128
$915$	−3497.70	−0.126372
$916$	2005.60	0.0723438
$917$	−28963.7	−1.04304
$918$	1855.38	0.0667067
$919$	−14697.9	−0.527571	−0.263786	−	0.964581i	$-0.584971\pi$
$919$	−14697.9	−0.527571	−0.263786	+	0.964581i	$0.584971\pi$
$920$	3780.86	0.135490
$921$	−20015.1	−0.716092
$922$	17934.6	0.640611
$923$	9965.13	0.355370
$924$	53548.2	1.90650
$925$	−6705.31	−0.238345
$926$	6005.31	0.213117
$927$	−35273.7	−1.24977
$928$	7185.18	0.254165
$929$	10488.0	0.370400	0.185200	−	0.982701i	$-0.440707\pi$
$929$	10488.0	0.370400	0.185200	+	0.982701i	$0.440707\pi$
$930$	0	0
$931$	−18477.1	−0.650441
$932$	60.3453	0.00212090
$933$	32826.6	1.15187
$934$	−6177.90	−0.216432
$935$	20141.7	0.704496
$936$	5036.87	0.175893
$937$	3135.35	0.109314	0.0546570	−	0.998505i	$-0.482593\pi$
$937$	3135.35	0.109314	0.0546570	+	0.998505i	$0.482593\pi$
$938$	28947.6	1.00765
$939$	69505.0	2.41556
$940$	−668.533	−0.0231970
$941$	−11316.6	−0.392042	−0.196021	−	0.980600i	$-0.562802\pi$
$941$	−11316.6	−0.392042	−0.196021	+	0.980600i	$0.562802\pi$
$942$	23436.8	0.810629
$943$	33033.3	1.14073
$944$	904.157	0.0311735
$945$	763.938	0.0262972
$946$	−26239.8	−0.901828
$947$	30537.4	1.04787	0.523935	−	0.851758i	$-0.324463\pi$
$947$	30537.4	1.04787	0.523935	+	0.851758i	$0.324463\pi$
$948$	−31848.7	−1.09114
$949$	−9888.57	−0.338247
$950$	−10004.7	−0.341680
$951$	−853.235	−0.0290936
$952$	22481.2	0.765357
$953$	22493.5	0.764572	0.382286	−	0.924044i	$-0.375137\pi$
$953$	22493.5	0.764572	0.382286	+	0.924044i	$0.375137\pi$
$954$	−19906.9	−0.675587
$955$	4790.35	0.162316
$956$	12231.0	0.413785
$957$	−108221.	−3.65547
$958$	−24764.3	−0.835177
$959$	−9870.35	−0.332357
$960$	−1394.13	−0.0468700
$961$	0	0
$962$	2828.06	0.0947821
$963$	37686.9	1.26110
$964$	−17785.9	−0.594236
$965$	10101.3	0.336965
$966$	−63560.6	−2.11701
$967$	−25244.5	−0.839514	−0.419757	−	0.907637i	$-0.637885\pi$
$967$	−25244.5	−0.839514	−0.419757	+	0.907637i	$0.637885\pi$
$968$	−24590.7	−0.816503
$969$	−31683.7	−1.05039
$970$	4220.44	0.139701
$971$	7656.12	0.253035	0.126517	−	0.991964i	$-0.459620\pi$
$971$	7656.12	0.253035	0.126517	+	0.991964i	$0.459620\pi$
$972$	21129.8	0.697263
$973$	−74287.6	−2.44764
$974$	−37567.8	−1.23588
$975$	−20607.9	−0.676903
$976$	2569.10	0.0842570
$977$	−6502.46	−0.212929	−0.106465	−	0.994316i	$-0.533953\pi$
$977$	−6502.46	−0.212929	−0.106465	+	0.994316i	$0.533953\pi$
$978$	−23222.1	−0.759266
$979$	−105482.	−3.44352
$980$	5140.99	0.167575
$981$	17668.1	0.575024
$982$	24622.4	0.800136
$983$	36630.4	1.18853	0.594267	−	0.804268i	$-0.297442\pi$
$983$	36630.4	1.18853	0.594267	+	0.804268i	$0.297442\pi$
$984$	−12180.5	−0.394612
$985$	−6109.87	−0.197641
$986$	−45434.6	−1.46748
$987$	11238.8	0.362447
$988$	4219.64	0.135875
$989$	31146.1	1.00140
$990$	10247.6	0.328979
$991$	23059.7	0.739169	0.369584	−	0.929197i	$-0.379500\pi$
$991$	23059.7	0.739169	0.369584	+	0.929197i	$0.379500\pi$
$992$	0	0
$993$	54811.2	1.75164
$994$	22629.1	0.722083
$995$	7951.06	0.253332
$996$	38564.7	1.22688
$997$	21458.2	0.681633	0.340817	−	0.940130i	$-0.389296\pi$
$997$	21458.2	0.681633	0.340817	+	0.940130i	$0.389296\pi$
$998$	24199.4	0.767552
$999$	530.013	0.0167857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1922.4.a.x.1.5	✓	32
31.30	odd	2	inner	1922.4.a.x.1.28	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1922.4.a.x.1.5	✓	32	1.1	even	1	trivial
1922.4.a.x.1.28	yes	32	31.30	odd	2	inner

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:	\( N \)	\(=\)	\( 1922 = 2 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1922.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.26205 q^{3} +4.00000 q^{4} +2.99959 q^{5} +14.5241 q^{6} +27.7754 q^{7} -8.00000 q^{8} +25.7374 q^{9} -5.99919 q^{10} -66.3689 q^{11} -29.0482 q^{12} -24.4628 q^{13} -55.5508 q^{14} -21.7832 q^{15} +16.0000 q^{16} -101.174 q^{17} -51.4747 q^{18} -43.1229 q^{19} +11.9984 q^{20} -201.707 q^{21} +132.738 q^{22} -157.557 q^{23} +58.0964 q^{24} -116.002 q^{25} +48.9257 q^{26} +9.16927 q^{27} +111.102 q^{28} -224.537 q^{29} +43.5664 q^{30} -32.0000 q^{32} +481.975 q^{33} +202.348 q^{34} +83.3150 q^{35} +102.949 q^{36} +57.8032 q^{37} +86.2458 q^{38} +177.650 q^{39} -23.9968 q^{40} -209.659 q^{41} +403.413 q^{42} -197.681 q^{43} -265.476 q^{44} +77.2017 q^{45} +315.114 q^{46} -55.7187 q^{47} -116.193 q^{48} +428.474 q^{49} +232.005 q^{50} +734.730 q^{51} -97.8514 q^{52} +386.731 q^{53} -18.3385 q^{54} -199.080 q^{55} -222.203 q^{56} +313.161 q^{57} +449.074 q^{58} +56.5098 q^{59} -87.1328 q^{60} +160.569 q^{61} +714.866 q^{63} +64.0000 q^{64} -73.3786 q^{65} -963.949 q^{66} -521.100 q^{67} -404.696 q^{68} +1144.19 q^{69} -166.630 q^{70} -407.358 q^{71} -205.899 q^{72} +404.228 q^{73} -115.606 q^{74} +842.415 q^{75} -172.492 q^{76} -1843.43 q^{77} -355.301 q^{78} +1096.41 q^{79} +47.9935 q^{80} -761.497 q^{81} +419.319 q^{82} -1327.61 q^{83} -806.826 q^{84} -303.481 q^{85} +395.362 q^{86} +1630.60 q^{87} +530.951 q^{88} +1589.32 q^{89} -154.403 q^{90} -679.466 q^{91} -630.228 q^{92} +111.437 q^{94} -129.351 q^{95} +232.386 q^{96} -703.502 q^{97} -856.948 q^{98} -1708.16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) \) 32 * q - 64 * q^2 + 128 * q^4 + 112 * q^7 - 256 * q^8 + 288 * q^9 - 224 * q^14 + 512 * q^16 - 576 * q^18 + 304 * q^19 + 1200 * q^25 + 448 * q^28 - 1024 * q^32 - 272 * q^33 + 1152 * q^36 - 608 * q^38 + 1616 * q^39 + 368 * q^41 + 1360 * q^45 + 784 * q^47 + 3088 * q^49 - 2400 * q^50 - 464 * q^51 - 896 * q^56 - 848 * q^59 + 6992 * q^63 + 2048 * q^64 + 544 * q^66 + 3216 * q^67 - 1664 * q^69 + 2688 * q^71 - 2304 * q^72 + 1216 * q^76 - 3232 * q^78 + 3584 * q^81 - 736 * q^82 + 4960 * q^87 - 2720 * q^90 - 1568 * q^94 - 6816 * q^95 + 7504 * q^97 - 6176 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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