LMFDB - Embedded newform 1922.4.a.x.1.2

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			1.2
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)

$f(q)$	$=$	\(q-2.00000 q^{2} -9.30256 q^{3} +4.00000 q^{4} +18.9675 q^{5} +18.6051 q^{6} +21.8384 q^{7} -8.00000 q^{8} +59.5377 q^{9} -37.9350 q^{10} +25.0859 q^{11} -37.2103 q^{12} +15.5621 q^{13} -43.6768 q^{14} -176.446 q^{15} +16.0000 q^{16} -124.008 q^{17} -119.075 q^{18} +64.5895 q^{19} +75.8699 q^{20} -203.153 q^{21} -50.1718 q^{22} -37.7192 q^{23} +74.4205 q^{24} +234.765 q^{25} -31.1241 q^{26} -302.684 q^{27} +87.3535 q^{28} +18.2913 q^{29} +352.892 q^{30} -32.0000 q^{32} -233.363 q^{33} +248.015 q^{34} +414.219 q^{35} +238.151 q^{36} +376.777 q^{37} -129.179 q^{38} -144.767 q^{39} -151.740 q^{40} -58.6849 q^{41} +406.306 q^{42} +475.071 q^{43} +100.344 q^{44} +1129.28 q^{45} +75.4383 q^{46} +27.7443 q^{47} -148.841 q^{48} +133.915 q^{49} -469.531 q^{50} +1153.59 q^{51} +62.2482 q^{52} -112.800 q^{53} +605.368 q^{54} +475.816 q^{55} -174.707 q^{56} -600.848 q^{57} -36.5825 q^{58} -767.752 q^{59} -705.785 q^{60} +133.657 q^{61} +1300.21 q^{63} +64.0000 q^{64} +295.173 q^{65} +466.726 q^{66} +526.105 q^{67} -496.031 q^{68} +350.885 q^{69} -828.438 q^{70} +153.082 q^{71} -476.301 q^{72} +442.646 q^{73} -753.554 q^{74} -2183.92 q^{75} +258.358 q^{76} +547.835 q^{77} +289.534 q^{78} +465.698 q^{79} +303.480 q^{80} +1208.22 q^{81} +117.370 q^{82} +725.948 q^{83} -812.612 q^{84} -2352.11 q^{85} -950.142 q^{86} -170.156 q^{87} -200.687 q^{88} -594.589 q^{89} -2258.56 q^{90} +339.850 q^{91} -150.877 q^{92} -55.4886 q^{94} +1225.10 q^{95} +297.682 q^{96} +1235.23 q^{97} -267.830 q^{98} +1493.56 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−9.30256	−1.79028	−0.895140	−	0.445786i	$-0.852924\pi$
$3$	−9.30256	−1.79028	−0.895140	+	0.445786i	$0.852924\pi$
$4$	4.00000	0.500000
$5$	18.9675	1.69650	0.848252	−	0.529593i	$-0.177655\pi$
$5$	18.9675	1.69650	0.848252	+	0.529593i	$0.177655\pi$
$6$	18.6051	1.26592
$7$	21.8384	1.17916	0.589581	−	0.807709i	$-0.299293\pi$
$7$	21.8384	1.17916	0.589581	+	0.807709i	$0.299293\pi$
$8$	−8.00000	−0.353553
$9$	59.5377	2.20510
$10$	−37.9350	−1.19961
$11$	25.0859	0.687607	0.343803	−	0.939042i	$-0.388285\pi$
$11$	25.0859	0.687607	0.343803	+	0.939042i	$0.388285\pi$
$12$	−37.2103	−0.895140
$13$	15.5621	0.332011	0.166005	−	0.986125i	$-0.446913\pi$
$13$	15.5621	0.332011	0.166005	+	0.986125i	$0.446913\pi$
$14$	−43.6768	−0.833793
$15$	−176.446	−3.03721
$16$	16.0000	0.250000
$17$	−124.008	−1.76919	−0.884596	−	0.466357i	$-0.845566\pi$
$17$	−124.008	−1.76919	−0.884596	+	0.466357i	$0.845566\pi$
$18$	−119.075	−1.55924
$19$	64.5895	0.779886	0.389943	−	0.920839i	$-0.372495\pi$
$19$	64.5895	0.779886	0.389943	+	0.920839i	$0.372495\pi$
$20$	75.8699	0.848252
$21$	−203.153	−2.11103
$22$	−50.1718	−0.486212
$23$	−37.7192	−0.341956	−0.170978	−	0.985275i	$-0.554693\pi$
$23$	−37.7192	−0.341956	−0.170978	+	0.985275i	$0.554693\pi$
$24$	74.4205	0.632959
$25$	234.765	1.87812
$26$	−31.1241	−0.234767
$27$	−302.684	−2.15746
$28$	87.3535	0.589581
$29$	18.2913	0.117124	0.0585621	−	0.998284i	$-0.481348\pi$
$29$	18.2913	0.117124	0.0585621	+	0.998284i	$0.481348\pi$
$30$	352.892	2.14763
$31$	0	0
$31$	0	0
$32$	−32.0000	−0.176777
$33$	−233.363	−1.23101
$34$	248.015	1.25101
$35$	414.219	2.00045
$36$	238.151	1.10255
$37$	376.777	1.67410	0.837051	−	0.547126i	$-0.184278\pi$
$37$	376.777	1.67410	0.837051	+	0.547126i	$0.184278\pi$
$38$	−129.179	−0.551463
$39$	−144.767	−0.594392
$40$	−151.740	−0.599805
$41$	−58.6849	−0.223538	−0.111769	−	0.993734i	$-0.535652\pi$
$41$	−58.6849	−0.223538	−0.111769	+	0.993734i	$0.535652\pi$
$42$	406.306	1.49272
$43$	475.071	1.68483	0.842415	−	0.538830i	$-0.181133\pi$
$43$	475.071	1.68483	0.842415	+	0.538830i	$0.181133\pi$
$44$	100.344	0.343803
$45$	1129.28	3.74096
$46$	75.4383	0.241799
$47$	27.7443	0.0861048	0.0430524	−	0.999073i	$-0.486292\pi$
$47$	27.7443	0.0861048	0.0430524	+	0.999073i	$0.486292\pi$
$48$	−148.841	−0.447570
$49$	133.915	0.390423
$50$	−469.531	−1.32803
$51$	1153.59	3.16735
$52$	62.2482	0.166005
$53$	−112.800	−0.292344	−0.146172	−	0.989259i	$-0.546695\pi$
$53$	−112.800	−0.292344	−0.146172	+	0.989259i	$0.546695\pi$
$54$	605.368	1.52556
$55$	475.816	1.16653
$56$	−174.707	−0.416897
$57$	−600.848	−1.39621
$58$	−36.5825	−0.0828193
$59$	−767.752	−1.69412	−0.847058	−	0.531501i	$-0.821628\pi$
$59$	−767.752	−1.69412	−0.847058	+	0.531501i	$0.821628\pi$
$60$	−705.785	−1.51861
$61$	133.657	0.280541	0.140270	−	0.990113i	$-0.455203\pi$
$61$	133.657	0.280541	0.140270	+	0.990113i	$0.455203\pi$
$62$	0	0
$63$	1300.21	2.60017
$64$	64.0000	0.125000
$65$	295.173	0.563257
$66$	466.726	0.870454
$67$	526.105	0.959313	0.479656	−	0.877456i	$-0.340761\pi$
$67$	526.105	0.959313	0.479656	+	0.877456i	$0.340761\pi$
$68$	−496.031	−0.884596
$69$	350.885	0.612197
$70$	−828.438	−1.41453
$71$	153.082	0.255881	0.127940	−	0.991782i	$-0.459163\pi$
$71$	153.082	0.255881	0.127940	+	0.991782i	$0.459163\pi$
$72$	−476.301	−0.779620
$73$	442.646	0.709695	0.354848	−	0.934924i	$-0.384533\pi$
$73$	442.646	0.709695	0.354848	+	0.934924i	$0.384533\pi$
$74$	−753.554	−1.18377
$75$	−2183.92	−3.36237
$76$	258.358	0.389943
$77$	547.835	0.810800
$78$	289.534	0.420298
$79$	465.698	0.663230	0.331615	−	0.943415i	$-0.392407\pi$
$79$	465.698	0.663230	0.331615	+	0.943415i	$0.392407\pi$
$80$	303.480	0.424126
$81$	1208.22	1.65736
$82$	117.370	0.158065
$83$	725.948	0.960038	0.480019	−	0.877258i	$-0.340630\pi$
$83$	725.948	0.960038	0.480019	+	0.877258i	$0.340630\pi$
$84$	−812.612	−1.05551
$85$	−2352.11	−3.00144
$86$	−950.142	−1.19135
$87$	−170.156	−0.209685
$88$	−200.687	−0.243106
$89$	−594.589	−0.708160	−0.354080	−	0.935215i	$-0.615206\pi$
$89$	−594.589	−0.708160	−0.354080	+	0.935215i	$0.615206\pi$
$90$	−2258.56	−2.64526
$91$	339.850	0.391494
$92$	−150.877	−0.170978
$93$	0	0
$94$	−55.4886	−0.0608853
$95$	1225.10	1.32308
$96$	297.682	0.316480
$97$	1235.23	1.29297	0.646487	−	0.762925i	$-0.276237\pi$
$97$	1235.23	1.29297	0.646487	+	0.762925i	$0.276237\pi$
$98$	−267.830	−0.276071
$99$	1493.56	1.51624
$100$	939.062	0.939062
$101$	−364.786	−0.359382	−0.179691	−	0.983723i	$-0.557510\pi$
$101$	−364.786	−0.359382	−0.179691	+	0.983723i	$0.557510\pi$
$102$	−2307.18	−2.23965
$103$	−205.239	−0.196338	−0.0981691	−	0.995170i	$-0.531299\pi$
$103$	−205.239	−0.196338	−0.0981691	+	0.995170i	$0.531299\pi$
$104$	−124.496	−0.117383
$105$	−3853.30	−3.58137
$106$	225.599	0.206718
$107$	582.226	0.526037	0.263018	−	0.964791i	$-0.415282\pi$
$107$	582.226	0.526037	0.263018	+	0.964791i	$0.415282\pi$
$108$	−1210.74	−1.07873
$109$	−1530.87	−1.34524	−0.672620	−	0.739988i	$-0.734831\pi$
$109$	−1530.87	−1.34524	−0.672620	+	0.739988i	$0.734831\pi$
$110$	−951.632	−0.824860
$111$	−3504.99	−2.99711
$112$	349.414	0.294790
$113$	−1785.59	−1.48650	−0.743249	−	0.669015i	$-0.766716\pi$
$113$	−1785.59	−1.48650	−0.743249	+	0.669015i	$0.766716\pi$
$114$	1201.70	0.987273
$115$	−715.438	−0.580130
$116$	73.1650	0.0585621
$117$	926.529	0.732116
$118$	1535.50	1.19792
$119$	−2708.13	−2.08616
$120$	1411.57	1.07382
$121$	−701.699	−0.527197
$122$	−267.313	−0.198372
$123$	545.920	0.400195
$124$	0	0
$125$	2081.97	1.48974
$126$	−2600.41	−1.83860
$127$	608.657	0.425272	0.212636	−	0.977131i	$-0.431795\pi$
$127$	608.657	0.425272	0.212636	+	0.977131i	$0.431795\pi$
$128$	−128.000	−0.0883883
$129$	−4419.38	−3.01631
$130$	−590.346	−0.398283
$131$	1386.49	0.924717	0.462359	−	0.886693i	$-0.347003\pi$
$131$	1386.49	0.924717	0.462359	+	0.886693i	$0.347003\pi$
$132$	−933.452	−0.615504
$133$	1410.53	0.919612
$134$	−1052.21	−0.678337
$135$	−5741.15	−3.66015
$136$	992.061	0.625504
$137$	1995.24	1.24427	0.622135	−	0.782910i	$-0.286265\pi$
$137$	1995.24	1.24427	0.622135	+	0.782910i	$0.286265\pi$
$138$	−701.770	−0.432889
$139$	2580.90	1.57489	0.787444	−	0.616387i	$-0.211404\pi$
$139$	2580.90	1.57489	0.787444	+	0.616387i	$0.211404\pi$
$140$	1656.88	1.00023
$141$	−258.093	−0.154152
$142$	−306.165	−0.180935
$143$	390.388	0.228293
$144$	952.603	0.551275
$145$	346.939	0.198701
$146$	−885.291	−0.501830
$147$	−1245.75	−0.698966
$148$	1507.11	0.837051
$149$	−669.372	−0.368034	−0.184017	−	0.982923i	$-0.558910\pi$
$149$	−669.372	−0.368034	−0.184017	+	0.982923i	$0.558910\pi$
$150$	4367.84	2.37755
$151$	−2676.03	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$151$	−2676.03	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$152$	−516.716	−0.275731
$153$	−7383.13	−3.90125
$154$	−1095.67	−0.573322
$155$	0	0
$156$	−579.068	−0.297196
$157$	1048.81	0.533149	0.266575	−	0.963814i	$-0.414108\pi$
$157$	1048.81	0.533149	0.266575	+	0.963814i	$0.414108\pi$
$158$	−931.397	−0.468974
$159$	1049.32	0.523377
$160$	−606.959	−0.299902
$161$	−823.726	−0.403222
$162$	−2416.44	−1.17193
$163$	3445.12	1.65548	0.827738	−	0.561114i	$-0.189627\pi$
$163$	3445.12	1.65548	0.827738	+	0.561114i	$0.189627\pi$
$164$	−234.740	−0.111769
$165$	−4426.31	−2.08841
$166$	−1451.90	−0.678849
$167$	2344.03	1.08615	0.543074	−	0.839685i	$-0.317260\pi$
$167$	2344.03	1.08615	0.543074	+	0.839685i	$0.317260\pi$
$168$	1625.22	0.746361
$169$	−1954.82	−0.889769
$170$	4704.23	2.12234
$171$	3845.51	1.71973
$172$	1900.28	0.842415
$173$	4008.38	1.76157	0.880785	−	0.473517i	$-0.157016\pi$
$173$	4008.38	1.76157	0.880785	+	0.473517i	$0.157016\pi$
$174$	340.311	0.148270
$175$	5126.90	2.21461
$176$	401.374	0.171902
$177$	7142.06	3.03294
$178$	1189.18	0.500745
$179$	−1351.91	−0.564506	−0.282253	−	0.959340i	$-0.591082\pi$
$179$	−1351.91	−0.564506	−0.282253	+	0.959340i	$0.591082\pi$
$180$	4517.12	1.87048
$181$	−3079.15	−1.26448	−0.632241	−	0.774772i	$-0.717865\pi$
$181$	−3079.15	−1.26448	−0.632241	+	0.774772i	$0.717865\pi$
$182$	−679.700	−0.276828
$183$	−1243.35	−0.502246
$184$	301.753	0.120900
$185$	7146.51	2.84012
$186$	0	0
$187$	−3110.84	−1.21651
$188$	110.977	0.0430524
$189$	−6610.13	−2.54400
$190$	−2450.20	−0.935559
$191$	−2232.93	−0.845912	−0.422956	−	0.906150i	$-0.639008\pi$
$191$	−2232.93	−0.845912	−0.422956	+	0.906150i	$0.639008\pi$
$192$	−595.364	−0.223785
$193$	−1348.32	−0.502870	−0.251435	−	0.967874i	$-0.580903\pi$
$193$	−1348.32	−0.502870	−0.251435	+	0.967874i	$0.580903\pi$
$194$	−2470.46	−0.914271
$195$	−2745.87	−1.00839
$196$	535.660	0.195211
$197$	−1794.32	−0.648932	−0.324466	−	0.945897i	$-0.605185\pi$
$197$	−1794.32	−0.648932	−0.324466	+	0.945897i	$0.605185\pi$
$198$	−2987.11	−1.07214
$199$	−117.294	−0.0417827	−0.0208914	−	0.999782i	$-0.506650\pi$
$199$	−117.294	−0.0417827	−0.0208914	+	0.999782i	$0.506650\pi$
$200$	−1878.12	−0.664017
$201$	−4894.13	−1.71744
$202$	729.572	0.254121
$203$	399.451	0.138108
$204$	4614.36	1.58367
$205$	−1113.11	−0.379233
$206$	410.479	0.138832
$207$	−2245.71	−0.754047
$208$	248.993	0.0830027
$209$	1620.28	0.536255
$210$	7706.60	2.53241
$211$	−798.694	−0.260589	−0.130295	−	0.991475i	$-0.541592\pi$
$211$	−798.694	−0.260589	−0.130295	+	0.991475i	$0.541592\pi$
$212$	−451.198	−0.146172
$213$	−1424.06	−0.458098
$214$	−1164.45	−0.371964
$215$	9010.90	2.85832
$216$	2421.47	0.762779
$217$	0	0
$218$	3061.75	0.951228
$219$	−4117.74	−1.27055
$220$	1903.26	0.583264
$221$	−1929.81	−0.587391
$222$	7009.98	2.11928
$223$	227.475	0.0683089	0.0341544	−	0.999417i	$-0.489126\pi$
$223$	227.475	0.0683089	0.0341544	+	0.999417i	$0.489126\pi$
$224$	−698.828	−0.208448
$225$	13977.4	4.14145
$226$	3571.18	1.05111
$227$	−332.128	−0.0971106	−0.0485553	−	0.998820i	$-0.515462\pi$
$227$	−332.128	−0.0971106	−0.0485553	+	0.998820i	$0.515462\pi$
$228$	−2403.39	−0.698107
$229$	−583.422	−0.168356	−0.0841781	−	0.996451i	$-0.526826\pi$
$229$	−583.422	−0.168356	−0.0841781	+	0.996451i	$0.526826\pi$
$230$	1430.88	0.410214
$231$	−5096.27	−1.45156
$232$	−146.330	−0.0414096
$233$	−3160.09	−0.888517	−0.444259	−	0.895899i	$-0.646533\pi$
$233$	−3160.09	−0.888517	−0.444259	+	0.895899i	$0.646533\pi$
$234$	−1853.06	−0.517684
$235$	526.240	0.146077
$236$	−3071.01	−0.847058
$237$	−4332.19	−1.18737
$238$	5416.25	1.47514
$239$	1518.84	0.411068	0.205534	−	0.978650i	$-0.434107\pi$
$239$	1518.84	0.411068	0.205534	+	0.978650i	$0.434107\pi$
$240$	−2823.14	−0.759304
$241$	992.640	0.265318	0.132659	−	0.991162i	$-0.457649\pi$
$241$	992.640	0.265318	0.132659	+	0.991162i	$0.457649\pi$
$242$	1403.40	0.372784
$243$	−3067.06	−0.809680
$244$	534.627	0.140270
$245$	2540.03	0.662354
$246$	−1091.84	−0.282981
$247$	1005.15	0.258931
$248$	0	0
$249$	−6753.18	−1.71874
$250$	−4163.95	−1.05340
$251$	−4305.84	−1.08280	−0.541399	−	0.840766i	$-0.682105\pi$
$251$	−4305.84	−1.08280	−0.541399	+	0.840766i	$0.682105\pi$
$252$	5200.83	1.30008
$253$	−946.219	−0.235131
$254$	−1217.31	−0.300713
$255$	21880.7	5.37342
$256$	256.000	0.0625000
$257$	−2895.77	−0.702854	−0.351427	−	0.936215i	$-0.614303\pi$
$257$	−2895.77	−0.702854	−0.351427	+	0.936215i	$0.614303\pi$
$258$	8838.76	2.13286
$259$	8228.20	1.97404
$260$	1180.69	0.281629
$261$	1089.02	0.258270
$262$	−2772.98	−0.653874
$263$	−4412.96	−1.03466	−0.517329	−	0.855787i	$-0.673074\pi$
$263$	−4412.96	−1.03466	−0.517329	+	0.855787i	$0.673074\pi$
$264$	1866.90	0.435227
$265$	−2139.52	−0.495962
$266$	−2821.06	−0.650264
$267$	5531.20	1.26780
$268$	2104.42	0.479656
$269$	8138.00	1.84455	0.922273	−	0.386540i	$-0.126330\pi$
$269$	8138.00	1.84455	0.922273	+	0.386540i	$0.126330\pi$
$270$	11482.3	2.58811
$271$	2388.72	0.535440	0.267720	−	0.963497i	$-0.413730\pi$
$271$	2388.72	0.535440	0.267720	+	0.963497i	$0.413730\pi$
$272$	−1984.12	−0.442298
$273$	−3161.48	−0.700884
$274$	−3990.48	−0.879832
$275$	5889.30	1.29141
$276$	1403.54	0.306098
$277$	−649.874	−0.140964	−0.0704822	−	0.997513i	$-0.522454\pi$
$277$	−649.874	−0.140964	−0.0704822	+	0.997513i	$0.522454\pi$
$278$	−5161.81	−1.11361
$279$	0	0
$280$	−3313.75	−0.707267
$281$	−5883.65	−1.24907	−0.624536	−	0.780996i	$-0.714712\pi$
$281$	−5883.65	−1.24907	−0.624536	+	0.780996i	$0.714712\pi$
$282$	516.187	0.109002
$283$	4596.12	0.965410	0.482705	−	0.875783i	$-0.339654\pi$
$283$	4596.12	0.965410	0.482705	+	0.875783i	$0.339654\pi$
$284$	612.330	0.127940
$285$	−11396.6	−2.36868
$286$	−780.776	−0.161427
$287$	−1281.58	−0.263587
$288$	−1905.21	−0.389810
$289$	10464.9	2.13004
$290$	−693.878	−0.140503
$291$	−11490.8	−2.31478
$292$	1770.58	0.354848
$293$	5288.10	1.05438	0.527192	−	0.849746i	$-0.323245\pi$
$293$	5288.10	1.05438	0.527192	+	0.849746i	$0.323245\pi$
$294$	2491.51	0.494243
$295$	−14562.3	−2.87407
$296$	−3014.22	−0.591884
$297$	−7593.09	−1.48349
$298$	1338.74	0.260239
$299$	−586.988	−0.113533
$300$	−8735.68	−1.68118
$301$	10374.8	1.98669
$302$	5352.07	1.01979
$303$	3393.44	0.643393
$304$	1033.43	0.194972
$305$	2535.13	0.475938
$306$	14766.3	2.75860
$307$	871.911	0.162093	0.0810466	−	0.996710i	$-0.474174\pi$
$307$	871.911	0.162093	0.0810466	+	0.996710i	$0.474174\pi$
$308$	2191.34	0.405400
$309$	1909.25	0.351500
$310$	0	0
$311$	6496.00	1.18442	0.592210	−	0.805784i	$-0.298256\pi$
$311$	6496.00	1.18442	0.592210	+	0.805784i	$0.298256\pi$
$312$	1158.14	0.210149
$313$	3021.30	0.545603	0.272802	−	0.962070i	$-0.412050\pi$
$313$	3021.30	0.545603	0.272802	+	0.962070i	$0.412050\pi$
$314$	−2097.63	−0.376994
$315$	24661.7	4.41120
$316$	1862.79	0.331615
$317$	−2402.80	−0.425725	−0.212862	−	0.977082i	$-0.568279\pi$
$317$	−2402.80	−0.425725	−0.212862	+	0.977082i	$0.568279\pi$
$318$	−2098.65	−0.370083
$319$	458.852	0.0805354
$320$	1213.92	0.212063
$321$	−5416.20	−0.941753
$322$	1647.45	0.285121
$323$	−8009.59	−1.37977
$324$	4832.87	0.828682
$325$	3653.43	0.623557
$326$	−6890.24	−1.17060
$327$	14241.0	2.40835
$328$	469.480	0.0790325
$329$	605.891	0.101531
$330$	8852.62	1.47673
$331$	−7141.13	−1.18584	−0.592919	−	0.805262i	$-0.702024\pi$
$331$	−7141.13	−1.18584	−0.592919	+	0.805262i	$0.702024\pi$
$332$	2903.79	0.480019
$333$	22432.4	3.69156
$334$	−4688.06	−0.768022
$335$	9978.89	1.62748
$336$	−3250.45	−0.527757
$337$	2276.13	0.367918	0.183959	−	0.982934i	$-0.441109\pi$
$337$	2276.13	0.367918	0.183959	+	0.982934i	$0.441109\pi$
$338$	3909.64	0.629162
$339$	16610.6	2.66125
$340$	−9408.45	−1.50072
$341$	0	0
$342$	−7691.02	−1.21603
$343$	−4566.08	−0.718790
$344$	−3800.57	−0.595677
$345$	6655.41	1.03859
$346$	−8016.76	−1.24562
$347$	5773.95	0.893262	0.446631	−	0.894718i	$-0.352624\pi$
$347$	5773.95	0.893262	0.446631	+	0.894718i	$0.352624\pi$
$348$	−680.622	−0.104842
$349$	9491.57	1.45579	0.727897	−	0.685686i	$-0.240498\pi$
$349$	9491.57	1.45579	0.727897	+	0.685686i	$0.240498\pi$
$350$	−10253.8	−1.56597
$351$	−4710.38	−0.716301
$352$	−802.748	−0.121553
$353$	−11012.8	−1.66049	−0.830246	−	0.557397i	$-0.811800\pi$
$353$	−11012.8	−1.66049	−0.830246	+	0.557397i	$0.811800\pi$
$354$	−14284.1	−2.14461
$355$	2903.59	0.434103
$356$	−2378.35	−0.354080
$357$	25192.5	3.73482
$358$	2703.82	0.399166
$359$	11646.8	1.71224	0.856119	−	0.516779i	$-0.172869\pi$
$359$	11646.8	1.71224	0.856119	+	0.516779i	$0.172869\pi$
$360$	−9034.24	−1.32263
$361$	−2687.20	−0.391777
$362$	6158.29	0.894123
$363$	6527.60	0.943829
$364$	1359.40	0.195747
$365$	8395.87	1.20400
$366$	2486.70	0.355142
$367$	2966.12	0.421880	0.210940	−	0.977499i	$-0.432347\pi$
$367$	2966.12	0.421880	0.210940	+	0.977499i	$0.432347\pi$
$368$	−603.507	−0.0854890
$369$	−3493.97	−0.492923
$370$	−14293.0	−2.00827
$371$	−2463.36	−0.344720
$372$	0	0
$373$	1235.77	0.171544	0.0857721	−	0.996315i	$-0.472664\pi$
$373$	1235.77	0.171544	0.0857721	+	0.996315i	$0.472664\pi$
$374$	6221.68	0.860202
$375$	−19367.7	−2.66705
$376$	−221.955	−0.0304426
$377$	284.649	0.0388865
$378$	13220.3	1.79888
$379$	5873.24	0.796011	0.398005	−	0.917383i	$-0.369703\pi$
$379$	5873.24	0.796011	0.398005	+	0.917383i	$0.369703\pi$
$380$	4900.40	0.661540
$381$	−5662.07	−0.761356
$382$	4465.86	0.598150
$383$	4313.92	0.575538	0.287769	−	0.957700i	$-0.407086\pi$
$383$	4313.92	0.575538	0.287769	+	0.957700i	$0.407086\pi$
$384$	1190.73	0.158240
$385$	10391.1	1.37552
$386$	2696.63	0.355583
$387$	28284.6	3.71522
$388$	4940.92	0.646487
$389$	−1530.70	−0.199511	−0.0997554	−	0.995012i	$-0.531806\pi$
$389$	−1530.70	−0.199511	−0.0997554	+	0.995012i	$0.531806\pi$
$390$	5491.73	0.713038
$391$	4677.47	0.604986
$392$	−1071.32	−0.138035
$393$	−12897.9	−1.65550
$394$	3588.63	0.458865
$395$	8833.13	1.12517
$396$	5974.22	0.758121
$397$	−9667.94	−1.22222	−0.611108	−	0.791547i	$-0.709276\pi$
$397$	−9667.94	−1.22222	−0.611108	+	0.791547i	$0.709276\pi$
$398$	234.588	0.0295448
$399$	−13121.5	−1.64636
$400$	3756.25	0.469531
$401$	9816.18	1.22243	0.611217	−	0.791463i	$-0.290680\pi$
$401$	9816.18	1.22243	0.611217	+	0.791463i	$0.290680\pi$
$402$	9788.25	1.21441
$403$	0	0
$404$	−1459.14	−0.179691
$405$	22916.9	2.81172
$406$	−798.903	−0.0976573
$407$	9451.78	1.15112
$408$	−9228.71	−1.11983
$409$	−8740.43	−1.05669	−0.528345	−	0.849030i	$-0.677187\pi$
$409$	−8740.43	−1.05669	−0.528345	+	0.849030i	$0.677187\pi$
$410$	2226.21	0.268158
$411$	−18560.9	−2.22759
$412$	−820.957	−0.0981691
$413$	−16766.5	−1.99764
$414$	4491.42	0.533192
$415$	13769.4	1.62871
$416$	−497.986	−0.0586917
$417$	−24009.0	−2.81949
$418$	−3240.57	−0.379190
$419$	4645.87	0.541684	0.270842	−	0.962624i	$-0.412698\pi$
$419$	4645.87	0.541684	0.270842	+	0.962624i	$0.412698\pi$
$420$	−15413.2	−1.79068
$421$	−11711.8	−1.35582	−0.677910	−	0.735145i	$-0.737114\pi$
$421$	−11711.8	−1.35582	−0.677910	+	0.735145i	$0.737114\pi$
$422$	1597.39	0.184264
$423$	1651.83	0.189870
$424$	902.396	0.103359
$425$	−29112.7	−3.32276
$426$	2848.12	0.323924
$427$	2918.85	0.330803
$428$	2328.90	0.263018
$429$	−3631.61	−0.408708
$430$	−18021.8	−2.02114
$431$	10459.7	1.16897	0.584486	−	0.811404i	$-0.301296\pi$
$431$	10459.7	1.16897	0.584486	+	0.811404i	$0.301296\pi$
$432$	−4842.94	−0.539366
$433$	−9768.26	−1.08414	−0.542070	−	0.840333i	$-0.682359\pi$
$433$	−9768.26	−1.08414	−0.542070	+	0.840333i	$0.682359\pi$
$434$	0	0
$435$	−3227.42	−0.355731
$436$	−6123.49	−0.672620
$437$	−2436.26	−0.266687
$438$	8235.48	0.898416
$439$	−9060.88	−0.985084	−0.492542	−	0.870289i	$-0.663932\pi$
$439$	−9060.88	−0.985084	−0.492542	+	0.870289i	$0.663932\pi$
$440$	−3806.53	−0.412430
$441$	7972.99	0.860921
$442$	3859.63	0.415348
$443$	8595.64	0.921877	0.460938	−	0.887432i	$-0.347513\pi$
$443$	8595.64	0.921877	0.460938	+	0.887432i	$0.347513\pi$
$444$	−14020.0	−1.49855
$445$	−11277.9	−1.20140
$446$	−454.951	−0.0483017
$447$	6226.88	0.658884
$448$	1397.66	0.147395
$449$	−18201.4	−1.91309	−0.956546	−	0.291580i	$-0.905819\pi$
$449$	−18201.4	−1.91309	−0.956546	+	0.291580i	$0.905819\pi$
$450$	−27954.8	−2.92845
$451$	−1472.16	−0.153706
$452$	−7142.37	−0.743249
$453$	24894.0	2.58194
$454$	664.256	0.0686676
$455$	6446.10	0.664171
$456$	4806.78	0.493636
$457$	−9393.41	−0.961499	−0.480750	−	0.876858i	$-0.659635\pi$
$457$	−9393.41	−0.961499	−0.480750	+	0.876858i	$0.659635\pi$
$458$	1166.84	0.119046
$459$	37535.1	3.81697
$460$	−2861.75	−0.290065
$461$	6976.95	0.704878	0.352439	−	0.935835i	$-0.385352\pi$
$461$	6976.95	0.704878	0.352439	+	0.935835i	$0.385352\pi$
$462$	10192.5	1.02641
$463$	−2767.72	−0.277812	−0.138906	−	0.990306i	$-0.544359\pi$
$463$	−2767.72	−0.277812	−0.138906	+	0.990306i	$0.544359\pi$
$464$	292.660	0.0292810
$465$	0	0
$466$	6320.18	0.628277
$467$	−846.452	−0.0838739	−0.0419370	−	0.999120i	$-0.513353\pi$
$467$	−846.452	−0.0838739	−0.0419370	+	0.999120i	$0.513353\pi$
$468$	3706.12	0.366058
$469$	11489.3	1.13119
$470$	−1052.48	−0.103292
$471$	−9756.66	−0.954486
$472$	6142.02	0.598960
$473$	11917.6	1.15850
$474$	8664.38	0.839595
$475$	15163.4	1.46472
$476$	−10832.5	−1.04308
$477$	−6715.82	−0.644647
$478$	−3037.67	−0.290669
$479$	15707.9	1.49836	0.749179	−	0.662368i	$-0.230448\pi$
$479$	15707.9	1.49836	0.749179	+	0.662368i	$0.230448\pi$
$480$	5646.28	0.536909
$481$	5863.42	0.555819
$482$	−1985.28	−0.187608
$483$	7662.76	0.721879
$484$	−2806.79	−0.263598
$485$	23429.2	2.19354
$486$	6134.12	0.572530
$487$	−18323.6	−1.70497	−0.852485	−	0.522752i	$-0.824906\pi$
$487$	−18323.6	−1.70497	−0.852485	+	0.522752i	$0.824906\pi$
$488$	−1069.25	−0.0991862
$489$	−32048.5	−2.96376
$490$	−5080.06	−0.468355
$491$	−9001.40	−0.827347	−0.413674	−	0.910425i	$-0.635755\pi$
$491$	−9001.40	−0.827347	−0.413674	+	0.910425i	$0.635755\pi$
$492$	2183.68	0.200098
$493$	−2268.26	−0.207215
$494$	−2010.29	−0.183092
$495$	28329.0	2.57231
$496$	0	0
$497$	3343.07	0.301725
$498$	13506.4	1.21533
$499$	−35.0587	−0.00314517	−0.00157259	−	0.999999i	$-0.500501\pi$
$499$	−35.0587	−0.00314517	−0.00157259	+	0.999999i	$0.500501\pi$
$500$	8327.90	0.744870
$501$	−21805.5	−1.94451
$502$	8611.68	0.765653
$503$	−12851.8	−1.13923	−0.569614	−	0.821912i	$-0.692907\pi$
$503$	−12851.8	−1.13923	−0.569614	+	0.821912i	$0.692907\pi$
$504$	−10401.7	−0.919299
$505$	−6919.07	−0.609692
$506$	1892.44	0.166263
$507$	18184.9	1.59293
$508$	2434.63	0.212636
$509$	18255.4	1.58970	0.794848	−	0.606809i	$-0.207551\pi$
$509$	18255.4	1.58970	0.794848	+	0.606809i	$0.207551\pi$
$510$	−43761.4	−3.79958
$511$	9666.66	0.836845
$512$	−512.000	−0.0441942
$513$	−19550.2	−1.68258
$514$	5791.55	0.496993
$515$	−3892.87	−0.333088
$516$	−17677.5	−1.50816
$517$	695.991	0.0592063
$518$	−16456.4	−1.39585
$519$	−37288.2	−3.15370
$520$	−2361.38	−0.199141
$521$	17386.3	1.46201	0.731006	−	0.682371i	$-0.239051\pi$
$521$	17386.3	1.46201	0.731006	+	0.682371i	$0.239051\pi$
$522$	−2178.04	−0.182625
$523$	5237.42	0.437890	0.218945	−	0.975737i	$-0.429739\pi$
$523$	5237.42	0.437890	0.218945	+	0.975737i	$0.429739\pi$
$524$	5545.95	0.462359
$525$	−47693.3	−3.96477
$526$	8825.93	0.731613
$527$	0	0
$528$	−3733.81	−0.307752
$529$	−10744.3	−0.883066
$530$	4279.05	0.350698
$531$	−45710.2	−3.73569
$532$	5642.12	0.459806
$533$	−913.259	−0.0742169
$534$	−11062.4	−0.896473
$535$	11043.4	0.892423
$536$	−4208.84	−0.339168
$537$	12576.2	1.01062
$538$	−16276.0	−1.30429
$539$	3359.38	0.268457
$540$	−22964.6	−1.83007
$541$	20273.6	1.61115	0.805575	−	0.592494i	$-0.201857\pi$
$541$	20273.6	1.61115	0.805575	+	0.592494i	$0.201857\pi$
$542$	−4777.44	−0.378614
$543$	28644.0	2.26377
$544$	3968.25	0.312752
$545$	−29036.8	−2.28220
$546$	6322.96	0.495600
$547$	−14019.5	−1.09585	−0.547927	−	0.836526i	$-0.684583\pi$
$547$	−14019.5	−1.09585	−0.547927	+	0.836526i	$0.684583\pi$
$548$	7980.97	0.622135
$549$	7957.61	0.618620
$550$	−11778.6	−0.913165
$551$	1181.42	0.0913435
$552$	−2807.08	−0.216444
$553$	10170.1	0.782055
$554$	1299.75	0.0996769
$555$	−66480.9	−5.08460
$556$	10323.6	0.787444
$557$	−7755.85	−0.589993	−0.294996	−	0.955498i	$-0.595318\pi$
$557$	−7755.85	−0.589993	−0.294996	+	0.955498i	$0.595318\pi$
$558$	0	0
$559$	7393.08	0.559381
$560$	6627.51	0.500113
$561$	28938.8	2.17789
$562$	11767.3	0.883227
$563$	−15219.7	−1.13931	−0.569657	−	0.821882i	$-0.692924\pi$
$563$	−15219.7	−1.13931	−0.569657	+	0.821882i	$0.692924\pi$
$564$	−1032.37	−0.0770758
$565$	−33868.2	−2.52185
$566$	−9192.24	−0.682648
$567$	26385.5	1.95430
$568$	−1224.66	−0.0904676
$569$	8097.99	0.596635	0.298318	−	0.954467i	$-0.403574\pi$
$569$	8097.99	0.596635	0.298318	+	0.954467i	$0.403574\pi$
$570$	22793.1	1.67491
$571$	−18568.4	−1.36088	−0.680441	−	0.732803i	$-0.738212\pi$
$571$	−18568.4	−1.36088	−0.680441	+	0.732803i	$0.738212\pi$
$572$	1561.55	0.114146
$573$	20772.0	1.51442
$574$	2563.17	0.186384
$575$	−8855.16	−0.642236
$576$	3810.41	0.275637
$577$	20991.7	1.51455	0.757274	−	0.653097i	$-0.226531\pi$
$577$	20991.7	1.51455	0.757274	+	0.653097i	$0.226531\pi$
$578$	−20929.8	−1.50617
$579$	12542.8	0.900278
$580$	1387.76	0.0993507
$581$	15853.5	1.13204
$582$	22981.6	1.63680
$583$	−2829.68	−0.201017
$584$	−3541.16	−0.250915
$585$	17573.9	1.24204
$586$	−10576.2	−0.745562
$587$	−17.6164	−0.00123868	−0.000619342	−	1.00000i	$-0.500197\pi$
$587$	−17.6164	−0.00123868	−0.000619342		1.00000i	$0.500197\pi$
$588$	−4983.01	−0.349483
$589$	0	0
$590$	29124.7	2.03228
$591$	16691.7	1.16177
$592$	6028.43	0.418525
$593$	−78.1858	−0.00541435	−0.00270717	−	0.999996i	$-0.500862\pi$
$593$	−78.1858	−0.00541435	−0.00270717	+	0.999996i	$0.500862\pi$
$594$	15186.2	1.04898
$595$	−51366.4	−3.53919
$596$	−2677.49	−0.184017
$597$	1091.14	0.0748027
$598$	1173.98	0.0802800
$599$	−9483.09	−0.646859	−0.323430	−	0.946252i	$-0.604836\pi$
$599$	−9483.09	−0.646859	−0.323430	+	0.946252i	$0.604836\pi$
$600$	17471.4	1.18878
$601$	−926.229	−0.0628647	−0.0314323	−	0.999506i	$-0.510007\pi$
$601$	−926.229	−0.0628647	−0.0314323	+	0.999506i	$0.510007\pi$
$602$	−20749.6	−1.40480
$603$	31323.1	2.11538
$604$	−10704.1	−0.721101
$605$	−13309.5	−0.894391
$606$	−6786.89	−0.454948
$607$	17485.7	1.16923	0.584616	−	0.811310i	$-0.301245\pi$
$607$	17485.7	1.16923	0.584616	+	0.811310i	$0.301245\pi$
$608$	−2066.86	−0.137866
$609$	−3715.92	−0.247252
$610$	−5070.26	−0.336539
$611$	431.759	0.0285877
$612$	−29532.5	−1.95062
$613$	25714.1	1.69426	0.847131	−	0.531383i	$-0.178328\pi$
$613$	25714.1	1.69426	0.847131	+	0.531383i	$0.178328\pi$
$614$	−1743.82	−0.114617
$615$	10354.7	0.678932
$616$	−4382.68	−0.286661
$617$	11716.5	0.764487	0.382244	−	0.924062i	$-0.375152\pi$
$617$	11716.5	0.764487	0.382244	+	0.924062i	$0.375152\pi$
$618$	−3818.50	−0.248548
$619$	1989.80	0.129203	0.0646017	−	0.997911i	$-0.479422\pi$
$619$	1989.80	0.129203	0.0646017	+	0.997911i	$0.479422\pi$
$620$	0	0
$621$	11417.0	0.737758
$622$	−12992.0	−0.837511
$623$	−12984.9	−0.835036
$624$	−2316.27	−0.148598
$625$	10144.1	0.649225
$626$	−6042.59	−0.385800
$627$	−15072.8	−0.960047
$628$	4195.26	0.266575
$629$	−46723.2	−2.96181
$630$	−49323.3	−3.11919
$631$	4620.99	0.291535	0.145768	−	0.989319i	$-0.453435\pi$
$631$	4620.99	0.291535	0.145768	+	0.989319i	$0.453435\pi$
$632$	−3725.59	−0.234487
$633$	7429.90	0.466528
$634$	4805.60	0.301033
$635$	11544.7	0.721476
$636$	4197.30	0.261688
$637$	2083.99	0.129625
$638$	−917.704	−0.0569471
$639$	9114.17	0.564243
$640$	−2427.84	−0.149951
$641$	−8759.36	−0.539741	−0.269870	−	0.962897i	$-0.586981\pi$
$641$	−8759.36	−0.539741	−0.269870	+	0.962897i	$0.586981\pi$
$642$	10832.4	0.665920
$643$	−7774.25	−0.476806	−0.238403	−	0.971166i	$-0.576624\pi$
$643$	−7774.25	−0.476806	−0.238403	+	0.971166i	$0.576624\pi$
$644$	−3294.90	−0.201611
$645$	−83824.5	−5.11719
$646$	16019.2	0.975644
$647$	−2402.26	−0.145970	−0.0729851	−	0.997333i	$-0.523253\pi$
$647$	−2402.26	−0.145970	−0.0729851	+	0.997333i	$0.523253\pi$
$648$	−9665.75	−0.585967
$649$	−19259.7	−1.16489
$650$	−7306.87	−0.440921
$651$	0	0
$652$	13780.5	0.827738
$653$	−15093.7	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$653$	−15093.7	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$654$	−28482.1	−1.70296
$655$	26298.2	1.56879
$656$	−938.959	−0.0558844
$657$	26354.1	1.56495
$658$	−1211.78	−0.0717936
$659$	−26356.7	−1.55798	−0.778991	−	0.627035i	$-0.784268\pi$
$659$	−26356.7	−1.55798	−0.778991	+	0.627035i	$0.784268\pi$
$660$	−17705.2	−1.04420
$661$	18450.5	1.08569	0.542846	−	0.839832i	$-0.317347\pi$
$661$	18450.5	1.08569	0.542846	+	0.839832i	$0.317347\pi$
$662$	14282.3	0.838513
$663$	17952.2	1.05159
$664$	−5807.58	−0.339425
$665$	26754.2	1.56013
$666$	−44864.8	−2.61033
$667$	−689.931	−0.0400513
$668$	9376.13	0.543074
$669$	−2116.10	−0.122292
$670$	−19957.8	−1.15080
$671$	3352.90	0.192902
$672$	6500.89	0.373181
$673$	12446.1	0.712873	0.356436	−	0.934320i	$-0.383992\pi$
$673$	12446.1	0.712873	0.356436	+	0.934320i	$0.383992\pi$
$674$	−4552.25	−0.260158
$675$	−71059.7	−4.05198
$676$	−7819.29	−0.444884
$677$	−3150.64	−0.178861	−0.0894304	−	0.995993i	$-0.528505\pi$
$677$	−3150.64	−0.178861	−0.0894304	+	0.995993i	$0.528505\pi$
$678$	−33221.2	−1.88179
$679$	26975.4	1.52463
$680$	18816.9	1.06117
$681$	3089.64	0.173855
$682$	0	0
$683$	14458.8	0.810028	0.405014	−	0.914310i	$-0.367267\pi$
$683$	14458.8	0.810028	0.405014	+	0.914310i	$0.367267\pi$
$684$	15382.0	0.859863
$685$	37844.7	2.11091
$686$	9132.16	0.508261
$687$	5427.32	0.301405
$688$	7601.14	0.421207
$689$	−1755.39	−0.0970612
$690$	−13310.8	−0.734397
$691$	31082.0	1.71116	0.855582	−	0.517668i	$-0.173200\pi$
$691$	31082.0	1.71116	0.855582	+	0.517668i	$0.173200\pi$
$692$	16033.5	0.880785
$693$	32616.8	1.78789
$694$	−11547.9	−0.631631
$695$	48953.2	2.67180
$696$	1361.24	0.0741348
$697$	7277.38	0.395481
$698$	−18983.1	−1.02940
$699$	29397.0	1.59069
$700$	20507.6	1.10731
$701$	−15840.0	−0.853449	−0.426725	−	0.904382i	$-0.640333\pi$
$701$	−15840.0	−0.853449	−0.426725	+	0.904382i	$0.640333\pi$
$702$	9420.77	0.506501
$703$	24335.8	1.30561
$704$	1605.50	0.0859509
$705$	−4895.38	−0.261519
$706$	22025.7	1.17415
$707$	−7966.33	−0.423769
$708$	28568.3	1.51647
$709$	−24026.9	−1.27271	−0.636354	−	0.771397i	$-0.719558\pi$
$709$	−24026.9	−1.27271	−0.636354	+	0.771397i	$0.719558\pi$
$710$	−5807.18	−0.306957
$711$	27726.6	1.46249
$712$	4756.71	0.250373
$713$	0	0
$714$	−50385.1	−2.64091
$715$	7404.68	0.387300
$716$	−5407.65	−0.282253
$717$	−14129.1	−0.735927
$718$	−23293.6	−1.21073
$719$	25697.5	1.33290	0.666450	−	0.745550i	$-0.267813\pi$
$719$	25697.5	1.33290	0.666450	+	0.745550i	$0.267813\pi$
$720$	18068.5	0.935240
$721$	−4482.10	−0.231515
$722$	5374.40	0.277028
$723$	−9234.10	−0.474993
$724$	−12316.6	−0.632241
$725$	4294.15	0.219974
$726$	−13055.2	−0.667388
$727$	−28437.5	−1.45074	−0.725369	−	0.688360i	$-0.758331\pi$
$727$	−28437.5	−1.45074	−0.725369	+	0.688360i	$0.758331\pi$
$728$	−2718.80	−0.138414
$729$	−4090.35	−0.207811
$730$	−16791.7	−0.851357
$731$	−58912.5	−2.98079
$732$	−4973.40	−0.251123
$733$	5168.75	0.260453	0.130226	−	0.991484i	$-0.458430\pi$
$733$	5168.75	0.260453	0.130226	+	0.991484i	$0.458430\pi$
$734$	−5932.23	−0.298314
$735$	−23628.8	−1.18580
$736$	1207.01	0.0604499
$737$	13197.8	0.659630
$738$	6987.93	0.348549
$739$	21290.6	1.05979	0.529896	−	0.848063i	$-0.322231\pi$
$739$	21290.6	1.05979	0.529896	+	0.848063i	$0.322231\pi$
$740$	28586.0	1.42006
$741$	−9350.43	−0.463558
$742$	4926.72	0.243754
$743$	33625.8	1.66031	0.830155	−	0.557533i	$-0.188252\pi$
$743$	33625.8	1.66031	0.830155	+	0.557533i	$0.188252\pi$
$744$	0	0
$745$	−12696.3	−0.624371
$746$	−2471.55	−0.121300
$747$	43221.2	2.11698
$748$	−12443.4	−0.608255
$749$	12714.9	0.620282
$750$	38735.4	1.88589
$751$	33731.6	1.63899	0.819496	−	0.573085i	$-0.194254\pi$
$751$	33731.6	1.63899	0.819496	+	0.573085i	$0.194254\pi$
$752$	443.909	0.0215262
$753$	40055.3	1.93851
$754$	−569.299	−0.0274969
$755$	−50757.6	−2.44670
$756$	−26440.5	−1.27200
$757$	−5859.05	−0.281309	−0.140654	−	0.990059i	$-0.544921\pi$
$757$	−5859.05	−0.281309	−0.140654	+	0.990059i	$0.544921\pi$
$758$	−11746.5	−0.562865
$759$	8802.26	0.420951
$760$	−9800.80	−0.467779
$761$	7835.81	0.373256	0.186628	−	0.982431i	$-0.440244\pi$
$761$	7835.81	0.373256	0.186628	+	0.982431i	$0.440244\pi$
$762$	11324.1	0.538360
$763$	−33431.8	−1.58625
$764$	−8931.72	−0.422956
$765$	−140039.	−6.61848
$766$	−8627.84	−0.406967
$767$	−11947.8	−0.562464
$768$	−2381.46	−0.111892
$769$	21548.5	1.01048	0.505240	−	0.862979i	$-0.331404\pi$
$769$	21548.5	1.01048	0.505240	+	0.862979i	$0.331404\pi$
$770$	−20782.1	−0.972643
$771$	26938.1	1.25830
$772$	−5393.27	−0.251435
$773$	14264.6	0.663729	0.331864	−	0.943327i	$-0.392322\pi$
$773$	14264.6	0.663729	0.331864	+	0.943327i	$0.392322\pi$
$774$	−56569.3	−2.62705
$775$	0	0
$776$	−9881.83	−0.457135
$777$	−76543.3	−3.53408
$778$	3061.41	0.141075
$779$	−3790.43	−0.174334
$780$	−10983.5	−0.504194
$781$	3840.21	0.175945
$782$	−9354.93	−0.427790
$783$	−5536.47	−0.252691
$784$	2142.64	0.0976057
$785$	19893.4	0.904490
$786$	25795.8	1.17062
$787$	3550.43	0.160812	0.0804061	−	0.996762i	$-0.474378\pi$
$787$	3550.43	0.160812	0.0804061	+	0.996762i	$0.474378\pi$
$788$	−7177.26	−0.324466
$789$	41051.9	1.85233
$790$	−17666.3	−0.795616
$791$	−38994.4	−1.75282
$792$	−11948.4	−0.536072
$793$	2079.97	0.0931425
$794$	19335.9	0.864238
$795$	19903.1	0.887910
$796$	−469.177	−0.0208914
$797$	−43580.8	−1.93690	−0.968450	−	0.249207i	$-0.919830\pi$
$797$	−43580.8	−1.93690	−0.968450	+	0.249207i	$0.919830\pi$
$798$	26243.1	1.16415
$799$	−3440.51	−0.152336
$800$	−7512.49	−0.332008
$801$	−35400.4	−1.56156
$802$	−19632.4	−0.864392
$803$	11104.2	0.487991
$804$	−19576.5	−0.858719
$805$	−15624.0	−0.684067
$806$	0	0
$807$	−75704.3	−3.30225
$808$	2918.29	0.127061
$809$	−1507.80	−0.0655271	−0.0327636	−	0.999463i	$-0.510431\pi$
$809$	−1507.80	−0.0655271	−0.0327636	+	0.999463i	$0.510431\pi$
$810$	−45833.7	−1.98819
$811$	201.334	0.00871736	0.00435868	−	0.999991i	$-0.498613\pi$
$811$	201.334	0.00871736	0.00435868	+	0.999991i	$0.498613\pi$
$812$	1597.81	0.0690542
$813$	−22221.2	−0.958588
$814$	−18903.6	−0.813967
$815$	65345.3	2.80852
$816$	18457.4	0.791837
$817$	30684.6	1.31398
$818$	17480.9	0.747193
$819$	20233.9	0.863284
$820$	−4452.42	−0.189616
$821$	−28841.4	−1.22603	−0.613016	−	0.790070i	$-0.710044\pi$
$821$	−28841.4	−1.22603	−0.613016	+	0.790070i	$0.710044\pi$
$822$	37121.7	1.57514
$823$	724.347	0.0306794	0.0153397	−	0.999882i	$-0.495117\pi$
$823$	724.347	0.0306794	0.0153397	+	0.999882i	$0.495117\pi$
$824$	1641.91	0.0694160
$825$	−54785.6	−2.31199
$826$	33532.9	1.41254
$827$	43558.9	1.83155	0.915775	−	0.401692i	$-0.131578\pi$
$827$	43558.9	1.83155	0.915775	+	0.401692i	$0.131578\pi$
$828$	−8982.85	−0.377024
$829$	20555.5	0.861185	0.430592	−	0.902546i	$-0.358305\pi$
$829$	20555.5	0.861185	0.430592	+	0.902546i	$0.358305\pi$
$830$	−27538.8	−1.15167
$831$	6045.49	0.252366
$832$	995.972	0.0415013
$833$	−16606.5	−0.690733
$834$	48018.0	1.99368
$835$	44460.4	1.84265
$836$	6481.13	0.268128
$837$	0	0
$838$	−9291.75	−0.383029
$839$	26258.1	1.08049	0.540244	−	0.841509i	$-0.318332\pi$
$839$	26258.1	1.08049	0.540244	+	0.841509i	$0.318332\pi$
$840$	30826.4	1.26620
$841$	−24054.4	−0.986282
$842$	23423.7	0.958709
$843$	54733.0	2.23619
$844$	−3194.78	−0.130295
$845$	−37078.1	−1.50950
$846$	−3303.67	−0.134258
$847$	−15324.0	−0.621650
$848$	−1804.79	−0.0730859
$849$	−42755.7	−1.72835
$850$	58225.4	2.34955
$851$	−14211.7	−0.572469
$852$	−5696.24	−0.229049
$853$	43343.6	1.73981	0.869905	−	0.493219i	$-0.164180\pi$
$853$	43343.6	1.73981	0.869905	+	0.493219i	$0.164180\pi$
$854$	−5837.69	−0.233913
$855$	72939.6	2.91752
$856$	−4657.81	−0.185982
$857$	8997.06	0.358616	0.179308	−	0.983793i	$-0.442614\pi$
$857$	8997.06	0.358616	0.179308	+	0.983793i	$0.442614\pi$
$858$	7263.22	0.289000
$859$	−11429.9	−0.453999	−0.226999	−	0.973895i	$-0.572891\pi$
$859$	−11429.9	−0.453999	−0.226999	+	0.973895i	$0.572891\pi$
$860$	36043.6	1.42916
$861$	11922.0	0.471895
$862$	−20919.5	−0.826589
$863$	−22401.8	−0.883622	−0.441811	−	0.897108i	$-0.645664\pi$
$863$	−22401.8	−0.883622	−0.441811	+	0.897108i	$0.645664\pi$
$864$	9685.88	0.381389
$865$	76028.9	2.98851
$866$	19536.5	0.766602
$867$	−97350.4	−3.81337
$868$	0	0
$869$	11682.5	0.456041
$870$	6454.84	0.251540
$871$	8187.28	0.318502
$872$	12247.0	0.475614
$873$	73542.7	2.85114
$874$	4872.52	0.188576
$875$	45467.0	1.75664
$876$	−16471.0	−0.635276
$877$	−37976.8	−1.46224	−0.731121	−	0.682247i	$-0.761003\pi$
$877$	−37976.8	−1.46224	−0.731121	+	0.682247i	$0.761003\pi$
$878$	18121.8	0.696560
$879$	−49192.9	−1.88764
$880$	7613.06	0.291632
$881$	12930.7	0.494489	0.247244	−	0.968953i	$-0.420475\pi$
$881$	12930.7	0.494489	0.247244	+	0.968953i	$0.420475\pi$
$882$	−15946.0	−0.608763
$883$	2136.97	0.0814438	0.0407219	−	0.999171i	$-0.487034\pi$
$883$	2136.97	0.0814438	0.0407219	+	0.999171i	$0.487034\pi$
$884$	−7719.26	−0.293695
$885$	135467.	5.14539
$886$	−17191.3	−0.651865
$887$	11911.7	0.450908	0.225454	−	0.974254i	$-0.427614\pi$
$887$	11911.7	0.450908	0.225454	+	0.974254i	$0.427614\pi$
$888$	28039.9	1.05964
$889$	13292.1	0.501465
$890$	22555.7	0.849516
$891$	30309.2	1.13961
$892$	909.902	0.0341544
$893$	1791.99	0.0671520
$894$	−12453.8	−0.465901
$895$	−25642.4	−0.957687
$896$	−2795.31	−0.104224
$897$	5460.49	0.203256
$898$	36402.9	1.35276
$899$	0	0
$900$	55909.6	2.07072
$901$	13988.0	0.517212
$902$	2944.33	0.108687
$903$	−96512.1	−3.55672
$904$	14284.7	0.525557
$905$	−58403.7	−2.14520
$906$	−49787.9	−1.82571
$907$	44982.2	1.64676	0.823380	−	0.567491i	$-0.192086\pi$
$907$	44982.2	1.64676	0.823380	+	0.567491i	$0.192086\pi$
$908$	−1328.51	−0.0485553
$909$	−21718.5	−0.792472
$910$	−12892.2	−0.469640
$911$	50390.0	1.83260	0.916299	−	0.400495i	$-0.131162\pi$
$911$	50390.0	1.83260	0.916299	+	0.400495i	$0.131162\pi$
$912$	−9613.56	−0.349054
$913$	18211.0	0.660129
$914$	18786.8	0.679883
$915$	−23583.2	−0.852063
$916$	−2333.69	−0.0841781
$917$	30278.6	1.09039
$918$	−75070.2	−2.69901
$919$	−27195.4	−0.976163	−0.488081	−	0.872798i	$-0.662303\pi$
$919$	−27195.4	−0.976163	−0.488081	+	0.872798i	$0.662303\pi$
$920$	5723.50	0.205107
$921$	−8111.01	−0.290192
$922$	−13953.9	−0.498424
$923$	2382.28	0.0849552
$924$	−20385.1	−0.725779
$925$	88454.2	3.14417
$926$	5535.43	0.196442
$927$	−12219.5	−0.432945
$928$	−585.320	−0.0207048
$929$	−26189.3	−0.924912	−0.462456	−	0.886642i	$-0.653032\pi$
$929$	−26189.3	−0.924912	−0.462456	+	0.886642i	$0.653032\pi$
$930$	0	0
$931$	8649.50	0.304485
$932$	−12640.4	−0.444259
$933$	−60429.5	−2.12044
$934$	1692.90	0.0593078
$935$	−59004.8	−2.06381
$936$	−7412.23	−0.258842
$937$	50513.0	1.76114	0.880569	−	0.473918i	$-0.157161\pi$
$937$	50513.0	1.76114	0.880569	+	0.473918i	$0.157161\pi$
$938$	−22978.6	−0.799869
$939$	−28105.8	−0.976782
$940$	2104.96	0.0730385
$941$	−34916.3	−1.20961	−0.604803	−	0.796375i	$-0.706748\pi$
$941$	−34916.3	−1.20961	−0.604803	+	0.796375i	$0.706748\pi$
$942$	19513.3	0.674924
$943$	2213.55	0.0764401
$944$	−12284.0	−0.423529
$945$	−125377.	−4.31590
$946$	−23835.1	−0.819183
$947$	32012.8	1.09850	0.549248	−	0.835659i	$-0.314914\pi$
$947$	32012.8	1.09850	0.549248	+	0.835659i	$0.314914\pi$
$948$	−17328.8	−0.593683
$949$	6888.48	0.235626
$950$	−30326.8	−1.03572
$951$	22352.2	0.762166
$952$	21665.0	0.737571
$953$	25598.0	0.870094	0.435047	−	0.900408i	$-0.356732\pi$
$953$	25598.0	0.870094	0.435047	+	0.900408i	$0.356732\pi$
$954$	13431.6	0.455834
$955$	−42353.1	−1.43509
$956$	6075.34	0.205534
$957$	−4268.50	−0.144181
$958$	−31415.9	−1.05950
$959$	43572.9	1.46720
$960$	−11292.6	−0.379652
$961$	0	0
$962$	−11726.8	−0.393024
$963$	34664.4	1.15996
$964$	3970.56	0.132659
$965$	−25574.2	−0.853121
$966$	−15325.5	−0.510446
$967$	22569.5	0.750555	0.375277	−	0.926913i	$-0.377547\pi$
$967$	22569.5	0.750555	0.375277	+	0.926913i	$0.377547\pi$
$968$	5613.59	0.186392
$969$	74509.7	2.47017
$970$	−46858.4	−1.55106
$971$	12779.3	0.422357	0.211178	−	0.977448i	$-0.432270\pi$
$971$	12779.3	0.422357	0.211178	+	0.977448i	$0.432270\pi$
$972$	−12268.2	−0.404840
$973$	56362.7	1.85705
$974$	36647.2	1.20560
$975$	−33986.3	−1.11634
$976$	2138.51	0.0701352
$977$	−5941.92	−0.194574	−0.0972871	−	0.995256i	$-0.531016\pi$
$977$	−5941.92	−0.194574	−0.0972871	+	0.995256i	$0.531016\pi$
$978$	64096.9	2.09570
$979$	−14915.8	−0.486936
$980$	10160.1	0.331177
$981$	−91144.6	−2.96639
$982$	18002.8	0.585023
$983$	8224.44	0.266856	0.133428	−	0.991059i	$-0.457402\pi$
$983$	8224.44	0.266856	0.133428	+	0.991059i	$0.457402\pi$
$984$	−4367.36	−0.141490
$985$	−34033.7	−1.10092
$986$	4536.51	0.146523
$987$	−5636.34	−0.181770
$988$	4020.58	0.129465
$989$	−17919.3	−0.576138
$990$	−56658.0	−1.81890
$991$	17785.8	0.570114	0.285057	−	0.958511i	$-0.407987\pi$
$991$	17785.8	0.570114	0.285057	+	0.958511i	$0.407987\pi$
$992$	0	0
$993$	66430.8	2.12298
$994$	−6686.15	−0.213352
$995$	−2224.77	−0.0708845
$996$	−27012.7	−0.859368
$997$	−8265.00	−0.262543	−0.131271	−	0.991346i	$-0.541906\pi$
$997$	−8265.00	−0.262543	−0.131271	+	0.991346i	$0.541906\pi$
$998$	70.1173	0.00222397
$999$	−114044.	−3.61181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1922.4.a.x.1.2	✓	32
31.30	odd	2	inner	1922.4.a.x.1.31	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1922.4.a.x.1.2	✓	32	1.1	even	1	trivial
1922.4.a.x.1.31	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -9.30256 q^{3} +4.00000 q^{4} +18.9675 q^{5} +18.6051 q^{6} +21.8384 q^{7} -8.00000 q^{8} +59.5377 q^{9} -37.9350 q^{10} +25.0859 q^{11} -37.2103 q^{12} +15.5621 q^{13} -43.6768 q^{14} -176.446 q^{15} +16.0000 q^{16} -124.008 q^{17} -119.075 q^{18} +64.5895 q^{19} +75.8699 q^{20} -203.153 q^{21} -50.1718 q^{22} -37.7192 q^{23} +74.4205 q^{24} +234.765 q^{25} -31.1241 q^{26} -302.684 q^{27} +87.3535 q^{28} +18.2913 q^{29} +352.892 q^{30} -32.0000 q^{32} -233.363 q^{33} +248.015 q^{34} +414.219 q^{35} +238.151 q^{36} +376.777 q^{37} -129.179 q^{38} -144.767 q^{39} -151.740 q^{40} -58.6849 q^{41} +406.306 q^{42} +475.071 q^{43} +100.344 q^{44} +1129.28 q^{45} +75.4383 q^{46} +27.7443 q^{47} -148.841 q^{48} +133.915 q^{49} -469.531 q^{50} +1153.59 q^{51} +62.2482 q^{52} -112.800 q^{53} +605.368 q^{54} +475.816 q^{55} -174.707 q^{56} -600.848 q^{57} -36.5825 q^{58} -767.752 q^{59} -705.785 q^{60} +133.657 q^{61} +1300.21 q^{63} +64.0000 q^{64} +295.173 q^{65} +466.726 q^{66} +526.105 q^{67} -496.031 q^{68} +350.885 q^{69} -828.438 q^{70} +153.082 q^{71} -476.301 q^{72} +442.646 q^{73} -753.554 q^{74} -2183.92 q^{75} +258.358 q^{76} +547.835 q^{77} +289.534 q^{78} +465.698 q^{79} +303.480 q^{80} +1208.22 q^{81} +117.370 q^{82} +725.948 q^{83} -812.612 q^{84} -2352.11 q^{85} -950.142 q^{86} -170.156 q^{87} -200.687 q^{88} -594.589 q^{89} -2258.56 q^{90} +339.850 q^{91} -150.877 q^{92} -55.4886 q^{94} +1225.10 q^{95} +297.682 q^{96} +1235.23 q^{97} -267.830 q^{98} +1493.56 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) \) 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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