LMFDB - Embedded newform 1922.4.a.x.1.10

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			1.10
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)

$f(q)$	$=$	\(q-2.00000 q^{2} -5.10931 q^{3} +4.00000 q^{4} -18.4069 q^{5} +10.2186 q^{6} +1.23685 q^{7} -8.00000 q^{8} -0.894964 q^{9} +36.8139 q^{10} +71.5290 q^{11} -20.4372 q^{12} +63.7199 q^{13} -2.47370 q^{14} +94.0467 q^{15} +16.0000 q^{16} +2.24508 q^{17} +1.78993 q^{18} +89.0186 q^{19} -73.6278 q^{20} -6.31944 q^{21} -143.058 q^{22} +48.7918 q^{23} +40.8745 q^{24} +213.815 q^{25} -127.440 q^{26} +142.524 q^{27} +4.94739 q^{28} +204.702 q^{29} -188.093 q^{30} -32.0000 q^{32} -365.464 q^{33} -4.49015 q^{34} -22.7666 q^{35} -3.57986 q^{36} +332.206 q^{37} -178.037 q^{38} -325.565 q^{39} +147.256 q^{40} +121.043 q^{41} +12.6389 q^{42} +527.799 q^{43} +286.116 q^{44} +16.4736 q^{45} -97.5836 q^{46} +75.7410 q^{47} -81.7489 q^{48} -341.470 q^{49} -427.631 q^{50} -11.4708 q^{51} +254.880 q^{52} -203.982 q^{53} -285.048 q^{54} -1316.63 q^{55} -9.89479 q^{56} -454.823 q^{57} -409.404 q^{58} -361.936 q^{59} +376.187 q^{60} +602.325 q^{61} -1.10694 q^{63} +64.0000 q^{64} -1172.89 q^{65} +730.928 q^{66} +192.721 q^{67} +8.98031 q^{68} -249.292 q^{69} +45.5332 q^{70} -226.007 q^{71} +7.15971 q^{72} -274.197 q^{73} -664.411 q^{74} -1092.45 q^{75} +356.074 q^{76} +88.4706 q^{77} +651.129 q^{78} -381.937 q^{79} -294.511 q^{80} -704.035 q^{81} -242.087 q^{82} +453.354 q^{83} -25.2778 q^{84} -41.3250 q^{85} -1055.60 q^{86} -1045.89 q^{87} -572.232 q^{88} +881.158 q^{89} -32.9471 q^{90} +78.8119 q^{91} +195.167 q^{92} -151.482 q^{94} -1638.56 q^{95} +163.498 q^{96} -132.635 q^{97} +682.940 q^{98} -64.0159 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.10931	−0.983287	−0.491643	−	0.870797i	$-0.663604\pi$
$3$	−5.10931	−0.983287	−0.491643	+	0.870797i	$0.663604\pi$
$4$	4.00000	0.500000
$5$	−18.4069	−1.64637	−0.823183	−	0.567776i	$-0.807804\pi$
$5$	−18.4069	−1.64637	−0.823183	+	0.567776i	$0.807804\pi$
$6$	10.2186	0.695289
$7$	1.23685	0.0667835	0.0333918	−	0.999442i	$-0.489369\pi$
$7$	1.23685	0.0667835	0.0333918	+	0.999442i	$0.489369\pi$
$8$	−8.00000	−0.353553
$9$	−0.894964	−0.0331468
$10$	36.8139	1.16416
$11$	71.5290	1.96062	0.980310	−	0.197466i	$-0.0632712\pi$
$11$	71.5290	1.96062	0.980310	+	0.197466i	$0.0632712\pi$
$12$	−20.4372	−0.491643
$13$	63.7199	1.35944	0.679720	−	0.733471i	$-0.262101\pi$
$13$	63.7199	1.35944	0.679720	+	0.733471i	$0.262101\pi$
$14$	−2.47370	−0.0472231
$15$	94.0467	1.61885
$16$	16.0000	0.250000
$17$	2.24508	0.0320301	0.0160150	−	0.999872i	$-0.494902\pi$
$17$	2.24508	0.0320301	0.0160150	+	0.999872i	$0.494902\pi$
$18$	1.78993	0.0234383
$19$	89.0186	1.07486	0.537428	−	0.843310i	$-0.319396\pi$
$19$	89.0186	1.07486	0.537428	+	0.843310i	$0.319396\pi$
$20$	−73.6278	−0.823183
$21$	−6.31944	−0.0656674
$22$	−143.058	−1.38637
$23$	48.7918	0.442339	0.221170	−	0.975235i	$-0.429013\pi$
$23$	48.7918	0.442339	0.221170	+	0.975235i	$0.429013\pi$
$24$	40.8745	0.347644
$25$	213.815	1.71052
$26$	−127.440	−0.961270
$27$	142.524	1.01588
$28$	4.94739	0.0333918
$29$	204.702	1.31077	0.655383	−	0.755297i	$-0.272507\pi$
$29$	204.702	1.31077	0.655383	+	0.755297i	$0.272507\pi$
$30$	−188.093	−1.14470
$31$	0	0
$31$	0	0
$32$	−32.0000	−0.176777
$33$	−365.464	−1.92785
$34$	−4.49015	−0.0226487
$35$	−22.7666	−0.109950
$36$	−3.57986	−0.0165734
$37$	332.206	1.47606	0.738031	−	0.674767i	$-0.235756\pi$
$37$	332.206	1.47606	0.738031	+	0.674767i	$0.235756\pi$
$38$	−178.037	−0.760038
$39$	−325.565	−1.33672
$40$	147.256	0.582079
$41$	121.043	0.461068	0.230534	−	0.973064i	$-0.425953\pi$
$41$	121.043	0.461068	0.230534	+	0.973064i	$0.425953\pi$
$42$	12.6389	0.0464338
$43$	527.799	1.87183	0.935913	−	0.352230i	$-0.114577\pi$
$43$	527.799	1.87183	0.935913	+	0.352230i	$0.114577\pi$
$44$	286.116	0.980310
$45$	16.4736	0.0545718
$46$	−97.5836	−0.312781
$47$	75.7410	0.235063	0.117531	−	0.993069i	$-0.462502\pi$
$47$	75.7410	0.235063	0.117531	+	0.993069i	$0.462502\pi$
$48$	−81.7489	−0.245822
$49$	−341.470	−0.995540
$50$	−427.631	−1.20952
$51$	−11.4708	−0.0314947
$52$	254.880	0.679720
$53$	−203.982	−0.528661	−0.264331	−	0.964432i	$-0.585151\pi$
$53$	−203.982	−0.528661	−0.264331	+	0.964432i	$0.585151\pi$
$54$	−285.048	−0.718335
$55$	−1316.63	−3.22790
$56$	−9.89479	−0.0236115
$57$	−454.823	−1.05689
$58$	−409.404	−0.926852
$59$	−361.936	−0.798646	−0.399323	−	0.916810i	$-0.630755\pi$
$59$	−361.936	−0.798646	−0.399323	+	0.916810i	$0.630755\pi$
$60$	376.187	0.809426
$61$	602.325	1.26426	0.632129	−	0.774863i	$-0.282181\pi$
$61$	602.325	1.26426	0.632129	+	0.774863i	$0.282181\pi$
$62$	0	0
$63$	−1.10694	−0.00221366
$64$	64.0000	0.125000
$65$	−1172.89	−2.23814
$66$	730.928	1.36320
$67$	192.721	0.351411	0.175706	−	0.984443i	$-0.443779\pi$
$67$	192.721	0.351411	0.175706	+	0.984443i	$0.443779\pi$
$68$	8.98031	0.0160150
$69$	−249.292	−0.434946
$70$	45.5332	0.0777465
$71$	−226.007	−0.377776	−0.188888	−	0.981999i	$-0.560488\pi$
$71$	−226.007	−0.377776	−0.188888	+	0.981999i	$0.560488\pi$
$72$	7.15971	0.0117192
$73$	−274.197	−0.439622	−0.219811	−	0.975543i	$-0.570544\pi$
$73$	−274.197	−0.439622	−0.219811	+	0.975543i	$0.570544\pi$
$74$	−664.411	−1.04373
$75$	−1092.45	−1.68194
$76$	356.074	0.537428
$77$	88.4706	0.130937
$78$	651.129	0.945204
$79$	−381.937	−0.543940	−0.271970	−	0.962306i	$-0.587675\pi$
$79$	−381.937	−0.543940	−0.271970	+	0.962306i	$0.587675\pi$
$80$	−294.511	−0.411592
$81$	−704.035	−0.965754
$82$	−242.087	−0.326024
$83$	453.354	0.599543	0.299771	−	0.954011i	$-0.403090\pi$
$83$	453.354	0.599543	0.299771	+	0.954011i	$0.403090\pi$
$84$	−25.2778	−0.0328337
$85$	−41.3250	−0.0527332
$86$	−1055.60	−1.32358
$87$	−1045.89	−1.28886
$88$	−572.232	−0.693184
$89$	881.158	1.04947	0.524733	−	0.851267i	$-0.324165\pi$
$89$	881.158	1.04947	0.524733	+	0.851267i	$0.324165\pi$
$90$	−32.9471	−0.0385881
$91$	78.8119	0.0907882
$92$	195.167	0.221170
$93$	0	0
$94$	−151.482	−0.166215
$95$	−1638.56	−1.76961
$96$	163.498	0.173822
$97$	−132.635	−0.138835	−0.0694177	−	0.997588i	$-0.522114\pi$
$97$	−132.635	−0.138835	−0.0694177	+	0.997588i	$0.522114\pi$
$98$	682.940	0.703953
$99$	−64.0159	−0.0649883
$100$	855.262	0.855262
$101$	1291.89	1.27275	0.636373	−	0.771381i	$-0.280434\pi$
$101$	1291.89	1.27275	0.636373	+	0.771381i	$0.280434\pi$
$102$	22.9416	0.0222702
$103$	1505.29	1.44001	0.720005	−	0.693969i	$-0.244140\pi$
$103$	1505.29	1.44001	0.720005	+	0.693969i	$0.244140\pi$
$104$	−509.759	−0.480635
$105$	116.322	0.108113
$106$	407.963	0.373820
$107$	−299.084	−0.270220	−0.135110	−	0.990831i	$-0.543139\pi$
$107$	−299.084	−0.270220	−0.135110	+	0.990831i	$0.543139\pi$
$108$	570.096	0.507940
$109$	345.139	0.303288	0.151644	−	0.988435i	$-0.451543\pi$
$109$	345.139	0.303288	0.151644	+	0.988435i	$0.451543\pi$
$110$	2633.26	2.28247
$111$	−1697.34	−1.45139
$112$	19.7896	0.0166959
$113$	−1760.17	−1.46533	−0.732666	−	0.680588i	$-0.761724\pi$
$113$	−1760.17	−1.46533	−0.732666	+	0.680588i	$0.761724\pi$
$114$	909.647	0.747335
$115$	−898.108	−0.728252
$116$	818.808	0.655383
$117$	−57.0271	−0.0450611
$118$	723.873	0.564728
$119$	2.77682	0.00213908
$120$	−752.374	−0.572350
$121$	3785.40	2.84403
$122$	−1204.65	−0.893966
$123$	−618.448	−0.453362
$124$	0	0
$125$	−1634.82	−1.16978
$126$	2.21387	0.00156530
$127$	−1880.64	−1.31402	−0.657009	−	0.753883i	$-0.728179\pi$
$127$	−1880.64	−1.31402	−0.657009	+	0.753883i	$0.728179\pi$
$128$	−128.000	−0.0883883
$129$	−2696.69	−1.84054
$130$	2345.78	1.58260
$131$	776.264	0.517729	0.258864	−	0.965914i	$-0.416652\pi$
$131$	776.264	0.517729	0.258864	+	0.965914i	$0.416652\pi$
$132$	−1461.86	−0.963926
$133$	110.102	0.0717827
$134$	−385.441	−0.248485
$135$	−2623.43	−1.67251
$136$	−17.9606	−0.0113243
$137$	1012.18	0.631214	0.315607	−	0.948890i	$-0.397792\pi$
$137$	1012.18	0.631214	0.315607	+	0.948890i	$0.397792\pi$
$138$	498.585	0.307553
$139$	936.821	0.571656	0.285828	−	0.958281i	$-0.407731\pi$
$139$	936.821	0.571656	0.285828	+	0.958281i	$0.407731\pi$
$140$	−91.0664	−0.0549751
$141$	−386.984	−0.231134
$142$	452.013	0.267128
$143$	4557.82	2.66535
$144$	−14.3194	−0.00828671
$145$	−3767.94	−2.15800
$146$	548.395	0.310860
$147$	1744.68	0.978901
$148$	1328.82	0.738031
$149$	−681.426	−0.374661	−0.187331	−	0.982297i	$-0.559984\pi$
$149$	−681.426	−0.374661	−0.187331	+	0.982297i	$0.559984\pi$
$150$	2184.90	1.18931
$151$	−2081.94	−1.12202	−0.561012	−	0.827808i	$-0.689588\pi$
$151$	−2081.94	−1.12202	−0.561012	+	0.827808i	$0.689588\pi$
$152$	−712.149	−0.380019
$153$	−2.00926	−0.00106170
$154$	−176.941	−0.0925865
$155$	0	0
$156$	−1302.26	−0.668360
$157$	3733.66	1.89795	0.948976	−	0.315347i	$-0.102121\pi$
$157$	3733.66	1.89795	0.948976	+	0.315347i	$0.102121\pi$
$158$	763.875	0.384624
$159$	1042.21	0.519826
$160$	589.022	0.291039
$161$	60.3481	0.0295410
$162$	1408.07	0.682892
$163$	712.093	0.342181	0.171090	−	0.985255i	$-0.445271\pi$
$163$	712.093	0.342181	0.171090	+	0.985255i	$0.445271\pi$
$164$	484.173	0.230534
$165$	6727.07	3.17395
$166$	−906.708	−0.423941
$167$	890.833	0.412783	0.206391	−	0.978470i	$-0.433828\pi$
$167$	890.833	0.412783	0.206391	+	0.978470i	$0.433828\pi$
$168$	50.5555	0.0232169
$169$	1863.23	0.848078
$170$	82.6500	0.0372880
$171$	−79.6684	−0.0356281
$172$	2111.19	0.935913
$173$	−3468.61	−1.52436	−0.762178	−	0.647368i	$-0.775870\pi$
$173$	−3468.61	−1.52436	−0.762178	+	0.647368i	$0.775870\pi$
$174$	2091.77	0.911361
$175$	264.457	0.114235
$176$	1144.46	0.490155
$177$	1849.24	0.785298
$178$	−1762.32	−0.742085
$179$	−188.106	−0.0785457	−0.0392728	−	0.999229i	$-0.512504\pi$
$179$	−188.106	−0.0785457	−0.0392728	+	0.999229i	$0.512504\pi$
$180$	65.8942	0.0272859
$181$	2769.88	1.13748	0.568738	−	0.822519i	$-0.307432\pi$
$181$	2769.88	1.13748	0.568738	+	0.822519i	$0.307432\pi$
$182$	−157.624	−0.0641970
$183$	−3077.46	−1.24313
$184$	−390.335	−0.156390
$185$	−6114.89	−2.43014
$186$	0	0
$187$	160.588	0.0627988
$188$	302.964	0.117531
$189$	176.281	0.0678440
$190$	3277.12	1.25130
$191$	−1238.47	−0.469175	−0.234588	−	0.972095i	$-0.575374\pi$
$191$	−1238.47	−0.469175	−0.234588	+	0.972095i	$0.575374\pi$
$192$	−326.996	−0.122911
$193$	2844.01	1.06070	0.530352	−	0.847777i	$-0.322060\pi$
$193$	2844.01	1.06070	0.530352	+	0.847777i	$0.322060\pi$
$194$	265.270	0.0981714
$195$	5992.65	2.20073
$196$	−1365.88	−0.497770
$197$	−395.753	−0.143128	−0.0715641	−	0.997436i	$-0.522799\pi$
$197$	−395.753	−0.143128	−0.0715641	+	0.997436i	$0.522799\pi$
$198$	128.032	0.0459537
$199$	−4306.58	−1.53410	−0.767049	−	0.641588i	$-0.778276\pi$
$199$	−4306.58	−1.53410	−0.767049	+	0.641588i	$0.778276\pi$
$200$	−1710.52	−0.604762
$201$	−984.669	−0.345538
$202$	−2583.77	−0.899967
$203$	253.185	0.0875376
$204$	−45.8832	−0.0157474
$205$	−2228.04	−0.759087
$206$	−3010.59	−1.01824
$207$	−43.6669	−0.0146621
$208$	1019.52	0.339860
$209$	6367.41	2.10738
$210$	−232.643	−0.0764472
$211$	−5420.99	−1.76870	−0.884352	−	0.466821i	$-0.845399\pi$
$211$	−5420.99	−1.76870	−0.884352	+	0.466821i	$0.845399\pi$
$212$	−815.927	−0.264331
$213$	1154.74	0.371462
$214$	598.167	0.191074
$215$	−9715.16	−3.08171
$216$	−1140.19	−0.359168
$217$	0	0
$218$	−690.279	−0.214457
$219$	1400.96	0.432274
$220$	−5266.52	−1.61395
$221$	143.056	0.0435430
$222$	3394.68	1.02629
$223$	−1946.15	−0.584412	−0.292206	−	0.956355i	$-0.594389\pi$
$223$	−1946.15	−0.584412	−0.292206	+	0.956355i	$0.594389\pi$
$224$	−39.5791	−0.0118058
$225$	−191.357	−0.0566984
$226$	3520.33	1.03615
$227$	−3880.84	−1.13471	−0.567357	−	0.823472i	$-0.692034\pi$
$227$	−3880.84	−1.13471	−0.567357	+	0.823472i	$0.692034\pi$
$228$	−1819.29	−0.528446
$229$	4754.47	1.37198	0.685992	−	0.727609i	$-0.259368\pi$
$229$	4754.47	1.37198	0.685992	+	0.727609i	$0.259368\pi$
$230$	1796.22	0.514952
$231$	−452.023	−0.128749
$232$	−1637.62	−0.463426
$233$	4420.49	1.24290	0.621451	−	0.783453i	$-0.286543\pi$
$233$	4420.49	1.24290	0.621451	+	0.783453i	$0.286543\pi$
$234$	114.054	0.0318630
$235$	−1394.16	−0.387000
$236$	−1447.75	−0.399323
$237$	1951.44	0.534850
$238$	−5.55364	−0.00151256
$239$	646.725	0.175034	0.0875170	−	0.996163i	$-0.472107\pi$
$239$	646.725	0.175034	0.0875170	+	0.996163i	$0.472107\pi$
$240$	1504.75	0.404713
$241$	−4657.70	−1.24493	−0.622466	−	0.782647i	$-0.713869\pi$
$241$	−4657.70	−1.24493	−0.622466	+	0.782647i	$0.713869\pi$
$242$	−7570.80	−2.01103
$243$	−251.015	−0.0662660
$244$	2409.30	0.632129
$245$	6285.42	1.63902
$246$	1236.90	0.320576
$247$	5672.26	1.46120
$248$	0	0
$249$	−2316.33	−0.589523
$250$	3269.64	0.827161
$251$	4592.00	1.15476	0.577380	−	0.816476i	$-0.304075\pi$
$251$	4592.00	1.15476	0.577380	+	0.816476i	$0.304075\pi$
$252$	−4.42774	−0.00110683
$253$	3490.03	0.867258
$254$	3761.29	0.929151
$255$	211.142	0.0518519
$256$	256.000	0.0625000
$257$	−4467.72	−1.08439	−0.542196	−	0.840252i	$-0.682407\pi$
$257$	−4467.72	−1.08439	−0.542196	+	0.840252i	$0.682407\pi$
$258$	5393.37	1.30146
$259$	410.888	0.0985766
$260$	−4691.56	−1.11907
$261$	−183.201	−0.0434477
$262$	−1552.53	−0.366090
$263$	1539.53	0.360957	0.180478	−	0.983579i	$-0.442235\pi$
$263$	1539.53	0.360957	0.180478	+	0.983579i	$0.442235\pi$
$264$	2923.71	0.681598
$265$	3754.68	0.870370
$266$	−220.205	−0.0507580
$267$	−4502.11	−1.03193
$268$	770.882	0.175706
$269$	−1025.30	−0.232393	−0.116196	−	0.993226i	$-0.537070\pi$
$269$	−1025.30	−0.232393	−0.116196	+	0.993226i	$0.537070\pi$
$270$	5246.86	1.18264
$271$	6618.60	1.48358	0.741792	−	0.670629i	$-0.233976\pi$
$271$	6618.60	1.48358	0.741792	+	0.670629i	$0.233976\pi$
$272$	35.9212	0.00800752
$273$	−402.674	−0.0892709
$274$	−2024.36	−0.446336
$275$	15294.0	3.35369
$276$	−997.170	−0.217473
$277$	1304.22	0.282899	0.141450	−	0.989945i	$-0.454824\pi$
$277$	1304.22	0.282899	0.141450	+	0.989945i	$0.454824\pi$
$278$	−1873.64	−0.404222
$279$	0	0
$280$	182.133	0.0388733
$281$	4630.87	0.983112	0.491556	−	0.870846i	$-0.336428\pi$
$281$	4630.87	0.983112	0.491556	+	0.870846i	$0.336428\pi$
$282$	773.968	0.163437
$283$	8131.30	1.70797	0.853985	−	0.520297i	$-0.174179\pi$
$283$	8131.30	1.70797	0.853985	+	0.520297i	$0.174179\pi$
$284$	−904.027	−0.188888
$285$	8371.91	1.74003
$286$	−9115.65	−1.88468
$287$	149.712	0.0307918
$288$	28.6389	0.00585959
$289$	−4907.96	−0.998974
$290$	7535.88	1.52594
$291$	677.673	0.136515
$292$	−1096.79	−0.219811
$293$	4682.90	0.933714	0.466857	−	0.884333i	$-0.345386\pi$
$293$	4682.90	0.933714	0.466857	+	0.884333i	$0.345386\pi$
$294$	−3489.35	−0.692188
$295$	6662.14	1.31486
$296$	−2657.64	−0.521866
$297$	10194.6	1.99175
$298$	1362.85	0.264926
$299$	3109.01	0.601334
$300$	−4369.80	−0.840968
$301$	652.807	0.125007
$302$	4163.87	0.793391
$303$	−6600.64	−1.25147
$304$	1424.30	0.268714
$305$	−11087.0	−2.08143
$306$	4.01853	0.000750732 0
$307$	−4529.99	−0.842151	−0.421075	−	0.907026i	$-0.638347\pi$
$307$	−4529.99	−0.842151	−0.421075	+	0.907026i	$0.638347\pi$
$308$	353.882	0.0654686
$309$	−7691.01	−1.41594
$310$	0	0
$311$	−8767.73	−1.59862	−0.799312	−	0.600916i	$-0.794803\pi$
$311$	−8767.73	−1.59862	−0.799312	+	0.600916i	$0.794803\pi$
$312$	2604.52	0.472602
$313$	1047.24	0.189116	0.0945580	−	0.995519i	$-0.469856\pi$
$313$	1047.24	0.189116	0.0945580	+	0.995519i	$0.469856\pi$
$314$	−7467.32	−1.34206
$315$	20.3753	0.00364450
$316$	−1527.75	−0.271970
$317$	1178.92	0.208878	0.104439	−	0.994531i	$-0.466695\pi$
$317$	1178.92	0.208878	0.104439	+	0.994531i	$0.466695\pi$
$318$	−2084.41	−0.367572
$319$	14642.1	2.56991
$320$	−1178.04	−0.205796
$321$	1528.11	0.265703
$322$	−120.696	−0.0208886
$323$	199.854	0.0344277
$324$	−2816.14	−0.482877
$325$	13624.3	2.32536
$326$	−1424.19	−0.241958
$327$	−1763.42	−0.298219
$328$	−968.347	−0.163012
$329$	93.6801	0.0156983
$330$	−13454.1	−2.24432
$331$	−5731.90	−0.951824	−0.475912	−	0.879493i	$-0.657882\pi$
$331$	−5731.90	−0.951824	−0.475912	+	0.879493i	$0.657882\pi$
$332$	1813.42	0.299771
$333$	−297.312	−0.0489267
$334$	−1781.67	−0.291881
$335$	−3547.40	−0.578552
$336$	−101.111	−0.0164168
$337$	4664.98	0.754058	0.377029	−	0.926202i	$-0.376946\pi$
$337$	4664.98	0.754058	0.377029	+	0.926202i	$0.376946\pi$
$338$	−3726.46	−0.599682
$339$	8993.24	1.44084
$340$	−165.300	−0.0263666
$341$	0	0
$342$	159.337	0.0251928
$343$	−846.586	−0.133269
$344$	−4222.39	−0.661791
$345$	4588.71	0.716081
$346$	6937.22	1.07788
$347$	8392.08	1.29830	0.649151	−	0.760660i	$-0.275125\pi$
$347$	8392.08	1.29830	0.649151	+	0.760660i	$0.275125\pi$
$348$	−4183.54	−0.644430
$349$	−8479.95	−1.30063	−0.650317	−	0.759663i	$-0.725364\pi$
$349$	−8479.95	−1.30063	−0.650317	+	0.759663i	$0.725364\pi$
$350$	−528.915	−0.0807762
$351$	9081.62	1.38103
$352$	−2288.93	−0.346592
$353$	6235.61	0.940192	0.470096	−	0.882615i	$-0.344219\pi$
$353$	6235.61	0.940192	0.470096	+	0.882615i	$0.344219\pi$
$354$	−3698.49	−0.555290
$355$	4160.09	0.621957
$356$	3524.63	0.524733
$357$	−14.1876	−0.00210333
$358$	376.211	0.0555402
$359$	−629.361	−0.0925248	−0.0462624	−	0.998929i	$-0.514731\pi$
$359$	−629.361	−0.0925248	−0.0462624	+	0.998929i	$0.514731\pi$
$360$	−131.788	−0.0192941
$361$	1065.31	0.155315
$362$	−5539.75	−0.804317
$363$	−19340.8	−2.79650
$364$	315.248	0.0453941
$365$	5047.14	0.723779
$366$	6154.93	0.879025
$367$	2297.01	0.326711	0.163355	−	0.986567i	$-0.447768\pi$
$367$	2297.01	0.326711	0.163355	+	0.986567i	$0.447768\pi$
$368$	780.669	0.110585
$369$	−108.329	−0.0152829
$370$	12229.8	1.71837
$371$	−252.294	−0.0353059
$372$	0	0
$373$	−4276.42	−0.593631	−0.296816	−	0.954935i	$-0.595925\pi$
$373$	−4276.42	−0.593631	−0.296816	+	0.954935i	$0.595925\pi$
$374$	−321.176	−0.0444054
$375$	8352.81	1.15023
$376$	−605.928	−0.0831073
$377$	13043.6	1.78191
$378$	−352.561	−0.0479730
$379$	5508.18	0.746533	0.373267	−	0.927724i	$-0.378238\pi$
$379$	5508.18	0.746533	0.373267	+	0.927724i	$0.378238\pi$
$380$	−6554.24	−0.884803
$381$	9608.79	1.29206
$382$	2476.94	0.331757
$383$	640.346	0.0854312	0.0427156	−	0.999087i	$-0.486399\pi$
$383$	640.346	0.0854312	0.0427156	+	0.999087i	$0.486399\pi$
$384$	653.992	0.0869111
$385$	−1628.47	−0.215571
$386$	−5688.01	−0.750031
$387$	−472.361	−0.0620451
$388$	−530.540	−0.0694177
$389$	−13343.2	−1.73914	−0.869570	−	0.493810i	$-0.835604\pi$
$389$	−13343.2	−1.73914	−0.869570	+	0.493810i	$0.835604\pi$
$390$	−11985.3	−1.55615
$391$	109.541	0.0141681
$392$	2731.76	0.351977
$393$	−3966.17	−0.509076
$394$	791.507	0.101207
$395$	7030.30	0.895526
$396$	−256.064	−0.0324942
$397$	11908.3	1.50544	0.752719	−	0.658342i	$-0.228742\pi$
$397$	11908.3	1.50544	0.752719	+	0.658342i	$0.228742\pi$
$398$	8613.17	1.08477
$399$	−562.548	−0.0705830
$400$	3421.05	0.427631
$401$	−12382.1	−1.54197	−0.770986	−	0.636852i	$-0.780236\pi$
$401$	−12382.1	−1.54197	−0.770986	+	0.636852i	$0.780236\pi$
$402$	1969.34	0.244332
$403$	0	0
$404$	5167.54	0.636373
$405$	12959.1	1.58999
$406$	−506.371	−0.0618984
$407$	23762.3	2.89399
$408$	91.7663	0.0111351
$409$	676.452	0.0817809	0.0408905	−	0.999164i	$-0.486981\pi$
$409$	676.452	0.0817809	0.0408905	+	0.999164i	$0.486981\pi$
$410$	4456.08	0.536756
$411$	−5171.54	−0.620664
$412$	6021.18	0.720005
$413$	−447.660	−0.0533364
$414$	87.3339	0.0103677
$415$	−8344.86	−0.987068
$416$	−2039.04	−0.240317
$417$	−4786.51	−0.562102
$418$	−12734.8	−1.49014
$419$	−5986.69	−0.698016	−0.349008	−	0.937120i	$-0.613481\pi$
$419$	−5986.69	−0.698016	−0.349008	+	0.937120i	$0.613481\pi$
$420$	465.286	0.0540563
$421$	9008.56	1.04287	0.521437	−	0.853290i	$-0.325396\pi$
$421$	9008.56	1.04287	0.521437	+	0.853290i	$0.325396\pi$
$422$	10842.0	1.25066
$423$	−67.7855	−0.00779159
$424$	1631.85	0.186910
$425$	480.032	0.0547882
$426$	−2309.48	−0.262663
$427$	744.984	0.0844317
$428$	−1196.33	−0.135110
$429$	−23287.3	−2.62080
$430$	19430.3	2.17910
$431$	−12819.1	−1.43265	−0.716325	−	0.697767i	$-0.754177\pi$
$431$	−12819.1	−1.43265	−0.716325	+	0.697767i	$0.754177\pi$
$432$	2280.38	0.253970
$433$	2431.51	0.269864	0.134932	−	0.990855i	$-0.456918\pi$
$433$	2431.51	0.269864	0.134932	+	0.990855i	$0.456918\pi$
$434$	0	0
$435$	19251.6	2.12194
$436$	1380.56	0.151644
$437$	4343.38	0.475451
$438$	−2801.92	−0.305664
$439$	−2759.91	−0.300053	−0.150026	−	0.988682i	$-0.547936\pi$
$439$	−2759.91	−0.300053	−0.150026	+	0.988682i	$0.547936\pi$
$440$	10533.0	1.14123
$441$	305.604	0.0329990
$442$	−286.112	−0.0307895
$443$	−16807.9	−1.80264	−0.901319	−	0.433156i	$-0.857400\pi$
$443$	−16807.9	−1.80264	−0.901319	+	0.433156i	$0.857400\pi$
$444$	−6789.36	−0.725696
$445$	−16219.4	−1.72781
$446$	3892.30	0.413242
$447$	3481.61	0.368400
$448$	79.1583	0.00834794
$449$	−8669.89	−0.911264	−0.455632	−	0.890168i	$-0.650587\pi$
$449$	−8669.89	−0.911264	−0.455632	+	0.890168i	$0.650587\pi$
$450$	382.714	0.0400919
$451$	8658.11	0.903979
$452$	−7040.67	−0.732666
$453$	10637.3	1.10327
$454$	7761.67	0.802364
$455$	−1450.69	−0.149471
$456$	3638.59	0.373668
$457$	1671.03	0.171045	0.0855223	−	0.996336i	$-0.472744\pi$
$457$	1671.03	0.171045	0.0855223	+	0.996336i	$0.472744\pi$
$458$	−9508.94	−0.970139
$459$	319.977	0.0325387
$460$	−3592.43	−0.364126
$461$	−6412.34	−0.647836	−0.323918	−	0.946085i	$-0.605000\pi$
$461$	−6412.34	−0.647836	−0.323918	+	0.946085i	$0.605000\pi$
$462$	904.047	0.0910391
$463$	10321.2	1.03600	0.517999	−	0.855381i	$-0.326677\pi$
$463$	10321.2	1.03600	0.517999	+	0.855381i	$0.326677\pi$
$464$	3275.23	0.327692
$465$	0	0
$466$	−8840.99	−0.878865
$467$	−3577.10	−0.354450	−0.177225	−	0.984170i	$-0.556712\pi$
$467$	−3577.10	−0.354450	−0.177225	+	0.984170i	$0.556712\pi$
$468$	−228.108	−0.0225306
$469$	238.366	0.0234685
$470$	2788.32	0.273650
$471$	−19076.4	−1.86623
$472$	2895.49	0.282364
$473$	37752.9	3.66994
$474$	−3902.87	−0.378196
$475$	19033.5	1.83857
$476$	11.1073	0.00106954
$477$	182.556	0.0175234
$478$	−1293.45	−0.123768
$479$	9330.82	0.890055	0.445027	−	0.895517i	$-0.353194\pi$
$479$	9330.82	0.890055	0.445027	+	0.895517i	$0.353194\pi$
$480$	−3009.50	−0.286175
$481$	21168.1	2.00662
$482$	9315.40	0.880300
$483$	−308.337	−0.0290472
$484$	15141.6	1.42201
$485$	2441.40	0.228574
$486$	502.031	0.0468572
$487$	6943.54	0.646082	0.323041	−	0.946385i	$-0.395295\pi$
$487$	6943.54	0.646082	0.323041	+	0.946385i	$0.395295\pi$
$488$	−4818.60	−0.446983
$489$	−3638.30	−0.336462
$490$	−12570.8	−1.15896
$491$	13671.8	1.25662	0.628309	−	0.777964i	$-0.283747\pi$
$491$	13671.8	1.25662	0.628309	+	0.777964i	$0.283747\pi$
$492$	−2473.79	−0.226681
$493$	459.572	0.0419839
$494$	−11344.5	−1.03323
$495$	1178.34	0.106995
$496$	0	0
$497$	−279.536	−0.0252292
$498$	4632.65	0.416856
$499$	−4233.11	−0.379759	−0.189880	−	0.981807i	$-0.560810\pi$
$499$	−4233.11	−0.379759	−0.189880	+	0.981807i	$0.560810\pi$
$500$	−6539.29	−0.584891
$501$	−4551.54	−0.405884
$502$	−9184.00	−0.816538
$503$	4147.64	0.367662	0.183831	−	0.982958i	$-0.441150\pi$
$503$	4147.64	0.367662	0.183831	+	0.982958i	$0.441150\pi$
$504$	8.85548	0.000782648 0
$505$	−23779.7	−2.09541
$506$	−6980.06	−0.613244
$507$	−9519.81	−0.833904
$508$	−7522.58	−0.657009
$509$	−9551.88	−0.831788	−0.415894	−	0.909413i	$-0.636531\pi$
$509$	−9551.88	−0.831788	−0.415894	+	0.909413i	$0.636531\pi$
$510$	−422.284	−0.0366648
$511$	−339.141	−0.0293595
$512$	−512.000	−0.0441942
$513$	12687.3	1.09192
$514$	8935.44	0.766781
$515$	−27707.9	−2.37078
$516$	−10786.7	−0.920271
$517$	5417.68	0.460869
$518$	−821.776	−0.0697042
$519$	17722.2	1.49888
$520$	9383.11	0.791301
$521$	−13773.6	−1.15822	−0.579109	−	0.815250i	$-0.696599\pi$
$521$	−13773.6	−1.15822	−0.579109	+	0.815250i	$0.696599\pi$
$522$	366.402	0.0307222
$523$	−2980.62	−0.249203	−0.124602	−	0.992207i	$-0.539765\pi$
$523$	−2980.62	−0.249203	−0.124602	+	0.992207i	$0.539765\pi$
$524$	3105.06	0.258864
$525$	−1351.19	−0.112326
$526$	−3079.06	−0.255235
$527$	0	0
$528$	−5847.42	−0.481963
$529$	−9786.36	−0.804336
$530$	−7509.36	−0.615445
$531$	323.920	0.0264726
$532$	440.410	0.0358913
$533$	7712.87	0.626795
$534$	9004.22	0.729683
$535$	5505.21	0.444881
$536$	−1541.76	−0.124243
$537$	961.090	0.0772330
$538$	2050.60	0.164326
$539$	−24425.0	−1.95188
$540$	−10493.7	−0.836255
$541$	−3572.56	−0.283912	−0.141956	−	0.989873i	$-0.545339\pi$
$541$	−3572.56	−0.283912	−0.141956	+	0.989873i	$0.545339\pi$
$542$	−13237.2	−1.04905
$543$	−14152.1	−1.11847
$544$	−71.8425	−0.00566217
$545$	−6352.96	−0.499323
$546$	805.348	0.0631241
$547$	5705.27	0.445960	0.222980	−	0.974823i	$-0.428422\pi$
$547$	5705.27	0.445960	0.222980	+	0.974823i	$0.428422\pi$
$548$	4048.72	0.315607
$549$	−539.059	−0.0419062
$550$	−30588.0	−2.37141
$551$	18222.3	1.40888
$552$	1994.34	0.153777
$553$	−472.398	−0.0363263
$554$	−2608.44	−0.200040
$555$	31242.9	2.38952
$556$	3747.29	0.285828
$557$	−15504.7	−1.17945	−0.589725	−	0.807604i	$-0.700764\pi$
$557$	−15504.7	−1.17945	−0.589725	+	0.807604i	$0.700764\pi$
$558$	0	0
$559$	33631.3	2.54464
$560$	−364.266	−0.0274876
$561$	−820.495	−0.0617492
$562$	−9261.74	−0.695165
$563$	−1416.77	−0.106056	−0.0530282	−	0.998593i	$-0.516887\pi$
$563$	−1416.77	−0.106056	−0.0530282	+	0.998593i	$0.516887\pi$
$564$	−1547.94	−0.115567
$565$	32399.3	2.41248
$566$	−16262.6	−1.20772
$567$	−870.785	−0.0644965
$568$	1808.05	0.133564
$569$	−16258.7	−1.19789	−0.598947	−	0.800789i	$-0.704414\pi$
$569$	−16258.7	−1.19789	−0.598947	+	0.800789i	$0.704414\pi$
$570$	−16743.8	−1.23039
$571$	1165.77	0.0854393	0.0427196	−	0.999087i	$-0.486398\pi$
$571$	1165.77	0.0854393	0.0427196	+	0.999087i	$0.486398\pi$
$572$	18231.3	1.33267
$573$	6327.72	0.461334
$574$	−299.425	−0.0217731
$575$	10432.4	0.756631
$576$	−57.2777	−0.00414335
$577$	9831.75	0.709361	0.354680	−	0.934988i	$-0.384590\pi$
$577$	9831.75	0.709361	0.354680	+	0.934988i	$0.384590\pi$
$578$	9815.92	0.706381
$579$	−14530.9	−1.04298
$580$	−15071.8	−1.07900
$581$	560.730	0.0400396
$582$	−1355.35	−0.0965307
$583$	−14590.6	−1.03650
$584$	2193.58	0.155430
$585$	1049.69	0.0741872
$586$	−9365.81	−0.660235
$587$	14252.7	1.00217	0.501083	−	0.865399i	$-0.332935\pi$
$587$	14252.7	1.00217	0.501083	+	0.865399i	$0.332935\pi$
$588$	6978.71	0.489451
$589$	0	0
$590$	−13324.3	−0.929749
$591$	2022.03	0.140736
$592$	5315.29	0.369015
$593$	−15398.3	−1.06633	−0.533163	−	0.846013i	$-0.678997\pi$
$593$	−15398.3	−1.06633	−0.533163	+	0.846013i	$0.678997\pi$
$594$	−20389.2	−1.40838
$595$	−51.1128	−0.00352171
$596$	−2725.70	−0.187331
$597$	22003.7	1.50846
$598$	−6218.02	−0.425207
$599$	2382.46	0.162512	0.0812560	−	0.996693i	$-0.474107\pi$
$599$	2382.46	0.162512	0.0812560	+	0.996693i	$0.474107\pi$
$600$	8739.59	0.594654
$601$	27262.6	1.85036	0.925180	−	0.379530i	$-0.123914\pi$
$601$	27262.6	1.85036	0.925180	+	0.379530i	$0.123914\pi$
$602$	−1305.61	−0.0883934
$603$	−172.478	−0.0116482
$604$	−8327.75	−0.561012
$605$	−69677.7	−4.68231
$606$	13201.3	0.884926
$607$	6974.72	0.466384	0.233192	−	0.972431i	$-0.425083\pi$
$607$	6974.72	0.466384	0.233192	+	0.972431i	$0.425083\pi$
$608$	−2848.59	−0.190009
$609$	−1293.60	−0.0860746
$610$	22173.9	1.47180
$611$	4826.21	0.319554
$612$	−8.03706	−0.000530848 0
$613$	−25337.5	−1.66945	−0.834726	−	0.550666i	$-0.814374\pi$
$613$	−25337.5	−1.66945	−0.834726	+	0.550666i	$0.814374\pi$
$614$	9059.99	0.595491
$615$	11383.7	0.746401
$616$	−707.765	−0.0462933
$617$	−15558.8	−1.01519	−0.507597	−	0.861594i	$-0.669466\pi$
$617$	−15558.8	−1.01519	−0.507597	+	0.861594i	$0.669466\pi$
$618$	15382.0	1.00122
$619$	7290.46	0.473390	0.236695	−	0.971584i	$-0.423936\pi$
$619$	7290.46	0.473390	0.236695	+	0.971584i	$0.423936\pi$
$620$	0	0
$621$	6954.00	0.449363
$622$	17535.5	1.13040
$623$	1089.86	0.0700871
$624$	−5209.04	−0.334180
$625$	3365.13	0.215368
$626$	−2094.47	−0.133725
$627$	−32533.1	−2.07216
$628$	14934.6	0.948976
$629$	745.827	0.0472783
$630$	−40.7506	−0.00257705
$631$	−7407.55	−0.467337	−0.233669	−	0.972316i	$-0.575073\pi$
$631$	−7407.55	−0.467337	−0.233669	+	0.972316i	$0.575073\pi$
$632$	3055.50	0.192312
$633$	27697.5	1.73914
$634$	−2357.83	−0.147699
$635$	34616.9	2.16335
$636$	4168.82	0.259913
$637$	−21758.5	−1.35338
$638$	−29284.3	−1.81720
$639$	202.268	0.0125221
$640$	2356.09	0.145520
$641$	6916.56	0.426190	0.213095	−	0.977031i	$-0.431646\pi$
$641$	6916.56	0.426190	0.213095	+	0.977031i	$0.431646\pi$
$642$	−3056.22	−0.187881
$643$	−3785.50	−0.232170	−0.116085	−	0.993239i	$-0.537035\pi$
$643$	−3785.50	−0.232170	−0.116085	+	0.993239i	$0.537035\pi$
$644$	241.392	0.0147705
$645$	49637.7	3.03021
$646$	−399.707	−0.0243441
$647$	−17649.1	−1.07242	−0.536212	−	0.844083i	$-0.680145\pi$
$647$	−17649.1	−1.07242	−0.536212	+	0.844083i	$0.680145\pi$
$648$	5632.28	0.341446
$649$	−25889.0	−1.56584
$650$	−27248.6	−1.64427
$651$	0	0
$652$	2848.37	0.171090
$653$	−2285.07	−0.136940	−0.0684699	−	0.997653i	$-0.521812\pi$
$653$	−2285.07	−0.136940	−0.0684699	+	0.997653i	$0.521812\pi$
$654$	3526.85	0.210873
$655$	−14288.6	−0.852372
$656$	1936.69	0.115267
$657$	245.397	0.0145721
$658$	−187.360	−0.0111004
$659$	−18327.1	−1.08334	−0.541671	−	0.840591i	$-0.682208\pi$
$659$	−18327.1	−1.08334	−0.541671	+	0.840591i	$0.682208\pi$
$660$	26908.3	1.58698
$661$	−5361.10	−0.315465	−0.157733	−	0.987482i	$-0.550418\pi$
$661$	−5361.10	−0.315465	−0.157733	+	0.987482i	$0.550418\pi$
$662$	11463.8	0.673041
$663$	−730.918	−0.0428152
$664$	−3626.83	−0.211970
$665$	−2026.65	−0.118181
$666$	594.624	0.0345964
$667$	9987.79	0.579803
$668$	3563.33	0.206391
$669$	9943.49	0.574645
$670$	7094.79	0.409098
$671$	43083.7	2.47873
$672$	202.222	0.0116085
$673$	9549.84	0.546982	0.273491	−	0.961875i	$-0.411822\pi$
$673$	9549.84	0.546982	0.273491	+	0.961875i	$0.411822\pi$
$674$	−9329.95	−0.533199
$675$	30473.8	1.73769
$676$	7452.91	0.424039
$677$	4768.08	0.270683	0.135341	−	0.990799i	$-0.456787\pi$
$677$	4768.08	0.270683	0.135341	+	0.990799i	$0.456787\pi$
$678$	−17986.5	−1.01883
$679$	−164.049	−0.00927192
$680$	330.600	0.0186440
$681$	19828.4	1.11575
$682$	0	0
$683$	−24923.5	−1.39630	−0.698149	−	0.715952i	$-0.745993\pi$
$683$	−24923.5	−1.39630	−0.698149	+	0.715952i	$0.745993\pi$
$684$	−318.674	−0.0178140
$685$	−18631.1	−1.03921
$686$	1693.17	0.0942356
$687$	−24292.1	−1.34905
$688$	8444.78	0.467957
$689$	−12997.7	−0.718683
$690$	−9177.42	−0.506346
$691$	446.948	0.0246059	0.0123030	−	0.999924i	$-0.496084\pi$
$691$	446.948	0.0246059	0.0123030	+	0.999924i	$0.496084\pi$
$692$	−13874.4	−0.762178
$693$	−79.1780	−0.00434015
$694$	−16784.2	−0.918037
$695$	−17244.0	−0.941155
$696$	8367.09	0.455681
$697$	271.752	0.0147680
$698$	16959.9	0.919687
$699$	−22585.7	−1.22213
$700$	1057.83	0.0571174
$701$	13677.3	0.736926	0.368463	−	0.929642i	$-0.379884\pi$
$701$	13677.3	0.736926	0.368463	+	0.929642i	$0.379884\pi$
$702$	−18163.2	−0.976534
$703$	29572.5	1.58655
$704$	4577.86	0.245077
$705$	7123.19	0.380532
$706$	−12471.2	−0.664816
$707$	1597.87	0.0849985
$708$	7396.98	0.392649
$709$	−33031.4	−1.74967	−0.874837	−	0.484417i	$-0.839032\pi$
$709$	−33031.4	−1.74967	−0.874837	+	0.484417i	$0.839032\pi$
$710$	−8320.19	−0.439790
$711$	341.820	0.0180299
$712$	−7049.26	−0.371043
$713$	0	0
$714$	28.3753	0.00148728
$715$	−83895.6	−4.38814
$716$	−752.423	−0.0392728
$717$	−3304.32	−0.172109
$718$	1258.72	0.0654249
$719$	19130.4	0.992271	0.496136	−	0.868245i	$-0.334752\pi$
$719$	19130.4	0.992271	0.496136	+	0.868245i	$0.334752\pi$
$720$	263.577	0.0136430
$721$	1861.82	0.0961690
$722$	−2130.61	−0.109824
$723$	23797.6	1.22413
$724$	11079.5	0.568738
$725$	43768.5	2.24210
$726$	38681.6	1.97742
$727$	−14556.7	−0.742609	−0.371305	−	0.928511i	$-0.621089\pi$
$727$	−14556.7	−0.742609	−0.371305	+	0.928511i	$0.621089\pi$
$728$	−630.495	−0.0320985
$729$	20291.5	1.03091
$730$	−10094.3	−0.511789
$731$	1184.95	0.0599547
$732$	−12309.9	−0.621565
$733$	−19943.5	−1.00495	−0.502475	−	0.864592i	$-0.667577\pi$
$733$	−19943.5	−1.00495	−0.502475	+	0.864592i	$0.667577\pi$
$734$	−4594.02	−0.231019
$735$	−32114.2	−1.61163
$736$	−1561.34	−0.0781952
$737$	13785.1	0.688984
$738$	216.659	0.0108067
$739$	−13368.6	−0.665457	−0.332728	−	0.943023i	$-0.607969\pi$
$739$	−13368.6	−0.665457	−0.332728	+	0.943023i	$0.607969\pi$
$740$	−24459.6	−1.21507
$741$	−28981.3	−1.43678
$742$	504.589	0.0249650
$743$	23187.4	1.14490	0.572452	−	0.819938i	$-0.305992\pi$
$743$	23187.4	1.14490	0.572452	+	0.819938i	$0.305992\pi$
$744$	0	0
$745$	12543.0	0.616830
$746$	8552.83	0.419761
$747$	−405.736	−0.0198729
$748$	642.353	0.0313994
$749$	−369.921	−0.0180462
$750$	−16705.6	−0.813337
$751$	−5785.58	−0.281117	−0.140558	−	0.990072i	$-0.544890\pi$
$751$	−5785.58	−0.281117	−0.140558	+	0.990072i	$0.544890\pi$
$752$	1211.86	0.0587657
$753$	−23462.0	−1.13546
$754$	−26087.2	−1.26000
$755$	38322.1	1.84726
$756$	705.122	0.0339220
$757$	32841.2	1.57679	0.788397	−	0.615167i	$-0.210911\pi$
$757$	32841.2	1.57679	0.788397	+	0.615167i	$0.210911\pi$
$758$	−11016.4	−0.527879
$759$	−17831.6	−0.852764
$760$	13108.5	0.625651
$761$	−33975.7	−1.61842	−0.809210	−	0.587520i	$-0.800104\pi$
$761$	−33975.7	−1.61842	−0.809210	+	0.587520i	$0.800104\pi$
$762$	−19217.6	−0.913622
$763$	426.885	0.0202546
$764$	−4953.88	−0.234588
$765$	36.9844	0.00174794
$766$	−1280.69	−0.0604090
$767$	−23062.6	−1.08571
$768$	−1307.98	−0.0614554
$769$	−15155.9	−0.710709	−0.355355	−	0.934732i	$-0.615640\pi$
$769$	−15155.9	−0.710709	−0.355355	+	0.934732i	$0.615640\pi$
$770$	3256.95	0.152431
$771$	22827.0	1.06627
$772$	11376.0	0.530352
$773$	−7946.70	−0.369758	−0.184879	−	0.982761i	$-0.559189\pi$
$773$	−7946.70	−0.369758	−0.184879	+	0.982761i	$0.559189\pi$
$774$	944.722	0.0438725
$775$	0	0
$776$	1061.08	0.0490857
$777$	−2099.35	−0.0969291
$778$	26686.3	1.22976
$779$	10775.1	0.495582
$780$	23970.6	1.10037
$781$	−16166.0	−0.740674
$782$	−219.083	−0.0100184
$783$	29175.0	1.33158
$784$	−5463.52	−0.248885
$785$	−68725.3	−3.12473
$786$	7932.34	0.359971
$787$	26688.8	1.20883	0.604416	−	0.796669i	$-0.293406\pi$
$787$	26688.8	1.20883	0.604416	+	0.796669i	$0.293406\pi$
$788$	−1583.01	−0.0715641
$789$	−7865.94	−0.354924
$790$	−14060.6	−0.633232
$791$	−2177.06	−0.0978601
$792$	512.127	0.0229768
$793$	38380.1	1.71868
$794$	−23816.5	−1.06451
$795$	−19183.8	−0.855824
$796$	−17226.3	−0.767049
$797$	−4914.80	−0.218433	−0.109217	−	0.994018i	$-0.534834\pi$
$797$	−4914.80	−0.218433	−0.109217	+	0.994018i	$0.534834\pi$
$798$	1125.10	0.0499097
$799$	170.044	0.00752908
$800$	−6842.10	−0.302381
$801$	−788.605	−0.0347865
$802$	24764.1	1.09034
$803$	−19613.1	−0.861931
$804$	−3938.67	−0.172769
$805$	−1110.82	−0.0486353
$806$	0	0
$807$	5238.57	0.228509
$808$	−10335.1	−0.449984
$809$	−33201.9	−1.44291	−0.721456	−	0.692460i	$-0.756527\pi$
$809$	−33201.9	−1.44291	−0.721456	+	0.692460i	$0.756527\pi$
$810$	−25918.3	−1.12429
$811$	−10480.4	−0.453781	−0.226891	−	0.973920i	$-0.572856\pi$
$811$	−10480.4	−0.453781	−0.226891	+	0.973920i	$0.572856\pi$
$812$	1012.74	0.0437688
$813$	−33816.5	−1.45879
$814$	−47524.7	−2.04636
$815$	−13107.5	−0.563355
$816$	−183.533	−0.00787369
$817$	46983.9	2.01194
$818$	−1352.90	−0.0578278
$819$	−70.5338	−0.00300934
$820$	−8912.15	−0.379544
$821$	24957.8	1.06094	0.530471	−	0.847703i	$-0.322015\pi$
$821$	24957.8	1.06094	0.530471	+	0.847703i	$0.322015\pi$
$822$	10343.1	0.438876
$823$	3465.99	0.146800	0.0734002	−	0.997303i	$-0.476615\pi$
$823$	3465.99	0.146800	0.0734002	+	0.997303i	$0.476615\pi$
$824$	−12042.4	−0.509120
$825$	−78141.8	−3.29764
$826$	895.321	0.0377145
$827$	−6211.01	−0.261158	−0.130579	−	0.991438i	$-0.541684\pi$
$827$	−6211.01	−0.261158	−0.130579	+	0.991438i	$0.541684\pi$
$828$	−174.668	−0.00733107
$829$	3808.39	0.159555	0.0797774	−	0.996813i	$-0.474579\pi$
$829$	3808.39	0.159555	0.0797774	+	0.996813i	$0.474579\pi$
$830$	16689.7	0.697962
$831$	−6663.67	−0.278171
$832$	4078.07	0.169930
$833$	−766.627	−0.0318872
$834$	9573.02	0.397466
$835$	−16397.5	−0.679592
$836$	25469.6	1.05369
$837$	0	0
$838$	11973.4	0.493572
$839$	−13545.9	−0.557397	−0.278699	−	0.960379i	$-0.589903\pi$
$839$	−13545.9	−0.557397	−0.278699	+	0.960379i	$0.589903\pi$
$840$	−930.573	−0.0382236
$841$	17513.9	0.718108
$842$	−18017.1	−0.737424
$843$	−23660.5	−0.966682
$844$	−21684.0	−0.884352
$845$	−34296.3	−1.39625
$846$	135.571	0.00550949
$847$	4681.97	0.189934
$848$	−3263.71	−0.132165
$849$	−41545.3	−1.67943
$850$	−960.064	−0.0387411
$851$	16208.9	0.652919
$852$	4618.95	0.185731
$853$	−4422.94	−0.177537	−0.0887683	−	0.996052i	$-0.528293\pi$
$853$	−4422.94	−0.177537	−0.0887683	+	0.996052i	$0.528293\pi$
$854$	−1489.97	−0.0597022
$855$	1466.45	0.0586569
$856$	2392.67	0.0955370
$857$	12367.5	0.492958	0.246479	−	0.969148i	$-0.420726\pi$
$857$	12367.5	0.492958	0.246479	+	0.969148i	$0.420726\pi$
$858$	46574.7	1.85318
$859$	42773.8	1.69898	0.849490	−	0.527604i	$-0.176910\pi$
$859$	42773.8	1.69898	0.849490	+	0.527604i	$0.176910\pi$
$860$	−38860.6	−1.54086
$861$	−764.926	−0.0302771
$862$	25638.1	1.01304
$863$	−14497.9	−0.571859	−0.285930	−	0.958251i	$-0.592302\pi$
$863$	−14497.9	−0.571859	−0.285930	+	0.958251i	$0.592302\pi$
$864$	−4560.77	−0.179584
$865$	63846.5	2.50965
$866$	−4863.03	−0.190823
$867$	25076.3	0.982278
$868$	0	0
$869$	−27319.6	−1.06646
$870$	−38503.1	−1.50043
$871$	12280.1	0.477723
$872$	−2761.11	−0.107228
$873$	118.704	0.00460195
$874$	−8686.76	−0.336194
$875$	−2022.03	−0.0781222
$876$	5603.84	0.216137
$877$	−4149.96	−0.159788	−0.0798941	−	0.996803i	$-0.525458\pi$
$877$	−4149.96	−0.159788	−0.0798941	+	0.996803i	$0.525458\pi$
$878$	5519.82	0.212170
$879$	−23926.4	−0.918108
$880$	−21066.1	−0.806975
$881$	−20067.4	−0.767411	−0.383706	−	0.923455i	$-0.625352\pi$
$881$	−20067.4	−0.767411	−0.383706	+	0.923455i	$0.625352\pi$
$882$	−611.207	−0.0233338
$883$	28822.4	1.09847	0.549236	−	0.835667i	$-0.314919\pi$
$883$	28822.4	1.09847	0.549236	+	0.835667i	$0.314919\pi$
$884$	572.225	0.0217715
$885$	−34038.9	−1.29289
$886$	33615.8	1.27466
$887$	−18786.4	−0.711146	−0.355573	−	0.934648i	$-0.615714\pi$
$887$	−18786.4	−0.711146	−0.355573	+	0.934648i	$0.615714\pi$
$888$	13578.7	0.513144
$889$	−2326.07	−0.0877547
$890$	32438.8	1.22174
$891$	−50358.9	−1.89348
$892$	−7784.61	−0.292206
$893$	6742.35	0.252659
$894$	−6963.23	−0.260498
$895$	3462.45	0.129315
$896$	−158.317	−0.00590289
$897$	−15884.9	−0.591283
$898$	17339.8	0.644361
$899$	0	0
$900$	−765.429	−0.0283492
$901$	−457.955	−0.0169331
$902$	−17316.2	−0.639210
$903$	−3335.39	−0.122918
$904$	14081.3	0.518073
$905$	−50984.9	−1.87270
$906$	−21274.5	−0.780131
$907$	−50237.1	−1.83914	−0.919568	−	0.392931i	$-0.871461\pi$
$907$	−50237.1	−1.83914	−0.919568	+	0.392931i	$0.871461\pi$
$908$	−15523.3	−0.567357
$909$	−1156.19	−0.0421875
$910$	2901.37	0.105692
$911$	17880.3	0.650276	0.325138	−	0.945667i	$-0.394589\pi$
$911$	17880.3	0.650276	0.325138	+	0.945667i	$0.394589\pi$
$912$	−7277.17	−0.264223
$913$	32428.0	1.17548
$914$	−3342.06	−0.120947
$915$	56646.7	2.04665
$916$	19017.9	0.685992
$917$	960.121	0.0345758
$918$	−639.955	−0.0230083
$919$	−12674.0	−0.454927	−0.227463	−	0.973787i	$-0.573043\pi$
$919$	−12674.0	−0.454927	−0.227463	+	0.973787i	$0.573043\pi$
$920$	7184.87	0.257476
$921$	23145.1	0.828076
$922$	12824.7	0.458089
$923$	−14401.1	−0.513563
$924$	−1808.09	−0.0643744
$925$	71030.7	2.52484
$926$	−20642.4	−0.732561
$927$	−1347.18	−0.0477318
$928$	−6550.47	−0.231713
$929$	11967.2	0.422638	0.211319	−	0.977417i	$-0.432224\pi$
$929$	11967.2	0.422638	0.211319	+	0.977417i	$0.432224\pi$
$930$	0	0
$931$	−30397.2	−1.07006
$932$	17682.0	0.621451
$933$	44797.0	1.57191
$934$	7154.20	0.250634
$935$	−2955.94	−0.103390
$936$	456.216	0.0159315
$937$	−3655.73	−0.127457	−0.0637286	−	0.997967i	$-0.520299\pi$
$937$	−3655.73	−0.127457	−0.0637286	+	0.997967i	$0.520299\pi$
$938$	−476.732	−0.0165947
$939$	−5350.66	−0.185955
$940$	−5576.64	−0.193500
$941$	24742.5	0.857155	0.428578	−	0.903505i	$-0.359015\pi$
$941$	24742.5	0.857155	0.428578	+	0.903505i	$0.359015\pi$
$942$	38152.9	1.31963
$943$	5905.92	0.203948
$944$	−5790.98	−0.199661
$945$	−3244.79	−0.111696
$946$	−75505.9	−2.59504
$947$	−9498.71	−0.325941	−0.162971	−	0.986631i	$-0.552108\pi$
$947$	−9498.71	−0.325941	−0.162971	+	0.986631i	$0.552108\pi$
$948$	7805.74	0.267425
$949$	−17471.8	−0.597640
$950$	−38067.1	−1.30006
$951$	−6023.44	−0.205387
$952$	−22.2146	−0.000756279 0
$953$	−9391.69	−0.319231	−0.159615	−	0.987179i	$-0.551025\pi$
$953$	−9391.69	−0.319231	−0.159615	+	0.987179i	$0.551025\pi$
$954$	−365.113	−0.0123909
$955$	22796.4	0.772434
$956$	2586.90	0.0875170
$957$	−74811.2	−2.52696
$958$	−18661.6	−0.629364
$959$	1251.91	0.0421547
$960$	6018.99	0.202356
$961$	0	0
$962$	−42336.2	−1.41889
$963$	267.669	0.00895692
$964$	−18630.8	−0.622466
$965$	−52349.4	−1.74631
$966$	616.674	0.0205395
$967$	−42697.0	−1.41990	−0.709950	−	0.704252i	$-0.751283\pi$
$967$	−42697.0	−1.41990	−0.709950	+	0.704252i	$0.751283\pi$
$968$	−30283.2	−1.00552
$969$	−1021.11	−0.0338523
$970$	−4882.81	−0.161626
$971$	3512.02	0.116072	0.0580361	−	0.998314i	$-0.481516\pi$
$971$	3512.02	0.116072	0.0580361	+	0.998314i	$0.481516\pi$
$972$	−1004.06	−0.0331330
$973$	1158.71	0.0381772
$974$	−13887.1	−0.456849
$975$	−69610.8	−2.28649
$976$	9637.20	0.316065
$977$	−15209.1	−0.498038	−0.249019	−	0.968499i	$-0.580108\pi$
$977$	−15209.1	−0.498038	−0.249019	+	0.968499i	$0.580108\pi$
$978$	7276.61	0.237914
$979$	63028.4	2.05761
$980$	25141.7	0.819512
$981$	−308.887	−0.0100530
$982$	−27343.6	−0.888563
$983$	13718.6	0.445124	0.222562	−	0.974919i	$-0.428558\pi$
$983$	13718.6	0.445124	0.222562	+	0.974919i	$0.428558\pi$
$984$	4947.58	0.160288
$985$	7284.61	0.235641
$986$	−919.144	−0.0296871
$987$	−478.641	−0.0154360
$988$	22689.0	0.730601
$989$	25752.3	0.827982
$990$	−2356.68	−0.0756566
$991$	−8851.99	−0.283747	−0.141873	−	0.989885i	$-0.545313\pi$
$991$	−8851.99	−0.283747	−0.141873	+	0.989885i	$0.545313\pi$
$992$	0	0
$993$	29286.0	0.935916
$994$	559.072	0.0178397
$995$	79271.1	2.52569
$996$	−9265.30	−0.294761
$997$	−51719.9	−1.64291	−0.821457	−	0.570271i	$-0.806838\pi$
$997$	−51719.9	−1.64291	−0.821457	+	0.570271i	$0.806838\pi$
$998$	8466.21	0.268530
$999$	47347.3	1.49950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1922.4.a.x.1.10	✓	32
31.30	odd	2	inner	1922.4.a.x.1.23	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1922.4.a.x.1.10	✓	32	1.1	even	1	trivial
1922.4.a.x.1.23	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -5.10931 q^{3} +4.00000 q^{4} -18.4069 q^{5} +10.2186 q^{6} +1.23685 q^{7} -8.00000 q^{8} -0.894964 q^{9} +36.8139 q^{10} +71.5290 q^{11} -20.4372 q^{12} +63.7199 q^{13} -2.47370 q^{14} +94.0467 q^{15} +16.0000 q^{16} +2.24508 q^{17} +1.78993 q^{18} +89.0186 q^{19} -73.6278 q^{20} -6.31944 q^{21} -143.058 q^{22} +48.7918 q^{23} +40.8745 q^{24} +213.815 q^{25} -127.440 q^{26} +142.524 q^{27} +4.94739 q^{28} +204.702 q^{29} -188.093 q^{30} -32.0000 q^{32} -365.464 q^{33} -4.49015 q^{34} -22.7666 q^{35} -3.57986 q^{36} +332.206 q^{37} -178.037 q^{38} -325.565 q^{39} +147.256 q^{40} +121.043 q^{41} +12.6389 q^{42} +527.799 q^{43} +286.116 q^{44} +16.4736 q^{45} -97.5836 q^{46} +75.7410 q^{47} -81.7489 q^{48} -341.470 q^{49} -427.631 q^{50} -11.4708 q^{51} +254.880 q^{52} -203.982 q^{53} -285.048 q^{54} -1316.63 q^{55} -9.89479 q^{56} -454.823 q^{57} -409.404 q^{58} -361.936 q^{59} +376.187 q^{60} +602.325 q^{61} -1.10694 q^{63} +64.0000 q^{64} -1172.89 q^{65} +730.928 q^{66} +192.721 q^{67} +8.98031 q^{68} -249.292 q^{69} +45.5332 q^{70} -226.007 q^{71} +7.15971 q^{72} -274.197 q^{73} -664.411 q^{74} -1092.45 q^{75} +356.074 q^{76} +88.4706 q^{77} +651.129 q^{78} -381.937 q^{79} -294.511 q^{80} -704.035 q^{81} -242.087 q^{82} +453.354 q^{83} -25.2778 q^{84} -41.3250 q^{85} -1055.60 q^{86} -1045.89 q^{87} -572.232 q^{88} +881.158 q^{89} -32.9471 q^{90} +78.8119 q^{91} +195.167 q^{92} -151.482 q^{94} -1638.56 q^{95} +163.498 q^{96} -132.635 q^{97} +682.940 q^{98} -64.0159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) \) 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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