LMFDB - Embedded newform 693.4.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("693.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-3,0,-3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.56155 q^{2} +4.68466 q^{4} +5.68466 q^{5} +7.00000 q^{7} +11.8078 q^{8} -20.2462 q^{10} +11.0000 q^{11} -39.3002 q^{13} -24.9309 q^{14} -79.5312 q^{16} +64.2462 q^{17} -82.9157 q^{19} +26.6307 q^{20} -39.1771 q^{22} +11.5076 q^{23} -92.6847 q^{25} +139.970 q^{26} +32.7926 q^{28} -128.501 q^{29} -101.400 q^{31} +188.793 q^{32} -228.816 q^{34} +39.7926 q^{35} +188.086 q^{37} +295.309 q^{38} +67.1231 q^{40} -198.294 q^{41} +231.386 q^{43} +51.5312 q^{44} -40.9848 q^{46} +285.533 q^{47} +49.0000 q^{49} +330.101 q^{50} -184.108 q^{52} -202.570 q^{53} +62.5312 q^{55} +82.6543 q^{56} +457.663 q^{58} -103.762 q^{59} -382.924 q^{61} +361.140 q^{62} -36.1449 q^{64} -223.408 q^{65} -259.348 q^{67} +300.972 q^{68} -141.723 q^{70} -352.553 q^{71} +490.012 q^{73} -669.879 q^{74} -388.432 q^{76} +77.0000 q^{77} +141.633 q^{79} -452.108 q^{80} +706.233 q^{82} +192.847 q^{83} +365.218 q^{85} -824.095 q^{86} +129.885 q^{88} +423.295 q^{89} -275.101 q^{91} +53.9091 q^{92} -1016.94 q^{94} -471.348 q^{95} -1095.89 q^{97} -174.516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 3 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} - 24 q^{10} + 22 q^{11} - 25 q^{13} - 21 q^{14} - 23 q^{16} + 112 q^{17} - 71 q^{19} + 78 q^{20} - 33 q^{22} + 56 q^{23} - 173 q^{25} + 148 q^{26} - 21 q^{28}+ \cdots - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 - 3 * q^4 - q^5 + 14 * q^7 + 3 * q^8 - 24 * q^10 + 22 * q^11 - 25 * q^13 - 21 * q^14 - 23 * q^16 + 112 * q^17 - 71 * q^19 + 78 * q^20 - 33 * q^22 + 56 * q^23 - 173 * q^25 + 148 * q^26 - 21 * q^28 + 11 * q^29 - 310 * q^31 + 291 * q^32 - 202 * q^34 - 7 * q^35 - 65 * q^37 + 302 * q^38 + 126 * q^40 - 42 * q^41 - 32 * q^43 - 33 * q^44 - 16 * q^46 - 101 * q^47 + 98 * q^49 + 285 * q^50 - 294 * q^52 - 166 * q^53 - 11 * q^55 + 21 * q^56 + 536 * q^58 + 11 * q^59 - 436 * q^61 + 244 * q^62 - 431 * q^64 - 319 * q^65 - 127 * q^67 - 66 * q^68 - 168 * q^70 - 936 * q^71 - 327 * q^73 - 812 * q^74 - 480 * q^76 + 154 * q^77 - 228 * q^79 - 830 * q^80 + 794 * q^82 + 262 * q^83 + 46 * q^85 - 972 * q^86 + 33 * q^88 - 44 * q^89 - 175 * q^91 - 288 * q^92 - 1234 * q^94 - 551 * q^95 - 2266 * q^97 - 147 * 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 $

$2$	−3.56155	−1.25920	−0.629600	−	0.776920i	$-0.716781\pi$
$2$	−3.56155	−1.25920	−0.629600	+	0.776920i	$0.716781\pi$
$3$	0	0
$3$	0	0
$4$	4.68466	0.585582
$5$	5.68466	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$5$	5.68466	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	11.8078	0.521834
$9$	0	0
$10$	−20.2462	−0.640241
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−39.3002	−0.838455	−0.419227	−	0.907881i	$-0.637699\pi$
$13$	−39.3002	−0.838455	−0.419227	+	0.907881i	$0.637699\pi$
$14$	−24.9309	−0.475933
$15$	0	0
$16$	−79.5312	−1.24268
$17$	64.2462	0.916588	0.458294	−	0.888801i	$-0.348461\pi$
$17$	64.2462	0.916588	0.458294	+	0.888801i	$0.348461\pi$
$18$	0	0
$19$	−82.9157	−1.00117	−0.500583	−	0.865688i	$-0.666881\pi$
$19$	−82.9157	−1.00117	−0.500583	+	0.865688i	$0.666881\pi$
$20$	26.6307	0.297740
$21$	0	0
$22$	−39.1771	−0.379663
$23$	11.5076	0.104326	0.0521630	−	0.998639i	$-0.483388\pi$
$23$	11.5076	0.104326	0.0521630	+	0.998639i	$0.483388\pi$
$24$	0	0
$25$	−92.6847	−0.741477
$26$	139.970	1.05578
$27$	0	0
$28$	32.7926	0.221329
$29$	−128.501	−0.822828	−0.411414	−	0.911448i	$-0.634965\pi$
$29$	−128.501	−0.822828	−0.411414	+	0.911448i	$0.634965\pi$
$30$	0	0
$31$	−101.400	−0.587481	−0.293740	−	0.955885i	$-0.594900\pi$
$31$	−101.400	−0.587481	−0.293740	+	0.955885i	$0.594900\pi$
$32$	188.793	1.04294
$33$	0	0
$34$	−228.816	−1.15417
$35$	39.7926	0.192177
$36$	0	0
$37$	188.086	0.835707	0.417854	−	0.908514i	$-0.362782\pi$
$37$	188.086	0.835707	0.417854	+	0.908514i	$0.362782\pi$
$38$	295.309	1.26067
$39$	0	0
$40$	67.1231	0.265327
$41$	−198.294	−0.755323	−0.377662	−	0.925944i	$-0.623272\pi$
$41$	−198.294	−0.755323	−0.377662	+	0.925944i	$0.623272\pi$
$42$	0	0
$43$	231.386	0.820607	0.410303	−	0.911949i	$-0.365423\pi$
$43$	231.386	0.820607	0.410303	+	0.911949i	$0.365423\pi$
$44$	51.5312	0.176560
$45$	0	0
$46$	−40.9848	−0.131367
$47$	285.533	0.886155	0.443077	−	0.896483i	$-0.353887\pi$
$47$	285.533	0.886155	0.443077	+	0.896483i	$0.353887\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	330.101	0.933667
$51$	0	0
$52$	−184.108	−0.490984
$53$	−202.570	−0.525003	−0.262501	−	0.964932i	$-0.584547\pi$
$53$	−202.570	−0.525003	−0.262501	+	0.964932i	$0.584547\pi$
$54$	0	0
$55$	62.5312	0.153304
$56$	82.6543	0.197235
$57$	0	0
$58$	457.663	1.03610
$59$	−103.762	−0.228961	−0.114481	−	0.993425i	$-0.536520\pi$
$59$	−103.762	−0.228961	−0.114481	+	0.993425i	$0.536520\pi$
$60$	0	0
$61$	−382.924	−0.803745	−0.401872	−	0.915696i	$-0.631640\pi$
$61$	−382.924	−0.803745	−0.401872	+	0.915696i	$0.631640\pi$
$62$	361.140	0.739756
$63$	0	0
$64$	−36.1449	−0.0705955
$65$	−223.408	−0.426313
$66$	0	0
$67$	−259.348	−0.472901	−0.236450	−	0.971644i	$-0.575984\pi$
$67$	−259.348	−0.472901	−0.236450	+	0.971644i	$0.575984\pi$
$68$	300.972	0.536738
$69$	0	0
$70$	−141.723	−0.241989
$71$	−352.553	−0.589301	−0.294650	−	0.955605i	$-0.595203\pi$
$71$	−352.553	−0.589301	−0.294650	+	0.955605i	$0.595203\pi$
$72$	0	0
$73$	490.012	0.785638	0.392819	−	0.919616i	$-0.371500\pi$
$73$	490.012	0.785638	0.392819	+	0.919616i	$0.371500\pi$
$74$	−669.879	−1.05232
$75$	0	0
$76$	−388.432	−0.586266
$77$	77.0000	0.113961
$78$	0	0
$79$	141.633	0.201708	0.100854	−	0.994901i	$-0.467843\pi$
$79$	141.633	0.201708	0.100854	+	0.994901i	$0.467843\pi$
$80$	−452.108	−0.631840
$81$	0	0
$82$	706.233	0.951102
$83$	192.847	0.255032	0.127516	−	0.991837i	$-0.459300\pi$
$83$	192.847	0.255032	0.127516	+	0.991837i	$0.459300\pi$
$84$	0	0
$85$	365.218	0.466040
$86$	−824.095	−1.03331
$87$	0	0
$88$	129.885	0.157339
$89$	423.295	0.504149	0.252074	−	0.967708i	$-0.418887\pi$
$89$	423.295	0.504149	0.252074	+	0.967708i	$0.418887\pi$
$90$	0	0
$91$	−275.101	−0.316906
$92$	53.9091	0.0610914
$93$	0	0
$94$	−1016.94	−1.11585
$95$	−471.348	−0.509045
$96$	0	0
$97$	−1095.89	−1.14712	−0.573562	−	0.819162i	$-0.694439\pi$
$97$	−1095.89	−1.14712	−0.573562	+	0.819162i	$0.694439\pi$
$98$	−174.516	−0.179886
$99$	0	0
$100$	−434.196	−0.434196
$101$	1781.30	1.75491	0.877455	−	0.479659i	$-0.159240\pi$
$101$	1781.30	1.75491	0.877455	+	0.479659i	$0.159240\pi$
$102$	0	0
$103$	10.3239	0.00987611	0.00493806	−	0.999988i	$-0.498428\pi$
$103$	10.3239	0.00987611	0.00493806	+	0.999988i	$0.498428\pi$
$104$	−464.047	−0.437534
$105$	0	0
$106$	721.464	0.661083
$107$	−1518.89	−1.37230	−0.686152	−	0.727458i	$-0.740702\pi$
$107$	−1518.89	−1.37230	−0.686152	+	0.727458i	$0.740702\pi$
$108$	0	0
$109$	−1648.93	−1.44898	−0.724489	−	0.689286i	$-0.757924\pi$
$109$	−1648.93	−1.44898	−0.724489	+	0.689286i	$0.757924\pi$
$110$	−222.708	−0.193040
$111$	0	0
$112$	−556.719	−0.469687
$113$	188.739	0.157124	0.0785621	−	0.996909i	$-0.474967\pi$
$113$	188.739	0.157124	0.0785621	+	0.996909i	$0.474967\pi$
$114$	0	0
$115$	65.4166	0.0530446
$116$	−601.983	−0.481834
$117$	0	0
$118$	369.555	0.288308
$119$	449.723	0.346438
$120$	0	0
$121$	121.000	0.0909091
$122$	1363.80	1.01207
$123$	0	0
$124$	−475.023	−0.344018
$125$	−1237.46	−0.885456
$126$	0	0
$127$	−2398.92	−1.67614	−0.838069	−	0.545564i	$-0.816316\pi$
$127$	−2398.92	−1.67614	−0.838069	+	0.545564i	$0.816316\pi$
$128$	−1381.61	−0.954048
$129$	0	0
$130$	795.680	0.536813
$131$	−1560.96	−1.04108	−0.520540	−	0.853837i	$-0.674269\pi$
$131$	−1560.96	−1.04108	−0.520540	+	0.853837i	$0.674269\pi$
$132$	0	0
$133$	−580.410	−0.378405
$134$	923.680	0.595476
$135$	0	0
$136$	758.604	0.478307
$137$	−821.595	−0.512362	−0.256181	−	0.966629i	$-0.582464\pi$
$137$	−821.595	−0.512362	−0.256181	+	0.966629i	$0.582464\pi$
$138$	0	0
$139$	−1158.44	−0.706886	−0.353443	−	0.935456i	$-0.614989\pi$
$139$	−1158.44	−0.706886	−0.353443	+	0.935456i	$0.614989\pi$
$140$	186.415	0.112535
$141$	0	0
$142$	1255.64	0.742047
$143$	−432.302	−0.252804
$144$	0	0
$145$	−730.484	−0.418368
$146$	−1745.20	−0.989275
$147$	0	0
$148$	881.119	0.489375
$149$	−1234.82	−0.678926	−0.339463	−	0.940619i	$-0.610245\pi$
$149$	−1234.82	−0.678926	−0.339463	+	0.940619i	$0.610245\pi$
$150$	0	0
$151$	−1154.58	−0.622242	−0.311121	−	0.950370i	$-0.600704\pi$
$151$	−1154.58	−0.622242	−0.311121	+	0.950370i	$0.600704\pi$
$152$	−979.049	−0.522443
$153$	0	0
$154$	−274.240	−0.143499
$155$	−576.422	−0.298705
$156$	0	0
$157$	−2830.17	−1.43868	−0.719339	−	0.694660i	$-0.755555\pi$
$157$	−2830.17	−1.43868	−0.719339	+	0.694660i	$0.755555\pi$
$158$	−504.432	−0.253990
$159$	0	0
$160$	1073.22	0.530285
$161$	80.5530	0.0394315
$162$	0	0
$163$	3418.64	1.64275	0.821377	−	0.570386i	$-0.193206\pi$
$163$	3418.64	1.64275	0.821377	+	0.570386i	$0.193206\pi$
$164$	−928.938	−0.442304
$165$	0	0
$166$	−686.833	−0.321136
$167$	−1575.50	−0.730033	−0.365017	−	0.931001i	$-0.618937\pi$
$167$	−1575.50	−0.730033	−0.365017	+	0.931001i	$0.618937\pi$
$168$	0	0
$169$	−652.495	−0.296994
$170$	−1300.74	−0.586838
$171$	0	0
$172$	1083.97	0.480533
$173$	−3597.33	−1.58092	−0.790462	−	0.612510i	$-0.790160\pi$
$173$	−3597.33	−1.58092	−0.790462	+	0.612510i	$0.790160\pi$
$174$	0	0
$175$	−648.793	−0.280252
$176$	−874.844	−0.374681
$177$	0	0
$178$	−1507.59	−0.634823
$179$	−1812.71	−0.756917	−0.378459	−	0.925618i	$-0.623546\pi$
$179$	−1812.71	−0.756917	−0.378459	+	0.925618i	$0.623546\pi$
$180$	0	0
$181$	2239.16	0.919535	0.459767	−	0.888039i	$-0.347933\pi$
$181$	2239.16	0.919535	0.459767	+	0.888039i	$0.347933\pi$
$182$	979.788	0.399048
$183$	0	0
$184$	135.879	0.0544408
$185$	1069.21	0.424917
$186$	0	0
$187$	706.708	0.276362
$188$	1337.63	0.518917
$189$	0	0
$190$	1678.73	0.640988
$191$	3960.32	1.50031	0.750153	−	0.661264i	$-0.229980\pi$
$191$	3960.32	1.50031	0.750153	+	0.661264i	$0.229980\pi$
$192$	0	0
$193$	−4740.22	−1.76792	−0.883961	−	0.467561i	$-0.845133\pi$
$193$	−4740.22	−1.76792	−0.883961	+	0.467561i	$0.845133\pi$
$194$	3903.08	1.44446
$195$	0	0
$196$	229.548	0.0836546
$197$	5144.18	1.86045	0.930223	−	0.366996i	$-0.119614\pi$
$197$	5144.18	1.86045	0.930223	+	0.366996i	$0.119614\pi$
$198$	0	0
$199$	−2232.02	−0.795094	−0.397547	−	0.917582i	$-0.630138\pi$
$199$	−2232.02	−0.795094	−0.397547	+	0.917582i	$0.630138\pi$
$200$	−1094.40	−0.386928
$201$	0	0
$202$	−6344.19	−2.20978
$203$	−899.507	−0.311000
$204$	0	0
$205$	−1127.23	−0.384045
$206$	−36.7689	−0.0124360
$207$	0	0
$208$	3125.59	1.04193
$209$	−912.073	−0.301863
$210$	0	0
$211$	−3119.05	−1.01765	−0.508825	−	0.860870i	$-0.669920\pi$
$211$	−3119.05	−1.01765	−0.508825	+	0.860870i	$0.669920\pi$
$212$	−948.972	−0.307432
$213$	0	0
$214$	5409.60	1.72800
$215$	1315.35	0.417239
$216$	0	0
$217$	−709.797	−0.222047
$218$	5872.74	1.82455
$219$	0	0
$220$	292.938	0.0897720
$221$	−2524.89	−0.768517
$222$	0	0
$223$	−1138.94	−0.342015	−0.171008	−	0.985270i	$-0.554702\pi$
$223$	−1138.94	−0.342015	−0.171008	+	0.985270i	$0.554702\pi$
$224$	1321.55	0.394195
$225$	0	0
$226$	−672.203	−0.197851
$227$	1754.97	0.513133	0.256567	−	0.966527i	$-0.417409\pi$
$227$	1754.97	0.513133	0.256567	+	0.966527i	$0.417409\pi$
$228$	0	0
$229$	304.271	0.0878025	0.0439013	−	0.999036i	$-0.486021\pi$
$229$	304.271	0.0878025	0.0439013	+	0.999036i	$0.486021\pi$
$230$	−232.985	−0.0667938
$231$	0	0
$232$	−1517.31	−0.429380
$233$	3908.74	1.09901	0.549506	−	0.835490i	$-0.314816\pi$
$233$	3908.74	1.09901	0.549506	+	0.835490i	$0.314816\pi$
$234$	0	0
$235$	1623.16	0.450567
$236$	−486.091	−0.134076
$237$	0	0
$238$	−1601.71	−0.436234
$239$	170.995	0.0462794	0.0231397	−	0.999732i	$-0.492634\pi$
$239$	170.995	0.0462794	0.0231397	+	0.999732i	$0.492634\pi$
$240$	0	0
$241$	−6654.70	−1.77870	−0.889351	−	0.457226i	$-0.848843\pi$
$241$	−6654.70	−1.77870	−0.889351	+	0.457226i	$0.848843\pi$
$242$	−430.948	−0.114473
$243$	0	0
$244$	−1793.87	−0.470659
$245$	278.548	0.0726359
$246$	0	0
$247$	3258.60	0.839433
$248$	−1197.30	−0.306568
$249$	0	0
$250$	4407.29	1.11497
$251$	−1019.50	−0.256376	−0.128188	−	0.991750i	$-0.540916\pi$
$251$	−1019.50	−0.256376	−0.128188	+	0.991750i	$0.540916\pi$
$252$	0	0
$253$	126.583	0.0314554
$254$	8543.87	2.11059
$255$	0	0
$256$	5209.83	1.27193
$257$	6742.31	1.63647	0.818237	−	0.574881i	$-0.194952\pi$
$257$	6742.31	1.63647	0.818237	+	0.574881i	$0.194952\pi$
$258$	0	0
$259$	1316.60	0.315868
$260$	−1046.59	−0.249642
$261$	0	0
$262$	5559.44	1.31093
$263$	−1282.62	−0.300721	−0.150360	−	0.988631i	$-0.548043\pi$
$263$	−1282.62	−0.300721	−0.150360	+	0.988631i	$0.548043\pi$
$264$	0	0
$265$	−1151.54	−0.266938
$266$	2067.16	0.476488
$267$	0	0
$268$	−1214.95	−0.276922
$269$	930.364	0.210875	0.105437	−	0.994426i	$-0.466376\pi$
$269$	930.364	0.210875	0.105437	+	0.994426i	$0.466376\pi$
$270$	0	0
$271$	8335.61	1.86846	0.934230	−	0.356672i	$-0.116089\pi$
$271$	8335.61	1.86846	0.934230	+	0.356672i	$0.116089\pi$
$272$	−5109.58	−1.13902
$273$	0	0
$274$	2926.15	0.645166
$275$	−1019.53	−0.223564
$276$	0	0
$277$	3493.35	0.757744	0.378872	−	0.925449i	$-0.376312\pi$
$277$	3493.35	0.757744	0.378872	+	0.925449i	$0.376312\pi$
$278$	4125.83	0.890111
$279$	0	0
$280$	469.862	0.100284
$281$	1031.25	0.218929	0.109465	−	0.993991i	$-0.465086\pi$
$281$	1031.25	0.218929	0.109465	+	0.993991i	$0.465086\pi$
$282$	0	0
$283$	−2380.71	−0.500066	−0.250033	−	0.968237i	$-0.580441\pi$
$283$	−2380.71	−0.500066	−0.250033	+	0.968237i	$0.580441\pi$
$284$	−1651.59	−0.345084
$285$	0	0
$286$	1539.67	0.318330
$287$	−1388.05	−0.285485
$288$	0	0
$289$	−785.424	−0.159867
$290$	2601.66	0.526809
$291$	0	0
$292$	2295.54	0.460056
$293$	−7189.97	−1.43359	−0.716796	−	0.697283i	$-0.754392\pi$
$293$	−7189.97	−1.43359	−0.716796	+	0.697283i	$0.754392\pi$
$294$	0	0
$295$	−589.853	−0.116416
$296$	2220.88	0.436101
$297$	0	0
$298$	4397.86	0.854903
$299$	−452.250	−0.0874725
$300$	0	0
$301$	1619.70	0.310160
$302$	4112.10	0.783526
$303$	0	0
$304$	6594.39	1.24413
$305$	−2176.79	−0.408665
$306$	0	0
$307$	−805.946	−0.149830	−0.0749150	−	0.997190i	$-0.523869\pi$
$307$	−805.946	−0.149830	−0.0749150	+	0.997190i	$0.523869\pi$
$308$	360.719	0.0667333
$309$	0	0
$310$	2052.96	0.376130
$311$	−1526.61	−0.278348	−0.139174	−	0.990268i	$-0.544445\pi$
$311$	−1526.61	−0.278348	−0.139174	+	0.990268i	$0.544445\pi$
$312$	0	0
$313$	5698.67	1.02910	0.514550	−	0.857461i	$-0.327959\pi$
$313$	5698.67	1.02910	0.514550	+	0.857461i	$0.327959\pi$
$314$	10079.8	1.81158
$315$	0	0
$316$	663.500	0.118116
$317$	4036.30	0.715146	0.357573	−	0.933885i	$-0.383604\pi$
$317$	4036.30	0.715146	0.357573	+	0.933885i	$0.383604\pi$
$318$	0	0
$319$	−1413.51	−0.248092
$320$	−205.471	−0.0358944
$321$	0	0
$322$	−286.894	−0.0496521
$323$	−5327.02	−0.917657
$324$	0	0
$325$	3642.52	0.621695
$326$	−12175.7	−2.06855
$327$	0	0
$328$	−2341.40	−0.394154
$329$	1998.73	0.334935
$330$	0	0
$331$	2666.48	0.442789	0.221394	−	0.975184i	$-0.428939\pi$
$331$	2666.48	0.442789	0.221394	+	0.975184i	$0.428939\pi$
$332$	903.420	0.149342
$333$	0	0
$334$	5611.21	0.919257
$335$	−1474.30	−0.240447
$336$	0	0
$337$	2057.65	0.332603	0.166302	−	0.986075i	$-0.446817\pi$
$337$	2057.65	0.332603	0.166302	+	0.986075i	$0.446817\pi$
$338$	2323.90	0.373974
$339$	0	0
$340$	1710.92	0.272905
$341$	−1115.40	−0.177132
$342$	0	0
$343$	343.000	0.0539949
$344$	2732.16	0.428221
$345$	0	0
$346$	12812.1	1.99070
$347$	−8556.95	−1.32381	−0.661904	−	0.749589i	$-0.730251\pi$
$347$	−8556.95	−1.32381	−0.661904	+	0.749589i	$0.730251\pi$
$348$	0	0
$349$	−9298.02	−1.42611	−0.713054	−	0.701109i	$-0.752689\pi$
$349$	−9298.02	−1.42611	−0.713054	+	0.701109i	$0.752689\pi$
$350$	2310.71	0.352893
$351$	0	0
$352$	2076.72	0.314459
$353$	−6315.48	−0.952236	−0.476118	−	0.879381i	$-0.657956\pi$
$353$	−6315.48	−0.952236	−0.476118	+	0.879381i	$0.657956\pi$
$354$	0	0
$355$	−2004.14	−0.299631
$356$	1982.99	0.295220
$357$	0	0
$358$	6456.06	0.953109
$359$	−4366.78	−0.641977	−0.320988	−	0.947083i	$-0.604015\pi$
$359$	−4366.78	−0.641977	−0.320988	+	0.947083i	$0.604015\pi$
$360$	0	0
$361$	16.0157	0.00233500
$362$	−7974.90	−1.15788
$363$	0	0
$364$	−1288.76	−0.185575
$365$	2785.55	0.399459
$366$	0	0
$367$	455.244	0.0647508	0.0323754	−	0.999476i	$-0.489693\pi$
$367$	455.244	0.0647508	0.0323754	+	0.999476i	$0.489693\pi$
$368$	−915.212	−0.129643
$369$	0	0
$370$	−3808.03	−0.535054
$371$	−1417.99	−0.198432
$372$	0	0
$373$	−5345.01	−0.741968	−0.370984	−	0.928639i	$-0.620980\pi$
$373$	−5345.01	−0.741968	−0.370984	+	0.928639i	$0.620980\pi$
$374$	−2516.98	−0.347994
$375$	0	0
$376$	3371.51	0.462426
$377$	5050.11	0.689904
$378$	0	0
$379$	9772.43	1.32448	0.662238	−	0.749294i	$-0.269607\pi$
$379$	9772.43	1.32448	0.662238	+	0.749294i	$0.269607\pi$
$380$	−2208.10	−0.298087
$381$	0	0
$382$	−14104.9	−1.88918
$383$	−821.560	−0.109608	−0.0548039	−	0.998497i	$-0.517453\pi$
$383$	−821.560	−0.109608	−0.0548039	+	0.998497i	$0.517453\pi$
$384$	0	0
$385$	437.719	0.0579434
$386$	16882.6	2.22617
$387$	0	0
$388$	−5133.88	−0.671735
$389$	3632.64	0.473476	0.236738	−	0.971574i	$-0.423922\pi$
$389$	3632.64	0.473476	0.236738	+	0.971574i	$0.423922\pi$
$390$	0	0
$391$	739.318	0.0956239
$392$	578.580	0.0745478
$393$	0	0
$394$	−18321.3	−2.34267
$395$	805.133	0.102559
$396$	0	0
$397$	4985.80	0.630302	0.315151	−	0.949041i	$-0.397945\pi$
$397$	4985.80	0.630302	0.315151	+	0.949041i	$0.397945\pi$
$398$	7949.45	1.00118
$399$	0	0
$400$	7371.33	0.921416
$401$	5585.37	0.695562	0.347781	−	0.937576i	$-0.386935\pi$
$401$	5585.37	0.695562	0.347781	+	0.937576i	$0.386935\pi$
$402$	0	0
$403$	3985.02	0.492576
$404$	8344.78	1.02764
$405$	0	0
$406$	3203.64	0.391611
$407$	2068.95	0.251975
$408$	0	0
$409$	−5307.38	−0.641645	−0.320823	−	0.947139i	$-0.603959\pi$
$409$	−5307.38	−0.641645	−0.320823	+	0.947139i	$0.603959\pi$
$410$	4014.69	0.483589
$411$	0	0
$412$	48.3637	0.00578328
$413$	−726.336	−0.0865391
$414$	0	0
$415$	1096.27	0.129671
$416$	−7419.58	−0.874459
$417$	0	0
$418$	3248.40	0.380106
$419$	−12248.3	−1.42808	−0.714041	−	0.700104i	$-0.753137\pi$
$419$	−12248.3	−1.42808	−0.714041	+	0.700104i	$0.753137\pi$
$420$	0	0
$421$	−6958.41	−0.805539	−0.402770	−	0.915301i	$-0.631952\pi$
$421$	−6958.41	−0.805539	−0.402770	+	0.915301i	$0.631952\pi$
$422$	11108.7	1.28142
$423$	0	0
$424$	−2391.90	−0.273964
$425$	−5954.64	−0.679629
$426$	0	0
$427$	−2680.47	−0.303787
$428$	−7115.48	−0.803597
$429$	0	0
$430$	−4684.70	−0.525386
$431$	−6245.52	−0.697996	−0.348998	−	0.937123i	$-0.613478\pi$
$431$	−6245.52	−0.697996	−0.348998	+	0.937123i	$0.613478\pi$
$432$	0	0
$433$	8035.72	0.891853	0.445926	−	0.895070i	$-0.352874\pi$
$433$	8035.72	0.891853	0.445926	+	0.895070i	$0.352874\pi$
$434$	2527.98	0.279601
$435$	0	0
$436$	−7724.66	−0.848496
$437$	−954.159	−0.104448
$438$	0	0
$439$	10775.4	1.17148	0.585742	−	0.810497i	$-0.300803\pi$
$439$	10775.4	1.17148	0.585742	+	0.810497i	$0.300803\pi$
$440$	738.354	0.0799992
$441$	0	0
$442$	8992.52	0.967716
$443$	−3395.81	−0.364198	−0.182099	−	0.983280i	$-0.558289\pi$
$443$	−3395.81	−0.364198	−0.182099	+	0.983280i	$0.558289\pi$
$444$	0	0
$445$	2406.29	0.256335
$446$	4056.41	0.430665
$447$	0	0
$448$	−253.014	−0.0266826
$449$	10625.6	1.11682	0.558409	−	0.829566i	$-0.311412\pi$
$449$	10625.6	1.11682	0.558409	+	0.829566i	$0.311412\pi$
$450$	0	0
$451$	−2181.23	−0.227738
$452$	884.176	0.0920092
$453$	0	0
$454$	−6250.41	−0.646137
$455$	−1563.86	−0.161131
$456$	0	0
$457$	3189.95	0.326520	0.163260	−	0.986583i	$-0.447799\pi$
$457$	3189.95	0.326520	0.163260	+	0.986583i	$0.447799\pi$
$458$	−1083.68	−0.110561
$459$	0	0
$460$	306.455	0.0310620
$461$	−11839.8	−1.19617	−0.598084	−	0.801433i	$-0.704071\pi$
$461$	−11839.8	−1.19617	−0.598084	+	0.801433i	$0.704071\pi$
$462$	0	0
$463$	−14847.0	−1.49027	−0.745137	−	0.666911i	$-0.767616\pi$
$463$	−14847.0	−1.49027	−0.745137	+	0.666911i	$0.767616\pi$
$464$	10219.8	1.02251
$465$	0	0
$466$	−13921.2	−1.38388
$467$	5678.07	0.562634	0.281317	−	0.959615i	$-0.409229\pi$
$467$	5678.07	0.562634	0.281317	+	0.959615i	$0.409229\pi$
$468$	0	0
$469$	−1815.43	−0.178740
$470$	−5780.96	−0.567353
$471$	0	0
$472$	−1225.20	−0.119480
$473$	2545.25	0.247422
$474$	0	0
$475$	7685.01	0.742342
$476$	2106.80	0.202868
$477$	0	0
$478$	−609.009	−0.0582750
$479$	−224.024	−0.0213694	−0.0106847	−	0.999943i	$-0.503401\pi$
$479$	−224.024	−0.0213694	−0.0106847	+	0.999943i	$0.503401\pi$
$480$	0	0
$481$	−7391.82	−0.700703
$482$	23701.1	2.23974
$483$	0	0
$484$	566.844	0.0532348
$485$	−6229.77	−0.583256
$486$	0	0
$487$	−6298.42	−0.586055	−0.293028	−	0.956104i	$-0.594663\pi$
$487$	−6298.42	−0.586055	−0.293028	+	0.956104i	$0.594663\pi$
$488$	−4521.48	−0.419422
$489$	0	0
$490$	−992.064	−0.0914631
$491$	−7973.25	−0.732847	−0.366424	−	0.930448i	$-0.619418\pi$
$491$	−7973.25	−0.732847	−0.366424	+	0.930448i	$0.619418\pi$
$492$	0	0
$493$	−8255.70	−0.754195
$494$	−11605.7	−1.05701
$495$	0	0
$496$	8064.44	0.730048
$497$	−2467.87	−0.222735
$498$	0	0
$499$	−15518.7	−1.39221	−0.696104	−	0.717941i	$-0.745085\pi$
$499$	−15518.7	−1.39221	−0.696104	+	0.717941i	$0.745085\pi$
$500$	−5797.09	−0.518508
$501$	0	0
$502$	3631.01	0.322829
$503$	−8420.95	−0.746465	−0.373232	−	0.927738i	$-0.621751\pi$
$503$	−8420.95	−0.746465	−0.373232	+	0.927738i	$0.621751\pi$
$504$	0	0
$505$	10126.1	0.892286
$506$	−450.833	−0.0396087
$507$	0	0
$508$	−11238.1	−0.981517
$509$	22582.1	1.96647	0.983234	−	0.182350i	$-0.0583704\pi$
$509$	22582.1	1.96647	0.983234	+	0.182350i	$0.0583704\pi$
$510$	0	0
$511$	3430.09	0.296943
$512$	−7502.22	−0.647567
$513$	0	0
$514$	−24013.1	−2.06065
$515$	58.6876	0.00502152
$516$	0	0
$517$	3140.86	0.267186
$518$	−4689.15	−0.397740
$519$	0	0
$520$	−2637.95	−0.222465
$521$	−2072.11	−0.174243	−0.0871215	−	0.996198i	$-0.527767\pi$
$521$	−2072.11	−0.174243	−0.0871215	+	0.996198i	$0.527767\pi$
$522$	0	0
$523$	−10531.2	−0.880491	−0.440245	−	0.897877i	$-0.645109\pi$
$523$	−10531.2	−0.880491	−0.440245	+	0.897877i	$0.645109\pi$
$524$	−7312.56	−0.609638
$525$	0	0
$526$	4568.11	0.378667
$527$	−6514.54	−0.538478
$528$	0	0
$529$	−12034.6	−0.989116
$530$	4101.28	0.336128
$531$	0	0
$532$	−2719.02	−0.221588
$533$	7792.97	0.633304
$534$	0	0
$535$	−8634.37	−0.697750
$536$	−3062.31	−0.246776
$537$	0	0
$538$	−3313.54	−0.265533
$539$	539.000	0.0430730
$540$	0	0
$541$	17730.0	1.40900	0.704502	−	0.709702i	$-0.251170\pi$
$541$	17730.0	1.40900	0.704502	+	0.709702i	$0.251170\pi$
$542$	−29687.7	−2.35276
$543$	0	0
$544$	12129.2	0.955948
$545$	−9373.59	−0.736735
$546$	0	0
$547$	−18082.8	−1.41346	−0.706730	−	0.707483i	$-0.749831\pi$
$547$	−18082.8	−1.41346	−0.706730	+	0.707483i	$0.749831\pi$
$548$	−3848.89	−0.300030
$549$	0	0
$550$	3631.11	0.281511
$551$	10654.7	0.823789
$552$	0	0
$553$	991.428	0.0762383
$554$	−12441.8	−0.954150
$555$	0	0
$556$	−5426.87	−0.413940
$557$	−18748.2	−1.42619	−0.713095	−	0.701068i	$-0.752707\pi$
$557$	−18748.2	−1.42619	−0.713095	+	0.701068i	$0.752707\pi$
$558$	0	0
$559$	−9093.53	−0.688041
$560$	−3164.76	−0.238813
$561$	0	0
$562$	−3672.85	−0.275676
$563$	9033.83	0.676253	0.338127	−	0.941101i	$-0.390207\pi$
$563$	9033.83	0.676253	0.338127	+	0.941101i	$0.390207\pi$
$564$	0	0
$565$	1072.91	0.0798900
$566$	8479.03	0.629682
$567$	0	0
$568$	−4162.86	−0.307517
$569$	−5852.87	−0.431222	−0.215611	−	0.976479i	$-0.569174\pi$
$569$	−5852.87	−0.431222	−0.215611	+	0.976479i	$0.569174\pi$
$570$	0	0
$571$	6521.25	0.477944	0.238972	−	0.971026i	$-0.423190\pi$
$571$	6521.25	0.477944	0.238972	+	0.971026i	$0.423190\pi$
$572$	−2025.19	−0.148037
$573$	0	0
$574$	4943.63	0.359483
$575$	−1066.58	−0.0773553
$576$	0	0
$577$	10542.3	0.760627	0.380314	−	0.924858i	$-0.375816\pi$
$577$	10542.3	0.760627	0.380314	+	0.924858i	$0.375816\pi$
$578$	2797.33	0.201304
$579$	0	0
$580$	−3422.07	−0.244989
$581$	1349.93	0.0963931
$582$	0	0
$583$	−2228.27	−0.158294
$584$	5785.95	0.409973
$585$	0	0
$586$	25607.5	1.80518
$587$	−20713.3	−1.45644	−0.728218	−	0.685346i	$-0.759651\pi$
$587$	−20713.3	−1.45644	−0.728218	+	0.685346i	$0.759651\pi$
$588$	0	0
$589$	8407.62	0.588166
$590$	2100.79	0.146590
$591$	0	0
$592$	−14958.7	−1.03851
$593$	−7202.25	−0.498754	−0.249377	−	0.968406i	$-0.580226\pi$
$593$	−7202.25	−0.498754	−0.249377	+	0.968406i	$0.580226\pi$
$594$	0	0
$595$	2556.52	0.176147
$596$	−5784.69	−0.397567
$597$	0	0
$598$	1610.71	0.110145
$599$	19894.2	1.35702	0.678510	−	0.734591i	$-0.262626\pi$
$599$	19894.2	1.35702	0.678510	+	0.734591i	$0.262626\pi$
$600$	0	0
$601$	10378.4	0.704397	0.352199	−	0.935925i	$-0.385434\pi$
$601$	10378.4	0.704397	0.352199	+	0.935925i	$0.385434\pi$
$602$	−5768.66	−0.390553
$603$	0	0
$604$	−5408.82	−0.364374
$605$	687.844	0.0462228
$606$	0	0
$607$	15224.0	1.01800	0.508998	−	0.860768i	$-0.330016\pi$
$607$	15224.0	1.01800	0.508998	+	0.860768i	$0.330016\pi$
$608$	−15653.9	−1.04416
$609$	0	0
$610$	7752.76	0.514591
$611$	−11221.5	−0.743001
$612$	0	0
$613$	−25967.7	−1.71097	−0.855485	−	0.517828i	$-0.826741\pi$
$613$	−25967.7	−1.71097	−0.855485	+	0.517828i	$0.826741\pi$
$614$	2870.42	0.188666
$615$	0	0
$616$	909.198	0.0594685
$617$	25500.4	1.66387	0.831935	−	0.554873i	$-0.187233\pi$
$617$	25500.4	1.66387	0.831935	+	0.554873i	$0.187233\pi$
$618$	0	0
$619$	−19794.4	−1.28531	−0.642654	−	0.766157i	$-0.722167\pi$
$619$	−19794.4	−1.28531	−0.642654	+	0.766157i	$0.722167\pi$
$620$	−2700.34	−0.174917
$621$	0	0
$622$	5437.11	0.350496
$623$	2963.07	0.190550
$624$	0	0
$625$	4551.03	0.291266
$626$	−20296.1	−1.29584
$627$	0	0
$628$	−13258.4	−0.842464
$629$	12083.8	0.765999
$630$	0	0
$631$	19940.5	1.25803	0.629017	−	0.777392i	$-0.283458\pi$
$631$	19940.5	1.25803	0.629017	+	0.777392i	$0.283458\pi$
$632$	1672.36	0.105258
$633$	0	0
$634$	−14375.5	−0.900511
$635$	−13637.0	−0.852235
$636$	0	0
$637$	−1925.71	−0.119779
$638$	5034.29	0.312397
$639$	0	0
$640$	−7853.97	−0.485087
$641$	22527.9	1.38814	0.694070	−	0.719907i	$-0.255816\pi$
$641$	22527.9	1.38814	0.694070	+	0.719907i	$0.255816\pi$
$642$	0	0
$643$	−11676.3	−0.716125	−0.358062	−	0.933698i	$-0.616562\pi$
$643$	−11676.3	−0.716125	−0.358062	+	0.933698i	$0.616562\pi$
$644$	377.363	0.0230904
$645$	0	0
$646$	18972.5	1.15551
$647$	9749.45	0.592412	0.296206	−	0.955124i	$-0.404279\pi$
$647$	9749.45	0.592412	0.296206	+	0.955124i	$0.404279\pi$
$648$	0	0
$649$	−1141.39	−0.0690343
$650$	−12973.0	−0.782838
$651$	0	0
$652$	16015.2	0.961968
$653$	−3439.33	−0.206112	−0.103056	−	0.994676i	$-0.532862\pi$
$653$	−3439.33	−0.206112	−0.103056	+	0.994676i	$0.532862\pi$
$654$	0	0
$655$	−8873.51	−0.529339
$656$	15770.5	0.938622
$657$	0	0
$658$	−7118.59	−0.421750
$659$	−16400.6	−0.969463	−0.484732	−	0.874663i	$-0.661083\pi$
$659$	−16400.6	−0.969463	−0.484732	+	0.874663i	$0.661083\pi$
$660$	0	0
$661$	−7571.56	−0.445537	−0.222768	−	0.974871i	$-0.571509\pi$
$661$	−7571.56	−0.445537	−0.222768	+	0.974871i	$0.571509\pi$
$662$	−9496.81	−0.557559
$663$	0	0
$664$	2277.09	0.133085
$665$	−3299.43	−0.192401
$666$	0	0
$667$	−1478.73	−0.0858423
$668$	−7380.66	−0.427494
$669$	0	0
$670$	5250.80	0.302771
$671$	−4212.17	−0.242338
$672$	0	0
$673$	−5876.65	−0.336595	−0.168297	−	0.985736i	$-0.553827\pi$
$673$	−5876.65	−0.336595	−0.168297	+	0.985736i	$0.553827\pi$
$674$	−7328.43	−0.418814
$675$	0	0
$676$	−3056.72	−0.173914
$677$	9870.55	0.560348	0.280174	−	0.959949i	$-0.409608\pi$
$677$	9870.55	0.560348	0.280174	+	0.959949i	$0.409608\pi$
$678$	0	0
$679$	−7671.24	−0.433572
$680$	4312.41	0.243196
$681$	0	0
$682$	3972.54	0.223045
$683$	9551.67	0.535116	0.267558	−	0.963542i	$-0.413783\pi$
$683$	9551.67	0.535116	0.267558	+	0.963542i	$0.413783\pi$
$684$	0	0
$685$	−4670.49	−0.260511
$686$	−1221.61	−0.0679904
$687$	0	0
$688$	−18402.4	−1.01975
$689$	7961.04	0.440191
$690$	0	0
$691$	−10329.4	−0.568665	−0.284333	−	0.958726i	$-0.591772\pi$
$691$	−10329.4	−0.568665	−0.284333	+	0.958726i	$0.591772\pi$
$692$	−16852.3	−0.925762
$693$	0	0
$694$	30476.0	1.66694
$695$	−6585.31	−0.359417
$696$	0	0
$697$	−12739.6	−0.692320
$698$	33115.4	1.79575
$699$	0	0
$700$	−3039.37	−0.164111
$701$	33570.9	1.80878	0.904390	−	0.426708i	$-0.140327\pi$
$701$	33570.9	1.80878	0.904390	+	0.426708i	$0.140327\pi$
$702$	0	0
$703$	−15595.3	−0.836682
$704$	−397.594	−0.0212853
$705$	0	0
$706$	22492.9	1.19905
$707$	12469.1	0.663294
$708$	0	0
$709$	22424.4	1.18782	0.593911	−	0.804531i	$-0.297583\pi$
$709$	22424.4	1.18782	0.593911	+	0.804531i	$0.297583\pi$
$710$	7137.86	0.377295
$711$	0	0
$712$	4998.17	0.263082
$713$	−1166.86	−0.0612895
$714$	0	0
$715$	−2457.49	−0.128538
$716$	−8491.92	−0.443237
$717$	0	0
$718$	15552.5	0.808377
$719$	12475.8	0.647107	0.323553	−	0.946210i	$-0.395122\pi$
$719$	12475.8	0.647107	0.323553	+	0.946210i	$0.395122\pi$
$720$	0	0
$721$	72.2670	0.00373282
$722$	−57.0409	−0.00294023
$723$	0	0
$724$	10489.7	0.538463
$725$	11910.1	0.610109
$726$	0	0
$727$	35619.7	1.81714	0.908569	−	0.417734i	$-0.137176\pi$
$727$	35619.7	1.81714	0.908569	+	0.417734i	$0.137176\pi$
$728$	−3248.33	−0.165372
$729$	0	0
$730$	−9920.89	−0.502998
$731$	14865.7	0.752158
$732$	0	0
$733$	−23587.0	−1.18855	−0.594274	−	0.804263i	$-0.702560\pi$
$733$	−23587.0	−1.18855	−0.594274	+	0.804263i	$0.702560\pi$
$734$	−1621.38	−0.0815341
$735$	0	0
$736$	2172.55	0.108806
$737$	−2852.82	−0.142585
$738$	0	0
$739$	21418.0	1.06613	0.533067	−	0.846073i	$-0.321039\pi$
$739$	21418.0	1.06613	0.533067	+	0.846073i	$0.321039\pi$
$740$	5008.86	0.248824
$741$	0	0
$742$	5050.25	0.249866
$743$	−25868.0	−1.27726	−0.638630	−	0.769514i	$-0.720499\pi$
$743$	−25868.0	−1.27726	−0.638630	+	0.769514i	$0.720499\pi$
$744$	0	0
$745$	−7019.50	−0.345201
$746$	19036.5	0.934286
$747$	0	0
$748$	3310.69	0.161832
$749$	−10632.2	−0.518682
$750$	0	0
$751$	−28345.7	−1.37730	−0.688648	−	0.725096i	$-0.741796\pi$
$751$	−28345.7	−1.37730	−0.688648	+	0.725096i	$0.741796\pi$
$752$	−22708.8	−1.10120
$753$	0	0
$754$	−17986.2	−0.868727
$755$	−6563.40	−0.316380
$756$	0	0
$757$	5065.33	0.243200	0.121600	−	0.992579i	$-0.461197\pi$
$757$	5065.33	0.243200	0.121600	+	0.992579i	$0.461197\pi$
$758$	−34805.0	−1.66778
$759$	0	0
$760$	−5565.56	−0.265637
$761$	24301.0	1.15757	0.578784	−	0.815481i	$-0.303527\pi$
$761$	24301.0	1.15757	0.578784	+	0.815481i	$0.303527\pi$
$762$	0	0
$763$	−11542.5	−0.547662
$764$	18552.7	0.878553
$765$	0	0
$766$	2926.03	0.138018
$767$	4077.88	0.191973
$768$	0	0
$769$	−1579.45	−0.0740656	−0.0370328	−	0.999314i	$-0.511791\pi$
$769$	−1579.45	−0.0740656	−0.0370328	+	0.999314i	$0.511791\pi$
$770$	−1558.96	−0.0729623
$771$	0	0
$772$	−22206.3	−1.03526
$773$	−7349.54	−0.341973	−0.170986	−	0.985273i	$-0.554695\pi$
$773$	−7349.54	−0.341973	−0.170986	+	0.985273i	$0.554695\pi$
$774$	0	0
$775$	9398.19	0.435604
$776$	−12940.0	−0.598608
$777$	0	0
$778$	−12937.8	−0.596200
$779$	16441.7	0.756204
$780$	0	0
$781$	−3878.08	−0.177681
$782$	−2633.12	−0.120409
$783$	0	0
$784$	−3897.03	−0.177525
$785$	−16088.6	−0.731497
$786$	0	0
$787$	6878.56	0.311555	0.155778	−	0.987792i	$-0.450212\pi$
$787$	6878.56	0.311555	0.155778	+	0.987792i	$0.450212\pi$
$788$	24098.7	1.08944
$789$	0	0
$790$	−2867.52	−0.129142
$791$	1321.17	0.0593874
$792$	0	0
$793$	15049.0	0.673903
$794$	−17757.2	−0.793676
$795$	0	0
$796$	−10456.2	−0.465593
$797$	−4830.47	−0.214685	−0.107343	−	0.994222i	$-0.534234\pi$
$797$	−4830.47	−0.214685	−0.107343	+	0.994222i	$0.534234\pi$
$798$	0	0
$799$	18344.4	0.812239
$800$	−17498.2	−0.773318
$801$	0	0
$802$	−19892.6	−0.875851
$803$	5390.13	0.236879
$804$	0	0
$805$	457.917	0.0200490
$806$	−14192.9	−0.620251
$807$	0	0
$808$	21033.2	0.915772
$809$	44538.0	1.93557	0.967783	−	0.251786i	$-0.0810178\pi$
$809$	44538.0	1.93557	0.967783	+	0.251786i	$0.0810178\pi$
$810$	0	0
$811$	40738.9	1.76391	0.881957	−	0.471329i	$-0.156226\pi$
$811$	40738.9	1.76391	0.881957	+	0.471329i	$0.156226\pi$
$812$	−4213.88	−0.182116
$813$	0	0
$814$	−7368.67	−0.317287
$815$	19433.8	0.835260
$816$	0	0
$817$	−19185.6	−0.821564
$818$	18902.5	0.807959
$819$	0	0
$820$	−5280.69	−0.224890
$821$	−29805.1	−1.26700	−0.633498	−	0.773744i	$-0.718382\pi$
$821$	−29805.1	−1.26700	−0.633498	+	0.773744i	$0.718382\pi$
$822$	0	0
$823$	3453.87	0.146287	0.0731435	−	0.997321i	$-0.476697\pi$
$823$	3453.87	0.146287	0.0731435	+	0.997321i	$0.476697\pi$
$824$	121.902	0.00515369
$825$	0	0
$826$	2586.88	0.108970
$827$	10258.1	0.431328	0.215664	−	0.976468i	$-0.430808\pi$
$827$	10258.1	0.431328	0.215664	+	0.976468i	$0.430808\pi$
$828$	0	0
$829$	33599.2	1.40766	0.703830	−	0.710369i	$-0.251472\pi$
$829$	33599.2	1.40766	0.703830	+	0.710369i	$0.251472\pi$
$830$	−3904.41	−0.163282
$831$	0	0
$832$	1420.50	0.0591911
$833$	3148.06	0.130941
$834$	0	0
$835$	−8956.16	−0.371186
$836$	−4272.75	−0.176766
$837$	0	0
$838$	43622.8	1.79824
$839$	24363.0	1.00251	0.501254	−	0.865300i	$-0.332872\pi$
$839$	24363.0	1.00251	0.501254	+	0.865300i	$0.332872\pi$
$840$	0	0
$841$	−7876.51	−0.322953
$842$	24782.7	1.01433
$843$	0	0
$844$	−14611.7	−0.595918
$845$	−3709.21	−0.151007
$846$	0	0
$847$	847.000	0.0343604
$848$	16110.6	0.652408
$849$	0	0
$850$	21207.8	0.855788
$851$	2164.42	0.0871859
$852$	0	0
$853$	12928.0	0.518928	0.259464	−	0.965753i	$-0.416454\pi$
$853$	12928.0	0.518928	0.259464	+	0.965753i	$0.416454\pi$
$854$	9546.63	0.382528
$855$	0	0
$856$	−17934.7	−0.716116
$857$	40734.2	1.62363	0.811817	−	0.583911i	$-0.198478\pi$
$857$	40734.2	1.62363	0.811817	+	0.583911i	$0.198478\pi$
$858$	0	0
$859$	32796.6	1.30269	0.651343	−	0.758784i	$-0.274206\pi$
$859$	32796.6	1.30269	0.651343	+	0.758784i	$0.274206\pi$
$860$	6161.98	0.244327
$861$	0	0
$862$	22243.8	0.878916
$863$	879.506	0.0346915	0.0173457	−	0.999850i	$-0.494478\pi$
$863$	879.506	0.0346915	0.0173457	+	0.999850i	$0.494478\pi$
$864$	0	0
$865$	−20449.6	−0.803823
$866$	−28619.7	−1.12302
$867$	0	0
$868$	−3325.16	−0.130027
$869$	1557.96	0.0608172
$870$	0	0
$871$	10192.4	0.396506
$872$	−19470.2	−0.756127
$873$	0	0
$874$	3398.29	0.131520
$875$	−8662.24	−0.334671
$876$	0	0
$877$	17607.7	0.677957	0.338978	−	0.940794i	$-0.389919\pi$
$877$	17607.7	0.677957	0.338978	+	0.940794i	$0.389919\pi$
$878$	−38377.2	−1.47513
$879$	0	0
$880$	−4973.19	−0.190507
$881$	24818.3	0.949093	0.474546	−	0.880230i	$-0.342612\pi$
$881$	24818.3	0.949093	0.474546	+	0.880230i	$0.342612\pi$
$882$	0	0
$883$	18274.6	0.696477	0.348238	−	0.937406i	$-0.386780\pi$
$883$	18274.6	0.696477	0.348238	+	0.937406i	$0.386780\pi$
$884$	−11828.2	−0.450030
$885$	0	0
$886$	12094.4	0.458598
$887$	21922.4	0.829856	0.414928	−	0.909854i	$-0.363807\pi$
$887$	21922.4	0.829856	0.414928	+	0.909854i	$0.363807\pi$
$888$	0	0
$889$	−16792.4	−0.633521
$890$	−8570.13	−0.322777
$891$	0	0
$892$	−5335.57	−0.200278
$893$	−23675.2	−0.887189
$894$	0	0
$895$	−10304.6	−0.384856
$896$	−9671.26	−0.360596
$897$	0	0
$898$	−37843.5	−1.40630
$899$	13029.9	0.483396
$900$	0	0
$901$	−13014.4	−0.481211
$902$	7768.56	0.286768
$903$	0	0
$904$	2228.58	0.0819928
$905$	12728.9	0.467539
$906$	0	0
$907$	−9704.52	−0.355274	−0.177637	−	0.984096i	$-0.556845\pi$
$907$	−9704.52	−0.355274	−0.177637	+	0.984096i	$0.556845\pi$
$908$	8221.42	0.300482
$909$	0	0
$910$	5569.76	0.202896
$911$	−40557.0	−1.47499	−0.737494	−	0.675354i	$-0.763991\pi$
$911$	−40557.0	−1.47499	−0.737494	+	0.675354i	$0.763991\pi$
$912$	0	0
$913$	2121.31	0.0768951
$914$	−11361.2	−0.411153
$915$	0	0
$916$	1425.40	0.0514156
$917$	−10926.7	−0.393491
$918$	0	0
$919$	−10182.1	−0.365482	−0.182741	−	0.983161i	$-0.558497\pi$
$919$	−10182.1	−0.365482	−0.182741	+	0.983161i	$0.558497\pi$
$920$	772.424	0.0276805
$921$	0	0
$922$	42168.0	1.50621
$923$	13855.4	0.494102
$924$	0	0
$925$	−17432.7	−0.619658
$926$	52878.3	1.87655
$927$	0	0
$928$	−24260.0	−0.858162
$929$	−35454.0	−1.25211	−0.626053	−	0.779780i	$-0.715331\pi$
$929$	−35454.0	−1.25211	−0.626053	+	0.779780i	$0.715331\pi$
$930$	0	0
$931$	−4062.87	−0.143024
$932$	18311.1	0.643562
$933$	0	0
$934$	−20222.8	−0.708468
$935$	4017.40	0.140516
$936$	0	0
$937$	35786.9	1.24771	0.623856	−	0.781539i	$-0.285565\pi$
$937$	35786.9	1.24771	0.623856	+	0.781539i	$0.285565\pi$
$938$	6465.76	0.225069
$939$	0	0
$940$	7603.94	0.263844
$941$	53299.3	1.84645	0.923223	−	0.384264i	$-0.125545\pi$
$941$	53299.3	1.84645	0.923223	+	0.384264i	$0.125545\pi$
$942$	0	0
$943$	−2281.88	−0.0787998
$944$	8252.34	0.284524
$945$	0	0
$946$	−9065.04	−0.311554
$947$	2396.55	0.0822359	0.0411180	−	0.999154i	$-0.486908\pi$
$947$	2396.55	0.0822359	0.0411180	+	0.999154i	$0.486908\pi$
$948$	0	0
$949$	−19257.6	−0.658722
$950$	−27370.6	−0.934757
$951$	0	0
$952$	5310.23	0.180783
$953$	−49723.2	−1.69013	−0.845064	−	0.534665i	$-0.820438\pi$
$953$	−49723.2	−1.69013	−0.845064	+	0.534665i	$0.820438\pi$
$954$	0	0
$955$	22513.0	0.762832
$956$	801.055	0.0271004
$957$	0	0
$958$	797.875	0.0269083
$959$	−5751.16	−0.193655
$960$	0	0
$961$	−19509.1	−0.654866
$962$	26326.4	0.882324
$963$	0	0
$964$	−31175.0	−1.04158
$965$	−26946.6	−0.898902
$966$	0	0
$967$	−34465.6	−1.14616	−0.573081	−	0.819499i	$-0.694252\pi$
$967$	−34465.6	−1.14616	−0.573081	+	0.819499i	$0.694252\pi$
$968$	1428.74	0.0474395
$969$	0	0
$970$	22187.7	0.734436
$971$	29569.6	0.977276	0.488638	−	0.872487i	$-0.337494\pi$
$971$	29569.6	0.977276	0.488638	+	0.872487i	$0.337494\pi$
$972$	0	0
$973$	−8109.05	−0.267178
$974$	22432.2	0.737960
$975$	0	0
$976$	30454.4	0.998794
$977$	−49863.2	−1.63282	−0.816411	−	0.577472i	$-0.804039\pi$
$977$	−49863.2	−1.63282	−0.816411	+	0.577472i	$0.804039\pi$
$978$	0	0
$979$	4656.25	0.152007
$980$	1304.90	0.0425343
$981$	0	0
$982$	28397.2	0.922800
$983$	−45383.3	−1.47254	−0.736268	−	0.676690i	$-0.763414\pi$
$983$	−45383.3	−1.47254	−0.736268	+	0.676690i	$0.763414\pi$
$984$	0	0
$985$	29242.9	0.945946
$986$	29403.1	0.949681
$987$	0	0
$988$	15265.4	0.491557
$989$	2662.70	0.0856105
$990$	0	0
$991$	−54822.7	−1.75732	−0.878658	−	0.477451i	$-0.841561\pi$
$991$	−54822.7	−1.75732	−0.878658	+	0.477451i	$0.841561\pi$
$992$	−19143.5	−0.612708
$993$	0	0
$994$	8789.45	0.280467
$995$	−12688.3	−0.404266
$996$	0	0
$997$	29649.9	0.941849	0.470924	−	0.882174i	$-0.343920\pi$
$997$	29649.9	0.941849	0.470924	+	0.882174i	$0.343920\pi$
$998$	55270.6	1.75307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.g.1.1		2
3.2	odd	2		231.4.a.h.1.2	✓	2
21.20	even	2		1617.4.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.h.1.2	✓	2	3.2	odd	2
693.4.a.g.1.1		2	1.1	even	1	trivial
1617.4.a.m.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-3.56155 q^{2} +4.68466 q^{4} +5.68466 q^{5} +7.00000 q^{7} +11.8078 q^{8} -20.2462 q^{10} +11.0000 q^{11} -39.3002 q^{13} -24.9309 q^{14} -79.5312 q^{16} +64.2462 q^{17} -82.9157 q^{19} +26.6307 q^{20} -39.1771 q^{22} +11.5076 q^{23} -92.6847 q^{25} +139.970 q^{26} +32.7926 q^{28} -128.501 q^{29} -101.400 q^{31} +188.793 q^{32} -228.816 q^{34} +39.7926 q^{35} +188.086 q^{37} +295.309 q^{38} +67.1231 q^{40} -198.294 q^{41} +231.386 q^{43} +51.5312 q^{44} -40.9848 q^{46} +285.533 q^{47} +49.0000 q^{49} +330.101 q^{50} -184.108 q^{52} -202.570 q^{53} +62.5312 q^{55} +82.6543 q^{56} +457.663 q^{58} -103.762 q^{59} -382.924 q^{61} +361.140 q^{62} -36.1449 q^{64} -223.408 q^{65} -259.348 q^{67} +300.972 q^{68} -141.723 q^{70} -352.553 q^{71} +490.012 q^{73} -669.879 q^{74} -388.432 q^{76} +77.0000 q^{77} +141.633 q^{79} -452.108 q^{80} +706.233 q^{82} +192.847 q^{83} +365.218 q^{85} -824.095 q^{86} +129.885 q^{88} +423.295 q^{89} -275.101 q^{91} +53.9091 q^{92} -1016.94 q^{94} -471.348 q^{95} -1095.89 q^{97} -174.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} - 24 q^{10} + 22 q^{11} - 25 q^{13} - 21 q^{14} - 23 q^{16} + 112 q^{17} - 71 q^{19} + 78 q^{20} - 33 q^{22} + 56 q^{23} - 173 q^{25} + 148 q^{26} - 21 q^{28}+ \cdots - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^4 - q^5 + 14 * q^7 + 3 * q^8 - 24 * q^10 + 22 * q^11 - 25 * q^13 - 21 * q^14 - 23 * q^16 + 112 * q^17 - 71 * q^19 + 78 * q^20 - 33 * q^22 + 56 * q^23 - 173 * q^25 + 148 * q^26 - 21 * q^28 + 11 * q^29 - 310 * q^31 + 291 * q^32 - 202 * q^34 - 7 * q^35 - 65 * q^37 + 302 * q^38 + 126 * q^40 - 42 * q^41 - 32 * q^43 - 33 * q^44 - 16 * q^46 - 101 * q^47 + 98 * q^49 + 285 * q^50 - 294 * q^52 - 166 * q^53 - 11 * q^55 + 21 * q^56 + 536 * q^58 + 11 * q^59 - 436 * q^61 + 244 * q^62 - 431 * q^64 - 319 * q^65 - 127 * q^67 - 66 * q^68 - 168 * q^70 - 936 * q^71 - 327 * q^73 - 812 * q^74 - 480 * q^76 + 154 * q^77 - 228 * q^79 - 830 * q^80 + 794 * q^82 + 262 * q^83 + 46 * q^85 - 972 * q^86 + 33 * q^88 - 44 * q^89 - 175 * q^91 - 288 * q^92 - 1234 * q^94 - 551 * q^95 - 2266 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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