LMFDB - Embedded newform 231.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [231,4,Mod(1,231)] 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("231.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$231 = 3 \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	231.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-1,-15,21,21] 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:	$13.6294412113$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162$ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.63074$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.63074 q^{2} -3.00000 q^{3} +5.18226 q^{4} +21.1113 q^{5} +10.8922 q^{6} +7.00000 q^{7} +10.2305 q^{8} +9.00000 q^{9} -76.6494 q^{10} +11.0000 q^{11} -15.5468 q^{12} +87.3602 q^{13} -25.4152 q^{14} -63.3338 q^{15} -78.6023 q^{16} -28.2691 q^{17} -32.6766 q^{18} -97.9284 q^{19} +109.404 q^{20} -21.0000 q^{21} -39.9381 q^{22} -112.065 q^{23} -30.6914 q^{24} +320.685 q^{25} -317.182 q^{26} -27.0000 q^{27} +36.2758 q^{28} +14.9213 q^{29} +229.948 q^{30} +138.440 q^{31} +203.540 q^{32} -33.0000 q^{33} +102.638 q^{34} +147.779 q^{35} +46.6403 q^{36} +206.944 q^{37} +355.552 q^{38} -262.081 q^{39} +215.978 q^{40} +321.063 q^{41} +76.2455 q^{42} +285.198 q^{43} +57.0048 q^{44} +190.001 q^{45} +406.877 q^{46} -303.300 q^{47} +235.807 q^{48} +49.0000 q^{49} -1164.32 q^{50} +84.8072 q^{51} +452.723 q^{52} -554.639 q^{53} +98.0299 q^{54} +232.224 q^{55} +71.6134 q^{56} +293.785 q^{57} -54.1752 q^{58} -693.110 q^{59} -328.212 q^{60} +156.761 q^{61} -502.638 q^{62} +63.0000 q^{63} -110.184 q^{64} +1844.28 q^{65} +119.814 q^{66} +584.667 q^{67} -146.498 q^{68} +336.194 q^{69} -536.546 q^{70} +363.745 q^{71} +92.0743 q^{72} -747.424 q^{73} -751.360 q^{74} -962.055 q^{75} -507.490 q^{76} +77.0000 q^{77} +951.546 q^{78} +419.344 q^{79} -1659.39 q^{80} +81.0000 q^{81} -1165.70 q^{82} +1178.20 q^{83} -108.827 q^{84} -596.796 q^{85} -1035.48 q^{86} -44.7638 q^{87} +112.535 q^{88} -397.908 q^{89} -689.845 q^{90} +611.521 q^{91} -580.748 q^{92} -415.319 q^{93} +1101.20 q^{94} -2067.39 q^{95} -610.621 q^{96} -1333.21 q^{97} -177.906 q^{98} +99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - q^{2} - 15 q^{3} + 21 q^{4} + 21 q^{5} + 3 q^{6} + 35 q^{7} - 42 q^{8} + 45 q^{9} - 23 q^{10} + 55 q^{11} - 63 q^{12} + 101 q^{13} - 7 q^{14} - 63 q^{15} - 7 q^{16} - 20 q^{17} - 9 q^{18} + 237 q^{19}+ \cdots + 495 q^{99}+O(q^{100})$ 5 * q - q^2 - 15 * q^3 + 21 * q^4 + 21 * q^5 + 3 * q^6 + 35 * q^7 - 42 * q^8 + 45 * q^9 - 23 * q^10 + 55 * q^11 - 63 * q^12 + 101 * q^13 - 7 * q^14 - 63 * q^15 - 7 * q^16 - 20 * q^17 - 9 * q^18 + 237 * q^19 + 85 * q^20 - 105 * q^21 - 11 * q^22 - 80 * q^23 + 126 * q^24 + 486 * q^25 + 165 * q^26 - 135 * q^27 + 147 * q^28 - 11 * q^29 + 69 * q^30 + 316 * q^31 + 453 * q^32 - 165 * q^33 + 936 * q^34 + 147 * q^35 + 189 * q^36 + 319 * q^37 + 89 * q^38 - 303 * q^39 + 624 * q^40 + 1190 * q^41 + 21 * q^42 + 88 * q^43 + 231 * q^44 + 189 * q^45 + 1000 * q^46 + 377 * q^47 + 21 * q^48 + 245 * q^49 - 644 * q^50 + 60 * q^51 + 1001 * q^52 - 992 * q^53 + 27 * q^54 + 231 * q^55 - 294 * q^56 - 711 * q^57 + 721 * q^58 + 71 * q^59 - 255 * q^60 - 574 * q^61 + 272 * q^62 + 315 * q^63 - 1380 * q^64 + 589 * q^65 + 33 * q^66 - 527 * q^67 - 2974 * q^68 + 240 * q^69 - 161 * q^70 - 1156 * q^71 - 378 * q^72 + 1061 * q^73 - 1609 * q^74 - 1458 * q^75 + 2399 * q^76 + 385 * q^77 - 495 * q^78 + 588 * q^79 - 1643 * q^80 + 405 * q^81 - 2602 * q^82 - 212 * q^83 - 441 * q^84 + 1918 * q^85 - 4760 * q^86 + 33 * q^87 - 462 * q^88 + 1030 * q^89 - 207 * q^90 + 707 * q^91 - 1174 * q^92 - 948 * q^93 - 1799 * q^94 - 3593 * q^95 - 1359 * q^96 + 2488 * q^97 - 49 * q^98 + 495 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−3.63074	−1.28366	−0.641830	−	0.766847i	$-0.721824\pi$
$2$	−3.63074	−1.28366	−0.641830	+	0.766847i	$0.721824\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	5.18226	0.647782
$5$	21.1113	1.88825	0.944124	−	0.329590i	$-0.106911\pi$
$5$	21.1113	1.88825	0.944124	+	0.329590i	$0.106911\pi$
$6$	10.8922	0.741121
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	10.2305	0.452128
$9$	9.00000	0.333333
$10$	−76.6494	−2.42387
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−15.5468	−0.373997
$13$	87.3602	1.86380	0.931898	−	0.362719i	$-0.118152\pi$
$13$	87.3602	1.86380	0.931898	+	0.362719i	$0.118152\pi$
$14$	−25.4152	−0.485178
$15$	−63.3338	−1.09018
$16$	−78.6023	−1.22816
$17$	−28.2691	−0.403309	−0.201655	−	0.979457i	$-0.564632\pi$
$17$	−28.2691	−0.403309	−0.201655	+	0.979457i	$0.564632\pi$
$18$	−32.6766	−0.427887
$19$	−97.9284	−1.18244	−0.591219	−	0.806511i	$-0.701353\pi$
$19$	−97.9284	−1.18244	−0.591219	+	0.806511i	$0.701353\pi$
$20$	109.404	1.22317
$21$	−21.0000	−0.218218
$22$	−39.9381	−0.387038
$23$	−112.065	−1.01596	−0.507980	−	0.861369i	$-0.669608\pi$
$23$	−112.065	−1.01596	−0.507980	+	0.861369i	$0.669608\pi$
$24$	−30.6914	−0.261036
$25$	320.685	2.56548
$26$	−317.182	−2.39248
$27$	−27.0000	−0.192450
$28$	36.2758	0.244839
$29$	14.9213	0.0955451	0.0477725	−	0.998858i	$-0.484788\pi$
$29$	14.9213	0.0955451	0.0477725	+	0.998858i	$0.484788\pi$
$30$	229.948	1.39942
$31$	138.440	0.802081	0.401040	−	0.916060i	$-0.368649\pi$
$31$	138.440	0.802081	0.401040	+	0.916060i	$0.368649\pi$
$32$	203.540	1.12441
$33$	−33.0000	−0.174078
$34$	102.638	0.517712
$35$	147.779	0.713691
$36$	46.6403	0.215927
$37$	206.944	0.919498	0.459749	−	0.888049i	$-0.347939\pi$
$37$	206.944	0.919498	0.459749	+	0.888049i	$0.347939\pi$
$38$	355.552	1.51785
$39$	−262.081	−1.07606
$40$	215.978	0.853729
$41$	321.063	1.22297	0.611483	−	0.791257i	$-0.290573\pi$
$41$	321.063	1.22297	0.611483	+	0.791257i	$0.290573\pi$
$42$	76.2455	0.280118
$43$	285.198	1.01145	0.505724	−	0.862696i	$-0.331226\pi$
$43$	285.198	1.01145	0.505724	+	0.862696i	$0.331226\pi$
$44$	57.0048	0.195314
$45$	190.001	0.629416
$46$	406.877	1.30415
$47$	−303.300	−0.941293	−0.470647	−	0.882322i	$-0.655979\pi$
$47$	−303.300	−0.941293	−0.470647	+	0.882322i	$0.655979\pi$
$48$	235.807	0.709079
$49$	49.0000	0.142857
$50$	−1164.32	−3.29320
$51$	84.8072	0.232851
$52$	452.723	1.20733
$53$	−554.639	−1.43746	−0.718732	−	0.695287i	$-0.755277\pi$
$53$	−554.639	−1.43746	−0.718732	+	0.695287i	$0.755277\pi$
$54$	98.0299	0.247040
$55$	232.224	0.569328
$56$	71.6134	0.170888
$57$	293.785	0.682681
$58$	−54.1752	−0.122647
$59$	−693.110	−1.52941	−0.764705	−	0.644380i	$-0.777115\pi$
$59$	−693.110	−1.52941	−0.764705	+	0.644380i	$0.777115\pi$
$60$	−328.212	−0.706200
$61$	156.761	0.329037	0.164518	−	0.986374i	$-0.447393\pi$
$61$	156.761	0.329037	0.164518	+	0.986374i	$0.447393\pi$
$62$	−502.638	−1.02960
$63$	63.0000	0.125988
$64$	−110.184	−0.215203
$65$	1844.28	3.51931
$66$	119.814	0.223456
$67$	584.667	1.06610	0.533048	−	0.846085i	$-0.321047\pi$
$67$	584.667	1.06610	0.533048	+	0.846085i	$0.321047\pi$
$68$	−146.498	−0.261257
$69$	336.194	0.586565
$70$	−536.546	−0.916136
$71$	363.745	0.608008	0.304004	−	0.952671i	$-0.401676\pi$
$71$	363.745	0.608008	0.304004	+	0.952671i	$0.401676\pi$
$72$	92.0743	0.150709
$73$	−747.424	−1.19835	−0.599173	−	0.800619i	$-0.704504\pi$
$73$	−747.424	−1.19835	−0.599173	+	0.800619i	$0.704504\pi$
$74$	−751.360	−1.18032
$75$	−962.055	−1.48118
$76$	−507.490	−0.765962
$77$	77.0000	0.113961
$78$	951.546	1.38130
$79$	419.344	0.597213	0.298607	−	0.954376i	$-0.403478\pi$
$79$	419.344	0.597213	0.298607	+	0.954376i	$0.403478\pi$
$80$	−1659.39	−2.31907
$81$	81.0000	0.111111
$82$	−1165.70	−1.56987
$83$	1178.20	1.55812	0.779061	−	0.626948i	$-0.215696\pi$
$83$	1178.20	1.55812	0.779061	+	0.626948i	$0.215696\pi$
$84$	−108.827	−0.141358
$85$	−596.796	−0.761548
$86$	−1035.48	−1.29835
$87$	−44.7638	−0.0551630
$88$	112.535	0.136322
$89$	−397.908	−0.473911	−0.236956	−	0.971520i	$-0.576150\pi$
$89$	−397.908	−0.473911	−0.236956	+	0.971520i	$0.576150\pi$
$90$	−689.845	−0.807956
$91$	611.521	0.704449
$92$	−580.748	−0.658121
$93$	−415.319	−0.463082
$94$	1101.20	1.20830
$95$	−2067.39	−2.23274
$96$	−610.621	−0.649180
$97$	−1333.21	−1.39553	−0.697766	−	0.716325i	$-0.745823\pi$
$97$	−1333.21	−1.39553	−0.697766	+	0.716325i	$0.745823\pi$
$98$	−177.906	−0.183380
$99$	99.0000	0.100504
$100$	1661.87	1.66187
$101$	747.399	0.736326	0.368163	−	0.929761i	$-0.379987\pi$
$101$	747.399	0.736326	0.368163	+	0.929761i	$0.379987\pi$
$102$	−307.913	−0.298901
$103$	548.716	0.524918	0.262459	−	0.964943i	$-0.415467\pi$
$103$	548.716	0.524918	0.262459	+	0.964943i	$0.415467\pi$
$104$	893.737	0.842674
$105$	−443.336	−0.412050
$106$	2013.75	1.84521
$107$	273.875	0.247444	0.123722	−	0.992317i	$-0.460517\pi$
$107$	273.875	0.247444	0.123722	+	0.992317i	$0.460517\pi$
$108$	−139.921	−0.124666
$109$	2026.77	1.78100	0.890500	−	0.454983i	$-0.150355\pi$
$109$	2026.77	1.78100	0.890500	+	0.454983i	$0.150355\pi$
$110$	−843.144	−0.730824
$111$	−620.833	−0.530872
$112$	−550.216	−0.464201
$113$	−1048.04	−0.872490	−0.436245	−	0.899828i	$-0.643692\pi$
$113$	−1048.04	−0.872490	−0.436245	+	0.899828i	$0.643692\pi$
$114$	−1066.66	−0.876330
$115$	−2365.83	−1.91839
$116$	77.3258	0.0618924
$117$	786.242	0.621266
$118$	2516.50	1.96324
$119$	−197.884	−0.152437
$120$	−647.935	−0.492901
$121$	121.000	0.0909091
$122$	−569.160	−0.422371
$123$	−963.189	−0.706080
$124$	717.431	0.519574
$125$	4131.16	2.95602
$126$	−228.736	−0.161726
$127$	−905.819	−0.632901	−0.316451	−	0.948609i	$-0.602491\pi$
$127$	−905.819	−0.632901	−0.316451	+	0.948609i	$0.602491\pi$
$128$	−1228.27	−0.848166
$129$	−855.593	−0.583959
$130$	−6696.11	−4.51760
$131$	1887.45	1.25883	0.629416	−	0.777069i	$-0.283294\pi$
$131$	1887.45	1.25883	0.629416	+	0.777069i	$0.283294\pi$
$132$	−171.015	−0.112764
$133$	−685.499	−0.446919
$134$	−2122.77	−1.36850
$135$	−570.004	−0.363394
$136$	−289.206	−0.182347
$137$	690.753	0.430767	0.215383	−	0.976530i	$-0.430900\pi$
$137$	690.753	0.430767	0.215383	+	0.976530i	$0.430900\pi$
$138$	−1220.63	−0.752950
$139$	−1466.05	−0.894592	−0.447296	−	0.894386i	$-0.647613\pi$
$139$	−1466.05	−0.894592	−0.447296	+	0.894386i	$0.647613\pi$
$140$	765.828	0.462316
$141$	909.899	0.543456
$142$	−1320.66	−0.780476
$143$	960.962	0.561956
$144$	−707.420	−0.409387
$145$	315.006	0.180413
$146$	2713.70	1.53827
$147$	−147.000	−0.0824786
$148$	1072.44	0.595634
$149$	1367.88	0.752088	0.376044	−	0.926602i	$-0.377284\pi$
$149$	1367.88	0.752088	0.376044	+	0.926602i	$0.377284\pi$
$150$	3492.97	1.90133
$151$	93.9161	0.0506145	0.0253072	−	0.999680i	$-0.491944\pi$
$151$	93.9161	0.0506145	0.0253072	+	0.999680i	$0.491944\pi$
$152$	−1001.85	−0.534613
$153$	−254.422	−0.134436
$154$	−279.567	−0.146287
$155$	2922.64	1.51453
$156$	−1358.17	−0.697055
$157$	−72.3063	−0.0367558	−0.0183779	−	0.999831i	$-0.505850\pi$
$157$	−72.3063	−0.0367558	−0.0183779	+	0.999831i	$0.505850\pi$
$158$	−1522.53	−0.766619
$159$	1663.92	0.829920
$160$	4296.99	2.12317
$161$	−784.453	−0.383997
$162$	−294.090	−0.142629
$163$	34.3598	0.0165108	0.00825542	−	0.999966i	$-0.497372\pi$
$163$	34.3598	0.0165108	0.00825542	+	0.999966i	$0.497372\pi$
$164$	1663.83	0.792216
$165$	−696.671	−0.328702
$166$	−4277.73	−2.00010
$167$	379.079	0.175653	0.0878263	−	0.996136i	$-0.472008\pi$
$167$	379.079	0.175653	0.0878263	+	0.996136i	$0.472008\pi$
$168$	−214.840	−0.0986623
$169$	5434.80	2.47374
$170$	2166.81	0.977569
$171$	−881.356	−0.394146
$172$	1477.97	0.655198
$173$	−1398.29	−0.614507	−0.307253	−	0.951628i	$-0.599410\pi$
$173$	−1398.29	−0.614507	−0.307253	+	0.951628i	$0.599410\pi$
$174$	162.525	0.0708105
$175$	2244.80	0.969661
$176$	−864.625	−0.370304
$177$	2079.33	0.883006
$178$	1444.70	0.608341
$179$	−2591.03	−1.08191	−0.540956	−	0.841051i	$-0.681938\pi$
$179$	−2591.03	−1.08191	−0.540956	+	0.841051i	$0.681938\pi$
$180$	984.636	0.407725
$181$	3408.35	1.39967	0.699837	−	0.714303i	$-0.253256\pi$
$181$	3408.35	1.39967	0.699837	+	0.714303i	$0.253256\pi$
$182$	−2220.27	−0.904273
$183$	−470.284	−0.189969
$184$	−1146.48	−0.459344
$185$	4368.85	1.73624
$186$	1507.92	0.594439
$187$	−310.960	−0.121602
$188$	−1571.78	−0.609753
$189$	−189.000	−0.0727393
$190$	7506.16	2.86607
$191$	−2785.74	−1.05533	−0.527667	−	0.849451i	$-0.676933\pi$
$191$	−2785.74	−1.05533	−0.527667	+	0.849451i	$0.676933\pi$
$192$	330.551	0.124247
$193$	−4931.75	−1.83935	−0.919677	−	0.392676i	$-0.871549\pi$
$193$	−4931.75	−1.83935	−0.919677	+	0.392676i	$0.871549\pi$
$194$	4840.53	1.79139
$195$	−5532.85	−2.03188
$196$	253.931	0.0925403
$197$	28.9860	0.0104831	0.00524154	−	0.999986i	$-0.498332\pi$
$197$	28.9860	0.0104831	0.00524154	+	0.999986i	$0.498332\pi$
$198$	−359.443	−0.129013
$199$	−499.588	−0.177964	−0.0889821	−	0.996033i	$-0.528361\pi$
$199$	−499.588	−0.177964	−0.0889821	+	0.996033i	$0.528361\pi$
$200$	3280.76	1.15992
$201$	−1754.00	−0.615511
$202$	−2713.61	−0.945192
$203$	104.449	0.0361126
$204$	439.493	0.150837
$205$	6778.04	2.30926
$206$	−1992.24	−0.673816
$207$	−1008.58	−0.338654
$208$	−6866.71	−2.28904
$209$	−1077.21	−0.356518
$210$	1609.64	0.528931
$211$	3435.50	1.12090	0.560450	−	0.828189i	$-0.310628\pi$
$211$	3435.50	1.12090	0.560450	+	0.828189i	$0.310628\pi$
$212$	−2874.29	−0.931164
$213$	−1091.23	−0.351034
$214$	−994.369	−0.317634
$215$	6020.88	1.90986
$216$	−276.223	−0.0870120
$217$	969.078	0.303158
$218$	−7358.66	−2.28620
$219$	2242.27	0.691866
$220$	1203.44	0.368801
$221$	−2469.59	−0.751687
$222$	2254.08	0.681459
$223$	−707.522	−0.212463	−0.106231	−	0.994341i	$-0.533878\pi$
$223$	−707.522	−0.212463	−0.106231	+	0.994341i	$0.533878\pi$
$224$	1424.78	0.424988
$225$	2886.17	0.855160
$226$	3805.16	1.11998
$227$	−500.427	−0.146319	−0.0731596	−	0.997320i	$-0.523308\pi$
$227$	−500.427	−0.146319	−0.0731596	+	0.997320i	$0.523308\pi$
$228$	1522.47	0.442228
$229$	3362.25	0.970236	0.485118	−	0.874449i	$-0.338777\pi$
$229$	3362.25	0.970236	0.485118	+	0.874449i	$0.338777\pi$
$230$	8589.69	2.46255
$231$	−231.000	−0.0657952
$232$	152.652	0.0431986
$233$	−2963.07	−0.833122	−0.416561	−	0.909108i	$-0.636765\pi$
$233$	−2963.07	−0.833122	−0.416561	+	0.909108i	$0.636765\pi$
$234$	−2854.64	−0.797494
$235$	−6403.03	−1.77740
$236$	−3591.88	−0.990725
$237$	−1258.03	−0.344801
$238$	718.463	0.195677
$239$	−844.179	−0.228474	−0.114237	−	0.993454i	$-0.536442\pi$
$239$	−844.179	−0.228474	−0.114237	+	0.993454i	$0.536442\pi$
$240$	4978.18	1.33892
$241$	3616.63	0.966672	0.483336	−	0.875435i	$-0.339425\pi$
$241$	3616.63	0.966672	0.483336	+	0.875435i	$0.339425\pi$
$242$	−439.319	−0.116696
$243$	−243.000	−0.0641500
$244$	812.378	0.213144
$245$	1034.45	0.269750
$246$	3497.09	0.906366
$247$	−8555.04	−2.20382
$248$	1416.30	0.362643
$249$	−3534.60	−0.899582
$250$	−14999.2	−3.79452
$251$	−6190.73	−1.55679	−0.778397	−	0.627772i	$-0.783967\pi$
$251$	−6190.73	−1.55679	−0.778397	+	0.627772i	$0.783967\pi$
$252$	326.482	0.0816129
$253$	−1232.71	−0.306324
$254$	3288.79	0.812430
$255$	1790.39	0.439680
$256$	5341.01	1.30396
$257$	−4990.16	−1.21120	−0.605598	−	0.795771i	$-0.707066\pi$
$257$	−4990.16	−1.21120	−0.605598	+	0.795771i	$0.707066\pi$
$258$	3106.43	0.749605
$259$	1448.61	0.347537
$260$	9557.55	2.27975
$261$	134.291	0.0318484
$262$	−6852.82	−1.61591
$263$	−672.993	−0.157789	−0.0788946	−	0.996883i	$-0.525139\pi$
$263$	−672.993	−0.157789	−0.0788946	+	0.996883i	$0.525139\pi$
$264$	−337.606	−0.0787053
$265$	−11709.1	−2.71429
$266$	2488.87	0.573692
$267$	1193.72	0.273613
$268$	3029.89	0.690598
$269$	−3558.01	−0.806453	−0.403226	−	0.915100i	$-0.632111\pi$
$269$	−3558.01	−0.806453	−0.403226	+	0.915100i	$0.632111\pi$
$270$	2069.53	0.466474
$271$	3305.52	0.740944	0.370472	−	0.928844i	$-0.379196\pi$
$271$	3305.52	0.740944	0.370472	+	0.928844i	$0.379196\pi$
$272$	2222.01	0.495328
$273$	−1834.56	−0.406714
$274$	−2507.94	−0.552958
$275$	3527.54	0.773522
$276$	1742.24	0.379967
$277$	1428.48	0.309851	0.154926	−	0.987926i	$-0.450486\pi$
$277$	1428.48	0.309851	0.154926	+	0.987926i	$0.450486\pi$
$278$	5322.83	1.14835
$279$	1245.96	0.267360
$280$	1511.85	0.322679
$281$	−7156.92	−1.51938	−0.759691	−	0.650284i	$-0.774650\pi$
$281$	−7156.92	−1.51938	−0.759691	+	0.650284i	$0.774650\pi$
$282$	−3303.60	−0.697612
$283$	−2470.58	−0.518942	−0.259471	−	0.965751i	$-0.583548\pi$
$283$	−2470.58	−0.518942	−0.259471	+	0.965751i	$0.583548\pi$
$284$	1885.02	0.393857
$285$	6202.17	1.28907
$286$	−3489.00	−0.721360
$287$	2247.44	0.462238
$288$	1831.86	0.374804
$289$	−4113.86	−0.837342
$290$	−1143.71	−0.231589
$291$	3999.62	0.805711
$292$	−3873.34	−0.776268
$293$	4544.57	0.906131	0.453066	−	0.891477i	$-0.350330\pi$
$293$	4544.57	0.906131	0.453066	+	0.891477i	$0.350330\pi$
$294$	533.718	0.105874
$295$	−14632.4	−2.88791
$296$	2117.14	0.415730
$297$	−297.000	−0.0580259
$298$	−4966.41	−0.965425
$299$	−9789.99	−1.89354
$300$	−4985.62	−0.959483
$301$	1996.38	0.382291
$302$	−340.985	−0.0649717
$303$	−2242.20	−0.425118
$304$	7697.39	1.45222
$305$	3309.43	0.621303
$306$	923.738	0.172571
$307$	6711.77	1.24776	0.623878	−	0.781522i	$-0.285556\pi$
$307$	6711.77	1.24776	0.623878	+	0.781522i	$0.285556\pi$
$308$	399.034	0.0738217
$309$	−1646.15	−0.303062
$310$	−10611.3	−1.94414
$311$	−3627.08	−0.661328	−0.330664	−	0.943749i	$-0.607273\pi$
$311$	−3627.08	−0.661328	−0.330664	+	0.943749i	$0.607273\pi$
$312$	−2681.21	−0.486518
$313$	4355.14	0.786475	0.393238	−	0.919437i	$-0.371355\pi$
$313$	4355.14	0.786475	0.393238	+	0.919437i	$0.371355\pi$
$314$	262.525	0.0471820
$315$	1330.01	0.237897
$316$	2173.15	0.386864
$317$	−9227.18	−1.63486	−0.817429	−	0.576029i	$-0.804602\pi$
$317$	−9227.18	−1.63486	−0.817429	+	0.576029i	$0.804602\pi$
$318$	−6041.25	−1.06534
$319$	164.134	0.0288079
$320$	−2326.12	−0.406356
$321$	−821.626	−0.142862
$322$	2848.14	0.492922
$323$	2768.35	0.476888
$324$	419.763	0.0719758
$325$	28015.1	4.78154
$326$	−124.751	−0.0211943
$327$	−6080.30	−1.02826
$328$	3284.63	0.552937
$329$	−2123.10	−0.355775
$330$	2529.43	0.421941
$331$	837.223	0.139027	0.0695135	−	0.997581i	$-0.477855\pi$
$331$	837.223	0.139027	0.0695135	+	0.997581i	$0.477855\pi$
$332$	6105.73	1.00932
$333$	1862.50	0.306499
$334$	−1376.34	−0.225478
$335$	12343.0	2.01305
$336$	1650.65	0.268007
$337$	−9563.00	−1.54579	−0.772893	−	0.634536i	$-0.781191\pi$
$337$	−9563.00	−1.54579	−0.772893	+	0.634536i	$0.781191\pi$
$338$	−19732.3	−3.17544
$339$	3144.12	0.503732
$340$	−3092.75	−0.493317
$341$	1522.84	0.241837
$342$	3199.97	0.505949
$343$	343.000	0.0539949
$344$	2917.71	0.457303
$345$	7097.48	1.10758
$346$	5076.81	0.788818
$347$	−7584.04	−1.17329	−0.586646	−	0.809843i	$-0.699552\pi$
$347$	−7584.04	−1.17329	−0.586646	+	0.809843i	$0.699552\pi$
$348$	−231.977	−0.0357336
$349$	−1800.32	−0.276128	−0.138064	−	0.990423i	$-0.544088\pi$
$349$	−1800.32	−0.276128	−0.138064	+	0.990423i	$0.544088\pi$
$350$	−8150.27	−1.24471
$351$	−2358.73	−0.358688
$352$	2238.94	0.339023
$353$	9042.84	1.36346	0.681731	−	0.731603i	$-0.261228\pi$
$353$	9042.84	1.36346	0.681731	+	0.731603i	$0.261228\pi$
$354$	−7549.50	−1.13348
$355$	7679.11	1.14807
$356$	−2062.06	−0.306991
$357$	593.651	0.0880093
$358$	9407.33	1.38881
$359$	8915.34	1.31068	0.655339	−	0.755335i	$-0.272526\pi$
$359$	8915.34	1.31068	0.655339	+	0.755335i	$0.272526\pi$
$360$	1943.80	0.284576
$361$	2730.97	0.398159
$362$	−12374.8	−1.79671
$363$	−363.000	−0.0524864
$364$	3169.06	0.456330
$365$	−15779.1	−2.26278
$366$	1707.48	0.243856
$367$	−11721.9	−1.66724	−0.833622	−	0.552335i	$-0.813737\pi$
$367$	−11721.9	−1.66724	−0.833622	+	0.552335i	$0.813737\pi$
$368$	8808.53	1.24776
$369$	2889.57	0.407655
$370$	−15862.2	−2.22874
$371$	−3882.48	−0.543310
$372$	−2152.29	−0.299976
$373$	−8089.21	−1.12291	−0.561453	−	0.827509i	$-0.689757\pi$
$373$	−8089.21	−1.12291	−0.561453	+	0.827509i	$0.689757\pi$
$374$	1129.01	0.156096
$375$	−12393.5	−1.70666
$376$	−3102.90	−0.425585
$377$	1303.52	0.178077
$378$	686.209	0.0933725
$379$	−10580.4	−1.43398	−0.716989	−	0.697084i	$-0.754480\pi$
$379$	−10580.4	−1.43398	−0.716989	+	0.697084i	$0.754480\pi$
$380$	−10713.8	−1.44633
$381$	2717.46	0.365406
$382$	10114.3	1.35469
$383$	10512.0	1.40245	0.701224	−	0.712941i	$-0.252637\pi$
$383$	10512.0	1.40245	0.701224	+	0.712941i	$0.252637\pi$
$384$	3684.82	0.489689
$385$	1625.57	0.215186
$386$	17905.9	2.36110
$387$	2566.78	0.337149
$388$	−6909.02	−0.904001
$389$	1019.47	0.132878	0.0664388	−	0.997790i	$-0.478836\pi$
$389$	1019.47	0.132878	0.0664388	+	0.997790i	$0.478836\pi$
$390$	20088.3	2.60824
$391$	3167.96	0.409746
$392$	501.294	0.0645897
$393$	−5662.34	−0.726787
$394$	−105.241	−0.0134567
$395$	8852.87	1.12769
$396$	513.044	0.0651046
$397$	11793.5	1.49093	0.745467	−	0.666543i	$-0.232227\pi$
$397$	11793.5	1.49093	0.745467	+	0.666543i	$0.232227\pi$
$398$	1813.87	0.228445
$399$	2056.50	0.258029
$400$	−25206.6	−3.15082
$401$	−5342.21	−0.665280	−0.332640	−	0.943054i	$-0.607939\pi$
$401$	−5342.21	−0.665280	−0.332640	+	0.943054i	$0.607939\pi$
$402$	6368.31	0.790106
$403$	12094.1	1.49492
$404$	3873.21	0.476979
$405$	1710.01	0.209805
$406$	−379.226	−0.0463563
$407$	2276.39	0.277239
$408$	867.619	0.105278
$409$	−8535.63	−1.03193	−0.515965	−	0.856610i	$-0.672567\pi$
$409$	−8535.63	−1.03193	−0.515965	+	0.856610i	$0.672567\pi$
$410$	−24609.3	−2.96431
$411$	−2072.26	−0.248703
$412$	2843.59	0.340033
$413$	−4851.77	−0.578063
$414$	3661.90	0.434716
$415$	24873.3	2.94212
$416$	17781.3	2.09568
$417$	4398.14	0.516493
$418$	3911.08	0.457648
$419$	4300.54	0.501420	0.250710	−	0.968062i	$-0.419336\pi$
$419$	4300.54	0.501420	0.250710	+	0.968062i	$0.419336\pi$
$420$	−2297.48	−0.266918
$421$	−5917.77	−0.685070	−0.342535	−	0.939505i	$-0.611285\pi$
$421$	−5917.77	−0.685070	−0.342535	+	0.939505i	$0.611285\pi$
$422$	−12473.4	−1.43885
$423$	−2729.70	−0.313764
$424$	−5674.23	−0.649917
$425$	−9065.47	−1.03468
$426$	3961.99	0.450608
$427$	1097.33	0.124364
$428$	1419.29	0.160290
$429$	−2882.89	−0.324445
$430$	−21860.2	−2.45161
$431$	8110.20	0.906391	0.453196	−	0.891411i	$-0.350284\pi$
$431$	8110.20	0.906391	0.453196	+	0.891411i	$0.350284\pi$
$432$	2122.26	0.236360
$433$	−4242.54	−0.470862	−0.235431	−	0.971891i	$-0.575650\pi$
$433$	−4242.54	−0.470862	−0.235431	+	0.971891i	$0.575650\pi$
$434$	−3518.47	−0.389152
$435$	−945.019	−0.104161
$436$	10503.2	1.15370
$437$	10974.3	1.20131
$438$	−8141.10	−0.888120
$439$	−595.983	−0.0647944	−0.0323972	−	0.999475i	$-0.510314\pi$
$439$	−595.983	−0.0647944	−0.0323972	+	0.999475i	$0.510314\pi$
$440$	2375.76	0.257409
$441$	441.000	0.0476190
$442$	8966.44	0.964910
$443$	−4284.60	−0.459521	−0.229760	−	0.973247i	$-0.573794\pi$
$443$	−4284.60	−0.459521	−0.229760	+	0.973247i	$0.573794\pi$
$444$	−3217.31	−0.343890
$445$	−8400.33	−0.894862
$446$	2568.83	0.272730
$447$	−4103.64	−0.434218
$448$	−771.286	−0.0813389
$449$	−704.286	−0.0740252	−0.0370126	−	0.999315i	$-0.511784\pi$
$449$	−704.286	−0.0740252	−0.0370126	+	0.999315i	$0.511784\pi$
$450$	−10478.9	−1.09773
$451$	3531.69	0.368738
$452$	−5431.22	−0.565184
$453$	−281.748	−0.0292223
$454$	1816.92	0.187824
$455$	12910.0	1.33017
$456$	3005.56	0.308659
$457$	−7013.21	−0.717865	−0.358932	−	0.933364i	$-0.616859\pi$
$457$	−7013.21	−0.717865	−0.358932	+	0.933364i	$0.616859\pi$
$458$	−12207.5	−1.24545
$459$	763.265	0.0776169
$460$	−12260.3	−1.24270
$461$	1032.99	0.104363	0.0521813	−	0.998638i	$-0.483383\pi$
$461$	1032.99	0.104363	0.0521813	+	0.998638i	$0.483383\pi$
$462$	838.700	0.0844586
$463$	12593.3	1.26406	0.632029	−	0.774945i	$-0.282222\pi$
$463$	12593.3	1.26406	0.632029	+	0.774945i	$0.282222\pi$
$464$	−1172.84	−0.117345
$465$	−8767.91	−0.874413
$466$	10758.1	1.06944
$467$	−18527.1	−1.83583	−0.917914	−	0.396779i	$-0.870128\pi$
$467$	−18527.1	−1.83583	−0.917914	+	0.396779i	$0.870128\pi$
$468$	4074.51	0.402445
$469$	4092.67	0.402946
$470$	23247.7	2.28157
$471$	216.919	0.0212210
$472$	−7090.85	−0.691489
$473$	3137.17	0.304963
$474$	4567.58	0.442607
$475$	−31404.2	−3.03352
$476$	−1025.48	−0.0987457
$477$	−4991.76	−0.479155
$478$	3064.99	0.293283
$479$	−15402.2	−1.46920	−0.734599	−	0.678501i	$-0.762630\pi$
$479$	−15402.2	−1.46920	−0.734599	+	0.678501i	$0.762630\pi$
$480$	−12891.0	−1.22581
$481$	18078.7	1.71376
$482$	−13131.0	−1.24088
$483$	2353.36	0.221701
$484$	627.053	0.0588893
$485$	−28145.7	−2.63511
$486$	882.269	0.0823468
$487$	6816.82	0.634291	0.317146	−	0.948377i	$-0.397276\pi$
$487$	6816.82	0.634291	0.317146	+	0.948377i	$0.397276\pi$
$488$	1603.74	0.148767
$489$	−103.079	−0.00953254
$490$	−3755.82	−0.346267
$491$	6232.66	0.572864	0.286432	−	0.958101i	$-0.407531\pi$
$491$	6232.66	0.572864	0.286432	+	0.958101i	$0.407531\pi$
$492$	−4991.49	−0.457386
$493$	−421.810	−0.0385342
$494$	31061.1	2.82896
$495$	2090.01	0.189776
$496$	−10881.7	−0.985084
$497$	2546.21	0.229806
$498$	12833.2	1.15476
$499$	−19244.6	−1.72646	−0.863232	−	0.504807i	$-0.831564\pi$
$499$	−19244.6	−1.72646	−0.863232	+	0.504807i	$0.831564\pi$
$500$	21408.7	1.91486
$501$	−1137.24	−0.101413
$502$	22476.9	1.99839
$503$	7177.94	0.636280	0.318140	−	0.948044i	$-0.396942\pi$
$503$	7177.94	0.636280	0.318140	+	0.948044i	$0.396942\pi$
$504$	644.520	0.0569627
$505$	15778.5	1.39037
$506$	4475.65	0.393215
$507$	−16304.4	−1.42821
$508$	−4694.19	−0.409982
$509$	15314.2	1.33357	0.666787	−	0.745248i	$-0.267669\pi$
$509$	15314.2	1.33357	0.666787	+	0.745248i	$0.267669\pi$
$510$	−6500.43	−0.564399
$511$	−5231.97	−0.452933
$512$	−9565.62	−0.825674
$513$	2644.07	0.227560
$514$	18118.0	1.55476
$515$	11584.1	0.991176
$516$	−4433.90	−0.378278
$517$	−3336.29	−0.283811
$518$	−5259.52	−0.446120
$519$	4194.86	0.354786
$520$	18867.9	1.59118
$521$	9650.62	0.811519	0.405760	−	0.913980i	$-0.367007\pi$
$521$	9650.62	0.811519	0.405760	+	0.913980i	$0.367007\pi$
$522$	−487.576	−0.0408825
$523$	−13116.1	−1.09661	−0.548305	−	0.836278i	$-0.684727\pi$
$523$	−13116.1	−1.09661	−0.548305	+	0.836278i	$0.684727\pi$
$524$	9781.23	0.815449
$525$	−6734.39	−0.559834
$526$	2443.46	0.202548
$527$	−3913.56	−0.323487
$528$	2593.87	0.213795
$529$	391.484	0.0321759
$530$	42512.8	3.48422
$531$	−6237.99	−0.509804
$532$	−3552.43	−0.289506
$533$	28048.1	2.27936
$534$	−4334.09	−0.351226
$535$	5781.85	0.467236
$536$	5981.42	0.482011
$537$	7773.08	0.624642
$538$	12918.2	1.03521
$539$	539.000	0.0430730
$540$	−2953.91	−0.235400
$541$	18625.5	1.48017	0.740086	−	0.672512i	$-0.234785\pi$
$541$	18625.5	1.48017	0.740086	+	0.672512i	$0.234785\pi$
$542$	−12001.5	−0.951121
$543$	−10225.1	−0.808102
$544$	−5753.90	−0.453486
$545$	42787.6	3.36297
$546$	6660.82	0.522082
$547$	−6967.57	−0.544628	−0.272314	−	0.962208i	$-0.587789\pi$
$547$	−6967.57	−0.544628	−0.272314	+	0.962208i	$0.587789\pi$
$548$	3579.66	0.279043
$549$	1410.85	0.109679
$550$	−12807.6	−0.992939
$551$	−1461.21	−0.112976
$552$	3439.43	0.265202
$553$	2935.41	0.225725
$554$	−5186.42	−0.397744
$555$	−13106.6	−1.00242
$556$	−7597.43	−0.579501
$557$	−17230.7	−1.31075	−0.655374	−	0.755305i	$-0.727489\pi$
$557$	−17230.7	−1.31075	−0.655374	+	0.755305i	$0.727489\pi$
$558$	−4523.75	−0.343200
$559$	24914.9	1.88513
$560$	−11615.7	−0.876527
$561$	932.879	0.0702071
$562$	25984.9	1.95037
$563$	−20751.0	−1.55338	−0.776688	−	0.629885i	$-0.783102\pi$
$563$	−20751.0	−1.55338	−0.776688	+	0.629885i	$0.783102\pi$
$564$	4715.33	0.352041
$565$	−22125.5	−1.64748
$566$	8970.01	0.666145
$567$	567.000	0.0419961
$568$	3721.29	0.274897
$569$	−15953.4	−1.17540	−0.587700	−	0.809079i	$-0.699966\pi$
$569$	−15953.4	−1.17540	−0.587700	+	0.809079i	$0.699966\pi$
$570$	−22518.5	−1.65473
$571$	10837.6	0.794291	0.397145	−	0.917756i	$-0.370001\pi$
$571$	10837.6	0.794291	0.397145	+	0.917756i	$0.370001\pi$
$572$	4979.95	0.364025
$573$	8357.21	0.609297
$574$	−8159.87	−0.593356
$575$	−35937.5	−2.60643
$576$	−991.654	−0.0717342
$577$	9613.09	0.693584	0.346792	−	0.937942i	$-0.387271\pi$
$577$	9613.09	0.693584	0.346792	+	0.937942i	$0.387271\pi$
$578$	14936.3	1.07486
$579$	14795.3	1.06195
$580$	1632.44	0.116868
$581$	8247.39	0.588915
$582$	−14521.6	−1.03426
$583$	−6101.03	−0.433412
$584$	−7646.50	−0.541806
$585$	16598.6	1.17310
$586$	−16500.1	−1.16316
$587$	2169.43	0.152542	0.0762708	−	0.997087i	$-0.475699\pi$
$587$	2169.43	0.152542	0.0762708	+	0.997087i	$0.475699\pi$
$588$	−761.792	−0.0534282
$589$	−13557.2	−0.948411
$590$	53126.5	3.70709
$591$	−86.9580	−0.00605241
$592$	−16266.3	−1.12929
$593$	4688.62	0.324686	0.162343	−	0.986734i	$-0.448095\pi$
$593$	4688.62	0.324686	0.162343	+	0.986734i	$0.448095\pi$
$594$	1078.33	0.0744855
$595$	−4177.57	−0.287838
$596$	7088.71	0.487189
$597$	1498.76	0.102748
$598$	35544.9	2.43067
$599$	941.884	0.0642476	0.0321238	−	0.999484i	$-0.489773\pi$
$599$	941.884	0.0642476	0.0321238	+	0.999484i	$0.489773\pi$
$600$	−9842.29	−0.669683
$601$	−8131.21	−0.551879	−0.275939	−	0.961175i	$-0.588989\pi$
$601$	−8131.21	−0.551879	−0.275939	+	0.961175i	$0.588989\pi$
$602$	−7248.34	−0.490732
$603$	5262.00	0.355365
$604$	486.698	0.0327872
$605$	2554.46	0.171659
$606$	8140.83	0.545707
$607$	−27616.5	−1.84666	−0.923328	−	0.384011i	$-0.874542\pi$
$607$	−27616.5	−1.84666	−0.923328	+	0.384011i	$0.874542\pi$
$608$	−19932.4	−1.32955
$609$	−313.346	−0.0208496
$610$	−12015.7	−0.797542
$611$	−26496.3	−1.75438
$612$	−1318.48	−0.0870855
$613$	7289.82	0.480315	0.240158	−	0.970734i	$-0.422801\pi$
$613$	7289.82	0.480315	0.240158	+	0.970734i	$0.422801\pi$
$614$	−24368.7	−1.60169
$615$	−20334.1	−1.33325
$616$	787.747	0.0515247
$617$	−13672.2	−0.892094	−0.446047	−	0.895010i	$-0.647169\pi$
$617$	−13672.2	−0.892094	−0.446047	+	0.895010i	$0.647169\pi$
$618$	5976.73	0.389028
$619$	20157.3	1.30887	0.654435	−	0.756118i	$-0.272906\pi$
$619$	20157.3	1.30887	0.654435	+	0.756118i	$0.272906\pi$
$620$	15145.9	0.981084
$621$	3025.75	0.195522
$622$	13169.0	0.848920
$623$	−2785.35	−0.179122
$624$	20600.1	1.32158
$625$	47128.3	3.01621
$626$	−15812.4	−1.00957
$627$	3231.64	0.205836
$628$	−374.710	−0.0238098
$629$	−5850.12	−0.370842
$630$	−4828.91	−0.305379
$631$	−21955.4	−1.38515	−0.692576	−	0.721345i	$-0.743524\pi$
$631$	−21955.4	−1.38515	−0.692576	+	0.721345i	$0.743524\pi$
$632$	4290.09	0.270017
$633$	−10306.5	−0.647151
$634$	33501.5	2.09860
$635$	−19123.0	−1.19507
$636$	8622.86	0.537608
$637$	4280.65	0.266257
$638$	−595.927	−0.0369796
$639$	3273.70	0.202669
$640$	−25930.4	−1.60155
$641$	3575.87	0.220340	0.110170	−	0.993913i	$-0.464860\pi$
$641$	3575.87	0.220340	0.110170	+	0.993913i	$0.464860\pi$
$642$	2983.11	0.183386
$643$	6676.04	0.409452	0.204726	−	0.978819i	$-0.434370\pi$
$643$	6676.04	0.409452	0.204726	+	0.978819i	$0.434370\pi$
$644$	−4065.24	−0.248746
$645$	−18062.6	−1.10266
$646$	−10051.1	−0.612162
$647$	−12366.4	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$647$	−12366.4	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$648$	828.669	0.0502364
$649$	−7624.21	−0.461135
$650$	−101716.	−6.13786
$651$	−2907.23	−0.175028
$652$	178.061	0.0106954
$653$	19797.3	1.18642	0.593208	−	0.805049i	$-0.297861\pi$
$653$	19797.3	1.18642	0.593208	+	0.805049i	$0.297861\pi$
$654$	22076.0	1.31994
$655$	39846.4	2.37699
$656$	−25236.3	−1.50200
$657$	−6726.81	−0.399449
$658$	7708.41	0.456695
$659$	−893.599	−0.0528220	−0.0264110	−	0.999651i	$-0.508408\pi$
$659$	−893.599	−0.0528220	−0.0264110	+	0.999651i	$0.508408\pi$
$660$	−3610.33	−0.212927
$661$	16418.8	0.966137	0.483068	−	0.875583i	$-0.339522\pi$
$661$	16418.8	0.966137	0.483068	+	0.875583i	$0.339522\pi$
$662$	−3039.74	−0.178463
$663$	7408.78	0.433986
$664$	12053.5	0.704470
$665$	−14471.7	−0.843895
$666$	−6762.24	−0.393441
$667$	−1672.14	−0.0970700
$668$	1964.48	0.113785
$669$	2122.57	0.122665
$670$	−44814.4	−2.58408
$671$	1724.38	0.0992083
$672$	−4274.35	−0.245367
$673$	−17447.6	−0.999339	−0.499670	−	0.866216i	$-0.666545\pi$
$673$	−17447.6	−0.999339	−0.499670	+	0.866216i	$0.666545\pi$
$674$	34720.8	1.98426
$675$	−8658.50	−0.493727
$676$	28164.6	1.60244
$677$	−23669.4	−1.34371	−0.671853	−	0.740685i	$-0.734501\pi$
$677$	−23669.4	−1.34371	−0.671853	+	0.740685i	$0.734501\pi$
$678$	−11415.5	−0.646621
$679$	−9332.45	−0.527462
$680$	−6105.51	−0.344317
$681$	1501.28	0.0844775
$682$	−5529.02	−0.310436
$683$	−15236.3	−0.853586	−0.426793	−	0.904349i	$-0.640357\pi$
$683$	−15236.3	−0.853586	−0.426793	+	0.904349i	$0.640357\pi$
$684$	−4567.41	−0.255321
$685$	14582.7	0.813394
$686$	−1245.34	−0.0693111
$687$	−10086.8	−0.560166
$688$	−22417.2	−1.24222
$689$	−48453.4	−2.67914
$690$	−25769.1	−1.42176
$691$	−8336.36	−0.458944	−0.229472	−	0.973315i	$-0.573700\pi$
$691$	−8336.36	−0.458944	−0.229472	+	0.973315i	$0.573700\pi$
$692$	−7246.28	−0.398067
$693$	693.000	0.0379869
$694$	27535.7	1.50611
$695$	−30950.1	−1.68921
$696$	−457.955	−0.0249407
$697$	−9076.15	−0.493234
$698$	6536.48	0.354455
$699$	8889.22	0.481003
$700$	11633.1	0.628129
$701$	798.786	0.0430381	0.0215191	−	0.999768i	$-0.493150\pi$
$701$	798.786	0.0430381	0.0215191	+	0.999768i	$0.493150\pi$
$702$	8563.91	0.460433
$703$	−20265.7	−1.08725
$704$	−1212.02	−0.0648860
$705$	19209.1	1.02618
$706$	−32832.2	−1.75022
$707$	5231.79	0.278305
$708$	10775.6	0.571995
$709$	−22352.2	−1.18400	−0.591999	−	0.805939i	$-0.701661\pi$
$709$	−22352.2	−1.18400	−0.591999	+	0.805939i	$0.701661\pi$
$710$	−27880.9	−1.47373
$711$	3774.09	0.199071
$712$	−4070.79	−0.214268
$713$	−15514.2	−0.814883
$714$	−2155.39	−0.112974
$715$	20287.1	1.06111
$716$	−13427.4	−0.700844
$717$	2532.54	0.131910
$718$	−32369.3	−1.68247
$719$	13763.8	0.713914	0.356957	−	0.934121i	$-0.383814\pi$
$719$	13763.8	0.713914	0.356957	+	0.934121i	$0.383814\pi$
$720$	−14934.5	−0.773024
$721$	3841.01	0.198400
$722$	−9915.44	−0.511100
$723$	−10849.9	−0.558108
$724$	17663.0	0.906684
$725$	4785.02	0.245119
$726$	1317.96	0.0673747
$727$	16014.2	0.816966	0.408483	−	0.912766i	$-0.366058\pi$
$727$	16014.2	0.816966	0.408483	+	0.912766i	$0.366058\pi$
$728$	6256.16	0.318501
$729$	729.000	0.0370370
$730$	57289.6	2.90463
$731$	−8062.27	−0.407926
$732$	−2437.13	−0.123059
$733$	21073.3	1.06188	0.530942	−	0.847408i	$-0.321838\pi$
$733$	21073.3	1.06188	0.530942	+	0.847408i	$0.321838\pi$
$734$	42559.2	2.14017
$735$	−3103.35	−0.155740
$736$	−22809.7	−1.14236
$737$	6431.33	0.321440
$738$	−10491.3	−0.523291
$739$	4570.10	0.227488	0.113744	−	0.993510i	$-0.463716\pi$
$739$	4570.10	0.227488	0.113744	+	0.993510i	$0.463716\pi$
$740$	22640.5	1.12471
$741$	25665.1	1.27238
$742$	14096.3	0.697426
$743$	7122.27	0.351670	0.175835	−	0.984420i	$-0.443737\pi$
$743$	7122.27	0.351670	0.175835	+	0.984420i	$0.443737\pi$
$744$	−4248.91	−0.209372
$745$	28877.7	1.42013
$746$	29369.8	1.44143
$747$	10603.8	0.519374
$748$	−1611.47	−0.0787718
$749$	1917.13	0.0935251
$750$	44997.5	2.19077
$751$	31096.8	1.51097	0.755485	−	0.655166i	$-0.227401\pi$
$751$	31096.8	1.51097	0.755485	+	0.655166i	$0.227401\pi$
$752$	23840.0	1.15606
$753$	18572.2	0.898816
$754$	−4732.75	−0.228590
$755$	1982.69	0.0955727
$756$	−979.447	−0.0471192
$757$	2783.78	0.133657	0.0668284	−	0.997764i	$-0.478712\pi$
$757$	2783.78	0.133657	0.0668284	+	0.997764i	$0.478712\pi$
$758$	38414.6	1.84074
$759$	3698.13	0.176856
$760$	−21150.4	−1.00948
$761$	22236.2	1.05922	0.529608	−	0.848243i	$-0.322339\pi$
$761$	22236.2	1.05922	0.529608	+	0.848243i	$0.322339\pi$
$762$	−9866.38	−0.469057
$763$	14187.4	0.673155
$764$	−14436.4	−0.683626
$765$	−5371.16	−0.253849
$766$	−38166.3	−1.80027
$767$	−60550.2	−2.85051
$768$	−16023.0	−0.752841
$769$	−28682.5	−1.34502	−0.672508	−	0.740090i	$-0.734783\pi$
$769$	−28682.5	−1.34502	−0.672508	+	0.740090i	$0.734783\pi$
$770$	−5902.01	−0.276225
$771$	14970.5	0.699284
$772$	−25557.6	−1.19150
$773$	27245.4	1.26772	0.633861	−	0.773447i	$-0.281469\pi$
$773$	27245.4	1.26772	0.633861	+	0.773447i	$0.281469\pi$
$774$	−9319.30	−0.432785
$775$	44395.6	2.05772
$776$	−13639.3	−0.630959
$777$	−4345.83	−0.200651
$778$	−3701.45	−0.170570
$779$	−31441.2	−1.44608
$780$	−28672.7	−1.31621
$781$	4001.19	0.183321
$782$	−11502.0	−0.525975
$783$	−402.874	−0.0183877
$784$	−3851.51	−0.175451
$785$	−1526.48	−0.0694042
$786$	20558.5	0.932947
$787$	−33427.2	−1.51404	−0.757020	−	0.653391i	$-0.773346\pi$
$787$	−33427.2	−1.51404	−0.757020	+	0.653391i	$0.773346\pi$
$788$	150.213	0.00679075
$789$	2018.98	0.0910996
$790$	−32142.5	−1.44757
$791$	−7336.29	−0.329770
$792$	1012.82	0.0454405
$793$	13694.7	0.613258
$794$	−42819.2	−1.91385
$795$	35127.4	1.56710
$796$	−2589.00	−0.115282
$797$	−6592.94	−0.293016	−0.146508	−	0.989209i	$-0.546803\pi$
$797$	−6592.94	−0.293016	−0.146508	+	0.989209i	$0.546803\pi$
$798$	−7466.60	−0.331221
$799$	8574.00	0.379632
$800$	65272.4	2.88466
$801$	−3581.17	−0.157970
$802$	19396.2	0.853993
$803$	−8221.66	−0.361315
$804$	−9089.68	−0.398717
$805$	−16560.8	−0.725082
$806$	−43910.6	−1.91896
$807$	10674.0	0.465606
$808$	7646.25	0.332913
$809$	22661.3	0.984830	0.492415	−	0.870361i	$-0.336114\pi$
$809$	22661.3	0.984830	0.492415	+	0.870361i	$0.336114\pi$
$810$	−6208.60	−0.269319
$811$	2938.33	0.127224	0.0636121	−	0.997975i	$-0.479738\pi$
$811$	2938.33	0.127224	0.0636121	+	0.997975i	$0.479738\pi$
$812$	541.281	0.0233931
$813$	−9916.56	−0.427784
$814$	−8264.96	−0.355881
$815$	725.379	0.0311766
$816$	−6666.04	−0.285978
$817$	−27928.9	−1.19597
$818$	30990.6	1.32465
$819$	5503.69	0.234816
$820$	35125.6	1.49590
$821$	27511.4	1.16949	0.584747	−	0.811216i	$-0.301194\pi$
$821$	27511.4	1.16949	0.584747	+	0.811216i	$0.301194\pi$
$822$	7523.83	0.319250
$823$	25092.9	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$823$	25092.9	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$824$	5613.63	0.237330
$825$	−10582.6	−0.446593
$826$	17615.5	0.742036
$827$	10259.3	0.431382	0.215691	−	0.976462i	$-0.430800\pi$
$827$	10259.3	0.431382	0.215691	+	0.976462i	$0.430800\pi$
$828$	−5226.73	−0.219374
$829$	−40456.9	−1.69497	−0.847483	−	0.530823i	$-0.821883\pi$
$829$	−40456.9	−1.69497	−0.847483	+	0.530823i	$0.821883\pi$
$830$	−90308.3	−3.77668
$831$	−4285.43	−0.178893
$832$	−9625.67	−0.401094
$833$	−1385.18	−0.0576156
$834$	−15968.5	−0.663001
$835$	8002.83	0.331676
$836$	−5582.39	−0.230946
$837$	−3737.87	−0.154361
$838$	−15614.1	−0.643652
$839$	11387.0	0.468563	0.234281	−	0.972169i	$-0.424726\pi$
$839$	11387.0	0.468563	0.234281	+	0.972169i	$0.424726\pi$
$840$	−4535.54	−0.186299
$841$	−24166.4	−0.990871
$842$	21485.9	0.879396
$843$	21470.8	0.877215
$844$	17803.7	0.726099
$845$	114736.	4.67103
$846$	9910.81	0.402767
$847$	847.000	0.0343604
$848$	43595.9	1.76544
$849$	7411.73	0.299611
$850$	32914.4	1.32818
$851$	−23191.1	−0.934173
$852$	−5655.06	−0.227393
$853$	8058.93	0.323485	0.161742	−	0.986833i	$-0.448289\pi$
$853$	8058.93	0.323485	0.161742	+	0.986833i	$0.448289\pi$
$854$	−3984.12	−0.159641
$855$	−18606.5	−0.744245
$856$	2801.87	0.111876
$857$	4025.02	0.160434	0.0802170	−	0.996777i	$-0.474439\pi$
$857$	4025.02	0.160434	0.0802170	+	0.996777i	$0.474439\pi$
$858$	10467.0	0.416477
$859$	−4319.84	−0.171584	−0.0857922	−	0.996313i	$-0.527342\pi$
$859$	−4319.84	−0.171584	−0.0857922	+	0.996313i	$0.527342\pi$
$860$	31201.8	1.23718
$861$	−6742.32	−0.266873
$862$	−29446.0	−1.16350
$863$	11967.7	0.472055	0.236028	−	0.971746i	$-0.424154\pi$
$863$	11967.7	0.472055	0.236028	+	0.971746i	$0.424154\pi$
$864$	−5495.59	−0.216393
$865$	−29519.6	−1.16034
$866$	15403.5	0.604427
$867$	12341.6	0.483439
$868$	5022.01	0.196380
$869$	4612.78	0.180067
$870$	3431.12	0.133708
$871$	51076.6	1.98699
$872$	20734.8	0.805239
$873$	−11998.9	−0.465178
$874$	−39844.8	−1.54207
$875$	28918.1	1.11727
$876$	11620.0	0.448179
$877$	−3248.36	−0.125074	−0.0625368	−	0.998043i	$-0.519919\pi$
$877$	−3248.36	−0.125074	−0.0625368	+	0.998043i	$0.519919\pi$
$878$	2163.86	0.0831740
$879$	−13633.7	−0.523155
$880$	−18253.3	−0.699226
$881$	1264.38	0.0483518	0.0241759	−	0.999708i	$-0.492304\pi$
$881$	1264.38	0.0483518	0.0241759	+	0.999708i	$0.492304\pi$
$882$	−1601.16	−0.0611267
$883$	−11524.4	−0.439216	−0.219608	−	0.975588i	$-0.570478\pi$
$883$	−11524.4	−0.439216	−0.219608	+	0.975588i	$0.570478\pi$
$884$	−12798.1	−0.486929
$885$	43897.3	1.66733
$886$	15556.3	0.589868
$887$	24949.8	0.944457	0.472228	−	0.881476i	$-0.343450\pi$
$887$	24949.8	0.944457	0.472228	+	0.881476i	$0.343450\pi$
$888$	−6351.41	−0.240022
$889$	−6340.73	−0.239214
$890$	30499.4	1.14870
$891$	891.000	0.0335013
$892$	−3666.56	−0.137630
$893$	29701.6	1.11302
$894$	14899.2	0.557388
$895$	−54699.8	−2.04292
$896$	−8597.92	−0.320576
$897$	29370.0	1.09324
$898$	2557.08	0.0950231
$899$	2065.69	0.0766349
$900$	14956.9	0.553958
$901$	15679.1	0.579743
$902$	−12822.7	−0.473334
$903$	−5989.15	−0.220716
$904$	−10722.0	−0.394477
$905$	71954.6	2.64293
$906$	1022.95	0.0375115
$907$	40615.8	1.48691	0.743454	−	0.668787i	$-0.233186\pi$
$907$	40615.8	1.48691	0.743454	+	0.668787i	$0.233186\pi$
$908$	−2593.34	−0.0947830
$909$	6726.59	0.245442
$910$	−46872.8	−1.70749
$911$	−14594.0	−0.530760	−0.265380	−	0.964144i	$-0.585497\pi$
$911$	−14594.0	−0.530760	−0.265380	+	0.964144i	$0.585497\pi$
$912$	−23092.2	−0.838441
$913$	12960.2	0.469792
$914$	25463.1	0.921494
$915$	−9928.29	−0.358710
$916$	17424.1	0.628502
$917$	13212.1	0.475794
$918$	−2771.22	−0.0996337
$919$	28339.1	1.01721	0.508607	−	0.860999i	$-0.330161\pi$
$919$	28339.1	1.01721	0.508607	+	0.860999i	$0.330161\pi$
$920$	−24203.5	−0.867355
$921$	−20135.3	−0.720392
$922$	−3750.51	−0.133966
$923$	31776.8	1.13320
$924$	−1197.10	−0.0426209
$925$	66363.9	2.35895
$926$	−45722.9	−1.62262
$927$	4938.44	0.174973
$928$	3037.08	0.107432
$929$	25916.5	0.915276	0.457638	−	0.889139i	$-0.348696\pi$
$929$	25916.5	0.915276	0.457638	+	0.889139i	$0.348696\pi$
$930$	31834.0	1.12245
$931$	−4798.49	−0.168920
$932$	−15355.4	−0.539682
$933$	10881.2	0.381818
$934$	67267.0	2.35658
$935$	−6564.75	−0.229615
$936$	8043.63	0.280891
$937$	−2004.67	−0.0698930	−0.0349465	−	0.999389i	$-0.511126\pi$
$937$	−2004.67	−0.0698930	−0.0349465	+	0.999389i	$0.511126\pi$
$938$	−14859.4	−0.517246
$939$	−13065.4	−0.454072
$940$	−33182.2	−1.15137
$941$	−36486.7	−1.26401	−0.632004	−	0.774965i	$-0.717767\pi$
$941$	−36486.7	−1.26401	−0.632004	+	0.774965i	$0.717767\pi$
$942$	−787.575	−0.0272405
$943$	−35979.8	−1.24249
$944$	54480.0	1.87836
$945$	−3990.03	−0.137350
$946$	−11390.3	−0.391468
$947$	−26275.1	−0.901611	−0.450806	−	0.892622i	$-0.648863\pi$
$947$	−26275.1	−0.901611	−0.450806	+	0.892622i	$0.648863\pi$
$948$	−6519.44	−0.223356
$949$	−65295.1	−2.23347
$950$	114020.	3.89401
$951$	27681.5	0.943886
$952$	−2024.44	−0.0689208
$953$	−25800.3	−0.876971	−0.438485	−	0.898738i	$-0.644485\pi$
$953$	−25800.3	−0.876971	−0.438485	+	0.898738i	$0.644485\pi$
$954$	18123.8	0.615072
$955$	−58810.4	−1.99273
$956$	−4374.75	−0.148002
$957$	−492.401	−0.0166323
$958$	55921.5	1.88595
$959$	4835.27	0.162814
$960$	6978.35	0.234610
$961$	−10625.4	−0.356666
$962$	−65639.0	−2.19988
$963$	2464.88	0.0824814
$964$	18742.3	0.626193
$965$	−104115.	−3.47316
$966$	−8544.42	−0.284588
$967$	−24454.9	−0.813254	−0.406627	−	0.913594i	$-0.633295\pi$
$967$	−24454.9	−0.813254	−0.406627	+	0.913594i	$0.633295\pi$
$968$	1237.89	0.0411025
$969$	−8305.04	−0.275331
$970$	102190.	3.38259
$971$	−57692.0	−1.90672	−0.953359	−	0.301839i	$-0.902400\pi$
$971$	−57692.0	−1.90672	−0.953359	+	0.301839i	$0.902400\pi$
$972$	−1259.29	−0.0415553
$973$	−10262.3	−0.338124
$974$	−24750.1	−0.814214
$975$	−84045.3	−2.76062
$976$	−12321.8	−0.404110
$977$	−6855.66	−0.224495	−0.112248	−	0.993680i	$-0.535805\pi$
$977$	−6855.66	−0.224495	−0.112248	+	0.993680i	$0.535805\pi$
$978$	374.254	0.0122365
$979$	−4376.98	−0.142890
$980$	5360.80	0.174739
$981$	18240.9	0.593667
$982$	−22629.2	−0.735362
$983$	−8142.33	−0.264191	−0.132096	−	0.991237i	$-0.542171\pi$
$983$	−8142.33	−0.264191	−0.132096	+	0.991237i	$0.542171\pi$
$984$	−9853.89	−0.319238
$985$	611.931	0.0197947
$986$	1531.48	0.0494648
$987$	6369.29	0.205407
$988$	−44334.4	−1.42760
$989$	−31960.6	−1.02759
$990$	−7588.29	−0.243608
$991$	11895.7	0.381312	0.190656	−	0.981657i	$-0.438938\pi$
$991$	11895.7	0.381312	0.190656	+	0.981657i	$0.438938\pi$
$992$	28178.1	0.901870
$993$	−2511.67	−0.0802672
$994$	−9244.64	−0.294992
$995$	−10546.9	−0.336041
$996$	−18317.2	−0.582734
$997$	−44608.5	−1.41702	−0.708509	−	0.705702i	$-0.750632\pi$
$997$	−44608.5	−1.41702	−0.708509	+	0.705702i	$0.750632\pi$
$998$	69872.0	2.21619
$999$	−5587.49	−0.176957

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.k.1.2	✓	5
3.2	odd	2		693.4.a.p.1.4		5
7.6	odd	2		1617.4.a.n.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.2	✓	5	1.1	even	1	trivial
693.4.a.p.1.4		5	3.2	odd	2
1617.4.a.n.1.2		5	7.6	odd	2

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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

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	231.4.a.k.1.2

Level	$231$
Weight	$4$
Character	231.1
Self dual	yes
Analytic conductor	$13.629$
Analytic rank	$0$
Dimension	$5$
CM	no
Inner twists	$1$