LMFDB - Embedded newform 231.4.a.g.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)

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")

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);

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

Newform invariants

comment:select newform

sage:traces = [2,-3,6,-3,25] 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:	$2$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+0.561553 q^{2} +3.00000 q^{3} -7.68466 q^{4} +18.6847 q^{5} +1.68466 q^{6} +7.00000 q^{7} -8.80776 q^{8} +9.00000 q^{9} +10.4924 q^{10} -11.0000 q^{11} -23.0540 q^{12} +36.4384 q^{13} +3.93087 q^{14} +56.0540 q^{15} +56.5312 q^{16} +41.1231 q^{17} +5.05398 q^{18} -23.6998 q^{19} -143.585 q^{20} +21.0000 q^{21} -6.17708 q^{22} -140.047 q^{23} -26.4233 q^{24} +224.116 q^{25} +20.4621 q^{26} +27.0000 q^{27} -53.7926 q^{28} +278.455 q^{29} +31.4773 q^{30} +191.447 q^{31} +102.207 q^{32} -33.0000 q^{33} +23.0928 q^{34} +130.793 q^{35} -69.1619 q^{36} +196.115 q^{37} -13.3087 q^{38} +109.315 q^{39} -164.570 q^{40} -322.695 q^{41} +11.7926 q^{42} -3.67615 q^{43} +84.5312 q^{44} +168.162 q^{45} -78.6440 q^{46} -397.596 q^{47} +169.594 q^{48} +49.0000 q^{49} +125.853 q^{50} +123.369 q^{51} -280.017 q^{52} +597.508 q^{53} +15.1619 q^{54} -205.531 q^{55} -61.6543 q^{56} -71.0994 q^{57} +156.367 q^{58} +668.779 q^{59} -430.756 q^{60} -667.788 q^{61} +107.508 q^{62} +63.0000 q^{63} -394.855 q^{64} +680.840 q^{65} -18.5312 q^{66} -730.762 q^{67} -316.017 q^{68} -420.142 q^{69} +73.4470 q^{70} -31.2651 q^{71} -79.2699 q^{72} -434.408 q^{73} +110.129 q^{74} +672.349 q^{75} +182.125 q^{76} -77.0000 q^{77} +61.3863 q^{78} -782.004 q^{79} +1056.27 q^{80} +81.0000 q^{81} -181.210 q^{82} -426.705 q^{83} -161.378 q^{84} +768.371 q^{85} -2.06435 q^{86} +835.366 q^{87} +96.8854 q^{88} -899.693 q^{89} +94.4318 q^{90} +255.069 q^{91} +1076.22 q^{92} +574.341 q^{93} -223.271 q^{94} -442.823 q^{95} +306.622 q^{96} -942.716 q^{97} +27.5161 q^{98} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} + 25 q^{5} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 12 q^{10} - 22 q^{11} - 9 q^{12} + 77 q^{13} - 21 q^{14} + 75 q^{15} - 23 q^{16} + 74 q^{17} - 27 q^{18} - 101 q^{19}+ \cdots - 198 q^{99}+O(q^{100})$ 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 + 25 * q^5 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9 - 12 * q^10 - 22 * q^11 - 9 * q^12 + 77 * q^13 - 21 * q^14 + 75 * q^15 - 23 * q^16 + 74 * q^17 - 27 * q^18 - 101 * q^19 - 114 * q^20 + 42 * q^21 + 33 * q^22 + 58 * q^23 + 9 * q^24 + 139 * q^25 - 124 * q^26 + 54 * q^27 - 21 * q^28 + 91 * q^29 - 36 * q^30 + 152 * q^31 + 291 * q^32 - 66 * q^33 - 94 * q^34 + 175 * q^35 - 27 * q^36 + 619 * q^37 + 262 * q^38 + 231 * q^39 - 90 * q^40 + 138 * q^41 - 63 * q^42 - 230 * q^43 + 33 * q^44 + 225 * q^45 - 784 * q^46 + 149 * q^47 - 69 * q^48 + 98 * q^49 + 429 * q^50 + 222 * q^51 - 90 * q^52 + 1228 * q^53 - 81 * q^54 - 275 * q^55 + 21 * q^56 - 303 * q^57 + 824 * q^58 + 649 * q^59 - 342 * q^60 - 412 * q^61 + 248 * q^62 + 126 * q^63 - 431 * q^64 + 937 * q^65 + 99 * q^66 - 1243 * q^67 - 162 * q^68 + 174 * q^69 - 84 * q^70 + 960 * q^71 + 27 * q^72 - 741 * q^73 - 1396 * q^74 + 417 * q^75 - 180 * q^76 - 154 * q^77 - 372 * q^78 - 492 * q^79 + 554 * q^80 + 162 * q^81 - 1822 * q^82 - 1744 * q^83 - 63 * q^84 + 976 * q^85 + 804 * q^86 + 273 * q^87 - 33 * q^88 - 1552 * q^89 - 108 * q^90 + 539 * q^91 + 2004 * q^92 + 456 * q^93 - 2170 * q^94 - 931 * q^95 + 873 * q^96 + 440 * q^97 - 147 * q^98 - 198 * 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

)

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

$2$	0.561553	0.198539	0.0992695	−	0.995061i	$-0.468349\pi$
$2$	0.561553	0.198539	0.0992695	+	0.995061i	$0.468349\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.68466	−0.960582
$5$	18.6847	1.67121	0.835603	−	0.549333i	$-0.185118\pi$
$5$	18.6847	1.67121	0.835603	+	0.549333i	$0.185118\pi$
$6$	1.68466	0.114626
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.80776	−0.389252
$9$	9.00000	0.333333
$10$	10.4924	0.331800
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−23.0540	−0.554592
$13$	36.4384	0.777401	0.388700	−	0.921364i	$-0.372924\pi$
$13$	36.4384	0.777401	0.388700	+	0.921364i	$0.372924\pi$
$14$	3.93087	0.0750407
$15$	56.0540	0.964872
$16$	56.5312	0.883301
$17$	41.1231	0.586695	0.293348	−	0.956006i	$-0.405231\pi$
$17$	41.1231	0.586695	0.293348	+	0.956006i	$0.405231\pi$
$18$	5.05398	0.0661796
$19$	−23.6998	−0.286164	−0.143082	−	0.989711i	$-0.545701\pi$
$19$	−23.6998	−0.286164	−0.143082	+	0.989711i	$0.545701\pi$
$20$	−143.585	−1.60533
$21$	21.0000	0.218218
$22$	−6.17708	−0.0598617
$23$	−140.047	−1.26965	−0.634824	−	0.772657i	$-0.718927\pi$
$23$	−140.047	−1.26965	−0.634824	+	0.772657i	$0.718927\pi$
$24$	−26.4233	−0.224735
$25$	224.116	1.79293
$26$	20.4621	0.154344
$27$	27.0000	0.192450
$28$	−53.7926	−0.363066
$29$	278.455	1.78303	0.891515	−	0.452991i	$-0.149643\pi$
$29$	278.455	1.78303	0.891515	+	0.452991i	$0.149643\pi$
$30$	31.4773	0.191565
$31$	191.447	1.10919	0.554595	−	0.832120i	$-0.312873\pi$
$31$	191.447	1.10919	0.554595	+	0.832120i	$0.312873\pi$
$32$	102.207	0.564621
$33$	−33.0000	−0.174078
$34$	23.0928	0.116482
$35$	130.793	0.631657
$36$	−69.1619	−0.320194
$37$	196.115	0.871379	0.435690	−	0.900097i	$-0.356504\pi$
$37$	196.115	0.871379	0.435690	+	0.900097i	$0.356504\pi$
$38$	−13.3087	−0.0568146
$39$	109.315	0.448832
$40$	−164.570	−0.650520
$41$	−322.695	−1.22918	−0.614591	−	0.788846i	$-0.710679\pi$
$41$	−322.695	−1.22918	−0.614591	+	0.788846i	$0.710679\pi$
$42$	11.7926	0.0433247
$43$	−3.67615	−0.0130374	−0.00651869	−	0.999979i	$-0.502075\pi$
$43$	−3.67615	−0.0130374	−0.00651869	+	0.999979i	$0.502075\pi$
$44$	84.5312	0.289626
$45$	168.162	0.557069
$46$	−78.6440	−0.252074
$47$	−397.596	−1.23394	−0.616971	−	0.786986i	$-0.711640\pi$
$47$	−397.596	−1.23394	−0.616971	+	0.786986i	$0.711640\pi$
$48$	169.594	0.509974
$49$	49.0000	0.142857
$50$	125.853	0.355967
$51$	123.369	0.338729
$52$	−280.017	−0.746757
$53$	597.508	1.54857	0.774283	−	0.632840i	$-0.218111\pi$
$53$	597.508	1.54857	0.774283	+	0.632840i	$0.218111\pi$
$54$	15.1619	0.0382088
$55$	−205.531	−0.503888
$56$	−61.6543	−0.147123
$57$	−71.0994	−0.165217
$58$	156.367	0.354001
$59$	668.779	1.47572	0.737861	−	0.674952i	$-0.235836\pi$
$59$	668.779	1.47572	0.737861	+	0.674952i	$0.235836\pi$
$60$	−430.756	−0.926839
$61$	−667.788	−1.40166	−0.700832	−	0.713327i	$-0.747188\pi$
$61$	−667.788	−1.40166	−0.700832	+	0.713327i	$0.747188\pi$
$62$	107.508	0.220217
$63$	63.0000	0.125988
$64$	−394.855	−0.771201
$65$	680.840	1.29920
$66$	−18.5312	−0.0345612
$67$	−730.762	−1.33249	−0.666245	−	0.745733i	$-0.732099\pi$
$67$	−730.762	−1.33249	−0.666245	+	0.745733i	$0.732099\pi$
$68$	−316.017	−0.563569
$69$	−420.142	−0.733031
$70$	73.4470	0.125408
$71$	−31.2651	−0.0522603	−0.0261302	−	0.999659i	$-0.508318\pi$
$71$	−31.2651	−0.0522603	−0.0261302	+	0.999659i	$0.508318\pi$
$72$	−79.2699	−0.129751
$73$	−434.408	−0.696488	−0.348244	−	0.937404i	$-0.613222\pi$
$73$	−434.408	−0.696488	−0.348244	+	0.937404i	$0.613222\pi$
$74$	110.129	0.173003
$75$	672.349	1.03515
$76$	182.125	0.274884
$77$	−77.0000	−0.113961
$78$	61.3863	0.0891107
$79$	−782.004	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$79$	−782.004	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$80$	1056.27	1.47618
$81$	81.0000	0.111111
$82$	−181.210	−0.244041
$83$	−426.705	−0.564300	−0.282150	−	0.959370i	$-0.591048\pi$
$83$	−426.705	−0.564300	−0.282150	+	0.959370i	$0.591048\pi$
$84$	−161.378	−0.209616
$85$	768.371	0.980489
$86$	−2.06435	−0.00258843
$87$	835.366	1.02943
$88$	96.8854	0.117364
$89$	−899.693	−1.07154	−0.535771	−	0.844363i	$-0.679979\pi$
$89$	−899.693	−1.07154	−0.535771	+	0.844363i	$0.679979\pi$
$90$	94.4318	0.110600
$91$	255.069	0.293830
$92$	1076.22	1.21960
$93$	574.341	0.640391
$94$	−223.271	−0.244986
$95$	−442.823	−0.478239
$96$	306.622	0.325984
$97$	−942.716	−0.986786	−0.493393	−	0.869806i	$-0.664244\pi$
$97$	−942.716	−0.986786	−0.493393	+	0.869806i	$0.664244\pi$
$98$	27.5161	0.0283627
$99$	−99.0000	−0.100504
$100$	−1722.26	−1.72226
$101$	366.945	0.361509	0.180754	−	0.983528i	$-0.442146\pi$
$101$	366.945	0.361509	0.180754	+	0.983528i	$0.442146\pi$
$102$	69.2784	0.0672508
$103$	1007.35	0.963658	0.481829	−	0.876265i	$-0.339973\pi$
$103$	1007.35	0.963658	0.481829	+	0.876265i	$0.339973\pi$
$104$	−320.941	−0.302605
$105$	392.378	0.364687
$106$	335.532	0.307451
$107$	−1690.80	−1.52763	−0.763814	−	0.645437i	$-0.776675\pi$
$107$	−1690.80	−1.52763	−0.763814	+	0.645437i	$0.776675\pi$
$108$	−207.486	−0.184864
$109$	−1808.57	−1.58926	−0.794631	−	0.607093i	$-0.792336\pi$
$109$	−1808.57	−1.58926	−0.794631	+	0.607093i	$0.792336\pi$
$110$	−115.417	−0.100041
$111$	588.344	0.503091
$112$	395.719	0.333856
$113$	952.557	0.793000	0.396500	−	0.918035i	$-0.370225\pi$
$113$	952.557	0.793000	0.396500	+	0.918035i	$0.370225\pi$
$114$	−39.9261	−0.0328019
$115$	−2616.74	−2.12184
$116$	−2139.84	−1.71275
$117$	327.946	0.259134
$118$	375.555	0.292988
$119$	287.862	0.221750
$120$	−493.710	−0.375578
$121$	121.000	0.0909091
$122$	−374.998	−0.278285
$123$	−968.085	−0.709669
$124$	−1471.20	−1.06547
$125$	1851.96	1.32515
$126$	35.3778	0.0250136
$127$	252.951	0.176738	0.0883691	−	0.996088i	$-0.471834\pi$
$127$	252.951	0.176738	0.0883691	+	0.996088i	$0.471834\pi$
$128$	−1039.39	−0.717735
$129$	−11.0284	−0.00752713
$130$	382.328	0.257941
$131$	−1497.38	−0.998679	−0.499339	−	0.866406i	$-0.666424\pi$
$131$	−1497.38	−0.998679	−0.499339	+	0.866406i	$0.666424\pi$
$132$	253.594	0.167216
$133$	−165.899	−0.108160
$134$	−410.362	−0.264551
$135$	504.486	0.321624
$136$	−362.203	−0.228372
$137$	−1207.56	−0.753056	−0.376528	−	0.926405i	$-0.622882\pi$
$137$	−1207.56	−0.753056	−0.376528	+	0.926405i	$0.622882\pi$
$138$	−235.932	−0.145535
$139$	212.519	0.129681	0.0648404	−	0.997896i	$-0.479346\pi$
$139$	212.519	0.129681	0.0648404	+	0.997896i	$0.479346\pi$
$140$	−1005.10	−0.606758
$141$	−1192.79	−0.712417
$142$	−17.5570	−0.0103757
$143$	−400.823	−0.234395
$144$	508.781	0.294434
$145$	5202.85	2.97981
$146$	−243.943	−0.138280
$147$	147.000	0.0824786
$148$	−1507.07	−0.837032
$149$	879.656	0.483653	0.241826	−	0.970320i	$-0.422254\pi$
$149$	879.656	0.483653	0.241826	+	0.970320i	$0.422254\pi$
$150$	377.560	0.205517
$151$	−1215.18	−0.654899	−0.327450	−	0.944869i	$-0.606189\pi$
$151$	−1215.18	−0.654899	−0.327450	+	0.944869i	$0.606189\pi$
$152$	208.742	0.111390
$153$	370.108	0.195565
$154$	−43.2396	−0.0226256
$155$	3577.12	1.85369
$156$	−840.051	−0.431140
$157$	1226.57	0.623507	0.311754	−	0.950163i	$-0.399084\pi$
$157$	1226.57	0.623507	0.311754	+	0.950163i	$0.399084\pi$
$158$	−439.136	−0.221113
$159$	1792.52	0.894065
$160$	1909.71	0.943599
$161$	−980.331	−0.479882
$162$	45.4858	0.0220599
$163$	−441.224	−0.212021	−0.106010	−	0.994365i	$-0.533808\pi$
$163$	−441.224	−0.212021	−0.106010	+	0.994365i	$0.533808\pi$
$164$	2479.80	1.18073
$165$	−616.594	−0.290920
$166$	−239.617	−0.112036
$167$	−1793.53	−0.831064	−0.415532	−	0.909579i	$-0.636405\pi$
$167$	−1793.53	−0.831064	−0.415532	+	0.909579i	$0.636405\pi$
$168$	−184.963	−0.0849417
$169$	−869.240	−0.395648
$170$	431.481	0.194665
$171$	−213.298	−0.0953879
$172$	28.2499	0.0125235
$173$	3825.04	1.68099	0.840497	−	0.541816i	$-0.182263\pi$
$173$	3825.04	1.68099	0.840497	+	0.541816i	$0.182263\pi$
$174$	469.102	0.204383
$175$	1568.82	0.677664
$176$	−621.844	−0.266325
$177$	2006.34	0.852009
$178$	−505.225	−0.212743
$179$	1315.65	0.549365	0.274683	−	0.961535i	$-0.411427\pi$
$179$	1315.65	0.549365	0.274683	+	0.961535i	$0.411427\pi$
$180$	−1292.27	−0.535111
$181$	674.682	0.277065	0.138532	−	0.990358i	$-0.455762\pi$
$181$	674.682	0.277065	0.138532	+	0.990358i	$0.455762\pi$
$182$	143.235	0.0583366
$183$	−2003.36	−0.809251
$184$	1233.50	0.494213
$185$	3664.33	1.45626
$186$	322.523	0.127143
$187$	−452.354	−0.176895
$188$	3055.39	1.18530
$189$	189.000	0.0727393
$190$	−248.668	−0.0949490
$191$	1915.69	0.725730	0.362865	−	0.931842i	$-0.381799\pi$
$191$	1915.69	0.725730	0.362865	+	0.931842i	$0.381799\pi$
$192$	−1184.57	−0.445253
$193$	−2394.71	−0.893134	−0.446567	−	0.894750i	$-0.647354\pi$
$193$	−2394.71	−0.893134	−0.446567	+	0.894750i	$0.647354\pi$
$194$	−529.385	−0.195915
$195$	2042.52	0.750092
$196$	−376.548	−0.137226
$197$	1356.81	0.490703	0.245351	−	0.969434i	$-0.421097\pi$
$197$	1356.81	0.490703	0.245351	+	0.969434i	$0.421097\pi$
$198$	−55.5937	−0.0199539
$199$	2121.45	0.755707	0.377854	−	0.925865i	$-0.376662\pi$
$199$	2121.45	0.755707	0.377854	+	0.925865i	$0.376662\pi$
$200$	−1973.96	−0.697902
$201$	−2192.29	−0.769313
$202$	206.059	0.0717736
$203$	1949.19	0.673922
$204$	−948.051	−0.325377
$205$	−6029.45	−2.05422
$206$	565.678	0.191324
$207$	−1260.43	−0.423216
$208$	2059.91	0.686678
$209$	260.698	0.0862816
$210$	220.341	0.0724046
$211$	4492.12	1.46564	0.732821	−	0.680421i	$-0.238203\pi$
$211$	4492.12	1.46564	0.732821	+	0.680421i	$0.238203\pi$
$212$	−4591.64	−1.48752
$213$	−93.7953	−0.0301725
$214$	−949.476	−0.303294
$215$	−68.6876	−0.0217882
$216$	−237.810	−0.0749116
$217$	1340.13	0.419234
$218$	−1015.61	−0.315530
$219$	−1303.22	−0.402118
$220$	1579.44	0.484026
$221$	1498.46	0.456097
$222$	330.386	0.0998832
$223$	5142.35	1.54420	0.772101	−	0.635499i	$-0.219206\pi$
$223$	5142.35	1.54420	0.772101	+	0.635499i	$0.219206\pi$
$224$	715.452	0.213407
$225$	2017.05	0.597644
$226$	534.911	0.157441
$227$	2987.86	0.873618	0.436809	−	0.899554i	$-0.356109\pi$
$227$	2987.86	0.873618	0.436809	+	0.899554i	$0.356109\pi$
$228$	546.375	0.158704
$229$	−5121.38	−1.47786	−0.738931	−	0.673781i	$-0.764669\pi$
$229$	−5121.38	−1.47786	−0.738931	+	0.673781i	$0.764669\pi$
$230$	−1469.44	−0.421268
$231$	−231.000	−0.0657952
$232$	−2452.57	−0.694048
$233$	1103.58	0.310291	0.155146	−	0.987892i	$-0.450415\pi$
$233$	1103.58	0.310291	0.155146	+	0.987892i	$0.450415\pi$
$234$	184.159	0.0514481
$235$	−7428.94	−2.06217
$236$	−5139.34	−1.41755
$237$	−2346.01	−0.642995
$238$	161.650	0.0440260
$239$	−1798.18	−0.486671	−0.243336	−	0.969942i	$-0.578242\pi$
$239$	−1798.18	−0.486671	−0.243336	+	0.969942i	$0.578242\pi$
$240$	3168.80	0.852272
$241$	−7342.09	−1.96243	−0.981215	−	0.192919i	$-0.938204\pi$
$241$	−7342.09	−1.96243	−0.981215	+	0.192919i	$0.938204\pi$
$242$	67.9479	0.0180490
$243$	243.000	0.0641500
$244$	5131.72	1.34641
$245$	915.548	0.238744
$246$	−543.631	−0.140897
$247$	−863.584	−0.222464
$248$	−1686.22	−0.431754
$249$	−1280.11	−0.325799
$250$	1039.97	0.263094
$251$	1799.05	0.452411	0.226206	−	0.974080i	$-0.427368\pi$
$251$	1799.05	0.452411	0.226206	+	0.974080i	$0.427368\pi$
$252$	−484.133	−0.121022
$253$	1540.52	0.382813
$254$	142.045	0.0350894
$255$	2305.11	0.566086
$256$	2575.17	0.628703
$257$	−5282.71	−1.28220	−0.641102	−	0.767455i	$-0.721523\pi$
$257$	−5282.71	−1.28220	−0.641102	+	0.767455i	$0.721523\pi$
$258$	−6.19305	−0.00149443
$259$	1372.80	0.329350
$260$	−5232.02	−1.24799
$261$	2506.10	0.594343
$262$	−840.859	−0.198277
$263$	3275.55	0.767981	0.383990	−	0.923337i	$-0.374550\pi$
$263$	3275.55	0.767981	0.383990	+	0.923337i	$0.374550\pi$
$264$	290.656	0.0677601
$265$	11164.2	2.58797
$266$	−93.1609	−0.0214739
$267$	−2699.08	−0.618655
$268$	5615.66	1.27997
$269$	3682.35	0.834636	0.417318	−	0.908761i	$-0.362970\pi$
$269$	3682.35	0.834636	0.417318	+	0.908761i	$0.362970\pi$
$270$	283.295	0.0638548
$271$	−7301.19	−1.63659	−0.818295	−	0.574799i	$-0.805080\pi$
$271$	−7301.19	−1.63659	−0.818295	+	0.574799i	$0.805080\pi$
$272$	2324.74	0.518228
$273$	765.207	0.169643
$274$	−678.108	−0.149511
$275$	−2465.28	−0.540589
$276$	3228.65	0.704137
$277$	9035.64	1.95993	0.979963	−	0.199182i	$-0.0638283\pi$
$277$	9035.64	1.95993	0.979963	+	0.199182i	$0.0638283\pi$
$278$	119.341	0.0257467
$279$	1723.02	0.369730
$280$	−1151.99	−0.245874
$281$	1651.82	0.350674	0.175337	−	0.984509i	$-0.443899\pi$
$281$	1651.82	0.350674	0.175337	+	0.984509i	$0.443899\pi$
$282$	−669.813	−0.141442
$283$	3091.62	0.649391	0.324695	−	0.945819i	$-0.394738\pi$
$283$	3091.62	0.649391	0.324695	+	0.945819i	$0.394738\pi$
$284$	240.262	0.0502004
$285$	−1328.47	−0.276111
$286$	−225.083	−0.0465365
$287$	−2258.87	−0.464587
$288$	919.867	0.188207
$289$	−3221.89	−0.655789
$290$	2921.67	0.591609
$291$	−2828.15	−0.569721
$292$	3338.28	0.669034
$293$	2231.84	0.445001	0.222500	−	0.974933i	$-0.428578\pi$
$293$	2231.84	0.445001	0.222500	+	0.974933i	$0.428578\pi$
$294$	82.5483	0.0163752
$295$	12495.9	2.46624
$296$	−1727.33	−0.339186
$297$	−297.000	−0.0580259
$298$	493.973	0.0960239
$299$	−5103.11	−0.987024
$300$	−5166.78	−0.994346
$301$	−25.7330	−0.00492767
$302$	−682.387	−0.130023
$303$	1100.83	0.208717
$304$	−1339.78	−0.252769
$305$	−12477.4	−2.34247
$306$	207.835	0.0388273
$307$	−9667.03	−1.79715	−0.898577	−	0.438816i	$-0.855398\pi$
$307$	−9667.03	−1.79715	−0.898577	+	0.438816i	$0.855398\pi$
$308$	591.719	0.109469
$309$	3022.04	0.556368
$310$	2008.74	0.368029
$311$	−7170.97	−1.30749	−0.653743	−	0.756717i	$-0.726802\pi$
$311$	−7170.97	−1.30749	−0.653743	+	0.756717i	$0.726802\pi$
$312$	−962.824	−0.174709
$313$	295.578	0.0533771	0.0266886	−	0.999644i	$-0.491504\pi$
$313$	295.578	0.0533771	0.0266886	+	0.999644i	$0.491504\pi$
$314$	688.782	0.123790
$315$	1177.13	0.210552
$316$	6009.43	1.06980
$317$	7189.81	1.27388	0.636940	−	0.770914i	$-0.280200\pi$
$317$	7189.81	1.27388	0.636940	+	0.770914i	$0.280200\pi$
$318$	1006.60	0.177507
$319$	−3063.01	−0.537604
$320$	−7377.73	−1.28884
$321$	−5072.41	−0.881976
$322$	−550.508	−0.0952752
$323$	−974.610	−0.167891
$324$	−622.457	−0.106731
$325$	8166.46	1.39383
$326$	−247.771	−0.0420943
$327$	−5425.71	−0.917561
$328$	2842.22	0.478462
$329$	−2783.17	−0.466386
$330$	−346.250	−0.0577589
$331$	−2878.87	−0.478058	−0.239029	−	0.971012i	$-0.576829\pi$
$331$	−2878.87	−0.478058	−0.239029	+	0.971012i	$0.576829\pi$
$332$	3279.08	0.542057
$333$	1765.03	0.290460
$334$	−1007.16	−0.164999
$335$	−13654.0	−2.22687
$336$	1187.16	0.192752
$337$	−11639.7	−1.88146	−0.940731	−	0.339153i	$-0.889860\pi$
$337$	−11639.7	−1.88146	−0.940731	+	0.339153i	$0.889860\pi$
$338$	−488.124	−0.0785516
$339$	2857.67	0.457839
$340$	−5904.67	−0.941840
$341$	−2105.92	−0.334433
$342$	−119.778	−0.0189382
$343$	343.000	0.0539949
$344$	32.3786	0.00507482
$345$	−7850.21	−1.22505
$346$	2147.96	0.333743
$347$	−4752.91	−0.735302	−0.367651	−	0.929964i	$-0.619838\pi$
$347$	−4752.91	−0.735302	−0.367651	+	0.929964i	$0.619838\pi$
$348$	−6419.51	−0.988855
$349$	−7420.08	−1.13807	−0.569037	−	0.822312i	$-0.692684\pi$
$349$	−7420.08	−1.13807	−0.569037	+	0.822312i	$0.692684\pi$
$350$	880.973	0.134543
$351$	983.838	0.149611
$352$	−1124.28	−0.170240
$353$	−517.851	−0.0780805	−0.0390403	−	0.999238i	$-0.512430\pi$
$353$	−517.851	−0.0780805	−0.0390403	+	0.999238i	$0.512430\pi$
$354$	1126.66	0.169157
$355$	−584.178	−0.0873378
$356$	6913.83	1.02930
$357$	863.585	0.128027
$358$	738.808	0.109070
$359$	−9874.35	−1.45167	−0.725833	−	0.687871i	$-0.758546\pi$
$359$	−9874.35	−1.45167	−0.725833	+	0.687871i	$0.758546\pi$
$360$	−1481.13	−0.216840
$361$	−6297.32	−0.918110
$362$	378.869	0.0550081
$363$	363.000	0.0524864
$364$	−1960.12	−0.282248
$365$	−8116.77	−1.16398
$366$	−1124.99	−0.160668
$367$	2098.51	0.298477	0.149239	−	0.988801i	$-0.452318\pi$
$367$	2098.51	0.298477	0.149239	+	0.988801i	$0.452318\pi$
$368$	−7917.05	−1.12148
$369$	−2904.26	−0.409728
$370$	2057.72	0.289123
$371$	4182.55	0.585303
$372$	−4413.61	−0.615148
$373$	−19.4150	−0.00269510	−0.00134755	−	0.999999i	$-0.500429\pi$
$373$	−19.4150	−0.00269510	−0.00134755	+	0.999999i	$0.500429\pi$
$374$	−254.021	−0.0351206
$375$	5555.87	0.765077
$376$	3501.93	0.480314
$377$	10146.5	1.38613
$378$	106.133	0.0144416
$379$	12541.3	1.69974	0.849869	−	0.526994i	$-0.176681\pi$
$379$	12541.3	1.69974	0.849869	+	0.526994i	$0.176681\pi$
$380$	3402.94	0.459388
$381$	758.852	0.102040
$382$	1075.76	0.144086
$383$	9131.73	1.21830	0.609151	−	0.793054i	$-0.291510\pi$
$383$	9131.73	1.21830	0.609151	+	0.793054i	$0.291510\pi$
$384$	−3118.17	−0.414384
$385$	−1438.72	−0.190452
$386$	−1344.76	−0.177322
$387$	−33.0853	−0.00434579
$388$	7244.45	0.947890
$389$	9542.57	1.24377	0.621886	−	0.783108i	$-0.286367\pi$
$389$	9542.57	1.24377	0.621886	+	0.783108i	$0.286367\pi$
$390$	1146.98	0.148922
$391$	−5759.18	−0.744896
$392$	−431.580	−0.0556074
$393$	−4492.15	−0.576587
$394$	761.919	0.0974236
$395$	−14611.5	−1.86122
$396$	760.781	0.0965422
$397$	5322.85	0.672912	0.336456	−	0.941699i	$-0.390772\pi$
$397$	5322.85	0.672912	0.336456	+	0.941699i	$0.390772\pi$
$398$	1191.31	0.150037
$399$	−497.696	−0.0624460
$400$	12669.6	1.58370
$401$	5875.92	0.731744	0.365872	−	0.930665i	$-0.380771\pi$
$401$	5875.92	0.731744	0.365872	+	0.930665i	$0.380771\pi$
$402$	−1231.08	−0.152739
$403$	6976.03	0.862285
$404$	−2819.85	−0.347259
$405$	1513.46	0.185690
$406$	1094.57	0.133800
$407$	−2157.26	−0.262731
$408$	−1086.61	−0.131851
$409$	−6658.52	−0.804994	−0.402497	−	0.915421i	$-0.631858\pi$
$409$	−6658.52	−0.804994	−0.402497	+	0.915421i	$0.631858\pi$
$410$	−3385.85	−0.407842
$411$	−3622.68	−0.434777
$412$	−7741.11	−0.925673
$413$	4681.46	0.557771
$414$	−707.796	−0.0840248
$415$	−7972.83	−0.943062
$416$	3724.28	0.438937
$417$	637.557	0.0748712
$418$	146.396	0.0171303
$419$	10913.7	1.27248	0.636238	−	0.771493i	$-0.280490\pi$
$419$	10913.7	1.27248	0.636238	+	0.771493i	$0.280490\pi$
$420$	−3015.29	−0.350312
$421$	3936.75	0.455737	0.227868	−	0.973692i	$-0.426824\pi$
$421$	3936.75	0.455737	0.227868	+	0.973692i	$0.426824\pi$
$422$	2522.56	0.290987
$423$	−3578.36	−0.411314
$424$	−5262.71	−0.602782
$425$	9216.36	1.05190
$426$	−52.6710	−0.00599042
$427$	−4674.51	−0.529779
$428$	12993.2	1.46741
$429$	−1202.47	−0.135328
$430$	−38.5717	−0.00432580
$431$	13769.3	1.53885	0.769424	−	0.638738i	$-0.220543\pi$
$431$	13769.3	1.53885	0.769424	+	0.638738i	$0.220543\pi$
$432$	1526.34	0.169991
$433$	2858.51	0.317254	0.158627	−	0.987339i	$-0.449293\pi$
$433$	2858.51	0.317254	0.158627	+	0.987339i	$0.449293\pi$
$434$	752.553	0.0832343
$435$	15608.5	1.72040
$436$	13898.2	1.52662
$437$	3319.10	0.363327
$438$	−731.829	−0.0798360
$439$	8408.98	0.914211	0.457106	−	0.889412i	$-0.348886\pi$
$439$	8408.98	0.914211	0.457106	+	0.889412i	$0.348886\pi$
$440$	1810.27	0.196139
$441$	441.000	0.0476190
$442$	841.466	0.0905530
$443$	−3537.39	−0.379383	−0.189692	−	0.981844i	$-0.560749\pi$
$443$	−3537.39	−0.379383	−0.189692	+	0.981844i	$0.560749\pi$
$444$	−4521.22	−0.483260
$445$	−16810.5	−1.79077
$446$	2887.70	0.306584
$447$	2638.97	0.279237
$448$	−2763.99	−0.291487
$449$	473.820	0.0498016	0.0249008	−	0.999690i	$-0.492073\pi$
$449$	473.820	0.0498016	0.0249008	+	0.999690i	$0.492073\pi$
$450$	1132.68	0.118656
$451$	3549.65	0.370613
$452$	−7320.07	−0.761742
$453$	−3645.53	−0.378106
$454$	1677.84	0.173447
$455$	4765.88	0.491050
$456$	626.227	0.0643109
$457$	535.534	0.0548167	0.0274083	−	0.999624i	$-0.491275\pi$
$457$	535.534	0.0548167	0.0274083	+	0.999624i	$0.491275\pi$
$458$	−2875.93	−0.293413
$459$	1110.32	0.112910
$460$	20108.7	2.03820
$461$	1075.05	0.108612	0.0543058	−	0.998524i	$-0.482705\pi$
$461$	1075.05	0.108612	0.0543058	+	0.998524i	$0.482705\pi$
$462$	−129.719	−0.0130629
$463$	−11373.8	−1.14165	−0.570826	−	0.821071i	$-0.693377\pi$
$463$	−11373.8	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$464$	15741.4	1.57495
$465$	10731.4	1.07023
$466$	619.718	0.0616049
$467$	−15171.1	−1.50328	−0.751642	−	0.659572i	$-0.770738\pi$
$467$	−15171.1	−1.50328	−0.751642	+	0.659572i	$0.770738\pi$
$468$	−2520.15	−0.248919
$469$	−5115.34	−0.503634
$470$	−4171.74	−0.409421
$471$	3679.70	0.359982
$472$	−5890.45	−0.574428
$473$	40.4376	0.00393092
$474$	−1317.41	−0.127660
$475$	−5311.52	−0.513072
$476$	−2212.12	−0.213009
$477$	5377.57	0.516189
$478$	−1009.77	−0.0966231
$479$	−3045.24	−0.290482	−0.145241	−	0.989396i	$-0.546396\pi$
$479$	−3045.24	−0.290482	−0.145241	+	0.989396i	$0.546396\pi$
$480$	5729.13	0.544787
$481$	7146.11	0.677411
$482$	−4122.97	−0.389619
$483$	−2940.99	−0.277060
$484$	−929.844	−0.0873257
$485$	−17614.3	−1.64912
$486$	136.457	0.0127363
$487$	−16956.1	−1.57773	−0.788866	−	0.614565i	$-0.789331\pi$
$487$	−16956.1	−1.57773	−0.788866	+	0.614565i	$0.789331\pi$
$488$	5881.72	0.545600
$489$	−1323.67	−0.122410
$490$	514.129	0.0473999
$491$	−13667.0	−1.25618	−0.628088	−	0.778142i	$-0.716162\pi$
$491$	−13667.0	−1.25618	−0.628088	+	0.778142i	$0.716162\pi$
$492$	7439.40	0.681696
$493$	11451.0	1.04610
$494$	−484.948	−0.0441677
$495$	−1849.78	−0.167963
$496$	10822.7	0.979748
$497$	−218.856	−0.0197526
$498$	−718.851	−0.0646837
$499$	−10939.4	−0.981394	−0.490697	−	0.871330i	$-0.663258\pi$
$499$	−10939.4	−0.981394	−0.490697	+	0.871330i	$0.663258\pi$
$500$	−14231.7	−1.27292
$501$	−5380.60	−0.479815
$502$	1010.26	0.0898213
$503$	−6194.02	−0.549061	−0.274530	−	0.961578i	$-0.588522\pi$
$503$	−6194.02	−0.549061	−0.274530	+	0.961578i	$0.588522\pi$
$504$	−554.889	−0.0490411
$505$	6856.24	0.604156
$506$	865.084	0.0760033
$507$	−2607.72	−0.228428
$508$	−1943.84	−0.169772
$509$	16742.7	1.45797	0.728987	−	0.684528i	$-0.239992\pi$
$509$	16742.7	1.45797	0.728987	+	0.684528i	$0.239992\pi$
$510$	1294.44	0.112390
$511$	−3040.86	−0.263248
$512$	9761.22	0.842557
$513$	−639.895	−0.0550722
$514$	−2966.52	−0.254567
$515$	18821.9	1.61047
$516$	84.7498	0.00723043
$517$	4373.55	0.372048
$518$	770.901	0.0653889
$519$	11475.1	0.970523
$520$	−5996.68	−0.505715
$521$	21184.3	1.78138	0.890691	−	0.454609i	$-0.150221\pi$
$521$	21184.3	1.78138	0.890691	+	0.454609i	$0.150221\pi$
$522$	1407.31	0.118000
$523$	7737.42	0.646910	0.323455	−	0.946244i	$-0.395156\pi$
$523$	7737.42	0.646910	0.323455	+	0.946244i	$0.395156\pi$
$524$	11506.9	0.959313
$525$	4706.45	0.391250
$526$	1839.39	0.152474
$527$	7872.89	0.650756
$528$	−1865.53	−0.153763
$529$	7446.25	0.612004
$530$	6269.30	0.513813
$531$	6019.01	0.491908
$532$	1274.87	0.103896
$533$	−11758.5	−0.955567
$534$	−1515.68	−0.122827
$535$	−31592.1	−2.55298
$536$	6436.38	0.518674
$537$	3946.95	0.317176
$538$	2067.84	0.165708
$539$	−539.000	−0.0430730
$540$	−3876.80	−0.308946
$541$	−12732.7	−1.01187	−0.505937	−	0.862571i	$-0.668853\pi$
$541$	−12732.7	−1.01187	−0.505937	+	0.862571i	$0.668853\pi$
$542$	−4100.00	−0.324927
$543$	2024.05	0.159963
$544$	4203.09	0.331261
$545$	−33792.5	−2.65599
$546$	429.704	0.0336807
$547$	4429.06	0.346202	0.173101	−	0.984904i	$-0.444621\pi$
$547$	4429.06	0.346202	0.173101	+	0.984904i	$0.444621\pi$
$548$	9279.68	0.723372
$549$	−6010.09	−0.467221
$550$	−1384.39	−0.107328
$551$	−6599.34	−0.510239
$552$	3700.51	0.285334
$553$	−5474.03	−0.420939
$554$	5073.99	0.389121
$555$	10993.0	0.840769
$556$	−1633.14	−0.124569
$557$	6080.68	0.462562	0.231281	−	0.972887i	$-0.425708\pi$
$557$	6080.68	0.462562	0.231281	+	0.972887i	$0.425708\pi$
$558$	967.568	0.0734058
$559$	−133.953	−0.0101353
$560$	7393.87	0.557943
$561$	−1357.06	−0.102131
$562$	927.584	0.0696223
$563$	−12141.8	−0.908907	−0.454453	−	0.890771i	$-0.650165\pi$
$563$	−12141.8	−0.908907	−0.454453	+	0.890771i	$0.650165\pi$
$564$	9166.16	0.684335
$565$	17798.2	1.32527
$566$	1736.11	0.128929
$567$	567.000	0.0419961
$568$	275.376	0.0203424
$569$	22369.0	1.64808	0.824039	−	0.566533i	$-0.191716\pi$
$569$	22369.0	1.64808	0.824039	+	0.566533i	$0.191716\pi$
$570$	−746.005	−0.0548188
$571$	−14445.2	−1.05869	−0.529346	−	0.848406i	$-0.677563\pi$
$571$	−14445.2	−1.05869	−0.529346	+	0.848406i	$0.677563\pi$
$572$	3080.19	0.225156
$573$	5747.07	0.419000
$574$	−1268.47	−0.0922387
$575$	−31386.9	−2.27639
$576$	−3553.70	−0.257067
$577$	1782.41	0.128601	0.0643004	−	0.997931i	$-0.479518\pi$
$577$	1782.41	0.128601	0.0643004	+	0.997931i	$0.479518\pi$
$578$	−1809.26	−0.130200
$579$	−7184.12	−0.515651
$580$	−39982.1	−2.86235
$581$	−2986.93	−0.213285
$582$	−1588.15	−0.113112
$583$	−6572.58	−0.466910
$584$	3826.16	0.271109
$585$	6127.56	0.433066
$586$	1253.29	0.0883499
$587$	23646.2	1.66266	0.831330	−	0.555779i	$-0.187580\pi$
$587$	23646.2	1.66266	0.831330	+	0.555779i	$0.187580\pi$
$588$	−1129.64	−0.0792275
$589$	−4537.26	−0.317410
$590$	7017.12	0.489644
$591$	4070.42	0.283307
$592$	11086.6	0.769690
$593$	−2871.03	−0.198818	−0.0994089	−	0.995047i	$-0.531695\pi$
$593$	−2871.03	−0.198818	−0.0994089	+	0.995047i	$0.531695\pi$
$594$	−166.781	−0.0115204
$595$	5378.60	0.370590
$596$	−6759.86	−0.464588
$597$	6364.36	0.436308
$598$	−2865.66	−0.195963
$599$	6685.11	0.456004	0.228002	−	0.973661i	$-0.426781\pi$
$599$	6685.11	0.456004	0.228002	+	0.973661i	$0.426781\pi$
$600$	−5921.89	−0.402934
$601$	8875.42	0.602389	0.301195	−	0.953563i	$-0.402615\pi$
$601$	8875.42	0.602389	0.301195	+	0.953563i	$0.402615\pi$
$602$	−14.4505	−0.000978333 0
$603$	−6576.86	−0.444163
$604$	9338.23	0.629085
$605$	2260.84	0.151928
$606$	618.177	0.0414385
$607$	379.477	0.0253748	0.0126874	−	0.999920i	$-0.495961\pi$
$607$	379.477	0.0253748	0.0126874	+	0.999920i	$0.495961\pi$
$608$	−2422.30	−0.161574
$609$	5847.56	0.389089
$610$	−7006.71	−0.465071
$611$	−14487.8	−0.959267
$612$	−2844.15	−0.187856
$613$	17480.5	1.15177	0.575883	−	0.817532i	$-0.304658\pi$
$613$	17480.5	1.15177	0.575883	+	0.817532i	$0.304658\pi$
$614$	−5428.55	−0.356805
$615$	−18088.3	−1.18600
$616$	678.198	0.0443594
$617$	−9901.31	−0.646048	−0.323024	−	0.946391i	$-0.604699\pi$
$617$	−9901.31	−0.646048	−0.323024	+	0.946391i	$0.604699\pi$
$618$	1697.03	0.110461
$619$	−152.433	−0.00989788	−0.00494894	−	0.999988i	$-0.501575\pi$
$619$	−152.433	−0.00989788	−0.00494894	+	0.999988i	$0.501575\pi$
$620$	−27489.0	−1.78062
$621$	−3781.28	−0.244344
$622$	−4026.88	−0.259587
$623$	−6297.85	−0.405005
$624$	6179.73	0.396454
$625$	6588.63	0.421672
$626$	165.983	0.0105974
$627$	782.094	0.0498147
$628$	−9425.74	−0.598930
$629$	8064.84	0.511234
$630$	661.023	0.0418028
$631$	−23661.7	−1.49280	−0.746399	−	0.665499i	$-0.768219\pi$
$631$	−23661.7	−1.49280	−0.746399	+	0.665499i	$0.768219\pi$
$632$	6887.70	0.433510
$633$	13476.4	0.846189
$634$	4037.46	0.252915
$635$	4726.30	0.295366
$636$	−13774.9	−0.858823
$637$	1785.48	0.111057
$638$	−1720.04	−0.106735
$639$	−281.386	−0.0174201
$640$	−19420.7	−1.19948
$641$	25340.3	1.56144	0.780719	−	0.624882i	$-0.214853\pi$
$641$	25340.3	1.56144	0.780719	+	0.624882i	$0.214853\pi$
$642$	−2848.43	−0.175107
$643$	−1774.90	−0.108858	−0.0544288	−	0.998518i	$-0.517334\pi$
$643$	−1774.90	−0.108858	−0.0544288	+	0.998518i	$0.517334\pi$
$644$	7533.51	0.460966
$645$	−206.063	−0.0125794
$646$	−547.295	−0.0333329
$647$	−23685.1	−1.43919	−0.719595	−	0.694394i	$-0.755672\pi$
$647$	−23685.1	−1.43919	−0.719595	+	0.694394i	$0.755672\pi$
$648$	−713.429	−0.0432502
$649$	−7356.57	−0.444947
$650$	4585.90	0.276729
$651$	4020.39	0.242045
$652$	3390.66	0.203663
$653$	−1969.99	−0.118058	−0.0590288	−	0.998256i	$-0.518800\pi$
$653$	−1969.99	−0.118058	−0.0590288	+	0.998256i	$0.518800\pi$
$654$	−3046.82	−0.182172
$655$	−27978.1	−1.66900
$656$	−18242.4	−1.08574
$657$	−3909.67	−0.232163
$658$	−1562.90	−0.0925958
$659$	15314.6	0.905270	0.452635	−	0.891696i	$-0.350484\pi$
$659$	15314.6	0.905270	0.452635	+	0.891696i	$0.350484\pi$
$660$	4738.31	0.279452
$661$	31443.3	1.85023	0.925115	−	0.379687i	$-0.123968\pi$
$661$	31443.3	1.85023	0.925115	+	0.379687i	$0.123968\pi$
$662$	−1616.64	−0.0949130
$663$	4495.39	0.263328
$664$	3758.31	0.219655
$665$	−3099.76	−0.180757
$666$	991.158	0.0576676
$667$	−38996.9	−2.26382
$668$	13782.7	0.798305
$669$	15427.0	0.891546
$670$	−7667.47	−0.442120
$671$	7345.67	0.422617
$672$	2146.36	0.123210
$673$	−8079.34	−0.462757	−0.231379	−	0.972864i	$-0.574324\pi$
$673$	−8079.34	−0.462757	−0.231379	+	0.972864i	$0.574324\pi$
$674$	−6536.29	−0.373543
$675$	6051.14	0.345050
$676$	6679.81	0.380053
$677$	−15701.7	−0.891383	−0.445692	−	0.895187i	$-0.647042\pi$
$677$	−15701.7	−0.891383	−0.445692	+	0.895187i	$0.647042\pi$
$678$	1604.73	0.0908988
$679$	−6599.01	−0.372970
$680$	−6767.63	−0.381657
$681$	8963.58	0.504384
$682$	−1182.58	−0.0663980
$683$	29752.9	1.66686	0.833430	−	0.552626i	$-0.186374\pi$
$683$	29752.9	1.66686	0.833430	+	0.552626i	$0.186374\pi$
$684$	1639.12	0.0916279
$685$	−22562.8	−1.25851
$686$	192.613	0.0107201
$687$	−15364.1	−0.853244
$688$	−207.817	−0.0115159
$689$	21772.2	1.20386
$690$	−4408.31	−0.243219
$691$	3082.57	0.169706	0.0848528	−	0.996393i	$-0.472958\pi$
$691$	3082.57	0.169706	0.0848528	+	0.996393i	$0.472958\pi$
$692$	−29394.1	−1.61473
$693$	−693.000	−0.0379869
$694$	−2669.01	−0.145986
$695$	3970.84	0.216723
$696$	−7357.71	−0.400709
$697$	−13270.2	−0.721156
$698$	−4166.77	−0.225952
$699$	3310.74	0.179147
$700$	−12055.8	−0.650953
$701$	19301.0	1.03993	0.519963	−	0.854189i	$-0.325946\pi$
$701$	19301.0	1.03993	0.519963	+	0.854189i	$0.325946\pi$
$702$	552.477	0.0297036
$703$	−4647.88	−0.249357
$704$	4343.41	0.232526
$705$	−22286.8	−1.19060
$706$	−290.801	−0.0155020
$707$	2568.61	0.136637
$708$	−15418.0	−0.818425
$709$	28531.3	1.51130	0.755652	−	0.654973i	$-0.227320\pi$
$709$	28531.3	1.51130	0.755652	+	0.654973i	$0.227320\pi$
$710$	−328.047	−0.0173400
$711$	−7038.03	−0.371233
$712$	7924.29	0.417100
$713$	−26811.6	−1.40828
$714$	484.949	0.0254184
$715$	−7489.24	−0.391723
$716$	−10110.3	−0.527711
$717$	−5394.53	−0.280980
$718$	−5544.97	−0.288212
$719$	3884.65	0.201492	0.100746	−	0.994912i	$-0.467877\pi$
$719$	3884.65	0.201492	0.100746	+	0.994912i	$0.467877\pi$
$720$	9506.40	0.492059
$721$	7051.43	0.364229
$722$	−3536.28	−0.182281
$723$	−22026.3	−1.13301
$724$	−5184.70	−0.266143
$725$	62406.5	3.19685
$726$	203.844	0.0104206
$727$	23489.8	1.19833	0.599167	−	0.800624i	$-0.295499\pi$
$727$	23489.8	1.19833	0.599167	+	0.800624i	$0.295499\pi$
$728$	−2246.59	−0.114374
$729$	729.000	0.0370370
$730$	−4557.99	−0.231094
$731$	−151.175	−0.00764897
$732$	15395.2	0.777352
$733$	30288.1	1.52622	0.763109	−	0.646270i	$-0.223672\pi$
$733$	30288.1	1.52622	0.763109	+	0.646270i	$0.223672\pi$
$734$	1178.42	0.0592593
$735$	2746.64	0.137839
$736$	−14313.9	−0.716870
$737$	8038.39	0.401761
$738$	−1630.89	−0.0813469
$739$	29765.0	1.48163	0.740815	−	0.671709i	$-0.234440\pi$
$739$	29765.0	1.48163	0.740815	+	0.671709i	$0.234440\pi$
$740$	−28159.2	−1.39885
$741$	−2590.75	−0.128440
$742$	2348.72	0.116205
$743$	−1200.21	−0.0592617	−0.0296309	−	0.999561i	$-0.509433\pi$
$743$	−1200.21	−0.0592617	−0.0296309	+	0.999561i	$0.509433\pi$
$744$	−5058.66	−0.249273
$745$	16436.1	0.808284
$746$	−10.9026	−0.000535082 0
$747$	−3840.34	−0.188100
$748$	3476.19	0.169922
$749$	−11835.6	−0.577389
$750$	3119.92	0.151898
$751$	441.096	0.0214325	0.0107163	−	0.999943i	$-0.496589\pi$
$751$	441.096	0.0214325	0.0107163	+	0.999943i	$0.496589\pi$
$752$	−22476.6	−1.08994
$753$	5397.16	0.261200
$754$	5697.79	0.275200
$755$	−22705.2	−1.09447
$756$	−1452.40	−0.0698721
$757$	−37080.3	−1.78032	−0.890162	−	0.455644i	$-0.849409\pi$
$757$	−37080.3	−1.78032	−0.890162	+	0.455644i	$0.849409\pi$
$758$	7042.58	0.337464
$759$	4621.56	0.221017
$760$	3900.28	0.186155
$761$	−3056.78	−0.145609	−0.0728044	−	0.997346i	$-0.523195\pi$
$761$	−3056.78	−0.145609	−0.0728044	+	0.997346i	$0.523195\pi$
$762$	426.136	0.0202589
$763$	−12660.0	−0.600685
$764$	−14721.4	−0.697123
$765$	6915.34	0.326830
$766$	5127.95	0.241880
$767$	24369.3	1.14723
$768$	7725.50	0.362982
$769$	31834.7	1.49283	0.746417	−	0.665479i	$-0.231773\pi$
$769$	31834.7	1.49283	0.746417	+	0.665479i	$0.231773\pi$
$770$	−807.917	−0.0378121
$771$	−15848.1	−0.740281
$772$	18402.5	0.857929
$773$	−2351.46	−0.109413	−0.0547064	−	0.998502i	$-0.517422\pi$
$773$	−2351.46	−0.109413	−0.0547064	+	0.998502i	$0.517422\pi$
$774$	−18.5792	−0.000862809 0
$775$	42906.4	1.98870
$776$	8303.22	0.384108
$777$	4118.41	0.190151
$778$	5358.65	0.246937
$779$	7647.81	0.351748
$780$	−15696.1	−0.720525
$781$	343.916	0.0157571
$782$	−3234.08	−0.147891
$783$	7518.30	0.343144
$784$	2770.03	0.126186
$785$	22918.0	1.04201
$786$	−2522.58	−0.114475
$787$	−41575.3	−1.88310	−0.941550	−	0.336874i	$-0.890630\pi$
$787$	−41575.3	−1.88310	−0.941550	+	0.336874i	$0.890630\pi$
$788$	−10426.6	−0.471361
$789$	9826.64	0.443394
$790$	−8205.11	−0.369525
$791$	6667.90	0.299726
$792$	871.969	0.0391213
$793$	−24333.2	−1.08965
$794$	2989.06	0.133599
$795$	33492.7	1.49417
$796$	−16302.6	−0.725919
$797$	−26199.7	−1.16442	−0.582208	−	0.813040i	$-0.697811\pi$
$797$	−26199.7	−1.16442	−0.582208	+	0.813040i	$0.697811\pi$
$798$	−279.483	−0.0123980
$799$	−16350.4	−0.723948
$800$	22906.4	1.01233
$801$	−8097.24	−0.357181
$802$	3299.64	0.145280
$803$	4778.49	0.209999
$804$	16847.0	0.738989
$805$	−18317.2	−0.801981
$806$	3917.41	0.171197
$807$	11047.1	0.481877
$808$	−3231.96	−0.140718
$809$	−2338.60	−0.101633	−0.0508164	−	0.998708i	$-0.516182\pi$
$809$	−2338.60	−0.101633	−0.0508164	+	0.998708i	$0.516182\pi$
$810$	849.886	0.0368666
$811$	−3032.48	−0.131301	−0.0656504	−	0.997843i	$-0.520912\pi$
$811$	−3032.48	−0.131301	−0.0656504	+	0.997843i	$0.520912\pi$
$812$	−14978.8	−0.647358
$813$	−21903.6	−0.944886
$814$	−1211.42	−0.0521623
$815$	−8244.13	−0.354330
$816$	6974.22	0.299199
$817$	87.1240	0.00373082
$818$	−3739.11	−0.159823
$819$	2295.62	0.0979433
$820$	46334.2	1.97325
$821$	−28900.3	−1.22854	−0.614268	−	0.789098i	$-0.710549\pi$
$821$	−28900.3	−1.22854	−0.614268	+	0.789098i	$0.710549\pi$
$822$	−2034.32	−0.0863202
$823$	−19810.2	−0.839054	−0.419527	−	0.907743i	$-0.637804\pi$
$823$	−19810.2	−0.839054	−0.419527	+	0.907743i	$0.637804\pi$
$824$	−8872.47	−0.375106
$825$	−7395.84	−0.312109
$826$	2628.88	0.110739
$827$	12280.7	0.516374	0.258187	−	0.966095i	$-0.416875\pi$
$827$	12280.7	0.516374	0.258187	+	0.966095i	$0.416875\pi$
$828$	9685.94	0.406534
$829$	−3499.72	−0.146623	−0.0733114	−	0.997309i	$-0.523357\pi$
$829$	−3499.72	−0.146623	−0.0733114	+	0.997309i	$0.523357\pi$
$830$	−4477.16	−0.187235
$831$	27106.9	1.13156
$832$	−14387.9	−0.599532
$833$	2015.03	0.0838136
$834$	358.022	0.0148649
$835$	−33511.5	−1.38888
$836$	−2003.37	−0.0828806
$837$	5169.07	0.213464
$838$	6128.60	0.252636
$839$	−11751.6	−0.483562	−0.241781	−	0.970331i	$-0.577732\pi$
$839$	−11751.6	−0.483562	−0.241781	+	0.970331i	$0.577732\pi$
$840$	−3455.97	−0.141955
$841$	53148.4	2.17920
$842$	2210.69	0.0904815
$843$	4955.46	0.202461
$844$	−34520.4	−1.40787
$845$	−16241.4	−0.661210
$846$	−2009.44	−0.0816618
$847$	847.000	0.0343604
$848$	33777.8	1.36785
$849$	9274.86	0.374926
$850$	5175.48	0.208844
$851$	−27465.3	−1.10634
$852$	720.785	0.0289832
$853$	1403.01	0.0563166	0.0281583	−	0.999603i	$-0.491036\pi$
$853$	1403.01	0.0563166	0.0281583	+	0.999603i	$0.491036\pi$
$854$	−2624.99	−0.105182
$855$	−3985.41	−0.159413
$856$	14892.2	0.594632
$857$	15111.0	0.602311	0.301156	−	0.953575i	$-0.402628\pi$
$857$	15111.0	0.602311	0.301156	+	0.953575i	$0.402628\pi$
$858$	−675.250	−0.0268679
$859$	18620.9	0.739623	0.369811	−	0.929107i	$-0.379422\pi$
$859$	18620.9	0.739623	0.369811	+	0.929107i	$0.379422\pi$
$860$	527.841	0.0209293
$861$	−6776.60	−0.268230
$862$	7732.19	0.305521
$863$	−33659.0	−1.32765	−0.663827	−	0.747886i	$-0.731069\pi$
$863$	−33659.0	−1.32765	−0.663827	+	0.747886i	$0.731069\pi$
$864$	2759.60	0.108661
$865$	71469.5	2.80929
$866$	1605.20	0.0629873
$867$	−9665.67	−0.378620
$868$	−10298.4	−0.402709
$869$	8602.04	0.335793
$870$	8765.02	0.341565
$871$	−26627.8	−1.03588
$872$	15929.5	0.618623
$873$	−8484.44	−0.328929
$874$	1863.85	0.0721345
$875$	12963.7	0.500861
$876$	10014.8	0.386267
$877$	−2730.45	−0.105132	−0.0525659	−	0.998617i	$-0.516740\pi$
$877$	−2730.45	−0.105132	−0.0525659	+	0.998617i	$0.516740\pi$
$878$	4722.09	0.181507
$879$	6695.51	0.256921
$880$	−11618.9	−0.445084
$881$	−16625.5	−0.635787	−0.317894	−	0.948126i	$-0.602975\pi$
$881$	−16625.5	−0.635787	−0.317894	+	0.948126i	$0.602975\pi$
$882$	247.645	0.00945423
$883$	10063.4	0.383536	0.191768	−	0.981440i	$-0.438578\pi$
$883$	10063.4	0.383536	0.191768	+	0.981440i	$0.438578\pi$
$884$	−11515.2	−0.438119
$885$	37487.7	1.42388
$886$	−1986.43	−0.0753223
$887$	29939.9	1.13335	0.566676	−	0.823940i	$-0.308229\pi$
$887$	29939.9	1.13335	0.566676	+	0.823940i	$0.308229\pi$
$888$	−5181.99	−0.195829
$889$	1770.66	0.0668008
$890$	−9439.96	−0.355537
$891$	−891.000	−0.0335013
$892$	−39517.2	−1.48333
$893$	9422.94	0.353109
$894$	1481.92	0.0554394
$895$	24582.5	0.918103
$896$	−7275.74	−0.271278
$897$	−15309.3	−0.569859
$898$	266.075	0.00988756
$899$	53309.5	1.97772
$900$	−15500.3	−0.574086
$901$	24571.4	0.908536
$902$	1993.31	0.0735810
$903$	−77.1991	−0.00284499
$904$	−8389.90	−0.308677
$905$	12606.2	0.463032
$906$	−2047.16	−0.0750688
$907$	296.554	0.0108566	0.00542829	−	0.999985i	$-0.498272\pi$
$907$	296.554	0.0108566	0.00542829	+	0.999985i	$0.498272\pi$
$908$	−22960.7	−0.839182
$909$	3302.50	0.120503
$910$	2676.29	0.0974926
$911$	−3552.96	−0.129215	−0.0646074	−	0.997911i	$-0.520580\pi$
$911$	−3552.96	−0.129215	−0.0646074	+	0.997911i	$0.520580\pi$
$912$	−4019.34	−0.145936
$913$	4693.75	0.170143
$914$	300.730	0.0108832
$915$	−37432.2	−1.35243
$916$	39356.1	1.41961
$917$	−10481.7	−0.377465
$918$	623.505	0.0224169
$919$	−32047.3	−1.15032	−0.575159	−	0.818042i	$-0.695060\pi$
$919$	−32047.3	−1.15032	−0.575159	+	0.818042i	$0.695060\pi$
$920$	23047.6	0.825931
$921$	−29001.1	−1.03759
$922$	603.696	0.0215636
$923$	−1139.25	−0.0406272
$924$	1775.16	0.0632017
$925$	43952.5	1.56232
$926$	−6386.98	−0.226662
$927$	9066.12	0.321219
$928$	28460.2	1.00674
$929$	5628.82	0.198790	0.0993949	−	0.995048i	$-0.468309\pi$
$929$	5628.82	0.198790	0.0993949	+	0.995048i	$0.468309\pi$
$930$	6026.23	0.212481
$931$	−1161.29	−0.0408805
$932$	−8480.63	−0.298060
$933$	−21512.9	−0.754877
$934$	−8519.36	−0.298460
$935$	−8452.08	−0.295629
$936$	−2888.47	−0.100868
$937$	−41269.1	−1.43885	−0.719425	−	0.694570i	$-0.755595\pi$
$937$	−41269.1	−1.43885	−0.719425	+	0.694570i	$0.755595\pi$
$938$	−2872.53	−0.0999909
$939$	886.733	0.0308173
$940$	57088.9	1.98089
$941$	2086.74	0.0722909	0.0361454	−	0.999347i	$-0.488492\pi$
$941$	2086.74	0.0722909	0.0361454	+	0.999347i	$0.488492\pi$
$942$	2066.35	0.0714705
$943$	45192.6	1.56063
$944$	37806.9	1.30351
$945$	3531.40	0.121562
$946$	22.7079	0.000780440 0
$947$	−21040.7	−0.721995	−0.360998	−	0.932567i	$-0.617564\pi$
$947$	−21040.7	−0.721995	−0.360998	+	0.932567i	$0.617564\pi$
$948$	18028.3	0.617650
$949$	−15829.2	−0.541450
$950$	−2982.70	−0.101865
$951$	21569.4	0.735475
$952$	−2535.42	−0.0863166
$953$	25644.0	0.871658	0.435829	−	0.900030i	$-0.356455\pi$
$953$	25644.0	0.871658	0.435829	+	0.900030i	$0.356455\pi$
$954$	3019.79	0.102484
$955$	35794.0	1.21284
$956$	13818.4	0.467488
$957$	−9189.03	−0.310386
$958$	−1710.06	−0.0576719
$959$	−8452.91	−0.284628
$960$	−22133.2	−0.744110
$961$	6860.94	0.230302
$962$	4012.92	0.134492
$963$	−15217.2	−0.509209
$964$	56421.4	1.88507
$965$	−44744.3	−1.49261
$966$	−1651.52	−0.0550071
$967$	28262.5	0.939878	0.469939	−	0.882699i	$-0.344276\pi$
$967$	28262.5	0.939878	0.469939	+	0.882699i	$0.344276\pi$
$968$	−1065.74	−0.0353865
$969$	−2923.83	−0.0969318
$970$	−9891.37	−0.327415
$971$	19180.0	0.633899	0.316950	−	0.948442i	$-0.397341\pi$
$971$	19180.0	0.633899	0.316950	+	0.948442i	$0.397341\pi$
$972$	−1867.37	−0.0616214
$973$	1487.63	0.0490147
$974$	−9521.76	−0.313241
$975$	24499.4	0.804726
$976$	−37750.9	−1.23809
$977$	−35269.8	−1.15495	−0.577473	−	0.816410i	$-0.695961\pi$
$977$	−35269.8	−1.15495	−0.577473	+	0.816410i	$0.695961\pi$
$978$	−743.312	−0.0243032
$979$	9896.62	0.323082
$980$	−7035.68	−0.229333
$981$	−16277.1	−0.529754
$982$	−7674.73	−0.249400
$983$	40100.1	1.30111	0.650557	−	0.759458i	$-0.274536\pi$
$983$	40100.1	1.30111	0.650557	+	0.759458i	$0.274536\pi$
$984$	8526.67	0.276240
$985$	25351.5	0.820066
$986$	6430.32	0.207691
$987$	−8349.51	−0.269268
$988$	6636.35	0.213695
$989$	514.835	0.0165529
$990$	−1038.75	−0.0333471
$991$	3191.81	0.102312	0.0511560	−	0.998691i	$-0.483709\pi$
$991$	3191.81	0.102312	0.0511560	+	0.998691i	$0.483709\pi$
$992$	19567.3	0.626272
$993$	−8636.61	−0.276007
$994$	−122.899	−0.00392165
$995$	39638.6	1.26294
$996$	9837.24	0.312957
$997$	8275.27	0.262869	0.131435	−	0.991325i	$-0.458042\pi$
$997$	8275.27	0.262869	0.131435	+	0.991325i	$0.458042\pi$
$998$	−6143.06	−0.194845
$999$	5295.09	0.167697

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Twists

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

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

Level	$231$
Weight	$4$
Character	231.1
Self dual	yes
Analytic conductor	$13.629$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$