LMFDB - Embedded newform 495.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.2059454528$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.87740$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.27945 q^{2} +2.75481 q^{4} -5.00000 q^{5} -33.3329 q^{7} +17.2014 q^{8} +O(q^{10})$	\(q-3.27945 q^{2} +2.75481 q^{4} -5.00000 q^{5} -33.3329 q^{7} +17.2014 q^{8} +16.3973 q^{10} +11.0000 q^{11} -24.2862 q^{13} +109.314 q^{14} -78.4495 q^{16} -69.7056 q^{17} -125.718 q^{19} -13.7740 q^{20} -36.0740 q^{22} -130.628 q^{23} +25.0000 q^{25} +79.6453 q^{26} -91.8258 q^{28} +238.181 q^{29} -133.663 q^{31} +119.661 q^{32} +228.596 q^{34} +166.665 q^{35} +166.505 q^{37} +412.285 q^{38} -86.0068 q^{40} -297.951 q^{41} -463.307 q^{43} +30.3029 q^{44} +428.387 q^{46} -585.981 q^{47} +768.085 q^{49} -81.9863 q^{50} -66.9037 q^{52} +40.0845 q^{53} -55.0000 q^{55} -573.372 q^{56} -781.105 q^{58} +312.763 q^{59} -391.339 q^{61} +438.342 q^{62} +235.175 q^{64} +121.431 q^{65} +858.232 q^{67} -192.026 q^{68} -546.569 q^{70} +583.028 q^{71} +368.303 q^{73} -546.044 q^{74} -346.328 q^{76} -366.662 q^{77} -438.935 q^{79} +392.247 q^{80} +977.117 q^{82} +877.643 q^{83} +348.528 q^{85} +1519.39 q^{86} +189.215 q^{88} +999.585 q^{89} +809.529 q^{91} -359.854 q^{92} +1921.70 q^{94} +628.588 q^{95} -1306.31 q^{97} -2518.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8	$ 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 42 q^{13} + 76 q^{14} - 85 q^{16} + 34 q^{17} - 280 q^{19} + 25 q^{20} - 11 q^{22} + 112 q^{23} + 75 q^{25} + 314 q^{26} - 280 q^{28} + 290 q^{29} - 392 q^{31} + 23 q^{32} + 310 q^{34} + 80 q^{35} + 570 q^{37} + 512 q^{38} - 15 q^{40} + 662 q^{41} - 68 q^{43} - 55 q^{44} + 728 q^{46} - 264 q^{47} + 731 q^{49} - 25 q^{50} + 582 q^{52} + 94 q^{53} - 165 q^{55} - 204 q^{56} - 54 q^{58} + 612 q^{59} - 582 q^{61} - 320 q^{62} + 347 q^{64} + 210 q^{65} + 940 q^{67} - 678 q^{68} - 380 q^{70} + 1616 q^{71} + 738 q^{73} + 642 q^{74} + 880 q^{76} - 176 q^{77} + 124 q^{79} + 425 q^{80} + 2126 q^{82} + 1232 q^{83} - 170 q^{85} + 1000 q^{86} + 33 q^{88} - 838 q^{89} - 1736 q^{91} - 1248 q^{92} + 2432 q^{94} + 1400 q^{95} - 90 q^{97} - 3481 q^{98}+O(q^{100}) $ 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 42 * q^13 + 76 * q^14 - 85 * q^16 + 34 * q^17 - 280 * q^19 + 25 * q^20 - 11 * q^22 + 112 * q^23 + 75 * q^25 + 314 * q^26 - 280 * q^28 + 290 * q^29 - 392 * q^31 + 23 * q^32 + 310 * q^34 + 80 * q^35 + 570 * q^37 + 512 * q^38 - 15 * q^40 + 662 * q^41 - 68 * q^43 - 55 * q^44 + 728 * q^46 - 264 * q^47 + 731 * q^49 - 25 * q^50 + 582 * q^52 + 94 * q^53 - 165 * q^55 - 204 * q^56 - 54 * q^58 + 612 * q^59 - 582 * q^61 - 320 * q^62 + 347 * q^64 + 210 * q^65 + 940 * q^67 - 678 * q^68 - 380 * q^70 + 1616 * q^71 + 738 * q^73 + 642 * q^74 + 880 * q^76 - 176 * q^77 + 124 * q^79 + 425 * q^80 + 2126 * q^82 + 1232 * q^83 - 170 * q^85 + 1000 * q^86 + 33 * q^88 - 838 * q^89 - 1736 * q^91 - 1248 * q^92 + 2432 * q^94 + 1400 * q^95 - 90 * q^97 - 3481 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−3.27945	−1.15946	−0.579731	−	0.814808i	$-0.696842\pi$
$2$	−3.27945	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$3$	0	0
$3$	0	0
$4$	2.75481	0.344351
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−33.3329	−1.79981	−0.899905	−	0.436086i	$-0.856364\pi$
$7$	−33.3329	−1.79981	−0.899905	+	0.436086i	$0.856364\pi$
$8$	17.2014	0.760200
$9$	0	0
$10$	16.3973	0.518527
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−24.2862	−0.518136	−0.259068	−	0.965859i	$-0.583415\pi$
$13$	−24.2862	−0.518136	−0.259068	+	0.965859i	$0.583415\pi$
$14$	109.314	2.08681
$15$	0	0
$16$	−78.4495	−1.22577
$17$	−69.7056	−0.994476	−0.497238	−	0.867614i	$-0.665652\pi$
$17$	−69.7056	−0.994476	−0.497238	+	0.867614i	$0.665652\pi$
$18$	0	0
$19$	−125.718	−1.51798	−0.758990	−	0.651102i	$-0.774307\pi$
$19$	−125.718	−1.51798	−0.758990	+	0.651102i	$0.774307\pi$
$20$	−13.7740	−0.153998
$21$	0	0
$22$	−36.0740	−0.349591
$23$	−130.628	−1.18425	−0.592125	−	0.805846i	$-0.701711\pi$
$23$	−130.628	−1.18425	−0.592125	+	0.805846i	$0.701711\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	79.6453	0.600759
$27$	0	0
$28$	−91.8258	−0.619766
$29$	238.181	1.52514	0.762572	−	0.646903i	$-0.223936\pi$
$29$	238.181	1.52514	0.762572	+	0.646903i	$0.223936\pi$
$30$	0	0
$31$	−133.663	−0.774408	−0.387204	−	0.921994i	$-0.626559\pi$
$31$	−133.663	−0.774408	−0.387204	+	0.921994i	$0.626559\pi$
$32$	119.661	0.661037
$33$	0	0
$34$	228.596	1.15306
$35$	166.665	0.804899
$36$	0	0
$37$	166.505	0.739815	0.369908	−	0.929068i	$-0.379389\pi$
$37$	166.505	0.739815	0.369908	+	0.929068i	$0.379389\pi$
$38$	412.285	1.76004
$39$	0	0
$40$	−86.0068	−0.339972
$41$	−297.951	−1.13493	−0.567466	−	0.823397i	$-0.692076\pi$
$41$	−297.951	−1.13493	−0.567466	+	0.823397i	$0.692076\pi$
$42$	0	0
$43$	−463.307	−1.64311	−0.821553	−	0.570132i	$-0.806892\pi$
$43$	−463.307	−1.64311	−0.821553	+	0.570132i	$0.806892\pi$
$44$	30.3029	0.103826
$45$	0	0
$46$	428.387	1.37309
$47$	−585.981	−1.81860	−0.909300	−	0.416142i	$-0.863382\pi$
$47$	−585.981	−1.81860	−0.909300	+	0.416142i	$0.863382\pi$
$48$	0	0
$49$	768.085	2.23931
$50$	−81.9863	−0.231892
$51$	0	0
$52$	−66.9037	−0.178421
$53$	40.0845	0.103887	0.0519436	−	0.998650i	$-0.483458\pi$
$53$	40.0845	0.103887	0.0519436	+	0.998650i	$0.483458\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	−573.372	−1.36822
$57$	0	0
$58$	−781.105	−1.76835
$59$	312.763	0.690140	0.345070	−	0.938577i	$-0.387855\pi$
$59$	312.763	0.690140	0.345070	+	0.938577i	$0.387855\pi$
$60$	0	0
$61$	−391.339	−0.821408	−0.410704	−	0.911769i	$-0.634717\pi$
$61$	−391.339	−0.821408	−0.410704	+	0.911769i	$0.634717\pi$
$62$	438.342	0.897896
$63$	0	0
$64$	235.175	0.459326
$65$	121.431	0.231718
$66$	0	0
$67$	858.232	1.56492	0.782461	−	0.622700i	$-0.213964\pi$
$67$	858.232	1.56492	0.782461	+	0.622700i	$0.213964\pi$
$68$	−192.026	−0.342449
$69$	0	0
$70$	−546.569	−0.933250
$71$	583.028	0.974546	0.487273	−	0.873250i	$-0.337992\pi$
$71$	583.028	0.974546	0.487273	+	0.873250i	$0.337992\pi$
$72$	0	0
$73$	368.303	0.590501	0.295251	−	0.955420i	$-0.404597\pi$
$73$	368.303	0.590501	0.295251	+	0.955420i	$0.404597\pi$
$74$	−546.044	−0.857788
$75$	0	0
$76$	−346.328	−0.522718
$77$	−366.662	−0.542663
$78$	0	0
$79$	−438.935	−0.625114	−0.312557	−	0.949899i	$-0.601186\pi$
$79$	−438.935	−0.625114	−0.312557	+	0.949899i	$0.601186\pi$
$80$	392.247	0.548183
$81$	0	0
$82$	977.117	1.31591
$83$	877.643	1.16065	0.580324	−	0.814386i	$-0.302926\pi$
$83$	877.643	1.16065	0.580324	+	0.814386i	$0.302926\pi$
$84$	0	0
$85$	348.528	0.444743
$86$	1519.39	1.90512
$87$	0	0
$88$	189.215	0.229209
$89$	999.585	1.19051	0.595257	−	0.803535i	$-0.297050\pi$
$89$	999.585	1.19051	0.595257	+	0.803535i	$0.297050\pi$
$90$	0	0
$91$	809.529	0.932546
$92$	−359.854	−0.407797
$93$	0	0
$94$	1921.70	2.10860
$95$	628.588	0.678861
$96$	0	0
$97$	−1306.31	−1.36738	−0.683688	−	0.729775i	$-0.739625\pi$
$97$	−1306.31	−1.36738	−0.683688	+	0.729775i	$0.739625\pi$
$98$	−2518.90	−2.59640
$99$	0	0
$100$	68.8702	0.0688702
$101$	−67.1570	−0.0661621	−0.0330810	−	0.999453i	$-0.510532\pi$
$101$	−67.1570	−0.0661621	−0.0330810	+	0.999453i	$0.510532\pi$
$102$	0	0
$103$	−813.283	−0.778011	−0.389005	−	0.921235i	$-0.627181\pi$
$103$	−813.283	−0.778011	−0.389005	+	0.921235i	$0.627181\pi$
$104$	−417.755	−0.393887
$105$	0	0
$106$	−131.455	−0.120453
$107$	597.911	0.540208	0.270104	−	0.962831i	$-0.412942\pi$
$107$	597.911	0.540208	0.270104	+	0.962831i	$0.412942\pi$
$108$	0	0
$109$	1176.85	1.03414	0.517072	−	0.855942i	$-0.327022\pi$
$109$	1176.85	1.03414	0.517072	+	0.855942i	$0.327022\pi$
$110$	180.370	0.156342
$111$	0	0
$112$	2614.95	2.20616
$113$	1344.44	1.11924	0.559621	−	0.828748i	$-0.310947\pi$
$113$	1344.44	1.11924	0.559621	+	0.828748i	$0.310947\pi$
$114$	0	0
$115$	653.138	0.529612
$116$	656.144	0.525185
$117$	0	0
$118$	−1025.69	−0.800191
$119$	2323.49	1.78987
$120$	0	0
$121$	121.000	0.0909091
$122$	1283.38	0.952391
$123$	0	0
$124$	−368.217	−0.266668
$125$	−125.000	−0.0894427
$126$	0	0
$127$	946.021	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$127$	946.021	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$128$	−1728.53	−1.19361
$129$	0	0
$130$	−398.227	−0.268668
$131$	974.351	0.649843	0.324922	−	0.945741i	$-0.394662\pi$
$131$	974.351	0.649843	0.324922	+	0.945741i	$0.394662\pi$
$132$	0	0
$133$	4190.54	2.73207
$134$	−2814.53	−1.81447
$135$	0	0
$136$	−1199.03	−0.756000
$137$	−452.696	−0.282310	−0.141155	−	0.989988i	$-0.545082\pi$
$137$	−452.696	−0.282310	−0.141155	+	0.989988i	$0.545082\pi$
$138$	0	0
$139$	−885.107	−0.540099	−0.270050	−	0.962846i	$-0.587040\pi$
$139$	−885.107	−0.540099	−0.270050	+	0.962846i	$0.587040\pi$
$140$	459.129	0.277168
$141$	0	0
$142$	−1912.01	−1.12995
$143$	−267.148	−0.156224
$144$	0	0
$145$	−1190.91	−0.682065
$146$	−1207.83	−0.684664
$147$	0	0
$148$	458.688	0.254756
$149$	2655.48	1.46004	0.730018	−	0.683428i	$-0.239512\pi$
$149$	2655.48	1.46004	0.730018	+	0.683428i	$0.239512\pi$
$150$	0	0
$151$	−356.398	−0.192075	−0.0960373	−	0.995378i	$-0.530617\pi$
$151$	−356.398	−0.192075	−0.0960373	+	0.995378i	$0.530617\pi$
$152$	−2162.52	−1.15397
$153$	0	0
$154$	1202.45	0.629197
$155$	668.316	0.346326
$156$	0	0
$157$	−2379.88	−1.20978	−0.604889	−	0.796310i	$-0.706782\pi$
$157$	−2379.88	−1.20978	−0.604889	+	0.796310i	$0.706782\pi$
$158$	1439.46	0.724795
$159$	0	0
$160$	−598.303	−0.295625
$161$	4354.20	2.13142
$162$	0	0
$163$	−651.774	−0.313195	−0.156598	−	0.987662i	$-0.550053\pi$
$163$	−651.774	−0.313195	−0.156598	+	0.987662i	$0.550053\pi$
$164$	−820.798	−0.390815
$165$	0	0
$166$	−2878.19	−1.34573
$167$	−2498.34	−1.15765	−0.578823	−	0.815453i	$-0.696488\pi$
$167$	−2498.34	−1.15765	−0.578823	+	0.815453i	$0.696488\pi$
$168$	0	0
$169$	−1607.18	−0.731535
$170$	−1142.98	−0.515663
$171$	0	0
$172$	−1276.32	−0.565805
$173$	−1094.34	−0.480931	−0.240465	−	0.970658i	$-0.577300\pi$
$173$	−1094.34	−0.480931	−0.240465	+	0.970658i	$0.577300\pi$
$174$	0	0
$175$	−833.324	−0.359962
$176$	−862.944	−0.369585
$177$	0	0
$178$	−3278.09	−1.38036
$179$	1794.49	0.749309	0.374654	−	0.927165i	$-0.377761\pi$
$179$	1794.49	0.749309	0.374654	+	0.927165i	$0.377761\pi$
$180$	0	0
$181$	1416.94	0.581881	0.290941	−	0.956741i	$-0.406032\pi$
$181$	1416.94	0.581881	0.290941	+	0.956741i	$0.406032\pi$
$182$	−2654.81	−1.08125
$183$	0	0
$184$	−2246.97	−0.900266
$185$	−832.523	−0.330856
$186$	0	0
$187$	−766.762	−0.299846
$188$	−1614.27	−0.626236
$189$	0	0
$190$	−2061.43	−0.787113
$191$	−1346.61	−0.510141	−0.255071	−	0.966922i	$-0.582099\pi$
$191$	−1346.61	−0.510141	−0.255071	+	0.966922i	$0.582099\pi$
$192$	0	0
$193$	−2333.09	−0.870151	−0.435076	−	0.900394i	$-0.643278\pi$
$193$	−2333.09	−0.870151	−0.435076	+	0.900394i	$0.643278\pi$
$194$	4283.97	1.58542
$195$	0	0
$196$	2115.93	0.771110
$197$	962.676	0.348162	0.174081	−	0.984731i	$-0.444305\pi$
$197$	962.676	0.348162	0.174081	+	0.984731i	$0.444305\pi$
$198$	0	0
$199$	1337.59	0.476477	0.238238	−	0.971207i	$-0.423430\pi$
$199$	1337.59	0.476477	0.238238	+	0.971207i	$0.423430\pi$
$200$	430.034	0.152040
$201$	0	0
$202$	220.238	0.0767124
$203$	−7939.29	−2.74497
$204$	0	0
$205$	1489.76	0.507557
$206$	2667.12	0.902074
$207$	0	0
$208$	1905.24	0.635117
$209$	−1382.89	−0.457688
$210$	0	0
$211$	4378.89	1.42870	0.714349	−	0.699790i	$-0.246723\pi$
$211$	4378.89	1.42870	0.714349	+	0.699790i	$0.246723\pi$
$212$	110.425	0.0357737
$213$	0	0
$214$	−1960.82	−0.626350
$215$	2316.53	0.734820
$216$	0	0
$217$	4455.39	1.39379
$218$	−3859.42	−1.19905
$219$	0	0
$220$	−151.514	−0.0464323
$221$	1692.88	0.515274
$222$	0	0
$223$	−920.657	−0.276465	−0.138233	−	0.990400i	$-0.544142\pi$
$223$	−920.657	−0.276465	−0.138233	+	0.990400i	$0.544142\pi$
$224$	−3988.64	−1.18974
$225$	0	0
$226$	−4409.03	−1.29772
$227$	−739.810	−0.216313	−0.108156	−	0.994134i	$-0.534495\pi$
$227$	−739.810	−0.216313	−0.108156	+	0.994134i	$0.534495\pi$
$228$	0	0
$229$	4230.73	1.22085	0.610425	−	0.792074i	$-0.290999\pi$
$229$	4230.73	1.22085	0.610425	+	0.792074i	$0.290999\pi$
$230$	−2141.93	−0.614065
$231$	0	0
$232$	4097.04	1.15941
$233$	−5318.96	−1.49552	−0.747761	−	0.663968i	$-0.768871\pi$
$233$	−5318.96	−1.49552	−0.747761	+	0.663968i	$0.768871\pi$
$234$	0	0
$235$	2929.91	0.813302
$236$	861.602	0.237650
$237$	0	0
$238$	−7619.78	−2.07528
$239$	−3924.28	−1.06209	−0.531047	−	0.847342i	$-0.678201\pi$
$239$	−3924.28	−1.06209	−0.531047	+	0.847342i	$0.678201\pi$
$240$	0	0
$241$	−4198.43	−1.12218	−0.561089	−	0.827756i	$-0.689617\pi$
$241$	−4198.43	−1.12218	−0.561089	+	0.827756i	$0.689617\pi$
$242$	−396.814	−0.105406
$243$	0	0
$244$	−1078.06	−0.282853
$245$	−3840.43	−1.00145
$246$	0	0
$247$	3053.20	0.786520
$248$	−2299.19	−0.588705
$249$	0	0
$250$	409.932	0.103705
$251$	654.822	0.164669	0.0823346	−	0.996605i	$-0.473762\pi$
$251$	654.822	0.164669	0.0823346	+	0.996605i	$0.473762\pi$
$252$	0	0
$253$	−1436.90	−0.357065
$254$	−3102.43	−0.766393
$255$	0	0
$256$	3787.23	0.924617
$257$	991.373	0.240623	0.120312	−	0.992736i	$-0.461611\pi$
$257$	991.373	0.240623	0.120312	+	0.992736i	$0.461611\pi$
$258$	0	0
$259$	−5550.09	−1.33153
$260$	334.519	0.0797921
$261$	0	0
$262$	−3195.34	−0.753468
$263$	−752.618	−0.176458	−0.0882289	−	0.996100i	$-0.528121\pi$
$263$	−752.618	−0.176458	−0.0882289	+	0.996100i	$0.528121\pi$
$264$	0	0
$265$	−200.422	−0.0464598
$266$	−13742.7	−3.16774
$267$	0	0
$268$	2364.26	0.538882
$269$	4673.88	1.05937	0.529687	−	0.848193i	$-0.322309\pi$
$269$	4673.88	1.05937	0.529687	+	0.848193i	$0.322309\pi$
$270$	0	0
$271$	−8145.45	−1.82583	−0.912917	−	0.408145i	$-0.866176\pi$
$271$	−8145.45	−1.82583	−0.912917	+	0.408145i	$0.866176\pi$
$272$	5468.37	1.21900
$273$	0	0
$274$	1484.60	0.327328
$275$	275.000	0.0603023
$276$	0	0
$277$	5804.28	1.25901	0.629504	−	0.776997i	$-0.283258\pi$
$277$	5804.28	1.25901	0.629504	+	0.776997i	$0.283258\pi$
$278$	2902.67	0.626225
$279$	0	0
$280$	2866.86	0.611884
$281$	−7850.68	−1.66666	−0.833332	−	0.552773i	$-0.813570\pi$
$281$	−7850.68	−1.66666	−0.833332	+	0.552773i	$0.813570\pi$
$282$	0	0
$283$	−7091.78	−1.48962	−0.744810	−	0.667277i	$-0.767460\pi$
$283$	−7091.78	−1.48962	−0.744810	+	0.667277i	$0.767460\pi$
$284$	1606.13	0.335586
$285$	0	0
$286$	876.098	0.181136
$287$	9931.59	2.04266
$288$	0	0
$289$	−54.1294	−0.0110176
$290$	3905.52	0.790828
$291$	0	0
$292$	1014.60	0.203340
$293$	−7796.02	−1.55443	−0.777216	−	0.629234i	$-0.783369\pi$
$293$	−7796.02	−1.55443	−0.777216	+	0.629234i	$0.783369\pi$
$294$	0	0
$295$	−1563.82	−0.308640
$296$	2864.10	0.562408
$297$	0	0
$298$	−8708.52	−1.69286
$299$	3172.44	0.613602
$300$	0	0
$301$	15443.4	2.95728
$302$	1168.79	0.222703
$303$	0	0
$304$	9862.49	1.86070
$305$	1956.70	0.367345
$306$	0	0
$307$	−5051.78	−0.939155	−0.469577	−	0.882891i	$-0.655594\pi$
$307$	−5051.78	−0.939155	−0.469577	+	0.882891i	$0.655594\pi$
$308$	−1010.08	−0.186867
$309$	0	0
$310$	−2191.71	−0.401551
$311$	−3533.10	−0.644193	−0.322096	−	0.946707i	$-0.604388\pi$
$311$	−3533.10	−0.644193	−0.322096	+	0.946707i	$0.604388\pi$
$312$	0	0
$313$	3376.33	0.609718	0.304859	−	0.952398i	$-0.401391\pi$
$313$	3376.33	0.609718	0.304859	+	0.952398i	$0.401391\pi$
$314$	7804.70	1.40269
$315$	0	0
$316$	−1209.18	−0.215259
$317$	−2924.54	−0.518165	−0.259083	−	0.965855i	$-0.583420\pi$
$317$	−2924.54	−0.518165	−0.259083	+	0.965855i	$0.583420\pi$
$318$	0	0
$319$	2620.00	0.459848
$320$	−1175.88	−0.205417
$321$	0	0
$322$	−14279.4	−2.47130
$323$	8763.23	1.50959
$324$	0	0
$325$	−607.154	−0.103627
$326$	2137.46	0.363138
$327$	0	0
$328$	−5125.17	−0.862774
$329$	19532.5	3.27313
$330$	0	0
$331$	10052.9	1.66935	0.834675	−	0.550743i	$-0.185655\pi$
$331$	10052.9	1.66935	0.834675	+	0.550743i	$0.185655\pi$
$332$	2417.74	0.399670
$333$	0	0
$334$	8193.17	1.34225
$335$	−4291.16	−0.699854
$336$	0	0
$337$	6091.66	0.984671	0.492335	−	0.870406i	$-0.336143\pi$
$337$	6091.66	0.984671	0.492335	+	0.870406i	$0.336143\pi$
$338$	5270.68	0.848187
$339$	0	0
$340$	960.128	0.153148
$341$	−1470.30	−0.233493
$342$	0	0
$343$	−14169.3	−2.23053
$344$	−7969.50	−1.24909
$345$	0	0
$346$	3588.83	0.557621
$347$	−1645.57	−0.254578	−0.127289	−	0.991866i	$-0.540628\pi$
$347$	−1645.57	−0.254578	−0.127289	+	0.991866i	$0.540628\pi$
$348$	0	0
$349$	−7753.75	−1.18925	−0.594625	−	0.804003i	$-0.702700\pi$
$349$	−7753.75	−1.18925	−0.594625	+	0.804003i	$0.702700\pi$
$350$	2732.84	0.417362
$351$	0	0
$352$	1316.27	0.199310
$353$	8040.32	1.21230	0.606152	−	0.795349i	$-0.292712\pi$
$353$	8040.32	1.21230	0.606152	+	0.795349i	$0.292712\pi$
$354$	0	0
$355$	−2915.14	−0.435830
$356$	2753.67	0.409955
$357$	0	0
$358$	−5884.94	−0.868795
$359$	−9483.44	−1.39420	−0.697099	−	0.716975i	$-0.745526\pi$
$359$	−9483.44	−1.39420	−0.697099	+	0.716975i	$0.745526\pi$
$360$	0	0
$361$	8945.94	1.30426
$362$	−4646.80	−0.674669
$363$	0	0
$364$	2230.10	0.321123
$365$	−1841.51	−0.264080
$366$	0	0
$367$	6572.60	0.934841	0.467421	−	0.884035i	$-0.345183\pi$
$367$	6572.60	0.934841	0.467421	+	0.884035i	$0.345183\pi$
$368$	10247.7	1.45162
$369$	0	0
$370$	2730.22	0.383614
$371$	−1336.13	−0.186977
$372$	0	0
$373$	4148.14	0.575825	0.287912	−	0.957657i	$-0.407039\pi$
$373$	4148.14	0.575825	0.287912	+	0.957657i	$0.407039\pi$
$374$	2514.56	0.347660
$375$	0	0
$376$	−10079.7	−1.38250
$377$	−5784.51	−0.790232
$378$	0	0
$379$	4256.57	0.576901	0.288451	−	0.957495i	$-0.406860\pi$
$379$	4256.57	0.576901	0.288451	+	0.957495i	$0.406860\pi$
$380$	1731.64	0.233766
$381$	0	0
$382$	4416.13	0.591489
$383$	5430.24	0.724471	0.362236	−	0.932087i	$-0.382014\pi$
$383$	5430.24	0.724471	0.362236	+	0.932087i	$0.382014\pi$
$384$	0	0
$385$	1833.31	0.242686
$386$	7651.24	1.00891
$387$	0	0
$388$	−3598.63	−0.470857
$389$	2574.81	0.335600	0.167800	−	0.985821i	$-0.446334\pi$
$389$	2574.81	0.335600	0.167800	+	0.985821i	$0.446334\pi$
$390$	0	0
$391$	9105.47	1.17771
$392$	13212.1	1.70233
$393$	0	0
$394$	−3157.05	−0.403680
$395$	2194.67	0.279559
$396$	0	0
$397$	−14389.3	−1.81909	−0.909545	−	0.415606i	$-0.863570\pi$
$397$	−14389.3	−1.81909	−0.909545	+	0.415606i	$0.863570\pi$
$398$	−4386.55	−0.552457
$399$	0	0
$400$	−1961.24	−0.245155
$401$	−1013.21	−0.126178	−0.0630888	−	0.998008i	$-0.520095\pi$
$401$	−1013.21	−0.126178	−0.0630888	+	0.998008i	$0.520095\pi$
$402$	0	0
$403$	3246.17	0.401249
$404$	−185.005	−0.0227830
$405$	0	0
$406$	26036.5	3.18269
$407$	1831.55	0.223063
$408$	0	0
$409$	1159.14	0.140136	0.0700682	−	0.997542i	$-0.477678\pi$
$409$	1159.14	0.140136	0.0700682	+	0.997542i	$0.477678\pi$
$410$	−4885.59	−0.588492
$411$	0	0
$412$	−2240.44	−0.267909
$413$	−10425.3	−1.24212
$414$	0	0
$415$	−4388.21	−0.519058
$416$	−2906.09	−0.342507
$417$	0	0
$418$	4535.14	0.530672
$419$	9711.04	1.13226	0.566128	−	0.824317i	$-0.308441\pi$
$419$	9711.04	1.13226	0.566128	+	0.824317i	$0.308441\pi$
$420$	0	0
$421$	3477.88	0.402616	0.201308	−	0.979528i	$-0.435481\pi$
$421$	3477.88	0.402616	0.201308	+	0.979528i	$0.435481\pi$
$422$	−14360.4	−1.65652
$423$	0	0
$424$	689.507	0.0789751
$425$	−1742.64	−0.198895
$426$	0	0
$427$	13044.5	1.47838
$428$	1647.13	0.186021
$429$	0	0
$430$	−7596.96	−0.851995
$431$	6748.27	0.754182	0.377091	−	0.926176i	$-0.376924\pi$
$431$	6748.27	0.754182	0.377091	+	0.926176i	$0.376924\pi$
$432$	0	0
$433$	5380.74	0.597186	0.298593	−	0.954380i	$-0.403483\pi$
$433$	5380.74	0.597186	0.298593	+	0.954380i	$0.403483\pi$
$434$	−14611.2	−1.61604
$435$	0	0
$436$	3241.99	0.356108
$437$	16422.2	1.79767
$438$	0	0
$439$	−1095.09	−0.119057	−0.0595283	−	0.998227i	$-0.518960\pi$
$439$	−1095.09	−0.119057	−0.0595283	+	0.998227i	$0.518960\pi$
$440$	−946.075	−0.102505
$441$	0	0
$442$	−5551.72	−0.597440
$443$	3167.11	0.339670	0.169835	−	0.985473i	$-0.445677\pi$
$443$	3167.11	0.339670	0.169835	+	0.985473i	$0.445677\pi$
$444$	0	0
$445$	−4997.93	−0.532414
$446$	3019.25	0.320551
$447$	0	0
$448$	−7839.08	−0.826700
$449$	391.054	0.0411024	0.0205512	−	0.999789i	$-0.493458\pi$
$449$	391.054	0.0411024	0.0205512	+	0.999789i	$0.493458\pi$
$450$	0	0
$451$	−3277.46	−0.342195
$452$	3703.68	0.385412
$453$	0	0
$454$	2426.17	0.250806
$455$	−4047.65	−0.417047
$456$	0	0
$457$	2563.42	0.262389	0.131194	−	0.991357i	$-0.458119\pi$
$457$	2563.42	0.262389	0.131194	+	0.991357i	$0.458119\pi$
$458$	−13874.5	−1.41553
$459$	0	0
$460$	1799.27	0.182373
$461$	6151.57	0.621491	0.310745	−	0.950493i	$-0.399421\pi$
$461$	6151.57	0.621491	0.310745	+	0.950493i	$0.399421\pi$
$462$	0	0
$463$	9833.63	0.987058	0.493529	−	0.869729i	$-0.335707\pi$
$463$	9833.63	0.987058	0.493529	+	0.869729i	$0.335707\pi$
$464$	−18685.2	−1.86948
$465$	0	0
$466$	17443.3	1.73400
$467$	7408.21	0.734071	0.367036	−	0.930207i	$-0.380373\pi$
$467$	7408.21	0.734071	0.367036	+	0.930207i	$0.380373\pi$
$468$	0	0
$469$	−28607.4	−2.81656
$470$	−9608.49	−0.942993
$471$	0	0
$472$	5379.95	0.524645
$473$	−5096.37	−0.495415
$474$	0	0
$475$	−3142.94	−0.303596
$476$	6400.78	0.616343
$477$	0	0
$478$	12869.5	1.23146
$479$	8257.19	0.787643	0.393821	−	0.919187i	$-0.371153\pi$
$479$	8257.19	0.787643	0.393821	+	0.919187i	$0.371153\pi$
$480$	0	0
$481$	−4043.76	−0.383325
$482$	13768.6	1.30112
$483$	0	0
$484$	333.332	0.0313046
$485$	6531.54	0.611509
$486$	0	0
$487$	−10086.8	−0.938552	−0.469276	−	0.883051i	$-0.655485\pi$
$487$	−10086.8	−0.938552	−0.469276	+	0.883051i	$0.655485\pi$
$488$	−6731.57	−0.624434
$489$	0	0
$490$	12594.5	1.16115
$491$	6343.10	0.583015	0.291507	−	0.956569i	$-0.405843\pi$
$491$	6343.10	0.583015	0.291507	+	0.956569i	$0.405843\pi$
$492$	0	0
$493$	−16602.6	−1.51672
$494$	−10012.8	−0.911940
$495$	0	0
$496$	10485.8	0.949248
$497$	−19434.0	−1.75400
$498$	0	0
$499$	−9862.39	−0.884772	−0.442386	−	0.896825i	$-0.645868\pi$
$499$	−9862.39	−0.884772	−0.442386	+	0.896825i	$0.645868\pi$
$500$	−344.351	−0.0307997
$501$	0	0
$502$	−2147.46	−0.190928
$503$	16774.2	1.48692	0.743462	−	0.668778i	$-0.233182\pi$
$503$	16774.2	1.48692	0.743462	+	0.668778i	$0.233182\pi$
$504$	0	0
$505$	335.785	0.0295886
$506$	4712.26	0.414003
$507$	0	0
$508$	2606.10	0.227613
$509$	−18723.8	−1.63049	−0.815244	−	0.579118i	$-0.803397\pi$
$509$	−18723.8	−1.63049	−0.815244	+	0.579118i	$0.803397\pi$
$510$	0	0
$511$	−12276.6	−1.06279
$512$	1408.20	0.121551
$513$	0	0
$514$	−3251.16	−0.278993
$515$	4066.41	0.347937
$516$	0	0
$517$	−6445.79	−0.548328
$518$	18201.2	1.54385
$519$	0	0
$520$	2088.78	0.176152
$521$	1321.90	0.111159	0.0555793	−	0.998454i	$-0.482299\pi$
$521$	1321.90	0.111159	0.0555793	+	0.998454i	$0.482299\pi$
$522$	0	0
$523$	4276.30	0.357533	0.178766	−	0.983892i	$-0.442789\pi$
$523$	4276.30	0.357533	0.178766	+	0.983892i	$0.442789\pi$
$524$	2684.15	0.223774
$525$	0	0
$526$	2468.17	0.204596
$527$	9317.08	0.770130
$528$	0	0
$529$	4896.57	0.402447
$530$	657.275	0.0538683
$531$	0	0
$532$	11544.1	0.940792
$533$	7236.09	0.588049
$534$	0	0
$535$	−2989.55	−0.241588
$536$	14762.8	1.18965
$537$	0	0
$538$	−15327.8	−1.22830
$539$	8448.94	0.675179
$540$	0	0
$541$	−2941.66	−0.233774	−0.116887	−	0.993145i	$-0.537292\pi$
$541$	−2941.66	−0.233774	−0.116887	+	0.993145i	$0.537292\pi$
$542$	26712.6	2.11698
$543$	0	0
$544$	−8341.01	−0.657386
$545$	−5884.24	−0.462483
$546$	0	0
$547$	8717.71	0.681430	0.340715	−	0.940167i	$-0.389331\pi$
$547$	8717.71	0.681430	0.340715	+	0.940167i	$0.389331\pi$
$548$	−1247.09	−0.0972137
$549$	0	0
$550$	−901.849	−0.0699182
$551$	−29943.6	−2.31514
$552$	0	0
$553$	14631.0	1.12509
$554$	−19034.9	−1.45977
$555$	0	0
$556$	−2438.30	−0.185984
$557$	4154.36	0.316025	0.158013	−	0.987437i	$-0.449491\pi$
$557$	4154.36	0.316025	0.158013	+	0.987437i	$0.449491\pi$
$558$	0	0
$559$	11251.9	0.851353
$560$	−13074.8	−0.986624
$561$	0	0
$562$	25745.9	1.93243
$563$	−8603.87	−0.644068	−0.322034	−	0.946728i	$-0.604366\pi$
$563$	−8603.87	−0.644068	−0.322034	+	0.946728i	$0.604366\pi$
$564$	0	0
$565$	−6722.21	−0.500541
$566$	23257.1	1.72716
$567$	0	0
$568$	10028.9	0.740849
$569$	−1598.57	−0.117778	−0.0588889	−	0.998265i	$-0.518756\pi$
$569$	−1598.57	−0.117778	−0.0588889	+	0.998265i	$0.518756\pi$
$570$	0	0
$571$	−21577.9	−1.58145	−0.790725	−	0.612171i	$-0.790296\pi$
$571$	−21577.9	−1.58145	−0.790725	+	0.612171i	$0.790296\pi$
$572$	−735.941	−0.0537959
$573$	0	0
$574$	−32570.2	−2.36839
$575$	−3265.69	−0.236850
$576$	0	0
$577$	7586.21	0.547345	0.273672	−	0.961823i	$-0.411762\pi$
$577$	7586.21	0.547345	0.273672	+	0.961823i	$0.411762\pi$
$578$	177.515	0.0127745
$579$	0	0
$580$	−3280.72	−0.234870
$581$	−29254.4	−2.08895
$582$	0	0
$583$	440.929	0.0313232
$584$	6335.31	0.448899
$585$	0	0
$586$	25566.7	1.80230
$587$	−4442.82	−0.312393	−0.156197	−	0.987726i	$-0.549923\pi$
$587$	−4442.82	−0.312393	−0.156197	+	0.987726i	$0.549923\pi$
$588$	0	0
$589$	16803.8	1.17554
$590$	5128.46	0.357856
$591$	0	0
$592$	−13062.2	−0.906846
$593$	15853.2	1.09783	0.548915	−	0.835878i	$-0.315041\pi$
$593$	15853.2	1.09783	0.548915	+	0.835878i	$0.315041\pi$
$594$	0	0
$595$	−11617.5	−0.800453
$596$	7315.33	0.502765
$597$	0	0
$598$	−10403.9	−0.711448
$599$	−3382.45	−0.230723	−0.115362	−	0.993324i	$-0.536803\pi$
$599$	−3382.45	−0.230723	−0.115362	+	0.993324i	$0.536803\pi$
$600$	0	0
$601$	−16494.3	−1.11950	−0.559748	−	0.828663i	$-0.689102\pi$
$601$	−16494.3	−1.11950	−0.559748	+	0.828663i	$0.689102\pi$
$602$	−50645.8	−3.42885
$603$	0	0
$604$	−981.808	−0.0661411
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−21074.0	−1.40917	−0.704585	−	0.709620i	$-0.748867\pi$
$607$	−21074.0	−1.40917	−0.704585	+	0.709620i	$0.748867\pi$
$608$	−15043.4	−1.00344
$609$	0	0
$610$	−6416.90	−0.425922
$611$	14231.2	0.942282
$612$	0	0
$613$	20645.9	1.36033	0.680165	−	0.733059i	$-0.261908\pi$
$613$	20645.9	1.36033	0.680165	+	0.733059i	$0.261908\pi$
$614$	16567.1	1.08891
$615$	0	0
$616$	−6307.09	−0.412532
$617$	3913.54	0.255353	0.127677	−	0.991816i	$-0.459248\pi$
$617$	3913.54	0.255353	0.127677	+	0.991816i	$0.459248\pi$
$618$	0	0
$619$	−27237.9	−1.76863	−0.884315	−	0.466891i	$-0.845374\pi$
$619$	−27237.9	−1.76863	−0.884315	+	0.466891i	$0.845374\pi$
$620$	1841.08	0.119258
$621$	0	0
$622$	11586.6	0.746917
$623$	−33319.1	−2.14270
$624$	0	0
$625$	625.000	0.0400000
$626$	−11072.5	−0.706944
$627$	0	0
$628$	−6556.11	−0.416588
$629$	−11606.3	−0.735729
$630$	0	0
$631$	−21440.7	−1.35268	−0.676341	−	0.736589i	$-0.736435\pi$
$631$	−21440.7	−1.35268	−0.676341	+	0.736589i	$0.736435\pi$
$632$	−7550.27	−0.475211
$633$	0	0
$634$	9590.89	0.600793
$635$	−4730.10	−0.295604
$636$	0	0
$637$	−18653.8	−1.16027
$638$	−8592.15	−0.533176
$639$	0	0
$640$	8642.65	0.533798
$641$	2173.52	0.133929	0.0669647	−	0.997755i	$-0.478669\pi$
$641$	2173.52	0.133929	0.0669647	+	0.997755i	$0.478669\pi$
$642$	0	0
$643$	26997.5	1.65579	0.827897	−	0.560880i	$-0.189537\pi$
$643$	26997.5	1.65579	0.827897	+	0.560880i	$0.189537\pi$
$644$	11995.0	0.733958
$645$	0	0
$646$	−28738.6	−1.75032
$647$	−8961.19	−0.544514	−0.272257	−	0.962225i	$-0.587770\pi$
$647$	−8961.19	−0.544514	−0.272257	+	0.962225i	$0.587770\pi$
$648$	0	0
$649$	3440.39	0.208085
$650$	1991.13	0.120152
$651$	0	0
$652$	−1795.51	−0.107849
$653$	27009.3	1.61862	0.809309	−	0.587384i	$-0.199842\pi$
$653$	27009.3	1.61862	0.809309	+	0.587384i	$0.199842\pi$
$654$	0	0
$655$	−4871.76	−0.290619
$656$	23374.1	1.39117
$657$	0	0
$658$	−64055.8	−3.79507
$659$	−8295.07	−0.490334	−0.245167	−	0.969481i	$-0.578843\pi$
$659$	−8295.07	−0.490334	−0.245167	+	0.969481i	$0.578843\pi$
$660$	0	0
$661$	2467.17	0.145177	0.0725883	−	0.997362i	$-0.476874\pi$
$661$	2467.17	0.145177	0.0725883	+	0.997362i	$0.476874\pi$
$662$	−32967.9	−1.93555
$663$	0	0
$664$	15096.6	0.882324
$665$	−20952.7	−1.22182
$666$	0	0
$667$	−31113.1	−1.80615
$668$	−6882.43	−0.398637
$669$	0	0
$670$	14072.7	0.811454
$671$	−4304.73	−0.247664
$672$	0	0
$673$	11725.8	0.671614	0.335807	−	0.941931i	$-0.390991\pi$
$673$	11725.8	0.671614	0.335807	+	0.941931i	$0.390991\pi$
$674$	−19977.3	−1.14169
$675$	0	0
$676$	−4427.48	−0.251905
$677$	24087.1	1.36742	0.683710	−	0.729754i	$-0.260365\pi$
$677$	24087.1	1.36742	0.683710	+	0.729754i	$0.260365\pi$
$678$	0	0
$679$	43543.1	2.46102
$680$	5995.16	0.338094
$681$	0	0
$682$	4821.77	0.270726
$683$	8183.23	0.458452	0.229226	−	0.973373i	$-0.426381\pi$
$683$	8183.23	0.458452	0.229226	+	0.973373i	$0.426381\pi$
$684$	0	0
$685$	2263.48	0.126253
$686$	46467.7	2.58621
$687$	0	0
$688$	36346.2	2.01408
$689$	−973.498	−0.0538277
$690$	0	0
$691$	17016.4	0.936810	0.468405	−	0.883514i	$-0.344829\pi$
$691$	17016.4	0.936810	0.468405	+	0.883514i	$0.344829\pi$
$692$	−3014.69	−0.165609
$693$	0	0
$694$	5396.56	0.295174
$695$	4425.54	0.241540
$696$	0	0
$697$	20768.9	1.12866
$698$	25428.0	1.37889
$699$	0	0
$700$	−2295.65	−0.123953
$701$	5983.09	0.322365	0.161183	−	0.986925i	$-0.448469\pi$
$701$	5983.09	0.322365	0.161183	+	0.986925i	$0.448469\pi$
$702$	0	0
$703$	−20932.6	−1.12302
$704$	2586.93	0.138492
$705$	0	0
$706$	−26367.8	−1.40562
$707$	2238.54	0.119079
$708$	0	0
$709$	−6361.76	−0.336983	−0.168491	−	0.985703i	$-0.553890\pi$
$709$	−6361.76	−0.336983	−0.168491	+	0.985703i	$0.553890\pi$
$710$	9560.07	0.505328
$711$	0	0
$712$	17194.2	0.905029
$713$	17460.1	0.917092
$714$	0	0
$715$	1335.74	0.0698655
$716$	4943.47	0.258025
$717$	0	0
$718$	31100.5	1.61652
$719$	4360.65	0.226182	0.113091	−	0.993585i	$-0.463925\pi$
$719$	4360.65	0.226182	0.113091	+	0.993585i	$0.463925\pi$
$720$	0	0
$721$	27109.1	1.40027
$722$	−29337.8	−1.51224
$723$	0	0
$724$	3903.40	0.200371
$725$	5954.53	0.305029
$726$	0	0
$727$	11314.3	0.577200	0.288600	−	0.957450i	$-0.406810\pi$
$727$	11314.3	0.577200	0.288600	+	0.957450i	$0.406810\pi$
$728$	13925.0	0.708922
$729$	0	0
$730$	6039.16	0.306191
$731$	32295.1	1.63403
$732$	0	0
$733$	22374.6	1.12745	0.563726	−	0.825962i	$-0.309367\pi$
$733$	22374.6	1.12745	0.563726	+	0.825962i	$0.309367\pi$
$734$	−21554.5	−1.08391
$735$	0	0
$736$	−15631.0	−0.782833
$737$	9440.55	0.471842
$738$	0	0
$739$	31277.2	1.55690	0.778451	−	0.627706i	$-0.216006\pi$
$739$	31277.2	1.55690	0.778451	+	0.627706i	$0.216006\pi$
$740$	−2293.44	−0.113930
$741$	0	0
$742$	4381.78	0.216793
$743$	−35409.4	−1.74838	−0.874189	−	0.485585i	$-0.838607\pi$
$743$	−35409.4	−1.74838	−0.874189	+	0.485585i	$0.838607\pi$
$744$	0	0
$745$	−13277.4	−0.652948
$746$	−13603.6	−0.667647
$747$	0	0
$748$	−2112.28	−0.103252
$749$	−19930.1	−0.972271
$750$	0	0
$751$	−24420.2	−1.18656	−0.593279	−	0.804997i	$-0.702167\pi$
$751$	−24420.2	−1.18656	−0.593279	+	0.804997i	$0.702167\pi$
$752$	45969.9	2.22919
$753$	0	0
$754$	18970.0	0.916244
$755$	1781.99	0.0858984
$756$	0	0
$757$	−1017.46	−0.0488512	−0.0244256	−	0.999702i	$-0.507776\pi$
$757$	−1017.46	−0.0488512	−0.0244256	+	0.999702i	$0.507776\pi$
$758$	−13959.2	−0.668895
$759$	0	0
$760$	10812.6	0.516070
$761$	−14539.8	−0.692600	−0.346300	−	0.938124i	$-0.612562\pi$
$761$	−14539.8	−0.692600	−0.346300	+	0.938124i	$0.612562\pi$
$762$	0	0
$763$	−39227.8	−1.86126
$764$	−3709.64	−0.175668
$765$	0	0
$766$	−17808.2	−0.839996
$767$	−7595.81	−0.357587
$768$	0	0
$769$	−31068.6	−1.45691	−0.728454	−	0.685095i	$-0.759761\pi$
$769$	−31068.6	−1.45691	−0.728454	+	0.685095i	$0.759761\pi$
$770$	−6012.26	−0.281385
$771$	0	0
$772$	−6427.20	−0.299637
$773$	−9031.49	−0.420233	−0.210117	−	0.977676i	$-0.567384\pi$
$773$	−9031.49	−0.420233	−0.210117	+	0.977676i	$0.567384\pi$
$774$	0	0
$775$	−3341.58	−0.154882
$776$	−22470.3	−1.03948
$777$	0	0
$778$	−8443.98	−0.389115
$779$	37457.7	1.72280
$780$	0	0
$781$	6413.31	0.293837
$782$	−29861.0	−1.36551
$783$	0	0
$784$	−60255.9	−2.74489
$785$	11899.4	0.541029
$786$	0	0
$787$	4975.83	0.225374	0.112687	−	0.993631i	$-0.464054\pi$
$787$	4975.83	0.225374	0.112687	+	0.993631i	$0.464054\pi$
$788$	2651.99	0.119890
$789$	0	0
$790$	−7197.32	−0.324138
$791$	−44814.2	−2.01442
$792$	0	0
$793$	9504.13	0.425601
$794$	47189.0	2.10916
$795$	0	0
$796$	3684.79	0.164075
$797$	13663.9	0.607276	0.303638	−	0.952787i	$-0.401799\pi$
$797$	13663.9	0.607276	0.303638	+	0.952787i	$0.401799\pi$
$798$	0	0
$799$	40846.2	1.80855
$800$	2991.51	0.132207
$801$	0	0
$802$	3322.77	0.146298
$803$	4051.33	0.178043
$804$	0	0
$805$	−21771.0	−0.953202
$806$	−10645.7	−0.465232
$807$	0	0
$808$	−1155.19	−0.0502964
$809$	536.654	0.0233223	0.0116612	−	0.999932i	$-0.496288\pi$
$809$	536.654	0.0233223	0.0116612	+	0.999932i	$0.496288\pi$
$810$	0	0
$811$	7711.19	0.333880	0.166940	−	0.985967i	$-0.446611\pi$
$811$	7711.19	0.333880	0.166940	+	0.985967i	$0.446611\pi$
$812$	−21871.2	−0.945233
$813$	0	0
$814$	−6006.48	−0.258633
$815$	3258.87	0.140065
$816$	0	0
$817$	58245.8	2.49420
$818$	−3801.34	−0.162483
$819$	0	0
$820$	4103.99	0.174778
$821$	−10451.9	−0.444305	−0.222152	−	0.975012i	$-0.571308\pi$
$821$	−10451.9	−0.444305	−0.222152	+	0.975012i	$0.571308\pi$
$822$	0	0
$823$	17400.3	0.736983	0.368491	−	0.929631i	$-0.379874\pi$
$823$	17400.3	0.736983	0.368491	+	0.929631i	$0.379874\pi$
$824$	−13989.6	−0.591444
$825$	0	0
$826$	34189.3	1.44019
$827$	18828.3	0.791684	0.395842	−	0.918319i	$-0.370453\pi$
$827$	18828.3	0.791684	0.395842	+	0.918319i	$0.370453\pi$
$828$	0	0
$829$	34431.9	1.44254	0.721272	−	0.692652i	$-0.243558\pi$
$829$	34431.9	1.44254	0.721272	+	0.692652i	$0.243558\pi$
$830$	14390.9	0.601827
$831$	0	0
$832$	−5711.50	−0.237994
$833$	−53539.8	−2.22694
$834$	0	0
$835$	12491.7	0.517715
$836$	−3809.61	−0.157605
$837$	0	0
$838$	−31846.9	−1.31281
$839$	−39094.5	−1.60869	−0.804346	−	0.594162i	$-0.797484\pi$
$839$	−39094.5	−1.60869	−0.804346	+	0.594162i	$0.797484\pi$
$840$	0	0
$841$	32341.4	1.32606
$842$	−11405.5	−0.466818
$843$	0	0
$844$	12063.0	0.491973
$845$	8035.91	0.327152
$846$	0	0
$847$	−4033.29	−0.163619
$848$	−3144.61	−0.127342
$849$	0	0
$850$	5714.90	0.230611
$851$	−21750.1	−0.876126
$852$	0	0
$853$	−38765.9	−1.55606	−0.778029	−	0.628228i	$-0.783780\pi$
$853$	−38765.9	−1.55606	−0.778029	+	0.628228i	$0.783780\pi$
$854$	−42778.8	−1.71412
$855$	0	0
$856$	10284.9	0.410666
$857$	4057.20	0.161717	0.0808584	−	0.996726i	$-0.474234\pi$
$857$	4057.20	0.161717	0.0808584	+	0.996726i	$0.474234\pi$
$858$	0	0
$859$	32753.0	1.30095	0.650477	−	0.759526i	$-0.274569\pi$
$859$	32753.0	1.30095	0.650477	+	0.759526i	$0.274569\pi$
$860$	6381.60	0.253036
$861$	0	0
$862$	−22130.6	−0.874445
$863$	−47506.8	−1.87387	−0.936936	−	0.349501i	$-0.886351\pi$
$863$	−47506.8	−1.87387	−0.936936	+	0.349501i	$0.886351\pi$
$864$	0	0
$865$	5471.69	0.215079
$866$	−17645.9	−0.692415
$867$	0	0
$868$	12273.7	0.479952
$869$	−4828.28	−0.188479
$870$	0	0
$871$	−20843.2	−0.810842
$872$	20243.4	0.786156
$873$	0	0
$874$	−53855.8	−2.08433
$875$	4166.62	0.160980
$876$	0	0
$877$	22933.4	0.883016	0.441508	−	0.897257i	$-0.354444\pi$
$877$	22933.4	0.883016	0.441508	+	0.897257i	$0.354444\pi$
$878$	3591.30	0.138042
$879$	0	0
$880$	4314.72	0.165283
$881$	−152.937	−0.00584854	−0.00292427	−	0.999996i	$-0.500931\pi$
$881$	−152.937	−0.00584854	−0.00292427	+	0.999996i	$0.500931\pi$
$882$	0	0
$883$	−41553.9	−1.58369	−0.791845	−	0.610722i	$-0.790879\pi$
$883$	−41553.9	−1.58369	−0.791845	+	0.610722i	$0.790879\pi$
$884$	4663.56	0.177435
$885$	0	0
$886$	−10386.4	−0.393834
$887$	−41795.1	−1.58212	−0.791062	−	0.611736i	$-0.790471\pi$
$887$	−41795.1	−1.58212	−0.791062	+	0.611736i	$0.790471\pi$
$888$	0	0
$889$	−31533.7	−1.18966
$890$	16390.5	0.617314
$891$	0	0
$892$	−2536.23	−0.0952011
$893$	73668.2	2.76060
$894$	0	0
$895$	−8972.44	−0.335101
$896$	57617.0	2.14827
$897$	0	0
$898$	−1282.44	−0.0476566
$899$	−31836.1	−1.18108
$900$	0	0
$901$	−2794.11	−0.103313
$902$	10748.3	0.396761
$903$	0	0
$904$	23126.2	0.850848
$905$	−7084.71	−0.260225
$906$	0	0
$907$	−11144.0	−0.407973	−0.203987	−	0.978974i	$-0.565390\pi$
$907$	−11144.0	−0.407973	−0.203987	+	0.978974i	$0.565390\pi$
$908$	−2038.04	−0.0744874
$909$	0	0
$910$	13274.1	0.483550
$911$	−38572.8	−1.40282	−0.701412	−	0.712756i	$-0.747447\pi$
$911$	−38572.8	−1.40282	−0.701412	+	0.712756i	$0.747447\pi$
$912$	0	0
$913$	9654.07	0.349949
$914$	−8406.61	−0.304230
$915$	0	0
$916$	11654.9	0.420401
$917$	−32478.0	−1.16959
$918$	0	0
$919$	−15020.5	−0.539152	−0.269576	−	0.962979i	$-0.586883\pi$
$919$	−15020.5	−0.539152	−0.269576	+	0.962979i	$0.586883\pi$
$920$	11234.9	0.402611
$921$	0	0
$922$	−20173.8	−0.720595
$923$	−14159.5	−0.504947
$924$	0	0
$925$	4162.61	0.147963
$926$	−32248.9	−1.14446
$927$	0	0
$928$	28500.9	1.00818
$929$	32208.5	1.13749	0.568744	−	0.822515i	$-0.307430\pi$
$929$	32208.5	1.13749	0.568744	+	0.822515i	$0.307430\pi$
$930$	0	0
$931$	−96561.9	−3.39923
$932$	−14652.7	−0.514984
$933$	0	0
$934$	−24294.9	−0.851127
$935$	3833.81	0.134095
$936$	0	0
$937$	−34189.4	−1.19202	−0.596008	−	0.802979i	$-0.703247\pi$
$937$	−34189.4	−1.19202	−0.596008	+	0.802979i	$0.703247\pi$
$938$	93816.6	3.26569
$939$	0	0
$940$	8071.33	0.280061
$941$	31309.9	1.08467	0.542334	−	0.840163i	$-0.317541\pi$
$941$	31309.9	1.08467	0.542334	+	0.840163i	$0.317541\pi$
$942$	0	0
$943$	38920.7	1.34404
$944$	−24536.1	−0.845956
$945$	0	0
$946$	16713.3	0.574415
$947$	14608.5	0.501281	0.250641	−	0.968080i	$-0.419359\pi$
$947$	14608.5	0.501281	0.250641	+	0.968080i	$0.419359\pi$
$948$	0	0
$949$	−8944.67	−0.305960
$950$	10307.1	0.352008
$951$	0	0
$952$	39967.2	1.36066
$953$	25778.5	0.876230	0.438115	−	0.898919i	$-0.355646\pi$
$953$	25778.5	0.876230	0.438115	+	0.898919i	$0.355646\pi$
$954$	0	0
$955$	6733.03	0.228142
$956$	−10810.6	−0.365733
$957$	0	0
$958$	−27079.1	−0.913241
$959$	15089.7	0.508104
$960$	0	0
$961$	−11925.1	−0.400293
$962$	13261.3	0.444451
$963$	0	0
$964$	−11565.9	−0.386423
$965$	11665.4	0.389143
$966$	0	0
$967$	−17308.2	−0.575588	−0.287794	−	0.957692i	$-0.592922\pi$
$967$	−17308.2	−0.575588	−0.287794	+	0.957692i	$0.592922\pi$
$968$	2081.36	0.0691091
$969$	0	0
$970$	−21419.9	−0.709021
$971$	−22356.6	−0.738886	−0.369443	−	0.929253i	$-0.620451\pi$
$971$	−22356.6	−0.738886	−0.369443	+	0.929253i	$0.620451\pi$
$972$	0	0
$973$	29503.2	0.972076
$974$	33079.1	1.08822
$975$	0	0
$976$	30700.4	1.00686
$977$	−3453.54	−0.113090	−0.0565449	−	0.998400i	$-0.518008\pi$
$977$	−3453.54	−0.113090	−0.0565449	+	0.998400i	$0.518008\pi$
$978$	0	0
$979$	10995.4	0.358954
$980$	−10579.6	−0.344851
$981$	0	0
$982$	−20801.9	−0.675983
$983$	16391.6	0.531853	0.265926	−	0.963993i	$-0.414322\pi$
$983$	16391.6	0.531853	0.265926	+	0.963993i	$0.414322\pi$
$984$	0	0
$985$	−4813.38	−0.155703
$986$	54447.4	1.75858
$987$	0	0
$988$	8410.98	0.270839
$989$	60520.6	1.94585
$990$	0	0
$991$	−36513.7	−1.17043	−0.585216	−	0.810878i	$-0.698990\pi$
$991$	−36513.7	−1.17043	−0.585216	+	0.810878i	$0.698990\pi$
$992$	−15994.2	−0.511912
$993$	0	0
$994$	63733.0	2.03369
$995$	−6687.93	−0.213087
$996$	0	0
$997$	−32452.1	−1.03086	−0.515430	−	0.856931i	$-0.672368\pi$
$997$	−32452.1	−1.03086	−0.515430	+	0.856931i	$0.672368\pi$
$998$	32343.2	1.02586
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.g.1.1		3
3.2	odd	2		165.4.a.f.1.3	✓	3
5.4	even	2		2475.4.a.w.1.3		3
15.2	even	4		825.4.c.o.199.6		6
15.8	even	4		825.4.c.o.199.1		6
15.14	odd	2		825.4.a.n.1.1		3
33.32	even	2		1815.4.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.f.1.3	✓	3	3.2	odd	2
495.4.a.g.1.1		3	1.1	even	1	trivial
825.4.a.n.1.1		3	15.14	odd	2
825.4.c.o.199.1		6	15.8	even	4
825.4.c.o.199.6		6	15.2	even	4
1815.4.a.p.1.1		3	33.32	even	2
2475.4.a.w.1.3		3	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-3.27945 q^{2} +2.75481 q^{4} -5.00000 q^{5} -33.3329 q^{7} +17.2014 q^{8} +O(q^{10})\)	\(q-3.27945 q^{2} +2.75481 q^{4} -5.00000 q^{5} -33.3329 q^{7} +17.2014 q^{8} +16.3973 q^{10} +11.0000 q^{11} -24.2862 q^{13} +109.314 q^{14} -78.4495 q^{16} -69.7056 q^{17} -125.718 q^{19} -13.7740 q^{20} -36.0740 q^{22} -130.628 q^{23} +25.0000 q^{25} +79.6453 q^{26} -91.8258 q^{28} +238.181 q^{29} -133.663 q^{31} +119.661 q^{32} +228.596 q^{34} +166.665 q^{35} +166.505 q^{37} +412.285 q^{38} -86.0068 q^{40} -297.951 q^{41} -463.307 q^{43} +30.3029 q^{44} +428.387 q^{46} -585.981 q^{47} +768.085 q^{49} -81.9863 q^{50} -66.9037 q^{52} +40.0845 q^{53} -55.0000 q^{55} -573.372 q^{56} -781.105 q^{58} +312.763 q^{59} -391.339 q^{61} +438.342 q^{62} +235.175 q^{64} +121.431 q^{65} +858.232 q^{67} -192.026 q^{68} -546.569 q^{70} +583.028 q^{71} +368.303 q^{73} -546.044 q^{74} -346.328 q^{76} -366.662 q^{77} -438.935 q^{79} +392.247 q^{80} +977.117 q^{82} +877.643 q^{83} +348.528 q^{85} +1519.39 q^{86} +189.215 q^{88} +999.585 q^{89} +809.529 q^{91} -359.854 q^{92} +1921.70 q^{94} +628.588 q^{95} -1306.31 q^{97} -2518.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8	\( 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 42 q^{13} + 76 q^{14} - 85 q^{16} + 34 q^{17} - 280 q^{19} + 25 q^{20} - 11 q^{22} + 112 q^{23} + 75 q^{25} + 314 q^{26} - 280 q^{28} + 290 q^{29} - 392 q^{31} + 23 q^{32} + 310 q^{34} + 80 q^{35} + 570 q^{37} + 512 q^{38} - 15 q^{40} + 662 q^{41} - 68 q^{43} - 55 q^{44} + 728 q^{46} - 264 q^{47} + 731 q^{49} - 25 q^{50} + 582 q^{52} + 94 q^{53} - 165 q^{55} - 204 q^{56} - 54 q^{58} + 612 q^{59} - 582 q^{61} - 320 q^{62} + 347 q^{64} + 210 q^{65} + 940 q^{67} - 678 q^{68} - 380 q^{70} + 1616 q^{71} + 738 q^{73} + 642 q^{74} + 880 q^{76} - 176 q^{77} + 124 q^{79} + 425 q^{80} + 2126 q^{82} + 1232 q^{83} - 170 q^{85} + 1000 q^{86} + 33 q^{88} - 838 q^{89} - 1736 q^{91} - 1248 q^{92} + 2432 q^{94} + 1400 q^{95} - 90 q^{97} - 3481 q^{98}+O(q^{100}) \) 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 42 * q^13 + 76 * q^14 - 85 * q^16 + 34 * q^17 - 280 * q^19 + 25 * q^20 - 11 * q^22 + 112 * q^23 + 75 * q^25 + 314 * q^26 - 280 * q^28 + 290 * q^29 - 392 * q^31 + 23 * q^32 + 310 * q^34 + 80 * q^35 + 570 * q^37 + 512 * q^38 - 15 * q^40 + 662 * q^41 - 68 * q^43 - 55 * q^44 + 728 * q^46 - 264 * q^47 + 731 * q^49 - 25 * q^50 + 582 * q^52 + 94 * q^53 - 165 * q^55 - 204 * q^56 - 54 * q^58 + 612 * q^59 - 582 * q^61 - 320 * q^62 + 347 * q^64 + 210 * q^65 + 940 * q^67 - 678 * q^68 - 380 * q^70 + 1616 * q^71 + 738 * q^73 + 642 * q^74 + 880 * q^76 - 176 * q^77 + 124 * q^79 + 425 * q^80 + 2126 * q^82 + 1232 * q^83 - 170 * q^85 + 1000 * q^86 + 33 * q^88 - 838 * q^89 - 1736 * q^91 - 1248 * q^92 + 2432 * q^94 + 1400 * q^95 - 90 * q^97 - 3481 * q^98

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