LMFDB - Embedded newform 2166.4.a.bh.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 350x^{6} + 948x^{5} + 37019x^{4} - 115308x^{3} - 1098530x^{2} + 2724222x + 7883581 $ x^8 - 2x^7 - 350x^6 + 948x^5 + 37019x^4 - 115308x^3 - 1098530x^2 + 2724222*x + 7883581
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 19 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-12.3217$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +9.05084 q^{5} -6.00000 q^{6} -16.7133 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +9.05084 q^{5} -6.00000 q^{6} -16.7133 q^{7} -8.00000 q^{8} +9.00000 q^{9} -18.1017 q^{10} +16.8916 q^{11} +12.0000 q^{12} +29.6964 q^{13} +33.4266 q^{14} +27.1525 q^{15} +16.0000 q^{16} +68.2165 q^{17} -18.0000 q^{18} +36.2033 q^{20} -50.1399 q^{21} -33.7832 q^{22} -193.365 q^{23} -24.0000 q^{24} -43.0824 q^{25} -59.3929 q^{26} +27.0000 q^{27} -66.8532 q^{28} -69.5133 q^{29} -54.3050 q^{30} -77.0080 q^{31} -32.0000 q^{32} +50.6748 q^{33} -136.433 q^{34} -151.269 q^{35} +36.0000 q^{36} -201.929 q^{37} +89.0893 q^{39} -72.4067 q^{40} +193.235 q^{41} +100.280 q^{42} -423.769 q^{43} +67.5663 q^{44} +81.4575 q^{45} +386.729 q^{46} +114.955 q^{47} +48.0000 q^{48} -63.6653 q^{49} +86.1648 q^{50} +204.650 q^{51} +118.786 q^{52} -618.547 q^{53} -54.0000 q^{54} +152.883 q^{55} +133.706 q^{56} +139.027 q^{58} +217.001 q^{59} +108.610 q^{60} -342.828 q^{61} +154.016 q^{62} -150.420 q^{63} +64.0000 q^{64} +268.778 q^{65} -101.350 q^{66} +539.647 q^{67} +272.866 q^{68} -580.094 q^{69} +302.539 q^{70} +557.019 q^{71} -72.0000 q^{72} +590.343 q^{73} +403.857 q^{74} -129.247 q^{75} -282.314 q^{77} -178.179 q^{78} -221.958 q^{79} +144.813 q^{80} +81.0000 q^{81} -386.469 q^{82} -215.074 q^{83} -200.560 q^{84} +617.417 q^{85} +847.538 q^{86} -208.540 q^{87} -135.133 q^{88} +592.110 q^{89} -162.915 q^{90} -496.326 q^{91} -773.458 q^{92} -231.024 q^{93} -229.909 q^{94} -96.0000 q^{96} -101.997 q^{97} +127.331 q^{98} +152.024 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9} + 60 q^{10} - 6 q^{11} + 96 q^{12} - 48 q^{13} + 68 q^{14} - 90 q^{15} + 128 q^{16} - 36 q^{17} - 144 q^{18} - 120 q^{20} - 102 q^{21} + 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} + 216 q^{27} - 136 q^{28} + 380 q^{29} + 180 q^{30} + 48 q^{31} - 256 q^{32} - 18 q^{33} + 72 q^{34} + 762 q^{35} + 288 q^{36} + 168 q^{37} - 144 q^{39} + 240 q^{40} - 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} + 564 q^{46} - 468 q^{47} + 384 q^{48} + 222 q^{49} - 792 q^{50} - 108 q^{51} - 192 q^{52} - 1682 q^{53} - 432 q^{54} - 46 q^{55} + 272 q^{56} - 760 q^{58} - 292 q^{59} - 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} + 1120 q^{65} + 36 q^{66} + 2484 q^{67} - 144 q^{68} - 846 q^{69} - 1524 q^{70} - 1182 q^{71} - 576 q^{72} + 182 q^{73} - 336 q^{74} + 1188 q^{75} - 504 q^{77} + 288 q^{78} + 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} - 408 q^{84} - 2238 q^{85} + 1576 q^{86} + 1140 q^{87} + 48 q^{88} - 1304 q^{89} + 540 q^{90} + 624 q^{91} - 1128 q^{92} + 144 q^{93} + 936 q^{94} - 768 q^{96} + 1248 q^{97} - 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9 + 60 * q^10 - 6 * q^11 + 96 * q^12 - 48 * q^13 + 68 * q^14 - 90 * q^15 + 128 * q^16 - 36 * q^17 - 144 * q^18 - 120 * q^20 - 102 * q^21 + 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 + 216 * q^27 - 136 * q^28 + 380 * q^29 + 180 * q^30 + 48 * q^31 - 256 * q^32 - 18 * q^33 + 72 * q^34 + 762 * q^35 + 288 * q^36 + 168 * q^37 - 144 * q^39 + 240 * q^40 - 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 + 564 * q^46 - 468 * q^47 + 384 * q^48 + 222 * q^49 - 792 * q^50 - 108 * q^51 - 192 * q^52 - 1682 * q^53 - 432 * q^54 - 46 * q^55 + 272 * q^56 - 760 * q^58 - 292 * q^59 - 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 + 1120 * q^65 + 36 * q^66 + 2484 * q^67 - 144 * q^68 - 846 * q^69 - 1524 * q^70 - 1182 * q^71 - 576 * q^72 + 182 * q^73 - 336 * q^74 + 1188 * q^75 - 504 * q^77 + 288 * q^78 + 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 - 408 * q^84 - 2238 * q^85 + 1576 * q^86 + 1140 * q^87 + 48 * q^88 - 1304 * q^89 + 540 * q^90 + 624 * q^91 - 1128 * q^92 + 144 * q^93 + 936 * q^94 - 768 * q^96 + 1248 * q^97 - 444 * q^98 - 54 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	9.05084	0.809531	0.404766	−	0.914420i	$-0.367353\pi$
$5$	9.05084	0.809531	0.404766	+	0.914420i	$0.367353\pi$
$6$	−6.00000	−0.408248
$7$	−16.7133	−0.902434	−0.451217	−	0.892414i	$-0.649010\pi$
$7$	−16.7133	−0.902434	−0.451217	+	0.892414i	$0.649010\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−18.1017	−0.572425
$11$	16.8916	0.463000	0.231500	−	0.972835i	$-0.425637\pi$
$11$	16.8916	0.463000	0.231500	+	0.972835i	$0.425637\pi$
$12$	12.0000	0.288675
$13$	29.6964	0.633562	0.316781	−	0.948499i	$-0.397398\pi$
$13$	29.6964	0.633562	0.316781	+	0.948499i	$0.397398\pi$
$14$	33.4266	0.638117
$15$	27.1525	0.467383
$16$	16.0000	0.250000
$17$	68.2165	0.973232	0.486616	−	0.873616i	$-0.338231\pi$
$17$	68.2165	0.973232	0.486616	+	0.873616i	$0.338231\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	36.2033	0.404766
$21$	−50.1399	−0.521020
$22$	−33.7832	−0.327391
$23$	−193.365	−1.75301	−0.876507	−	0.481390i	$-0.840132\pi$
$23$	−193.365	−1.75301	−0.876507	+	0.481390i	$0.840132\pi$
$24$	−24.0000	−0.204124
$25$	−43.0824	−0.344659
$26$	−59.3929	−0.447996
$27$	27.0000	0.192450
$28$	−66.8532	−0.451217
$29$	−69.5133	−0.445114	−0.222557	−	0.974920i	$-0.571440\pi$
$29$	−69.5133	−0.445114	−0.222557	+	0.974920i	$0.571440\pi$
$30$	−54.3050	−0.330490
$31$	−77.0080	−0.446163	−0.223081	−	0.974800i	$-0.571612\pi$
$31$	−77.0080	−0.446163	−0.223081	+	0.974800i	$0.571612\pi$
$32$	−32.0000	−0.176777
$33$	50.6748	0.267313
$34$	−136.433	−0.688179
$35$	−151.269	−0.730548
$36$	36.0000	0.166667
$37$	−201.929	−0.897213	−0.448606	−	0.893729i	$-0.648080\pi$
$37$	−201.929	−0.897213	−0.448606	+	0.893729i	$0.648080\pi$
$38$	0	0
$39$	89.0893	0.365787
$40$	−72.4067	−0.286213
$41$	193.235	0.736053	0.368026	−	0.929815i	$-0.380034\pi$
$41$	193.235	0.736053	0.368026	+	0.929815i	$0.380034\pi$
$42$	100.280	0.368417
$43$	−423.769	−1.50289	−0.751443	−	0.659798i	$-0.770642\pi$
$43$	−423.769	−1.50289	−0.751443	+	0.659798i	$0.770642\pi$
$44$	67.5663	0.231500
$45$	81.4575	0.269844
$46$	386.729	1.23957
$47$	114.955	0.356763	0.178381	−	0.983961i	$-0.442914\pi$
$47$	114.955	0.356763	0.178381	+	0.983961i	$0.442914\pi$
$48$	48.0000	0.144338
$49$	−63.6653	−0.185613
$50$	86.1648	0.243711
$51$	204.650	0.561896
$52$	118.786	0.316781
$53$	−618.547	−1.60309	−0.801546	−	0.597933i	$-0.795989\pi$
$53$	−618.547	−1.60309	−0.801546	+	0.597933i	$0.795989\pi$
$54$	−54.0000	−0.136083
$55$	152.883	0.374813
$56$	133.706	0.319059
$57$	0	0
$58$	139.027	0.314743
$59$	217.001	0.478832	0.239416	−	0.970917i	$-0.423044\pi$
$59$	217.001	0.478832	0.239416	+	0.970917i	$0.423044\pi$
$60$	108.610	0.233692
$61$	−342.828	−0.719584	−0.359792	−	0.933032i	$-0.617152\pi$
$61$	−342.828	−0.719584	−0.359792	+	0.933032i	$0.617152\pi$
$62$	154.016	0.315485
$63$	−150.420	−0.300811
$64$	64.0000	0.125000
$65$	268.778	0.512889
$66$	−101.350	−0.189019
$67$	539.647	0.984006	0.492003	−	0.870593i	$-0.336265\pi$
$67$	539.647	0.984006	0.492003	+	0.870593i	$0.336265\pi$
$68$	272.866	0.486616
$69$	−580.094	−1.01210
$70$	302.539	0.516576
$71$	557.019	0.931071	0.465535	−	0.885029i	$-0.345862\pi$
$71$	557.019	0.931071	0.465535	+	0.885029i	$0.345862\pi$
$72$	−72.0000	−0.117851
$73$	590.343	0.946499	0.473250	−	0.880928i	$-0.343081\pi$
$73$	590.343	0.946499	0.473250	+	0.880928i	$0.343081\pi$
$74$	403.857	0.634425
$75$	−129.247	−0.198989
$76$	0	0
$77$	−282.314	−0.417827
$78$	−178.179	−0.258651
$79$	−221.958	−0.316105	−0.158052	−	0.987431i	$-0.550521\pi$
$79$	−221.958	−0.316105	−0.158052	+	0.987431i	$0.550521\pi$
$80$	144.813	0.202383
$81$	81.0000	0.111111
$82$	−386.469	−0.520468
$83$	−215.074	−0.284427	−0.142213	−	0.989836i	$-0.545422\pi$
$83$	−215.074	−0.284427	−0.142213	+	0.989836i	$0.545422\pi$
$84$	−200.560	−0.260510
$85$	617.417	0.787861
$86$	847.538	1.06270
$87$	−208.540	−0.256987
$88$	−135.133	−0.163695
$89$	592.110	0.705208	0.352604	−	0.935773i	$-0.385296\pi$
$89$	592.110	0.705208	0.352604	+	0.935773i	$0.385296\pi$
$90$	−162.915	−0.190808
$91$	−496.326	−0.571748
$92$	−773.458	−0.876507
$93$	−231.024	−0.257592
$94$	−229.909	−0.252269
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−101.997	−0.106765	−0.0533825	−	0.998574i	$-0.517000\pi$
$97$	−101.997	−0.106765	−0.0533825	+	0.998574i	$0.517000\pi$
$98$	127.331	0.131248
$99$	152.024	0.154333
$100$	−172.330	−0.172330
$101$	−1699.06	−1.67389	−0.836947	−	0.547284i	$-0.815662\pi$
$101$	−1699.06	−1.67389	−0.836947	+	0.547284i	$0.815662\pi$
$102$	−409.299	−0.397320
$103$	−252.852	−0.241886	−0.120943	−	0.992659i	$-0.538592\pi$
$103$	−252.852	−0.241886	−0.120943	+	0.992659i	$0.538592\pi$
$104$	−237.572	−0.223998
$105$	−453.808	−0.421782
$106$	1237.09	1.13356
$107$	469.089	0.423818	0.211909	−	0.977289i	$-0.432032\pi$
$107$	469.089	0.423818	0.211909	+	0.977289i	$0.432032\pi$
$108$	108.000	0.0962250
$109$	−2263.06	−1.98864	−0.994319	−	0.106441i	$-0.966054\pi$
$109$	−2263.06	−1.98864	−0.994319	+	0.106441i	$0.966054\pi$
$110$	−305.766	−0.265033
$111$	−605.786	−0.518006
$112$	−267.413	−0.225608
$113$	250.012	0.208134	0.104067	−	0.994570i	$-0.466814\pi$
$113$	250.012	0.208134	0.104067	+	0.994570i	$0.466814\pi$
$114$	0	0
$115$	−1750.11	−1.41912
$116$	−278.053	−0.222557
$117$	267.268	0.211187
$118$	−434.001	−0.338585
$119$	−1140.12	−0.878277
$120$	−217.220	−0.165245
$121$	−1045.67	−0.785631
$122$	685.656	0.508823
$123$	579.704	0.424960
$124$	−308.032	−0.223081
$125$	−1521.29	−1.08854
$126$	300.840	0.212706
$127$	1771.80	1.23797	0.618985	−	0.785403i	$-0.287544\pi$
$127$	1771.80	1.23797	0.618985	+	0.785403i	$0.287544\pi$
$128$	−128.000	−0.0883883
$129$	−1271.31	−0.867692
$130$	−537.555	−0.362667
$131$	973.066	0.648986	0.324493	−	0.945888i	$-0.394806\pi$
$131$	973.066	0.648986	0.324493	+	0.945888i	$0.394806\pi$
$132$	202.699	0.133657
$133$	0	0
$134$	−1079.29	−0.695798
$135$	244.373	0.155794
$136$	−545.732	−0.344089
$137$	1129.41	0.704321	0.352160	−	0.935940i	$-0.385447\pi$
$137$	1129.41	0.704321	0.352160	+	0.935940i	$0.385447\pi$
$138$	1160.19	0.715665
$139$	−710.495	−0.433550	−0.216775	−	0.976222i	$-0.569554\pi$
$139$	−710.495	−0.433550	−0.216775	+	0.976222i	$0.569554\pi$
$140$	−605.078	−0.365274
$141$	344.864	0.205977
$142$	−1114.04	−0.658367
$143$	501.620	0.293340
$144$	144.000	0.0833333
$145$	−629.154	−0.360334
$146$	−1180.69	−0.669276
$147$	−190.996	−0.107164
$148$	−807.715	−0.448606
$149$	−2932.35	−1.61226	−0.806131	−	0.591737i	$-0.798443\pi$
$149$	−2932.35	−1.61226	−0.806131	+	0.591737i	$0.798443\pi$
$150$	258.494	0.140706
$151$	−698.600	−0.376498	−0.188249	−	0.982121i	$-0.560281\pi$
$151$	−698.600	−0.376498	−0.188249	+	0.982121i	$0.560281\pi$
$152$	0	0
$153$	613.949	0.324411
$154$	564.629	0.295448
$155$	−696.986	−0.361183
$156$	356.357	0.182894
$157$	−1275.79	−0.648531	−0.324265	−	0.945966i	$-0.605117\pi$
$157$	−1275.79	−0.648531	−0.324265	+	0.945966i	$0.605117\pi$
$158$	443.917	0.223520
$159$	−1855.64	−0.925546
$160$	−289.627	−0.143106
$161$	3231.76	1.58198
$162$	−162.000	−0.0785674
$163$	1574.05	0.756373	0.378186	−	0.925729i	$-0.376548\pi$
$163$	1574.05	0.756373	0.378186	+	0.925729i	$0.376548\pi$
$164$	772.938	0.368026
$165$	458.649	0.216399
$166$	430.148	0.201120
$167$	3290.46	1.52469	0.762345	−	0.647171i	$-0.224048\pi$
$167$	3290.46	1.52469	0.762345	+	0.647171i	$0.224048\pi$
$168$	401.119	0.184209
$169$	−1315.12	−0.598599
$170$	−1234.83	−0.557102
$171$	0	0
$172$	−1695.08	−0.751443
$173$	−1310.58	−0.575961	−0.287981	−	0.957636i	$-0.592984\pi$
$173$	−1310.58	−0.575961	−0.287981	+	0.957636i	$0.592984\pi$
$174$	417.080	0.181717
$175$	720.049	0.311032
$176$	270.265	0.115750
$177$	651.002	0.276454
$178$	−1184.22	−0.498657
$179$	1300.28	0.542947	0.271473	−	0.962446i	$-0.412489\pi$
$179$	1300.28	0.542947	0.271473	+	0.962446i	$0.412489\pi$
$180$	325.830	0.134922
$181$	1279.12	0.525285	0.262642	−	0.964893i	$-0.415406\pi$
$181$	1279.12	0.525285	0.262642	+	0.964893i	$0.415406\pi$
$182$	992.652	0.404287
$183$	−1028.48	−0.415452
$184$	1546.92	0.619784
$185$	−1827.62	−0.726322
$186$	462.048	0.182145
$187$	1152.29	0.450607
$188$	459.818	0.178381
$189$	−451.259	−0.173673
$190$	0	0
$191$	−2466.67	−0.934461	−0.467231	−	0.884135i	$-0.654748\pi$
$191$	−2466.67	−0.934461	−0.467231	+	0.884135i	$0.654748\pi$
$192$	192.000	0.0721688
$193$	−1734.04	−0.646730	−0.323365	−	0.946274i	$-0.604814\pi$
$193$	−1734.04	−0.646730	−0.323365	+	0.946274i	$0.604814\pi$
$194$	203.994	0.0754942
$195$	806.333	0.296116
$196$	−254.661	−0.0928066
$197$	−4389.16	−1.58739	−0.793693	−	0.608319i	$-0.791844\pi$
$197$	−4389.16	−1.58739	−0.793693	+	0.608319i	$0.791844\pi$
$198$	−304.049	−0.109130
$199$	5269.56	1.87713	0.938565	−	0.345103i	$-0.112156\pi$
$199$	5269.56	1.87713	0.938565	+	0.345103i	$0.112156\pi$
$200$	344.659	0.121855
$201$	1618.94	0.568116
$202$	3398.13	1.18362
$203$	1161.80	0.401686
$204$	818.598	0.280948
$205$	1748.93	0.595858
$206$	505.704	0.171039
$207$	−1740.28	−0.584338
$208$	475.143	0.158391
$209$	0	0
$210$	907.616	0.298245
$211$	−4300.05	−1.40298	−0.701488	−	0.712682i	$-0.747480\pi$
$211$	−4300.05	−1.40298	−0.701488	+	0.712682i	$0.747480\pi$
$212$	−2474.19	−0.801546
$213$	1671.06	0.537554
$214$	−938.178	−0.299685
$215$	−3835.46	−1.21663
$216$	−216.000	−0.0680414
$217$	1287.06	0.402632
$218$	4526.11	1.40618
$219$	1771.03	0.546462
$220$	611.532	0.187407
$221$	2025.79	0.616603
$222$	1211.57	0.366286
$223$	6063.30	1.82076	0.910378	−	0.413778i	$-0.135791\pi$
$223$	6063.30	1.82076	0.910378	+	0.413778i	$0.135791\pi$
$224$	534.826	0.159529
$225$	−387.741	−0.114886
$226$	−500.023	−0.147173
$227$	1159.73	0.339092	0.169546	−	0.985522i	$-0.445770\pi$
$227$	1159.73	0.339092	0.169546	+	0.985522i	$0.445770\pi$
$228$	0	0
$229$	−1936.25	−0.558738	−0.279369	−	0.960184i	$-0.590125\pi$
$229$	−1936.25	−0.558738	−0.279369	+	0.960184i	$0.590125\pi$
$230$	3500.22	1.00347
$231$	−846.943	−0.241233
$232$	556.107	0.157371
$233$	3360.75	0.944936	0.472468	−	0.881348i	$-0.343363\pi$
$233$	3360.75	0.944936	0.472468	+	0.881348i	$0.343363\pi$
$234$	−534.536	−0.149332
$235$	1040.43	0.288811
$236$	868.002	0.239416
$237$	−665.875	−0.182503
$238$	2280.25	0.621036
$239$	−5863.02	−1.58681	−0.793405	−	0.608694i	$-0.791694\pi$
$239$	−5863.02	−1.58681	−0.793405	+	0.608694i	$0.791694\pi$
$240$	434.440	0.116846
$241$	5964.25	1.59415	0.797077	−	0.603878i	$-0.206379\pi$
$241$	5964.25	1.59415	0.797077	+	0.603878i	$0.206379\pi$
$242$	2091.35	0.555525
$243$	243.000	0.0641500
$244$	−1371.31	−0.359792
$245$	−576.224	−0.150260
$246$	−1159.41	−0.300492
$247$	0	0
$248$	616.064	0.157742
$249$	−645.222	−0.164214
$250$	3042.57	0.769717
$251$	−5665.40	−1.42469	−0.712344	−	0.701830i	$-0.752367\pi$
$251$	−5665.40	−1.42469	−0.712344	+	0.701830i	$0.752367\pi$
$252$	−601.679	−0.150406
$253$	−3266.23	−0.811646
$254$	−3543.61	−0.875377
$255$	1852.25	0.454872
$256$	256.000	0.0625000
$257$	24.7316	0.00600278	0.00300139	−	0.999995i	$-0.499045\pi$
$257$	24.7316	0.00600278	0.00300139	+	0.999995i	$0.499045\pi$
$258$	2542.61	0.613551
$259$	3374.90	0.809675
$260$	1075.11	0.256444
$261$	−625.620	−0.148371
$262$	−1946.13	−0.458902
$263$	−2951.24	−0.691944	−0.345972	−	0.938245i	$-0.612451\pi$
$263$	−2951.24	−0.691944	−0.345972	+	0.938245i	$0.612451\pi$
$264$	−405.398	−0.0945096
$265$	−5598.36	−1.29775
$266$	0	0
$267$	1776.33	0.407152
$268$	2158.59	0.492003
$269$	−4589.02	−1.04014	−0.520069	−	0.854124i	$-0.674094\pi$
$269$	−4589.02	−1.04014	−0.520069	+	0.854124i	$0.674094\pi$
$270$	−488.745	−0.110163
$271$	6457.39	1.44745	0.723724	−	0.690090i	$-0.242429\pi$
$271$	6457.39	1.44745	0.723724	+	0.690090i	$0.242429\pi$
$272$	1091.46	0.243308
$273$	−1488.98	−0.330099
$274$	−2258.82	−0.498030
$275$	−727.730	−0.159577
$276$	−2320.38	−0.506051
$277$	1023.77	0.222065	0.111033	−	0.993817i	$-0.464584\pi$
$277$	1023.77	0.222065	0.111033	+	0.993817i	$0.464584\pi$
$278$	1420.99	0.306566
$279$	−693.072	−0.148721
$280$	1210.16	0.258288
$281$	−5839.29	−1.23965	−0.619827	−	0.784739i	$-0.712797\pi$
$281$	−5839.29	−1.23965	−0.619827	+	0.784739i	$0.712797\pi$
$282$	−689.727	−0.145648
$283$	−7670.54	−1.61119	−0.805594	−	0.592468i	$-0.798154\pi$
$283$	−7670.54	−1.61119	−0.805594	+	0.592468i	$0.798154\pi$
$284$	2228.08	0.465535
$285$	0	0
$286$	−1003.24	−0.207422
$287$	−3229.59	−0.664239
$288$	−288.000	−0.0589256
$289$	−259.506	−0.0528202
$290$	1258.31	0.254794
$291$	−305.990	−0.0616408
$292$	2361.37	0.473250
$293$	1099.39	0.219205	0.109602	−	0.993976i	$-0.465042\pi$
$293$	1099.39	0.219205	0.109602	+	0.993976i	$0.465042\pi$
$294$	381.992	0.0757763
$295$	1964.04	0.387629
$296$	1615.43	0.317213
$297$	456.073	0.0891045
$298$	5864.69	1.14004
$299$	−5742.24	−1.11064
$300$	−516.989	−0.0994945
$301$	7082.58	1.35626
$302$	1397.20	0.266225
$303$	−5097.19	−0.966423
$304$	0	0
$305$	−3102.88	−0.582526
$306$	−1227.90	−0.229393
$307$	−8098.83	−1.50562	−0.752809	−	0.658239i	$-0.771302\pi$
$307$	−8098.83	−1.50562	−0.752809	+	0.658239i	$0.771302\pi$
$308$	−1129.26	−0.208914
$309$	−758.556	−0.139653
$310$	1393.97	0.255395
$311$	−1610.96	−0.293728	−0.146864	−	0.989157i	$-0.546918\pi$
$311$	−1610.96	−0.293728	−0.146864	+	0.989157i	$0.546918\pi$
$312$	−712.715	−0.129325
$313$	2822.19	0.509647	0.254823	−	0.966988i	$-0.417983\pi$
$313$	2822.19	0.509647	0.254823	+	0.966988i	$0.417983\pi$
$314$	2551.58	0.458580
$315$	−1361.42	−0.243516
$316$	−887.833	−0.158052
$317$	916.487	0.162382	0.0811909	−	0.996699i	$-0.474128\pi$
$317$	916.487	0.162382	0.0811909	+	0.996699i	$0.474128\pi$
$318$	3711.28	0.654460
$319$	−1174.19	−0.206088
$320$	579.253	0.101191
$321$	1407.27	0.244692
$322$	−6463.52	−1.11863
$323$	0	0
$324$	324.000	0.0555556
$325$	−1279.39	−0.218363
$326$	−3148.09	−0.534836
$327$	−6789.17	−1.14814
$328$	−1545.88	−0.260234
$329$	−1921.27	−0.321955
$330$	−917.298	−0.153017
$331$	1218.53	0.202346	0.101173	−	0.994869i	$-0.467740\pi$
$331$	1218.53	0.202346	0.101173	+	0.994869i	$0.467740\pi$
$332$	−860.296	−0.142213
$333$	−1817.36	−0.299071
$334$	−6580.92	−1.07812
$335$	4884.26	0.796584
$336$	−802.239	−0.130255
$337$	4739.60	0.766119	0.383060	−	0.923724i	$-0.374870\pi$
$337$	4739.60	0.766119	0.383060	+	0.923724i	$0.374870\pi$
$338$	2630.24	0.423273
$339$	750.035	0.120166
$340$	2469.67	0.393931
$341$	−1300.79	−0.206573
$342$	0	0
$343$	6796.72	1.06994
$344$	3390.15	0.531351
$345$	−5250.33	−0.819329
$346$	2621.15	0.407266
$347$	11736.1	1.81564	0.907819	−	0.419362i	$-0.137746\pi$
$347$	11736.1	1.81564	0.907819	+	0.419362i	$0.137746\pi$
$348$	−834.160	−0.128493
$349$	−11749.1	−1.80204	−0.901022	−	0.433774i	$-0.857182\pi$
$349$	−11749.1	−1.80204	−0.901022	+	0.433774i	$0.857182\pi$
$350$	−1440.10	−0.219933
$351$	801.804	0.121929
$352$	−540.531	−0.0818477
$353$	−4186.40	−0.631217	−0.315608	−	0.948890i	$-0.602209\pi$
$353$	−4186.40	−0.631217	−0.315608	+	0.948890i	$0.602209\pi$
$354$	−1302.00	−0.195482
$355$	5041.49	0.753731
$356$	2368.44	0.352604
$357$	−3420.37	−0.507074
$358$	−2600.56	−0.383921
$359$	−1074.75	−0.158003	−0.0790017	−	0.996874i	$-0.525173\pi$
$359$	−1074.75	−0.158003	−0.0790017	+	0.996874i	$0.525173\pi$
$360$	−651.660	−0.0954042
$361$	0	0
$362$	−2558.25	−0.371432
$363$	−3137.02	−0.453584
$364$	−1985.30	−0.285874
$365$	5343.10	0.766221
$366$	2056.97	0.293769
$367$	−1441.55	−0.205036	−0.102518	−	0.994731i	$-0.532690\pi$
$367$	−1441.55	−0.205036	−0.102518	+	0.994731i	$0.532690\pi$
$368$	−3093.83	−0.438253
$369$	1739.11	0.245351
$370$	3655.25	0.513587
$371$	10338.0	1.44669
$372$	−924.096	−0.128796
$373$	−6996.28	−0.971189	−0.485595	−	0.874184i	$-0.661397\pi$
$373$	−6996.28	−0.971189	−0.485595	+	0.874184i	$0.661397\pi$
$374$	−2304.57	−0.318627
$375$	−4563.86	−0.628471
$376$	−919.636	−0.126135
$377$	−2064.30	−0.282007
$378$	902.519	0.122806
$379$	13643.3	1.84910	0.924550	−	0.381061i	$-0.124441\pi$
$379$	13643.3	1.84910	0.924550	+	0.381061i	$0.124441\pi$
$380$	0	0
$381$	5315.41	0.714742
$382$	4933.34	0.660764
$383$	−12008.8	−1.60215	−0.801074	−	0.598565i	$-0.795738\pi$
$383$	−12008.8	−1.60215	−0.801074	+	0.598565i	$0.795738\pi$
$384$	−384.000	−0.0510310
$385$	−2555.18	−0.338244
$386$	3468.08	0.457307
$387$	−3813.92	−0.500962
$388$	−407.987	−0.0533825
$389$	−11418.3	−1.48825	−0.744125	−	0.668041i	$-0.767133\pi$
$389$	−11418.3	−1.48825	−0.744125	+	0.668041i	$0.767133\pi$
$390$	−1612.67	−0.209386
$391$	−13190.7	−1.70609
$392$	509.323	0.0656242
$393$	2919.20	0.374692
$394$	8778.33	1.12245
$395$	−2008.91	−0.255897
$396$	608.097	0.0771667
$397$	−11135.2	−1.40771	−0.703855	−	0.710344i	$-0.748539\pi$
$397$	−11135.2	−1.40771	−0.703855	+	0.710344i	$0.748539\pi$
$398$	−10539.1	−1.32733
$399$	0	0
$400$	−689.318	−0.0861648
$401$	−5641.65	−0.702570	−0.351285	−	0.936269i	$-0.614255\pi$
$401$	−5641.65	−0.702570	−0.351285	+	0.936269i	$0.614255\pi$
$402$	−3237.88	−0.401719
$403$	−2286.86	−0.282672
$404$	−6796.26	−0.836947
$405$	733.118	0.0899479
$406$	−2323.59	−0.284035
$407$	−3410.90	−0.415410
$408$	−1637.20	−0.198660
$409$	9406.69	1.13724	0.568620	−	0.822601i	$-0.307478\pi$
$409$	9406.69	1.13724	0.568620	+	0.822601i	$0.307478\pi$
$410$	−3497.87	−0.421335
$411$	3388.23	0.406640
$412$	−1011.41	−0.120943
$413$	−3626.80	−0.432114
$414$	3480.56	0.413189
$415$	−1946.60	−0.230253
$416$	−950.286	−0.111999
$417$	−2131.49	−0.250310
$418$	0	0
$419$	−14623.8	−1.70505	−0.852527	−	0.522684i	$-0.824931\pi$
$419$	−14623.8	−1.70505	−0.852527	+	0.522684i	$0.824931\pi$
$420$	−1815.23	−0.210891
$421$	2856.94	0.330733	0.165367	−	0.986232i	$-0.447119\pi$
$421$	2856.94	0.330733	0.165367	+	0.986232i	$0.447119\pi$
$422$	8600.10	0.992053
$423$	1034.59	0.118921
$424$	4948.37	0.566779
$425$	−2938.93	−0.335433
$426$	−3342.12	−0.380108
$427$	5729.79	0.649377
$428$	1876.36	0.211909
$429$	1504.86	0.169360
$430$	7670.92	0.860290
$431$	6888.19	0.769820	0.384910	−	0.922954i	$-0.374233\pi$
$431$	6888.19	0.769820	0.384910	+	0.922954i	$0.374233\pi$
$432$	432.000	0.0481125
$433$	−13008.4	−1.44375	−0.721874	−	0.692025i	$-0.756719\pi$
$433$	−13008.4	−1.44375	−0.721874	+	0.692025i	$0.756719\pi$
$434$	−2574.12	−0.284704
$435$	−1887.46	−0.208039
$436$	−9052.23	−0.994319
$437$	0	0
$438$	−3542.06	−0.386407
$439$	16474.9	1.79112	0.895561	−	0.444939i	$-0.146775\pi$
$439$	16474.9	1.79112	0.895561	+	0.444939i	$0.146775\pi$
$440$	−1223.06	−0.132517
$441$	−572.988	−0.0618711
$442$	−4051.58	−0.436004
$443$	329.183	0.0353046	0.0176523	−	0.999844i	$-0.494381\pi$
$443$	329.183	0.0353046	0.0176523	+	0.999844i	$0.494381\pi$
$444$	−2423.14	−0.259003
$445$	5359.09	0.570888
$446$	−12126.6	−1.28747
$447$	−8797.04	−0.930840
$448$	−1069.65	−0.112804
$449$	4410.36	0.463558	0.231779	−	0.972768i	$-0.425545\pi$
$449$	4410.36	0.463558	0.231779	+	0.972768i	$0.425545\pi$
$450$	775.483	0.0812369
$451$	3264.04	0.340793
$452$	1000.05	0.104067
$453$	−2095.80	−0.217371
$454$	−2319.45	−0.239774
$455$	−4492.16	−0.462848
$456$	0	0
$457$	−15212.6	−1.55715	−0.778573	−	0.627554i	$-0.784056\pi$
$457$	−15212.6	−1.55715	−0.778573	+	0.627554i	$0.784056\pi$
$458$	3872.50	0.395087
$459$	1841.85	0.187299
$460$	−7000.45	−0.709560
$461$	−7446.99	−0.752366	−0.376183	−	0.926545i	$-0.622764\pi$
$461$	−7446.99	−0.752366	−0.376183	+	0.926545i	$0.622764\pi$
$462$	1693.89	0.170577
$463$	12336.0	1.23823	0.619115	−	0.785300i	$-0.287491\pi$
$463$	12336.0	1.23823	0.619115	+	0.785300i	$0.287491\pi$
$464$	−1112.21	−0.111278
$465$	−2090.96	−0.208529
$466$	−6721.50	−0.668170
$467$	−9050.18	−0.896772	−0.448386	−	0.893840i	$-0.648001\pi$
$467$	−9050.18	−0.896772	−0.448386	+	0.893840i	$0.648001\pi$
$468$	1069.07	0.105594
$469$	−9019.29	−0.888001
$470$	−2080.87	−0.204220
$471$	−3827.38	−0.374429
$472$	−1736.00	−0.169293
$473$	−7158.13	−0.695837
$474$	1331.75	0.129049
$475$	0	0
$476$	−4560.50	−0.439139
$477$	−5566.92	−0.534364
$478$	11726.0	1.12204
$479$	6510.19	0.620998	0.310499	−	0.950574i	$-0.399504\pi$
$479$	6510.19	0.620998	0.310499	+	0.950574i	$0.399504\pi$
$480$	−868.880	−0.0826224
$481$	−5996.56	−0.568440
$482$	−11928.5	−1.12724
$483$	9695.29	0.913356
$484$	−4182.70	−0.392815
$485$	−923.156	−0.0864296
$486$	−486.000	−0.0453609
$487$	3835.40	0.356876	0.178438	−	0.983951i	$-0.442896\pi$
$487$	3835.40	0.356876	0.178438	+	0.983951i	$0.442896\pi$
$488$	2742.63	0.254412
$489$	4722.14	0.436692
$490$	1152.45	0.106250
$491$	−8445.05	−0.776212	−0.388106	−	0.921615i	$-0.626870\pi$
$491$	−8445.05	−0.776212	−0.388106	+	0.921615i	$0.626870\pi$
$492$	2318.81	0.212480
$493$	−4741.96	−0.433199
$494$	0	0
$495$	1375.95	0.124938
$496$	−1232.13	−0.111541
$497$	−9309.64	−0.840230
$498$	1290.44	0.116117
$499$	10634.7	0.954054	0.477027	−	0.878889i	$-0.341714\pi$
$499$	10634.7	0.954054	0.477027	+	0.878889i	$0.341714\pi$
$500$	−6085.14	−0.544272
$501$	9871.37	0.880280
$502$	11330.8	1.00741
$503$	−7549.24	−0.669193	−0.334596	−	0.942361i	$-0.608600\pi$
$503$	−7549.24	−0.669193	−0.334596	+	0.942361i	$0.608600\pi$
$504$	1203.36	0.106353
$505$	−15378.0	−1.35507
$506$	6532.47	0.573920
$507$	−3945.36	−0.345601
$508$	7087.22	0.618985
$509$	−11386.1	−0.991514	−0.495757	−	0.868461i	$-0.665109\pi$
$509$	−11386.1	−0.991514	−0.495757	+	0.868461i	$0.665109\pi$
$510$	−3704.50	−0.321643
$511$	−9866.59	−0.854153
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−49.4632	−0.00424460
$515$	−2288.52	−0.195814
$516$	−5085.23	−0.433846
$517$	1941.76	0.165181
$518$	−6749.79	−0.572527
$519$	−3931.73	−0.332531
$520$	−2150.22	−0.181334
$521$	−19651.7	−1.65251	−0.826254	−	0.563297i	$-0.809533\pi$
$521$	−19651.7	−1.65251	−0.826254	+	0.563297i	$0.809533\pi$
$522$	1251.24	0.104914
$523$	−2096.20	−0.175259	−0.0876294	−	0.996153i	$-0.527929\pi$
$523$	−2096.20	−0.175259	−0.0876294	+	0.996153i	$0.527929\pi$
$524$	3892.26	0.324493
$525$	2160.15	0.179574
$526$	5902.48	0.489279
$527$	−5253.22	−0.434219
$528$	810.796	0.0668284
$529$	25222.9	2.07306
$530$	11196.7	0.917651
$531$	1953.00	0.159611
$532$	0	0
$533$	5738.38	0.466335
$534$	−3552.66	−0.287900
$535$	4245.65	0.343094
$536$	−4317.18	−0.347899
$537$	3900.84	0.313470
$538$	9178.04	0.735489
$539$	−1075.41	−0.0859390
$540$	977.490	0.0778972
$541$	19853.0	1.57772	0.788859	−	0.614574i	$-0.210672\pi$
$541$	19853.0	1.57772	0.788859	+	0.614574i	$0.210672\pi$
$542$	−12914.8	−1.02350
$543$	3837.37	0.303273
$544$	−2182.93	−0.172045
$545$	−20482.6	−1.60986
$546$	2977.95	0.233415
$547$	−9480.46	−0.741052	−0.370526	−	0.928822i	$-0.620823\pi$
$547$	−9480.46	−0.741052	−0.370526	+	0.928822i	$0.620823\pi$
$548$	4517.64	0.352160
$549$	−3085.45	−0.239861
$550$	1455.46	0.112838
$551$	0	0
$552$	4640.75	0.357832
$553$	3709.66	0.285264
$554$	−2047.53	−0.157024
$555$	−5482.87	−0.419342
$556$	−2841.98	−0.216775
$557$	4181.28	0.318073	0.159036	−	0.987273i	$-0.449161\pi$
$557$	4181.28	0.318073	0.159036	+	0.987273i	$0.449161\pi$
$558$	1386.14	0.105162
$559$	−12584.4	−0.952173
$560$	−2420.31	−0.182637
$561$	3456.86	0.260158
$562$	11678.6	0.876567
$563$	4466.37	0.334343	0.167171	−	0.985928i	$-0.446537\pi$
$563$	4466.37	0.334343	0.167171	+	0.985928i	$0.446537\pi$
$564$	1379.45	0.102989
$565$	2262.81	0.168491
$566$	15341.1	1.13928
$567$	−1353.78	−0.100270
$568$	−4456.15	−0.329183
$569$	3484.56	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$569$	3484.56	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$570$	0	0
$571$	−11795.4	−0.864490	−0.432245	−	0.901756i	$-0.642278\pi$
$571$	−11795.4	−0.864490	−0.432245	+	0.901756i	$0.642278\pi$
$572$	2006.48	0.146670
$573$	−7400.02	−0.539512
$574$	6459.18	0.469688
$575$	8330.61	0.604192
$576$	576.000	0.0416667
$577$	13009.9	0.938667	0.469333	−	0.883021i	$-0.344494\pi$
$577$	13009.9	0.938667	0.469333	+	0.883021i	$0.344494\pi$
$578$	519.012	0.0373495
$579$	−5202.12	−0.373390
$580$	−2516.61	−0.180167
$581$	3594.60	0.256677
$582$	611.981	0.0435866
$583$	−10448.2	−0.742233
$584$	−4722.75	−0.334638
$585$	2419.00	0.170963
$586$	−2198.78	−0.155001
$587$	−11239.4	−0.790292	−0.395146	−	0.918618i	$-0.629306\pi$
$587$	−11239.4	−0.790292	−0.395146	+	0.918618i	$0.629306\pi$
$588$	−763.984	−0.0535819
$589$	0	0
$590$	−3928.07	−0.274095
$591$	−13167.5	−0.916478
$592$	−3230.86	−0.224303
$593$	18382.0	1.27295	0.636474	−	0.771298i	$-0.280392\pi$
$593$	18382.0	1.27295	0.636474	+	0.771298i	$0.280392\pi$
$594$	−912.146	−0.0630064
$595$	−10319.1	−0.710993
$596$	−11729.4	−0.806131
$597$	15808.7	1.08376
$598$	11484.5	0.785344
$599$	−15005.3	−1.02354	−0.511771	−	0.859122i	$-0.671010\pi$
$599$	−15005.3	−1.02354	−0.511771	+	0.859122i	$0.671010\pi$
$600$	1033.98	0.0703532
$601$	−3839.83	−0.260616	−0.130308	−	0.991474i	$-0.541597\pi$
$601$	−3839.83	−0.260616	−0.130308	+	0.991474i	$0.541597\pi$
$602$	−14165.2	−0.959018
$603$	4856.83	0.328002
$604$	−2794.40	−0.188249
$605$	−9464.23	−0.635993
$606$	10194.4	0.683364
$607$	−157.835	−0.0105541	−0.00527704	−	0.999986i	$-0.501680\pi$
$607$	−157.835	−0.0105541	−0.00527704	+	0.999986i	$0.501680\pi$
$608$	0	0
$609$	3485.39	0.231913
$610$	6205.76	0.411908
$611$	3413.74	0.226031
$612$	2455.79	0.162205
$613$	17082.1	1.12551	0.562757	−	0.826622i	$-0.309741\pi$
$613$	17082.1	1.12551	0.562757	+	0.826622i	$0.309741\pi$
$614$	16197.7	1.06463
$615$	5246.80	0.344019
$616$	2258.51	0.147724
$617$	25072.0	1.63591	0.817957	−	0.575279i	$-0.195107\pi$
$617$	25072.0	1.63591	0.817957	+	0.575279i	$0.195107\pi$
$618$	1517.11	0.0987495
$619$	11370.7	0.738330	0.369165	−	0.929364i	$-0.379644\pi$
$619$	11370.7	0.738330	0.369165	+	0.929364i	$0.379644\pi$
$620$	−2787.95	−0.180591
$621$	−5220.84	−0.337368
$622$	3221.93	0.207697
$623$	−9896.11	−0.636403
$624$	1425.43	0.0914469
$625$	−8383.61	−0.536551
$626$	−5644.38	−0.360375
$627$	0	0
$628$	−5103.17	−0.324265
$629$	−13774.9	−0.873196
$630$	2722.85	0.172192
$631$	−23451.5	−1.47954	−0.739771	−	0.672859i	$-0.765066\pi$
$631$	−23451.5	−1.47954	−0.739771	+	0.672859i	$0.765066\pi$
$632$	1775.67	0.111760
$633$	−12900.2	−0.810008
$634$	−1832.97	−0.114821
$635$	16036.3	1.00218
$636$	−7422.56	−0.462773
$637$	−1890.63	−0.117598
$638$	2348.38	0.145726
$639$	5013.17	0.310357
$640$	−1158.51	−0.0715531
$641$	−29712.8	−1.83086	−0.915432	−	0.402473i	$-0.868151\pi$
$641$	−29712.8	−1.83086	−0.915432	+	0.402473i	$0.868151\pi$
$642$	−2814.54	−0.173023
$643$	8038.46	0.493011	0.246505	−	0.969141i	$-0.420718\pi$
$643$	8038.46	0.493011	0.246505	+	0.969141i	$0.420718\pi$
$644$	12927.0	0.790989
$645$	−11506.4	−0.702424
$646$	0	0
$647$	13295.8	0.807902	0.403951	−	0.914781i	$-0.367637\pi$
$647$	13295.8	0.807902	0.403951	+	0.914781i	$0.367637\pi$
$648$	−648.000	−0.0392837
$649$	3665.48	0.221699
$650$	2558.79	0.154406
$651$	3861.17	0.232460
$652$	6296.18	0.378186
$653$	9048.52	0.542260	0.271130	−	0.962543i	$-0.412603\pi$
$653$	9048.52	0.542260	0.271130	+	0.962543i	$0.412603\pi$
$654$	13578.3	0.811858
$655$	8807.06	0.525374
$656$	3091.75	0.184013
$657$	5313.09	0.315500
$658$	3842.54	0.227656
$659$	8447.02	0.499316	0.249658	−	0.968334i	$-0.419682\pi$
$659$	8447.02	0.499316	0.249658	+	0.968334i	$0.419682\pi$
$660$	1834.60	0.108199
$661$	−2352.29	−0.138417	−0.0692084	−	0.997602i	$-0.522047\pi$
$661$	−2352.29	−0.138417	−0.0692084	+	0.997602i	$0.522047\pi$
$662$	−2437.06	−0.143080
$663$	6077.36	0.355996
$664$	1720.59	0.100560
$665$	0	0
$666$	3634.72	0.211475
$667$	13441.4	0.780290
$668$	13161.8	0.762345
$669$	18189.9	1.05121
$670$	−9768.52	−0.563270
$671$	−5790.91	−0.333168
$672$	1604.48	0.0921043
$673$	730.456	0.0418381	0.0209190	−	0.999781i	$-0.493341\pi$
$673$	730.456	0.0418381	0.0209190	+	0.999781i	$0.493341\pi$
$674$	−9479.19	−0.541728
$675$	−1163.22	−0.0663297
$676$	−5260.48	−0.299299
$677$	−29625.1	−1.68181	−0.840905	−	0.541183i	$-0.817977\pi$
$677$	−29625.1	−1.68181	−0.840905	+	0.541183i	$0.817977\pi$
$678$	−1500.07	−0.0849703
$679$	1704.70	0.0963483
$680$	−4939.33	−0.278551
$681$	3479.18	0.195775
$682$	2601.57	0.146069
$683$	1105.63	0.0619408	0.0309704	−	0.999520i	$-0.490140\pi$
$683$	1105.63	0.0619408	0.0309704	+	0.999520i	$0.490140\pi$
$684$	0	0
$685$	10222.1	0.570170
$686$	−13593.4	−0.756560
$687$	−5808.75	−0.322587
$688$	−6780.30	−0.375722
$689$	−18368.6	−1.01566
$690$	10500.7	0.579353
$691$	20737.1	1.14165	0.570824	−	0.821073i	$-0.306624\pi$
$691$	20737.1	1.14165	0.570824	+	0.821073i	$0.306624\pi$
$692$	−5242.31	−0.287981
$693$	−2540.83	−0.139276
$694$	−23472.2	−1.28385
$695$	−6430.58	−0.350972
$696$	1668.32	0.0908585
$697$	13181.8	0.716350
$698$	23498.1	1.27424
$699$	10082.2	0.545559
$700$	2880.20	0.155516
$701$	11297.2	0.608689	0.304344	−	0.952562i	$-0.401563\pi$
$701$	11297.2	0.608689	0.304344	+	0.952562i	$0.401563\pi$
$702$	−1603.61	−0.0862169
$703$	0	0
$704$	1081.06	0.0578751
$705$	3121.30	0.166745
$706$	8372.80	0.446338
$707$	28397.0	1.51058
$708$	2604.01	0.138227
$709$	−21030.4	−1.11398	−0.556991	−	0.830518i	$-0.688044\pi$
$709$	−21030.4	−1.11398	−0.556991	+	0.830518i	$0.688044\pi$
$710$	−10083.0	−0.532968
$711$	−1997.63	−0.105368
$712$	−4736.88	−0.249329
$713$	14890.6	0.782129
$714$	6840.74	0.358555
$715$	4540.08	0.237468
$716$	5201.12	0.271473
$717$	−17589.1	−0.916145
$718$	2149.50	0.111725
$719$	17759.9	0.921186	0.460593	−	0.887611i	$-0.347637\pi$
$719$	17759.9	0.921186	0.460593	+	0.887611i	$0.347637\pi$
$720$	1303.32	0.0674609
$721$	4225.99	0.218286
$722$	0	0
$723$	17892.7	0.920385
$724$	5116.49	0.262642
$725$	2994.80	0.153412
$726$	6274.05	0.320732
$727$	−13733.2	−0.700602	−0.350301	−	0.936637i	$-0.613921\pi$
$727$	−13733.2	−0.700602	−0.350301	+	0.936637i	$0.613921\pi$
$728$	3970.61	0.202143
$729$	729.000	0.0370370
$730$	−10686.2	−0.541800
$731$	−28908.0	−1.46266
$732$	−4113.94	−0.207726
$733$	−48.6820	−0.00245308	−0.00122654	−	0.999999i	$-0.500390\pi$
$733$	−48.6820	−0.00245308	−0.00122654	+	0.999999i	$0.500390\pi$
$734$	2883.09	0.144982
$735$	−1728.67	−0.0867525
$736$	6187.67	0.309892
$737$	9115.50	0.455595
$738$	−3478.22	−0.173489
$739$	9219.16	0.458907	0.229453	−	0.973320i	$-0.426306\pi$
$739$	9219.16	0.458907	0.229453	+	0.973320i	$0.426306\pi$
$740$	−7310.49	−0.363161
$741$	0	0
$742$	−20675.9	−1.02296
$743$	−24973.1	−1.23307	−0.616536	−	0.787327i	$-0.711465\pi$
$743$	−24973.1	−1.23307	−0.616536	+	0.787327i	$0.711465\pi$
$744$	1848.19	0.0910725
$745$	−26540.2	−1.30518
$746$	13992.6	0.686735
$747$	−1935.67	−0.0948090
$748$	4609.14	0.225303
$749$	−7840.03	−0.382468
$750$	9127.72	0.444396
$751$	−11716.5	−0.569296	−0.284648	−	0.958632i	$-0.591877\pi$
$751$	−11716.5	−0.569296	−0.284648	+	0.958632i	$0.591877\pi$
$752$	1839.27	0.0891907
$753$	−16996.2	−0.822544
$754$	4128.60	0.199409
$755$	−6322.91	−0.304787
$756$	−1805.04	−0.0868367
$757$	27102.4	1.30126	0.650630	−	0.759395i	$-0.274505\pi$
$757$	27102.4	1.30126	0.650630	+	0.759395i	$0.274505\pi$
$758$	−27286.6	−1.30751
$759$	−9798.70	−0.468604
$760$	0	0
$761$	7731.53	0.368289	0.184144	−	0.982899i	$-0.441049\pi$
$761$	7731.53	0.368289	0.184144	+	0.982899i	$0.441049\pi$
$762$	−10630.8	−0.505399
$763$	37823.2	1.79461
$764$	−9866.69	−0.467231
$765$	5556.75	0.262620
$766$	24017.7	1.13289
$767$	6444.14	0.303370
$768$	768.000	0.0360844
$769$	35693.1	1.67377	0.836883	−	0.547381i	$-0.184375\pi$
$769$	35693.1	1.67377	0.836883	+	0.547381i	$0.184375\pi$
$770$	5110.36	0.239175
$771$	74.1947	0.00346570
$772$	−6936.15	−0.323365
$773$	−31253.9	−1.45424	−0.727119	−	0.686511i	$-0.759141\pi$
$773$	−31253.9	−1.45424	−0.727119	+	0.686511i	$0.759141\pi$
$774$	7627.84	0.354234
$775$	3317.69	0.153774
$776$	815.974	0.0377471
$777$	10124.7	0.467466
$778$	22836.5	1.05235
$779$	0	0
$780$	3225.33	0.148058
$781$	9408.94	0.431086
$782$	26381.3	1.20639
$783$	−1876.86	−0.0856622
$784$	−1018.65	−0.0464033
$785$	−11547.0	−0.525006
$786$	−5838.39	−0.264947
$787$	−32847.0	−1.48776	−0.743881	−	0.668312i	$-0.767017\pi$
$787$	−32847.0	−1.48776	−0.743881	+	0.668312i	$0.767017\pi$
$788$	−17556.7	−0.793693
$789$	−8853.73	−0.399494
$790$	4017.82	0.180946
$791$	−4178.52	−0.187827
$792$	−1216.19	−0.0545651
$793$	−10180.8	−0.455902
$794$	22270.4	0.995401
$795$	−16795.1	−0.749259
$796$	21078.2	0.938565
$797$	33775.6	1.50112	0.750561	−	0.660802i	$-0.229784\pi$
$797$	33775.6	1.50112	0.750561	+	0.660802i	$0.229784\pi$
$798$	0	0
$799$	7841.80	0.347213
$800$	1378.64	0.0609277
$801$	5328.99	0.235069
$802$	11283.3	0.496792
$803$	9971.83	0.438230
$804$	6475.77	0.284058
$805$	29250.2	1.28066
$806$	4573.73	0.199879
$807$	−13767.1	−0.600524
$808$	13592.5	0.591811
$809$	15428.5	0.670502	0.335251	−	0.942129i	$-0.391179\pi$
$809$	15428.5	0.670502	0.335251	+	0.942129i	$0.391179\pi$
$810$	−1466.24	−0.0636028
$811$	14846.9	0.642840	0.321420	−	0.946937i	$-0.395840\pi$
$811$	14846.9	0.642840	0.321420	+	0.946937i	$0.395840\pi$
$812$	4647.19	0.200843
$813$	19372.2	0.835684
$814$	6821.79	0.293739
$815$	14246.4	0.612308
$816$	3274.39	0.140474
$817$	0	0
$818$	−18813.4	−0.804150
$819$	−4466.93	−0.190583
$820$	6995.74	0.297929
$821$	15407.5	0.654962	0.327481	−	0.944858i	$-0.393800\pi$
$821$	15407.5	0.654962	0.327481	+	0.944858i	$0.393800\pi$
$822$	−6776.46	−0.287538
$823$	7459.31	0.315936	0.157968	−	0.987444i	$-0.449506\pi$
$823$	7459.31	0.315936	0.157968	+	0.987444i	$0.449506\pi$
$824$	2022.82	0.0855195
$825$	−2183.19	−0.0921320
$826$	7253.59	0.305551
$827$	44706.3	1.87980	0.939898	−	0.341454i	$-0.110919\pi$
$827$	44706.3	1.87980	0.939898	+	0.341454i	$0.110919\pi$
$828$	−6961.13	−0.292169
$829$	−21693.2	−0.908848	−0.454424	−	0.890786i	$-0.650155\pi$
$829$	−21693.2	−0.908848	−0.454424	+	0.890786i	$0.650155\pi$
$830$	3893.20	0.162813
$831$	3071.30	0.128210
$832$	1900.57	0.0791953
$833$	−4343.03	−0.180645
$834$	4262.97	0.176996
$835$	29781.4	1.23428
$836$	0	0
$837$	−2079.21	−0.0858640
$838$	29247.5	1.20565
$839$	−11183.1	−0.460171	−0.230086	−	0.973170i	$-0.573901\pi$
$839$	−11183.1	−0.460171	−0.230086	+	0.973170i	$0.573901\pi$
$840$	3630.47	0.149123
$841$	−19556.9	−0.801874
$842$	−5713.88	−0.233864
$843$	−17517.9	−0.715714
$844$	−17200.2	−0.701488
$845$	−11902.9	−0.484584
$846$	−2069.18	−0.0840898
$847$	17476.7	0.708980
$848$	−9896.75	−0.400773
$849$	−23011.6	−0.930220
$850$	5877.86	0.237187
$851$	39045.9	1.57283
$852$	6684.23	0.268777
$853$	22510.2	0.903559	0.451780	−	0.892130i	$-0.350789\pi$
$853$	22510.2	0.903559	0.451780	+	0.892130i	$0.350789\pi$
$854$	−11459.6	−0.459179
$855$	0	0
$856$	−3752.71	−0.149842
$857$	−1048.90	−0.0418085	−0.0209043	−	0.999781i	$-0.506655\pi$
$857$	−1048.90	−0.0418085	−0.0209043	+	0.999781i	$0.506655\pi$
$858$	−3009.72	−0.119755
$859$	−31366.2	−1.24587	−0.622935	−	0.782274i	$-0.714060\pi$
$859$	−31366.2	−1.24587	−0.622935	+	0.782274i	$0.714060\pi$
$860$	−15341.8	−0.608317
$861$	−9688.76	−0.383499
$862$	−13776.4	−0.544345
$863$	−5111.22	−0.201608	−0.100804	−	0.994906i	$-0.532142\pi$
$863$	−5111.22	−0.201608	−0.100804	+	0.994906i	$0.532142\pi$
$864$	−864.000	−0.0340207
$865$	−11861.8	−0.466259
$866$	26016.8	1.02088
$867$	−778.517	−0.0304958
$868$	5148.23	0.201316
$869$	−3749.23	−0.146357
$870$	3774.92	0.147106
$871$	16025.6	0.623430
$872$	18104.5	0.703090
$873$	−917.971	−0.0355883
$874$	0	0
$875$	25425.7	0.982339
$876$	7084.12	0.273231
$877$	−6164.24	−0.237345	−0.118672	−	0.992933i	$-0.537864\pi$
$877$	−6164.24	−0.237345	−0.118672	+	0.992933i	$0.537864\pi$
$878$	−32949.7	−1.26651
$879$	3298.17	0.126558
$880$	2446.13	0.0937033
$881$	−1886.25	−0.0721332	−0.0360666	−	0.999349i	$-0.511483\pi$
$881$	−1886.25	−0.0721332	−0.0360666	+	0.999349i	$0.511483\pi$
$882$	1145.98	0.0437495
$883$	−14634.9	−0.557762	−0.278881	−	0.960326i	$-0.589964\pi$
$883$	−14634.9	−0.557762	−0.278881	+	0.960326i	$0.589964\pi$
$884$	8103.15	0.308301
$885$	5892.11	0.223798
$886$	−658.366	−0.0249641
$887$	−11706.7	−0.443147	−0.221573	−	0.975144i	$-0.571119\pi$
$887$	−11706.7	−0.443147	−0.221573	+	0.975144i	$0.571119\pi$
$888$	4846.29	0.183143
$889$	−29612.7	−1.11719
$890$	−10718.2	−0.403679
$891$	1368.22	0.0514445
$892$	24253.2	0.910378
$893$	0	0
$894$	17594.1	0.658204
$895$	11768.6	0.439532
$896$	2139.30	0.0797646
$897$	−17226.7	−0.641230
$898$	−8820.71	−0.327785
$899$	5353.08	0.198593
$900$	−1550.97	−0.0574432
$901$	−42195.1	−1.56018
$902$	−6528.07	−0.240977
$903$	21247.7	0.783035
$904$	−2000.09	−0.0735864
$905$	11577.1	0.425234
$906$	4191.60	0.153705
$907$	−38946.9	−1.42581	−0.712906	−	0.701259i	$-0.752622\pi$
$907$	−38946.9	−1.42581	−0.712906	+	0.701259i	$0.752622\pi$
$908$	4638.91	0.169546
$909$	−15291.6	−0.557965
$910$	8984.33	0.327283
$911$	44693.2	1.62541	0.812707	−	0.582672i	$-0.197993\pi$
$911$	44693.2	1.62541	0.812707	+	0.582672i	$0.197993\pi$
$912$	0	0
$913$	−3632.94	−0.131690
$914$	30425.2	1.10107
$915$	−9308.65	−0.336322
$916$	−7745.00	−0.279369
$917$	−16263.1	−0.585667
$918$	−3683.69	−0.132440
$919$	10299.3	0.369687	0.184843	−	0.982768i	$-0.440822\pi$
$919$	10299.3	0.369687	0.184843	+	0.982768i	$0.440822\pi$
$920$	14000.9	0.501734
$921$	−24296.5	−0.869269
$922$	14894.0	0.532003
$923$	16541.5	0.589892
$924$	−3387.77	−0.120616
$925$	8699.57	0.309233
$926$	−24671.9	−0.875561
$927$	−2275.67	−0.0806286
$928$	2224.43	0.0786857
$929$	25828.5	0.912171	0.456085	−	0.889936i	$-0.349251\pi$
$929$	25828.5	0.912171	0.456085	+	0.889936i	$0.349251\pi$
$930$	4181.92	0.147452
$931$	0	0
$932$	13443.0	0.472468
$933$	−4832.89	−0.169584
$934$	18100.4	0.634113
$935$	10429.1	0.364780
$936$	−2138.14	−0.0746660
$937$	52480.0	1.82972	0.914859	−	0.403773i	$-0.132302\pi$
$937$	52480.0	1.82972	0.914859	+	0.403773i	$0.132302\pi$
$938$	18038.6	0.627911
$939$	8466.56	0.294245
$940$	4161.74	0.144405
$941$	−8313.79	−0.288015	−0.144007	−	0.989577i	$-0.545999\pi$
$941$	−8313.79	−0.288015	−0.144007	+	0.989577i	$0.545999\pi$
$942$	7654.75	0.264761
$943$	−37364.7	−1.29031
$944$	3472.01	0.119708
$945$	−4084.27	−0.140594
$946$	14316.3	0.492031
$947$	−26999.1	−0.926454	−0.463227	−	0.886240i	$-0.653309\pi$
$947$	−26999.1	−0.926454	−0.463227	+	0.886240i	$0.653309\pi$
$948$	−2663.50	−0.0912516
$949$	17531.1	0.599666
$950$	0	0
$951$	2749.46	0.0937512
$952$	9120.99	0.310518
$953$	−25759.1	−0.875570	−0.437785	−	0.899080i	$-0.644237\pi$
$953$	−25759.1	−0.875570	−0.437785	+	0.899080i	$0.644237\pi$
$954$	11133.8	0.377853
$955$	−22325.4	−0.756476
$956$	−23452.1	−0.793405
$957$	−3522.57	−0.118985
$958$	−13020.4	−0.439112
$959$	−18876.2	−0.635603
$960$	1737.76	0.0584229
$961$	−23860.8	−0.800939
$962$	11993.1	0.401948
$963$	4221.80	0.141273
$964$	23857.0	0.797077
$965$	−15694.5	−0.523548
$966$	−19390.6	−0.645840
$967$	−57320.9	−1.90622	−0.953111	−	0.302621i	$-0.902138\pi$
$967$	−57320.9	−1.90622	−0.953111	+	0.302621i	$0.902138\pi$
$968$	8365.39	0.277762
$969$	0	0
$970$	1846.31	0.0611149
$971$	−26256.0	−0.867762	−0.433881	−	0.900970i	$-0.642856\pi$
$971$	−26256.0	−0.867762	−0.433881	+	0.900970i	$0.642856\pi$
$972$	972.000	0.0320750
$973$	11874.7	0.391250
$974$	−7670.79	−0.252349
$975$	−3838.18	−0.126072
$976$	−5485.25	−0.179896
$977$	38506.3	1.26093	0.630463	−	0.776219i	$-0.282865\pi$
$977$	38506.3	1.26093	0.630463	+	0.776219i	$0.282865\pi$
$978$	−9444.27	−0.308788
$979$	10001.7	0.326512
$980$	−2304.90	−0.0751299
$981$	−20367.5	−0.662879
$982$	16890.1	0.548864
$983$	−33423.2	−1.08447	−0.542236	−	0.840226i	$-0.682422\pi$
$983$	−33423.2	−1.08447	−0.542236	+	0.840226i	$0.682422\pi$
$984$	−4637.63	−0.150246
$985$	−39725.6	−1.28504
$986$	9483.91	0.306318
$987$	−5763.81	−0.185881
$988$	0	0
$989$	81941.9	2.63458
$990$	−2751.89	−0.0883444
$991$	44125.6	1.41443	0.707213	−	0.707001i	$-0.249952\pi$
$991$	44125.6	1.41443	0.707213	+	0.707001i	$0.249952\pi$
$992$	2464.25	0.0788711
$993$	3655.60	0.116825
$994$	18619.3	0.594132
$995$	47693.9	1.51960
$996$	−2580.89	−0.0821070
$997$	35412.4	1.12490	0.562449	−	0.826832i	$-0.309859\pi$
$997$	35412.4	1.12490	0.562449	+	0.826832i	$0.309859\pi$
$998$	−21269.3	−0.674618
$999$	−5452.08	−0.172669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bh.1.6	✓	8
19.18	odd	2		2166.4.a.bi.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.bh.1.6	✓	8	1.1	even	1	trivial
2166.4.a.bi.1.6	yes	8	19.18	odd	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +9.05084 q^{5} -6.00000 q^{6} -16.7133 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +9.05084 q^{5} -6.00000 q^{6} -16.7133 q^{7} -8.00000 q^{8} +9.00000 q^{9} -18.1017 q^{10} +16.8916 q^{11} +12.0000 q^{12} +29.6964 q^{13} +33.4266 q^{14} +27.1525 q^{15} +16.0000 q^{16} +68.2165 q^{17} -18.0000 q^{18} +36.2033 q^{20} -50.1399 q^{21} -33.7832 q^{22} -193.365 q^{23} -24.0000 q^{24} -43.0824 q^{25} -59.3929 q^{26} +27.0000 q^{27} -66.8532 q^{28} -69.5133 q^{29} -54.3050 q^{30} -77.0080 q^{31} -32.0000 q^{32} +50.6748 q^{33} -136.433 q^{34} -151.269 q^{35} +36.0000 q^{36} -201.929 q^{37} +89.0893 q^{39} -72.4067 q^{40} +193.235 q^{41} +100.280 q^{42} -423.769 q^{43} +67.5663 q^{44} +81.4575 q^{45} +386.729 q^{46} +114.955 q^{47} +48.0000 q^{48} -63.6653 q^{49} +86.1648 q^{50} +204.650 q^{51} +118.786 q^{52} -618.547 q^{53} -54.0000 q^{54} +152.883 q^{55} +133.706 q^{56} +139.027 q^{58} +217.001 q^{59} +108.610 q^{60} -342.828 q^{61} +154.016 q^{62} -150.420 q^{63} +64.0000 q^{64} +268.778 q^{65} -101.350 q^{66} +539.647 q^{67} +272.866 q^{68} -580.094 q^{69} +302.539 q^{70} +557.019 q^{71} -72.0000 q^{72} +590.343 q^{73} +403.857 q^{74} -129.247 q^{75} -282.314 q^{77} -178.179 q^{78} -221.958 q^{79} +144.813 q^{80} +81.0000 q^{81} -386.469 q^{82} -215.074 q^{83} -200.560 q^{84} +617.417 q^{85} +847.538 q^{86} -208.540 q^{87} -135.133 q^{88} +592.110 q^{89} -162.915 q^{90} -496.326 q^{91} -773.458 q^{92} -231.024 q^{93} -229.909 q^{94} -96.0000 q^{96} -101.997 q^{97} +127.331 q^{98} +152.024 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} + 24 q^{3} + 32 q^{4} - 30 q^{5} - 48 q^{6} - 34 q^{7} - 64 q^{8} + 72 q^{9} + 60 q^{10} - 6 q^{11} + 96 q^{12} - 48 q^{13} + 68 q^{14} - 90 q^{15} + 128 q^{16} - 36 q^{17} - 144 q^{18} - 120 q^{20} - 102 q^{21} + 12 q^{22} - 282 q^{23} - 192 q^{24} + 396 q^{25} + 96 q^{26} + 216 q^{27} - 136 q^{28} + 380 q^{29} + 180 q^{30} + 48 q^{31} - 256 q^{32} - 18 q^{33} + 72 q^{34} + 762 q^{35} + 288 q^{36} + 168 q^{37} - 144 q^{39} + 240 q^{40} - 342 q^{41} + 204 q^{42} - 788 q^{43} - 24 q^{44} - 270 q^{45} + 564 q^{46} - 468 q^{47} + 384 q^{48} + 222 q^{49} - 792 q^{50} - 108 q^{51} - 192 q^{52} - 1682 q^{53} - 432 q^{54} - 46 q^{55} + 272 q^{56} - 760 q^{58} - 292 q^{59} - 360 q^{60} + 522 q^{61} - 96 q^{62} - 306 q^{63} + 512 q^{64} + 1120 q^{65} + 36 q^{66} + 2484 q^{67} - 144 q^{68} - 846 q^{69} - 1524 q^{70} - 1182 q^{71} - 576 q^{72} + 182 q^{73} - 336 q^{74} + 1188 q^{75} - 504 q^{77} + 288 q^{78} + 2232 q^{79} - 480 q^{80} + 648 q^{81} + 684 q^{82} - 750 q^{83} - 408 q^{84} - 2238 q^{85} + 1576 q^{86} + 1140 q^{87} + 48 q^{88} - 1304 q^{89} + 540 q^{90} + 624 q^{91} - 1128 q^{92} + 144 q^{93} + 936 q^{94} - 768 q^{96} + 1248 q^{97} - 444 q^{98} - 54 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 + 24 * q^3 + 32 * q^4 - 30 * q^5 - 48 * q^6 - 34 * q^7 - 64 * q^8 + 72 * q^9 + 60 * q^10 - 6 * q^11 + 96 * q^12 - 48 * q^13 + 68 * q^14 - 90 * q^15 + 128 * q^16 - 36 * q^17 - 144 * q^18 - 120 * q^20 - 102 * q^21 + 12 * q^22 - 282 * q^23 - 192 * q^24 + 396 * q^25 + 96 * q^26 + 216 * q^27 - 136 * q^28 + 380 * q^29 + 180 * q^30 + 48 * q^31 - 256 * q^32 - 18 * q^33 + 72 * q^34 + 762 * q^35 + 288 * q^36 + 168 * q^37 - 144 * q^39 + 240 * q^40 - 342 * q^41 + 204 * q^42 - 788 * q^43 - 24 * q^44 - 270 * q^45 + 564 * q^46 - 468 * q^47 + 384 * q^48 + 222 * q^49 - 792 * q^50 - 108 * q^51 - 192 * q^52 - 1682 * q^53 - 432 * q^54 - 46 * q^55 + 272 * q^56 - 760 * q^58 - 292 * q^59 - 360 * q^60 + 522 * q^61 - 96 * q^62 - 306 * q^63 + 512 * q^64 + 1120 * q^65 + 36 * q^66 + 2484 * q^67 - 144 * q^68 - 846 * q^69 - 1524 * q^70 - 1182 * q^71 - 576 * q^72 + 182 * q^73 - 336 * q^74 + 1188 * q^75 - 504 * q^77 + 288 * q^78 + 2232 * q^79 - 480 * q^80 + 648 * q^81 + 684 * q^82 - 750 * q^83 - 408 * q^84 - 2238 * q^85 + 1576 * q^86 + 1140 * q^87 + 48 * q^88 - 1304 * q^89 + 540 * q^90 + 624 * q^91 - 1128 * q^92 + 144 * q^93 + 936 * q^94 - 768 * q^96 + 1248 * q^97 - 444 * q^98 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more