LMFDB - Embedded newform 2166.4.a.w.1.2

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:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.27307$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.21359 q^{5} +6.00000 q^{6} +31.4905 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.21359 q^{5} +6.00000 q^{6} +31.4905 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.4272 q^{10} -21.2113 q^{11} +12.0000 q^{12} -56.2769 q^{13} +62.9809 q^{14} -15.6408 q^{15} +16.0000 q^{16} +17.2792 q^{17} +18.0000 q^{18} -20.8543 q^{20} +94.4714 q^{21} -42.4225 q^{22} -206.316 q^{23} +24.0000 q^{24} -97.8185 q^{25} -112.554 q^{26} +27.0000 q^{27} +125.962 q^{28} -206.513 q^{29} -31.2815 q^{30} -129.624 q^{31} +32.0000 q^{32} -63.6338 q^{33} +34.5584 q^{34} -164.178 q^{35} +36.0000 q^{36} -440.212 q^{37} -168.831 q^{39} -41.7087 q^{40} -435.454 q^{41} +188.943 q^{42} +129.997 q^{43} -84.8451 q^{44} -46.9223 q^{45} -412.633 q^{46} +108.482 q^{47} +48.0000 q^{48} +648.649 q^{49} -195.637 q^{50} +51.8376 q^{51} -225.107 q^{52} +407.280 q^{53} +54.0000 q^{54} +110.587 q^{55} +251.924 q^{56} -413.027 q^{58} +116.245 q^{59} -62.5630 q^{60} -340.684 q^{61} -259.248 q^{62} +283.414 q^{63} +64.0000 q^{64} +293.404 q^{65} -127.268 q^{66} +210.745 q^{67} +69.1168 q^{68} -618.949 q^{69} -328.356 q^{70} -158.017 q^{71} +72.0000 q^{72} +573.547 q^{73} -880.424 q^{74} -293.456 q^{75} -667.952 q^{77} -337.661 q^{78} +885.297 q^{79} -83.4174 q^{80} +81.0000 q^{81} -870.907 q^{82} -573.632 q^{83} +377.886 q^{84} -90.0866 q^{85} +259.994 q^{86} -619.540 q^{87} -169.690 q^{88} -215.235 q^{89} -93.8446 q^{90} -1772.18 q^{91} -825.266 q^{92} -388.872 q^{93} +216.964 q^{94} +96.0000 q^{96} -528.974 q^{97} +1297.30 q^{98} -190.901 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 6 * q^2 + 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} + 24 q^{8} + 27 q^{9} + 4 q^{10} - 52 q^{11} + 36 q^{12} - 75 q^{13} - 34 q^{14} + 6 q^{15} + 48 q^{16} - 48 q^{17} + 54 q^{18} + 8 q^{20} - 51 q^{21} - 104 q^{22} - 238 q^{23} + 72 q^{24} + 229 q^{25} - 150 q^{26} + 81 q^{27} - 68 q^{28} + 8 q^{29} + 12 q^{30} - 107 q^{31} + 96 q^{32} - 156 q^{33} - 96 q^{34} - 294 q^{35} + 108 q^{36} - 305 q^{37} - 225 q^{39} + 16 q^{40} - 16 q^{41} - 102 q^{42} - 331 q^{43} - 208 q^{44} + 18 q^{45} - 476 q^{46} - 766 q^{47} + 144 q^{48} + 1142 q^{49} + 458 q^{50} - 144 q^{51} - 300 q^{52} + 118 q^{53} + 162 q^{54} - 1400 q^{55} - 136 q^{56} + 16 q^{58} - 936 q^{59} + 24 q^{60} - 399 q^{61} - 214 q^{62} - 153 q^{63} + 192 q^{64} - 370 q^{65} - 312 q^{66} - 61 q^{67} - 192 q^{68} - 714 q^{69} - 588 q^{70} - 974 q^{71} + 216 q^{72} + 91 q^{73} - 610 q^{74} + 687 q^{75} - 36 q^{77} - 450 q^{78} + 321 q^{79} + 32 q^{80} + 243 q^{81} - 32 q^{82} - 2148 q^{83} - 204 q^{84} - 1680 q^{85} - 662 q^{86} + 24 q^{87} - 416 q^{88} - 1116 q^{89} + 36 q^{90} - 1367 q^{91} - 952 q^{92} - 321 q^{93} - 1532 q^{94} + 288 q^{96} - 1382 q^{97} + 2284 q^{98} - 468 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 + 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 + 24 * q^8 + 27 * q^9 + 4 * q^10 - 52 * q^11 + 36 * q^12 - 75 * q^13 - 34 * q^14 + 6 * q^15 + 48 * q^16 - 48 * q^17 + 54 * q^18 + 8 * q^20 - 51 * q^21 - 104 * q^22 - 238 * q^23 + 72 * q^24 + 229 * q^25 - 150 * q^26 + 81 * q^27 - 68 * q^28 + 8 * q^29 + 12 * q^30 - 107 * q^31 + 96 * q^32 - 156 * q^33 - 96 * q^34 - 294 * q^35 + 108 * q^36 - 305 * q^37 - 225 * q^39 + 16 * q^40 - 16 * q^41 - 102 * q^42 - 331 * q^43 - 208 * q^44 + 18 * q^45 - 476 * q^46 - 766 * q^47 + 144 * q^48 + 1142 * q^49 + 458 * q^50 - 144 * q^51 - 300 * q^52 + 118 * q^53 + 162 * q^54 - 1400 * q^55 - 136 * q^56 + 16 * q^58 - 936 * q^59 + 24 * q^60 - 399 * q^61 - 214 * q^62 - 153 * q^63 + 192 * q^64 - 370 * q^65 - 312 * q^66 - 61 * q^67 - 192 * q^68 - 714 * q^69 - 588 * q^70 - 974 * q^71 + 216 * q^72 + 91 * q^73 - 610 * q^74 + 687 * q^75 - 36 * q^77 - 450 * q^78 + 321 * q^79 + 32 * q^80 + 243 * q^81 - 32 * q^82 - 2148 * q^83 - 204 * q^84 - 1680 * q^85 - 662 * q^86 + 24 * q^87 - 416 * q^88 - 1116 * q^89 + 36 * q^90 - 1367 * q^91 - 952 * q^92 - 321 * q^93 - 1532 * q^94 + 288 * q^96 - 1382 * q^97 + 2284 * q^98 - 468 * 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
$5$	−5.21359	−0.466317	−0.233159	−	0.972439i	$-0.574906\pi$
$5$	−5.21359	−0.466317	−0.233159	+	0.972439i	$0.574906\pi$
$6$	6.00000	0.408248
$7$	31.4905	1.70033	0.850163	−	0.526520i	$-0.176504\pi$
$7$	31.4905	1.70033	0.850163	+	0.526520i	$0.176504\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.4272	−0.329736
$11$	−21.2113	−0.581403	−0.290702	−	0.956814i	$-0.593889\pi$
$11$	−21.2113	−0.581403	−0.290702	+	0.956814i	$0.593889\pi$
$12$	12.0000	0.288675
$13$	−56.2769	−1.20065	−0.600323	−	0.799758i	$-0.704961\pi$
$13$	−56.2769	−1.20065	−0.600323	+	0.799758i	$0.704961\pi$
$14$	62.9809	1.20231
$15$	−15.6408	−0.269228
$16$	16.0000	0.250000
$17$	17.2792	0.246519	0.123259	−	0.992374i	$-0.460665\pi$
$17$	17.2792	0.246519	0.123259	+	0.992374i	$0.460665\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−20.8543	−0.233159
$21$	94.4714	0.981683
$22$	−42.4225	−0.411114
$23$	−206.316	−1.87043	−0.935216	−	0.354077i	$-0.884795\pi$
$23$	−206.316	−1.87043	−0.935216	+	0.354077i	$0.884795\pi$
$24$	24.0000	0.204124
$25$	−97.8185	−0.782548
$26$	−112.554	−0.848985
$27$	27.0000	0.192450
$28$	125.962	0.850163
$29$	−206.513	−1.32236	−0.661182	−	0.750226i	$-0.729945\pi$
$29$	−206.513	−1.32236	−0.661182	+	0.750226i	$0.729945\pi$
$30$	−31.2815	−0.190373
$31$	−129.624	−0.751005	−0.375503	−	0.926821i	$-0.622530\pi$
$31$	−129.624	−0.751005	−0.375503	+	0.926821i	$0.622530\pi$
$32$	32.0000	0.176777
$33$	−63.6338	−0.335673
$34$	34.5584	0.174315
$35$	−164.178	−0.792891
$36$	36.0000	0.166667
$37$	−440.212	−1.95596	−0.977979	−	0.208704i	$-0.933075\pi$
$37$	−440.212	−1.95596	−0.977979	+	0.208704i	$0.933075\pi$
$38$	0	0
$39$	−168.831	−0.693193
$40$	−41.7087	−0.164868
$41$	−435.454	−1.65869	−0.829347	−	0.558734i	$-0.811287\pi$
$41$	−435.454	−1.65869	−0.829347	+	0.558734i	$0.811287\pi$
$42$	188.943	0.694155
$43$	129.997	0.461031	0.230515	−	0.973069i	$-0.425959\pi$
$43$	129.997	0.461031	0.230515	+	0.973069i	$0.425959\pi$
$44$	−84.8451	−0.290702
$45$	−46.9223	−0.155439
$46$	−412.633	−1.32260
$47$	108.482	0.336675	0.168338	−	0.985729i	$-0.446160\pi$
$47$	108.482	0.336675	0.168338	+	0.985729i	$0.446160\pi$
$48$	48.0000	0.144338
$49$	648.649	1.89111
$50$	−195.637	−0.553345
$51$	51.8376	0.142328
$52$	−225.107	−0.600323
$53$	407.280	1.05555	0.527776	−	0.849384i	$-0.323026\pi$
$53$	407.280	1.05555	0.527776	+	0.849384i	$0.323026\pi$
$54$	54.0000	0.136083
$55$	110.587	0.271118
$56$	251.924	0.601156
$57$	0	0
$58$	−413.027	−0.935052
$59$	116.245	0.256505	0.128252	−	0.991742i	$-0.459063\pi$
$59$	116.245	0.256505	0.128252	+	0.991742i	$0.459063\pi$
$60$	−62.5630	−0.134614
$61$	−340.684	−0.715085	−0.357542	−	0.933897i	$-0.616385\pi$
$61$	−340.684	−0.715085	−0.357542	+	0.933897i	$0.616385\pi$
$62$	−259.248	−0.531041
$63$	283.414	0.566775
$64$	64.0000	0.125000
$65$	293.404	0.559882
$66$	−127.268	−0.237357
$67$	210.745	0.384278	0.192139	−	0.981368i	$-0.438457\pi$
$67$	210.745	0.384278	0.192139	+	0.981368i	$0.438457\pi$
$68$	69.1168	0.123259
$69$	−618.949	−1.07989
$70$	−328.356	−0.560659
$71$	−158.017	−0.264130	−0.132065	−	0.991241i	$-0.542161\pi$
$71$	−158.017	−0.264130	−0.132065	+	0.991241i	$0.542161\pi$
$72$	72.0000	0.117851
$73$	573.547	0.919570	0.459785	−	0.888030i	$-0.347926\pi$
$73$	573.547	0.919570	0.459785	+	0.888030i	$0.347926\pi$
$74$	−880.424	−1.38307
$75$	−293.456	−0.451804
$76$	0	0
$77$	−667.952	−0.988575
$78$	−337.661	−0.490162
$79$	885.297	1.26081	0.630403	−	0.776268i	$-0.282890\pi$
$79$	885.297	1.26081	0.630403	+	0.776268i	$0.282890\pi$
$80$	−83.4174	−0.116579
$81$	81.0000	0.111111
$82$	−870.907	−1.17287
$83$	−573.632	−0.758606	−0.379303	−	0.925273i	$-0.623836\pi$
$83$	−573.632	−0.758606	−0.379303	+	0.925273i	$0.623836\pi$
$84$	377.886	0.490842
$85$	−90.0866	−0.114956
$86$	259.994	0.325998
$87$	−619.540	−0.763467
$88$	−169.690	−0.205557
$89$	−215.235	−0.256347	−0.128174	−	0.991752i	$-0.540911\pi$
$89$	−215.235	−0.256347	−0.128174	+	0.991752i	$0.540911\pi$
$90$	−93.8446	−0.109912
$91$	−1772.18	−2.04149
$92$	−825.266	−0.935216
$93$	−388.872	−0.433593
$94$	216.964	0.238065
$95$	0	0
$96$	96.0000	0.102062
$97$	−528.974	−0.553703	−0.276851	−	0.960913i	$-0.589291\pi$
$97$	−528.974	−0.553703	−0.276851	+	0.960913i	$0.589291\pi$
$98$	1297.30	1.33721
$99$	−190.901	−0.193801
$100$	−391.274	−0.391274
$101$	1028.74	1.01350	0.506748	−	0.862094i	$-0.330848\pi$
$101$	1028.74	1.01350	0.506748	+	0.862094i	$0.330848\pi$
$102$	103.675	0.100641
$103$	−108.300	−0.103603	−0.0518016	−	0.998657i	$-0.516496\pi$
$103$	−108.300	−0.103603	−0.0518016	+	0.998657i	$0.516496\pi$
$104$	−450.215	−0.424492
$105$	−492.535	−0.457776
$106$	814.560	0.746388
$107$	1820.02	1.64438	0.822188	−	0.569215i	$-0.192753\pi$
$107$	1820.02	1.64438	0.822188	+	0.569215i	$0.192753\pi$
$108$	108.000	0.0962250
$109$	119.143	0.104696	0.0523480	−	0.998629i	$-0.483329\pi$
$109$	119.143	0.104696	0.0523480	+	0.998629i	$0.483329\pi$
$110$	221.174	0.191710
$111$	−1320.64	−1.12927
$112$	503.847	0.425081
$113$	−1014.96	−0.844949	−0.422474	−	0.906375i	$-0.638838\pi$
$113$	−1014.96	−0.844949	−0.422474	+	0.906375i	$0.638838\pi$
$114$	0	0
$115$	1075.65	0.872215
$116$	−826.053	−0.661182
$117$	−506.492	−0.400215
$118$	232.490	0.181376
$119$	544.130	0.419162
$120$	−125.126	−0.0951866
$121$	−881.082	−0.661970
$122$	−681.369	−0.505641
$123$	−1306.36	−0.957647
$124$	−518.496	−0.375503
$125$	1161.68	0.831233
$126$	566.828	0.400770
$127$	−1028.20	−0.718410	−0.359205	−	0.933259i	$-0.616952\pi$
$127$	−1028.20	−0.718410	−0.359205	+	0.933259i	$0.616952\pi$
$128$	128.000	0.0883883
$129$	389.990	0.266176
$130$	586.809	0.395896
$131$	−1589.17	−1.05990	−0.529948	−	0.848030i	$-0.677788\pi$
$131$	−1589.17	−1.05990	−0.529948	+	0.848030i	$0.677788\pi$
$132$	−254.535	−0.167837
$133$	0	0
$134$	421.491	0.271726
$135$	−140.767	−0.0897428
$136$	138.234	0.0871576
$137$	−2325.07	−1.44995	−0.724977	−	0.688773i	$-0.758150\pi$
$137$	−2325.07	−1.44995	−0.724977	+	0.688773i	$0.758150\pi$
$138$	−1237.90	−0.763601
$139$	635.056	0.387516	0.193758	−	0.981049i	$-0.437932\pi$
$139$	635.056	0.387516	0.193758	+	0.981049i	$0.437932\pi$
$140$	−656.713	−0.396446
$141$	325.446	0.194380
$142$	−316.034	−0.186768
$143$	1193.70	0.698060
$144$	144.000	0.0833333
$145$	1076.67	0.616641
$146$	1147.09	0.650234
$147$	1945.95	1.09183
$148$	−1760.85	−0.977979
$149$	1698.31	0.933767	0.466883	−	0.884319i	$-0.345377\pi$
$149$	1698.31	0.933767	0.466883	+	0.884319i	$0.345377\pi$
$150$	−586.911	−0.319474
$151$	1632.38	0.879745	0.439873	−	0.898060i	$-0.355024\pi$
$151$	1632.38	0.879745	0.439873	+	0.898060i	$0.355024\pi$
$152$	0	0
$153$	155.513	0.0821730
$154$	−1335.90	−0.699028
$155$	675.806	0.350207
$156$	−675.322	−0.346597
$157$	2962.57	1.50598	0.752990	−	0.658032i	$-0.228611\pi$
$157$	2962.57	1.50598	0.752990	+	0.658032i	$0.228611\pi$
$158$	1770.59	0.891525
$159$	1221.84	0.609423
$160$	−166.835	−0.0824340
$161$	−6497.00	−3.18034
$162$	162.000	0.0785674
$163$	626.972	0.301277	0.150639	−	0.988589i	$-0.451867\pi$
$163$	626.972	0.301277	0.150639	+	0.988589i	$0.451867\pi$
$164$	−1741.81	−0.829347
$165$	331.760	0.156530
$166$	−1147.26	−0.536415
$167$	−1602.16	−0.742387	−0.371193	−	0.928556i	$-0.621051\pi$
$167$	−1602.16	−0.742387	−0.371193	+	0.928556i	$0.621051\pi$
$168$	755.771	0.347077
$169$	970.087	0.441551
$170$	−180.173	−0.0812862
$171$	0	0
$172$	519.987	0.230515
$173$	−2290.69	−1.00669	−0.503346	−	0.864085i	$-0.667898\pi$
$173$	−2290.69	−1.00669	−0.503346	+	0.864085i	$0.667898\pi$
$174$	−1239.08	−0.539853
$175$	−3080.35	−1.33059
$176$	−339.380	−0.145351
$177$	348.734	0.148093
$178$	−430.471	−0.181265
$179$	218.545	0.0912558	0.0456279	−	0.998959i	$-0.485471\pi$
$179$	218.545	0.0912558	0.0456279	+	0.998959i	$0.485471\pi$
$180$	−187.689	−0.0777196
$181$	3748.44	1.53933	0.769667	−	0.638446i	$-0.220422\pi$
$181$	3748.44	1.53933	0.769667	+	0.638446i	$0.220422\pi$
$182$	−3544.37	−1.44355
$183$	−1022.05	−0.412854
$184$	−1650.53	−0.661298
$185$	2295.08	0.912097
$186$	−777.744	−0.306597
$187$	−366.514	−0.143327
$188$	433.928	0.168338
$189$	850.242	0.327228
$190$	0	0
$191$	−141.733	−0.0536935	−0.0268467	−	0.999640i	$-0.508547\pi$
$191$	−141.733	−0.0536935	−0.0268467	+	0.999640i	$0.508547\pi$
$192$	192.000	0.0721688
$193$	3909.89	1.45824	0.729120	−	0.684386i	$-0.239930\pi$
$193$	3909.89	1.45824	0.729120	+	0.684386i	$0.239930\pi$
$194$	−1057.95	−0.391527
$195$	880.213	0.323248
$196$	2594.60	0.945553
$197$	1203.16	0.435134	0.217567	−	0.976045i	$-0.430188\pi$
$197$	1203.16	0.435134	0.217567	+	0.976045i	$0.430188\pi$
$198$	−381.803	−0.137038
$199$	3703.46	1.31925	0.659626	−	0.751594i	$-0.270715\pi$
$199$	3703.46	1.31925	0.659626	+	0.751594i	$0.270715\pi$
$200$	−782.548	−0.276673
$201$	632.236	0.221863
$202$	2057.47	0.716649
$203$	−6503.20	−2.24845
$204$	207.350	0.0711639
$205$	2270.28	0.773478
$206$	−216.600	−0.0732586
$207$	−1856.85	−0.623478
$208$	−900.430	−0.300161
$209$	0	0
$210$	−985.069	−0.323696
$211$	25.6998	0.00838506	0.00419253	−	0.999991i	$-0.498665\pi$
$211$	25.6998	0.00838506	0.00419253	+	0.999991i	$0.498665\pi$
$212$	1629.12	0.527776
$213$	−474.052	−0.152495
$214$	3640.05	1.16275
$215$	−677.749	−0.214987
$216$	216.000	0.0680414
$217$	−4081.92	−1.27695
$218$	238.287	0.0740312
$219$	1720.64	0.530914
$220$	442.347	0.135559
$221$	−972.419	−0.295982
$222$	−2641.27	−0.798516
$223$	3105.19	0.932461	0.466230	−	0.884663i	$-0.345612\pi$
$223$	3105.19	0.932461	0.466230	+	0.884663i	$0.345612\pi$
$224$	1007.69	0.300578
$225$	−880.367	−0.260849
$226$	−2029.92	−0.597469
$227$	−664.566	−0.194312	−0.0971560	−	0.995269i	$-0.530975\pi$
$227$	−664.566	−0.194312	−0.0971560	+	0.995269i	$0.530975\pi$
$228$	0	0
$229$	−6031.99	−1.74063	−0.870317	−	0.492492i	$-0.836086\pi$
$229$	−6031.99	−1.74063	−0.870317	+	0.492492i	$0.836086\pi$
$230$	2151.30	0.616749
$231$	−2003.86	−0.570754
$232$	−1652.11	−0.467526
$233$	−5758.16	−1.61901	−0.809506	−	0.587112i	$-0.800265\pi$
$233$	−5758.16	−1.61901	−0.809506	+	0.587112i	$0.800265\pi$
$234$	−1012.98	−0.282995
$235$	−565.581	−0.156998
$236$	464.979	0.128252
$237$	2655.89	0.727927
$238$	1088.26	0.296392
$239$	−741.406	−0.200659	−0.100330	−	0.994954i	$-0.531990\pi$
$239$	−741.406	−0.200659	−0.100330	+	0.994954i	$0.531990\pi$
$240$	−250.252	−0.0673071
$241$	−5274.07	−1.40968	−0.704840	−	0.709367i	$-0.748981\pi$
$241$	−5274.07	−1.40968	−0.704840	+	0.709367i	$0.748981\pi$
$242$	−1762.16	−0.468084
$243$	243.000	0.0641500
$244$	−1362.74	−0.357542
$245$	−3381.79	−0.881855
$246$	−2612.72	−0.677159
$247$	0	0
$248$	−1036.99	−0.265520
$249$	−1720.90	−0.437981
$250$	2323.37	0.587771
$251$	2846.36	0.715778	0.357889	−	0.933764i	$-0.383497\pi$
$251$	2846.36	0.715778	0.357889	+	0.933764i	$0.383497\pi$
$252$	1133.66	0.283388
$253$	4376.23	1.08748
$254$	−2056.40	−0.507993
$255$	−270.260	−0.0663699
$256$	256.000	0.0625000
$257$	−6440.71	−1.56327	−0.781635	−	0.623736i	$-0.785614\pi$
$257$	−6440.71	−1.56327	−0.781635	+	0.623736i	$0.785614\pi$
$258$	779.981	0.188215
$259$	−13862.5	−3.32576
$260$	1173.62	0.279941
$261$	−1858.62	−0.440788
$262$	−3178.34	−0.749459
$263$	4140.40	0.970754	0.485377	−	0.874305i	$-0.338682\pi$
$263$	4140.40	0.970754	0.485377	+	0.874305i	$0.338682\pi$
$264$	−509.070	−0.118678
$265$	−2123.39	−0.492222
$266$	0	0
$267$	−645.706	−0.148002
$268$	842.982	0.192139
$269$	−3165.99	−0.717599	−0.358799	−	0.933415i	$-0.616814\pi$
$269$	−3165.99	−0.717599	−0.358799	+	0.933415i	$0.616814\pi$
$270$	−281.534	−0.0634578
$271$	6444.23	1.44450	0.722249	−	0.691633i	$-0.243109\pi$
$271$	6444.23	1.44450	0.722249	+	0.691633i	$0.243109\pi$
$272$	276.467	0.0616297
$273$	−5316.55	−1.17865
$274$	−4650.13	−1.02527
$275$	2074.85	0.454976
$276$	−2475.80	−0.539947
$277$	399.738	0.0867073	0.0433537	−	0.999060i	$-0.486196\pi$
$277$	399.738	0.0867073	0.0433537	+	0.999060i	$0.486196\pi$
$278$	1270.11	0.274015
$279$	−1166.62	−0.250335
$280$	−1313.43	−0.280329
$281$	−630.367	−0.133824	−0.0669120	−	0.997759i	$-0.521315\pi$
$281$	−630.367	−0.133824	−0.0669120	+	0.997759i	$0.521315\pi$
$282$	650.892	0.137447
$283$	1996.63	0.419391	0.209695	−	0.977767i	$-0.432753\pi$
$283$	1996.63	0.419391	0.209695	+	0.977767i	$0.432753\pi$
$284$	−632.069	−0.132065
$285$	0	0
$286$	2387.41	0.493603
$287$	−13712.6	−2.82032
$288$	288.000	0.0589256
$289$	−4614.43	−0.939228
$290$	2153.35	0.436031
$291$	−1586.92	−0.319680
$292$	2294.19	0.459785
$293$	−9676.85	−1.92945	−0.964723	−	0.263268i	$-0.915200\pi$
$293$	−9676.85	−1.92945	−0.964723	+	0.263268i	$0.915200\pi$
$294$	3891.89	0.772041
$295$	−606.052	−0.119613
$296$	−3521.70	−0.691535
$297$	−572.704	−0.111891
$298$	3396.63	0.660273
$299$	11610.8	2.24573
$300$	−1173.82	−0.225902
$301$	4093.66	0.783902
$302$	3264.77	0.622074
$303$	3086.21	0.585142
$304$	0	0
$305$	1776.19	0.333456
$306$	311.026	0.0581051
$307$	−6795.52	−1.26332	−0.631662	−	0.775244i	$-0.717627\pi$
$307$	−6795.52	−1.26332	−0.631662	+	0.775244i	$0.717627\pi$
$308$	−2671.81	−0.494287
$309$	−324.901	−0.0598154
$310$	1351.61	0.247634
$311$	−4418.47	−0.805622	−0.402811	−	0.915283i	$-0.631967\pi$
$311$	−4418.47	−0.805622	−0.402811	+	0.915283i	$0.631967\pi$
$312$	−1350.64	−0.245081
$313$	−1391.56	−0.251296	−0.125648	−	0.992075i	$-0.540101\pi$
$313$	−1391.56	−0.251296	−0.125648	+	0.992075i	$0.540101\pi$
$314$	5925.14	1.06489
$315$	−1477.60	−0.264297
$316$	3541.19	0.630403
$317$	−5998.68	−1.06284	−0.531419	−	0.847109i	$-0.678341\pi$
$317$	−5998.68	−1.06284	−0.531419	+	0.847109i	$0.678341\pi$
$318$	2443.68	0.430927
$319$	4380.41	0.768827
$320$	−333.670	−0.0582897
$321$	5460.07	0.949381
$322$	−12994.0	−2.24884
$323$	0	0
$324$	324.000	0.0555556
$325$	5504.92	0.939563
$326$	1253.94	0.213035
$327$	357.430	0.0604462
$328$	−3483.63	−0.586437
$329$	3416.15	0.572457
$330$	663.521	0.110684
$331$	6604.73	1.09676	0.548382	−	0.836228i	$-0.315244\pi$
$331$	6604.73	1.09676	0.548382	+	0.836228i	$0.315244\pi$
$332$	−2294.53	−0.379303
$333$	−3961.91	−0.651986
$334$	−3204.31	−0.524947
$335$	−1098.74	−0.179196
$336$	1511.54	0.245421
$337$	4792.90	0.774735	0.387367	−	0.921925i	$-0.373384\pi$
$337$	4792.90	0.774735	0.387367	+	0.921925i	$0.373384\pi$
$338$	1940.17	0.312223
$339$	−3044.87	−0.487831
$340$	−360.346	−0.0574780
$341$	2749.49	0.436637
$342$	0	0
$343$	9625.03	1.51517
$344$	1039.97	0.162999
$345$	3226.95	0.503574
$346$	−4581.37	−0.711839
$347$	4871.36	0.753627	0.376813	−	0.926289i	$-0.377020\pi$
$347$	4871.36	0.753627	0.376813	+	0.926289i	$0.377020\pi$
$348$	−2478.16	−0.381734
$349$	−2676.44	−0.410506	−0.205253	−	0.978709i	$-0.565802\pi$
$349$	−2676.44	−0.410506	−0.205253	+	0.978709i	$0.565802\pi$
$350$	−6160.70	−0.940867
$351$	−1519.48	−0.231064
$352$	−678.760	−0.102779
$353$	3111.41	0.469132	0.234566	−	0.972100i	$-0.424633\pi$
$353$	3111.41	0.469132	0.234566	+	0.972100i	$0.424633\pi$
$354$	697.469	0.104718
$355$	823.837	0.123168
$356$	−860.942	−0.128174
$357$	1632.39	0.242003
$358$	437.089	0.0645276
$359$	−935.758	−0.137569	−0.0687847	−	0.997632i	$-0.521912\pi$
$359$	−935.758	−0.137569	−0.0687847	+	0.997632i	$0.521912\pi$
$360$	−375.378	−0.0549560
$361$	0	0
$362$	7496.88	1.08847
$363$	−2643.25	−0.382189
$364$	−7088.74	−1.02074
$365$	−2990.24	−0.428812
$366$	−2044.11	−0.291932
$367$	11009.2	1.56588	0.782940	−	0.622097i	$-0.213719\pi$
$367$	11009.2	1.56588	0.782940	+	0.622097i	$0.213719\pi$
$368$	−3301.06	−0.467608
$369$	−3919.08	−0.552898
$370$	4590.17	0.644950
$371$	12825.4	1.79478
$372$	−1555.49	−0.216797
$373$	6151.39	0.853906	0.426953	−	0.904274i	$-0.359587\pi$
$373$	6151.39	0.853906	0.426953	+	0.904274i	$0.359587\pi$
$374$	−733.027	−0.101347
$375$	3485.05	0.479913
$376$	867.857	0.119033
$377$	11621.9	1.58769
$378$	1700.48	0.231385
$379$	−13458.6	−1.82407	−0.912037	−	0.410109i	$-0.865491\pi$
$379$	−13458.6	−1.82407	−0.912037	+	0.410109i	$0.865491\pi$
$380$	0	0
$381$	−3084.60	−0.414774
$382$	−283.466	−0.0379670
$383$	11090.4	1.47961	0.739807	−	0.672819i	$-0.234917\pi$
$383$	11090.4	1.47961	0.739807	+	0.672819i	$0.234917\pi$
$384$	384.000	0.0510310
$385$	3482.43	0.460990
$386$	7819.79	1.03113
$387$	1169.97	0.153677
$388$	−2115.90	−0.276851
$389$	1273.09	0.165933	0.0829667	−	0.996552i	$-0.473560\pi$
$389$	1273.09	0.165933	0.0829667	+	0.996552i	$0.473560\pi$
$390$	1760.43	0.228571
$391$	−3564.98	−0.461097
$392$	5189.19	0.668607
$393$	−4767.51	−0.611931
$394$	2406.32	0.307687
$395$	−4615.57	−0.587936
$396$	−763.605	−0.0969006
$397$	−12328.9	−1.55862	−0.779309	−	0.626640i	$-0.784430\pi$
$397$	−12328.9	−1.55862	−0.779309	+	0.626640i	$0.784430\pi$
$398$	7406.92	0.932853
$399$	0	0
$400$	−1565.10	−0.195637
$401$	−9799.54	−1.22036	−0.610181	−	0.792262i	$-0.708903\pi$
$401$	−9799.54	−1.22036	−0.610181	+	0.792262i	$0.708903\pi$
$402$	1264.47	0.156881
$403$	7294.83	0.901691
$404$	4114.94	0.506748
$405$	−422.301	−0.0518130
$406$	−13006.4	−1.58989
$407$	9337.46	1.13720
$408$	414.701	0.0503205
$409$	−8084.65	−0.977409	−0.488705	−	0.872449i	$-0.662530\pi$
$409$	−8084.65	−0.977409	−0.488705	+	0.872449i	$0.662530\pi$
$410$	4540.55	0.546931
$411$	−6975.20	−0.837132
$412$	−433.201	−0.0518016
$413$	3660.60	0.436141
$414$	−3713.70	−0.440865
$415$	2990.68	0.353751
$416$	−1800.86	−0.212246
$417$	1905.17	0.223733
$418$	0	0
$419$	−78.3882	−0.00913965	−0.00456983	−	0.999990i	$-0.501455\pi$
$419$	−78.3882	−0.00913965	−0.00456983	+	0.999990i	$0.501455\pi$
$420$	−1970.14	−0.228888
$421$	−7085.19	−0.820216	−0.410108	−	0.912037i	$-0.634509\pi$
$421$	−7085.19	−0.820216	−0.410108	+	0.912037i	$0.634509\pi$
$422$	51.3996	0.00592914
$423$	976.339	0.112225
$424$	3258.24	0.373194
$425$	−1690.23	−0.192913
$426$	−948.103	−0.107830
$427$	−10728.3	−1.21588
$428$	7280.09	0.822188
$429$	3581.11	0.403025
$430$	−1355.50	−0.152019
$431$	14901.4	1.66537	0.832686	−	0.553745i	$-0.186802\pi$
$431$	14901.4	1.66537	0.832686	+	0.553745i	$0.186802\pi$
$432$	432.000	0.0481125
$433$	−4229.48	−0.469413	−0.234707	−	0.972066i	$-0.575413\pi$
$433$	−4229.48	−0.469413	−0.234707	+	0.972066i	$0.575413\pi$
$434$	−8163.84	−0.902942
$435$	3230.02	0.356018
$436$	476.573	0.0523480
$437$	0	0
$438$	3441.28	0.375413
$439$	−10521.5	−1.14389	−0.571943	−	0.820293i	$-0.693810\pi$
$439$	−10521.5	−1.14389	−0.571943	+	0.820293i	$0.693810\pi$
$440$	884.694	0.0958549
$441$	5837.84	0.630368
$442$	−1944.84	−0.209291
$443$	13867.4	1.48727	0.743636	−	0.668585i	$-0.233100\pi$
$443$	13867.4	1.48727	0.743636	+	0.668585i	$0.233100\pi$
$444$	−5282.55	−0.564636
$445$	1122.15	0.119539
$446$	6210.38	0.659349
$447$	5094.94	0.539111
$448$	2015.39	0.212541
$449$	−9599.89	−1.00901	−0.504507	−	0.863408i	$-0.668326\pi$
$449$	−9599.89	−1.00901	−0.504507	+	0.863408i	$0.668326\pi$
$450$	−1760.73	−0.184448
$451$	9236.52	0.964370
$452$	−4059.83	−0.422474
$453$	4897.15	0.507921
$454$	−1329.13	−0.137399
$455$	9239.44	0.951981
$456$	0	0
$457$	−6261.10	−0.640880	−0.320440	−	0.947269i	$-0.603831\pi$
$457$	−6261.10	−0.640880	−0.320440	+	0.947269i	$0.603831\pi$
$458$	−12064.0	−1.23081
$459$	466.538	0.0474426
$460$	4302.59	0.436108
$461$	−9825.55	−0.992671	−0.496336	−	0.868131i	$-0.665321\pi$
$461$	−9825.55	−0.992671	−0.496336	+	0.868131i	$0.665321\pi$
$462$	−4007.71	−0.403584
$463$	9276.57	0.931142	0.465571	−	0.885011i	$-0.345849\pi$
$463$	9276.57	0.931142	0.465571	+	0.885011i	$0.345849\pi$
$464$	−3304.21	−0.330591
$465$	2027.42	0.202192
$466$	−11516.3	−1.14481
$467$	18209.4	1.80435	0.902174	−	0.431372i	$-0.141970\pi$
$467$	18209.4	1.80435	0.902174	+	0.431372i	$0.141970\pi$
$468$	−2025.97	−0.200108
$469$	6636.47	0.653398
$470$	−1131.16	−0.111014
$471$	8887.71	0.869478
$472$	929.958	0.0906881
$473$	−2757.40	−0.268045
$474$	5311.78	0.514722
$475$	0	0
$476$	2176.52	0.209581
$477$	3665.52	0.351850
$478$	−1482.81	−0.141887
$479$	12741.7	1.21541	0.607706	−	0.794162i	$-0.292090\pi$
$479$	12741.7	1.21541	0.607706	+	0.794162i	$0.292090\pi$
$480$	−500.504	−0.0475933
$481$	24773.8	2.34841
$482$	−10548.1	−0.996794
$483$	−19491.0	−1.83617
$484$	−3524.33	−0.330985
$485$	2757.85	0.258201
$486$	486.000	0.0453609
$487$	5006.72	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$487$	5006.72	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$488$	−2725.48	−0.252821
$489$	1880.92	0.173943
$490$	−6763.58	−0.623566
$491$	963.312	0.0885411	0.0442705	−	0.999020i	$-0.485904\pi$
$491$	963.312	0.0885411	0.0442705	+	0.999020i	$0.485904\pi$
$492$	−5225.44	−0.478824
$493$	−3568.38	−0.325988
$494$	0	0
$495$	995.281	0.0903728
$496$	−2073.98	−0.187751
$497$	−4976.04	−0.449106
$498$	−3441.79	−0.309700
$499$	4964.84	0.445405	0.222702	−	0.974886i	$-0.428512\pi$
$499$	4964.84	0.445405	0.222702	+	0.974886i	$0.428512\pi$
$500$	4646.73	0.415617
$501$	−4806.47	−0.428617
$502$	5692.71	0.506132
$503$	−14768.5	−1.30913	−0.654567	−	0.756004i	$-0.727149\pi$
$503$	−14768.5	−1.30913	−0.654567	+	0.756004i	$0.727149\pi$
$504$	2267.31	0.200385
$505$	−5363.40	−0.472610
$506$	8752.46	0.768962
$507$	2910.26	0.254929
$508$	−4112.80	−0.359205
$509$	−6015.92	−0.523872	−0.261936	−	0.965085i	$-0.584361\pi$
$509$	−6015.92	−0.523872	−0.261936	+	0.965085i	$0.584361\pi$
$510$	−540.520	−0.0469306
$511$	18061.3	1.56357
$512$	512.000	0.0441942
$513$	0	0
$514$	−12881.4	−1.10540
$515$	564.632	0.0483120
$516$	1559.96	0.133088
$517$	−2301.04	−0.195744
$518$	−27725.0	−2.35167
$519$	−6872.06	−0.581214
$520$	2347.23	0.197948
$521$	−10009.4	−0.841685	−0.420842	−	0.907134i	$-0.638265\pi$
$521$	−10009.4	−0.841685	−0.420842	+	0.907134i	$0.638265\pi$
$522$	−3717.24	−0.311684
$523$	−2488.09	−0.208024	−0.104012	−	0.994576i	$-0.533168\pi$
$523$	−2488.09	−0.208024	−0.104012	+	0.994576i	$0.533168\pi$
$524$	−6356.67	−0.529948
$525$	−9241.05	−0.768214
$526$	8280.81	0.686427
$527$	−2239.80	−0.185137
$528$	−1018.14	−0.0839183
$529$	30399.5	2.49852
$530$	−4246.78	−0.348053
$531$	1046.20	0.0855016
$532$	0	0
$533$	24506.0	1.99150
$534$	−1291.41	−0.104653
$535$	−9488.85	−0.766802
$536$	1685.96	0.135863
$537$	655.634	0.0526866
$538$	−6331.99	−0.507419
$539$	−13758.7	−1.09949
$540$	−563.067	−0.0448714
$541$	−9732.04	−0.773407	−0.386703	−	0.922204i	$-0.626386\pi$
$541$	−9732.04	−0.773407	−0.386703	+	0.922204i	$0.626386\pi$
$542$	12888.5	1.02141
$543$	11245.3	0.888735
$544$	552.934	0.0435788
$545$	−621.164	−0.0488216
$546$	−10633.1	−0.833434
$547$	−12432.4	−0.971791	−0.485895	−	0.874017i	$-0.661506\pi$
$547$	−12432.4	−0.971791	−0.485895	+	0.874017i	$0.661506\pi$
$548$	−9300.26	−0.724977
$549$	−3066.16	−0.238362
$550$	4149.71	0.321717
$551$	0	0
$552$	−4951.59	−0.381800
$553$	27878.4	2.14378
$554$	799.476	0.0613113
$555$	6885.25	0.526599
$556$	2540.22	0.193758
$557$	−17905.5	−1.36208	−0.681041	−	0.732245i	$-0.738472\pi$
$557$	−17905.5	−1.36208	−0.681041	+	0.732245i	$0.738472\pi$
$558$	−2333.23	−0.177014
$559$	−7315.81	−0.553535
$560$	−2626.85	−0.198223
$561$	−1099.54	−0.0827498
$562$	−1260.73	−0.0946278
$563$	1825.07	0.136621	0.0683106	−	0.997664i	$-0.478239\pi$
$563$	1825.07	0.136621	0.0683106	+	0.997664i	$0.478239\pi$
$564$	1301.78	0.0971898
$565$	5291.57	0.394014
$566$	3993.27	0.296554
$567$	2550.73	0.188925
$568$	−1264.14	−0.0933839
$569$	4974.94	0.366538	0.183269	−	0.983063i	$-0.441332\pi$
$569$	4974.94	0.366538	0.183269	+	0.983063i	$0.441332\pi$
$570$	0	0
$571$	14203.4	1.04097	0.520485	−	0.853871i	$-0.325751\pi$
$571$	14203.4	1.04097	0.520485	+	0.853871i	$0.325751\pi$
$572$	4774.81	0.349030
$573$	−425.200	−0.0309999
$574$	−27425.3	−1.99427
$575$	20181.6	1.46370
$576$	576.000	0.0416667
$577$	5279.89	0.380944	0.190472	−	0.981693i	$-0.438998\pi$
$577$	5279.89	0.380944	0.190472	+	0.981693i	$0.438998\pi$
$578$	−9228.86	−0.664135
$579$	11729.7	0.841915
$580$	4306.70	0.308321
$581$	−18063.9	−1.28988
$582$	−3173.84	−0.226048
$583$	−8638.93	−0.613701
$584$	4588.38	0.325117
$585$	2640.64	0.186627
$586$	−19353.7	−1.36432
$587$	13228.6	0.930156	0.465078	−	0.885270i	$-0.346026\pi$
$587$	13228.6	0.930156	0.465078	+	0.885270i	$0.346026\pi$
$588$	7783.79	0.545915
$589$	0	0
$590$	−1212.10	−0.0845789
$591$	3609.47	0.251225
$592$	−7043.39	−0.488989
$593$	−696.480	−0.0482310	−0.0241155	−	0.999709i	$-0.507677\pi$
$593$	−696.480	−0.0482310	−0.0241155	+	0.999709i	$0.507677\pi$
$594$	−1145.41	−0.0791190
$595$	−2836.87	−0.195463
$596$	6793.25	0.466883
$597$	11110.4	0.761671
$598$	23221.7	1.58797
$599$	−22697.1	−1.54821	−0.774105	−	0.633058i	$-0.781800\pi$
$599$	−22697.1	−1.54821	−0.774105	+	0.633058i	$0.781800\pi$
$600$	−2347.64	−0.159737
$601$	−15591.5	−1.05822	−0.529110	−	0.848553i	$-0.677474\pi$
$601$	−15591.5	−1.05822	−0.529110	+	0.848553i	$0.677474\pi$
$602$	8187.32	0.554302
$603$	1896.71	0.128093
$604$	6529.54	0.439873
$605$	4593.60	0.308688
$606$	6172.41	0.413758
$607$	−13728.2	−0.917975	−0.458988	−	0.888443i	$-0.651788\pi$
$607$	−13728.2	−0.917975	−0.458988	+	0.888443i	$0.651788\pi$
$608$	0	0
$609$	−19509.6	−1.29814
$610$	3552.38	0.235789
$611$	−6105.03	−0.404228
$612$	622.051	0.0410865
$613$	21971.3	1.44766	0.723829	−	0.689979i	$-0.242380\pi$
$613$	21971.3	1.44766	0.723829	+	0.689979i	$0.242380\pi$
$614$	−13591.0	−0.893305
$615$	6810.83	0.446568
$616$	−5343.62	−0.349514
$617$	−18518.7	−1.20832	−0.604160	−	0.796863i	$-0.706491\pi$
$617$	−18518.7	−1.20832	−0.604160	+	0.796863i	$0.706491\pi$
$618$	−649.801	−0.0422958
$619$	−7363.53	−0.478134	−0.239067	−	0.971003i	$-0.576842\pi$
$619$	−7363.53	−0.478134	−0.239067	+	0.971003i	$0.576842\pi$
$620$	2703.22	0.175103
$621$	−5570.54	−0.359965
$622$	−8836.95	−0.569661
$623$	−6777.86	−0.435874
$624$	−2701.29	−0.173298
$625$	6170.78	0.394930
$626$	−2783.12	−0.177693
$627$	0	0
$628$	11850.3	0.752990
$629$	−7606.51	−0.482181
$630$	−2955.21	−0.186886
$631$	−20379.0	−1.28570	−0.642848	−	0.765993i	$-0.722247\pi$
$631$	−20379.0	−1.28570	−0.642848	+	0.765993i	$0.722247\pi$
$632$	7082.38	0.445763
$633$	77.0995	0.00484112
$634$	−11997.4	−0.751540
$635$	5360.61	0.335007
$636$	4887.36	0.304711
$637$	−36503.9	−2.27055
$638$	8760.81	0.543643
$639$	−1422.16	−0.0880432
$640$	−667.339	−0.0412170
$641$	−1986.25	−0.122390	−0.0611951	−	0.998126i	$-0.519491\pi$
$641$	−1986.25	−0.122390	−0.0611951	+	0.998126i	$0.519491\pi$
$642$	10920.1	0.671314
$643$	−11156.6	−0.684254	−0.342127	−	0.939654i	$-0.611147\pi$
$643$	−11156.6	−0.684254	−0.342127	+	0.939654i	$0.611147\pi$
$644$	−25988.0	−1.59017
$645$	−2033.25	−0.124123
$646$	0	0
$647$	−18657.7	−1.13371	−0.566854	−	0.823818i	$-0.691840\pi$
$647$	−18657.7	−1.13371	−0.566854	+	0.823818i	$0.691840\pi$
$648$	648.000	0.0392837
$649$	−2465.70	−0.149133
$650$	11009.8	0.664371
$651$	−12245.8	−0.737249
$652$	2507.89	0.150639
$653$	7972.68	0.477787	0.238894	−	0.971046i	$-0.423215\pi$
$653$	7972.68	0.477787	0.238894	+	0.971046i	$0.423215\pi$
$654$	714.860	0.0427420
$655$	8285.27	0.494248
$656$	−6967.26	−0.414673
$657$	5161.93	0.306523
$658$	6832.30	0.404789
$659$	−5327.61	−0.314923	−0.157461	−	0.987525i	$-0.550331\pi$
$659$	−5327.61	−0.314923	−0.157461	+	0.987525i	$0.550331\pi$
$660$	1327.04	0.0782652
$661$	6945.06	0.408671	0.204336	−	0.978901i	$-0.434497\pi$
$661$	6945.06	0.408671	0.204336	+	0.978901i	$0.434497\pi$
$662$	13209.5	0.775530
$663$	−2917.26	−0.170885
$664$	−4589.06	−0.268208
$665$	0	0
$666$	−7923.82	−0.461024
$667$	42607.1	2.47339
$668$	−6408.63	−0.371193
$669$	9315.57	0.538357
$670$	−2197.48	−0.126710
$671$	7226.35	0.415753
$672$	3023.08	0.173539
$673$	8261.87	0.473212	0.236606	−	0.971606i	$-0.423965\pi$
$673$	8261.87	0.473212	0.236606	+	0.971606i	$0.423965\pi$
$674$	9585.79	0.547820
$675$	−2641.10	−0.150601
$676$	3880.35	0.220775
$677$	−12128.3	−0.688523	−0.344261	−	0.938874i	$-0.611871\pi$
$677$	−12128.3	−0.688523	−0.344261	+	0.938874i	$0.611871\pi$
$678$	−6089.75	−0.344949
$679$	−16657.6	−0.941475
$680$	−720.693	−0.0406431
$681$	−1993.70	−0.112186
$682$	5498.98	0.308749
$683$	−3609.67	−0.202226	−0.101113	−	0.994875i	$-0.532240\pi$
$683$	−3609.67	−0.202226	−0.101113	+	0.994875i	$0.532240\pi$
$684$	0	0
$685$	12121.9	0.676139
$686$	19250.1	1.07139
$687$	−18096.0	−1.00496
$688$	2079.95	0.115258
$689$	−22920.5	−1.26734
$690$	6453.89	0.356080
$691$	−375.152	−0.0206534	−0.0103267	−	0.999947i	$-0.503287\pi$
$691$	−375.152	−0.0206534	−0.0103267	+	0.999947i	$0.503287\pi$
$692$	−9162.75	−0.503346
$693$	−6011.57	−0.329525
$694$	9742.73	0.532895
$695$	−3310.92	−0.180705
$696$	−4956.32	−0.269926
$697$	−7524.29	−0.408899
$698$	−5352.88	−0.290271
$699$	−17274.5	−0.934737
$700$	−12321.4	−0.665293
$701$	13800.4	0.743558	0.371779	−	0.928321i	$-0.378748\pi$
$701$	13800.4	0.743558	0.371779	+	0.928321i	$0.378748\pi$
$702$	−3038.95	−0.163387
$703$	0	0
$704$	−1357.52	−0.0726754
$705$	−1696.74	−0.0906426
$706$	6222.82	0.331727
$707$	32395.4	1.72327
$708$	1394.94	0.0740465
$709$	−25345.2	−1.34254	−0.671270	−	0.741213i	$-0.734251\pi$
$709$	−25345.2	−1.34254	−0.671270	+	0.741213i	$0.734251\pi$
$710$	1647.67	0.0870931
$711$	7967.68	0.420269
$712$	−1721.88	−0.0906325
$713$	26743.6	1.40470
$714$	3264.78	0.171122
$715$	−6223.48	−0.325517
$716$	874.179	0.0456279
$717$	−2224.22	−0.115851
$718$	−1871.52	−0.0972763
$719$	−19927.8	−1.03363	−0.516816	−	0.856097i	$-0.672883\pi$
$719$	−19927.8	−1.03363	−0.516816	+	0.856097i	$0.672883\pi$
$720$	−750.756	−0.0388598
$721$	−3410.42	−0.176159
$722$	0	0
$723$	−15822.2	−0.813879
$724$	14993.8	0.769667
$725$	20200.8	1.03481
$726$	−5286.49	−0.270248
$727$	6658.48	0.339683	0.169841	−	0.985471i	$-0.445674\pi$
$727$	6658.48	0.339683	0.169841	+	0.985471i	$0.445674\pi$
$728$	−14177.5	−0.721775
$729$	729.000	0.0370370
$730$	−5980.48	−0.303216
$731$	2246.24	0.113653
$732$	−4088.21	−0.206427
$733$	27502.1	1.38583	0.692914	−	0.721021i	$-0.256327\pi$
$733$	27502.1	1.38583	0.692914	+	0.721021i	$0.256327\pi$
$734$	22018.5	1.10724
$735$	−10145.4	−0.509139
$736$	−6602.13	−0.330649
$737$	−4470.18	−0.223421
$738$	−7838.17	−0.390958
$739$	12843.7	0.639327	0.319663	−	0.947531i	$-0.396430\pi$
$739$	12843.7	0.639327	0.319663	+	0.947531i	$0.396430\pi$
$740$	9180.34	0.456049
$741$	0	0
$742$	25650.9	1.26910
$743$	−4806.49	−0.237326	−0.118663	−	0.992935i	$-0.537861\pi$
$743$	−4806.49	−0.237326	−0.118663	+	0.992935i	$0.537861\pi$
$744$	−3110.98	−0.153298
$745$	−8854.31	−0.435432
$746$	12302.8	0.603803
$747$	−5162.69	−0.252869
$748$	−1466.05	−0.0716634
$749$	57313.4	2.79598
$750$	6970.10	0.339350
$751$	4295.77	0.208728	0.104364	−	0.994539i	$-0.466719\pi$
$751$	4295.77	0.208728	0.104364	+	0.994539i	$0.466719\pi$
$752$	1735.71	0.0841688
$753$	8539.07	0.413255
$754$	23243.8	1.12267
$755$	−8510.58	−0.410241
$756$	3400.97	0.163614
$757$	20825.0	0.999866	0.499933	−	0.866064i	$-0.333358\pi$
$757$	20825.0	0.999866	0.499933	+	0.866064i	$0.333358\pi$
$758$	−26917.3	−1.28981
$759$	13128.7	0.627854
$760$	0	0
$761$	−19534.2	−0.930507	−0.465253	−	0.885178i	$-0.654037\pi$
$761$	−19534.2	−0.930507	−0.465253	+	0.885178i	$0.654037\pi$
$762$	−6169.21	−0.293290
$763$	3751.88	0.178017
$764$	−566.933	−0.0268467
$765$	−810.779	−0.0383187
$766$	22180.8	1.04625
$767$	−6541.89	−0.307971
$768$	768.000	0.0360844
$769$	568.457	0.0266568	0.0133284	−	0.999911i	$-0.495757\pi$
$769$	568.457	0.0266568	0.0133284	+	0.999911i	$0.495757\pi$
$770$	6964.86	0.325969
$771$	−19322.1	−0.902555
$772$	15639.6	0.729120
$773$	−96.9965	−0.00451322	−0.00225661	−	0.999997i	$-0.500718\pi$
$773$	−96.9965	−0.00451322	−0.00225661	+	0.999997i	$0.500718\pi$
$774$	2339.94	0.108666
$775$	12679.6	0.587698
$776$	−4231.79	−0.195763
$777$	−41587.5	−1.92013
$778$	2546.17	0.117333
$779$	0	0
$780$	3520.85	0.161624
$781$	3351.75	0.153566
$782$	−7129.96	−0.326045
$783$	−5575.86	−0.254489
$784$	10378.4	0.472776
$785$	−15445.6	−0.702265
$786$	−9535.01	−0.432700
$787$	−4906.36	−0.222227	−0.111114	−	0.993808i	$-0.535442\pi$
$787$	−4906.36	−0.222227	−0.111114	+	0.993808i	$0.535442\pi$
$788$	4812.63	0.217567
$789$	12421.2	0.560465
$790$	−9231.15	−0.415734
$791$	−31961.5	−1.43669
$792$	−1527.21	−0.0685190
$793$	19172.7	0.858564
$794$	−24657.9	−1.10211
$795$	−6370.17	−0.284184
$796$	14813.8	0.659626
$797$	32043.3	1.42413	0.712066	−	0.702113i	$-0.247760\pi$
$797$	32043.3	1.42413	0.712066	+	0.702113i	$0.247760\pi$
$798$	0	0
$799$	1874.48	0.0829968
$800$	−3130.19	−0.138336
$801$	−1937.12	−0.0854491
$802$	−19599.1	−0.862927
$803$	−12165.7	−0.534641
$804$	2528.95	0.110932
$805$	33872.7	1.48305
$806$	14589.7	0.637592
$807$	−9497.98	−0.414306
$808$	8229.89	0.358325
$809$	18714.9	0.813326	0.406663	−	0.913578i	$-0.366692\pi$
$809$	18714.9	0.813326	0.406663	+	0.913578i	$0.366692\pi$
$810$	−844.601	−0.0366374
$811$	9129.34	0.395283	0.197641	−	0.980274i	$-0.436672\pi$
$811$	9129.34	0.395283	0.197641	+	0.980274i	$0.436672\pi$
$812$	−26012.8	−1.12422
$813$	19332.7	0.833981
$814$	18674.9	0.804122
$815$	−3268.77	−0.140491
$816$	829.401	0.0355819
$817$	0	0
$818$	−16169.3	−0.691133
$819$	−15949.7	−0.680496
$820$	9081.10	0.386739
$821$	22828.7	0.970436	0.485218	−	0.874393i	$-0.338740\pi$
$821$	22828.7	0.970436	0.485218	+	0.874393i	$0.338740\pi$
$822$	−13950.4	−0.591942
$823$	−37877.8	−1.60430	−0.802149	−	0.597124i	$-0.796310\pi$
$823$	−37877.8	−1.60430	−0.802149	+	0.597124i	$0.796310\pi$
$824$	−866.401	−0.0366293
$825$	6224.56	0.262681
$826$	7321.20	0.308399
$827$	4120.90	0.173274	0.0866370	−	0.996240i	$-0.472388\pi$
$827$	4120.90	0.173274	0.0866370	+	0.996240i	$0.472388\pi$
$828$	−7427.39	−0.311739
$829$	17959.7	0.752431	0.376216	−	0.926532i	$-0.377225\pi$
$829$	17959.7	0.752431	0.376216	+	0.926532i	$0.377225\pi$
$830$	5981.36	0.250140
$831$	1199.21	0.0500605
$832$	−3601.72	−0.150081
$833$	11208.1	0.466193
$834$	3810.33	0.158203
$835$	8352.98	0.346188
$836$	0	0
$837$	−3499.85	−0.144531
$838$	−156.776	−0.00646271
$839$	48200.8	1.98340	0.991702	−	0.128559i	$-0.0410353\pi$
$839$	48200.8	1.98340	0.991702	+	0.128559i	$0.0410353\pi$
$840$	−3940.28	−0.161848
$841$	18258.7	0.748646
$842$	−14170.4	−0.579980
$843$	−1891.10	−0.0772633
$844$	102.799	0.00419253
$845$	−5057.63	−0.205903
$846$	1952.68	0.0793551
$847$	−27745.7	−1.12556
$848$	6516.48	0.263888
$849$	5989.90	0.242135
$850$	−3380.45	−0.136410
$851$	90823.0	3.65849
$852$	−1896.21	−0.0762476
$853$	36871.2	1.48001	0.740004	−	0.672603i	$-0.234824\pi$
$853$	36871.2	1.48001	0.740004	+	0.672603i	$0.234824\pi$
$854$	−21456.6	−0.859755
$855$	0	0
$856$	14560.2	0.581375
$857$	−33670.9	−1.34209	−0.671047	−	0.741415i	$-0.734155\pi$
$857$	−33670.9	−1.34209	−0.671047	+	0.741415i	$0.734155\pi$
$858$	7162.22	0.284982
$859$	20846.5	0.828025	0.414013	−	0.910271i	$-0.364127\pi$
$859$	20846.5	0.828025	0.414013	+	0.910271i	$0.364127\pi$
$860$	−2711.00	−0.107493
$861$	−41137.9	−1.62831
$862$	29802.8	1.17760
$863$	47623.4	1.87847	0.939235	−	0.343275i	$-0.111536\pi$
$863$	47623.4	1.87847	0.939235	+	0.343275i	$0.111536\pi$
$864$	864.000	0.0340207
$865$	11942.7	0.469438
$866$	−8458.97	−0.331925
$867$	−13843.3	−0.542264
$868$	−16327.7	−0.638476
$869$	−18778.3	−0.733037
$870$	6460.05	0.251743
$871$	−11860.1	−0.461382
$872$	953.147	0.0370156
$873$	−4760.77	−0.184568
$874$	0	0
$875$	36582.0	1.41337
$876$	6882.57	0.265457
$877$	9397.35	0.361831	0.180916	−	0.983499i	$-0.442094\pi$
$877$	9397.35	0.361831	0.180916	+	0.983499i	$0.442094\pi$
$878$	−21043.1	−0.808850
$879$	−29030.5	−1.11397
$880$	1769.39	0.0677796
$881$	49897.4	1.90816	0.954079	−	0.299555i	$-0.0968384\pi$
$881$	49897.4	1.90816	0.954079	+	0.299555i	$0.0968384\pi$
$882$	11675.7	0.445738
$883$	−3263.13	−0.124364	−0.0621818	−	0.998065i	$-0.519806\pi$
$883$	−3263.13	−0.124364	−0.0621818	+	0.998065i	$0.519806\pi$
$884$	−3889.68	−0.147991
$885$	−1818.16	−0.0690584
$886$	27734.8	1.05166
$887$	−38617.6	−1.46184	−0.730921	−	0.682462i	$-0.760909\pi$
$887$	−38617.6	−1.46184	−0.730921	+	0.682462i	$0.760909\pi$
$888$	−10565.1	−0.399258
$889$	−32378.5	−1.22153
$890$	2244.30	0.0845270
$891$	−1718.11	−0.0646004
$892$	12420.8	0.466230
$893$	0	0
$894$	10189.9	0.381209
$895$	−1139.40	−0.0425542
$896$	4030.78	0.150289
$897$	34832.5	1.29657
$898$	−19199.8	−0.713480
$899$	26769.1	0.993102
$900$	−3521.47	−0.130425
$901$	7037.47	0.260213
$902$	18473.0	0.681913
$903$	12281.0	0.452586
$904$	−8119.66	−0.298735
$905$	−19542.8	−0.717818
$906$	9794.31	0.359155
$907$	−580.444	−0.0212496	−0.0106248	−	0.999944i	$-0.503382\pi$
$907$	−580.444	−0.0212496	−0.0106248	+	0.999944i	$0.503382\pi$
$908$	−2658.27	−0.0971560
$909$	9258.62	0.337832
$910$	18478.9	0.673153
$911$	16704.1	0.607498	0.303749	−	0.952752i	$-0.401762\pi$
$911$	16704.1	0.607498	0.303749	+	0.952752i	$0.401762\pi$
$912$	0	0
$913$	12167.5	0.441056
$914$	−12522.2	−0.453170
$915$	5328.56	0.192521
$916$	−24128.0	−0.870317
$917$	−50043.6	−1.80217
$918$	933.077	0.0335470
$919$	46725.6	1.67719	0.838594	−	0.544756i	$-0.183378\pi$
$919$	46725.6	1.67719	0.838594	+	0.544756i	$0.183378\pi$
$920$	8605.19	0.308375
$921$	−20386.6	−0.729381
$922$	−19651.1	−0.701925
$923$	8892.72	0.317126
$924$	−8015.43	−0.285377
$925$	43060.9	1.53063
$926$	18553.1	0.658417
$927$	−974.702	−0.0345344
$928$	−6608.42	−0.233763
$929$	−18196.1	−0.642621	−0.321310	−	0.946974i	$-0.604123\pi$
$929$	−18196.1	−0.642621	−0.321310	+	0.946974i	$0.604123\pi$
$930$	4054.84	0.142971
$931$	0	0
$932$	−23032.6	−0.809506
$933$	−13255.4	−0.465126
$934$	36418.8	1.27587
$935$	1910.85	0.0668358
$936$	−4051.93	−0.141497
$937$	−28744.6	−1.00218	−0.501091	−	0.865395i	$-0.667068\pi$
$937$	−28744.6	−1.00218	−0.501091	+	0.865395i	$0.667068\pi$
$938$	13272.9	0.462022
$939$	−4174.68	−0.145086
$940$	−2262.32	−0.0784988
$941$	−11102.6	−0.384628	−0.192314	−	0.981333i	$-0.561599\pi$
$941$	−11102.6	−0.384628	−0.192314	+	0.981333i	$0.561599\pi$
$942$	17775.4	0.614814
$943$	89841.3	3.10248
$944$	1859.92	0.0641262
$945$	−4432.81	−0.152592
$946$	−5514.79	−0.189536
$947$	−34810.8	−1.19451	−0.597254	−	0.802052i	$-0.703741\pi$
$947$	−34810.8	−1.19451	−0.597254	+	0.802052i	$0.703741\pi$
$948$	10623.6	0.363964
$949$	−32277.4	−1.10408
$950$	0	0
$951$	−17996.0	−0.613629
$952$	4353.04	0.148196
$953$	−33236.8	−1.12974	−0.564872	−	0.825178i	$-0.691075\pi$
$953$	−33236.8	−1.12974	−0.564872	+	0.825178i	$0.691075\pi$
$954$	7331.04	0.248796
$955$	738.938	0.0250382
$956$	−2965.62	−0.100330
$957$	13141.2	0.443882
$958$	25483.4	0.859426
$959$	−73217.4	−2.46539
$960$	−1001.01	−0.0336536
$961$	−12988.6	−0.435991
$962$	49547.5	1.66058
$963$	16380.2	0.548126
$964$	−21096.3	−0.704840
$965$	−20384.6	−0.680003
$966$	−38982.0	−1.29837
$967$	−5203.54	−0.173045	−0.0865225	−	0.996250i	$-0.527575\pi$
$967$	−5203.54	−0.173045	−0.0865225	+	0.996250i	$0.527575\pi$
$968$	−7048.66	−0.234042
$969$	0	0
$970$	5515.70	0.182576
$971$	37036.4	1.22405	0.612026	−	0.790838i	$-0.290355\pi$
$971$	37036.4	1.22405	0.612026	+	0.790838i	$0.290355\pi$
$972$	972.000	0.0320750
$973$	19998.2	0.658903
$974$	10013.4	0.329416
$975$	16514.8	0.542457
$976$	−5450.95	−0.178771
$977$	−28630.1	−0.937520	−0.468760	−	0.883326i	$-0.655299\pi$
$977$	−28630.1	−0.937520	−0.468760	+	0.883326i	$0.655299\pi$
$978$	3761.83	0.122996
$979$	4565.42	0.149041
$980$	−13527.2	−0.440928
$981$	1072.29	0.0348987
$982$	1926.62	0.0626080
$983$	25662.6	0.832664	0.416332	−	0.909213i	$-0.363315\pi$
$983$	25662.6	0.832664	0.416332	+	0.909213i	$0.363315\pi$
$984$	−10450.9	−0.338579
$985$	−6272.77	−0.202911
$986$	−7136.77	−0.230508
$987$	10248.5	0.330508
$988$	0	0
$989$	−26820.5	−0.862327
$990$	1990.56	0.0639032
$991$	23629.7	0.757440	0.378720	−	0.925511i	$-0.376364\pi$
$991$	23629.7	0.757440	0.378720	+	0.925511i	$0.376364\pi$
$992$	−4147.97	−0.132760
$993$	19814.2	0.633217
$994$	−9952.07	−0.317566
$995$	−19308.3	−0.615190
$996$	−6883.58	−0.218991
$997$	51303.6	1.62969	0.814845	−	0.579679i	$-0.196822\pi$
$997$	51303.6	1.62969	0.814845	+	0.579679i	$0.196822\pi$
$998$	9929.69	0.314949
$999$	−11885.7	−0.376424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.w.1.2		3
19.8	odd	6		114.4.e.e.7.2	✓	6
19.12	odd	6		114.4.e.e.49.2	yes	6
19.18	odd	2		2166.4.a.s.1.2		3
57.8	even	6		342.4.g.g.235.2		6
57.50	even	6		342.4.g.g.163.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.e.7.2	✓	6	19.8	odd	6
114.4.e.e.49.2	yes	6	19.12	odd	6
342.4.g.g.163.2		6	57.50	even	6
342.4.g.g.235.2		6	57.8	even	6
2166.4.a.s.1.2		3	19.18	odd	2
2166.4.a.w.1.2		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.21359 q^{5} +6.00000 q^{6} +31.4905 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.21359 q^{5} +6.00000 q^{6} +31.4905 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.4272 q^{10} -21.2113 q^{11} +12.0000 q^{12} -56.2769 q^{13} +62.9809 q^{14} -15.6408 q^{15} +16.0000 q^{16} +17.2792 q^{17} +18.0000 q^{18} -20.8543 q^{20} +94.4714 q^{21} -42.4225 q^{22} -206.316 q^{23} +24.0000 q^{24} -97.8185 q^{25} -112.554 q^{26} +27.0000 q^{27} +125.962 q^{28} -206.513 q^{29} -31.2815 q^{30} -129.624 q^{31} +32.0000 q^{32} -63.6338 q^{33} +34.5584 q^{34} -164.178 q^{35} +36.0000 q^{36} -440.212 q^{37} -168.831 q^{39} -41.7087 q^{40} -435.454 q^{41} +188.943 q^{42} +129.997 q^{43} -84.8451 q^{44} -46.9223 q^{45} -412.633 q^{46} +108.482 q^{47} +48.0000 q^{48} +648.649 q^{49} -195.637 q^{50} +51.8376 q^{51} -225.107 q^{52} +407.280 q^{53} +54.0000 q^{54} +110.587 q^{55} +251.924 q^{56} -413.027 q^{58} +116.245 q^{59} -62.5630 q^{60} -340.684 q^{61} -259.248 q^{62} +283.414 q^{63} +64.0000 q^{64} +293.404 q^{65} -127.268 q^{66} +210.745 q^{67} +69.1168 q^{68} -618.949 q^{69} -328.356 q^{70} -158.017 q^{71} +72.0000 q^{72} +573.547 q^{73} -880.424 q^{74} -293.456 q^{75} -667.952 q^{77} -337.661 q^{78} +885.297 q^{79} -83.4174 q^{80} +81.0000 q^{81} -870.907 q^{82} -573.632 q^{83} +377.886 q^{84} -90.0866 q^{85} +259.994 q^{86} -619.540 q^{87} -169.690 q^{88} -215.235 q^{89} -93.8446 q^{90} -1772.18 q^{91} -825.266 q^{92} -388.872 q^{93} +216.964 q^{94} +96.0000 q^{96} -528.974 q^{97} +1297.30 q^{98} -190.901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 6 * q^2 + 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 + 24 * q^8 + 27 * q^9	\( 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} + 2 q^{5} + 18 q^{6} - 17 q^{7} + 24 q^{8} + 27 q^{9} + 4 q^{10} - 52 q^{11} + 36 q^{12} - 75 q^{13} - 34 q^{14} + 6 q^{15} + 48 q^{16} - 48 q^{17} + 54 q^{18} + 8 q^{20} - 51 q^{21} - 104 q^{22} - 238 q^{23} + 72 q^{24} + 229 q^{25} - 150 q^{26} + 81 q^{27} - 68 q^{28} + 8 q^{29} + 12 q^{30} - 107 q^{31} + 96 q^{32} - 156 q^{33} - 96 q^{34} - 294 q^{35} + 108 q^{36} - 305 q^{37} - 225 q^{39} + 16 q^{40} - 16 q^{41} - 102 q^{42} - 331 q^{43} - 208 q^{44} + 18 q^{45} - 476 q^{46} - 766 q^{47} + 144 q^{48} + 1142 q^{49} + 458 q^{50} - 144 q^{51} - 300 q^{52} + 118 q^{53} + 162 q^{54} - 1400 q^{55} - 136 q^{56} + 16 q^{58} - 936 q^{59} + 24 q^{60} - 399 q^{61} - 214 q^{62} - 153 q^{63} + 192 q^{64} - 370 q^{65} - 312 q^{66} - 61 q^{67} - 192 q^{68} - 714 q^{69} - 588 q^{70} - 974 q^{71} + 216 q^{72} + 91 q^{73} - 610 q^{74} + 687 q^{75} - 36 q^{77} - 450 q^{78} + 321 q^{79} + 32 q^{80} + 243 q^{81} - 32 q^{82} - 2148 q^{83} - 204 q^{84} - 1680 q^{85} - 662 q^{86} + 24 q^{87} - 416 q^{88} - 1116 q^{89} + 36 q^{90} - 1367 q^{91} - 952 q^{92} - 321 q^{93} - 1532 q^{94} + 288 q^{96} - 1382 q^{97} + 2284 q^{98} - 468 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 + 9 * q^3 + 12 * q^4 + 2 * q^5 + 18 * q^6 - 17 * q^7 + 24 * q^8 + 27 * q^9 + 4 * q^10 - 52 * q^11 + 36 * q^12 - 75 * q^13 - 34 * q^14 + 6 * q^15 + 48 * q^16 - 48 * q^17 + 54 * q^18 + 8 * q^20 - 51 * q^21 - 104 * q^22 - 238 * q^23 + 72 * q^24 + 229 * q^25 - 150 * q^26 + 81 * q^27 - 68 * q^28 + 8 * q^29 + 12 * q^30 - 107 * q^31 + 96 * q^32 - 156 * q^33 - 96 * q^34 - 294 * q^35 + 108 * q^36 - 305 * q^37 - 225 * q^39 + 16 * q^40 - 16 * q^41 - 102 * q^42 - 331 * q^43 - 208 * q^44 + 18 * q^45 - 476 * q^46 - 766 * q^47 + 144 * q^48 + 1142 * q^49 + 458 * q^50 - 144 * q^51 - 300 * q^52 + 118 * q^53 + 162 * q^54 - 1400 * q^55 - 136 * q^56 + 16 * q^58 - 936 * q^59 + 24 * q^60 - 399 * q^61 - 214 * q^62 - 153 * q^63 + 192 * q^64 - 370 * q^65 - 312 * q^66 - 61 * q^67 - 192 * q^68 - 714 * q^69 - 588 * q^70 - 974 * q^71 + 216 * q^72 + 91 * q^73 - 610 * q^74 + 687 * q^75 - 36 * q^77 - 450 * q^78 + 321 * q^79 + 32 * q^80 + 243 * q^81 - 32 * q^82 - 2148 * q^83 - 204 * q^84 - 1680 * q^85 - 662 * q^86 + 24 * q^87 - 416 * q^88 - 1116 * q^89 + 36 * q^90 - 1367 * q^91 - 952 * q^92 - 321 * q^93 - 1532 * q^94 + 288 * q^96 - 1382 * q^97 + 2284 * q^98 - 468 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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