LMFDB - Embedded newform 2166.4.a.n.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.36932 q^{5} -6.00000 q^{6} +26.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.36932 q^{5} -6.00000 q^{6} +26.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} +6.73863 q^{10} +5.26137 q^{11} +12.0000 q^{12} -66.1080 q^{13} -53.4773 q^{14} -10.1080 q^{15} +16.0000 q^{16} +23.2614 q^{17} -18.0000 q^{18} -13.4773 q^{20} +80.2159 q^{21} -10.5227 q^{22} -2.63068 q^{23} -24.0000 q^{24} -113.648 q^{25} +132.216 q^{26} +27.0000 q^{27} +106.955 q^{28} +27.0455 q^{29} +20.2159 q^{30} +32.9375 q^{31} -32.0000 q^{32} +15.7841 q^{33} -46.5227 q^{34} -90.0909 q^{35} +36.0000 q^{36} -7.58522 q^{37} -198.324 q^{39} +26.9545 q^{40} +371.080 q^{41} -160.432 q^{42} -25.0455 q^{43} +21.0455 q^{44} -30.3239 q^{45} +5.26137 q^{46} -268.938 q^{47} +48.0000 q^{48} +371.955 q^{49} +227.295 q^{50} +69.7841 q^{51} -264.432 q^{52} -536.773 q^{53} -54.0000 q^{54} -17.7272 q^{55} -213.909 q^{56} -54.0909 q^{58} +845.727 q^{59} -40.4318 q^{60} +834.250 q^{61} -65.8750 q^{62} +240.648 q^{63} +64.0000 q^{64} +222.739 q^{65} -31.5682 q^{66} +391.602 q^{67} +93.0455 q^{68} -7.89205 q^{69} +180.182 q^{70} +833.727 q^{71} -72.0000 q^{72} +134.830 q^{73} +15.1704 q^{74} -340.943 q^{75} +140.682 q^{77} +396.648 q^{78} +381.403 q^{79} -53.9091 q^{80} +81.0000 q^{81} -742.159 q^{82} -28.6137 q^{83} +320.864 q^{84} -78.3749 q^{85} +50.0909 q^{86} +81.1364 q^{87} -42.0909 q^{88} +511.477 q^{89} +60.6477 q^{90} -1767.64 q^{91} -10.5227 q^{92} +98.8125 q^{93} +537.875 q^{94} -96.0000 q^{96} -1430.10 q^{97} -743.909 q^{98} +47.3523 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9} - 36 q^{10} + 60 q^{11} + 24 q^{12} - 58 q^{13} - 8 q^{14} + 54 q^{15} + 32 q^{16} + 96 q^{17} - 36 q^{18} + 72 q^{20} + 12 q^{21} - 120 q^{22} - 30 q^{23} - 48 q^{24} + 218 q^{25} + 116 q^{26} + 54 q^{27} + 16 q^{28} + 252 q^{29} - 108 q^{30} + 338 q^{31} - 64 q^{32} + 180 q^{33} - 192 q^{34} - 576 q^{35} + 72 q^{36} + 158 q^{37} - 174 q^{39} - 144 q^{40} - 24 q^{42} - 248 q^{43} + 240 q^{44} + 162 q^{45} + 60 q^{46} - 810 q^{47} + 96 q^{48} + 546 q^{49} - 436 q^{50} + 288 q^{51} - 232 q^{52} - 84 q^{53} - 108 q^{54} + 1152 q^{55} - 32 q^{56} - 504 q^{58} + 504 q^{59} + 216 q^{60} + 580 q^{61} - 676 q^{62} + 36 q^{63} + 128 q^{64} + 396 q^{65} - 360 q^{66} + 140 q^{67} + 384 q^{68} - 90 q^{69} + 1152 q^{70} + 480 q^{71} - 144 q^{72} + 616 q^{73} - 316 q^{74} + 654 q^{75} - 1104 q^{77} + 348 q^{78} - 202 q^{79} + 288 q^{80} + 162 q^{81} - 552 q^{83} + 48 q^{84} + 1476 q^{85} + 496 q^{86} + 756 q^{87} - 480 q^{88} + 924 q^{89} - 324 q^{90} - 1952 q^{91} - 120 q^{92} + 1014 q^{93} + 1620 q^{94} - 192 q^{96} - 40 q^{97} - 1092 q^{98} + 540 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 - 16 * q^8 + 18 * q^9 - 36 * q^10 + 60 * q^11 + 24 * q^12 - 58 * q^13 - 8 * q^14 + 54 * q^15 + 32 * q^16 + 96 * q^17 - 36 * q^18 + 72 * q^20 + 12 * q^21 - 120 * q^22 - 30 * q^23 - 48 * q^24 + 218 * q^25 + 116 * q^26 + 54 * q^27 + 16 * q^28 + 252 * q^29 - 108 * q^30 + 338 * q^31 - 64 * q^32 + 180 * q^33 - 192 * q^34 - 576 * q^35 + 72 * q^36 + 158 * q^37 - 174 * q^39 - 144 * q^40 - 24 * q^42 - 248 * q^43 + 240 * q^44 + 162 * q^45 + 60 * q^46 - 810 * q^47 + 96 * q^48 + 546 * q^49 - 436 * q^50 + 288 * q^51 - 232 * q^52 - 84 * q^53 - 108 * q^54 + 1152 * q^55 - 32 * q^56 - 504 * q^58 + 504 * q^59 + 216 * q^60 + 580 * q^61 - 676 * q^62 + 36 * q^63 + 128 * q^64 + 396 * q^65 - 360 * q^66 + 140 * q^67 + 384 * q^68 - 90 * q^69 + 1152 * q^70 + 480 * q^71 - 144 * q^72 + 616 * q^73 - 316 * q^74 + 654 * q^75 - 1104 * q^77 + 348 * q^78 - 202 * q^79 + 288 * q^80 + 162 * q^81 - 552 * q^83 + 48 * q^84 + 1476 * q^85 + 496 * q^86 + 756 * q^87 - 480 * q^88 + 924 * q^89 - 324 * q^90 - 1952 * q^91 - 120 * q^92 + 1014 * q^93 + 1620 * q^94 - 192 * q^96 - 40 * q^97 - 1092 * q^98 + 540 * 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
$5$	−3.36932	−0.301361	−0.150680	−	0.988583i	$-0.548146\pi$
$5$	−3.36932	−0.301361	−0.150680	+	0.988583i	$0.548146\pi$
$6$	−6.00000	−0.408248
$7$	26.7386	1.44375	0.721875	−	0.692023i	$-0.243280\pi$
$7$	26.7386	1.44375	0.721875	+	0.692023i	$0.243280\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	6.73863	0.213094
$11$	5.26137	0.144215	0.0721073	−	0.997397i	$-0.477028\pi$
$11$	5.26137	0.144215	0.0721073	+	0.997397i	$0.477028\pi$
$12$	12.0000	0.288675
$13$	−66.1080	−1.41039	−0.705194	−	0.709014i	$-0.749140\pi$
$13$	−66.1080	−1.41039	−0.705194	+	0.709014i	$0.749140\pi$
$14$	−53.4773	−1.02089
$15$	−10.1080	−0.173991
$16$	16.0000	0.250000
$17$	23.2614	0.331865	0.165933	−	0.986137i	$-0.446937\pi$
$17$	23.2614	0.331865	0.165933	+	0.986137i	$0.446937\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	−13.4773	−0.150680
$21$	80.2159	0.833550
$22$	−10.5227	−0.101975
$23$	−2.63068	−0.0238494	−0.0119247	−	0.999929i	$-0.503796\pi$
$23$	−2.63068	−0.0238494	−0.0119247	+	0.999929i	$0.503796\pi$
$24$	−24.0000	−0.204124
$25$	−113.648	−0.909182
$26$	132.216	0.997295
$27$	27.0000	0.192450
$28$	106.955	0.721875
$29$	27.0455	0.173180	0.0865899	−	0.996244i	$-0.472403\pi$
$29$	27.0455	0.173180	0.0865899	+	0.996244i	$0.472403\pi$
$30$	20.2159	0.123030
$31$	32.9375	0.190831	0.0954154	−	0.995438i	$-0.469582\pi$
$31$	32.9375	0.190831	0.0954154	+	0.995438i	$0.469582\pi$
$32$	−32.0000	−0.176777
$33$	15.7841	0.0832624
$34$	−46.5227	−0.234664
$35$	−90.0909	−0.435090
$36$	36.0000	0.166667
$37$	−7.58522	−0.0337028	−0.0168514	−	0.999858i	$-0.505364\pi$
$37$	−7.58522	−0.0337028	−0.0168514	+	0.999858i	$0.505364\pi$
$38$	0	0
$39$	−198.324	−0.814288
$40$	26.9545	0.106547
$41$	371.080	1.41348	0.706742	−	0.707471i	$-0.250164\pi$
$41$	371.080	1.41348	0.706742	+	0.707471i	$0.250164\pi$
$42$	−160.432	−0.589409
$43$	−25.0455	−0.0888232	−0.0444116	−	0.999013i	$-0.514141\pi$
$43$	−25.0455	−0.0888232	−0.0444116	+	0.999013i	$0.514141\pi$
$44$	21.0455	0.0721073
$45$	−30.3239	−0.100454
$46$	5.26137	0.0168640
$47$	−268.938	−0.834650	−0.417325	−	0.908757i	$-0.637032\pi$
$47$	−268.938	−0.834650	−0.417325	+	0.908757i	$0.637032\pi$
$48$	48.0000	0.144338
$49$	371.955	1.08442
$50$	227.295	0.642888
$51$	69.7841	0.191603
$52$	−264.432	−0.705194
$53$	−536.773	−1.39116	−0.695579	−	0.718449i	$-0.744852\pi$
$53$	−536.773	−1.39116	−0.695579	+	0.718449i	$0.744852\pi$
$54$	−54.0000	−0.136083
$55$	−17.7272	−0.0434607
$56$	−213.909	−0.510443
$57$	0	0
$58$	−54.0909	−0.122457
$59$	845.727	1.86617	0.933087	−	0.359650i	$-0.117104\pi$
$59$	845.727	1.86617	0.933087	+	0.359650i	$0.117104\pi$
$60$	−40.4318	−0.0869954
$61$	834.250	1.75106	0.875531	−	0.483162i	$-0.160512\pi$
$61$	834.250	1.75106	0.875531	+	0.483162i	$0.160512\pi$
$62$	−65.8750	−0.134938
$63$	240.648	0.481250
$64$	64.0000	0.125000
$65$	222.739	0.425036
$66$	−31.5682	−0.0588754
$67$	391.602	0.714057	0.357029	−	0.934093i	$-0.383790\pi$
$67$	391.602	0.714057	0.357029	+	0.934093i	$0.383790\pi$
$68$	93.0455	0.165933
$69$	−7.89205	−0.0137694
$70$	180.182	0.307655
$71$	833.727	1.39359	0.696797	−	0.717268i	$-0.254608\pi$
$71$	833.727	1.39359	0.696797	+	0.717268i	$0.254608\pi$
$72$	−72.0000	−0.117851
$73$	134.830	0.216173	0.108086	−	0.994142i	$-0.465528\pi$
$73$	134.830	0.216173	0.108086	+	0.994142i	$0.465528\pi$
$74$	15.1704	0.0238315
$75$	−340.943	−0.524916
$76$	0	0
$77$	140.682	0.208210
$78$	396.648	0.575789
$79$	381.403	0.543180	0.271590	−	0.962413i	$-0.412451\pi$
$79$	381.403	0.543180	0.271590	+	0.962413i	$0.412451\pi$
$80$	−53.9091	−0.0753402
$81$	81.0000	0.111111
$82$	−742.159	−0.999485
$83$	−28.6137	−0.0378405	−0.0189202	−	0.999821i	$-0.506023\pi$
$83$	−28.6137	−0.0378405	−0.0189202	+	0.999821i	$0.506023\pi$
$84$	320.864	0.416775
$85$	−78.3749	−0.100011
$86$	50.0909	0.0628075
$87$	81.1364	0.0999855
$88$	−42.0909	−0.0509876
$89$	511.477	0.609174	0.304587	−	0.952485i	$-0.401482\pi$
$89$	511.477	0.609174	0.304587	+	0.952485i	$0.401482\pi$
$90$	60.6477	0.0710314
$91$	−1767.64	−2.03625
$92$	−10.5227	−0.0119247
$93$	98.8125	0.110176
$94$	537.875	0.590187
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1430.10	−1.49696	−0.748479	−	0.663159i	$-0.769215\pi$
$97$	−1430.10	−1.49696	−0.748479	+	0.663159i	$0.769215\pi$
$98$	−743.909	−0.766798
$99$	47.3523	0.0480716
$100$	−454.591	−0.454591
$101$	1697.31	1.67217	0.836084	−	0.548602i	$-0.184840\pi$
$101$	1697.31	1.67217	0.836084	+	0.548602i	$0.184840\pi$
$102$	−139.568	−0.135483
$103$	−1251.53	−1.19725	−0.598625	−	0.801029i	$-0.704286\pi$
$103$	−1251.53	−1.19725	−0.598625	+	0.801029i	$0.704286\pi$
$104$	528.864	0.498648
$105$	−270.273	−0.251199
$106$	1073.55	0.983698
$107$	481.477	0.435011	0.217505	−	0.976059i	$-0.430208\pi$
$107$	481.477	0.435011	0.217505	+	0.976059i	$0.430208\pi$
$108$	108.000	0.0962250
$109$	−234.756	−0.206289	−0.103145	−	0.994666i	$-0.532890\pi$
$109$	−234.756	−0.206289	−0.103145	+	0.994666i	$0.532890\pi$
$110$	35.4544	0.0307313
$111$	−22.7557	−0.0194583
$112$	427.818	0.360938
$113$	481.852	0.401140	0.200570	−	0.979679i	$-0.435721\pi$
$113$	481.852	0.401140	0.200570	+	0.979679i	$0.435721\pi$
$114$	0	0
$115$	8.86361	0.00718726
$116$	108.182	0.0865899
$117$	−594.972	−0.470129
$118$	−1691.45	−1.31958
$119$	621.977	0.479131
$120$	80.8636	0.0615150
$121$	−1303.32	−0.979202
$122$	−1668.50	−1.23819
$123$	1113.24	0.816076
$124$	131.750	0.0954154
$125$	804.080	0.575353
$126$	−481.295	−0.340295
$127$	2480.21	1.73294	0.866469	−	0.499231i	$-0.166384\pi$
$127$	2480.21	1.73294	0.866469	+	0.499231i	$0.166384\pi$
$128$	−128.000	−0.0883883
$129$	−75.1364	−0.0512821
$130$	−445.477	−0.300546
$131$	1845.05	1.23055	0.615276	−	0.788312i	$-0.289044\pi$
$131$	1845.05	1.23055	0.615276	+	0.788312i	$0.289044\pi$
$132$	63.1364	0.0416312
$133$	0	0
$134$	−783.204	−0.504915
$135$	−90.9716	−0.0579969
$136$	−186.091	−0.117332
$137$	−1860.19	−1.16005	−0.580026	−	0.814598i	$-0.696957\pi$
$137$	−1860.19	−1.16005	−0.580026	+	0.814598i	$0.696957\pi$
$138$	15.7841	0.00973646
$139$	−1442.34	−0.880128	−0.440064	−	0.897966i	$-0.645044\pi$
$139$	−1442.34	−0.880128	−0.440064	+	0.897966i	$0.645044\pi$
$140$	−360.364	−0.217545
$141$	−806.813	−0.481886
$142$	−1667.45	−0.985420
$143$	−347.818	−0.203399
$144$	144.000	0.0833333
$145$	−91.1247	−0.0521896
$146$	−269.659	−0.152857
$147$	1115.86	0.626088
$148$	−30.3409	−0.0168514
$149$	1145.96	0.630072	0.315036	−	0.949080i	$-0.397983\pi$
$149$	1145.96	0.630072	0.315036	+	0.949080i	$0.397983\pi$
$150$	681.886	0.371172
$151$	3319.56	1.78902	0.894510	−	0.447047i	$-0.147524\pi$
$151$	3319.56	1.78902	0.894510	+	0.447047i	$0.147524\pi$
$152$	0	0
$153$	209.352	0.110622
$154$	−281.363	−0.147227
$155$	−110.977	−0.0575089
$156$	−793.295	−0.407144
$157$	−3468.87	−1.76335	−0.881676	−	0.471855i	$-0.843585\pi$
$157$	−3468.87	−1.76335	−0.881676	+	0.471855i	$0.843585\pi$
$158$	−762.807	−0.384086
$159$	−1610.32	−0.803186
$160$	107.818	0.0532736
$161$	−70.3409	−0.0344325
$162$	−162.000	−0.0785674
$163$	−47.3863	−0.0227705	−0.0113852	−	0.999935i	$-0.503624\pi$
$163$	−47.3863	−0.0227705	−0.0113852	+	0.999935i	$0.503624\pi$
$164$	1484.32	0.706742
$165$	−53.1816	−0.0250920
$166$	57.2273	0.0267572
$167$	−1422.92	−0.659335	−0.329667	−	0.944097i	$-0.606937\pi$
$167$	−1422.92	−0.659335	−0.329667	+	0.944097i	$0.606937\pi$
$168$	−641.727	−0.294704
$169$	2173.26	0.989195
$170$	156.750	0.0707186
$171$	0	0
$172$	−100.182	−0.0444116
$173$	3590.78	1.57805	0.789024	−	0.614363i	$-0.210587\pi$
$173$	3590.78	1.57805	0.789024	+	0.614363i	$0.210587\pi$
$174$	−162.273	−0.0707004
$175$	−3038.78	−1.31263
$176$	84.1819	0.0360537
$177$	2537.18	1.07744
$178$	−1022.95	−0.430751
$179$	742.886	0.310201	0.155100	−	0.987899i	$-0.450430\pi$
$179$	742.886	0.310201	0.155100	+	0.987899i	$0.450430\pi$
$180$	−121.295	−0.0502268
$181$	−3362.32	−1.38077	−0.690386	−	0.723442i	$-0.742559\pi$
$181$	−3362.32	−1.38077	−0.690386	+	0.723442i	$0.742559\pi$
$182$	3535.27	1.43985
$183$	2502.75	1.01098
$184$	21.0455	0.00843202
$185$	25.5570	0.0101567
$186$	−197.625	−0.0779063
$187$	122.387	0.0478599
$188$	−1075.75	−0.417325
$189$	721.943	0.277850
$190$	0	0
$191$	583.324	0.220984	0.110492	−	0.993877i	$-0.464757\pi$
$191$	583.324	0.220984	0.110492	+	0.993877i	$0.464757\pi$
$192$	192.000	0.0721688
$193$	1953.17	0.728457	0.364229	−	0.931310i	$-0.381333\pi$
$193$	1953.17	0.728457	0.364229	+	0.931310i	$0.381333\pi$
$194$	2860.20	1.05851
$195$	668.216	0.245395
$196$	1487.82	0.542208
$197$	−2552.99	−0.923316	−0.461658	−	0.887058i	$-0.652745\pi$
$197$	−2552.99	−0.923316	−0.461658	+	0.887058i	$0.652745\pi$
$198$	−94.7046	−0.0339917
$199$	620.102	0.220894	0.110447	−	0.993882i	$-0.464772\pi$
$199$	620.102	0.220894	0.110447	+	0.993882i	$0.464772\pi$
$200$	909.182	0.321444
$201$	1174.81	0.412261
$202$	−3394.62	−1.18240
$203$	723.159	0.250029
$204$	279.136	0.0958013
$205$	−1250.28	−0.425969
$206$	2503.06	0.846584
$207$	−23.6761	−0.00794979
$208$	−1057.73	−0.352597
$209$	0	0
$210$	540.546	0.177625
$211$	2036.43	0.664425	0.332213	−	0.943205i	$-0.392205\pi$
$211$	2036.43	0.664425	0.332213	+	0.943205i	$0.392205\pi$
$212$	−2147.09	−0.695579
$213$	2501.18	0.804592
$214$	−962.955	−0.307599
$215$	84.3861	0.0267678
$216$	−216.000	−0.0680414
$217$	880.704	0.275512
$218$	469.511	0.145868
$219$	404.489	0.124807
$220$	−70.9088	−0.0217303
$221$	−1537.76	−0.468059
$222$	45.5113	0.0137591
$223$	1899.96	0.570541	0.285271	−	0.958447i	$-0.407916\pi$
$223$	1899.96	0.570541	0.285271	+	0.958447i	$0.407916\pi$
$224$	−855.636	−0.255221
$225$	−1022.83	−0.303061
$226$	−963.704	−0.283649
$227$	3042.48	0.889587	0.444794	−	0.895633i	$-0.353277\pi$
$227$	3042.48	0.889587	0.444794	+	0.895633i	$0.353277\pi$
$228$	0	0
$229$	1968.89	0.568156	0.284078	−	0.958801i	$-0.408313\pi$
$229$	1968.89	0.568156	0.284078	+	0.958801i	$0.408313\pi$
$230$	−17.7272	−0.00508216
$231$	422.045	0.120210
$232$	−216.364	−0.0612283
$233$	−2068.14	−0.581494	−0.290747	−	0.956800i	$-0.593904\pi$
$233$	−2068.14	−0.581494	−0.290747	+	0.956800i	$0.593904\pi$
$234$	1189.94	0.332432
$235$	906.136	0.251531
$236$	3382.91	0.933087
$237$	1144.21	0.313605
$238$	−1243.95	−0.338797
$239$	5041.54	1.36448	0.682239	−	0.731130i	$-0.261007\pi$
$239$	5041.54	1.36448	0.682239	+	0.731130i	$0.261007\pi$
$240$	−161.727	−0.0434977
$241$	−5214.90	−1.39386	−0.696932	−	0.717137i	$-0.745452\pi$
$241$	−5214.90	−1.39386	−0.696932	+	0.717137i	$0.745452\pi$
$242$	2606.64	0.692400
$243$	243.000	0.0641500
$244$	3337.00	0.875531
$245$	−1253.23	−0.326800
$246$	−2226.48	−0.577053
$247$	0	0
$248$	−263.500	−0.0674688
$249$	−85.8410	−0.0218472
$250$	−1608.16	−0.406836
$251$	3989.34	1.00321	0.501604	−	0.865098i	$-0.332744\pi$
$251$	3989.34	1.00321	0.501604	+	0.865098i	$0.332744\pi$
$252$	962.591	0.240625
$253$	−13.8410	−0.00343943
$254$	−4960.42	−1.22537
$255$	−235.125	−0.0577415
$256$	256.000	0.0625000
$257$	−7224.30	−1.75346	−0.876730	−	0.480983i	$-0.840280\pi$
$257$	−7224.30	−1.75346	−0.876730	+	0.480983i	$0.840280\pi$
$258$	150.273	0.0362619
$259$	−202.818	−0.0486584
$260$	890.955	0.212518
$261$	243.409	0.0577266
$262$	−3690.09	−0.870132
$263$	3236.93	0.758925	0.379463	−	0.925207i	$-0.376109\pi$
$263$	3236.93	0.758925	0.379463	+	0.925207i	$0.376109\pi$
$264$	−126.273	−0.0294377
$265$	1808.56	0.419241
$266$	0	0
$267$	1534.43	0.351707
$268$	1566.41	0.357029
$269$	4768.44	1.08081	0.540404	−	0.841406i	$-0.318272\pi$
$269$	4768.44	1.08081	0.540404	+	0.841406i	$0.318272\pi$
$270$	181.943	0.0410100
$271$	−1057.51	−0.237045	−0.118523	−	0.992951i	$-0.537816\pi$
$271$	−1057.51	−0.237045	−0.118523	+	0.992951i	$0.537816\pi$
$272$	372.182	0.0829663
$273$	−5302.91	−1.17563
$274$	3720.39	0.820280
$275$	−597.942	−0.131117
$276$	−31.5682	−0.00688472
$277$	3345.02	0.725570	0.362785	−	0.931873i	$-0.381826\pi$
$277$	3345.02	0.725570	0.362785	+	0.931873i	$0.381826\pi$
$278$	2884.68	0.622344
$279$	296.438	0.0636102
$280$	720.727	0.153828
$281$	−4429.85	−0.940437	−0.470219	−	0.882550i	$-0.655825\pi$
$281$	−4429.85	−0.940437	−0.470219	+	0.882550i	$0.655825\pi$
$282$	1613.63	0.340745
$283$	3548.00	0.745253	0.372627	−	0.927981i	$-0.378457\pi$
$283$	3548.00	0.745253	0.372627	+	0.927981i	$0.378457\pi$
$284$	3334.91	0.696797
$285$	0	0
$286$	695.636	0.143825
$287$	9922.16	2.04072
$288$	−288.000	−0.0589256
$289$	−4371.91	−0.889865
$290$	182.249	0.0369036
$291$	−4290.31	−0.864269
$292$	539.318	0.108086
$293$	5562.47	1.10909	0.554544	−	0.832154i	$-0.312893\pi$
$293$	5562.47	1.10909	0.554544	+	0.832154i	$0.312893\pi$
$294$	−2231.73	−0.442711
$295$	−2849.52	−0.562392
$296$	60.6817	0.0119157
$297$	142.057	0.0277541
$298$	−2291.92	−0.445528
$299$	173.909	0.0336369
$300$	−1363.77	−0.262458
$301$	−669.682	−0.128239
$302$	−6639.12	−1.26503
$303$	5091.94	0.965426
$304$	0	0
$305$	−2810.85	−0.527701
$306$	−418.705	−0.0782214
$307$	−6335.24	−1.17776	−0.588878	−	0.808222i	$-0.700430\pi$
$307$	−6335.24	−1.17776	−0.588878	+	0.808222i	$0.700430\pi$
$308$	562.727	0.104105
$309$	−3754.58	−0.691233
$310$	221.954	0.0406649
$311$	8480.79	1.54631	0.773153	−	0.634219i	$-0.218678\pi$
$311$	8480.79	1.54631	0.773153	+	0.634219i	$0.218678\pi$
$312$	1586.59	0.287894
$313$	8714.85	1.57378	0.786889	−	0.617094i	$-0.211690\pi$
$313$	8714.85	1.57378	0.786889	+	0.617094i	$0.211690\pi$
$314$	6937.75	1.24688
$315$	−810.818	−0.145030
$316$	1525.61	0.271590
$317$	−9466.03	−1.67718	−0.838589	−	0.544765i	$-0.816619\pi$
$317$	−9466.03	−1.67718	−0.838589	+	0.544765i	$0.816619\pi$
$318$	3220.64	0.567938
$319$	142.296	0.0249751
$320$	−215.636	−0.0376701
$321$	1444.43	0.251154
$322$	140.682	0.0243475
$323$	0	0
$324$	324.000	0.0555556
$325$	7513.02	1.28230
$326$	94.7727	0.0161011
$327$	−704.267	−0.119101
$328$	−2968.64	−0.499742
$329$	−7191.02	−1.20503
$330$	106.363	0.0177427
$331$	−6685.28	−1.11014	−0.555070	−	0.831804i	$-0.687309\pi$
$331$	−6685.28	−1.11014	−0.555070	+	0.831804i	$0.687309\pi$
$332$	−114.455	−0.0189202
$333$	−68.2670	−0.0112343
$334$	2845.84	0.466220
$335$	−1319.43	−0.215189
$336$	1283.45	0.208387
$337$	980.137	0.158432	0.0792158	−	0.996857i	$-0.474758\pi$
$337$	980.137	0.158432	0.0792158	+	0.996857i	$0.474758\pi$
$338$	−4346.52	−0.699466
$339$	1445.56	0.231598
$340$	−313.500	−0.0500056
$341$	173.296	0.0275206
$342$	0	0
$343$	774.205	0.121875
$344$	200.364	0.0314037
$345$	26.5908	0.00414957
$346$	−7181.57	−1.11585
$347$	−1556.08	−0.240734	−0.120367	−	0.992729i	$-0.538407\pi$
$347$	−1556.08	−0.240734	−0.120367	+	0.992729i	$0.538407\pi$
$348$	324.546	0.0499927
$349$	−5477.35	−0.840103	−0.420051	−	0.907500i	$-0.637988\pi$
$349$	−5477.35	−0.840103	−0.420051	+	0.907500i	$0.637988\pi$
$350$	6077.57	0.928171
$351$	−1784.91	−0.271429
$352$	−168.364	−0.0254938
$353$	−1017.89	−0.153475	−0.0767374	−	0.997051i	$-0.524450\pi$
$353$	−1017.89	−0.153475	−0.0767374	+	0.997051i	$0.524450\pi$
$354$	−5074.36	−0.761863
$355$	−2809.09	−0.419975
$356$	2045.91	0.304587
$357$	1865.93	0.276626
$358$	−1485.77	−0.219345
$359$	1419.28	0.208654	0.104327	−	0.994543i	$-0.466731\pi$
$359$	1419.28	0.208654	0.104327	+	0.994543i	$0.466731\pi$
$360$	242.591	0.0355157
$361$	0	0
$362$	6724.65	0.976353
$363$	−3909.95	−0.565343
$364$	−7070.55	−1.01812
$365$	−454.284	−0.0651460
$366$	−5005.50	−0.714868
$367$	−9545.99	−1.35776	−0.678878	−	0.734251i	$-0.737534\pi$
$367$	−9545.99	−1.35776	−0.678878	+	0.734251i	$0.737534\pi$
$368$	−42.0909	−0.00596234
$369$	3339.72	0.471162
$370$	−51.1140	−0.00718187
$371$	−14352.6	−2.00849
$372$	395.250	0.0550881
$373$	3139.05	0.435748	0.217874	−	0.975977i	$-0.430088\pi$
$373$	3139.05	0.435748	0.217874	+	0.975977i	$0.430088\pi$
$374$	−244.773	−0.0338420
$375$	2412.24	0.332180
$376$	2151.50	0.295093
$377$	−1787.92	−0.244251
$378$	−1443.89	−0.196470
$379$	12114.1	1.64185	0.820925	−	0.571037i	$-0.193459\pi$
$379$	12114.1	1.64185	0.820925	+	0.571037i	$0.193459\pi$
$380$	0	0
$381$	7440.63	1.00051
$382$	−1166.65	−0.156259
$383$	−1585.09	−0.211474	−0.105737	−	0.994394i	$-0.533720\pi$
$383$	−1585.09	−0.211474	−0.105737	+	0.994394i	$0.533720\pi$
$384$	−384.000	−0.0510310
$385$	−474.001	−0.0627464
$386$	−3906.34	−0.515097
$387$	−225.409	−0.0296077
$388$	−5720.41	−0.748479
$389$	8870.97	1.15624	0.578118	−	0.815953i	$-0.303787\pi$
$389$	8870.97	1.15624	0.578118	+	0.815953i	$0.303787\pi$
$390$	−1336.43	−0.173520
$391$	−61.1933	−0.00791478
$392$	−2975.64	−0.383399
$393$	5535.14	0.710460
$394$	5105.99	0.652883
$395$	−1285.07	−0.163693
$396$	189.409	0.0240358
$397$	11197.0	1.41552	0.707762	−	0.706451i	$-0.249705\pi$
$397$	11197.0	1.41552	0.707762	+	0.706451i	$0.249705\pi$
$398$	−1240.20	−0.156196
$399$	0	0
$400$	−1818.36	−0.227295
$401$	−624.420	−0.0777607	−0.0388803	−	0.999244i	$-0.512379\pi$
$401$	−624.420	−0.0777607	−0.0388803	+	0.999244i	$0.512379\pi$
$402$	−2349.61	−0.291513
$403$	−2177.43	−0.269145
$404$	6789.25	0.836084
$405$	−272.915	−0.0334845
$406$	−1446.32	−0.176797
$407$	−39.9086	−0.00486043
$408$	−558.273	−0.0677417
$409$	4488.35	0.542628	0.271314	−	0.962491i	$-0.412542\pi$
$409$	4488.35	0.542628	0.271314	+	0.962491i	$0.412542\pi$
$410$	2500.57	0.301206
$411$	−5580.58	−0.669756
$412$	−5006.11	−0.598625
$413$	22613.6	2.69429
$414$	47.3523	0.00562135
$415$	96.4085	0.0114036
$416$	2115.45	0.249324
$417$	−4327.02	−0.508142
$418$	0	0
$419$	16235.8	1.89301	0.946503	−	0.322694i	$-0.104588\pi$
$419$	16235.8	1.89301	0.946503	+	0.322694i	$0.104588\pi$
$420$	−1081.09	−0.125600
$421$	−6949.33	−0.804489	−0.402244	−	0.915532i	$-0.631770\pi$
$421$	−6949.33	−0.804489	−0.402244	+	0.915532i	$0.631770\pi$
$422$	−4072.86	−0.469820
$423$	−2420.44	−0.278217
$424$	4294.18	0.491849
$425$	−2643.60	−0.301726
$426$	−5002.36	−0.568933
$427$	22306.7	2.52810
$428$	1925.91	0.217505
$429$	−1043.45	−0.117432
$430$	−168.772	−0.0189277
$431$	10665.4	1.19196	0.595979	−	0.803000i	$-0.296764\pi$
$431$	10665.4	1.19196	0.595979	+	0.803000i	$0.296764\pi$
$432$	432.000	0.0481125
$433$	15703.4	1.74285	0.871426	−	0.490527i	$-0.163196\pi$
$433$	15703.4	1.74285	0.871426	+	0.490527i	$0.163196\pi$
$434$	−1761.41	−0.194816
$435$	−273.374	−0.0301317
$436$	−939.023	−0.103145
$437$	0	0
$438$	−808.977	−0.0882521
$439$	5701.39	0.619846	0.309923	−	0.950762i	$-0.399697\pi$
$439$	5701.39	0.619846	0.309923	+	0.950762i	$0.399697\pi$
$440$	141.818	0.0153657
$441$	3347.59	0.361472
$442$	3075.52	0.330968
$443$	9262.15	0.993359	0.496679	−	0.867934i	$-0.334553\pi$
$443$	9262.15	0.993359	0.496679	+	0.867934i	$0.334553\pi$
$444$	−91.0226	−0.00972915
$445$	−1723.33	−0.183581
$446$	−3799.92	−0.403434
$447$	3437.88	0.363772
$448$	1711.27	0.180469
$449$	−11341.6	−1.19208	−0.596040	−	0.802955i	$-0.703260\pi$
$449$	−11341.6	−1.19208	−0.596040	+	0.802955i	$0.703260\pi$
$450$	2045.66	0.214296
$451$	1952.39	0.203845
$452$	1927.41	0.200570
$453$	9958.69	1.03289
$454$	−6084.95	−0.629033
$455$	5955.73	0.613646
$456$	0	0
$457$	5838.61	0.597634	0.298817	−	0.954310i	$-0.403408\pi$
$457$	5838.61	0.597634	0.298817	+	0.954310i	$0.403408\pi$
$458$	−3937.77	−0.401747
$459$	628.057	0.0638675
$460$	35.4544	0.00359363
$461$	−18437.6	−1.86274	−0.931372	−	0.364069i	$-0.881387\pi$
$461$	−18437.6	−1.86274	−0.931372	+	0.364069i	$0.881387\pi$
$462$	−844.090	−0.0850014
$463$	−9458.81	−0.949434	−0.474717	−	0.880138i	$-0.657450\pi$
$463$	−9458.81	−0.949434	−0.474717	+	0.880138i	$0.657450\pi$
$464$	432.727	0.0432950
$465$	−332.931	−0.0332028
$466$	4136.27	0.411178
$467$	5344.64	0.529594	0.264797	−	0.964304i	$-0.414695\pi$
$467$	5344.64	0.529594	0.264797	+	0.964304i	$0.414695\pi$
$468$	−2379.89	−0.235065
$469$	10470.9	1.03092
$470$	−1812.27	−0.177859
$471$	−10406.6	−1.01807
$472$	−6765.82	−0.659792
$473$	−131.773	−0.0128096
$474$	−2288.42	−0.221752
$475$	0	0
$476$	2487.91	0.239565
$477$	−4830.95	−0.463719
$478$	−10083.1	−0.964831
$479$	18654.8	1.77946	0.889728	−	0.456491i	$-0.150894\pi$
$479$	18654.8	1.77946	0.889728	+	0.456491i	$0.150894\pi$
$480$	323.454	0.0307575
$481$	501.443	0.0475340
$482$	10429.8	0.985610
$483$	−211.023	−0.0198796
$484$	−5213.27	−0.489601
$485$	4818.47	0.451124
$486$	−486.000	−0.0453609
$487$	7734.88	0.719714	0.359857	−	0.933007i	$-0.382825\pi$
$487$	7734.88	0.719714	0.359857	+	0.933007i	$0.382825\pi$
$488$	−6674.00	−0.619094
$489$	−142.159	−0.0131465
$490$	2506.47	0.231083
$491$	18582.0	1.70793	0.853963	−	0.520333i	$-0.174192\pi$
$491$	18582.0	1.70793	0.853963	+	0.520333i	$0.174192\pi$
$492$	4452.95	0.408038
$493$	629.114	0.0574724
$494$	0	0
$495$	−159.545	−0.0144869
$496$	527.000	0.0477077
$497$	22292.7	2.01200
$498$	171.682	0.0154483
$499$	5684.20	0.509940	0.254970	−	0.966949i	$-0.417934\pi$
$499$	5684.20	0.509940	0.254970	+	0.966949i	$0.417934\pi$
$500$	3216.32	0.287676
$501$	−4268.76	−0.380667
$502$	−7978.68	−0.709375
$503$	1540.49	0.136555	0.0682776	−	0.997666i	$-0.478250\pi$
$503$	1540.49	0.136555	0.0682776	+	0.997666i	$0.478250\pi$
$504$	−1925.18	−0.170148
$505$	−5718.78	−0.503926
$506$	27.6820	0.00243204
$507$	6519.78	0.571112
$508$	9920.84	0.866469
$509$	−11411.5	−0.993728	−0.496864	−	0.867829i	$-0.665515\pi$
$509$	−11411.5	−0.993728	−0.496864	+	0.867829i	$0.665515\pi$
$510$	470.249	0.0408294
$511$	3605.16	0.312099
$512$	−512.000	−0.0441942
$513$	0	0
$514$	14448.6	1.23988
$515$	4216.80	0.360804
$516$	−300.546	−0.0256410
$517$	−1414.98	−0.120369
$518$	405.637	0.0344067
$519$	10772.4	0.911086
$520$	−1781.91	−0.150273
$521$	15346.7	1.29050	0.645252	−	0.763970i	$-0.276753\pi$
$521$	15346.7	1.29050	0.645252	+	0.763970i	$0.276753\pi$
$522$	−486.818	−0.0408189
$523$	8617.21	0.720467	0.360234	−	0.932862i	$-0.382697\pi$
$523$	8617.21	0.720467	0.360234	+	0.932862i	$0.382697\pi$
$524$	7380.18	0.615276
$525$	−9116.35	−0.757848
$526$	−6473.85	−0.536641
$527$	766.172	0.0633301
$528$	252.546	0.0208156
$529$	−12160.1	−0.999431
$530$	−3617.11	−0.296448
$531$	7611.54	0.622058
$532$	0	0
$533$	−24531.3	−1.99356
$534$	−3068.86	−0.248694
$535$	−1622.25	−0.131095
$536$	−3132.82	−0.252457
$537$	2228.66	0.179095
$538$	−9536.89	−0.764246
$539$	1956.99	0.156389
$540$	−363.886	−0.0289985
$541$	12948.6	1.02903	0.514513	−	0.857483i	$-0.327973\pi$
$541$	12948.6	1.02903	0.514513	+	0.857483i	$0.327973\pi$
$542$	2115.02	0.167616
$543$	−10087.0	−0.797189
$544$	−744.364	−0.0586660
$545$	790.966	0.0621675
$546$	10605.8	0.831295
$547$	11870.4	0.927867	0.463934	−	0.885870i	$-0.346438\pi$
$547$	11870.4	0.927867	0.463934	+	0.885870i	$0.346438\pi$
$548$	−7440.77	−0.580026
$549$	7508.25	0.583687
$550$	1195.88	0.0927140
$551$	0	0
$552$	63.1364	0.00486823
$553$	10198.2	0.784217
$554$	−6690.05	−0.513056
$555$	76.6710	0.00586397
$556$	−5769.36	−0.440064
$557$	−15448.6	−1.17518	−0.587591	−	0.809158i	$-0.699924\pi$
$557$	−15448.6	−1.17518	−0.587591	+	0.809158i	$0.699924\pi$
$558$	−592.875	−0.0449792
$559$	1655.70	0.125275
$560$	−1441.45	−0.108772
$561$	367.160	0.0276319
$562$	8859.70	0.664989
$563$	−4092.36	−0.306346	−0.153173	−	0.988199i	$-0.548949\pi$
$563$	−4092.36	−0.306346	−0.153173	+	0.988199i	$0.548949\pi$
$564$	−3227.25	−0.240943
$565$	−1623.51	−0.120888
$566$	−7096.00	−0.526974
$567$	2165.83	0.160417
$568$	−6669.82	−0.492710
$569$	−2382.57	−0.175540	−0.0877702	−	0.996141i	$-0.527974\pi$
$569$	−2382.57	−0.175540	−0.0877702	+	0.996141i	$0.527974\pi$
$570$	0	0
$571$	−1378.84	−0.101055	−0.0505277	−	0.998723i	$-0.516090\pi$
$571$	−1378.84	−0.101055	−0.0505277	+	0.998723i	$0.516090\pi$
$572$	−1391.27	−0.101699
$573$	1749.97	0.127585
$574$	−19844.3	−1.44301
$575$	298.971	0.0216834
$576$	576.000	0.0416667
$577$	21757.2	1.56978	0.784890	−	0.619635i	$-0.212719\pi$
$577$	21757.2	1.56978	0.784890	+	0.619635i	$0.212719\pi$
$578$	8743.82	0.629230
$579$	5859.51	0.420575
$580$	−364.499	−0.0260948
$581$	−765.090	−0.0546322
$582$	8580.61	0.611130
$583$	−2824.16	−0.200625
$584$	−1078.64	−0.0764286
$585$	2004.65	0.141679
$586$	−11124.9	−0.784244
$587$	−18746.5	−1.31815	−0.659073	−	0.752079i	$-0.729051\pi$
$587$	−18746.5	−1.31815	−0.659073	+	0.752079i	$0.729051\pi$
$588$	4463.45	0.313044
$589$	0	0
$590$	5699.05	0.397671
$591$	−7658.98	−0.533077
$592$	−121.363	−0.00842569
$593$	10329.0	0.715282	0.357641	−	0.933859i	$-0.383581\pi$
$593$	10329.0	0.715282	0.357641	+	0.933859i	$0.383581\pi$
$594$	−284.114	−0.0196251
$595$	−2095.64	−0.144391
$596$	4583.84	0.315036
$597$	1860.31	0.127533
$598$	−347.818	−0.0237849
$599$	19872.6	1.35555	0.677773	−	0.735272i	$-0.262945\pi$
$599$	19872.6	1.35555	0.677773	+	0.735272i	$0.262945\pi$
$600$	2727.54	0.185586
$601$	−21770.9	−1.47763	−0.738813	−	0.673910i	$-0.764614\pi$
$601$	−21770.9	−1.47763	−0.738813	+	0.673910i	$0.764614\pi$
$602$	1339.36	0.0906783
$603$	3524.42	0.238019
$604$	13278.2	0.894510
$605$	4391.29	0.295093
$606$	−10183.9	−0.682659
$607$	16597.2	1.10982	0.554909	−	0.831911i	$-0.312753\pi$
$607$	16597.2	1.10982	0.554909	+	0.831911i	$0.312753\pi$
$608$	0	0
$609$	2169.48	0.144354
$610$	5621.70	0.373141
$611$	17778.9	1.17718
$612$	837.409	0.0553109
$613$	−15932.4	−1.04976	−0.524881	−	0.851176i	$-0.675890\pi$
$613$	−15932.4	−1.04976	−0.524881	+	0.851176i	$0.675890\pi$
$614$	12670.5	0.832799
$615$	−3750.85	−0.245933
$616$	−1125.45	−0.0736134
$617$	6415.84	0.418626	0.209313	−	0.977849i	$-0.432877\pi$
$617$	6415.84	0.418626	0.209313	+	0.977849i	$0.432877\pi$
$618$	7509.17	0.488775
$619$	11499.5	0.746692	0.373346	−	0.927692i	$-0.378211\pi$
$619$	11499.5	0.746692	0.373346	+	0.927692i	$0.378211\pi$
$620$	−443.908	−0.0287545
$621$	−71.0284	−0.00458981
$622$	−16961.6	−1.09340
$623$	13676.2	0.879495
$624$	−3173.18	−0.203572
$625$	11496.8	0.735793
$626$	−17429.7	−1.11283
$627$	0	0
$628$	−13875.5	−0.881676
$629$	−176.443	−0.0111848
$630$	1621.64	0.102552
$631$	−8541.27	−0.538863	−0.269432	−	0.963020i	$-0.586836\pi$
$631$	−8541.27	−0.538863	−0.269432	+	0.963020i	$0.586836\pi$
$632$	−3051.23	−0.192043
$633$	6109.30	0.383606
$634$	18932.1	1.18594
$635$	−8356.61	−0.522240
$636$	−6441.27	−0.401593
$637$	−24589.2	−1.52945
$638$	−284.592	−0.0176601
$639$	7503.54	0.464532
$640$	431.273	0.0266368
$641$	−16663.5	−1.02678	−0.513392	−	0.858154i	$-0.671611\pi$
$641$	−16663.5	−1.02678	−0.513392	+	0.858154i	$0.671611\pi$
$642$	−2888.86	−0.177592
$643$	−7012.20	−0.430069	−0.215034	−	0.976606i	$-0.568986\pi$
$643$	−7012.20	−0.430069	−0.215034	+	0.976606i	$0.568986\pi$
$644$	−281.363	−0.0172163
$645$	253.158	0.0154544
$646$	0	0
$647$	−15676.2	−0.952544	−0.476272	−	0.879298i	$-0.658012\pi$
$647$	−15676.2	−0.952544	−0.476272	+	0.879298i	$0.658012\pi$
$648$	−648.000	−0.0392837
$649$	4449.68	0.269130
$650$	−15026.0	−0.906722
$651$	2642.11	0.159067
$652$	−189.545	−0.0113852
$653$	13923.6	0.834411	0.417206	−	0.908812i	$-0.363009\pi$
$653$	13923.6	0.834411	0.417206	+	0.908812i	$0.363009\pi$
$654$	1408.53	0.0842172
$655$	−6216.54	−0.370840
$656$	5937.27	0.353371
$657$	1213.47	0.0720576
$658$	14382.0	0.852083
$659$	−29862.2	−1.76520	−0.882600	−	0.470124i	$-0.844209\pi$
$659$	−29862.2	−1.76520	−0.882600	+	0.470124i	$0.844209\pi$
$660$	−212.727	−0.0125460
$661$	−6445.81	−0.379294	−0.189647	−	0.981852i	$-0.560734\pi$
$661$	−6445.81	−0.379294	−0.189647	+	0.981852i	$0.560734\pi$
$662$	13370.6	0.784988
$663$	−4613.28	−0.270234
$664$	228.909	0.0133786
$665$	0	0
$666$	136.534	0.00794382
$667$	−71.1480	−0.00413023
$668$	−5691.68	−0.329667
$669$	5699.88	0.329402
$670$	2638.86	0.152162
$671$	4389.29	0.252529
$672$	−2566.91	−0.147352
$673$	−17013.1	−0.974450	−0.487225	−	0.873276i	$-0.661991\pi$
$673$	−17013.1	−0.974450	−0.487225	+	0.873276i	$0.661991\pi$
$674$	−1960.27	−0.112028
$675$	−3068.49	−0.174972
$676$	8693.04	0.494597
$677$	−7.47727	−0.000424483 0	−0.000212241	−	1.00000i	$-0.500068\pi$
$677$	−7.47727	−0.000424483 0	−0.000212241		1.00000i	$0.500068\pi$
$678$	−2891.11	−0.163765
$679$	−38239.0	−2.16123
$680$	626.999	0.0353593
$681$	9127.43	0.513603
$682$	−346.593	−0.0194600
$683$	289.388	0.0162125	0.00810623	−	0.999967i	$-0.497420\pi$
$683$	289.388	0.0162125	0.00810623	+	0.999967i	$0.497420\pi$
$684$	0	0
$685$	6267.58	0.349594
$686$	−1548.41	−0.0861786
$687$	5906.66	0.328025
$688$	−400.727	−0.0222058
$689$	35484.9	1.96207
$690$	−53.1816	−0.00293419
$691$	33964.2	1.86984	0.934919	−	0.354861i	$-0.115472\pi$
$691$	33964.2	1.86984	0.934919	+	0.354861i	$0.115472\pi$
$692$	14363.1	0.789024
$693$	1266.14	0.0694033
$694$	3112.16	0.170225
$695$	4859.70	0.265236
$696$	−649.091	−0.0353502
$697$	8631.82	0.469087
$698$	10954.7	0.594042
$699$	−6204.41	−0.335726
$700$	−12155.1	−0.656316
$701$	9861.88	0.531353	0.265676	−	0.964062i	$-0.414405\pi$
$701$	9861.88	0.531353	0.265676	+	0.964062i	$0.414405\pi$
$702$	3569.83	0.191930
$703$	0	0
$704$	336.727	0.0180268
$705$	2718.41	0.145221
$706$	2035.77	0.108523
$707$	45383.8	2.41419
$708$	10148.7	0.538718
$709$	−8935.54	−0.473316	−0.236658	−	0.971593i	$-0.576052\pi$
$709$	−8935.54	−0.473316	−0.236658	+	0.971593i	$0.576052\pi$
$710$	5618.18	0.296967
$711$	3432.63	0.181060
$712$	−4091.82	−0.215376
$713$	−86.6482	−0.00455119
$714$	−3731.86	−0.195604
$715$	1171.91	0.0612964
$716$	2971.55	0.155100
$717$	15124.6	0.787781
$718$	−2838.56	−0.147540
$719$	−7575.87	−0.392952	−0.196476	−	0.980509i	$-0.562950\pi$
$719$	−7575.87	−0.392952	−0.196476	+	0.980509i	$0.562950\pi$
$720$	−485.182	−0.0251134
$721$	−33464.2	−1.72853
$722$	0	0
$723$	−15644.7	−0.804747
$724$	−13449.3	−0.690386
$725$	−3073.65	−0.157452
$726$	7819.91	0.399758
$727$	−17958.5	−0.916152	−0.458076	−	0.888913i	$-0.651461\pi$
$727$	−17958.5	−0.916152	−0.458076	+	0.888913i	$0.651461\pi$
$728$	14141.1	0.719923
$729$	729.000	0.0370370
$730$	908.567	0.0460652
$731$	−582.592	−0.0294773
$732$	10011.0	0.505488
$733$	3017.92	0.152073	0.0760365	−	0.997105i	$-0.475773\pi$
$733$	3017.92	0.152073	0.0760365	+	0.997105i	$0.475773\pi$
$734$	19092.0	0.960079
$735$	−3759.70	−0.188678
$736$	84.1819	0.00421601
$737$	2060.36	0.102978
$738$	−6679.43	−0.333162
$739$	−11885.3	−0.591618	−0.295809	−	0.955247i	$-0.595589\pi$
$739$	−11885.3	−0.591618	−0.295809	+	0.955247i	$0.595589\pi$
$740$	102.228	0.00507835
$741$	0	0
$742$	28705.1	1.42021
$743$	23819.1	1.17610	0.588048	−	0.808826i	$-0.299897\pi$
$743$	23819.1	1.17610	0.588048	+	0.808826i	$0.299897\pi$
$744$	−790.500	−0.0389532
$745$	−3861.10	−0.189879
$746$	−6278.10	−0.308120
$747$	−257.523	−0.0126135
$748$	489.546	0.0239299
$749$	12874.0	0.628047
$750$	−4824.48	−0.234887
$751$	−37361.9	−1.81539	−0.907693	−	0.419634i	$-0.862158\pi$
$751$	−37361.9	−1.81539	−0.907693	+	0.419634i	$0.862158\pi$
$752$	−4303.00	−0.208663
$753$	11968.0	0.579202
$754$	3575.84	0.172711
$755$	−11184.7	−0.539141
$756$	2887.77	0.138925
$757$	−11620.9	−0.557951	−0.278975	−	0.960298i	$-0.589995\pi$
$757$	−11620.9	−0.557951	−0.278975	+	0.960298i	$0.589995\pi$
$758$	−24228.2	−1.16096
$759$	−41.5230	−0.00198575
$760$	0	0
$761$	−30746.1	−1.46458	−0.732289	−	0.680994i	$-0.761548\pi$
$761$	−30746.1	−1.46458	−0.732289	+	0.680994i	$0.761548\pi$
$762$	−14881.3	−0.707469
$763$	−6277.05	−0.297830
$764$	2333.30	0.110492
$765$	−705.374	−0.0333371
$766$	3170.18	0.149534
$767$	−55909.3	−2.63203
$768$	768.000	0.0360844
$769$	−30276.0	−1.41974	−0.709871	−	0.704332i	$-0.751247\pi$
$769$	−30276.0	−1.41974	−0.709871	+	0.704332i	$0.751247\pi$
$770$	948.003	0.0443684
$771$	−21672.9	−1.01236
$772$	7812.68	0.364229
$773$	−273.057	−0.0127053	−0.00635263	−	0.999980i	$-0.502022\pi$
$773$	−273.057	−0.0127053	−0.00635263	+	0.999980i	$0.502022\pi$
$774$	450.818	0.0209358
$775$	−3743.27	−0.173500
$776$	11440.8	0.529254
$777$	−608.455	−0.0280929
$778$	−17741.9	−0.817583
$779$	0	0
$780$	2672.86	0.122697
$781$	4386.54	0.200977
$782$	122.387	0.00559659
$783$	730.228	0.0333285
$784$	5951.27	0.271104
$785$	11687.7	0.531405
$786$	−11070.3	−0.502371
$787$	−24071.8	−1.09030	−0.545150	−	0.838339i	$-0.683527\pi$
$787$	−24071.8	−1.09030	−0.545150	+	0.838339i	$0.683527\pi$
$788$	−10212.0	−0.461658
$789$	9710.78	0.438166
$790$	2570.14	0.115749
$791$	12884.1	0.579146
$792$	−378.818	−0.0169959
$793$	−55150.6	−2.46968
$794$	−22394.1	−1.00093
$795$	5425.67	0.242049
$796$	2480.41	0.110447
$797$	39851.1	1.77114	0.885570	−	0.464505i	$-0.153768\pi$
$797$	39851.1	1.77114	0.885570	+	0.464505i	$0.153768\pi$
$798$	0	0
$799$	−6255.85	−0.276991
$800$	3636.73	0.160722
$801$	4603.30	0.203058
$802$	1248.84	0.0549851
$803$	709.388	0.0311753
$804$	4699.23	0.206131
$805$	237.001	0.0103766
$806$	4354.86	0.190315
$807$	14305.3	0.624004
$808$	−13578.5	−0.591200
$809$	−11830.3	−0.514130	−0.257065	−	0.966394i	$-0.582756\pi$
$809$	−11830.3	−0.514130	−0.257065	+	0.966394i	$0.582756\pi$
$810$	545.829	0.0236771
$811$	33656.6	1.45727	0.728633	−	0.684904i	$-0.240156\pi$
$811$	33656.6	1.45727	0.728633	+	0.684904i	$0.240156\pi$
$812$	2892.64	0.125014
$813$	−3172.53	−0.136858
$814$	79.8172	0.00343685
$815$	159.660	0.00686212
$816$	1116.55	0.0479006
$817$	0	0
$818$	−8976.70	−0.383696
$819$	−15908.7	−0.678750
$820$	−5001.14	−0.212985
$821$	170.610	0.00725252	0.00362626	−	0.999993i	$-0.498846\pi$
$821$	170.610	0.00725252	0.00362626	+	0.999993i	$0.498846\pi$
$822$	11161.2	0.473589
$823$	−23528.0	−0.996516	−0.498258	−	0.867029i	$-0.666027\pi$
$823$	−23528.0	−0.996516	−0.498258	+	0.867029i	$0.666027\pi$
$824$	10012.2	0.423292
$825$	−1793.83	−0.0757006
$826$	−45227.2	−1.90515
$827$	40832.7	1.71692	0.858461	−	0.512880i	$-0.171421\pi$
$827$	40832.7	1.71692	0.858461	+	0.512880i	$0.171421\pi$
$828$	−94.7046	−0.00397489
$829$	17815.0	0.746371	0.373185	−	0.927757i	$-0.378266\pi$
$829$	17815.0	0.746371	0.373185	+	0.927757i	$0.378266\pi$
$830$	−192.817	−0.00806358
$831$	10035.1	0.418908
$832$	−4230.91	−0.176299
$833$	8652.17	0.359880
$834$	8654.05	0.359311
$835$	4794.27	0.198698
$836$	0	0
$837$	889.313	0.0367254
$838$	−32471.6	−1.33856
$839$	40098.6	1.65001	0.825004	−	0.565126i	$-0.191173\pi$
$839$	40098.6	1.65001	0.825004	+	0.565126i	$0.191173\pi$
$840$	2162.18	0.0888124
$841$	−23657.5	−0.970009
$842$	13898.7	0.568860
$843$	−13289.6	−0.542962
$844$	8145.73	0.332213
$845$	−7322.41	−0.298105
$846$	4840.88	0.196729
$847$	−34848.9	−1.41372
$848$	−8588.36	−0.347790
$849$	10644.0	0.430272
$850$	5287.20	0.213352
$851$	19.9543	0.000803789 0
$852$	10004.7	0.402296
$853$	18623.6	0.747549	0.373775	−	0.927520i	$-0.378063\pi$
$853$	18623.6	0.747549	0.373775	+	0.927520i	$0.378063\pi$
$854$	−44613.4	−1.78763
$855$	0	0
$856$	−3851.82	−0.153800
$857$	−41820.2	−1.66692	−0.833460	−	0.552580i	$-0.813643\pi$
$857$	−41820.2	−1.66692	−0.833460	+	0.552580i	$0.813643\pi$
$858$	2086.91	0.0830372
$859$	−31734.1	−1.26048	−0.630240	−	0.776400i	$-0.717044\pi$
$859$	−31734.1	−1.26048	−0.630240	+	0.776400i	$0.717044\pi$
$860$	337.544	0.0133839
$861$	29766.5	1.17821
$862$	−21330.8	−0.842842
$863$	−41420.3	−1.63379	−0.816897	−	0.576784i	$-0.804307\pi$
$863$	−41420.3	−1.63379	−0.816897	+	0.576784i	$0.804307\pi$
$864$	−864.000	−0.0340207
$865$	−12098.5	−0.475562
$866$	−31406.7	−1.23238
$867$	−13115.7	−0.513764
$868$	3522.82	0.137756
$869$	2006.70	0.0783346
$870$	546.748	0.0213063
$871$	−25888.0	−1.00710
$872$	1878.05	0.0729342
$873$	−12870.9	−0.498986
$874$	0	0
$875$	21500.0	0.830666
$876$	1617.95	0.0624037
$877$	−15683.1	−0.603853	−0.301927	−	0.953331i	$-0.597630\pi$
$877$	−15683.1	−0.603853	−0.301927	+	0.953331i	$0.597630\pi$
$878$	−11402.8	−0.438298
$879$	16687.4	0.640332
$880$	−283.635	−0.0108652
$881$	8031.86	0.307151	0.153576	−	0.988137i	$-0.450921\pi$
$881$	8031.86	0.307151	0.153576	+	0.988137i	$0.450921\pi$
$882$	−6695.18	−0.255599
$883$	30115.2	1.14774	0.573871	−	0.818946i	$-0.305441\pi$
$883$	30115.2	1.14774	0.573871	+	0.818946i	$0.305441\pi$
$884$	−6151.05	−0.234029
$885$	−8548.57	−0.324697
$886$	−18524.3	−0.702411
$887$	10727.2	0.406071	0.203036	−	0.979171i	$-0.434919\pi$
$887$	10727.2	0.406071	0.203036	+	0.979171i	$0.434919\pi$
$888$	182.045	0.00687955
$889$	66317.4	2.50193
$890$	3446.66	0.129812
$891$	426.171	0.0160239
$892$	7599.84	0.285271
$893$	0	0
$894$	−6875.76	−0.257226
$895$	−2503.02	−0.0934824
$896$	−3422.55	−0.127611
$897$	521.727	0.0194203
$898$	22683.2	0.842928
$899$	890.810	0.0330480
$900$	−4091.32	−0.151530
$901$	−12486.1	−0.461677
$902$	−3904.77	−0.144140
$903$	−2009.04	−0.0740386
$904$	−3854.82	−0.141824
$905$	11328.7	0.416110
$906$	−19917.4	−0.730365
$907$	−45248.5	−1.65651	−0.828254	−	0.560353i	$-0.810665\pi$
$907$	−45248.5	−1.65651	−0.828254	+	0.560353i	$0.810665\pi$
$908$	12169.9	0.444794
$909$	15275.8	0.557389
$910$	−11911.5	−0.433913
$911$	−28890.6	−1.05070	−0.525351	−	0.850886i	$-0.676066\pi$
$911$	−28890.6	−1.05070	−0.525351	+	0.850886i	$0.676066\pi$
$912$	0	0
$913$	−150.547	−0.00545715
$914$	−11677.2	−0.422591
$915$	−8432.56	−0.304669
$916$	7875.55	0.284078
$917$	49334.0	1.77661
$918$	−1256.11	−0.0451611
$919$	−9958.68	−0.357461	−0.178730	−	0.983898i	$-0.557199\pi$
$919$	−9958.68	−0.357461	−0.178730	+	0.983898i	$0.557199\pi$
$920$	−70.9088	−0.00254108
$921$	−19005.7	−0.679978
$922$	36875.2	1.31716
$923$	−55116.0	−1.96551
$924$	1688.18	0.0601051
$925$	862.043	0.0306419
$926$	18917.6	0.671351
$927$	−11263.8	−0.399083
$928$	−865.455	−0.0306142
$929$	31344.4	1.10697	0.553486	−	0.832859i	$-0.313297\pi$
$929$	31344.4	1.10697	0.553486	+	0.832859i	$0.313297\pi$
$930$	665.862	0.0234779
$931$	0	0
$932$	−8272.55	−0.290747
$933$	25442.4	0.892761
$934$	−10689.3	−0.374479
$935$	−412.359	−0.0144231
$936$	4759.77	0.166216
$937$	−48523.3	−1.69177	−0.845885	−	0.533366i	$-0.820927\pi$
$937$	−48523.3	−1.69177	−0.845885	+	0.533366i	$0.820927\pi$
$938$	−20941.8	−0.728971
$939$	26144.6	0.908621
$940$	3624.54	0.125765
$941$	−45816.6	−1.58722	−0.793612	−	0.608424i	$-0.791802\pi$
$941$	−45816.6	−1.58722	−0.793612	+	0.608424i	$0.791802\pi$
$942$	20813.2	0.719886
$943$	−976.193	−0.0337107
$944$	13531.6	0.466544
$945$	−2432.46	−0.0837331
$946$	263.547	0.00905776
$947$	−28705.5	−0.985008	−0.492504	−	0.870310i	$-0.663918\pi$
$947$	−28705.5	−0.985008	−0.492504	+	0.870310i	$0.663918\pi$
$948$	4576.84	0.156803
$949$	−8913.31	−0.304887
$950$	0	0
$951$	−28398.1	−0.968319
$952$	−4975.82	−0.169398
$953$	−11489.4	−0.390534	−0.195267	−	0.980750i	$-0.562557\pi$
$953$	−11489.4	−0.390534	−0.195267	+	0.980750i	$0.562557\pi$
$954$	9661.91	0.327899
$955$	−1965.40	−0.0665958
$956$	20166.2	0.682239
$957$	426.888	0.0144194
$958$	−37309.6	−1.25827
$959$	−49739.0	−1.67482
$960$	−646.909	−0.0217488
$961$	−28706.1	−0.963584
$962$	−1002.89	−0.0336116
$963$	4333.30	0.145004
$964$	−20859.6	−0.696932
$965$	−6580.85	−0.219529
$966$	422.045	0.0140570
$967$	−2401.60	−0.0798659	−0.0399329	−	0.999202i	$-0.512714\pi$
$967$	−2401.60	−0.0798659	−0.0399329	+	0.999202i	$0.512714\pi$
$968$	10426.5	0.346200
$969$	0	0
$970$	−9636.93	−0.318993
$971$	−24309.4	−0.803425	−0.401712	−	0.915766i	$-0.631585\pi$
$971$	−24309.4	−0.803425	−0.401712	+	0.915766i	$0.631585\pi$
$972$	972.000	0.0320750
$973$	−38566.2	−1.27068
$974$	−15469.8	−0.508915
$975$	22539.1	0.740336
$976$	13348.0	0.437765
$977$	−36258.3	−1.18731	−0.593657	−	0.804718i	$-0.702317\pi$
$977$	−36258.3	−1.18731	−0.593657	+	0.804718i	$0.702317\pi$
$978$	284.318	0.00929600
$979$	2691.07	0.0878518
$980$	−5012.93	−0.163400
$981$	−2112.80	−0.0687630
$982$	−37163.9	−1.20769
$983$	−52269.3	−1.69596	−0.847981	−	0.530026i	$-0.822182\pi$
$983$	−52269.3	−1.69596	−0.847981	+	0.530026i	$0.822182\pi$
$984$	−8905.91	−0.288526
$985$	8601.85	0.278251
$986$	−1258.23	−0.0406391
$987$	−21573.1	−0.695723
$988$	0	0
$989$	65.8867	0.00211838
$990$	319.090	0.0102438
$991$	13918.4	0.446147	0.223073	−	0.974802i	$-0.428391\pi$
$991$	13918.4	0.446147	0.223073	+	0.974802i	$0.428391\pi$
$992$	−1054.00	−0.0337344
$993$	−20055.9	−0.640940
$994$	−44585.5	−1.42270
$995$	−2089.32	−0.0665688
$996$	−343.364	−0.0109236
$997$	1886.44	0.0599240	0.0299620	−	0.999551i	$-0.490461\pi$
$997$	1886.44	0.0599240	0.0299620	+	0.999551i	$0.490461\pi$
$998$	−11368.4	−0.360582
$999$	−204.801	−0.00648610

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.n.1.1		2
19.18	odd	2		114.4.a.e.1.1	✓	2
57.56	even	2		342.4.a.f.1.2		2
76.75	even	2		912.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.e.1.1	✓	2	19.18	odd	2
342.4.a.f.1.2		2	57.56	even	2
912.4.a.m.1.1		2	76.75	even	2
2166.4.a.n.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.36932 q^{5} -6.00000 q^{6} +26.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -3.36932 q^{5} -6.00000 q^{6} +26.7386 q^{7} -8.00000 q^{8} +9.00000 q^{9} +6.73863 q^{10} +5.26137 q^{11} +12.0000 q^{12} -66.1080 q^{13} -53.4773 q^{14} -10.1080 q^{15} +16.0000 q^{16} +23.2614 q^{17} -18.0000 q^{18} -13.4773 q^{20} +80.2159 q^{21} -10.5227 q^{22} -2.63068 q^{23} -24.0000 q^{24} -113.648 q^{25} +132.216 q^{26} +27.0000 q^{27} +106.955 q^{28} +27.0455 q^{29} +20.2159 q^{30} +32.9375 q^{31} -32.0000 q^{32} +15.7841 q^{33} -46.5227 q^{34} -90.0909 q^{35} +36.0000 q^{36} -7.58522 q^{37} -198.324 q^{39} +26.9545 q^{40} +371.080 q^{41} -160.432 q^{42} -25.0455 q^{43} +21.0455 q^{44} -30.3239 q^{45} +5.26137 q^{46} -268.938 q^{47} +48.0000 q^{48} +371.955 q^{49} +227.295 q^{50} +69.7841 q^{51} -264.432 q^{52} -536.773 q^{53} -54.0000 q^{54} -17.7272 q^{55} -213.909 q^{56} -54.0909 q^{58} +845.727 q^{59} -40.4318 q^{60} +834.250 q^{61} -65.8750 q^{62} +240.648 q^{63} +64.0000 q^{64} +222.739 q^{65} -31.5682 q^{66} +391.602 q^{67} +93.0455 q^{68} -7.89205 q^{69} +180.182 q^{70} +833.727 q^{71} -72.0000 q^{72} +134.830 q^{73} +15.1704 q^{74} -340.943 q^{75} +140.682 q^{77} +396.648 q^{78} +381.403 q^{79} -53.9091 q^{80} +81.0000 q^{81} -742.159 q^{82} -28.6137 q^{83} +320.864 q^{84} -78.3749 q^{85} +50.0909 q^{86} +81.1364 q^{87} -42.0909 q^{88} +511.477 q^{89} +60.6477 q^{90} -1767.64 q^{91} -10.5227 q^{92} +98.8125 q^{93} +537.875 q^{94} -96.0000 q^{96} -1430.10 q^{97} -743.909 q^{98} +47.3523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} - 16 q^{8} + 18 q^{9} - 36 q^{10} + 60 q^{11} + 24 q^{12} - 58 q^{13} - 8 q^{14} + 54 q^{15} + 32 q^{16} + 96 q^{17} - 36 q^{18} + 72 q^{20} + 12 q^{21} - 120 q^{22} - 30 q^{23} - 48 q^{24} + 218 q^{25} + 116 q^{26} + 54 q^{27} + 16 q^{28} + 252 q^{29} - 108 q^{30} + 338 q^{31} - 64 q^{32} + 180 q^{33} - 192 q^{34} - 576 q^{35} + 72 q^{36} + 158 q^{37} - 174 q^{39} - 144 q^{40} - 24 q^{42} - 248 q^{43} + 240 q^{44} + 162 q^{45} + 60 q^{46} - 810 q^{47} + 96 q^{48} + 546 q^{49} - 436 q^{50} + 288 q^{51} - 232 q^{52} - 84 q^{53} - 108 q^{54} + 1152 q^{55} - 32 q^{56} - 504 q^{58} + 504 q^{59} + 216 q^{60} + 580 q^{61} - 676 q^{62} + 36 q^{63} + 128 q^{64} + 396 q^{65} - 360 q^{66} + 140 q^{67} + 384 q^{68} - 90 q^{69} + 1152 q^{70} + 480 q^{71} - 144 q^{72} + 616 q^{73} - 316 q^{74} + 654 q^{75} - 1104 q^{77} + 348 q^{78} - 202 q^{79} + 288 q^{80} + 162 q^{81} - 552 q^{83} + 48 q^{84} + 1476 q^{85} + 496 q^{86} + 756 q^{87} - 480 q^{88} + 924 q^{89} - 324 q^{90} - 1952 q^{91} - 120 q^{92} + 1014 q^{93} + 1620 q^{94} - 192 q^{96} - 40 q^{97} - 1092 q^{98} + 540 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 - 16 * q^8 + 18 * q^9 - 36 * q^10 + 60 * q^11 + 24 * q^12 - 58 * q^13 - 8 * q^14 + 54 * q^15 + 32 * q^16 + 96 * q^17 - 36 * q^18 + 72 * q^20 + 12 * q^21 - 120 * q^22 - 30 * q^23 - 48 * q^24 + 218 * q^25 + 116 * q^26 + 54 * q^27 + 16 * q^28 + 252 * q^29 - 108 * q^30 + 338 * q^31 - 64 * q^32 + 180 * q^33 - 192 * q^34 - 576 * q^35 + 72 * q^36 + 158 * q^37 - 174 * q^39 - 144 * q^40 - 24 * q^42 - 248 * q^43 + 240 * q^44 + 162 * q^45 + 60 * q^46 - 810 * q^47 + 96 * q^48 + 546 * q^49 - 436 * q^50 + 288 * q^51 - 232 * q^52 - 84 * q^53 - 108 * q^54 + 1152 * q^55 - 32 * q^56 - 504 * q^58 + 504 * q^59 + 216 * q^60 + 580 * q^61 - 676 * q^62 + 36 * q^63 + 128 * q^64 + 396 * q^65 - 360 * q^66 + 140 * q^67 + 384 * q^68 - 90 * q^69 + 1152 * q^70 + 480 * q^71 - 144 * q^72 + 616 * q^73 - 316 * q^74 + 654 * q^75 - 1104 * q^77 + 348 * q^78 - 202 * q^79 + 288 * q^80 + 162 * q^81 - 552 * q^83 + 48 * q^84 + 1476 * q^85 + 496 * q^86 + 756 * q^87 - 480 * q^88 + 924 * q^89 - 324 * q^90 - 1952 * q^91 - 120 * q^92 + 1014 * q^93 + 1620 * q^94 - 192 * q^96 - 40 * q^97 - 1092 * q^98 + 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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