LMFDB - Embedded newform 2166.4.a.ba.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:	$4$
Coefficient field:	4.4.184225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 45x^{2} + 46x + 449 $ x^4 - 2x^3 - 45x^2 + 46*x + 449
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17400$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0134 q^{5} +6.00000 q^{6} +1.83039 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0134 q^{5} +6.00000 q^{6} +1.83039 q^{7} +8.00000 q^{8} +9.00000 q^{9} -28.0269 q^{10} -65.5442 q^{11} +12.0000 q^{12} +80.7791 q^{13} +3.66079 q^{14} -42.0403 q^{15} +16.0000 q^{16} +17.9597 q^{17} +18.0000 q^{18} -56.0538 q^{20} +5.49118 q^{21} -131.088 q^{22} +107.049 q^{23} +24.0000 q^{24} +71.3768 q^{25} +161.558 q^{26} +27.0000 q^{27} +7.32158 q^{28} -183.335 q^{29} -84.0807 q^{30} +105.498 q^{31} +32.0000 q^{32} -196.633 q^{33} +35.9193 q^{34} -25.6501 q^{35} +36.0000 q^{36} -294.966 q^{37} +242.337 q^{39} -112.108 q^{40} -325.731 q^{41} +10.9824 q^{42} +88.2853 q^{43} -262.177 q^{44} -126.121 q^{45} +214.097 q^{46} +416.954 q^{47} +48.0000 q^{48} -339.650 q^{49} +142.754 q^{50} +53.8790 q^{51} +323.117 q^{52} +292.522 q^{53} +54.0000 q^{54} +918.501 q^{55} +14.6432 q^{56} -366.670 q^{58} +43.7141 q^{59} -168.161 q^{60} -711.651 q^{61} +210.996 q^{62} +16.4735 q^{63} +64.0000 q^{64} -1131.99 q^{65} -393.265 q^{66} +953.869 q^{67} +71.8387 q^{68} +321.146 q^{69} -51.3003 q^{70} -753.109 q^{71} +72.0000 q^{72} +201.722 q^{73} -589.932 q^{74} +214.130 q^{75} -119.972 q^{77} +484.675 q^{78} -865.863 q^{79} -224.215 q^{80} +81.0000 q^{81} -651.462 q^{82} -642.216 q^{83} +21.9647 q^{84} -251.677 q^{85} +176.571 q^{86} -550.006 q^{87} -524.354 q^{88} -1423.94 q^{89} -252.242 q^{90} +147.858 q^{91} +428.195 q^{92} +316.494 q^{93} +833.908 q^{94} +96.0000 q^{96} -264.678 q^{97} -679.299 q^{98} -589.898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9} - 56 q^{10} - 72 q^{11} + 48 q^{12} - 8 q^{13} - 34 q^{14} - 84 q^{15} + 64 q^{16} + 33 q^{17} + 72 q^{18} - 112 q^{20} - 51 q^{21} - 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} + 108 q^{27} - 68 q^{28} - 75 q^{29} - 168 q^{30} + 419 q^{31} + 128 q^{32} - 216 q^{33} + 66 q^{34} + 331 q^{35} + 144 q^{36} + 131 q^{37} - 24 q^{39} - 224 q^{40} - 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} - 176 q^{46} + 395 q^{47} + 192 q^{48} - 1033 q^{49} + 188 q^{50} + 99 q^{51} - 32 q^{52} - 625 q^{53} + 216 q^{54} + 413 q^{55} - 136 q^{56} - 150 q^{58} - 1328 q^{59} - 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} - 1817 q^{65} - 432 q^{66} - 99 q^{67} + 132 q^{68} - 264 q^{69} + 662 q^{70} - 1307 q^{71} + 288 q^{72} + 183 q^{73} + 262 q^{74} + 282 q^{75} - 513 q^{77} - 48 q^{78} - 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} - 204 q^{84} - 867 q^{85} + 152 q^{86} - 225 q^{87} - 576 q^{88} + 207 q^{89} - 504 q^{90} - 183 q^{91} - 352 q^{92} + 1257 q^{93} + 790 q^{94} + 384 q^{96} - 1331 q^{97} - 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9 - 56 * q^10 - 72 * q^11 + 48 * q^12 - 8 * q^13 - 34 * q^14 - 84 * q^15 + 64 * q^16 + 33 * q^17 + 72 * q^18 - 112 * q^20 - 51 * q^21 - 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 + 108 * q^27 - 68 * q^28 - 75 * q^29 - 168 * q^30 + 419 * q^31 + 128 * q^32 - 216 * q^33 + 66 * q^34 + 331 * q^35 + 144 * q^36 + 131 * q^37 - 24 * q^39 - 224 * q^40 - 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 - 176 * q^46 + 395 * q^47 + 192 * q^48 - 1033 * q^49 + 188 * q^50 + 99 * q^51 - 32 * q^52 - 625 * q^53 + 216 * q^54 + 413 * q^55 - 136 * q^56 - 150 * q^58 - 1328 * q^59 - 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 - 1817 * q^65 - 432 * q^66 - 99 * q^67 + 132 * q^68 - 264 * q^69 + 662 * q^70 - 1307 * q^71 + 288 * q^72 + 183 * q^73 + 262 * q^74 + 282 * q^75 - 513 * q^77 - 48 * q^78 - 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 - 204 * q^84 - 867 * q^85 + 152 * q^86 - 225 * q^87 - 576 * q^88 + 207 * q^89 - 504 * q^90 - 183 * q^91 - 352 * q^92 + 1257 * q^93 + 790 * q^94 + 384 * q^96 - 1331 * q^97 - 2066 * q^98 - 648 * 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
$5$	−14.0134	−1.25340	−0.626701	−	0.779260i	$-0.715595\pi$
$5$	−14.0134	−1.25340	−0.626701	+	0.779260i	$0.715595\pi$
$6$	6.00000	0.408248
$7$	1.83039	0.0988320	0.0494160	−	0.998778i	$-0.484264\pi$
$7$	1.83039	0.0988320	0.0494160	+	0.998778i	$0.484264\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−28.0269	−0.886288
$11$	−65.5442	−1.79658	−0.898288	−	0.439408i	$-0.855188\pi$
$11$	−65.5442	−1.79658	−0.898288	+	0.439408i	$0.855188\pi$
$12$	12.0000	0.288675
$13$	80.7791	1.72339	0.861696	−	0.507424i	$-0.169402\pi$
$13$	80.7791	1.72339	0.861696	+	0.507424i	$0.169402\pi$
$14$	3.66079	0.0698848
$15$	−42.0403	−0.723651
$16$	16.0000	0.250000
$17$	17.9597	0.256227	0.128114	−	0.991760i	$-0.459108\pi$
$17$	17.9597	0.256227	0.128114	+	0.991760i	$0.459108\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	−56.0538	−0.626701
$21$	5.49118	0.0570607
$22$	−131.088	−1.27037
$23$	107.049	0.970486	0.485243	−	0.874379i	$-0.338731\pi$
$23$	107.049	0.970486	0.485243	+	0.874379i	$0.338731\pi$
$24$	24.0000	0.204124
$25$	71.3768	0.571014
$26$	161.558	1.21862
$27$	27.0000	0.192450
$28$	7.32158	0.0494160
$29$	−183.335	−1.17395	−0.586974	−	0.809606i	$-0.699681\pi$
$29$	−183.335	−1.17395	−0.586974	+	0.809606i	$0.699681\pi$
$30$	−84.0807	−0.511699
$31$	105.498	0.611226	0.305613	−	0.952156i	$-0.401139\pi$
$31$	105.498	0.611226	0.305613	+	0.952156i	$0.401139\pi$
$32$	32.0000	0.176777
$33$	−196.633	−1.03725
$34$	35.9193	0.181180
$35$	−25.6501	−0.123876
$36$	36.0000	0.166667
$37$	−294.966	−1.31060	−0.655299	−	0.755370i	$-0.727457\pi$
$37$	−294.966	−1.31060	−0.655299	+	0.755370i	$0.727457\pi$
$38$	0	0
$39$	242.337	0.995001
$40$	−112.108	−0.443144
$41$	−325.731	−1.24075	−0.620374	−	0.784306i	$-0.713019\pi$
$41$	−325.731	−1.24075	−0.620374	+	0.784306i	$0.713019\pi$
$42$	10.9824	0.0403480
$43$	88.2853	0.313102	0.156551	−	0.987670i	$-0.449962\pi$
$43$	88.2853	0.313102	0.156551	+	0.987670i	$0.449962\pi$
$44$	−262.177	−0.898288
$45$	−126.121	−0.417800
$46$	214.097	0.686237
$47$	416.954	1.29402	0.647011	−	0.762481i	$-0.276019\pi$
$47$	416.954	1.29402	0.647011	+	0.762481i	$0.276019\pi$
$48$	48.0000	0.144338
$49$	−339.650	−0.990232
$50$	142.754	0.403768
$51$	53.8790	0.147933
$52$	323.117	0.861696
$53$	292.522	0.758131	0.379066	−	0.925370i	$-0.376245\pi$
$53$	292.522	0.758131	0.379066	+	0.925370i	$0.376245\pi$
$54$	54.0000	0.136083
$55$	918.501	2.25183
$56$	14.6432	0.0349424
$57$	0	0
$58$	−366.670	−0.830107
$59$	43.7141	0.0964591	0.0482296	−	0.998836i	$-0.484642\pi$
$59$	43.7141	0.0964591	0.0482296	+	0.998836i	$0.484642\pi$
$60$	−168.161	−0.361826
$61$	−711.651	−1.49373	−0.746865	−	0.664976i	$-0.768442\pi$
$61$	−711.651	−1.49373	−0.746865	+	0.664976i	$0.768442\pi$
$62$	210.996	0.432202
$63$	16.4735	0.0329440
$64$	64.0000	0.125000
$65$	−1131.99	−2.16010
$66$	−393.265	−0.733449
$67$	953.869	1.73931	0.869654	−	0.493662i	$-0.164342\pi$
$67$	953.869	1.73931	0.869654	+	0.493662i	$0.164342\pi$
$68$	71.8387	0.128114
$69$	321.146	0.560311
$70$	−51.3003	−0.0875936
$71$	−753.109	−1.25884	−0.629420	−	0.777066i	$-0.716707\pi$
$71$	−753.109	−1.25884	−0.629420	+	0.777066i	$0.716707\pi$
$72$	72.0000	0.117851
$73$	201.722	0.323422	0.161711	−	0.986838i	$-0.448299\pi$
$73$	201.722	0.323422	0.161711	+	0.986838i	$0.448299\pi$
$74$	−589.932	−0.926733
$75$	214.130	0.329675
$76$	0	0
$77$	−119.972	−0.177559
$78$	484.675	0.703572
$79$	−865.863	−1.23313	−0.616565	−	0.787304i	$-0.711476\pi$
$79$	−865.863	−1.23313	−0.616565	+	0.787304i	$0.711476\pi$
$80$	−224.215	−0.313350
$81$	81.0000	0.111111
$82$	−651.462	−0.877341
$83$	−642.216	−0.849305	−0.424653	−	0.905356i	$-0.639604\pi$
$83$	−642.216	−0.849305	−0.424653	+	0.905356i	$0.639604\pi$
$84$	21.9647	0.0285303
$85$	−251.677	−0.321155
$86$	176.571	0.221396
$87$	−550.006	−0.677779
$88$	−524.354	−0.635185
$89$	−1423.94	−1.69592	−0.847961	−	0.530058i	$-0.822170\pi$
$89$	−1423.94	−1.69592	−0.847961	+	0.530058i	$0.822170\pi$
$90$	−252.242	−0.295429
$91$	147.858	0.170326
$92$	428.195	0.485243
$93$	316.494	0.352892
$94$	833.908	0.915011
$95$	0	0
$96$	96.0000	0.102062
$97$	−264.678	−0.277052	−0.138526	−	0.990359i	$-0.544236\pi$
$97$	−264.678	−0.277052	−0.138526	+	0.990359i	$0.544236\pi$
$98$	−679.299	−0.700200
$99$	−589.898	−0.598859
$100$	285.507	0.285507
$101$	−953.180	−0.939059	−0.469529	−	0.882917i	$-0.655576\pi$
$101$	−953.180	−0.939059	−0.469529	+	0.882917i	$0.655576\pi$
$102$	107.758	0.104604
$103$	66.1075	0.0632404	0.0316202	−	0.999500i	$-0.489933\pi$
$103$	66.1075	0.0632404	0.0316202	+	0.999500i	$0.489933\pi$
$104$	646.233	0.609311
$105$	−76.9504	−0.0715199
$106$	585.043	0.536080
$107$	−1827.15	−1.65081	−0.825406	−	0.564539i	$-0.809054\pi$
$107$	−1827.15	−1.65081	−0.825406	+	0.564539i	$0.809054\pi$
$108$	108.000	0.0962250
$109$	91.2444	0.0801800	0.0400900	−	0.999196i	$-0.487236\pi$
$109$	91.2444	0.0801800	0.0400900	+	0.999196i	$0.487236\pi$
$110$	1837.00	1.59228
$111$	−884.898	−0.756674
$112$	29.2863	0.0247080
$113$	−403.769	−0.336136	−0.168068	−	0.985775i	$-0.553753\pi$
$113$	−403.769	−0.336136	−0.168068	+	0.985775i	$0.553753\pi$
$114$	0	0
$115$	−1500.12	−1.21641
$116$	−733.341	−0.586974
$117$	727.012	0.574464
$118$	87.4282	0.0682069
$119$	32.8733	0.0253234
$120$	−336.323	−0.255849
$121$	2965.05	2.22768
$122$	−1423.30	−1.05623
$123$	−977.193	−0.716346
$124$	421.992	0.305613
$125$	751.447	0.537691
$126$	32.9471	0.0232949
$127$	43.6835	0.0305219	0.0152610	−	0.999884i	$-0.495142\pi$
$127$	43.6835	0.0305219	0.0152610	+	0.999884i	$0.495142\pi$
$128$	128.000	0.0883883
$129$	264.856	0.180769
$130$	−2263.99	−1.52742
$131$	2019.60	1.34697	0.673485	−	0.739201i	$-0.264797\pi$
$131$	2019.60	1.34697	0.673485	+	0.739201i	$0.264797\pi$
$132$	−786.531	−0.518627
$133$	0	0
$134$	1907.74	1.22988
$135$	−378.363	−0.241217
$136$	143.677	0.0905899
$137$	−1327.21	−0.827673	−0.413836	−	0.910351i	$-0.635811\pi$
$137$	−1327.21	−0.827673	−0.413836	+	0.910351i	$0.635811\pi$
$138$	642.292	0.396199
$139$	−432.830	−0.264116	−0.132058	−	0.991242i	$-0.542159\pi$
$139$	−432.830	−0.264116	−0.132058	+	0.991242i	$0.542159\pi$
$140$	−102.601	−0.0619381
$141$	1250.86	0.747104
$142$	−1506.22	−0.890134
$143$	−5294.61	−3.09620
$144$	144.000	0.0833333
$145$	2569.16	1.47143
$146$	403.445	0.228694
$147$	−1018.95	−0.571711
$148$	−1179.86	−0.655299
$149$	−1271.50	−0.699095	−0.349547	−	0.936919i	$-0.613665\pi$
$149$	−1271.50	−0.699095	−0.349547	+	0.936919i	$0.613665\pi$
$150$	428.261	0.233116
$151$	−797.383	−0.429736	−0.214868	−	0.976643i	$-0.568932\pi$
$151$	−797.383	−0.429736	−0.214868	+	0.976643i	$0.568932\pi$
$152$	0	0
$153$	161.637	0.0854090
$154$	−239.944	−0.125553
$155$	−1478.39	−0.766111
$156$	969.350	0.497501
$157$	−2418.11	−1.22921	−0.614606	−	0.788835i	$-0.710685\pi$
$157$	−2418.11	−1.22921	−0.614606	+	0.788835i	$0.710685\pi$
$158$	−1731.73	−0.871954
$159$	877.565	0.437707
$160$	−448.430	−0.221572
$161$	195.941	0.0959151
$162$	162.000	0.0785674
$163$	−1186.19	−0.569998	−0.284999	−	0.958528i	$-0.591993\pi$
$163$	−1186.19	−0.569998	−0.284999	+	0.958528i	$0.591993\pi$
$164$	−1302.92	−0.620374
$165$	2755.50	1.30009
$166$	−1284.43	−0.600550
$167$	−3345.69	−1.55028	−0.775141	−	0.631788i	$-0.782321\pi$
$167$	−3345.69	−1.55028	−0.775141	+	0.631788i	$0.782321\pi$
$168$	43.9295	0.0201740
$169$	4328.27	1.97008
$170$	−503.354	−0.227091
$171$	0	0
$172$	353.141	0.156551
$173$	−3510.12	−1.54260	−0.771299	−	0.636473i	$-0.780393\pi$
$173$	−3510.12	−1.54260	−0.771299	+	0.636473i	$0.780393\pi$
$174$	−1100.01	−0.479262
$175$	130.648	0.0564345
$176$	−1048.71	−0.449144
$177$	131.142	0.0556907
$178$	−2847.88	−1.19920
$179$	3743.73	1.56324	0.781619	−	0.623756i	$-0.214394\pi$
$179$	3743.73	1.56324	0.781619	+	0.623756i	$0.214394\pi$
$180$	−504.484	−0.208900
$181$	−3407.01	−1.39912	−0.699561	−	0.714573i	$-0.746621\pi$
$181$	−3407.01	−1.39912	−0.699561	+	0.714573i	$0.746621\pi$
$182$	295.715	0.120439
$183$	−2134.95	−0.862405
$184$	856.389	0.343119
$185$	4133.49	1.64270
$186$	632.988	0.249532
$187$	−1177.15	−0.460331
$188$	1667.82	0.647011
$189$	49.4206	0.0190202
$190$	0	0
$191$	−1439.15	−0.545200	−0.272600	−	0.962127i	$-0.587884\pi$
$191$	−1439.15	−0.545200	−0.272600	+	0.962127i	$0.587884\pi$
$192$	192.000	0.0721688
$193$	2431.88	0.906999	0.453500	−	0.891257i	$-0.350175\pi$
$193$	2431.88	0.906999	0.453500	+	0.891257i	$0.350175\pi$
$194$	−529.356	−0.195905
$195$	−3395.98	−1.24714
$196$	−1358.60	−0.495116
$197$	−2279.30	−0.824331	−0.412165	−	0.911109i	$-0.635227\pi$
$197$	−2279.30	−0.824331	−0.412165	+	0.911109i	$0.635227\pi$
$198$	−1179.80	−0.423457
$199$	−3162.71	−1.12663	−0.563313	−	0.826244i	$-0.690473\pi$
$199$	−3162.71	−1.12663	−0.563313	+	0.826244i	$0.690473\pi$
$200$	571.014	0.201884
$201$	2861.61	1.00419
$202$	−1906.36	−0.664015
$203$	−335.576	−0.116024
$204$	215.516	0.0739664
$205$	4564.61	1.55515
$206$	132.215	0.0447177
$207$	963.438	0.323495
$208$	1292.47	0.430848
$209$	0	0
$210$	−153.901	−0.0505722
$211$	3674.97	1.19903	0.599516	−	0.800363i	$-0.295360\pi$
$211$	3674.97	1.19903	0.599516	+	0.800363i	$0.295360\pi$
$212$	1170.09	0.379066
$213$	−2259.33	−0.726791
$214$	−3654.29	−1.16730
$215$	−1237.18	−0.392442
$216$	216.000	0.0680414
$217$	193.103	0.0604087
$218$	182.489	0.0566959
$219$	605.167	0.186728
$220$	3674.00	1.12591
$221$	1450.77	0.441580
$222$	−1769.80	−0.535049
$223$	978.191	0.293742	0.146871	−	0.989156i	$-0.453080\pi$
$223$	978.191	0.293742	0.146871	+	0.989156i	$0.453080\pi$
$224$	58.5726	0.0174712
$225$	642.391	0.190338
$226$	−807.538	−0.237684
$227$	3346.74	0.978550	0.489275	−	0.872129i	$-0.337261\pi$
$227$	3346.74	0.978550	0.489275	+	0.872129i	$0.337261\pi$
$228$	0	0
$229$	6443.10	1.85927	0.929634	−	0.368484i	$-0.120123\pi$
$229$	6443.10	1.85927	0.929634	+	0.368484i	$0.120123\pi$
$230$	−3000.24	−0.860131
$231$	−359.915	−0.102514
$232$	−1466.68	−0.415053
$233$	1530.79	0.430410	0.215205	−	0.976569i	$-0.430958\pi$
$233$	1530.79	0.430410	0.215205	+	0.976569i	$0.430958\pi$
$234$	1454.02	0.406208
$235$	−5842.96	−1.62193
$236$	174.856	0.0482296
$237$	−2597.59	−0.711948
$238$	65.7465	0.0179064
$239$	−5565.42	−1.50627	−0.753133	−	0.657869i	$-0.771458\pi$
$239$	−5565.42	−1.50627	−0.753133	+	0.657869i	$0.771458\pi$
$240$	−672.646	−0.180913
$241$	−3645.48	−0.974382	−0.487191	−	0.873295i	$-0.661978\pi$
$241$	−3645.48	−0.974382	−0.487191	+	0.873295i	$0.661978\pi$
$242$	5930.09	1.57521
$243$	243.000	0.0641500
$244$	−2846.60	−0.746865
$245$	4759.66	1.24116
$246$	−1954.39	−0.506533
$247$	0	0
$248$	843.984	0.216101
$249$	−1926.65	−0.490347
$250$	1502.89	0.380205
$251$	474.441	0.119309	0.0596543	−	0.998219i	$-0.481000\pi$
$251$	474.441	0.119309	0.0596543	+	0.998219i	$0.481000\pi$
$252$	65.8942	0.0164720
$253$	−7016.42	−1.74355
$254$	87.3670	0.0215822
$255$	−755.031	−0.185419
$256$	256.000	0.0625000
$257$	−2968.26	−0.720448	−0.360224	−	0.932866i	$-0.617300\pi$
$257$	−2968.26	−0.720448	−0.360224	+	0.932866i	$0.617300\pi$
$258$	529.712	0.127823
$259$	−539.904	−0.129529
$260$	−4527.98	−1.08005
$261$	−1650.02	−0.391316
$262$	4039.20	0.952452
$263$	−1668.10	−0.391101	−0.195550	−	0.980694i	$-0.562649\pi$
$263$	−1668.10	−0.391101	−0.195550	+	0.980694i	$0.562649\pi$
$264$	−1573.06	−0.366724
$265$	−4099.24	−0.950242
$266$	0	0
$267$	−4271.81	−0.979142
$268$	3815.47	0.869654
$269$	−930.925	−0.211002	−0.105501	−	0.994419i	$-0.533645\pi$
$269$	−930.925	−0.211002	−0.105501	+	0.994419i	$0.533645\pi$
$270$	−756.726	−0.170566
$271$	−175.488	−0.0393362	−0.0196681	−	0.999807i	$-0.506261\pi$
$271$	−175.488	−0.0393362	−0.0196681	+	0.999807i	$0.506261\pi$
$272$	287.355	0.0640568
$273$	443.573	0.0983379
$274$	−2654.42	−0.585253
$275$	−4678.34	−1.02587
$276$	1284.58	0.280155
$277$	3765.27	0.816726	0.408363	−	0.912820i	$-0.366100\pi$
$277$	3765.27	0.816726	0.408363	+	0.912820i	$0.366100\pi$
$278$	−865.659	−0.186758
$279$	949.482	0.203742
$280$	−205.201	−0.0437968
$281$	2391.60	0.507726	0.253863	−	0.967240i	$-0.418299\pi$
$281$	2391.60	0.507726	0.253863	+	0.967240i	$0.418299\pi$
$282$	2501.72	0.528282
$283$	5776.31	1.21331	0.606654	−	0.794966i	$-0.292512\pi$
$283$	5776.31	1.21331	0.606654	+	0.794966i	$0.292512\pi$
$284$	−3012.44	−0.629420
$285$	0	0
$286$	−10589.2	−2.18935
$287$	−596.216	−0.122626
$288$	288.000	0.0589256
$289$	−4590.45	−0.934348
$290$	5138.32	1.04046
$291$	−794.035	−0.159956
$292$	806.890	0.161711
$293$	884.445	0.176348	0.0881738	−	0.996105i	$-0.471897\pi$
$293$	884.445	0.176348	0.0881738	+	0.996105i	$0.471897\pi$
$294$	−2037.90	−0.404261
$295$	−612.585	−0.120902
$296$	−2359.73	−0.463366
$297$	−1769.69	−0.345751
$298$	−2542.99	−0.494335
$299$	8647.30	1.67253
$300$	856.521	0.164838
$301$	161.597	0.0309445
$302$	−1594.77	−0.303869
$303$	−2859.54	−0.542166
$304$	0	0
$305$	9972.68	1.87224
$306$	323.274	0.0603933
$307$	1103.40	0.205127	0.102564	−	0.994726i	$-0.467295\pi$
$307$	1103.40	0.205127	0.102564	+	0.994726i	$0.467295\pi$
$308$	−479.887	−0.0887796
$309$	198.322	0.0365119
$310$	−2956.78	−0.541723
$311$	−3782.84	−0.689728	−0.344864	−	0.938653i	$-0.612075\pi$
$311$	−3782.84	−0.689728	−0.344864	+	0.938653i	$0.612075\pi$
$312$	1938.70	0.351786
$313$	−4767.45	−0.860934	−0.430467	−	0.902606i	$-0.641651\pi$
$313$	−4767.45	−0.860934	−0.430467	+	0.902606i	$0.641651\pi$
$314$	−4836.22	−0.869184
$315$	−230.851	−0.0412920
$316$	−3463.45	−0.616565
$317$	7410.30	1.31295	0.656473	−	0.754350i	$-0.272048\pi$
$317$	7410.30	1.31295	0.656473	+	0.754350i	$0.272048\pi$
$318$	1755.13	0.309506
$319$	12016.6	2.10909
$320$	−896.861	−0.156675
$321$	−5481.44	−0.953097
$322$	391.882	0.0678222
$323$	0	0
$324$	324.000	0.0555556
$325$	5765.75	0.984081
$326$	−2372.38	−0.403050
$327$	273.733	0.0462920
$328$	−2605.85	−0.438670
$329$	763.190	0.127891
$330$	5511.00	0.919306
$331$	−6809.98	−1.13085	−0.565424	−	0.824801i	$-0.691287\pi$
$331$	−6809.98	−1.13085	−0.565424	+	0.824801i	$0.691287\pi$
$332$	−2568.86	−0.424653
$333$	−2654.69	−0.436866
$334$	−6691.38	−1.09622
$335$	−13367.0	−2.18005
$336$	87.8589	0.0142652
$337$	5805.17	0.938361	0.469180	−	0.883102i	$-0.344549\pi$
$337$	5805.17	0.938361	0.469180	+	0.883102i	$0.344549\pi$
$338$	8656.54	1.39306
$339$	−1211.31	−0.194068
$340$	−1006.71	−0.160578
$341$	−6914.79	−1.09811
$342$	0	0
$343$	−1249.52	−0.196699
$344$	706.282	0.110698
$345$	−4500.36	−0.702294
$346$	−7020.24	−1.09078
$347$	−2107.51	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$347$	−2107.51	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$348$	−2200.02	−0.338890
$349$	−3504.71	−0.537544	−0.268772	−	0.963204i	$-0.586618\pi$
$349$	−3504.71	−0.537544	−0.268772	+	0.963204i	$0.586618\pi$
$350$	261.295	0.0399052
$351$	2181.04	0.331667
$352$	−2097.42	−0.317593
$353$	−9324.77	−1.40597	−0.702985	−	0.711204i	$-0.748150\pi$
$353$	−9324.77	−1.40597	−0.702985	+	0.711204i	$0.748150\pi$
$354$	262.285	0.0393793
$355$	10553.7	1.57783
$356$	−5695.75	−0.847961
$357$	98.6198	0.0146205
$358$	7487.47	1.10538
$359$	2594.86	0.381481	0.190741	−	0.981640i	$-0.438911\pi$
$359$	2594.86	0.381481	0.190741	+	0.981640i	$0.438911\pi$
$360$	−1008.97	−0.147715
$361$	0	0
$362$	−6814.02	−0.989328
$363$	8895.14	1.28615
$364$	591.431	0.0851632
$365$	−2826.83	−0.405378
$366$	−4269.90	−0.609813
$367$	11307.9	1.60836	0.804180	−	0.594385i	$-0.202605\pi$
$367$	11307.9	1.60836	0.804180	+	0.594385i	$0.202605\pi$
$368$	1712.78	0.242622
$369$	−2931.58	−0.413582
$370$	8266.98	1.16157
$371$	535.430	0.0749276
$372$	1265.98	0.176446
$373$	5136.10	0.712969	0.356484	−	0.934301i	$-0.383975\pi$
$373$	5136.10	0.712969	0.356484	+	0.934301i	$0.383975\pi$
$374$	−2354.31	−0.325503
$375$	2254.34	0.310436
$376$	3335.63	0.457506
$377$	−14809.7	−2.02317
$378$	98.8413	0.0134493
$379$	−4189.99	−0.567877	−0.283939	−	0.958842i	$-0.591641\pi$
$379$	−4189.99	−0.567877	−0.283939	+	0.958842i	$0.591641\pi$
$380$	0	0
$381$	131.050	0.0176218
$382$	−2878.30	−0.385515
$383$	−13268.6	−1.77022	−0.885111	−	0.465379i	$-0.845918\pi$
$383$	−13268.6	−1.77022	−0.885111	+	0.465379i	$0.845918\pi$
$384$	384.000	0.0510310
$385$	1681.22	0.222553
$386$	4863.77	0.641345
$387$	794.568	0.104367
$388$	−1058.71	−0.138526
$389$	12080.4	1.57455	0.787276	−	0.616601i	$-0.211491\pi$
$389$	12080.4	1.57455	0.787276	+	0.616601i	$0.211491\pi$
$390$	−6791.97	−0.881858
$391$	1922.56	0.248665
$392$	−2717.20	−0.350100
$393$	6058.79	0.777674
$394$	−4558.59	−0.582890
$395$	12133.7	1.54561
$396$	−2359.59	−0.299429
$397$	14183.9	1.79312	0.896562	−	0.442918i	$-0.146057\pi$
$397$	14183.9	1.79312	0.896562	+	0.442918i	$0.146057\pi$
$398$	−6325.42	−0.796645
$399$	0	0
$400$	1142.03	0.142754
$401$	5745.99	0.715564	0.357782	−	0.933805i	$-0.383533\pi$
$401$	5745.99	0.715564	0.357782	+	0.933805i	$0.383533\pi$
$402$	5723.21	0.710069
$403$	8522.04	1.05338
$404$	−3812.72	−0.469529
$405$	−1135.09	−0.139267
$406$	−671.151	−0.0820411
$407$	19333.3	2.35459
$408$	431.032	0.0523021
$409$	10788.1	1.30425	0.652126	−	0.758110i	$-0.273877\pi$
$409$	10788.1	1.30425	0.652126	+	0.758110i	$0.273877\pi$
$410$	9129.23	1.09966
$411$	−3981.63	−0.477857
$412$	264.430	0.0316202
$413$	80.0140	0.00953325
$414$	1926.88	0.228746
$415$	8999.66	1.06452
$416$	2584.93	0.304656
$417$	−1298.49	−0.152487
$418$	0	0
$419$	7348.24	0.856767	0.428383	−	0.903597i	$-0.359083\pi$
$419$	7348.24	0.856767	0.428383	+	0.903597i	$0.359083\pi$
$420$	−307.802	−0.0357600
$421$	−4594.00	−0.531824	−0.265912	−	0.963997i	$-0.585673\pi$
$421$	−4594.00	−0.531824	−0.265912	+	0.963997i	$0.585673\pi$
$422$	7349.95	0.847843
$423$	3752.59	0.431340
$424$	2340.17	0.268040
$425$	1281.90	0.146309
$426$	−4518.65	−0.513919
$427$	−1302.60	−0.147628
$428$	−7308.59	−0.825406
$429$	−15883.8	−1.78759
$430$	−2474.36	−0.277499
$431$	−2291.80	−0.256131	−0.128065	−	0.991766i	$-0.540877\pi$
$431$	−2291.80	−0.256131	−0.128065	+	0.991766i	$0.540877\pi$
$432$	432.000	0.0481125
$433$	−15973.9	−1.77288	−0.886440	−	0.462844i	$-0.846829\pi$
$433$	−15973.9	−1.77288	−0.886440	+	0.462844i	$0.846829\pi$
$434$	386.206	0.0427154
$435$	7707.48	0.849529
$436$	364.977	0.0400900
$437$	0	0
$438$	1210.33	0.132037
$439$	8671.73	0.942777	0.471388	−	0.881926i	$-0.343753\pi$
$439$	8671.73	0.942777	0.471388	+	0.881926i	$0.343753\pi$
$440$	7348.01	0.796142
$441$	−3056.85	−0.330077
$442$	2901.53	0.312244
$443$	4850.23	0.520183	0.260092	−	0.965584i	$-0.416247\pi$
$443$	4850.23	0.520183	0.260092	+	0.965584i	$0.416247\pi$
$444$	−3539.59	−0.378337
$445$	19954.3	2.12567
$446$	1956.38	0.207707
$447$	−3814.49	−0.403623
$448$	117.145	0.0123540
$449$	14685.8	1.54357	0.771786	−	0.635883i	$-0.219364\pi$
$449$	14685.8	1.54357	0.771786	+	0.635883i	$0.219364\pi$
$450$	1284.78	0.134589
$451$	21349.8	2.22910
$452$	−1615.08	−0.168068
$453$	−2392.15	−0.248108
$454$	6693.48	0.691940
$455$	−2072.00	−0.213487
$456$	0	0
$457$	−3324.42	−0.340284	−0.170142	−	0.985420i	$-0.554423\pi$
$457$	−3324.42	−0.340284	−0.170142	+	0.985420i	$0.554423\pi$
$458$	12886.2	1.31470
$459$	484.911	0.0493109
$460$	−6000.48	−0.608204
$461$	7026.45	0.709880	0.354940	−	0.934889i	$-0.384501\pi$
$461$	7026.45	0.709880	0.354940	+	0.934889i	$0.384501\pi$
$462$	−719.831	−0.0724882
$463$	−12854.1	−1.29024	−0.645122	−	0.764080i	$-0.723193\pi$
$463$	−12854.1	−1.29024	−0.645122	+	0.764080i	$0.723193\pi$
$464$	−2933.36	−0.293487
$465$	−4435.17	−0.442315
$466$	3061.58	0.304346
$467$	748.937	0.0742113	0.0371057	−	0.999311i	$-0.488186\pi$
$467$	748.937	0.0742113	0.0371057	+	0.999311i	$0.488186\pi$
$468$	2908.05	0.287232
$469$	1745.96	0.171899
$470$	−11685.9	−1.14688
$471$	−7254.33	−0.709685
$472$	349.713	0.0341035
$473$	−5786.59	−0.562511
$474$	−5195.18	−0.503423
$475$	0	0
$476$	131.493	0.0126617
$477$	2632.70	0.252710
$478$	−11130.8	−1.06509
$479$	−18022.1	−1.71910	−0.859550	−	0.511052i	$-0.829256\pi$
$479$	−18022.1	−1.71910	−0.859550	+	0.511052i	$0.829256\pi$
$480$	−1345.29	−0.127925
$481$	−23827.1	−2.25867
$482$	−7290.96	−0.688992
$483$	587.824	0.0553766
$484$	11860.2	1.11384
$485$	3709.06	0.347257
$486$	486.000	0.0453609
$487$	10429.4	0.970437	0.485218	−	0.874393i	$-0.338740\pi$
$487$	10429.4	0.970437	0.485218	+	0.874393i	$0.338740\pi$
$488$	−5693.21	−0.528113
$489$	−3558.58	−0.329089
$490$	9519.33	0.877631
$491$	−7371.55	−0.677542	−0.338771	−	0.940869i	$-0.610011\pi$
$491$	−7371.55	−0.677542	−0.338771	+	0.940869i	$0.610011\pi$
$492$	−3908.77	−0.358173
$493$	−3292.64	−0.300797
$494$	0	0
$495$	8266.51	0.750610
$496$	1687.97	0.152807
$497$	−1378.49	−0.124414
$498$	−3853.29	−0.346727
$499$	186.487	0.0167300	0.00836502	−	0.999965i	$-0.497337\pi$
$499$	186.487	0.0167300	0.00836502	+	0.999965i	$0.497337\pi$
$500$	3005.79	0.268846
$501$	−10037.1	−0.895056
$502$	948.882	0.0843639
$503$	−1935.38	−0.171560	−0.0857798	−	0.996314i	$-0.527338\pi$
$503$	−1935.38	−0.171560	−0.0857798	+	0.996314i	$0.527338\pi$
$504$	131.788	0.0116475
$505$	13357.3	1.17702
$506$	−14032.8	−1.23288
$507$	12984.8	1.13743
$508$	174.734	0.0152610
$509$	21790.7	1.89755	0.948777	−	0.315947i	$-0.102322\pi$
$509$	21790.7	1.89755	0.948777	+	0.315947i	$0.102322\pi$
$510$	−1510.06	−0.131111
$511$	369.231	0.0319645
$512$	512.000	0.0441942
$513$	0	0
$514$	−5936.53	−0.509434
$515$	−926.394	−0.0792656
$516$	1059.42	0.0903847
$517$	−27328.9	−2.32481
$518$	−1079.81	−0.0915908
$519$	−10530.4	−0.890620
$520$	−9055.96	−0.763711
$521$	3981.03	0.334764	0.167382	−	0.985892i	$-0.446469\pi$
$521$	3981.03	0.334764	0.167382	+	0.985892i	$0.446469\pi$
$522$	−3300.03	−0.276702
$523$	2431.38	0.203282	0.101641	−	0.994821i	$-0.467591\pi$
$523$	2431.38	0.203282	0.101641	+	0.994821i	$0.467591\pi$
$524$	8078.39	0.673485
$525$	391.943	0.0325824
$526$	−3336.20	−0.276550
$527$	1894.71	0.156613
$528$	−3146.12	−0.259313
$529$	−707.589	−0.0581564
$530$	−8198.48	−0.671923
$531$	393.427	0.0321530
$532$	0	0
$533$	−26312.3	−2.13829
$534$	−8543.63	−0.692358
$535$	25604.6	2.06913
$536$	7630.95	0.614938
$537$	11231.2	0.902536
$538$	−1861.85	−0.149201
$539$	22262.1	1.77903
$540$	−1513.45	−0.120609
$541$	−8340.50	−0.662821	−0.331410	−	0.943487i	$-0.607524\pi$
$541$	−8340.50	−0.662821	−0.331410	+	0.943487i	$0.607524\pi$
$542$	−350.975	−0.0278149
$543$	−10221.0	−0.807783
$544$	574.709	0.0452950
$545$	−1278.65	−0.100498
$546$	887.146	0.0695354
$547$	−21608.2	−1.68903	−0.844514	−	0.535534i	$-0.820111\pi$
$547$	−21608.2	−1.68903	−0.844514	+	0.535534i	$0.820111\pi$
$548$	−5308.84	−0.413836
$549$	−6404.86	−0.497910
$550$	−9356.67	−0.725400
$551$	0	0
$552$	2569.17	0.198100
$553$	−1584.87	−0.121873
$554$	7530.54	0.577512
$555$	12400.5	0.948416
$556$	−1731.32	−0.132058
$557$	−8305.90	−0.631835	−0.315918	−	0.948787i	$-0.602312\pi$
$557$	−8305.90	−0.631835	−0.315918	+	0.948787i	$0.602312\pi$
$558$	1898.96	0.144067
$559$	7131.61	0.539597
$560$	−410.402	−0.0309690
$561$	−3531.46	−0.265772
$562$	4783.20	0.359016
$563$	−15160.0	−1.13484	−0.567421	−	0.823428i	$-0.692059\pi$
$563$	−15160.0	−1.13484	−0.567421	+	0.823428i	$0.692059\pi$
$564$	5003.45	0.373552
$565$	5658.19	0.421313
$566$	11552.6	0.857938
$567$	148.262	0.0109813
$568$	−6024.87	−0.445067
$569$	16867.9	1.24278	0.621389	−	0.783502i	$-0.286569\pi$
$569$	16867.9	1.24278	0.621389	+	0.783502i	$0.286569\pi$
$570$	0	0
$571$	−2058.18	−0.150844	−0.0754222	−	0.997152i	$-0.524030\pi$
$571$	−2058.18	−0.150844	−0.0754222	+	0.997152i	$0.524030\pi$
$572$	−21178.4	−1.54810
$573$	−4317.45	−0.314771
$574$	−1192.43	−0.0867093
$575$	7640.79	0.554161
$576$	576.000	0.0416667
$577$	−26942.0	−1.94386	−0.971931	−	0.235266i	$-0.924404\pi$
$577$	−26942.0	−1.94386	−0.971931	+	0.235266i	$0.924404\pi$
$578$	−9180.90	−0.660684
$579$	7295.65	0.523656
$580$	10276.6	0.735714
$581$	−1175.51	−0.0839385
$582$	−1588.07	−0.113106
$583$	−19173.1	−1.36204
$584$	1613.78	0.114347
$585$	−10187.9	−0.720034
$586$	1768.89	0.124697
$587$	−7346.77	−0.516582	−0.258291	−	0.966067i	$-0.583159\pi$
$587$	−7346.77	−0.516582	−0.258291	+	0.966067i	$0.583159\pi$
$588$	−4075.80	−0.285855
$589$	0	0
$590$	−1225.17	−0.0854906
$591$	−6837.89	−0.475928
$592$	−4719.46	−0.327649
$593$	16572.5	1.14764	0.573820	−	0.818981i	$-0.305461\pi$
$593$	16572.5	1.14764	0.573820	+	0.818981i	$0.305461\pi$
$594$	−3539.39	−0.244483
$595$	−460.668	−0.0317404
$596$	−5085.99	−0.349547
$597$	−9488.13	−0.650458
$598$	17294.6	1.18266
$599$	229.143	0.0156303	0.00781515	−	0.999969i	$-0.497512\pi$
$599$	229.143	0.0156303	0.00781515	+	0.999969i	$0.497512\pi$
$600$	1713.04	0.116558
$601$	−10435.8	−0.708297	−0.354148	−	0.935189i	$-0.615229\pi$
$601$	−10435.8	−0.708297	−0.354148	+	0.935189i	$0.615229\pi$
$602$	323.194	0.0218811
$603$	8584.82	0.579769
$604$	−3189.53	−0.214868
$605$	−41550.5	−2.79218
$606$	−5719.08	−0.383369
$607$	20594.1	1.37708	0.688541	−	0.725197i	$-0.258251\pi$
$607$	20594.1	1.37708	0.688541	+	0.725197i	$0.258251\pi$
$608$	0	0
$609$	−1006.73	−0.0669863
$610$	19945.4	1.32388
$611$	33681.2	2.23011
$612$	646.548	0.0427045
$613$	17748.1	1.16939	0.584696	−	0.811252i	$-0.301214\pi$
$613$	17748.1	1.16939	0.584696	+	0.811252i	$0.301214\pi$
$614$	2206.79	0.145047
$615$	13693.8	0.897868
$616$	−959.774	−0.0627766
$617$	−8364.48	−0.545772	−0.272886	−	0.962046i	$-0.587978\pi$
$617$	−8364.48	−0.545772	−0.272886	+	0.962046i	$0.587978\pi$
$618$	396.645	0.0258178
$619$	−14662.9	−0.952101	−0.476050	−	0.879418i	$-0.657932\pi$
$619$	−14662.9	−0.952101	−0.476050	+	0.879418i	$0.657932\pi$
$620$	−5913.57	−0.383056
$621$	2890.31	0.186770
$622$	−7565.69	−0.487711
$623$	−2606.37	−0.167611
$624$	3877.40	0.248750
$625$	−19452.5	−1.24496
$626$	−9534.91	−0.608773
$627$	0	0
$628$	−9672.44	−0.614606
$629$	−5297.49	−0.335811
$630$	−461.702	−0.0291979
$631$	−24031.0	−1.51610	−0.758049	−	0.652198i	$-0.773847\pi$
$631$	−24031.0	−1.51610	−0.758049	+	0.652198i	$0.773847\pi$
$632$	−6926.91	−0.435977
$633$	11024.9	0.692261
$634$	14820.6	0.928393
$635$	−612.156	−0.0382562
$636$	3510.26	0.218854
$637$	−27436.6	−1.70656
$638$	24033.1	1.49135
$639$	−6777.98	−0.419613
$640$	−1793.72	−0.110786
$641$	−22998.7	−1.41715	−0.708576	−	0.705635i	$-0.750662\pi$
$641$	−22998.7	−1.41715	−0.708576	+	0.705635i	$0.750662\pi$
$642$	−10962.9	−0.673942
$643$	15702.0	0.963029	0.481515	−	0.876438i	$-0.340087\pi$
$643$	15702.0	0.963029	0.481515	+	0.876438i	$0.340087\pi$
$644$	783.765	0.0479575
$645$	−3711.54	−0.226577
$646$	0	0
$647$	23662.6	1.43782	0.718912	−	0.695101i	$-0.244640\pi$
$647$	23662.6	1.43782	0.718912	+	0.695101i	$0.244640\pi$
$648$	648.000	0.0392837
$649$	−2865.21	−0.173296
$650$	11531.5	0.695851
$651$	579.309	0.0348770
$652$	−4744.77	−0.284999
$653$	16344.2	0.979474	0.489737	−	0.871870i	$-0.337093\pi$
$653$	16344.2	0.979474	0.489737	+	0.871870i	$0.337093\pi$
$654$	547.466	0.0327334
$655$	−28301.5	−1.68829
$656$	−5211.70	−0.310187
$657$	1815.50	0.107807
$658$	1526.38	0.0904324
$659$	22546.0	1.33273	0.666364	−	0.745626i	$-0.267850\pi$
$659$	22546.0	1.33273	0.666364	+	0.745626i	$0.267850\pi$
$660$	11022.0	0.650047
$661$	−7835.86	−0.461088	−0.230544	−	0.973062i	$-0.574051\pi$
$661$	−7835.86	−0.461088	−0.230544	+	0.973062i	$0.574051\pi$
$662$	−13620.0	−0.799630
$663$	4352.30	0.254946
$664$	−5137.73	−0.300275
$665$	0	0
$666$	−5309.39	−0.308911
$667$	−19625.8	−1.13930
$668$	−13382.8	−0.775141
$669$	2934.57	0.169592
$670$	−26734.0	−1.54153
$671$	46644.6	2.68360
$672$	175.718	0.0100870
$673$	10712.4	0.613571	0.306785	−	0.951779i	$-0.400747\pi$
$673$	10712.4	0.613571	0.306785	+	0.951779i	$0.400747\pi$
$674$	11610.3	0.663521
$675$	1927.17	0.109892
$676$	17313.1	0.985041
$677$	3313.18	0.188088	0.0940442	−	0.995568i	$-0.470020\pi$
$677$	3313.18	0.188088	0.0940442	+	0.995568i	$0.470020\pi$
$678$	−2422.61	−0.137227
$679$	−484.465	−0.0273816
$680$	−2013.42	−0.113546
$681$	10040.2	0.564966
$682$	−13829.6	−0.776484
$683$	−29238.4	−1.63803	−0.819016	−	0.573771i	$-0.805480\pi$
$683$	−29238.4	−1.63803	−0.819016	+	0.573771i	$0.805480\pi$
$684$	0	0
$685$	18598.8	1.03741
$686$	−2499.04	−0.139087
$687$	19329.3	1.07345
$688$	1412.56	0.0782755
$689$	23629.7	1.30656
$690$	−9000.72	−0.496597
$691$	−27329.6	−1.50458	−0.752291	−	0.658831i	$-0.771051\pi$
$691$	−27329.6	−1.50458	−0.752291	+	0.658831i	$0.771051\pi$
$692$	−14040.5	−0.771299
$693$	−1079.75	−0.0591864
$694$	−4215.03	−0.230548
$695$	6065.44	0.331043
$696$	−4400.05	−0.239631
$697$	−5850.02	−0.317913
$698$	−7009.42	−0.380101
$699$	4592.38	0.248497
$700$	522.590	0.0282172
$701$	10867.9	0.585558	0.292779	−	0.956180i	$-0.405420\pi$
$701$	10867.9	0.585558	0.292779	+	0.956180i	$0.405420\pi$
$702$	4362.07	0.234524
$703$	0	0
$704$	−4194.83	−0.224572
$705$	−17528.9	−0.936420
$706$	−18649.5	−0.994171
$707$	−1744.69	−0.0928090
$708$	524.569	0.0278454
$709$	−18657.7	−0.988299	−0.494150	−	0.869377i	$-0.664520\pi$
$709$	−18657.7	−0.988299	−0.494150	+	0.869377i	$0.664520\pi$
$710$	21107.3	1.11569
$711$	−7792.77	−0.411043
$712$	−11391.5	−0.599599
$713$	11293.4	0.593187
$714$	197.240	0.0103382
$715$	74195.7	3.88079
$716$	14974.9	0.781619
$717$	−16696.3	−0.869643
$718$	5189.73	0.269748
$719$	18958.6	0.983360	0.491680	−	0.870776i	$-0.336383\pi$
$719$	18958.6	0.983360	0.491680	+	0.870776i	$0.336383\pi$
$720$	−2017.94	−0.104450
$721$	121.003	0.00625017
$722$	0	0
$723$	−10936.4	−0.562560
$724$	−13628.0	−0.699561
$725$	−13085.9	−0.670341
$726$	17790.3	0.909448
$727$	−19580.5	−0.998900	−0.499450	−	0.866343i	$-0.666465\pi$
$727$	−19580.5	−0.998900	−0.499450	+	0.866343i	$0.666465\pi$
$728$	1182.86	0.0602194
$729$	729.000	0.0370370
$730$	−5653.65	−0.286645
$731$	1585.57	0.0802252
$732$	−8539.81	−0.431203
$733$	13711.4	0.690918	0.345459	−	0.938434i	$-0.387723\pi$
$733$	13711.4	0.690918	0.345459	+	0.938434i	$0.387723\pi$
$734$	22615.8	1.13728
$735$	14279.0	0.716583
$736$	3425.56	0.171559
$737$	−62520.6	−3.12480
$738$	−5863.16	−0.292447
$739$	21564.9	1.07345	0.536724	−	0.843758i	$-0.319662\pi$
$739$	21564.9	1.07345	0.536724	+	0.843758i	$0.319662\pi$
$740$	16534.0	0.821352
$741$	0	0
$742$	1070.86	0.0529818
$743$	−7713.09	−0.380842	−0.190421	−	0.981702i	$-0.560985\pi$
$743$	−7713.09	−0.380842	−0.190421	+	0.981702i	$0.560985\pi$
$744$	2531.95	0.124766
$745$	17818.1	0.876246
$746$	10272.2	0.504145
$747$	−5779.94	−0.283102
$748$	−4708.61	−0.230166
$749$	−3344.40	−0.163153
$750$	4508.68	0.219512
$751$	17486.2	0.849642	0.424821	−	0.905277i	$-0.360337\pi$
$751$	17486.2	0.849642	0.424821	+	0.905277i	$0.360337\pi$
$752$	6671.27	0.323505
$753$	1423.32	0.0688828
$754$	−29619.3	−1.43060
$755$	11174.1	0.538631
$756$	197.683	0.00951011
$757$	7683.97	0.368928	0.184464	−	0.982839i	$-0.440945\pi$
$757$	7683.97	0.368928	0.184464	+	0.982839i	$0.440945\pi$
$758$	−8379.98	−0.401550
$759$	−21049.3	−1.00664
$760$	0	0
$761$	6508.24	0.310018	0.155009	−	0.987913i	$-0.450459\pi$
$761$	6508.24	0.310018	0.155009	+	0.987913i	$0.450459\pi$
$762$	262.101	0.0124605
$763$	167.013	0.00792435
$764$	−5756.60	−0.272600
$765$	−2265.09	−0.107052
$766$	−26537.3	−1.25174
$767$	3531.19	0.166237
$768$	768.000	0.0360844
$769$	37777.1	1.77149	0.885745	−	0.464172i	$-0.153648\pi$
$769$	37777.1	1.77149	0.885745	+	0.464172i	$0.153648\pi$
$770$	3362.44	0.157369
$771$	−8904.79	−0.415951
$772$	9727.54	0.453500
$773$	−11686.9	−0.543790	−0.271895	−	0.962327i	$-0.587650\pi$
$773$	−11686.9	−0.543790	−0.271895	+	0.962327i	$0.587650\pi$
$774$	1589.14	0.0737988
$775$	7530.11	0.349019
$776$	−2117.43	−0.0979525
$777$	−1619.71	−0.0747836
$778$	24160.8	1.11338
$779$	0	0
$780$	−13583.9	−0.623568
$781$	49361.9	2.26160
$782$	3845.12	0.175833
$783$	−4950.05	−0.225926
$784$	−5434.39	−0.247558
$785$	33886.1	1.54069
$786$	12117.6	0.549898
$787$	−17644.4	−0.799182	−0.399591	−	0.916693i	$-0.630848\pi$
$787$	−17644.4	−0.799182	−0.399591	+	0.916693i	$0.630848\pi$
$788$	−9117.19	−0.412165
$789$	−5004.30	−0.225802
$790$	24267.5	1.09291
$791$	−739.056	−0.0332210
$792$	−4719.18	−0.211728
$793$	−57486.5	−2.57428
$794$	28367.8	1.26793
$795$	−12297.7	−0.548623
$796$	−12650.8	−0.563313
$797$	27458.5	1.22036	0.610181	−	0.792262i	$-0.291097\pi$
$797$	27458.5	1.22036	0.610181	+	0.792262i	$0.291097\pi$
$798$	0	0
$799$	7488.36	0.331563
$800$	2284.06	0.100942
$801$	−12815.4	−0.565308
$802$	11492.0	0.505980
$803$	−13221.7	−0.581052
$804$	11446.4	0.502095
$805$	−2745.81	−0.120220
$806$	17044.1	0.744854
$807$	−2792.77	−0.121822
$808$	−7625.44	−0.332007
$809$	−37355.0	−1.62340	−0.811701	−	0.584073i	$-0.801458\pi$
$809$	−37355.0	−1.62340	−0.811701	+	0.584073i	$0.801458\pi$
$810$	−2270.18	−0.0984765
$811$	−32385.7	−1.40224	−0.701120	−	0.713043i	$-0.747316\pi$
$811$	−32385.7	−1.40224	−0.701120	+	0.713043i	$0.747316\pi$
$812$	−1342.30	−0.0580118
$813$	−526.463	−0.0227108
$814$	38666.6	1.66495
$815$	16622.6	0.714436
$816$	862.064	0.0369832
$817$	0	0
$818$	21576.3	0.922246
$819$	1330.72	0.0567754
$820$	18258.5	0.777577
$821$	15002.5	0.637748	0.318874	−	0.947797i	$-0.396695\pi$
$821$	15002.5	0.637748	0.318874	+	0.947797i	$0.396695\pi$
$822$	−7963.26	−0.337896
$823$	44985.5	1.90534	0.952672	−	0.304001i	$-0.0983228\pi$
$823$	44985.5	1.90534	0.952672	+	0.304001i	$0.0983228\pi$
$824$	528.860	0.0223589
$825$	−14035.0	−0.592286
$826$	160.028	0.00674102
$827$	14036.3	0.590195	0.295098	−	0.955467i	$-0.404648\pi$
$827$	14036.3	0.590195	0.295098	+	0.955467i	$0.404648\pi$
$828$	3853.75	0.161748
$829$	8350.83	0.349863	0.174931	−	0.984581i	$-0.444030\pi$
$829$	8350.83	0.349863	0.174931	+	0.984581i	$0.444030\pi$
$830$	17999.3	0.752729
$831$	11295.8	0.471537
$832$	5169.87	0.215424
$833$	−6100.00	−0.253724
$834$	−2596.98	−0.107825
$835$	46884.7	1.94313
$836$	0	0
$837$	2848.45	0.117631
$838$	14696.5	0.605826
$839$	−40309.1	−1.65867	−0.829334	−	0.558752i	$-0.811280\pi$
$839$	−40309.1	−1.65867	−0.829334	+	0.558752i	$0.811280\pi$
$840$	−615.603	−0.0252861
$841$	9222.80	0.378154
$842$	−9188.00	−0.376056
$843$	7174.80	0.293136
$844$	14699.9	0.599516
$845$	−60654.0	−2.46930
$846$	7505.17	0.305004
$847$	5427.20	0.220166
$848$	4680.35	0.189533
$849$	17328.9	0.700503
$850$	2563.81	0.103456
$851$	−31575.7	−1.27192
$852$	−9037.31	−0.363396
$853$	38931.0	1.56269	0.781344	−	0.624101i	$-0.214534\pi$
$853$	38931.0	1.56269	0.781344	+	0.624101i	$0.214534\pi$
$854$	−2605.20	−0.104389
$855$	0	0
$856$	−14617.2	−0.583650
$857$	−3222.59	−0.128450	−0.0642250	−	0.997935i	$-0.520458\pi$
$857$	−3222.59	−0.128450	−0.0642250	+	0.997935i	$0.520458\pi$
$858$	−31767.6	−1.26402
$859$	−20371.0	−0.809136	−0.404568	−	0.914508i	$-0.632578\pi$
$859$	−20371.0	−0.809136	−0.404568	+	0.914508i	$0.632578\pi$
$860$	−4948.73	−0.196221
$861$	−1788.65	−0.0707979
$862$	−4583.61	−0.181112
$863$	46749.0	1.84398	0.921990	−	0.387213i	$-0.126562\pi$
$863$	46749.0	1.84398	0.921990	+	0.387213i	$0.126562\pi$
$864$	864.000	0.0340207
$865$	49188.9	1.93349
$866$	−31947.8	−1.25362
$867$	−13771.4	−0.539446
$868$	772.412	0.0302043
$869$	56752.4	2.21541
$870$	15415.0	0.600708
$871$	77052.7	2.99751
$872$	729.955	0.0283479
$873$	−2382.10	−0.0923505
$874$	0	0
$875$	1375.44	0.0531411
$876$	2420.67	0.0933639
$877$	22006.8	0.847338	0.423669	−	0.905817i	$-0.360742\pi$
$877$	22006.8	0.847338	0.423669	+	0.905817i	$0.360742\pi$
$878$	17343.5	0.666644
$879$	2653.34	0.101814
$880$	14696.0	0.562957
$881$	29928.6	1.14452	0.572259	−	0.820073i	$-0.306067\pi$
$881$	29928.6	1.14452	0.572259	+	0.820073i	$0.306067\pi$
$882$	−6113.69	−0.233400
$883$	18171.2	0.692537	0.346269	−	0.938135i	$-0.387449\pi$
$883$	18171.2	0.692537	0.346269	+	0.938135i	$0.387449\pi$
$884$	5803.07	0.220790
$885$	−1837.76	−0.0698028
$886$	9700.45	0.367825
$887$	−19701.2	−0.745774	−0.372887	−	0.927877i	$-0.621632\pi$
$887$	−19701.2	−0.745774	−0.372887	+	0.927877i	$0.621632\pi$
$888$	−7079.18	−0.267525
$889$	79.9580	0.00301654
$890$	39908.6	1.50308
$891$	−5309.08	−0.199620
$892$	3912.76	0.146871
$893$	0	0
$894$	−7628.98	−0.285404
$895$	−52462.6	−1.95936
$896$	234.290	0.00873560
$897$	25941.9	0.965635
$898$	29371.5	1.09147
$899$	−19341.5	−0.717548
$900$	2569.56	0.0951690
$901$	5253.59	0.194254
$902$	42699.6	1.57621
$903$	484.791	0.0178658
$904$	−3230.15	−0.118842
$905$	47743.9	1.75366
$906$	−4784.30	−0.175439
$907$	−21281.0	−0.779079	−0.389539	−	0.921010i	$-0.627366\pi$
$907$	−21281.0	−0.779079	−0.389539	+	0.921010i	$0.627366\pi$
$908$	13387.0	0.489275
$909$	−8578.62	−0.313020
$910$	−4143.99	−0.150958
$911$	22360.9	0.813228	0.406614	−	0.913600i	$-0.366709\pi$
$911$	22360.9	0.813228	0.406614	+	0.913600i	$0.366709\pi$
$912$	0	0
$913$	42093.5	1.52584
$914$	−6648.85	−0.240617
$915$	29918.0	1.08094
$916$	25772.4	0.929634
$917$	3696.66	0.133124
$918$	969.822	0.0348681
$919$	27942.6	1.00298	0.501492	−	0.865162i	$-0.332785\pi$
$919$	27942.6	1.00298	0.501492	+	0.865162i	$0.332785\pi$
$920$	−12001.0	−0.430065
$921$	3310.19	0.118430
$922$	14052.9	0.501961
$923$	−60835.5	−2.16947
$924$	−1439.66	−0.0512569
$925$	−21053.7	−0.748370
$926$	−25708.3	−0.912340
$927$	594.967	0.0210801
$928$	−5866.73	−0.207527
$929$	−31966.3	−1.12894	−0.564468	−	0.825455i	$-0.690919\pi$
$929$	−31966.3	−1.12894	−0.564468	+	0.825455i	$0.690919\pi$
$930$	−8870.35	−0.312764
$931$	0	0
$932$	6123.17	0.215205
$933$	−11348.5	−0.398215
$934$	1497.87	0.0524753
$935$	16496.0	0.576980
$936$	5816.10	0.203104
$937$	−8032.71	−0.280061	−0.140031	−	0.990147i	$-0.544720\pi$
$937$	−8032.71	−0.280061	−0.140031	+	0.990147i	$0.544720\pi$
$938$	3491.91	0.121551
$939$	−14302.4	−0.497061
$940$	−23371.9	−0.810964
$941$	1529.23	0.0529773	0.0264886	−	0.999649i	$-0.491567\pi$
$941$	1529.23	0.0529773	0.0264886	+	0.999649i	$0.491567\pi$
$942$	−14508.7	−0.501823
$943$	−34869.1	−1.20413
$944$	699.425	0.0241148
$945$	−692.554	−0.0238400
$946$	−11573.2	−0.397755
$947$	1851.92	0.0635473	0.0317737	−	0.999495i	$-0.489884\pi$
$947$	1851.92	0.0635473	0.0317737	+	0.999495i	$0.489884\pi$
$948$	−10390.4	−0.355974
$949$	16295.0	0.557383
$950$	0	0
$951$	22230.9	0.758030
$952$	262.986	0.00895318
$953$	−9271.51	−0.315145	−0.157573	−	0.987507i	$-0.550367\pi$
$953$	−9271.51	−0.315145	−0.157573	+	0.987507i	$0.550367\pi$
$954$	5265.39	0.178693
$955$	20167.4	0.683354
$956$	−22261.7	−0.753133
$957$	36049.7	1.21768
$958$	−36044.1	−1.21559
$959$	−2429.32	−0.0818005
$960$	−2690.58	−0.0904564
$961$	−18661.2	−0.626403
$962$	−47654.2	−1.59712
$963$	−16444.3	−0.550271
$964$	−14581.9	−0.487191
$965$	−34079.1	−1.13683
$966$	1175.65	0.0391572
$967$	−36171.2	−1.20288	−0.601442	−	0.798917i	$-0.705407\pi$
$967$	−36171.2	−1.20288	−0.601442	+	0.798917i	$0.705407\pi$
$968$	23720.4	0.787605
$969$	0	0
$970$	7418.11	0.245548
$971$	−12600.9	−0.416460	−0.208230	−	0.978080i	$-0.566770\pi$
$971$	−12600.9	−0.416460	−0.208230	+	0.978080i	$0.566770\pi$
$972$	972.000	0.0320750
$973$	−792.249	−0.0261031
$974$	20858.9	0.686203
$975$	17297.3	0.568160
$976$	−11386.4	−0.373433
$977$	−59123.4	−1.93605	−0.968027	−	0.250845i	$-0.919292\pi$
$977$	−59123.4	−1.93605	−0.968027	+	0.250845i	$0.919292\pi$
$978$	−7117.15	−0.232701
$979$	93330.9	3.04685
$980$	19038.7	0.620579
$981$	821.199	0.0267267
$982$	−14743.1	−0.479095
$983$	−17874.0	−0.579950	−0.289975	−	0.957034i	$-0.593647\pi$
$983$	−17874.0	−0.579950	−0.289975	+	0.957034i	$0.593647\pi$
$984$	−7817.54	−0.253266
$985$	31940.8	1.03322
$986$	−6585.28	−0.212696
$987$	2289.57	0.0738377
$988$	0	0
$989$	9450.82	0.303861
$990$	16533.0	0.530761
$991$	−43618.6	−1.39817	−0.699087	−	0.715037i	$-0.746410\pi$
$991$	−43618.6	−1.39817	−0.699087	+	0.715037i	$0.746410\pi$
$992$	3375.94	0.108051
$993$	−20430.0	−0.652895
$994$	−2756.97	−0.0879737
$995$	44320.5	1.41211
$996$	−7706.59	−0.245173
$997$	−17147.8	−0.544711	−0.272355	−	0.962197i	$-0.587803\pi$
$997$	−17147.8	−0.544711	−0.272355	+	0.962197i	$0.587803\pi$
$998$	372.973	0.0118299
$999$	−7964.08	−0.252225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.ba.1.2	yes	4
19.18	odd	2		2166.4.a.x.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.x.1.2	✓	4	19.18	odd	2
2166.4.a.ba.1.2	yes	4	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0134 q^{5} +6.00000 q^{6} +1.83039 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0134 q^{5} +6.00000 q^{6} +1.83039 q^{7} +8.00000 q^{8} +9.00000 q^{9} -28.0269 q^{10} -65.5442 q^{11} +12.0000 q^{12} +80.7791 q^{13} +3.66079 q^{14} -42.0403 q^{15} +16.0000 q^{16} +17.9597 q^{17} +18.0000 q^{18} -56.0538 q^{20} +5.49118 q^{21} -131.088 q^{22} +107.049 q^{23} +24.0000 q^{24} +71.3768 q^{25} +161.558 q^{26} +27.0000 q^{27} +7.32158 q^{28} -183.335 q^{29} -84.0807 q^{30} +105.498 q^{31} +32.0000 q^{32} -196.633 q^{33} +35.9193 q^{34} -25.6501 q^{35} +36.0000 q^{36} -294.966 q^{37} +242.337 q^{39} -112.108 q^{40} -325.731 q^{41} +10.9824 q^{42} +88.2853 q^{43} -262.177 q^{44} -126.121 q^{45} +214.097 q^{46} +416.954 q^{47} +48.0000 q^{48} -339.650 q^{49} +142.754 q^{50} +53.8790 q^{51} +323.117 q^{52} +292.522 q^{53} +54.0000 q^{54} +918.501 q^{55} +14.6432 q^{56} -366.670 q^{58} +43.7141 q^{59} -168.161 q^{60} -711.651 q^{61} +210.996 q^{62} +16.4735 q^{63} +64.0000 q^{64} -1131.99 q^{65} -393.265 q^{66} +953.869 q^{67} +71.8387 q^{68} +321.146 q^{69} -51.3003 q^{70} -753.109 q^{71} +72.0000 q^{72} +201.722 q^{73} -589.932 q^{74} +214.130 q^{75} -119.972 q^{77} +484.675 q^{78} -865.863 q^{79} -224.215 q^{80} +81.0000 q^{81} -651.462 q^{82} -642.216 q^{83} +21.9647 q^{84} -251.677 q^{85} +176.571 q^{86} -550.006 q^{87} -524.354 q^{88} -1423.94 q^{89} -252.242 q^{90} +147.858 q^{91} +428.195 q^{92} +316.494 q^{93} +833.908 q^{94} +96.0000 q^{96} -264.678 q^{97} -679.299 q^{98} -589.898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9	\( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 28 q^{5} + 24 q^{6} - 17 q^{7} + 32 q^{8} + 36 q^{9} - 56 q^{10} - 72 q^{11} + 48 q^{12} - 8 q^{13} - 34 q^{14} - 84 q^{15} + 64 q^{16} + 33 q^{17} + 72 q^{18} - 112 q^{20} - 51 q^{21} - 144 q^{22} - 88 q^{23} + 96 q^{24} + 94 q^{25} - 16 q^{26} + 108 q^{27} - 68 q^{28} - 75 q^{29} - 168 q^{30} + 419 q^{31} + 128 q^{32} - 216 q^{33} + 66 q^{34} + 331 q^{35} + 144 q^{36} + 131 q^{37} - 24 q^{39} - 224 q^{40} - 1134 q^{41} - 102 q^{42} + 76 q^{43} - 288 q^{44} - 252 q^{45} - 176 q^{46} + 395 q^{47} + 192 q^{48} - 1033 q^{49} + 188 q^{50} + 99 q^{51} - 32 q^{52} - 625 q^{53} + 216 q^{54} + 413 q^{55} - 136 q^{56} - 150 q^{58} - 1328 q^{59} - 336 q^{60} - 1887 q^{61} + 838 q^{62} - 153 q^{63} + 256 q^{64} - 1817 q^{65} - 432 q^{66} - 99 q^{67} + 132 q^{68} - 264 q^{69} + 662 q^{70} - 1307 q^{71} + 288 q^{72} + 183 q^{73} + 262 q^{74} + 282 q^{75} - 513 q^{77} - 48 q^{78} - 2064 q^{79} - 448 q^{80} + 324 q^{81} - 2268 q^{82} + 816 q^{83} - 204 q^{84} - 867 q^{85} + 152 q^{86} - 225 q^{87} - 576 q^{88} + 207 q^{89} - 504 q^{90} - 183 q^{91} - 352 q^{92} + 1257 q^{93} + 790 q^{94} + 384 q^{96} - 1331 q^{97} - 2066 q^{98} - 648 q^{99}+O(q^{100}) \) 4 * q + 8 * q^2 + 12 * q^3 + 16 * q^4 - 28 * q^5 + 24 * q^6 - 17 * q^7 + 32 * q^8 + 36 * q^9 - 56 * q^10 - 72 * q^11 + 48 * q^12 - 8 * q^13 - 34 * q^14 - 84 * q^15 + 64 * q^16 + 33 * q^17 + 72 * q^18 - 112 * q^20 - 51 * q^21 - 144 * q^22 - 88 * q^23 + 96 * q^24 + 94 * q^25 - 16 * q^26 + 108 * q^27 - 68 * q^28 - 75 * q^29 - 168 * q^30 + 419 * q^31 + 128 * q^32 - 216 * q^33 + 66 * q^34 + 331 * q^35 + 144 * q^36 + 131 * q^37 - 24 * q^39 - 224 * q^40 - 1134 * q^41 - 102 * q^42 + 76 * q^43 - 288 * q^44 - 252 * q^45 - 176 * q^46 + 395 * q^47 + 192 * q^48 - 1033 * q^49 + 188 * q^50 + 99 * q^51 - 32 * q^52 - 625 * q^53 + 216 * q^54 + 413 * q^55 - 136 * q^56 - 150 * q^58 - 1328 * q^59 - 336 * q^60 - 1887 * q^61 + 838 * q^62 - 153 * q^63 + 256 * q^64 - 1817 * q^65 - 432 * q^66 - 99 * q^67 + 132 * q^68 - 264 * q^69 + 662 * q^70 - 1307 * q^71 + 288 * q^72 + 183 * q^73 + 262 * q^74 + 282 * q^75 - 513 * q^77 - 48 * q^78 - 2064 * q^79 - 448 * q^80 + 324 * q^81 - 2268 * q^82 + 816 * q^83 - 204 * q^84 - 867 * q^85 + 152 * q^86 - 225 * q^87 - 576 * q^88 + 207 * q^89 - 504 * q^90 - 183 * q^91 - 352 * q^92 + 1257 * q^93 + 790 * q^94 + 384 * q^96 - 1331 * q^97 - 2066 * q^98 - 648 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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