LMFDB - Embedded newform 4018.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4018,2,Mod(1,4018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4018, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4018 = 2 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4018.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:	$32.0838915322$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.23607 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.23607 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -2.23607 q^{10} +4.47214 q^{11} -1.00000 q^{12} -6.47214 q^{13} -2.23607 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +4.00000 q^{19} +2.23607 q^{20} -4.47214 q^{22} -0.472136 q^{23} +1.00000 q^{24} +6.47214 q^{26} +5.00000 q^{27} +6.23607 q^{29} +2.23607 q^{30} -1.76393 q^{31} -1.00000 q^{32} -4.47214 q^{33} +3.00000 q^{34} -2.00000 q^{36} -8.47214 q^{37} -4.00000 q^{38} +6.47214 q^{39} -2.23607 q^{40} +1.00000 q^{41} +9.00000 q^{43} +4.47214 q^{44} -4.47214 q^{45} +0.472136 q^{46} -4.94427 q^{47} -1.00000 q^{48} +3.00000 q^{51} -6.47214 q^{52} -6.70820 q^{53} -5.00000 q^{54} +10.0000 q^{55} -4.00000 q^{57} -6.23607 q^{58} -3.52786 q^{59} -2.23607 q^{60} -1.29180 q^{61} +1.76393 q^{62} +1.00000 q^{64} -14.4721 q^{65} +4.47214 q^{66} -11.4164 q^{67} -3.00000 q^{68} +0.472136 q^{69} +2.23607 q^{71} +2.00000 q^{72} -6.00000 q^{73} +8.47214 q^{74} +4.00000 q^{76} -6.47214 q^{78} -2.70820 q^{79} +2.23607 q^{80} +1.00000 q^{81} -1.00000 q^{82} +2.94427 q^{83} -6.70820 q^{85} -9.00000 q^{86} -6.23607 q^{87} -4.47214 q^{88} +3.00000 q^{89} +4.47214 q^{90} -0.472136 q^{92} +1.76393 q^{93} +4.94427 q^{94} +8.94427 q^{95} +1.00000 q^{96} -1.00000 q^{97} -8.94427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 4 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 4 q^{9} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 6 q^{17} + 4 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} + 4 q^{26} + 10 q^{27} + 8 q^{29} - 8 q^{31} - 2 q^{32} + 6 q^{34} - 4 q^{36} - 8 q^{37} - 8 q^{38} + 4 q^{39} + 2 q^{41} + 18 q^{43} - 8 q^{46} + 8 q^{47} - 2 q^{48} + 6 q^{51} - 4 q^{52} - 10 q^{54} + 20 q^{55} - 8 q^{57} - 8 q^{58} - 16 q^{59} - 16 q^{61} + 8 q^{62} + 2 q^{64} - 20 q^{65} + 4 q^{67} - 6 q^{68} - 8 q^{69} + 4 q^{72} - 12 q^{73} + 8 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 2 q^{82} - 12 q^{83} - 18 q^{86} - 8 q^{87} + 6 q^{89} + 8 q^{92} + 8 q^{93} - 8 q^{94} + 2 q^{96} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 4 * q^9 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 6 * q^17 + 4 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 + 4 * q^26 + 10 * q^27 + 8 * q^29 - 8 * q^31 - 2 * q^32 + 6 * q^34 - 4 * q^36 - 8 * q^37 - 8 * q^38 + 4 * q^39 + 2 * q^41 + 18 * q^43 - 8 * q^46 + 8 * q^47 - 2 * q^48 + 6 * q^51 - 4 * q^52 - 10 * q^54 + 20 * q^55 - 8 * q^57 - 8 * q^58 - 16 * q^59 - 16 * q^61 + 8 * q^62 + 2 * q^64 - 20 * q^65 + 4 * q^67 - 6 * q^68 - 8 * q^69 + 4 * q^72 - 12 * q^73 + 8 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 2 * q^82 - 12 * q^83 - 18 * q^86 - 8 * q^87 + 6 * q^89 + 8 * q^92 + 8 * q^93 - 8 * q^94 + 2 * q^96 - 2 * q^97

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−2.23607	−0.707107
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	−1.00000	−0.288675
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	0	0
$15$	−2.23607	−0.577350
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.00000	0.471405
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	2.23607	0.500000
$21$	0	0
$22$	−4.47214	−0.953463
$23$	−0.472136	−0.0984472	−0.0492236	−	0.998788i	$-0.515675\pi$
$23$	−0.472136	−0.0984472	−0.0492236	+	0.998788i	$0.515675\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.47214	1.26929
$27$	5.00000	0.962250
$28$	0	0
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	2.23607	0.408248
$31$	−1.76393	−0.316812	−0.158406	−	0.987374i	$-0.550635\pi$
$31$	−1.76393	−0.316812	−0.158406	+	0.987374i	$0.550635\pi$
$32$	−1.00000	−0.176777
$33$	−4.47214	−0.778499
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	−4.00000	−0.648886
$39$	6.47214	1.03637
$40$	−2.23607	−0.353553
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	4.47214	0.674200
$45$	−4.47214	−0.666667
$46$	0.472136	0.0696126
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	−6.47214	−0.897524
$53$	−6.70820	−0.921443	−0.460721	−	0.887545i	$-0.652409\pi$
$53$	−6.70820	−0.921443	−0.460721	+	0.887545i	$0.652409\pi$
$54$	−5.00000	−0.680414
$55$	10.0000	1.34840
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−6.23607	−0.818836
$59$	−3.52786	−0.459289	−0.229644	−	0.973275i	$-0.573756\pi$
$59$	−3.52786	−0.459289	−0.229644	+	0.973275i	$0.573756\pi$
$60$	−2.23607	−0.288675
$61$	−1.29180	−0.165398	−0.0826988	−	0.996575i	$-0.526354\pi$
$61$	−1.29180	−0.165398	−0.0826988	+	0.996575i	$0.526354\pi$
$62$	1.76393	0.224020
$63$	0	0
$64$	1.00000	0.125000
$65$	−14.4721	−1.79505
$66$	4.47214	0.550482
$67$	−11.4164	−1.39474	−0.697368	−	0.716713i	$-0.745646\pi$
$67$	−11.4164	−1.39474	−0.697368	+	0.716713i	$0.745646\pi$
$68$	−3.00000	−0.363803
$69$	0.472136	0.0568385
$70$	0	0
$71$	2.23607	0.265372	0.132686	−	0.991158i	$-0.457640\pi$
$71$	2.23607	0.265372	0.132686	+	0.991158i	$0.457640\pi$
$72$	2.00000	0.235702
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	8.47214	0.984866
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	−6.47214	−0.732825
$79$	−2.70820	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$79$	−2.70820	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$80$	2.23607	0.250000
$81$	1.00000	0.111111
$82$	−1.00000	−0.110432
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	0	0
$85$	−6.70820	−0.727607
$86$	−9.00000	−0.970495
$87$	−6.23607	−0.668577
$88$	−4.47214	−0.476731
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	4.47214	0.471405
$91$	0	0
$92$	−0.472136	−0.0492236
$93$	1.76393	0.182911
$94$	4.94427	0.509963
$95$	8.94427	0.917663
$96$	1.00000	0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	−8.94427	−0.898933
$100$	0	0
$101$	0.472136	0.0469793	0.0234896	−	0.999724i	$-0.492522\pi$
$101$	0.472136	0.0469793	0.0234896	+	0.999724i	$0.492522\pi$
$102$	−3.00000	−0.297044
$103$	−18.7082	−1.84337	−0.921687	−	0.387934i	$-0.873189\pi$
$103$	−18.7082	−1.84337	−0.921687	+	0.387934i	$0.873189\pi$
$104$	6.47214	0.634645
$105$	0	0
$106$	6.70820	0.651558
$107$	1.94427	0.187960	0.0939799	−	0.995574i	$-0.470041\pi$
$107$	1.94427	0.187960	0.0939799	+	0.995574i	$0.470041\pi$
$108$	5.00000	0.481125
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	−10.0000	−0.953463
$111$	8.47214	0.804140
$112$	0	0
$113$	13.9443	1.31177	0.655883	−	0.754862i	$-0.272296\pi$
$113$	13.9443	1.31177	0.655883	+	0.754862i	$0.272296\pi$
$114$	4.00000	0.374634
$115$	−1.05573	−0.0984472
$116$	6.23607	0.579004
$117$	12.9443	1.19670
$118$	3.52786	0.324766
$119$	0	0
$120$	2.23607	0.204124
$121$	9.00000	0.818182
$122$	1.29180	0.116954
$123$	−1.00000	−0.0901670
$124$	−1.76393	−0.158406
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−9.52786	−0.845461	−0.422731	−	0.906255i	$-0.638928\pi$
$127$	−9.52786	−0.845461	−0.422731	+	0.906255i	$0.638928\pi$
$128$	−1.00000	−0.0883883
$129$	−9.00000	−0.792406
$130$	14.4721	1.26929
$131$	−13.5279	−1.18193	−0.590967	−	0.806695i	$-0.701254\pi$
$131$	−13.5279	−1.18193	−0.590967	+	0.806695i	$0.701254\pi$
$132$	−4.47214	−0.389249
$133$	0	0
$134$	11.4164	0.986227
$135$	11.1803	0.962250
$136$	3.00000	0.257248
$137$	16.4721	1.40731	0.703655	−	0.710542i	$-0.251550\pi$
$137$	16.4721	1.40731	0.703655	+	0.710542i	$0.251550\pi$
$138$	−0.472136	−0.0401909
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	−2.23607	−0.187647
$143$	−28.9443	−2.42044
$144$	−2.00000	−0.166667
$145$	13.9443	1.15801
$146$	6.00000	0.496564
$147$	0	0
$148$	−8.47214	−0.696405
$149$	19.1803	1.57131	0.785657	−	0.618662i	$-0.212325\pi$
$149$	19.1803	1.57131	0.785657	+	0.618662i	$0.212325\pi$
$150$	0	0
$151$	5.76393	0.469062	0.234531	−	0.972109i	$-0.424645\pi$
$151$	5.76393	0.469062	0.234531	+	0.972109i	$0.424645\pi$
$152$	−4.00000	−0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	−3.94427	−0.316812
$156$	6.47214	0.518186
$157$	−7.41641	−0.591894	−0.295947	−	0.955204i	$-0.595635\pi$
$157$	−7.41641	−0.591894	−0.295947	+	0.955204i	$0.595635\pi$
$158$	2.70820	0.215453
$159$	6.70820	0.531995
$160$	−2.23607	−0.176777
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	16.9443	1.32718	0.663589	−	0.748097i	$-0.269032\pi$
$163$	16.9443	1.32718	0.663589	+	0.748097i	$0.269032\pi$
$164$	1.00000	0.0780869
$165$	−10.0000	−0.778499
$166$	−2.94427	−0.228520
$167$	19.4164	1.50249	0.751243	−	0.660025i	$-0.229454\pi$
$167$	19.4164	1.50249	0.751243	+	0.660025i	$0.229454\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	6.70820	0.514496
$171$	−8.00000	−0.611775
$172$	9.00000	0.686244
$173$	−18.7082	−1.42236	−0.711179	−	0.703011i	$-0.751839\pi$
$173$	−18.7082	−1.42236	−0.711179	+	0.703011i	$0.751839\pi$
$174$	6.23607	0.472755
$175$	0	0
$176$	4.47214	0.337100
$177$	3.52786	0.265170
$178$	−3.00000	−0.224860
$179$	−4.47214	−0.334263	−0.167132	−	0.985935i	$-0.553450\pi$
$179$	−4.47214	−0.334263	−0.167132	+	0.985935i	$0.553450\pi$
$180$	−4.47214	−0.333333
$181$	5.05573	0.375789	0.187895	−	0.982189i	$-0.439834\pi$
$181$	5.05573	0.375789	0.187895	+	0.982189i	$0.439834\pi$
$182$	0	0
$183$	1.29180	0.0954923
$184$	0.472136	0.0348063
$185$	−18.9443	−1.39281
$186$	−1.76393	−0.129338
$187$	−13.4164	−0.981105
$188$	−4.94427	−0.360598
$189$	0	0
$190$	−8.94427	−0.648886
$191$	−6.70820	−0.485389	−0.242694	−	0.970103i	$-0.578031\pi$
$191$	−6.70820	−0.485389	−0.242694	+	0.970103i	$0.578031\pi$
$192$	−1.00000	−0.0721688
$193$	−24.4721	−1.76154	−0.880771	−	0.473542i	$-0.842975\pi$
$193$	−24.4721	−1.76154	−0.880771	+	0.473542i	$0.842975\pi$
$194$	1.00000	0.0717958
$195$	14.4721	1.03637
$196$	0	0
$197$	−8.94427	−0.637253	−0.318626	−	0.947880i	$-0.603222\pi$
$197$	−8.94427	−0.637253	−0.318626	+	0.947880i	$0.603222\pi$
$198$	8.94427	0.635642
$199$	6.47214	0.458798	0.229399	−	0.973333i	$-0.426324\pi$
$199$	6.47214	0.458798	0.229399	+	0.973333i	$0.426324\pi$
$200$	0	0
$201$	11.4164	0.805251
$202$	−0.472136	−0.0332194
$203$	0	0
$204$	3.00000	0.210042
$205$	2.23607	0.156174
$206$	18.7082	1.30346
$207$	0.944272	0.0656314
$208$	−6.47214	−0.448762
$209$	17.8885	1.23738
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−6.70820	−0.460721
$213$	−2.23607	−0.153213
$214$	−1.94427	−0.132908
$215$	20.1246	1.37249
$216$	−5.00000	−0.340207
$217$	0	0
$218$	13.4164	0.908674
$219$	6.00000	0.405442
$220$	10.0000	0.674200
$221$	19.4164	1.30609
$222$	−8.47214	−0.568613
$223$	−11.6525	−0.780307	−0.390154	−	0.920750i	$-0.627578\pi$
$223$	−11.6525	−0.780307	−0.390154	+	0.920750i	$0.627578\pi$
$224$	0	0
$225$	0	0
$226$	−13.9443	−0.927559
$227$	−21.9443	−1.45649	−0.728246	−	0.685316i	$-0.759664\pi$
$227$	−21.9443	−1.45649	−0.728246	+	0.685316i	$0.759664\pi$
$228$	−4.00000	−0.264906
$229$	−15.8885	−1.04994	−0.524972	−	0.851119i	$-0.675924\pi$
$229$	−15.8885	−1.04994	−0.524972	+	0.851119i	$0.675924\pi$
$230$	1.05573	0.0696126
$231$	0	0
$232$	−6.23607	−0.409418
$233$	11.8885	0.778844	0.389422	−	0.921059i	$-0.372675\pi$
$233$	11.8885	0.778844	0.389422	+	0.921059i	$0.372675\pi$
$234$	−12.9443	−0.846194
$235$	−11.0557	−0.721196
$236$	−3.52786	−0.229644
$237$	2.70820	0.175917
$238$	0	0
$239$	0.944272	0.0610799	0.0305399	−	0.999534i	$-0.490277\pi$
$239$	0.944272	0.0610799	0.0305399	+	0.999534i	$0.490277\pi$
$240$	−2.23607	−0.144338
$241$	8.47214	0.545738	0.272869	−	0.962051i	$-0.412027\pi$
$241$	8.47214	0.545738	0.272869	+	0.962051i	$0.412027\pi$
$242$	−9.00000	−0.578542
$243$	−16.0000	−1.02640
$244$	−1.29180	−0.0826988
$245$	0	0
$246$	1.00000	0.0637577
$247$	−25.8885	−1.64725
$248$	1.76393	0.112010
$249$	−2.94427	−0.186586
$250$	11.1803	0.707107
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−2.11146	−0.132746
$254$	9.52786	0.597831
$255$	6.70820	0.420084
$256$	1.00000	0.0625000
$257$	−31.9443	−1.99263	−0.996314	−	0.0857758i	$-0.972663\pi$
$257$	−31.9443	−1.99263	−0.996314	+	0.0857758i	$0.972663\pi$
$258$	9.00000	0.560316
$259$	0	0
$260$	−14.4721	−0.897524
$261$	−12.4721	−0.772006
$262$	13.5279	0.835754
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	4.47214	0.275241
$265$	−15.0000	−0.921443
$266$	0	0
$267$	−3.00000	−0.183597
$268$	−11.4164	−0.697368
$269$	−28.4721	−1.73598	−0.867988	−	0.496584i	$-0.834587\pi$
$269$	−28.4721	−1.73598	−0.867988	+	0.496584i	$0.834587\pi$
$270$	−11.1803	−0.680414
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−16.4721	−0.995118
$275$	0	0
$276$	0.472136	0.0284192
$277$	2.47214	0.148536	0.0742681	−	0.997238i	$-0.476338\pi$
$277$	2.47214	0.148536	0.0742681	+	0.997238i	$0.476338\pi$
$278$	13.4164	0.804663
$279$	3.52786	0.211208
$280$	0	0
$281$	13.4164	0.800356	0.400178	−	0.916437i	$-0.368948\pi$
$281$	13.4164	0.800356	0.400178	+	0.916437i	$0.368948\pi$
$282$	−4.94427	−0.294427
$283$	19.4164	1.15419	0.577093	−	0.816679i	$-0.304187\pi$
$283$	19.4164	1.15419	0.577093	+	0.816679i	$0.304187\pi$
$284$	2.23607	0.132686
$285$	−8.94427	−0.529813
$286$	28.9443	1.71151
$287$	0	0
$288$	2.00000	0.117851
$289$	−8.00000	−0.470588
$290$	−13.9443	−0.818836
$291$	1.00000	0.0586210
$292$	−6.00000	−0.351123
$293$	−24.9443	−1.45726	−0.728630	−	0.684908i	$-0.759843\pi$
$293$	−24.9443	−1.45726	−0.728630	+	0.684908i	$0.759843\pi$
$294$	0	0
$295$	−7.88854	−0.459289
$296$	8.47214	0.492433
$297$	22.3607	1.29750
$298$	−19.1803	−1.11109
$299$	3.05573	0.176717
$300$	0	0
$301$	0	0
$302$	−5.76393	−0.331677
$303$	−0.472136	−0.0271235
$304$	4.00000	0.229416
$305$	−2.88854	−0.165398
$306$	−6.00000	−0.342997
$307$	21.8885	1.24925	0.624623	−	0.780927i	$-0.285253\pi$
$307$	21.8885	1.24925	0.624623	+	0.780927i	$0.285253\pi$
$308$	0	0
$309$	18.7082	1.06427
$310$	3.94427	0.224020
$311$	−18.9443	−1.07423	−0.537116	−	0.843509i	$-0.680486\pi$
$311$	−18.9443	−1.07423	−0.537116	+	0.843509i	$0.680486\pi$
$312$	−6.47214	−0.366413
$313$	31.8885	1.80245	0.901224	−	0.433355i	$-0.142670\pi$
$313$	31.8885	1.80245	0.901224	+	0.433355i	$0.142670\pi$
$314$	7.41641	0.418532
$315$	0	0
$316$	−2.70820	−0.152348
$317$	1.41641	0.0795534	0.0397767	−	0.999209i	$-0.487335\pi$
$317$	1.41641	0.0795534	0.0397767	+	0.999209i	$0.487335\pi$
$318$	−6.70820	−0.376177
$319$	27.8885	1.56146
$320$	2.23607	0.125000
$321$	−1.94427	−0.108519
$322$	0	0
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.9443	−0.938456
$327$	13.4164	0.741929
$328$	−1.00000	−0.0552158
$329$	0	0
$330$	10.0000	0.550482
$331$	−2.47214	−0.135881	−0.0679404	−	0.997689i	$-0.521643\pi$
$331$	−2.47214	−0.135881	−0.0679404	+	0.997689i	$0.521643\pi$
$332$	2.94427	0.161588
$333$	16.9443	0.928540
$334$	−19.4164	−1.06242
$335$	−25.5279	−1.39474
$336$	0	0
$337$	−0.888544	−0.0484021	−0.0242010	−	0.999707i	$-0.507704\pi$
$337$	−0.888544	−0.0484021	−0.0242010	+	0.999707i	$0.507704\pi$
$338$	−28.8885	−1.57133
$339$	−13.9443	−0.757349
$340$	−6.70820	−0.363803
$341$	−7.88854	−0.427189
$342$	8.00000	0.432590
$343$	0	0
$344$	−9.00000	−0.485247
$345$	1.05573	0.0568385
$346$	18.7082	1.00576
$347$	−29.8885	−1.60450	−0.802251	−	0.596987i	$-0.796364\pi$
$347$	−29.8885	−1.60450	−0.802251	+	0.596987i	$0.796364\pi$
$348$	−6.23607	−0.334288
$349$	5.41641	0.289934	0.144967	−	0.989437i	$-0.453692\pi$
$349$	5.41641	0.289934	0.144967	+	0.989437i	$0.453692\pi$
$350$	0	0
$351$	−32.3607	−1.72729
$352$	−4.47214	−0.238366
$353$	26.3607	1.40304	0.701519	−	0.712651i	$-0.252506\pi$
$353$	26.3607	1.40304	0.701519	+	0.712651i	$0.252506\pi$
$354$	−3.52786	−0.187504
$355$	5.00000	0.265372
$356$	3.00000	0.159000
$357$	0	0
$358$	4.47214	0.236360
$359$	−26.3607	−1.39126	−0.695632	−	0.718399i	$-0.744875\pi$
$359$	−26.3607	−1.39126	−0.695632	+	0.718399i	$0.744875\pi$
$360$	4.47214	0.235702
$361$	−3.00000	−0.157895
$362$	−5.05573	−0.265723
$363$	−9.00000	−0.472377
$364$	0	0
$365$	−13.4164	−0.702247
$366$	−1.29180	−0.0675233
$367$	18.7082	0.976560	0.488280	−	0.872687i	$-0.337624\pi$
$367$	18.7082	0.976560	0.488280	+	0.872687i	$0.337624\pi$
$368$	−0.472136	−0.0246118
$369$	−2.00000	−0.104116
$370$	18.9443	0.984866
$371$	0	0
$372$	1.76393	0.0914556
$373$	7.88854	0.408453	0.204227	−	0.978924i	$-0.434532\pi$
$373$	7.88854	0.408453	0.204227	+	0.978924i	$0.434532\pi$
$374$	13.4164	0.693746
$375$	11.1803	0.577350
$376$	4.94427	0.254981
$377$	−40.3607	−2.07868
$378$	0	0
$379$	21.8328	1.12148	0.560738	−	0.827993i	$-0.310517\pi$
$379$	21.8328	1.12148	0.560738	+	0.827993i	$0.310517\pi$
$380$	8.94427	0.458831
$381$	9.52786	0.488127
$382$	6.70820	0.343222
$383$	−23.8885	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$383$	−23.8885	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	24.4721	1.24560
$387$	−18.0000	−0.914991
$388$	−1.00000	−0.0507673
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−14.4721	−0.732825
$391$	1.41641	0.0716308
$392$	0	0
$393$	13.5279	0.682390
$394$	8.94427	0.450606
$395$	−6.05573	−0.304697
$396$	−8.94427	−0.449467
$397$	−30.8328	−1.54745	−0.773727	−	0.633519i	$-0.781610\pi$
$397$	−30.8328	−1.54745	−0.773727	+	0.633519i	$0.781610\pi$
$398$	−6.47214	−0.324419
$399$	0	0
$400$	0	0
$401$	28.8885	1.44263	0.721313	−	0.692610i	$-0.243539\pi$
$401$	28.8885	1.44263	0.721313	+	0.692610i	$0.243539\pi$
$402$	−11.4164	−0.569399
$403$	11.4164	0.568692
$404$	0.472136	0.0234896
$405$	2.23607	0.111111
$406$	0	0
$407$	−37.8885	−1.87807
$408$	−3.00000	−0.148522
$409$	−27.8885	−1.37900	−0.689500	−	0.724286i	$-0.742170\pi$
$409$	−27.8885	−1.37900	−0.689500	+	0.724286i	$0.742170\pi$
$410$	−2.23607	−0.110432
$411$	−16.4721	−0.812511
$412$	−18.7082	−0.921687
$413$	0	0
$414$	−0.944272	−0.0464084
$415$	6.58359	0.323176
$416$	6.47214	0.317323
$417$	13.4164	0.657004
$418$	−17.8885	−0.874957
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	−1.29180	−0.0629583	−0.0314791	−	0.999504i	$-0.510022\pi$
$421$	−1.29180	−0.0629583	−0.0314791	+	0.999504i	$0.510022\pi$
$422$	0	0
$423$	9.88854	0.480797
$424$	6.70820	0.325779
$425$	0	0
$426$	2.23607	0.108338
$427$	0	0
$428$	1.94427	0.0939799
$429$	28.9443	1.39744
$430$	−20.1246	−0.970495
$431$	−19.4164	−0.935255	−0.467628	−	0.883926i	$-0.654891\pi$
$431$	−19.4164	−0.935255	−0.467628	+	0.883926i	$0.654891\pi$
$432$	5.00000	0.240563
$433$	0.472136	0.0226894	0.0113447	−	0.999936i	$-0.496389\pi$
$433$	0.472136	0.0226894	0.0113447	+	0.999936i	$0.496389\pi$
$434$	0	0
$435$	−13.9443	−0.668577
$436$	−13.4164	−0.642529
$437$	−1.88854	−0.0903413
$438$	−6.00000	−0.286691
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	−10.0000	−0.476731
$441$	0	0
$442$	−19.4164	−0.923544
$443$	17.9443	0.852558	0.426279	−	0.904592i	$-0.359824\pi$
$443$	17.9443	0.852558	0.426279	+	0.904592i	$0.359824\pi$
$444$	8.47214	0.402070
$445$	6.70820	0.317999
$446$	11.6525	0.551761
$447$	−19.1803	−0.907199
$448$	0	0
$449$	−40.7771	−1.92439	−0.962195	−	0.272362i	$-0.912195\pi$
$449$	−40.7771	−1.92439	−0.962195	+	0.272362i	$0.912195\pi$
$450$	0	0
$451$	4.47214	0.210585
$452$	13.9443	0.655883
$453$	−5.76393	−0.270813
$454$	21.9443	1.02990
$455$	0	0
$456$	4.00000	0.187317
$457$	−19.4164	−0.908261	−0.454131	−	0.890935i	$-0.650050\pi$
$457$	−19.4164	−0.908261	−0.454131	+	0.890935i	$0.650050\pi$
$458$	15.8885	0.742423
$459$	−15.0000	−0.700140
$460$	−1.05573	−0.0492236
$461$	−12.5967	−0.586689	−0.293345	−	0.956007i	$-0.594768\pi$
$461$	−12.5967	−0.586689	−0.293345	+	0.956007i	$0.594768\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.23607	0.289502
$465$	3.94427	0.182911
$466$	−11.8885	−0.550726
$467$	1.88854	0.0873914	0.0436957	−	0.999045i	$-0.486087\pi$
$467$	1.88854	0.0873914	0.0436957	+	0.999045i	$0.486087\pi$
$468$	12.9443	0.598349
$469$	0	0
$470$	11.0557	0.509963
$471$	7.41641	0.341730
$472$	3.52786	0.162383
$473$	40.2492	1.85066
$474$	−2.70820	−0.124392
$475$	0	0
$476$	0	0
$477$	13.4164	0.614295
$478$	−0.944272	−0.0431900
$479$	−31.4164	−1.43545	−0.717726	−	0.696325i	$-0.754817\pi$
$479$	−31.4164	−1.43545	−0.717726	+	0.696325i	$0.754817\pi$
$480$	2.23607	0.102062
$481$	54.8328	2.50016
$482$	−8.47214	−0.385895
$483$	0	0
$484$	9.00000	0.409091
$485$	−2.23607	−0.101535
$486$	16.0000	0.725775
$487$	38.8328	1.75968	0.879841	−	0.475267i	$-0.157649\pi$
$487$	38.8328	1.75968	0.879841	+	0.475267i	$0.157649\pi$
$488$	1.29180	0.0584769
$489$	−16.9443	−0.766246
$490$	0	0
$491$	−32.7771	−1.47921	−0.739605	−	0.673042i	$-0.764987\pi$
$491$	−32.7771	−1.47921	−0.739605	+	0.673042i	$0.764987\pi$
$492$	−1.00000	−0.0450835
$493$	−18.7082	−0.842575
$494$	25.8885	1.16478
$495$	−20.0000	−0.898933
$496$	−1.76393	−0.0792029
$497$	0	0
$498$	2.94427	0.131936
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	−11.1803	−0.500000
$501$	−19.4164	−0.867461
$502$	6.00000	0.267793
$503$	27.3050	1.21747	0.608734	−	0.793375i	$-0.291678\pi$
$503$	27.3050	1.21747	0.608734	+	0.793375i	$0.291678\pi$
$504$	0	0
$505$	1.05573	0.0469793
$506$	2.11146	0.0938657
$507$	−28.8885	−1.28299
$508$	−9.52786	−0.422731
$509$	17.0557	0.755982	0.377991	−	0.925809i	$-0.376615\pi$
$509$	17.0557	0.755982	0.377991	+	0.925809i	$0.376615\pi$
$510$	−6.70820	−0.297044
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	20.0000	0.883022
$514$	31.9443	1.40900
$515$	−41.8328	−1.84337
$516$	−9.00000	−0.396203
$517$	−22.1115	−0.972461
$518$	0	0
$519$	18.7082	0.821199
$520$	14.4721	0.634645
$521$	−9.05573	−0.396739	−0.198369	−	0.980127i	$-0.563565\pi$
$521$	−9.05573	−0.396739	−0.198369	+	0.980127i	$0.563565\pi$
$522$	12.4721	0.545891
$523$	38.3607	1.67740	0.838698	−	0.544597i	$-0.183317\pi$
$523$	38.3607	1.67740	0.838698	+	0.544597i	$0.183317\pi$
$524$	−13.5279	−0.590967
$525$	0	0
$526$	12.0000	0.523225
$527$	5.29180	0.230514
$528$	−4.47214	−0.194625
$529$	−22.7771	−0.990308
$530$	15.0000	0.651558
$531$	7.05573	0.306193
$532$	0	0
$533$	−6.47214	−0.280339
$534$	3.00000	0.129823
$535$	4.34752	0.187960
$536$	11.4164	0.493114
$537$	4.47214	0.192987
$538$	28.4721	1.22752
$539$	0	0
$540$	11.1803	0.481125
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	4.00000	0.171815
$543$	−5.05573	−0.216962
$544$	3.00000	0.128624
$545$	−30.0000	−1.28506
$546$	0	0
$547$	26.8328	1.14729	0.573644	−	0.819105i	$-0.305529\pi$
$547$	26.8328	1.14729	0.573644	+	0.819105i	$0.305529\pi$
$548$	16.4721	0.703655
$549$	2.58359	0.110265
$550$	0	0
$551$	24.9443	1.06266
$552$	−0.472136	−0.0200954
$553$	0	0
$554$	−2.47214	−0.105031
$555$	18.9443	0.804140
$556$	−13.4164	−0.568982
$557$	6.70820	0.284236	0.142118	−	0.989850i	$-0.454609\pi$
$557$	6.70820	0.284236	0.142118	+	0.989850i	$0.454609\pi$
$558$	−3.52786	−0.149346
$559$	−58.2492	−2.46368
$560$	0	0
$561$	13.4164	0.566441
$562$	−13.4164	−0.565937
$563$	−13.8885	−0.585332	−0.292666	−	0.956215i	$-0.594542\pi$
$563$	−13.8885	−0.585332	−0.292666	+	0.956215i	$0.594542\pi$
$564$	4.94427	0.208191
$565$	31.1803	1.31177
$566$	−19.4164	−0.816132
$567$	0	0
$568$	−2.23607	−0.0938233
$569$	−4.05573	−0.170025	−0.0850125	−	0.996380i	$-0.527093\pi$
$569$	−4.05573	−0.170025	−0.0850125	+	0.996380i	$0.527093\pi$
$570$	8.94427	0.374634
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	−28.9443	−1.21022
$573$	6.70820	0.280239
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	46.9443	1.95432	0.977158	−	0.212515i	$-0.0681654\pi$
$577$	46.9443	1.95432	0.977158	+	0.212515i	$0.0681654\pi$
$578$	8.00000	0.332756
$579$	24.4721	1.01703
$580$	13.9443	0.579004
$581$	0	0
$582$	−1.00000	−0.0414513
$583$	−30.0000	−1.24247
$584$	6.00000	0.248282
$585$	28.9443	1.19670
$586$	24.9443	1.03044
$587$	−10.0557	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$587$	−10.0557	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$588$	0	0
$589$	−7.05573	−0.290726
$590$	7.88854	0.324766
$591$	8.94427	0.367918
$592$	−8.47214	−0.348203
$593$	44.7771	1.83877	0.919387	−	0.393354i	$-0.128685\pi$
$593$	44.7771	1.83877	0.919387	+	0.393354i	$0.128685\pi$
$594$	−22.3607	−0.917470
$595$	0	0
$596$	19.1803	0.785657
$597$	−6.47214	−0.264887
$598$	−3.05573	−0.124958
$599$	3.05573	0.124854	0.0624268	−	0.998050i	$-0.480116\pi$
$599$	3.05573	0.124854	0.0624268	+	0.998050i	$0.480116\pi$
$600$	0	0
$601$	11.8328	0.482671	0.241335	−	0.970442i	$-0.422415\pi$
$601$	11.8328	0.482671	0.241335	+	0.970442i	$0.422415\pi$
$602$	0	0
$603$	22.8328	0.929824
$604$	5.76393	0.234531
$605$	20.1246	0.818182
$606$	0.472136	0.0191792
$607$	−30.7082	−1.24641	−0.623204	−	0.782059i	$-0.714169\pi$
$607$	−30.7082	−1.24641	−0.623204	+	0.782059i	$0.714169\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	2.88854	0.116954
$611$	32.0000	1.29458
$612$	6.00000	0.242536
$613$	10.5836	0.427467	0.213734	−	0.976892i	$-0.431438\pi$
$613$	10.5836	0.427467	0.213734	+	0.976892i	$0.431438\pi$
$614$	−21.8885	−0.883350
$615$	−2.23607	−0.0901670
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−18.7082	−0.752554
$619$	−33.4164	−1.34312	−0.671559	−	0.740951i	$-0.734375\pi$
$619$	−33.4164	−1.34312	−0.671559	+	0.740951i	$0.734375\pi$
$620$	−3.94427	−0.158406
$621$	−2.36068	−0.0947308
$622$	18.9443	0.759596
$623$	0	0
$624$	6.47214	0.259093
$625$	−25.0000	−1.00000
$626$	−31.8885	−1.27452
$627$	−17.8885	−0.714400
$628$	−7.41641	−0.295947
$629$	25.4164	1.01342
$630$	0	0
$631$	26.8328	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$631$	26.8328	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$632$	2.70820	0.107727
$633$	0	0
$634$	−1.41641	−0.0562527
$635$	−21.3050	−0.845461
$636$	6.70820	0.265998
$637$	0	0
$638$	−27.8885	−1.10412
$639$	−4.47214	−0.176915
$640$	−2.23607	−0.0883883
$641$	20.9443	0.827249	0.413625	−	0.910448i	$-0.364263\pi$
$641$	20.9443	0.827249	0.413625	+	0.910448i	$0.364263\pi$
$642$	1.94427	0.0767343
$643$	−11.8328	−0.466641	−0.233320	−	0.972400i	$-0.574959\pi$
$643$	−11.8328	−0.466641	−0.233320	+	0.972400i	$0.574959\pi$
$644$	0	0
$645$	−20.1246	−0.792406
$646$	12.0000	0.472134
$647$	−5.76393	−0.226604	−0.113302	−	0.993561i	$-0.536143\pi$
$647$	−5.76393	−0.226604	−0.113302	+	0.993561i	$0.536143\pi$
$648$	−1.00000	−0.0392837
$649$	−15.7771	−0.619305
$650$	0	0
$651$	0	0
$652$	16.9443	0.663589
$653$	−33.5410	−1.31256	−0.656281	−	0.754517i	$-0.727871\pi$
$653$	−33.5410	−1.31256	−0.656281	+	0.754517i	$0.727871\pi$
$654$	−13.4164	−0.524623
$655$	−30.2492	−1.18193
$656$	1.00000	0.0390434
$657$	12.0000	0.468165
$658$	0	0
$659$	26.3607	1.02687	0.513433	−	0.858130i	$-0.328373\pi$
$659$	26.3607	1.02687	0.513433	+	0.858130i	$0.328373\pi$
$660$	−10.0000	−0.389249
$661$	25.4164	0.988584	0.494292	−	0.869296i	$-0.335427\pi$
$661$	25.4164	0.988584	0.494292	+	0.869296i	$0.335427\pi$
$662$	2.47214	0.0960823
$663$	−19.4164	−0.754071
$664$	−2.94427	−0.114260
$665$	0	0
$666$	−16.9443	−0.656577
$667$	−2.94427	−0.114003
$668$	19.4164	0.751243
$669$	11.6525	0.450511
$670$	25.5279	0.986227
$671$	−5.77709	−0.223022
$672$	0	0
$673$	41.4164	1.59649	0.798243	−	0.602336i	$-0.205763\pi$
$673$	41.4164	1.59649	0.798243	+	0.602336i	$0.205763\pi$
$674$	0.888544	0.0342254
$675$	0	0
$676$	28.8885	1.11110
$677$	−1.41641	−0.0544370	−0.0272185	−	0.999630i	$-0.508665\pi$
$677$	−1.41641	−0.0544370	−0.0272185	+	0.999630i	$0.508665\pi$
$678$	13.9443	0.535527
$679$	0	0
$680$	6.70820	0.257248
$681$	21.9443	0.840906
$682$	7.88854	0.302068
$683$	−34.2492	−1.31051	−0.655255	−	0.755408i	$-0.727439\pi$
$683$	−34.2492	−1.31051	−0.655255	+	0.755408i	$0.727439\pi$
$684$	−8.00000	−0.305888
$685$	36.8328	1.40731
$686$	0	0
$687$	15.8885	0.606186
$688$	9.00000	0.343122
$689$	43.4164	1.65403
$690$	−1.05573	−0.0401909
$691$	42.7771	1.62732	0.813659	−	0.581343i	$-0.197473\pi$
$691$	42.7771	1.62732	0.813659	+	0.581343i	$0.197473\pi$
$692$	−18.7082	−0.711179
$693$	0	0
$694$	29.8885	1.13455
$695$	−30.0000	−1.13796
$696$	6.23607	0.236378
$697$	−3.00000	−0.113633
$698$	−5.41641	−0.205014
$699$	−11.8885	−0.449666
$700$	0	0
$701$	16.4721	0.622144	0.311072	−	0.950386i	$-0.399312\pi$
$701$	16.4721	0.622144	0.311072	+	0.950386i	$0.399312\pi$
$702$	32.3607	1.22138
$703$	−33.8885	−1.27813
$704$	4.47214	0.168550
$705$	11.0557	0.416383
$706$	−26.3607	−0.992097
$707$	0	0
$708$	3.52786	0.132585
$709$	9.29180	0.348961	0.174480	−	0.984661i	$-0.444175\pi$
$709$	9.29180	0.348961	0.174480	+	0.984661i	$0.444175\pi$
$710$	−5.00000	−0.187647
$711$	5.41641	0.203131
$712$	−3.00000	−0.112430
$713$	0.832816	0.0311892
$714$	0	0
$715$	−64.7214	−2.42044
$716$	−4.47214	−0.167132
$717$	−0.944272	−0.0352645
$718$	26.3607	0.983772
$719$	25.3050	0.943715	0.471858	−	0.881675i	$-0.343584\pi$
$719$	25.3050	0.943715	0.471858	+	0.881675i	$0.343584\pi$
$720$	−4.47214	−0.166667
$721$	0	0
$722$	3.00000	0.111648
$723$	−8.47214	−0.315082
$724$	5.05573	0.187895
$725$	0	0
$726$	9.00000	0.334021
$727$	−15.5279	−0.575897	−0.287948	−	0.957646i	$-0.592973\pi$
$727$	−15.5279	−0.575897	−0.287948	+	0.957646i	$0.592973\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	13.4164	0.496564
$731$	−27.0000	−0.998631
$732$	1.29180	0.0477462
$733$	18.7082	0.691003	0.345502	−	0.938418i	$-0.387709\pi$
$733$	18.7082	0.691003	0.345502	+	0.938418i	$0.387709\pi$
$734$	−18.7082	−0.690532
$735$	0	0
$736$	0.472136	0.0174032
$737$	−51.0557	−1.88066
$738$	2.00000	0.0736210
$739$	−27.9443	−1.02795	−0.513973	−	0.857806i	$-0.671827\pi$
$739$	−27.9443	−1.02795	−0.513973	+	0.857806i	$0.671827\pi$
$740$	−18.9443	−0.696405
$741$	25.8885	0.951039
$742$	0	0
$743$	8.11146	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$743$	8.11146	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$744$	−1.76393	−0.0646689
$745$	42.8885	1.57131
$746$	−7.88854	−0.288820
$747$	−5.88854	−0.215451
$748$	−13.4164	−0.490552
$749$	0	0
$750$	−11.1803	−0.408248
$751$	−26.8328	−0.979143	−0.489572	−	0.871963i	$-0.662847\pi$
$751$	−26.8328	−0.979143	−0.489572	+	0.871963i	$0.662847\pi$
$752$	−4.94427	−0.180299
$753$	6.00000	0.218652
$754$	40.3607	1.46985
$755$	12.8885	0.469062
$756$	0	0
$757$	0.347524	0.0126310	0.00631549	−	0.999980i	$-0.497990\pi$
$757$	0.347524	0.0126310	0.00631549	+	0.999980i	$0.497990\pi$
$758$	−21.8328	−0.793004
$759$	2.11146	0.0766410
$760$	−8.94427	−0.324443
$761$	10.4721	0.379615	0.189807	−	0.981821i	$-0.439214\pi$
$761$	10.4721	0.379615	0.189807	+	0.981821i	$0.439214\pi$
$762$	−9.52786	−0.345158
$763$	0	0
$764$	−6.70820	−0.242694
$765$	13.4164	0.485071
$766$	23.8885	0.863128
$767$	22.8328	0.824445
$768$	−1.00000	−0.0360844
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	31.9443	1.15044
$772$	−24.4721	−0.880771
$773$	3.05573	0.109907	0.0549535	−	0.998489i	$-0.482499\pi$
$773$	3.05573	0.109907	0.0549535	+	0.998489i	$0.482499\pi$
$774$	18.0000	0.646997
$775$	0	0
$776$	1.00000	0.0358979
$777$	0	0
$778$	−12.0000	−0.430221
$779$	4.00000	0.143315
$780$	14.4721	0.518186
$781$	10.0000	0.357828
$782$	−1.41641	−0.0506506
$783$	31.1803	1.11429
$784$	0	0
$785$	−16.5836	−0.591894
$786$	−13.5279	−0.482523
$787$	−0.583592	−0.0208028	−0.0104014	−	0.999946i	$-0.503311\pi$
$787$	−0.583592	−0.0208028	−0.0104014	+	0.999946i	$0.503311\pi$
$788$	−8.94427	−0.318626
$789$	12.0000	0.427211
$790$	6.05573	0.215453
$791$	0	0
$792$	8.94427	0.317821
$793$	8.36068	0.296896
$794$	30.8328	1.09422
$795$	15.0000	0.531995
$796$	6.47214	0.229399
$797$	12.5967	0.446200	0.223100	−	0.974796i	$-0.428382\pi$
$797$	12.5967	0.446200	0.223100	+	0.974796i	$0.428382\pi$
$798$	0	0
$799$	14.8328	0.524747
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−28.8885	−1.02009
$803$	−26.8328	−0.946910
$804$	11.4164	0.402626
$805$	0	0
$806$	−11.4164	−0.402126
$807$	28.4721	1.00227
$808$	−0.472136	−0.0166097
$809$	−37.4164	−1.31549	−0.657745	−	0.753240i	$-0.728490\pi$
$809$	−37.4164	−1.31549	−0.657745	+	0.753240i	$0.728490\pi$
$810$	−2.23607	−0.0785674
$811$	26.8328	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$811$	26.8328	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	37.8885	1.32799
$815$	37.8885	1.32718
$816$	3.00000	0.105021
$817$	36.0000	1.25948
$818$	27.8885	0.975100
$819$	0	0
$820$	2.23607	0.0780869
$821$	50.3607	1.75760	0.878800	−	0.477190i	$-0.158345\pi$
$821$	50.3607	1.75760	0.878800	+	0.477190i	$0.158345\pi$
$822$	16.4721	0.574532
$823$	44.1246	1.53809	0.769044	−	0.639196i	$-0.220733\pi$
$823$	44.1246	1.53809	0.769044	+	0.639196i	$0.220733\pi$
$824$	18.7082	0.651731
$825$	0	0
$826$	0	0
$827$	22.9443	0.797851	0.398925	−	0.916983i	$-0.369383\pi$
$827$	22.9443	0.797851	0.398925	+	0.916983i	$0.369383\pi$
$828$	0.944272	0.0328157
$829$	34.2361	1.18907	0.594534	−	0.804071i	$-0.297337\pi$
$829$	34.2361	1.18907	0.594534	+	0.804071i	$0.297337\pi$
$830$	−6.58359	−0.228520
$831$	−2.47214	−0.0857574
$832$	−6.47214	−0.224381
$833$	0	0
$834$	−13.4164	−0.464572
$835$	43.4164	1.50249
$836$	17.8885	0.618688
$837$	−8.81966	−0.304852
$838$	29.8885	1.03248
$839$	13.8885	0.479486	0.239743	−	0.970836i	$-0.422937\pi$
$839$	13.8885	0.479486	0.239743	+	0.970836i	$0.422937\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	1.29180	0.0445182
$843$	−13.4164	−0.462086
$844$	0	0
$845$	64.5967	2.22220
$846$	−9.88854	−0.339975
$847$	0	0
$848$	−6.70820	−0.230361
$849$	−19.4164	−0.666369
$850$	0	0
$851$	4.00000	0.137118
$852$	−2.23607	−0.0766064
$853$	−6.70820	−0.229685	−0.114842	−	0.993384i	$-0.536636\pi$
$853$	−6.70820	−0.229685	−0.114842	+	0.993384i	$0.536636\pi$
$854$	0	0
$855$	−17.8885	−0.611775
$856$	−1.94427	−0.0664538
$857$	23.3050	0.796082	0.398041	−	0.917368i	$-0.369690\pi$
$857$	23.3050	0.796082	0.398041	+	0.917368i	$0.369690\pi$
$858$	−28.9443	−0.988141
$859$	43.7771	1.49365	0.746827	−	0.665018i	$-0.231576\pi$
$859$	43.7771	1.49365	0.746827	+	0.665018i	$0.231576\pi$
$860$	20.1246	0.686244
$861$	0	0
$862$	19.4164	0.661325
$863$	46.4721	1.58193	0.790965	−	0.611861i	$-0.209579\pi$
$863$	46.4721	1.58193	0.790965	+	0.611861i	$0.209579\pi$
$864$	−5.00000	−0.170103
$865$	−41.8328	−1.42236
$866$	−0.472136	−0.0160438
$867$	8.00000	0.271694
$868$	0	0
$869$	−12.1115	−0.410853
$870$	13.9443	0.472755
$871$	73.8885	2.50362
$872$	13.4164	0.454337
$873$	2.00000	0.0676897
$874$	1.88854	0.0638809
$875$	0	0
$876$	6.00000	0.202721
$877$	−32.3607	−1.09274	−0.546371	−	0.837543i	$-0.683991\pi$
$877$	−32.3607	−1.09274	−0.546371	+	0.837543i	$0.683991\pi$
$878$	28.0000	0.944954
$879$	24.9443	0.841349
$880$	10.0000	0.337100
$881$	21.0557	0.709386	0.354693	−	0.934983i	$-0.384585\pi$
$881$	21.0557	0.709386	0.354693	+	0.934983i	$0.384585\pi$
$882$	0	0
$883$	16.9443	0.570220	0.285110	−	0.958495i	$-0.407970\pi$
$883$	16.9443	0.570220	0.285110	+	0.958495i	$0.407970\pi$
$884$	19.4164	0.653044
$885$	7.88854	0.265170
$886$	−17.9443	−0.602850
$887$	−34.3607	−1.15372	−0.576859	−	0.816843i	$-0.695722\pi$
$887$	−34.3607	−1.15372	−0.576859	+	0.816843i	$0.695722\pi$
$888$	−8.47214	−0.284306
$889$	0	0
$890$	−6.70820	−0.224860
$891$	4.47214	0.149822
$892$	−11.6525	−0.390154
$893$	−19.7771	−0.661815
$894$	19.1803	0.641487
$895$	−10.0000	−0.334263
$896$	0	0
$897$	−3.05573	−0.102028
$898$	40.7771	1.36075
$899$	−11.0000	−0.366871
$900$	0	0
$901$	20.1246	0.670448
$902$	−4.47214	−0.148906
$903$	0	0
$904$	−13.9443	−0.463780
$905$	11.3050	0.375789
$906$	5.76393	0.191494
$907$	51.8328	1.72108	0.860540	−	0.509383i	$-0.170126\pi$
$907$	51.8328	1.72108	0.860540	+	0.509383i	$0.170126\pi$
$908$	−21.9443	−0.728246
$909$	−0.944272	−0.0313195
$910$	0	0
$911$	−7.41641	−0.245717	−0.122858	−	0.992424i	$-0.539206\pi$
$911$	−7.41641	−0.245717	−0.122858	+	0.992424i	$0.539206\pi$
$912$	−4.00000	−0.132453
$913$	13.1672	0.435770
$914$	19.4164	0.642238
$915$	2.88854	0.0954923
$916$	−15.8885	−0.524972
$917$	0	0
$918$	15.0000	0.495074
$919$	29.5410	0.974468	0.487234	−	0.873271i	$-0.338006\pi$
$919$	29.5410	0.974468	0.487234	+	0.873271i	$0.338006\pi$
$920$	1.05573	0.0348063
$921$	−21.8885	−0.721252
$922$	12.5967	0.414852
$923$	−14.4721	−0.476356
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	37.4164	1.22892
$928$	−6.23607	−0.204709
$929$	27.8885	0.914993	0.457497	−	0.889211i	$-0.348746\pi$
$929$	27.8885	0.914993	0.457497	+	0.889211i	$0.348746\pi$
$930$	−3.94427	−0.129338
$931$	0	0
$932$	11.8885	0.389422
$933$	18.9443	0.620208
$934$	−1.88854	−0.0617950
$935$	−30.0000	−0.981105
$936$	−12.9443	−0.423097
$937$	−27.8328	−0.909258	−0.454629	−	0.890681i	$-0.650228\pi$
$937$	−27.8328	−0.909258	−0.454629	+	0.890681i	$0.650228\pi$
$938$	0	0
$939$	−31.8885	−1.04064
$940$	−11.0557	−0.360598
$941$	−46.3607	−1.51131	−0.755657	−	0.654967i	$-0.772683\pi$
$941$	−46.3607	−1.51131	−0.755657	+	0.654967i	$0.772683\pi$
$942$	−7.41641	−0.241640
$943$	−0.472136	−0.0153749
$944$	−3.52786	−0.114822
$945$	0	0
$946$	−40.2492	−1.30862
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	2.70820	0.0879584
$949$	38.8328	1.26057
$950$	0	0
$951$	−1.41641	−0.0459302
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−13.4164	−0.434372
$955$	−15.0000	−0.485389
$956$	0.944272	0.0305399
$957$	−27.8885	−0.901509
$958$	31.4164	1.01502
$959$	0	0
$960$	−2.23607	−0.0721688
$961$	−27.8885	−0.899630
$962$	−54.8328	−1.76788
$963$	−3.88854	−0.125307
$964$	8.47214	0.272869
$965$	−54.7214	−1.76154
$966$	0	0
$967$	40.1246	1.29032	0.645160	−	0.764047i	$-0.276791\pi$
$967$	40.1246	1.29032	0.645160	+	0.764047i	$0.276791\pi$
$968$	−9.00000	−0.289271
$969$	12.0000	0.385496
$970$	2.23607	0.0717958
$971$	−41.7214	−1.33890	−0.669451	−	0.742856i	$-0.733471\pi$
$971$	−41.7214	−1.33890	−0.669451	+	0.742856i	$0.733471\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	−38.8328	−1.24428
$975$	0	0
$976$	−1.29180	−0.0413494
$977$	−8.47214	−0.271048	−0.135524	−	0.990774i	$-0.543272\pi$
$977$	−8.47214	−0.271048	−0.135524	+	0.990774i	$0.543272\pi$
$978$	16.9443	0.541818
$979$	13.4164	0.428790
$980$	0	0
$981$	26.8328	0.856706
$982$	32.7771	1.04596
$983$	−38.4853	−1.22749	−0.613745	−	0.789504i	$-0.710338\pi$
$983$	−38.4853	−1.22749	−0.613745	+	0.789504i	$0.710338\pi$
$984$	1.00000	0.0318788
$985$	−20.0000	−0.637253
$986$	18.7082	0.595791
$987$	0	0
$988$	−25.8885	−0.823624
$989$	−4.24922	−0.135117
$990$	20.0000	0.635642
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	1.76393	0.0560049
$993$	2.47214	0.0784509
$994$	0	0
$995$	14.4721	0.458798
$996$	−2.94427	−0.0932928
$997$	15.8885	0.503195	0.251598	−	0.967832i	$-0.419044\pi$
$997$	15.8885	0.503195	0.251598	+	0.967832i	$0.419044\pi$
$998$	38.0000	1.20287
$999$	−42.3607	−1.34023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4018.2.a.t.1.2	✓	2
7.6	odd	2		4018.2.a.w.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4018.2.a.t.1.2	✓	2	1.1	even	1	trivial
4018.2.a.w.1.1	yes	2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4018.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.23607 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.23607 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -2.23607 q^{10} +4.47214 q^{11} -1.00000 q^{12} -6.47214 q^{13} -2.23607 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +4.00000 q^{19} +2.23607 q^{20} -4.47214 q^{22} -0.472136 q^{23} +1.00000 q^{24} +6.47214 q^{26} +5.00000 q^{27} +6.23607 q^{29} +2.23607 q^{30} -1.76393 q^{31} -1.00000 q^{32} -4.47214 q^{33} +3.00000 q^{34} -2.00000 q^{36} -8.47214 q^{37} -4.00000 q^{38} +6.47214 q^{39} -2.23607 q^{40} +1.00000 q^{41} +9.00000 q^{43} +4.47214 q^{44} -4.47214 q^{45} +0.472136 q^{46} -4.94427 q^{47} -1.00000 q^{48} +3.00000 q^{51} -6.47214 q^{52} -6.70820 q^{53} -5.00000 q^{54} +10.0000 q^{55} -4.00000 q^{57} -6.23607 q^{58} -3.52786 q^{59} -2.23607 q^{60} -1.29180 q^{61} +1.76393 q^{62} +1.00000 q^{64} -14.4721 q^{65} +4.47214 q^{66} -11.4164 q^{67} -3.00000 q^{68} +0.472136 q^{69} +2.23607 q^{71} +2.00000 q^{72} -6.00000 q^{73} +8.47214 q^{74} +4.00000 q^{76} -6.47214 q^{78} -2.70820 q^{79} +2.23607 q^{80} +1.00000 q^{81} -1.00000 q^{82} +2.94427 q^{83} -6.70820 q^{85} -9.00000 q^{86} -6.23607 q^{87} -4.47214 q^{88} +3.00000 q^{89} +4.47214 q^{90} -0.472136 q^{92} +1.76393 q^{93} +4.94427 q^{94} +8.94427 q^{95} +1.00000 q^{96} -1.00000 q^{97} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 4 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} - 4 q^{9} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 6 q^{17} + 4 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} + 4 q^{26} + 10 q^{27} + 8 q^{29} - 8 q^{31} - 2 q^{32} + 6 q^{34} - 4 q^{36} - 8 q^{37} - 8 q^{38} + 4 q^{39} + 2 q^{41} + 18 q^{43} - 8 q^{46} + 8 q^{47} - 2 q^{48} + 6 q^{51} - 4 q^{52} - 10 q^{54} + 20 q^{55} - 8 q^{57} - 8 q^{58} - 16 q^{59} - 16 q^{61} + 8 q^{62} + 2 q^{64} - 20 q^{65} + 4 q^{67} - 6 q^{68} - 8 q^{69} + 4 q^{72} - 12 q^{73} + 8 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 2 q^{82} - 12 q^{83} - 18 q^{86} - 8 q^{87} + 6 q^{89} + 8 q^{92} + 8 q^{93} - 8 q^{94} + 2 q^{96} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 - 4 * q^9 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 6 * q^17 + 4 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 + 4 * q^26 + 10 * q^27 + 8 * q^29 - 8 * q^31 - 2 * q^32 + 6 * q^34 - 4 * q^36 - 8 * q^37 - 8 * q^38 + 4 * q^39 + 2 * q^41 + 18 * q^43 - 8 * q^46 + 8 * q^47 - 2 * q^48 + 6 * q^51 - 4 * q^52 - 10 * q^54 + 20 * q^55 - 8 * q^57 - 8 * q^58 - 16 * q^59 - 16 * q^61 + 8 * q^62 + 2 * q^64 - 20 * q^65 + 4 * q^67 - 6 * q^68 - 8 * q^69 + 4 * q^72 - 12 * q^73 + 8 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 2 * q^82 - 12 * q^83 - 18 * q^86 - 8 * q^87 + 6 * q^89 + 8 * q^92 + 8 * q^93 - 8 * q^94 + 2 * q^96 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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