LMFDB - Embedded newform 4018.2.a.bg.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 574)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.12489$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.484862 q^{3} +1.00000 q^{4} -3.12489 q^{5} -0.484862 q^{6} +1.00000 q^{8} -2.76491 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.484862 q^{3} +1.00000 q^{4} -3.12489 q^{5} -0.484862 q^{6} +1.00000 q^{8} -2.76491 q^{9} -3.12489 q^{10} +4.64002 q^{11} -0.484862 q^{12} +1.51514 q^{15} +1.00000 q^{16} +2.48486 q^{17} -2.76491 q^{18} -1.03028 q^{19} -3.12489 q^{20} +4.64002 q^{22} -4.24977 q^{23} -0.484862 q^{24} +4.76491 q^{25} +2.79518 q^{27} -1.12489 q^{29} +1.51514 q^{30} -2.48486 q^{31} +1.00000 q^{32} -2.24977 q^{33} +2.48486 q^{34} -2.76491 q^{36} +2.00000 q^{37} -1.03028 q^{38} -3.12489 q^{40} +1.00000 q^{41} -3.45459 q^{43} +4.64002 q^{44} +8.64002 q^{45} -4.24977 q^{46} -2.00000 q^{47} -0.484862 q^{48} +4.76491 q^{50} -1.20482 q^{51} -10.4049 q^{53} +2.79518 q^{54} -14.4995 q^{55} +0.499542 q^{57} -1.12489 q^{58} +8.88979 q^{59} +1.51514 q^{60} +1.84484 q^{61} -2.48486 q^{62} +1.00000 q^{64} -2.24977 q^{66} +10.5795 q^{67} +2.48486 q^{68} +2.06055 q^{69} +13.0450 q^{71} -2.76491 q^{72} +9.03028 q^{73} +2.00000 q^{74} -2.31032 q^{75} -1.03028 q^{76} +11.7649 q^{79} -3.12489 q^{80} +6.93945 q^{81} +1.00000 q^{82} +7.60975 q^{83} -7.76491 q^{85} -3.45459 q^{86} +0.545414 q^{87} +4.64002 q^{88} -2.23509 q^{89} +8.64002 q^{90} -4.24977 q^{92} +1.20482 q^{93} -2.00000 q^{94} +3.21949 q^{95} -0.484862 q^{96} +13.7649 q^{97} -12.8292 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 8 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 8 q^{9} - q^{10} + 6 q^{11} - q^{12} + 5 q^{15} + 3 q^{16} + 7 q^{17} + 8 q^{18} - 4 q^{19} - q^{20} + 6 q^{22} + 4 q^{23} - q^{24} - 2 q^{25} - 7 q^{27} + 5 q^{29} + 5 q^{30} - 7 q^{31} + 3 q^{32} + 10 q^{33} + 7 q^{34} + 8 q^{36} + 6 q^{37} - 4 q^{38} - q^{40} + 3 q^{41} - 9 q^{43} + 6 q^{44} + 18 q^{45} + 4 q^{46} - 6 q^{47} - q^{48} - 2 q^{50} - 19 q^{51} - 7 q^{53} - 7 q^{54} - 10 q^{55} - 32 q^{57} + 5 q^{58} + 2 q^{59} + 5 q^{60} + 13 q^{61} - 7 q^{62} + 3 q^{64} + 10 q^{66} + 22 q^{67} + 7 q^{68} + 8 q^{69} + 7 q^{71} + 8 q^{72} + 28 q^{73} + 6 q^{74} + 8 q^{75} - 4 q^{76} + 19 q^{79} - q^{80} + 19 q^{81} + 3 q^{82} + 14 q^{83} - 7 q^{85} - 9 q^{86} + 3 q^{87} + 6 q^{88} - 23 q^{89} + 18 q^{90} + 4 q^{92} + 19 q^{93} - 6 q^{94} - 8 q^{95} - q^{96} + 25 q^{97} - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 8 * q^9 - q^10 + 6 * q^11 - q^12 + 5 * q^15 + 3 * q^16 + 7 * q^17 + 8 * q^18 - 4 * q^19 - q^20 + 6 * q^22 + 4 * q^23 - q^24 - 2 * q^25 - 7 * q^27 + 5 * q^29 + 5 * q^30 - 7 * q^31 + 3 * q^32 + 10 * q^33 + 7 * q^34 + 8 * q^36 + 6 * q^37 - 4 * q^38 - q^40 + 3 * q^41 - 9 * q^43 + 6 * q^44 + 18 * q^45 + 4 * q^46 - 6 * q^47 - q^48 - 2 * q^50 - 19 * q^51 - 7 * q^53 - 7 * q^54 - 10 * q^55 - 32 * q^57 + 5 * q^58 + 2 * q^59 + 5 * q^60 + 13 * q^61 - 7 * q^62 + 3 * q^64 + 10 * q^66 + 22 * q^67 + 7 * q^68 + 8 * q^69 + 7 * q^71 + 8 * q^72 + 28 * q^73 + 6 * q^74 + 8 * q^75 - 4 * q^76 + 19 * q^79 - q^80 + 19 * q^81 + 3 * q^82 + 14 * q^83 - 7 * q^85 - 9 * q^86 + 3 * q^87 + 6 * q^88 - 23 * q^89 + 18 * q^90 + 4 * q^92 + 19 * q^93 - 6 * q^94 - 8 * q^95 - q^96 + 25 * q^97 - 12 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.484862	−0.279935	−0.139968	−	0.990156i	$-0.544700\pi$
$3$	−0.484862	−0.279935	−0.139968	+	0.990156i	$0.544700\pi$
$4$	1.00000	0.500000
$5$	−3.12489	−1.39749	−0.698746	−	0.715370i	$-0.746258\pi$
$5$	−3.12489	−1.39749	−0.698746	+	0.715370i	$0.746258\pi$
$6$	−0.484862	−0.197944
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.76491	−0.921636
$10$	−3.12489	−0.988176
$11$	4.64002	1.39902	0.699510	−	0.714623i	$-0.253402\pi$
$11$	4.64002	1.39902	0.699510	+	0.714623i	$0.253402\pi$
$12$	−0.484862	−0.139968
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.51514	0.391207
$16$	1.00000	0.250000
$17$	2.48486	0.602668	0.301334	−	0.953519i	$-0.402568\pi$
$17$	2.48486	0.602668	0.301334	+	0.953519i	$0.402568\pi$
$18$	−2.76491	−0.651695
$19$	−1.03028	−0.236362	−0.118181	−	0.992992i	$-0.537706\pi$
$19$	−1.03028	−0.236362	−0.118181	+	0.992992i	$0.537706\pi$
$20$	−3.12489	−0.698746
$21$	0	0
$22$	4.64002	0.989256
$23$	−4.24977	−0.886138	−0.443069	−	0.896487i	$-0.646110\pi$
$23$	−4.24977	−0.886138	−0.443069	+	0.896487i	$0.646110\pi$
$24$	−0.484862	−0.0989720
$25$	4.76491	0.952982
$26$	0	0
$27$	2.79518	0.537934
$28$	0	0
$29$	−1.12489	−0.208886	−0.104443	−	0.994531i	$-0.533306\pi$
$29$	−1.12489	−0.208886	−0.104443	+	0.994531i	$0.533306\pi$
$30$	1.51514	0.276625
$31$	−2.48486	−0.446294	−0.223147	−	0.974785i	$-0.571633\pi$
$31$	−2.48486	−0.446294	−0.223147	+	0.974785i	$0.571633\pi$
$32$	1.00000	0.176777
$33$	−2.24977	−0.391635
$34$	2.48486	0.426150
$35$	0	0
$36$	−2.76491	−0.460818
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.03028	−0.167133
$39$	0	0
$40$	−3.12489	−0.494088
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−3.45459	−0.526819	−0.263410	−	0.964684i	$-0.584847\pi$
$43$	−3.45459	−0.526819	−0.263410	+	0.964684i	$0.584847\pi$
$44$	4.64002	0.699510
$45$	8.64002	1.28798
$46$	−4.24977	−0.626595
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−0.484862	−0.0699838
$49$	0	0
$50$	4.76491	0.673860
$51$	−1.20482	−0.168708
$52$	0	0
$53$	−10.4049	−1.42923	−0.714614	−	0.699519i	$-0.753397\pi$
$53$	−10.4049	−1.42923	−0.714614	+	0.699519i	$0.753397\pi$
$54$	2.79518	0.380376
$55$	−14.4995	−1.95512
$56$	0	0
$57$	0.499542	0.0661659
$58$	−1.12489	−0.147705
$59$	8.88979	1.15735	0.578676	−	0.815557i	$-0.303569\pi$
$59$	8.88979	1.15735	0.578676	+	0.815557i	$0.303569\pi$
$60$	1.51514	0.195603
$61$	1.84484	0.236207	0.118104	−	0.993001i	$-0.462318\pi$
$61$	1.84484	0.236207	0.118104	+	0.993001i	$0.462318\pi$
$62$	−2.48486	−0.315578
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.24977	−0.276928
$67$	10.5795	1.29249	0.646244	−	0.763131i	$-0.276339\pi$
$67$	10.5795	1.29249	0.646244	+	0.763131i	$0.276339\pi$
$68$	2.48486	0.301334
$69$	2.06055	0.248061
$70$	0	0
$71$	13.0450	1.54815	0.774076	−	0.633093i	$-0.218215\pi$
$71$	13.0450	1.54815	0.774076	+	0.633093i	$0.218215\pi$
$72$	−2.76491	−0.325848
$73$	9.03028	1.05691	0.528457	−	0.848960i	$-0.322771\pi$
$73$	9.03028	1.05691	0.528457	+	0.848960i	$0.322771\pi$
$74$	2.00000	0.232495
$75$	−2.31032	−0.266773
$76$	−1.03028	−0.118181
$77$	0	0
$78$	0	0
$79$	11.7649	1.32366	0.661828	−	0.749656i	$-0.269781\pi$
$79$	11.7649	1.32366	0.661828	+	0.749656i	$0.269781\pi$
$80$	−3.12489	−0.349373
$81$	6.93945	0.771050
$82$	1.00000	0.110432
$83$	7.60975	0.835278	0.417639	−	0.908613i	$-0.362858\pi$
$83$	7.60975	0.835278	0.417639	+	0.908613i	$0.362858\pi$
$84$	0	0
$85$	−7.76491	−0.842223
$86$	−3.45459	−0.372518
$87$	0.545414	0.0584745
$88$	4.64002	0.494628
$89$	−2.23509	−0.236919	−0.118460	−	0.992959i	$-0.537796\pi$
$89$	−2.23509	−0.236919	−0.118460	+	0.992959i	$0.537796\pi$
$90$	8.64002	0.910738
$91$	0	0
$92$	−4.24977	−0.443069
$93$	1.20482	0.124933
$94$	−2.00000	−0.206284
$95$	3.21949	0.330313
$96$	−0.484862	−0.0494860
$97$	13.7649	1.39761	0.698807	−	0.715310i	$-0.253714\pi$
$97$	13.7649	1.39761	0.698807	+	0.715310i	$0.253714\pi$
$98$	0	0
$99$	−12.8292	−1.28939
$100$	4.76491	0.476491
$101$	10.7493	1.06960	0.534798	−	0.844980i	$-0.320388\pi$
$101$	10.7493	1.06960	0.534798	+	0.844980i	$0.320388\pi$
$102$	−1.20482	−0.119294
$103$	11.9541	1.17788	0.588938	−	0.808179i	$-0.299546\pi$
$103$	11.9541	1.17788	0.588938	+	0.808179i	$0.299546\pi$
$104$	0	0
$105$	0	0
$106$	−10.4049	−1.01062
$107$	16.2645	1.57234	0.786172	−	0.618008i	$-0.212060\pi$
$107$	16.2645	1.57234	0.786172	+	0.618008i	$0.212060\pi$
$108$	2.79518	0.268967
$109$	−19.4499	−1.86296	−0.931481	−	0.363791i	$-0.881482\pi$
$109$	−19.4499	−1.86296	−0.931481	+	0.363791i	$0.881482\pi$
$110$	−14.4995	−1.38248
$111$	−0.969724	−0.0920421
$112$	0	0
$113$	−9.82546	−0.924302	−0.462151	−	0.886801i	$-0.652922\pi$
$113$	−9.82546	−0.924302	−0.462151	+	0.886801i	$0.652922\pi$
$114$	0.499542	0.0467864
$115$	13.2800	1.23837
$116$	−1.12489	−0.104443
$117$	0	0
$118$	8.88979	0.818372
$119$	0	0
$120$	1.51514	0.138313
$121$	10.5298	0.957256
$122$	1.84484	0.167024
$123$	−0.484862	−0.0437185
$124$	−2.48486	−0.223147
$125$	0.734633	0.0657076
$126$	0	0
$127$	−11.2800	−1.00094	−0.500471	−	0.865753i	$-0.666840\pi$
$127$	−11.2800	−1.00094	−0.500471	+	0.865753i	$0.666840\pi$
$128$	1.00000	0.0883883
$129$	1.67500	0.147475
$130$	0	0
$131$	13.0790	1.14272	0.571359	−	0.820700i	$-0.306416\pi$
$131$	13.0790	1.14272	0.571359	+	0.820700i	$0.306416\pi$
$132$	−2.24977	−0.195817
$133$	0	0
$134$	10.5795	0.913927
$135$	−8.73463	−0.751757
$136$	2.48486	0.213075
$137$	18.8099	1.60704	0.803518	−	0.595281i	$-0.202959\pi$
$137$	18.8099	1.60704	0.803518	+	0.595281i	$0.202959\pi$
$138$	2.06055	0.175406
$139$	−8.10929	−0.687821	−0.343910	−	0.939002i	$-0.611752\pi$
$139$	−8.10929	−0.687821	−0.343910	+	0.939002i	$0.611752\pi$
$140$	0	0
$141$	0.969724	0.0816655
$142$	13.0450	1.09471
$143$	0	0
$144$	−2.76491	−0.230409
$145$	3.51514	0.291916
$146$	9.03028	0.747351
$147$	0	0
$148$	2.00000	0.164399
$149$	−1.62443	−0.133078	−0.0665391	−	0.997784i	$-0.521196\pi$
$149$	−1.62443	−0.133078	−0.0665391	+	0.997784i	$0.521196\pi$
$150$	−2.31032	−0.188637
$151$	7.45459	0.606646	0.303323	−	0.952888i	$-0.401904\pi$
$151$	7.45459	0.606646	0.303323	+	0.952888i	$0.401904\pi$
$152$	−1.03028	−0.0835664
$153$	−6.87042	−0.555440
$154$	0	0
$155$	7.76491	0.623692
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	11.7649	0.935966
$159$	5.04496	0.400091
$160$	−3.12489	−0.247044
$161$	0	0
$162$	6.93945	0.545215
$163$	15.2195	1.19208	0.596041	−	0.802954i	$-0.296739\pi$
$163$	15.2195	1.19208	0.596041	+	0.802954i	$0.296739\pi$
$164$	1.00000	0.0780869
$165$	7.03028	0.547306
$166$	7.60975	0.590631
$167$	6.56009	0.507635	0.253818	−	0.967252i	$-0.418314\pi$
$167$	6.56009	0.507635	0.253818	+	0.967252i	$0.418314\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−7.76491	−0.595541
$171$	2.84862	0.217839
$172$	−3.45459	−0.263410
$173$	−18.9650	−1.44188	−0.720942	−	0.692995i	$-0.756291\pi$
$173$	−18.9650	−1.44188	−0.720942	+	0.692995i	$0.756291\pi$
$174$	0.545414	0.0413477
$175$	0	0
$176$	4.64002	0.349755
$177$	−4.31032	−0.323984
$178$	−2.23509	−0.167527
$179$	4.32970	0.323617	0.161809	−	0.986822i	$-0.448267\pi$
$179$	4.32970	0.323617	0.161809	+	0.986822i	$0.448267\pi$
$180$	8.64002	0.643989
$181$	−6.37088	−0.473543	−0.236772	−	0.971565i	$-0.576089\pi$
$181$	−6.37088	−0.473543	−0.236772	+	0.971565i	$0.576089\pi$
$182$	0	0
$183$	−0.894492	−0.0661228
$184$	−4.24977	−0.313297
$185$	−6.24977	−0.459492
$186$	1.20482	0.0883413
$187$	11.5298	0.843144
$188$	−2.00000	−0.145865
$189$	0	0
$190$	3.21949	0.233567
$191$	14.4849	1.04809	0.524044	−	0.851691i	$-0.324423\pi$
$191$	14.4849	1.04809	0.524044	+	0.851691i	$0.324423\pi$
$192$	−0.484862	−0.0349919
$193$	13.6509	0.982615	0.491307	−	0.870986i	$-0.336519\pi$
$193$	13.6509	0.982615	0.491307	+	0.870986i	$0.336519\pi$
$194$	13.7649	0.988263
$195$	0	0
$196$	0	0
$197$	8.24977	0.587772	0.293886	−	0.955841i	$-0.405051\pi$
$197$	8.24977	0.587772	0.293886	+	0.955841i	$0.405051\pi$
$198$	−12.8292	−0.911735
$199$	−14.0294	−0.994515	−0.497257	−	0.867603i	$-0.665660\pi$
$199$	−14.0294	−0.994515	−0.497257	+	0.867603i	$0.665660\pi$
$200$	4.76491	0.336930
$201$	−5.12958	−0.361813
$202$	10.7493	0.756319
$203$	0	0
$204$	−1.20482	−0.0843539
$205$	−3.12489	−0.218251
$206$	11.9541	0.832884
$207$	11.7502	0.816697
$208$	0	0
$209$	−4.78051	−0.330674
$210$	0	0
$211$	−10.1093	−0.695952	−0.347976	−	0.937503i	$-0.613131\pi$
$211$	−10.1093	−0.695952	−0.347976	+	0.937503i	$0.613131\pi$
$212$	−10.4049	−0.714614
$213$	−6.32500	−0.433382
$214$	16.2645	1.11181
$215$	10.7952	0.736226
$216$	2.79518	0.190188
$217$	0	0
$218$	−19.4499	−1.31731
$219$	−4.37844	−0.295867
$220$	−14.4995	−0.977559
$221$	0	0
$222$	−0.969724	−0.0650836
$223$	17.2342	1.15409	0.577043	−	0.816714i	$-0.304207\pi$
$223$	17.2342	1.15409	0.577043	+	0.816714i	$0.304207\pi$
$224$	0	0
$225$	−13.1745	−0.878303
$226$	−9.82546	−0.653580
$227$	0.795185	0.0527783	0.0263891	−	0.999652i	$-0.491599\pi$
$227$	0.795185	0.0527783	0.0263891	+	0.999652i	$0.491599\pi$
$228$	0.499542	0.0330830
$229$	−11.2195	−0.741405	−0.370703	−	0.928752i	$-0.620883\pi$
$229$	−11.2195	−0.741405	−0.370703	+	0.928752i	$0.620883\pi$
$230$	13.2800	0.875660
$231$	0	0
$232$	−1.12489	−0.0738523
$233$	1.87890	0.123091	0.0615453	−	0.998104i	$-0.480397\pi$
$233$	1.87890	0.123091	0.0615453	+	0.998104i	$0.480397\pi$
$234$	0	0
$235$	6.24977	0.407690
$236$	8.88979	0.578676
$237$	−5.70436	−0.370538
$238$	0	0
$239$	15.7190	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$239$	15.7190	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$240$	1.51514	0.0978017
$241$	4.06055	0.261563	0.130782	−	0.991411i	$-0.458251\pi$
$241$	4.06055	0.261563	0.130782	+	0.991411i	$0.458251\pi$
$242$	10.5298	0.676882
$243$	−11.7502	−0.753778
$244$	1.84484	0.118104
$245$	0	0
$246$	−0.484862	−0.0309137
$247$	0	0
$248$	−2.48486	−0.157789
$249$	−3.68968	−0.233824
$250$	0.734633	0.0464623
$251$	−22.4802	−1.41894	−0.709468	−	0.704738i	$-0.751065\pi$
$251$	−22.4802	−1.41894	−0.709468	+	0.704738i	$0.751065\pi$
$252$	0	0
$253$	−19.7190	−1.23973
$254$	−11.2800	−0.707773
$255$	3.76491	0.235768
$256$	1.00000	0.0625000
$257$	−24.8245	−1.54851	−0.774256	−	0.632872i	$-0.781876\pi$
$257$	−24.8245	−1.54851	−0.774256	+	0.632872i	$0.781876\pi$
$258$	1.67500	0.104281
$259$	0	0
$260$	0	0
$261$	3.11021	0.192517
$262$	13.0790	0.808024
$263$	5.43991	0.335439	0.167719	−	0.985835i	$-0.446360\pi$
$263$	5.43991	0.335439	0.167719	+	0.985835i	$0.446360\pi$
$264$	−2.24977	−0.138464
$265$	32.5142	1.99733
$266$	0	0
$267$	1.08371	0.0663220
$268$	10.5795	0.646244
$269$	−1.79897	−0.109685	−0.0548424	−	0.998495i	$-0.517466\pi$
$269$	−1.79897	−0.109685	−0.0548424	+	0.998495i	$0.517466\pi$
$270$	−8.73463	−0.531573
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	2.48486	0.150667
$273$	0	0
$274$	18.8099	1.13635
$275$	22.1093	1.33324
$276$	2.06055	0.124031
$277$	15.1589	0.910813	0.455406	−	0.890284i	$-0.349494\pi$
$277$	15.1589	0.910813	0.455406	+	0.890284i	$0.349494\pi$
$278$	−8.10929	−0.486363
$279$	6.87042	0.411321
$280$	0	0
$281$	−2.49954	−0.149110	−0.0745551	−	0.997217i	$-0.523754\pi$
$281$	−2.49954	−0.149110	−0.0745551	+	0.997217i	$0.523754\pi$
$282$	0.969724	0.0577462
$283$	−10.8292	−0.643732	−0.321866	−	0.946785i	$-0.604310\pi$
$283$	−10.8292	−0.643732	−0.321866	+	0.946785i	$0.604310\pi$
$284$	13.0450	0.774076
$285$	−1.56101	−0.0924663
$286$	0	0
$287$	0	0
$288$	−2.76491	−0.162924
$289$	−10.8255	−0.636792
$290$	3.51514	0.206416
$291$	−6.67408	−0.391242
$292$	9.03028	0.528457
$293$	−11.6509	−0.680654	−0.340327	−	0.940307i	$-0.610538\pi$
$293$	−11.6509	−0.680654	−0.340327	+	0.940307i	$0.610538\pi$
$294$	0	0
$295$	−27.7796	−1.61739
$296$	2.00000	0.116248
$297$	12.9697	0.752580
$298$	−1.62443	−0.0941005
$299$	0	0
$300$	−2.31032	−0.133387
$301$	0	0
$302$	7.45459	0.428963
$303$	−5.21193	−0.299418
$304$	−1.03028	−0.0590904
$305$	−5.76491	−0.330098
$306$	−6.87042	−0.392756
$307$	21.6997	1.23846	0.619232	−	0.785208i	$-0.287444\pi$
$307$	21.6997	1.23846	0.619232	+	0.785208i	$0.287444\pi$
$308$	0	0
$309$	−5.79610	−0.329729
$310$	7.76491	0.441017
$311$	−24.9385	−1.41413	−0.707067	−	0.707146i	$-0.749982\pi$
$311$	−24.9385	−1.41413	−0.707067	+	0.707146i	$0.749982\pi$
$312$	0	0
$313$	−33.3094	−1.88276	−0.941379	−	0.337349i	$-0.890470\pi$
$313$	−33.3094	−1.88276	−0.941379	+	0.337349i	$0.890470\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	11.7649	0.661828
$317$	−11.4499	−0.643090	−0.321545	−	0.946894i	$-0.604202\pi$
$317$	−11.4499	−0.643090	−0.321545	+	0.946894i	$0.604202\pi$
$318$	5.04496	0.282907
$319$	−5.21949	−0.292236
$320$	−3.12489	−0.174686
$321$	−7.88601	−0.440154
$322$	0	0
$323$	−2.56009	−0.142447
$324$	6.93945	0.385525
$325$	0	0
$326$	15.2195	0.842930
$327$	9.43051	0.521508
$328$	1.00000	0.0552158
$329$	0	0
$330$	7.03028	0.387004
$331$	6.10929	0.335797	0.167898	−	0.985804i	$-0.446302\pi$
$331$	6.10929	0.335797	0.167898	+	0.985804i	$0.446302\pi$
$332$	7.60975	0.417639
$333$	−5.52982	−0.303032
$334$	6.56009	0.358952
$335$	−33.0596	−1.80624
$336$	0	0
$337$	−0.916289	−0.0499135	−0.0249567	−	0.999689i	$-0.507945\pi$
$337$	−0.916289	−0.0499135	−0.0249567	+	0.999689i	$0.507945\pi$
$338$	−13.0000	−0.707107
$339$	4.76399	0.258745
$340$	−7.76491	−0.421111
$341$	−11.5298	−0.624375
$342$	2.84862	0.154036
$343$	0	0
$344$	−3.45459	−0.186259
$345$	−6.43899	−0.346664
$346$	−18.9650	−1.01957
$347$	−23.3893	−1.25561	−0.627803	−	0.778373i	$-0.716046\pi$
$347$	−23.3893	−1.25561	−0.627803	+	0.778373i	$0.716046\pi$
$348$	0.545414	0.0292373
$349$	18.0799	0.967796	0.483898	−	0.875124i	$-0.339220\pi$
$349$	18.0799	0.967796	0.483898	+	0.875124i	$0.339220\pi$
$350$	0	0
$351$	0	0
$352$	4.64002	0.247314
$353$	21.2489	1.13096	0.565481	−	0.824761i	$-0.308690\pi$
$353$	21.2489	1.13096	0.565481	+	0.824761i	$0.308690\pi$
$354$	−4.31032	−0.229091
$355$	−40.7640	−2.16353
$356$	−2.23509	−0.118460
$357$	0	0
$358$	4.32970	0.228832
$359$	16.9991	0.897177	0.448589	−	0.893738i	$-0.351927\pi$
$359$	16.9991	0.897177	0.448589	+	0.893738i	$0.351927\pi$
$360$	8.64002	0.455369
$361$	−17.9385	−0.944133
$362$	−6.37088	−0.334846
$363$	−5.10551	−0.267970
$364$	0	0
$365$	−28.2186	−1.47703
$366$	−0.894492	−0.0467558
$367$	−14.7952	−0.772302	−0.386151	−	0.922436i	$-0.626196\pi$
$367$	−14.7952	−0.772302	−0.386151	+	0.922436i	$0.626196\pi$
$368$	−4.24977	−0.221535
$369$	−2.76491	−0.143935
$370$	−6.24977	−0.324910
$371$	0	0
$372$	1.20482	0.0624667
$373$	29.5592	1.53052	0.765258	−	0.643724i	$-0.222611\pi$
$373$	29.5592	1.53052	0.765258	+	0.643724i	$0.222611\pi$
$374$	11.5298	0.596193
$375$	−0.356195	−0.0183939
$376$	−2.00000	−0.103142
$377$	0	0
$378$	0	0
$379$	3.29473	0.169239	0.0846194	−	0.996413i	$-0.473033\pi$
$379$	3.29473	0.169239	0.0846194	+	0.996413i	$0.473033\pi$
$380$	3.21949	0.165157
$381$	5.46927	0.280199
$382$	14.4849	0.741110
$383$	−13.4693	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$383$	−13.4693	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$384$	−0.484862	−0.0247430
$385$	0	0
$386$	13.6509	0.694814
$387$	9.55162	0.485536
$388$	13.7649	0.698807
$389$	12.6812	0.642962	0.321481	−	0.946916i	$-0.395819\pi$
$389$	12.6812	0.642962	0.321481	+	0.946916i	$0.395819\pi$
$390$	0	0
$391$	−10.5601	−0.534047
$392$	0	0
$393$	−6.34152	−0.319887
$394$	8.24977	0.415617
$395$	−36.7640	−1.84980
$396$	−12.8292	−0.644694
$397$	−19.7190	−0.989670	−0.494835	−	0.868987i	$-0.664772\pi$
$397$	−19.7190	−0.989670	−0.494835	+	0.868987i	$0.664772\pi$
$398$	−14.0294	−0.703228
$399$	0	0
$400$	4.76491	0.238245
$401$	11.6438	0.581464	0.290732	−	0.956805i	$-0.406101\pi$
$401$	11.6438	0.581464	0.290732	+	0.956805i	$0.406101\pi$
$402$	−5.12958	−0.255840
$403$	0	0
$404$	10.7493	0.534798
$405$	−21.6850	−1.07754
$406$	0	0
$407$	9.28005	0.459995
$408$	−1.20482	−0.0596472
$409$	27.1202	1.34101	0.670503	−	0.741906i	$-0.266078\pi$
$409$	27.1202	1.34101	0.670503	+	0.741906i	$0.266078\pi$
$410$	−3.12489	−0.154327
$411$	−9.12019	−0.449866
$412$	11.9541	0.588938
$413$	0	0
$414$	11.7502	0.577492
$415$	−23.7796	−1.16729
$416$	0	0
$417$	3.93189	0.192545
$418$	−4.78051	−0.233822
$419$	25.6997	1.25551	0.627755	−	0.778411i	$-0.283974\pi$
$419$	25.6997	1.25551	0.627755	+	0.778411i	$0.283974\pi$
$420$	0	0
$421$	14.2451	0.694262	0.347131	−	0.937817i	$-0.387156\pi$
$421$	14.2451	0.694262	0.347131	+	0.937817i	$0.387156\pi$
$422$	−10.1093	−0.492112
$423$	5.52982	0.268869
$424$	−10.4049	−0.505308
$425$	11.8401	0.574331
$426$	−6.32500	−0.306447
$427$	0	0
$428$	16.2645	0.786172
$429$	0	0
$430$	10.7952	0.520590
$431$	−5.68968	−0.274062	−0.137031	−	0.990567i	$-0.543756\pi$
$431$	−5.68968	−0.274062	−0.137031	+	0.990567i	$0.543756\pi$
$432$	2.79518	0.134483
$433$	37.4087	1.79775	0.898874	−	0.438207i	$-0.144386\pi$
$433$	37.4087	1.79775	0.898874	+	0.438207i	$0.144386\pi$
$434$	0	0
$435$	−1.70436	−0.0817176
$436$	−19.4499	−0.931481
$437$	4.37844	0.209449
$438$	−4.37844	−0.209210
$439$	14.9991	0.715867	0.357934	−	0.933747i	$-0.383481\pi$
$439$	14.9991	0.715867	0.357934	+	0.933747i	$0.383481\pi$
$440$	−14.4995	−0.691239
$441$	0	0
$442$	0	0
$443$	−12.5748	−0.597446	−0.298723	−	0.954340i	$-0.596561\pi$
$443$	−12.5748	−0.597446	−0.298723	+	0.954340i	$0.596561\pi$
$444$	−0.969724	−0.0460211
$445$	6.98440	0.331092
$446$	17.2342	0.816062
$447$	0.787623	0.0372533
$448$	0	0
$449$	26.8557	1.26740	0.633700	−	0.773579i	$-0.281535\pi$
$449$	26.8557	1.26740	0.633700	+	0.773579i	$0.281535\pi$
$450$	−13.1745	−0.621054
$451$	4.64002	0.218490
$452$	−9.82546	−0.462151
$453$	−3.61445	−0.169821
$454$	0.795185	0.0373199
$455$	0	0
$456$	0.499542	0.0233932
$457$	−30.4995	−1.42671	−0.713354	−	0.700804i	$-0.752825\pi$
$457$	−30.4995	−1.42671	−0.713354	+	0.700804i	$0.752825\pi$
$458$	−11.2195	−0.524253
$459$	6.94565	0.324195
$460$	13.2800	0.619185
$461$	36.4343	1.69691	0.848457	−	0.529264i	$-0.177532\pi$
$461$	36.4343	1.69691	0.848457	+	0.529264i	$0.177532\pi$
$462$	0	0
$463$	−26.5601	−1.23435	−0.617176	−	0.786825i	$-0.711723\pi$
$463$	−26.5601	−1.23435	−0.617176	+	0.786825i	$0.711723\pi$
$464$	−1.12489	−0.0522215
$465$	−3.76491	−0.174593
$466$	1.87890	0.0870382
$467$	21.0109	0.972268	0.486134	−	0.873884i	$-0.338407\pi$
$467$	21.0109	0.972268	0.486134	+	0.873884i	$0.338407\pi$
$468$	0	0
$469$	0	0
$470$	6.24977	0.288280
$471$	−3.87890	−0.178730
$472$	8.88979	0.409186
$473$	−16.0294	−0.737031
$474$	−5.70436	−0.262010
$475$	−4.90917	−0.225248
$476$	0	0
$477$	28.7687	1.31723
$478$	15.7190	0.718972
$479$	8.68120	0.396654	0.198327	−	0.980136i	$-0.436449\pi$
$479$	8.68120	0.396654	0.198327	+	0.980136i	$0.436449\pi$
$480$	1.51514	0.0691563
$481$	0	0
$482$	4.06055	0.184953
$483$	0	0
$484$	10.5298	0.478628
$485$	−43.0138	−1.95315
$486$	−11.7502	−0.533001
$487$	8.99908	0.407787	0.203894	−	0.978993i	$-0.434640\pi$
$487$	8.99908	0.407787	0.203894	+	0.978993i	$0.434640\pi$
$488$	1.84484	0.0835119
$489$	−7.37935	−0.333706
$490$	0	0
$491$	38.0147	1.71558	0.857789	−	0.514002i	$-0.171837\pi$
$491$	38.0147	1.71558	0.857789	+	0.514002i	$0.171837\pi$
$492$	−0.484862	−0.0218593
$493$	−2.79518	−0.125889
$494$	0	0
$495$	40.0899	1.80191
$496$	−2.48486	−0.111574
$497$	0	0
$498$	−3.68968	−0.165338
$499$	−6.70058	−0.299959	−0.149979	−	0.988689i	$-0.547921\pi$
$499$	−6.70058	−0.299959	−0.149979	+	0.988689i	$0.547921\pi$
$500$	0.734633	0.0328538
$501$	−3.18074	−0.142105
$502$	−22.4802	−1.00334
$503$	−31.0596	−1.38488	−0.692440	−	0.721475i	$-0.743464\pi$
$503$	−31.0596	−1.38488	−0.692440	+	0.721475i	$0.743464\pi$
$504$	0	0
$505$	−33.5904	−1.49475
$506$	−19.7190	−0.876618
$507$	6.30321	0.279935
$508$	−11.2800	−0.500471
$509$	−32.7181	−1.45021	−0.725103	−	0.688641i	$-0.758208\pi$
$509$	−32.7181	−1.45021	−0.725103	+	0.688641i	$0.758208\pi$
$510$	3.76491	0.166713
$511$	0	0
$512$	1.00000	0.0441942
$513$	−2.87981	−0.127147
$514$	−24.8245	−1.09496
$515$	−37.3553	−1.64607
$516$	1.67500	0.0737376
$517$	−9.28005	−0.408136
$518$	0	0
$519$	9.19542	0.403634
$520$	0	0
$521$	−30.2498	−1.32527	−0.662633	−	0.748944i	$-0.730561\pi$
$521$	−30.2498	−1.32527	−0.662633	+	0.748944i	$0.730561\pi$
$522$	3.11021	0.136130
$523$	−14.3591	−0.627878	−0.313939	−	0.949443i	$-0.601649\pi$
$523$	−14.3591	−0.627878	−0.313939	+	0.949443i	$0.601649\pi$
$524$	13.0790	0.571359
$525$	0	0
$526$	5.43991	0.237191
$527$	−6.17454	−0.268967
$528$	−2.24977	−0.0979087
$529$	−4.93945	−0.214759
$530$	32.5142	1.41233
$531$	−24.5795	−1.06666
$532$	0	0
$533$	0	0
$534$	1.08371	0.0468967
$535$	−50.8245	−2.19734
$536$	10.5795	0.456964
$537$	−2.09931	−0.0905918
$538$	−1.79897	−0.0775589
$539$	0	0
$540$	−8.73463	−0.375879
$541$	−5.50046	−0.236483	−0.118242	−	0.992985i	$-0.537726\pi$
$541$	−5.50046	−0.236483	−0.118242	+	0.992985i	$0.537726\pi$
$542$	−20.0000	−0.859074
$543$	3.08899	0.132561
$544$	2.48486	0.106538
$545$	60.7787	2.60347
$546$	0	0
$547$	−31.5786	−1.35020	−0.675101	−	0.737726i	$-0.735900\pi$
$547$	−31.5786	−1.35020	−0.675101	+	0.737726i	$0.735900\pi$
$548$	18.8099	0.803518
$549$	−5.10081	−0.217697
$550$	22.1093	0.942743
$551$	1.15894	0.0493726
$552$	2.06055	0.0877029
$553$	0	0
$554$	15.1589	0.644042
$555$	3.03028	0.128628
$556$	−8.10929	−0.343910
$557$	14.7834	0.626391	0.313196	−	0.949689i	$-0.398600\pi$
$557$	14.7834	0.626391	0.313196	+	0.949689i	$0.398600\pi$
$558$	6.87042	0.290848
$559$	0	0
$560$	0	0
$561$	−5.59037	−0.236026
$562$	−2.49954	−0.105437
$563$	45.5298	1.91885	0.959427	−	0.281959i	$-0.0909841\pi$
$563$	45.5298	1.91885	0.959427	+	0.281959i	$0.0909841\pi$
$564$	0.969724	0.0408327
$565$	30.7034	1.29170
$566$	−10.8292	−0.455187
$567$	0	0
$568$	13.0450	0.547354
$569$	−8.23509	−0.345233	−0.172616	−	0.984989i	$-0.555222\pi$
$569$	−8.23509	−0.345233	−0.172616	+	0.984989i	$0.555222\pi$
$570$	−1.56101	−0.0653835
$571$	14.1093	0.590455	0.295228	−	0.955427i	$-0.404604\pi$
$571$	14.1093	0.590455	0.295228	+	0.955427i	$0.404604\pi$
$572$	0	0
$573$	−7.02316	−0.293397
$574$	0	0
$575$	−20.2498	−0.844474
$576$	−2.76491	−0.115205
$577$	13.7502	0.572430	0.286215	−	0.958165i	$-0.407603\pi$
$577$	13.7502	0.572430	0.286215	+	0.958165i	$0.407603\pi$
$578$	−10.8255	−0.450280
$579$	−6.61881	−0.275068
$580$	3.51514	0.145958
$581$	0	0
$582$	−6.67408	−0.276650
$583$	−48.2791	−1.99952
$584$	9.03028	0.373675
$585$	0	0
$586$	−11.6509	−0.481295
$587$	−32.5142	−1.34201	−0.671003	−	0.741455i	$-0.734136\pi$
$587$	−32.5142	−1.34201	−0.671003	+	0.741455i	$0.734136\pi$
$588$	0	0
$589$	2.56009	0.105487
$590$	−27.7796	−1.14367
$591$	−4.00000	−0.164538
$592$	2.00000	0.0821995
$593$	28.1745	1.15699	0.578495	−	0.815686i	$-0.303640\pi$
$593$	28.1745	1.15699	0.578495	+	0.815686i	$0.303640\pi$
$594$	12.9697	0.532154
$595$	0	0
$596$	−1.62443	−0.0665391
$597$	6.80230	0.278400
$598$	0	0
$599$	8.99908	0.367693	0.183846	−	0.982955i	$-0.441145\pi$
$599$	8.99908	0.367693	0.183846	+	0.982955i	$0.441145\pi$
$600$	−2.31032	−0.0943185
$601$	−4.45367	−0.181669	−0.0908345	−	0.995866i	$-0.528953\pi$
$601$	−4.45367	−0.181669	−0.0908345	+	0.995866i	$0.528953\pi$
$602$	0	0
$603$	−29.2513	−1.19120
$604$	7.45459	0.303323
$605$	−32.9045	−1.33776
$606$	−5.21193	−0.211720
$607$	−45.8842	−1.86238	−0.931191	−	0.364532i	$-0.881229\pi$
$607$	−45.8842	−1.86238	−0.931191	+	0.364532i	$0.881229\pi$
$608$	−1.03028	−0.0417832
$609$	0	0
$610$	−5.76491	−0.233414
$611$	0	0
$612$	−6.87042	−0.277720
$613$	−25.1277	−1.01490	−0.507450	−	0.861681i	$-0.669412\pi$
$613$	−25.1277	−1.01490	−0.507450	+	0.861681i	$0.669412\pi$
$614$	21.6997	0.875727
$615$	1.51514	0.0610963
$616$	0	0
$617$	42.5895	1.71459	0.857293	−	0.514828i	$-0.172144\pi$
$617$	42.5895	1.71459	0.857293	+	0.514828i	$0.172144\pi$
$618$	−5.79610	−0.233153
$619$	9.98062	0.401155	0.200578	−	0.979678i	$-0.435718\pi$
$619$	9.98062	0.401155	0.200578	+	0.979678i	$0.435718\pi$
$620$	7.76491	0.311846
$621$	−11.8789	−0.476684
$622$	−24.9385	−0.999944
$623$	0	0
$624$	0	0
$625$	−26.1202	−1.04481
$626$	−33.3094	−1.33131
$627$	2.31789	0.0925674
$628$	8.00000	0.319235
$629$	4.96972	0.198156
$630$	0	0
$631$	−31.6585	−1.26030	−0.630152	−	0.776472i	$-0.717008\pi$
$631$	−31.6585	−1.26030	−0.630152	+	0.776472i	$0.717008\pi$
$632$	11.7649	0.467983
$633$	4.90161	0.194821
$634$	−11.4499	−0.454733
$635$	35.2489	1.39881
$636$	5.04496	0.200046
$637$	0	0
$638$	−5.21949	−0.206642
$639$	−36.0681	−1.42683
$640$	−3.12489	−0.123522
$641$	−2.37844	−0.0939426	−0.0469713	−	0.998896i	$-0.514957\pi$
$641$	−2.37844	−0.0939426	−0.0469713	+	0.998896i	$0.514957\pi$
$642$	−7.88601	−0.311236
$643$	27.8548	1.09849	0.549243	−	0.835662i	$-0.314916\pi$
$643$	27.8548	1.09849	0.549243	+	0.835662i	$0.314916\pi$
$644$	0	0
$645$	−5.23417	−0.206095
$646$	−2.56009	−0.100726
$647$	39.5133	1.55343	0.776714	−	0.629853i	$-0.216885\pi$
$647$	39.5133	1.55343	0.776714	+	0.629853i	$0.216885\pi$
$648$	6.93945	0.272607
$649$	41.2489	1.61916
$650$	0	0
$651$	0	0
$652$	15.2195	0.596041
$653$	23.8742	0.934270	0.467135	−	0.884186i	$-0.345286\pi$
$653$	23.8742	0.934270	0.467135	+	0.884186i	$0.345286\pi$
$654$	9.43051	0.368762
$655$	−40.8704	−1.59694
$656$	1.00000	0.0390434
$657$	−24.9679	−0.974090
$658$	0	0
$659$	−0.708138	−0.0275851	−0.0137926	−	0.999905i	$-0.504390\pi$
$659$	−0.708138	−0.0275851	−0.0137926	+	0.999905i	$0.504390\pi$
$660$	7.03028	0.273653
$661$	−1.58039	−0.0614700	−0.0307350	−	0.999528i	$-0.509785\pi$
$661$	−1.58039	−0.0614700	−0.0307350	+	0.999528i	$0.509785\pi$
$662$	6.10929	0.237444
$663$	0	0
$664$	7.60975	0.295315
$665$	0	0
$666$	−5.52982	−0.214276
$667$	4.78051	0.185102
$668$	6.56009	0.253818
$669$	−8.35620	−0.323069
$670$	−33.0596	−1.27721
$671$	8.56009	0.330459
$672$	0	0
$673$	1.46170	0.0563445	0.0281723	−	0.999603i	$-0.491031\pi$
$673$	1.46170	0.0563445	0.0281723	+	0.999603i	$0.491031\pi$
$674$	−0.916289	−0.0352941
$675$	13.3188	0.512641
$676$	−13.0000	−0.500000
$677$	40.8586	1.57032	0.785162	−	0.619291i	$-0.212580\pi$
$677$	40.8586	1.57032	0.785162	+	0.619291i	$0.212580\pi$
$678$	4.76399	0.182960
$679$	0	0
$680$	−7.76491	−0.297771
$681$	−0.385555	−0.0147745
$682$	−11.5298	−0.441500
$683$	−5.23131	−0.200171	−0.100085	−	0.994979i	$-0.531912\pi$
$683$	−5.23131	−0.200171	−0.100085	+	0.994979i	$0.531912\pi$
$684$	2.84862	0.108920
$685$	−58.7787	−2.24582
$686$	0	0
$687$	5.43991	0.207545
$688$	−3.45459	−0.131705
$689$	0	0
$690$	−6.43899	−0.245128
$691$	32.5142	1.23690	0.618450	−	0.785824i	$-0.287761\pi$
$691$	32.5142	1.23690	0.618450	+	0.785824i	$0.287761\pi$
$692$	−18.9650	−0.720942
$693$	0	0
$694$	−23.3893	−0.887847
$695$	25.3406	0.961224
$696$	0.545414	0.0206739
$697$	2.48486	0.0941209
$698$	18.0799	0.684335
$699$	−0.911005	−0.0344574
$700$	0	0
$701$	33.4087	1.26183	0.630915	−	0.775852i	$-0.282680\pi$
$701$	33.4087	1.26183	0.630915	+	0.775852i	$0.282680\pi$
$702$	0	0
$703$	−2.06055	−0.0777152
$704$	4.64002	0.174877
$705$	−3.03028	−0.114127
$706$	21.2489	0.799711
$707$	0	0
$708$	−4.31032	−0.161992
$709$	49.2753	1.85057	0.925287	−	0.379267i	$-0.123824\pi$
$709$	49.2753	1.85057	0.925287	+	0.379267i	$0.123824\pi$
$710$	−40.7640	−1.52985
$711$	−32.5289	−1.21993
$712$	−2.23509	−0.0837636
$713$	10.5601	0.395479
$714$	0	0
$715$	0	0
$716$	4.32970	0.161809
$717$	−7.62156	−0.284632
$718$	16.9991	0.634400
$719$	34.8174	1.29847	0.649235	−	0.760587i	$-0.275089\pi$
$719$	34.8174	1.29847	0.649235	+	0.760587i	$0.275089\pi$
$720$	8.64002	0.321995
$721$	0	0
$722$	−17.9385	−0.667603
$723$	−1.96881	−0.0732207
$724$	−6.37088	−0.236772
$725$	−5.35998	−0.199065
$726$	−5.10551	−0.189483
$727$	−7.87890	−0.292212	−0.146106	−	0.989269i	$-0.546674\pi$
$727$	−7.87890	−0.292212	−0.146106	+	0.989269i	$0.546674\pi$
$728$	0	0
$729$	−15.1211	−0.560041
$730$	−28.2186	−1.04442
$731$	−8.58417	−0.317497
$732$	−0.894492	−0.0330614
$733$	12.7446	0.470733	0.235367	−	0.971907i	$-0.424371\pi$
$733$	12.7446	0.470733	0.235367	+	0.971907i	$0.424371\pi$
$734$	−14.7952	−0.546100
$735$	0	0
$736$	−4.24977	−0.156649
$737$	49.0890	1.80822
$738$	−2.76491	−0.101778
$739$	−53.4528	−1.96629	−0.983146	−	0.182824i	$-0.941476\pi$
$739$	−53.4528	−1.96629	−0.983146	+	0.182824i	$0.941476\pi$
$740$	−6.24977	−0.229746
$741$	0	0
$742$	0	0
$743$	−1.93945	−0.0711514	−0.0355757	−	0.999367i	$-0.511326\pi$
$743$	−1.93945	−0.0711514	−0.0355757	+	0.999367i	$0.511326\pi$
$744$	1.20482	0.0441707
$745$	5.07615	0.185976
$746$	29.5592	1.08224
$747$	−21.0403	−0.769823
$748$	11.5298	0.421572
$749$	0	0
$750$	−0.356195	−0.0130064
$751$	−26.4002	−0.963358	−0.481679	−	0.876348i	$-0.659973\pi$
$751$	−26.4002	−0.963358	−0.481679	+	0.876348i	$0.659973\pi$
$752$	−2.00000	−0.0729325
$753$	10.8998	0.397210
$754$	0	0
$755$	−23.2947	−0.847782
$756$	0	0
$757$	24.8733	0.904035	0.452017	−	0.892009i	$-0.350704\pi$
$757$	24.8733	0.904035	0.452017	+	0.892009i	$0.350704\pi$
$758$	3.29473	0.119670
$759$	9.56101	0.347043
$760$	3.21949	0.116783
$761$	−31.7796	−1.15201	−0.576005	−	0.817446i	$-0.695389\pi$
$761$	−31.7796	−1.15201	−0.576005	+	0.817446i	$0.695389\pi$
$762$	5.46927	0.198131
$763$	0	0
$764$	14.4849	0.524044
$765$	21.4693	0.776223
$766$	−13.4693	−0.486664
$767$	0	0
$768$	−0.484862	−0.0174959
$769$	−37.9612	−1.36892	−0.684459	−	0.729052i	$-0.739961\pi$
$769$	−37.9612	−1.36892	−0.684459	+	0.729052i	$0.739961\pi$
$770$	0	0
$771$	12.0365	0.433483
$772$	13.6509	0.491307
$773$	39.9083	1.43540	0.717700	−	0.696352i	$-0.245195\pi$
$773$	39.9083	1.43540	0.717700	+	0.696352i	$0.245195\pi$
$774$	9.55162	0.343326
$775$	−11.8401	−0.425310
$776$	13.7649	0.494131
$777$	0	0
$778$	12.6812	0.454643
$779$	−1.03028	−0.0369135
$780$	0	0
$781$	60.5289	2.16589
$782$	−10.5601	−0.377628
$783$	−3.14426	−0.112367
$784$	0	0
$785$	−24.9991	−0.892256
$786$	−6.34152	−0.226194
$787$	6.79988	0.242390	0.121195	−	0.992629i	$-0.461327\pi$
$787$	6.79988	0.242390	0.121195	+	0.992629i	$0.461327\pi$
$788$	8.24977	0.293886
$789$	−2.63760	−0.0939012
$790$	−36.7640	−1.30800
$791$	0	0
$792$	−12.8292	−0.455867
$793$	0	0
$794$	−19.7190	−0.699802
$795$	−15.7649	−0.559124
$796$	−14.0294	−0.497257
$797$	10.3444	0.366417	0.183208	−	0.983074i	$-0.441352\pi$
$797$	10.3444	0.366417	0.183208	+	0.983074i	$0.441352\pi$
$798$	0	0
$799$	−4.96972	−0.175816
$800$	4.76491	0.168465
$801$	6.17982	0.218353
$802$	11.6438	0.411157
$803$	41.9007	1.47864
$804$	−5.12958	−0.180906
$805$	0	0
$806$	0	0
$807$	0.872250	0.0307047
$808$	10.7493	0.378159
$809$	1.37935	0.0484955	0.0242478	−	0.999706i	$-0.492281\pi$
$809$	1.37935	0.0484955	0.0242478	+	0.999706i	$0.492281\pi$
$810$	−21.6850	−0.761933
$811$	6.01938	0.211369	0.105684	−	0.994400i	$-0.466297\pi$
$811$	6.01938	0.211369	0.105684	+	0.994400i	$0.466297\pi$
$812$	0	0
$813$	9.69724	0.340097
$814$	9.28005	0.325265
$815$	−47.5592	−1.66593
$816$	−1.20482	−0.0421770
$817$	3.55918	0.124520
$818$	27.1202	0.948235
$819$	0	0
$820$	−3.12489	−0.109126
$821$	19.3094	0.673903	0.336951	−	0.941522i	$-0.390604\pi$
$821$	19.3094	0.673903	0.336951	+	0.941522i	$0.390604\pi$
$822$	−9.12019	−0.318103
$823$	14.4849	0.504911	0.252455	−	0.967609i	$-0.418762\pi$
$823$	14.4849	0.504911	0.252455	+	0.967609i	$0.418762\pi$
$824$	11.9541	0.416442
$825$	−10.7200	−0.373221
$826$	0	0
$827$	4.79988	0.166908	0.0834541	−	0.996512i	$-0.473405\pi$
$827$	4.79988	0.166908	0.0834541	+	0.996512i	$0.473405\pi$
$828$	11.7502	0.408349
$829$	−32.6235	−1.13306	−0.566531	−	0.824041i	$-0.691715\pi$
$829$	−32.6235	−1.13306	−0.566531	+	0.824041i	$0.691715\pi$
$830$	−23.7796	−0.825402
$831$	−7.34999	−0.254968
$832$	0	0
$833$	0	0
$834$	3.93189	0.136150
$835$	−20.4995	−0.709416
$836$	−4.78051	−0.165337
$837$	−6.94565	−0.240077
$838$	25.6997	0.887780
$839$	−33.0596	−1.14135	−0.570673	−	0.821178i	$-0.693317\pi$
$839$	−33.0596	−1.14135	−0.570673	+	0.821178i	$0.693317\pi$
$840$	0	0
$841$	−27.7346	−0.956367
$842$	14.2451	0.490918
$843$	1.21193	0.0417412
$844$	−10.1093	−0.347976
$845$	40.6235	1.39749
$846$	5.52982	0.190119
$847$	0	0
$848$	−10.4049	−0.357307
$849$	5.25069	0.180203
$850$	11.8401	0.406113
$851$	−8.49954	−0.291361
$852$	−6.32500	−0.216691
$853$	−24.7834	−0.848566	−0.424283	−	0.905530i	$-0.639474\pi$
$853$	−24.7834	−0.848566	−0.424283	+	0.905530i	$0.639474\pi$
$854$	0	0
$855$	−8.90161	−0.304429
$856$	16.2645	0.555907
$857$	−19.1202	−0.653133	−0.326567	−	0.945174i	$-0.605892\pi$
$857$	−19.1202	−0.653133	−0.326567	+	0.945174i	$0.605892\pi$
$858$	0	0
$859$	−14.2616	−0.486599	−0.243300	−	0.969951i	$-0.578230\pi$
$859$	−14.2616	−0.486599	−0.243300	+	0.969951i	$0.578230\pi$
$860$	10.7952	0.368113
$861$	0	0
$862$	−5.68968	−0.193791
$863$	−12.8780	−0.438372	−0.219186	−	0.975683i	$-0.570340\pi$
$863$	−12.8780	−0.438372	−0.219186	+	0.975683i	$0.570340\pi$
$864$	2.79518	0.0950941
$865$	59.2635	2.01502
$866$	37.4087	1.27120
$867$	5.24885	0.178260
$868$	0	0
$869$	54.5895	1.85182
$870$	−1.70436	−0.0577831
$871$	0	0
$872$	−19.4499	−0.658656
$873$	−38.0587	−1.28809
$874$	4.37844	0.148103
$875$	0	0
$876$	−4.37844	−0.147934
$877$	−3.87133	−0.130726	−0.0653628	−	0.997862i	$-0.520820\pi$
$877$	−3.87133	−0.130726	−0.0653628	+	0.997862i	$0.520820\pi$
$878$	14.9991	0.506195
$879$	5.64909	0.190539
$880$	−14.4995	−0.488779
$881$	−27.1883	−0.915997	−0.457999	−	0.888953i	$-0.651434\pi$
$881$	−27.1883	−0.915997	−0.457999	+	0.888953i	$0.651434\pi$
$882$	0	0
$883$	8.39781	0.282609	0.141304	−	0.989966i	$-0.454870\pi$
$883$	8.39781	0.282609	0.141304	+	0.989966i	$0.454870\pi$
$884$	0	0
$885$	13.4693	0.452764
$886$	−12.5748	−0.422458
$887$	12.4096	0.416675	0.208337	−	0.978057i	$-0.433195\pi$
$887$	12.4096	0.416675	0.208337	+	0.978057i	$0.433195\pi$
$888$	−0.969724	−0.0325418
$889$	0	0
$890$	6.98440	0.234118
$891$	32.1992	1.07871
$892$	17.2342	0.577043
$893$	2.06055	0.0689537
$894$	0.787623	0.0263421
$895$	−13.5298	−0.452252
$896$	0	0
$897$	0	0
$898$	26.8557	0.896188
$899$	2.79518	0.0932246
$900$	−13.1745	−0.439151
$901$	−25.8548	−0.861349
$902$	4.64002	0.154496
$903$	0	0
$904$	−9.82546	−0.326790
$905$	19.9083	0.661773
$906$	−3.61445	−0.120082
$907$	−12.9532	−0.430104	−0.215052	−	0.976603i	$-0.568992\pi$
$907$	−12.9532	−0.430104	−0.215052	+	0.976603i	$0.568992\pi$
$908$	0.795185	0.0263891
$909$	−29.7209	−0.985779
$910$	0	0
$911$	58.2186	1.92887	0.964434	−	0.264325i	$-0.0851490\pi$
$911$	58.2186	1.92887	0.964434	+	0.264325i	$0.0851490\pi$
$912$	0.499542	0.0165415
$913$	35.3094	1.16857
$914$	−30.4995	−1.00884
$915$	2.79518	0.0924060
$916$	−11.2195	−0.370703
$917$	0	0
$918$	6.94565	0.229241
$919$	−5.66560	−0.186891	−0.0934455	−	0.995624i	$-0.529788\pi$
$919$	−5.66560	−0.186891	−0.0934455	+	0.995624i	$0.529788\pi$
$920$	13.2800	0.437830
$921$	−10.5213	−0.346690
$922$	36.4343	1.19990
$923$	0	0
$924$	0	0
$925$	9.52982	0.313338
$926$	−26.5601	−0.872819
$927$	−33.0521	−1.08557
$928$	−1.12489	−0.0369262
$929$	20.0606	0.658165	0.329083	−	0.944301i	$-0.393261\pi$
$929$	20.0606	0.658165	0.329083	+	0.944301i	$0.393261\pi$
$930$	−3.76491	−0.123456
$931$	0	0
$932$	1.87890	0.0615453
$933$	12.0917	0.395866
$934$	21.0109	0.687498
$935$	−36.0294	−1.17829
$936$	0	0
$937$	40.8245	1.33368	0.666840	−	0.745201i	$-0.267646\pi$
$937$	40.8245	1.33368	0.666840	+	0.745201i	$0.267646\pi$
$938$	0	0
$939$	16.1505	0.527050
$940$	6.24977	0.203845
$941$	−21.7990	−0.710626	−0.355313	−	0.934747i	$-0.615626\pi$
$941$	−21.7990	−0.710626	−0.355313	+	0.934747i	$0.615626\pi$
$942$	−3.87890	−0.126381
$943$	−4.24977	−0.138392
$944$	8.88979	0.289338
$945$	0	0
$946$	−16.0294	−0.521159
$947$	32.0587	1.04177	0.520884	−	0.853627i	$-0.325602\pi$
$947$	32.0587	1.04177	0.520884	+	0.853627i	$0.325602\pi$
$948$	−5.70436	−0.185269
$949$	0	0
$950$	−4.90917	−0.159275
$951$	5.55162	0.180023
$952$	0	0
$953$	−35.7872	−1.15926	−0.579630	−	0.814880i	$-0.696803\pi$
$953$	−35.7872	−1.15926	−0.579630	+	0.814880i	$0.696803\pi$
$954$	28.7687	0.931421
$955$	−45.2635	−1.46469
$956$	15.7190	0.508390
$957$	2.53073	0.0818070
$958$	8.68120	0.280477
$959$	0	0
$960$	1.51514	0.0489009
$961$	−24.8255	−0.800821
$962$	0	0
$963$	−44.9697	−1.44913
$964$	4.06055	0.130782
$965$	−42.6576	−1.37320
$966$	0	0
$967$	−29.6126	−0.952277	−0.476139	−	0.879370i	$-0.657964\pi$
$967$	−29.6126	−0.952277	−0.476139	+	0.879370i	$0.657964\pi$
$968$	10.5298	0.338441
$969$	1.24129	0.0398760
$970$	−43.0138	−1.38109
$971$	−33.3241	−1.06942	−0.534710	−	0.845035i	$-0.679579\pi$
$971$	−33.3241	−1.06942	−0.534710	+	0.845035i	$0.679579\pi$
$972$	−11.7502	−0.376889
$973$	0	0
$974$	8.99908	0.288349
$975$	0	0
$976$	1.84484	0.0590518
$977$	16.2791	0.520816	0.260408	−	0.965499i	$-0.416143\pi$
$977$	16.2791	0.520816	0.260408	+	0.965499i	$0.416143\pi$
$978$	−7.37935	−0.235966
$979$	−10.3709	−0.331455
$980$	0	0
$981$	53.7772	1.71697
$982$	38.0147	1.21310
$983$	−21.5445	−0.687163	−0.343581	−	0.939123i	$-0.611640\pi$
$983$	−21.5445	−0.687163	−0.343581	+	0.939123i	$0.611640\pi$
$984$	−0.484862	−0.0154568
$985$	−25.7796	−0.821406
$986$	−2.79518	−0.0890168
$987$	0	0
$988$	0	0
$989$	14.6812	0.466835
$990$	40.0899	1.27414
$991$	26.1580	0.830937	0.415469	−	0.909608i	$-0.363618\pi$
$991$	26.1580	0.830937	0.415469	+	0.909608i	$0.363618\pi$
$992$	−2.48486	−0.0788944
$993$	−2.96216	−0.0940014
$994$	0	0
$995$	43.8401	1.38983
$996$	−3.68968	−0.116912
$997$	24.2186	0.767010	0.383505	−	0.923539i	$-0.374717\pi$
$997$	24.2186	0.767010	0.383505	+	0.923539i	$0.374717\pi$
$998$	−6.70058	−0.212103
$999$	5.59037	0.176871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4018.2.a.bg.1.2		3
7.6	odd	2		574.2.a.l.1.2	✓	3
21.20	even	2		5166.2.a.bt.1.1		3
28.27	even	2		4592.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.a.l.1.2	✓	3	7.6	odd	2
4018.2.a.bg.1.2		3	1.1	even	1	trivial
4592.2.a.s.1.2		3	28.27	even	2
5166.2.a.bt.1.1		3	21.20	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.484862 q^{3} +1.00000 q^{4} -3.12489 q^{5} -0.484862 q^{6} +1.00000 q^{8} -2.76491 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.484862 q^{3} +1.00000 q^{4} -3.12489 q^{5} -0.484862 q^{6} +1.00000 q^{8} -2.76491 q^{9} -3.12489 q^{10} +4.64002 q^{11} -0.484862 q^{12} +1.51514 q^{15} +1.00000 q^{16} +2.48486 q^{17} -2.76491 q^{18} -1.03028 q^{19} -3.12489 q^{20} +4.64002 q^{22} -4.24977 q^{23} -0.484862 q^{24} +4.76491 q^{25} +2.79518 q^{27} -1.12489 q^{29} +1.51514 q^{30} -2.48486 q^{31} +1.00000 q^{32} -2.24977 q^{33} +2.48486 q^{34} -2.76491 q^{36} +2.00000 q^{37} -1.03028 q^{38} -3.12489 q^{40} +1.00000 q^{41} -3.45459 q^{43} +4.64002 q^{44} +8.64002 q^{45} -4.24977 q^{46} -2.00000 q^{47} -0.484862 q^{48} +4.76491 q^{50} -1.20482 q^{51} -10.4049 q^{53} +2.79518 q^{54} -14.4995 q^{55} +0.499542 q^{57} -1.12489 q^{58} +8.88979 q^{59} +1.51514 q^{60} +1.84484 q^{61} -2.48486 q^{62} +1.00000 q^{64} -2.24977 q^{66} +10.5795 q^{67} +2.48486 q^{68} +2.06055 q^{69} +13.0450 q^{71} -2.76491 q^{72} +9.03028 q^{73} +2.00000 q^{74} -2.31032 q^{75} -1.03028 q^{76} +11.7649 q^{79} -3.12489 q^{80} +6.93945 q^{81} +1.00000 q^{82} +7.60975 q^{83} -7.76491 q^{85} -3.45459 q^{86} +0.545414 q^{87} +4.64002 q^{88} -2.23509 q^{89} +8.64002 q^{90} -4.24977 q^{92} +1.20482 q^{93} -2.00000 q^{94} +3.21949 q^{95} -0.484862 q^{96} +13.7649 q^{97} -12.8292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 8 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 8 q^{9} - q^{10} + 6 q^{11} - q^{12} + 5 q^{15} + 3 q^{16} + 7 q^{17} + 8 q^{18} - 4 q^{19} - q^{20} + 6 q^{22} + 4 q^{23} - q^{24} - 2 q^{25} - 7 q^{27} + 5 q^{29} + 5 q^{30} - 7 q^{31} + 3 q^{32} + 10 q^{33} + 7 q^{34} + 8 q^{36} + 6 q^{37} - 4 q^{38} - q^{40} + 3 q^{41} - 9 q^{43} + 6 q^{44} + 18 q^{45} + 4 q^{46} - 6 q^{47} - q^{48} - 2 q^{50} - 19 q^{51} - 7 q^{53} - 7 q^{54} - 10 q^{55} - 32 q^{57} + 5 q^{58} + 2 q^{59} + 5 q^{60} + 13 q^{61} - 7 q^{62} + 3 q^{64} + 10 q^{66} + 22 q^{67} + 7 q^{68} + 8 q^{69} + 7 q^{71} + 8 q^{72} + 28 q^{73} + 6 q^{74} + 8 q^{75} - 4 q^{76} + 19 q^{79} - q^{80} + 19 q^{81} + 3 q^{82} + 14 q^{83} - 7 q^{85} - 9 q^{86} + 3 q^{87} + 6 q^{88} - 23 q^{89} + 18 q^{90} + 4 q^{92} + 19 q^{93} - 6 q^{94} - 8 q^{95} - q^{96} + 25 q^{97} - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 8 * q^9 - q^10 + 6 * q^11 - q^12 + 5 * q^15 + 3 * q^16 + 7 * q^17 + 8 * q^18 - 4 * q^19 - q^20 + 6 * q^22 + 4 * q^23 - q^24 - 2 * q^25 - 7 * q^27 + 5 * q^29 + 5 * q^30 - 7 * q^31 + 3 * q^32 + 10 * q^33 + 7 * q^34 + 8 * q^36 + 6 * q^37 - 4 * q^38 - q^40 + 3 * q^41 - 9 * q^43 + 6 * q^44 + 18 * q^45 + 4 * q^46 - 6 * q^47 - q^48 - 2 * q^50 - 19 * q^51 - 7 * q^53 - 7 * q^54 - 10 * q^55 - 32 * q^57 + 5 * q^58 + 2 * q^59 + 5 * q^60 + 13 * q^61 - 7 * q^62 + 3 * q^64 + 10 * q^66 + 22 * q^67 + 7 * q^68 + 8 * q^69 + 7 * q^71 + 8 * q^72 + 28 * q^73 + 6 * q^74 + 8 * q^75 - 4 * q^76 + 19 * q^79 - q^80 + 19 * q^81 + 3 * q^82 + 14 * q^83 - 7 * q^85 - 9 * q^86 + 3 * q^87 + 6 * q^88 - 23 * q^89 + 18 * q^90 + 4 * q^92 + 19 * q^93 - 6 * q^94 - 8 * q^95 - q^96 + 25 * q^97 - 12 * q^99

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