LMFDB - Embedded newform 4018.2.a.bg.1.3

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			1.3
Root			$-1.76156$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.62620 q^{3} +1.00000 q^{4} +1.76156 q^{5} +2.62620 q^{6} +1.00000 q^{8} +3.89692 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.62620 q^{3} +1.00000 q^{4} +1.76156 q^{5} +2.62620 q^{6} +1.00000 q^{8} +3.89692 q^{9} +1.76156 q^{10} +2.86464 q^{11} +2.62620 q^{12} +4.62620 q^{15} +1.00000 q^{16} -0.626198 q^{17} +3.89692 q^{18} -7.25240 q^{19} +1.76156 q^{20} +2.86464 q^{22} +5.52311 q^{23} +2.62620 q^{24} -1.89692 q^{25} +2.35548 q^{27} +3.76156 q^{29} +4.62620 q^{30} +0.626198 q^{31} +1.00000 q^{32} +7.52311 q^{33} -0.626198 q^{34} +3.89692 q^{36} +2.00000 q^{37} -7.25240 q^{38} +1.76156 q^{40} +1.00000 q^{41} +5.87859 q^{43} +2.86464 q^{44} +6.86464 q^{45} +5.52311 q^{46} -2.00000 q^{47} +2.62620 q^{48} -1.89692 q^{50} -1.64452 q^{51} -1.96772 q^{53} +2.35548 q^{54} +5.04623 q^{55} -19.0462 q^{57} +3.76156 q^{58} -2.65847 q^{59} +4.62620 q^{60} +0.509161 q^{61} +0.626198 q^{62} +1.00000 q^{64} +7.52311 q^{66} -3.64015 q^{67} -0.626198 q^{68} +14.5048 q^{69} +2.83237 q^{71} +3.89692 q^{72} +15.2524 q^{73} +2.00000 q^{74} -4.98168 q^{75} -7.25240 q^{76} +5.10308 q^{79} +1.76156 q^{80} -5.50479 q^{81} +1.00000 q^{82} -0.387755 q^{83} -1.10308 q^{85} +5.87859 q^{86} +9.87859 q^{87} +2.86464 q^{88} -8.89692 q^{89} +6.86464 q^{90} +5.52311 q^{92} +1.64452 q^{93} -2.00000 q^{94} -12.7755 q^{95} +2.62620 q^{96} +7.10308 q^{97} +11.1633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.62620	1.51624	0.758118	−	0.652117i	$-0.226119\pi$
$3$	2.62620	1.51624	0.758118	+	0.652117i	$0.226119\pi$
$4$	1.00000	0.500000
$5$	1.76156	0.787792	0.393896	−	0.919155i	$-0.371127\pi$
$5$	1.76156	0.787792	0.393896	+	0.919155i	$0.371127\pi$
$6$	2.62620	1.07214
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	3.89692	1.29897
$10$	1.76156	0.557053
$11$	2.86464	0.863722	0.431861	−	0.901940i	$-0.357857\pi$
$11$	2.86464	0.863722	0.431861	+	0.901940i	$0.357857\pi$
$12$	2.62620	0.758118
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	4.62620	1.19448
$16$	1.00000	0.250000
$17$	−0.626198	−0.151875	−0.0759377	−	0.997113i	$-0.524195\pi$
$17$	−0.626198	−0.151875	−0.0759377	+	0.997113i	$0.524195\pi$
$18$	3.89692	0.918512
$19$	−7.25240	−1.66381	−0.831907	−	0.554915i	$-0.812751\pi$
$19$	−7.25240	−1.66381	−0.831907	+	0.554915i	$0.812751\pi$
$20$	1.76156	0.393896
$21$	0	0
$22$	2.86464	0.610743
$23$	5.52311	1.15165	0.575824	−	0.817573i	$-0.304681\pi$
$23$	5.52311	1.15165	0.575824	+	0.817573i	$0.304681\pi$
$24$	2.62620	0.536070
$25$	−1.89692	−0.379383
$26$	0	0
$27$	2.35548	0.453312
$28$	0	0
$29$	3.76156	0.698504	0.349252	−	0.937029i	$-0.386436\pi$
$29$	3.76156	0.698504	0.349252	+	0.937029i	$0.386436\pi$
$30$	4.62620	0.844624
$31$	0.626198	0.112468	0.0562342	−	0.998418i	$-0.482091\pi$
$31$	0.626198	0.112468	0.0562342	+	0.998418i	$0.482091\pi$
$32$	1.00000	0.176777
$33$	7.52311	1.30961
$34$	−0.626198	−0.107392
$35$	0	0
$36$	3.89692	0.649486
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−7.25240	−1.17649
$39$	0	0
$40$	1.76156	0.278527
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.87859	0.896477	0.448239	−	0.893914i	$-0.352051\pi$
$43$	5.87859	0.896477	0.448239	+	0.893914i	$0.352051\pi$
$44$	2.86464	0.431861
$45$	6.86464	1.02332
$46$	5.52311	0.814339
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	2.62620	0.379059
$49$	0	0
$50$	−1.89692	−0.268264
$51$	−1.64452	−0.230279
$52$	0	0
$53$	−1.96772	−0.270288	−0.135144	−	0.990826i	$-0.543150\pi$
$53$	−1.96772	−0.270288	−0.135144	+	0.990826i	$0.543150\pi$
$54$	2.35548	0.320540
$55$	5.04623	0.680433
$56$	0	0
$57$	−19.0462	−2.52273
$58$	3.76156	0.493917
$59$	−2.65847	−0.346104	−0.173052	−	0.984913i	$-0.555363\pi$
$59$	−2.65847	−0.346104	−0.173052	+	0.984913i	$0.555363\pi$
$60$	4.62620	0.597240
$61$	0.509161	0.0651914	0.0325957	−	0.999469i	$-0.489623\pi$
$61$	0.509161	0.0651914	0.0325957	+	0.999469i	$0.489623\pi$
$62$	0.626198	0.0795272
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	7.52311	0.926031
$67$	−3.64015	−0.444715	−0.222358	−	0.974965i	$-0.571375\pi$
$67$	−3.64015	−0.444715	−0.222358	+	0.974965i	$0.571375\pi$
$68$	−0.626198	−0.0759377
$69$	14.5048	1.74617
$70$	0	0
$71$	2.83237	0.336140	0.168070	−	0.985775i	$-0.446247\pi$
$71$	2.83237	0.336140	0.168070	+	0.985775i	$0.446247\pi$
$72$	3.89692	0.459256
$73$	15.2524	1.78516	0.892579	−	0.450891i	$-0.148894\pi$
$73$	15.2524	1.78516	0.892579	+	0.450891i	$0.148894\pi$
$74$	2.00000	0.232495
$75$	−4.98168	−0.575235
$76$	−7.25240	−0.831907
$77$	0	0
$78$	0	0
$79$	5.10308	0.574142	0.287071	−	0.957909i	$-0.407318\pi$
$79$	5.10308	0.574142	0.287071	+	0.957909i	$0.407318\pi$
$80$	1.76156	0.196948
$81$	−5.50479	−0.611644
$82$	1.00000	0.110432
$83$	−0.387755	−0.0425617	−0.0212808	−	0.999774i	$-0.506774\pi$
$83$	−0.387755	−0.0425617	−0.0212808	+	0.999774i	$0.506774\pi$
$84$	0	0
$85$	−1.10308	−0.119646
$86$	5.87859	0.633905
$87$	9.87859	1.05910
$88$	2.86464	0.305372
$89$	−8.89692	−0.943071	−0.471536	−	0.881847i	$-0.656300\pi$
$89$	−8.89692	−0.943071	−0.471536	+	0.881847i	$0.656300\pi$
$90$	6.86464	0.723597
$91$	0	0
$92$	5.52311	0.575824
$93$	1.64452	0.170529
$94$	−2.00000	−0.206284
$95$	−12.7755	−1.31074
$96$	2.62620	0.268035
$97$	7.10308	0.721209	0.360604	−	0.932719i	$-0.382570\pi$
$97$	7.10308	0.721209	0.360604	+	0.932719i	$0.382570\pi$
$98$	0	0
$99$	11.1633	1.12195
$100$	−1.89692	−0.189692
$101$	−18.5693	−1.84772	−0.923859	−	0.382732i	$-0.874983\pi$
$101$	−18.5693	−1.84772	−0.923859	+	0.382732i	$0.874983\pi$
$102$	−1.64452	−0.162832
$103$	−16.9248	−1.66765	−0.833826	−	0.552027i	$-0.813854\pi$
$103$	−16.9248	−1.66765	−0.833826	+	0.552027i	$0.813854\pi$
$104$	0	0
$105$	0	0
$106$	−1.96772	−0.191122
$107$	−9.94315	−0.961240	−0.480620	−	0.876929i	$-0.659588\pi$
$107$	−9.94315	−0.961240	−0.480620	+	0.876929i	$0.659588\pi$
$108$	2.35548	0.226656
$109$	−0.800090	−0.0766347	−0.0383174	−	0.999266i	$-0.512200\pi$
$109$	−0.800090	−0.0766347	−0.0383174	+	0.999266i	$0.512200\pi$
$110$	5.04623	0.481139
$111$	5.25240	0.498535
$112$	0	0
$113$	−15.6079	−1.46827	−0.734133	−	0.679006i	$-0.762411\pi$
$113$	−15.6079	−1.46827	−0.734133	+	0.679006i	$0.762411\pi$
$114$	−19.0462	−1.78384
$115$	9.72928	0.907260
$116$	3.76156	0.349252
$117$	0	0
$118$	−2.65847	−0.244732
$119$	0	0
$120$	4.62620	0.422312
$121$	−2.79383	−0.253985
$122$	0.509161	0.0460973
$123$	2.62620	0.236796
$124$	0.626198	0.0562342
$125$	−12.1493	−1.08667
$126$	0	0
$127$	−7.72928	−0.685863	−0.342931	−	0.939360i	$-0.611420\pi$
$127$	−7.72928	−0.685863	−0.342931	+	0.939360i	$0.611420\pi$
$128$	1.00000	0.0883883
$129$	15.4384	1.35927
$130$	0	0
$131$	−20.6864	−1.80738	−0.903689	−	0.428190i	$-0.859151\pi$
$131$	−20.6864	−1.80738	−0.903689	+	0.428190i	$0.859151\pi$
$132$	7.52311	0.654803
$133$	0	0
$134$	−3.64015	−0.314461
$135$	4.14931	0.357116
$136$	−0.626198	−0.0536960
$137$	1.93545	0.165357	0.0826783	−	0.996576i	$-0.473653\pi$
$137$	1.93545	0.165357	0.0826783	+	0.996576i	$0.473653\pi$
$138$	14.5048	1.23473
$139$	19.4340	1.64837	0.824185	−	0.566321i	$-0.191634\pi$
$139$	19.4340	1.64837	0.824185	+	0.566321i	$0.191634\pi$
$140$	0	0
$141$	−5.25240	−0.442332
$142$	2.83237	0.237687
$143$	0	0
$144$	3.89692	0.324743
$145$	6.62620	0.550276
$146$	15.2524	1.26230
$147$	0	0
$148$	2.00000	0.164399
$149$	22.8078	1.86849	0.934243	−	0.356636i	$-0.116076\pi$
$149$	22.8078	1.86849	0.934243	+	0.356636i	$0.116076\pi$
$150$	−4.98168	−0.406752
$151$	−1.87859	−0.152878	−0.0764389	−	0.997074i	$-0.524355\pi$
$151$	−1.87859	−0.152878	−0.0764389	+	0.997074i	$0.524355\pi$
$152$	−7.25240	−0.588247
$153$	−2.44024	−0.197282
$154$	0	0
$155$	1.10308	0.0886018
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	5.10308	0.405980
$159$	−5.16763	−0.409820
$160$	1.76156	0.139263
$161$	0	0
$162$	−5.50479	−0.432497
$163$	−0.775511	−0.0607427	−0.0303713	−	0.999539i	$-0.509669\pi$
$163$	−0.775511	−0.0607427	−0.0303713	+	0.999539i	$0.509669\pi$
$164$	1.00000	0.0780869
$165$	13.2524	1.03170
$166$	−0.387755	−0.0300956
$167$	−0.541436	−0.0418976	−0.0209488	−	0.999781i	$-0.506669\pi$
$167$	−0.541436	−0.0418976	−0.0209488	+	0.999781i	$0.506669\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−1.10308	−0.0846027
$171$	−28.2620	−2.16125
$172$	5.87859	0.448239
$173$	−3.42629	−0.260496	−0.130248	−	0.991481i	$-0.541577\pi$
$173$	−3.42629	−0.260496	−0.130248	+	0.991481i	$0.541577\pi$
$174$	9.87859	0.748894
$175$	0	0
$176$	2.86464	0.215930
$177$	−6.98168	−0.524775
$178$	−8.89692	−0.666852
$179$	−0.117037	−0.00874776	−0.00437388	−	0.999990i	$-0.501392\pi$
$179$	−0.117037	−0.00874776	−0.00437388	+	0.999990i	$0.501392\pi$
$180$	6.86464	0.511660
$181$	−21.4865	−1.59708	−0.798538	−	0.601944i	$-0.794393\pi$
$181$	−21.4865	−1.59708	−0.798538	+	0.601944i	$0.794393\pi$
$182$	0	0
$183$	1.33716	0.0988455
$184$	5.52311	0.407169
$185$	3.52311	0.259025
$186$	1.64452	0.120582
$187$	−1.79383	−0.131178
$188$	−2.00000	−0.145865
$189$	0	0
$190$	−12.7755	−0.926833
$191$	11.3738	0.822979	0.411490	−	0.911414i	$-0.365009\pi$
$191$	11.3738	0.822979	0.411490	+	0.911414i	$0.365009\pi$
$192$	2.62620	0.189530
$193$	25.2158	1.81507	0.907535	−	0.419977i	$-0.137962\pi$
$193$	25.2158	1.81507	0.907535	+	0.419977i	$0.137962\pi$
$194$	7.10308	0.509972
$195$	0	0
$196$	0	0
$197$	−1.52311	−0.108517	−0.0542587	−	0.998527i	$-0.517280\pi$
$197$	−1.52311	−0.108517	−0.0542587	+	0.998527i	$0.517280\pi$
$198$	11.1633	0.793339
$199$	18.8401	1.33554	0.667768	−	0.744369i	$-0.267250\pi$
$199$	18.8401	1.33554	0.667768	+	0.744369i	$0.267250\pi$
$200$	−1.89692	−0.134132
$201$	−9.55976	−0.674293
$202$	−18.5693	−1.30653
$203$	0	0
$204$	−1.64452	−0.115139
$205$	1.76156	0.123032
$206$	−16.9248	−1.17921
$207$	21.5231	1.49596
$208$	0	0
$209$	−20.7755	−1.43707
$210$	0	0
$211$	17.4340	1.20020	0.600102	−	0.799923i	$-0.295126\pi$
$211$	17.4340	1.20020	0.600102	+	0.799923i	$0.295126\pi$
$212$	−1.96772	−0.135144
$213$	7.43835	0.509668
$214$	−9.94315	−0.679699
$215$	10.3555	0.706238
$216$	2.35548	0.160270
$217$	0	0
$218$	−0.800090	−0.0541889
$219$	40.0558	2.70672
$220$	5.04623	0.340217
$221$	0	0
$222$	5.25240	0.352518
$223$	−15.1955	−1.01757	−0.508784	−	0.860894i	$-0.669905\pi$
$223$	−15.1955	−1.01757	−0.508784	+	0.860894i	$0.669905\pi$
$224$	0	0
$225$	−7.39212	−0.492808
$226$	−15.6079	−1.03822
$227$	0.355480	0.0235940	0.0117970	−	0.999930i	$-0.496245\pi$
$227$	0.355480	0.0235940	0.0117970	+	0.999930i	$0.496245\pi$
$228$	−19.0462	−1.26137
$229$	4.77551	0.315575	0.157787	−	0.987473i	$-0.449564\pi$
$229$	4.77551	0.315575	0.157787	+	0.987473i	$0.449564\pi$
$230$	9.72928	0.641530
$231$	0	0
$232$	3.76156	0.246958
$233$	−23.0096	−1.50741	−0.753704	−	0.657214i	$-0.771735\pi$
$233$	−23.0096	−1.50741	−0.753704	+	0.657214i	$0.771735\pi$
$234$	0	0
$235$	−3.52311	−0.229823
$236$	−2.65847	−0.173052
$237$	13.4017	0.870535
$238$	0	0
$239$	−19.8217	−1.28216	−0.641081	−	0.767473i	$-0.721514\pi$
$239$	−19.8217	−1.28216	−0.641081	+	0.767473i	$0.721514\pi$
$240$	4.62620	0.298620
$241$	16.5048	1.06317	0.531584	−	0.847006i	$-0.321597\pi$
$241$	16.5048	1.06317	0.531584	+	0.847006i	$0.321597\pi$
$242$	−2.79383	−0.179594
$243$	−21.5231	−1.38071
$244$	0.509161	0.0325957
$245$	0	0
$246$	2.62620	0.167440
$247$	0	0
$248$	0.626198	0.0397636
$249$	−1.01832	−0.0645335
$250$	−12.1493	−0.768390
$251$	−10.0525	−0.634507	−0.317254	−	0.948341i	$-0.602761\pi$
$251$	−10.0525	−0.634507	−0.317254	+	0.948341i	$0.602761\pi$
$252$	0	0
$253$	15.8217	0.994704
$254$	−7.72928	−0.484978
$255$	−2.89692	−0.181412
$256$	1.00000	0.0625000
$257$	8.48458	0.529254	0.264627	−	0.964351i	$-0.414751\pi$
$257$	8.48458	0.529254	0.264627	+	0.964351i	$0.414751\pi$
$258$	15.4384	0.961150
$259$	0	0
$260$	0	0
$261$	14.6585	0.907337
$262$	−20.6864	−1.27801
$263$	12.5414	0.773338	0.386669	−	0.922219i	$-0.373626\pi$
$263$	12.5414	0.773338	0.386669	+	0.922219i	$0.373626\pi$
$264$	7.52311	0.463016
$265$	−3.46626	−0.212931
$266$	0	0
$267$	−23.3651	−1.42992
$268$	−3.64015	−0.222358
$269$	28.4157	1.73253	0.866267	−	0.499582i	$-0.166513\pi$
$269$	28.4157	1.73253	0.866267	+	0.499582i	$0.166513\pi$
$270$	4.14931	0.252519
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−0.626198	−0.0379688
$273$	0	0
$274$	1.93545	0.116925
$275$	−5.43398	−0.327682
$276$	14.5048	0.873086
$277$	−13.2803	−0.797936	−0.398968	−	0.916965i	$-0.630632\pi$
$277$	−13.2803	−0.797936	−0.398968	+	0.916965i	$0.630632\pi$
$278$	19.4340	1.16557
$279$	2.44024	0.146093
$280$	0	0
$281$	17.0462	1.01689	0.508446	−	0.861094i	$-0.330220\pi$
$281$	17.0462	1.01689	0.508446	+	0.861094i	$0.330220\pi$
$282$	−5.25240	−0.312776
$283$	13.1633	0.782475	0.391237	−	0.920290i	$-0.372047\pi$
$283$	13.1633	0.782475	0.391237	+	0.920290i	$0.372047\pi$
$284$	2.83237	0.168070
$285$	−33.5510	−1.98739
$286$	0	0
$287$	0	0
$288$	3.89692	0.229628
$289$	−16.6079	−0.976934
$290$	6.62620	0.389104
$291$	18.6541	1.09352
$292$	15.2524	0.892579
$293$	−23.2158	−1.35628	−0.678139	−	0.734933i	$-0.737213\pi$
$293$	−23.2158	−1.35628	−0.678139	+	0.734933i	$0.737213\pi$
$294$	0	0
$295$	−4.68305	−0.272658
$296$	2.00000	0.116248
$297$	6.74760	0.391536
$298$	22.8078	1.32122
$299$	0	0
$300$	−4.98168	−0.287617
$301$	0	0
$302$	−1.87859	−0.108101
$303$	−48.7668	−2.80158
$304$	−7.25240	−0.415953
$305$	0.896916	0.0513573
$306$	−2.44024	−0.139499
$307$	−6.72302	−0.383703	−0.191852	−	0.981424i	$-0.561449\pi$
$307$	−6.72302	−0.383703	−0.191852	+	0.981424i	$0.561449\pi$
$308$	0	0
$309$	−44.4479	−2.52855
$310$	1.10308	0.0626509
$311$	26.5972	1.50819	0.754096	−	0.656764i	$-0.228075\pi$
$311$	26.5972	1.50819	0.754096	+	0.656764i	$0.228075\pi$
$312$	0	0
$313$	3.11078	0.175832	0.0879158	−	0.996128i	$-0.471979\pi$
$313$	3.11078	0.175832	0.0879158	+	0.996128i	$0.471979\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	5.10308	0.287071
$317$	7.19991	0.404387	0.202194	−	0.979346i	$-0.435193\pi$
$317$	7.19991	0.404387	0.202194	+	0.979346i	$0.435193\pi$
$318$	−5.16763	−0.289787
$319$	10.7755	0.603313
$320$	1.76156	0.0984740
$321$	−26.1127	−1.45747
$322$	0	0
$323$	4.54144	0.252692
$324$	−5.50479	−0.305822
$325$	0	0
$326$	−0.775511	−0.0429516
$327$	−2.10119	−0.116196
$328$	1.00000	0.0552158
$329$	0	0
$330$	13.2524	0.729520
$331$	−21.4340	−1.17812	−0.589059	−	0.808090i	$-0.700502\pi$
$331$	−21.4340	−1.17812	−0.589059	+	0.808090i	$0.700502\pi$
$332$	−0.387755	−0.0212808
$333$	7.79383	0.427099
$334$	−0.541436	−0.0296261
$335$	−6.41233	−0.350343
$336$	0	0
$337$	−25.3651	−1.38172	−0.690862	−	0.722987i	$-0.742769\pi$
$337$	−25.3651	−1.38172	−0.690862	+	0.722987i	$0.742769\pi$
$338$	−13.0000	−0.707107
$339$	−40.9894	−2.22624
$340$	−1.10308	−0.0598231
$341$	1.79383	0.0971415
$342$	−28.2620	−1.52823
$343$	0	0
$344$	5.87859	0.316953
$345$	25.5510	1.37562
$346$	−3.42629	−0.184198
$347$	7.70470	0.413610	0.206805	−	0.978382i	$-0.433693\pi$
$347$	7.70470	0.413610	0.206805	+	0.978382i	$0.433693\pi$
$348$	9.87859	0.529548
$349$	23.4061	1.25290	0.626449	−	0.779462i	$-0.284508\pi$
$349$	23.4061	1.25290	0.626449	+	0.779462i	$0.284508\pi$
$350$	0	0
$351$	0	0
$352$	2.86464	0.152686
$353$	−27.6156	−1.46983	−0.734914	−	0.678160i	$-0.762778\pi$
$353$	−27.6156	−1.46983	−0.734914	+	0.678160i	$0.762778\pi$
$354$	−6.98168	−0.371072
$355$	4.98937	0.264808
$356$	−8.89692	−0.471536
$357$	0	0
$358$	−0.117037	−0.00618560
$359$	−22.0925	−1.16600	−0.582998	−	0.812474i	$-0.698120\pi$
$359$	−22.0925	−1.16600	−0.582998	+	0.812474i	$0.698120\pi$
$360$	6.86464	0.361798
$361$	33.5972	1.76828
$362$	−21.4865	−1.12930
$363$	−7.33716	−0.385101
$364$	0	0
$365$	26.8680	1.40633
$366$	1.33716	0.0698944
$367$	−14.3555	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$367$	−14.3555	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$368$	5.52311	0.287912
$369$	3.89692	0.202865
$370$	3.52311	0.183158
$371$	0	0
$372$	1.64452	0.0852644
$373$	−16.6339	−0.861270	−0.430635	−	0.902526i	$-0.641710\pi$
$373$	−16.6339	−0.861270	−0.430635	+	0.902526i	$0.641710\pi$
$374$	−1.79383	−0.0927569
$375$	−31.9065	−1.64764
$376$	−2.00000	−0.103142
$377$	0	0
$378$	0	0
$379$	−16.6907	−0.857346	−0.428673	−	0.903460i	$-0.641019\pi$
$379$	−16.6907	−0.857346	−0.428673	+	0.903460i	$0.641019\pi$
$380$	−12.7755	−0.655370
$381$	−20.2986	−1.03993
$382$	11.3738	0.581934
$383$	12.2986	0.628430	0.314215	−	0.949352i	$-0.398259\pi$
$383$	12.2986	0.628430	0.314215	+	0.949352i	$0.398259\pi$
$384$	2.62620	0.134018
$385$	0	0
$386$	25.2158	1.28345
$387$	22.9084	1.16450
$388$	7.10308	0.360604
$389$	30.4681	1.54480	0.772398	−	0.635139i	$-0.219057\pi$
$389$	30.4681	1.54480	0.772398	+	0.635139i	$0.219057\pi$
$390$	0	0
$391$	−3.45856	−0.174907
$392$	0	0
$393$	−54.3265	−2.74041
$394$	−1.52311	−0.0767334
$395$	8.98937	0.452304
$396$	11.1633	0.560975
$397$	15.8217	0.794070	0.397035	−	0.917803i	$-0.370039\pi$
$397$	15.8217	0.794070	0.397035	+	0.917803i	$0.370039\pi$
$398$	18.8401	0.944367
$399$	0	0
$400$	−1.89692	−0.0948458
$401$	−19.9065	−0.994083	−0.497042	−	0.867727i	$-0.665580\pi$
$401$	−19.9065	−0.994083	−0.497042	+	0.867727i	$0.665580\pi$
$402$	−9.55976	−0.476797
$403$	0	0
$404$	−18.5693	−0.923859
$405$	−9.69701	−0.481848
$406$	0	0
$407$	5.72928	0.283990
$408$	−1.64452	−0.0814159
$409$	12.9171	0.638711	0.319355	−	0.947635i	$-0.396534\pi$
$409$	12.9171	0.638711	0.319355	+	0.947635i	$0.396534\pi$
$410$	1.76156	0.0869971
$411$	5.08287	0.250720
$412$	−16.9248	−0.833826
$413$	0	0
$414$	21.5231	1.05780
$415$	−0.683053	−0.0335298
$416$	0	0
$417$	51.0375	2.49932
$418$	−20.7755	−1.01616
$419$	−2.72302	−0.133028	−0.0665142	−	0.997785i	$-0.521188\pi$
$419$	−2.72302	−0.133028	−0.0665142	+	0.997785i	$0.521188\pi$
$420$	0	0
$421$	−4.84443	−0.236103	−0.118052	−	0.993007i	$-0.537665\pi$
$421$	−4.84443	−0.236103	−0.118052	+	0.993007i	$0.537665\pi$
$422$	17.4340	0.848673
$423$	−7.79383	−0.378949
$424$	−1.96772	−0.0955612
$425$	1.18785	0.0576190
$426$	7.43835	0.360389
$427$	0	0
$428$	−9.94315	−0.480620
$429$	0	0
$430$	10.3555	0.499386
$431$	−3.01832	−0.145387	−0.0726937	−	0.997354i	$-0.523160\pi$
$431$	−3.01832	−0.145387	−0.0726937	+	0.997354i	$0.523160\pi$
$432$	2.35548	0.113328
$433$	−0.803417	−0.0386098	−0.0193049	−	0.999814i	$-0.506145\pi$
$433$	−0.803417	−0.0386098	−0.0193049	+	0.999814i	$0.506145\pi$
$434$	0	0
$435$	17.4017	0.834348
$436$	−0.800090	−0.0383174
$437$	−40.0558	−1.91613
$438$	40.0558	1.91394
$439$	−24.0925	−1.14987	−0.574935	−	0.818199i	$-0.694973\pi$
$439$	−24.0925	−1.14987	−0.574935	+	0.818199i	$0.694973\pi$
$440$	5.04623	0.240570
$441$	0	0
$442$	0	0
$443$	10.9615	0.520795	0.260398	−	0.965501i	$-0.416146\pi$
$443$	10.9615	0.520795	0.260398	+	0.965501i	$0.416146\pi$
$444$	5.25240	0.249268
$445$	−15.6724	−0.742944
$446$	−15.1955	−0.719530
$447$	59.8978	2.83307
$448$	0	0
$449$	38.8603	1.83393	0.916965	−	0.398968i	$-0.130632\pi$
$449$	38.8603	1.83393	0.916965	+	0.398968i	$0.130632\pi$
$450$	−7.39212	−0.348468
$451$	2.86464	0.134891
$452$	−15.6079	−0.734133
$453$	−4.93356	−0.231799
$454$	0.355480	0.0166835
$455$	0	0
$456$	−19.0462	−0.891921
$457$	−10.9538	−0.512396	−0.256198	−	0.966624i	$-0.582470\pi$
$457$	−10.9538	−0.512396	−0.256198	+	0.966624i	$0.582470\pi$
$458$	4.77551	0.223145
$459$	−1.47500	−0.0688470
$460$	9.72928	0.453630
$461$	−4.87234	−0.226927	−0.113464	−	0.993542i	$-0.536195\pi$
$461$	−4.87234	−0.226927	−0.113464	+	0.993542i	$0.536195\pi$
$462$	0	0
$463$	−19.4586	−0.904316	−0.452158	−	0.891938i	$-0.649346\pi$
$463$	−19.4586	−0.904316	−0.452158	+	0.891938i	$0.649346\pi$
$464$	3.76156	0.174626
$465$	2.89692	0.134341
$466$	−23.0096	−1.06590
$467$	34.3511	1.58958	0.794790	−	0.606885i	$-0.207581\pi$
$467$	34.3511	1.58958	0.794790	+	0.606885i	$0.207581\pi$
$468$	0	0
$469$	0	0
$470$	−3.52311	−0.162509
$471$	21.0096	0.968071
$472$	−2.65847	−0.122366
$473$	16.8401	0.774307
$474$	13.4017	0.615561
$475$	13.7572	0.631223
$476$	0	0
$477$	−7.66806	−0.351096
$478$	−19.8217	−0.906625
$479$	26.4681	1.20936	0.604680	−	0.796468i	$-0.293301\pi$
$479$	26.4681	1.20936	0.604680	+	0.796468i	$0.293301\pi$
$480$	4.62620	0.211156
$481$	0	0
$482$	16.5048	0.751773
$483$	0	0
$484$	−2.79383	−0.126992
$485$	12.5125	0.568163
$486$	−21.5231	−0.976308
$487$	−30.0925	−1.36362	−0.681810	−	0.731530i	$-0.738807\pi$
$487$	−30.0925	−1.36362	−0.681810	+	0.731530i	$0.738807\pi$
$488$	0.509161	0.0230486
$489$	−2.03664	−0.0921002
$490$	0	0
$491$	21.5800	0.973890	0.486945	−	0.873433i	$-0.338111\pi$
$491$	21.5800	0.973890	0.486945	+	0.873433i	$0.338111\pi$
$492$	2.62620	0.118398
$493$	−2.35548	−0.106085
$494$	0	0
$495$	19.6647	0.883864
$496$	0.626198	0.0281171
$497$	0	0
$498$	−1.01832	−0.0456321
$499$	−17.3694	−0.777563	−0.388781	−	0.921330i	$-0.627104\pi$
$499$	−17.3694	−0.777563	−0.388781	+	0.921330i	$0.627104\pi$
$500$	−12.1493	−0.543334
$501$	−1.42192	−0.0635267
$502$	−10.0525	−0.448664
$503$	−4.41233	−0.196736	−0.0983681	−	0.995150i	$-0.531362\pi$
$503$	−4.41233	−0.196736	−0.0983681	+	0.995150i	$0.531362\pi$
$504$	0	0
$505$	−32.7110	−1.45562
$506$	15.8217	0.703362
$507$	−34.1406	−1.51624
$508$	−7.72928	−0.342931
$509$	41.9142	1.85781	0.928907	−	0.370313i	$-0.120750\pi$
$509$	41.9142	1.85781	0.928907	+	0.370313i	$0.120750\pi$
$510$	−2.89692	−0.128278
$511$	0	0
$512$	1.00000	0.0441942
$513$	−17.0829	−0.754227
$514$	8.48458	0.374239
$515$	−29.8140	−1.31376
$516$	15.4384	0.679636
$517$	−5.72928	−0.251974
$518$	0	0
$519$	−8.99811	−0.394973
$520$	0	0
$521$	−20.4769	−0.897109	−0.448554	−	0.893756i	$-0.648061\pi$
$521$	−20.4769	−0.897109	−0.448554	+	0.893756i	$0.648061\pi$
$522$	14.6585	0.641584
$523$	22.9571	1.00384	0.501922	−	0.864913i	$-0.332626\pi$
$523$	22.9571	1.00384	0.501922	+	0.864913i	$0.332626\pi$
$524$	−20.6864	−0.903689
$525$	0	0
$526$	12.5414	0.546833
$527$	−0.392124	−0.0170812
$528$	7.52311	0.327402
$529$	7.50479	0.326295
$530$	−3.46626	−0.150565
$531$	−10.3598	−0.449579
$532$	0	0
$533$	0	0
$534$	−23.3651	−1.01111
$535$	−17.5154	−0.757258
$536$	−3.64015	−0.157231
$537$	−0.307362	−0.0132637
$538$	28.4157	1.22509
$539$	0	0
$540$	4.14931	0.178558
$541$	−25.0462	−1.07682	−0.538411	−	0.842683i	$-0.680975\pi$
$541$	−25.0462	−1.07682	−0.538411	+	0.842683i	$0.680975\pi$
$542$	−20.0000	−0.859074
$543$	−56.4277	−2.42155
$544$	−0.626198	−0.0268480
$545$	−1.40940	−0.0603723
$546$	0	0
$547$	21.7326	0.929219	0.464610	−	0.885516i	$-0.346195\pi$
$547$	21.7326	0.929219	0.464610	+	0.885516i	$0.346195\pi$
$548$	1.93545	0.0826783
$549$	1.98416	0.0846818
$550$	−5.43398	−0.231706
$551$	−27.2803	−1.16218
$552$	14.5048	0.617365
$553$	0	0
$554$	−13.2803	−0.564226
$555$	9.25240	0.392742
$556$	19.4340	0.824185
$557$	−38.0881	−1.61384	−0.806922	−	0.590658i	$-0.798868\pi$
$557$	−38.0881	−1.61384	−0.806922	+	0.590658i	$0.798868\pi$
$558$	2.44024	0.103304
$559$	0	0
$560$	0	0
$561$	−4.71096	−0.198897
$562$	17.0462	0.719052
$563$	32.2062	1.35733	0.678664	−	0.734449i	$-0.262559\pi$
$563$	32.2062	1.35733	0.678664	+	0.734449i	$0.262559\pi$
$564$	−5.25240	−0.221166
$565$	−27.4942	−1.15669
$566$	13.1633	0.553293
$567$	0	0
$568$	2.83237	0.118843
$569$	−14.8969	−0.624511	−0.312256	−	0.949998i	$-0.601085\pi$
$569$	−14.8969	−0.624511	−0.312256	+	0.949998i	$0.601085\pi$
$570$	−33.5510	−1.40530
$571$	−13.4340	−0.562195	−0.281097	−	0.959679i	$-0.590698\pi$
$571$	−13.4340	−0.562195	−0.281097	+	0.959679i	$0.590698\pi$
$572$	0	0
$573$	29.8699	1.24783
$574$	0	0
$575$	−10.4769	−0.436916
$576$	3.89692	0.162372
$577$	23.5231	0.979280	0.489640	−	0.871925i	$-0.337128\pi$
$577$	23.5231	0.979280	0.489640	+	0.871925i	$0.337128\pi$
$578$	−16.6079	−0.690797
$579$	66.2216	2.75207
$580$	6.62620	0.275138
$581$	0	0
$582$	18.6541	0.773238
$583$	−5.63682	−0.233453
$584$	15.2524	0.631149
$585$	0	0
$586$	−23.2158	−0.959034
$587$	3.46626	0.143068	0.0715339	−	0.997438i	$-0.477211\pi$
$587$	3.46626	0.143068	0.0715339	+	0.997438i	$0.477211\pi$
$588$	0	0
$589$	−4.54144	−0.187127
$590$	−4.68305	−0.192798
$591$	−4.00000	−0.164538
$592$	2.00000	0.0821995
$593$	22.3921	0.919534	0.459767	−	0.888040i	$-0.347933\pi$
$593$	22.3921	0.919534	0.459767	+	0.888040i	$0.347933\pi$
$594$	6.74760	0.276858
$595$	0	0
$596$	22.8078	0.934243
$597$	49.4777	2.02499
$598$	0	0
$599$	−30.0925	−1.22954	−0.614772	−	0.788705i	$-0.710752\pi$
$599$	−30.0925	−1.22954	−0.614772	+	0.788705i	$0.710752\pi$
$600$	−4.98168	−0.203376
$601$	43.9711	1.79362	0.896808	−	0.442419i	$-0.145880\pi$
$601$	43.9711	1.79362	0.896808	+	0.442419i	$0.145880\pi$
$602$	0	0
$603$	−14.1854	−0.577673
$604$	−1.87859	−0.0764389
$605$	−4.92150	−0.200087
$606$	−48.7668	−1.98101
$607$	14.0722	0.571175	0.285587	−	0.958353i	$-0.407811\pi$
$607$	14.0722	0.571175	0.285587	+	0.958353i	$0.407811\pi$
$608$	−7.25240	−0.294124
$609$	0	0
$610$	0.896916	0.0363151
$611$	0	0
$612$	−2.44024	−0.0986409
$613$	48.6252	1.96395	0.981976	−	0.189007i	$-0.0605268\pi$
$613$	48.6252	1.96395	0.981976	+	0.189007i	$0.0605268\pi$
$614$	−6.72302	−0.271319
$615$	4.62620	0.186546
$616$	0	0
$617$	2.61850	0.105417	0.0527085	−	0.998610i	$-0.483215\pi$
$617$	2.61850	0.105417	0.0527085	+	0.998610i	$0.483215\pi$
$618$	−44.4479	−1.78796
$619$	17.0987	0.687255	0.343628	−	0.939106i	$-0.388344\pi$
$619$	17.0987	0.687255	0.343628	+	0.939106i	$0.388344\pi$
$620$	1.10308	0.0443009
$621$	13.0096	0.522057
$622$	26.5972	1.06645
$623$	0	0
$624$	0	0
$625$	−11.9171	−0.476685
$626$	3.11078	0.124332
$627$	−54.5606	−2.17894
$628$	8.00000	0.319235
$629$	−1.25240	−0.0499363
$630$	0	0
$631$	16.3265	0.649949	0.324974	−	0.945723i	$-0.394644\pi$
$631$	16.3265	0.649949	0.324974	+	0.945723i	$0.394644\pi$
$632$	5.10308	0.202990
$633$	45.7851	1.81979
$634$	7.19991	0.285945
$635$	−13.6156	−0.540317
$636$	−5.16763	−0.204910
$637$	0	0
$638$	10.7755	0.426607
$639$	11.0375	0.436636
$640$	1.76156	0.0696317
$641$	42.0558	1.66110	0.830552	−	0.556941i	$-0.188025\pi$
$641$	42.0558	1.66110	0.830552	+	0.556941i	$0.188025\pi$
$642$	−26.1127	−1.03058
$643$	0.767815	0.0302797	0.0151398	−	0.999885i	$-0.495181\pi$
$643$	0.767815	0.0302797	0.0151398	+	0.999885i	$0.495181\pi$
$644$	0	0
$645$	27.1955	1.07082
$646$	4.54144	0.178680
$647$	−35.5587	−1.39796	−0.698979	−	0.715142i	$-0.746362\pi$
$647$	−35.5587	−1.39796	−0.698979	+	0.715142i	$0.746362\pi$
$648$	−5.50479	−0.216249
$649$	−7.61557	−0.298937
$650$	0	0
$651$	0	0
$652$	−0.775511	−0.0303713
$653$	−10.3309	−0.404279	−0.202140	−	0.979357i	$-0.564789\pi$
$653$	−10.3309	−0.404279	−0.202140	+	0.979357i	$0.564789\pi$
$654$	−2.10119	−0.0821632
$655$	−36.4402	−1.42384
$656$	1.00000	0.0390434
$657$	59.4373	2.31887
$658$	0	0
$659$	48.1729	1.87655	0.938274	−	0.345893i	$-0.112424\pi$
$659$	48.1729	1.87655	0.938274	+	0.345893i	$0.112424\pi$
$660$	13.2524	0.515849
$661$	−26.4523	−1.02888	−0.514438	−	0.857528i	$-0.671999\pi$
$661$	−26.4523	−1.02888	−0.514438	+	0.857528i	$0.671999\pi$
$662$	−21.4340	−0.833055
$663$	0	0
$664$	−0.387755	−0.0150478
$665$	0	0
$666$	7.79383	0.302005
$667$	20.7755	0.804431
$668$	−0.541436	−0.0209488
$669$	−39.9065	−1.54287
$670$	−6.41233	−0.247730
$671$	1.45856	0.0563072
$672$	0	0
$673$	35.2437	1.35854	0.679272	−	0.733887i	$-0.262296\pi$
$673$	35.2437	1.35854	0.679272	+	0.733887i	$0.262296\pi$
$674$	−25.3651	−0.977026
$675$	−4.46815	−0.171979
$676$	−13.0000	−0.500000
$677$	−16.0033	−0.615058	−0.307529	−	0.951539i	$-0.599502\pi$
$677$	−16.0033	−0.615058	−0.307529	+	0.951539i	$0.599502\pi$
$678$	−40.9894	−1.57419
$679$	0	0
$680$	−1.10308	−0.0423013
$681$	0.933560	0.0357741
$682$	1.79383	0.0686894
$683$	−41.6681	−1.59438	−0.797192	−	0.603726i	$-0.793682\pi$
$683$	−41.6681	−1.59438	−0.797192	+	0.603726i	$0.793682\pi$
$684$	−28.2620	−1.08062
$685$	3.40940	0.130267
$686$	0	0
$687$	12.5414	0.478486
$688$	5.87859	0.224119
$689$	0	0
$690$	25.5510	0.972711
$691$	−3.46626	−0.131863	−0.0659314	−	0.997824i	$-0.521002\pi$
$691$	−3.46626	−0.131863	−0.0659314	+	0.997824i	$0.521002\pi$
$692$	−3.42629	−0.130248
$693$	0	0
$694$	7.70470	0.292466
$695$	34.2341	1.29857
$696$	9.87859	0.374447
$697$	−0.626198	−0.0237189
$698$	23.4061	0.885933
$699$	−60.4277	−2.28559
$700$	0	0
$701$	−4.80342	−0.181423	−0.0907113	−	0.995877i	$-0.528914\pi$
$701$	−4.80342	−0.181423	−0.0907113	+	0.995877i	$0.528914\pi$
$702$	0	0
$703$	−14.5048	−0.547059
$704$	2.86464	0.107965
$705$	−9.25240	−0.348465
$706$	−27.6156	−1.03933
$707$	0	0
$708$	−6.98168	−0.262388
$709$	36.4080	1.36733	0.683665	−	0.729796i	$-0.260385\pi$
$709$	36.4080	1.36733	0.683665	+	0.729796i	$0.260385\pi$
$710$	4.98937	0.187248
$711$	19.8863	0.745794
$712$	−8.89692	−0.333426
$713$	3.45856	0.129524
$714$	0	0
$715$	0	0
$716$	−0.117037	−0.00437388
$717$	−52.0558	−1.94406
$718$	−22.0925	−0.824483
$719$	−41.6068	−1.55167	−0.775837	−	0.630934i	$-0.782672\pi$
$719$	−41.6068	−1.55167	−0.775837	+	0.630934i	$0.782672\pi$
$720$	6.86464	0.255830
$721$	0	0
$722$	33.5972	1.25036
$723$	43.3449	1.61201
$724$	−21.4865	−0.798538
$725$	−7.13536	−0.265001
$726$	−7.33716	−0.272307
$727$	17.0096	0.630851	0.315425	−	0.948950i	$-0.397853\pi$
$727$	17.0096	0.630851	0.315425	+	0.948950i	$0.397853\pi$
$728$	0	0
$729$	−40.0096	−1.48184
$730$	26.8680	0.994428
$731$	−3.68116	−0.136153
$732$	1.33716	0.0494228
$733$	−25.8907	−0.956293	−0.478147	−	0.878280i	$-0.658691\pi$
$733$	−25.8907	−0.956293	−0.478147	+	0.878280i	$0.658691\pi$
$734$	−14.3555	−0.529870
$735$	0	0
$736$	5.52311	0.203585
$737$	−10.4277	−0.384110
$738$	3.89692	0.143447
$739$	34.0635	1.25305	0.626523	−	0.779403i	$-0.284477\pi$
$739$	34.0635	1.25305	0.626523	+	0.779403i	$0.284477\pi$
$740$	3.52311	0.129512
$741$	0	0
$742$	0	0
$743$	10.5048	0.385383	0.192692	−	0.981259i	$-0.438278\pi$
$743$	10.5048	0.385383	0.192692	+	0.981259i	$0.438278\pi$
$744$	1.64452	0.0602910
$745$	40.1772	1.47198
$746$	−16.6339	−0.609010
$747$	−1.51105	−0.0552864
$748$	−1.79383	−0.0655890
$749$	0	0
$750$	−31.9065	−1.16506
$751$	−8.64641	−0.315512	−0.157756	−	0.987478i	$-0.550426\pi$
$751$	−8.64641	−0.315512	−0.157756	+	0.987478i	$0.550426\pi$
$752$	−2.00000	−0.0729325
$753$	−26.3998	−0.962063
$754$	0	0
$755$	−3.30925	−0.120436
$756$	0	0
$757$	−48.4234	−1.75998	−0.879988	−	0.474995i	$-0.842450\pi$
$757$	−48.4234	−1.75998	−0.879988	+	0.474995i	$0.842450\pi$
$758$	−16.6907	−0.606235
$759$	41.5510	1.50821
$760$	−12.7755	−0.463416
$761$	−8.68305	−0.314760	−0.157380	−	0.987538i	$-0.550305\pi$
$761$	−8.68305	−0.314760	−0.157380	+	0.987538i	$0.550305\pi$
$762$	−20.2986	−0.735342
$763$	0	0
$764$	11.3738	0.411490
$765$	−4.29862	−0.155417
$766$	12.2986	0.444367
$767$	0	0
$768$	2.62620	0.0947648
$769$	−52.1974	−1.88229	−0.941144	−	0.338007i	$-0.890247\pi$
$769$	−52.1974	−1.88229	−0.941144	+	0.338007i	$0.890247\pi$
$770$	0	0
$771$	22.2822	0.802474
$772$	25.2158	0.907535
$773$	−17.8496	−0.642007	−0.321004	−	0.947078i	$-0.604020\pi$
$773$	−17.8496	−0.642007	−0.321004	+	0.947078i	$0.604020\pi$
$774$	22.9084	0.823425
$775$	−1.18785	−0.0426687
$776$	7.10308	0.254986
$777$	0	0
$778$	30.4681	1.09234
$779$	−7.25240	−0.259844
$780$	0	0
$781$	8.11371	0.290331
$782$	−3.45856	−0.123678
$783$	8.86027	0.316640
$784$	0	0
$785$	14.0925	0.502981
$786$	−54.3265	−1.93776
$787$	15.6768	0.558817	0.279409	−	0.960172i	$-0.409862\pi$
$787$	15.6768	0.558817	0.279409	+	0.960172i	$0.409862\pi$
$788$	−1.52311	−0.0542587
$789$	32.9363	1.17256
$790$	8.98937	0.319828
$791$	0	0
$792$	11.1633	0.396669
$793$	0	0
$794$	15.8217	0.561493
$795$	−9.10308	−0.322853
$796$	18.8401	0.667768
$797$	−10.5371	−0.373242	−0.186621	−	0.982432i	$-0.559754\pi$
$797$	−10.5371	−0.373242	−0.186621	+	0.982432i	$0.559754\pi$
$798$	0	0
$799$	1.25240	0.0443066
$800$	−1.89692	−0.0670661
$801$	−34.6705	−1.22502
$802$	−19.9065	−0.702923
$803$	43.6926	1.54188
$804$	−9.55976	−0.337147
$805$	0	0
$806$	0	0
$807$	74.6252	2.62693
$808$	−18.5693	−0.653267
$809$	−3.96336	−0.139344	−0.0696721	−	0.997570i	$-0.522195\pi$
$809$	−3.96336	−0.139344	−0.0696721	+	0.997570i	$0.522195\pi$
$810$	−9.69701	−0.340718
$811$	−1.09871	−0.0385811	−0.0192905	−	0.999814i	$-0.506141\pi$
$811$	−1.09871	−0.0385811	−0.0192905	+	0.999814i	$0.506141\pi$
$812$	0	0
$813$	−52.5240	−1.84210
$814$	5.72928	0.200811
$815$	−1.36611	−0.0478526
$816$	−1.64452	−0.0575697
$817$	−42.6339	−1.49157
$818$	12.9171	0.451637
$819$	0	0
$820$	1.76156	0.0615162
$821$	−17.1108	−0.597170	−0.298585	−	0.954383i	$-0.596515\pi$
$821$	−17.1108	−0.597170	−0.298585	+	0.954383i	$0.596515\pi$
$822$	5.08287	0.177286
$823$	11.3738	0.396466	0.198233	−	0.980155i	$-0.436480\pi$
$823$	11.3738	0.396466	0.198233	+	0.980155i	$0.436480\pi$
$824$	−16.9248	−0.589604
$825$	−14.2707	−0.496843
$826$	0	0
$827$	13.6768	0.475589	0.237794	−	0.971316i	$-0.423576\pi$
$827$	13.6768	0.475589	0.237794	+	0.971316i	$0.423576\pi$
$828$	21.5231	0.747980
$829$	30.9002	1.07321	0.536605	−	0.843834i	$-0.319707\pi$
$829$	30.9002	1.07321	0.536605	+	0.843834i	$0.319707\pi$
$830$	−0.683053	−0.0237091
$831$	−34.8767	−1.20986
$832$	0	0
$833$	0	0
$834$	51.0375	1.76728
$835$	−0.953771	−0.0330066
$836$	−20.7755	−0.718536
$837$	1.47500	0.0509834
$838$	−2.72302	−0.0940653
$839$	−6.41233	−0.221378	−0.110689	−	0.993855i	$-0.535306\pi$
$839$	−6.41233	−0.221378	−0.110689	+	0.993855i	$0.535306\pi$
$840$	0	0
$841$	−14.8507	−0.512093
$842$	−4.84443	−0.166950
$843$	44.7668	1.54185
$844$	17.4340	0.600102
$845$	−22.9002	−0.787792
$846$	−7.79383	−0.267957
$847$	0	0
$848$	−1.96772	−0.0675719
$849$	34.5693	1.18642
$850$	1.18785	0.0407428
$851$	11.0462	0.378660
$852$	7.43835	0.254834
$853$	28.0881	0.961718	0.480859	−	0.876798i	$-0.340325\pi$
$853$	28.0881	0.961718	0.480859	+	0.876798i	$0.340325\pi$
$854$	0	0
$855$	−49.7851	−1.70261
$856$	−9.94315	−0.339850
$857$	−4.91713	−0.167966	−0.0839829	−	0.996467i	$-0.526764\pi$
$857$	−4.91713	−0.167966	−0.0839829	+	0.996467i	$0.526764\pi$
$858$	0	0
$859$	−56.9205	−1.94210	−0.971050	−	0.238875i	$-0.923222\pi$
$859$	−56.9205	−1.94210	−0.971050	+	0.238875i	$0.923222\pi$
$860$	10.3555	0.353119
$861$	0	0
$862$	−3.01832	−0.102804
$863$	51.1020	1.73953	0.869767	−	0.493463i	$-0.164269\pi$
$863$	51.1020	1.73953	0.869767	+	0.493463i	$0.164269\pi$
$864$	2.35548	0.0801351
$865$	−6.03560	−0.205217
$866$	−0.803417	−0.0273012
$867$	−43.6156	−1.48126
$868$	0	0
$869$	14.6185	0.495899
$870$	17.4017	0.589973
$871$	0	0
$872$	−0.800090	−0.0270945
$873$	27.6801	0.936830
$874$	−40.0558	−1.35491
$875$	0	0
$876$	40.0558	1.35336
$877$	−38.5327	−1.30116	−0.650578	−	0.759439i	$-0.725473\pi$
$877$	−38.5327	−1.30116	−0.650578	+	0.759439i	$0.725473\pi$
$878$	−24.0925	−0.813081
$879$	−60.9692	−2.05644
$880$	5.04623	0.170108
$881$	34.1204	1.14954	0.574772	−	0.818314i	$-0.305091\pi$
$881$	34.1204	1.14954	0.574772	+	0.818314i	$0.305091\pi$
$882$	0	0
$883$	−43.1545	−1.45227	−0.726133	−	0.687555i	$-0.758684\pi$
$883$	−43.1545	−1.45227	−0.726133	+	0.687555i	$0.758684\pi$
$884$	0	0
$885$	−12.2986	−0.413414
$886$	10.9615	0.368258
$887$	13.2890	0.446202	0.223101	−	0.974795i	$-0.428382\pi$
$887$	13.2890	0.446202	0.223101	+	0.974795i	$0.428382\pi$
$888$	5.25240	0.176259
$889$	0	0
$890$	−15.6724	−0.525341
$891$	−15.7693	−0.528290
$892$	−15.1955	−0.508784
$893$	14.5048	0.485384
$894$	59.8978	2.00328
$895$	−0.206167	−0.00689142
$896$	0	0
$897$	0	0
$898$	38.8603	1.29678
$899$	2.35548	0.0785597
$900$	−7.39212	−0.246404
$901$	1.23219	0.0410500
$902$	2.86464	0.0953821
$903$	0	0
$904$	−15.6079	−0.519110
$905$	−37.8496	−1.25816
$906$	−4.93356	−0.163907
$907$	55.0173	1.82682	0.913409	−	0.407042i	$-0.133440\pi$
$907$	55.0173	1.82682	0.913409	+	0.407042i	$0.133440\pi$
$908$	0.355480	0.0117970
$909$	−72.3632	−2.40014
$910$	0	0
$911$	3.13203	0.103769	0.0518844	−	0.998653i	$-0.483477\pi$
$911$	3.13203	0.103769	0.0518844	+	0.998653i	$0.483477\pi$
$912$	−19.0462	−0.630684
$913$	−1.11078	−0.0367614
$914$	−10.9538	−0.362319
$915$	2.35548	0.0778698
$916$	4.77551	0.157787
$917$	0	0
$918$	−1.47500	−0.0486822
$919$	−0.795721	−0.0262484	−0.0131242	−	0.999914i	$-0.504178\pi$
$919$	−0.795721	−0.0262484	−0.0131242	+	0.999914i	$0.504178\pi$
$920$	9.72928	0.320765
$921$	−17.6560	−0.581785
$922$	−4.87234	−0.160462
$923$	0	0
$924$	0	0
$925$	−3.79383	−0.124740
$926$	−19.4586	−0.639448
$927$	−65.9546	−2.16623
$928$	3.76156	0.123479
$929$	32.5048	1.06645	0.533224	−	0.845974i	$-0.320980\pi$
$929$	32.5048	1.06645	0.533224	+	0.845974i	$0.320980\pi$
$930$	2.89692	0.0949936
$931$	0	0
$932$	−23.0096	−0.753704
$933$	69.8496	2.28677
$934$	34.3511	1.12400
$935$	−3.15994	−0.103341
$936$	0	0
$937$	7.51542	0.245518	0.122759	−	0.992437i	$-0.460826\pi$
$937$	7.51542	0.245518	0.122759	+	0.992437i	$0.460826\pi$
$938$	0	0
$939$	8.16952	0.266602
$940$	−3.52311	−0.114911
$941$	8.41566	0.274343	0.137171	−	0.990547i	$-0.456199\pi$
$941$	8.41566	0.274343	0.137171	+	0.990547i	$0.456199\pi$
$942$	21.0096	0.684529
$943$	5.52311	0.179857
$944$	−2.65847	−0.0865259
$945$	0	0
$946$	16.8401	0.547518
$947$	−33.6801	−1.09446	−0.547228	−	0.836983i	$-0.684317\pi$
$947$	−33.6801	−1.09446	−0.547228	+	0.836983i	$0.684317\pi$
$948$	13.4017	0.435267
$949$	0	0
$950$	13.7572	0.446342
$951$	18.9084	0.613147
$952$	0	0
$953$	46.8592	1.51792	0.758960	−	0.651138i	$-0.225708\pi$
$953$	46.8592	1.51792	0.758960	+	0.651138i	$0.225708\pi$
$954$	−7.66806	−0.248263
$955$	20.0356	0.648337
$956$	−19.8217	−0.641081
$957$	28.2986	0.914765
$958$	26.4681	0.855147
$959$	0	0
$960$	4.62620	0.149310
$961$	−30.6079	−0.987351
$962$	0	0
$963$	−38.7476	−1.24862
$964$	16.5048	0.531584
$965$	44.4190	1.42990
$966$	0	0
$967$	47.2514	1.51950	0.759751	−	0.650215i	$-0.225321\pi$
$967$	47.2514	1.51950	0.759751	+	0.650215i	$0.225321\pi$
$968$	−2.79383	−0.0897972
$969$	11.9267	0.383141
$970$	12.5125	0.401752
$971$	19.5308	0.626774	0.313387	−	0.949626i	$-0.398536\pi$
$971$	19.5308	0.626774	0.313387	+	0.949626i	$0.398536\pi$
$972$	−21.5231	−0.690354
$973$	0	0
$974$	−30.0925	−0.964225
$975$	0	0
$976$	0.509161	0.0162978
$977$	−26.3632	−0.843433	−0.421716	−	0.906728i	$-0.638572\pi$
$977$	−26.3632	−0.843433	−0.421716	+	0.906728i	$0.638572\pi$
$978$	−2.03664	−0.0651247
$979$	−25.4865	−0.814551
$980$	0	0
$981$	−3.11788	−0.0995464
$982$	21.5800	0.688644
$983$	8.21386	0.261982	0.130991	−	0.991384i	$-0.458184\pi$
$983$	8.21386	0.261982	0.130991	+	0.991384i	$0.458184\pi$
$984$	2.62620	0.0837201
$985$	−2.68305	−0.0854892
$986$	−2.35548	−0.0750138
$987$	0	0
$988$	0	0
$989$	32.4681	1.03243
$990$	19.6647	0.624986
$991$	−41.3728	−1.31425	−0.657125	−	0.753782i	$-0.728227\pi$
$991$	−41.3728	−1.31425	−0.657125	+	0.753782i	$0.728227\pi$
$992$	0.626198	0.0198818
$993$	−56.2899	−1.78631
$994$	0	0
$995$	33.1878	1.05213
$996$	−1.01832	−0.0322668
$997$	−30.8680	−0.977598	−0.488799	−	0.872396i	$-0.662565\pi$
$997$	−30.8680	−0.977598	−0.488799	+	0.872396i	$0.662565\pi$
$998$	−17.3694	−0.549820
$999$	4.71096	0.149048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4018.2.a.bg.1.3		3
7.6	odd	2		574.2.a.l.1.1	✓	3
21.20	even	2		5166.2.a.bt.1.3		3
28.27	even	2		4592.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
574.2.a.l.1.1	✓	3	7.6	odd	2
4018.2.a.bg.1.3		3	1.1	even	1	trivial
4592.2.a.s.1.3		3	28.27	even	2
5166.2.a.bt.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.62620 q^{3} +1.00000 q^{4} +1.76156 q^{5} +2.62620 q^{6} +1.00000 q^{8} +3.89692 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.62620 q^{3} +1.00000 q^{4} +1.76156 q^{5} +2.62620 q^{6} +1.00000 q^{8} +3.89692 q^{9} +1.76156 q^{10} +2.86464 q^{11} +2.62620 q^{12} +4.62620 q^{15} +1.00000 q^{16} -0.626198 q^{17} +3.89692 q^{18} -7.25240 q^{19} +1.76156 q^{20} +2.86464 q^{22} +5.52311 q^{23} +2.62620 q^{24} -1.89692 q^{25} +2.35548 q^{27} +3.76156 q^{29} +4.62620 q^{30} +0.626198 q^{31} +1.00000 q^{32} +7.52311 q^{33} -0.626198 q^{34} +3.89692 q^{36} +2.00000 q^{37} -7.25240 q^{38} +1.76156 q^{40} +1.00000 q^{41} +5.87859 q^{43} +2.86464 q^{44} +6.86464 q^{45} +5.52311 q^{46} -2.00000 q^{47} +2.62620 q^{48} -1.89692 q^{50} -1.64452 q^{51} -1.96772 q^{53} +2.35548 q^{54} +5.04623 q^{55} -19.0462 q^{57} +3.76156 q^{58} -2.65847 q^{59} +4.62620 q^{60} +0.509161 q^{61} +0.626198 q^{62} +1.00000 q^{64} +7.52311 q^{66} -3.64015 q^{67} -0.626198 q^{68} +14.5048 q^{69} +2.83237 q^{71} +3.89692 q^{72} +15.2524 q^{73} +2.00000 q^{74} -4.98168 q^{75} -7.25240 q^{76} +5.10308 q^{79} +1.76156 q^{80} -5.50479 q^{81} +1.00000 q^{82} -0.387755 q^{83} -1.10308 q^{85} +5.87859 q^{86} +9.87859 q^{87} +2.86464 q^{88} -8.89692 q^{89} +6.86464 q^{90} +5.52311 q^{92} +1.64452 q^{93} -2.00000 q^{94} -12.7755 q^{95} +2.62620 q^{96} +7.10308 q^{97} +11.1633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 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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