LMFDB - Embedded newform 1338.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1338,2,Mod(1,1338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1338, 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("1338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1338 = 2 \cdot 3 \cdot 223$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1338.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:	$10.6839837904$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.23607 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -4.85410 q^{11} -1.00000 q^{12} +1.85410 q^{13} +1.23607 q^{15} +1.00000 q^{16} -2.47214 q^{17} -1.00000 q^{18} +1.38197 q^{19} -1.23607 q^{20} +4.85410 q^{22} +2.00000 q^{23} +1.00000 q^{24} -3.47214 q^{25} -1.85410 q^{26} -1.00000 q^{27} +4.09017 q^{29} -1.23607 q^{30} +6.00000 q^{31} -1.00000 q^{32} +4.85410 q^{33} +2.47214 q^{34} +1.00000 q^{36} -2.47214 q^{37} -1.38197 q^{38} -1.85410 q^{39} +1.23607 q^{40} -10.0000 q^{41} +6.38197 q^{43} -4.85410 q^{44} -1.23607 q^{45} -2.00000 q^{46} +5.61803 q^{47} -1.00000 q^{48} -7.00000 q^{49} +3.47214 q^{50} +2.47214 q^{51} +1.85410 q^{52} +3.09017 q^{53} +1.00000 q^{54} +6.00000 q^{55} -1.38197 q^{57} -4.09017 q^{58} +0.145898 q^{59} +1.23607 q^{60} +13.7984 q^{61} -6.00000 q^{62} +1.00000 q^{64} -2.29180 q^{65} -4.85410 q^{66} -2.94427 q^{67} -2.47214 q^{68} -2.00000 q^{69} +15.4164 q^{71} -1.00000 q^{72} +0.909830 q^{73} +2.47214 q^{74} +3.47214 q^{75} +1.38197 q^{76} +1.85410 q^{78} -2.38197 q^{79} -1.23607 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} +3.05573 q^{85} -6.38197 q^{86} -4.09017 q^{87} +4.85410 q^{88} -7.23607 q^{89} +1.23607 q^{90} +2.00000 q^{92} -6.00000 q^{93} -5.61803 q^{94} -1.70820 q^{95} +1.00000 q^{96} +7.52786 q^{97} +7.00000 q^{98} -4.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 3 q^{11} - 2 q^{12} - 3 q^{13} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{20} + 3 q^{22} + 4 q^{23}+ \cdots - 3 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 3 * q^11 - 2 * q^12 - 3 * q^13 - 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^20 + 3 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^25 + 3 * q^26 - 2 * q^27 - 3 * q^29 + 2 * q^30 + 12 * q^31 - 2 * q^32 + 3 * q^33 - 4 * q^34 + 2 * q^36 + 4 * q^37 - 5 * q^38 + 3 * q^39 - 2 * q^40 - 20 * q^41 + 15 * q^43 - 3 * q^44 + 2 * q^45 - 4 * q^46 + 9 * q^47 - 2 * q^48 - 14 * q^49 - 2 * q^50 - 4 * q^51 - 3 * q^52 - 5 * q^53 + 2 * q^54 + 12 * q^55 - 5 * q^57 + 3 * q^58 + 7 * q^59 - 2 * q^60 + 3 * q^61 - 12 * q^62 + 2 * q^64 - 18 * q^65 - 3 * q^66 + 12 * q^67 + 4 * q^68 - 4 * q^69 + 4 * q^71 - 2 * q^72 + 13 * q^73 - 4 * q^74 - 2 * q^75 + 5 * q^76 - 3 * q^78 - 7 * q^79 + 2 * q^80 + 2 * q^81 + 20 * q^82 + 24 * q^83 + 24 * q^85 - 15 * q^86 + 3 * q^87 + 3 * q^88 - 10 * q^89 - 2 * q^90 + 4 * q^92 - 12 * q^93 - 9 * q^94 + 10 * q^95 + 2 * q^96 + 24 * q^97 + 14 * q^98 - 3 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.23607	0.390879
$11$	−4.85410	−1.46357	−0.731783	−	0.681537i	$-0.761312\pi$
$11$	−4.85410	−1.46357	−0.731783	+	0.681537i	$0.761312\pi$
$12$	−1.00000	−0.288675
$13$	1.85410	0.514235	0.257118	−	0.966380i	$-0.417227\pi$
$13$	1.85410	0.514235	0.257118	+	0.966380i	$0.417227\pi$
$14$	0	0
$15$	1.23607	0.319151
$16$	1.00000	0.250000
$17$	−2.47214	−0.599581	−0.299791	−	0.954005i	$-0.596917\pi$
$17$	−2.47214	−0.599581	−0.299791	+	0.954005i	$0.596917\pi$
$18$	−1.00000	−0.235702
$19$	1.38197	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$19$	1.38197	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$20$	−1.23607	−0.276393
$21$	0	0
$22$	4.85410	1.03490
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	1.00000	0.204124
$25$	−3.47214	−0.694427
$26$	−1.85410	−0.363619
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.09017	0.759525	0.379763	−	0.925084i	$-0.376006\pi$
$29$	4.09017	0.759525	0.379763	+	0.925084i	$0.376006\pi$
$30$	−1.23607	−0.225674
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	4.85410	0.844991
$34$	2.47214	0.423968
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.47214	−0.406417	−0.203208	−	0.979136i	$-0.565137\pi$
$37$	−2.47214	−0.406417	−0.203208	+	0.979136i	$0.565137\pi$
$38$	−1.38197	−0.224184
$39$	−1.85410	−0.296894
$40$	1.23607	0.195440
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	6.38197	0.973241	0.486620	−	0.873614i	$-0.338230\pi$
$43$	6.38197	0.973241	0.486620	+	0.873614i	$0.338230\pi$
$44$	−4.85410	−0.731783
$45$	−1.23607	−0.184262
$46$	−2.00000	−0.294884
$47$	5.61803	0.819474	0.409737	−	0.912204i	$-0.365620\pi$
$47$	5.61803	0.819474	0.409737	+	0.912204i	$0.365620\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	3.47214	0.491034
$51$	2.47214	0.346168
$52$	1.85410	0.257118
$53$	3.09017	0.424467	0.212234	−	0.977219i	$-0.431926\pi$
$53$	3.09017	0.424467	0.212234	+	0.977219i	$0.431926\pi$
$54$	1.00000	0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	−1.38197	−0.183046
$58$	−4.09017	−0.537066
$59$	0.145898	0.0189943	0.00949715	−	0.999955i	$-0.496977\pi$
$59$	0.145898	0.0189943	0.00949715	+	0.999955i	$0.496977\pi$
$60$	1.23607	0.159576
$61$	13.7984	1.76670	0.883350	−	0.468713i	$-0.155282\pi$
$61$	13.7984	1.76670	0.883350	+	0.468713i	$0.155282\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.29180	−0.284262
$66$	−4.85410	−0.597499
$67$	−2.94427	−0.359700	−0.179850	−	0.983694i	$-0.557561\pi$
$67$	−2.94427	−0.359700	−0.179850	+	0.983694i	$0.557561\pi$
$68$	−2.47214	−0.299791
$69$	−2.00000	−0.240772
$70$	0	0
$71$	15.4164	1.82959	0.914796	−	0.403917i	$-0.132352\pi$
$71$	15.4164	1.82959	0.914796	+	0.403917i	$0.132352\pi$
$72$	−1.00000	−0.117851
$73$	0.909830	0.106488	0.0532438	−	0.998582i	$-0.483044\pi$
$73$	0.909830	0.106488	0.0532438	+	0.998582i	$0.483044\pi$
$74$	2.47214	0.287380
$75$	3.47214	0.400928
$76$	1.38197	0.158522
$77$	0	0
$78$	1.85410	0.209936
$79$	−2.38197	−0.267992	−0.133996	−	0.990982i	$-0.542781\pi$
$79$	−2.38197	−0.267992	−0.133996	+	0.990982i	$0.542781\pi$
$80$	−1.23607	−0.138197
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	3.05573	0.331440
$86$	−6.38197	−0.688185
$87$	−4.09017	−0.438512
$88$	4.85410	0.517449
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	1.23607	0.130293
$91$	0	0
$92$	2.00000	0.208514
$93$	−6.00000	−0.622171
$94$	−5.61803	−0.579456
$95$	−1.70820	−0.175258
$96$	1.00000	0.102062
$97$	7.52786	0.764339	0.382169	−	0.924092i	$-0.375177\pi$
$97$	7.52786	0.764339	0.382169	+	0.924092i	$0.375177\pi$
$98$	7.00000	0.707107
$99$	−4.85410	−0.487856
$100$	−3.47214	−0.347214
$101$	−1.85410	−0.184490	−0.0922450	−	0.995736i	$-0.529404\pi$
$101$	−1.85410	−0.184490	−0.0922450	+	0.995736i	$0.529404\pi$
$102$	−2.47214	−0.244778
$103$	3.56231	0.351004	0.175502	−	0.984479i	$-0.443845\pi$
$103$	3.56231	0.351004	0.175502	+	0.984479i	$0.443845\pi$
$104$	−1.85410	−0.181810
$105$	0	0
$106$	−3.09017	−0.300144
$107$	9.38197	0.906989	0.453494	−	0.891259i	$-0.350177\pi$
$107$	9.38197	0.906989	0.453494	+	0.891259i	$0.350177\pi$
$108$	−1.00000	−0.0962250
$109$	12.7639	1.22256	0.611281	−	0.791413i	$-0.290654\pi$
$109$	12.7639	1.22256	0.611281	+	0.791413i	$0.290654\pi$
$110$	−6.00000	−0.572078
$111$	2.47214	0.234645
$112$	0	0
$113$	9.85410	0.926996	0.463498	−	0.886098i	$-0.346594\pi$
$113$	9.85410	0.926996	0.463498	+	0.886098i	$0.346594\pi$
$114$	1.38197	0.129433
$115$	−2.47214	−0.230528
$116$	4.09017	0.379763
$117$	1.85410	0.171412
$118$	−0.145898	−0.0134310
$119$	0	0
$120$	−1.23607	−0.112837
$121$	12.5623	1.14203
$122$	−13.7984	−1.24925
$123$	10.0000	0.901670
$124$	6.00000	0.538816
$125$	10.4721	0.936656
$126$	0	0
$127$	17.7082	1.57135	0.785675	−	0.618640i	$-0.212316\pi$
$127$	17.7082	1.57135	0.785675	+	0.618640i	$0.212316\pi$
$128$	−1.00000	−0.0883883
$129$	−6.38197	−0.561901
$130$	2.29180	0.201004
$131$	−3.70820	−0.323987	−0.161994	−	0.986792i	$-0.551792\pi$
$131$	−3.70820	−0.323987	−0.161994	+	0.986792i	$0.551792\pi$
$132$	4.85410	0.422495
$133$	0	0
$134$	2.94427	0.254346
$135$	1.23607	0.106384
$136$	2.47214	0.211984
$137$	0.673762	0.0575634	0.0287817	−	0.999586i	$-0.490837\pi$
$137$	0.673762	0.0575634	0.0287817	+	0.999586i	$0.490837\pi$
$138$	2.00000	0.170251
$139$	0.0901699	0.00764811	0.00382406	−	0.999993i	$-0.498783\pi$
$139$	0.0901699	0.00764811	0.00382406	+	0.999993i	$0.498783\pi$
$140$	0	0
$141$	−5.61803	−0.473124
$142$	−15.4164	−1.29372
$143$	−9.00000	−0.752618
$144$	1.00000	0.0833333
$145$	−5.05573	−0.419855
$146$	−0.909830	−0.0752981
$147$	7.00000	0.577350
$148$	−2.47214	−0.203208
$149$	−8.47214	−0.694064	−0.347032	−	0.937853i	$-0.612811\pi$
$149$	−8.47214	−0.694064	−0.347032	+	0.937853i	$0.612811\pi$
$150$	−3.47214	−0.283499
$151$	13.3820	1.08901	0.544504	−	0.838758i	$-0.316718\pi$
$151$	13.3820	1.08901	0.544504	+	0.838758i	$0.316718\pi$
$152$	−1.38197	−0.112092
$153$	−2.47214	−0.199860
$154$	0	0
$155$	−7.41641	−0.595700
$156$	−1.85410	−0.148447
$157$	5.61803	0.448368	0.224184	−	0.974547i	$-0.428028\pi$
$157$	5.61803	0.448368	0.224184	+	0.974547i	$0.428028\pi$
$158$	2.38197	0.189499
$159$	−3.09017	−0.245066
$160$	1.23607	0.0977198
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−10.0000	−0.780869
$165$	−6.00000	−0.467099
$166$	−12.0000	−0.931381
$167$	−21.7082	−1.67983	−0.839916	−	0.542717i	$-0.817396\pi$
$167$	−21.7082	−1.67983	−0.839916	+	0.542717i	$0.817396\pi$
$168$	0	0
$169$	−9.56231	−0.735562
$170$	−3.05573	−0.234364
$171$	1.38197	0.105682
$172$	6.38197	0.486620
$173$	15.8885	1.20798	0.603992	−	0.796991i	$-0.293576\pi$
$173$	15.8885	1.20798	0.603992	+	0.796991i	$0.293576\pi$
$174$	4.09017	0.310075
$175$	0	0
$176$	−4.85410	−0.365892
$177$	−0.145898	−0.0109664
$178$	7.23607	0.542366
$179$	15.1246	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$179$	15.1246	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$180$	−1.23607	−0.0921311
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	−13.7984	−1.02001
$184$	−2.00000	−0.147442
$185$	3.05573	0.224662
$186$	6.00000	0.439941
$187$	12.0000	0.877527
$188$	5.61803	0.409737
$189$	0	0
$190$	1.70820	0.123926
$191$	−16.1803	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$191$	−16.1803	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$192$	−1.00000	−0.0721688
$193$	−22.4721	−1.61758	−0.808790	−	0.588098i	$-0.799877\pi$
$193$	−22.4721	−1.61758	−0.808790	+	0.588098i	$0.799877\pi$
$194$	−7.52786	−0.540469
$195$	2.29180	0.164119
$196$	−7.00000	−0.500000
$197$	3.32624	0.236985	0.118492	−	0.992955i	$-0.462194\pi$
$197$	3.32624	0.236985	0.118492	+	0.992955i	$0.462194\pi$
$198$	4.85410	0.344966
$199$	4.76393	0.337706	0.168853	−	0.985641i	$-0.445994\pi$
$199$	4.76393	0.337706	0.168853	+	0.985641i	$0.445994\pi$
$200$	3.47214	0.245517
$201$	2.94427	0.207673
$202$	1.85410	0.130454
$203$	0	0
$204$	2.47214	0.173084
$205$	12.3607	0.863307
$206$	−3.56231	−0.248198
$207$	2.00000	0.139010
$208$	1.85410	0.128559
$209$	−6.70820	−0.464016
$210$	0	0
$211$	−6.38197	−0.439353	−0.219676	−	0.975573i	$-0.570500\pi$
$211$	−6.38197	−0.439353	−0.219676	+	0.975573i	$0.570500\pi$
$212$	3.09017	0.212234
$213$	−15.4164	−1.05631
$214$	−9.38197	−0.641338
$215$	−7.88854	−0.537994
$216$	1.00000	0.0680414
$217$	0	0
$218$	−12.7639	−0.864483
$219$	−0.909830	−0.0614806
$220$	6.00000	0.404520
$221$	−4.58359	−0.308326
$222$	−2.47214	−0.165919
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	−3.47214	−0.231476
$226$	−9.85410	−0.655485
$227$	10.4721	0.695060	0.347530	−	0.937669i	$-0.387020\pi$
$227$	10.4721	0.695060	0.347530	+	0.937669i	$0.387020\pi$
$228$	−1.38197	−0.0915229
$229$	−9.41641	−0.622254	−0.311127	−	0.950368i	$-0.600706\pi$
$229$	−9.41641	−0.622254	−0.311127	+	0.950368i	$0.600706\pi$
$230$	2.47214	0.163008
$231$	0	0
$232$	−4.09017	−0.268533
$233$	15.1459	0.992241	0.496120	−	0.868254i	$-0.334757\pi$
$233$	15.1459	0.992241	0.496120	+	0.868254i	$0.334757\pi$
$234$	−1.85410	−0.121206
$235$	−6.94427	−0.452994
$236$	0.145898	0.00949715
$237$	2.38197	0.154725
$238$	0	0
$239$	7.56231	0.489165	0.244582	−	0.969628i	$-0.421349\pi$
$239$	7.56231	0.489165	0.244582	+	0.969628i	$0.421349\pi$
$240$	1.23607	0.0797878
$241$	−12.2705	−0.790413	−0.395207	−	0.918592i	$-0.629327\pi$
$241$	−12.2705	−0.790413	−0.395207	+	0.918592i	$0.629327\pi$
$242$	−12.5623	−0.807536
$243$	−1.00000	−0.0641500
$244$	13.7984	0.883350
$245$	8.65248	0.552786
$246$	−10.0000	−0.637577
$247$	2.56231	0.163036
$248$	−6.00000	−0.381000
$249$	−12.0000	−0.760469
$250$	−10.4721	−0.662316
$251$	−7.23607	−0.456737	−0.228368	−	0.973575i	$-0.573339\pi$
$251$	−7.23607	−0.456737	−0.228368	+	0.973575i	$0.573339\pi$
$252$	0	0
$253$	−9.70820	−0.610350
$254$	−17.7082	−1.11111
$255$	−3.05573	−0.191357
$256$	1.00000	0.0625000
$257$	18.3607	1.14531	0.572654	−	0.819797i	$-0.305914\pi$
$257$	18.3607	1.14531	0.572654	+	0.819797i	$0.305914\pi$
$258$	6.38197	0.397324
$259$	0	0
$260$	−2.29180	−0.142131
$261$	4.09017	0.253175
$262$	3.70820	0.229094
$263$	−21.2361	−1.30947	−0.654736	−	0.755858i	$-0.727220\pi$
$263$	−21.2361	−1.30947	−0.654736	+	0.755858i	$0.727220\pi$
$264$	−4.85410	−0.298749
$265$	−3.81966	−0.234640
$266$	0	0
$267$	7.23607	0.442840
$268$	−2.94427	−0.179850
$269$	−9.41641	−0.574129	−0.287064	−	0.957911i	$-0.592679\pi$
$269$	−9.41641	−0.574129	−0.287064	+	0.957911i	$0.592679\pi$
$270$	−1.23607	−0.0752247
$271$	−4.67376	−0.283911	−0.141955	−	0.989873i	$-0.545339\pi$
$271$	−4.67376	−0.283911	−0.141955	+	0.989873i	$0.545339\pi$
$272$	−2.47214	−0.149895
$273$	0	0
$274$	−0.673762	−0.0407035
$275$	16.8541	1.01634
$276$	−2.00000	−0.120386
$277$	−19.5279	−1.17332	−0.586658	−	0.809835i	$-0.699557\pi$
$277$	−19.5279	−1.17332	−0.586658	+	0.809835i	$0.699557\pi$
$278$	−0.0901699	−0.00540803
$279$	6.00000	0.359211
$280$	0	0
$281$	−23.7082	−1.41431	−0.707156	−	0.707057i	$-0.750022\pi$
$281$	−23.7082	−1.41431	−0.707156	+	0.707057i	$0.750022\pi$
$282$	5.61803	0.334549
$283$	8.79837	0.523009	0.261505	−	0.965202i	$-0.415781\pi$
$283$	8.79837	0.523009	0.261505	+	0.965202i	$0.415781\pi$
$284$	15.4164	0.914796
$285$	1.70820	0.101185
$286$	9.00000	0.532181
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−10.8885	−0.640503
$290$	5.05573	0.296883
$291$	−7.52786	−0.441291
$292$	0.909830	0.0532438
$293$	25.5967	1.49538	0.747689	−	0.664049i	$-0.231163\pi$
$293$	25.5967	1.49538	0.747689	+	0.664049i	$0.231163\pi$
$294$	−7.00000	−0.408248
$295$	−0.180340	−0.0104998
$296$	2.47214	0.143690
$297$	4.85410	0.281664
$298$	8.47214	0.490778
$299$	3.70820	0.214451
$300$	3.47214	0.200464
$301$	0	0
$302$	−13.3820	−0.770046
$303$	1.85410	0.106515
$304$	1.38197	0.0792612
$305$	−17.0557	−0.976608
$306$	2.47214	0.141323
$307$	6.65248	0.379677	0.189838	−	0.981815i	$-0.439204\pi$
$307$	6.65248	0.379677	0.189838	+	0.981815i	$0.439204\pi$
$308$	0	0
$309$	−3.56231	−0.202653
$310$	7.41641	0.421224
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	1.85410	0.104968
$313$	−10.6525	−0.602114	−0.301057	−	0.953606i	$-0.597339\pi$
$313$	−10.6525	−0.602114	−0.301057	+	0.953606i	$0.597339\pi$
$314$	−5.61803	−0.317044
$315$	0	0
$316$	−2.38197	−0.133996
$317$	−9.56231	−0.537073	−0.268536	−	0.963270i	$-0.586540\pi$
$317$	−9.56231	−0.537073	−0.268536	+	0.963270i	$0.586540\pi$
$318$	3.09017	0.173288
$319$	−19.8541	−1.11162
$320$	−1.23607	−0.0690983
$321$	−9.38197	−0.523650
$322$	0	0
$323$	−3.41641	−0.190094
$324$	1.00000	0.0555556
$325$	−6.43769	−0.357099
$326$	−14.0000	−0.775388
$327$	−12.7639	−0.705847
$328$	10.0000	0.552158
$329$	0	0
$330$	6.00000	0.330289
$331$	−5.41641	−0.297713	−0.148856	−	0.988859i	$-0.547559\pi$
$331$	−5.41641	−0.297713	−0.148856	+	0.988859i	$0.547559\pi$
$332$	12.0000	0.658586
$333$	−2.47214	−0.135472
$334$	21.7082	1.18782
$335$	3.63932	0.198837
$336$	0	0
$337$	32.0689	1.74690	0.873452	−	0.486911i	$-0.161876\pi$
$337$	32.0689	1.74690	0.873452	+	0.486911i	$0.161876\pi$
$338$	9.56231	0.520121
$339$	−9.85410	−0.535201
$340$	3.05573	0.165720
$341$	−29.1246	−1.57719
$342$	−1.38197	−0.0747282
$343$	0	0
$344$	−6.38197	−0.344093
$345$	2.47214	0.133095
$346$	−15.8885	−0.854173
$347$	−15.4164	−0.827596	−0.413798	−	0.910369i	$-0.635798\pi$
$347$	−15.4164	−0.827596	−0.413798	+	0.910369i	$0.635798\pi$
$348$	−4.09017	−0.219256
$349$	8.94427	0.478776	0.239388	−	0.970924i	$-0.423053\pi$
$349$	8.94427	0.478776	0.239388	+	0.970924i	$0.423053\pi$
$350$	0	0
$351$	−1.85410	−0.0989646
$352$	4.85410	0.258725
$353$	−23.7082	−1.26186	−0.630930	−	0.775840i	$-0.717327\pi$
$353$	−23.7082	−1.26186	−0.630930	+	0.775840i	$0.717327\pi$
$354$	0.145898	0.00775439
$355$	−19.0557	−1.01137
$356$	−7.23607	−0.383511
$357$	0	0
$358$	−15.1246	−0.799361
$359$	9.74265	0.514197	0.257099	−	0.966385i	$-0.417233\pi$
$359$	9.74265	0.514197	0.257099	+	0.966385i	$0.417233\pi$
$360$	1.23607	0.0651465
$361$	−17.0902	−0.899483
$362$	−8.00000	−0.420471
$363$	−12.5623	−0.659350
$364$	0	0
$365$	−1.12461	−0.0588649
$366$	13.7984	0.721253
$367$	−25.1246	−1.31149	−0.655747	−	0.754981i	$-0.727646\pi$
$367$	−25.1246	−1.31149	−0.655747	+	0.754981i	$0.727646\pi$
$368$	2.00000	0.104257
$369$	−10.0000	−0.520579
$370$	−3.05573	−0.158860
$371$	0	0
$372$	−6.00000	−0.311086
$373$	−26.3607	−1.36490	−0.682452	−	0.730930i	$-0.739086\pi$
$373$	−26.3607	−1.36490	−0.682452	+	0.730930i	$0.739086\pi$
$374$	−12.0000	−0.620505
$375$	−10.4721	−0.540779
$376$	−5.61803	−0.289728
$377$	7.58359	0.390575
$378$	0	0
$379$	−18.0344	−0.926367	−0.463184	−	0.886262i	$-0.653293\pi$
$379$	−18.0344	−0.926367	−0.463184	+	0.886262i	$0.653293\pi$
$380$	−1.70820	−0.0876290
$381$	−17.7082	−0.907219
$382$	16.1803	0.827858
$383$	13.5279	0.691242	0.345621	−	0.938374i	$-0.387668\pi$
$383$	13.5279	0.691242	0.345621	+	0.938374i	$0.387668\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.4721	1.14380
$387$	6.38197	0.324414
$388$	7.52786	0.382169
$389$	32.2705	1.63618	0.818090	−	0.575090i	$-0.195033\pi$
$389$	32.2705	1.63618	0.818090	+	0.575090i	$0.195033\pi$
$390$	−2.29180	−0.116050
$391$	−4.94427	−0.250043
$392$	7.00000	0.353553
$393$	3.70820	0.187054
$394$	−3.32624	−0.167573
$395$	2.94427	0.148142
$396$	−4.85410	−0.243928
$397$	−15.1459	−0.760151	−0.380075	−	0.924956i	$-0.624102\pi$
$397$	−15.1459	−0.760151	−0.380075	+	0.924956i	$0.624102\pi$
$398$	−4.76393	−0.238794
$399$	0	0
$400$	−3.47214	−0.173607
$401$	22.6525	1.13121	0.565605	−	0.824676i	$-0.308643\pi$
$401$	22.6525	1.13121	0.565605	+	0.824676i	$0.308643\pi$
$402$	−2.94427	−0.146847
$403$	11.1246	0.554156
$404$	−1.85410	−0.0922450
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	12.0000	0.594818
$408$	−2.47214	−0.122389
$409$	−24.6525	−1.21899	−0.609493	−	0.792791i	$-0.708627\pi$
$409$	−24.6525	−1.21899	−0.609493	+	0.792791i	$0.708627\pi$
$410$	−12.3607	−0.610450
$411$	−0.673762	−0.0332342
$412$	3.56231	0.175502
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	−14.8328	−0.728114
$416$	−1.85410	−0.0909048
$417$	−0.0901699	−0.00441564
$418$	6.70820	0.328109
$419$	7.88854	0.385381	0.192690	−	0.981260i	$-0.438279\pi$
$419$	7.88854	0.385381	0.192690	+	0.981260i	$0.438279\pi$
$420$	0	0
$421$	−6.79837	−0.331332	−0.165666	−	0.986182i	$-0.552977\pi$
$421$	−6.79837	−0.331332	−0.165666	+	0.986182i	$0.552977\pi$
$422$	6.38197	0.310669
$423$	5.61803	0.273158
$424$	−3.09017	−0.150072
$425$	8.58359	0.416365
$426$	15.4164	0.746927
$427$	0	0
$428$	9.38197	0.453494
$429$	9.00000	0.434524
$430$	7.88854	0.380419
$431$	−8.47214	−0.408088	−0.204044	−	0.978962i	$-0.565409\pi$
$431$	−8.47214	−0.408088	−0.204044	+	0.978962i	$0.565409\pi$
$432$	−1.00000	−0.0481125
$433$	6.79837	0.326709	0.163354	−	0.986567i	$-0.447769\pi$
$433$	6.79837	0.326709	0.163354	+	0.986567i	$0.447769\pi$
$434$	0	0
$435$	5.05573	0.242404
$436$	12.7639	0.611281
$437$	2.76393	0.132217
$438$	0.909830	0.0434734
$439$	26.4721	1.26345	0.631723	−	0.775194i	$-0.282348\pi$
$439$	26.4721	1.26345	0.631723	+	0.775194i	$0.282348\pi$
$440$	−6.00000	−0.286039
$441$	−7.00000	−0.333333
$442$	4.58359	0.218019
$443$	15.7082	0.746319	0.373160	−	0.927767i	$-0.378274\pi$
$443$	15.7082	0.746319	0.373160	+	0.927767i	$0.378274\pi$
$444$	2.47214	0.117322
$445$	8.94427	0.423999
$446$	−1.00000	−0.0473514
$447$	8.47214	0.400718
$448$	0	0
$449$	−17.4164	−0.821931	−0.410966	−	0.911651i	$-0.634808\pi$
$449$	−17.4164	−0.821931	−0.410966	+	0.911651i	$0.634808\pi$
$450$	3.47214	0.163678
$451$	48.5410	2.28571
$452$	9.85410	0.463498
$453$	−13.3820	−0.628740
$454$	−10.4721	−0.491482
$455$	0	0
$456$	1.38197	0.0647165
$457$	−15.2361	−0.712713	−0.356357	−	0.934350i	$-0.615981\pi$
$457$	−15.2361	−0.712713	−0.356357	+	0.934350i	$0.615981\pi$
$458$	9.41641	0.440000
$459$	2.47214	0.115389
$460$	−2.47214	−0.115264
$461$	−12.9098	−0.601271	−0.300635	−	0.953739i	$-0.597199\pi$
$461$	−12.9098	−0.601271	−0.300635	+	0.953739i	$0.597199\pi$
$462$	0	0
$463$	42.2492	1.96349	0.981744	−	0.190207i	$-0.0609160\pi$
$463$	42.2492	1.96349	0.981744	+	0.190207i	$0.0609160\pi$
$464$	4.09017	0.189881
$465$	7.41641	0.343928
$466$	−15.1459	−0.701620
$467$	34.9787	1.61862	0.809311	−	0.587380i	$-0.199841\pi$
$467$	34.9787	1.61862	0.809311	+	0.587380i	$0.199841\pi$
$468$	1.85410	0.0857059
$469$	0	0
$470$	6.94427	0.320315
$471$	−5.61803	−0.258865
$472$	−0.145898	−0.00671550
$473$	−30.9787	−1.42440
$474$	−2.38197	−0.109407
$475$	−4.79837	−0.220164
$476$	0	0
$477$	3.09017	0.141489
$478$	−7.56231	−0.345892
$479$	6.38197	0.291599	0.145800	−	0.989314i	$-0.453424\pi$
$479$	6.38197	0.291599	0.145800	+	0.989314i	$0.453424\pi$
$480$	−1.23607	−0.0564185
$481$	−4.58359	−0.208994
$482$	12.2705	0.558906
$483$	0	0
$484$	12.5623	0.571014
$485$	−9.30495	−0.422516
$486$	1.00000	0.0453609
$487$	11.8885	0.538721	0.269361	−	0.963039i	$-0.413188\pi$
$487$	11.8885	0.538721	0.269361	+	0.963039i	$0.413188\pi$
$488$	−13.7984	−0.624623
$489$	−14.0000	−0.633102
$490$	−8.65248	−0.390879
$491$	−29.6180	−1.33664	−0.668322	−	0.743872i	$-0.732987\pi$
$491$	−29.6180	−1.33664	−0.668322	+	0.743872i	$0.732987\pi$
$492$	10.0000	0.450835
$493$	−10.1115	−0.455397
$494$	−2.56231	−0.115284
$495$	6.00000	0.269680
$496$	6.00000	0.269408
$497$	0	0
$498$	12.0000	0.537733
$499$	30.9230	1.38430	0.692151	−	0.721752i	$-0.256663\pi$
$499$	30.9230	1.38430	0.692151	+	0.721752i	$0.256663\pi$
$500$	10.4721	0.468328
$501$	21.7082	0.969851
$502$	7.23607	0.322962
$503$	−4.76393	−0.212413	−0.106207	−	0.994344i	$-0.533870\pi$
$503$	−4.76393	−0.212413	−0.106207	+	0.994344i	$0.533870\pi$
$504$	0	0
$505$	2.29180	0.101984
$506$	9.70820	0.431582
$507$	9.56231	0.424677
$508$	17.7082	0.785675
$509$	7.90983	0.350597	0.175299	−	0.984515i	$-0.443911\pi$
$509$	7.90983	0.350597	0.175299	+	0.984515i	$0.443911\pi$
$510$	3.05573	0.135310
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−1.38197	−0.0610153
$514$	−18.3607	−0.809855
$515$	−4.40325	−0.194030
$516$	−6.38197	−0.280950
$517$	−27.2705	−1.19936
$518$	0	0
$519$	−15.8885	−0.697430
$520$	2.29180	0.100502
$521$	13.4164	0.587784	0.293892	−	0.955839i	$-0.405049\pi$
$521$	13.4164	0.587784	0.293892	+	0.955839i	$0.405049\pi$
$522$	−4.09017	−0.179022
$523$	11.3475	0.496193	0.248096	−	0.968735i	$-0.420195\pi$
$523$	11.3475	0.496193	0.248096	+	0.968735i	$0.420195\pi$
$524$	−3.70820	−0.161994
$525$	0	0
$526$	21.2361	0.925937
$527$	−14.8328	−0.646128
$528$	4.85410	0.211248
$529$	−19.0000	−0.826087
$530$	3.81966	0.165915
$531$	0.145898	0.00633144
$532$	0	0
$533$	−18.5410	−0.803101
$534$	−7.23607	−0.313135
$535$	−11.5967	−0.501371
$536$	2.94427	0.127173
$537$	−15.1246	−0.652675
$538$	9.41641	0.405970
$539$	33.9787	1.46357
$540$	1.23607	0.0531919
$541$	8.61803	0.370518	0.185259	−	0.982690i	$-0.440688\pi$
$541$	8.61803	0.370518	0.185259	+	0.982690i	$0.440688\pi$
$542$	4.67376	0.200755
$543$	−8.00000	−0.343313
$544$	2.47214	0.105992
$545$	−15.7771	−0.675816
$546$	0	0
$547$	1.09017	0.0466123	0.0233062	−	0.999728i	$-0.492581\pi$
$547$	1.09017	0.0466123	0.0233062	+	0.999728i	$0.492581\pi$
$548$	0.673762	0.0287817
$549$	13.7984	0.588900
$550$	−16.8541	−0.718661
$551$	5.65248	0.240804
$552$	2.00000	0.0851257
$553$	0	0
$554$	19.5279	0.829659
$555$	−3.05573	−0.129708
$556$	0.0901699	0.00382406
$557$	−24.0689	−1.01983	−0.509916	−	0.860224i	$-0.670323\pi$
$557$	−24.0689	−1.01983	−0.509916	+	0.860224i	$0.670323\pi$
$558$	−6.00000	−0.254000
$559$	11.8328	0.500475
$560$	0	0
$561$	−12.0000	−0.506640
$562$	23.7082	1.00007
$563$	34.7426	1.46423	0.732114	−	0.681182i	$-0.238534\pi$
$563$	34.7426	1.46423	0.732114	+	0.681182i	$0.238534\pi$
$564$	−5.61803	−0.236562
$565$	−12.1803	−0.512431
$566$	−8.79837	−0.369823
$567$	0	0
$568$	−15.4164	−0.646858
$569$	−0.618034	−0.0259093	−0.0129547	−	0.999916i	$-0.504124\pi$
$569$	−0.618034	−0.0259093	−0.0129547	+	0.999916i	$0.504124\pi$
$570$	−1.70820	−0.0715488
$571$	−0.875388	−0.0366339	−0.0183169	−	0.999832i	$-0.505831\pi$
$571$	−0.875388	−0.0366339	−0.0183169	+	0.999832i	$0.505831\pi$
$572$	−9.00000	−0.376309
$573$	16.1803	0.675943
$574$	0	0
$575$	−6.94427	−0.289596
$576$	1.00000	0.0416667
$577$	−13.4377	−0.559419	−0.279709	−	0.960085i	$-0.590238\pi$
$577$	−13.4377	−0.559419	−0.279709	+	0.960085i	$0.590238\pi$
$578$	10.8885	0.452904
$579$	22.4721	0.933910
$580$	−5.05573	−0.209928
$581$	0	0
$582$	7.52786	0.312040
$583$	−15.0000	−0.621237
$584$	−0.909830	−0.0376490
$585$	−2.29180	−0.0947541
$586$	−25.5967	−1.05739
$587$	33.4508	1.38066	0.690332	−	0.723493i	$-0.257464\pi$
$587$	33.4508	1.38066	0.690332	+	0.723493i	$0.257464\pi$
$588$	7.00000	0.288675
$589$	8.29180	0.341658
$590$	0.180340	0.00742448
$591$	−3.32624	−0.136823
$592$	−2.47214	−0.101604
$593$	47.3050	1.94258	0.971291	−	0.237895i	$-0.0764576\pi$
$593$	47.3050	1.94258	0.971291	+	0.237895i	$0.0764576\pi$
$594$	−4.85410	−0.199166
$595$	0	0
$596$	−8.47214	−0.347032
$597$	−4.76393	−0.194975
$598$	−3.70820	−0.151640
$599$	−5.79837	−0.236915	−0.118458	−	0.992959i	$-0.537795\pi$
$599$	−5.79837	−0.236915	−0.118458	+	0.992959i	$0.537795\pi$
$600$	−3.47214	−0.141749
$601$	−18.7639	−0.765397	−0.382698	−	0.923873i	$-0.625005\pi$
$601$	−18.7639	−0.765397	−0.382698	+	0.923873i	$0.625005\pi$
$602$	0	0
$603$	−2.94427	−0.119900
$604$	13.3820	0.544504
$605$	−15.5279	−0.631297
$606$	−1.85410	−0.0753177
$607$	34.8328	1.41382	0.706910	−	0.707303i	$-0.250088\pi$
$607$	34.8328	1.41382	0.706910	+	0.707303i	$0.250088\pi$
$608$	−1.38197	−0.0560461
$609$	0	0
$610$	17.0557	0.690566
$611$	10.4164	0.421403
$612$	−2.47214	−0.0999302
$613$	32.2705	1.30339	0.651697	−	0.758480i	$-0.274057\pi$
$613$	32.2705	1.30339	0.651697	+	0.758480i	$0.274057\pi$
$614$	−6.65248	−0.268472
$615$	−12.3607	−0.498431
$616$	0	0
$617$	7.70820	0.310321	0.155160	−	0.987889i	$-0.450411\pi$
$617$	7.70820	0.310321	0.155160	+	0.987889i	$0.450411\pi$
$618$	3.56231	0.143297
$619$	27.2361	1.09471	0.547355	−	0.836901i	$-0.315635\pi$
$619$	27.2361	1.09471	0.547355	+	0.836901i	$0.315635\pi$
$620$	−7.41641	−0.297850
$621$	−2.00000	−0.0802572
$622$	6.00000	0.240578
$623$	0	0
$624$	−1.85410	−0.0742235
$625$	4.41641	0.176656
$626$	10.6525	0.425759
$627$	6.70820	0.267900
$628$	5.61803	0.224184
$629$	6.11146	0.243680
$630$	0	0
$631$	−34.7426	−1.38308	−0.691541	−	0.722337i	$-0.743068\pi$
$631$	−34.7426	−1.38308	−0.691541	+	0.722337i	$0.743068\pi$
$632$	2.38197	0.0947495
$633$	6.38197	0.253660
$634$	9.56231	0.379768
$635$	−21.8885	−0.868620
$636$	−3.09017	−0.122533
$637$	−12.9787	−0.514235
$638$	19.8541	0.786031
$639$	15.4164	0.609864
$640$	1.23607	0.0488599
$641$	28.9787	1.14459	0.572295	−	0.820048i	$-0.306053\pi$
$641$	28.9787	1.14459	0.572295	+	0.820048i	$0.306053\pi$
$642$	9.38197	0.370277
$643$	−29.9787	−1.18225	−0.591123	−	0.806582i	$-0.701315\pi$
$643$	−29.9787	−1.18225	−0.591123	+	0.806582i	$0.701315\pi$
$644$	0	0
$645$	7.88854	0.310611
$646$	3.41641	0.134417
$647$	46.0344	1.80980	0.904900	−	0.425624i	$-0.139945\pi$
$647$	46.0344	1.80980	0.904900	+	0.425624i	$0.139945\pi$
$648$	−1.00000	−0.0392837
$649$	−0.708204	−0.0277994
$650$	6.43769	0.252507
$651$	0	0
$652$	14.0000	0.548282
$653$	31.3050	1.22506	0.612529	−	0.790448i	$-0.290152\pi$
$653$	31.3050	1.22506	0.612529	+	0.790448i	$0.290152\pi$
$654$	12.7639	0.499109
$655$	4.58359	0.179096
$656$	−10.0000	−0.390434
$657$	0.909830	0.0354959
$658$	0	0
$659$	−3.88854	−0.151476	−0.0757381	−	0.997128i	$-0.524131\pi$
$659$	−3.88854	−0.151476	−0.0757381	+	0.997128i	$0.524131\pi$
$660$	−6.00000	−0.233550
$661$	−45.3951	−1.76567	−0.882833	−	0.469687i	$-0.844367\pi$
$661$	−45.3951	−1.76567	−0.882833	+	0.469687i	$0.844367\pi$
$662$	5.41641	0.210515
$663$	4.58359	0.178012
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.47214	0.0957933
$667$	8.18034	0.316744
$668$	−21.7082	−0.839916
$669$	−1.00000	−0.0386622
$670$	−3.63932	−0.140599
$671$	−66.9787	−2.58568
$672$	0	0
$673$	40.1033	1.54587	0.772935	−	0.634485i	$-0.218788\pi$
$673$	40.1033	1.54587	0.772935	+	0.634485i	$0.218788\pi$
$674$	−32.0689	−1.23525
$675$	3.47214	0.133643
$676$	−9.56231	−0.367781
$677$	18.3607	0.705658	0.352829	−	0.935688i	$-0.385220\pi$
$677$	18.3607	0.705658	0.352829	+	0.935688i	$0.385220\pi$
$678$	9.85410	0.378445
$679$	0	0
$680$	−3.05573	−0.117182
$681$	−10.4721	−0.401293
$682$	29.1246	1.11524
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	1.38197	0.0528408
$685$	−0.832816	−0.0318203
$686$	0	0
$687$	9.41641	0.359258
$688$	6.38197	0.243310
$689$	5.72949	0.218276
$690$	−2.47214	−0.0941126
$691$	26.9443	1.02501	0.512504	−	0.858685i	$-0.328718\pi$
$691$	26.9443	1.02501	0.512504	+	0.858685i	$0.328718\pi$
$692$	15.8885	0.603992
$693$	0	0
$694$	15.4164	0.585199
$695$	−0.111456	−0.00422777
$696$	4.09017	0.155037
$697$	24.7214	0.936388
$698$	−8.94427	−0.338546
$699$	−15.1459	−0.572870
$700$	0	0
$701$	7.88854	0.297946	0.148973	−	0.988841i	$-0.452403\pi$
$701$	7.88854	0.297946	0.148973	+	0.988841i	$0.452403\pi$
$702$	1.85410	0.0699786
$703$	−3.41641	−0.128852
$704$	−4.85410	−0.182946
$705$	6.94427	0.261536
$706$	23.7082	0.892270
$707$	0	0
$708$	−0.145898	−0.00548318
$709$	−23.3050	−0.875236	−0.437618	−	0.899161i	$-0.644178\pi$
$709$	−23.3050	−0.875236	−0.437618	+	0.899161i	$0.644178\pi$
$710$	19.0557	0.715149
$711$	−2.38197	−0.0893307
$712$	7.23607	0.271183
$713$	12.0000	0.449404
$714$	0	0
$715$	11.1246	0.416037
$716$	15.1246	0.565233
$717$	−7.56231	−0.282419
$718$	−9.74265	−0.363592
$719$	21.8885	0.816305	0.408152	−	0.912914i	$-0.366173\pi$
$719$	21.8885	0.816305	0.408152	+	0.912914i	$0.366173\pi$
$720$	−1.23607	−0.0460655
$721$	0	0
$722$	17.0902	0.636030
$723$	12.2705	0.456345
$724$	8.00000	0.297318
$725$	−14.2016	−0.527435
$726$	12.5623	0.466231
$727$	−16.6525	−0.617606	−0.308803	−	0.951126i	$-0.599928\pi$
$727$	−16.6525	−0.617606	−0.308803	+	0.951126i	$0.599928\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.12461	0.0416238
$731$	−15.7771	−0.583537
$732$	−13.7984	−0.510003
$733$	−10.6525	−0.393458	−0.196729	−	0.980458i	$-0.563032\pi$
$733$	−10.6525	−0.393458	−0.196729	+	0.980458i	$0.563032\pi$
$734$	25.1246	0.927366
$735$	−8.65248	−0.319151
$736$	−2.00000	−0.0737210
$737$	14.2918	0.526445
$738$	10.0000	0.368105
$739$	−35.4853	−1.30535	−0.652674	−	0.757639i	$-0.726353\pi$
$739$	−35.4853	−1.30535	−0.652674	+	0.757639i	$0.726353\pi$
$740$	3.05573	0.112331
$741$	−2.56231	−0.0941287
$742$	0	0
$743$	−37.7426	−1.38464	−0.692322	−	0.721589i	$-0.743412\pi$
$743$	−37.7426	−1.38464	−0.692322	+	0.721589i	$0.743412\pi$
$744$	6.00000	0.219971
$745$	10.4721	0.383669
$746$	26.3607	0.965133
$747$	12.0000	0.439057
$748$	12.0000	0.438763
$749$	0	0
$750$	10.4721	0.382388
$751$	−18.9443	−0.691286	−0.345643	−	0.938366i	$-0.612339\pi$
$751$	−18.9443	−0.691286	−0.345643	+	0.938366i	$0.612339\pi$
$752$	5.61803	0.204869
$753$	7.23607	0.263697
$754$	−7.58359	−0.276178
$755$	−16.5410	−0.601989
$756$	0	0
$757$	−0.0344419	−0.00125181	−0.000625905	−	1.00000i	$-0.500199\pi$
$757$	−0.0344419	−0.00125181	−0.000625905		1.00000i	$0.500199\pi$
$758$	18.0344	0.655040
$759$	9.70820	0.352385
$760$	1.70820	0.0619631
$761$	18.5066	0.670863	0.335431	−	0.942065i	$-0.391118\pi$
$761$	18.5066	0.670863	0.335431	+	0.942065i	$0.391118\pi$
$762$	17.7082	0.641501
$763$	0	0
$764$	−16.1803	−0.585384
$765$	3.05573	0.110480
$766$	−13.5279	−0.488782
$767$	0.270510	0.00976754
$768$	−1.00000	−0.0360844
$769$	−35.3262	−1.27390	−0.636948	−	0.770906i	$-0.719804\pi$
$769$	−35.3262	−1.27390	−0.636948	+	0.770906i	$0.719804\pi$
$770$	0	0
$771$	−18.3607	−0.661244
$772$	−22.4721	−0.808790
$773$	21.4164	0.770295	0.385147	−	0.922855i	$-0.374151\pi$
$773$	21.4164	0.770295	0.385147	+	0.922855i	$0.374151\pi$
$774$	−6.38197	−0.229395
$775$	−20.8328	−0.748337
$776$	−7.52786	−0.270235
$777$	0	0
$778$	−32.2705	−1.15695
$779$	−13.8197	−0.495141
$780$	2.29180	0.0820595
$781$	−74.8328	−2.67773
$782$	4.94427	0.176807
$783$	−4.09017	−0.146171
$784$	−7.00000	−0.250000
$785$	−6.94427	−0.247852
$786$	−3.70820	−0.132267
$787$	1.63932	0.0584355	0.0292177	−	0.999573i	$-0.490698\pi$
$787$	1.63932	0.0584355	0.0292177	+	0.999573i	$0.490698\pi$
$788$	3.32624	0.118492
$789$	21.2361	0.756024
$790$	−2.94427	−0.104752
$791$	0	0
$792$	4.85410	0.172483
$793$	25.5836	0.908500
$794$	15.1459	0.537508
$795$	3.81966	0.135469
$796$	4.76393	0.168853
$797$	39.3050	1.39225	0.696126	−	0.717919i	$-0.254905\pi$
$797$	39.3050	1.39225	0.696126	+	0.717919i	$0.254905\pi$
$798$	0	0
$799$	−13.8885	−0.491341
$800$	3.47214	0.122759
$801$	−7.23607	−0.255674
$802$	−22.6525	−0.799887
$803$	−4.41641	−0.155852
$804$	2.94427	0.103836
$805$	0	0
$806$	−11.1246	−0.391848
$807$	9.41641	0.331473
$808$	1.85410	0.0652271
$809$	36.8328	1.29497	0.647486	−	0.762077i	$-0.275820\pi$
$809$	36.8328	1.29497	0.647486	+	0.762077i	$0.275820\pi$
$810$	1.23607	0.0434310
$811$	1.05573	0.0370716	0.0185358	−	0.999828i	$-0.494100\pi$
$811$	1.05573	0.0370716	0.0185358	+	0.999828i	$0.494100\pi$
$812$	0	0
$813$	4.67376	0.163916
$814$	−12.0000	−0.420600
$815$	−17.3050	−0.606166
$816$	2.47214	0.0865421
$817$	8.81966	0.308561
$818$	24.6525	0.861954
$819$	0	0
$820$	12.3607	0.431654
$821$	−41.7984	−1.45877	−0.729387	−	0.684102i	$-0.760194\pi$
$821$	−41.7984	−1.45877	−0.729387	+	0.684102i	$0.760194\pi$
$822$	0.673762	0.0235002
$823$	−15.4164	−0.537382	−0.268691	−	0.963226i	$-0.586591\pi$
$823$	−15.4164	−0.537382	−0.268691	+	0.963226i	$0.586591\pi$
$824$	−3.56231	−0.124099
$825$	−16.8541	−0.586785
$826$	0	0
$827$	55.2705	1.92194	0.960972	−	0.276646i	$-0.0892229\pi$
$827$	55.2705	1.92194	0.960972	+	0.276646i	$0.0892229\pi$
$828$	2.00000	0.0695048
$829$	15.3262	0.532302	0.266151	−	0.963931i	$-0.414248\pi$
$829$	15.3262	0.532302	0.266151	+	0.963931i	$0.414248\pi$
$830$	14.8328	0.514855
$831$	19.5279	0.677414
$832$	1.85410	0.0642794
$833$	17.3050	0.599581
$834$	0.0901699	0.00312233
$835$	26.8328	0.928588
$836$	−6.70820	−0.232008
$837$	−6.00000	−0.207390
$838$	−7.88854	−0.272505
$839$	9.70820	0.335164	0.167582	−	0.985858i	$-0.446404\pi$
$839$	9.70820	0.335164	0.167582	+	0.985858i	$0.446404\pi$
$840$	0	0
$841$	−12.2705	−0.423121
$842$	6.79837	0.234287
$843$	23.7082	0.816554
$844$	−6.38197	−0.219676
$845$	11.8197	0.406609
$846$	−5.61803	−0.193152
$847$	0	0
$848$	3.09017	0.106117
$849$	−8.79837	−0.301959
$850$	−8.58359	−0.294415
$851$	−4.94427	−0.169487
$852$	−15.4164	−0.528157
$853$	26.7426	0.915651	0.457825	−	0.889042i	$-0.348628\pi$
$853$	26.7426	0.915651	0.457825	+	0.889042i	$0.348628\pi$
$854$	0	0
$855$	−1.70820	−0.0584193
$856$	−9.38197	−0.320669
$857$	−6.94427	−0.237212	−0.118606	−	0.992941i	$-0.537843\pi$
$857$	−6.94427	−0.237212	−0.118606	+	0.992941i	$0.537843\pi$
$858$	−9.00000	−0.307255
$859$	−29.8197	−1.01743	−0.508717	−	0.860934i	$-0.669880\pi$
$859$	−29.8197	−1.01743	−0.508717	+	0.860934i	$0.669880\pi$
$860$	−7.88854	−0.268997
$861$	0	0
$862$	8.47214	0.288562
$863$	−15.1246	−0.514848	−0.257424	−	0.966299i	$-0.582874\pi$
$863$	−15.1246	−0.514848	−0.257424	+	0.966299i	$0.582874\pi$
$864$	1.00000	0.0340207
$865$	−19.6393	−0.667757
$866$	−6.79837	−0.231018
$867$	10.8885	0.369794
$868$	0	0
$869$	11.5623	0.392224
$870$	−5.05573	−0.171405
$871$	−5.45898	−0.184970
$872$	−12.7639	−0.432241
$873$	7.52786	0.254780
$874$	−2.76393	−0.0934914
$875$	0	0
$876$	−0.909830	−0.0307403
$877$	28.8541	0.974334	0.487167	−	0.873309i	$-0.338030\pi$
$877$	28.8541	0.974334	0.487167	+	0.873309i	$0.338030\pi$
$878$	−26.4721	−0.893391
$879$	−25.5967	−0.863357
$880$	6.00000	0.202260
$881$	−16.0689	−0.541374	−0.270687	−	0.962667i	$-0.587251\pi$
$881$	−16.0689	−0.541374	−0.270687	+	0.962667i	$0.587251\pi$
$882$	7.00000	0.235702
$883$	−20.3607	−0.685191	−0.342596	−	0.939483i	$-0.611306\pi$
$883$	−20.3607	−0.685191	−0.342596	+	0.939483i	$0.611306\pi$
$884$	−4.58359	−0.154163
$885$	0.180340	0.00606206
$886$	−15.7082	−0.527727
$887$	−13.9656	−0.468918	−0.234459	−	0.972126i	$-0.575332\pi$
$887$	−13.9656	−0.468918	−0.234459	+	0.972126i	$0.575332\pi$
$888$	−2.47214	−0.0829595
$889$	0	0
$890$	−8.94427	−0.299813
$891$	−4.85410	−0.162619
$892$	1.00000	0.0334825
$893$	7.76393	0.259810
$894$	−8.47214	−0.283351
$895$	−18.6950	−0.624907
$896$	0	0
$897$	−3.70820	−0.123813
$898$	17.4164	0.581193
$899$	24.5410	0.818489
$900$	−3.47214	−0.115738
$901$	−7.63932	−0.254503
$902$	−48.5410	−1.61624
$903$	0	0
$904$	−9.85410	−0.327743
$905$	−9.88854	−0.328706
$906$	13.3820	0.444586
$907$	−28.8673	−0.958522	−0.479261	−	0.877673i	$-0.659095\pi$
$907$	−28.8673	−0.958522	−0.479261	+	0.877673i	$0.659095\pi$
$908$	10.4721	0.347530
$909$	−1.85410	−0.0614967
$910$	0	0
$911$	−16.5836	−0.549439	−0.274719	−	0.961524i	$-0.588585\pi$
$911$	−16.5836	−0.549439	−0.274719	+	0.961524i	$0.588585\pi$
$912$	−1.38197	−0.0457615
$913$	−58.2492	−1.92777
$914$	15.2361	0.503964
$915$	17.0557	0.563845
$916$	−9.41641	−0.311127
$917$	0	0
$918$	−2.47214	−0.0815926
$919$	49.7426	1.64086	0.820429	−	0.571748i	$-0.193735\pi$
$919$	49.7426	1.64086	0.820429	+	0.571748i	$0.193735\pi$
$920$	2.47214	0.0815039
$921$	−6.65248	−0.219207
$922$	12.9098	0.425163
$923$	28.5836	0.940840
$924$	0	0
$925$	8.58359	0.282227
$926$	−42.2492	−1.38840
$927$	3.56231	0.117001
$928$	−4.09017	−0.134266
$929$	−0.291796	−0.00957352	−0.00478676	−	0.999989i	$-0.501524\pi$
$929$	−0.291796	−0.00957352	−0.00478676	+	0.999989i	$0.501524\pi$
$930$	−7.41641	−0.243194
$931$	−9.67376	−0.317045
$932$	15.1459	0.496120
$933$	6.00000	0.196431
$934$	−34.9787	−1.14454
$935$	−14.8328	−0.485085
$936$	−1.85410	−0.0606032
$937$	2.47214	0.0807612	0.0403806	−	0.999184i	$-0.487143\pi$
$937$	2.47214	0.0807612	0.0403806	+	0.999184i	$0.487143\pi$
$938$	0	0
$939$	10.6525	0.347630
$940$	−6.94427	−0.226497
$941$	25.2705	0.823795	0.411898	−	0.911230i	$-0.364866\pi$
$941$	25.2705	0.823795	0.411898	+	0.911230i	$0.364866\pi$
$942$	5.61803	0.183045
$943$	−20.0000	−0.651290
$944$	0.145898	0.00474858
$945$	0	0
$946$	30.9787	1.00720
$947$	−38.4721	−1.25018	−0.625088	−	0.780554i	$-0.714937\pi$
$947$	−38.4721	−1.25018	−0.625088	+	0.780554i	$0.714937\pi$
$948$	2.38197	0.0773627
$949$	1.68692	0.0547597
$950$	4.79837	0.155680
$951$	9.56231	0.310079
$952$	0	0
$953$	−55.3050	−1.79150	−0.895752	−	0.444555i	$-0.853362\pi$
$953$	−55.3050	−1.79150	−0.895752	+	0.444555i	$0.853362\pi$
$954$	−3.09017	−0.100048
$955$	20.0000	0.647185
$956$	7.56231	0.244582
$957$	19.8541	0.641792
$958$	−6.38197	−0.206192
$959$	0	0
$960$	1.23607	0.0398939
$961$	5.00000	0.161290
$962$	4.58359	0.147781
$963$	9.38197	0.302330
$964$	−12.2705	−0.395207
$965$	27.7771	0.894176
$966$	0	0
$967$	0.437694	0.0140753	0.00703765	−	0.999975i	$-0.497760\pi$
$967$	0.437694	0.0140753	0.00703765	+	0.999975i	$0.497760\pi$
$968$	−12.5623	−0.403768
$969$	3.41641	0.109751
$970$	9.30495	0.298764
$971$	39.6869	1.27361	0.636807	−	0.771023i	$-0.280255\pi$
$971$	39.6869	1.27361	0.636807	+	0.771023i	$0.280255\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−11.8885	−0.380934
$975$	6.43769	0.206171
$976$	13.7984	0.441675
$977$	15.5066	0.496099	0.248050	−	0.968747i	$-0.420210\pi$
$977$	15.5066	0.496099	0.248050	+	0.968747i	$0.420210\pi$
$978$	14.0000	0.447671
$979$	35.1246	1.12259
$980$	8.65248	0.276393
$981$	12.7639	0.407521
$982$	29.6180	0.945149
$983$	−27.8197	−0.887309	−0.443655	−	0.896198i	$-0.646318\pi$
$983$	−27.8197	−0.887309	−0.443655	+	0.896198i	$0.646318\pi$
$984$	−10.0000	−0.318788
$985$	−4.11146	−0.131002
$986$	10.1115	0.322014
$987$	0	0
$988$	2.56231	0.0815178
$989$	12.7639	0.405869
$990$	−6.00000	−0.190693
$991$	48.7984	1.55013	0.775066	−	0.631881i	$-0.217717\pi$
$991$	48.7984	1.55013	0.775066	+	0.631881i	$0.217717\pi$
$992$	−6.00000	−0.190500
$993$	5.41641	0.171885
$994$	0	0
$995$	−5.88854	−0.186679
$996$	−12.0000	−0.380235
$997$	10.9443	0.346609	0.173304	−	0.984868i	$-0.444556\pi$
$997$	10.9443	0.346609	0.173304	+	0.984868i	$0.444556\pi$
$998$	−30.9230	−0.978850
$999$	2.47214	0.0782149

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.b.1.1	✓	2
3.2	odd	2		4014.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.b.1.1	✓	2	1.1	even	1	trivial
4014.2.a.k.1.2		2	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Groups

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1338.2.a.b.1.1

Level	$1338$
Weight	$2$
Character	1338.1
Self dual	yes
Analytic conductor	$10.684$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$