LMFDB - Embedded newform 1386.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.23607 q^{10} -1.00000 q^{11} -3.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} -7.23607 q^{19} +1.23607 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.47214 q^{25} +3.23607 q^{26} +1.00000 q^{28} -4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.47214 q^{34} +1.23607 q^{35} +6.94427 q^{37} +7.23607 q^{38} -1.23607 q^{40} +2.47214 q^{41} -10.4721 q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} -8.47214 q^{53} -1.23607 q^{55} -1.00000 q^{56} +4.47214 q^{58} -2.76393 q^{59} -0.763932 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +11.4164 q^{67} -2.47214 q^{68} -1.23607 q^{70} -6.47214 q^{71} +12.9443 q^{73} -6.94427 q^{74} -7.23607 q^{76} -1.00000 q^{77} +1.23607 q^{80} -2.47214 q^{82} +12.1803 q^{83} -3.05573 q^{85} +10.4721 q^{86} +1.00000 q^{88} -10.0000 q^{89} -3.23607 q^{91} -4.00000 q^{92} -2.00000 q^{94} -8.94427 q^{95} +12.4721 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 10 q^{38} + 2 q^{40} - 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} - 2 q^{56} - 10 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 4 q^{68} + 2 q^{70} - 4 q^{71} + 8 q^{73} + 4 q^{74} - 10 q^{76} - 2 q^{77} - 2 q^{80} + 4 q^{82} + 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} - 20 q^{89} - 2 q^{91} - 8 q^{92} - 4 q^{94} + 16 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 + 4 * q^31 - 2 * q^32 - 4 * q^34 - 2 * q^35 - 4 * q^37 + 10 * q^38 + 2 * q^40 - 4 * q^41 - 12 * q^43 - 2 * q^44 + 8 * q^46 + 4 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 8 * q^53 + 2 * q^55 - 2 * q^56 - 10 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^65 - 4 * q^67 + 4 * q^68 + 2 * q^70 - 4 * q^71 + 8 * q^73 + 4 * q^74 - 10 * q^76 - 2 * q^77 - 2 * q^80 + 4 * q^82 + 2 * q^83 - 24 * q^85 + 12 * q^86 + 2 * q^88 - 20 * q^89 - 2 * q^91 - 8 * q^92 - 4 * q^94 + 16 * q^97 - 2 * q^98

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	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.23607	−0.390879
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	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	$0.596917\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	1.23607	0.276393
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	3.23607	0.634645
$27$	0	0
$28$	1.00000	0.188982
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.47214	0.423968
$35$	1.23607	0.208934
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	7.23607	1.17385
$39$	0	0
$40$	−1.23607	−0.195440
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.47214	0.491034
$51$	0	0
$52$	−3.23607	−0.448762
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	−1.23607	−0.166671
$56$	−1.00000	−0.133631
$57$	0	0
$58$	4.47214	0.587220
$59$	−2.76393	−0.359833	−0.179917	−	0.983682i	$-0.557583\pi$
$59$	−2.76393	−0.359833	−0.179917	+	0.983682i	$0.557583\pi$
$60$	0	0
$61$	−0.763932	−0.0978115	−0.0489057	−	0.998803i	$-0.515573\pi$
$61$	−0.763932	−0.0978115	−0.0489057	+	0.998803i	$0.515573\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	11.4164	1.39474	0.697368	−	0.716713i	$-0.254354\pi$
$67$	11.4164	1.39474	0.697368	+	0.716713i	$0.254354\pi$
$68$	−2.47214	−0.299791
$69$	0	0
$70$	−1.23607	−0.147738
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	12.9443	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$73$	12.9443	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$74$	−6.94427	−0.807255
$75$	0	0
$76$	−7.23607	−0.830034
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.23607	0.138197
$81$	0	0
$82$	−2.47214	−0.273002
$83$	12.1803	1.33697	0.668483	−	0.743727i	$-0.266944\pi$
$83$	12.1803	1.33697	0.668483	+	0.743727i	$0.266944\pi$
$84$	0	0
$85$	−3.05573	−0.331440
$86$	10.4721	1.12924
$87$	0	0
$88$	1.00000	0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−3.23607	−0.339232
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−2.00000	−0.206284
$95$	−8.94427	−0.917663
$96$	0	0
$97$	12.4721	1.26635	0.633177	−	0.774007i	$-0.281751\pi$
$97$	12.4721	1.26635	0.633177	+	0.774007i	$0.281751\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.47214	−0.347214
$101$	−8.18034	−0.813974	−0.406987	−	0.913434i	$-0.633421\pi$
$101$	−8.18034	−0.813974	−0.406987	+	0.913434i	$0.633421\pi$
$102$	0	0
$103$	−14.9443	−1.47250	−0.736251	−	0.676708i	$-0.763406\pi$
$103$	−14.9443	−1.47250	−0.736251	+	0.676708i	$0.763406\pi$
$104$	3.23607	0.317323
$105$	0	0
$106$	8.47214	0.822887
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	1.23607	0.117854
$111$	0	0
$112$	1.00000	0.0944911
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	0	0
$115$	−4.94427	−0.461056
$116$	−4.47214	−0.415227
$117$	0	0
$118$	2.76393	0.254441
$119$	−2.47214	−0.226620
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.763932	0.0691632
$123$	0	0
$124$	2.00000	0.179605
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−4.76393	−0.416227	−0.208113	−	0.978105i	$-0.566732\pi$
$131$	−4.76393	−0.416227	−0.208113	+	0.978105i	$0.566732\pi$
$132$	0	0
$133$	−7.23607	−0.627447
$134$	−11.4164	−0.986227
$135$	0	0
$136$	2.47214	0.211984
$137$	19.8885	1.69919	0.849596	−	0.527433i	$-0.176846\pi$
$137$	19.8885	1.69919	0.849596	+	0.527433i	$0.176846\pi$
$138$	0	0
$139$	−21.7082	−1.84127	−0.920633	−	0.390429i	$-0.872327\pi$
$139$	−21.7082	−1.84127	−0.920633	+	0.390429i	$0.872327\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	6.47214	0.543130
$143$	3.23607	0.270614
$144$	0	0
$145$	−5.52786	−0.459064
$146$	−12.9443	−1.07128
$147$	0	0
$148$	6.94427	0.570816
$149$	22.3607	1.83186	0.915929	−	0.401340i	$-0.131455\pi$
$149$	22.3607	1.83186	0.915929	+	0.401340i	$0.131455\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	7.23607	0.586923
$153$	0	0
$154$	1.00000	0.0805823
$155$	2.47214	0.198567
$156$	0	0
$157$	−12.6525	−1.00978	−0.504889	−	0.863184i	$-0.668466\pi$
$157$	−12.6525	−1.00978	−0.504889	+	0.863184i	$0.668466\pi$
$158$	0	0
$159$	0	0
$160$	−1.23607	−0.0977198
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−19.4164	−1.52081	−0.760405	−	0.649449i	$-0.775000\pi$
$163$	−19.4164	−1.52081	−0.760405	+	0.649449i	$0.775000\pi$
$164$	2.47214	0.193041
$165$	0	0
$166$	−12.1803	−0.945378
$167$	−11.4164	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$167$	−11.4164	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	3.05573	0.234364
$171$	0	0
$172$	−10.4721	−0.798493
$173$	3.23607	0.246034	0.123017	−	0.992405i	$-0.460743\pi$
$173$	3.23607	0.246034	0.123017	+	0.992405i	$0.460743\pi$
$174$	0	0
$175$	−3.47214	−0.262469
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	10.0000	0.749532
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	9.23607	0.686512	0.343256	−	0.939242i	$-0.388470\pi$
$181$	9.23607	0.686512	0.343256	+	0.939242i	$0.388470\pi$
$182$	3.23607	0.239873
$183$	0	0
$184$	4.00000	0.294884
$185$	8.58359	0.631078
$186$	0	0
$187$	2.47214	0.180780
$188$	2.00000	0.145865
$189$	0	0
$190$	8.94427	0.648886
$191$	2.47214	0.178877	0.0894387	−	0.995992i	$-0.471493\pi$
$191$	2.47214	0.178877	0.0894387	+	0.995992i	$0.471493\pi$
$192$	0	0
$193$	−14.9443	−1.07571	−0.537856	−	0.843037i	$-0.680766\pi$
$193$	−14.9443	−1.07571	−0.537856	+	0.843037i	$0.680766\pi$
$194$	−12.4721	−0.895447
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	18.9443	1.34292	0.671462	−	0.741039i	$-0.265667\pi$
$199$	18.9443	1.34292	0.671462	+	0.741039i	$0.265667\pi$
$200$	3.47214	0.245517
$201$	0	0
$202$	8.18034	0.575567
$203$	−4.47214	−0.313882
$204$	0	0
$205$	3.05573	0.213421
$206$	14.9443	1.04122
$207$	0	0
$208$	−3.23607	−0.224381
$209$	7.23607	0.500529
$210$	0	0
$211$	−13.5279	−0.931297	−0.465648	−	0.884970i	$-0.654179\pi$
$211$	−13.5279	−0.931297	−0.465648	+	0.884970i	$0.654179\pi$
$212$	−8.47214	−0.581869
$213$	0	0
$214$	2.47214	0.168992
$215$	−12.9443	−0.882792
$216$	0	0
$217$	2.00000	0.135769
$218$	10.0000	0.677285
$219$	0	0
$220$	−1.23607	−0.0833357
$221$	8.00000	0.538138
$222$	0	0
$223$	−0.472136	−0.0316166	−0.0158083	−	0.999875i	$-0.505032\pi$
$223$	−0.472136	−0.0316166	−0.0158083	+	0.999875i	$0.505032\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−0.472136	−0.0314060
$227$	19.2361	1.27674	0.638371	−	0.769729i	$-0.279608\pi$
$227$	19.2361	1.27674	0.638371	+	0.769729i	$0.279608\pi$
$228$	0	0
$229$	17.2361	1.13899	0.569496	−	0.821994i	$-0.307139\pi$
$229$	17.2361	1.13899	0.569496	+	0.821994i	$0.307139\pi$
$230$	4.94427	0.326016
$231$	0	0
$232$	4.47214	0.293610
$233$	14.9443	0.979032	0.489516	−	0.871994i	$-0.337174\pi$
$233$	14.9443	0.979032	0.489516	+	0.871994i	$0.337174\pi$
$234$	0	0
$235$	2.47214	0.161264
$236$	−2.76393	−0.179917
$237$	0	0
$238$	2.47214	0.160245
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	15.4164	0.993058	0.496529	−	0.868020i	$-0.334608\pi$
$241$	15.4164	0.993058	0.496529	+	0.868020i	$0.334608\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−0.763932	−0.0489057
$245$	1.23607	0.0789695
$246$	0	0
$247$	23.4164	1.48995
$248$	−2.00000	−0.127000
$249$	0	0
$250$	10.4721	0.662316
$251$	−29.2361	−1.84536	−0.922682	−	0.385562i	$-0.874008\pi$
$251$	−29.2361	−1.84536	−0.922682	+	0.385562i	$0.874008\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.94427	−0.433172	−0.216586	−	0.976264i	$-0.569492\pi$
$257$	−6.94427	−0.433172	−0.216586	+	0.976264i	$0.569492\pi$
$258$	0	0
$259$	6.94427	0.431496
$260$	−4.00000	−0.248069
$261$	0	0
$262$	4.76393	0.294317
$263$	4.94427	0.304877	0.152438	−	0.988313i	$-0.451287\pi$
$263$	4.94427	0.304877	0.152438	+	0.988313i	$0.451287\pi$
$264$	0	0
$265$	−10.4721	−0.643298
$266$	7.23607	0.443672
$267$	0	0
$268$	11.4164	0.697368
$269$	22.7639	1.38794	0.693971	−	0.720003i	$-0.255860\pi$
$269$	22.7639	1.38794	0.693971	+	0.720003i	$0.255860\pi$
$270$	0	0
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	−2.47214	−0.149895
$273$	0	0
$274$	−19.8885	−1.20151
$275$	3.47214	0.209378
$276$	0	0
$277$	3.52786	0.211969	0.105984	−	0.994368i	$-0.466201\pi$
$277$	3.52786	0.211969	0.105984	+	0.994368i	$0.466201\pi$
$278$	21.7082	1.30197
$279$	0	0
$280$	−1.23607	−0.0738692
$281$	−28.8328	−1.72002	−0.860011	−	0.510276i	$-0.829543\pi$
$281$	−28.8328	−1.72002	−0.860011	+	0.510276i	$0.829543\pi$
$282$	0	0
$283$	14.6525	0.870999	0.435500	−	0.900189i	$-0.356572\pi$
$283$	14.6525	0.870999	0.435500	+	0.900189i	$0.356572\pi$
$284$	−6.47214	−0.384051
$285$	0	0
$286$	−3.23607	−0.191353
$287$	2.47214	0.145926
$288$	0	0
$289$	−10.8885	−0.640503
$290$	5.52786	0.324607
$291$	0	0
$292$	12.9443	0.757506
$293$	26.6525	1.55705	0.778527	−	0.627611i	$-0.215967\pi$
$293$	26.6525	1.55705	0.778527	+	0.627611i	$0.215967\pi$
$294$	0	0
$295$	−3.41641	−0.198911
$296$	−6.94427	−0.403628
$297$	0	0
$298$	−22.3607	−1.29532
$299$	12.9443	0.748587
$300$	0	0
$301$	−10.4721	−0.603604
$302$	−12.0000	−0.690522
$303$	0	0
$304$	−7.23607	−0.415017
$305$	−0.944272	−0.0540689
$306$	0	0
$307$	−26.0689	−1.48783	−0.743915	−	0.668274i	$-0.767033\pi$
$307$	−26.0689	−1.48783	−0.743915	+	0.668274i	$0.767033\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−2.47214	−0.140408
$311$	21.4164	1.21441	0.607207	−	0.794544i	$-0.292290\pi$
$311$	21.4164	1.21441	0.607207	+	0.794544i	$0.292290\pi$
$312$	0	0
$313$	19.5279	1.10378	0.551890	−	0.833917i	$-0.313907\pi$
$313$	19.5279	1.10378	0.551890	+	0.833917i	$0.313907\pi$
$314$	12.6525	0.714021
$315$	0	0
$316$	0	0
$317$	30.9443	1.73800	0.869002	−	0.494809i	$-0.164762\pi$
$317$	30.9443	1.73800	0.869002	+	0.494809i	$0.164762\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	1.23607	0.0690983
$321$	0	0
$322$	4.00000	0.222911
$323$	17.8885	0.995345
$324$	0	0
$325$	11.2361	0.623265
$326$	19.4164	1.07538
$327$	0	0
$328$	−2.47214	−0.136501
$329$	2.00000	0.110264
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	12.1803	0.668483
$333$	0	0
$334$	11.4164	0.624678
$335$	14.1115	0.770991
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	2.52786	0.137498
$339$	0	0
$340$	−3.05573	−0.165720
$341$	−2.00000	−0.108306
$342$	0	0
$343$	1.00000	0.0539949
$344$	10.4721	0.564620
$345$	0	0
$346$	−3.23607	−0.173972
$347$	−2.47214	−0.132711	−0.0663556	−	0.997796i	$-0.521137\pi$
$347$	−2.47214	−0.132711	−0.0663556	+	0.997796i	$0.521137\pi$
$348$	0	0
$349$	21.7082	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$349$	21.7082	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$350$	3.47214	0.185593
$351$	0	0
$352$	1.00000	0.0533002
$353$	17.0557	0.907785	0.453892	−	0.891056i	$-0.350035\pi$
$353$	17.0557	0.907785	0.453892	+	0.891056i	$0.350035\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−8.94427	−0.472719
$359$	−26.8328	−1.41618	−0.708091	−	0.706121i	$-0.750443\pi$
$359$	−26.8328	−1.41618	−0.708091	+	0.706121i	$0.750443\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	−9.23607	−0.485437
$363$	0	0
$364$	−3.23607	−0.169616
$365$	16.0000	0.837478
$366$	0	0
$367$	−5.41641	−0.282734	−0.141367	−	0.989957i	$-0.545150\pi$
$367$	−5.41641	−0.282734	−0.141367	+	0.989957i	$0.545150\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−8.58359	−0.446240
$371$	−8.47214	−0.439851
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−2.47214	−0.127831
$375$	0	0
$376$	−2.00000	−0.103142
$377$	14.4721	0.745353
$378$	0	0
$379$	14.4721	0.743384	0.371692	−	0.928356i	$-0.378778\pi$
$379$	14.4721	0.743384	0.371692	+	0.928356i	$0.378778\pi$
$380$	−8.94427	−0.458831
$381$	0	0
$382$	−2.47214	−0.126485
$383$	23.8885	1.22065	0.610324	−	0.792152i	$-0.291039\pi$
$383$	23.8885	1.22065	0.610324	+	0.792152i	$0.291039\pi$
$384$	0	0
$385$	−1.23607	−0.0629959
$386$	14.9443	0.760643
$387$	0	0
$388$	12.4721	0.633177
$389$	−33.4164	−1.69428	−0.847140	−	0.531370i	$-0.821677\pi$
$389$	−33.4164	−1.69428	−0.847140	+	0.531370i	$0.821677\pi$
$390$	0	0
$391$	9.88854	0.500085
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	−23.7082	−1.18988	−0.594940	−	0.803770i	$-0.702824\pi$
$397$	−23.7082	−1.18988	−0.594940	+	0.803770i	$0.702824\pi$
$398$	−18.9443	−0.949591
$399$	0	0
$400$	−3.47214	−0.173607
$401$	−14.3607	−0.717138	−0.358569	−	0.933503i	$-0.616735\pi$
$401$	−14.3607	−0.717138	−0.358569	+	0.933503i	$0.616735\pi$
$402$	0	0
$403$	−6.47214	−0.322400
$404$	−8.18034	−0.406987
$405$	0	0
$406$	4.47214	0.221948
$407$	−6.94427	−0.344215
$408$	0	0
$409$	3.41641	0.168930	0.0844652	−	0.996426i	$-0.473082\pi$
$409$	3.41641	0.168930	0.0844652	+	0.996426i	$0.473082\pi$
$410$	−3.05573	−0.150912
$411$	0	0
$412$	−14.9443	−0.736251
$413$	−2.76393	−0.136004
$414$	0	0
$415$	15.0557	0.739057
$416$	3.23607	0.158661
$417$	0	0
$418$	−7.23607	−0.353928
$419$	−17.2361	−0.842037	−0.421019	−	0.907052i	$-0.638327\pi$
$419$	−17.2361	−0.842037	−0.421019	+	0.907052i	$0.638327\pi$
$420$	0	0
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	13.5279	0.658526
$423$	0	0
$424$	8.47214	0.411443
$425$	8.58359	0.416365
$426$	0	0
$427$	−0.763932	−0.0369693
$428$	−2.47214	−0.119495
$429$	0	0
$430$	12.9443	0.624228
$431$	−23.0557	−1.11056	−0.555278	−	0.831665i	$-0.687388\pi$
$431$	−23.0557	−1.11056	−0.555278	+	0.831665i	$0.687388\pi$
$432$	0	0
$433$	28.4721	1.36828	0.684142	−	0.729349i	$-0.260177\pi$
$433$	28.4721	1.36828	0.684142	+	0.729349i	$0.260177\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−10.0000	−0.478913
$437$	28.9443	1.38459
$438$	0	0
$439$	8.94427	0.426887	0.213443	−	0.976955i	$-0.431532\pi$
$439$	8.94427	0.426887	0.213443	+	0.976955i	$0.431532\pi$
$440$	1.23607	0.0589272
$441$	0	0
$442$	−8.00000	−0.380521
$443$	24.9443	1.18514	0.592569	−	0.805520i	$-0.298114\pi$
$443$	24.9443	1.18514	0.592569	+	0.805520i	$0.298114\pi$
$444$	0	0
$445$	−12.3607	−0.585952
$446$	0.472136	0.0223563
$447$	0	0
$448$	1.00000	0.0472456
$449$	−18.9443	−0.894035	−0.447018	−	0.894525i	$-0.647514\pi$
$449$	−18.9443	−0.894035	−0.447018	+	0.894525i	$0.647514\pi$
$450$	0	0
$451$	−2.47214	−0.116408
$452$	0.472136	0.0222074
$453$	0	0
$454$	−19.2361	−0.902793
$455$	−4.00000	−0.187523
$456$	0	0
$457$	26.9443	1.26040	0.630200	−	0.776433i	$-0.282973\pi$
$457$	26.9443	1.26040	0.630200	+	0.776433i	$0.282973\pi$
$458$	−17.2361	−0.805389
$459$	0	0
$460$	−4.94427	−0.230528
$461$	−24.7639	−1.15337	−0.576686	−	0.816966i	$-0.695654\pi$
$461$	−24.7639	−1.15337	−0.576686	+	0.816966i	$0.695654\pi$
$462$	0	0
$463$	−30.4721	−1.41616	−0.708080	−	0.706132i	$-0.750438\pi$
$463$	−30.4721	−1.41616	−0.708080	+	0.706132i	$0.750438\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−14.9443	−0.692280
$467$	27.1246	1.25518	0.627589	−	0.778545i	$-0.284042\pi$
$467$	27.1246	1.25518	0.627589	+	0.778545i	$0.284042\pi$
$468$	0	0
$469$	11.4164	0.527161
$470$	−2.47214	−0.114031
$471$	0	0
$472$	2.76393	0.127220
$473$	10.4721	0.481509
$474$	0	0
$475$	25.1246	1.15280
$476$	−2.47214	−0.113310
$477$	0	0
$478$	20.0000	0.914779
$479$	−12.3607	−0.564774	−0.282387	−	0.959301i	$-0.591126\pi$
$479$	−12.3607	−0.564774	−0.282387	+	0.959301i	$0.591126\pi$
$480$	0	0
$481$	−22.4721	−1.02464
$482$	−15.4164	−0.702198
$483$	0	0
$484$	1.00000	0.0454545
$485$	15.4164	0.700023
$486$	0	0
$487$	16.9443	0.767818	0.383909	−	0.923371i	$-0.374578\pi$
$487$	16.9443	0.767818	0.383909	+	0.923371i	$0.374578\pi$
$488$	0.763932	0.0345816
$489$	0	0
$490$	−1.23607	−0.0558399
$491$	16.9443	0.764684	0.382342	−	0.924021i	$-0.375118\pi$
$491$	16.9443	0.764684	0.382342	+	0.924021i	$0.375118\pi$
$492$	0	0
$493$	11.0557	0.497925
$494$	−23.4164	−1.05355
$495$	0	0
$496$	2.00000	0.0898027
$497$	−6.47214	−0.290315
$498$	0	0
$499$	−32.3607	−1.44866	−0.724331	−	0.689452i	$-0.757851\pi$
$499$	−32.3607	−1.44866	−0.724331	+	0.689452i	$0.757851\pi$
$500$	−10.4721	−0.468328
$501$	0	0
$502$	29.2361	1.30487
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	−10.1115	−0.449954
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−12.0000	−0.532414
$509$	−24.0689	−1.06683	−0.533417	−	0.845852i	$-0.679092\pi$
$509$	−24.0689	−1.06683	−0.533417	+	0.845852i	$0.679092\pi$
$510$	0	0
$511$	12.9443	0.572621
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.94427	0.306299
$515$	−18.4721	−0.813980
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	−6.94427	−0.305114
$519$	0	0
$520$	4.00000	0.175412
$521$	10.3607	0.453910	0.226955	−	0.973905i	$-0.427123\pi$
$521$	10.3607	0.453910	0.226955	+	0.973905i	$0.427123\pi$
$522$	0	0
$523$	−14.2918	−0.624937	−0.312468	−	0.949928i	$-0.601156\pi$
$523$	−14.2918	−0.624937	−0.312468	+	0.949928i	$0.601156\pi$
$524$	−4.76393	−0.208113
$525$	0	0
$526$	−4.94427	−0.215580
$527$	−4.94427	−0.215376
$528$	0	0
$529$	−7.00000	−0.304348
$530$	10.4721	0.454881
$531$	0	0
$532$	−7.23607	−0.313723
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−3.05573	−0.132111
$536$	−11.4164	−0.493114
$537$	0	0
$538$	−22.7639	−0.981423
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−26.9443	−1.15842	−0.579212	−	0.815177i	$-0.696640\pi$
$541$	−26.9443	−1.15842	−0.579212	+	0.815177i	$0.696640\pi$
$542$	−0.944272	−0.0405600
$543$	0	0
$544$	2.47214	0.105992
$545$	−12.3607	−0.529473
$546$	0	0
$547$	−0.944272	−0.0403742	−0.0201871	−	0.999796i	$-0.506426\pi$
$547$	−0.944272	−0.0403742	−0.0201871	+	0.999796i	$0.506426\pi$
$548$	19.8885	0.849596
$549$	0	0
$550$	−3.47214	−0.148052
$551$	32.3607	1.37861
$552$	0	0
$553$	0	0
$554$	−3.52786	−0.149885
$555$	0	0
$556$	−21.7082	−0.920633
$557$	−24.8328	−1.05220	−0.526100	−	0.850423i	$-0.676346\pi$
$557$	−24.8328	−1.05220	−0.526100	+	0.850423i	$0.676346\pi$
$558$	0	0
$559$	33.8885	1.43333
$560$	1.23607	0.0522334
$561$	0	0
$562$	28.8328	1.21624
$563$	−31.2361	−1.31644	−0.658222	−	0.752824i	$-0.728691\pi$
$563$	−31.2361	−1.31644	−0.658222	+	0.752824i	$0.728691\pi$
$564$	0	0
$565$	0.583592	0.0245519
$566$	−14.6525	−0.615889
$567$	0	0
$568$	6.47214	0.271565
$569$	36.8328	1.54411	0.772056	−	0.635555i	$-0.219228\pi$
$569$	36.8328	1.54411	0.772056	+	0.635555i	$0.219228\pi$
$570$	0	0
$571$	−10.1115	−0.423151	−0.211576	−	0.977362i	$-0.567859\pi$
$571$	−10.1115	−0.423151	−0.211576	+	0.977362i	$0.567859\pi$
$572$	3.23607	0.135307
$573$	0	0
$574$	−2.47214	−0.103185
$575$	13.8885	0.579192
$576$	0	0
$577$	26.9443	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$577$	26.9443	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$578$	10.8885	0.452904
$579$	0	0
$580$	−5.52786	−0.229532
$581$	12.1803	0.505326
$582$	0	0
$583$	8.47214	0.350880
$584$	−12.9443	−0.535638
$585$	0	0
$586$	−26.6525	−1.10100
$587$	5.81966	0.240203	0.120102	−	0.992762i	$-0.461678\pi$
$587$	5.81966	0.240203	0.120102	+	0.992762i	$0.461678\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	3.41641	0.140651
$591$	0	0
$592$	6.94427	0.285408
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	−3.05573	−0.125273
$596$	22.3607	0.915929
$597$	0	0
$598$	−12.9443	−0.529331
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	−34.8328	−1.42086	−0.710430	−	0.703768i	$-0.751500\pi$
$601$	−34.8328	−1.42086	−0.710430	+	0.703768i	$0.751500\pi$
$602$	10.4721	0.426812
$603$	0	0
$604$	12.0000	0.488273
$605$	1.23607	0.0502533
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	7.23607	0.293461
$609$	0	0
$610$	0.944272	0.0382325
$611$	−6.47214	−0.261835
$612$	0	0
$613$	28.4721	1.14998	0.574989	−	0.818161i	$-0.305006\pi$
$613$	28.4721	1.14998	0.574989	+	0.818161i	$0.305006\pi$
$614$	26.0689	1.05205
$615$	0	0
$616$	1.00000	0.0402911
$617$	−21.4164	−0.862192	−0.431096	−	0.902306i	$-0.641873\pi$
$617$	−21.4164	−0.862192	−0.431096	+	0.902306i	$0.641873\pi$
$618$	0	0
$619$	18.5410	0.745227	0.372613	−	0.927987i	$-0.378462\pi$
$619$	18.5410	0.745227	0.372613	+	0.927987i	$0.378462\pi$
$620$	2.47214	0.0992834
$621$	0	0
$622$	−21.4164	−0.858720
$623$	−10.0000	−0.400642
$624$	0	0
$625$	4.41641	0.176656
$626$	−19.5279	−0.780490
$627$	0	0
$628$	−12.6525	−0.504889
$629$	−17.1672	−0.684500
$630$	0	0
$631$	−31.4164	−1.25067	−0.625334	−	0.780357i	$-0.715037\pi$
$631$	−31.4164	−1.25067	−0.625334	+	0.780357i	$0.715037\pi$
$632$	0	0
$633$	0	0
$634$	−30.9443	−1.22895
$635$	−14.8328	−0.588622
$636$	0	0
$637$	−3.23607	−0.128218
$638$	−4.47214	−0.177054
$639$	0	0
$640$	−1.23607	−0.0488599
$641$	−27.5279	−1.08729	−0.543643	−	0.839317i	$-0.682955\pi$
$641$	−27.5279	−1.08729	−0.543643	+	0.839317i	$0.682955\pi$
$642$	0	0
$643$	−18.7639	−0.739977	−0.369989	−	0.929036i	$-0.620638\pi$
$643$	−18.7639	−0.739977	−0.369989	+	0.929036i	$0.620638\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−17.8885	−0.703815
$647$	28.8328	1.13353	0.566767	−	0.823878i	$-0.308194\pi$
$647$	28.8328	1.13353	0.566767	+	0.823878i	$0.308194\pi$
$648$	0	0
$649$	2.76393	0.108494
$650$	−11.2361	−0.440715
$651$	0	0
$652$	−19.4164	−0.760405
$653$	−46.3607	−1.81423	−0.907117	−	0.420879i	$-0.861722\pi$
$653$	−46.3607	−1.81423	−0.907117	+	0.420879i	$0.861722\pi$
$654$	0	0
$655$	−5.88854	−0.230084
$656$	2.47214	0.0965207
$657$	0	0
$658$	−2.00000	−0.0779681
$659$	−16.5836	−0.646005	−0.323003	−	0.946398i	$-0.604692\pi$
$659$	−16.5836	−0.646005	−0.323003	+	0.946398i	$0.604692\pi$
$660$	0	0
$661$	−3.12461	−0.121533	−0.0607667	−	0.998152i	$-0.519355\pi$
$661$	−3.12461	−0.121533	−0.0607667	+	0.998152i	$0.519355\pi$
$662$	16.9443	0.658558
$663$	0	0
$664$	−12.1803	−0.472689
$665$	−8.94427	−0.346844
$666$	0	0
$667$	17.8885	0.692647
$668$	−11.4164	−0.441714
$669$	0	0
$670$	−14.1115	−0.545173
$671$	0.763932	0.0294913
$672$	0	0
$673$	−3.88854	−0.149892	−0.0749462	−	0.997188i	$-0.523878\pi$
$673$	−3.88854	−0.149892	−0.0749462	+	0.997188i	$0.523878\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−2.52786	−0.0972255
$677$	26.0689	1.00191	0.500954	−	0.865474i	$-0.332982\pi$
$677$	26.0689	1.00191	0.500954	+	0.865474i	$0.332982\pi$
$678$	0	0
$679$	12.4721	0.478637
$680$	3.05573	0.117182
$681$	0	0
$682$	2.00000	0.0765840
$683$	−32.9443	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$684$	0	0
$685$	24.5836	0.939291
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−10.4721	−0.399246
$689$	27.4164	1.04448
$690$	0	0
$691$	12.6525	0.481323	0.240661	−	0.970609i	$-0.422636\pi$
$691$	12.6525	0.481323	0.240661	+	0.970609i	$0.422636\pi$
$692$	3.23607	0.123017
$693$	0	0
$694$	2.47214	0.0938410
$695$	−26.8328	−1.01783
$696$	0	0
$697$	−6.11146	−0.231488
$698$	−21.7082	−0.821668
$699$	0	0
$700$	−3.47214	−0.131234
$701$	42.7214	1.61356	0.806782	−	0.590850i	$-0.201207\pi$
$701$	42.7214	1.61356	0.806782	+	0.590850i	$0.201207\pi$
$702$	0	0
$703$	−50.2492	−1.89519
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−17.0557	−0.641901
$707$	−8.18034	−0.307653
$708$	0	0
$709$	4.47214	0.167955	0.0839773	−	0.996468i	$-0.473238\pi$
$709$	4.47214	0.167955	0.0839773	+	0.996468i	$0.473238\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	10.0000	0.374766
$713$	−8.00000	−0.299602
$714$	0	0
$715$	4.00000	0.149592
$716$	8.94427	0.334263
$717$	0	0
$718$	26.8328	1.00139
$719$	−16.8328	−0.627758	−0.313879	−	0.949463i	$-0.601629\pi$
$719$	−16.8328	−0.627758	−0.313879	+	0.949463i	$0.601629\pi$
$720$	0	0
$721$	−14.9443	−0.556554
$722$	−33.3607	−1.24156
$723$	0	0
$724$	9.23607	0.343256
$725$	15.5279	0.576690
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	3.23607	0.119937
$729$	0	0
$730$	−16.0000	−0.592187
$731$	25.8885	0.957522
$732$	0	0
$733$	49.1246	1.81446	0.907229	−	0.420636i	$-0.138193\pi$
$733$	49.1246	1.81446	0.907229	+	0.420636i	$0.138193\pi$
$734$	5.41641	0.199923
$735$	0	0
$736$	4.00000	0.147442
$737$	−11.4164	−0.420529
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	8.58359	0.315539
$741$	0	0
$742$	8.47214	0.311022
$743$	−21.8885	−0.803013	−0.401506	−	0.915856i	$-0.631513\pi$
$743$	−21.8885	−0.803013	−0.401506	+	0.915856i	$0.631513\pi$
$744$	0	0
$745$	27.6393	1.01263
$746$	6.00000	0.219676
$747$	0	0
$748$	2.47214	0.0903902
$749$	−2.47214	−0.0903299
$750$	0	0
$751$	−16.9443	−0.618305	−0.309153	−	0.951012i	$-0.600045\pi$
$751$	−16.9443	−0.618305	−0.309153	+	0.951012i	$0.600045\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	−14.4721	−0.527044
$755$	14.8328	0.539821
$756$	0	0
$757$	−23.3050	−0.847033	−0.423516	−	0.905888i	$-0.639204\pi$
$757$	−23.3050	−0.847033	−0.423516	+	0.905888i	$0.639204\pi$
$758$	−14.4721	−0.525652
$759$	0	0
$760$	8.94427	0.324443
$761$	11.4164	0.413844	0.206922	−	0.978357i	$-0.433655\pi$
$761$	11.4164	0.413844	0.206922	+	0.978357i	$0.433655\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	2.47214	0.0894387
$765$	0	0
$766$	−23.8885	−0.863128
$767$	8.94427	0.322959
$768$	0	0
$769$	−43.4164	−1.56564	−0.782818	−	0.622251i	$-0.786218\pi$
$769$	−43.4164	−1.56564	−0.782818	+	0.622251i	$0.786218\pi$
$770$	1.23607	0.0445448
$771$	0	0
$772$	−14.9443	−0.537856
$773$	−15.7082	−0.564985	−0.282492	−	0.959270i	$-0.591161\pi$
$773$	−15.7082	−0.564985	−0.282492	+	0.959270i	$0.591161\pi$
$774$	0	0
$775$	−6.94427	−0.249446
$776$	−12.4721	−0.447724
$777$	0	0
$778$	33.4164	1.19804
$779$	−17.8885	−0.640924
$780$	0	0
$781$	6.47214	0.231591
$782$	−9.88854	−0.353614
$783$	0	0
$784$	1.00000	0.0357143
$785$	−15.6393	−0.558191
$786$	0	0
$787$	−28.1803	−1.00452	−0.502260	−	0.864716i	$-0.667498\pi$
$787$	−28.1803	−1.00452	−0.502260	+	0.864716i	$0.667498\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	0	0
$791$	0.472136	0.0167872
$792$	0	0
$793$	2.47214	0.0877881
$794$	23.7082	0.841373
$795$	0	0
$796$	18.9443	0.671462
$797$	41.5967	1.47343	0.736716	−	0.676202i	$-0.236375\pi$
$797$	41.5967	1.47343	0.736716	+	0.676202i	$0.236375\pi$
$798$	0	0
$799$	−4.94427	−0.174916
$800$	3.47214	0.122759
$801$	0	0
$802$	14.3607	0.507093
$803$	−12.9443	−0.456793
$804$	0	0
$805$	−4.94427	−0.174263
$806$	6.47214	0.227971
$807$	0	0
$808$	8.18034	0.287783
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	0	0
$811$	4.76393	0.167284	0.0836421	−	0.996496i	$-0.473345\pi$
$811$	4.76393	0.167284	0.0836421	+	0.996496i	$0.473345\pi$
$812$	−4.47214	−0.156941
$813$	0	0
$814$	6.94427	0.243397
$815$	−24.0000	−0.840683
$816$	0	0
$817$	75.7771	2.65110
$818$	−3.41641	−0.119452
$819$	0	0
$820$	3.05573	0.106711
$821$	1.41641	0.0494330	0.0247165	−	0.999695i	$-0.492132\pi$
$821$	1.41641	0.0494330	0.0247165	+	0.999695i	$0.492132\pi$
$822$	0	0
$823$	−46.2492	−1.61215	−0.806073	−	0.591816i	$-0.798411\pi$
$823$	−46.2492	−1.61215	−0.806073	+	0.591816i	$0.798411\pi$
$824$	14.9443	0.520608
$825$	0	0
$826$	2.76393	0.0961695
$827$	−16.9443	−0.589210	−0.294605	−	0.955619i	$-0.595188\pi$
$827$	−16.9443	−0.589210	−0.294605	+	0.955619i	$0.595188\pi$
$828$	0	0
$829$	11.7082	0.406643	0.203321	−	0.979112i	$-0.434826\pi$
$829$	11.7082	0.406643	0.203321	+	0.979112i	$0.434826\pi$
$830$	−15.0557	−0.522592
$831$	0	0
$832$	−3.23607	−0.112190
$833$	−2.47214	−0.0856544
$834$	0	0
$835$	−14.1115	−0.488347
$836$	7.23607	0.250265
$837$	0	0
$838$	17.2361	0.595410
$839$	−16.8328	−0.581133	−0.290567	−	0.956855i	$-0.593844\pi$
$839$	−16.8328	−0.581133	−0.290567	+	0.956855i	$0.593844\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−16.4721	−0.567667
$843$	0	0
$844$	−13.5279	−0.465648
$845$	−3.12461	−0.107490
$846$	0	0
$847$	1.00000	0.0343604
$848$	−8.47214	−0.290934
$849$	0	0
$850$	−8.58359	−0.294415
$851$	−27.7771	−0.952186
$852$	0	0
$853$	32.5410	1.11418	0.557092	−	0.830451i	$-0.311917\pi$
$853$	32.5410	1.11418	0.557092	+	0.830451i	$0.311917\pi$
$854$	0.763932	0.0261412
$855$	0	0
$856$	2.47214	0.0844959
$857$	46.4721	1.58746	0.793729	−	0.608272i	$-0.208137\pi$
$857$	46.4721	1.58746	0.793729	+	0.608272i	$0.208137\pi$
$858$	0	0
$859$	15.1246	0.516045	0.258023	−	0.966139i	$-0.416929\pi$
$859$	15.1246	0.516045	0.258023	+	0.966139i	$0.416929\pi$
$860$	−12.9443	−0.441396
$861$	0	0
$862$	23.0557	0.785281
$863$	−0.583592	−0.0198657	−0.00993285	−	0.999951i	$-0.503162\pi$
$863$	−0.583592	−0.0198657	−0.00993285	+	0.999951i	$0.503162\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	−28.4721	−0.967523
$867$	0	0
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−36.9443	−1.25181
$872$	10.0000	0.338643
$873$	0	0
$874$	−28.9443	−0.979055
$875$	−10.4721	−0.354023
$876$	0	0
$877$	9.05573	0.305790	0.152895	−	0.988242i	$-0.451140\pi$
$877$	9.05573	0.305790	0.152895	+	0.988242i	$0.451140\pi$
$878$	−8.94427	−0.301855
$879$	0	0
$880$	−1.23607	−0.0416678
$881$	−28.8328	−0.971402	−0.485701	−	0.874125i	$-0.661436\pi$
$881$	−28.8328	−0.971402	−0.485701	+	0.874125i	$0.661436\pi$
$882$	0	0
$883$	−2.83282	−0.0953318	−0.0476659	−	0.998863i	$-0.515178\pi$
$883$	−2.83282	−0.0953318	−0.0476659	+	0.998863i	$0.515178\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−24.9443	−0.838019
$887$	44.3607	1.48949	0.744743	−	0.667351i	$-0.232572\pi$
$887$	44.3607	1.48949	0.744743	+	0.667351i	$0.232572\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	12.3607	0.414331
$891$	0	0
$892$	−0.472136	−0.0158083
$893$	−14.4721	−0.484292
$894$	0	0
$895$	11.0557	0.369552
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.9443	0.632179
$899$	−8.94427	−0.298308
$900$	0	0
$901$	20.9443	0.697755
$902$	2.47214	0.0823131
$903$	0	0
$904$	−0.472136	−0.0157030
$905$	11.4164	0.379494
$906$	0	0
$907$	−24.3607	−0.808883	−0.404442	−	0.914564i	$-0.632534\pi$
$907$	−24.3607	−0.808883	−0.404442	+	0.914564i	$0.632534\pi$
$908$	19.2361	0.638371
$909$	0	0
$910$	4.00000	0.132599
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	−12.1803	−0.403110
$914$	−26.9443	−0.891237
$915$	0	0
$916$	17.2361	0.569496
$917$	−4.76393	−0.157319
$918$	0	0
$919$	−22.1115	−0.729390	−0.364695	−	0.931127i	$-0.618827\pi$
$919$	−22.1115	−0.729390	−0.364695	+	0.931127i	$0.618827\pi$
$920$	4.94427	0.163008
$921$	0	0
$922$	24.7639	0.815557
$923$	20.9443	0.689389
$924$	0	0
$925$	−24.1115	−0.792780
$926$	30.4721	1.00138
$927$	0	0
$928$	4.47214	0.146805
$929$	−40.2492	−1.32053	−0.660267	−	0.751031i	$-0.729557\pi$
$929$	−40.2492	−1.32053	−0.660267	+	0.751031i	$0.729557\pi$
$930$	0	0
$931$	−7.23607	−0.237153
$932$	14.9443	0.489516
$933$	0	0
$934$	−27.1246	−0.887544
$935$	3.05573	0.0999330
$936$	0	0
$937$	−3.05573	−0.0998263	−0.0499131	−	0.998754i	$-0.515894\pi$
$937$	−3.05573	−0.0998263	−0.0499131	+	0.998754i	$0.515894\pi$
$938$	−11.4164	−0.372759
$939$	0	0
$940$	2.47214	0.0806322
$941$	11.8197	0.385310	0.192655	−	0.981267i	$-0.438290\pi$
$941$	11.8197	0.385310	0.192655	+	0.981267i	$0.438290\pi$
$942$	0	0
$943$	−9.88854	−0.322015
$944$	−2.76393	−0.0899583
$945$	0	0
$946$	−10.4721	−0.340479
$947$	−16.9443	−0.550615	−0.275307	−	0.961356i	$-0.588780\pi$
$947$	−16.9443	−0.550615	−0.275307	+	0.961356i	$0.588780\pi$
$948$	0	0
$949$	−41.8885	−1.35976
$950$	−25.1246	−0.815150
$951$	0	0
$952$	2.47214	0.0801224
$953$	−22.9443	−0.743238	−0.371619	−	0.928385i	$-0.621197\pi$
$953$	−22.9443	−0.743238	−0.371619	+	0.928385i	$0.621197\pi$
$954$	0	0
$955$	3.05573	0.0988810
$956$	−20.0000	−0.646846
$957$	0	0
$958$	12.3607	0.399355
$959$	19.8885	0.642235
$960$	0	0
$961$	−27.0000	−0.870968
$962$	22.4721	0.724531
$963$	0	0
$964$	15.4164	0.496529
$965$	−18.4721	−0.594639
$966$	0	0
$967$	45.8885	1.47568	0.737838	−	0.674978i	$-0.235847\pi$
$967$	45.8885	1.47568	0.737838	+	0.674978i	$0.235847\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−15.4164	−0.494991
$971$	−50.5410	−1.62194	−0.810969	−	0.585089i	$-0.801060\pi$
$971$	−50.5410	−1.62194	−0.810969	+	0.585089i	$0.801060\pi$
$972$	0	0
$973$	−21.7082	−0.695933
$974$	−16.9443	−0.542929
$975$	0	0
$976$	−0.763932	−0.0244529
$977$	28.8328	0.922443	0.461222	−	0.887285i	$-0.347411\pi$
$977$	28.8328	0.922443	0.461222	+	0.887285i	$0.347411\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	1.23607	0.0394847
$981$	0	0
$982$	−16.9443	−0.540713
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	0	0
$985$	−22.2492	−0.708919
$986$	−11.0557	−0.352086
$987$	0	0
$988$	23.4164	0.744975
$989$	41.8885	1.33198
$990$	0	0
$991$	−0.360680	−0.0114574	−0.00572869	−	0.999984i	$-0.501824\pi$
$991$	−0.360680	−0.0114574	−0.00572869	+	0.999984i	$0.501824\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	6.47214	0.205284
$995$	23.4164	0.742350
$996$	0	0
$997$	24.1803	0.765799	0.382900	−	0.923790i	$-0.374926\pi$
$997$	24.1803	0.765799	0.382900	+	0.923790i	$0.374926\pi$
$998$	32.3607	1.02436
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.m.1.2		2
3.2	odd	2		154.2.a.d.1.2	✓	2
7.6	odd	2		9702.2.a.cu.1.1		2
12.11	even	2		1232.2.a.p.1.1		2
15.2	even	4		3850.2.c.q.1849.3		4
15.8	even	4		3850.2.c.q.1849.2		4
15.14	odd	2		3850.2.a.bj.1.1		2
21.2	odd	6		1078.2.e.q.67.1		4
21.5	even	6		1078.2.e.n.67.2		4
21.11	odd	6		1078.2.e.q.177.1		4
21.17	even	6		1078.2.e.n.177.2		4
21.20	even	2		1078.2.a.w.1.1		2
24.5	odd	2		4928.2.a.bt.1.1		2
24.11	even	2		4928.2.a.bk.1.2		2
33.32	even	2		1694.2.a.l.1.2		2
84.83	odd	2		8624.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.d.1.2	✓	2	3.2	odd	2
1078.2.a.w.1.1		2	21.20	even	2
1078.2.e.n.67.2		4	21.5	even	6
1078.2.e.n.177.2		4	21.17	even	6
1078.2.e.q.67.1		4	21.2	odd	6
1078.2.e.q.177.1		4	21.11	odd	6
1232.2.a.p.1.1		2	12.11	even	2
1386.2.a.m.1.2		2	1.1	even	1	trivial
1694.2.a.l.1.2		2	33.32	even	2
3850.2.a.bj.1.1		2	15.14	odd	2
3850.2.c.q.1849.2		4	15.8	even	4
3850.2.c.q.1849.3		4	15.2	even	4
4928.2.a.bk.1.2		2	24.11	even	2
4928.2.a.bt.1.1		2	24.5	odd	2
8624.2.a.bf.1.2		2	84.83	odd	2
9702.2.a.cu.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.23607 q^{10} -1.00000 q^{11} -3.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} -7.23607 q^{19} +1.23607 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.47214 q^{25} +3.23607 q^{26} +1.00000 q^{28} -4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.47214 q^{34} +1.23607 q^{35} +6.94427 q^{37} +7.23607 q^{38} -1.23607 q^{40} +2.47214 q^{41} -10.4721 q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} -8.47214 q^{53} -1.23607 q^{55} -1.00000 q^{56} +4.47214 q^{58} -2.76393 q^{59} -0.763932 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +11.4164 q^{67} -2.47214 q^{68} -1.23607 q^{70} -6.47214 q^{71} +12.9443 q^{73} -6.94427 q^{74} -7.23607 q^{76} -1.00000 q^{77} +1.23607 q^{80} -2.47214 q^{82} +12.1803 q^{83} -3.05573 q^{85} +10.4721 q^{86} +1.00000 q^{88} -10.0000 q^{89} -3.23607 q^{91} -4.00000 q^{92} -2.00000 q^{94} -8.94427 q^{95} +12.4721 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 10 q^{38} + 2 q^{40} - 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} - 2 q^{56} - 10 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 4 q^{68} + 2 q^{70} - 4 q^{71} + 8 q^{73} + 4 q^{74} - 10 q^{76} - 2 q^{77} - 2 q^{80} + 4 q^{82} + 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} - 20 q^{89} - 2 q^{91} - 8 q^{92} - 4 q^{94} + 16 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 + 4 * q^31 - 2 * q^32 - 4 * q^34 - 2 * q^35 - 4 * q^37 + 10 * q^38 + 2 * q^40 - 4 * q^41 - 12 * q^43 - 2 * q^44 + 8 * q^46 + 4 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 8 * q^53 + 2 * q^55 - 2 * q^56 - 10 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^65 - 4 * q^67 + 4 * q^68 + 2 * q^70 - 4 * q^71 + 8 * q^73 + 4 * q^74 - 10 * q^76 - 2 * q^77 - 2 * q^80 + 4 * q^82 + 2 * q^83 - 24 * q^85 + 12 * q^86 + 2 * q^88 - 20 * q^89 - 2 * q^91 - 8 * q^92 - 4 * q^94 + 16 * q^97 - 2 * q^98

Introduction

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