LMFDB - Embedded newform 3450.2.d.b.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$27.5483886973$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.b.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +3.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} -3.00000 q^{21} -3.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} -3.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} -3.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +3.00000i q^{38} -3.00000 q^{39} -9.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} -1.00000 q^{46} +2.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} -3.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} -8.00000 q^{59} +12.0000 q^{61} -10.0000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +4.00000i q^{68} -1.00000 q^{69} -14.0000 q^{71} +1.00000i q^{72} -13.0000i q^{73} +8.00000 q^{74} -3.00000 q^{76} -9.00000i q^{77} -3.00000i q^{78} +17.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -1.00000i q^{83} +3.00000 q^{84} +1.00000 q^{86} -3.00000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -9.00000 q^{91} -1.00000i q^{92} -10.0000i q^{93} -2.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} -2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{19} - 6 q^{21} + 2 q^{24} - 6 q^{26} - 6 q^{29} - 20 q^{31} + 8 q^{34} + 2 q^{36} - 6 q^{39} - 18 q^{41} + 6 q^{44} - 2 q^{46} - 4 q^{49} + 8 q^{51} + 2 q^{54} + 6 q^{56} - 16 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} - 2 q^{69} - 28 q^{71} + 16 q^{74} - 6 q^{76} + 34 q^{79} + 2 q^{81} + 6 q^{84} + 2 q^{86} + 12 q^{89} - 18 q^{91} - 4 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^19 - 6 * q^21 + 2 * q^24 - 6 * q^26 - 6 * q^29 - 20 * q^31 + 8 * q^34 + 2 * q^36 - 6 * q^39 - 18 * q^41 + 6 * q^44 - 2 * q^46 - 4 * q^49 + 8 * q^51 + 2 * q^54 + 6 * q^56 - 16 * q^59 + 24 * q^61 - 2 * q^64 + 6 * q^66 - 2 * q^69 - 28 * q^71 + 16 * q^74 - 6 * q^76 + 34 * q^79 + 2 * q^81 + 6 * q^84 + 2 * q^86 + 12 * q^89 - 18 * q^91 - 4 * q^94 - 2 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 1.00000i	− 0.288675i
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	− 3.00000i	− 0.639602i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	1.00000	0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	− 1.00000i	− 0.192450i
$28$	− 3.00000i	− 0.566947i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000i	0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	3.00000i	0.486664i
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	− 3.00000i	− 0.462910i
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−1.00000	−0.147442
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	1.00000i	0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	4.00000	0.560112
$52$	− 3.00000i	− 0.416025i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	3.00000i	0.397360i
$58$	− 3.00000i	− 0.393919i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	− 10.0000i	− 1.27000i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	4.00000i	0.485071i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	1.00000i	0.117851i
$73$	− 13.0000i	− 1.52153i	−0.649025	−	0.760767i	$-0.724823\pi$
$73$	− 13.0000i	− 1.52153i	0.649025	−	0.760767i	$-0.275177\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−3.00000	−0.344124
$77$	− 9.00000i	− 1.02565i
$78$	− 3.00000i	− 0.339683i
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 9.00000i	− 0.993884i
$83$	− 1.00000i	− 0.109764i	−0.998493	−	0.0548821i	$-0.982522\pi$
$83$	− 1.00000i	− 0.109764i	0.998493	−	0.0548821i	$-0.0174783\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	1.00000	0.107833
$87$	− 3.00000i	− 0.321634i
$88$	3.00000i	0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−9.00000	−0.943456
$92$	− 1.00000i	− 0.104257i
$93$	− 10.0000i	− 1.03695i
$94$	−2.00000	−0.206284
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 2.00000i	− 0.202031i
$99$	3.00000	0.301511
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	4.00000i	0.396059i
$103$	− 13.0000i	− 1.28093i	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	− 13.0000i	− 1.28093i	0.767988	−	0.640464i	$-0.221258\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	3.00000i	0.283473i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	3.00000	0.278543
$117$	− 3.00000i	− 0.277350i
$118$	− 8.00000i	− 0.736460i
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	12.0000i	1.08643i
$123$	− 9.00000i	− 0.811503i
$124$	10.0000	0.898027
$125$	0	0
$126$	3.00000	0.267261
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	3.00000i	0.261116i
$133$	9.00000i	0.780399i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	− 14.0000i	− 1.17485i
$143$	− 9.00000i	− 0.752618i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	13.0000	1.07589
$147$	− 2.00000i	− 0.164957i
$148$	8.00000i	0.657596i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	− 3.00000i	− 0.243332i
$153$	4.00000i	0.323381i
$154$	9.00000	0.725241
$155$	0	0
$156$	3.00000	0.240192
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	17.0000i	1.35245i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.00000i	0.0785674i
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	1.00000	0.0776151
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	3.00000i	0.231455i
$169$	4.00000	0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	1.00000i	0.0762493i
$173$	− 15.0000i	− 1.14043i	−0.821496	−	0.570214i	$-0.806860\pi$
$173$	− 15.0000i	− 1.14043i	0.821496	−	0.570214i	$-0.193140\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 8.00000i	− 0.601317i
$178$	6.00000i	0.449719i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 9.00000i	− 0.667124i
$183$	12.0000i	0.887066i
$184$	1.00000	0.0737210
$185$	0	0
$186$	10.0000	0.733236
$187$	12.0000i	0.877527i
$188$	− 2.00000i	− 0.145865i
$189$	3.00000	0.218218
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	2.00000	0.142857
$197$	− 17.0000i	− 1.21120i	−0.795769	−	0.605600i	$-0.792933\pi$
$197$	− 17.0000i	− 1.21120i	0.795769	−	0.605600i	$-0.207067\pi$
$198$	3.00000i	0.213201i
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	− 6.00000i	− 0.422159i
$203$	− 9.00000i	− 0.631676i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	13.0000	0.905753
$207$	− 1.00000i	− 0.0695048i
$208$	3.00000i	0.208013i
$209$	−9.00000	−0.622543
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 14.0000i	− 0.959264i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 30.0000i	− 2.03653i
$218$	0	0
$219$	13.0000	0.878459
$220$	0	0
$221$	12.0000	0.807207
$222$	8.00000i	0.536925i
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	6.00000	0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 3.00000i	− 0.198680i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	3.00000i	0.196960i
$233$	1.00000i	0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$	1.00000i	0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	8.00000	0.520756
$237$	17.0000i	1.10427i
$238$	12.0000i	0.777844i
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 2.00000i	− 0.128565i
$243$	1.00000i	0.0641500i
$244$	−12.0000	−0.768221
$245$	0	0
$246$	9.00000	0.573819
$247$	9.00000i	0.572656i
$248$	10.0000i	0.635001i
$249$	1.00000	0.0633724
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	3.00000i	0.188982i
$253$	− 3.00000i	− 0.188608i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	1.00000i	0.0622573i
$259$	24.0000	1.49129
$260$	0	0
$261$	3.00000	0.185695
$262$	10.0000i	0.617802i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	−9.00000	−0.551825
$267$	6.00000i	0.367194i
$268$	− 8.00000i	− 0.488678i
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 9.00000i	− 0.544705i
$274$	18.0000	1.08742
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 13.0000i	− 0.781094i	−0.920583	−	0.390547i	$-0.872286\pi$
$277$	− 13.0000i	− 0.781094i	0.920583	−	0.390547i	$-0.127714\pi$
$278$	− 8.00000i	− 0.479808i
$279$	10.0000	0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	− 2.00000i	− 0.119098i
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	9.00000	0.532181
$287$	− 27.0000i	− 1.59376i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	13.0000i	0.760767i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	−8.00000	−0.464991
$297$	3.00000i	0.174078i
$298$	6.00000i	0.347571i
$299$	−3.00000	−0.173494
$300$	0	0
$301$	3.00000	0.172917
$302$	− 22.0000i	− 1.26596i
$303$	− 6.00000i	− 0.344691i
$304$	3.00000	0.172062
$305$	0	0
$306$	−4.00000	−0.228665
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	9.00000i	0.512823i
$309$	13.0000	0.739544
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	3.00000i	0.169842i
$313$	− 24.0000i	− 1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	− 24.0000i	− 1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−17.0000	−0.956325
$317$	31.0000i	1.74113i	0.492050	+	0.870567i	$0.336248\pi$
$317$	31.0000i	1.74113i	−0.492050	+	0.870567i	$0.663752\pi$
$318$	− 6.00000i	− 0.336463i
$319$	9.00000	0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	− 3.00000i	− 0.167183i
$323$	− 12.0000i	− 0.667698i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	9.00000i	0.496942i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	1.00000i	0.0548821i
$333$	8.00000i	0.438397i
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−3.00000	−0.163663
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	4.00000i	0.217571i
$339$	6.00000	0.325875
$340$	0	0
$341$	30.0000	1.62459
$342$	− 3.00000i	− 0.162221i
$343$	15.0000i	0.809924i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	15.0000	0.806405
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	3.00000i	0.160817i
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	− 3.00000i	− 0.159901i
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−6.00000	−0.317999
$357$	12.0000i	0.635107i
$358$	10.0000i	0.528516i
$359$	−31.0000	−1.63612	−0.818059	−	0.575135i	$-0.804950\pi$
$359$	−31.0000	−1.63612	−0.818059	+	0.575135i	$0.804950\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	− 16.0000i	− 0.840941i
$363$	− 2.00000i	− 0.104973i
$364$	9.00000	0.471728
$365$	0	0
$366$	−12.0000	−0.627250
$367$	27.0000i	1.40939i	0.709511	+	0.704694i	$0.248916\pi$
$367$	27.0000i	1.40939i	−0.709511	+	0.704694i	$0.751084\pi$
$368$	1.00000i	0.0521286i
$369$	9.00000	0.468521
$370$	0	0
$371$	−18.0000	−0.934513
$372$	10.0000i	0.518476i
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	2.00000	0.103142
$377$	− 9.00000i	− 0.463524i
$378$	3.00000i	0.154303i
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 15.0000i	− 0.767467i
$383$	29.0000i	1.48183i	0.671598	+	0.740915i	$0.265608\pi$
$383$	29.0000i	1.48183i	−0.671598	+	0.740915i	$0.734392\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	1.00000i	0.0508329i
$388$	− 6.00000i	− 0.304604i
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	2.00000i	0.101015i
$393$	10.0000i	0.504433i
$394$	17.0000	0.856448
$395$	0	0
$396$	−3.00000	−0.150756
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	− 11.0000i	− 0.551380i
$399$	−9.00000	−0.450564
$400$	0	0
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 30.0000i	− 1.49441i
$404$	6.00000	0.298511
$405$	0	0
$406$	9.00000	0.446663
$407$	24.0000i	1.18964i
$408$	− 4.00000i	− 0.198030i
$409$	23.0000	1.13728	0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000	1.13728	0.568638	+	0.822588i	$0.307470\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	13.0000i	0.640464i
$413$	− 24.0000i	− 1.18096i
$414$	1.00000	0.0491473
$415$	0	0
$416$	−3.00000	−0.147087
$417$	− 8.00000i	− 0.391762i
$418$	− 9.00000i	− 0.440204i
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	22.0000i	1.07094i
$423$	− 2.00000i	− 0.0972433i
$424$	6.00000	0.291386
$425$	0	0
$426$	14.0000	0.678302
$427$	36.0000i	1.74216i
$428$	12.0000i	0.580042i
$429$	9.00000	0.434524
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	30.0000	1.44005
$435$	0	0
$436$	0	0
$437$	3.00000i	0.143509i
$438$	13.0000i	0.621164i
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	12.0000i	0.570782i
$443$	− 26.0000i	− 1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$	− 26.0000i	− 1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−6.00000	−0.284108
$447$	6.00000i	0.283790i
$448$	− 3.00000i	− 0.141737i
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	27.0000	1.27138
$452$	6.00000i	0.282216i
$453$	− 22.0000i	− 1.03365i
$454$	12.0000	0.563188
$455$	0	0
$456$	3.00000	0.140488
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	− 6.00000i	− 0.280362i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	9.00000i	0.418718i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	11.0000i	0.509019i	0.967070	+	0.254510i	$0.0819141\pi$
$467$	11.0000i	0.509019i	−0.967070	+	0.254510i	$0.918086\pi$
$468$	3.00000i	0.138675i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	8.00000i	0.368230i
$473$	3.00000i	0.137940i
$474$	−17.0000	−0.780836
$475$	0	0
$476$	−12.0000	−0.550019
$477$	− 6.00000i	− 0.274721i
$478$	10.0000i	0.457389i
$479$	−33.0000	−1.50781	−0.753904	−	0.656984i	$-0.771832\pi$
$479$	−33.0000	−1.50781	−0.753904	+	0.656984i	$0.771832\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	− 10.0000i	− 0.455488i
$483$	− 3.00000i	− 0.136505i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	− 12.0000i	− 0.543214i
$489$	6.00000	0.271329
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	9.00000i	0.405751i
$493$	12.0000i	0.540453i
$494$	−9.00000	−0.404929
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 42.0000i	− 1.88396i
$498$	1.00000i	0.0448111i
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	− 16.0000i	− 0.714115i
$503$	35.0000i	1.56057i	0.625422	+	0.780286i	$0.284927\pi$
$503$	35.0000i	1.56057i	−0.625422	+	0.780286i	$0.715073\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	3.00000	0.133366
$507$	4.00000i	0.177646i
$508$	− 16.0000i	− 0.709885i
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	39.0000	1.72526
$512$	1.00000i	0.0441942i
$513$	− 3.00000i	− 0.132453i
$514$	18.0000	0.793946
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	− 6.00000i	− 0.263880i
$518$	24.0000i	1.05450i
$519$	15.0000	0.658427
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	3.00000i	0.131306i
$523$	− 11.0000i	− 0.480996i	−0.970650	−	0.240498i	$-0.922689\pi$
$523$	− 11.0000i	− 0.480996i	0.970650	−	0.240498i	$-0.0773108\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	−8.00000	−0.348817
$527$	40.0000i	1.74243i
$528$	− 3.00000i	− 0.130558i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	8.00000	0.347170
$532$	− 9.00000i	− 0.390199i
$533$	− 27.0000i	− 1.16950i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	8.00000	0.345547
$537$	10.0000i	0.431532i
$538$	5.00000i	0.215565i
$539$	6.00000	0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	22.0000i	0.944981i
$543$	− 16.0000i	− 0.686626i
$544$	4.00000	0.171499
$545$	0	0
$546$	9.00000	0.385164
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	18.0000i	0.768922i
$549$	−12.0000	−0.512148
$550$	0	0
$551$	−9.00000	−0.383413
$552$	1.00000i	0.0425628i
$553$	51.0000i	2.16874i
$554$	13.0000	0.552317
$555$	0	0
$556$	8.00000	0.339276
$557$	− 32.0000i	− 1.35588i	−0.735116	−	0.677942i	$-0.762872\pi$
$557$	− 32.0000i	− 1.35588i	0.735116	−	0.677942i	$-0.237128\pi$
$558$	10.0000i	0.423334i
$559$	3.00000	0.126886
$560$	0	0
$561$	−12.0000	−0.506640
$562$	− 10.0000i	− 0.421825i
$563$	− 5.00000i	− 0.210725i	−0.994434	−	0.105362i	$-0.966400\pi$
$563$	− 5.00000i	− 0.210725i	0.994434	−	0.105362i	$-0.0336003\pi$
$564$	2.00000	0.0842152
$565$	0	0
$566$	0	0
$567$	3.00000i	0.125988i
$568$	14.0000i	0.587427i
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	9.00000i	0.376309i
$573$	− 15.0000i	− 0.626634i
$574$	27.0000	1.12696
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 7.00000i	− 0.291414i	−0.989328	−	0.145707i	$-0.953454\pi$
$577$	− 7.00000i	− 0.291414i	0.989328	−	0.145707i	$-0.0465456\pi$
$578$	1.00000i	0.0415945i
$579$	−6.00000	−0.249351
$580$	0	0
$581$	3.00000	0.124461
$582$	− 6.00000i	− 0.248708i
$583$	− 18.0000i	− 0.745484i
$584$	−13.0000	−0.537944
$585$	0	0
$586$	0	0
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	2.00000i	0.0824786i
$589$	−30.0000	−1.23613
$590$	0	0
$591$	17.0000	0.699287
$592$	− 8.00000i	− 0.328798i
$593$	21.0000i	0.862367i	0.902264	+	0.431183i	$0.141904\pi$
$593$	21.0000i	0.862367i	−0.902264	+	0.431183i	$0.858096\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 11.0000i	− 0.450200i
$598$	− 3.00000i	− 0.122679i
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	3.00000i	0.122271i
$603$	− 8.00000i	− 0.325785i
$604$	22.0000	0.895167
$605$	0	0
$606$	6.00000	0.243733
$607$	2.00000i	0.0811775i	0.999176	+	0.0405887i	$0.0129233\pi$
$607$	2.00000i	0.0811775i	−0.999176	+	0.0405887i	$0.987077\pi$
$608$	3.00000i	0.121666i
$609$	9.00000	0.364698
$610$	0	0
$611$	−6.00000	−0.242734
$612$	− 4.00000i	− 0.161690i
$613$	− 20.0000i	− 0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$	− 20.0000i	− 0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−9.00000	−0.362620
$617$	− 16.0000i	− 0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$	− 16.0000i	− 0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$	13.0000i	0.522937i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	24.0000i	0.962312i
$623$	18.0000i	0.721155i
$624$	−3.00000	−0.120096
$625$	0	0
$626$	24.0000	0.959233
$627$	− 9.00000i	− 0.359425i
$628$	− 6.00000i	− 0.239426i
$629$	−32.0000	−1.27592
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	− 17.0000i	− 0.676224i
$633$	22.0000i	0.874421i
$634$	−31.0000	−1.23117
$635$	0	0
$636$	6.00000	0.237915
$637$	− 6.00000i	− 0.237729i
$638$	9.00000i	0.356313i
$639$	14.0000	0.553831
$640$	0	0
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	12.0000i	0.473602i
$643$	31.0000i	1.22252i	0.791430	+	0.611260i	$0.209337\pi$
$643$	31.0000i	1.22252i	−0.791430	+	0.611260i	$0.790663\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	12.0000	0.472134
$647$	− 18.0000i	− 0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$	− 18.0000i	− 0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	24.0000	0.942082
$650$	0	0
$651$	30.0000	1.17579
$652$	6.00000i	0.234978i
$653$	3.00000i	0.117399i	0.998276	+	0.0586995i	$0.0186954\pi$
$653$	3.00000i	0.117399i	−0.998276	+	0.0586995i	$0.981305\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	13.0000i	0.507178i
$658$	− 6.00000i	− 0.233904i
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 26.0000i	− 1.01052i
$663$	12.0000i	0.466041i
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	−8.00000	−0.309994
$667$	− 3.00000i	− 0.116160i
$668$	− 2.00000i	− 0.0773823i
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−36.0000	−1.38976
$672$	− 3.00000i	− 0.115728i
$673$	− 3.00000i	− 0.115642i	−0.998327	−	0.0578208i	$-0.981585\pi$
$673$	− 3.00000i	− 0.115642i	0.998327	−	0.0578208i	$-0.0184152\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−4.00000	−0.153846
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	6.00000i	0.230429i
$679$	−18.0000	−0.690777
$680$	0	0
$681$	12.0000	0.459841
$682$	30.0000i	1.14876i
$683$	− 38.0000i	− 1.45403i	−0.686622	−	0.727015i	$-0.740907\pi$
$683$	− 38.0000i	− 1.45403i	0.686622	−	0.727015i	$-0.259093\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	−15.0000	−0.572703
$687$	− 6.00000i	− 0.228914i
$688$	− 1.00000i	− 0.0381246i
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	15.0000i	0.570214i
$693$	9.00000i	0.341882i
$694$	0	0
$695$	0	0
$696$	−3.00000	−0.113715
$697$	36.0000i	1.36360i
$698$	5.00000i	0.189253i
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	3.00000i	0.113228i
$703$	− 24.0000i	− 0.905177i
$704$	3.00000	0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	− 18.0000i	− 0.676960i
$708$	8.00000i	0.300658i
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−17.0000	−0.637550
$712$	− 6.00000i	− 0.224860i
$713$	− 10.0000i	− 0.374503i
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−10.0000	−0.373718
$717$	10.0000i	0.373457i
$718$	− 31.0000i	− 1.15691i
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	39.0000	1.45244
$722$	− 10.0000i	− 0.372161i
$723$	− 10.0000i	− 0.371904i
$724$	16.0000	0.594635
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	9.00000i	0.333562i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	− 12.0000i	− 0.443533i
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−27.0000	−0.996588
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	− 24.0000i	− 0.884051i
$738$	9.00000i	0.331295i
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	−9.00000	−0.330623
$742$	− 18.0000i	− 0.660801i
$743$	29.0000i	1.06391i	0.846774	+	0.531953i	$0.178542\pi$
$743$	29.0000i	1.06391i	−0.846774	+	0.531953i	$0.821458\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	−16.0000	−0.585802
$747$	1.00000i	0.0365881i
$748$	− 12.0000i	− 0.438763i
$749$	36.0000	1.31541
$750$	0	0
$751$	45.0000	1.64207	0.821037	−	0.570875i	$-0.193396\pi$
$751$	45.0000	1.64207	0.821037	+	0.570875i	$0.193396\pi$
$752$	2.00000i	0.0729325i
$753$	− 16.0000i	− 0.583072i
$754$	9.00000	0.327761
$755$	0	0
$756$	−3.00000	−0.109109
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	− 12.0000i	− 0.435860i
$759$	3.00000	0.108893
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	− 16.0000i	− 0.579619i
$763$	0	0
$764$	15.0000	0.542681
$765$	0	0
$766$	−29.0000	−1.04781
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	−36.0000	−1.29819	−0.649097	−	0.760706i	$-0.724853\pi$
$769$	−36.0000	−1.29819	−0.649097	+	0.760706i	$0.724853\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 6.00000i	− 0.215945i
$773$	40.0000i	1.43870i	0.694648	+	0.719350i	$0.255560\pi$
$773$	40.0000i	1.43870i	−0.694648	+	0.719350i	$0.744440\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	6.00000	0.215387
$777$	24.0000i	0.860995i
$778$	− 38.0000i	− 1.36237i
$779$	−27.0000	−0.967375
$780$	0	0
$781$	42.0000	1.50288
$782$	4.00000i	0.143040i
$783$	3.00000i	0.107211i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	−10.0000	−0.356688
$787$	− 5.00000i	− 0.178231i	−0.996021	−	0.0891154i	$-0.971596\pi$
$787$	− 5.00000i	− 0.178231i	0.996021	−	0.0891154i	$-0.0284040\pi$
$788$	17.0000i	0.605600i
$789$	−8.00000	−0.284808
$790$	0	0
$791$	18.0000	0.640006
$792$	− 3.00000i	− 0.106600i
$793$	36.0000i	1.27840i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	11.0000	0.389885
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	− 9.00000i	− 0.318597i
$799$	8.00000	0.283020
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 28.0000i	− 0.988714i
$803$	39.0000i	1.37628i
$804$	8.00000	0.282138
$805$	0	0
$806$	30.0000	1.05670
$807$	5.00000i	0.176008i
$808$	6.00000i	0.211079i
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	9.00000i	0.315838i
$813$	22.0000i	0.771574i
$814$	−24.0000	−0.841200
$815$	0	0
$816$	4.00000	0.140028
$817$	− 3.00000i	− 0.104957i
$818$	23.0000i	0.804176i
$819$	9.00000	0.314485
$820$	0	0
$821$	−55.0000	−1.91951	−0.959757	−	0.280833i	$-0.909389\pi$
$821$	−55.0000	−1.91951	−0.959757	+	0.280833i	$0.909389\pi$
$822$	18.0000i	0.627822i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	24.0000	0.835067
$827$	− 9.00000i	− 0.312961i	−0.987681	−	0.156480i	$-0.949985\pi$
$827$	− 9.00000i	− 0.312961i	0.987681	−	0.156480i	$-0.0500148\pi$
$828$	1.00000i	0.0347524i
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	0	0
$831$	13.0000	0.450965
$832$	− 3.00000i	− 0.104006i
$833$	8.00000i	0.277184i
$834$	8.00000	0.277017
$835$	0	0
$836$	9.00000	0.311272
$837$	10.0000i	0.345651i
$838$	− 19.0000i	− 0.656344i
$839$	−33.0000	−1.13929	−0.569643	−	0.821892i	$-0.692919\pi$
$839$	−33.0000	−1.13929	−0.569643	+	0.821892i	$0.692919\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 10.0000i	− 0.344623i
$843$	− 10.0000i	− 0.344418i
$844$	−22.0000	−0.757271
$845$	0	0
$846$	2.00000	0.0687614
$847$	− 6.00000i	− 0.206162i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	8.00000	0.274236
$852$	14.0000i	0.479632i
$853$	23.0000i	0.787505i	0.919216	+	0.393753i	$0.128823\pi$
$853$	23.0000i	0.787505i	−0.919216	+	0.393753i	$0.871177\pi$
$854$	−36.0000	−1.23189
$855$	0	0
$856$	−12.0000	−0.410152
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	9.00000i	0.307255i
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	27.0000	0.920158
$862$	16.0000i	0.544962i
$863$	− 54.0000i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$	− 54.0000i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	1.00000i	0.0339618i
$868$	30.0000i	1.01827i
$869$	−51.0000	−1.73006
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−13.0000	−0.439229
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	4.00000i	0.134993i
$879$	0	0
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	2.00000i	0.0673435i
$883$	48.0000i	1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$	48.0000i	1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	26.0000	0.873487
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	− 8.00000i	− 0.268462i
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 6.00000i	− 0.200895i
$893$	6.00000i	0.200782i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	3.00000	0.100223
$897$	− 3.00000i	− 0.100167i
$898$	26.0000i	0.867631i
$899$	30.0000	1.00056
$900$	0	0
$901$	24.0000	0.799556
$902$	27.0000i	0.899002i
$903$	3.00000i	0.0998337i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	22.0000	0.730901
$907$	− 1.00000i	− 0.0332045i	−0.999862	−	0.0166022i	$-0.994715\pi$
$907$	− 1.00000i	− 0.0332045i	0.999862	−	0.0166022i	$-0.00528490\pi$
$908$	12.0000i	0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	55.0000	1.82223	0.911116	−	0.412151i	$-0.135222\pi$
$911$	55.0000	1.82223	0.911116	+	0.412151i	$0.135222\pi$
$912$	3.00000i	0.0993399i
$913$	3.00000i	0.0992855i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	6.00000	0.198246
$917$	30.0000i	0.990687i
$918$	− 4.00000i	− 0.132020i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	− 9.00000i	− 0.296399i
$923$	− 42.0000i	− 1.38245i
$924$	−9.00000	−0.296078
$925$	0	0
$926$	−4.00000	−0.131448
$927$	13.0000i	0.426976i
$928$	− 3.00000i	− 0.0984798i
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 1.00000i	− 0.0327561i
$933$	24.0000i	0.785725i
$934$	−11.0000	−0.359931
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	− 24.0000i	− 0.783628i
$939$	24.0000	0.783210
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	− 6.00000i	− 0.195491i
$943$	− 9.00000i	− 0.293080i
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	2.00000i	0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$	2.00000i	0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$	− 17.0000i	− 0.552134i
$949$	39.0000	1.26599
$950$	0	0
$951$	−31.0000	−1.00524
$952$	− 12.0000i	− 0.388922i
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−10.0000	−0.323423
$957$	9.00000i	0.290929i
$958$	− 33.0000i	− 1.06618i
$959$	54.0000	1.74375
$960$	0	0
$961$	69.0000	2.22581
$962$	24.0000i	0.773791i
$963$	12.0000i	0.386695i
$964$	10.0000	0.322078
$965$	0	0
$966$	3.00000	0.0965234
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	2.00000i	0.0642824i
$969$	12.0000	0.385496
$970$	0	0
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	2.00000	0.0640841
$975$	0	0
$976$	12.0000	0.384111
$977$	38.0000i	1.21573i	0.794041	+	0.607864i	$0.207973\pi$
$977$	38.0000i	1.21573i	−0.794041	+	0.607864i	$0.792027\pi$
$978$	6.00000i	0.191859i
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	12.0000i	0.382935i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	−12.0000	−0.382158
$987$	− 6.00000i	− 0.190982i
$988$	− 9.00000i	− 0.286328i
$989$	1.00000	0.0317982
$990$	0	0
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	− 10.0000i	− 0.317500i
$993$	− 26.0000i	− 0.825085i
$994$	42.0000	1.33216
$995$	0	0
$996$	−1.00000	−0.0316862
$997$	− 49.0000i	− 1.55185i	−0.630828	−	0.775923i	$-0.717285\pi$
$997$	− 49.0000i	− 1.55185i	0.630828	−	0.775923i	$-0.282715\pi$
$998$	− 34.0000i	− 1.07625i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.b.2899.2		2
5.2	odd	4		3450.2.a.h.1.1	✓	1
5.3	odd	4		3450.2.a.s.1.1	yes	1
5.4	even	2	inner	3450.2.d.b.2899.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.h.1.1	✓	1	5.2	odd	4
3450.2.a.s.1.1	yes	1	5.3	odd	4
3450.2.d.b.2899.1		2	5.4	even	2	inner
3450.2.d.b.2899.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +3.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} -3.00000 q^{21} -3.00000i q^{22} +1.00000i q^{23} +1.00000 q^{24} -3.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} -3.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +3.00000i q^{38} -3.00000 q^{39} -9.00000 q^{41} -3.00000i q^{42} -1.00000i q^{43} +3.00000 q^{44} -1.00000 q^{46} +2.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} -3.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} +3.00000i q^{57} -3.00000i q^{58} -8.00000 q^{59} +12.0000 q^{61} -10.0000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +8.00000i q^{67} +4.00000i q^{68} -1.00000 q^{69} -14.0000 q^{71} +1.00000i q^{72} -13.0000i q^{73} +8.00000 q^{74} -3.00000 q^{76} -9.00000i q^{77} -3.00000i q^{78} +17.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -1.00000i q^{83} +3.00000 q^{84} +1.00000 q^{86} -3.00000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -9.00000 q^{91} -1.00000i q^{92} -10.0000i q^{93} -2.00000 q^{94} -1.00000 q^{96} +6.00000i q^{97} -2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{19} - 6 q^{21} + 2 q^{24} - 6 q^{26} - 6 q^{29} - 20 q^{31} + 8 q^{34} + 2 q^{36} - 6 q^{39} - 18 q^{41} + 6 q^{44} - 2 q^{46} - 4 q^{49} + 8 q^{51} + 2 q^{54} + 6 q^{56} - 16 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} - 2 q^{69} - 28 q^{71} + 16 q^{74} - 6 q^{76} + 34 q^{79} + 2 q^{81} + 6 q^{84} + 2 q^{86} + 12 q^{89} - 18 q^{91} - 4 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^19 - 6 * q^21 + 2 * q^24 - 6 * q^26 - 6 * q^29 - 20 * q^31 + 8 * q^34 + 2 * q^36 - 6 * q^39 - 18 * q^41 + 6 * q^44 - 2 * q^46 - 4 * q^49 + 8 * q^51 + 2 * q^54 + 6 * q^56 - 16 * q^59 + 24 * q^61 - 2 * q^64 + 6 * q^66 - 2 * q^69 - 28 * q^71 + 16 * q^74 - 6 * q^76 + 34 * q^79 + 2 * q^81 + 6 * q^84 + 2 * q^86 + 12 * q^89 - 18 * q^91 - 4 * q^94 - 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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