LMFDB - Embedded newform 1950.2.e.p.1249.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.e (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:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.p.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.70156 q^{11} +1.00000i q^{12} +1.00000i q^{13} -4.70156 q^{14} +1.00000 q^{16} -0.701562i q^{17} -1.00000i q^{18} +1.70156 q^{19} +4.70156 q^{21} +4.70156i q^{22} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -4.70156i q^{28} +6.40312 q^{29} -10.1047 q^{31} +1.00000i q^{32} -4.70156i q^{33} +0.701562 q^{34} +1.00000 q^{36} -1.70156i q^{37} +1.70156i q^{38} +1.00000 q^{39} -3.70156 q^{41} +4.70156i q^{42} +11.4031i q^{43} -4.70156 q^{44} +7.00000i q^{47} -1.00000i q^{48} -15.1047 q^{49} -0.701562 q^{51} -1.00000i q^{52} -2.40312i q^{53} -1.00000 q^{54} +4.70156 q^{56} -1.70156i q^{57} +6.40312i q^{58} -2.70156 q^{59} +14.1047 q^{61} -10.1047i q^{62} -4.70156i q^{63} -1.00000 q^{64} +4.70156 q^{66} -6.40312i q^{67} +0.701562i q^{68} -1.70156 q^{71} +1.00000i q^{72} +12.0000i q^{73} +1.70156 q^{74} -1.70156 q^{76} +22.1047i q^{77} +1.00000i q^{78} +5.70156 q^{79} +1.00000 q^{81} -3.70156i q^{82} +10.7016i q^{83} -4.70156 q^{84} -11.4031 q^{86} -6.40312i q^{87} -4.70156i q^{88} -11.4031 q^{89} -4.70156 q^{91} +10.1047i q^{93} -7.00000 q^{94} +1.00000 q^{96} -2.59688i q^{97} -15.1047i q^{98} -4.70156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 6 q^{21} - 4 q^{24} - 4 q^{26} - 2 q^{31} - 10 q^{34} + 4 q^{36} + 4 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{49} + 10 q^{51} - 4 q^{54} + 6 q^{56} + 2 q^{59} + 18 q^{61} - 4 q^{64} + 6 q^{66} + 6 q^{71} - 6 q^{74} + 6 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} - 20 q^{86} - 20 q^{89} - 6 q^{91} - 28 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 6 * q^11 - 6 * q^14 + 4 * q^16 - 6 * q^19 + 6 * q^21 - 4 * q^24 - 4 * q^26 - 2 * q^31 - 10 * q^34 + 4 * q^36 + 4 * q^39 - 2 * q^41 - 6 * q^44 - 22 * q^49 + 10 * q^51 - 4 * q^54 + 6 * q^56 + 2 * q^59 + 18 * q^61 - 4 * q^64 + 6 * q^66 + 6 * q^71 - 6 * q^74 + 6 * q^76 + 10 * q^79 + 4 * q^81 - 6 * q^84 - 20 * q^86 - 20 * q^89 - 6 * q^91 - 28 * q^94 + 4 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	4.70156i	1.77702i	0.458854	+	0.888512i	$0.348260\pi$
$7$	4.70156i	1.77702i	−0.458854	+	0.888512i	$0.651740\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.70156	1.41757	0.708787	−	0.705422i	$-0.249243\pi$
$11$	4.70156	1.41757	0.708787	+	0.705422i	$0.249243\pi$
$12$	1.00000i	0.288675i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−4.70156	−1.25655
$15$	0	0
$16$	1.00000	0.250000
$17$	− 0.701562i	− 0.170154i	−0.996374	−	0.0850769i	$-0.972886\pi$
$17$	− 0.701562i	− 0.170154i	0.996374	−	0.0850769i	$-0.0271136\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.70156	0.390365	0.195183	−	0.980767i	$-0.437470\pi$
$19$	1.70156	0.390365	0.195183	+	0.980767i	$0.437470\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	4.70156i	1.00238i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	− 4.70156i	− 0.888512i
$29$	6.40312	1.18903	0.594515	−	0.804084i	$-0.297344\pi$
$29$	6.40312	1.18903	0.594515	+	0.804084i	$0.297344\pi$
$30$	0	0
$31$	−10.1047	−1.81486	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−10.1047	−1.81486	−0.907428	+	0.420208i	$0.861957\pi$
$32$	1.00000i	0.176777i
$33$	− 4.70156i	− 0.818437i
$34$	0.701562	0.120317
$35$	0	0
$36$	1.00000	0.166667
$37$	− 1.70156i	− 0.279735i	−0.990170	−	0.139868i	$-0.955332\pi$
$37$	− 1.70156i	− 0.279735i	0.990170	−	0.139868i	$-0.0446677\pi$
$38$	1.70156i	0.276030i
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.70156	−0.578087	−0.289043	−	0.957316i	$-0.593337\pi$
$41$	−3.70156	−0.578087	−0.289043	+	0.957316i	$0.593337\pi$
$42$	4.70156i	0.725467i
$43$	11.4031i	1.73896i	0.493968	+	0.869480i	$0.335546\pi$
$43$	11.4031i	1.73896i	−0.493968	+	0.869480i	$0.664454\pi$
$44$	−4.70156	−0.708787
$45$	0	0
$46$	0	0
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−15.1047	−2.15781
$50$	0	0
$51$	−0.701562	−0.0982383
$52$	− 1.00000i	− 0.138675i
$53$	− 2.40312i	− 0.330095i	−0.986286	−	0.165047i	$-0.947222\pi$
$53$	− 2.40312i	− 0.330095i	0.986286	−	0.165047i	$-0.0527777\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.70156	0.628273
$57$	− 1.70156i	− 0.225377i
$58$	6.40312i	0.840771i
$59$	−2.70156	−0.351713	−0.175857	−	0.984416i	$-0.556270\pi$
$59$	−2.70156	−0.351713	−0.175857	+	0.984416i	$0.556270\pi$
$60$	0	0
$61$	14.1047	1.80592	0.902960	−	0.429725i	$-0.141389\pi$
$61$	14.1047	1.80592	0.902960	+	0.429725i	$0.141389\pi$
$62$	− 10.1047i	− 1.28330i
$63$	− 4.70156i	− 0.592341i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.70156	0.578722
$67$	− 6.40312i	− 0.782266i	−0.920334	−	0.391133i	$-0.872083\pi$
$67$	− 6.40312i	− 0.782266i	0.920334	−	0.391133i	$-0.127917\pi$
$68$	0.701562i	0.0850769i
$69$	0	0
$70$	0	0
$71$	−1.70156	−0.201938	−0.100969	−	0.994890i	$-0.532194\pi$
$71$	−1.70156	−0.201938	−0.100969	+	0.994890i	$0.532194\pi$
$72$	1.00000i	0.117851i
$73$	12.0000i	1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$	12.0000i	1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$	1.70156	0.197803
$75$	0	0
$76$	−1.70156	−0.195183
$77$	22.1047i	2.51906i
$78$	1.00000i	0.113228i
$79$	5.70156	0.641476	0.320738	−	0.947168i	$-0.396069\pi$
$79$	5.70156	0.641476	0.320738	+	0.947168i	$0.396069\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.70156i	− 0.408769i
$83$	10.7016i	1.17465i	0.809352	+	0.587325i	$0.199819\pi$
$83$	10.7016i	1.17465i	−0.809352	+	0.587325i	$0.800181\pi$
$84$	−4.70156	−0.512982
$85$	0	0
$86$	−11.4031	−1.22963
$87$	− 6.40312i	− 0.686487i
$88$	− 4.70156i	− 0.501188i
$89$	−11.4031	−1.20873	−0.604364	−	0.796708i	$-0.706573\pi$
$89$	−11.4031	−1.20873	−0.604364	+	0.796708i	$0.706573\pi$
$90$	0	0
$91$	−4.70156	−0.492858
$92$	0	0
$93$	10.1047i	1.04781i
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.59688i	− 0.263673i	−0.991271	−	0.131836i	$-0.957913\pi$
$97$	− 2.59688i	− 0.263673i	0.991271	−	0.131836i	$-0.0420874\pi$
$98$	− 15.1047i	− 1.52580i
$99$	−4.70156	−0.472525
$100$	0	0
$101$	6.70156	0.666830	0.333415	−	0.942780i	$-0.391799\pi$
$101$	6.70156	0.666830	0.333415	+	0.942780i	$0.391799\pi$
$102$	− 0.701562i	− 0.0694650i
$103$	− 1.40312i	− 0.138254i	−0.997608	−	0.0691270i	$-0.977979\pi$
$103$	− 1.40312i	− 0.138254i	0.997608	−	0.0691270i	$-0.0220214\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	2.40312	0.233412
$107$	19.1047i	1.84692i	0.383695	+	0.923460i	$0.374651\pi$
$107$	19.1047i	1.84692i	−0.383695	+	0.923460i	$0.625349\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−4.29844	−0.411716	−0.205858	−	0.978582i	$-0.565998\pi$
$109$	−4.29844	−0.411716	−0.205858	+	0.978582i	$0.565998\pi$
$110$	0	0
$111$	−1.70156	−0.161505
$112$	4.70156i	0.444256i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	1.70156	0.159366
$115$	0	0
$116$	−6.40312	−0.594515
$117$	− 1.00000i	− 0.0924500i
$118$	− 2.70156i	− 0.248699i
$119$	3.29844	0.302367
$120$	0	0
$121$	11.1047	1.00952
$122$	14.1047i	1.27698i
$123$	3.70156i	0.333759i
$124$	10.1047	0.907428
$125$	0	0
$126$	4.70156	0.418848
$127$	− 6.29844i	− 0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$127$	− 6.29844i	− 0.558896i	0.960161	−	0.279448i	$-0.0901515\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	11.4031	1.00399
$130$	0	0
$131$	22.5078	1.96652	0.983258	−	0.182217i	$-0.0583275\pi$
$131$	22.5078	1.96652	0.983258	+	0.182217i	$0.0583275\pi$
$132$	4.70156i	0.409218i
$133$	8.00000i	0.693688i
$134$	6.40312	0.553146
$135$	0	0
$136$	−0.701562	−0.0601585
$137$	13.7016i	1.17060i	0.810816	+	0.585302i	$0.199024\pi$
$137$	13.7016i	1.17060i	−0.810816	+	0.585302i	$0.800976\pi$
$138$	0	0
$139$	−9.40312	−0.797563	−0.398781	−	0.917046i	$-0.630567\pi$
$139$	−9.40312	−0.797563	−0.398781	+	0.917046i	$0.630567\pi$
$140$	0	0
$141$	7.00000	0.589506
$142$	− 1.70156i	− 0.142792i
$143$	4.70156i	0.393164i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−12.0000	−0.993127
$147$	15.1047i	1.24581i
$148$	1.70156i	0.139868i
$149$	−6.59688	−0.540437	−0.270219	−	0.962799i	$-0.587096\pi$
$149$	−6.59688	−0.540437	−0.270219	+	0.962799i	$0.587096\pi$
$150$	0	0
$151$	−14.1047	−1.14782	−0.573912	−	0.818917i	$-0.694575\pi$
$151$	−14.1047	−1.14782	−0.573912	+	0.818917i	$0.694575\pi$
$152$	− 1.70156i	− 0.138015i
$153$	0.701562i	0.0567179i
$154$	−22.1047	−1.78125
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	− 22.7016i	− 1.81178i	−0.423511	−	0.905891i	$-0.639203\pi$
$157$	− 22.7016i	− 1.81178i	0.423511	−	0.905891i	$-0.360797\pi$
$158$	5.70156i	0.453592i
$159$	−2.40312	−0.190580
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	18.8062i	1.47302i	0.676427	+	0.736510i	$0.263527\pi$
$163$	18.8062i	1.47302i	−0.676427	+	0.736510i	$0.736473\pi$
$164$	3.70156	0.289043
$165$	0	0
$166$	−10.7016	−0.830602
$167$	− 1.10469i	− 0.0854832i	−0.999086	−	0.0427416i	$-0.986391\pi$
$167$	− 1.10469i	− 0.0854832i	0.999086	−	0.0427416i	$-0.0136092\pi$
$168$	− 4.70156i	− 0.362733i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−1.70156	−0.130122
$172$	− 11.4031i	− 0.869480i
$173$	0.193752i	0.0147307i	0.999973	+	0.00736533i	$0.00234448\pi$
$173$	0.193752i	0.0147307i	−0.999973	+	0.00736533i	$0.997656\pi$
$174$	6.40312	0.485420
$175$	0	0
$176$	4.70156	0.354394
$177$	2.70156i	0.203062i
$178$	− 11.4031i	− 0.854700i
$179$	24.2094	1.80949	0.904747	−	0.425950i	$-0.140060\pi$
$179$	24.2094	1.80949	0.904747	+	0.425950i	$0.140060\pi$
$180$	0	0
$181$	11.2984	0.839806	0.419903	−	0.907569i	$-0.362064\pi$
$181$	11.2984	0.839806	0.419903	+	0.907569i	$0.362064\pi$
$182$	− 4.70156i	− 0.348503i
$183$	− 14.1047i	− 1.04265i
$184$	0	0
$185$	0	0
$186$	−10.1047	−0.740912
$187$	− 3.29844i	− 0.241206i
$188$	− 7.00000i	− 0.510527i
$189$	−4.70156	−0.341988
$190$	0	0
$191$	−12.8062	−0.926628	−0.463314	−	0.886194i	$-0.653340\pi$
$191$	−12.8062	−0.926628	−0.463314	+	0.886194i	$0.653340\pi$
$192$	1.00000i	0.0721688i
$193$	− 16.2094i	− 1.16678i	−0.812194	−	0.583388i	$-0.801727\pi$
$193$	− 16.2094i	− 1.16678i	0.812194	−	0.583388i	$-0.198273\pi$
$194$	2.59688	0.186445
$195$	0	0
$196$	15.1047	1.07891
$197$	− 5.40312i	− 0.384957i	−0.981301	−	0.192478i	$-0.938347\pi$
$197$	− 5.40312i	− 0.384957i	0.981301	−	0.192478i	$-0.0616525\pi$
$198$	− 4.70156i	− 0.334125i
$199$	−8.29844	−0.588261	−0.294130	−	0.955765i	$-0.595030\pi$
$199$	−8.29844	−0.588261	−0.294130	+	0.955765i	$0.595030\pi$
$200$	0	0
$201$	−6.40312	−0.451642
$202$	6.70156i	0.471520i
$203$	30.1047i	2.11293i
$204$	0.701562	0.0491192
$205$	0	0
$206$	1.40312	0.0977603
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	8.00000	0.553372
$210$	0	0
$211$	6.80625	0.468561	0.234281	−	0.972169i	$-0.424727\pi$
$211$	6.80625	0.468561	0.234281	+	0.972169i	$0.424727\pi$
$212$	2.40312i	0.165047i
$213$	1.70156i	0.116589i
$214$	−19.1047	−1.30597
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 47.5078i	− 3.22504i
$218$	− 4.29844i	− 0.291127i
$219$	12.0000	0.810885
$220$	0	0
$221$	0.701562	0.0471922
$222$	− 1.70156i	− 0.114201i
$223$	− 11.4031i	− 0.763610i	−0.924243	−	0.381805i	$-0.875303\pi$
$223$	− 11.4031i	− 0.763610i	0.924243	−	0.381805i	$-0.124697\pi$
$224$	−4.70156	−0.314136
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 16.1047i	− 1.06891i	−0.845198	−	0.534453i	$-0.820518\pi$
$227$	− 16.1047i	− 1.06891i	0.845198	−	0.534453i	$-0.179482\pi$
$228$	1.70156i	0.112689i
$229$	−15.7016	−1.03759	−0.518794	−	0.854899i	$-0.673619\pi$
$229$	−15.7016	−1.03759	−0.518794	+	0.854899i	$0.673619\pi$
$230$	0	0
$231$	22.1047	1.45438
$232$	− 6.40312i	− 0.420386i
$233$	20.2094i	1.32396i	0.749521	+	0.661980i	$0.230284\pi$
$233$	20.2094i	1.32396i	−0.749521	+	0.661980i	$0.769716\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	2.70156	0.175857
$237$	− 5.70156i	− 0.370356i
$238$	3.29844i	0.213806i
$239$	−6.10469	−0.394879	−0.197440	−	0.980315i	$-0.563263\pi$
$239$	−6.10469	−0.394879	−0.197440	+	0.980315i	$0.563263\pi$
$240$	0	0
$241$	−2.59688	−0.167279	−0.0836397	−	0.996496i	$-0.526654\pi$
$241$	−2.59688	−0.167279	−0.0836397	+	0.996496i	$0.526654\pi$
$242$	11.1047i	0.713836i
$243$	− 1.00000i	− 0.0641500i
$244$	−14.1047	−0.902960
$245$	0	0
$246$	−3.70156	−0.236003
$247$	1.70156i	0.108268i
$248$	10.1047i	0.641648i
$249$	10.7016	0.678184
$250$	0	0
$251$	0.298438	0.0188372	0.00941862	−	0.999956i	$-0.497002\pi$
$251$	0.298438	0.0188372	0.00941862	+	0.999956i	$0.497002\pi$
$252$	4.70156i	0.296171i
$253$	0	0
$254$	6.29844	0.395199
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 15.2984i	− 0.954290i	−0.878824	−	0.477145i	$-0.841672\pi$
$257$	− 15.2984i	− 0.954290i	0.878824	−	0.477145i	$-0.158328\pi$
$258$	11.4031i	0.709928i
$259$	8.00000	0.497096
$260$	0	0
$261$	−6.40312	−0.396343
$262$	22.5078i	1.39054i
$263$	− 2.00000i	− 0.123325i	−0.998097	−	0.0616626i	$-0.980360\pi$
$263$	− 2.00000i	− 0.123325i	0.998097	−	0.0616626i	$-0.0196403\pi$
$264$	−4.70156	−0.289361
$265$	0	0
$266$	−8.00000	−0.490511
$267$	11.4031i	0.697860i
$268$	6.40312i	0.391133i
$269$	−31.2094	−1.90287	−0.951435	−	0.307851i	$-0.900390\pi$
$269$	−31.2094	−1.90287	−0.951435	+	0.307851i	$0.900390\pi$
$270$	0	0
$271$	19.5078	1.18502	0.592508	−	0.805565i	$-0.298138\pi$
$271$	19.5078	1.18502	0.592508	+	0.805565i	$0.298138\pi$
$272$	− 0.701562i	− 0.0425385i
$273$	4.70156i	0.284551i
$274$	−13.7016	−0.827742
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	− 9.40312i	− 0.563962i
$279$	10.1047	0.604952
$280$	0	0
$281$	−21.9109	−1.30710	−0.653548	−	0.756885i	$-0.726720\pi$
$281$	−21.9109	−1.30710	−0.653548	+	0.756885i	$0.726720\pi$
$282$	7.00000i	0.416844i
$283$	− 21.4031i	− 1.27228i	−0.771572	−	0.636142i	$-0.780529\pi$
$283$	− 21.4031i	− 1.27228i	0.771572	−	0.636142i	$-0.219471\pi$
$284$	1.70156	0.100969
$285$	0	0
$286$	−4.70156	−0.278009
$287$	− 17.4031i	− 1.02727i
$288$	− 1.00000i	− 0.0589256i
$289$	16.5078	0.971048
$290$	0	0
$291$	−2.59688	−0.152232
$292$	− 12.0000i	− 0.702247i
$293$	− 21.4031i	− 1.25038i	−0.780471	−	0.625192i	$-0.785021\pi$
$293$	− 21.4031i	− 1.25038i	0.780471	−	0.625192i	$-0.214979\pi$
$294$	−15.1047	−0.880923
$295$	0	0
$296$	−1.70156	−0.0989013
$297$	4.70156i	0.272812i
$298$	− 6.59688i	− 0.382147i
$299$	0	0
$300$	0	0
$301$	−53.6125	−3.09017
$302$	− 14.1047i	− 0.811633i
$303$	− 6.70156i	− 0.384995i
$304$	1.70156	0.0975913
$305$	0	0
$306$	−0.701562	−0.0401056
$307$	− 5.70156i	− 0.325405i	−0.986675	−	0.162703i	$-0.947979\pi$
$307$	− 5.70156i	− 0.325405i	0.986675	−	0.162703i	$-0.0520211\pi$
$308$	− 22.1047i	− 1.25953i
$309$	−1.40312	−0.0798209
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	21.2094i	1.19882i	0.800440	+	0.599412i	$0.204599\pi$
$313$	21.2094i	1.19882i	−0.800440	+	0.599412i	$0.795401\pi$
$314$	22.7016	1.28112
$315$	0	0
$316$	−5.70156	−0.320738
$317$	1.19375i	0.0670478i	0.999438	+	0.0335239i	$0.0106730\pi$
$317$	1.19375i	0.0670478i	−0.999438	+	0.0335239i	$0.989327\pi$
$318$	− 2.40312i	− 0.134761i
$319$	30.1047	1.68554
$320$	0	0
$321$	19.1047	1.06632
$322$	0	0
$323$	− 1.19375i	− 0.0664221i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−18.8062	−1.04158
$327$	4.29844i	0.237704i
$328$	3.70156i	0.204385i
$329$	−32.9109	−1.81444
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 10.7016i	− 0.587325i
$333$	1.70156i	0.0932450i
$334$	1.10469	0.0604457
$335$	0	0
$336$	4.70156	0.256491
$337$	8.10469i	0.441490i	0.975332	+	0.220745i	$0.0708489\pi$
$337$	8.10469i	0.441490i	−0.975332	+	0.220745i	$0.929151\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	14.0000	0.760376
$340$	0	0
$341$	−47.5078	−2.57269
$342$	− 1.70156i	− 0.0920099i
$343$	− 38.1047i	− 2.05746i
$344$	11.4031	0.614815
$345$	0	0
$346$	−0.193752	−0.0104161
$347$	10.5078i	0.564089i	0.959401	+	0.282044i	$0.0910126\pi$
$347$	10.5078i	0.564089i	−0.959401	+	0.282044i	$0.908987\pi$
$348$	6.40312i	0.343243i
$349$	11.4031	0.610395	0.305198	−	0.952289i	$-0.401277\pi$
$349$	11.4031	0.610395	0.305198	+	0.952289i	$0.401277\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	4.70156i	0.250594i
$353$	− 14.5078i	− 0.772173i	−0.922463	−	0.386086i	$-0.873827\pi$
$353$	− 14.5078i	− 0.772173i	0.922463	−	0.386086i	$-0.126173\pi$
$354$	−2.70156	−0.143586
$355$	0	0
$356$	11.4031	0.604364
$357$	− 3.29844i	− 0.174572i
$358$	24.2094i	1.27951i
$359$	34.6125	1.82678	0.913389	−	0.407088i	$-0.133456\pi$
$359$	34.6125	1.82678	0.913389	+	0.407088i	$0.133456\pi$
$360$	0	0
$361$	−16.1047	−0.847615
$362$	11.2984i	0.593833i
$363$	− 11.1047i	− 0.582845i
$364$	4.70156	0.246429
$365$	0	0
$366$	14.1047	0.737264
$367$	− 2.29844i	− 0.119977i	−0.998199	−	0.0599887i	$-0.980894\pi$
$367$	− 2.29844i	− 0.119977i	0.998199	−	0.0599887i	$-0.0191065\pi$
$368$	0	0
$369$	3.70156	0.192696
$370$	0	0
$371$	11.2984	0.586586
$372$	− 10.1047i	− 0.523904i
$373$	− 5.29844i	− 0.274343i	−0.990547	−	0.137171i	$-0.956199\pi$
$373$	− 5.29844i	− 0.274343i	0.990547	−	0.137171i	$-0.0438011\pi$
$374$	3.29844	0.170558
$375$	0	0
$376$	7.00000	0.360997
$377$	6.40312i	0.329778i
$378$	− 4.70156i	− 0.241822i
$379$	13.8953	0.713754	0.356877	−	0.934151i	$-0.383841\pi$
$379$	13.8953	0.713754	0.356877	+	0.934151i	$0.383841\pi$
$380$	0	0
$381$	−6.29844	−0.322679
$382$	− 12.8062i	− 0.655225i
$383$	− 32.5078i	− 1.66107i	−0.556965	−	0.830536i	$-0.688034\pi$
$383$	− 32.5078i	− 1.66107i	0.556965	−	0.830536i	$-0.311966\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	16.2094	0.825035
$387$	− 11.4031i	− 0.579653i
$388$	2.59688i	0.131836i
$389$	27.1047	1.37426	0.687131	−	0.726533i	$-0.258870\pi$
$389$	27.1047	1.37426	0.687131	+	0.726533i	$0.258870\pi$
$390$	0	0
$391$	0	0
$392$	15.1047i	0.762902i
$393$	− 22.5078i	− 1.13537i
$394$	5.40312	0.272205
$395$	0	0
$396$	4.70156	0.236262
$397$	− 21.9109i	− 1.09968i	−0.835271	−	0.549839i	$-0.814689\pi$
$397$	− 21.9109i	− 1.09968i	0.835271	−	0.549839i	$-0.185311\pi$
$398$	− 8.29844i	− 0.415963i
$399$	8.00000	0.400501
$400$	0	0
$401$	18.2094	0.909333	0.454666	−	0.890662i	$-0.349759\pi$
$401$	18.2094	0.909333	0.454666	+	0.890662i	$0.349759\pi$
$402$	− 6.40312i	− 0.319359i
$403$	− 10.1047i	− 0.503350i
$404$	−6.70156	−0.333415
$405$	0	0
$406$	−30.1047	−1.49407
$407$	− 8.00000i	− 0.396545i
$408$	0.701562i	0.0347325i
$409$	−29.4031	−1.45389	−0.726945	−	0.686695i	$-0.759061\pi$
$409$	−29.4031	−1.45389	−0.726945	+	0.686695i	$0.759061\pi$
$410$	0	0
$411$	13.7016	0.675848
$412$	1.40312i	0.0691270i
$413$	− 12.7016i	− 0.625003i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	9.40312i	0.460473i
$418$	8.00000i	0.391293i
$419$	−9.91093	−0.484181	−0.242090	−	0.970254i	$-0.577833\pi$
$419$	−9.91093	−0.484181	−0.242090	+	0.970254i	$0.577833\pi$
$420$	0	0
$421$	−0.596876	−0.0290899	−0.0145450	−	0.999894i	$-0.504630\pi$
$421$	−0.596876	−0.0290899	−0.0145450	+	0.999894i	$0.504630\pi$
$422$	6.80625i	0.331323i
$423$	− 7.00000i	− 0.340352i
$424$	−2.40312	−0.116706
$425$	0	0
$426$	−1.70156	−0.0824410
$427$	66.3141i	3.20916i
$428$	− 19.1047i	− 0.923460i
$429$	4.70156	0.226994
$430$	0	0
$431$	35.3141	1.70102	0.850509	−	0.525960i	$-0.176294\pi$
$431$	35.3141	1.70102	0.850509	+	0.525960i	$0.176294\pi$
$432$	1.00000i	0.0481125i
$433$	11.1047i	0.533657i	0.963744	+	0.266829i	$0.0859758\pi$
$433$	11.1047i	0.533657i	−0.963744	+	0.266829i	$0.914024\pi$
$434$	47.5078	2.28045
$435$	0	0
$436$	4.29844	0.205858
$437$	0	0
$438$	12.0000i	0.573382i
$439$	10.2984	0.491518	0.245759	−	0.969331i	$-0.420963\pi$
$439$	10.2984	0.491518	0.245759	+	0.969331i	$0.420963\pi$
$440$	0	0
$441$	15.1047	0.719271
$442$	0.701562i	0.0333699i
$443$	− 32.5078i	− 1.54449i	−0.635323	−	0.772246i	$-0.719133\pi$
$443$	− 32.5078i	− 1.54449i	0.635323	−	0.772246i	$-0.280867\pi$
$444$	1.70156	0.0807526
$445$	0	0
$446$	11.4031	0.539954
$447$	6.59688i	0.312022i
$448$	− 4.70156i	− 0.222128i
$449$	−2.29844	−0.108470	−0.0542350	−	0.998528i	$-0.517272\pi$
$449$	−2.29844	−0.108470	−0.0542350	+	0.998528i	$0.517272\pi$
$450$	0	0
$451$	−17.4031	−0.819481
$452$	− 14.0000i	− 0.658505i
$453$	14.1047i	0.662696i
$454$	16.1047	0.755830
$455$	0	0
$456$	−1.70156	−0.0796829
$457$	− 2.59688i	− 0.121477i	−0.998154	−	0.0607384i	$-0.980654\pi$
$457$	− 2.59688i	− 0.121477i	0.998154	−	0.0607384i	$-0.0193455\pi$
$458$	− 15.7016i	− 0.733686i
$459$	0.701562	0.0327461
$460$	0	0
$461$	36.2094	1.68644	0.843219	−	0.537570i	$-0.180658\pi$
$461$	36.2094	1.68644	0.843219	+	0.537570i	$0.180658\pi$
$462$	22.1047i	1.02840i
$463$	21.2984i	0.989822i	0.868944	+	0.494911i	$0.164799\pi$
$463$	21.2984i	0.989822i	−0.868944	+	0.494911i	$0.835201\pi$
$464$	6.40312	0.297258
$465$	0	0
$466$	−20.2094	−0.936181
$467$	30.2984i	1.40204i	0.713139	+	0.701022i	$0.247273\pi$
$467$	30.2984i	1.40204i	−0.713139	+	0.701022i	$0.752727\pi$
$468$	1.00000i	0.0462250i
$469$	30.1047	1.39011
$470$	0	0
$471$	−22.7016	−1.04603
$472$	2.70156i	0.124349i
$473$	53.6125i	2.46511i
$474$	5.70156	0.261881
$475$	0	0
$476$	−3.29844	−0.151184
$477$	2.40312i	0.110032i
$478$	− 6.10469i	− 0.279222i
$479$	40.6125	1.85563	0.927816	−	0.373038i	$-0.121684\pi$
$479$	40.6125	1.85563	0.927816	+	0.373038i	$0.121684\pi$
$480$	0	0
$481$	1.70156	0.0775846
$482$	− 2.59688i	− 0.118284i
$483$	0	0
$484$	−11.1047	−0.504758
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 7.89531i	− 0.357771i	−0.983870	−	0.178885i	$-0.942751\pi$
$487$	− 7.89531i	− 0.357771i	0.983870	−	0.178885i	$-0.0572491\pi$
$488$	− 14.1047i	− 0.638489i
$489$	18.8062	0.850448
$490$	0	0
$491$	33.4031	1.50746	0.753731	−	0.657183i	$-0.228252\pi$
$491$	33.4031	1.50746	0.753731	+	0.657183i	$0.228252\pi$
$492$	− 3.70156i	− 0.166879i
$493$	− 4.49219i	− 0.202318i
$494$	−1.70156	−0.0765569
$495$	0	0
$496$	−10.1047	−0.453714
$497$	− 8.00000i	− 0.358849i
$498$	10.7016i	0.479548i
$499$	−17.2094	−0.770397	−0.385199	−	0.922834i	$-0.625867\pi$
$499$	−17.2094	−0.770397	−0.385199	+	0.922834i	$0.625867\pi$
$500$	0	0
$501$	−1.10469	−0.0493537
$502$	0.298438i	0.0133199i
$503$	12.2094i	0.544389i	0.962242	+	0.272195i	$0.0877494\pi$
$503$	12.2094i	0.544389i	−0.962242	+	0.272195i	$0.912251\pi$
$504$	−4.70156	−0.209424
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	6.29844i	0.279448i
$509$	5.40312	0.239489	0.119745	−	0.992805i	$-0.461792\pi$
$509$	5.40312	0.239489	0.119745	+	0.992805i	$0.461792\pi$
$510$	0	0
$511$	−56.4187	−2.49582
$512$	1.00000i	0.0441942i
$513$	1.70156i	0.0751258i
$514$	15.2984	0.674785
$515$	0	0
$516$	−11.4031	−0.501995
$517$	32.9109i	1.44742i
$518$	8.00000i	0.351500i
$519$	0.193752	0.00850475
$520$	0	0
$521$	−36.2094	−1.58636	−0.793181	−	0.608986i	$-0.791576\pi$
$521$	−36.2094	−1.58636	−0.793181	+	0.608986i	$0.791576\pi$
$522$	− 6.40312i	− 0.280257i
$523$	24.8062i	1.08470i	0.840152	+	0.542351i	$0.182466\pi$
$523$	24.8062i	1.08470i	−0.840152	+	0.542351i	$0.817534\pi$
$524$	−22.5078	−0.983258
$525$	0	0
$526$	2.00000	0.0872041
$527$	7.08907i	0.308805i
$528$	− 4.70156i	− 0.204609i
$529$	23.0000	1.00000
$530$	0	0
$531$	2.70156	0.117238
$532$	− 8.00000i	− 0.346844i
$533$	− 3.70156i	− 0.160332i
$534$	−11.4031	−0.493461
$535$	0	0
$536$	−6.40312	−0.276573
$537$	− 24.2094i	− 1.04471i
$538$	− 31.2094i	− 1.34553i
$539$	−71.0156	−3.05886
$540$	0	0
$541$	−1.79063	−0.0769851	−0.0384925	−	0.999259i	$-0.512256\pi$
$541$	−1.79063	−0.0769851	−0.0384925	+	0.999259i	$0.512256\pi$
$542$	19.5078i	0.837932i
$543$	− 11.2984i	− 0.484862i
$544$	0.701562	0.0300792
$545$	0	0
$546$	−4.70156	−0.201208
$547$	− 31.6125i	− 1.35165i	−0.737061	−	0.675826i	$-0.763787\pi$
$547$	− 31.6125i	− 1.35165i	0.737061	−	0.675826i	$-0.236213\pi$
$548$	− 13.7016i	− 0.585302i
$549$	−14.1047	−0.601973
$550$	0	0
$551$	10.8953	0.464156
$552$	0	0
$553$	26.8062i	1.13992i
$554$	22.0000	0.934690
$555$	0	0
$556$	9.40312	0.398781
$557$	16.8062i	0.712104i	0.934466	+	0.356052i	$0.115877\pi$
$557$	16.8062i	0.712104i	−0.934466	+	0.356052i	$0.884123\pi$
$558$	10.1047i	0.427765i
$559$	−11.4031	−0.482301
$560$	0	0
$561$	−3.29844	−0.139260
$562$	− 21.9109i	− 0.924257i
$563$	32.5078i	1.37004i	0.728524	+	0.685020i	$0.240207\pi$
$563$	32.5078i	1.37004i	−0.728524	+	0.685020i	$0.759793\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	21.4031	0.899640
$567$	4.70156i	0.197447i
$568$	1.70156i	0.0713960i
$569$	11.2984	0.473655	0.236828	−	0.971552i	$-0.423892\pi$
$569$	11.2984	0.473655	0.236828	+	0.971552i	$0.423892\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	− 4.70156i	− 0.196582i
$573$	12.8062i	0.534989i
$574$	17.4031	0.726392
$575$	0	0
$576$	1.00000	0.0416667
$577$	24.2094i	1.00785i	0.863748	+	0.503925i	$0.168111\pi$
$577$	24.2094i	1.00785i	−0.863748	+	0.503925i	$0.831889\pi$
$578$	16.5078i	0.686634i
$579$	−16.2094	−0.673639
$580$	0	0
$581$	−50.3141	−2.08738
$582$	− 2.59688i	− 0.107644i
$583$	− 11.2984i	− 0.467933i
$584$	12.0000	0.496564
$585$	0	0
$586$	21.4031	0.884155
$587$	− 7.50781i	− 0.309881i	−0.987924	−	0.154940i	$-0.950481\pi$
$587$	− 7.50781i	− 0.309881i	0.987924	−	0.154940i	$-0.0495185\pi$
$588$	− 15.1047i	− 0.622907i
$589$	−17.1938	−0.708456
$590$	0	0
$591$	−5.40312	−0.222255
$592$	− 1.70156i	− 0.0699338i
$593$	− 17.9109i	− 0.735514i	−0.929922	−	0.367757i	$-0.880126\pi$
$593$	− 17.9109i	− 0.735514i	0.929922	−	0.367757i	$-0.119874\pi$
$594$	−4.70156	−0.192907
$595$	0	0
$596$	6.59688	0.270219
$597$	8.29844i	0.339632i
$598$	0	0
$599$	−8.20937	−0.335426	−0.167713	−	0.985836i	$-0.553638\pi$
$599$	−8.20937	−0.335426	−0.167713	+	0.985836i	$0.553638\pi$
$600$	0	0
$601$	10.6125	0.432893	0.216446	−	0.976295i	$-0.430553\pi$
$601$	10.6125	0.432893	0.216446	+	0.976295i	$0.430553\pi$
$602$	− 53.6125i	− 2.18508i
$603$	6.40312i	0.260755i
$604$	14.1047	0.573912
$605$	0	0
$606$	6.70156	0.272232
$607$	− 35.1047i	− 1.42486i	−0.701746	−	0.712428i	$-0.747596\pi$
$607$	− 35.1047i	− 1.42486i	0.701746	−	0.712428i	$-0.252404\pi$
$608$	1.70156i	0.0690075i
$609$	30.1047	1.21990
$610$	0	0
$611$	−7.00000	−0.283190
$612$	− 0.701562i	− 0.0283590i
$613$	12.8062i	0.517240i	0.965979	+	0.258620i	$0.0832677\pi$
$613$	12.8062i	0.517240i	−0.965979	+	0.258620i	$0.916732\pi$
$614$	5.70156	0.230096
$615$	0	0
$616$	22.1047	0.890623
$617$	− 27.1047i	− 1.09119i	−0.838048	−	0.545597i	$-0.816303\pi$
$617$	− 27.1047i	− 1.09119i	0.838048	−	0.545597i	$-0.183697\pi$
$618$	− 1.40312i	− 0.0564419i
$619$	13.1938	0.530302	0.265151	−	0.964207i	$-0.414578\pi$
$619$	13.1938	0.530302	0.265151	+	0.964207i	$0.414578\pi$
$620$	0	0
$621$	0	0
$622$	30.0000i	1.20289i
$623$	− 53.6125i	− 2.14794i
$624$	1.00000	0.0400320
$625$	0	0
$626$	−21.2094	−0.847697
$627$	− 8.00000i	− 0.319489i
$628$	22.7016i	0.905891i
$629$	−1.19375	−0.0475980
$630$	0	0
$631$	33.6125	1.33809	0.669046	−	0.743221i	$-0.266703\pi$
$631$	33.6125	1.33809	0.669046	+	0.743221i	$0.266703\pi$
$632$	− 5.70156i	− 0.226796i
$633$	− 6.80625i	− 0.270524i
$634$	−1.19375	−0.0474099
$635$	0	0
$636$	2.40312	0.0952901
$637$	− 15.1047i	− 0.598469i
$638$	30.1047i	1.19186i
$639$	1.70156	0.0673128
$640$	0	0
$641$	5.50781	0.217545	0.108773	−	0.994067i	$-0.465308\pi$
$641$	5.50781	0.217545	0.108773	+	0.994067i	$0.465308\pi$
$642$	19.1047i	0.754002i
$643$	10.2984i	0.406131i	0.979165	+	0.203065i	$0.0650904\pi$
$643$	10.2984i	0.406131i	−0.979165	+	0.203065i	$0.934910\pi$
$644$	0	0
$645$	0	0
$646$	1.19375	0.0469675
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−12.7016	−0.498580
$650$	0	0
$651$	−47.5078	−1.86198
$652$	− 18.8062i	− 0.736510i
$653$	21.2984i	0.833472i	0.909027	+	0.416736i	$0.136826\pi$
$653$	21.2984i	0.833472i	−0.909027	+	0.416736i	$0.863174\pi$
$654$	−4.29844	−0.168082
$655$	0	0
$656$	−3.70156	−0.144522
$657$	− 12.0000i	− 0.468165i
$658$	− 32.9109i	− 1.28300i
$659$	19.3141	0.752369	0.376184	−	0.926545i	$-0.377236\pi$
$659$	19.3141	0.752369	0.376184	+	0.926545i	$0.377236\pi$
$660$	0	0
$661$	−40.1203	−1.56050	−0.780250	−	0.625468i	$-0.784908\pi$
$661$	−40.1203	−1.56050	−0.780250	+	0.625468i	$0.784908\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 0.701562i	− 0.0272464i
$664$	10.7016	0.415301
$665$	0	0
$666$	−1.70156	−0.0659342
$667$	0	0
$668$	1.10469i	0.0427416i
$669$	−11.4031	−0.440870
$670$	0	0
$671$	66.3141	2.56003
$672$	4.70156i	0.181367i
$673$	− 36.0156i	− 1.38830i	−0.719830	−	0.694150i	$-0.755780\pi$
$673$	− 36.0156i	− 1.38830i	0.719830	−	0.694150i	$-0.244220\pi$
$674$	−8.10469	−0.312181
$675$	0	0
$676$	1.00000	0.0384615
$677$	31.6125i	1.21497i	0.794332	+	0.607483i	$0.207821\pi$
$677$	31.6125i	1.21497i	−0.794332	+	0.607483i	$0.792179\pi$
$678$	14.0000i	0.537667i
$679$	12.2094	0.468553
$680$	0	0
$681$	−16.1047	−0.617133
$682$	− 47.5078i	− 1.81917i
$683$	− 44.7016i	− 1.71046i	−0.518251	−	0.855229i	$-0.673417\pi$
$683$	− 44.7016i	− 1.71046i	0.518251	−	0.855229i	$-0.326583\pi$
$684$	1.70156	0.0650609
$685$	0	0
$686$	38.1047	1.45484
$687$	15.7016i	0.599052i
$688$	11.4031i	0.434740i
$689$	2.40312	0.0915517
$690$	0	0
$691$	−22.1938	−0.844290	−0.422145	−	0.906528i	$-0.638723\pi$
$691$	−22.1938	−0.844290	−0.422145	+	0.906528i	$0.638723\pi$
$692$	− 0.193752i	− 0.00736533i
$693$	− 22.1047i	− 0.839688i
$694$	−10.5078	−0.398871
$695$	0	0
$696$	−6.40312	−0.242710
$697$	2.59688i	0.0983637i
$698$	11.4031i	0.431615i
$699$	20.2094	0.764389
$700$	0	0
$701$	30.9109	1.16749	0.583745	−	0.811937i	$-0.301587\pi$
$701$	30.9109	1.16749	0.583745	+	0.811937i	$0.301587\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	− 2.89531i	− 0.109199i
$704$	−4.70156	−0.177197
$705$	0	0
$706$	14.5078	0.546009
$707$	31.5078i	1.18497i
$708$	− 2.70156i	− 0.101531i
$709$	−15.6125	−0.586340	−0.293170	−	0.956060i	$-0.594710\pi$
$709$	−15.6125	−0.586340	−0.293170	+	0.956060i	$0.594710\pi$
$710$	0	0
$711$	−5.70156	−0.213825
$712$	11.4031i	0.427350i
$713$	0	0
$714$	3.29844	0.123441
$715$	0	0
$716$	−24.2094	−0.904747
$717$	6.10469i	0.227984i
$718$	34.6125i	1.29173i
$719$	11.0156	0.410813	0.205407	−	0.978677i	$-0.434148\pi$
$719$	11.0156	0.410813	0.205407	+	0.978677i	$0.434148\pi$
$720$	0	0
$721$	6.59688	0.245680
$722$	− 16.1047i	− 0.599354i
$723$	2.59688i	0.0965788i
$724$	−11.2984	−0.419903
$725$	0	0
$726$	11.1047	0.412134
$727$	7.79063i	0.288938i	0.989509	+	0.144469i	$0.0461475\pi$
$727$	7.79063i	0.288938i	−0.989509	+	0.144469i	$0.953853\pi$
$728$	4.70156i	0.174251i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	14.1047i	0.521324i
$733$	− 47.9109i	− 1.76963i	−0.465942	−	0.884815i	$-0.654284\pi$
$733$	− 47.9109i	− 1.76963i	0.465942	−	0.884815i	$-0.345716\pi$
$734$	2.29844	0.0848369
$735$	0	0
$736$	0	0
$737$	− 30.1047i	− 1.10892i
$738$	3.70156i	0.136256i
$739$	−4.61250	−0.169673	−0.0848367	−	0.996395i	$-0.527037\pi$
$739$	−4.61250	−0.169673	−0.0848367	+	0.996395i	$0.527037\pi$
$740$	0	0
$741$	1.70156	0.0625084
$742$	11.2984i	0.414779i
$743$	30.6125i	1.12306i	0.827455	+	0.561532i	$0.189788\pi$
$743$	30.6125i	1.12306i	−0.827455	+	0.561532i	$0.810212\pi$
$744$	10.1047	0.370456
$745$	0	0
$746$	5.29844	0.193990
$747$	− 10.7016i	− 0.391550i
$748$	3.29844i	0.120603i
$749$	−89.8219	−3.28202
$750$	0	0
$751$	8.50781	0.310454	0.155227	−	0.987879i	$-0.450389\pi$
$751$	8.50781	0.310454	0.155227	+	0.987879i	$0.450389\pi$
$752$	7.00000i	0.255264i
$753$	− 0.298438i	− 0.0108757i
$754$	−6.40312	−0.233188
$755$	0	0
$756$	4.70156	0.170994
$757$	− 17.8953i	− 0.650416i	−0.945642	−	0.325208i	$-0.894566\pi$
$757$	− 17.8953i	− 0.650416i	0.945642	−	0.325208i	$-0.105434\pi$
$758$	13.8953i	0.504701i
$759$	0	0
$760$	0	0
$761$	−26.7172	−0.968497	−0.484249	−	0.874930i	$-0.660907\pi$
$761$	−26.7172	−0.968497	−0.484249	+	0.874930i	$0.660907\pi$
$762$	− 6.29844i	− 0.228168i
$763$	− 20.2094i	− 0.731628i
$764$	12.8062	0.463314
$765$	0	0
$766$	32.5078	1.17455
$767$	− 2.70156i	− 0.0975478i
$768$	− 1.00000i	− 0.0360844i
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	−15.2984	−0.550960
$772$	16.2094i	0.583388i
$773$	− 26.5969i	− 0.956623i	−0.878190	−	0.478312i	$-0.841249\pi$
$773$	− 26.5969i	− 0.956623i	0.878190	−	0.478312i	$-0.158751\pi$
$774$	11.4031	0.409877
$775$	0	0
$776$	−2.59688	−0.0932224
$777$	− 8.00000i	− 0.286998i
$778$	27.1047i	0.971750i
$779$	−6.29844	−0.225665
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	6.40312i	0.228829i
$784$	−15.1047	−0.539453
$785$	0	0
$786$	22.5078	0.802827
$787$	33.8953i	1.20824i	0.796894	+	0.604119i	$0.206475\pi$
$787$	33.8953i	1.20824i	−0.796894	+	0.604119i	$0.793525\pi$
$788$	5.40312i	0.192478i
$789$	−2.00000	−0.0712019
$790$	0	0
$791$	−65.8219	−2.34036
$792$	4.70156i	0.167063i
$793$	14.1047i	0.500872i
$794$	21.9109	0.777590
$795$	0	0
$796$	8.29844	0.294130
$797$	2.91093i	0.103111i	0.998670	+	0.0515553i	$0.0164178\pi$
$797$	2.91093i	0.103111i	−0.998670	+	0.0515553i	$0.983582\pi$
$798$	8.00000i	0.283197i
$799$	4.91093	0.173736
$800$	0	0
$801$	11.4031	0.402910
$802$	18.2094i	0.642995i
$803$	56.4187i	1.99097i
$804$	6.40312	0.225821
$805$	0	0
$806$	10.1047	0.355922
$807$	31.2094i	1.09862i
$808$	− 6.70156i	− 0.235760i
$809$	−7.19375	−0.252919	−0.126459	−	0.991972i	$-0.540361\pi$
$809$	−7.19375	−0.252919	−0.126459	+	0.991972i	$0.540361\pi$
$810$	0	0
$811$	48.7016	1.71014	0.855072	−	0.518510i	$-0.173513\pi$
$811$	48.7016	1.71014	0.855072	+	0.518510i	$0.173513\pi$
$812$	− 30.1047i	− 1.05647i
$813$	− 19.5078i	− 0.684169i
$814$	8.00000	0.280400
$815$	0	0
$816$	−0.701562	−0.0245596
$817$	19.4031i	0.678829i
$818$	− 29.4031i	− 1.02806i
$819$	4.70156	0.164286
$820$	0	0
$821$	−20.5969	−0.718836	−0.359418	−	0.933177i	$-0.617025\pi$
$821$	−20.5969	−0.718836	−0.359418	+	0.933177i	$0.617025\pi$
$822$	13.7016i	0.477897i
$823$	− 17.1047i	− 0.596232i	−0.954530	−	0.298116i	$-0.903642\pi$
$823$	− 17.1047i	− 0.596232i	0.954530	−	0.298116i	$-0.0963582\pi$
$824$	−1.40312	−0.0488801
$825$	0	0
$826$	12.7016	0.441944
$827$	20.7016i	0.719864i	0.932979	+	0.359932i	$0.117200\pi$
$827$	20.7016i	0.719864i	−0.932979	+	0.359932i	$0.882800\pi$
$828$	0	0
$829$	5.50781	0.191294	0.0956471	−	0.995415i	$-0.469508\pi$
$829$	5.50781	0.191294	0.0956471	+	0.995415i	$0.469508\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	− 1.00000i	− 0.0346688i
$833$	10.5969i	0.367160i
$834$	−9.40312	−0.325604
$835$	0	0
$836$	−8.00000	−0.276686
$837$	− 10.1047i	− 0.349269i
$838$	− 9.91093i	− 0.342368i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	12.0000	0.413793
$842$	− 0.596876i	− 0.0205697i
$843$	21.9109i	0.754653i
$844$	−6.80625	−0.234281
$845$	0	0
$846$	7.00000	0.240665
$847$	52.2094i	1.79394i
$848$	− 2.40312i	− 0.0825236i
$849$	−21.4031	−0.734553
$850$	0	0
$851$	0	0
$852$	− 1.70156i	− 0.0582946i
$853$	− 0.507811i	− 0.0173871i	−0.999962	−	0.00869355i	$-0.997233\pi$
$853$	− 0.507811i	− 0.0173871i	0.999962	−	0.00869355i	$-0.00276728\pi$
$854$	−66.3141	−2.26922
$855$	0	0
$856$	19.1047	0.652985
$857$	− 7.61250i	− 0.260038i	−0.991512	−	0.130019i	$-0.958496\pi$
$857$	− 7.61250i	− 0.260038i	0.991512	−	0.130019i	$-0.0415038\pi$
$858$	4.70156i	0.160509i
$859$	6.20937	0.211861	0.105931	−	0.994374i	$-0.466218\pi$
$859$	6.20937	0.211861	0.105931	+	0.994374i	$0.466218\pi$
$860$	0	0
$861$	−17.4031	−0.593097
$862$	35.3141i	1.20280i
$863$	− 54.2250i	− 1.84584i	−0.384991	−	0.922920i	$-0.625796\pi$
$863$	− 54.2250i	− 1.84584i	0.384991	−	0.922920i	$-0.374204\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−11.1047	−0.377353
$867$	− 16.5078i	− 0.560635i
$868$	47.5078i	1.61252i
$869$	26.8062	0.909340
$870$	0	0
$871$	6.40312	0.216962
$872$	4.29844i	0.145563i
$873$	2.59688i	0.0878909i
$874$	0	0
$875$	0	0
$876$	−12.0000	−0.405442
$877$	− 42.7172i	− 1.44246i	−0.692697	−	0.721228i	$-0.743578\pi$
$877$	− 42.7172i	− 1.44246i	0.692697	−	0.721228i	$-0.256422\pi$
$878$	10.2984i	0.347555i
$879$	−21.4031	−0.721909
$880$	0	0
$881$	−39.7172	−1.33811	−0.669053	−	0.743215i	$-0.733300\pi$
$881$	−39.7172	−1.33811	−0.669053	+	0.743215i	$0.733300\pi$
$882$	15.1047i	0.508601i
$883$	− 45.6125i	− 1.53498i	−0.641059	−	0.767491i	$-0.721505\pi$
$883$	− 45.6125i	− 1.53498i	0.641059	−	0.767491i	$-0.278495\pi$
$884$	−0.701562	−0.0235961
$885$	0	0
$886$	32.5078	1.09212
$887$	− 20.0000i	− 0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$	− 20.0000i	− 0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$	1.70156i	0.0571007i
$889$	29.6125	0.993171
$890$	0	0
$891$	4.70156	0.157508
$892$	11.4031i	0.381805i
$893$	11.9109i	0.398584i
$894$	−6.59688	−0.220633
$895$	0	0
$896$	4.70156	0.157068
$897$	0	0
$898$	− 2.29844i	− 0.0766999i
$899$	−64.7016	−2.15792
$900$	0	0
$901$	−1.68594	−0.0561668
$902$	− 17.4031i	− 0.579461i
$903$	53.6125i	1.78411i
$904$	14.0000	0.465633
$905$	0	0
$906$	−14.1047	−0.468597
$907$	− 25.0156i	− 0.830630i	−0.909678	−	0.415315i	$-0.863671\pi$
$907$	− 25.0156i	− 0.830630i	0.909678	−	0.415315i	$-0.136329\pi$
$908$	16.1047i	0.534453i
$909$	−6.70156	−0.222277
$910$	0	0
$911$	42.2094	1.39846	0.699229	−	0.714897i	$-0.253527\pi$
$911$	42.2094	1.39846	0.699229	+	0.714897i	$0.253527\pi$
$912$	− 1.70156i	− 0.0563444i
$913$	50.3141i	1.66515i
$914$	2.59688	0.0858970
$915$	0	0
$916$	15.7016	0.518794
$917$	105.822i	3.49455i
$918$	0.701562i	0.0231550i
$919$	4.89531	0.161481	0.0807407	−	0.996735i	$-0.474271\pi$
$919$	4.89531	0.161481	0.0807407	+	0.996735i	$0.474271\pi$
$920$	0	0
$921$	−5.70156	−0.187873
$922$	36.2094i	1.19249i
$923$	− 1.70156i	− 0.0560076i
$924$	−22.1047	−0.727191
$925$	0	0
$926$	−21.2984	−0.699910
$927$	1.40312i	0.0460846i
$928$	6.40312i	0.210193i
$929$	−52.2984	−1.71586	−0.857928	−	0.513770i	$-0.828249\pi$
$929$	−52.2984	−1.71586	−0.857928	+	0.513770i	$0.828249\pi$
$930$	0	0
$931$	−25.7016	−0.842335
$932$	− 20.2094i	− 0.661980i
$933$	− 30.0000i	− 0.982156i
$934$	−30.2984	−0.991395
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	14.9109i	0.487119i	0.969886	+	0.243560i	$0.0783151\pi$
$937$	14.9109i	0.487119i	−0.969886	+	0.243560i	$0.921685\pi$
$938$	30.1047i	0.982953i
$939$	21.2094	0.692142
$940$	0	0
$941$	28.4187	0.926425	0.463212	−	0.886247i	$-0.346697\pi$
$941$	28.4187	0.926425	0.463212	+	0.886247i	$0.346697\pi$
$942$	− 22.7016i	− 0.739657i
$943$	0	0
$944$	−2.70156	−0.0879284
$945$	0	0
$946$	−53.6125	−1.74309
$947$	47.5078i	1.54380i	0.635746	+	0.771898i	$0.280693\pi$
$947$	47.5078i	1.54380i	−0.635746	+	0.771898i	$0.719307\pi$
$948$	5.70156i	0.185178i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	1.19375	0.0387100
$952$	− 3.29844i	− 0.106903i
$953$	31.2984i	1.01386i	0.861988	+	0.506928i	$0.169219\pi$
$953$	31.2984i	1.01386i	−0.861988	+	0.506928i	$0.830781\pi$
$954$	−2.40312	−0.0778040
$955$	0	0
$956$	6.10469	0.197440
$957$	− 30.1047i	− 0.973146i
$958$	40.6125i	1.31213i
$959$	−64.4187	−2.08019
$960$	0	0
$961$	71.1047	2.29370
$962$	1.70156i	0.0548606i
$963$	− 19.1047i	− 0.615640i
$964$	2.59688	0.0836397
$965$	0	0
$966$	0	0
$967$	29.8953i	0.961368i	0.876894	+	0.480684i	$0.159612\pi$
$967$	29.8953i	0.961368i	−0.876894	+	0.480684i	$0.840388\pi$
$968$	− 11.1047i	− 0.356918i
$969$	−1.19375	−0.0383488
$970$	0	0
$971$	−36.8953	−1.18403	−0.592013	−	0.805928i	$-0.701667\pi$
$971$	−36.8953	−1.18403	−0.592013	+	0.805928i	$0.701667\pi$
$972$	1.00000i	0.0320750i
$973$	− 44.2094i	− 1.41729i
$974$	7.89531	0.252982
$975$	0	0
$976$	14.1047	0.451480
$977$	− 23.4031i	− 0.748732i	−0.927281	−	0.374366i	$-0.877860\pi$
$977$	− 23.4031i	− 0.748732i	0.927281	−	0.374366i	$-0.122140\pi$
$978$	18.8062i	0.601358i
$979$	−53.6125	−1.71346
$980$	0	0
$981$	4.29844	0.137239
$982$	33.4031i	1.06594i
$983$	7.50781i	0.239462i	0.992806	+	0.119731i	$0.0382032\pi$
$983$	7.50781i	0.239462i	−0.992806	+	0.119731i	$0.961797\pi$
$984$	3.70156	0.118001
$985$	0	0
$986$	4.49219	0.143060
$987$	32.9109i	1.04757i
$988$	− 1.70156i	− 0.0541339i
$989$	0	0
$990$	0	0
$991$	−9.10469	−0.289220	−0.144610	−	0.989489i	$-0.546193\pi$
$991$	−9.10469	−0.289220	−0.144610	+	0.989489i	$0.546193\pi$
$992$	− 10.1047i	− 0.320824i
$993$	12.0000i	0.380808i
$994$	8.00000	0.253745
$995$	0	0
$996$	−10.7016	−0.339092
$997$	− 10.4922i	− 0.332291i	−0.986101	−	0.166145i	$-0.946868\pi$
$997$	− 10.4922i	− 0.332291i	0.986101	−	0.166145i	$-0.0531321\pi$
$998$	− 17.2094i	− 0.544753i
$999$	1.70156	0.0538350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.p.1249.4		4
3.2	odd	2		5850.2.e.bi.5149.2		4
5.2	odd	4		1950.2.a.bc.1.1	✓	2
5.3	odd	4		1950.2.a.bg.1.2	yes	2
5.4	even	2	inner	1950.2.e.p.1249.1		4
15.2	even	4		5850.2.a.cj.1.1		2
15.8	even	4		5850.2.a.cg.1.2		2
15.14	odd	2		5850.2.e.bi.5149.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.bc.1.1	✓	2	5.2	odd	4
1950.2.a.bg.1.2	yes	2	5.3	odd	4
1950.2.e.p.1249.1		4	5.4	even	2	inner
1950.2.e.p.1249.4		4	1.1	even	1	trivial
5850.2.a.cg.1.2		2	15.8	even	4
5850.2.a.cj.1.1		2	15.2	even	4
5850.2.e.bi.5149.2		4	3.2	odd	2
5850.2.e.bi.5149.3		4	15.14	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.70156 q^{11} +1.00000i q^{12} +1.00000i q^{13} -4.70156 q^{14} +1.00000 q^{16} -0.701562i q^{17} -1.00000i q^{18} +1.70156 q^{19} +4.70156 q^{21} +4.70156i q^{22} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -4.70156i q^{28} +6.40312 q^{29} -10.1047 q^{31} +1.00000i q^{32} -4.70156i q^{33} +0.701562 q^{34} +1.00000 q^{36} -1.70156i q^{37} +1.70156i q^{38} +1.00000 q^{39} -3.70156 q^{41} +4.70156i q^{42} +11.4031i q^{43} -4.70156 q^{44} +7.00000i q^{47} -1.00000i q^{48} -15.1047 q^{49} -0.701562 q^{51} -1.00000i q^{52} -2.40312i q^{53} -1.00000 q^{54} +4.70156 q^{56} -1.70156i q^{57} +6.40312i q^{58} -2.70156 q^{59} +14.1047 q^{61} -10.1047i q^{62} -4.70156i q^{63} -1.00000 q^{64} +4.70156 q^{66} -6.40312i q^{67} +0.701562i q^{68} -1.70156 q^{71} +1.00000i q^{72} +12.0000i q^{73} +1.70156 q^{74} -1.70156 q^{76} +22.1047i q^{77} +1.00000i q^{78} +5.70156 q^{79} +1.00000 q^{81} -3.70156i q^{82} +10.7016i q^{83} -4.70156 q^{84} -11.4031 q^{86} -6.40312i q^{87} -4.70156i q^{88} -11.4031 q^{89} -4.70156 q^{91} +10.1047i q^{93} -7.00000 q^{94} +1.00000 q^{96} -2.59688i q^{97} -15.1047i q^{98} -4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 6 q^{21} - 4 q^{24} - 4 q^{26} - 2 q^{31} - 10 q^{34} + 4 q^{36} + 4 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{49} + 10 q^{51} - 4 q^{54} + 6 q^{56} + 2 q^{59} + 18 q^{61} - 4 q^{64} + 6 q^{66} + 6 q^{71} - 6 q^{74} + 6 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} - 20 q^{86} - 20 q^{89} - 6 q^{91} - 28 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 6 * q^11 - 6 * q^14 + 4 * q^16 - 6 * q^19 + 6 * q^21 - 4 * q^24 - 4 * q^26 - 2 * q^31 - 10 * q^34 + 4 * q^36 + 4 * q^39 - 2 * q^41 - 6 * q^44 - 22 * q^49 + 10 * q^51 - 4 * q^54 + 6 * q^56 + 2 * q^59 + 18 * q^61 - 4 * q^64 + 6 * q^66 + 6 * q^71 - 6 * q^74 + 6 * q^76 + 10 * q^79 + 4 * q^81 - 6 * q^84 - 20 * q^86 - 20 * q^89 - 6 * q^91 - 28 * q^94 + 4 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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