LMFDB - Embedded newform 169.2.b.a.168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	169.b (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:	$1.34947179416$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.2.b.a.168.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.73205i q^{5} +3.46410i q^{6} +1.73205i q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{12} -3.46410i q^{15} -5.00000 q^{16} -3.00000 q^{17} +1.73205i q^{18} -3.46410i q^{19} +1.73205i q^{20} -6.00000 q^{23} +3.46410i q^{24} +2.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} +6.00000 q^{30} +3.46410i q^{31} -5.19615i q^{32} -5.19615i q^{34} -1.00000 q^{36} -8.66025i q^{37} +6.00000 q^{38} +3.00000 q^{40} +5.19615i q^{41} +8.00000 q^{43} -1.73205i q^{45} -10.3923i q^{46} -3.46410i q^{47} -10.0000 q^{48} +7.00000 q^{49} +3.46410i q^{50} -6.00000 q^{51} -3.00000 q^{53} -6.92820i q^{54} -6.92820i q^{57} +5.19615i q^{58} +6.92820i q^{59} +3.46410i q^{60} +1.00000 q^{61} -6.00000 q^{62} -1.00000 q^{64} -3.46410i q^{67} +3.00000 q^{68} -12.0000 q^{69} +3.46410i q^{71} +1.73205i q^{72} +1.73205i q^{73} +15.0000 q^{74} +4.00000 q^{75} +3.46410i q^{76} +4.00000 q^{79} +8.66025i q^{80} -11.0000 q^{81} -9.00000 q^{82} +13.8564i q^{83} +5.19615i q^{85} +13.8564i q^{86} +6.00000 q^{87} +6.92820i q^{89} +3.00000 q^{90} +6.00000 q^{92} +6.92820i q^{93} +6.00000 q^{94} -6.00000 q^{95} -10.3923i q^{96} +6.92820i q^{97} +12.1244i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 4 q^{12} - 10 q^{16} - 6 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{27} + 6 q^{29} + 12 q^{30} - 2 q^{36} + 12 q^{38} + 6 q^{40} + 16 q^{43} - 20 q^{48} + 14 q^{49}+ \cdots - 12 q^{95}+O(q^{100})$ 2 * q + 4 * q^3 - 2 * q^4 + 2 * q^9 + 6 * q^10 - 4 * q^12 - 10 * q^16 - 6 * q^17 - 12 * q^23 + 4 * q^25 - 8 * q^27 + 6 * q^29 + 12 * q^30 - 2 * q^36 + 12 * q^38 + 6 * q^40 + 16 * q^43 - 20 * q^48 + 14 * q^49 - 12 * q^51 - 6 * q^53 + 2 * q^61 - 12 * q^62 - 2 * q^64 + 6 * q^68 - 24 * q^69 + 30 * q^74 + 8 * q^75 + 8 * q^79 - 22 * q^81 - 18 * q^82 + 12 * q^87 + 6 * q^90 + 12 * q^92 + 12 * q^94 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$ .

$n$	$2$
$\chi(n)$	$-1$

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

)

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

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	3.46410i	1.41421i
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.73205i	0.612372i
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.46410i	− 0.894427i
$16$	−5.00000	−1.25000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.73205i	0.408248i
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	1.73205i	0.387298i
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	3.46410i	0.707107i
$25$	2.00000	0.400000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	6.00000	1.09545
$31$	3.46410i	0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$	3.46410i	0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	− 5.19615i	− 0.918559i
$33$	0	0
$34$	− 5.19615i	− 0.891133i
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 8.66025i	− 1.42374i	−0.702313	−	0.711868i	$-0.747849\pi$
$37$	− 8.66025i	− 1.42374i	0.702313	−	0.711868i	$-0.252151\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	5.19615i	0.811503i	0.913984	+	0.405751i	$0.132990\pi$
$41$	5.19615i	0.811503i	−0.913984	+	0.405751i	$0.867010\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	− 1.73205i	− 0.258199i
$46$	− 10.3923i	− 1.53226i
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	−10.0000	−1.44338
$49$	7.00000	1.00000
$50$	3.46410i	0.489898i
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	− 6.92820i	− 0.942809i
$55$	0	0
$56$	0	0
$57$	− 6.92820i	− 0.917663i
$58$	5.19615i	0.682288i
$59$	6.92820i	0.901975i	0.892530	+	0.450988i	$0.148928\pi$
$59$	6.92820i	0.901975i	−0.892530	+	0.450988i	$0.851072\pi$
$60$	3.46410i	0.447214i
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 3.46410i	− 0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	− 3.46410i	− 0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$	3.00000	0.363803
$69$	−12.0000	−1.44463
$70$	0	0
$71$	3.46410i	0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$	3.46410i	0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$	1.73205i	0.204124i
$73$	1.73205i	0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	1.73205i	0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$	15.0000	1.74371
$75$	4.00000	0.461880
$76$	3.46410i	0.397360i
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	8.66025i	0.968246i
$81$	−11.0000	−1.22222
$82$	−9.00000	−0.993884
$83$	13.8564i	1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$	13.8564i	1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$	0	0
$85$	5.19615i	0.563602i
$86$	13.8564i	1.49417i
$87$	6.00000	0.643268
$88$	0	0
$89$	6.92820i	0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$	6.92820i	0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	6.00000	0.625543
$93$	6.92820i	0.718421i
$94$	6.00000	0.618853
$95$	−6.00000	−0.615587
$96$	− 10.3923i	− 1.06066i
$97$	6.92820i	0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$	6.92820i	0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	12.1244i	1.22474i
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	− 10.3923i	− 1.02899i
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	0	0
$106$	− 5.19615i	− 0.504695i
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	− 13.8564i	− 1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$	− 13.8564i	− 1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$	0	0
$111$	− 17.3205i	− 1.64399i
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	12.0000	1.12390
$115$	10.3923i	0.969087i
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	6.00000	0.547723
$121$	11.0000	1.00000
$122$	1.73205i	0.156813i
$123$	10.3923i	0.937043i
$124$	− 3.46410i	− 0.311086i
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	− 12.1244i	− 1.07165i
$129$	16.0000	1.40872
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	6.00000	0.518321
$135$	6.92820i	0.596285i
$136$	− 5.19615i	− 0.445566i
$137$	− 15.5885i	− 1.33181i	−0.746036	−	0.665906i	$-0.768045\pi$
$137$	− 15.5885i	− 1.33181i	0.746036	−	0.665906i	$-0.231955\pi$
$138$	− 20.7846i	− 1.76930i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	− 6.92820i	− 0.583460i
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−5.00000	−0.416667
$145$	− 5.19615i	− 0.431517i
$146$	−3.00000	−0.248282
$147$	14.0000	1.15470
$148$	8.66025i	0.711868i
$149$	− 19.0526i	− 1.56085i	−0.625252	−	0.780423i	$-0.715004\pi$
$149$	− 19.0526i	− 1.56085i	0.625252	−	0.780423i	$-0.284996\pi$
$150$	6.92820i	0.565685i
$151$	17.3205i	1.40952i	0.709444	+	0.704761i	$0.248946\pi$
$151$	17.3205i	1.40952i	−0.709444	+	0.704761i	$0.751054\pi$
$152$	6.00000	0.486664
$153$	−3.00000	−0.242536
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	6.92820i	0.551178i
$159$	−6.00000	−0.475831
$160$	−9.00000	−0.711512
$161$	0	0
$162$	− 19.0526i	− 1.49691i
$163$	20.7846i	1.62798i	0.580881	+	0.813988i	$0.302708\pi$
$163$	20.7846i	1.62798i	−0.580881	+	0.813988i	$0.697292\pi$
$164$	− 5.19615i	− 0.405751i
$165$	0	0
$166$	−24.0000	−1.86276
$167$	13.8564i	1.07224i	0.844141	+	0.536120i	$0.180111\pi$
$167$	13.8564i	1.07224i	−0.844141	+	0.536120i	$0.819889\pi$
$168$	0	0
$169$	0	0
$170$	−9.00000	−0.690268
$171$	− 3.46410i	− 0.264906i
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	10.3923i	0.787839i
$175$	0	0
$176$	0	0
$177$	13.8564i	1.04151i
$178$	−12.0000	−0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.73205i	0.129099i
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	− 10.3923i	− 0.766131i
$185$	−15.0000	−1.10282
$186$	−12.0000	−0.879883
$187$	0	0
$188$	3.46410i	0.252646i
$189$	0	0
$190$	− 10.3923i	− 0.753937i
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	−2.00000	−0.144338
$193$	5.19615i	0.374027i	0.982357	+	0.187014i	$0.0598809\pi$
$193$	5.19615i	0.374027i	−0.982357	+	0.187014i	$0.940119\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	3.46410i	0.244949i
$201$	− 6.92820i	− 0.488678i
$202$	− 5.19615i	− 0.365600i
$203$	0	0
$204$	6.00000	0.420084
$205$	9.00000	0.628587
$206$	− 17.3205i	− 1.20678i
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	3.00000	0.206041
$213$	6.92820i	0.474713i
$214$	10.3923i	0.710403i
$215$	− 13.8564i	− 0.944999i
$216$	− 6.92820i	− 0.471405i
$217$	0	0
$218$	24.0000	1.62549
$219$	3.46410i	0.234082i
$220$	0	0
$221$	0	0
$222$	30.0000	2.01347
$223$	10.3923i	0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$	10.3923i	0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	− 25.9808i	− 1.72821i
$227$	24.2487i	1.60944i	0.593652	+	0.804722i	$0.297686\pi$
$227$	24.2487i	1.60944i	−0.593652	+	0.804722i	$0.702314\pi$
$228$	6.92820i	0.458831i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	5.19615i	0.341144i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	− 6.92820i	− 0.450988i
$237$	8.00000	0.519656
$238$	0	0
$239$	− 20.7846i	− 1.34444i	−0.740349	−	0.672222i	$-0.765340\pi$
$239$	− 20.7846i	− 1.34444i	0.740349	−	0.672222i	$-0.234660\pi$
$240$	17.3205i	1.11803i
$241$	− 1.73205i	− 0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	− 1.73205i	− 0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$	19.0526i	1.22474i
$243$	−10.0000	−0.641500
$244$	−1.00000	−0.0640184
$245$	− 12.1244i	− 0.774597i
$246$	−18.0000	−1.14764
$247$	0	0
$248$	−6.00000	−0.381000
$249$	27.7128i	1.75623i
$250$	21.0000	1.32816
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	− 3.46410i	− 0.217357i
$255$	10.3923i	0.650791i
$256$	19.0000	1.18750
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	27.7128i	1.72532i
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	31.1769i	1.92612i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	5.19615i	0.319197i
$266$	0	0
$267$	13.8564i	0.847998i
$268$	3.46410i	0.211604i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−12.0000	−0.730297
$271$	− 20.7846i	− 1.26258i	−0.775549	−	0.631288i	$-0.782527\pi$
$271$	− 20.7846i	− 1.26258i	0.775549	−	0.631288i	$-0.217473\pi$
$272$	15.0000	0.909509
$273$	0	0
$274$	27.0000	1.63113
$275$	0	0
$276$	12.0000	0.722315
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	− 6.92820i	− 0.415526i
$279$	3.46410i	0.207390i
$280$	0	0
$281$	− 22.5167i	− 1.34323i	−0.740900	−	0.671616i	$-0.765601\pi$
$281$	− 22.5167i	− 1.34323i	0.740900	−	0.671616i	$-0.234399\pi$
$282$	12.0000	0.714590
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	− 3.46410i	− 0.205557i
$285$	−12.0000	−0.710819
$286$	0	0
$287$	0	0
$288$	− 5.19615i	− 0.306186i
$289$	−8.00000	−0.470588
$290$	9.00000	0.528498
$291$	13.8564i	0.812277i
$292$	− 1.73205i	− 0.101361i
$293$	5.19615i	0.303562i	0.988414	+	0.151781i	$0.0485009\pi$
$293$	5.19615i	0.303562i	−0.988414	+	0.151781i	$0.951499\pi$
$294$	24.2487i	1.41421i
$295$	12.0000	0.698667
$296$	15.0000	0.871857
$297$	0	0
$298$	33.0000	1.91164
$299$	0	0
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−30.0000	−1.72631
$303$	−6.00000	−0.344691
$304$	17.3205i	0.993399i
$305$	− 1.73205i	− 0.0991769i
$306$	− 5.19615i	− 0.297044i
$307$	− 17.3205i	− 0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$	− 17.3205i	− 0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$	0	0
$309$	−20.0000	−1.13776
$310$	10.3923i	0.590243i
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	− 22.5167i	− 1.27069i
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 5.19615i	− 0.291845i	−0.989296	−	0.145922i	$-0.953385\pi$
$317$	− 5.19615i	− 0.291845i	0.989296	−	0.145922i	$-0.0466150\pi$
$318$	− 10.3923i	− 0.582772i
$319$	0	0
$320$	1.73205i	0.0968246i
$321$	12.0000	0.669775
$322$	0	0
$323$	10.3923i	0.578243i
$324$	11.0000	0.611111
$325$	0	0
$326$	−36.0000	−1.99386
$327$	− 27.7128i	− 1.53252i
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	− 27.7128i	− 1.52323i	−0.648027	−	0.761617i	$-0.724406\pi$
$331$	− 27.7128i	− 1.52323i	0.648027	−	0.761617i	$-0.275594\pi$
$332$	− 13.8564i	− 0.760469i
$333$	− 8.66025i	− 0.474579i
$334$	−24.0000	−1.31322
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	−30.0000	−1.62938
$340$	− 5.19615i	− 0.281801i
$341$	0	0
$342$	6.00000	0.324443
$343$	0	0
$344$	13.8564i	0.747087i
$345$	20.7846i	1.11901i
$346$	10.3923i	0.558694i
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	−6.00000	−0.321634
$349$	− 13.8564i	− 0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$	− 13.8564i	− 0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 32.9090i	− 1.75157i	−0.482704	−	0.875784i	$-0.660345\pi$
$353$	− 32.9090i	− 1.75157i	0.482704	−	0.875784i	$-0.339655\pi$
$354$	−24.0000	−1.27559
$355$	6.00000	0.318447
$356$	− 6.92820i	− 0.367194i
$357$	0	0
$358$	0	0
$359$	− 6.92820i	− 0.365657i	−0.983145	−	0.182828i	$-0.941475\pi$
$359$	− 6.92820i	− 0.365657i	0.983145	−	0.182828i	$-0.0585252\pi$
$360$	3.00000	0.158114
$361$	7.00000	0.368421
$362$	19.0526i	1.00138i
$363$	22.0000	1.15470
$364$	0	0
$365$	3.00000	0.157027
$366$	3.46410i	0.181071i
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	30.0000	1.56386
$369$	5.19615i	0.270501i
$370$	− 25.9808i	− 1.35068i
$371$	0	0
$372$	− 6.92820i	− 0.359211i
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	− 24.2487i	− 1.25220i
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	24.2487i	1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$	24.2487i	1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$	6.00000	0.307794
$381$	−4.00000	−0.204926
$382$	31.1769i	1.59515i
$383$	− 20.7846i	− 1.06204i	−0.847358	−	0.531022i	$-0.821808\pi$
$383$	− 20.7846i	− 1.06204i	0.847358	−	0.531022i	$-0.178192\pi$
$384$	− 24.2487i	− 1.23744i
$385$	0	0
$386$	−9.00000	−0.458088
$387$	8.00000	0.406663
$388$	− 6.92820i	− 0.351726i
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	12.1244i	0.612372i
$393$	36.0000	1.81596
$394$	24.0000	1.20910
$395$	− 6.92820i	− 0.348596i
$396$	0	0
$397$	− 13.8564i	− 0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$	− 13.8564i	− 0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$	− 3.46410i	− 0.173640i
$399$	0	0
$400$	−10.0000	−0.500000
$401$	1.73205i	0.0864945i	0.999064	+	0.0432472i	$0.0137703\pi$
$401$	1.73205i	0.0864945i	−0.999064	+	0.0432472i	$0.986230\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	3.00000	0.149256
$405$	19.0526i	0.946729i
$406$	0	0
$407$	0	0
$408$	− 10.3923i	− 0.514496i
$409$	15.5885i	0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$	15.5885i	0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$	15.5885i	0.769859i
$411$	− 31.1769i	− 1.53784i
$412$	10.0000	0.492665
$413$	0	0
$414$	− 10.3923i	− 0.510754i
$415$	24.0000	1.17811
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	15.5885i	0.759735i	0.925041	+	0.379867i	$0.124030\pi$
$421$	15.5885i	0.759735i	−0.925041	+	0.379867i	$0.875970\pi$
$422$	17.3205i	0.843149i
$423$	− 3.46410i	− 0.168430i
$424$	− 5.19615i	− 0.252347i
$425$	−6.00000	−0.291043
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	24.0000	1.15738
$431$	6.92820i	0.333720i	0.985981	+	0.166860i	$0.0533628\pi$
$431$	6.92820i	0.333720i	−0.985981	+	0.166860i	$0.946637\pi$
$432$	20.0000	0.962250
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	− 10.3923i	− 0.498273i
$436$	13.8564i	0.663602i
$437$	20.7846i	0.994263i
$438$	−6.00000	−0.286691
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	17.3205i	0.821995i
$445$	12.0000	0.568855
$446$	−18.0000	−0.852325
$447$	− 38.1051i	− 1.80231i
$448$	0	0
$449$	− 6.92820i	− 0.326962i	−0.986546	−	0.163481i	$-0.947728\pi$
$449$	− 6.92820i	− 0.326962i	0.986546	−	0.163481i	$-0.0522723\pi$
$450$	3.46410i	0.163299i
$451$	0	0
$452$	15.0000	0.705541
$453$	34.6410i	1.62758i
$454$	−42.0000	−1.97116
$455$	0	0
$456$	12.0000	0.561951
$457$	− 1.73205i	− 0.0810219i	−0.999179	−	0.0405110i	$-0.987101\pi$
$457$	− 1.73205i	− 0.0810219i	0.999179	−	0.0405110i	$-0.0128986\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	− 10.3923i	− 0.484544i
$461$	22.5167i	1.04871i	0.851501	+	0.524353i	$0.175693\pi$
$461$	22.5167i	1.04871i	−0.851501	+	0.524353i	$0.824307\pi$
$462$	0	0
$463$	13.8564i	0.643962i	0.946746	+	0.321981i	$0.104349\pi$
$463$	13.8564i	0.643962i	−0.946746	+	0.321981i	$0.895651\pi$
$464$	−15.0000	−0.696358
$465$	12.0000	0.556487
$466$	10.3923i	0.481414i
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	− 10.3923i	− 0.479361i
$471$	−26.0000	−1.19802
$472$	−12.0000	−0.552345
$473$	0	0
$474$	13.8564i	0.636446i
$475$	− 6.92820i	− 0.317888i
$476$	0	0
$477$	−3.00000	−0.137361
$478$	36.0000	1.64660
$479$	− 24.2487i	− 1.10795i	−0.832533	−	0.553976i	$-0.813110\pi$
$479$	− 24.2487i	− 1.10795i	0.832533	−	0.553976i	$-0.186890\pi$
$480$	−18.0000	−0.821584
$481$	0	0
$482$	3.00000	0.136646
$483$	0	0
$484$	−11.0000	−0.500000
$485$	12.0000	0.544892
$486$	− 17.3205i	− 0.785674i
$487$	− 6.92820i	− 0.313947i	−0.987603	−	0.156973i	$-0.949826\pi$
$487$	− 6.92820i	− 0.313947i	0.987603	−	0.156973i	$-0.0501737\pi$
$488$	1.73205i	0.0784063i
$489$	41.5692i	1.87983i
$490$	21.0000	0.948683
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 10.3923i	− 0.468521i
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	− 17.3205i	− 0.777714i
$497$	0	0
$498$	−48.0000	−2.15093
$499$	31.1769i	1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$	31.1769i	1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	12.1244i	0.542218i
$501$	27.7128i	1.23812i
$502$	− 31.1769i	− 1.39149i
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	5.19615i	0.231226i
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	− 19.0526i	− 0.844490i	−0.906482	−	0.422245i	$-0.861242\pi$
$509$	− 19.0526i	− 0.844490i	0.906482	−	0.422245i	$-0.138758\pi$
$510$	−18.0000	−0.797053
$511$	0	0
$512$	8.66025i	0.382733i
$513$	13.8564i	0.611775i
$514$	5.19615i	0.229192i
$515$	17.3205i	0.763233i
$516$	−16.0000	−0.704361
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	5.19615i	0.227429i
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	20.7846i	0.906252i
$527$	− 10.3923i	− 0.452696i
$528$	0	0
$529$	13.0000	0.565217
$530$	−9.00000	−0.390935
$531$	6.92820i	0.300658i
$532$	0	0
$533$	0	0
$534$	−24.0000	−1.03858
$535$	− 10.3923i	− 0.449299i
$536$	6.00000	0.259161
$537$	0	0
$538$	− 10.3923i	− 0.448044i
$539$	0	0
$540$	− 6.92820i	− 0.298142i
$541$	29.4449i	1.26593i	0.774179	+	0.632967i	$0.218163\pi$
$541$	29.4449i	1.26593i	−0.774179	+	0.632967i	$0.781837\pi$
$542$	36.0000	1.54633
$543$	22.0000	0.944110
$544$	15.5885i	0.668350i
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	15.5885i	0.665906i
$549$	1.00000	0.0426790
$550$	0	0
$551$	− 10.3923i	− 0.442727i
$552$	− 20.7846i	− 0.884652i
$553$	0	0
$554$	− 12.1244i	− 0.515115i
$555$	−30.0000	−1.27343
$556$	4.00000	0.169638
$557$	15.5885i	0.660504i	0.943893	+	0.330252i	$0.107134\pi$
$557$	15.5885i	0.660504i	−0.943893	+	0.330252i	$0.892866\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	0	0
$561$	0	0
$562$	39.0000	1.64512
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	6.92820i	0.291730i
$565$	25.9808i	1.09302i
$566$	6.92820i	0.291214i
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	− 20.7846i	− 0.870572i
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	−12.0000	−0.500435
$576$	−1.00000	−0.0416667
$577$	− 19.0526i	− 0.793168i	−0.917998	−	0.396584i	$-0.870195\pi$
$577$	− 19.0526i	− 0.793168i	0.917998	−	0.396584i	$-0.129805\pi$
$578$	− 13.8564i	− 0.576351i
$579$	10.3923i	0.431889i
$580$	5.19615i	0.215758i
$581$	0	0
$582$	−24.0000	−0.994832
$583$	0	0
$584$	−3.00000	−0.124141
$585$	0	0
$586$	−9.00000	−0.371787
$587$	20.7846i	0.857873i	0.903335	+	0.428936i	$0.141112\pi$
$587$	20.7846i	0.857873i	−0.903335	+	0.428936i	$0.858888\pi$
$588$	−14.0000	−0.577350
$589$	12.0000	0.494451
$590$	20.7846i	0.855689i
$591$	− 27.7128i	− 1.13995i
$592$	43.3013i	1.77967i
$593$	− 25.9808i	− 1.06690i	−0.845831	−	0.533451i	$-0.820895\pi$
$593$	− 25.9808i	− 1.06690i	0.845831	−	0.533451i	$-0.179105\pi$
$594$	0	0
$595$	0	0
$596$	19.0526i	0.780423i
$597$	−4.00000	−0.163709
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	6.92820i	0.282843i
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	− 3.46410i	− 0.141069i
$604$	− 17.3205i	− 0.704761i
$605$	− 19.0526i	− 0.774597i
$606$	− 10.3923i	− 0.422159i
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−18.0000	−0.729996
$609$	0	0
$610$	3.00000	0.121466
$611$	0	0
$612$	3.00000	0.121268
$613$	12.1244i	0.489698i	0.969561	+	0.244849i	$0.0787384\pi$
$613$	12.1244i	0.489698i	−0.969561	+	0.244849i	$0.921262\pi$
$614$	30.0000	1.21070
$615$	18.0000	0.725830
$616$	0	0
$617$	− 22.5167i	− 0.906487i	−0.891387	−	0.453243i	$-0.850267\pi$
$617$	− 22.5167i	− 0.906487i	0.891387	−	0.453243i	$-0.149733\pi$
$618$	− 34.6410i	− 1.39347i
$619$	− 20.7846i	− 0.835404i	−0.908584	−	0.417702i	$-0.862836\pi$
$619$	− 20.7846i	− 0.835404i	0.908584	−	0.417702i	$-0.137164\pi$
$620$	−6.00000	−0.240966
$621$	24.0000	0.963087
$622$	− 51.9615i	− 2.08347i
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	17.3205i	0.692267i
$627$	0	0
$628$	13.0000	0.518756
$629$	25.9808i	1.03592i
$630$	0	0
$631$	− 48.4974i	− 1.93065i	−0.261048	−	0.965326i	$-0.584068\pi$
$631$	− 48.4974i	− 1.93065i	0.261048	−	0.965326i	$-0.415932\pi$
$632$	6.92820i	0.275589i
$633$	20.0000	0.794929
$634$	9.00000	0.357436
$635$	3.46410i	0.137469i
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	3.46410i	0.137038i
$640$	−21.0000	−0.830098
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	20.7846i	0.820303i
$643$	13.8564i	0.546443i	0.961951	+	0.273222i	$0.0880892\pi$
$643$	13.8564i	0.546443i	−0.961951	+	0.273222i	$0.911911\pi$
$644$	0	0
$645$	− 27.7128i	− 1.09119i
$646$	−18.0000	−0.708201
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	− 19.0526i	− 0.748455i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 20.7846i	− 0.813988i
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	48.0000	1.87695
$655$	− 31.1769i	− 1.21818i
$656$	− 25.9808i	− 1.01438i
$657$	1.73205i	0.0675737i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	46.7654i	1.81896i	0.415745	+	0.909481i	$0.363521\pi$
$661$	46.7654i	1.81896i	−0.415745	+	0.909481i	$0.636479\pi$
$662$	48.0000	1.86557
$663$	0	0
$664$	−24.0000	−0.931381
$665$	0	0
$666$	15.0000	0.581238
$667$	−18.0000	−0.696963
$668$	− 13.8564i	− 0.536120i
$669$	20.7846i	0.803579i
$670$	− 10.3923i	− 0.401490i
$671$	0	0
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	− 39.8372i	− 1.53447i
$675$	−8.00000	−0.307920
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	− 51.9615i	− 1.99557i
$679$	0	0
$680$	−9.00000	−0.345134
$681$	48.4974i	1.85843i
$682$	0	0
$683$	24.2487i	0.927851i	0.885874	+	0.463926i	$0.153559\pi$
$683$	24.2487i	0.927851i	−0.885874	+	0.463926i	$0.846441\pi$
$684$	3.46410i	0.132453i
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	−40.0000	−1.52499
$689$	0	0
$690$	−36.0000	−1.37050
$691$	− 13.8564i	− 0.527123i	−0.964643	−	0.263561i	$-0.915103\pi$
$691$	− 13.8564i	− 0.527123i	0.964643	−	0.263561i	$-0.0848971\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	− 51.9615i	− 1.97243i
$695$	6.92820i	0.262802i
$696$	10.3923i	0.393919i
$697$	− 15.5885i	− 0.590455i
$698$	24.0000	0.908413
$699$	12.0000	0.453882
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	0	0
$705$	−12.0000	−0.451946
$706$	57.0000	2.14522
$707$	0	0
$708$	− 13.8564i	− 0.520756i
$709$	− 5.19615i	− 0.195146i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	− 5.19615i	− 0.195146i	0.995228	−	0.0975728i	$-0.0311079\pi$
$710$	10.3923i	0.390016i
$711$	4.00000	0.150012
$712$	−12.0000	−0.449719
$713$	− 20.7846i	− 0.778390i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 41.5692i	− 1.55243i
$718$	12.0000	0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	8.66025i	0.322749i
$721$	0	0
$722$	12.1244i	0.451222i
$723$	− 3.46410i	− 0.128831i
$724$	−11.0000	−0.408812
$725$	6.00000	0.222834
$726$	38.1051i	1.41421i
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	5.19615i	0.192318i
$731$	−24.0000	−0.887672
$732$	−2.00000	−0.0739221
$733$	12.1244i	0.447823i	0.974609	+	0.223912i	$0.0718827\pi$
$733$	12.1244i	0.447823i	−0.974609	+	0.223912i	$0.928117\pi$
$734$	− 38.1051i	− 1.40649i
$735$	− 24.2487i	− 0.894427i
$736$	31.1769i	1.14920i
$737$	0	0
$738$	−9.00000	−0.331295
$739$	20.7846i	0.764574i	0.924044	+	0.382287i	$0.124863\pi$
$739$	20.7846i	0.764574i	−0.924044	+	0.382287i	$0.875137\pi$
$740$	15.0000	0.551411
$741$	0	0
$742$	0	0
$743$	34.6410i	1.27086i	0.772160	+	0.635428i	$0.219176\pi$
$743$	34.6410i	1.27086i	−0.772160	+	0.635428i	$0.780824\pi$
$744$	−12.0000	−0.439941
$745$	−33.0000	−1.20903
$746$	32.9090i	1.20488i
$747$	13.8564i	0.506979i
$748$	0	0
$749$	0	0
$750$	42.0000	1.53362
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	17.3205i	0.631614i
$753$	−36.0000	−1.31191
$754$	0	0
$755$	30.0000	1.09181
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−42.0000	−1.52551
$759$	0	0
$760$	− 10.3923i	− 0.376969i
$761$	34.6410i	1.25574i	0.778320	+	0.627868i	$0.216072\pi$
$761$	34.6410i	1.25574i	−0.778320	+	0.627868i	$0.783928\pi$
$762$	− 6.92820i	− 0.250982i
$763$	0	0
$764$	−18.0000	−0.651217
$765$	5.19615i	0.187867i
$766$	36.0000	1.30073
$767$	0	0
$768$	38.0000	1.37121
$769$	− 6.92820i	− 0.249837i	−0.992167	−	0.124919i	$-0.960133\pi$
$769$	− 6.92820i	− 0.249837i	0.992167	−	0.124919i	$-0.0398670\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	− 5.19615i	− 0.187014i
$773$	− 34.6410i	− 1.24595i	−0.782241	−	0.622975i	$-0.785924\pi$
$773$	− 34.6410i	− 1.24595i	0.782241	−	0.622975i	$-0.214076\pi$
$774$	13.8564i	0.498058i
$775$	6.92820i	0.248868i
$776$	−12.0000	−0.430775
$777$	0	0
$778$	− 15.5885i	− 0.558873i
$779$	18.0000	0.644917
$780$	0	0
$781$	0	0
$782$	31.1769i	1.11488i
$783$	−12.0000	−0.428845
$784$	−35.0000	−1.25000
$785$	22.5167i	0.803654i
$786$	62.3538i	2.22409i
$787$	38.1051i	1.35830i	0.733999	+	0.679150i	$0.237652\pi$
$787$	38.1051i	1.35830i	−0.733999	+	0.679150i	$0.762348\pi$
$788$	13.8564i	0.493614i
$789$	24.0000	0.854423
$790$	12.0000	0.426941
$791$	0	0
$792$	0	0
$793$	0	0
$794$	24.0000	0.851728
$795$	10.3923i	0.368577i
$796$	2.00000	0.0708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	10.3923i	0.367653i
$800$	− 10.3923i	− 0.367423i
$801$	6.92820i	0.244796i
$802$	−3.00000	−0.105934
$803$	0	0
$804$	6.92820i	0.244339i
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	− 5.19615i	− 0.182800i
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	−33.0000	−1.15950
$811$	− 38.1051i	− 1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$	− 38.1051i	− 1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	0	0
$813$	− 41.5692i	− 1.45790i
$814$	0	0
$815$	36.0000	1.26102
$816$	30.0000	1.05021
$817$	− 27.7128i	− 0.969549i
$818$	−27.0000	−0.944033
$819$	0	0
$820$	−9.00000	−0.314294
$821$	41.5692i	1.45078i	0.688340	+	0.725388i	$0.258340\pi$
$821$	41.5692i	1.45078i	−0.688340	+	0.725388i	$0.741660\pi$
$822$	54.0000	1.88347
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	− 17.3205i	− 0.603388i
$825$	0	0
$826$	0	0
$827$	20.7846i	0.722752i	0.932420	+	0.361376i	$0.117693\pi$
$827$	20.7846i	0.722752i	−0.932420	+	0.361376i	$0.882307\pi$
$828$	6.00000	0.208514
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	41.5692i	1.44289i
$831$	−14.0000	−0.485655
$832$	0	0
$833$	−21.0000	−0.727607
$834$	− 13.8564i	− 0.479808i
$835$	24.0000	0.830554
$836$	0	0
$837$	− 13.8564i	− 0.478947i
$838$	31.1769i	1.07699i
$839$	45.0333i	1.55472i	0.629054	+	0.777361i	$0.283442\pi$
$839$	45.0333i	1.55472i	−0.629054	+	0.777361i	$0.716558\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−27.0000	−0.930481
$843$	− 45.0333i	− 1.55103i
$844$	−10.0000	−0.344214
$845$	0	0
$846$	6.00000	0.206284
$847$	0	0
$848$	15.0000	0.515102
$849$	8.00000	0.274559
$850$	− 10.3923i	− 0.356453i
$851$	51.9615i	1.78122i
$852$	− 6.92820i	− 0.237356i
$853$	25.9808i	0.889564i	0.895639	+	0.444782i	$0.146719\pi$
$853$	25.9808i	0.889564i	−0.895639	+	0.444782i	$0.853281\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	10.3923i	0.355202i
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	13.8564i	0.472500i
$861$	0	0
$862$	−12.0000	−0.408722
$863$	− 27.7128i	− 0.943355i	−0.881771	−	0.471678i	$-0.843649\pi$
$863$	− 27.7128i	− 0.943355i	0.881771	−	0.471678i	$-0.156351\pi$
$864$	20.7846i	0.707107i
$865$	− 10.3923i	− 0.353349i
$866$	− 29.4449i	− 1.00058i
$867$	−16.0000	−0.543388
$868$	0	0
$869$	0	0
$870$	18.0000	0.610257
$871$	0	0
$872$	24.0000	0.812743
$873$	6.92820i	0.234484i
$874$	−36.0000	−1.21772
$875$	0	0
$876$	− 3.46410i	− 0.117041i
$877$	12.1244i	0.409410i	0.978824	+	0.204705i	$0.0656236\pi$
$877$	12.1244i	0.409410i	−0.978824	+	0.204705i	$0.934376\pi$
$878$	− 48.4974i	− 1.63671i
$879$	10.3923i	0.350524i
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	12.1244i	0.408248i
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	− 20.7846i	− 0.698273i
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	30.0000	1.00673
$889$	0	0
$890$	20.7846i	0.696702i
$891$	0	0
$892$	− 10.3923i	− 0.347960i
$893$	−12.0000	−0.401565
$894$	66.0000	2.20737
$895$	0	0
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	10.3923i	0.346603i
$900$	−2.00000	−0.0666667
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	− 25.9808i	− 0.864107i
$905$	− 19.0526i	− 0.633328i
$906$	−60.0000	−1.99337
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	− 24.2487i	− 0.804722i
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	34.6410i	1.14708i
$913$	0	0
$914$	3.00000	0.0992312
$915$	− 3.46410i	− 0.114520i
$916$	0	0
$917$	0	0
$918$	20.7846i	0.685994i
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	−18.0000	−0.593442
$921$	− 34.6410i	− 1.14146i
$922$	−39.0000	−1.28440
$923$	0	0
$924$	0	0
$925$	− 17.3205i	− 0.569495i
$926$	−24.0000	−0.788689
$927$	−10.0000	−0.328443
$928$	− 15.5885i	− 0.511716i
$929$	− 46.7654i	− 1.53432i	−0.641455	−	0.767161i	$-0.721669\pi$
$929$	− 46.7654i	− 1.53432i	0.641455	−	0.767161i	$-0.278331\pi$
$930$	20.7846i	0.681554i
$931$	− 24.2487i	− 0.794719i
$932$	−6.00000	−0.196537
$933$	−60.0000	−1.96431
$934$	20.7846i	0.680093i
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	6.00000	0.195698
$941$	20.7846i	0.677559i	0.940866	+	0.338779i	$0.110014\pi$
$941$	20.7846i	0.677559i	−0.940866	+	0.338779i	$0.889986\pi$
$942$	− 45.0333i	− 1.46726i
$943$	− 31.1769i	− 1.01526i
$944$	− 34.6410i	− 1.12747i
$945$	0	0
$946$	0	0
$947$	17.3205i	0.562841i	0.959585	+	0.281420i	$0.0908056\pi$
$947$	17.3205i	0.562841i	−0.959585	+	0.281420i	$0.909194\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	12.0000	0.389331
$951$	− 10.3923i	− 0.336994i
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	− 5.19615i	− 0.168232i
$955$	− 31.1769i	− 1.00886i
$956$	20.7846i	0.672222i
$957$	0	0
$958$	42.0000	1.35696
$959$	0	0
$960$	3.46410i	0.111803i
$961$	19.0000	0.612903
$962$	0	0
$963$	6.00000	0.193347
$964$	1.73205i	0.0557856i
$965$	9.00000	0.289720
$966$	0	0
$967$	58.8897i	1.89377i	0.321578	+	0.946883i	$0.395787\pi$
$967$	58.8897i	1.89377i	−0.321578	+	0.946883i	$0.604213\pi$
$968$	19.0526i	0.612372i
$969$	20.7846i	0.667698i
$970$	20.7846i	0.667354i
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	12.0000	0.384505
$975$	0	0
$976$	−5.00000	−0.160046
$977$	− 43.3013i	− 1.38533i	−0.721259	−	0.692665i	$-0.756436\pi$
$977$	− 43.3013i	− 1.38533i	0.721259	−	0.692665i	$-0.243564\pi$
$978$	−72.0000	−2.30231
$979$	0	0
$980$	12.1244i	0.387298i
$981$	− 13.8564i	− 0.442401i
$982$	20.7846i	0.663264i
$983$	51.9615i	1.65732i	0.559756	+	0.828658i	$0.310895\pi$
$983$	51.9615i	1.65732i	−0.559756	+	0.828658i	$0.689105\pi$
$984$	−18.0000	−0.573819
$985$	−24.0000	−0.764704
$986$	− 15.5885i	− 0.496438i
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	18.0000	0.571501
$993$	− 55.4256i	− 1.75888i
$994$	0	0
$995$	3.46410i	0.109819i
$996$	− 27.7128i	− 0.878114i
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	−54.0000	−1.70934
$999$	34.6410i	1.09599i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.b.a.168.2		2
3.2	odd	2		1521.2.b.a.1351.1		2
4.3	odd	2		2704.2.f.b.337.1		2
13.2	odd	12		169.2.c.a.22.1		4
13.3	even	3		13.2.e.a.4.1	✓	2
13.4	even	6		13.2.e.a.10.1	yes	2
13.5	odd	4		169.2.a.a.1.2		2
13.6	odd	12		169.2.c.a.146.1		4
13.7	odd	12		169.2.c.a.146.2		4
13.8	odd	4		169.2.a.a.1.1		2
13.9	even	3		169.2.e.a.23.1		2
13.10	even	6		169.2.e.a.147.1		2
13.11	odd	12		169.2.c.a.22.2		4
13.12	even	2	inner	169.2.b.a.168.1		2
39.5	even	4		1521.2.a.k.1.1		2
39.8	even	4		1521.2.a.k.1.2		2
39.17	odd	6		117.2.q.c.10.1		2
39.29	odd	6		117.2.q.c.82.1		2
39.38	odd	2		1521.2.b.a.1351.2		2
52.3	odd	6		208.2.w.b.17.1		2
52.31	even	4		2704.2.a.o.1.1		2
52.43	odd	6		208.2.w.b.49.1		2
52.47	even	4		2704.2.a.o.1.2		2
52.51	odd	2		2704.2.f.b.337.2		2
65.3	odd	12		325.2.m.a.199.1		4
65.4	even	6		325.2.n.a.101.1		2
65.17	odd	12		325.2.m.a.49.1		4
65.29	even	6		325.2.n.a.251.1		2
65.34	odd	4		4225.2.a.v.1.2		2
65.42	odd	12		325.2.m.a.199.2		4
65.43	odd	12		325.2.m.a.49.2		4
65.44	odd	4		4225.2.a.v.1.1		2
91.3	odd	6		637.2.u.b.30.1		2
91.4	even	6		637.2.k.a.569.1		2
91.16	even	3		637.2.k.a.459.1		2
91.17	odd	6		637.2.k.c.569.1		2
91.30	even	6		637.2.u.c.361.1		2
91.34	even	4		8281.2.a.q.1.1		2
91.55	odd	6		637.2.q.a.589.1		2
91.68	odd	6		637.2.k.c.459.1		2
91.69	odd	6		637.2.q.a.491.1		2
91.81	even	3		637.2.u.c.30.1		2
91.82	odd	6		637.2.u.b.361.1		2
91.83	even	4		8281.2.a.q.1.2		2
104.3	odd	6		832.2.w.a.641.1		2
104.29	even	6		832.2.w.d.641.1		2
104.43	odd	6		832.2.w.a.257.1		2
104.69	even	6		832.2.w.d.257.1		2
156.95	even	6		1872.2.by.d.1297.1		2
156.107	even	6		1872.2.by.d.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	13.3	even	3
13.2.e.a.10.1	yes	2	13.4	even	6
117.2.q.c.10.1		2	39.17	odd	6
117.2.q.c.82.1		2	39.29	odd	6
169.2.a.a.1.1		2	13.8	odd	4
169.2.a.a.1.2		2	13.5	odd	4
169.2.b.a.168.1		2	13.12	even	2	inner
169.2.b.a.168.2		2	1.1	even	1	trivial
169.2.c.a.22.1		4	13.2	odd	12
169.2.c.a.22.2		4	13.11	odd	12
169.2.c.a.146.1		4	13.6	odd	12
169.2.c.a.146.2		4	13.7	odd	12
169.2.e.a.23.1		2	13.9	even	3
169.2.e.a.147.1		2	13.10	even	6
208.2.w.b.17.1		2	52.3	odd	6
208.2.w.b.49.1		2	52.43	odd	6
325.2.m.a.49.1		4	65.17	odd	12
325.2.m.a.49.2		4	65.43	odd	12
325.2.m.a.199.1		4	65.3	odd	12
325.2.m.a.199.2		4	65.42	odd	12
325.2.n.a.101.1		2	65.4	even	6
325.2.n.a.251.1		2	65.29	even	6
637.2.k.a.459.1		2	91.16	even	3
637.2.k.a.569.1		2	91.4	even	6
637.2.k.c.459.1		2	91.68	odd	6
637.2.k.c.569.1		2	91.17	odd	6
637.2.q.a.491.1		2	91.69	odd	6
637.2.q.a.589.1		2	91.55	odd	6
637.2.u.b.30.1		2	91.3	odd	6
637.2.u.b.361.1		2	91.82	odd	6
637.2.u.c.30.1		2	91.81	even	3
637.2.u.c.361.1		2	91.30	even	6
832.2.w.a.257.1		2	104.43	odd	6
832.2.w.a.641.1		2	104.3	odd	6
832.2.w.d.257.1		2	104.69	even	6
832.2.w.d.641.1		2	104.29	even	6
1521.2.a.k.1.1		2	39.5	even	4
1521.2.a.k.1.2		2	39.8	even	4
1521.2.b.a.1351.1		2	3.2	odd	2
1521.2.b.a.1351.2		2	39.38	odd	2
1872.2.by.d.433.1		2	156.107	even	6
1872.2.by.d.1297.1		2	156.95	even	6
2704.2.a.o.1.1		2	52.31	even	4
2704.2.a.o.1.2		2	52.47	even	4
2704.2.f.b.337.1		2	4.3	odd	2
2704.2.f.b.337.2		2	52.51	odd	2
4225.2.a.v.1.1		2	65.44	odd	4
4225.2.a.v.1.2		2	65.34	odd	4
8281.2.a.q.1.1		2	91.34	even	4
8281.2.a.q.1.2		2	91.83	even	4

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

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

Label	169.2.b.a.168.2

Level	$169$
Weight	$2$
Character	169.168
Analytic conductor	$1.349$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$