LMFDB - Embedded newform 169.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(1,169)]

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([0]))

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.34947179416$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	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
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	3.46410	1.41421
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.46410	−0.894427
$16$	−5.00000	−1.25000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.73205	0.408248
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	−1.73205	−0.387298
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−3.46410	−0.707107
$25$	−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
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	5.19615	0.891133
$35$	0	0
$36$	1.00000	0.166667
$37$	8.66025	1.42374	0.711868	−	0.702313i	$-0.247849\pi$
$37$	8.66025	1.42374	0.711868	+	0.702313i	$0.247849\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	5.19615	0.811503	0.405751	−	0.913984i	$-0.367010\pi$
$41$	5.19615	0.811503	0.405751	+	0.913984i	$0.367010\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−1.73205	−0.258199
$46$	10.3923	1.53226
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	−10.0000	−1.44338
$49$	−7.00000	−1.00000
$50$	−3.46410	−0.489898
$51$	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	$0.566058\pi$
$54$	−6.92820	−0.942809
$55$	0	0
$56$	0	0
$57$	−6.92820	−0.917663
$58$	5.19615	0.682288
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	−3.46410	−0.447214
$61$	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
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	3.00000	0.363803
$69$	12.0000	1.44463
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	−1.73205	−0.204124
$73$	−1.73205	−0.202721	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	−1.73205	−0.202721	−0.101361	+	0.994850i	$0.532320\pi$
$74$	15.0000	1.74371
$75$	−4.00000	−0.461880
$76$	−3.46410	−0.397360
$77$	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	$0.427756\pi$
$80$	8.66025	0.968246
$81$	−11.0000	−1.22222
$82$	9.00000	0.993884
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	−5.19615	−0.563602
$86$	−13.8564	−1.49417
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	6.00000	0.625543
$93$	6.92820	0.718421
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	−10.3923	−1.06066
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	−12.1244	−1.22474
$99$	0	0
$100$	−2.00000	−0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	10.3923	1.02899
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	−5.19615	−0.504695
$107$	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
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	17.3205	1.64399
$112$	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
$115$	−10.3923	−0.969087
$116$	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
$122$	1.73205	0.156813
$123$	10.3923	0.937043
$124$	3.46410	0.311086
$125$	12.1244	1.08444
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	12.1244	1.07165
$129$	−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
$135$	6.92820	0.596285
$136$	−5.19615	−0.445566
$137$	15.5885	1.33181	0.665906	−	0.746036i	$-0.268045\pi$
$137$	15.5885	1.33181	0.665906	+	0.746036i	$0.268045\pi$
$138$	20.7846	1.76930
$139$	−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
$141$	6.92820	0.583460
$142$	6.00000	0.503509
$143$	0	0
$144$	−5.00000	−0.416667
$145$	−5.19615	−0.431517
$146$	−3.00000	−0.248282
$147$	−14.0000	−1.15470
$148$	8.66025	0.711868
$149$	−19.0526	−1.56085	−0.780423	−	0.625252i	$-0.784996\pi$
$149$	−19.0526	−1.56085	−0.780423	+	0.625252i	$0.784996\pi$
$150$	−6.92820	−0.565685
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	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	$0.673605\pi$
$158$	6.92820	0.551178
$159$	−6.00000	−0.475831
$160$	9.00000	0.711512
$161$	0	0
$162$	−19.0526	−1.49691
$163$	−20.7846	−1.62798	−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846	−1.62798	−0.813988	+	0.580881i	$0.802708\pi$
$164$	5.19615	0.405751
$165$	0	0
$166$	24.0000	1.86276
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	0	0
$170$	−9.00000	−0.690268
$171$	−3.46410	−0.264906
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	10.3923	0.787839
$175$	0	0
$176$	0	0
$177$	−13.8564	−1.04151
$178$	−12.0000	−0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.73205	−0.129099
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−10.3923	−0.766131
$185$	−15.0000	−1.10282
$186$	12.0000	0.879883
$187$	0	0
$188$	3.46410	0.252646
$189$	0	0
$190$	10.3923	0.753937
$191$	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
$193$	−5.19615	−0.374027	−0.187014	−	0.982357i	$-0.559881\pi$
$193$	−5.19615	−0.374027	−0.187014	+	0.982357i	$0.559881\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	3.46410	0.244949
$201$	−6.92820	−0.488678
$202$	5.19615	0.365600
$203$	0	0
$204$	6.00000	0.420084
$205$	−9.00000	−0.628587
$206$	17.3205	1.20678
$207$	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
$213$	6.92820	0.474713
$214$	10.3923	0.710403
$215$	13.8564	0.944999
$216$	6.92820	0.471405
$217$	0	0
$218$	−24.0000	−1.62549
$219$	−3.46410	−0.234082
$220$	0	0
$221$	0	0
$222$	30.0000	2.01347
$223$	10.3923	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	−25.9808	−1.72821
$227$	24.2487	1.60944	0.804722	−	0.593652i	$-0.202314\pi$
$227$	24.2487	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$228$	−6.92820	−0.458831
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	−5.19615	−0.341144
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−6.92820	−0.450988
$237$	8.00000	0.519656
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	17.3205	1.11803
$241$	1.73205	0.111571	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	1.73205	0.111571	0.0557856	+	0.998443i	$0.482234\pi$
$242$	−19.0526	−1.22474
$243$	−10.0000	−0.641500
$244$	1.00000	0.0640184
$245$	12.1244	0.774597
$246$	18.0000	1.14764
$247$	0	0
$248$	−6.00000	−0.381000
$249$	27.7128	1.75623
$250$	21.0000	1.32816
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	3.46410	0.217357
$255$	−10.3923	−0.650791
$256$	19.0000	1.18750
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−27.7128	−1.72532
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	31.1769	1.92612
$263$	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
$265$	5.19615	0.319197
$266$	0	0
$267$	−13.8564	−0.847998
$268$	−3.46410	−0.211604
$269$	−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
$271$	20.7846	1.26258	0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846	1.26258	0.631288	+	0.775549i	$0.282527\pi$
$272$	−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	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	−6.92820	−0.415526
$279$	3.46410	0.207390
$280$	0	0
$281$	22.5167	1.34323	0.671616	−	0.740900i	$-0.265601\pi$
$281$	22.5167	1.34323	0.671616	+	0.740900i	$0.265601\pi$
$282$	12.0000	0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	3.46410	0.205557
$285$	12.0000	0.710819
$286$	0	0
$287$	0	0
$288$	−5.19615	−0.306186
$289$	−8.00000	−0.470588
$290$	−9.00000	−0.528498
$291$	13.8564	0.812277
$292$	−1.73205	−0.101361
$293$	−5.19615	−0.303562	−0.151781	−	0.988414i	$-0.548501\pi$
$293$	−5.19615	−0.303562	−0.151781	+	0.988414i	$0.548501\pi$
$294$	−24.2487	−1.41421
$295$	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
$304$	17.3205	0.993399
$305$	−1.73205	−0.0991769
$306$	5.19615	0.297044
$307$	17.3205	0.988534	0.494267	−	0.869310i	$-0.335437\pi$
$307$	17.3205	0.988534	0.494267	+	0.869310i	$0.335437\pi$
$308$	0	0
$309$	20.0000	1.13776
$310$	−10.3923	−0.590243
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−22.5167	−1.27069
$315$	0	0
$316$	4.00000	0.225018
$317$	−5.19615	−0.291845	−0.145922	−	0.989296i	$-0.546615\pi$
$317$	−5.19615	−0.291845	−0.145922	+	0.989296i	$0.546615\pi$
$318$	−10.3923	−0.582772
$319$	0	0
$320$	−1.73205	−0.0968246
$321$	12.0000	0.669775
$322$	0	0
$323$	−10.3923	−0.578243
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−36.0000	−1.99386
$327$	−27.7128	−1.53252
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−27.7128	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$331$	−27.7128	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$332$	13.8564	0.760469
$333$	8.66025	0.474579
$334$	−24.0000	−1.31322
$335$	6.00000	0.327815
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	−30.0000	−1.62938
$340$	−5.19615	−0.281801
$341$	0	0
$342$	−6.00000	−0.324443
$343$	0	0
$344$	13.8564	0.747087
$345$	−20.7846	−1.11901
$346$	−10.3923	−0.558694
$347$	−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
$349$	13.8564	0.741716	0.370858	−	0.928689i	$-0.379064\pi$
$349$	13.8564	0.741716	0.370858	+	0.928689i	$0.379064\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.9090	−1.75157	−0.875784	−	0.482704i	$-0.839655\pi$
$353$	−32.9090	−1.75157	−0.875784	+	0.482704i	$0.839655\pi$
$354$	−24.0000	−1.27559
$355$	−6.00000	−0.318447
$356$	−6.92820	−0.367194
$357$	0	0
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	3.00000	0.158114
$361$	−7.00000	−0.368421
$362$	−19.0526	−1.00138
$363$	−22.0000	−1.15470
$364$	0	0
$365$	3.00000	0.157027
$366$	3.46410	0.181071
$367$	−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
$369$	5.19615	0.270501
$370$	−25.9808	−1.35068
$371$	0	0
$372$	6.92820	0.359211
$373$	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
$375$	24.2487	1.25220
$376$	−6.00000	−0.309426
$377$	0	0
$378$	0	0
$379$	24.2487	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$379$	24.2487	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$380$	6.00000	0.307794
$381$	4.00000	0.204926
$382$	31.1769	1.59515
$383$	−20.7846	−1.06204	−0.531022	−	0.847358i	$-0.678192\pi$
$383$	−20.7846	−1.06204	−0.531022	+	0.847358i	$0.678192\pi$
$384$	24.2487	1.23744
$385$	0	0
$386$	−9.00000	−0.458088
$387$	−8.00000	−0.406663
$388$	6.92820	0.351726
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	12.1244	0.612372
$393$	36.0000	1.81596
$394$	−24.0000	−1.20910
$395$	−6.92820	−0.348596
$396$	0	0
$397$	13.8564	0.695433	0.347717	−	0.937600i	$-0.386957\pi$
$397$	13.8564	0.695433	0.347717	+	0.937600i	$0.386957\pi$
$398$	3.46410	0.173640
$399$	0	0
$400$	10.0000	0.500000
$401$	−1.73205	−0.0864945	−0.0432472	−	0.999064i	$-0.513770\pi$
$401$	−1.73205	−0.0864945	−0.0432472	+	0.999064i	$0.513770\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	3.00000	0.149256
$405$	19.0526	0.946729
$406$	0	0
$407$	0	0
$408$	−10.3923	−0.514496
$409$	15.5885	0.770800	0.385400	−	0.922750i	$-0.374064\pi$
$409$	15.5885	0.770800	0.385400	+	0.922750i	$0.374064\pi$
$410$	−15.5885	−0.769859
$411$	31.1769	1.53784
$412$	10.0000	0.492665
$413$	0	0
$414$	10.3923	0.510754
$415$	−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
$421$	15.5885	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$421$	15.5885	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$422$	17.3205	0.843149
$423$	3.46410	0.168430
$424$	5.19615	0.252347
$425$	−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
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	20.0000	0.962250
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	0	0
$435$	−10.3923	−0.498273
$436$	−13.8564	−0.663602
$437$	−20.7846	−0.994263
$438$	−6.00000	−0.286691
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	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	$0.592016\pi$
$444$	17.3205	0.821995
$445$	12.0000	0.568855
$446$	18.0000	0.852325
$447$	−38.1051	−1.80231
$448$	0	0
$449$	6.92820	0.326962	0.163481	−	0.986546i	$-0.447728\pi$
$449$	6.92820	0.326962	0.163481	+	0.986546i	$0.447728\pi$
$450$	−3.46410	−0.163299
$451$	0	0
$452$	−15.0000	−0.705541
$453$	−34.6410	−1.62758
$454$	42.0000	1.97116
$455$	0	0
$456$	12.0000	0.561951
$457$	−1.73205	−0.0810219	−0.0405110	−	0.999179i	$-0.512899\pi$
$457$	−1.73205	−0.0810219	−0.0405110	+	0.999179i	$0.512899\pi$
$458$	0	0
$459$	−12.0000	−0.560112
$460$	−10.3923	−0.484544
$461$	22.5167	1.04871	0.524353	−	0.851501i	$-0.324307\pi$
$461$	22.5167	1.04871	0.524353	+	0.851501i	$0.324307\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	−15.0000	−0.696358
$465$	−12.0000	−0.556487
$466$	−10.3923	−0.481414
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	−10.3923	−0.479361
$471$	−26.0000	−1.19802
$472$	12.0000	0.552345
$473$	0	0
$474$	13.8564	0.636446
$475$	6.92820	0.317888
$476$	0	0
$477$	−3.00000	−0.137361
$478$	−36.0000	−1.64660
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	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
$486$	−17.3205	−0.785674
$487$	−6.92820	−0.313947	−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820	−0.313947	−0.156973	+	0.987603i	$0.550174\pi$
$488$	−1.73205	−0.0784063
$489$	−41.5692	−1.87983
$490$	21.0000	0.948683
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	10.3923	0.468521
$493$	9.00000	0.405340
$494$	0	0
$495$	0	0
$496$	−17.3205	−0.777714
$497$	0	0
$498$	48.0000	2.15093
$499$	31.1769	1.39567	0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769	1.39567	0.697835	+	0.716258i	$0.254147\pi$
$500$	12.1244	0.542218
$501$	−27.7128	−1.23812
$502$	31.1769	1.39149
$503$	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
$505$	−5.19615	−0.231226
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	−19.0526	−0.844490	−0.422245	−	0.906482i	$-0.638758\pi$
$509$	−19.0526	−0.844490	−0.422245	+	0.906482i	$0.638758\pi$
$510$	−18.0000	−0.797053
$511$	0	0
$512$	8.66025	0.382733
$513$	13.8564	0.611775
$514$	−5.19615	−0.229192
$515$	−17.3205	−0.763233
$516$	−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	$0.436832\pi$
$522$	5.19615	0.227429
$523$	−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
$526$	20.7846	0.906252
$527$	10.3923	0.452696
$528$	0	0
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	−6.92820	−0.300658
$532$	0	0
$533$	0	0
$534$	−24.0000	−1.03858
$535$	−10.3923	−0.449299
$536$	6.00000	0.259161
$537$	0	0
$538$	−10.3923	−0.448044
$539$	0	0
$540$	6.92820	0.298142
$541$	−29.4449	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$541$	−29.4449	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$542$	36.0000	1.54633
$543$	−22.0000	−0.944110
$544$	−15.5885	−0.668350
$545$	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	$0.655864\pi$
$548$	15.5885	0.665906
$549$	1.00000	0.0426790
$550$	0	0
$551$	−10.3923	−0.442727
$552$	−20.7846	−0.884652
$553$	0	0
$554$	12.1244	0.515115
$555$	−30.0000	−1.27343
$556$	−4.00000	−0.169638
$557$	−15.5885	−0.660504	−0.330252	−	0.943893i	$-0.607134\pi$
$557$	−15.5885	−0.660504	−0.330252	+	0.943893i	$0.607134\pi$
$558$	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	$0.5\pi$
$564$	6.92820	0.291730
$565$	25.9808	1.09302
$566$	−6.92820	−0.291214
$567$	0	0
$568$	−6.00000	−0.251754
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	20.7846	0.870572
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	−19.0526	−0.793168	−0.396584	−	0.917998i	$-0.629805\pi$
$577$	−19.0526	−0.793168	−0.396584	+	0.917998i	$0.629805\pi$
$578$	−13.8564	−0.576351
$579$	−10.3923	−0.431889
$580$	−5.19615	−0.215758
$581$	0	0
$582$	24.0000	0.994832
$583$	0	0
$584$	3.00000	0.124141
$585$	0	0
$586$	−9.00000	−0.371787
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	−14.0000	−0.577350
$589$	−12.0000	−0.494451
$590$	20.7846	0.855689
$591$	−27.7128	−1.13995
$592$	−43.3013	−1.77967
$593$	25.9808	1.06690	0.533451	−	0.845831i	$-0.320895\pi$
$593$	25.9808	1.06690	0.533451	+	0.845831i	$0.320895\pi$
$594$	0	0
$595$	0	0
$596$	−19.0526	−0.780423
$597$	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	$0.290010\pi$
$600$	6.92820	0.282843
$601$	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
$603$	−3.46410	−0.141069
$604$	−17.3205	−0.704761
$605$	19.0526	0.774597
$606$	10.3923	0.422159
$607$	−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
$613$	12.1244	0.489698	0.244849	−	0.969561i	$-0.421262\pi$
$613$	12.1244	0.489698	0.244849	+	0.969561i	$0.421262\pi$
$614$	30.0000	1.21070
$615$	−18.0000	−0.725830
$616$	0	0
$617$	−22.5167	−0.906487	−0.453243	−	0.891387i	$-0.649733\pi$
$617$	−22.5167	−0.906487	−0.453243	+	0.891387i	$0.649733\pi$
$618$	34.6410	1.39347
$619$	20.7846	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$620$	−6.00000	−0.240966
$621$	−24.0000	−0.963087
$622$	51.9615	2.08347
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	17.3205	0.692267
$627$	0	0
$628$	−13.0000	−0.518756
$629$	25.9808	1.03592
$630$	0	0
$631$	48.4974	1.93065	0.965326	−	0.261048i	$-0.0840679\pi$
$631$	48.4974	1.93065	0.965326	+	0.261048i	$0.0840679\pi$
$632$	−6.92820	−0.275589
$633$	20.0000	0.794929
$634$	−9.00000	−0.357436
$635$	−3.46410	−0.137469
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	3.46410	0.137038
$640$	−21.0000	−0.830098
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	20.7846	0.820303
$643$	13.8564	0.546443	0.273222	−	0.961951i	$-0.411911\pi$
$643$	13.8564	0.546443	0.273222	+	0.961951i	$0.411911\pi$
$644$	0	0
$645$	27.7128	1.09119
$646$	−18.0000	−0.708201
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	19.0526	0.748455
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−20.7846	−0.813988
$653$	−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
$655$	−31.1769	−1.21818
$656$	−25.9808	−1.01438
$657$	−1.73205	−0.0675737
$658$	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
$661$	−46.7654	−1.81896	−0.909481	−	0.415745i	$-0.863521\pi$
$661$	−46.7654	−1.81896	−0.909481	+	0.415745i	$0.863521\pi$
$662$	−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
$668$	−13.8564	−0.536120
$669$	20.7846	0.803579
$670$	10.3923	0.401490
$671$	0	0
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	39.8372	1.53447
$675$	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	$0.536783\pi$
$678$	−51.9615	−1.99557
$679$	0	0
$680$	9.00000	0.345134
$681$	48.4974	1.85843
$682$	0	0
$683$	−24.2487	−0.927851	−0.463926	−	0.885874i	$-0.653559\pi$
$683$	−24.2487	−0.927851	−0.463926	+	0.885874i	$0.653559\pi$
$684$	−3.46410	−0.132453
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	40.0000	1.52499
$689$	0	0
$690$	−36.0000	−1.37050
$691$	−13.8564	−0.527123	−0.263561	−	0.964643i	$-0.584897\pi$
$691$	−13.8564	−0.527123	−0.263561	+	0.964643i	$0.584897\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−51.9615	−1.97243
$695$	6.92820	0.262802
$696$	−10.3923	−0.393919
$697$	15.5885	0.590455
$698$	24.0000	0.908413
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	0	0
$705$	−12.0000	−0.451946
$706$	−57.0000	−2.14522
$707$	0	0
$708$	−13.8564	−0.520756
$709$	5.19615	0.195146	0.0975728	−	0.995228i	$-0.468892\pi$
$709$	5.19615	0.195146	0.0975728	+	0.995228i	$0.468892\pi$
$710$	−10.3923	−0.390016
$711$	4.00000	0.150012
$712$	12.0000	0.449719
$713$	20.7846	0.778390
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−41.5692	−1.55243
$718$	12.0000	0.447836
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	8.66025	0.322749
$721$	0	0
$722$	−12.1244	−0.451222
$723$	3.46410	0.128831
$724$	−11.0000	−0.408812
$725$	−6.00000	−0.222834
$726$	−38.1051	−1.41421
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	5.19615	0.192318
$731$	−24.0000	−0.887672
$732$	2.00000	0.0739221
$733$	12.1244	0.447823	0.223912	−	0.974609i	$-0.428117\pi$
$733$	12.1244	0.447823	0.223912	+	0.974609i	$0.428117\pi$
$734$	−38.1051	−1.40649
$735$	24.2487	0.894427
$736$	−31.1769	−1.14920
$737$	0	0
$738$	9.00000	0.331295
$739$	−20.7846	−0.764574	−0.382287	−	0.924044i	$-0.624863\pi$
$739$	−20.7846	−0.764574	−0.382287	+	0.924044i	$0.624863\pi$
$740$	−15.0000	−0.551411
$741$	0	0
$742$	0	0
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	−12.0000	−0.439941
$745$	33.0000	1.20903
$746$	32.9090	1.20488
$747$	13.8564	0.506979
$748$	0	0
$749$	0	0
$750$	42.0000	1.53362
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−17.3205	−0.631614
$753$	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
$760$	−10.3923	−0.376969
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	6.92820	0.250982
$763$	0	0
$764$	18.0000	0.651217
$765$	−5.19615	−0.187867
$766$	−36.0000	−1.30073
$767$	0	0
$768$	38.0000	1.37121
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−5.19615	−0.187014
$773$	−34.6410	−1.24595	−0.622975	−	0.782241i	$-0.714076\pi$
$773$	−34.6410	−1.24595	−0.622975	+	0.782241i	$0.714076\pi$
$774$	−13.8564	−0.498058
$775$	−6.92820	−0.248868
$776$	−12.0000	−0.430775
$777$	0	0
$778$	15.5885	0.558873
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	31.1769	1.11488
$783$	−12.0000	−0.428845
$784$	35.0000	1.25000
$785$	22.5167	0.803654
$786$	62.3538	2.22409
$787$	−38.1051	−1.35830	−0.679150	−	0.733999i	$-0.737652\pi$
$787$	−38.1051	−1.35830	−0.679150	+	0.733999i	$0.737652\pi$
$788$	−13.8564	−0.493614
$789$	24.0000	0.854423
$790$	−12.0000	−0.426941
$791$	0	0
$792$	0	0
$793$	0	0
$794$	24.0000	0.851728
$795$	10.3923	0.368577
$796$	2.00000	0.0708881
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	10.3923	0.367653
$800$	10.3923	0.367423
$801$	−6.92820	−0.244796
$802$	−3.00000	−0.105934
$803$	0	0
$804$	−6.92820	−0.244339
$805$	0	0
$806$	0	0
$807$	−12.0000	−0.422420
$808$	−5.19615	−0.182800
$809$	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
$811$	−38.1051	−1.33805	−0.669026	−	0.743239i	$-0.733288\pi$
$811$	−38.1051	−1.33805	−0.669026	+	0.743239i	$0.733288\pi$
$812$	0	0
$813$	41.5692	1.45790
$814$	0	0
$815$	36.0000	1.26102
$816$	−30.0000	−1.05021
$817$	27.7128	0.969549
$818$	27.0000	0.944033
$819$	0	0
$820$	−9.00000	−0.314294
$821$	41.5692	1.45078	0.725388	−	0.688340i	$-0.241660\pi$
$821$	41.5692	1.45078	0.725388	+	0.688340i	$0.241660\pi$
$822$	54.0000	1.88347
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−17.3205	−0.603388
$825$	0	0
$826$	0	0
$827$	−20.7846	−0.722752	−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846	−0.722752	−0.361376	+	0.932420i	$0.617693\pi$
$828$	6.00000	0.208514
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	−41.5692	−1.44289
$831$	14.0000	0.485655
$832$	0	0
$833$	−21.0000	−0.727607
$834$	−13.8564	−0.479808
$835$	24.0000	0.830554
$836$	0	0
$837$	−13.8564	−0.478947
$838$	31.1769	1.07699
$839$	−45.0333	−1.55472	−0.777361	−	0.629054i	$-0.783442\pi$
$839$	−45.0333	−1.55472	−0.777361	+	0.629054i	$0.783442\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	27.0000	0.930481
$843$	45.0333	1.55103
$844$	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
$850$	−10.3923	−0.356453
$851$	51.9615	1.78122
$852$	6.92820	0.237356
$853$	−25.9808	−0.889564	−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808	−0.889564	−0.444782	+	0.895639i	$0.646719\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	−10.3923	−0.355202
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	13.8564	0.472500
$861$	0	0
$862$	12.0000	0.408722
$863$	−27.7128	−0.943355	−0.471678	−	0.881771i	$-0.656351\pi$
$863$	−27.7128	−0.943355	−0.471678	+	0.881771i	$0.656351\pi$
$864$	20.7846	0.707107
$865$	10.3923	0.353349
$866$	29.4449	1.00058
$867$	−16.0000	−0.543388
$868$	0	0
$869$	0	0
$870$	−18.0000	−0.610257
$871$	0	0
$872$	24.0000	0.812743
$873$	6.92820	0.234484
$874$	−36.0000	−1.21772
$875$	0	0
$876$	−3.46410	−0.117041
$877$	12.1244	0.409410	0.204705	−	0.978824i	$-0.434376\pi$
$877$	12.1244	0.409410	0.204705	+	0.978824i	$0.434376\pi$
$878$	48.4974	1.63671
$879$	−10.3923	−0.350524
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	−12.1244	−0.408248
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	−20.7846	−0.698273
$887$	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
$890$	20.7846	0.696702
$891$	0	0
$892$	10.3923	0.347960
$893$	−12.0000	−0.401565
$894$	−66.0000	−2.20737
$895$	0	0
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	10.3923	0.346603
$900$	−2.00000	−0.0666667
$901$	−9.00000	−0.299833
$902$	0	0
$903$	0	0
$904$	25.9808	0.864107
$905$	19.0526	0.633328
$906$	−60.0000	−1.99337
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.2487	0.804722
$909$	3.00000	0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	34.6410	1.14708
$913$	0	0
$914$	−3.00000	−0.0992312
$915$	−3.46410	−0.114520
$916$	0	0
$917$	0	0
$918$	−20.7846	−0.685994
$919$	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
$921$	34.6410	1.14146
$922$	39.0000	1.28440
$923$	0	0
$924$	0	0
$925$	−17.3205	−0.569495
$926$	−24.0000	−0.788689
$927$	10.0000	0.328443
$928$	−15.5885	−0.511716
$929$	−46.7654	−1.53432	−0.767161	−	0.641455i	$-0.778331\pi$
$929$	−46.7654	−1.53432	−0.767161	+	0.641455i	$0.778331\pi$
$930$	−20.7846	−0.681554
$931$	24.2487	0.794719
$932$	−6.00000	−0.196537
$933$	60.0000	1.96431
$934$	−20.7846	−0.680093
$935$	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
$941$	20.7846	0.677559	0.338779	−	0.940866i	$-0.389986\pi$
$941$	20.7846	0.677559	0.338779	+	0.940866i	$0.389986\pi$
$942$	−45.0333	−1.46726
$943$	31.1769	1.01526
$944$	34.6410	1.12747
$945$	0	0
$946$	0	0
$947$	−17.3205	−0.562841	−0.281420	−	0.959585i	$-0.590806\pi$
$947$	−17.3205	−0.562841	−0.281420	+	0.959585i	$0.590806\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	12.0000	0.389331
$951$	−10.3923	−0.336994
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−5.19615	−0.168232
$955$	−31.1769	−1.00886
$956$	−20.7846	−0.672222
$957$	0	0
$958$	42.0000	1.35696
$959$	0	0
$960$	−3.46410	−0.111803
$961$	−19.0000	−0.612903
$962$	0	0
$963$	6.00000	0.193347
$964$	1.73205	0.0557856
$965$	9.00000	0.289720
$966$	0	0
$967$	58.8897	1.89377	0.946883	−	0.321578i	$-0.104213\pi$
$967$	58.8897	1.89377	0.946883	+	0.321578i	$0.104213\pi$
$968$	19.0526	0.612372
$969$	−20.7846	−0.667698
$970$	−20.7846	−0.667354
$971$	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
$977$	−43.3013	−1.38533	−0.692665	−	0.721259i	$-0.743564\pi$
$977$	−43.3013	−1.38533	−0.692665	+	0.721259i	$0.743564\pi$
$978$	−72.0000	−2.30231
$979$	0	0
$980$	12.1244	0.387298
$981$	−13.8564	−0.442401
$982$	−20.7846	−0.663264
$983$	−51.9615	−1.65732	−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615	−1.65732	−0.828658	+	0.559756i	$0.810895\pi$
$984$	−18.0000	−0.573819
$985$	24.0000	0.764704
$986$	15.5885	0.496438
$987$	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
$993$	−55.4256	−1.75888
$994$	0	0
$995$	−3.46410	−0.109819
$996$	27.7128	0.878114
$997$	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
$999$	−34.6410	−1.09599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.a.a.1.2		2
3.2	odd	2		1521.2.a.k.1.1		2
4.3	odd	2		2704.2.a.o.1.1		2
5.4	even	2		4225.2.a.v.1.1		2
7.6	odd	2		8281.2.a.q.1.2		2
13.2	odd	12		169.2.e.a.147.1		2
13.3	even	3		169.2.c.a.22.1		4
13.4	even	6		169.2.c.a.146.2		4
13.5	odd	4		169.2.b.a.168.1		2
13.6	odd	12		13.2.e.a.10.1	yes	2
13.7	odd	12		169.2.e.a.23.1		2
13.8	odd	4		169.2.b.a.168.2		2
13.9	even	3		169.2.c.a.146.1		4
13.10	even	6		169.2.c.a.22.2		4
13.11	odd	12		13.2.e.a.4.1	✓	2
13.12	even	2	inner	169.2.a.a.1.1		2
39.5	even	4		1521.2.b.a.1351.2		2
39.8	even	4		1521.2.b.a.1351.1		2
39.11	even	12		117.2.q.c.82.1		2
39.32	even	12		117.2.q.c.10.1		2
39.38	odd	2		1521.2.a.k.1.2		2
52.11	even	12		208.2.w.b.17.1		2
52.19	even	12		208.2.w.b.49.1		2
52.31	even	4		2704.2.f.b.337.2		2
52.47	even	4		2704.2.f.b.337.1		2
52.51	odd	2		2704.2.a.o.1.2		2
65.19	odd	12		325.2.n.a.101.1		2
65.24	odd	12		325.2.n.a.251.1		2
65.32	even	12		325.2.m.a.49.1		4
65.37	even	12		325.2.m.a.199.2		4
65.58	even	12		325.2.m.a.49.2		4
65.63	even	12		325.2.m.a.199.1		4
65.64	even	2		4225.2.a.v.1.2		2
91.6	even	12		637.2.q.a.491.1		2
91.11	odd	12		637.2.u.c.30.1		2
91.19	even	12		637.2.u.b.361.1		2
91.24	even	12		637.2.u.b.30.1		2
91.32	odd	12		637.2.k.a.569.1		2
91.37	odd	12		637.2.k.a.459.1		2
91.45	even	12		637.2.k.c.569.1		2
91.58	odd	12		637.2.u.c.361.1		2
91.76	even	12		637.2.q.a.589.1		2
91.89	even	12		637.2.k.c.459.1		2
91.90	odd	2		8281.2.a.q.1.1		2
104.11	even	12		832.2.w.a.641.1		2
104.19	even	12		832.2.w.a.257.1		2
104.37	odd	12		832.2.w.d.641.1		2
104.45	odd	12		832.2.w.d.257.1		2
156.11	odd	12		1872.2.by.d.433.1		2
156.71	odd	12		1872.2.by.d.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	13.11	odd	12
13.2.e.a.10.1	yes	2	13.6	odd	12
117.2.q.c.10.1		2	39.32	even	12
117.2.q.c.82.1		2	39.11	even	12
169.2.a.a.1.1		2	13.12	even	2	inner
169.2.a.a.1.2		2	1.1	even	1	trivial
169.2.b.a.168.1		2	13.5	odd	4
169.2.b.a.168.2		2	13.8	odd	4
169.2.c.a.22.1		4	13.3	even	3
169.2.c.a.22.2		4	13.10	even	6
169.2.c.a.146.1		4	13.9	even	3
169.2.c.a.146.2		4	13.4	even	6
169.2.e.a.23.1		2	13.7	odd	12
169.2.e.a.147.1		2	13.2	odd	12
208.2.w.b.17.1		2	52.11	even	12
208.2.w.b.49.1		2	52.19	even	12
325.2.m.a.49.1		4	65.32	even	12
325.2.m.a.49.2		4	65.58	even	12
325.2.m.a.199.1		4	65.63	even	12
325.2.m.a.199.2		4	65.37	even	12
325.2.n.a.101.1		2	65.19	odd	12
325.2.n.a.251.1		2	65.24	odd	12
637.2.k.a.459.1		2	91.37	odd	12
637.2.k.a.569.1		2	91.32	odd	12
637.2.k.c.459.1		2	91.89	even	12
637.2.k.c.569.1		2	91.45	even	12
637.2.q.a.491.1		2	91.6	even	12
637.2.q.a.589.1		2	91.76	even	12
637.2.u.b.30.1		2	91.24	even	12
637.2.u.b.361.1		2	91.19	even	12
637.2.u.c.30.1		2	91.11	odd	12
637.2.u.c.361.1		2	91.58	odd	12
832.2.w.a.257.1		2	104.19	even	12
832.2.w.a.641.1		2	104.11	even	12
832.2.w.d.257.1		2	104.45	odd	12
832.2.w.d.641.1		2	104.37	odd	12
1521.2.a.k.1.1		2	3.2	odd	2
1521.2.a.k.1.2		2	39.38	odd	2
1521.2.b.a.1351.1		2	39.8	even	4
1521.2.b.a.1351.2		2	39.5	even	4
1872.2.by.d.433.1		2	156.11	odd	12
1872.2.by.d.1297.1		2	156.71	odd	12
2704.2.a.o.1.1		2	4.3	odd	2
2704.2.a.o.1.2		2	52.51	odd	2
2704.2.f.b.337.1		2	52.47	even	4
2704.2.f.b.337.2		2	52.31	even	4
4225.2.a.v.1.1		2	5.4	even	2
4225.2.a.v.1.2		2	65.64	even	2
8281.2.a.q.1.1		2	91.90	odd	2
8281.2.a.q.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

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

Introduction

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