LMFDB - Embedded newform 45.14.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,14,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-131] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.2539180284$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3121}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 780 $ x^2 - x - 780
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$28.4330$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-149.299 q^{2} +14098.2 q^{4} +15625.0 q^{5} +384225. q^{7} -881782. q^{8} -2.33279e6 q^{10} -6.10216e6 q^{11} +1.13332e6 q^{13} -5.73644e7 q^{14} +1.61570e7 q^{16} +2.69912e7 q^{17} -3.29943e8 q^{19} +2.20284e8 q^{20} +9.11046e8 q^{22} -4.70204e8 q^{23} +2.44141e8 q^{25} -1.69203e8 q^{26} +5.41687e9 q^{28} +4.24890e9 q^{29} +7.67054e9 q^{31} +4.81134e9 q^{32} -4.02976e9 q^{34} +6.00352e9 q^{35} -2.83184e10 q^{37} +4.92601e10 q^{38} -1.37778e10 q^{40} +2.11336e10 q^{41} -6.93928e10 q^{43} -8.60292e10 q^{44} +7.02010e10 q^{46} -1.01233e11 q^{47} +5.07401e10 q^{49} -3.64499e10 q^{50} +1.59777e10 q^{52} +1.81846e11 q^{53} -9.53463e10 q^{55} -3.38803e11 q^{56} -6.34357e11 q^{58} -3.10938e11 q^{59} +1.76596e11 q^{61} -1.14520e12 q^{62} -8.50686e11 q^{64} +1.77081e10 q^{65} +6.25757e10 q^{67} +3.80526e11 q^{68} -8.96319e11 q^{70} -1.13419e12 q^{71} -1.17191e11 q^{73} +4.22790e12 q^{74} -4.65158e12 q^{76} -2.34461e12 q^{77} +3.99281e12 q^{79} +2.52452e11 q^{80} -3.15523e12 q^{82} -8.58275e11 q^{83} +4.21738e11 q^{85} +1.03603e13 q^{86} +5.38077e12 q^{88} +9.67991e11 q^{89} +4.35450e11 q^{91} -6.62901e12 q^{92} +1.51139e13 q^{94} -5.15536e12 q^{95} +2.50052e12 q^{97} -7.57544e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 131 q^{2} + 6241 q^{4} + 31250 q^{5} + 496272 q^{7} - 1175463 q^{8} - 2046875 q^{10} - 6245888 q^{11} + 1761164 q^{13} - 55314084 q^{14} + 75148705 q^{16} - 48151604 q^{17} + 38475344 q^{19} + 97515625 q^{20}+ \cdots - 9118668656251 q^{98}+O(q^{100}) $ 2 * q - 131 * q^2 + 6241 * q^4 + 31250 * q^5 + 496272 * q^7 - 1175463 * q^8 - 2046875 * q^10 - 6245888 * q^11 + 1761164 * q^13 - 55314084 * q^14 + 75148705 * q^16 - 48151604 * q^17 + 38475344 * q^19 + 97515625 * q^20 + 908415728 * q^22 - 734264616 * q^23 + 488281250 * q^25 - 157714454 * q^26 + 4536499884 * q^28 - 1126474708 * q^29 + 6100104312 * q^31 + 8296660073 * q^32 - 5404788238 * q^34 + 7754250000 * q^35 - 46819674548 * q^37 + 56001723484 * q^38 - 18366609375 * q^40 + 29607909788 * q^41 - 63189768520 * q^43 - 84899911888 * q^44 + 65368954992 * q^46 - 105004415960 * q^47 - 33594444590 * q^49 - 31982421875 * q^50 + 11044638034 * q^52 - 48425833796 * q^53 - 97592000000 * q^55 - 371708899380 * q^56 - 732719858498 * q^58 - 722768180096 * q^59 - 229984402180 * q^61 - 1173940563384 * q^62 - 1270168541071 * q^64 + 27518187500 * q^65 + 359062188984 * q^67 + 970935017018 * q^68 - 864282562500 * q^70 - 736545668224 * q^71 - 1494470740204 * q^73 + 3889351112906 * q^74 - 7546301154644 * q^76 - 2360709208704 * q^77 + 3526348590120 * q^79 + 1174198515625 * q^80 - 3000156702386 * q^82 - 5631050986968 * q^83 - 752368812500 * q^85 + 10473766954252 * q^86 + 5422984137648 * q^88 - 5913121459764 * q^89 + 505797926592 * q^91 - 4554248300592 * q^92 + 15044889696176 * q^94 + 601177250000 * q^95 + 16383233322244 * q^97 - 9118668656251 * q^98

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 $

$2$	−149.299	−1.64954	−0.824768	−	0.565472i	$-0.808694\pi$
$2$	−149.299	−1.64954	−0.824768	+	0.565472i	$0.808694\pi$
$3$	0	0
$3$	0	0
$4$	14098.2	1.72097
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	384225.	1.23438	0.617190	−	0.786814i	$-0.288271\pi$
$7$	384225.	1.23438	0.617190	+	0.786814i	$0.288271\pi$
$8$	−881782.	−1.18926
$9$	0	0
$10$	−2.33279e6	−0.737694
$11$	−6.10216e6	−1.03856	−0.519280	−	0.854604i	$-0.673800\pi$
$11$	−6.10216e6	−1.03856	−0.519280	+	0.854604i	$0.673800\pi$
$12$	0	0
$13$	1.13332e6	0.0651209	0.0325605	−	0.999470i	$-0.489634\pi$
$13$	1.13332e6	0.0651209	0.0325605	+	0.999470i	$0.489634\pi$
$14$	−5.73644e7	−2.03615
$15$	0	0
$16$	1.61570e7	0.240757
$17$	2.69912e7	0.271209	0.135605	−	0.990763i	$-0.456702\pi$
$17$	2.69912e7	0.271209	0.135605	+	0.990763i	$0.456702\pi$
$18$	0	0
$19$	−3.29943e8	−1.60894	−0.804471	−	0.593992i	$-0.797551\pi$
$19$	−3.29943e8	−1.60894	−0.804471	+	0.593992i	$0.797551\pi$
$20$	2.20284e8	0.769639
$21$	0	0
$22$	9.11046e8	1.71314
$23$	−4.70204e8	−0.662301	−0.331151	−	0.943578i	$-0.607437\pi$
$23$	−4.70204e8	−0.662301	−0.331151	+	0.943578i	$0.607437\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	−1.69203e8	−0.107419
$27$	0	0
$28$	5.41687e9	2.12433
$29$	4.24890e9	1.32645	0.663224	−	0.748421i	$-0.269188\pi$
$29$	4.24890e9	1.32645	0.663224	+	0.748421i	$0.269188\pi$
$30$	0	0
$31$	7.67054e9	1.55230	0.776149	−	0.630550i	$-0.217170\pi$
$31$	7.67054e9	1.55230	0.776149	+	0.630550i	$0.217170\pi$
$32$	4.81134e9	0.792121
$33$	0	0
$34$	−4.02976e9	−0.447369
$35$	6.00352e9	0.552031
$36$	0	0
$37$	−2.83184e10	−1.81450	−0.907251	−	0.420590i	$-0.861823\pi$
$37$	−2.83184e10	−1.81450	−0.907251	+	0.420590i	$0.861823\pi$
$38$	4.92601e10	2.65401
$39$	0	0
$40$	−1.37778e10	−0.531852
$41$	2.11336e10	0.694831	0.347415	−	0.937711i	$-0.387059\pi$
$41$	2.11336e10	0.694831	0.347415	+	0.937711i	$0.387059\pi$
$42$	0	0
$43$	−6.93928e10	−1.67405	−0.837026	−	0.547162i	$-0.815708\pi$
$43$	−6.93928e10	−1.67405	−0.837026	+	0.547162i	$0.815708\pi$
$44$	−8.60292e10	−1.78732
$45$	0	0
$46$	7.02010e10	1.09249
$47$	−1.01233e11	−1.36989	−0.684943	−	0.728597i	$-0.740173\pi$
$47$	−1.01233e11	−1.36989	−0.684943	+	0.728597i	$0.740173\pi$
$48$	0	0
$49$	5.07401e10	0.523693
$50$	−3.64499e10	−0.329907
$51$	0	0
$52$	1.59777e10	0.112071
$53$	1.81846e11	1.12697	0.563484	−	0.826127i	$-0.309461\pi$
$53$	1.81846e11	1.12697	0.563484	+	0.826127i	$0.309461\pi$
$54$	0	0
$55$	−9.53463e10	−0.464458
$56$	−3.38803e11	−1.46800
$57$	0	0
$58$	−6.34357e11	−2.18802
$59$	−3.10938e11	−0.959701	−0.479850	−	0.877350i	$-0.659309\pi$
$59$	−3.10938e11	−0.959701	−0.479850	+	0.877350i	$0.659309\pi$
$60$	0	0
$61$	1.76596e11	0.438870	0.219435	−	0.975627i	$-0.429579\pi$
$61$	1.76596e11	0.438870	0.219435	+	0.975627i	$0.429579\pi$
$62$	−1.14520e12	−2.56057
$63$	0	0
$64$	−8.50686e11	−1.54739
$65$	1.77081e10	0.0291230
$66$	0	0
$67$	6.25757e10	0.0845123	0.0422561	−	0.999107i	$-0.486545\pi$
$67$	6.25757e10	0.0845123	0.0422561	+	0.999107i	$0.486545\pi$
$68$	3.80526e11	0.466742
$69$	0	0
$70$	−8.96319e11	−0.910595
$71$	−1.13419e12	−1.05076	−0.525382	−	0.850867i	$-0.676077\pi$
$71$	−1.13419e12	−1.05076	−0.525382	+	0.850867i	$0.676077\pi$
$72$	0	0
$73$	−1.17191e11	−0.0906350	−0.0453175	−	0.998973i	$-0.514430\pi$
$73$	−1.17191e11	−0.0906350	−0.0453175	+	0.998973i	$0.514430\pi$
$74$	4.22790e12	2.99308
$75$	0	0
$76$	−4.65158e12	−2.76893
$77$	−2.34461e12	−1.28198
$78$	0	0
$79$	3.99281e12	1.84800	0.924002	−	0.382387i	$-0.124898\pi$
$79$	3.99281e12	1.84800	0.924002	+	0.382387i	$0.124898\pi$
$80$	2.52452e11	0.107670
$81$	0	0
$82$	−3.15523e12	−1.14615
$83$	−8.58275e11	−0.288150	−0.144075	−	0.989567i	$-0.546021\pi$
$83$	−8.58275e11	−0.288150	−0.144075	+	0.989567i	$0.546021\pi$
$84$	0	0
$85$	4.21738e11	0.121289
$86$	1.03603e13	2.76141
$87$	0	0
$88$	5.38077e12	1.23512
$89$	9.67991e11	0.206460	0.103230	−	0.994657i	$-0.467082\pi$
$89$	9.67991e11	0.206460	0.103230	+	0.994657i	$0.467082\pi$
$90$	0	0
$91$	4.35450e11	0.0803839
$92$	−6.62901e12	−1.13980
$93$	0	0
$94$	1.51139e13	2.25967
$95$	−5.15536e12	−0.719541
$96$	0	0
$97$	2.50052e12	0.304799	0.152400	−	0.988319i	$-0.451300\pi$
$97$	2.50052e12	0.304799	0.152400	+	0.988319i	$0.451300\pi$
$98$	−7.57544e12	−0.863850
$99$	0	0
$100$	3.44193e12	0.344193
$101$	3.51016e12	0.329032	0.164516	−	0.986374i	$-0.447394\pi$
$101$	3.51016e12	0.329032	0.164516	+	0.986374i	$0.447394\pi$
$102$	0	0
$103$	−1.08186e13	−0.892747	−0.446373	−	0.894847i	$-0.647285\pi$
$103$	−1.08186e13	−0.892747	−0.446373	+	0.894847i	$0.647285\pi$
$104$	−9.99340e11	−0.0774456
$105$	0	0
$106$	−2.71495e13	−1.85897
$107$	−1.49010e13	−0.959892	−0.479946	−	0.877298i	$-0.659344\pi$
$107$	−1.49010e13	−0.959892	−0.479946	+	0.877298i	$0.659344\pi$
$108$	0	0
$109$	−2.01251e13	−1.14939	−0.574694	−	0.818369i	$-0.694879\pi$
$109$	−2.01251e13	−1.14939	−0.574694	+	0.818369i	$0.694879\pi$
$110$	1.42351e13	0.766139
$111$	0	0
$112$	6.20791e12	0.297186
$113$	−2.87905e13	−1.30088	−0.650442	−	0.759556i	$-0.725417\pi$
$113$	−2.87905e13	−1.30088	−0.650442	+	0.759556i	$0.725417\pi$
$114$	0	0
$115$	−7.34694e12	−0.296190
$116$	5.99017e13	2.28277
$117$	0	0
$118$	4.64227e13	1.58306
$119$	1.03707e13	0.334775
$120$	0	0
$121$	2.71366e12	0.0786051
$122$	−2.63655e13	−0.723932
$123$	0	0
$124$	1.08140e14	2.67145
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	7.01983e13	1.48458	0.742288	−	0.670081i	$-0.233741\pi$
$127$	7.01983e13	1.48458	0.742288	+	0.670081i	$0.233741\pi$
$128$	8.75919e13	1.76035
$129$	0	0
$130$	−2.64380e12	−0.0480393
$131$	−9.26965e13	−1.60251	−0.801254	−	0.598325i	$-0.795833\pi$
$131$	−9.26965e13	−1.60251	−0.801254	+	0.598325i	$0.795833\pi$
$132$	0	0
$133$	−1.26772e14	−1.98604
$134$	−9.34249e12	−0.139406
$135$	0	0
$136$	−2.38004e13	−0.322538
$137$	4.89202e13	0.632127	0.316064	−	0.948738i	$-0.397639\pi$
$137$	4.89202e13	0.632127	0.316064	+	0.948738i	$0.397639\pi$
$138$	0	0
$139$	4.48733e13	0.527706	0.263853	−	0.964563i	$-0.415007\pi$
$139$	4.48733e13	0.527706	0.263853	+	0.964563i	$0.415007\pi$
$140$	8.46386e13	0.950027
$141$	0	0
$142$	1.69333e14	1.73327
$143$	−6.91570e12	−0.0676319
$144$	0	0
$145$	6.63891e13	0.593205
$146$	1.74965e13	0.149506
$147$	0	0
$148$	−3.99237e14	−3.12270
$149$	−1.30203e14	−0.974789	−0.487395	−	0.873182i	$-0.662053\pi$
$149$	−1.30203e14	−0.974789	−0.487395	+	0.873182i	$0.662053\pi$
$150$	0	0
$151$	−1.94653e14	−1.33632	−0.668160	−	0.744018i	$-0.732918\pi$
$151$	−1.94653e14	−1.33632	−0.668160	+	0.744018i	$0.732918\pi$
$152$	2.90938e14	1.91345
$153$	0	0
$154$	3.50047e14	2.11466
$155$	1.19852e14	0.694209
$156$	0	0
$157$	−7.93587e13	−0.422909	−0.211454	−	0.977388i	$-0.567820\pi$
$157$	−7.93587e13	−0.422909	−0.211454	+	0.977388i	$0.567820\pi$
$158$	−5.96123e14	−3.04835
$159$	0	0
$160$	7.51772e13	0.354247
$161$	−1.80664e14	−0.817531
$162$	0	0
$163$	−1.62086e14	−0.676904	−0.338452	−	0.940984i	$-0.609903\pi$
$163$	−1.62086e14	−0.676904	−0.338452	+	0.940984i	$0.609903\pi$
$164$	2.97945e14	1.19578
$165$	0	0
$166$	1.28139e14	0.475314
$167$	−2.62269e14	−0.935600	−0.467800	−	0.883834i	$-0.654953\pi$
$167$	−2.62269e14	−0.935600	−0.467800	+	0.883834i	$0.654953\pi$
$168$	0	0
$169$	−3.01591e14	−0.995759
$170$	−6.29650e13	−0.200070
$171$	0	0
$172$	−9.78310e14	−2.88099
$173$	−7.87595e13	−0.223359	−0.111679	−	0.993744i	$-0.535623\pi$
$173$	−7.87595e13	−0.223359	−0.111679	+	0.993744i	$0.535623\pi$
$174$	0	0
$175$	9.38050e13	0.246876
$176$	−9.85923e13	−0.250041
$177$	0	0
$178$	−1.44520e14	−0.340563
$179$	7.53329e14	1.71175	0.855874	−	0.517184i	$-0.173020\pi$
$179$	7.53329e14	1.71175	0.855874	+	0.517184i	$0.173020\pi$
$180$	0	0
$181$	−4.79674e14	−1.01399	−0.506997	−	0.861948i	$-0.669245\pi$
$181$	−4.79674e14	−1.01399	−0.506997	+	0.861948i	$0.669245\pi$
$182$	−6.50122e13	−0.132596
$183$	0	0
$184$	4.14618e14	0.787647
$185$	−4.42475e14	−0.811470
$186$	0	0
$187$	−1.64705e14	−0.281667
$188$	−1.42719e15	−2.35753
$189$	0	0
$190$	7.69689e14	1.18691
$191$	−4.70960e14	−0.701888	−0.350944	−	0.936397i	$-0.614139\pi$
$191$	−4.70960e14	−0.701888	−0.350944	+	0.936397i	$0.614139\pi$
$192$	0	0
$193$	−1.24350e14	−0.173191	−0.0865954	−	0.996244i	$-0.527599\pi$
$193$	−1.24350e14	−0.173191	−0.0865954	+	0.996244i	$0.527599\pi$
$194$	−3.73324e14	−0.502777
$195$	0	0
$196$	7.15342e14	0.901258
$197$	4.20205e12	0.00512190	0.00256095	−	0.999997i	$-0.499185\pi$
$197$	4.20205e12	0.00512190	0.00256095	+	0.999997i	$0.499185\pi$
$198$	0	0
$199$	1.09925e15	1.25474	0.627370	−	0.778722i	$-0.284132\pi$
$199$	1.09925e15	1.25474	0.627370	+	0.778722i	$0.284132\pi$
$200$	−2.15279e14	−0.237852
$201$	0	0
$202$	−5.24063e14	−0.542749
$203$	1.63254e15	1.63734
$204$	0	0
$205$	3.30213e14	0.310738
$206$	1.61520e15	1.47262
$207$	0	0
$208$	1.83110e13	0.0156783
$209$	2.01336e15	1.67098
$210$	0	0
$211$	−4.43736e14	−0.346170	−0.173085	−	0.984907i	$-0.555373\pi$
$211$	−4.43736e14	−0.346170	−0.173085	+	0.984907i	$0.555373\pi$
$212$	2.56370e15	1.93947
$213$	0	0
$214$	2.22471e15	1.58337
$215$	−1.08426e15	−0.748659
$216$	0	0
$217$	2.94722e15	1.91613
$218$	3.00466e15	1.89595
$219$	0	0
$220$	−1.34421e15	−0.799316
$221$	3.05897e13	0.0176614
$222$	0	0
$223$	−2.15493e15	−1.17342	−0.586708	−	0.809799i	$-0.699576\pi$
$223$	−2.15493e15	−1.17342	−0.586708	+	0.809799i	$0.699576\pi$
$224$	1.84864e15	0.977777
$225$	0	0
$226$	4.29838e15	2.14586
$227$	−3.86902e15	−1.87686	−0.938431	−	0.345466i	$-0.887721\pi$
$227$	−3.86902e15	−1.87686	−0.938431	+	0.345466i	$0.887721\pi$
$228$	0	0
$229$	−1.09844e15	−0.503324	−0.251662	−	0.967815i	$-0.580977\pi$
$229$	−1.09844e15	−0.503324	−0.251662	+	0.967815i	$0.580977\pi$
$230$	1.09689e15	0.488576
$231$	0	0
$232$	−3.74661e15	−1.57749
$233$	−4.34127e15	−1.77747	−0.888737	−	0.458417i	$-0.848417\pi$
$233$	−4.34127e15	−1.77747	−0.888737	+	0.458417i	$0.848417\pi$
$234$	0	0
$235$	−1.58176e15	−0.612631
$236$	−4.38365e15	−1.65161
$237$	0	0
$238$	−1.54834e15	−0.552224
$239$	1.70803e15	0.592801	0.296400	−	0.955064i	$-0.404214\pi$
$239$	1.70803e15	0.592801	0.296400	+	0.955064i	$0.404214\pi$
$240$	0	0
$241$	2.87167e15	0.944114	0.472057	−	0.881568i	$-0.343512\pi$
$241$	2.87167e15	0.944114	0.472057	+	0.881568i	$0.343512\pi$
$242$	−4.05146e14	−0.129662
$243$	0	0
$244$	2.48967e15	0.755281
$245$	7.92814e14	0.234203
$246$	0	0
$247$	−3.73931e14	−0.104776
$248$	−6.76374e15	−1.84608
$249$	0	0
$250$	−5.69530e14	−0.147539
$251$	9.41402e14	0.237627	0.118813	−	0.992917i	$-0.462091\pi$
$251$	9.41402e14	0.237627	0.118813	+	0.992917i	$0.462091\pi$
$252$	0	0
$253$	2.86926e15	0.687839
$254$	−1.04805e16	−2.44886
$255$	0	0
$256$	−6.10855e15	−1.35637
$257$	3.31316e15	0.717260	0.358630	−	0.933480i	$-0.383244\pi$
$257$	3.31316e15	0.717260	0.358630	+	0.933480i	$0.383244\pi$
$258$	0	0
$259$	−1.08806e16	−2.23978
$260$	2.49652e14	0.0501196
$261$	0	0
$262$	1.38395e16	2.64339
$263$	3.12580e15	0.582437	0.291219	−	0.956657i	$-0.405939\pi$
$263$	3.12580e15	0.582437	0.291219	+	0.956657i	$0.405939\pi$
$264$	0	0
$265$	2.84135e15	0.503995
$266$	1.89270e16	3.27605
$267$	0	0
$268$	8.82202e14	0.145443
$269$	−2.51897e15	−0.405352	−0.202676	−	0.979246i	$-0.564964\pi$
$269$	−2.51897e15	−0.405352	−0.202676	+	0.979246i	$0.564964\pi$
$270$	0	0
$271$	−9.26522e14	−0.142087	−0.0710436	−	0.997473i	$-0.522633\pi$
$271$	−9.26522e14	−0.142087	−0.0710436	+	0.997473i	$0.522633\pi$
$272$	4.36096e14	0.0652957
$273$	0	0
$274$	−7.30373e15	−1.04272
$275$	−1.48979e15	−0.207712
$276$	0	0
$277$	−4.03418e14	−0.0536583	−0.0268292	−	0.999640i	$-0.508541\pi$
$277$	−4.03418e14	−0.0536583	−0.0268292	+	0.999640i	$0.508541\pi$
$278$	−6.69953e15	−0.870469
$279$	0	0
$280$	−5.29380e15	−0.656508
$281$	8.56272e15	1.03758	0.518789	−	0.854903i	$-0.326383\pi$
$281$	8.56272e15	1.03758	0.518789	+	0.854903i	$0.326383\pi$
$282$	0	0
$283$	−3.74874e15	−0.433784	−0.216892	−	0.976196i	$-0.569592\pi$
$283$	−3.74874e15	−0.433784	−0.216892	+	0.976196i	$0.569592\pi$
$284$	−1.59899e16	−1.80833
$285$	0	0
$286$	1.03251e15	0.111561
$287$	8.12008e15	0.857685
$288$	0	0
$289$	−9.17605e15	−0.926445
$290$	−9.91182e15	−0.978513
$291$	0	0
$292$	−1.65218e15	−0.155980
$293$	−2.47408e15	−0.228441	−0.114221	−	0.993455i	$-0.536437\pi$
$293$	−2.47408e15	−0.228441	−0.114221	+	0.993455i	$0.536437\pi$
$294$	0	0
$295$	−4.85841e15	−0.429191
$296$	2.49706e16	2.15791
$297$	0	0
$298$	1.94392e16	1.60795
$299$	−5.32892e14	−0.0431297
$300$	0	0
$301$	−2.66625e16	−2.06642
$302$	2.90614e16	2.20431
$303$	0	0
$304$	−5.33087e15	−0.387365
$305$	2.75931e15	0.196269
$306$	0	0
$307$	3.12228e15	0.212849	0.106425	−	0.994321i	$-0.466060\pi$
$307$	3.12228e15	0.212849	0.106425	+	0.994321i	$0.466060\pi$
$308$	−3.30546e16	−2.20624
$309$	0	0
$310$	−1.78938e16	−1.14512
$311$	−8.71900e15	−0.546417	−0.273209	−	0.961955i	$-0.588085\pi$
$311$	−8.71900e15	−0.546417	−0.273209	+	0.961955i	$0.588085\pi$
$312$	0	0
$313$	1.94308e16	1.16803	0.584013	−	0.811744i	$-0.301482\pi$
$313$	1.94308e16	1.16803	0.584013	+	0.811744i	$0.301482\pi$
$314$	1.18482e16	0.697603
$315$	0	0
$316$	5.62913e16	3.18035
$317$	−2.73728e16	−1.51508	−0.757538	−	0.652791i	$-0.773598\pi$
$317$	−2.73728e16	−1.51508	−0.757538	+	0.652791i	$0.773598\pi$
$318$	0	0
$319$	−2.59275e16	−1.37759
$320$	−1.32920e16	−0.692013
$321$	0	0
$322$	2.69730e16	1.34855
$323$	−8.90556e15	−0.436360
$324$	0	0
$325$	2.76689e14	0.0130242
$326$	2.41993e16	1.11658
$327$	0	0
$328$	−1.86352e16	−0.826333
$329$	−3.88961e16	−1.69096
$330$	0	0
$331$	1.48805e16	0.621923	0.310961	−	0.950423i	$-0.399349\pi$
$331$	1.48805e16	0.621923	0.310961	+	0.950423i	$0.399349\pi$
$332$	−1.21001e16	−0.495897
$333$	0	0
$334$	3.91565e16	1.54330
$335$	9.77746e14	0.0377950
$336$	0	0
$337$	−9.51124e15	−0.353706	−0.176853	−	0.984237i	$-0.556592\pi$
$337$	−9.51124e15	−0.353706	−0.176853	+	0.984237i	$0.556592\pi$
$338$	4.50271e16	1.64254
$339$	0	0
$340$	5.94573e15	0.208733
$341$	−4.68069e16	−1.61215
$342$	0	0
$343$	−1.77316e16	−0.587943
$344$	6.11893e16	1.99088
$345$	0	0
$346$	1.17587e16	0.368438
$347$	−2.02295e16	−0.622076	−0.311038	−	0.950398i	$-0.600677\pi$
$347$	−2.02295e16	−0.622076	−0.311038	+	0.950398i	$0.600677\pi$
$348$	0	0
$349$	8.80222e15	0.260751	0.130376	−	0.991465i	$-0.458382\pi$
$349$	8.80222e15	0.260751	0.130376	+	0.991465i	$0.458382\pi$
$350$	−1.40050e16	−0.407231
$351$	0	0
$352$	−2.93596e16	−0.822664
$353$	−1.67276e16	−0.460148	−0.230074	−	0.973173i	$-0.573897\pi$
$353$	−1.67276e16	−0.460148	−0.230074	+	0.973173i	$0.573897\pi$
$354$	0	0
$355$	−1.77217e16	−0.469916
$356$	1.36469e16	0.355311
$357$	0	0
$358$	−1.12471e17	−2.82359
$359$	7.94820e16	1.95954	0.979772	−	0.200119i	$-0.0641328\pi$
$359$	7.94820e16	1.95954	0.979772	+	0.200119i	$0.0641328\pi$
$360$	0	0
$361$	6.68093e16	1.58869
$362$	7.16148e16	1.67262
$363$	0	0
$364$	6.13904e15	0.138338
$365$	−1.83111e15	−0.0405332
$366$	0	0
$367$	−6.97003e16	−1.48904	−0.744518	−	0.667602i	$-0.767321\pi$
$367$	−6.97003e16	−1.48904	−0.744518	+	0.667602i	$0.767321\pi$
$368$	−7.59707e15	−0.159454
$369$	0	0
$370$	6.60610e16	1.33855
$371$	6.98700e16	1.39111
$372$	0	0
$373$	2.07893e16	0.399698	0.199849	−	0.979827i	$-0.435955\pi$
$373$	2.07893e16	0.399698	0.199849	+	0.979827i	$0.435955\pi$
$374$	2.45902e16	0.464619
$375$	0	0
$376$	8.92650e16	1.62915
$377$	4.81537e15	0.0863795
$378$	0	0
$379$	7.77269e16	1.34715	0.673576	−	0.739118i	$-0.264757\pi$
$379$	7.77269e16	1.34715	0.673576	+	0.739118i	$0.264757\pi$
$380$	−7.26810e16	−1.23830
$381$	0	0
$382$	7.03139e16	1.15779
$383$	−6.62740e16	−1.07288	−0.536440	−	0.843938i	$-0.680231\pi$
$383$	−6.62740e16	−1.07288	−0.536440	+	0.843938i	$0.680231\pi$
$384$	0	0
$385$	−3.66345e16	−0.573317
$386$	1.85654e16	0.285684
$387$	0	0
$388$	3.52527e16	0.524549
$389$	3.03230e15	0.0443711	0.0221855	−	0.999754i	$-0.492938\pi$
$389$	3.03230e15	0.0443711	0.0221855	+	0.999754i	$0.492938\pi$
$390$	0	0
$391$	−1.26914e16	−0.179622
$392$	−4.47417e16	−0.622806
$393$	0	0
$394$	−6.27362e14	−0.00844876
$395$	6.23877e16	0.826453
$396$	0	0
$397$	3.47691e16	0.445712	0.222856	−	0.974851i	$-0.428462\pi$
$397$	3.47691e16	0.445712	0.222856	+	0.974851i	$0.428462\pi$
$398$	−1.64118e17	−2.06974
$399$	0	0
$400$	3.94457e15	0.0481515
$401$	−4.47291e16	−0.537220	−0.268610	−	0.963249i	$-0.586564\pi$
$401$	−4.47291e16	−0.537220	−0.268610	+	0.963249i	$0.586564\pi$
$402$	0	0
$403$	8.69317e15	0.101087
$404$	4.94867e16	0.566252
$405$	0	0
$406$	−2.43736e17	−2.70085
$407$	1.72803e17	1.88447
$408$	0	0
$409$	4.58740e16	0.484579	0.242290	−	0.970204i	$-0.422102\pi$
$409$	4.58740e16	0.484579	0.242290	+	0.970204i	$0.422102\pi$
$410$	−4.93004e16	−0.512573
$411$	0	0
$412$	−1.52522e17	−1.53639
$413$	−1.19470e17	−1.18463
$414$	0	0
$415$	−1.34105e16	−0.128865
$416$	5.45279e15	0.0515836
$417$	0	0
$418$	−3.00593e17	−2.75634
$419$	−7.04588e16	−0.636128	−0.318064	−	0.948069i	$-0.603033\pi$
$419$	−7.04588e16	−0.636128	−0.318064	+	0.948069i	$0.603033\pi$
$420$	0	0
$421$	−2.11625e16	−0.185239	−0.0926196	−	0.995702i	$-0.529524\pi$
$421$	−2.11625e16	−0.185239	−0.0926196	+	0.995702i	$0.529524\pi$
$422$	6.62493e16	0.571019
$423$	0	0
$424$	−1.60349e17	−1.34026
$425$	6.58966e15	0.0542419
$426$	0	0
$427$	6.78526e16	0.541733
$428$	−2.10077e17	−1.65194
$429$	0	0
$430$	1.61879e17	1.23494
$431$	−4.09241e16	−0.307523	−0.153761	−	0.988108i	$-0.549139\pi$
$431$	−4.09241e16	−0.307523	−0.153761	+	0.988108i	$0.549139\pi$
$432$	0	0
$433$	8.57212e16	0.625053	0.312527	−	0.949909i	$-0.398825\pi$
$433$	8.57212e16	0.625053	0.312527	+	0.949909i	$0.398825\pi$
$434$	−4.40016e17	−3.16072
$435$	0	0
$436$	−2.83727e17	−1.97806
$437$	1.55141e17	1.06560
$438$	0	0
$439$	3.70635e16	0.247131	0.123565	−	0.992336i	$-0.460567\pi$
$439$	3.70635e16	0.247131	0.123565	+	0.992336i	$0.460567\pi$
$440$	8.40746e16	0.552360
$441$	0	0
$442$	−4.56700e15	−0.0291331
$443$	1.55015e16	0.0974431	0.0487215	−	0.998812i	$-0.484485\pi$
$443$	1.55015e16	0.0974431	0.0487215	+	0.998812i	$0.484485\pi$
$444$	0	0
$445$	1.51249e16	0.0923318
$446$	3.21729e17	1.93559
$447$	0	0
$448$	−3.26855e17	−1.91006
$449$	2.75973e17	1.58952	0.794759	−	0.606925i	$-0.207597\pi$
$449$	2.75973e17	1.58952	0.794759	+	0.606925i	$0.207597\pi$
$450$	0	0
$451$	−1.28961e17	−0.721623
$452$	−4.05892e17	−2.23878
$453$	0	0
$454$	5.77640e17	3.09595
$455$	6.80391e15	0.0359488
$456$	0	0
$457$	2.05141e17	1.05341	0.526703	−	0.850049i	$-0.323428\pi$
$457$	2.05141e17	1.05341	0.526703	+	0.850049i	$0.323428\pi$
$458$	1.63997e17	0.830250
$459$	0	0
$460$	−1.03578e17	−0.509733
$461$	−2.29523e17	−1.11370	−0.556852	−	0.830612i	$-0.687991\pi$
$461$	−2.29523e17	−1.11370	−0.556852	+	0.830612i	$0.687991\pi$
$462$	0	0
$463$	1.90465e17	0.898544	0.449272	−	0.893395i	$-0.351683\pi$
$463$	1.90465e17	0.898544	0.449272	+	0.893395i	$0.351683\pi$
$464$	6.86493e16	0.319352
$465$	0	0
$466$	6.48147e17	2.93201
$467$	1.67739e17	0.748297	0.374148	−	0.927369i	$-0.377935\pi$
$467$	1.67739e17	0.748297	0.374148	+	0.927369i	$0.377935\pi$
$468$	0	0
$469$	2.40432e16	0.104320
$470$	2.36155e17	1.01056
$471$	0	0
$472$	2.74179e17	1.14133
$473$	4.23446e17	1.73860
$474$	0	0
$475$	−8.05524e16	−0.321788
$476$	1.46208e17	0.576137
$477$	0	0
$478$	−2.55007e17	−0.977845
$479$	3.71727e17	1.40619	0.703095	−	0.711096i	$-0.251801\pi$
$479$	3.71727e17	1.40619	0.703095	+	0.711096i	$0.251801\pi$
$480$	0	0
$481$	−3.20938e16	−0.118162
$482$	−4.28738e17	−1.55735
$483$	0	0
$484$	3.82576e16	0.135277
$485$	3.90706e16	0.136310
$486$	0	0
$487$	4.87760e16	0.165679	0.0828397	−	0.996563i	$-0.473601\pi$
$487$	4.87760e16	0.165679	0.0828397	+	0.996563i	$0.473601\pi$
$488$	−1.55719e17	−0.521930
$489$	0	0
$490$	−1.18366e17	−0.386326
$491$	6.19444e16	0.199513	0.0997567	−	0.995012i	$-0.468194\pi$
$491$	6.19444e16	0.199513	0.0997567	+	0.995012i	$0.468194\pi$
$492$	0	0
$493$	1.14683e17	0.359745
$494$	5.58274e16	0.172831
$495$	0	0
$496$	1.23933e17	0.373727
$497$	−4.35783e17	−1.29704
$498$	0	0
$499$	−2.79726e17	−0.811108	−0.405554	−	0.914071i	$-0.632921\pi$
$499$	−2.79726e17	−0.811108	−0.405554	+	0.914071i	$0.632921\pi$
$500$	5.37802e16	0.153928
$501$	0	0
$502$	−1.40550e17	−0.391974
$503$	−3.38759e17	−0.932609	−0.466304	−	0.884624i	$-0.654415\pi$
$503$	−3.38759e17	−0.932609	−0.466304	+	0.884624i	$0.654415\pi$
$504$	0	0
$505$	5.48462e16	0.147147
$506$	−4.28378e17	−1.13461
$507$	0	0
$508$	9.89666e17	2.55490
$509$	3.81496e17	0.972354	0.486177	−	0.873860i	$-0.338391\pi$
$509$	3.81496e17	0.972354	0.486177	+	0.873860i	$0.338391\pi$
$510$	0	0
$511$	−4.50278e16	−0.111878
$512$	1.94447e17	0.477032
$513$	0	0
$514$	−4.94651e17	−1.18315
$515$	−1.69040e17	−0.399249
$516$	0	0
$517$	6.17738e17	1.42271
$518$	1.62447e18	3.69460
$519$	0	0
$520$	−1.56147e16	−0.0346347
$521$	3.97604e17	0.870975	0.435487	−	0.900195i	$-0.356576\pi$
$521$	3.97604e17	0.870975	0.435487	+	0.900195i	$0.356576\pi$
$522$	0	0
$523$	6.78177e16	0.144905	0.0724523	−	0.997372i	$-0.476918\pi$
$523$	6.78177e16	0.144905	0.0724523	+	0.997372i	$0.476918\pi$
$524$	−1.30685e18	−2.75786
$525$	0	0
$526$	−4.66679e17	−0.960750
$527$	2.07037e17	0.420998
$528$	0	0
$529$	−2.82944e17	−0.561357
$530$	−4.24210e17	−0.831358
$531$	0	0
$532$	−1.78726e18	−3.41792
$533$	2.39511e16	0.0452480
$534$	0	0
$535$	−2.32829e17	−0.429277
$536$	−5.51781e16	−0.100507
$537$	0	0
$538$	3.76079e17	0.668643
$539$	−3.09624e17	−0.543886
$540$	0	0
$541$	−9.53886e17	−1.63574	−0.817870	−	0.575403i	$-0.804845\pi$
$541$	−9.53886e17	−1.63574	−0.817870	+	0.575403i	$0.804845\pi$
$542$	1.38329e17	0.234378
$543$	0	0
$544$	1.29864e17	0.214830
$545$	−3.14455e17	−0.514022
$546$	0	0
$547$	5.54825e17	0.885601	0.442800	−	0.896620i	$-0.353985\pi$
$547$	5.54825e17	0.885601	0.442800	+	0.896620i	$0.353985\pi$
$548$	6.89684e17	1.08787
$549$	0	0
$550$	2.22423e17	0.342628
$551$	−1.40190e18	−2.13418
$552$	0	0
$553$	1.53414e18	2.28114
$554$	6.02299e16	0.0885113
$555$	0	0
$556$	6.32630e17	0.908163
$557$	−8.83747e17	−1.25392	−0.626959	−	0.779052i	$-0.715701\pi$
$557$	−8.83747e17	−1.25392	−0.626959	+	0.779052i	$0.715701\pi$
$558$	0	0
$559$	−7.86442e16	−0.109016
$560$	9.69986e16	0.132906
$561$	0	0
$562$	−1.27840e18	−1.71152
$563$	−1.34034e18	−1.77383	−0.886913	−	0.461937i	$-0.847154\pi$
$563$	−1.34034e18	−1.77383	−0.886913	+	0.461937i	$0.847154\pi$
$564$	0	0
$565$	−4.49851e17	−0.581773
$566$	5.59682e17	0.715541
$567$	0	0
$568$	1.00010e18	1.24963
$569$	2.62408e16	0.0324151	0.0162075	−	0.999869i	$-0.494841\pi$
$569$	2.62408e16	0.0324151	0.0162075	+	0.999869i	$0.494841\pi$
$570$	0	0
$571$	6.67128e17	0.805517	0.402758	−	0.915306i	$-0.368051\pi$
$571$	6.67128e17	0.805517	0.402758	+	0.915306i	$0.368051\pi$
$572$	−9.74986e16	−0.116392
$573$	0	0
$574$	−1.21232e18	−1.41478
$575$	−1.14796e17	−0.132460
$576$	0	0
$577$	−1.37034e17	−0.154591	−0.0772955	−	0.997008i	$-0.524628\pi$
$577$	−1.37034e17	−0.154591	−0.0772955	+	0.997008i	$0.524628\pi$
$578$	1.36997e18	1.52820
$579$	0	0
$580$	9.35964e17	1.02089
$581$	−3.29771e17	−0.355687
$582$	0	0
$583$	−1.10966e18	−1.17042
$584$	1.03337e17	0.107788
$585$	0	0
$586$	3.69378e17	0.376822
$587$	−1.19496e17	−0.120561	−0.0602805	−	0.998181i	$-0.519200\pi$
$587$	−1.19496e17	−0.120561	−0.0602805	+	0.998181i	$0.519200\pi$
$588$	0	0
$589$	−2.53084e18	−2.49756
$590$	7.25355e17	0.707966
$591$	0	0
$592$	−4.57539e17	−0.436855
$593$	1.87526e18	1.77095	0.885475	−	0.464687i	$-0.153833\pi$
$593$	1.87526e18	1.77095	0.885475	+	0.464687i	$0.153833\pi$
$594$	0	0
$595$	1.62042e17	0.149716
$596$	−1.83562e18	−1.67758
$597$	0	0
$598$	7.95601e16	0.0711439
$599$	5.19643e17	0.459654	0.229827	−	0.973231i	$-0.426184\pi$
$599$	5.19643e17	0.459654	0.229827	+	0.973231i	$0.426184\pi$
$600$	0	0
$601$	5.26704e17	0.455913	0.227957	−	0.973671i	$-0.426796\pi$
$601$	5.26704e17	0.455913	0.227957	+	0.973671i	$0.426796\pi$
$602$	3.98067e18	3.40863
$603$	0	0
$604$	−2.74424e18	−2.29976
$605$	4.24009e16	0.0351533
$606$	0	0
$607$	1.87571e18	1.52209	0.761043	−	0.648701i	$-0.224688\pi$
$607$	1.87571e18	1.52209	0.761043	+	0.648701i	$0.224688\pi$
$608$	−1.58747e18	−1.27448
$609$	0	0
$610$	−4.11962e17	−0.323752
$611$	−1.14729e17	−0.0892082
$612$	0	0
$613$	−4.47233e17	−0.340440	−0.170220	−	0.985406i	$-0.554448\pi$
$613$	−4.47233e17	−0.340440	−0.170220	+	0.985406i	$0.554448\pi$
$614$	−4.66152e17	−0.351102
$615$	0	0
$616$	2.06743e18	1.52460
$617$	8.44420e17	0.616176	0.308088	−	0.951358i	$-0.400311\pi$
$617$	8.44420e17	0.616176	0.308088	+	0.951358i	$0.400311\pi$
$618$	0	0
$619$	2.36925e18	1.69286	0.846430	−	0.532500i	$-0.178747\pi$
$619$	2.36925e18	1.69286	0.846430	+	0.532500i	$0.178747\pi$
$620$	1.68969e18	1.19471
$621$	0	0
$622$	1.30174e18	0.901334
$623$	3.71927e17	0.254850
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−2.90100e18	−1.92670
$627$	0	0
$628$	−1.11881e18	−0.727811
$629$	−7.64348e17	−0.492110
$630$	0	0
$631$	−2.91193e18	−1.83649	−0.918247	−	0.396009i	$-0.870395\pi$
$631$	−2.91193e18	−1.83649	−0.918247	+	0.396009i	$0.870395\pi$
$632$	−3.52079e18	−2.19775
$633$	0	0
$634$	4.08673e18	2.49917
$635$	1.09685e18	0.663922
$636$	0	0
$637$	5.75048e16	0.0341034
$638$	3.87095e18	2.27239
$639$	0	0
$640$	1.36862e18	0.787252
$641$	3.49512e17	0.199014	0.0995072	−	0.995037i	$-0.468273\pi$
$641$	3.49512e17	0.199014	0.0995072	+	0.995037i	$0.468273\pi$
$642$	0	0
$643$	−3.11686e18	−1.73919	−0.869593	−	0.493769i	$-0.835619\pi$
$643$	−3.11686e18	−1.73919	−0.869593	+	0.493769i	$0.835619\pi$
$644$	−2.54703e18	−1.40694
$645$	0	0
$646$	1.32959e18	0.719791
$647$	−1.99766e18	−1.07064	−0.535321	−	0.844649i	$-0.679809\pi$
$647$	−1.99766e18	−1.07064	−0.535321	+	0.844649i	$0.679809\pi$
$648$	0	0
$649$	1.89739e18	0.996706
$650$	−4.13094e16	−0.0214838
$651$	0	0
$652$	−2.28512e18	−1.16493
$653$	1.08626e18	0.548276	0.274138	−	0.961690i	$-0.411607\pi$
$653$	1.08626e18	0.548276	0.274138	+	0.961690i	$0.411607\pi$
$654$	0	0
$655$	−1.44838e18	−0.716663
$656$	3.41455e17	0.167286
$657$	0	0
$658$	5.80715e18	2.78930
$659$	1.64776e17	0.0783678	0.0391839	−	0.999232i	$-0.487524\pi$
$659$	1.64776e17	0.0783678	0.0391839	+	0.999232i	$0.487524\pi$
$660$	0	0
$661$	−4.20993e18	−1.96320	−0.981601	−	0.190942i	$-0.938846\pi$
$661$	−4.20993e18	−1.96320	−0.981601	+	0.190942i	$0.938846\pi$
$662$	−2.22165e18	−1.02588
$663$	0	0
$664$	7.56811e17	0.342685
$665$	−1.98082e18	−0.888186
$666$	0	0
$667$	−1.99785e18	−0.878508
$668$	−3.69751e18	−1.61013
$669$	0	0
$670$	−1.45976e17	−0.0623442
$671$	−1.07762e18	−0.455793
$672$	0	0
$673$	−2.30935e18	−0.958057	−0.479029	−	0.877799i	$-0.659011\pi$
$673$	−2.30935e18	−0.958057	−0.479029	+	0.877799i	$0.659011\pi$
$674$	1.42002e18	0.583451
$675$	0	0
$676$	−4.25187e18	−1.71367
$677$	4.16216e18	1.66147	0.830736	−	0.556667i	$-0.187920\pi$
$677$	4.16216e18	1.66147	0.830736	+	0.556667i	$0.187920\pi$
$678$	0	0
$679$	9.60762e17	0.376238
$680$	−3.71881e17	−0.144243
$681$	0	0
$682$	6.98822e18	2.65930
$683$	5.67684e17	0.213979	0.106990	−	0.994260i	$-0.465879\pi$
$683$	5.67684e17	0.213979	0.106990	+	0.994260i	$0.465879\pi$
$684$	0	0
$685$	7.64378e17	0.282696
$686$	2.64730e18	0.969833
$687$	0	0
$688$	−1.12118e18	−0.403041
$689$	2.06090e17	0.0733892
$690$	0	0
$691$	1.60571e18	0.561126	0.280563	−	0.959836i	$-0.409479\pi$
$691$	1.60571e18	0.561126	0.280563	+	0.959836i	$0.409479\pi$
$692$	−1.11036e18	−0.384393
$693$	0	0
$694$	3.02024e18	1.02614
$695$	7.01145e17	0.235997
$696$	0	0
$697$	5.70423e17	0.188445
$698$	−1.31416e18	−0.430119
$699$	0	0
$700$	1.32248e18	0.424865
$701$	2.88624e18	0.918681	0.459341	−	0.888260i	$-0.348086\pi$
$701$	2.88624e18	0.918681	0.459341	+	0.888260i	$0.348086\pi$
$702$	0	0
$703$	9.34345e18	2.91943
$704$	5.19102e18	1.60705
$705$	0	0
$706$	2.49741e18	0.759030
$707$	1.34869e18	0.406150
$708$	0	0
$709$	3.41704e18	1.01030	0.505150	−	0.863032i	$-0.331437\pi$
$709$	3.41704e18	1.01030	0.505150	+	0.863032i	$0.331437\pi$
$710$	2.64582e18	0.775142
$711$	0	0
$712$	−8.53557e17	−0.245535
$713$	−3.60672e18	−1.02809
$714$	0	0
$715$	−1.08058e17	−0.0302459
$716$	1.06205e19	2.94586
$717$	0	0
$718$	−1.18666e19	−3.23234
$719$	4.01107e18	1.08274	0.541368	−	0.840786i	$-0.317907\pi$
$719$	4.01107e18	1.08274	0.541368	+	0.840786i	$0.317907\pi$
$720$	0	0
$721$	−4.15677e18	−1.10199
$722$	−9.97455e18	−2.62060
$723$	0	0
$724$	−6.76252e18	−1.74505
$725$	1.03733e18	0.265289
$726$	0	0
$727$	−2.32433e18	−0.583881	−0.291941	−	0.956436i	$-0.594301\pi$
$727$	−2.32433e18	−0.583881	−0.291941	+	0.956436i	$0.594301\pi$
$728$	−3.83972e17	−0.0955973
$729$	0	0
$730$	2.73383e17	0.0668609
$731$	−1.87300e18	−0.454019
$732$	0	0
$733$	−2.96621e17	−0.0706359	−0.0353180	−	0.999376i	$-0.511244\pi$
$733$	−2.96621e17	−0.0706359	−0.0353180	+	0.999376i	$0.511244\pi$
$734$	1.04062e19	2.45622
$735$	0	0
$736$	−2.26231e18	−0.524622
$737$	−3.81847e17	−0.0877710
$738$	0	0
$739$	8.38574e18	1.89388	0.946940	−	0.321410i	$-0.104157\pi$
$739$	8.38574e18	1.89388	0.946940	+	0.321410i	$0.104157\pi$
$740$	−6.23808e18	−1.39651
$741$	0	0
$742$	−1.04315e19	−2.29468
$743$	−1.06547e18	−0.232335	−0.116168	−	0.993230i	$-0.537061\pi$
$743$	−1.06547e18	−0.232335	−0.116168	+	0.993230i	$0.537061\pi$
$744$	0	0
$745$	−2.03442e18	−0.435939
$746$	−3.10382e18	−0.659316
$747$	0	0
$748$	−2.32203e18	−0.484739
$749$	−5.72536e18	−1.18487
$750$	0	0
$751$	−9.57457e18	−1.94742	−0.973710	−	0.227792i	$-0.926849\pi$
$751$	−9.57457e18	−1.94742	−0.973710	+	0.227792i	$0.926849\pi$
$752$	−1.63561e18	−0.329810
$753$	0	0
$754$	−7.18929e17	−0.142486
$755$	−3.04145e18	−0.597620
$756$	0	0
$757$	4.59607e18	0.887695	0.443847	−	0.896102i	$-0.353613\pi$
$757$	4.59607e18	0.887695	0.443847	+	0.896102i	$0.353613\pi$
$758$	−1.16045e19	−2.22217
$759$	0	0
$760$	4.54590e18	0.855720
$761$	−3.11769e18	−0.581879	−0.290940	−	0.956741i	$-0.593968\pi$
$761$	−3.11769e18	−0.581879	−0.290940	+	0.956741i	$0.593968\pi$
$762$	0	0
$763$	−7.73258e18	−1.41878
$764$	−6.63967e18	−1.20792
$765$	0	0
$766$	9.89464e18	1.76975
$767$	−3.52392e17	−0.0624966
$768$	0	0
$769$	4.30427e18	0.750547	0.375274	−	0.926914i	$-0.377549\pi$
$769$	4.30427e18	0.750547	0.375274	+	0.926914i	$0.377549\pi$
$770$	5.46948e18	0.945707
$771$	0	0
$772$	−1.75311e18	−0.298056
$773$	6.51519e18	1.09840	0.549200	−	0.835691i	$-0.314933\pi$
$773$	6.51519e18	1.09840	0.549200	+	0.835691i	$0.314933\pi$
$774$	0	0
$775$	1.87269e18	0.310460
$776$	−2.20491e18	−0.362485
$777$	0	0
$778$	−4.52719e17	−0.0731916
$779$	−6.97289e18	−1.11794
$780$	0	0
$781$	6.92099e18	1.09128
$782$	1.89481e18	0.296293
$783$	0	0
$784$	8.19806e17	0.126083
$785$	−1.23998e18	−0.189131
$786$	0	0
$787$	−7.49401e18	−1.12429	−0.562145	−	0.827039i	$-0.690024\pi$
$787$	−7.49401e18	−1.12429	−0.562145	+	0.827039i	$0.690024\pi$
$788$	5.92412e16	0.00881462
$789$	0	0
$790$	−9.31442e18	−1.36326
$791$	−1.10620e19	−1.60579
$792$	0	0
$793$	2.00139e17	0.0285796
$794$	−5.19098e18	−0.735218
$795$	0	0
$796$	1.54975e19	2.15936
$797$	8.43590e18	1.16588	0.582938	−	0.812516i	$-0.301903\pi$
$797$	8.43590e18	1.16588	0.582938	+	0.812516i	$0.301903\pi$
$798$	0	0
$799$	−2.73239e18	−0.371526
$800$	1.17464e18	0.158424
$801$	0	0
$802$	6.67801e18	0.886163
$803$	7.15119e17	0.0941298
$804$	0	0
$805$	−2.82288e18	−0.365611
$806$	−1.29788e18	−0.166747
$807$	0	0
$808$	−3.09519e18	−0.391304
$809$	2.98965e18	0.374934	0.187467	−	0.982271i	$-0.439972\pi$
$809$	2.98965e18	0.374934	0.187467	+	0.982271i	$0.439972\pi$
$810$	0	0
$811$	−1.10858e19	−1.36814	−0.684072	−	0.729415i	$-0.739792\pi$
$811$	−1.10858e19	−1.36814	−0.684072	+	0.729415i	$0.739792\pi$
$812$	2.30157e19	2.81780
$813$	0	0
$814$	−2.57993e19	−3.10849
$815$	−2.53260e18	−0.302720
$816$	0	0
$817$	2.28956e19	2.69345
$818$	−6.84893e18	−0.799330
$819$	0	0
$820$	4.65539e18	0.534769
$821$	−1.97742e18	−0.225356	−0.112678	−	0.993632i	$-0.535943\pi$
$821$	−1.97742e18	−0.225356	−0.112678	+	0.993632i	$0.535943\pi$
$822$	0	0
$823$	−1.12620e19	−1.26333	−0.631665	−	0.775242i	$-0.717628\pi$
$823$	−1.12620e19	−1.26333	−0.631665	+	0.775242i	$0.717628\pi$
$824$	9.53963e18	1.06171
$825$	0	0
$826$	1.78368e19	1.95410
$827$	1.16025e19	1.26115	0.630574	−	0.776129i	$-0.282819\pi$
$827$	1.16025e19	1.26115	0.630574	+	0.776129i	$0.282819\pi$
$828$	0	0
$829$	−4.67038e18	−0.499745	−0.249872	−	0.968279i	$-0.580389\pi$
$829$	−4.67038e18	−0.499745	−0.249872	+	0.968279i	$0.580389\pi$
$830$	2.00218e18	0.212567
$831$	0	0
$832$	−9.64098e17	−0.100767
$833$	1.36954e18	0.142030
$834$	0	0
$835$	−4.09795e18	−0.418413
$836$	2.83847e19	2.87570
$837$	0	0
$838$	1.05194e19	1.04931
$839$	8.73017e17	0.0864112	0.0432056	−	0.999066i	$-0.486243\pi$
$839$	8.73017e17	0.0864112	0.0432056	+	0.999066i	$0.486243\pi$
$840$	0	0
$841$	7.79256e18	0.759462
$842$	3.15953e18	0.305559
$843$	0	0
$844$	−6.25586e18	−0.595746
$845$	−4.71235e18	−0.445317
$846$	0	0
$847$	1.04266e18	0.0970285
$848$	2.93808e18	0.271326
$849$	0	0
$850$	−9.83828e17	−0.0894739
$851$	1.33154e19	1.20175
$852$	0	0
$853$	1.24079e19	1.10288	0.551442	−	0.834213i	$-0.314078\pi$
$853$	1.24079e19	1.10288	0.551442	+	0.834213i	$0.314078\pi$
$854$	−1.01303e19	−0.893607
$855$	0	0
$856$	1.31395e19	1.14156
$857$	−7.04934e18	−0.607817	−0.303909	−	0.952701i	$-0.598292\pi$
$857$	−7.04934e18	−0.607817	−0.303909	+	0.952701i	$0.598292\pi$
$858$	0	0
$859$	−2.37855e17	−0.0202003	−0.0101001	−	0.999949i	$-0.503215\pi$
$859$	−2.37855e17	−0.0202003	−0.0101001	+	0.999949i	$0.503215\pi$
$860$	−1.52861e19	−1.28842
$861$	0	0
$862$	6.10992e18	0.507269
$863$	−1.21210e19	−0.998774	−0.499387	−	0.866379i	$-0.666441\pi$
$863$	−1.21210e19	−0.998774	−0.499387	+	0.866379i	$0.666441\pi$
$864$	0	0
$865$	−1.23062e18	−0.0998892
$866$	−1.27981e19	−1.03105
$867$	0	0
$868$	4.15503e19	3.29759
$869$	−2.43648e19	−1.91926
$870$	0	0
$871$	7.09183e16	0.00550352
$872$	1.77460e19	1.36692
$873$	0	0
$874$	−2.31623e19	−1.75775
$875$	1.46570e18	0.110406
$876$	0	0
$877$	−3.74100e18	−0.277645	−0.138823	−	0.990317i	$-0.544332\pi$
$877$	−3.74100e18	−0.277645	−0.138823	+	0.990317i	$0.544332\pi$
$878$	−5.53354e18	−0.407651
$879$	0	0
$880$	−1.54051e18	−0.111822
$881$	1.36085e19	0.980544	0.490272	−	0.871570i	$-0.336898\pi$
$881$	1.36085e19	0.980544	0.490272	+	0.871570i	$0.336898\pi$
$882$	0	0
$883$	9.87005e18	0.700769	0.350384	−	0.936606i	$-0.386051\pi$
$883$	9.87005e18	0.700769	0.350384	+	0.936606i	$0.386051\pi$
$884$	4.31258e17	0.0303947
$885$	0	0
$886$	−2.31436e18	−0.160736
$887$	−1.35100e19	−0.931432	−0.465716	−	0.884934i	$-0.654203\pi$
$887$	−1.35100e19	−0.931432	−0.465716	+	0.884934i	$0.654203\pi$
$888$	0	0
$889$	2.69720e19	1.83253
$890$	−2.25813e18	−0.152305
$891$	0	0
$892$	−3.03805e19	−2.01941
$893$	3.34010e19	2.20407
$894$	0	0
$895$	1.17708e19	0.765517
$896$	3.36550e19	2.17294
$897$	0	0
$898$	−4.12025e19	−2.62197
$899$	3.25914e19	2.05904
$900$	0	0
$901$	4.90826e18	0.305644
$902$	1.92537e19	1.19034
$903$	0	0
$904$	2.53869e19	1.54709
$905$	−7.49491e18	−0.453472
$906$	0	0
$907$	−7.78812e18	−0.464500	−0.232250	−	0.972656i	$-0.574609\pi$
$907$	−7.78812e18	−0.464500	−0.232250	+	0.972656i	$0.574609\pi$
$908$	−5.45460e19	−3.23002
$909$	0	0
$910$	−1.01582e18	−0.0592988
$911$	−6.24185e17	−0.0361779	−0.0180890	−	0.999836i	$-0.505758\pi$
$911$	−6.24185e17	−0.0361779	−0.0180890	+	0.999836i	$0.505758\pi$
$912$	0	0
$913$	5.23733e18	0.299261
$914$	−3.06272e19	−1.73763
$915$	0	0
$916$	−1.54860e19	−0.866203
$917$	−3.56164e19	−1.97810
$918$	0	0
$919$	−3.08457e19	−1.68906	−0.844528	−	0.535511i	$-0.820119\pi$
$919$	−3.08457e19	−1.68906	−0.844528	+	0.535511i	$0.820119\pi$
$920$	6.47840e18	0.352247
$921$	0	0
$922$	3.42675e19	1.83709
$923$	−1.28539e18	−0.0684267
$924$	0	0
$925$	−6.91367e18	−0.362900
$926$	−2.84363e19	−1.48218
$927$	0	0
$928$	2.04429e19	1.05071
$929$	2.60942e19	1.33181	0.665905	−	0.746037i	$-0.268046\pi$
$929$	2.60942e19	1.33181	0.665905	+	0.746037i	$0.268046\pi$
$930$	0	0
$931$	−1.67413e19	−0.842592
$932$	−6.12039e19	−3.05897
$933$	0	0
$934$	−2.50432e19	−1.23434
$935$	−2.57351e18	−0.125965
$936$	0	0
$937$	−1.43219e19	−0.691340	−0.345670	−	0.938356i	$-0.612348\pi$
$937$	−1.43219e19	−0.691340	−0.345670	+	0.938356i	$0.612348\pi$
$938$	−3.58962e18	−0.172080
$939$	0	0
$940$	−2.22999e19	−1.05432
$941$	6.74040e18	0.316485	0.158242	−	0.987400i	$-0.449417\pi$
$941$	6.74040e18	0.316485	0.158242	+	0.987400i	$0.449417\pi$
$942$	0	0
$943$	−9.93712e18	−0.460187
$944$	−5.02381e18	−0.231055
$945$	0	0
$946$	−6.32200e19	−2.86789
$947$	4.26475e18	0.192140	0.0960701	−	0.995375i	$-0.469373\pi$
$947$	4.26475e18	0.192140	0.0960701	+	0.995375i	$0.469373\pi$
$948$	0	0
$949$	−1.32815e17	−0.00590223
$950$	1.20264e19	0.530801
$951$	0	0
$952$	−9.14471e18	−0.398134
$953$	−2.39618e17	−0.0103613	−0.00518067	−	0.999987i	$-0.501649\pi$
$953$	−2.39618e17	−0.0103613	−0.00518067	+	0.999987i	$0.501649\pi$
$954$	0	0
$955$	−7.35876e18	−0.313894
$956$	2.40800e19	1.02019
$957$	0	0
$958$	−5.54985e19	−2.31956
$959$	1.87964e19	0.780285
$960$	0	0
$961$	3.44197e19	1.40963
$962$	4.79157e18	0.194912
$963$	0	0
$964$	4.04853e19	1.62479
$965$	−1.94298e18	−0.0774533
$966$	0	0
$967$	1.50588e19	0.592268	0.296134	−	0.955146i	$-0.404302\pi$
$967$	1.50588e19	0.592268	0.296134	+	0.955146i	$0.404302\pi$
$968$	−2.39286e18	−0.0934817
$969$	0	0
$970$	−5.83319e18	−0.224849
$971$	−1.51669e19	−0.580728	−0.290364	−	0.956916i	$-0.593776\pi$
$971$	−1.51669e19	−0.580728	−0.290364	+	0.956916i	$0.593776\pi$
$972$	0	0
$973$	1.72415e19	0.651389
$974$	−7.28221e18	−0.273294
$975$	0	0
$976$	2.85325e18	0.105661
$977$	−5.43050e18	−0.199768	−0.0998838	−	0.994999i	$-0.531847\pi$
$977$	−5.43050e18	−0.199768	−0.0998838	+	0.994999i	$0.531847\pi$
$978$	0	0
$979$	−5.90684e18	−0.214421
$980$	1.11772e19	0.403055
$981$	0	0
$982$	−9.24822e18	−0.329104
$983$	4.45646e19	1.57540	0.787702	−	0.616057i	$-0.211271\pi$
$983$	4.45646e19	1.57540	0.787702	+	0.616057i	$0.211271\pi$
$984$	0	0
$985$	6.56571e16	0.00229058
$986$	−1.71221e19	−0.593412
$987$	0	0
$988$	−5.27173e18	−0.180315
$989$	3.26288e19	1.10873
$990$	0	0
$991$	8.11044e18	0.271998	0.135999	−	0.990709i	$-0.456576\pi$
$991$	8.11044e18	0.271998	0.135999	+	0.990709i	$0.456576\pi$
$992$	3.69056e19	1.22961
$993$	0	0
$994$	6.50619e19	2.13951
$995$	1.71759e19	0.561136
$996$	0	0
$997$	3.75569e19	1.21107	0.605537	−	0.795817i	$-0.292958\pi$
$997$	3.75569e19	1.21107	0.605537	+	0.795817i	$0.292958\pi$
$998$	4.17627e19	1.33795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.a.b.1.1		2
3.2	odd	2		15.14.a.c.1.2	✓	2
15.2	even	4		75.14.b.e.49.4		4
15.8	even	4		75.14.b.e.49.1		4
15.14	odd	2		75.14.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.14.a.c.1.2	✓	2	3.2	odd	2
45.14.a.b.1.1		2	1.1	even	1	trivial
75.14.a.c.1.1		2	15.14	odd	2
75.14.b.e.49.1		4	15.8	even	4
75.14.b.e.49.4		4	15.2	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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-149.299 q^{2} +14098.2 q^{4} +15625.0 q^{5} +384225. q^{7} -881782. q^{8} -2.33279e6 q^{10} -6.10216e6 q^{11} +1.13332e6 q^{13} -5.73644e7 q^{14} +1.61570e7 q^{16} +2.69912e7 q^{17} -3.29943e8 q^{19} +2.20284e8 q^{20} +9.11046e8 q^{22} -4.70204e8 q^{23} +2.44141e8 q^{25} -1.69203e8 q^{26} +5.41687e9 q^{28} +4.24890e9 q^{29} +7.67054e9 q^{31} +4.81134e9 q^{32} -4.02976e9 q^{34} +6.00352e9 q^{35} -2.83184e10 q^{37} +4.92601e10 q^{38} -1.37778e10 q^{40} +2.11336e10 q^{41} -6.93928e10 q^{43} -8.60292e10 q^{44} +7.02010e10 q^{46} -1.01233e11 q^{47} +5.07401e10 q^{49} -3.64499e10 q^{50} +1.59777e10 q^{52} +1.81846e11 q^{53} -9.53463e10 q^{55} -3.38803e11 q^{56} -6.34357e11 q^{58} -3.10938e11 q^{59} +1.76596e11 q^{61} -1.14520e12 q^{62} -8.50686e11 q^{64} +1.77081e10 q^{65} +6.25757e10 q^{67} +3.80526e11 q^{68} -8.96319e11 q^{70} -1.13419e12 q^{71} -1.17191e11 q^{73} +4.22790e12 q^{74} -4.65158e12 q^{76} -2.34461e12 q^{77} +3.99281e12 q^{79} +2.52452e11 q^{80} -3.15523e12 q^{82} -8.58275e11 q^{83} +4.21738e11 q^{85} +1.03603e13 q^{86} +5.38077e12 q^{88} +9.67991e11 q^{89} +4.35450e11 q^{91} -6.62901e12 q^{92} +1.51139e13 q^{94} -5.15536e12 q^{95} +2.50052e12 q^{97} -7.57544e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 131 q^{2} + 6241 q^{4} + 31250 q^{5} + 496272 q^{7} - 1175463 q^{8} - 2046875 q^{10} - 6245888 q^{11} + 1761164 q^{13} - 55314084 q^{14} + 75148705 q^{16} - 48151604 q^{17} + 38475344 q^{19} + 97515625 q^{20}+ \cdots - 9118668656251 q^{98}+O(q^{100}) \) 2 * q - 131 * q^2 + 6241 * q^4 + 31250 * q^5 + 496272 * q^7 - 1175463 * q^8 - 2046875 * q^10 - 6245888 * q^11 + 1761164 * q^13 - 55314084 * q^14 + 75148705 * q^16 - 48151604 * q^17 + 38475344 * q^19 + 97515625 * q^20 + 908415728 * q^22 - 734264616 * q^23 + 488281250 * q^25 - 157714454 * q^26 + 4536499884 * q^28 - 1126474708 * q^29 + 6100104312 * q^31 + 8296660073 * q^32 - 5404788238 * q^34 + 7754250000 * q^35 - 46819674548 * q^37 + 56001723484 * q^38 - 18366609375 * q^40 + 29607909788 * q^41 - 63189768520 * q^43 - 84899911888 * q^44 + 65368954992 * q^46 - 105004415960 * q^47 - 33594444590 * q^49 - 31982421875 * q^50 + 11044638034 * q^52 - 48425833796 * q^53 - 97592000000 * q^55 - 371708899380 * q^56 - 732719858498 * q^58 - 722768180096 * q^59 - 229984402180 * q^61 - 1173940563384 * q^62 - 1270168541071 * q^64 + 27518187500 * q^65 + 359062188984 * q^67 + 970935017018 * q^68 - 864282562500 * q^70 - 736545668224 * q^71 - 1494470740204 * q^73 + 3889351112906 * q^74 - 7546301154644 * q^76 - 2360709208704 * q^77 + 3526348590120 * q^79 + 1174198515625 * q^80 - 3000156702386 * q^82 - 5631050986968 * q^83 - 752368812500 * q^85 + 10473766954252 * q^86 + 5422984137648 * q^88 - 5913121459764 * q^89 + 505797926592 * q^91 - 4554248300592 * q^92 + 15044889696176 * q^94 + 601177250000 * q^95 + 16383233322244 * q^97 - 9118668656251 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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