LMFDB - Embedded newform 27.10.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,10,Mod(1,27)] 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("27.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3] 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:	$13.9059675764$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.177113.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 118x + 136 $ x^3 - x^2 - 118*x + 136
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.15428$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.46285 q^{2} -500.009 q^{4} -1404.01 q^{5} +1665.57 q^{7} +3504.44 q^{8} +4861.88 q^{10} +63618.6 q^{11} -111210. q^{13} -5767.62 q^{14} +243869. q^{16} +383783. q^{17} +7441.82 q^{19} +702017. q^{20} -220302. q^{22} +2.61722e6 q^{23} +18115.8 q^{25} +385102. q^{26} -832799. q^{28} -579091. q^{29} +3.51831e6 q^{31} -2.63875e6 q^{32} -1.32898e6 q^{34} -2.33847e6 q^{35} -9.82783e6 q^{37} -25769.9 q^{38} -4.92026e6 q^{40} -1.77760e7 q^{41} +3.91298e7 q^{43} -3.18098e7 q^{44} -9.06304e6 q^{46} -2.67867e7 q^{47} -3.75795e7 q^{49} -62732.5 q^{50} +5.56058e7 q^{52} +6.04430e7 q^{53} -8.93210e7 q^{55} +5.83689e6 q^{56} +2.00531e6 q^{58} +1.34817e8 q^{59} +1.40769e8 q^{61} -1.21834e7 q^{62} -1.15723e8 q^{64} +1.56139e8 q^{65} +7.31341e7 q^{67} -1.91895e8 q^{68} +8.09780e6 q^{70} +2.72910e8 q^{71} -2.64531e8 q^{73} +3.40323e7 q^{74} -3.72097e6 q^{76} +1.05961e8 q^{77} +5.44662e8 q^{79} -3.42394e8 q^{80} +6.15557e7 q^{82} -1.73988e8 q^{83} -5.38834e8 q^{85} -1.35501e8 q^{86} +2.22947e8 q^{88} -5.69208e8 q^{89} -1.85227e8 q^{91} -1.30863e9 q^{92} +9.27586e7 q^{94} -1.04484e7 q^{95} -5.86520e8 q^{97} +1.30132e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 597 q^{4} + 1983 q^{5} - 3693 q^{7} + 4503 q^{8} - 18981 q^{10} + 16863 q^{11} + 116916 q^{13} + 503463 q^{14} - 239919 q^{16} + 1014048 q^{17} - 15222 q^{19} + 2548407 q^{20} + 305721 q^{22}+ \cdots - 2592182286 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 597 * q^4 + 1983 * q^5 - 3693 * q^7 + 4503 * q^8 - 18981 * q^10 + 16863 * q^11 + 116916 * q^13 + 503463 * q^14 - 239919 * q^16 + 1014048 * q^17 - 15222 * q^19 + 2548407 * q^20 + 305721 * q^22 + 2927118 * q^23 + 2133732 * q^25 + 4765116 * q^26 - 3535725 * q^28 + 5768790 * q^29 - 6575223 * q^31 - 2687697 * q^32 + 17098128 * q^34 - 17340537 * q^35 - 11686026 * q^37 - 50473374 * q^38 - 4130811 * q^40 - 22213518 * q^41 + 45384414 * q^43 - 57206991 * q^44 - 95638590 * q^46 + 12392034 * q^47 + 18933462 * q^49 - 82984044 * q^50 + 182736492 * q^52 + 80579637 * q^53 - 174735333 * q^55 + 21850521 * q^56 - 102758922 * q^58 + 244026660 * q^59 + 369729960 * q^61 + 166297341 * q^62 - 420692127 * q^64 + 492225684 * q^65 - 252614586 * q^67 + 162270144 * q^68 + 935092377 * q^70 + 403193088 * q^71 - 406626717 * q^73 - 641373558 * q^74 - 39499590 * q^76 + 360443199 * q^77 + 265451856 * q^79 - 1168359837 * q^80 - 884408166 * q^82 + 121625871 * q^83 + 316251216 * q^85 - 2368956570 * q^86 + 219257739 * q^88 + 377904006 * q^89 + 245059140 * q^91 - 1178754558 * q^92 + 86583438 * q^94 + 536878770 * q^95 - 438907539 * q^97 - 2592182286 * 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 $

$2$	−3.46285	−0.153038	−0.0765190	−	0.997068i	$-0.524381\pi$
$2$	−3.46285	−0.153038	−0.0765190	+	0.997068i	$0.524381\pi$
$3$	0	0
$3$	0	0
$4$	−500.009	−0.976579
$5$	−1404.01	−1.00463	−0.502313	−	0.864686i	$-0.667518\pi$
$5$	−1404.01	−1.00463	−0.502313	+	0.864686i	$0.667518\pi$
$6$	0	0
$7$	1665.57	0.262193	0.131097	−	0.991370i	$-0.458150\pi$
$7$	1665.57	0.262193	0.131097	+	0.991370i	$0.458150\pi$
$8$	3504.44	0.302492
$9$	0	0
$10$	4861.88	0.153746
$11$	63618.6	1.31014	0.655069	−	0.755569i	$-0.272640\pi$
$11$	63618.6	1.31014	0.655069	+	0.755569i	$0.272640\pi$
$12$	0	0
$13$	−111210.	−1.07993	−0.539967	−	0.841686i	$-0.681563\pi$
$13$	−111210.	−1.07993	−0.539967	+	0.841686i	$0.681563\pi$
$14$	−5767.62	−0.0401255
$15$	0	0
$16$	243869.	0.930287
$17$	383783.	1.11446	0.557231	−	0.830358i	$-0.311864\pi$
$17$	383783.	1.11446	0.557231	+	0.830358i	$0.311864\pi$
$18$	0	0
$19$	7441.82	0.0131005	0.00655025	−	0.999979i	$-0.497915\pi$
$19$	7441.82	0.0131005	0.00655025	+	0.999979i	$0.497915\pi$
$20$	702017.	0.981098
$21$	0	0
$22$	−220302.	−0.200501
$23$	2.61722e6	1.95013	0.975067	−	0.221910i	$-0.0712290\pi$
$23$	2.61722e6	1.95013	0.975067	+	0.221910i	$0.0712290\pi$
$24$	0	0
$25$	18115.8	0.00927531
$26$	385102.	0.165271
$27$	0	0
$28$	−832799.	−0.256053
$29$	−579091.	−0.152039	−0.0760196	−	0.997106i	$-0.524221\pi$
$29$	−579091.	−0.152039	−0.0760196	+	0.997106i	$0.524221\pi$
$30$	0	0
$31$	3.51831e6	0.684237	0.342118	−	0.939657i	$-0.388856\pi$
$31$	3.51831e6	0.684237	0.342118	+	0.939657i	$0.388856\pi$
$32$	−2.63875e6	−0.444861
$33$	0	0
$34$	−1.32898e6	−0.170555
$35$	−2.33847e6	−0.263407
$36$	0	0
$37$	−9.82783e6	−0.862084	−0.431042	−	0.902332i	$-0.641854\pi$
$37$	−9.82783e6	−0.862084	−0.431042	+	0.902332i	$0.641854\pi$
$38$	−25769.9	−0.00200487
$39$	0	0
$40$	−4.92026e6	−0.303891
$41$	−1.77760e7	−0.982441	−0.491221	−	0.871035i	$-0.663449\pi$
$41$	−1.77760e7	−0.982441	−0.491221	+	0.871035i	$0.663449\pi$
$42$	0	0
$43$	3.91298e7	1.74542	0.872709	−	0.488240i	$-0.162361\pi$
$43$	3.91298e7	1.74542	0.872709	+	0.488240i	$0.162361\pi$
$44$	−3.18098e7	−1.27945
$45$	0	0
$46$	−9.06304e6	−0.298445
$47$	−2.67867e7	−0.800718	−0.400359	−	0.916358i	$-0.631115\pi$
$47$	−2.67867e7	−0.800718	−0.400359	+	0.916358i	$0.631115\pi$
$48$	0	0
$49$	−3.75795e7	−0.931255
$50$	−62732.5	−0.00141947
$51$	0	0
$52$	5.56058e7	1.05464
$53$	6.04430e7	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$53$	6.04430e7	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$54$	0	0
$55$	−8.93210e7	−1.31620
$56$	5.83689e6	0.0793113
$57$	0	0
$58$	2.00531e6	0.0232678
$59$	1.34817e8	1.44847	0.724237	−	0.689551i	$-0.242192\pi$
$59$	1.34817e8	1.44847	0.724237	+	0.689551i	$0.242192\pi$
$60$	0	0
$61$	1.40769e8	1.30174	0.650869	−	0.759190i	$-0.274405\pi$
$61$	1.40769e8	1.30174	0.650869	+	0.759190i	$0.274405\pi$
$62$	−1.21834e7	−0.104714
$63$	0	0
$64$	−1.15723e8	−0.862206
$65$	1.56139e8	1.08493
$66$	0	0
$67$	7.31341e7	0.443387	0.221694	−	0.975116i	$-0.428841\pi$
$67$	7.31341e7	0.443387	0.221694	+	0.975116i	$0.428841\pi$
$68$	−1.91895e8	−1.08836
$69$	0	0
$70$	8.09780e6	0.0403112
$71$	2.72910e8	1.27455	0.637276	−	0.770636i	$-0.280061\pi$
$71$	2.72910e8	1.27455	0.637276	+	0.770636i	$0.280061\pi$
$72$	0	0
$73$	−2.64531e8	−1.09025	−0.545123	−	0.838356i	$-0.683517\pi$
$73$	−2.64531e8	−1.09025	−0.545123	+	0.838356i	$0.683517\pi$
$74$	3.40323e7	0.131932
$75$	0	0
$76$	−3.72097e6	−0.0127937
$77$	1.05961e8	0.343509
$78$	0	0
$79$	5.44662e8	1.57328	0.786639	−	0.617413i	$-0.211819\pi$
$79$	5.44662e8	1.57328	0.786639	+	0.617413i	$0.211819\pi$
$80$	−3.42394e8	−0.934591
$81$	0	0
$82$	6.15557e7	0.150351
$83$	−1.73988e8	−0.402408	−0.201204	−	0.979549i	$-0.564485\pi$
$83$	−1.73988e8	−0.402408	−0.201204	+	0.979549i	$0.564485\pi$
$84$	0	0
$85$	−5.38834e8	−1.11962
$86$	−1.35501e8	−0.267115
$87$	0	0
$88$	2.22947e8	0.396306
$89$	−5.69208e8	−0.961647	−0.480823	−	0.876817i	$-0.659662\pi$
$89$	−5.69208e8	−0.961647	−0.480823	+	0.876817i	$0.659662\pi$
$90$	0	0
$91$	−1.85227e8	−0.283151
$92$	−1.30863e9	−1.90446
$93$	0	0
$94$	9.27586e7	0.122540
$95$	−1.04484e7	−0.0131611
$96$	0	0
$97$	−5.86520e8	−0.672682	−0.336341	−	0.941740i	$-0.609189\pi$
$97$	−5.86520e8	−0.672682	−0.336341	+	0.941740i	$0.609189\pi$
$98$	1.30132e8	0.142517
$99$	0	0
$100$	−9.05808e6	−0.00905808
$101$	1.62439e9	1.55326	0.776629	−	0.629959i	$-0.216928\pi$
$101$	1.62439e9	1.55326	0.776629	+	0.629959i	$0.216928\pi$
$102$	0	0
$103$	5.92705e8	0.518885	0.259442	−	0.965759i	$-0.416461\pi$
$103$	5.92705e8	0.518885	0.259442	+	0.965759i	$0.416461\pi$
$104$	−3.89727e8	−0.326671
$105$	0	0
$106$	−2.09305e8	−0.161029
$107$	4.69405e8	0.346195	0.173098	−	0.984905i	$-0.444622\pi$
$107$	4.69405e8	0.346195	0.173098	+	0.984905i	$0.444622\pi$
$108$	0	0
$109$	2.10211e9	1.42639	0.713193	−	0.700968i	$-0.247248\pi$
$109$	2.10211e9	1.42639	0.713193	+	0.700968i	$0.247248\pi$
$110$	3.09306e8	0.201428
$111$	0	0
$112$	4.06181e8	0.243915
$113$	1.25356e9	0.723255	0.361628	−	0.932323i	$-0.382221\pi$
$113$	1.25356e9	0.723255	0.361628	+	0.932323i	$0.382221\pi$
$114$	0	0
$115$	−3.67460e9	−1.95916
$116$	2.89550e8	0.148478
$117$	0	0
$118$	−4.66851e8	−0.221671
$119$	6.39217e8	0.292204
$120$	0	0
$121$	1.68938e9	0.716460
$122$	−4.87463e8	−0.199215
$123$	0	0
$124$	−1.75919e9	−0.668212
$125$	2.71677e9	0.995309
$126$	0	0
$127$	3.81433e9	1.30107	0.650536	−	0.759476i	$-0.274544\pi$
$127$	3.81433e9	1.30107	0.650536	+	0.759476i	$0.274544\pi$
$128$	1.75178e9	0.576811
$129$	0	0
$130$	−5.40687e8	−0.166036
$131$	1.93974e9	0.575471	0.287735	−	0.957710i	$-0.407098\pi$
$131$	1.93974e9	0.575471	0.287735	+	0.957710i	$0.407098\pi$
$132$	0	0
$133$	1.23949e7	0.00343486
$134$	−2.53253e8	−0.0678551
$135$	0	0
$136$	1.34494e9	0.337115
$137$	−5.94264e9	−1.44124	−0.720621	−	0.693329i	$-0.756143\pi$
$137$	−5.94264e9	−1.44124	−0.720621	+	0.693329i	$0.756143\pi$
$138$	0	0
$139$	−5.55150e9	−1.26137	−0.630687	−	0.776038i	$-0.717227\pi$
$139$	−5.55150e9	−1.26137	−0.630687	+	0.776038i	$0.717227\pi$
$140$	1.16926e9	0.257237
$141$	0	0
$142$	−9.45049e8	−0.195055
$143$	−7.07500e9	−1.41486
$144$	0	0
$145$	8.13049e8	0.152743
$146$	9.16033e8	0.166849
$147$	0	0
$148$	4.91400e9	0.841894
$149$	−9.72401e9	−1.61624	−0.808122	−	0.589015i	$-0.799516\pi$
$149$	−9.72401e9	−1.61624	−0.808122	+	0.589015i	$0.799516\pi$
$150$	0	0
$151$	−3.21852e9	−0.503802	−0.251901	−	0.967753i	$-0.581056\pi$
$151$	−3.21852e9	−0.503802	−0.251901	+	0.967753i	$0.581056\pi$
$152$	2.60794e7	0.00396279
$153$	0	0
$154$	−3.66928e8	−0.0525700
$155$	−4.93974e9	−0.687403
$156$	0	0
$157$	1.28576e9	0.168893	0.0844465	−	0.996428i	$-0.473088\pi$
$157$	1.28576e9	0.168893	0.0844465	+	0.996428i	$0.473088\pi$
$158$	−1.88609e9	−0.240771
$159$	0	0
$160$	3.70484e9	0.446919
$161$	4.35916e9	0.511312
$162$	0	0
$163$	4.38269e9	0.486291	0.243146	−	0.969990i	$-0.421821\pi$
$163$	4.38269e9	0.486291	0.243146	+	0.969990i	$0.421821\pi$
$164$	8.88815e9	0.959432
$165$	0	0
$166$	6.02494e8	0.0615837
$167$	−1.53534e9	−0.152750	−0.0763750	−	0.997079i	$-0.524335\pi$
$167$	−1.53534e9	−0.152750	−0.0763750	+	0.997079i	$0.524335\pi$
$168$	0	0
$169$	1.76307e9	0.166257
$170$	1.86590e9	0.171344
$171$	0	0
$172$	−1.95652e10	−1.70454
$173$	7.58197e8	0.0643538	0.0321769	−	0.999482i	$-0.489756\pi$
$173$	7.58197e8	0.0643538	0.0321769	+	0.999482i	$0.489756\pi$
$174$	0	0
$175$	3.01732e7	0.00243193
$176$	1.55146e10	1.21880
$177$	0	0
$178$	1.97108e9	0.147168
$179$	−1.17238e10	−0.853549	−0.426774	−	0.904358i	$-0.640350\pi$
$179$	−1.17238e10	−0.853549	−0.426774	+	0.904358i	$0.640350\pi$
$180$	0	0
$181$	−8.83265e9	−0.611698	−0.305849	−	0.952080i	$-0.598940\pi$
$181$	−8.83265e9	−0.611698	−0.305849	+	0.952080i	$0.598940\pi$
$182$	6.41415e8	0.0433329
$183$	0	0
$184$	9.17188e9	0.589899
$185$	1.37984e10	0.866073
$186$	0	0
$187$	2.44157e10	1.46010
$188$	1.33936e10	0.781965
$189$	0	0
$190$	3.61812e7	0.00201415
$191$	3.14017e9	0.170727	0.0853636	−	0.996350i	$-0.472795\pi$
$191$	3.14017e9	0.170727	0.0853636	+	0.996350i	$0.472795\pi$
$192$	0	0
$193$	−3.06350e10	−1.58932	−0.794658	−	0.607057i	$-0.792350\pi$
$193$	−3.06350e10	−1.58932	−0.794658	+	0.607057i	$0.792350\pi$
$194$	2.03103e9	0.102946
$195$	0	0
$196$	1.87901e10	0.909444
$197$	2.16444e10	1.02388	0.511939	−	0.859022i	$-0.328927\pi$
$197$	2.16444e10	1.02388	0.511939	+	0.859022i	$0.328927\pi$
$198$	0	0
$199$	−3.61495e9	−0.163404	−0.0817021	−	0.996657i	$-0.526036\pi$
$199$	−3.61495e9	−0.163404	−0.0817021	+	0.996657i	$0.526036\pi$
$200$	6.34858e7	0.00280570
$201$	0	0
$202$	−5.62501e9	−0.237707
$203$	−9.64516e8	−0.0398637
$204$	0	0
$205$	2.49576e10	0.986987
$206$	−2.05245e9	−0.0794091
$207$	0	0
$208$	−2.71206e10	−1.00465
$209$	4.73438e8	0.0171635
$210$	0	0
$211$	−1.18587e10	−0.411877	−0.205938	−	0.978565i	$-0.566025\pi$
$211$	−1.18587e10	−0.411877	−0.205938	+	0.978565i	$0.566025\pi$
$212$	−3.02220e10	−1.02757
$213$	0	0
$214$	−1.62548e9	−0.0529810
$215$	−5.49386e10	−1.75349
$216$	0	0
$217$	5.85999e9	0.179402
$218$	−7.27931e9	−0.218291
$219$	0	0
$220$	4.46613e10	1.28537
$221$	−4.26803e10	−1.20355
$222$	0	0
$223$	−1.88757e10	−0.511129	−0.255565	−	0.966792i	$-0.582261\pi$
$223$	−1.88757e10	−0.511129	−0.255565	+	0.966792i	$0.582261\pi$
$224$	−4.39503e9	−0.116640
$225$	0	0
$226$	−4.34089e9	−0.110685
$227$	2.04771e9	0.0511862	0.0255931	−	0.999672i	$-0.491853\pi$
$227$	2.04771e9	0.0511862	0.0255931	+	0.999672i	$0.491853\pi$
$228$	0	0
$229$	2.13934e9	0.0514068	0.0257034	−	0.999670i	$-0.491817\pi$
$229$	2.13934e9	0.0514068	0.0257034	+	0.999670i	$0.491817\pi$
$230$	1.27246e10	0.299825
$231$	0	0
$232$	−2.02939e9	−0.0459906
$233$	4.02632e10	0.894966	0.447483	−	0.894292i	$-0.352321\pi$
$233$	4.02632e10	0.894966	0.447483	+	0.894292i	$0.352321\pi$
$234$	0	0
$235$	3.76088e10	0.804423
$236$	−6.74097e10	−1.41455
$237$	0	0
$238$	−2.21351e9	−0.0447184
$239$	9.26326e9	0.183643	0.0918213	−	0.995776i	$-0.470731\pi$
$239$	9.26326e9	0.183643	0.0918213	+	0.995776i	$0.470731\pi$
$240$	0	0
$241$	−7.80952e10	−1.49124	−0.745620	−	0.666371i	$-0.767847\pi$
$241$	−7.80952e10	−1.49124	−0.745620	+	0.666371i	$0.767847\pi$
$242$	−5.85006e9	−0.109646
$243$	0	0
$244$	−7.03859e10	−1.27125
$245$	5.27619e10	0.935564
$246$	0	0
$247$	−8.27602e8	−0.0141477
$248$	1.23297e10	0.206976
$249$	0	0
$250$	−9.40778e9	−0.152320
$251$	1.54732e10	0.246064	0.123032	−	0.992403i	$-0.460738\pi$
$251$	1.54732e10	0.246064	0.123032	+	0.992403i	$0.460738\pi$
$252$	0	0
$253$	1.66504e11	2.55494
$254$	−1.32085e10	−0.199113
$255$	0	0
$256$	5.31842e10	0.773932
$257$	9.34772e10	1.33662	0.668308	−	0.743885i	$-0.267019\pi$
$257$	9.34772e10	1.33662	0.668308	+	0.743885i	$0.267019\pi$
$258$	0	0
$259$	−1.63689e10	−0.226033
$260$	−7.80710e10	−1.05952
$261$	0	0
$262$	−6.71704e9	−0.0880689
$263$	7.90979e9	0.101945	0.0509723	−	0.998700i	$-0.483768\pi$
$263$	7.90979e9	0.101945	0.0509723	+	0.998700i	$0.483768\pi$
$264$	0	0
$265$	−8.48625e10	−1.05708
$266$	−4.29216e7	−0.000525665 0
$267$	0	0
$268$	−3.65677e10	−0.433003
$269$	1.42705e11	1.66171	0.830853	−	0.556492i	$-0.187853\pi$
$269$	1.42705e11	1.66171	0.830853	+	0.556492i	$0.187853\pi$
$270$	0	0
$271$	6.60775e10	0.744203	0.372102	−	0.928192i	$-0.378637\pi$
$271$	6.60775e10	0.744203	0.372102	+	0.928192i	$0.378637\pi$
$272$	9.35927e10	1.03677
$273$	0	0
$274$	2.05785e10	0.220565
$275$	1.15250e9	0.0121519
$276$	0	0
$277$	−4.30135e10	−0.438981	−0.219491	−	0.975615i	$-0.570440\pi$
$277$	−4.30135e10	−0.438981	−0.219491	+	0.975615i	$0.570440\pi$
$278$	1.92240e10	0.193038
$279$	0	0
$280$	−8.19504e9	−0.0796783
$281$	1.01031e11	0.966665	0.483333	−	0.875437i	$-0.339426\pi$
$281$	1.01031e11	0.966665	0.483333	+	0.875437i	$0.339426\pi$
$282$	0	0
$283$	−8.17953e10	−0.758036	−0.379018	−	0.925389i	$-0.623738\pi$
$283$	−8.17953e10	−0.758036	−0.379018	+	0.925389i	$0.623738\pi$
$284$	−1.36458e11	−1.24470
$285$	0	0
$286$	2.44997e10	0.216528
$287$	−2.96072e10	−0.257590
$288$	0	0
$289$	2.87013e10	0.242025
$290$	−2.81547e9	−0.0233754
$291$	0	0
$292$	1.32268e11	1.06471
$293$	8.09103e10	0.641357	0.320678	−	0.947188i	$-0.396089\pi$
$293$	8.09103e10	0.641357	0.320678	+	0.947188i	$0.396089\pi$
$294$	0	0
$295$	−1.89284e11	−1.45518
$296$	−3.44410e10	−0.260773
$297$	0	0
$298$	3.36728e10	0.247347
$299$	−2.91060e11	−2.10602
$300$	0	0
$301$	6.51734e10	0.457637
$302$	1.11453e10	0.0771008
$303$	0	0
$304$	1.81483e9	0.0121872
$305$	−1.97641e11	−1.30776
$306$	0	0
$307$	7.58062e10	0.487060	0.243530	−	0.969893i	$-0.421695\pi$
$307$	7.58062e10	0.487060	0.243530	+	0.969893i	$0.421695\pi$
$308$	−5.29815e10	−0.335464
$309$	0	0
$310$	1.71056e10	0.105199
$311$	−1.77651e11	−1.07683	−0.538414	−	0.842680i	$-0.680976\pi$
$311$	−1.77651e11	−1.07683	−0.538414	+	0.842680i	$0.680976\pi$
$312$	0	0
$313$	8.37424e10	0.493169	0.246585	−	0.969121i	$-0.420692\pi$
$313$	8.37424e10	0.493169	0.246585	+	0.969121i	$0.420692\pi$
$314$	−4.45240e9	−0.0258470
$315$	0	0
$316$	−2.72336e11	−1.53643
$317$	5.30032e10	0.294805	0.147403	−	0.989077i	$-0.452909\pi$
$317$	5.30032e10	0.294805	0.147403	+	0.989077i	$0.452909\pi$
$318$	0	0
$319$	−3.68409e10	−0.199192
$320$	1.62477e11	0.866195
$321$	0	0
$322$	−1.50951e10	−0.0782502
$323$	2.85604e9	0.0146000
$324$	0	0
$325$	−2.01466e9	−0.0100167
$326$	−1.51766e10	−0.0744210
$327$	0	0
$328$	−6.22949e10	−0.297180
$329$	−4.46152e10	−0.209943
$330$	0	0
$331$	2.21979e11	1.01645	0.508225	−	0.861225i	$-0.330302\pi$
$331$	2.21979e11	1.01645	0.508225	+	0.861225i	$0.330302\pi$
$332$	8.69953e10	0.392984
$333$	0	0
$334$	5.31667e9	0.0233765
$335$	−1.02681e11	−0.445439
$336$	0	0
$337$	−3.13439e8	−0.00132379	−0.000661895	−	1.00000i	$-0.500211\pi$
$337$	−3.13439e8	−0.00132379	−0.000661895		1.00000i	$0.500211\pi$
$338$	−6.10526e9	−0.0254436
$339$	0	0
$340$	2.69422e11	1.09340
$341$	2.23830e11	0.896444
$342$	0	0
$343$	−1.29803e11	−0.506362
$344$	1.37128e11	0.527974
$345$	0	0
$346$	−2.62552e9	−0.00984858
$347$	−2.96272e11	−1.09700	−0.548502	−	0.836149i	$-0.684802\pi$
$347$	−2.96272e11	−1.09700	−0.548502	+	0.836149i	$0.684802\pi$
$348$	0	0
$349$	2.21560e11	0.799424	0.399712	−	0.916641i	$-0.369110\pi$
$349$	2.21560e11	0.799424	0.399712	+	0.916641i	$0.369110\pi$
$350$	−1.04485e8	−0.000372177 0
$351$	0	0
$352$	−1.67874e11	−0.582829
$353$	3.81073e11	1.30624	0.653119	−	0.757255i	$-0.273460\pi$
$353$	3.81073e11	1.30624	0.653119	+	0.757255i	$0.273460\pi$
$354$	0	0
$355$	−3.83169e11	−1.28045
$356$	2.84609e11	0.939125
$357$	0	0
$358$	4.05977e10	0.130625
$359$	−3.34181e11	−1.06183	−0.530917	−	0.847424i	$-0.678153\pi$
$359$	−3.34181e11	−1.06183	−0.530917	+	0.847424i	$0.678153\pi$
$360$	0	0
$361$	−3.22632e11	−0.999828
$362$	3.05862e10	0.0936130
$363$	0	0
$364$	9.26153e10	0.276520
$365$	3.71404e11	1.09529
$366$	0	0
$367$	4.62773e11	1.33159	0.665795	−	0.746135i	$-0.268092\pi$
$367$	4.62773e11	1.33159	0.665795	+	0.746135i	$0.268092\pi$
$368$	6.38258e11	1.81418
$369$	0	0
$370$	−4.77817e10	−0.132542
$371$	1.00672e11	0.275884
$372$	0	0
$373$	−2.32740e11	−0.622559	−0.311280	−	0.950318i	$-0.600758\pi$
$373$	−2.32740e11	−0.622559	−0.311280	+	0.950318i	$0.600758\pi$
$374$	−8.45480e10	−0.223450
$375$	0	0
$376$	−9.38725e10	−0.242211
$377$	6.44005e10	0.164192
$378$	0	0
$379$	5.92570e10	0.147524	0.0737621	−	0.997276i	$-0.476499\pi$
$379$	5.92570e10	0.147524	0.0737621	+	0.997276i	$0.476499\pi$
$380$	5.22428e9	0.0128529
$381$	0	0
$382$	−1.08739e10	−0.0261277
$383$	−1.10580e11	−0.262593	−0.131296	−	0.991343i	$-0.541914\pi$
$383$	−1.10580e11	−0.262593	−0.131296	+	0.991343i	$0.541914\pi$
$384$	0	0
$385$	−1.48770e11	−0.345099
$386$	1.06085e11	0.243226
$387$	0	0
$388$	2.93265e11	0.656928
$389$	−5.19724e11	−1.15080	−0.575400	−	0.817872i	$-0.695154\pi$
$389$	−5.19724e11	−1.15080	−0.575400	+	0.817872i	$0.695154\pi$
$390$	0	0
$391$	1.00444e12	2.17335
$392$	−1.31695e11	−0.281697
$393$	0	0
$394$	−7.49515e10	−0.156692
$395$	−7.64711e11	−1.58056
$396$	0	0
$397$	−4.49123e11	−0.907420	−0.453710	−	0.891149i	$-0.649900\pi$
$397$	−4.49123e11	−0.907420	−0.453710	+	0.891149i	$0.649900\pi$
$398$	1.25180e10	0.0250071
$399$	0	0
$400$	4.41789e9	0.00862870
$401$	4.75290e11	0.917929	0.458964	−	0.888455i	$-0.348221\pi$
$401$	4.75290e11	0.917929	0.458964	+	0.888455i	$0.348221\pi$
$402$	0	0
$403$	−3.91270e11	−0.738930
$404$	−8.12208e11	−1.51688
$405$	0	0
$406$	3.33998e9	0.00610066
$407$	−6.25232e11	−1.12945
$408$	0	0
$409$	−4.31939e11	−0.763251	−0.381626	−	0.924317i	$-0.624636\pi$
$409$	−4.31939e11	−0.763251	−0.381626	+	0.924317i	$0.624636\pi$
$410$	−8.64247e10	−0.151046
$411$	0	0
$412$	−2.96358e11	−0.506732
$413$	2.24547e11	0.379780
$414$	0	0
$415$	2.44280e11	0.404270
$416$	2.93455e11	0.480420
$417$	0	0
$418$	−1.63945e9	−0.00262666
$419$	−9.04463e11	−1.43360	−0.716799	−	0.697280i	$-0.754394\pi$
$419$	−9.04463e11	−1.43360	−0.716799	+	0.697280i	$0.754394\pi$
$420$	0	0
$421$	1.53568e11	0.238250	0.119125	−	0.992879i	$-0.461991\pi$
$421$	1.53568e11	0.238250	0.119125	+	0.992879i	$0.461991\pi$
$422$	4.10651e10	0.0630328
$423$	0	0
$424$	2.11819e11	0.318286
$425$	6.95255e9	0.0103370
$426$	0	0
$427$	2.34461e11	0.341307
$428$	−2.34707e11	−0.338087
$429$	0	0
$430$	1.90244e11	0.268351
$431$	1.01001e12	1.40987	0.704933	−	0.709274i	$-0.250977\pi$
$431$	1.01001e12	1.40987	0.704933	+	0.709274i	$0.250977\pi$
$432$	0	0
$433$	1.38892e12	1.89881	0.949407	−	0.314048i	$-0.101685\pi$
$433$	1.38892e12	1.89881	0.949407	+	0.314048i	$0.101685\pi$
$434$	−2.02923e10	−0.0274554
$435$	0	0
$436$	−1.05107e12	−1.39298
$437$	1.94769e10	0.0255477
$438$	0	0
$439$	−1.13642e12	−1.46032	−0.730159	−	0.683277i	$-0.760554\pi$
$439$	−1.13642e12	−1.46032	−0.730159	+	0.683277i	$0.760554\pi$
$440$	−3.13020e11	−0.398139
$441$	0	0
$442$	1.47796e11	0.184188
$443$	9.02042e11	1.11278	0.556391	−	0.830921i	$-0.312186\pi$
$443$	9.02042e11	1.11278	0.556391	+	0.830921i	$0.312186\pi$
$444$	0	0
$445$	7.99172e11	0.966096
$446$	6.53637e10	0.0782222
$447$	0	0
$448$	−1.92745e11	−0.226065
$449$	−4.28311e11	−0.497337	−0.248668	−	0.968589i	$-0.579993\pi$
$449$	−4.28311e11	−0.497337	−0.248668	+	0.968589i	$0.579993\pi$
$450$	0	0
$451$	−1.13088e12	−1.28713
$452$	−6.26790e11	−0.706316
$453$	0	0
$454$	−7.09093e9	−0.00783343
$455$	2.60061e11	0.284462
$456$	0	0
$457$	−5.75246e11	−0.616922	−0.308461	−	0.951237i	$-0.599814\pi$
$457$	−5.75246e11	−0.616922	−0.308461	+	0.951237i	$0.599814\pi$
$458$	−7.40822e9	−0.00786719
$459$	0	0
$460$	1.83733e12	1.91327
$461$	1.06514e12	1.09838	0.549192	−	0.835697i	$-0.314936\pi$
$461$	1.06514e12	1.09838	0.549192	+	0.835697i	$0.314936\pi$
$462$	0	0
$463$	−6.91661e11	−0.699486	−0.349743	−	0.936846i	$-0.613731\pi$
$463$	−6.91661e11	−0.699486	−0.349743	+	0.936846i	$0.613731\pi$
$464$	−1.41222e11	−0.141440
$465$	0	0
$466$	−1.39426e11	−0.136964
$467$	1.39973e12	1.36181	0.680907	−	0.732370i	$-0.261586\pi$
$467$	1.39973e12	1.36181	0.680907	+	0.732370i	$0.261586\pi$
$468$	0	0
$469$	1.21810e11	0.116253
$470$	−1.30234e11	−0.123107
$471$	0	0
$472$	4.72458e11	0.438151
$473$	2.48938e12	2.28674
$474$	0	0
$475$	1.34815e8	0.000121511 0
$476$	−3.19614e11	−0.285361
$477$	0	0
$478$	−3.20773e10	−0.0281043
$479$	−9.56803e11	−0.830448	−0.415224	−	0.909719i	$-0.636297\pi$
$479$	−9.56803e11	−0.830448	−0.415224	+	0.909719i	$0.636297\pi$
$480$	0	0
$481$	1.09295e12	0.930994
$482$	2.70432e11	0.228216
$483$	0	0
$484$	−8.44702e11	−0.699680
$485$	8.23479e11	0.675795
$486$	0	0
$487$	−2.88361e11	−0.232303	−0.116152	−	0.993231i	$-0.537056\pi$
$487$	−2.88361e11	−0.232303	−0.116152	+	0.993231i	$0.537056\pi$
$488$	4.93317e11	0.393765
$489$	0	0
$490$	−1.82707e11	−0.143177
$491$	−2.34624e11	−0.182182	−0.0910910	−	0.995843i	$-0.529035\pi$
$491$	−2.34624e11	−0.182182	−0.0910910	+	0.995843i	$0.529035\pi$
$492$	0	0
$493$	−2.22245e11	−0.169442
$494$	2.86586e9	0.00216513
$495$	0	0
$496$	8.58007e11	0.636536
$497$	4.54551e11	0.334179
$498$	0	0
$499$	2.52955e12	1.82638	0.913189	−	0.407536i	$-0.133612\pi$
$499$	2.52955e12	1.82638	0.913189	+	0.407536i	$0.133612\pi$
$500$	−1.35841e12	−0.971998
$501$	0	0
$502$	−5.35814e10	−0.0376571
$503$	−3.09347e11	−0.215471	−0.107736	−	0.994180i	$-0.534360\pi$
$503$	−3.09347e11	−0.215471	−0.107736	+	0.994180i	$0.534360\pi$
$504$	0	0
$505$	−2.28065e12	−1.56044
$506$	−5.76578e11	−0.391003
$507$	0	0
$508$	−1.90720e12	−1.27060
$509$	−2.54357e12	−1.67963	−0.839816	−	0.542871i	$-0.817337\pi$
$509$	−2.54357e12	−1.67963	−0.839816	+	0.542871i	$0.817337\pi$
$510$	0	0
$511$	−4.40595e11	−0.285855
$512$	−1.08108e12	−0.695252
$513$	0	0
$514$	−3.23698e11	−0.204553
$515$	−8.32163e11	−0.521286
$516$	0	0
$517$	−1.70413e12	−1.04905
$518$	5.66832e10	0.0345916
$519$	0	0
$520$	5.47180e11	0.328182
$521$	−1.75471e12	−1.04336	−0.521682	−	0.853140i	$-0.674695\pi$
$521$	−1.75471e12	−1.04336	−0.521682	+	0.853140i	$0.674695\pi$
$522$	0	0
$523$	9.65849e11	0.564484	0.282242	−	0.959343i	$-0.408922\pi$
$523$	9.65849e11	0.564484	0.282242	+	0.959343i	$0.408922\pi$
$524$	−9.69888e11	−0.561993
$525$	0	0
$526$	−2.73904e10	−0.0156014
$527$	1.35027e12	0.762556
$528$	0	0
$529$	5.04867e12	2.80302
$530$	2.93866e11	0.161774
$531$	0	0
$532$	−6.19754e9	−0.00335442
$533$	1.97686e12	1.06097
$534$	0	0
$535$	−6.59049e11	−0.347797
$536$	2.56294e11	0.134121
$537$	0	0
$538$	−4.94167e11	−0.254304
$539$	−2.39075e12	−1.22007
$540$	0	0
$541$	−2.09700e12	−1.05247	−0.526235	−	0.850339i	$-0.676397\pi$
$541$	−2.09700e12	−1.05247	−0.526235	+	0.850339i	$0.676397\pi$
$542$	−2.28817e11	−0.113891
$543$	0	0
$544$	−1.01271e12	−0.495780
$545$	−2.95139e12	−1.43299
$546$	0	0
$547$	1.50322e12	0.717926	0.358963	−	0.933352i	$-0.383130\pi$
$547$	1.50322e12	0.717926	0.358963	+	0.933352i	$0.383130\pi$
$548$	2.97137e12	1.40749
$549$	0	0
$550$	−3.99095e9	−0.00185971
$551$	−4.30949e9	−0.00199179
$552$	0	0
$553$	9.07173e11	0.412503
$554$	1.48949e11	0.0671808
$555$	0	0
$556$	2.77580e12	1.23183
$557$	4.83664e11	0.212910	0.106455	−	0.994318i	$-0.466050\pi$
$557$	4.83664e11	0.212910	0.106455	+	0.994318i	$0.466050\pi$
$558$	0	0
$559$	−4.35161e12	−1.88494
$560$	−5.70282e11	−0.245044
$561$	0	0
$562$	−3.49855e11	−0.147936
$563$	2.47754e12	1.03928	0.519641	−	0.854385i	$-0.326066\pi$
$563$	2.47754e12	1.03928	0.519641	+	0.854385i	$0.326066\pi$
$564$	0	0
$565$	−1.76001e12	−0.726602
$566$	2.83245e11	0.116008
$567$	0	0
$568$	9.56398e11	0.385541
$569$	−3.50996e12	−1.40377	−0.701886	−	0.712289i	$-0.747658\pi$
$569$	−3.50996e12	−1.40377	−0.701886	+	0.712289i	$0.747658\pi$
$570$	0	0
$571$	2.57255e11	0.101275	0.0506375	−	0.998717i	$-0.483875\pi$
$571$	2.57255e11	0.101275	0.0506375	+	0.998717i	$0.483875\pi$
$572$	3.53756e12	1.38172
$573$	0	0
$574$	1.02525e11	0.0394210
$575$	4.74131e10	0.0180881
$576$	0	0
$577$	7.55281e11	0.283673	0.141836	−	0.989890i	$-0.454699\pi$
$577$	7.55281e11	0.283673	0.141836	+	0.989890i	$0.454699\pi$
$578$	−9.93883e10	−0.0370390
$579$	0	0
$580$	−4.06531e11	−0.149165
$581$	−2.89789e11	−0.105509
$582$	0	0
$583$	3.84530e12	1.37855
$584$	−9.27034e11	−0.329790
$585$	0	0
$586$	−2.80181e11	−0.0981519
$587$	5.13111e12	1.78378	0.891888	−	0.452257i	$-0.149381\pi$
$587$	5.13111e12	1.78378	0.891888	+	0.452257i	$0.149381\pi$
$588$	0	0
$589$	2.61826e10	0.00896384
$590$	6.55464e11	0.222697
$591$	0	0
$592$	−2.39670e12	−0.801986
$593$	1.98175e12	0.658116	0.329058	−	0.944310i	$-0.393269\pi$
$593$	1.98175e12	0.658116	0.329058	+	0.944310i	$0.393269\pi$
$594$	0	0
$595$	−8.97466e11	−0.293557
$596$	4.86209e12	1.57839
$597$	0	0
$598$	1.00790e12	0.322300
$599$	1.03251e12	0.327699	0.163850	−	0.986485i	$-0.447609\pi$
$599$	1.03251e12	0.327699	0.163850	+	0.986485i	$0.447609\pi$
$600$	0	0
$601$	−1.06818e12	−0.333971	−0.166986	−	0.985959i	$-0.553403\pi$
$601$	−1.06818e12	−0.333971	−0.166986	+	0.985959i	$0.553403\pi$
$602$	−2.25686e11	−0.0700358
$603$	0	0
$604$	1.60929e12	0.492002
$605$	−2.37190e12	−0.719775
$606$	0	0
$607$	3.33870e12	0.998225	0.499112	−	0.866537i	$-0.333660\pi$
$607$	3.33870e12	0.998225	0.499112	+	0.866537i	$0.333660\pi$
$608$	−1.96371e10	−0.00582790
$609$	0	0
$610$	6.84403e11	0.200137
$611$	2.97894e12	0.864722
$612$	0	0
$613$	−4.12503e12	−1.17993	−0.589964	−	0.807430i	$-0.700858\pi$
$613$	−4.12503e12	−1.17993	−0.589964	+	0.807430i	$0.700858\pi$
$614$	−2.62506e11	−0.0745386
$615$	0	0
$616$	3.71334e11	0.103909
$617$	−4.32528e12	−1.20152	−0.600759	−	0.799430i	$-0.705135\pi$
$617$	−4.32528e12	−1.20152	−0.600759	+	0.799430i	$0.705135\pi$
$618$	0	0
$619$	−6.31375e12	−1.72854	−0.864271	−	0.503027i	$-0.832220\pi$
$619$	−6.31375e12	−1.72854	−0.864271	+	0.503027i	$0.832220\pi$
$620$	2.46991e12	0.671303
$621$	0	0
$622$	6.15180e11	0.164796
$623$	−9.48055e11	−0.252137
$624$	0	0
$625$	−3.84975e12	−1.00919
$626$	−2.89988e11	−0.0754736
$627$	0	0
$628$	−6.42891e11	−0.164937
$629$	−3.77175e12	−0.960760
$630$	0	0
$631$	6.69921e12	1.68225	0.841127	−	0.540838i	$-0.181893\pi$
$631$	6.69921e12	1.68225	0.841127	+	0.540838i	$0.181893\pi$
$632$	1.90873e12	0.475903
$633$	0	0
$634$	−1.83542e11	−0.0451164
$635$	−5.35535e12	−1.30709
$636$	0	0
$637$	4.17920e12	1.00569
$638$	1.27575e11	0.0304840
$639$	0	0
$640$	−2.45951e12	−0.579480
$641$	−7.06833e12	−1.65370	−0.826849	−	0.562425i	$-0.809869\pi$
$641$	−7.06833e12	−1.65370	−0.826849	+	0.562425i	$0.809869\pi$
$642$	0	0
$643$	1.70634e12	0.393656	0.196828	−	0.980438i	$-0.436936\pi$
$643$	1.70634e12	0.393656	0.196828	+	0.980438i	$0.436936\pi$
$644$	−2.17962e12	−0.499337
$645$	0	0
$646$	−9.89005e9	−0.00223436
$647$	−3.85704e12	−0.865336	−0.432668	−	0.901553i	$-0.642428\pi$
$647$	−3.85704e12	−0.865336	−0.432668	+	0.901553i	$0.642428\pi$
$648$	0	0
$649$	8.57687e12	1.89770
$650$	6.97646e9	0.00153294
$651$	0	0
$652$	−2.19138e12	−0.474902
$653$	4.62412e12	0.995221	0.497610	−	0.867401i	$-0.334211\pi$
$653$	4.62412e12	0.995221	0.497610	+	0.867401i	$0.334211\pi$
$654$	0	0
$655$	−2.72342e12	−0.578133
$656$	−4.33501e12	−0.913952
$657$	0	0
$658$	1.54496e11	0.0321292
$659$	−8.16802e11	−0.168707	−0.0843533	−	0.996436i	$-0.526882\pi$
$659$	−8.16802e11	−0.168707	−0.0843533	+	0.996436i	$0.526882\pi$
$660$	0	0
$661$	−1.14771e12	−0.233843	−0.116921	−	0.993141i	$-0.537303\pi$
$661$	−1.14771e12	−0.233843	−0.116921	+	0.993141i	$0.537303\pi$
$662$	−7.68680e11	−0.155555
$663$	0	0
$664$	−6.09729e11	−0.121725
$665$	−1.74025e10	−0.00345076
$666$	0	0
$667$	−1.51561e12	−0.296497
$668$	7.67684e11	0.149172
$669$	0	0
$670$	3.55569e11	0.0681691
$671$	8.95554e12	1.70546
$672$	0	0
$673$	−7.17515e12	−1.34823	−0.674114	−	0.738627i	$-0.735474\pi$
$673$	−7.17515e12	−1.34823	−0.674114	+	0.738627i	$0.735474\pi$
$674$	1.08539e9	0.000202590 0
$675$	0	0
$676$	−8.81552e11	−0.162363
$677$	−5.21004e12	−0.953217	−0.476609	−	0.879116i	$-0.658134\pi$
$677$	−5.21004e12	−0.953217	−0.476609	+	0.879116i	$0.658134\pi$
$678$	0	0
$679$	−9.76890e11	−0.176373
$680$	−1.88831e12	−0.338675
$681$	0	0
$682$	−7.75090e11	−0.137190
$683$	2.81291e12	0.494611	0.247305	−	0.968938i	$-0.420455\pi$
$683$	2.81291e12	0.494611	0.247305	+	0.968938i	$0.420455\pi$
$684$	0	0
$685$	8.34352e12	1.44791
$686$	4.49489e11	0.0774926
$687$	0	0
$688$	9.54255e12	1.62374
$689$	−6.72184e12	−1.13632
$690$	0	0
$691$	−7.74991e12	−1.29314	−0.646570	−	0.762855i	$-0.723797\pi$
$691$	−7.74991e12	−1.29314	−0.646570	+	0.762855i	$0.723797\pi$
$692$	−3.79105e11	−0.0628466
$693$	0	0
$694$	1.02595e12	0.167883
$695$	7.79435e12	1.26721
$696$	0	0
$697$	−6.82212e12	−1.09489
$698$	−7.67230e11	−0.122342
$699$	0	0
$700$	−1.50869e10	−0.00237497
$701$	1.86576e12	0.291826	0.145913	−	0.989297i	$-0.453388\pi$
$701$	1.86576e12	0.291826	0.145913	+	0.989297i	$0.453388\pi$
$702$	0	0
$703$	−7.31369e10	−0.0112937
$704$	−7.36215e12	−1.12961
$705$	0	0
$706$	−1.31960e12	−0.199904
$707$	2.70553e12	0.407254
$708$	0	0
$709$	2.27617e11	0.0338295	0.0169148	−	0.999857i	$-0.494616\pi$
$709$	2.27617e11	0.0338295	0.0169148	+	0.999857i	$0.494616\pi$
$710$	1.32686e12	0.195957
$711$	0	0
$712$	−1.99475e12	−0.290890
$713$	9.20818e12	1.33435
$714$	0	0
$715$	9.93336e12	1.42141
$716$	5.86198e12	0.833558
$717$	0	0
$718$	1.15722e12	0.162501
$719$	2.12785e12	0.296934	0.148467	−	0.988917i	$-0.452566\pi$
$719$	2.12785e12	0.296934	0.148467	+	0.988917i	$0.452566\pi$
$720$	0	0
$721$	9.87191e11	0.136048
$722$	1.11723e12	0.153012
$723$	0	0
$724$	4.41640e12	0.597372
$725$	−1.04907e10	−0.00141021
$726$	0	0
$727$	−1.57756e12	−0.209450	−0.104725	−	0.994501i	$-0.533396\pi$
$727$	−1.57756e12	−0.209450	−0.104725	+	0.994501i	$0.533396\pi$
$728$	−6.49118e11	−0.0856510
$729$	0	0
$730$	−1.28612e12	−0.167621
$731$	1.50173e13	1.94520
$732$	0	0
$733$	−4.60083e12	−0.588665	−0.294333	−	0.955703i	$-0.595097\pi$
$733$	−4.60083e12	−0.588665	−0.294333	+	0.955703i	$0.595097\pi$
$734$	−1.60251e12	−0.203784
$735$	0	0
$736$	−6.90620e12	−0.867538
$737$	4.65269e12	0.580899
$738$	0	0
$739$	−2.25557e12	−0.278199	−0.139099	−	0.990278i	$-0.544421\pi$
$739$	−2.25557e12	−0.278199	−0.139099	+	0.990278i	$0.544421\pi$
$740$	−6.89930e12	−0.845789
$741$	0	0
$742$	−3.48612e11	−0.0422207
$743$	1.07745e13	1.29702	0.648508	−	0.761208i	$-0.275393\pi$
$743$	1.07745e13	1.29702	0.648508	+	0.761208i	$0.275393\pi$
$744$	0	0
$745$	1.36526e13	1.62372
$746$	8.05943e11	0.0952752
$747$	0	0
$748$	−1.22081e13	−1.42590
$749$	7.81827e11	0.0907700
$750$	0	0
$751$	−5.32047e12	−0.610338	−0.305169	−	0.952298i	$-0.598713\pi$
$751$	−5.32047e12	−0.610338	−0.305169	+	0.952298i	$0.598713\pi$
$752$	−6.53246e12	−0.744897
$753$	0	0
$754$	−2.23009e11	−0.0251277
$755$	4.51883e12	0.506133
$756$	0	0
$757$	8.65829e12	0.958298	0.479149	−	0.877734i	$-0.340945\pi$
$757$	8.65829e12	0.958298	0.479149	+	0.877734i	$0.340945\pi$
$758$	−2.05198e11	−0.0225768
$759$	0	0
$760$	−3.66157e10	−0.00398113
$761$	−7.04534e12	−0.761502	−0.380751	−	0.924678i	$-0.624334\pi$
$761$	−7.04534e12	−0.761502	−0.380751	+	0.924678i	$0.624334\pi$
$762$	0	0
$763$	3.50122e12	0.373989
$764$	−1.57011e12	−0.166729
$765$	0	0
$766$	3.82923e11	0.0401866
$767$	−1.49929e13	−1.56426
$768$	0	0
$769$	−6.58271e12	−0.678791	−0.339395	−	0.940644i	$-0.610222\pi$
$769$	−6.58271e12	−0.678791	−0.339395	+	0.940644i	$0.610222\pi$
$770$	5.15170e11	0.0528132
$771$	0	0
$772$	1.53178e13	1.55209
$773$	−1.34877e13	−1.35872	−0.679360	−	0.733805i	$-0.737742\pi$
$773$	−1.34877e13	−1.35872	−0.679360	+	0.733805i	$0.737742\pi$
$774$	0	0
$775$	6.37371e10	0.00634651
$776$	−2.05542e12	−0.203481
$777$	0	0
$778$	1.79973e12	0.176116
$779$	−1.32286e11	−0.0128705
$780$	0	0
$781$	1.73622e13	1.66984
$782$	−3.47824e12	−0.332605
$783$	0	0
$784$	−9.16447e12	−0.866334
$785$	−1.80522e12	−0.169674
$786$	0	0
$787$	−1.23113e13	−1.14398	−0.571988	−	0.820262i	$-0.693828\pi$
$787$	−1.23113e13	−1.14398	−0.571988	+	0.820262i	$0.693828\pi$
$788$	−1.08224e13	−0.999899
$789$	0	0
$790$	2.64808e12	0.241885
$791$	2.08789e12	0.189633
$792$	0	0
$793$	−1.56549e13	−1.40579
$794$	1.55525e12	0.138870
$795$	0	0
$796$	1.80751e12	0.159577
$797$	1.13882e13	0.999753	0.499877	−	0.866097i	$-0.333379\pi$
$797$	1.13882e13	0.999753	0.499877	+	0.866097i	$0.333379\pi$
$798$	0	0
$799$	−1.02803e13	−0.892370
$800$	−4.78033e10	−0.00412622
$801$	0	0
$802$	−1.64586e12	−0.140478
$803$	−1.68291e13	−1.42837
$804$	0	0
$805$	−6.12030e12	−0.513678
$806$	1.35491e12	0.113084
$807$	0	0
$808$	5.69256e12	0.469847
$809$	−9.39455e12	−0.771095	−0.385547	−	0.922688i	$-0.625987\pi$
$809$	−9.39455e12	−0.771095	−0.385547	+	0.922688i	$0.625987\pi$
$810$	0	0
$811$	8.85202e12	0.718536	0.359268	−	0.933234i	$-0.383026\pi$
$811$	8.85202e12	0.718536	0.359268	+	0.933234i	$0.383026\pi$
$812$	4.82266e11	0.0389300
$813$	0	0
$814$	2.16509e12	0.172849
$815$	−6.15334e12	−0.488541
$816$	0	0
$817$	2.91197e11	0.0228659
$818$	1.49574e12	0.116806
$819$	0	0
$820$	−1.24790e13	−0.963871
$821$	1.33940e13	1.02888	0.514441	−	0.857526i	$-0.328000\pi$
$821$	1.33940e13	1.02888	0.514441	+	0.857526i	$0.328000\pi$
$822$	0	0
$823$	1.86595e13	1.41775	0.708876	−	0.705333i	$-0.249203\pi$
$823$	1.86595e13	1.41775	0.708876	+	0.705333i	$0.249203\pi$
$824$	2.07710e12	0.156958
$825$	0	0
$826$	−7.77574e11	−0.0581208
$827$	1.27791e13	0.950006	0.475003	−	0.879984i	$-0.342447\pi$
$827$	1.27791e13	0.950006	0.475003	+	0.879984i	$0.342447\pi$
$828$	0	0
$829$	−8.09935e12	−0.595600	−0.297800	−	0.954628i	$-0.596253\pi$
$829$	−8.09935e12	−0.595600	−0.297800	+	0.954628i	$0.596253\pi$
$830$	−8.45906e11	−0.0618687
$831$	0	0
$832$	1.28695e13	0.931126
$833$	−1.44224e13	−1.03785
$834$	0	0
$835$	2.15563e12	0.153457
$836$	−2.36723e11	−0.0167615
$837$	0	0
$838$	3.13202e12	0.219395
$839$	1.17750e13	0.820409	0.410205	−	0.911994i	$-0.365457\pi$
$839$	1.17750e13	0.820409	0.410205	+	0.911994i	$0.365457\pi$
$840$	0	0
$841$	−1.41718e13	−0.976884
$842$	−5.31784e11	−0.0364612
$843$	0	0
$844$	5.92947e12	0.402230
$845$	−2.47537e12	−0.167026
$846$	0	0
$847$	2.81377e12	0.187851
$848$	1.47402e13	0.978862
$849$	0	0
$850$	−2.40756e10	−0.00158195
$851$	−2.57216e13	−1.68118
$852$	0	0
$853$	6.84220e11	0.0442512	0.0221256	−	0.999755i	$-0.492957\pi$
$853$	6.84220e11	0.0442512	0.0221256	+	0.999755i	$0.492957\pi$
$854$	−8.11904e11	−0.0522329
$855$	0	0
$856$	1.64500e12	0.104721
$857$	−1.15644e13	−0.732337	−0.366168	−	0.930549i	$-0.619331\pi$
$857$	−1.15644e13	−0.732337	−0.366168	+	0.930549i	$0.619331\pi$
$858$	0	0
$859$	1.50261e13	0.941623	0.470812	−	0.882234i	$-0.343961\pi$
$859$	1.50261e13	0.941623	0.470812	+	0.882234i	$0.343961\pi$
$860$	2.74698e13	1.71243
$861$	0	0
$862$	−3.49752e12	−0.215763
$863$	1.54066e13	0.945490	0.472745	−	0.881199i	$-0.343263\pi$
$863$	1.54066e13	0.945490	0.472745	+	0.881199i	$0.343263\pi$
$864$	0	0
$865$	−1.06451e12	−0.0646516
$866$	−4.80964e12	−0.290591
$867$	0	0
$868$	−2.93005e12	−0.175201
$869$	3.46506e13	2.06121
$870$	0	0
$871$	−8.13322e12	−0.478829
$872$	7.36673e12	0.431470
$873$	0	0
$874$	−6.74455e10	−0.00390977
$875$	4.52497e12	0.260963
$876$	0	0
$877$	−2.57641e13	−1.47068	−0.735338	−	0.677700i	$-0.762977\pi$
$877$	−2.57641e13	−1.47068	−0.735338	+	0.677700i	$0.762977\pi$
$878$	3.93525e12	0.223484
$879$	0	0
$880$	−2.17826e13	−1.22444
$881$	−8.67327e12	−0.485055	−0.242528	−	0.970144i	$-0.577977\pi$
$881$	−8.67327e12	−0.485055	−0.242528	+	0.970144i	$0.577977\pi$
$882$	0	0
$883$	1.27283e13	0.704607	0.352304	−	0.935886i	$-0.385398\pi$
$883$	1.27283e13	0.704607	0.352304	+	0.935886i	$0.385398\pi$
$884$	2.13405e13	1.17536
$885$	0	0
$886$	−3.12364e12	−0.170298
$887$	3.99323e12	0.216605	0.108302	−	0.994118i	$-0.465459\pi$
$887$	3.99323e12	0.216605	0.108302	+	0.994118i	$0.465459\pi$
$888$	0	0
$889$	6.35303e12	0.341132
$890$	−2.76742e12	−0.147849
$891$	0	0
$892$	9.43801e12	0.499159
$893$	−1.99342e11	−0.0104898
$894$	0	0
$895$	1.64603e13	0.857498
$896$	2.91770e12	0.151236
$897$	0	0
$898$	1.48318e12	0.0761114
$899$	−2.03742e12	−0.104031
$900$	0	0
$901$	2.31970e13	1.17265
$902$	3.91608e12	0.196980
$903$	0	0
$904$	4.39302e12	0.218779
$905$	1.24011e13	0.614529
$906$	0	0
$907$	−8.79778e12	−0.431659	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−8.79778e12	−0.431659	−0.215829	+	0.976431i	$0.569245\pi$
$908$	−1.02387e12	−0.0499874
$909$	0	0
$910$	−9.00552e11	−0.0435334
$911$	−2.95612e13	−1.42197	−0.710983	−	0.703209i	$-0.751750\pi$
$911$	−2.95612e13	−1.42197	−0.710983	+	0.703209i	$0.751750\pi$
$912$	0	0
$913$	−1.10688e13	−0.527210
$914$	1.99199e12	0.0944125
$915$	0	0
$916$	−1.06969e12	−0.0502028
$917$	3.23078e12	0.150885
$918$	0	0
$919$	−1.05646e13	−0.488577	−0.244288	−	0.969703i	$-0.578554\pi$
$919$	−1.05646e13	−0.488577	−0.244288	+	0.969703i	$0.578554\pi$
$920$	−1.28774e13	−0.592629
$921$	0	0
$922$	−3.68843e12	−0.168094
$923$	−3.03503e13	−1.37643
$924$	0	0
$925$	−1.78039e11	−0.00799610
$926$	2.39512e12	0.107048
$927$	0	0
$928$	1.52808e12	0.0676363
$929$	−1.99951e13	−0.880752	−0.440376	−	0.897814i	$-0.645155\pi$
$929$	−1.99951e13	−0.880752	−0.440376	+	0.897814i	$0.645155\pi$
$930$	0	0
$931$	−2.79660e11	−0.0121999
$932$	−2.01319e13	−0.874005
$933$	0	0
$934$	−4.84705e12	−0.208409
$935$	−3.42799e13	−1.46685
$936$	0	0
$937$	3.60094e13	1.52612	0.763058	−	0.646330i	$-0.223697\pi$
$937$	3.60094e13	1.52612	0.763058	+	0.646330i	$0.223697\pi$
$938$	−4.21810e11	−0.0177912
$939$	0	0
$940$	−1.88047e13	−0.785583
$941$	−2.20084e13	−0.915028	−0.457514	−	0.889202i	$-0.651260\pi$
$941$	−2.20084e13	−0.915028	−0.457514	+	0.889202i	$0.651260\pi$
$942$	0	0
$943$	−4.65236e13	−1.91589
$944$	3.28777e13	1.34750
$945$	0	0
$946$	−8.62036e12	−0.349958
$947$	2.27142e13	0.917746	0.458873	−	0.888502i	$-0.348253\pi$
$947$	2.27142e13	0.917746	0.458873	+	0.888502i	$0.348253\pi$
$948$	0	0
$949$	2.94184e13	1.17739
$950$	−4.66844e8	−1.85958e−5 0
$951$	0	0
$952$	2.24010e12	0.0883894
$953$	−1.64449e13	−0.645824	−0.322912	−	0.946429i	$-0.604662\pi$
$953$	−1.64449e13	−0.645824	−0.322912	+	0.946429i	$0.604662\pi$
$954$	0	0
$955$	−4.40882e12	−0.171517
$956$	−4.63171e12	−0.179342
$957$	0	0
$958$	3.31327e12	0.127090
$959$	−9.89789e12	−0.377884
$960$	0	0
$961$	−1.40611e13	−0.531820
$962$	−3.78472e12	−0.142477
$963$	0	0
$964$	3.90483e13	1.45631
$965$	4.30118e13	1.59667
$966$	0	0
$967$	2.49306e13	0.916883	0.458442	−	0.888725i	$-0.348408\pi$
$967$	2.49306e13	0.916883	0.458442	+	0.888725i	$0.348408\pi$
$968$	5.92031e12	0.216723
$969$	0	0
$970$	−2.85159e12	−0.103422
$971$	−4.38342e13	−1.58244	−0.791218	−	0.611534i	$-0.790553\pi$
$971$	−4.38342e13	−1.58244	−0.791218	+	0.611534i	$0.790553\pi$
$972$	0	0
$973$	−9.24641e12	−0.330724
$974$	9.98550e11	0.0355512
$975$	0	0
$976$	3.43293e13	1.21099
$977$	4.57358e12	0.160595	0.0802973	−	0.996771i	$-0.474413\pi$
$977$	4.57358e12	0.160595	0.0802973	+	0.996771i	$0.474413\pi$
$978$	0	0
$979$	−3.62122e13	−1.25989
$980$	−2.63814e13	−0.913652
$981$	0	0
$982$	8.12468e11	0.0278808
$983$	−4.78406e13	−1.63420	−0.817101	−	0.576494i	$-0.804420\pi$
$983$	−4.78406e13	−1.63420	−0.817101	+	0.576494i	$0.804420\pi$
$984$	0	0
$985$	−3.03890e13	−1.02862
$986$	7.69602e11	0.0259310
$987$	0	0
$988$	4.13808e11	0.0138163
$989$	1.02411e14	3.40380
$990$	0	0
$991$	−1.94011e13	−0.638993	−0.319496	−	0.947588i	$-0.603514\pi$
$991$	−1.94011e13	−0.638993	−0.319496	+	0.947588i	$0.603514\pi$
$992$	−9.28396e12	−0.304390
$993$	0	0
$994$	−1.57405e12	−0.0511421
$995$	5.07542e12	0.164160
$996$	0	0
$997$	4.73855e13	1.51886	0.759429	−	0.650591i	$-0.225479\pi$
$997$	4.73855e13	1.51886	0.759429	+	0.650591i	$0.225479\pi$
$998$	−8.75946e12	−0.279505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.b.1.2	✓	3
3.2	odd	2		27.10.a.c.1.2	yes	3
9.2	odd	6		81.10.c.g.28.2		6
9.4	even	3		81.10.c.h.55.2		6
9.5	odd	6		81.10.c.g.55.2		6
9.7	even	3		81.10.c.h.28.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.b.1.2	✓	3	1.1	even	1	trivial
27.10.a.c.1.2	yes	3	3.2	odd	2
81.10.c.g.28.2		6	9.2	odd	6
81.10.c.g.55.2		6	9.5	odd	6
81.10.c.h.28.2		6	9.7	even	3
81.10.c.h.55.2		6	9.4	even	3

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-3.46285 q^{2} -500.009 q^{4} -1404.01 q^{5} +1665.57 q^{7} +3504.44 q^{8} +4861.88 q^{10} +63618.6 q^{11} -111210. q^{13} -5767.62 q^{14} +243869. q^{16} +383783. q^{17} +7441.82 q^{19} +702017. q^{20} -220302. q^{22} +2.61722e6 q^{23} +18115.8 q^{25} +385102. q^{26} -832799. q^{28} -579091. q^{29} +3.51831e6 q^{31} -2.63875e6 q^{32} -1.32898e6 q^{34} -2.33847e6 q^{35} -9.82783e6 q^{37} -25769.9 q^{38} -4.92026e6 q^{40} -1.77760e7 q^{41} +3.91298e7 q^{43} -3.18098e7 q^{44} -9.06304e6 q^{46} -2.67867e7 q^{47} -3.75795e7 q^{49} -62732.5 q^{50} +5.56058e7 q^{52} +6.04430e7 q^{53} -8.93210e7 q^{55} +5.83689e6 q^{56} +2.00531e6 q^{58} +1.34817e8 q^{59} +1.40769e8 q^{61} -1.21834e7 q^{62} -1.15723e8 q^{64} +1.56139e8 q^{65} +7.31341e7 q^{67} -1.91895e8 q^{68} +8.09780e6 q^{70} +2.72910e8 q^{71} -2.64531e8 q^{73} +3.40323e7 q^{74} -3.72097e6 q^{76} +1.05961e8 q^{77} +5.44662e8 q^{79} -3.42394e8 q^{80} +6.15557e7 q^{82} -1.73988e8 q^{83} -5.38834e8 q^{85} -1.35501e8 q^{86} +2.22947e8 q^{88} -5.69208e8 q^{89} -1.85227e8 q^{91} -1.30863e9 q^{92} +9.27586e7 q^{94} -1.04484e7 q^{95} -5.86520e8 q^{97} +1.30132e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 597 q^{4} + 1983 q^{5} - 3693 q^{7} + 4503 q^{8} - 18981 q^{10} + 16863 q^{11} + 116916 q^{13} + 503463 q^{14} - 239919 q^{16} + 1014048 q^{17} - 15222 q^{19} + 2548407 q^{20} + 305721 q^{22}+ \cdots - 2592182286 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 597 * q^4 + 1983 * q^5 - 3693 * q^7 + 4503 * q^8 - 18981 * q^10 + 16863 * q^11 + 116916 * q^13 + 503463 * q^14 - 239919 * q^16 + 1014048 * q^17 - 15222 * q^19 + 2548407 * q^20 + 305721 * q^22 + 2927118 * q^23 + 2133732 * q^25 + 4765116 * q^26 - 3535725 * q^28 + 5768790 * q^29 - 6575223 * q^31 - 2687697 * q^32 + 17098128 * q^34 - 17340537 * q^35 - 11686026 * q^37 - 50473374 * q^38 - 4130811 * q^40 - 22213518 * q^41 + 45384414 * q^43 - 57206991 * q^44 - 95638590 * q^46 + 12392034 * q^47 + 18933462 * q^49 - 82984044 * q^50 + 182736492 * q^52 + 80579637 * q^53 - 174735333 * q^55 + 21850521 * q^56 - 102758922 * q^58 + 244026660 * q^59 + 369729960 * q^61 + 166297341 * q^62 - 420692127 * q^64 + 492225684 * q^65 - 252614586 * q^67 + 162270144 * q^68 + 935092377 * q^70 + 403193088 * q^71 - 406626717 * q^73 - 641373558 * q^74 - 39499590 * q^76 + 360443199 * q^77 + 265451856 * q^79 - 1168359837 * q^80 - 884408166 * q^82 + 121625871 * q^83 + 316251216 * q^85 - 2368956570 * q^86 + 219257739 * q^88 + 377904006 * q^89 + 245059140 * q^91 - 1178754558 * q^92 + 86583438 * q^94 + 536878770 * q^95 - 438907539 * q^97 - 2592182286 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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