LMFDB - Embedded newform 27.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,8,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	27.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:	$8.43439568807$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.53113$ 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)

$f(q)$	$=$	\(q-7.59339 q^{2} -70.3405 q^{4} +65.8132 q^{5} -738.405 q^{7} +1506.08 q^{8} -499.745 q^{10} +4961.26 q^{11} +5967.24 q^{13} +5606.99 q^{14} -2432.64 q^{16} +36651.6 q^{17} +22378.9 q^{19} -4629.33 q^{20} -37672.8 q^{22} -51473.6 q^{23} -73793.6 q^{25} -45311.5 q^{26} +51939.7 q^{28} -68495.7 q^{29} +150655. q^{31} -174306. q^{32} -278310. q^{34} -48596.8 q^{35} +489027. q^{37} -169932. q^{38} +99119.7 q^{40} +590635. q^{41} -842643. q^{43} -348978. q^{44} +390859. q^{46} +1.22637e6 q^{47} -278301. q^{49} +560343. q^{50} -419738. q^{52} +958904. q^{53} +326517. q^{55} -1.11209e6 q^{56} +520114. q^{58} -316269. q^{59} -29722.9 q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +392723. q^{65} +293025. q^{67} -2.57809e6 q^{68} +369014. q^{70} +714537. q^{71} -3.96273e6 q^{73} -3.71337e6 q^{74} -1.57415e6 q^{76} -3.66342e6 q^{77} +2.53805e6 q^{79} -160100. q^{80} -4.48492e6 q^{82} +1.66311e6 q^{83} +2.41216e6 q^{85} +6.39851e6 q^{86} +7.47204e6 q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +3.62068e6 q^{92} -9.31228e6 q^{94} +1.47283e6 q^{95} +1.47010e7 q^{97} +2.11325e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) $ 2 * q + 9 * q^2 + 77 * q^4 + 180 * q^5 + 700 * q^7 + 1827 * q^8 + 1395 * q^10 + 10890 * q^11 - 5480 * q^13 + 29475 * q^14 - 15967 * q^16 + 16416 * q^17 + 16024 * q^19 + 12195 * q^20 + 60705 * q^22 + 24372 * q^23 - 138880 * q^25 - 235260 * q^26 + 263875 * q^28 - 143280 * q^29 - 38708 * q^31 - 439965 * q^32 - 614088 * q^34 + 115650 * q^35 + 455620 * q^37 - 275382 * q^38 + 135765 * q^40 + 731880 * q^41 - 1088840 * q^43 + 524565 * q^44 + 1649394 * q^46 + 1561500 * q^47 + 967164 * q^49 - 519660 * q^50 - 2106380 * q^52 + 2610468 * q^53 + 1003500 * q^55 - 650475 * q^56 - 720810 * q^58 + 1731960 * q^59 - 620192 * q^61 - 4286151 * q^62 - 1040839 * q^64 - 914400 * q^65 + 346600 * q^67 - 5559624 * q^68 + 3094425 * q^70 - 4242240 * q^71 - 3145190 * q^73 - 4267710 * q^74 - 2510486 * q^76 + 4864500 * q^77 + 10110616 * q^79 - 1705545 * q^80 - 2141190 * q^82 + 644202 * q^83 + 101520 * q^85 + 2313270 * q^86 + 9374715 * q^88 + 6021000 * q^89 - 20872000 * q^91 + 14795802 * q^92 - 3751290 * q^94 + 747180 * q^95 + 4098670 * q^97 + 22779738 * 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 $

$2$	−7.59339	−0.671167	−0.335583	−	0.942010i	$-0.608933\pi$
$2$	−7.59339	−0.671167	−0.335583	+	0.942010i	$0.608933\pi$
$3$	0	0
$3$	0	0
$4$	−70.3405	−0.549535
$5$	65.8132	0.235461	0.117730	−	0.993046i	$-0.462438\pi$
$5$	65.8132	0.235461	0.117730	+	0.993046i	$0.462438\pi$
$6$	0	0
$7$	−738.405	−0.813676	−0.406838	−	0.913500i	$-0.633369\pi$
$7$	−738.405	−0.813676	−0.406838	+	0.913500i	$0.633369\pi$
$8$	1506.08	1.04000
$9$	0	0
$10$	−499.745	−0.158033
$11$	4961.26	1.12387	0.561937	−	0.827180i	$-0.310056\pi$
$11$	4961.26	1.12387	0.561937	+	0.827180i	$0.310056\pi$
$12$	0	0
$13$	5967.24	0.753306	0.376653	−	0.926354i	$-0.377075\pi$
$13$	5967.24	0.753306	0.376653	+	0.926354i	$0.377075\pi$
$14$	5606.99	0.546112
$15$	0	0
$16$	−2432.64	−0.148476
$17$	36651.6	1.80935	0.904674	−	0.426104i	$-0.140114\pi$
$17$	36651.6	1.80935	0.904674	+	0.426104i	$0.140114\pi$
$18$	0	0
$19$	22378.9	0.748518	0.374259	−	0.927324i	$-0.377897\pi$
$19$	22378.9	0.748518	0.374259	+	0.927324i	$0.377897\pi$
$20$	−4629.33	−0.129394
$21$	0	0
$22$	−37672.8	−0.754308
$23$	−51473.6	−0.882139	−0.441069	−	0.897473i	$-0.645401\pi$
$23$	−51473.6	−0.882139	−0.441069	+	0.897473i	$0.645401\pi$
$24$	0	0
$25$	−73793.6	−0.944558
$26$	−45311.5	−0.505594
$27$	0	0
$28$	51939.7	0.447143
$29$	−68495.7	−0.521519	−0.260760	−	0.965404i	$-0.583973\pi$
$29$	−68495.7	−0.521519	−0.260760	+	0.965404i	$0.583973\pi$
$30$	0	0
$31$	150655.	0.908275	0.454137	−	0.890932i	$-0.349948\pi$
$31$	150655.	0.908275	0.454137	+	0.890932i	$0.349948\pi$
$32$	−174306.	−0.940344
$33$	0	0
$34$	−278310.	−1.21437
$35$	−48596.8	−0.191589
$36$	0	0
$37$	489027.	1.58718	0.793591	−	0.608451i	$-0.208209\pi$
$37$	489027.	1.58718	0.793591	+	0.608451i	$0.208209\pi$
$38$	−169932.	−0.502380
$39$	0	0
$40$	99119.7	0.244878
$41$	590635.	1.33837	0.669184	−	0.743096i	$-0.266644\pi$
$41$	590635.	1.33837	0.669184	+	0.743096i	$0.266644\pi$
$42$	0	0
$43$	−842643.	−1.61623	−0.808117	−	0.589023i	$-0.799513\pi$
$43$	−842643.	−1.61623	−0.808117	+	0.589023i	$0.799513\pi$
$44$	−348978.	−0.617609
$45$	0	0
$46$	390859.	0.592062
$47$	1.22637e6	1.72297	0.861486	−	0.507782i	$-0.169534\pi$
$47$	1.22637e6	1.72297	0.861486	+	0.507782i	$0.169534\pi$
$48$	0	0
$49$	−278301.	−0.337932
$50$	560343.	0.633956
$51$	0	0
$52$	−419738.	−0.413968
$53$	958904.	0.884727	0.442364	−	0.896836i	$-0.354140\pi$
$53$	958904.	0.884727	0.442364	+	0.896836i	$0.354140\pi$
$54$	0	0
$55$	326517.	0.264628
$56$	−1.11209e6	−0.846220
$57$	0	0
$58$	520114.	0.350027
$59$	−316269.	−0.200482	−0.100241	−	0.994963i	$-0.531961\pi$
$59$	−316269.	−0.200482	−0.100241	+	0.994963i	$0.531961\pi$
$60$	0	0
$61$	−29722.9	−0.0167663	−0.00838315	−	0.999965i	$-0.502668\pi$
$61$	−29722.9	−0.0167663	−0.00838315	+	0.999965i	$0.502668\pi$
$62$	−1.14398e6	−0.609604
$63$	0	0
$64$	1.63495e6	0.779604
$65$	392723.	0.177374
$66$	0	0
$67$	293025.	0.119026	0.0595130	−	0.998228i	$-0.481045\pi$
$67$	293025.	0.119026	0.0595130	+	0.998228i	$0.481045\pi$
$68$	−2.57809e6	−0.994300
$69$	0	0
$70$	369014.	0.128588
$71$	714537.	0.236930	0.118465	−	0.992958i	$-0.462203\pi$
$71$	714537.	0.236930	0.118465	+	0.992958i	$0.462203\pi$
$72$	0	0
$73$	−3.96273e6	−1.19224	−0.596121	−	0.802894i	$-0.703292\pi$
$73$	−3.96273e6	−1.19224	−0.596121	+	0.802894i	$0.703292\pi$
$74$	−3.71337e6	−1.06526
$75$	0	0
$76$	−1.57415e6	−0.411337
$77$	−3.66342e6	−0.914470
$78$	0	0
$79$	2.53805e6	0.579168	0.289584	−	0.957153i	$-0.406483\pi$
$79$	2.53805e6	0.579168	0.289584	+	0.957153i	$0.406483\pi$
$80$	−160100.	−0.0349603
$81$	0	0
$82$	−4.48492e6	−0.898269
$83$	1.66311e6	0.319263	0.159631	−	0.987177i	$-0.448969\pi$
$83$	1.66311e6	0.319263	0.159631	+	0.987177i	$0.448969\pi$
$84$	0	0
$85$	2.41216e6	0.426030
$86$	6.39851e6	1.08476
$87$	0	0
$88$	7.47204e6	1.16883
$89$	4.64819e6	0.698906	0.349453	−	0.936954i	$-0.386368\pi$
$89$	4.64819e6	0.698906	0.349453	+	0.936954i	$0.386368\pi$
$90$	0	0
$91$	−4.40624e6	−0.612947
$92$	3.62068e6	0.484766
$93$	0	0
$94$	−9.31228e6	−1.15640
$95$	1.47283e6	0.176246
$96$	0	0
$97$	1.47010e7	1.63548	0.817738	−	0.575590i	$-0.195228\pi$
$97$	1.47010e7	1.63548	0.817738	+	0.575590i	$0.195228\pi$
$98$	2.11325e6	0.226809
$99$	0	0
$100$	5.19068e6	0.519068
$101$	9.28380e6	0.896604	0.448302	−	0.893882i	$-0.352029\pi$
$101$	9.28380e6	0.896604	0.448302	+	0.893882i	$0.352029\pi$
$102$	0	0
$103$	4.11049e6	0.370650	0.185325	−	0.982677i	$-0.440666\pi$
$103$	4.11049e6	0.370650	0.185325	+	0.982677i	$0.440666\pi$
$104$	8.98711e6	0.783436
$105$	0	0
$106$	−7.28133e6	−0.593800
$107$	−1.77931e7	−1.40414	−0.702068	−	0.712110i	$-0.747740\pi$
$107$	−1.77931e7	−1.40414	−0.702068	+	0.712110i	$0.747740\pi$
$108$	0	0
$109$	−1.72244e7	−1.27394	−0.636972	−	0.770887i	$-0.719813\pi$
$109$	−1.72244e7	−1.27394	−0.636972	+	0.770887i	$0.719813\pi$
$110$	−2.47937e6	−0.177610
$111$	0	0
$112$	1.79627e6	0.120812
$113$	−2.16751e6	−0.141315	−0.0706573	−	0.997501i	$-0.522510\pi$
$113$	−2.16751e6	−0.141315	−0.0706573	+	0.997501i	$0.522510\pi$
$114$	0	0
$115$	−3.38764e6	−0.207709
$116$	4.81802e6	0.286593
$117$	0	0
$118$	2.40156e6	0.134557
$119$	−2.70638e7	−1.47222
$120$	0	0
$121$	5.12697e6	0.263095
$122$	225698.	0.0112530
$123$	0	0
$124$	−1.05971e7	−0.499129
$125$	−9.99825e6	−0.457867
$126$	0	0
$127$	9.66827e6	0.418828	0.209414	−	0.977827i	$-0.432844\pi$
$127$	9.66827e6	0.418828	0.209414	+	0.977827i	$0.432844\pi$
$128$	9.89634e6	0.417100
$129$	0	0
$130$	−2.98210e6	−0.119048
$131$	−4.22431e7	−1.64175	−0.820873	−	0.571111i	$-0.806513\pi$
$131$	−4.22431e7	−1.64175	−0.820873	+	0.571111i	$0.806513\pi$
$132$	0	0
$133$	−1.65247e7	−0.609051
$134$	−2.22505e6	−0.0798864
$135$	0	0
$136$	5.52002e7	1.88172
$137$	−8.01451e6	−0.266290	−0.133145	−	0.991097i	$-0.542508\pi$
$137$	−8.01451e6	−0.266290	−0.133145	+	0.991097i	$0.542508\pi$
$138$	0	0
$139$	1.46010e7	0.461138	0.230569	−	0.973056i	$-0.425941\pi$
$139$	1.46010e7	0.461138	0.230569	+	0.973056i	$0.425941\pi$
$140$	3.41832e6	0.105285
$141$	0	0
$142$	−5.42576e6	−0.159020
$143$	2.96050e7	0.846622
$144$	0	0
$145$	−4.50792e6	−0.122797
$146$	3.00906e7	0.800194
$147$	0	0
$148$	−3.43984e7	−0.872212
$149$	4.07087e7	1.00817	0.504086	−	0.863653i	$-0.331829\pi$
$149$	4.07087e7	1.00817	0.504086	+	0.863653i	$0.331829\pi$
$150$	0	0
$151$	−5.29940e7	−1.25259	−0.626293	−	0.779588i	$-0.715428\pi$
$151$	−5.29940e7	−1.25259	−0.626293	+	0.779588i	$0.715428\pi$
$152$	3.37044e7	0.778456
$153$	0	0
$154$	2.78178e7	0.613762
$155$	9.91508e6	0.213863
$156$	0	0
$157$	4.10669e7	0.846922	0.423461	−	0.905914i	$-0.360815\pi$
$157$	4.10669e7	0.846922	0.423461	+	0.905914i	$0.360815\pi$
$158$	−1.92724e7	−0.388718
$159$	0	0
$160$	−1.14716e7	−0.221414
$161$	3.80083e7	0.717775
$162$	0	0
$163$	−4.33772e7	−0.784522	−0.392261	−	0.919854i	$-0.628307\pi$
$163$	−4.33772e7	−0.784522	−0.392261	+	0.919854i	$0.628307\pi$
$164$	−4.15456e7	−0.735480
$165$	0	0
$166$	−1.26287e7	−0.214279
$167$	3.86845e7	0.642731	0.321365	−	0.946955i	$-0.395858\pi$
$167$	3.86845e7	0.642731	0.321365	+	0.946955i	$0.395858\pi$
$168$	0	0
$169$	−2.71406e7	−0.432529
$170$	−1.83165e7	−0.285937
$171$	0	0
$172$	5.92719e7	0.888177
$173$	5.30547e7	0.779045	0.389523	−	0.921017i	$-0.372640\pi$
$173$	5.30547e7	0.779045	0.389523	+	0.921017i	$0.372640\pi$
$174$	0	0
$175$	5.44896e7	0.768564
$176$	−1.20689e7	−0.166869
$177$	0	0
$178$	−3.52955e7	−0.469082
$179$	−1.34132e8	−1.74802	−0.874012	−	0.485904i	$-0.838491\pi$
$179$	−1.34132e8	−1.74802	−0.874012	+	0.485904i	$0.838491\pi$
$180$	0	0
$181$	6.35105e6	0.0796105	0.0398053	−	0.999207i	$-0.487326\pi$
$181$	6.35105e6	0.0796105	0.0398053	+	0.999207i	$0.487326\pi$
$182$	3.34583e7	0.411390
$183$	0	0
$184$	−7.75231e7	−0.917422
$185$	3.21845e7	0.373719
$186$	0	0
$187$	1.81839e8	2.03348
$188$	−8.62633e7	−0.946833
$189$	0	0
$190$	−1.11838e7	−0.118291
$191$	−1.15843e8	−1.20297	−0.601483	−	0.798885i	$-0.705423\pi$
$191$	−1.15843e8	−1.20297	−0.601483	+	0.798885i	$0.705423\pi$
$192$	0	0
$193$	5.73856e7	0.574582	0.287291	−	0.957843i	$-0.407245\pi$
$193$	5.73856e7	0.574582	0.287291	+	0.957843i	$0.407245\pi$
$194$	−1.11630e8	−1.09768
$195$	0	0
$196$	1.95759e7	0.185705
$197$	1.04846e8	0.977059	0.488530	−	0.872547i	$-0.337533\pi$
$197$	1.04846e8	0.977059	0.488530	+	0.872547i	$0.337533\pi$
$198$	0	0
$199$	−2.10623e8	−1.89461	−0.947304	−	0.320335i	$-0.896205\pi$
$199$	−2.10623e8	−1.89461	−0.947304	+	0.320335i	$0.896205\pi$
$200$	−1.11139e8	−0.982337
$201$	0	0
$202$	−7.04955e7	−0.601771
$203$	5.05776e7	0.424348
$204$	0	0
$205$	3.88716e7	0.315133
$206$	−3.12125e7	−0.248768
$207$	0	0
$208$	−1.45161e7	−0.111848
$209$	1.11028e8	0.841240
$210$	0	0
$211$	−6.49193e7	−0.475757	−0.237878	−	0.971295i	$-0.576452\pi$
$211$	−6.49193e7	−0.475757	−0.237878	+	0.971295i	$0.576452\pi$
$212$	−6.74497e7	−0.486189
$213$	0	0
$214$	1.35110e8	0.942410
$215$	−5.54570e7	−0.380559
$216$	0	0
$217$	−1.11244e8	−0.739041
$218$	1.30791e8	0.855030
$219$	0	0
$220$	−2.29674e7	−0.145422
$221$	2.18709e8	1.36299
$222$	0	0
$223$	1.40879e8	0.850705	0.425352	−	0.905028i	$-0.360150\pi$
$223$	1.40879e8	0.850705	0.425352	+	0.905028i	$0.360150\pi$
$224$	1.28708e8	0.765135
$225$	0	0
$226$	1.64588e7	0.0948457
$227$	−3.40163e6	−0.0193017	−0.00965086	−	0.999953i	$-0.503072\pi$
$227$	−3.40163e6	−0.0193017	−0.00965086	+	0.999953i	$0.503072\pi$
$228$	0	0
$229$	−2.76169e7	−0.151967	−0.0759836	−	0.997109i	$-0.524210\pi$
$229$	−2.76169e7	−0.151967	−0.0759836	+	0.997109i	$0.524210\pi$
$230$	2.57237e7	0.139407
$231$	0	0
$232$	−1.03160e8	−0.542378
$233$	3.51458e8	1.82023	0.910117	−	0.414351i	$-0.135992\pi$
$233$	3.51458e8	1.82023	0.910117	+	0.414351i	$0.135992\pi$
$234$	0	0
$235$	8.07112e7	0.405692
$236$	2.22465e7	0.110172
$237$	0	0
$238$	2.05506e8	0.988107
$239$	−3.79195e8	−1.79668	−0.898339	−	0.439302i	$-0.855226\pi$
$239$	−3.79195e8	−1.79668	−0.898339	+	0.439302i	$0.855226\pi$
$240$	0	0
$241$	2.99276e7	0.137725	0.0688624	−	0.997626i	$-0.478063\pi$
$241$	2.99276e7	0.137725	0.0688624	+	0.997626i	$0.478063\pi$
$242$	−3.89311e7	−0.176581
$243$	0	0
$244$	2.09072e6	0.00921367
$245$	−1.83159e7	−0.0795696
$246$	0	0
$247$	1.33540e8	0.563863
$248$	2.26898e8	0.944602
$249$	0	0
$250$	7.59206e7	0.307305
$251$	−1.71456e7	−0.0684374	−0.0342187	−	0.999414i	$-0.510894\pi$
$251$	−1.71456e7	−0.0684374	−0.0342187	+	0.999414i	$0.510894\pi$
$252$	0	0
$253$	−2.55374e8	−0.991414
$254$	−7.34149e7	−0.281103
$255$	0	0
$256$	−2.84420e8	−1.05955
$257$	−9.79449e7	−0.359928	−0.179964	−	0.983673i	$-0.557598\pi$
$257$	−9.79449e7	−0.359928	−0.179964	+	0.983673i	$0.557598\pi$
$258$	0	0
$259$	−3.61100e8	−1.29145
$260$	−2.76243e7	−0.0974732
$261$	0	0
$262$	3.20768e8	1.10189
$263$	−8.92540e7	−0.302540	−0.151270	−	0.988492i	$-0.548336\pi$
$263$	−8.92540e7	−0.302540	−0.151270	+	0.988492i	$0.548336\pi$
$264$	0	0
$265$	6.31085e7	0.208318
$266$	1.25479e8	0.408775
$267$	0	0
$268$	−2.06115e7	−0.0654090
$269$	−5.34653e8	−1.67471	−0.837354	−	0.546662i	$-0.815898\pi$
$269$	−5.34653e8	−1.67471	−0.837354	+	0.546662i	$0.815898\pi$
$270$	0	0
$271$	2.11501e8	0.645535	0.322768	−	0.946478i	$-0.395387\pi$
$271$	2.11501e8	0.645535	0.322768	+	0.946478i	$0.395387\pi$
$272$	−8.91601e7	−0.268645
$273$	0	0
$274$	6.08573e7	0.178725
$275$	−3.66110e8	−1.06157
$276$	0	0
$277$	−3.24017e8	−0.915986	−0.457993	−	0.888956i	$-0.651432\pi$
$277$	−3.24017e8	−0.915986	−0.457993	+	0.888956i	$0.651432\pi$
$278$	−1.10871e8	−0.309501
$279$	0	0
$280$	−7.31905e7	−0.199251
$281$	4.91210e7	0.132067	0.0660336	−	0.997817i	$-0.478966\pi$
$281$	4.91210e7	0.132067	0.0660336	+	0.997817i	$0.478966\pi$
$282$	0	0
$283$	−4.03017e8	−1.05699	−0.528495	−	0.848936i	$-0.677244\pi$
$283$	−4.03017e8	−1.05699	−0.528495	+	0.848936i	$0.677244\pi$
$284$	−5.02609e7	−0.130202
$285$	0	0
$286$	−2.24803e8	−0.568225
$287$	−4.36128e8	−1.08900
$288$	0	0
$289$	9.33004e8	2.27374
$290$	3.42304e7	0.0824174
$291$	0	0
$292$	2.78740e8	0.655179
$293$	4.85906e8	1.12854	0.564268	−	0.825592i	$-0.309159\pi$
$293$	4.85906e8	1.12854	0.564268	+	0.825592i	$0.309159\pi$
$294$	0	0
$295$	−2.08147e7	−0.0472056
$296$	7.36512e8	1.65066
$297$	0	0
$298$	−3.09117e8	−0.676652
$299$	−3.07155e8	−0.664521
$300$	0	0
$301$	6.22212e8	1.31509
$302$	4.02404e8	0.840694
$303$	0	0
$304$	−5.44398e7	−0.111137
$305$	−1.95616e6	−0.00394780
$306$	0	0
$307$	2.82215e8	0.556668	0.278334	−	0.960484i	$-0.410218\pi$
$307$	2.82215e8	0.556668	0.278334	+	0.960484i	$0.410218\pi$
$308$	2.57687e8	0.502533
$309$	0	0
$310$	−7.52890e7	−0.143538
$311$	7.04440e7	0.132795	0.0663977	−	0.997793i	$-0.478849\pi$
$311$	7.04440e7	0.132795	0.0663977	+	0.997793i	$0.478849\pi$
$312$	0	0
$313$	−4.78096e8	−0.881273	−0.440636	−	0.897686i	$-0.645247\pi$
$313$	−4.78096e8	−0.881273	−0.440636	+	0.897686i	$0.645247\pi$
$314$	−3.11837e8	−0.568426
$315$	0	0
$316$	−1.78527e8	−0.318273
$317$	6.89470e8	1.21565	0.607824	−	0.794072i	$-0.292043\pi$
$317$	6.89470e8	1.21565	0.607824	+	0.794072i	$0.292043\pi$
$318$	0	0
$319$	−3.39825e8	−0.586123
$320$	1.07601e8	0.183566
$321$	0	0
$322$	−2.88612e8	−0.481747
$323$	8.20225e8	1.35433
$324$	0	0
$325$	−4.40344e8	−0.711542
$326$	3.29380e8	0.526545
$327$	0	0
$328$	8.89541e8	1.39190
$329$	−9.05555e8	−1.40194
$330$	0	0
$331$	5.36833e8	0.813657	0.406829	−	0.913504i	$-0.366635\pi$
$331$	5.36833e8	0.813657	0.406829	+	0.913504i	$0.366635\pi$
$332$	−1.16984e8	−0.175446
$333$	0	0
$334$	−2.93746e8	−0.431379
$335$	1.92849e7	0.0280259
$336$	0	0
$337$	−1.24011e9	−1.76505	−0.882524	−	0.470268i	$-0.844157\pi$
$337$	−1.24011e9	−1.76505	−0.882524	+	0.470268i	$0.844157\pi$
$338$	2.06089e8	0.290299
$339$	0	0
$340$	−1.69673e8	−0.234119
$341$	7.47438e8	1.02079
$342$	0	0
$343$	8.13607e8	1.08864
$344$	−1.26908e9	−1.68088
$345$	0	0
$346$	−4.02865e8	−0.522869
$347$	4.86766e7	0.0625413	0.0312706	−	0.999511i	$-0.490045\pi$
$347$	4.86766e7	0.0625413	0.0312706	+	0.999511i	$0.490045\pi$
$348$	0	0
$349$	1.23665e9	1.55725	0.778626	−	0.627488i	$-0.215917\pi$
$349$	1.23665e9	1.55725	0.778626	+	0.627488i	$0.215917\pi$
$350$	−4.13760e8	−0.515835
$351$	0	0
$352$	−8.64777e8	−1.05683
$353$	6.06858e7	0.0734304	0.0367152	−	0.999326i	$-0.488311\pi$
$353$	6.06858e7	0.0734304	0.0367152	+	0.999326i	$0.488311\pi$
$354$	0	0
$355$	4.70260e7	0.0557878
$356$	−3.26956e8	−0.384073
$357$	0	0
$358$	1.01852e9	1.17322
$359$	−5.36061e8	−0.611482	−0.305741	−	0.952115i	$-0.598904\pi$
$359$	−5.36061e8	−0.611482	−0.305741	+	0.952115i	$0.598904\pi$
$360$	0	0
$361$	−3.93055e8	−0.439722
$362$	−4.82260e7	−0.0534320
$363$	0	0
$364$	3.09937e8	0.336836
$365$	−2.60800e8	−0.280726
$366$	0	0
$367$	−4.31327e8	−0.455487	−0.227743	−	0.973721i	$-0.573135\pi$
$367$	−4.31327e8	−0.455487	−0.227743	+	0.973721i	$0.573135\pi$
$368$	1.25216e8	0.130977
$369$	0	0
$370$	−2.44389e8	−0.250828
$371$	−7.08059e8	−0.719881
$372$	0	0
$373$	2.69689e8	0.269080	0.134540	−	0.990908i	$-0.457044\pi$
$373$	2.69689e8	0.269080	0.134540	+	0.990908i	$0.457044\pi$
$374$	−1.38077e9	−1.36481
$375$	0	0
$376$	1.84700e9	1.79188
$377$	−4.08730e8	−0.392864
$378$	0	0
$379$	−8.03807e8	−0.758429	−0.379214	−	0.925309i	$-0.623806\pi$
$379$	−8.03807e8	−0.758429	−0.379214	+	0.925309i	$0.623806\pi$
$380$	−1.03600e8	−0.0968535
$381$	0	0
$382$	8.79642e8	0.807392
$383$	9.83923e8	0.894882	0.447441	−	0.894314i	$-0.352336\pi$
$383$	9.83923e8	0.894882	0.447441	+	0.894314i	$0.352336\pi$
$384$	0	0
$385$	−2.41102e8	−0.215322
$386$	−4.35751e8	−0.385641
$387$	0	0
$388$	−1.03407e9	−0.898752
$389$	−2.16250e8	−0.186265	−0.0931326	−	0.995654i	$-0.529688\pi$
$389$	−2.16250e8	−0.186265	−0.0931326	+	0.995654i	$0.529688\pi$
$390$	0	0
$391$	−1.88659e9	−1.59610
$392$	−4.19143e8	−0.351448
$393$	0	0
$394$	−7.96138e8	−0.655770
$395$	1.67037e8	0.136371
$396$	0	0
$397$	4.33533e8	0.347741	0.173870	−	0.984769i	$-0.444373\pi$
$397$	4.33533e8	0.347741	0.173870	+	0.984769i	$0.444373\pi$
$398$	1.59934e9	1.27160
$399$	0	0
$400$	1.79513e8	0.140245
$401$	2.42713e9	1.87970	0.939849	−	0.341591i	$-0.110966\pi$
$401$	2.42713e9	1.87970	0.939849	+	0.341591i	$0.110966\pi$
$402$	0	0
$403$	8.98993e8	0.684209
$404$	−6.53027e8	−0.492715
$405$	0	0
$406$	−3.84055e8	−0.284808
$407$	2.42619e9	1.78379
$408$	0	0
$409$	9.18976e8	0.664160	0.332080	−	0.943251i	$-0.392250\pi$
$409$	9.18976e8	0.664160	0.332080	+	0.943251i	$0.392250\pi$
$410$	−2.95167e8	−0.211507
$411$	0	0
$412$	−2.89134e8	−0.203685
$413$	2.33535e8	0.163127
$414$	0	0
$415$	1.09455e8	0.0751738
$416$	−1.04012e9	−0.708367
$417$	0	0
$418$	−8.43077e8	−0.564612
$419$	1.10024e9	0.730699	0.365349	−	0.930870i	$-0.380950\pi$
$419$	1.10024e9	0.730699	0.365349	+	0.930870i	$0.380950\pi$
$420$	0	0
$421$	−2.14909e9	−1.40367	−0.701837	−	0.712337i	$-0.747637\pi$
$421$	−2.14909e9	−1.40367	−0.701837	+	0.712337i	$0.747637\pi$
$422$	4.92957e8	0.319312
$423$	0	0
$424$	1.44418e9	0.920113
$425$	−2.70466e9	−1.70904
$426$	0	0
$427$	2.19476e7	0.0136423
$428$	1.25158e9	0.771622
$429$	0	0
$430$	4.21107e8	0.255419
$431$	1.61850e9	0.973737	0.486869	−	0.873475i	$-0.338139\pi$
$431$	1.61850e9	0.973737	0.486869	+	0.873475i	$0.338139\pi$
$432$	0	0
$433$	1.16527e9	0.689794	0.344897	−	0.938641i	$-0.387914\pi$
$433$	1.16527e9	0.689794	0.344897	+	0.938641i	$0.387914\pi$
$434$	8.44721e8	0.496020
$435$	0	0
$436$	1.21157e9	0.700077
$437$	−1.15192e9	−0.660296
$438$	0	0
$439$	2.08096e9	1.17392	0.586958	−	0.809617i	$-0.300325\pi$
$439$	2.08096e9	1.17392	0.586958	+	0.809617i	$0.300325\pi$
$440$	4.91759e8	0.275212
$441$	0	0
$442$	−1.66074e9	−0.914796
$443$	2.23100e9	1.21923	0.609616	−	0.792697i	$-0.291324\pi$
$443$	2.23100e9	1.21923	0.609616	+	0.792697i	$0.291324\pi$
$444$	0	0
$445$	3.05912e8	0.164565
$446$	−1.06975e9	−0.570965
$447$	0	0
$448$	−1.20725e9	−0.634345
$449$	−2.52631e9	−1.31712	−0.658559	−	0.752529i	$-0.728834\pi$
$449$	−2.52631e9	−1.31712	−0.658559	+	0.752529i	$0.728834\pi$
$450$	0	0
$451$	2.93030e9	1.50416
$452$	1.52464e8	0.0776573
$453$	0	0
$454$	2.58299e7	0.0129547
$455$	−2.89989e8	−0.144325
$456$	0	0
$457$	−1.87627e9	−0.919579	−0.459790	−	0.888028i	$-0.652075\pi$
$457$	−1.87627e9	−0.919579	−0.459790	+	0.888028i	$0.652075\pi$
$458$	2.09705e8	0.101995
$459$	0	0
$460$	2.38288e8	0.114143
$461$	−1.97321e9	−0.938036	−0.469018	−	0.883189i	$-0.655392\pi$
$461$	−1.97321e9	−0.938036	−0.469018	+	0.883189i	$0.655392\pi$
$462$	0	0
$463$	−1.60344e9	−0.750790	−0.375395	−	0.926865i	$-0.622493\pi$
$463$	−1.60344e9	−0.750790	−0.375395	+	0.926865i	$0.622493\pi$
$464$	1.66625e8	0.0774333
$465$	0	0
$466$	−2.66875e9	−1.22168
$467$	−1.32390e9	−0.601515	−0.300757	−	0.953701i	$-0.597239\pi$
$467$	−1.32390e9	−0.601515	−0.300757	+	0.953701i	$0.597239\pi$
$468$	0	0
$469$	−2.16371e8	−0.0968486
$470$	−6.12871e8	−0.272287
$471$	0	0
$472$	−4.76326e8	−0.208501
$473$	−4.18057e9	−1.81644
$474$	0	0
$475$	−1.65142e9	−0.707018
$476$	1.90368e9	0.809038
$477$	0	0
$478$	2.87938e9	1.20587
$479$	−2.44225e9	−1.01535	−0.507675	−	0.861548i	$-0.669495\pi$
$479$	−2.44225e9	−1.01535	−0.507675	+	0.861548i	$0.669495\pi$
$480$	0	0
$481$	2.91814e9	1.19563
$482$	−2.27252e8	−0.0924363
$483$	0	0
$484$	−3.60634e8	−0.144580
$485$	9.67517e8	0.385090
$486$	0	0
$487$	−1.37309e7	−0.00538701	−0.00269350	−	0.999996i	$-0.500857\pi$
$487$	−1.37309e7	−0.00538701	−0.00269350	+	0.999996i	$0.500857\pi$
$488$	−4.47650e7	−0.0174369
$489$	0	0
$490$	1.39080e8	0.0534045
$491$	−6.32242e8	−0.241045	−0.120522	−	0.992711i	$-0.538457\pi$
$491$	−6.32242e8	−0.241045	−0.120522	+	0.992711i	$0.538457\pi$
$492$	0	0
$493$	−2.51048e9	−0.943610
$494$	−1.01402e9	−0.378446
$495$	0	0
$496$	−3.66488e8	−0.134857
$497$	−5.27618e8	−0.192784
$498$	0	0
$499$	−2.43365e9	−0.876812	−0.438406	−	0.898777i	$-0.644457\pi$
$499$	−2.43365e9	−0.876812	−0.438406	+	0.898777i	$0.644457\pi$
$500$	7.03282e8	0.251614
$501$	0	0
$502$	1.30193e8	0.0459329
$503$	2.00281e9	0.701700	0.350850	−	0.936432i	$-0.385893\pi$
$503$	2.00281e9	0.701700	0.350850	+	0.936432i	$0.385893\pi$
$504$	0	0
$505$	6.10997e8	0.211115
$506$	1.93915e9	0.665404
$507$	0	0
$508$	−6.80071e8	−0.230161
$509$	−5.28309e8	−0.177573	−0.0887863	−	0.996051i	$-0.528299\pi$
$509$	−5.28309e8	−0.177573	−0.0887863	+	0.996051i	$0.528299\pi$
$510$	0	0
$511$	2.92610e9	0.970099
$512$	8.92980e8	0.294034
$513$	0	0
$514$	7.43734e8	0.241572
$515$	2.70525e8	0.0872733
$516$	0	0
$517$	6.08433e9	1.93640
$518$	2.74197e9	0.866780
$519$	0	0
$520$	5.91471e8	0.184468
$521$	−1.76283e9	−0.546109	−0.273055	−	0.961999i	$-0.588034\pi$
$521$	−1.76283e9	−0.546109	−0.273055	+	0.961999i	$0.588034\pi$
$522$	0	0
$523$	−3.39318e9	−1.03717	−0.518586	−	0.855026i	$-0.673541\pi$
$523$	−3.39318e9	−1.03717	−0.518586	+	0.855026i	$0.673541\pi$
$524$	2.97140e9	0.902197
$525$	0	0
$526$	6.77740e8	0.203055
$527$	5.52175e9	1.64339
$528$	0	0
$529$	−7.55295e8	−0.221831
$530$	−4.79208e8	−0.139816
$531$	0	0
$532$	1.16236e9	0.334695
$533$	3.52446e9	1.00820
$534$	0	0
$535$	−1.17102e9	−0.330619
$536$	4.41317e8	0.123787
$537$	0	0
$538$	4.05983e9	1.12401
$539$	−1.38073e9	−0.379793
$540$	0	0
$541$	−3.83907e9	−1.04240	−0.521202	−	0.853433i	$-0.674516\pi$
$541$	−3.83907e9	−1.04240	−0.521202	+	0.853433i	$0.674516\pi$
$542$	−1.60601e9	−0.433262
$543$	0	0
$544$	−6.38859e9	−1.70141
$545$	−1.13359e9	−0.299964
$546$	0	0
$547$	4.31235e9	1.12657	0.563285	−	0.826263i	$-0.309537\pi$
$547$	4.31235e9	1.12657	0.563285	+	0.826263i	$0.309537\pi$
$548$	5.63744e8	0.146336
$549$	0	0
$550$	2.78001e9	0.712488
$551$	−1.53286e9	−0.390366
$552$	0	0
$553$	−1.87410e9	−0.471255
$554$	2.46039e9	0.614779
$555$	0	0
$556$	−1.02704e9	−0.253412
$557$	−1.84109e9	−0.451421	−0.225710	−	0.974194i	$-0.572470\pi$
$557$	−1.84109e9	−0.451421	−0.225710	+	0.974194i	$0.572470\pi$
$558$	0	0
$559$	−5.02825e9	−1.21752
$560$	1.18218e8	0.0284464
$561$	0	0
$562$	−3.72994e8	−0.0886391
$563$	−1.16055e9	−0.274085	−0.137042	−	0.990565i	$-0.543760\pi$
$563$	−1.16055e9	−0.274085	−0.137042	+	0.990565i	$0.543760\pi$
$564$	0	0
$565$	−1.42651e8	−0.0332740
$566$	3.06027e9	0.709417
$567$	0	0
$568$	1.07615e9	0.246407
$569$	7.02920e8	0.159960	0.0799802	−	0.996796i	$-0.474514\pi$
$569$	7.02920e8	0.159960	0.0799802	+	0.996796i	$0.474514\pi$
$570$	0	0
$571$	−7.95929e8	−0.178915	−0.0894577	−	0.995991i	$-0.528513\pi$
$571$	−7.95929e8	−0.178915	−0.0894577	+	0.995991i	$0.528513\pi$
$572$	−2.08243e9	−0.465249
$573$	0	0
$574$	3.31169e9	0.730900
$575$	3.79842e9	0.833232
$576$	0	0
$577$	−4.44712e9	−0.963749	−0.481874	−	0.876240i	$-0.660044\pi$
$577$	−4.44712e9	−0.963749	−0.481874	+	0.876240i	$0.660044\pi$
$578$	−7.08466e9	−1.52606
$579$	0	0
$580$	3.17090e8	0.0674814
$581$	−1.22805e9	−0.259776
$582$	0	0
$583$	4.75737e9	0.994323
$584$	−5.96818e9	−1.23993
$585$	0	0
$586$	−3.68967e9	−0.757436
$587$	5.50846e9	1.12408	0.562039	−	0.827111i	$-0.310017\pi$
$587$	5.50846e9	1.12408	0.562039	+	0.827111i	$0.310017\pi$
$588$	0	0
$589$	3.37150e9	0.679859
$590$	1.58054e8	0.0316828
$591$	0	0
$592$	−1.18962e9	−0.235659
$593$	−8.05850e9	−1.58695	−0.793475	−	0.608603i	$-0.791730\pi$
$593$	−8.05850e9	−1.58695	−0.793475	+	0.608603i	$0.791730\pi$
$594$	0	0
$595$	−1.78115e9	−0.346650
$596$	−2.86347e9	−0.554026
$597$	0	0
$598$	2.33235e9	0.446004
$599$	−3.39566e9	−0.645551	−0.322775	−	0.946476i	$-0.604616\pi$
$599$	−3.39566e9	−0.645551	−0.322775	+	0.946476i	$0.604616\pi$
$600$	0	0
$601$	1.35177e9	0.254005	0.127002	−	0.991902i	$-0.459464\pi$
$601$	1.35177e9	0.254005	0.127002	+	0.991902i	$0.459464\pi$
$602$	−4.72469e9	−0.882645
$603$	0	0
$604$	3.72762e9	0.688339
$605$	3.37423e8	0.0619485
$606$	0	0
$607$	4.91297e9	0.891627	0.445814	−	0.895126i	$-0.352914\pi$
$607$	4.91297e9	0.891627	0.445814	+	0.895126i	$0.352914\pi$
$608$	−3.90078e9	−0.703864
$609$	0	0
$610$	1.48539e7	0.00264963
$611$	7.31802e9	1.29793
$612$	0	0
$613$	−2.06110e9	−0.361399	−0.180700	−	0.983538i	$-0.557836\pi$
$613$	−2.06110e9	−0.361399	−0.180700	+	0.983538i	$0.557836\pi$
$614$	−2.14297e9	−0.373617
$615$	0	0
$616$	−5.51739e9	−0.951045
$617$	−3.72277e9	−0.638071	−0.319035	−	0.947743i	$-0.603359\pi$
$617$	−3.72277e9	−0.638071	−0.319035	+	0.947743i	$0.603359\pi$
$618$	0	0
$619$	3.30066e9	0.559349	0.279675	−	0.960095i	$-0.409773\pi$
$619$	3.30066e9	0.559349	0.279675	+	0.960095i	$0.409773\pi$
$620$	−6.97432e8	−0.117525
$621$	0	0
$622$	−5.34909e8	−0.0891278
$623$	−3.43224e9	−0.568683
$624$	0	0
$625$	5.10711e9	0.836749
$626$	3.63037e9	0.591481
$627$	0	0
$628$	−2.88867e9	−0.465413
$629$	1.79236e10	2.87177
$630$	0	0
$631$	5.12709e9	0.812397	0.406199	−	0.913785i	$-0.366854\pi$
$631$	5.12709e9	0.812397	0.406199	+	0.913785i	$0.366854\pi$
$632$	3.82249e9	0.602333
$633$	0	0
$634$	−5.23541e9	−0.815903
$635$	6.36300e8	0.0986174
$636$	0	0
$637$	−1.66069e9	−0.254566
$638$	2.58043e9	0.393386
$639$	0	0
$640$	6.51310e8	0.0982105
$641$	−4.51842e9	−0.677615	−0.338808	−	0.940856i	$-0.610024\pi$
$641$	−4.51842e9	−0.677615	−0.338808	+	0.940856i	$0.610024\pi$
$642$	0	0
$643$	−9.70600e9	−1.43980	−0.719900	−	0.694078i	$-0.755812\pi$
$643$	−9.70600e9	−1.43980	−0.719900	+	0.694078i	$0.755812\pi$
$644$	−2.67353e9	−0.394443
$645$	0	0
$646$	−6.22829e9	−0.908981
$647$	−6.12536e9	−0.889132	−0.444566	−	0.895746i	$-0.646642\pi$
$647$	−6.12536e9	−0.889132	−0.444566	+	0.895746i	$0.646642\pi$
$648$	0	0
$649$	−1.56910e9	−0.225317
$650$	3.34370e9	0.477563
$651$	0	0
$652$	3.05117e9	0.431122
$653$	1.30103e10	1.82849	0.914243	−	0.405166i	$-0.132786\pi$
$653$	1.30103e10	1.82849	0.914243	+	0.405166i	$0.132786\pi$
$654$	0	0
$655$	−2.78015e9	−0.386566
$656$	−1.43680e9	−0.198716
$657$	0	0
$658$	6.87623e9	0.940936
$659$	1.23137e10	1.67606	0.838032	−	0.545621i	$-0.183706\pi$
$659$	1.23137e10	1.67606	0.838032	+	0.545621i	$0.183706\pi$
$660$	0	0
$661$	6.97776e9	0.939746	0.469873	−	0.882734i	$-0.344300\pi$
$661$	6.97776e9	0.939746	0.469873	+	0.882734i	$0.344300\pi$
$662$	−4.07638e9	−0.546100
$663$	0	0
$664$	2.50477e9	0.332032
$665$	−1.08755e9	−0.143407
$666$	0	0
$667$	3.52572e9	0.460053
$668$	−2.72108e9	−0.353203
$669$	0	0
$670$	−1.46438e8	−0.0188101
$671$	−1.47463e8	−0.0188432
$672$	0	0
$673$	−1.06462e10	−1.34630	−0.673152	−	0.739504i	$-0.735060\pi$
$673$	−1.06462e10	−1.34630	−0.673152	+	0.739504i	$0.735060\pi$
$674$	9.41665e9	1.18464
$675$	0	0
$676$	1.90908e9	0.237690
$677$	4.33348e9	0.536756	0.268378	−	0.963314i	$-0.413512\pi$
$677$	4.33348e9	0.536756	0.268378	+	0.963314i	$0.413512\pi$
$678$	0	0
$679$	−1.08553e10	−1.33075
$680$	3.63290e9	0.443070
$681$	0	0
$682$	−5.67559e9	−0.685118
$683$	−6.16980e9	−0.740966	−0.370483	−	0.928839i	$-0.620808\pi$
$683$	−6.16980e9	−0.740966	−0.370483	+	0.928839i	$0.620808\pi$
$684$	0	0
$685$	−5.27461e8	−0.0627008
$686$	−6.17803e9	−0.730661
$687$	0	0
$688$	2.04984e9	0.239972
$689$	5.72201e9	0.666471
$690$	0	0
$691$	7.57273e9	0.873131	0.436565	−	0.899672i	$-0.356195\pi$
$691$	7.57273e9	0.873131	0.436565	+	0.899672i	$0.356195\pi$
$692$	−3.73190e9	−0.428113
$693$	0	0
$694$	−3.69620e8	−0.0419756
$695$	9.60939e8	0.108580
$696$	0	0
$697$	2.16477e10	2.42158
$698$	−9.39039e9	−1.04518
$699$	0	0
$700$	−3.83282e9	−0.422353
$701$	−4.42696e9	−0.485391	−0.242696	−	0.970102i	$-0.578032\pi$
$701$	−4.42696e9	−0.485391	−0.242696	+	0.970102i	$0.578032\pi$
$702$	0	0
$703$	1.09439e10	1.18803
$704$	8.11141e9	0.876178
$705$	0	0
$706$	−4.60811e8	−0.0492841
$707$	−6.85520e9	−0.729545
$708$	0	0
$709$	−1.13232e10	−1.19319	−0.596594	−	0.802544i	$-0.703480\pi$
$709$	−1.13232e10	−1.19319	−0.596594	+	0.802544i	$0.703480\pi$
$710$	−3.57087e8	−0.0374429
$711$	0	0
$712$	7.00052e9	0.726859
$713$	−7.75474e9	−0.801224
$714$	0	0
$715$	1.94840e9	0.199346
$716$	9.43492e9	0.960601
$717$	0	0
$718$	4.07052e9	0.410407
$719$	−1.02156e10	−1.02497	−0.512487	−	0.858695i	$-0.671276\pi$
$719$	−1.02156e10	−1.02497	−0.512487	+	0.858695i	$0.671276\pi$
$720$	0	0
$721$	−3.03521e9	−0.301589
$722$	2.98462e9	0.295127
$723$	0	0
$724$	−4.46736e8	−0.0437488
$725$	5.05455e9	0.492605
$726$	0	0
$727$	−1.95355e10	−1.88562	−0.942811	−	0.333329i	$-0.891828\pi$
$727$	−1.95355e10	−1.88562	−0.942811	+	0.333329i	$0.891828\pi$
$728$	−6.63613e9	−0.637463
$729$	0	0
$730$	1.98036e9	0.188414
$731$	−3.08842e10	−2.92433
$732$	0	0
$733$	−1.32745e10	−1.24496	−0.622478	−	0.782637i	$-0.713874\pi$
$733$	−1.32745e10	−1.24496	−0.622478	+	0.782637i	$0.713874\pi$
$734$	3.27523e9	0.305708
$735$	0	0
$736$	8.97214e9	0.829514
$737$	1.45377e9	0.133770
$738$	0	0
$739$	1.69495e9	0.154490	0.0772452	−	0.997012i	$-0.475388\pi$
$739$	1.69495e9	0.154490	0.0772452	+	0.997012i	$0.475388\pi$
$740$	−2.26387e9	−0.205372
$741$	0	0
$742$	5.37657e9	0.483160
$743$	−9.34493e9	−0.835824	−0.417912	−	0.908487i	$-0.637238\pi$
$743$	−9.34493e9	−0.835824	−0.417912	+	0.908487i	$0.637238\pi$
$744$	0	0
$745$	2.67917e9	0.237385
$746$	−2.04785e9	−0.180598
$747$	0	0
$748$	−1.27906e10	−1.11747
$749$	1.31385e10	1.14251
$750$	0	0
$751$	−1.31194e10	−1.13025	−0.565124	−	0.825006i	$-0.691172\pi$
$751$	−1.31194e10	−1.13025	−0.565124	+	0.825006i	$0.691172\pi$
$752$	−2.98330e9	−0.255820
$753$	0	0
$754$	3.10365e9	0.263677
$755$	−3.48770e9	−0.294934
$756$	0	0
$757$	5.72593e9	0.479745	0.239873	−	0.970804i	$-0.422894\pi$
$757$	5.72593e9	0.479745	0.239873	+	0.970804i	$0.422894\pi$
$758$	6.10362e9	0.509032
$759$	0	0
$760$	2.21819e9	0.183296
$761$	−1.47621e10	−1.21423	−0.607115	−	0.794614i	$-0.707673\pi$
$761$	−1.47621e10	−1.21423	−0.607115	+	0.794614i	$0.707673\pi$
$762$	0	0
$763$	1.27186e10	1.03658
$764$	8.14847e9	0.661072
$765$	0	0
$766$	−7.47131e9	−0.600615
$767$	−1.88726e9	−0.151024
$768$	0	0
$769$	5.63551e9	0.446880	0.223440	−	0.974718i	$-0.428271\pi$
$769$	5.63551e9	0.446880	0.223440	+	0.974718i	$0.428271\pi$
$770$	1.83078e9	0.144517
$771$	0	0
$772$	−4.03653e9	−0.315753
$773$	4.65338e9	0.362360	0.181180	−	0.983450i	$-0.442008\pi$
$773$	4.65338e9	0.362360	0.181180	+	0.983450i	$0.442008\pi$
$774$	0	0
$775$	−1.11174e10	−0.857918
$776$	2.21408e10	1.70089
$777$	0	0
$778$	1.64207e9	0.125015
$779$	1.32178e10	1.00179
$780$	0	0
$781$	3.54501e9	0.266280
$782$	1.43256e10	1.07125
$783$	0	0
$784$	6.77006e8	0.0501749
$785$	2.70275e9	0.199417
$786$	0	0
$787$	−1.23587e10	−0.903780	−0.451890	−	0.892074i	$-0.649250\pi$
$787$	−1.23587e10	−0.903780	−0.451890	+	0.892074i	$0.649250\pi$
$788$	−7.37493e9	−0.536928
$789$	0	0
$790$	−1.26838e9	−0.0915279
$791$	1.60050e9	0.114984
$792$	0	0
$793$	−1.77364e8	−0.0126302
$794$	−3.29198e9	−0.233392
$795$	0	0
$796$	1.48153e10	1.04115
$797$	1.70444e10	1.19256	0.596278	−	0.802778i	$-0.296646\pi$
$797$	1.70444e10	1.19256	0.596278	+	0.802778i	$0.296646\pi$
$798$	0	0
$799$	4.49484e10	3.11746
$800$	1.28627e10	0.888210
$801$	0	0
$802$	−1.84301e10	−1.26159
$803$	−1.96602e10	−1.33993
$804$	0	0
$805$	2.50145e9	0.169008
$806$	−6.82640e9	−0.459218
$807$	0	0
$808$	1.39821e10	0.932465
$809$	1.00497e10	0.667315	0.333658	−	0.942694i	$-0.391717\pi$
$809$	1.00497e10	0.667315	0.333658	+	0.942694i	$0.391717\pi$
$810$	0	0
$811$	5.26821e9	0.346809	0.173404	−	0.984851i	$-0.444523\pi$
$811$	5.26821e9	0.346809	0.173404	+	0.984851i	$0.444523\pi$
$812$	−3.55765e9	−0.233194
$813$	0	0
$814$	−1.84230e10	−1.19722
$815$	−2.85480e9	−0.184724
$816$	0	0
$817$	−1.88575e10	−1.20978
$818$	−6.97814e9	−0.445762
$819$	0	0
$820$	−2.73425e9	−0.173177
$821$	1.82202e10	1.14909	0.574543	−	0.818474i	$-0.305180\pi$
$821$	1.82202e10	1.14909	0.574543	+	0.818474i	$0.305180\pi$
$822$	0	0
$823$	−1.35868e9	−0.0849603	−0.0424802	−	0.999097i	$-0.513526\pi$
$823$	−1.35868e9	−0.0849603	−0.0424802	+	0.999097i	$0.513526\pi$
$824$	6.19071e9	0.385474
$825$	0	0
$826$	−1.77332e9	−0.109486
$827$	8.90171e9	0.547273	0.273637	−	0.961833i	$-0.411773\pi$
$827$	8.90171e9	0.547273	0.273637	+	0.961833i	$0.411773\pi$
$828$	0	0
$829$	6.14205e9	0.374432	0.187216	−	0.982319i	$-0.440054\pi$
$829$	6.14205e9	0.374432	0.187216	+	0.982319i	$0.440054\pi$
$830$	−8.31133e8	−0.0504542
$831$	0	0
$832$	9.75613e9	0.587281
$833$	−1.02002e10	−0.611436
$834$	0	0
$835$	2.54595e9	0.151338
$836$	−7.80975e9	−0.462291
$837$	0	0
$838$	−8.35455e9	−0.490421
$839$	1.05464e10	0.616504	0.308252	−	0.951305i	$-0.400256\pi$
$839$	1.05464e10	0.616504	0.308252	+	0.951305i	$0.400256\pi$
$840$	0	0
$841$	−1.25582e10	−0.728018
$842$	1.63188e10	0.942100
$843$	0	0
$844$	4.56645e9	0.261445
$845$	−1.78621e9	−0.101844
$846$	0	0
$847$	−3.78578e9	−0.214074
$848$	−2.33266e9	−0.131361
$849$	0	0
$850$	2.05375e10	1.14705
$851$	−2.51720e10	−1.40012
$852$	0	0
$853$	3.41878e9	0.188604	0.0943018	−	0.995544i	$-0.469938\pi$
$853$	3.41878e9	0.188604	0.0943018	+	0.995544i	$0.469938\pi$
$854$	−1.66656e8	−0.00915628
$855$	0	0
$856$	−2.67978e10	−1.46030
$857$	−2.17037e10	−1.17788	−0.588941	−	0.808176i	$-0.700455\pi$
$857$	−2.17037e10	−1.17788	−0.588941	+	0.808176i	$0.700455\pi$
$858$	0	0
$859$	−2.19211e10	−1.18001	−0.590005	−	0.807400i	$-0.700874\pi$
$859$	−2.19211e10	−1.18001	−0.590005	+	0.807400i	$0.700874\pi$
$860$	3.90088e9	0.209131
$861$	0	0
$862$	−1.22899e10	−0.653540
$863$	−1.35347e10	−0.716822	−0.358411	−	0.933564i	$-0.616681\pi$
$863$	−1.35347e10	−0.716822	−0.358411	+	0.933564i	$0.616681\pi$
$864$	0	0
$865$	3.49170e9	0.183434
$866$	−8.84836e9	−0.462967
$867$	0	0
$868$	7.82497e9	0.406129
$869$	1.25919e10	0.650912
$870$	0	0
$871$	1.74855e9	0.0896631
$872$	−2.59412e10	−1.32490
$873$	0	0
$874$	8.74701e9	0.443169
$875$	7.38276e9	0.372555
$876$	0	0
$877$	2.45197e10	1.22749	0.613743	−	0.789506i	$-0.289663\pi$
$877$	2.45197e10	1.22749	0.613743	+	0.789506i	$0.289663\pi$
$878$	−1.58015e10	−0.787894
$879$	0	0
$880$	−7.94296e8	−0.0392910
$881$	−2.13536e10	−1.05210	−0.526048	−	0.850455i	$-0.676327\pi$
$881$	−2.13536e10	−1.05210	−0.526048	+	0.850455i	$0.676327\pi$
$882$	0	0
$883$	−1.64504e9	−0.0804109	−0.0402054	−	0.999191i	$-0.512801\pi$
$883$	−1.64504e9	−0.0804109	−0.0402054	+	0.999191i	$0.512801\pi$
$884$	−1.53841e10	−0.749013
$885$	0	0
$886$	−1.69408e10	−0.818309
$887$	9.65301e9	0.464441	0.232220	−	0.972663i	$-0.425401\pi$
$887$	9.65301e9	0.464441	0.232220	+	0.972663i	$0.425401\pi$
$888$	0	0
$889$	−7.13909e9	−0.340790
$890$	−2.32291e9	−0.110450
$891$	0	0
$892$	−9.90949e9	−0.467492
$893$	2.74448e10	1.28967
$894$	0	0
$895$	−8.82767e9	−0.411591
$896$	−7.30751e9	−0.339384
$897$	0	0
$898$	1.91833e10	0.884006
$899$	−1.03192e10	−0.473683
$900$	0	0
$901$	3.51454e10	1.60078
$902$	−2.22509e10	−1.00954
$903$	0	0
$904$	−3.26444e9	−0.146967
$905$	4.17983e8	0.0187451
$906$	0	0
$907$	−8.05417e9	−0.358423	−0.179211	−	0.983811i	$-0.557355\pi$
$907$	−8.05417e9	−0.358423	−0.179211	+	0.983811i	$0.557355\pi$
$908$	2.39272e8	0.0106070
$909$	0	0
$910$	2.20200e9	0.0968661
$911$	5.73799e9	0.251446	0.125723	−	0.992065i	$-0.459875\pi$
$911$	5.73799e9	0.251446	0.125723	+	0.992065i	$0.459875\pi$
$912$	0	0
$913$	8.25114e9	0.358811
$914$	1.42473e10	0.617191
$915$	0	0
$916$	1.94258e9	0.0835113
$917$	3.11925e10	1.33585
$918$	0	0
$919$	3.67326e10	1.56116	0.780579	−	0.625057i	$-0.214924\pi$
$919$	3.67326e10	1.56116	0.780579	+	0.625057i	$0.214924\pi$
$920$	−5.10205e9	−0.216017
$921$	0	0
$922$	1.49833e10	0.629579
$923$	4.26381e9	0.178481
$924$	0	0
$925$	−3.60871e10	−1.49919
$926$	1.21755e10	0.503906
$927$	0	0
$928$	1.19392e10	0.490408
$929$	−3.50768e9	−0.143537	−0.0717687	−	0.997421i	$-0.522864\pi$
$929$	−3.50768e9	−0.143537	−0.0717687	+	0.997421i	$0.522864\pi$
$930$	0	0
$931$	−6.22809e9	−0.252948
$932$	−2.47217e10	−1.00028
$933$	0	0
$934$	1.00529e10	0.403717
$935$	1.19674e10	0.478805
$936$	0	0
$937$	−1.68124e10	−0.667638	−0.333819	−	0.942637i	$-0.608337\pi$
$937$	−1.68124e10	−0.667638	−0.333819	+	0.942637i	$0.608337\pi$
$938$	1.64299e9	0.0650016
$939$	0	0
$940$	−5.67726e9	−0.222942
$941$	3.51454e10	1.37501	0.687503	−	0.726182i	$-0.258707\pi$
$941$	3.51454e10	1.37501	0.687503	+	0.726182i	$0.258707\pi$
$942$	0	0
$943$	−3.04021e10	−1.18063
$944$	7.69368e8	0.0297668
$945$	0	0
$946$	3.17447e10	1.21914
$947$	3.30590e10	1.26493	0.632463	−	0.774591i	$-0.282044\pi$
$947$	3.30590e10	1.26493	0.632463	+	0.774591i	$0.282044\pi$
$948$	0	0
$949$	−2.36466e10	−0.898124
$950$	1.25399e10	0.474527
$951$	0	0
$952$	−4.07601e10	−1.53111
$953$	6.24805e9	0.233840	0.116920	−	0.993141i	$-0.462698\pi$
$953$	6.24805e9	0.233840	0.116920	+	0.993141i	$0.462698\pi$
$954$	0	0
$955$	−7.62401e9	−0.283251
$956$	2.66728e10	0.987338
$957$	0	0
$958$	1.85450e10	0.681470
$959$	5.91795e9	0.216674
$960$	0	0
$961$	−4.81574e9	−0.175037
$962$	−2.21586e10	−0.802471
$963$	0	0
$964$	−2.10512e9	−0.0756846
$965$	3.77673e9	0.135291
$966$	0	0
$967$	2.21264e10	0.786899	0.393449	−	0.919346i	$-0.371282\pi$
$967$	2.21264e10	0.786899	0.393449	+	0.919346i	$0.371282\pi$
$968$	7.72161e9	0.273618
$969$	0	0
$970$	−7.34673e9	−0.258460
$971$	5.45446e9	0.191199	0.0955993	−	0.995420i	$-0.469523\pi$
$971$	5.45446e9	0.191199	0.0955993	+	0.995420i	$0.469523\pi$
$972$	0	0
$973$	−1.07815e10	−0.375217
$974$	1.04264e8	0.00361558
$975$	0	0
$976$	7.23050e7	0.00248940
$977$	−4.95373e10	−1.69942	−0.849711	−	0.527248i	$-0.823224\pi$
$977$	−4.95373e10	−1.69942	−0.849711	+	0.527248i	$0.823224\pi$
$978$	0	0
$979$	2.30609e10	0.785482
$980$	1.28835e9	0.0437263
$981$	0	0
$982$	4.80086e9	0.161781
$983$	−8.72042e9	−0.292820	−0.146410	−	0.989224i	$-0.546772\pi$
$983$	−8.72042e9	−0.292820	−0.146410	+	0.989224i	$0.546772\pi$
$984$	0	0
$985$	6.90027e9	0.230059
$986$	1.90631e10	0.633320
$987$	0	0
$988$	−9.39330e9	−0.309862
$989$	4.33739e10	1.42574
$990$	0	0
$991$	1.64359e10	0.536458	0.268229	−	0.963355i	$-0.413562\pi$
$991$	1.64359e10	0.536458	0.268229	+	0.963355i	$0.413562\pi$
$992$	−2.62600e10	−0.854091
$993$	0	0
$994$	4.00641e9	0.129391
$995$	−1.38618e10	−0.446106
$996$	0	0
$997$	−3.18717e10	−1.01853	−0.509264	−	0.860611i	$-0.670082\pi$
$997$	−3.18717e10	−1.01853	−0.509264	+	0.860611i	$0.670082\pi$
$998$	1.84797e10	0.588487
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.a.e.1.1	yes	2
3.2	odd	2		27.8.a.b.1.2	✓	2
4.3	odd	2		432.8.a.q.1.1		2
9.2	odd	6		81.8.c.h.28.1		4
9.4	even	3		81.8.c.d.55.2		4
9.5	odd	6		81.8.c.h.55.1		4
9.7	even	3		81.8.c.d.28.2		4
12.11	even	2		432.8.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.b.1.2	✓	2	3.2	odd	2
27.8.a.e.1.1	yes	2	1.1	even	1	trivial
81.8.c.d.28.2		4	9.7	even	3
81.8.c.d.55.2		4	9.4	even	3
81.8.c.h.28.1		4	9.2	odd	6
81.8.c.h.55.1		4	9.5	odd	6
432.8.a.j.1.2		2	12.11	even	2
432.8.a.q.1.1		2	4.3	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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-7.59339 q^{2} -70.3405 q^{4} +65.8132 q^{5} -738.405 q^{7} +1506.08 q^{8} -499.745 q^{10} +4961.26 q^{11} +5967.24 q^{13} +5606.99 q^{14} -2432.64 q^{16} +36651.6 q^{17} +22378.9 q^{19} -4629.33 q^{20} -37672.8 q^{22} -51473.6 q^{23} -73793.6 q^{25} -45311.5 q^{26} +51939.7 q^{28} -68495.7 q^{29} +150655. q^{31} -174306. q^{32} -278310. q^{34} -48596.8 q^{35} +489027. q^{37} -169932. q^{38} +99119.7 q^{40} +590635. q^{41} -842643. q^{43} -348978. q^{44} +390859. q^{46} +1.22637e6 q^{47} -278301. q^{49} +560343. q^{50} -419738. q^{52} +958904. q^{53} +326517. q^{55} -1.11209e6 q^{56} +520114. q^{58} -316269. q^{59} -29722.9 q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +392723. q^{65} +293025. q^{67} -2.57809e6 q^{68} +369014. q^{70} +714537. q^{71} -3.96273e6 q^{73} -3.71337e6 q^{74} -1.57415e6 q^{76} -3.66342e6 q^{77} +2.53805e6 q^{79} -160100. q^{80} -4.48492e6 q^{82} +1.66311e6 q^{83} +2.41216e6 q^{85} +6.39851e6 q^{86} +7.47204e6 q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +3.62068e6 q^{92} -9.31228e6 q^{94} +1.47283e6 q^{95} +1.47010e7 q^{97} +2.11325e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) \) 2 * q + 9 * q^2 + 77 * q^4 + 180 * q^5 + 700 * q^7 + 1827 * q^8 + 1395 * q^10 + 10890 * q^11 - 5480 * q^13 + 29475 * q^14 - 15967 * q^16 + 16416 * q^17 + 16024 * q^19 + 12195 * q^20 + 60705 * q^22 + 24372 * q^23 - 138880 * q^25 - 235260 * q^26 + 263875 * q^28 - 143280 * q^29 - 38708 * q^31 - 439965 * q^32 - 614088 * q^34 + 115650 * q^35 + 455620 * q^37 - 275382 * q^38 + 135765 * q^40 + 731880 * q^41 - 1088840 * q^43 + 524565 * q^44 + 1649394 * q^46 + 1561500 * q^47 + 967164 * q^49 - 519660 * q^50 - 2106380 * q^52 + 2610468 * q^53 + 1003500 * q^55 - 650475 * q^56 - 720810 * q^58 + 1731960 * q^59 - 620192 * q^61 - 4286151 * q^62 - 1040839 * q^64 - 914400 * q^65 + 346600 * q^67 - 5559624 * q^68 + 3094425 * q^70 - 4242240 * q^71 - 3145190 * q^73 - 4267710 * q^74 - 2510486 * q^76 + 4864500 * q^77 + 10110616 * q^79 - 1705545 * q^80 - 2141190 * q^82 + 644202 * q^83 + 101520 * q^85 + 2313270 * q^86 + 9374715 * q^88 + 6021000 * q^89 - 20872000 * q^91 + 14795802 * q^92 - 3751290 * q^94 + 747180 * q^95 + 4098670 * q^97 + 22779738 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more