LMFDB - Embedded newform 9.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,8,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, 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("9.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	9.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:	$2.81146522936$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.9737 q^{2} +232.000 q^{4} +303.579 q^{5} +260.000 q^{7} -1973.26 q^{8} +O(q^{10})$	\(q-18.9737 q^{2} +232.000 q^{4} +303.579 q^{5} +260.000 q^{7} -1973.26 q^{8} -5760.00 q^{10} +6071.57 q^{11} +6890.00 q^{13} -4933.15 q^{14} +7744.00 q^{16} -23679.1 q^{17} +33176.0 q^{19} +70430.2 q^{20} -115200. q^{22} -31572.2 q^{23} +14035.0 q^{25} -130729. q^{26} +60320.0 q^{28} +138128. q^{29} +1508.00 q^{31} +105645. q^{32} +449280. q^{34} +78930.5 q^{35} -380770. q^{37} -629470. q^{38} -599040. q^{40} -88037.8 q^{41} +7640.00 q^{43} +1.40860e6 q^{44} +599040. q^{46} -565871. q^{47} -755943. q^{49} -266295. q^{50} +1.59848e6 q^{52} -1.03004e6 q^{53} +1.84320e6 q^{55} -513048. q^{56} -2.62080e6 q^{58} +2.70792e6 q^{59} -988858. q^{61} -28612.3 q^{62} -2.99571e6 q^{64} +2.09166e6 q^{65} +3.85736e6 q^{67} -5.49356e6 q^{68} -1.49760e6 q^{70} -4.22581e6 q^{71} -2.00473e6 q^{73} +7.22460e6 q^{74} +7.69683e6 q^{76} +1.57861e6 q^{77} +2.69968e6 q^{79} +2.35091e6 q^{80} +1.67040e6 q^{82} -2.71156e6 q^{83} -7.18848e6 q^{85} -144959. q^{86} -1.19808e7 q^{88} -7.74126e6 q^{89} +1.79140e6 q^{91} -7.32475e6 q^{92} +1.07366e7 q^{94} +1.00715e7 q^{95} -1.29575e7 q^{97} +1.43430e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 464 q^{4} + 520 q^{7}+O(q^{10}) $ 2 * q + 464 * q^4 + 520 * q^7	$ 2 q + 464 q^{4} + 520 q^{7} - 11520 q^{10} + 13780 q^{13} + 15488 q^{16} + 66352 q^{19} - 230400 q^{22} + 28070 q^{25} + 120640 q^{28} + 3016 q^{31} + 898560 q^{34} - 761540 q^{37} - 1198080 q^{40} + 15280 q^{43} + 1198080 q^{46} - 1511886 q^{49} + 3196960 q^{52} + 3686400 q^{55} - 5241600 q^{58} - 1977716 q^{61} - 5991424 q^{64} + 7714720 q^{67} - 2995200 q^{70} - 4009460 q^{73} + 15393664 q^{76} + 5399368 q^{79} + 3340800 q^{82} - 14376960 q^{85} - 23961600 q^{88} + 3582800 q^{91} + 21473280 q^{94} - 25914980 q^{97}+O(q^{100}) $ 2 * q + 464 * q^4 + 520 * q^7 - 11520 * q^10 + 13780 * q^13 + 15488 * q^16 + 66352 * q^19 - 230400 * q^22 + 28070 * q^25 + 120640 * q^28 + 3016 * q^31 + 898560 * q^34 - 761540 * q^37 - 1198080 * q^40 + 15280 * q^43 + 1198080 * q^46 - 1511886 * q^49 + 3196960 * q^52 + 3686400 * q^55 - 5241600 * q^58 - 1977716 * q^61 - 5991424 * q^64 + 7714720 * q^67 - 2995200 * q^70 - 4009460 * q^73 + 15393664 * q^76 + 5399368 * q^79 + 3340800 * q^82 - 14376960 * q^85 - 23961600 * q^88 + 3582800 * q^91 + 21473280 * q^94 - 25914980 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−18.9737	−1.67705	−0.838525	−	0.544862i	$-0.816582\pi$
$2$	−18.9737	−1.67705	−0.838525	+	0.544862i	$0.816582\pi$
$3$	0	0
$3$	0	0
$4$	232.000	1.81250
$5$	303.579	1.08612	0.543058	−	0.839695i	$-0.317266\pi$
$5$	303.579	1.08612	0.543058	+	0.839695i	$0.317266\pi$
$6$	0	0
$7$	260.000	0.286504	0.143252	−	0.989686i	$-0.454244\pi$
$7$	260.000	0.286504	0.143252	+	0.989686i	$0.454244\pi$
$8$	−1973.26	−1.36260
$9$	0	0
$10$	−5760.00	−1.82147
$11$	6071.57	1.37539	0.687697	−	0.725998i	$-0.258622\pi$
$11$	6071.57	1.37539	0.687697	+	0.725998i	$0.258622\pi$
$12$	0	0
$13$	6890.00	0.869796	0.434898	−	0.900480i	$-0.356784\pi$
$13$	6890.00	0.869796	0.434898	+	0.900480i	$0.356784\pi$
$14$	−4933.15	−0.480481
$15$	0	0
$16$	7744.00	0.472656
$17$	−23679.1	−1.16895	−0.584473	−	0.811413i	$-0.698699\pi$
$17$	−23679.1	−1.16895	−0.584473	+	0.811413i	$0.698699\pi$
$18$	0	0
$19$	33176.0	1.10965	0.554826	−	0.831967i	$-0.312785\pi$
$19$	33176.0	1.10965	0.554826	+	0.831967i	$0.312785\pi$
$20$	70430.2	1.96859
$21$	0	0
$22$	−115200.	−2.30660
$23$	−31572.2	−0.541075	−0.270537	−	0.962709i	$-0.587201\pi$
$23$	−31572.2	−0.541075	−0.270537	+	0.962709i	$0.587201\pi$
$24$	0	0
$25$	14035.0	0.179648
$26$	−130729.	−1.45869
$27$	0	0
$28$	60320.0	0.519288
$29$	138128.	1.05169	0.525847	−	0.850579i	$-0.323748\pi$
$29$	138128.	1.05169	0.525847	+	0.850579i	$0.323748\pi$
$30$	0	0
$31$	1508.00	0.00909150	0.00454575	−	0.999990i	$-0.498553\pi$
$31$	1508.00	0.00909150	0.00454575	+	0.999990i	$0.498553\pi$
$32$	105645.	0.569935
$33$	0	0
$34$	449280.	1.96038
$35$	78930.5	0.311176
$36$	0	0
$37$	−380770.	−1.23582	−0.617912	−	0.786247i	$-0.712021\pi$
$37$	−380770.	−1.23582	−0.617912	+	0.786247i	$0.712021\pi$
$38$	−629470.	−1.86094
$39$	0	0
$40$	−599040.	−1.47995
$41$	−88037.8	−0.199492	−0.0997461	−	0.995013i	$-0.531803\pi$
$41$	−88037.8	−0.199492	−0.0997461	+	0.995013i	$0.531803\pi$
$42$	0	0
$43$	7640.00	0.0146539	0.00732696	−	0.999973i	$-0.497668\pi$
$43$	7640.00	0.0146539	0.00732696	+	0.999973i	$0.497668\pi$
$44$	1.40860e6	2.49290
$45$	0	0
$46$	599040.	0.907410
$47$	−565871.	−0.795014	−0.397507	−	0.917599i	$-0.630125\pi$
$47$	−565871.	−0.795014	−0.397507	+	0.917599i	$0.630125\pi$
$48$	0	0
$49$	−755943.	−0.917916
$50$	−266295.	−0.301279
$51$	0	0
$52$	1.59848e6	1.57651
$53$	−1.03004e6	−0.950363	−0.475182	−	0.879888i	$-0.657618\pi$
$53$	−1.03004e6	−0.950363	−0.475182	+	0.879888i	$0.657618\pi$
$54$	0	0
$55$	1.84320e6	1.49384
$56$	−513048.	−0.390391
$57$	0	0
$58$	−2.62080e6	−1.76375
$59$	2.70792e6	1.71654	0.858270	−	0.513198i	$-0.171539\pi$
$59$	2.70792e6	1.71654	0.858270	+	0.513198i	$0.171539\pi$
$60$	0	0
$61$	−988858.	−0.557801	−0.278901	−	0.960320i	$-0.589970\pi$
$61$	−988858.	−0.557801	−0.278901	+	0.960320i	$0.589970\pi$
$62$	−28612.3	−0.0152469
$63$	0	0
$64$	−2.99571e6	−1.42847
$65$	2.09166e6	0.944700
$66$	0	0
$67$	3.85736e6	1.56685	0.783427	−	0.621484i	$-0.213470\pi$
$67$	3.85736e6	1.56685	0.783427	+	0.621484i	$0.213470\pi$
$68$	−5.49356e6	−2.11872
$69$	0	0
$70$	−1.49760e6	−0.521858
$71$	−4.22581e6	−1.40122	−0.700610	−	0.713545i	$-0.747089\pi$
$71$	−4.22581e6	−1.40122	−0.700610	+	0.713545i	$0.747089\pi$
$72$	0	0
$73$	−2.00473e6	−0.603151	−0.301575	−	0.953442i	$-0.597512\pi$
$73$	−2.00473e6	−0.603151	−0.301575	+	0.953442i	$0.597512\pi$
$74$	7.22460e6	2.07254
$75$	0	0
$76$	7.69683e6	2.01124
$77$	1.57861e6	0.394055
$78$	0	0
$79$	2.69968e6	0.616053	0.308027	−	0.951378i	$-0.400331\pi$
$79$	2.69968e6	0.616053	0.308027	+	0.951378i	$0.400331\pi$
$80$	2.35091e6	0.513360
$81$	0	0
$82$	1.67040e6	0.334558
$83$	−2.71156e6	−0.520531	−0.260266	−	0.965537i	$-0.583810\pi$
$83$	−2.71156e6	−0.520531	−0.260266	+	0.965537i	$0.583810\pi$
$84$	0	0
$85$	−7.18848e6	−1.26961
$86$	−144959.	−0.0245754
$87$	0	0
$88$	−1.19808e7	−1.87412
$89$	−7.74126e6	−1.16398	−0.581991	−	0.813195i	$-0.697726\pi$
$89$	−7.74126e6	−1.16398	−0.581991	+	0.813195i	$0.697726\pi$
$90$	0	0
$91$	1.79140e6	0.249200
$92$	−7.32475e6	−0.980698
$93$	0	0
$94$	1.07366e7	1.33328
$95$	1.00715e7	1.20521
$96$	0	0
$97$	−1.29575e7	−1.44152	−0.720759	−	0.693186i	$-0.756206\pi$
$97$	−1.29575e7	−1.44152	−0.720759	+	0.693186i	$0.756206\pi$
$98$	1.43430e7	1.53939
$99$	0	0
$100$	3.25612e6	0.325612
$101$	5.58433e6	0.539320	0.269660	−	0.962956i	$-0.413089\pi$
$101$	5.58433e6	0.539320	0.269660	+	0.962956i	$0.413089\pi$
$102$	0	0
$103$	5.07326e6	0.457464	0.228732	−	0.973489i	$-0.426542\pi$
$103$	5.07326e6	0.457464	0.228732	+	0.973489i	$0.426542\pi$
$104$	−1.35958e7	−1.18519
$105$	0	0
$106$	1.95437e7	1.59381
$107$	3.69395e6	0.291506	0.145753	−	0.989321i	$-0.453440\pi$
$107$	3.69395e6	0.291506	0.145753	+	0.989321i	$0.453440\pi$
$108$	0	0
$109$	6.59383e6	0.487692	0.243846	−	0.969814i	$-0.421591\pi$
$109$	6.59383e6	0.487692	0.243846	+	0.969814i	$0.421591\pi$
$110$	−3.49723e7	−2.50524
$111$	0	0
$112$	2.01344e6	0.135418
$113$	4.75161e6	0.309789	0.154895	−	0.987931i	$-0.450496\pi$
$113$	4.75161e6	0.309789	0.154895	+	0.987931i	$0.450496\pi$
$114$	0	0
$115$	−9.58464e6	−0.587670
$116$	3.20458e7	1.90620
$117$	0	0
$118$	−5.13792e7	−2.87873
$119$	−6.15658e6	−0.334907
$120$	0	0
$121$	1.73768e7	0.891706
$122$	1.87623e7	0.935461
$123$	0	0
$124$	349856.	0.0164783
$125$	−1.94564e7	−0.890997
$126$	0	0
$127$	1.95361e7	0.846303	0.423151	−	0.906059i	$-0.360924\pi$
$127$	1.95361e7	0.846303	0.423151	+	0.906059i	$0.360924\pi$
$128$	4.33170e7	1.82568
$129$	0	0
$130$	−3.96864e7	−1.58431
$131$	−3.72552e7	−1.44789	−0.723947	−	0.689855i	$-0.757674\pi$
$131$	−3.72552e7	−1.44789	−0.723947	+	0.689855i	$0.757674\pi$
$132$	0	0
$133$	8.62576e6	0.317919
$134$	−7.31883e7	−2.62769
$135$	0	0
$136$	4.67251e7	1.59281
$137$	2.12171e7	0.704960	0.352480	−	0.935819i	$-0.385339\pi$
$137$	2.12171e7	0.704960	0.352480	+	0.935819i	$0.385339\pi$
$138$	0	0
$139$	−2.44669e7	−0.772728	−0.386364	−	0.922346i	$-0.626269\pi$
$139$	−2.44669e7	−0.772728	−0.386364	+	0.922346i	$0.626269\pi$
$140$	1.83119e7	0.564007
$141$	0	0
$142$	8.01792e7	2.34992
$143$	4.18331e7	1.19631
$144$	0	0
$145$	4.19328e7	1.14226
$146$	3.80371e7	1.01151
$147$	0	0
$148$	−8.83386e7	−2.23993
$149$	1.85320e7	0.458954	0.229477	−	0.973314i	$-0.426298\pi$
$149$	1.85320e7	0.458954	0.229477	+	0.973314i	$0.426298\pi$
$150$	0	0
$151$	−5.12102e7	−1.21042	−0.605211	−	0.796065i	$-0.706911\pi$
$151$	−5.12102e7	−1.21042	−0.605211	+	0.796065i	$0.706911\pi$
$152$	−6.54649e7	−1.51201
$153$	0	0
$154$	−2.99520e7	−0.660851
$155$	457797.	0.00987442
$156$	0	0
$157$	9.63815e6	0.198767	0.0993836	−	0.995049i	$-0.468313\pi$
$157$	9.63815e6	0.198767	0.0993836	+	0.995049i	$0.468313\pi$
$158$	−5.12229e7	−1.03315
$159$	0	0
$160$	3.20717e7	0.619016
$161$	−8.20877e6	−0.155020
$162$	0	0
$163$	−7.18666e7	−1.29978	−0.649891	−	0.760027i	$-0.725185\pi$
$163$	−7.18666e7	−1.29978	−0.649891	+	0.760027i	$0.725185\pi$
$164$	−2.04248e7	−0.361579
$165$	0	0
$166$	5.14483e7	0.872957
$167$	8.56578e6	0.142318	0.0711589	−	0.997465i	$-0.477330\pi$
$167$	8.56578e6	0.142318	0.0711589	+	0.997465i	$0.477330\pi$
$168$	0	0
$169$	−1.52764e7	−0.243455
$170$	1.36392e8	2.12920
$171$	0	0
$172$	1.77248e6	0.0265602
$173$	−4.97065e7	−0.729880	−0.364940	−	0.931031i	$-0.618910\pi$
$173$	−4.97065e7	−0.729880	−0.364940	+	0.931031i	$0.618910\pi$
$174$	0	0
$175$	3.64910e6	0.0514698
$176$	4.70183e7	0.650088
$177$	0	0
$178$	1.46880e8	1.95206
$179$	5.73035e7	0.746785	0.373393	−	0.927673i	$-0.378194\pi$
$179$	5.73035e7	0.746785	0.373393	+	0.927673i	$0.378194\pi$
$180$	0	0
$181$	8.15744e7	1.02254	0.511268	−	0.859421i	$-0.329176\pi$
$181$	8.15744e7	1.02254	0.511268	+	0.859421i	$0.329176\pi$
$182$	−3.39894e7	−0.417921
$183$	0	0
$184$	6.23002e7	0.737270
$185$	−1.15594e8	−1.34225
$186$	0	0
$187$	−1.43770e8	−1.60776
$188$	−1.31282e8	−1.44096
$189$	0	0
$190$	−1.91094e8	−2.02120
$191$	−1.60229e8	−1.66389	−0.831944	−	0.554860i	$-0.812772\pi$
$191$	−1.60229e8	−1.66389	−0.831944	+	0.554860i	$0.812772\pi$
$192$	0	0
$193$	6.00372e7	0.601133	0.300566	−	0.953761i	$-0.402824\pi$
$193$	6.00372e7	0.601133	0.300566	+	0.953761i	$0.402824\pi$
$194$	2.45851e8	2.41750
$195$	0	0
$196$	−1.75379e8	−1.66372
$197$	1.90188e8	1.77236	0.886179	−	0.463343i	$-0.153350\pi$
$197$	1.90188e8	1.77236	0.886179	+	0.463343i	$0.153350\pi$
$198$	0	0
$199$	1.94001e8	1.74509	0.872545	−	0.488533i	$-0.162468\pi$
$199$	1.94001e8	1.74509	0.872545	+	0.488533i	$0.162468\pi$
$200$	−2.76947e7	−0.244789
$201$	0	0
$202$	−1.05955e8	−0.904466
$203$	3.59134e7	0.301314
$204$	0	0
$205$	−2.67264e7	−0.216672
$206$	−9.62583e7	−0.767190
$207$	0	0
$208$	5.33562e7	0.411115
$209$	2.01431e8	1.52621
$210$	0	0
$211$	−1.04896e7	−0.0768727	−0.0384364	−	0.999261i	$-0.512238\pi$
$211$	−1.04896e7	−0.0768727	−0.0384364	+	0.999261i	$0.512238\pi$
$212$	−2.38970e8	−1.72253
$213$	0	0
$214$	−7.00877e7	−0.488870
$215$	2.31934e6	0.0159159
$216$	0	0
$217$	392080.	0.00260475
$218$	−1.25109e8	−0.817884
$219$	0	0
$220$	4.27622e8	2.70758
$221$	−1.63149e8	−1.01675
$222$	0	0
$223$	2.25160e8	1.35964	0.679820	−	0.733379i	$-0.262058\pi$
$223$	2.25160e8	1.35964	0.679820	+	0.733379i	$0.262058\pi$
$224$	2.74678e7	0.163289
$225$	0	0
$226$	−9.01555e7	−0.519533
$227$	1.69490e8	0.961733	0.480867	−	0.876794i	$-0.340322\pi$
$227$	1.69490e8	0.961733	0.480867	+	0.876794i	$0.340322\pi$
$228$	0	0
$229$	−6.52595e7	−0.359103	−0.179552	−	0.983749i	$-0.557465\pi$
$229$	−6.52595e7	−0.359103	−0.179552	+	0.983749i	$0.557465\pi$
$230$	1.81856e8	0.985552
$231$	0	0
$232$	−2.72563e8	−1.43304
$233$	−1.86355e8	−0.965150	−0.482575	−	0.875855i	$-0.660298\pi$
$233$	−1.86355e8	−0.965150	−0.482575	+	0.875855i	$0.660298\pi$
$234$	0	0
$235$	−1.71786e8	−0.863477
$236$	6.28238e8	3.11123
$237$	0	0
$238$	1.16813e8	0.561657
$239$	−4.26103e7	−0.201893	−0.100947	−	0.994892i	$-0.532187\pi$
$239$	−4.26103e7	−0.201893	−0.100947	+	0.994892i	$0.532187\pi$
$240$	0	0
$241$	−2.56943e8	−1.18243	−0.591217	−	0.806513i	$-0.701352\pi$
$241$	−2.56943e8	−1.18243	−0.591217	+	0.806513i	$0.701352\pi$
$242$	−3.29702e8	−1.49544
$243$	0	0
$244$	−2.29415e8	−1.01101
$245$	−2.29488e8	−0.996963
$246$	0	0
$247$	2.28583e8	0.965170
$248$	−2.97568e6	−0.0123881
$249$	0	0
$250$	3.69158e8	1.49425
$251$	−2.73968e8	−1.09356	−0.546778	−	0.837278i	$-0.684146\pi$
$251$	−2.73968e8	−1.09356	−0.546778	+	0.837278i	$0.684146\pi$
$252$	0	0
$253$	−1.91693e8	−0.744190
$254$	−3.70672e8	−1.41929
$255$	0	0
$256$	−4.38432e8	−1.63329
$257$	3.19542e8	1.17425	0.587127	−	0.809495i	$-0.300259\pi$
$257$	3.19542e8	1.17425	0.587127	+	0.809495i	$0.300259\pi$
$258$	0	0
$259$	−9.90002e7	−0.354068
$260$	4.85264e8	1.71227
$261$	0	0
$262$	7.06867e8	2.42819
$263$	2.71016e8	0.918648	0.459324	−	0.888269i	$-0.348092\pi$
$263$	2.71016e8	0.918648	0.459324	+	0.888269i	$0.348092\pi$
$264$	0	0
$265$	−3.12699e8	−1.03220
$266$	−1.63662e8	−0.533167
$267$	0	0
$268$	8.94908e8	2.83992
$269$	1.09930e8	0.344338	0.172169	−	0.985067i	$-0.444923\pi$
$269$	1.09930e8	0.344338	0.172169	+	0.985067i	$0.444923\pi$
$270$	0	0
$271$	4.35740e8	1.32995	0.664974	−	0.746866i	$-0.268443\pi$
$271$	4.35740e8	1.32995	0.664974	+	0.746866i	$0.268443\pi$
$272$	−1.83371e8	−0.552510
$273$	0	0
$274$	−4.02566e8	−1.18225
$275$	8.52145e7	0.247087
$276$	0	0
$277$	−2.35120e8	−0.664676	−0.332338	−	0.943160i	$-0.607837\pi$
$277$	−2.35120e8	−0.664676	−0.332338	+	0.943160i	$0.607837\pi$
$278$	4.64226e8	1.29590
$279$	0	0
$280$	−1.55750e8	−0.424010
$281$	−2.13052e8	−0.572812	−0.286406	−	0.958108i	$-0.592461\pi$
$281$	−2.13052e8	−0.572812	−0.286406	+	0.958108i	$0.592461\pi$
$282$	0	0
$283$	−6.90664e7	−0.181140	−0.0905700	−	0.995890i	$-0.528869\pi$
$283$	−6.90664e7	−0.181140	−0.0905700	+	0.995890i	$0.528869\pi$
$284$	−9.80389e8	−2.53971
$285$	0	0
$286$	−7.93728e8	−2.00628
$287$	−2.28898e7	−0.0571552
$288$	0	0
$289$	1.50363e8	0.366436
$290$	−7.95619e8	−1.91563
$291$	0	0
$292$	−4.65097e8	−1.09321
$293$	6.91892e8	1.60695	0.803474	−	0.595340i	$-0.202983\pi$
$293$	6.91892e8	1.60695	0.803474	+	0.595340i	$0.202983\pi$
$294$	0	0
$295$	8.22067e8	1.86436
$296$	7.51359e8	1.68394
$297$	0	0
$298$	−3.51619e8	−0.769690
$299$	−2.17532e8	−0.470625
$300$	0	0
$301$	1.98640e6	0.00419840
$302$	9.71645e8	2.02994
$303$	0	0
$304$	2.56915e8	0.524483
$305$	−3.00196e8	−0.605837
$306$	0	0
$307$	−8.97049e8	−1.76942	−0.884712	−	0.466139i	$-0.845645\pi$
$307$	−8.97049e8	−1.76942	−0.884712	+	0.466139i	$0.845645\pi$
$308$	3.66237e8	0.714225
$309$	0	0
$310$	−8.68608e6	−0.0165599
$311$	−1.63228e8	−0.307704	−0.153852	−	0.988094i	$-0.549168\pi$
$311$	−1.63228e8	−0.307704	−0.153852	+	0.988094i	$0.549168\pi$
$312$	0	0
$313$	6.86242e8	1.26495	0.632473	−	0.774583i	$-0.282040\pi$
$313$	6.86242e8	1.26495	0.632473	+	0.774583i	$0.282040\pi$
$314$	−1.82871e8	−0.333343
$315$	0	0
$316$	6.26327e8	1.11660
$317$	−5.23972e8	−0.923849	−0.461924	−	0.886919i	$-0.652841\pi$
$317$	−5.23972e8	−0.923849	−0.461924	+	0.886919i	$0.652841\pi$
$318$	0	0
$319$	8.38656e8	1.44649
$320$	−9.09434e8	−1.55148
$321$	0	0
$322$	1.55750e8	0.259976
$323$	−7.85579e8	−1.29712
$324$	0	0
$325$	9.67012e7	0.156257
$326$	1.36357e9	2.17980
$327$	0	0
$328$	1.73722e8	0.271829
$329$	−1.47126e8	−0.227774
$330$	0	0
$331$	−5.72280e8	−0.867382	−0.433691	−	0.901062i	$-0.642789\pi$
$331$	−5.72280e8	−0.867382	−0.433691	+	0.901062i	$0.642789\pi$
$332$	−6.29083e8	−0.943462
$333$	0	0
$334$	−1.62524e8	−0.238674
$335$	1.17101e9	1.70178
$336$	0	0
$337$	−5.56883e8	−0.792609	−0.396305	−	0.918119i	$-0.629708\pi$
$337$	−5.56883e8	−0.792609	−0.396305	+	0.918119i	$0.629708\pi$
$338$	2.89850e8	0.408286
$339$	0	0
$340$	−1.66773e9	−2.30117
$341$	9.15593e6	0.0125044
$342$	0	0
$343$	−4.10666e8	−0.549490
$344$	−1.50757e7	−0.0199675
$345$	0	0
$346$	9.43114e8	1.22405
$347$	8.89499e8	1.14286	0.571429	−	0.820652i	$-0.306389\pi$
$347$	8.89499e8	1.14286	0.571429	+	0.820652i	$0.306389\pi$
$348$	0	0
$349$	7.80706e8	0.983102	0.491551	−	0.870849i	$-0.336430\pi$
$349$	7.80706e8	0.983102	0.491551	+	0.870849i	$0.336430\pi$
$350$	−6.92368e7	−0.0863175
$351$	0	0
$352$	6.41434e8	0.783885
$353$	5.40517e8	0.654030	0.327015	−	0.945019i	$-0.393957\pi$
$353$	5.40517e8	0.654030	0.327015	+	0.945019i	$0.393957\pi$
$354$	0	0
$355$	−1.28287e9	−1.52189
$356$	−1.79597e9	−2.10972
$357$	0	0
$358$	−1.08726e9	−1.25240
$359$	6.93617e7	0.0791205	0.0395602	−	0.999217i	$-0.487404\pi$
$359$	6.93617e7	0.0791205	0.0395602	+	0.999217i	$0.487404\pi$
$360$	0	0
$361$	2.06775e8	0.231325
$362$	−1.54777e9	−1.71485
$363$	0	0
$364$	4.15605e8	0.451675
$365$	−6.08593e8	−0.655092
$366$	0	0
$367$	−3.73436e8	−0.394353	−0.197177	−	0.980368i	$-0.563177\pi$
$367$	−3.73436e8	−0.394353	−0.197177	+	0.980368i	$0.563177\pi$
$368$	−2.44495e8	−0.255742
$369$	0	0
$370$	2.19324e9	2.25102
$371$	−2.67811e8	−0.272283
$372$	0	0
$373$	2.78207e8	0.277579	0.138790	−	0.990322i	$-0.455679\pi$
$373$	2.78207e8	0.277579	0.138790	+	0.990322i	$0.455679\pi$
$374$	2.72784e9	2.69630
$375$	0	0
$376$	1.11661e9	1.08329
$377$	9.51704e8	0.914760
$378$	0	0
$379$	−1.36851e9	−1.29125	−0.645626	−	0.763654i	$-0.723404\pi$
$379$	−1.36851e9	−1.29125	−0.645626	+	0.763654i	$0.723404\pi$
$380$	2.33659e9	2.18444
$381$	0	0
$382$	3.04013e9	2.79042
$383$	−2.07950e9	−1.89131	−0.945655	−	0.325173i	$-0.894578\pi$
$383$	−2.07950e9	−1.89131	−0.945655	+	0.325173i	$0.894578\pi$
$384$	0	0
$385$	4.79232e8	0.427990
$386$	−1.13913e9	−1.00813
$387$	0	0
$388$	−3.00614e9	−2.61275
$389$	1.96793e8	0.169507	0.0847533	−	0.996402i	$-0.472990\pi$
$389$	1.96793e8	0.169507	0.0847533	+	0.996402i	$0.472990\pi$
$390$	0	0
$391$	7.47602e8	0.632487
$392$	1.49167e9	1.25076
$393$	0	0
$394$	−3.60857e9	−2.97234
$395$	8.19566e8	0.669105
$396$	0	0
$397$	2.99153e8	0.239953	0.119977	−	0.992777i	$-0.461718\pi$
$397$	2.99153e8	0.239953	0.119977	+	0.992777i	$0.461718\pi$
$398$	−3.68091e9	−2.92661
$399$	0	0
$400$	1.08687e8	0.0849117
$401$	−8.45142e8	−0.654522	−0.327261	−	0.944934i	$-0.606126\pi$
$401$	−8.45142e8	−0.654522	−0.327261	+	0.944934i	$0.606126\pi$
$402$	0	0
$403$	1.03901e7	0.00790775
$404$	1.29556e9	0.977517
$405$	0	0
$406$	−6.81408e8	−0.505320
$407$	−2.31187e9	−1.69974
$408$	0	0
$409$	1.07268e9	0.775248	0.387624	−	0.921818i	$-0.373296\pi$
$409$	1.07268e9	0.775248	0.387624	+	0.921818i	$0.373296\pi$
$410$	5.07098e8	0.363369
$411$	0	0
$412$	1.17700e9	0.829153
$413$	7.04060e8	0.491795
$414$	0	0
$415$	−8.23173e8	−0.565357
$416$	7.27897e8	0.495728
$417$	0	0
$418$	−3.82188e9	−2.55953
$419$	1.41451e9	0.939416	0.469708	−	0.882822i	$-0.344359\pi$
$419$	1.41451e9	0.939416	0.469708	+	0.882822i	$0.344359\pi$
$420$	0	0
$421$	1.41562e9	0.924613	0.462307	−	0.886720i	$-0.347022\pi$
$421$	1.41562e9	0.924613	0.462307	+	0.886720i	$0.347022\pi$
$422$	1.99027e8	0.128919
$423$	0	0
$424$	2.03254e9	1.29497
$425$	−3.32337e8	−0.209999
$426$	0	0
$427$	−2.57103e8	−0.159812
$428$	8.56995e8	0.528354
$429$	0	0
$430$	−4.40064e7	−0.0266917
$431$	3.27067e9	1.96773	0.983867	−	0.178900i	$-0.0572540\pi$
$431$	3.27067e9	1.96773	0.983867	+	0.178900i	$0.0572540\pi$
$432$	0	0
$433$	1.30708e9	0.773741	0.386870	−	0.922134i	$-0.373556\pi$
$433$	1.30708e9	0.773741	0.386870	+	0.922134i	$0.373556\pi$
$434$	−7.43919e6	−0.00436829
$435$	0	0
$436$	1.52977e9	0.883941
$437$	−1.04744e9	−0.600404
$438$	0	0
$439$	2.54539e9	1.43591	0.717957	−	0.696087i	$-0.245077\pi$
$439$	2.54539e9	1.43591	0.717957	+	0.696087i	$0.245077\pi$
$440$	−3.63712e9	−2.03551
$441$	0	0
$442$	3.09554e9	1.70513
$443$	2.69926e8	0.147514	0.0737568	−	0.997276i	$-0.476501\pi$
$443$	2.69926e8	0.147514	0.0737568	+	0.997276i	$0.476501\pi$
$444$	0	0
$445$	−2.35008e9	−1.26422
$446$	−4.27211e9	−2.28018
$447$	0	0
$448$	−7.78885e8	−0.409261
$449$	−9.77302e8	−0.509526	−0.254763	−	0.967004i	$-0.581997\pi$
$449$	−9.77302e8	−0.509526	−0.254763	+	0.967004i	$0.581997\pi$
$450$	0	0
$451$	−5.34528e8	−0.274380
$452$	1.10237e9	0.561493
$453$	0	0
$454$	−3.21585e9	−1.61288
$455$	5.43831e8	0.270660
$456$	0	0
$457$	−2.83182e9	−1.38790	−0.693951	−	0.720022i	$-0.744132\pi$
$457$	−2.83182e9	−1.38790	−0.693951	+	0.720022i	$0.744132\pi$
$458$	1.23821e9	0.602234
$459$	0	0
$460$	−2.22364e9	−1.06515
$461$	3.52138e9	1.67402	0.837008	−	0.547190i	$-0.184303\pi$
$461$	3.52138e9	1.67402	0.837008	+	0.547190i	$0.184303\pi$
$462$	0	0
$463$	1.59163e9	0.745261	0.372631	−	0.927980i	$-0.378456\pi$
$463$	1.59163e9	0.745261	0.372631	+	0.927980i	$0.378456\pi$
$464$	1.06967e9	0.497090
$465$	0	0
$466$	3.53583e9	1.61861
$467$	−1.77030e9	−0.804336	−0.402168	−	0.915566i	$-0.631743\pi$
$467$	−1.77030e9	−0.804336	−0.402168	+	0.915566i	$0.631743\pi$
$468$	0	0
$469$	1.00291e9	0.448909
$470$	3.25941e9	1.44810
$471$	0	0
$472$	−5.34344e9	−2.33896
$473$	4.63868e7	0.0201549
$474$	0	0
$475$	4.65625e8	0.199347
$476$	−1.42833e9	−0.607020
$477$	0	0
$478$	8.08474e8	0.338585
$479$	−2.80890e9	−1.16778	−0.583892	−	0.811831i	$-0.698471\pi$
$479$	−2.80890e9	−1.16778	−0.583892	+	0.811831i	$0.698471\pi$
$480$	0	0
$481$	−2.62351e9	−1.07492
$482$	4.87515e9	1.98300
$483$	0	0
$484$	4.03142e9	1.61622
$485$	−3.93362e9	−1.56565
$486$	0	0
$487$	−3.42599e9	−1.34411	−0.672054	−	0.740502i	$-0.734588\pi$
$487$	−3.42599e9	−1.34411	−0.672054	+	0.740502i	$0.734588\pi$
$488$	1.95128e9	0.760062
$489$	0	0
$490$	4.35423e9	1.67196
$491$	−1.72132e9	−0.656259	−0.328129	−	0.944633i	$-0.606418\pi$
$491$	−1.72132e9	−0.656259	−0.328129	+	0.944633i	$0.606418\pi$
$492$	0	0
$493$	−3.27076e9	−1.22937
$494$	−4.33705e9	−1.61864
$495$	0	0
$496$	1.16780e7	0.00429715
$497$	−1.09871e9	−0.401455
$498$	0	0
$499$	4.16845e9	1.50184	0.750919	−	0.660395i	$-0.229611\pi$
$499$	4.16845e9	1.50184	0.750919	+	0.660395i	$0.229611\pi$
$500$	−4.51387e9	−1.61493
$501$	0	0
$502$	5.19817e9	1.83395
$503$	2.73826e9	0.959370	0.479685	−	0.877441i	$-0.340751\pi$
$503$	2.73826e9	0.959370	0.479685	+	0.877441i	$0.340751\pi$
$504$	0	0
$505$	1.69528e9	0.585764
$506$	3.63712e9	1.24804
$507$	0	0
$508$	4.53238e9	1.53392
$509$	1.76205e9	0.592253	0.296126	−	0.955149i	$-0.404305\pi$
$509$	1.76205e9	0.592253	0.296126	+	0.955149i	$0.404305\pi$
$510$	0	0
$511$	−5.21230e8	−0.172805
$512$	2.77408e9	0.913427
$513$	0	0
$514$	−6.06288e9	−1.96928
$515$	1.54013e9	0.496859
$516$	0	0
$517$	−3.43572e9	−1.09346
$518$	1.87840e9	0.593790
$519$	0	0
$520$	−4.12739e9	−1.28725
$521$	5.30709e9	1.64408	0.822042	−	0.569427i	$-0.192835\pi$
$521$	5.30709e9	1.64408	0.822042	+	0.569427i	$0.192835\pi$
$522$	0	0
$523$	4.17075e9	1.27485	0.637424	−	0.770513i	$-0.280000\pi$
$523$	4.17075e9	1.27485	0.637424	+	0.770513i	$0.280000\pi$
$524$	−8.64320e9	−2.62431
$525$	0	0
$526$	−5.14216e9	−1.54062
$527$	−3.57081e7	−0.0106275
$528$	0	0
$529$	−2.40802e9	−0.707238
$530$	5.93304e9	1.73106
$531$	0	0
$532$	2.00118e9	0.576228
$533$	−6.06581e8	−0.173517
$534$	0	0
$535$	1.12140e9	0.316609
$536$	−7.61158e9	−2.13500
$537$	0	0
$538$	−2.08578e9	−0.577472
$539$	−4.58976e9	−1.26249
$540$	0	0
$541$	1.78986e9	0.485991	0.242996	−	0.970027i	$-0.421870\pi$
$541$	1.78986e9	0.485991	0.242996	+	0.970027i	$0.421870\pi$
$542$	−8.26758e9	−2.23039
$543$	0	0
$544$	−2.50159e9	−0.666224
$545$	2.00175e9	0.529690
$546$	0	0
$547$	2.59509e9	0.677948	0.338974	−	0.940796i	$-0.389920\pi$
$547$	2.59509e9	0.677948	0.338974	+	0.940796i	$0.389920\pi$
$548$	4.92237e9	1.27774
$549$	0	0
$550$	−1.61683e9	−0.414377
$551$	4.58254e9	1.16701
$552$	0	0
$553$	7.01918e8	0.176501
$554$	4.46108e9	1.11470
$555$	0	0
$556$	−5.67631e9	−1.40057
$557$	−3.59472e9	−0.881397	−0.440698	−	0.897655i	$-0.645269\pi$
$557$	−3.59472e9	−0.881397	−0.440698	+	0.897655i	$0.645269\pi$
$558$	0	0
$559$	5.26396e7	0.0127459
$560$	6.11237e8	0.147079
$561$	0	0
$562$	4.04237e9	0.960636
$563$	2.58729e9	0.611035	0.305518	−	0.952186i	$-0.401171\pi$
$563$	2.58729e9	0.611035	0.305518	+	0.952186i	$0.401171\pi$
$564$	0	0
$565$	1.44249e9	0.336467
$566$	1.31044e9	0.303781
$567$	0	0
$568$	8.33864e9	1.90931
$569$	−4.24097e9	−0.965100	−0.482550	−	0.875868i	$-0.660289\pi$
$569$	−4.24097e9	−0.965100	−0.482550	+	0.875868i	$0.660289\pi$
$570$	0	0
$571$	5.88953e9	1.32390	0.661949	−	0.749549i	$-0.269730\pi$
$571$	5.88953e9	1.32390	0.661949	+	0.749549i	$0.269730\pi$
$572$	9.70529e9	2.16831
$573$	0	0
$574$	4.34304e8	0.0958522
$575$	−4.43116e8	−0.0972030
$576$	0	0
$577$	4.36032e8	0.0944936	0.0472468	−	0.998883i	$-0.484955\pi$
$577$	4.36032e8	0.0944936	0.0472468	+	0.998883i	$0.484955\pi$
$578$	−2.85293e9	−0.614531
$579$	0	0
$580$	9.72841e9	2.07035
$581$	−7.05007e8	−0.149134
$582$	0	0
$583$	−6.25398e9	−1.30712
$584$	3.95586e9	0.821856
$585$	0	0
$586$	−1.31277e10	−2.69493
$587$	−5.28188e8	−0.107784	−0.0538921	−	0.998547i	$-0.517163\pi$
$587$	−5.28188e8	−0.107784	−0.0538921	+	0.998547i	$0.517163\pi$
$588$	0	0
$589$	5.00294e7	0.0100884
$590$	−1.55976e10	−3.12663
$591$	0	0
$592$	−2.94868e9	−0.584120
$593$	−8.24070e9	−1.62283	−0.811414	−	0.584471i	$-0.801302\pi$
$593$	−8.24070e9	−1.62283	−0.811414	+	0.584471i	$0.801302\pi$
$594$	0	0
$595$	−1.86900e9	−0.363748
$596$	4.29941e9	0.831855
$597$	0	0
$598$	4.12739e9	0.789261
$599$	−1.08340e9	−0.205966	−0.102983	−	0.994683i	$-0.532839\pi$
$599$	−1.08340e9	−0.205966	−0.102983	+	0.994683i	$0.532839\pi$
$600$	0	0
$601$	9.28702e6	0.00174508	0.000872541	−	1.00000i	$-0.499722\pi$
$601$	9.28702e6	0.00174508	0.000872541		1.00000i	$0.499722\pi$
$602$	−3.76893e7	−0.00704093
$603$	0	0
$604$	−1.18808e10	−2.19389
$605$	5.27523e9	0.968496
$606$	0	0
$607$	−3.72021e9	−0.675161	−0.337580	−	0.941297i	$-0.609608\pi$
$607$	−3.72021e9	−0.675161	−0.337580	+	0.941297i	$0.609608\pi$
$608$	3.50489e9	0.632429
$609$	0	0
$610$	5.69582e9	1.01602
$611$	−3.89885e9	−0.691500
$612$	0	0
$613$	8.10111e9	1.42047	0.710236	−	0.703963i	$-0.248588\pi$
$613$	8.10111e9	1.42047	0.710236	+	0.703963i	$0.248588\pi$
$614$	1.70203e10	2.96741
$615$	0	0
$616$	−3.11501e9	−0.536941
$617$	5.74845e9	0.985264	0.492632	−	0.870238i	$-0.336035\pi$
$617$	5.74845e9	0.985264	0.492632	+	0.870238i	$0.336035\pi$
$618$	0	0
$619$	−2.12511e9	−0.360134	−0.180067	−	0.983654i	$-0.557632\pi$
$619$	−2.12511e9	−0.360134	−0.180067	+	0.983654i	$0.557632\pi$
$620$	1.06209e8	0.0178974
$621$	0	0
$622$	3.09704e9	0.516036
$623$	−2.01273e9	−0.333485
$624$	0	0
$625$	−7.00302e9	−1.14737
$626$	−1.30205e10	−2.12138
$627$	0	0
$628$	2.23605e9	0.360266
$629$	9.01630e9	1.44461
$630$	0	0
$631$	−1.09353e10	−1.73272	−0.866362	−	0.499417i	$-0.833548\pi$
$631$	−1.09353e10	−1.73272	−0.866362	+	0.499417i	$0.833548\pi$
$632$	−5.32718e9	−0.839436
$633$	0	0
$634$	9.94167e9	1.54934
$635$	5.93076e9	0.919183
$636$	0	0
$637$	−5.20845e9	−0.798400
$638$	−1.59124e10	−2.42584
$639$	0	0
$640$	1.31501e10	1.98290
$641$	1.71030e9	0.256490	0.128245	−	0.991743i	$-0.459066\pi$
$641$	1.71030e9	0.256490	0.128245	+	0.991743i	$0.459066\pi$
$642$	0	0
$643$	6.49295e9	0.963172	0.481586	−	0.876399i	$-0.340061\pi$
$643$	6.49295e9	0.963172	0.481586	+	0.876399i	$0.340061\pi$
$644$	−1.90443e9	−0.280974
$645$	0	0
$646$	1.49053e10	2.17534
$647$	−4.41995e9	−0.641582	−0.320791	−	0.947150i	$-0.603949\pi$
$647$	−4.41995e9	−0.641582	−0.320791	+	0.947150i	$0.603949\pi$
$648$	0	0
$649$	1.64413e10	2.36092
$650$	−1.83478e9	−0.262051
$651$	0	0
$652$	−1.66731e10	−2.35585
$653$	−2.06198e9	−0.289793	−0.144896	−	0.989447i	$-0.546285\pi$
$653$	−2.06198e9	−0.289793	−0.144896	+	0.989447i	$0.546285\pi$
$654$	0	0
$655$	−1.13099e10	−1.57258
$656$	−6.81765e8	−0.0942912
$657$	0	0
$658$	2.79153e9	0.381989
$659$	5.97125e9	0.812767	0.406383	−	0.913703i	$-0.366790\pi$
$659$	5.97125e9	0.812767	0.406383	+	0.913703i	$0.366790\pi$
$660$	0	0
$661$	−5.91741e8	−0.0796942	−0.0398471	−	0.999206i	$-0.512687\pi$
$661$	−5.91741e8	−0.0796942	−0.0398471	+	0.999206i	$0.512687\pi$
$662$	1.08582e10	1.45464
$663$	0	0
$664$	5.35063e9	0.709278
$665$	2.61860e9	0.345297
$666$	0	0
$667$	−4.36101e9	−0.569045
$668$	1.98726e9	0.257951
$669$	0	0
$670$	−2.22184e10	−2.85398
$671$	−6.00392e9	−0.767196
$672$	0	0
$673$	2.50848e9	0.317219	0.158609	−	0.987341i	$-0.449299\pi$
$673$	2.50848e9	0.317219	0.158609	+	0.987341i	$0.449299\pi$
$674$	1.05661e10	1.32925
$675$	0	0
$676$	−3.54413e9	−0.441262
$677$	−1.17984e10	−1.46138	−0.730689	−	0.682711i	$-0.760801\pi$
$677$	−1.17984e10	−1.46138	−0.730689	+	0.682711i	$0.760801\pi$
$678$	0	0
$679$	−3.36895e9	−0.413000
$680$	1.41847e10	1.72998
$681$	0	0
$682$	−1.73722e8	−0.0209705
$683$	5.40513e9	0.649133	0.324567	−	0.945863i	$-0.394782\pi$
$683$	5.40513e9	0.649133	0.324567	+	0.945863i	$0.394782\pi$
$684$	0	0
$685$	6.44106e9	0.765668
$686$	7.79185e9	0.921523
$687$	0	0
$688$	5.91642e7	0.00692627
$689$	−7.09699e9	−0.826622
$690$	0	0
$691$	1.38617e10	1.59825	0.799125	−	0.601165i	$-0.205296\pi$
$691$	1.38617e10	1.59825	0.799125	+	0.601165i	$0.205296\pi$
$692$	−1.15319e10	−1.32291
$693$	0	0
$694$	−1.68771e10	−1.91663
$695$	−7.42762e9	−0.839272
$696$	0	0
$697$	2.08466e9	0.233196
$698$	−1.48129e10	−1.64871
$699$	0	0
$700$	8.46591e8	0.0932890
$701$	−6.02060e9	−0.660126	−0.330063	−	0.943959i	$-0.607070\pi$
$701$	−6.02060e9	−0.660126	−0.330063	+	0.943959i	$0.607070\pi$
$702$	0	0
$703$	−1.26324e10	−1.37133
$704$	−1.81887e10	−1.96470
$705$	0	0
$706$	−1.02556e10	−1.09684
$707$	1.45193e9	0.154517
$708$	0	0
$709$	6.36497e9	0.670710	0.335355	−	0.942092i	$-0.391144\pi$
$709$	6.36497e9	0.670710	0.335355	+	0.942092i	$0.391144\pi$
$710$	2.43407e10	2.55228
$711$	0	0
$712$	1.52755e10	1.58605
$713$	−4.76108e7	−0.00491918
$714$	0	0
$715$	1.26996e10	1.29933
$716$	1.32944e10	1.35355
$717$	0	0
$718$	−1.31604e9	−0.132689
$719$	1.25812e10	1.26232	0.631162	−	0.775651i	$-0.282578\pi$
$719$	1.25812e10	1.26232	0.631162	+	0.775651i	$0.282578\pi$
$720$	0	0
$721$	1.31905e9	0.131065
$722$	−3.92328e9	−0.387944
$723$	0	0
$724$	1.89253e10	1.85335
$725$	1.93863e9	0.188935
$726$	0	0
$727$	2.10511e9	0.203191	0.101596	−	0.994826i	$-0.467605\pi$
$727$	2.10511e9	0.203191	0.101596	+	0.994826i	$0.467605\pi$
$728$	−3.53490e9	−0.339561
$729$	0	0
$730$	1.15472e10	1.09862
$731$	−1.80909e8	−0.0171296
$732$	0	0
$733$	−4.15189e9	−0.389388	−0.194694	−	0.980864i	$-0.562371\pi$
$733$	−4.15189e9	−0.389388	−0.194694	+	0.980864i	$0.562371\pi$
$734$	7.08546e9	0.661350
$735$	0	0
$736$	−3.33545e9	−0.308378
$737$	2.34202e10	2.15504
$738$	0	0
$739$	−1.05151e10	−0.958429	−0.479214	−	0.877698i	$-0.659078\pi$
$739$	−1.05151e10	−0.958429	−0.479214	+	0.877698i	$0.659078\pi$
$740$	−2.68177e10	−2.43283
$741$	0	0
$742$	5.08136e9	0.456632
$743$	−3.59871e9	−0.321874	−0.160937	−	0.986965i	$-0.551452\pi$
$743$	−3.59871e9	−0.321874	−0.160937	+	0.986965i	$0.551452\pi$
$744$	0	0
$745$	5.62591e9	0.498478
$746$	−5.27861e9	−0.465515
$747$	0	0
$748$	−3.33545e10	−2.91407
$749$	9.60426e8	0.0835175
$750$	0	0
$751$	7.44144e8	0.0641087	0.0320544	−	0.999486i	$-0.489795\pi$
$751$	7.44144e8	0.0641087	0.0320544	+	0.999486i	$0.489795\pi$
$752$	−4.38210e9	−0.375768
$753$	0	0
$754$	−1.80573e10	−1.53410
$755$	−1.55463e10	−1.31466
$756$	0	0
$757$	9.80850e9	0.821802	0.410901	−	0.911680i	$-0.365214\pi$
$757$	9.80850e9	0.821802	0.410901	+	0.911680i	$0.365214\pi$
$758$	2.59657e10	2.16550
$759$	0	0
$760$	−1.98738e10	−1.64222
$761$	−1.73200e10	−1.42463	−0.712315	−	0.701860i	$-0.752353\pi$
$761$	−1.73200e10	−1.42463	−0.712315	+	0.701860i	$0.752353\pi$
$762$	0	0
$763$	1.71440e9	0.139725
$764$	−3.71731e10	−3.01580
$765$	0	0
$766$	3.94557e10	3.17182
$767$	1.86576e10	1.49304
$768$	0	0
$769$	−2.35092e10	−1.86421	−0.932107	−	0.362183i	$-0.882032\pi$
$769$	−2.35092e10	−1.86421	−0.932107	+	0.362183i	$0.882032\pi$
$770$	−9.09279e9	−0.717760
$771$	0	0
$772$	1.39286e10	1.08955
$773$	1.71140e10	1.33268	0.666338	−	0.745650i	$-0.267861\pi$
$773$	1.71140e10	1.33268	0.666338	+	0.745650i	$0.267861\pi$
$774$	0	0
$775$	2.11648e7	0.00163327
$776$	2.55685e10	1.96422
$777$	0	0
$778$	−3.73389e9	−0.284271
$779$	−2.92074e9	−0.221367
$780$	0	0
$781$	−2.56573e10	−1.92723
$782$	−1.41847e10	−1.06071
$783$	0	0
$784$	−5.85402e9	−0.433859
$785$	2.92594e9	0.215884
$786$	0	0
$787$	1.03332e10	0.755654	0.377827	−	0.925876i	$-0.376671\pi$
$787$	1.03332e10	0.755654	0.377827	+	0.925876i	$0.376671\pi$
$788$	4.41236e10	3.21240
$789$	0	0
$790$	−1.55502e10	−1.12212
$791$	1.23542e9	0.0887558
$792$	0	0
$793$	−6.81323e9	−0.485173
$794$	−5.67603e9	−0.402414
$795$	0	0
$796$	4.50082e10	3.16298
$797$	−2.96320e9	−0.207328	−0.103664	−	0.994612i	$-0.533057\pi$
$797$	−2.96320e9	−0.207328	−0.103664	+	0.994612i	$0.533057\pi$
$798$	0	0
$799$	1.33993e10	0.929328
$800$	1.48273e9	0.102388
$801$	0	0
$802$	1.60354e10	1.09767
$803$	−1.21719e10	−0.829569
$804$	0	0
$805$	−2.49201e9	−0.168370
$806$	−1.97139e8	−0.0132617
$807$	0	0
$808$	−1.10193e10	−0.734879
$809$	−2.47434e10	−1.64301	−0.821504	−	0.570203i	$-0.806864\pi$
$809$	−2.47434e10	−1.64301	−0.821504	+	0.570203i	$0.806864\pi$
$810$	0	0
$811$	2.30066e8	0.0151454	0.00757269	−	0.999971i	$-0.497590\pi$
$811$	2.30066e8	0.0151454	0.00757269	+	0.999971i	$0.497590\pi$
$812$	8.33190e9	0.546132
$813$	0	0
$814$	4.38647e10	2.85056
$815$	−2.18172e10	−1.41171
$816$	0	0
$817$	2.53465e8	0.0162607
$818$	−2.03528e10	−1.30013
$819$	0	0
$820$	−6.20052e9	−0.392717
$821$	−3.04369e9	−0.191955	−0.0959774	−	0.995384i	$-0.530598\pi$
$821$	−3.04369e9	−0.191955	−0.0959774	+	0.995384i	$0.530598\pi$
$822$	0	0
$823$	−2.37505e9	−0.148516	−0.0742582	−	0.997239i	$-0.523659\pi$
$823$	−2.37505e9	−0.148516	−0.0742582	+	0.997239i	$0.523659\pi$
$824$	−1.00109e10	−0.623342
$825$	0	0
$826$	−1.33586e10	−0.824766
$827$	5.58209e9	0.343184	0.171592	−	0.985168i	$-0.445109\pi$
$827$	5.58209e9	0.343184	0.171592	+	0.985168i	$0.445109\pi$
$828$	0	0
$829$	−2.06578e10	−1.25934	−0.629670	−	0.776863i	$-0.716810\pi$
$829$	−2.06578e10	−1.25934	−0.629670	+	0.776863i	$0.716810\pi$
$830$	1.56186e10	0.948133
$831$	0	0
$832$	−2.06405e10	−1.24247
$833$	1.79001e10	1.07299
$834$	0	0
$835$	2.60039e9	0.154574
$836$	4.67319e10	2.76625
$837$	0	0
$838$	−2.68385e10	−1.57545
$839$	7.55935e9	0.441894	0.220947	−	0.975286i	$-0.429085\pi$
$839$	7.55935e9	0.441894	0.220947	+	0.975286i	$0.429085\pi$
$840$	0	0
$841$	1.82955e9	0.106061
$842$	−2.68595e10	−1.55062
$843$	0	0
$844$	−2.43360e9	−0.139332
$845$	−4.63759e9	−0.264420
$846$	0	0
$847$	4.51798e9	0.255477
$848$	−7.97665e9	−0.449195
$849$	0	0
$850$	6.30564e9	0.352179
$851$	1.20217e10	0.668673
$852$	0	0
$853$	2.47919e10	1.36769	0.683846	−	0.729626i	$-0.260306\pi$
$853$	2.47919e10	1.36769	0.683846	+	0.729626i	$0.260306\pi$
$854$	4.87819e9	0.268013
$855$	0	0
$856$	−7.28912e9	−0.397207
$857$	2.93883e9	0.159493	0.0797464	−	0.996815i	$-0.474589\pi$
$857$	2.93883e9	0.159493	0.0797464	+	0.996815i	$0.474589\pi$
$858$	0	0
$859$	2.08095e10	1.12017	0.560086	−	0.828435i	$-0.310768\pi$
$859$	2.08095e10	1.12017	0.560086	+	0.828435i	$0.310768\pi$
$860$	5.38087e8	0.0288475
$861$	0	0
$862$	−6.20566e10	−3.29999
$863$	2.36529e10	1.25270	0.626350	−	0.779542i	$-0.284548\pi$
$863$	2.36529e10	1.25270	0.626350	+	0.779542i	$0.284548\pi$
$864$	0	0
$865$	−1.50898e10	−0.792734
$866$	−2.48001e10	−1.29760
$867$	0	0
$868$	9.09626e7	0.00472110
$869$	1.63913e10	0.847315
$870$	0	0
$871$	2.65772e10	1.36284
$872$	−1.30114e10	−0.664530
$873$	0	0
$874$	1.98738e10	1.00691
$875$	−5.05865e9	−0.255274
$876$	0	0
$877$	−1.67295e10	−0.837501	−0.418751	−	0.908101i	$-0.637532\pi$
$877$	−1.67295e10	−0.837501	−0.418751	+	0.908101i	$0.637532\pi$
$878$	−4.82954e10	−2.40810
$879$	0	0
$880$	1.42737e10	0.706071
$881$	−2.71046e10	−1.33545	−0.667724	−	0.744409i	$-0.732731\pi$
$881$	−2.71046e10	−1.33545	−0.667724	+	0.744409i	$0.732731\pi$
$882$	0	0
$883$	−1.22708e10	−0.599805	−0.299903	−	0.953970i	$-0.596954\pi$
$883$	−1.22708e10	−0.599805	−0.299903	+	0.953970i	$0.596954\pi$
$884$	−3.78506e10	−1.84285
$885$	0	0
$886$	−5.12149e9	−0.247388
$887$	3.38351e10	1.62793	0.813963	−	0.580917i	$-0.197306\pi$
$887$	3.38351e10	1.62793	0.813963	+	0.580917i	$0.197306\pi$
$888$	0	0
$889$	5.07940e9	0.242469
$890$	4.45896e10	2.12016
$891$	0	0
$892$	5.22371e10	2.46435
$893$	−1.87733e10	−0.882188
$894$	0	0
$895$	1.73961e10	0.811095
$896$	1.12624e10	0.523063
$897$	0	0
$898$	1.85430e10	0.854501
$899$	2.08297e8	0.00956148
$900$	0	0
$901$	2.43905e10	1.11092
$902$	1.01420e10	0.460149
$903$	0	0
$904$	−9.37617e9	−0.422120
$905$	2.47643e10	1.11059
$906$	0	0
$907$	5.42470e9	0.241407	0.120704	−	0.992689i	$-0.461485\pi$
$907$	5.42470e9	0.241407	0.120704	+	0.992689i	$0.461485\pi$
$908$	3.93218e10	1.74314
$909$	0	0
$910$	−1.03185e10	−0.453910
$911$	−3.91181e10	−1.71421	−0.857104	−	0.515144i	$-0.827738\pi$
$911$	−3.91181e10	−1.71421	−0.857104	+	0.515144i	$0.827738\pi$
$912$	0	0
$913$	−1.64635e10	−0.715935
$914$	5.37300e10	2.32758
$915$	0	0
$916$	−1.51402e10	−0.650875
$917$	−9.68634e9	−0.414827
$918$	0	0
$919$	−1.11381e10	−0.473375	−0.236688	−	0.971586i	$-0.576062\pi$
$919$	−1.11381e10	−0.473375	−0.236688	+	0.971586i	$0.576062\pi$
$920$	1.89130e10	0.800761
$921$	0	0
$922$	−6.68135e10	−2.80741
$923$	−2.91159e10	−1.21878
$924$	0	0
$925$	−5.34411e9	−0.222013
$926$	−3.01990e10	−1.24984
$927$	0	0
$928$	1.45926e10	0.599398
$929$	4.22587e10	1.72926	0.864631	−	0.502408i	$-0.167552\pi$
$929$	4.22587e10	1.72926	0.864631	+	0.502408i	$0.167552\pi$
$930$	0	0
$931$	−2.50792e10	−1.01857
$932$	−4.32343e10	−1.74933
$933$	0	0
$934$	3.35891e10	1.34891
$935$	−4.36454e10	−1.74621
$936$	0	0
$937$	3.74502e10	1.48719	0.743594	−	0.668631i	$-0.233120\pi$
$937$	3.74502e10	1.48719	0.743594	+	0.668631i	$0.233120\pi$
$938$	−1.90289e10	−0.752844
$939$	0	0
$940$	−3.98544e10	−1.56505
$941$	−1.04380e10	−0.408369	−0.204185	−	0.978932i	$-0.565454\pi$
$941$	−1.04380e10	−0.408369	−0.204185	+	0.978932i	$0.565454\pi$
$942$	0	0
$943$	2.77955e9	0.107940
$944$	2.09701e10	0.811334
$945$	0	0
$946$	−8.80128e8	−0.0338008
$947$	9.12893e9	0.349297	0.174649	−	0.984631i	$-0.444121\pi$
$947$	9.12893e9	0.349297	0.174649	+	0.984631i	$0.444121\pi$
$948$	0	0
$949$	−1.38126e10	−0.524618
$950$	−8.83462e9	−0.334314
$951$	0	0
$952$	1.21485e10	0.456346
$953$	−1.00452e10	−0.375954	−0.187977	−	0.982173i	$-0.560193\pi$
$953$	−1.00452e10	−0.375954	−0.187977	+	0.982173i	$0.560193\pi$
$954$	0	0
$955$	−4.86420e10	−1.80717
$956$	−9.88559e9	−0.365932
$957$	0	0
$958$	5.32952e10	1.95843
$959$	5.51645e9	0.201974
$960$	0	0
$961$	−2.75103e10	−0.999917
$962$	4.97775e10	1.80269
$963$	0	0
$964$	−5.96107e10	−2.14316
$965$	1.82260e10	0.652900
$966$	0	0
$967$	3.17355e10	1.12863	0.564316	−	0.825559i	$-0.309140\pi$
$967$	3.17355e10	1.12863	0.564316	+	0.825559i	$0.309140\pi$
$968$	−3.42890e10	−1.21504
$969$	0	0
$970$	7.46351e10	2.62568
$971$	3.26126e10	1.14319	0.571594	−	0.820537i	$-0.306325\pi$
$971$	3.26126e10	1.14319	0.571594	+	0.820537i	$0.306325\pi$
$972$	0	0
$973$	−6.36138e9	−0.221389
$974$	6.50035e10	2.25414
$975$	0	0
$976$	−7.65772e9	−0.263648
$977$	−3.72925e10	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$977$	−3.72925e10	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$978$	0	0
$979$	−4.70016e10	−1.60093
$980$	−5.32413e10	−1.80700
$981$	0	0
$982$	3.26597e10	1.10058
$983$	9.42117e9	0.316350	0.158175	−	0.987411i	$-0.449439\pi$
$983$	9.42117e9	0.316350	0.158175	+	0.987411i	$0.449439\pi$
$984$	0	0
$985$	5.77370e10	1.92499
$986$	6.20583e10	2.06172
$987$	0	0
$988$	5.30312e10	1.74937
$989$	−2.41211e8	−0.00792886
$990$	0	0
$991$	1.16481e9	0.0380188	0.0190094	−	0.999819i	$-0.493949\pi$
$991$	1.16481e9	0.0380188	0.0190094	+	0.999819i	$0.493949\pi$
$992$	1.59313e8	0.00518157
$993$	0	0
$994$	2.08466e10	0.673260
$995$	5.88945e10	1.89537
$996$	0	0
$997$	2.18933e10	0.699645	0.349822	−	0.936816i	$-0.386242\pi$
$997$	2.18933e10	0.699645	0.349822	+	0.936816i	$0.386242\pi$
$998$	−7.90908e10	−2.51866
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.8.a.b.1.1	✓	2
3.2	odd	2	inner	9.8.a.b.1.2	yes	2
4.3	odd	2		144.8.a.m.1.2		2
5.2	odd	4		225.8.b.k.199.2		4
5.3	odd	4		225.8.b.k.199.3		4
5.4	even	2		225.8.a.q.1.2		2
7.6	odd	2		441.8.a.k.1.1		2
8.3	odd	2		576.8.a.bi.1.1		2
8.5	even	2		576.8.a.bj.1.1		2
9.2	odd	6		81.8.c.f.28.1		4
9.4	even	3		81.8.c.f.55.2		4
9.5	odd	6		81.8.c.f.55.1		4
9.7	even	3		81.8.c.f.28.2		4
12.11	even	2		144.8.a.m.1.1		2
15.2	even	4		225.8.b.k.199.4		4
15.8	even	4		225.8.b.k.199.1		4
15.14	odd	2		225.8.a.q.1.1		2
21.20	even	2		441.8.a.k.1.2		2
24.5	odd	2		576.8.a.bj.1.2		2
24.11	even	2		576.8.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.8.a.b.1.1	✓	2	1.1	even	1	trivial
9.8.a.b.1.2	yes	2	3.2	odd	2	inner
81.8.c.f.28.1		4	9.2	odd	6
81.8.c.f.28.2		4	9.7	even	3
81.8.c.f.55.1		4	9.5	odd	6
81.8.c.f.55.2		4	9.4	even	3
144.8.a.m.1.1		2	12.11	even	2
144.8.a.m.1.2		2	4.3	odd	2
225.8.a.q.1.1		2	15.14	odd	2
225.8.a.q.1.2		2	5.4	even	2
225.8.b.k.199.1		4	15.8	even	4
225.8.b.k.199.2		4	5.2	odd	4
225.8.b.k.199.3		4	5.3	odd	4
225.8.b.k.199.4		4	15.2	even	4
441.8.a.k.1.1		2	7.6	odd	2
441.8.a.k.1.2		2	21.20	even	2
576.8.a.bi.1.1		2	8.3	odd	2
576.8.a.bi.1.2		2	24.11	even	2
576.8.a.bj.1.1		2	8.5	even	2
576.8.a.bj.1.2		2	24.5	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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q-18.9737 q^{2} +232.000 q^{4} +303.579 q^{5} +260.000 q^{7} -1973.26 q^{8} +O(q^{10})\)	\(q-18.9737 q^{2} +232.000 q^{4} +303.579 q^{5} +260.000 q^{7} -1973.26 q^{8} -5760.00 q^{10} +6071.57 q^{11} +6890.00 q^{13} -4933.15 q^{14} +7744.00 q^{16} -23679.1 q^{17} +33176.0 q^{19} +70430.2 q^{20} -115200. q^{22} -31572.2 q^{23} +14035.0 q^{25} -130729. q^{26} +60320.0 q^{28} +138128. q^{29} +1508.00 q^{31} +105645. q^{32} +449280. q^{34} +78930.5 q^{35} -380770. q^{37} -629470. q^{38} -599040. q^{40} -88037.8 q^{41} +7640.00 q^{43} +1.40860e6 q^{44} +599040. q^{46} -565871. q^{47} -755943. q^{49} -266295. q^{50} +1.59848e6 q^{52} -1.03004e6 q^{53} +1.84320e6 q^{55} -513048. q^{56} -2.62080e6 q^{58} +2.70792e6 q^{59} -988858. q^{61} -28612.3 q^{62} -2.99571e6 q^{64} +2.09166e6 q^{65} +3.85736e6 q^{67} -5.49356e6 q^{68} -1.49760e6 q^{70} -4.22581e6 q^{71} -2.00473e6 q^{73} +7.22460e6 q^{74} +7.69683e6 q^{76} +1.57861e6 q^{77} +2.69968e6 q^{79} +2.35091e6 q^{80} +1.67040e6 q^{82} -2.71156e6 q^{83} -7.18848e6 q^{85} -144959. q^{86} -1.19808e7 q^{88} -7.74126e6 q^{89} +1.79140e6 q^{91} -7.32475e6 q^{92} +1.07366e7 q^{94} +1.00715e7 q^{95} -1.29575e7 q^{97} +1.43430e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 464 q^{4} + 520 q^{7}+O(q^{10}) \) 2 * q + 464 * q^4 + 520 * q^7	\( 2 q + 464 q^{4} + 520 q^{7} - 11520 q^{10} + 13780 q^{13} + 15488 q^{16} + 66352 q^{19} - 230400 q^{22} + 28070 q^{25} + 120640 q^{28} + 3016 q^{31} + 898560 q^{34} - 761540 q^{37} - 1198080 q^{40} + 15280 q^{43} + 1198080 q^{46} - 1511886 q^{49} + 3196960 q^{52} + 3686400 q^{55} - 5241600 q^{58} - 1977716 q^{61} - 5991424 q^{64} + 7714720 q^{67} - 2995200 q^{70} - 4009460 q^{73} + 15393664 q^{76} + 5399368 q^{79} + 3340800 q^{82} - 14376960 q^{85} - 23961600 q^{88} + 3582800 q^{91} + 21473280 q^{94} - 25914980 q^{97}+O(q^{100}) \) 2 * q + 464 * q^4 + 520 * q^7 - 11520 * q^10 + 13780 * q^13 + 15488 * q^16 + 66352 * q^19 - 230400 * q^22 + 28070 * q^25 + 120640 * q^28 + 3016 * q^31 + 898560 * q^34 - 761540 * q^37 - 1198080 * q^40 + 15280 * q^43 + 1198080 * q^46 - 1511886 * q^49 + 3196960 * q^52 + 3686400 * q^55 - 5241600 * q^58 - 1977716 * q^61 - 5991424 * q^64 + 7714720 * q^67 - 2995200 * q^70 - 4009460 * q^73 + 15393664 * q^76 + 5399368 * q^79 + 3340800 * q^82 - 14376960 * q^85 - 23961600 * q^88 + 3582800 * q^91 + 21473280 * q^94 - 25914980 * q^97

Introduction

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Database

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