LMFDB - Embedded newform 9.5.b.a.8.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,5,Mod(8,9)]

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([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("9.8");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	9.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$0.930329667755$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			8.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	9.8
Dual form			9.5.b.a.8.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{2} -2.00000 q^{4} +29.6985i q^{5} -28.0000 q^{7} -59.3970i q^{8} +O(q^{10})$	\(q-4.24264i q^{2} -2.00000 q^{4} +29.6985i q^{5} -28.0000 q^{7} -59.3970i q^{8} +126.000 q^{10} -16.9706i q^{11} -112.000 q^{13} +118.794i q^{14} -284.000 q^{16} +89.0955i q^{17} +560.000 q^{19} -59.3970i q^{20} -72.0000 q^{22} -797.616i q^{23} -257.000 q^{25} +475.176i q^{26} +56.0000 q^{28} +988.535i q^{29} -364.000 q^{31} +254.558i q^{32} +378.000 q^{34} -831.558i q^{35} -826.000 q^{37} -2375.88i q^{38} +1764.00 q^{40} +1811.61i q^{41} +1736.00 q^{43} +33.9411i q^{44} -3384.00 q^{46} +1306.73i q^{47} -1617.00 q^{49} +1090.36i q^{50} +224.000 q^{52} -1794.64i q^{53} +504.000 q^{55} +1663.12i q^{56} +4194.00 q^{58} -4514.17i q^{59} +2618.00 q^{61} +1544.32i q^{62} -3464.00 q^{64} -3326.23i q^{65} -3784.00 q^{67} -178.191i q^{68} -3528.00 q^{70} +8604.08i q^{71} +6608.00 q^{73} +3504.42i q^{74} -1120.00 q^{76} +475.176i q^{77} -4276.00 q^{79} -8434.37i q^{80} +7686.00 q^{82} -118.794i q^{83} -2646.00 q^{85} -7365.22i q^{86} -1008.00 q^{88} -4365.68i q^{89} +3136.00 q^{91} +1595.23i q^{92} +5544.00 q^{94} +16631.2i q^{95} -5824.00 q^{97} +6860.35i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 56 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 - 56 * q^7	$ 2 q - 4 q^{4} - 56 q^{7} + 252 q^{10} - 224 q^{13} - 568 q^{16} + 1120 q^{19} - 144 q^{22} - 514 q^{25} + 112 q^{28} - 728 q^{31} + 756 q^{34} - 1652 q^{37} + 3528 q^{40} + 3472 q^{43} - 6768 q^{46} - 3234 q^{49} + 448 q^{52} + 1008 q^{55} + 8388 q^{58} + 5236 q^{61} - 6928 q^{64} - 7568 q^{67} - 7056 q^{70} + 13216 q^{73} - 2240 q^{76} - 8552 q^{79} + 15372 q^{82} - 5292 q^{85} - 2016 q^{88} + 6272 q^{91} + 11088 q^{94} - 11648 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 56 * q^7 + 252 * q^10 - 224 * q^13 - 568 * q^16 + 1120 * q^19 - 144 * q^22 - 514 * q^25 + 112 * q^28 - 728 * q^31 + 756 * q^34 - 1652 * q^37 + 3528 * q^40 + 3472 * q^43 - 6768 * q^46 - 3234 * q^49 + 448 * q^52 + 1008 * q^55 + 8388 * q^58 + 5236 * q^61 - 6928 * q^64 - 7568 * q^67 - 7056 * q^70 + 13216 * q^73 - 2240 * q^76 - 8552 * q^79 + 15372 * q^82 - 5292 * q^85 - 2016 * q^88 + 6272 * q^91 + 11088 * q^94 - 11648 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 4.24264i	− 1.06066i	−0.847791	−	0.530330i	$-0.822068\pi$
$2$	− 4.24264i	− 1.06066i	0.847791	−	0.530330i	$-0.177932\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.125000
$5$	29.6985i	1.18794i	0.804487	+	0.593970i	$0.202440\pi$
$5$	29.6985i	1.18794i	−0.804487	+	0.593970i	$0.797560\pi$
$6$	0	0
$7$	−28.0000	−0.571429	−0.285714	−	0.958315i	$-0.592231\pi$
$7$	−28.0000	−0.571429	−0.285714	+	0.958315i	$0.592231\pi$
$8$	− 59.3970i	− 0.928078i
$9$	0	0
$10$	126.000	1.26000
$11$	− 16.9706i	− 0.140253i	−0.997538	−	0.0701263i	$-0.977660\pi$
$11$	− 16.9706i	− 0.140253i	0.997538	−	0.0701263i	$-0.0223402\pi$
$12$	0	0
$13$	−112.000	−0.662722	−0.331361	−	0.943504i	$-0.607508\pi$
$13$	−112.000	−0.662722	−0.331361	+	0.943504i	$0.607508\pi$
$14$	118.794i	0.606092i
$15$	0	0
$16$	−284.000	−1.10938
$17$	89.0955i	0.308289i	0.988048	+	0.154144i	$0.0492621\pi$
$17$	89.0955i	0.308289i	−0.988048	+	0.154144i	$0.950738\pi$
$18$	0	0
$19$	560.000	1.55125	0.775623	−	0.631196i	$-0.217436\pi$
$19$	560.000	1.55125	0.775623	+	0.631196i	$0.217436\pi$
$20$	− 59.3970i	− 0.148492i
$21$	0	0
$22$	−72.0000	−0.148760
$23$	− 797.616i	− 1.50778i	−0.657000	−	0.753891i	$-0.728175\pi$
$23$	− 797.616i	− 1.50778i	0.657000	−	0.753891i	$-0.271825\pi$
$24$	0	0
$25$	−257.000	−0.411200
$26$	475.176i	0.702923i
$27$	0	0
$28$	56.0000	0.0714286
$29$	988.535i	1.17543i	0.809069	+	0.587714i	$0.199972\pi$
$29$	988.535i	1.17543i	−0.809069	+	0.587714i	$0.800028\pi$
$30$	0	0
$31$	−364.000	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$31$	−364.000	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$32$	254.558i	0.248592i
$33$	0	0
$34$	378.000	0.326990
$35$	− 831.558i	− 0.678823i
$36$	0	0
$37$	−826.000	−0.603360	−0.301680	−	0.953409i	$-0.597547\pi$
$37$	−826.000	−0.603360	−0.301680	+	0.953409i	$0.597547\pi$
$38$	− 2375.88i	− 1.64535i
$39$	0	0
$40$	1764.00	1.10250
$41$	1811.61i	1.07770i	0.842403	+	0.538848i	$0.181140\pi$
$41$	1811.61i	1.07770i	−0.842403	+	0.538848i	$0.818860\pi$
$42$	0	0
$43$	1736.00	0.938886	0.469443	−	0.882963i	$-0.344455\pi$
$43$	1736.00	0.938886	0.469443	+	0.882963i	$0.344455\pi$
$44$	33.9411i	0.0175316i
$45$	0	0
$46$	−3384.00	−1.59924
$47$	1306.73i	0.591550i	0.955258	+	0.295775i	$0.0955778\pi$
$47$	1306.73i	0.591550i	−0.955258	+	0.295775i	$0.904422\pi$
$48$	0	0
$49$	−1617.00	−0.673469
$50$	1090.36i	0.436143i
$51$	0	0
$52$	224.000	0.0828402
$53$	− 1794.64i	− 0.638888i	−0.947605	−	0.319444i	$-0.896504\pi$
$53$	− 1794.64i	− 0.638888i	0.947605	−	0.319444i	$-0.103496\pi$
$54$	0	0
$55$	504.000	0.166612
$56$	1663.12i	0.530330i
$57$	0	0
$58$	4194.00	1.24673
$59$	− 4514.17i	− 1.29680i	−0.761299	−	0.648401i	$-0.775438\pi$
$59$	− 4514.17i	− 1.29680i	0.761299	−	0.648401i	$-0.224562\pi$
$60$	0	0
$61$	2618.00	0.703574	0.351787	−	0.936080i	$-0.385574\pi$
$61$	2618.00	0.703574	0.351787	+	0.936080i	$0.385574\pi$
$62$	1544.32i	0.401748i
$63$	0	0
$64$	−3464.00	−0.845703
$65$	− 3326.23i	− 0.787273i
$66$	0	0
$67$	−3784.00	−0.842949	−0.421475	−	0.906840i	$-0.638487\pi$
$67$	−3784.00	−0.842949	−0.421475	+	0.906840i	$0.638487\pi$
$68$	− 178.191i	− 0.0385361i
$69$	0	0
$70$	−3528.00	−0.720000
$71$	8604.08i	1.70682i	0.521241	+	0.853410i	$0.325469\pi$
$71$	8604.08i	1.70682i	−0.521241	+	0.853410i	$0.674531\pi$
$72$	0	0
$73$	6608.00	1.24001	0.620004	−	0.784599i	$-0.287131\pi$
$73$	6608.00	1.24001	0.620004	+	0.784599i	$0.287131\pi$
$74$	3504.42i	0.639960i
$75$	0	0
$76$	−1120.00	−0.193906
$77$	475.176i	0.0801443i
$78$	0	0
$79$	−4276.00	−0.685147	−0.342573	−	0.939491i	$-0.611299\pi$
$79$	−4276.00	−0.685147	−0.342573	+	0.939491i	$0.611299\pi$
$80$	− 8434.37i	− 1.31787i
$81$	0	0
$82$	7686.00	1.14307
$83$	− 118.794i	− 0.0172440i	−0.999963	−	0.00862200i	$-0.997255\pi$
$83$	− 118.794i	− 0.0172440i	0.999963	−	0.00862200i	$-0.00274450\pi$
$84$	0	0
$85$	−2646.00	−0.366228
$86$	− 7365.22i	− 0.995839i
$87$	0	0
$88$	−1008.00	−0.130165
$89$	− 4365.68i	− 0.551152i	−0.961279	−	0.275576i	$-0.911131\pi$
$89$	− 4365.68i	− 0.551152i	0.961279	−	0.275576i	$-0.0888686\pi$
$90$	0	0
$91$	3136.00	0.378698
$92$	1595.23i	0.188473i
$93$	0	0
$94$	5544.00	0.627433
$95$	16631.2i	1.84279i
$96$	0	0
$97$	−5824.00	−0.618982	−0.309491	−	0.950902i	$-0.600159\pi$
$97$	−5824.00	−0.618982	−0.309491	+	0.950902i	$0.600159\pi$
$98$	6860.35i	0.714322i
$99$	0	0
$100$	514.000	0.0514000
$101$	− 8048.29i	− 0.788971i	−0.918902	−	0.394485i	$-0.870923\pi$
$101$	− 8048.29i	− 0.788971i	0.918902	−	0.394485i	$-0.129077\pi$
$102$	0	0
$103$	−14980.0	−1.41201	−0.706004	−	0.708208i	$-0.749504\pi$
$103$	−14980.0	−1.41201	−0.706004	+	0.708208i	$0.749504\pi$
$104$	6652.46i	0.615057i
$105$	0	0
$106$	−7614.00	−0.677643
$107$	− 9775.04i	− 0.853790i	−0.904301	−	0.426895i	$-0.859607\pi$
$107$	− 9775.04i	− 0.853790i	0.904301	−	0.426895i	$-0.140393\pi$
$108$	0	0
$109$	10640.0	0.895548	0.447774	−	0.894147i	$-0.352217\pi$
$109$	10640.0	0.895548	0.447774	+	0.894147i	$0.352217\pi$
$110$	− 2138.29i	− 0.176718i
$111$	0	0
$112$	7952.00	0.633929
$113$	− 2872.27i	− 0.224941i	−0.993655	−	0.112470i	$-0.964124\pi$
$113$	− 2872.27i	− 0.224941i	0.993655	−	0.112470i	$-0.0358763\pi$
$114$	0	0
$115$	23688.0	1.79115
$116$	− 1977.07i	− 0.146929i
$117$	0	0
$118$	−19152.0	−1.37547
$119$	− 2494.67i	− 0.176165i
$120$	0	0
$121$	14353.0	0.980329
$122$	− 11107.2i	− 0.746253i
$123$	0	0
$124$	728.000	0.0473465
$125$	10929.0i	0.699459i
$126$	0	0
$127$	−19420.0	−1.20404	−0.602021	−	0.798480i	$-0.705638\pi$
$127$	−19420.0	−1.20404	−0.602021	+	0.798480i	$0.705638\pi$
$128$	18769.4i	1.14560i
$129$	0	0
$130$	−14112.0	−0.835030
$131$	− 20195.0i	− 1.17679i	−0.808572	−	0.588397i	$-0.799759\pi$
$131$	− 20195.0i	− 1.17679i	0.808572	−	0.588397i	$-0.200241\pi$
$132$	0	0
$133$	−15680.0	−0.886427
$134$	16054.2i	0.894083i
$135$	0	0
$136$	5292.00	0.286116
$137$	17331.2i	0.923394i	0.887038	+	0.461697i	$0.152759\pi$
$137$	17331.2i	0.923394i	−0.887038	+	0.461697i	$0.847241\pi$
$138$	0	0
$139$	18536.0	0.959371	0.479685	−	0.877441i	$-0.340751\pi$
$139$	18536.0	0.959371	0.479685	+	0.877441i	$0.340751\pi$
$140$	1663.12i	0.0848528i
$141$	0	0
$142$	36504.0	1.81036
$143$	1900.70i	0.0929485i
$144$	0	0
$145$	−29358.0	−1.39634
$146$	− 28035.4i	− 1.31523i
$147$	0	0
$148$	1652.00	0.0754200
$149$	− 3152.28i	− 0.141988i	−0.997477	−	0.0709941i	$-0.977383\pi$
$149$	− 3152.28i	− 0.141988i	0.997477	−	0.0709941i	$-0.0226172\pi$
$150$	0	0
$151$	7028.00	0.308232	0.154116	−	0.988053i	$-0.450747\pi$
$151$	7028.00	0.308232	0.154116	+	0.988053i	$0.450747\pi$
$152$	− 33262.3i	− 1.43968i
$153$	0	0
$154$	2016.00	0.0850059
$155$	− 10810.2i	− 0.449958i
$156$	0	0
$157$	−5530.00	−0.224350	−0.112175	−	0.993688i	$-0.535782\pi$
$157$	−5530.00	−0.224350	−0.112175	+	0.993688i	$0.535782\pi$
$158$	18141.5i	0.726708i
$159$	0	0
$160$	−7560.00	−0.295312
$161$	22333.3i	0.861589i
$162$	0	0
$163$	10856.0	0.408596	0.204298	−	0.978909i	$-0.434509\pi$
$163$	10856.0	0.408596	0.204298	+	0.978909i	$0.434509\pi$
$164$	− 3623.22i	− 0.134712i
$165$	0	0
$166$	−504.000	−0.0182900
$167$	29698.5i	1.06488i	0.846467	+	0.532441i	$0.178725\pi$
$167$	29698.5i	1.06488i	−0.846467	+	0.532441i	$0.821275\pi$
$168$	0	0
$169$	−16017.0	−0.560800
$170$	11226.0i	0.388444i
$171$	0	0
$172$	−3472.00	−0.117361
$173$	27589.9i	0.921845i	0.887440	+	0.460922i	$0.152481\pi$
$173$	27589.9i	0.921845i	−0.887440	+	0.460922i	$0.847519\pi$
$174$	0	0
$175$	7196.00	0.234971
$176$	4819.64i	0.155593i
$177$	0	0
$178$	−18522.0	−0.584585
$179$	− 54781.0i	− 1.70971i	−0.518863	−	0.854857i	$-0.673644\pi$
$179$	− 54781.0i	− 1.70971i	0.518863	−	0.854857i	$-0.326356\pi$
$180$	0	0
$181$	18704.0	0.570923	0.285461	−	0.958390i	$-0.407853\pi$
$181$	18704.0	0.570923	0.285461	+	0.958390i	$0.407853\pi$
$182$	− 13304.9i	− 0.401670i
$183$	0	0
$184$	−47376.0	−1.39934
$185$	− 24530.9i	− 0.716755i
$186$	0	0
$187$	1512.00	0.0432383
$188$	− 2613.47i	− 0.0739437i
$189$	0	0
$190$	70560.0	1.95457
$191$	− 8519.22i	− 0.233525i	−0.993160	−	0.116762i	$-0.962748\pi$
$191$	− 8519.22i	− 0.233525i	0.993160	−	0.116762i	$-0.0372516\pi$
$192$	0	0
$193$	48398.0	1.29931	0.649655	−	0.760229i	$-0.274913\pi$
$193$	48398.0	1.29931	0.649655	+	0.760229i	$0.274913\pi$
$194$	24709.1i	0.656529i
$195$	0	0
$196$	3234.00	0.0841837
$197$	− 10016.9i	− 0.258107i	−0.991638	−	0.129054i	$-0.958806\pi$
$197$	− 10016.9i	− 0.258107i	0.991638	−	0.129054i	$-0.0411939\pi$
$198$	0	0
$199$	−13300.0	−0.335850	−0.167925	−	0.985800i	$-0.553707\pi$
$199$	−13300.0	−0.335850	−0.167925	+	0.985800i	$0.553707\pi$
$200$	15265.0i	0.381626i
$201$	0	0
$202$	−34146.0	−0.836830
$203$	− 27679.0i	− 0.671673i
$204$	0	0
$205$	−53802.0	−1.28024
$206$	63554.8i	1.49766i
$207$	0	0
$208$	31808.0	0.735207
$209$	− 9503.52i	− 0.217566i
$210$	0	0
$211$	−36904.0	−0.828912	−0.414456	−	0.910069i	$-0.636028\pi$
$211$	−36904.0	−0.828912	−0.414456	+	0.910069i	$0.636028\pi$
$212$	3589.27i	0.0798610i
$213$	0	0
$214$	−41472.0	−0.905581
$215$	51556.6i	1.11534i
$216$	0	0
$217$	10192.0	0.216441
$218$	− 45141.7i	− 0.949872i
$219$	0	0
$220$	−1008.00	−0.0208264
$221$	− 9978.69i	− 0.204310i
$222$	0	0
$223$	51380.0	1.03320	0.516600	−	0.856227i	$-0.327197\pi$
$223$	51380.0	1.03320	0.516600	+	0.856227i	$0.327197\pi$
$224$	− 7127.64i	− 0.142053i
$225$	0	0
$226$	−12186.0	−0.238586
$227$	90402.2i	1.75439i	0.480130	+	0.877197i	$0.340589\pi$
$227$	90402.2i	1.75439i	−0.480130	+	0.877197i	$0.659411\pi$
$228$	0	0
$229$	78512.0	1.49715	0.748575	−	0.663051i	$-0.230739\pi$
$229$	78512.0	1.49715	0.748575	+	0.663051i	$0.230739\pi$
$230$	− 100500.i	− 1.89980i
$231$	0	0
$232$	58716.0	1.09089
$233$	− 52451.8i	− 0.966158i	−0.875577	−	0.483079i	$-0.839518\pi$
$233$	− 52451.8i	− 0.966158i	0.875577	−	0.483079i	$-0.160482\pi$
$234$	0	0
$235$	−38808.0	−0.702725
$236$	9028.34i	0.162100i
$237$	0	0
$238$	−10584.0	−0.186851
$239$	22265.4i	0.389793i	0.980824	+	0.194897i	$0.0624371\pi$
$239$	22265.4i	0.389793i	−0.980824	+	0.194897i	$0.937563\pi$
$240$	0	0
$241$	−53200.0	−0.915962	−0.457981	−	0.888962i	$-0.651427\pi$
$241$	−53200.0	−0.915962	−0.457981	+	0.888962i	$0.651427\pi$
$242$	− 60894.6i	− 1.03980i
$243$	0	0
$244$	−5236.00	−0.0879468
$245$	− 48022.4i	− 0.800041i
$246$	0	0
$247$	−62720.0	−1.02805
$248$	21620.5i	0.351530i
$249$	0	0
$250$	46368.0	0.741888
$251$	83037.0i	1.31803i	0.752131	+	0.659013i	$0.229026\pi$
$251$	83037.0i	1.31803i	−0.752131	+	0.659013i	$0.770974\pi$
$252$	0	0
$253$	−13536.0	−0.211470
$254$	82392.1i	1.27708i
$255$	0	0
$256$	24208.0	0.369385
$257$	84670.4i	1.28193i	0.767569	+	0.640966i	$0.221466\pi$
$257$	84670.4i	1.28193i	−0.767569	+	0.640966i	$0.778534\pi$
$258$	0	0
$259$	23128.0	0.344777
$260$	6652.46i	0.0984092i
$261$	0	0
$262$	−85680.0	−1.24818
$263$	− 110343.i	− 1.59526i	−0.603146	−	0.797630i	$-0.706087\pi$
$263$	− 110343.i	− 1.59526i	0.603146	−	0.797630i	$-0.293913\pi$
$264$	0	0
$265$	53298.0	0.758960
$266$	66524.6i	0.940197i
$267$	0	0
$268$	7568.00	0.105369
$269$	− 27530.5i	− 0.380460i	−0.981740	−	0.190230i	$-0.939077\pi$
$269$	− 27530.5i	− 0.380460i	0.981740	−	0.190230i	$-0.0609234\pi$
$270$	0	0
$271$	−73276.0	−0.997753	−0.498877	−	0.866673i	$-0.666254\pi$
$271$	−73276.0	−0.997753	−0.498877	+	0.866673i	$0.666254\pi$
$272$	− 25303.1i	− 0.342008i
$273$	0	0
$274$	73530.0	0.979408
$275$	4361.43i	0.0576719i
$276$	0	0
$277$	46736.0	0.609105	0.304552	−	0.952496i	$-0.401493\pi$
$277$	46736.0	0.609105	0.304552	+	0.952496i	$0.401493\pi$
$278$	− 78641.6i	− 1.01757i
$279$	0	0
$280$	−49392.0	−0.630000
$281$	− 112774.i	− 1.42822i	−0.700034	−	0.714110i	$-0.746832\pi$
$281$	− 112774.i	− 1.42822i	0.700034	−	0.714110i	$-0.253168\pi$
$282$	0	0
$283$	−24304.0	−0.303462	−0.151731	−	0.988422i	$-0.548485\pi$
$283$	−24304.0	−0.303462	−0.151731	+	0.988422i	$0.548485\pi$
$284$	− 17208.2i	− 0.213352i
$285$	0	0
$286$	8064.00	0.0985867
$287$	− 50725.0i	− 0.615826i
$288$	0	0
$289$	75583.0	0.904958
$290$	124555.i	1.48104i
$291$	0	0
$292$	−13216.0	−0.155001
$293$	9473.82i	0.110354i	0.998477	+	0.0551772i	$0.0175724\pi$
$293$	9473.82i	0.110354i	−0.998477	+	0.0551772i	$0.982428\pi$
$294$	0	0
$295$	134064.	1.54052
$296$	49061.9i	0.559965i
$297$	0	0
$298$	−13374.0	−0.150601
$299$	89333.0i	0.999240i
$300$	0	0
$301$	−48608.0	−0.536506
$302$	− 29817.3i	− 0.326930i
$303$	0	0
$304$	−159040.	−1.72091
$305$	77750.6i	0.835804i
$306$	0	0
$307$	19208.0	0.203801	0.101900	−	0.994795i	$-0.467508\pi$
$307$	19208.0	0.203801	0.101900	+	0.994795i	$0.467508\pi$
$308$	− 950.352i	− 0.0100180i
$309$	0	0
$310$	−45864.0	−0.477253
$311$	− 78641.6i	− 0.813077i	−0.913634	−	0.406538i	$-0.866736\pi$
$311$	− 78641.6i	− 0.813077i	0.913634	−	0.406538i	$-0.133264\pi$
$312$	0	0
$313$	−127918.	−1.30570	−0.652849	−	0.757488i	$-0.726427\pi$
$313$	−127918.	−1.30570	−0.652849	+	0.757488i	$0.726427\pi$
$314$	23461.8i	0.237959i
$315$	0	0
$316$	8552.00	0.0856433
$317$	65875.5i	0.655549i	0.944756	+	0.327775i	$0.106299\pi$
$317$	65875.5i	0.655549i	−0.944756	+	0.327775i	$0.893701\pi$
$318$	0	0
$319$	16776.0	0.164857
$320$	− 102876.i	− 1.00464i
$321$	0	0
$322$	94752.0	0.913854
$323$	49893.5i	0.478232i
$324$	0	0
$325$	28784.0	0.272511
$326$	− 46058.1i	− 0.433382i
$327$	0	0
$328$	107604.	1.00019
$329$	− 36588.5i	− 0.338028i
$330$	0	0
$331$	15848.0	0.144650	0.0723250	−	0.997381i	$-0.476958\pi$
$331$	15848.0	0.144650	0.0723250	+	0.997381i	$0.476958\pi$
$332$	237.588i	0.00215550i
$333$	0	0
$334$	126000.	1.12948
$335$	− 112379.i	− 1.00137i
$336$	0	0
$337$	−127120.	−1.11932	−0.559660	−	0.828722i	$-0.689068\pi$
$337$	−127120.	−1.11932	−0.559660	+	0.828722i	$0.689068\pi$
$338$	67954.4i	0.594818i
$339$	0	0
$340$	5292.00	0.0457785
$341$	6177.28i	0.0531238i
$342$	0	0
$343$	112504.	0.956268
$344$	− 103113.i	− 0.871359i
$345$	0	0
$346$	117054.	0.977764
$347$	− 21875.1i	− 0.181673i	−0.995866	−	0.0908365i	$-0.971046\pi$
$347$	− 21875.1i	− 0.181673i	0.995866	−	0.0908365i	$-0.0289541\pi$
$348$	0	0
$349$	−200998.	−1.65022	−0.825108	−	0.564975i	$-0.808886\pi$
$349$	−200998.	−1.65022	−0.825108	+	0.564975i	$0.808886\pi$
$350$	− 30530.0i	− 0.249225i
$351$	0	0
$352$	4320.00	0.0348657
$353$	241775.i	1.94027i	0.242561	+	0.970136i	$0.422012\pi$
$353$	241775.i	1.94027i	−0.242561	+	0.970136i	$0.577988\pi$
$354$	0	0
$355$	−255528.	−2.02760
$356$	8731.35i	0.0688940i
$357$	0	0
$358$	−232416.	−1.81343
$359$	− 108187.i	− 0.839436i	−0.907655	−	0.419718i	$-0.862129\pi$
$359$	− 108187.i	− 0.839436i	0.907655	−	0.419718i	$-0.137871\pi$
$360$	0	0
$361$	183279.	1.40637
$362$	− 79354.4i	− 0.605555i
$363$	0	0
$364$	−6272.00	−0.0473373
$365$	196248.i	1.47305i
$366$	0	0
$367$	221732.	1.64625	0.823126	−	0.567859i	$-0.192228\pi$
$367$	221732.	1.64625	0.823126	+	0.567859i	$0.192228\pi$
$368$	226523.i	1.67270i
$369$	0	0
$370$	−104076.	−0.760234
$371$	50249.8i	0.365079i
$372$	0	0
$373$	87674.0	0.630163	0.315082	−	0.949065i	$-0.397968\pi$
$373$	87674.0	0.630163	0.315082	+	0.949065i	$0.397968\pi$
$374$	− 6414.87i	− 0.0458611i
$375$	0	0
$376$	77616.0	0.549004
$377$	− 110716.i	− 0.778982i
$378$	0	0
$379$	−40768.0	−0.283819	−0.141909	−	0.989880i	$-0.545324\pi$
$379$	−40768.0	−0.283819	−0.141909	+	0.989880i	$0.545324\pi$
$380$	− 33262.3i	− 0.230348i
$381$	0	0
$382$	−36144.0	−0.247691
$383$	− 75077.8i	− 0.511816i	−0.966701	−	0.255908i	$-0.917626\pi$
$383$	− 75077.8i	− 0.511816i	0.966701	−	0.255908i	$-0.0823744\pi$
$384$	0	0
$385$	−14112.0	−0.0952066
$386$	− 205335.i	− 1.37813i
$387$	0	0
$388$	11648.0	0.0773727
$389$	− 123236.i	− 0.814401i	−0.913339	−	0.407201i	$-0.866505\pi$
$389$	− 123236.i	− 0.814401i	0.913339	−	0.407201i	$-0.133495\pi$
$390$	0	0
$391$	71064.0	0.464832
$392$	96044.9i	0.625032i
$393$	0	0
$394$	−42498.0	−0.273764
$395$	− 126991.i	− 0.813913i
$396$	0	0
$397$	−8134.00	−0.0516087	−0.0258044	−	0.999667i	$-0.508215\pi$
$397$	−8134.00	−0.0516087	−0.0258044	+	0.999667i	$0.508215\pi$
$398$	56427.1i	0.356223i
$399$	0	0
$400$	72988.0	0.456175
$401$	− 133639.i	− 0.831083i	−0.909574	−	0.415541i	$-0.863592\pi$
$401$	− 133639.i	− 0.831083i	0.909574	−	0.415541i	$-0.136408\pi$
$402$	0	0
$403$	40768.0	0.251021
$404$	16096.6i	0.0986213i
$405$	0	0
$406$	−117432.	−0.712417
$407$	14017.7i	0.0846228i
$408$	0	0
$409$	35168.0	0.210233	0.105117	−	0.994460i	$-0.466478\pi$
$409$	35168.0	0.210233	0.105117	+	0.994460i	$0.466478\pi$
$410$	228263.i	1.35790i
$411$	0	0
$412$	29960.0	0.176501
$413$	126397.i	0.741030i
$414$	0	0
$415$	3528.00	0.0204848
$416$	− 28510.5i	− 0.164748i
$417$	0	0
$418$	−40320.0	−0.230764
$419$	157640.i	0.897919i	0.893552	+	0.448959i	$0.148205\pi$
$419$	157640.i	0.897919i	−0.893552	+	0.448959i	$0.851795\pi$
$420$	0	0
$421$	18800.0	0.106070	0.0530351	−	0.998593i	$-0.483110\pi$
$421$	18800.0	0.106070	0.0530351	+	0.998593i	$0.483110\pi$
$422$	156570.i	0.879194i
$423$	0	0
$424$	−106596.	−0.592938
$425$	− 22897.5i	− 0.126768i
$426$	0	0
$427$	−73304.0	−0.402042
$428$	19550.1i	0.106724i
$429$	0	0
$430$	218736.	1.18300
$431$	72243.7i	0.388907i	0.980912	+	0.194453i	$0.0622933\pi$
$431$	72243.7i	0.388907i	−0.980912	+	0.194453i	$0.937707\pi$
$432$	0	0
$433$	−351106.	−1.87268	−0.936338	−	0.351101i	$-0.885807\pi$
$433$	−351106.	−1.87268	−0.936338	+	0.351101i	$0.885807\pi$
$434$	− 43241.0i	− 0.229571i
$435$	0	0
$436$	−21280.0	−0.111943
$437$	− 446665.i	− 2.33894i
$438$	0	0
$439$	−60508.0	−0.313967	−0.156983	−	0.987601i	$-0.550177\pi$
$439$	−60508.0	−0.313967	−0.156983	+	0.987601i	$0.550177\pi$
$440$	− 29936.1i	− 0.154628i
$441$	0	0
$442$	−42336.0	−0.216703
$443$	57105.9i	0.290987i	0.989359	+	0.145494i	$0.0464770\pi$
$443$	57105.9i	0.290987i	−0.989359	+	0.145494i	$0.953523\pi$
$444$	0	0
$445$	129654.	0.654736
$446$	− 217987.i	− 1.09587i
$447$	0	0
$448$	96992.0	0.483259
$449$	210482.i	1.04405i	0.852930	+	0.522025i	$0.174823\pi$
$449$	210482.i	1.04405i	−0.852930	+	0.522025i	$0.825177\pi$
$450$	0	0
$451$	30744.0	0.151150
$452$	5744.54i	0.0281176i
$453$	0	0
$454$	383544.	1.86082
$455$	93134.4i	0.449871i
$456$	0	0
$457$	324800.	1.55519	0.777595	−	0.628765i	$-0.216439\pi$
$457$	324800.	1.55519	0.777595	+	0.628765i	$0.216439\pi$
$458$	− 333098.i	− 1.58797i
$459$	0	0
$460$	−47376.0	−0.223894
$461$	165331.i	0.777954i	0.921248	+	0.388977i	$0.127171\pi$
$461$	165331.i	0.777954i	−0.921248	+	0.388977i	$0.872829\pi$
$462$	0	0
$463$	−117220.	−0.546814	−0.273407	−	0.961898i	$-0.588151\pi$
$463$	−117220.	−0.546814	−0.273407	+	0.961898i	$0.588151\pi$
$464$	− 280744.i	− 1.30399i
$465$	0	0
$466$	−222534.	−1.02477
$467$	170707.i	0.782740i	0.920233	+	0.391370i	$0.127999\pi$
$467$	170707.i	0.782740i	−0.920233	+	0.391370i	$0.872001\pi$
$468$	0	0
$469$	105952.	0.481685
$470$	164648.i	0.745353i
$471$	0	0
$472$	−268128.	−1.20353
$473$	− 29460.9i	− 0.131681i
$474$	0	0
$475$	−143920.	−0.637873
$476$	4989.35i	0.0220206i
$477$	0	0
$478$	94464.0	0.413438
$479$	99668.1i	0.434395i	0.976128	+	0.217198i	$0.0696916\pi$
$479$	99668.1i	0.434395i	−0.976128	+	0.217198i	$0.930308\pi$
$480$	0	0
$481$	92512.0	0.399860
$482$	225708.i	0.971525i
$483$	0	0
$484$	−28706.0	−0.122541
$485$	− 172964.i	− 0.735313i
$486$	0	0
$487$	−416500.	−1.75613	−0.878066	−	0.478540i	$-0.841166\pi$
$487$	−416500.	−1.75613	−0.878066	+	0.478540i	$0.841166\pi$
$488$	− 155501.i	− 0.652972i
$489$	0	0
$490$	−203742.	−0.848571
$491$	355024.i	1.47263i	0.676637	+	0.736317i	$0.263437\pi$
$491$	355024.i	1.47263i	−0.676637	+	0.736317i	$0.736563\pi$
$492$	0	0
$493$	−88074.0	−0.362371
$494$	266098.i	1.09041i
$495$	0	0
$496$	103376.	0.420200
$497$	− 240914.i	− 0.975325i
$498$	0	0
$499$	135968.	0.546054	0.273027	−	0.962006i	$-0.411975\pi$
$499$	135968.	0.546054	0.273027	+	0.962006i	$0.411975\pi$
$500$	− 21858.1i	− 0.0874323i
$501$	0	0
$502$	352296.	1.39798
$503$	− 170707.i	− 0.674707i	−0.941378	−	0.337353i	$-0.890468\pi$
$503$	− 170707.i	− 0.674707i	0.941378	−	0.337353i	$-0.109532\pi$
$504$	0	0
$505$	239022.	0.937249
$506$	57428.4i	0.224298i
$507$	0	0
$508$	38840.0	0.150505
$509$	193367.i	0.746357i	0.927760	+	0.373178i	$0.121732\pi$
$509$	193367.i	0.746357i	−0.927760	+	0.373178i	$0.878268\pi$
$510$	0	0
$511$	−185024.	−0.708576
$512$	197605.i	0.753804i
$513$	0	0
$514$	359226.	1.35970
$515$	− 444883.i	− 1.67738i
$516$	0	0
$517$	22176.0	0.0829664
$518$	− 98123.8i	− 0.365691i
$519$	0	0
$520$	−197568.	−0.730651
$521$	− 391931.i	− 1.44389i	−0.691951	−	0.721945i	$-0.743248\pi$
$521$	− 391931.i	− 1.44389i	0.691951	−	0.721945i	$-0.256752\pi$
$522$	0	0
$523$	410816.	1.50191	0.750955	−	0.660353i	$-0.229593\pi$
$523$	410816.	1.50191	0.750955	+	0.660353i	$0.229593\pi$
$524$	40389.9i	0.147099i
$525$	0	0
$526$	−468144.	−1.69203
$527$	− 32430.7i	− 0.116771i
$528$	0	0
$529$	−356351.	−1.27341
$530$	− 226124.i	− 0.804999i
$531$	0	0
$532$	31360.0	0.110803
$533$	− 202900.i	− 0.714213i
$534$	0	0
$535$	290304.	1.01425
$536$	224758.i	0.782323i
$537$	0	0
$538$	−116802.	−0.403539
$539$	27441.4i	0.0944558i
$540$	0	0
$541$	−57616.0	−0.196856	−0.0984280	−	0.995144i	$-0.531381\pi$
$541$	−57616.0	−0.196856	−0.0984280	+	0.995144i	$0.531381\pi$
$542$	310884.i	1.05828i
$543$	0	0
$544$	−22680.0	−0.0766382
$545$	315992.i	1.06386i
$546$	0	0
$547$	78920.0	0.263762	0.131881	−	0.991266i	$-0.457898\pi$
$547$	78920.0	0.263762	0.131881	+	0.991266i	$0.457898\pi$
$548$	− 34662.4i	− 0.115424i
$549$	0	0
$550$	18504.0	0.0611702
$551$	553580.i	1.82338i
$552$	0	0
$553$	119728.	0.391512
$554$	− 198284.i	− 0.646053i
$555$	0	0
$556$	−37072.0	−0.119921
$557$	178025.i	0.573815i	0.957958	+	0.286907i	$0.0926272\pi$
$557$	178025.i	0.573815i	−0.957958	+	0.286907i	$0.907373\pi$
$558$	0	0
$559$	−194432.	−0.622220
$560$	236162.i	0.753069i
$561$	0	0
$562$	−478458.	−1.51486
$563$	− 431816.i	− 1.36233i	−0.732131	−	0.681164i	$-0.761474\pi$
$563$	− 431816.i	− 1.36233i	0.732131	−	0.681164i	$-0.238526\pi$
$564$	0	0
$565$	85302.0	0.267216
$566$	103113.i	0.321870i
$567$	0	0
$568$	511056.	1.58406
$569$	− 65756.7i	− 0.203103i	−0.994830	−	0.101551i	$-0.967619\pi$
$569$	− 65756.7i	− 0.203103i	0.994830	−	0.101551i	$-0.0323806\pi$
$570$	0	0
$571$	111632.	0.342386	0.171193	−	0.985237i	$-0.445238\pi$
$571$	111632.	0.342386	0.171193	+	0.985237i	$0.445238\pi$
$572$	− 3801.41i	− 0.0116186i
$573$	0	0
$574$	−215208.	−0.653183
$575$	204987.i	0.620000i
$576$	0	0
$577$	−164878.	−0.495235	−0.247617	−	0.968858i	$-0.579648\pi$
$577$	−164878.	−0.495235	−0.247617	+	0.968858i	$0.579648\pi$
$578$	− 320672.i	− 0.959853i
$579$	0	0
$580$	58716.0	0.174542
$581$	3326.23i	0.00985372i
$582$	0	0
$583$	−30456.0	−0.0896057
$584$	− 392495.i	− 1.15082i
$585$	0	0
$586$	40194.0	0.117049
$587$	− 568548.i	− 1.65003i	−0.565114	−	0.825013i	$-0.691168\pi$
$587$	− 568548.i	− 1.65003i	0.565114	−	0.825013i	$-0.308832\pi$
$588$	0	0
$589$	−203840.	−0.587569
$590$	− 568785.i	− 1.63397i
$591$	0	0
$592$	234584.	0.669353
$593$	580279.i	1.65016i	0.565013	+	0.825082i	$0.308871\pi$
$593$	580279.i	1.65016i	−0.565013	+	0.825082i	$0.691129\pi$
$594$	0	0
$595$	74088.0	0.209273
$596$	6304.56i	0.0177485i
$597$	0	0
$598$	379008.	1.05985
$599$	5719.08i	0.0159394i	0.999968	+	0.00796971i	$0.00253686\pi$
$599$	5719.08i	0.0159394i	−0.999968	+	0.00796971i	$0.997463\pi$
$600$	0	0
$601$	233198.	0.645618	0.322809	−	0.946464i	$-0.395373\pi$
$601$	233198.	0.645618	0.322809	+	0.946464i	$0.395373\pi$
$602$	206226.i	0.569051i
$603$	0	0
$604$	−14056.0	−0.0385290
$605$	426262.i	1.16457i
$606$	0	0
$607$	154868.	0.420324	0.210162	−	0.977667i	$-0.432601\pi$
$607$	154868.	0.420324	0.210162	+	0.977667i	$0.432601\pi$
$608$	142553.i	0.385628i
$609$	0	0
$610$	329868.	0.886504
$611$	− 146354.i	− 0.392033i
$612$	0	0
$613$	−333862.	−0.888477	−0.444238	−	0.895909i	$-0.646526\pi$
$613$	−333862.	−0.888477	−0.444238	+	0.895909i	$0.646526\pi$
$614$	− 81492.6i	− 0.216163i
$615$	0	0
$616$	28224.0	0.0743802
$617$	19325.2i	0.0507638i	0.999678	+	0.0253819i	$0.00808018\pi$
$617$	19325.2i	0.0507638i	−0.999678	+	0.0253819i	$0.991920\pi$
$618$	0	0
$619$	−473536.	−1.23587	−0.617933	−	0.786230i	$-0.712030\pi$
$619$	−473536.	−1.23587	−0.617933	+	0.786230i	$0.712030\pi$
$620$	21620.5i	0.0562448i
$621$	0	0
$622$	−333648.	−0.862398
$623$	122239.i	0.314944i
$624$	0	0
$625$	−485201.	−1.24211
$626$	542710.i	1.38490i
$627$	0	0
$628$	11060.0	0.0280437
$629$	− 73592.8i	− 0.186009i
$630$	0	0
$631$	−384820.	−0.966493	−0.483247	−	0.875484i	$-0.660543\pi$
$631$	−384820.	−0.966493	−0.483247	+	0.875484i	$0.660543\pi$
$632$	253981.i	0.635869i
$633$	0	0
$634$	279486.	0.695315
$635$	− 576745.i	− 1.43033i
$636$	0	0
$637$	181104.	0.446323
$638$	− 71174.5i	− 0.174857i
$639$	0	0
$640$	−557424.	−1.36090
$641$	145832.i	0.354926i	0.984127	+	0.177463i	$0.0567890\pi$
$641$	145832.i	0.354926i	−0.984127	+	0.177463i	$0.943211\pi$
$642$	0	0
$643$	528584.	1.27847	0.639237	−	0.769010i	$-0.279250\pi$
$643$	528584.	1.27847	0.639237	+	0.769010i	$0.279250\pi$
$644$	− 44666.5i	− 0.107699i
$645$	0	0
$646$	211680.	0.507242
$647$	176409.i	0.421417i	0.977549	+	0.210709i	$0.0675771\pi$
$647$	176409.i	0.421417i	−0.977549	+	0.210709i	$0.932423\pi$
$648$	0	0
$649$	−76608.0	−0.181880
$650$	− 122120.i	− 0.289042i
$651$	0	0
$652$	−21712.0	−0.0510746
$653$	403250.i	0.945689i	0.881146	+	0.472844i	$0.156773\pi$
$653$	403250.i	0.945689i	−0.881146	+	0.472844i	$0.843227\pi$
$654$	0	0
$655$	599760.	1.39796
$656$	− 514497.i	− 1.19557i
$657$	0	0
$658$	−155232.	−0.358533
$659$	− 137020.i	− 0.315511i	−0.987478	−	0.157755i	$-0.949574\pi$
$659$	− 137020.i	− 0.315511i	0.987478	−	0.157755i	$-0.0504257\pi$
$660$	0	0
$661$	286790.	0.656389	0.328194	−	0.944610i	$-0.393560\pi$
$661$	286790.	0.656389	0.328194	+	0.944610i	$0.393560\pi$
$662$	− 67237.4i	− 0.153425i
$663$	0	0
$664$	−7056.00	−0.0160038
$665$	− 465672.i	− 1.05302i
$666$	0	0
$667$	788472.	1.77229
$668$	− 59397.0i	− 0.133110i
$669$	0	0
$670$	−476784.	−1.06212
$671$	− 44428.9i	− 0.0986781i
$672$	0	0
$673$	121058.	0.267278	0.133639	−	0.991030i	$-0.457334\pi$
$673$	121058.	0.267278	0.133639	+	0.991030i	$0.457334\pi$
$674$	539324.i	1.18722i
$675$	0	0
$676$	32034.0	0.0701000
$677$	− 377260.i	− 0.823120i	−0.911383	−	0.411560i	$-0.864984\pi$
$677$	− 377260.i	− 0.823120i	0.911383	−	0.411560i	$-0.135016\pi$
$678$	0	0
$679$	163072.	0.353704
$680$	157164.i	0.339888i
$681$	0	0
$682$	26208.0	0.0563463
$683$	− 335457.i	− 0.719110i	−0.933124	−	0.359555i	$-0.882928\pi$
$683$	− 335457.i	− 0.719110i	0.933124	−	0.359555i	$-0.117072\pi$
$684$	0	0
$685$	−514710.	−1.09694
$686$	− 477314.i	− 1.01428i
$687$	0	0
$688$	−493024.	−1.04158
$689$	200999.i	0.423405i
$690$	0	0
$691$	−578872.	−1.21235	−0.606173	−	0.795333i	$-0.707296\pi$
$691$	−578872.	−1.21235	−0.606173	+	0.795333i	$0.707296\pi$
$692$	− 55179.8i	− 0.115231i
$693$	0	0
$694$	−92808.0	−0.192693
$695$	550491.i	1.13967i
$696$	0	0
$697$	−161406.	−0.332242
$698$	852762.i	1.75032i
$699$	0	0
$700$	−14392.0	−0.0293714
$701$	− 654279.i	− 1.33146i	−0.746194	−	0.665728i	$-0.768121\pi$
$701$	− 654279.i	− 1.33146i	0.746194	−	0.665728i	$-0.231879\pi$
$702$	0	0
$703$	−462560.	−0.935960
$704$	58786.0i	0.118612i
$705$	0	0
$706$	1.02577e6	2.05797
$707$	225352.i	0.450840i
$708$	0	0
$709$	779408.	1.55050	0.775251	−	0.631653i	$-0.217623\pi$
$709$	779408.	1.55050	0.775251	+	0.631653i	$0.217623\pi$
$710$	1.08411e6i	2.15059i
$711$	0	0
$712$	−259308.	−0.511512
$713$	290332.i	0.571106i
$714$	0	0
$715$	−56448.0	−0.110417
$716$	109562.i	0.213714i
$717$	0	0
$718$	−459000.	−0.890356
$719$	147898.i	0.286092i	0.989716	+	0.143046i	$0.0456897\pi$
$719$	147898.i	0.286092i	−0.989716	+	0.143046i	$0.954310\pi$
$720$	0	0
$721$	419440.	0.806862
$722$	− 777587.i	− 1.49168i
$723$	0	0
$724$	−37408.0	−0.0713653
$725$	− 254054.i	− 0.483336i
$726$	0	0
$727$	−713188.	−1.34938	−0.674691	−	0.738100i	$-0.735723\pi$
$727$	−713188.	−1.34938	−0.674691	+	0.738100i	$0.735723\pi$
$728$	− 186269.i	− 0.351461i
$729$	0	0
$730$	832608.	1.56241
$731$	154670.i	0.289448i
$732$	0	0
$733$	−826672.	−1.53860	−0.769299	−	0.638889i	$-0.779394\pi$
$733$	−826672.	−1.53860	−0.769299	+	0.638889i	$0.779394\pi$
$734$	− 940729.i	− 1.74611i
$735$	0	0
$736$	203040.	0.374823
$737$	64216.6i	0.118226i
$738$	0	0
$739$	703280.	1.28777	0.643887	−	0.765121i	$-0.277321\pi$
$739$	703280.	1.28777	0.643887	+	0.765121i	$0.277321\pi$
$740$	49061.9i	0.0895944i
$741$	0	0
$742$	213192.	0.387225
$743$	− 469643.i	− 0.850728i	−0.905022	−	0.425364i	$-0.860146\pi$
$743$	− 469643.i	− 0.850728i	0.905022	−	0.425364i	$-0.139854\pi$
$744$	0	0
$745$	93618.0	0.168673
$746$	− 371969.i	− 0.668389i
$747$	0	0
$748$	−3024.00	−0.00540479
$749$	273701.i	0.487880i
$750$	0	0
$751$	−963004.	−1.70745	−0.853725	−	0.520723i	$-0.825662\pi$
$751$	−963004.	−1.70745	−0.853725	+	0.520723i	$0.825662\pi$
$752$	− 371112.i	− 0.656250i
$753$	0	0
$754$	−469728.	−0.826235
$755$	208721.i	0.366161i
$756$	0	0
$757$	899696.	1.57002	0.785008	−	0.619486i	$-0.212659\pi$
$757$	899696.	1.57002	0.785008	+	0.619486i	$0.212659\pi$
$758$	172964.i	0.301035i
$759$	0	0
$760$	987840.	1.71025
$761$	− 886292.i	− 1.53041i	−0.643787	−	0.765204i	$-0.722638\pi$
$761$	− 886292.i	− 1.53041i	0.643787	−	0.765204i	$-0.277362\pi$
$762$	0	0
$763$	−297920.	−0.511741
$764$	17038.4i	0.0291906i
$765$	0	0
$766$	−318528.	−0.542863
$767$	505587.i	0.859419i
$768$	0	0
$769$	183218.	0.309824	0.154912	−	0.987928i	$-0.450491\pi$
$769$	183218.	0.309824	0.154912	+	0.987928i	$0.450491\pi$
$770$	59872.1i	0.100982i
$771$	0	0
$772$	−96796.0	−0.162414
$773$	1.11521e6i	1.86637i	0.359402	+	0.933183i	$0.382981\pi$
$773$	1.11521e6i	1.86637i	−0.359402	+	0.933183i	$0.617019\pi$
$774$	0	0
$775$	93548.0	0.155751
$776$	345928.i	0.574463i
$777$	0	0
$778$	−522846.	−0.863803
$779$	1.01450e6i	1.67177i
$780$	0	0
$781$	146016.	0.239386
$782$	− 301499.i	− 0.493029i
$783$	0	0
$784$	459228.	0.747130
$785$	− 164233.i	− 0.266514i
$786$	0	0
$787$	−276304.	−0.446106	−0.223053	−	0.974806i	$-0.571602\pi$
$787$	−276304.	−0.446106	−0.223053	+	0.974806i	$0.571602\pi$
$788$	20033.7i	0.0322634i
$789$	0	0
$790$	−538776.	−0.863285
$791$	80423.5i	0.128538i
$792$	0	0
$793$	−293216.	−0.466274
$794$	34509.6i	0.0547393i
$795$	0	0
$796$	26600.0	0.0419813
$797$	− 249438.i	− 0.392686i	−0.980535	−	0.196343i	$-0.937093\pi$
$797$	− 249438.i	− 0.392686i	0.980535	−	0.196343i	$-0.0629066\pi$
$798$	0	0
$799$	−116424.	−0.182368
$800$	− 65421.5i	− 0.102221i
$801$	0	0
$802$	−566982.	−0.881496
$803$	− 112141.i	− 0.173914i
$804$	0	0
$805$	−663264.	−1.02352
$806$	− 172964.i	− 0.266248i
$807$	0	0
$808$	−478044.	−0.732226
$809$	− 657868.i	− 1.00518i	−0.864526	−	0.502588i	$-0.832381\pi$
$809$	− 657868.i	− 1.00518i	0.864526	−	0.502588i	$-0.167619\pi$
$810$	0	0
$811$	660800.	1.00468	0.502341	−	0.864670i	$-0.332472\pi$
$811$	660800.	1.00468	0.502341	+	0.864670i	$0.332472\pi$
$812$	55358.0i	0.0839592i
$813$	0	0
$814$	59472.0	0.0897561
$815$	322407.i	0.485388i
$816$	0	0
$817$	972160.	1.45644
$818$	− 149205.i	− 0.222986i
$819$	0	0
$820$	107604.	0.160030
$821$	− 1.23522e6i	− 1.83256i	−0.400533	−	0.916282i	$-0.631175\pi$
$821$	− 1.23522e6i	− 1.83256i	0.400533	−	0.916282i	$-0.368825\pi$
$822$	0	0
$823$	1.07330e6	1.58461	0.792303	−	0.610127i	$-0.208882\pi$
$823$	1.07330e6	1.58461	0.792303	+	0.610127i	$0.208882\pi$
$824$	889767.i	1.31045i
$825$	0	0
$826$	536256.	0.785981
$827$	356687.i	0.521527i	0.965403	+	0.260763i	$0.0839742\pi$
$827$	356687.i	0.521527i	−0.965403	+	0.260763i	$0.916026\pi$
$828$	0	0
$829$	91280.0	0.132821	0.0664105	−	0.997792i	$-0.478845\pi$
$829$	91280.0	0.132821	0.0664105	+	0.997792i	$0.478845\pi$
$830$	− 14968.0i	− 0.0217274i
$831$	0	0
$832$	387968.	0.560466
$833$	− 144067.i	− 0.207623i
$834$	0	0
$835$	−882000.	−1.26501
$836$	19007.0i	0.0271958i
$837$	0	0
$838$	668808.	0.952387
$839$	− 1.18473e6i	− 1.68305i	−0.540221	−	0.841523i	$-0.681659\pi$
$839$	− 1.18473e6i	− 1.68305i	0.540221	−	0.841523i	$-0.318341\pi$
$840$	0	0
$841$	−269921.	−0.381632
$842$	− 79761.6i	− 0.112505i
$843$	0	0
$844$	73808.0	0.103614
$845$	− 475681.i	− 0.666196i
$846$	0	0
$847$	−401884.	−0.560188
$848$	509677.i	0.708767i
$849$	0	0
$850$	−97146.0	−0.134458
$851$	658831.i	0.909735i
$852$	0	0
$853$	429254.	0.589951	0.294976	−	0.955505i	$-0.404688\pi$
$853$	429254.	0.589951	0.294976	+	0.955505i	$0.404688\pi$
$854$	311003.i	0.426430i
$855$	0	0
$856$	−580608.	−0.792384
$857$	337286.i	0.459236i	0.973281	+	0.229618i	$0.0737477\pi$
$857$	337286.i	0.459236i	−0.973281	+	0.229618i	$0.926252\pi$
$858$	0	0
$859$	−535864.	−0.726220	−0.363110	−	0.931746i	$-0.618285\pi$
$859$	−535864.	−0.726220	−0.363110	+	0.931746i	$0.618285\pi$
$860$	− 103113.i	− 0.139417i
$861$	0	0
$862$	306504.	0.412498
$863$	− 894722.i	− 1.20134i	−0.799496	−	0.600671i	$-0.794900\pi$
$863$	− 894722.i	− 1.20134i	0.799496	−	0.600671i	$-0.205100\pi$
$864$	0	0
$865$	−819378.	−1.09510
$866$	1.48962e6i	1.98627i
$867$	0	0
$868$	−20384.0	−0.0270552
$869$	72566.1i	0.0960936i
$870$	0	0
$871$	423808.	0.558641
$872$	− 631984.i	− 0.831138i
$873$	0	0
$874$	−1.89504e6	−2.48082
$875$	− 306013.i	− 0.399691i
$876$	0	0
$877$	−179878.	−0.233872	−0.116936	−	0.993139i	$-0.537307\pi$
$877$	−179878.	−0.233872	−0.116936	+	0.993139i	$0.537307\pi$
$878$	256714.i	0.333012i
$879$	0	0
$880$	−143136.	−0.184835
$881$	1.27451e6i	1.64207i	0.570878	+	0.821035i	$0.306603\pi$
$881$	1.27451e6i	1.64207i	−0.570878	+	0.821035i	$0.693397\pi$
$882$	0	0
$883$	383096.	0.491345	0.245672	−	0.969353i	$-0.420991\pi$
$883$	383096.	0.491345	0.245672	+	0.969353i	$0.420991\pi$
$884$	19957.4i	0.0255387i
$885$	0	0
$886$	242280.	0.308639
$887$	450229.i	0.572251i	0.958192	+	0.286125i	$0.0923674\pi$
$887$	450229.i	0.572251i	−0.958192	+	0.286125i	$0.907633\pi$
$888$	0	0
$889$	543760.	0.688024
$890$	− 550075.i	− 0.694452i
$891$	0	0
$892$	−102760.	−0.129150
$893$	731771.i	0.917639i
$894$	0	0
$895$	1.62691e6	2.03104
$896$	− 525544.i	− 0.654626i
$897$	0	0
$898$	892998.	1.10738
$899$	− 359827.i	− 0.445219i
$900$	0	0
$901$	159894.	0.196962
$902$	− 130436.i	− 0.160318i
$903$	0	0
$904$	−170604.	−0.208762
$905$	555480.i	0.678222i
$906$	0	0
$907$	−1.09900e6	−1.33593	−0.667964	−	0.744193i	$-0.732834\pi$
$907$	−1.09900e6	−1.33593	−0.667964	+	0.744193i	$0.732834\pi$
$908$	− 180804.i	− 0.219299i
$909$	0	0
$910$	395136.	0.477160
$911$	− 284461.i	− 0.342756i	−0.985205	−	0.171378i	$-0.945178\pi$
$911$	− 284461.i	− 0.342756i	0.985205	−	0.171378i	$-0.0548220\pi$
$912$	0	0
$913$	−2016.00	−0.00241852
$914$	− 1.37801e6i	− 1.64953i
$915$	0	0
$916$	−157024.	−0.187144
$917$	565459.i	0.672454i
$918$	0	0
$919$	−490588.	−0.580879	−0.290440	−	0.956893i	$-0.593802\pi$
$919$	−490588.	−0.580879	−0.290440	+	0.956893i	$0.593802\pi$
$920$	− 1.40700e6i	− 1.66233i
$921$	0	0
$922$	701442.	0.825144
$923$	− 963656.i	− 1.13115i
$924$	0	0
$925$	212282.	0.248102
$926$	497322.i	0.579984i
$927$	0	0
$928$	−251640.	−0.292202
$929$	− 651852.i	− 0.755297i	−0.925949	−	0.377648i	$-0.876733\pi$
$929$	− 651852.i	− 0.755297i	0.925949	−	0.377648i	$-0.123267\pi$
$930$	0	0
$931$	−905520.	−1.04472
$932$	104904.i	0.120770i
$933$	0	0
$934$	724248.	0.830221
$935$	44904.1i	0.0513645i
$936$	0	0
$937$	−947842.	−1.07958	−0.539792	−	0.841798i	$-0.681497\pi$
$937$	−947842.	−1.07958	−0.539792	+	0.841798i	$0.681497\pi$
$938$	− 449516.i	− 0.510905i
$939$	0	0
$940$	77616.0	0.0878407
$941$	214750.i	0.242523i	0.992621	+	0.121262i	$0.0386940\pi$
$941$	214750.i	0.242523i	−0.992621	+	0.121262i	$0.961306\pi$
$942$	0	0
$943$	1.44497e6	1.62493
$944$	1.28202e6i	1.43864i
$945$	0	0
$946$	−124992.	−0.139669
$947$	1.18220e6i	1.31823i	0.752041	+	0.659117i	$0.229070\pi$
$947$	1.18220e6i	1.31823i	−0.752041	+	0.659117i	$0.770930\pi$
$948$	0	0
$949$	−740096.	−0.821780
$950$	610601.i	0.676566i
$951$	0	0
$952$	−148176.	−0.163495
$953$	426729.i	0.469858i	0.972013	+	0.234929i	$0.0754857\pi$
$953$	426729.i	0.469858i	−0.972013	+	0.234929i	$0.924514\pi$
$954$	0	0
$955$	253008.	0.277413
$956$	− 44530.8i	− 0.0487242i
$957$	0	0
$958$	422856.	0.460746
$959$	− 485273.i	− 0.527654i
$960$	0	0
$961$	−791025.	−0.856532
$962$	− 392495.i	− 0.424116i
$963$	0	0
$964$	106400.	0.114495
$965$	1.43735e6i	1.54350i
$966$	0	0
$967$	392828.	0.420097	0.210048	−	0.977691i	$-0.432638\pi$
$967$	392828.	0.420097	0.210048	+	0.977691i	$0.432638\pi$
$968$	− 852525.i	− 0.909822i
$969$	0	0
$970$	−733824.	−0.779917
$971$	567716.i	0.602134i	0.953603	+	0.301067i	$0.0973427\pi$
$971$	567716.i	0.602134i	−0.953603	+	0.301067i	$0.902657\pi$
$972$	0	0
$973$	−519008.	−0.548212
$974$	1.76706e6i	1.86266i
$975$	0	0
$976$	−743512.	−0.780528
$977$	122370.i	0.128200i	0.997943	+	0.0640999i	$0.0204176\pi$
$977$	122370.i	0.128200i	−0.997943	+	0.0640999i	$0.979582\pi$
$978$	0	0
$979$	−74088.0	−0.0773005
$980$	96044.9i	0.100005i
$981$	0	0
$982$	1.50624e6	1.56196
$983$	− 1.58495e6i	− 1.64024i	−0.572189	−	0.820121i	$-0.693906\pi$
$983$	− 1.58495e6i	− 1.64024i	0.572189	−	0.820121i	$-0.306094\pi$
$984$	0	0
$985$	297486.	0.306615
$986$	373666.i	0.384353i
$987$	0	0
$988$	125440.	0.128506
$989$	− 1.38466e6i	− 1.41563i
$990$	0	0
$991$	7364.00	0.00749836	0.00374918	−	0.999993i	$-0.498807\pi$
$991$	7364.00	0.00749836	0.00374918	+	0.999993i	$0.498807\pi$
$992$	− 92659.3i	− 0.0941598i
$993$	0	0
$994$	−1.02211e6	−1.03449
$995$	− 394990.i	− 0.398970i
$996$	0	0
$997$	−1.43479e6	−1.44344	−0.721719	−	0.692186i	$-0.756648\pi$
$997$	−1.43479e6	−1.44344	−0.721719	+	0.692186i	$0.756648\pi$
$998$	− 576863.i	− 0.579178i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.5.b.a.8.1	✓	2
3.2	odd	2	inner	9.5.b.a.8.2	yes	2
4.3	odd	2		144.5.e.c.17.2		2
5.2	odd	4		225.5.d.a.224.3		4
5.3	odd	4		225.5.d.a.224.2		4
5.4	even	2		225.5.c.a.26.2		2
7.6	odd	2		441.5.b.a.197.1		2
8.3	odd	2		576.5.e.g.449.1		2
8.5	even	2		576.5.e.d.449.1		2
9.2	odd	6		81.5.d.c.53.2		4
9.4	even	3		81.5.d.c.26.2		4
9.5	odd	6		81.5.d.c.26.1		4
9.7	even	3		81.5.d.c.53.1		4
12.11	even	2		144.5.e.c.17.1		2
15.2	even	4		225.5.d.a.224.1		4
15.8	even	4		225.5.d.a.224.4		4
15.14	odd	2		225.5.c.a.26.1		2
21.20	even	2		441.5.b.a.197.2		2
24.5	odd	2		576.5.e.d.449.2		2
24.11	even	2		576.5.e.g.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.b.a.8.1	✓	2	1.1	even	1	trivial
9.5.b.a.8.2	yes	2	3.2	odd	2	inner
81.5.d.c.26.1		4	9.5	odd	6
81.5.d.c.26.2		4	9.4	even	3
81.5.d.c.53.1		4	9.7	even	3
81.5.d.c.53.2		4	9.2	odd	6
144.5.e.c.17.1		2	12.11	even	2
144.5.e.c.17.2		2	4.3	odd	2
225.5.c.a.26.1		2	15.14	odd	2
225.5.c.a.26.2		2	5.4	even	2
225.5.d.a.224.1		4	15.2	even	4
225.5.d.a.224.2		4	5.3	odd	4
225.5.d.a.224.3		4	5.2	odd	4
225.5.d.a.224.4		4	15.8	even	4
441.5.b.a.197.1		2	7.6	odd	2
441.5.b.a.197.2		2	21.20	even	2
576.5.e.d.449.1		2	8.5	even	2
576.5.e.d.449.2		2	24.5	odd	2
576.5.e.g.449.1		2	8.3	odd	2
576.5.e.g.449.2		2	24.11	even	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	9.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{2} -2.00000 q^{4} +29.6985i q^{5} -28.0000 q^{7} -59.3970i q^{8} +O(q^{10})\)	\(q-4.24264i q^{2} -2.00000 q^{4} +29.6985i q^{5} -28.0000 q^{7} -59.3970i q^{8} +126.000 q^{10} -16.9706i q^{11} -112.000 q^{13} +118.794i q^{14} -284.000 q^{16} +89.0955i q^{17} +560.000 q^{19} -59.3970i q^{20} -72.0000 q^{22} -797.616i q^{23} -257.000 q^{25} +475.176i q^{26} +56.0000 q^{28} +988.535i q^{29} -364.000 q^{31} +254.558i q^{32} +378.000 q^{34} -831.558i q^{35} -826.000 q^{37} -2375.88i q^{38} +1764.00 q^{40} +1811.61i q^{41} +1736.00 q^{43} +33.9411i q^{44} -3384.00 q^{46} +1306.73i q^{47} -1617.00 q^{49} +1090.36i q^{50} +224.000 q^{52} -1794.64i q^{53} +504.000 q^{55} +1663.12i q^{56} +4194.00 q^{58} -4514.17i q^{59} +2618.00 q^{61} +1544.32i q^{62} -3464.00 q^{64} -3326.23i q^{65} -3784.00 q^{67} -178.191i q^{68} -3528.00 q^{70} +8604.08i q^{71} +6608.00 q^{73} +3504.42i q^{74} -1120.00 q^{76} +475.176i q^{77} -4276.00 q^{79} -8434.37i q^{80} +7686.00 q^{82} -118.794i q^{83} -2646.00 q^{85} -7365.22i q^{86} -1008.00 q^{88} -4365.68i q^{89} +3136.00 q^{91} +1595.23i q^{92} +5544.00 q^{94} +16631.2i q^{95} -5824.00 q^{97} +6860.35i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 56 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 - 56 * q^7	\( 2 q - 4 q^{4} - 56 q^{7} + 252 q^{10} - 224 q^{13} - 568 q^{16} + 1120 q^{19} - 144 q^{22} - 514 q^{25} + 112 q^{28} - 728 q^{31} + 756 q^{34} - 1652 q^{37} + 3528 q^{40} + 3472 q^{43} - 6768 q^{46} - 3234 q^{49} + 448 q^{52} + 1008 q^{55} + 8388 q^{58} + 5236 q^{61} - 6928 q^{64} - 7568 q^{67} - 7056 q^{70} + 13216 q^{73} - 2240 q^{76} - 8552 q^{79} + 15372 q^{82} - 5292 q^{85} - 2016 q^{88} + 6272 q^{91} + 11088 q^{94} - 11648 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 56 * q^7 + 252 * q^10 - 224 * q^13 - 568 * q^16 + 1120 * q^19 - 144 * q^22 - 514 * q^25 + 112 * q^28 - 728 * q^31 + 756 * q^34 - 1652 * q^37 + 3528 * q^40 + 3472 * q^43 - 6768 * q^46 - 3234 * q^49 + 448 * q^52 + 1008 * q^55 + 8388 * q^58 + 5236 * q^61 - 6928 * q^64 - 7568 * q^67 - 7056 * q^70 + 13216 * q^73 - 2240 * q^76 - 8552 * q^79 + 15372 * q^82 - 5292 * q^85 - 2016 * q^88 + 6272 * q^91 + 11088 * q^94 - 11648 * q^97

Introduction

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