LMFDB - Embedded newform 45.8.b.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,8,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	45.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:	$14.0573261468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 29 $ x^2 + 29
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-5.38516i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.8.b.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.7703i q^{2} +12.0000 q^{4} +(-75.0000 + 269.258i) q^{5} -420.043i q^{7} -1507.85i q^{8} +O(q^{10})$	\(q-10.7703i q^{2} +12.0000 q^{4} +(-75.0000 + 269.258i) q^{5} -420.043i q^{7} -1507.85i q^{8} +(2900.00 + 807.775i) q^{10} +6828.00 q^{11} -10145.7i q^{13} -4524.00 q^{14} -14704.0 q^{16} -15681.6i q^{17} +6860.00 q^{19} +(-900.000 + 3231.10i) q^{20} -73539.8i q^{22} -29219.9i q^{23} +(-66875.0 - 40388.7i) q^{25} -109272. q^{26} -5040.51i q^{28} -25590.0 q^{29} +82112.0 q^{31} -34637.4i q^{32} -168896. q^{34} +(113100. + 31503.2i) q^{35} -223527. i q^{37} -73884.5i q^{38} +(406000. + 113088. i) q^{40} +533118. q^{41} +708935. i q^{43} +81936.0 q^{44} -314708. q^{46} -5826.75i q^{47} +647107. q^{49} +(-435000. + 720266. i) q^{50} -121748. i q^{52} +589374. i q^{53} +(-512100. + 1.83850e6i) q^{55} -633360. q^{56} +275613. i q^{58} -1.43898e6 q^{59} +1.38102e6 q^{61} -884373. i q^{62} -2.25517e6 q^{64} +(2.73180e6 + 760924. i) q^{65} -2.71487e6i q^{67} -188179. i q^{68} +(339300. - 1.21812e6i) q^{70} +481608. q^{71} +1.48618e6i q^{73} -2.40746e6 q^{74} +82320.0 q^{76} -2.86805e6i q^{77} -1.05976e6 q^{79} +(1.10280e6 - 3.95917e6i) q^{80} -5.74186e6i q^{82} +2.60380e6i q^{83} +(4.22240e6 + 1.17612e6i) q^{85} +7.63547e6 q^{86} -1.02956e7i q^{88} -5.64417e6 q^{89} -4.26161e6 q^{91} -350639. i q^{92} -62756.0 q^{94} +(-514500. + 1.84711e6i) q^{95} +1.20091e7i q^{97} -6.96956e6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{4} - 150 q^{5}+O(q^{10}) $ 2 * q + 24 * q^4 - 150 * q^5	$ 2 q + 24 q^{4} - 150 q^{5} + 5800 q^{10} + 13656 q^{11} - 9048 q^{14} - 29408 q^{16} + 13720 q^{19} - 1800 q^{20} - 133750 q^{25} - 218544 q^{26} - 51180 q^{29} + 164224 q^{31} - 337792 q^{34} + 226200 q^{35} + 812000 q^{40} + 1066236 q^{41} + 163872 q^{44} - 629416 q^{46} + 1294214 q^{49} - 870000 q^{50} - 1024200 q^{55} - 1266720 q^{56} - 2877960 q^{59} + 2762044 q^{61} - 4510336 q^{64} + 5463600 q^{65} + 678600 q^{70} + 963216 q^{71} - 4814928 q^{74} + 164640 q^{76} - 2119520 q^{79} + 2205600 q^{80} + 8444800 q^{85} + 15270936 q^{86} - 11288340 q^{89} - 8523216 q^{91} - 125512 q^{94} - 1029000 q^{95}+O(q^{100}) $ 2 * q + 24 * q^4 - 150 * q^5 + 5800 * q^10 + 13656 * q^11 - 9048 * q^14 - 29408 * q^16 + 13720 * q^19 - 1800 * q^20 - 133750 * q^25 - 218544 * q^26 - 51180 * q^29 + 164224 * q^31 - 337792 * q^34 + 226200 * q^35 + 812000 * q^40 + 1066236 * q^41 + 163872 * q^44 - 629416 * q^46 + 1294214 * q^49 - 870000 * q^50 - 1024200 * q^55 - 1266720 * q^56 - 2877960 * q^59 + 2762044 * q^61 - 4510336 * q^64 + 5463600 * q^65 + 678600 * q^70 + 963216 * q^71 - 4814928 * q^74 + 164640 * q^76 - 2119520 * q^79 + 2205600 * q^80 + 8444800 * q^85 + 15270936 * q^86 - 11288340 * q^89 - 8523216 * q^91 - 125512 * q^94 - 1029000 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	10.7703i		−	0.951972i	−0.879453	−	0.475986i	$-0.842091\pi$
$2$		−	10.7703i		−	0.951972i	0.879453	−	0.475986i	$-0.157909\pi$
$3$		0			0
$3$		0			0
$4$	12.0000			0.0937500
$5$	−75.0000	+	269.258i	−0.268328	+	0.963328i
$5$	−75.0000	+	269.258i	−0.268328	+	0.963328i
$6$		0			0
$7$		−	420.043i		−	0.462861i	−0.972851	−	0.231430i	$-0.925659\pi$
$7$		−	420.043i		−	0.462861i	0.972851	−	0.231430i	$-0.0743406\pi$
$8$		−	1507.85i		−	1.04122i
$9$		0			0
$10$	2900.00	+	807.775i	0.917061	+	0.255441i
$11$	6828.00			1.54675			0.773373	−	0.633951i	$-0.218568\pi$
$11$	6828.00			1.54675			0.773373	+	0.633951i	$0.218568\pi$
$12$		0			0
$13$		−	10145.7i		−	1.28079i	−0.768045	−	0.640395i	$-0.778771\pi$
$13$		−	10145.7i		−	1.28079i	0.768045	−	0.640395i	$-0.221229\pi$
$14$	−4524.00			−0.440630
$15$		0			0
$16$	−14704.0			−0.897461
$17$		−	15681.6i		−	0.774139i	−0.922050	−	0.387070i	$-0.873487\pi$
$17$		−	15681.6i		−	0.774139i	0.922050	−	0.387070i	$-0.126513\pi$
$18$		0			0
$19$	6860.00			0.229449			0.114725	−	0.993397i	$-0.463401\pi$
$19$	6860.00			0.229449			0.114725	+	0.993397i	$0.463401\pi$
$20$	−900.000	+	3231.10i	−0.0251558	+	0.0903120i
$21$		0			0
$22$		−	73539.8i		−	1.47246i
$23$		−	29219.9i		−	0.500762i	−0.968147	−	0.250381i	$-0.919444\pi$
$23$		−	29219.9i		−	0.500762i	0.968147	−	0.250381i	$-0.0805559\pi$
$24$		0			0
$25$	−66875.0	−	40388.7i	−0.856000	−	0.516976i
$26$	−109272.			−1.21928
$27$		0			0
$28$		−	5040.51i		−	0.0433932i
$29$	−25590.0			−0.194840			−0.0974198	−	0.995243i	$-0.531059\pi$
$29$	−25590.0			−0.194840			−0.0974198	+	0.995243i	$0.531059\pi$
$30$		0			0
$31$	82112.0			0.495040			0.247520	−	0.968883i	$-0.420384\pi$
$31$	82112.0			0.495040			0.247520	+	0.968883i	$0.420384\pi$
$32$		−	34637.4i		−	0.186862i
$33$		0			0
$34$	−168896.			−0.736959
$35$	113100.	+	31503.2i	0.445887	+	0.124199i
$36$		0			0
$37$		−	223527.i		−	0.725479i	−0.931891	−	0.362739i	$-0.881842\pi$
$37$		−	223527.i		−	0.725479i	0.931891	−	0.362739i	$-0.118158\pi$
$38$		−	73884.5i		−	0.218429i
$39$		0			0
$40$	406000.	+	113088.i	1.00303	+	0.279388i
$41$	533118.			1.20804			0.604018	−	0.796971i	$-0.293566\pi$
$41$	533118.			1.20804			0.604018	+	0.796971i	$0.293566\pi$
$42$		0			0
$43$			708935.i			1.35978i	0.733316	+	0.679888i	$0.237971\pi$
$43$			708935.i			1.35978i	−0.733316	+	0.679888i	$0.762029\pi$
$44$	81936.0			0.145007
$45$		0			0
$46$	−314708.			−0.476711
$47$		−	5826.75i		−	0.00818623i	−0.999992	−	0.00409311i	$-0.998697\pi$
$47$		−	5826.75i		−	0.00818623i	0.999992	−	0.00409311i	$-0.00130288\pi$
$48$		0			0
$49$	647107.			0.785760
$50$	−435000.	+	720266.i	−0.492146	+	0.814888i
$51$		0			0
$52$		−	121748.i		−	0.120074i
$53$			589374.i			0.543783i	0.962328	+	0.271891i	$0.0876491\pi$
$53$			589374.i			0.543783i	−0.962328	+	0.271891i	$0.912351\pi$
$54$		0			0
$55$	−512100.	+	1.83850e6i	−0.415036	+	1.49002i
$56$	−633360.			−0.481940
$57$		0			0
$58$			275613.i			0.185482i
$59$	−1.43898e6			−0.912164			−0.456082	−	0.889938i	$-0.650748\pi$
$59$	−1.43898e6			−0.912164			−0.456082	+	0.889938i	$0.650748\pi$
$60$		0			0
$61$	1.38102e6			0.779016			0.389508	−	0.921023i	$-0.372645\pi$
$61$	1.38102e6			0.779016			0.389508	+	0.921023i	$0.372645\pi$
$62$		−	884373.i		−	0.471264i
$63$		0			0
$64$	−2.25517e6			−1.07535
$65$	2.73180e6	+	760924.i	1.23382	+	0.343672i
$66$		0			0
$67$		−	2.71487e6i		−	1.10277i	−0.834250	−	0.551387i	$-0.814099\pi$
$67$		−	2.71487e6i		−	1.10277i	0.834250	−	0.551387i	$-0.185901\pi$
$68$		−	188179.i		−	0.0725756i
$69$		0			0
$70$	339300.	−	1.21812e6i	0.118234	−	0.424471i
$71$	481608.			0.159694			0.0798472	−	0.996807i	$-0.474557\pi$
$71$	481608.			0.159694			0.0798472	+	0.996807i	$0.474557\pi$
$72$		0			0
$73$			1.48618e6i			0.447137i	0.974688	+	0.223568i	$0.0717706\pi$
$73$			1.48618e6i			0.447137i	−0.974688	+	0.223568i	$0.928229\pi$
$74$	−2.40746e6			−0.690635
$75$		0			0
$76$	82320.0			0.0215109
$77$		−	2.86805e6i		−	0.715928i
$78$		0			0
$79$	−1.05976e6			−0.241831			−0.120916	−	0.992663i	$-0.538583\pi$
$79$	−1.05976e6			−0.241831			−0.120916	+	0.992663i	$0.538583\pi$
$80$	1.10280e6	−	3.95917e6i	0.240814	−	0.864549i
$81$		0			0
$82$		−	5.74186e6i		−	1.15002i
$83$			2.60380e6i			0.499844i	0.968266	+	0.249922i	$0.0804050\pi$
$83$			2.60380e6i			0.499844i	−0.968266	+	0.249922i	$0.919595\pi$
$84$		0			0
$85$	4.22240e6	+	1.17612e6i	0.745750	+	0.207723i
$86$	7.63547e6			1.29447
$87$		0			0
$88$		−	1.02956e7i		−	1.61050i
$89$	−5.64417e6			−0.848663			−0.424331	−	0.905507i	$-0.639491\pi$
$89$	−5.64417e6			−0.848663			−0.424331	+	0.905507i	$0.639491\pi$
$90$		0			0
$91$	−4.26161e6			−0.592828
$92$		−	350639.i		−	0.0469464i
$93$		0			0
$94$	−62756.0			−0.00779306
$95$	−514500.	+	1.84711e6i	−0.0615677	+	0.221035i
$96$		0			0
$97$			1.20091e7i			1.33601i	0.744158	+	0.668004i	$0.232851\pi$
$97$			1.20091e7i			1.33601i	−0.744158	+	0.668004i	$0.767149\pi$
$98$		−	6.96956e6i		−	0.748021i
$99$		0			0
$100$	−802500.	−	484665.i	−0.0802500	−	0.0484665i
$101$	−5.14270e6			−0.496668			−0.248334	−	0.968674i	$-0.579883\pi$
$101$	−5.14270e6			−0.496668			−0.248334	+	0.968674i	$0.579883\pi$
$102$		0			0
$103$			3.48477e6i			0.314227i	0.987581	+	0.157114i	$0.0502189\pi$
$103$			3.48477e6i			0.314227i	−0.987581	+	0.157114i	$0.949781\pi$
$104$	−1.52981e7			−1.33358
$105$		0			0
$106$	6.34775e6			0.517666
$107$			1.48640e7i			1.17299i	0.809954	+	0.586493i	$0.199492\pi$
$107$			1.48640e7i			1.17299i	−0.809954	+	0.586493i	$0.800508\pi$
$108$		0			0
$109$	−2.01124e7			−1.48755			−0.743773	−	0.668432i	$-0.766966\pi$
$109$	−2.01124e7			−1.48755			−0.743773	+	0.668432i	$0.766966\pi$
$110$	1.98012e7	+	5.51549e6i	1.41846	+	0.395102i
$111$		0			0
$112$			6.17631e6i			0.415400i
$113$			5.62633e6i			0.366818i	0.983037	+	0.183409i	$0.0587133\pi$
$113$			5.62633e6i			0.366818i	−0.983037	+	0.183409i	$0.941287\pi$
$114$		0			0
$115$	7.86770e6	+	2.19149e6i	0.482398	+	0.134369i
$116$	−307080.			−0.0182662
$117$		0			0
$118$			1.54983e7i			0.868354i
$119$	−6.58694e6			−0.358319
$120$		0			0
$121$	2.71344e7			1.39242
$122$		−	1.48741e7i		−	0.741601i
$123$		0			0
$124$	985344.			0.0464100
$125$	1.58906e7	−	1.49775e7i	0.727706	−	0.685889i
$126$		0			0
$127$			2.85360e7i			1.23618i	0.786109	+	0.618088i	$0.212092\pi$
$127$			2.85360e7i			1.23618i	−0.786109	+	0.618088i	$0.787908\pi$
$128$			1.98553e7i			0.836839i
$129$		0			0
$130$	8.19540e6	−	2.94224e7i	0.327166	−	1.17456i
$131$	3.33132e7			1.29469			0.647346	−	0.762196i	$-0.275879\pi$
$131$	3.33132e7			1.29469			0.647346	+	0.762196i	$0.275879\pi$
$132$		0			0
$133$		−	2.88149e6i		−	0.106203i
$134$	−2.92400e7			−1.04981
$135$		0			0
$136$	−2.36454e7			−0.806049
$137$			4.28099e7i			1.42240i	0.702988	+	0.711202i	$0.251849\pi$
$137$			4.28099e7i			1.42240i	−0.702988	+	0.711202i	$0.748151\pi$
$138$		0			0
$139$	1.13808e7			0.359436			0.179718	−	0.983718i	$-0.442481\pi$
$139$	1.13808e7			0.359436			0.179718	+	0.983718i	$0.442481\pi$
$140$	1.35720e6	+	378039.i	0.0418019	+	0.0116436i
$141$		0			0
$142$		−	5.18708e6i		−	0.152024i
$143$		−	6.92745e7i		−	1.98106i
$144$		0			0
$145$	1.91925e6	−	6.89032e6i	0.0522810	−	0.187694i
$146$	1.60066e7			0.425661
$147$		0			0
$148$		−	2.68233e6i		−	0.0680136i
$149$	−4.00070e7			−0.990794			−0.495397	−	0.868667i	$-0.664977\pi$
$149$	−4.00070e7			−0.990794			−0.495397	+	0.868667i	$0.664977\pi$
$150$		0			0
$151$	−2.86594e7			−0.677405			−0.338703	−	0.940893i	$-0.609988\pi$
$151$	−2.86594e7			−0.677405			−0.338703	+	0.940893i	$0.609988\pi$
$152$		−	1.03438e7i		−	0.238907i
$153$		0			0
$154$	−3.08899e7			−0.681544
$155$	−6.15840e6	+	2.21093e7i	−0.132833	+	0.476886i
$156$		0			0
$157$		−	3.01958e7i		−	0.622728i	−0.950291	−	0.311364i	$-0.899214\pi$
$157$		−	3.01958e7i		−	0.622728i	0.950291	−	0.311364i	$-0.100786\pi$
$158$			1.14140e7i			0.230217i
$159$		0			0
$160$	9.32640e6	+	2.59780e6i	0.180009	+	0.0501402i
$161$	−1.22736e7			−0.231783
$162$		0			0
$163$		−	9.35416e7i		−	1.69180i	−0.533345	−	0.845898i	$-0.679065\pi$
$163$		−	9.35416e7i		−	1.69180i	0.533345	−	0.845898i	$-0.320935\pi$
$164$	6.39742e6			0.113253
$165$		0			0
$166$	2.80438e7			0.475838
$167$		−	5.73507e7i		−	0.952865i	−0.879211	−	0.476432i	$-0.841930\pi$
$167$		−	5.73507e7i		−	0.952865i	0.879211	−	0.476432i	$-0.158070\pi$
$168$		0			0
$169$	−4.01857e7			−0.640425
$170$	1.26672e7	−	4.54766e7i	0.197747	−	0.709933i
$171$		0			0
$172$			8.50722e6i			0.127479i
$173$			4.87192e7i			0.715383i	0.933840	+	0.357691i	$0.116436\pi$
$173$			4.87192e7i			0.715383i	−0.933840	+	0.357691i	$0.883564\pi$
$174$		0			0
$175$	−1.69650e7	+	2.80904e7i	−0.239288	+	0.396209i
$176$	−1.00399e8			−1.38814
$177$		0			0
$178$			6.07896e7i			0.807903i
$179$	−1.93505e7			−0.252178			−0.126089	−	0.992019i	$-0.540243\pi$
$179$	−1.93505e7			−0.252178			−0.126089	+	0.992019i	$0.540243\pi$
$180$		0			0
$181$	7.82617e7			0.981011			0.490506	−	0.871438i	$-0.336812\pi$
$181$	7.82617e7			0.981011			0.490506	+	0.871438i	$0.336812\pi$
$182$			4.58989e7i			0.564355i
$183$		0			0
$184$	−4.40591e7			−0.521403
$185$	6.01866e7	+	1.67646e7i	0.698874	+	0.194666i
$186$		0			0
$187$		−	1.07074e8i		−	1.19740i
$188$		−	69921.0i		−	0.000767459i
$189$		0			0
$190$	1.98940e7	+	5.54133e6i	0.210419	+	0.0586107i
$191$	1.19454e8			1.24046			0.620229	−	0.784420i	$-0.287040\pi$
$191$	1.19454e8			1.24046			0.620229	+	0.784420i	$0.287040\pi$
$192$		0			0
$193$			5.98469e7i			0.599227i	0.954061	+	0.299613i	$0.0968577\pi$
$193$			5.98469e7i			0.599227i	−0.954061	+	0.299613i	$0.903142\pi$
$194$	1.29342e8			1.27184
$195$		0			0
$196$	7.76528e6			0.0736650
$197$		−	1.22964e8i		−	1.14590i	−0.819589	−	0.572952i	$-0.805798\pi$
$197$		−	1.22964e8i		−	1.14590i	0.819589	−	0.572952i	$-0.194202\pi$
$198$		0			0
$199$	1.69053e8			1.52067			0.760337	−	0.649529i	$-0.225034\pi$
$199$	1.69053e8			1.52067			0.760337	+	0.649529i	$0.225034\pi$
$200$	−6.09000e7	+	1.00837e8i	−0.538285	+	0.891283i
$201$		0			0
$202$			5.53886e7i			0.472814i
$203$			1.07489e7i			0.0901836i
$204$		0			0
$205$	−3.99838e7	+	1.43546e8i	−0.324150	+	1.16373i
$206$	3.75321e7			0.299136
$207$		0			0
$208$			1.49182e8i			1.14946i
$209$	4.68401e7			0.354900
$210$		0			0
$211$	−2.67605e8			−1.96113			−0.980565	−	0.196195i	$-0.937141\pi$
$211$	−2.67605e8			−1.96113			−0.980565	+	0.196195i	$0.937141\pi$
$212$			7.07249e6i			0.0509796i
$213$		0			0
$214$	1.60090e8			1.11665
$215$	−1.90887e8	−	5.31702e7i	−1.30991	−	0.364866i
$216$		0			0
$217$		−	3.44906e7i		−	0.229135i
$218$			2.16617e8i			1.41610i
$219$		0			0
$220$	−6.14520e6	+	2.20619e7i	−0.0389096	+	0.139690i
$221$	−1.59100e8			−0.991510
$222$		0			0
$223$		−	1.49333e8i		−	0.901753i	−0.892586	−	0.450877i	$-0.851111\pi$
$223$		−	1.49333e8i		−	0.901753i	0.892586	−	0.450877i	$-0.148889\pi$
$224$	−1.45492e7			−0.0864909
$225$		0			0
$226$	6.05975e7			0.349201
$227$			2.34185e8i			1.32883i	0.747365	+	0.664414i	$0.231319\pi$
$227$			2.34185e8i			1.32883i	−0.747365	+	0.664414i	$0.768681\pi$
$228$		0			0
$229$	−1.31882e8			−0.725706			−0.362853	−	0.931846i	$-0.618197\pi$
$229$	−1.31882e8			−0.725706			−0.362853	+	0.931846i	$0.618197\pi$
$230$	2.36031e7	−	8.47377e7i	0.127915	−	0.459229i
$231$		0			0
$232$			3.85858e7i			0.202871i
$233$		−	1.83419e8i		−	0.949948i	−0.880000	−	0.474974i	$-0.842458\pi$
$233$		−	1.83419e8i		−	0.949948i	0.880000	−	0.474974i	$-0.157542\pi$
$234$		0			0
$235$	1.56890e6	+	437006.i	0.00788602	+	0.00219660i
$236$	−1.72678e7			−0.0855153
$237$		0			0
$238$			7.09436e7i			0.341109i
$239$	1.05117e8			0.498058			0.249029	−	0.968496i	$-0.419889\pi$
$239$	1.05117e8			0.498058			0.249029	+	0.968496i	$0.419889\pi$
$240$		0			0
$241$	1.94216e8			0.893770			0.446885	−	0.894591i	$-0.352533\pi$
$241$	1.94216e8			0.893770			0.446885	+	0.894591i	$0.352533\pi$
$242$		−	2.92247e8i		−	1.32555i
$243$		0			0
$244$	1.65723e7			0.0730327
$245$	−4.85330e7	+	1.74239e8i	−0.210841	+	0.756944i
$246$		0			0
$247$		−	6.95992e7i		−	0.293876i
$248$		−	1.23812e8i		−	0.515446i
$249$		0			0
$250$	−1.61312e8	−	1.71147e8i	−0.652947	−	0.692755i
$251$	−2.02689e8			−0.809045			−0.404523	−	0.914528i	$-0.632562\pi$
$251$	−2.02689e8			−0.809045			−0.404523	+	0.914528i	$0.632562\pi$
$252$		0			0
$253$		−	1.99514e8i		−	0.774552i
$254$	3.07342e8			1.17680
$255$		0			0
$256$	−7.48132e7			−0.278701
$257$			1.34593e7i			0.0494603i	0.999694	+	0.0247301i	$0.00787265\pi$
$257$			1.34593e7i			0.0494603i	−0.999694	+	0.0247301i	$0.992127\pi$
$258$		0			0
$259$	−9.38911e7			−0.335796
$260$	3.27816e7	+	9.13109e6i	0.115671	+	0.0322193i
$261$		0			0
$262$		−	3.58794e8i		−	1.23251i
$263$		−	1.42205e8i		−	0.482026i	−0.970522	−	0.241013i	$-0.922520\pi$
$263$		−	1.42205e8i		−	0.482026i	0.970522	−	0.241013i	$-0.0774797\pi$
$264$		0			0
$265$	−1.58694e8	−	4.42030e7i	−0.523841	−	0.145912i
$266$	−3.10346e7			−0.101102
$267$		0			0
$268$		−	3.25784e7i		−	0.103385i
$269$	5.07548e8			1.58981			0.794903	−	0.606737i	$-0.207522\pi$
$269$	5.07548e8			1.58981			0.794903	+	0.606737i	$0.207522\pi$
$270$		0			0
$271$	1.12836e8			0.344393			0.172197	−	0.985063i	$-0.444914\pi$
$271$	1.12836e8			0.344393			0.172197	+	0.985063i	$0.444914\pi$
$272$			2.30582e8i			0.694760i
$273$		0			0
$274$	4.61077e8			1.35409
$275$	−4.56622e8	−	2.75774e8i	−1.32401	−	0.799630i
$276$		0			0
$277$			5.10728e8i			1.44381i	0.691991	+	0.721906i	$0.256734\pi$
$277$			5.10728e8i			1.44381i	−0.691991	+	0.721906i	$0.743266\pi$
$278$		−	1.22575e8i		−	0.342173i
$279$		0			0
$280$	4.75020e7	−	1.70537e8i	0.129318	−	0.464266i
$281$	1.70459e8			0.458297			0.229148	−	0.973391i	$-0.426406\pi$
$281$	1.70459e8			0.458297			0.229148	+	0.973391i	$0.426406\pi$
$282$		0			0
$283$		−	1.62144e8i		−	0.425253i	−0.977134	−	0.212626i	$-0.931798\pi$
$283$		−	1.62144e8i		−	0.425253i	0.977134	−	0.212626i	$-0.0682017\pi$
$284$	5.77930e6			0.0149713
$285$		0			0
$286$	−7.46109e8			−1.88591
$287$		−	2.23932e8i		−	0.559153i
$288$		0			0
$289$	1.64426e8			0.400708
$290$	−7.42110e7	−	2.06710e7i	−0.178680	−	0.0497700i
$291$		0			0
$292$			1.78341e7i			0.0419191i
$293$			3.85845e8i			0.896141i	0.893998	+	0.448070i	$0.147889\pi$
$293$			3.85845e8i			0.896141i	−0.893998	+	0.448070i	$0.852111\pi$
$294$		0			0
$295$	1.07924e8	−	3.87457e8i	0.244759	−	0.878712i
$296$	−3.37045e8			−0.755382
$297$		0			0
$298$			4.30888e8i			0.943208i
$299$	−2.96455e8			−0.641371
$300$		0			0
$301$	2.97783e8			0.629387
$302$			3.08672e8i			0.644871i
$303$		0			0
$304$	−1.00869e8			−0.205922
$305$	−1.03577e8	+	3.71852e8i	−0.209032	+	0.750447i
$306$		0			0
$307$			6.37817e8i			1.25809i	0.777369	+	0.629045i	$0.216554\pi$
$307$			6.37817e8i			1.25809i	−0.777369	+	0.629045i	$0.783446\pi$
$308$		−	3.44166e7i		−	0.0671183i
$309$		0			0
$310$	2.38125e8	+	6.63280e7i	0.453982	+	0.126454i
$311$	7.27817e8			1.37202			0.686010	−	0.727592i	$-0.259360\pi$
$311$	7.27817e8			1.37202			0.686010	+	0.727592i	$0.259360\pi$
$312$		0			0
$313$			4.46869e8i			0.823711i	0.911249	+	0.411855i	$0.135119\pi$
$313$			4.46869e8i			0.823711i	−0.911249	+	0.411855i	$0.864881\pi$
$314$	−3.25219e8			−0.592819
$315$		0			0
$316$	−1.27171e7			−0.0226717
$317$			5.91083e8i			1.04218i	0.853503	+	0.521088i	$0.174474\pi$
$317$			5.91083e8i			1.04218i	−0.853503	+	0.521088i	$0.825526\pi$
$318$		0			0
$319$	−1.74729e8			−0.301367
$320$	1.69138e8	−	6.07223e8i	0.288546	−	1.03591i
$321$		0			0
$322$			1.32191e8i			0.220651i
$323$		−	1.07576e8i		−	0.177626i
$324$		0			0
$325$	−4.09770e8	+	6.78490e8i	−0.662138	+	1.09636i
$326$	−1.00747e9			−1.61054
$327$		0			0
$328$		−	8.03860e8i		−	1.25783i
$329$	−2.44748e6			−0.00378908
$330$		0			0
$331$	5.84868e8			0.886462			0.443231	−	0.896407i	$-0.353832\pi$
$331$	5.84868e8			0.886462			0.443231	+	0.896407i	$0.353832\pi$
$332$			3.12456e7i			0.0468604i
$333$		0			0
$334$	−6.17686e8			−0.907100
$335$	7.31000e8	+	2.03615e8i	1.06233	+	0.295905i
$336$		0			0
$337$			7.39373e8i			1.05235i	0.850377	+	0.526174i	$0.176374\pi$
$337$			7.39373e8i			1.05235i	−0.850377	+	0.526174i	$0.823626\pi$
$338$			4.32813e8i			0.609666i
$339$		0			0
$340$	5.06688e7	+	1.41134e7i	0.0699140	+	0.0194741i
$341$	5.60661e8			0.765702
$342$		0			0
$343$		−	6.17736e8i		−	0.826558i
$344$	1.06897e9			1.41582
$345$		0			0
$346$	5.24722e8			0.681024
$347$		−	3.70870e8i		−	0.476506i	−0.971203	−	0.238253i	$-0.923425\pi$
$347$		−	3.70870e8i		−	0.476506i	0.971203	−	0.238253i	$-0.0765747\pi$
$348$		0			0
$349$	1.13274e9			1.42640			0.713199	−	0.700962i	$-0.247246\pi$
$349$	1.13274e9			1.42640			0.713199	+	0.700962i	$0.247246\pi$
$350$	3.02542e8	+	1.82719e8i	0.377180	+	0.227795i
$351$		0			0
$352$		−	2.36504e8i		−	0.289028i
$353$		−	8.32858e8i		−	1.00777i	−0.863772	−	0.503883i	$-0.831904\pi$
$353$		−	8.32858e8i		−	1.00777i	0.863772	−	0.503883i	$-0.168096\pi$
$354$		0			0
$355$	−3.61206e7	+	1.29677e8i	−0.0428505	+	0.153838i
$356$	−6.77300e7			−0.0795621
$357$		0			0
$358$			2.08412e8i			0.240066i
$359$	−6.75318e8			−0.770332			−0.385166	−	0.922847i	$-0.625856\pi$
$359$	−6.75318e8			−0.770332			−0.385166	+	0.922847i	$0.625856\pi$
$360$		0			0
$361$	−8.46812e8			−0.947353
$362$		−	8.42904e8i		−	0.933895i
$363$		0			0
$364$	−5.11393e7			−0.0555776
$365$	−4.00165e8	−	1.11463e8i	−0.430739	−	0.119979i
$366$		0			0
$367$		−	1.80237e9i		−	1.90333i	−0.307141	−	0.951664i	$-0.599372\pi$
$367$		−	1.80237e9i		−	1.90333i	0.307141	−	0.951664i	$-0.400628\pi$
$368$			4.29649e8i			0.449414i
$369$		0			0
$370$	1.80560e8	−	6.48230e8i	0.185317	−	0.665308i
$371$	2.47562e8			0.251696
$372$		0			0
$373$			9.29928e8i			0.927830i	0.885880	+	0.463915i	$0.153556\pi$
$373$			9.29928e8i			0.927830i	−0.885880	+	0.463915i	$0.846444\pi$
$374$	−1.15322e9			−1.13989
$375$		0			0
$376$	−8.78584e6			−0.00852365
$377$			2.59627e8i			0.249549i
$378$		0			0
$379$	−1.43545e9			−1.35441			−0.677206	−	0.735794i	$-0.736809\pi$
$379$	−1.43545e9			−1.35441			−0.677206	+	0.735794i	$0.736809\pi$
$380$	−6.17400e6	+	2.21653e7i	−0.00577197	+	0.0207220i
$381$		0			0
$382$		−	1.28655e9i		−	1.18088i
$383$			1.57707e9i			1.43435i	0.696894	+	0.717174i	$0.254565\pi$
$383$			1.57707e9i			1.43435i	−0.696894	+	0.717174i	$0.745435\pi$
$384$		0			0
$385$	7.72247e8	+	2.15104e8i	0.689674	+	0.192104i
$386$	6.44571e8			0.570447
$387$		0			0
$388$			1.44109e8i			0.125251i
$389$	−2.24425e9			−1.93307			−0.966537	−	0.256528i	$-0.917421\pi$
$389$	−2.24425e9			−1.93307			−0.966537	+	0.256528i	$0.917421\pi$
$390$		0			0
$391$	−4.58215e8			−0.387660
$392$		−	9.75738e8i		−	0.818148i
$393$		0			0
$394$	−1.32437e9			−1.09087
$395$	7.94820e7	−	2.85349e8i	0.0648902	−	0.232963i
$396$		0			0
$397$		−	2.26641e8i		−	0.181791i	−0.995860	−	0.0908956i	$-0.971027\pi$
$397$		−	2.26641e8i		−	0.181791i	0.995860	−	0.0908956i	$-0.0289729\pi$
$398$		−	1.82075e9i		−	1.44764i
$399$		0			0
$400$	9.83330e8	+	5.93876e8i	0.768227	+	0.463966i
$401$	9.11721e8			0.706085			0.353042	−	0.935607i	$-0.385147\pi$
$401$	9.11721e8			0.706085			0.353042	+	0.935607i	$0.385147\pi$
$402$		0			0
$403$		−	8.33080e8i		−	0.634043i
$404$	−6.17124e7			−0.0465627
$405$		0			0
$406$	1.15769e8			0.0858523
$407$		−	1.52625e9i		−	1.12213i
$408$		0			0
$409$	−2.55215e8			−0.184449			−0.0922243	−	0.995738i	$-0.529398\pi$
$409$	−2.55215e8			−0.184449			−0.0922243	+	0.995738i	$0.529398\pi$
$410$	1.54604e9	+	4.30639e8i	1.10784	+	0.308582i
$411$		0			0
$412$			4.18173e7i			0.0294588i
$413$			6.04433e8i			0.422205i
$414$		0			0
$415$	−7.01095e8	−	1.95285e8i	−0.481514	−	0.134122i
$416$	−3.51419e8			−0.239331
$417$		0			0
$418$		−	5.04483e8i		−	0.337854i
$419$	−2.96316e8			−0.196791			−0.0983957	−	0.995147i	$-0.531371\pi$
$419$	−2.96316e8			−0.196791			−0.0983957	+	0.995147i	$0.531371\pi$
$420$		0			0
$421$	1.06676e9			0.696754			0.348377	−	0.937354i	$-0.386733\pi$
$421$	1.06676e9			0.696754			0.348377	+	0.937354i	$0.386733\pi$
$422$			2.88220e9i			1.86694i
$423$		0			0
$424$	8.88685e8			0.566197
$425$	−6.33360e8	+	1.04871e9i	−0.400211	+	0.662663i
$426$		0			0
$427$		−	5.80088e8i		−	0.360576i
$428$			1.78368e8i			0.109967i
$429$		0			0
$430$	−5.72660e8	+	2.05591e9i	−0.347342	+	1.24700i
$431$	−9.53169e7			−0.0573455			−0.0286728	−	0.999589i	$-0.509128\pi$
$431$	−9.53169e7			−0.0573455			−0.0286728	+	0.999589i	$0.509128\pi$
$432$		0			0
$433$			1.89973e9i			1.12456i	0.826946	+	0.562281i	$0.190076\pi$
$433$			1.89973e9i			1.12456i	−0.826946	+	0.562281i	$0.809924\pi$
$434$	−3.71475e8			−0.218130
$435$		0			0
$436$	−2.41348e8			−0.139457
$437$		−	2.00449e8i		−	0.114899i
$438$		0			0
$439$	−1.11226e9			−0.627450			−0.313725	−	0.949514i	$-0.601577\pi$
$439$	−1.11226e9			−0.627450			−0.313725	+	0.949514i	$0.601577\pi$
$440$	2.77217e9	+	7.72168e8i	1.55144	+	0.432143i
$441$		0			0
$442$			1.71356e9i			0.943890i
$443$		−	3.22249e9i		−	1.76108i	−0.473972	−	0.880540i	$-0.657180\pi$
$443$		−	3.22249e9i		−	1.76108i	0.473972	−	0.880540i	$-0.342820\pi$
$444$		0			0
$445$	4.23313e8	−	1.51974e9i	0.227720	−	0.817540i
$446$	−1.60836e9			−0.858443
$447$		0			0
$448$			9.47267e8i			0.497736i
$449$	7.29482e7			0.0380323			0.0190161	−	0.999819i	$-0.493947\pi$
$449$	7.29482e7			0.0380323			0.0190161	+	0.999819i	$0.493947\pi$
$450$		0			0
$451$	3.64013e9			1.86853
$452$			6.75160e7i			0.0343892i
$453$		0			0
$454$	2.52225e9			1.26501
$455$	3.19621e8	−	1.14747e9i	0.159072	−	0.571087i
$456$		0			0
$457$			2.45286e8i			0.120217i	0.998192	+	0.0601085i	$0.0191447\pi$
$457$			2.45286e8i			0.120217i	−0.998192	+	0.0601085i	$0.980855\pi$
$458$			1.42041e9i			0.690852i
$459$		0			0
$460$	9.44124e7	+	2.62979e7i	0.0452248	+	0.0125971i
$461$	3.25654e9			1.54812			0.774058	−	0.633115i	$-0.218224\pi$
$461$	3.25654e9			1.54812			0.774058	+	0.633115i	$0.218224\pi$
$462$		0			0
$463$			5.48463e8i			0.256811i	0.991722	+	0.128406i	$0.0409859\pi$
$463$			5.48463e8i			0.256811i	−0.991722	+	0.128406i	$0.959014\pi$
$464$	3.76275e8			0.174861
$465$		0			0
$466$	−1.97549e9			−0.904323
$467$			1.31891e9i			0.599245i	0.954058	+	0.299623i	$0.0968607\pi$
$467$			1.31891e9i			0.599245i	−0.954058	+	0.299623i	$0.903139\pi$
$468$		0			0
$469$	−1.14036e9			−0.510431
$470$	4.70670e6	−	1.68976e7i	0.00209110	−	0.00750727i
$471$		0			0
$472$			2.16976e9i			0.949762i
$473$			4.84061e9i			2.10323i
$474$		0			0
$475$	−4.58762e8	−	2.77067e8i	−0.196408	−	0.118620i
$476$	−7.90433e7			−0.0335924
$477$		0			0
$478$		−	1.13214e9i		−	0.474137i
$479$	−1.59989e9			−0.665144			−0.332572	−	0.943078i	$-0.607916\pi$
$479$	−1.59989e9			−0.665144			−0.332572	+	0.943078i	$0.607916\pi$
$480$		0			0
$481$	−2.26783e9			−0.929187
$482$		−	2.09177e9i		−	0.850844i
$483$		0			0
$484$	3.25613e8			0.130540
$485$	−3.23355e9	−	9.00682e8i	−1.28701	−	0.358489i
$486$		0			0
$487$		−	1.95948e9i		−	0.768759i	−0.923175	−	0.384380i	$-0.874415\pi$
$487$		−	1.95948e9i		−	0.768759i	0.923175	−	0.384380i	$-0.125585\pi$
$488$		−	2.08237e9i		−	0.811126i
$489$		0			0
$490$	1.87661e9	+	5.22717e8i	0.720589	+	0.200715i
$491$	−2.38785e8			−0.0910376			−0.0455188	−	0.998963i	$-0.514494\pi$
$491$	−2.38785e8			−0.0910376			−0.0455188	+	0.998963i	$0.514494\pi$
$492$		0			0
$493$			4.01292e8i			0.150833i
$494$	−7.49606e8			−0.279762
$495$		0			0
$496$	−1.20737e9			−0.444279
$497$		−	2.02296e8i		−	0.0739163i
$498$		0			0
$499$	−3.06642e9			−1.10479			−0.552394	−	0.833583i	$-0.686286\pi$
$499$	−3.06642e9			−1.10479			−0.552394	+	0.833583i	$0.686286\pi$
$500$	1.90688e8	−	1.79730e8i	0.0682224	−	0.0643021i
$501$		0			0
$502$			2.18303e9i			0.770188i
$503$		−	1.60348e9i		−	0.561793i	−0.959738	−	0.280897i	$-0.909368\pi$
$503$		−	1.60348e9i		−	0.561793i	0.959738	−	0.280897i	$-0.0906318\pi$
$504$		0			0
$505$	3.85703e8	−	1.38471e9i	0.133270	−	0.478454i
$506$	−2.14883e9			−0.737351
$507$		0			0
$508$			3.42432e8i			0.115891i
$509$	1.21742e9			0.409192			0.204596	−	0.978847i	$-0.434412\pi$
$509$	1.21742e9			0.409192			0.204596	+	0.978847i	$0.434412\pi$
$510$		0			0
$511$	6.24258e8			0.206962
$512$			3.34724e9i			1.10215i
$513$		0			0
$514$	1.44961e8			0.0470848
$515$	−9.38304e8	−	2.61358e8i	−0.302704	−	0.0843161i
$516$		0			0
$517$		−	3.97850e7i		−	0.0126620i
$518$			1.01124e9i			0.319668i
$519$		0			0
$520$	1.14736e9	−	4.11913e9i	0.357838	−	1.28468i
$521$	2.08635e9			0.646331			0.323166	−	0.946342i	$-0.395253\pi$
$521$	2.08635e9			0.646331			0.323166	+	0.946342i	$0.395253\pi$
$522$		0			0
$523$		−	4.28922e9i		−	1.31106i	−0.755169	−	0.655531i	$-0.772445\pi$
$523$		−	4.28922e9i		−	1.31106i	0.755169	−	0.655531i	$-0.227555\pi$
$524$	3.99758e8			0.121377
$525$		0			0
$526$	−1.53160e9			−0.458875
$527$		−	1.28765e9i		−	0.383230i
$528$		0			0
$529$	2.55102e9			0.749237
$530$	−4.76081e8	+	1.70918e9i	−0.138904	+	0.498682i
$531$		0			0
$532$		−	3.45779e7i		−	0.00995654i
$533$		−	5.40883e9i		−	1.54724i
$534$		0			0
$535$	−4.00226e9	−	1.11480e9i	−1.12997	−	0.314745i
$536$	−4.09360e9			−1.14823
$537$		0			0
$538$		−	5.46646e9i		−	1.51345i
$539$	4.41845e9			1.21537
$540$		0			0
$541$	4.91116e9			1.33350			0.666751	−	0.745280i	$-0.267684\pi$
$541$	4.91116e9			1.33350			0.666751	+	0.745280i	$0.267684\pi$
$542$		−	1.21528e9i		−	0.327852i
$543$		0			0
$544$	−5.43170e8			−0.144657
$545$	1.50843e9	−	5.41542e9i	0.399151	−	1.43299i
$546$		0			0
$547$			1.76451e9i			0.460965i	0.973077	+	0.230482i	$0.0740304\pi$
$547$			1.76451e9i			0.460965i	−0.973077	+	0.230482i	$0.925970\pi$
$548$			5.13719e8i			0.133350i
$549$		0			0
$550$	−2.97018e9	+	4.91797e9i	−0.761226	+	1.26042i
$551$	−1.75547e8			−0.0447058
$552$		0			0
$553$			4.45145e8i			0.111934i
$554$	5.50071e9			1.37447
$555$		0			0
$556$	1.36570e8			0.0336971
$557$			4.13406e8i			0.101364i	0.998715	+	0.0506820i	$0.0161395\pi$
$557$			4.13406e8i			0.101364i	−0.998715	+	0.0506820i	$0.983860\pi$
$558$		0			0
$559$	7.19261e9			1.74159
$560$	−1.66302e9	−	4.63223e8i	−0.400166	−	0.111463i
$561$		0			0
$562$		−	1.83590e9i		−	0.436286i
$563$			7.57073e9i			1.78796i	0.448104	+	0.893982i	$0.352100\pi$
$563$			7.57073e9i			1.78796i	−0.448104	+	0.893982i	$0.647900\pi$
$564$		0			0
$565$	−1.51494e9	−	4.21975e8i	−0.353366	−	0.0984277i
$566$	−1.74634e9			−0.404829
$567$		0			0
$568$		−	7.26191e8i		−	0.166277i
$569$	8.90287e8			0.202599			0.101299	−	0.994856i	$-0.467700\pi$
$569$	8.90287e8			0.202599			0.101299	+	0.994856i	$0.467700\pi$
$570$		0			0
$571$	−4.96089e9			−1.11515			−0.557575	−	0.830126i	$-0.688268\pi$
$571$	−4.96089e9			−1.11515			−0.557575	+	0.830126i	$0.688268\pi$
$572$		−	8.31294e8i		−	0.185724i
$573$		0			0
$574$	−2.41183e9			−0.532297
$575$	−1.18016e9	+	1.95408e9i	−0.258882	+	0.428652i
$576$		0			0
$577$			1.53066e9i			0.331713i	0.986150	+	0.165856i	$0.0530388\pi$
$577$			1.53066e9i			0.331713i	−0.986150	+	0.165856i	$0.946961\pi$
$578$		−	1.77092e9i		−	0.381463i
$579$		0			0
$580$	2.30310e7	−	8.26838e7i	0.00490134	−	0.0175963i
$581$	1.09371e9			0.231358
$582$		0			0
$583$			4.02425e9i			0.841094i
$584$	2.24093e9			0.465567
$585$		0			0
$586$	4.15568e9			0.853100
$587$			4.39564e9i			0.896992i	0.893785	+	0.448496i	$0.148040\pi$
$587$			4.39564e9i			0.896992i	−0.893785	+	0.448496i	$0.851960\pi$
$588$		0			0
$589$	5.63288e8			0.113587
$590$	−4.17304e9	−	1.16237e9i	−0.836509	−	0.233004i
$591$		0			0
$592$			3.28675e9i			0.651089i
$593$		−	3.32990e9i		−	0.655753i	−0.944721	−	0.327877i	$-0.893667\pi$
$593$		−	3.32990e9i		−	0.655753i	0.944721	−	0.327877i	$-0.106333\pi$
$594$		0			0
$595$	4.94021e8	−	1.77359e9i	0.0961470	−	0.345178i
$596$	−4.80083e8			−0.0928870
$597$		0			0
$598$			3.19292e9i			0.610567i
$599$	−4.53030e9			−0.861258			−0.430629	−	0.902529i	$-0.641708\pi$
$599$	−4.53030e9			−0.861258			−0.430629	+	0.902529i	$0.641708\pi$
$600$		0			0
$601$	−4.70479e9			−0.884056			−0.442028	−	0.897001i	$-0.645741\pi$
$601$	−4.70479e9			−0.884056			−0.442028	+	0.897001i	$0.645741\pi$
$602$		−	3.20722e9i		−	0.599158i
$603$		0			0
$604$	−3.43913e8			−0.0635067
$605$	−2.03508e9	+	7.30616e9i	−0.373627	+	1.34136i
$606$		0			0
$607$			2.24429e9i			0.407303i	0.979043	+	0.203652i	$0.0652810\pi$
$607$			2.24429e9i			0.407303i	−0.979043	+	0.203652i	$0.934719\pi$
$608$		−	2.37612e8i		−	0.0428752i
$609$		0			0
$610$	4.00496e9	+	1.11555e9i	0.714404	+	0.198992i
$611$	−5.91162e7			−0.0104848
$612$		0			0
$613$			8.74415e9i			1.53323i	0.642109	+	0.766613i	$0.278059\pi$
$613$			8.74415e9i			1.53323i	−0.642109	+	0.766613i	$0.721941\pi$
$614$	6.86950e9			1.19767
$615$		0			0
$616$	−4.32458e9			−0.745438
$617$			4.49031e9i			0.769623i	0.922995	+	0.384812i	$0.125734\pi$
$617$			4.49031e9i			0.769623i	−0.922995	+	0.384812i	$0.874266\pi$
$618$		0			0
$619$	3.74101e9			0.633974			0.316987	−	0.948430i	$-0.397329\pi$
$619$	3.74101e9			0.633974			0.316987	+	0.948430i	$0.397329\pi$
$620$	−7.39008e7	+	2.65312e8i	−0.0124531	+	0.0447081i
$621$		0			0
$622$		−	7.83883e9i		−	1.30612i
$623$			2.37079e9i			0.392813i
$624$		0			0
$625$	2.84102e9	+	5.40199e9i	0.465472	+	0.885063i
$626$	4.81292e9			0.784149
$627$		0			0
$628$		−	3.62350e8i		−	0.0583808i
$629$	−3.50527e9			−0.561622
$630$		0			0
$631$	1.93545e9			0.306675			0.153337	−	0.988174i	$-0.450998\pi$
$631$	1.93545e9			0.306675			0.153337	+	0.988174i	$0.450998\pi$
$632$			1.59796e9i			0.251799i
$633$		0			0
$634$	6.36616e9			0.992122
$635$	−7.68355e9	−	2.14020e9i	−1.19084	−	0.331701i
$636$		0			0
$637$		−	6.56532e9i		−	1.00639i
$638$			1.88188e9i			0.286893i
$639$		0			0
$640$	−5.34621e9	−	1.48915e9i	−0.806150	−	0.224547i
$641$	−5.89076e9			−0.883422			−0.441711	−	0.897157i	$-0.645628\pi$
$641$	−5.89076e9			−0.883422			−0.441711	+	0.897157i	$0.645628\pi$
$642$		0			0
$643$			3.16008e9i			0.468770i	0.972144	+	0.234385i	$0.0753076\pi$
$643$			3.16008e9i			0.468770i	−0.972144	+	0.234385i	$0.924692\pi$
$644$	−1.47283e8			−0.0217297
$645$		0			0
$646$	−1.15863e9			−0.169095
$647$		−	1.27557e10i		−	1.85157i	−0.378054	−	0.925783i	$-0.623407\pi$
$647$		−	1.27557e10i		−	1.85157i	0.378054	−	0.925783i	$-0.376593\pi$
$648$		0			0
$649$	−9.82536e9			−1.41089
$650$	7.30756e9	+	4.41336e9i	1.04370	+	0.630336i
$651$		0			0
$652$		−	1.12250e9i		−	0.158606i
$653$			4.43892e9i			0.623852i	0.950106	+	0.311926i	$0.100974\pi$
$653$			4.43892e9i			0.623852i	−0.950106	+	0.311926i	$0.899026\pi$
$654$		0			0
$655$	−2.49849e9	+	8.96985e9i	−0.347402	+	1.24721i
$656$	−7.83897e9			−1.08417
$657$		0			0
$658$			2.63602e7i			0.00360710i
$659$	1.08526e10			1.47719			0.738595	−	0.674149i	$-0.235489\pi$
$659$	1.08526e10			1.47719			0.738595	+	0.674149i	$0.235489\pi$
$660$		0			0
$661$	8.49307e9			1.14382			0.571912	−	0.820315i	$-0.306202\pi$
$661$	8.49307e9			1.14382			0.571912	+	0.820315i	$0.306202\pi$
$662$		−	6.29922e9i		−	0.843887i
$663$		0			0
$664$	3.92613e9			0.520447
$665$	7.75866e8	+	2.16112e8i	0.102308	+	0.0284973i
$666$		0			0
$667$			7.47737e8i			0.0975683i
$668$		−	6.88209e8i		−	0.0893311i
$669$		0			0
$670$	2.19300e9	−	7.87311e9i	0.281694	−	1.01131i
$671$	9.42962e9			1.20494
$672$		0			0
$673$		−	4.20188e9i		−	0.531362i	−0.964061	−	0.265681i	$-0.914403\pi$
$673$		−	4.20188e9i		−	0.531362i	0.964061	−	0.265681i	$-0.0855969\pi$
$674$	7.96330e9			1.00180
$675$		0			0
$676$	−4.82228e8			−0.0600398
$677$			6.56755e9i			0.813472i	0.913546	+	0.406736i	$0.133333\pi$
$677$			6.56755e9i			0.813472i	−0.913546	+	0.406736i	$0.866667\pi$
$678$		0			0
$679$	5.04433e9			0.618386
$680$	1.77341e9	−	6.36673e9i	0.216286	−	0.776489i
$681$		0			0
$682$		−	6.03850e9i		−	0.728927i
$683$		−	6.31484e9i		−	0.758386i	−0.925318	−	0.379193i	$-0.876202\pi$
$683$		−	6.31484e9i		−	0.758386i	0.925318	−	0.379193i	$-0.123798\pi$
$684$		0			0
$685$	−1.15269e10	−	3.21075e9i	−1.37024	−	0.381671i
$686$	−6.65322e9			−0.786860
$687$		0			0
$688$		−	1.04242e10i		−	1.22035i
$689$	5.97958e9			0.696472
$690$		0			0
$691$	3.76447e9			0.434041			0.217020	−	0.976167i	$-0.430366\pi$
$691$	3.76447e9			0.434041			0.217020	+	0.976167i	$0.430366\pi$
$692$			5.84630e8i			0.0670672i
$693$		0			0
$694$	−3.99439e9			−0.453620
$695$	−8.53562e8	+	3.06438e9i	−0.0964468	+	0.346255i
$696$		0			0
$697$		−	8.36014e9i		−	0.935188i
$698$		−	1.22000e10i		−	1.35789i
$699$		0			0
$700$	−2.03580e8	+	3.37084e8i	−0.0224332	+	0.0371446i
$701$	−1.97083e9			−0.216090			−0.108045	−	0.994146i	$-0.534459\pi$
$701$	−1.97083e9			−0.216090			−0.108045	+	0.994146i	$0.534459\pi$
$702$		0			0
$703$		−	1.53340e9i		−	0.166461i
$704$	−1.53983e10			−1.66329
$705$		0			0
$706$	−8.97016e9			−0.959364
$707$			2.16016e9i			0.229888i
$708$		0			0
$709$	−9.62853e9			−1.01461			−0.507304	−	0.861767i	$-0.669358\pi$
$709$	−9.62853e9			−1.01461			−0.507304	+	0.861767i	$0.669358\pi$
$710$	1.39666e9	+	3.89031e8i	0.146449	+	0.0407924i
$711$		0			0
$712$			8.51054e9i			0.883644i
$713$		−	2.39930e9i		−	0.247897i
$714$		0			0
$715$	1.86527e10	+	5.19559e9i	1.90841	+	0.531574i
$716$	−2.32206e8			−0.0236417
$717$		0			0
$718$			7.27340e9i			0.733334i
$719$	−1.89490e10			−1.90123			−0.950614	−	0.310376i	$-0.899545\pi$
$719$	−1.89490e10			−1.90123			−0.950614	+	0.310376i	$0.899545\pi$
$720$		0			0
$721$	1.46375e9			0.145444
$722$			9.12045e9i			0.901853i
$723$		0			0
$724$	9.39140e8			0.0919698
$725$	1.71133e9	+	1.03355e9i	0.166783	+	0.100727i
$726$		0			0
$727$		−	1.44446e9i		−	0.139423i	−0.997567	−	0.0697116i	$-0.977792\pi$
$727$		−	1.44446e9i		−	0.139423i	0.997567	−	0.0697116i	$-0.0222079\pi$
$728$			6.42585e9i			0.617264i
$729$		0			0
$730$	−1.20050e9	+	4.30991e9i	−0.114217	+	0.410051i
$731$	1.11172e10			1.05266
$732$		0			0
$733$		−	1.38939e10i		−	1.30305i	−0.758627	−	0.651525i	$-0.774129\pi$
$733$		−	1.38939e10i		−	1.30305i	0.758627	−	0.651525i	$-0.225871\pi$
$734$	−1.94122e10			−1.81191
$735$		0			0
$736$	−1.01210e9			−0.0935732
$737$		−	1.85371e10i		−	1.70571i
$738$		0			0
$739$	−1.19008e10			−1.08473			−0.542366	−	0.840142i	$-0.682471\pi$
$739$	−1.19008e10			−1.08473			−0.542366	+	0.840142i	$0.682471\pi$
$740$	7.22239e8	+	2.01175e8i	0.0655194	+	0.0182500i
$741$		0			0
$742$		−	2.66633e9i		−	0.239607i
$743$			1.57512e10i			1.40882i	0.709796	+	0.704408i	$0.248787\pi$
$743$			1.57512e10i			1.40882i	−0.709796	+	0.704408i	$0.751213\pi$
$744$		0			0
$745$	3.00052e9	−	1.07722e10i	0.265858	−	0.954459i
$746$	1.00156e10			0.883268
$747$		0			0
$748$		−	1.28489e9i		−	0.112256i
$749$	6.24352e9			0.542929
$750$		0			0
$751$	−1.60645e10			−1.38397			−0.691984	−	0.721912i	$-0.743263\pi$
$751$	−1.60645e10			−1.38397			−0.691984	+	0.721912i	$0.743263\pi$
$752$			8.56765e7i			0.00734682i
$753$		0			0
$754$	2.79627e9			0.237563
$755$	2.14946e9	−	7.71679e9i	0.181767	−	0.652563i
$756$		0			0
$757$			1.88969e10i			1.58327i	0.610993	+	0.791636i	$0.290771\pi$
$757$			1.88969e10i			1.58327i	−0.610993	+	0.791636i	$0.709229\pi$
$758$			1.54603e10i			1.28936i
$759$		0			0
$760$	2.78516e9	+	7.75787e8i	0.230146	+	0.0641054i
$761$	−1.01100e10			−0.831584			−0.415792	−	0.909460i	$-0.636496\pi$
$761$	−1.01100e10			−0.831584			−0.415792	+	0.909460i	$0.636496\pi$
$762$		0			0
$763$			8.44806e9i			0.688527i
$764$	1.43344e9			0.116293
$765$		0			0
$766$	1.69855e10			1.36546
$767$			1.45994e10i			1.16829i
$768$		0			0
$769$	2.00677e10			1.59132			0.795658	−	0.605746i	$-0.207125\pi$
$769$	2.00677e10			1.59132			0.795658	+	0.605746i	$0.207125\pi$
$770$	2.31674e9	−	8.31735e9i	0.182877	−	0.656550i
$771$		0			0
$772$			7.18163e8i			0.0561775i
$773$		−	2.15770e10i		−	1.68020i	−0.542428	−	0.840102i	$-0.682495\pi$
$773$		−	2.15770e10i		−	1.68020i	0.542428	−	0.840102i	$-0.317505\pi$
$774$		0			0
$775$	−5.49124e9	−	3.31640e9i	−0.423755	−	0.255924i
$776$	1.81079e10			1.39108
$777$		0			0
$778$			2.41714e10i			1.84023i
$779$	3.65719e9			0.277183
$780$

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.7703i q^{2} +12.0000 q^{4} +(-75.0000 + 269.258i) q^{5} -420.043i q^{7} -1507.85i q^{8} +O(q^{10})\)	\(q-10.7703i q^{2} +12.0000 q^{4} +(-75.0000 + 269.258i) q^{5} -420.043i q^{7} -1507.85i q^{8} +(2900.00 + 807.775i) q^{10} +6828.00 q^{11} -10145.7i q^{13} -4524.00 q^{14} -14704.0 q^{16} -15681.6i q^{17} +6860.00 q^{19} +(-900.000 + 3231.10i) q^{20} -73539.8i q^{22} -29219.9i q^{23} +(-66875.0 - 40388.7i) q^{25} -109272. q^{26} -5040.51i q^{28} -25590.0 q^{29} +82112.0 q^{31} -34637.4i q^{32} -168896. q^{34} +(113100. + 31503.2i) q^{35} -223527. i q^{37} -73884.5i q^{38} +(406000. + 113088. i) q^{40} +533118. q^{41} +708935. i q^{43} +81936.0 q^{44} -314708. q^{46} -5826.75i q^{47} +647107. q^{49} +(-435000. + 720266. i) q^{50} -121748. i q^{52} +589374. i q^{53} +(-512100. + 1.83850e6i) q^{55} -633360. q^{56} +275613. i q^{58} -1.43898e6 q^{59} +1.38102e6 q^{61} -884373. i q^{62} -2.25517e6 q^{64} +(2.73180e6 + 760924. i) q^{65} -2.71487e6i q^{67} -188179. i q^{68} +(339300. - 1.21812e6i) q^{70} +481608. q^{71} +1.48618e6i q^{73} -2.40746e6 q^{74} +82320.0 q^{76} -2.86805e6i q^{77} -1.05976e6 q^{79} +(1.10280e6 - 3.95917e6i) q^{80} -5.74186e6i q^{82} +2.60380e6i q^{83} +(4.22240e6 + 1.17612e6i) q^{85} +7.63547e6 q^{86} -1.02956e7i q^{88} -5.64417e6 q^{89} -4.26161e6 q^{91} -350639. i q^{92} -62756.0 q^{94} +(-514500. + 1.84711e6i) q^{95} +1.20091e7i q^{97} -6.96956e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{4} - 150 q^{5}+O(q^{10}) \) 2 * q + 24 * q^4 - 150 * q^5	\( 2 q + 24 q^{4} - 150 q^{5} + 5800 q^{10} + 13656 q^{11} - 9048 q^{14} - 29408 q^{16} + 13720 q^{19} - 1800 q^{20} - 133750 q^{25} - 218544 q^{26} - 51180 q^{29} + 164224 q^{31} - 337792 q^{34} + 226200 q^{35} + 812000 q^{40} + 1066236 q^{41} + 163872 q^{44} - 629416 q^{46} + 1294214 q^{49} - 870000 q^{50} - 1024200 q^{55} - 1266720 q^{56} - 2877960 q^{59} + 2762044 q^{61} - 4510336 q^{64} + 5463600 q^{65} + 678600 q^{70} + 963216 q^{71} - 4814928 q^{74} + 164640 q^{76} - 2119520 q^{79} + 2205600 q^{80} + 8444800 q^{85} + 15270936 q^{86} - 11288340 q^{89} - 8523216 q^{91} - 125512 q^{94} - 1029000 q^{95}+O(q^{100}) \) 2 * q + 24 * q^4 - 150 * q^5 + 5800 * q^10 + 13656 * q^11 - 9048 * q^14 - 29408 * q^16 + 13720 * q^19 - 1800 * q^20 - 133750 * q^25 - 218544 * q^26 - 51180 * q^29 + 164224 * q^31 - 337792 * q^34 + 226200 * q^35 + 812000 * q^40 + 1066236 * q^41 + 163872 * q^44 - 629416 * q^46 + 1294214 * q^49 - 870000 * q^50 - 1024200 * q^55 - 1266720 * q^56 - 2877960 * q^59 + 2762044 * q^61 - 4510336 * q^64 + 5463600 * q^65 + 678600 * q^70 + 963216 * q^71 - 4814928 * q^74 + 164640 * q^76 - 2119520 * q^79 + 2205600 * q^80 + 8444800 * q^85 + 15270936 * q^86 - 11288340 * q^89 - 8523216 * q^91 - 125512 * q^94 - 1029000 * q^95

Introduction

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