LMFDB - Embedded newform 800.6.c.n.449.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not 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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 456x^{8} + 72528x^{6} + 4625233x^{4} + 92116896x^{2} + 21678336 $ x^10 + 456x^8 + 72528x^6 + 4625233x^4 + 92116896x^2 + 21678336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.8
Root			$-10.4009i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.n.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.8018i q^{3} -160.939i q^{7} -149.111 q^{9} +O(q^{10})$	$q+19.8018i q^{3} -160.939i q^{7} -149.111 q^{9} -404.231 q^{11} -249.498i q^{13} +1251.45i q^{17} +905.566 q^{19} +3186.88 q^{21} +2150.12i q^{23} +1859.18i q^{27} +2792.21 q^{29} +3313.11 q^{31} -8004.49i q^{33} +13197.0i q^{37} +4940.51 q^{39} -5075.65 q^{41} -11840.0i q^{43} +1615.00i q^{47} -9094.33 q^{49} -24781.0 q^{51} -33580.6i q^{53} +17931.8i q^{57} -22150.2 q^{59} -43258.9 q^{61} +23997.7i q^{63} -44075.6i q^{67} -42576.2 q^{69} +38109.7 q^{71} +37225.1i q^{73} +65056.4i q^{77} +10876.2 q^{79} -73048.9 q^{81} -75737.5i q^{83} +55290.6i q^{87} -121533. q^{89} -40154.0 q^{91} +65605.4i q^{93} +157807. i q^{97} +60275.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 1212 q^{9}+O(q^{10}) $ 10 * q - 1212 * q^9	$ 10 q - 1212 q^{9} - 10 q^{11} + 8970 q^{19} - 836 q^{21} + 6504 q^{29} - 16500 q^{31} + 17600 q^{39} - 22830 q^{41} - 52450 q^{49} - 5930 q^{51} - 78560 q^{59} + 69708 q^{61} + 81316 q^{69} - 102400 q^{71} - 1100 q^{79} - 30438 q^{81} + 207658 q^{89} - 370560 q^{91} + 488780 q^{99}+O(q^{100}) $ 10 * q - 1212 * q^9 - 10 * q^11 + 8970 * q^19 - 836 * q^21 + 6504 * q^29 - 16500 * q^31 + 17600 * q^39 - 22830 * q^41 - 52450 * q^49 - 5930 * q^51 - 78560 * q^59 + 69708 * q^61 + 81316 * q^69 - 102400 * q^71 - 1100 * q^79 - 30438 * q^81 + 207658 * q^89 - 370560 * q^91 + 488780 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	19.8018i	1.27028i	0.772395	+	0.635142i	$0.219059\pi$
$3$	19.8018i	1.27028i	−0.772395	+	0.635142i	$0.780941\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 160.939i	− 1.24141i	−0.784043	−	0.620706i	$-0.786846\pi$
$7$	− 160.939i	− 1.24141i	0.784043	−	0.620706i	$-0.213154\pi$
$8$	0	0
$9$	−149.111	−0.613623
$10$	0	0
$11$	−404.231	−1.00727	−0.503637	−	0.863915i	$-0.668005\pi$
$11$	−404.231	−1.00727	−0.503637	+	0.863915i	$0.668005\pi$
$12$	0	0
$13$	− 249.498i	− 0.409458i	−0.978819	−	0.204729i	$-0.934369\pi$
$13$	− 249.498i	− 0.409458i	0.978819	−	0.204729i	$-0.0656313\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1251.45i	1.05025i	0.851025	+	0.525125i	$0.175981\pi$
$17$	1251.45i	1.05025i	−0.851025	+	0.525125i	$0.824019\pi$
$18$	0	0
$19$	905.566	0.575488	0.287744	−	0.957707i	$-0.407095\pi$
$19$	905.566	0.575488	0.287744	+	0.957707i	$0.407095\pi$
$20$	0	0
$21$	3186.88	1.57695
$22$	0	0
$23$	2150.12i	0.847507i	0.905778	+	0.423754i	$0.139288\pi$
$23$	2150.12i	0.847507i	−0.905778	+	0.423754i	$0.860712\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1859.18i	0.490808i
$28$	0	0
$29$	2792.21	0.616527	0.308264	−	0.951301i	$-0.400252\pi$
$29$	2792.21	0.616527	0.308264	+	0.951301i	$0.400252\pi$
$30$	0	0
$31$	3313.11	0.619201	0.309600	−	0.950867i	$-0.399805\pi$
$31$	3313.11	0.619201	0.309600	+	0.950867i	$0.399805\pi$
$32$	0	0
$33$	− 8004.49i	− 1.27952i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	13197.0i	1.58478i	0.610013	+	0.792392i	$0.291164\pi$
$37$	13197.0i	1.58478i	−0.610013	+	0.792392i	$0.708836\pi$
$38$	0	0
$39$	4940.51	0.520128
$40$	0	0
$41$	−5075.65	−0.471555	−0.235777	−	0.971807i	$-0.575764\pi$
$41$	−5075.65	−0.471555	−0.235777	+	0.971807i	$0.575764\pi$
$42$	0	0
$43$	− 11840.0i	− 0.976521i	−0.872698	−	0.488261i	$-0.837632\pi$
$43$	− 11840.0i	− 0.976521i	0.872698	−	0.488261i	$-0.162368\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1615.00i	0.106642i	0.998577	+	0.0533208i	$0.0169806\pi$
$47$	1615.00i	0.106642i	−0.998577	+	0.0533208i	$0.983019\pi$
$48$	0	0
$49$	−9094.33	−0.541104
$50$	0	0
$51$	−24781.0	−1.33412
$52$	0	0
$53$	− 33580.6i	− 1.64210i	−0.570857	−	0.821049i	$-0.693389\pi$
$53$	− 33580.6i	− 1.64210i	0.570857	−	0.821049i	$-0.306611\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17931.8i	0.731033i
$58$	0	0
$59$	−22150.2	−0.828413	−0.414206	−	0.910183i	$-0.635941\pi$
$59$	−22150.2	−0.828413	−0.414206	+	0.910183i	$0.635941\pi$
$60$	0	0
$61$	−43258.9	−1.48851	−0.744254	−	0.667896i	$-0.767195\pi$
$61$	−43258.9	−1.48851	−0.744254	+	0.667896i	$0.767195\pi$
$62$	0	0
$63$	23997.7i	0.761759i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 44075.6i	− 1.19953i	−0.800176	−	0.599766i	$-0.795260\pi$
$67$	− 44075.6i	− 1.19953i	0.800176	−	0.599766i	$-0.204740\pi$
$68$	0	0
$69$	−42576.2	−1.07658
$70$	0	0
$71$	38109.7	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$71$	38109.7	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$72$	0	0
$73$	37225.1i	0.817577i	0.912629	+	0.408788i	$0.134049\pi$
$73$	37225.1i	0.817577i	−0.912629	+	0.408788i	$0.865951\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	65056.4i	1.25044i
$78$	0	0
$79$	10876.2	0.196069	0.0980345	−	0.995183i	$-0.468744\pi$
$79$	10876.2	0.196069	0.0980345	+	0.995183i	$0.468744\pi$
$80$	0	0
$81$	−73048.9	−1.23709
$82$	0	0
$83$	− 75737.5i	− 1.20675i	−0.797459	−	0.603373i	$-0.793823\pi$
$83$	− 75737.5i	− 1.20675i	0.797459	−	0.603373i	$-0.206177\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	55290.6i	0.783165i
$88$	0	0
$89$	−121533.	−1.62637	−0.813183	−	0.582008i	$-0.802267\pi$
$89$	−121533.	−1.62637	−0.813183	+	0.582008i	$0.802267\pi$
$90$	0	0
$91$	−40154.0	−0.508306
$92$	0	0
$93$	65605.4i	0.786561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	157807.i	1.70293i	0.524411	+	0.851465i	$0.324286\pi$
$97$	157807.i	1.70293i	−0.524411	+	0.851465i	$0.675714\pi$
$98$	0	0
$99$	60275.0	0.618087
$100$	0	0
$101$	−32558.5	−0.317586	−0.158793	−	0.987312i	$-0.550760\pi$
$101$	−32558.5	−0.317586	−0.158793	+	0.987312i	$0.550760\pi$
$102$	0	0
$103$	1240.18i	0.0115184i	0.999983	+	0.00575920i	$0.00183322\pi$
$103$	1240.18i	0.0115184i	−0.999983	+	0.00575920i	$0.998167\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 170351.i	− 1.43842i	−0.694794	−	0.719209i	$-0.744504\pi$
$107$	− 170351.i	− 1.43842i	0.694794	−	0.719209i	$-0.255496\pi$
$108$	0	0
$109$	−127565.	−1.02841	−0.514204	−	0.857668i	$-0.671912\pi$
$109$	−127565.	−1.02841	−0.514204	+	0.857668i	$0.671912\pi$
$110$	0	0
$111$	−261323.	−2.01313
$112$	0	0
$113$	− 110843.i	− 0.816602i	−0.912847	−	0.408301i	$-0.866121\pi$
$113$	− 110843.i	− 0.816602i	0.912847	−	0.408301i	$-0.133879\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37202.8i	0.251253i
$118$	0	0
$119$	201408.	1.30379
$120$	0	0
$121$	2351.45	0.0146007
$122$	0	0
$123$	− 100507.i	− 0.599009i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	115536.i	0.635634i	0.948152	+	0.317817i	$0.102950\pi$
$127$	115536.i	0.635634i	−0.948152	+	0.317817i	$0.897050\pi$
$128$	0	0
$129$	234454.	1.24046
$130$	0	0
$131$	29750.5	0.151466	0.0757332	−	0.997128i	$-0.475870\pi$
$131$	29750.5	0.151466	0.0757332	+	0.997128i	$0.475870\pi$
$132$	0	0
$133$	− 145741.i	− 0.714417i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 238138.i	− 1.08400i	−0.840380	−	0.541998i	$-0.817668\pi$
$137$	− 238138.i	− 1.08400i	0.840380	−	0.541998i	$-0.182332\pi$
$138$	0	0
$139$	−406539.	−1.78470	−0.892351	−	0.451342i	$-0.850945\pi$
$139$	−406539.	−1.78470	−0.892351	+	0.451342i	$0.850945\pi$
$140$	0	0
$141$	−31979.8	−0.135465
$142$	0	0
$143$	100855.i	0.412436i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 180084.i	− 0.687356i
$148$	0	0
$149$	178599.	0.659043	0.329521	−	0.944148i	$-0.393113\pi$
$149$	178599.	0.659043	0.329521	+	0.944148i	$0.393113\pi$
$150$	0	0
$151$	−432317.	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$151$	−432317.	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$152$	0	0
$153$	− 186605.i	− 0.644458i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	198160.i	0.641603i	0.947146	+	0.320802i	$0.103952\pi$
$157$	198160.i	0.641603i	−0.947146	+	0.320802i	$0.896048\pi$
$158$	0	0
$159$	664957.	2.08593
$160$	0	0
$161$	346038.	1.05211
$162$	0	0
$163$	− 450882.i	− 1.32921i	−0.747195	−	0.664605i	$-0.768600\pi$
$163$	− 450882.i	− 1.32921i	0.747195	−	0.664605i	$-0.231400\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 519661.i	− 1.44188i	−0.692998	−	0.720940i	$-0.743710\pi$
$167$	− 519661.i	− 1.44188i	0.692998	−	0.720940i	$-0.256290\pi$
$168$	0	0
$169$	309044.	0.832344
$170$	0	0
$171$	−135029.	−0.353133
$172$	0	0
$173$	− 27351.8i	− 0.0694818i	−0.999396	−	0.0347409i	$-0.988939\pi$
$173$	− 27351.8i	− 0.0694818i	0.999396	−	0.0347409i	$-0.0110606\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 438613.i	− 1.05232i
$178$	0	0
$179$	−100983.	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$179$	−100983.	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$180$	0	0
$181$	124851.	0.283267	0.141633	−	0.989919i	$-0.454765\pi$
$181$	124851.	0.283267	0.141633	+	0.989919i	$0.454765\pi$
$182$	0	0
$183$	− 856604.i	− 1.89083i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 505876.i	− 1.05789i
$188$	0	0
$189$	299214.	0.609295
$190$	0	0
$191$	−829467.	−1.64519	−0.822594	−	0.568629i	$-0.807474\pi$
$191$	−829467.	−1.64519	−0.822594	+	0.568629i	$0.807474\pi$
$192$	0	0
$193$	712128.i	1.37615i	0.725641	+	0.688073i	$0.241543\pi$
$193$	712128.i	1.37615i	−0.725641	+	0.688073i	$0.758457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 970619.i	− 1.78190i	−0.454102	−	0.890950i	$-0.650040\pi$
$197$	− 970619.i	− 1.78190i	0.454102	−	0.890950i	$-0.349960\pi$
$198$	0	0
$199$	516230.	0.924082	0.462041	−	0.886858i	$-0.347117\pi$
$199$	516230.	0.924082	0.462041	+	0.886858i	$0.347117\pi$
$200$	0	0
$201$	872776.	1.52375
$202$	0	0
$203$	− 449375.i	− 0.765365i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 320606.i	− 0.520050i
$208$	0	0
$209$	−366057.	−0.579674
$210$	0	0
$211$	−664118.	−1.02693	−0.513463	−	0.858112i	$-0.671638\pi$
$211$	−664118.	−1.02693	−0.513463	+	0.858112i	$0.671638\pi$
$212$	0	0
$213$	754640.i	1.13970i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 533208.i	− 0.768683i
$218$	0	0
$219$	−737123.	−1.03856
$220$	0	0
$221$	312236.	0.430033
$222$	0	0
$223$	990032.i	1.33317i	0.745427	+	0.666587i	$0.232246\pi$
$223$	990032.i	1.33317i	−0.745427	+	0.666587i	$0.767754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15566.7i	0.0200509i	0.999950	+	0.0100254i	$0.00319125\pi$
$227$	15566.7i	0.0200509i	−0.999950	+	0.0100254i	$0.996809\pi$
$228$	0	0
$229$	−359522.	−0.453040	−0.226520	−	0.974006i	$-0.572735\pi$
$229$	−359522.	−0.453040	−0.226520	+	0.974006i	$0.572735\pi$
$230$	0	0
$231$	−1.28823e6	−1.58842
$232$	0	0
$233$	1.21546e6i	1.46673i	0.679833	+	0.733367i	$0.262052\pi$
$233$	1.21546e6i	1.46673i	−0.679833	+	0.733367i	$0.737948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	215368.i	0.249064i
$238$	0	0
$239$	1.00004e6	1.13246	0.566231	−	0.824247i	$-0.308401\pi$
$239$	1.00004e6	1.13246	0.566231	+	0.824247i	$0.308401\pi$
$240$	0	0
$241$	−1.25841e6	−1.39566	−0.697831	−	0.716263i	$-0.745851\pi$
$241$	−1.25841e6	−1.39566	−0.697831	+	0.716263i	$0.745851\pi$
$242$	0	0
$243$	− 994718.i	− 1.08065i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 225937.i	− 0.235638i
$248$	0	0
$249$	1.49974e6	1.53291
$250$	0	0
$251$	−990751.	−0.992614	−0.496307	−	0.868147i	$-0.665311\pi$
$251$	−990751.	−0.992614	−0.496307	+	0.868147i	$0.665311\pi$
$252$	0	0
$253$	− 869145.i	− 0.853672i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.71639e6i	− 1.62100i	−0.585741	−	0.810498i	$-0.699196\pi$
$257$	− 1.71639e6i	− 1.62100i	0.585741	−	0.810498i	$-0.300804\pi$
$258$	0	0
$259$	2.12391e6	1.96737
$260$	0	0
$261$	−416347.	−0.378316
$262$	0	0
$263$	− 1.59528e6i	− 1.42216i	−0.703112	−	0.711080i	$-0.748207\pi$
$263$	− 1.59528e6i	− 1.42216i	0.703112	−	0.711080i	$-0.251793\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.40657e6i	− 2.06595i
$268$	0	0
$269$	−895598.	−0.754627	−0.377313	−	0.926086i	$-0.623152\pi$
$269$	−895598.	−0.754627	−0.377313	+	0.926086i	$0.623152\pi$
$270$	0	0
$271$	−648702.	−0.536565	−0.268282	−	0.963340i	$-0.586456\pi$
$271$	−648702.	−0.536565	−0.268282	+	0.963340i	$0.586456\pi$
$272$	0	0
$273$	− 795120.i	− 0.645693i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.35736e6i	1.84597i	0.384830	+	0.922987i	$0.374260\pi$
$277$	2.35736e6i	1.84597i	−0.384830	+	0.922987i	$0.625740\pi$
$278$	0	0
$279$	−494019.	−0.379956
$280$	0	0
$281$	586738.	0.443280	0.221640	−	0.975129i	$-0.428859\pi$
$281$	586738.	0.443280	0.221640	+	0.975129i	$0.428859\pi$
$282$	0	0
$283$	− 1.42618e6i	− 1.05855i	−0.848452	−	0.529273i	$-0.822465\pi$
$283$	− 1.42618e6i	− 1.05855i	0.848452	−	0.529273i	$-0.177535\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	816870.i	0.585394i
$288$	0	0
$289$	−146279.	−0.103024
$290$	0	0
$291$	−3.12486e6	−2.16321
$292$	0	0
$293$	− 931708.i	− 0.634032i	−0.948420	−	0.317016i	$-0.897319\pi$
$293$	− 931708.i	− 0.634032i	0.948420	−	0.317016i	$-0.102681\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 751537.i	− 0.494378i
$298$	0	0
$299$	536451.	0.347018
$300$	0	0
$301$	−1.90552e6	−1.21227
$302$	0	0
$303$	− 644716.i	− 0.403424i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.51715e6i	− 0.918716i	−0.888251	−	0.459358i	$-0.848079\pi$
$307$	− 1.51715e6i	− 0.918716i	0.888251	−	0.459358i	$-0.151921\pi$
$308$	0	0
$309$	−24557.8	−0.0146317
$310$	0	0
$311$	366257.	0.214726	0.107363	−	0.994220i	$-0.465759\pi$
$311$	366257.	0.214726	0.107363	+	0.994220i	$0.465759\pi$
$312$	0	0
$313$	− 1.19487e6i	− 0.689381i	−0.938716	−	0.344691i	$-0.887984\pi$
$313$	− 1.19487e6i	− 0.689381i	0.938716	−	0.344691i	$-0.112016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.14809e6i	1.20062i	0.799768	+	0.600309i	$0.204956\pi$
$317$	2.14809e6i	1.20062i	−0.799768	+	0.600309i	$0.795044\pi$
$318$	0	0
$319$	−1.12870e6	−0.621012
$320$	0	0
$321$	3.37325e6	1.82720
$322$	0	0
$323$	1.13327e6i	0.604406i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.52601e6i	− 1.30637i
$328$	0	0
$329$	259916.	0.132386
$330$	0	0
$331$	442778.	0.222135	0.111067	−	0.993813i	$-0.464573\pi$
$331$	442778.	0.222135	0.111067	+	0.993813i	$0.464573\pi$
$332$	0	0
$333$	− 1.96781e6i	− 0.972460i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.01258e6i	1.92464i	0.271923	+	0.962319i	$0.412340\pi$
$337$	4.01258e6i	1.92464i	−0.271923	+	0.962319i	$0.587660\pi$
$338$	0	0
$339$	2.19488e6	1.03732
$340$	0	0
$341$	−1.33926e6	−0.623705
$342$	0	0
$343$	− 1.24127e6i	− 0.569679i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.81191e6i	0.807816i	0.914800	+	0.403908i	$0.132348\pi$
$347$	1.81191e6i	0.807816i	−0.914800	+	0.403908i	$0.867652\pi$
$348$	0	0
$349$	−3.69069e6	−1.62197	−0.810987	−	0.585065i	$-0.801069\pi$
$349$	−3.69069e6	−1.62197	−0.810987	+	0.585065i	$0.801069\pi$
$350$	0	0
$351$	463862.	0.200965
$352$	0	0
$353$	119137.i	0.0508876i	0.999676	+	0.0254438i	$0.00809988\pi$
$353$	119137.i	0.0508876i	−0.999676	+	0.0254438i	$0.991900\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.98823e6i	1.65619i
$358$	0	0
$359$	−3.65995e6	−1.49879	−0.749393	−	0.662126i	$-0.769654\pi$
$359$	−3.65995e6	−1.49879	−0.749393	+	0.662126i	$0.769654\pi$
$360$	0	0
$361$	−1.65605e6	−0.668814
$362$	0	0
$363$	46563.0i	0.0185470i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.04946e6i	− 0.406726i	−0.979103	−	0.203363i	$-0.934813\pi$
$367$	− 1.04946e6i	− 0.406726i	0.979103	−	0.203363i	$-0.0651871\pi$
$368$	0	0
$369$	756833.	0.289357
$370$	0	0
$371$	−5.40443e6	−2.03852
$372$	0	0
$373$	− 150748.i	− 0.0561020i	−0.999606	−	0.0280510i	$-0.991070\pi$
$373$	− 150748.i	− 0.0561020i	0.999606	−	0.0280510i	$-0.00893008\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 696651.i	− 0.252442i
$378$	0	0
$379$	416359.	0.148892	0.0744458	−	0.997225i	$-0.476281\pi$
$379$	416359.	0.148892	0.0744458	+	0.997225i	$0.476281\pi$
$380$	0	0
$381$	−2.28781e6	−0.807436
$382$	0	0
$383$	− 2.08322e6i	− 0.725669i	−0.931854	−	0.362834i	$-0.881809\pi$
$383$	− 2.08322e6i	− 0.725669i	0.931854	−	0.362834i	$-0.118191\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.76547e6i	0.599216i
$388$	0	0
$389$	−1.37343e6	−0.460186	−0.230093	−	0.973169i	$-0.573903\pi$
$389$	−1.37343e6	−0.460186	−0.230093	+	0.973169i	$0.573903\pi$
$390$	0	0
$391$	−2.69078e6	−0.890094
$392$	0	0
$393$	589113.i	0.192405i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.99424e6i	0.635040i	0.948252	+	0.317520i	$0.102850\pi$
$397$	1.99424e6i	0.635040i	−0.948252	+	0.317520i	$0.897150\pi$
$398$	0	0
$399$	2.88593e6	0.907513
$400$	0	0
$401$	−23001.8	−0.00714333	−0.00357166	−	0.999994i	$-0.501137\pi$
$401$	−23001.8	−0.00714333	−0.00357166	+	0.999994i	$0.501137\pi$
$402$	0	0
$403$	− 826615.i	− 0.253537i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.33462e6i	− 1.59631i
$408$	0	0
$409$	−2.27406e6	−0.672193	−0.336097	−	0.941828i	$-0.609107\pi$
$409$	−2.27406e6	−0.672193	−0.336097	+	0.941828i	$0.609107\pi$
$410$	0	0
$411$	4.71556e6	1.37698
$412$	0	0
$413$	3.56482e6i	1.02840i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.05020e6i	− 2.26708i
$418$	0	0
$419$	3.94280e6	1.09716	0.548579	−	0.836099i	$-0.315169\pi$
$419$	3.94280e6	1.09716	0.548579	+	0.836099i	$0.315169\pi$
$420$	0	0
$421$	−4.20370e6	−1.15592	−0.577958	−	0.816067i	$-0.696150\pi$
$421$	−4.20370e6	−1.15592	−0.577958	+	0.816067i	$0.696150\pi$
$422$	0	0
$423$	− 240813.i	− 0.0654378i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.96205e6i	1.84785i
$428$	0	0
$429$	−1.99711e6	−0.523911
$430$	0	0
$431$	5.90469e6	1.53110	0.765550	−	0.643376i	$-0.222467\pi$
$431$	5.90469e6	1.53110	0.765550	+	0.643376i	$0.222467\pi$
$432$	0	0
$433$	3.98952e6i	1.02259i	0.859406	+	0.511294i	$0.170834\pi$
$433$	3.98952e6i	1.02259i	−0.859406	+	0.511294i	$0.829166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.94708e6i	0.487730i
$438$	0	0
$439$	10775.1	0.00266846	0.00133423	−	0.999999i	$-0.499575\pi$
$439$	10775.1	0.00266846	0.00133423	+	0.999999i	$0.499575\pi$
$440$	0	0
$441$	1.35606e6	0.332034
$442$	0	0
$443$	6.72453e6i	1.62799i	0.580870	+	0.813996i	$0.302712\pi$
$443$	6.72453e6i	1.62799i	−0.580870	+	0.813996i	$0.697288\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.53658e6i	0.837172i
$448$	0	0
$449$	6.39615e6	1.49728	0.748640	−	0.662977i	$-0.230707\pi$
$449$	6.39615e6	1.49728	0.748640	+	0.662977i	$0.230707\pi$
$450$	0	0
$451$	2.05173e6	0.474985
$452$	0	0
$453$	− 8.56065e6i	− 1.96002i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.40846e6i	− 1.65935i	−0.558248	−	0.829674i	$-0.688526\pi$
$457$	− 7.40846e6i	− 1.65935i	0.558248	−	0.829674i	$-0.311474\pi$
$458$	0	0
$459$	−2.32668e6	−0.515471
$460$	0	0
$461$	−4.34835e6	−0.952954	−0.476477	−	0.879187i	$-0.658086\pi$
$461$	−4.34835e6	−0.952954	−0.476477	+	0.879187i	$0.658086\pi$
$462$	0	0
$463$	2.15517e6i	0.467229i	0.972329	+	0.233614i	$0.0750553\pi$
$463$	2.15517e6i	0.467229i	−0.972329	+	0.233614i	$0.924945\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.33954e6i	− 0.284226i	−0.989850	−	0.142113i	$-0.954610\pi$
$467$	− 1.33954e6i	− 0.284226i	0.989850	−	0.142113i	$-0.0453896\pi$
$468$	0	0
$469$	−7.09349e6	−1.48911
$470$	0	0
$471$	−3.92392e6	−0.815019
$472$	0	0
$473$	4.78610e6i	0.983624i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.00723e6i	1.00763i
$478$	0	0
$479$	−9.34745e6	−1.86146	−0.930731	−	0.365704i	$-0.880828\pi$
$479$	−9.34745e6	−1.86146	−0.930731	+	0.365704i	$0.880828\pi$
$480$	0	0
$481$	3.29262e6	0.648902
$482$	0	0
$483$	6.85217e6i	1.33647i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.51352e6i	− 0.671305i	−0.941986	−	0.335652i	$-0.891043\pi$
$487$	− 3.51352e6i	− 0.671305i	0.941986	−	0.335652i	$-0.108957\pi$
$488$	0	0
$489$	8.92826e6	1.68847
$490$	0	0
$491$	5.22485e6	0.978070	0.489035	−	0.872264i	$-0.337349\pi$
$491$	5.22485e6	0.978070	0.489035	+	0.872264i	$0.337349\pi$
$492$	0	0
$493$	3.49432e6i	0.647508i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.13333e6i	− 1.11380i
$498$	0	0
$499$	−1.86243e6	−0.334834	−0.167417	−	0.985886i	$-0.553543\pi$
$499$	−1.86243e6	−0.334834	−0.167417	+	0.985886i	$0.553543\pi$
$500$	0	0
$501$	1.02902e7	1.83160
$502$	0	0
$503$	− 6.06780e6i	− 1.06933i	−0.845065	−	0.534664i	$-0.820438\pi$
$503$	− 6.06780e6i	− 1.06933i	0.845065	−	0.534664i	$-0.179562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.11961e6i	1.05731i
$508$	0	0
$509$	6.24842e6	1.06899	0.534497	−	0.845170i	$-0.320501\pi$
$509$	6.24842e6	1.06899	0.534497	+	0.845170i	$0.320501\pi$
$510$	0	0
$511$	5.99096e6	1.01495
$512$	0	0
$513$	1.68361e6i	0.282454i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 652831.i	− 0.107417i
$518$	0	0
$519$	541615.	0.0882617
$520$	0	0
$521$	9.23593e6	1.49069	0.745343	−	0.666681i	$-0.232286\pi$
$521$	9.23593e6	1.49069	0.745343	+	0.666681i	$0.232286\pi$
$522$	0	0
$523$	1.05161e6i	0.168113i	0.996461	+	0.0840566i	$0.0267877\pi$
$523$	1.05161e6i	0.168113i	−0.996461	+	0.0840566i	$0.973212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.14620e6i	0.650315i
$528$	0	0
$529$	1.81332e6	0.281732
$530$	0	0
$531$	3.30282e6	0.508334
$532$	0	0
$533$	1.26637e6i	0.193082i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.99965e6i	− 0.299239i
$538$	0	0
$539$	3.67621e6	0.545040
$540$	0	0
$541$	4.34911e6	0.638862	0.319431	−	0.947610i	$-0.396508\pi$
$541$	4.34911e6	0.638862	0.319431	+	0.947610i	$0.396508\pi$
$542$	0	0
$543$	2.47227e6i	0.359830i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.98339e6i	− 0.426326i	−0.977017	−	0.213163i	$-0.931624\pi$
$547$	− 2.98339e6i	− 0.426326i	0.977017	−	0.213163i	$-0.0683765\pi$
$548$	0	0
$549$	6.45036e6	0.913384
$550$	0	0
$551$	2.52853e6	0.354804
$552$	0	0
$553$	− 1.75040e6i	− 0.243402i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.74611e6i	− 0.648186i	−0.946025	−	0.324093i	$-0.894941\pi$
$557$	− 4.74611e6i	− 0.648186i	0.946025	−	0.324093i	$-0.105059\pi$
$558$	0	0
$559$	−2.95407e6	−0.399844
$560$	0	0
$561$	1.00172e7	1.34382
$562$	0	0
$563$	− 4.22516e6i	− 0.561787i	−0.959739	−	0.280894i	$-0.909369\pi$
$563$	− 4.22516e6i	− 0.561787i	0.959739	−	0.280894i	$-0.0906308\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.17564e7i	1.53574i
$568$	0	0
$569$	−1.95043e6	−0.252551	−0.126275	−	0.991995i	$-0.540302\pi$
$569$	−1.95043e6	−0.252551	−0.126275	+	0.991995i	$0.540302\pi$
$570$	0	0
$571$	1.24377e7	1.59643	0.798215	−	0.602372i	$-0.205778\pi$
$571$	1.24377e7	1.59643	0.798215	+	0.602372i	$0.205778\pi$
$572$	0	0
$573$	− 1.64249e7i	− 2.08986i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.21199e7i	− 1.51552i	−0.652536	−	0.757758i	$-0.726295\pi$
$577$	− 1.21199e7i	− 1.51552i	0.652536	−	0.757758i	$-0.273705\pi$
$578$	0	0
$579$	−1.41014e7	−1.74810
$580$	0	0
$581$	−1.21891e7	−1.49807
$582$	0	0
$583$	1.35743e7i	1.65404i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.40591e6i	0.168407i	0.996449	+	0.0842036i	$0.0268346\pi$
$587$	1.40591e6i	0.168407i	−0.996449	+	0.0842036i	$0.973165\pi$
$588$	0	0
$589$	3.00024e6	0.356342
$590$	0	0
$591$	1.92200e7	2.26352
$592$	0	0
$593$	8.82785e6i	1.03090i	0.856919	+	0.515452i	$0.172376\pi$
$593$	8.82785e6i	1.03090i	−0.856919	+	0.515452i	$0.827624\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.02223e7i	1.17385i
$598$	0	0
$599$	4.21509e6	0.479999	0.239999	−	0.970773i	$-0.422853\pi$
$599$	4.21509e6	0.479999	0.239999	+	0.970773i	$0.422853\pi$
$600$	0	0
$601$	9.00063e6	1.01645	0.508226	−	0.861224i	$-0.330302\pi$
$601$	9.00063e6	1.01645	0.508226	+	0.861224i	$0.330302\pi$
$602$	0	0
$603$	6.57214e6i	0.736061i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.74038e7i	1.91722i	0.284721	+	0.958611i	$0.408099\pi$
$607$	1.74038e7i	1.91722i	−0.284721	+	0.958611i	$0.591901\pi$
$608$	0	0
$609$	8.89842e6	0.972231
$610$	0	0
$611$	402939.	0.0436653
$612$	0	0
$613$	7.31051e6i	0.785772i	0.919587	+	0.392886i	$0.128523\pi$
$613$	7.31051e6i	0.785772i	−0.919587	+	0.392886i	$0.871477\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.06016e6i	− 0.323617i	−0.986822	−	0.161809i	$-0.948267\pi$
$617$	− 3.06016e6i	− 0.323617i	0.986822	−	0.161809i	$-0.0517327\pi$
$618$	0	0
$619$	−1.94950e6	−0.204502	−0.102251	−	0.994759i	$-0.532604\pi$
$619$	−1.94950e6	−0.204502	−0.102251	+	0.994759i	$0.532604\pi$
$620$	0	0
$621$	−3.99746e6	−0.415963
$622$	0	0
$623$	1.95593e7i	2.01899i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.24859e6i	− 0.736351i
$628$	0	0
$629$	−1.65154e7	−1.66442
$630$	0	0
$631$	−5.34043e6	−0.533953	−0.266976	−	0.963703i	$-0.586025\pi$
$631$	−5.34043e6	−0.533953	−0.266976	+	0.963703i	$0.586025\pi$
$632$	0	0
$633$	− 1.31507e7i	− 1.30449i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.26902e6i	0.221559i
$638$	0	0
$639$	−5.68255e6	−0.550543
$640$	0	0
$641$	−1.70992e7	−1.64373	−0.821864	−	0.569684i	$-0.807066\pi$
$641$	−1.70992e7	−1.64373	−0.821864	+	0.569684i	$0.807066\pi$
$642$	0	0
$643$	− 1.21597e7i	− 1.15983i	−0.814676	−	0.579917i	$-0.803085\pi$
$643$	− 1.21597e7i	− 1.15983i	0.814676	−	0.579917i	$-0.196915\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.68331e6i	0.158089i	0.996871	+	0.0790447i	$0.0251870\pi$
$647$	1.68331e6i	0.158089i	−0.996871	+	0.0790447i	$0.974813\pi$
$648$	0	0
$649$	8.95377e6	0.834439
$650$	0	0
$651$	1.05585e7	0.976446
$652$	0	0
$653$	1.29171e7i	1.18545i	0.805404	+	0.592726i	$0.201948\pi$
$653$	1.29171e7i	1.18545i	−0.805404	+	0.592726i	$0.798052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.55065e6i	− 0.501684i
$658$	0	0
$659$	3.20405e6	0.287399	0.143700	−	0.989621i	$-0.454100\pi$
$659$	3.20405e6	0.287399	0.143700	+	0.989621i	$0.454100\pi$
$660$	0	0
$661$	−1.82464e7	−1.62433	−0.812165	−	0.583428i	$-0.801711\pi$
$661$	−1.82464e7	−1.62433	−0.812165	+	0.583428i	$0.801711\pi$
$662$	0	0
$663$	6.18282e6i	0.546264i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00358e6i	0.522511i
$668$	0	0
$669$	−1.96044e7	−1.69351
$670$	0	0
$671$	1.74866e7	1.49934
$672$	0	0
$673$	1.17328e7i	0.998538i	0.866447	+	0.499269i	$0.166398\pi$
$673$	1.17328e7i	0.998538i	−0.866447	+	0.499269i	$0.833602\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.27923e7i	1.07270i	0.843997	+	0.536348i	$0.180197\pi$
$677$	1.27923e7i	1.07270i	−0.843997	+	0.536348i	$0.819803\pi$
$678$	0	0
$679$	2.53973e7	2.11404
$680$	0	0
$681$	−308249.	−0.0254703
$682$	0	0
$683$	1.34398e7i	1.10240i	0.834372	+	0.551202i	$0.185831\pi$
$683$	1.34398e7i	1.10240i	−0.834372	+	0.551202i	$0.814169\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.11918e6i	− 0.575490i
$688$	0	0
$689$	−8.37831e6	−0.672370
$690$	0	0
$691$	−1.79328e7	−1.42874	−0.714371	−	0.699767i	$-0.753287\pi$
$691$	−1.79328e7	−1.42874	−0.714371	+	0.699767i	$0.753287\pi$
$692$	0	0
$693$	− 9.70060e6i	− 0.767300i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.35194e6i	− 0.495250i
$698$	0	0
$699$	−2.40683e7	−1.86317
$700$	0	0
$701$	2.36981e6	0.182146	0.0910728	−	0.995844i	$-0.470970\pi$
$701$	2.36981e6	0.182146	0.0910728	+	0.995844i	$0.470970\pi$
$702$	0	0
$703$	1.19507e7i	0.912023i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.23993e6i	0.394255i
$708$	0	0
$709$	1.86295e6	0.139183	0.0695913	−	0.997576i	$-0.477830\pi$
$709$	1.86295e6	0.139183	0.0695913	+	0.997576i	$0.477830\pi$
$710$	0	0
$711$	−1.62175e6	−0.120313
$712$	0	0
$713$	7.12358e6i	0.524777i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.98026e7i	1.43855i
$718$	0	0
$719$	−1.14819e7	−0.828308	−0.414154	−	0.910207i	$-0.635923\pi$
$719$	−1.14819e7	−0.828308	−0.414154	+	0.910207i	$0.635923\pi$
$720$	0	0
$721$	199594.	0.0142991
$722$	0	0
$723$	− 2.49188e7i	− 1.77289i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.08210e7i	1.46105i	0.682887	+	0.730524i	$0.260724\pi$
$727$	2.08210e7i	1.46105i	−0.682887	+	0.730524i	$0.739276\pi$
$728$	0	0
$729$	1.94630e6	0.135641
$730$	0	0
$731$	1.48172e7	1.02559
$732$	0	0
$733$	8.47296e6i	0.582472i	0.956651	+	0.291236i	$0.0940665\pi$
$733$	8.47296e6i	0.582472i	−0.956651	+	0.291236i	$0.905934\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.78167e7i	1.20826i
$738$	0	0
$739$	−1.57784e7	−1.06280	−0.531402	−	0.847120i	$-0.678335\pi$
$739$	−1.57784e7	−1.06280	−0.531402	+	0.847120i	$0.678335\pi$
$740$	0	0
$741$	4.47396e6	0.299327
$742$	0	0
$743$	− 1.54066e7i	− 1.02384i	−0.859032	−	0.511922i	$-0.828934\pi$
$743$	− 1.54066e7i	− 1.02384i	0.859032	−	0.511922i	$-0.171066\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.12933e7i	0.740487i
$748$	0	0
$749$	−2.74161e7	−1.78567
$750$	0	0
$751$	1.60309e7	1.03719	0.518594	−	0.855021i	$-0.326456\pi$
$751$	1.60309e7	1.03719	0.518594	+	0.855021i	$0.326456\pi$
$752$	0	0
$753$	− 1.96186e7i	− 1.26090i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.65387e6i	− 0.104896i	−0.998624	−	0.0524482i	$-0.983298\pi$
$757$	− 1.65387e6i	− 0.104896i	0.998624	−	0.0524482i	$-0.0167024\pi$
$758$	0	0
$759$	1.72106e7	1.08441
$760$	0	0
$761$	−1.00105e7	−0.626605	−0.313302	−	0.949653i	$-0.601435\pi$
$761$	−1.00105e7	−0.626605	−0.313302	+	0.949653i	$0.601435\pi$
$762$	0	0
$763$	2.05302e7i	1.27668i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.52643e6i	0.339200i
$768$	0	0
$769$	−3.06629e7	−1.86981	−0.934906	−	0.354895i	$-0.884517\pi$
$769$	−3.06629e7	−1.86981	−0.934906	+	0.354895i	$0.884517\pi$
$770$	0	0
$771$	3.39875e7	2.05913
$772$	0	0
$773$	− 2.29138e7i	− 1.37927i	−0.724159	−	0.689633i	$-0.757772\pi$
$773$	− 2.29138e7i	− 1.37927i	0.724159	−	0.689633i	$-0.242228\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.20571e7i	2.49912i
$778$	0	0
$779$	−4.59634e6	−0.271374
$780$	0	0
$781$	−1.54051e7	−0.903727
$782$	0	0
$783$	5.19121e6i	0.302597i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.27348e7i	− 1.88396i	−0.335663	−	0.941982i	$-0.608960\pi$
$787$	− 3.27348e7i	− 1.88396i	0.335663	−	0.941982i	$-0.391040\pi$
$788$	0	0
$789$	3.15894e7	1.80655
$790$	0	0
$791$	−1.78389e7	−1.01374
$792$	0	0
$793$	1.07930e7i	0.609481i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.16718e6i	0.232379i	0.993227	+	0.116189i	$0.0370680\pi$
$797$	4.16718e6i	0.232379i	−0.993227	+	0.116189i	$0.962932\pi$
$798$	0	0
$799$	−2.02109e6	−0.112000
$800$	0	0
$801$	1.81218e7	0.997976
$802$	0	0
$803$	− 1.50475e7i	− 0.823524i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.77344e7i	− 0.958591i
$808$	0	0
$809$	−1.28521e6	−0.0690405	−0.0345202	−	0.999404i	$-0.510990\pi$
$809$	−1.28521e6	−0.0690405	−0.0345202	+	0.999404i	$0.510990\pi$
$810$	0	0
$811$	−6.16431e6	−0.329103	−0.164552	−	0.986368i	$-0.552618\pi$
$811$	−6.16431e6	−0.329103	−0.164552	+	0.986368i	$0.552618\pi$
$812$	0	0
$813$	− 1.28455e7i	− 0.681590i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.07219e7i	− 0.561976i
$818$	0	0
$819$	5.98738e6	0.311908
$820$	0	0
$821$	1.50757e7	0.780581	0.390291	−	0.920692i	$-0.372375\pi$
$821$	1.50757e7	0.780581	0.390291	+	0.920692i	$0.372375\pi$
$822$	0	0
$823$	7.55088e6i	0.388596i	0.980943	+	0.194298i	$0.0622428\pi$
$823$	7.55088e6i	0.388596i	−0.980943	+	0.194298i	$0.937757\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.02532e6i	0.204662i	0.994750	+	0.102331i	$0.0326300\pi$
$827$	4.02532e6i	0.204662i	−0.994750	+	0.102331i	$0.967370\pi$
$828$	0	0
$829$	1.48566e7	0.750813	0.375407	−	0.926860i	$-0.377503\pi$
$829$	1.48566e7	0.750813	0.375407	+	0.926860i	$0.377503\pi$
$830$	0	0
$831$	−4.66798e7	−2.34491
$832$	0	0
$833$	− 1.13811e7i	− 0.568294i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.15966e6i	0.303909i
$838$	0	0
$839$	−9.59062e6	−0.470372	−0.235186	−	0.971950i	$-0.575570\pi$
$839$	−9.59062e6	−0.470372	−0.235186	+	0.971950i	$0.575570\pi$
$840$	0	0
$841$	−1.27147e7	−0.619894
$842$	0	0
$843$	1.16185e7i	0.563092i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 378440.i	− 0.0181255i
$848$	0	0
$849$	2.82410e7	1.34465
$850$	0	0
$851$	−2.83751e7	−1.34311
$852$	0	0
$853$	− 6.24756e6i	− 0.293994i	−0.989137	−	0.146997i	$-0.953039\pi$
$853$	− 6.24756e6i	− 0.293994i	0.989137	−	0.146997i	$-0.0469607\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.19127e7i	− 1.01916i	−0.860423	−	0.509581i	$-0.829800\pi$
$857$	− 2.19127e7i	− 1.01916i	0.860423	−	0.509581i	$-0.170200\pi$
$858$	0	0
$859$	8.90797e6	0.411904	0.205952	−	0.978562i	$-0.433971\pi$
$859$	8.90797e6	0.411904	0.205952	+	0.978562i	$0.433971\pi$
$860$	0	0
$861$	−1.61755e7	−0.743617
$862$	0	0
$863$	− 1.28474e7i	− 0.587204i	−0.955928	−	0.293602i	$-0.905146\pi$
$863$	− 1.28474e7i	− 0.587204i	0.955928	−	0.293602i	$-0.0948540\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.89659e6i	− 0.130870i
$868$	0	0
$869$	−4.39649e6	−0.197495
$870$	0	0
$871$	−1.09968e7	−0.491158
$872$	0	0
$873$	− 2.35307e7i	− 1.04496i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.36870e7i	− 1.03995i	−0.854183	−	0.519973i	$-0.825942\pi$
$877$	− 2.36870e7i	− 1.03995i	0.854183	−	0.519973i	$-0.174058\pi$
$878$	0	0
$879$	1.84495e7	0.805401
$880$	0	0
$881$	−1.77488e7	−0.770423	−0.385212	−	0.922828i	$-0.625872\pi$
$881$	−1.77488e7	−0.770423	−0.385212	+	0.922828i	$0.625872\pi$
$882$	0	0
$883$	293447.i	0.0126656i	0.999980	+	0.00633282i	$0.00201581\pi$
$883$	293447.i	0.0126656i	−0.999980	+	0.00633282i	$0.997984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.50694e7i	− 0.643112i	−0.946891	−	0.321556i	$-0.895794\pi$
$887$	− 1.50694e7i	− 0.643112i	0.946891	−	0.321556i	$-0.104206\pi$
$888$	0	0
$889$	1.85942e7	0.789083
$890$	0	0
$891$	2.95286e7	1.24609
$892$	0	0
$893$	1.46249e6i	0.0613710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.06227e7i	0.440812i
$898$	0	0
$899$	9.25088e6	0.381754
$900$	0	0
$901$	4.20246e7	1.72461
$902$	0	0
$903$	− 3.77327e7i	− 1.53992i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.62189e6i	0.186553i	0.995640	+	0.0932764i	$0.0297340\pi$
$907$	4.62189e6i	0.186553i	−0.995640	+	0.0932764i	$0.970266\pi$
$908$	0	0
$909$	4.85481e6	0.194878
$910$	0	0
$911$	9.02270e6	0.360197	0.180099	−	0.983649i	$-0.442358\pi$
$911$	9.02270e6	0.360197	0.180099	+	0.983649i	$0.442358\pi$
$912$	0	0
$913$	3.06154e7i	1.21552i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.78801e6i	− 0.188032i
$918$	0	0
$919$	1.81033e7	0.707081	0.353540	−	0.935419i	$-0.384978\pi$
$919$	1.81033e7	0.707081	0.353540	+	0.935419i	$0.384978\pi$
$920$	0	0
$921$	3.00422e7	1.16703
$922$	0	0
$923$	− 9.50830e6i	− 0.367366i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 184924.i	− 0.00706797i
$928$	0	0
$929$	−3.59953e7	−1.36838	−0.684190	−	0.729304i	$-0.739844\pi$
$929$	−3.59953e7	−1.36838	−0.684190	+	0.729304i	$0.739844\pi$
$930$	0	0
$931$	−8.23551e6	−0.311398
$932$	0	0
$933$	7.25253e6i	0.272763i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.79708e7i	− 1.04077i	−0.853931	−	0.520387i	$-0.825788\pi$
$937$	− 2.79708e7i	− 1.04077i	0.853931	−	0.520387i	$-0.174212\pi$
$938$	0	0
$939$	2.36605e7	0.875710
$940$	0	0
$941$	−1.35773e7	−0.499851	−0.249926	−	0.968265i	$-0.580406\pi$
$941$	−1.35773e7	−0.499851	−0.249926	+	0.968265i	$0.580406\pi$
$942$	0	0
$943$	− 1.09133e7i	− 0.399646i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.32801e6i	0.193059i	0.995330	+	0.0965296i	$0.0307742\pi$
$947$	5.32801e6i	0.193059i	−0.995330	+	0.0965296i	$0.969226\pi$
$948$	0	0
$949$	9.28759e6	0.334763
$950$	0	0
$951$	−4.25360e7	−1.52513
$952$	0	0
$953$	1.02144e7i	0.364316i	0.983269	+	0.182158i	$0.0583083\pi$
$953$	1.02144e7i	0.364316i	−0.983269	+	0.182158i	$0.941692\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.23502e7i	− 0.788862i
$958$	0	0
$959$	−3.83257e7	−1.34569
$960$	0	0
$961$	−1.76525e7	−0.616591
$962$	0	0
$963$	2.54011e7i	0.882647i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.17005e7i	− 0.746285i	−0.927774	−	0.373142i	$-0.878280\pi$
$967$	− 2.17005e7i	− 0.746285i	0.927774	−	0.373142i	$-0.121720\pi$
$968$	0	0
$969$	−2.24408e7	−0.767767
$970$	0	0
$971$	−4.93228e7	−1.67880	−0.839401	−	0.543512i	$-0.817094\pi$
$971$	−4.93228e7	−1.67880	−0.839401	+	0.543512i	$0.817094\pi$
$972$	0	0
$973$	6.54280e7i	2.21555i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.04782e7i	− 0.686366i	−0.939269	−	0.343183i	$-0.888495\pi$
$977$	− 2.04782e7i	− 0.686366i	0.939269	−	0.343183i	$-0.111505\pi$
$978$	0	0
$979$	4.91273e7	1.63820
$980$	0	0
$981$	1.90213e7	0.631055
$982$	0	0
$983$	− 1.58306e7i	− 0.522533i	−0.965267	−	0.261267i	$-0.915860\pi$
$983$	− 1.58306e7i	− 0.522533i	0.965267	−	0.261267i	$-0.0841402\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.14679e6i	0.168168i
$988$	0	0
$989$	2.54575e7	0.827609
$990$	0	0
$991$	1.75681e7	0.568250	0.284125	−	0.958787i	$-0.408297\pi$
$991$	1.75681e7	0.568250	0.284125	+	0.958787i	$0.408297\pi$
$992$	0	0
$993$	8.76779e6i	0.282174i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.21502e7i	− 1.34296i	−0.741025	−	0.671478i	$-0.765660\pi$
$997$	− 4.21502e7i	− 1.34296i	0.741025	−	0.671478i	$-0.234340\pi$
$998$	0	0
$999$	−2.45355e7	−0.777825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.n.449.8		10
4.3	odd	2		800.6.c.o.449.3		10
5.2	odd	4		800.6.a.s.1.4	✓	5
5.3	odd	4		800.6.a.u.1.2	yes	5
5.4	even	2	inner	800.6.c.n.449.3		10
20.3	even	4		800.6.a.t.1.4	yes	5
20.7	even	4		800.6.a.v.1.2	yes	5
20.19	odd	2		800.6.c.o.449.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.6.a.s.1.4	✓	5	5.2	odd	4
800.6.a.t.1.4	yes	5	20.3	even	4
800.6.a.u.1.2	yes	5	5.3	odd	4
800.6.a.v.1.2	yes	5	20.7	even	4
800.6.c.n.449.3		10	5.4	even	2	inner
800.6.c.n.449.8		10	1.1	even	1	trivial
800.6.c.o.449.3		10	4.3	odd	2
800.6.c.o.449.8		10	20.19	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+19.8018i q^{3} -160.939i q^{7} -149.111 q^{9} +O(q^{10})\)	\(q+19.8018i q^{3} -160.939i q^{7} -149.111 q^{9} -404.231 q^{11} -249.498i q^{13} +1251.45i q^{17} +905.566 q^{19} +3186.88 q^{21} +2150.12i q^{23} +1859.18i q^{27} +2792.21 q^{29} +3313.11 q^{31} -8004.49i q^{33} +13197.0i q^{37} +4940.51 q^{39} -5075.65 q^{41} -11840.0i q^{43} +1615.00i q^{47} -9094.33 q^{49} -24781.0 q^{51} -33580.6i q^{53} +17931.8i q^{57} -22150.2 q^{59} -43258.9 q^{61} +23997.7i q^{63} -44075.6i q^{67} -42576.2 q^{69} +38109.7 q^{71} +37225.1i q^{73} +65056.4i q^{77} +10876.2 q^{79} -73048.9 q^{81} -75737.5i q^{83} +55290.6i q^{87} -121533. q^{89} -40154.0 q^{91} +65605.4i q^{93} +157807. i q^{97} +60275.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 1212 q^{9}+O(q^{10}) \) 10 * q - 1212 * q^9	\( 10 q - 1212 q^{9} - 10 q^{11} + 8970 q^{19} - 836 q^{21} + 6504 q^{29} - 16500 q^{31} + 17600 q^{39} - 22830 q^{41} - 52450 q^{49} - 5930 q^{51} - 78560 q^{59} + 69708 q^{61} + 81316 q^{69} - 102400 q^{71} - 1100 q^{79} - 30438 q^{81} + 207658 q^{89} - 370560 q^{91} + 488780 q^{99}+O(q^{100}) \) 10 * q - 1212 * q^9 - 10 * q^11 + 8970 * q^19 - 836 * q^21 + 6504 * q^29 - 16500 * q^31 + 17600 * q^39 - 22830 * q^41 - 52450 * q^49 - 5930 * q^51 - 78560 * q^59 + 69708 * q^61 + 81316 * q^69 - 102400 * q^71 - 1100 * q^79 - 30438 * q^81 + 207658 * q^89 - 370560 * q^91 + 488780 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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