LMFDB - Embedded newform 800.6.f.c.49.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{93}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.20
Root			$3.46430 + 1.99965i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.6.f.c.49.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+29.2080 q^{3} +168.173i q^{7} +610.110 q^{9} +514.493i q^{11} +491.622 q^{13} +183.094i q^{17} -1250.96i q^{19} +4912.01i q^{21} -423.498i q^{23} +10722.5 q^{27} +3463.40i q^{29} -2343.92 q^{31} +15027.3i q^{33} +7388.25 q^{37} +14359.3 q^{39} +4240.39 q^{41} -15159.4 q^{43} +15357.8i q^{47} -11475.3 q^{49} +5347.82i q^{51} +11393.9 q^{53} -36538.2i q^{57} -11978.0i q^{59} -41454.0i q^{61} +102604. i q^{63} -66524.9 q^{67} -12369.6i q^{69} +26214.5 q^{71} +86291.9i q^{73} -86524.0 q^{77} -19799.4 q^{79} +164928. q^{81} -8370.24 q^{83} +101159. i q^{87} +3824.45 q^{89} +82677.7i q^{91} -68461.3 q^{93} +35158.5i q^{97} +313897. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{3} + 1620 q^{9} + 11664 q^{27} - 7160 q^{31} + 3608 q^{37} - 44904 q^{39} + 11608 q^{41} - 51772 q^{43} - 18756 q^{49} - 928 q^{53} - 161604 q^{67} + 200312 q^{71} - 26008 q^{77} + 282080 q^{79}+ \cdots - 293472 q^{93}+O(q^{100}) $ 20 * q + 36 * q^3 + 1620 * q^9 + 11664 * q^27 - 7160 * q^31 + 3608 * q^37 - 44904 * q^39 + 11608 * q^41 - 51772 * q^43 - 18756 * q^49 - 928 * q^53 - 161604 * q^67 + 200312 * q^71 - 26008 * q^77 + 282080 * q^79 + 65172 * q^81 - 99092 * q^83 + 3160 * q^89 - 293472 * q^93

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
$3$	29.2080	1.87370	0.936848	−	0.349736i	$-0.113729\pi$
$3$	29.2080	1.87370	0.936848	+	0.349736i	$0.113729\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	168.173i	1.29722i	0.761123	+	0.648608i	$0.224648\pi$
$7$	168.173i	1.29722i	−0.761123	+	0.648608i	$0.775352\pi$
$8$	0	0
$9$	610.110	2.51074
$10$	0	0
$11$	514.493i	1.28203i	0.767529	+	0.641014i	$0.221486\pi$
$11$	514.493i	1.28203i	−0.767529	+	0.641014i	$0.778514\pi$
$12$	0	0
$13$	491.622	0.806813	0.403406	−	0.915021i	$-0.367826\pi$
$13$	491.622	0.806813	0.403406	+	0.915021i	$0.367826\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	183.094i	0.153657i	0.997044	+	0.0768284i	$0.0244794\pi$
$17$	183.094i	0.153657i	−0.997044	+	0.0768284i	$0.975521\pi$
$18$	0	0
$19$	− 1250.96i	− 0.794988i	−0.917605	−	0.397494i	$-0.869880\pi$
$19$	− 1250.96i	− 0.794988i	0.917605	−	0.397494i	$-0.130120\pi$
$20$	0	0
$21$	4912.01i	2.43059i
$22$	0	0
$23$	− 423.498i	− 0.166929i	−0.996511	−	0.0834646i	$-0.973401\pi$
$23$	− 423.498i	− 0.166929i	0.996511	−	0.0834646i	$-0.0265985\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10722.5	2.83067
$28$	0	0
$29$	3463.40i	0.764729i	0.924012	+	0.382364i	$0.124890\pi$
$29$	3463.40i	0.764729i	−0.924012	+	0.382364i	$0.875110\pi$
$30$	0	0
$31$	−2343.92	−0.438065	−0.219032	−	0.975718i	$-0.570290\pi$
$31$	−2343.92	−0.438065	−0.219032	+	0.975718i	$0.570290\pi$
$32$	0	0
$33$	15027.3i	2.40213i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7388.25	0.887232	0.443616	−	0.896217i	$-0.353695\pi$
$37$	7388.25	0.887232	0.443616	+	0.896217i	$0.353695\pi$
$38$	0	0
$39$	14359.3	1.51172
$40$	0	0
$41$	4240.39	0.393954	0.196977	−	0.980408i	$-0.436888\pi$
$41$	4240.39	0.393954	0.196977	+	0.980408i	$0.436888\pi$
$42$	0	0
$43$	−15159.4	−1.25029	−0.625143	−	0.780510i	$-0.714960\pi$
$43$	−15159.4	−1.25029	−0.625143	+	0.780510i	$0.714960\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	15357.8i	1.01411i	0.861914	+	0.507055i	$0.169266\pi$
$47$	15357.8i	1.01411i	−0.861914	+	0.507055i	$0.830734\pi$
$48$	0	0
$49$	−11475.3	−0.682768
$50$	0	0
$51$	5347.82i	0.287906i
$52$	0	0
$53$	11393.9	0.557162	0.278581	−	0.960413i	$-0.410136\pi$
$53$	11393.9	0.557162	0.278581	+	0.960413i	$0.410136\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 36538.2i	− 1.48957i
$58$	0	0
$59$	− 11978.0i	− 0.447974i	−0.974592	−	0.223987i	$-0.928093\pi$
$59$	− 11978.0i	− 0.447974i	0.974592	−	0.223987i	$-0.0719074\pi$
$60$	0	0
$61$	− 41454.0i	− 1.42640i	−0.700960	−	0.713200i	$-0.747245\pi$
$61$	− 41454.0i	− 1.42640i	0.700960	−	0.713200i	$-0.252755\pi$
$62$	0	0
$63$	102604.i	3.25697i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−66524.9	−1.81049	−0.905247	−	0.424885i	$-0.860315\pi$
$67$	−66524.9	−1.81049	−0.905247	+	0.424885i	$0.860315\pi$
$68$	0	0
$69$	− 12369.6i	− 0.312775i
$70$	0	0
$71$	26214.5	0.617157	0.308579	−	0.951199i	$-0.400147\pi$
$71$	26214.5	0.617157	0.308579	+	0.951199i	$0.400147\pi$
$72$	0	0
$73$	86291.9i	1.89523i	0.319409	+	0.947617i	$0.396516\pi$
$73$	86291.9i	1.89523i	−0.319409	+	0.947617i	$0.603484\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−86524.0	−1.66307
$78$	0	0
$79$	−19799.4	−0.356931	−0.178466	−	0.983946i	$-0.557113\pi$
$79$	−19799.4	−0.356931	−0.178466	+	0.983946i	$0.557113\pi$
$80$	0	0
$81$	164928.	2.79307
$82$	0	0
$83$	−8370.24	−0.133365	−0.0666826	−	0.997774i	$-0.521241\pi$
$83$	−8370.24	−0.133365	−0.0666826	+	0.997774i	$0.521241\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	101159.i	1.43287i
$88$	0	0
$89$	3824.45	0.0511793	0.0255896	−	0.999673i	$-0.491854\pi$
$89$	3824.45	0.0511793	0.0255896	+	0.999673i	$0.491854\pi$
$90$	0	0
$91$	82677.7i	1.04661i
$92$	0	0
$93$	−68461.3	−0.820801
$94$	0	0
$95$	0	0
$96$	0	0
$97$	35158.5i	0.379403i	0.981842	+	0.189702i	$0.0607520\pi$
$97$	35158.5i	0.379403i	−0.981842	+	0.189702i	$0.939248\pi$
$98$	0	0
$99$	313897.i	3.21884i
$100$	0	0
$101$	99536.3i	0.970908i	0.874262	+	0.485454i	$0.161346\pi$
$101$	99536.3i	0.970908i	−0.874262	+	0.485454i	$0.838654\pi$
$102$	0	0
$103$	46921.2i	0.435789i	0.975972	+	0.217895i	$0.0699189\pi$
$103$	46921.2i	0.435789i	−0.975972	+	0.217895i	$0.930081\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	38381.3	0.324086	0.162043	−	0.986784i	$-0.448192\pi$
$107$	38381.3	0.324086	0.162043	+	0.986784i	$0.448192\pi$
$108$	0	0
$109$	− 14288.7i	− 0.115193i	−0.998340	−	0.0575966i	$-0.981656\pi$
$109$	− 14288.7i	− 0.115193i	0.998340	−	0.0575966i	$-0.0183437\pi$
$110$	0	0
$111$	215796.	1.66240
$112$	0	0
$113$	− 130552.i	− 0.961804i	−0.876774	−	0.480902i	$-0.840309\pi$
$113$	− 130552.i	− 0.961804i	0.876774	−	0.480902i	$-0.159691\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	299943.	2.02570
$118$	0	0
$119$	−30791.5	−0.199326
$120$	0	0
$121$	−103652.	−0.643596
$122$	0	0
$123$	123853.	0.738151
$124$	0	0
$125$	0	0
$126$	0	0
$127$	327594.i	1.80230i	0.433508	+	0.901150i	$0.357276\pi$
$127$	327594.i	1.80230i	−0.433508	+	0.901150i	$0.642724\pi$
$128$	0	0
$129$	−442775.	−2.34266
$130$	0	0
$131$	116999.i	0.595669i	0.954618	+	0.297834i	$0.0962643\pi$
$131$	116999.i	0.595669i	−0.954618	+	0.297834i	$0.903736\pi$
$132$	0	0
$133$	210379.	1.03127
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 74409.1i	− 0.338707i	−0.985555	−	0.169354i	$-0.945832\pi$
$137$	− 74409.1i	− 0.338707i	0.985555	−	0.169354i	$-0.0541680\pi$
$138$	0	0
$139$	− 80434.5i	− 0.353106i	−0.984291	−	0.176553i	$-0.943505\pi$
$139$	− 80434.5i	− 0.353106i	0.984291	−	0.176553i	$-0.0564947\pi$
$140$	0	0
$141$	448572.i	1.90013i
$142$	0	0
$143$	252936.i	1.03436i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−335171.	−1.27930
$148$	0	0
$149$	− 57917.4i	− 0.213719i	−0.994274	−	0.106860i	$-0.965920\pi$
$149$	− 57917.4i	− 0.213719i	0.994274	−	0.106860i	$-0.0340795\pi$
$150$	0	0
$151$	−450813.	−1.60899	−0.804497	−	0.593957i	$-0.797565\pi$
$151$	−450813.	−1.60899	−0.804497	+	0.593957i	$0.797565\pi$
$152$	0	0
$153$	111707.i	0.385792i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	72067.3	0.233340	0.116670	−	0.993171i	$-0.462778\pi$
$157$	72067.3	0.233340	0.116670	+	0.993171i	$0.462778\pi$
$158$	0	0
$159$	332792.	1.04395
$160$	0	0
$161$	71221.2	0.216543
$162$	0	0
$163$	471144.	1.38895	0.694473	−	0.719519i	$-0.255638\pi$
$163$	471144.	1.38895	0.694473	+	0.719519i	$0.255638\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 519164.i	− 1.44050i	−0.693715	−	0.720250i	$-0.744027\pi$
$167$	− 519164.i	− 1.44050i	0.693715	−	0.720250i	$-0.255973\pi$
$168$	0	0
$169$	−129601.	−0.349053
$170$	0	0
$171$	− 763225.i	− 1.99601i
$172$	0	0
$173$	726898.	1.84654	0.923269	−	0.384153i	$-0.125506\pi$
$173$	726898.	1.84654	0.923269	+	0.384153i	$0.125506\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 349853.i	− 0.839368i
$178$	0	0
$179$	− 327278.i	− 0.763457i	−0.924274	−	0.381729i	$-0.875329\pi$
$179$	− 327278.i	− 0.763457i	0.924274	−	0.381729i	$-0.124671\pi$
$180$	0	0
$181$	− 651584.i	− 1.47834i	−0.673519	−	0.739170i	$-0.735218\pi$
$181$	− 651584.i	− 1.47834i	0.673519	−	0.739170i	$-0.264782\pi$
$182$	0	0
$183$	− 1.21079e6i	− 2.67264i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−94200.5	−0.196992
$188$	0	0
$189$	1.80325e6i	3.67198i
$190$	0	0
$191$	427987.	0.848882	0.424441	−	0.905456i	$-0.360471\pi$
$191$	427987.	0.848882	0.424441	+	0.905456i	$0.360471\pi$
$192$	0	0
$193$	− 560696.i	− 1.08351i	−0.840535	−	0.541757i	$-0.817759\pi$
$193$	− 560696.i	− 1.08351i	0.840535	−	0.541757i	$-0.182241\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−268824.	−0.493518	−0.246759	−	0.969077i	$-0.579366\pi$
$197$	−268824.	−0.493518	−0.246759	+	0.969077i	$0.579366\pi$
$198$	0	0
$199$	512715.	0.917789	0.458895	−	0.888491i	$-0.348246\pi$
$199$	512715.	0.917789	0.458895	+	0.888491i	$0.348246\pi$
$200$	0	0
$201$	−1.94306e6	−3.39232
$202$	0	0
$203$	−582451.	−0.992018
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 258380.i	− 0.419116i
$208$	0	0
$209$	643612.	1.01920
$210$	0	0
$211$	661872.i	1.02345i	0.859148	+	0.511726i	$0.170994\pi$
$211$	661872.i	1.02345i	−0.859148	+	0.511726i	$0.829006\pi$
$212$	0	0
$213$	765674.	1.15637
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 394185.i	− 0.568265i
$218$	0	0
$219$	2.52042e6i	3.55109i
$220$	0	0
$221$	90013.0i	0.123972i
$222$	0	0
$223$	1.06479e6i	1.43384i	0.697153	+	0.716922i	$0.254450\pi$
$223$	1.06479e6i	1.43384i	−0.697153	+	0.716922i	$0.745550\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	619730.	0.798248	0.399124	−	0.916897i	$-0.369314\pi$
$227$	619730.	0.798248	0.399124	+	0.916897i	$0.369314\pi$
$228$	0	0
$229$	− 434907.i	− 0.548035i	−0.961725	−	0.274017i	$-0.911647\pi$
$229$	− 434907.i	− 0.548035i	0.961725	−	0.274017i	$-0.0883525\pi$
$230$	0	0
$231$	−2.52720e6	−3.11608
$232$	0	0
$233$	− 793810.i	− 0.957915i	−0.877838	−	0.478957i	$-0.841015\pi$
$233$	− 793810.i	− 0.957915i	0.877838	−	0.478957i	$-0.158985\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−578302.	−0.668781
$238$	0	0
$239$	−1.64777e6	−1.86596	−0.932978	−	0.359933i	$-0.882800\pi$
$239$	−1.64777e6	−1.86596	−0.932978	+	0.359933i	$0.882800\pi$
$240$	0	0
$241$	592599.	0.657231	0.328616	−	0.944464i	$-0.393418\pi$
$241$	592599.	0.657231	0.328616	+	0.944464i	$0.393418\pi$
$242$	0	0
$243$	2.21164e6	2.40270
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 615001.i	− 0.641406i
$248$	0	0
$249$	−244478.	−0.249886
$250$	0	0
$251$	1.64581e6i	1.64890i	0.565933	+	0.824451i	$0.308516\pi$
$251$	1.64581e6i	1.64890i	−0.565933	+	0.824451i	$0.691484\pi$
$252$	0	0
$253$	217887.	0.214008
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.18756e6i	− 1.12156i	−0.827966	−	0.560779i	$-0.810502\pi$
$257$	− 1.18756e6i	− 1.12156i	0.827966	−	0.560779i	$-0.189498\pi$
$258$	0	0
$259$	1.24251e6i	1.15093i
$260$	0	0
$261$	2.11305e6i	1.92003i
$262$	0	0
$263$	− 1.62916e6i	− 1.45236i	−0.687505	−	0.726180i	$-0.741294\pi$
$263$	− 1.62916e6i	− 1.45236i	0.687505	−	0.726180i	$-0.258706\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	111705.	0.0958944
$268$	0	0
$269$	895226.i	0.754314i	0.926149	+	0.377157i	$0.123098\pi$
$269$	895226.i	0.754314i	−0.926149	+	0.377157i	$0.876902\pi$
$270$	0	0
$271$	−16721.4	−0.0138309	−0.00691545	−	0.999976i	$-0.502201\pi$
$271$	−16721.4	−0.0138309	−0.00691545	+	0.999976i	$0.502201\pi$
$272$	0	0
$273$	2.41485e6i	1.96103i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	573914.	0.449415	0.224708	−	0.974426i	$-0.427857\pi$
$277$	573914.	0.449415	0.224708	+	0.974426i	$0.427857\pi$
$278$	0	0
$279$	−1.43005e6	−1.09987
$280$	0	0
$281$	1.95965e6	1.48052	0.740258	−	0.672322i	$-0.234703\pi$
$281$	1.95965e6	1.48052	0.740258	+	0.672322i	$0.234703\pi$
$282$	0	0
$283$	2.04454e6	1.51751	0.758753	−	0.651378i	$-0.225809\pi$
$283$	2.04454e6	1.51751	0.758753	+	0.651378i	$0.225809\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	713120.i	0.511044i
$288$	0	0
$289$	1.38633e6	0.976390
$290$	0	0
$291$	1.02691e6i	0.710886i
$292$	0	0
$293$	−2.25113e6	−1.53190	−0.765951	−	0.642899i	$-0.777731\pi$
$293$	−2.25113e6	−1.53190	−0.765951	+	0.642899i	$0.777731\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.51667e6i	3.62899i
$298$	0	0
$299$	− 208201.i	− 0.134681i
$300$	0	0
$301$	− 2.54940e6i	− 1.62189i
$302$	0	0
$303$	2.90726e6i	1.81919i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	572436.	0.346642	0.173321	−	0.984865i	$-0.444550\pi$
$307$	572436.	0.346642	0.173321	+	0.984865i	$0.444550\pi$
$308$	0	0
$309$	1.37048e6i	0.816537i
$310$	0	0
$311$	2.86177e6	1.67778	0.838889	−	0.544303i	$-0.183206\pi$
$311$	2.86177e6	1.67778	0.838889	+	0.544303i	$0.183206\pi$
$312$	0	0
$313$	345643.i	0.199419i	0.995017	+	0.0997095i	$0.0317913\pi$
$313$	345643.i	0.199419i	−0.995017	+	0.0997095i	$0.968209\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.84370e6	−1.58941	−0.794706	−	0.606995i	$-0.792375\pi$
$317$	−2.84370e6	−1.58941	−0.794706	+	0.606995i	$0.792375\pi$
$318$	0	0
$319$	−1.78189e6	−0.980404
$320$	0	0
$321$	1.12104e6	0.607238
$322$	0	0
$323$	229044.	0.122155
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 417345.i	− 0.215837i
$328$	0	0
$329$	−2.58278e6	−1.31552
$330$	0	0
$331$	− 2.20630e6i	− 1.10686i	−0.832895	−	0.553431i	$-0.813318\pi$
$331$	− 2.20630e6i	− 1.10686i	0.832895	−	0.553431i	$-0.186682\pi$
$332$	0	0
$333$	4.50764e6	2.22761
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 210820.i	− 0.101120i	−0.998721	−	0.0505599i	$-0.983899\pi$
$337$	− 210820.i	− 0.101120i	0.998721	−	0.0505599i	$-0.0161006\pi$
$338$	0	0
$339$	− 3.81316e6i	− 1.80213i
$340$	0	0
$341$	− 1.20593e6i	− 0.561611i
$342$	0	0
$343$	896652.i	0.411518i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.39916e6	1.06963	0.534817	−	0.844968i	$-0.320381\pi$
$347$	2.39916e6	1.06963	0.534817	+	0.844968i	$0.320381\pi$
$348$	0	0
$349$	1.76207e6i	0.774392i	0.921997	+	0.387196i	$0.126556\pi$
$349$	1.76207e6i	0.774392i	−0.921997	+	0.387196i	$0.873444\pi$
$350$	0	0
$351$	5.27144e6	2.28382
$352$	0	0
$353$	2.11387e6i	0.902904i	0.892296	+	0.451452i	$0.149094\pi$
$353$	2.11387e6i	0.902904i	−0.892296	+	0.451452i	$0.850906\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−899360.	−0.373476
$358$	0	0
$359$	−25391.5	−0.0103981	−0.00519903	−	0.999986i	$-0.501655\pi$
$359$	−25391.5	−0.0103981	−0.00519903	+	0.999986i	$0.501655\pi$
$360$	0	0
$361$	911190.	0.367994
$362$	0	0
$363$	−3.02747e6	−1.20590
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.55872e6i	− 0.604091i	−0.953293	−	0.302046i	$-0.902330\pi$
$367$	− 1.55872e6i	− 0.604091i	0.953293	−	0.302046i	$-0.0976695\pi$
$368$	0	0
$369$	2.58710e6	0.989116
$370$	0	0
$371$	1.91614e6i	0.722759i
$372$	0	0
$373$	741743.	0.276046	0.138023	−	0.990429i	$-0.455925\pi$
$373$	741743.	0.276046	0.138023	+	0.990429i	$0.455925\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.70268e6i	0.616993i
$378$	0	0
$379$	2.61039e6i	0.933487i	0.884393	+	0.466743i	$0.154573\pi$
$379$	2.61039e6i	0.933487i	−0.884393	+	0.466743i	$0.845427\pi$
$380$	0	0
$381$	9.56839e6i	3.37696i
$382$	0	0
$383$	− 3.05949e6i	− 1.06574i	−0.846197	−	0.532871i	$-0.821113\pi$
$383$	− 3.05949e6i	− 1.06574i	0.846197	−	0.532871i	$-0.178887\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.24887e6	−3.13914
$388$	0	0
$389$	− 498203.i	− 0.166929i	−0.996511	−	0.0834646i	$-0.973401\pi$
$389$	− 498203.i	− 0.166929i	0.996511	−	0.0834646i	$-0.0265985\pi$
$390$	0	0
$391$	77540.0	0.0256498
$392$	0	0
$393$	3.41732e6i	1.11610i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.95290e6	1.25875	0.629376	−	0.777101i	$-0.283311\pi$
$397$	3.95290e6	1.25875	0.629376	+	0.777101i	$0.283311\pi$
$398$	0	0
$399$	6.14475e6	1.93229
$400$	0	0
$401$	−1.15248e6	−0.357909	−0.178955	−	0.983857i	$-0.557272\pi$
$401$	−1.15248e6	−0.357909	−0.178955	+	0.983857i	$0.557272\pi$
$402$	0	0
$403$	−1.15232e6	−0.353436
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.80120e6i	1.13746i
$408$	0	0
$409$	−2.54459e6	−0.752159	−0.376079	−	0.926587i	$-0.622728\pi$
$409$	−2.54459e6	−0.752159	−0.376079	+	0.926587i	$0.622728\pi$
$410$	0	0
$411$	− 2.17334e6i	− 0.634635i
$412$	0	0
$413$	2.01438e6	0.581119
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.34933e6i	− 0.661614i
$418$	0	0
$419$	62820.4i	0.0174810i	0.999962	+	0.00874049i	$0.00278222\pi$
$419$	62820.4i	0.0174810i	−0.999962	+	0.00874049i	$0.997218\pi$
$420$	0	0
$421$	− 1.89940e6i	− 0.522289i	−0.965300	−	0.261145i	$-0.915900\pi$
$421$	− 1.89940e6i	− 0.522289i	0.965300	−	0.261145i	$-0.0841000\pi$
$422$	0	0
$423$	9.36995e6i	2.54616i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.97145e6	1.85035
$428$	0	0
$429$	7.38776e6i	1.93807i
$430$	0	0
$431$	−5.91659e6	−1.53419	−0.767094	−	0.641535i	$-0.778298\pi$
$431$	−5.91659e6	−1.53419	−0.767094	+	0.641535i	$0.778298\pi$
$432$	0	0
$433$	1.64308e6i	0.421153i	0.977577	+	0.210577i	$0.0675341\pi$
$433$	1.64308e6i	0.421153i	−0.977577	+	0.210577i	$0.932466\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−529781.	−0.132707
$438$	0	0
$439$	−4.34976e6	−1.07722	−0.538610	−	0.842555i	$-0.681050\pi$
$439$	−4.34976e6	−1.07722	−0.538610	+	0.842555i	$0.681050\pi$
$440$	0	0
$441$	−7.00118e6	−1.71425
$442$	0	0
$443$	−2.27280e6	−0.550239	−0.275119	−	0.961410i	$-0.588717\pi$
$443$	−2.27280e6	−0.550239	−0.275119	+	0.961410i	$0.588717\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.69165e6i	− 0.400445i
$448$	0	0
$449$	−7.36304e6	−1.72362	−0.861809	−	0.507233i	$-0.830669\pi$
$449$	−7.36304e6	−1.72362	−0.861809	+	0.507233i	$0.830669\pi$
$450$	0	0
$451$	2.18165e6i	0.505060i
$452$	0	0
$453$	−1.31674e7	−3.01477
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.23984e6i	− 1.39760i	−0.715317	−	0.698800i	$-0.753718\pi$
$457$	− 6.23984e6i	− 1.39760i	0.715317	−	0.698800i	$-0.246282\pi$
$458$	0	0
$459$	1.96323e6i	0.434951i
$460$	0	0
$461$	− 832291.i	− 0.182399i	−0.995833	−	0.0911996i	$-0.970930\pi$
$461$	− 832291.i	− 0.182399i	0.995833	−	0.0911996i	$-0.0290701\pi$
$462$	0	0
$463$	− 3.61852e6i	− 0.784474i	−0.919864	−	0.392237i	$-0.871701\pi$
$463$	− 3.61852e6i	− 0.784474i	0.919864	−	0.392237i	$-0.128299\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−286010.	−0.0606861	−0.0303431	−	0.999540i	$-0.509660\pi$
$467$	−286010.	−0.0606861	−0.0303431	+	0.999540i	$0.509660\pi$
$468$	0	0
$469$	− 1.11877e7i	− 2.34860i
$470$	0	0
$471$	2.10494e6	0.437208
$472$	0	0
$473$	− 7.79938e6i	− 1.60290i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.95150e6	1.39889
$478$	0	0
$479$	−2.28996e6	−0.456026	−0.228013	−	0.973658i	$-0.573223\pi$
$479$	−2.28996e6	−0.456026	−0.228013	+	0.973658i	$0.573223\pi$
$480$	0	0
$481$	3.63222e6	0.715830
$482$	0	0
$483$	2.08023e6	0.405736
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 6.55915e6i	− 1.25321i	−0.779335	−	0.626607i	$-0.784443\pi$
$487$	− 6.55915e6i	− 1.25321i	0.779335	−	0.626607i	$-0.215557\pi$
$488$	0	0
$489$	1.37612e7	2.60246
$490$	0	0
$491$	− 3.50573e6i	− 0.656257i	−0.944633	−	0.328129i	$-0.893582\pi$
$491$	− 3.50573e6i	− 0.656257i	0.944633	−	0.328129i	$-0.106418\pi$
$492$	0	0
$493$	−634127.	−0.117506
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.40858e6i	0.800586i
$498$	0	0
$499$	1.11789e6i	0.200977i	0.994938	+	0.100489i	$0.0320406\pi$
$499$	1.11789e6i	0.200977i	−0.994938	+	0.100489i	$0.967959\pi$
$500$	0	0
$501$	− 1.51638e7i	− 2.69906i
$502$	0	0
$503$	− 3.97264e6i	− 0.700098i	−0.936731	−	0.350049i	$-0.886165\pi$
$503$	− 3.97264e6i	− 0.700098i	0.936731	−	0.350049i	$-0.113835\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.78539e6	−0.654020
$508$	0	0
$509$	− 5.19239e6i	− 0.888327i	−0.895946	−	0.444164i	$-0.853501\pi$
$509$	− 5.19239e6i	− 0.888327i	0.895946	−	0.444164i	$-0.146499\pi$
$510$	0	0
$511$	−1.45120e7	−2.45853
$512$	0	0
$513$	− 1.34135e7i	− 2.25035i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.90148e6	−1.30012
$518$	0	0
$519$	2.12313e7	3.45985
$520$	0	0
$521$	1.09827e6	0.177262	0.0886311	−	0.996065i	$-0.471751\pi$
$521$	1.09827e6	0.177262	0.0886311	+	0.996065i	$0.471751\pi$
$522$	0	0
$523$	8.67189e6	1.38631	0.693154	−	0.720790i	$-0.256221\pi$
$523$	8.67189e6	1.38631	0.693154	+	0.720790i	$0.256221\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 429158.i	− 0.0673116i
$528$	0	0
$529$	6.25699e6	0.972135
$530$	0	0
$531$	− 7.30787e6i	− 1.12475i
$532$	0	0
$533$	2.08467e6	0.317847
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.55916e6i	− 1.43049i
$538$	0	0
$539$	− 5.90395e6i	− 0.875328i
$540$	0	0
$541$	746497.i	0.109657i	0.998496	+	0.0548283i	$0.0174612\pi$
$541$	746497.i	0.109657i	−0.998496	+	0.0548283i	$0.982539\pi$
$542$	0	0
$543$	− 1.90315e7i	− 2.76996i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.52087e6	0.360231	0.180116	−	0.983645i	$-0.442353\pi$
$547$	2.52087e6	0.360231	0.180116	+	0.983645i	$0.442353\pi$
$548$	0	0
$549$	− 2.52915e7i	− 3.58132i
$550$	0	0
$551$	4.33258e6	0.607950
$552$	0	0
$553$	− 3.32973e6i	− 0.463017i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	186236.	0.0254347	0.0127173	−	0.999919i	$-0.495952\pi$
$557$	186236.	0.0254347	0.0127173	+	0.999919i	$0.495952\pi$
$558$	0	0
$559$	−7.45267e6	−1.00875
$560$	0	0
$561$	−2.75141e6	−0.369104
$562$	0	0
$563$	358662.	0.0476886	0.0238443	−	0.999716i	$-0.492409\pi$
$563$	358662.	0.0476886	0.0238443	+	0.999716i	$0.492409\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.77365e7i	3.62321i
$568$	0	0
$569$	−1.08964e7	−1.41093	−0.705463	−	0.708747i	$-0.749261\pi$
$569$	−1.08964e7	−1.41093	−0.705463	+	0.708747i	$0.749261\pi$
$570$	0	0
$571$	1.93042e6i	0.247778i	0.992296	+	0.123889i	$0.0395366\pi$
$571$	1.93042e6i	0.247778i	−0.992296	+	0.123889i	$0.960463\pi$
$572$	0	0
$573$	1.25007e7	1.59055
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.33310e7i	− 1.66695i	−0.552555	−	0.833477i	$-0.686347\pi$
$577$	− 1.33310e7i	− 1.66695i	0.552555	−	0.833477i	$-0.313653\pi$
$578$	0	0
$579$	− 1.63768e7i	− 2.03018i
$580$	0	0
$581$	− 1.40765e6i	− 0.173003i
$582$	0	0
$583$	5.86206e6i	0.714297i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.15350e6	−0.856887	−0.428443	−	0.903569i	$-0.640938\pi$
$587$	−7.15350e6	−0.856887	−0.428443	+	0.903569i	$0.640938\pi$
$588$	0	0
$589$	2.93216e6i	0.348256i
$590$	0	0
$591$	−7.85183e6	−0.924703
$592$	0	0
$593$	− 1.46858e7i	− 1.71499i	−0.514490	−	0.857496i	$-0.672019\pi$
$593$	− 1.46858e7i	− 1.71499i	0.514490	−	0.857496i	$-0.327981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.49754e7	1.71966
$598$	0	0
$599$	−9.83597e6	−1.12008	−0.560042	−	0.828465i	$-0.689215\pi$
$599$	−9.83597e6	−1.12008	−0.560042	+	0.828465i	$0.689215\pi$
$600$	0	0
$601$	2.68643e6	0.303381	0.151691	−	0.988428i	$-0.451528\pi$
$601$	2.68643e6	0.303381	0.151691	+	0.988428i	$0.451528\pi$
$602$	0	0
$603$	−4.05875e7	−4.54568
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 387079.i	− 0.0426411i	−0.999773	−	0.0213205i	$-0.993213\pi$
$607$	− 387079.i	− 0.0426411i	0.999773	−	0.0213205i	$-0.00678705\pi$
$608$	0	0
$609$	−1.70123e7	−1.85874
$610$	0	0
$611$	7.55024e6i	0.818196i
$612$	0	0
$613$	−1.37500e7	−1.47792	−0.738962	−	0.673748i	$-0.764684\pi$
$613$	−1.37500e7	−1.47792	−0.738962	+	0.673748i	$0.764684\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.13239e7i	− 1.19752i	−0.800930	−	0.598758i	$-0.795661\pi$
$617$	− 1.13239e7i	− 1.19752i	0.800930	−	0.598758i	$-0.204339\pi$
$618$	0	0
$619$	6.62505e6i	0.694965i	0.937687	+	0.347482i	$0.112963\pi$
$619$	6.62505e6i	0.694965i	−0.937687	+	0.347482i	$0.887037\pi$
$620$	0	0
$621$	− 4.54098e6i	− 0.472521i
$622$	0	0
$623$	643171.i	0.0663905i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.87986e7	1.90967
$628$	0	0
$629$	1.35274e6i	0.136329i
$630$	0	0
$631$	1.12049e7	1.12030	0.560152	−	0.828390i	$-0.310743\pi$
$631$	1.12049e7	1.12030	0.560152	+	0.828390i	$0.310743\pi$
$632$	0	0
$633$	1.93320e7i	1.91764i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.64150e6	−0.550866
$638$	0	0
$639$	1.59937e7	1.54952
$640$	0	0
$641$	−5.35866e6	−0.515123	−0.257562	−	0.966262i	$-0.582919\pi$
$641$	−5.35866e6	−0.515123	−0.257562	+	0.966262i	$0.582919\pi$
$642$	0	0
$643$	−5.64079e6	−0.538038	−0.269019	−	0.963135i	$-0.586699\pi$
$643$	−5.64079e6	−0.538038	−0.269019	+	0.963135i	$0.586699\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.67405e6i	− 0.532884i	−0.963851	−	0.266442i	$-0.914152\pi$
$647$	− 5.67405e6i	− 0.532884i	0.963851	−	0.266442i	$-0.0858480\pi$
$648$	0	0
$649$	6.16258e6	0.574316
$650$	0	0
$651$	− 1.15134e7i	− 1.06476i
$652$	0	0
$653$	1.50064e7	1.37719	0.688593	−	0.725148i	$-0.258229\pi$
$653$	1.50064e7	1.37719	0.688593	+	0.725148i	$0.258229\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.26475e7i	4.75844i
$658$	0	0
$659$	− 7.48265e6i	− 0.671184i	−0.942007	−	0.335592i	$-0.891064\pi$
$659$	− 7.48265e6i	− 0.671184i	0.942007	−	0.335592i	$-0.108936\pi$
$660$	0	0
$661$	1.85142e7i	1.64817i	0.566467	+	0.824084i	$0.308310\pi$
$661$	1.85142e7i	1.64817i	−0.566467	+	0.824084i	$0.691690\pi$
$662$	0	0
$663$	2.62910e6i	0.232286i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.46674e6	0.127656
$668$	0	0
$669$	3.11004e7i	2.68659i
$670$	0	0
$671$	2.13278e7	1.82869
$672$	0	0
$673$	4.62662e6i	0.393755i	0.980428	+	0.196878i	$0.0630802\pi$
$673$	4.62662e6i	0.393755i	−0.980428	+	0.196878i	$0.936920\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.21437e7	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$677$	1.21437e7	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$678$	0	0
$679$	−5.91272e6	−0.492168
$680$	0	0
$681$	1.81011e7	1.49567
$682$	0	0
$683$	9.19906e6	0.754557	0.377278	−	0.926100i	$-0.376860\pi$
$683$	9.19906e6	0.754557	0.377278	+	0.926100i	$0.376860\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.27028e7i	− 1.02685i
$688$	0	0
$689$	5.60147e6	0.449525
$690$	0	0
$691$	1.02344e7i	0.815394i	0.913117	+	0.407697i	$0.133668\pi$
$691$	1.02344e7i	0.815394i	−0.913117	+	0.407697i	$0.866332\pi$
$692$	0	0
$693$	−5.27891e7	−4.17553
$694$	0	0
$695$	0	0
$696$	0	0
$697$	776389.i	0.0605338i
$698$	0	0
$699$	− 2.31856e7i	− 1.79484i
$700$	0	0
$701$	1.23863e6i	0.0952023i	0.998866	+	0.0476011i	$0.0151576\pi$
$701$	1.23863e6i	0.0952023i	−0.998866	+	0.0476011i	$0.984842\pi$
$702$	0	0
$703$	− 9.24242e6i	− 0.705339i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.67394e7	−1.25948
$708$	0	0
$709$	− 2.52759e7i	− 1.88839i	−0.329386	−	0.944195i	$-0.606842\pi$
$709$	− 2.52759e7i	− 1.88839i	0.329386	−	0.944195i	$-0.393158\pi$
$710$	0	0
$711$	−1.20798e7	−0.896161
$712$	0	0
$713$	992646.i	0.0731258i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.81281e7	−3.49624
$718$	0	0
$719$	−5.23203e6	−0.377440	−0.188720	−	0.982031i	$-0.560434\pi$
$719$	−5.23203e6	−0.377440	−0.188720	+	0.982031i	$0.560434\pi$
$720$	0	0
$721$	−7.89090e6	−0.565313
$722$	0	0
$723$	1.73087e7	1.23145
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.79530e7i	− 1.25980i	−0.776677	−	0.629899i	$-0.783096\pi$
$727$	− 1.79530e7i	− 1.25980i	0.776677	−	0.629899i	$-0.216904\pi$
$728$	0	0
$729$	2.45203e7	1.70886
$730$	0	0
$731$	− 2.77559e6i	− 0.192115i
$732$	0	0
$733$	−2.06019e6	−0.141627	−0.0708137	−	0.997490i	$-0.522560\pi$
$733$	−2.06019e6	−0.141627	−0.0708137	+	0.997490i	$0.522560\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.42266e7i	− 2.32111i
$738$	0	0
$739$	1.47746e6i	0.0995187i	0.998761	+	0.0497594i	$0.0158454\pi$
$739$	1.47746e6i	0.0995187i	−0.998761	+	0.0497594i	$0.984155\pi$
$740$	0	0
$741$	− 1.79630e7i	− 1.20180i
$742$	0	0
$743$	2.46543e7i	1.63840i	0.573508	+	0.819200i	$0.305582\pi$
$743$	2.46543e7i	1.63840i	−0.573508	+	0.819200i	$0.694418\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.10676e6	−0.334845
$748$	0	0
$749$	6.45471e6i	0.420409i
$750$	0	0
$751$	6.30342e6	0.407827	0.203914	−	0.978989i	$-0.434634\pi$
$751$	6.30342e6	0.407827	0.203914	+	0.978989i	$0.434634\pi$
$752$	0	0
$753$	4.80708e7i	3.08954i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.44827e7	0.918567	0.459284	−	0.888290i	$-0.348106\pi$
$757$	1.44827e7	0.918567	0.459284	+	0.888290i	$0.348106\pi$
$758$	0	0
$759$	6.36405e6	0.400986
$760$	0	0
$761$	9.50962e6	0.595253	0.297626	−	0.954682i	$-0.403805\pi$
$761$	9.50962e6	0.595253	0.297626	+	0.954682i	$0.403805\pi$
$762$	0	0
$763$	2.40298e6	0.149430
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.88863e6i	− 0.361431i
$768$	0	0
$769$	−2.88208e6	−0.175748	−0.0878741	−	0.996132i	$-0.528007\pi$
$769$	−2.88208e6	−0.175748	−0.0878741	+	0.996132i	$0.528007\pi$
$770$	0	0
$771$	− 3.46862e7i	− 2.10146i
$772$	0	0
$773$	1.83423e7	1.10409	0.552044	−	0.833815i	$-0.313848\pi$
$773$	1.83423e7	1.10409	0.552044	+	0.833815i	$0.313848\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.62912e7i	2.15650i
$778$	0	0
$779$	− 5.30457e6i	− 0.313189i
$780$	0	0
$781$	1.34872e7i	0.791213i
$782$	0	0
$783$	3.71365e7i	2.16469i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.35279e7	−0.778564	−0.389282	−	0.921119i	$-0.627277\pi$
$787$	−1.35279e7	−0.778564	−0.389282	+	0.921119i	$0.627277\pi$
$788$	0	0
$789$	− 4.75846e7i	− 2.72128i
$790$	0	0
$791$	2.19553e7	1.24767
$792$	0	0
$793$	− 2.03797e7i	− 1.15084i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.10994e7	1.73423	0.867115	−	0.498109i	$-0.165972\pi$
$797$	3.10994e7	1.73423	0.867115	+	0.498109i	$0.165972\pi$
$798$	0	0
$799$	−2.81192e6	−0.155825
$800$	0	0
$801$	2.33333e6	0.128498
$802$	0	0
$803$	−4.43965e7	−2.42974
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.61478e7i	1.41336i
$808$	0	0
$809$	−1.19594e6	−0.0642449	−0.0321225	−	0.999484i	$-0.510227\pi$
$809$	−1.19594e6	−0.0642449	−0.0321225	+	0.999484i	$0.510227\pi$
$810$	0	0
$811$	5.59211e6i	0.298554i	0.988795	+	0.149277i	$0.0476947\pi$
$811$	5.59211e6i	0.298554i	−0.988795	+	0.149277i	$0.952305\pi$
$812$	0	0
$813$	−488400.	−0.0259149
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.89638e7i	0.993963i
$818$	0	0
$819$	5.04424e7i	2.62776i
$820$	0	0
$821$	− 2.42817e7i	− 1.25725i	−0.777708	−	0.628625i	$-0.783618\pi$
$821$	− 2.42817e7i	− 1.25725i	0.777708	−	0.628625i	$-0.216382\pi$
$822$	0	0
$823$	− 1.21094e7i	− 0.623193i	−0.950214	−	0.311597i	$-0.899136\pi$
$823$	− 1.21094e7i	− 0.623193i	0.950214	−	0.311597i	$-0.100864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.18004e7	1.10841	0.554205	−	0.832380i	$-0.313023\pi$
$827$	2.18004e7	1.10841	0.554205	+	0.832380i	$0.313023\pi$
$828$	0	0
$829$	2.81218e7i	1.42121i	0.703593	+	0.710603i	$0.251578\pi$
$829$	2.81218e7i	1.42121i	−0.703593	+	0.710603i	$0.748422\pi$
$830$	0	0
$831$	1.67629e7	0.842067
$832$	0	0
$833$	− 2.10106e6i	− 0.104912i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.51328e7	−1.24002
$838$	0	0
$839$	1.77035e6	0.0868270	0.0434135	−	0.999057i	$-0.486177\pi$
$839$	1.77035e6	0.0868270	0.0434135	+	0.999057i	$0.486177\pi$
$840$	0	0
$841$	8.51602e6	0.415190
$842$	0	0
$843$	5.72376e7	2.77404
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.74315e7i	− 0.834883i
$848$	0	0
$849$	5.97171e7	2.84335
$850$	0	0
$851$	− 3.12891e6i	− 0.148105i
$852$	0	0
$853$	−2.98606e7	−1.40516	−0.702581	−	0.711604i	$-0.747969\pi$
$853$	−2.98606e7	−1.40516	−0.702581	+	0.711604i	$0.747969\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.59538e7i	− 0.742012i	−0.928630	−	0.371006i	$-0.879013\pi$
$857$	− 1.59538e7i	− 0.742012i	0.928630	−	0.371006i	$-0.120987\pi$
$858$	0	0
$859$	− 2.68382e7i	− 1.24099i	−0.784209	−	0.620497i	$-0.786931\pi$
$859$	− 2.68382e7i	− 1.24099i	0.784209	−	0.620497i	$-0.213069\pi$
$860$	0	0
$861$	2.08288e7i	0.957541i
$862$	0	0
$863$	1.99512e7i	0.911891i	0.890008	+	0.455946i	$0.150699\pi$
$863$	1.99512e7i	0.911891i	−0.890008	+	0.455946i	$0.849301\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.04921e7	1.82946
$868$	0	0
$869$	− 1.01867e7i	− 0.457596i
$870$	0	0
$871$	−3.27051e7	−1.46073
$872$	0	0
$873$	2.14505e7i	0.952582i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.73500e7	1.20077	0.600384	−	0.799712i	$-0.295014\pi$
$877$	2.73500e7	1.20077	0.600384	+	0.799712i	$0.295014\pi$
$878$	0	0
$879$	−6.57510e7	−2.87032
$880$	0	0
$881$	−3.29524e7	−1.43037	−0.715184	−	0.698937i	$-0.753657\pi$
$881$	−3.29524e7	−1.43037	−0.715184	+	0.698937i	$0.753657\pi$
$882$	0	0
$883$	−1.45135e7	−0.626429	−0.313214	−	0.949682i	$-0.601406\pi$
$883$	−1.45135e7	−0.626429	−0.313214	+	0.949682i	$0.601406\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.89058e7i	− 1.23360i	−0.787118	−	0.616802i	$-0.788428\pi$
$887$	− 2.89058e7i	− 1.23360i	0.787118	−	0.616802i	$-0.211572\pi$
$888$	0	0
$889$	−5.50926e7	−2.33797
$890$	0	0
$891$	8.48543e7i	3.58079i
$892$	0	0
$893$	1.92121e7	0.806205
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.08114e6i	− 0.252351i
$898$	0	0
$899$	− 8.11793e6i	− 0.335001i
$900$	0	0
$901$	2.08615e6i	0.0856117i
$902$	0	0
$903$	− 7.44630e7i	− 3.03893i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.75219e7	0.707233	0.353617	−	0.935390i	$-0.384952\pi$
$907$	1.75219e7	0.707233	0.353617	+	0.935390i	$0.384952\pi$
$908$	0	0
$909$	6.07280e7i	2.43770i
$910$	0	0
$911$	5.49598e6	0.219406	0.109703	−	0.993964i	$-0.465010\pi$
$911$	5.49598e6	0.219406	0.109703	+	0.993964i	$0.465010\pi$
$912$	0	0
$913$	− 4.30643e6i	− 0.170978i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.96762e7	−0.772711
$918$	0	0
$919$	1.89582e6	0.0740472	0.0370236	−	0.999314i	$-0.488212\pi$
$919$	1.89582e6	0.0740472	0.0370236	+	0.999314i	$0.488212\pi$
$920$	0	0
$921$	1.67197e7	0.649502
$922$	0	0
$923$	1.28876e7	0.497930
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.86271e7i	1.09415i
$928$	0	0
$929$	−3.44700e7	−1.31040	−0.655198	−	0.755457i	$-0.727415\pi$
$929$	−3.44700e7	−1.31040	−0.655198	+	0.755457i	$0.727415\pi$
$930$	0	0
$931$	1.43552e7i	0.542793i
$932$	0	0
$933$	8.35868e7	3.14365
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.96544e7i	1.47551i	0.675068	+	0.737756i	$0.264114\pi$
$937$	3.96544e7i	1.47551i	−0.675068	+	0.737756i	$0.735886\pi$
$938$	0	0
$939$	1.00955e7i	0.373651i
$940$	0	0
$941$	1.88367e7i	0.693474i	0.937962	+	0.346737i	$0.112710\pi$
$941$	1.88367e7i	0.693474i	−0.937962	+	0.346737i	$0.887290\pi$
$942$	0	0
$943$	− 1.79580e6i	− 0.0657625i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.04648e6	−0.146623	−0.0733115	−	0.997309i	$-0.523357\pi$
$947$	−4.04648e6	−0.146623	−0.0733115	+	0.997309i	$0.523357\pi$
$948$	0	0
$949$	4.24230e7i	1.52910i
$950$	0	0
$951$	−8.30590e7	−2.97807
$952$	0	0
$953$	− 4.74092e7i	− 1.69095i	−0.534016	−	0.845474i	$-0.679318\pi$
$953$	− 4.74092e7i	− 1.69095i	0.534016	−	0.845474i	$-0.320682\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.20456e7	−1.83698
$958$	0	0
$959$	1.25136e7	0.439376
$960$	0	0
$961$	−2.31352e7	−0.808099
$962$	0	0
$963$	2.34168e7	0.813695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.05815e7i	0.363898i	0.983308	+	0.181949i	$0.0582406\pi$
$967$	1.05815e7i	0.363898i	−0.983308	+	0.181949i	$0.941759\pi$
$968$	0	0
$969$	6.68992e6	0.228882
$970$	0	0
$971$	− 3.45206e7i	− 1.17498i	−0.809232	−	0.587489i	$-0.800116\pi$
$971$	− 3.45206e7i	− 1.17498i	0.809232	−	0.587489i	$-0.199884\pi$
$972$	0	0
$973$	1.35269e7	0.458055
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.80039e7i	− 0.603433i	−0.953398	−	0.301717i	$-0.902440\pi$
$977$	− 1.80039e7i	− 0.603433i	0.953398	−	0.301717i	$-0.0975596\pi$
$978$	0	0
$979$	1.96765e6i	0.0656133i
$980$	0	0
$981$	− 8.71768e6i	− 0.289220i
$982$	0	0
$983$	− 2.71815e7i	− 0.897200i	−0.893733	−	0.448600i	$-0.851923\pi$
$983$	− 2.71815e7i	− 0.897200i	0.893733	−	0.448600i	$-0.148077\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.54378e7	−2.46488
$988$	0	0
$989$	6.41996e6i	0.208709i
$990$	0	0
$991$	2.26954e7	0.734097	0.367048	−	0.930202i	$-0.380368\pi$
$991$	2.26954e7	0.734097	0.367048	+	0.930202i	$0.380368\pi$
$992$	0	0
$993$	− 6.44416e7i	− 2.07393i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.28102e7	0.726762	0.363381	−	0.931641i	$-0.381622\pi$
$997$	2.28102e7	0.726762	0.363381	+	0.931641i	$0.381622\pi$
$998$	0	0
$999$	7.92208e7	2.51146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.c.49.20		20
4.3	odd	2		200.6.f.b.149.4		20
5.2	odd	4		800.6.d.c.401.1		20
5.3	odd	4		160.6.d.a.81.20		20
5.4	even	2		800.6.f.b.49.1		20
8.3	odd	2		200.6.f.c.149.18		20
8.5	even	2		800.6.f.b.49.2		20
20.3	even	4		40.6.d.a.21.14	yes	20
20.7	even	4		200.6.d.b.101.7		20
20.19	odd	2		200.6.f.c.149.17		20
40.3	even	4		40.6.d.a.21.13	✓	20
40.13	odd	4		160.6.d.a.81.1		20
40.19	odd	2		200.6.f.b.149.3		20
40.27	even	4		200.6.d.b.101.8		20
40.29	even	2	inner	800.6.f.c.49.19		20
40.37	odd	4		800.6.d.c.401.20		20
60.23	odd	4		360.6.k.b.181.7		20
120.83	odd	4		360.6.k.b.181.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.13	✓	20	40.3	even	4
40.6.d.a.21.14	yes	20	20.3	even	4
160.6.d.a.81.1		20	40.13	odd	4
160.6.d.a.81.20		20	5.3	odd	4
200.6.d.b.101.7		20	20.7	even	4
200.6.d.b.101.8		20	40.27	even	4
200.6.f.b.149.3		20	40.19	odd	2
200.6.f.b.149.4		20	4.3	odd	2
200.6.f.c.149.17		20	20.19	odd	2
200.6.f.c.149.18		20	8.3	odd	2
360.6.k.b.181.7		20	60.23	odd	4
360.6.k.b.181.8		20	120.83	odd	4
800.6.d.c.401.1		20	5.2	odd	4
800.6.d.c.401.20		20	40.37	odd	4
800.6.f.b.49.1		20	5.4	even	2
800.6.f.b.49.2		20	8.5	even	2
800.6.f.c.49.19		20	40.29	even	2	inner
800.6.f.c.49.20		20	1.1	even	1	trivial

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+29.2080 q^{3} +168.173i q^{7} +610.110 q^{9} +514.493i q^{11} +491.622 q^{13} +183.094i q^{17} -1250.96i q^{19} +4912.01i q^{21} -423.498i q^{23} +10722.5 q^{27} +3463.40i q^{29} -2343.92 q^{31} +15027.3i q^{33} +7388.25 q^{37} +14359.3 q^{39} +4240.39 q^{41} -15159.4 q^{43} +15357.8i q^{47} -11475.3 q^{49} +5347.82i q^{51} +11393.9 q^{53} -36538.2i q^{57} -11978.0i q^{59} -41454.0i q^{61} +102604. i q^{63} -66524.9 q^{67} -12369.6i q^{69} +26214.5 q^{71} +86291.9i q^{73} -86524.0 q^{77} -19799.4 q^{79} +164928. q^{81} -8370.24 q^{83} +101159. i q^{87} +3824.45 q^{89} +82677.7i q^{91} -68461.3 q^{93} +35158.5i q^{97} +313897. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{3} + 1620 q^{9} + 11664 q^{27} - 7160 q^{31} + 3608 q^{37} - 44904 q^{39} + 11608 q^{41} - 51772 q^{43} - 18756 q^{49} - 928 q^{53} - 161604 q^{67} + 200312 q^{71} - 26008 q^{77} + 282080 q^{79}+ \cdots - 293472 q^{93}+O(q^{100}) \) 20 * q + 36 * q^3 + 1620 * q^9 + 11664 * q^27 - 7160 * q^31 + 3608 * q^37 - 44904 * q^39 + 11608 * q^41 - 51772 * q^43 - 18756 * q^49 - 928 * q^53 - 161604 * q^67 + 200312 * q^71 - 26008 * q^77 + 282080 * q^79 + 65172 * q^81 - 99092 * q^83 + 3160 * q^89 - 293472 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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