LMFDB - Embedded newform 800.6.c.k.449.2

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:	$6$
Coefficient field:	6.0.6140289600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 32x^{4} + 116x^{3} + 256x^{2} + 2778x + 7605 $ x^6 - 2x^5 + 32x^4 + 116x^3 + 256x^2 + 2778*x + 7605
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$3.05894 - 4.88658i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.k.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.4715i q^{3} -121.899i q^{7} -98.1968 q^{9} +O(q^{10})$	$q-18.4715i q^{3} -121.899i q^{7} -98.1968 q^{9} +438.282 q^{11} +758.285i q^{13} -1534.86i q^{17} +75.8518 q^{19} -2251.66 q^{21} +3694.27i q^{23} -2674.73i q^{27} -6323.09 q^{29} -2691.35 q^{31} -8095.74i q^{33} -7252.19i q^{37} +14006.7 q^{39} +4913.34 q^{41} +2533.81i q^{43} -11380.6i q^{47} +1947.64 q^{49} -28351.1 q^{51} -29442.3i q^{53} -1401.10i q^{57} -5681.08 q^{59} -48039.8 q^{61} +11970.1i q^{63} -39570.6i q^{67} +68238.7 q^{69} -12622.3 q^{71} -57959.2i q^{73} -53426.2i q^{77} -29504.0 q^{79} -73268.2 q^{81} +112209. i q^{83} +116797. i q^{87} -66969.5 q^{89} +92434.2 q^{91} +49713.4i q^{93} +131833. i q^{97} -43037.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 934 q^{9}+O(q^{10}) $ 6 * q - 934 * q^9	$ 6 q - 934 q^{9} + 792 q^{11} + 6384 q^{19} - 1640 q^{21} - 852 q^{29} - 6552 q^{31} + 42696 q^{39} + 24900 q^{41} + 7698 q^{49} - 142888 q^{51} + 70080 q^{59} - 48276 q^{61} + 54072 q^{69} - 176184 q^{71} - 185904 q^{79} + 302782 q^{81} - 345372 q^{89} + 213624 q^{91} - 702872 q^{99}+O(q^{100}) $ 6 * q - 934 * q^9 + 792 * q^11 + 6384 * q^19 - 1640 * q^21 - 852 * q^29 - 6552 * q^31 + 42696 * q^39 + 24900 * q^41 + 7698 * q^49 - 142888 * q^51 + 70080 * q^59 - 48276 * q^61 + 54072 * q^69 - 176184 * q^71 - 185904 * q^79 + 302782 * q^81 - 345372 * q^89 + 213624 * q^91 - 702872 * 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
$3$	− 18.4715i	− 1.18495i	−0.805590	−	0.592474i	$-0.798151\pi$
$3$	− 18.4715i	− 1.18495i	0.805590	−	0.592474i	$-0.201849\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 121.899i	− 0.940275i	−0.882593	−	0.470138i	$-0.844204\pi$
$7$	− 121.899i	− 0.940275i	0.882593	−	0.470138i	$-0.155796\pi$
$8$	0	0
$9$	−98.1968	−0.404102
$10$	0	0
$11$	438.282	1.09213	0.546063	−	0.837744i	$-0.316126\pi$
$11$	438.282	1.09213	0.546063	+	0.837744i	$0.316126\pi$
$12$	0	0
$13$	758.285i	1.24444i	0.782842	+	0.622220i	$0.213769\pi$
$13$	758.285i	1.24444i	−0.782842	+	0.622220i	$0.786231\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1534.86i	− 1.28809i	−0.764989	−	0.644044i	$-0.777256\pi$
$17$	− 1534.86i	− 1.28809i	0.764989	−	0.644044i	$-0.222744\pi$
$18$	0	0
$19$	75.8518	0.0482039	0.0241019	−	0.999710i	$-0.492327\pi$
$19$	75.8518	0.0482039	0.0241019	+	0.999710i	$0.492327\pi$
$20$	0	0
$21$	−2251.66	−1.11418
$22$	0	0
$23$	3694.27i	1.45616i	0.685493	+	0.728079i	$0.259587\pi$
$23$	3694.27i	1.45616i	−0.685493	+	0.728079i	$0.740413\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2674.73i	− 0.706108i
$28$	0	0
$29$	−6323.09	−1.39616	−0.698079	−	0.716021i	$-0.745961\pi$
$29$	−6323.09	−1.39616	−0.698079	+	0.716021i	$0.745961\pi$
$30$	0	0
$31$	−2691.35	−0.502998	−0.251499	−	0.967857i	$-0.580924\pi$
$31$	−2691.35	−0.502998	−0.251499	+	0.967857i	$0.580924\pi$
$32$	0	0
$33$	− 8095.74i	− 1.29411i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7252.19i	− 0.870893i	−0.900215	−	0.435446i	$-0.856591\pi$
$37$	− 7252.19i	− 0.870893i	0.900215	−	0.435446i	$-0.143409\pi$
$38$	0	0
$39$	14006.7	1.47460
$40$	0	0
$41$	4913.34	0.456475	0.228238	−	0.973605i	$-0.426704\pi$
$41$	4913.34	0.456475	0.228238	+	0.973605i	$0.426704\pi$
$42$	0	0
$43$	2533.81i	0.208979i	0.994526	+	0.104490i	$0.0333209\pi$
$43$	2533.81i	0.208979i	−0.994526	+	0.104490i	$0.966679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11380.6i	− 0.751487i	−0.926724	−	0.375743i	$-0.877387\pi$
$47$	− 11380.6i	− 0.751487i	0.926724	−	0.375743i	$-0.122613\pi$
$48$	0	0
$49$	1947.64	0.115882
$50$	0	0
$51$	−28351.1	−1.52632
$52$	0	0
$53$	− 29442.3i	− 1.43973i	−0.694113	−	0.719866i	$-0.744203\pi$
$53$	− 29442.3i	− 1.43973i	0.694113	−	0.719866i	$-0.255797\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1401.10i	− 0.0571191i
$58$	0	0
$59$	−5681.08	−0.212472	−0.106236	−	0.994341i	$-0.533880\pi$
$59$	−5681.08	−0.212472	−0.106236	+	0.994341i	$0.533880\pi$
$60$	0	0
$61$	−48039.8	−1.65301	−0.826507	−	0.562926i	$-0.809676\pi$
$61$	−48039.8	−1.65301	−0.826507	+	0.562926i	$0.809676\pi$
$62$	0	0
$63$	11970.1i	0.379967i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 39570.6i	− 1.07693i	−0.842649	−	0.538463i	$-0.819005\pi$
$67$	− 39570.6i	− 1.07693i	0.842649	−	0.538463i	$-0.180995\pi$
$68$	0	0
$69$	68238.7	1.72547
$70$	0	0
$71$	−12622.3	−0.297161	−0.148581	−	0.988900i	$-0.547470\pi$
$71$	−12622.3	−0.297161	−0.148581	+	0.988900i	$0.547470\pi$
$72$	0	0
$73$	− 57959.2i	− 1.27296i	−0.771292	−	0.636481i	$-0.780389\pi$
$73$	− 57959.2i	− 1.27296i	0.771292	−	0.636481i	$-0.219611\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 53426.2i	− 1.02690i
$78$	0	0
$79$	−29504.0	−0.531879	−0.265939	−	0.963990i	$-0.585682\pi$
$79$	−29504.0	−0.531879	−0.265939	+	0.963990i	$0.585682\pi$
$80$	0	0
$81$	−73268.2	−1.24080
$82$	0	0
$83$	112209.i	1.78786i	0.448211	+	0.893928i	$0.352061\pi$
$83$	112209.i	1.78786i	−0.448211	+	0.893928i	$0.647939\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	116797.i	1.65437i
$88$	0	0
$89$	−66969.5	−0.896194	−0.448097	−	0.893985i	$-0.647898\pi$
$89$	−66969.5	−0.896194	−0.448097	+	0.893985i	$0.647898\pi$
$90$	0	0
$91$	92434.2	1.17012
$92$	0	0
$93$	49713.4i	0.596027i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	131833.i	1.42264i	0.702867	+	0.711322i	$0.251903\pi$
$97$	131833.i	1.42264i	−0.702867	+	0.711322i	$0.748097\pi$
$98$	0	0
$99$	−43037.9	−0.441330
$100$	0	0
$101$	−30688.2	−0.299343	−0.149671	−	0.988736i	$-0.547822\pi$
$101$	−30688.2	−0.299343	−0.149671	+	0.988736i	$0.547822\pi$
$102$	0	0
$103$	− 140744.i	− 1.30718i	−0.756848	−	0.653591i	$-0.773262\pi$
$103$	− 140744.i	− 1.30718i	0.756848	−	0.653591i	$-0.226738\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14196.9i	− 0.119876i	−0.998202	−	0.0599381i	$-0.980910\pi$
$107$	− 14196.9i	− 0.119876i	0.998202	−	0.0599381i	$-0.0190903\pi$
$108$	0	0
$109$	−9873.09	−0.0795952	−0.0397976	−	0.999208i	$-0.512671\pi$
$109$	−9873.09	−0.0795952	−0.0397976	+	0.999208i	$0.512671\pi$
$110$	0	0
$111$	−133959.	−1.03196
$112$	0	0
$113$	− 245866.i	− 1.81135i	−0.423973	−	0.905675i	$-0.639365\pi$
$113$	− 245866.i	− 1.81135i	0.423973	−	0.905675i	$-0.360635\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 74461.2i	− 0.502881i
$118$	0	0
$119$	−187097.	−1.21116
$120$	0	0
$121$	31040.5	0.192737
$122$	0	0
$123$	− 90756.9i	− 0.540900i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 88403.3i	− 0.486361i	−0.969981	−	0.243181i	$-0.921809\pi$
$127$	− 88403.3i	− 0.486361i	0.969981	−	0.243181i	$-0.0781908\pi$
$128$	0	0
$129$	46803.3	0.247630
$130$	0	0
$131$	274297.	1.39651	0.698253	−	0.715851i	$-0.253961\pi$
$131$	274297.	1.39651	0.698253	+	0.715851i	$0.253961\pi$
$132$	0	0
$133$	− 9246.26i	− 0.0453249i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 309183.i	− 1.40739i	−0.710503	−	0.703694i	$-0.751533\pi$
$137$	− 309183.i	− 1.40739i	0.710503	−	0.703694i	$-0.248467\pi$
$138$	0	0
$139$	−381380.	−1.67425	−0.837125	−	0.547011i	$-0.815765\pi$
$139$	−381380.	−1.67425	−0.837125	+	0.547011i	$0.815765\pi$
$140$	0	0
$141$	−210217.	−0.890473
$142$	0	0
$143$	332343.i	1.35908i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 35975.8i	− 0.137315i
$148$	0	0
$149$	101742.	0.375436	0.187718	−	0.982223i	$-0.439891\pi$
$149$	101742.	0.375436	0.187718	+	0.982223i	$0.439891\pi$
$150$	0	0
$151$	−338843.	−1.20936	−0.604680	−	0.796469i	$-0.706699\pi$
$151$	−338843.	−1.20936	−0.604680	+	0.796469i	$0.706699\pi$
$152$	0	0
$153$	150718.i	0.520519i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	101650.i	0.329123i	0.986367	+	0.164561i	$0.0526208\pi$
$157$	101650.i	0.329123i	−0.986367	+	0.164561i	$0.947379\pi$
$158$	0	0
$159$	−543843.	−1.70601
$160$	0	0
$161$	450327.	1.36919
$162$	0	0
$163$	214136.i	0.631279i	0.948879	+	0.315640i	$0.102219\pi$
$163$	214136.i	0.631279i	−0.948879	+	0.315640i	$0.897781\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	606370.i	1.68247i	0.540672	+	0.841234i	$0.318170\pi$
$167$	606370.i	1.68247i	−0.540672	+	0.841234i	$0.681830\pi$
$168$	0	0
$169$	−203703.	−0.548633
$170$	0	0
$171$	−7448.41	−0.0194793
$172$	0	0
$173$	426958.i	1.08460i	0.840184	+	0.542301i	$0.182447\pi$
$173$	426958.i	1.08460i	−0.840184	+	0.542301i	$0.817553\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	104938.i	0.251768i
$178$	0	0
$179$	682764.	1.59271	0.796357	−	0.604826i	$-0.206757\pi$
$179$	682764.	1.59271	0.796357	+	0.604826i	$0.206757\pi$
$180$	0	0
$181$	725666.	1.64642	0.823209	−	0.567738i	$-0.192181\pi$
$181$	725666.	1.64642	0.823209	+	0.567738i	$0.192181\pi$
$182$	0	0
$183$	887368.i	1.95874i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 672700.i	− 1.40675i
$188$	0	0
$189$	−326047.	−0.663936
$190$	0	0
$191$	−662000.	−1.31303	−0.656515	−	0.754313i	$-0.727970\pi$
$191$	−662000.	−1.31303	−0.656515	+	0.754313i	$0.727970\pi$
$192$	0	0
$193$	481557.i	0.930582i	0.885158	+	0.465291i	$0.154050\pi$
$193$	481557.i	0.930582i	−0.885158	+	0.465291i	$0.845950\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	569264.i	1.04508i	0.852616	+	0.522538i	$0.175015\pi$
$197$	569264.i	1.04508i	−0.852616	+	0.522538i	$0.824985\pi$
$198$	0	0
$199$	−850151.	−1.52182	−0.760911	−	0.648857i	$-0.775247\pi$
$199$	−850151.	−1.52182	−0.760911	+	0.648857i	$0.775247\pi$
$200$	0	0
$201$	−730930.	−1.27610
$202$	0	0
$203$	770778.i	1.31277i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 362765.i	− 0.588437i
$208$	0	0
$209$	33244.5	0.0526447
$210$	0	0
$211$	−285552.	−0.441548	−0.220774	−	0.975325i	$-0.570858\pi$
$211$	−285552.	−0.441548	−0.220774	+	0.975325i	$0.570858\pi$
$212$	0	0
$213$	233153.i	0.352120i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	328073.i	0.472957i
$218$	0	0
$219$	−1.07059e6	−1.50839
$220$	0	0
$221$	1.16386e6	1.60295
$222$	0	0
$223$	279669.i	0.376602i	0.982111	+	0.188301i	$0.0602981\pi$
$223$	279669.i	0.376602i	−0.982111	+	0.188301i	$0.939702\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.27295e6i	− 1.63964i	−0.572623	−	0.819819i	$-0.694074\pi$
$227$	− 1.27295e6i	− 1.63964i	0.572623	−	0.819819i	$-0.305926\pi$
$228$	0	0
$229$	−584351.	−0.736352	−0.368176	−	0.929756i	$-0.620018\pi$
$229$	−584351.	−0.736352	−0.368176	+	0.929756i	$0.620018\pi$
$230$	0	0
$231$	−986863.	−1.21682
$232$	0	0
$233$	658722.i	0.794900i	0.917624	+	0.397450i	$0.130105\pi$
$233$	658722.i	0.794900i	−0.917624	+	0.397450i	$0.869895\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	544983.i	0.630248i
$238$	0	0
$239$	−664166.	−0.752111	−0.376055	−	0.926597i	$-0.622720\pi$
$239$	−664166.	−0.752111	−0.376055	+	0.926597i	$0.622720\pi$
$240$	0	0
$241$	−192140.	−0.213095	−0.106548	−	0.994308i	$-0.533980\pi$
$241$	−192140.	−0.213095	−0.106548	+	0.994308i	$0.533980\pi$
$242$	0	0
$243$	703414.i	0.764180i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	57517.3i	0.0599869i
$248$	0	0
$249$	2.07267e6	2.11852
$250$	0	0
$251$	−1.20021e6	−1.20246	−0.601231	−	0.799075i	$-0.705323\pi$
$251$	−1.20021e6	−1.20246	−0.601231	+	0.799075i	$0.705323\pi$
$252$	0	0
$253$	1.61913e6i	1.59031i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 494440.i	− 0.466961i	−0.972361	−	0.233481i	$-0.924988\pi$
$257$	− 494440.i	− 0.466961i	0.972361	−	0.233481i	$-0.0750115\pi$
$258$	0	0
$259$	−884034.	−0.818879
$260$	0	0
$261$	620907.	0.564190
$262$	0	0
$263$	535899.i	0.477742i	0.971051	+	0.238871i	$0.0767773\pi$
$263$	535899.i	0.477742i	−0.971051	+	0.238871i	$0.923223\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.23703e6i	1.06194i
$268$	0	0
$269$	886621.	0.747063	0.373532	−	0.927617i	$-0.378147\pi$
$269$	886621.	0.747063	0.373532	+	0.927617i	$0.378147\pi$
$270$	0	0
$271$	1.31305e6	1.08607	0.543036	−	0.839710i	$-0.317275\pi$
$271$	1.31305e6	1.08607	0.543036	+	0.839710i	$0.317275\pi$
$272$	0	0
$273$	− 1.70740e6i	− 1.38653i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	175211.i	0.137203i	0.997644	+	0.0686013i	$0.0218536\pi$
$277$	175211.i	0.137203i	−0.997644	+	0.0686013i	$0.978146\pi$
$278$	0	0
$279$	264282.	0.203263
$280$	0	0
$281$	−1.78398e6	−1.34779	−0.673897	−	0.738826i	$-0.735381\pi$
$281$	−1.78398e6	−1.34779	−0.673897	+	0.738826i	$0.735381\pi$
$282$	0	0
$283$	1.98478e6i	1.47315i	0.676358	+	0.736573i	$0.263557\pi$
$283$	1.98478e6i	1.47315i	−0.676358	+	0.736573i	$0.736443\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 598931.i	− 0.429212i
$288$	0	0
$289$	−935925.	−0.659168
$290$	0	0
$291$	2.43516e6	1.68576
$292$	0	0
$293$	− 2.44146e6i	− 1.66143i	−0.556700	−	0.830714i	$-0.687933\pi$
$293$	− 2.44146e6i	− 1.66143i	0.556700	−	0.830714i	$-0.312067\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.17229e6i	− 0.771158i
$298$	0	0
$299$	−2.80131e6	−1.81210
$300$	0	0
$301$	308869.	0.196498
$302$	0	0
$303$	566858.i	0.354706i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.53943e6i	1.53777i	0.639390	+	0.768883i	$0.279187\pi$
$307$	2.53943e6i	1.53777i	−0.639390	+	0.768883i	$0.720813\pi$
$308$	0	0
$309$	−2.59975e6	−1.54894
$310$	0	0
$311$	−1.86583e6	−1.09388	−0.546941	−	0.837171i	$-0.684208\pi$
$311$	−1.86583e6	−1.09388	−0.546941	+	0.837171i	$0.684208\pi$
$312$	0	0
$313$	− 2.17459e6i	− 1.25463i	−0.778765	−	0.627316i	$-0.784153\pi$
$313$	− 2.17459e6i	− 1.25463i	0.778765	−	0.627316i	$-0.215847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.48421e6i	− 0.829557i	−0.909922	−	0.414779i	$-0.863859\pi$
$317$	− 1.48421e6i	− 0.829557i	0.909922	−	0.414779i	$-0.136141\pi$
$318$	0	0
$319$	−2.77130e6	−1.52478
$320$	0	0
$321$	−262238.	−0.142047
$322$	0	0
$323$	− 116422.i	− 0.0620908i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	182371.i	0.0943162i
$328$	0	0
$329$	−1.38729e6	−0.706604
$330$	0	0
$331$	−2.02880e6	−1.01782	−0.508908	−	0.860821i	$-0.669950\pi$
$331$	−2.02880e6	−1.01782	−0.508908	+	0.860821i	$0.669950\pi$
$332$	0	0
$333$	712141.i	0.351930i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.34355e6i	− 0.644435i	−0.946666	−	0.322218i	$-0.895572\pi$
$337$	− 1.34355e6i	− 0.644435i	0.946666	−	0.322218i	$-0.104428\pi$
$338$	0	0
$339$	−4.54152e6	−2.14636
$340$	0	0
$341$	−1.17957e6	−0.549337
$342$	0	0
$343$	− 2.28617e6i	− 1.04924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.52038e6i	− 0.677843i	−0.940815	−	0.338921i	$-0.889938\pi$
$347$	− 1.52038e6i	− 0.677843i	0.940815	−	0.338921i	$-0.110062\pi$
$348$	0	0
$349$	1.80948e6	0.795224	0.397612	−	0.917554i	$-0.369839\pi$
$349$	1.80948e6	0.795224	0.397612	+	0.917554i	$0.369839\pi$
$350$	0	0
$351$	2.02821e6	0.878710
$352$	0	0
$353$	− 2.48843e6i	− 1.06289i	−0.847092	−	0.531446i	$-0.821649\pi$
$353$	− 2.48843e6i	− 1.06289i	0.847092	−	0.531446i	$-0.178351\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.45597e6i	1.43516i
$358$	0	0
$359$	1.44154e6	0.590324	0.295162	−	0.955447i	$-0.404626\pi$
$359$	1.44154e6	0.590324	0.295162	+	0.955447i	$0.404626\pi$
$360$	0	0
$361$	−2.47035e6	−0.997676
$362$	0	0
$363$	− 573365.i	− 0.228383i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 257353.i	− 0.0997388i	−0.998756	−	0.0498694i	$-0.984119\pi$
$367$	− 257353.i	− 0.0997388i	0.998756	−	0.0498694i	$-0.0158805\pi$
$368$	0	0
$369$	−482474.	−0.184463
$370$	0	0
$371$	−3.58898e6	−1.35374
$372$	0	0
$373$	4.55163e6i	1.69393i	0.531651	+	0.846964i	$0.321572\pi$
$373$	4.55163e6i	1.69393i	−0.531651	+	0.846964i	$0.678428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.79471e6i	− 1.73743i
$378$	0	0
$379$	2.96634e6	1.06077	0.530387	−	0.847756i	$-0.322047\pi$
$379$	2.96634e6	1.06077	0.530387	+	0.847756i	$0.322047\pi$
$380$	0	0
$381$	−1.63294e6	−0.576313
$382$	0	0
$383$	3.57107e6i	1.24395i	0.783039	+	0.621973i	$0.213669\pi$
$383$	3.57107e6i	1.24395i	−0.783039	+	0.621973i	$0.786331\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 248812.i	− 0.0844489i
$388$	0	0
$389$	881685.	0.295420	0.147710	−	0.989031i	$-0.452810\pi$
$389$	881685.	0.295420	0.147710	+	0.989031i	$0.452810\pi$
$390$	0	0
$391$	5.67017e6	1.87566
$392$	0	0
$393$	− 5.06668e6i	− 1.65479i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.56117e6i	− 0.815570i	−0.913078	−	0.407785i	$-0.866301\pi$
$397$	− 2.56117e6i	− 0.815570i	0.913078	−	0.407785i	$-0.133699\pi$
$398$	0	0
$399$	−170792.	−0.0537077
$400$	0	0
$401$	−3.37596e6	−1.04842	−0.524211	−	0.851589i	$-0.675640\pi$
$401$	−3.37596e6	−1.04842	−0.524211	+	0.851589i	$0.675640\pi$
$402$	0	0
$403$	− 2.04081e6i	− 0.625952i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.17851e6i	− 0.951124i
$408$	0	0
$409$	2.56437e6	0.758006	0.379003	−	0.925395i	$-0.376267\pi$
$409$	2.56437e6	0.758006	0.379003	+	0.925395i	$0.376267\pi$
$410$	0	0
$411$	−5.71107e6	−1.66768
$412$	0	0
$413$	692518.i	0.199782i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.04466e6i	1.98390i
$418$	0	0
$419$	6.64881e6	1.85016	0.925079	−	0.379775i	$-0.123999\pi$
$419$	6.64881e6	1.85016	0.925079	+	0.379775i	$0.123999\pi$
$420$	0	0
$421$	4.98840e6	1.37169	0.685846	−	0.727747i	$-0.259432\pi$
$421$	4.98840e6	1.37169	0.685846	+	0.727747i	$0.259432\pi$
$422$	0	0
$423$	1.11754e6i	0.303677i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.85600e6i	1.55429i
$428$	0	0
$429$	6.13888e6	1.61045
$430$	0	0
$431$	−802200.	−0.208012	−0.104006	−	0.994577i	$-0.533166\pi$
$431$	−802200.	−0.208012	−0.104006	+	0.994577i	$0.533166\pi$
$432$	0	0
$433$	− 5.80978e6i	− 1.48916i	−0.667535	−	0.744578i	$-0.732651\pi$
$433$	− 5.80978e6i	− 1.48916i	0.667535	−	0.744578i	$-0.267349\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	280217.i	0.0701925i
$438$	0	0
$439$	585972.	0.145116	0.0725581	−	0.997364i	$-0.476884\pi$
$439$	585972.	0.145116	0.0725581	+	0.997364i	$0.476884\pi$
$440$	0	0
$441$	−191252.	−0.0468283
$442$	0	0
$443$	4.41675e6i	1.06929i	0.845078	+	0.534643i	$0.179554\pi$
$443$	4.41675e6i	1.06929i	−0.845078	+	0.534643i	$0.820446\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.87934e6i	− 0.444872i
$448$	0	0
$449$	−2.28408e6	−0.534681	−0.267340	−	0.963602i	$-0.586145\pi$
$449$	−2.28408e6	−0.534681	−0.267340	+	0.963602i	$0.586145\pi$
$450$	0	0
$451$	2.15343e6	0.498528
$452$	0	0
$453$	6.25893e6i	1.43303i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.43086e6i	0.768444i	0.923241	+	0.384222i	$0.125530\pi$
$457$	3.43086e6i	0.768444i	−0.923241	+	0.384222i	$0.874470\pi$
$458$	0	0
$459$	−4.10533e6	−0.909529
$460$	0	0
$461$	674699.	0.147862	0.0739311	−	0.997263i	$-0.476445\pi$
$461$	674699.	0.147862	0.0739311	+	0.997263i	$0.476445\pi$
$462$	0	0
$463$	5.78627e6i	1.25443i	0.778846	+	0.627215i	$0.215805\pi$
$463$	5.78627e6i	1.25443i	−0.778846	+	0.627215i	$0.784195\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.88097e6i	− 1.46002i	−0.683439	−	0.730008i	$-0.739516\pi$
$467$	− 6.88097e6i	− 1.46002i	0.683439	−	0.730008i	$-0.260484\pi$
$468$	0	0
$469$	−4.82362e6	−1.01261
$470$	0	0
$471$	1.87763e6	0.389993
$472$	0	0
$473$	1.11052e6i	0.228231i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.89114e6i	0.581798i
$478$	0	0
$479$	4.94000e6	0.983757	0.491879	−	0.870664i	$-0.336310\pi$
$479$	4.94000e6	0.983757	0.491879	+	0.870664i	$0.336310\pi$
$480$	0	0
$481$	5.49923e6	1.08377
$482$	0	0
$483$	− 8.31823e6i	− 1.62242i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.21802e6i	− 0.996973i	−0.866897	−	0.498487i	$-0.833889\pi$
$487$	− 5.21802e6i	− 0.996973i	0.866897	−	0.498487i	$-0.166111\pi$
$488$	0	0
$489$	3.95542e6	0.748033
$490$	0	0
$491$	4.20805e6	0.787730	0.393865	−	0.919168i	$-0.371138\pi$
$491$	4.20805e6	0.787730	0.393865	+	0.919168i	$0.371138\pi$
$492$	0	0
$493$	9.70503e6i	1.79837i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.53864e6i	0.279413i
$498$	0	0
$499$	7.74013e6	1.39154	0.695771	−	0.718263i	$-0.255063\pi$
$499$	7.74013e6	1.39154	0.695771	+	0.718263i	$0.255063\pi$
$500$	0	0
$501$	1.12006e7	1.99364
$502$	0	0
$503$	− 6.20893e6i	− 1.09420i	−0.837067	−	0.547100i	$-0.815732\pi$
$503$	− 6.20893e6i	− 1.09420i	0.837067	−	0.547100i	$-0.184268\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.76271e6i	0.650101i
$508$	0	0
$509$	−8.43892e6	−1.44375	−0.721876	−	0.692022i	$-0.756720\pi$
$509$	−8.43892e6	−1.44375	−0.721876	+	0.692022i	$0.756720\pi$
$510$	0	0
$511$	−7.06517e6	−1.19693
$512$	0	0
$513$	− 202883.i	− 0.0340372i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.98793e6i	− 0.820717i
$518$	0	0
$519$	7.88657e6	1.28520
$520$	0	0
$521$	−6.61888e6	−1.06829	−0.534146	−	0.845392i	$-0.679367\pi$
$521$	−6.61888e6	−1.06829	−0.534146	+	0.845392i	$0.679367\pi$
$522$	0	0
$523$	− 5.48657e6i	− 0.877095i	−0.898708	−	0.438548i	$-0.855493\pi$
$523$	− 5.48657e6i	− 0.877095i	0.898708	−	0.438548i	$-0.144507\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.13084e6i	0.647906i
$528$	0	0
$529$	−7.21127e6	−1.12040
$530$	0	0
$531$	557864.	0.0858602
$532$	0	0
$533$	3.72571e6i	0.568057i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.26117e7i	− 1.88728i
$538$	0	0
$539$	853614.	0.126558
$540$	0	0
$541$	−427673.	−0.0628230	−0.0314115	−	0.999507i	$-0.510000\pi$
$541$	−427673.	−0.0628230	−0.0314115	+	0.999507i	$0.510000\pi$
$542$	0	0
$543$	− 1.34041e7i	− 1.95092i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.94393e6i	1.27809i	0.769171	+	0.639043i	$0.220669\pi$
$547$	8.94393e6i	1.27809i	−0.769171	+	0.639043i	$0.779331\pi$
$548$	0	0
$549$	4.71736e6	0.667987
$550$	0	0
$551$	−479618.	−0.0673002
$552$	0	0
$553$	3.59650e6i	0.500112i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.93887e6i	1.08423i	0.840305	+	0.542114i	$0.182376\pi$
$557$	7.93887e6i	1.08423i	−0.840305	+	0.542114i	$0.817624\pi$
$558$	0	0
$559$	−1.92135e6	−0.260062
$560$	0	0
$561$	−1.24258e7	−1.66693
$562$	0	0
$563$	− 1.61543e6i	− 0.214792i	−0.994216	−	0.107396i	$-0.965749\pi$
$563$	− 1.61543e6i	− 0.214792i	0.994216	−	0.107396i	$-0.0342513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8.93132e6i	1.16670i
$568$	0	0
$569$	−1.97353e6	−0.255543	−0.127771	−	0.991804i	$-0.540782\pi$
$569$	−1.97353e6	−0.255543	−0.127771	+	0.991804i	$0.540782\pi$
$570$	0	0
$571$	7.18140e6	0.921762	0.460881	−	0.887462i	$-0.347533\pi$
$571$	7.18140e6	0.921762	0.460881	+	0.887462i	$0.347533\pi$
$572$	0	0
$573$	1.22281e7i	1.55587i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 9.61498e6i	− 1.20229i	−0.799140	−	0.601145i	$-0.794712\pi$
$577$	− 9.61498e6i	− 1.20229i	0.799140	−	0.601145i	$-0.205288\pi$
$578$	0	0
$579$	8.89509e6	1.10269
$580$	0	0
$581$	1.36782e7	1.68108
$582$	0	0
$583$	− 1.29040e7i	− 1.57237i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.66522e6i	− 0.319255i	−0.987177	−	0.159627i	$-0.948971\pi$
$587$	− 2.66522e6i	− 0.319255i	0.987177	−	0.159627i	$-0.0510293\pi$
$588$	0	0
$589$	−204144.	−0.0242465
$590$	0	0
$591$	1.05152e7	1.23836
$592$	0	0
$593$	− 5.83235e6i	− 0.681093i	−0.940228	−	0.340547i	$-0.889388\pi$
$593$	− 5.83235e6i	− 0.681093i	0.940228	−	0.340547i	$-0.110612\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.57036e7i	1.80328i
$598$	0	0
$599$	6.39890e6	0.728683	0.364341	−	0.931265i	$-0.381294\pi$
$599$	6.39890e6	0.728683	0.364341	+	0.931265i	$0.381294\pi$
$600$	0	0
$601$	1.00505e7	1.13502	0.567509	−	0.823367i	$-0.307907\pi$
$601$	1.00505e7	1.13502	0.567509	+	0.823367i	$0.307907\pi$
$602$	0	0
$603$	3.88571e6i	0.435188i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.83636e6i	− 0.202295i	−0.994871	−	0.101148i	$-0.967749\pi$
$607$	− 1.83636e6i	− 0.202295i	0.994871	−	0.101148i	$-0.0322515\pi$
$608$	0	0
$609$	1.42374e7	1.55557
$610$	0	0
$611$	8.62976e6	0.935181
$612$	0	0
$613$	− 1.33155e7i	− 1.43122i	−0.698499	−	0.715611i	$-0.746148\pi$
$613$	− 1.33155e7i	− 1.43122i	0.698499	−	0.715611i	$-0.253852\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.09296e6i	0.327085i	0.986536	+	0.163543i	$0.0522921\pi$
$617$	3.09296e6i	0.327085i	−0.986536	+	0.163543i	$0.947708\pi$
$618$	0	0
$619$	−9.64802e6	−1.01207	−0.506036	−	0.862512i	$-0.668890\pi$
$619$	−9.64802e6	−1.01207	−0.506036	+	0.862512i	$0.668890\pi$
$620$	0	0
$621$	9.88118e6	1.02821
$622$	0	0
$623$	8.16352e6i	0.842669i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 614077.i	− 0.0623812i
$628$	0	0
$629$	−1.11311e7	−1.12179
$630$	0	0
$631$	1.72048e7	1.72019	0.860096	−	0.510132i	$-0.170403\pi$
$631$	1.72048e7	1.72019	0.860096	+	0.510132i	$0.170403\pi$
$632$	0	0
$633$	5.27457e6i	0.523212i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.47686e6i	0.144209i
$638$	0	0
$639$	1.23947e6	0.120083
$640$	0	0
$641$	1.86292e6	0.179081	0.0895403	−	0.995983i	$-0.471460\pi$
$641$	1.86292e6	0.179081	0.0895403	+	0.995983i	$0.471460\pi$
$642$	0	0
$643$	− 1.39819e7i	− 1.33364i	−0.745218	−	0.666821i	$-0.767655\pi$
$643$	− 1.39819e7i	− 1.33364i	0.745218	−	0.666821i	$-0.232345\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.28310e7i	1.20504i	0.798104	+	0.602520i	$0.205837\pi$
$647$	1.28310e7i	1.20504i	−0.798104	+	0.602520i	$0.794163\pi$
$648$	0	0
$649$	−2.48992e6	−0.232045
$650$	0	0
$651$	6.06001e6	0.560430
$652$	0	0
$653$	− 423926.i	− 0.0389051i	−0.999811	−	0.0194526i	$-0.993808\pi$
$653$	− 423926.i	− 0.0389051i	0.999811	−	0.0194526i	$-0.00619233\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.69141e6i	0.514406i
$658$	0	0
$659$	1.64781e7	1.47807	0.739035	−	0.673668i	$-0.235282\pi$
$659$	1.64781e7	1.47807	0.739035	+	0.673668i	$0.235282\pi$
$660$	0	0
$661$	−1.71556e7	−1.52723	−0.763613	−	0.645674i	$-0.776577\pi$
$661$	−1.71556e7	−1.52723	−0.763613	+	0.645674i	$0.776577\pi$
$662$	0	0
$663$	− 2.14982e7i	− 1.89941i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.33592e7i	− 2.03303i
$668$	0	0
$669$	5.16592e6	0.446254
$670$	0	0
$671$	−2.10550e7	−1.80530
$672$	0	0
$673$	− 2.24172e7i	− 1.90785i	−0.300053	−	0.953923i	$-0.597004\pi$
$673$	− 2.24172e7i	− 1.90785i	0.300053	−	0.953923i	$-0.402996\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.63329e6i	− 0.472379i	−0.971707	−	0.236190i	$-0.924101\pi$
$677$	− 5.63329e6i	− 0.472379i	0.971707	−	0.236190i	$-0.0758986\pi$
$678$	0	0
$679$	1.60704e7	1.33768
$680$	0	0
$681$	−2.35134e7	−1.94289
$682$	0	0
$683$	− 1.27876e6i	− 0.104891i	−0.998624	−	0.0524453i	$-0.983298\pi$
$683$	− 1.27876e6i	− 0.104891i	0.998624	−	0.0524453i	$-0.0167015\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.07939e7i	0.872539i
$688$	0	0
$689$	2.23256e7	1.79166
$690$	0	0
$691$	712345.	0.0567539	0.0283769	−	0.999597i	$-0.490966\pi$
$691$	712345.	0.0567539	0.0283769	+	0.999597i	$0.490966\pi$
$692$	0	0
$693$	5.24628e6i	0.414972i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.54127e6i	− 0.587980i
$698$	0	0
$699$	1.21676e7	0.941915
$700$	0	0
$701$	7.89586e6	0.606882	0.303441	−	0.952850i	$-0.401864\pi$
$701$	7.89586e6	0.606882	0.303441	+	0.952850i	$0.401864\pi$
$702$	0	0
$703$	− 550091.i	− 0.0419804i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.74087e6i	0.281464i
$708$	0	0
$709$	812363.	0.0606924	0.0303462	−	0.999539i	$-0.490339\pi$
$709$	812363.	0.0606924	0.0303462	+	0.999539i	$0.490339\pi$
$710$	0	0
$711$	2.89719e6	0.214933
$712$	0	0
$713$	− 9.94258e6i	− 0.732446i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.22681e7i	0.891212i
$718$	0	0
$719$	−1.46975e7	−1.06028	−0.530142	−	0.847909i	$-0.677861\pi$
$719$	−1.46975e7	−1.06028	−0.530142	+	0.847909i	$0.677861\pi$
$720$	0	0
$721$	−1.71565e7	−1.22911
$722$	0	0
$723$	3.54911e6i	0.252507i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 946478.i	− 0.0664163i	−0.999448	−	0.0332081i	$-0.989428\pi$
$727$	− 946478.i	− 0.0664163i	0.999448	−	0.0332081i	$-0.0105724\pi$
$728$	0	0
$729$	−4.81105e6	−0.335290
$730$	0	0
$731$	3.88903e6	0.269183
$732$	0	0
$733$	− 2.33370e6i	− 0.160430i	−0.996778	−	0.0802150i	$-0.974439\pi$
$733$	− 2.33370e6i	− 0.160430i	0.996778	−	0.0802150i	$-0.0255607\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.73431e7i	− 1.17614i
$738$	0	0
$739$	1.17842e7	0.793757	0.396878	−	0.917871i	$-0.370094\pi$
$739$	1.17842e7	0.793757	0.396878	+	0.917871i	$0.370094\pi$
$740$	0	0
$741$	1.06243e6	0.0710813
$742$	0	0
$743$	1.02537e7i	0.681413i	0.940170	+	0.340706i	$0.110666\pi$
$743$	1.02537e7i	0.681413i	−0.940170	+	0.340706i	$0.889334\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.10186e7i	− 0.722476i
$748$	0	0
$749$	−1.73058e6	−0.112717
$750$	0	0
$751$	−2.02684e7	−1.31135	−0.655676	−	0.755043i	$-0.727616\pi$
$751$	−2.02684e7	−1.31135	−0.655676	+	0.755043i	$0.727616\pi$
$752$	0	0
$753$	2.21696e7i	1.42486i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.54478e7i	− 0.979776i	−0.871785	−	0.489888i	$-0.837038\pi$
$757$	− 1.54478e7i	− 0.979776i	0.871785	−	0.489888i	$-0.162962\pi$
$758$	0	0
$759$	2.99078e7	1.88443
$760$	0	0
$761$	−1.68238e7	−1.05308	−0.526540	−	0.850150i	$-0.676511\pi$
$761$	−1.68238e7	−1.05308	−0.526540	+	0.850150i	$0.676511\pi$
$762$	0	0
$763$	1.20352e6i	0.0748414i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.30788e6i	− 0.264408i
$768$	0	0
$769$	−2.25563e7	−1.37547	−0.687737	−	0.725960i	$-0.741396\pi$
$769$	−2.25563e7	−1.37547	−0.687737	+	0.725960i	$0.741396\pi$
$770$	0	0
$771$	−9.13305e6	−0.553325
$772$	0	0
$773$	5.87038e6i	0.353360i	0.984268	+	0.176680i	$0.0565358\pi$
$773$	5.87038e6i	0.353360i	−0.984268	+	0.176680i	$0.943464\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.63294e7i	0.970329i
$778$	0	0
$779$	372686.	0.0220039
$780$	0	0
$781$	−5.53212e6	−0.324537
$782$	0	0
$783$	1.69126e7i	0.985838i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.45154e6i	0.141092i	0.997509	+	0.0705459i	$0.0224741\pi$
$787$	2.45154e6i	0.141092i	−0.997509	+	0.0705459i	$0.977526\pi$
$788$	0	0
$789$	9.89886e6	0.566099
$790$	0	0
$791$	−2.99708e7	−1.70317
$792$	0	0
$793$	− 3.64279e7i	− 2.05708i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.18622e7i	− 1.21913i	−0.792737	−	0.609564i	$-0.791345\pi$
$797$	− 2.18622e7i	− 1.21913i	0.792737	−	0.609564i	$-0.208655\pi$
$798$	0	0
$799$	−1.74676e7	−0.967980
$800$	0	0
$801$	6.57619e6	0.362154
$802$	0	0
$803$	− 2.54025e7i	− 1.39023i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.63772e7i	− 0.885231i
$808$	0	0
$809$	−2.66048e7	−1.42919	−0.714593	−	0.699541i	$-0.753388\pi$
$809$	−2.66048e7	−1.42919	−0.714593	+	0.699541i	$0.753388\pi$
$810$	0	0
$811$	1.44067e7	0.769152	0.384576	−	0.923093i	$-0.374348\pi$
$811$	1.44067e7	0.769152	0.384576	+	0.923093i	$0.374348\pi$
$812$	0	0
$813$	− 2.42540e7i	− 1.28694i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	192194.i	0.0100736i
$818$	0	0
$819$	−9.07674e6	−0.472847
$820$	0	0
$821$	1.88597e6	0.0976508	0.0488254	−	0.998807i	$-0.484452\pi$
$821$	1.88597e6	0.0976508	0.0488254	+	0.998807i	$0.484452\pi$
$822$	0	0
$823$	1.52832e7i	0.786530i	0.919425	+	0.393265i	$0.128654\pi$
$823$	1.52832e7i	0.786530i	−0.919425	+	0.393265i	$0.871346\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.78935e6i	0.141821i	0.997483	+	0.0709103i	$0.0225904\pi$
$827$	2.78935e6i	0.141821i	−0.997483	+	0.0709103i	$0.977410\pi$
$828$	0	0
$829$	1.87077e6	0.0945440	0.0472720	−	0.998882i	$-0.484947\pi$
$829$	1.87077e6	0.0945440	0.0472720	+	0.998882i	$0.484947\pi$
$830$	0	0
$831$	3.23641e6	0.162578
$832$	0	0
$833$	− 2.98934e6i	− 0.149267i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.19866e6i	0.355171i
$838$	0	0
$839$	−2.01293e6	−0.0987244	−0.0493622	−	0.998781i	$-0.515719\pi$
$839$	−2.01293e6	−0.0987244	−0.0493622	+	0.998781i	$0.515719\pi$
$840$	0	0
$841$	1.94703e7	0.949255
$842$	0	0
$843$	3.29527e7i	1.59706i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.78381e6i	− 0.181226i
$848$	0	0
$849$	3.66618e7	1.74560
$850$	0	0
$851$	2.67915e7	1.26816
$852$	0	0
$853$	− 1.09443e7i	− 0.515008i	−0.966277	−	0.257504i	$-0.917100\pi$
$853$	− 1.09443e7i	− 0.515008i	0.966277	−	0.257504i	$-0.0829000\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.17692e7i	− 1.01249i	−0.862390	−	0.506245i	$-0.831033\pi$
$857$	− 2.17692e7i	− 1.01249i	0.862390	−	0.506245i	$-0.168967\pi$
$858$	0	0
$859$	−6.86741e6	−0.317549	−0.158774	−	0.987315i	$-0.550754\pi$
$859$	−6.86741e6	−0.317549	−0.158774	+	0.987315i	$0.550754\pi$
$860$	0	0
$861$	−1.10632e7	−0.508595
$862$	0	0
$863$	− 1.75635e7i	− 0.802755i	−0.915913	−	0.401378i	$-0.868532\pi$
$863$	− 1.75635e7i	− 0.802755i	0.915913	−	0.401378i	$-0.131468\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.72880e7i	0.781080i
$868$	0	0
$869$	−1.29311e7	−0.580878
$870$	0	0
$871$	3.00058e7	1.34017
$872$	0	0
$873$	− 1.29456e7i	− 0.574893i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.95517e6i	− 0.129743i	−0.997894	−	0.0648715i	$-0.979336\pi$
$877$	− 2.95517e6i	− 0.129743i	0.997894	−	0.0648715i	$-0.0206638\pi$
$878$	0	0
$879$	−4.50975e7	−1.96870
$880$	0	0
$881$	3.53773e7	1.53562	0.767811	−	0.640676i	$-0.221346\pi$
$881$	3.53773e7	1.53562	0.767811	+	0.640676i	$0.221346\pi$
$882$	0	0
$883$	2.26106e7i	0.975912i	0.872868	+	0.487956i	$0.162257\pi$
$883$	2.26106e7i	0.975912i	−0.872868	+	0.487956i	$0.837743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.57410e6i	0.365914i	0.983121	+	0.182957i	$0.0585670\pi$
$887$	8.57410e6i	0.365914i	−0.983121	+	0.182957i	$0.941433\pi$
$888$	0	0
$889$	−1.07763e7	−0.457314
$890$	0	0
$891$	−3.21122e7	−1.35511
$892$	0	0
$893$	− 863241.i	− 0.0362246i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.17444e7i	2.14725i
$898$	0	0
$899$	1.70177e7	0.702265
$900$	0	0
$901$	−4.51896e7	−1.85450
$902$	0	0
$903$	− 5.70528e6i	− 0.232840i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.76363e7i	− 0.711852i	−0.934514	−	0.355926i	$-0.884166\pi$
$907$	− 1.76363e7i	− 0.711852i	0.934514	−	0.355926i	$-0.115834\pi$
$908$	0	0
$909$	3.01349e6	0.120965
$910$	0	0
$911$	1.48438e7	0.592585	0.296292	−	0.955097i	$-0.404250\pi$
$911$	1.48438e7	0.592585	0.296292	+	0.955097i	$0.404250\pi$
$912$	0	0
$913$	4.91792e7i	1.95256i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.34365e7i	− 1.31310i
$918$	0	0
$919$	−448602.	−0.0175215	−0.00876077	−	0.999962i	$-0.502789\pi$
$919$	−448602.	−0.0175215	−0.00876077	+	0.999962i	$0.502789\pi$
$920$	0	0
$921$	4.69071e7	1.82217
$922$	0	0
$923$	− 9.57129e6i	− 0.369799i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.38206e7i	0.528235i
$928$	0	0
$929$	−2.11976e6	−0.0805836	−0.0402918	−	0.999188i	$-0.512829\pi$
$929$	−2.11976e6	−0.0805836	−0.0402918	+	0.999188i	$0.512829\pi$
$930$	0	0
$931$	147732.	0.00558598
$932$	0	0
$933$	3.44647e7i	1.29619i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.54945e6i	0.206491i	0.994656	+	0.103245i	$0.0329227\pi$
$937$	5.54945e6i	0.206491i	−0.994656	+	0.103245i	$0.967077\pi$
$938$	0	0
$939$	−4.01680e7	−1.48667
$940$	0	0
$941$	6.00433e6	0.221050	0.110525	−	0.993873i	$-0.464747\pi$
$941$	6.00433e6	0.221050	0.110525	+	0.993873i	$0.464747\pi$
$942$	0	0
$943$	1.81512e7i	0.664701i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.19597e7i	0.433357i	0.976243	+	0.216679i	$0.0695224\pi$
$947$	1.19597e7i	0.433357i	−0.976243	+	0.216679i	$0.930478\pi$
$948$	0	0
$949$	4.39496e7	1.58413
$950$	0	0
$951$	−2.74156e7	−0.982982
$952$	0	0
$953$	2.84885e7i	1.01610i	0.861327	+	0.508051i	$0.169634\pi$
$953$	2.84885e7i	1.01610i	−0.861327	+	0.508051i	$0.830366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.11901e7i	1.80678i
$958$	0	0
$959$	−3.76891e7	−1.32333
$960$	0	0
$961$	−2.13858e7	−0.746993
$962$	0	0
$963$	1.39409e6i	0.0484422i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.44173e6i	− 0.0839714i	−0.999118	−	0.0419857i	$-0.986632\pi$
$967$	− 2.44173e6i	− 0.0839714i	0.999118	−	0.0419857i	$-0.0133684\pi$
$968$	0	0
$969$	−2.15048e6	−0.0735744
$970$	0	0
$971$	−2.63888e7	−0.898198	−0.449099	−	0.893482i	$-0.648255\pi$
$971$	−2.63888e7	−0.898198	−0.449099	+	0.893482i	$0.648255\pi$
$972$	0	0
$973$	4.64898e7i	1.57426i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.67324e6i	− 0.223666i	−0.993727	−	0.111833i	$-0.964328\pi$
$977$	− 6.67324e6i	− 0.223666i	0.993727	−	0.111833i	$-0.0356722\pi$
$978$	0	0
$979$	−2.93516e7	−0.978756
$980$	0	0
$981$	969506.	0.0321646
$982$	0	0
$983$	− 1.21646e7i	− 0.401528i	−0.979640	−	0.200764i	$-0.935658\pi$
$983$	− 1.21646e7i	− 0.401528i	0.979640	−	0.200764i	$-0.0643424\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.56253e7i	0.837289i
$988$	0	0
$989$	−9.36057e6	−0.304307
$990$	0	0
$991$	1.16204e7	0.375869	0.187935	−	0.982182i	$-0.439821\pi$
$991$	1.16204e7	0.375869	0.187935	+	0.982182i	$0.439821\pi$
$992$	0	0
$993$	3.74750e7i	1.20606i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.61789e7i	− 1.47132i	−0.677354	−	0.735658i	$-0.736873\pi$
$997$	− 4.61789e7i	− 1.47132i	0.677354	−	0.735658i	$-0.263127\pi$
$998$	0	0
$999$	−1.93977e7	−0.614944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.k.449.2		6
4.3	odd	2		800.6.c.j.449.5		6
5.2	odd	4		800.6.a.o.1.1		3
5.3	odd	4		160.6.a.f.1.3	✓	3
5.4	even	2	inner	800.6.c.k.449.5		6
20.3	even	4		160.6.a.g.1.1	yes	3
20.7	even	4		800.6.a.n.1.3		3
20.19	odd	2		800.6.c.j.449.2		6
40.3	even	4		320.6.a.x.1.3		3
40.13	odd	4		320.6.a.y.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.f.1.3	✓	3	5.3	odd	4
160.6.a.g.1.1	yes	3	20.3	even	4
320.6.a.x.1.3		3	40.3	even	4
320.6.a.y.1.1		3	40.13	odd	4
800.6.a.n.1.3		3	20.7	even	4
800.6.a.o.1.1		3	5.2	odd	4
800.6.c.j.449.2		6	20.19	odd	2
800.6.c.j.449.5		6	4.3	odd	2
800.6.c.k.449.2		6	1.1	even	1	trivial
800.6.c.k.449.5		6	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-18.4715i q^{3} -121.899i q^{7} -98.1968 q^{9} +O(q^{10})\)	\(q-18.4715i q^{3} -121.899i q^{7} -98.1968 q^{9} +438.282 q^{11} +758.285i q^{13} -1534.86i q^{17} +75.8518 q^{19} -2251.66 q^{21} +3694.27i q^{23} -2674.73i q^{27} -6323.09 q^{29} -2691.35 q^{31} -8095.74i q^{33} -7252.19i q^{37} +14006.7 q^{39} +4913.34 q^{41} +2533.81i q^{43} -11380.6i q^{47} +1947.64 q^{49} -28351.1 q^{51} -29442.3i q^{53} -1401.10i q^{57} -5681.08 q^{59} -48039.8 q^{61} +11970.1i q^{63} -39570.6i q^{67} +68238.7 q^{69} -12622.3 q^{71} -57959.2i q^{73} -53426.2i q^{77} -29504.0 q^{79} -73268.2 q^{81} +112209. i q^{83} +116797. i q^{87} -66969.5 q^{89} +92434.2 q^{91} +49713.4i q^{93} +131833. i q^{97} -43037.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 934 q^{9}+O(q^{10}) \) 6 * q - 934 * q^9	\( 6 q - 934 q^{9} + 792 q^{11} + 6384 q^{19} - 1640 q^{21} - 852 q^{29} - 6552 q^{31} + 42696 q^{39} + 24900 q^{41} + 7698 q^{49} - 142888 q^{51} + 70080 q^{59} - 48276 q^{61} + 54072 q^{69} - 176184 q^{71} - 185904 q^{79} + 302782 q^{81} - 345372 q^{89} + 213624 q^{91} - 702872 q^{99}+O(q^{100}) \) 6 * q - 934 * q^9 + 792 * q^11 + 6384 * q^19 - 1640 * q^21 - 852 * q^29 - 6552 * q^31 + 42696 * q^39 + 24900 * q^41 + 7698 * q^49 - 142888 * q^51 + 70080 * q^59 - 48276 * q^61 + 54072 * q^69 - 176184 * q^71 - 185904 * q^79 + 302782 * q^81 - 345372 * q^89 + 213624 * q^91 - 702872 * q^99

Introduction

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