LMFDB - Embedded newform 800.5.g.i.751.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,5,Mod(751,800)]

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([1, 1, 0]))

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

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

chi := DirichletCharacter("800.751");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	800.g (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:	$82.6959704671$
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} - 10 x^{18} - 20 x^{16} - 2640 x^{14} + 22400 x^{12} + 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 $ x^20 - 10x^18 - 20x^16 - 2640x^14 + 22400x^12 + 652288x^10 + 5734400x^8 - 173015040x^6 - 335544320x^4 - 42949672960*x^2 + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{71}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.15
Root			$-3.96917 + 0.495691i$ of defining polynomial
Character	$\chi$	$=$	800.751
Dual form			800.5.g.i.751.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.2033 q^{3} -84.8600i q^{7} +23.1073 q^{9} -71.3055 q^{11} +109.279i q^{13} +151.344 q^{17} -368.470 q^{19} -865.852i q^{21} +358.031i q^{23} -590.697 q^{27} +387.610i q^{29} -1687.73i q^{31} -727.551 q^{33} -1158.77i q^{37} +1115.01i q^{39} -2545.99 q^{41} +1354.04 q^{43} -901.424i q^{47} -4800.23 q^{49} +1544.21 q^{51} +3409.47i q^{53} -3759.61 q^{57} -1454.37 q^{59} +5324.84i q^{61} -1960.88i q^{63} +657.048 q^{67} +3653.09i q^{69} +6741.79i q^{71} -4130.26 q^{73} +6050.98i q^{77} -2307.14i q^{79} -7898.74 q^{81} -2146.14 q^{83} +3954.90i q^{87} -5117.03 q^{89} +9273.44 q^{91} -17220.4i q^{93} -9169.66 q^{97} -1647.68 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 652 q^{9} + 168 q^{11} + 728 q^{19} - 4728 q^{41} - 540 q^{49} - 4352 q^{51} + 8280 q^{59} - 10956 q^{81} - 22248 q^{89} + 39760 q^{91} - 62504 q^{99}+O(q^{100}) $ 20 * q + 652 * q^9 + 168 * q^11 + 728 * q^19 - 4728 * q^41 - 540 * q^49 - 4352 * q^51 + 8280 * q^59 - 10956 * q^81 - 22248 * q^89 + 39760 * q^91 - 62504 * q^99

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
$3$	10.2033	1.13370	0.566850	−	0.823821i	$-0.308162\pi$
$3$	10.2033	1.13370	0.566850	+	0.823821i	$0.308162\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 84.8600i	− 1.73184i	−0.500185	−	0.865919i	$-0.666735\pi$
$7$	− 84.8600i	− 1.73184i	0.500185	−	0.865919i	$-0.333265\pi$
$8$	0	0
$9$	23.1073	0.285275
$10$	0	0
$11$	−71.3055	−0.589301	−0.294651	−	0.955605i	$-0.595203\pi$
$11$	−71.3055	−0.589301	−0.294651	+	0.955605i	$0.595203\pi$
$12$	0	0
$13$	109.279i	0.646623i	0.946293	+	0.323311i	$0.104796\pi$
$13$	109.279i	0.646623i	−0.946293	+	0.323311i	$0.895204\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	151.344	0.523681	0.261840	−	0.965111i	$-0.415671\pi$
$17$	151.344	0.523681	0.261840	+	0.965111i	$0.415671\pi$
$18$	0	0
$19$	−368.470	−1.02069	−0.510346	−	0.859969i	$-0.670483\pi$
$19$	−368.470	−1.02069	−0.510346	+	0.859969i	$0.670483\pi$
$20$	0	0
$21$	− 865.852i	− 1.96338i
$22$	0	0
$23$	358.031i	0.676807i	0.941001	+	0.338403i	$0.109887\pi$
$23$	358.031i	0.676807i	−0.941001	+	0.338403i	$0.890113\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−590.697	−0.810283
$28$	0	0
$29$	387.610i	0.460892i	0.973085	+	0.230446i	$0.0740185\pi$
$29$	387.610i	0.460892i	−0.973085	+	0.230446i	$0.925982\pi$
$30$	0	0
$31$	− 1687.73i	− 1.75622i	−0.478457	−	0.878111i	$-0.658804\pi$
$31$	− 1687.73i	− 1.75622i	0.478457	−	0.878111i	$-0.341196\pi$
$32$	0	0
$33$	−727.551	−0.668091
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1158.77i	− 0.846434i	−0.906028	−	0.423217i	$-0.860901\pi$
$37$	− 1158.77i	− 0.846434i	0.906028	−	0.423217i	$-0.139099\pi$
$38$	0	0
$39$	1115.01i	0.733076i
$40$	0	0
$41$	−2545.99	−1.51457	−0.757285	−	0.653084i	$-0.773475\pi$
$41$	−2545.99	−1.51457	−0.757285	+	0.653084i	$0.773475\pi$
$42$	0	0
$43$	1354.04	0.732308	0.366154	−	0.930554i	$-0.380674\pi$
$43$	1354.04	0.732308	0.366154	+	0.930554i	$0.380674\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 901.424i	− 0.408069i	−0.978964	−	0.204034i	$-0.934595\pi$
$47$	− 901.424i	− 0.408069i	0.978964	−	0.204034i	$-0.0654054\pi$
$48$	0	0
$49$	−4800.23	−1.99926
$50$	0	0
$51$	1544.21	0.593697
$52$	0	0
$53$	3409.47i	1.21377i	0.794791	+	0.606883i	$0.207580\pi$
$53$	3409.47i	1.21377i	−0.794791	+	0.606883i	$0.792420\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3759.61	−1.15716
$58$	0	0
$59$	−1454.37	−0.417803	−0.208902	−	0.977937i	$-0.566989\pi$
$59$	−1454.37	−0.417803	−0.208902	+	0.977937i	$0.566989\pi$
$60$	0	0
$61$	5324.84i	1.43102i	0.698601	+	0.715512i	$0.253806\pi$
$61$	5324.84i	1.43102i	−0.698601	+	0.715512i	$0.746194\pi$
$62$	0	0
$63$	− 1960.88i	− 0.494050i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	657.048	0.146368	0.0731842	−	0.997318i	$-0.476684\pi$
$67$	657.048	0.146368	0.0731842	+	0.997318i	$0.476684\pi$
$68$	0	0
$69$	3653.09i	0.767295i
$70$	0	0
$71$	6741.79i	1.33739i	0.743536	+	0.668696i	$0.233147\pi$
$71$	6741.79i	1.33739i	−0.743536	+	0.668696i	$0.766853\pi$
$72$	0	0
$73$	−4130.26	−0.775054	−0.387527	−	0.921858i	$-0.626671\pi$
$73$	−4130.26	−0.775054	−0.387527	+	0.921858i	$0.626671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6050.98i	1.02057i
$78$	0	0
$79$	− 2307.14i	− 0.369675i	−0.982769	−	0.184837i	$-0.940824\pi$
$79$	− 2307.14i	− 0.369675i	0.982769	−	0.184837i	$-0.0591758\pi$
$80$	0	0
$81$	−7898.74	−1.20389
$82$	0	0
$83$	−2146.14	−0.311532	−0.155766	−	0.987794i	$-0.549785\pi$
$83$	−2146.14	−0.311532	−0.155766	+	0.987794i	$0.549785\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3954.90i	0.522513i
$88$	0	0
$89$	−5117.03	−0.646008	−0.323004	−	0.946398i	$-0.604693\pi$
$89$	−5117.03	−0.646008	−0.323004	+	0.946398i	$0.604693\pi$
$90$	0	0
$91$	9273.44	1.11985
$92$	0	0
$93$	− 17220.4i	− 1.99103i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9169.66	−0.974563	−0.487281	−	0.873245i	$-0.662011\pi$
$97$	−9169.66	−0.974563	−0.487281	+	0.873245i	$0.662011\pi$
$98$	0	0
$99$	−1647.68	−0.168113
$100$	0	0
$101$	536.675i	0.0526100i	0.999654	+	0.0263050i	$0.00837411\pi$
$101$	536.675i	0.0526100i	−0.999654	+	0.0263050i	$0.991626\pi$
$102$	0	0
$103$	3140.33i	0.296007i	0.988987	+	0.148003i	$0.0472846\pi$
$103$	3140.33i	0.296007i	−0.988987	+	0.148003i	$0.952715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16703.3	−1.45893	−0.729466	−	0.684017i	$-0.760231\pi$
$107$	−16703.3	−1.45893	−0.729466	+	0.684017i	$0.760231\pi$
$108$	0	0
$109$	− 15790.1i	− 1.32902i	−0.747279	−	0.664510i	$-0.768640\pi$
$109$	− 15790.1i	− 1.32902i	0.747279	−	0.664510i	$-0.231360\pi$
$110$	0	0
$111$	− 11823.3i	− 0.959602i
$112$	0	0
$113$	4460.76	0.349343	0.174672	−	0.984627i	$-0.444114\pi$
$113$	4460.76	0.349343	0.174672	+	0.984627i	$0.444114\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2525.15i	0.184465i
$118$	0	0
$119$	− 12843.0i	− 0.906930i
$120$	0	0
$121$	−9556.53	−0.652724
$122$	0	0
$123$	−25977.5	−1.71707
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10888.2i	0.675070i	0.941313	+	0.337535i	$0.109593\pi$
$127$	10888.2i	0.675070i	−0.941313	+	0.337535i	$0.890407\pi$
$128$	0	0
$129$	13815.7	0.830218
$130$	0	0
$131$	9105.25	0.530578	0.265289	−	0.964169i	$-0.414533\pi$
$131$	9105.25	0.530578	0.265289	+	0.964169i	$0.414533\pi$
$132$	0	0
$133$	31268.4i	1.76767i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14976.2	−0.797921	−0.398960	−	0.916968i	$-0.630629\pi$
$137$	−14976.2	−0.797921	−0.398960	+	0.916968i	$0.630629\pi$
$138$	0	0
$139$	25598.6	1.32491	0.662456	−	0.749101i	$-0.269514\pi$
$139$	25598.6	1.32491	0.662456	+	0.749101i	$0.269514\pi$
$140$	0	0
$141$	− 9197.49i	− 0.462627i
$142$	0	0
$143$	− 7792.21i	− 0.381056i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−48978.1	−2.26656
$148$	0	0
$149$	10269.8i	0.462585i	0.972884	+	0.231292i	$0.0742954\pi$
$149$	10269.8i	0.462585i	−0.972884	+	0.231292i	$0.925705\pi$
$150$	0	0
$151$	− 14739.4i	− 0.646435i	−0.946325	−	0.323217i	$-0.895236\pi$
$151$	− 14739.4i	− 0.646435i	0.946325	−	0.323217i	$-0.104764\pi$
$152$	0	0
$153$	3497.14	0.149393
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 27740.2i	− 1.12541i	−0.826658	−	0.562705i	$-0.809761\pi$
$157$	− 27740.2i	− 1.12541i	0.826658	−	0.562705i	$-0.190239\pi$
$158$	0	0
$159$	34787.8i	1.37605i
$160$	0	0
$161$	30382.5	1.17212
$162$	0	0
$163$	−3732.97	−0.140501	−0.0702505	−	0.997529i	$-0.522380\pi$
$163$	−3732.97	−0.140501	−0.0702505	+	0.997529i	$0.522380\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20717.4i	− 0.742853i	−0.928462	−	0.371427i	$-0.878869\pi$
$167$	− 20717.4i	− 0.742853i	0.928462	−	0.371427i	$-0.121131\pi$
$168$	0	0
$169$	16619.0	0.581879
$170$	0	0
$171$	−8514.34	−0.291178
$172$	0	0
$173$	− 35335.7i	− 1.18065i	−0.807165	−	0.590326i	$-0.798999\pi$
$173$	− 35335.7i	− 1.18065i	0.807165	−	0.590326i	$-0.201001\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14839.4	−0.473664
$178$	0	0
$179$	23669.8	0.738734	0.369367	−	0.929284i	$-0.379575\pi$
$179$	23669.8	0.738734	0.369367	+	0.929284i	$0.379575\pi$
$180$	0	0
$181$	− 7525.59i	− 0.229712i	−0.993382	−	0.114856i	$-0.963359\pi$
$181$	− 7525.59i	− 0.229712i	0.993382	−	0.114856i	$-0.0366406\pi$
$182$	0	0
$183$	54330.9i	1.62235i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10791.6	−0.308606
$188$	0	0
$189$	50126.5i	1.40328i
$190$	0	0
$191$	28925.8i	0.792901i	0.918056	+	0.396450i	$0.129758\pi$
$191$	28925.8i	0.792901i	−0.918056	+	0.396450i	$0.870242\pi$
$192$	0	0
$193$	−48944.4	−1.31398	−0.656990	−	0.753900i	$-0.728171\pi$
$193$	−48944.4	−1.31398	−0.656990	+	0.753900i	$0.728171\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	47883.5i	1.23382i	0.787032	+	0.616912i	$0.211617\pi$
$197$	47883.5i	1.23382i	−0.787032	+	0.616912i	$0.788383\pi$
$198$	0	0
$199$	− 29644.2i	− 0.748573i	−0.927313	−	0.374286i	$-0.877888\pi$
$199$	− 29644.2i	− 0.748573i	0.927313	−	0.374286i	$-0.122112\pi$
$200$	0	0
$201$	6704.06	0.165938
$202$	0	0
$203$	32892.6	0.798190
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8273.12i	0.193076i
$208$	0	0
$209$	26273.9	0.601496
$210$	0	0
$211$	−19653.5	−0.441444	−0.220722	−	0.975337i	$-0.570841\pi$
$211$	−19653.5	−0.441444	−0.220722	+	0.975337i	$0.570841\pi$
$212$	0	0
$213$	68788.5i	1.51620i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−143221.	−3.04149
$218$	0	0
$219$	−42142.3	−0.878678
$220$	0	0
$221$	16538.7i	0.338624i
$222$	0	0
$223$	− 60977.7i	− 1.22620i	−0.790005	−	0.613100i	$-0.789922\pi$
$223$	− 60977.7i	− 1.22620i	0.790005	−	0.613100i	$-0.210078\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2698.14	−0.0523616	−0.0261808	−	0.999657i	$-0.508335\pi$
$227$	−2698.14	−0.0523616	−0.0261808	+	0.999657i	$0.508335\pi$
$228$	0	0
$229$	− 68901.3i	− 1.31388i	−0.753942	−	0.656941i	$-0.771850\pi$
$229$	− 68901.3i	− 1.31388i	0.753942	−	0.656941i	$-0.228150\pi$
$230$	0	0
$231$	61740.0i	1.15702i
$232$	0	0
$233$	−83770.9	−1.54305	−0.771527	−	0.636196i	$-0.780507\pi$
$233$	−83770.9	−1.54305	−0.771527	+	0.636196i	$0.780507\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 23540.4i	− 0.419100i
$238$	0	0
$239$	36602.8i	0.640795i	0.947283	+	0.320397i	$0.103816\pi$
$239$	36602.8i	0.640795i	−0.947283	+	0.320397i	$0.896184\pi$
$240$	0	0
$241$	48755.9	0.839447	0.419724	−	0.907652i	$-0.362127\pi$
$241$	48755.9	0.839447	0.419724	+	0.907652i	$0.362127\pi$
$242$	0	0
$243$	−32746.8	−0.554570
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 40266.1i	− 0.660003i
$248$	0	0
$249$	−21897.8	−0.353184
$250$	0	0
$251$	−75319.3	−1.19553	−0.597763	−	0.801673i	$-0.703944\pi$
$251$	−75319.3	−1.19553	−0.597763	+	0.801673i	$0.703944\pi$
$252$	0	0
$253$	− 25529.5i	− 0.398843i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27190.5	0.411672	0.205836	−	0.978587i	$-0.434009\pi$
$257$	27190.5	0.411672	0.205836	+	0.978587i	$0.434009\pi$
$258$	0	0
$259$	−98333.1	−1.46589
$260$	0	0
$261$	8956.62i	0.131481i
$262$	0	0
$263$	− 48135.0i	− 0.695905i	−0.937512	−	0.347952i	$-0.886877\pi$
$263$	− 48135.0i	− 0.695905i	0.937512	−	0.347952i	$-0.113123\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−52210.6	−0.732379
$268$	0	0
$269$	− 57856.4i	− 0.799553i	−0.916613	−	0.399776i	$-0.869088\pi$
$269$	− 57856.4i	− 0.799553i	0.916613	−	0.399776i	$-0.130912\pi$
$270$	0	0
$271$	− 135141.i	− 1.84013i	−0.391765	−	0.920065i	$-0.628136\pi$
$271$	− 135141.i	− 1.84013i	0.391765	−	0.920065i	$-0.371864\pi$
$272$	0	0
$273$	94619.7	1.26957
$274$	0	0
$275$	0	0
$276$	0	0
$277$	66626.3i	0.868333i	0.900833	+	0.434166i	$0.142957\pi$
$277$	66626.3i	0.868333i	−0.900833	+	0.434166i	$0.857043\pi$
$278$	0	0
$279$	− 38998.8i	− 0.501006i
$280$	0	0
$281$	70518.3	0.893078	0.446539	−	0.894764i	$-0.352656\pi$
$281$	70518.3	0.893078	0.446539	+	0.894764i	$0.352656\pi$
$282$	0	0
$283$	5580.69	0.0696811	0.0348406	−	0.999393i	$-0.488908\pi$
$283$	5580.69	0.0696811	0.0348406	+	0.999393i	$0.488908\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	216053.i	2.62299i
$288$	0	0
$289$	−60616.1	−0.725758
$290$	0	0
$291$	−93560.8	−1.10486
$292$	0	0
$293$	− 85817.1i	− 0.999628i	−0.866133	−	0.499814i	$-0.833402\pi$
$293$	− 85817.1i	− 0.999628i	0.866133	−	0.499814i	$-0.166598\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	42119.9	0.477501
$298$	0	0
$299$	−39125.3	−0.437639
$300$	0	0
$301$	− 114904.i	− 1.26824i
$302$	0	0
$303$	5475.85i	0.0596440i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24380.3	0.258680	0.129340	−	0.991600i	$-0.458714\pi$
$307$	24380.3	0.258680	0.129340	+	0.991600i	$0.458714\pi$
$308$	0	0
$309$	32041.8i	0.335583i
$310$	0	0
$311$	145304.i	1.50230i	0.660133	+	0.751149i	$0.270500\pi$
$311$	145304.i	1.50230i	−0.660133	+	0.751149i	$0.729500\pi$
$312$	0	0
$313$	−15.3803	−0.000156991 0	−7.84956e−5	−	1.00000i	$-0.500025\pi$
$313$	−15.3803	−0.000156991 0	−7.84956e−5		1.00000i	$0.500025\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 19873.1i	− 0.197764i	−0.995099	−	0.0988819i	$-0.968473\pi$
$317$	− 19873.1i	− 0.197764i	0.995099	−	0.0988819i	$-0.0315266\pi$
$318$	0	0
$319$	− 27638.7i	− 0.271604i
$320$	0	0
$321$	−170429.	−1.65399
$322$	0	0
$323$	−55765.7	−0.534517
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 161111.i	− 1.50671i
$328$	0	0
$329$	−76494.8	−0.706708
$330$	0	0
$331$	55183.9	0.503682	0.251841	−	0.967769i	$-0.418964\pi$
$331$	55183.9	0.503682	0.251841	+	0.967769i	$0.418964\pi$
$332$	0	0
$333$	− 26776.0i	− 0.241466i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	172509.	1.51898	0.759489	−	0.650520i	$-0.225449\pi$
$337$	172509.	1.51898	0.759489	+	0.650520i	$0.225449\pi$
$338$	0	0
$339$	45514.5	0.396050
$340$	0	0
$341$	120344.i	1.03494i
$342$	0	0
$343$	203598.i	1.73056i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−53756.2	−0.446447	−0.223223	−	0.974767i	$-0.571658\pi$
$347$	−53756.2	−0.446447	−0.223223	+	0.974767i	$0.571658\pi$
$348$	0	0
$349$	101720.i	0.835136i	0.908646	+	0.417568i	$0.137117\pi$
$349$	101720.i	0.835136i	−0.908646	+	0.417568i	$0.862883\pi$
$350$	0	0
$351$	− 64550.9i	− 0.523948i
$352$	0	0
$353$	119472.	0.958778	0.479389	−	0.877603i	$-0.340858\pi$
$353$	119472.	0.958778	0.479389	+	0.877603i	$0.340858\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 131041.i	− 1.02819i
$358$	0	0
$359$	− 66348.7i	− 0.514806i	−0.966304	−	0.257403i	$-0.917133\pi$
$359$	− 66348.7i	− 0.514806i	0.966304	−	0.257403i	$-0.0828668\pi$
$360$	0	0
$361$	5449.20	0.0418137
$362$	0	0
$363$	−97508.1	−0.739993
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 146492.i	− 1.08763i	−0.839204	−	0.543817i	$-0.816979\pi$
$367$	− 146492.i	− 1.08763i	0.839204	−	0.543817i	$-0.183021\pi$
$368$	0	0
$369$	−58831.0	−0.432069
$370$	0	0
$371$	289328.	2.10205
$372$	0	0
$373$	113125.i	0.813092i	0.913630	+	0.406546i	$0.133267\pi$
$373$	113125.i	0.813092i	−0.913630	+	0.406546i	$0.866733\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−42357.7	−0.298023
$378$	0	0
$379$	43592.1	0.303480	0.151740	−	0.988420i	$-0.451512\pi$
$379$	43592.1	0.303480	0.151740	+	0.988420i	$0.451512\pi$
$380$	0	0
$381$	111096.i	0.765327i
$382$	0	0
$383$	− 99396.2i	− 0.677598i	−0.940859	−	0.338799i	$-0.889979\pi$
$383$	− 99396.2i	− 0.677598i	0.940859	−	0.338799i	$-0.110021\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	31288.2	0.208909
$388$	0	0
$389$	99896.0i	0.660159i	0.943953	+	0.330080i	$0.107076\pi$
$389$	99896.0i	0.660159i	−0.943953	+	0.330080i	$0.892924\pi$
$390$	0	0
$391$	54185.7i	0.354431i
$392$	0	0
$393$	92903.6	0.601516
$394$	0	0
$395$	0	0
$396$	0	0
$397$	55234.5i	0.350453i	0.984528	+	0.175226i	$0.0560657\pi$
$397$	55234.5i	0.350453i	−0.984528	+	0.175226i	$0.943934\pi$
$398$	0	0
$399$	319041.i	2.00401i
$400$	0	0
$401$	−40688.8	−0.253038	−0.126519	−	0.991964i	$-0.540380\pi$
$401$	−40688.8	−0.253038	−0.126519	+	0.991964i	$0.540380\pi$
$402$	0	0
$403$	184434.	1.13561
$404$	0	0
$405$	0	0
$406$	0	0
$407$	82626.5i	0.498804i
$408$	0	0
$409$	313804.	1.87591	0.937955	−	0.346757i	$-0.112717\pi$
$409$	313804.	1.87591	0.937955	+	0.346757i	$0.112717\pi$
$410$	0	0
$411$	−152806.	−0.904603
$412$	0	0
$413$	123418.i	0.723567i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	261190.	1.50205
$418$	0	0
$419$	264287.	1.50538	0.752692	−	0.658373i	$-0.228755\pi$
$419$	264287.	1.50538	0.752692	+	0.658373i	$0.228755\pi$
$420$	0	0
$421$	− 179764.i	− 1.01424i	−0.861877	−	0.507118i	$-0.830711\pi$
$421$	− 179764.i	− 1.01424i	0.861877	−	0.507118i	$-0.169289\pi$
$422$	0	0
$423$	− 20829.4i	− 0.116412i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	451866.	2.47830
$428$	0	0
$429$	− 79506.2i	− 0.432003i
$430$	0	0
$431$	− 151073.i	− 0.813266i	−0.913592	−	0.406633i	$-0.866703\pi$
$431$	− 151073.i	− 0.813266i	0.913592	−	0.406633i	$-0.133297\pi$
$432$	0	0
$433$	−133641.	−0.712796	−0.356398	−	0.934334i	$-0.615995\pi$
$433$	−133641.	−0.712796	−0.356398	+	0.934334i	$0.615995\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 131924.i	− 0.690812i
$438$	0	0
$439$	74613.0i	0.387156i	0.981085	+	0.193578i	$0.0620092\pi$
$439$	74613.0i	0.387156i	−0.981085	+	0.193578i	$0.937991\pi$
$440$	0	0
$441$	−110920.	−0.570339
$442$	0	0
$443$	110554.	0.563334	0.281667	−	0.959512i	$-0.409113\pi$
$443$	110554.	0.563334	0.281667	+	0.959512i	$0.409113\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	104786.i	0.524432i
$448$	0	0
$449$	−9722.73	−0.0482276	−0.0241138	−	0.999709i	$-0.507676\pi$
$449$	−9722.73	−0.0482276	−0.0241138	+	0.999709i	$0.507676\pi$
$450$	0	0
$451$	181543.	0.892539
$452$	0	0
$453$	− 150390.i	− 0.732863i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−331629.	−1.58789	−0.793945	−	0.607990i	$-0.791976\pi$
$457$	−331629.	−1.58789	−0.793945	+	0.607990i	$0.791976\pi$
$458$	0	0
$459$	−89398.3	−0.424330
$460$	0	0
$461$	86774.7i	0.408311i	0.978938	+	0.204156i	$0.0654449\pi$
$461$	86774.7i	0.408311i	−0.978938	+	0.204156i	$0.934555\pi$
$462$	0	0
$463$	− 35664.8i	− 0.166371i	−0.996534	−	0.0831856i	$-0.973491\pi$
$463$	− 35664.8i	− 0.166371i	0.996534	−	0.0831856i	$-0.0265094\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	293718.	1.34678	0.673389	−	0.739288i	$-0.264838\pi$
$467$	293718.	1.34678	0.673389	+	0.739288i	$0.264838\pi$
$468$	0	0
$469$	− 55757.1i	− 0.253486i
$470$	0	0
$471$	− 283042.i	− 1.27588i
$472$	0	0
$473$	−96550.3	−0.431550
$474$	0	0
$475$	0	0
$476$	0	0
$477$	78783.6i	0.346257i
$478$	0	0
$479$	− 435529.i	− 1.89822i	−0.314945	−	0.949110i	$-0.601986\pi$
$479$	− 435529.i	− 1.89822i	0.314945	−	0.949110i	$-0.398014\pi$
$480$	0	0
$481$	126629.	0.547323
$482$	0	0
$483$	310002.	1.32883
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 187286.i	− 0.789672i	−0.918752	−	0.394836i	$-0.870801\pi$
$487$	− 187286.i	− 0.789672i	0.918752	−	0.394836i	$-0.129199\pi$
$488$	0	0
$489$	−38088.6	−0.159286
$490$	0	0
$491$	287674.	1.19327	0.596634	−	0.802513i	$-0.296504\pi$
$491$	287674.	1.19327	0.596634	+	0.802513i	$0.296504\pi$
$492$	0	0
$493$	58662.4i	0.241360i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	572109.	2.31615
$498$	0	0
$499$	175430.	0.704537	0.352268	−	0.935899i	$-0.385410\pi$
$499$	175430.	0.704537	0.352268	+	0.935899i	$0.385410\pi$
$500$	0	0
$501$	− 211386.i	− 0.842172i
$502$	0	0
$503$	− 95682.5i	− 0.378178i	−0.981960	−	0.189089i	$-0.939447\pi$
$503$	− 95682.5i	− 0.378178i	0.981960	−	0.189089i	$-0.0605535\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	169569.	0.659676
$508$	0	0
$509$	− 370388.i	− 1.42962i	−0.699318	−	0.714811i	$-0.746513\pi$
$509$	− 370388.i	− 1.42962i	0.699318	−	0.714811i	$-0.253487\pi$
$510$	0	0
$511$	350494.i	1.34227i
$512$	0	0
$513$	217654.	0.827050
$514$	0	0
$515$	0	0
$516$	0	0
$517$	64276.4i	0.240475i
$518$	0	0
$519$	− 360541.i	− 1.33850i
$520$	0	0
$521$	−259622.	−0.956458	−0.478229	−	0.878235i	$-0.658721\pi$
$521$	−259622.	−0.956458	−0.478229	+	0.878235i	$0.658721\pi$
$522$	0	0
$523$	108562.	0.396895	0.198448	−	0.980111i	$-0.436410\pi$
$523$	108562.	0.396895	0.198448	+	0.980111i	$0.436410\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 255427.i	− 0.919700i
$528$	0	0
$529$	151655.	0.541933
$530$	0	0
$531$	−33606.6	−0.119189
$532$	0	0
$533$	− 278224.i	− 0.979356i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	241510.	0.837502
$538$	0	0
$539$	342282.	1.17817
$540$	0	0
$541$	280006.i	0.956694i	0.878171	+	0.478347i	$0.158764\pi$
$541$	280006.i	0.956694i	−0.878171	+	0.478347i	$0.841236\pi$
$542$	0	0
$543$	− 76785.8i	− 0.260424i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−72377.7	−0.241897	−0.120948	−	0.992659i	$-0.538594\pi$
$547$	−72377.7	−0.241897	−0.120948	+	0.992659i	$0.538594\pi$
$548$	0	0
$549$	123043.i	0.408235i
$550$	0	0
$551$	− 142823.i	− 0.470429i
$552$	0	0
$553$	−195784.	−0.640216
$554$	0	0
$555$	0	0
$556$	0	0
$557$	402974.i	1.29887i	0.760416	+	0.649437i	$0.224995\pi$
$557$	402974.i	1.29887i	−0.760416	+	0.649437i	$0.775005\pi$
$558$	0	0
$559$	147968.i	0.473527i
$560$	0	0
$561$	−110110.	−0.349866
$562$	0	0
$563$	−244892.	−0.772606	−0.386303	−	0.922372i	$-0.626248\pi$
$563$	−244892.	−0.772606	−0.386303	+	0.922372i	$0.626248\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	670288.i	2.08495i
$568$	0	0
$569$	−644997.	−1.99220	−0.996100	−	0.0882331i	$-0.971878\pi$
$569$	−644997.	−1.99220	−0.996100	+	0.0882331i	$0.971878\pi$
$570$	0	0
$571$	−198216.	−0.607947	−0.303973	−	0.952681i	$-0.598313\pi$
$571$	−198216.	−0.607947	−0.303973	+	0.952681i	$0.598313\pi$
$572$	0	0
$573$	295139.i	0.898911i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	218459.	0.656172	0.328086	−	0.944648i	$-0.393596\pi$
$577$	218459.	0.656172	0.328086	+	0.944648i	$0.393596\pi$
$578$	0	0
$579$	−499394.	−1.48966
$580$	0	0
$581$	182122.i	0.539523i
$582$	0	0
$583$	− 243114.i	− 0.715274i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−330673.	−0.959673	−0.479836	−	0.877358i	$-0.659304\pi$
$587$	−330673.	−0.959673	−0.479836	+	0.877358i	$0.659304\pi$
$588$	0	0
$589$	621878.i	1.79256i
$590$	0	0
$591$	488569.i	1.39879i
$592$	0	0
$593$	499960.	1.42176	0.710879	−	0.703314i	$-0.248297\pi$
$593$	499960.	1.42176	0.710879	+	0.703314i	$0.248297\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 302469.i	− 0.848657i
$598$	0	0
$599$	407848.i	1.13670i	0.822788	+	0.568348i	$0.192417\pi$
$599$	407848.i	1.13670i	−0.822788	+	0.568348i	$0.807583\pi$
$600$	0	0
$601$	441217.	1.22153	0.610763	−	0.791813i	$-0.290863\pi$
$601$	441217.	1.22153	0.610763	+	0.791813i	$0.290863\pi$
$602$	0	0
$603$	15182.6	0.0417553
$604$	0	0
$605$	0	0
$606$	0	0
$607$	71921.7i	0.195201i	0.995226	+	0.0976006i	$0.0311168\pi$
$607$	71921.7i	0.195201i	−0.995226	+	0.0976006i	$0.968883\pi$
$608$	0	0
$609$	335613.	0.904908
$610$	0	0
$611$	98506.9	0.263866
$612$	0	0
$613$	− 527274.i	− 1.40319i	−0.712577	−	0.701593i	$-0.752472\pi$
$613$	− 527274.i	− 1.40319i	0.712577	−	0.701593i	$-0.247528\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−91867.0	−0.241318	−0.120659	−	0.992694i	$-0.538501\pi$
$617$	−91867.0	−0.241318	−0.120659	+	0.992694i	$0.538501\pi$
$618$	0	0
$619$	−633381.	−1.65304	−0.826521	−	0.562906i	$-0.809683\pi$
$619$	−633381.	−1.65304	−0.826521	+	0.562906i	$0.809683\pi$
$620$	0	0
$621$	− 211488.i	− 0.548405i
$622$	0	0
$623$	434231.i	1.11878i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	268081.	0.681915
$628$	0	0
$629$	− 175372.i	− 0.443261i
$630$	0	0
$631$	128753.i	0.323368i	0.986843	+	0.161684i	$0.0516926\pi$
$631$	128753.i	0.323368i	−0.986843	+	0.161684i	$0.948307\pi$
$632$	0	0
$633$	−200531.	−0.500465
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 524565.i	− 1.29277i
$638$	0	0
$639$	155785.i	0.381525i
$640$	0	0
$641$	391151.	0.951980	0.475990	−	0.879451i	$-0.342090\pi$
$641$	391151.	0.951980	0.475990	+	0.879451i	$0.342090\pi$
$642$	0	0
$643$	−720581.	−1.74285	−0.871427	−	0.490525i	$-0.836805\pi$
$643$	−720581.	−1.74285	−0.871427	+	0.490525i	$0.836805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 403682.i	− 0.964341i	−0.876077	−	0.482171i	$-0.839849\pi$
$647$	− 403682.i	− 0.964341i	0.876077	−	0.482171i	$-0.160151\pi$
$648$	0	0
$649$	103705.	0.246212
$650$	0	0
$651$	−1.46132e6	−3.44814
$652$	0	0
$653$	404377.i	0.948332i	0.880436	+	0.474166i	$0.157250\pi$
$653$	404377.i	0.948332i	−0.880436	+	0.474166i	$0.842750\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−95439.1	−0.221104
$658$	0	0
$659$	179720.	0.413834	0.206917	−	0.978359i	$-0.433657\pi$
$659$	179720.	0.413834	0.206917	+	0.978359i	$0.433657\pi$
$660$	0	0
$661$	512413.i	1.17278i	0.810028	+	0.586391i	$0.199452\pi$
$661$	512413.i	1.17278i	−0.810028	+	0.586391i	$0.800548\pi$
$662$	0	0
$663$	168750.i	0.383898i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−138776.	−0.311935
$668$	0	0
$669$	− 622174.i	− 1.39014i
$670$	0	0
$671$	− 379690.i	− 0.843304i
$672$	0	0
$673$	136431.	0.301220	0.150610	−	0.988593i	$-0.451876\pi$
$673$	136431.	0.301220	0.150610	+	0.988593i	$0.451876\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 185710.i	− 0.405190i	−0.979263	−	0.202595i	$-0.935063\pi$
$677$	− 185710.i	− 0.405190i	0.979263	−	0.202595i	$-0.0649374\pi$
$678$	0	0
$679$	778138.i	1.68778i
$680$	0	0
$681$	−27529.9	−0.0593623
$682$	0	0
$683$	521505.	1.11794	0.558968	−	0.829189i	$-0.311198\pi$
$683$	521505.	1.11794	0.558968	+	0.829189i	$0.311198\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 703020.i	− 1.48955i
$688$	0	0
$689$	−372584.	−0.784849
$690$	0	0
$691$	159485.	0.334012	0.167006	−	0.985956i	$-0.446590\pi$
$691$	159485.	0.334012	0.167006	+	0.985956i	$0.446590\pi$
$692$	0	0
$693$	139822.i	0.291144i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−385320.	−0.793152
$698$	0	0
$699$	−854739.	−1.74936
$700$	0	0
$701$	− 924458.i	− 1.88127i	−0.339418	−	0.940635i	$-0.610230\pi$
$701$	− 924458.i	− 1.88127i	0.339418	−	0.940635i	$-0.389770\pi$
$702$	0	0
$703$	426971.i	0.863949i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45542.2	0.0911120
$708$	0	0
$709$	49979.9i	0.0994267i	0.998764	+	0.0497134i	$0.0158308\pi$
$709$	49979.9i	0.0994267i	−0.998764	+	0.0497134i	$0.984169\pi$
$710$	0	0
$711$	− 53311.7i	− 0.105459i
$712$	0	0
$713$	604259.	1.18862
$714$	0	0
$715$	0	0
$716$	0	0
$717$	373470.i	0.726469i
$718$	0	0
$719$	− 823502.i	− 1.59297i	−0.604659	−	0.796484i	$-0.706691\pi$
$719$	− 823502.i	− 1.59297i	0.604659	−	0.796484i	$-0.293309\pi$
$720$	0	0
$721$	266489.	0.512635
$722$	0	0
$723$	497471.	0.951681
$724$	0	0
$725$	0	0
$726$	0	0
$727$	791865.i	1.49824i	0.662432	+	0.749122i	$0.269524\pi$
$727$	791865.i	1.49824i	−0.662432	+	0.749122i	$0.730476\pi$
$728$	0	0
$729$	305673.	0.575177
$730$	0	0
$731$	204925.	0.383496
$732$	0	0
$733$	586208.i	1.09105i	0.838095	+	0.545524i	$0.183669\pi$
$733$	586208.i	1.09105i	−0.838095	+	0.545524i	$0.816331\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−46851.1	−0.0862551
$738$	0	0
$739$	51496.3	0.0942947	0.0471474	−	0.998888i	$-0.484987\pi$
$739$	51496.3	0.0942947	0.0471474	+	0.998888i	$0.484987\pi$
$740$	0	0
$741$	− 410847.i	− 0.748245i
$742$	0	0
$743$	− 333510.i	− 0.604132i	−0.953287	−	0.302066i	$-0.902324\pi$
$743$	− 333510.i	− 0.604132i	0.953287	−	0.302066i	$-0.0976763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−49591.6	−0.0888724
$748$	0	0
$749$	1.41744e6i	2.52663i
$750$	0	0
$751$	− 315513.i	− 0.559419i	−0.960085	−	0.279710i	$-0.909762\pi$
$751$	− 315513.i	− 0.559419i	0.960085	−	0.279710i	$-0.0902382\pi$
$752$	0	0
$753$	−768505.	−1.35537
$754$	0	0
$755$	0	0
$756$	0	0
$757$	559184.i	0.975806i	0.872898	+	0.487903i	$0.162238\pi$
$757$	559184.i	0.975806i	−0.872898	+	0.487903i	$0.837762\pi$
$758$	0	0
$759$	− 260486.i	− 0.452168i
$760$	0	0
$761$	−432702.	−0.747170	−0.373585	−	0.927596i	$-0.621872\pi$
$761$	−432702.	−0.747170	−0.373585	+	0.927596i	$0.621872\pi$
$762$	0	0
$763$	−1.33995e6	−2.30165
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 158933.i	− 0.270161i
$768$	0	0
$769$	−258310.	−0.436805	−0.218403	−	0.975859i	$-0.570085\pi$
$769$	−258310.	−0.436805	−0.218403	+	0.975859i	$0.570085\pi$
$770$	0	0
$771$	277433.	0.466712
$772$	0	0
$773$	− 280428.i	− 0.469312i	−0.972078	−	0.234656i	$-0.924604\pi$
$773$	− 280428.i	− 0.469312i	0.972078	−	0.234656i	$-0.0753964\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.00332e6	−1.66187
$778$	0	0
$779$	938122.	1.54591
$780$	0	0
$781$	− 480727.i	− 0.788127i
$782$	0	0
$783$	− 228960.i	− 0.373453i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.17964e6	−1.90459	−0.952296	−	0.305177i	$-0.901284\pi$
$787$	−1.17964e6	−1.90459	−0.952296	+	0.305177i	$0.901284\pi$
$788$	0	0
$789$	− 491136.i	− 0.788947i
$790$	0	0
$791$	− 378540.i	− 0.605005i
$792$	0	0
$793$	−581894.	−0.925332
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 627511.i	− 0.987881i	−0.869496	−	0.493940i	$-0.835556\pi$
$797$	− 627511.i	− 0.987881i	0.869496	−	0.493940i	$-0.164444\pi$
$798$	0	0
$799$	− 136425.i	− 0.213698i
$800$	0	0
$801$	−118241.	−0.184290
$802$	0	0
$803$	294510.	0.456740
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 590326.i	− 0.906453i
$808$	0	0
$809$	328656.	0.502163	0.251081	−	0.967966i	$-0.419214\pi$
$809$	328656.	0.502163	0.251081	+	0.967966i	$0.419214\pi$
$810$	0	0
$811$	−510808.	−0.776634	−0.388317	−	0.921526i	$-0.626943\pi$
$811$	−510808.	−0.776634	−0.388317	+	0.921526i	$0.626943\pi$
$812$	0	0
$813$	− 1.37888e6i	− 2.08616i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−498923.	−0.747462
$818$	0	0
$819$	214284.	0.319464
$820$	0	0
$821$	− 37920.6i	− 0.0562586i	−0.999604	−	0.0281293i	$-0.991045\pi$
$821$	− 37920.6i	− 0.0562586i	0.999604	−	0.0281293i	$-0.00895502\pi$
$822$	0	0
$823$	− 449975.i	− 0.664337i	−0.943220	−	0.332168i	$-0.892220\pi$
$823$	− 449975.i	− 0.664337i	0.943220	−	0.332168i	$-0.107780\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41920.8	−0.0612940	−0.0306470	−	0.999530i	$-0.509757\pi$
$827$	−41920.8	−0.0612940	−0.0306470	+	0.999530i	$0.509757\pi$
$828$	0	0
$829$	− 1.28858e6i	− 1.87500i	−0.347985	−	0.937500i	$-0.613134\pi$
$829$	− 1.28858e6i	− 1.87500i	0.347985	−	0.937500i	$-0.386866\pi$
$830$	0	0
$831$	679808.i	0.984429i
$832$	0	0
$833$	−726484.	−1.04697
$834$	0	0
$835$	0	0
$836$	0	0
$837$	996936.i	1.42304i
$838$	0	0
$839$	506198.i	0.719112i	0.933124	+	0.359556i	$0.117072\pi$
$839$	506198.i	0.719112i	−0.933124	+	0.359556i	$0.882928\pi$
$840$	0	0
$841$	557039.	0.787579
$842$	0	0
$843$	719519.	1.01248
$844$	0	0
$845$	0	0
$846$	0	0
$847$	810968.i	1.13041i
$848$	0	0
$849$	56941.5	0.0789975
$850$	0	0
$851$	414874.	0.572872
$852$	0	0
$853$	656590.i	0.902395i	0.892424	+	0.451197i	$0.149003\pi$
$853$	656590.i	0.902395i	−0.892424	+	0.451197i	$0.850997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−388067.	−0.528379	−0.264189	−	0.964471i	$-0.585104\pi$
$857$	−388067.	−0.528379	−0.264189	+	0.964471i	$0.585104\pi$
$858$	0	0
$859$	282145.	0.382372	0.191186	−	0.981554i	$-0.438767\pi$
$859$	282145.	0.382372	0.191186	+	0.981554i	$0.438767\pi$
$860$	0	0
$861$	2.20445e6i	2.97368i
$862$	0	0
$863$	1.01110e6i	1.35761i	0.734320	+	0.678804i	$0.237501\pi$
$863$	1.01110e6i	1.35761i	−0.734320	+	0.678804i	$0.762499\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−618484.	−0.822792
$868$	0	0
$869$	164512.i	0.217850i
$870$	0	0
$871$	71801.7i	0.0946452i
$872$	0	0
$873$	−211886.	−0.278018
$874$	0	0
$875$	0	0
$876$	0	0
$877$	190192.i	0.247283i	0.992327	+	0.123641i	$0.0394572\pi$
$877$	190192.i	0.247283i	−0.992327	+	0.123641i	$0.960543\pi$
$878$	0	0
$879$	− 875617.i	− 1.13328i
$880$	0	0
$881$	237801.	0.306380	0.153190	−	0.988197i	$-0.451045\pi$
$881$	237801.	0.306380	0.153190	+	0.988197i	$0.451045\pi$
$882$	0	0
$883$	−175861.	−0.225552	−0.112776	−	0.993620i	$-0.535974\pi$
$883$	−175861.	−0.225552	−0.112776	+	0.993620i	$0.535974\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.36691e6i	1.73738i	0.495360	+	0.868688i	$0.335036\pi$
$887$	1.36691e6i	1.73738i	−0.495360	+	0.868688i	$0.664964\pi$
$888$	0	0
$889$	923974.	1.16911
$890$	0	0
$891$	563224.	0.709456
$892$	0	0
$893$	332148.i	0.416513i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−399207.	−0.496151
$898$	0	0
$899$	654181.	0.809428
$900$	0	0
$901$	516002.i	0.635626i
$902$	0	0
$903$	− 1.17240e6i	− 1.43780i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−699123.	−0.849843	−0.424922	−	0.905230i	$-0.639698\pi$
$907$	−699123.	−0.849843	−0.424922	+	0.905230i	$0.639698\pi$
$908$	0	0
$909$	12401.1i	0.0150083i
$910$	0	0
$911$	1.27217e6i	1.53288i	0.642315	+	0.766441i	$0.277974\pi$
$911$	1.27217e6i	1.53288i	−0.642315	+	0.766441i	$0.722026\pi$
$912$	0	0
$913$	153032.	0.183586
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 772672.i	− 0.918875i
$918$	0	0
$919$	− 507149.i	− 0.600488i	−0.953862	−	0.300244i	$-0.902932\pi$
$919$	− 507149.i	− 0.600488i	0.953862	−	0.300244i	$-0.0970681\pi$
$920$	0	0
$921$	248760.	0.293265
$922$	0	0
$923$	−736738.	−0.864788
$924$	0	0
$925$	0	0
$926$	0	0
$927$	72564.6i	0.0844433i
$928$	0	0
$929$	31736.2	0.0367725	0.0183862	−	0.999831i	$-0.494147\pi$
$929$	31736.2	0.0367725	0.0183862	+	0.999831i	$0.494147\pi$
$930$	0	0
$931$	1.76874e6	2.04063
$932$	0	0
$933$	1.48258e6i	1.70315i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.23459e6	1.40619	0.703094	−	0.711096i	$-0.251801\pi$
$937$	1.23459e6	1.40619	0.703094	+	0.711096i	$0.251801\pi$
$938$	0	0
$939$	−156.930	−0.000177981 0
$940$	0	0
$941$	− 126651.i	− 0.143031i	−0.997439	−	0.0715156i	$-0.977216\pi$
$941$	− 126651.i	− 0.143031i	0.997439	−	0.0715156i	$-0.0227836\pi$
$942$	0	0
$943$	− 911544.i	− 1.02507i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−213933.	−0.238549	−0.119275	−	0.992861i	$-0.538057\pi$
$947$	−213933.	−0.238549	−0.119275	+	0.992861i	$0.538057\pi$
$948$	0	0
$949$	− 451352.i	− 0.501168i
$950$	0	0
$951$	− 202771.i	− 0.224205i
$952$	0	0
$953$	281174.	0.309592	0.154796	−	0.987946i	$-0.450528\pi$
$953$	281174.	0.309592	0.154796	+	0.987946i	$0.450528\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 282006.i	− 0.307918i
$958$	0	0
$959$	1.27088e6i	1.38187i
$960$	0	0
$961$	−1.92491e6	−2.08431
$962$	0	0
$963$	−385968.	−0.416197
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.15290e6i	− 1.23293i	−0.787381	−	0.616467i	$-0.788563\pi$
$967$	− 1.15290e6i	− 1.23293i	0.787381	−	0.616467i	$-0.211437\pi$
$968$	0	0
$969$	−568994.	−0.605982
$970$	0	0
$971$	−857201.	−0.909169	−0.454584	−	0.890704i	$-0.650212\pi$
$971$	−857201.	−0.909169	−0.454584	+	0.890704i	$0.650212\pi$
$972$	0	0
$973$	− 2.17230e6i	− 2.29453i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.68459e6	−1.76484	−0.882419	−	0.470464i	$-0.844086\pi$
$977$	−1.68459e6	−1.76484	−0.882419	+	0.470464i	$0.844086\pi$
$978$	0	0
$979$	364872.	0.380693
$980$	0	0
$981$	− 364866.i	− 0.379136i
$982$	0	0
$983$	501269.i	0.518756i	0.965776	+	0.259378i	$0.0835176\pi$
$983$	501269.i	0.518756i	−0.965776	+	0.259378i	$0.916482\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−780500.	−0.801195
$988$	0	0
$989$	484787.i	0.495631i
$990$	0	0
$991$	− 650776.i	− 0.662650i	−0.943517	−	0.331325i	$-0.892504\pi$
$991$	− 650776.i	− 0.662650i	0.943517	−	0.331325i	$-0.107496\pi$
$992$	0	0
$993$	563058.	0.571024
$994$	0	0
$995$	0	0
$996$	0	0
$997$	724549.i	0.728916i	0.931220	+	0.364458i	$0.118746\pi$
$997$	724549.i	0.728916i	−0.931220	+	0.364458i	$0.881254\pi$
$998$	0	0
$999$	684480.i	0.685851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.g.i.751.15		20
4.3	odd	2		200.5.g.i.51.1		20
5.2	odd	4		160.5.e.c.79.5		20
5.3	odd	4		160.5.e.c.79.16		20
5.4	even	2	inner	800.5.g.i.751.6		20
8.3	odd	2	inner	800.5.g.i.751.16		20
8.5	even	2		200.5.g.i.51.2		20
20.3	even	4		40.5.e.c.19.10	yes	20
20.7	even	4		40.5.e.c.19.11	yes	20
20.19	odd	2		200.5.g.i.51.20		20
40.3	even	4		160.5.e.c.79.15		20
40.13	odd	4		40.5.e.c.19.12	yes	20
40.19	odd	2	inner	800.5.g.i.751.5		20
40.27	even	4		160.5.e.c.79.6		20
40.29	even	2		200.5.g.i.51.19		20
40.37	odd	4		40.5.e.c.19.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.e.c.19.9	✓	20	40.37	odd	4
40.5.e.c.19.10	yes	20	20.3	even	4
40.5.e.c.19.11	yes	20	20.7	even	4
40.5.e.c.19.12	yes	20	40.13	odd	4
160.5.e.c.79.5		20	5.2	odd	4
160.5.e.c.79.6		20	40.27	even	4
160.5.e.c.79.15		20	40.3	even	4
160.5.e.c.79.16		20	5.3	odd	4
200.5.g.i.51.1		20	4.3	odd	2
200.5.g.i.51.2		20	8.5	even	2
200.5.g.i.51.19		20	40.29	even	2
200.5.g.i.51.20		20	20.19	odd	2
800.5.g.i.751.5		20	40.19	odd	2	inner
800.5.g.i.751.6		20	5.4	even	2	inner
800.5.g.i.751.15		20	1.1	even	1	trivial
800.5.g.i.751.16		20	8.3	odd	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.2033 q^{3} -84.8600i q^{7} +23.1073 q^{9} -71.3055 q^{11} +109.279i q^{13} +151.344 q^{17} -368.470 q^{19} -865.852i q^{21} +358.031i q^{23} -590.697 q^{27} +387.610i q^{29} -1687.73i q^{31} -727.551 q^{33} -1158.77i q^{37} +1115.01i q^{39} -2545.99 q^{41} +1354.04 q^{43} -901.424i q^{47} -4800.23 q^{49} +1544.21 q^{51} +3409.47i q^{53} -3759.61 q^{57} -1454.37 q^{59} +5324.84i q^{61} -1960.88i q^{63} +657.048 q^{67} +3653.09i q^{69} +6741.79i q^{71} -4130.26 q^{73} +6050.98i q^{77} -2307.14i q^{79} -7898.74 q^{81} -2146.14 q^{83} +3954.90i q^{87} -5117.03 q^{89} +9273.44 q^{91} -17220.4i q^{93} -9169.66 q^{97} -1647.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 652 q^{9} + 168 q^{11} + 728 q^{19} - 4728 q^{41} - 540 q^{49} - 4352 q^{51} + 8280 q^{59} - 10956 q^{81} - 22248 q^{89} + 39760 q^{91} - 62504 q^{99}+O(q^{100}) \) 20 * q + 652 * q^9 + 168 * q^11 + 728 * q^19 - 4728 * q^41 - 540 * q^49 - 4352 * q^51 + 8280 * q^59 - 10956 * q^81 - 22248 * q^89 + 39760 * q^91 - 62504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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