LMFDB - Embedded newform 800.4.d.d.401.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(401,800)]

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, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.d (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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 $ x^12 - 4x^11 + 7x^10 - 12x^9 + 21x^8 - 68x^6 + 336x^4 - 768x^3 + 1792x^2 - 4096*x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{32}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.12
Root			$1.98839 - 0.215211i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.4.d.d.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.57890i q^{3} +21.5703 q^{7} -64.7554 q^{9} +37.2871i q^{11} +24.1097i q^{13} +14.4940 q^{17} -26.6887i q^{19} +206.620i q^{21} -8.36366 q^{23} -361.655i q^{27} +104.553i q^{29} -204.288 q^{31} -357.170 q^{33} +130.923i q^{37} -230.945 q^{39} -437.963 q^{41} +308.764i q^{43} +97.1081 q^{47} +122.277 q^{49} +138.836i q^{51} +154.502i q^{53} +255.648 q^{57} -544.088i q^{59} +374.601i q^{61} -1396.79 q^{63} +19.3982i q^{67} -80.1147i q^{69} +955.725 q^{71} -127.417 q^{73} +804.293i q^{77} -973.871 q^{79} +1715.87 q^{81} -55.0210i q^{83} -1001.50 q^{87} -919.812 q^{89} +520.054i q^{91} -1956.85i q^{93} -297.986 q^{97} -2414.54i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 28 q^{7} - 108 q^{9} + 604 q^{23} + 264 q^{31} + 232 q^{33} - 600 q^{39} + 40 q^{41} - 940 q^{47} + 1308 q^{49} + 680 q^{57} - 1300 q^{63} + 1592 q^{71} - 432 q^{73} - 2016 q^{79} + 2508 q^{81} - 1968 q^{87}+ \cdots + 1584 q^{97}+O(q^{100}) $ 12 * q + 28 * q^7 - 108 * q^9 + 604 * q^23 + 264 * q^31 + 232 * q^33 - 600 * q^39 + 40 * q^41 - 940 * q^47 + 1308 * q^49 + 680 * q^57 - 1300 * q^63 + 1592 * q^71 - 432 * q^73 - 2016 * q^79 + 2508 * q^81 - 1968 * q^87 - 424 * q^89 + 1584 * q^97

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
$3$	9.57890i	1.84346i	0.387831	+	0.921730i	$0.373224\pi$
$3$	9.57890i	1.84346i	−0.387831	+	0.921730i	$0.626776\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.5703	1.16469	0.582343	−	0.812943i	$-0.302136\pi$
$7$	21.5703	1.16469	0.582343	+	0.812943i	$0.302136\pi$
$8$	0	0
$9$	−64.7554	−2.39835
$10$	0	0
$11$	37.2871i	1.02204i	0.859568	+	0.511022i	$0.170733\pi$
$11$	37.2871i	1.02204i	−0.859568	+	0.511022i	$0.829267\pi$
$12$	0	0
$13$	24.1097i	0.514372i	0.966362	+	0.257186i	$0.0827954\pi$
$13$	24.1097i	0.514372i	−0.966362	+	0.257186i	$0.917205\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.4940	0.206783	0.103391	−	0.994641i	$-0.467031\pi$
$17$	14.4940	0.206783	0.103391	+	0.994641i	$0.467031\pi$
$18$	0	0
$19$	− 26.6887i	− 0.322253i	−0.986934	−	0.161126i	$-0.948487\pi$
$19$	− 26.6887i	− 0.322253i	0.986934	−	0.161126i	$-0.0515127\pi$
$20$	0	0
$21$	206.620i	2.14705i
$22$	0	0
$23$	−8.36366	−0.0758236	−0.0379118	−	0.999281i	$-0.512071\pi$
$23$	−8.36366	−0.0758236	−0.0379118	+	0.999281i	$0.512071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 361.655i	− 2.57780i
$28$	0	0
$29$	104.553i	0.669481i	0.942310	+	0.334741i	$0.108649\pi$
$29$	104.553i	0.669481i	−0.942310	+	0.334741i	$0.891351\pi$
$30$	0	0
$31$	−204.288	−1.18359	−0.591793	−	0.806090i	$-0.701580\pi$
$31$	−204.288	−1.18359	−0.591793	+	0.806090i	$0.701580\pi$
$32$	0	0
$33$	−357.170	−1.88410
$34$	0	0
$35$	0	0
$36$	0	0
$37$	130.923i	0.581720i	0.956766	+	0.290860i	$0.0939414\pi$
$37$	130.923i	0.581720i	−0.956766	+	0.290860i	$0.906059\pi$
$38$	0	0
$39$	−230.945	−0.948225
$40$	0	0
$41$	−437.963	−1.66825	−0.834126	−	0.551574i	$-0.814027\pi$
$41$	−437.963	−1.66825	−0.834126	+	0.551574i	$0.814027\pi$
$42$	0	0
$43$	308.764i	1.09503i	0.836797	+	0.547513i	$0.184425\pi$
$43$	308.764i	1.09503i	−0.836797	+	0.547513i	$0.815575\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	97.1081	0.301376	0.150688	−	0.988581i	$-0.451851\pi$
$47$	97.1081	0.301376	0.150688	+	0.988581i	$0.451851\pi$
$48$	0	0
$49$	122.277	0.356492
$50$	0	0
$51$	138.836i	0.381196i
$52$	0	0
$53$	154.502i	0.400425i	0.979753	+	0.200212i	$0.0641632\pi$
$53$	154.502i	0.400425i	−0.979753	+	0.200212i	$0.935837\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	255.648	0.594060
$58$	0	0
$59$	− 544.088i	− 1.20058i	−0.799782	−	0.600290i	$-0.795052\pi$
$59$	− 544.088i	− 1.20058i	0.799782	−	0.600290i	$-0.204948\pi$
$60$	0	0
$61$	374.601i	0.786275i	0.919480	+	0.393137i	$0.128610\pi$
$61$	374.601i	0.786275i	−0.919480	+	0.393137i	$0.871390\pi$
$62$	0	0
$63$	−1396.79	−2.79332
$64$	0	0
$65$	0	0
$66$	0	0
$67$	19.3982i	0.0353711i	0.999844	+	0.0176855i	$0.00562977\pi$
$67$	19.3982i	0.0353711i	−0.999844	+	0.0176855i	$0.994370\pi$
$68$	0	0
$69$	− 80.1147i	− 0.139778i
$70$	0	0
$71$	955.725	1.59752	0.798759	−	0.601651i	$-0.205490\pi$
$71$	955.725	1.59752	0.798759	+	0.601651i	$0.205490\pi$
$72$	0	0
$73$	−127.417	−0.204287	−0.102144	−	0.994770i	$-0.532570\pi$
$73$	−127.417	−0.204287	−0.102144	+	0.994770i	$0.532570\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	804.293i	1.19036i
$78$	0	0
$79$	−973.871	−1.38695	−0.693475	−	0.720481i	$-0.743921\pi$
$79$	−973.871	−1.38695	−0.693475	+	0.720481i	$0.743921\pi$
$80$	0	0
$81$	1715.87	2.35373
$82$	0	0
$83$	− 55.0210i	− 0.0727631i	−0.999338	−	0.0363816i	$-0.988417\pi$
$83$	− 55.0210i	− 0.0727631i	0.999338	−	0.0363816i	$-0.0115832\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1001.50	−1.23416
$88$	0	0
$89$	−919.812	−1.09550	−0.547752	−	0.836641i	$-0.684516\pi$
$89$	−919.812	−1.09550	−0.547752	+	0.836641i	$0.684516\pi$
$90$	0	0
$91$	520.054i	0.599082i
$92$	0	0
$93$	− 1956.85i	− 2.18189i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−297.986	−0.311916	−0.155958	−	0.987764i	$-0.549846\pi$
$97$	−297.986	−0.311916	−0.155958	+	0.987764i	$0.549846\pi$
$98$	0	0
$99$	− 2414.54i	− 2.45122i
$100$	0	0
$101$	− 1793.53i	− 1.76696i	−0.468469	−	0.883480i	$-0.655194\pi$
$101$	− 1793.53i	− 1.76696i	0.468469	−	0.883480i	$-0.344806\pi$
$102$	0	0
$103$	2084.11	1.99372	0.996862	−	0.0791611i	$-0.0252241\pi$
$103$	2084.11	1.99372	0.996862	+	0.0791611i	$0.0252241\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 67.2542i	− 0.0607637i	−0.999538	−	0.0303818i	$-0.990328\pi$
$107$	− 67.2542i	− 0.0607637i	0.999538	−	0.0303818i	$-0.00967233\pi$
$108$	0	0
$109$	− 1459.15i	− 1.28222i	−0.767451	−	0.641108i	$-0.778475\pi$
$109$	− 1459.15i	− 1.28222i	0.767451	−	0.641108i	$-0.221525\pi$
$110$	0	0
$111$	−1254.10	−1.07238
$112$	0	0
$113$	−458.263	−0.381502	−0.190751	−	0.981638i	$-0.561092\pi$
$113$	−458.263	−0.381502	−0.190751	+	0.981638i	$0.561092\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1561.24i	− 1.23364i
$118$	0	0
$119$	312.639	0.240837
$120$	0	0
$121$	−59.3283	−0.0445743
$122$	0	0
$123$	− 4195.21i	− 3.07536i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	208.685	0.145809	0.0729047	−	0.997339i	$-0.476773\pi$
$127$	208.685	0.145809	0.0729047	+	0.997339i	$0.476773\pi$
$128$	0	0
$129$	−2957.62	−2.01864
$130$	0	0
$131$	− 1784.69i	− 1.19030i	−0.803616	−	0.595148i	$-0.797094\pi$
$131$	− 1784.69i	− 1.19030i	0.803616	−	0.595148i	$-0.202906\pi$
$132$	0	0
$133$	− 575.682i	− 0.375323i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2380.60	−1.48459	−0.742293	−	0.670075i	$-0.766262\pi$
$137$	−2380.60	−1.48459	−0.742293	+	0.670075i	$0.766262\pi$
$138$	0	0
$139$	2858.52i	1.74429i	0.489245	+	0.872146i	$0.337272\pi$
$139$	2858.52i	1.74429i	−0.489245	+	0.872146i	$0.662728\pi$
$140$	0	0
$141$	930.189i	0.555575i
$142$	0	0
$143$	−898.983	−0.525711
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1171.28i	0.657179i
$148$	0	0
$149$	897.894i	0.493680i	0.969056	+	0.246840i	$0.0793922\pi$
$149$	897.894i	0.493680i	−0.969056	+	0.246840i	$0.920608\pi$
$150$	0	0
$151$	3431.32	1.84925	0.924625	−	0.380878i	$-0.124378\pi$
$151$	3431.32	1.84925	0.924625	+	0.380878i	$0.124378\pi$
$152$	0	0
$153$	−938.563	−0.495937
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 571.369i	− 0.290447i	−0.989399	−	0.145224i	$-0.953610\pi$
$157$	− 571.369i	− 0.290447i	0.989399	−	0.145224i	$-0.0463901\pi$
$158$	0	0
$159$	−1479.96	−0.738167
$160$	0	0
$161$	−180.406	−0.0883106
$162$	0	0
$163$	− 236.962i	− 0.113867i	−0.998378	−	0.0569333i	$-0.981868\pi$
$163$	− 236.962i	− 0.113867i	0.998378	−	0.0569333i	$-0.0181322\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1448.46	0.671169	0.335585	−	0.942010i	$-0.391066\pi$
$167$	1448.46	0.671169	0.335585	+	0.942010i	$0.391066\pi$
$168$	0	0
$169$	1615.72	0.735421
$170$	0	0
$171$	1728.24i	0.772874i
$172$	0	0
$173$	− 2529.74i	− 1.11175i	−0.831266	−	0.555875i	$-0.812383\pi$
$173$	− 2529.74i	− 1.11175i	0.831266	−	0.555875i	$-0.187617\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5211.77	2.21322
$178$	0	0
$179$	141.524i	0.0590949i	0.999563	+	0.0295475i	$0.00940662\pi$
$179$	141.524i	0.0590949i	−0.999563	+	0.0295475i	$0.990593\pi$
$180$	0	0
$181$	3457.79i	1.41997i	0.704215	+	0.709987i	$0.251299\pi$
$181$	3457.79i	1.41997i	−0.704215	+	0.709987i	$0.748701\pi$
$182$	0	0
$183$	−3588.27	−1.44947
$184$	0	0
$185$	0	0
$186$	0	0
$187$	540.438i	0.211341i
$188$	0	0
$189$	− 7801.01i	− 3.00233i
$190$	0	0
$191$	1416.99	0.536804	0.268402	−	0.963307i	$-0.413504\pi$
$191$	1416.99	0.536804	0.268402	+	0.963307i	$0.413504\pi$
$192$	0	0
$193$	2681.12	0.999955	0.499978	−	0.866038i	$-0.333342\pi$
$193$	2681.12	0.999955	0.499978	+	0.866038i	$0.333342\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2697.74i	0.975664i	0.872938	+	0.487832i	$0.162212\pi$
$197$	2697.74i	0.975664i	−0.872938	+	0.487832i	$0.837788\pi$
$198$	0	0
$199$	−3543.25	−1.26218	−0.631091	−	0.775709i	$-0.717393\pi$
$199$	−3543.25	−1.26218	−0.631091	+	0.775709i	$0.717393\pi$
$200$	0	0
$201$	−185.813	−0.0652052
$202$	0	0
$203$	2255.23i	0.779735i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	541.592	0.181851
$208$	0	0
$209$	995.143	0.329356
$210$	0	0
$211$	152.205i	0.0496598i	0.999692	+	0.0248299i	$0.00790441\pi$
$211$	152.205i	0.0496598i	−0.999692	+	0.0248299i	$0.992096\pi$
$212$	0	0
$213$	9154.80i	2.94496i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4406.54	−1.37851
$218$	0	0
$219$	− 1220.51i	− 0.376596i
$220$	0	0
$221$	349.446i	0.106363i
$222$	0	0
$223$	−2065.40	−0.620222	−0.310111	−	0.950700i	$-0.600366\pi$
$223$	−2065.40	−0.620222	−0.310111	+	0.950700i	$0.600366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2940.45i	0.859757i	0.902887	+	0.429878i	$0.141444\pi$
$227$	2940.45i	0.859757i	−0.902887	+	0.429878i	$0.858556\pi$
$228$	0	0
$229$	4949.77i	1.42834i	0.699971	+	0.714171i	$0.253196\pi$
$229$	4949.77i	1.42834i	−0.699971	+	0.714171i	$0.746804\pi$
$230$	0	0
$231$	−7704.25	−2.19438
$232$	0	0
$233$	−5193.31	−1.46019	−0.730096	−	0.683344i	$-0.760525\pi$
$233$	−5193.31	−1.46019	−0.730096	+	0.683344i	$0.760525\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9328.62i	− 2.55679i
$238$	0	0
$239$	2066.30	0.559238	0.279619	−	0.960111i	$-0.409792\pi$
$239$	2066.30	0.559238	0.279619	+	0.960111i	$0.409792\pi$
$240$	0	0
$241$	−2309.08	−0.617181	−0.308591	−	0.951195i	$-0.599857\pi$
$241$	−2309.08	−0.617181	−0.308591	+	0.951195i	$0.599857\pi$
$242$	0	0
$243$	6671.42i	1.76120i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	643.457	0.165758
$248$	0	0
$249$	527.041	0.134136
$250$	0	0
$251$	3802.60i	0.956246i	0.878293	+	0.478123i	$0.158683\pi$
$251$	3802.60i	0.956246i	−0.878293	+	0.478123i	$0.841317\pi$
$252$	0	0
$253$	− 311.857i	− 0.0774951i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6168.56	−1.49721	−0.748607	−	0.663013i	$-0.769277\pi$
$257$	−6168.56	−1.49721	−0.748607	+	0.663013i	$0.769277\pi$
$258$	0	0
$259$	2824.05i	0.677521i
$260$	0	0
$261$	− 6770.36i	− 1.60565i
$262$	0	0
$263$	−1636.11	−0.383599	−0.191800	−	0.981434i	$-0.561432\pi$
$263$	−1636.11	−0.383599	−0.191800	+	0.981434i	$0.561432\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8810.79i	− 2.01952i
$268$	0	0
$269$	− 2262.24i	− 0.512756i	−0.966577	−	0.256378i	$-0.917471\pi$
$269$	− 2262.24i	− 0.512756i	0.966577	−	0.256378i	$-0.0825292\pi$
$270$	0	0
$271$	−41.6380	−0.00933332	−0.00466666	−	0.999989i	$-0.501485\pi$
$271$	−41.6380	−0.00933332	−0.00466666	+	0.999989i	$0.501485\pi$
$272$	0	0
$273$	−4981.55	−1.10438
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2139.00i	0.463971i	0.972719	+	0.231986i	$0.0745223\pi$
$277$	2139.00i	0.463971i	−0.972719	+	0.231986i	$0.925478\pi$
$278$	0	0
$279$	13228.7	2.83865
$280$	0	0
$281$	−1493.53	−0.317069	−0.158534	−	0.987353i	$-0.550677\pi$
$281$	−1493.53	−0.317069	−0.158534	+	0.987353i	$0.550677\pi$
$282$	0	0
$283$	− 1090.20i	− 0.228995i	−0.993424	−	0.114498i	$-0.963474\pi$
$283$	− 1090.20i	− 0.228995i	0.993424	−	0.114498i	$-0.0365259\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9446.98	−1.94299
$288$	0	0
$289$	−4702.92	−0.957241
$290$	0	0
$291$	− 2854.38i	− 0.575005i
$292$	0	0
$293$	4067.51i	0.811011i	0.914093	+	0.405506i	$0.132905\pi$
$293$	4067.51i	0.811011i	−0.914093	+	0.405506i	$0.867095\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13485.1	2.63463
$298$	0	0
$299$	− 201.646i	− 0.0390016i
$300$	0	0
$301$	6660.13i	1.27536i
$302$	0	0
$303$	17180.1	3.25732
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6736.05i	1.25227i	0.779715	+	0.626135i	$0.215364\pi$
$307$	6736.05i	1.25227i	−0.779715	+	0.626135i	$0.784636\pi$
$308$	0	0
$309$	19963.5i	3.67535i
$310$	0	0
$311$	−3274.35	−0.597014	−0.298507	−	0.954407i	$-0.596489\pi$
$311$	−3274.35	−0.597014	−0.298507	+	0.954407i	$0.596489\pi$
$312$	0	0
$313$	9357.60	1.68985	0.844925	−	0.534885i	$-0.179645\pi$
$313$	9357.60	1.68985	0.844925	+	0.534885i	$0.179645\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9064.68i	1.60607i	0.595934	+	0.803033i	$0.296782\pi$
$317$	9064.68i	1.60607i	−0.595934	+	0.803033i	$0.703218\pi$
$318$	0	0
$319$	−3898.47	−0.684239
$320$	0	0
$321$	644.222	0.112015
$322$	0	0
$323$	− 386.825i	− 0.0666362i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13977.1	2.36371
$328$	0	0
$329$	2094.65	0.351008
$330$	0	0
$331$	− 280.089i	− 0.0465108i	−0.999730	−	0.0232554i	$-0.992597\pi$
$331$	− 280.089i	− 0.0465108i	0.999730	−	0.0232554i	$-0.00740310\pi$
$332$	0	0
$333$	− 8477.98i	− 1.39517i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5748.75	0.929242	0.464621	−	0.885510i	$-0.346191\pi$
$337$	5748.75	0.929242	0.464621	+	0.885510i	$0.346191\pi$
$338$	0	0
$339$	− 4389.65i	− 0.703284i
$340$	0	0
$341$	− 7617.30i	− 1.20968i
$342$	0	0
$343$	−4761.06	−0.749484
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2911.52i	0.450428i	0.974309	+	0.225214i	$0.0723081\pi$
$347$	2911.52i	0.450428i	−0.974309	+	0.225214i	$0.927692\pi$
$348$	0	0
$349$	− 5665.17i	− 0.868910i	−0.900693	−	0.434455i	$-0.856941\pi$
$349$	− 5665.17i	− 0.868910i	0.900693	−	0.434455i	$-0.143059\pi$
$350$	0	0
$351$	8719.42	1.32595
$352$	0	0
$353$	9428.18	1.42156	0.710781	−	0.703414i	$-0.248342\pi$
$353$	9428.18	1.42156	0.710781	+	0.703414i	$0.248342\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2994.74i	0.443973i
$358$	0	0
$359$	−4573.16	−0.672318	−0.336159	−	0.941805i	$-0.609128\pi$
$359$	−4573.16	−0.672318	−0.336159	+	0.941805i	$0.609128\pi$
$360$	0	0
$361$	6146.71	0.896153
$362$	0	0
$363$	− 568.300i	− 0.0821709i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2717.97	−0.386585	−0.193293	−	0.981141i	$-0.561917\pi$
$367$	−2717.97	−0.386585	−0.193293	+	0.981141i	$0.561917\pi$
$368$	0	0
$369$	28360.5	4.00105
$370$	0	0
$371$	3332.65i	0.466369i
$372$	0	0
$373$	− 5627.51i	− 0.781183i	−0.920564	−	0.390592i	$-0.872270\pi$
$373$	− 5627.51i	− 0.781183i	0.920564	−	0.390592i	$-0.127730\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2520.74	−0.344363
$378$	0	0
$379$	8066.74i	1.09330i	0.837361	+	0.546650i	$0.184097\pi$
$379$	8066.74i	1.09330i	−0.837361	+	0.546650i	$0.815903\pi$
$380$	0	0
$381$	1998.97i	0.268794i
$382$	0	0
$383$	10607.8	1.41523	0.707613	−	0.706600i	$-0.249772\pi$
$383$	10607.8	1.41523	0.707613	+	0.706600i	$0.249772\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 19994.1i	− 2.62625i
$388$	0	0
$389$	8992.58i	1.17209i	0.810280	+	0.586044i	$0.199315\pi$
$389$	8992.58i	1.17209i	−0.810280	+	0.586044i	$0.800685\pi$
$390$	0	0
$391$	−121.223	−0.0156790
$392$	0	0
$393$	17095.3	2.19426
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8977.17i	1.13489i	0.823411	+	0.567445i	$0.192068\pi$
$397$	8977.17i	1.13489i	−0.823411	+	0.567445i	$0.807932\pi$
$398$	0	0
$399$	5514.40	0.691893
$400$	0	0
$401$	−3165.72	−0.394236	−0.197118	−	0.980380i	$-0.563158\pi$
$401$	−3165.72	−0.394236	−0.197118	+	0.980380i	$0.563158\pi$
$402$	0	0
$403$	− 4925.33i	− 0.608804i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4881.75	−0.594543
$408$	0	0
$409$	−5417.61	−0.654972	−0.327486	−	0.944856i	$-0.606202\pi$
$409$	−5417.61	−0.654972	−0.327486	+	0.944856i	$0.606202\pi$
$410$	0	0
$411$	− 22803.5i	− 2.73678i
$412$	0	0
$413$	− 11736.1i	− 1.39830i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−27381.5	−3.21554
$418$	0	0
$419$	6198.14i	0.722670i	0.932436	+	0.361335i	$0.117679\pi$
$419$	6198.14i	0.722670i	−0.932436	+	0.361335i	$0.882321\pi$
$420$	0	0
$421$	− 4344.57i	− 0.502949i	−0.967864	−	0.251474i	$-0.919085\pi$
$421$	− 4344.57i	− 0.502949i	0.967864	−	0.251474i	$-0.0809154\pi$
$422$	0	0
$423$	−6288.27	−0.722804
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8080.25i	0.915763i
$428$	0	0
$429$	− 8611.27i	− 0.969128i
$430$	0	0
$431$	−4750.74	−0.530940	−0.265470	−	0.964119i	$-0.585527\pi$
$431$	−4750.74	−0.530940	−0.265470	+	0.964119i	$0.585527\pi$
$432$	0	0
$433$	−4210.26	−0.467280	−0.233640	−	0.972323i	$-0.575064\pi$
$433$	−4210.26	−0.467280	−0.233640	+	0.972323i	$0.575064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	223.215i	0.0244344i
$438$	0	0
$439$	5776.50	0.628012	0.314006	−	0.949421i	$-0.398329\pi$
$439$	5776.50	0.628012	0.314006	+	0.949421i	$0.398329\pi$
$440$	0	0
$441$	−7918.08	−0.854992
$442$	0	0
$443$	11981.6i	1.28502i	0.766278	+	0.642510i	$0.222107\pi$
$443$	11981.6i	1.28502i	−0.766278	+	0.642510i	$0.777893\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8600.84	−0.910080
$448$	0	0
$449$	−5567.58	−0.585190	−0.292595	−	0.956236i	$-0.594519\pi$
$449$	−5567.58	−0.585190	−0.292595	+	0.956236i	$0.594519\pi$
$450$	0	0
$451$	− 16330.4i	− 1.70503i
$452$	0	0
$453$	32868.3i	3.40902i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7283.56	0.745538	0.372769	−	0.927924i	$-0.378408\pi$
$457$	7283.56	0.745538	0.372769	+	0.927924i	$0.378408\pi$
$458$	0	0
$459$	− 5241.82i	− 0.533044i
$460$	0	0
$461$	− 11901.1i	− 1.20236i	−0.799113	−	0.601181i	$-0.794697\pi$
$461$	− 11901.1i	− 1.20236i	0.799113	−	0.601181i	$-0.205303\pi$
$462$	0	0
$463$	4915.73	0.493419	0.246710	−	0.969089i	$-0.420651\pi$
$463$	4915.73	0.493419	0.246710	+	0.969089i	$0.420651\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3110.15i	− 0.308181i	−0.988057	−	0.154091i	$-0.950755\pi$
$467$	− 3110.15i	− 0.308181i	0.988057	−	0.154091i	$-0.0492447\pi$
$468$	0	0
$469$	418.424i	0.0411962i
$470$	0	0
$471$	5473.09	0.535428
$472$	0	0
$473$	−11512.9	−1.11916
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10004.8i	− 0.960358i
$478$	0	0
$479$	−6590.43	−0.628652	−0.314326	−	0.949315i	$-0.601779\pi$
$479$	−6590.43	−0.628652	−0.314326	+	0.949315i	$0.601779\pi$
$480$	0	0
$481$	−3156.52	−0.299221
$482$	0	0
$483$	− 1728.10i	− 0.162797i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3139.99	0.292169	0.146085	−	0.989272i	$-0.453333\pi$
$487$	3139.99	0.292169	0.146085	+	0.989272i	$0.453333\pi$
$488$	0	0
$489$	2269.83	0.209909
$490$	0	0
$491$	12057.6i	1.10825i	0.832433	+	0.554126i	$0.186947\pi$
$491$	12057.6i	1.10825i	−0.832433	+	0.554126i	$0.813053\pi$
$492$	0	0
$493$	1515.38i	0.138437i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20615.3	1.86061
$498$	0	0
$499$	− 8458.64i	− 0.758839i	−0.925225	−	0.379420i	$-0.876124\pi$
$499$	− 8458.64i	− 0.758839i	0.925225	−	0.379420i	$-0.123876\pi$
$500$	0	0
$501$	13874.7i	1.23727i
$502$	0	0
$503$	16896.2	1.49774	0.748869	−	0.662718i	$-0.230597\pi$
$503$	16896.2	1.49774	0.748869	+	0.662718i	$0.230597\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15476.8i	1.35572i
$508$	0	0
$509$	6849.13i	0.596429i	0.954499	+	0.298214i	$0.0963910\pi$
$509$	6849.13i	0.596429i	−0.954499	+	0.298214i	$0.903609\pi$
$510$	0	0
$511$	−2748.41	−0.237931
$512$	0	0
$513$	−9652.10	−0.830703
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3620.88i	0.308019i
$518$	0	0
$519$	24232.1	2.04947
$520$	0	0
$521$	11044.1	0.928696	0.464348	−	0.885653i	$-0.346289\pi$
$521$	11044.1	0.928696	0.464348	+	0.885653i	$0.346289\pi$
$522$	0	0
$523$	3115.75i	0.260502i	0.991481	+	0.130251i	$0.0415782\pi$
$523$	3115.75i	0.260502i	−0.991481	+	0.130251i	$0.958422\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2960.94	−0.244745
$528$	0	0
$529$	−12097.0	−0.994251
$530$	0	0
$531$	35232.6i	2.87941i
$532$	0	0
$533$	− 10559.2i	− 0.858103i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1355.64	−0.108939
$538$	0	0
$539$	4559.35i	0.364350i
$540$	0	0
$541$	14840.6i	1.17938i	0.807629	+	0.589691i	$0.200751\pi$
$541$	14840.6i	1.17938i	−0.807629	+	0.589691i	$0.799249\pi$
$542$	0	0
$543$	−33121.8	−2.61767
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12622.8i	0.986679i	0.869837	+	0.493340i	$0.164224\pi$
$547$	12622.8i	0.986679i	−0.869837	+	0.493340i	$0.835776\pi$
$548$	0	0
$549$	− 24257.5i	− 1.88576i
$550$	0	0
$551$	2790.37	0.215742
$552$	0	0
$553$	−21006.7	−1.61536
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13382.1i	− 1.01798i	−0.860771	−	0.508992i	$-0.830018\pi$
$557$	− 13382.1i	− 1.01798i	0.860771	−	0.508992i	$-0.169982\pi$
$558$	0	0
$559$	−7444.23	−0.563251
$560$	0	0
$561$	−5176.80	−0.389599
$562$	0	0
$563$	− 153.895i	− 0.0115203i	−0.999983	−	0.00576014i	$-0.998166\pi$
$563$	− 153.895i	− 0.0115203i	0.999983	−	0.00576014i	$-0.00183352\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	37011.7	2.74135
$568$	0	0
$569$	−5395.71	−0.397539	−0.198770	−	0.980046i	$-0.563695\pi$
$569$	−5395.71	−0.397539	−0.198770	+	0.980046i	$0.563695\pi$
$570$	0	0
$571$	22390.5i	1.64100i	0.571646	+	0.820500i	$0.306305\pi$
$571$	22390.5i	1.64100i	−0.571646	+	0.820500i	$0.693695\pi$
$572$	0	0
$573$	13573.2i	0.989578i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1935.38	0.139638	0.0698188	−	0.997560i	$-0.477758\pi$
$577$	1935.38	0.139638	0.0698188	+	0.997560i	$0.477758\pi$
$578$	0	0
$579$	25682.2i	1.84338i
$580$	0	0
$581$	− 1186.82i	− 0.0847461i
$582$	0	0
$583$	−5760.94	−0.409252
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7905.44i	− 0.555865i	−0.960601	−	0.277932i	$-0.910351\pi$
$587$	− 7905.44i	− 0.555865i	0.960601	−	0.277932i	$-0.0896491\pi$
$588$	0	0
$589$	5452.17i	0.381414i
$590$	0	0
$591$	−25841.4	−1.79860
$592$	0	0
$593$	11141.6	0.771552	0.385776	−	0.922592i	$-0.373934\pi$
$593$	11141.6	0.771552	0.385776	+	0.922592i	$0.373934\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 33940.4i	− 2.32678i
$598$	0	0
$599$	−278.847	−0.0190207	−0.00951033	−	0.999955i	$-0.503027\pi$
$599$	−278.847	−0.0190207	−0.00951033	+	0.999955i	$0.503027\pi$
$600$	0	0
$601$	18890.7	1.28214	0.641071	−	0.767482i	$-0.278490\pi$
$601$	18890.7	1.28214	0.641071	+	0.767482i	$0.278490\pi$
$602$	0	0
$603$	− 1256.14i	− 0.0848321i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5398.05	0.360956	0.180478	−	0.983579i	$-0.442236\pi$
$607$	5398.05	0.360956	0.180478	+	0.983579i	$0.442236\pi$
$608$	0	0
$609$	−21602.6	−1.43741
$610$	0	0
$611$	2341.25i	0.155019i
$612$	0	0
$613$	14412.8i	0.949635i	0.880084	+	0.474817i	$0.157486\pi$
$613$	14412.8i	0.949635i	−0.880084	+	0.474817i	$0.842514\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13226.3	−0.863001	−0.431500	−	0.902113i	$-0.642016\pi$
$617$	−13226.3	−0.863001	−0.431500	+	0.902113i	$0.642016\pi$
$618$	0	0
$619$	− 14179.4i	− 0.920705i	−0.887736	−	0.460352i	$-0.847723\pi$
$619$	− 14179.4i	− 0.920705i	0.887736	−	0.460352i	$-0.152277\pi$
$620$	0	0
$621$	3024.76i	0.195458i
$622$	0	0
$623$	−19840.6	−1.27592
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9532.38i	0.607156i
$628$	0	0
$629$	1897.60i	0.120290i
$630$	0	0
$631$	−1533.79	−0.0967657	−0.0483828	−	0.998829i	$-0.515407\pi$
$631$	−1533.79	−0.0967657	−0.0483828	+	0.998829i	$0.515407\pi$
$632$	0	0
$633$	−1457.96	−0.0915459
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2948.06i	0.183370i
$638$	0	0
$639$	−61888.4	−3.83140
$640$	0	0
$641$	−26682.6	−1.64415	−0.822074	−	0.569381i	$-0.807183\pi$
$641$	−26682.6	−1.64415	−0.822074	+	0.569381i	$0.807183\pi$
$642$	0	0
$643$	− 16498.3i	− 1.01186i	−0.862574	−	0.505931i	$-0.831149\pi$
$643$	− 16498.3i	− 1.01186i	0.862574	−	0.505931i	$-0.168851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20127.5	−1.22302	−0.611510	−	0.791236i	$-0.709438\pi$
$647$	−20127.5	−1.22302	−0.611510	+	0.791236i	$0.709438\pi$
$648$	0	0
$649$	20287.5	1.22705
$650$	0	0
$651$	− 42209.9i	− 2.54122i
$652$	0	0
$653$	− 21447.4i	− 1.28530i	−0.766160	−	0.642650i	$-0.777835\pi$
$653$	− 21447.4i	− 1.28530i	0.766160	−	0.642650i	$-0.222165\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8250.91	0.489953
$658$	0	0
$659$	19402.5i	1.14691i	0.819237	+	0.573455i	$0.194397\pi$
$659$	19402.5i	1.14691i	−0.819237	+	0.573455i	$0.805603\pi$
$660$	0	0
$661$	− 9326.38i	− 0.548796i	−0.961616	−	0.274398i	$-0.911521\pi$
$661$	− 9326.38i	− 0.548796i	0.961616	−	0.274398i	$-0.0884786\pi$
$662$	0	0
$663$	−3347.31	−0.196076
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 874.443i	− 0.0507625i
$668$	0	0
$669$	− 19784.3i	− 1.14336i
$670$	0	0
$671$	−13967.8	−0.803608
$672$	0	0
$673$	15027.4	0.860718	0.430359	−	0.902658i	$-0.358387\pi$
$673$	15027.4	0.860718	0.430359	+	0.902658i	$0.358387\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9717.31i	− 0.551649i	−0.961208	−	0.275825i	$-0.911049\pi$
$677$	− 9717.31i	− 0.551649i	0.961208	−	0.275825i	$-0.0889509\pi$
$678$	0	0
$679$	−6427.63	−0.363284
$680$	0	0
$681$	−28166.3	−1.58493
$682$	0	0
$683$	11473.9i	0.642804i	0.946943	+	0.321402i	$0.104154\pi$
$683$	11473.9i	0.642804i	−0.946943	+	0.321402i	$0.895846\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−47413.4	−2.63309
$688$	0	0
$689$	−3725.01	−0.205967
$690$	0	0
$691$	28412.6i	1.56421i	0.623149	+	0.782103i	$0.285853\pi$
$691$	28412.6i	1.56421i	−0.623149	+	0.782103i	$0.714147\pi$
$692$	0	0
$693$	− 52082.3i	− 2.85490i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6347.82	−0.344965
$698$	0	0
$699$	− 49746.2i	− 2.69181i
$700$	0	0
$701$	− 12291.3i	− 0.662250i	−0.943587	−	0.331125i	$-0.892572\pi$
$701$	− 12291.3i	− 0.662250i	0.943587	−	0.331125i	$-0.107428\pi$
$702$	0	0
$703$	3494.17	0.187461
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 38686.9i	− 2.05795i
$708$	0	0
$709$	− 12774.2i	− 0.676650i	−0.941029	−	0.338325i	$-0.890140\pi$
$709$	− 12774.2i	− 0.676650i	0.941029	−	0.338325i	$-0.109860\pi$
$710$	0	0
$711$	63063.4	3.32639
$712$	0	0
$713$	1708.59	0.0897438
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19792.9i	1.03093i
$718$	0	0
$719$	15748.6	0.816861	0.408430	−	0.912790i	$-0.366076\pi$
$719$	15748.6	0.816861	0.408430	+	0.912790i	$0.366076\pi$
$720$	0	0
$721$	44954.8	2.32206
$722$	0	0
$723$	− 22118.4i	− 1.13775i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9536.44	0.486502	0.243251	−	0.969963i	$-0.421786\pi$
$727$	9536.44	0.486502	0.243251	+	0.969963i	$0.421786\pi$
$728$	0	0
$729$	−17576.5	−0.892980
$730$	0	0
$731$	4475.22i	0.226432i
$732$	0	0
$733$	34913.5i	1.75929i	0.475631	+	0.879645i	$0.342220\pi$
$733$	34913.5i	1.75929i	−0.475631	+	0.879645i	$0.657780\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−723.301	−0.0361508
$738$	0	0
$739$	− 19324.3i	− 0.961918i	−0.876743	−	0.480959i	$-0.840289\pi$
$739$	− 19324.3i	− 0.961918i	0.876743	−	0.480959i	$-0.159711\pi$
$740$	0	0
$741$	6163.61i	0.305568i
$742$	0	0
$743$	−7053.57	−0.348278	−0.174139	−	0.984721i	$-0.555714\pi$
$743$	−7053.57	−0.348278	−0.174139	+	0.984721i	$0.555714\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3562.91i	0.174511i
$748$	0	0
$749$	− 1450.69i	− 0.0707705i
$750$	0	0
$751$	36810.6	1.78860	0.894299	−	0.447471i	$-0.147675\pi$
$751$	36810.6	1.78860	0.894299	+	0.447471i	$0.147675\pi$
$752$	0	0
$753$	−36424.7	−1.76280
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28515.3i	1.36910i	0.728967	+	0.684549i	$0.240001\pi$
$757$	28515.3i	1.36910i	−0.728967	+	0.684549i	$0.759999\pi$
$758$	0	0
$759$	2987.24	0.142859
$760$	0	0
$761$	1517.92	0.0723057	0.0361528	−	0.999346i	$-0.488490\pi$
$761$	1517.92	0.0723057	0.0361528	+	0.999346i	$0.488490\pi$
$762$	0	0
$763$	− 31474.3i	− 1.49338i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13117.8	0.617545
$768$	0	0
$769$	−10413.9	−0.488341	−0.244171	−	0.969732i	$-0.578516\pi$
$769$	−10413.9	−0.488341	−0.244171	+	0.969732i	$0.578516\pi$
$770$	0	0
$771$	− 59088.0i	− 2.76006i
$772$	0	0
$773$	− 32909.8i	− 1.53129i	−0.643266	−	0.765643i	$-0.722421\pi$
$773$	− 32909.8i	− 1.53129i	0.643266	−	0.765643i	$-0.277579\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−27051.3	−1.24898
$778$	0	0
$779$	11688.6i	0.537599i
$780$	0	0
$781$	35636.2i	1.63273i
$782$	0	0
$783$	37812.1	1.72579
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35511.6i	1.60845i	0.594324	+	0.804226i	$0.297420\pi$
$787$	35511.6i	1.60845i	−0.594324	+	0.804226i	$0.702580\pi$
$788$	0	0
$789$	− 15672.1i	− 0.707150i
$790$	0	0
$791$	−9884.85	−0.444330
$792$	0	0
$793$	−9031.54	−0.404438
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22692.3i	1.00853i	0.863548	+	0.504267i	$0.168237\pi$
$797$	22692.3i	1.00853i	−0.863548	+	0.504267i	$0.831763\pi$
$798$	0	0
$799$	1407.48	0.0623193
$800$	0	0
$801$	59562.8	2.62740
$802$	0	0
$803$	− 4751.00i	− 0.208791i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21669.8	0.945246
$808$	0	0
$809$	−22716.0	−0.987209	−0.493604	−	0.869687i	$-0.664321\pi$
$809$	−22716.0	−0.987209	−0.493604	+	0.869687i	$0.664321\pi$
$810$	0	0
$811$	− 17237.0i	− 0.746327i	−0.927766	−	0.373164i	$-0.878273\pi$
$811$	− 17237.0i	− 0.746327i	0.927766	−	0.373164i	$-0.121727\pi$
$812$	0	0
$813$	− 398.847i	− 0.0172056i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8240.51	0.352875
$818$	0	0
$819$	− 33676.3i	− 1.43681i
$820$	0	0
$821$	15695.7i	0.667217i	0.942712	+	0.333609i	$0.108266\pi$
$821$	15695.7i	0.667217i	−0.942712	+	0.333609i	$0.891734\pi$
$822$	0	0
$823$	16856.0	0.713927	0.356963	−	0.934118i	$-0.383812\pi$
$823$	16856.0	0.713927	0.356963	+	0.934118i	$0.383812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3712.67i	0.156109i	0.996949	+	0.0780544i	$0.0248708\pi$
$827$	3712.67i	0.156109i	−0.996949	+	0.0780544i	$0.975129\pi$
$828$	0	0
$829$	22469.6i	0.941376i	0.882300	+	0.470688i	$0.155994\pi$
$829$	22469.6i	0.941376i	−0.882300	+	0.470688i	$0.844006\pi$
$830$	0	0
$831$	−20489.3	−0.855313
$832$	0	0
$833$	1772.27	0.0737163
$834$	0	0
$835$	0	0
$836$	0	0
$837$	73881.8i	3.05105i
$838$	0	0
$839$	31935.2	1.31409	0.657046	−	0.753850i	$-0.271806\pi$
$839$	31935.2	1.31409	0.657046	+	0.753850i	$0.271806\pi$
$840$	0	0
$841$	13457.7	0.551795
$842$	0	0
$843$	− 14306.3i	− 0.584504i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1279.73	−0.0519150
$848$	0	0
$849$	10442.9	0.422144
$850$	0	0
$851$	− 1095.00i	− 0.0441081i
$852$	0	0
$853$	− 14782.2i	− 0.593355i	−0.954978	−	0.296678i	$-0.904121\pi$
$853$	− 14782.2i	− 0.593355i	0.954978	−	0.296678i	$-0.0958787\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7222.91	0.287900	0.143950	−	0.989585i	$-0.454020\pi$
$857$	7222.91	0.287900	0.143950	+	0.989585i	$0.454020\pi$
$858$	0	0
$859$	− 4646.96i	− 0.184578i	−0.995732	−	0.0922889i	$-0.970582\pi$
$859$	− 4646.96i	− 0.184578i	0.995732	−	0.0922889i	$-0.0294183\pi$
$860$	0	0
$861$	− 90491.7i	− 3.58182i
$862$	0	0
$863$	8122.19	0.320374	0.160187	−	0.987087i	$-0.448790\pi$
$863$	8122.19	0.320374	0.160187	+	0.987087i	$0.448790\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 45048.9i	− 1.76464i
$868$	0	0
$869$	− 36312.8i	− 1.41752i
$870$	0	0
$871$	−467.685	−0.0181939
$872$	0	0
$873$	19296.2	0.748083
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31986.1i	− 1.23158i	−0.787910	−	0.615790i	$-0.788837\pi$
$877$	− 31986.1i	− 1.23158i	0.787910	−	0.615790i	$-0.211163\pi$
$878$	0	0
$879$	−38962.3	−1.49507
$880$	0	0
$881$	−29450.6	−1.12624	−0.563118	−	0.826376i	$-0.690398\pi$
$881$	−29450.6	−1.12624	−0.563118	+	0.826376i	$0.690398\pi$
$882$	0	0
$883$	46156.5i	1.75911i	0.475801	+	0.879553i	$0.342158\pi$
$883$	46156.5i	1.75911i	−0.475801	+	0.879553i	$0.657842\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41051.2	1.55396	0.776982	−	0.629523i	$-0.216750\pi$
$887$	41051.2	1.55396	0.776982	+	0.629523i	$0.216750\pi$
$888$	0	0
$889$	4501.39	0.169822
$890$	0	0
$891$	63979.7i	2.40561i
$892$	0	0
$893$	− 2591.69i	− 0.0971192i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1931.54	0.0718979
$898$	0	0
$899$	− 21358.8i	− 0.792389i
$900$	0	0
$901$	2239.35i	0.0828008i
$902$	0	0
$903$	−63796.7	−2.35108
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27915.8i	− 1.02197i	−0.859588	−	0.510987i	$-0.829280\pi$
$907$	− 27915.8i	− 1.02197i	0.859588	−	0.510987i	$-0.170720\pi$
$908$	0	0
$909$	116141.i	4.23778i
$910$	0	0
$911$	6557.22	0.238475	0.119237	−	0.992866i	$-0.461955\pi$
$911$	6557.22	0.238475	0.119237	+	0.992866i	$0.461955\pi$
$912$	0	0
$913$	2051.57	0.0743671
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 38496.2i	− 1.38632i
$918$	0	0
$919$	11493.0	0.412534	0.206267	−	0.978496i	$-0.433869\pi$
$919$	11493.0	0.412534	0.206267	+	0.978496i	$0.433869\pi$
$920$	0	0
$921$	−64524.0	−2.30851
$922$	0	0
$923$	23042.3i	0.821719i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−134957.	−4.78164
$928$	0	0
$929$	56597.2	1.99881	0.999405	−	0.0345020i	$-0.0109845\pi$
$929$	56597.2	1.99881	0.999405	+	0.0345020i	$0.0109845\pi$
$930$	0	0
$931$	− 3263.40i	− 0.114880i
$932$	0	0
$933$	− 31364.7i	− 1.10057i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37901.5	1.32144	0.660720	−	0.750633i	$-0.270251\pi$
$937$	37901.5	1.32144	0.660720	+	0.750633i	$0.270251\pi$
$938$	0	0
$939$	89635.5i	3.11517i
$940$	0	0
$941$	− 4449.85i	− 0.154156i	−0.997025	−	0.0770781i	$-0.975441\pi$
$941$	− 4449.85i	− 0.154156i	0.997025	−	0.0770781i	$-0.0245591\pi$
$942$	0	0
$943$	3662.97	0.126493
$944$	0	0
$945$	0	0
$946$	0	0
$947$	225.160i	0.00772620i	0.999993	+	0.00386310i	$0.00122967\pi$
$947$	225.160i	0.00772620i	−0.999993	+	0.00386310i	$0.998770\pi$
$948$	0	0
$949$	− 3071.98i	− 0.105080i
$950$	0	0
$951$	−86829.7	−2.96072
$952$	0	0
$953$	−4845.81	−0.164713	−0.0823563	−	0.996603i	$-0.526245\pi$
$953$	−4845.81	−0.164713	−0.0823563	+	0.996603i	$0.526245\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 37343.1i	− 1.26137i
$958$	0	0
$959$	−51350.2	−1.72908
$960$	0	0
$961$	11942.5	0.400876
$962$	0	0
$963$	4355.07i	0.145732i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40394.8	1.34334	0.671670	−	0.740851i	$-0.265577\pi$
$967$	40394.8	1.34334	0.671670	+	0.740851i	$0.265577\pi$
$968$	0	0
$969$	3705.36	0.122841
$970$	0	0
$971$	− 29066.2i	− 0.960637i	−0.877094	−	0.480318i	$-0.840521\pi$
$971$	− 29066.2i	− 0.960637i	0.877094	−	0.480318i	$-0.159479\pi$
$972$	0	0
$973$	61659.1i	2.03155i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42580.5	−1.39434	−0.697170	−	0.716906i	$-0.745558\pi$
$977$	−42580.5	−1.39434	−0.697170	+	0.716906i	$0.745558\pi$
$978$	0	0
$979$	− 34297.1i	− 1.11965i
$980$	0	0
$981$	94488.0i	3.07520i
$982$	0	0
$983$	−40586.0	−1.31688	−0.658440	−	0.752633i	$-0.728783\pi$
$983$	−40586.0	−1.31688	−0.658440	+	0.752633i	$0.728783\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	20064.4i	0.647070i
$988$	0	0
$989$	− 2582.40i	− 0.0830288i
$990$	0	0
$991$	50370.4	1.61460	0.807299	−	0.590142i	$-0.200928\pi$
$991$	50370.4	1.61460	0.807299	+	0.590142i	$0.200928\pi$
$992$	0	0
$993$	2682.95	0.0857409
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22467.5i	− 0.713694i	−0.934163	−	0.356847i	$-0.883852\pi$
$997$	− 22467.5i	− 0.713694i	0.934163	−	0.356847i	$-0.116148\pi$
$998$	0	0
$999$	47349.1	1.49956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.d.d.401.12		12
4.3	odd	2		200.4.d.b.101.4		12
5.2	odd	4		800.4.f.c.49.12		12
5.3	odd	4		800.4.f.b.49.1		12
5.4	even	2		160.4.d.a.81.1		12
8.3	odd	2		200.4.d.b.101.3		12
8.5	even	2	inner	800.4.d.d.401.1		12
15.14	odd	2		1440.4.k.c.721.2		12
20.3	even	4		200.4.f.c.149.10		12
20.7	even	4		200.4.f.b.149.3		12
20.19	odd	2		40.4.d.a.21.9	✓	12
40.3	even	4		200.4.f.b.149.4		12
40.13	odd	4		800.4.f.c.49.11		12
40.19	odd	2		40.4.d.a.21.10	yes	12
40.27	even	4		200.4.f.c.149.9		12
40.29	even	2		160.4.d.a.81.12		12
40.37	odd	4		800.4.f.b.49.2		12
60.59	even	2		360.4.k.c.181.4		12
80.19	odd	4		1280.4.a.bb.1.1		6
80.29	even	4		1280.4.a.bd.1.6		6
80.59	odd	4		1280.4.a.bc.1.6		6
80.69	even	4		1280.4.a.ba.1.1		6
120.29	odd	2		1440.4.k.c.721.8		12
120.59	even	2		360.4.k.c.181.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.9	✓	12	20.19	odd	2
40.4.d.a.21.10	yes	12	40.19	odd	2
160.4.d.a.81.1		12	5.4	even	2
160.4.d.a.81.12		12	40.29	even	2
200.4.d.b.101.3		12	8.3	odd	2
200.4.d.b.101.4		12	4.3	odd	2
200.4.f.b.149.3		12	20.7	even	4
200.4.f.b.149.4		12	40.3	even	4
200.4.f.c.149.9		12	40.27	even	4
200.4.f.c.149.10		12	20.3	even	4
360.4.k.c.181.3		12	120.59	even	2
360.4.k.c.181.4		12	60.59	even	2
800.4.d.d.401.1		12	8.5	even	2	inner
800.4.d.d.401.12		12	1.1	even	1	trivial
800.4.f.b.49.1		12	5.3	odd	4
800.4.f.b.49.2		12	40.37	odd	4
800.4.f.c.49.11		12	40.13	odd	4
800.4.f.c.49.12		12	5.2	odd	4
1280.4.a.ba.1.1		6	80.69	even	4
1280.4.a.bb.1.1		6	80.19	odd	4
1280.4.a.bc.1.6		6	80.59	odd	4
1280.4.a.bd.1.6		6	80.29	even	4
1440.4.k.c.721.2		12	15.14	odd	2
1440.4.k.c.721.8		12	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.57890i q^{3} +21.5703 q^{7} -64.7554 q^{9} +37.2871i q^{11} +24.1097i q^{13} +14.4940 q^{17} -26.6887i q^{19} +206.620i q^{21} -8.36366 q^{23} -361.655i q^{27} +104.553i q^{29} -204.288 q^{31} -357.170 q^{33} +130.923i q^{37} -230.945 q^{39} -437.963 q^{41} +308.764i q^{43} +97.1081 q^{47} +122.277 q^{49} +138.836i q^{51} +154.502i q^{53} +255.648 q^{57} -544.088i q^{59} +374.601i q^{61} -1396.79 q^{63} +19.3982i q^{67} -80.1147i q^{69} +955.725 q^{71} -127.417 q^{73} +804.293i q^{77} -973.871 q^{79} +1715.87 q^{81} -55.0210i q^{83} -1001.50 q^{87} -919.812 q^{89} +520.054i q^{91} -1956.85i q^{93} -297.986 q^{97} -2414.54i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 28 q^{7} - 108 q^{9} + 604 q^{23} + 264 q^{31} + 232 q^{33} - 600 q^{39} + 40 q^{41} - 940 q^{47} + 1308 q^{49} + 680 q^{57} - 1300 q^{63} + 1592 q^{71} - 432 q^{73} - 2016 q^{79} + 2508 q^{81} - 1968 q^{87}+ \cdots + 1584 q^{97}+O(q^{100}) \) 12 * q + 28 * q^7 - 108 * q^9 + 604 * q^23 + 264 * q^31 + 232 * q^33 - 600 * q^39 + 40 * q^41 - 940 * q^47 + 1308 * q^49 + 680 * q^57 - 1300 * q^63 + 1592 * q^71 - 432 * q^73 - 2016 * q^79 + 2508 * q^81 - 1968 * q^87 - 424 * q^89 + 1584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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