LMFDB - Embedded newform 800.4.f.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.2015280046$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.98839 + 0.215211i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.4.f.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.57890 q^{3} +21.5703i q^{7} +64.7554 q^{9} -37.2871i q^{11} -24.1097 q^{13} +14.4940i q^{17} -26.6887i q^{19} -206.620i q^{21} +8.36366i q^{23} -361.655 q^{27} +104.553i q^{29} -204.288 q^{31} +357.170i q^{33} +130.923 q^{37} +230.945 q^{39} -437.963 q^{41} -308.764 q^{43} +97.1081i q^{47} -122.277 q^{49} -138.836i q^{51} -154.502 q^{53} +255.648i q^{57} -544.088i q^{59} -374.601i q^{61} +1396.79i q^{63} +19.3982 q^{67} -80.1147i q^{69} +955.725 q^{71} +127.417i q^{73} +804.293 q^{77} +973.871 q^{79} +1715.87 q^{81} +55.0210 q^{83} -1001.50i q^{87} +919.812 q^{89} -520.054i q^{91} +1956.85 q^{93} -297.986i q^{97} -2414.54i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 108 q^{9} - 432 q^{27} + 264 q^{31} + 136 q^{37} + 600 q^{39} + 40 q^{41} + 1204 q^{43} - 1308 q^{49} + 1056 q^{53} + 2412 q^{67} + 1592 q^{71} + 824 q^{77} + 2016 q^{79} + 2508 q^{81}+ \cdots + 2784 q^{93}+O(q^{100}) $ 12 * q - 12 * q^3 + 108 * q^9 - 432 * q^27 + 264 * q^31 + 136 * q^37 + 600 * q^39 + 40 * q^41 + 1204 * q^43 - 1308 * q^49 + 1056 * q^53 + 2412 * q^67 + 1592 * q^71 + 824 * q^77 + 2016 * q^79 + 2508 * q^81 - 3556 * q^83 + 424 * q^89 + 2784 * q^93

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.57890	−1.84346	−0.921730	−	0.387831i	$-0.873224\pi$
$3$	−9.57890	−1.84346	−0.921730	+	0.387831i	$0.873224\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.5703i	1.16469i	0.812943	+	0.582343i	$0.197864\pi$
$7$	21.5703i	1.16469i	−0.812943	+	0.582343i	$0.802136\pi$
$8$	0	0
$9$	64.7554	2.39835
$10$	0	0
$11$	− 37.2871i	− 1.02204i	−0.859568	−	0.511022i	$-0.829267\pi$
$11$	− 37.2871i	− 1.02204i	0.859568	−	0.511022i	$-0.170733\pi$
$12$	0	0
$13$	−24.1097	−0.514372	−0.257186	−	0.966362i	$-0.582795\pi$
$13$	−24.1097	−0.514372	−0.257186	+	0.966362i	$0.582795\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.4940i	0.206783i	0.994641	+	0.103391i	$0.0329694\pi$
$17$	14.4940i	0.206783i	−0.994641	+	0.103391i	$0.967031\pi$
$18$	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
$23$	8.36366i	0.0758236i	0.999281	+	0.0379118i	$0.0120706\pi$
$23$	8.36366i	0.0758236i	−0.999281	+	0.0379118i	$0.987929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−361.655	−2.57780
$28$	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
$33$	357.170i	1.88410i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	130.923	0.581720	0.290860	−	0.956766i	$-0.406059\pi$
$37$	130.923	0.581720	0.290860	+	0.956766i	$0.406059\pi$
$38$	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
$43$	−308.764	−1.09503	−0.547513	−	0.836797i	$-0.684425\pi$
$43$	−308.764	−1.09503	−0.547513	+	0.836797i	$0.684425\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	97.1081i	0.301376i	0.988581	+	0.150688i	$0.0481489\pi$
$47$	97.1081i	0.301376i	−0.988581	+	0.150688i	$0.951851\pi$
$48$	0	0
$49$	−122.277	−0.356492
$50$	0	0
$51$	− 138.836i	− 0.381196i
$52$	0	0
$53$	−154.502	−0.400425	−0.200212	−	0.979753i	$-0.564163\pi$
$53$	−154.502	−0.400425	−0.200212	+	0.979753i	$0.564163\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	255.648i	0.594060i
$58$	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	$-0.871390\pi$
$61$	− 374.601i	− 0.786275i	0.919480	−	0.393137i	$-0.128610\pi$
$62$	0	0
$63$	1396.79i	2.79332i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	19.3982	0.0353711	0.0176855	−	0.999844i	$-0.494370\pi$
$67$	19.3982	0.0353711	0.0176855	+	0.999844i	$0.494370\pi$
$68$	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
$73$	127.417i	0.204287i	0.994770	+	0.102144i	$0.0325702\pi$
$73$	127.417i	0.204287i	−0.994770	+	0.102144i	$0.967430\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	804.293	1.19036
$78$	0	0
$79$	973.871	1.38695	0.693475	−	0.720481i	$-0.256079\pi$
$79$	973.871	1.38695	0.693475	+	0.720481i	$0.256079\pi$
$80$	0	0
$81$	1715.87	2.35373
$82$	0	0
$83$	55.0210	0.0727631	0.0363816	−	0.999338i	$-0.488417\pi$
$83$	55.0210	0.0727631	0.0363816	+	0.999338i	$0.488417\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1001.50i	− 1.23416i
$88$	0	0
$89$	919.812	1.09550	0.547752	−	0.836641i	$-0.315484\pi$
$89$	919.812	1.09550	0.547752	+	0.836641i	$0.315484\pi$
$90$	0	0
$91$	− 520.054i	− 0.599082i
$92$	0	0
$93$	1956.85	2.18189
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 297.986i	− 0.311916i	−0.987764	−	0.155958i	$-0.950154\pi$
$97$	− 297.986i	− 0.311916i	0.987764	−	0.155958i	$-0.0498465\pi$
$98$	0	0
$99$	− 2414.54i	− 2.45122i
$100$	0	0
$101$	1793.53i	1.76696i	0.468469	+	0.883480i	$0.344806\pi$
$101$	1793.53i	1.76696i	−0.468469	+	0.883480i	$0.655194\pi$
$102$	0	0
$103$	− 2084.11i	− 1.99372i	−0.0791611	−	0.996862i	$-0.525224\pi$
$103$	− 2084.11i	− 1.99372i	0.0791611	−	0.996862i	$-0.474776\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−67.2542	−0.0607637	−0.0303818	−	0.999538i	$-0.509672\pi$
$107$	−67.2542	−0.0607637	−0.0303818	+	0.999538i	$0.509672\pi$
$108$	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
$113$	458.263i	0.381502i	0.981638	+	0.190751i	$0.0610923\pi$
$113$	458.263i	0.381502i	−0.981638	+	0.190751i	$0.938908\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1561.24	−1.23364
$118$	0	0
$119$	−312.639	−0.240837
$120$	0	0
$121$	−59.3283	−0.0445743
$122$	0	0
$123$	4195.21	3.07536
$124$	0	0
$125$	0	0
$126$	0	0
$127$	208.685i	0.145809i	0.997339	+	0.0729047i	$0.0232269\pi$
$127$	208.685i	0.145809i	−0.997339	+	0.0729047i	$0.976773\pi$
$128$	0	0
$129$	2957.62	2.01864
$130$	0	0
$131$	1784.69i	1.19030i	0.803616	+	0.595148i	$0.202906\pi$
$131$	1784.69i	1.19030i	−0.803616	+	0.595148i	$0.797094\pi$
$132$	0	0
$133$	575.682	0.375323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2380.60i	− 1.48459i	−0.670075	−	0.742293i	$-0.733738\pi$
$137$	− 2380.60i	− 1.48459i	0.670075	−	0.742293i	$-0.266262\pi$
$138$	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
$143$	898.983i	0.525711i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1171.28	0.657179
$148$	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
$153$	938.563i	0.495937i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−571.369	−0.290447	−0.145224	−	0.989399i	$-0.546390\pi$
$157$	−571.369	−0.290447	−0.145224	+	0.989399i	$0.546390\pi$
$158$	0	0
$159$	1479.96	0.738167
$160$	0	0
$161$	−180.406	−0.0883106
$162$	0	0
$163$	236.962	0.113867	0.0569333	−	0.998378i	$-0.481868\pi$
$163$	236.962	0.113867	0.0569333	+	0.998378i	$0.481868\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1448.46i	0.671169i	0.942010	+	0.335585i	$0.108934\pi$
$167$	1448.46i	0.671169i	−0.942010	+	0.335585i	$0.891066\pi$
$168$	0	0
$169$	−1615.72	−0.735421
$170$	0	0
$171$	− 1728.24i	− 0.772874i
$172$	0	0
$173$	2529.74	1.11175	0.555875	−	0.831266i	$-0.312383\pi$
$173$	2529.74	1.11175	0.555875	+	0.831266i	$0.312383\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5211.77i	2.21322i
$178$	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	$-0.748701\pi$
$181$	− 3457.79i	− 1.41997i	0.704215	−	0.709987i	$-0.251299\pi$
$182$	0	0
$183$	3588.27i	1.44947i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	540.438	0.211341
$188$	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
$193$	− 2681.12i	− 0.999955i	−0.866038	−	0.499978i	$-0.833342\pi$
$193$	− 2681.12i	− 0.999955i	0.866038	−	0.499978i	$-0.166658\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2697.74	0.975664	0.487832	−	0.872938i	$-0.337788\pi$
$197$	2697.74	0.975664	0.487832	+	0.872938i	$0.337788\pi$
$198$	0	0
$199$	3543.25	1.26218	0.631091	−	0.775709i	$-0.282607\pi$
$199$	3543.25	1.26218	0.631091	+	0.775709i	$0.282607\pi$
$200$	0	0
$201$	−185.813	−0.0652052
$202$	0	0
$203$	−2255.23	−0.779735
$204$	0	0
$205$	0	0
$206$	0	0
$207$	541.592i	0.181851i
$208$	0	0
$209$	−995.143	−0.329356
$210$	0	0
$211$	− 152.205i	− 0.0496598i	−0.999692	−	0.0248299i	$-0.992096\pi$
$211$	− 152.205i	− 0.0496598i	0.999692	−	0.0248299i	$-0.00790441\pi$
$212$	0	0
$213$	−9154.80	−2.94496
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4406.54i	− 1.37851i
$218$	0	0
$219$	− 1220.51i	− 0.376596i
$220$	0	0
$221$	− 349.446i	− 0.106363i
$222$	0	0
$223$	2065.40i	0.620222i	0.950700	+	0.310111i	$0.100366\pi$
$223$	2065.40i	0.620222i	−0.950700	+	0.310111i	$0.899634\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2940.45	0.859757	0.429878	−	0.902887i	$-0.358556\pi$
$227$	2940.45	0.859757	0.429878	+	0.902887i	$0.358556\pi$
$228$	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
$233$	5193.31i	1.46019i	0.683344	+	0.730096i	$0.260525\pi$
$233$	5193.31i	1.46019i	−0.683344	+	0.730096i	$0.739475\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−9328.62	−2.55679
$238$	0	0
$239$	−2066.30	−0.559238	−0.279619	−	0.960111i	$-0.590208\pi$
$239$	−2066.30	−0.559238	−0.279619	+	0.960111i	$0.590208\pi$
$240$	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
$243$	−6671.42	−1.76120
$244$	0	0
$245$	0	0
$246$	0	0
$247$	643.457i	0.165758i
$248$	0	0
$249$	−527.041	−0.134136
$250$	0	0
$251$	− 3802.60i	− 0.956246i	−0.878293	−	0.478123i	$-0.841317\pi$
$251$	− 3802.60i	− 0.956246i	0.878293	−	0.478123i	$-0.158683\pi$
$252$	0	0
$253$	311.857	0.0774951
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6168.56i	− 1.49721i	−0.663013	−	0.748607i	$-0.730723\pi$
$257$	− 6168.56i	− 1.49721i	0.663013	−	0.748607i	$-0.269277\pi$
$258$	0	0
$259$	2824.05i	0.677521i
$260$	0	0
$261$	6770.36i	1.60565i
$262$	0	0
$263$	1636.11i	0.383599i	0.981434	+	0.191800i	$0.0614324\pi$
$263$	1636.11i	0.383599i	−0.981434	+	0.191800i	$0.938568\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8810.79	−2.01952
$268$	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
$273$	4981.55i	1.10438i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2139.00	0.463971	0.231986	−	0.972719i	$-0.425478\pi$
$277$	2139.00	0.463971	0.231986	+	0.972719i	$0.425478\pi$
$278$	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
$283$	1090.20	0.228995	0.114498	−	0.993424i	$-0.463474\pi$
$283$	1090.20	0.228995	0.114498	+	0.993424i	$0.463474\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 9446.98i	− 1.94299i
$288$	0	0
$289$	4702.92	0.957241
$290$	0	0
$291$	2854.38i	0.575005i
$292$	0	0
$293$	−4067.51	−0.811011	−0.405506	−	0.914093i	$-0.632905\pi$
$293$	−4067.51	−0.811011	−0.405506	+	0.914093i	$0.632905\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13485.1i	2.63463i
$298$	0	0
$299$	− 201.646i	− 0.0390016i
$300$	0	0
$301$	− 6660.13i	− 1.27536i
$302$	0	0
$303$	− 17180.1i	− 3.25732i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6736.05	1.25227	0.626135	−	0.779715i	$-0.284636\pi$
$307$	6736.05	1.25227	0.626135	+	0.779715i	$0.284636\pi$
$308$	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
$313$	− 9357.60i	− 1.68985i	−0.534885	−	0.844925i	$-0.679645\pi$
$313$	− 9357.60i	− 1.68985i	0.534885	−	0.844925i	$-0.320355\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9064.68	1.60607	0.803033	−	0.595934i	$-0.203218\pi$
$317$	9064.68	1.60607	0.803033	+	0.595934i	$0.203218\pi$
$318$	0	0
$319$	3898.47	0.684239
$320$	0	0
$321$	644.222	0.112015
$322$	0	0
$323$	386.825	0.0666362
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13977.1i	2.36371i
$328$	0	0
$329$	−2094.65	−0.351008
$330$	0	0
$331$	280.089i	0.0465108i	0.999730	+	0.0232554i	$0.00740310\pi$
$331$	280.089i	0.0465108i	−0.999730	+	0.0232554i	$0.992597\pi$
$332$	0	0
$333$	8477.98	1.39517
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5748.75i	0.929242i	0.885510	+	0.464621i	$0.153809\pi$
$337$	5748.75i	0.929242i	−0.885510	+	0.464621i	$0.846191\pi$
$338$	0	0
$339$	− 4389.65i	− 0.703284i
$340$	0	0
$341$	7617.30i	1.20968i
$342$	0	0
$343$	4761.06i	0.749484i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2911.52	0.450428	0.225214	−	0.974309i	$-0.427692\pi$
$347$	2911.52	0.450428	0.225214	+	0.974309i	$0.427692\pi$
$348$	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
$353$	− 9428.18i	− 1.42156i	−0.703414	−	0.710781i	$-0.748342\pi$
$353$	− 9428.18i	− 1.42156i	0.703414	−	0.710781i	$-0.251658\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2994.74	0.443973
$358$	0	0
$359$	4573.16	0.672318	0.336159	−	0.941805i	$-0.390872\pi$
$359$	4573.16	0.672318	0.336159	+	0.941805i	$0.390872\pi$
$360$	0	0
$361$	6146.71	0.896153
$362$	0	0
$363$	568.300	0.0821709
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2717.97i	− 0.386585i	−0.981141	−	0.193293i	$-0.938083\pi$
$367$	− 2717.97i	− 0.386585i	0.981141	−	0.193293i	$-0.0619166\pi$
$368$	0	0
$369$	−28360.5	−4.00105
$370$	0	0
$371$	− 3332.65i	− 0.466369i
$372$	0	0
$373$	5627.51	0.781183	0.390592	−	0.920564i	$-0.372270\pi$
$373$	5627.51	0.781183	0.390592	+	0.920564i	$0.372270\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2520.74i	− 0.344363i
$378$	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
$383$	− 10607.8i	− 1.41523i	−0.706600	−	0.707613i	$-0.749772\pi$
$383$	− 10607.8i	− 1.41523i	0.706600	−	0.707613i	$-0.250228\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19994.1	−2.62625
$388$	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
$393$	− 17095.3i	− 2.19426i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8977.17	1.13489	0.567445	−	0.823411i	$-0.307932\pi$
$397$	8977.17	1.13489	0.567445	+	0.823411i	$0.307932\pi$
$398$	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
$403$	4925.33	0.608804
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4881.75i	− 0.594543i
$408$	0	0
$409$	5417.61	0.654972	0.327486	−	0.944856i	$-0.393798\pi$
$409$	5417.61	0.654972	0.327486	+	0.944856i	$0.393798\pi$
$410$	0	0
$411$	22803.5i	2.73678i
$412$	0	0
$413$	11736.1	1.39830
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 27381.5i	− 3.21554i
$418$	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	$0.0809154\pi$
$421$	4344.57i	0.502949i	−0.967864	+	0.251474i	$0.919085\pi$
$422$	0	0
$423$	6288.27i	0.722804i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8080.25	0.915763
$428$	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
$433$	4210.26i	0.467280i	0.972323	+	0.233640i	$0.0750638\pi$
$433$	4210.26i	0.467280i	−0.972323	+	0.233640i	$0.924936\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	223.215	0.0244344
$438$	0	0
$439$	−5776.50	−0.628012	−0.314006	−	0.949421i	$-0.601671\pi$
$439$	−5776.50	−0.628012	−0.314006	+	0.949421i	$0.601671\pi$
$440$	0	0
$441$	−7918.08	−0.854992
$442$	0	0
$443$	−11981.6	−1.28502	−0.642510	−	0.766278i	$-0.722107\pi$
$443$	−11981.6	−1.28502	−0.642510	+	0.766278i	$0.722107\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8600.84i	− 0.910080i
$448$	0	0
$449$	5567.58	0.585190	0.292595	−	0.956236i	$-0.405481\pi$
$449$	5567.58	0.585190	0.292595	+	0.956236i	$0.405481\pi$
$450$	0	0
$451$	16330.4i	1.70503i
$452$	0	0
$453$	−32868.3	−3.40902
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7283.56i	0.745538i	0.927924	+	0.372769i	$0.121592\pi$
$457$	7283.56i	0.745538i	−0.927924	+	0.372769i	$0.878408\pi$
$458$	0	0
$459$	− 5241.82i	− 0.533044i
$460$	0	0
$461$	11901.1i	1.20236i	0.799113	+	0.601181i	$0.205303\pi$
$461$	11901.1i	1.20236i	−0.799113	+	0.601181i	$0.794697\pi$
$462$	0	0
$463$	− 4915.73i	− 0.493419i	−0.969089	−	0.246710i	$-0.920651\pi$
$463$	− 4915.73i	− 0.493419i	0.969089	−	0.246710i	$-0.0793494\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3110.15	−0.308181	−0.154091	−	0.988057i	$-0.549245\pi$
$467$	−3110.15	−0.308181	−0.154091	+	0.988057i	$0.549245\pi$
$468$	0	0
$469$	418.424i	0.0411962i
$470$	0	0
$471$	5473.09	0.535428
$472$	0	0
$473$	11512.9i	1.11916i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10004.8	−0.960358
$478$	0	0
$479$	6590.43	0.628652	0.314326	−	0.949315i	$-0.398221\pi$
$479$	6590.43	0.628652	0.314326	+	0.949315i	$0.398221\pi$
$480$	0	0
$481$	−3156.52	−0.299221
$482$	0	0
$483$	1728.10	0.162797
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3139.99i	0.292169i	0.989272	+	0.146085i	$0.0466672\pi$
$487$	3139.99i	0.292169i	−0.989272	+	0.146085i	$0.953333\pi$
$488$	0	0
$489$	−2269.83	−0.209909
$490$	0	0
$491$	− 12057.6i	− 1.10825i	−0.832433	−	0.554126i	$-0.813053\pi$
$491$	− 12057.6i	− 1.10825i	0.832433	−	0.554126i	$-0.186947\pi$
$492$	0	0
$493$	−1515.38	−0.138437
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20615.3i	1.86061i
$498$	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
$503$	− 16896.2i	− 1.49774i	−0.662718	−	0.748869i	$-0.730597\pi$
$503$	− 16896.2i	− 1.49774i	0.662718	−	0.748869i	$-0.269403\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15476.8	1.35572
$508$	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
$513$	9652.10i	0.830703i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3620.88	0.308019
$518$	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
$523$	−3115.75	−0.260502	−0.130251	−	0.991481i	$-0.541578\pi$
$523$	−3115.75	−0.260502	−0.130251	+	0.991481i	$0.541578\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2960.94i	− 0.244745i
$528$	0	0
$529$	12097.0	0.994251
$530$	0	0
$531$	− 35232.6i	− 2.87941i
$532$	0	0
$533$	10559.2	0.858103
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1355.64i	− 0.108939i
$538$	0	0
$539$	4559.35i	0.364350i
$540$	0	0
$541$	− 14840.6i	− 1.17938i	−0.807629	−	0.589691i	$-0.799249\pi$
$541$	− 14840.6i	− 1.17938i	0.807629	−	0.589691i	$-0.200751\pi$
$542$	0	0
$543$	33121.8i	2.61767i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12622.8	0.986679	0.493340	−	0.869837i	$-0.335776\pi$
$547$	12622.8	0.986679	0.493340	+	0.869837i	$0.335776\pi$
$548$	0	0
$549$	− 24257.5i	− 1.88576i
$550$	0	0
$551$	2790.37	0.215742
$552$	0	0
$553$	21006.7i	1.61536i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13382.1	−1.01798	−0.508992	−	0.860771i	$-0.669982\pi$
$557$	−13382.1	−1.01798	−0.508992	+	0.860771i	$0.669982\pi$
$558$	0	0
$559$	7444.23	0.563251
$560$	0	0
$561$	−5176.80	−0.389599
$562$	0	0
$563$	153.895	0.0115203	0.00576014	−	0.999983i	$-0.498166\pi$
$563$	153.895	0.0115203	0.00576014	+	0.999983i	$0.498166\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	37011.7i	2.74135i
$568$	0	0
$569$	5395.71	0.397539	0.198770	−	0.980046i	$-0.436305\pi$
$569$	5395.71	0.397539	0.198770	+	0.980046i	$0.436305\pi$
$570$	0	0
$571$	− 22390.5i	− 1.64100i	−0.571646	−	0.820500i	$-0.693695\pi$
$571$	− 22390.5i	− 1.64100i	0.571646	−	0.820500i	$-0.306305\pi$
$572$	0	0
$573$	−13573.2	−0.989578
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1935.38i	0.139638i	0.997560	+	0.0698188i	$0.0222421\pi$
$577$	1935.38i	0.139638i	−0.997560	+	0.0698188i	$0.977758\pi$
$578$	0	0
$579$	25682.2i	1.84338i
$580$	0	0
$581$	1186.82i	0.0847461i
$582$	0	0
$583$	5760.94i	0.409252i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7905.44	−0.555865	−0.277932	−	0.960601i	$-0.589649\pi$
$587$	−7905.44	−0.555865	−0.277932	+	0.960601i	$0.589649\pi$
$588$	0	0
$589$	5452.17i	0.381414i
$590$	0	0
$591$	−25841.4	−1.79860
$592$	0	0
$593$	− 11141.6i	− 0.771552i	−0.922592	−	0.385776i	$-0.873934\pi$
$593$	− 11141.6i	− 0.771552i	0.922592	−	0.385776i	$-0.126066\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−33940.4	−2.32678
$598$	0	0
$599$	278.847	0.0190207	0.00951033	−	0.999955i	$-0.496973\pi$
$599$	278.847	0.0190207	0.00951033	+	0.999955i	$0.496973\pi$
$600$	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
$603$	1256.14	0.0848321
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5398.05i	0.360956i	0.983579	+	0.180478i	$0.0577644\pi$
$607$	5398.05i	0.360956i	−0.983579	+	0.180478i	$0.942236\pi$
$608$	0	0
$609$	21602.6	1.43741
$610$	0	0
$611$	− 2341.25i	− 0.155019i
$612$	0	0
$613$	−14412.8	−0.949635	−0.474817	−	0.880084i	$-0.657486\pi$
$613$	−14412.8	−0.949635	−0.474817	+	0.880084i	$0.657486\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13226.3i	− 0.863001i	−0.902113	−	0.431500i	$-0.857984\pi$
$617$	− 13226.3i	− 0.863001i	0.902113	−	0.431500i	$-0.142016\pi$
$618$	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
$623$	19840.6i	1.27592i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9532.38	0.607156
$628$	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
$633$	1457.96i	0.0915459i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2948.06	0.183370
$638$	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
$643$	16498.3	1.01186	0.505931	−	0.862574i	$-0.331149\pi$
$643$	16498.3	1.01186	0.505931	+	0.862574i	$0.331149\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 20127.5i	− 1.22302i	−0.791236	−	0.611510i	$-0.790562\pi$
$647$	− 20127.5i	− 1.22302i	0.791236	−	0.611510i	$-0.209438\pi$
$648$	0	0
$649$	−20287.5	−1.22705
$650$	0	0
$651$	42209.9i	2.54122i
$652$	0	0
$653$	21447.4	1.28530	0.642650	−	0.766160i	$-0.277835\pi$
$653$	21447.4	1.28530	0.642650	+	0.766160i	$0.277835\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8250.91i	0.489953i
$658$	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	$0.0884786\pi$
$661$	9326.38i	0.548796i	−0.961616	+	0.274398i	$0.911521\pi$
$662$	0	0
$663$	3347.31i	0.196076i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−874.443	−0.0507625
$668$	0	0
$669$	− 19784.3i	− 1.14336i
$670$	0	0
$671$	−13967.8	−0.803608
$672$	0	0
$673$	− 15027.4i	− 0.860718i	−0.902658	−	0.430359i	$-0.858387\pi$
$673$	− 15027.4i	− 0.860718i	0.902658	−	0.430359i	$-0.141613\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9717.31	−0.551649	−0.275825	−	0.961208i	$-0.588951\pi$
$677$	−9717.31	−0.551649	−0.275825	+	0.961208i	$0.588951\pi$
$678$	0	0
$679$	6427.63	0.363284
$680$	0	0
$681$	−28166.3	−1.58493
$682$	0	0
$683$	−11473.9	−0.642804	−0.321402	−	0.946943i	$-0.604154\pi$
$683$	−11473.9	−0.642804	−0.321402	+	0.946943i	$0.604154\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 47413.4i	− 2.63309i
$688$	0	0
$689$	3725.01	0.205967
$690$	0	0
$691$	− 28412.6i	− 1.56421i	−0.623149	−	0.782103i	$-0.714147\pi$
$691$	− 28412.6i	− 1.56421i	0.623149	−	0.782103i	$-0.285853\pi$
$692$	0	0
$693$	52082.3	2.85490
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6347.82i	− 0.344965i
$698$	0	0
$699$	− 49746.2i	− 2.69181i
$700$	0	0
$701$	12291.3i	0.662250i	0.943587	+	0.331125i	$0.107428\pi$
$701$	12291.3i	0.662250i	−0.943587	+	0.331125i	$0.892572\pi$
$702$	0	0
$703$	− 3494.17i	− 0.187461i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38686.9	−2.05795
$708$	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
$713$	− 1708.59i	− 0.0897438i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19792.9	1.03093
$718$	0	0
$719$	−15748.6	−0.816861	−0.408430	−	0.912790i	$-0.633924\pi$
$719$	−15748.6	−0.816861	−0.408430	+	0.912790i	$0.633924\pi$
$720$	0	0
$721$	44954.8	2.32206
$722$	0	0
$723$	22118.4	1.13775
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9536.44i	0.486502i	0.969963	+	0.243251i	$0.0782138\pi$
$727$	9536.44i	0.486502i	−0.969963	+	0.243251i	$0.921786\pi$
$728$	0	0
$729$	17576.5	0.892980
$730$	0	0
$731$	− 4475.22i	− 0.226432i
$732$	0	0
$733$	−34913.5	−1.75929	−0.879645	−	0.475631i	$-0.842220\pi$
$733$	−34913.5	−1.75929	−0.879645	+	0.475631i	$0.842220\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 723.301i	− 0.0361508i
$738$	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
$743$	7053.57i	0.348278i	0.984721	+	0.174139i	$0.0557142\pi$
$743$	7053.57i	0.348278i	−0.984721	+	0.174139i	$0.944286\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3562.91	0.174511
$748$	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
$753$	36424.7i	1.76280i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28515.3	1.36910	0.684549	−	0.728967i	$-0.259999\pi$
$757$	28515.3	1.36910	0.684549	+	0.728967i	$0.259999\pi$
$758$	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
$763$	31474.3	1.49338
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13117.8i	0.617545i
$768$	0	0
$769$	10413.9	0.488341	0.244171	−	0.969732i	$-0.421484\pi$
$769$	10413.9	0.488341	0.244171	+	0.969732i	$0.421484\pi$
$770$	0	0
$771$	59088.0i	2.76006i
$772$	0	0
$773$	32909.8	1.53129	0.765643	−	0.643266i	$-0.222421\pi$
$773$	32909.8	1.53129	0.765643	+	0.643266i	$0.222421\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 27051.3i	− 1.24898i
$778$	0	0
$779$	11688.6i	0.537599i
$780$	0	0
$781$	− 35636.2i	− 1.63273i
$782$	0	0
$783$	− 37812.1i	− 1.72579i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35511.6	1.60845	0.804226	−	0.594324i	$-0.202580\pi$
$787$	35511.6	1.60845	0.804226	+	0.594324i	$0.202580\pi$
$788$	0	0
$789$	− 15672.1i	− 0.707150i
$790$	0	0
$791$	−9884.85	−0.444330
$792$	0	0
$793$	9031.54i	0.404438i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22692.3	1.00853	0.504267	−	0.863548i	$-0.331763\pi$
$797$	22692.3	1.00853	0.504267	+	0.863548i	$0.331763\pi$
$798$	0	0
$799$	−1407.48	−0.0623193
$800$	0	0
$801$	59562.8	2.62740
$802$	0	0
$803$	4751.00	0.208791
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21669.8i	0.945246i
$808$	0	0
$809$	22716.0	0.987209	0.493604	−	0.869687i	$-0.335679\pi$
$809$	22716.0	0.987209	0.493604	+	0.869687i	$0.335679\pi$
$810$	0	0
$811$	17237.0i	0.746327i	0.927766	+	0.373164i	$0.121727\pi$
$811$	17237.0i	0.746327i	−0.927766	+	0.373164i	$0.878273\pi$
$812$	0	0
$813$	398.847	0.0172056
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8240.51i	0.352875i
$818$	0	0
$819$	− 33676.3i	− 1.43681i
$820$	0	0
$821$	− 15695.7i	− 0.667217i	−0.942712	−	0.333609i	$-0.891734\pi$
$821$	− 15695.7i	− 0.667217i	0.942712	−	0.333609i	$-0.108266\pi$
$822$	0	0
$823$	− 16856.0i	− 0.713927i	−0.934118	−	0.356963i	$-0.883812\pi$
$823$	− 16856.0i	− 0.713927i	0.934118	−	0.356963i	$-0.116188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3712.67	0.156109	0.0780544	−	0.996949i	$-0.475129\pi$
$827$	3712.67	0.156109	0.0780544	+	0.996949i	$0.475129\pi$
$828$	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
$833$	− 1772.27i	− 0.0737163i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	73881.8	3.05105
$838$	0	0
$839$	−31935.2	−1.31409	−0.657046	−	0.753850i	$-0.728194\pi$
$839$	−31935.2	−1.31409	−0.657046	+	0.753850i	$0.728194\pi$
$840$	0	0
$841$	13457.7	0.551795
$842$	0	0
$843$	14306.3	0.584504
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1279.73i	− 0.0519150i
$848$	0	0
$849$	−10442.9	−0.422144
$850$	0	0
$851$	1095.00i	0.0441081i
$852$	0	0
$853$	14782.2	0.593355	0.296678	−	0.954978i	$-0.404121\pi$
$853$	14782.2	0.593355	0.296678	+	0.954978i	$0.404121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7222.91i	0.287900i	0.989585	+	0.143950i	$0.0459804\pi$
$857$	7222.91i	0.287900i	−0.989585	+	0.143950i	$0.954020\pi$
$858$	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
$863$	− 8122.19i	− 0.320374i	−0.987087	−	0.160187i	$-0.948790\pi$
$863$	− 8122.19i	− 0.320374i	0.987087	−	0.160187i	$-0.0512097\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−45048.9	−1.76464
$868$	0	0
$869$	− 36312.8i	− 1.41752i
$870$	0	0
$871$	−467.685	−0.0181939
$872$	0	0
$873$	− 19296.2i	− 0.748083i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31986.1	−1.23158	−0.615790	−	0.787910i	$-0.711163\pi$
$877$	−31986.1	−1.23158	−0.615790	+	0.787910i	$0.711163\pi$
$878$	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
$883$	−46156.5	−1.75911	−0.879553	−	0.475801i	$-0.842158\pi$
$883$	−46156.5	−1.75911	−0.879553	+	0.475801i	$0.842158\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41051.2i	1.55396i	0.629523	+	0.776982i	$0.283250\pi$
$887$	41051.2i	1.55396i	−0.629523	+	0.776982i	$0.716750\pi$
$888$	0	0
$889$	−4501.39	−0.169822
$890$	0	0
$891$	− 63979.7i	− 2.40561i
$892$	0	0
$893$	2591.69	0.0971192
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1931.54i	0.0718979i
$898$	0	0
$899$	− 21358.8i	− 0.792389i
$900$	0	0
$901$	− 2239.35i	− 0.0828008i
$902$	0	0
$903$	63796.7i	2.35108i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27915.8	−1.02197	−0.510987	−	0.859588i	$-0.670720\pi$
$907$	−27915.8	−1.02197	−0.510987	+	0.859588i	$0.670720\pi$
$908$	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
$913$	− 2051.57i	− 0.0743671i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38496.2	−1.38632
$918$	0	0
$919$	−11493.0	−0.412534	−0.206267	−	0.978496i	$-0.566131\pi$
$919$	−11493.0	−0.412534	−0.206267	+	0.978496i	$0.566131\pi$
$920$	0	0
$921$	−64524.0	−2.30851
$922$	0	0
$923$	−23042.3	−0.821719
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 134957.i	− 4.78164i
$928$	0	0
$929$	−56597.2	−1.99881	−0.999405	−	0.0345020i	$-0.989015\pi$
$929$	−56597.2	−1.99881	−0.999405	+	0.0345020i	$0.989015\pi$
$930$	0	0
$931$	3263.40i	0.114880i
$932$	0	0
$933$	31364.7	1.10057
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37901.5i	1.32144i	0.750633	+	0.660720i	$0.229749\pi$
$937$	37901.5i	1.32144i	−0.750633	+	0.660720i	$0.770251\pi$
$938$	0	0
$939$	89635.5i	3.11517i
$940$	0	0
$941$	4449.85i	0.154156i	0.997025	+	0.0770781i	$0.0245591\pi$
$941$	4449.85i	0.154156i	−0.997025	+	0.0770781i	$0.975441\pi$
$942$	0	0
$943$	− 3662.97i	− 0.126493i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	225.160	0.00772620	0.00386310	−	0.999993i	$-0.498770\pi$
$947$	225.160	0.00772620	0.00386310	+	0.999993i	$0.498770\pi$
$948$	0	0
$949$	− 3071.98i	− 0.105080i
$950$	0	0
$951$	−86829.7	−2.96072
$952$	0	0
$953$	4845.81i	0.164713i	0.996603	+	0.0823563i	$0.0262445\pi$
$953$	4845.81i	0.164713i	−0.996603	+	0.0823563i	$0.973755\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−37343.1	−1.26137
$958$	0	0
$959$	51350.2	1.72908
$960$	0	0
$961$	11942.5	0.400876
$962$	0	0
$963$	−4355.07	−0.145732
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40394.8i	1.34334i	0.740851	+	0.671670i	$0.234423\pi$
$967$	40394.8i	1.34334i	−0.740851	+	0.671670i	$0.765577\pi$
$968$	0	0
$969$	−3705.36	−0.122841
$970$	0	0
$971$	29066.2i	0.960637i	0.877094	+	0.480318i	$0.159479\pi$
$971$	29066.2i	0.960637i	−0.877094	+	0.480318i	$0.840521\pi$
$972$	0	0
$973$	−61659.1	−2.03155
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42580.5i	− 1.39434i	−0.716906	−	0.697170i	$-0.754442\pi$
$977$	− 42580.5i	− 1.39434i	0.716906	−	0.697170i	$-0.245558\pi$
$978$	0	0
$979$	− 34297.1i	− 1.11965i
$980$	0	0
$981$	− 94488.0i	− 3.07520i
$982$	0	0
$983$	40586.0i	1.31688i	0.752633	+	0.658440i	$0.228783\pi$
$983$	40586.0i	1.31688i	−0.752633	+	0.658440i	$0.771217\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	20064.4	0.647070
$988$	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
$993$	− 2682.95i	− 0.0857409i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22467.5	−0.713694	−0.356847	−	0.934163i	$-0.616148\pi$
$997$	−22467.5	−0.713694	−0.356847	+	0.934163i	$0.616148\pi$
$998$	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.f.b.49.2		12
4.3	odd	2		200.4.f.c.149.9		12
5.2	odd	4		160.4.d.a.81.12		12
5.3	odd	4		800.4.d.d.401.1		12
5.4	even	2		800.4.f.c.49.11		12
8.3	odd	2		200.4.f.b.149.3		12
8.5	even	2		800.4.f.c.49.12		12
15.2	even	4		1440.4.k.c.721.8		12
20.3	even	4		200.4.d.b.101.3		12
20.7	even	4		40.4.d.a.21.10	yes	12
20.19	odd	2		200.4.f.b.149.4		12
40.3	even	4		200.4.d.b.101.4		12
40.13	odd	4		800.4.d.d.401.12		12
40.19	odd	2		200.4.f.c.149.10		12
40.27	even	4		40.4.d.a.21.9	✓	12
40.29	even	2	inner	800.4.f.b.49.1		12
40.37	odd	4		160.4.d.a.81.1		12
60.47	odd	4		360.4.k.c.181.3		12
80.27	even	4		1280.4.a.bb.1.1		6
80.37	odd	4		1280.4.a.bd.1.6		6
80.67	even	4		1280.4.a.bc.1.6		6
80.77	odd	4		1280.4.a.ba.1.1		6
120.77	even	4		1440.4.k.c.721.2		12
120.107	odd	4		360.4.k.c.181.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.9	✓	12	40.27	even	4
40.4.d.a.21.10	yes	12	20.7	even	4
160.4.d.a.81.1		12	40.37	odd	4
160.4.d.a.81.12		12	5.2	odd	4
200.4.d.b.101.3		12	20.3	even	4
200.4.d.b.101.4		12	40.3	even	4
200.4.f.b.149.3		12	8.3	odd	2
200.4.f.b.149.4		12	20.19	odd	2
200.4.f.c.149.9		12	4.3	odd	2
200.4.f.c.149.10		12	40.19	odd	2
360.4.k.c.181.3		12	60.47	odd	4
360.4.k.c.181.4		12	120.107	odd	4
800.4.d.d.401.1		12	5.3	odd	4
800.4.d.d.401.12		12	40.13	odd	4
800.4.f.b.49.1		12	40.29	even	2	inner
800.4.f.b.49.2		12	1.1	even	1	trivial
800.4.f.c.49.11		12	5.4	even	2
800.4.f.c.49.12		12	8.5	even	2
1280.4.a.ba.1.1		6	80.77	odd	4
1280.4.a.bb.1.1		6	80.27	even	4
1280.4.a.bc.1.6		6	80.67	even	4
1280.4.a.bd.1.6		6	80.37	odd	4
1440.4.k.c.721.2		12	120.77	even	4
1440.4.k.c.721.8		12	15.2	even	4

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

\(f(q)\)	\(=\)	\(q-9.57890 q^{3} +21.5703i q^{7} +64.7554 q^{9} -37.2871i q^{11} -24.1097 q^{13} +14.4940i q^{17} -26.6887i q^{19} -206.620i q^{21} +8.36366i q^{23} -361.655 q^{27} +104.553i q^{29} -204.288 q^{31} +357.170i q^{33} +130.923 q^{37} +230.945 q^{39} -437.963 q^{41} -308.764 q^{43} +97.1081i q^{47} -122.277 q^{49} -138.836i q^{51} -154.502 q^{53} +255.648i q^{57} -544.088i q^{59} -374.601i q^{61} +1396.79i q^{63} +19.3982 q^{67} -80.1147i q^{69} +955.725 q^{71} +127.417i q^{73} +804.293 q^{77} +973.871 q^{79} +1715.87 q^{81} +55.0210 q^{83} -1001.50i q^{87} +919.812 q^{89} -520.054i q^{91} +1956.85 q^{93} -297.986i q^{97} -2414.54i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 108 q^{9} - 432 q^{27} + 264 q^{31} + 136 q^{37} + 600 q^{39} + 40 q^{41} + 1204 q^{43} - 1308 q^{49} + 1056 q^{53} + 2412 q^{67} + 1592 q^{71} + 824 q^{77} + 2016 q^{79} + 2508 q^{81}+ \cdots + 2784 q^{93}+O(q^{100}) \) 12 * q - 12 * q^3 + 108 * q^9 - 432 * q^27 + 264 * q^31 + 136 * q^37 + 600 * q^39 + 40 * q^41 + 1204 * q^43 - 1308 * q^49 + 1056 * q^53 + 2412 * q^67 + 1592 * q^71 + 824 * q^77 + 2016 * q^79 + 2508 * q^81 - 3556 * q^83 + 424 * q^89 + 2784 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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