LMFDB - Embedded newform 800.4.c.m.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2068430400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 23x^{4} + 133x^{2} + 36 $ x^6 + 23x^4 + 133x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.533386i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.m.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06677i q^{3} +28.8620i q^{7} +22.7285 q^{9} +O(q^{10})$	$q-2.06677i q^{3} +28.8620i q^{7} +22.7285 q^{9} +18.7952 q^{11} +86.7195i q^{13} -64.7968i q^{17} -27.2566 q^{19} +59.6512 q^{21} +102.125i q^{23} -102.777i q^{27} -8.87408 q^{29} -272.771 q^{31} -38.8455i q^{33} -82.4677i q^{37} +179.230 q^{39} -249.119 q^{41} -137.227i q^{43} +439.114i q^{47} -490.015 q^{49} -133.920 q^{51} -490.724i q^{53} +56.3332i q^{57} +530.319 q^{59} -407.580 q^{61} +655.989i q^{63} +595.664i q^{67} +211.068 q^{69} -569.274 q^{71} +435.927i q^{73} +542.468i q^{77} -678.589 q^{79} +401.251 q^{81} +1277.18i q^{83} +18.3407i q^{87} +711.431 q^{89} -2502.90 q^{91} +563.756i q^{93} -1741.08i q^{97} +427.186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 36 q^{9}+O(q^{10}) $ 6 * q - 36 * q^9	$ 6 q - 36 q^{9} - 62 q^{11} + 174 q^{19} - 140 q^{21} - 168 q^{29} - 588 q^{31} - 64 q^{39} - 690 q^{41} - 718 q^{49} - 2702 q^{51} + 2080 q^{59} + 964 q^{61} + 1228 q^{69} - 4096 q^{71} + 1996 q^{79} - 1098 q^{81} - 2378 q^{89} - 8064 q^{91} + 3908 q^{99}+O(q^{100}) $ 6 * q - 36 * q^9 - 62 * q^11 + 174 * q^19 - 140 * q^21 - 168 * q^29 - 588 * q^31 - 64 * q^39 - 690 * q^41 - 718 * q^49 - 2702 * q^51 + 2080 * q^59 + 964 * q^61 + 1228 * q^69 - 4096 * q^71 + 1996 * q^79 - 1098 * q^81 - 2378 * q^89 - 8064 * q^91 + 3908 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 2.06677i	− 0.397751i	−0.980025	−	0.198875i	$-0.936271\pi$
$3$	− 2.06677i	− 0.397751i	0.980025	−	0.198875i	$-0.0637289\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.8620i	1.55840i	0.626775	+	0.779201i	$0.284375\pi$
$7$	28.8620i	1.55840i	−0.626775	+	0.779201i	$0.715625\pi$
$8$	0	0
$9$	22.7285	0.841795
$10$	0	0
$11$	18.7952	0.515179	0.257590	−	0.966254i	$-0.417072\pi$
$11$	18.7952	0.515179	0.257590	+	0.966254i	$0.417072\pi$
$12$	0	0
$13$	86.7195i	1.85013i	0.379811	+	0.925064i	$0.375989\pi$
$13$	86.7195i	1.85013i	−0.379811	+	0.925064i	$0.624011\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 64.7968i	− 0.924443i	−0.886764	−	0.462222i	$-0.847052\pi$
$17$	− 64.7968i	− 0.924443i	0.886764	−	0.462222i	$-0.152948\pi$
$18$	0	0
$19$	−27.2566	−0.329110	−0.164555	−	0.986368i	$-0.552619\pi$
$19$	−27.2566	−0.329110	−0.164555	+	0.986368i	$0.552619\pi$
$20$	0	0
$21$	59.6512	0.619855
$22$	0	0
$23$	102.125i	0.925846i	0.886399	+	0.462923i	$0.153199\pi$
$23$	102.125i	0.925846i	−0.886399	+	0.462923i	$0.846801\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 102.777i	− 0.732575i
$28$	0	0
$29$	−8.87408	−0.0568233	−0.0284117	−	0.999596i	$-0.509045\pi$
$29$	−8.87408	−0.0568233	−0.0284117	+	0.999596i	$0.509045\pi$
$30$	0	0
$31$	−272.771	−1.58036	−0.790180	−	0.612874i	$-0.790013\pi$
$31$	−272.771	−1.58036	−0.790180	+	0.612874i	$0.790013\pi$
$32$	0	0
$33$	− 38.8455i	− 0.204913i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 82.4677i	− 0.366422i	−0.983074	−	0.183211i	$-0.941351\pi$
$37$	− 82.4677i	− 0.366422i	0.983074	−	0.183211i	$-0.0586491\pi$
$38$	0	0
$39$	179.230	0.735890
$40$	0	0
$41$	−249.119	−0.948923	−0.474461	−	0.880276i	$-0.657357\pi$
$41$	−249.119	−0.948923	−0.474461	+	0.880276i	$0.657357\pi$
$42$	0	0
$43$	− 137.227i	− 0.486672i	−0.969942	−	0.243336i	$-0.921758\pi$
$43$	− 137.227i	− 0.486672i	0.969942	−	0.243336i	$-0.0782418\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	439.114i	1.36280i	0.731913	+	0.681398i	$0.238628\pi$
$47$	439.114i	1.36280i	−0.731913	+	0.681398i	$0.761372\pi$
$48$	0	0
$49$	−490.015	−1.42861
$50$	0	0
$51$	−133.920	−0.367698
$52$	0	0
$53$	− 490.724i	− 1.27181i	−0.771766	−	0.635906i	$-0.780626\pi$
$53$	− 490.724i	− 1.27181i	0.771766	−	0.635906i	$-0.219374\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	56.3332i	0.130904i
$58$	0	0
$59$	530.319	1.17020	0.585099	−	0.810962i	$-0.301056\pi$
$59$	530.319	1.17020	0.585099	+	0.810962i	$0.301056\pi$
$60$	0	0
$61$	−407.580	−0.855496	−0.427748	−	0.903898i	$-0.640693\pi$
$61$	−407.580	−0.855496	−0.427748	+	0.903898i	$0.640693\pi$
$62$	0	0
$63$	655.989i	1.31185i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	595.664i	1.08615i	0.839685	+	0.543074i	$0.182740\pi$
$67$	595.664i	1.08615i	−0.839685	+	0.543074i	$0.817260\pi$
$68$	0	0
$69$	211.068	0.368256
$70$	0	0
$71$	−569.274	−0.951555	−0.475778	−	0.879566i	$-0.657833\pi$
$71$	−569.274	−0.951555	−0.475778	+	0.879566i	$0.657833\pi$
$72$	0	0
$73$	435.927i	0.698923i	0.936951	+	0.349461i	$0.113635\pi$
$73$	435.927i	0.698923i	−0.936951	+	0.349461i	$0.886365\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	542.468i	0.802856i
$78$	0	0
$79$	−678.589	−0.966421	−0.483210	−	0.875504i	$-0.660529\pi$
$79$	−678.589	−0.966421	−0.483210	+	0.875504i	$0.660529\pi$
$80$	0	0
$81$	401.251	0.550413
$82$	0	0
$83$	1277.18i	1.68902i	0.535543	+	0.844508i	$0.320107\pi$
$83$	1277.18i	1.68902i	−0.535543	+	0.844508i	$0.679893\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.3407i	0.0226015i
$88$	0	0
$89$	711.431	0.847321	0.423660	−	0.905821i	$-0.360745\pi$
$89$	711.431	0.847321	0.423660	+	0.905821i	$0.360745\pi$
$90$	0	0
$91$	−2502.90	−2.88324
$92$	0	0
$93$	563.756i	0.628589i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1741.08i	− 1.82247i	−0.411882	−	0.911237i	$-0.635129\pi$
$97$	− 1741.08i	− 1.82247i	0.411882	−	0.911237i	$-0.364871\pi$
$98$	0	0
$99$	427.186	0.433675
$100$	0	0
$101$	1768.54	1.74234	0.871170	−	0.490981i	$-0.163361\pi$
$101$	1768.54	1.74234	0.871170	+	0.490981i	$0.163361\pi$
$102$	0	0
$103$	− 24.9858i	− 0.0239022i	−0.999929	−	0.0119511i	$-0.996196\pi$
$103$	− 24.9858i	− 0.0239022i	0.999929	−	0.0119511i	$-0.00380424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1027.18i	0.928052i	0.885821	+	0.464026i	$0.153596\pi$
$107$	1027.18i	0.928052i	−0.885821	+	0.464026i	$0.846404\pi$
$108$	0	0
$109$	418.035	0.367344	0.183672	−	0.982988i	$-0.441202\pi$
$109$	418.035	0.367344	0.183672	+	0.982988i	$0.441202\pi$
$110$	0	0
$111$	−170.442	−0.145744
$112$	0	0
$113$	527.349i	0.439016i	0.975611	+	0.219508i	$0.0704452\pi$
$113$	527.349i	0.439016i	−0.975611	+	0.219508i	$0.929555\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1971.00i	1.55743i
$118$	0	0
$119$	1870.17	1.44065
$120$	0	0
$121$	−977.740	−0.734590
$122$	0	0
$123$	514.872i	0.377435i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1958.66i	1.36853i	0.729234	+	0.684264i	$0.239877\pi$
$127$	1958.66i	1.36853i	−0.729234	+	0.684264i	$0.760123\pi$
$128$	0	0
$129$	−283.617	−0.193574
$130$	0	0
$131$	−1566.40	−1.04471	−0.522354	−	0.852729i	$-0.674946\pi$
$131$	−1566.40	−1.04471	−0.522354	+	0.852729i	$0.674946\pi$
$132$	0	0
$133$	− 786.680i	− 0.512885i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	496.635i	0.309711i	0.987937	+	0.154856i	$0.0494912\pi$
$137$	496.635i	0.309711i	−0.987937	+	0.154856i	$0.950509\pi$
$138$	0	0
$139$	2675.60	1.63267	0.816335	−	0.577579i	$-0.196003\pi$
$139$	2675.60	1.63267	0.816335	+	0.577579i	$0.196003\pi$
$140$	0	0
$141$	907.550	0.542053
$142$	0	0
$143$	1629.91i	0.953148i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1012.75i	0.568232i
$148$	0	0
$149$	1454.05	0.799465	0.399732	−	0.916632i	$-0.369103\pi$
$149$	1454.05	0.799465	0.399732	+	0.916632i	$0.369103\pi$
$150$	0	0
$151$	−2359.49	−1.27160	−0.635802	−	0.771852i	$-0.719331\pi$
$151$	−2359.49	−1.27160	−0.635802	+	0.771852i	$0.719331\pi$
$152$	0	0
$153$	− 1472.73i	− 0.778191i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	533.577i	0.271236i	0.990761	+	0.135618i	$0.0433020\pi$
$157$	533.577i	0.271236i	−0.990761	+	0.135618i	$0.956698\pi$
$158$	0	0
$159$	−1014.21	−0.505864
$160$	0	0
$161$	−2947.52	−1.44284
$162$	0	0
$163$	1190.37i	0.572004i	0.958229	+	0.286002i	$0.0923264\pi$
$163$	1190.37i	0.572004i	−0.958229	+	0.286002i	$0.907674\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 294.777i	− 0.136590i	−0.997665	−	0.0682950i	$-0.978244\pi$
$167$	− 294.777i	− 0.136590i	0.997665	−	0.0682950i	$-0.0217559\pi$
$168$	0	0
$169$	−5323.28	−2.42298
$170$	0	0
$171$	−619.500	−0.277043
$172$	0	0
$173$	− 2806.61i	− 1.23343i	−0.787188	−	0.616713i	$-0.788464\pi$
$173$	− 2806.61i	− 1.23343i	0.787188	−	0.616713i	$-0.211536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1096.05i	− 0.465447i
$178$	0	0
$179$	−4593.72	−1.91816	−0.959079	−	0.283137i	$-0.908625\pi$
$179$	−4593.72	−1.91816	−0.959079	+	0.283137i	$0.908625\pi$
$180$	0	0
$181$	−1411.60	−0.579689	−0.289844	−	0.957074i	$-0.593604\pi$
$181$	−1411.60	−0.579689	−0.289844	+	0.957074i	$0.593604\pi$
$182$	0	0
$183$	842.375i	0.340274i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1217.87i	− 0.476254i
$188$	0	0
$189$	2966.36	1.14165
$190$	0	0
$191$	−3918.19	−1.48435	−0.742174	−	0.670207i	$-0.766205\pi$
$191$	−3918.19	−1.48435	−0.742174	+	0.670207i	$0.766205\pi$
$192$	0	0
$193$	2227.06i	0.830606i	0.909683	+	0.415303i	$0.136325\pi$
$193$	2227.06i	0.830606i	−0.909683	+	0.415303i	$0.863675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3321.98i	1.20143i	0.799464	+	0.600714i	$0.205117\pi$
$197$	3321.98i	1.20143i	−0.799464	+	0.600714i	$0.794883\pi$
$198$	0	0
$199$	1939.22	0.690792	0.345396	−	0.938457i	$-0.387745\pi$
$199$	1939.22	0.690792	0.345396	+	0.938457i	$0.387745\pi$
$200$	0	0
$201$	1231.10	0.432016
$202$	0	0
$203$	− 256.124i	− 0.0885535i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2321.13i	0.779372i
$208$	0	0
$209$	−512.294	−0.169551
$210$	0	0
$211$	4774.66	1.55783	0.778913	−	0.627132i	$-0.215771\pi$
$211$	4774.66	1.55783	0.778913	+	0.627132i	$0.215771\pi$
$212$	0	0
$213$	1176.56i	0.378482i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7872.73i	− 2.46284i
$218$	0	0
$219$	900.961	0.277997
$220$	0	0
$221$	5619.15	1.71034
$222$	0	0
$223$	1874.06i	0.562764i	0.959596	+	0.281382i	$0.0907928\pi$
$223$	1874.06i	0.562764i	−0.959596	+	0.281382i	$0.909207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1933.07i	0.565209i	0.959237	+	0.282604i	$0.0911983\pi$
$227$	1933.07i	0.565209i	−0.959237	+	0.282604i	$0.908802\pi$
$228$	0	0
$229$	162.819	0.0469843	0.0234921	−	0.999724i	$-0.492522\pi$
$229$	162.819	0.0469843	0.0234921	+	0.999724i	$0.492522\pi$
$230$	0	0
$231$	1121.16	0.319337
$232$	0	0
$233$	3496.60i	0.983133i	0.870840	+	0.491566i	$0.163576\pi$
$233$	3496.60i	0.983133i	−0.870840	+	0.491566i	$0.836424\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1402.49i	0.384394i
$238$	0	0
$239$	3571.35	0.966576	0.483288	−	0.875462i	$-0.339442\pi$
$239$	3571.35	0.966576	0.483288	+	0.875462i	$0.339442\pi$
$240$	0	0
$241$	−2751.24	−0.735365	−0.367682	−	0.929951i	$-0.619849\pi$
$241$	−2751.24	−0.735365	−0.367682	+	0.929951i	$0.619849\pi$
$242$	0	0
$243$	− 3604.28i	− 0.951502i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2363.68i	− 0.608896i
$248$	0	0
$249$	2639.63	0.671807
$250$	0	0
$251$	2136.31	0.537222	0.268611	−	0.963249i	$-0.413435\pi$
$251$	2136.31	0.537222	0.268611	+	0.963249i	$0.413435\pi$
$252$	0	0
$253$	1919.46i	0.476977i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4446.43i	− 1.07922i	−0.841914	−	0.539612i	$-0.818571\pi$
$257$	− 4446.43i	− 1.07922i	0.841914	−	0.539612i	$-0.181429\pi$
$258$	0	0
$259$	2380.18	0.571032
$260$	0	0
$261$	−201.694	−0.0478336
$262$	0	0
$263$	− 6431.62i	− 1.50795i	−0.656903	−	0.753975i	$-0.728134\pi$
$263$	− 6431.62i	− 1.50795i	0.656903	−	0.753975i	$-0.271866\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1470.37i	− 0.337022i
$268$	0	0
$269$	−1026.67	−0.232703	−0.116352	−	0.993208i	$-0.537120\pi$
$269$	−1026.67	−0.232703	−0.116352	+	0.993208i	$0.537120\pi$
$270$	0	0
$271$	3555.80	0.797046	0.398523	−	0.917158i	$-0.369523\pi$
$271$	3555.80	0.797046	0.398523	+	0.917158i	$0.369523\pi$
$272$	0	0
$273$	5172.92i	1.14681i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4641.61i	1.00681i	0.864050	+	0.503406i	$0.167920\pi$
$277$	4641.61i	1.00681i	−0.864050	+	0.503406i	$0.832080\pi$
$278$	0	0
$279$	−6199.67	−1.33034
$280$	0	0
$281$	−5560.83	−1.18054	−0.590269	−	0.807206i	$-0.700978\pi$
$281$	−5560.83	−1.18054	−0.590269	+	0.807206i	$0.700978\pi$
$282$	0	0
$283$	960.808i	0.201817i	0.994896	+	0.100908i	$0.0321749\pi$
$283$	960.808i	0.201817i	−0.994896	+	0.100908i	$0.967825\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7190.07i	− 1.47880i
$288$	0	0
$289$	714.373	0.145405
$290$	0	0
$291$	−3598.42	−0.724890
$292$	0	0
$293$	− 6120.18i	− 1.22029i	−0.792290	−	0.610144i	$-0.791111\pi$
$293$	− 6120.18i	− 1.22029i	0.792290	−	0.610144i	$-0.208889\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1931.72i	− 0.377407i
$298$	0	0
$299$	−8856.20	−1.71293
$300$	0	0
$301$	3960.64	0.758431
$302$	0	0
$303$	− 3655.17i	− 0.693017i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8419.04i	− 1.56515i	−0.622558	−	0.782574i	$-0.713907\pi$
$307$	− 8419.04i	− 1.56515i	0.622558	−	0.782574i	$-0.286093\pi$
$308$	0	0
$309$	−51.6400	−0.00950711
$310$	0	0
$311$	10558.8	1.92519	0.962594	−	0.270947i	$-0.0873369\pi$
$311$	10558.8	1.92519	0.962594	+	0.270947i	$0.0873369\pi$
$312$	0	0
$313$	− 7401.02i	− 1.33652i	−0.743928	−	0.668259i	$-0.767040\pi$
$313$	− 7401.02i	− 1.33652i	0.743928	−	0.668259i	$-0.232960\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5845.54i	1.03570i	0.855470	+	0.517852i	$0.173268\pi$
$317$	5845.54i	1.03570i	−0.855470	+	0.517852i	$0.826732\pi$
$318$	0	0
$319$	−166.790	−0.0292742
$320$	0	0
$321$	2122.96	0.369133
$322$	0	0
$323$	1766.14i	0.304244i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 863.984i	− 0.146111i
$328$	0	0
$329$	−12673.7	−2.12378
$330$	0	0
$331$	2795.86	0.464273	0.232137	−	0.972683i	$-0.425428\pi$
$331$	2795.86	0.464273	0.232137	+	0.972683i	$0.425428\pi$
$332$	0	0
$333$	− 1874.36i	− 0.308452i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4087.70i	0.660745i	0.943851	+	0.330372i	$0.107174\pi$
$337$	4087.70i	0.660745i	−0.943851	+	0.330372i	$0.892826\pi$
$338$	0	0
$339$	1089.91	0.174619
$340$	0	0
$341$	−5126.80	−0.814169
$342$	0	0
$343$	− 4243.14i	− 0.667954i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8513.81i	1.31713i	0.752523	+	0.658566i	$0.228837\pi$
$347$	8513.81i	1.31713i	−0.752523	+	0.658566i	$0.771163\pi$
$348$	0	0
$349$	10514.7	1.61271	0.806356	−	0.591430i	$-0.201437\pi$
$349$	10514.7	1.61271	0.806356	+	0.591430i	$0.201437\pi$
$350$	0	0
$351$	8912.81	1.35536
$352$	0	0
$353$	− 1812.42i	− 0.273273i	−0.990621	−	0.136636i	$-0.956371\pi$
$353$	− 1812.42i	− 0.273273i	0.990621	−	0.136636i	$-0.0436292\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3865.21i	− 0.573021i
$358$	0	0
$359$	−5373.97	−0.790048	−0.395024	−	0.918671i	$-0.629264\pi$
$359$	−5373.97	−0.790048	−0.395024	+	0.918671i	$0.629264\pi$
$360$	0	0
$361$	−6116.08	−0.891687
$362$	0	0
$363$	2020.77i	0.292184i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4596.67i	− 0.653799i	−0.945059	−	0.326899i	$-0.893996\pi$
$367$	− 4596.67i	− 0.653799i	0.945059	−	0.326899i	$-0.106004\pi$
$368$	0	0
$369$	−5662.09	−0.798798
$370$	0	0
$371$	14163.3	1.98199
$372$	0	0
$373$	− 7835.69i	− 1.08771i	−0.839179	−	0.543856i	$-0.816964\pi$
$373$	− 7835.69i	− 1.08771i	0.839179	−	0.543856i	$-0.183036\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 769.556i	− 0.105130i
$378$	0	0
$379$	2494.50	0.338084	0.169042	−	0.985609i	$-0.445933\pi$
$379$	2494.50	0.338084	0.169042	+	0.985609i	$0.445933\pi$
$380$	0	0
$381$	4048.11	0.544333
$382$	0	0
$383$	− 7274.12i	− 0.970470i	−0.874384	−	0.485235i	$-0.838734\pi$
$383$	− 7274.12i	− 0.970470i	0.874384	−	0.485235i	$-0.161266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3118.96i	− 0.409678i
$388$	0	0
$389$	3306.03	0.430906	0.215453	−	0.976514i	$-0.430877\pi$
$389$	3306.03	0.430906	0.215453	+	0.976514i	$0.430877\pi$
$390$	0	0
$391$	6617.35	0.855892
$392$	0	0
$393$	3237.39i	0.415533i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8077.86i	1.02120i	0.859819	+	0.510600i	$0.170577\pi$
$397$	8077.86i	1.02120i	−0.859819	+	0.510600i	$0.829423\pi$
$398$	0	0
$399$	−1625.89	−0.204000
$400$	0	0
$401$	12015.5	1.49632	0.748160	−	0.663518i	$-0.230937\pi$
$401$	12015.5	1.49632	0.748160	+	0.663518i	$0.230937\pi$
$402$	0	0
$403$	− 23654.6i	− 2.92387i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1550.00i	− 0.188773i
$408$	0	0
$409$	7199.69	0.870420	0.435210	−	0.900329i	$-0.356674\pi$
$409$	7199.69	0.870420	0.435210	+	0.900329i	$0.356674\pi$
$410$	0	0
$411$	1026.43	0.123188
$412$	0	0
$413$	15306.1i	1.82364i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5529.85i	− 0.649395i
$418$	0	0
$419$	1429.38	0.166658	0.0833292	−	0.996522i	$-0.473445\pi$
$419$	1429.38	0.166658	0.0833292	+	0.996522i	$0.473445\pi$
$420$	0	0
$421$	13142.9	1.52148	0.760741	−	0.649056i	$-0.224836\pi$
$421$	13142.9	1.52148	0.760741	+	0.649056i	$0.224836\pi$
$422$	0	0
$423$	9980.39i	1.14719i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 11763.6i	− 1.33321i
$428$	0	0
$429$	3368.66	0.379115
$430$	0	0
$431$	−7241.89	−0.809349	−0.404675	−	0.914461i	$-0.632615\pi$
$431$	−7241.89	−0.809349	−0.404675	+	0.914461i	$0.632615\pi$
$432$	0	0
$433$	5596.76i	0.621162i	0.950547	+	0.310581i	$0.100524\pi$
$433$	5596.76i	0.621162i	−0.950547	+	0.310581i	$0.899476\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2783.57i	− 0.304705i
$438$	0	0
$439$	−5214.92	−0.566958	−0.283479	−	0.958978i	$-0.591489\pi$
$439$	−5214.92	−0.566958	−0.283479	+	0.958978i	$0.591489\pi$
$440$	0	0
$441$	−11137.3	−1.20260
$442$	0	0
$443$	2732.92i	0.293104i	0.989203	+	0.146552i	$0.0468176\pi$
$443$	2732.92i	0.293104i	−0.989203	+	0.146552i	$0.953182\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3005.19i	− 0.317988i
$448$	0	0
$449$	15217.8	1.59949	0.799747	−	0.600338i	$-0.204967\pi$
$449$	15217.8	1.59949	0.799747	+	0.600338i	$0.204967\pi$
$450$	0	0
$451$	−4682.25	−0.488865
$452$	0	0
$453$	4876.52i	0.505781i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12772.1i	1.30734i	0.756780	+	0.653670i	$0.226771\pi$
$457$	12772.1i	1.30734i	−0.756780	+	0.653670i	$0.773229\pi$
$458$	0	0
$459$	−6659.65	−0.677224
$460$	0	0
$461$	−10121.5	−1.02257	−0.511285	−	0.859411i	$-0.670830\pi$
$461$	−10121.5	−1.02257	−0.511285	+	0.859411i	$0.670830\pi$
$462$	0	0
$463$	− 5422.15i	− 0.544252i	−0.962262	−	0.272126i	$-0.912273\pi$
$463$	− 5422.15i	− 0.544252i	0.962262	−	0.272126i	$-0.0877267\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16663.3i	− 1.65115i	−0.564292	−	0.825575i	$-0.690851\pi$
$467$	− 16663.3i	− 1.65115i	0.564292	−	0.825575i	$-0.309149\pi$
$468$	0	0
$469$	−17192.0	−1.69265
$470$	0	0
$471$	1102.78	0.107884
$472$	0	0
$473$	− 2579.21i	− 0.250724i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11153.4i	− 1.07060i
$478$	0	0
$479$	6745.65	0.643458	0.321729	−	0.946832i	$-0.395736\pi$
$479$	6745.65	0.643458	0.321729	+	0.946832i	$0.395736\pi$
$480$	0	0
$481$	7151.56	0.677927
$482$	0	0
$483$	6091.85i	0.573890i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3148.75i	0.292985i	0.989212	+	0.146492i	$0.0467984\pi$
$487$	3148.75i	0.292985i	−0.989212	+	0.146492i	$0.953202\pi$
$488$	0	0
$489$	2460.22	0.227515
$490$	0	0
$491$	16871.1	1.55067	0.775336	−	0.631548i	$-0.217580\pi$
$491$	16871.1	1.55067	0.775336	+	0.631548i	$0.217580\pi$
$492$	0	0
$493$	575.012i	0.0525299i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16430.4i	− 1.48290i
$498$	0	0
$499$	6873.07	0.616595	0.308297	−	0.951290i	$-0.400241\pi$
$499$	6873.07	0.616595	0.308297	+	0.951290i	$0.400241\pi$
$500$	0	0
$501$	−609.237	−0.0543288
$502$	0	0
$503$	12534.1i	1.11107i	0.831492	+	0.555536i	$0.187487\pi$
$503$	12534.1i	1.11107i	−0.831492	+	0.555536i	$0.812513\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11002.0i	0.963740i
$508$	0	0
$509$	−6046.18	−0.526507	−0.263254	−	0.964727i	$-0.584796\pi$
$509$	−6046.18	−0.526507	−0.263254	+	0.964727i	$0.584796\pi$
$510$	0	0
$511$	−12581.7	−1.08920
$512$	0	0
$513$	2801.36i	0.241098i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8253.25i	0.702085i
$518$	0	0
$519$	−5800.63	−0.490596
$520$	0	0
$521$	3643.76	0.306403	0.153202	−	0.988195i	$-0.451042\pi$
$521$	3643.76	0.306403	0.153202	+	0.988195i	$0.451042\pi$
$522$	0	0
$523$	− 14510.8i	− 1.21322i	−0.794999	−	0.606611i	$-0.792529\pi$
$523$	− 14510.8i	− 1.21322i	0.794999	−	0.606611i	$-0.207471\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17674.7i	1.46095i
$528$	0	0
$529$	1737.56	0.142809
$530$	0	0
$531$	12053.3	0.985067
$532$	0	0
$533$	− 21603.5i	− 1.75563i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9494.17i	0.762949i
$538$	0	0
$539$	−9209.94	−0.735993
$540$	0	0
$541$	−5064.47	−0.402475	−0.201237	−	0.979543i	$-0.564496\pi$
$541$	−5064.47	−0.402475	−0.201237	+	0.979543i	$0.564496\pi$
$542$	0	0
$543$	2917.46i	0.230571i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11557.1i	0.903376i	0.892176	+	0.451688i	$0.149178\pi$
$547$	11557.1i	0.903376i	−0.892176	+	0.451688i	$0.850822\pi$
$548$	0	0
$549$	−9263.66	−0.720152
$550$	0	0
$551$	241.877	0.0187011
$552$	0	0
$553$	− 19585.4i	− 1.50607i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13511.4i	− 1.02782i	−0.857843	−	0.513911i	$-0.828196\pi$
$557$	− 13511.4i	− 1.02782i	0.857843	−	0.513911i	$-0.171804\pi$
$558$	0	0
$559$	11900.3	0.900407
$560$	0	0
$561$	−2517.06	−0.189430
$562$	0	0
$563$	9383.50i	0.702429i	0.936295	+	0.351214i	$0.114231\pi$
$563$	9383.50i	0.702429i	−0.936295	+	0.351214i	$0.885769\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11580.9i	0.857764i
$568$	0	0
$569$	−20395.3	−1.50266	−0.751332	−	0.659924i	$-0.770588\pi$
$569$	−20395.3	−1.50266	−0.751332	+	0.659924i	$0.770588\pi$
$570$	0	0
$571$	−24939.7	−1.82783	−0.913917	−	0.405902i	$-0.866957\pi$
$571$	−24939.7	−1.82783	−0.913917	+	0.405902i	$0.866957\pi$
$572$	0	0
$573$	8098.02i	0.590400i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13633.6i	0.983665i	0.870690	+	0.491832i	$0.163673\pi$
$577$	13633.6i	0.983665i	−0.870690	+	0.491832i	$0.836327\pi$
$578$	0	0
$579$	4602.82	0.330374
$580$	0	0
$581$	−36861.9	−2.63217
$582$	0	0
$583$	− 9223.26i	− 0.655212i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1261.57i	0.0887065i	0.999016	+	0.0443532i	$0.0141227\pi$
$587$	1261.57i	0.0887065i	−0.999016	+	0.0443532i	$0.985877\pi$
$588$	0	0
$589$	7434.82	0.520112
$590$	0	0
$591$	6865.78	0.477869
$592$	0	0
$593$	9137.25i	0.632752i	0.948634	+	0.316376i	$0.102466\pi$
$593$	9137.25i	0.632752i	−0.948634	+	0.316376i	$0.897534\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4007.92i	− 0.274763i
$598$	0	0
$599$	−5793.88	−0.395211	−0.197606	−	0.980282i	$-0.563317\pi$
$599$	−5793.88	−0.395211	−0.197606	+	0.980282i	$0.563317\pi$
$600$	0	0
$601$	10763.6	0.730545	0.365272	−	0.930901i	$-0.380976\pi$
$601$	10763.6	0.730545	0.365272	+	0.930901i	$0.380976\pi$
$602$	0	0
$603$	13538.5i	0.914314i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10668.7i	− 0.713393i	−0.934220	−	0.356697i	$-0.883903\pi$
$607$	− 10668.7i	− 0.713393i	0.934220	−	0.356697i	$-0.116097\pi$
$608$	0	0
$609$	−529.350	−0.0352222
$610$	0	0
$611$	−38079.8	−2.52135
$612$	0	0
$613$	− 1699.04i	− 0.111947i	−0.998432	−	0.0559735i	$-0.982174\pi$
$613$	− 1699.04i	− 0.111947i	0.998432	−	0.0559735i	$-0.0178262\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13951.6i	0.910324i	0.890409	+	0.455162i	$0.150419\pi$
$617$	13951.6i	0.910324i	−0.890409	+	0.455162i	$0.849581\pi$
$618$	0	0
$619$	20560.1	1.33502	0.667512	−	0.744599i	$-0.267359\pi$
$619$	20560.1	1.33502	0.667512	+	0.744599i	$0.267359\pi$
$620$	0	0
$621$	10496.1	0.678251
$622$	0	0
$623$	20533.3i	1.32047i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1058.79i	0.0674389i
$628$	0	0
$629$	−5343.64	−0.338736
$630$	0	0
$631$	−14838.5	−0.936154	−0.468077	−	0.883688i	$-0.655053\pi$
$631$	−14838.5	−0.936154	−0.468077	+	0.883688i	$0.655053\pi$
$632$	0	0
$633$	− 9868.14i	− 0.619626i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 42493.9i	− 2.64312i
$638$	0	0
$639$	−12938.7	−0.801014
$640$	0	0
$641$	7766.13	0.478539	0.239270	−	0.970953i	$-0.423092\pi$
$641$	7766.13	0.478539	0.239270	+	0.970953i	$0.423092\pi$
$642$	0	0
$643$	22756.2i	1.39567i	0.716259	+	0.697834i	$0.245853\pi$
$643$	22756.2i	1.39567i	−0.716259	+	0.697834i	$0.754147\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9541.64i	− 0.579785i	−0.957059	−	0.289892i	$-0.906381\pi$
$647$	− 9541.64i	− 0.579785i	0.957059	−	0.289892i	$-0.0936195\pi$
$648$	0	0
$649$	9967.47	0.602862
$650$	0	0
$651$	−16271.1	−0.979594
$652$	0	0
$653$	23220.2i	1.39154i	0.718264	+	0.695770i	$0.244937\pi$
$653$	23220.2i	1.39154i	−0.718264	+	0.695770i	$0.755063\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9907.94i	0.588349i
$658$	0	0
$659$	−3704.40	−0.218973	−0.109486	−	0.993988i	$-0.534921\pi$
$659$	−3704.40	−0.218973	−0.109486	+	0.993988i	$0.534921\pi$
$660$	0	0
$661$	−7879.78	−0.463673	−0.231837	−	0.972755i	$-0.574473\pi$
$661$	−7879.78	−0.463673	−0.231837	+	0.972755i	$0.574473\pi$
$662$	0	0
$663$	− 11613.5i	− 0.680288i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 906.263i	− 0.0526096i
$668$	0	0
$669$	3873.26	0.223840
$670$	0	0
$671$	−7660.55	−0.440734
$672$	0	0
$673$	22045.2i	1.26267i	0.775509	+	0.631336i	$0.217493\pi$
$673$	22045.2i	1.26267i	−0.775509	+	0.631336i	$0.782507\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4640.91i	0.263463i	0.991285	+	0.131732i	$0.0420537\pi$
$677$	4640.91i	0.263463i	−0.991285	+	0.131732i	$0.957946\pi$
$678$	0	0
$679$	50251.1	2.84015
$680$	0	0
$681$	3995.22	0.224812
$682$	0	0
$683$	2734.15i	0.153176i	0.997063	+	0.0765880i	$0.0244026\pi$
$683$	2734.15i	0.153176i	−0.997063	+	0.0765880i	$0.975597\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 336.510i	− 0.0186880i
$688$	0	0
$689$	42555.3	2.35302
$690$	0	0
$691$	1039.34	0.0572189	0.0286095	−	0.999591i	$-0.490892\pi$
$691$	1039.34	0.0572189	0.0286095	+	0.999591i	$0.490892\pi$
$692$	0	0
$693$	12329.5i	0.675840i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16142.1i	0.877225i
$698$	0	0
$699$	7226.68	0.391042
$700$	0	0
$701$	29579.2	1.59371	0.796856	−	0.604169i	$-0.206495\pi$
$701$	29579.2	1.59371	0.796856	+	0.604169i	$0.206495\pi$
$702$	0	0
$703$	2247.79i	0.120593i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51043.6i	2.71527i
$708$	0	0
$709$	−2106.06	−0.111558	−0.0557790	−	0.998443i	$-0.517764\pi$
$709$	−2106.06	−0.111558	−0.0557790	+	0.998443i	$0.517764\pi$
$710$	0	0
$711$	−15423.3	−0.813528
$712$	0	0
$713$	− 27856.7i	− 1.46317i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7381.17i	− 0.384456i
$718$	0	0
$719$	−26617.9	−1.38064	−0.690321	−	0.723503i	$-0.742531\pi$
$719$	−26617.9	−1.38064	−0.690321	+	0.723503i	$0.742531\pi$
$720$	0	0
$721$	721.141	0.0372492
$722$	0	0
$723$	5686.19i	0.292492i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2710.31i	− 0.138266i	−0.997607	−	0.0691332i	$-0.977977\pi$
$727$	− 2710.31i	− 0.138266i	0.997607	−	0.0691332i	$-0.0220234\pi$
$728$	0	0
$729$	3384.54	0.171952
$730$	0	0
$731$	−8891.87	−0.449901
$732$	0	0
$733$	31768.6i	1.60082i	0.599455	+	0.800408i	$0.295384\pi$
$733$	31768.6i	1.60082i	−0.599455	+	0.800408i	$0.704616\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11195.6i	0.559561i
$738$	0	0
$739$	7867.06	0.391603	0.195801	−	0.980644i	$-0.437269\pi$
$739$	7867.06	0.391603	0.195801	+	0.980644i	$0.437269\pi$
$740$	0	0
$741$	−4885.19	−0.242189
$742$	0	0
$743$	6820.78i	0.336783i	0.985720	+	0.168392i	$0.0538573\pi$
$743$	6820.78i	0.336783i	−0.985720	+	0.168392i	$0.946143\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	29028.2i	1.42180i
$748$	0	0
$749$	−29646.6	−1.44628
$750$	0	0
$751$	−10534.9	−0.511884	−0.255942	−	0.966692i	$-0.582386\pi$
$751$	−10534.9	−0.511884	−0.255942	+	0.966692i	$0.582386\pi$
$752$	0	0
$753$	− 4415.27i	− 0.213680i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12358.9i	− 0.593384i	−0.954973	−	0.296692i	$-0.904116\pi$
$757$	− 12358.9i	− 0.593384i	0.954973	−	0.296692i	$-0.0958835\pi$
$758$	0	0
$759$	3967.08	0.189718
$760$	0	0
$761$	19823.6	0.944292	0.472146	−	0.881520i	$-0.343480\pi$
$761$	19823.6	0.944292	0.472146	+	0.881520i	$0.343480\pi$
$762$	0	0
$763$	12065.3i	0.572470i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45989.1i	2.16502i
$768$	0	0
$769$	−1342.30	−0.0629449	−0.0314725	−	0.999505i	$-0.510020\pi$
$769$	−1342.30	−0.0629449	−0.0314725	+	0.999505i	$0.510020\pi$
$770$	0	0
$771$	−9189.76	−0.429262
$772$	0	0
$773$	− 25002.9i	− 1.16338i	−0.813411	−	0.581689i	$-0.802392\pi$
$773$	− 25002.9i	− 1.16338i	0.813411	−	0.581689i	$-0.197608\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4919.30i	− 0.227128i
$778$	0	0
$779$	6790.13	0.312300
$780$	0	0
$781$	−10699.6	−0.490222
$782$	0	0
$783$	912.055i	0.0416273i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9270.25i	− 0.419884i	−0.977714	−	0.209942i	$-0.932672\pi$
$787$	− 9270.25i	− 0.419884i	0.977714	−	0.209942i	$-0.0673275\pi$
$788$	0	0
$789$	−13292.7	−0.599788
$790$	0	0
$791$	−15220.3	−0.684163
$792$	0	0
$793$	− 35345.1i	− 1.58278i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4918.58i	0.218601i	0.994009	+	0.109301i	$0.0348611\pi$
$797$	4918.58i	0.218601i	−0.994009	+	0.109301i	$0.965139\pi$
$798$	0	0
$799$	28453.2	1.25983
$800$	0	0
$801$	16169.7	0.713270
$802$	0	0
$803$	8193.34i	0.360071i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2121.89i	0.0925578i
$808$	0	0
$809$	23789.6	1.03387	0.516933	−	0.856026i	$-0.327074\pi$
$809$	23789.6	1.03387	0.516933	+	0.856026i	$0.327074\pi$
$810$	0	0
$811$	1656.24	0.0717122	0.0358561	−	0.999357i	$-0.488584\pi$
$811$	1656.24	0.0717122	0.0358561	+	0.999357i	$0.488584\pi$
$812$	0	0
$813$	− 7349.03i	− 0.317025i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3740.34i	0.160169i
$818$	0	0
$819$	−56887.0	−2.42710
$820$	0	0
$821$	−33677.4	−1.43161	−0.715804	−	0.698301i	$-0.753940\pi$
$821$	−33677.4	−1.43161	−0.715804	+	0.698301i	$0.753940\pi$
$822$	0	0
$823$	20918.5i	0.885995i	0.896523	+	0.442998i	$0.146085\pi$
$823$	20918.5i	0.885995i	−0.896523	+	0.442998i	$0.853915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46941.5i	− 1.97378i	−0.161391	−	0.986891i	$-0.551598\pi$
$827$	− 46941.5i	− 1.97378i	0.161391	−	0.986891i	$-0.448402\pi$
$828$	0	0
$829$	1512.56	0.0633695	0.0316848	−	0.999498i	$-0.489913\pi$
$829$	1512.56	0.0633695	0.0316848	+	0.999498i	$0.489913\pi$
$830$	0	0
$831$	9593.14	0.400460
$832$	0	0
$833$	31751.4i	1.32067i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	28034.7i	1.15773i
$838$	0	0
$839$	−8785.71	−0.361521	−0.180761	−	0.983527i	$-0.557856\pi$
$839$	−8785.71	−0.361521	−0.180761	+	0.983527i	$0.557856\pi$
$840$	0	0
$841$	−24310.3	−0.996771
$842$	0	0
$843$	11493.0i	0.469560i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 28219.5i	− 1.14479i
$848$	0	0
$849$	1985.77	0.0802726
$850$	0	0
$851$	8421.98	0.339250
$852$	0	0
$853$	− 17742.0i	− 0.712162i	−0.934455	−	0.356081i	$-0.884113\pi$
$853$	− 17742.0i	− 0.712162i	0.934455	−	0.356081i	$-0.115887\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48201.1i	1.92126i	0.277835	+	0.960629i	$0.410383\pi$
$857$	48201.1i	1.92126i	−0.277835	+	0.960629i	$0.589617\pi$
$858$	0	0
$859$	27332.1	1.08563	0.542817	−	0.839851i	$-0.317358\pi$
$859$	27332.1	1.08563	0.542817	+	0.839851i	$0.317358\pi$
$860$	0	0
$861$	−14860.2	−0.588194
$862$	0	0
$863$	21577.8i	0.851119i	0.904930	+	0.425560i	$0.139923\pi$
$863$	21577.8i	0.851119i	−0.904930	+	0.425560i	$0.860077\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1476.45i	− 0.0578348i
$868$	0	0
$869$	−12754.2	−0.497880
$870$	0	0
$871$	−51655.7	−2.00951
$872$	0	0
$873$	− 39572.1i	− 1.53415i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 25212.9i	− 0.970785i	−0.874296	−	0.485392i	$-0.838677\pi$
$877$	− 25212.9i	− 0.970785i	0.874296	−	0.485392i	$-0.161323\pi$
$878$	0	0
$879$	−12649.0	−0.485370
$880$	0	0
$881$	33688.3	1.28829	0.644147	−	0.764902i	$-0.277213\pi$
$881$	33688.3	1.28829	0.644147	+	0.764902i	$0.277213\pi$
$882$	0	0
$883$	− 25749.9i	− 0.981372i	−0.871336	−	0.490686i	$-0.836746\pi$
$883$	− 25749.9i	− 0.981372i	0.871336	−	0.490686i	$-0.163254\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25524.9i	0.966227i	0.875558	+	0.483114i	$0.160494\pi$
$887$	25524.9i	0.966227i	−0.875558	+	0.483114i	$0.839506\pi$
$888$	0	0
$889$	−56530.9	−2.13272
$890$	0	0
$891$	7541.60	0.283561
$892$	0	0
$893$	− 11968.8i	− 0.448510i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18303.7i	0.681320i
$898$	0	0
$899$	2420.60	0.0898013
$900$	0	0
$901$	−31797.3	−1.17572
$902$	0	0
$903$	− 8185.75i	− 0.301666i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 24092.9i	− 0.882018i	−0.897503	−	0.441009i	$-0.854621\pi$
$907$	− 24092.9i	− 0.882018i	0.897503	−	0.441009i	$-0.145379\pi$
$908$	0	0
$909$	40196.2	1.46669
$910$	0	0
$911$	9445.56	0.343519	0.171759	−	0.985139i	$-0.445055\pi$
$911$	9445.56	0.343519	0.171759	+	0.985139i	$0.445055\pi$
$912$	0	0
$913$	24004.8i	0.870146i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 45209.4i	− 1.62808i
$918$	0	0
$919$	47739.0	1.71356	0.856782	−	0.515679i	$-0.172460\pi$
$919$	47739.0	1.71356	0.856782	+	0.515679i	$0.172460\pi$
$920$	0	0
$921$	−17400.2	−0.622538
$922$	0	0
$923$	− 49367.2i	− 1.76050i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 567.889i	− 0.0201207i
$928$	0	0
$929$	30097.7	1.06294	0.531471	−	0.847076i	$-0.321639\pi$
$929$	30097.7	1.06294	0.531471	+	0.847076i	$0.321639\pi$
$930$	0	0
$931$	13356.1	0.470171
$932$	0	0
$933$	− 21822.6i	− 0.765745i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22895.9i	− 0.798266i	−0.916893	−	0.399133i	$-0.869311\pi$
$937$	− 22895.9i	− 0.798266i	0.916893	−	0.399133i	$-0.130689\pi$
$938$	0	0
$939$	−15296.2	−0.531601
$940$	0	0
$941$	31370.1	1.08676	0.543378	−	0.839488i	$-0.317145\pi$
$941$	31370.1	1.08676	0.543378	+	0.839488i	$0.317145\pi$
$942$	0	0
$943$	− 25441.2i	− 0.878556i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26224.3i	0.899867i	0.893062	+	0.449934i	$0.148552\pi$
$947$	26224.3i	0.899867i	−0.893062	+	0.449934i	$0.851448\pi$
$948$	0	0
$949$	−37803.4	−1.29310
$950$	0	0
$951$	12081.4	0.411952
$952$	0	0
$953$	57163.4i	1.94303i	0.236988	+	0.971513i	$0.423840\pi$
$953$	57163.4i	1.94303i	−0.236988	+	0.971513i	$0.576160\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	344.718i	0.0116438i
$958$	0	0
$959$	−14333.9	−0.482654
$960$	0	0
$961$	44613.2	1.49754
$962$	0	0
$963$	23346.3i	0.781229i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12182.2i	− 0.405122i	−0.979270	−	0.202561i	$-0.935074\pi$
$967$	− 12182.2i	− 0.405122i	0.979270	−	0.202561i	$-0.0649264\pi$
$968$	0	0
$969$	3650.21	0.121013
$970$	0	0
$971$	39736.0	1.31327	0.656637	−	0.754207i	$-0.271978\pi$
$971$	39736.0	1.31327	0.656637	+	0.754207i	$0.271978\pi$
$972$	0	0
$973$	77223.0i	2.54435i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15841.8i	− 0.518756i	−0.965776	−	0.259378i	$-0.916482\pi$
$977$	− 15841.8i	− 0.518756i	0.965776	−	0.259378i	$-0.0835176\pi$
$978$	0	0
$979$	13371.5	0.436522
$980$	0	0
$981$	9501.30	0.309228
$982$	0	0
$983$	− 19382.2i	− 0.628887i	−0.949276	−	0.314444i	$-0.898182\pi$
$983$	− 19382.2i	− 0.628887i	0.949276	−	0.314444i	$-0.101818\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	26193.7i	0.844736i
$988$	0	0
$989$	14014.3	0.450584
$990$	0	0
$991$	18065.8	0.579090	0.289545	−	0.957164i	$-0.406496\pi$
$991$	18065.8	0.579090	0.289545	+	0.957164i	$0.406496\pi$
$992$	0	0
$993$	− 5778.41i	− 0.184665i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 23743.9i	− 0.754241i	−0.926164	−	0.377120i	$-0.876914\pi$
$997$	− 23743.9i	− 0.754241i	0.926164	−	0.377120i	$-0.123086\pi$
$998$	0	0
$999$	−8475.82	−0.268431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.m.449.3		6
4.3	odd	2		800.4.c.n.449.4		6
5.2	odd	4		800.4.a.u.1.2	✓	3
5.3	odd	4		800.4.a.w.1.2	yes	3
5.4	even	2	inner	800.4.c.m.449.4		6
20.3	even	4		800.4.a.v.1.2	yes	3
20.7	even	4		800.4.a.x.1.2	yes	3
20.19	odd	2		800.4.c.n.449.3		6
40.3	even	4		1600.4.a.cs.1.2		3
40.13	odd	4		1600.4.a.cr.1.2		3
40.27	even	4		1600.4.a.cq.1.2		3
40.37	odd	4		1600.4.a.ct.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.u.1.2	✓	3	5.2	odd	4
800.4.a.v.1.2	yes	3	20.3	even	4
800.4.a.w.1.2	yes	3	5.3	odd	4
800.4.a.x.1.2	yes	3	20.7	even	4
800.4.c.m.449.3		6	1.1	even	1	trivial
800.4.c.m.449.4		6	5.4	even	2	inner
800.4.c.n.449.3		6	20.19	odd	2
800.4.c.n.449.4		6	4.3	odd	2
1600.4.a.cq.1.2		3	40.27	even	4
1600.4.a.cr.1.2		3	40.13	odd	4
1600.4.a.cs.1.2		3	40.3	even	4
1600.4.a.ct.1.2		3	40.37	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.06677i q^{3} +28.8620i q^{7} +22.7285 q^{9} +O(q^{10})\)	\(q-2.06677i q^{3} +28.8620i q^{7} +22.7285 q^{9} +18.7952 q^{11} +86.7195i q^{13} -64.7968i q^{17} -27.2566 q^{19} +59.6512 q^{21} +102.125i q^{23} -102.777i q^{27} -8.87408 q^{29} -272.771 q^{31} -38.8455i q^{33} -82.4677i q^{37} +179.230 q^{39} -249.119 q^{41} -137.227i q^{43} +439.114i q^{47} -490.015 q^{49} -133.920 q^{51} -490.724i q^{53} +56.3332i q^{57} +530.319 q^{59} -407.580 q^{61} +655.989i q^{63} +595.664i q^{67} +211.068 q^{69} -569.274 q^{71} +435.927i q^{73} +542.468i q^{77} -678.589 q^{79} +401.251 q^{81} +1277.18i q^{83} +18.3407i q^{87} +711.431 q^{89} -2502.90 q^{91} +563.756i q^{93} -1741.08i q^{97} +427.186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 36 q^{9}+O(q^{10}) \) 6 * q - 36 * q^9	\( 6 q - 36 q^{9} - 62 q^{11} + 174 q^{19} - 140 q^{21} - 168 q^{29} - 588 q^{31} - 64 q^{39} - 690 q^{41} - 718 q^{49} - 2702 q^{51} + 2080 q^{59} + 964 q^{61} + 1228 q^{69} - 4096 q^{71} + 1996 q^{79} - 1098 q^{81} - 2378 q^{89} - 8064 q^{91} + 3908 q^{99}+O(q^{100}) \) 6 * q - 36 * q^9 - 62 * q^11 + 174 * q^19 - 140 * q^21 - 168 * q^29 - 588 * q^31 - 64 * q^39 - 690 * q^41 - 718 * q^49 - 2702 * q^51 + 2080 * q^59 + 964 * q^61 + 1228 * q^69 - 4096 * q^71 + 1996 * q^79 - 1098 * q^81 - 2378 * q^89 - 8064 * q^91 + 3908 * q^99

Introduction

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