LMFDB - Embedded newform 5202.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.a (trivial)

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:	yes
Analytic conductor:	$41.5381791315$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1734)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5202.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.00000 q^{11} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} +1.00000 q^{20} +3.00000 q^{22} -6.00000 q^{23} -4.00000 q^{25} +6.00000 q^{26} +4.00000 q^{28} +7.00000 q^{29} -3.00000 q^{31} +1.00000 q^{32} +4.00000 q^{35} +8.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +2.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} +6.00000 q^{52} -13.0000 q^{53} +3.00000 q^{55} +4.00000 q^{56} +7.00000 q^{58} -9.00000 q^{59} -10.0000 q^{61} -3.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{67} +4.00000 q^{70} +12.0000 q^{71} -7.00000 q^{73} +8.00000 q^{74} +4.00000 q^{76} +12.0000 q^{77} -5.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{86} +3.00000 q^{88} -4.00000 q^{89} +24.0000 q^{91} -6.00000 q^{92} +2.00000 q^{94} +4.00000 q^{95} +3.00000 q^{97} +9.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.00000	0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	6.00000	1.17670
$27$	0	0
$28$	4.00000	0.755929
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	6.00000	0.832050
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	4.00000	0.534522
$57$	0	0
$58$	7.00000	0.919145
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−3.00000	−0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	0	0
$70$	4.00000	0.478091
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	4.00000	0.458831
$77$	12.0000	1.36753
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	−6.00000	−0.625543
$93$	0	0
$94$	2.00000	0.206284
$95$	4.00000	0.410391
$96$	0	0
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−13.0000	−1.26267
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	4.00000	0.377964
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	7.00000	0.649934
$117$	0	0
$118$	−9.00000	−0.828517
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−3.00000	−0.269408
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	12.0000	1.00702
$143$	18.0000	1.50524
$144$	0	0
$145$	7.00000	0.581318
$146$	−7.00000	−0.579324
$147$	0	0
$148$	8.00000	0.657596
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	12.0000	0.966988
$155$	−3.00000	−0.240966
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	1.00000	0.0790569
$161$	−24.0000	−1.89146
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	3.00000	0.226134
$177$	0	0
$178$	−4.00000	−0.299813
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	24.0000	1.77900
$183$	0	0
$184$	−6.00000	−0.442326
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	2.00000	0.145865
$189$	0	0
$190$	4.00000	0.290191
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	3.00000	0.215387
$195$	0	0
$196$	9.00000	0.642857
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−17.0000	−1.19612
$203$	28.0000	1.96521
$204$	0	0
$205$	−2.00000	−0.139686
$206$	13.0000	0.905753
$207$	0	0
$208$	6.00000	0.416025
$209$	12.0000	0.830057
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−13.0000	−0.892844
$213$	0	0
$214$	−15.0000	−1.02538
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−12.0000	−0.814613
$218$	10.0000	0.677285
$219$	0	0
$220$	3.00000	0.202260
$221$	0	0
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	10.0000	0.665190
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	7.00000	0.459573
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	−9.00000	−0.585850
$237$	0	0
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−10.0000	−0.640184
$245$	9.00000	0.574989
$246$	0	0
$247$	24.0000	1.52708
$248$	−3.00000	−0.190500
$249$	0	0
$250$	−9.00000	−0.569210
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	6.00000	0.372104
$261$	0	0
$262$	−20.0000	−1.23560
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−13.0000	−0.798584
$266$	16.0000	0.981023
$267$	0	0
$268$	−6.00000	−0.366508
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−12.0000	−0.723627
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	4.00000	0.239046
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	18.0000	1.06436
$287$	−8.00000	−0.472225
$288$	0	0
$289$	0	0
$290$	7.00000	0.411054
$291$	0	0
$292$	−7.00000	−0.409644
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	8.00000	0.464991
$297$	0	0
$298$	−15.0000	−0.868927
$299$	−36.0000	−2.08193
$300$	0	0
$301$	−16.0000	−0.922225
$302$	−9.00000	−0.517892
$303$	0	0
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	−3.00000	−0.170389
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	21.0000	1.17577
$320$	1.00000	0.0559017
$321$	0	0
$322$	−24.0000	−1.33747
$323$	0	0
$324$	0	0
$325$	−24.0000	−1.33128
$326$	6.00000	0.332309
$327$	0	0
$328$	−2.00000	−0.110432
$329$	8.00000	0.441054
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	8.00000	0.437741
$335$	−6.00000	−0.327815
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	0	0
$341$	−9.00000	−0.487377
$342$	0	0
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	13.0000	0.698884
$347$	−17.0000	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$347$	−17.0000	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	−16.0000	−0.855236
$351$	0	0
$352$	3.00000	0.159901
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−4.00000	−0.212000
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000	0.525588
$363$	0	0
$364$	24.0000	1.25794
$365$	−7.00000	−0.366397
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	8.00000	0.415900
$371$	−52.0000	−2.69971
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	0	0
$376$	2.00000	0.103142
$377$	42.0000	2.16311
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−14.0000	−0.716302
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	9.00000	0.458088
$387$	0	0
$388$	3.00000	0.152302
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	0	0
$394$	10.0000	0.503793
$395$	−5.00000	−0.251577
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−1.00000	−0.0501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	−18.0000	−0.896644
$404$	−17.0000	−0.845782
$405$	0	0
$406$	28.0000	1.38962
$407$	24.0000	1.18964
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	13.0000	0.640464
$413$	−36.0000	−1.77144
$414$	0	0
$415$	−12.0000	−0.589057
$416$	6.00000	0.294174
$417$	0	0
$418$	12.0000	0.586939
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	−13.0000	−0.631336
$425$	0	0
$426$	0	0
$427$	−40.0000	−1.93574
$428$	−15.0000	−0.725052
$429$	0	0
$430$	−4.00000	−0.192897
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−3.00000	−0.144171	−0.0720854	−	0.997398i	$-0.522965\pi$
$433$	−3.00000	−0.144171	−0.0720854	+	0.997398i	$0.522965\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	10.0000	0.478913
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	17.0000	0.804973
$447$	0	0
$448$	4.00000	0.188982
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	10.0000	0.470360
$453$	0	0
$454$	20.0000	0.938647
$455$	24.0000	1.12514
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	−6.00000	−0.279751
$461$	−35.0000	−1.63011	−0.815056	−	0.579382i	$-0.803294\pi$
$461$	−35.0000	−1.63011	−0.815056	+	0.579382i	$0.803294\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	7.00000	0.324967
$465$	0	0
$466$	−24.0000	−1.11178
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	2.00000	0.0922531
$471$	0	0
$472$	−9.00000	−0.414259
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−16.0000	−0.734130
$476$	0	0
$477$	0	0
$478$	26.0000	1.18921
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	15.0000	0.683231
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	3.00000	0.136223
$486$	0	0
$487$	25.0000	1.13286	0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000	1.13286	0.566429	+	0.824110i	$0.308325\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	9.00000	0.406579
$491$	23.0000	1.03798	0.518988	−	0.854782i	$-0.326309\pi$
$491$	23.0000	1.03798	0.518988	+	0.854782i	$0.326309\pi$
$492$	0	0
$493$	0	0
$494$	24.0000	1.07981
$495$	0	0
$496$	−3.00000	−0.134704
$497$	48.0000	2.15309
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	15.0000	0.669483
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−17.0000	−0.756490
$506$	−18.0000	−0.800198
$507$	0	0
$508$	−16.0000	−0.709885
$509$	31.0000	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$509$	31.0000	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$510$	0	0
$511$	−28.0000	−1.23865
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	13.0000	0.572848
$516$	0	0
$517$	6.00000	0.263880
$518$	32.0000	1.40600
$519$	0	0
$520$	6.00000	0.263117
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	18.0000	0.784837
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−13.0000	−0.564684
$531$	0	0
$532$	16.0000	0.693688
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−15.0000	−0.648507
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−9.00000	−0.388018
$539$	27.0000	1.16297
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	5.00000	0.214768
$543$	0	0
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−12.0000	−0.511682
$551$	28.0000	1.19284
$552$	0	0
$553$	−20.0000	−0.850487
$554$	22.0000	0.934690
$555$	0	0
$556$	8.00000	0.339276
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	0	0
$562$	0	0
$563$	−23.0000	−0.969334	−0.484667	−	0.874699i	$-0.661059\pi$
$563$	−23.0000	−0.969334	−0.484667	+	0.874699i	$0.661059\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	−4.00000	−0.168133
$567$	0	0
$568$	12.0000	0.503509
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	18.0000	0.752618
$573$	0	0
$574$	−8.00000	−0.333914
$575$	24.0000	1.00087
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	0	0
$580$	7.00000	0.290659
$581$	−48.0000	−1.99138
$582$	0	0
$583$	−39.0000	−1.61521
$584$	−7.00000	−0.289662
$585$	0	0
$586$	19.0000	0.784883
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	−9.00000	−0.370524
$591$	0	0
$592$	8.00000	0.328798
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	0	0
$598$	−36.0000	−1.47215
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	−9.00000	−0.366205
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−10.0000	−0.404888
$611$	12.0000	0.485468
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	12.0000	0.483494
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	11.0000	0.440000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	12.0000	0.478852
$629$	0	0
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−5.00000	−0.198889
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	−16.0000	−0.634941
$636$	0	0
$637$	54.0000	2.13956
$638$	21.0000	0.831398
$639$	0	0
$640$	1.00000	0.0395285
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	0	0
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	0	0
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	−27.0000	−1.05984
$650$	−24.0000	−0.941357
$651$	0	0
$652$	6.00000	0.234978
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	8.00000	0.311872
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−22.0000	−0.855054
$663$	0	0
$664$	−12.0000	−0.465690
$665$	16.0000	0.620453
$666$	0	0
$667$	−42.0000	−1.62625
$668$	8.00000	0.309529
$669$	0	0
$670$	−6.00000	−0.231800
$671$	−30.0000	−1.15814
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	23.0000	0.884615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	0	0
$682$	−9.00000	−0.344628
$683$	−27.0000	−1.03313	−0.516563	−	0.856249i	$-0.672789\pi$
$683$	−27.0000	−1.03313	−0.516563	+	0.856249i	$0.672789\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	8.00000	0.305441
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−78.0000	−2.97156
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	13.0000	0.494186
$693$	0	0
$694$	−17.0000	−0.645311
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	12.0000	0.454207
$699$	0	0
$700$	−16.0000	−0.604743
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	3.00000	0.113067
$705$	0	0
$706$	4.00000	0.150542
$707$	−68.0000	−2.55740
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	−4.00000	−0.149906
$713$	18.0000	0.674105
$714$	0	0
$715$	18.0000	0.673162
$716$	0	0
$717$	0	0
$718$	30.0000	1.11959
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	52.0000	1.93658
$722$	−3.00000	−0.111648
$723$	0	0
$724$	10.0000	0.371647
$725$	−28.0000	−1.03989
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	24.0000	0.889499
$729$	0	0
$730$	−7.00000	−0.259082
$731$	0	0
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−18.0000	−0.663039
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	−52.0000	−1.90898
$743$	10.0000	0.366864	0.183432	−	0.983032i	$-0.441279\pi$
$743$	10.0000	0.366864	0.183432	+	0.983032i	$0.441279\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	−32.0000	−1.17160
$747$	0	0
$748$	0	0
$749$	−60.0000	−2.19235
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	42.0000	1.52955
$755$	−9.00000	−0.327544
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	4.00000	0.145095
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	0	0
$763$	40.0000	1.44810
$764$	−14.0000	−0.506502
$765$	0	0
$766$	2.00000	0.0722629
$767$	−54.0000	−1.94983
$768$	0	0
$769$	−55.0000	−1.98335	−0.991675	−	0.128763i	$-0.958899\pi$
$769$	−55.0000	−1.98335	−0.991675	+	0.128763i	$0.958899\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	9.00000	0.323917
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	3.00000	0.107694
$777$	0	0
$778$	5.00000	0.179259
$779$	−8.00000	−0.286630
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	12.0000	0.428298
$786$	0	0
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	−5.00000	−0.177892
$791$	40.0000	1.42224
$792$	0	0
$793$	−60.0000	−2.13066
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−1.00000	−0.0354441
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	12.0000	0.423735
$803$	−21.0000	−0.741074
$804$	0	0
$805$	−24.0000	−0.845889
$806$	−18.0000	−0.634023
$807$	0	0
$808$	−17.0000	−0.598058
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	28.0000	0.982607
$813$	0	0
$814$	24.0000	0.841200
$815$	6.00000	0.210171
$816$	0	0
$817$	−16.0000	−0.559769
$818$	19.0000	0.664319
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	−36.0000	−1.25260
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	0	0
$829$	−12.0000	−0.416777	−0.208389	−	0.978046i	$-0.566822\pi$
$829$	−12.0000	−0.416777	−0.208389	+	0.978046i	$0.566822\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	6.00000	0.208013
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	12.0000	0.415029
$837$	0	0
$838$	3.00000	0.103633
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	22.0000	0.758170
$843$	0	0
$844$	8.00000	0.275371
$845$	23.0000	0.791224
$846$	0	0
$847$	−8.00000	−0.274883
$848$	−13.0000	−0.446422
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	32.0000	1.09566	0.547830	−	0.836590i	$-0.315454\pi$
$853$	32.0000	1.09566	0.547830	+	0.836590i	$0.315454\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	−15.0000	−0.512689
$857$	−50.0000	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$857$	−50.0000	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	10.0000	0.340601
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	13.0000	0.442013
$866$	−3.00000	−0.101944
$867$	0	0
$868$	−12.0000	−0.407307
$869$	−15.0000	−0.508840
$870$	0	0
$871$	−36.0000	−1.21981
$872$	10.0000	0.338643
$873$	0	0
$874$	−24.0000	−0.811812
$875$	−36.0000	−1.21702
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	3.00000	0.101130
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	−64.0000	−2.14649
$890$	−4.00000	−0.134080
$891$	0	0
$892$	17.0000	0.569202
$893$	8.00000	0.267710
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	−4.00000	−0.133482
$899$	−21.0000	−0.700389
$900$	0	0
$901$	0	0
$902$	−6.00000	−0.199778
$903$	0	0
$904$	10.0000	0.332595
$905$	10.0000	0.332411
$906$	0	0
$907$	−54.0000	−1.79304	−0.896520	−	0.443003i	$-0.853913\pi$
$907$	−54.0000	−1.79304	−0.896520	+	0.443003i	$0.853913\pi$
$908$	20.0000	0.663723
$909$	0	0
$910$	24.0000	0.795592
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	−17.0000	−0.562310
$915$	0	0
$916$	−14.0000	−0.462573
$917$	−80.0000	−2.64183
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	−35.0000	−1.15266
$923$	72.0000	2.36991
$924$	0	0
$925$	−32.0000	−1.05215
$926$	5.00000	0.164310
$927$	0	0
$928$	7.00000	0.229786
$929$	−8.00000	−0.262471	−0.131236	−	0.991351i	$-0.541894\pi$
$929$	−8.00000	−0.262471	−0.131236	+	0.991351i	$0.541894\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−24.0000	−0.786146
$933$	0	0
$934$	3.00000	0.0981630
$935$	0	0
$936$	0	0
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	−24.0000	−0.783628
$939$	0	0
$940$	2.00000	0.0652328
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	−9.00000	−0.292925
$945$	0	0
$946$	−12.0000	−0.390154
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	−42.0000	−1.36338
$950$	−16.0000	−0.519109
$951$	0	0
$952$	0	0
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	−14.0000	−0.453029
$956$	26.0000	0.840900
$957$	0	0
$958$	14.0000	0.452319
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	48.0000	1.54758
$963$	0	0
$964$	15.0000	0.483117
$965$	9.00000	0.289720
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	3.00000	0.0963242
$971$	19.0000	0.609739	0.304870	−	0.952394i	$-0.401387\pi$
$971$	19.0000	0.609739	0.304870	+	0.952394i	$0.401387\pi$
$972$	0	0
$973$	32.0000	1.02587
$974$	25.0000	0.801052
$975$	0	0
$976$	−10.0000	−0.320092
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	9.00000	0.287494
$981$	0	0
$982$	23.0000	0.733959
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	0	0
$988$	24.0000	0.763542
$989$	24.0000	0.763156
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−3.00000	−0.0952501
$993$	0	0
$994$	48.0000	1.52247
$995$	−1.00000	−0.0317021
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	8.00000	0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.m.1.1		1
3.2	odd	2		1734.2.a.a.1.1	✓	1
17.16	even	2		5202.2.a.i.1.1		1
51.2	odd	8		1734.2.f.a.1483.1		4
51.8	odd	8		1734.2.f.a.829.2		4
51.26	odd	8		1734.2.f.a.829.1		4
51.32	odd	8		1734.2.f.a.1483.2		4
51.38	odd	4		1734.2.b.g.577.2		2
51.47	odd	4		1734.2.b.g.577.1		2
51.50	odd	2		1734.2.a.g.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1734.2.a.a.1.1	✓	1	3.2	odd	2
1734.2.a.g.1.1	yes	1	51.50	odd	2
1734.2.b.g.577.1		2	51.47	odd	4
1734.2.b.g.577.2		2	51.38	odd	4
1734.2.f.a.829.1		4	51.26	odd	8
1734.2.f.a.829.2		4	51.8	odd	8
1734.2.f.a.1483.1		4	51.2	odd	8
1734.2.f.a.1483.2		4	51.32	odd	8
5202.2.a.i.1.1		1	17.16	even	2
5202.2.a.m.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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