LMFDB - Embedded newform 755.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(1,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	755.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +3.00000 q^{7} -2.00000 q^{9} +2.00000 q^{10} +2.00000 q^{12} +6.00000 q^{13} +6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} -4.00000 q^{17} -4.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} +12.0000 q^{26} -5.00000 q^{27} +6.00000 q^{28} -3.00000 q^{29} +2.00000 q^{30} +7.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} +3.00000 q^{35} -4.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +6.00000 q^{39} +8.00000 q^{41} +6.00000 q^{42} +8.00000 q^{43} -2.00000 q^{45} -6.00000 q^{46} -8.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} -4.00000 q^{51} +12.0000 q^{52} -10.0000 q^{53} -10.0000 q^{54} -1.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +14.0000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} -4.00000 q^{67} -8.00000 q^{68} -3.00000 q^{69} +6.00000 q^{70} -12.0000 q^{71} -7.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} +12.0000 q^{78} +4.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} +12.0000 q^{83} +6.00000 q^{84} -4.00000 q^{85} +16.0000 q^{86} -3.00000 q^{87} -4.00000 q^{89} -4.00000 q^{90} +18.0000 q^{91} -6.00000 q^{92} +7.00000 q^{93} -16.0000 q^{94} -1.00000 q^{95} -8.00000 q^{96} +16.0000 q^{97} +4.00000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.00000	0.816497
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.00000	0.577350
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	6.00000	1.60357
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−4.00000	−0.942809
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	2.00000	0.447214
$21$	3.00000	0.654654
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	12.0000	2.35339
$27$	−5.00000	−0.962250
$28$	6.00000	1.13389
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	2.00000	0.365148
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−8.00000	−1.37199
$35$	3.00000	0.507093
$36$	−4.00000	−0.666667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	6.00000	0.960769
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	6.00000	0.925820
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−6.00000	−0.884652
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	2.00000	0.282843
$51$	−4.00000	−0.560112
$52$	12.0000	1.66410
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−10.0000	−1.36083
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−6.00000	−0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	2.00000	0.258199
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	14.0000	1.77800
$63$	−6.00000	−0.755929
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−8.00000	−0.970143
$69$	−3.00000	−0.361158
$70$	6.00000	0.717137
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	−2.00000	−0.229416
$77$	0	0
$78$	12.0000	1.35873
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	16.0000	1.76690
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	6.00000	0.654654
$85$	−4.00000	−0.433861
$86$	16.0000	1.72532
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	−4.00000	−0.421637
$91$	18.0000	1.88691
$92$	−6.00000	−0.625543
$93$	7.00000	0.725866
$94$	−16.0000	−1.65027
$95$	−1.00000	−0.102598
$96$	−8.00000	−0.816497
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	4.00000	0.404061
$99$	0	0
$100$	2.00000	0.200000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−8.00000	−0.792118
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	−20.0000	−1.94257
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	−10.0000	−0.962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−12.0000	−1.13389
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	−2.00000	−0.187317
$115$	−3.00000	−0.279751
$116$	−6.00000	−0.557086
$117$	−12.0000	−1.10940
$118$	−8.00000	−0.736460
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	8.00000	0.721336
$124$	14.0000	1.25724
$125$	1.00000	0.0894427
$126$	−12.0000	−1.06904
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	12.0000	1.05247
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−8.00000	−0.691095
$135$	−5.00000	−0.430331
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−6.00000	−0.510754
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	6.00000	0.507093
$141$	−8.00000	−0.673722
$142$	−24.0000	−2.01404
$143$	0	0
$144$	8.00000	0.666667
$145$	−3.00000	−0.249136
$146$	−14.0000	−1.15865
$147$	2.00000	0.164957
$148$	−4.00000	−0.328798
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	2.00000	0.163299
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	7.00000	0.562254
$156$	12.0000	0.960769
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	8.00000	0.636446
$159$	−10.0000	−0.793052
$160$	−8.00000	−0.632456
$161$	−9.00000	−0.709299
$162$	2.00000	0.157135
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	16.0000	1.24939
$165$	0	0
$166$	24.0000	1.86276
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−8.00000	−0.613572
$171$	2.00000	0.152944
$172$	16.0000	1.21999
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	−6.00000	−0.454859
$175$	3.00000	0.226779
$176$	0	0
$177$	−4.00000	−0.300658
$178$	−8.00000	−0.599625
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	−4.00000	−0.298142
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	36.0000	2.66850
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	14.0000	1.02653
$187$	0	0
$188$	−16.0000	−1.16692
$189$	−15.0000	−1.09109
$190$	−2.00000	−0.145095
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	−8.00000	−0.577350
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	32.0000	2.29747
$195$	6.00000	0.429669
$196$	4.00000	0.285714
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	20.0000	1.40720
$203$	−9.00000	−0.631676
$204$	−8.00000	−0.560112
$205$	8.00000	0.558744
$206$	12.0000	0.836080
$207$	6.00000	0.417029
$208$	−24.0000	−1.66410
$209$	0	0
$210$	6.00000	0.414039
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−20.0000	−1.37361
$213$	−12.0000	−0.822226
$214$	−34.0000	−2.32419
$215$	8.00000	0.545595
$216$	0	0
$217$	21.0000	1.42557
$218$	−12.0000	−0.812743
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−24.0000	−1.61441
$222$	−4.00000	−0.268462
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−24.0000	−1.60357
$225$	−2.00000	−0.133333
$226$	−22.0000	−1.46342
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−2.00000	−0.132453
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−24.0000	−1.56893
$235$	−8.00000	−0.521862
$236$	−8.00000	−0.520756
$237$	4.00000	0.259828
$238$	−24.0000	−1.55569
$239$	13.0000	0.840900	0.420450	−	0.907316i	$-0.361872\pi$
$239$	13.0000	0.840900	0.420450	+	0.907316i	$0.361872\pi$
$240$	−4.00000	−0.258199
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	−22.0000	−1.41421
$243$	16.0000	1.02640
$244$	0	0
$245$	2.00000	0.127775
$246$	16.0000	1.02012
$247$	−6.00000	−0.381771
$248$	0	0
$249$	12.0000	0.760469
$250$	2.00000	0.126491
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	−12.0000	−0.755929
$253$	0	0
$254$	−4.00000	−0.250982
$255$	−4.00000	−0.250490
$256$	16.0000	1.00000
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	16.0000	0.996116
$259$	−6.00000	−0.372822
$260$	12.0000	0.744208
$261$	6.00000	0.371391
$262$	36.0000	2.22409
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	−6.00000	−0.367884
$267$	−4.00000	−0.244796
$268$	−8.00000	−0.488678
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	−10.0000	−0.608581
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	16.0000	0.970143
$273$	18.0000	1.08941
$274$	24.0000	1.44989
$275$	0	0
$276$	−6.00000	−0.361158
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	6.00000	0.359856
$279$	−14.0000	−0.838158
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−16.0000	−0.952786
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	−24.0000	−1.42414
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	24.0000	1.41668
$288$	16.0000	0.942809
$289$	−1.00000	−0.0588235
$290$	−6.00000	−0.352332
$291$	16.0000	0.937937
$292$	−14.0000	−0.819288
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	4.00000	0.233285
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	12.0000	0.695141
$299$	−18.0000	−1.04097
$300$	2.00000	0.115470
$301$	24.0000	1.38334
$302$	−2.00000	−0.115087
$303$	10.0000	0.574485
$304$	4.00000	0.229416
$305$	0	0
$306$	16.0000	0.914659
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	14.0000	0.795147
$311$	−17.0000	−0.963982	−0.481991	−	0.876176i	$-0.660086\pi$
$311$	−17.0000	−0.963982	−0.481991	+	0.876176i	$0.660086\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−26.0000	−1.46726
$315$	−6.00000	−0.338062
$316$	8.00000	0.450035
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	−20.0000	−1.12154
$319$	0	0
$320$	−8.00000	−0.447214
$321$	−17.0000	−0.948847
$322$	−18.0000	−1.00310
$323$	4.00000	0.222566
$324$	2.00000	0.111111
$325$	6.00000	0.332820
$326$	−10.0000	−0.553849
$327$	−6.00000	−0.331801
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	24.0000	1.31717
$333$	4.00000	0.219199
$334$	36.0000	1.96983
$335$	−4.00000	−0.218543
$336$	−12.0000	−0.654654
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	46.0000	2.50207
$339$	−11.0000	−0.597438
$340$	−8.00000	−0.433861
$341$	0	0
$342$	4.00000	0.216295
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−3.00000	−0.161515
$346$	−16.0000	−0.860165
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	−6.00000	−0.321634
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	6.00000	0.320713
$351$	−30.0000	−1.60128
$352$	0	0
$353$	37.0000	1.96931	0.984656	−	0.174509i	$-0.0558337\pi$
$353$	37.0000	1.96931	0.984656	+	0.174509i	$0.0558337\pi$
$354$	−8.00000	−0.425195
$355$	−12.0000	−0.636894
$356$	−8.00000	−0.423999
$357$	−12.0000	−0.635107
$358$	−36.0000	−1.90266
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−4.00000	−0.210235
$363$	−11.0000	−0.577350
$364$	36.0000	1.88691
$365$	−7.00000	−0.366397
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	12.0000	0.625543
$369$	−16.0000	−0.832927
$370$	−4.00000	−0.207950
$371$	−30.0000	−1.55752
$372$	14.0000	0.725866
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−18.0000	−0.927047
$378$	−30.0000	−1.54303
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	−2.00000	−0.102598
$381$	−2.00000	−0.102463
$382$	26.0000	1.33028
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	0	0
$386$	8.00000	0.407189
$387$	−16.0000	−0.813326
$388$	32.0000	1.62455
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	12.0000	0.607644
$391$	12.0000	0.606866
$392$	0	0
$393$	18.0000	0.907980
$394$	18.0000	0.906827
$395$	4.00000	0.201262
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−40.0000	−2.00502
$399$	−3.00000	−0.150188
$400$	−4.00000	−0.200000
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	−8.00000	−0.399004
$403$	42.0000	2.09217
$404$	20.0000	0.995037
$405$	1.00000	0.0496904
$406$	−18.0000	−0.893325
$407$	0	0
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	16.0000	0.790184
$411$	12.0000	0.591916
$412$	12.0000	0.591198
$413$	−12.0000	−0.590481
$414$	12.0000	0.589768
$415$	12.0000	0.589057
$416$	−48.0000	−2.35339
$417$	3.00000	0.146911
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	6.00000	0.292770
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	44.0000	2.14189
$423$	16.0000	0.777947
$424$	0	0
$425$	−4.00000	−0.194029
$426$	−24.0000	−1.16280
$427$	0	0
$428$	−34.0000	−1.64345
$429$	0	0
$430$	16.0000	0.771589
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	20.0000	0.962250
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	42.0000	2.01606
$435$	−3.00000	−0.143839
$436$	−12.0000	−0.574696
$437$	3.00000	0.143509
$438$	−14.0000	−0.668946
$439$	−31.0000	−1.47955	−0.739775	−	0.672855i	$-0.765068\pi$
$439$	−31.0000	−1.47955	−0.739775	+	0.672855i	$0.765068\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	−48.0000	−2.28313
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	−4.00000	−0.189832
$445$	−4.00000	−0.189618
$446$	−20.0000	−0.947027
$447$	6.00000	0.283790
$448$	−24.0000	−1.13389
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	−22.0000	−1.03479
$453$	−1.00000	−0.0469841
$454$	12.0000	0.563188
$455$	18.0000	0.843853
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	12.0000	0.560723
$459$	20.0000	0.933520
$460$	−6.00000	−0.279751
$461$	−5.00000	−0.232873	−0.116437	−	0.993198i	$-0.537147\pi$
$461$	−5.00000	−0.232873	−0.116437	+	0.993198i	$0.537147\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	12.0000	0.557086
$465$	7.00000	0.324617
$466$	36.0000	1.66767
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−24.0000	−1.10940
$469$	−12.0000	−0.554109
$470$	−16.0000	−0.738025
$471$	−13.0000	−0.599008
$472$	0	0
$473$	0	0
$474$	8.00000	0.367452
$475$	−1.00000	−0.0458831
$476$	−24.0000	−1.10004
$477$	20.0000	0.915737
$478$	26.0000	1.18921
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	−8.00000	−0.365148
$481$	−12.0000	−0.547153
$482$	34.0000	1.54866
$483$	−9.00000	−0.409514
$484$	−22.0000	−1.00000
$485$	16.0000	0.726523
$486$	32.0000	1.45155
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	−5.00000	−0.226108
$490$	4.00000	0.180702
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	16.0000	0.721336
$493$	12.0000	0.540453
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−28.0000	−1.25724
$497$	−36.0000	−1.61482
$498$	24.0000	1.07547
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	2.00000	0.0894427
$501$	18.0000	0.804181
$502$	−18.0000	−0.803379
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	23.0000	1.02147
$508$	−4.00000	−0.177471
$509$	32.0000	1.41838	0.709188	−	0.705020i	$-0.249062\pi$
$509$	32.0000	1.41838	0.709188	+	0.705020i	$0.249062\pi$
$510$	−8.00000	−0.354246
$511$	−21.0000	−0.928985
$512$	32.0000	1.41421
$513$	5.00000	0.220755
$514$	54.0000	2.38184
$515$	6.00000	0.264392
$516$	16.0000	0.704361
$517$	0	0
$518$	−12.0000	−0.527250
$519$	−8.00000	−0.351161
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	12.0000	0.525226
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	36.0000	1.57267
$525$	3.00000	0.130931
$526$	42.0000	1.83129
$527$	−28.0000	−1.21970
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−20.0000	−0.868744
$531$	8.00000	0.347170
$532$	−6.00000	−0.260133
$533$	48.0000	2.07911
$534$	−8.00000	−0.346194
$535$	−17.0000	−0.734974
$536$	0	0
$537$	−18.0000	−0.776757
$538$	−10.0000	−0.431131
$539$	0	0
$540$	−10.0000	−0.430331
$541$	−15.0000	−0.644900	−0.322450	−	0.946586i	$-0.604506\pi$
$541$	−15.0000	−0.644900	−0.322450	+	0.946586i	$0.604506\pi$
$542$	−28.0000	−1.20270
$543$	−2.00000	−0.0858282
$544$	32.0000	1.37199
$545$	−6.00000	−0.257012
$546$	36.0000	1.54066
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	24.0000	1.02523
$549$	0	0
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	12.0000	0.510292
$554$	12.0000	0.509831
$555$	−2.00000	−0.0848953
$556$	6.00000	0.254457
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−28.0000	−1.18533
$559$	48.0000	2.03018
$560$	−12.0000	−0.507093
$561$	0	0
$562$	36.0000	1.51857
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−16.0000	−0.673722
$565$	−11.0000	−0.462773
$566$	−26.0000	−1.09286
$567$	3.00000	0.125988
$568$	0	0
$569$	−23.0000	−0.964210	−0.482105	−	0.876113i	$-0.660128\pi$
$569$	−23.0000	−0.964210	−0.482105	+	0.876113i	$0.660128\pi$
$570$	−2.00000	−0.0837708
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	13.0000	0.543083
$574$	48.0000	2.00348
$575$	−3.00000	−0.125109
$576$	16.0000	0.666667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−2.00000	−0.0831890
$579$	4.00000	0.166234
$580$	−6.00000	−0.249136
$581$	36.0000	1.49353
$582$	32.0000	1.32644
$583$	0	0
$584$	0	0
$585$	−12.0000	−0.496139
$586$	−42.0000	−1.73500
$587$	−45.0000	−1.85735	−0.928674	−	0.370896i	$-0.879051\pi$
$587$	−45.0000	−1.85735	−0.928674	+	0.370896i	$0.879051\pi$
$588$	4.00000	0.164957
$589$	−7.00000	−0.288430
$590$	−8.00000	−0.329355
$591$	9.00000	0.370211
$592$	8.00000	0.328798
$593$	−21.0000	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	12.0000	0.491539
$597$	−20.0000	−0.818546
$598$	−36.0000	−1.47215
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	−43.0000	−1.75401	−0.877003	−	0.480484i	$-0.840461\pi$
$601$	−43.0000	−1.75401	−0.877003	+	0.480484i	$0.840461\pi$
$602$	48.0000	1.95633
$603$	8.00000	0.325785
$604$	−2.00000	−0.0813788
$605$	−11.0000	−0.447214
$606$	20.0000	0.812444
$607$	−43.0000	−1.74532	−0.872658	−	0.488332i	$-0.837606\pi$
$607$	−43.0000	−1.74532	−0.872658	+	0.488332i	$0.837606\pi$
$608$	8.00000	0.324443
$609$	−9.00000	−0.364698
$610$	0	0
$611$	−48.0000	−1.94187
$612$	16.0000	0.646762
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	−8.00000	−0.322854
$615$	8.00000	0.322591
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	12.0000	0.482711
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	14.0000	0.562254
$621$	15.0000	0.601929
$622$	−34.0000	−1.36328
$623$	−12.0000	−0.480770
$624$	−24.0000	−0.960769
$625$	1.00000	0.0400000
$626$	−12.0000	−0.479616
$627$	0	0
$628$	−26.0000	−1.03751
$629$	8.00000	0.318981
$630$	−12.0000	−0.478091
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	66.0000	2.62119
$635$	−2.00000	−0.0793676
$636$	−20.0000	−0.793052
$637$	12.0000	0.475457
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	−34.0000	−1.34187
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−18.0000	−0.709299
$645$	8.00000	0.315000
$646$	8.00000	0.314756
$647$	−26.0000	−1.02217	−0.511083	−	0.859532i	$-0.670755\pi$
$647$	−26.0000	−1.02217	−0.511083	+	0.859532i	$0.670755\pi$
$648$	0	0
$649$	0	0
$650$	12.0000	0.470679
$651$	21.0000	0.823055
$652$	−10.0000	−0.391630
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−12.0000	−0.469237
$655$	18.0000	0.703318
$656$	−32.0000	−1.24939
$657$	14.0000	0.546192
$658$	−48.0000	−1.87123
$659$	−3.00000	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$659$	−3.00000	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	−8.00000	−0.310929
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−3.00000	−0.116335
$666$	8.00000	0.309994
$667$	9.00000	0.348481
$668$	36.0000	1.39288
$669$	−10.0000	−0.386622
$670$	−8.00000	−0.309067
$671$	0	0
$672$	−24.0000	−0.925820
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	46.0000	1.77185
$675$	−5.00000	−0.192450
$676$	46.0000	1.76923
$677$	−15.0000	−0.576497	−0.288248	−	0.957556i	$-0.593073\pi$
$677$	−15.0000	−0.576497	−0.288248	+	0.957556i	$0.593073\pi$
$678$	−22.0000	−0.844905
$679$	48.0000	1.84207
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	3.00000	0.114792	0.0573959	−	0.998351i	$-0.481720\pi$
$683$	3.00000	0.114792	0.0573959	+	0.998351i	$0.481720\pi$
$684$	4.00000	0.152944
$685$	12.0000	0.458496
$686$	−30.0000	−1.14541
$687$	6.00000	0.228914
$688$	−32.0000	−1.21999
$689$	−60.0000	−2.28582
$690$	−6.00000	−0.228416
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	−16.0000	−0.608229
$693$	0	0
$694$	−28.0000	−1.06287
$695$	3.00000	0.113796
$696$	0	0
$697$	−32.0000	−1.21209
$698$	38.0000	1.43832
$699$	18.0000	0.680823
$700$	6.00000	0.226779
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	−60.0000	−2.26455
$703$	2.00000	0.0754314
$704$	0	0
$705$	−8.00000	−0.301297
$706$	74.0000	2.78503
$707$	30.0000	1.12827
$708$	−8.00000	−0.300658
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	−24.0000	−0.900704
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−21.0000	−0.786456
$714$	−24.0000	−0.898177
$715$	0	0
$716$	−36.0000	−1.34538
$717$	13.0000	0.485494
$718$	20.0000	0.746393
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	8.00000	0.298142
$721$	18.0000	0.670355
$722$	−36.0000	−1.33978
$723$	17.0000	0.632237
$724$	−4.00000	−0.148659
$725$	−3.00000	−0.111417
$726$	−22.0000	−0.816497
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−14.0000	−0.518163
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−3.00000	−0.110808	−0.0554038	−	0.998464i	$-0.517645\pi$
$733$	−3.00000	−0.110808	−0.0554038	+	0.998464i	$0.517645\pi$
$734$	−24.0000	−0.885856
$735$	2.00000	0.0737711
$736$	24.0000	0.884652
$737$	0	0
$738$	−32.0000	−1.17794
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	−4.00000	−0.147043
$741$	−6.00000	−0.220416
$742$	−60.0000	−2.20267
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	58.0000	2.12353
$747$	−24.0000	−0.878114
$748$	0	0
$749$	−51.0000	−1.86350
$750$	2.00000	0.0730297
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	32.0000	1.16692
$753$	−9.00000	−0.327978
$754$	−36.0000	−1.31104
$755$	−1.00000	−0.0363937
$756$	−30.0000	−1.09109
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	68.0000	2.46987
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−4.00000	−0.144905
$763$	−18.0000	−0.651644
$764$	26.0000	0.940647
$765$	8.00000	0.289241
$766$	28.0000	1.01168
$767$	−24.0000	−0.866590
$768$	16.0000	0.577350
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	8.00000	0.287926
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−32.0000	−1.15022
$775$	7.00000	0.251447
$776$	0	0
$777$	−6.00000	−0.215249
$778$	−20.0000	−0.717035
$779$	−8.00000	−0.286630
$780$	12.0000	0.429669
$781$	0	0
$782$	24.0000	0.858238
$783$	15.0000	0.536056
$784$	−8.00000	−0.285714
$785$	−13.0000	−0.463990
$786$	36.0000	1.28408
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	18.0000	0.641223
$789$	21.0000	0.747620
$790$	8.00000	0.284627
$791$	−33.0000	−1.17334
$792$	0	0
$793$	0	0
$794$	52.0000	1.84541
$795$	−10.0000	−0.354663
$796$	−40.0000	−1.41776
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	−6.00000	−0.212398
$799$	32.0000	1.13208
$800$	−8.00000	−0.282843
$801$	8.00000	0.282666
$802$	18.0000	0.635602
$803$	0	0
$804$	−8.00000	−0.282138
$805$	−9.00000	−0.317208
$806$	84.0000	2.95877
$807$	−5.00000	−0.176008
$808$	0	0
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	2.00000	0.0702728
$811$	−48.0000	−1.68551	−0.842754	−	0.538299i	$-0.819067\pi$
$811$	−48.0000	−1.68551	−0.842754	+	0.538299i	$0.819067\pi$
$812$	−18.0000	−0.631676
$813$	−14.0000	−0.491001
$814$	0	0
$815$	−5.00000	−0.175142
$816$	16.0000	0.560112
$817$	−8.00000	−0.279885
$818$	24.0000	0.839140
$819$	−36.0000	−1.25794
$820$	16.0000	0.558744
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	24.0000	0.837096
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	0	0
$825$	0	0
$826$	−24.0000	−0.835067
$827$	−40.0000	−1.39094	−0.695468	−	0.718557i	$-0.744803\pi$
$827$	−40.0000	−1.39094	−0.695468	+	0.718557i	$0.744803\pi$
$828$	12.0000	0.417029
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	24.0000	0.833052
$831$	6.00000	0.208138
$832$	−48.0000	−1.66410
$833$	−8.00000	−0.277184
$834$	6.00000	0.207763
$835$	18.0000	0.622916
$836$	0	0
$837$	−35.0000	−1.20978
$838$	−48.0000	−1.65813
$839$	19.0000	0.655953	0.327976	−	0.944686i	$-0.393633\pi$
$839$	19.0000	0.655953	0.327976	+	0.944686i	$0.393633\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−56.0000	−1.92989
$843$	18.0000	0.619953
$844$	44.0000	1.51454
$845$	23.0000	0.791224
$846$	32.0000	1.10018
$847$	−33.0000	−1.13389
$848$	40.0000	1.37361
$849$	−13.0000	−0.446159
$850$	−8.00000	−0.274398
$851$	6.00000	0.205677
$852$	−24.0000	−0.822226
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	16.0000	0.545595
$861$	24.0000	0.817918
$862$	−20.0000	−0.681203
$863$	−55.0000	−1.87222	−0.936111	−	0.351705i	$-0.885602\pi$
$863$	−55.0000	−1.87222	−0.936111	+	0.351705i	$0.885602\pi$
$864$	40.0000	1.36083
$865$	−8.00000	−0.272008
$866$	34.0000	1.15537
$867$	−1.00000	−0.0339618
$868$	42.0000	1.42557
$869$	0	0
$870$	−6.00000	−0.203419
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−32.0000	−1.08304
$874$	6.00000	0.202953
$875$	3.00000	0.101419
$876$	−14.0000	−0.473016
$877$	37.0000	1.24940	0.624701	−	0.780864i	$-0.285221\pi$
$877$	37.0000	1.24940	0.624701	+	0.780864i	$0.285221\pi$
$878$	−62.0000	−2.09240
$879$	−21.0000	−0.708312
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	−8.00000	−0.269374
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	−48.0000	−1.61441
$885$	−4.00000	−0.134459
$886$	−2.00000	−0.0671913
$887$	−21.0000	−0.705111	−0.352555	−	0.935791i	$-0.614687\pi$
$887$	−21.0000	−0.705111	−0.352555	+	0.935791i	$0.614687\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	−8.00000	−0.268161
$891$	0	0
$892$	−20.0000	−0.669650
$893$	8.00000	0.267710
$894$	12.0000	0.401340
$895$	−18.0000	−0.601674
$896$	0	0
$897$	−18.0000	−0.601003
$898$	−52.0000	−1.73526
$899$	−21.0000	−0.700389
$900$	−4.00000	−0.133333
$901$	40.0000	1.33259
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	−2.00000	−0.0664455
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	12.0000	0.398234
$909$	−20.0000	−0.663358
$910$	36.0000	1.19339
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−16.0000	−0.529233
$915$	0	0
$916$	12.0000	0.396491
$917$	54.0000	1.78324
$918$	40.0000	1.32020
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	−10.0000	−0.329332
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	40.0000	1.31448
$927$	−12.0000	−0.394132
$928$	24.0000	0.787839
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	14.0000	0.459078
$931$	−2.00000	−0.0655474
$932$	36.0000	1.17922
$933$	−17.0000	−0.556555
$934$	−72.0000	−2.35591
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−24.0000	−0.783628
$939$	−6.00000	−0.195803
$940$	−16.0000	−0.521862
$941$	16.0000	0.521585	0.260793	−	0.965395i	$-0.416016\pi$
$941$	16.0000	0.521585	0.260793	+	0.965395i	$0.416016\pi$
$942$	−26.0000	−0.847126
$943$	−24.0000	−0.781548
$944$	16.0000	0.520756
$945$	−15.0000	−0.487950
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	8.00000	0.259828
$949$	−42.0000	−1.36338
$950$	−2.00000	−0.0648886
$951$	33.0000	1.07010
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	40.0000	1.29505
$955$	13.0000	0.420670
$956$	26.0000	0.840900
$957$	0	0
$958$	20.0000	0.646171
$959$	36.0000	1.16250
$960$	−8.00000	−0.258199
$961$	18.0000	0.580645
$962$	−24.0000	−0.773791
$963$	34.0000	1.09563
$964$	34.0000	1.09507
$965$	4.00000	0.128765
$966$	−18.0000	−0.579141
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	32.0000	1.02746
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	32.0000	1.02640
$973$	9.00000	0.288527
$974$	−16.0000	−0.512673
$975$	6.00000	0.192154
$976$	0	0
$977$	−37.0000	−1.18373	−0.591867	−	0.806035i	$-0.701609\pi$
$977$	−37.0000	−1.18373	−0.591867	+	0.806035i	$0.701609\pi$
$978$	−10.0000	−0.319765
$979$	0	0
$980$	4.00000	0.127775
$981$	12.0000	0.383131
$982$	30.0000	0.957338
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	24.0000	0.764316
$987$	−24.0000	−0.763928
$988$	−12.0000	−0.381771
$989$	−24.0000	−0.763156
$990$	0	0
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−56.0000	−1.77800
$993$	−4.00000	−0.126936
$994$	−72.0000	−2.28370
$995$	−20.0000	−0.634043
$996$	24.0000	0.760469
$997$	−56.0000	−1.77354	−0.886769	−	0.462213i	$-0.847056\pi$
$997$	−56.0000	−1.77354	−0.886769	+	0.462213i	$0.847056\pi$
$998$	−48.0000	−1.51941
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.e.1.1	✓	1
3.2	odd	2		6795.2.a.e.1.1		1
5.4	even	2		3775.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.e.1.1	✓	1	1.1	even	1	trivial
3775.2.a.b.1.1		1	5.4	even	2
6795.2.a.e.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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