LMFDB - Embedded newform 1425.2.c.p.799.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-12,0,0,0,0,-6,0,-6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.4
Root			$0.523976i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.p.799.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.523976i q^{2} -1.00000i q^{3} +1.72545 q^{4} +0.523976 q^{6} -3.20147i q^{7} +1.95205i q^{8} -1.00000 q^{9} +2.20147 q^{11} -1.72545i q^{12} -3.04795i q^{13} +1.67750 q^{14} +2.42807 q^{16} -0.523976i q^{18} -1.00000 q^{19} -3.20147 q^{21} +1.15352i q^{22} -3.24943i q^{23} +1.95205 q^{24} +1.59706 q^{26} +1.00000i q^{27} -5.52398i q^{28} -3.00000 q^{29} -1.24943 q^{31} +5.17635i q^{32} -2.20147i q^{33} -1.72545 q^{36} -3.45090i q^{37} -0.523976i q^{38} -3.04795 q^{39} +8.55646 q^{41} -1.67750i q^{42} -8.00000i q^{43} +3.79853 q^{44} +1.70262 q^{46} +3.35499i q^{47} -2.42807i q^{48} -3.24943 q^{49} -5.25909i q^{52} -0.904094i q^{53} -0.523976 q^{54} +6.24943 q^{56} +1.00000i q^{57} -1.57193i q^{58} -11.2015 q^{59} +3.40294 q^{61} -0.654669i q^{62} +3.20147i q^{63} +2.14386 q^{64} +1.15352 q^{66} -2.29738i q^{67} -3.24943 q^{69} +7.29738 q^{71} -1.95205i q^{72} -1.09591i q^{73} +1.80819 q^{74} -1.72545 q^{76} -7.04795i q^{77} -1.59706i q^{78} -13.5565 q^{79} +1.00000 q^{81} +4.48339i q^{82} +8.20147i q^{83} -5.52398 q^{84} +4.19181 q^{86} +3.00000i q^{87} +4.29738i q^{88} +7.40294 q^{89} -9.75794 q^{91} -5.60672i q^{92} +1.24943i q^{93} -1.75794 q^{94} +5.17635 q^{96} -7.14386i q^{97} -1.70262i q^{98} -2.20147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0.523976i	0.370507i	0.982691	+	0.185254i	$0.0593107\pi$
$2$	0.523976i	0.370507i	−0.982691	+	0.185254i	$0.940689\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.72545	0.862724
$5$	0	0
$5$	0	0
$6$	0.523976	0.213912
$7$	− 3.20147i	− 1.21004i	−0.796209	−	0.605021i	$-0.793165\pi$
$7$	− 3.20147i	− 1.21004i	0.796209	−	0.605021i	$-0.206835\pi$
$8$	1.95205i	0.690153i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.20147	0.663769	0.331884	−	0.943320i	$-0.392316\pi$
$11$	2.20147	0.663769	0.331884	+	0.943320i	$0.392316\pi$
$12$	− 1.72545i	− 0.498094i
$13$	− 3.04795i	− 0.845350i	−0.906281	−	0.422675i	$-0.861091\pi$
$13$	− 3.04795i	− 0.845350i	0.906281	−	0.422675i	$-0.138909\pi$
$14$	1.67750	0.448330
$15$	0	0
$16$	2.42807	0.607018
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 0.523976i	− 0.123502i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.20147	−0.698619
$22$	1.15352i	0.245931i
$23$	− 3.24943i	− 0.677552i	−0.940867	−	0.338776i	$-0.889987\pi$
$23$	− 3.24943i	− 0.677552i	0.940867	−	0.338776i	$-0.110013\pi$
$24$	1.95205	0.398460
$25$	0	0
$26$	1.59706	0.313208
$27$	1.00000i	0.192450i
$28$	− 5.52398i	− 1.04393i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−1.24943	−0.224403	−0.112202	−	0.993685i	$-0.535790\pi$
$31$	−1.24943	−0.224403	−0.112202	+	0.993685i	$0.535790\pi$
$32$	5.17635i	0.915057i
$33$	− 2.20147i	− 0.383227i
$34$	0	0
$35$	0	0
$36$	−1.72545	−0.287575
$37$	− 3.45090i	− 0.567324i	−0.958924	−	0.283662i	$-0.908451\pi$
$37$	− 3.45090i	− 0.567324i	0.958924	−	0.283662i	$-0.0915494\pi$
$38$	− 0.523976i	− 0.0850002i
$39$	−3.04795	−0.488063
$40$	0	0
$41$	8.55646	1.33630	0.668148	−	0.744029i	$-0.267087\pi$
$41$	8.55646	1.33630	0.668148	+	0.744029i	$0.267087\pi$
$42$	− 1.67750i	− 0.258843i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	3.79853	0.572650
$45$	0	0
$46$	1.70262	0.251038
$47$	3.35499i	0.489376i	0.969602	+	0.244688i	$0.0786855\pi$
$47$	3.35499i	0.489376i	−0.969602	+	0.244688i	$0.921314\pi$
$48$	− 2.42807i	− 0.350462i
$49$	−3.24943	−0.464204
$50$	0	0
$51$	0	0
$52$	− 5.25909i	− 0.729304i
$53$	− 0.904094i	− 0.124187i	−0.998070	−	0.0620935i	$-0.980222\pi$
$53$	− 0.904094i	− 0.124187i	0.998070	−	0.0620935i	$-0.0197777\pi$
$54$	−0.523976	−0.0713042
$55$	0	0
$56$	6.24943	0.835115
$57$	1.00000i	0.132453i
$58$	− 1.57193i	− 0.206404i
$59$	−11.2015	−1.45831	−0.729154	−	0.684350i	$-0.760086\pi$
$59$	−11.2015	−1.45831	−0.729154	+	0.684350i	$0.760086\pi$
$60$	0	0
$61$	3.40294	0.435702	0.217851	−	0.975982i	$-0.430095\pi$
$61$	3.40294	0.435702	0.217851	+	0.975982i	$0.430095\pi$
$62$	− 0.654669i	− 0.0831431i
$63$	3.20147i	0.403348i
$64$	2.14386	0.267982
$65$	0	0
$66$	1.15352	0.141988
$67$	− 2.29738i	− 0.280669i	−0.990104	−	0.140335i	$-0.955182\pi$
$67$	− 2.29738i	− 0.280669i	0.990104	−	0.140335i	$-0.0448179\pi$
$68$	0	0
$69$	−3.24943	−0.391185
$70$	0	0
$71$	7.29738	0.866039	0.433020	−	0.901384i	$-0.357448\pi$
$71$	7.29738	0.866039	0.433020	+	0.901384i	$0.357448\pi$
$72$	− 1.95205i	− 0.230051i
$73$	− 1.09591i	− 0.128266i	−0.997941	−	0.0641330i	$-0.979572\pi$
$73$	− 1.09591i	− 0.128266i	0.997941	−	0.0641330i	$-0.0204282\pi$
$74$	1.80819	0.210198
$75$	0	0
$76$	−1.72545	−0.197923
$77$	− 7.04795i	− 0.803189i
$78$	− 1.59706i	− 0.180831i
$79$	−13.5565	−1.52522	−0.762611	−	0.646858i	$-0.776083\pi$
$79$	−13.5565	−1.52522	−0.762611	+	0.646858i	$0.776083\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.48339i	0.495107i
$83$	8.20147i	0.900229i	0.892971	+	0.450114i	$0.148617\pi$
$83$	8.20147i	0.900229i	−0.892971	+	0.450114i	$0.851383\pi$
$84$	−5.52398	−0.602715
$85$	0	0
$86$	4.19181	0.452015
$87$	3.00000i	0.321634i
$88$	4.29738i	0.458102i
$89$	7.40294	0.784711	0.392355	−	0.919814i	$-0.371660\pi$
$89$	7.40294	0.784711	0.392355	+	0.919814i	$0.371660\pi$
$90$	0	0
$91$	−9.75794	−1.02291
$92$	− 5.60672i	− 0.584541i
$93$	1.24943i	0.129559i
$94$	−1.75794	−0.181317
$95$	0	0
$96$	5.17635	0.528309
$97$	− 7.14386i	− 0.725349i	−0.931916	−	0.362674i	$-0.881864\pi$
$97$	− 7.14386i	− 0.725349i	0.931916	−	0.362674i	$-0.118136\pi$
$98$	− 1.70262i	− 0.171991i
$99$	−2.20147	−0.221256
$100$	0	0
$101$	13.7579	1.36897	0.684483	−	0.729029i	$-0.260028\pi$
$101$	13.7579	1.36897	0.684483	+	0.729029i	$0.260028\pi$
$102$	0	0
$103$	6.20147i	0.611049i	0.952184	+	0.305525i	$0.0988318\pi$
$103$	6.20147i	0.611049i	−0.952184	+	0.305525i	$0.901168\pi$
$104$	5.94975	0.583421
$105$	0	0
$106$	0.473724	0.0460122
$107$	− 15.6044i	− 1.50854i	−0.656567	−	0.754268i	$-0.727992\pi$
$107$	− 15.6044i	− 1.50854i	0.656567	−	0.754268i	$-0.272008\pi$
$108$	1.72545i	0.166031i
$109$	−7.45090	−0.713667	−0.356833	−	0.934168i	$-0.616144\pi$
$109$	−7.45090	−0.713667	−0.356833	+	0.934168i	$0.616144\pi$
$110$	0	0
$111$	−3.45090	−0.327545
$112$	− 7.77340i	− 0.734517i
$113$	15.2494i	1.43455i	0.696793	+	0.717273i	$0.254610\pi$
$113$	15.2494i	1.43455i	−0.696793	+	0.717273i	$0.745390\pi$
$114$	−0.523976	−0.0490749
$115$	0	0
$116$	−5.17635	−0.480612
$117$	3.04795i	0.281783i
$118$	− 5.86931i	− 0.540314i
$119$	0	0
$120$	0	0
$121$	−6.15352	−0.559411
$122$	1.78306i	0.161431i
$123$	− 8.55646i	− 0.771510i
$124$	−2.15582	−0.193598
$125$	0	0
$126$	−1.67750	−0.149443
$127$	14.0553i	1.24721i	0.781741	+	0.623604i	$0.214332\pi$
$127$	14.0553i	1.24721i	−0.781741	+	0.623604i	$0.785668\pi$
$128$	11.4760i	1.01435i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	4.34763	0.379854	0.189927	−	0.981798i	$-0.439175\pi$
$131$	4.34763	0.379854	0.189927	+	0.981798i	$0.439175\pi$
$132$	− 3.79853i	− 0.330619i
$133$	3.20147i	0.277603i
$134$	1.20377	0.103990
$135$	0	0
$136$	0	0
$137$	− 19.7077i	− 1.68374i	−0.539680	−	0.841871i	$-0.681455\pi$
$137$	− 19.7077i	− 1.68374i	0.539680	−	0.841871i	$-0.318545\pi$
$138$	− 1.70262i	− 0.144937i
$139$	15.9115	1.34959	0.674796	−	0.738004i	$-0.264232\pi$
$139$	15.9115	1.34959	0.674796	+	0.738004i	$0.264232\pi$
$140$	0	0
$141$	3.35499	0.282541
$142$	3.82365i	0.320874i
$143$	− 6.70998i	− 0.561117i
$144$	−2.42807	−0.202339
$145$	0	0
$146$	0.574229	0.0475235
$147$	3.24943i	0.268008i
$148$	− 5.95435i	− 0.489444i
$149$	15.8538	1.29880	0.649399	−	0.760448i	$-0.275021\pi$
$149$	15.8538	1.29880	0.649399	+	0.760448i	$0.275021\pi$
$150$	0	0
$151$	18.9018	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$151$	18.9018	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$152$	− 1.95205i	− 0.158332i
$153$	0	0
$154$	3.69296	0.297587
$155$	0	0
$156$	−5.25909	−0.421064
$157$	14.4606i	1.15408i	0.816717	+	0.577039i	$0.195792\pi$
$157$	14.4606i	1.15408i	−0.816717	+	0.577039i	$0.804208\pi$
$158$	− 7.10327i	− 0.565106i
$159$	−0.904094	−0.0716994
$160$	0	0
$161$	−10.4029	−0.819867
$162$	0.523976i	0.0411675i
$163$	− 22.0074i	− 1.72375i	−0.507121	−	0.861875i	$-0.669290\pi$
$163$	− 22.0074i	− 1.72375i	0.507121	−	0.861875i	$-0.330710\pi$
$164$	14.7637	1.15285
$165$	0	0
$166$	−4.29738	−0.333541
$167$	3.10557i	0.240316i	0.992755	+	0.120158i	$0.0383401\pi$
$167$	3.10557i	0.240316i	−0.992755	+	0.120158i	$0.961660\pi$
$168$	− 6.24943i	− 0.482154i
$169$	3.70998	0.285383
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 13.8036i	− 1.05251i
$173$	15.9977i	1.21628i	0.793829	+	0.608141i	$0.208085\pi$
$173$	15.9977i	1.21628i	−0.793829	+	0.608141i	$0.791915\pi$
$174$	−1.57193	−0.119168
$175$	0	0
$176$	5.34533	0.402919
$177$	11.2015i	0.841954i
$178$	3.87897i	0.290741i
$179$	−2.89443	−0.216340	−0.108170	−	0.994132i	$-0.534499\pi$
$179$	−2.89443	−0.216340	−0.108170	+	0.994132i	$0.534499\pi$
$180$	0	0
$181$	6.90179	0.513006	0.256503	−	0.966543i	$-0.417430\pi$
$181$	6.90179	0.513006	0.256503	+	0.966543i	$0.417430\pi$
$182$	− 5.11293i	− 0.378995i
$183$	− 3.40294i	− 0.251553i
$184$	6.34303	0.467614
$185$	0	0
$186$	−0.654669	−0.0480027
$187$	0	0
$188$	5.78887i	0.422196i
$189$	3.20147	0.232873
$190$	0	0
$191$	−9.24943	−0.669265	−0.334632	−	0.942349i	$-0.608612\pi$
$191$	−9.24943	−0.669265	−0.334632	+	0.942349i	$0.608612\pi$
$192$	− 2.14386i	− 0.154720i
$193$	1.19411i	0.0859540i	0.999076	+	0.0429770i	$0.0136842\pi$
$193$	1.19411i	0.0859540i	−0.999076	+	0.0429770i	$0.986316\pi$
$194$	3.74321	0.268747
$195$	0	0
$196$	−5.60672	−0.400480
$197$	17.4509i	1.24332i	0.783285	+	0.621662i	$0.213542\pi$
$197$	17.4509i	1.24332i	−0.783285	+	0.621662i	$0.786458\pi$
$198$	− 1.15352i	− 0.0819771i
$199$	−2.79853	−0.198382	−0.0991912	−	0.995068i	$-0.531626\pi$
$199$	−2.79853	−0.198382	−0.0991912	+	0.995068i	$0.531626\pi$
$200$	0	0
$201$	−2.29738	−0.162045
$202$	7.20883i	0.507212i
$203$	9.60442i	0.674098i
$204$	0	0
$205$	0	0
$206$	−3.24943	−0.226398
$207$	3.24943i	0.225851i
$208$	− 7.40065i	− 0.513142i
$209$	−2.20147	−0.152279
$210$	0	0
$211$	−26.2471	−1.80693	−0.903463	−	0.428665i	$-0.858984\pi$
$211$	−26.2471	−1.80693	−0.903463	+	0.428665i	$0.858984\pi$
$212$	− 1.55997i	− 0.107139i
$213$	− 7.29738i	− 0.500008i
$214$	8.17635	0.558924
$215$	0	0
$216$	−1.95205	−0.132820
$217$	4.00000i	0.271538i
$218$	− 3.90409i	− 0.264419i
$219$	−1.09591	−0.0740544
$220$	0	0
$221$	0	0
$222$	− 1.80819i	− 0.121358i
$223$	13.2494i	0.887247i	0.896213	+	0.443624i	$0.146307\pi$
$223$	13.2494i	0.887247i	−0.896213	+	0.443624i	$0.853693\pi$
$224$	16.5719	1.10726
$225$	0	0
$226$	−7.99034	−0.531509
$227$	7.29738i	0.484344i	0.970233	+	0.242172i	$0.0778598\pi$
$227$	7.29738i	0.484344i	−0.970233	+	0.242172i	$0.922140\pi$
$228$	1.72545i	0.114271i
$229$	−15.8635	−1.04829	−0.524145	−	0.851629i	$-0.675615\pi$
$229$	−15.8635	−1.04829	−0.524145	+	0.851629i	$0.675615\pi$
$230$	0	0
$231$	−7.04795	−0.463721
$232$	− 5.85614i	− 0.384475i
$233$	8.64501i	0.566353i	0.959068	+	0.283177i	$0.0913883\pi$
$233$	8.64501i	0.566353i	−0.959068	+	0.283177i	$0.908612\pi$
$234$	−1.59706	−0.104403
$235$	0	0
$236$	−19.3276	−1.25812
$237$	13.5565i	0.880587i
$238$	0	0
$239$	−25.7579	−1.66614	−0.833071	−	0.553166i	$-0.813420\pi$
$239$	−25.7579	−1.66614	−0.833071	+	0.553166i	$0.813420\pi$
$240$	0	0
$241$	9.59706	0.618201	0.309100	−	0.951029i	$-0.399972\pi$
$241$	9.59706	0.618201	0.309100	+	0.951029i	$0.399972\pi$
$242$	− 3.22430i	− 0.207266i
$243$	− 1.00000i	− 0.0641500i
$244$	5.87161	0.375891
$245$	0	0
$246$	4.48339	0.285850
$247$	3.04795i	0.193937i
$248$	− 2.43894i	− 0.154873i
$249$	8.20147	0.519747
$250$	0	0
$251$	17.6118	1.11165	0.555823	−	0.831301i	$-0.312403\pi$
$251$	17.6118	1.11165	0.555823	+	0.831301i	$0.312403\pi$
$252$	5.52398i	0.347978i
$253$	− 7.15352i	− 0.449738i
$254$	−7.36465	−0.462099
$255$	0	0
$256$	−1.72545	−0.107841
$257$	0.692961i	0.0432257i	0.999766	+	0.0216129i	$0.00688012\pi$
$257$	0.692961i	0.0432257i	−0.999766	+	0.0216129i	$0.993120\pi$
$258$	− 4.19181i	− 0.260971i
$259$	−11.0480	−0.686486
$260$	0	0
$261$	3.00000	0.185695
$262$	2.27806i	0.140739i
$263$	23.5062i	1.44946i	0.689036	+	0.724728i	$0.258034\pi$
$263$	23.5062i	1.44946i	−0.689036	+	0.724728i	$0.741966\pi$
$264$	4.29738	0.264485
$265$	0	0
$266$	−1.67750	−0.102854
$267$	− 7.40294i	− 0.453053i
$268$	− 3.96401i	− 0.242140i
$269$	10.6141	0.647152	0.323576	−	0.946202i	$-0.395115\pi$
$269$	10.6141	0.647152	0.323576	+	0.946202i	$0.395115\pi$
$270$	0	0
$271$	15.2974	0.929250	0.464625	−	0.885508i	$-0.346189\pi$
$271$	15.2974	0.929250	0.464625	+	0.885508i	$0.346189\pi$
$272$	0	0
$273$	9.75794i	0.590577i
$274$	10.3264	0.623838
$275$	0	0
$276$	−5.60672	−0.337485
$277$	19.5948i	1.17733i	0.808375	+	0.588667i	$0.200347\pi$
$277$	19.5948i	1.17733i	−0.808375	+	0.588667i	$0.799653\pi$
$278$	8.33723i	0.500034i
$279$	1.24943	0.0748011
$280$	0	0
$281$	−30.5542	−1.82271	−0.911354	−	0.411623i	$-0.864962\pi$
$281$	−30.5542	−1.82271	−0.911354	+	0.411623i	$0.864962\pi$
$282$	1.75794i	0.104684i
$283$	8.61408i	0.512054i	0.966670	+	0.256027i	$0.0824136\pi$
$283$	8.61408i	0.512054i	−0.966670	+	0.256027i	$0.917586\pi$
$284$	12.5913	0.747153
$285$	0	0
$286$	3.51587	0.207898
$287$	− 27.3933i	− 1.61697i
$288$	− 5.17635i	− 0.305019i
$289$	17.0000	1.00000
$290$	0	0
$291$	−7.14386	−0.418780
$292$	− 1.89093i	− 0.110658i
$293$	29.5565i	1.72671i	0.504600	+	0.863354i	$0.331640\pi$
$293$	29.5565i	1.72671i	−0.504600	+	0.863354i	$0.668360\pi$
$294$	−1.70262	−0.0992989
$295$	0	0
$296$	6.73631	0.391540
$297$	2.20147i	0.127742i
$298$	8.30704i	0.481214i
$299$	−9.90409	−0.572769
$300$	0	0
$301$	−25.6118	−1.47624
$302$	9.90409i	0.569917i
$303$	− 13.7579i	− 0.790373i
$304$	−2.42807	−0.139259
$305$	0	0
$306$	0	0
$307$	16.9115i	0.965188i	0.875844	+	0.482594i	$0.160305\pi$
$307$	16.9115i	0.965188i	−0.875844	+	0.482594i	$0.839695\pi$
$308$	− 12.1609i	− 0.692931i
$309$	6.20147	0.352789
$310$	0	0
$311$	14.0959	0.799305	0.399653	−	0.916667i	$-0.369131\pi$
$311$	14.0959	0.799305	0.399653	+	0.916667i	$0.369131\pi$
$312$	− 5.94975i	− 0.336838i
$313$	4.05531i	0.229220i	0.993411	+	0.114610i	$0.0365618\pi$
$313$	4.05531i	0.229220i	−0.993411	+	0.114610i	$0.963438\pi$
$314$	−7.57699	−0.427594
$315$	0	0
$316$	−23.3910	−1.31585
$317$	21.9977i	1.23551i	0.786369	+	0.617757i	$0.211958\pi$
$317$	21.9977i	1.23551i	−0.786369	+	0.617757i	$0.788042\pi$
$318$	− 0.473724i	− 0.0265651i
$319$	−6.60442	−0.369776
$320$	0	0
$321$	−15.6044	−0.870954
$322$	− 5.45090i	− 0.303767i
$323$	0	0
$324$	1.72545	0.0958583
$325$	0	0
$326$	11.5313	0.638662
$327$	7.45090i	0.412036i
$328$	16.7026i	0.922248i
$329$	10.7409	0.592166
$330$	0	0
$331$	12.4583	0.684768	0.342384	−	0.939560i	$-0.388766\pi$
$331$	12.4583	0.684768	0.342384	+	0.939560i	$0.388766\pi$
$332$	14.1512i	0.776649i
$333$	3.45090i	0.189108i
$334$	−1.62724	−0.0890388
$335$	0	0
$336$	−7.77340	−0.424074
$337$	20.4989i	1.11664i	0.829624	+	0.558322i	$0.188555\pi$
$337$	20.4989i	1.11664i	−0.829624	+	0.558322i	$0.811445\pi$
$338$	1.94394i	0.105737i
$339$	15.2494	0.828235
$340$	0	0
$341$	−2.75057	−0.148952
$342$	0.523976i	0.0283334i
$343$	− 12.0074i	− 0.648337i
$344$	15.6164	0.841979
$345$	0	0
$346$	−8.38242	−0.450642
$347$	− 4.45320i	− 0.239060i	−0.992831	−	0.119530i	$-0.961861\pi$
$347$	− 4.45320i	− 0.239060i	0.992831	−	0.119530i	$-0.0381388\pi$
$348$	5.17635i	0.277481i
$349$	9.30704	0.498194	0.249097	−	0.968478i	$-0.419866\pi$
$349$	9.30704	0.498194	0.249097	+	0.968478i	$0.419866\pi$
$350$	0	0
$351$	3.04795	0.162688
$352$	11.3956i	0.607387i
$353$	16.6141i	0.884278i	0.896947	+	0.442139i	$0.145780\pi$
$353$	16.6141i	0.884278i	−0.896947	+	0.442139i	$0.854220\pi$
$354$	−5.86931	−0.311950
$355$	0	0
$356$	12.7734	0.676989
$357$	0	0
$358$	− 1.51661i	− 0.0801556i
$359$	−29.9497	−1.58069	−0.790344	−	0.612664i	$-0.790098\pi$
$359$	−29.9497	−1.58069	−0.790344	+	0.612664i	$0.790098\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	3.61638i	0.190073i
$363$	6.15352i	0.322976i
$364$	−16.8368	−0.882489
$365$	0	0
$366$	1.78306	0.0932022
$367$	− 28.1106i	− 1.46736i	−0.679494	−	0.733681i	$-0.737800\pi$
$367$	− 28.1106i	− 1.46736i	0.679494	−	0.733681i	$-0.262200\pi$
$368$	− 7.88983i	− 0.411286i
$369$	−8.55646	−0.445432
$370$	0	0
$371$	−2.89443	−0.150271
$372$	2.15582i	0.111774i
$373$	− 10.5948i	− 0.548576i	−0.961648	−	0.274288i	$-0.911558\pi$
$373$	− 10.5948i	− 0.548576i	0.961648	−	0.274288i	$-0.0884421\pi$
$374$	0	0
$375$	0	0
$376$	−6.54910	−0.337744
$377$	9.14386i	0.470933i
$378$	1.67750i	0.0862811i
$379$	−3.04795	−0.156563	−0.0782814	−	0.996931i	$-0.524943\pi$
$379$	−3.04795	−0.156563	−0.0782814	+	0.996931i	$0.524943\pi$
$380$	0	0
$381$	14.0553	0.720076
$382$	− 4.84648i	− 0.247968i
$383$	16.6021i	0.848329i	0.905585	+	0.424164i	$0.139432\pi$
$383$	16.6021i	0.848329i	−0.905585	+	0.424164i	$0.860568\pi$
$384$	11.4760	0.585633
$385$	0	0
$386$	−0.625686	−0.0318466
$387$	8.00000i	0.406663i
$388$	− 12.3264i	− 0.625776i
$389$	−7.25909	−0.368050	−0.184025	−	0.982922i	$-0.558913\pi$
$389$	−7.25909	−0.368050	−0.184025	+	0.982922i	$0.558913\pi$
$390$	0	0
$391$	0	0
$392$	− 6.34303i	− 0.320371i
$393$	− 4.34763i	− 0.219309i
$394$	−9.14386	−0.460661
$395$	0	0
$396$	−3.79853	−0.190883
$397$	20.2664i	1.01714i	0.861020	+	0.508572i	$0.169826\pi$
$397$	20.2664i	1.01714i	−0.861020	+	0.508572i	$0.830174\pi$
$398$	− 1.46636i	− 0.0735021i
$399$	3.20147	0.160274
$400$	0	0
$401$	−3.24943	−0.162269	−0.0811343	−	0.996703i	$-0.525854\pi$
$401$	−3.24943	−0.162269	−0.0811343	+	0.996703i	$0.525854\pi$
$402$	− 1.20377i	− 0.0600387i
$403$	3.80819i	0.189699i
$404$	23.7386	1.18104
$405$	0	0
$406$	−5.03249	−0.249758
$407$	− 7.59706i	− 0.376572i
$408$	0	0
$409$	−34.0457	−1.68345	−0.841725	−	0.539907i	$-0.818459\pi$
$409$	−34.0457	−1.68345	−0.841725	+	0.539907i	$0.818459\pi$
$410$	0	0
$411$	−19.7077	−0.972108
$412$	10.7003i	0.527167i
$413$	35.8612i	1.76461i
$414$	−1.70262	−0.0836793
$415$	0	0
$416$	15.7773	0.773544
$417$	− 15.9115i	− 0.779187i
$418$	− 1.15352i	− 0.0564205i
$419$	−26.2568	−1.28273	−0.641364	−	0.767237i	$-0.721631\pi$
$419$	−26.2568	−1.28273	−0.641364	+	0.767237i	$0.721631\pi$
$420$	0	0
$421$	−11.7579	−0.573047	−0.286523	−	0.958073i	$-0.592500\pi$
$421$	−11.7579	−0.573047	−0.286523	+	0.958073i	$0.592500\pi$
$422$	− 13.7529i	− 0.669479i
$423$	− 3.35499i	− 0.163125i
$424$	1.76483	0.0857080
$425$	0	0
$426$	3.82365	0.185257
$427$	− 10.8944i	− 0.527219i
$428$	− 26.9246i	− 1.30145i
$429$	−6.70998	−0.323961
$430$	0	0
$431$	−27.0051	−1.30079	−0.650394	−	0.759597i	$-0.725396\pi$
$431$	−27.0051	−1.30079	−0.650394	+	0.759597i	$0.725396\pi$
$432$	2.42807i	0.116821i
$433$	− 14.6597i	− 0.704502i	−0.935906	−	0.352251i	$-0.885416\pi$
$433$	− 14.6597i	− 0.704502i	0.935906	−	0.352251i	$-0.114584\pi$
$434$	−2.09591	−0.100607
$435$	0	0
$436$	−12.8561	−0.615698
$437$	3.24943i	0.155441i
$438$	− 0.574229i	− 0.0274377i
$439$	22.7653	1.08653	0.543264	−	0.839562i	$-0.317188\pi$
$439$	22.7653	1.08653	0.543264	+	0.839562i	$0.317188\pi$
$440$	0	0
$441$	3.24943	0.154735
$442$	0	0
$443$	− 39.3601i	− 1.87005i	−0.354578	−	0.935026i	$-0.615376\pi$
$443$	− 39.3601i	− 1.87005i	0.354578	−	0.935026i	$-0.384624\pi$
$444$	−5.95435	−0.282581
$445$	0	0
$446$	−6.94239	−0.328732
$447$	− 15.8538i	− 0.749861i
$448$	− 6.86350i	− 0.324270i
$449$	12.3047	0.580697	0.290348	−	0.956921i	$-0.406229\pi$
$449$	12.3047	0.580697	0.290348	+	0.956921i	$0.406229\pi$
$450$	0	0
$451$	18.8368	0.886991
$452$	26.3121i	1.23762i
$453$	− 18.9018i	− 0.888084i
$454$	−3.82365	−0.179453
$455$	0	0
$456$	−1.95205	−0.0914130
$457$	− 3.15122i	− 0.147408i	−0.997280	−	0.0737039i	$-0.976518\pi$
$457$	− 3.15122i	− 0.147408i	0.997280	−	0.0737039i	$-0.0234820\pi$
$458$	− 8.31210i	− 0.388399i
$459$	0	0
$460$	0	0
$461$	−32.0457	−1.49251	−0.746257	−	0.665657i	$-0.768151\pi$
$461$	−32.0457	−1.49251	−0.746257	+	0.665657i	$0.768151\pi$
$462$	− 3.69296i	− 0.171812i
$463$	19.0936i	0.887355i	0.896186	+	0.443678i	$0.146326\pi$
$463$	19.0936i	0.887355i	−0.896186	+	0.443678i	$0.853674\pi$
$464$	−7.28421	−0.338161
$465$	0	0
$466$	−4.52978	−0.209838
$467$	− 0.105567i	− 0.00488505i	−0.999997	−	0.00244252i	$-0.999223\pi$
$467$	− 0.105567i	− 0.00488505i	0.999997	−	0.00244252i	$-0.000777480\pi$
$468$	5.25909i	0.243101i
$469$	−7.35499	−0.339622
$470$	0	0
$471$	14.4606	0.666307
$472$	− 21.8658i	− 1.00646i
$473$	− 17.6118i	− 0.809790i
$474$	−7.10327	−0.326264
$475$	0	0
$476$	0	0
$477$	0.904094i	0.0413956i
$478$	− 13.4966i	− 0.617318i
$479$	29.2951	1.33853	0.669263	−	0.743025i	$-0.266610\pi$
$479$	29.2951	1.33853	0.669263	+	0.743025i	$0.266610\pi$
$480$	0	0
$481$	−10.5182	−0.479587
$482$	5.02863i	0.229048i
$483$	10.4029i	0.473350i
$484$	−10.6176	−0.482617
$485$	0	0
$486$	0.523976	0.0237681
$487$	6.24206i	0.282855i	0.989949	+	0.141427i	$0.0451692\pi$
$487$	6.24206i	0.282855i	−0.989949	+	0.141427i	$0.954831\pi$
$488$	6.64271i	0.300701i
$489$	−22.0074	−0.995207
$490$	0	0
$491$	−14.5182	−0.655196	−0.327598	−	0.944817i	$-0.606239\pi$
$491$	−14.5182	−0.655196	−0.327598	+	0.944817i	$0.606239\pi$
$492$	− 14.7637i	− 0.665601i
$493$	0	0
$494$	−1.59706	−0.0718549
$495$	0	0
$496$	−3.03369	−0.136217
$497$	− 23.3624i	− 1.04794i
$498$	4.29738i	0.192570i
$499$	−0.913756	−0.0409053	−0.0204527	−	0.999791i	$-0.506511\pi$
$499$	−0.913756	−0.0409053	−0.0204527	+	0.999791i	$0.506511\pi$
$500$	0	0
$501$	3.10557	0.138746
$502$	9.22816i	0.411873i
$503$	27.8538i	1.24194i	0.783834	+	0.620971i	$0.213261\pi$
$503$	27.8538i	1.24194i	−0.783834	+	0.620971i	$0.786739\pi$
$504$	−6.24943	−0.278372
$505$	0	0
$506$	3.74828	0.166631
$507$	− 3.70998i	− 0.164766i
$508$	24.2517i	1.07600i
$509$	31.5136	1.39681	0.698407	−	0.715701i	$-0.253892\pi$
$509$	31.5136	1.39681	0.698407	+	0.715701i	$0.253892\pi$
$510$	0	0
$511$	−3.50851	−0.155207
$512$	22.0480i	0.974391i
$513$	− 1.00000i	− 0.0441511i
$514$	−0.363095	−0.0160154
$515$	0	0
$516$	−13.8036	−0.607669
$517$	7.38592i	0.324832i
$518$	− 5.78887i	− 0.254348i
$519$	15.9977	0.702221
$520$	0	0
$521$	−9.49885	−0.416152	−0.208076	−	0.978113i	$-0.566720\pi$
$521$	−9.49885	−0.416152	−0.208076	+	0.978113i	$0.566720\pi$
$522$	1.57193i	0.0688015i
$523$	23.9691i	1.04809i	0.851689	+	0.524047i	$0.175578\pi$
$523$	23.9691i	1.04809i	−0.851689	+	0.524047i	$0.824422\pi$
$524$	7.50161	0.327709
$525$	0	0
$526$	−12.3167	−0.537034
$527$	0	0
$528$	− 5.34533i	− 0.232626i
$529$	12.4412	0.540923
$530$	0	0
$531$	11.2015	0.486102
$532$	5.52398i	0.239495i
$533$	− 26.0797i	− 1.12964i
$534$	3.87897	0.167859
$535$	0	0
$536$	4.48459	0.193705
$537$	2.89443i	0.124904i
$538$	5.56153i	0.239774i
$539$	−7.15352	−0.308124
$540$	0	0
$541$	21.9401	0.943278	0.471639	−	0.881792i	$-0.343663\pi$
$541$	21.9401	0.943278	0.471639	+	0.881792i	$0.343663\pi$
$542$	8.01546i	0.344294i
$543$	− 6.90179i	− 0.296184i
$544$	0	0
$545$	0	0
$546$	−5.11293	−0.218813
$547$	− 39.0074i	− 1.66783i	−0.551890	−	0.833917i	$-0.686093\pi$
$547$	− 39.0074i	− 1.66783i	0.551890	−	0.833917i	$-0.313907\pi$
$548$	− 34.0046i	− 1.45260i
$549$	−3.40294	−0.145234
$550$	0	0
$551$	3.00000	0.127804
$552$	− 6.34303i	− 0.269977i
$553$	43.4006i	1.84558i
$554$	−10.2672	−0.436211
$555$	0	0
$556$	27.4544	1.16433
$557$	− 26.0959i	− 1.10572i	−0.833274	−	0.552860i	$-0.813537\pi$
$557$	− 26.0959i	− 1.10572i	0.833274	−	0.552860i	$-0.186463\pi$
$558$	0.654669i	0.0277144i
$559$	−24.3836	−1.03132
$560$	0	0
$561$	0	0
$562$	− 16.0097i	− 0.675327i
$563$	− 20.8944i	− 0.880595i	−0.897852	−	0.440298i	$-0.854873\pi$
$563$	− 20.8944i	− 0.880595i	0.897852	−	0.440298i	$-0.145127\pi$
$564$	5.78887	0.243755
$565$	0	0
$566$	−4.51357	−0.189720
$567$	− 3.20147i	− 0.134449i
$568$	14.2448i	0.597700i
$569$	39.8419	1.67026	0.835129	−	0.550054i	$-0.185393\pi$
$569$	39.8419	1.67026	0.835129	+	0.550054i	$0.185393\pi$
$570$	0	0
$571$	16.7173	0.699599	0.349800	−	0.936825i	$-0.386250\pi$
$571$	16.7173	0.699599	0.349800	+	0.936825i	$0.386250\pi$
$572$	− 11.5777i	− 0.484089i
$573$	9.24943i	0.386400i
$574$	14.3534	0.599101
$575$	0	0
$576$	−2.14386	−0.0893274
$577$	36.0360i	1.50020i	0.661326	+	0.750099i	$0.269994\pi$
$577$	36.0360i	1.50020i	−0.661326	+	0.750099i	$0.730006\pi$
$578$	8.90760i	0.370507i
$579$	1.19411	0.0496255
$580$	0	0
$581$	26.2568	1.08932
$582$	− 3.74321i	− 0.155161i
$583$	− 1.99034i	− 0.0824314i
$584$	2.13926	0.0885232
$585$	0	0
$586$	−15.4869	−0.639758
$587$	3.95941i	0.163422i	0.996656	+	0.0817111i	$0.0260385\pi$
$587$	3.95941i	0.163422i	−0.996656	+	0.0817111i	$0.973962\pi$
$588$	5.60672i	0.231217i
$589$	1.24943	0.0514817
$590$	0	0
$591$	17.4509	0.717834
$592$	− 8.37902i	− 0.344376i
$593$	− 21.1439i	− 0.868274i	−0.900847	−	0.434137i	$-0.857053\pi$
$593$	− 21.1439i	− 0.868274i	0.900847	−	0.434137i	$-0.142947\pi$
$594$	−1.15352	−0.0473295
$595$	0	0
$596$	27.3550	1.12050
$597$	2.79853i	0.114536i
$598$	− 5.18951i	− 0.212215i
$599$	−4.19181	−0.171273	−0.0856364	−	0.996326i	$-0.527292\pi$
$599$	−4.19181	−0.171273	−0.0856364	+	0.996326i	$0.527292\pi$
$600$	0	0
$601$	−4.54910	−0.185562	−0.0927809	−	0.995687i	$-0.529576\pi$
$601$	−4.54910	−0.185562	−0.0927809	+	0.995687i	$0.529576\pi$
$602$	− 13.4200i	− 0.546957i
$603$	2.29738i	0.0935565i
$604$	32.6141	1.32705
$605$	0	0
$606$	7.20883	0.292839
$607$	− 34.6044i	− 1.40455i	−0.711906	−	0.702275i	$-0.752168\pi$
$607$	− 34.6044i	− 1.40455i	0.711906	−	0.702275i	$-0.247832\pi$
$608$	− 5.17635i	− 0.209929i
$609$	9.60442	0.389191
$610$	0	0
$611$	10.2259	0.413694
$612$	0	0
$613$	− 4.13420i	− 0.166979i	−0.996509	−	0.0834893i	$-0.973394\pi$
$613$	− 4.13420i	− 0.166979i	0.996509	−	0.0834893i	$-0.0266064\pi$
$614$	−8.86120	−0.357609
$615$	0	0
$616$	13.7579	0.554323
$617$	− 17.2398i	− 0.694047i	−0.937856	−	0.347023i	$-0.887192\pi$
$617$	− 17.2398i	− 0.694047i	0.937856	−	0.347023i	$-0.112808\pi$
$618$	3.24943i	0.130711i
$619$	39.0244	1.56852	0.784261	−	0.620431i	$-0.213042\pi$
$619$	39.0244	1.56852	0.784261	+	0.620431i	$0.213042\pi$
$620$	0	0
$621$	3.24943	0.130395
$622$	7.38592i	0.296148i
$623$	− 23.7003i	− 0.949533i
$624$	−7.40065	−0.296263
$625$	0	0
$626$	−2.12489	−0.0849276
$627$	2.20147i	0.0879183i
$628$	24.9510i	0.995651i
$629$	0	0
$630$	0	0
$631$	34.0959	1.35734	0.678668	−	0.734445i	$-0.262557\pi$
$631$	34.0959	1.35734	0.678668	+	0.734445i	$0.262557\pi$
$632$	− 26.4629i	− 1.05264i
$633$	26.2471i	1.04323i
$634$	−11.5263	−0.457767
$635$	0	0
$636$	−1.55997	−0.0618568
$637$	9.90409i	0.392415i
$638$	− 3.46056i	− 0.137005i
$639$	−7.29738	−0.288680
$640$	0	0
$641$	27.8036	1.09818	0.549088	−	0.835765i	$-0.314975\pi$
$641$	27.8036	1.09818	0.549088	+	0.835765i	$0.314975\pi$
$642$	− 8.17635i	− 0.322695i
$643$	− 40.2185i	− 1.58606i	−0.609181	−	0.793031i	$-0.708502\pi$
$643$	− 40.2185i	− 1.58606i	0.609181	−	0.793031i	$-0.291498\pi$
$644$	−17.9497	−0.707319
$645$	0	0
$646$	0	0
$647$	20.1512i	0.792226i	0.918202	+	0.396113i	$0.129641\pi$
$647$	20.1512i	0.792226i	−0.918202	+	0.396113i	$0.870359\pi$
$648$	1.95205i	0.0766837i
$649$	−24.6597	−0.967979
$650$	0	0
$651$	4.00000	0.156772
$652$	− 37.9726i	− 1.48712i
$653$	− 26.7556i	− 1.04703i	−0.852017	−	0.523514i	$-0.824621\pi$
$653$	− 26.7556i	− 1.04703i	0.852017	−	0.523514i	$-0.175379\pi$
$654$	−3.90409	−0.152662
$655$	0	0
$656$	20.7757	0.811155
$657$	1.09591i	0.0427553i
$658$	5.62799i	0.219402i
$659$	−17.6118	−0.686057	−0.343029	−	0.939325i	$-0.611453\pi$
$659$	−17.6118	−0.686057	−0.343029	+	0.939325i	$0.611453\pi$
$660$	0	0
$661$	11.5159	0.447916	0.223958	−	0.974599i	$-0.428102\pi$
$661$	11.5159	0.447916	0.223958	+	0.974599i	$0.428102\pi$
$662$	6.52783i	0.253711i
$663$	0	0
$664$	−16.0097	−0.621295
$665$	0	0
$666$	−1.80819	−0.0700659
$667$	9.74828i	0.377455i
$668$	5.35850i	0.207326i
$669$	13.2494	0.512252
$670$	0	0
$671$	7.49149	0.289206
$672$	− 16.5719i	− 0.639276i
$673$	7.85384i	0.302743i	0.988477	+	0.151372i	$0.0483690\pi$
$673$	7.85384i	0.302743i	−0.988477	+	0.151372i	$0.951631\pi$
$674$	−10.7409	−0.413725
$675$	0	0
$676$	6.40139	0.246207
$677$	43.5136i	1.67236i	0.548453	+	0.836181i	$0.315217\pi$
$677$	43.5136i	1.67236i	−0.548453	+	0.836181i	$0.684783\pi$
$678$	7.99034i	0.306867i
$679$	−22.8709	−0.877703
$680$	0	0
$681$	7.29738	0.279636
$682$	− 1.44124i	− 0.0551878i
$683$	4.60212i	0.176095i	0.996116	+	0.0880476i	$0.0280627\pi$
$683$	4.60212i	0.176095i	−0.996116	+	0.0880476i	$0.971937\pi$
$684$	1.72545	0.0659742
$685$	0	0
$686$	6.29157	0.240213
$687$	15.8635i	0.605230i
$688$	− 19.4246i	− 0.740555i
$689$	−2.75564	−0.104981
$690$	0	0
$691$	28.5948	1.08780	0.543898	−	0.839151i	$-0.316948\pi$
$691$	28.5948	1.08780	0.543898	+	0.839151i	$0.316948\pi$
$692$	27.6032i	1.04932i
$693$	7.04795i	0.267730i
$694$	2.33337	0.0885735
$695$	0	0
$696$	−5.85614	−0.221976
$697$	0	0
$698$	4.87667i	0.184585i
$699$	8.64501	0.326984
$700$	0	0
$701$	−18.2111	−0.687825	−0.343913	−	0.939002i	$-0.611752\pi$
$701$	−18.2111	−0.687825	−0.343913	+	0.939002i	$0.611752\pi$
$702$	1.59706i	0.0602770i
$703$	3.45090i	0.130153i
$704$	4.71964	0.177878
$705$	0	0
$706$	−8.70538	−0.327631
$707$	− 44.0457i	− 1.65651i
$708$	19.3276i	0.726374i
$709$	10.6930	0.401583	0.200791	−	0.979634i	$-0.435649\pi$
$709$	10.6930	0.401583	0.200791	+	0.979634i	$0.435649\pi$
$710$	0	0
$711$	13.5565	0.508407
$712$	14.4509i	0.541570i
$713$	4.05991i	0.152045i
$714$	0	0
$715$	0	0
$716$	−4.99420	−0.186642
$717$	25.7579i	0.961948i
$718$	− 15.6930i	− 0.585656i
$719$	41.9548	1.56465	0.782325	−	0.622870i	$-0.214034\pi$
$719$	41.9548	1.56465	0.782325	+	0.622870i	$0.214034\pi$
$720$	0	0
$721$	19.8538	0.739396
$722$	0.523976i	0.0195004i
$723$	− 9.59706i	− 0.356918i
$724$	11.9087	0.442583
$725$	0	0
$726$	−3.22430	−0.119665
$727$	− 28.4103i	− 1.05368i	−0.849964	−	0.526840i	$-0.823377\pi$
$727$	− 28.4103i	− 1.05368i	0.849964	−	0.526840i	$-0.176623\pi$
$728$	− 19.0480i	− 0.705964i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 5.87161i	− 0.217021i
$733$	− 14.6930i	− 0.542697i	−0.962481	−	0.271348i	$-0.912530\pi$
$733$	− 14.6930i	− 0.542697i	0.962481	−	0.271348i	$-0.0874696\pi$
$734$	14.7293	0.543669
$735$	0	0
$736$	16.8201	0.619999
$737$	− 5.05761i	− 0.186300i
$738$	− 4.48339i	− 0.165036i
$739$	−14.4103	−0.530092	−0.265046	−	0.964236i	$-0.585387\pi$
$739$	−14.4103	−0.530092	−0.265046	+	0.964236i	$0.585387\pi$
$740$	0	0
$741$	3.04795	0.111969
$742$	− 1.51661i	− 0.0556767i
$743$	− 31.5851i	− 1.15874i	−0.815063	−	0.579372i	$-0.803298\pi$
$743$	− 31.5851i	− 1.15874i	0.815063	−	0.579372i	$-0.196702\pi$
$744$	−2.43894	−0.0894158
$745$	0	0
$746$	5.55140	0.203251
$747$	− 8.20147i	− 0.300076i
$748$	0	0
$749$	−49.9571	−1.82539
$750$	0	0
$751$	25.4509	0.928716	0.464358	−	0.885647i	$-0.346285\pi$
$751$	25.4509	0.928716	0.464358	+	0.885647i	$0.346285\pi$
$752$	8.14616i	0.297060i
$753$	− 17.6118i	− 0.641809i
$754$	−4.79117	−0.174484
$755$	0	0
$756$	5.52398	0.200905
$757$	14.0170i	0.509457i	0.967013	+	0.254729i	$0.0819862\pi$
$757$	14.0170i	0.509457i	−0.967013	+	0.254729i	$0.918014\pi$
$758$	− 1.59706i	− 0.0580077i
$759$	−7.15352	−0.259656
$760$	0	0
$761$	−47.2738	−1.71367	−0.856837	−	0.515587i	$-0.827574\pi$
$761$	−47.2738	−1.71367	−0.856837	+	0.515587i	$0.827574\pi$
$762$	7.36465i	0.266793i
$763$	23.8538i	0.863567i
$764$	−15.9594	−0.577391
$765$	0	0
$766$	−8.69912	−0.314312
$767$	34.1416i	1.23278i
$768$	1.72545i	0.0622618i
$769$	−26.0936	−0.940960	−0.470480	−	0.882411i	$-0.655919\pi$
$769$	−26.0936	−0.940960	−0.470480	+	0.882411i	$0.655919\pi$
$770$	0	0
$771$	0.692961	0.0249564
$772$	2.06038i	0.0741546i
$773$	− 23.6501i	− 0.850634i	−0.905044	−	0.425317i	$-0.860163\pi$
$773$	− 23.6501i	− 0.850634i	0.905044	−	0.425317i	$-0.139837\pi$
$774$	−4.19181	−0.150672
$775$	0	0
$776$	13.9451	0.500602
$777$	11.0480i	0.396343i
$778$	− 3.80359i	− 0.136365i
$779$	−8.55646	−0.306567
$780$	0	0
$781$	16.0650	0.574850
$782$	0	0
$783$	− 3.00000i	− 0.107211i
$784$	−7.88983	−0.281780
$785$	0	0
$786$	2.27806	0.0812556
$787$	− 22.8156i	− 0.813287i	−0.913587	−	0.406643i	$-0.866699\pi$
$787$	− 22.8156i	− 0.813287i	0.913587	−	0.406643i	$-0.133301\pi$
$788$	30.1106i	1.07265i
$789$	23.5062	0.836843
$790$	0	0
$791$	48.8206	1.73586
$792$	− 4.29738i	− 0.152701i
$793$	− 10.3720i	− 0.368321i
$794$	−10.6191	−0.376859
$795$	0	0
$796$	−4.82872	−0.171149
$797$	0.748275i	0.0265053i	0.999912	+	0.0132526i	$0.00421857\pi$
$797$	0.748275i	0.0265053i	−0.999912	+	0.0132526i	$0.995781\pi$
$798$	1.67750i	0.0593827i
$799$	0	0
$800$	0	0
$801$	−7.40294	−0.261570
$802$	− 1.70262i	− 0.0601217i
$803$	− 2.41261i	− 0.0851390i
$804$	−3.96401	−0.139800
$805$	0	0
$806$	−1.99540	−0.0702850
$807$	− 10.6141i	− 0.373633i
$808$	26.8561i	0.944796i
$809$	−50.1563	−1.76340	−0.881700	−	0.471810i	$-0.843601\pi$
$809$	−50.1563	−1.76340	−0.881700	+	0.471810i	$0.843601\pi$
$810$	0	0
$811$	27.9903	0.982874	0.491437	−	0.870913i	$-0.336472\pi$
$811$	27.9903	0.982874	0.491437	+	0.870913i	$0.336472\pi$
$812$	16.5719i	0.581561i
$813$	− 15.2974i	− 0.536502i
$814$	3.98068	0.139523
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	− 17.8391i	− 0.623730i
$819$	9.75794	0.340970
$820$	0	0
$821$	−42.3218	−1.47704	−0.738520	−	0.674232i	$-0.764475\pi$
$821$	−42.3218	−1.47704	−0.738520	+	0.674232i	$0.764475\pi$
$822$	− 10.3264i	− 0.360173i
$823$	− 35.6044i	− 1.24109i	−0.784170	−	0.620546i	$-0.786911\pi$
$823$	− 35.6044i	− 1.24109i	0.784170	−	0.620546i	$-0.213089\pi$
$824$	−12.1056	−0.421717
$825$	0	0
$826$	−18.7904	−0.653802
$827$	51.4154i	1.78789i	0.448179	+	0.893944i	$0.352073\pi$
$827$	51.4154i	1.78789i	−0.448179	+	0.893944i	$0.647927\pi$
$828$	5.60672i	0.194847i
$829$	26.9520	0.936083	0.468042	−	0.883706i	$-0.344960\pi$
$829$	26.9520	0.936083	0.468042	+	0.883706i	$0.344960\pi$
$830$	0	0
$831$	19.5948	0.679735
$832$	− 6.53438i	− 0.226539i
$833$	0	0
$834$	8.33723	0.288695
$835$	0	0
$836$	−3.79853	−0.131375
$837$	− 1.24943i	− 0.0431865i
$838$	− 13.7579i	− 0.475260i
$839$	−34.7026	−1.19807	−0.599034	−	0.800724i	$-0.704448\pi$
$839$	−34.7026	−1.19807	−0.599034	+	0.800724i	$0.704448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 6.16088i	− 0.212318i
$843$	30.5542i	1.05234i
$844$	−45.2881	−1.55888
$845$	0	0
$846$	1.75794	0.0604391
$847$	19.7003i	0.676911i
$848$	− 2.19521i	− 0.0753837i
$849$	8.61408	0.295634
$850$	0	0
$851$	−11.2134	−0.384392
$852$	− 12.5913i	− 0.431369i
$853$	0.134197i	0.00459483i	0.999997	+	0.00229741i	$0.000731290\pi$
$853$	0.134197i	0.00459483i	−0.999997	+	0.00229741i	$0.999269\pi$
$854$	5.70843	0.195338
$855$	0	0
$856$	30.4606	1.04112
$857$	− 30.5371i	− 1.04313i	−0.853212	−	0.521564i	$-0.825349\pi$
$857$	− 30.5371i	− 1.04313i	0.853212	−	0.521564i	$-0.174651\pi$
$858$	− 3.51587i	− 0.120030i
$859$	57.1010	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$859$	57.1010	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$860$	0	0
$861$	−27.3933	−0.933561
$862$	− 14.1500i	− 0.481951i
$863$	− 0.376261i	− 0.0128081i	−0.999979	−	0.00640403i	$-0.997962\pi$
$863$	− 0.376261i	− 0.0128081i	0.999979	−	0.00640403i	$-0.00203848\pi$
$864$	−5.17635	−0.176103
$865$	0	0
$866$	7.68135	0.261023
$867$	− 17.0000i	− 0.577350i
$868$	6.90179i	0.234262i
$869$	−29.8442	−1.01239
$870$	0	0
$871$	−7.00230	−0.237264
$872$	− 14.5445i	− 0.492539i
$873$	7.14386i	0.241783i
$874$	−1.70262	−0.0575921
$875$	0	0
$876$	−1.89093	−0.0638886
$877$	− 5.54680i	− 0.187302i	−0.995605	−	0.0936511i	$-0.970146\pi$
$877$	− 5.54680i	− 0.187302i	0.995605	−	0.0936511i	$-0.0298538\pi$
$878$	11.9285i	0.402567i
$879$	29.5565	0.996915
$880$	0	0
$881$	13.3357	0.449290	0.224645	−	0.974441i	$-0.427878\pi$
$881$	13.3357	0.449290	0.224645	+	0.974441i	$0.427878\pi$
$882$	1.70262i	0.0573303i
$883$	− 11.6044i	− 0.390520i	−0.980752	−	0.195260i	$-0.937445\pi$
$883$	− 11.6044i	− 0.390520i	0.980752	−	0.195260i	$-0.0625550\pi$
$884$	0	0
$885$	0	0
$886$	20.6237	0.692868
$887$	− 33.2282i	− 1.11569i	−0.829944	−	0.557846i	$-0.811628\pi$
$887$	− 33.2282i	− 1.11569i	0.829944	−	0.557846i	$-0.188372\pi$
$888$	− 6.73631i	− 0.226056i
$889$	44.9977	1.50917
$890$	0	0
$891$	2.20147	0.0737521
$892$	22.8612i	0.765450i
$893$	− 3.35499i	− 0.112271i
$894$	8.30704	0.277829
$895$	0	0
$896$	36.7402	1.22740
$897$	9.90409i	0.330688i
$898$	6.44739i	0.215152i
$899$	3.74828	0.125012
$900$	0	0
$901$	0	0
$902$	9.87005i	0.328637i
$903$	25.6118i	0.852307i
$904$	−29.7676	−0.990056
$905$	0	0
$906$	9.90409	0.329041
$907$	18.9018i	0.627624i	0.949485	+	0.313812i	$0.101606\pi$
$907$	18.9018i	0.627624i	−0.949485	+	0.313812i	$0.898394\pi$
$908$	12.5913i	0.417855i
$909$	−13.7579	−0.456322
$910$	0	0
$911$	−10.3910	−0.344269	−0.172134	−	0.985073i	$-0.555066\pi$
$911$	−10.3910	−0.344269	−0.172134	+	0.985073i	$0.555066\pi$
$912$	2.42807i	0.0804015i
$913$	18.0553i	0.597544i
$914$	1.65116	0.0546157
$915$	0	0
$916$	−27.3717	−0.904385
$917$	− 13.9188i	− 0.459640i
$918$	0	0
$919$	0.894433	0.0295046	0.0147523	−	0.999891i	$-0.495304\pi$
$919$	0.894433	0.0295046	0.0147523	+	0.999891i	$0.495304\pi$
$920$	0	0
$921$	16.9115	0.557251
$922$	− 16.7912i	− 0.552988i
$923$	− 22.2421i	− 0.732106i
$924$	−12.1609	−0.400064
$925$	0	0
$926$	−10.0046	−0.328772
$927$	− 6.20147i	− 0.203683i
$928$	− 15.5290i	− 0.509766i
$929$	3.69296	0.121162	0.0605811	−	0.998163i	$-0.480705\pi$
$929$	3.69296	0.121162	0.0605811	+	0.998163i	$0.480705\pi$
$930$	0	0
$931$	3.24943	0.106496
$932$	14.9165i	0.488607i
$933$	− 14.0959i	− 0.461479i
$934$	0.0553145	0.00180995
$935$	0	0
$936$	−5.94975	−0.194474
$937$	− 16.2494i	− 0.530846i	−0.964132	−	0.265423i	$-0.914488\pi$
$937$	− 16.2494i	− 0.530846i	0.964132	−	0.265423i	$-0.0855115\pi$
$938$	− 3.85384i	− 0.125832i
$939$	4.05531	0.132340
$940$	0	0
$941$	13.1535	0.428792	0.214396	−	0.976747i	$-0.431222\pi$
$941$	13.1535	0.428792	0.214396	+	0.976747i	$0.431222\pi$
$942$	7.57699i	0.246872i
$943$	− 27.8036i	− 0.905409i
$944$	−27.1980	−0.885218
$945$	0	0
$946$	9.22816	0.300033
$947$	− 40.5136i	− 1.31651i	−0.752793	−	0.658257i	$-0.771294\pi$
$947$	− 40.5136i	− 1.31651i	0.752793	−	0.658257i	$-0.228706\pi$
$948$	23.3910i	0.759704i
$949$	−3.34027	−0.108430
$950$	0	0
$951$	21.9977	0.713324
$952$	0	0
$953$	44.6884i	1.44760i	0.690011	+	0.723799i	$0.257606\pi$
$953$	44.6884i	1.44760i	−0.690011	+	0.723799i	$0.742394\pi$
$954$	−0.473724	−0.0153374
$955$	0	0
$956$	−44.4440	−1.43742
$957$	6.60442i	0.213490i
$958$	15.3499i	0.495934i
$959$	−63.0936	−2.03740
$960$	0	0
$961$	−29.4389	−0.949643
$962$	− 5.51127i	− 0.177691i
$963$	15.6044i	0.502845i
$964$	16.5592	0.533337
$965$	0	0
$966$	−5.45090	−0.175380
$967$	− 38.3910i	− 1.23457i	−0.786739	−	0.617285i	$-0.788232\pi$
$967$	− 38.3910i	− 1.23457i	0.786739	−	0.617285i	$-0.211768\pi$
$968$	− 12.0120i	− 0.386079i
$969$	0	0
$970$	0	0
$971$	−12.6980	−0.407499	−0.203749	−	0.979023i	$-0.565313\pi$
$971$	−12.6980	−0.407499	−0.203749	+	0.979023i	$0.565313\pi$
$972$	− 1.72545i	− 0.0553438i
$973$	− 50.9401i	− 1.63306i
$974$	−3.27069	−0.104800
$975$	0	0
$976$	8.26259	0.264479
$977$	− 3.48183i	− 0.111394i	−0.998448	−	0.0556968i	$-0.982262\pi$
$977$	− 3.48183i	− 0.111394i	0.998448	−	0.0556968i	$-0.0177380\pi$
$978$	− 11.5313i	− 0.368732i
$979$	16.2974	0.520866
$980$	0	0
$981$	7.45090	0.237889
$982$	− 7.60718i	− 0.242755i
$983$	− 10.1152i	− 0.322626i	−0.986903	−	0.161313i	$-0.948427\pi$
$983$	− 10.1152i	− 0.322626i	0.986903	−	0.161313i	$-0.0515728\pi$
$984$	16.7026	0.532460
$985$	0	0
$986$	0	0
$987$	− 10.7409i	− 0.341887i
$988$	5.25909i	0.167314i
$989$	−25.9954	−0.826606
$990$	0	0
$991$	23.7985	0.755985	0.377993	−	0.925809i	$-0.376614\pi$
$991$	23.7985	0.755985	0.377993	+	0.925809i	$0.376614\pi$
$992$	− 6.46746i	− 0.205342i
$993$	− 12.4583i	− 0.395351i
$994$	12.2413	0.388271
$995$	0	0
$996$	14.1512	0.448399
$997$	− 38.9571i	− 1.23378i	−0.787048	−	0.616892i	$-0.788392\pi$
$997$	− 38.9571i	− 1.23378i	0.787048	−	0.616892i	$-0.211608\pi$
$998$	− 0.478786i	− 0.0151557i
$999$	3.45090	0.109182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.p.799.4		6
5.2	odd	4		1425.2.a.u.1.2	✓	3
5.3	odd	4		1425.2.a.v.1.2	yes	3
5.4	even	2	inner	1425.2.c.p.799.3		6
15.2	even	4		4275.2.a.be.1.2		3
15.8	even	4		4275.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.2	✓	3	5.2	odd	4
1425.2.a.v.1.2	yes	3	5.3	odd	4
1425.2.c.p.799.3		6	5.4	even	2	inner
1425.2.c.p.799.4		6	1.1	even	1	trivial
4275.2.a.be.1.2		3	15.2	even	4
4275.2.a.bh.1.2		3	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.523976i q^{2} -1.00000i q^{3} +1.72545 q^{4} +0.523976 q^{6} -3.20147i q^{7} +1.95205i q^{8} -1.00000 q^{9} +2.20147 q^{11} -1.72545i q^{12} -3.04795i q^{13} +1.67750 q^{14} +2.42807 q^{16} -0.523976i q^{18} -1.00000 q^{19} -3.20147 q^{21} +1.15352i q^{22} -3.24943i q^{23} +1.95205 q^{24} +1.59706 q^{26} +1.00000i q^{27} -5.52398i q^{28} -3.00000 q^{29} -1.24943 q^{31} +5.17635i q^{32} -2.20147i q^{33} -1.72545 q^{36} -3.45090i q^{37} -0.523976i q^{38} -3.04795 q^{39} +8.55646 q^{41} -1.67750i q^{42} -8.00000i q^{43} +3.79853 q^{44} +1.70262 q^{46} +3.35499i q^{47} -2.42807i q^{48} -3.24943 q^{49} -5.25909i q^{52} -0.904094i q^{53} -0.523976 q^{54} +6.24943 q^{56} +1.00000i q^{57} -1.57193i q^{58} -11.2015 q^{59} +3.40294 q^{61} -0.654669i q^{62} +3.20147i q^{63} +2.14386 q^{64} +1.15352 q^{66} -2.29738i q^{67} -3.24943 q^{69} +7.29738 q^{71} -1.95205i q^{72} -1.09591i q^{73} +1.80819 q^{74} -1.72545 q^{76} -7.04795i q^{77} -1.59706i q^{78} -13.5565 q^{79} +1.00000 q^{81} +4.48339i q^{82} +8.20147i q^{83} -5.52398 q^{84} +4.19181 q^{86} +3.00000i q^{87} +4.29738i q^{88} +7.40294 q^{89} -9.75794 q^{91} -5.60672i q^{92} +1.24943i q^{93} -1.75794 q^{94} +5.17635 q^{96} -7.14386i q^{97} -1.70262i q^{98} -2.20147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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