LMFDB - Embedded newform 1425.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.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:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.523976 q^{2} +1.00000 q^{3} -1.72545 q^{4} +0.523976 q^{6} -3.20147 q^{7} -1.95205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.523976 q^{2} +1.00000 q^{3} -1.72545 q^{4} +0.523976 q^{6} -3.20147 q^{7} -1.95205 q^{8} +1.00000 q^{9} +2.20147 q^{11} -1.72545 q^{12} +3.04795 q^{13} -1.67750 q^{14} +2.42807 q^{16} +0.523976 q^{18} +1.00000 q^{19} -3.20147 q^{21} +1.15352 q^{22} +3.24943 q^{23} -1.95205 q^{24} +1.59706 q^{26} +1.00000 q^{27} +5.52398 q^{28} +3.00000 q^{29} -1.24943 q^{31} +5.17635 q^{32} +2.20147 q^{33} -1.72545 q^{36} -3.45090 q^{37} +0.523976 q^{38} +3.04795 q^{39} +8.55646 q^{41} -1.67750 q^{42} +8.00000 q^{43} -3.79853 q^{44} +1.70262 q^{46} +3.35499 q^{47} +2.42807 q^{48} +3.24943 q^{49} -5.25909 q^{52} +0.904094 q^{53} +0.523976 q^{54} +6.24943 q^{56} +1.00000 q^{57} +1.57193 q^{58} +11.2015 q^{59} +3.40294 q^{61} -0.654669 q^{62} -3.20147 q^{63} -2.14386 q^{64} +1.15352 q^{66} -2.29738 q^{67} +3.24943 q^{69} +7.29738 q^{71} -1.95205 q^{72} +1.09591 q^{73} -1.80819 q^{74} -1.72545 q^{76} -7.04795 q^{77} +1.59706 q^{78} +13.5565 q^{79} +1.00000 q^{81} +4.48339 q^{82} -8.20147 q^{83} +5.52398 q^{84} +4.19181 q^{86} +3.00000 q^{87} -4.29738 q^{88} -7.40294 q^{89} -9.75794 q^{91} -5.60672 q^{92} -1.24943 q^{93} +1.75794 q^{94} +5.17635 q^{96} -7.14386 q^{97} +1.70262 q^{98} +2.20147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0.523976	0.370507	0.185254	−	0.982691i	$-0.440689\pi$
$2$	0.523976	0.370507	0.185254	+	0.982691i	$0.440689\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.72545	−0.862724
$5$	0	0
$5$	0	0
$6$	0.523976	0.213912
$7$	−3.20147	−1.21004	−0.605021	−	0.796209i	$-0.706835\pi$
$7$	−3.20147	−1.21004	−0.605021	+	0.796209i	$0.706835\pi$
$8$	−1.95205	−0.690153
$9$	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	$0.392316\pi$
$12$	−1.72545	−0.498094
$13$	3.04795	0.845350	0.422675	−	0.906281i	$-0.361091\pi$
$13$	3.04795	0.845350	0.422675	+	0.906281i	$0.361091\pi$
$14$	−1.67750	−0.448330
$15$	0	0
$16$	2.42807	0.607018
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0.523976	0.123502
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.20147	−0.698619
$22$	1.15352	0.245931
$23$	3.24943	0.677552	0.338776	−	0.940867i	$-0.389987\pi$
$23$	3.24943	0.677552	0.338776	+	0.940867i	$0.389987\pi$
$24$	−1.95205	−0.398460
$25$	0	0
$26$	1.59706	0.313208
$27$	1.00000	0.192450
$28$	5.52398	1.04393
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−1.24943	−0.224403	−0.112202	−	0.993685i	$-0.535790\pi$
$31$	−1.24943	−0.224403	−0.112202	+	0.993685i	$0.535790\pi$
$32$	5.17635	0.915057
$33$	2.20147	0.383227
$34$	0	0
$35$	0	0
$36$	−1.72545	−0.287575
$37$	−3.45090	−0.567324	−0.283662	−	0.958924i	$-0.591549\pi$
$37$	−3.45090	−0.567324	−0.283662	+	0.958924i	$0.591549\pi$
$38$	0.523976	0.0850002
$39$	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	$0.267087\pi$
$42$	−1.67750	−0.258843
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−3.79853	−0.572650
$45$	0	0
$46$	1.70262	0.251038
$47$	3.35499	0.489376	0.244688	−	0.969602i	$-0.421314\pi$
$47$	3.35499	0.489376	0.244688	+	0.969602i	$0.421314\pi$
$48$	2.42807	0.350462
$49$	3.24943	0.464204
$50$	0	0
$51$	0	0
$52$	−5.25909	−0.729304
$53$	0.904094	0.124187	0.0620935	−	0.998070i	$-0.480222\pi$
$53$	0.904094	0.124187	0.0620935	+	0.998070i	$0.480222\pi$
$54$	0.523976	0.0713042
$55$	0	0
$56$	6.24943	0.835115
$57$	1.00000	0.132453
$58$	1.57193	0.206404
$59$	11.2015	1.45831	0.729154	−	0.684350i	$-0.239914\pi$
$59$	11.2015	1.45831	0.729154	+	0.684350i	$0.239914\pi$
$60$	0	0
$61$	3.40294	0.435702	0.217851	−	0.975982i	$-0.430095\pi$
$61$	3.40294	0.435702	0.217851	+	0.975982i	$0.430095\pi$
$62$	−0.654669	−0.0831431
$63$	−3.20147	−0.403348
$64$	−2.14386	−0.267982
$65$	0	0
$66$	1.15352	0.141988
$67$	−2.29738	−0.280669	−0.140335	−	0.990104i	$-0.544818\pi$
$67$	−2.29738	−0.280669	−0.140335	+	0.990104i	$0.544818\pi$
$68$	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	$0.357448\pi$
$72$	−1.95205	−0.230051
$73$	1.09591	0.128266	0.0641330	−	0.997941i	$-0.479572\pi$
$73$	1.09591	0.128266	0.0641330	+	0.997941i	$0.479572\pi$
$74$	−1.80819	−0.210198
$75$	0	0
$76$	−1.72545	−0.197923
$77$	−7.04795	−0.803189
$78$	1.59706	0.180831
$79$	13.5565	1.52522	0.762611	−	0.646858i	$-0.223917\pi$
$79$	13.5565	1.52522	0.762611	+	0.646858i	$0.223917\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.48339	0.495107
$83$	−8.20147	−0.900229	−0.450114	−	0.892971i	$-0.648617\pi$
$83$	−8.20147	−0.900229	−0.450114	+	0.892971i	$0.648617\pi$
$84$	5.52398	0.602715
$85$	0	0
$86$	4.19181	0.452015
$87$	3.00000	0.321634
$88$	−4.29738	−0.458102
$89$	−7.40294	−0.784711	−0.392355	−	0.919814i	$-0.628340\pi$
$89$	−7.40294	−0.784711	−0.392355	+	0.919814i	$0.628340\pi$
$90$	0	0
$91$	−9.75794	−1.02291
$92$	−5.60672	−0.584541
$93$	−1.24943	−0.129559
$94$	1.75794	0.181317
$95$	0	0
$96$	5.17635	0.528309
$97$	−7.14386	−0.725349	−0.362674	−	0.931916i	$-0.618136\pi$
$97$	−7.14386	−0.725349	−0.362674	+	0.931916i	$0.618136\pi$
$98$	1.70262	0.171991
$99$	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
$103$	−6.20147	−0.611049	−0.305525	−	0.952184i	$-0.598832\pi$
$103$	−6.20147	−0.611049	−0.305525	+	0.952184i	$0.598832\pi$
$104$	−5.94975	−0.583421
$105$	0	0
$106$	0.473724	0.0460122
$107$	−15.6044	−1.50854	−0.754268	−	0.656567i	$-0.772008\pi$
$107$	−15.6044	−1.50854	−0.754268	+	0.656567i	$0.772008\pi$
$108$	−1.72545	−0.166031
$109$	7.45090	0.713667	0.356833	−	0.934168i	$-0.383856\pi$
$109$	7.45090	0.713667	0.356833	+	0.934168i	$0.383856\pi$
$110$	0	0
$111$	−3.45090	−0.327545
$112$	−7.77340	−0.734517
$113$	−15.2494	−1.43455	−0.717273	−	0.696793i	$-0.754610\pi$
$113$	−15.2494	−1.43455	−0.717273	+	0.696793i	$0.754610\pi$
$114$	0.523976	0.0490749
$115$	0	0
$116$	−5.17635	−0.480612
$117$	3.04795	0.281783
$118$	5.86931	0.540314
$119$	0	0
$120$	0	0
$121$	−6.15352	−0.559411
$122$	1.78306	0.161431
$123$	8.55646	0.771510
$124$	2.15582	0.193598
$125$	0	0
$126$	−1.67750	−0.149443
$127$	14.0553	1.24721	0.623604	−	0.781741i	$-0.285668\pi$
$127$	14.0553	1.24721	0.623604	+	0.781741i	$0.285668\pi$
$128$	−11.4760	−1.01435
$129$	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	$0.439175\pi$
$132$	−3.79853	−0.330619
$133$	−3.20147	−0.277603
$134$	−1.20377	−0.103990
$135$	0	0
$136$	0	0
$137$	−19.7077	−1.68374	−0.841871	−	0.539680i	$-0.818545\pi$
$137$	−19.7077	−1.68374	−0.841871	+	0.539680i	$0.818545\pi$
$138$	1.70262	0.144937
$139$	−15.9115	−1.34959	−0.674796	−	0.738004i	$-0.735768\pi$
$139$	−15.9115	−1.34959	−0.674796	+	0.738004i	$0.735768\pi$
$140$	0	0
$141$	3.35499	0.282541
$142$	3.82365	0.320874
$143$	6.70998	0.561117
$144$	2.42807	0.202339
$145$	0	0
$146$	0.574229	0.0475235
$147$	3.24943	0.268008
$148$	5.95435	0.489444
$149$	−15.8538	−1.29880	−0.649399	−	0.760448i	$-0.724979\pi$
$149$	−15.8538	−1.29880	−0.649399	+	0.760448i	$0.724979\pi$
$150$	0	0
$151$	18.9018	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$151$	18.9018	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$152$	−1.95205	−0.158332
$153$	0	0
$154$	−3.69296	−0.297587
$155$	0	0
$156$	−5.25909	−0.421064
$157$	14.4606	1.15408	0.577039	−	0.816717i	$-0.304208\pi$
$157$	14.4606	1.15408	0.577039	+	0.816717i	$0.304208\pi$
$158$	7.10327	0.565106
$159$	0.904094	0.0716994
$160$	0	0
$161$	−10.4029	−0.819867
$162$	0.523976	0.0411675
$163$	22.0074	1.72375	0.861875	−	0.507121i	$-0.169290\pi$
$163$	22.0074	1.72375	0.861875	+	0.507121i	$0.169290\pi$
$164$	−14.7637	−1.15285
$165$	0	0
$166$	−4.29738	−0.333541
$167$	3.10557	0.240316	0.120158	−	0.992755i	$-0.461660\pi$
$167$	3.10557	0.240316	0.120158	+	0.992755i	$0.461660\pi$
$168$	6.24943	0.482154
$169$	−3.70998	−0.285383
$170$	0	0
$171$	1.00000	0.0764719
$172$	−13.8036	−1.05251
$173$	−15.9977	−1.21628	−0.608141	−	0.793829i	$-0.708085\pi$
$173$	−15.9977	−1.21628	−0.608141	+	0.793829i	$0.708085\pi$
$174$	1.57193	0.119168
$175$	0	0
$176$	5.34533	0.402919
$177$	11.2015	0.841954
$178$	−3.87897	−0.290741
$179$	2.89443	0.216340	0.108170	−	0.994132i	$-0.465501\pi$
$179$	2.89443	0.216340	0.108170	+	0.994132i	$0.465501\pi$
$180$	0	0
$181$	6.90179	0.513006	0.256503	−	0.966543i	$-0.417430\pi$
$181$	6.90179	0.513006	0.256503	+	0.966543i	$0.417430\pi$
$182$	−5.11293	−0.378995
$183$	3.40294	0.251553
$184$	−6.34303	−0.467614
$185$	0	0
$186$	−0.654669	−0.0480027
$187$	0	0
$188$	−5.78887	−0.422196
$189$	−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	$0.608612\pi$
$192$	−2.14386	−0.154720
$193$	−1.19411	−0.0859540	−0.0429770	−	0.999076i	$-0.513684\pi$
$193$	−1.19411	−0.0859540	−0.0429770	+	0.999076i	$0.513684\pi$
$194$	−3.74321	−0.268747
$195$	0	0
$196$	−5.60672	−0.400480
$197$	17.4509	1.24332	0.621662	−	0.783285i	$-0.286458\pi$
$197$	17.4509	1.24332	0.621662	+	0.783285i	$0.286458\pi$
$198$	1.15352	0.0819771
$199$	2.79853	0.198382	0.0991912	−	0.995068i	$-0.468374\pi$
$199$	2.79853	0.198382	0.0991912	+	0.995068i	$0.468374\pi$
$200$	0	0
$201$	−2.29738	−0.162045
$202$	7.20883	0.507212
$203$	−9.60442	−0.674098
$204$	0	0
$205$	0	0
$206$	−3.24943	−0.226398
$207$	3.24943	0.225851
$208$	7.40065	0.513142
$209$	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	$0.858984\pi$
$212$	−1.55997	−0.107139
$213$	7.29738	0.500008
$214$	−8.17635	−0.558924
$215$	0	0
$216$	−1.95205	−0.132820
$217$	4.00000	0.271538
$218$	3.90409	0.264419
$219$	1.09591	0.0740544
$220$	0	0
$221$	0	0
$222$	−1.80819	−0.121358
$223$	−13.2494	−0.887247	−0.443624	−	0.896213i	$-0.646307\pi$
$223$	−13.2494	−0.887247	−0.443624	+	0.896213i	$0.646307\pi$
$224$	−16.5719	−1.10726
$225$	0	0
$226$	−7.99034	−0.531509
$227$	7.29738	0.484344	0.242172	−	0.970233i	$-0.422140\pi$
$227$	7.29738	0.484344	0.242172	+	0.970233i	$0.422140\pi$
$228$	−1.72545	−0.114271
$229$	15.8635	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$229$	15.8635	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$230$	0	0
$231$	−7.04795	−0.463721
$232$	−5.85614	−0.384475
$233$	−8.64501	−0.566353	−0.283177	−	0.959068i	$-0.591388\pi$
$233$	−8.64501	−0.566353	−0.283177	+	0.959068i	$0.591388\pi$
$234$	1.59706	0.104403
$235$	0	0
$236$	−19.3276	−1.25812
$237$	13.5565	0.880587
$238$	0	0
$239$	25.7579	1.66614	0.833071	−	0.553166i	$-0.186580\pi$
$239$	25.7579	1.66614	0.833071	+	0.553166i	$0.186580\pi$
$240$	0	0
$241$	9.59706	0.618201	0.309100	−	0.951029i	$-0.399972\pi$
$241$	9.59706	0.618201	0.309100	+	0.951029i	$0.399972\pi$
$242$	−3.22430	−0.207266
$243$	1.00000	0.0641500
$244$	−5.87161	−0.375891
$245$	0	0
$246$	4.48339	0.285850
$247$	3.04795	0.193937
$248$	2.43894	0.154873
$249$	−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	$0.312403\pi$
$252$	5.52398	0.347978
$253$	7.15352	0.449738
$254$	7.36465	0.462099
$255$	0	0
$256$	−1.72545	−0.107841
$257$	0.692961	0.0432257	0.0216129	−	0.999766i	$-0.493120\pi$
$257$	0.692961	0.0432257	0.0216129	+	0.999766i	$0.493120\pi$
$258$	4.19181	0.260971
$259$	11.0480	0.686486
$260$	0	0
$261$	3.00000	0.185695
$262$	2.27806	0.140739
$263$	−23.5062	−1.44946	−0.724728	−	0.689036i	$-0.758034\pi$
$263$	−23.5062	−1.44946	−0.724728	+	0.689036i	$0.758034\pi$
$264$	−4.29738	−0.264485
$265$	0	0
$266$	−1.67750	−0.102854
$267$	−7.40294	−0.453053
$268$	3.96401	0.242140
$269$	−10.6141	−0.647152	−0.323576	−	0.946202i	$-0.604885\pi$
$269$	−10.6141	−0.647152	−0.323576	+	0.946202i	$0.604885\pi$
$270$	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
$273$	−9.75794	−0.590577
$274$	−10.3264	−0.623838
$275$	0	0
$276$	−5.60672	−0.337485
$277$	19.5948	1.17733	0.588667	−	0.808375i	$-0.299653\pi$
$277$	19.5948	1.17733	0.588667	+	0.808375i	$0.299653\pi$
$278$	−8.33723	−0.500034
$279$	−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	$0.864962\pi$
$282$	1.75794	0.104684
$283$	−8.61408	−0.512054	−0.256027	−	0.966670i	$-0.582414\pi$
$283$	−8.61408	−0.512054	−0.256027	+	0.966670i	$0.582414\pi$
$284$	−12.5913	−0.747153
$285$	0	0
$286$	3.51587	0.207898
$287$	−27.3933	−1.61697
$288$	5.17635	0.305019
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−7.14386	−0.418780
$292$	−1.89093	−0.110658
$293$	−29.5565	−1.72671	−0.863354	−	0.504600i	$-0.831640\pi$
$293$	−29.5565	−1.72671	−0.863354	+	0.504600i	$0.831640\pi$
$294$	1.70262	0.0992989
$295$	0	0
$296$	6.73631	0.391540
$297$	2.20147	0.127742
$298$	−8.30704	−0.481214
$299$	9.90409	0.572769
$300$	0	0
$301$	−25.6118	−1.47624
$302$	9.90409	0.569917
$303$	13.7579	0.790373
$304$	2.42807	0.139259
$305$	0	0
$306$	0	0
$307$	16.9115	0.965188	0.482594	−	0.875844i	$-0.339695\pi$
$307$	16.9115	0.965188	0.482594	+	0.875844i	$0.339695\pi$
$308$	12.1609	0.692931
$309$	−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	$0.369131\pi$
$312$	−5.94975	−0.336838
$313$	−4.05531	−0.229220	−0.114610	−	0.993411i	$-0.536562\pi$
$313$	−4.05531	−0.229220	−0.114610	+	0.993411i	$0.536562\pi$
$314$	7.57699	0.427594
$315$	0	0
$316$	−23.3910	−1.31585
$317$	21.9977	1.23551	0.617757	−	0.786369i	$-0.288042\pi$
$317$	21.9977	1.23551	0.617757	+	0.786369i	$0.288042\pi$
$318$	0.473724	0.0265651
$319$	6.60442	0.369776
$320$	0	0
$321$	−15.6044	−0.870954
$322$	−5.45090	−0.303767
$323$	0	0
$324$	−1.72545	−0.0958583
$325$	0	0
$326$	11.5313	0.638662
$327$	7.45090	0.412036
$328$	−16.7026	−0.922248
$329$	−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	$0.388766\pi$
$332$	14.1512	0.776649
$333$	−3.45090	−0.189108
$334$	1.62724	0.0890388
$335$	0	0
$336$	−7.77340	−0.424074
$337$	20.4989	1.11664	0.558322	−	0.829624i	$-0.311445\pi$
$337$	20.4989	1.11664	0.558322	+	0.829624i	$0.311445\pi$
$338$	−1.94394	−0.105737
$339$	−15.2494	−0.828235
$340$	0	0
$341$	−2.75057	−0.148952
$342$	0.523976	0.0283334
$343$	12.0074	0.648337
$344$	−15.6164	−0.841979
$345$	0	0
$346$	−8.38242	−0.450642
$347$	−4.45320	−0.239060	−0.119530	−	0.992831i	$-0.538139\pi$
$347$	−4.45320	−0.239060	−0.119530	+	0.992831i	$0.538139\pi$
$348$	−5.17635	−0.277481
$349$	−9.30704	−0.498194	−0.249097	−	0.968478i	$-0.580134\pi$
$349$	−9.30704	−0.498194	−0.249097	+	0.968478i	$0.580134\pi$
$350$	0	0
$351$	3.04795	0.162688
$352$	11.3956	0.607387
$353$	−16.6141	−0.884278	−0.442139	−	0.896947i	$-0.645780\pi$
$353$	−16.6141	−0.884278	−0.442139	+	0.896947i	$0.645780\pi$
$354$	5.86931	0.311950
$355$	0	0
$356$	12.7734	0.676989
$357$	0	0
$358$	1.51661	0.0801556
$359$	29.9497	1.58069	0.790344	−	0.612664i	$-0.209902\pi$
$359$	29.9497	1.58069	0.790344	+	0.612664i	$0.209902\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	3.61638	0.190073
$363$	−6.15352	−0.322976
$364$	16.8368	0.882489
$365$	0	0
$366$	1.78306	0.0932022
$367$	−28.1106	−1.46736	−0.733681	−	0.679494i	$-0.762200\pi$
$367$	−28.1106	−1.46736	−0.733681	+	0.679494i	$0.762200\pi$
$368$	7.88983	0.411286
$369$	8.55646	0.445432
$370$	0	0
$371$	−2.89443	−0.150271
$372$	2.15582	0.111774
$373$	10.5948	0.548576	0.274288	−	0.961648i	$-0.411558\pi$
$373$	10.5948	0.548576	0.274288	+	0.961648i	$0.411558\pi$
$374$	0	0
$375$	0	0
$376$	−6.54910	−0.337744
$377$	9.14386	0.470933
$378$	−1.67750	−0.0862811
$379$	3.04795	0.156563	0.0782814	−	0.996931i	$-0.475057\pi$
$379$	3.04795	0.156563	0.0782814	+	0.996931i	$0.475057\pi$
$380$	0	0
$381$	14.0553	0.720076
$382$	−4.84648	−0.247968
$383$	−16.6021	−0.848329	−0.424164	−	0.905585i	$-0.639432\pi$
$383$	−16.6021	−0.848329	−0.424164	+	0.905585i	$0.639432\pi$
$384$	−11.4760	−0.585633
$385$	0	0
$386$	−0.625686	−0.0318466
$387$	8.00000	0.406663
$388$	12.3264	0.625776
$389$	7.25909	0.368050	0.184025	−	0.982922i	$-0.441087\pi$
$389$	7.25909	0.368050	0.184025	+	0.982922i	$0.441087\pi$
$390$	0	0
$391$	0	0
$392$	−6.34303	−0.320371
$393$	4.34763	0.219309
$394$	9.14386	0.460661
$395$	0	0
$396$	−3.79853	−0.190883
$397$	20.2664	1.01714	0.508572	−	0.861020i	$-0.330174\pi$
$397$	20.2664	1.01714	0.508572	+	0.861020i	$0.330174\pi$
$398$	1.46636	0.0735021
$399$	−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	$0.525854\pi$
$402$	−1.20377	−0.0600387
$403$	−3.80819	−0.189699
$404$	−23.7386	−1.18104
$405$	0	0
$406$	−5.03249	−0.249758
$407$	−7.59706	−0.376572
$408$	0	0
$409$	34.0457	1.68345	0.841725	−	0.539907i	$-0.181541\pi$
$409$	34.0457	1.68345	0.841725	+	0.539907i	$0.181541\pi$
$410$	0	0
$411$	−19.7077	−0.972108
$412$	10.7003	0.527167
$413$	−35.8612	−1.76461
$414$	1.70262	0.0836793
$415$	0	0
$416$	15.7773	0.773544
$417$	−15.9115	−0.779187
$418$	1.15352	0.0564205
$419$	26.2568	1.28273	0.641364	−	0.767237i	$-0.278369\pi$
$419$	26.2568	1.28273	0.641364	+	0.767237i	$0.278369\pi$
$420$	0	0
$421$	−11.7579	−0.573047	−0.286523	−	0.958073i	$-0.592500\pi$
$421$	−11.7579	−0.573047	−0.286523	+	0.958073i	$0.592500\pi$
$422$	−13.7529	−0.669479
$423$	3.35499	0.163125
$424$	−1.76483	−0.0857080
$425$	0	0
$426$	3.82365	0.185257
$427$	−10.8944	−0.527219
$428$	26.9246	1.30145
$429$	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	$0.725396\pi$
$432$	2.42807	0.116821
$433$	14.6597	0.704502	0.352251	−	0.935906i	$-0.385416\pi$
$433$	14.6597	0.704502	0.352251	+	0.935906i	$0.385416\pi$
$434$	2.09591	0.100607
$435$	0	0
$436$	−12.8561	−0.615698
$437$	3.24943	0.155441
$438$	0.574229	0.0274377
$439$	−22.7653	−1.08653	−0.543264	−	0.839562i	$-0.682812\pi$
$439$	−22.7653	−1.08653	−0.543264	+	0.839562i	$0.682812\pi$
$440$	0	0
$441$	3.24943	0.154735
$442$	0	0
$443$	39.3601	1.87005	0.935026	−	0.354578i	$-0.115376\pi$
$443$	39.3601	1.87005	0.935026	+	0.354578i	$0.115376\pi$
$444$	5.95435	0.282581
$445$	0	0
$446$	−6.94239	−0.328732
$447$	−15.8538	−0.749861
$448$	6.86350	0.324270
$449$	−12.3047	−0.580697	−0.290348	−	0.956921i	$-0.593771\pi$
$449$	−12.3047	−0.580697	−0.290348	+	0.956921i	$0.593771\pi$
$450$	0	0
$451$	18.8368	0.886991
$452$	26.3121	1.23762
$453$	18.9018	0.888084
$454$	3.82365	0.179453
$455$	0	0
$456$	−1.95205	−0.0914130
$457$	−3.15122	−0.147408	−0.0737039	−	0.997280i	$-0.523482\pi$
$457$	−3.15122	−0.147408	−0.0737039	+	0.997280i	$0.523482\pi$
$458$	8.31210	0.388399
$459$	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	$0.768151\pi$
$462$	−3.69296	−0.171812
$463$	−19.0936	−0.887355	−0.443678	−	0.896186i	$-0.646326\pi$
$463$	−19.0936	−0.887355	−0.443678	+	0.896186i	$0.646326\pi$
$464$	7.28421	0.338161
$465$	0	0
$466$	−4.52978	−0.209838
$467$	−0.105567	−0.00488505	−0.00244252	−	0.999997i	$-0.500777\pi$
$467$	−0.105567	−0.00488505	−0.00244252	+	0.999997i	$0.500777\pi$
$468$	−5.25909	−0.243101
$469$	7.35499	0.339622
$470$	0	0
$471$	14.4606	0.666307
$472$	−21.8658	−1.00646
$473$	17.6118	0.809790
$474$	7.10327	0.326264
$475$	0	0
$476$	0	0
$477$	0.904094	0.0413956
$478$	13.4966	0.617318
$479$	−29.2951	−1.33853	−0.669263	−	0.743025i	$-0.733390\pi$
$479$	−29.2951	−1.33853	−0.669263	+	0.743025i	$0.733390\pi$
$480$	0	0
$481$	−10.5182	−0.479587
$482$	5.02863	0.229048
$483$	−10.4029	−0.473350
$484$	10.6176	0.482617
$485$	0	0
$486$	0.523976	0.0237681
$487$	6.24206	0.282855	0.141427	−	0.989949i	$-0.454831\pi$
$487$	6.24206	0.282855	0.141427	+	0.989949i	$0.454831\pi$
$488$	−6.64271	−0.300701
$489$	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	$0.606239\pi$
$492$	−14.7637	−0.665601
$493$	0	0
$494$	1.59706	0.0718549
$495$	0	0
$496$	−3.03369	−0.136217
$497$	−23.3624	−1.04794
$498$	−4.29738	−0.192570
$499$	0.913756	0.0409053	0.0204527	−	0.999791i	$-0.493489\pi$
$499$	0.913756	0.0409053	0.0204527	+	0.999791i	$0.493489\pi$
$500$	0	0
$501$	3.10557	0.138746
$502$	9.22816	0.411873
$503$	−27.8538	−1.24194	−0.620971	−	0.783834i	$-0.713261\pi$
$503$	−27.8538	−1.24194	−0.620971	+	0.783834i	$0.713261\pi$
$504$	6.24943	0.278372
$505$	0	0
$506$	3.74828	0.166631
$507$	−3.70998	−0.164766
$508$	−24.2517	−1.07600
$509$	−31.5136	−1.39681	−0.698407	−	0.715701i	$-0.746108\pi$
$509$	−31.5136	−1.39681	−0.698407	+	0.715701i	$0.746108\pi$
$510$	0	0
$511$	−3.50851	−0.155207
$512$	22.0480	0.974391
$513$	1.00000	0.0441511
$514$	0.363095	0.0160154
$515$	0	0
$516$	−13.8036	−0.607669
$517$	7.38592	0.324832
$518$	5.78887	0.254348
$519$	−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	$0.566720\pi$
$522$	1.57193	0.0688015
$523$	−23.9691	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$523$	−23.9691	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$524$	−7.50161	−0.327709
$525$	0	0
$526$	−12.3167	−0.537034
$527$	0	0
$528$	5.34533	0.232626
$529$	−12.4412	−0.540923
$530$	0	0
$531$	11.2015	0.486102
$532$	5.52398	0.239495
$533$	26.0797	1.12964
$534$	−3.87897	−0.167859
$535$	0	0
$536$	4.48459	0.193705
$537$	2.89443	0.124904
$538$	−5.56153	−0.239774
$539$	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	$0.343663\pi$
$542$	8.01546	0.344294
$543$	6.90179	0.296184
$544$	0	0
$545$	0	0
$546$	−5.11293	−0.218813
$547$	−39.0074	−1.66783	−0.833917	−	0.551890i	$-0.813907\pi$
$547$	−39.0074	−1.66783	−0.833917	+	0.551890i	$0.813907\pi$
$548$	34.0046	1.45260
$549$	3.40294	0.145234
$550$	0	0
$551$	3.00000	0.127804
$552$	−6.34303	−0.269977
$553$	−43.4006	−1.84558
$554$	10.2672	0.436211
$555$	0	0
$556$	27.4544	1.16433
$557$	−26.0959	−1.10572	−0.552860	−	0.833274i	$-0.686463\pi$
$557$	−26.0959	−1.10572	−0.552860	+	0.833274i	$0.686463\pi$
$558$	−0.654669	−0.0277144
$559$	24.3836	1.03132
$560$	0	0
$561$	0	0
$562$	−16.0097	−0.675327
$563$	20.8944	0.880595	0.440298	−	0.897852i	$-0.354873\pi$
$563$	20.8944	0.880595	0.440298	+	0.897852i	$0.354873\pi$
$564$	−5.78887	−0.243755
$565$	0	0
$566$	−4.51357	−0.189720
$567$	−3.20147	−0.134449
$568$	−14.2448	−0.597700
$569$	−39.8419	−1.67026	−0.835129	−	0.550054i	$-0.814607\pi$
$569$	−39.8419	−1.67026	−0.835129	+	0.550054i	$0.814607\pi$
$570$	0	0
$571$	16.7173	0.699599	0.349800	−	0.936825i	$-0.386250\pi$
$571$	16.7173	0.699599	0.349800	+	0.936825i	$0.386250\pi$
$572$	−11.5777	−0.484089
$573$	−9.24943	−0.386400
$574$	−14.3534	−0.599101
$575$	0	0
$576$	−2.14386	−0.0893274
$577$	36.0360	1.50020	0.750099	−	0.661326i	$-0.230006\pi$
$577$	36.0360	1.50020	0.750099	+	0.661326i	$0.230006\pi$
$578$	−8.90760	−0.370507
$579$	−1.19411	−0.0496255
$580$	0	0
$581$	26.2568	1.08932
$582$	−3.74321	−0.155161
$583$	1.99034	0.0824314
$584$	−2.13926	−0.0885232
$585$	0	0
$586$	−15.4869	−0.639758
$587$	3.95941	0.163422	0.0817111	−	0.996656i	$-0.473962\pi$
$587$	3.95941	0.163422	0.0817111	+	0.996656i	$0.473962\pi$
$588$	−5.60672	−0.231217
$589$	−1.24943	−0.0514817
$590$	0	0
$591$	17.4509	0.717834
$592$	−8.37902	−0.344376
$593$	21.1439	0.868274	0.434137	−	0.900847i	$-0.357053\pi$
$593$	21.1439	0.868274	0.434137	+	0.900847i	$0.357053\pi$
$594$	1.15352	0.0473295
$595$	0	0
$596$	27.3550	1.12050
$597$	2.79853	0.114536
$598$	5.18951	0.212215
$599$	4.19181	0.171273	0.0856364	−	0.996326i	$-0.472708\pi$
$599$	4.19181	0.171273	0.0856364	+	0.996326i	$0.472708\pi$
$600$	0	0
$601$	−4.54910	−0.185562	−0.0927809	−	0.995687i	$-0.529576\pi$
$601$	−4.54910	−0.185562	−0.0927809	+	0.995687i	$0.529576\pi$
$602$	−13.4200	−0.546957
$603$	−2.29738	−0.0935565
$604$	−32.6141	−1.32705
$605$	0	0
$606$	7.20883	0.292839
$607$	−34.6044	−1.40455	−0.702275	−	0.711906i	$-0.747832\pi$
$607$	−34.6044	−1.40455	−0.702275	+	0.711906i	$0.747832\pi$
$608$	5.17635	0.209929
$609$	−9.60442	−0.389191
$610$	0	0
$611$	10.2259	0.413694
$612$	0	0
$613$	4.13420	0.166979	0.0834893	−	0.996509i	$-0.473394\pi$
$613$	4.13420	0.166979	0.0834893	+	0.996509i	$0.473394\pi$
$614$	8.86120	0.357609
$615$	0	0
$616$	13.7579	0.554323
$617$	−17.2398	−0.694047	−0.347023	−	0.937856i	$-0.612808\pi$
$617$	−17.2398	−0.694047	−0.347023	+	0.937856i	$0.612808\pi$
$618$	−3.24943	−0.130711
$619$	−39.0244	−1.56852	−0.784261	−	0.620431i	$-0.786958\pi$
$619$	−39.0244	−1.56852	−0.784261	+	0.620431i	$0.786958\pi$
$620$	0	0
$621$	3.24943	0.130395
$622$	7.38592	0.296148
$623$	23.7003	0.949533
$624$	7.40065	0.296263
$625$	0	0
$626$	−2.12489	−0.0849276
$627$	2.20147	0.0879183
$628$	−24.9510	−0.995651
$629$	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	$0.262557\pi$
$632$	−26.4629	−1.05264
$633$	−26.2471	−1.04323
$634$	11.5263	0.457767
$635$	0	0
$636$	−1.55997	−0.0618568
$637$	9.90409	0.392415
$638$	3.46056	0.137005
$639$	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	$0.314975\pi$
$642$	−8.17635	−0.322695
$643$	40.2185	1.58606	0.793031	−	0.609181i	$-0.208502\pi$
$643$	40.2185	1.58606	0.793031	+	0.609181i	$0.208502\pi$
$644$	17.9497	0.707319
$645$	0	0
$646$	0	0
$647$	20.1512	0.792226	0.396113	−	0.918202i	$-0.370359\pi$
$647$	20.1512	0.792226	0.396113	+	0.918202i	$0.370359\pi$
$648$	−1.95205	−0.0766837
$649$	24.6597	0.967979
$650$	0	0
$651$	4.00000	0.156772
$652$	−37.9726	−1.48712
$653$	26.7556	1.04703	0.523514	−	0.852017i	$-0.324621\pi$
$653$	26.7556	1.04703	0.523514	+	0.852017i	$0.324621\pi$
$654$	3.90409	0.152662
$655$	0	0
$656$	20.7757	0.811155
$657$	1.09591	0.0427553
$658$	−5.62799	−0.219402
$659$	17.6118	0.686057	0.343029	−	0.939325i	$-0.388547\pi$
$659$	17.6118	0.686057	0.343029	+	0.939325i	$0.388547\pi$
$660$	0	0
$661$	11.5159	0.447916	0.223958	−	0.974599i	$-0.428102\pi$
$661$	11.5159	0.447916	0.223958	+	0.974599i	$0.428102\pi$
$662$	6.52783	0.253711
$663$	0	0
$664$	16.0097	0.621295
$665$	0	0
$666$	−1.80819	−0.0700659
$667$	9.74828	0.377455
$668$	−5.35850	−0.207326
$669$	−13.2494	−0.512252
$670$	0	0
$671$	7.49149	0.289206
$672$	−16.5719	−0.639276
$673$	−7.85384	−0.302743	−0.151372	−	0.988477i	$-0.548369\pi$
$673$	−7.85384	−0.302743	−0.151372	+	0.988477i	$0.548369\pi$
$674$	10.7409	0.413725
$675$	0	0
$676$	6.40139	0.246207
$677$	43.5136	1.67236	0.836181	−	0.548453i	$-0.184783\pi$
$677$	43.5136	1.67236	0.836181	+	0.548453i	$0.184783\pi$
$678$	−7.99034	−0.306867
$679$	22.8709	0.877703
$680$	0	0
$681$	7.29738	0.279636
$682$	−1.44124	−0.0551878
$683$	−4.60212	−0.176095	−0.0880476	−	0.996116i	$-0.528063\pi$
$683$	−4.60212	−0.176095	−0.0880476	+	0.996116i	$0.528063\pi$
$684$	−1.72545	−0.0659742
$685$	0	0
$686$	6.29157	0.240213
$687$	15.8635	0.605230
$688$	19.4246	0.740555
$689$	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	$0.316948\pi$
$692$	27.6032	1.04932
$693$	−7.04795	−0.267730
$694$	−2.33337	−0.0885735
$695$	0	0
$696$	−5.85614	−0.221976
$697$	0	0
$698$	−4.87667	−0.184585
$699$	−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	$0.611752\pi$
$702$	1.59706	0.0602770
$703$	−3.45090	−0.130153
$704$	−4.71964	−0.177878
$705$	0	0
$706$	−8.70538	−0.327631
$707$	−44.0457	−1.65651
$708$	−19.3276	−0.726374
$709$	−10.6930	−0.401583	−0.200791	−	0.979634i	$-0.564351\pi$
$709$	−10.6930	−0.401583	−0.200791	+	0.979634i	$0.564351\pi$
$710$	0	0
$711$	13.5565	0.508407
$712$	14.4509	0.541570
$713$	−4.05991	−0.152045
$714$	0	0
$715$	0	0
$716$	−4.99420	−0.186642
$717$	25.7579	0.961948
$718$	15.6930	0.585656
$719$	−41.9548	−1.56465	−0.782325	−	0.622870i	$-0.785966\pi$
$719$	−41.9548	−1.56465	−0.782325	+	0.622870i	$0.785966\pi$
$720$	0	0
$721$	19.8538	0.739396
$722$	0.523976	0.0195004
$723$	9.59706	0.356918
$724$	−11.9087	−0.442583
$725$	0	0
$726$	−3.22430	−0.119665
$727$	−28.4103	−1.05368	−0.526840	−	0.849964i	$-0.676623\pi$
$727$	−28.4103	−1.05368	−0.526840	+	0.849964i	$0.676623\pi$
$728$	19.0480	0.705964
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−5.87161	−0.217021
$733$	14.6930	0.542697	0.271348	−	0.962481i	$-0.412530\pi$
$733$	14.6930	0.542697	0.271348	+	0.962481i	$0.412530\pi$
$734$	−14.7293	−0.543669
$735$	0	0
$736$	16.8201	0.619999
$737$	−5.05761	−0.186300
$738$	4.48339	0.165036
$739$	14.4103	0.530092	0.265046	−	0.964236i	$-0.414613\pi$
$739$	14.4103	0.530092	0.265046	+	0.964236i	$0.414613\pi$
$740$	0	0
$741$	3.04795	0.111969
$742$	−1.51661	−0.0556767
$743$	31.5851	1.15874	0.579372	−	0.815063i	$-0.303298\pi$
$743$	31.5851	1.15874	0.579372	+	0.815063i	$0.303298\pi$
$744$	2.43894	0.0894158
$745$	0	0
$746$	5.55140	0.203251
$747$	−8.20147	−0.300076
$748$	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	$0.346285\pi$
$752$	8.14616	0.297060
$753$	17.6118	0.641809
$754$	4.79117	0.174484
$755$	0	0
$756$	5.52398	0.200905
$757$	14.0170	0.509457	0.254729	−	0.967013i	$-0.418014\pi$
$757$	14.0170	0.509457	0.254729	+	0.967013i	$0.418014\pi$
$758$	1.59706	0.0580077
$759$	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	$0.827574\pi$
$762$	7.36465	0.266793
$763$	−23.8538	−0.863567
$764$	15.9594	0.577391
$765$	0	0
$766$	−8.69912	−0.314312
$767$	34.1416	1.23278
$768$	−1.72545	−0.0622618
$769$	26.0936	0.940960	0.470480	−	0.882411i	$-0.344081\pi$
$769$	26.0936	0.940960	0.470480	+	0.882411i	$0.344081\pi$
$770$	0	0
$771$	0.692961	0.0249564
$772$	2.06038	0.0741546
$773$	23.6501	0.850634	0.425317	−	0.905044i	$-0.360163\pi$
$773$	23.6501	0.850634	0.425317	+	0.905044i	$0.360163\pi$
$774$	4.19181	0.150672
$775$	0	0
$776$	13.9451	0.500602
$777$	11.0480	0.396343
$778$	3.80359	0.136365
$779$	8.55646	0.306567
$780$	0	0
$781$	16.0650	0.574850
$782$	0	0
$783$	3.00000	0.107211
$784$	7.88983	0.281780
$785$	0	0
$786$	2.27806	0.0812556
$787$	−22.8156	−0.813287	−0.406643	−	0.913587i	$-0.633301\pi$
$787$	−22.8156	−0.813287	−0.406643	+	0.913587i	$0.633301\pi$
$788$	−30.1106	−1.07265
$789$	−23.5062	−0.836843
$790$	0	0
$791$	48.8206	1.73586
$792$	−4.29738	−0.152701
$793$	10.3720	0.368321
$794$	10.6191	0.376859
$795$	0	0
$796$	−4.82872	−0.171149
$797$	0.748275	0.0265053	0.0132526	−	0.999912i	$-0.495781\pi$
$797$	0.748275	0.0265053	0.0132526	+	0.999912i	$0.495781\pi$
$798$	−1.67750	−0.0593827
$799$	0	0
$800$	0	0
$801$	−7.40294	−0.261570
$802$	−1.70262	−0.0601217
$803$	2.41261	0.0851390
$804$	3.96401	0.139800
$805$	0	0
$806$	−1.99540	−0.0702850
$807$	−10.6141	−0.373633
$808$	−26.8561	−0.944796
$809$	50.1563	1.76340	0.881700	−	0.471810i	$-0.156399\pi$
$809$	50.1563	1.76340	0.881700	+	0.471810i	$0.156399\pi$
$810$	0	0
$811$	27.9903	0.982874	0.491437	−	0.870913i	$-0.336472\pi$
$811$	27.9903	0.982874	0.491437	+	0.870913i	$0.336472\pi$
$812$	16.5719	0.581561
$813$	15.2974	0.536502
$814$	−3.98068	−0.139523
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	17.8391	0.623730
$819$	−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	$0.764475\pi$
$822$	−10.3264	−0.360173
$823$	35.6044	1.24109	0.620546	−	0.784170i	$-0.286911\pi$
$823$	35.6044	1.24109	0.620546	+	0.784170i	$0.286911\pi$
$824$	12.1056	0.421717
$825$	0	0
$826$	−18.7904	−0.653802
$827$	51.4154	1.78789	0.893944	−	0.448179i	$-0.147927\pi$
$827$	51.4154	1.78789	0.893944	+	0.448179i	$0.147927\pi$
$828$	−5.60672	−0.194847
$829$	−26.9520	−0.936083	−0.468042	−	0.883706i	$-0.655040\pi$
$829$	−26.9520	−0.936083	−0.468042	+	0.883706i	$0.655040\pi$
$830$	0	0
$831$	19.5948	0.679735
$832$	−6.53438	−0.226539
$833$	0	0
$834$	−8.33723	−0.288695
$835$	0	0
$836$	−3.79853	−0.131375
$837$	−1.24943	−0.0431865
$838$	13.7579	0.475260
$839$	34.7026	1.19807	0.599034	−	0.800724i	$-0.295552\pi$
$839$	34.7026	1.19807	0.599034	+	0.800724i	$0.295552\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−6.16088	−0.212318
$843$	−30.5542	−1.05234
$844$	45.2881	1.55888
$845$	0	0
$846$	1.75794	0.0604391
$847$	19.7003	0.676911
$848$	2.19521	0.0753837
$849$	−8.61408	−0.295634
$850$	0	0
$851$	−11.2134	−0.384392
$852$	−12.5913	−0.431369
$853$	−0.134197	−0.00459483	−0.00229741	−	0.999997i	$-0.500731\pi$
$853$	−0.134197	−0.00459483	−0.00229741	+	0.999997i	$0.500731\pi$
$854$	−5.70843	−0.195338
$855$	0	0
$856$	30.4606	1.04112
$857$	−30.5371	−1.04313	−0.521564	−	0.853212i	$-0.674651\pi$
$857$	−30.5371	−1.04313	−0.521564	+	0.853212i	$0.674651\pi$
$858$	3.51587	0.120030
$859$	−57.1010	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$859$	−57.1010	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$860$	0	0
$861$	−27.3933	−0.933561
$862$	−14.1500	−0.481951
$863$	0.376261	0.0128081	0.00640403	−	0.999979i	$-0.497962\pi$
$863$	0.376261	0.0128081	0.00640403	+	0.999979i	$0.497962\pi$
$864$	5.17635	0.176103
$865$	0	0
$866$	7.68135	0.261023
$867$	−17.0000	−0.577350
$868$	−6.90179	−0.234262
$869$	29.8442	1.01239
$870$	0	0
$871$	−7.00230	−0.237264
$872$	−14.5445	−0.492539
$873$	−7.14386	−0.241783
$874$	1.70262	0.0575921
$875$	0	0
$876$	−1.89093	−0.0638886
$877$	−5.54680	−0.187302	−0.0936511	−	0.995605i	$-0.529854\pi$
$877$	−5.54680	−0.187302	−0.0936511	+	0.995605i	$0.529854\pi$
$878$	−11.9285	−0.402567
$879$	−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	$0.427878\pi$
$882$	1.70262	0.0573303
$883$	11.6044	0.390520	0.195260	−	0.980752i	$-0.437445\pi$
$883$	11.6044	0.390520	0.195260	+	0.980752i	$0.437445\pi$
$884$	0	0
$885$	0	0
$886$	20.6237	0.692868
$887$	−33.2282	−1.11569	−0.557846	−	0.829944i	$-0.688372\pi$
$887$	−33.2282	−1.11569	−0.557846	+	0.829944i	$0.688372\pi$
$888$	6.73631	0.226056
$889$	−44.9977	−1.50917
$890$	0	0
$891$	2.20147	0.0737521
$892$	22.8612	0.765450
$893$	3.35499	0.112271
$894$	−8.30704	−0.277829
$895$	0	0
$896$	36.7402	1.22740
$897$	9.90409	0.330688
$898$	−6.44739	−0.215152
$899$	−3.74828	−0.125012
$900$	0	0
$901$	0	0
$902$	9.87005	0.328637
$903$	−25.6118	−0.852307
$904$	29.7676	0.990056
$905$	0	0
$906$	9.90409	0.329041
$907$	18.9018	0.627624	0.313812	−	0.949485i	$-0.398394\pi$
$907$	18.9018	0.627624	0.313812	+	0.949485i	$0.398394\pi$
$908$	−12.5913	−0.417855
$909$	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	$0.555066\pi$
$912$	2.42807	0.0804015
$913$	−18.0553	−0.597544
$914$	−1.65116	−0.0546157
$915$	0	0
$916$	−27.3717	−0.904385
$917$	−13.9188	−0.459640
$918$	0	0
$919$	−0.894433	−0.0295046	−0.0147523	−	0.999891i	$-0.504696\pi$
$919$	−0.894433	−0.0295046	−0.0147523	+	0.999891i	$0.504696\pi$
$920$	0	0
$921$	16.9115	0.557251
$922$	−16.7912	−0.552988
$923$	22.2421	0.732106
$924$	12.1609	0.400064
$925$	0	0
$926$	−10.0046	−0.328772
$927$	−6.20147	−0.203683
$928$	15.5290	0.509766
$929$	−3.69296	−0.121162	−0.0605811	−	0.998163i	$-0.519295\pi$
$929$	−3.69296	−0.121162	−0.0605811	+	0.998163i	$0.519295\pi$
$930$	0	0
$931$	3.24943	0.106496
$932$	14.9165	0.488607
$933$	14.0959	0.461479
$934$	−0.0553145	−0.00180995
$935$	0	0
$936$	−5.94975	−0.194474
$937$	−16.2494	−0.530846	−0.265423	−	0.964132i	$-0.585512\pi$
$937$	−16.2494	−0.530846	−0.265423	+	0.964132i	$0.585512\pi$
$938$	3.85384	0.125832
$939$	−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	$0.431222\pi$
$942$	7.57699	0.246872
$943$	27.8036	0.905409
$944$	27.1980	0.885218
$945$	0	0
$946$	9.22816	0.300033
$947$	−40.5136	−1.31651	−0.658257	−	0.752793i	$-0.728706\pi$
$947$	−40.5136	−1.31651	−0.658257	+	0.752793i	$0.728706\pi$
$948$	−23.3910	−0.759704
$949$	3.34027	0.108430
$950$	0	0
$951$	21.9977	0.713324
$952$	0	0
$953$	−44.6884	−1.44760	−0.723799	−	0.690011i	$-0.757606\pi$
$953$	−44.6884	−1.44760	−0.723799	+	0.690011i	$0.757606\pi$
$954$	0.473724	0.0153374
$955$	0	0
$956$	−44.4440	−1.43742
$957$	6.60442	0.213490
$958$	−15.3499	−0.495934
$959$	63.0936	2.03740
$960$	0	0
$961$	−29.4389	−0.949643
$962$	−5.51127	−0.177691
$963$	−15.6044	−0.502845
$964$	−16.5592	−0.533337
$965$	0	0
$966$	−5.45090	−0.175380
$967$	−38.3910	−1.23457	−0.617285	−	0.786739i	$-0.711768\pi$
$967$	−38.3910	−1.23457	−0.617285	+	0.786739i	$0.711768\pi$
$968$	12.0120	0.386079
$969$	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	$0.565313\pi$
$972$	−1.72545	−0.0553438
$973$	50.9401	1.63306
$974$	3.27069	0.104800
$975$	0	0
$976$	8.26259	0.264479
$977$	−3.48183	−0.111394	−0.0556968	−	0.998448i	$-0.517738\pi$
$977$	−3.48183	−0.111394	−0.0556968	+	0.998448i	$0.517738\pi$
$978$	11.5313	0.368732
$979$	−16.2974	−0.520866
$980$	0	0
$981$	7.45090	0.237889
$982$	−7.60718	−0.242755
$983$	10.1152	0.322626	0.161313	−	0.986903i	$-0.448427\pi$
$983$	10.1152	0.322626	0.161313	+	0.986903i	$0.448427\pi$
$984$	−16.7026	−0.532460
$985$	0	0
$986$	0	0
$987$	−10.7409	−0.341887
$988$	−5.25909	−0.167314
$989$	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	$0.376614\pi$
$992$	−6.46746	−0.205342
$993$	12.4583	0.395351
$994$	−12.2413	−0.388271
$995$	0	0
$996$	14.1512	0.448399
$997$	−38.9571	−1.23378	−0.616892	−	0.787048i	$-0.711608\pi$
$997$	−38.9571	−1.23378	−0.616892	+	0.787048i	$0.711608\pi$
$998$	0.478786	0.0151557
$999$	−3.45090	−0.109182

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.2	✓	3	5.4	even	2
1425.2.a.v.1.2	yes	3	1.1	even	1	trivial
1425.2.c.p.799.3		6	5.3	odd	4
1425.2.c.p.799.4		6	5.2	odd	4
4275.2.a.be.1.2		3	15.14	odd	2
4275.2.a.bh.1.2		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q+0.523976 q^{2} +1.00000 q^{3} -1.72545 q^{4} +0.523976 q^{6} -3.20147 q^{7} -1.95205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.523976 q^{2} +1.00000 q^{3} -1.72545 q^{4} +0.523976 q^{6} -3.20147 q^{7} -1.95205 q^{8} +1.00000 q^{9} +2.20147 q^{11} -1.72545 q^{12} +3.04795 q^{13} -1.67750 q^{14} +2.42807 q^{16} +0.523976 q^{18} +1.00000 q^{19} -3.20147 q^{21} +1.15352 q^{22} +3.24943 q^{23} -1.95205 q^{24} +1.59706 q^{26} +1.00000 q^{27} +5.52398 q^{28} +3.00000 q^{29} -1.24943 q^{31} +5.17635 q^{32} +2.20147 q^{33} -1.72545 q^{36} -3.45090 q^{37} +0.523976 q^{38} +3.04795 q^{39} +8.55646 q^{41} -1.67750 q^{42} +8.00000 q^{43} -3.79853 q^{44} +1.70262 q^{46} +3.35499 q^{47} +2.42807 q^{48} +3.24943 q^{49} -5.25909 q^{52} +0.904094 q^{53} +0.523976 q^{54} +6.24943 q^{56} +1.00000 q^{57} +1.57193 q^{58} +11.2015 q^{59} +3.40294 q^{61} -0.654669 q^{62} -3.20147 q^{63} -2.14386 q^{64} +1.15352 q^{66} -2.29738 q^{67} +3.24943 q^{69} +7.29738 q^{71} -1.95205 q^{72} +1.09591 q^{73} -1.80819 q^{74} -1.72545 q^{76} -7.04795 q^{77} +1.59706 q^{78} +13.5565 q^{79} +1.00000 q^{81} +4.48339 q^{82} -8.20147 q^{83} +5.52398 q^{84} +4.19181 q^{86} +3.00000 q^{87} -4.29738 q^{88} -7.40294 q^{89} -9.75794 q^{91} -5.60672 q^{92} -1.24943 q^{93} +1.75794 q^{94} +5.17635 q^{96} -7.14386 q^{97} +1.70262 q^{98} +2.20147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more