LMFDB - Embedded newform 5775.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.61803 q^{12} -0.381966 q^{13} -0.618034 q^{14} +1.85410 q^{16} -5.23607 q^{17} +0.618034 q^{18} +1.38197 q^{19} +1.00000 q^{21} +0.618034 q^{22} +5.47214 q^{23} +2.23607 q^{24} -0.236068 q^{26} -1.00000 q^{27} +1.61803 q^{28} -0.236068 q^{31} +5.61803 q^{32} -1.00000 q^{33} -3.23607 q^{34} -1.61803 q^{36} -1.09017 q^{37} +0.854102 q^{38} +0.381966 q^{39} +8.70820 q^{41} +0.618034 q^{42} +0.472136 q^{43} -1.61803 q^{44} +3.38197 q^{46} -0.236068 q^{47} -1.85410 q^{48} +1.00000 q^{49} +5.23607 q^{51} +0.618034 q^{52} -9.00000 q^{53} -0.618034 q^{54} +2.23607 q^{56} -1.38197 q^{57} -3.09017 q^{59} +13.1803 q^{61} -0.145898 q^{62} -1.00000 q^{63} -0.236068 q^{64} -0.618034 q^{66} -15.2361 q^{67} +8.47214 q^{68} -5.47214 q^{69} +2.52786 q^{71} -2.23607 q^{72} +15.4721 q^{73} -0.673762 q^{74} -2.23607 q^{76} -1.00000 q^{77} +0.236068 q^{78} -9.47214 q^{79} +1.00000 q^{81} +5.38197 q^{82} +6.32624 q^{83} -1.61803 q^{84} +0.291796 q^{86} -2.23607 q^{88} -3.29180 q^{89} +0.381966 q^{91} -8.85410 q^{92} +0.236068 q^{93} -0.145898 q^{94} -5.61803 q^{96} +3.38197 q^{97} +0.618034 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 2 * q^7 + 2 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{11} + q^{12} - 3 q^{13} + q^{14} - 3 q^{16} - 6 q^{17} - q^{18} + 5 q^{19} + 2 q^{21} - q^{22} + 2 q^{23} + 4 q^{26} - 2 q^{27} + q^{28} + 4 q^{31} + 9 q^{32} - 2 q^{33} - 2 q^{34} - q^{36} + 9 q^{37} - 5 q^{38} + 3 q^{39} + 4 q^{41} - q^{42} - 8 q^{43} - q^{44} + 9 q^{46} + 4 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{51} - q^{52} - 18 q^{53} + q^{54} - 5 q^{57} + 5 q^{59} + 4 q^{61} - 7 q^{62} - 2 q^{63} + 4 q^{64} + q^{66} - 26 q^{67} + 8 q^{68} - 2 q^{69} + 14 q^{71} + 22 q^{73} - 17 q^{74} - 2 q^{77} - 4 q^{78} - 10 q^{79} + 2 q^{81} + 13 q^{82} - 3 q^{83} - q^{84} + 14 q^{86} - 20 q^{89} + 3 q^{91} - 11 q^{92} - 4 q^{93} - 7 q^{94} - 9 q^{96} + 9 q^{97} - q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 2 * q^7 + 2 * q^9 + 2 * q^11 + q^12 - 3 * q^13 + q^14 - 3 * q^16 - 6 * q^17 - q^18 + 5 * q^19 + 2 * q^21 - q^22 + 2 * q^23 + 4 * q^26 - 2 * q^27 + q^28 + 4 * q^31 + 9 * q^32 - 2 * q^33 - 2 * q^34 - q^36 + 9 * q^37 - 5 * q^38 + 3 * q^39 + 4 * q^41 - q^42 - 8 * q^43 - q^44 + 9 * q^46 + 4 * q^47 + 3 * q^48 + 2 * q^49 + 6 * q^51 - q^52 - 18 * q^53 + q^54 - 5 * q^57 + 5 * q^59 + 4 * q^61 - 7 * q^62 - 2 * q^63 + 4 * q^64 + q^66 - 26 * q^67 + 8 * q^68 - 2 * q^69 + 14 * q^71 + 22 * q^73 - 17 * q^74 - 2 * q^77 - 4 * q^78 - 10 * q^79 + 2 * q^81 + 13 * q^82 - 3 * q^83 - q^84 + 14 * q^86 - 20 * q^89 + 3 * q^91 - 11 * q^92 - 4 * q^93 - 7 * q^94 - 9 * q^96 + 9 * q^97 - q^98 + 2 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.61803	0.467086
$13$	−0.381966	−0.105938	−0.0529692	−	0.998596i	$-0.516869\pi$
$13$	−0.381966	−0.105938	−0.0529692	+	0.998596i	$0.516869\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	0.618034	0.145672
$19$	1.38197	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$19$	1.38197	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0.618034	0.131765
$23$	5.47214	1.14102	0.570510	−	0.821291i	$-0.306746\pi$
$23$	5.47214	1.14102	0.570510	+	0.821291i	$0.306746\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−0.236068	−0.0462967
$27$	−1.00000	−0.192450
$28$	1.61803	0.305780
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−0.236068	−0.0423991	−0.0211995	−	0.999775i	$-0.506749\pi$
$31$	−0.236068	−0.0423991	−0.0211995	+	0.999775i	$0.506749\pi$
$32$	5.61803	0.993137
$33$	−1.00000	−0.174078
$34$	−3.23607	−0.554981
$35$	0	0
$36$	−1.61803	−0.269672
$37$	−1.09017	−0.179223	−0.0896114	−	0.995977i	$-0.528563\pi$
$37$	−1.09017	−0.179223	−0.0896114	+	0.995977i	$0.528563\pi$
$38$	0.854102	0.138554
$39$	0.381966	0.0611635
$40$	0	0
$41$	8.70820	1.35999	0.679996	−	0.733215i	$-0.261981\pi$
$41$	8.70820	1.35999	0.679996	+	0.733215i	$0.261981\pi$
$42$	0.618034	0.0953647
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	−1.61803	−0.243928
$45$	0	0
$46$	3.38197	0.498644
$47$	−0.236068	−0.0344341	−0.0172170	−	0.999852i	$-0.505481\pi$
$47$	−0.236068	−0.0344341	−0.0172170	+	0.999852i	$0.505481\pi$
$48$	−1.85410	−0.267617
$49$	1.00000	0.142857
$50$	0	0
$51$	5.23607	0.733196
$52$	0.618034	0.0857059
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	2.23607	0.298807
$57$	−1.38197	−0.183046
$58$	0	0
$59$	−3.09017	−0.402306	−0.201153	−	0.979560i	$-0.564469\pi$
$59$	−3.09017	−0.402306	−0.201153	+	0.979560i	$0.564469\pi$
$60$	0	0
$61$	13.1803	1.68757	0.843785	−	0.536682i	$-0.180322\pi$
$61$	13.1803	1.68757	0.843785	+	0.536682i	$0.180322\pi$
$62$	−0.145898	−0.0185291
$63$	−1.00000	−0.125988
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−0.618034	−0.0760747
$67$	−15.2361	−1.86138	−0.930691	−	0.365806i	$-0.880793\pi$
$67$	−15.2361	−1.86138	−0.930691	+	0.365806i	$0.880793\pi$
$68$	8.47214	1.02740
$69$	−5.47214	−0.658768
$70$	0	0
$71$	2.52786	0.300002	0.150001	−	0.988686i	$-0.452072\pi$
$71$	2.52786	0.300002	0.150001	+	0.988686i	$0.452072\pi$
$72$	−2.23607	−0.263523
$73$	15.4721	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$73$	15.4721	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$74$	−0.673762	−0.0783233
$75$	0	0
$76$	−2.23607	−0.256495
$77$	−1.00000	−0.113961
$78$	0.236068	0.0267294
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.38197	0.594339
$83$	6.32624	0.694395	0.347197	−	0.937792i	$-0.387133\pi$
$83$	6.32624	0.694395	0.347197	+	0.937792i	$0.387133\pi$
$84$	−1.61803	−0.176542
$85$	0	0
$86$	0.291796	0.0314652
$87$	0	0
$88$	−2.23607	−0.238366
$89$	−3.29180	−0.348930	−0.174465	−	0.984663i	$-0.555820\pi$
$89$	−3.29180	−0.348930	−0.174465	+	0.984663i	$0.555820\pi$
$90$	0	0
$91$	0.381966	0.0400409
$92$	−8.85410	−0.923104
$93$	0.236068	0.0244791
$94$	−0.145898	−0.0150482
$95$	0	0
$96$	−5.61803	−0.573388
$97$	3.38197	0.343387	0.171693	−	0.985150i	$-0.445076\pi$
$97$	3.38197	0.343387	0.171693	+	0.985150i	$0.445076\pi$
$98$	0.618034	0.0624309
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.18034	−0.913478	−0.456739	−	0.889601i	$-0.650983\pi$
$101$	−9.18034	−0.913478	−0.456739	+	0.889601i	$0.650983\pi$
$102$	3.23607	0.320418
$103$	−8.14590	−0.802639	−0.401320	−	0.915938i	$-0.631448\pi$
$103$	−8.14590	−0.802639	−0.401320	+	0.915938i	$0.631448\pi$
$104$	0.854102	0.0837516
$105$	0	0
$106$	−5.56231	−0.540259
$107$	20.4164	1.97373	0.986864	−	0.161551i	$-0.0516497\pi$
$107$	20.4164	1.97373	0.986864	+	0.161551i	$0.0516497\pi$
$108$	1.61803	0.155695
$109$	−10.3262	−0.989074	−0.494537	−	0.869157i	$-0.664662\pi$
$109$	−10.3262	−0.989074	−0.494537	+	0.869157i	$0.664662\pi$
$110$	0	0
$111$	1.09017	0.103474
$112$	−1.85410	−0.175196
$113$	−8.79837	−0.827681	−0.413841	−	0.910349i	$-0.635813\pi$
$113$	−8.79837	−0.827681	−0.413841	+	0.910349i	$0.635813\pi$
$114$	−0.854102	−0.0799940
$115$	0	0
$116$	0	0
$117$	−0.381966	−0.0353128
$118$	−1.90983	−0.175814
$119$	5.23607	0.479990
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.14590	0.737495
$123$	−8.70820	−0.785192
$124$	0.381966	0.0343016
$125$	0	0
$126$	−0.618034	−0.0550588
$127$	−1.61803	−0.143577	−0.0717886	−	0.997420i	$-0.522871\pi$
$127$	−1.61803	−0.143577	−0.0717886	+	0.997420i	$0.522871\pi$
$128$	−11.3820	−1.00603
$129$	−0.472136	−0.0415693
$130$	0	0
$131$	−6.41641	−0.560604	−0.280302	−	0.959912i	$-0.590435\pi$
$131$	−6.41641	−0.560604	−0.280302	+	0.959912i	$0.590435\pi$
$132$	1.61803	0.140832
$133$	−1.38197	−0.119832
$134$	−9.41641	−0.813454
$135$	0	0
$136$	11.7082	1.00397
$137$	−14.7082	−1.25661	−0.628303	−	0.777968i	$-0.716250\pi$
$137$	−14.7082	−1.25661	−0.628303	+	0.777968i	$0.716250\pi$
$138$	−3.38197	−0.287892
$139$	−11.7082	−0.993077	−0.496538	−	0.868015i	$-0.665396\pi$
$139$	−11.7082	−0.993077	−0.496538	+	0.868015i	$0.665396\pi$
$140$	0	0
$141$	0.236068	0.0198805
$142$	1.56231	0.131106
$143$	−0.381966	−0.0319416
$144$	1.85410	0.154508
$145$	0	0
$146$	9.56231	0.791382
$147$	−1.00000	−0.0824786
$148$	1.76393	0.144994
$149$	11.9098	0.975691	0.487846	−	0.872930i	$-0.337783\pi$
$149$	11.9098	0.975691	0.487846	+	0.872930i	$0.337783\pi$
$150$	0	0
$151$	5.09017	0.414232	0.207116	−	0.978316i	$-0.433592\pi$
$151$	5.09017	0.414232	0.207116	+	0.978316i	$0.433592\pi$
$152$	−3.09017	−0.250646
$153$	−5.23607	−0.423311
$154$	−0.618034	−0.0498026
$155$	0	0
$156$	−0.618034	−0.0494823
$157$	−11.9443	−0.953257	−0.476628	−	0.879105i	$-0.658141\pi$
$157$	−11.9443	−0.953257	−0.476628	+	0.879105i	$0.658141\pi$
$158$	−5.85410	−0.465727
$159$	9.00000	0.713746
$160$	0	0
$161$	−5.47214	−0.431265
$162$	0.618034	0.0485573
$163$	2.70820	0.212123	0.106061	−	0.994360i	$-0.466176\pi$
$163$	2.70820	0.212123	0.106061	+	0.994360i	$0.466176\pi$
$164$	−14.0902	−1.10026
$165$	0	0
$166$	3.90983	0.303462
$167$	−7.79837	−0.603456	−0.301728	−	0.953394i	$-0.597563\pi$
$167$	−7.79837	−0.603456	−0.301728	+	0.953394i	$0.597563\pi$
$168$	−2.23607	−0.172516
$169$	−12.8541	−0.988777
$170$	0	0
$171$	1.38197	0.105682
$172$	−0.763932	−0.0582493
$173$	5.79837	0.440842	0.220421	−	0.975405i	$-0.429257\pi$
$173$	5.79837	0.440842	0.220421	+	0.975405i	$0.429257\pi$
$174$	0	0
$175$	0	0
$176$	1.85410	0.139758
$177$	3.09017	0.232271
$178$	−2.03444	−0.152488
$179$	15.8541	1.18499	0.592496	−	0.805574i	$-0.298143\pi$
$179$	15.8541	1.18499	0.592496	+	0.805574i	$0.298143\pi$
$180$	0	0
$181$	14.2361	1.05816	0.529079	−	0.848572i	$-0.322537\pi$
$181$	14.2361	1.05816	0.529079	+	0.848572i	$0.322537\pi$
$182$	0.236068	0.0174985
$183$	−13.1803	−0.974319
$184$	−12.2361	−0.902055
$185$	0	0
$186$	0.145898	0.0106978
$187$	−5.23607	−0.382899
$188$	0.381966	0.0278577
$189$	1.00000	0.0727393
$190$	0	0
$191$	−13.8541	−1.00245	−0.501224	−	0.865318i	$-0.667117\pi$
$191$	−13.8541	−1.00245	−0.501224	+	0.865318i	$0.667117\pi$
$192$	0.236068	0.0170367
$193$	−22.7426	−1.63705	−0.818526	−	0.574470i	$-0.805208\pi$
$193$	−22.7426	−1.63705	−0.818526	+	0.574470i	$0.805208\pi$
$194$	2.09017	0.150065
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−17.7984	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$197$	−17.7984	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$198$	0.618034	0.0439218
$199$	−16.1803	−1.14699	−0.573497	−	0.819208i	$-0.694414\pi$
$199$	−16.1803	−1.14699	−0.573497	+	0.819208i	$0.694414\pi$
$200$	0	0
$201$	15.2361	1.07467
$202$	−5.67376	−0.399205
$203$	0	0
$204$	−8.47214	−0.593168
$205$	0	0
$206$	−5.03444	−0.350766
$207$	5.47214	0.380340
$208$	−0.708204	−0.0491051
$209$	1.38197	0.0955926
$210$	0	0
$211$	10.4164	0.717095	0.358548	−	0.933511i	$-0.383272\pi$
$211$	10.4164	0.717095	0.358548	+	0.933511i	$0.383272\pi$
$212$	14.5623	1.00014
$213$	−2.52786	−0.173206
$214$	12.6180	0.862551
$215$	0	0
$216$	2.23607	0.152145
$217$	0.236068	0.0160253
$218$	−6.38197	−0.432241
$219$	−15.4721	−1.04551
$220$	0	0
$221$	2.00000	0.134535
$222$	0.673762	0.0452199
$223$	7.18034	0.480831	0.240416	−	0.970670i	$-0.422716\pi$
$223$	7.18034	0.480831	0.240416	+	0.970670i	$0.422716\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	−5.43769	−0.361710
$227$	−9.18034	−0.609321	−0.304660	−	0.952461i	$-0.598543\pi$
$227$	−9.18034	−0.609321	−0.304660	+	0.952461i	$0.598543\pi$
$228$	2.23607	0.148087
$229$	11.5066	0.760376	0.380188	−	0.924909i	$-0.375859\pi$
$229$	11.5066	0.760376	0.380188	+	0.924909i	$0.375859\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−24.3262	−1.59366	−0.796832	−	0.604200i	$-0.793493\pi$
$233$	−24.3262	−1.59366	−0.796832	+	0.604200i	$0.793493\pi$
$234$	−0.236068	−0.0154322
$235$	0	0
$236$	5.00000	0.325472
$237$	9.47214	0.615281
$238$	3.23607	0.209763
$239$	−4.14590	−0.268176	−0.134088	−	0.990969i	$-0.542810\pi$
$239$	−4.14590	−0.268176	−0.134088	+	0.990969i	$0.542810\pi$
$240$	0	0
$241$	10.9443	0.704983	0.352491	−	0.935815i	$-0.385335\pi$
$241$	10.9443	0.704983	0.352491	+	0.935815i	$0.385335\pi$
$242$	0.618034	0.0397287
$243$	−1.00000	−0.0641500
$244$	−21.3262	−1.36527
$245$	0	0
$246$	−5.38197	−0.343142
$247$	−0.527864	−0.0335872
$248$	0.527864	0.0335194
$249$	−6.32624	−0.400909
$250$	0	0
$251$	−22.5967	−1.42629	−0.713147	−	0.701014i	$-0.752731\pi$
$251$	−22.5967	−1.42629	−0.713147	+	0.701014i	$0.752731\pi$
$252$	1.61803	0.101927
$253$	5.47214	0.344030
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−27.2705	−1.70109	−0.850544	−	0.525904i	$-0.823727\pi$
$257$	−27.2705	−1.70109	−0.850544	+	0.525904i	$0.823727\pi$
$258$	−0.291796	−0.0181664
$259$	1.09017	0.0677399
$260$	0	0
$261$	0	0
$262$	−3.96556	−0.244993
$263$	−4.32624	−0.266767	−0.133384	−	0.991064i	$-0.542584\pi$
$263$	−4.32624	−0.266767	−0.133384	+	0.991064i	$0.542584\pi$
$264$	2.23607	0.137620
$265$	0	0
$266$	−0.854102	−0.0523684
$267$	3.29180	0.201455
$268$	24.6525	1.50589
$269$	4.14590	0.252780	0.126390	−	0.991981i	$-0.459661\pi$
$269$	4.14590	0.252780	0.126390	+	0.991981i	$0.459661\pi$
$270$	0	0
$271$	2.65248	0.161126	0.0805632	−	0.996750i	$-0.474328\pi$
$271$	2.65248	0.161126	0.0805632	+	0.996750i	$0.474328\pi$
$272$	−9.70820	−0.588646
$273$	−0.381966	−0.0231176
$274$	−9.09017	−0.549157
$275$	0	0
$276$	8.85410	0.532954
$277$	14.5623	0.874964	0.437482	−	0.899227i	$-0.355870\pi$
$277$	14.5623	0.874964	0.437482	+	0.899227i	$0.355870\pi$
$278$	−7.23607	−0.433991
$279$	−0.236068	−0.0141330
$280$	0	0
$281$	7.65248	0.456508	0.228254	−	0.973602i	$-0.426698\pi$
$281$	7.65248	0.456508	0.228254	+	0.973602i	$0.426698\pi$
$282$	0.145898	0.00868810
$283$	−25.5066	−1.51621	−0.758104	−	0.652133i	$-0.773874\pi$
$283$	−25.5066	−1.51621	−0.758104	+	0.652133i	$0.773874\pi$
$284$	−4.09017	−0.242707
$285$	0	0
$286$	−0.236068	−0.0139590
$287$	−8.70820	−0.514029
$288$	5.61803	0.331046
$289$	10.4164	0.612730
$290$	0	0
$291$	−3.38197	−0.198254
$292$	−25.0344	−1.46503
$293$	−27.0902	−1.58262	−0.791312	−	0.611412i	$-0.790602\pi$
$293$	−27.0902	−1.58262	−0.791312	+	0.611412i	$0.790602\pi$
$294$	−0.618034	−0.0360445
$295$	0	0
$296$	2.43769	0.141688
$297$	−1.00000	−0.0580259
$298$	7.36068	0.426393
$299$	−2.09017	−0.120878
$300$	0	0
$301$	−0.472136	−0.0272135
$302$	3.14590	0.181026
$303$	9.18034	0.527397
$304$	2.56231	0.146958
$305$	0	0
$306$	−3.23607	−0.184994
$307$	−26.4164	−1.50766	−0.753832	−	0.657067i	$-0.771797\pi$
$307$	−26.4164	−1.50766	−0.753832	+	0.657067i	$0.771797\pi$
$308$	1.61803	0.0921960
$309$	8.14590	0.463404
$310$	0	0
$311$	8.70820	0.493797	0.246898	−	0.969041i	$-0.420589\pi$
$311$	8.70820	0.493797	0.246898	+	0.969041i	$0.420589\pi$
$312$	−0.854102	−0.0483540
$313$	26.9787	1.52493	0.762464	−	0.647031i	$-0.223990\pi$
$313$	26.9787	1.52493	0.762464	+	0.647031i	$0.223990\pi$
$314$	−7.38197	−0.416588
$315$	0	0
$316$	15.3262	0.862168
$317$	13.9098	0.781254	0.390627	−	0.920549i	$-0.372258\pi$
$317$	13.9098	0.781254	0.390627	+	0.920549i	$0.372258\pi$
$318$	5.56231	0.311919
$319$	0	0
$320$	0	0
$321$	−20.4164	−1.13953
$322$	−3.38197	−0.188470
$323$	−7.23607	−0.402626
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	1.67376	0.0927011
$327$	10.3262	0.571042
$328$	−19.4721	−1.07517
$329$	0.236068	0.0130148
$330$	0	0
$331$	−2.14590	−0.117949	−0.0589746	−	0.998259i	$-0.518783\pi$
$331$	−2.14590	−0.117949	−0.0589746	+	0.998259i	$0.518783\pi$
$332$	−10.2361	−0.561777
$333$	−1.09017	−0.0597409
$334$	−4.81966	−0.263720
$335$	0	0
$336$	1.85410	0.101150
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−7.94427	−0.432111
$339$	8.79837	0.477862
$340$	0	0
$341$	−0.236068	−0.0127838
$342$	0.854102	0.0461845
$343$	−1.00000	−0.0539949
$344$	−1.05573	−0.0569210
$345$	0	0
$346$	3.58359	0.192655
$347$	−14.1803	−0.761241	−0.380620	−	0.924731i	$-0.624289\pi$
$347$	−14.1803	−0.761241	−0.380620	+	0.924731i	$0.624289\pi$
$348$	0	0
$349$	−6.05573	−0.324156	−0.162078	−	0.986778i	$-0.551820\pi$
$349$	−6.05573	−0.324156	−0.162078	+	0.986778i	$0.551820\pi$
$350$	0	0
$351$	0.381966	0.0203878
$352$	5.61803	0.299442
$353$	−15.7082	−0.836063	−0.418032	−	0.908432i	$-0.637280\pi$
$353$	−15.7082	−0.836063	−0.418032	+	0.908432i	$0.637280\pi$
$354$	1.90983	0.101506
$355$	0	0
$356$	5.32624	0.282290
$357$	−5.23607	−0.277122
$358$	9.79837	0.517860
$359$	−1.58359	−0.0835788	−0.0417894	−	0.999126i	$-0.513306\pi$
$359$	−1.58359	−0.0835788	−0.0417894	+	0.999126i	$0.513306\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	8.79837	0.462432
$363$	−1.00000	−0.0524864
$364$	−0.618034	−0.0323938
$365$	0	0
$366$	−8.14590	−0.425793
$367$	−16.7426	−0.873959	−0.436979	−	0.899471i	$-0.643952\pi$
$367$	−16.7426	−0.873959	−0.436979	+	0.899471i	$0.643952\pi$
$368$	10.1459	0.528891
$369$	8.70820	0.453331
$370$	0	0
$371$	9.00000	0.467257
$372$	−0.381966	−0.0198040
$373$	−28.4721	−1.47423	−0.737116	−	0.675767i	$-0.763813\pi$
$373$	−28.4721	−1.47423	−0.737116	+	0.675767i	$0.763813\pi$
$374$	−3.23607	−0.167333
$375$	0	0
$376$	0.527864	0.0272225
$377$	0	0
$378$	0.618034	0.0317882
$379$	−24.2705	−1.24669	−0.623346	−	0.781946i	$-0.714227\pi$
$379$	−24.2705	−1.24669	−0.623346	+	0.781946i	$0.714227\pi$
$380$	0	0
$381$	1.61803	0.0828944
$382$	−8.56231	−0.438086
$383$	−22.9443	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$383$	−22.9443	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	−14.0557	−0.715418
$387$	0.472136	0.0240000
$388$	−5.47214	−0.277806
$389$	19.5967	0.993595	0.496797	−	0.867867i	$-0.334509\pi$
$389$	19.5967	0.993595	0.496797	+	0.867867i	$0.334509\pi$
$390$	0	0
$391$	−28.6525	−1.44902
$392$	−2.23607	−0.112938
$393$	6.41641	0.323665
$394$	−11.0000	−0.554172
$395$	0	0
$396$	−1.61803	−0.0813093
$397$	30.4164	1.52656	0.763278	−	0.646070i	$-0.223589\pi$
$397$	30.4164	1.52656	0.763278	+	0.646070i	$0.223589\pi$
$398$	−10.0000	−0.501255
$399$	1.38197	0.0691848
$400$	0	0
$401$	−7.79837	−0.389432	−0.194716	−	0.980860i	$-0.562379\pi$
$401$	−7.79837	−0.389432	−0.194716	+	0.980860i	$0.562379\pi$
$402$	9.41641	0.469648
$403$	0.0901699	0.00449168
$404$	14.8541	0.739019
$405$	0	0
$406$	0	0
$407$	−1.09017	−0.0540377
$408$	−11.7082	−0.579642
$409$	−22.0344	−1.08953	−0.544767	−	0.838588i	$-0.683382\pi$
$409$	−22.0344	−1.08953	−0.544767	+	0.838588i	$0.683382\pi$
$410$	0	0
$411$	14.7082	0.725502
$412$	13.1803	0.649349
$413$	3.09017	0.152057
$414$	3.38197	0.166215
$415$	0	0
$416$	−2.14590	−0.105211
$417$	11.7082	0.573353
$418$	0.854102	0.0417755
$419$	31.1803	1.52326	0.761630	−	0.648013i	$-0.224400\pi$
$419$	31.1803	1.52326	0.761630	+	0.648013i	$0.224400\pi$
$420$	0	0
$421$	−19.9098	−0.970346	−0.485173	−	0.874418i	$-0.661243\pi$
$421$	−19.9098	−0.970346	−0.485173	+	0.874418i	$0.661243\pi$
$422$	6.43769	0.313382
$423$	−0.236068	−0.0114780
$424$	20.1246	0.977338
$425$	0	0
$426$	−1.56231	−0.0756940
$427$	−13.1803	−0.637841
$428$	−33.0344	−1.59678
$429$	0.381966	0.0184415
$430$	0	0
$431$	22.9787	1.10685	0.553423	−	0.832900i	$-0.313321\pi$
$431$	22.9787	1.10685	0.553423	+	0.832900i	$0.313321\pi$
$432$	−1.85410	−0.0892055
$433$	8.56231	0.411478	0.205739	−	0.978607i	$-0.434040\pi$
$433$	8.56231	0.411478	0.205739	+	0.978607i	$0.434040\pi$
$434$	0.145898	0.00700333
$435$	0	0
$436$	16.7082	0.800178
$437$	7.56231	0.361754
$438$	−9.56231	−0.456905
$439$	1.90983	0.0911512	0.0455756	−	0.998961i	$-0.485488\pi$
$439$	1.90983	0.0911512	0.0455756	+	0.998961i	$0.485488\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	1.23607	0.0587938
$443$	−26.5623	−1.26201	−0.631007	−	0.775777i	$-0.717358\pi$
$443$	−26.5623	−1.26201	−0.631007	+	0.775777i	$0.717358\pi$
$444$	−1.76393	−0.0837125
$445$	0	0
$446$	4.43769	0.210131
$447$	−11.9098	−0.563316
$448$	0.236068	0.0111532
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	0	0
$451$	8.70820	0.410053
$452$	14.2361	0.669608
$453$	−5.09017	−0.239157
$454$	−5.67376	−0.266283
$455$	0	0
$456$	3.09017	0.144710
$457$	−12.1459	−0.568161	−0.284081	−	0.958800i	$-0.591688\pi$
$457$	−12.1459	−0.568161	−0.284081	+	0.958800i	$0.591688\pi$
$458$	7.11146	0.332297
$459$	5.23607	0.244399
$460$	0	0
$461$	8.18034	0.380996	0.190498	−	0.981688i	$-0.438990\pi$
$461$	8.18034	0.380996	0.190498	+	0.981688i	$0.438990\pi$
$462$	0.618034	0.0287535
$463$	14.8197	0.688728	0.344364	−	0.938836i	$-0.388095\pi$
$463$	14.8197	0.688728	0.344364	+	0.938836i	$0.388095\pi$
$464$	0	0
$465$	0	0
$466$	−15.0344	−0.696457
$467$	10.0902	0.466917	0.233459	−	0.972367i	$-0.424996\pi$
$467$	10.0902	0.466917	0.233459	+	0.972367i	$0.424996\pi$
$468$	0.618034	0.0285686
$469$	15.2361	0.703536
$470$	0	0
$471$	11.9443	0.550363
$472$	6.90983	0.318051
$473$	0.472136	0.0217088
$474$	5.85410	0.268888
$475$	0	0
$476$	−8.47214	−0.388320
$477$	−9.00000	−0.412082
$478$	−2.56231	−0.117197
$479$	−11.3820	−0.520055	−0.260028	−	0.965601i	$-0.583732\pi$
$479$	−11.3820	−0.520055	−0.260028	+	0.965601i	$0.583732\pi$
$480$	0	0
$481$	0.416408	0.0189866
$482$	6.76393	0.308089
$483$	5.47214	0.248991
$484$	−1.61803	−0.0735470
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	−1.49342	−0.0676734	−0.0338367	−	0.999427i	$-0.510773\pi$
$487$	−1.49342	−0.0676734	−0.0338367	+	0.999427i	$0.510773\pi$
$488$	−29.4721	−1.33414
$489$	−2.70820	−0.122469
$490$	0	0
$491$	17.3262	0.781922	0.390961	−	0.920407i	$-0.372143\pi$
$491$	17.3262	0.781922	0.390961	+	0.920407i	$0.372143\pi$
$492$	14.0902	0.635234
$493$	0	0
$494$	−0.326238	−0.0146781
$495$	0	0
$496$	−0.437694	−0.0196530
$497$	−2.52786	−0.113390
$498$	−3.90983	−0.175204
$499$	15.1246	0.677071	0.338535	−	0.940954i	$-0.390069\pi$
$499$	15.1246	0.677071	0.338535	+	0.940954i	$0.390069\pi$
$500$	0	0
$501$	7.79837	0.348406
$502$	−13.9656	−0.623313
$503$	33.8885	1.51102	0.755508	−	0.655140i	$-0.227390\pi$
$503$	33.8885	1.51102	0.755508	+	0.655140i	$0.227390\pi$
$504$	2.23607	0.0996024
$505$	0	0
$506$	3.38197	0.150347
$507$	12.8541	0.570871
$508$	2.61803	0.116156
$509$	12.3607	0.547877	0.273939	−	0.961747i	$-0.411673\pi$
$509$	12.3607	0.547877	0.273939	+	0.961747i	$0.411673\pi$
$510$	0	0
$511$	−15.4721	−0.684447
$512$	18.7082	0.826794
$513$	−1.38197	−0.0610153
$514$	−16.8541	−0.743403
$515$	0	0
$516$	0.763932	0.0336302
$517$	−0.236068	−0.0103823
$518$	0.673762	0.0296034
$519$	−5.79837	−0.254520
$520$	0	0
$521$	35.9443	1.57475	0.787374	−	0.616476i	$-0.211440\pi$
$521$	35.9443	1.57475	0.787374	+	0.616476i	$0.211440\pi$
$522$	0	0
$523$	8.11146	0.354689	0.177345	−	0.984149i	$-0.443249\pi$
$523$	8.11146	0.354689	0.177345	+	0.984149i	$0.443249\pi$
$524$	10.3820	0.453538
$525$	0	0
$526$	−2.67376	−0.116582
$527$	1.23607	0.0538440
$528$	−1.85410	−0.0806894
$529$	6.94427	0.301925
$530$	0	0
$531$	−3.09017	−0.134102
$532$	2.23607	0.0969458
$533$	−3.32624	−0.144075
$534$	2.03444	0.0880389
$535$	0	0
$536$	34.0689	1.47155
$537$	−15.8541	−0.684155
$538$	2.56231	0.110469
$539$	1.00000	0.0430730
$540$	0	0
$541$	−41.9443	−1.80333	−0.901663	−	0.432440i	$-0.857653\pi$
$541$	−41.9443	−1.80333	−0.901663	+	0.432440i	$0.857653\pi$
$542$	1.63932	0.0704148
$543$	−14.2361	−0.610928
$544$	−29.4164	−1.26122
$545$	0	0
$546$	−0.236068	−0.0101028
$547$	−11.2918	−0.482802	−0.241401	−	0.970425i	$-0.577607\pi$
$547$	−11.2918	−0.482802	−0.241401	+	0.970425i	$0.577607\pi$
$548$	23.7984	1.01662
$549$	13.1803	0.562523
$550$	0	0
$551$	0	0
$552$	12.2361	0.520802
$553$	9.47214	0.402796
$554$	9.00000	0.382373
$555$	0	0
$556$	18.9443	0.803416
$557$	6.79837	0.288056	0.144028	−	0.989574i	$-0.453994\pi$
$557$	6.79837	0.288056	0.144028	+	0.989574i	$0.453994\pi$
$558$	−0.145898	−0.00617636
$559$	−0.180340	−0.00762756
$560$	0	0
$561$	5.23607	0.221067
$562$	4.72949	0.199502
$563$	25.0689	1.05653	0.528264	−	0.849080i	$-0.322843\pi$
$563$	25.0689	1.05653	0.528264	+	0.849080i	$0.322843\pi$
$564$	−0.381966	−0.0160837
$565$	0	0
$566$	−15.7639	−0.662607
$567$	−1.00000	−0.0419961
$568$	−5.65248	−0.237173
$569$	−23.8197	−0.998572	−0.499286	−	0.866437i	$-0.666404\pi$
$569$	−23.8197	−0.998572	−0.499286	+	0.866437i	$0.666404\pi$
$570$	0	0
$571$	16.7984	0.702990	0.351495	−	0.936190i	$-0.385673\pi$
$571$	16.7984	0.702990	0.351495	+	0.936190i	$0.385673\pi$
$572$	0.618034	0.0258413
$573$	13.8541	0.578763
$574$	−5.38197	−0.224639
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−7.79837	−0.324651	−0.162325	−	0.986737i	$-0.551899\pi$
$577$	−7.79837	−0.324651	−0.162325	+	0.986737i	$0.551899\pi$
$578$	6.43769	0.267773
$579$	22.7426	0.945152
$580$	0	0
$581$	−6.32624	−0.262457
$582$	−2.09017	−0.0866403
$583$	−9.00000	−0.372742
$584$	−34.5967	−1.43162
$585$	0	0
$586$	−16.7426	−0.691632
$587$	25.6180	1.05737	0.528685	−	0.848818i	$-0.322685\pi$
$587$	25.6180	1.05737	0.528685	+	0.848818i	$0.322685\pi$
$588$	1.61803	0.0667266
$589$	−0.326238	−0.0134424
$590$	0	0
$591$	17.7984	0.732127
$592$	−2.02129	−0.0830744
$593$	−12.6180	−0.518161	−0.259080	−	0.965856i	$-0.583419\pi$
$593$	−12.6180	−0.518161	−0.259080	+	0.965856i	$0.583419\pi$
$594$	−0.618034	−0.0253582
$595$	0	0
$596$	−19.2705	−0.789351
$597$	16.1803	0.662217
$598$	−1.29180	−0.0528255
$599$	12.5623	0.513282	0.256641	−	0.966507i	$-0.417384\pi$
$599$	12.5623	0.513282	0.256641	+	0.966507i	$0.417384\pi$
$600$	0	0
$601$	−31.5410	−1.28659	−0.643293	−	0.765620i	$-0.722432\pi$
$601$	−31.5410	−1.28659	−0.643293	+	0.765620i	$0.722432\pi$
$602$	−0.291796	−0.0118927
$603$	−15.2361	−0.620461
$604$	−8.23607	−0.335121
$605$	0	0
$606$	5.67376	0.230481
$607$	35.5410	1.44257	0.721283	−	0.692641i	$-0.243553\pi$
$607$	35.5410	1.44257	0.721283	+	0.692641i	$0.243553\pi$
$608$	7.76393	0.314869
$609$	0	0
$610$	0	0
$611$	0.0901699	0.00364789
$612$	8.47214	0.342466
$613$	−32.4164	−1.30929	−0.654643	−	0.755938i	$-0.727181\pi$
$613$	−32.4164	−1.30929	−0.654643	+	0.755938i	$0.727181\pi$
$614$	−16.3262	−0.658873
$615$	0	0
$616$	2.23607	0.0900937
$617$	19.8885	0.800683	0.400341	−	0.916366i	$-0.368892\pi$
$617$	19.8885	0.800683	0.400341	+	0.916366i	$0.368892\pi$
$618$	5.03444	0.202515
$619$	30.6525	1.23203	0.616014	−	0.787736i	$-0.288747\pi$
$619$	30.6525	1.23203	0.616014	+	0.787736i	$0.288747\pi$
$620$	0	0
$621$	−5.47214	−0.219589
$622$	5.38197	0.215797
$623$	3.29180	0.131883
$624$	0.708204	0.0283508
$625$	0	0
$626$	16.6738	0.666418
$627$	−1.38197	−0.0551904
$628$	19.3262	0.771201
$629$	5.70820	0.227601
$630$	0	0
$631$	−21.0902	−0.839586	−0.419793	−	0.907620i	$-0.637897\pi$
$631$	−21.0902	−0.839586	−0.419793	+	0.907620i	$0.637897\pi$
$632$	21.1803	0.842509
$633$	−10.4164	−0.414015
$634$	8.59675	0.341420
$635$	0	0
$636$	−14.5623	−0.577433
$637$	−0.381966	−0.0151340
$638$	0	0
$639$	2.52786	0.100001
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−12.6180	−0.497994
$643$	17.5066	0.690392	0.345196	−	0.938531i	$-0.387812\pi$
$643$	17.5066	0.690392	0.345196	+	0.938531i	$0.387812\pi$
$644$	8.85410	0.348900
$645$	0	0
$646$	−4.47214	−0.175954
$647$	−10.8885	−0.428073	−0.214036	−	0.976826i	$-0.568661\pi$
$647$	−10.8885	−0.428073	−0.214036	+	0.976826i	$0.568661\pi$
$648$	−2.23607	−0.0878410
$649$	−3.09017	−0.121300
$650$	0	0
$651$	−0.236068	−0.00925223
$652$	−4.38197	−0.171611
$653$	−24.3262	−0.951959	−0.475980	−	0.879456i	$-0.657906\pi$
$653$	−24.3262	−0.951959	−0.475980	+	0.879456i	$0.657906\pi$
$654$	6.38197	0.249555
$655$	0	0
$656$	16.1459	0.630391
$657$	15.4721	0.603626
$658$	0.145898	0.00568770
$659$	14.2705	0.555900	0.277950	−	0.960596i	$-0.410345\pi$
$659$	14.2705	0.555900	0.277950	+	0.960596i	$0.410345\pi$
$660$	0	0
$661$	4.36068	0.169611	0.0848054	−	0.996398i	$-0.472973\pi$
$661$	4.36068	0.169611	0.0848054	+	0.996398i	$0.472973\pi$
$662$	−1.32624	−0.0515457
$663$	−2.00000	−0.0776736
$664$	−14.1459	−0.548967
$665$	0	0
$666$	−0.673762	−0.0261078
$667$	0	0
$668$	12.6180	0.488206
$669$	−7.18034	−0.277608
$670$	0	0
$671$	13.1803	0.508821
$672$	5.61803	0.216720
$673$	−6.23607	−0.240383	−0.120191	−	0.992751i	$-0.538351\pi$
$673$	−6.23607	−0.240383	−0.120191	+	0.992751i	$0.538351\pi$
$674$	−8.03444	−0.309475
$675$	0	0
$676$	20.7984	0.799937
$677$	4.23607	0.162805	0.0814027	−	0.996681i	$-0.474060\pi$
$677$	4.23607	0.162805	0.0814027	+	0.996681i	$0.474060\pi$
$678$	5.43769	0.208833
$679$	−3.38197	−0.129788
$680$	0	0
$681$	9.18034	0.351791
$682$	−0.145898	−0.00558672
$683$	−24.6525	−0.943301	−0.471650	−	0.881786i	$-0.656342\pi$
$683$	−24.6525	−0.943301	−0.471650	+	0.881786i	$0.656342\pi$
$684$	−2.23607	−0.0854982
$685$	0	0
$686$	−0.618034	−0.0235966
$687$	−11.5066	−0.439003
$688$	0.875388	0.0333739
$689$	3.43769	0.130966
$690$	0	0
$691$	13.1803	0.501404	0.250702	−	0.968064i	$-0.419339\pi$
$691$	13.1803	0.501404	0.250702	+	0.968064i	$0.419339\pi$
$692$	−9.38197	−0.356649
$693$	−1.00000	−0.0379869
$694$	−8.76393	−0.332674
$695$	0	0
$696$	0	0
$697$	−45.5967	−1.72710
$698$	−3.74265	−0.141661
$699$	24.3262	0.920103
$700$	0	0
$701$	−44.7082	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$701$	−44.7082	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$702$	0.236068	0.00890981
$703$	−1.50658	−0.0568217
$704$	−0.236068	−0.00889715
$705$	0	0
$706$	−9.70820	−0.365373
$707$	9.18034	0.345262
$708$	−5.00000	−0.187912
$709$	−41.6312	−1.56349	−0.781746	−	0.623597i	$-0.785671\pi$
$709$	−41.6312	−1.56349	−0.781746	+	0.623597i	$0.785671\pi$
$710$	0	0
$711$	−9.47214	−0.355233
$712$	7.36068	0.275853
$713$	−1.29180	−0.0483781
$714$	−3.23607	−0.121107
$715$	0	0
$716$	−25.6525	−0.958678
$717$	4.14590	0.154831
$718$	−0.978714	−0.0365253
$719$	−19.5967	−0.730835	−0.365418	−	0.930844i	$-0.619074\pi$
$719$	−19.5967	−0.730835	−0.365418	+	0.930844i	$0.619074\pi$
$720$	0	0
$721$	8.14590	0.303369
$722$	−10.5623	−0.393088
$723$	−10.9443	−0.407022
$724$	−23.0344	−0.856068
$725$	0	0
$726$	−0.618034	−0.0229374
$727$	27.4508	1.01810	0.509048	−	0.860738i	$-0.329998\pi$
$727$	27.4508	1.01810	0.509048	+	0.860738i	$0.329998\pi$
$728$	−0.854102	−0.0316551
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.47214	−0.0914353
$732$	21.3262	0.788240
$733$	16.6525	0.615073	0.307537	−	0.951536i	$-0.400495\pi$
$733$	16.6525	0.615073	0.307537	+	0.951536i	$0.400495\pi$
$734$	−10.3475	−0.381934
$735$	0	0
$736$	30.7426	1.13319
$737$	−15.2361	−0.561228
$738$	5.38197	0.198113
$739$	−16.8328	−0.619205	−0.309603	−	0.950866i	$-0.600196\pi$
$739$	−16.8328	−0.619205	−0.309603	+	0.950866i	$0.600196\pi$
$740$	0	0
$741$	0.527864	0.0193916
$742$	5.56231	0.204199
$743$	4.09017	0.150054	0.0750269	−	0.997182i	$-0.476096\pi$
$743$	4.09017	0.150054	0.0750269	+	0.997182i	$0.476096\pi$
$744$	−0.527864	−0.0193524
$745$	0	0
$746$	−17.5967	−0.644263
$747$	6.32624	0.231465
$748$	8.47214	0.309772
$749$	−20.4164	−0.745999
$750$	0	0
$751$	−21.4164	−0.781496	−0.390748	−	0.920498i	$-0.627784\pi$
$751$	−21.4164	−0.781496	−0.390748	+	0.920498i	$0.627784\pi$
$752$	−0.437694	−0.0159611
$753$	22.5967	0.823471
$754$	0	0
$755$	0	0
$756$	−1.61803	−0.0588473
$757$	−1.09017	−0.0396229	−0.0198115	−	0.999804i	$-0.506307\pi$
$757$	−1.09017	−0.0396229	−0.0198115	+	0.999804i	$0.506307\pi$
$758$	−15.0000	−0.544825
$759$	−5.47214	−0.198626
$760$	0	0
$761$	3.05573	0.110770	0.0553850	−	0.998465i	$-0.482361\pi$
$761$	3.05573	0.110770	0.0553850	+	0.998465i	$0.482361\pi$
$762$	1.00000	0.0362262
$763$	10.3262	0.373835
$764$	22.4164	0.810997
$765$	0	0
$766$	−14.1803	−0.512357
$767$	1.18034	0.0426196
$768$	6.56231	0.236797
$769$	−42.2361	−1.52307	−0.761536	−	0.648123i	$-0.775554\pi$
$769$	−42.2361	−1.52307	−0.761536	+	0.648123i	$0.775554\pi$
$770$	0	0
$771$	27.2705	0.982123
$772$	36.7984	1.32440
$773$	47.8328	1.72043	0.860213	−	0.509934i	$-0.170330\pi$
$773$	47.8328	1.72043	0.860213	+	0.509934i	$0.170330\pi$
$774$	0.291796	0.0104884
$775$	0	0
$776$	−7.56231	−0.271471
$777$	−1.09017	−0.0391096
$778$	12.1115	0.434217
$779$	12.0344	0.431179
$780$	0	0
$781$	2.52786	0.0904541
$782$	−17.7082	−0.633244
$783$	0	0
$784$	1.85410	0.0662179
$785$	0	0
$786$	3.96556	0.141447
$787$	19.8885	0.708950	0.354475	−	0.935065i	$-0.384660\pi$
$787$	19.8885	0.708950	0.354475	+	0.935065i	$0.384660\pi$
$788$	28.7984	1.02590
$789$	4.32624	0.154018
$790$	0	0
$791$	8.79837	0.312834
$792$	−2.23607	−0.0794552
$793$	−5.03444	−0.178778
$794$	18.7984	0.667129
$795$	0	0
$796$	26.1803	0.927938
$797$	−20.5623	−0.728354	−0.364177	−	0.931330i	$-0.618650\pi$
$797$	−20.5623	−0.728354	−0.364177	+	0.931330i	$0.618650\pi$
$798$	0.854102	0.0302349
$799$	1.23607	0.0437289
$800$	0	0
$801$	−3.29180	−0.116310
$802$	−4.81966	−0.170188
$803$	15.4721	0.546000
$804$	−24.6525	−0.869426
$805$	0	0
$806$	0.0557281	0.00196294
$807$	−4.14590	−0.145943
$808$	20.5279	0.722168
$809$	48.4164	1.70223	0.851115	−	0.524979i	$-0.175927\pi$
$809$	48.4164	1.70223	0.851115	+	0.524979i	$0.175927\pi$
$810$	0	0
$811$	−1.16718	−0.0409854	−0.0204927	−	0.999790i	$-0.506523\pi$
$811$	−1.16718	−0.0409854	−0.0204927	+	0.999790i	$0.506523\pi$
$812$	0	0
$813$	−2.65248	−0.0930264
$814$	−0.673762	−0.0236153
$815$	0	0
$816$	9.70820	0.339855
$817$	0.652476	0.0228272
$818$	−13.6180	−0.476143
$819$	0.381966	0.0133470
$820$	0	0
$821$	−27.4721	−0.958784	−0.479392	−	0.877601i	$-0.659143\pi$
$821$	−27.4721	−0.958784	−0.479392	+	0.877601i	$0.659143\pi$
$822$	9.09017	0.317056
$823$	50.2705	1.75232	0.876160	−	0.482021i	$-0.160097\pi$
$823$	50.2705	1.75232	0.876160	+	0.482021i	$0.160097\pi$
$824$	18.2148	0.634542
$825$	0	0
$826$	1.90983	0.0664515
$827$	−14.7082	−0.511454	−0.255727	−	0.966749i	$-0.582315\pi$
$827$	−14.7082	−0.511454	−0.255727	+	0.966749i	$0.582315\pi$
$828$	−8.85410	−0.307701
$829$	−42.0344	−1.45992	−0.729958	−	0.683492i	$-0.760461\pi$
$829$	−42.0344	−1.45992	−0.729958	+	0.683492i	$0.760461\pi$
$830$	0	0
$831$	−14.5623	−0.505161
$832$	0.0901699	0.00312608
$833$	−5.23607	−0.181419
$834$	7.23607	0.250565
$835$	0	0
$836$	−2.23607	−0.0773360
$837$	0.236068	0.00815970
$838$	19.2705	0.665689
$839$	43.5410	1.50320	0.751601	−	0.659617i	$-0.229282\pi$
$839$	43.5410	1.50320	0.751601	+	0.659617i	$0.229282\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−12.3050	−0.424057
$843$	−7.65248	−0.263565
$844$	−16.8541	−0.580142
$845$	0	0
$846$	−0.145898	−0.00501608
$847$	−1.00000	−0.0343604
$848$	−16.6869	−0.573031
$849$	25.5066	0.875383
$850$	0	0
$851$	−5.96556	−0.204497
$852$	4.09017	0.140127
$853$	−36.0344	−1.23380	−0.616898	−	0.787043i	$-0.711611\pi$
$853$	−36.0344	−1.23380	−0.616898	+	0.787043i	$0.711611\pi$
$854$	−8.14590	−0.278747
$855$	0	0
$856$	−45.6525	−1.56037
$857$	−9.38197	−0.320482	−0.160241	−	0.987078i	$-0.551227\pi$
$857$	−9.38197	−0.320482	−0.160241	+	0.987078i	$0.551227\pi$
$858$	0.236068	0.00805923
$859$	55.4508	1.89196	0.945979	−	0.324227i	$-0.105104\pi$
$859$	55.4508	1.89196	0.945979	+	0.324227i	$0.105104\pi$
$860$	0	0
$861$	8.70820	0.296775
$862$	14.2016	0.483709
$863$	21.7771	0.741301	0.370650	−	0.928772i	$-0.379135\pi$
$863$	21.7771	0.741301	0.370650	+	0.928772i	$0.379135\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	5.29180	0.179823
$867$	−10.4164	−0.353760
$868$	−0.381966	−0.0129648
$869$	−9.47214	−0.321320
$870$	0	0
$871$	5.81966	0.197192
$872$	23.0902	0.781932
$873$	3.38197	0.114462
$874$	4.67376	0.158092
$875$	0	0
$876$	25.0344	0.845835
$877$	−0.763932	−0.0257962	−0.0128981	−	0.999917i	$-0.504106\pi$
$877$	−0.763932	−0.0257962	−0.0128981	+	0.999917i	$0.504106\pi$
$878$	1.18034	0.0398345
$879$	27.0902	0.913729
$880$	0	0
$881$	−5.11146	−0.172209	−0.0861047	−	0.996286i	$-0.527442\pi$
$881$	−5.11146	−0.172209	−0.0861047	+	0.996286i	$0.527442\pi$
$882$	0.618034	0.0208103
$883$	−8.27051	−0.278325	−0.139162	−	0.990270i	$-0.544441\pi$
$883$	−8.27051	−0.278325	−0.139162	+	0.990270i	$0.544441\pi$
$884$	−3.23607	−0.108841
$885$	0	0
$886$	−16.4164	−0.551520
$887$	−6.94427	−0.233166	−0.116583	−	0.993181i	$-0.537194\pi$
$887$	−6.94427	−0.233166	−0.116583	+	0.993181i	$0.537194\pi$
$888$	−2.43769	−0.0818037
$889$	1.61803	0.0542671
$890$	0	0
$891$	1.00000	0.0335013
$892$	−11.6180	−0.389001
$893$	−0.326238	−0.0109171
$894$	−7.36068	−0.246178
$895$	0	0
$896$	11.3820	0.380245
$897$	2.09017	0.0697887
$898$	3.09017	0.103120
$899$	0	0
$900$	0	0
$901$	47.1246	1.56995
$902$	5.38197	0.179200
$903$	0.472136	0.0157117
$904$	19.6738	0.654340
$905$	0	0
$906$	−3.14590	−0.104515
$907$	−31.5410	−1.04730	−0.523651	−	0.851933i	$-0.675430\pi$
$907$	−31.5410	−1.04730	−0.523651	+	0.851933i	$0.675430\pi$
$908$	14.8541	0.492951
$909$	−9.18034	−0.304493
$910$	0	0
$911$	42.7771	1.41727	0.708634	−	0.705576i	$-0.249312\pi$
$911$	42.7771	1.41727	0.708634	+	0.705576i	$0.249312\pi$
$912$	−2.56231	−0.0848464
$913$	6.32624	0.209368
$914$	−7.50658	−0.248296
$915$	0	0
$916$	−18.6180	−0.615157
$917$	6.41641	0.211888
$918$	3.23607	0.106806
$919$	−29.6738	−0.978847	−0.489424	−	0.872046i	$-0.662793\pi$
$919$	−29.6738	−0.978847	−0.489424	+	0.872046i	$0.662793\pi$
$920$	0	0
$921$	26.4164	0.870450
$922$	5.05573	0.166502
$923$	−0.965558	−0.0317817
$924$	−1.61803	−0.0532294
$925$	0	0
$926$	9.15905	0.300985
$927$	−8.14590	−0.267546
$928$	0	0
$929$	35.5279	1.16563	0.582816	−	0.812604i	$-0.301951\pi$
$929$	35.5279	1.16563	0.582816	+	0.812604i	$0.301951\pi$
$930$	0	0
$931$	1.38197	0.0452921
$932$	39.3607	1.28930
$933$	−8.70820	−0.285094
$934$	6.23607	0.204050
$935$	0	0
$936$	0.854102	0.0279172
$937$	−6.29180	−0.205544	−0.102772	−	0.994705i	$-0.532771\pi$
$937$	−6.29180	−0.205544	−0.102772	+	0.994705i	$0.532771\pi$
$938$	9.41641	0.307457
$939$	−26.9787	−0.880417
$940$	0	0
$941$	13.1803	0.429667	0.214833	−	0.976651i	$-0.431079\pi$
$941$	13.1803	0.429667	0.214833	+	0.976651i	$0.431079\pi$
$942$	7.38197	0.240517
$943$	47.6525	1.55178
$944$	−5.72949	−0.186479
$945$	0	0
$946$	0.291796	0.00948711
$947$	39.6869	1.28965	0.644826	−	0.764330i	$-0.276930\pi$
$947$	39.6869	1.28965	0.644826	+	0.764330i	$0.276930\pi$
$948$	−15.3262	−0.497773
$949$	−5.90983	−0.191841
$950$	0	0
$951$	−13.9098	−0.451057
$952$	−11.7082	−0.379465
$953$	−18.5967	−0.602408	−0.301204	−	0.953560i	$-0.597388\pi$
$953$	−18.5967	−0.602408	−0.301204	+	0.953560i	$0.597388\pi$
$954$	−5.56231	−0.180086
$955$	0	0
$956$	6.70820	0.216959
$957$	0	0
$958$	−7.03444	−0.227273
$959$	14.7082	0.474953
$960$	0	0
$961$	−30.9443	−0.998202
$962$	0.257354	0.00829743
$963$	20.4164	0.657910
$964$	−17.7082	−0.570343
$965$	0	0
$966$	3.38197	0.108813
$967$	−22.7984	−0.733146	−0.366573	−	0.930389i	$-0.619469\pi$
$967$	−22.7984	−0.733146	−0.366573	+	0.930389i	$0.619469\pi$
$968$	−2.23607	−0.0718699
$969$	7.23607	0.232456
$970$	0	0
$971$	10.9443	0.351218	0.175609	−	0.984460i	$-0.443810\pi$
$971$	10.9443	0.351218	0.175609	+	0.984460i	$0.443810\pi$
$972$	1.61803	0.0518985
$973$	11.7082	0.375348
$974$	−0.922986	−0.0295744
$975$	0	0
$976$	24.4377	0.782232
$977$	17.4508	0.558302	0.279151	−	0.960247i	$-0.409947\pi$
$977$	17.4508	0.558302	0.279151	+	0.960247i	$0.409947\pi$
$978$	−1.67376	−0.0535210
$979$	−3.29180	−0.105206
$980$	0	0
$981$	−10.3262	−0.329691
$982$	10.7082	0.341713
$983$	−15.5066	−0.494583	−0.247292	−	0.968941i	$-0.579541\pi$
$983$	−15.5066	−0.494583	−0.247292	+	0.968941i	$0.579541\pi$
$984$	19.4721	0.620749
$985$	0	0
$986$	0	0
$987$	−0.236068	−0.00751413
$988$	0.854102	0.0271726
$989$	2.58359	0.0821535
$990$	0	0
$991$	−4.18034	−0.132793	−0.0663964	−	0.997793i	$-0.521150\pi$
$991$	−4.18034	−0.132793	−0.0663964	+	0.997793i	$0.521150\pi$
$992$	−1.32624	−0.0421081
$993$	2.14590	0.0680980
$994$	−1.56231	−0.0495533
$995$	0	0
$996$	10.2361	0.324342
$997$	−24.5836	−0.778570	−0.389285	−	0.921117i	$-0.627278\pi$
$997$	−24.5836	−0.778570	−0.389285	+	0.921117i	$0.627278\pi$
$998$	9.34752	0.295891
$999$	1.09017	0.0344915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bd.1.2	✓	2
5.4	even	2		5775.2.a.bk.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5775.2.a.bd.1.2	✓	2	1.1	even	1	trivial
5775.2.a.bk.1.1	yes	2	5.4	even	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:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.61803 q^{12} -0.381966 q^{13} -0.618034 q^{14} +1.85410 q^{16} -5.23607 q^{17} +0.618034 q^{18} +1.38197 q^{19} +1.00000 q^{21} +0.618034 q^{22} +5.47214 q^{23} +2.23607 q^{24} -0.236068 q^{26} -1.00000 q^{27} +1.61803 q^{28} -0.236068 q^{31} +5.61803 q^{32} -1.00000 q^{33} -3.23607 q^{34} -1.61803 q^{36} -1.09017 q^{37} +0.854102 q^{38} +0.381966 q^{39} +8.70820 q^{41} +0.618034 q^{42} +0.472136 q^{43} -1.61803 q^{44} +3.38197 q^{46} -0.236068 q^{47} -1.85410 q^{48} +1.00000 q^{49} +5.23607 q^{51} +0.618034 q^{52} -9.00000 q^{53} -0.618034 q^{54} +2.23607 q^{56} -1.38197 q^{57} -3.09017 q^{59} +13.1803 q^{61} -0.145898 q^{62} -1.00000 q^{63} -0.236068 q^{64} -0.618034 q^{66} -15.2361 q^{67} +8.47214 q^{68} -5.47214 q^{69} +2.52786 q^{71} -2.23607 q^{72} +15.4721 q^{73} -0.673762 q^{74} -2.23607 q^{76} -1.00000 q^{77} +0.236068 q^{78} -9.47214 q^{79} +1.00000 q^{81} +5.38197 q^{82} +6.32624 q^{83} -1.61803 q^{84} +0.291796 q^{86} -2.23607 q^{88} -3.29180 q^{89} +0.381966 q^{91} -8.85410 q^{92} +0.236068 q^{93} -0.145898 q^{94} -5.61803 q^{96} +3.38197 q^{97} +0.618034 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 2 * q^7 + 2 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{11} + q^{12} - 3 q^{13} + q^{14} - 3 q^{16} - 6 q^{17} - q^{18} + 5 q^{19} + 2 q^{21} - q^{22} + 2 q^{23} + 4 q^{26} - 2 q^{27} + q^{28} + 4 q^{31} + 9 q^{32} - 2 q^{33} - 2 q^{34} - q^{36} + 9 q^{37} - 5 q^{38} + 3 q^{39} + 4 q^{41} - q^{42} - 8 q^{43} - q^{44} + 9 q^{46} + 4 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{51} - q^{52} - 18 q^{53} + q^{54} - 5 q^{57} + 5 q^{59} + 4 q^{61} - 7 q^{62} - 2 q^{63} + 4 q^{64} + q^{66} - 26 q^{67} + 8 q^{68} - 2 q^{69} + 14 q^{71} + 22 q^{73} - 17 q^{74} - 2 q^{77} - 4 q^{78} - 10 q^{79} + 2 q^{81} + 13 q^{82} - 3 q^{83} - q^{84} + 14 q^{86} - 20 q^{89} + 3 q^{91} - 11 q^{92} - 4 q^{93} - 7 q^{94} - 9 q^{96} + 9 q^{97} - q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 2 * q^7 + 2 * q^9 + 2 * q^11 + q^12 - 3 * q^13 + q^14 - 3 * q^16 - 6 * q^17 - q^18 + 5 * q^19 + 2 * q^21 - q^22 + 2 * q^23 + 4 * q^26 - 2 * q^27 + q^28 + 4 * q^31 + 9 * q^32 - 2 * q^33 - 2 * q^34 - q^36 + 9 * q^37 - 5 * q^38 + 3 * q^39 + 4 * q^41 - q^42 - 8 * q^43 - q^44 + 9 * q^46 + 4 * q^47 + 3 * q^48 + 2 * q^49 + 6 * q^51 - q^52 - 18 * q^53 + q^54 - 5 * q^57 + 5 * q^59 + 4 * q^61 - 7 * q^62 - 2 * q^63 + 4 * q^64 + q^66 - 26 * q^67 + 8 * q^68 - 2 * q^69 + 14 * q^71 + 22 * q^73 - 17 * q^74 - 2 * q^77 - 4 * q^78 - 10 * q^79 + 2 * q^81 + 13 * q^82 - 3 * q^83 - q^84 + 14 * q^86 - 20 * q^89 + 3 * q^91 - 11 * q^92 - 4 * q^93 - 7 * q^94 - 9 * q^96 + 9 * q^97 - q^98 + 2 * q^99

Introduction

L-functions

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Database

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