LMFDB - Embedded newform 1425.2.c.m.799.1

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 = [4,0,0,2,0,-2,0,0,-4,0,-16,0,0,-10] 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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.m.799.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -2.23607i q^{7} -2.23607i q^{8} -1.00000 q^{9} -4.00000 q^{11} +0.618034i q^{12} -2.47214i q^{13} -3.61803 q^{14} -4.85410 q^{16} +3.23607i q^{17} +1.61803i q^{18} +1.00000 q^{19} -2.23607 q^{21} +6.47214i q^{22} -1.23607i q^{23} -2.23607 q^{24} -4.00000 q^{26} +1.00000i q^{27} +1.38197i q^{28} +1.47214 q^{29} -1.52786 q^{31} +3.38197i q^{32} +4.00000i q^{33} +5.23607 q^{34} +0.618034 q^{36} -7.23607i q^{37} -1.61803i q^{38} -2.47214 q^{39} -5.00000 q^{41} +3.61803i q^{42} +4.00000i q^{43} +2.47214 q^{44} -2.00000 q^{46} +8.47214i q^{47} +4.85410i q^{48} +2.00000 q^{49} +3.23607 q^{51} +1.52786i q^{52} -5.00000i q^{53} +1.61803 q^{54} -5.00000 q^{56} -1.00000i q^{57} -2.38197i q^{58} +1.29180 q^{59} -9.94427 q^{61} +2.47214i q^{62} +2.23607i q^{63} -4.23607 q^{64} +6.47214 q^{66} -6.94427i q^{67} -2.00000i q^{68} -1.23607 q^{69} -13.1803 q^{71} +2.23607i q^{72} -5.47214i q^{73} -11.7082 q^{74} -0.618034 q^{76} +8.94427i q^{77} +4.00000i q^{78} +15.7082 q^{79} +1.00000 q^{81} +8.09017i q^{82} +10.9443i q^{83} +1.38197 q^{84} +6.47214 q^{86} -1.47214i q^{87} +8.94427i q^{88} -7.94427 q^{89} -5.52786 q^{91} +0.763932i q^{92} +1.52786i q^{93} +13.7082 q^{94} +3.38197 q^{96} -11.7082i q^{97} -3.23607i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} - 6 q^{16} + 4 q^{19} - 16 q^{26} - 12 q^{29} - 24 q^{31} + 12 q^{34} - 2 q^{36} + 8 q^{39} - 20 q^{41} - 8 q^{44} - 8 q^{46} + 8 q^{49} + 4 q^{51}+ \cdots + 16 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 - 6 * q^16 + 4 * q^19 - 16 * q^26 - 12 * q^29 - 24 * q^31 + 12 * q^34 - 2 * q^36 + 8 * q^39 - 20 * q^41 - 8 * q^44 - 8 * q^46 + 8 * q^49 + 4 * q^51 + 2 * q^54 - 20 * q^56 + 32 * q^59 - 4 * q^61 - 8 * q^64 + 8 * q^66 + 4 * q^69 - 8 * q^71 - 20 * q^74 + 2 * q^76 + 36 * q^79 + 4 * q^81 + 10 * q^84 + 8 * q^86 + 4 * q^89 - 40 * q^91 + 28 * q^94 + 18 * q^96 + 16 * 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 $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	− 2.23607i	− 0.845154i	−0.906327	−	0.422577i	$-0.861126\pi$
$7$	− 2.23607i	− 0.845154i	0.906327	−	0.422577i	$-0.138874\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0.618034i	0.178411i
$13$	− 2.47214i	− 0.685647i	−0.939400	−	0.342824i	$-0.888617\pi$
$13$	− 2.47214i	− 0.685647i	0.939400	−	0.342824i	$-0.111383\pi$
$14$	−3.61803	−0.966960
$15$	0	0
$16$	−4.85410	−1.21353
$17$	3.23607i	0.784862i	0.919781	+	0.392431i	$0.128366\pi$
$17$	3.23607i	0.784862i	−0.919781	+	0.392431i	$0.871634\pi$
$18$	1.61803i	0.381374i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.23607	−0.487950
$22$	6.47214i	1.37986i
$23$	− 1.23607i	− 0.257738i	−0.991662	−	0.128869i	$-0.958865\pi$
$23$	− 1.23607i	− 0.257738i	0.991662	−	0.128869i	$-0.0411347\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−4.00000	−0.784465
$27$	1.00000i	0.192450i
$28$	1.38197i	0.261167i
$29$	1.47214	0.273369	0.136684	−	0.990615i	$-0.456355\pi$
$29$	1.47214	0.273369	0.136684	+	0.990615i	$0.456355\pi$
$30$	0	0
$31$	−1.52786	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$32$	3.38197i	0.597853i
$33$	4.00000i	0.696311i
$34$	5.23607	0.897978
$35$	0	0
$36$	0.618034	0.103006
$37$	− 7.23607i	− 1.18960i	−0.803873	−	0.594801i	$-0.797231\pi$
$37$	− 7.23607i	− 1.18960i	0.803873	−	0.594801i	$-0.202769\pi$
$38$	− 1.61803i	− 0.262480i
$39$	−2.47214	−0.395859
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	3.61803i	0.558275i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	2.47214	0.372689
$45$	0	0
$46$	−2.00000	−0.294884
$47$	8.47214i	1.23579i	0.786261	+	0.617894i	$0.212014\pi$
$47$	8.47214i	1.23579i	−0.786261	+	0.617894i	$0.787986\pi$
$48$	4.85410i	0.700629i
$49$	2.00000	0.285714
$50$	0	0
$51$	3.23607	0.453140
$52$	1.52786i	0.211877i
$53$	− 5.00000i	− 0.686803i	−0.939189	−	0.343401i	$-0.888421\pi$
$53$	− 5.00000i	− 0.686803i	0.939189	−	0.343401i	$-0.111579\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	−5.00000	−0.668153
$57$	− 1.00000i	− 0.132453i
$58$	− 2.38197i	− 0.312767i
$59$	1.29180	0.168178	0.0840888	−	0.996458i	$-0.473202\pi$
$59$	1.29180	0.168178	0.0840888	+	0.996458i	$0.473202\pi$
$60$	0	0
$61$	−9.94427	−1.27323	−0.636617	−	0.771180i	$-0.719667\pi$
$61$	−9.94427	−1.27323	−0.636617	+	0.771180i	$0.719667\pi$
$62$	2.47214i	0.313962i
$63$	2.23607i	0.281718i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	6.47214	0.796665
$67$	− 6.94427i	− 0.848378i	−0.905574	−	0.424189i	$-0.860559\pi$
$67$	− 6.94427i	− 0.848378i	0.905574	−	0.424189i	$-0.139441\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−1.23607	−0.148805
$70$	0	0
$71$	−13.1803	−1.56422	−0.782109	−	0.623141i	$-0.785856\pi$
$71$	−13.1803	−1.56422	−0.782109	+	0.623141i	$0.785856\pi$
$72$	2.23607i	0.263523i
$73$	− 5.47214i	− 0.640465i	−0.947339	−	0.320233i	$-0.896239\pi$
$73$	− 5.47214i	− 0.640465i	0.947339	−	0.320233i	$-0.103761\pi$
$74$	−11.7082	−1.36105
$75$	0	0
$76$	−0.618034	−0.0708934
$77$	8.94427i	1.01929i
$78$	4.00000i	0.452911i
$79$	15.7082	1.76731	0.883656	−	0.468138i	$-0.155075\pi$
$79$	15.7082	1.76731	0.883656	+	0.468138i	$0.155075\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.09017i	0.893410i
$83$	10.9443i	1.20129i	0.799516	+	0.600645i	$0.205089\pi$
$83$	10.9443i	1.20129i	−0.799516	+	0.600645i	$0.794911\pi$
$84$	1.38197	0.150785
$85$	0	0
$86$	6.47214	0.697908
$87$	− 1.47214i	− 0.157830i
$88$	8.94427i	0.953463i
$89$	−7.94427	−0.842091	−0.421046	−	0.907039i	$-0.638337\pi$
$89$	−7.94427	−0.842091	−0.421046	+	0.907039i	$0.638337\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	0.763932i	0.0796454i
$93$	1.52786i	0.158432i
$94$	13.7082	1.41389
$95$	0	0
$96$	3.38197	0.345170
$97$	− 11.7082i	− 1.18879i	−0.804174	−	0.594394i	$-0.797392\pi$
$97$	− 11.7082i	− 1.18879i	0.804174	−	0.594394i	$-0.202608\pi$
$98$	− 3.23607i	− 0.326892i
$99$	4.00000	0.402015
$100$	0	0
$101$	8.18034	0.813974	0.406987	−	0.913434i	$-0.366579\pi$
$101$	8.18034	0.813974	0.406987	+	0.913434i	$0.366579\pi$
$102$	− 5.23607i	− 0.518448i
$103$	− 0.944272i	− 0.0930419i	−0.998917	−	0.0465209i	$-0.985187\pi$
$103$	− 0.944272i	− 0.0930419i	0.998917	−	0.0465209i	$-0.0148134\pi$
$104$	−5.52786	−0.542052
$105$	0	0
$106$	−8.09017	−0.785787
$107$	11.7639i	1.13726i	0.822593	+	0.568631i	$0.192527\pi$
$107$	11.7639i	1.13726i	−0.822593	+	0.568631i	$0.807473\pi$
$108$	− 0.618034i	− 0.0594703i
$109$	−5.70820	−0.546747	−0.273373	−	0.961908i	$-0.588139\pi$
$109$	−5.70820	−0.546747	−0.273373	+	0.961908i	$0.588139\pi$
$110$	0	0
$111$	−7.23607	−0.686817
$112$	10.8541i	1.02562i
$113$	− 3.52786i	− 0.331874i	−0.986136	−	0.165937i	$-0.946935\pi$
$113$	− 3.52786i	− 0.331874i	0.986136	−	0.165937i	$-0.0530648\pi$
$114$	−1.61803	−0.151543
$115$	0	0
$116$	−0.909830	−0.0844756
$117$	2.47214i	0.228549i
$118$	− 2.09017i	− 0.192416i
$119$	7.23607	0.663329
$120$	0	0
$121$	5.00000	0.454545
$122$	16.0902i	1.45674i
$123$	5.00000i	0.450835i
$124$	0.944272	0.0847981
$125$	0	0
$126$	3.61803	0.322320
$127$	− 15.2361i	− 1.35198i	−0.736910	−	0.675991i	$-0.763716\pi$
$127$	− 15.2361i	− 1.35198i	0.736910	−	0.675991i	$-0.236284\pi$
$128$	13.6180i	1.20368i
$129$	4.00000	0.352180
$130$	0	0
$131$	10.9443	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$131$	10.9443	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$132$	− 2.47214i	− 0.215172i
$133$	− 2.23607i	− 0.193892i
$134$	−11.2361	−0.970648
$135$	0	0
$136$	7.23607	0.620488
$137$	− 17.7082i	− 1.51291i	−0.654043	−	0.756457i	$-0.726929\pi$
$137$	− 17.7082i	− 1.51291i	0.654043	−	0.756457i	$-0.273071\pi$
$138$	2.00000i	0.170251i
$139$	20.2361	1.71640	0.858200	−	0.513315i	$-0.171583\pi$
$139$	20.2361	1.71640	0.858200	+	0.513315i	$0.171583\pi$
$140$	0	0
$141$	8.47214	0.713483
$142$	21.3262i	1.78966i
$143$	9.88854i	0.826922i
$144$	4.85410	0.404508
$145$	0	0
$146$	−8.85410	−0.732771
$147$	− 2.00000i	− 0.164957i
$148$	4.47214i	0.367607i
$149$	−7.41641	−0.607576	−0.303788	−	0.952740i	$-0.598251\pi$
$149$	−7.41641	−0.607576	−0.303788	+	0.952740i	$0.598251\pi$
$150$	0	0
$151$	−1.70820	−0.139012	−0.0695058	−	0.997582i	$-0.522142\pi$
$151$	−1.70820	−0.139012	−0.0695058	+	0.997582i	$0.522142\pi$
$152$	− 2.23607i	− 0.181369i
$153$	− 3.23607i	− 0.261621i
$154$	14.4721	1.16620
$155$	0	0
$156$	1.52786	0.122327
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	− 25.4164i	− 2.02202i
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−2.76393	−0.217828
$162$	− 1.61803i	− 0.127125i
$163$	11.1803i	0.875712i	0.899045	+	0.437856i	$0.144262\pi$
$163$	11.1803i	0.875712i	−0.899045	+	0.437856i	$0.855738\pi$
$164$	3.09017	0.241302
$165$	0	0
$166$	17.7082	1.37442
$167$	− 24.5967i	− 1.90335i	−0.307102	−	0.951677i	$-0.599359\pi$
$167$	− 24.5967i	− 1.90335i	0.307102	−	0.951677i	$-0.400641\pi$
$168$	5.00000i	0.385758i
$169$	6.88854	0.529888
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 2.47214i	− 0.188499i
$173$	− 4.52786i	− 0.344247i	−0.985075	−	0.172124i	$-0.944937\pi$
$173$	− 4.52786i	− 0.344247i	0.985075	−	0.172124i	$-0.0550628\pi$
$174$	−2.38197	−0.180576
$175$	0	0
$176$	19.4164	1.46357
$177$	− 1.29180i	− 0.0970973i
$178$	12.8541i	0.963456i
$179$	18.1246	1.35470	0.677349	−	0.735662i	$-0.263129\pi$
$179$	18.1246	1.35470	0.677349	+	0.735662i	$0.263129\pi$
$180$	0	0
$181$	1.52786	0.113565	0.0567826	−	0.998387i	$-0.481916\pi$
$181$	1.52786	0.113565	0.0567826	+	0.998387i	$0.481916\pi$
$182$	8.94427i	0.662994i
$183$	9.94427i	0.735102i
$184$	−2.76393	−0.203760
$185$	0	0
$186$	2.47214	0.181266
$187$	− 12.9443i	− 0.946579i
$188$	− 5.23607i	− 0.381880i
$189$	2.23607	0.162650
$190$	0	0
$191$	−19.4164	−1.40492	−0.702461	−	0.711722i	$-0.747915\pi$
$191$	−19.4164	−1.40492	−0.702461	+	0.711722i	$0.747915\pi$
$192$	4.23607i	0.305712i
$193$	1.23607i	0.0889741i	0.999010	+	0.0444871i	$0.0141653\pi$
$193$	1.23607i	0.0889741i	−0.999010	+	0.0444871i	$0.985835\pi$
$194$	−18.9443	−1.36012
$195$	0	0
$196$	−1.23607	−0.0882906
$197$	− 8.65248i	− 0.616463i	−0.951311	−	0.308232i	$-0.900263\pi$
$197$	− 8.65248i	− 0.616463i	0.951311	−	0.308232i	$-0.0997372\pi$
$198$	− 6.47214i	− 0.459955i
$199$	−6.23607	−0.442063	−0.221032	−	0.975267i	$-0.570942\pi$
$199$	−6.23607	−0.442063	−0.221032	+	0.975267i	$0.570942\pi$
$200$	0	0
$201$	−6.94427	−0.489811
$202$	− 13.2361i	− 0.931286i
$203$	− 3.29180i	− 0.231039i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−1.52786	−0.106451
$207$	1.23607i	0.0859127i
$208$	12.0000i	0.832050i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	28.1803	1.94001	0.970007	−	0.243076i	$-0.0781564\pi$
$211$	28.1803	1.94001	0.970007	+	0.243076i	$0.0781564\pi$
$212$	3.09017i	0.212234i
$213$	13.1803i	0.903102i
$214$	19.0344	1.30117
$215$	0	0
$216$	2.23607	0.152145
$217$	3.41641i	0.231921i
$218$	9.23607i	0.625545i
$219$	−5.47214	−0.369773
$220$	0	0
$221$	8.00000	0.538138
$222$	11.7082i	0.785803i
$223$	4.76393i	0.319016i	0.987197	+	0.159508i	$0.0509909\pi$
$223$	4.76393i	0.319016i	−0.987197	+	0.159508i	$0.949009\pi$
$224$	7.56231	0.505278
$225$	0	0
$226$	−5.70820	−0.379704
$227$	12.1246i	0.804739i	0.915477	+	0.402369i	$0.131813\pi$
$227$	12.1246i	0.804739i	−0.915477	+	0.402369i	$0.868187\pi$
$228$	0.618034i	0.0409303i
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	8.94427	0.588490
$232$	− 3.29180i	− 0.216117i
$233$	− 14.9443i	− 0.979032i	−0.871994	−	0.489516i	$-0.837174\pi$
$233$	− 14.9443i	− 0.979032i	0.871994	−	0.489516i	$-0.162826\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−0.798374	−0.0519697
$237$	− 15.7082i	− 1.02036i
$238$	− 11.7082i	− 0.758930i
$239$	−13.7082	−0.886710	−0.443355	−	0.896346i	$-0.646212\pi$
$239$	−13.7082	−0.886710	−0.443355	+	0.896346i	$0.646212\pi$
$240$	0	0
$241$	6.18034	0.398111	0.199055	−	0.979988i	$-0.436213\pi$
$241$	6.18034	0.398111	0.199055	+	0.979988i	$0.436213\pi$
$242$	− 8.09017i	− 0.520056i
$243$	− 1.00000i	− 0.0641500i
$244$	6.14590	0.393451
$245$	0	0
$246$	8.09017	0.515810
$247$	− 2.47214i	− 0.157298i
$248$	3.41641i	0.216942i
$249$	10.9443	0.693565
$250$	0	0
$251$	−9.70820	−0.612776	−0.306388	−	0.951907i	$-0.599121\pi$
$251$	−9.70820	−0.612776	−0.306388	+	0.951907i	$0.599121\pi$
$252$	− 1.38197i	− 0.0870557i
$253$	4.94427i	0.310844i
$254$	−24.6525	−1.54683
$255$	0	0
$256$	13.5623	0.847644
$257$	13.0000i	0.810918i	0.914113	+	0.405459i	$0.132888\pi$
$257$	13.0000i	0.810918i	−0.914113	+	0.405459i	$0.867112\pi$
$258$	− 6.47214i	− 0.402938i
$259$	−16.1803	−1.00540
$260$	0	0
$261$	−1.47214	−0.0911229
$262$	− 17.7082i	− 1.09402i
$263$	− 23.7082i	− 1.46191i	−0.682425	−	0.730955i	$-0.739075\pi$
$263$	− 23.7082i	− 1.46191i	0.682425	−	0.730955i	$-0.260925\pi$
$264$	8.94427	0.550482
$265$	0	0
$266$	−3.61803	−0.221836
$267$	7.94427i	0.486182i
$268$	4.29180i	0.262163i
$269$	−11.5279	−0.702866	−0.351433	−	0.936213i	$-0.614306\pi$
$269$	−11.5279	−0.702866	−0.351433	+	0.936213i	$0.614306\pi$
$270$	0	0
$271$	−19.1803	−1.16512	−0.582561	−	0.812787i	$-0.697949\pi$
$271$	−19.1803	−1.16512	−0.582561	+	0.812787i	$0.697949\pi$
$272$	− 15.7082i	− 0.952450i
$273$	5.52786i	0.334562i
$274$	−28.6525	−1.73096
$275$	0	0
$276$	0.763932	0.0459833
$277$	− 32.4164i	− 1.94771i	−0.227164	−	0.973857i	$-0.572945\pi$
$277$	− 32.4164i	− 1.94771i	0.227164	−	0.973857i	$-0.427055\pi$
$278$	− 32.7426i	− 1.96377i
$279$	1.52786	0.0914708
$280$	0	0
$281$	−27.8885	−1.66369	−0.831846	−	0.555007i	$-0.812715\pi$
$281$	−27.8885	−1.66369	−0.831846	+	0.555007i	$0.812715\pi$
$282$	− 13.7082i	− 0.816312i
$283$	− 25.8885i	− 1.53891i	−0.638699	−	0.769457i	$-0.720527\pi$
$283$	− 25.8885i	− 1.53891i	0.638699	−	0.769457i	$-0.279473\pi$
$284$	8.14590	0.483370
$285$	0	0
$286$	16.0000	0.946100
$287$	11.1803i	0.659955i
$288$	− 3.38197i	− 0.199284i
$289$	6.52786	0.383992
$290$	0	0
$291$	−11.7082	−0.686347
$292$	3.38197i	0.197915i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	−3.23607	−0.188731
$295$	0	0
$296$	−16.1803	−0.940463
$297$	− 4.00000i	− 0.232104i
$298$	12.0000i	0.695141i
$299$	−3.05573	−0.176717
$300$	0	0
$301$	8.94427	0.515539
$302$	2.76393i	0.159046i
$303$	− 8.18034i	− 0.469948i
$304$	−4.85410	−0.278402
$305$	0	0
$306$	−5.23607	−0.299326
$307$	20.9443i	1.19535i	0.801737	+	0.597676i	$0.203909\pi$
$307$	20.9443i	1.19535i	−0.801737	+	0.597676i	$0.796091\pi$
$308$	− 5.52786i	− 0.314979i
$309$	−0.944272	−0.0537178
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	5.52786i	0.312954i
$313$	− 13.0557i	− 0.737953i	−0.929439	−	0.368977i	$-0.879708\pi$
$313$	− 13.0557i	− 0.737953i	0.929439	−	0.368977i	$-0.120292\pi$
$314$	−11.3262	−0.639177
$315$	0	0
$316$	−9.70820	−0.546129
$317$	− 9.94427i	− 0.558526i	−0.960215	−	0.279263i	$-0.909910\pi$
$317$	− 9.94427i	− 0.558526i	0.960215	−	0.279263i	$-0.0900901\pi$
$318$	8.09017i	0.453674i
$319$	−5.88854	−0.329695
$320$	0	0
$321$	11.7639	0.656599
$322$	4.47214i	0.249222i
$323$	3.23607i	0.180060i
$324$	−0.618034	−0.0343352
$325$	0	0
$326$	18.0902	1.00192
$327$	5.70820i	0.315664i
$328$	11.1803i	0.617331i
$329$	18.9443	1.04443
$330$	0	0
$331$	18.3607	1.00919	0.504597	−	0.863355i	$-0.331641\pi$
$331$	18.3607	1.00919	0.504597	+	0.863355i	$0.331641\pi$
$332$	− 6.76393i	− 0.371219i
$333$	7.23607i	0.396534i
$334$	−39.7984	−2.17767
$335$	0	0
$336$	10.8541	0.592140
$337$	− 0.472136i	− 0.0257189i	−0.999917	−	0.0128594i	$-0.995907\pi$
$337$	− 0.472136i	− 0.0257189i	0.999917	−	0.0128594i	$-0.00409340\pi$
$338$	− 11.1459i	− 0.606257i
$339$	−3.52786	−0.191607
$340$	0	0
$341$	6.11146	0.330954
$342$	1.61803i	0.0874933i
$343$	− 20.1246i	− 1.08663i
$344$	8.94427	0.482243
$345$	0	0
$346$	−7.32624	−0.393861
$347$	− 11.7082i	− 0.628529i	−0.949335	−	0.314265i	$-0.898242\pi$
$347$	− 11.7082i	− 0.628529i	0.949335	−	0.314265i	$-0.101758\pi$
$348$	0.909830i	0.0487720i
$349$	−35.8328	−1.91809	−0.959043	−	0.283259i	$-0.908584\pi$
$349$	−35.8328	−1.91809	−0.959043	+	0.283259i	$0.908584\pi$
$350$	0	0
$351$	2.47214	0.131953
$352$	− 13.5279i	− 0.721038i
$353$	− 17.5279i	− 0.932914i	−0.884544	−	0.466457i	$-0.845530\pi$
$353$	− 17.5279i	− 0.932914i	0.884544	−	0.466457i	$-0.154470\pi$
$354$	−2.09017	−0.111091
$355$	0	0
$356$	4.90983	0.260220
$357$	− 7.23607i	− 0.382973i
$358$	− 29.3262i	− 1.54994i
$359$	20.7639	1.09588	0.547939	−	0.836518i	$-0.315413\pi$
$359$	20.7639	1.09588	0.547939	+	0.836518i	$0.315413\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 2.47214i	− 0.129933i
$363$	− 5.00000i	− 0.262432i
$364$	3.41641	0.179068
$365$	0	0
$366$	16.0902	0.841047
$367$	29.3050i	1.52971i	0.644205	+	0.764853i	$0.277188\pi$
$367$	29.3050i	1.52971i	−0.644205	+	0.764853i	$0.722812\pi$
$368$	6.00000i	0.312772i
$369$	5.00000	0.260290
$370$	0	0
$371$	−11.1803	−0.580454
$372$	− 0.944272i	− 0.0489582i
$373$	− 10.1803i	− 0.527118i	−0.964643	−	0.263559i	$-0.915104\pi$
$373$	− 10.1803i	− 0.527118i	0.964643	−	0.263559i	$-0.0848964\pi$
$374$	−20.9443	−1.08300
$375$	0	0
$376$	18.9443	0.976976
$377$	− 3.63932i	− 0.187435i
$378$	− 3.61803i	− 0.186092i
$379$	−11.2361	−0.577158	−0.288579	−	0.957456i	$-0.593183\pi$
$379$	−11.2361	−0.577158	−0.288579	+	0.957456i	$0.593183\pi$
$380$	0	0
$381$	−15.2361	−0.780567
$382$	31.4164i	1.60740i
$383$	− 24.1246i	− 1.23271i	−0.787468	−	0.616355i	$-0.788609\pi$
$383$	− 24.1246i	− 1.23271i	0.787468	−	0.616355i	$-0.211391\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	2.00000	0.101797
$387$	− 4.00000i	− 0.203331i
$388$	7.23607i	0.367356i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	− 4.47214i	− 0.225877i
$393$	− 10.9443i	− 0.552065i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	−2.47214	−0.124230
$397$	22.3607i	1.12225i	0.827731	+	0.561125i	$0.189631\pi$
$397$	22.3607i	1.12225i	−0.827731	+	0.561125i	$0.810369\pi$
$398$	10.0902i	0.505775i
$399$	−2.23607	−0.111943
$400$	0	0
$401$	32.8328	1.63959	0.819796	−	0.572655i	$-0.194087\pi$
$401$	32.8328	1.63959	0.819796	+	0.572655i	$0.194087\pi$
$402$	11.2361i	0.560404i
$403$	3.77709i	0.188150i
$404$	−5.05573	−0.251532
$405$	0	0
$406$	−5.32624	−0.264337
$407$	28.9443i	1.43471i
$408$	− 7.23607i	− 0.358239i
$409$	−13.2361	−0.654481	−0.327241	−	0.944941i	$-0.606119\pi$
$409$	−13.2361	−0.654481	−0.327241	+	0.944941i	$0.606119\pi$
$410$	0	0
$411$	−17.7082	−0.873481
$412$	0.583592i	0.0287515i
$413$	− 2.88854i	− 0.142136i
$414$	2.00000	0.0982946
$415$	0	0
$416$	8.36068	0.409916
$417$	− 20.2361i	− 0.990965i
$418$	6.47214i	0.316563i
$419$	23.3050	1.13852	0.569261	−	0.822157i	$-0.307230\pi$
$419$	23.3050	1.13852	0.569261	+	0.822157i	$0.307230\pi$
$420$	0	0
$421$	20.4721	0.997751	0.498875	−	0.866674i	$-0.333747\pi$
$421$	20.4721	0.997751	0.498875	+	0.866674i	$0.333747\pi$
$422$	− 45.5967i	− 2.21961i
$423$	− 8.47214i	− 0.411929i
$424$	−11.1803	−0.542965
$425$	0	0
$426$	21.3262	1.03326
$427$	22.2361i	1.07608i
$428$	− 7.27051i	− 0.351433i
$429$	9.88854	0.477423
$430$	0	0
$431$	17.1803	0.827548	0.413774	−	0.910380i	$-0.364210\pi$
$431$	17.1803	0.827548	0.413774	+	0.910380i	$0.364210\pi$
$432$	− 4.85410i	− 0.233543i
$433$	− 28.4721i	− 1.36828i	−0.729349	−	0.684142i	$-0.760177\pi$
$433$	− 28.4721i	− 1.36828i	0.729349	−	0.684142i	$-0.239823\pi$
$434$	5.52786	0.265346
$435$	0	0
$436$	3.52786	0.168954
$437$	− 1.23607i	− 0.0591292i
$438$	8.85410i	0.423065i
$439$	21.2361	1.01354	0.506771	−	0.862081i	$-0.330839\pi$
$439$	21.2361	1.01354	0.506771	+	0.862081i	$0.330839\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	− 12.9443i	− 0.615696i
$443$	− 13.4164i	− 0.637433i	−0.947850	−	0.318716i	$-0.896748\pi$
$443$	− 13.4164i	− 0.637433i	0.947850	−	0.318716i	$-0.103252\pi$
$444$	4.47214	0.212238
$445$	0	0
$446$	7.70820	0.364994
$447$	7.41641i	0.350784i
$448$	9.47214i	0.447516i
$449$	16.5279	0.779998	0.389999	−	0.920815i	$-0.372475\pi$
$449$	16.5279	0.779998	0.389999	+	0.920815i	$0.372475\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	2.18034i	0.102555i
$453$	1.70820i	0.0802584i
$454$	19.6180	0.920720
$455$	0	0
$456$	−2.23607	−0.104713
$457$	26.8885i	1.25779i	0.777489	+	0.628897i	$0.216493\pi$
$457$	26.8885i	1.25779i	−0.777489	+	0.628897i	$0.783507\pi$
$458$	7.23607i	0.338119i
$459$	−3.23607	−0.151047
$460$	0	0
$461$	−24.1803	−1.12619	−0.563095	−	0.826392i	$-0.690390\pi$
$461$	−24.1803	−1.12619	−0.563095	+	0.826392i	$0.690390\pi$
$462$	− 14.4721i	− 0.673305i
$463$	35.4164i	1.64594i	0.568085	+	0.822970i	$0.307685\pi$
$463$	35.4164i	1.64594i	−0.568085	+	0.822970i	$0.692315\pi$
$464$	−7.14590	−0.331740
$465$	0	0
$466$	−24.1803	−1.12013
$467$	3.81966i	0.176753i	0.996087	+	0.0883764i	$0.0281678\pi$
$467$	3.81966i	0.176753i	−0.996087	+	0.0883764i	$0.971832\pi$
$468$	− 1.52786i	− 0.0706255i
$469$	−15.5279	−0.717010
$470$	0	0
$471$	−7.00000	−0.322543
$472$	− 2.88854i	− 0.132956i
$473$	− 16.0000i	− 0.735681i
$474$	−25.4164	−1.16741
$475$	0	0
$476$	−4.47214	−0.204980
$477$	5.00000i	0.228934i
$478$	22.1803i	1.01451i
$479$	43.2361	1.97551	0.987753	−	0.156025i	$-0.0498679\pi$
$479$	43.2361	1.97551	0.987753	+	0.156025i	$0.0498679\pi$
$480$	0	0
$481$	−17.8885	−0.815647
$482$	− 10.0000i	− 0.455488i
$483$	2.76393i	0.125763i
$484$	−3.09017	−0.140462
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	− 24.0000i	− 1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	− 24.0000i	− 1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$	22.2361i	1.00658i
$489$	11.1803	0.505592
$490$	0	0
$491$	−21.7082	−0.979678	−0.489839	−	0.871813i	$-0.662944\pi$
$491$	−21.7082	−0.979678	−0.489839	+	0.871813i	$0.662944\pi$
$492$	− 3.09017i	− 0.139316i
$493$	4.76393i	0.214557i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	7.41641	0.333007
$497$	29.4721i	1.32201i
$498$	− 17.7082i	− 0.793524i
$499$	8.81966	0.394822	0.197411	−	0.980321i	$-0.436747\pi$
$499$	8.81966	0.394822	0.197411	+	0.980321i	$0.436747\pi$
$500$	0	0
$501$	−24.5967	−1.09890
$502$	15.7082i	0.701091i
$503$	− 18.4721i	− 0.823632i	−0.911267	−	0.411816i	$-0.864895\pi$
$503$	− 18.4721i	− 0.823632i	0.911267	−	0.411816i	$-0.135105\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	8.00000	0.355643
$507$	− 6.88854i	− 0.305931i
$508$	9.41641i	0.417786i
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−12.2361	−0.541292
$512$	5.29180i	0.233867i
$513$	1.00000i	0.0441511i
$514$	21.0344	0.927789
$515$	0	0
$516$	−2.47214	−0.108830
$517$	− 33.8885i	− 1.49042i
$518$	26.1803i	1.15030i
$519$	−4.52786	−0.198751
$520$	0	0
$521$	17.8328	0.781270	0.390635	−	0.920546i	$-0.372255\pi$
$521$	17.8328	0.781270	0.390635	+	0.920546i	$0.372255\pi$
$522$	2.38197i	0.104256i
$523$	0.875388i	0.0382781i	0.999817	+	0.0191390i	$0.00609251\pi$
$523$	0.875388i	0.0382781i	−0.999817	+	0.0191390i	$0.993907\pi$
$524$	−6.76393	−0.295484
$525$	0	0
$526$	−38.3607	−1.67261
$527$	− 4.94427i	− 0.215376i
$528$	− 19.4164i	− 0.844991i
$529$	21.4721	0.933571
$530$	0	0
$531$	−1.29180	−0.0560592
$532$	1.38197i	0.0599158i
$533$	12.3607i	0.535400i
$534$	12.8541	0.556251
$535$	0	0
$536$	−15.5279	−0.670702
$537$	− 18.1246i	− 0.782135i
$538$	18.6525i	0.804165i
$539$	−8.00000	−0.344584
$540$	0	0
$541$	−28.8328	−1.23962	−0.619810	−	0.784752i	$-0.712790\pi$
$541$	−28.8328	−1.23962	−0.619810	+	0.784752i	$0.712790\pi$
$542$	31.0344i	1.33304i
$543$	− 1.52786i	− 0.0655669i
$544$	−10.9443	−0.469232
$545$	0	0
$546$	8.94427	0.382780
$547$	− 5.70820i	− 0.244065i	−0.992526	−	0.122033i	$-0.961059\pi$
$547$	− 5.70820i	− 0.244065i	0.992526	−	0.122033i	$-0.0389412\pi$
$548$	10.9443i	0.467516i
$549$	9.94427	0.424411
$550$	0	0
$551$	1.47214	0.0627151
$552$	2.76393i	0.117641i
$553$	− 35.1246i	− 1.49365i
$554$	−52.4508	−2.22842
$555$	0	0
$556$	−12.5066	−0.530397
$557$	− 1.88854i	− 0.0800202i	−0.999199	−	0.0400101i	$-0.987261\pi$
$557$	− 1.88854i	− 0.0800202i	0.999199	−	0.0400101i	$-0.0127390\pi$
$558$	− 2.47214i	− 0.104654i
$559$	9.88854	0.418241
$560$	0	0
$561$	−12.9443	−0.546508
$562$	45.1246i	1.90347i
$563$	− 7.65248i	− 0.322513i	−0.986912	−	0.161257i	$-0.948445\pi$
$563$	− 7.65248i	− 0.322513i	0.986912	−	0.161257i	$-0.0515547\pi$
$564$	−5.23607	−0.220478
$565$	0	0
$566$	−41.8885	−1.76071
$567$	− 2.23607i	− 0.0939060i
$568$	29.4721i	1.23662i
$569$	−14.8885	−0.624160	−0.312080	−	0.950056i	$-0.601026\pi$
$569$	−14.8885	−0.624160	−0.312080	+	0.950056i	$0.601026\pi$
$570$	0	0
$571$	15.6525	0.655036	0.327518	−	0.944845i	$-0.393788\pi$
$571$	15.6525	0.655036	0.327518	+	0.944845i	$0.393788\pi$
$572$	− 6.11146i	− 0.255533i
$573$	19.4164i	0.811132i
$574$	18.0902	0.755069
$575$	0	0
$576$	4.23607	0.176503
$577$	− 25.4164i	− 1.05810i	−0.848591	−	0.529049i	$-0.822549\pi$
$577$	− 25.4164i	− 1.05810i	0.848591	−	0.529049i	$-0.177451\pi$
$578$	− 10.5623i	− 0.439334i
$579$	1.23607	0.0513692
$580$	0	0
$581$	24.4721	1.01528
$582$	18.9443i	0.785265i
$583$	20.0000i	0.828315i
$584$	−12.2361	−0.506332
$585$	0	0
$586$	3.23607	0.133681
$587$	18.6525i	0.769870i	0.922944	+	0.384935i	$0.125776\pi$
$587$	18.6525i	0.769870i	−0.922944	+	0.384935i	$0.874224\pi$
$588$	1.23607i	0.0509746i
$589$	−1.52786	−0.0629545
$590$	0	0
$591$	−8.65248	−0.355915
$592$	35.1246i	1.44361i
$593$	21.8885i	0.898855i	0.893317	+	0.449427i	$0.148372\pi$
$593$	21.8885i	0.898855i	−0.893317	+	0.449427i	$0.851628\pi$
$594$	−6.47214	−0.265555
$595$	0	0
$596$	4.58359	0.187751
$597$	6.23607i	0.255225i
$598$	4.94427i	0.202186i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	35.7771	1.45938	0.729689	−	0.683779i	$-0.239665\pi$
$601$	35.7771	1.45938	0.729689	+	0.683779i	$0.239665\pi$
$602$	− 14.4721i	− 0.589840i
$603$	6.94427i	0.282793i
$604$	1.05573	0.0429570
$605$	0	0
$606$	−13.2361	−0.537679
$607$	− 21.1246i	− 0.857422i	−0.903442	−	0.428711i	$-0.858968\pi$
$607$	− 21.1246i	− 0.857422i	0.903442	−	0.428711i	$-0.141032\pi$
$608$	3.38197i	0.137157i
$609$	−3.29180	−0.133390
$610$	0	0
$611$	20.9443	0.847315
$612$	2.00000i	0.0808452i
$613$	23.0000i	0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$613$	23.0000i	0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$614$	33.8885	1.36763
$615$	0	0
$616$	20.0000	0.805823
$617$	− 22.0000i	− 0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	− 22.0000i	− 0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	1.52786i	0.0614597i
$619$	−17.2918	−0.695016	−0.347508	−	0.937677i	$-0.612972\pi$
$619$	−17.2918	−0.695016	−0.347508	+	0.937677i	$0.612972\pi$
$620$	0	0
$621$	1.23607	0.0496017
$622$	− 6.47214i	− 0.259509i
$623$	17.7639i	0.711697i
$624$	12.0000	0.480384
$625$	0	0
$626$	−21.1246	−0.844309
$627$	4.00000i	0.159745i
$628$	4.32624i	0.172636i
$629$	23.4164	0.933673
$630$	0	0
$631$	−16.5836	−0.660182	−0.330091	−	0.943949i	$-0.607080\pi$
$631$	−16.5836	−0.660182	−0.330091	+	0.943949i	$0.607080\pi$
$632$	− 35.1246i	− 1.39718i
$633$	− 28.1803i	− 1.12007i
$634$	−16.0902	−0.639022
$635$	0	0
$636$	3.09017	0.122533
$637$	− 4.94427i	− 0.195899i
$638$	9.52786i	0.377212i
$639$	13.1803	0.521406
$640$	0	0
$641$	−10.9443	−0.432273	−0.216136	−	0.976363i	$-0.569346\pi$
$641$	−10.9443	−0.432273	−0.216136	+	0.976363i	$0.569346\pi$
$642$	− 19.0344i	− 0.751229i
$643$	5.18034i	0.204293i	0.994769	+	0.102146i	$0.0325710\pi$
$643$	5.18034i	0.204293i	−0.994769	+	0.102146i	$0.967429\pi$
$644$	1.70820	0.0673127
$645$	0	0
$646$	5.23607	0.206010
$647$	45.3050i	1.78112i	0.454864	+	0.890561i	$0.349688\pi$
$647$	45.3050i	1.78112i	−0.454864	+	0.890561i	$0.650312\pi$
$648$	− 2.23607i	− 0.0878410i
$649$	−5.16718	−0.202830
$650$	0	0
$651$	3.41641	0.133900
$652$	− 6.90983i	− 0.270610i
$653$	− 9.88854i	− 0.386969i	−0.981103	−	0.193484i	$-0.938021\pi$
$653$	− 9.88854i	− 0.386969i	0.981103	−	0.193484i	$-0.0619789\pi$
$654$	9.23607	0.361159
$655$	0	0
$656$	24.2705	0.947604
$657$	5.47214i	0.213488i
$658$	− 30.6525i	− 1.19496i
$659$	−7.05573	−0.274852	−0.137426	−	0.990512i	$-0.543883\pi$
$659$	−7.05573	−0.274852	−0.137426	+	0.990512i	$0.543883\pi$
$660$	0	0
$661$	17.1246	0.666070	0.333035	−	0.942914i	$-0.391927\pi$
$661$	17.1246	0.666070	0.333035	+	0.942914i	$0.391927\pi$
$662$	− 29.7082i	− 1.15464i
$663$	− 8.00000i	− 0.310694i
$664$	24.4721	0.949703
$665$	0	0
$666$	11.7082	0.453684
$667$	− 1.81966i	− 0.0704575i
$668$	15.2016i	0.588169i
$669$	4.76393	0.184184
$670$	0	0
$671$	39.7771	1.53558
$672$	− 7.56231i	− 0.291722i
$673$	46.2492i	1.78278i	0.453240	+	0.891388i	$0.350268\pi$
$673$	46.2492i	1.78278i	−0.453240	+	0.891388i	$0.649732\pi$
$674$	−0.763932	−0.0294256
$675$	0	0
$676$	−4.25735	−0.163744
$677$	47.8328i	1.83836i	0.393833	+	0.919182i	$0.371149\pi$
$677$	47.8328i	1.83836i	−0.393833	+	0.919182i	$0.628851\pi$
$678$	5.70820i	0.219222i
$679$	−26.1803	−1.00471
$680$	0	0
$681$	12.1246	0.464616
$682$	− 9.88854i	− 0.378652i
$683$	39.6525i	1.51726i	0.651522	+	0.758630i	$0.274131\pi$
$683$	39.6525i	1.51726i	−0.651522	+	0.758630i	$0.725869\pi$
$684$	0.618034	0.0236311
$685$	0	0
$686$	−32.5623	−1.24323
$687$	4.47214i	0.170623i
$688$	− 19.4164i	− 0.740244i
$689$	−12.3607	−0.470904
$690$	0	0
$691$	−24.9443	−0.948925	−0.474462	−	0.880276i	$-0.657358\pi$
$691$	−24.9443	−0.948925	−0.474462	+	0.880276i	$0.657358\pi$
$692$	2.79837i	0.106378i
$693$	− 8.94427i	− 0.339765i
$694$	−18.9443	−0.719115
$695$	0	0
$696$	−3.29180	−0.124775
$697$	− 16.1803i	− 0.612874i
$698$	57.9787i	2.19453i
$699$	−14.9443	−0.565244
$700$	0	0
$701$	48.6525	1.83758	0.918789	−	0.394748i	$-0.129168\pi$
$701$	48.6525	1.83758	0.918789	+	0.394748i	$0.129168\pi$
$702$	− 4.00000i	− 0.150970i
$703$	− 7.23607i	− 0.272913i
$704$	16.9443	0.638611
$705$	0	0
$706$	−28.3607	−1.06737
$707$	− 18.2918i	− 0.687934i
$708$	0.798374i	0.0300047i
$709$	13.5836	0.510143	0.255071	−	0.966922i	$-0.417901\pi$
$709$	13.5836	0.510143	0.255071	+	0.966922i	$0.417901\pi$
$710$	0	0
$711$	−15.7082	−0.589104
$712$	17.7639i	0.665731i
$713$	1.88854i	0.0707265i
$714$	−11.7082	−0.438169
$715$	0	0
$716$	−11.2016	−0.418624
$717$	13.7082i	0.511942i
$718$	− 33.5967i	− 1.25382i
$719$	46.4721	1.73312	0.866559	−	0.499074i	$-0.166327\pi$
$719$	46.4721	1.73312	0.866559	+	0.499074i	$0.166327\pi$
$720$	0	0
$721$	−2.11146	−0.0786347
$722$	− 1.61803i	− 0.0602170i
$723$	− 6.18034i	− 0.229849i
$724$	−0.944272	−0.0350936
$725$	0	0
$726$	−8.09017	−0.300254
$727$	20.2361i	0.750514i	0.926921	+	0.375257i	$0.122446\pi$
$727$	20.2361i	0.750514i	−0.926921	+	0.375257i	$0.877554\pi$
$728$	12.3607i	0.458117i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.9443	−0.478761
$732$	− 6.14590i	− 0.227159i
$733$	14.4164i	0.532482i	0.963906	+	0.266241i	$0.0857817\pi$
$733$	14.4164i	0.532482i	−0.963906	+	0.266241i	$0.914218\pi$
$734$	47.4164	1.75017
$735$	0	0
$736$	4.18034	0.154089
$737$	27.7771i	1.02318i
$738$	− 8.09017i	− 0.297803i
$739$	−44.5967	−1.64052	−0.820259	−	0.571992i	$-0.806171\pi$
$739$	−44.5967	−1.64052	−0.820259	+	0.571992i	$0.806171\pi$
$740$	0	0
$741$	−2.47214	−0.0908162
$742$	18.0902i	0.664111i
$743$	1.87539i	0.0688013i	0.999408	+	0.0344007i	$0.0109522\pi$
$743$	1.87539i	0.0688013i	−0.999408	+	0.0344007i	$0.989048\pi$
$744$	3.41641	0.125252
$745$	0	0
$746$	−16.4721	−0.603088
$747$	− 10.9443i	− 0.400430i
$748$	8.00000i	0.292509i
$749$	26.3050	0.961162
$750$	0	0
$751$	−11.5279	−0.420658	−0.210329	−	0.977631i	$-0.567453\pi$
$751$	−11.5279	−0.420658	−0.210329	+	0.977631i	$0.567453\pi$
$752$	− 41.1246i	− 1.49966i
$753$	9.70820i	0.353787i
$754$	−5.88854	−0.214448
$755$	0	0
$756$	−1.38197	−0.0502616
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	18.1803i	0.660340i
$759$	4.94427	0.179466
$760$	0	0
$761$	15.1246	0.548267	0.274133	−	0.961692i	$-0.411609\pi$
$761$	15.1246	0.548267	0.274133	+	0.961692i	$0.411609\pi$
$762$	24.6525i	0.893065i
$763$	12.7639i	0.462085i
$764$	12.0000	0.434145
$765$	0	0
$766$	−39.0344	−1.41037
$767$	− 3.19350i	− 0.115310i
$768$	− 13.5623i	− 0.489388i
$769$	32.4164	1.16897	0.584483	−	0.811406i	$-0.301297\pi$
$769$	32.4164	1.16897	0.584483	+	0.811406i	$0.301297\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	− 0.763932i	− 0.0274945i
$773$	13.9443i	0.501541i	0.968047	+	0.250770i	$0.0806839\pi$
$773$	13.9443i	0.501541i	−0.968047	+	0.250770i	$0.919316\pi$
$774$	−6.47214	−0.232636
$775$	0	0
$776$	−26.1803	−0.939819
$777$	16.1803i	0.580466i
$778$	12.9443i	0.464075i
$779$	−5.00000	−0.179144
$780$	0	0
$781$	52.7214	1.88652
$782$	− 6.47214i	− 0.231443i
$783$	1.47214i	0.0526098i
$784$	−9.70820	−0.346722
$785$	0	0
$786$	−17.7082	−0.631631
$787$	50.6525i	1.80557i	0.430097	+	0.902783i	$0.358480\pi$
$787$	50.6525i	1.80557i	−0.430097	+	0.902783i	$0.641520\pi$
$788$	5.34752i	0.190498i
$789$	−23.7082	−0.844034
$790$	0	0
$791$	−7.88854	−0.280484
$792$	− 8.94427i	− 0.317821i
$793$	24.5836i	0.872989i
$794$	36.1803	1.28399
$795$	0	0
$796$	3.85410	0.136605
$797$	− 13.3607i	− 0.473260i	−0.971600	−	0.236630i	$-0.923957\pi$
$797$	− 13.3607i	− 0.473260i	0.971600	−	0.236630i	$-0.0760428\pi$
$798$	3.61803i	0.128077i
$799$	−27.4164	−0.969923
$800$	0	0
$801$	7.94427	0.280697
$802$	− 53.1246i	− 1.87590i
$803$	21.8885i	0.772430i
$804$	4.29180	0.151360
$805$	0	0
$806$	6.11146	0.215267
$807$	11.5279i	0.405800i
$808$	− 18.2918i	− 0.643503i
$809$	−36.8328	−1.29497	−0.647486	−	0.762077i	$-0.724180\pi$
$809$	−36.8328	−1.29497	−0.647486	+	0.762077i	$0.724180\pi$
$810$	0	0
$811$	−42.7214	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$811$	−42.7214	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$812$	2.03444i	0.0713949i
$813$	19.1803i	0.672684i
$814$	46.8328	1.64149
$815$	0	0
$816$	−15.7082	−0.549897
$817$	4.00000i	0.139942i
$818$	21.4164i	0.748807i
$819$	5.52786	0.193159
$820$	0	0
$821$	−35.1246	−1.22586	−0.612929	−	0.790138i	$-0.710009\pi$
$821$	−35.1246	−1.22586	−0.612929	+	0.790138i	$0.710009\pi$
$822$	28.6525i	0.999370i
$823$	− 10.8197i	− 0.377150i	−0.982059	−	0.188575i	$-0.939613\pi$
$823$	− 10.8197i	− 0.377150i	0.982059	−	0.188575i	$-0.0603868\pi$
$824$	−2.11146	−0.0735561
$825$	0	0
$826$	−4.67376	−0.162621
$827$	19.0557i	0.662633i	0.943520	+	0.331316i	$0.107493\pi$
$827$	19.0557i	0.662633i	−0.943520	+	0.331316i	$0.892507\pi$
$828$	− 0.763932i	− 0.0265485i
$829$	51.6656	1.79442	0.897211	−	0.441603i	$-0.145590\pi$
$829$	51.6656	1.79442	0.897211	+	0.441603i	$0.145590\pi$
$830$	0	0
$831$	−32.4164	−1.12451
$832$	10.4721i	0.363056i
$833$	6.47214i	0.224246i
$834$	−32.7426	−1.13379
$835$	0	0
$836$	2.47214	0.0855006
$837$	− 1.52786i	− 0.0528107i
$838$	− 37.7082i	− 1.30261i
$839$	33.7639	1.16566	0.582830	−	0.812594i	$-0.301945\pi$
$839$	33.7639	1.16566	0.582830	+	0.812594i	$0.301945\pi$
$840$	0	0
$841$	−26.8328	−0.925270
$842$	− 33.1246i	− 1.14155i
$843$	27.8885i	0.960532i
$844$	−17.4164	−0.599497
$845$	0	0
$846$	−13.7082	−0.471298
$847$	− 11.1803i	− 0.384161i
$848$	24.2705i	0.833453i
$849$	−25.8885	−0.888493
$850$	0	0
$851$	−8.94427	−0.306606
$852$	− 8.14590i	− 0.279074i
$853$	− 1.11146i	− 0.0380555i	−0.999819	−	0.0190278i	$-0.993943\pi$
$853$	− 1.11146i	− 0.0380555i	0.999819	−	0.0190278i	$-0.00605709\pi$
$854$	35.9787	1.23117
$855$	0	0
$856$	26.3050	0.899085
$857$	38.7771i	1.32460i	0.749239	+	0.662300i	$0.230420\pi$
$857$	38.7771i	1.32460i	−0.749239	+	0.662300i	$0.769580\pi$
$858$	− 16.0000i	− 0.546231i
$859$	34.7082	1.18423	0.592114	−	0.805854i	$-0.298293\pi$
$859$	34.7082	1.18423	0.592114	+	0.805854i	$0.298293\pi$
$860$	0	0
$861$	11.1803	0.381025
$862$	− 27.7984i	− 0.946816i
$863$	57.1803i	1.94644i	0.229873	+	0.973221i	$0.426169\pi$
$863$	57.1803i	1.94644i	−0.229873	+	0.973221i	$0.573831\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	−46.0689	−1.56548
$867$	− 6.52786i	− 0.221698i
$868$	− 2.11146i	− 0.0716675i
$869$	−62.8328	−2.13146
$870$	0	0
$871$	−17.1672	−0.581688
$872$	12.7639i	0.432241i
$873$	11.7082i	0.396263i
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	3.38197	0.114266
$877$	27.0557i	0.913607i	0.889568	+	0.456804i	$0.151006\pi$
$877$	27.0557i	0.913607i	−0.889568	+	0.456804i	$0.848994\pi$
$878$	− 34.3607i	− 1.15962i
$879$	2.00000	0.0674583
$880$	0	0
$881$	16.9443	0.570867	0.285434	−	0.958399i	$-0.407862\pi$
$881$	16.9443	0.570867	0.285434	+	0.958399i	$0.407862\pi$
$882$	3.23607i	0.108964i
$883$	14.3475i	0.482833i	0.970422	+	0.241416i	$0.0776119\pi$
$883$	14.3475i	0.482833i	−0.970422	+	0.241416i	$0.922388\pi$
$884$	−4.94427	−0.166294
$885$	0	0
$886$	−21.7082	−0.729301
$887$	− 34.8328i	− 1.16957i	−0.811188	−	0.584786i	$-0.801179\pi$
$887$	− 34.8328i	− 1.16957i	0.811188	−	0.584786i	$-0.198821\pi$
$888$	16.1803i	0.542977i
$889$	−34.0689	−1.14263
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 2.94427i	− 0.0985815i
$893$	8.47214i	0.283509i
$894$	12.0000	0.401340
$895$	0	0
$896$	30.4508	1.01729
$897$	3.05573i	0.102028i
$898$	− 26.7426i	− 0.892414i
$899$	−2.24922	−0.0750158
$900$	0	0
$901$	16.1803	0.539045
$902$	− 32.3607i	− 1.07749i
$903$	− 8.94427i	− 0.297647i
$904$	−7.88854	−0.262369
$905$	0	0
$906$	2.76393	0.0918255
$907$	52.1803i	1.73262i	0.499507	+	0.866310i	$0.333515\pi$
$907$	52.1803i	1.73262i	−0.499507	+	0.866310i	$0.666485\pi$
$908$	− 7.49342i	− 0.248678i
$909$	−8.18034	−0.271325
$910$	0	0
$911$	16.2361	0.537925	0.268962	−	0.963151i	$-0.413319\pi$
$911$	16.2361	0.537925	0.268962	+	0.963151i	$0.413319\pi$
$912$	4.85410i	0.160735i
$913$	− 43.7771i	− 1.44881i
$914$	43.5066	1.43907
$915$	0	0
$916$	2.76393	0.0913229
$917$	− 24.4721i	− 0.808141i
$918$	5.23607i	0.172816i
$919$	−54.7082	−1.80466	−0.902329	−	0.431049i	$-0.858144\pi$
$919$	−54.7082	−1.80466	−0.902329	+	0.431049i	$0.858144\pi$
$920$	0	0
$921$	20.9443	0.690137
$922$	39.1246i	1.28850i
$923$	32.5836i	1.07250i
$924$	−5.52786	−0.181853
$925$	0	0
$926$	57.3050	1.88316
$927$	0.944272i	0.0310140i
$928$	4.97871i	0.163434i
$929$	1.59675	0.0523876	0.0261938	−	0.999657i	$-0.491661\pi$
$929$	1.59675	0.0523876	0.0261938	+	0.999657i	$0.491661\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	9.23607i	0.302537i
$933$	− 4.00000i	− 0.130954i
$934$	6.18034	0.202227
$935$	0	0
$936$	5.52786	0.180684
$937$	− 22.0557i	− 0.720529i	−0.932850	−	0.360265i	$-0.882686\pi$
$937$	− 22.0557i	− 0.720529i	0.932850	−	0.360265i	$-0.117314\pi$
$938$	25.1246i	0.820348i
$939$	−13.0557	−0.426058
$940$	0	0
$941$	27.5279	0.897383	0.448691	−	0.893687i	$-0.351890\pi$
$941$	27.5279	0.897383	0.448691	+	0.893687i	$0.351890\pi$
$942$	11.3262i	0.369029i
$943$	6.18034i	0.201260i
$944$	−6.27051	−0.204088
$945$	0	0
$946$	−25.8885	−0.841709
$947$	19.0557i	0.619228i	0.950862	+	0.309614i	$0.100200\pi$
$947$	19.0557i	0.619228i	−0.950862	+	0.309614i	$0.899800\pi$
$948$	9.70820i	0.315308i
$949$	−13.5279	−0.439133
$950$	0	0
$951$	−9.94427	−0.322465
$952$	− 16.1803i	− 0.524408i
$953$	− 27.2492i	− 0.882689i	−0.897338	−	0.441344i	$-0.854502\pi$
$953$	− 27.2492i	− 0.882689i	0.897338	−	0.441344i	$-0.145498\pi$
$954$	8.09017	0.261929
$955$	0	0
$956$	8.47214	0.274008
$957$	5.88854i	0.190350i
$958$	− 69.9574i	− 2.26022i
$959$	−39.5967	−1.27865
$960$	0	0
$961$	−28.6656	−0.924698
$962$	28.9443i	0.933201i
$963$	− 11.7639i	− 0.379087i
$964$	−3.81966	−0.123023
$965$	0	0
$966$	4.47214	0.143889
$967$	26.5967i	0.855294i	0.903946	+	0.427647i	$0.140657\pi$
$967$	26.5967i	0.855294i	−0.903946	+	0.427647i	$0.859343\pi$
$968$	− 11.1803i	− 0.359350i
$969$	3.23607	0.103957
$970$	0	0
$971$	−37.0689	−1.18960	−0.594799	−	0.803875i	$-0.702768\pi$
$971$	−37.0689	−1.18960	−0.594799	+	0.803875i	$0.702768\pi$
$972$	0.618034i	0.0198234i
$973$	− 45.2492i	− 1.45062i
$974$	−38.8328	−1.24428
$975$	0	0
$976$	48.2705	1.54510
$977$	− 22.9443i	− 0.734052i	−0.930211	−	0.367026i	$-0.880376\pi$
$977$	− 22.9443i	− 0.734052i	0.930211	−	0.367026i	$-0.119624\pi$
$978$	− 18.0902i	− 0.578460i
$979$	31.7771	1.01560
$980$	0	0
$981$	5.70820	0.182249
$982$	35.1246i	1.12087i
$983$	− 5.30495i	− 0.169202i	−0.996415	−	0.0846008i	$-0.973038\pi$
$983$	− 5.30495i	− 0.169202i	0.996415	−	0.0846008i	$-0.0269615\pi$
$984$	11.1803	0.356416
$985$	0	0
$986$	7.70820	0.245479
$987$	− 18.9443i	− 0.603003i
$988$	1.52786i	0.0486078i
$989$	4.94427	0.157219
$990$	0	0
$991$	−54.9443	−1.74536	−0.872681	−	0.488290i	$-0.837621\pi$
$991$	−54.9443	−1.74536	−0.872681	+	0.488290i	$0.837621\pi$
$992$	− 5.16718i	− 0.164058i
$993$	− 18.3607i	− 0.582659i
$994$	47.6869	1.51254
$995$	0	0
$996$	−6.76393	−0.214323
$997$	− 3.52786i	− 0.111729i	−0.998438	−	0.0558643i	$-0.982209\pi$
$997$	− 3.52786i	− 0.111729i	0.998438	−	0.0558643i	$-0.0177914\pi$
$998$	− 14.2705i	− 0.451725i
$999$	7.23607	0.228939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.m.799.1		4
5.2	odd	4		1425.2.a.q.1.2	yes	2
5.3	odd	4		1425.2.a.n.1.1	✓	2
5.4	even	2	inner	1425.2.c.m.799.4		4
15.2	even	4		4275.2.a.s.1.1		2
15.8	even	4		4275.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.n.1.1	✓	2	5.3	odd	4
1425.2.a.q.1.2	yes	2	5.2	odd	4
1425.2.c.m.799.1		4	1.1	even	1	trivial
1425.2.c.m.799.4		4	5.4	even	2	inner
4275.2.a.s.1.1		2	15.2	even	4
4275.2.a.v.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} -2.23607i q^{7} -2.23607i q^{8} -1.00000 q^{9} -4.00000 q^{11} +0.618034i q^{12} -2.47214i q^{13} -3.61803 q^{14} -4.85410 q^{16} +3.23607i q^{17} +1.61803i q^{18} +1.00000 q^{19} -2.23607 q^{21} +6.47214i q^{22} -1.23607i q^{23} -2.23607 q^{24} -4.00000 q^{26} +1.00000i q^{27} +1.38197i q^{28} +1.47214 q^{29} -1.52786 q^{31} +3.38197i q^{32} +4.00000i q^{33} +5.23607 q^{34} +0.618034 q^{36} -7.23607i q^{37} -1.61803i q^{38} -2.47214 q^{39} -5.00000 q^{41} +3.61803i q^{42} +4.00000i q^{43} +2.47214 q^{44} -2.00000 q^{46} +8.47214i q^{47} +4.85410i q^{48} +2.00000 q^{49} +3.23607 q^{51} +1.52786i q^{52} -5.00000i q^{53} +1.61803 q^{54} -5.00000 q^{56} -1.00000i q^{57} -2.38197i q^{58} +1.29180 q^{59} -9.94427 q^{61} +2.47214i q^{62} +2.23607i q^{63} -4.23607 q^{64} +6.47214 q^{66} -6.94427i q^{67} -2.00000i q^{68} -1.23607 q^{69} -13.1803 q^{71} +2.23607i q^{72} -5.47214i q^{73} -11.7082 q^{74} -0.618034 q^{76} +8.94427i q^{77} +4.00000i q^{78} +15.7082 q^{79} +1.00000 q^{81} +8.09017i q^{82} +10.9443i q^{83} +1.38197 q^{84} +6.47214 q^{86} -1.47214i q^{87} +8.94427i q^{88} -7.94427 q^{89} -5.52786 q^{91} +0.763932i q^{92} +1.52786i q^{93} +13.7082 q^{94} +3.38197 q^{96} -11.7082i q^{97} -3.23607i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} - 6 q^{16} + 4 q^{19} - 16 q^{26} - 12 q^{29} - 24 q^{31} + 12 q^{34} - 2 q^{36} + 8 q^{39} - 20 q^{41} - 8 q^{44} - 8 q^{46} + 8 q^{49} + 4 q^{51}+ \cdots + 16 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 - 6 * q^16 + 4 * q^19 - 16 * q^26 - 12 * q^29 - 24 * q^31 + 12 * q^34 - 2 * q^36 + 8 * q^39 - 20 * q^41 - 8 * q^44 - 8 * q^46 + 8 * q^49 + 4 * q^51 + 2 * q^54 - 20 * q^56 + 32 * q^59 - 4 * q^61 - 8 * q^64 + 8 * q^66 + 4 * q^69 - 8 * q^71 - 20 * q^74 + 2 * q^76 + 36 * q^79 + 4 * q^81 + 10 * q^84 + 8 * q^86 + 4 * q^89 - 40 * q^91 + 28 * q^94 + 18 * q^96 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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