LMFDB - Embedded newform 425.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(1,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} -4.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +4.46410 q^{9} +4.73205 q^{11} -2.73205 q^{12} +4.00000 q^{13} +4.73205 q^{14} -5.00000 q^{16} +1.00000 q^{17} +7.73205 q^{18} -1.46410 q^{19} -7.46410 q^{21} +8.19615 q^{22} +8.19615 q^{23} +4.73205 q^{24} +6.92820 q^{26} -4.00000 q^{27} +2.73205 q^{28} -3.46410 q^{29} +3.26795 q^{31} -5.19615 q^{32} -12.9282 q^{33} +1.73205 q^{34} +4.46410 q^{36} +0.535898 q^{37} -2.53590 q^{38} -10.9282 q^{39} -3.46410 q^{41} -12.9282 q^{42} +0.535898 q^{43} +4.73205 q^{44} +14.1962 q^{46} -12.9282 q^{47} +13.6603 q^{48} +0.464102 q^{49} -2.73205 q^{51} +4.00000 q^{52} -6.00000 q^{53} -6.92820 q^{54} -4.73205 q^{56} +4.00000 q^{57} -6.00000 q^{58} +2.53590 q^{59} -4.92820 q^{61} +5.66025 q^{62} +12.1962 q^{63} +1.00000 q^{64} -22.3923 q^{66} +10.0000 q^{67} +1.00000 q^{68} -22.3923 q^{69} +11.6603 q^{71} -7.73205 q^{72} -6.39230 q^{73} +0.928203 q^{74} -1.46410 q^{76} +12.9282 q^{77} -18.9282 q^{78} +14.5885 q^{79} -2.46410 q^{81} -6.00000 q^{82} -8.53590 q^{83} -7.46410 q^{84} +0.928203 q^{86} +9.46410 q^{87} -8.19615 q^{88} +4.39230 q^{89} +10.9282 q^{91} +8.19615 q^{92} -8.92820 q^{93} -22.3923 q^{94} +14.1962 q^{96} +4.92820 q^{97} +0.803848 q^{98} +21.1244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 8 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{17} + 12 q^{18} + 4 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 6 q^{24} - 8 q^{27} + 2 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 8 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^17 + 12 * q^18 + 4 * q^19 - 8 * q^21 + 6 * q^22 + 6 * q^23 + 6 * q^24 - 8 * q^27 + 2 * q^28 + 10 * q^31 - 12 * q^33 + 2 * q^36 + 8 * q^37 - 12 * q^38 - 8 * q^39 - 12 * q^42 + 8 * q^43 + 6 * q^44 + 18 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 2 * q^51 + 8 * q^52 - 12 * q^53 - 6 * q^56 + 8 * q^57 - 12 * q^58 + 12 * q^59 + 4 * q^61 - 6 * q^62 + 14 * q^63 + 2 * q^64 - 24 * q^66 + 20 * q^67 + 2 * q^68 - 24 * q^69 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 12 * q^74 + 4 * q^76 + 12 * q^77 - 24 * q^78 - 2 * q^79 + 2 * q^81 - 12 * q^82 - 24 * q^83 - 8 * q^84 - 12 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 24 * q^94 + 18 * q^96 - 4 * q^97 + 12 * q^98 + 18 * q^99

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−4.73205	−1.93185
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	−1.73205	−0.612372
$9$	4.46410	1.48803
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	−2.73205	−0.788675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	4.73205	1.26469
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	7.73205	1.82246
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	0	0
$21$	−7.46410	−1.62880
$22$	8.19615	1.74743
$23$	8.19615	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$24$	4.73205	0.965926
$25$	0	0
$26$	6.92820	1.35873
$27$	−4.00000	−0.769800
$28$	2.73205	0.516309
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	−5.19615	−0.918559
$33$	−12.9282	−2.25051
$34$	1.73205	0.297044
$35$	0	0
$36$	4.46410	0.744017
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	−2.53590	−0.411377
$39$	−10.9282	−1.74991
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	−12.9282	−1.99487
$43$	0.535898	0.0817237	0.0408619	−	0.999165i	$-0.486990\pi$
$43$	0.535898	0.0817237	0.0408619	+	0.999165i	$0.486990\pi$
$44$	4.73205	0.713384
$45$	0	0
$46$	14.1962	2.09311
$47$	−12.9282	−1.88577	−0.942886	−	0.333115i	$-0.891900\pi$
$47$	−12.9282	−1.88577	−0.942886	+	0.333115i	$0.891900\pi$
$48$	13.6603	1.97169
$49$	0.464102	0.0663002
$50$	0	0
$51$	−2.73205	−0.382564
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−6.92820	−0.942809
$55$	0	0
$56$	−4.73205	−0.632347
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	5.66025	0.718853
$63$	12.1962	1.53657
$64$	1.00000	0.125000
$65$	0	0
$66$	−22.3923	−2.75630
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	1.00000	0.121268
$69$	−22.3923	−2.69572
$70$	0	0
$71$	11.6603	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$71$	11.6603	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$72$	−7.73205	−0.911231
$73$	−6.39230	−0.748163	−0.374081	−	0.927396i	$-0.622042\pi$
$73$	−6.39230	−0.748163	−0.374081	+	0.927396i	$0.622042\pi$
$74$	0.928203	0.107901
$75$	0	0
$76$	−1.46410	−0.167944
$77$	12.9282	1.47331
$78$	−18.9282	−2.14320
$79$	14.5885	1.64133	0.820665	−	0.571410i	$-0.193603\pi$
$79$	14.5885	1.64133	0.820665	+	0.571410i	$0.193603\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	−6.00000	−0.662589
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	−7.46410	−0.814400
$85$	0	0
$86$	0.928203	0.100091
$87$	9.46410	1.01466
$88$	−8.19615	−0.873713
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	8.19615	0.854508
$93$	−8.92820	−0.925812
$94$	−22.3923	−2.30959
$95$	0	0
$96$	14.1962	1.44889
$97$	4.92820	0.500383	0.250192	−	0.968196i	$-0.419506\pi$
$97$	4.92820	0.500383	0.250192	+	0.968196i	$0.419506\pi$
$98$	0.803848	0.0812009
$99$	21.1244	2.12308
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	−4.73205	−0.468543
$103$	−8.92820	−0.879722	−0.439861	−	0.898066i	$-0.644972\pi$
$103$	−8.92820	−0.879722	−0.439861	+	0.898066i	$0.644972\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	−10.3923	−1.00939
$107$	−17.6603	−1.70728	−0.853641	−	0.520862i	$-0.825610\pi$
$107$	−17.6603	−1.70728	−0.853641	+	0.520862i	$0.825610\pi$
$108$	−4.00000	−0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−1.46410	−0.138966
$112$	−13.6603	−1.29077
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	6.92820	0.648886
$115$	0	0
$116$	−3.46410	−0.321634
$117$	17.8564	1.65083
$118$	4.39230	0.404344
$119$	2.73205	0.250447
$120$	0	0
$121$	11.3923	1.03566
$122$	−8.53590	−0.772804
$123$	9.46410	0.853349
$124$	3.26795	0.293471
$125$	0	0
$126$	21.1244	1.88191
$127$	14.3923	1.27711	0.638555	−	0.769576i	$-0.279532\pi$
$127$	14.3923	1.27711	0.638555	+	0.769576i	$0.279532\pi$
$128$	12.1244	1.07165
$129$	−1.46410	−0.128907
$130$	0	0
$131$	−2.19615	−0.191879	−0.0959394	−	0.995387i	$-0.530585\pi$
$131$	−2.19615	−0.191879	−0.0959394	+	0.995387i	$0.530585\pi$
$132$	−12.9282	−1.12526
$133$	−4.00000	−0.346844
$134$	17.3205	1.49626
$135$	0	0
$136$	−1.73205	−0.148522
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−38.7846	−3.30157
$139$	−3.66025	−0.310459	−0.155229	−	0.987878i	$-0.549612\pi$
$139$	−3.66025	−0.310459	−0.155229	+	0.987878i	$0.549612\pi$
$140$	0	0
$141$	35.3205	2.97452
$142$	20.1962	1.69482
$143$	18.9282	1.58286
$144$	−22.3205	−1.86004
$145$	0	0
$146$	−11.0718	−0.916308
$147$	−1.26795	−0.104579
$148$	0.535898	0.0440506
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	2.53590	0.205689
$153$	4.46410	0.360901
$154$	22.3923	1.80442
$155$	0	0
$156$	−10.9282	−0.874957
$157$	−8.92820	−0.712548	−0.356274	−	0.934381i	$-0.615953\pi$
$157$	−8.92820	−0.712548	−0.356274	+	0.934381i	$0.615953\pi$
$158$	25.2679	2.01021
$159$	16.3923	1.29999
$160$	0	0
$161$	22.3923	1.76476
$162$	−4.26795	−0.335322
$163$	0.196152	0.0153638	0.00768192	−	0.999970i	$-0.497555\pi$
$163$	0.196152	0.0153638	0.00768192	+	0.999970i	$0.497555\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	−12.5885	−0.974124	−0.487062	−	0.873367i	$-0.661931\pi$
$167$	−12.5885	−0.974124	−0.487062	+	0.873367i	$0.661931\pi$
$168$	12.9282	0.997433
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.53590	−0.499813
$172$	0.535898	0.0408619
$173$	−3.46410	−0.263371	−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410	−0.263371	−0.131685	+	0.991292i	$0.542039\pi$
$174$	16.3923	1.24270
$175$	0	0
$176$	−23.6603	−1.78346
$177$	−6.92820	−0.520756
$178$	7.60770	0.570221
$179$	−11.3205	−0.846135	−0.423067	−	0.906098i	$-0.639047\pi$
$179$	−11.3205	−0.846135	−0.423067	+	0.906098i	$0.639047\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	18.9282	1.40305
$183$	13.4641	0.995295
$184$	−14.1962	−1.04655
$185$	0	0
$186$	−15.4641	−1.13388
$187$	4.73205	0.346042
$188$	−12.9282	−0.942886
$189$	−10.9282	−0.794910
$190$	0	0
$191$	1.85641	0.134325	0.0671624	−	0.997742i	$-0.478605\pi$
$191$	1.85641	0.134325	0.0671624	+	0.997742i	$0.478605\pi$
$192$	−2.73205	−0.197169
$193$	−16.5359	−1.19028	−0.595140	−	0.803622i	$-0.702903\pi$
$193$	−16.5359	−1.19028	−0.595140	+	0.803622i	$0.702903\pi$
$194$	8.53590	0.612842
$195$	0	0
$196$	0.464102	0.0331501
$197$	−17.3205	−1.23404	−0.617018	−	0.786949i	$-0.711659\pi$
$197$	−17.3205	−1.23404	−0.617018	+	0.786949i	$0.711659\pi$
$198$	36.5885	2.60023
$199$	10.1962	0.722786	0.361393	−	0.932414i	$-0.382301\pi$
$199$	10.1962	0.722786	0.361393	+	0.932414i	$0.382301\pi$
$200$	0	0
$201$	−27.3205	−1.92704
$202$	−16.3923	−1.15336
$203$	−9.46410	−0.664250
$204$	−2.73205	−0.191282
$205$	0	0
$206$	−15.4641	−1.07744
$207$	36.5885	2.54307
$208$	−20.0000	−1.38675
$209$	−6.92820	−0.479234
$210$	0	0
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	−6.00000	−0.412082
$213$	−31.8564	−2.18277
$214$	−30.5885	−2.09098
$215$	0	0
$216$	6.92820	0.471405
$217$	8.92820	0.606086
$218$	−17.3205	−1.17309
$219$	17.4641	1.18011
$220$	0	0
$221$	4.00000	0.269069
$222$	−2.53590	−0.170198
$223$	26.3923	1.76736	0.883680	−	0.468092i	$-0.155058\pi$
$223$	26.3923	1.76736	0.883680	+	0.468092i	$0.155058\pi$
$224$	−14.1962	−0.948520
$225$	0	0
$226$	30.0000	1.99557
$227$	−22.7321	−1.50878	−0.754390	−	0.656427i	$-0.772067\pi$
$227$	−22.7321	−1.50878	−0.754390	+	0.656427i	$0.772067\pi$
$228$	4.00000	0.264906
$229$	−8.39230	−0.554579	−0.277290	−	0.960786i	$-0.589436\pi$
$229$	−8.39230	−0.554579	−0.277290	+	0.960786i	$0.589436\pi$
$230$	0	0
$231$	−35.3205	−2.32392
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	30.9282	2.02184
$235$	0	0
$236$	2.53590	0.165073
$237$	−39.8564	−2.58895
$238$	4.73205	0.306733
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−5.60770	−0.361223	−0.180612	−	0.983554i	$-0.557808\pi$
$241$	−5.60770	−0.361223	−0.180612	+	0.983554i	$0.557808\pi$
$242$	19.7321	1.26842
$243$	18.7321	1.20166
$244$	−4.92820	−0.315496
$245$	0	0
$246$	16.3923	1.04514
$247$	−5.85641	−0.372634
$248$	−5.66025	−0.359426
$249$	23.3205	1.47788
$250$	0	0
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	12.1962	0.768285
$253$	38.7846	2.43837
$254$	24.9282	1.56413
$255$	0	0
$256$	19.0000	1.18750
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	−2.53590	−0.157878
$259$	1.46410	0.0909748
$260$	0	0
$261$	−15.4641	−0.957204
$262$	−3.80385	−0.235002
$263$	1.60770	0.0991347	0.0495674	−	0.998771i	$-0.484216\pi$
$263$	1.60770	0.0991347	0.0495674	+	0.998771i	$0.484216\pi$
$264$	22.3923	1.37815
$265$	0	0
$266$	−6.92820	−0.424795
$267$	−12.0000	−0.734388
$268$	10.0000	0.610847
$269$	0.928203	0.0565935	0.0282968	−	0.999600i	$-0.490992\pi$
$269$	0.928203	0.0565935	0.0282968	+	0.999600i	$0.490992\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	−5.00000	−0.303170
$273$	−29.8564	−1.80699
$274$	0	0
$275$	0	0
$276$	−22.3923	−1.34786
$277$	−20.9282	−1.25745	−0.628727	−	0.777626i	$-0.716424\pi$
$277$	−20.9282	−1.25745	−0.628727	+	0.777626i	$0.716424\pi$
$278$	−6.33975	−0.380233
$279$	14.5885	0.873388
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	61.1769	3.64303
$283$	5.26795	0.313147	0.156574	−	0.987666i	$-0.449955\pi$
$283$	5.26795	0.313147	0.156574	+	0.987666i	$0.449955\pi$
$284$	11.6603	0.691909
$285$	0	0
$286$	32.7846	1.93859
$287$	−9.46410	−0.558648
$288$	−23.1962	−1.36685
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.4641	−0.789280
$292$	−6.39230	−0.374081
$293$	0.928203	0.0542262	0.0271131	−	0.999632i	$-0.491369\pi$
$293$	0.928203	0.0542262	0.0271131	+	0.999632i	$0.491369\pi$
$294$	−2.19615	−0.128082
$295$	0	0
$296$	−0.928203	−0.0539507
$297$	−18.9282	−1.09833
$298$	−10.3923	−0.602010
$299$	32.7846	1.89598
$300$	0	0
$301$	1.46410	0.0843894
$302$	−2.53590	−0.145925
$303$	25.8564	1.48541
$304$	7.32051	0.419860
$305$	0	0
$306$	7.73205	0.442012
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	12.9282	0.736653
$309$	24.3923	1.38763
$310$	0	0
$311$	−16.0526	−0.910257	−0.455129	−	0.890426i	$-0.650407\pi$
$311$	−16.0526	−0.910257	−0.455129	+	0.890426i	$0.650407\pi$
$312$	18.9282	1.07160
$313$	26.3923	1.49178	0.745891	−	0.666068i	$-0.232024\pi$
$313$	26.3923	1.49178	0.745891	+	0.666068i	$0.232024\pi$
$314$	−15.4641	−0.872690
$315$	0	0
$316$	14.5885	0.820665
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	28.3923	1.59216
$319$	−16.3923	−0.917793
$320$	0	0
$321$	48.2487	2.69298
$322$	38.7846	2.16138
$323$	−1.46410	−0.0814648
$324$	−2.46410	−0.136895
$325$	0	0
$326$	0.339746	0.0188168
$327$	27.3205	1.51083
$328$	6.00000	0.331295
$329$	−35.3205	−1.94728
$330$	0	0
$331$	−6.53590	−0.359245	−0.179623	−	0.983736i	$-0.557488\pi$
$331$	−6.53590	−0.359245	−0.179623	+	0.983736i	$0.557488\pi$
$332$	−8.53590	−0.468468
$333$	2.39230	0.131097
$334$	−21.8038	−1.19305
$335$	0	0
$336$	37.3205	2.03600
$337$	6.78461	0.369581	0.184791	−	0.982778i	$-0.440839\pi$
$337$	6.78461	0.369581	0.184791	+	0.982778i	$0.440839\pi$
$338$	5.19615	0.282633
$339$	−47.3205	−2.57010
$340$	0	0
$341$	15.4641	0.837428
$342$	−11.3205	−0.612143
$343$	−17.8564	−0.964155
$344$	−0.928203	−0.0500454
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−3.80385	−0.204201	−0.102101	−	0.994774i	$-0.532556\pi$
$347$	−3.80385	−0.204201	−0.102101	+	0.994774i	$0.532556\pi$
$348$	9.46410	0.507329
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	−24.5885	−1.31057
$353$	−26.7846	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$353$	−26.7846	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	4.39230	0.232792
$357$	−7.46410	−0.395042
$358$	−19.6077	−1.03630
$359$	−21.4641	−1.13283	−0.566416	−	0.824119i	$-0.691670\pi$
$359$	−21.4641	−1.13283	−0.566416	+	0.824119i	$0.691670\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	−4.14359	−0.217782
$363$	−31.1244	−1.63361
$364$	10.9282	0.572793
$365$	0	0
$366$	23.3205	1.21898
$367$	7.80385	0.407358	0.203679	−	0.979038i	$-0.434710\pi$
$367$	7.80385	0.407358	0.203679	+	0.979038i	$0.434710\pi$
$368$	−40.9808	−2.13627
$369$	−15.4641	−0.805029
$370$	0	0
$371$	−16.3923	−0.851046
$372$	−8.92820	−0.462906
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	8.19615	0.423813
$375$	0	0
$376$	22.3923	1.15479
$377$	−13.8564	−0.713641
$378$	−18.9282	−0.973562
$379$	17.8038	0.914522	0.457261	−	0.889332i	$-0.348830\pi$
$379$	17.8038	0.914522	0.457261	+	0.889332i	$0.348830\pi$
$380$	0	0
$381$	−39.3205	−2.01445
$382$	3.21539	0.164514
$383$	−15.4641	−0.790179	−0.395089	−	0.918643i	$-0.629286\pi$
$383$	−15.4641	−0.790179	−0.395089	+	0.918643i	$0.629286\pi$
$384$	−33.1244	−1.69037
$385$	0	0
$386$	−28.6410	−1.45779
$387$	2.39230	0.121608
$388$	4.92820	0.250192
$389$	−4.39230	−0.222699	−0.111349	−	0.993781i	$-0.535517\pi$
$389$	−4.39230	−0.222699	−0.111349	+	0.993781i	$0.535517\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	−0.803848	−0.0406004
$393$	6.00000	0.302660
$394$	−30.0000	−1.51138
$395$	0	0
$396$	21.1244	1.06154
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	17.6603	0.885229
$399$	10.9282	0.547094
$400$	0	0
$401$	12.9282	0.645604	0.322802	−	0.946467i	$-0.395375\pi$
$401$	12.9282	0.645604	0.322802	+	0.946467i	$0.395375\pi$
$402$	−47.3205	−2.36013
$403$	13.0718	0.651153
$404$	−9.46410	−0.470857
$405$	0	0
$406$	−16.3923	−0.813536
$407$	2.53590	0.125700
$408$	4.73205	0.234271
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	−8.92820	−0.439861
$413$	6.92820	0.340915
$414$	63.3731	3.11462
$415$	0	0
$416$	−20.7846	−1.01905
$417$	10.0000	0.489702
$418$	−12.0000	−0.586939
$419$	38.1962	1.86600	0.933002	−	0.359871i	$-0.117179\pi$
$419$	38.1962	1.86600	0.933002	+	0.359871i	$0.117179\pi$
$420$	0	0
$421$	5.46410	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$422$	17.6603	0.859688
$423$	−57.7128	−2.80609
$424$	10.3923	0.504695
$425$	0	0
$426$	−55.1769	−2.67333
$427$	−13.4641	−0.651574
$428$	−17.6603	−0.853641
$429$	−51.7128	−2.49672
$430$	0	0
$431$	9.80385	0.472235	0.236117	−	0.971725i	$-0.424125\pi$
$431$	9.80385	0.472235	0.236117	+	0.971725i	$0.424125\pi$
$432$	20.0000	0.962250
$433$	17.8564	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$433$	17.8564	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$434$	15.4641	0.742301
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−12.0000	−0.574038
$438$	30.2487	1.44534
$439$	−20.0526	−0.957056	−0.478528	−	0.878072i	$-0.658830\pi$
$439$	−20.0526	−0.957056	−0.478528	+	0.878072i	$0.658830\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	6.92820	0.329541
$443$	12.9282	0.614237	0.307119	−	0.951671i	$-0.400635\pi$
$443$	12.9282	0.614237	0.307119	+	0.951671i	$0.400635\pi$
$444$	−1.46410	−0.0694832
$445$	0	0
$446$	45.7128	2.16456
$447$	16.3923	0.775329
$448$	2.73205	0.129077
$449$	−34.3923	−1.62307	−0.811537	−	0.584302i	$-0.801369\pi$
$449$	−34.3923	−1.62307	−0.811537	+	0.584302i	$0.801369\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	17.3205	0.814688
$453$	4.00000	0.187936
$454$	−39.3731	−1.84787
$455$	0	0
$456$	−6.92820	−0.324443
$457$	36.7846	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$457$	36.7846	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$458$	−14.5359	−0.679218
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−24.9282	−1.16102	−0.580511	−	0.814252i	$-0.697147\pi$
$461$	−24.9282	−1.16102	−0.580511	+	0.814252i	$0.697147\pi$
$462$	−61.1769	−2.84621
$463$	23.8564	1.10870	0.554351	−	0.832283i	$-0.312967\pi$
$463$	23.8564	1.10870	0.554351	+	0.832283i	$0.312967\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	−10.3923	−0.481414
$467$	−1.60770	−0.0743953	−0.0371976	−	0.999308i	$-0.511843\pi$
$467$	−1.60770	−0.0743953	−0.0371976	+	0.999308i	$0.511843\pi$
$468$	17.8564	0.825413
$469$	27.3205	1.26154
$470$	0	0
$471$	24.3923	1.12394
$472$	−4.39230	−0.202172
$473$	2.53590	0.116601
$474$	−69.0333	−3.17081
$475$	0	0
$476$	2.73205	0.125223
$477$	−26.7846	−1.22638
$478$	−36.0000	−1.64660
$479$	−11.6603	−0.532771	−0.266385	−	0.963867i	$-0.585829\pi$
$479$	−11.6603	−0.532771	−0.266385	+	0.963867i	$0.585829\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	−9.71281	−0.442407
$483$	−61.1769	−2.78365
$484$	11.3923	0.517832
$485$	0	0
$486$	32.4449	1.47173
$487$	−24.9808	−1.13199	−0.565993	−	0.824410i	$-0.691507\pi$
$487$	−24.9808	−1.13199	−0.565993	+	0.824410i	$0.691507\pi$
$488$	8.53590	0.386402
$489$	−0.535898	−0.0242342
$490$	0	0
$491$	−19.6077	−0.884883	−0.442441	−	0.896797i	$-0.645888\pi$
$491$	−19.6077	−0.884883	−0.442441	+	0.896797i	$0.645888\pi$
$492$	9.46410	0.426675
$493$	−3.46410	−0.156015
$494$	−10.1436	−0.456382
$495$	0	0
$496$	−16.3397	−0.733676
$497$	31.8564	1.42896
$498$	40.3923	1.81002
$499$	−15.6603	−0.701049	−0.350525	−	0.936554i	$-0.613997\pi$
$499$	−15.6603	−0.701049	−0.350525	+	0.936554i	$0.613997\pi$
$500$	0	0
$501$	34.3923	1.53653
$502$	12.0000	0.535586
$503$	−15.1244	−0.674362	−0.337181	−	0.941440i	$-0.609473\pi$
$503$	−15.1244	−0.674362	−0.337181	+	0.941440i	$0.609473\pi$
$504$	−21.1244	−0.940954
$505$	0	0
$506$	67.1769	2.98638
$507$	−8.19615	−0.364004
$508$	14.3923	0.638555
$509$	19.8564	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$509$	19.8564	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$510$	0	0
$511$	−17.4641	−0.772566
$512$	8.66025	0.382733
$513$	5.85641	0.258567
$514$	12.0000	0.529297
$515$	0	0
$516$	−1.46410	−0.0644535
$517$	−61.1769	−2.69056
$518$	2.53590	0.111421
$519$	9.46410	0.415428
$520$	0	0
$521$	4.14359	0.181534	0.0907671	−	0.995872i	$-0.471068\pi$
$521$	4.14359	0.181534	0.0907671	+	0.995872i	$0.471068\pi$
$522$	−26.7846	−1.17233
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−2.19615	−0.0959394
$525$	0	0
$526$	2.78461	0.121415
$527$	3.26795	0.142354
$528$	64.6410	2.81314
$529$	44.1769	1.92074
$530$	0	0
$531$	11.3205	0.491268
$532$	−4.00000	−0.173422
$533$	−13.8564	−0.600188
$534$	−20.7846	−0.899438
$535$	0	0
$536$	−17.3205	−0.748132
$537$	30.9282	1.33465
$538$	1.60770	0.0693127
$539$	2.19615	0.0945950
$540$	0	0
$541$	39.1769	1.68435	0.842174	−	0.539207i	$-0.181276\pi$
$541$	39.1769	1.68435	0.842174	+	0.539207i	$0.181276\pi$
$542$	5.07180	0.217852
$543$	6.53590	0.280482
$544$	−5.19615	−0.222783
$545$	0	0
$546$	−51.7128	−2.21310
$547$	39.9090	1.70638	0.853192	−	0.521597i	$-0.174663\pi$
$547$	39.9090	1.70638	0.853192	+	0.521597i	$0.174663\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	5.07180	0.216066
$552$	38.7846	1.65078
$553$	39.8564	1.69487
$554$	−36.2487	−1.54006
$555$	0	0
$556$	−3.66025	−0.155229
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	25.2679	1.06968
$559$	2.14359	0.0906643
$560$	0	0
$561$	−12.9282	−0.545829
$562$	−22.3923	−0.944562
$563$	−27.4641	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$563$	−27.4641	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$564$	35.3205	1.48726
$565$	0	0
$566$	9.12436	0.383525
$567$	−6.73205	−0.282720
$568$	−20.1962	−0.847412
$569$	40.6410	1.70376	0.851880	−	0.523737i	$-0.175463\pi$
$569$	40.6410	1.70376	0.851880	+	0.523737i	$0.175463\pi$
$570$	0	0
$571$	−36.4449	−1.52517	−0.762585	−	0.646888i	$-0.776070\pi$
$571$	−36.4449	−1.52517	−0.762585	+	0.646888i	$0.776070\pi$
$572$	18.9282	0.791428
$573$	−5.07180	−0.211877
$574$	−16.3923	−0.684202
$575$	0	0
$576$	4.46410	0.186004
$577$	38.6410	1.60865	0.804323	−	0.594192i	$-0.202528\pi$
$577$	38.6410	1.60865	0.804323	+	0.594192i	$0.202528\pi$
$578$	1.73205	0.0720438
$579$	45.1769	1.87749
$580$	0	0
$581$	−23.3205	−0.967498
$582$	−23.3205	−0.966666
$583$	−28.3923	−1.17589
$584$	11.0718	0.458154
$585$	0	0
$586$	1.60770	0.0664133
$587$	−46.3923	−1.91482	−0.957408	−	0.288740i	$-0.906764\pi$
$587$	−46.3923	−1.91482	−0.957408	+	0.288740i	$0.906764\pi$
$588$	−1.26795	−0.0522893
$589$	−4.78461	−0.197146
$590$	0	0
$591$	47.3205	1.94651
$592$	−2.67949	−0.110126
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	−32.7846	−1.34517
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−27.8564	−1.14009
$598$	56.7846	2.32210
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	8.24871	0.336472	0.168236	−	0.985747i	$-0.446193\pi$
$601$	8.24871	0.336472	0.168236	+	0.985747i	$0.446193\pi$
$602$	2.53590	0.103356
$603$	44.6410	1.81792
$604$	−1.46410	−0.0595734
$605$	0	0
$606$	44.7846	1.81925
$607$	21.6603	0.879163	0.439581	−	0.898203i	$-0.355127\pi$
$607$	21.6603	0.879163	0.439581	+	0.898203i	$0.355127\pi$
$608$	7.60770	0.308533
$609$	25.8564	1.04775
$610$	0	0
$611$	−51.7128	−2.09208
$612$	4.46410	0.180451
$613$	−15.8564	−0.640434	−0.320217	−	0.947344i	$-0.603756\pi$
$613$	−15.8564	−0.640434	−0.320217	+	0.947344i	$0.603756\pi$
$614$	17.3205	0.698999
$615$	0	0
$616$	−22.3923	−0.902212
$617$	27.4641	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$617$	27.4641	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$618$	42.2487	1.69949
$619$	38.5885	1.55100	0.775501	−	0.631347i	$-0.217498\pi$
$619$	38.5885	1.55100	0.775501	+	0.631347i	$0.217498\pi$
$620$	0	0
$621$	−32.7846	−1.31560
$622$	−27.8038	−1.11483
$623$	12.0000	0.480770
$624$	54.6410	2.18739
$625$	0	0
$626$	45.7128	1.82705
$627$	18.9282	0.755920
$628$	−8.92820	−0.356274
$629$	0.535898	0.0213677
$630$	0	0
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	−25.2679	−1.00511
$633$	−27.8564	−1.10719
$634$	43.1769	1.71477
$635$	0	0
$636$	16.3923	0.649997
$637$	1.85641	0.0735535
$638$	−28.3923	−1.12406
$639$	52.0526	2.05917
$640$	0	0
$641$	31.1769	1.23141	0.615707	−	0.787975i	$-0.288870\pi$
$641$	31.1769	1.23141	0.615707	+	0.787975i	$0.288870\pi$
$642$	83.5692	3.29821
$643$	24.1962	0.954203	0.477102	−	0.878848i	$-0.341687\pi$
$643$	24.1962	0.954203	0.477102	+	0.878848i	$0.341687\pi$
$644$	22.3923	0.882380
$645$	0	0
$646$	−2.53590	−0.0997736
$647$	38.7846	1.52478	0.762390	−	0.647118i	$-0.224026\pi$
$647$	38.7846	1.52478	0.762390	+	0.647118i	$0.224026\pi$
$648$	4.26795	0.167661
$649$	12.0000	0.471041
$650$	0	0
$651$	−24.3923	−0.956010
$652$	0.196152	0.00768192
$653$	25.6077	1.00211	0.501053	−	0.865416i	$-0.332946\pi$
$653$	25.6077	1.00211	0.501053	+	0.865416i	$0.332946\pi$
$654$	47.3205	1.85038
$655$	0	0
$656$	17.3205	0.676252
$657$	−28.5359	−1.11329
$658$	−61.1769	−2.38492
$659$	−32.7846	−1.27711	−0.638554	−	0.769577i	$-0.720467\pi$
$659$	−32.7846	−1.27711	−0.638554	+	0.769577i	$0.720467\pi$
$660$	0	0
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	−11.3205	−0.439984
$663$	−10.9282	−0.424416
$664$	14.7846	0.573754
$665$	0	0
$666$	4.14359	0.160561
$667$	−28.3923	−1.09935
$668$	−12.5885	−0.487062
$669$	−72.1051	−2.78774
$670$	0	0
$671$	−23.3205	−0.900278
$672$	38.7846	1.49615
$673$	−23.4641	−0.904475	−0.452237	−	0.891898i	$-0.649374\pi$
$673$	−23.4641	−0.904475	−0.452237	+	0.891898i	$0.649374\pi$
$674$	11.7513	0.452643
$675$	0	0
$676$	3.00000	0.115385
$677$	−2.78461	−0.107021	−0.0535106	−	0.998567i	$-0.517041\pi$
$677$	−2.78461	−0.107021	−0.0535106	+	0.998567i	$0.517041\pi$
$678$	−81.9615	−3.14771
$679$	13.4641	0.516705
$680$	0	0
$681$	62.1051	2.37987
$682$	26.7846	1.02564
$683$	30.8372	1.17995	0.589976	−	0.807421i	$-0.299137\pi$
$683$	30.8372	1.17995	0.589976	+	0.807421i	$0.299137\pi$
$684$	−6.53590	−0.249906
$685$	0	0
$686$	−30.9282	−1.18084
$687$	22.9282	0.874766
$688$	−2.67949	−0.102155
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−38.9808	−1.48290	−0.741449	−	0.671009i	$-0.765861\pi$
$691$	−38.9808	−1.48290	−0.741449	+	0.671009i	$0.765861\pi$
$692$	−3.46410	−0.131685
$693$	57.7128	2.19233
$694$	−6.58846	−0.250094
$695$	0	0
$696$	−16.3923	−0.621349
$697$	−3.46410	−0.131212
$698$	18.6795	0.707029
$699$	16.3923	0.620014
$700$	0	0
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	−27.7128	−1.04595
$703$	−0.784610	−0.0295921
$704$	4.73205	0.178346
$705$	0	0
$706$	−46.3923	−1.74600
$707$	−25.8564	−0.972430
$708$	−6.92820	−0.260378
$709$	4.53590	0.170349	0.0851746	−	0.996366i	$-0.472855\pi$
$709$	4.53590	0.170349	0.0851746	+	0.996366i	$0.472855\pi$
$710$	0	0
$711$	65.1244	2.44235
$712$	−7.60770	−0.285110
$713$	26.7846	1.00309
$714$	−12.9282	−0.483826
$715$	0	0
$716$	−11.3205	−0.423067
$717$	56.7846	2.12066
$718$	−37.1769	−1.38743
$719$	−5.41154	−0.201816	−0.100908	−	0.994896i	$-0.532175\pi$
$719$	−5.41154	−0.201816	−0.100908	+	0.994896i	$0.532175\pi$
$720$	0	0
$721$	−24.3923	−0.908417
$722$	−29.1962	−1.08657
$723$	15.3205	0.569776
$724$	−2.39230	−0.0889093
$725$	0	0
$726$	−53.9090	−2.00075
$727$	−0.143594	−0.00532559	−0.00266279	−	0.999996i	$-0.500848\pi$
$727$	−0.143594	−0.00532559	−0.00266279	+	0.999996i	$0.500848\pi$
$728$	−18.9282	−0.701526
$729$	−43.7846	−1.62165
$730$	0	0
$731$	0.535898	0.0198209
$732$	13.4641	0.497648
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	13.5167	0.498909
$735$	0	0
$736$	−42.5885	−1.56983
$737$	47.3205	1.74307
$738$	−26.7846	−0.985955
$739$	−11.6077	−0.426996	−0.213498	−	0.976944i	$-0.568486\pi$
$739$	−11.6077	−0.426996	−0.213498	+	0.976944i	$0.568486\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	−28.3923	−1.04231
$743$	22.7321	0.833958	0.416979	−	0.908916i	$-0.363089\pi$
$743$	22.7321	0.833958	0.416979	+	0.908916i	$0.363089\pi$
$744$	15.4641	0.566941
$745$	0	0
$746$	−34.6410	−1.26830
$747$	−38.1051	−1.39419
$748$	4.73205	0.173021
$749$	−48.2487	−1.76297
$750$	0	0
$751$	43.6603	1.59319	0.796593	−	0.604516i	$-0.206634\pi$
$751$	43.6603	1.59319	0.796593	+	0.604516i	$0.206634\pi$
$752$	64.6410	2.35722
$753$	−18.9282	−0.689782
$754$	−24.0000	−0.874028
$755$	0	0
$756$	−10.9282	−0.397455
$757$	−30.6410	−1.11367	−0.556833	−	0.830624i	$-0.687984\pi$
$757$	−30.6410	−1.11367	−0.556833	+	0.830624i	$0.687984\pi$
$758$	30.8372	1.12006
$759$	−105.962	−3.84616
$760$	0	0
$761$	−28.3923	−1.02922	−0.514610	−	0.857424i	$-0.672063\pi$
$761$	−28.3923	−1.02922	−0.514610	+	0.857424i	$0.672063\pi$
$762$	−68.1051	−2.46719
$763$	−27.3205	−0.989069
$764$	1.85641	0.0671624
$765$	0	0
$766$	−26.7846	−0.967767
$767$	10.1436	0.366264
$768$	−51.9090	−1.87310
$769$	36.3923	1.31234	0.656170	−	0.754613i	$-0.272175\pi$
$769$	36.3923	1.31234	0.656170	+	0.754613i	$0.272175\pi$
$770$	0	0
$771$	−18.9282	−0.681683
$772$	−16.5359	−0.595140
$773$	46.6410	1.67756	0.838780	−	0.544470i	$-0.183269\pi$
$773$	46.6410	1.67756	0.838780	+	0.544470i	$0.183269\pi$
$774$	4.14359	0.148938
$775$	0	0
$776$	−8.53590	−0.306421
$777$	−4.00000	−0.143499
$778$	−7.60770	−0.272749
$779$	5.07180	0.181716
$780$	0	0
$781$	55.1769	1.97439
$782$	14.1962	0.507653
$783$	13.8564	0.495188
$784$	−2.32051	−0.0828753
$785$	0	0
$786$	10.3923	0.370681
$787$	−43.9090	−1.56519	−0.782593	−	0.622534i	$-0.786103\pi$
$787$	−43.9090	−1.56519	−0.782593	+	0.622534i	$0.786103\pi$
$788$	−17.3205	−0.617018
$789$	−4.39230	−0.156370
$790$	0	0
$791$	47.3205	1.68252
$792$	−36.5885	−1.30011
$793$	−19.7128	−0.700023
$794$	−24.2487	−0.860555
$795$	0	0
$796$	10.1962	0.361393
$797$	24.9282	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$797$	24.9282	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$798$	18.9282	0.670051
$799$	−12.9282	−0.457367
$800$	0	0
$801$	19.6077	0.692804
$802$	22.3923	0.790700
$803$	−30.2487	−1.06745
$804$	−27.3205	−0.963520
$805$	0	0
$806$	22.6410	0.797496
$807$	−2.53590	−0.0892679
$808$	16.3923	0.576679
$809$	−55.8564	−1.96381	−0.981903	−	0.189383i	$-0.939351\pi$
$809$	−55.8564	−1.96381	−0.981903	+	0.189383i	$0.939351\pi$
$810$	0	0
$811$	44.8372	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$811$	44.8372	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$812$	−9.46410	−0.332125
$813$	−8.00000	−0.280572
$814$	4.39230	0.153950
$815$	0	0
$816$	13.6603	0.478205
$817$	−0.784610	−0.0274500
$818$	45.0333	1.57455
$819$	48.7846	1.70467
$820$	0	0
$821$	−24.9282	−0.870000	−0.435000	−	0.900430i	$-0.643252\pi$
$821$	−24.9282	−0.870000	−0.435000	+	0.900430i	$0.643252\pi$
$822$	0	0
$823$	8.98076	0.313050	0.156525	−	0.987674i	$-0.449971\pi$
$823$	8.98076	0.313050	0.156525	+	0.987674i	$0.449971\pi$
$824$	15.4641	0.538718
$825$	0	0
$826$	12.0000	0.417533
$827$	15.8038	0.549554	0.274777	−	0.961508i	$-0.411396\pi$
$827$	15.8038	0.549554	0.274777	+	0.961508i	$0.411396\pi$
$828$	36.5885	1.27154
$829$	17.7128	0.615191	0.307596	−	0.951517i	$-0.400476\pi$
$829$	17.7128	0.615191	0.307596	+	0.951517i	$0.400476\pi$
$830$	0	0
$831$	57.1769	1.98345
$832$	4.00000	0.138675
$833$	0.464102	0.0160802
$834$	17.3205	0.599760
$835$	0	0
$836$	−6.92820	−0.239617
$837$	−13.0718	−0.451827
$838$	66.1577	2.28538
$839$	19.2679	0.665203	0.332602	−	0.943067i	$-0.392074\pi$
$839$	19.2679	0.665203	0.332602	+	0.943067i	$0.392074\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	9.46410	0.326154
$843$	35.3205	1.21650
$844$	10.1962	0.350966
$845$	0	0
$846$	−99.9615	−3.43675
$847$	31.1244	1.06945
$848$	30.0000	1.03020
$849$	−14.3923	−0.493943
$850$	0	0
$851$	4.39230	0.150566
$852$	−31.8564	−1.09138
$853$	23.1769	0.793562	0.396781	−	0.917913i	$-0.370127\pi$
$853$	23.1769	0.793562	0.396781	+	0.917913i	$0.370127\pi$
$854$	−23.3205	−0.798011
$855$	0	0
$856$	30.5885	1.04549
$857$	−31.1769	−1.06498	−0.532492	−	0.846435i	$-0.678744\pi$
$857$	−31.1769	−1.06498	−0.532492	+	0.846435i	$0.678744\pi$
$858$	−89.5692	−3.05784
$859$	−25.4641	−0.868824	−0.434412	−	0.900714i	$-0.643044\pi$
$859$	−25.4641	−0.868824	−0.434412	+	0.900714i	$0.643044\pi$
$860$	0	0
$861$	25.8564	0.881184
$862$	16.9808	0.578367
$863$	−23.0718	−0.785373	−0.392687	−	0.919672i	$-0.628454\pi$
$863$	−23.0718	−0.785373	−0.392687	+	0.919672i	$0.628454\pi$
$864$	20.7846	0.707107
$865$	0	0
$866$	30.9282	1.05098
$867$	−2.73205	−0.0927853
$868$	8.92820	0.303043
$869$	69.0333	2.34180
$870$	0	0
$871$	40.0000	1.35535
$872$	17.3205	0.586546
$873$	22.0000	0.744587
$874$	−20.7846	−0.703050
$875$	0	0
$876$	17.4641	0.590057
$877$	1.21539	0.0410408	0.0205204	−	0.999789i	$-0.493468\pi$
$877$	1.21539	0.0410408	0.0205204	+	0.999789i	$0.493468\pi$
$878$	−34.7321	−1.17215
$879$	−2.53590	−0.0855337
$880$	0	0
$881$	−41.3205	−1.39212	−0.696062	−	0.717982i	$-0.745066\pi$
$881$	−41.3205	−1.39212	−0.696062	+	0.717982i	$0.745066\pi$
$882$	3.58846	0.120830
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	22.3923	0.752284
$887$	−3.12436	−0.104906	−0.0524528	−	0.998623i	$-0.516704\pi$
$887$	−3.12436	−0.104906	−0.0524528	+	0.998623i	$0.516704\pi$
$888$	2.53590	0.0850992
$889$	39.3205	1.31877
$890$	0	0
$891$	−11.6603	−0.390633
$892$	26.3923	0.883680
$893$	18.9282	0.633408
$894$	28.3923	0.949581
$895$	0	0
$896$	33.1244	1.10661
$897$	−89.5692	−2.99063
$898$	−59.5692	−1.98785
$899$	−11.3205	−0.377560
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−28.3923	−0.945360
$903$	−4.00000	−0.133112
$904$	−30.0000	−0.997785
$905$	0	0
$906$	6.92820	0.230174
$907$	−30.7321	−1.02044	−0.510220	−	0.860044i	$-0.670436\pi$
$907$	−30.7321	−1.02044	−0.510220	+	0.860044i	$0.670436\pi$
$908$	−22.7321	−0.754390
$909$	−42.2487	−1.40130
$910$	0	0
$911$	36.3397	1.20399	0.601995	−	0.798500i	$-0.294373\pi$
$911$	36.3397	1.20399	0.601995	+	0.798500i	$0.294373\pi$
$912$	−20.0000	−0.662266
$913$	−40.3923	−1.33679
$914$	63.7128	2.10743
$915$	0	0
$916$	−8.39230	−0.277290
$917$	−6.00000	−0.198137
$918$	−6.92820	−0.228665
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−27.3205	−0.900241
$922$	−43.1769	−1.42196
$923$	46.6410	1.53521
$924$	−35.3205	−1.16196
$925$	0	0
$926$	41.3205	1.35788
$927$	−39.8564	−1.30906
$928$	18.0000	0.590879
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	−0.679492	−0.0222694
$932$	−6.00000	−0.196537
$933$	43.8564	1.43579
$934$	−2.78461	−0.0911152
$935$	0	0
$936$	−30.9282	−1.01092
$937$	32.6410	1.06634	0.533168	−	0.846010i	$-0.321001\pi$
$937$	32.6410	1.06634	0.533168	+	0.846010i	$0.321001\pi$
$938$	47.3205	1.54507
$939$	−72.1051	−2.35306
$940$	0	0
$941$	48.2487	1.57286	0.786432	−	0.617677i	$-0.211926\pi$
$941$	48.2487	1.57286	0.786432	+	0.617677i	$0.211926\pi$
$942$	42.2487	1.37654
$943$	−28.3923	−0.924581
$944$	−12.6795	−0.412682
$945$	0	0
$946$	4.39230	0.142806
$947$	−8.19615	−0.266339	−0.133170	−	0.991093i	$-0.542515\pi$
$947$	−8.19615	−0.266339	−0.133170	+	0.991093i	$0.542515\pi$
$948$	−39.8564	−1.29448
$949$	−25.5692	−0.830012
$950$	0	0
$951$	−68.1051	−2.20846
$952$	−4.73205	−0.153367
$953$	8.78461	0.284561	0.142281	−	0.989826i	$-0.454556\pi$
$953$	8.78461	0.284561	0.142281	+	0.989826i	$0.454556\pi$
$954$	−46.3923	−1.50201
$955$	0	0
$956$	−20.7846	−0.672222
$957$	44.7846	1.44768
$958$	−20.1962	−0.652508
$959$	0	0
$960$	0	0
$961$	−20.3205	−0.655500
$962$	3.71281	0.119706
$963$	−78.8372	−2.54049
$964$	−5.60770	−0.180612
$965$	0	0
$966$	−105.962	−3.40926
$967$	−39.1769	−1.25984	−0.629922	−	0.776658i	$-0.716913\pi$
$967$	−39.1769	−1.25984	−0.629922	+	0.776658i	$0.716913\pi$
$968$	−19.7321	−0.634212
$969$	4.00000	0.128499
$970$	0	0
$971$	54.2487	1.74092	0.870462	−	0.492236i	$-0.163820\pi$
$971$	54.2487	1.74092	0.870462	+	0.492236i	$0.163820\pi$
$972$	18.7321	0.600831
$973$	−10.0000	−0.320585
$974$	−43.2679	−1.38639
$975$	0	0
$976$	24.6410	0.788740
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−0.928203	−0.0296807
$979$	20.7846	0.664279
$980$	0	0
$981$	−44.6410	−1.42528
$982$	−33.9615	−1.08376
$983$	−58.7321	−1.87326	−0.936631	−	0.350318i	$-0.886073\pi$
$983$	−58.7321	−1.87326	−0.936631	+	0.350318i	$0.886073\pi$
$984$	−16.3923	−0.522568
$985$	0	0
$986$	−6.00000	−0.191079
$987$	96.4974	3.07155
$988$	−5.85641	−0.186317
$989$	4.39230	0.139667
$990$	0	0
$991$	−2.98076	−0.0946870	−0.0473435	−	0.998879i	$-0.515076\pi$
$991$	−2.98076	−0.0946870	−0.0473435	+	0.998879i	$0.515076\pi$
$992$	−16.9808	−0.539140
$993$	17.8564	0.566656
$994$	55.1769	1.75011
$995$	0	0
$996$	23.3205	0.738939
$997$	2.39230	0.0757651	0.0378825	−	0.999282i	$-0.487939\pi$
$997$	2.39230	0.0757651	0.0378825	+	0.999282i	$0.487939\pi$
$998$	−27.1244	−0.858606
$999$	−2.14359	−0.0678203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.e.1.2		2
3.2	odd	2		3825.2.a.v.1.1		2
4.3	odd	2		6800.2.a.bg.1.2		2
5.2	odd	4		425.2.b.d.324.4		4
5.3	odd	4		425.2.b.d.324.1		4
5.4	even	2		85.2.a.c.1.1	✓	2
15.14	odd	2		765.2.a.g.1.2		2
17.16	even	2		7225.2.a.l.1.2		2
20.19	odd	2		1360.2.a.k.1.1		2
35.34	odd	2		4165.2.a.t.1.1		2
40.19	odd	2		5440.2.a.bl.1.2		2
40.29	even	2		5440.2.a.bb.1.1		2
85.4	even	4		1445.2.d.e.866.3		4
85.64	even	4		1445.2.d.e.866.4		4
85.84	even	2		1445.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.1	✓	2	5.4	even	2
425.2.a.e.1.2		2	1.1	even	1	trivial
425.2.b.d.324.1		4	5.3	odd	4
425.2.b.d.324.4		4	5.2	odd	4
765.2.a.g.1.2		2	15.14	odd	2
1360.2.a.k.1.1		2	20.19	odd	2
1445.2.a.g.1.1		2	85.84	even	2
1445.2.d.e.866.3		4	85.4	even	4
1445.2.d.e.866.4		4	85.64	even	4
3825.2.a.v.1.1		2	3.2	odd	2
4165.2.a.t.1.1		2	35.34	odd	2
5440.2.a.bb.1.1		2	40.29	even	2
5440.2.a.bl.1.2		2	40.19	odd	2
6800.2.a.bg.1.2		2	4.3	odd	2
7225.2.a.l.1.2		2	17.16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} -4.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +4.46410 q^{9} +4.73205 q^{11} -2.73205 q^{12} +4.00000 q^{13} +4.73205 q^{14} -5.00000 q^{16} +1.00000 q^{17} +7.73205 q^{18} -1.46410 q^{19} -7.46410 q^{21} +8.19615 q^{22} +8.19615 q^{23} +4.73205 q^{24} +6.92820 q^{26} -4.00000 q^{27} +2.73205 q^{28} -3.46410 q^{29} +3.26795 q^{31} -5.19615 q^{32} -12.9282 q^{33} +1.73205 q^{34} +4.46410 q^{36} +0.535898 q^{37} -2.53590 q^{38} -10.9282 q^{39} -3.46410 q^{41} -12.9282 q^{42} +0.535898 q^{43} +4.73205 q^{44} +14.1962 q^{46} -12.9282 q^{47} +13.6603 q^{48} +0.464102 q^{49} -2.73205 q^{51} +4.00000 q^{52} -6.00000 q^{53} -6.92820 q^{54} -4.73205 q^{56} +4.00000 q^{57} -6.00000 q^{58} +2.53590 q^{59} -4.92820 q^{61} +5.66025 q^{62} +12.1962 q^{63} +1.00000 q^{64} -22.3923 q^{66} +10.0000 q^{67} +1.00000 q^{68} -22.3923 q^{69} +11.6603 q^{71} -7.73205 q^{72} -6.39230 q^{73} +0.928203 q^{74} -1.46410 q^{76} +12.9282 q^{77} -18.9282 q^{78} +14.5885 q^{79} -2.46410 q^{81} -6.00000 q^{82} -8.53590 q^{83} -7.46410 q^{84} +0.928203 q^{86} +9.46410 q^{87} -8.19615 q^{88} +4.39230 q^{89} +10.9282 q^{91} +8.19615 q^{92} -8.92820 q^{93} -22.3923 q^{94} +14.1962 q^{96} +4.92820 q^{97} +0.803848 q^{98} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 8 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{17} + 12 q^{18} + 4 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 6 q^{24} - 8 q^{27} + 2 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 8 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^17 + 12 * q^18 + 4 * q^19 - 8 * q^21 + 6 * q^22 + 6 * q^23 + 6 * q^24 - 8 * q^27 + 2 * q^28 + 10 * q^31 - 12 * q^33 + 2 * q^36 + 8 * q^37 - 12 * q^38 - 8 * q^39 - 12 * q^42 + 8 * q^43 + 6 * q^44 + 18 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 2 * q^51 + 8 * q^52 - 12 * q^53 - 6 * q^56 + 8 * q^57 - 12 * q^58 + 12 * q^59 + 4 * q^61 - 6 * q^62 + 14 * q^63 + 2 * q^64 - 24 * q^66 + 20 * q^67 + 2 * q^68 - 24 * q^69 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 12 * q^74 + 4 * q^76 + 12 * q^77 - 24 * q^78 - 2 * q^79 + 2 * q^81 - 12 * q^82 - 24 * q^83 - 8 * q^84 - 12 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 24 * q^94 + 18 * q^96 - 4 * q^97 + 12 * q^98 + 18 * q^99

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