LMFDB - Embedded newform 520.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.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:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +0.449490 q^{11} -1.00000 q^{13} +2.44949 q^{15} -2.89898 q^{17} -4.44949 q^{19} +4.89898 q^{21} +1.55051 q^{23} +1.00000 q^{25} +4.00000 q^{29} +0.449490 q^{31} +1.10102 q^{33} +2.00000 q^{35} -4.89898 q^{37} -2.44949 q^{39} +1.10102 q^{41} -3.34847 q^{43} +3.00000 q^{45} -2.00000 q^{47} -3.00000 q^{49} -7.10102 q^{51} +10.8990 q^{53} +0.449490 q^{55} -10.8990 q^{57} -5.34847 q^{59} +13.7980 q^{61} +6.00000 q^{63} -1.00000 q^{65} -14.8990 q^{67} +3.79796 q^{69} -8.44949 q^{71} +14.6969 q^{73} +2.44949 q^{75} +0.898979 q^{77} -4.89898 q^{79} -9.00000 q^{81} +2.00000 q^{83} -2.89898 q^{85} +9.79796 q^{87} +6.00000 q^{89} -2.00000 q^{91} +1.10102 q^{93} -4.44949 q^{95} -11.7980 q^{97} +1.34847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9	$ 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} + 12 q^{33} + 4 q^{35} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 24 q^{51} + 12 q^{53} - 4 q^{55} - 12 q^{57} + 4 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{77} - 18 q^{81} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 4 q^{91} + 12 q^{93} - 4 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 + 12 * q^33 + 4 * q^35 + 12 * q^41 + 8 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 24 * q^51 + 12 * q^53 - 4 * q^55 - 12 * q^57 + 4 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^77 - 18 * q^81 + 4 * q^83 + 4 * q^85 + 12 * q^89 - 4 * q^91 + 12 * q^93 - 4 * q^95 - 4 * q^97 - 12 * 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	0
$2$	0	0
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	0.449490	0.135526	0.0677631	−	0.997701i	$-0.478414\pi$
$11$	0.449490	0.135526	0.0677631	+	0.997701i	$0.478414\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.44949	0.632456
$16$	0	0
$17$	−2.89898	−0.703106	−0.351553	−	0.936168i	$-0.614346\pi$
$17$	−2.89898	−0.703106	−0.351553	+	0.936168i	$0.614346\pi$
$18$	0	0
$19$	−4.44949	−1.02078	−0.510391	−	0.859942i	$-0.670499\pi$
$19$	−4.44949	−1.02078	−0.510391	+	0.859942i	$0.670499\pi$
$20$	0	0
$21$	4.89898	1.06904
$22$	0	0
$23$	1.55051	0.323304	0.161652	−	0.986848i	$-0.448318\pi$
$23$	1.55051	0.323304	0.161652	+	0.986848i	$0.448318\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0.449490	0.0807307	0.0403654	−	0.999185i	$-0.487148\pi$
$31$	0.449490	0.0807307	0.0403654	+	0.999185i	$0.487148\pi$
$32$	0	0
$33$	1.10102	0.191663
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−4.89898	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$38$	0	0
$39$	−2.44949	−0.392232
$40$	0	0
$41$	1.10102	0.171951	0.0859753	−	0.996297i	$-0.472599\pi$
$41$	1.10102	0.171951	0.0859753	+	0.996297i	$0.472599\pi$
$42$	0	0
$43$	−3.34847	−0.510637	−0.255318	−	0.966857i	$-0.582180\pi$
$43$	−3.34847	−0.510637	−0.255318	+	0.966857i	$0.582180\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−7.10102	−0.994342
$52$	0	0
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	0	0
$55$	0.449490	0.0606092
$56$	0	0
$57$	−10.8990	−1.44361
$58$	0	0
$59$	−5.34847	−0.696311	−0.348156	−	0.937437i	$-0.613192\pi$
$59$	−5.34847	−0.696311	−0.348156	+	0.937437i	$0.613192\pi$
$60$	0	0
$61$	13.7980	1.76665	0.883324	−	0.468763i	$-0.155300\pi$
$61$	13.7980	1.76665	0.883324	+	0.468763i	$0.155300\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−14.8990	−1.82020	−0.910100	−	0.414389i	$-0.863995\pi$
$67$	−14.8990	−1.82020	−0.910100	+	0.414389i	$0.863995\pi$
$68$	0	0
$69$	3.79796	0.457221
$70$	0	0
$71$	−8.44949	−1.00277	−0.501385	−	0.865224i	$-0.667176\pi$
$71$	−8.44949	−1.00277	−0.501385	+	0.865224i	$0.667176\pi$
$72$	0	0
$73$	14.6969	1.72015	0.860073	−	0.510171i	$-0.170418\pi$
$73$	14.6969	1.72015	0.860073	+	0.510171i	$0.170418\pi$
$74$	0	0
$75$	2.44949	0.282843
$76$	0	0
$77$	0.898979	0.102448
$78$	0	0
$79$	−4.89898	−0.551178	−0.275589	−	0.961276i	$-0.588873\pi$
$79$	−4.89898	−0.551178	−0.275589	+	0.961276i	$0.588873\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−2.89898	−0.314438
$86$	0	0
$87$	9.79796	1.05045
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	1.10102	0.114171
$94$	0	0
$95$	−4.44949	−0.456508
$96$	0	0
$97$	−11.7980	−1.19790	−0.598951	−	0.800786i	$-0.704415\pi$
$97$	−11.7980	−1.19790	−0.598951	+	0.800786i	$0.704415\pi$
$98$	0	0
$99$	1.34847	0.135526
$100$	0	0
$101$	3.79796	0.377911	0.188956	−	0.981986i	$-0.439490\pi$
$101$	3.79796	0.377911	0.188956	+	0.981986i	$0.439490\pi$
$102$	0	0
$103$	−3.34847	−0.329934	−0.164967	−	0.986299i	$-0.552752\pi$
$103$	−3.34847	−0.329934	−0.164967	+	0.986299i	$0.552752\pi$
$104$	0	0
$105$	4.89898	0.478091
$106$	0	0
$107$	12.2474	1.18401	0.592003	−	0.805936i	$-0.298337\pi$
$107$	12.2474	1.18401	0.592003	+	0.805936i	$0.298337\pi$
$108$	0	0
$109$	−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
$111$	−12.0000	−1.13899
$112$	0	0
$113$	10.8990	1.02529	0.512645	−	0.858601i	$-0.328666\pi$
$113$	10.8990	1.02529	0.512645	+	0.858601i	$0.328666\pi$
$114$	0	0
$115$	1.55051	0.144586
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	−5.79796	−0.531498
$120$	0	0
$121$	−10.7980	−0.981633
$122$	0	0
$123$	2.69694	0.243175
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.3485	1.71690	0.858450	−	0.512898i	$-0.171428\pi$
$127$	19.3485	1.71690	0.858450	+	0.512898i	$0.171428\pi$
$128$	0	0
$129$	−8.20204	−0.722149
$130$	0	0
$131$	−19.5959	−1.71210	−0.856052	−	0.516890i	$-0.827090\pi$
$131$	−19.5959	−1.71210	−0.856052	+	0.516890i	$0.827090\pi$
$132$	0	0
$133$	−8.89898	−0.771639
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−7.10102	−0.602301	−0.301150	−	0.953577i	$-0.597371\pi$
$139$	−7.10102	−0.602301	−0.301150	+	0.953577i	$0.597371\pi$
$140$	0	0
$141$	−4.89898	−0.412568
$142$	0	0
$143$	−0.449490	−0.0375882
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−7.34847	−0.606092
$148$	0	0
$149$	−7.79796	−0.638834	−0.319417	−	0.947614i	$-0.603487\pi$
$149$	−7.79796	−0.638834	−0.319417	+	0.947614i	$0.603487\pi$
$150$	0	0
$151$	−17.3485	−1.41180	−0.705899	−	0.708312i	$-0.749457\pi$
$151$	−17.3485	−1.41180	−0.705899	+	0.708312i	$0.749457\pi$
$152$	0	0
$153$	−8.69694	−0.703106
$154$	0	0
$155$	0.449490	0.0361039
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	26.6969	2.11720
$160$	0	0
$161$	3.10102	0.244395
$162$	0	0
$163$	−18.8990	−1.48028	−0.740141	−	0.672452i	$-0.765241\pi$
$163$	−18.8990	−1.48028	−0.740141	+	0.672452i	$0.765241\pi$
$164$	0	0
$165$	1.10102	0.0857143
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−13.3485	−1.02078
$172$	0	0
$173$	−2.89898	−0.220405	−0.110203	−	0.993909i	$-0.535150\pi$
$173$	−2.89898	−0.220405	−0.110203	+	0.993909i	$0.535150\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−13.1010	−0.984733
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	−13.7980	−1.02559	−0.512797	−	0.858510i	$-0.671391\pi$
$181$	−13.7980	−1.02559	−0.512797	+	0.858510i	$0.671391\pi$
$182$	0	0
$183$	33.7980	2.49842
$184$	0	0
$185$	−4.89898	−0.360180
$186$	0	0
$187$	−1.30306	−0.0952893
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.7980	1.28782	0.643908	−	0.765103i	$-0.277312\pi$
$191$	17.7980	1.28782	0.643908	+	0.765103i	$0.277312\pi$
$192$	0	0
$193$	19.7980	1.42509	0.712544	−	0.701627i	$-0.247543\pi$
$193$	19.7980	1.42509	0.712544	+	0.701627i	$0.247543\pi$
$194$	0	0
$195$	−2.44949	−0.175412
$196$	0	0
$197$	−4.20204	−0.299383	−0.149692	−	0.988733i	$-0.547828\pi$
$197$	−4.20204	−0.299383	−0.149692	+	0.988733i	$0.547828\pi$
$198$	0	0
$199$	−13.7980	−0.978111	−0.489056	−	0.872253i	$-0.662659\pi$
$199$	−13.7980	−0.978111	−0.489056	+	0.872253i	$0.662659\pi$
$200$	0	0
$201$	−36.4949	−2.57415
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	1.10102	0.0768986
$206$	0	0
$207$	4.65153	0.323304
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−6.20204	−0.426966	−0.213483	−	0.976947i	$-0.568481\pi$
$211$	−6.20204	−0.426966	−0.213483	+	0.976947i	$0.568481\pi$
$212$	0	0
$213$	−20.6969	−1.41813
$214$	0	0
$215$	−3.34847	−0.228364
$216$	0	0
$217$	0.898979	0.0610267
$218$	0	0
$219$	36.0000	2.43265
$220$	0	0
$221$	2.89898	0.195006
$222$	0	0
$223$	21.5959	1.44617	0.723085	−	0.690759i	$-0.242724\pi$
$223$	21.5959	1.44617	0.723085	+	0.690759i	$0.242724\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	13.1010	0.869545	0.434773	−	0.900540i	$-0.356829\pi$
$227$	13.1010	0.869545	0.434773	+	0.900540i	$0.356829\pi$
$228$	0	0
$229$	−9.10102	−0.601412	−0.300706	−	0.953717i	$-0.597222\pi$
$229$	−9.10102	−0.601412	−0.300706	+	0.953717i	$0.597222\pi$
$230$	0	0
$231$	2.20204	0.144884
$232$	0	0
$233$	13.5959	0.890698	0.445349	−	0.895357i	$-0.353080\pi$
$233$	13.5959	0.890698	0.445349	+	0.895357i	$0.353080\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	23.1464	1.49722	0.748609	−	0.663012i	$-0.230722\pi$
$239$	23.1464	1.49722	0.748609	+	0.663012i	$0.230722\pi$
$240$	0	0
$241$	28.6969	1.84853	0.924266	−	0.381749i	$-0.124678\pi$
$241$	28.6969	1.84853	0.924266	+	0.381749i	$0.124678\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	4.44949	0.283114
$248$	0	0
$249$	4.89898	0.310460
$250$	0	0
$251$	14.6969	0.927663	0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969	0.927663	0.463831	+	0.885924i	$0.346474\pi$
$252$	0	0
$253$	0.696938	0.0438161
$254$	0	0
$255$	−7.10102	−0.444683
$256$	0	0
$257$	15.7980	0.985450	0.492725	−	0.870185i	$-0.336001\pi$
$257$	15.7980	0.985450	0.492725	+	0.870185i	$0.336001\pi$
$258$	0	0
$259$	−9.79796	−0.608816
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	21.5505	1.32886	0.664431	−	0.747350i	$-0.268674\pi$
$263$	21.5505	1.32886	0.664431	+	0.747350i	$0.268674\pi$
$264$	0	0
$265$	10.8990	0.669519
$266$	0	0
$267$	14.6969	0.899438
$268$	0	0
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	8.44949	0.513270	0.256635	−	0.966508i	$-0.417386\pi$
$271$	8.44949	0.513270	0.256635	+	0.966508i	$0.417386\pi$
$272$	0	0
$273$	−4.89898	−0.296500
$274$	0	0
$275$	0.449490	0.0271053
$276$	0	0
$277$	26.4949	1.59192	0.795962	−	0.605347i	$-0.206965\pi$
$277$	26.4949	1.59192	0.795962	+	0.605347i	$0.206965\pi$
$278$	0	0
$279$	1.34847	0.0807307
$280$	0	0
$281$	2.89898	0.172939	0.0864693	−	0.996255i	$-0.472442\pi$
$281$	2.89898	0.172939	0.0864693	+	0.996255i	$0.472442\pi$
$282$	0	0
$283$	−1.55051	−0.0921683	−0.0460841	−	0.998938i	$-0.514674\pi$
$283$	−1.55051	−0.0921683	−0.0460841	+	0.998938i	$0.514674\pi$
$284$	0	0
$285$	−10.8990	−0.645600
$286$	0	0
$287$	2.20204	0.129982
$288$	0	0
$289$	−8.59592	−0.505642
$290$	0	0
$291$	−28.8990	−1.69409
$292$	0	0
$293$	−20.4949	−1.19732	−0.598662	−	0.801001i	$-0.704301\pi$
$293$	−20.4949	−1.19732	−0.598662	+	0.801001i	$0.704301\pi$
$294$	0	0
$295$	−5.34847	−0.311400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.55051	−0.0896683
$300$	0	0
$301$	−6.69694	−0.386005
$302$	0	0
$303$	9.30306	0.534447
$304$	0	0
$305$	13.7980	0.790069
$306$	0	0
$307$	−11.7980	−0.673345	−0.336673	−	0.941622i	$-0.609302\pi$
$307$	−11.7980	−0.673345	−0.336673	+	0.941622i	$0.609302\pi$
$308$	0	0
$309$	−8.20204	−0.466598
$310$	0	0
$311$	12.8990	0.731434	0.365717	−	0.930726i	$-0.380824\pi$
$311$	12.8990	0.731434	0.365717	+	0.930726i	$0.380824\pi$
$312$	0	0
$313$	26.4949	1.49758	0.748790	−	0.662807i	$-0.230635\pi$
$313$	26.4949	1.49758	0.748790	+	0.662807i	$0.230635\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	−2.69694	−0.151475	−0.0757376	−	0.997128i	$-0.524131\pi$
$317$	−2.69694	−0.151475	−0.0757376	+	0.997128i	$0.524131\pi$
$318$	0	0
$319$	1.79796	0.100666
$320$	0	0
$321$	30.0000	1.67444
$322$	0	0
$323$	12.8990	0.717718
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−24.4949	−1.35457
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−8.44949	−0.464426	−0.232213	−	0.972665i	$-0.574597\pi$
$331$	−8.44949	−0.464426	−0.232213	+	0.972665i	$0.574597\pi$
$332$	0	0
$333$	−14.6969	−0.805387
$334$	0	0
$335$	−14.8990	−0.814018
$336$	0	0
$337$	12.6969	0.691646	0.345823	−	0.938300i	$-0.387600\pi$
$337$	12.6969	0.691646	0.345823	+	0.938300i	$0.387600\pi$
$338$	0	0
$339$	26.6969	1.44998
$340$	0	0
$341$	0.202041	0.0109411
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	3.79796	0.204475
$346$	0	0
$347$	−4.24745	−0.228015	−0.114007	−	0.993480i	$-0.536369\pi$
$347$	−4.24745	−0.228015	−0.114007	+	0.993480i	$0.536369\pi$
$348$	0	0
$349$	−8.69694	−0.465536	−0.232768	−	0.972532i	$-0.574778\pi$
$349$	−8.69694	−0.465536	−0.232768	+	0.972532i	$0.574778\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.4949	1.09083	0.545417	−	0.838165i	$-0.316371\pi$
$353$	20.4949	1.09083	0.545417	+	0.838165i	$0.316371\pi$
$354$	0	0
$355$	−8.44949	−0.448452
$356$	0	0
$357$	−14.2020	−0.751652
$358$	0	0
$359$	−30.2474	−1.59640	−0.798200	−	0.602393i	$-0.794214\pi$
$359$	−30.2474	−1.59640	−0.798200	+	0.602393i	$0.794214\pi$
$360$	0	0
$361$	0.797959	0.0419978
$362$	0	0
$363$	−26.4495	−1.38824
$364$	0	0
$365$	14.6969	0.769273
$366$	0	0
$367$	−18.9444	−0.988889	−0.494444	−	0.869209i	$-0.664628\pi$
$367$	−18.9444	−0.988889	−0.494444	+	0.869209i	$0.664628\pi$
$368$	0	0
$369$	3.30306	0.171951
$370$	0	0
$371$	21.7980	1.13169
$372$	0	0
$373$	−9.59592	−0.496858	−0.248429	−	0.968650i	$-0.579914\pi$
$373$	−9.59592	−0.496858	−0.248429	+	0.968650i	$0.579914\pi$
$374$	0	0
$375$	2.44949	0.126491
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−11.5505	−0.593310	−0.296655	−	0.954985i	$-0.595871\pi$
$379$	−11.5505	−0.593310	−0.296655	+	0.954985i	$0.595871\pi$
$380$	0	0
$381$	47.3939	2.42806
$382$	0	0
$383$	13.5959	0.694719	0.347359	−	0.937732i	$-0.387078\pi$
$383$	13.5959	0.694719	0.347359	+	0.937732i	$0.387078\pi$
$384$	0	0
$385$	0.898979	0.0458162
$386$	0	0
$387$	−10.0454	−0.510637
$388$	0	0
$389$	−37.5959	−1.90619	−0.953094	−	0.302673i	$-0.902121\pi$
$389$	−37.5959	−1.90619	−0.953094	+	0.302673i	$0.902121\pi$
$390$	0	0
$391$	−4.49490	−0.227317
$392$	0	0
$393$	−48.0000	−2.42128
$394$	0	0
$395$	−4.89898	−0.246494
$396$	0	0
$397$	1.30306	0.0653988	0.0326994	−	0.999465i	$-0.489590\pi$
$397$	1.30306	0.0653988	0.0326994	+	0.999465i	$0.489590\pi$
$398$	0	0
$399$	−21.7980	−1.09126
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−0.449490	−0.0223907
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−2.20204	−0.109151
$408$	0	0
$409$	2.89898	0.143345	0.0716727	−	0.997428i	$-0.477166\pi$
$409$	2.89898	0.143345	0.0716727	+	0.997428i	$0.477166\pi$
$410$	0	0
$411$	−34.2929	−1.69154
$412$	0	0
$413$	−10.6969	−0.526362
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	−17.3939	−0.851782
$418$	0	0
$419$	9.30306	0.454484	0.227242	−	0.973838i	$-0.427029\pi$
$419$	9.30306	0.454484	0.227242	+	0.973838i	$0.427029\pi$
$420$	0	0
$421$	15.7980	0.769945	0.384973	−	0.922928i	$-0.374211\pi$
$421$	15.7980	0.769945	0.384973	+	0.922928i	$0.374211\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−2.89898	−0.140621
$426$	0	0
$427$	27.5959	1.33546
$428$	0	0
$429$	−1.10102	−0.0531578
$430$	0	0
$431$	12.0454	0.580207	0.290103	−	0.956995i	$-0.406310\pi$
$431$	12.0454	0.580207	0.290103	+	0.956995i	$0.406310\pi$
$432$	0	0
$433$	−7.79796	−0.374746	−0.187373	−	0.982289i	$-0.559997\pi$
$433$	−7.79796	−0.374746	−0.187373	+	0.982289i	$0.559997\pi$
$434$	0	0
$435$	9.79796	0.469776
$436$	0	0
$437$	−6.89898	−0.330023
$438$	0	0
$439$	1.79796	0.0858119	0.0429059	−	0.999079i	$-0.486338\pi$
$439$	1.79796	0.0858119	0.0429059	+	0.999079i	$0.486338\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	22.0454	1.04741	0.523704	−	0.851900i	$-0.324550\pi$
$443$	22.0454	1.04741	0.523704	+	0.851900i	$0.324550\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−19.1010	−0.903447
$448$	0	0
$449$	8.69694	0.410434	0.205217	−	0.978717i	$-0.434210\pi$
$449$	8.69694	0.410434	0.205217	+	0.978717i	$0.434210\pi$
$450$	0	0
$451$	0.494897	0.0233038
$452$	0	0
$453$	−42.4949	−1.99658
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.30306	−0.153839	−0.0769195	−	0.997037i	$-0.524508\pi$
$461$	−3.30306	−0.153839	−0.0769195	+	0.997037i	$0.524508\pi$
$462$	0	0
$463$	−13.1010	−0.608856	−0.304428	−	0.952535i	$-0.598465\pi$
$463$	−13.1010	−0.608856	−0.304428	+	0.952535i	$0.598465\pi$
$464$	0	0
$465$	1.10102	0.0510586
$466$	0	0
$467$	−30.9444	−1.43194	−0.715968	−	0.698133i	$-0.754014\pi$
$467$	−30.9444	−1.43194	−0.715968	+	0.698133i	$0.754014\pi$
$468$	0	0
$469$	−29.7980	−1.37594
$470$	0	0
$471$	24.4949	1.12867
$472$	0	0
$473$	−1.50510	−0.0692047
$474$	0	0
$475$	−4.44949	−0.204157
$476$	0	0
$477$	32.6969	1.49709
$478$	0	0
$479$	5.34847	0.244378	0.122189	−	0.992507i	$-0.461009\pi$
$479$	5.34847	0.244378	0.122189	+	0.992507i	$0.461009\pi$
$480$	0	0
$481$	4.89898	0.223374
$482$	0	0
$483$	7.59592	0.345626
$484$	0	0
$485$	−11.7980	−0.535718
$486$	0	0
$487$	2.89898	0.131365	0.0656826	−	0.997841i	$-0.479078\pi$
$487$	2.89898	0.131365	0.0656826	+	0.997841i	$0.479078\pi$
$488$	0	0
$489$	−46.2929	−2.09344
$490$	0	0
$491$	34.2929	1.54761	0.773807	−	0.633421i	$-0.218350\pi$
$491$	34.2929	1.54761	0.773807	+	0.633421i	$0.218350\pi$
$492$	0	0
$493$	−11.5959	−0.522254
$494$	0	0
$495$	1.34847	0.0606092
$496$	0	0
$497$	−16.8990	−0.758023
$498$	0	0
$499$	21.3485	0.955689	0.477844	−	0.878445i	$-0.341418\pi$
$499$	21.3485	0.955689	0.477844	+	0.878445i	$0.341418\pi$
$500$	0	0
$501$	44.0908	1.96983
$502$	0	0
$503$	−26.9444	−1.20139	−0.600695	−	0.799478i	$-0.705110\pi$
$503$	−26.9444	−1.20139	−0.600695	+	0.799478i	$0.705110\pi$
$504$	0	0
$505$	3.79796	0.169007
$506$	0	0
$507$	2.44949	0.108786
$508$	0	0
$509$	26.8990	1.19228	0.596138	−	0.802882i	$-0.296701\pi$
$509$	26.8990	1.19228	0.596138	+	0.802882i	$0.296701\pi$
$510$	0	0
$511$	29.3939	1.30031
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.34847	−0.147551
$516$	0	0
$517$	−0.898979	−0.0395371
$518$	0	0
$519$	−7.10102	−0.311700
$520$	0	0
$521$	6.20204	0.271716	0.135858	−	0.990728i	$-0.456621\pi$
$521$	6.20204	0.271716	0.135858	+	0.990728i	$0.456621\pi$
$522$	0	0
$523$	−39.3485	−1.72059	−0.860294	−	0.509798i	$-0.829720\pi$
$523$	−39.3485	−1.72059	−0.860294	+	0.509798i	$0.829720\pi$
$524$	0	0
$525$	4.89898	0.213809
$526$	0	0
$527$	−1.30306	−0.0567623
$528$	0	0
$529$	−20.5959	−0.895475
$530$	0	0
$531$	−16.0454	−0.696311
$532$	0	0
$533$	−1.10102	−0.0476905
$534$	0	0
$535$	12.2474	0.529503
$536$	0	0
$537$	19.5959	0.845626
$538$	0	0
$539$	−1.34847	−0.0580827
$540$	0	0
$541$	5.10102	0.219310	0.109655	−	0.993970i	$-0.465025\pi$
$541$	5.10102	0.219310	0.109655	+	0.993970i	$0.465025\pi$
$542$	0	0
$543$	−33.7980	−1.45041
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	0.651531	0.0278574	0.0139287	−	0.999903i	$-0.495566\pi$
$547$	0.651531	0.0278574	0.0139287	+	0.999903i	$0.495566\pi$
$548$	0	0
$549$	41.3939	1.76665
$550$	0	0
$551$	−17.7980	−0.758219
$552$	0	0
$553$	−9.79796	−0.416652
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	−1.30306	−0.0552125	−0.0276062	−	0.999619i	$-0.508788\pi$
$557$	−1.30306	−0.0552125	−0.0276062	+	0.999619i	$0.508788\pi$
$558$	0	0
$559$	3.34847	0.141625
$560$	0	0
$561$	−3.19184	−0.134759
$562$	0	0
$563$	−7.75255	−0.326731	−0.163366	−	0.986566i	$-0.552235\pi$
$563$	−7.75255	−0.326731	−0.163366	+	0.986566i	$0.552235\pi$
$564$	0	0
$565$	10.8990	0.458524
$566$	0	0
$567$	−18.0000	−0.755929
$568$	0	0
$569$	10.2020	0.427692	0.213846	−	0.976867i	$-0.431401\pi$
$569$	10.2020	0.427692	0.213846	+	0.976867i	$0.431401\pi$
$570$	0	0
$571$	28.4949	1.19247	0.596237	−	0.802808i	$-0.296662\pi$
$571$	28.4949	1.19247	0.596237	+	0.802808i	$0.296662\pi$
$572$	0	0
$573$	43.5959	1.82125
$574$	0	0
$575$	1.55051	0.0646607
$576$	0	0
$577$	27.1010	1.12823	0.564115	−	0.825696i	$-0.309217\pi$
$577$	27.1010	1.12823	0.564115	+	0.825696i	$0.309217\pi$
$578$	0	0
$579$	48.4949	2.01538
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	4.89898	0.202895
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	−10.8990	−0.449849	−0.224925	−	0.974376i	$-0.572214\pi$
$587$	−10.8990	−0.449849	−0.224925	+	0.974376i	$0.572214\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−10.2929	−0.423392
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−5.79796	−0.237693
$596$	0	0
$597$	−33.7980	−1.38326
$598$	0	0
$599$	−28.8990	−1.18078	−0.590390	−	0.807118i	$-0.701026\pi$
$599$	−28.8990	−1.18078	−0.590390	+	0.807118i	$0.701026\pi$
$600$	0	0
$601$	−33.5959	−1.37041	−0.685203	−	0.728352i	$-0.740287\pi$
$601$	−33.5959	−1.37041	−0.685203	+	0.728352i	$0.740287\pi$
$602$	0	0
$603$	−44.6969	−1.82020
$604$	0	0
$605$	−10.7980	−0.438999
$606$	0	0
$607$	21.5505	0.874708	0.437354	−	0.899289i	$-0.355916\pi$
$607$	21.5505	0.874708	0.437354	+	0.899289i	$0.355916\pi$
$608$	0	0
$609$	19.5959	0.794067
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	45.1918	1.82528	0.912641	−	0.408763i	$-0.134040\pi$
$613$	45.1918	1.82528	0.912641	+	0.408763i	$0.134040\pi$
$614$	0	0
$615$	2.69694	0.108751
$616$	0	0
$617$	−35.3939	−1.42490	−0.712452	−	0.701721i	$-0.752415\pi$
$617$	−35.3939	−1.42490	−0.712452	+	0.701721i	$0.752415\pi$
$618$	0	0
$619$	−5.34847	−0.214973	−0.107487	−	0.994207i	$-0.534280\pi$
$619$	−5.34847	−0.214973	−0.107487	+	0.994207i	$0.534280\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.89898	−0.195646
$628$	0	0
$629$	14.2020	0.566272
$630$	0	0
$631$	−16.4495	−0.654844	−0.327422	−	0.944878i	$-0.606180\pi$
$631$	−16.4495	−0.654844	−0.327422	+	0.944878i	$0.606180\pi$
$632$	0	0
$633$	−15.1918	−0.603821
$634$	0	0
$635$	19.3485	0.767821
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	−25.3485	−1.00277
$640$	0	0
$641$	−35.3939	−1.39797	−0.698987	−	0.715134i	$-0.746366\pi$
$641$	−35.3939	−1.39797	−0.698987	+	0.715134i	$0.746366\pi$
$642$	0	0
$643$	−30.0000	−1.18308	−0.591542	−	0.806274i	$-0.701481\pi$
$643$	−30.0000	−1.18308	−0.591542	+	0.806274i	$0.701481\pi$
$644$	0	0
$645$	−8.20204	−0.322955
$646$	0	0
$647$	14.9444	0.587524	0.293762	−	0.955879i	$-0.405093\pi$
$647$	14.9444	0.587524	0.293762	+	0.955879i	$0.405093\pi$
$648$	0	0
$649$	−2.40408	−0.0943685
$650$	0	0
$651$	2.20204	0.0863048
$652$	0	0
$653$	−43.7980	−1.71395	−0.856973	−	0.515361i	$-0.827658\pi$
$653$	−43.7980	−1.71395	−0.856973	+	0.515361i	$0.827658\pi$
$654$	0	0
$655$	−19.5959	−0.765676
$656$	0	0
$657$	44.0908	1.72015
$658$	0	0
$659$	14.6969	0.572511	0.286256	−	0.958153i	$-0.407589\pi$
$659$	14.6969	0.572511	0.286256	+	0.958153i	$0.407589\pi$
$660$	0	0
$661$	39.7980	1.54796	0.773981	−	0.633209i	$-0.218263\pi$
$661$	39.7980	1.54796	0.773981	+	0.633209i	$0.218263\pi$
$662$	0	0
$663$	7.10102	0.275781
$664$	0	0
$665$	−8.89898	−0.345088
$666$	0	0
$667$	6.20204	0.240144
$668$	0	0
$669$	52.8990	2.04519
$670$	0	0
$671$	6.20204	0.239427
$672$	0	0
$673$	−1.10102	−0.0424412	−0.0212206	−	0.999775i	$-0.506755\pi$
$673$	−1.10102	−0.0424412	−0.0212206	+	0.999775i	$0.506755\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−40.6969	−1.56411	−0.782055	−	0.623209i	$-0.785829\pi$
$677$	−40.6969	−1.56411	−0.782055	+	0.623209i	$0.785829\pi$
$678$	0	0
$679$	−23.5959	−0.905528
$680$	0	0
$681$	32.0908	1.22972
$682$	0	0
$683$	−42.4949	−1.62602	−0.813011	−	0.582248i	$-0.802173\pi$
$683$	−42.4949	−1.62602	−0.813011	+	0.582248i	$0.802173\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	−22.2929	−0.850526
$688$	0	0
$689$	−10.8990	−0.415218
$690$	0	0
$691$	−14.6515	−0.557370	−0.278685	−	0.960382i	$-0.589899\pi$
$691$	−14.6515	−0.557370	−0.278685	+	0.960382i	$0.589899\pi$
$692$	0	0
$693$	2.69694	0.102448
$694$	0	0
$695$	−7.10102	−0.269357
$696$	0	0
$697$	−3.19184	−0.120899
$698$	0	0
$699$	33.3031	1.25964
$700$	0	0
$701$	−37.1918	−1.40472	−0.702358	−	0.711824i	$-0.747869\pi$
$701$	−37.1918	−1.40472	−0.702358	+	0.711824i	$0.747869\pi$
$702$	0	0
$703$	21.7980	0.822126
$704$	0	0
$705$	−4.89898	−0.184506
$706$	0	0
$707$	7.59592	0.285674
$708$	0	0
$709$	−32.2929	−1.21278	−0.606392	−	0.795166i	$-0.707384\pi$
$709$	−32.2929	−1.21278	−0.606392	+	0.795166i	$0.707384\pi$
$710$	0	0
$711$	−14.6969	−0.551178
$712$	0	0
$713$	0.696938	0.0261006
$714$	0	0
$715$	−0.449490	−0.0168100
$716$	0	0
$717$	56.6969	2.11739
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−6.69694	−0.249407
$722$	0	0
$723$	70.2929	2.61422
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	18.0454	0.669267	0.334634	−	0.942348i	$-0.391387\pi$
$727$	18.0454	0.669267	0.334634	+	0.942348i	$0.391387\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	9.70714	0.359032
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	−7.34847	−0.271052
$736$	0	0
$737$	−6.69694	−0.246685
$738$	0	0
$739$	−6.24745	−0.229816	−0.114908	−	0.993376i	$-0.536657\pi$
$739$	−6.24745	−0.229816	−0.114908	+	0.993376i	$0.536657\pi$
$740$	0	0
$741$	10.8990	0.400384
$742$	0	0
$743$	−35.3939	−1.29848	−0.649238	−	0.760586i	$-0.724912\pi$
$743$	−35.3939	−1.29848	−0.649238	+	0.760586i	$0.724912\pi$
$744$	0	0
$745$	−7.79796	−0.285695
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	24.4949	0.895024
$750$	0	0
$751$	31.1010	1.13489	0.567446	−	0.823410i	$-0.307931\pi$
$751$	31.1010	1.13489	0.567446	+	0.823410i	$0.307931\pi$
$752$	0	0
$753$	36.0000	1.31191
$754$	0	0
$755$	−17.3485	−0.631375
$756$	0	0
$757$	26.4949	0.962973	0.481487	−	0.876453i	$-0.340097\pi$
$757$	26.4949	0.962973	0.481487	+	0.876453i	$0.340097\pi$
$758$	0	0
$759$	1.70714	0.0619654
$760$	0	0
$761$	−10.4041	−0.377148	−0.188574	−	0.982059i	$-0.560386\pi$
$761$	−10.4041	−0.377148	−0.188574	+	0.982059i	$0.560386\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	−8.69694	−0.314438
$766$	0	0
$767$	5.34847	0.193122
$768$	0	0
$769$	39.3939	1.42058	0.710290	−	0.703909i	$-0.248564\pi$
$769$	39.3939	1.42058	0.710290	+	0.703909i	$0.248564\pi$
$770$	0	0
$771$	38.6969	1.39364
$772$	0	0
$773$	44.0908	1.58584	0.792918	−	0.609328i	$-0.208561\pi$
$773$	44.0908	1.58584	0.792918	+	0.609328i	$0.208561\pi$
$774$	0	0
$775$	0.449490	0.0161461
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	−4.89898	−0.175524
$780$	0	0
$781$	−3.79796	−0.135902
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	52.7878	1.87929
$790$	0	0
$791$	21.7980	0.775046
$792$	0	0
$793$	−13.7980	−0.489980
$794$	0	0
$795$	26.6969	0.946843
$796$	0	0
$797$	27.7980	0.984654	0.492327	−	0.870410i	$-0.336146\pi$
$797$	27.7980	0.984654	0.492327	+	0.870410i	$0.336146\pi$
$798$	0	0
$799$	5.79796	0.205117
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	6.60612	0.233125
$804$	0	0
$805$	3.10102	0.109297
$806$	0	0
$807$	63.6867	2.24188
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−53.8434	−1.89070	−0.945348	−	0.326063i	$-0.894278\pi$
$811$	−53.8434	−1.89070	−0.945348	+	0.326063i	$0.894278\pi$
$812$	0	0
$813$	20.6969	0.725873
$814$	0	0
$815$	−18.8990	−0.662002
$816$	0	0
$817$	14.8990	0.521249
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−53.1918	−1.85641	−0.928204	−	0.372072i	$-0.878648\pi$
$821$	−53.1918	−1.85641	−0.928204	+	0.372072i	$0.878648\pi$
$822$	0	0
$823$	30.4495	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$823$	30.4495	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$824$	0	0
$825$	1.10102	0.0383326
$826$	0	0
$827$	57.1918	1.98875	0.994377	−	0.105893i	$-0.0337702\pi$
$827$	57.1918	1.98875	0.994377	+	0.105893i	$0.0337702\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	64.8990	2.25132
$832$	0	0
$833$	8.69694	0.301331
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.5505	−0.674959	−0.337479	−	0.941333i	$-0.609574\pi$
$839$	−19.5505	−0.674959	−0.337479	+	0.941333i	$0.609574\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	7.10102	0.244572
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−21.5959	−0.742045
$848$	0	0
$849$	−3.79796	−0.130346
$850$	0	0
$851$	−7.59592	−0.260385
$852$	0	0
$853$	−18.6969	−0.640171	−0.320085	−	0.947389i	$-0.603712\pi$
$853$	−18.6969	−0.640171	−0.320085	+	0.947389i	$0.603712\pi$
$854$	0	0
$855$	−13.3485	−0.456508
$856$	0	0
$857$	35.3939	1.20903	0.604516	−	0.796593i	$-0.293367\pi$
$857$	35.3939	1.20903	0.604516	+	0.796593i	$0.293367\pi$
$858$	0	0
$859$	−30.6969	−1.04737	−0.523683	−	0.851913i	$-0.675442\pi$
$859$	−30.6969	−1.04737	−0.523683	+	0.851913i	$0.675442\pi$
$860$	0	0
$861$	5.39388	0.183823
$862$	0	0
$863$	23.3939	0.796337	0.398168	−	0.917312i	$-0.369646\pi$
$863$	23.3939	0.796337	0.398168	+	0.917312i	$0.369646\pi$
$864$	0	0
$865$	−2.89898	−0.0985683
$866$	0	0
$867$	−21.0556	−0.715086
$868$	0	0
$869$	−2.20204	−0.0746991
$870$	0	0
$871$	14.8990	0.504833
$872$	0	0
$873$	−35.3939	−1.19790
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	17.5959	0.594172	0.297086	−	0.954851i	$-0.403985\pi$
$877$	17.5959	0.594172	0.297086	+	0.954851i	$0.403985\pi$
$878$	0	0
$879$	−50.2020	−1.69327
$880$	0	0
$881$	9.79796	0.330102	0.165051	−	0.986285i	$-0.447221\pi$
$881$	9.79796	0.330102	0.165051	+	0.986285i	$0.447221\pi$
$882$	0	0
$883$	−11.3485	−0.381906	−0.190953	−	0.981599i	$-0.561158\pi$
$883$	−11.3485	−0.381906	−0.190953	+	0.981599i	$0.561158\pi$
$884$	0	0
$885$	−13.1010	−0.440386
$886$	0	0
$887$	4.24745	0.142615	0.0713077	−	0.997454i	$-0.477283\pi$
$887$	4.24745	0.142615	0.0713077	+	0.997454i	$0.477283\pi$
$888$	0	0
$889$	38.6969	1.29785
$890$	0	0
$891$	−4.04541	−0.135526
$892$	0	0
$893$	8.89898	0.297793
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	−3.79796	−0.126810
$898$	0	0
$899$	1.79796	0.0599653
$900$	0	0
$901$	−31.5959	−1.05261
$902$	0	0
$903$	−16.4041	−0.545894
$904$	0	0
$905$	−13.7980	−0.458660
$906$	0	0
$907$	43.8434	1.45580	0.727898	−	0.685686i	$-0.240498\pi$
$907$	43.8434	1.45580	0.727898	+	0.685686i	$0.240498\pi$
$908$	0	0
$909$	11.3939	0.377911
$910$	0	0
$911$	−49.3939	−1.63649	−0.818246	−	0.574868i	$-0.805053\pi$
$911$	−49.3939	−1.63649	−0.818246	+	0.574868i	$0.805053\pi$
$912$	0	0
$913$	0.898979	0.0297519
$914$	0	0
$915$	33.7980	1.11733
$916$	0	0
$917$	−39.1918	−1.29423
$918$	0	0
$919$	−11.1010	−0.366189	−0.183094	−	0.983095i	$-0.558611\pi$
$919$	−11.1010	−0.366189	−0.183094	+	0.983095i	$0.558611\pi$
$920$	0	0
$921$	−28.8990	−0.952254
$922$	0	0
$923$	8.44949	0.278118
$924$	0	0
$925$	−4.89898	−0.161077
$926$	0	0
$927$	−10.0454	−0.329934
$928$	0	0
$929$	24.6969	0.810280	0.405140	−	0.914255i	$-0.367223\pi$
$929$	24.6969	0.810280	0.405140	+	0.914255i	$0.367223\pi$
$930$	0	0
$931$	13.3485	0.437478
$932$	0	0
$933$	31.5959	1.03440
$934$	0	0
$935$	−1.30306	−0.0426147
$936$	0	0
$937$	−19.3939	−0.633570	−0.316785	−	0.948497i	$-0.602603\pi$
$937$	−19.3939	−0.633570	−0.316785	+	0.948497i	$0.602603\pi$
$938$	0	0
$939$	64.8990	2.11790
$940$	0	0
$941$	−23.3031	−0.759658	−0.379829	−	0.925057i	$-0.624017\pi$
$941$	−23.3031	−0.759658	−0.379829	+	0.925057i	$0.624017\pi$
$942$	0	0
$943$	1.70714	0.0555922
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.7980	−1.55322	−0.776612	−	0.629979i	$-0.783064\pi$
$947$	−47.7980	−1.55322	−0.776612	+	0.629979i	$0.783064\pi$
$948$	0	0
$949$	−14.6969	−0.477083
$950$	0	0
$951$	−6.60612	−0.214218
$952$	0	0
$953$	−57.1918	−1.85263	−0.926313	−	0.376756i	$-0.877040\pi$
$953$	−57.1918	−1.85263	−0.926313	+	0.376756i	$0.877040\pi$
$954$	0	0
$955$	17.7980	0.575928
$956$	0	0
$957$	4.40408	0.142364
$958$	0	0
$959$	−28.0000	−0.904167
$960$	0	0
$961$	−30.7980	−0.993483
$962$	0	0
$963$	36.7423	1.18401
$964$	0	0
$965$	19.7980	0.637319
$966$	0	0
$967$	25.1010	0.807194	0.403597	−	0.914937i	$-0.367760\pi$
$967$	25.1010	0.807194	0.403597	+	0.914937i	$0.367760\pi$
$968$	0	0
$969$	31.5959	1.01501
$970$	0	0
$971$	27.5959	0.885595	0.442798	−	0.896622i	$-0.353986\pi$
$971$	27.5959	0.885595	0.442798	+	0.896622i	$0.353986\pi$
$972$	0	0
$973$	−14.2020	−0.455297
$974$	0	0
$975$	−2.44949	−0.0784465
$976$	0	0
$977$	−1.30306	−0.0416886	−0.0208443	−	0.999783i	$-0.506635\pi$
$977$	−1.30306	−0.0416886	−0.0208443	+	0.999783i	$0.506635\pi$
$978$	0	0
$979$	2.69694	0.0861945
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	58.8990	1.87859	0.939293	−	0.343117i	$-0.111483\pi$
$983$	58.8990	1.87859	0.939293	+	0.343117i	$0.111483\pi$
$984$	0	0
$985$	−4.20204	−0.133888
$986$	0	0
$987$	−9.79796	−0.311872
$988$	0	0
$989$	−5.19184	−0.165091
$990$	0	0
$991$	−30.2020	−0.959399	−0.479700	−	0.877433i	$-0.659254\pi$
$991$	−30.2020	−0.959399	−0.479700	+	0.877433i	$0.659254\pi$
$992$	0	0
$993$	−20.6969	−0.656797
$994$	0	0
$995$	−13.7980	−0.437425
$996$	0	0
$997$	25.1010	0.794957	0.397479	−	0.917611i	$-0.369885\pi$
$997$	25.1010	0.794957	0.397479	+	0.917611i	$0.369885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.a.f.1.2	✓	2
3.2	odd	2		4680.2.a.z.1.1		2
4.3	odd	2		1040.2.a.k.1.1		2
5.2	odd	4		2600.2.d.h.1249.2		4
5.3	odd	4		2600.2.d.h.1249.3		4
5.4	even	2		2600.2.a.q.1.1		2
8.3	odd	2		4160.2.a.ba.1.2		2
8.5	even	2		4160.2.a.bg.1.1		2
12.11	even	2		9360.2.a.cc.1.2		2
13.12	even	2		6760.2.a.s.1.2		2
20.19	odd	2		5200.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.f.1.2	✓	2	1.1	even	1	trivial
1040.2.a.k.1.1		2	4.3	odd	2
2600.2.a.q.1.1		2	5.4	even	2
2600.2.d.h.1249.2		4	5.2	odd	4
2600.2.d.h.1249.3		4	5.3	odd	4
4160.2.a.ba.1.2		2	8.3	odd	2
4160.2.a.bg.1.1		2	8.5	even	2
4680.2.a.z.1.1		2	3.2	odd	2
5200.2.a.bv.1.2		2	20.19	odd	2
6760.2.a.s.1.2		2	13.12	even	2
9360.2.a.cc.1.2		2	12.11	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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{9} +0.449490 q^{11} -1.00000 q^{13} +2.44949 q^{15} -2.89898 q^{17} -4.44949 q^{19} +4.89898 q^{21} +1.55051 q^{23} +1.00000 q^{25} +4.00000 q^{29} +0.449490 q^{31} +1.10102 q^{33} +2.00000 q^{35} -4.89898 q^{37} -2.44949 q^{39} +1.10102 q^{41} -3.34847 q^{43} +3.00000 q^{45} -2.00000 q^{47} -3.00000 q^{49} -7.10102 q^{51} +10.8990 q^{53} +0.449490 q^{55} -10.8990 q^{57} -5.34847 q^{59} +13.7980 q^{61} +6.00000 q^{63} -1.00000 q^{65} -14.8990 q^{67} +3.79796 q^{69} -8.44949 q^{71} +14.6969 q^{73} +2.44949 q^{75} +0.898979 q^{77} -4.89898 q^{79} -9.00000 q^{81} +2.00000 q^{83} -2.89898 q^{85} +9.79796 q^{87} +6.00000 q^{89} -2.00000 q^{91} +1.10102 q^{93} -4.44949 q^{95} -11.7980 q^{97} +1.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9	\( 2 q + 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} + 12 q^{33} + 4 q^{35} + 12 q^{41} + 8 q^{43} + 6 q^{45} - 4 q^{47} - 6 q^{49} - 24 q^{51} + 12 q^{53} - 4 q^{55} - 12 q^{57} + 4 q^{59} + 8 q^{61} + 12 q^{63} - 2 q^{65} - 20 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{77} - 18 q^{81} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 4 q^{91} + 12 q^{93} - 4 q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 + 4 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 + 12 * q^33 + 4 * q^35 + 12 * q^41 + 8 * q^43 + 6 * q^45 - 4 * q^47 - 6 * q^49 - 24 * q^51 + 12 * q^53 - 4 * q^55 - 12 * q^57 + 4 * q^59 + 8 * q^61 + 12 * q^63 - 2 * q^65 - 20 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^77 - 18 * q^81 + 4 * q^83 + 4 * q^85 + 12 * q^89 - 4 * q^91 + 12 * q^93 - 4 * q^95 - 4 * q^97 - 12 * q^99

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