LMFDB - Embedded newform 4000.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.16400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 41 $ x^4 - 13*x^2 + 41
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.76008$ of defining polynomial
Character	$\chi$	$=$	4000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} -5.52016 q^{11} +5.52016 q^{13} -3.41164 q^{17} +3.41164 q^{19} -1.00000 q^{21} -2.38197 q^{23} +3.47214 q^{27} +1.09017 q^{29} +5.52016 q^{31} +3.41164 q^{33} +8.93180 q^{37} -3.41164 q^{39} +5.85410 q^{41} -12.3262 q^{43} +1.09017 q^{47} -4.38197 q^{49} +2.10851 q^{51} -11.0403 q^{53} -2.10851 q^{57} -12.3434 q^{59} +1.14590 q^{61} -4.23607 q^{63} -10.4721 q^{67} +1.47214 q^{69} +3.41164 q^{71} -12.3434 q^{73} -8.93180 q^{77} +8.93180 q^{79} +5.70820 q^{81} -12.5623 q^{83} -0.673762 q^{87} +3.14590 q^{89} +8.93180 q^{91} -3.41164 q^{93} +12.3434 q^{97} +14.4520 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 2 * q^7 - 6 * q^9	$ 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9} - 4 q^{21} - 14 q^{23} - 4 q^{27} - 18 q^{29} + 10 q^{41} - 18 q^{43} - 18 q^{47} - 22 q^{49} + 18 q^{61} - 8 q^{63} - 24 q^{67} - 12 q^{69} - 4 q^{81} - 10 q^{83} - 34 q^{87} + 26 q^{89}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^7 - 6 * q^9 - 4 * q^21 - 14 * q^23 - 4 * q^27 - 18 * q^29 + 10 * q^41 - 18 * q^43 - 18 * q^47 - 22 * q^49 + 18 * q^61 - 8 * q^63 - 24 * q^67 - 12 * q^69 - 4 * q^81 - 10 * q^83 - 34 * q^87 + 26 * q^89

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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.61803	0.611559	0.305780	−	0.952102i	$-0.401083\pi$
$7$	1.61803	0.611559	0.305780	+	0.952102i	$0.401083\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−5.52016	−1.66439	−0.832195	−	0.554483i	$-0.812916\pi$
$11$	−5.52016	−1.66439	−0.832195	+	0.554483i	$0.812916\pi$
$12$	0	0
$13$	5.52016	1.53102	0.765508	−	0.643426i	$-0.222488\pi$
$13$	5.52016	1.53102	0.765508	+	0.643426i	$0.222488\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.41164	−0.827445	−0.413723	−	0.910403i	$-0.635772\pi$
$17$	−3.41164	−0.827445	−0.413723	+	0.910403i	$0.635772\pi$
$18$	0	0
$19$	3.41164	0.782685	0.391342	−	0.920245i	$-0.372011\pi$
$19$	3.41164	0.782685	0.391342	+	0.920245i	$0.372011\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.38197	−0.496674	−0.248337	−	0.968674i	$-0.579884\pi$
$23$	−2.38197	−0.496674	−0.248337	+	0.968674i	$0.579884\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	1.09017	0.202439	0.101220	−	0.994864i	$-0.467725\pi$
$29$	1.09017	0.202439	0.101220	+	0.994864i	$0.467725\pi$
$30$	0	0
$31$	5.52016	0.991450	0.495725	−	0.868480i	$-0.334902\pi$
$31$	5.52016	0.991450	0.495725	+	0.868480i	$0.334902\pi$
$32$	0	0
$33$	3.41164	0.593891
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.93180	1.46838	0.734190	−	0.678944i	$-0.237562\pi$
$37$	8.93180	1.46838	0.734190	+	0.678944i	$0.237562\pi$
$38$	0	0
$39$	−3.41164	−0.546300
$40$	0	0
$41$	5.85410	0.914257	0.457129	−	0.889401i	$-0.348878\pi$
$41$	5.85410	0.914257	0.457129	+	0.889401i	$0.348878\pi$
$42$	0	0
$43$	−12.3262	−1.87973	−0.939867	−	0.341541i	$-0.889051\pi$
$43$	−12.3262	−1.87973	−0.939867	+	0.341541i	$0.889051\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.09017	0.159018	0.0795088	−	0.996834i	$-0.474665\pi$
$47$	1.09017	0.159018	0.0795088	+	0.996834i	$0.474665\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	0	0
$51$	2.10851	0.295251
$52$	0	0
$53$	−11.0403	−1.51650	−0.758252	−	0.651962i	$-0.773946\pi$
$53$	−11.0403	−1.51650	−0.758252	+	0.651962i	$0.773946\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.10851	−0.279279
$58$	0	0
$59$	−12.3434	−1.60698	−0.803490	−	0.595318i	$-0.797026\pi$
$59$	−12.3434	−1.60698	−0.803490	+	0.595318i	$0.797026\pi$
$60$	0	0
$61$	1.14590	0.146717	0.0733586	−	0.997306i	$-0.476628\pi$
$61$	1.14590	0.146717	0.0733586	+	0.997306i	$0.476628\pi$
$62$	0	0
$63$	−4.23607	−0.533694
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	1.47214	0.177224
$70$	0	0
$71$	3.41164	0.404888	0.202444	−	0.979294i	$-0.435112\pi$
$71$	3.41164	0.404888	0.202444	+	0.979294i	$0.435112\pi$
$72$	0	0
$73$	−12.3434	−1.44469	−0.722346	−	0.691532i	$-0.756936\pi$
$73$	−12.3434	−1.44469	−0.722346	+	0.691532i	$0.756936\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.93180	−1.01787
$78$	0	0
$79$	8.93180	1.00491	0.502453	−	0.864604i	$-0.332431\pi$
$79$	8.93180	1.00491	0.502453	+	0.864604i	$0.332431\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−12.5623	−1.37889	−0.689446	−	0.724337i	$-0.742146\pi$
$83$	−12.5623	−1.37889	−0.689446	+	0.724337i	$0.742146\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.673762	−0.0722349
$88$	0	0
$89$	3.14590	0.333465	0.166732	−	0.986002i	$-0.446678\pi$
$89$	3.14590	0.333465	0.166732	+	0.986002i	$0.446678\pi$
$90$	0	0
$91$	8.93180	0.936307
$92$	0	0
$93$	−3.41164	−0.353771
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.3434	1.25329	0.626644	−	0.779306i	$-0.284428\pi$
$97$	12.3434	1.25329	0.626644	+	0.779306i	$0.284428\pi$
$98$	0	0
$99$	14.4520	1.45248
$100$	0	0
$101$	−14.0902	−1.40202	−0.701012	−	0.713149i	$-0.747268\pi$
$101$	−14.0902	−1.40202	−0.701012	+	0.713149i	$0.747268\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.1459	−0.980841	−0.490420	−	0.871486i	$-0.663157\pi$
$107$	−10.1459	−0.980841	−0.490420	+	0.871486i	$0.663157\pi$
$108$	0	0
$109$	6.32624	0.605944	0.302972	−	0.953000i	$-0.402021\pi$
$109$	6.32624	0.605944	0.302972	+	0.953000i	$0.402021\pi$
$110$	0	0
$111$	−5.52016	−0.523950
$112$	0	0
$113$	5.52016	0.519293	0.259646	−	0.965704i	$-0.416394\pi$
$113$	5.52016	0.519293	0.259646	+	0.965704i	$0.416394\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−14.4520	−1.33608
$118$	0	0
$119$	−5.52016	−0.506032
$120$	0	0
$121$	19.4721	1.77019
$122$	0	0
$123$	−3.61803	−0.326227
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.09017	0.806622	0.403311	−	0.915063i	$-0.367859\pi$
$127$	9.09017	0.806622	0.403311	+	0.915063i	$0.367859\pi$
$128$	0	0
$129$	7.61803	0.670730
$130$	0	0
$131$	−2.10851	−0.184222	−0.0921108	−	0.995749i	$-0.529361\pi$
$131$	−2.10851	−0.184222	−0.0921108	+	0.995749i	$0.529361\pi$
$132$	0	0
$133$	5.52016	0.478658
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.41164	−0.291476	−0.145738	−	0.989323i	$-0.546556\pi$
$137$	−3.41164	−0.291476	−0.145738	+	0.989323i	$0.546556\pi$
$138$	0	0
$139$	−2.10851	−0.178842	−0.0894208	−	0.995994i	$-0.528502\pi$
$139$	−2.10851	−0.178842	−0.0894208	+	0.995994i	$0.528502\pi$
$140$	0	0
$141$	−0.673762	−0.0567410
$142$	0	0
$143$	−30.4721	−2.54821
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.70820	0.223369
$148$	0	0
$149$	−9.14590	−0.749261	−0.374631	−	0.927174i	$-0.622230\pi$
$149$	−9.14590	−0.749261	−0.374631	+	0.927174i	$0.622230\pi$
$150$	0	0
$151$	−19.9721	−1.62531	−0.812654	−	0.582747i	$-0.801978\pi$
$151$	−19.9721	−1.62531	−0.812654	+	0.582747i	$0.801978\pi$
$152$	0	0
$153$	8.93180	0.722093
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.62867	0.608834	0.304417	−	0.952539i	$-0.401538\pi$
$157$	7.62867	0.608834	0.304417	+	0.952539i	$0.401538\pi$
$158$	0	0
$159$	6.82329	0.541122
$160$	0	0
$161$	−3.85410	−0.303746
$162$	0	0
$163$	−18.2705	−1.43106	−0.715528	−	0.698584i	$-0.753814\pi$
$163$	−18.2705	−1.43106	−0.715528	+	0.698584i	$0.753814\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.6180	−1.13118	−0.565589	−	0.824687i	$-0.691351\pi$
$167$	−14.6180	−1.13118	−0.565589	+	0.824687i	$0.691351\pi$
$168$	0	0
$169$	17.4721	1.34401
$170$	0	0
$171$	−8.93180	−0.683032
$172$	0	0
$173$	10.2349	0.778148	0.389074	−	0.921207i	$-0.372795\pi$
$173$	10.2349	0.778148	0.389074	+	0.921207i	$0.372795\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.62867	0.573406
$178$	0	0
$179$	−12.3434	−0.922593	−0.461296	−	0.887246i	$-0.652615\pi$
$179$	−12.3434	−0.922593	−0.461296	+	0.887246i	$0.652615\pi$
$180$	0	0
$181$	−21.0902	−1.56762	−0.783810	−	0.621001i	$-0.786726\pi$
$181$	−21.0902	−1.56762	−0.783810	+	0.621001i	$0.786726\pi$
$182$	0	0
$183$	−0.708204	−0.0523519
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.8328	1.37719
$188$	0	0
$189$	5.61803	0.408652
$190$	0	0
$191$	−10.2349	−0.740574	−0.370287	−	0.928917i	$-0.620741\pi$
$191$	−10.2349	−0.740574	−0.370287	+	0.928917i	$0.620741\pi$
$192$	0	0
$193$	8.93180	0.642925	0.321463	−	0.946922i	$-0.395826\pi$
$193$	8.93180	0.642925	0.321463	+	0.946922i	$0.395826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.41164	0.243070	0.121535	−	0.992587i	$-0.461218\pi$
$197$	3.41164	0.243070	0.121535	+	0.992587i	$0.461218\pi$
$198$	0	0
$199$	−14.4520	−1.02447	−0.512236	−	0.858845i	$-0.671183\pi$
$199$	−14.4520	−1.02447	−0.512236	+	0.858845i	$0.671183\pi$
$200$	0	0
$201$	6.47214	0.456509
$202$	0	0
$203$	1.76393	0.123804
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.23607	0.433437
$208$	0	0
$209$	−18.8328	−1.30269
$210$	0	0
$211$	13.1488	0.905203	0.452601	−	0.891713i	$-0.350496\pi$
$211$	13.1488	0.905203	0.452601	+	0.891713i	$0.350496\pi$
$212$	0	0
$213$	−2.10851	−0.144473
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.93180	0.606330
$218$	0	0
$219$	7.62867	0.515498
$220$	0	0
$221$	−18.8328	−1.26683
$222$	0	0
$223$	−18.3262	−1.22722	−0.613608	−	0.789611i	$-0.710282\pi$
$223$	−18.3262	−1.22722	−0.613608	+	0.789611i	$0.710282\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.85410	−0.189433	−0.0947167	−	0.995504i	$-0.530195\pi$
$227$	−2.85410	−0.189433	−0.0947167	+	0.995504i	$0.530195\pi$
$228$	0	0
$229$	−25.3262	−1.67360	−0.836802	−	0.547505i	$-0.815578\pi$
$229$	−25.3262	−1.67360	−0.836802	+	0.547505i	$0.815578\pi$
$230$	0	0
$231$	5.52016	0.363200
$232$	0	0
$233$	−3.41164	−0.223504	−0.111752	−	0.993736i	$-0.535646\pi$
$233$	−3.41164	−0.223504	−0.111752	+	0.993736i	$0.535646\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.52016	−0.358573
$238$	0	0
$239$	8.93180	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$239$	8.93180	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$240$	0	0
$241$	14.5623	0.938041	0.469020	−	0.883187i	$-0.344607\pi$
$241$	14.5623	0.938041	0.469020	+	0.883187i	$0.344607\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.8328	1.19830
$248$	0	0
$249$	7.76393	0.492019
$250$	0	0
$251$	24.6869	1.55822	0.779111	−	0.626885i	$-0.215671\pi$
$251$	24.6869	1.55822	0.779111	+	0.626885i	$0.215671\pi$
$252$	0	0
$253$	13.1488	0.826660
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.8636	−1.11430	−0.557151	−	0.830412i	$-0.688105\pi$
$257$	−17.8636	−1.11430	−0.557151	+	0.830412i	$0.688105\pi$
$258$	0	0
$259$	14.4520	0.898001
$260$	0	0
$261$	−2.85410	−0.176664
$262$	0	0
$263$	−18.3262	−1.13004	−0.565022	−	0.825076i	$-0.691132\pi$
$263$	−18.3262	−1.13004	−0.565022	+	0.825076i	$0.691132\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.94427	−0.118988
$268$	0	0
$269$	3.52786	0.215098	0.107549	−	0.994200i	$-0.465700\pi$
$269$	3.52786	0.215098	0.107549	+	0.994200i	$0.465700\pi$
$270$	0	0
$271$	11.0403	0.670651	0.335326	−	0.942102i	$-0.391154\pi$
$271$	11.0403	0.670651	0.335326	+	0.942102i	$0.391154\pi$
$272$	0	0
$273$	−5.52016	−0.334095
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.41164	0.204986	0.102493	−	0.994734i	$-0.467318\pi$
$277$	3.41164	0.204986	0.102493	+	0.994734i	$0.467318\pi$
$278$	0	0
$279$	−14.4520	−0.865216
$280$	0	0
$281$	22.5623	1.34595	0.672977	−	0.739663i	$-0.265015\pi$
$281$	22.5623	1.34595	0.672977	+	0.739663i	$0.265015\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.47214	0.559123
$288$	0	0
$289$	−5.36068	−0.315334
$290$	0	0
$291$	−7.62867	−0.447201
$292$	0	0
$293$	−1.30313	−0.0761298	−0.0380649	−	0.999275i	$-0.512119\pi$
$293$	−1.30313	−0.0761298	−0.0380649	+	0.999275i	$0.512119\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−19.1667	−1.11217
$298$	0	0
$299$	−13.1488	−0.760416
$300$	0	0
$301$	−19.9443	−1.14957
$302$	0	0
$303$	8.70820	0.500273
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.6180	−0.891368	−0.445684	−	0.895190i	$-0.647040\pi$
$307$	−15.6180	−0.891368	−0.445684	+	0.895190i	$0.647040\pi$
$308$	0	0
$309$	4.94427	0.281270
$310$	0	0
$311$	23.3838	1.32597	0.662986	−	0.748632i	$-0.269289\pi$
$311$	23.3838	1.32597	0.662986	+	0.748632i	$0.269289\pi$
$312$	0	0
$313$	−22.0806	−1.24807	−0.624035	−	0.781396i	$-0.714508\pi$
$313$	−22.0806	−1.24807	−0.624035	+	0.781396i	$0.714508\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.6187	1.88821	0.944107	−	0.329639i	$-0.106927\pi$
$317$	33.6187	1.88821	0.944107	+	0.329639i	$0.106927\pi$
$318$	0	0
$319$	−6.01791	−0.336938
$320$	0	0
$321$	6.27051	0.349986
$322$	0	0
$323$	−11.6393	−0.647629
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.90983	−0.216214
$328$	0	0
$329$	1.76393	0.0972487
$330$	0	0
$331$	−26.7954	−1.47281	−0.736404	−	0.676542i	$-0.763478\pi$
$331$	−26.7954	−1.47281	−0.736404	+	0.676542i	$0.763478\pi$
$332$	0	0
$333$	−23.3838	−1.28142
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.1488	−0.716262	−0.358131	−	0.933671i	$-0.616586\pi$
$337$	−13.1488	−0.716262	−0.358131	+	0.933671i	$0.616586\pi$
$338$	0	0
$339$	−3.41164	−0.185295
$340$	0	0
$341$	−30.4721	−1.65016
$342$	0	0
$343$	−18.4164	−0.994393
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.67376	−0.358266	−0.179133	−	0.983825i	$-0.557329\pi$
$347$	−6.67376	−0.358266	−0.179133	+	0.983825i	$0.557329\pi$
$348$	0	0
$349$	−8.38197	−0.448676	−0.224338	−	0.974511i	$-0.572022\pi$
$349$	−8.38197	−0.448676	−0.224338	+	0.974511i	$0.572022\pi$
$350$	0	0
$351$	19.1667	1.02304
$352$	0	0
$353$	−12.3434	−0.656975	−0.328488	−	0.944508i	$-0.606539\pi$
$353$	−12.3434	−0.656975	−0.328488	+	0.944508i	$0.606539\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.41164	0.180563
$358$	0	0
$359$	34.4241	1.81683	0.908417	−	0.418066i	$-0.137292\pi$
$359$	34.4241	1.81683	0.908417	+	0.418066i	$0.137292\pi$
$360$	0	0
$361$	−7.36068	−0.387404
$362$	0	0
$363$	−12.0344	−0.631644
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.9787	−0.729683	−0.364841	−	0.931070i	$-0.618877\pi$
$367$	−13.9787	−0.729683	−0.364841	+	0.931070i	$0.618877\pi$
$368$	0	0
$369$	−15.3262	−0.797852
$370$	0	0
$371$	−17.8636	−0.927432
$372$	0	0
$373$	−4.71478	−0.244122	−0.122061	−	0.992523i	$-0.538950\pi$
$373$	−4.71478	−0.244122	−0.122061	+	0.992523i	$0.538950\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.01791	0.309938
$378$	0	0
$379$	17.8636	0.917592	0.458796	−	0.888542i	$-0.348281\pi$
$379$	17.8636	0.917592	0.458796	+	0.888542i	$0.348281\pi$
$380$	0	0
$381$	−5.61803	−0.287821
$382$	0	0
$383$	22.9098	1.17064	0.585319	−	0.810803i	$-0.300969\pi$
$383$	22.9098	1.17064	0.585319	+	0.810803i	$0.300969\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.2705	1.64040
$388$	0	0
$389$	11.8541	0.601027	0.300513	−	0.953778i	$-0.402842\pi$
$389$	11.8541	0.601027	0.300513	+	0.953778i	$0.402842\pi$
$390$	0	0
$391$	8.12642	0.410971
$392$	0	0
$393$	1.30313	0.0657343
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−30.2071	−1.51605	−0.758024	−	0.652226i	$-0.773835\pi$
$397$	−30.2071	−1.51605	−0.758024	+	0.652226i	$0.773835\pi$
$398$	0	0
$399$	−3.41164	−0.170796
$400$	0	0
$401$	−0.0901699	−0.00450287	−0.00225144	−	0.999997i	$-0.500717\pi$
$401$	−0.0901699	−0.00450287	−0.00225144	+	0.999997i	$0.500717\pi$
$402$	0	0
$403$	30.4721	1.51793
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−49.3050	−2.44396
$408$	0	0
$409$	−7.50658	−0.371176	−0.185588	−	0.982628i	$-0.559419\pi$
$409$	−7.50658	−0.371176	−0.185588	+	0.982628i	$0.559419\pi$
$410$	0	0
$411$	2.10851	0.104005
$412$	0	0
$413$	−19.9721	−0.982764
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.30313	0.0638147
$418$	0	0
$419$	−15.2573	−0.745370	−0.372685	−	0.927958i	$-0.621563\pi$
$419$	−15.2573	−0.745370	−0.372685	+	0.927958i	$0.621563\pi$
$420$	0	0
$421$	−1.38197	−0.0673529	−0.0336765	−	0.999433i	$-0.510722\pi$
$421$	−1.38197	−0.0673529	−0.0336765	+	0.999433i	$0.510722\pi$
$422$	0	0
$423$	−2.85410	−0.138771
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.85410	0.0897263
$428$	0	0
$429$	18.8328	0.909257
$430$	0	0
$431$	−19.9721	−0.962023	−0.481012	−	0.876714i	$-0.659730\pi$
$431$	−19.9721	−0.962023	−0.481012	+	0.876714i	$0.659730\pi$
$432$	0	0
$433$	−8.93180	−0.429235	−0.214618	−	0.976698i	$-0.568850\pi$
$433$	−8.93180	−0.429235	−0.214618	+	0.976698i	$0.568850\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.12642	−0.388739
$438$	0	0
$439$	10.2349	0.488487	0.244243	−	0.969714i	$-0.421460\pi$
$439$	10.2349	0.488487	0.244243	+	0.969714i	$0.421460\pi$
$440$	0	0
$441$	11.4721	0.546292
$442$	0	0
$443$	−8.20163	−0.389671	−0.194836	−	0.980836i	$-0.562417\pi$
$443$	−8.20163	−0.389671	−0.194836	+	0.980836i	$0.562417\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.65248	0.267353
$448$	0	0
$449$	−3.52786	−0.166490	−0.0832451	−	0.996529i	$-0.526528\pi$
$449$	−3.52786	−0.166490	−0.0832451	+	0.996529i	$0.526528\pi$
$450$	0	0
$451$	−32.3156	−1.52168
$452$	0	0
$453$	12.3434	0.579946
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.1209	1.54933	0.774666	−	0.632370i	$-0.217918\pi$
$457$	33.1209	1.54933	0.774666	+	0.632370i	$0.217918\pi$
$458$	0	0
$459$	−11.8457	−0.552910
$460$	0	0
$461$	−5.50658	−0.256467	−0.128233	−	0.991744i	$-0.540931\pi$
$461$	−5.50658	−0.256467	−0.128233	+	0.991744i	$0.540931\pi$
$462$	0	0
$463$	−26.7426	−1.24284	−0.621418	−	0.783479i	$-0.713443\pi$
$463$	−26.7426	−1.24284	−0.621418	+	0.783479i	$0.713443\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.4508	−1.22400	−0.612000	−	0.790858i	$-0.709635\pi$
$467$	−26.4508	−1.22400	−0.612000	+	0.790858i	$0.709635\pi$
$468$	0	0
$469$	−16.9443	−0.782414
$470$	0	0
$471$	−4.71478	−0.217245
$472$	0	0
$473$	68.0428	3.12861
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.9039	1.32342
$478$	0	0
$479$	−2.10851	−0.0963404	−0.0481702	−	0.998839i	$-0.515339\pi$
$479$	−2.10851	−0.0963404	−0.0481702	+	0.998839i	$0.515339\pi$
$480$	0	0
$481$	49.3050	2.24811
$482$	0	0
$483$	2.38197	0.108383
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.2705	1.23574	0.617872	−	0.786278i	$-0.287995\pi$
$487$	27.2705	1.23574	0.617872	+	0.786278i	$0.287995\pi$
$488$	0	0
$489$	11.2918	0.510633
$490$	0	0
$491$	−17.0582	−0.769827	−0.384913	−	0.922953i	$-0.625769\pi$
$491$	−17.0582	−0.769827	−0.384913	+	0.922953i	$0.625769\pi$
$492$	0	0
$493$	−3.71927	−0.167508
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.52016	0.247613
$498$	0	0
$499$	−17.0582	−0.763631	−0.381815	−	0.924239i	$-0.624701\pi$
$499$	−17.0582	−0.763631	−0.381815	+	0.924239i	$0.624701\pi$
$500$	0	0
$501$	9.03444	0.403629
$502$	0	0
$503$	38.1591	1.70143	0.850714	−	0.525629i	$-0.176170\pi$
$503$	38.1591	1.70143	0.850714	+	0.525629i	$0.176170\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.7984	−0.479573
$508$	0	0
$509$	43.3050	1.91946	0.959729	−	0.280927i	$-0.0906419\pi$
$509$	43.3050	1.91946	0.959729	+	0.280927i	$0.0906419\pi$
$510$	0	0
$511$	−19.9721	−0.883514
$512$	0	0
$513$	11.8457	0.523000
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.01791	−0.264667
$518$	0	0
$519$	−6.32554	−0.277660
$520$	0	0
$521$	7.09017	0.310626	0.155313	−	0.987865i	$-0.450361\pi$
$521$	7.09017	0.310626	0.155313	+	0.987865i	$0.450361\pi$
$522$	0	0
$523$	12.9787	0.567520	0.283760	−	0.958895i	$-0.408418\pi$
$523$	12.9787	0.567520	0.283760	+	0.958895i	$0.408418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.8328	−0.820370
$528$	0	0
$529$	−17.3262	−0.753315
$530$	0	0
$531$	32.3156	1.40238
$532$	0	0
$533$	32.3156	1.39974
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.62867	0.329201
$538$	0	0
$539$	24.1891	1.04190
$540$	0	0
$541$	−27.5623	−1.18500	−0.592498	−	0.805572i	$-0.701858\pi$
$541$	−27.5623	−1.18500	−0.592498	+	0.805572i	$0.701858\pi$
$542$	0	0
$543$	13.0344	0.559361
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.9098	0.765769	0.382885	−	0.923796i	$-0.374931\pi$
$547$	17.9098	0.765769	0.382885	+	0.923796i	$0.374931\pi$
$548$	0	0
$549$	−3.00000	−0.128037
$550$	0	0
$551$	3.71927	0.158446
$552$	0	0
$553$	14.4520	0.614560
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.7093	−1.25882	−0.629412	−	0.777072i	$-0.716704\pi$
$557$	−29.7093	−1.25882	−0.629412	+	0.777072i	$0.716704\pi$
$558$	0	0
$559$	−68.0428	−2.87790
$560$	0	0
$561$	−11.6393	−0.491412
$562$	0	0
$563$	−8.94427	−0.376956	−0.188478	−	0.982077i	$-0.560355\pi$
$563$	−8.94427	−0.376956	−0.188478	+	0.982077i	$0.560355\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.23607	0.387878
$568$	0	0
$569$	15.0902	0.632613	0.316306	−	0.948657i	$-0.397557\pi$
$569$	15.0902	0.632613	0.316306	+	0.948657i	$0.397557\pi$
$570$	0	0
$571$	42.5505	1.78068	0.890341	−	0.455293i	$-0.150466\pi$
$571$	42.5505	1.78068	0.890341	+	0.455293i	$0.150466\pi$
$572$	0	0
$573$	6.32554	0.264253
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−35.7272	−1.48734	−0.743672	−	0.668545i	$-0.766917\pi$
$577$	−35.7272	−1.48734	−0.743672	+	0.668545i	$0.766917\pi$
$578$	0	0
$579$	−5.52016	−0.229410
$580$	0	0
$581$	−20.3262	−0.843274
$582$	0	0
$583$	60.9443	2.52405
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.7771	1.47668	0.738339	−	0.674430i	$-0.235610\pi$
$587$	35.7771	1.47668	0.738339	+	0.674430i	$0.235610\pi$
$588$	0	0
$589$	18.8328	0.775993
$590$	0	0
$591$	−2.10851	−0.0867326
$592$	0	0
$593$	−33.6187	−1.38055	−0.690277	−	0.723545i	$-0.742511\pi$
$593$	−33.6187	−1.38055	−0.690277	+	0.723545i	$0.742511\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.93180	0.365554
$598$	0	0
$599$	42.0527	1.71823	0.859114	−	0.511784i	$-0.171015\pi$
$599$	42.0527	1.71823	0.859114	+	0.511784i	$0.171015\pi$
$600$	0	0
$601$	6.43769	0.262599	0.131300	−	0.991343i	$-0.458085\pi$
$601$	6.43769	0.262599	0.131300	+	0.991343i	$0.458085\pi$
$602$	0	0
$603$	27.4164	1.11648
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.36068	−0.176995	−0.0884973	−	0.996076i	$-0.528206\pi$
$607$	−4.36068	−0.176995	−0.0884973	+	0.996076i	$0.528206\pi$
$608$	0	0
$609$	−1.09017	−0.0441759
$610$	0	0
$611$	6.01791	0.243459
$612$	0	0
$613$	11.0403	0.445914	0.222957	−	0.974828i	$-0.428429\pi$
$613$	11.0403	0.445914	0.222957	+	0.974828i	$0.428429\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.62867	0.307119	0.153559	−	0.988139i	$-0.450926\pi$
$617$	7.62867	0.307119	0.153559	+	0.988139i	$0.450926\pi$
$618$	0	0
$619$	18.6690	0.750370	0.375185	−	0.926950i	$-0.377579\pi$
$619$	18.6690	0.750370	0.375185	+	0.926950i	$0.377579\pi$
$620$	0	0
$621$	−8.27051	−0.331884
$622$	0	0
$623$	5.09017	0.203933
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.6393	0.464830
$628$	0	0
$629$	−30.4721	−1.21500
$630$	0	0
$631$	−5.52016	−0.219754	−0.109877	−	0.993945i	$-0.535046\pi$
$631$	−5.52016	−0.219754	−0.109877	+	0.993945i	$0.535046\pi$
$632$	0	0
$633$	−8.12642	−0.322996
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.1891	−0.958409
$638$	0	0
$639$	−8.93180	−0.353337
$640$	0	0
$641$	41.2705	1.63009	0.815044	−	0.579400i	$-0.196713\pi$
$641$	41.2705	1.63009	0.815044	+	0.579400i	$0.196713\pi$
$642$	0	0
$643$	−18.0902	−0.713407	−0.356703	−	0.934218i	$-0.616099\pi$
$643$	−18.0902	−0.713407	−0.356703	+	0.934218i	$0.616099\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.05573	−0.120133	−0.0600665	−	0.998194i	$-0.519131\pi$
$647$	−3.05573	−0.120133	−0.0600665	+	0.998194i	$0.519131\pi$
$648$	0	0
$649$	68.1378	2.67464
$650$	0	0
$651$	−5.52016	−0.216352
$652$	0	0
$653$	−4.71478	−0.184503	−0.0922517	−	0.995736i	$-0.529406\pi$
$653$	−4.71478	−0.184503	−0.0922517	+	0.995736i	$0.529406\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.3156	1.26075
$658$	0	0
$659$	25.4923	0.993038	0.496519	−	0.868026i	$-0.334611\pi$
$659$	25.4923	0.993038	0.496519	+	0.868026i	$0.334611\pi$
$660$	0	0
$661$	27.5623	1.07205	0.536025	−	0.844202i	$-0.319925\pi$
$661$	27.5623	1.07205	0.536025	+	0.844202i	$0.319925\pi$
$662$	0	0
$663$	11.6393	0.452034
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.59675	−0.100546
$668$	0	0
$669$	11.3262	0.437898
$670$	0	0
$671$	−6.32554	−0.244195
$672$	0	0
$673$	−27.6008	−1.06393	−0.531966	−	0.846766i	$-0.678547\pi$
$673$	−27.6008	−1.06393	−0.531966	+	0.846766i	$0.678547\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.8636	−0.686554	−0.343277	−	0.939234i	$-0.611537\pi$
$677$	−17.8636	−0.686554	−0.343277	+	0.939234i	$0.611537\pi$
$678$	0	0
$679$	19.9721	0.766459
$680$	0	0
$681$	1.76393	0.0675940
$682$	0	0
$683$	48.6869	1.86295	0.931477	−	0.363801i	$-0.118521\pi$
$683$	48.6869	1.86295	0.931477	+	0.363801i	$0.118521\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.6525	0.597179
$688$	0	0
$689$	−60.9443	−2.32179
$690$	0	0
$691$	33.6187	1.27892	0.639458	−	0.768826i	$-0.279159\pi$
$691$	33.6187	1.27892	0.639458	+	0.768826i	$0.279159\pi$
$692$	0	0
$693$	23.3838	0.888276
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.9721	−0.756498
$698$	0	0
$699$	2.10851	0.0797513
$700$	0	0
$701$	26.9443	1.01767	0.508836	−	0.860864i	$-0.330076\pi$
$701$	26.9443	1.01767	0.508836	+	0.860864i	$0.330076\pi$
$702$	0	0
$703$	30.4721	1.14928
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.7984	−0.857421
$708$	0	0
$709$	−36.3820	−1.36635	−0.683177	−	0.730253i	$-0.739402\pi$
$709$	−36.3820	−1.36635	−0.683177	+	0.730253i	$0.739402\pi$
$710$	0	0
$711$	−23.3838	−0.876960
$712$	0	0
$713$	−13.1488	−0.492427
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.52016	−0.206154
$718$	0	0
$719$	29.4017	1.09650	0.548249	−	0.836315i	$-0.315295\pi$
$719$	29.4017	1.09650	0.548249	+	0.836315i	$0.315295\pi$
$720$	0	0
$721$	−12.9443	−0.482070
$722$	0	0
$723$	−9.00000	−0.334714
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.9098	1.29473	0.647367	−	0.762178i	$-0.275870\pi$
$727$	34.9098	1.29473	0.647367	+	0.762178i	$0.275870\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	42.0527	1.55538
$732$	0	0
$733$	−9.73718	−0.359651	−0.179826	−	0.983699i	$-0.557553\pi$
$733$	−9.73718	−0.359651	−0.179826	+	0.983699i	$0.557553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	57.8078	2.12938
$738$	0	0
$739$	−37.8357	−1.39181	−0.695905	−	0.718134i	$-0.744996\pi$
$739$	−37.8357	−1.39181	−0.695905	+	0.718134i	$0.744996\pi$
$740$	0	0
$741$	−11.6393	−0.427581
$742$	0	0
$743$	−10.1115	−0.370953	−0.185477	−	0.982649i	$-0.559383\pi$
$743$	−10.1115	−0.370953	−0.185477	+	0.982649i	$0.559383\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.8885	1.20333
$748$	0	0
$749$	−16.4164	−0.599842
$750$	0	0
$751$	37.8357	1.38065	0.690323	−	0.723502i	$-0.257469\pi$
$751$	37.8357	1.38065	0.690323	+	0.723502i	$0.257469\pi$
$752$	0	0
$753$	−15.2573	−0.556008
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.91389	−0.105907	−0.0529536	−	0.998597i	$-0.516864\pi$
$757$	−2.91389	−0.105907	−0.0529536	+	0.998597i	$0.516864\pi$
$758$	0	0
$759$	−8.12642	−0.294970
$760$	0	0
$761$	0.854102	0.0309612	0.0154806	−	0.999880i	$-0.495072\pi$
$761$	0.854102	0.0309612	0.0154806	+	0.999880i	$0.495072\pi$
$762$	0	0
$763$	10.2361	0.370571
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−68.1378	−2.46031
$768$	0	0
$769$	14.7984	0.533643	0.266822	−	0.963746i	$-0.414027\pi$
$769$	14.7984	0.533643	0.266822	+	0.963746i	$0.414027\pi$
$770$	0	0
$771$	11.0403	0.397607
$772$	0	0
$773$	−50.9845	−1.83379	−0.916893	−	0.399132i	$-0.869311\pi$
$773$	−50.9845	−1.83379	−0.916893	+	0.399132i	$0.869311\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.93180	−0.320427
$778$	0	0
$779$	19.9721	0.715575
$780$	0	0
$781$	−18.8328	−0.673891
$782$	0	0
$783$	3.78522	0.135273
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.5623	1.19637	0.598184	−	0.801359i	$-0.295889\pi$
$787$	33.5623	1.19637	0.598184	+	0.801359i	$0.295889\pi$
$788$	0	0
$789$	11.3262	0.403225
$790$	0	0
$791$	8.93180	0.317578
$792$	0	0
$793$	6.32554	0.224626
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.2877	1.85212	0.926062	−	0.377371i	$-0.123172\pi$
$797$	52.2877	1.85212	0.926062	+	0.377371i	$0.123172\pi$
$798$	0	0
$799$	−3.71927	−0.131578
$800$	0	0
$801$	−8.23607	−0.291007
$802$	0	0
$803$	68.1378	2.40453
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.18034	−0.0767516
$808$	0	0
$809$	−2.38197	−0.0837455	−0.0418727	−	0.999123i	$-0.513332\pi$
$809$	−2.38197	−0.0837455	−0.0418727	+	0.999123i	$0.513332\pi$
$810$	0	0
$811$	−52.2877	−1.83607	−0.918034	−	0.396501i	$-0.870224\pi$
$811$	−52.2877	−1.83607	−0.918034	+	0.396501i	$0.870224\pi$
$812$	0	0
$813$	−6.82329	−0.239303
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−42.0527	−1.47124
$818$	0	0
$819$	−23.3838	−0.817095
$820$	0	0
$821$	24.5623	0.857230	0.428615	−	0.903487i	$-0.359002\pi$
$821$	24.5623	0.857230	0.428615	+	0.903487i	$0.359002\pi$
$822$	0	0
$823$	54.2492	1.89101	0.945505	−	0.325609i	$-0.105569\pi$
$823$	54.2492	1.89101	0.945505	+	0.325609i	$0.105569\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.8328	−1.62854	−0.814268	−	0.580489i	$-0.802862\pi$
$827$	−46.8328	−1.62854	−0.814268	+	0.580489i	$0.802862\pi$
$828$	0	0
$829$	7.85410	0.272784	0.136392	−	0.990655i	$-0.456449\pi$
$829$	7.85410	0.272784	0.136392	+	0.990655i	$0.456449\pi$
$830$	0	0
$831$	−2.10851	−0.0731435
$832$	0	0
$833$	14.9497	0.517977
$834$	0	0
$835$	0	0
$836$	0	0
$837$	19.1667	0.662499
$838$	0	0
$839$	−51.4823	−1.77737	−0.888683	−	0.458522i	$-0.848379\pi$
$839$	−51.4823	−1.77737	−0.888683	+	0.458522i	$0.848379\pi$
$840$	0	0
$841$	−27.8115	−0.959018
$842$	0	0
$843$	−13.9443	−0.480266
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.5066	1.08258
$848$	0	0
$849$	−2.47214	−0.0848435
$850$	0	0
$851$	−21.2752	−0.729306
$852$	0	0
$853$	−15.2573	−0.522401	−0.261201	−	0.965285i	$-0.584118\pi$
$853$	−15.2573	−0.522401	−0.261201	+	0.965285i	$0.584118\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.2474	−1.40898	−0.704492	−	0.709712i	$-0.748825\pi$
$857$	−41.2474	−1.40898	−0.704492	+	0.709712i	$0.748825\pi$
$858$	0	0
$859$	−6.82329	−0.232808	−0.116404	−	0.993202i	$-0.537137\pi$
$859$	−6.82329	−0.232808	−0.116404	+	0.993202i	$0.537137\pi$
$860$	0	0
$861$	−5.85410	−0.199507
$862$	0	0
$863$	15.5066	0.527850	0.263925	−	0.964543i	$-0.414983\pi$
$863$	15.5066	0.527850	0.263925	+	0.964543i	$0.414983\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.31308	0.112518
$868$	0	0
$869$	−49.3050	−1.67256
$870$	0	0
$871$	−57.8078	−1.95874
$872$	0	0
$873$	−32.3156	−1.09372
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.9721	0.674410	0.337205	−	0.941431i	$-0.390518\pi$
$877$	19.9721	0.674410	0.337205	+	0.941431i	$0.390518\pi$
$878$	0	0
$879$	0.805380	0.0271648
$880$	0	0
$881$	29.9787	1.01001	0.505004	−	0.863117i	$-0.331491\pi$
$881$	29.9787	1.01001	0.505004	+	0.863117i	$0.331491\pi$
$882$	0	0
$883$	38.8541	1.30754	0.653772	−	0.756691i	$-0.273185\pi$
$883$	38.8541	1.30754	0.653772	+	0.756691i	$0.273185\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.43769	0.182580	0.0912899	−	0.995824i	$-0.470901\pi$
$887$	5.43769	0.182580	0.0912899	+	0.995824i	$0.470901\pi$
$888$	0	0
$889$	14.7082	0.493297
$890$	0	0
$891$	−31.5102	−1.05563
$892$	0	0
$893$	3.71927	0.124461
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.12642	0.271333
$898$	0	0
$899$	6.01791	0.200709
$900$	0	0
$901$	37.6656	1.25482
$902$	0	0
$903$	12.3262	0.410192
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.7984	−0.923030	−0.461515	−	0.887132i	$-0.652694\pi$
$907$	−27.7984	−0.923030	−0.461515	+	0.887132i	$0.652694\pi$
$908$	0	0
$909$	36.8885	1.22352
$910$	0	0
$911$	−8.12642	−0.269240	−0.134620	−	0.990897i	$-0.542981\pi$
$911$	−8.12642	−0.269240	−0.134620	+	0.990897i	$0.542981\pi$
$912$	0	0
$913$	69.3459	2.29501
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.41164	−0.112662
$918$	0	0
$919$	39.1389	1.29107	0.645536	−	0.763730i	$-0.276634\pi$
$919$	39.1389	1.29107	0.645536	+	0.763730i	$0.276634\pi$
$920$	0	0
$921$	9.65248	0.318060
$922$	0	0
$923$	18.8328	0.619890
$924$	0	0
$925$	0	0
$926$	0	0
$927$	20.9443	0.687900
$928$	0	0
$929$	−1.03444	−0.0339389	−0.0169695	−	0.999856i	$-0.505402\pi$
$929$	−1.03444	−0.0339389	−0.0169695	+	0.999856i	$0.505402\pi$
$930$	0	0
$931$	−14.9497	−0.489957
$932$	0	0
$933$	−14.4520	−0.473136
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0806	−0.721343	−0.360671	−	0.932693i	$-0.617452\pi$
$937$	−22.0806	−0.721343	−0.360671	+	0.932693i	$0.617452\pi$
$938$	0	0
$939$	13.6466	0.445339
$940$	0	0
$941$	0.111456	0.00363337	0.00181668	−	0.999998i	$-0.499422\pi$
$941$	0.111456	0.00363337	0.00181668	+	0.999998i	$0.499422\pi$
$942$	0	0
$943$	−13.9443	−0.454088
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.2705	−1.21113	−0.605564	−	0.795796i	$-0.707053\pi$
$947$	−37.2705	−1.21113	−0.605564	+	0.795796i	$0.707053\pi$
$948$	0	0
$949$	−68.1378	−2.21185
$950$	0	0
$951$	−20.7775	−0.673756
$952$	0	0
$953$	−53.5908	−1.73598	−0.867988	−	0.496585i	$-0.834587\pi$
$953$	−53.5908	−1.73598	−0.867988	+	0.496585i	$0.834587\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.71927	0.120227
$958$	0	0
$959$	−5.52016	−0.178255
$960$	0	0
$961$	−0.527864	−0.0170279
$962$	0	0
$963$	26.5623	0.855958
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.3951	−1.36334	−0.681668	−	0.731662i	$-0.738745\pi$
$967$	−42.3951	−1.36334	−0.681668	+	0.731662i	$0.738745\pi$
$968$	0	0
$969$	7.19350	0.231088
$970$	0	0
$971$	2.10851	0.0676654	0.0338327	−	0.999428i	$-0.489229\pi$
$971$	2.10851	0.0676654	0.0338327	+	0.999428i	$0.489229\pi$
$972$	0	0
$973$	−3.41164	−0.109372
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.8457	0.378977	0.189489	−	0.981883i	$-0.439317\pi$
$977$	11.8457	0.378977	0.189489	+	0.981883i	$0.439317\pi$
$978$	0	0
$979$	−17.3659	−0.555015
$980$	0	0
$981$	−16.5623	−0.528794
$982$	0	0
$983$	7.41641	0.236547	0.118273	−	0.992981i	$-0.462264\pi$
$983$	7.41641	0.236547	0.118273	+	0.992981i	$0.462264\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.09017	−0.0347005
$988$	0	0
$989$	29.3607	0.933615
$990$	0	0
$991$	37.3380	1.18608	0.593040	−	0.805173i	$-0.297928\pi$
$991$	37.3380	1.18608	0.593040	+	0.805173i	$0.297928\pi$
$992$	0	0
$993$	16.5605	0.525531
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0985	0.889890	0.444945	−	0.895558i	$-0.353223\pi$
$997$	28.0985	0.889890	0.444945	+	0.895558i	$0.353223\pi$
$998$	0	0
$999$	31.0124	0.981190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.a.h.1.1	yes	4
4.3	odd	2		4000.2.a.c.1.4	yes	4
5.2	odd	4		4000.2.c.e.1249.5		8
5.3	odd	4		4000.2.c.e.1249.3		8
5.4	even	2		4000.2.a.c.1.3	✓	4
8.3	odd	2		8000.2.a.bp.1.1		4
8.5	even	2		8000.2.a.bc.1.4		4
20.3	even	4		4000.2.c.e.1249.6		8
20.7	even	4		4000.2.c.e.1249.4		8
20.19	odd	2	inner	4000.2.a.h.1.2	yes	4
40.19	odd	2		8000.2.a.bc.1.3		4
40.29	even	2		8000.2.a.bp.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.c.1.3	✓	4	5.4	even	2
4000.2.a.c.1.4	yes	4	4.3	odd	2
4000.2.a.h.1.1	yes	4	1.1	even	1	trivial
4000.2.a.h.1.2	yes	4	20.19	odd	2	inner
4000.2.c.e.1249.3		8	5.3	odd	4
4000.2.c.e.1249.4		8	20.7	even	4
4000.2.c.e.1249.5		8	5.2	odd	4
4000.2.c.e.1249.6		8	20.3	even	4
8000.2.a.bc.1.3		4	40.19	odd	2
8000.2.a.bc.1.4		4	8.5	even	2
8000.2.a.bp.1.1		4	8.3	odd	2
8000.2.a.bp.1.2		4	40.29	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +1.61803 q^{7} -2.61803 q^{9} -5.52016 q^{11} +5.52016 q^{13} -3.41164 q^{17} +3.41164 q^{19} -1.00000 q^{21} -2.38197 q^{23} +3.47214 q^{27} +1.09017 q^{29} +5.52016 q^{31} +3.41164 q^{33} +8.93180 q^{37} -3.41164 q^{39} +5.85410 q^{41} -12.3262 q^{43} +1.09017 q^{47} -4.38197 q^{49} +2.10851 q^{51} -11.0403 q^{53} -2.10851 q^{57} -12.3434 q^{59} +1.14590 q^{61} -4.23607 q^{63} -10.4721 q^{67} +1.47214 q^{69} +3.41164 q^{71} -12.3434 q^{73} -8.93180 q^{77} +8.93180 q^{79} +5.70820 q^{81} -12.5623 q^{83} -0.673762 q^{87} +3.14590 q^{89} +8.93180 q^{91} -3.41164 q^{93} +12.3434 q^{97} +14.4520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 2 * q^7 - 6 * q^9	\( 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9} - 4 q^{21} - 14 q^{23} - 4 q^{27} - 18 q^{29} + 10 q^{41} - 18 q^{43} - 18 q^{47} - 22 q^{49} + 18 q^{61} - 8 q^{63} - 24 q^{67} - 12 q^{69} - 4 q^{81} - 10 q^{83} - 34 q^{87} + 26 q^{89}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^7 - 6 * q^9 - 4 * q^21 - 14 * q^23 - 4 * q^27 - 18 * q^29 + 10 * q^41 - 18 * q^43 - 18 * q^47 - 22 * q^49 + 18 * q^61 - 8 * q^63 - 24 * q^67 - 12 * q^69 - 4 * q^81 - 10 * q^83 - 34 * q^87 + 26 * q^89

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