LMFDB - Embedded newform 4000.2.a.g.1.3

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:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
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			1.3
Root			$1.17557$ 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)

$f(q)$	$=$	$q+1.17557 q^{3} +0.726543 q^{7} -1.61803 q^{9} +O(q^{10})$	$q+1.17557 q^{3} +0.726543 q^{7} -1.61803 q^{9} -3.07768 q^{11} +0.381966 q^{13} +0.236068 q^{17} +4.97980 q^{19} +0.854102 q^{21} -7.60845 q^{23} -5.42882 q^{27} -5.00000 q^{29} +6.60440 q^{31} -3.61803 q^{33} -2.47214 q^{37} +0.449028 q^{39} -0.527864 q^{41} -6.88191 q^{43} -0.449028 q^{47} -6.47214 q^{49} +0.277515 q^{51} +6.32624 q^{53} +5.85410 q^{57} -11.3067 q^{59} +0.854102 q^{61} -1.17557 q^{63} +10.9637 q^{67} -8.94427 q^{69} -3.07768 q^{71} -2.61803 q^{73} -2.23607 q^{77} -5.98385 q^{79} -1.52786 q^{81} -5.87785 q^{87} -14.4721 q^{89} +0.277515 q^{91} +7.76393 q^{93} -3.85410 q^{97} +4.97980 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^9	$ 4 q - 2 q^{9} + 6 q^{13} - 8 q^{17} - 10 q^{21} - 20 q^{29} - 10 q^{33} + 8 q^{37} - 20 q^{41} - 8 q^{49} - 6 q^{53} + 10 q^{57} - 10 q^{61} - 6 q^{73} - 24 q^{81} - 40 q^{89} + 40 q^{93} - 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^9 + 6 * q^13 - 8 * q^17 - 10 * q^21 - 20 * q^29 - 10 * q^33 + 8 * q^37 - 20 * q^41 - 8 * q^49 - 6 * q^53 + 10 * q^57 - 10 * q^61 - 6 * q^73 - 24 * q^81 - 40 * q^89 + 40 * q^93 - 2 * q^97

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
$3$	1.17557	0.678716	0.339358	−	0.940657i	$-0.389790\pi$
$3$	1.17557	0.678716	0.339358	+	0.940657i	$0.389790\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.726543	0.274607	0.137304	−	0.990529i	$-0.456156\pi$
$7$	0.726543	0.274607	0.137304	+	0.990529i	$0.456156\pi$
$8$	0	0
$9$	−1.61803	−0.539345
$10$	0	0
$11$	−3.07768	−0.927957	−0.463978	−	0.885847i	$-0.653578\pi$
$11$	−3.07768	−0.927957	−0.463978	+	0.885847i	$0.653578\pi$
$12$	0	0
$13$	0.381966	0.105938	0.0529692	−	0.998596i	$-0.483131\pi$
$13$	0.381966	0.105938	0.0529692	+	0.998596i	$0.483131\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	4.97980	1.14244	0.571222	−	0.820796i	$-0.306470\pi$
$19$	4.97980	1.14244	0.571222	+	0.820796i	$0.306470\pi$
$20$	0	0
$21$	0.854102	0.186380
$22$	0	0
$23$	−7.60845	−1.58647	−0.793236	−	0.608914i	$-0.791605\pi$
$23$	−7.60845	−1.58647	−0.793236	+	0.608914i	$0.791605\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.42882	−1.04478
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	6.60440	1.18618	0.593092	−	0.805135i	$-0.297907\pi$
$31$	6.60440	1.18618	0.593092	+	0.805135i	$0.297907\pi$
$32$	0	0
$33$	−3.61803	−0.629819
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.47214	−0.406417	−0.203208	−	0.979136i	$-0.565137\pi$
$37$	−2.47214	−0.406417	−0.203208	+	0.979136i	$0.565137\pi$
$38$	0	0
$39$	0.449028	0.0719020
$40$	0	0
$41$	−0.527864	−0.0824385	−0.0412193	−	0.999150i	$-0.513124\pi$
$41$	−0.527864	−0.0824385	−0.0412193	+	0.999150i	$0.513124\pi$
$42$	0	0
$43$	−6.88191	−1.04948	−0.524741	−	0.851262i	$-0.675838\pi$
$43$	−6.88191	−1.04948	−0.524741	+	0.851262i	$0.675838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.449028	−0.0654975	−0.0327487	−	0.999464i	$-0.510426\pi$
$47$	−0.449028	−0.0654975	−0.0327487	+	0.999464i	$0.510426\pi$
$48$	0	0
$49$	−6.47214	−0.924591
$50$	0	0
$51$	0.277515	0.0388598
$52$	0	0
$53$	6.32624	0.868976	0.434488	−	0.900678i	$-0.356929\pi$
$53$	6.32624	0.868976	0.434488	+	0.900678i	$0.356929\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410	0.775395
$58$	0	0
$59$	−11.3067	−1.47200	−0.736002	−	0.676979i	$-0.763289\pi$
$59$	−11.3067	−1.47200	−0.736002	+	0.676979i	$0.763289\pi$
$60$	0	0
$61$	0.854102	0.109357	0.0546783	−	0.998504i	$-0.482587\pi$
$61$	0.854102	0.109357	0.0546783	+	0.998504i	$0.482587\pi$
$62$	0	0
$63$	−1.17557	−0.148108
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.9637	1.33942	0.669712	−	0.742621i	$-0.266418\pi$
$67$	10.9637	1.33942	0.669712	+	0.742621i	$0.266418\pi$
$68$	0	0
$69$	−8.94427	−1.07676
$70$	0	0
$71$	−3.07768	−0.365254	−0.182627	−	0.983182i	$-0.558460\pi$
$71$	−3.07768	−0.365254	−0.182627	+	0.983182i	$0.558460\pi$
$72$	0	0
$73$	−2.61803	−0.306418	−0.153209	−	0.988194i	$-0.548961\pi$
$73$	−2.61803	−0.306418	−0.153209	+	0.988194i	$0.548961\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.23607	−0.254824
$78$	0	0
$79$	−5.98385	−0.673236	−0.336618	−	0.941641i	$-0.609283\pi$
$79$	−5.98385	−0.673236	−0.336618	+	0.941641i	$0.609283\pi$
$80$	0	0
$81$	−1.52786	−0.169763
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.87785	−0.630172
$88$	0	0
$89$	−14.4721	−1.53404	−0.767022	−	0.641621i	$-0.778262\pi$
$89$	−14.4721	−1.53404	−0.767022	+	0.641621i	$0.778262\pi$
$90$	0	0
$91$	0.277515	0.0290914
$92$	0	0
$93$	7.76393	0.805082
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.85410	−0.391325	−0.195662	−	0.980671i	$-0.562686\pi$
$97$	−3.85410	−0.391325	−0.195662	+	0.980671i	$0.562686\pi$
$98$	0	0
$99$	4.97980	0.500488
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	2.17963	0.214765	0.107383	−	0.994218i	$-0.465753\pi$
$103$	2.17963	0.214765	0.107383	+	0.994218i	$0.465753\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.17557	0.113647	0.0568233	−	0.998384i	$-0.481903\pi$
$107$	1.17557	0.113647	0.0568233	+	0.998384i	$0.481903\pi$
$108$	0	0
$109$	−12.0902	−1.15803	−0.579014	−	0.815318i	$-0.696562\pi$
$109$	−12.0902	−1.15803	−0.579014	+	0.815318i	$0.696562\pi$
$110$	0	0
$111$	−2.90617	−0.275841
$112$	0	0
$113$	−17.1803	−1.61619	−0.808095	−	0.589052i	$-0.799501\pi$
$113$	−17.1803	−1.61619	−0.808095	+	0.589052i	$0.799501\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.618034	−0.0571373
$118$	0	0
$119$	0.171513	0.0157226
$120$	0	0
$121$	−1.52786	−0.138897
$122$	0	0
$123$	−0.620541	−0.0559523
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.60845	0.675141	0.337570	−	0.941300i	$-0.390395\pi$
$127$	7.60845	0.675141	0.337570	+	0.941300i	$0.390395\pi$
$128$	0	0
$129$	−8.09017	−0.712300
$130$	0	0
$131$	−15.4944	−1.35375	−0.676877	−	0.736096i	$-0.736667\pi$
$131$	−15.4944	−1.35375	−0.676877	+	0.736096i	$0.736667\pi$
$132$	0	0
$133$	3.61803	0.313723
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.67376	−0.228435	−0.114217	−	0.993456i	$-0.536436\pi$
$137$	−2.67376	−0.228435	−0.114217	+	0.993456i	$0.536436\pi$
$138$	0	0
$139$	7.43694	0.630793	0.315396	−	0.948960i	$-0.397863\pi$
$139$	7.43694	0.630793	0.315396	+	0.948960i	$0.397863\pi$
$140$	0	0
$141$	−0.527864	−0.0444542
$142$	0	0
$143$	−1.17557	−0.0983061
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.60845	−0.627535
$148$	0	0
$149$	−15.4721	−1.26753	−0.633763	−	0.773527i	$-0.718491\pi$
$149$	−15.4721	−1.26753	−0.633763	+	0.773527i	$0.718491\pi$
$150$	0	0
$151$	16.2210	1.32004	0.660022	−	0.751247i	$-0.270547\pi$
$151$	16.2210	1.32004	0.660022	+	0.751247i	$0.270547\pi$
$152$	0	0
$153$	−0.381966	−0.0308801
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.8541	1.50472	0.752361	−	0.658752i	$-0.228915\pi$
$157$	18.8541	1.50472	0.752361	+	0.658752i	$0.228915\pi$
$158$	0	0
$159$	7.43694	0.589788
$160$	0	0
$161$	−5.52786	−0.435657
$162$	0	0
$163$	−8.05748	−0.631111	−0.315555	−	0.948907i	$-0.602191\pi$
$163$	−8.05748	−0.631111	−0.315555	+	0.948907i	$0.602191\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.81234	−0.449772	−0.224886	−	0.974385i	$-0.572201\pi$
$167$	−5.81234	−0.449772	−0.224886	+	0.974385i	$0.572201\pi$
$168$	0	0
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−8.05748	−0.616171
$172$	0	0
$173$	21.6525	1.64621	0.823104	−	0.567891i	$-0.192241\pi$
$173$	21.6525	1.64621	0.823104	+	0.567891i	$0.192241\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.2918	−0.999073
$178$	0	0
$179$	0.171513	0.0128195	0.00640976	−	0.999979i	$-0.497960\pi$
$179$	0.171513	0.0128195	0.00640976	+	0.999979i	$0.497960\pi$
$180$	0	0
$181$	9.88854	0.735010	0.367505	−	0.930022i	$-0.380212\pi$
$181$	9.88854	0.735010	0.367505	+	0.930022i	$0.380212\pi$
$182$	0	0
$183$	1.00406	0.0742220
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.726543	−0.0531301
$188$	0	0
$189$	−3.94427	−0.286904
$190$	0	0
$191$	22.8254	1.65158	0.825792	−	0.563974i	$-0.190728\pi$
$191$	22.8254	1.65158	0.825792	+	0.563974i	$0.190728\pi$
$192$	0	0
$193$	−11.8541	−0.853277	−0.426638	−	0.904422i	$-0.640302\pi$
$193$	−11.8541	−0.853277	−0.426638	+	0.904422i	$0.640302\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.65248	−0.260228	−0.130114	−	0.991499i	$-0.541534\pi$
$197$	−3.65248	−0.260228	−0.130114	+	0.991499i	$0.541534\pi$
$198$	0	0
$199$	−5.81234	−0.412026	−0.206013	−	0.978549i	$-0.566049\pi$
$199$	−5.81234	−0.412026	−0.206013	+	0.978549i	$0.566049\pi$
$200$	0	0
$201$	12.8885	0.909088
$202$	0	0
$203$	−3.63271	−0.254966
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.3107	0.855655
$208$	0	0
$209$	−15.3262	−1.06014
$210$	0	0
$211$	1.79611	0.123649	0.0618247	−	0.998087i	$-0.480308\pi$
$211$	1.79611	0.123649	0.0618247	+	0.998087i	$0.480308\pi$
$212$	0	0
$213$	−3.61803	−0.247904
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.79837	0.325735
$218$	0	0
$219$	−3.07768	−0.207971
$220$	0	0
$221$	0.0901699	0.00606549
$222$	0	0
$223$	22.1643	1.48423	0.742117	−	0.670271i	$-0.233822\pi$
$223$	22.1643	1.48423	0.742117	+	0.670271i	$0.233822\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.85359	0.654006	0.327003	−	0.945023i	$-0.393961\pi$
$227$	9.85359	0.654006	0.327003	+	0.945023i	$0.393961\pi$
$228$	0	0
$229$	−17.5623	−1.16055	−0.580275	−	0.814421i	$-0.697055\pi$
$229$	−17.5623	−1.16055	−0.580275	+	0.814421i	$0.697055\pi$
$230$	0	0
$231$	−2.62866	−0.172953
$232$	0	0
$233$	−21.6525	−1.41850	−0.709250	−	0.704957i	$-0.750966\pi$
$233$	−21.6525	−1.41850	−0.709250	+	0.704957i	$0.750966\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.03444	−0.456936
$238$	0	0
$239$	20.9232	1.35341	0.676706	−	0.736253i	$-0.263407\pi$
$239$	20.9232	1.35341	0.676706	+	0.736253i	$0.263407\pi$
$240$	0	0
$241$	9.47214	0.610154	0.305077	−	0.952328i	$-0.401318\pi$
$241$	9.47214	0.610154	0.305077	+	0.952328i	$0.401318\pi$
$242$	0	0
$243$	14.4904	0.929557
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.90211	0.121029
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.88597	0.497758	0.248879	−	0.968535i	$-0.419938\pi$
$251$	7.88597	0.497758	0.248879	+	0.968535i	$0.419938\pi$
$252$	0	0
$253$	23.4164	1.47218
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.2361	0.638508	0.319254	−	0.947669i	$-0.396568\pi$
$257$	10.2361	0.638508	0.319254	+	0.947669i	$0.396568\pi$
$258$	0	0
$259$	−1.79611	−0.111605
$260$	0	0
$261$	8.09017	0.500769
$262$	0	0
$263$	21.2663	1.31133	0.655667	−	0.755050i	$-0.272387\pi$
$263$	21.2663	1.31133	0.655667	+	0.755050i	$0.272387\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−17.0130	−1.04118
$268$	0	0
$269$	−7.90983	−0.482271	−0.241135	−	0.970491i	$-0.577520\pi$
$269$	−7.90983	−0.482271	−0.241135	+	0.970491i	$0.577520\pi$
$270$	0	0
$271$	−25.4540	−1.54622	−0.773111	−	0.634271i	$-0.781300\pi$
$271$	−25.4540	−1.54622	−0.773111	+	0.634271i	$0.781300\pi$
$272$	0	0
$273$	0.326238	0.0197448
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.2361	−0.855362	−0.427681	−	0.903930i	$-0.640669\pi$
$277$	−14.2361	−0.855362	−0.427681	+	0.903930i	$0.640669\pi$
$278$	0	0
$279$	−10.6861	−0.639762
$280$	0	0
$281$	−17.8885	−1.06714	−0.533571	−	0.845756i	$-0.679150\pi$
$281$	−17.8885	−1.06714	−0.533571	+	0.845756i	$0.679150\pi$
$282$	0	0
$283$	26.4176	1.57036	0.785181	−	0.619266i	$-0.212570\pi$
$283$	26.4176	1.57036	0.785181	+	0.619266i	$0.212570\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.383516	−0.0226382
$288$	0	0
$289$	−16.9443	−0.996722
$290$	0	0
$291$	−4.53077	−0.265598
$292$	0	0
$293$	10.7082	0.625580	0.312790	−	0.949822i	$-0.398736\pi$
$293$	10.7082	0.625580	0.312790	+	0.949822i	$0.398736\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.7082	0.969508
$298$	0	0
$299$	−2.90617	−0.168068
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	−8.22899	−0.472743
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.0009	−1.36981	−0.684903	−	0.728635i	$-0.740155\pi$
$307$	−24.0009	−1.36981	−0.684903	+	0.728635i	$0.740155\pi$
$308$	0	0
$309$	2.56231	0.145764
$310$	0	0
$311$	−16.0090	−0.907785	−0.453892	−	0.891057i	$-0.649965\pi$
$311$	−16.0090	−0.907785	−0.453892	+	0.891057i	$0.649965\pi$
$312$	0	0
$313$	−22.8328	−1.29059	−0.645294	−	0.763935i	$-0.723265\pi$
$313$	−22.8328	−1.29059	−0.645294	+	0.763935i	$0.723265\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9443	0.951685	0.475843	−	0.879530i	$-0.342143\pi$
$317$	16.9443	0.951685	0.475843	+	0.879530i	$0.342143\pi$
$318$	0	0
$319$	15.3884	0.861586
$320$	0	0
$321$	1.38197	0.0771338
$322$	0	0
$323$	1.17557	0.0654105
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.2128	−0.785972
$328$	0	0
$329$	−0.326238	−0.0179861
$330$	0	0
$331$	18.0171	0.990308	0.495154	−	0.868805i	$-0.335112\pi$
$331$	18.0171	0.990308	0.495154	+	0.868805i	$0.335112\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.6525	−1.83317	−0.916584	−	0.399843i	$-0.869065\pi$
$337$	−33.6525	−1.83317	−0.916584	+	0.399843i	$0.869065\pi$
$338$	0	0
$339$	−20.1967	−1.09693
$340$	0	0
$341$	−20.3262	−1.10073
$342$	0	0
$343$	−9.78808	−0.528507
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.8464	1.70960	0.854802	−	0.518954i	$-0.173678\pi$
$347$	31.8464	1.70960	0.854802	+	0.518954i	$0.173678\pi$
$348$	0	0
$349$	−12.8885	−0.689908	−0.344954	−	0.938620i	$-0.612105\pi$
$349$	−12.8885	−0.689908	−0.344954	+	0.938620i	$0.612105\pi$
$350$	0	0
$351$	−2.07363	−0.110682
$352$	0	0
$353$	−0.583592	−0.0310615	−0.0155307	−	0.999879i	$-0.504944\pi$
$353$	−0.583592	−0.0310615	−0.0155307	+	0.999879i	$0.504944\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.201626	0.0106712
$358$	0	0
$359$	−31.9524	−1.68638	−0.843192	−	0.537613i	$-0.819326\pi$
$359$	−31.9524	−1.68638	−0.843192	+	0.537613i	$0.819326\pi$
$360$	0	0
$361$	5.79837	0.305178
$362$	0	0
$363$	−1.79611	−0.0942714
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.0413	−0.732952	−0.366476	−	0.930428i	$-0.619436\pi$
$367$	−14.0413	−0.732952	−0.366476	+	0.930428i	$0.619436\pi$
$368$	0	0
$369$	0.854102	0.0444628
$370$	0	0
$371$	4.59628	0.238627
$372$	0	0
$373$	35.2705	1.82624	0.913119	−	0.407693i	$-0.133667\pi$
$373$	35.2705	1.82624	0.913119	+	0.407693i	$0.133667\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.90983	−0.0983613
$378$	0	0
$379$	−18.9151	−0.971605	−0.485802	−	0.874069i	$-0.661473\pi$
$379$	−18.9151	−0.971605	−0.485802	+	0.874069i	$0.661473\pi$
$380$	0	0
$381$	8.94427	0.458229
$382$	0	0
$383$	−31.5034	−1.60975	−0.804874	−	0.593446i	$-0.797767\pi$
$383$	−31.5034	−1.60975	−0.804874	+	0.593446i	$0.797767\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.1352	0.566032
$388$	0	0
$389$	14.4164	0.730941	0.365470	−	0.930823i	$-0.380908\pi$
$389$	14.4164	0.730941	0.365470	+	0.930823i	$0.380908\pi$
$390$	0	0
$391$	−1.79611	−0.0908333
$392$	0	0
$393$	−18.2148	−0.918814
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.7639	0.991923	0.495962	−	0.868344i	$-0.334816\pi$
$397$	19.7639	0.991923	0.495962	+	0.868344i	$0.334816\pi$
$398$	0	0
$399$	4.25325	0.212929
$400$	0	0
$401$	−4.58359	−0.228894	−0.114447	−	0.993429i	$-0.536510\pi$
$401$	−4.58359	−0.228894	−0.114447	+	0.993429i	$0.536510\pi$
$402$	0	0
$403$	2.52265	0.125662
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.60845	0.377137
$408$	0	0
$409$	0.583592	0.0288568	0.0144284	−	0.999896i	$-0.495407\pi$
$409$	0.583592	0.0288568	0.0144284	+	0.999896i	$0.495407\pi$
$410$	0	0
$411$	−3.14320	−0.155042
$412$	0	0
$413$	−8.21478	−0.404223
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.74265	0.428129
$418$	0	0
$419$	38.5568	1.88362	0.941811	−	0.336142i	$-0.109122\pi$
$419$	38.5568	1.88362	0.941811	+	0.336142i	$0.109122\pi$
$420$	0	0
$421$	8.61803	0.420017	0.210009	−	0.977700i	$-0.432651\pi$
$421$	8.61803	0.420017	0.210009	+	0.977700i	$0.432651\pi$
$422$	0	0
$423$	0.726543	0.0353257
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.620541	0.0300301
$428$	0	0
$429$	−1.38197	−0.0667219
$430$	0	0
$431$	1.51860	0.0731483	0.0365741	−	0.999331i	$-0.488355\pi$
$431$	1.51860	0.0731483	0.0365741	+	0.999331i	$0.488355\pi$
$432$	0	0
$433$	−29.5279	−1.41902	−0.709509	−	0.704696i	$-0.751083\pi$
$433$	−29.5279	−1.41902	−0.709509	+	0.704696i	$0.751083\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−37.8885	−1.81245
$438$	0	0
$439$	−17.1845	−0.820173	−0.410086	−	0.912047i	$-0.634501\pi$
$439$	−17.1845	−0.820173	−0.410086	+	0.912047i	$0.634501\pi$
$440$	0	0
$441$	10.4721	0.498673
$442$	0	0
$443$	30.0503	1.42773	0.713866	−	0.700282i	$-0.246942\pi$
$443$	30.0503	1.42773	0.713866	+	0.700282i	$0.246942\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.1886	−0.860291
$448$	0	0
$449$	16.9787	0.801275	0.400638	−	0.916237i	$-0.368789\pi$
$449$	16.9787	0.801275	0.400638	+	0.916237i	$0.368789\pi$
$450$	0	0
$451$	1.62460	0.0764994
$452$	0	0
$453$	19.0689	0.895934
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.3050	1.37083	0.685414	−	0.728154i	$-0.259621\pi$
$457$	29.3050	1.37083	0.685414	+	0.728154i	$0.259621\pi$
$458$	0	0
$459$	−1.28157	−0.0598186
$460$	0	0
$461$	−16.2705	−0.757793	−0.378897	−	0.925439i	$-0.623696\pi$
$461$	−16.2705	−0.757793	−0.378897	+	0.925439i	$0.623696\pi$
$462$	0	0
$463$	3.97574	0.184768	0.0923841	−	0.995723i	$-0.470551\pi$
$463$	3.97574	0.184768	0.0923841	+	0.995723i	$0.470551\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.1231	0.838636	0.419318	−	0.907839i	$-0.362269\pi$
$467$	18.1231	0.838636	0.419318	+	0.907839i	$0.362269\pi$
$468$	0	0
$469$	7.96556	0.367815
$470$	0	0
$471$	22.1643	1.02128
$472$	0	0
$473$	21.1803	0.973873
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.2361	−0.468677
$478$	0	0
$479$	8.44100	0.385679	0.192839	−	0.981230i	$-0.438230\pi$
$479$	8.44100	0.385679	0.192839	+	0.981230i	$0.438230\pi$
$480$	0	0
$481$	−0.944272	−0.0430551
$482$	0	0
$483$	−6.49839	−0.295687
$484$	0	0
$485$	0	0
$486$	0	0
$487$	35.1361	1.59217	0.796084	−	0.605186i	$-0.206901\pi$
$487$	35.1361	1.59217	0.796084	+	0.605186i	$0.206901\pi$
$488$	0	0
$489$	−9.47214	−0.428345
$490$	0	0
$491$	17.8456	0.805359	0.402679	−	0.915341i	$-0.368079\pi$
$491$	17.8456	0.805359	0.402679	+	0.915341i	$0.368079\pi$
$492$	0	0
$493$	−1.18034	−0.0531598
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.23607	−0.100301
$498$	0	0
$499$	23.5114	1.05252	0.526258	−	0.850325i	$-0.323595\pi$
$499$	23.5114	1.05252	0.526258	+	0.850325i	$0.323595\pi$
$500$	0	0
$501$	−6.83282	−0.305268
$502$	0	0
$503$	24.0009	1.07015	0.535074	−	0.844805i	$-0.320284\pi$
$503$	24.0009	1.07015	0.535074	+	0.844805i	$0.320284\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−15.1109	−0.671099
$508$	0	0
$509$	8.94427	0.396448	0.198224	−	0.980157i	$-0.436483\pi$
$509$	8.94427	0.396448	0.198224	+	0.980157i	$0.436483\pi$
$510$	0	0
$511$	−1.90211	−0.0841445
$512$	0	0
$513$	−27.0344	−1.19360
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.38197	0.0607788
$518$	0	0
$519$	25.4540	1.11731
$520$	0	0
$521$	−39.6869	−1.73872	−0.869358	−	0.494183i	$-0.835467\pi$
$521$	−39.6869	−1.73872	−0.869358	+	0.494183i	$0.835467\pi$
$522$	0	0
$523$	−15.9030	−0.695388	−0.347694	−	0.937608i	$-0.613035\pi$
$523$	−15.9030	−0.695388	−0.347694	+	0.937608i	$0.613035\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.55909	0.0679149
$528$	0	0
$529$	34.8885	1.51689
$530$	0	0
$531$	18.2946	0.793917
$532$	0	0
$533$	−0.201626	−0.00873340
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.201626	0.00870081
$538$	0	0
$539$	19.9192	0.857980
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	11.6247	0.498863
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.5559	−0.622364	−0.311182	−	0.950350i	$-0.600725\pi$
$547$	−14.5559	−0.622364	−0.311182	+	0.950350i	$0.600725\pi$
$548$	0	0
$549$	−1.38197	−0.0589809
$550$	0	0
$551$	−24.8990	−1.06073
$552$	0	0
$553$	−4.34752	−0.184876
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.4721	1.12166	0.560830	−	0.827931i	$-0.310482\pi$
$557$	26.4721	1.12166	0.560830	+	0.827931i	$0.310482\pi$
$558$	0	0
$559$	−2.62866	−0.111180
$560$	0	0
$561$	−0.854102	−0.0360602
$562$	0	0
$563$	−27.5276	−1.16015	−0.580076	−	0.814562i	$-0.696977\pi$
$563$	−27.5276	−1.16015	−0.580076	+	0.814562i	$0.696977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.11006	−0.0466181
$568$	0	0
$569$	8.36068	0.350498	0.175249	−	0.984524i	$-0.443927\pi$
$569$	8.36068	0.350498	0.175249	+	0.984524i	$0.443927\pi$
$570$	0	0
$571$	−32.0584	−1.34160	−0.670801	−	0.741637i	$-0.734050\pi$
$571$	−32.0584	−1.34160	−0.670801	+	0.741637i	$0.734050\pi$
$572$	0	0
$573$	26.8328	1.12096
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−33.3820	−1.38971	−0.694855	−	0.719150i	$-0.744531\pi$
$577$	−33.3820	−1.38971	−0.694855	+	0.719150i	$0.744531\pi$
$578$	0	0
$579$	−13.9353	−0.579133
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.4702	−0.806372
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.2007	−0.462301	−0.231151	−	0.972918i	$-0.574249\pi$
$587$	−11.2007	−0.462301	−0.231151	+	0.972918i	$0.574249\pi$
$588$	0	0
$589$	32.8885	1.35515
$590$	0	0
$591$	−4.29374	−0.176621
$592$	0	0
$593$	−8.59675	−0.353026	−0.176513	−	0.984298i	$-0.556482\pi$
$593$	−8.59675	−0.353026	−0.176513	+	0.984298i	$0.556482\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.83282	−0.279649
$598$	0	0
$599$	5.53483	0.226147	0.113073	−	0.993587i	$-0.463930\pi$
$599$	5.53483	0.226147	0.113073	+	0.993587i	$0.463930\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	−17.7396	−0.722411
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.90617	0.117958	0.0589789	−	0.998259i	$-0.481216\pi$
$607$	2.90617	0.117958	0.0589789	+	0.998259i	$0.481216\pi$
$608$	0	0
$609$	−4.27051	−0.173050
$610$	0	0
$611$	−0.171513	−0.00693869
$612$	0	0
$613$	−7.18034	−0.290011	−0.145006	−	0.989431i	$-0.546320\pi$
$613$	−7.18034	−0.290011	−0.145006	+	0.989431i	$0.546320\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.0689	1.49234	0.746169	−	0.665757i	$-0.231891\pi$
$617$	37.0689	1.49234	0.746169	+	0.665757i	$0.231891\pi$
$618$	0	0
$619$	−33.5115	−1.34694	−0.673470	−	0.739214i	$-0.735197\pi$
$619$	−33.5115	−1.34694	−0.673470	+	0.739214i	$0.735197\pi$
$620$	0	0
$621$	41.3050	1.65751
$622$	0	0
$623$	−10.5146	−0.421259
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18.0171	−0.719533
$628$	0	0
$629$	−0.583592	−0.0232693
$630$	0	0
$631$	−35.8221	−1.42606	−0.713029	−	0.701135i	$-0.752677\pi$
$631$	−35.8221	−1.42606	−0.713029	+	0.701135i	$0.752677\pi$
$632$	0	0
$633$	2.11146	0.0839228
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.47214	−0.0979496
$638$	0	0
$639$	4.97980	0.196998
$640$	0	0
$641$	−0.326238	−0.0128856	−0.00644281	−	0.999979i	$-0.502051\pi$
$641$	−0.326238	−0.0128856	−0.00644281	+	0.999979i	$0.502051\pi$
$642$	0	0
$643$	−19.2986	−0.761064	−0.380532	−	0.924768i	$-0.624259\pi$
$643$	−19.2986	−0.761064	−0.380532	+	0.924768i	$0.624259\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.08580	0.199943	0.0999717	−	0.994990i	$-0.468125\pi$
$647$	5.08580	0.199943	0.0999717	+	0.994990i	$0.468125\pi$
$648$	0	0
$649$	34.7984	1.36596
$650$	0	0
$651$	5.64083	0.221081
$652$	0	0
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.23607	0.165265
$658$	0	0
$659$	16.9070	0.658604	0.329302	−	0.944225i	$-0.393187\pi$
$659$	16.9070	0.658604	0.329302	+	0.944225i	$0.393187\pi$
$660$	0	0
$661$	11.0557	0.430018	0.215009	−	0.976612i	$-0.431022\pi$
$661$	11.0557	0.430018	0.215009	+	0.976612i	$0.431022\pi$
$662$	0	0
$663$	0.106001	0.00411674
$664$	0	0
$665$	0	0
$666$	0	0
$667$	38.0423	1.47300
$668$	0	0
$669$	26.0557	1.00737
$670$	0	0
$671$	−2.62866	−0.101478
$672$	0	0
$673$	33.6869	1.29854	0.649268	−	0.760560i	$-0.275076\pi$
$673$	33.6869	1.29854	0.649268	+	0.760560i	$0.275076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−45.8885	−1.76364	−0.881820	−	0.471586i	$-0.843682\pi$
$677$	−45.8885	−1.76364	−0.881820	+	0.471586i	$0.843682\pi$
$678$	0	0
$679$	−2.80017	−0.107461
$680$	0	0
$681$	11.5836	0.443884
$682$	0	0
$683$	−31.1604	−1.19232	−0.596159	−	0.802867i	$-0.703307\pi$
$683$	−31.1604	−1.19232	−0.596159	+	0.802867i	$0.703307\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20.6457	−0.787684
$688$	0	0
$689$	2.41641	0.0920578
$690$	0	0
$691$	35.1361	1.33664	0.668320	−	0.743874i	$-0.267014\pi$
$691$	35.1361	1.33664	0.668320	+	0.743874i	$0.267014\pi$
$692$	0	0
$693$	3.61803	0.137438
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.124612	−0.00472001
$698$	0	0
$699$	−25.4540	−0.962759
$700$	0	0
$701$	−47.1591	−1.78117	−0.890586	−	0.454814i	$-0.849706\pi$
$701$	−47.1591	−1.78117	−0.890586	+	0.454814i	$0.849706\pi$
$702$	0	0
$703$	−12.3107	−0.464308
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.08580	−0.191271
$708$	0	0
$709$	45.9787	1.72677	0.863383	−	0.504548i	$-0.168341\pi$
$709$	45.9787	1.72677	0.863383	+	0.504548i	$0.168341\pi$
$710$	0	0
$711$	9.68208	0.363106
$712$	0	0
$713$	−50.2492	−1.88185
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.5967	0.918582
$718$	0	0
$719$	−46.4428	−1.73202	−0.866011	−	0.500024i	$-0.833324\pi$
$719$	−46.4428	−1.73202	−0.866011	+	0.500024i	$0.833324\pi$
$720$	0	0
$721$	1.58359	0.0589761
$722$	0	0
$723$	11.1352	0.414121
$724$	0	0
$725$	0	0
$726$	0	0
$727$	38.0423	1.41091	0.705455	−	0.708755i	$-0.250743\pi$
$727$	38.0423	1.41091	0.705455	+	0.708755i	$0.250743\pi$
$728$	0	0
$729$	21.6180	0.800668
$730$	0	0
$731$	−1.62460	−0.0600879
$732$	0	0
$733$	−26.6869	−0.985704	−0.492852	−	0.870113i	$-0.664046\pi$
$733$	−26.6869	−0.985704	−0.492852	+	0.870113i	$0.664046\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.7426	−1.24293
$738$	0	0
$739$	20.8577	0.767264	0.383632	−	0.923486i	$-0.374673\pi$
$739$	20.8577	0.767264	0.383632	+	0.923486i	$0.374673\pi$
$740$	0	0
$741$	2.23607	0.0821440
$742$	0	0
$743$	−33.9605	−1.24589	−0.622945	−	0.782265i	$-0.714064\pi$
$743$	−33.9605	−1.24589	−0.622945	+	0.782265i	$0.714064\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.854102	0.0312082
$750$	0	0
$751$	−0.938545	−0.0342480	−0.0171240	−	0.999853i	$-0.505451\pi$
$751$	−0.938545	−0.0342480	−0.0171240	+	0.999853i	$0.505451\pi$
$752$	0	0
$753$	9.27051	0.337836
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.1803	1.42403	0.712017	−	0.702162i	$-0.247782\pi$
$757$	39.1803	1.42403	0.712017	+	0.702162i	$0.247782\pi$
$758$	0	0
$759$	27.5276	0.999190
$760$	0	0
$761$	−2.47214	−0.0896149	−0.0448074	−	0.998996i	$-0.514267\pi$
$761$	−2.47214	−0.0896149	−0.0448074	+	0.998996i	$0.514267\pi$
$762$	0	0
$763$	−8.78402	−0.318003
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.31877	−0.155942
$768$	0	0
$769$	13.9443	0.502843	0.251422	−	0.967878i	$-0.419102\pi$
$769$	13.9443	0.502843	0.251422	+	0.967878i	$0.419102\pi$
$770$	0	0
$771$	12.0332	0.433366
$772$	0	0
$773$	25.3050	0.910156	0.455078	−	0.890452i	$-0.349611\pi$
$773$	25.3050	0.910156	0.455078	+	0.890452i	$0.349611\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.11146	−0.0757481
$778$	0	0
$779$	−2.62866	−0.0941814
$780$	0	0
$781$	9.47214	0.338940
$782$	0	0
$783$	27.1441	0.970052
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.63271	0.129492	0.0647461	−	0.997902i	$-0.479376\pi$
$787$	3.63271	0.129492	0.0647461	+	0.997902i	$0.479376\pi$
$788$	0	0
$789$	25.0000	0.890024
$790$	0	0
$791$	−12.4822	−0.443818
$792$	0	0
$793$	0.326238	0.0115850
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.3820	0.863654	0.431827	−	0.901957i	$-0.357869\pi$
$797$	24.3820	0.863654	0.431827	+	0.901957i	$0.357869\pi$
$798$	0	0
$799$	−0.106001	−0.00375005
$800$	0	0
$801$	23.4164	0.827378
$802$	0	0
$803$	8.05748	0.284342
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.29856	−0.327325
$808$	0	0
$809$	30.2492	1.06351	0.531753	−	0.846899i	$-0.321533\pi$
$809$	30.2492	1.06351	0.531753	+	0.846899i	$0.321533\pi$
$810$	0	0
$811$	14.1068	0.495358	0.247679	−	0.968842i	$-0.420332\pi$
$811$	14.1068	0.495358	0.247679	+	0.968842i	$0.420332\pi$
$812$	0	0
$813$	−29.9230	−1.04944
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−34.2705	−1.19897
$818$	0	0
$819$	−0.449028	−0.0156903
$820$	0	0
$821$	−3.09017	−0.107848	−0.0539238	−	0.998545i	$-0.517173\pi$
$821$	−3.09017	−0.107848	−0.0539238	+	0.998545i	$0.517173\pi$
$822$	0	0
$823$	−17.0535	−0.594448	−0.297224	−	0.954808i	$-0.596061\pi$
$823$	−17.0535	−0.594448	−0.297224	+	0.954808i	$0.596061\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.4993	−1.06057	−0.530283	−	0.847821i	$-0.677914\pi$
$827$	−30.4993	−1.06057	−0.530283	+	0.847821i	$0.677914\pi$
$828$	0	0
$829$	−0.549150	−0.0190728	−0.00953639	−	0.999955i	$-0.503036\pi$
$829$	−0.549150	−0.0190728	−0.00953639	+	0.999955i	$0.503036\pi$
$830$	0	0
$831$	−16.7355	−0.580548
$832$	0	0
$833$	−1.52786	−0.0529374
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−35.8541	−1.23930
$838$	0	0
$839$	−34.3035	−1.18429	−0.592145	−	0.805831i	$-0.701719\pi$
$839$	−34.3035	−1.18429	−0.592145	+	0.805831i	$0.701719\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−21.0292	−0.724286
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.11006	−0.0381421
$848$	0	0
$849$	31.0557	1.06583
$850$	0	0
$851$	18.8091	0.644769
$852$	0	0
$853$	27.5410	0.942987	0.471493	−	0.881870i	$-0.343715\pi$
$853$	27.5410	0.942987	0.471493	+	0.881870i	$0.343715\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.4721	1.58746	0.793729	−	0.608272i	$-0.208137\pi$
$857$	46.4721	1.58746	0.793729	+	0.608272i	$0.208137\pi$
$858$	0	0
$859$	17.0130	0.580477	0.290238	−	0.956954i	$-0.406265\pi$
$859$	17.0130	0.580477	0.290238	+	0.956954i	$0.406265\pi$
$860$	0	0
$861$	−0.450850	−0.0153649
$862$	0	0
$863$	23.5114	0.800338	0.400169	−	0.916441i	$-0.368951\pi$
$863$	23.5114	0.800338	0.400169	+	0.916441i	$0.368951\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−19.9192	−0.676491
$868$	0	0
$869$	18.4164	0.624734
$870$	0	0
$871$	4.18774	0.141896
$872$	0	0
$873$	6.23607	0.211059
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.9230	0.942892	0.471446	−	0.881895i	$-0.343732\pi$
$877$	27.9230	0.942892	0.471446	+	0.881895i	$0.343732\pi$
$878$	0	0
$879$	12.5882	0.424591
$880$	0	0
$881$	−45.2492	−1.52449	−0.762243	−	0.647292i	$-0.775902\pi$
$881$	−45.2492	−1.52449	−0.762243	+	0.647292i	$0.775902\pi$
$882$	0	0
$883$	53.9857	1.81676	0.908382	−	0.418142i	$-0.137318\pi$
$883$	53.9857	1.81676	0.908382	+	0.418142i	$0.137318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.9937	−1.54432	−0.772159	−	0.635429i	$-0.780823\pi$
$887$	−45.9937	−1.54432	−0.772159	+	0.635429i	$0.780823\pi$
$888$	0	0
$889$	5.52786	0.185399
$890$	0	0
$891$	4.70228	0.157532
$892$	0	0
$893$	−2.23607	−0.0748272
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.41641	−0.114071
$898$	0	0
$899$	−33.0220	−1.10134
$900$	0	0
$901$	1.49342	0.0497531
$902$	0	0
$903$	−5.87785	−0.195603
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50.9735	1.69255	0.846274	−	0.532748i	$-0.178840\pi$
$907$	50.9735	1.69255	0.846274	+	0.532748i	$0.178840\pi$
$908$	0	0
$909$	11.3262	0.375668
$910$	0	0
$911$	38.0423	1.26040	0.630198	−	0.776434i	$-0.282974\pi$
$911$	38.0423	1.26040	0.630198	+	0.776434i	$0.282974\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.2574	−0.371751
$918$	0	0
$919$	−4.70228	−0.155114	−0.0775570	−	0.996988i	$-0.524712\pi$
$919$	−4.70228	−0.155114	−0.0775570	+	0.996988i	$0.524712\pi$
$920$	0	0
$921$	−28.2148	−0.929709
$922$	0	0
$923$	−1.17557	−0.0386944
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.52671	−0.115832
$928$	0	0
$929$	37.9443	1.24491	0.622456	−	0.782655i	$-0.286135\pi$
$929$	37.9443	1.24491	0.622456	+	0.782655i	$0.286135\pi$
$930$	0	0
$931$	−32.2299	−1.05629
$932$	0	0
$933$	−18.8197	−0.616128
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.8541	−0.746611	−0.373305	−	0.927708i	$-0.621776\pi$
$937$	−22.8541	−0.746611	−0.373305	+	0.927708i	$0.621776\pi$
$938$	0	0
$939$	−26.8416	−0.875942
$940$	0	0
$941$	−41.3394	−1.34763	−0.673813	−	0.738902i	$-0.735345\pi$
$941$	−41.3394	−1.34763	−0.673813	+	0.738902i	$0.735345\pi$
$942$	0	0
$943$	4.01623	0.130786
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.99197	0.259704	0.129852	−	0.991533i	$-0.458550\pi$
$947$	7.99197	0.259704	0.129852	+	0.991533i	$0.458550\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	0	0
$951$	19.9192	0.645924
$952$	0	0
$953$	43.2705	1.40167	0.700835	−	0.713324i	$-0.252811\pi$
$953$	43.2705	1.40167	0.700835	+	0.713324i	$0.252811\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	18.0902	0.584772
$958$	0	0
$959$	−1.94260	−0.0627299
$960$	0	0
$961$	12.6180	0.407033
$962$	0	0
$963$	−1.90211	−0.0612947
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.9848	−0.964246	−0.482123	−	0.876104i	$-0.660134\pi$
$967$	−29.9848	−0.964246	−0.482123	+	0.876104i	$0.660134\pi$
$968$	0	0
$969$	1.38197	0.0443951
$970$	0	0
$971$	8.89002	0.285294	0.142647	−	0.989774i	$-0.454439\pi$
$971$	8.89002	0.285294	0.142647	+	0.989774i	$0.454439\pi$
$972$	0	0
$973$	5.40325	0.173220
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−35.5623	−1.13774	−0.568869	−	0.822428i	$-0.692619\pi$
$977$	−35.5623	−1.13774	−0.568869	+	0.822428i	$0.692619\pi$
$978$	0	0
$979$	44.5407	1.42353
$980$	0	0
$981$	19.5623	0.624576
$982$	0	0
$983$	31.0543	0.990480	0.495240	−	0.868756i	$-0.335080\pi$
$983$	31.0543	0.990480	0.495240	+	0.868756i	$0.335080\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.383516	−0.0122074
$988$	0	0
$989$	52.3607	1.66497
$990$	0	0
$991$	−54.1167	−1.71907	−0.859537	−	0.511073i	$-0.829248\pi$
$991$	−54.1167	−1.71907	−0.859537	+	0.511073i	$0.829248\pi$
$992$	0	0
$993$	21.1803	0.672138
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.3262	−0.548727	−0.274364	−	0.961626i	$-0.588467\pi$
$997$	−17.3262	−0.548727	−0.274364	+	0.961626i	$0.588467\pi$
$998$	0	0
$999$	13.4208	0.424615

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.d.1.2	✓	4	5.4	even	2
4000.2.a.d.1.3	yes	4	20.19	odd	2
4000.2.a.g.1.2	yes	4	4.3	odd	2	inner
4000.2.a.g.1.3	yes	4	1.1	even	1	trivial
4000.2.c.c.1249.3		8	5.2	odd	4
4000.2.c.c.1249.4		8	20.3	even	4
4000.2.c.c.1249.5		8	5.3	odd	4
4000.2.c.c.1249.6		8	20.7	even	4
8000.2.a.bf.1.2		4	8.5	even	2
8000.2.a.bf.1.3		4	8.3	odd	2
8000.2.a.bi.1.2		4	40.19	odd	2
8000.2.a.bi.1.3		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)

\(f(q)\)	\(=\)	\(q+1.17557 q^{3} +0.726543 q^{7} -1.61803 q^{9} +O(q^{10})\)	\(q+1.17557 q^{3} +0.726543 q^{7} -1.61803 q^{9} -3.07768 q^{11} +0.381966 q^{13} +0.236068 q^{17} +4.97980 q^{19} +0.854102 q^{21} -7.60845 q^{23} -5.42882 q^{27} -5.00000 q^{29} +6.60440 q^{31} -3.61803 q^{33} -2.47214 q^{37} +0.449028 q^{39} -0.527864 q^{41} -6.88191 q^{43} -0.449028 q^{47} -6.47214 q^{49} +0.277515 q^{51} +6.32624 q^{53} +5.85410 q^{57} -11.3067 q^{59} +0.854102 q^{61} -1.17557 q^{63} +10.9637 q^{67} -8.94427 q^{69} -3.07768 q^{71} -2.61803 q^{73} -2.23607 q^{77} -5.98385 q^{79} -1.52786 q^{81} -5.87785 q^{87} -14.4721 q^{89} +0.277515 q^{91} +7.76393 q^{93} -3.85410 q^{97} +4.97980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^9	\( 4 q - 2 q^{9} + 6 q^{13} - 8 q^{17} - 10 q^{21} - 20 q^{29} - 10 q^{33} + 8 q^{37} - 20 q^{41} - 8 q^{49} - 6 q^{53} + 10 q^{57} - 10 q^{61} - 6 q^{73} - 24 q^{81} - 40 q^{89} + 40 q^{93} - 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^9 + 6 * q^13 - 8 * q^17 - 10 * q^21 - 20 * q^29 - 10 * q^33 + 8 * q^37 - 20 * q^41 - 8 * q^49 - 6 * q^53 + 10 * q^57 - 10 * q^61 - 6 * q^73 - 24 * q^81 - 40 * q^89 + 40 * q^93 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more