LMFDB - Embedded newform 8004.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.39711$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +0.338239 q^{11} +3.11150 q^{13} -1.20115 q^{15} -0.639136 q^{17} +0.596248 q^{19} +1.61263 q^{21} -1.00000 q^{23} -3.55724 q^{25} -1.00000 q^{27} +1.00000 q^{29} +6.81971 q^{31} -0.338239 q^{33} -1.93701 q^{35} -9.39475 q^{37} -3.11150 q^{39} -7.25774 q^{41} -10.1340 q^{43} +1.20115 q^{45} +8.35438 q^{47} -4.39942 q^{49} +0.639136 q^{51} -3.58270 q^{53} +0.406277 q^{55} -0.596248 q^{57} -7.07731 q^{59} +2.15381 q^{61} -1.61263 q^{63} +3.73739 q^{65} +14.3862 q^{67} +1.00000 q^{69} +2.94972 q^{71} -2.80159 q^{73} +3.55724 q^{75} -0.545455 q^{77} -0.0498286 q^{79} +1.00000 q^{81} -1.56713 q^{83} -0.767698 q^{85} -1.00000 q^{87} +1.00920 q^{89} -5.01771 q^{91} -6.81971 q^{93} +0.716184 q^{95} +6.80229 q^{97} +0.338239 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.20115	0.537171	0.268585	−	0.963256i	$-0.413444\pi$
$5$	1.20115	0.537171	0.268585	+	0.963256i	$0.413444\pi$
$6$	0	0
$7$	−1.61263	−0.609517	−0.304759	−	0.952430i	$-0.598576\pi$
$7$	−1.61263	−0.609517	−0.304759	+	0.952430i	$0.598576\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.338239	0.101983	0.0509915	−	0.998699i	$-0.483762\pi$
$11$	0.338239	0.101983	0.0509915	+	0.998699i	$0.483762\pi$
$12$	0	0
$13$	3.11150	0.862976	0.431488	−	0.902119i	$-0.357989\pi$
$13$	3.11150	0.862976	0.431488	+	0.902119i	$0.357989\pi$
$14$	0	0
$15$	−1.20115	−0.310136
$16$	0	0
$17$	−0.639136	−0.155013	−0.0775066	−	0.996992i	$-0.524696\pi$
$17$	−0.639136	−0.155013	−0.0775066	+	0.996992i	$0.524696\pi$
$18$	0	0
$19$	0.596248	0.136789	0.0683944	−	0.997658i	$-0.478212\pi$
$19$	0.596248	0.136789	0.0683944	+	0.997658i	$0.478212\pi$
$20$	0	0
$21$	1.61263	0.351905
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.55724	−0.711447
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.81971	1.22486	0.612428	−	0.790527i	$-0.290193\pi$
$31$	6.81971	1.22486	0.612428	+	0.790527i	$0.290193\pi$
$32$	0	0
$33$	−0.338239	−0.0588799
$34$	0	0
$35$	−1.93701	−0.327415
$36$	0	0
$37$	−9.39475	−1.54449	−0.772243	−	0.635327i	$-0.780865\pi$
$37$	−9.39475	−1.54449	−0.772243	+	0.635327i	$0.780865\pi$
$38$	0	0
$39$	−3.11150	−0.498239
$40$	0	0
$41$	−7.25774	−1.13347	−0.566734	−	0.823901i	$-0.691793\pi$
$41$	−7.25774	−1.13347	−0.566734	+	0.823901i	$0.691793\pi$
$42$	0	0
$43$	−10.1340	−1.54543	−0.772713	−	0.634756i	$-0.781101\pi$
$43$	−10.1340	−1.54543	−0.772713	+	0.634756i	$0.781101\pi$
$44$	0	0
$45$	1.20115	0.179057
$46$	0	0
$47$	8.35438	1.21861	0.609306	−	0.792935i	$-0.291448\pi$
$47$	8.35438	1.21861	0.609306	+	0.792935i	$0.291448\pi$
$48$	0	0
$49$	−4.39942	−0.628489
$50$	0	0
$51$	0.639136	0.0894969
$52$	0	0
$53$	−3.58270	−0.492122	−0.246061	−	0.969254i	$-0.579136\pi$
$53$	−3.58270	−0.492122	−0.246061	+	0.969254i	$0.579136\pi$
$54$	0	0
$55$	0.406277	0.0547823
$56$	0	0
$57$	−0.596248	−0.0789750
$58$	0	0
$59$	−7.07731	−0.921388	−0.460694	−	0.887559i	$-0.652399\pi$
$59$	−7.07731	−0.921388	−0.460694	+	0.887559i	$0.652399\pi$
$60$	0	0
$61$	2.15381	0.275767	0.137883	−	0.990448i	$-0.455970\pi$
$61$	2.15381	0.275767	0.137883	+	0.990448i	$0.455970\pi$
$62$	0	0
$63$	−1.61263	−0.203172
$64$	0	0
$65$	3.73739	0.463566
$66$	0	0
$67$	14.3862	1.75756	0.878780	−	0.477227i	$-0.158358\pi$
$67$	14.3862	1.75756	0.878780	+	0.477227i	$0.158358\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	2.94972	0.350067	0.175034	−	0.984562i	$-0.443997\pi$
$71$	2.94972	0.350067	0.175034	+	0.984562i	$0.443997\pi$
$72$	0	0
$73$	−2.80159	−0.327901	−0.163951	−	0.986469i	$-0.552424\pi$
$73$	−2.80159	−0.327901	−0.163951	+	0.986469i	$0.552424\pi$
$74$	0	0
$75$	3.55724	0.410754
$76$	0	0
$77$	−0.545455	−0.0621604
$78$	0	0
$79$	−0.0498286	−0.00560616	−0.00280308	−	0.999996i	$-0.500892\pi$
$79$	−0.0498286	−0.00560616	−0.00280308	+	0.999996i	$0.500892\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.56713	−0.172015	−0.0860074	−	0.996294i	$-0.527411\pi$
$83$	−1.56713	−0.172015	−0.0860074	+	0.996294i	$0.527411\pi$
$84$	0	0
$85$	−0.767698	−0.0832685
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	1.00920	0.106975	0.0534876	−	0.998569i	$-0.482966\pi$
$89$	1.00920	0.106975	0.0534876	+	0.998569i	$0.482966\pi$
$90$	0	0
$91$	−5.01771	−0.525999
$92$	0	0
$93$	−6.81971	−0.707171
$94$	0	0
$95$	0.716184	0.0734789
$96$	0	0
$97$	6.80229	0.690667	0.345334	−	0.938480i	$-0.387766\pi$
$97$	6.80229	0.690667	0.345334	+	0.938480i	$0.387766\pi$
$98$	0	0
$99$	0.338239	0.0339943
$100$	0	0
$101$	−17.8893	−1.78005	−0.890024	−	0.455913i	$-0.849313\pi$
$101$	−17.8893	−1.78005	−0.890024	+	0.455913i	$0.849313\pi$
$102$	0	0
$103$	0.803404	0.0791618	0.0395809	−	0.999216i	$-0.487398\pi$
$103$	0.803404	0.0791618	0.0395809	+	0.999216i	$0.487398\pi$
$104$	0	0
$105$	1.93701	0.189033
$106$	0	0
$107$	6.26733	0.605886	0.302943	−	0.953009i	$-0.402031\pi$
$107$	6.26733	0.605886	0.302943	+	0.953009i	$0.402031\pi$
$108$	0	0
$109$	5.86541	0.561804	0.280902	−	0.959736i	$-0.409366\pi$
$109$	5.86541	0.561804	0.280902	+	0.959736i	$0.409366\pi$
$110$	0	0
$111$	9.39475	0.891710
$112$	0	0
$113$	18.1926	1.71141	0.855707	−	0.517461i	$-0.173123\pi$
$113$	18.1926	1.71141	0.855707	+	0.517461i	$0.173123\pi$
$114$	0	0
$115$	−1.20115	−0.112008
$116$	0	0
$117$	3.11150	0.287659
$118$	0	0
$119$	1.03069	0.0944832
$120$	0	0
$121$	−10.8856	−0.989599
$122$	0	0
$123$	7.25774	0.654408
$124$	0	0
$125$	−10.2785	−0.919340
$126$	0	0
$127$	4.35561	0.386498	0.193249	−	0.981150i	$-0.438097\pi$
$127$	4.35561	0.386498	0.193249	+	0.981150i	$0.438097\pi$
$128$	0	0
$129$	10.1340	0.892252
$130$	0	0
$131$	8.07738	0.705724	0.352862	−	0.935675i	$-0.385209\pi$
$131$	8.07738	0.705724	0.352862	+	0.935675i	$0.385209\pi$
$132$	0	0
$133$	−0.961528	−0.0833751
$134$	0	0
$135$	−1.20115	−0.103379
$136$	0	0
$137$	−21.7999	−1.86249	−0.931245	−	0.364392i	$-0.881277\pi$
$137$	−21.7999	−1.86249	−0.931245	+	0.364392i	$0.881277\pi$
$138$	0	0
$139$	8.45828	0.717422	0.358711	−	0.933449i	$-0.383216\pi$
$139$	8.45828	0.717422	0.358711	+	0.933449i	$0.383216\pi$
$140$	0	0
$141$	−8.35438	−0.703566
$142$	0	0
$143$	1.05243	0.0880089
$144$	0	0
$145$	1.20115	0.0997501
$146$	0	0
$147$	4.39942	0.362858
$148$	0	0
$149$	−2.71469	−0.222396	−0.111198	−	0.993798i	$-0.535469\pi$
$149$	−2.71469	−0.222396	−0.111198	+	0.993798i	$0.535469\pi$
$150$	0	0
$151$	−15.0192	−1.22225	−0.611123	−	0.791536i	$-0.709282\pi$
$151$	−15.0192	−1.22225	−0.611123	+	0.791536i	$0.709282\pi$
$152$	0	0
$153$	−0.639136	−0.0516710
$154$	0	0
$155$	8.19150	0.657957
$156$	0	0
$157$	−14.5542	−1.16155	−0.580775	−	0.814064i	$-0.697251\pi$
$157$	−14.5542	−1.16155	−0.580775	+	0.814064i	$0.697251\pi$
$158$	0	0
$159$	3.58270	0.284127
$160$	0	0
$161$	1.61263	0.127093
$162$	0	0
$163$	0.681185	0.0533545	0.0266773	−	0.999644i	$-0.491507\pi$
$163$	0.681185	0.0533545	0.0266773	+	0.999644i	$0.491507\pi$
$164$	0	0
$165$	−0.406277	−0.0316286
$166$	0	0
$167$	−11.6805	−0.903863	−0.451931	−	0.892053i	$-0.649265\pi$
$167$	−11.6805	−0.903863	−0.451931	+	0.892053i	$0.649265\pi$
$168$	0	0
$169$	−3.31854	−0.255272
$170$	0	0
$171$	0.596248	0.0455962
$172$	0	0
$173$	−9.91936	−0.754155	−0.377077	−	0.926182i	$-0.623071\pi$
$173$	−9.91936	−0.754155	−0.377077	+	0.926182i	$0.623071\pi$
$174$	0	0
$175$	5.73651	0.433639
$176$	0	0
$177$	7.07731	0.531963
$178$	0	0
$179$	0.993143	0.0742310	0.0371155	−	0.999311i	$-0.488183\pi$
$179$	0.993143	0.0742310	0.0371155	+	0.999311i	$0.488183\pi$
$180$	0	0
$181$	−11.2303	−0.834738	−0.417369	−	0.908737i	$-0.637048\pi$
$181$	−11.2303	−0.834738	−0.417369	+	0.908737i	$0.637048\pi$
$182$	0	0
$183$	−2.15381	−0.159214
$184$	0	0
$185$	−11.2845	−0.829653
$186$	0	0
$187$	−0.216181	−0.0158087
$188$	0	0
$189$	1.61263	0.117302
$190$	0	0
$191$	−6.98963	−0.505752	−0.252876	−	0.967499i	$-0.581376\pi$
$191$	−6.98963	−0.505752	−0.252876	+	0.967499i	$0.581376\pi$
$192$	0	0
$193$	23.6537	1.70263	0.851316	−	0.524653i	$-0.175805\pi$
$193$	23.6537	1.70263	0.851316	+	0.524653i	$0.175805\pi$
$194$	0	0
$195$	−3.73739	−0.267640
$196$	0	0
$197$	−12.5658	−0.895279	−0.447640	−	0.894214i	$-0.647735\pi$
$197$	−12.5658	−0.895279	−0.447640	+	0.894214i	$0.647735\pi$
$198$	0	0
$199$	−13.2327	−0.938043	−0.469022	−	0.883187i	$-0.655393\pi$
$199$	−13.2327	−0.938043	−0.469022	+	0.883187i	$0.655393\pi$
$200$	0	0
$201$	−14.3862	−1.01473
$202$	0	0
$203$	−1.61263	−0.113184
$204$	0	0
$205$	−8.71764	−0.608866
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0.201675	0.0139501
$210$	0	0
$211$	17.0732	1.17537	0.587684	−	0.809091i	$-0.300040\pi$
$211$	17.0732	1.17537	0.587684	+	0.809091i	$0.300040\pi$
$212$	0	0
$213$	−2.94972	−0.202111
$214$	0	0
$215$	−12.1725	−0.830158
$216$	0	0
$217$	−10.9977	−0.746570
$218$	0	0
$219$	2.80159	0.189314
$220$	0	0
$221$	−1.98867	−0.133773
$222$	0	0
$223$	−2.73917	−0.183428	−0.0917141	−	0.995785i	$-0.529235\pi$
$223$	−2.73917	−0.183428	−0.0917141	+	0.995785i	$0.529235\pi$
$224$	0	0
$225$	−3.55724	−0.237149
$226$	0	0
$227$	15.3610	1.01954	0.509772	−	0.860309i	$-0.329730\pi$
$227$	15.3610	1.01954	0.509772	+	0.860309i	$0.329730\pi$
$228$	0	0
$229$	−13.5282	−0.893966	−0.446983	−	0.894543i	$-0.647502\pi$
$229$	−13.5282	−0.893966	−0.446983	+	0.894543i	$0.647502\pi$
$230$	0	0
$231$	0.545455	0.0358883
$232$	0	0
$233$	2.16344	0.141732	0.0708660	−	0.997486i	$-0.477424\pi$
$233$	2.16344	0.141732	0.0708660	+	0.997486i	$0.477424\pi$
$234$	0	0
$235$	10.0349	0.654603
$236$	0	0
$237$	0.0498286	0.00323672
$238$	0	0
$239$	−0.642665	−0.0415705	−0.0207853	−	0.999784i	$-0.506617\pi$
$239$	−0.642665	−0.0415705	−0.0207853	+	0.999784i	$0.506617\pi$
$240$	0	0
$241$	−19.7133	−1.26985	−0.634924	−	0.772575i	$-0.718969\pi$
$241$	−19.7133	−1.26985	−0.634924	+	0.772575i	$0.718969\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.28437	−0.337606
$246$	0	0
$247$	1.85523	0.118045
$248$	0	0
$249$	1.56713	0.0993128
$250$	0	0
$251$	22.4803	1.41895	0.709473	−	0.704732i	$-0.248933\pi$
$251$	22.4803	1.41895	0.709473	+	0.704732i	$0.248933\pi$
$252$	0	0
$253$	−0.338239	−0.0212649
$254$	0	0
$255$	0.767698	0.0480751
$256$	0	0
$257$	3.21026	0.200250	0.100125	−	0.994975i	$-0.468076\pi$
$257$	3.21026	0.200250	0.100125	+	0.994975i	$0.468076\pi$
$258$	0	0
$259$	15.1503	0.941391
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	11.0263	0.679911	0.339955	−	0.940442i	$-0.389588\pi$
$263$	11.0263	0.679911	0.339955	+	0.940442i	$0.389588\pi$
$264$	0	0
$265$	−4.30337	−0.264354
$266$	0	0
$267$	−1.00920	−0.0617621
$268$	0	0
$269$	29.7003	1.81086	0.905429	−	0.424498i	$-0.139549\pi$
$269$	29.7003	1.81086	0.905429	+	0.424498i	$0.139549\pi$
$270$	0	0
$271$	−5.89141	−0.357877	−0.178939	−	0.983860i	$-0.557266\pi$
$271$	−5.89141	−0.357877	−0.178939	+	0.983860i	$0.557266\pi$
$272$	0	0
$273$	5.01771	0.303685
$274$	0	0
$275$	−1.20320	−0.0725556
$276$	0	0
$277$	2.94396	0.176886	0.0884428	−	0.996081i	$-0.471811\pi$
$277$	2.94396	0.176886	0.0884428	+	0.996081i	$0.471811\pi$
$278$	0	0
$279$	6.81971	0.408285
$280$	0	0
$281$	−4.01684	−0.239625	−0.119812	−	0.992797i	$-0.538229\pi$
$281$	−4.01684	−0.239625	−0.119812	+	0.992797i	$0.538229\pi$
$282$	0	0
$283$	−4.45516	−0.264831	−0.132416	−	0.991194i	$-0.542273\pi$
$283$	−4.45516	−0.264831	−0.132416	+	0.991194i	$0.542273\pi$
$284$	0	0
$285$	−0.716184	−0.0424231
$286$	0	0
$287$	11.7041	0.690868
$288$	0	0
$289$	−16.5915	−0.975971
$290$	0	0
$291$	−6.80229	−0.398757
$292$	0	0
$293$	−21.7830	−1.27258	−0.636288	−	0.771452i	$-0.719531\pi$
$293$	−21.7830	−1.27258	−0.636288	+	0.771452i	$0.719531\pi$
$294$	0	0
$295$	−8.50092	−0.494943
$296$	0	0
$297$	−0.338239	−0.0196266
$298$	0	0
$299$	−3.11150	−0.179943
$300$	0	0
$301$	16.3425	0.941963
$302$	0	0
$303$	17.8893	1.02771
$304$	0	0
$305$	2.58705	0.148134
$306$	0	0
$307$	−9.41091	−0.537109	−0.268554	−	0.963265i	$-0.586546\pi$
$307$	−9.41091	−0.537109	−0.268554	+	0.963265i	$0.586546\pi$
$308$	0	0
$309$	−0.803404	−0.0457041
$310$	0	0
$311$	−27.3265	−1.54954	−0.774770	−	0.632243i	$-0.782135\pi$
$311$	−27.3265	−1.54954	−0.774770	+	0.632243i	$0.782135\pi$
$312$	0	0
$313$	−4.68271	−0.264682	−0.132341	−	0.991204i	$-0.542249\pi$
$313$	−4.68271	−0.264682	−0.132341	+	0.991204i	$0.542249\pi$
$314$	0	0
$315$	−1.93701	−0.109138
$316$	0	0
$317$	10.0704	0.565608	0.282804	−	0.959178i	$-0.408735\pi$
$317$	10.0704	0.565608	0.282804	+	0.959178i	$0.408735\pi$
$318$	0	0
$319$	0.338239	0.0189378
$320$	0	0
$321$	−6.26733	−0.349808
$322$	0	0
$323$	−0.381083	−0.0212041
$324$	0	0
$325$	−11.0684	−0.613962
$326$	0	0
$327$	−5.86541	−0.324358
$328$	0	0
$329$	−13.4725	−0.742765
$330$	0	0
$331$	−11.1249	−0.611481	−0.305741	−	0.952115i	$-0.598904\pi$
$331$	−11.1249	−0.611481	−0.305741	+	0.952115i	$0.598904\pi$
$332$	0	0
$333$	−9.39475	−0.514829
$334$	0	0
$335$	17.2800	0.944110
$336$	0	0
$337$	−6.15480	−0.335273	−0.167637	−	0.985849i	$-0.553614\pi$
$337$	−6.15480	−0.335273	−0.167637	+	0.985849i	$0.553614\pi$
$338$	0	0
$339$	−18.1926	−0.988085
$340$	0	0
$341$	2.30669	0.124914
$342$	0	0
$343$	18.3831	0.992592
$344$	0	0
$345$	1.20115	0.0646678
$346$	0	0
$347$	−21.9824	−1.18008	−0.590039	−	0.807375i	$-0.700888\pi$
$347$	−21.9824	−1.18008	−0.590039	+	0.807375i	$0.700888\pi$
$348$	0	0
$349$	−3.02475	−0.161911	−0.0809557	−	0.996718i	$-0.525797\pi$
$349$	−3.02475	−0.161911	−0.0809557	+	0.996718i	$0.525797\pi$
$350$	0	0
$351$	−3.11150	−0.166080
$352$	0	0
$353$	−7.90540	−0.420762	−0.210381	−	0.977619i	$-0.567470\pi$
$353$	−7.90540	−0.420762	−0.210381	+	0.977619i	$0.567470\pi$
$354$	0	0
$355$	3.54306	0.188046
$356$	0	0
$357$	−1.03069	−0.0545499
$358$	0	0
$359$	−31.1058	−1.64170	−0.820850	−	0.571143i	$-0.806500\pi$
$359$	−31.1058	−1.64170	−0.820850	+	0.571143i	$0.806500\pi$
$360$	0	0
$361$	−18.6445	−0.981289
$362$	0	0
$363$	10.8856	0.571346
$364$	0	0
$365$	−3.36513	−0.176139
$366$	0	0
$367$	−27.8879	−1.45574	−0.727868	−	0.685717i	$-0.759489\pi$
$367$	−27.8879	−1.45574	−0.727868	+	0.685717i	$0.759489\pi$
$368$	0	0
$369$	−7.25774	−0.377823
$370$	0	0
$371$	5.77758	0.299957
$372$	0	0
$373$	8.19547	0.424345	0.212173	−	0.977232i	$-0.431946\pi$
$373$	8.19547	0.424345	0.212173	+	0.977232i	$0.431946\pi$
$374$	0	0
$375$	10.2785	0.530781
$376$	0	0
$377$	3.11150	0.160251
$378$	0	0
$379$	−0.785868	−0.0403673	−0.0201837	−	0.999796i	$-0.506425\pi$
$379$	−0.785868	−0.0403673	−0.0201837	+	0.999796i	$0.506425\pi$
$380$	0	0
$381$	−4.35561	−0.223145
$382$	0	0
$383$	1.00046	0.0511212	0.0255606	−	0.999673i	$-0.491863\pi$
$383$	1.00046	0.0511212	0.0255606	+	0.999673i	$0.491863\pi$
$384$	0	0
$385$	−0.655174	−0.0333908
$386$	0	0
$387$	−10.1340	−0.515142
$388$	0	0
$389$	1.73378	0.0879061	0.0439531	−	0.999034i	$-0.486005\pi$
$389$	1.73378	0.0879061	0.0439531	+	0.999034i	$0.486005\pi$
$390$	0	0
$391$	0.639136	0.0323225
$392$	0	0
$393$	−8.07738	−0.407450
$394$	0	0
$395$	−0.0598517	−0.00301147
$396$	0	0
$397$	4.10969	0.206259	0.103130	−	0.994668i	$-0.467114\pi$
$397$	4.10969	0.206259	0.103130	+	0.994668i	$0.467114\pi$
$398$	0	0
$399$	0.961528	0.0481366
$400$	0	0
$401$	−25.7878	−1.28778	−0.643890	−	0.765118i	$-0.722680\pi$
$401$	−25.7878	−1.28778	−0.643890	+	0.765118i	$0.722680\pi$
$402$	0	0
$403$	21.2195	1.05702
$404$	0	0
$405$	1.20115	0.0596857
$406$	0	0
$407$	−3.17767	−0.157511
$408$	0	0
$409$	16.9038	0.835837	0.417919	−	0.908484i	$-0.362760\pi$
$409$	16.9038	0.835837	0.417919	+	0.908484i	$0.362760\pi$
$410$	0	0
$411$	21.7999	1.07531
$412$	0	0
$413$	11.4131	0.561602
$414$	0	0
$415$	−1.88236	−0.0924013
$416$	0	0
$417$	−8.45828	−0.414204
$418$	0	0
$419$	−23.4626	−1.14622	−0.573111	−	0.819478i	$-0.694264\pi$
$419$	−23.4626	−1.14622	−0.573111	+	0.819478i	$0.694264\pi$
$420$	0	0
$421$	−7.82889	−0.381557	−0.190778	−	0.981633i	$-0.561101\pi$
$421$	−7.82889	−0.381557	−0.190778	+	0.981633i	$0.561101\pi$
$422$	0	0
$423$	8.35438	0.406204
$424$	0	0
$425$	2.27356	0.110284
$426$	0	0
$427$	−3.47329	−0.168085
$428$	0	0
$429$	−1.05243	−0.0508120
$430$	0	0
$431$	−3.45370	−0.166359	−0.0831795	−	0.996535i	$-0.526507\pi$
$431$	−3.45370	−0.166359	−0.0831795	+	0.996535i	$0.526507\pi$
$432$	0	0
$433$	−7.84877	−0.377188	−0.188594	−	0.982055i	$-0.560393\pi$
$433$	−7.84877	−0.377188	−0.188594	+	0.982055i	$0.560393\pi$
$434$	0	0
$435$	−1.20115	−0.0575908
$436$	0	0
$437$	−0.596248	−0.0285224
$438$	0	0
$439$	−40.5186	−1.93385	−0.966925	−	0.255062i	$-0.917904\pi$
$439$	−40.5186	−1.93385	−0.966925	+	0.255062i	$0.917904\pi$
$440$	0	0
$441$	−4.39942	−0.209496
$442$	0	0
$443$	−29.7763	−1.41472	−0.707358	−	0.706856i	$-0.750113\pi$
$443$	−29.7763	−1.41472	−0.707358	+	0.706856i	$0.750113\pi$
$444$	0	0
$445$	1.21220	0.0574639
$446$	0	0
$447$	2.71469	0.128401
$448$	0	0
$449$	33.0566	1.56004	0.780018	−	0.625757i	$-0.215210\pi$
$449$	33.0566	1.56004	0.780018	+	0.625757i	$0.215210\pi$
$450$	0	0
$451$	−2.45485	−0.115595
$452$	0	0
$453$	15.0192	0.705664
$454$	0	0
$455$	−6.02702	−0.282551
$456$	0	0
$457$	15.8554	0.741682	0.370841	−	0.928696i	$-0.379069\pi$
$457$	15.8554	0.741682	0.370841	+	0.928696i	$0.379069\pi$
$458$	0	0
$459$	0.639136	0.0298323
$460$	0	0
$461$	−1.78319	−0.0830516	−0.0415258	−	0.999137i	$-0.513222\pi$
$461$	−1.78319	−0.0830516	−0.0415258	+	0.999137i	$0.513222\pi$
$462$	0	0
$463$	34.3951	1.59848	0.799238	−	0.601015i	$-0.205237\pi$
$463$	34.3951	1.59848	0.799238	+	0.601015i	$0.205237\pi$
$464$	0	0
$465$	−8.19150	−0.379872
$466$	0	0
$467$	−17.8508	−0.826038	−0.413019	−	0.910722i	$-0.635526\pi$
$467$	−17.8508	−0.826038	−0.413019	+	0.910722i	$0.635526\pi$
$468$	0	0
$469$	−23.1997	−1.07126
$470$	0	0
$471$	14.5542	0.670622
$472$	0	0
$473$	−3.42773	−0.157607
$474$	0	0
$475$	−2.12100	−0.0973180
$476$	0	0
$477$	−3.58270	−0.164041
$478$	0	0
$479$	−14.8933	−0.680492	−0.340246	−	0.940337i	$-0.610510\pi$
$479$	−14.8933	−0.680492	−0.340246	+	0.940337i	$0.610510\pi$
$480$	0	0
$481$	−29.2318	−1.33285
$482$	0	0
$483$	−1.61263	−0.0733772
$484$	0	0
$485$	8.17057	0.371006
$486$	0	0
$487$	31.3135	1.41895	0.709476	−	0.704730i	$-0.248932\pi$
$487$	31.3135	1.41895	0.709476	+	0.704730i	$0.248932\pi$
$488$	0	0
$489$	−0.681185	−0.0308042
$490$	0	0
$491$	0.662539	0.0299000	0.0149500	−	0.999888i	$-0.495241\pi$
$491$	0.662539	0.0299000	0.0149500	+	0.999888i	$0.495241\pi$
$492$	0	0
$493$	−0.639136	−0.0287852
$494$	0	0
$495$	0.406277	0.0182608
$496$	0	0
$497$	−4.75681	−0.213372
$498$	0	0
$499$	5.89940	0.264093	0.132047	−	0.991244i	$-0.457845\pi$
$499$	5.89940	0.264093	0.132047	+	0.991244i	$0.457845\pi$
$500$	0	0
$501$	11.6805	0.521845
$502$	0	0
$503$	−5.24014	−0.233646	−0.116823	−	0.993153i	$-0.537271\pi$
$503$	−5.24014	−0.233646	−0.116823	+	0.993153i	$0.537271\pi$
$504$	0	0
$505$	−21.4877	−0.956190
$506$	0	0
$507$	3.31854	0.147382
$508$	0	0
$509$	4.61806	0.204692	0.102346	−	0.994749i	$-0.467365\pi$
$509$	4.61806	0.204692	0.102346	+	0.994749i	$0.467365\pi$
$510$	0	0
$511$	4.51793	0.199862
$512$	0	0
$513$	−0.596248	−0.0263250
$514$	0	0
$515$	0.965010	0.0425234
$516$	0	0
$517$	2.82578	0.124278
$518$	0	0
$519$	9.91936	0.435412
$520$	0	0
$521$	−42.4262	−1.85873	−0.929363	−	0.369168i	$-0.879643\pi$
$521$	−42.4262	−1.85873	−0.929363	+	0.369168i	$0.879643\pi$
$522$	0	0
$523$	8.74382	0.382341	0.191170	−	0.981557i	$-0.438772\pi$
$523$	8.74382	0.382341	0.191170	+	0.981557i	$0.438772\pi$
$524$	0	0
$525$	−5.73651	−0.250362
$526$	0	0
$527$	−4.35872	−0.189869
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.07731	−0.307129
$532$	0	0
$533$	−22.5825	−0.978156
$534$	0	0
$535$	7.52801	0.325464
$536$	0	0
$537$	−0.993143	−0.0428573
$538$	0	0
$539$	−1.48806	−0.0640952
$540$	0	0
$541$	−23.6159	−1.01533	−0.507663	−	0.861556i	$-0.669490\pi$
$541$	−23.6159	−1.01533	−0.507663	+	0.861556i	$0.669490\pi$
$542$	0	0
$543$	11.2303	0.481936
$544$	0	0
$545$	7.04524	0.301785
$546$	0	0
$547$	6.05500	0.258893	0.129447	−	0.991586i	$-0.458680\pi$
$547$	6.05500	0.258893	0.129447	+	0.991586i	$0.458680\pi$
$548$	0	0
$549$	2.15381	0.0919222
$550$	0	0
$551$	0.596248	0.0254010
$552$	0	0
$553$	0.0803552	0.00341705
$554$	0	0
$555$	11.2845	0.479001
$556$	0	0
$557$	−39.7879	−1.68587	−0.842935	−	0.538016i	$-0.819174\pi$
$557$	−39.7879	−1.68587	−0.842935	+	0.538016i	$0.819174\pi$
$558$	0	0
$559$	−31.5321	−1.33366
$560$	0	0
$561$	0.216181	0.00912716
$562$	0	0
$563$	16.8818	0.711483	0.355741	−	0.934584i	$-0.384228\pi$
$563$	16.8818	0.711483	0.355741	+	0.934584i	$0.384228\pi$
$564$	0	0
$565$	21.8520	0.919321
$566$	0	0
$567$	−1.61263	−0.0677241
$568$	0	0
$569$	−5.00577	−0.209853	−0.104926	−	0.994480i	$-0.533461\pi$
$569$	−5.00577	−0.209853	−0.104926	+	0.994480i	$0.533461\pi$
$570$	0	0
$571$	42.7880	1.79062	0.895310	−	0.445444i	$-0.146954\pi$
$571$	42.7880	1.79062	0.895310	+	0.445444i	$0.146954\pi$
$572$	0	0
$573$	6.98963	0.291996
$574$	0	0
$575$	3.55724	0.148347
$576$	0	0
$577$	17.5686	0.731391	0.365696	−	0.930734i	$-0.380831\pi$
$577$	17.5686	0.731391	0.365696	+	0.930734i	$0.380831\pi$
$578$	0	0
$579$	−23.6537	−0.983015
$580$	0	0
$581$	2.52720	0.104846
$582$	0	0
$583$	−1.21181	−0.0501881
$584$	0	0
$585$	3.73739	0.154522
$586$	0	0
$587$	−21.9948	−0.907821	−0.453910	−	0.891047i	$-0.649971\pi$
$587$	−21.9948	−0.907821	−0.453910	+	0.891047i	$0.649971\pi$
$588$	0	0
$589$	4.06624	0.167546
$590$	0	0
$591$	12.5658	0.516890
$592$	0	0
$593$	−20.4911	−0.841468	−0.420734	−	0.907184i	$-0.638227\pi$
$593$	−20.4911	−0.841468	−0.420734	+	0.907184i	$0.638227\pi$
$594$	0	0
$595$	1.23801	0.0507536
$596$	0	0
$597$	13.2327	0.541580
$598$	0	0
$599$	−30.4985	−1.24613	−0.623067	−	0.782169i	$-0.714113\pi$
$599$	−30.4985	−1.24613	−0.623067	+	0.782169i	$0.714113\pi$
$600$	0	0
$601$	32.9470	1.34394	0.671969	−	0.740580i	$-0.265449\pi$
$601$	32.9470	1.34394	0.671969	+	0.740580i	$0.265449\pi$
$602$	0	0
$603$	14.3862	0.585853
$604$	0	0
$605$	−13.0752	−0.531584
$606$	0	0
$607$	15.9288	0.646531	0.323266	−	0.946308i	$-0.395219\pi$
$607$	15.9288	0.646531	0.323266	+	0.946308i	$0.395219\pi$
$608$	0	0
$609$	1.61263	0.0653471
$610$	0	0
$611$	25.9947	1.05163
$612$	0	0
$613$	41.2801	1.66729	0.833643	−	0.552303i	$-0.186251\pi$
$613$	41.2801	1.66729	0.833643	+	0.552303i	$0.186251\pi$
$614$	0	0
$615$	8.71764	0.351529
$616$	0	0
$617$	24.0064	0.966460	0.483230	−	0.875493i	$-0.339464\pi$
$617$	24.0064	0.966460	0.483230	+	0.875493i	$0.339464\pi$
$618$	0	0
$619$	−26.4600	−1.06352	−0.531759	−	0.846896i	$-0.678469\pi$
$619$	−26.4600	−1.06352	−0.531759	+	0.846896i	$0.678469\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−1.62747	−0.0652032
$624$	0	0
$625$	5.44012	0.217605
$626$	0	0
$627$	−0.201675	−0.00805411
$628$	0	0
$629$	6.00452	0.239416
$630$	0	0
$631$	−3.25857	−0.129722	−0.0648609	−	0.997894i	$-0.520660\pi$
$631$	−3.25857	−0.129722	−0.0648609	+	0.997894i	$0.520660\pi$
$632$	0	0
$633$	−17.0732	−0.678599
$634$	0	0
$635$	5.23175	0.207615
$636$	0	0
$637$	−13.6888	−0.542371
$638$	0	0
$639$	2.94972	0.116689
$640$	0	0
$641$	0.165723	0.00654567	0.00327283	−	0.999995i	$-0.498958\pi$
$641$	0.165723	0.00654567	0.00327283	+	0.999995i	$0.498958\pi$
$642$	0	0
$643$	−30.6352	−1.20813	−0.604067	−	0.796934i	$-0.706454\pi$
$643$	−30.6352	−1.20813	−0.604067	+	0.796934i	$0.706454\pi$
$644$	0	0
$645$	12.1725	0.479292
$646$	0	0
$647$	−5.14154	−0.202135	−0.101067	−	0.994880i	$-0.532226\pi$
$647$	−5.14154	−0.202135	−0.101067	+	0.994880i	$0.532226\pi$
$648$	0	0
$649$	−2.39383	−0.0939659
$650$	0	0
$651$	10.9977	0.431033
$652$	0	0
$653$	18.3580	0.718402	0.359201	−	0.933260i	$-0.383049\pi$
$653$	18.3580	0.718402	0.359201	+	0.933260i	$0.383049\pi$
$654$	0	0
$655$	9.70215	0.379094
$656$	0	0
$657$	−2.80159	−0.109300
$658$	0	0
$659$	−41.1689	−1.60371	−0.801857	−	0.597516i	$-0.796154\pi$
$659$	−41.1689	−1.60371	−0.801857	+	0.597516i	$0.796154\pi$
$660$	0	0
$661$	33.4208	1.29992	0.649958	−	0.759970i	$-0.274786\pi$
$661$	33.4208	1.29992	0.649958	+	0.759970i	$0.274786\pi$
$662$	0	0
$663$	1.98867	0.0772337
$664$	0	0
$665$	−1.15494	−0.0447867
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	2.73917	0.105902
$670$	0	0
$671$	0.728502	0.0281235
$672$	0	0
$673$	35.2889	1.36029	0.680143	−	0.733080i	$-0.261918\pi$
$673$	35.2889	1.36029	0.680143	+	0.733080i	$0.261918\pi$
$674$	0	0
$675$	3.55724	0.136918
$676$	0	0
$677$	−16.2575	−0.624827	−0.312414	−	0.949946i	$-0.601137\pi$
$677$	−16.2575	−0.624827	−0.312414	+	0.949946i	$0.601137\pi$
$678$	0	0
$679$	−10.9696	−0.420974
$680$	0	0
$681$	−15.3610	−0.588634
$682$	0	0
$683$	−38.3035	−1.46564	−0.732821	−	0.680421i	$-0.761797\pi$
$683$	−38.3035	−1.46564	−0.732821	+	0.680421i	$0.761797\pi$
$684$	0	0
$685$	−26.1850	−1.00048
$686$	0	0
$687$	13.5282	0.516131
$688$	0	0
$689$	−11.1476	−0.424690
$690$	0	0
$691$	5.64282	0.214663	0.107331	−	0.994223i	$-0.465769\pi$
$691$	5.64282	0.214663	0.107331	+	0.994223i	$0.465769\pi$
$692$	0	0
$693$	−0.545455	−0.0207201
$694$	0	0
$695$	10.1597	0.385378
$696$	0	0
$697$	4.63868	0.175702
$698$	0	0
$699$	−2.16344	−0.0818290
$700$	0	0
$701$	14.2253	0.537283	0.268641	−	0.963240i	$-0.413425\pi$
$701$	14.2253	0.537283	0.268641	+	0.963240i	$0.413425\pi$
$702$	0	0
$703$	−5.60160	−0.211268
$704$	0	0
$705$	−10.0349	−0.377935
$706$	0	0
$707$	28.8488	1.08497
$708$	0	0
$709$	−19.2154	−0.721650	−0.360825	−	0.932633i	$-0.617505\pi$
$709$	−19.2154	−0.721650	−0.360825	+	0.932633i	$0.617505\pi$
$710$	0	0
$711$	−0.0498286	−0.00186872
$712$	0	0
$713$	−6.81971	−0.255400
$714$	0	0
$715$	1.26413	0.0472758
$716$	0	0
$717$	0.642665	0.0240008
$718$	0	0
$719$	−3.53093	−0.131682	−0.0658408	−	0.997830i	$-0.520973\pi$
$719$	−3.53093	−0.131682	−0.0658408	+	0.997830i	$0.520973\pi$
$720$	0	0
$721$	−1.29559	−0.0482505
$722$	0	0
$723$	19.7133	0.733147
$724$	0	0
$725$	−3.55724	−0.132112
$726$	0	0
$727$	4.63590	0.171936	0.0859679	−	0.996298i	$-0.472602\pi$
$727$	4.63590	0.171936	0.0859679	+	0.996298i	$0.472602\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.47702	0.239561
$732$	0	0
$733$	−45.6038	−1.68442	−0.842208	−	0.539153i	$-0.818745\pi$
$733$	−45.6038	−1.68442	−0.842208	+	0.539153i	$0.818745\pi$
$734$	0	0
$735$	5.28437	0.194917
$736$	0	0
$737$	4.86600	0.179241
$738$	0	0
$739$	4.44161	0.163387	0.0816936	−	0.996657i	$-0.473967\pi$
$739$	4.44161	0.163387	0.0816936	+	0.996657i	$0.473967\pi$
$740$	0	0
$741$	−1.85523	−0.0681535
$742$	0	0
$743$	−1.66431	−0.0610578	−0.0305289	−	0.999534i	$-0.509719\pi$
$743$	−1.66431	−0.0610578	−0.0305289	+	0.999534i	$0.509719\pi$
$744$	0	0
$745$	−3.26076	−0.119465
$746$	0	0
$747$	−1.56713	−0.0573383
$748$	0	0
$749$	−10.1069	−0.369298
$750$	0	0
$751$	28.9356	1.05588	0.527938	−	0.849283i	$-0.322965\pi$
$751$	28.9356	1.05588	0.527938	+	0.849283i	$0.322965\pi$
$752$	0	0
$753$	−22.4803	−0.819229
$754$	0	0
$755$	−18.0403	−0.656555
$756$	0	0
$757$	−10.0491	−0.365240	−0.182620	−	0.983184i	$-0.558458\pi$
$757$	−10.0491	−0.365240	−0.182620	+	0.983184i	$0.558458\pi$
$758$	0	0
$759$	0.338239	0.0122773
$760$	0	0
$761$	−4.78753	−0.173548	−0.0867740	−	0.996228i	$-0.527656\pi$
$761$	−4.78753	−0.173548	−0.0867740	+	0.996228i	$0.527656\pi$
$762$	0	0
$763$	−9.45873	−0.342429
$764$	0	0
$765$	−0.767698	−0.0277562
$766$	0	0
$767$	−22.0211	−0.795135
$768$	0	0
$769$	−27.8260	−1.00343	−0.501715	−	0.865033i	$-0.667297\pi$
$769$	−27.8260	−1.00343	−0.501715	+	0.865033i	$0.667297\pi$
$770$	0	0
$771$	−3.21026	−0.115615
$772$	0	0
$773$	39.4542	1.41907	0.709534	−	0.704672i	$-0.248906\pi$
$773$	39.4542	1.41907	0.709534	+	0.704672i	$0.248906\pi$
$774$	0	0
$775$	−24.2593	−0.871420
$776$	0	0
$777$	−15.1503	−0.543512
$778$	0	0
$779$	−4.32741	−0.155046
$780$	0	0
$781$	0.997711	0.0357009
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−17.4818	−0.623951
$786$	0	0
$787$	31.7369	1.13130	0.565649	−	0.824646i	$-0.308626\pi$
$787$	31.7369	1.13130	0.565649	+	0.824646i	$0.308626\pi$
$788$	0	0
$789$	−11.0263	−0.392547
$790$	0	0
$791$	−29.3379	−1.04314
$792$	0	0
$793$	6.70158	0.237980
$794$	0	0
$795$	4.30337	0.152625
$796$	0	0
$797$	2.81985	0.0998843	0.0499422	−	0.998752i	$-0.484096\pi$
$797$	2.81985	0.0998843	0.0499422	+	0.998752i	$0.484096\pi$
$798$	0	0
$799$	−5.33958	−0.188901
$800$	0	0
$801$	1.00920	0.0356584
$802$	0	0
$803$	−0.947609	−0.0334404
$804$	0	0
$805$	1.93701	0.0682707
$806$	0	0
$807$	−29.7003	−1.04550
$808$	0	0
$809$	−21.1839	−0.744788	−0.372394	−	0.928075i	$-0.621463\pi$
$809$	−21.1839	−0.744788	−0.372394	+	0.928075i	$0.621463\pi$
$810$	0	0
$811$	13.9950	0.491432	0.245716	−	0.969342i	$-0.420977\pi$
$811$	13.9950	0.491432	0.245716	+	0.969342i	$0.420977\pi$
$812$	0	0
$813$	5.89141	0.206621
$814$	0	0
$815$	0.818206	0.0286605
$816$	0	0
$817$	−6.04240	−0.211397
$818$	0	0
$819$	−5.01771	−0.175333
$820$	0	0
$821$	0.808941	0.0282322	0.0141161	−	0.999900i	$-0.495507\pi$
$821$	0.808941	0.0282322	0.0141161	+	0.999900i	$0.495507\pi$
$822$	0	0
$823$	16.6231	0.579444	0.289722	−	0.957111i	$-0.406437\pi$
$823$	16.6231	0.579444	0.289722	+	0.957111i	$0.406437\pi$
$824$	0	0
$825$	1.20320	0.0418900
$826$	0	0
$827$	−46.0968	−1.60294	−0.801472	−	0.598033i	$-0.795949\pi$
$827$	−46.0968	−1.60294	−0.801472	+	0.598033i	$0.795949\pi$
$828$	0	0
$829$	−33.7905	−1.17359	−0.586796	−	0.809735i	$-0.699611\pi$
$829$	−33.7905	−1.17359	−0.586796	+	0.809735i	$0.699611\pi$
$830$	0	0
$831$	−2.94396	−0.102125
$832$	0	0
$833$	2.81183	0.0974240
$834$	0	0
$835$	−14.0300	−0.485529
$836$	0	0
$837$	−6.81971	−0.235724
$838$	0	0
$839$	1.78694	0.0616919	0.0308460	−	0.999524i	$-0.490180\pi$
$839$	1.78694	0.0616919	0.0308460	+	0.999524i	$0.490180\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	4.01684	0.138347
$844$	0	0
$845$	−3.98607	−0.137125
$846$	0	0
$847$	17.5544	0.603178
$848$	0	0
$849$	4.45516	0.152901
$850$	0	0
$851$	9.39475	0.322048
$852$	0	0
$853$	56.6350	1.93915	0.969573	−	0.244802i	$-0.0787231\pi$
$853$	56.6350	1.93915	0.969573	+	0.244802i	$0.0787231\pi$
$854$	0	0
$855$	0.716184	0.0244930
$856$	0	0
$857$	−23.1805	−0.791830	−0.395915	−	0.918287i	$-0.629573\pi$
$857$	−23.1805	−0.791830	−0.395915	+	0.918287i	$0.629573\pi$
$858$	0	0
$859$	−10.7068	−0.365312	−0.182656	−	0.983177i	$-0.558469\pi$
$859$	−10.7068	−0.365312	−0.182656	+	0.983177i	$0.558469\pi$
$860$	0	0
$861$	−11.7041	−0.398873
$862$	0	0
$863$	−12.3716	−0.421133	−0.210567	−	0.977580i	$-0.567531\pi$
$863$	−12.3716	−0.421133	−0.210567	+	0.977580i	$0.567531\pi$
$864$	0	0
$865$	−11.9146	−0.405110
$866$	0	0
$867$	16.5915	0.563477
$868$	0	0
$869$	−0.0168540	−0.000571733 0
$870$	0	0
$871$	44.7629	1.51673
$872$	0	0
$873$	6.80229	0.230222
$874$	0	0
$875$	16.5755	0.560353
$876$	0	0
$877$	−41.1193	−1.38850	−0.694250	−	0.719734i	$-0.744264\pi$
$877$	−41.1193	−1.38850	−0.694250	+	0.719734i	$0.744264\pi$
$878$	0	0
$879$	21.7830	0.734722
$880$	0	0
$881$	−33.6496	−1.13368	−0.566842	−	0.823826i	$-0.691835\pi$
$881$	−33.6496	−1.13368	−0.566842	+	0.823826i	$0.691835\pi$
$882$	0	0
$883$	−36.7557	−1.23693	−0.618463	−	0.785814i	$-0.712244\pi$
$883$	−36.7557	−1.23693	−0.618463	+	0.785814i	$0.712244\pi$
$884$	0	0
$885$	8.50092	0.285755
$886$	0	0
$887$	39.9939	1.34287	0.671433	−	0.741066i	$-0.265679\pi$
$887$	39.9939	1.34287	0.671433	+	0.741066i	$0.265679\pi$
$888$	0	0
$889$	−7.02399	−0.235577
$890$	0	0
$891$	0.338239	0.0113314
$892$	0	0
$893$	4.98129	0.166692
$894$	0	0
$895$	1.19291	0.0398747
$896$	0	0
$897$	3.11150	0.103890
$898$	0	0
$899$	6.81971	0.227450
$900$	0	0
$901$	2.28983	0.0762854
$902$	0	0
$903$	−16.3425	−0.543843
$904$	0	0
$905$	−13.4892	−0.448397
$906$	0	0
$907$	29.5832	0.982294	0.491147	−	0.871077i	$-0.336578\pi$
$907$	29.5832	0.982294	0.491147	+	0.871077i	$0.336578\pi$
$908$	0	0
$909$	−17.8893	−0.593350
$910$	0	0
$911$	−6.07788	−0.201369	−0.100685	−	0.994918i	$-0.532103\pi$
$911$	−6.07788	−0.201369	−0.100685	+	0.994918i	$0.532103\pi$
$912$	0	0
$913$	−0.530065	−0.0175426
$914$	0	0
$915$	−2.58705	−0.0855251
$916$	0	0
$917$	−13.0258	−0.430151
$918$	0	0
$919$	28.9627	0.955392	0.477696	−	0.878525i	$-0.341472\pi$
$919$	28.9627	0.955392	0.477696	+	0.878525i	$0.341472\pi$
$920$	0	0
$921$	9.41091	0.310100
$922$	0	0
$923$	9.17806	0.302100
$924$	0	0
$925$	33.4193	1.09882
$926$	0	0
$927$	0.803404	0.0263873
$928$	0	0
$929$	−8.76667	−0.287625	−0.143813	−	0.989605i	$-0.545936\pi$
$929$	−8.76667	−0.287625	−0.143813	+	0.989605i	$0.545936\pi$
$930$	0	0
$931$	−2.62315	−0.0859702
$932$	0	0
$933$	27.3265	0.894628
$934$	0	0
$935$	−0.259666	−0.00849198
$936$	0	0
$937$	−2.90935	−0.0950442	−0.0475221	−	0.998870i	$-0.515132\pi$
$937$	−2.90935	−0.0950442	−0.0475221	+	0.998870i	$0.515132\pi$
$938$	0	0
$939$	4.68271	0.152814
$940$	0	0
$941$	10.4834	0.341748	0.170874	−	0.985293i	$-0.445341\pi$
$941$	10.4834	0.341748	0.170874	+	0.985293i	$0.445341\pi$
$942$	0	0
$943$	7.25774	0.236344
$944$	0	0
$945$	1.93701	0.0630110
$946$	0	0
$947$	−31.4324	−1.02142	−0.510708	−	0.859754i	$-0.670617\pi$
$947$	−31.4324	−1.02142	−0.510708	+	0.859754i	$0.670617\pi$
$948$	0	0
$949$	−8.71716	−0.282971
$950$	0	0
$951$	−10.0704	−0.326554
$952$	0	0
$953$	−29.1720	−0.944973	−0.472487	−	0.881338i	$-0.656643\pi$
$953$	−29.1720	−0.944973	−0.472487	+	0.881338i	$0.656643\pi$
$954$	0	0
$955$	−8.39560	−0.271675
$956$	0	0
$957$	−0.338239	−0.0109337
$958$	0	0
$959$	35.1552	1.13522
$960$	0	0
$961$	15.5084	0.500271
$962$	0	0
$963$	6.26733	0.201962
$964$	0	0
$965$	28.4117	0.914605
$966$	0	0
$967$	45.4493	1.46155	0.730775	−	0.682618i	$-0.239159\pi$
$967$	45.4493	1.46155	0.730775	+	0.682618i	$0.239159\pi$
$968$	0	0
$969$	0.381083	0.0122422
$970$	0	0
$971$	−3.45579	−0.110902	−0.0554508	−	0.998461i	$-0.517660\pi$
$971$	−3.45579	−0.110902	−0.0554508	+	0.998461i	$0.517660\pi$
$972$	0	0
$973$	−13.6401	−0.437281
$974$	0	0
$975$	11.0684	0.354471
$976$	0	0
$977$	−30.5352	−0.976907	−0.488453	−	0.872590i	$-0.662439\pi$
$977$	−30.5352	−0.976907	−0.488453	+	0.872590i	$0.662439\pi$
$978$	0	0
$979$	0.341352	0.0109097
$980$	0	0
$981$	5.86541	0.187268
$982$	0	0
$983$	−42.9381	−1.36951	−0.684756	−	0.728773i	$-0.740091\pi$
$983$	−42.9381	−1.36951	−0.684756	+	0.728773i	$0.740091\pi$
$984$	0	0
$985$	−15.0935	−0.480918
$986$	0	0
$987$	13.4725	0.428836
$988$	0	0
$989$	10.1340	0.322243
$990$	0	0
$991$	−0.0808830	−0.00256933	−0.00128467	−	0.999999i	$-0.500409\pi$
$991$	−0.0808830	−0.00256933	−0.00128467	+	0.999999i	$0.500409\pi$
$992$	0	0
$993$	11.1249	0.353039
$994$	0	0
$995$	−15.8945	−0.503890
$996$	0	0
$997$	41.0203	1.29912	0.649562	−	0.760308i	$-0.274952\pi$
$997$	41.0203	1.29912	0.649562	+	0.760308i	$0.274952\pi$
$998$	0	0
$999$	9.39475	0.297237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.e.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.7	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.20115 q^{5} -1.61263 q^{7} +1.00000 q^{9} +0.338239 q^{11} +3.11150 q^{13} -1.20115 q^{15} -0.639136 q^{17} +0.596248 q^{19} +1.61263 q^{21} -1.00000 q^{23} -3.55724 q^{25} -1.00000 q^{27} +1.00000 q^{29} +6.81971 q^{31} -0.338239 q^{33} -1.93701 q^{35} -9.39475 q^{37} -3.11150 q^{39} -7.25774 q^{41} -10.1340 q^{43} +1.20115 q^{45} +8.35438 q^{47} -4.39942 q^{49} +0.639136 q^{51} -3.58270 q^{53} +0.406277 q^{55} -0.596248 q^{57} -7.07731 q^{59} +2.15381 q^{61} -1.61263 q^{63} +3.73739 q^{65} +14.3862 q^{67} +1.00000 q^{69} +2.94972 q^{71} -2.80159 q^{73} +3.55724 q^{75} -0.545455 q^{77} -0.0498286 q^{79} +1.00000 q^{81} -1.56713 q^{83} -0.767698 q^{85} -1.00000 q^{87} +1.00920 q^{89} -5.01771 q^{91} -6.81971 q^{93} +0.716184 q^{95} +6.80229 q^{97} +0.338239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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