LMFDB - Embedded newform 8004.2.a.k.1.13

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.40801$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.40801 q^{5} +3.93112 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.40801 q^{5} +3.93112 q^{7} +1.00000 q^{9} -5.83783 q^{11} +4.21687 q^{13} +2.40801 q^{15} +0.537546 q^{17} +5.08224 q^{19} +3.93112 q^{21} -1.00000 q^{23} +0.798489 q^{25} +1.00000 q^{27} +1.00000 q^{29} +1.84865 q^{31} -5.83783 q^{33} +9.46615 q^{35} -6.96593 q^{37} +4.21687 q^{39} -0.514856 q^{41} +10.1214 q^{43} +2.40801 q^{45} +4.15643 q^{47} +8.45367 q^{49} +0.537546 q^{51} -10.8627 q^{53} -14.0575 q^{55} +5.08224 q^{57} +1.94316 q^{59} +5.79793 q^{61} +3.93112 q^{63} +10.1543 q^{65} +15.2528 q^{67} -1.00000 q^{69} -3.37225 q^{71} +4.51139 q^{73} +0.798489 q^{75} -22.9492 q^{77} -6.51894 q^{79} +1.00000 q^{81} -15.5940 q^{83} +1.29441 q^{85} +1.00000 q^{87} +5.84936 q^{89} +16.5770 q^{91} +1.84865 q^{93} +12.2381 q^{95} -5.46771 q^{97} -5.83783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$5$	2.40801	1.07689	0.538446	−	0.842660i	$-0.319011\pi$
$5$	2.40801	1.07689	0.538446	+	0.842660i	$0.319011\pi$
$6$	0	0
$7$	3.93112	1.48582	0.742911	−	0.669390i	$-0.233445\pi$
$7$	3.93112	1.48582	0.742911	+	0.669390i	$0.233445\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.83783	−1.76017	−0.880086	−	0.474814i	$-0.842515\pi$
$11$	−5.83783	−1.76017	−0.880086	+	0.474814i	$0.842515\pi$
$12$	0	0
$13$	4.21687	1.16955	0.584775	−	0.811196i	$-0.301183\pi$
$13$	4.21687	1.16955	0.584775	+	0.811196i	$0.301183\pi$
$14$	0	0
$15$	2.40801	0.621744
$16$	0	0
$17$	0.537546	0.130374	0.0651870	−	0.997873i	$-0.479236\pi$
$17$	0.537546	0.130374	0.0651870	+	0.997873i	$0.479236\pi$
$18$	0	0
$19$	5.08224	1.16595	0.582973	−	0.812492i	$-0.301889\pi$
$19$	5.08224	1.16595	0.582973	+	0.812492i	$0.301889\pi$
$20$	0	0
$21$	3.93112	0.857840
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.798489	0.159698
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.84865	0.332027	0.166014	−	0.986123i	$-0.446910\pi$
$31$	1.84865	0.332027	0.166014	+	0.986123i	$0.446910\pi$
$32$	0	0
$33$	−5.83783	−1.01624
$34$	0	0
$35$	9.46615	1.60007
$36$	0	0
$37$	−6.96593	−1.14519	−0.572596	−	0.819838i	$-0.694064\pi$
$37$	−6.96593	−1.14519	−0.572596	+	0.819838i	$0.694064\pi$
$38$	0	0
$39$	4.21687	0.675240
$40$	0	0
$41$	−0.514856	−0.0804070	−0.0402035	−	0.999192i	$-0.512801\pi$
$41$	−0.514856	−0.0804070	−0.0402035	+	0.999192i	$0.512801\pi$
$42$	0	0
$43$	10.1214	1.54349	0.771746	−	0.635931i	$-0.219384\pi$
$43$	10.1214	1.54349	0.771746	+	0.635931i	$0.219384\pi$
$44$	0	0
$45$	2.40801	0.358964
$46$	0	0
$47$	4.15643	0.606278	0.303139	−	0.952946i	$-0.401965\pi$
$47$	4.15643	0.606278	0.303139	+	0.952946i	$0.401965\pi$
$48$	0	0
$49$	8.45367	1.20767
$50$	0	0
$51$	0.537546	0.0752715
$52$	0	0
$53$	−10.8627	−1.49211	−0.746054	−	0.665885i	$-0.768054\pi$
$53$	−10.8627	−1.49211	−0.746054	+	0.665885i	$0.768054\pi$
$54$	0	0
$55$	−14.0575	−1.89552
$56$	0	0
$57$	5.08224	0.673159
$58$	0	0
$59$	1.94316	0.252978	0.126489	−	0.991968i	$-0.459629\pi$
$59$	1.94316	0.252978	0.126489	+	0.991968i	$0.459629\pi$
$60$	0	0
$61$	5.79793	0.742349	0.371174	−	0.928563i	$-0.378955\pi$
$61$	5.79793	0.742349	0.371174	+	0.928563i	$0.378955\pi$
$62$	0	0
$63$	3.93112	0.495274
$64$	0	0
$65$	10.1543	1.25948
$66$	0	0
$67$	15.2528	1.86342	0.931712	−	0.363199i	$-0.118316\pi$
$67$	15.2528	1.86342	0.931712	+	0.363199i	$0.118316\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.37225	−0.400212	−0.200106	−	0.979774i	$-0.564129\pi$
$71$	−3.37225	−0.400212	−0.200106	+	0.979774i	$0.564129\pi$
$72$	0	0
$73$	4.51139	0.528018	0.264009	−	0.964520i	$-0.414955\pi$
$73$	4.51139	0.528018	0.264009	+	0.964520i	$0.414955\pi$
$74$	0	0
$75$	0.798489	0.0922016
$76$	0	0
$77$	−22.9492	−2.61530
$78$	0	0
$79$	−6.51894	−0.733438	−0.366719	−	0.930332i	$-0.619519\pi$
$79$	−6.51894	−0.733438	−0.366719	+	0.930332i	$0.619519\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.5940	−1.71166	−0.855830	−	0.517257i	$-0.826953\pi$
$83$	−15.5940	−1.71166	−0.855830	+	0.517257i	$0.826953\pi$
$84$	0	0
$85$	1.29441	0.140399
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	5.84936	0.620031	0.310015	−	0.950732i	$-0.399666\pi$
$89$	5.84936	0.620031	0.310015	+	0.950732i	$0.399666\pi$
$90$	0	0
$91$	16.5770	1.73774
$92$	0	0
$93$	1.84865	0.191696
$94$	0	0
$95$	12.2381	1.25560
$96$	0	0
$97$	−5.46771	−0.555162	−0.277581	−	0.960702i	$-0.589533\pi$
$97$	−5.46771	−0.555162	−0.277581	+	0.960702i	$0.589533\pi$
$98$	0	0
$99$	−5.83783	−0.586724
$100$	0	0
$101$	11.6666	1.16087	0.580433	−	0.814308i	$-0.302883\pi$
$101$	11.6666	1.16087	0.580433	+	0.814308i	$0.302883\pi$
$102$	0	0
$103$	−3.15254	−0.310629	−0.155315	−	0.987865i	$-0.549639\pi$
$103$	−3.15254	−0.310629	−0.155315	+	0.987865i	$0.549639\pi$
$104$	0	0
$105$	9.46615	0.923801
$106$	0	0
$107$	10.2317	0.989136	0.494568	−	0.869139i	$-0.335326\pi$
$107$	10.2317	0.989136	0.494568	+	0.869139i	$0.335326\pi$
$108$	0	0
$109$	13.5923	1.30190	0.650951	−	0.759120i	$-0.274370\pi$
$109$	13.5923	1.30190	0.650951	+	0.759120i	$0.274370\pi$
$110$	0	0
$111$	−6.96593	−0.661177
$112$	0	0
$113$	9.96009	0.936967	0.468483	−	0.883472i	$-0.344801\pi$
$113$	9.96009	0.936967	0.468483	+	0.883472i	$0.344801\pi$
$114$	0	0
$115$	−2.40801	−0.224548
$116$	0	0
$117$	4.21687	0.389850
$118$	0	0
$119$	2.11316	0.193713
$120$	0	0
$121$	23.0803	2.09821
$122$	0	0
$123$	−0.514856	−0.0464230
$124$	0	0
$125$	−10.1173	−0.904915
$126$	0	0
$127$	12.5637	1.11485	0.557424	−	0.830228i	$-0.311790\pi$
$127$	12.5637	1.11485	0.557424	+	0.830228i	$0.311790\pi$
$128$	0	0
$129$	10.1214	0.891135
$130$	0	0
$131$	−19.2085	−1.67825	−0.839126	−	0.543937i	$-0.816933\pi$
$131$	−19.2085	−1.67825	−0.839126	+	0.543937i	$0.816933\pi$
$132$	0	0
$133$	19.9789	1.73239
$134$	0	0
$135$	2.40801	0.207248
$136$	0	0
$137$	−15.8316	−1.35259	−0.676293	−	0.736633i	$-0.736415\pi$
$137$	−15.8316	−1.35259	−0.676293	+	0.736633i	$0.736415\pi$
$138$	0	0
$139$	−20.0499	−1.70061	−0.850307	−	0.526288i	$-0.823583\pi$
$139$	−20.0499	−1.70061	−0.850307	+	0.526288i	$0.823583\pi$
$140$	0	0
$141$	4.15643	0.350035
$142$	0	0
$143$	−24.6174	−2.05861
$144$	0	0
$145$	2.40801	0.199974
$146$	0	0
$147$	8.45367	0.697247
$148$	0	0
$149$	12.7729	1.04640	0.523199	−	0.852211i	$-0.324739\pi$
$149$	12.7729	1.04640	0.523199	+	0.852211i	$0.324739\pi$
$150$	0	0
$151$	−10.2496	−0.834101	−0.417050	−	0.908883i	$-0.636936\pi$
$151$	−10.2496	−0.834101	−0.417050	+	0.908883i	$0.636936\pi$
$152$	0	0
$153$	0.537546	0.0434580
$154$	0	0
$155$	4.45156	0.357558
$156$	0	0
$157$	−22.6930	−1.81110	−0.905550	−	0.424240i	$-0.860541\pi$
$157$	−22.6930	−1.81110	−0.905550	+	0.424240i	$0.860541\pi$
$158$	0	0
$159$	−10.8627	−0.861469
$160$	0	0
$161$	−3.93112	−0.309815
$162$	0	0
$163$	−3.52878	−0.276396	−0.138198	−	0.990405i	$-0.544131\pi$
$163$	−3.52878	−0.276396	−0.138198	+	0.990405i	$0.544131\pi$
$164$	0	0
$165$	−14.0575	−1.09438
$166$	0	0
$167$	−17.6254	−1.36390	−0.681948	−	0.731401i	$-0.738867\pi$
$167$	−17.6254	−1.36390	−0.681948	+	0.731401i	$0.738867\pi$
$168$	0	0
$169$	4.78201	0.367847
$170$	0	0
$171$	5.08224	0.388649
$172$	0	0
$173$	22.4634	1.70786	0.853931	−	0.520387i	$-0.174212\pi$
$173$	22.4634	1.70786	0.853931	+	0.520387i	$0.174212\pi$
$174$	0	0
$175$	3.13895	0.237283
$176$	0	0
$177$	1.94316	0.146057
$178$	0	0
$179$	6.32345	0.472637	0.236319	−	0.971676i	$-0.424059\pi$
$179$	6.32345	0.472637	0.236319	+	0.971676i	$0.424059\pi$
$180$	0	0
$181$	25.0156	1.85940	0.929698	−	0.368322i	$-0.120067\pi$
$181$	25.0156	1.85940	0.929698	+	0.368322i	$0.120067\pi$
$182$	0	0
$183$	5.79793	0.428595
$184$	0	0
$185$	−16.7740	−1.23325
$186$	0	0
$187$	−3.13810	−0.229481
$188$	0	0
$189$	3.93112	0.285947
$190$	0	0
$191$	14.4387	1.04475	0.522373	−	0.852717i	$-0.325047\pi$
$191$	14.4387	1.04475	0.522373	+	0.852717i	$0.325047\pi$
$192$	0	0
$193$	15.0308	1.08194	0.540969	−	0.841042i	$-0.318058\pi$
$193$	15.0308	1.08194	0.540969	+	0.841042i	$0.318058\pi$
$194$	0	0
$195$	10.1543	0.727161
$196$	0	0
$197$	7.61703	0.542691	0.271345	−	0.962482i	$-0.412531\pi$
$197$	7.61703	0.542691	0.271345	+	0.962482i	$0.412531\pi$
$198$	0	0
$199$	22.8783	1.62180	0.810899	−	0.585186i	$-0.198979\pi$
$199$	22.8783	1.62180	0.810899	+	0.585186i	$0.198979\pi$
$200$	0	0
$201$	15.2528	1.07585
$202$	0	0
$203$	3.93112	0.275910
$204$	0	0
$205$	−1.23978	−0.0865897
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−29.6693	−2.05227
$210$	0	0
$211$	−6.32707	−0.435573	−0.217787	−	0.975996i	$-0.569884\pi$
$211$	−6.32707	−0.435573	−0.217787	+	0.975996i	$0.569884\pi$
$212$	0	0
$213$	−3.37225	−0.231063
$214$	0	0
$215$	24.3723	1.66217
$216$	0	0
$217$	7.26725	0.493333
$218$	0	0
$219$	4.51139	0.304851
$220$	0	0
$221$	2.26676	0.152479
$222$	0	0
$223$	−18.9647	−1.26997	−0.634986	−	0.772523i	$-0.718994\pi$
$223$	−18.9647	−1.26997	−0.634986	+	0.772523i	$0.718994\pi$
$224$	0	0
$225$	0.798489	0.0532326
$226$	0	0
$227$	−8.04418	−0.533911	−0.266955	−	0.963709i	$-0.586018\pi$
$227$	−8.04418	−0.533911	−0.266955	+	0.963709i	$0.586018\pi$
$228$	0	0
$229$	6.16701	0.407528	0.203764	−	0.979020i	$-0.434683\pi$
$229$	6.16701	0.407528	0.203764	+	0.979020i	$0.434683\pi$
$230$	0	0
$231$	−22.9492	−1.50995
$232$	0	0
$233$	13.7895	0.903377	0.451689	−	0.892176i	$-0.350822\pi$
$233$	13.7895	0.903377	0.451689	+	0.892176i	$0.350822\pi$
$234$	0	0
$235$	10.0087	0.652896
$236$	0	0
$237$	−6.51894	−0.423451
$238$	0	0
$239$	28.1819	1.82294	0.911468	−	0.411372i	$-0.134950\pi$
$239$	28.1819	1.82294	0.911468	+	0.411372i	$0.134950\pi$
$240$	0	0
$241$	26.0602	1.67869	0.839344	−	0.543601i	$-0.182940\pi$
$241$	26.0602	1.67869	0.839344	+	0.543601i	$0.182940\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.3565	1.30053
$246$	0	0
$247$	21.4312	1.36363
$248$	0	0
$249$	−15.5940	−0.988227
$250$	0	0
$251$	−30.5916	−1.93093	−0.965463	−	0.260540i	$-0.916100\pi$
$251$	−30.5916	−1.93093	−0.965463	+	0.260540i	$0.916100\pi$
$252$	0	0
$253$	5.83783	0.367021
$254$	0	0
$255$	1.29441	0.0810593
$256$	0	0
$257$	3.87752	0.241873	0.120937	−	0.992660i	$-0.461410\pi$
$257$	3.87752	0.241873	0.120937	+	0.992660i	$0.461410\pi$
$258$	0	0
$259$	−27.3839	−1.70155
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	6.52284	0.402216	0.201108	−	0.979569i	$-0.435546\pi$
$263$	6.52284	0.402216	0.201108	+	0.979569i	$0.435546\pi$
$264$	0	0
$265$	−26.1575	−1.60684
$266$	0	0
$267$	5.84936	0.357975
$268$	0	0
$269$	−14.6981	−0.896161	−0.448081	−	0.893993i	$-0.647892\pi$
$269$	−14.6981	−0.896161	−0.448081	+	0.893993i	$0.647892\pi$
$270$	0	0
$271$	−25.7822	−1.56615	−0.783077	−	0.621924i	$-0.786351\pi$
$271$	−25.7822	−1.56615	−0.783077	+	0.621924i	$0.786351\pi$
$272$	0	0
$273$	16.5770	1.00329
$274$	0	0
$275$	−4.66145	−0.281096
$276$	0	0
$277$	−29.5944	−1.77816	−0.889079	−	0.457754i	$-0.848654\pi$
$277$	−29.5944	−1.77816	−0.889079	+	0.457754i	$0.848654\pi$
$278$	0	0
$279$	1.84865	0.110676
$280$	0	0
$281$	−24.3861	−1.45475	−0.727375	−	0.686240i	$-0.759260\pi$
$281$	−24.3861	−1.45475	−0.727375	+	0.686240i	$0.759260\pi$
$282$	0	0
$283$	8.38994	0.498730	0.249365	−	0.968410i	$-0.419778\pi$
$283$	8.38994	0.498730	0.249365	+	0.968410i	$0.419778\pi$
$284$	0	0
$285$	12.2381	0.724920
$286$	0	0
$287$	−2.02396	−0.119470
$288$	0	0
$289$	−16.7110	−0.983003
$290$	0	0
$291$	−5.46771	−0.320523
$292$	0	0
$293$	4.99584	0.291860	0.145930	−	0.989295i	$-0.453383\pi$
$293$	4.99584	0.291860	0.145930	+	0.989295i	$0.453383\pi$
$294$	0	0
$295$	4.67914	0.272430
$296$	0	0
$297$	−5.83783	−0.338745
$298$	0	0
$299$	−4.21687	−0.243868
$300$	0	0
$301$	39.7882	2.29335
$302$	0	0
$303$	11.6666	0.670227
$304$	0	0
$305$	13.9614	0.799430
$306$	0	0
$307$	−22.6342	−1.29180	−0.645900	−	0.763422i	$-0.723518\pi$
$307$	−22.6342	−1.29180	−0.645900	+	0.763422i	$0.723518\pi$
$308$	0	0
$309$	−3.15254	−0.179342
$310$	0	0
$311$	8.48155	0.480945	0.240472	−	0.970656i	$-0.422698\pi$
$311$	8.48155	0.480945	0.240472	+	0.970656i	$0.422698\pi$
$312$	0	0
$313$	−27.3378	−1.54522	−0.772612	−	0.634879i	$-0.781050\pi$
$313$	−27.3378	−1.54522	−0.772612	+	0.634879i	$0.781050\pi$
$314$	0	0
$315$	9.46615	0.533357
$316$	0	0
$317$	−26.4046	−1.48303	−0.741516	−	0.670936i	$-0.765893\pi$
$317$	−26.4046	−1.48303	−0.741516	+	0.670936i	$0.765893\pi$
$318$	0	0
$319$	−5.83783	−0.326856
$320$	0	0
$321$	10.2317	0.571078
$322$	0	0
$323$	2.73194	0.152009
$324$	0	0
$325$	3.36713	0.186775
$326$	0	0
$327$	13.5923	0.751653
$328$	0	0
$329$	16.3394	0.900821
$330$	0	0
$331$	−26.9291	−1.48016	−0.740079	−	0.672520i	$-0.765212\pi$
$331$	−26.9291	−1.48016	−0.740079	+	0.672520i	$0.765212\pi$
$332$	0	0
$333$	−6.96593	−0.381730
$334$	0	0
$335$	36.7288	2.00671
$336$	0	0
$337$	−5.80377	−0.316152	−0.158076	−	0.987427i	$-0.550529\pi$
$337$	−5.80377	−0.316152	−0.158076	+	0.987427i	$0.550529\pi$
$338$	0	0
$339$	9.96009	0.540958
$340$	0	0
$341$	−10.7921	−0.584425
$342$	0	0
$343$	5.71455	0.308557
$344$	0	0
$345$	−2.40801	−0.129643
$346$	0	0
$347$	2.99116	0.160574	0.0802870	−	0.996772i	$-0.474416\pi$
$347$	2.99116	0.160574	0.0802870	+	0.996772i	$0.474416\pi$
$348$	0	0
$349$	26.3099	1.40834	0.704168	−	0.710033i	$-0.251320\pi$
$349$	26.3099	1.40834	0.704168	+	0.710033i	$0.251320\pi$
$350$	0	0
$351$	4.21687	0.225080
$352$	0	0
$353$	23.1859	1.23406	0.617031	−	0.786939i	$-0.288335\pi$
$353$	23.1859	1.23406	0.617031	+	0.786939i	$0.288335\pi$
$354$	0	0
$355$	−8.12040	−0.430986
$356$	0	0
$357$	2.11316	0.111840
$358$	0	0
$359$	−19.4230	−1.02510	−0.512552	−	0.858656i	$-0.671300\pi$
$359$	−19.4230	−1.02510	−0.512552	+	0.858656i	$0.671300\pi$
$360$	0	0
$361$	6.82916	0.359429
$362$	0	0
$363$	23.0803	1.21140
$364$	0	0
$365$	10.8635	0.568619
$366$	0	0
$367$	−35.3869	−1.84718	−0.923591	−	0.383380i	$-0.874760\pi$
$367$	−35.3869	−1.84718	−0.923591	+	0.383380i	$0.874760\pi$
$368$	0	0
$369$	−0.514856	−0.0268023
$370$	0	0
$371$	−42.7026	−2.21701
$372$	0	0
$373$	3.05489	0.158176	0.0790881	−	0.996868i	$-0.474799\pi$
$373$	3.05489	0.158176	0.0790881	+	0.996868i	$0.474799\pi$
$374$	0	0
$375$	−10.1173	−0.522453
$376$	0	0
$377$	4.21687	0.217180
$378$	0	0
$379$	−30.2137	−1.55197	−0.775987	−	0.630749i	$-0.782748\pi$
$379$	−30.2137	−1.55197	−0.775987	+	0.630749i	$0.782748\pi$
$380$	0	0
$381$	12.5637	0.643658
$382$	0	0
$383$	12.6131	0.644501	0.322250	−	0.946654i	$-0.395561\pi$
$383$	12.6131	0.644501	0.322250	+	0.946654i	$0.395561\pi$
$384$	0	0
$385$	−55.2618	−2.81640
$386$	0	0
$387$	10.1214	0.514497
$388$	0	0
$389$	−7.46984	−0.378736	−0.189368	−	0.981906i	$-0.560644\pi$
$389$	−7.46984	−0.378736	−0.189368	+	0.981906i	$0.560644\pi$
$390$	0	0
$391$	−0.537546	−0.0271849
$392$	0	0
$393$	−19.2085	−0.968940
$394$	0	0
$395$	−15.6976	−0.789834
$396$	0	0
$397$	25.0460	1.25702	0.628511	−	0.777801i	$-0.283665\pi$
$397$	25.0460	1.25702	0.628511	+	0.777801i	$0.283665\pi$
$398$	0	0
$399$	19.9789	1.00019
$400$	0	0
$401$	−6.99740	−0.349434	−0.174717	−	0.984619i	$-0.555901\pi$
$401$	−6.99740	−0.349434	−0.174717	+	0.984619i	$0.555901\pi$
$402$	0	0
$403$	7.79552	0.388322
$404$	0	0
$405$	2.40801	0.119655
$406$	0	0
$407$	40.6659	2.01573
$408$	0	0
$409$	−3.07078	−0.151840	−0.0759201	−	0.997114i	$-0.524189\pi$
$409$	−3.07078	−0.151840	−0.0759201	+	0.997114i	$0.524189\pi$
$410$	0	0
$411$	−15.8316	−0.780915
$412$	0	0
$413$	7.63879	0.375880
$414$	0	0
$415$	−37.5503	−1.84327
$416$	0	0
$417$	−20.0499	−0.981849
$418$	0	0
$419$	30.9948	1.51420	0.757099	−	0.653301i	$-0.226616\pi$
$419$	30.9948	1.51420	0.757099	+	0.653301i	$0.226616\pi$
$420$	0	0
$421$	−30.7182	−1.49712	−0.748558	−	0.663070i	$-0.769253\pi$
$421$	−30.7182	−1.49712	−0.748558	+	0.663070i	$0.769253\pi$
$422$	0	0
$423$	4.15643	0.202093
$424$	0	0
$425$	0.429225	0.0208205
$426$	0	0
$427$	22.7923	1.10300
$428$	0	0
$429$	−24.6174	−1.18854
$430$	0	0
$431$	−21.8615	−1.05303	−0.526516	−	0.850165i	$-0.676502\pi$
$431$	−21.8615	−1.05303	−0.526516	+	0.850165i	$0.676502\pi$
$432$	0	0
$433$	2.95225	0.141876	0.0709381	−	0.997481i	$-0.477401\pi$
$433$	2.95225	0.141876	0.0709381	+	0.997481i	$0.477401\pi$
$434$	0	0
$435$	2.40801	0.115455
$436$	0	0
$437$	−5.08224	−0.243116
$438$	0	0
$439$	5.82547	0.278034	0.139017	−	0.990290i	$-0.455606\pi$
$439$	5.82547	0.278034	0.139017	+	0.990290i	$0.455606\pi$
$440$	0	0
$441$	8.45367	0.402556
$442$	0	0
$443$	−23.5246	−1.11769	−0.558845	−	0.829272i	$-0.688755\pi$
$443$	−23.5246	−1.11769	−0.558845	+	0.829272i	$0.688755\pi$
$444$	0	0
$445$	14.0853	0.667706
$446$	0	0
$447$	12.7729	0.604138
$448$	0	0
$449$	−22.0982	−1.04288	−0.521438	−	0.853289i	$-0.674604\pi$
$449$	−22.0982	−1.04288	−0.521438	+	0.853289i	$0.674604\pi$
$450$	0	0
$451$	3.00564	0.141530
$452$	0	0
$453$	−10.2496	−0.481568
$454$	0	0
$455$	39.9175	1.87136
$456$	0	0
$457$	2.95469	0.138214	0.0691072	−	0.997609i	$-0.477985\pi$
$457$	2.95469	0.138214	0.0691072	+	0.997609i	$0.477985\pi$
$458$	0	0
$459$	0.537546	0.0250905
$460$	0	0
$461$	28.5466	1.32955	0.664774	−	0.747045i	$-0.268528\pi$
$461$	28.5466	1.32955	0.664774	+	0.747045i	$0.268528\pi$
$462$	0	0
$463$	−22.9086	−1.06465	−0.532325	−	0.846540i	$-0.678682\pi$
$463$	−22.9086	−1.06465	−0.532325	+	0.846540i	$0.678682\pi$
$464$	0	0
$465$	4.45156	0.206436
$466$	0	0
$467$	−2.39763	−0.110949	−0.0554746	−	0.998460i	$-0.517667\pi$
$467$	−2.39763	−0.110949	−0.0554746	+	0.998460i	$0.517667\pi$
$468$	0	0
$469$	59.9604	2.76872
$470$	0	0
$471$	−22.6930	−1.04564
$472$	0	0
$473$	−59.0867	−2.71681
$474$	0	0
$475$	4.05811	0.186199
$476$	0	0
$477$	−10.8627	−0.497369
$478$	0	0
$479$	29.9805	1.36985	0.684923	−	0.728615i	$-0.259836\pi$
$479$	29.9805	1.36985	0.684923	+	0.728615i	$0.259836\pi$
$480$	0	0
$481$	−29.3744	−1.33936
$482$	0	0
$483$	−3.93112	−0.178872
$484$	0	0
$485$	−13.1663	−0.597850
$486$	0	0
$487$	13.2372	0.599835	0.299918	−	0.953965i	$-0.403041\pi$
$487$	13.2372	0.599835	0.299918	+	0.953965i	$0.403041\pi$
$488$	0	0
$489$	−3.52878	−0.159577
$490$	0	0
$491$	22.7609	1.02719	0.513594	−	0.858034i	$-0.328314\pi$
$491$	22.7609	1.02719	0.513594	+	0.858034i	$0.328314\pi$
$492$	0	0
$493$	0.537546	0.0242099
$494$	0	0
$495$	−14.0575	−0.631839
$496$	0	0
$497$	−13.2567	−0.594645
$498$	0	0
$499$	−43.0049	−1.92516	−0.962581	−	0.270995i	$-0.912647\pi$
$499$	−43.0049	−1.92516	−0.962581	+	0.270995i	$0.912647\pi$
$500$	0	0
$501$	−17.6254	−0.787446
$502$	0	0
$503$	31.8680	1.42092	0.710461	−	0.703737i	$-0.248487\pi$
$503$	31.8680	1.42092	0.710461	+	0.703737i	$0.248487\pi$
$504$	0	0
$505$	28.0932	1.25013
$506$	0	0
$507$	4.78201	0.212377
$508$	0	0
$509$	−27.5494	−1.22111	−0.610554	−	0.791975i	$-0.709053\pi$
$509$	−27.5494	−1.22111	−0.610554	+	0.791975i	$0.709053\pi$
$510$	0	0
$511$	17.7348	0.784541
$512$	0	0
$513$	5.08224	0.224386
$514$	0	0
$515$	−7.59134	−0.334514
$516$	0	0
$517$	−24.2646	−1.06715
$518$	0	0
$519$	22.4634	0.986034
$520$	0	0
$521$	−7.90743	−0.346431	−0.173215	−	0.984884i	$-0.555416\pi$
$521$	−7.90743	−0.346431	−0.173215	+	0.984884i	$0.555416\pi$
$522$	0	0
$523$	−18.3635	−0.802980	−0.401490	−	0.915863i	$-0.631508\pi$
$523$	−18.3635	−0.802980	−0.401490	+	0.915863i	$0.631508\pi$
$524$	0	0
$525$	3.13895	0.136995
$526$	0	0
$527$	0.993734	0.0432877
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.94316	0.0843260
$532$	0	0
$533$	−2.17108	−0.0940400
$534$	0	0
$535$	24.6380	1.06519
$536$	0	0
$537$	6.32345	0.272877
$538$	0	0
$539$	−49.3511	−2.12570
$540$	0	0
$541$	−1.46054	−0.0627934	−0.0313967	−	0.999507i	$-0.509996\pi$
$541$	−1.46054	−0.0627934	−0.0313967	+	0.999507i	$0.509996\pi$
$542$	0	0
$543$	25.0156	1.07352
$544$	0	0
$545$	32.7302	1.40201
$546$	0	0
$547$	32.0344	1.36969	0.684847	−	0.728687i	$-0.259869\pi$
$547$	32.0344	1.36969	0.684847	+	0.728687i	$0.259869\pi$
$548$	0	0
$549$	5.79793	0.247450
$550$	0	0
$551$	5.08224	0.216511
$552$	0	0
$553$	−25.6267	−1.08976
$554$	0	0
$555$	−16.7740	−0.712016
$556$	0	0
$557$	−9.47836	−0.401611	−0.200805	−	0.979631i	$-0.564356\pi$
$557$	−9.47836	−0.401611	−0.200805	+	0.979631i	$0.564356\pi$
$558$	0	0
$559$	42.6804	1.80519
$560$	0	0
$561$	−3.13810	−0.132491
$562$	0	0
$563$	30.5895	1.28919	0.644596	−	0.764523i	$-0.277025\pi$
$563$	30.5895	1.28919	0.644596	+	0.764523i	$0.277025\pi$
$564$	0	0
$565$	23.9840	1.00901
$566$	0	0
$567$	3.93112	0.165091
$568$	0	0
$569$	−8.13074	−0.340858	−0.170429	−	0.985370i	$-0.554515\pi$
$569$	−8.13074	−0.340858	−0.170429	+	0.985370i	$0.554515\pi$
$570$	0	0
$571$	31.9786	1.33826	0.669131	−	0.743144i	$-0.266666\pi$
$571$	31.9786	1.33826	0.669131	+	0.743144i	$0.266666\pi$
$572$	0	0
$573$	14.4387	0.603184
$574$	0	0
$575$	−0.798489	−0.0332993
$576$	0	0
$577$	12.5210	0.521256	0.260628	−	0.965439i	$-0.416070\pi$
$577$	12.5210	0.521256	0.260628	+	0.965439i	$0.416070\pi$
$578$	0	0
$579$	15.0308	0.624658
$580$	0	0
$581$	−61.3017	−2.54322
$582$	0	0
$583$	63.4147	2.62637
$584$	0	0
$585$	10.1543	0.419827
$586$	0	0
$587$	−24.7313	−1.02077	−0.510385	−	0.859946i	$-0.670497\pi$
$587$	−24.7313	−1.02077	−0.510385	+	0.859946i	$0.670497\pi$
$588$	0	0
$589$	9.39528	0.387126
$590$	0	0
$591$	7.61703	0.313323
$592$	0	0
$593$	22.8210	0.937148	0.468574	−	0.883424i	$-0.344768\pi$
$593$	22.8210	0.937148	0.468574	+	0.883424i	$0.344768\pi$
$594$	0	0
$595$	5.08849	0.208608
$596$	0	0
$597$	22.8783	0.936346
$598$	0	0
$599$	38.2377	1.56235	0.781175	−	0.624312i	$-0.214621\pi$
$599$	38.2377	1.56235	0.781175	+	0.624312i	$0.214621\pi$
$600$	0	0
$601$	27.3936	1.11741	0.558703	−	0.829368i	$-0.311299\pi$
$601$	27.3936	1.11741	0.558703	+	0.829368i	$0.311299\pi$
$602$	0	0
$603$	15.2528	0.621141
$604$	0	0
$605$	55.5774	2.25954
$606$	0	0
$607$	−20.8004	−0.844260	−0.422130	−	0.906535i	$-0.638717\pi$
$607$	−20.8004	−0.844260	−0.422130	+	0.906535i	$0.638717\pi$
$608$	0	0
$609$	3.93112	0.159297
$610$	0	0
$611$	17.5271	0.709073
$612$	0	0
$613$	20.7521	0.838171	0.419086	−	0.907947i	$-0.362351\pi$
$613$	20.7521	0.838171	0.419086	+	0.907947i	$0.362351\pi$
$614$	0	0
$615$	−1.23978	−0.0499926
$616$	0	0
$617$	10.1109	0.407051	0.203526	−	0.979070i	$-0.434760\pi$
$617$	10.1109	0.407051	0.203526	+	0.979070i	$0.434760\pi$
$618$	0	0
$619$	22.8862	0.919875	0.459937	−	0.887951i	$-0.347872\pi$
$619$	22.8862	0.919875	0.459937	+	0.887951i	$0.347872\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	22.9945	0.921255
$624$	0	0
$625$	−28.3549	−1.13419
$626$	0	0
$627$	−29.6693	−1.18488
$628$	0	0
$629$	−3.74451	−0.149303
$630$	0	0
$631$	−26.6736	−1.06186	−0.530929	−	0.847416i	$-0.678157\pi$
$631$	−26.6736	−1.06186	−0.530929	+	0.847416i	$0.678157\pi$
$632$	0	0
$633$	−6.32707	−0.251478
$634$	0	0
$635$	30.2535	1.20057
$636$	0	0
$637$	35.6480	1.41243
$638$	0	0
$639$	−3.37225	−0.133404
$640$	0	0
$641$	−9.83336	−0.388394	−0.194197	−	0.980963i	$-0.562210\pi$
$641$	−9.83336	−0.388394	−0.194197	+	0.980963i	$0.562210\pi$
$642$	0	0
$643$	31.2285	1.23153	0.615766	−	0.787929i	$-0.288846\pi$
$643$	31.2285	1.23153	0.615766	+	0.787929i	$0.288846\pi$
$644$	0	0
$645$	24.3723	0.959657
$646$	0	0
$647$	−47.4207	−1.86430	−0.932150	−	0.362071i	$-0.882070\pi$
$647$	−47.4207	−1.86430	−0.932150	+	0.362071i	$0.882070\pi$
$648$	0	0
$649$	−11.3438	−0.445285
$650$	0	0
$651$	7.26725	0.284826
$652$	0	0
$653$	41.9724	1.64251	0.821253	−	0.570564i	$-0.193275\pi$
$653$	41.9724	1.64251	0.821253	+	0.570564i	$0.193275\pi$
$654$	0	0
$655$	−46.2541	−1.80730
$656$	0	0
$657$	4.51139	0.176006
$658$	0	0
$659$	−13.1121	−0.510775	−0.255388	−	0.966839i	$-0.582203\pi$
$659$	−13.1121	−0.510775	−0.255388	+	0.966839i	$0.582203\pi$
$660$	0	0
$661$	47.9695	1.86580	0.932899	−	0.360138i	$-0.117270\pi$
$661$	47.9695	1.86580	0.932899	+	0.360138i	$0.117270\pi$
$662$	0	0
$663$	2.26676	0.0880338
$664$	0	0
$665$	48.1092	1.86560
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−18.9647	−0.733219
$670$	0	0
$671$	−33.8473	−1.30666
$672$	0	0
$673$	−18.3634	−0.707856	−0.353928	−	0.935273i	$-0.615154\pi$
$673$	−18.3634	−0.707856	−0.353928	+	0.935273i	$0.615154\pi$
$674$	0	0
$675$	0.798489	0.0307339
$676$	0	0
$677$	16.7309	0.643019	0.321509	−	0.946906i	$-0.395810\pi$
$677$	16.7309	0.643019	0.321509	+	0.946906i	$0.395810\pi$
$678$	0	0
$679$	−21.4942	−0.824872
$680$	0	0
$681$	−8.04418	−0.308254
$682$	0	0
$683$	1.20147	0.0459728	0.0229864	−	0.999736i	$-0.492683\pi$
$683$	1.20147	0.0459728	0.0229864	+	0.999736i	$0.492683\pi$
$684$	0	0
$685$	−38.1226	−1.45659
$686$	0	0
$687$	6.16701	0.235286
$688$	0	0
$689$	−45.8067	−1.74510
$690$	0	0
$691$	9.67227	0.367950	0.183975	−	0.982931i	$-0.441103\pi$
$691$	9.67227	0.367950	0.183975	+	0.982931i	$0.441103\pi$
$692$	0	0
$693$	−22.9492	−0.871768
$694$	0	0
$695$	−48.2803	−1.83138
$696$	0	0
$697$	−0.276759	−0.0104830
$698$	0	0
$699$	13.7895	0.521565
$700$	0	0
$701$	42.8165	1.61716	0.808579	−	0.588388i	$-0.200237\pi$
$701$	42.8165	1.61716	0.808579	+	0.588388i	$0.200237\pi$
$702$	0	0
$703$	−35.4025	−1.33523
$704$	0	0
$705$	10.0087	0.376950
$706$	0	0
$707$	45.8626	1.72484
$708$	0	0
$709$	31.6034	1.18689	0.593445	−	0.804874i	$-0.297767\pi$
$709$	31.6034	1.18689	0.593445	+	0.804874i	$0.297767\pi$
$710$	0	0
$711$	−6.51894	−0.244479
$712$	0	0
$713$	−1.84865	−0.0692324
$714$	0	0
$715$	−59.2788	−2.21690
$716$	0	0
$717$	28.1819	1.05247
$718$	0	0
$719$	−3.96415	−0.147838	−0.0739189	−	0.997264i	$-0.523551\pi$
$719$	−3.96415	−0.147838	−0.0739189	+	0.997264i	$0.523551\pi$
$720$	0	0
$721$	−12.3930	−0.461540
$722$	0	0
$723$	26.0602	0.969191
$724$	0	0
$725$	0.798489	0.0296551
$726$	0	0
$727$	17.9711	0.666511	0.333255	−	0.942837i	$-0.391853\pi$
$727$	17.9711	0.666511	0.333255	+	0.942837i	$0.391853\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.44069	0.201231
$732$	0	0
$733$	20.2813	0.749107	0.374554	−	0.927205i	$-0.377796\pi$
$733$	20.2813	0.749107	0.374554	+	0.927205i	$0.377796\pi$
$734$	0	0
$735$	20.3565	0.750860
$736$	0	0
$737$	−89.0431	−3.27995
$738$	0	0
$739$	2.19943	0.0809074	0.0404537	−	0.999181i	$-0.487120\pi$
$739$	2.19943	0.0809074	0.0404537	+	0.999181i	$0.487120\pi$
$740$	0	0
$741$	21.4312	0.787293
$742$	0	0
$743$	−23.8285	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$743$	−23.8285	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$744$	0	0
$745$	30.7572	1.12686
$746$	0	0
$747$	−15.5940	−0.570553
$748$	0	0
$749$	40.2220	1.46968
$750$	0	0
$751$	−28.1086	−1.02570	−0.512849	−	0.858479i	$-0.671410\pi$
$751$	−28.1086	−1.02570	−0.512849	+	0.858479i	$0.671410\pi$
$752$	0	0
$753$	−30.5916	−1.11482
$754$	0	0
$755$	−24.6811	−0.898237
$756$	0	0
$757$	−16.7462	−0.608652	−0.304326	−	0.952568i	$-0.598431\pi$
$757$	−16.7462	−0.608652	−0.304326	+	0.952568i	$0.598431\pi$
$758$	0	0
$759$	5.83783	0.211900
$760$	0	0
$761$	24.7164	0.895968	0.447984	−	0.894042i	$-0.352142\pi$
$761$	24.7164	0.895968	0.447984	+	0.894042i	$0.352142\pi$
$762$	0	0
$763$	53.4327	1.93439
$764$	0	0
$765$	1.29441	0.0467996
$766$	0	0
$767$	8.19406	0.295870
$768$	0	0
$769$	−24.4414	−0.881381	−0.440690	−	0.897659i	$-0.645266\pi$
$769$	−24.4414	−0.881381	−0.440690	+	0.897659i	$0.645266\pi$
$770$	0	0
$771$	3.87752	0.139646
$772$	0	0
$773$	37.7484	1.35771	0.678857	−	0.734270i	$-0.262476\pi$
$773$	37.7484	1.35771	0.678857	+	0.734270i	$0.262476\pi$
$774$	0	0
$775$	1.47613	0.0530240
$776$	0	0
$777$	−27.3839	−0.982391
$778$	0	0
$779$	−2.61662	−0.0937502
$780$	0	0
$781$	19.6866	0.704443
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−54.6449	−1.95036
$786$	0	0
$787$	−0.661797	−0.0235905	−0.0117953	−	0.999930i	$-0.503755\pi$
$787$	−0.661797	−0.0235905	−0.0117953	+	0.999930i	$0.503755\pi$
$788$	0	0
$789$	6.52284	0.232219
$790$	0	0
$791$	39.1543	1.39217
$792$	0	0
$793$	24.4491	0.868214
$794$	0	0
$795$	−26.1575	−0.927710
$796$	0	0
$797$	−9.30582	−0.329629	−0.164814	−	0.986325i	$-0.552703\pi$
$797$	−9.30582	−0.329629	−0.164814	+	0.986325i	$0.552703\pi$
$798$	0	0
$799$	2.23427	0.0790429
$800$	0	0
$801$	5.84936	0.206677
$802$	0	0
$803$	−26.3367	−0.929403
$804$	0	0
$805$	−9.46615	−0.333638
$806$	0	0
$807$	−14.6981	−0.517399
$808$	0	0
$809$	−6.40290	−0.225114	−0.112557	−	0.993645i	$-0.535904\pi$
$809$	−6.40290	−0.225114	−0.112557	+	0.993645i	$0.535904\pi$
$810$	0	0
$811$	−36.1685	−1.27005	−0.635025	−	0.772492i	$-0.719010\pi$
$811$	−36.1685	−1.27005	−0.635025	+	0.772492i	$0.719010\pi$
$812$	0	0
$813$	−25.7822	−0.904220
$814$	0	0
$815$	−8.49733	−0.297648
$816$	0	0
$817$	51.4391	1.79963
$818$	0	0
$819$	16.5770	0.579248
$820$	0	0
$821$	−9.15920	−0.319658	−0.159829	−	0.987145i	$-0.551094\pi$
$821$	−9.15920	−0.319658	−0.159829	+	0.987145i	$0.551094\pi$
$822$	0	0
$823$	−31.9672	−1.11431	−0.557154	−	0.830409i	$-0.688107\pi$
$823$	−31.9672	−1.11431	−0.557154	+	0.830409i	$0.688107\pi$
$824$	0	0
$825$	−4.66145	−0.162291
$826$	0	0
$827$	18.3499	0.638088	0.319044	−	0.947740i	$-0.396638\pi$
$827$	18.3499	0.638088	0.319044	+	0.947740i	$0.396638\pi$
$828$	0	0
$829$	−23.9932	−0.833317	−0.416658	−	0.909063i	$-0.636799\pi$
$829$	−23.9932	−0.833317	−0.416658	+	0.909063i	$0.636799\pi$
$830$	0	0
$831$	−29.5944	−1.02662
$832$	0	0
$833$	4.54424	0.157448
$834$	0	0
$835$	−42.4421	−1.46877
$836$	0	0
$837$	1.84865	0.0638987
$838$	0	0
$839$	−28.0289	−0.967663	−0.483832	−	0.875161i	$-0.660755\pi$
$839$	−28.0289	−0.967663	−0.483832	+	0.875161i	$0.660755\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−24.3861	−0.839900
$844$	0	0
$845$	11.5151	0.396132
$846$	0	0
$847$	90.7312	3.11756
$848$	0	0
$849$	8.38994	0.287942
$850$	0	0
$851$	6.96593	0.238789
$852$	0	0
$853$	−3.83088	−0.131167	−0.0655833	−	0.997847i	$-0.520891\pi$
$853$	−3.83088	−0.131167	−0.0655833	+	0.997847i	$0.520891\pi$
$854$	0	0
$855$	12.2381	0.418533
$856$	0	0
$857$	32.2231	1.10072	0.550360	−	0.834927i	$-0.314490\pi$
$857$	32.2231	1.10072	0.550360	+	0.834927i	$0.314490\pi$
$858$	0	0
$859$	−12.7497	−0.435012	−0.217506	−	0.976059i	$-0.569792\pi$
$859$	−12.7497	−0.435012	−0.217506	+	0.976059i	$0.569792\pi$
$860$	0	0
$861$	−2.02396	−0.0689763
$862$	0	0
$863$	−34.3571	−1.16953	−0.584764	−	0.811203i	$-0.698813\pi$
$863$	−34.3571	−1.16953	−0.584764	+	0.811203i	$0.698813\pi$
$864$	0	0
$865$	54.0920	1.83918
$866$	0	0
$867$	−16.7110	−0.567537
$868$	0	0
$869$	38.0565	1.29098
$870$	0	0
$871$	64.3190	2.17937
$872$	0	0
$873$	−5.46771	−0.185054
$874$	0	0
$875$	−39.7721	−1.34454
$876$	0	0
$877$	55.6241	1.87829	0.939146	−	0.343519i	$-0.111619\pi$
$877$	55.6241	1.87829	0.939146	+	0.343519i	$0.111619\pi$
$878$	0	0
$879$	4.99584	0.168506
$880$	0	0
$881$	−50.5141	−1.70186	−0.850931	−	0.525277i	$-0.823962\pi$
$881$	−50.5141	−1.70186	−0.850931	+	0.525277i	$0.823962\pi$
$882$	0	0
$883$	40.9299	1.37740	0.688700	−	0.725046i	$-0.258182\pi$
$883$	40.9299	1.37740	0.688700	+	0.725046i	$0.258182\pi$
$884$	0	0
$885$	4.67914	0.157288
$886$	0	0
$887$	25.5654	0.858402	0.429201	−	0.903209i	$-0.358795\pi$
$887$	25.5654	0.858402	0.429201	+	0.903209i	$0.358795\pi$
$888$	0	0
$889$	49.3894	1.65647
$890$	0	0
$891$	−5.83783	−0.195575
$892$	0	0
$893$	21.1240	0.706887
$894$	0	0
$895$	15.2269	0.508979
$896$	0	0
$897$	−4.21687	−0.140797
$898$	0	0
$899$	1.84865	0.0616559
$900$	0	0
$901$	−5.83921	−0.194532
$902$	0	0
$903$	39.7882	1.32407
$904$	0	0
$905$	60.2378	2.00237
$906$	0	0
$907$	19.8511	0.659145	0.329572	−	0.944130i	$-0.393095\pi$
$907$	19.8511	0.659145	0.329572	+	0.944130i	$0.393095\pi$
$908$	0	0
$909$	11.6666	0.386956
$910$	0	0
$911$	−35.0478	−1.16119	−0.580593	−	0.814194i	$-0.697179\pi$
$911$	−35.0478	−1.16119	−0.580593	+	0.814194i	$0.697179\pi$
$912$	0	0
$913$	91.0349	3.01282
$914$	0	0
$915$	13.9614	0.461551
$916$	0	0
$917$	−75.5108	−2.49358
$918$	0	0
$919$	−13.9449	−0.459999	−0.229999	−	0.973191i	$-0.573872\pi$
$919$	−13.9449	−0.459999	−0.229999	+	0.973191i	$0.573872\pi$
$920$	0	0
$921$	−22.6342	−0.745821
$922$	0	0
$923$	−14.2203	−0.468069
$924$	0	0
$925$	−5.56222	−0.182885
$926$	0	0
$927$	−3.15254	−0.103543
$928$	0	0
$929$	45.0613	1.47841	0.739206	−	0.673479i	$-0.235201\pi$
$929$	45.0613	1.47841	0.739206	+	0.673479i	$0.235201\pi$
$930$	0	0
$931$	42.9636	1.40807
$932$	0	0
$933$	8.48155	0.277673
$934$	0	0
$935$	−7.55657	−0.247126
$936$	0	0
$937$	45.5504	1.48807	0.744034	−	0.668142i	$-0.232910\pi$
$937$	45.5504	1.48807	0.744034	+	0.668142i	$0.232910\pi$
$938$	0	0
$939$	−27.3378	−0.892135
$940$	0	0
$941$	−9.67000	−0.315233	−0.157616	−	0.987500i	$-0.550381\pi$
$941$	−9.67000	−0.315233	−0.157616	+	0.987500i	$0.550381\pi$
$942$	0	0
$943$	0.514856	0.0167660
$944$	0	0
$945$	9.46615	0.307934
$946$	0	0
$947$	−25.6485	−0.833466	−0.416733	−	0.909029i	$-0.636825\pi$
$947$	−25.6485	−0.833466	−0.416733	+	0.909029i	$0.636825\pi$
$948$	0	0
$949$	19.0240	0.617544
$950$	0	0
$951$	−26.4046	−0.856228
$952$	0	0
$953$	−8.78177	−0.284469	−0.142235	−	0.989833i	$-0.545429\pi$
$953$	−8.78177	−0.284469	−0.142235	+	0.989833i	$0.545429\pi$
$954$	0	0
$955$	34.7684	1.12508
$956$	0	0
$957$	−5.83783	−0.188710
$958$	0	0
$959$	−62.2359	−2.00970
$960$	0	0
$961$	−27.5825	−0.889758
$962$	0	0
$963$	10.2317	0.329712
$964$	0	0
$965$	36.1942	1.16513
$966$	0	0
$967$	−14.9412	−0.480477	−0.240238	−	0.970714i	$-0.577226\pi$
$967$	−14.9412	−0.480477	−0.240238	+	0.970714i	$0.577226\pi$
$968$	0	0
$969$	2.73194	0.0877625
$970$	0	0
$971$	−48.9659	−1.57139	−0.785695	−	0.618614i	$-0.787694\pi$
$971$	−48.9659	−1.57139	−0.785695	+	0.618614i	$0.787694\pi$
$972$	0	0
$973$	−78.8186	−2.52681
$974$	0	0
$975$	3.36713	0.107834
$976$	0	0
$977$	13.1365	0.420275	0.210138	−	0.977672i	$-0.432609\pi$
$977$	13.1365	0.420275	0.210138	+	0.977672i	$0.432609\pi$
$978$	0	0
$979$	−34.1476	−1.09136
$980$	0	0
$981$	13.5923	0.433967
$982$	0	0
$983$	−41.5399	−1.32492	−0.662458	−	0.749099i	$-0.730487\pi$
$983$	−41.5399	−1.32492	−0.662458	+	0.749099i	$0.730487\pi$
$984$	0	0
$985$	18.3418	0.584420
$986$	0	0
$987$	16.3394	0.520089
$988$	0	0
$989$	−10.1214	−0.321840
$990$	0	0
$991$	−26.5818	−0.844399	−0.422200	−	0.906503i	$-0.638742\pi$
$991$	−26.5818	−0.844399	−0.422200	+	0.906503i	$0.638742\pi$
$992$	0	0
$993$	−26.9291	−0.854570
$994$	0	0
$995$	55.0910	1.74650
$996$	0	0
$997$	−9.71650	−0.307725	−0.153862	−	0.988092i	$-0.549171\pi$
$997$	−9.71650	−0.307725	−0.153862	+	0.988092i	$0.549171\pi$
$998$	0	0
$999$	−6.96593	−0.220392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.13	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.13	✓	18	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.40801 q^{5} +3.93112 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.40801 q^{5} +3.93112 q^{7} +1.00000 q^{9} -5.83783 q^{11} +4.21687 q^{13} +2.40801 q^{15} +0.537546 q^{17} +5.08224 q^{19} +3.93112 q^{21} -1.00000 q^{23} +0.798489 q^{25} +1.00000 q^{27} +1.00000 q^{29} +1.84865 q^{31} -5.83783 q^{33} +9.46615 q^{35} -6.96593 q^{37} +4.21687 q^{39} -0.514856 q^{41} +10.1214 q^{43} +2.40801 q^{45} +4.15643 q^{47} +8.45367 q^{49} +0.537546 q^{51} -10.8627 q^{53} -14.0575 q^{55} +5.08224 q^{57} +1.94316 q^{59} +5.79793 q^{61} +3.93112 q^{63} +10.1543 q^{65} +15.2528 q^{67} -1.00000 q^{69} -3.37225 q^{71} +4.51139 q^{73} +0.798489 q^{75} -22.9492 q^{77} -6.51894 q^{79} +1.00000 q^{81} -15.5940 q^{83} +1.29441 q^{85} +1.00000 q^{87} +5.84936 q^{89} +16.5770 q^{91} +1.84865 q^{93} +12.2381 q^{95} -5.46771 q^{97} -5.83783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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