LMFDB - Embedded newform 8004.2.a.k.1.16

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			1.16
Root			$3.50248$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.50248 q^{5} -4.71048 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.50248 q^{5} -4.71048 q^{7} +1.00000 q^{9} +2.19631 q^{11} +6.58278 q^{13} +3.50248 q^{15} +3.39516 q^{17} +5.90929 q^{19} -4.71048 q^{21} -1.00000 q^{23} +7.26733 q^{25} +1.00000 q^{27} +1.00000 q^{29} -5.12948 q^{31} +2.19631 q^{33} -16.4984 q^{35} -7.87537 q^{37} +6.58278 q^{39} +2.36031 q^{41} +3.79900 q^{43} +3.50248 q^{45} -10.9253 q^{47} +15.1887 q^{49} +3.39516 q^{51} +11.7776 q^{53} +7.69252 q^{55} +5.90929 q^{57} -0.680031 q^{59} +0.133410 q^{61} -4.71048 q^{63} +23.0560 q^{65} +5.55008 q^{67} -1.00000 q^{69} -3.13652 q^{71} +6.86140 q^{73} +7.26733 q^{75} -10.3457 q^{77} -6.61212 q^{79} +1.00000 q^{81} +11.2304 q^{83} +11.8915 q^{85} +1.00000 q^{87} -8.97644 q^{89} -31.0081 q^{91} -5.12948 q^{93} +20.6971 q^{95} -13.5780 q^{97} +2.19631 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.50248	1.56635	0.783177	−	0.621798i	$-0.213598\pi$
$5$	3.50248	1.56635	0.783177	+	0.621798i	$0.213598\pi$
$6$	0	0
$7$	−4.71048	−1.78040	−0.890198	−	0.455574i	$-0.849434\pi$
$7$	−4.71048	−1.78040	−0.890198	+	0.455574i	$0.849434\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.19631	0.662212	0.331106	−	0.943594i	$-0.392578\pi$
$11$	2.19631	0.662212	0.331106	+	0.943594i	$0.392578\pi$
$12$	0	0
$13$	6.58278	1.82573	0.912867	−	0.408258i	$-0.133864\pi$
$13$	6.58278	1.82573	0.912867	+	0.408258i	$0.133864\pi$
$14$	0	0
$15$	3.50248	0.904335
$16$	0	0
$17$	3.39516	0.823447	0.411724	−	0.911309i	$-0.364927\pi$
$17$	3.39516	0.823447	0.411724	+	0.911309i	$0.364927\pi$
$18$	0	0
$19$	5.90929	1.35568	0.677842	−	0.735208i	$-0.262915\pi$
$19$	5.90929	1.35568	0.677842	+	0.735208i	$0.262915\pi$
$20$	0	0
$21$	−4.71048	−1.02791
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.26733	1.45347
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.12948	−0.921282	−0.460641	−	0.887587i	$-0.652380\pi$
$31$	−5.12948	−0.921282	−0.460641	+	0.887587i	$0.652380\pi$
$32$	0	0
$33$	2.19631	0.382328
$34$	0	0
$35$	−16.4984	−2.78873
$36$	0	0
$37$	−7.87537	−1.29470	−0.647351	−	0.762192i	$-0.724123\pi$
$37$	−7.87537	−1.29470	−0.647351	+	0.762192i	$0.724123\pi$
$38$	0	0
$39$	6.58278	1.05409
$40$	0	0
$41$	2.36031	0.368619	0.184310	−	0.982868i	$-0.440995\pi$
$41$	2.36031	0.368619	0.184310	+	0.982868i	$0.440995\pi$
$42$	0	0
$43$	3.79900	0.579341	0.289671	−	0.957126i	$-0.406454\pi$
$43$	3.79900	0.579341	0.289671	+	0.957126i	$0.406454\pi$
$44$	0	0
$45$	3.50248	0.522118
$46$	0	0
$47$	−10.9253	−1.59363	−0.796813	−	0.604226i	$-0.793482\pi$
$47$	−10.9253	−1.59363	−0.796813	+	0.604226i	$0.793482\pi$
$48$	0	0
$49$	15.1887	2.16981
$50$	0	0
$51$	3.39516	0.475417
$52$	0	0
$53$	11.7776	1.61778	0.808892	−	0.587958i	$-0.200068\pi$
$53$	11.7776	1.61778	0.808892	+	0.587958i	$0.200068\pi$
$54$	0	0
$55$	7.69252	1.03726
$56$	0	0
$57$	5.90929	0.782704
$58$	0	0
$59$	−0.680031	−0.0885324	−0.0442662	−	0.999020i	$-0.514095\pi$
$59$	−0.680031	−0.0885324	−0.0442662	+	0.999020i	$0.514095\pi$
$60$	0	0
$61$	0.133410	0.0170813	0.00854067	−	0.999964i	$-0.497281\pi$
$61$	0.133410	0.0170813	0.00854067	+	0.999964i	$0.497281\pi$
$62$	0	0
$63$	−4.71048	−0.593465
$64$	0	0
$65$	23.0560	2.85975
$66$	0	0
$67$	5.55008	0.678050	0.339025	−	0.940777i	$-0.389903\pi$
$67$	5.55008	0.678050	0.339025	+	0.940777i	$0.389903\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.13652	−0.372237	−0.186118	−	0.982527i	$-0.559591\pi$
$71$	−3.13652	−0.372237	−0.186118	+	0.982527i	$0.559591\pi$
$72$	0	0
$73$	6.86140	0.803066	0.401533	−	0.915844i	$-0.368477\pi$
$73$	6.86140	0.803066	0.401533	+	0.915844i	$0.368477\pi$
$74$	0	0
$75$	7.26733	0.839159
$76$	0	0
$77$	−10.3457	−1.17900
$78$	0	0
$79$	−6.61212	−0.743922	−0.371961	−	0.928248i	$-0.621314\pi$
$79$	−6.61212	−0.743922	−0.371961	+	0.928248i	$0.621314\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.2304	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$83$	11.2304	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$84$	0	0
$85$	11.8915	1.28981
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−8.97644	−0.951501	−0.475750	−	0.879580i	$-0.657823\pi$
$89$	−8.97644	−0.951501	−0.475750	+	0.879580i	$0.657823\pi$
$90$	0	0
$91$	−31.0081	−3.25053
$92$	0	0
$93$	−5.12948	−0.531902
$94$	0	0
$95$	20.6971	2.12348
$96$	0	0
$97$	−13.5780	−1.37864	−0.689319	−	0.724458i	$-0.742090\pi$
$97$	−13.5780	−1.37864	−0.689319	+	0.724458i	$0.742090\pi$
$98$	0	0
$99$	2.19631	0.220737
$100$	0	0
$101$	−0.150025	−0.0149280	−0.00746401	−	0.999972i	$-0.502376\pi$
$101$	−0.150025	−0.0149280	−0.00746401	+	0.999972i	$0.502376\pi$
$102$	0	0
$103$	−11.0507	−1.08886	−0.544430	−	0.838807i	$-0.683254\pi$
$103$	−11.0507	−1.08886	−0.544430	+	0.838807i	$0.683254\pi$
$104$	0	0
$105$	−16.4984	−1.61007
$106$	0	0
$107$	14.8538	1.43597	0.717986	−	0.696058i	$-0.245064\pi$
$107$	14.8538	1.43597	0.717986	+	0.696058i	$0.245064\pi$
$108$	0	0
$109$	−0.149928	−0.0143605	−0.00718023	−	0.999974i	$-0.502286\pi$
$109$	−0.149928	−0.0143605	−0.00718023	+	0.999974i	$0.502286\pi$
$110$	0	0
$111$	−7.87537	−0.747497
$112$	0	0
$113$	−4.49187	−0.422559	−0.211280	−	0.977426i	$-0.567763\pi$
$113$	−4.49187	−0.422559	−0.211280	+	0.977426i	$0.567763\pi$
$114$	0	0
$115$	−3.50248	−0.326608
$116$	0	0
$117$	6.58278	0.608578
$118$	0	0
$119$	−15.9928	−1.46606
$120$	0	0
$121$	−6.17623	−0.561475
$122$	0	0
$123$	2.36031	0.212822
$124$	0	0
$125$	7.94128	0.710290
$126$	0	0
$127$	−19.2929	−1.71197	−0.855984	−	0.517002i	$-0.827048\pi$
$127$	−19.2929	−1.71197	−0.855984	+	0.517002i	$0.827048\pi$
$128$	0	0
$129$	3.79900	0.334483
$130$	0	0
$131$	3.27196	0.285872	0.142936	−	0.989732i	$-0.454346\pi$
$131$	3.27196	0.285872	0.142936	+	0.989732i	$0.454346\pi$
$132$	0	0
$133$	−27.8356	−2.41365
$134$	0	0
$135$	3.50248	0.301445
$136$	0	0
$137$	−6.84748	−0.585019	−0.292510	−	0.956263i	$-0.594490\pi$
$137$	−6.84748	−0.585019	−0.292510	+	0.956263i	$0.594490\pi$
$138$	0	0
$139$	4.03549	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$139$	4.03549	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$140$	0	0
$141$	−10.9253	−0.920080
$142$	0	0
$143$	14.4578	1.20902
$144$	0	0
$145$	3.50248	0.290865
$146$	0	0
$147$	15.1887	1.25274
$148$	0	0
$149$	8.29404	0.679474	0.339737	−	0.940520i	$-0.389662\pi$
$149$	8.29404	0.679474	0.339737	+	0.940520i	$0.389662\pi$
$150$	0	0
$151$	−23.4311	−1.90680	−0.953398	−	0.301715i	$-0.902441\pi$
$151$	−23.4311	−1.90680	−0.953398	+	0.301715i	$0.902441\pi$
$152$	0	0
$153$	3.39516	0.274482
$154$	0	0
$155$	−17.9659	−1.44305
$156$	0	0
$157$	1.40312	0.111981	0.0559907	−	0.998431i	$-0.482168\pi$
$157$	1.40312	0.111981	0.0559907	+	0.998431i	$0.482168\pi$
$158$	0	0
$159$	11.7776	0.934027
$160$	0	0
$161$	4.71048	0.371238
$162$	0	0
$163$	22.5865	1.76911	0.884557	−	0.466433i	$-0.154461\pi$
$163$	22.5865	1.76911	0.884557	+	0.466433i	$0.154461\pi$
$164$	0	0
$165$	7.69252	0.598862
$166$	0	0
$167$	24.2793	1.87879	0.939395	−	0.342837i	$-0.111388\pi$
$167$	24.2793	1.87879	0.939395	+	0.342837i	$0.111388\pi$
$168$	0	0
$169$	30.3329	2.33330
$170$	0	0
$171$	5.90929	0.451894
$172$	0	0
$173$	13.4810	1.02494	0.512471	−	0.858705i	$-0.328730\pi$
$173$	13.4810	1.02494	0.512471	+	0.858705i	$0.328730\pi$
$174$	0	0
$175$	−34.2327	−2.58775
$176$	0	0
$177$	−0.680031	−0.0511142
$178$	0	0
$179$	5.50916	0.411774	0.205887	−	0.978576i	$-0.433992\pi$
$179$	5.50916	0.411774	0.205887	+	0.978576i	$0.433992\pi$
$180$	0	0
$181$	−8.20613	−0.609957	−0.304979	−	0.952359i	$-0.598649\pi$
$181$	−8.20613	−0.609957	−0.304979	+	0.952359i	$0.598649\pi$
$182$	0	0
$183$	0.133410	0.00986191
$184$	0	0
$185$	−27.5833	−2.02796
$186$	0	0
$187$	7.45682	0.545297
$188$	0	0
$189$	−4.71048	−0.342637
$190$	0	0
$191$	−6.45372	−0.466975	−0.233487	−	0.972360i	$-0.575014\pi$
$191$	−6.45372	−0.466975	−0.233487	+	0.972360i	$0.575014\pi$
$192$	0	0
$193$	16.5824	1.19363	0.596813	−	0.802381i	$-0.296434\pi$
$193$	16.5824	1.19363	0.596813	+	0.802381i	$0.296434\pi$
$194$	0	0
$195$	23.0560	1.65108
$196$	0	0
$197$	12.9392	0.921882	0.460941	−	0.887431i	$-0.347512\pi$
$197$	12.9392	0.921882	0.460941	+	0.887431i	$0.347512\pi$
$198$	0	0
$199$	18.5875	1.31763	0.658817	−	0.752303i	$-0.271057\pi$
$199$	18.5875	1.31763	0.658817	+	0.752303i	$0.271057\pi$
$200$	0	0
$201$	5.55008	0.391473
$202$	0	0
$203$	−4.71048	−0.330611
$204$	0	0
$205$	8.26694	0.577388
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	12.9786	0.897750
$210$	0	0
$211$	2.73196	0.188076	0.0940379	−	0.995569i	$-0.470023\pi$
$211$	2.73196	0.188076	0.0940379	+	0.995569i	$0.470023\pi$
$212$	0	0
$213$	−3.13652	−0.214911
$214$	0	0
$215$	13.3059	0.907454
$216$	0	0
$217$	24.1623	1.64025
$218$	0	0
$219$	6.86140	0.463651
$220$	0	0
$221$	22.3496	1.50339
$222$	0	0
$223$	0.148745	0.00996072	0.00498036	−	0.999988i	$-0.498415\pi$
$223$	0.148745	0.00996072	0.00498036	+	0.999988i	$0.498415\pi$
$224$	0	0
$225$	7.26733	0.484489
$226$	0	0
$227$	−20.8869	−1.38631	−0.693156	−	0.720787i	$-0.743780\pi$
$227$	−20.8869	−1.38631	−0.693156	+	0.720787i	$0.743780\pi$
$228$	0	0
$229$	8.56006	0.565665	0.282832	−	0.959169i	$-0.408726\pi$
$229$	8.56006	0.565665	0.282832	+	0.959169i	$0.408726\pi$
$230$	0	0
$231$	−10.3457	−0.680696
$232$	0	0
$233$	23.6208	1.54745	0.773725	−	0.633522i	$-0.218391\pi$
$233$	23.6208	1.54745	0.773725	+	0.633522i	$0.218391\pi$
$234$	0	0
$235$	−38.2658	−2.49618
$236$	0	0
$237$	−6.61212	−0.429503
$238$	0	0
$239$	−0.581913	−0.0376408	−0.0188204	−	0.999823i	$-0.505991\pi$
$239$	−0.581913	−0.0376408	−0.0188204	+	0.999823i	$0.505991\pi$
$240$	0	0
$241$	−16.8532	−1.08561	−0.542804	−	0.839860i	$-0.682637\pi$
$241$	−16.8532	−1.08561	−0.542804	+	0.839860i	$0.682637\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	53.1979	3.39869
$246$	0	0
$247$	38.8995	2.47512
$248$	0	0
$249$	11.2304	0.711701
$250$	0	0
$251$	21.8767	1.38084	0.690421	−	0.723408i	$-0.257425\pi$
$251$	21.8767	1.38084	0.690421	+	0.723408i	$0.257425\pi$
$252$	0	0
$253$	−2.19631	−0.138081
$254$	0	0
$255$	11.8915	0.744672
$256$	0	0
$257$	−18.4551	−1.15120	−0.575599	−	0.817732i	$-0.695231\pi$
$257$	−18.4551	−1.15120	−0.575599	+	0.817732i	$0.695231\pi$
$258$	0	0
$259$	37.0968	2.30508
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−29.6605	−1.82895	−0.914473	−	0.404647i	$-0.867394\pi$
$263$	−29.6605	−1.82895	−0.914473	+	0.404647i	$0.867394\pi$
$264$	0	0
$265$	41.2509	2.53402
$266$	0	0
$267$	−8.97644	−0.549349
$268$	0	0
$269$	4.97768	0.303495	0.151747	−	0.988419i	$-0.451510\pi$
$269$	4.97768	0.303495	0.151747	+	0.988419i	$0.451510\pi$
$270$	0	0
$271$	−27.0333	−1.64216	−0.821078	−	0.570816i	$-0.806627\pi$
$271$	−27.0333	−1.64216	−0.821078	+	0.570816i	$0.806627\pi$
$272$	0	0
$273$	−31.0081	−1.87669
$274$	0	0
$275$	15.9613	0.962503
$276$	0	0
$277$	14.0645	0.845053	0.422527	−	0.906350i	$-0.361143\pi$
$277$	14.0645	0.845053	0.422527	+	0.906350i	$0.361143\pi$
$278$	0	0
$279$	−5.12948	−0.307094
$280$	0	0
$281$	26.1880	1.56225	0.781123	−	0.624378i	$-0.214647\pi$
$281$	26.1880	1.56225	0.781123	+	0.624378i	$0.214647\pi$
$282$	0	0
$283$	−19.1249	−1.13686	−0.568429	−	0.822733i	$-0.692448\pi$
$283$	−19.1249	−1.13686	−0.568429	+	0.822733i	$0.692448\pi$
$284$	0	0
$285$	20.6971	1.22599
$286$	0	0
$287$	−11.1182	−0.656288
$288$	0	0
$289$	−5.47289	−0.321935
$290$	0	0
$291$	−13.5780	−0.795957
$292$	0	0
$293$	−19.2844	−1.12661	−0.563304	−	0.826250i	$-0.690470\pi$
$293$	−19.2844	−1.12661	−0.563304	+	0.826250i	$0.690470\pi$
$294$	0	0
$295$	−2.38179	−0.138673
$296$	0	0
$297$	2.19631	0.127443
$298$	0	0
$299$	−6.58278	−0.380692
$300$	0	0
$301$	−17.8951	−1.03146
$302$	0	0
$303$	−0.150025	−0.00861870
$304$	0	0
$305$	0.467263	0.0267554
$306$	0	0
$307$	−15.7343	−0.898002	−0.449001	−	0.893531i	$-0.648220\pi$
$307$	−15.7343	−0.898002	−0.449001	+	0.893531i	$0.648220\pi$
$308$	0	0
$309$	−11.0507	−0.628653
$310$	0	0
$311$	21.1330	1.19834	0.599171	−	0.800621i	$-0.295497\pi$
$311$	21.1330	1.19834	0.599171	+	0.800621i	$0.295497\pi$
$312$	0	0
$313$	−26.3521	−1.48951	−0.744755	−	0.667338i	$-0.767434\pi$
$313$	−26.3521	−1.48951	−0.744755	+	0.667338i	$0.767434\pi$
$314$	0	0
$315$	−16.4984	−0.929577
$316$	0	0
$317$	−30.9139	−1.73630	−0.868149	−	0.496304i	$-0.834690\pi$
$317$	−30.9139	−1.73630	−0.868149	+	0.496304i	$0.834690\pi$
$318$	0	0
$319$	2.19631	0.122970
$320$	0	0
$321$	14.8538	0.829058
$322$	0	0
$323$	20.0630	1.11633
$324$	0	0
$325$	47.8392	2.65364
$326$	0	0
$327$	−0.149928	−0.00829101
$328$	0	0
$329$	51.4637	2.83728
$330$	0	0
$331$	27.4531	1.50896	0.754480	−	0.656323i	$-0.227889\pi$
$331$	27.4531	1.50896	0.754480	+	0.656323i	$0.227889\pi$
$332$	0	0
$333$	−7.87537	−0.431567
$334$	0	0
$335$	19.4390	1.06207
$336$	0	0
$337$	0.671756	0.0365929	0.0182964	−	0.999833i	$-0.494176\pi$
$337$	0.671756	0.0365929	0.0182964	+	0.999833i	$0.494176\pi$
$338$	0	0
$339$	−4.49187	−0.243965
$340$	0	0
$341$	−11.2659	−0.610084
$342$	0	0
$343$	−38.5725	−2.08272
$344$	0	0
$345$	−3.50248	−0.188567
$346$	0	0
$347$	−24.7761	−1.33005	−0.665025	−	0.746821i	$-0.731579\pi$
$347$	−24.7761	−1.33005	−0.665025	+	0.746821i	$0.731579\pi$
$348$	0	0
$349$	−18.2390	−0.976314	−0.488157	−	0.872756i	$-0.662331\pi$
$349$	−18.2390	−0.976314	−0.488157	+	0.872756i	$0.662331\pi$
$350$	0	0
$351$	6.58278	0.351363
$352$	0	0
$353$	13.1302	0.698848	0.349424	−	0.936965i	$-0.386377\pi$
$353$	13.1302	0.698848	0.349424	+	0.936965i	$0.386377\pi$
$354$	0	0
$355$	−10.9856	−0.583055
$356$	0	0
$357$	−15.9928	−0.846431
$358$	0	0
$359$	−24.5763	−1.29708	−0.648542	−	0.761178i	$-0.724621\pi$
$359$	−24.5763	−1.29708	−0.648542	+	0.761178i	$0.724621\pi$
$360$	0	0
$361$	15.9197	0.837877
$362$	0	0
$363$	−6.17623	−0.324168
$364$	0	0
$365$	24.0319	1.25789
$366$	0	0
$367$	35.3903	1.84736	0.923679	−	0.383168i	$-0.125167\pi$
$367$	35.3903	1.84736	0.923679	+	0.383168i	$0.125167\pi$
$368$	0	0
$369$	2.36031	0.122873
$370$	0	0
$371$	−55.4784	−2.88029
$372$	0	0
$373$	28.8983	1.49630	0.748150	−	0.663530i	$-0.230942\pi$
$373$	28.8983	1.49630	0.748150	+	0.663530i	$0.230942\pi$
$374$	0	0
$375$	7.94128	0.410086
$376$	0	0
$377$	6.58278	0.339030
$378$	0	0
$379$	24.4336	1.25507	0.627536	−	0.778588i	$-0.284064\pi$
$379$	24.4336	1.25507	0.627536	+	0.778588i	$0.284064\pi$
$380$	0	0
$381$	−19.2929	−0.988405
$382$	0	0
$383$	8.95662	0.457662	0.228831	−	0.973466i	$-0.426510\pi$
$383$	8.95662	0.457662	0.228831	+	0.973466i	$0.426510\pi$
$384$	0	0
$385$	−36.2355	−1.84673
$386$	0	0
$387$	3.79900	0.193114
$388$	0	0
$389$	9.99964	0.507002	0.253501	−	0.967335i	$-0.418418\pi$
$389$	9.99964	0.507002	0.253501	+	0.967335i	$0.418418\pi$
$390$	0	0
$391$	−3.39516	−0.171701
$392$	0	0
$393$	3.27196	0.165049
$394$	0	0
$395$	−23.1588	−1.16525
$396$	0	0
$397$	−17.7152	−0.889100	−0.444550	−	0.895754i	$-0.646636\pi$
$397$	−17.7152	−0.889100	−0.444550	+	0.895754i	$0.646636\pi$
$398$	0	0
$399$	−27.8356	−1.39352
$400$	0	0
$401$	23.9546	1.19624	0.598118	−	0.801408i	$-0.295915\pi$
$401$	23.9546	1.19624	0.598118	+	0.801408i	$0.295915\pi$
$402$	0	0
$403$	−33.7662	−1.68201
$404$	0	0
$405$	3.50248	0.174039
$406$	0	0
$407$	−17.2967	−0.857368
$408$	0	0
$409$	6.38530	0.315733	0.157866	−	0.987460i	$-0.449538\pi$
$409$	6.38530	0.315733	0.157866	+	0.987460i	$0.449538\pi$
$410$	0	0
$411$	−6.84748	−0.337761
$412$	0	0
$413$	3.20327	0.157623
$414$	0	0
$415$	39.3344	1.93085
$416$	0	0
$417$	4.03549	0.197619
$418$	0	0
$419$	16.4743	0.804822	0.402411	−	0.915459i	$-0.368172\pi$
$419$	16.4743	0.804822	0.402411	+	0.915459i	$0.368172\pi$
$420$	0	0
$421$	14.4008	0.701850	0.350925	−	0.936404i	$-0.385867\pi$
$421$	14.4008	0.701850	0.350925	+	0.936404i	$0.385867\pi$
$422$	0	0
$423$	−10.9253	−0.531209
$424$	0	0
$425$	24.6738	1.19685
$426$	0	0
$427$	−0.628423	−0.0304115
$428$	0	0
$429$	14.4578	0.698030
$430$	0	0
$431$	38.5590	1.85732	0.928660	−	0.370933i	$-0.120962\pi$
$431$	38.5590	1.85732	0.928660	+	0.370933i	$0.120962\pi$
$432$	0	0
$433$	15.3671	0.738498	0.369249	−	0.929331i	$-0.379615\pi$
$433$	15.3671	0.738498	0.369249	+	0.929331i	$0.379615\pi$
$434$	0	0
$435$	3.50248	0.167931
$436$	0	0
$437$	−5.90929	−0.282680
$438$	0	0
$439$	−11.0659	−0.528149	−0.264074	−	0.964502i	$-0.585066\pi$
$439$	−11.0659	−0.528149	−0.264074	+	0.964502i	$0.585066\pi$
$440$	0	0
$441$	15.1887	0.723269
$442$	0	0
$443$	−14.0936	−0.669605	−0.334802	−	0.942288i	$-0.608670\pi$
$443$	−14.0936	−0.669605	−0.334802	+	0.942288i	$0.608670\pi$
$444$	0	0
$445$	−31.4398	−1.49039
$446$	0	0
$447$	8.29404	0.392295
$448$	0	0
$449$	25.2964	1.19381	0.596905	−	0.802312i	$-0.296397\pi$
$449$	25.2964	1.19381	0.596905	+	0.802312i	$0.296397\pi$
$450$	0	0
$451$	5.18398	0.244104
$452$	0	0
$453$	−23.4311	−1.10089
$454$	0	0
$455$	−108.605	−5.09148
$456$	0	0
$457$	−10.1374	−0.474206	−0.237103	−	0.971485i	$-0.576198\pi$
$457$	−10.1374	−0.474206	−0.237103	+	0.971485i	$0.576198\pi$
$458$	0	0
$459$	3.39516	0.158472
$460$	0	0
$461$	−22.1387	−1.03110	−0.515552	−	0.856858i	$-0.672413\pi$
$461$	−22.1387	−1.03110	−0.515552	+	0.856858i	$0.672413\pi$
$462$	0	0
$463$	37.5666	1.74587	0.872934	−	0.487839i	$-0.162215\pi$
$463$	37.5666	1.74587	0.872934	+	0.487839i	$0.162215\pi$
$464$	0	0
$465$	−17.9659	−0.833148
$466$	0	0
$467$	−13.4819	−0.623866	−0.311933	−	0.950104i	$-0.600977\pi$
$467$	−13.4819	−0.623866	−0.311933	+	0.950104i	$0.600977\pi$
$468$	0	0
$469$	−26.1436	−1.20720
$470$	0	0
$471$	1.40312	0.0646525
$472$	0	0
$473$	8.34377	0.383647
$474$	0	0
$475$	42.9448	1.97044
$476$	0	0
$477$	11.7776	0.539261
$478$	0	0
$479$	−13.9697	−0.638294	−0.319147	−	0.947705i	$-0.603396\pi$
$479$	−13.9697	−0.638294	−0.319147	+	0.947705i	$0.603396\pi$
$480$	0	0
$481$	−51.8418	−2.36378
$482$	0	0
$483$	4.71048	0.214334
$484$	0	0
$485$	−47.5566	−2.15944
$486$	0	0
$487$	19.2797	0.873645	0.436822	−	0.899548i	$-0.356104\pi$
$487$	19.2797	0.873645	0.436822	+	0.899548i	$0.356104\pi$
$488$	0	0
$489$	22.5865	1.02140
$490$	0	0
$491$	6.40047	0.288849	0.144425	−	0.989516i	$-0.453867\pi$
$491$	6.40047	0.288849	0.144425	+	0.989516i	$0.453867\pi$
$492$	0	0
$493$	3.39516	0.152910
$494$	0	0
$495$	7.69252	0.345753
$496$	0	0
$497$	14.7745	0.662729
$498$	0	0
$499$	−7.94119	−0.355497	−0.177748	−	0.984076i	$-0.556881\pi$
$499$	−7.94119	−0.355497	−0.177748	+	0.984076i	$0.556881\pi$
$500$	0	0
$501$	24.2793	1.08472
$502$	0	0
$503$	33.5601	1.49637	0.748185	−	0.663490i	$-0.230925\pi$
$503$	33.5601	1.49637	0.748185	+	0.663490i	$0.230925\pi$
$504$	0	0
$505$	−0.525458	−0.0233826
$506$	0	0
$507$	30.3329	1.34713
$508$	0	0
$509$	28.8207	1.27745	0.638727	−	0.769433i	$-0.279461\pi$
$509$	28.8207	1.27745	0.638727	+	0.769433i	$0.279461\pi$
$510$	0	0
$511$	−32.3205	−1.42978
$512$	0	0
$513$	5.90929	0.260901
$514$	0	0
$515$	−38.7049	−1.70554
$516$	0	0
$517$	−23.9954	−1.05532
$518$	0	0
$519$	13.4810	0.591750
$520$	0	0
$521$	−20.4931	−0.897817	−0.448909	−	0.893578i	$-0.648187\pi$
$521$	−20.4931	−0.897817	−0.448909	+	0.893578i	$0.648187\pi$
$522$	0	0
$523$	16.8958	0.738802	0.369401	−	0.929270i	$-0.379563\pi$
$523$	16.8958	0.738802	0.369401	+	0.929270i	$0.379563\pi$
$524$	0	0
$525$	−34.2327	−1.49404
$526$	0	0
$527$	−17.4154	−0.758627
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.680031	−0.0295108
$532$	0	0
$533$	15.5374	0.673000
$534$	0	0
$535$	52.0251	2.24924
$536$	0	0
$537$	5.50916	0.237738
$538$	0	0
$539$	33.3590	1.43687
$540$	0	0
$541$	23.4735	1.00921	0.504603	−	0.863352i	$-0.331639\pi$
$541$	23.4735	1.00921	0.504603	+	0.863352i	$0.331639\pi$
$542$	0	0
$543$	−8.20613	−0.352159
$544$	0	0
$545$	−0.525118	−0.0224936
$546$	0	0
$547$	−23.8627	−1.02030	−0.510149	−	0.860086i	$-0.670410\pi$
$547$	−23.8627	−1.02030	−0.510149	+	0.860086i	$0.670410\pi$
$548$	0	0
$549$	0.133410	0.00569378
$550$	0	0
$551$	5.90929	0.251744
$552$	0	0
$553$	31.1463	1.32448
$554$	0	0
$555$	−27.5833	−1.17084
$556$	0	0
$557$	−3.00361	−0.127267	−0.0636335	−	0.997973i	$-0.520269\pi$
$557$	−3.00361	−0.127267	−0.0636335	+	0.997973i	$0.520269\pi$
$558$	0	0
$559$	25.0079	1.05772
$560$	0	0
$561$	7.45682	0.314827
$562$	0	0
$563$	−35.6429	−1.50217	−0.751085	−	0.660205i	$-0.770469\pi$
$563$	−35.6429	−1.50217	−0.751085	+	0.660205i	$0.770469\pi$
$564$	0	0
$565$	−15.7326	−0.661878
$566$	0	0
$567$	−4.71048	−0.197822
$568$	0	0
$569$	−7.91023	−0.331614	−0.165807	−	0.986158i	$-0.553023\pi$
$569$	−7.91023	−0.331614	−0.165807	+	0.986158i	$0.553023\pi$
$570$	0	0
$571$	10.9023	0.456248	0.228124	−	0.973632i	$-0.426741\pi$
$571$	10.9023	0.456248	0.228124	+	0.973632i	$0.426741\pi$
$572$	0	0
$573$	−6.45372	−0.269608
$574$	0	0
$575$	−7.26733	−0.303069
$576$	0	0
$577$	−36.6468	−1.52563	−0.762813	−	0.646619i	$-0.776183\pi$
$577$	−36.6468	−1.52563	−0.762813	+	0.646619i	$0.776183\pi$
$578$	0	0
$579$	16.5824	0.689140
$580$	0	0
$581$	−52.9008	−2.19470
$582$	0	0
$583$	25.8673	1.07132
$584$	0	0
$585$	23.0560	0.953249
$586$	0	0
$587$	−44.3187	−1.82923	−0.914614	−	0.404327i	$-0.867506\pi$
$587$	−44.3187	−1.82923	−0.914614	+	0.404327i	$0.867506\pi$
$588$	0	0
$589$	−30.3116	−1.24897
$590$	0	0
$591$	12.9392	0.532249
$592$	0	0
$593$	−4.13642	−0.169862	−0.0849311	−	0.996387i	$-0.527067\pi$
$593$	−4.13642	−0.169862	−0.0849311	+	0.996387i	$0.527067\pi$
$594$	0	0
$595$	−56.0145	−2.29637
$596$	0	0
$597$	18.5875	0.760737
$598$	0	0
$599$	−4.30916	−0.176068	−0.0880338	−	0.996117i	$-0.528058\pi$
$599$	−4.30916	−0.176068	−0.0880338	+	0.996117i	$0.528058\pi$
$600$	0	0
$601$	13.3675	0.545272	0.272636	−	0.962117i	$-0.412105\pi$
$601$	13.3675	0.545272	0.272636	+	0.962117i	$0.412105\pi$
$602$	0	0
$603$	5.55008	0.226017
$604$	0	0
$605$	−21.6321	−0.879469
$606$	0	0
$607$	−22.8889	−0.929034	−0.464517	−	0.885564i	$-0.653772\pi$
$607$	−22.8889	−0.929034	−0.464517	+	0.885564i	$0.653772\pi$
$608$	0	0
$609$	−4.71048	−0.190878
$610$	0	0
$611$	−71.9191	−2.90954
$612$	0	0
$613$	−24.0016	−0.969416	−0.484708	−	0.874676i	$-0.661074\pi$
$613$	−24.0016	−0.969416	−0.484708	+	0.874676i	$0.661074\pi$
$614$	0	0
$615$	8.26694	0.333355
$616$	0	0
$617$	−18.1841	−0.732062	−0.366031	−	0.930603i	$-0.619284\pi$
$617$	−18.1841	−0.732062	−0.366031	+	0.930603i	$0.619284\pi$
$618$	0	0
$619$	−4.31450	−0.173414	−0.0867072	−	0.996234i	$-0.527634\pi$
$619$	−4.31450	−0.173414	−0.0867072	+	0.996234i	$0.527634\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	42.2834	1.69405
$624$	0	0
$625$	−8.52253	−0.340901
$626$	0	0
$627$	12.9786	0.518316
$628$	0	0
$629$	−26.7381	−1.06612
$630$	0	0
$631$	−25.2975	−1.00708	−0.503538	−	0.863973i	$-0.667969\pi$
$631$	−25.2975	−1.00708	−0.503538	+	0.863973i	$0.667969\pi$
$632$	0	0
$633$	2.73196	0.108586
$634$	0	0
$635$	−67.5729	−2.68155
$636$	0	0
$637$	99.9835	3.96149
$638$	0	0
$639$	−3.13652	−0.124079
$640$	0	0
$641$	−24.3564	−0.962020	−0.481010	−	0.876715i	$-0.659730\pi$
$641$	−24.3564	−0.962020	−0.481010	+	0.876715i	$0.659730\pi$
$642$	0	0
$643$	−29.1419	−1.14924	−0.574622	−	0.818419i	$-0.694851\pi$
$643$	−29.1419	−1.14924	−0.574622	+	0.818419i	$0.694851\pi$
$644$	0	0
$645$	13.3059	0.523919
$646$	0	0
$647$	−4.74081	−0.186381	−0.0931903	−	0.995648i	$-0.529706\pi$
$647$	−4.74081	−0.186381	−0.0931903	+	0.995648i	$0.529706\pi$
$648$	0	0
$649$	−1.49356	−0.0586273
$650$	0	0
$651$	24.1623	0.946996
$652$	0	0
$653$	12.1898	0.477024	0.238512	−	0.971140i	$-0.423340\pi$
$653$	12.1898	0.477024	0.238512	+	0.971140i	$0.423340\pi$
$654$	0	0
$655$	11.4600	0.447778
$656$	0	0
$657$	6.86140	0.267689
$658$	0	0
$659$	32.1555	1.25260	0.626301	−	0.779581i	$-0.284568\pi$
$659$	32.1555	1.25260	0.626301	+	0.779581i	$0.284568\pi$
$660$	0	0
$661$	−27.6799	−1.07662	−0.538311	−	0.842746i	$-0.680937\pi$
$661$	−27.6799	−1.07662	−0.538311	+	0.842746i	$0.680937\pi$
$662$	0	0
$663$	22.3496	0.867985
$664$	0	0
$665$	−97.4935	−3.78064
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	0.148745	0.00575082
$670$	0	0
$671$	0.293009	0.0113115
$672$	0	0
$673$	12.5528	0.483873	0.241937	−	0.970292i	$-0.422217\pi$
$673$	12.5528	0.483873	0.241937	+	0.970292i	$0.422217\pi$
$674$	0	0
$675$	7.26733	0.279720
$676$	0	0
$677$	19.2529	0.739950	0.369975	−	0.929042i	$-0.379366\pi$
$677$	19.2529	0.739950	0.369975	+	0.929042i	$0.379366\pi$
$678$	0	0
$679$	63.9590	2.45452
$680$	0	0
$681$	−20.8869	−0.800388
$682$	0	0
$683$	−41.8208	−1.60023	−0.800114	−	0.599847i	$-0.795228\pi$
$683$	−41.8208	−1.60023	−0.800114	+	0.599847i	$0.795228\pi$
$684$	0	0
$685$	−23.9831	−0.916348
$686$	0	0
$687$	8.56006	0.326587
$688$	0	0
$689$	77.5295	2.95364
$690$	0	0
$691$	26.0082	0.989397	0.494699	−	0.869065i	$-0.335278\pi$
$691$	26.0082	0.989397	0.494699	+	0.869065i	$0.335278\pi$
$692$	0	0
$693$	−10.3457	−0.393000
$694$	0	0
$695$	14.1342	0.536141
$696$	0	0
$697$	8.01364	0.303538
$698$	0	0
$699$	23.6208	0.893420
$700$	0	0
$701$	−12.9211	−0.488023	−0.244011	−	0.969772i	$-0.578463\pi$
$701$	−12.9211	−0.488023	−0.244011	+	0.969772i	$0.578463\pi$
$702$	0	0
$703$	−46.5378	−1.75521
$704$	0	0
$705$	−38.2658	−1.44117
$706$	0	0
$707$	0.706689	0.0265778
$708$	0	0
$709$	5.45491	0.204863	0.102432	−	0.994740i	$-0.467338\pi$
$709$	5.45491	0.204863	0.102432	+	0.994740i	$0.467338\pi$
$710$	0	0
$711$	−6.61212	−0.247974
$712$	0	0
$713$	5.12948	0.192101
$714$	0	0
$715$	50.6381	1.89376
$716$	0	0
$717$	−0.581913	−0.0217319
$718$	0	0
$719$	−45.8190	−1.70876	−0.854380	−	0.519648i	$-0.826063\pi$
$719$	−45.8190	−1.70876	−0.854380	+	0.519648i	$0.826063\pi$
$720$	0	0
$721$	52.0542	1.93860
$722$	0	0
$723$	−16.8532	−0.626776
$724$	0	0
$725$	7.26733	0.269902
$726$	0	0
$727$	41.3643	1.53412	0.767059	−	0.641577i	$-0.221720\pi$
$727$	41.3643	1.53412	0.767059	+	0.641577i	$0.221720\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.8982	0.477057
$732$	0	0
$733$	2.49787	0.0922610	0.0461305	−	0.998935i	$-0.485311\pi$
$733$	2.49787	0.0922610	0.0461305	+	0.998935i	$0.485311\pi$
$734$	0	0
$735$	53.1979	1.96223
$736$	0	0
$737$	12.1897	0.449013
$738$	0	0
$739$	−39.7649	−1.46278	−0.731388	−	0.681961i	$-0.761127\pi$
$739$	−39.7649	−1.46278	−0.731388	+	0.681961i	$0.761127\pi$
$740$	0	0
$741$	38.8995	1.42901
$742$	0	0
$743$	−17.4052	−0.638534	−0.319267	−	0.947665i	$-0.603437\pi$
$743$	−17.4052	−0.638534	−0.319267	+	0.947665i	$0.603437\pi$
$744$	0	0
$745$	29.0497	1.06430
$746$	0	0
$747$	11.2304	0.410901
$748$	0	0
$749$	−69.9686	−2.55660
$750$	0	0
$751$	14.4215	0.526249	0.263124	−	0.964762i	$-0.415247\pi$
$751$	14.4215	0.526249	0.263124	+	0.964762i	$0.415247\pi$
$752$	0	0
$753$	21.8767	0.797230
$754$	0	0
$755$	−82.0669	−2.98672
$756$	0	0
$757$	−39.4057	−1.43223	−0.716113	−	0.697984i	$-0.754080\pi$
$757$	−39.4057	−1.43223	−0.716113	+	0.697984i	$0.754080\pi$
$758$	0	0
$759$	−2.19631	−0.0797210
$760$	0	0
$761$	22.1304	0.802228	0.401114	−	0.916028i	$-0.368623\pi$
$761$	22.1304	0.802228	0.401114	+	0.916028i	$0.368623\pi$
$762$	0	0
$763$	0.706231	0.0255673
$764$	0	0
$765$	11.8915	0.429937
$766$	0	0
$767$	−4.47649	−0.161637
$768$	0	0
$769$	−17.8742	−0.644562	−0.322281	−	0.946644i	$-0.604450\pi$
$769$	−17.8742	−0.644562	−0.322281	+	0.946644i	$0.604450\pi$
$770$	0	0
$771$	−18.4551	−0.664644
$772$	0	0
$773$	−4.13208	−0.148621	−0.0743104	−	0.997235i	$-0.523676\pi$
$773$	−4.13208	−0.148621	−0.0743104	+	0.997235i	$0.523676\pi$
$774$	0	0
$775$	−37.2776	−1.33905
$776$	0	0
$777$	37.0968	1.33084
$778$	0	0
$779$	13.9478	0.499731
$780$	0	0
$781$	−6.88878	−0.246500
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	4.91440	0.175403
$786$	0	0
$787$	17.0752	0.608665	0.304333	−	0.952566i	$-0.401567\pi$
$787$	17.0752	0.608665	0.304333	+	0.952566i	$0.401567\pi$
$788$	0	0
$789$	−29.6605	−1.05594
$790$	0	0
$791$	21.1589	0.752322
$792$	0	0
$793$	0.878205	0.0311860
$794$	0	0
$795$	41.2509	1.46302
$796$	0	0
$797$	4.45794	0.157908	0.0789541	−	0.996878i	$-0.474842\pi$
$797$	4.45794	0.157908	0.0789541	+	0.996878i	$0.474842\pi$
$798$	0	0
$799$	−37.0933	−1.31227
$800$	0	0
$801$	−8.97644	−0.317167
$802$	0	0
$803$	15.0698	0.531800
$804$	0	0
$805$	16.4984	0.581491
$806$	0	0
$807$	4.97768	0.175223
$808$	0	0
$809$	−33.3250	−1.17164	−0.585822	−	0.810440i	$-0.699228\pi$
$809$	−33.3250	−1.17164	−0.585822	+	0.810440i	$0.699228\pi$
$810$	0	0
$811$	−42.5402	−1.49379	−0.746895	−	0.664942i	$-0.768456\pi$
$811$	−42.5402	−1.49379	−0.746895	+	0.664942i	$0.768456\pi$
$812$	0	0
$813$	−27.0333	−0.948100
$814$	0	0
$815$	79.1088	2.77106
$816$	0	0
$817$	22.4494	0.785404
$818$	0	0
$819$	−31.0081	−1.08351
$820$	0	0
$821$	−42.1105	−1.46967	−0.734834	−	0.678247i	$-0.762740\pi$
$821$	−42.1105	−1.46967	−0.734834	+	0.678247i	$0.762740\pi$
$822$	0	0
$823$	36.0536	1.25675	0.628374	−	0.777911i	$-0.283721\pi$
$823$	36.0536	1.25675	0.628374	+	0.777911i	$0.283721\pi$
$824$	0	0
$825$	15.9613	0.555702
$826$	0	0
$827$	26.6198	0.925661	0.462831	−	0.886447i	$-0.346834\pi$
$827$	26.6198	0.925661	0.462831	+	0.886447i	$0.346834\pi$
$828$	0	0
$829$	37.1344	1.28973	0.644866	−	0.764296i	$-0.276913\pi$
$829$	37.1344	1.28973	0.644866	+	0.764296i	$0.276913\pi$
$830$	0	0
$831$	14.0645	0.487892
$832$	0	0
$833$	51.5679	1.78672
$834$	0	0
$835$	85.0377	2.94285
$836$	0	0
$837$	−5.12948	−0.177301
$838$	0	0
$839$	12.4418	0.429539	0.214770	−	0.976665i	$-0.431100\pi$
$839$	12.4418	0.429539	0.214770	+	0.976665i	$0.431100\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	26.1880	0.901963
$844$	0	0
$845$	106.240	3.65478
$846$	0	0
$847$	29.0930	0.999648
$848$	0	0
$849$	−19.1249	−0.656365
$850$	0	0
$851$	7.87537	0.269964
$852$	0	0
$853$	21.9194	0.750505	0.375252	−	0.926923i	$-0.377556\pi$
$853$	21.9194	0.750505	0.375252	+	0.926923i	$0.377556\pi$
$854$	0	0
$855$	20.6971	0.707827
$856$	0	0
$857$	−19.9201	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$857$	−19.9201	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$858$	0	0
$859$	0.537401	0.0183359	0.00916795	−	0.999958i	$-0.497082\pi$
$859$	0.537401	0.0183359	0.00916795	+	0.999958i	$0.497082\pi$
$860$	0	0
$861$	−11.1182	−0.378908
$862$	0	0
$863$	−23.8065	−0.810382	−0.405191	−	0.914232i	$-0.632795\pi$
$863$	−23.8065	−0.810382	−0.405191	+	0.914232i	$0.632795\pi$
$864$	0	0
$865$	47.2169	1.60542
$866$	0	0
$867$	−5.47289	−0.185869
$868$	0	0
$869$	−14.5223	−0.492634
$870$	0	0
$871$	36.5349	1.23794
$872$	0	0
$873$	−13.5780	−0.459546
$874$	0	0
$875$	−37.4073	−1.26460
$876$	0	0
$877$	−32.3863	−1.09361	−0.546804	−	0.837261i	$-0.684156\pi$
$877$	−32.3863	−1.09361	−0.546804	+	0.837261i	$0.684156\pi$
$878$	0	0
$879$	−19.2844	−0.650448
$880$	0	0
$881$	10.8622	0.365955	0.182978	−	0.983117i	$-0.441426\pi$
$881$	10.8622	0.365955	0.182978	+	0.983117i	$0.441426\pi$
$882$	0	0
$883$	34.1191	1.14820	0.574099	−	0.818786i	$-0.305352\pi$
$883$	34.1191	1.14820	0.574099	+	0.818786i	$0.305352\pi$
$884$	0	0
$885$	−2.38179	−0.0800630
$886$	0	0
$887$	−16.3192	−0.547945	−0.273973	−	0.961737i	$-0.588338\pi$
$887$	−16.3192	−0.547945	−0.273973	+	0.961737i	$0.588338\pi$
$888$	0	0
$889$	90.8789	3.04798
$890$	0	0
$891$	2.19631	0.0735791
$892$	0	0
$893$	−64.5610	−2.16045
$894$	0	0
$895$	19.2957	0.644984
$896$	0	0
$897$	−6.58278	−0.219792
$898$	0	0
$899$	−5.12948	−0.171078
$900$	0	0
$901$	39.9870	1.33216
$902$	0	0
$903$	−17.8951	−0.595512
$904$	0	0
$905$	−28.7418	−0.955409
$906$	0	0
$907$	14.3045	0.474972	0.237486	−	0.971391i	$-0.423677\pi$
$907$	14.3045	0.474972	0.237486	+	0.971391i	$0.423677\pi$
$908$	0	0
$909$	−0.150025	−0.00497601
$910$	0	0
$911$	−4.40592	−0.145975	−0.0729874	−	0.997333i	$-0.523253\pi$
$911$	−4.40592	−0.145975	−0.0729874	+	0.997333i	$0.523253\pi$
$912$	0	0
$913$	24.6655	0.816310
$914$	0	0
$915$	0.467263	0.0154473
$916$	0	0
$917$	−15.4125	−0.508966
$918$	0	0
$919$	53.5829	1.76754	0.883768	−	0.467925i	$-0.154998\pi$
$919$	53.5829	1.76754	0.883768	+	0.467925i	$0.154998\pi$
$920$	0	0
$921$	−15.7343	−0.518462
$922$	0	0
$923$	−20.6470	−0.679605
$924$	0	0
$925$	−57.2329	−1.88181
$926$	0	0
$927$	−11.0507	−0.362953
$928$	0	0
$929$	−36.6839	−1.20356	−0.601780	−	0.798662i	$-0.705542\pi$
$929$	−36.6839	−1.20356	−0.601780	+	0.798662i	$0.705542\pi$
$930$	0	0
$931$	89.7541	2.94157
$932$	0	0
$933$	21.1330	0.691863
$934$	0	0
$935$	26.1173	0.854128
$936$	0	0
$937$	16.8008	0.548857	0.274428	−	0.961608i	$-0.411511\pi$
$937$	16.8008	0.548857	0.274428	+	0.961608i	$0.411511\pi$
$938$	0	0
$939$	−26.3521	−0.859969
$940$	0	0
$941$	46.3604	1.51131	0.755654	−	0.654971i	$-0.227319\pi$
$941$	46.3604	1.51131	0.755654	+	0.654971i	$0.227319\pi$
$942$	0	0
$943$	−2.36031	−0.0768624
$944$	0	0
$945$	−16.4984	−0.536691
$946$	0	0
$947$	−32.5851	−1.05887	−0.529437	−	0.848350i	$-0.677597\pi$
$947$	−32.5851	−1.05887	−0.529437	+	0.848350i	$0.677597\pi$
$948$	0	0
$949$	45.1671	1.46618
$950$	0	0
$951$	−30.9139	−1.00245
$952$	0	0
$953$	−28.4616	−0.921961	−0.460980	−	0.887410i	$-0.652502\pi$
$953$	−28.4616	−0.921961	−0.460980	+	0.887410i	$0.652502\pi$
$954$	0	0
$955$	−22.6040	−0.731448
$956$	0	0
$957$	2.19631	0.0709966
$958$	0	0
$959$	32.2549	1.04157
$960$	0	0
$961$	−4.68843	−0.151240
$962$	0	0
$963$	14.8538	0.478657
$964$	0	0
$965$	58.0794	1.86964
$966$	0	0
$967$	−44.9483	−1.44544	−0.722720	−	0.691141i	$-0.757108\pi$
$967$	−44.9483	−1.44544	−0.722720	+	0.691141i	$0.757108\pi$
$968$	0	0
$969$	20.0630	0.644516
$970$	0	0
$971$	−12.2350	−0.392641	−0.196320	−	0.980540i	$-0.562899\pi$
$971$	−12.2350	−0.392641	−0.196320	+	0.980540i	$0.562899\pi$
$972$	0	0
$973$	−19.0091	−0.609405
$974$	0	0
$975$	47.8392	1.53208
$976$	0	0
$977$	−39.3166	−1.25785	−0.628924	−	0.777466i	$-0.716505\pi$
$977$	−39.3166	−1.25785	−0.628924	+	0.777466i	$0.716505\pi$
$978$	0	0
$979$	−19.7150	−0.630095
$980$	0	0
$981$	−0.149928	−0.00478682
$982$	0	0
$983$	−6.02574	−0.192191	−0.0960957	−	0.995372i	$-0.530635\pi$
$983$	−6.02574	−0.192191	−0.0960957	+	0.995372i	$0.530635\pi$
$984$	0	0
$985$	45.3193	1.44399
$986$	0	0
$987$	51.4637	1.63811
$988$	0	0
$989$	−3.79900	−0.120801
$990$	0	0
$991$	1.30366	0.0414120	0.0207060	−	0.999786i	$-0.493409\pi$
$991$	1.30366	0.0414120	0.0207060	+	0.999786i	$0.493409\pi$
$992$	0	0
$993$	27.4531	0.871199
$994$	0	0
$995$	65.1023	2.06388
$996$	0	0
$997$	−0.0770244	−0.00243939	−0.00121969	−	0.999999i	$-0.500388\pi$
$997$	−0.0770244	−0.00243939	−0.00121969	+	0.999999i	$0.500388\pi$
$998$	0	0
$999$	−7.87537	−0.249166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.50248 q^{5} -4.71048 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.50248 q^{5} -4.71048 q^{7} +1.00000 q^{9} +2.19631 q^{11} +6.58278 q^{13} +3.50248 q^{15} +3.39516 q^{17} +5.90929 q^{19} -4.71048 q^{21} -1.00000 q^{23} +7.26733 q^{25} +1.00000 q^{27} +1.00000 q^{29} -5.12948 q^{31} +2.19631 q^{33} -16.4984 q^{35} -7.87537 q^{37} +6.58278 q^{39} +2.36031 q^{41} +3.79900 q^{43} +3.50248 q^{45} -10.9253 q^{47} +15.1887 q^{49} +3.39516 q^{51} +11.7776 q^{53} +7.69252 q^{55} +5.90929 q^{57} -0.680031 q^{59} +0.133410 q^{61} -4.71048 q^{63} +23.0560 q^{65} +5.55008 q^{67} -1.00000 q^{69} -3.13652 q^{71} +6.86140 q^{73} +7.26733 q^{75} -10.3457 q^{77} -6.61212 q^{79} +1.00000 q^{81} +11.2304 q^{83} +11.8915 q^{85} +1.00000 q^{87} -8.97644 q^{89} -31.0081 q^{91} -5.12948 q^{93} +20.6971 q^{95} -13.5780 q^{97} +2.19631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 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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