LMFDB - Embedded newform 8008.2.a.z.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-3.11300$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11300 q^{3} -2.63092 q^{5} -1.00000 q^{7} +6.69079 q^{9} +O(q^{10})$	$q+3.11300 q^{3} -2.63092 q^{5} -1.00000 q^{7} +6.69079 q^{9} -1.00000 q^{11} +1.00000 q^{13} -8.19007 q^{15} -4.45177 q^{17} -8.24091 q^{19} -3.11300 q^{21} +7.28341 q^{23} +1.92175 q^{25} +11.4894 q^{27} -5.34027 q^{29} +0.963653 q^{31} -3.11300 q^{33} +2.63092 q^{35} +9.18755 q^{37} +3.11300 q^{39} +10.2515 q^{41} +7.33078 q^{43} -17.6029 q^{45} -1.94078 q^{47} +1.00000 q^{49} -13.8584 q^{51} +14.5075 q^{53} +2.63092 q^{55} -25.6540 q^{57} +3.19315 q^{59} +13.7420 q^{61} -6.69079 q^{63} -2.63092 q^{65} +9.69712 q^{67} +22.6733 q^{69} -0.878412 q^{71} -6.04717 q^{73} +5.98240 q^{75} +1.00000 q^{77} -6.05877 q^{79} +15.6943 q^{81} -10.8659 q^{83} +11.7122 q^{85} -16.6243 q^{87} +5.01272 q^{89} -1.00000 q^{91} +2.99985 q^{93} +21.6812 q^{95} -3.71285 q^{97} -6.69079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	3.11300	1.79729	0.898647	−	0.438673i	$-0.144551\pi$
$3$	3.11300	1.79729	0.898647	+	0.438673i	$0.144551\pi$
$4$	0	0
$5$	−2.63092	−1.17658	−0.588292	−	0.808649i	$-0.700199\pi$
$5$	−2.63092	−1.17658	−0.588292	+	0.808649i	$0.700199\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.69079	2.23026
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−8.19007	−2.11467
$16$	0	0
$17$	−4.45177	−1.07971	−0.539856	−	0.841757i	$-0.681521\pi$
$17$	−4.45177	−1.07971	−0.539856	+	0.841757i	$0.681521\pi$
$18$	0	0
$19$	−8.24091	−1.89060	−0.945298	−	0.326209i	$-0.894229\pi$
$19$	−8.24091	−1.89060	−0.945298	+	0.326209i	$0.894229\pi$
$20$	0	0
$21$	−3.11300	−0.679313
$22$	0	0
$23$	7.28341	1.51870	0.759348	−	0.650684i	$-0.225518\pi$
$23$	7.28341	1.51870	0.759348	+	0.650684i	$0.225518\pi$
$24$	0	0
$25$	1.92175	0.384349
$26$	0	0
$27$	11.4894	2.21114
$28$	0	0
$29$	−5.34027	−0.991663	−0.495831	−	0.868419i	$-0.665137\pi$
$29$	−5.34027	−0.991663	−0.495831	+	0.868419i	$0.665137\pi$
$30$	0	0
$31$	0.963653	0.173077	0.0865386	−	0.996249i	$-0.472419\pi$
$31$	0.963653	0.173077	0.0865386	+	0.996249i	$0.472419\pi$
$32$	0	0
$33$	−3.11300	−0.541904
$34$	0	0
$35$	2.63092	0.444707
$36$	0	0
$37$	9.18755	1.51042	0.755212	−	0.655481i	$-0.227534\pi$
$37$	9.18755	1.51042	0.755212	+	0.655481i	$0.227534\pi$
$38$	0	0
$39$	3.11300	0.498479
$40$	0	0
$41$	10.2515	1.60101	0.800506	−	0.599324i	$-0.204564\pi$
$41$	10.2515	1.60101	0.800506	+	0.599324i	$0.204564\pi$
$42$	0	0
$43$	7.33078	1.11793	0.558967	−	0.829190i	$-0.311198\pi$
$43$	7.33078	1.11793	0.558967	+	0.829190i	$0.311198\pi$
$44$	0	0
$45$	−17.6029	−2.62409
$46$	0	0
$47$	−1.94078	−0.283091	−0.141546	−	0.989932i	$-0.545207\pi$
$47$	−1.94078	−0.283091	−0.141546	+	0.989932i	$0.545207\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−13.8584	−1.94056
$52$	0	0
$53$	14.5075	1.99276	0.996380	−	0.0850063i	$-0.0270910\pi$
$53$	14.5075	1.99276	0.996380	+	0.0850063i	$0.0270910\pi$
$54$	0	0
$55$	2.63092	0.354753
$56$	0	0
$57$	−25.6540	−3.39795
$58$	0	0
$59$	3.19315	0.415713	0.207856	−	0.978159i	$-0.433351\pi$
$59$	3.19315	0.415713	0.207856	+	0.978159i	$0.433351\pi$
$60$	0	0
$61$	13.7420	1.75948	0.879739	−	0.475456i	$-0.157717\pi$
$61$	13.7420	1.75948	0.879739	+	0.475456i	$0.157717\pi$
$62$	0	0
$63$	−6.69079	−0.842960
$64$	0	0
$65$	−2.63092	−0.326326
$66$	0	0
$67$	9.69712	1.18469	0.592346	−	0.805684i	$-0.298202\pi$
$67$	9.69712	1.18469	0.592346	+	0.805684i	$0.298202\pi$
$68$	0	0
$69$	22.6733	2.72954
$70$	0	0
$71$	−0.878412	−0.104248	−0.0521242	−	0.998641i	$-0.516599\pi$
$71$	−0.878412	−0.104248	−0.0521242	+	0.998641i	$0.516599\pi$
$72$	0	0
$73$	−6.04717	−0.707768	−0.353884	−	0.935289i	$-0.615139\pi$
$73$	−6.04717	−0.707768	−0.353884	+	0.935289i	$0.615139\pi$
$74$	0	0
$75$	5.98240	0.690788
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−6.05877	−0.681665	−0.340833	−	0.940124i	$-0.610709\pi$
$79$	−6.05877	−0.681665	−0.340833	+	0.940124i	$0.610709\pi$
$80$	0	0
$81$	15.6943	1.74381
$82$	0	0
$83$	−10.8659	−1.19269	−0.596344	−	0.802729i	$-0.703381\pi$
$83$	−10.8659	−1.19269	−0.596344	+	0.802729i	$0.703381\pi$
$84$	0	0
$85$	11.7122	1.27037
$86$	0	0
$87$	−16.6243	−1.78231
$88$	0	0
$89$	5.01272	0.531348	0.265674	−	0.964063i	$-0.414406\pi$
$89$	5.01272	0.531348	0.265674	+	0.964063i	$0.414406\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.99985	0.311070
$94$	0	0
$95$	21.6812	2.22444
$96$	0	0
$97$	−3.71285	−0.376983	−0.188491	−	0.982075i	$-0.560360\pi$
$97$	−3.71285	−0.376983	−0.188491	+	0.982075i	$0.560360\pi$
$98$	0	0
$99$	−6.69079	−0.672450
$100$	0	0
$101$	12.3816	1.23202	0.616009	−	0.787739i	$-0.288748\pi$
$101$	12.3816	1.23202	0.616009	+	0.787739i	$0.288748\pi$
$102$	0	0
$103$	4.19157	0.413008	0.206504	−	0.978446i	$-0.433791\pi$
$103$	4.19157	0.413008	0.206504	+	0.978446i	$0.433791\pi$
$104$	0	0
$105$	8.19007	0.799269
$106$	0	0
$107$	13.8316	1.33715	0.668576	−	0.743643i	$-0.266904\pi$
$107$	13.8316	1.33715	0.668576	+	0.743643i	$0.266904\pi$
$108$	0	0
$109$	−9.05377	−0.867194	−0.433597	−	0.901107i	$-0.642756\pi$
$109$	−9.05377	−0.867194	−0.433597	+	0.901107i	$0.642756\pi$
$110$	0	0
$111$	28.6009	2.71467
$112$	0	0
$113$	4.96943	0.467485	0.233742	−	0.972299i	$-0.424903\pi$
$113$	4.96943	0.467485	0.233742	+	0.972299i	$0.424903\pi$
$114$	0	0
$115$	−19.1621	−1.78687
$116$	0	0
$117$	6.69079	0.618564
$118$	0	0
$119$	4.45177	0.408093
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	31.9129	2.87749
$124$	0	0
$125$	8.09865	0.724365
$126$	0	0
$127$	13.5549	1.20280	0.601401	−	0.798947i	$-0.294609\pi$
$127$	13.5549	1.20280	0.601401	+	0.798947i	$0.294609\pi$
$128$	0	0
$129$	22.8207	2.00925
$130$	0	0
$131$	−6.47545	−0.565763	−0.282881	−	0.959155i	$-0.591290\pi$
$131$	−6.47545	−0.565763	−0.282881	+	0.959155i	$0.591290\pi$
$132$	0	0
$133$	8.24091	0.714578
$134$	0	0
$135$	−30.2278	−2.60160
$136$	0	0
$137$	−6.38850	−0.545807	−0.272903	−	0.962041i	$-0.587984\pi$
$137$	−6.38850	−0.545807	−0.272903	+	0.962041i	$0.587984\pi$
$138$	0	0
$139$	−16.2552	−1.37875	−0.689374	−	0.724406i	$-0.742114\pi$
$139$	−16.2552	−1.37875	−0.689374	+	0.724406i	$0.742114\pi$
$140$	0	0
$141$	−6.04164	−0.508798
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	14.0498	1.16677
$146$	0	0
$147$	3.11300	0.256756
$148$	0	0
$149$	16.4992	1.35167	0.675833	−	0.737054i	$-0.263784\pi$
$149$	16.4992	1.35167	0.675833	+	0.737054i	$0.263784\pi$
$150$	0	0
$151$	5.06309	0.412029	0.206014	−	0.978549i	$-0.433951\pi$
$151$	5.06309	0.412029	0.206014	+	0.978549i	$0.433951\pi$
$152$	0	0
$153$	−29.7858	−2.40804
$154$	0	0
$155$	−2.53529	−0.203640
$156$	0	0
$157$	23.7661	1.89674	0.948372	−	0.317161i	$-0.102729\pi$
$157$	23.7661	1.89674	0.948372	+	0.317161i	$0.102729\pi$
$158$	0	0
$159$	45.1620	3.58158
$160$	0	0
$161$	−7.28341	−0.574013
$162$	0	0
$163$	1.97076	0.154362	0.0771808	−	0.997017i	$-0.475408\pi$
$163$	1.97076	0.154362	0.0771808	+	0.997017i	$0.475408\pi$
$164$	0	0
$165$	8.19007	0.637596
$166$	0	0
$167$	22.8335	1.76691	0.883455	−	0.468517i	$-0.155211\pi$
$167$	22.8335	1.76691	0.883455	+	0.468517i	$0.155211\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−55.1382	−4.21653
$172$	0	0
$173$	−21.1624	−1.60895	−0.804473	−	0.593990i	$-0.797552\pi$
$173$	−21.1624	−1.60895	−0.804473	+	0.593990i	$0.797552\pi$
$174$	0	0
$175$	−1.92175	−0.145270
$176$	0	0
$177$	9.94029	0.747158
$178$	0	0
$179$	14.3460	1.07227	0.536134	−	0.844133i	$-0.319884\pi$
$179$	14.3460	1.07227	0.536134	+	0.844133i	$0.319884\pi$
$180$	0	0
$181$	−20.3360	−1.51157	−0.755783	−	0.654822i	$-0.772744\pi$
$181$	−20.3360	−1.51157	−0.755783	+	0.654822i	$0.772744\pi$
$182$	0	0
$183$	42.7788	3.16230
$184$	0	0
$185$	−24.1717	−1.77714
$186$	0	0
$187$	4.45177	0.325546
$188$	0	0
$189$	−11.4894	−0.835734
$190$	0	0
$191$	7.18631	0.519983	0.259991	−	0.965611i	$-0.416280\pi$
$191$	7.18631	0.519983	0.259991	+	0.965611i	$0.416280\pi$
$192$	0	0
$193$	2.34037	0.168463	0.0842316	−	0.996446i	$-0.473156\pi$
$193$	2.34037	0.168463	0.0842316	+	0.996446i	$0.473156\pi$
$194$	0	0
$195$	−8.19007	−0.586503
$196$	0	0
$197$	−26.2114	−1.86748	−0.933741	−	0.357950i	$-0.883476\pi$
$197$	−26.2114	−1.86748	−0.933741	+	0.357950i	$0.883476\pi$
$198$	0	0
$199$	−11.1550	−0.790759	−0.395379	−	0.918518i	$-0.629387\pi$
$199$	−11.1550	−0.790759	−0.395379	+	0.918518i	$0.629387\pi$
$200$	0	0
$201$	30.1872	2.12924
$202$	0	0
$203$	5.34027	0.374813
$204$	0	0
$205$	−26.9708	−1.88373
$206$	0	0
$207$	48.7318	3.38709
$208$	0	0
$209$	8.24091	0.570036
$210$	0	0
$211$	−4.03241	−0.277603	−0.138801	−	0.990320i	$-0.544325\pi$
$211$	−4.03241	−0.277603	−0.138801	+	0.990320i	$0.544325\pi$
$212$	0	0
$213$	−2.73450	−0.187365
$214$	0	0
$215$	−19.2867	−1.31534
$216$	0	0
$217$	−0.963653	−0.0654170
$218$	0	0
$219$	−18.8249	−1.27207
$220$	0	0
$221$	−4.45177	−0.299458
$222$	0	0
$223$	−15.9906	−1.07081	−0.535405	−	0.844595i	$-0.679841\pi$
$223$	−15.9906	−1.07081	−0.535405	+	0.844595i	$0.679841\pi$
$224$	0	0
$225$	12.8580	0.857199
$226$	0	0
$227$	−25.1259	−1.66766	−0.833831	−	0.552020i	$-0.813857\pi$
$227$	−25.1259	−1.66766	−0.833831	+	0.552020i	$0.813857\pi$
$228$	0	0
$229$	11.0423	0.729695	0.364848	−	0.931067i	$-0.381121\pi$
$229$	11.0423	0.729695	0.364848	+	0.931067i	$0.381121\pi$
$230$	0	0
$231$	3.11300	0.204821
$232$	0	0
$233$	5.37367	0.352041	0.176020	−	0.984387i	$-0.443678\pi$
$233$	5.37367	0.352041	0.176020	+	0.984387i	$0.443678\pi$
$234$	0	0
$235$	5.10603	0.333081
$236$	0	0
$237$	−18.8610	−1.22515
$238$	0	0
$239$	9.15656	0.592288	0.296144	−	0.955143i	$-0.404299\pi$
$239$	9.15656	0.592288	0.296144	+	0.955143i	$0.404299\pi$
$240$	0	0
$241$	−16.8626	−1.08621	−0.543107	−	0.839663i	$-0.682752\pi$
$241$	−16.8626	−1.08621	−0.543107	+	0.839663i	$0.682752\pi$
$242$	0	0
$243$	14.3881	0.922996
$244$	0	0
$245$	−2.63092	−0.168083
$246$	0	0
$247$	−8.24091	−0.524357
$248$	0	0
$249$	−33.8256	−2.14361
$250$	0	0
$251$	11.7347	0.740688	0.370344	−	0.928895i	$-0.379240\pi$
$251$	11.7347	0.740688	0.370344	+	0.928895i	$0.379240\pi$
$252$	0	0
$253$	−7.28341	−0.457904
$254$	0	0
$255$	36.4603	2.28323
$256$	0	0
$257$	−11.4223	−0.712505	−0.356253	−	0.934390i	$-0.615946\pi$
$257$	−11.4223	−0.712505	−0.356253	+	0.934390i	$0.615946\pi$
$258$	0	0
$259$	−9.18755	−0.570887
$260$	0	0
$261$	−35.7306	−2.21167
$262$	0	0
$263$	−4.19057	−0.258402	−0.129201	−	0.991618i	$-0.541241\pi$
$263$	−4.19057	−0.258402	−0.129201	+	0.991618i	$0.541241\pi$
$264$	0	0
$265$	−38.1681	−2.34465
$266$	0	0
$267$	15.6046	0.954987
$268$	0	0
$269$	−10.2633	−0.625764	−0.312882	−	0.949792i	$-0.601294\pi$
$269$	−10.2633	−0.625764	−0.312882	+	0.949792i	$0.601294\pi$
$270$	0	0
$271$	11.1525	0.677467	0.338734	−	0.940882i	$-0.390001\pi$
$271$	11.1525	0.677467	0.338734	+	0.940882i	$0.390001\pi$
$272$	0	0
$273$	−3.11300	−0.188408
$274$	0	0
$275$	−1.92175	−0.115886
$276$	0	0
$277$	−10.8809	−0.653771	−0.326886	−	0.945064i	$-0.605999\pi$
$277$	−10.8809	−0.653771	−0.326886	+	0.945064i	$0.605999\pi$
$278$	0	0
$279$	6.44760	0.386008
$280$	0	0
$281$	−0.675140	−0.0402755	−0.0201377	−	0.999797i	$-0.506410\pi$
$281$	−0.675140	−0.0402755	−0.0201377	+	0.999797i	$0.506410\pi$
$282$	0	0
$283$	−3.91819	−0.232912	−0.116456	−	0.993196i	$-0.537153\pi$
$283$	−3.91819	−0.232912	−0.116456	+	0.993196i	$0.537153\pi$
$284$	0	0
$285$	67.4936	3.99798
$286$	0	0
$287$	−10.2515	−0.605126
$288$	0	0
$289$	2.81824	0.165779
$290$	0	0
$291$	−11.5581	−0.677548
$292$	0	0
$293$	17.8251	1.04136	0.520678	−	0.853753i	$-0.325679\pi$
$293$	17.8251	1.04136	0.520678	+	0.853753i	$0.325679\pi$
$294$	0	0
$295$	−8.40093	−0.489121
$296$	0	0
$297$	−11.4894	−0.666685
$298$	0	0
$299$	7.28341	0.421211
$300$	0	0
$301$	−7.33078	−0.422539
$302$	0	0
$303$	38.5441	2.21430
$304$	0	0
$305$	−36.1540	−2.07017
$306$	0	0
$307$	12.9279	0.737837	0.368918	−	0.929462i	$-0.379728\pi$
$307$	12.9279	0.737837	0.368918	+	0.929462i	$0.379728\pi$
$308$	0	0
$309$	13.0484	0.742296
$310$	0	0
$311$	13.0398	0.739419	0.369709	−	0.929147i	$-0.379457\pi$
$311$	13.0398	0.739419	0.369709	+	0.929147i	$0.379457\pi$
$312$	0	0
$313$	−15.4636	−0.874055	−0.437027	−	0.899448i	$-0.643969\pi$
$313$	−15.4636	−0.874055	−0.437027	+	0.899448i	$0.643969\pi$
$314$	0	0
$315$	17.6029	0.991813
$316$	0	0
$317$	−15.9771	−0.897362	−0.448681	−	0.893692i	$-0.648106\pi$
$317$	−15.9771	−0.897362	−0.448681	+	0.893692i	$0.648106\pi$
$318$	0	0
$319$	5.34027	0.298998
$320$	0	0
$321$	43.0579	2.40326
$322$	0	0
$323$	36.6866	2.04130
$324$	0	0
$325$	1.92175	0.106599
$326$	0	0
$327$	−28.1844	−1.55860
$328$	0	0
$329$	1.94078	0.106998
$330$	0	0
$331$	−17.9258	−0.985292	−0.492646	−	0.870230i	$-0.663970\pi$
$331$	−17.9258	−0.985292	−0.492646	+	0.870230i	$0.663970\pi$
$332$	0	0
$333$	61.4720	3.36864
$334$	0	0
$335$	−25.5124	−1.39389
$336$	0	0
$337$	−5.07110	−0.276241	−0.138120	−	0.990415i	$-0.544106\pi$
$337$	−5.07110	−0.276241	−0.138120	+	0.990415i	$0.544106\pi$
$338$	0	0
$339$	15.4699	0.840208
$340$	0	0
$341$	−0.963653	−0.0521847
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−59.6516	−3.21154
$346$	0	0
$347$	−26.8000	−1.43870	−0.719350	−	0.694648i	$-0.755560\pi$
$347$	−26.8000	−1.43870	−0.719350	+	0.694648i	$0.755560\pi$
$348$	0	0
$349$	24.7834	1.32662	0.663311	−	0.748343i	$-0.269150\pi$
$349$	24.7834	1.32662	0.663311	+	0.748343i	$0.269150\pi$
$350$	0	0
$351$	11.4894	0.613261
$352$	0	0
$353$	−20.9086	−1.11285	−0.556426	−	0.830897i	$-0.687828\pi$
$353$	−20.9086	−1.11285	−0.556426	+	0.830897i	$0.687828\pi$
$354$	0	0
$355$	2.31103	0.122657
$356$	0	0
$357$	13.8584	0.733463
$358$	0	0
$359$	36.0759	1.90401	0.952005	−	0.306081i	$-0.0990179\pi$
$359$	36.0759	1.90401	0.952005	+	0.306081i	$0.0990179\pi$
$360$	0	0
$361$	48.9127	2.57435
$362$	0	0
$363$	3.11300	0.163390
$364$	0	0
$365$	15.9096	0.832748
$366$	0	0
$367$	−2.81555	−0.146970	−0.0734851	−	0.997296i	$-0.523412\pi$
$367$	−2.81555	−0.146970	−0.0734851	+	0.997296i	$0.523412\pi$
$368$	0	0
$369$	68.5905	3.57068
$370$	0	0
$371$	−14.5075	−0.753193
$372$	0	0
$373$	28.0199	1.45082	0.725408	−	0.688319i	$-0.241651\pi$
$373$	28.0199	1.45082	0.725408	+	0.688319i	$0.241651\pi$
$374$	0	0
$375$	25.2111	1.30190
$376$	0	0
$377$	−5.34027	−0.275038
$378$	0	0
$379$	27.3676	1.40578	0.702889	−	0.711300i	$-0.251893\pi$
$379$	27.3676	1.40578	0.702889	+	0.711300i	$0.251893\pi$
$380$	0	0
$381$	42.1964	2.16179
$382$	0	0
$383$	22.8273	1.16642	0.583211	−	0.812321i	$-0.301796\pi$
$383$	22.8273	1.16642	0.583211	+	0.812321i	$0.301796\pi$
$384$	0	0
$385$	−2.63092	−0.134084
$386$	0	0
$387$	49.0487	2.49329
$388$	0	0
$389$	−22.6458	−1.14819	−0.574093	−	0.818790i	$-0.694645\pi$
$389$	−22.6458	−1.14819	−0.574093	+	0.818790i	$0.694645\pi$
$390$	0	0
$391$	−32.4241	−1.63976
$392$	0	0
$393$	−20.1581	−1.01684
$394$	0	0
$395$	15.9402	0.802036
$396$	0	0
$397$	−19.7344	−0.990443	−0.495221	−	0.868767i	$-0.664913\pi$
$397$	−19.7344	−0.990443	−0.495221	+	0.868767i	$0.664913\pi$
$398$	0	0
$399$	25.6540	1.28431
$400$	0	0
$401$	7.29937	0.364513	0.182256	−	0.983251i	$-0.441660\pi$
$401$	7.29937	0.364513	0.182256	+	0.983251i	$0.441660\pi$
$402$	0	0
$403$	0.963653	0.0480030
$404$	0	0
$405$	−41.2905	−2.05174
$406$	0	0
$407$	−9.18755	−0.455410
$408$	0	0
$409$	14.6665	0.725210	0.362605	−	0.931943i	$-0.381887\pi$
$409$	14.6665	0.725210	0.362605	+	0.931943i	$0.381887\pi$
$410$	0	0
$411$	−19.8874	−0.980975
$412$	0	0
$413$	−3.19315	−0.157125
$414$	0	0
$415$	28.5874	1.40330
$416$	0	0
$417$	−50.6025	−2.47801
$418$	0	0
$419$	1.09311	0.0534017	0.0267008	−	0.999643i	$-0.491500\pi$
$419$	1.09311	0.0534017	0.0267008	+	0.999643i	$0.491500\pi$
$420$	0	0
$421$	−11.7196	−0.571176	−0.285588	−	0.958352i	$-0.592189\pi$
$421$	−11.7196	−0.571176	−0.285588	+	0.958352i	$0.592189\pi$
$422$	0	0
$423$	−12.9853	−0.631368
$424$	0	0
$425$	−8.55516	−0.414986
$426$	0	0
$427$	−13.7420	−0.665020
$428$	0	0
$429$	−3.11300	−0.150297
$430$	0	0
$431$	19.4208	0.935466	0.467733	−	0.883870i	$-0.345071\pi$
$431$	19.4208	0.935466	0.467733	+	0.883870i	$0.345071\pi$
$432$	0	0
$433$	21.7194	1.04377	0.521884	−	0.853016i	$-0.325229\pi$
$433$	21.7194	1.04377	0.521884	+	0.853016i	$0.325229\pi$
$434$	0	0
$435$	43.7371	2.09704
$436$	0	0
$437$	−60.0220	−2.87124
$438$	0	0
$439$	−13.6276	−0.650408	−0.325204	−	0.945644i	$-0.605433\pi$
$439$	−13.6276	−0.650408	−0.325204	+	0.945644i	$0.605433\pi$
$440$	0	0
$441$	6.69079	0.318609
$442$	0	0
$443$	−10.7077	−0.508737	−0.254369	−	0.967107i	$-0.581868\pi$
$443$	−10.7077	−0.508737	−0.254369	+	0.967107i	$0.581868\pi$
$444$	0	0
$445$	−13.1881	−0.625175
$446$	0	0
$447$	51.3620	2.42934
$448$	0	0
$449$	7.91499	0.373531	0.186766	−	0.982404i	$-0.440199\pi$
$449$	7.91499	0.373531	0.186766	+	0.982404i	$0.440199\pi$
$450$	0	0
$451$	−10.2515	−0.482724
$452$	0	0
$453$	15.7614	0.740536
$454$	0	0
$455$	2.63092	0.123339
$456$	0	0
$457$	27.1293	1.26906	0.634529	−	0.772899i	$-0.281194\pi$
$457$	27.1293	1.26906	0.634529	+	0.772899i	$0.281194\pi$
$458$	0	0
$459$	−51.1483	−2.38740
$460$	0	0
$461$	−17.2096	−0.801530	−0.400765	−	0.916181i	$-0.631256\pi$
$461$	−17.2096	−0.801530	−0.400765	+	0.916181i	$0.631256\pi$
$462$	0	0
$463$	−33.0526	−1.53608	−0.768042	−	0.640399i	$-0.778769\pi$
$463$	−33.0526	−1.53608	−0.768042	+	0.640399i	$0.778769\pi$
$464$	0	0
$465$	−7.89238	−0.366000
$466$	0	0
$467$	−29.5299	−1.36648	−0.683240	−	0.730194i	$-0.739430\pi$
$467$	−29.5299	−1.36648	−0.683240	+	0.730194i	$0.739430\pi$
$468$	0	0
$469$	−9.69712	−0.447772
$470$	0	0
$471$	73.9840	3.40900
$472$	0	0
$473$	−7.33078	−0.337070
$474$	0	0
$475$	−15.8369	−0.726648
$476$	0	0
$477$	97.0668	4.44438
$478$	0	0
$479$	33.9276	1.55019	0.775096	−	0.631843i	$-0.217701\pi$
$479$	33.9276	1.55019	0.775096	+	0.631843i	$0.217701\pi$
$480$	0	0
$481$	9.18755	0.418916
$482$	0	0
$483$	−22.6733	−1.03167
$484$	0	0
$485$	9.76821	0.443552
$486$	0	0
$487$	−31.0504	−1.40703	−0.703514	−	0.710682i	$-0.748387\pi$
$487$	−31.0504	−1.40703	−0.703514	+	0.710682i	$0.748387\pi$
$488$	0	0
$489$	6.13497	0.277433
$490$	0	0
$491$	42.0257	1.89659	0.948297	−	0.317384i	$-0.102804\pi$
$491$	42.0257	1.89659	0.948297	+	0.317384i	$0.102804\pi$
$492$	0	0
$493$	23.7736	1.07071
$494$	0	0
$495$	17.6029	0.791193
$496$	0	0
$497$	0.878412	0.0394022
$498$	0	0
$499$	−17.5198	−0.784296	−0.392148	−	0.919902i	$-0.628268\pi$
$499$	−17.5198	−0.784296	−0.392148	+	0.919902i	$0.628268\pi$
$500$	0	0
$501$	71.0808	3.17565
$502$	0	0
$503$	12.2557	0.546456	0.273228	−	0.961949i	$-0.411909\pi$
$503$	12.2557	0.546456	0.273228	+	0.961949i	$0.411909\pi$
$504$	0	0
$505$	−32.5751	−1.44957
$506$	0	0
$507$	3.11300	0.138253
$508$	0	0
$509$	−4.09600	−0.181552	−0.0907761	−	0.995871i	$-0.528935\pi$
$509$	−4.09600	−0.181552	−0.0907761	+	0.995871i	$0.528935\pi$
$510$	0	0
$511$	6.04717	0.267511
$512$	0	0
$513$	−94.6835	−4.18038
$514$	0	0
$515$	−11.0277	−0.485938
$516$	0	0
$517$	1.94078	0.0853552
$518$	0	0
$519$	−65.8785	−2.89175
$520$	0	0
$521$	19.2708	0.844270	0.422135	−	0.906533i	$-0.361281\pi$
$521$	19.2708	0.844270	0.422135	+	0.906533i	$0.361281\pi$
$522$	0	0
$523$	−14.8939	−0.651264	−0.325632	−	0.945497i	$-0.605577\pi$
$523$	−14.8939	−0.651264	−0.325632	+	0.945497i	$0.605577\pi$
$524$	0	0
$525$	−5.98240	−0.261093
$526$	0	0
$527$	−4.28996	−0.186873
$528$	0	0
$529$	30.0481	1.30644
$530$	0	0
$531$	21.3647	0.927149
$532$	0	0
$533$	10.2515	0.444041
$534$	0	0
$535$	−36.3899	−1.57327
$536$	0	0
$537$	44.6591	1.92718
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	24.5032	1.05348	0.526738	−	0.850028i	$-0.323415\pi$
$541$	24.5032	1.05348	0.526738	+	0.850028i	$0.323415\pi$
$542$	0	0
$543$	−63.3062	−2.71673
$544$	0	0
$545$	23.8198	1.02033
$546$	0	0
$547$	12.8818	0.550788	0.275394	−	0.961331i	$-0.411192\pi$
$547$	12.8818	0.550788	0.275394	+	0.961331i	$0.411192\pi$
$548$	0	0
$549$	91.9446	3.92410
$550$	0	0
$551$	44.0087	1.87483
$552$	0	0
$553$	6.05877	0.257645
$554$	0	0
$555$	−75.2466	−3.19404
$556$	0	0
$557$	1.08019	0.0457690	0.0228845	−	0.999738i	$-0.492715\pi$
$557$	1.08019	0.0457690	0.0228845	+	0.999738i	$0.492715\pi$
$558$	0	0
$559$	7.33078	0.310059
$560$	0	0
$561$	13.8584	0.585101
$562$	0	0
$563$	32.6589	1.37641	0.688205	−	0.725516i	$-0.258399\pi$
$563$	32.6589	1.37641	0.688205	+	0.725516i	$0.258399\pi$
$564$	0	0
$565$	−13.0742	−0.550035
$566$	0	0
$567$	−15.6943	−0.659099
$568$	0	0
$569$	36.5568	1.53254	0.766271	−	0.642518i	$-0.222110\pi$
$569$	36.5568	1.53254	0.766271	+	0.642518i	$0.222110\pi$
$570$	0	0
$571$	−39.4148	−1.64946	−0.824729	−	0.565529i	$-0.808672\pi$
$571$	−39.4148	−1.64946	−0.824729	+	0.565529i	$0.808672\pi$
$572$	0	0
$573$	22.3710	0.934562
$574$	0	0
$575$	13.9969	0.583710
$576$	0	0
$577$	24.6880	1.02778	0.513888	−	0.857858i	$-0.328205\pi$
$577$	24.6880	1.02778	0.513888	+	0.857858i	$0.328205\pi$
$578$	0	0
$579$	7.28557	0.302778
$580$	0	0
$581$	10.8659	0.450794
$582$	0	0
$583$	−14.5075	−0.600840
$584$	0	0
$585$	−17.6029	−0.727792
$586$	0	0
$587$	0.720181	0.0297251	0.0148625	−	0.999890i	$-0.495269\pi$
$587$	0.720181	0.0297251	0.0148625	+	0.999890i	$0.495269\pi$
$588$	0	0
$589$	−7.94138	−0.327219
$590$	0	0
$591$	−81.5960	−3.35641
$592$	0	0
$593$	−3.68650	−0.151386	−0.0756932	−	0.997131i	$-0.524117\pi$
$593$	−3.68650	−0.151386	−0.0756932	+	0.997131i	$0.524117\pi$
$594$	0	0
$595$	−11.7122	−0.480155
$596$	0	0
$597$	−34.7256	−1.42123
$598$	0	0
$599$	−12.7406	−0.520565	−0.260283	−	0.965532i	$-0.583816\pi$
$599$	−12.7406	−0.520565	−0.260283	+	0.965532i	$0.583816\pi$
$600$	0	0
$601$	−2.65538	−0.108315	−0.0541576	−	0.998532i	$-0.517247\pi$
$601$	−2.65538	−0.108315	−0.0541576	+	0.998532i	$0.517247\pi$
$602$	0	0
$603$	64.8814	2.64218
$604$	0	0
$605$	−2.63092	−0.106962
$606$	0	0
$607$	−33.4725	−1.35861	−0.679304	−	0.733857i	$-0.737718\pi$
$607$	−33.4725	−1.35861	−0.679304	+	0.733857i	$0.737718\pi$
$608$	0	0
$609$	16.6243	0.673649
$610$	0	0
$611$	−1.94078	−0.0785154
$612$	0	0
$613$	−8.98307	−0.362823	−0.181411	−	0.983407i	$-0.558067\pi$
$613$	−8.98307	−0.362823	−0.181411	+	0.983407i	$0.558067\pi$
$614$	0	0
$615$	−83.9603	−3.38561
$616$	0	0
$617$	20.1741	0.812177	0.406089	−	0.913834i	$-0.366892\pi$
$617$	20.1741	0.812177	0.406089	+	0.913834i	$0.366892\pi$
$618$	0	0
$619$	−4.97047	−0.199780	−0.0998900	−	0.994998i	$-0.531849\pi$
$619$	−4.97047	−0.199780	−0.0998900	+	0.994998i	$0.531849\pi$
$620$	0	0
$621$	83.6823	3.35806
$622$	0	0
$623$	−5.01272	−0.200830
$624$	0	0
$625$	−30.9156	−1.23662
$626$	0	0
$627$	25.6540	1.02452
$628$	0	0
$629$	−40.9008	−1.63082
$630$	0	0
$631$	−15.5460	−0.618878	−0.309439	−	0.950919i	$-0.600141\pi$
$631$	−15.5460	−0.618878	−0.309439	+	0.950919i	$0.600141\pi$
$632$	0	0
$633$	−12.5529	−0.498933
$634$	0	0
$635$	−35.6618	−1.41520
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−5.87727	−0.232501
$640$	0	0
$641$	40.5382	1.60116	0.800582	−	0.599223i	$-0.204524\pi$
$641$	40.5382	1.60116	0.800582	+	0.599223i	$0.204524\pi$
$642$	0	0
$643$	−6.81122	−0.268608	−0.134304	−	0.990940i	$-0.542880\pi$
$643$	−6.81122	−0.268608	−0.134304	+	0.990940i	$0.542880\pi$
$644$	0	0
$645$	−60.0396	−2.36406
$646$	0	0
$647$	−31.3983	−1.23439	−0.617197	−	0.786809i	$-0.711732\pi$
$647$	−31.3983	−1.23439	−0.617197	+	0.786809i	$0.711732\pi$
$648$	0	0
$649$	−3.19315	−0.125342
$650$	0	0
$651$	−2.99985	−0.117574
$652$	0	0
$653$	−4.40062	−0.172209	−0.0861047	−	0.996286i	$-0.527442\pi$
$653$	−4.40062	−0.172209	−0.0861047	+	0.996286i	$0.527442\pi$
$654$	0	0
$655$	17.0364	0.665667
$656$	0	0
$657$	−40.4604	−1.57851
$658$	0	0
$659$	7.48535	0.291588	0.145794	−	0.989315i	$-0.453426\pi$
$659$	7.48535	0.291588	0.145794	+	0.989315i	$0.453426\pi$
$660$	0	0
$661$	20.4273	0.794528	0.397264	−	0.917704i	$-0.369960\pi$
$661$	20.4273	0.794528	0.397264	+	0.917704i	$0.369960\pi$
$662$	0	0
$663$	−13.8584	−0.538214
$664$	0	0
$665$	−21.6812	−0.840761
$666$	0	0
$667$	−38.8954	−1.50603
$668$	0	0
$669$	−49.7788	−1.92456
$670$	0	0
$671$	−13.7420	−0.530503
$672$	0	0
$673$	18.8861	0.728005	0.364002	−	0.931398i	$-0.381410\pi$
$673$	18.8861	0.728005	0.364002	+	0.931398i	$0.381410\pi$
$674$	0	0
$675$	22.0798	0.849851
$676$	0	0
$677$	12.1053	0.465246	0.232623	−	0.972567i	$-0.425269\pi$
$677$	12.1053	0.465246	0.232623	+	0.972567i	$0.425269\pi$
$678$	0	0
$679$	3.71285	0.142486
$680$	0	0
$681$	−78.2169	−2.99728
$682$	0	0
$683$	4.38385	0.167743	0.0838716	−	0.996477i	$-0.473271\pi$
$683$	4.38385	0.167743	0.0838716	+	0.996477i	$0.473271\pi$
$684$	0	0
$685$	16.8076	0.642187
$686$	0	0
$687$	34.3747	1.31148
$688$	0	0
$689$	14.5075	0.552692
$690$	0	0
$691$	−13.0522	−0.496530	−0.248265	−	0.968692i	$-0.579860\pi$
$691$	−13.0522	−0.496530	−0.248265	+	0.968692i	$0.579860\pi$
$692$	0	0
$693$	6.69079	0.254162
$694$	0	0
$695$	42.7661	1.62221
$696$	0	0
$697$	−45.6372	−1.72863
$698$	0	0
$699$	16.7282	0.632720
$700$	0	0
$701$	34.9604	1.32044	0.660218	−	0.751074i	$-0.270464\pi$
$701$	34.9604	1.32044	0.660218	+	0.751074i	$0.270464\pi$
$702$	0	0
$703$	−75.7138	−2.85560
$704$	0	0
$705$	15.8951	0.598644
$706$	0	0
$707$	−12.3816	−0.465659
$708$	0	0
$709$	−45.2039	−1.69767	−0.848834	−	0.528659i	$-0.822695\pi$
$709$	−45.2039	−1.69767	−0.848834	+	0.528659i	$0.822695\pi$
$710$	0	0
$711$	−40.5380	−1.52029
$712$	0	0
$713$	7.01868	0.262852
$714$	0	0
$715$	2.63092	0.0983909
$716$	0	0
$717$	28.5044	1.06452
$718$	0	0
$719$	21.1747	0.789682	0.394841	−	0.918750i	$-0.370800\pi$
$719$	21.1747	0.789682	0.394841	+	0.918750i	$0.370800\pi$
$720$	0	0
$721$	−4.19157	−0.156102
$722$	0	0
$723$	−52.4933	−1.95225
$724$	0	0
$725$	−10.2626	−0.381145
$726$	0	0
$727$	46.4242	1.72178	0.860889	−	0.508793i	$-0.169908\pi$
$727$	46.4242	1.72178	0.860889	+	0.508793i	$0.169908\pi$
$728$	0	0
$729$	−2.29275	−0.0849168
$730$	0	0
$731$	−32.6349	−1.20705
$732$	0	0
$733$	−49.4921	−1.82803	−0.914016	−	0.405678i	$-0.867035\pi$
$733$	−49.4921	−1.82803	−0.914016	+	0.405678i	$0.867035\pi$
$734$	0	0
$735$	−8.19007	−0.302095
$736$	0	0
$737$	−9.69712	−0.357198
$738$	0	0
$739$	−47.4181	−1.74430	−0.872151	−	0.489236i	$-0.837276\pi$
$739$	−47.4181	−1.74430	−0.872151	+	0.489236i	$0.837276\pi$
$740$	0	0
$741$	−25.6540	−0.942423
$742$	0	0
$743$	39.1541	1.43643	0.718213	−	0.695823i	$-0.244960\pi$
$743$	39.1541	1.43643	0.718213	+	0.695823i	$0.244960\pi$
$744$	0	0
$745$	−43.4081	−1.59035
$746$	0	0
$747$	−72.7015	−2.66001
$748$	0	0
$749$	−13.8316	−0.505396
$750$	0	0
$751$	19.6697	0.717757	0.358879	−	0.933384i	$-0.383159\pi$
$751$	19.6697	0.717757	0.358879	+	0.933384i	$0.383159\pi$
$752$	0	0
$753$	36.5302	1.33123
$754$	0	0
$755$	−13.3206	−0.484786
$756$	0	0
$757$	9.90605	0.360042	0.180021	−	0.983663i	$-0.442383\pi$
$757$	9.90605	0.360042	0.180021	+	0.983663i	$0.442383\pi$
$758$	0	0
$759$	−22.6733	−0.822988
$760$	0	0
$761$	4.37919	0.158746	0.0793728	−	0.996845i	$-0.474708\pi$
$761$	4.37919	0.158746	0.0793728	+	0.996845i	$0.474708\pi$
$762$	0	0
$763$	9.05377	0.327769
$764$	0	0
$765$	78.3642	2.83326
$766$	0	0
$767$	3.19315	0.115298
$768$	0	0
$769$	−26.2253	−0.945709	−0.472855	−	0.881140i	$-0.656776\pi$
$769$	−26.2253	−0.945709	−0.472855	+	0.881140i	$0.656776\pi$
$770$	0	0
$771$	−35.5577	−1.28058
$772$	0	0
$773$	−19.7263	−0.709505	−0.354753	−	0.934960i	$-0.615435\pi$
$773$	−19.7263	−0.709505	−0.354753	+	0.934960i	$0.615435\pi$
$774$	0	0
$775$	1.85189	0.0665220
$776$	0	0
$777$	−28.6009	−1.02605
$778$	0	0
$779$	−84.4816	−3.02687
$780$	0	0
$781$	0.878412	0.0314320
$782$	0	0
$783$	−61.3567	−2.19271
$784$	0	0
$785$	−62.5268	−2.23168
$786$	0	0
$787$	−27.1160	−0.966582	−0.483291	−	0.875460i	$-0.660559\pi$
$787$	−27.1160	−0.966582	−0.483291	+	0.875460i	$0.660559\pi$
$788$	0	0
$789$	−13.0453	−0.464424
$790$	0	0
$791$	−4.96943	−0.176693
$792$	0	0
$793$	13.7420	0.487992
$794$	0	0
$795$	−118.818	−4.21402
$796$	0	0
$797$	20.9533	0.742202	0.371101	−	0.928592i	$-0.378980\pi$
$797$	20.9533	0.742202	0.371101	+	0.928592i	$0.378980\pi$
$798$	0	0
$799$	8.63989	0.305657
$800$	0	0
$801$	33.5391	1.18504
$802$	0	0
$803$	6.04717	0.213400
$804$	0	0
$805$	19.1621	0.675375
$806$	0	0
$807$	−31.9497	−1.12468
$808$	0	0
$809$	−4.07136	−0.143142	−0.0715708	−	0.997436i	$-0.522801\pi$
$809$	−4.07136	−0.143142	−0.0715708	+	0.997436i	$0.522801\pi$
$810$	0	0
$811$	41.6406	1.46220	0.731100	−	0.682270i	$-0.239007\pi$
$811$	41.6406	1.46220	0.731100	+	0.682270i	$0.239007\pi$
$812$	0	0
$813$	34.7178	1.21761
$814$	0	0
$815$	−5.18491	−0.181619
$816$	0	0
$817$	−60.4123	−2.11356
$818$	0	0
$819$	−6.69079	−0.233795
$820$	0	0
$821$	−12.0936	−0.422068	−0.211034	−	0.977479i	$-0.567683\pi$
$821$	−12.0936	−0.422068	−0.211034	+	0.977479i	$0.567683\pi$
$822$	0	0
$823$	34.9317	1.21764	0.608821	−	0.793307i	$-0.291643\pi$
$823$	34.9317	1.21764	0.608821	+	0.793307i	$0.291643\pi$
$824$	0	0
$825$	−5.98240	−0.208280
$826$	0	0
$827$	−18.4416	−0.641278	−0.320639	−	0.947202i	$-0.603898\pi$
$827$	−18.4416	−0.641278	−0.320639	+	0.947202i	$0.603898\pi$
$828$	0	0
$829$	20.3896	0.708161	0.354081	−	0.935215i	$-0.384794\pi$
$829$	20.3896	0.708161	0.354081	+	0.935215i	$0.384794\pi$
$830$	0	0
$831$	−33.8724	−1.17502
$832$	0	0
$833$	−4.45177	−0.154245
$834$	0	0
$835$	−60.0731	−2.07892
$836$	0	0
$837$	11.0718	0.382698
$838$	0	0
$839$	−17.3301	−0.598302	−0.299151	−	0.954206i	$-0.596703\pi$
$839$	−17.3301	−0.598302	−0.299151	+	0.954206i	$0.596703\pi$
$840$	0	0
$841$	−0.481542	−0.0166049
$842$	0	0
$843$	−2.10171	−0.0723868
$844$	0	0
$845$	−2.63092	−0.0905064
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−12.1973	−0.418612
$850$	0	0
$851$	66.9167	2.29388
$852$	0	0
$853$	20.6053	0.705513	0.352757	−	0.935715i	$-0.385244\pi$
$853$	20.6053	0.705513	0.352757	+	0.935715i	$0.385244\pi$
$854$	0	0
$855$	145.064	4.96109
$856$	0	0
$857$	−48.2301	−1.64751	−0.823755	−	0.566946i	$-0.808125\pi$
$857$	−48.2301	−1.64751	−0.823755	+	0.566946i	$0.808125\pi$
$858$	0	0
$859$	17.0481	0.581675	0.290838	−	0.956772i	$-0.406066\pi$
$859$	17.0481	0.581675	0.290838	+	0.956772i	$0.406066\pi$
$860$	0	0
$861$	−31.9129	−1.08759
$862$	0	0
$863$	12.5116	0.425901	0.212950	−	0.977063i	$-0.431693\pi$
$863$	12.5116	0.425901	0.212950	+	0.977063i	$0.431693\pi$
$864$	0	0
$865$	55.6765	1.89306
$866$	0	0
$867$	8.77319	0.297953
$868$	0	0
$869$	6.05877	0.205530
$870$	0	0
$871$	9.69712	0.328574
$872$	0	0
$873$	−24.8419	−0.840770
$874$	0	0
$875$	−8.09865	−0.273784
$876$	0	0
$877$	−4.14509	−0.139970	−0.0699849	−	0.997548i	$-0.522295\pi$
$877$	−4.14509	−0.139970	−0.0699849	+	0.997548i	$0.522295\pi$
$878$	0	0
$879$	55.4897	1.87162
$880$	0	0
$881$	21.7830	0.733888	0.366944	−	0.930243i	$-0.380404\pi$
$881$	21.7830	0.733888	0.366944	+	0.930243i	$0.380404\pi$
$882$	0	0
$883$	4.91508	0.165406	0.0827028	−	0.996574i	$-0.473645\pi$
$883$	4.91508	0.165406	0.0827028	+	0.996574i	$0.473645\pi$
$884$	0	0
$885$	−26.1521	−0.879094
$886$	0	0
$887$	35.8083	1.20233	0.601163	−	0.799126i	$-0.294704\pi$
$887$	35.8083	1.20233	0.601163	+	0.799126i	$0.294704\pi$
$888$	0	0
$889$	−13.5549	−0.454616
$890$	0	0
$891$	−15.6943	−0.525779
$892$	0	0
$893$	15.9938	0.535211
$894$	0	0
$895$	−37.7431	−1.26161
$896$	0	0
$897$	22.6733	0.757039
$898$	0	0
$899$	−5.14616	−0.171634
$900$	0	0
$901$	−64.5841	−2.15161
$902$	0	0
$903$	−22.8207	−0.759427
$904$	0	0
$905$	53.5025	1.77848
$906$	0	0
$907$	−3.94616	−0.131030	−0.0655150	−	0.997852i	$-0.520869\pi$
$907$	−3.94616	−0.131030	−0.0655150	+	0.997852i	$0.520869\pi$
$908$	0	0
$909$	82.8429	2.74773
$910$	0	0
$911$	44.1632	1.46319	0.731596	−	0.681739i	$-0.238776\pi$
$911$	44.1632	1.46319	0.731596	+	0.681739i	$0.238776\pi$
$912$	0	0
$913$	10.8659	0.359609
$914$	0	0
$915$	−112.548	−3.72071
$916$	0	0
$917$	6.47545	0.213838
$918$	0	0
$919$	−2.96808	−0.0979079	−0.0489539	−	0.998801i	$-0.515589\pi$
$919$	−2.96808	−0.0979079	−0.0489539	+	0.998801i	$0.515589\pi$
$920$	0	0
$921$	40.2447	1.32611
$922$	0	0
$923$	−0.878412	−0.0289133
$924$	0	0
$925$	17.6561	0.580530
$926$	0	0
$927$	28.0449	0.921116
$928$	0	0
$929$	−31.8112	−1.04369	−0.521847	−	0.853039i	$-0.674757\pi$
$929$	−31.8112	−1.04369	−0.521847	+	0.853039i	$0.674757\pi$
$930$	0	0
$931$	−8.24091	−0.270085
$932$	0	0
$933$	40.5929	1.32895
$934$	0	0
$935$	−11.7122	−0.383032
$936$	0	0
$937$	−20.8593	−0.681444	−0.340722	−	0.940164i	$-0.610671\pi$
$937$	−20.8593	−0.681444	−0.340722	+	0.940164i	$0.610671\pi$
$938$	0	0
$939$	−48.1383	−1.57093
$940$	0	0
$941$	32.5949	1.06256	0.531282	−	0.847195i	$-0.321710\pi$
$941$	32.5949	1.06256	0.531282	+	0.847195i	$0.321710\pi$
$942$	0	0
$943$	74.6658	2.43145
$944$	0	0
$945$	30.2278	0.983311
$946$	0	0
$947$	−38.5217	−1.25179	−0.625893	−	0.779909i	$-0.715265\pi$
$947$	−38.5217	−1.25179	−0.625893	+	0.779909i	$0.715265\pi$
$948$	0	0
$949$	−6.04717	−0.196300
$950$	0	0
$951$	−49.7367	−1.61282
$952$	0	0
$953$	51.2072	1.65876	0.829381	−	0.558683i	$-0.188693\pi$
$953$	51.2072	1.65876	0.829381	+	0.558683i	$0.188693\pi$
$954$	0	0
$955$	−18.9066	−0.611803
$956$	0	0
$957$	16.6243	0.537386
$958$	0	0
$959$	6.38850	0.206296
$960$	0	0
$961$	−30.0714	−0.970044
$962$	0	0
$963$	92.5444	2.98220
$964$	0	0
$965$	−6.15732	−0.198211
$966$	0	0
$967$	8.24991	0.265299	0.132650	−	0.991163i	$-0.457652\pi$
$967$	8.24991	0.265299	0.132650	+	0.991163i	$0.457652\pi$
$968$	0	0
$969$	114.206	3.66881
$970$	0	0
$971$	22.5968	0.725166	0.362583	−	0.931952i	$-0.381895\pi$
$971$	22.5968	0.725166	0.362583	+	0.931952i	$0.381895\pi$
$972$	0	0
$973$	16.2552	0.521118
$974$	0	0
$975$	5.98240	0.191590
$976$	0	0
$977$	1.90104	0.0608197	0.0304098	−	0.999538i	$-0.490319\pi$
$977$	1.90104	0.0608197	0.0304098	+	0.999538i	$0.490319\pi$
$978$	0	0
$979$	−5.01272	−0.160207
$980$	0	0
$981$	−60.5769	−1.93407
$982$	0	0
$983$	−13.6018	−0.433829	−0.216914	−	0.976191i	$-0.569599\pi$
$983$	−13.6018	−0.433829	−0.216914	+	0.976191i	$0.569599\pi$
$984$	0	0
$985$	68.9600	2.19725
$986$	0	0
$987$	6.04164	0.192308
$988$	0	0
$989$	53.3931	1.69780
$990$	0	0
$991$	−24.0609	−0.764319	−0.382160	−	0.924096i	$-0.624820\pi$
$991$	−24.0609	−0.764319	−0.382160	+	0.924096i	$0.624820\pi$
$992$	0	0
$993$	−55.8032	−1.77086
$994$	0	0
$995$	29.3480	0.930394
$996$	0	0
$997$	−8.18960	−0.259367	−0.129684	−	0.991555i	$-0.541396\pi$
$997$	−8.18960	−0.259367	−0.129684	+	0.991555i	$0.541396\pi$
$998$	0	0
$999$	105.560	3.33976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.14	✓	15	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.11300 q^{3} -2.63092 q^{5} -1.00000 q^{7} +6.69079 q^{9} +O(q^{10})\)	\(q+3.11300 q^{3} -2.63092 q^{5} -1.00000 q^{7} +6.69079 q^{9} -1.00000 q^{11} +1.00000 q^{13} -8.19007 q^{15} -4.45177 q^{17} -8.24091 q^{19} -3.11300 q^{21} +7.28341 q^{23} +1.92175 q^{25} +11.4894 q^{27} -5.34027 q^{29} +0.963653 q^{31} -3.11300 q^{33} +2.63092 q^{35} +9.18755 q^{37} +3.11300 q^{39} +10.2515 q^{41} +7.33078 q^{43} -17.6029 q^{45} -1.94078 q^{47} +1.00000 q^{49} -13.8584 q^{51} +14.5075 q^{53} +2.63092 q^{55} -25.6540 q^{57} +3.19315 q^{59} +13.7420 q^{61} -6.69079 q^{63} -2.63092 q^{65} +9.69712 q^{67} +22.6733 q^{69} -0.878412 q^{71} -6.04717 q^{73} +5.98240 q^{75} +1.00000 q^{77} -6.05877 q^{79} +15.6943 q^{81} -10.8659 q^{83} +11.7122 q^{85} -16.6243 q^{87} +5.01272 q^{89} -1.00000 q^{91} +2.99985 q^{93} +21.6812 q^{95} -3.71285 q^{97} -6.69079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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