LMFDB - Embedded newform 8008.2.a.l.1.2

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.15747$ 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)

$f(q)$	$=$	$q-2.59521 q^{3} +0.486965 q^{5} +1.00000 q^{7} +3.73512 q^{9} +O(q^{10})$	$q-2.59521 q^{3} +0.486965 q^{5} +1.00000 q^{7} +3.73512 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.26378 q^{15} +4.76972 q^{17} -5.64455 q^{19} -2.59521 q^{21} -5.89812 q^{23} -4.76286 q^{25} -1.90780 q^{27} +5.02403 q^{29} +7.10351 q^{31} -2.59521 q^{33} +0.486965 q^{35} +3.80930 q^{37} -2.59521 q^{39} -5.77763 q^{41} -1.96647 q^{43} +1.81887 q^{45} -3.82892 q^{47} +1.00000 q^{49} -12.3784 q^{51} +1.43886 q^{53} +0.486965 q^{55} +14.6488 q^{57} -5.08231 q^{59} -11.4424 q^{61} +3.73512 q^{63} +0.486965 q^{65} +7.40072 q^{67} +15.3069 q^{69} -16.5246 q^{71} -10.6963 q^{73} +12.3606 q^{75} +1.00000 q^{77} +4.42282 q^{79} -6.25423 q^{81} -10.8596 q^{83} +2.32269 q^{85} -13.0384 q^{87} +14.7624 q^{89} +1.00000 q^{91} -18.4351 q^{93} -2.74870 q^{95} +9.76630 q^{97} +3.73512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * 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
$3$	−2.59521	−1.49835	−0.749173	−	0.662374i	$-0.769549\pi$
$3$	−2.59521	−1.49835	−0.749173	+	0.662374i	$0.769549\pi$
$4$	0	0
$5$	0.486965	0.217777	0.108889	−	0.994054i	$-0.465271\pi$
$5$	0.486965	0.217777	0.108889	+	0.994054i	$0.465271\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.73512	1.24504
$10$	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
$15$	−1.26378	−0.326306
$16$	0	0
$17$	4.76972	1.15683	0.578414	−	0.815744i	$-0.303672\pi$
$17$	4.76972	1.15683	0.578414	+	0.815744i	$0.303672\pi$
$18$	0	0
$19$	−5.64455	−1.29495	−0.647474	−	0.762088i	$-0.724175\pi$
$19$	−5.64455	−1.29495	−0.647474	+	0.762088i	$0.724175\pi$
$20$	0	0
$21$	−2.59521	−0.566322
$22$	0	0
$23$	−5.89812	−1.22984	−0.614921	−	0.788589i	$-0.710812\pi$
$23$	−5.89812	−1.22984	−0.614921	+	0.788589i	$0.710812\pi$
$24$	0	0
$25$	−4.76286	−0.952573
$26$	0	0
$27$	−1.90780	−0.367156
$28$	0	0
$29$	5.02403	0.932940	0.466470	−	0.884537i	$-0.345526\pi$
$29$	5.02403	0.932940	0.466470	+	0.884537i	$0.345526\pi$
$30$	0	0
$31$	7.10351	1.27583	0.637914	−	0.770108i	$-0.279798\pi$
$31$	7.10351	1.27583	0.637914	+	0.770108i	$0.279798\pi$
$32$	0	0
$33$	−2.59521	−0.451768
$34$	0	0
$35$	0.486965	0.0823121
$36$	0	0
$37$	3.80930	0.626244	0.313122	−	0.949713i	$-0.398625\pi$
$37$	3.80930	0.626244	0.313122	+	0.949713i	$0.398625\pi$
$38$	0	0
$39$	−2.59521	−0.415566
$40$	0	0
$41$	−5.77763	−0.902314	−0.451157	−	0.892445i	$-0.648989\pi$
$41$	−5.77763	−0.902314	−0.451157	+	0.892445i	$0.648989\pi$
$42$	0	0
$43$	−1.96647	−0.299884	−0.149942	−	0.988695i	$-0.547909\pi$
$43$	−1.96647	−0.299884	−0.149942	+	0.988695i	$0.547909\pi$
$44$	0	0
$45$	1.81887	0.271142
$46$	0	0
$47$	−3.82892	−0.558505	−0.279252	−	0.960218i	$-0.590087\pi$
$47$	−3.82892	−0.558505	−0.279252	+	0.960218i	$0.590087\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.3784	−1.73333
$52$	0	0
$53$	1.43886	0.197643	0.0988216	−	0.995105i	$-0.468493\pi$
$53$	1.43886	0.197643	0.0988216	+	0.995105i	$0.468493\pi$
$54$	0	0
$55$	0.486965	0.0656624
$56$	0	0
$57$	14.6488	1.94028
$58$	0	0
$59$	−5.08231	−0.661660	−0.330830	−	0.943690i	$-0.607329\pi$
$59$	−5.08231	−0.661660	−0.330830	+	0.943690i	$0.607329\pi$
$60$	0	0
$61$	−11.4424	−1.46505	−0.732523	−	0.680742i	$-0.761657\pi$
$61$	−11.4424	−1.46505	−0.732523	+	0.680742i	$0.761657\pi$
$62$	0	0
$63$	3.73512	0.470581
$64$	0	0
$65$	0.486965	0.0604006
$66$	0	0
$67$	7.40072	0.904141	0.452071	−	0.891982i	$-0.350685\pi$
$67$	7.40072	0.904141	0.452071	+	0.891982i	$0.350685\pi$
$68$	0	0
$69$	15.3069	1.84273
$70$	0	0
$71$	−16.5246	−1.96111	−0.980556	−	0.196238i	$-0.937128\pi$
$71$	−16.5246	−1.96111	−0.980556	+	0.196238i	$0.937128\pi$
$72$	0	0
$73$	−10.6963	−1.25191	−0.625954	−	0.779860i	$-0.715290\pi$
$73$	−10.6963	−1.25191	−0.625954	+	0.779860i	$0.715290\pi$
$74$	0	0
$75$	12.3606	1.42728
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	4.42282	0.497606	0.248803	−	0.968554i	$-0.419963\pi$
$79$	4.42282	0.497606	0.248803	+	0.968554i	$0.419963\pi$
$80$	0	0
$81$	−6.25423	−0.694914
$82$	0	0
$83$	−10.8596	−1.19200	−0.596000	−	0.802984i	$-0.703244\pi$
$83$	−10.8596	−1.19200	−0.596000	+	0.802984i	$0.703244\pi$
$84$	0	0
$85$	2.32269	0.251931
$86$	0	0
$87$	−13.0384	−1.39787
$88$	0	0
$89$	14.7624	1.56481	0.782407	−	0.622767i	$-0.213992\pi$
$89$	14.7624	1.56481	0.782407	+	0.622767i	$0.213992\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−18.4351	−1.91163
$94$	0	0
$95$	−2.74870	−0.282010
$96$	0	0
$97$	9.76630	0.991617	0.495809	−	0.868432i	$-0.334872\pi$
$97$	9.76630	0.991617	0.495809	+	0.868432i	$0.334872\pi$
$98$	0	0
$99$	3.73512	0.375394
$100$	0	0
$101$	−8.07038	−0.803033	−0.401517	−	0.915852i	$-0.631517\pi$
$101$	−8.07038	−0.803033	−0.401517	+	0.915852i	$0.631517\pi$
$102$	0	0
$103$	7.32408	0.721663	0.360832	−	0.932631i	$-0.382493\pi$
$103$	7.32408	0.721663	0.360832	+	0.932631i	$0.382493\pi$
$104$	0	0
$105$	−1.26378	−0.123332
$106$	0	0
$107$	−10.2643	−0.992286	−0.496143	−	0.868241i	$-0.665251\pi$
$107$	−10.2643	−0.992286	−0.496143	+	0.868241i	$0.665251\pi$
$108$	0	0
$109$	3.47938	0.333264	0.166632	−	0.986019i	$-0.446711\pi$
$109$	3.47938	0.333264	0.166632	+	0.986019i	$0.446711\pi$
$110$	0	0
$111$	−9.88593	−0.938331
$112$	0	0
$113$	19.6968	1.85292	0.926460	−	0.376393i	$-0.122836\pi$
$113$	19.6968	1.85292	0.926460	+	0.376393i	$0.122836\pi$
$114$	0	0
$115$	−2.87218	−0.267832
$116$	0	0
$117$	3.73512	0.345312
$118$	0	0
$119$	4.76972	0.437240
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	14.9942	1.35198
$124$	0	0
$125$	−4.75417	−0.425226
$126$	0	0
$127$	−10.8768	−0.965159	−0.482579	−	0.875852i	$-0.660300\pi$
$127$	−10.8768	−0.965159	−0.482579	+	0.875852i	$0.660300\pi$
$128$	0	0
$129$	5.10340	0.449329
$130$	0	0
$131$	10.0205	0.875495	0.437748	−	0.899098i	$-0.355776\pi$
$131$	10.0205	0.875495	0.437748	+	0.899098i	$0.355776\pi$
$132$	0	0
$133$	−5.64455	−0.489444
$134$	0	0
$135$	−0.929031	−0.0799583
$136$	0	0
$137$	5.91877	0.505675	0.252837	−	0.967509i	$-0.418636\pi$
$137$	5.91877	0.505675	0.252837	+	0.967509i	$0.418636\pi$
$138$	0	0
$139$	6.97797	0.591864	0.295932	−	0.955209i	$-0.404370\pi$
$139$	6.97797	0.591864	0.295932	+	0.955209i	$0.404370\pi$
$140$	0	0
$141$	9.93685	0.836834
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	2.44653	0.203173
$146$	0	0
$147$	−2.59521	−0.214049
$148$	0	0
$149$	15.1513	1.24124	0.620622	−	0.784110i	$-0.286880\pi$
$149$	15.1513	1.24124	0.620622	+	0.784110i	$0.286880\pi$
$150$	0	0
$151$	12.5452	1.02091	0.510455	−	0.859904i	$-0.329477\pi$
$151$	12.5452	1.02091	0.510455	+	0.859904i	$0.329477\pi$
$152$	0	0
$153$	17.8155	1.44030
$154$	0	0
$155$	3.45916	0.277846
$156$	0	0
$157$	−8.51981	−0.679955	−0.339978	−	0.940434i	$-0.610419\pi$
$157$	−8.51981	−0.679955	−0.339978	+	0.940434i	$0.610419\pi$
$158$	0	0
$159$	−3.73416	−0.296138
$160$	0	0
$161$	−5.89812	−0.464837
$162$	0	0
$163$	−23.6806	−1.85481	−0.927406	−	0.374058i	$-0.877966\pi$
$163$	−23.6806	−1.85481	−0.927406	+	0.374058i	$0.877966\pi$
$164$	0	0
$165$	−1.26378	−0.0983849
$166$	0	0
$167$	−14.6312	−1.13219	−0.566097	−	0.824339i	$-0.691547\pi$
$167$	−14.6312	−1.13219	−0.566097	+	0.824339i	$0.691547\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−21.0831	−1.61226
$172$	0	0
$173$	11.3769	0.864973	0.432486	−	0.901640i	$-0.357636\pi$
$173$	11.3769	0.864973	0.432486	+	0.901640i	$0.357636\pi$
$174$	0	0
$175$	−4.76286	−0.360039
$176$	0	0
$177$	13.1897	0.991396
$178$	0	0
$179$	−4.24178	−0.317045	−0.158523	−	0.987355i	$-0.550673\pi$
$179$	−4.24178	−0.317045	−0.158523	+	0.987355i	$0.550673\pi$
$180$	0	0
$181$	−23.4042	−1.73962	−0.869811	−	0.493385i	$-0.835759\pi$
$181$	−23.4042	−1.73962	−0.869811	+	0.493385i	$0.835759\pi$
$182$	0	0
$183$	29.6954	2.19515
$184$	0	0
$185$	1.85499	0.136382
$186$	0	0
$187$	4.76972	0.348797
$188$	0	0
$189$	−1.90780	−0.138772
$190$	0	0
$191$	17.6219	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$191$	17.6219	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$192$	0	0
$193$	16.3413	1.17627	0.588136	−	0.808762i	$-0.299862\pi$
$193$	16.3413	1.17627	0.588136	+	0.808762i	$0.299862\pi$
$194$	0	0
$195$	−1.26378	−0.0905010
$196$	0	0
$197$	6.59285	0.469721	0.234861	−	0.972029i	$-0.424537\pi$
$197$	6.59285	0.469721	0.234861	+	0.972029i	$0.424537\pi$
$198$	0	0
$199$	5.30502	0.376063	0.188031	−	0.982163i	$-0.439789\pi$
$199$	5.30502	0.376063	0.188031	+	0.982163i	$0.439789\pi$
$200$	0	0
$201$	−19.2064	−1.35472
$202$	0	0
$203$	5.02403	0.352618
$204$	0	0
$205$	−2.81350	−0.196504
$206$	0	0
$207$	−22.0302	−1.53120
$208$	0	0
$209$	−5.64455	−0.390441
$210$	0	0
$211$	−15.9974	−1.10130	−0.550652	−	0.834735i	$-0.685621\pi$
$211$	−15.9974	−1.10130	−0.550652	+	0.834735i	$0.685621\pi$
$212$	0	0
$213$	42.8849	2.93843
$214$	0	0
$215$	−0.957601	−0.0653079
$216$	0	0
$217$	7.10351	0.482218
$218$	0	0
$219$	27.7592	1.87579
$220$	0	0
$221$	4.76972	0.320846
$222$	0	0
$223$	17.9305	1.20071	0.600357	−	0.799732i	$-0.295025\pi$
$223$	17.9305	1.20071	0.600357	+	0.799732i	$0.295025\pi$
$224$	0	0
$225$	−17.7899	−1.18599
$226$	0	0
$227$	−5.01065	−0.332569	−0.166284	−	0.986078i	$-0.553177\pi$
$227$	−5.01065	−0.332569	−0.166284	+	0.986078i	$0.553177\pi$
$228$	0	0
$229$	−24.9987	−1.65196	−0.825981	−	0.563698i	$-0.809378\pi$
$229$	−24.9987	−1.65196	−0.825981	+	0.563698i	$0.809378\pi$
$230$	0	0
$231$	−2.59521	−0.170752
$232$	0	0
$233$	−7.86422	−0.515202	−0.257601	−	0.966251i	$-0.582932\pi$
$233$	−7.86422	−0.515202	−0.257601	+	0.966251i	$0.582932\pi$
$234$	0	0
$235$	−1.86455	−0.121630
$236$	0	0
$237$	−11.4782	−0.745586
$238$	0	0
$239$	15.5940	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$239$	15.5940	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$240$	0	0
$241$	26.2508	1.69097	0.845483	−	0.534003i	$-0.179313\pi$
$241$	26.2508	1.69097	0.845483	+	0.534003i	$0.179313\pi$
$242$	0	0
$243$	21.9544	1.40838
$244$	0	0
$245$	0.486965	0.0311111
$246$	0	0
$247$	−5.64455	−0.359154
$248$	0	0
$249$	28.1831	1.78603
$250$	0	0
$251$	2.19241	0.138383	0.0691917	−	0.997603i	$-0.477958\pi$
$251$	2.19241	0.138383	0.0691917	+	0.997603i	$0.477958\pi$
$252$	0	0
$253$	−5.89812	−0.370811
$254$	0	0
$255$	−6.02787	−0.377480
$256$	0	0
$257$	−4.65888	−0.290613	−0.145306	−	0.989387i	$-0.546417\pi$
$257$	−4.65888	−0.290613	−0.145306	+	0.989387i	$0.546417\pi$
$258$	0	0
$259$	3.80930	0.236698
$260$	0	0
$261$	18.7654	1.16155
$262$	0	0
$263$	10.0222	0.617997	0.308998	−	0.951063i	$-0.400006\pi$
$263$	10.0222	0.617997	0.308998	+	0.951063i	$0.400006\pi$
$264$	0	0
$265$	0.700677	0.0430422
$266$	0	0
$267$	−38.3116	−2.34463
$268$	0	0
$269$	3.05164	0.186062	0.0930308	−	0.995663i	$-0.470345\pi$
$269$	3.05164	0.186062	0.0930308	+	0.995663i	$0.470345\pi$
$270$	0	0
$271$	−28.1152	−1.70788	−0.853939	−	0.520373i	$-0.825793\pi$
$271$	−28.1152	−1.70788	−0.853939	+	0.520373i	$0.825793\pi$
$272$	0	0
$273$	−2.59521	−0.157069
$274$	0	0
$275$	−4.76286	−0.287212
$276$	0	0
$277$	−3.73275	−0.224279	−0.112140	−	0.993692i	$-0.535770\pi$
$277$	−3.73275	−0.224279	−0.112140	+	0.993692i	$0.535770\pi$
$278$	0	0
$279$	26.5325	1.58846
$280$	0	0
$281$	−12.6837	−0.756645	−0.378322	−	0.925674i	$-0.623499\pi$
$281$	−12.6837	−0.756645	−0.378322	+	0.925674i	$0.623499\pi$
$282$	0	0
$283$	3.96817	0.235883	0.117942	−	0.993021i	$-0.462370\pi$
$283$	3.96817	0.235883	0.117942	+	0.993021i	$0.462370\pi$
$284$	0	0
$285$	7.13345	0.422549
$286$	0	0
$287$	−5.77763	−0.341043
$288$	0	0
$289$	5.75025	0.338250
$290$	0	0
$291$	−25.3456	−1.48579
$292$	0	0
$293$	−16.9622	−0.990943	−0.495472	−	0.868624i	$-0.665005\pi$
$293$	−16.9622	−0.990943	−0.495472	+	0.868624i	$0.665005\pi$
$294$	0	0
$295$	−2.47491	−0.144095
$296$	0	0
$297$	−1.90780	−0.110702
$298$	0	0
$299$	−5.89812	−0.341097
$300$	0	0
$301$	−1.96647	−0.113345
$302$	0	0
$303$	20.9444	1.20322
$304$	0	0
$305$	−5.57204	−0.319054
$306$	0	0
$307$	−31.6818	−1.80818	−0.904088	−	0.427347i	$-0.859448\pi$
$307$	−31.6818	−1.80818	−0.904088	+	0.427347i	$0.859448\pi$
$308$	0	0
$309$	−19.0075	−1.08130
$310$	0	0
$311$	11.1568	0.632642	0.316321	−	0.948652i	$-0.397552\pi$
$311$	11.1568	0.632642	0.316321	+	0.948652i	$0.397552\pi$
$312$	0	0
$313$	−32.7238	−1.84966	−0.924829	−	0.380384i	$-0.875792\pi$
$313$	−32.7238	−1.84966	−0.924829	+	0.380384i	$0.875792\pi$
$314$	0	0
$315$	1.81887	0.102482
$316$	0	0
$317$	−4.49832	−0.252651	−0.126325	−	0.991989i	$-0.540318\pi$
$317$	−4.49832	−0.252651	−0.126325	+	0.991989i	$0.540318\pi$
$318$	0	0
$319$	5.02403	0.281292
$320$	0	0
$321$	26.6380	1.48679
$322$	0	0
$323$	−26.9229	−1.49803
$324$	0	0
$325$	−4.76286	−0.264196
$326$	0	0
$327$	−9.02972	−0.499345
$328$	0	0
$329$	−3.82892	−0.211095
$330$	0	0
$331$	4.60374	0.253045	0.126522	−	0.991964i	$-0.459618\pi$
$331$	4.60374	0.253045	0.126522	+	0.991964i	$0.459618\pi$
$332$	0	0
$333$	14.2282	0.779700
$334$	0	0
$335$	3.60389	0.196902
$336$	0	0
$337$	−18.7606	−1.02195	−0.510977	−	0.859594i	$-0.670716\pi$
$337$	−18.7606	−1.02195	−0.510977	+	0.859594i	$0.670716\pi$
$338$	0	0
$339$	−51.1174	−2.77632
$340$	0	0
$341$	7.10351	0.384677
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.45391	0.401305
$346$	0	0
$347$	1.53525	0.0824164	0.0412082	−	0.999151i	$-0.486879\pi$
$347$	1.53525	0.0824164	0.0412082	+	0.999151i	$0.486879\pi$
$348$	0	0
$349$	−26.8894	−1.43936	−0.719679	−	0.694307i	$-0.755711\pi$
$349$	−26.8894	−1.43936	−0.719679	+	0.694307i	$0.755711\pi$
$350$	0	0
$351$	−1.90780	−0.101831
$352$	0	0
$353$	−9.50972	−0.506151	−0.253076	−	0.967446i	$-0.581442\pi$
$353$	−9.50972	−0.506151	−0.253076	+	0.967446i	$0.581442\pi$
$354$	0	0
$355$	−8.04692	−0.427086
$356$	0	0
$357$	−12.3784	−0.655136
$358$	0	0
$359$	−16.8655	−0.890129	−0.445065	−	0.895498i	$-0.646819\pi$
$359$	−16.8655	−0.890129	−0.445065	+	0.895498i	$0.646819\pi$
$360$	0	0
$361$	12.8609	0.676890
$362$	0	0
$363$	−2.59521	−0.136213
$364$	0	0
$365$	−5.20873	−0.272637
$366$	0	0
$367$	−12.1067	−0.631965	−0.315982	−	0.948765i	$-0.602334\pi$
$367$	−12.1067	−0.631965	−0.315982	+	0.948765i	$0.602334\pi$
$368$	0	0
$369$	−21.5802	−1.12342
$370$	0	0
$371$	1.43886	0.0747021
$372$	0	0
$373$	−3.47254	−0.179802	−0.0899008	−	0.995951i	$-0.528655\pi$
$373$	−3.47254	−0.179802	−0.0899008	+	0.995951i	$0.528655\pi$
$374$	0	0
$375$	12.3381	0.637136
$376$	0	0
$377$	5.02403	0.258751
$378$	0	0
$379$	−19.8099	−1.01756	−0.508782	−	0.860895i	$-0.669904\pi$
$379$	−19.8099	−1.01756	−0.508782	+	0.860895i	$0.669904\pi$
$380$	0	0
$381$	28.2276	1.44614
$382$	0	0
$383$	−14.2350	−0.727377	−0.363688	−	0.931521i	$-0.618483\pi$
$383$	−14.2350	−0.727377	−0.363688	+	0.931521i	$0.618483\pi$
$384$	0	0
$385$	0.486965	0.0248180
$386$	0	0
$387$	−7.34500	−0.373367
$388$	0	0
$389$	−19.7049	−0.999078	−0.499539	−	0.866291i	$-0.666497\pi$
$389$	−19.7049	−0.999078	−0.499539	+	0.866291i	$0.666497\pi$
$390$	0	0
$391$	−28.1324	−1.42272
$392$	0	0
$393$	−26.0053	−1.31180
$394$	0	0
$395$	2.15376	0.108367
$396$	0	0
$397$	18.9940	0.953283	0.476641	−	0.879098i	$-0.341854\pi$
$397$	18.9940	0.953283	0.476641	+	0.879098i	$0.341854\pi$
$398$	0	0
$399$	14.6488	0.733357
$400$	0	0
$401$	−27.6196	−1.37926	−0.689629	−	0.724162i	$-0.742227\pi$
$401$	−27.6196	−1.37926	−0.689629	+	0.724162i	$0.742227\pi$
$402$	0	0
$403$	7.10351	0.353851
$404$	0	0
$405$	−3.04559	−0.151337
$406$	0	0
$407$	3.80930	0.188820
$408$	0	0
$409$	20.1552	0.996612	0.498306	−	0.867001i	$-0.333956\pi$
$409$	20.1552	0.996612	0.498306	+	0.867001i	$0.333956\pi$
$410$	0	0
$411$	−15.3605	−0.757676
$412$	0	0
$413$	−5.08231	−0.250084
$414$	0	0
$415$	−5.28827	−0.259591
$416$	0	0
$417$	−18.1093	−0.886816
$418$	0	0
$419$	−36.4996	−1.78312	−0.891561	−	0.452901i	$-0.850389\pi$
$419$	−36.4996	−1.78312	−0.891561	+	0.452901i	$0.850389\pi$
$420$	0	0
$421$	10.5480	0.514078	0.257039	−	0.966401i	$-0.417253\pi$
$421$	10.5480	0.514078	0.257039	+	0.966401i	$0.417253\pi$
$422$	0	0
$423$	−14.3015	−0.695361
$424$	0	0
$425$	−22.7175	−1.10196
$426$	0	0
$427$	−11.4424	−0.553735
$428$	0	0
$429$	−2.59521	−0.125298
$430$	0	0
$431$	11.1827	0.538652	0.269326	−	0.963049i	$-0.413199\pi$
$431$	11.1827	0.538652	0.269326	+	0.963049i	$0.413199\pi$
$432$	0	0
$433$	−4.75039	−0.228289	−0.114145	−	0.993464i	$-0.536413\pi$
$433$	−4.75039	−0.228289	−0.114145	+	0.993464i	$0.536413\pi$
$434$	0	0
$435$	−6.34926	−0.304424
$436$	0	0
$437$	33.2922	1.59258
$438$	0	0
$439$	−20.8048	−0.992961	−0.496481	−	0.868048i	$-0.665375\pi$
$439$	−20.8048	−0.992961	−0.496481	+	0.868048i	$0.665375\pi$
$440$	0	0
$441$	3.73512	0.177863
$442$	0	0
$443$	−16.1713	−0.768323	−0.384161	−	0.923266i	$-0.625509\pi$
$443$	−16.1713	−0.768323	−0.384161	+	0.923266i	$0.625509\pi$
$444$	0	0
$445$	7.18879	0.340781
$446$	0	0
$447$	−39.3209	−1.85981
$448$	0	0
$449$	38.6243	1.82279	0.911396	−	0.411531i	$-0.135006\pi$
$449$	38.6243	1.82279	0.911396	+	0.411531i	$0.135006\pi$
$450$	0	0
$451$	−5.77763	−0.272058
$452$	0	0
$453$	−32.5573	−1.52968
$454$	0	0
$455$	0.486965	0.0228293
$456$	0	0
$457$	40.7371	1.90560	0.952801	−	0.303597i	$-0.0981876\pi$
$457$	40.7371	1.90560	0.952801	+	0.303597i	$0.0981876\pi$
$458$	0	0
$459$	−9.09967	−0.424736
$460$	0	0
$461$	−15.4966	−0.721749	−0.360874	−	0.932614i	$-0.617522\pi$
$461$	−15.4966	−0.721749	−0.360874	+	0.932614i	$0.617522\pi$
$462$	0	0
$463$	−17.5492	−0.815579	−0.407790	−	0.913076i	$-0.633700\pi$
$463$	−17.5492	−0.815579	−0.407790	+	0.913076i	$0.633700\pi$
$464$	0	0
$465$	−8.97725	−0.416310
$466$	0	0
$467$	−29.6675	−1.37285	−0.686425	−	0.727201i	$-0.740821\pi$
$467$	−29.6675	−1.37285	−0.686425	+	0.727201i	$0.740821\pi$
$468$	0	0
$469$	7.40072	0.341733
$470$	0	0
$471$	22.1107	1.01881
$472$	0	0
$473$	−1.96647	−0.0904183
$474$	0	0
$475$	26.8842	1.23353
$476$	0	0
$477$	5.37434	0.246074
$478$	0	0
$479$	−22.4009	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$479$	−22.4009	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$480$	0	0
$481$	3.80930	0.173689
$482$	0	0
$483$	15.3069	0.696486
$484$	0	0
$485$	4.75585	0.215952
$486$	0	0
$487$	7.29827	0.330716	0.165358	−	0.986234i	$-0.447122\pi$
$487$	7.29827	0.330716	0.165358	+	0.986234i	$0.447122\pi$
$488$	0	0
$489$	61.4563	2.77915
$490$	0	0
$491$	−16.5352	−0.746224	−0.373112	−	0.927786i	$-0.621709\pi$
$491$	−16.5352	−0.746224	−0.373112	+	0.927786i	$0.621709\pi$
$492$	0	0
$493$	23.9632	1.07925
$494$	0	0
$495$	1.81887	0.0817523
$496$	0	0
$497$	−16.5246	−0.741231
$498$	0	0
$499$	29.1426	1.30460	0.652301	−	0.757960i	$-0.273804\pi$
$499$	29.1426	1.30460	0.652301	+	0.757960i	$0.273804\pi$
$500$	0	0
$501$	37.9710	1.69642
$502$	0	0
$503$	−19.8856	−0.886654	−0.443327	−	0.896360i	$-0.646202\pi$
$503$	−19.8856	−0.886654	−0.443327	+	0.896360i	$0.646202\pi$
$504$	0	0
$505$	−3.92999	−0.174882
$506$	0	0
$507$	−2.59521	−0.115257
$508$	0	0
$509$	−35.0889	−1.55529	−0.777643	−	0.628706i	$-0.783585\pi$
$509$	−35.0889	−1.55529	−0.777643	+	0.628706i	$0.783585\pi$
$510$	0	0
$511$	−10.6963	−0.473177
$512$	0	0
$513$	10.7687	0.475448
$514$	0	0
$515$	3.56657	0.157162
$516$	0	0
$517$	−3.82892	−0.168396
$518$	0	0
$519$	−29.5256	−1.29603
$520$	0	0
$521$	−1.64490	−0.0720645	−0.0360323	−	0.999351i	$-0.511472\pi$
$521$	−1.64490	−0.0720645	−0.0360323	+	0.999351i	$0.511472\pi$
$522$	0	0
$523$	−1.29838	−0.0567743	−0.0283872	−	0.999597i	$-0.509037\pi$
$523$	−1.29838	−0.0567743	−0.0283872	+	0.999597i	$0.509037\pi$
$524$	0	0
$525$	12.3606	0.539463
$526$	0	0
$527$	33.8818	1.47591
$528$	0	0
$529$	11.7878	0.512512
$530$	0	0
$531$	−18.9830	−0.823794
$532$	0	0
$533$	−5.77763	−0.250257
$534$	0	0
$535$	−4.99835	−0.216097
$536$	0	0
$537$	11.0083	0.475043
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−15.6859	−0.674388	−0.337194	−	0.941435i	$-0.609478\pi$
$541$	−15.6859	−0.674388	−0.337194	+	0.941435i	$0.609478\pi$
$542$	0	0
$543$	60.7389	2.60656
$544$	0	0
$545$	1.69434	0.0725774
$546$	0	0
$547$	−28.7555	−1.22949	−0.614747	−	0.788724i	$-0.710742\pi$
$547$	−28.7555	−1.22949	−0.614747	+	0.788724i	$0.710742\pi$
$548$	0	0
$549$	−42.7387	−1.82404
$550$	0	0
$551$	−28.3584	−1.20811
$552$	0	0
$553$	4.42282	0.188077
$554$	0	0
$555$	−4.81410	−0.204347
$556$	0	0
$557$	27.2823	1.15599	0.577993	−	0.816042i	$-0.303836\pi$
$557$	27.2823	1.15599	0.577993	+	0.816042i	$0.303836\pi$
$558$	0	0
$559$	−1.96647	−0.0831727
$560$	0	0
$561$	−12.3784	−0.522618
$562$	0	0
$563$	12.8820	0.542910	0.271455	−	0.962451i	$-0.412495\pi$
$563$	12.8820	0.542910	0.271455	+	0.962451i	$0.412495\pi$
$564$	0	0
$565$	9.59166	0.403524
$566$	0	0
$567$	−6.25423	−0.262653
$568$	0	0
$569$	9.32883	0.391085	0.195542	−	0.980695i	$-0.437353\pi$
$569$	9.32883	0.391085	0.195542	+	0.980695i	$0.437353\pi$
$570$	0	0
$571$	−3.37175	−0.141103	−0.0705517	−	0.997508i	$-0.522476\pi$
$571$	−3.37175	−0.141103	−0.0705517	+	0.997508i	$0.522476\pi$
$572$	0	0
$573$	−45.7326	−1.91050
$574$	0	0
$575$	28.0919	1.17151
$576$	0	0
$577$	−7.71712	−0.321268	−0.160634	−	0.987014i	$-0.551354\pi$
$577$	−7.71712	−0.321268	−0.160634	+	0.987014i	$0.551354\pi$
$578$	0	0
$579$	−42.4091	−1.76246
$580$	0	0
$581$	−10.8596	−0.450534
$582$	0	0
$583$	1.43886	0.0595917
$584$	0	0
$585$	1.81887	0.0752012
$586$	0	0
$587$	−9.22189	−0.380628	−0.190314	−	0.981723i	$-0.560951\pi$
$587$	−9.22189	−0.380628	−0.190314	+	0.981723i	$0.560951\pi$
$588$	0	0
$589$	−40.0961	−1.65213
$590$	0	0
$591$	−17.1098	−0.703805
$592$	0	0
$593$	−6.88523	−0.282743	−0.141371	−	0.989957i	$-0.545151\pi$
$593$	−6.88523	−0.282743	−0.141371	+	0.989957i	$0.545151\pi$
$594$	0	0
$595$	2.32269	0.0952209
$596$	0	0
$597$	−13.7676	−0.563472
$598$	0	0
$599$	−21.9279	−0.895950	−0.447975	−	0.894046i	$-0.647855\pi$
$599$	−21.9279	−0.895950	−0.447975	+	0.894046i	$0.647855\pi$
$600$	0	0
$601$	25.8735	1.05540	0.527700	−	0.849431i	$-0.323055\pi$
$601$	25.8735	1.05540	0.527700	+	0.849431i	$0.323055\pi$
$602$	0	0
$603$	27.6426	1.12569
$604$	0	0
$605$	0.486965	0.0197979
$606$	0	0
$607$	−20.5943	−0.835898	−0.417949	−	0.908471i	$-0.637251\pi$
$607$	−20.5943	−0.835898	−0.417949	+	0.908471i	$0.637251\pi$
$608$	0	0
$609$	−13.0384	−0.528344
$610$	0	0
$611$	−3.82892	−0.154901
$612$	0	0
$613$	−27.1260	−1.09561	−0.547805	−	0.836606i	$-0.684536\pi$
$613$	−27.1260	−1.09561	−0.547805	+	0.836606i	$0.684536\pi$
$614$	0	0
$615$	7.30164	0.294431
$616$	0	0
$617$	12.1155	0.487751	0.243876	−	0.969807i	$-0.421581\pi$
$617$	12.1155	0.487751	0.243876	+	0.969807i	$0.421581\pi$
$618$	0	0
$619$	21.0397	0.845659	0.422829	−	0.906209i	$-0.361037\pi$
$619$	21.0397	0.845659	0.422829	+	0.906209i	$0.361037\pi$
$620$	0	0
$621$	11.2524	0.451544
$622$	0	0
$623$	14.7624	0.591444
$624$	0	0
$625$	21.4992	0.859968
$626$	0	0
$627$	14.6488	0.585016
$628$	0	0
$629$	18.1693	0.724457
$630$	0	0
$631$	22.4358	0.893157	0.446578	−	0.894745i	$-0.352642\pi$
$631$	22.4358	0.893157	0.446578	+	0.894745i	$0.352642\pi$
$632$	0	0
$633$	41.5166	1.65013
$634$	0	0
$635$	−5.29662	−0.210190
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−61.7215	−2.44167
$640$	0	0
$641$	−2.19247	−0.0865973	−0.0432986	−	0.999062i	$-0.513787\pi$
$641$	−2.19247	−0.0865973	−0.0432986	+	0.999062i	$0.513787\pi$
$642$	0	0
$643$	−18.3672	−0.724333	−0.362166	−	0.932113i	$-0.617963\pi$
$643$	−18.3672	−0.724333	−0.362166	+	0.932113i	$0.617963\pi$
$644$	0	0
$645$	2.48518	0.0978538
$646$	0	0
$647$	30.1567	1.18558	0.592791	−	0.805356i	$-0.298026\pi$
$647$	30.1567	1.18558	0.592791	+	0.805356i	$0.298026\pi$
$648$	0	0
$649$	−5.08231	−0.199498
$650$	0	0
$651$	−18.4351	−0.722529
$652$	0	0
$653$	2.96939	0.116201	0.0581006	−	0.998311i	$-0.481496\pi$
$653$	2.96939	0.116201	0.0581006	+	0.998311i	$0.481496\pi$
$654$	0	0
$655$	4.87964	0.190663
$656$	0	0
$657$	−39.9520	−1.55868
$658$	0	0
$659$	38.2598	1.49039	0.745195	−	0.666846i	$-0.232356\pi$
$659$	38.2598	1.49039	0.745195	+	0.666846i	$0.232356\pi$
$660$	0	0
$661$	−23.0562	−0.896782	−0.448391	−	0.893838i	$-0.648003\pi$
$661$	−23.0562	−0.896782	−0.448391	+	0.893838i	$0.648003\pi$
$662$	0	0
$663$	−12.3784	−0.480739
$664$	0	0
$665$	−2.74870	−0.106590
$666$	0	0
$667$	−29.6323	−1.14737
$668$	0	0
$669$	−46.5334	−1.79909
$670$	0	0
$671$	−11.4424	−0.441728
$672$	0	0
$673$	20.3894	0.785954	0.392977	−	0.919548i	$-0.371445\pi$
$673$	20.3894	0.785954	0.392977	+	0.919548i	$0.371445\pi$
$674$	0	0
$675$	9.08659	0.349743
$676$	0	0
$677$	−23.9718	−0.921311	−0.460655	−	0.887579i	$-0.652386\pi$
$677$	−23.9718	−0.921311	−0.460655	+	0.887579i	$0.652386\pi$
$678$	0	0
$679$	9.76630	0.374796
$680$	0	0
$681$	13.0037	0.498303
$682$	0	0
$683$	−40.8124	−1.56164	−0.780821	−	0.624755i	$-0.785199\pi$
$683$	−40.8124	−1.56164	−0.780821	+	0.624755i	$0.785199\pi$
$684$	0	0
$685$	2.88223	0.110125
$686$	0	0
$687$	64.8770	2.47521
$688$	0	0
$689$	1.43886	0.0548164
$690$	0	0
$691$	38.4357	1.46216	0.731081	−	0.682291i	$-0.239016\pi$
$691$	38.4357	1.46216	0.731081	+	0.682291i	$0.239016\pi$
$692$	0	0
$693$	3.73512	0.141886
$694$	0	0
$695$	3.39803	0.128895
$696$	0	0
$697$	−27.5577	−1.04382
$698$	0	0
$699$	20.4093	0.771951
$700$	0	0
$701$	11.9212	0.450259	0.225130	−	0.974329i	$-0.427719\pi$
$701$	11.9212	0.450259	0.225130	+	0.974329i	$0.427719\pi$
$702$	0	0
$703$	−21.5017	−0.810954
$704$	0	0
$705$	4.83890	0.182243
$706$	0	0
$707$	−8.07038	−0.303518
$708$	0	0
$709$	41.8255	1.57079	0.785395	−	0.618995i	$-0.212460\pi$
$709$	41.8255	1.57079	0.785395	+	0.618995i	$0.212460\pi$
$710$	0	0
$711$	16.5198	0.619540
$712$	0	0
$713$	−41.8973	−1.56907
$714$	0	0
$715$	0.486965	0.0182115
$716$	0	0
$717$	−40.4698	−1.51137
$718$	0	0
$719$	−5.11648	−0.190812	−0.0954062	−	0.995438i	$-0.530415\pi$
$719$	−5.11648	−0.190812	−0.0954062	+	0.995438i	$0.530415\pi$
$720$	0	0
$721$	7.32408	0.272763
$722$	0	0
$723$	−68.1265	−2.53365
$724$	0	0
$725$	−23.9288	−0.888693
$726$	0	0
$727$	−32.3696	−1.20052	−0.600260	−	0.799805i	$-0.704936\pi$
$727$	−32.3696	−1.20052	−0.600260	+	0.799805i	$0.704936\pi$
$728$	0	0
$729$	−38.2137	−1.41532
$730$	0	0
$731$	−9.37950	−0.346914
$732$	0	0
$733$	33.2719	1.22892	0.614462	−	0.788946i	$-0.289373\pi$
$733$	33.2719	1.22892	0.614462	+	0.788946i	$0.289373\pi$
$734$	0	0
$735$	−1.26378	−0.0466151
$736$	0	0
$737$	7.40072	0.272609
$738$	0	0
$739$	39.5124	1.45349	0.726744	−	0.686909i	$-0.241033\pi$
$739$	39.5124	1.45349	0.726744	+	0.686909i	$0.241033\pi$
$740$	0	0
$741$	14.6488	0.538137
$742$	0	0
$743$	−45.8604	−1.68245	−0.841227	−	0.540682i	$-0.818166\pi$
$743$	−45.8604	−1.68245	−0.841227	+	0.540682i	$0.818166\pi$
$744$	0	0
$745$	7.37817	0.270315
$746$	0	0
$747$	−40.5621	−1.48409
$748$	0	0
$749$	−10.2643	−0.375049
$750$	0	0
$751$	23.1980	0.846509	0.423254	−	0.906011i	$-0.360888\pi$
$751$	23.1980	0.846509	0.423254	+	0.906011i	$0.360888\pi$
$752$	0	0
$753$	−5.68976	−0.207346
$754$	0	0
$755$	6.10905	0.222331
$756$	0	0
$757$	−48.9242	−1.77818	−0.889089	−	0.457734i	$-0.848661\pi$
$757$	−48.9242	−1.77818	−0.889089	+	0.457734i	$0.848661\pi$
$758$	0	0
$759$	15.3069	0.555604
$760$	0	0
$761$	−12.3899	−0.449133	−0.224566	−	0.974459i	$-0.572097\pi$
$761$	−12.3899	−0.449133	−0.224566	+	0.974459i	$0.572097\pi$
$762$	0	0
$763$	3.47938	0.125962
$764$	0	0
$765$	8.67552	0.313664
$766$	0	0
$767$	−5.08231	−0.183512
$768$	0	0
$769$	−18.4356	−0.664804	−0.332402	−	0.943138i	$-0.607859\pi$
$769$	−18.4356	−0.664804	−0.332402	+	0.943138i	$0.607859\pi$
$770$	0	0
$771$	12.0908	0.435439
$772$	0	0
$773$	−2.11011	−0.0758953	−0.0379476	−	0.999280i	$-0.512082\pi$
$773$	−2.11011	−0.0758953	−0.0379476	+	0.999280i	$0.512082\pi$
$774$	0	0
$775$	−33.8331	−1.21532
$776$	0	0
$777$	−9.88593	−0.354656
$778$	0	0
$779$	32.6121	1.16845
$780$	0	0
$781$	−16.5246	−0.591298
$782$	0	0
$783$	−9.58484	−0.342534
$784$	0	0
$785$	−4.14885	−0.148079
$786$	0	0
$787$	−30.1767	−1.07568	−0.537841	−	0.843046i	$-0.680760\pi$
$787$	−30.1767	−1.07568	−0.537841	+	0.843046i	$0.680760\pi$
$788$	0	0
$789$	−26.0098	−0.925973
$790$	0	0
$791$	19.6968	0.700338
$792$	0	0
$793$	−11.4424	−0.406331
$794$	0	0
$795$	−1.81840	−0.0644922
$796$	0	0
$797$	−39.3379	−1.39342	−0.696710	−	0.717353i	$-0.745353\pi$
$797$	−39.3379	−1.39342	−0.696710	+	0.717353i	$0.745353\pi$
$798$	0	0
$799$	−18.2629	−0.646094
$800$	0	0
$801$	55.1395	1.94826
$802$	0	0
$803$	−10.6963	−0.377465
$804$	0	0
$805$	−2.87218	−0.101231
$806$	0	0
$807$	−7.91964	−0.278785
$808$	0	0
$809$	−6.63213	−0.233173	−0.116587	−	0.993181i	$-0.537195\pi$
$809$	−6.63213	−0.233173	−0.116587	+	0.993181i	$0.537195\pi$
$810$	0	0
$811$	44.3096	1.55592	0.777961	−	0.628313i	$-0.216254\pi$
$811$	44.3096	1.55592	0.777961	+	0.628313i	$0.216254\pi$
$812$	0	0
$813$	72.9650	2.55899
$814$	0	0
$815$	−11.5316	−0.403936
$816$	0	0
$817$	11.0998	0.388334
$818$	0	0
$819$	3.73512	0.130516
$820$	0	0
$821$	8.97713	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$821$	8.97713	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$822$	0	0
$823$	38.0984	1.32803	0.664013	−	0.747721i	$-0.268852\pi$
$823$	38.0984	1.32803	0.664013	+	0.747721i	$0.268852\pi$
$824$	0	0
$825$	12.3606	0.430342
$826$	0	0
$827$	−12.5931	−0.437904	−0.218952	−	0.975736i	$-0.570264\pi$
$827$	−12.5931	−0.437904	−0.218952	+	0.975736i	$0.570264\pi$
$828$	0	0
$829$	54.7983	1.90322	0.951612	−	0.307301i	$-0.0994259\pi$
$829$	54.7983	1.90322	0.951612	+	0.307301i	$0.0994259\pi$
$830$	0	0
$831$	9.68727	0.336048
$832$	0	0
$833$	4.76972	0.165261
$834$	0	0
$835$	−7.12487	−0.246566
$836$	0	0
$837$	−13.5521	−0.468428
$838$	0	0
$839$	−42.0271	−1.45094	−0.725468	−	0.688256i	$-0.758376\pi$
$839$	−42.0271	−1.45094	−0.725468	+	0.688256i	$0.758376\pi$
$840$	0	0
$841$	−3.75909	−0.129624
$842$	0	0
$843$	32.9168	1.13372
$844$	0	0
$845$	0.486965	0.0167521
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−10.2982	−0.353435
$850$	0	0
$851$	−22.4677	−0.770182
$852$	0	0
$853$	−54.8358	−1.87754	−0.938771	−	0.344541i	$-0.888035\pi$
$853$	−54.8358	−1.87754	−0.938771	+	0.344541i	$0.888035\pi$
$854$	0	0
$855$	−10.2667	−0.351114
$856$	0	0
$857$	34.7728	1.18782	0.593909	−	0.804533i	$-0.297584\pi$
$857$	34.7728	1.18782	0.593909	+	0.804533i	$0.297584\pi$
$858$	0	0
$859$	−29.9915	−1.02330	−0.511649	−	0.859194i	$-0.670965\pi$
$859$	−29.9915	−1.02330	−0.511649	+	0.859194i	$0.670965\pi$
$860$	0	0
$861$	14.9942	0.511000
$862$	0	0
$863$	35.7424	1.21669	0.608343	−	0.793674i	$-0.291835\pi$
$863$	35.7424	1.21669	0.608343	+	0.793674i	$0.291835\pi$
$864$	0	0
$865$	5.54017	0.188372
$866$	0	0
$867$	−14.9231	−0.506815
$868$	0	0
$869$	4.42282	0.150034
$870$	0	0
$871$	7.40072	0.250764
$872$	0	0
$873$	36.4783	1.23460
$874$	0	0
$875$	−4.75417	−0.160720
$876$	0	0
$877$	−13.4777	−0.455111	−0.227555	−	0.973765i	$-0.573073\pi$
$877$	−13.4777	−0.455111	−0.227555	+	0.973765i	$0.573073\pi$
$878$	0	0
$879$	44.0205	1.48478
$880$	0	0
$881$	−54.9209	−1.85033	−0.925166	−	0.379562i	$-0.876075\pi$
$881$	−54.9209	−1.85033	−0.925166	+	0.379562i	$0.876075\pi$
$882$	0	0
$883$	54.0719	1.81966	0.909832	−	0.414976i	$-0.136210\pi$
$883$	54.0719	1.81966	0.909832	+	0.414976i	$0.136210\pi$
$884$	0	0
$885$	6.42291	0.215904
$886$	0	0
$887$	−22.8302	−0.766562	−0.383281	−	0.923632i	$-0.625206\pi$
$887$	−22.8302	−0.766562	−0.383281	+	0.923632i	$0.625206\pi$
$888$	0	0
$889$	−10.8768	−0.364796
$890$	0	0
$891$	−6.25423	−0.209525
$892$	0	0
$893$	21.6125	0.723235
$894$	0	0
$895$	−2.06560	−0.0690453
$896$	0	0
$897$	15.3069	0.511081
$898$	0	0
$899$	35.6883	1.19027
$900$	0	0
$901$	6.86298	0.228639
$902$	0	0
$903$	5.10340	0.169831
$904$	0	0
$905$	−11.3970	−0.378850
$906$	0	0
$907$	8.35715	0.277495	0.138747	−	0.990328i	$-0.455692\pi$
$907$	8.35715	0.277495	0.138747	+	0.990328i	$0.455692\pi$
$908$	0	0
$909$	−30.1439	−0.999809
$910$	0	0
$911$	17.6063	0.583323	0.291661	−	0.956522i	$-0.405792\pi$
$911$	17.6063	0.583323	0.291661	+	0.956522i	$0.405792\pi$
$912$	0	0
$913$	−10.8596	−0.359402
$914$	0	0
$915$	14.4606	0.478053
$916$	0	0
$917$	10.0205	0.330906
$918$	0	0
$919$	−7.89728	−0.260507	−0.130254	−	0.991481i	$-0.541579\pi$
$919$	−7.89728	−0.260507	−0.130254	+	0.991481i	$0.541579\pi$
$920$	0	0
$921$	82.2210	2.70927
$922$	0	0
$923$	−16.5246	−0.543915
$924$	0	0
$925$	−18.1432	−0.596543
$926$	0	0
$927$	27.3564	0.898500
$928$	0	0
$929$	−45.6526	−1.49781	−0.748907	−	0.662675i	$-0.769421\pi$
$929$	−45.6526	−1.49781	−0.748907	+	0.662675i	$0.769421\pi$
$930$	0	0
$931$	−5.64455	−0.184993
$932$	0	0
$933$	−28.9542	−0.947917
$934$	0	0
$935$	2.32269	0.0759600
$936$	0	0
$937$	−3.08766	−0.100869	−0.0504347	−	0.998727i	$-0.516061\pi$
$937$	−3.08766	−0.100869	−0.0504347	+	0.998727i	$0.516061\pi$
$938$	0	0
$939$	84.9251	2.77143
$940$	0	0
$941$	−2.07863	−0.0677614	−0.0338807	−	0.999426i	$-0.510787\pi$
$941$	−2.07863	−0.0677614	−0.0338807	+	0.999426i	$0.510787\pi$
$942$	0	0
$943$	34.0771	1.10970
$944$	0	0
$945$	−0.929031	−0.0302214
$946$	0	0
$947$	−50.6194	−1.64491	−0.822455	−	0.568831i	$-0.807396\pi$
$947$	−50.6194	−1.64491	−0.822455	+	0.568831i	$0.807396\pi$
$948$	0	0
$949$	−10.6963	−0.347217
$950$	0	0
$951$	11.6741	0.378559
$952$	0	0
$953$	−10.6956	−0.346463	−0.173231	−	0.984881i	$-0.555421\pi$
$953$	−10.6956	−0.346463	−0.173231	+	0.984881i	$0.555421\pi$
$954$	0	0
$955$	8.58125	0.277683
$956$	0	0
$957$	−13.0384	−0.421473
$958$	0	0
$959$	5.91877	0.191127
$960$	0	0
$961$	19.4598	0.627736
$962$	0	0
$963$	−38.3383	−1.23544
$964$	0	0
$965$	7.95763	0.256165
$966$	0	0
$967$	−60.1284	−1.93360	−0.966799	−	0.255539i	$-0.917747\pi$
$967$	−60.1284	−1.93360	−0.966799	+	0.255539i	$0.917747\pi$
$968$	0	0
$969$	69.8707	2.24457
$970$	0	0
$971$	11.6918	0.375207	0.187604	−	0.982245i	$-0.439928\pi$
$971$	11.6918	0.375207	0.187604	+	0.982245i	$0.439928\pi$
$972$	0	0
$973$	6.97797	0.223703
$974$	0	0
$975$	12.3606	0.395857
$976$	0	0
$977$	−28.9642	−0.926645	−0.463323	−	0.886190i	$-0.653343\pi$
$977$	−28.9642	−0.926645	−0.463323	+	0.886190i	$0.653343\pi$
$978$	0	0
$979$	14.7624	0.471809
$980$	0	0
$981$	12.9959	0.414927
$982$	0	0
$983$	−40.9042	−1.30464	−0.652321	−	0.757943i	$-0.726204\pi$
$983$	−40.9042	−1.30464	−0.652321	+	0.757943i	$0.726204\pi$
$984$	0	0
$985$	3.21049	0.102295
$986$	0	0
$987$	9.93685	0.316293
$988$	0	0
$989$	11.5985	0.368809
$990$	0	0
$991$	−35.6460	−1.13233	−0.566166	−	0.824291i	$-0.691574\pi$
$991$	−35.6460	−1.13233	−0.566166	+	0.824291i	$0.691574\pi$
$992$	0	0
$993$	−11.9477	−0.379148
$994$	0	0
$995$	2.58336	0.0818980
$996$	0	0
$997$	15.1943	0.481208	0.240604	−	0.970623i	$-0.422654\pi$
$997$	15.1943	0.481208	0.240604	+	0.970623i	$0.422654\pi$
$998$	0	0
$999$	−7.26737	−0.229929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.2	✓	8	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)

\(f(q)\)	\(=\)	\(q-2.59521 q^{3} +0.486965 q^{5} +1.00000 q^{7} +3.73512 q^{9} +O(q^{10})\)	\(q-2.59521 q^{3} +0.486965 q^{5} +1.00000 q^{7} +3.73512 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.26378 q^{15} +4.76972 q^{17} -5.64455 q^{19} -2.59521 q^{21} -5.89812 q^{23} -4.76286 q^{25} -1.90780 q^{27} +5.02403 q^{29} +7.10351 q^{31} -2.59521 q^{33} +0.486965 q^{35} +3.80930 q^{37} -2.59521 q^{39} -5.77763 q^{41} -1.96647 q^{43} +1.81887 q^{45} -3.82892 q^{47} +1.00000 q^{49} -12.3784 q^{51} +1.43886 q^{53} +0.486965 q^{55} +14.6488 q^{57} -5.08231 q^{59} -11.4424 q^{61} +3.73512 q^{63} +0.486965 q^{65} +7.40072 q^{67} +15.3069 q^{69} -16.5246 q^{71} -10.6963 q^{73} +12.3606 q^{75} +1.00000 q^{77} +4.42282 q^{79} -6.25423 q^{81} -10.8596 q^{83} +2.32269 q^{85} -13.0384 q^{87} +14.7624 q^{89} +1.00000 q^{91} -18.4351 q^{93} -2.74870 q^{95} +9.76630 q^{97} +3.73512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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