LMFDB - Embedded newform 8008.2.a.q.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.61891$ 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)

$f(q)$	$=$	$q+2.61891 q^{3} -2.59083 q^{5} +1.00000 q^{7} +3.85871 q^{9} +O(q^{10})$	$q+2.61891 q^{3} -2.59083 q^{5} +1.00000 q^{7} +3.85871 q^{9} +1.00000 q^{11} -1.00000 q^{13} -6.78516 q^{15} +3.03107 q^{17} -2.34281 q^{19} +2.61891 q^{21} -1.27075 q^{23} +1.71240 q^{25} +2.24890 q^{27} -0.481768 q^{29} +4.91110 q^{31} +2.61891 q^{33} -2.59083 q^{35} -1.63209 q^{37} -2.61891 q^{39} +1.58784 q^{41} +11.7462 q^{43} -9.99728 q^{45} +1.27075 q^{47} +1.00000 q^{49} +7.93812 q^{51} -1.51284 q^{53} -2.59083 q^{55} -6.13563 q^{57} +7.19716 q^{59} -1.26999 q^{61} +3.85871 q^{63} +2.59083 q^{65} +8.44610 q^{67} -3.32800 q^{69} +12.9170 q^{71} +7.71888 q^{73} +4.48464 q^{75} +1.00000 q^{77} -11.2346 q^{79} -5.68647 q^{81} +0.628980 q^{83} -7.85300 q^{85} -1.26171 q^{87} +4.05223 q^{89} -1.00000 q^{91} +12.8618 q^{93} +6.06983 q^{95} +12.1825 q^{97} +3.85871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * 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
$3$	2.61891	1.51203	0.756016	−	0.654554i	$-0.227143\pi$
$3$	2.61891	1.51203	0.756016	+	0.654554i	$0.227143\pi$
$4$	0	0
$5$	−2.59083	−1.15865	−0.579327	−	0.815095i	$-0.696685\pi$
$5$	−2.59083	−1.15865	−0.579327	+	0.815095i	$0.696685\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.85871	1.28624
$10$	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
$15$	−6.78516	−1.75192
$16$	0	0
$17$	3.03107	0.735143	0.367572	−	0.929995i	$-0.380189\pi$
$17$	3.03107	0.735143	0.367572	+	0.929995i	$0.380189\pi$
$18$	0	0
$19$	−2.34281	−0.537478	−0.268739	−	0.963213i	$-0.586607\pi$
$19$	−2.34281	−0.537478	−0.268739	+	0.963213i	$0.586607\pi$
$20$	0	0
$21$	2.61891	0.571494
$22$	0	0
$23$	−1.27075	−0.264971	−0.132485	−	0.991185i	$-0.542296\pi$
$23$	−1.27075	−0.264971	−0.132485	+	0.991185i	$0.542296\pi$
$24$	0	0
$25$	1.71240	0.342481
$26$	0	0
$27$	2.24890	0.432801
$28$	0	0
$29$	−0.481768	−0.0894621	−0.0447310	−	0.998999i	$-0.514243\pi$
$29$	−0.481768	−0.0894621	−0.0447310	+	0.998999i	$0.514243\pi$
$30$	0	0
$31$	4.91110	0.882060	0.441030	−	0.897492i	$-0.354613\pi$
$31$	4.91110	0.882060	0.441030	+	0.897492i	$0.354613\pi$
$32$	0	0
$33$	2.61891	0.455895
$34$	0	0
$35$	−2.59083	−0.437930
$36$	0	0
$37$	−1.63209	−0.268314	−0.134157	−	0.990960i	$-0.542833\pi$
$37$	−1.63209	−0.268314	−0.134157	+	0.990960i	$0.542833\pi$
$38$	0	0
$39$	−2.61891	−0.419362
$40$	0	0
$41$	1.58784	0.247979	0.123990	−	0.992284i	$-0.460431\pi$
$41$	1.58784	0.247979	0.123990	+	0.992284i	$0.460431\pi$
$42$	0	0
$43$	11.7462	1.79127	0.895637	−	0.444787i	$-0.146720\pi$
$43$	11.7462	1.79127	0.895637	+	0.444787i	$0.146720\pi$
$44$	0	0
$45$	−9.99728	−1.49031
$46$	0	0
$47$	1.27075	0.185359	0.0926793	−	0.995696i	$-0.470457\pi$
$47$	1.27075	0.185359	0.0926793	+	0.995696i	$0.470457\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.93812	1.11156
$52$	0	0
$53$	−1.51284	−0.207805	−0.103902	−	0.994587i	$-0.533133\pi$
$53$	−1.51284	−0.207805	−0.103902	+	0.994587i	$0.533133\pi$
$54$	0	0
$55$	−2.59083	−0.349348
$56$	0	0
$57$	−6.13563	−0.812684
$58$	0	0
$59$	7.19716	0.936991	0.468495	−	0.883466i	$-0.344796\pi$
$59$	7.19716	0.936991	0.468495	+	0.883466i	$0.344796\pi$
$60$	0	0
$61$	−1.26999	−0.162606	−0.0813028	−	0.996689i	$-0.525908\pi$
$61$	−1.26999	−0.162606	−0.0813028	+	0.996689i	$0.525908\pi$
$62$	0	0
$63$	3.85871	0.486152
$64$	0	0
$65$	2.59083	0.321353
$66$	0	0
$67$	8.44610	1.03186	0.515928	−	0.856632i	$-0.327447\pi$
$67$	8.44610	1.03186	0.515928	+	0.856632i	$0.327447\pi$
$68$	0	0
$69$	−3.32800	−0.400644
$70$	0	0
$71$	12.9170	1.53296	0.766481	−	0.642267i	$-0.222006\pi$
$71$	12.9170	1.53296	0.766481	+	0.642267i	$0.222006\pi$
$72$	0	0
$73$	7.71888	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$73$	7.71888	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$74$	0	0
$75$	4.48464	0.517841
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−11.2346	−1.26399	−0.631995	−	0.774972i	$-0.717764\pi$
$79$	−11.2346	−1.26399	−0.631995	+	0.774972i	$0.717764\pi$
$80$	0	0
$81$	−5.68647	−0.631830
$82$	0	0
$83$	0.628980	0.0690395	0.0345198	−	0.999404i	$-0.489010\pi$
$83$	0.628980	0.0690395	0.0345198	+	0.999404i	$0.489010\pi$
$84$	0	0
$85$	−7.85300	−0.851777
$86$	0	0
$87$	−1.26171	−0.135269
$88$	0	0
$89$	4.05223	0.429535	0.214768	−	0.976665i	$-0.431101\pi$
$89$	4.05223	0.429535	0.214768	+	0.976665i	$0.431101\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	12.8618	1.33370
$94$	0	0
$95$	6.06983	0.622752
$96$	0	0
$97$	12.1825	1.23694	0.618470	−	0.785808i	$-0.287753\pi$
$97$	12.1825	1.23694	0.618470	+	0.785808i	$0.287753\pi$
$98$	0	0
$99$	3.85871	0.387815
$100$	0	0
$101$	7.88802	0.784887	0.392443	−	0.919776i	$-0.371630\pi$
$101$	7.88802	0.784887	0.392443	+	0.919776i	$0.371630\pi$
$102$	0	0
$103$	−11.5117	−1.13428	−0.567142	−	0.823620i	$-0.691951\pi$
$103$	−11.5117	−1.13428	−0.567142	+	0.823620i	$0.691951\pi$
$104$	0	0
$105$	−6.78516	−0.662164
$106$	0	0
$107$	−5.15590	−0.498440	−0.249220	−	0.968447i	$-0.580174\pi$
$107$	−5.15590	−0.498440	−0.249220	+	0.968447i	$0.580174\pi$
$108$	0	0
$109$	11.0754	1.06083	0.530415	−	0.847738i	$-0.322036\pi$
$109$	11.0754	1.06083	0.530415	+	0.847738i	$0.322036\pi$
$110$	0	0
$111$	−4.27431	−0.405700
$112$	0	0
$113$	2.55056	0.239937	0.119968	−	0.992778i	$-0.461721\pi$
$113$	2.55056	0.239937	0.119968	+	0.992778i	$0.461721\pi$
$114$	0	0
$115$	3.29231	0.307009
$116$	0	0
$117$	−3.85871	−0.356738
$118$	0	0
$119$	3.03107	0.277858
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.15842	0.374952
$124$	0	0
$125$	8.51761	0.761838
$126$	0	0
$127$	−4.75148	−0.421625	−0.210813	−	0.977526i	$-0.567611\pi$
$127$	−4.75148	−0.421625	−0.210813	+	0.977526i	$0.567611\pi$
$128$	0	0
$129$	30.7622	2.70846
$130$	0	0
$131$	−2.02947	−0.177316	−0.0886579	−	0.996062i	$-0.528258\pi$
$131$	−2.02947	−0.177316	−0.0886579	+	0.996062i	$0.528258\pi$
$132$	0	0
$133$	−2.34281	−0.203148
$134$	0	0
$135$	−5.82652	−0.501467
$136$	0	0
$137$	2.00248	0.171083	0.0855415	−	0.996335i	$-0.472738\pi$
$137$	2.00248	0.171083	0.0855415	+	0.996335i	$0.472738\pi$
$138$	0	0
$139$	2.42429	0.205626	0.102813	−	0.994701i	$-0.467216\pi$
$139$	2.42429	0.205626	0.102813	+	0.994701i	$0.467216\pi$
$140$	0	0
$141$	3.32800	0.280268
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	1.24818	0.103656
$146$	0	0
$147$	2.61891	0.216004
$148$	0	0
$149$	19.1537	1.56913	0.784566	−	0.620045i	$-0.212886\pi$
$149$	19.1537	1.56913	0.784566	+	0.620045i	$0.212886\pi$
$150$	0	0
$151$	−19.2688	−1.56807	−0.784036	−	0.620715i	$-0.786842\pi$
$151$	−19.2688	−1.56807	−0.784036	+	0.620715i	$0.786842\pi$
$152$	0	0
$153$	11.6960	0.945570
$154$	0	0
$155$	−12.7238	−1.02200
$156$	0	0
$157$	3.81664	0.304601	0.152301	−	0.988334i	$-0.451332\pi$
$157$	3.81664	0.304601	0.152301	+	0.988334i	$0.451332\pi$
$158$	0	0
$159$	−3.96200	−0.314207
$160$	0	0
$161$	−1.27075	−0.100149
$162$	0	0
$163$	6.19751	0.485426	0.242713	−	0.970098i	$-0.421963\pi$
$163$	6.19751	0.485426	0.242713	+	0.970098i	$0.421963\pi$
$164$	0	0
$165$	−6.78516	−0.528224
$166$	0	0
$167$	0.391831	0.0303208	0.0151604	−	0.999885i	$-0.495174\pi$
$167$	0.391831	0.0303208	0.0151604	+	0.999885i	$0.495174\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−9.04025	−0.691325
$172$	0	0
$173$	−2.61007	−0.198440	−0.0992198	−	0.995066i	$-0.531635\pi$
$173$	−2.61007	−0.198440	−0.0992198	+	0.995066i	$0.531635\pi$
$174$	0	0
$175$	1.71240	0.129446
$176$	0	0
$177$	18.8488	1.41676
$178$	0	0
$179$	7.37023	0.550877	0.275438	−	0.961319i	$-0.411177\pi$
$179$	7.37023	0.550877	0.275438	+	0.961319i	$0.411177\pi$
$180$	0	0
$181$	−18.1494	−1.34904	−0.674518	−	0.738259i	$-0.735648\pi$
$181$	−18.1494	−1.34904	−0.674518	+	0.738259i	$0.735648\pi$
$182$	0	0
$183$	−3.32600	−0.245865
$184$	0	0
$185$	4.22847	0.310884
$186$	0	0
$187$	3.03107	0.221654
$188$	0	0
$189$	2.24890	0.163583
$190$	0	0
$191$	0.951315	0.0688347	0.0344174	−	0.999408i	$-0.489042\pi$
$191$	0.951315	0.0688347	0.0344174	+	0.999408i	$0.489042\pi$
$192$	0	0
$193$	4.73527	0.340852	0.170426	−	0.985370i	$-0.445486\pi$
$193$	4.73527	0.340852	0.170426	+	0.985370i	$0.445486\pi$
$194$	0	0
$195$	6.78516	0.485896
$196$	0	0
$197$	26.7229	1.90393	0.951965	−	0.306207i	$-0.0990599\pi$
$197$	26.7229	1.90393	0.951965	+	0.306207i	$0.0990599\pi$
$198$	0	0
$199$	11.6934	0.828920	0.414460	−	0.910068i	$-0.363971\pi$
$199$	11.6934	0.828920	0.414460	+	0.910068i	$0.363971\pi$
$200$	0	0
$201$	22.1196	1.56020
$202$	0	0
$203$	−0.481768	−0.0338135
$204$	0	0
$205$	−4.11383	−0.287322
$206$	0	0
$207$	−4.90348	−0.340815
$208$	0	0
$209$	−2.34281	−0.162056
$210$	0	0
$211$	2.45785	0.169205	0.0846027	−	0.996415i	$-0.473038\pi$
$211$	2.45785	0.169205	0.0846027	+	0.996415i	$0.473038\pi$
$212$	0	0
$213$	33.8284	2.31789
$214$	0	0
$215$	−30.4323	−2.07547
$216$	0	0
$217$	4.91110	0.333387
$218$	0	0
$219$	20.2151	1.36601
$220$	0	0
$221$	−3.03107	−0.203892
$222$	0	0
$223$	−18.0588	−1.20931	−0.604653	−	0.796489i	$-0.706688\pi$
$223$	−18.0588	−1.20931	−0.604653	+	0.796489i	$0.706688\pi$
$224$	0	0
$225$	6.60768	0.440512
$226$	0	0
$227$	14.1413	0.938589	0.469294	−	0.883042i	$-0.344508\pi$
$227$	14.1413	0.938589	0.469294	+	0.883042i	$0.344508\pi$
$228$	0	0
$229$	−21.0681	−1.39222	−0.696109	−	0.717937i	$-0.745087\pi$
$229$	−21.0681	−1.39222	−0.696109	+	0.717937i	$0.745087\pi$
$230$	0	0
$231$	2.61891	0.172312
$232$	0	0
$233$	−28.3509	−1.85733	−0.928666	−	0.370917i	$-0.879043\pi$
$233$	−28.3509	−1.85733	−0.928666	+	0.370917i	$0.879043\pi$
$234$	0	0
$235$	−3.29231	−0.214767
$236$	0	0
$237$	−29.4224	−1.91119
$238$	0	0
$239$	7.95182	0.514361	0.257180	−	0.966363i	$-0.417207\pi$
$239$	7.95182	0.514361	0.257180	+	0.966363i	$0.417207\pi$
$240$	0	0
$241$	8.30049	0.534681	0.267341	−	0.963602i	$-0.413855\pi$
$241$	8.30049	0.534681	0.267341	+	0.963602i	$0.413855\pi$
$242$	0	0
$243$	−21.6391	−1.38815
$244$	0	0
$245$	−2.59083	−0.165522
$246$	0	0
$247$	2.34281	0.149070
$248$	0	0
$249$	1.64724	0.104390
$250$	0	0
$251$	−5.97498	−0.377138	−0.188569	−	0.982060i	$-0.560385\pi$
$251$	−5.97498	−0.377138	−0.188569	+	0.982060i	$0.560385\pi$
$252$	0	0
$253$	−1.27075	−0.0798916
$254$	0	0
$255$	−20.5663	−1.28791
$256$	0	0
$257$	−14.5749	−0.909155	−0.454577	−	0.890707i	$-0.650210\pi$
$257$	−14.5749	−0.909155	−0.454577	+	0.890707i	$0.650210\pi$
$258$	0	0
$259$	−1.63209	−0.101413
$260$	0	0
$261$	−1.85901	−0.115070
$262$	0	0
$263$	−0.332630	−0.0205109	−0.0102554	−	0.999947i	$-0.503264\pi$
$263$	−0.332630	−0.0205109	−0.0102554	+	0.999947i	$0.503264\pi$
$264$	0	0
$265$	3.91952	0.240774
$266$	0	0
$267$	10.6124	0.649471
$268$	0	0
$269$	25.0787	1.52907	0.764537	−	0.644580i	$-0.222968\pi$
$269$	25.0787	1.52907	0.764537	+	0.644580i	$0.222968\pi$
$270$	0	0
$271$	−15.8679	−0.963908	−0.481954	−	0.876197i	$-0.660073\pi$
$271$	−15.8679	−0.963908	−0.481954	+	0.876197i	$0.660073\pi$
$272$	0	0
$273$	−2.61891	−0.158504
$274$	0	0
$275$	1.71240	0.103262
$276$	0	0
$277$	13.4963	0.810916	0.405458	−	0.914114i	$-0.367112\pi$
$277$	13.4963	0.810916	0.405458	+	0.914114i	$0.367112\pi$
$278$	0	0
$279$	18.9505	1.13454
$280$	0	0
$281$	−5.18594	−0.309367	−0.154684	−	0.987964i	$-0.549436\pi$
$281$	−5.18594	−0.309367	−0.154684	+	0.987964i	$0.549436\pi$
$282$	0	0
$283$	0.0682025	0.00405422	0.00202711	−	0.999998i	$-0.499355\pi$
$283$	0.0682025	0.00405422	0.00202711	+	0.999998i	$0.499355\pi$
$284$	0	0
$285$	15.8964	0.941620
$286$	0	0
$287$	1.58784	0.0937273
$288$	0	0
$289$	−7.81259	−0.459564
$290$	0	0
$291$	31.9048	1.87029
$292$	0	0
$293$	15.4619	0.903295	0.451648	−	0.892196i	$-0.350836\pi$
$293$	15.4619	0.903295	0.451648	+	0.892196i	$0.350836\pi$
$294$	0	0
$295$	−18.6466	−1.08565
$296$	0	0
$297$	2.24890	0.130494
$298$	0	0
$299$	1.27075	0.0734896
$300$	0	0
$301$	11.7462	0.677038
$302$	0	0
$303$	20.6580	1.18677
$304$	0	0
$305$	3.29033	0.188404
$306$	0	0
$307$	−5.21734	−0.297769	−0.148885	−	0.988855i	$-0.547568\pi$
$307$	−5.21734	−0.297769	−0.148885	+	0.988855i	$0.547568\pi$
$308$	0	0
$309$	−30.1482	−1.71507
$310$	0	0
$311$	15.1168	0.857197	0.428599	−	0.903495i	$-0.359007\pi$
$311$	15.1168	0.857197	0.428599	+	0.903495i	$0.359007\pi$
$312$	0	0
$313$	32.1992	1.82001	0.910004	−	0.414599i	$-0.136078\pi$
$313$	32.1992	1.82001	0.910004	+	0.414599i	$0.136078\pi$
$314$	0	0
$315$	−9.99728	−0.563283
$316$	0	0
$317$	−24.9010	−1.39858	−0.699291	−	0.714837i	$-0.746501\pi$
$317$	−24.9010	−1.39858	−0.699291	+	0.714837i	$0.746501\pi$
$318$	0	0
$319$	−0.481768	−0.0269738
$320$	0	0
$321$	−13.5029	−0.753656
$322$	0	0
$323$	−7.10124	−0.395124
$324$	0	0
$325$	−1.71240	−0.0949870
$326$	0	0
$327$	29.0055	1.60401
$328$	0	0
$329$	1.27075	0.0700589
$330$	0	0
$331$	11.5124	0.632780	0.316390	−	0.948629i	$-0.397529\pi$
$331$	11.5124	0.632780	0.316390	+	0.948629i	$0.397529\pi$
$332$	0	0
$333$	−6.29778	−0.345116
$334$	0	0
$335$	−21.8824	−1.19556
$336$	0	0
$337$	−13.3808	−0.728899	−0.364449	−	0.931223i	$-0.618743\pi$
$337$	−13.3808	−0.728899	−0.364449	+	0.931223i	$0.618743\pi$
$338$	0	0
$339$	6.67970	0.362792
$340$	0	0
$341$	4.91110	0.265951
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.62228	0.464208
$346$	0	0
$347$	−19.2642	−1.03415	−0.517077	−	0.855939i	$-0.672980\pi$
$347$	−19.2642	−1.03415	−0.517077	+	0.855939i	$0.672980\pi$
$348$	0	0
$349$	9.97922	0.534175	0.267088	−	0.963672i	$-0.413939\pi$
$349$	9.97922	0.534175	0.267088	+	0.963672i	$0.413939\pi$
$350$	0	0
$351$	−2.24890	−0.120037
$352$	0	0
$353$	31.1436	1.65761	0.828803	−	0.559540i	$-0.189022\pi$
$353$	31.1436	1.65761	0.828803	+	0.559540i	$0.189022\pi$
$354$	0	0
$355$	−33.4657	−1.77617
$356$	0	0
$357$	7.93812	0.420130
$358$	0	0
$359$	17.4186	0.919316	0.459658	−	0.888096i	$-0.347972\pi$
$359$	17.4186	0.919316	0.459658	+	0.888096i	$0.347972\pi$
$360$	0	0
$361$	−13.5112	−0.711117
$362$	0	0
$363$	2.61891	0.137457
$364$	0	0
$365$	−19.9983	−1.04676
$366$	0	0
$367$	3.16615	0.165272	0.0826359	−	0.996580i	$-0.473666\pi$
$367$	3.16615	0.165272	0.0826359	+	0.996580i	$0.473666\pi$
$368$	0	0
$369$	6.12702	0.318960
$370$	0	0
$371$	−1.51284	−0.0785429
$372$	0	0
$373$	7.29358	0.377647	0.188824	−	0.982011i	$-0.439533\pi$
$373$	7.29358	0.377647	0.188824	+	0.982011i	$0.439533\pi$
$374$	0	0
$375$	22.3069	1.15192
$376$	0	0
$377$	0.481768	0.0248123
$378$	0	0
$379$	27.2671	1.40062	0.700309	−	0.713840i	$-0.253045\pi$
$379$	27.2671	1.40062	0.700309	+	0.713840i	$0.253045\pi$
$380$	0	0
$381$	−12.4437	−0.637511
$382$	0	0
$383$	−12.6391	−0.645827	−0.322914	−	0.946428i	$-0.604662\pi$
$383$	−12.6391	−0.645827	−0.322914	+	0.946428i	$0.604662\pi$
$384$	0	0
$385$	−2.59083	−0.132041
$386$	0	0
$387$	45.3251	2.30400
$388$	0	0
$389$	14.9896	0.760004	0.380002	−	0.924986i	$-0.375923\pi$
$389$	14.9896	0.760004	0.380002	+	0.924986i	$0.375923\pi$
$390$	0	0
$391$	−3.85175	−0.194791
$392$	0	0
$393$	−5.31501	−0.268107
$394$	0	0
$395$	29.1069	1.46453
$396$	0	0
$397$	10.2960	0.516743	0.258372	−	0.966046i	$-0.416814\pi$
$397$	10.2960	0.516743	0.258372	+	0.966046i	$0.416814\pi$
$398$	0	0
$399$	−6.13563	−0.307166
$400$	0	0
$401$	−15.7484	−0.786438	−0.393219	−	0.919445i	$-0.628639\pi$
$401$	−15.7484	−0.786438	−0.393219	+	0.919445i	$0.628639\pi$
$402$	0	0
$403$	−4.91110	−0.244640
$404$	0	0
$405$	14.7327	0.732072
$406$	0	0
$407$	−1.63209	−0.0808998
$408$	0	0
$409$	37.2859	1.84367	0.921835	−	0.387582i	$-0.126690\pi$
$409$	37.2859	1.84367	0.921835	+	0.387582i	$0.126690\pi$
$410$	0	0
$411$	5.24431	0.258683
$412$	0	0
$413$	7.19716	0.354149
$414$	0	0
$415$	−1.62958	−0.0799929
$416$	0	0
$417$	6.34901	0.310912
$418$	0	0
$419$	22.1180	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$419$	22.1180	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$420$	0	0
$421$	−32.2592	−1.57222	−0.786109	−	0.618088i	$-0.787908\pi$
$421$	−32.2592	−1.57222	−0.786109	+	0.618088i	$0.787908\pi$
$422$	0	0
$423$	4.90348	0.238415
$424$	0	0
$425$	5.19042	0.251772
$426$	0	0
$427$	−1.26999	−0.0614592
$428$	0	0
$429$	−2.61891	−0.126442
$430$	0	0
$431$	−27.3251	−1.31620	−0.658102	−	0.752929i	$-0.728640\pi$
$431$	−27.3251	−1.31620	−0.658102	+	0.752929i	$0.728640\pi$
$432$	0	0
$433$	29.0723	1.39712	0.698562	−	0.715549i	$-0.253824\pi$
$433$	29.0723	1.39712	0.698562	+	0.715549i	$0.253824\pi$
$434$	0	0
$435$	3.26888	0.156731
$436$	0	0
$437$	2.97714	0.142416
$438$	0	0
$439$	2.49827	0.119236	0.0596180	−	0.998221i	$-0.481012\pi$
$439$	2.49827	0.119236	0.0596180	+	0.998221i	$0.481012\pi$
$440$	0	0
$441$	3.85871	0.183748
$442$	0	0
$443$	3.02974	0.143947	0.0719735	−	0.997407i	$-0.477070\pi$
$443$	3.02974	0.143947	0.0719735	+	0.997407i	$0.477070\pi$
$444$	0	0
$445$	−10.4986	−0.497683
$446$	0	0
$447$	50.1619	2.37258
$448$	0	0
$449$	−2.56494	−0.121047	−0.0605235	−	0.998167i	$-0.519277\pi$
$449$	−2.56494	−0.121047	−0.0605235	+	0.998167i	$0.519277\pi$
$450$	0	0
$451$	1.58784	0.0747685
$452$	0	0
$453$	−50.4633	−2.37097
$454$	0	0
$455$	2.59083	0.121460
$456$	0	0
$457$	15.6689	0.732958	0.366479	−	0.930426i	$-0.380563\pi$
$457$	15.6689	0.732958	0.366479	+	0.930426i	$0.380563\pi$
$458$	0	0
$459$	6.81658	0.318171
$460$	0	0
$461$	16.1167	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$461$	16.1167	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$462$	0	0
$463$	−7.13079	−0.331396	−0.165698	−	0.986177i	$-0.552988\pi$
$463$	−7.13079	−0.331396	−0.165698	+	0.986177i	$0.552988\pi$
$464$	0	0
$465$	−33.3226	−1.54530
$466$	0	0
$467$	−0.266202	−0.0123184	−0.00615919	−	0.999981i	$-0.501961\pi$
$467$	−0.266202	−0.0123184	−0.00615919	+	0.999981i	$0.501961\pi$
$468$	0	0
$469$	8.44610	0.390005
$470$	0	0
$471$	9.99546	0.460566
$472$	0	0
$473$	11.7462	0.540089
$474$	0	0
$475$	−4.01184	−0.184076
$476$	0	0
$477$	−5.83763	−0.267287
$478$	0	0
$479$	−34.6862	−1.58485	−0.792426	−	0.609968i	$-0.791182\pi$
$479$	−34.6862	−1.58485	−0.792426	+	0.609968i	$0.791182\pi$
$480$	0	0
$481$	1.63209	0.0744170
$482$	0	0
$483$	−3.32800	−0.151429
$484$	0	0
$485$	−31.5627	−1.43319
$486$	0	0
$487$	−4.80410	−0.217695	−0.108847	−	0.994058i	$-0.534716\pi$
$487$	−4.80410	−0.217695	−0.108847	+	0.994058i	$0.534716\pi$
$488$	0	0
$489$	16.2307	0.733980
$490$	0	0
$491$	36.1932	1.63338	0.816689	−	0.577078i	$-0.195807\pi$
$491$	36.1932	1.63338	0.816689	+	0.577078i	$0.195807\pi$
$492$	0	0
$493$	−1.46027	−0.0657675
$494$	0	0
$495$	−9.99728	−0.449344
$496$	0	0
$497$	12.9170	0.579405
$498$	0	0
$499$	−8.85884	−0.396576	−0.198288	−	0.980144i	$-0.563538\pi$
$499$	−8.85884	−0.396576	−0.198288	+	0.980144i	$0.563538\pi$
$500$	0	0
$501$	1.02617	0.0458459
$502$	0	0
$503$	11.6780	0.520697	0.260348	−	0.965515i	$-0.416163\pi$
$503$	11.6780	0.520697	0.260348	+	0.965515i	$0.416163\pi$
$504$	0	0
$505$	−20.4365	−0.909413
$506$	0	0
$507$	2.61891	0.116310
$508$	0	0
$509$	−6.15035	−0.272609	−0.136305	−	0.990667i	$-0.543523\pi$
$509$	−6.15035	−0.272609	−0.136305	+	0.990667i	$0.543523\pi$
$510$	0	0
$511$	7.71888	0.341463
$512$	0	0
$513$	−5.26875	−0.232621
$514$	0	0
$515$	29.8249	1.31424
$516$	0	0
$517$	1.27075	0.0558877
$518$	0	0
$519$	−6.83554	−0.300047
$520$	0	0
$521$	−1.28179	−0.0561561	−0.0280781	−	0.999606i	$-0.508939\pi$
$521$	−1.28179	−0.0561561	−0.0280781	+	0.999606i	$0.508939\pi$
$522$	0	0
$523$	15.6500	0.684327	0.342163	−	0.939640i	$-0.388840\pi$
$523$	15.6500	0.684327	0.342163	+	0.939640i	$0.388840\pi$
$524$	0	0
$525$	4.48464	0.195726
$526$	0	0
$527$	14.8859	0.648441
$528$	0	0
$529$	−21.3852	−0.929791
$530$	0	0
$531$	27.7718	1.20519
$532$	0	0
$533$	−1.58784	−0.0687770
$534$	0	0
$535$	13.3581	0.577519
$536$	0	0
$537$	19.3020	0.832943
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	10.7653	0.462838	0.231419	−	0.972854i	$-0.425663\pi$
$541$	10.7653	0.462838	0.231419	+	0.972854i	$0.425663\pi$
$542$	0	0
$543$	−47.5318	−2.03978
$544$	0	0
$545$	−28.6944	−1.22913
$546$	0	0
$547$	37.3786	1.59819	0.799096	−	0.601203i	$-0.205312\pi$
$547$	37.3786	1.59819	0.799096	+	0.601203i	$0.205312\pi$
$548$	0	0
$549$	−4.90053	−0.209150
$550$	0	0
$551$	1.12869	0.0480839
$552$	0	0
$553$	−11.2346	−0.477743
$554$	0	0
$555$	11.0740	0.470066
$556$	0	0
$557$	−17.6906	−0.749577	−0.374788	−	0.927110i	$-0.622285\pi$
$557$	−17.6906	−0.749577	−0.374788	+	0.927110i	$0.622285\pi$
$558$	0	0
$559$	−11.7462	−0.496810
$560$	0	0
$561$	7.93812	0.335148
$562$	0	0
$563$	−0.00366526	−0.000154472 0	−7.72361e−5	−	1.00000i	$-0.500025\pi$
$563$	−0.00366526	−0.000154472 0	−7.72361e−5		1.00000i	$0.500025\pi$
$564$	0	0
$565$	−6.60807	−0.278004
$566$	0	0
$567$	−5.68647	−0.238809
$568$	0	0
$569$	7.38211	0.309474	0.154737	−	0.987956i	$-0.450547\pi$
$569$	7.38211	0.309474	0.154737	+	0.987956i	$0.450547\pi$
$570$	0	0
$571$	−1.38230	−0.0578477	−0.0289238	−	0.999582i	$-0.509208\pi$
$571$	−1.38230	−0.0578477	−0.0289238	+	0.999582i	$0.509208\pi$
$572$	0	0
$573$	2.49141	0.104080
$574$	0	0
$575$	−2.17604	−0.0907473
$576$	0	0
$577$	−34.3436	−1.42974	−0.714871	−	0.699256i	$-0.753515\pi$
$577$	−34.3436	−1.42974	−0.714871	+	0.699256i	$0.753515\pi$
$578$	0	0
$579$	12.4013	0.515379
$580$	0	0
$581$	0.628980	0.0260945
$582$	0	0
$583$	−1.51284	−0.0626555
$584$	0	0
$585$	9.99728	0.413336
$586$	0	0
$587$	−8.31283	−0.343107	−0.171554	−	0.985175i	$-0.554879\pi$
$587$	−8.31283	−0.343107	−0.171554	+	0.985175i	$0.554879\pi$
$588$	0	0
$589$	−11.5058	−0.474088
$590$	0	0
$591$	69.9851	2.87880
$592$	0	0
$593$	22.5314	0.925255	0.462628	−	0.886553i	$-0.346907\pi$
$593$	22.5314	0.925255	0.462628	+	0.886553i	$0.346907\pi$
$594$	0	0
$595$	−7.85300	−0.321942
$596$	0	0
$597$	30.6239	1.25335
$598$	0	0
$599$	−1.74860	−0.0714460	−0.0357230	−	0.999362i	$-0.511373\pi$
$599$	−1.74860	−0.0714460	−0.0357230	+	0.999362i	$0.511373\pi$
$600$	0	0
$601$	−21.2223	−0.865677	−0.432838	−	0.901472i	$-0.642488\pi$
$601$	−21.2223	−0.865677	−0.432838	+	0.901472i	$0.642488\pi$
$602$	0	0
$603$	32.5911	1.32721
$604$	0	0
$605$	−2.59083	−0.105332
$606$	0	0
$607$	1.14609	0.0465186	0.0232593	−	0.999729i	$-0.492596\pi$
$607$	1.14609	0.0465186	0.0232593	+	0.999729i	$0.492596\pi$
$608$	0	0
$609$	−1.26171	−0.0511270
$610$	0	0
$611$	−1.27075	−0.0514092
$612$	0	0
$613$	−2.86633	−0.115770	−0.0578850	−	0.998323i	$-0.518436\pi$
$613$	−2.86633	−0.115770	−0.0578850	+	0.998323i	$0.518436\pi$
$614$	0	0
$615$	−10.7738	−0.434440
$616$	0	0
$617$	−37.4978	−1.50960	−0.754801	−	0.655953i	$-0.772267\pi$
$617$	−37.4978	−1.50960	−0.754801	+	0.655953i	$0.772267\pi$
$618$	0	0
$619$	−11.7512	−0.472320	−0.236160	−	0.971714i	$-0.575889\pi$
$619$	−11.7512	−0.472320	−0.236160	+	0.971714i	$0.575889\pi$
$620$	0	0
$621$	−2.85780	−0.114679
$622$	0	0
$623$	4.05223	0.162349
$624$	0	0
$625$	−30.6297	−1.22519
$626$	0	0
$627$	−6.13563	−0.245033
$628$	0	0
$629$	−4.94699	−0.197250
$630$	0	0
$631$	−14.1570	−0.563583	−0.281791	−	0.959476i	$-0.590929\pi$
$631$	−14.1570	−0.563583	−0.281791	+	0.959476i	$0.590929\pi$
$632$	0	0
$633$	6.43690	0.255844
$634$	0	0
$635$	12.3103	0.488518
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	49.8429	1.97175
$640$	0	0
$641$	20.6175	0.814344	0.407172	−	0.913352i	$-0.366515\pi$
$641$	20.6175	0.814344	0.407172	+	0.913352i	$0.366515\pi$
$642$	0	0
$643$	9.39593	0.370539	0.185270	−	0.982688i	$-0.440684\pi$
$643$	9.39593	0.370539	0.185270	+	0.982688i	$0.440684\pi$
$644$	0	0
$645$	−79.6997	−3.13817
$646$	0	0
$647$	−43.3402	−1.70388	−0.851940	−	0.523640i	$-0.824574\pi$
$647$	−43.3402	−1.70388	−0.851940	+	0.523640i	$0.824574\pi$
$648$	0	0
$649$	7.19716	0.282513
$650$	0	0
$651$	12.8618	0.504092
$652$	0	0
$653$	−2.83326	−0.110874	−0.0554370	−	0.998462i	$-0.517655\pi$
$653$	−2.83326	−0.110874	−0.0554370	+	0.998462i	$0.517655\pi$
$654$	0	0
$655$	5.25802	0.205448
$656$	0	0
$657$	29.7849	1.16202
$658$	0	0
$659$	−13.5128	−0.526384	−0.263192	−	0.964743i	$-0.584775\pi$
$659$	−13.5128	−0.526384	−0.263192	+	0.964743i	$0.584775\pi$
$660$	0	0
$661$	−13.6327	−0.530252	−0.265126	−	0.964214i	$-0.585414\pi$
$661$	−13.6327	−0.530252	−0.265126	+	0.964214i	$0.585414\pi$
$662$	0	0
$663$	−7.93812	−0.308291
$664$	0	0
$665$	6.06983	0.235378
$666$	0	0
$667$	0.612209	0.0237048
$668$	0	0
$669$	−47.2944	−1.82851
$670$	0	0
$671$	−1.26999	−0.0490275
$672$	0	0
$673$	−10.1606	−0.391661	−0.195831	−	0.980638i	$-0.562740\pi$
$673$	−10.1606	−0.391661	−0.195831	+	0.980638i	$0.562740\pi$
$674$	0	0
$675$	3.85102	0.148226
$676$	0	0
$677$	12.8077	0.492238	0.246119	−	0.969240i	$-0.420845\pi$
$677$	12.8077	0.492238	0.246119	+	0.969240i	$0.420845\pi$
$678$	0	0
$679$	12.1825	0.467520
$680$	0	0
$681$	37.0348	1.41918
$682$	0	0
$683$	7.51081	0.287393	0.143697	−	0.989622i	$-0.454101\pi$
$683$	7.51081	0.287393	0.143697	+	0.989622i	$0.454101\pi$
$684$	0	0
$685$	−5.18807	−0.198226
$686$	0	0
$687$	−55.1755	−2.10508
$688$	0	0
$689$	1.51284	0.0576347
$690$	0	0
$691$	−11.2939	−0.429640	−0.214820	−	0.976654i	$-0.568917\pi$
$691$	−11.2939	−0.429640	−0.214820	+	0.976654i	$0.568917\pi$
$692$	0	0
$693$	3.85871	0.146580
$694$	0	0
$695$	−6.28092	−0.238249
$696$	0	0
$697$	4.81286	0.182300
$698$	0	0
$699$	−74.2487	−2.80834
$700$	0	0
$701$	4.24295	0.160254	0.0801270	−	0.996785i	$-0.474467\pi$
$701$	4.24295	0.160254	0.0801270	+	0.996785i	$0.474467\pi$
$702$	0	0
$703$	3.82369	0.144213
$704$	0	0
$705$	−8.62228	−0.324734
$706$	0	0
$707$	7.88802	0.296659
$708$	0	0
$709$	−20.8514	−0.783092	−0.391546	−	0.920159i	$-0.628060\pi$
$709$	−20.8514	−0.783092	−0.391546	+	0.920159i	$0.628060\pi$
$710$	0	0
$711$	−43.3511	−1.62579
$712$	0	0
$713$	−6.24081	−0.233720
$714$	0	0
$715$	2.59083	0.0968916
$716$	0	0
$717$	20.8251	0.777729
$718$	0	0
$719$	−29.0994	−1.08522	−0.542612	−	0.839983i	$-0.682565\pi$
$719$	−29.0994	−1.08522	−0.542612	+	0.839983i	$0.682565\pi$
$720$	0	0
$721$	−11.5117	−0.428719
$722$	0	0
$723$	21.7383	0.808455
$724$	0	0
$725$	−0.824981	−0.0306390
$726$	0	0
$727$	1.52612	0.0566006	0.0283003	−	0.999599i	$-0.490991\pi$
$727$	1.52612	0.0566006	0.0283003	+	0.999599i	$0.490991\pi$
$728$	0	0
$729$	−39.6115	−1.46709
$730$	0	0
$731$	35.6035	1.31684
$732$	0	0
$733$	−33.9214	−1.25292	−0.626459	−	0.779455i	$-0.715496\pi$
$733$	−33.9214	−1.25292	−0.626459	+	0.779455i	$0.715496\pi$
$734$	0	0
$735$	−6.78516	−0.250275
$736$	0	0
$737$	8.44610	0.311116
$738$	0	0
$739$	1.85422	0.0682084	0.0341042	−	0.999418i	$-0.489142\pi$
$739$	1.85422	0.0682084	0.0341042	+	0.999418i	$0.489142\pi$
$740$	0	0
$741$	6.13563	0.225398
$742$	0	0
$743$	−42.3014	−1.55189	−0.775945	−	0.630801i	$-0.782726\pi$
$743$	−42.3014	−1.55189	−0.775945	+	0.630801i	$0.782726\pi$
$744$	0	0
$745$	−49.6240	−1.81808
$746$	0	0
$747$	2.42705	0.0888012
$748$	0	0
$749$	−5.15590	−0.188392
$750$	0	0
$751$	7.50196	0.273750	0.136875	−	0.990588i	$-0.456294\pi$
$751$	7.50196	0.273750	0.136875	+	0.990588i	$0.456294\pi$
$752$	0	0
$753$	−15.6480	−0.570244
$754$	0	0
$755$	49.9222	1.81685
$756$	0	0
$757$	−52.8941	−1.92247	−0.961235	−	0.275731i	$-0.911080\pi$
$757$	−52.8941	−1.92247	−0.961235	+	0.275731i	$0.911080\pi$
$758$	0	0
$759$	−3.32800	−0.120799
$760$	0	0
$761$	1.34330	0.0486946	0.0243473	−	0.999704i	$-0.492249\pi$
$761$	1.34330	0.0486946	0.0243473	+	0.999704i	$0.492249\pi$
$762$	0	0
$763$	11.0754	0.400956
$764$	0	0
$765$	−30.3025	−1.09559
$766$	0	0
$767$	−7.19716	−0.259874
$768$	0	0
$769$	−26.8874	−0.969584	−0.484792	−	0.874630i	$-0.661105\pi$
$769$	−26.8874	−0.969584	−0.484792	+	0.874630i	$0.661105\pi$
$770$	0	0
$771$	−38.1703	−1.37467
$772$	0	0
$773$	−1.99400	−0.0717192	−0.0358596	−	0.999357i	$-0.511417\pi$
$773$	−1.99400	−0.0717192	−0.0358596	+	0.999357i	$0.511417\pi$
$774$	0	0
$775$	8.40979	0.302089
$776$	0	0
$777$	−4.27431	−0.153340
$778$	0	0
$779$	−3.72001	−0.133283
$780$	0	0
$781$	12.9170	0.462205
$782$	0	0
$783$	−1.08345	−0.0387193
$784$	0	0
$785$	−9.88827	−0.352927
$786$	0	0
$787$	−46.7034	−1.66479	−0.832397	−	0.554179i	$-0.813032\pi$
$787$	−46.7034	−1.66479	−0.832397	+	0.554179i	$0.813032\pi$
$788$	0	0
$789$	−0.871131	−0.0310131
$790$	0	0
$791$	2.55056	0.0906876
$792$	0	0
$793$	1.26999	0.0450987
$794$	0	0
$795$	10.2649	0.364058
$796$	0	0
$797$	2.56153	0.0907342	0.0453671	−	0.998970i	$-0.485554\pi$
$797$	2.56153	0.0907342	0.0453671	+	0.998970i	$0.485554\pi$
$798$	0	0
$799$	3.85175	0.136265
$800$	0	0
$801$	15.6364	0.552485
$802$	0	0
$803$	7.71888	0.272393
$804$	0	0
$805$	3.29231	0.116039
$806$	0	0
$807$	65.6789	2.31201
$808$	0	0
$809$	−10.6653	−0.374974	−0.187487	−	0.982267i	$-0.560034\pi$
$809$	−10.6653	−0.374974	−0.187487	+	0.982267i	$0.560034\pi$
$810$	0	0
$811$	46.2424	1.62379	0.811895	−	0.583803i	$-0.198436\pi$
$811$	46.2424	1.62379	0.811895	+	0.583803i	$0.198436\pi$
$812$	0	0
$813$	−41.5567	−1.45746
$814$	0	0
$815$	−16.0567	−0.562441
$816$	0	0
$817$	−27.5191	−0.962771
$818$	0	0
$819$	−3.85871	−0.134834
$820$	0	0
$821$	6.91694	0.241403	0.120701	−	0.992689i	$-0.461486\pi$
$821$	6.91694	0.241403	0.120701	+	0.992689i	$0.461486\pi$
$822$	0	0
$823$	10.6104	0.369854	0.184927	−	0.982752i	$-0.440795\pi$
$823$	10.6104	0.369854	0.184927	+	0.982752i	$0.440795\pi$
$824$	0	0
$825$	4.48464	0.156135
$826$	0	0
$827$	9.08271	0.315837	0.157918	−	0.987452i	$-0.449522\pi$
$827$	9.08271	0.315837	0.157918	+	0.987452i	$0.449522\pi$
$828$	0	0
$829$	−32.2055	−1.11854	−0.559271	−	0.828985i	$-0.688919\pi$
$829$	−32.2055	−1.11854	−0.559271	+	0.828985i	$0.688919\pi$
$830$	0	0
$831$	35.3457	1.22613
$832$	0	0
$833$	3.03107	0.105020
$834$	0	0
$835$	−1.01517	−0.0351313
$836$	0	0
$837$	11.0446	0.381757
$838$	0	0
$839$	11.4661	0.395853	0.197927	−	0.980217i	$-0.436579\pi$
$839$	11.4661	0.395853	0.197927	+	0.980217i	$0.436579\pi$
$840$	0	0
$841$	−28.7679	−0.991997
$842$	0	0
$843$	−13.5815	−0.467773
$844$	0	0
$845$	−2.59083	−0.0891273
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	0.178617	0.00613010
$850$	0	0
$851$	2.07399	0.0710954
$852$	0	0
$853$	−15.3632	−0.526025	−0.263013	−	0.964792i	$-0.584716\pi$
$853$	−15.3632	−0.526025	−0.263013	+	0.964792i	$0.584716\pi$
$854$	0	0
$855$	23.4218	0.801007
$856$	0	0
$857$	18.2912	0.624817	0.312409	−	0.949948i	$-0.398864\pi$
$857$	18.2912	0.624817	0.312409	+	0.949948i	$0.398864\pi$
$858$	0	0
$859$	−37.9082	−1.29341	−0.646706	−	0.762739i	$-0.723854\pi$
$859$	−37.9082	−1.29341	−0.646706	+	0.762739i	$0.723854\pi$
$860$	0	0
$861$	4.15842	0.141719
$862$	0	0
$863$	−15.8291	−0.538830	−0.269415	−	0.963024i	$-0.586830\pi$
$863$	−15.8291	−0.538830	−0.269415	+	0.963024i	$0.586830\pi$
$864$	0	0
$865$	6.76224	0.229923
$866$	0	0
$867$	−20.4605	−0.694875
$868$	0	0
$869$	−11.2346	−0.381107
$870$	0	0
$871$	−8.44610	−0.286185
$872$	0	0
$873$	47.0086	1.59100
$874$	0	0
$875$	8.51761	0.287948
$876$	0	0
$877$	−24.0375	−0.811687	−0.405844	−	0.913943i	$-0.633022\pi$
$877$	−24.0375	−0.811687	−0.405844	+	0.913943i	$0.633022\pi$
$878$	0	0
$879$	40.4935	1.36581
$880$	0	0
$881$	−9.62428	−0.324250	−0.162125	−	0.986770i	$-0.551835\pi$
$881$	−9.62428	−0.324250	−0.162125	+	0.986770i	$0.551835\pi$
$882$	0	0
$883$	28.6983	0.965774	0.482887	−	0.875683i	$-0.339588\pi$
$883$	28.6983	0.965774	0.482887	+	0.875683i	$0.339588\pi$
$884$	0	0
$885$	−48.8339	−1.64153
$886$	0	0
$887$	13.3521	0.448321	0.224161	−	0.974552i	$-0.428036\pi$
$887$	13.3521	0.448321	0.224161	+	0.974552i	$0.428036\pi$
$888$	0	0
$889$	−4.75148	−0.159359
$890$	0	0
$891$	−5.68647	−0.190504
$892$	0	0
$893$	−2.97714	−0.0996262
$894$	0	0
$895$	−19.0950	−0.638276
$896$	0	0
$897$	3.32800	0.111119
$898$	0	0
$899$	−2.36601	−0.0789109
$900$	0	0
$901$	−4.58554	−0.152766
$902$	0	0
$903$	30.7622	1.02370
$904$	0	0
$905$	47.0221	1.56307
$906$	0	0
$907$	21.6965	0.720421	0.360211	−	0.932871i	$-0.382705\pi$
$907$	21.6965	0.720421	0.360211	+	0.932871i	$0.382705\pi$
$908$	0	0
$909$	30.4376	1.00955
$910$	0	0
$911$	3.26878	0.108300	0.0541498	−	0.998533i	$-0.482755\pi$
$911$	3.26878	0.108300	0.0541498	+	0.998533i	$0.482755\pi$
$912$	0	0
$913$	0.628980	0.0208162
$914$	0	0
$915$	8.61710	0.284872
$916$	0	0
$917$	−2.02947	−0.0670191
$918$	0	0
$919$	−32.7346	−1.07982	−0.539908	−	0.841724i	$-0.681541\pi$
$919$	−32.7346	−1.07982	−0.539908	+	0.841724i	$0.681541\pi$
$920$	0	0
$921$	−13.6638	−0.450236
$922$	0	0
$923$	−12.9170	−0.425167
$924$	0	0
$925$	−2.79480	−0.0918925
$926$	0	0
$927$	−44.4205	−1.45896
$928$	0	0
$929$	56.5921	1.85673	0.928363	−	0.371676i	$-0.121217\pi$
$929$	56.5921	1.85673	0.928363	+	0.371676i	$0.121217\pi$
$930$	0	0
$931$	−2.34281	−0.0767826
$932$	0	0
$933$	39.5897	1.29611
$934$	0	0
$935$	−7.85300	−0.256821
$936$	0	0
$937$	−50.6315	−1.65406	−0.827030	−	0.562158i	$-0.809971\pi$
$937$	−50.6315	−1.65406	−0.827030	+	0.562158i	$0.809971\pi$
$938$	0	0
$939$	84.3271	2.75191
$940$	0	0
$941$	51.4443	1.67704	0.838519	−	0.544873i	$-0.183422\pi$
$941$	51.4443	1.67704	0.838519	+	0.544873i	$0.183422\pi$
$942$	0	0
$943$	−2.01775	−0.0657071
$944$	0	0
$945$	−5.82652	−0.189537
$946$	0	0
$947$	−60.8358	−1.97690	−0.988449	−	0.151554i	$-0.951572\pi$
$947$	−60.8358	−1.97690	−0.988449	+	0.151554i	$0.951572\pi$
$948$	0	0
$949$	−7.71888	−0.250565
$950$	0	0
$951$	−65.2137	−2.11470
$952$	0	0
$953$	−30.3025	−0.981595	−0.490797	−	0.871274i	$-0.663295\pi$
$953$	−30.3025	−0.981595	−0.490797	+	0.871274i	$0.663295\pi$
$954$	0	0
$955$	−2.46470	−0.0797557
$956$	0	0
$957$	−1.26171	−0.0407853
$958$	0	0
$959$	2.00248	0.0646633
$960$	0	0
$961$	−6.88106	−0.221970
$962$	0	0
$963$	−19.8951	−0.641112
$964$	0	0
$965$	−12.2683	−0.394930
$966$	0	0
$967$	−37.1944	−1.19609	−0.598046	−	0.801462i	$-0.704056\pi$
$967$	−37.1944	−1.19609	−0.598046	+	0.801462i	$0.704056\pi$
$968$	0	0
$969$	−18.5975	−0.597439
$970$	0	0
$971$	−1.92404	−0.0617454	−0.0308727	−	0.999523i	$-0.509829\pi$
$971$	−1.92404	−0.0617454	−0.0308727	+	0.999523i	$0.509829\pi$
$972$	0	0
$973$	2.42429	0.0777192
$974$	0	0
$975$	−4.48464	−0.143623
$976$	0	0
$977$	8.99309	0.287714	0.143857	−	0.989598i	$-0.454049\pi$
$977$	8.99309	0.287714	0.143857	+	0.989598i	$0.454049\pi$
$978$	0	0
$979$	4.05223	0.129510
$980$	0	0
$981$	42.7367	1.36448
$982$	0	0
$983$	−4.67258	−0.149032	−0.0745160	−	0.997220i	$-0.523741\pi$
$983$	−4.67258	−0.149032	−0.0745160	+	0.997220i	$0.523741\pi$
$984$	0	0
$985$	−69.2346	−2.20600
$986$	0	0
$987$	3.32800	0.105931
$988$	0	0
$989$	−14.9265	−0.474635
$990$	0	0
$991$	−40.1389	−1.27506	−0.637528	−	0.770428i	$-0.720043\pi$
$991$	−40.1389	−1.27506	−0.637528	+	0.770428i	$0.720043\pi$
$992$	0	0
$993$	30.1501	0.956783
$994$	0	0
$995$	−30.2955	−0.960432
$996$	0	0
$997$	−31.8737	−1.00945	−0.504726	−	0.863280i	$-0.668406\pi$
$997$	−31.8737	−1.00945	−0.504726	+	0.863280i	$0.668406\pi$
$998$	0	0
$999$	−3.67041	−0.116127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.8	✓	9	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.61891 q^{3} -2.59083 q^{5} +1.00000 q^{7} +3.85871 q^{9} +O(q^{10})\)	\(q+2.61891 q^{3} -2.59083 q^{5} +1.00000 q^{7} +3.85871 q^{9} +1.00000 q^{11} -1.00000 q^{13} -6.78516 q^{15} +3.03107 q^{17} -2.34281 q^{19} +2.61891 q^{21} -1.27075 q^{23} +1.71240 q^{25} +2.24890 q^{27} -0.481768 q^{29} +4.91110 q^{31} +2.61891 q^{33} -2.59083 q^{35} -1.63209 q^{37} -2.61891 q^{39} +1.58784 q^{41} +11.7462 q^{43} -9.99728 q^{45} +1.27075 q^{47} +1.00000 q^{49} +7.93812 q^{51} -1.51284 q^{53} -2.59083 q^{55} -6.13563 q^{57} +7.19716 q^{59} -1.26999 q^{61} +3.85871 q^{63} +2.59083 q^{65} +8.44610 q^{67} -3.32800 q^{69} +12.9170 q^{71} +7.71888 q^{73} +4.48464 q^{75} +1.00000 q^{77} -11.2346 q^{79} -5.68647 q^{81} +0.628980 q^{83} -7.85300 q^{85} -1.26171 q^{87} +4.05223 q^{89} -1.00000 q^{91} +12.8618 q^{93} +6.06983 q^{95} +12.1825 q^{97} +3.85871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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