LMFDB - Embedded newform 8016.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.284897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 $ x^5 - 2x^4 - 6x^3 + 5x^2 + 10x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.65065$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.457884 q^{5} +2.37531 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.457884 q^{5} +2.37531 q^{7} +1.00000 q^{9} +1.86534 q^{11} +1.48384 q^{13} +0.457884 q^{15} -0.676771 q^{17} -4.18757 q^{19} -2.37531 q^{21} +1.60277 q^{23} -4.79034 q^{25} -1.00000 q^{27} +8.15542 q^{29} +1.46812 q^{31} -1.86534 q^{33} -1.08761 q^{35} -8.61076 q^{37} -1.48384 q^{39} -12.1631 q^{41} -1.80872 q^{43} -0.457884 q^{45} -4.53272 q^{47} -1.35793 q^{49} +0.676771 q^{51} -7.07284 q^{53} -0.854111 q^{55} +4.18757 q^{57} -15.0829 q^{59} +6.30014 q^{61} +2.37531 q^{63} -0.679427 q^{65} +10.7228 q^{67} -1.60277 q^{69} +1.23296 q^{71} -10.3160 q^{73} +4.79034 q^{75} +4.43076 q^{77} +6.40327 q^{79} +1.00000 q^{81} +10.9254 q^{83} +0.309883 q^{85} -8.15542 q^{87} -2.83435 q^{89} +3.52458 q^{91} -1.46812 q^{93} +1.91742 q^{95} -6.73223 q^{97} +1.86534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.457884	−0.204772	−0.102386	−	0.994745i	$-0.532648\pi$
$5$	−0.457884	−0.204772	−0.102386	+	0.994745i	$0.532648\pi$
$6$	0	0
$7$	2.37531	0.897781	0.448890	−	0.893587i	$-0.351819\pi$
$7$	2.37531	0.897781	0.448890	+	0.893587i	$0.351819\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.86534	0.562423	0.281211	−	0.959646i	$-0.409264\pi$
$11$	1.86534	0.562423	0.281211	+	0.959646i	$0.409264\pi$
$12$	0	0
$13$	1.48384	0.411543	0.205772	−	0.978600i	$-0.434030\pi$
$13$	1.48384	0.411543	0.205772	+	0.978600i	$0.434030\pi$
$14$	0	0
$15$	0.457884	0.118225
$16$	0	0
$17$	−0.676771	−0.164141	−0.0820706	−	0.996627i	$-0.526153\pi$
$17$	−0.676771	−0.164141	−0.0820706	+	0.996627i	$0.526153\pi$
$18$	0	0
$19$	−4.18757	−0.960695	−0.480347	−	0.877078i	$-0.659489\pi$
$19$	−4.18757	−0.960695	−0.480347	+	0.877078i	$0.659489\pi$
$20$	0	0
$21$	−2.37531	−0.518334
$22$	0	0
$23$	1.60277	0.334201	0.167100	−	0.985940i	$-0.446560\pi$
$23$	1.60277	0.334201	0.167100	+	0.985940i	$0.446560\pi$
$24$	0	0
$25$	−4.79034	−0.958068
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.15542	1.51442	0.757211	−	0.653170i	$-0.226561\pi$
$29$	8.15542	1.51442	0.757211	+	0.653170i	$0.226561\pi$
$30$	0	0
$31$	1.46812	0.263681	0.131841	−	0.991271i	$-0.457911\pi$
$31$	1.46812	0.263681	0.131841	+	0.991271i	$0.457911\pi$
$32$	0	0
$33$	−1.86534	−0.324715
$34$	0	0
$35$	−1.08761	−0.183840
$36$	0	0
$37$	−8.61076	−1.41560	−0.707800	−	0.706413i	$-0.750312\pi$
$37$	−8.61076	−1.41560	−0.707800	+	0.706413i	$0.750312\pi$
$38$	0	0
$39$	−1.48384	−0.237605
$40$	0	0
$41$	−12.1631	−1.89956	−0.949779	−	0.312921i	$-0.898692\pi$
$41$	−12.1631	−1.89956	−0.949779	+	0.312921i	$0.898692\pi$
$42$	0	0
$43$	−1.80872	−0.275828	−0.137914	−	0.990444i	$-0.544040\pi$
$43$	−1.80872	−0.275828	−0.137914	+	0.990444i	$0.544040\pi$
$44$	0	0
$45$	−0.457884	−0.0682573
$46$	0	0
$47$	−4.53272	−0.661166	−0.330583	−	0.943777i	$-0.607245\pi$
$47$	−4.53272	−0.661166	−0.330583	+	0.943777i	$0.607245\pi$
$48$	0	0
$49$	−1.35793	−0.193989
$50$	0	0
$51$	0.676771	0.0947670
$52$	0	0
$53$	−7.07284	−0.971529	−0.485764	−	0.874090i	$-0.661459\pi$
$53$	−7.07284	−0.971529	−0.485764	+	0.874090i	$0.661459\pi$
$54$	0	0
$55$	−0.854111	−0.115168
$56$	0	0
$57$	4.18757	0.554657
$58$	0	0
$59$	−15.0829	−1.96363	−0.981814	−	0.189847i	$-0.939201\pi$
$59$	−15.0829	−1.96363	−0.981814	+	0.189847i	$0.939201\pi$
$60$	0	0
$61$	6.30014	0.806650	0.403325	−	0.915057i	$-0.367854\pi$
$61$	6.30014	0.806650	0.403325	+	0.915057i	$0.367854\pi$
$62$	0	0
$63$	2.37531	0.299260
$64$	0	0
$65$	−0.679427	−0.0842725
$66$	0	0
$67$	10.7228	1.31000	0.655002	−	0.755628i	$-0.272668\pi$
$67$	10.7228	1.31000	0.655002	+	0.755628i	$0.272668\pi$
$68$	0	0
$69$	−1.60277	−0.192951
$70$	0	0
$71$	1.23296	0.146325	0.0731627	−	0.997320i	$-0.476691\pi$
$71$	1.23296	0.146325	0.0731627	+	0.997320i	$0.476691\pi$
$72$	0	0
$73$	−10.3160	−1.20740	−0.603700	−	0.797212i	$-0.706307\pi$
$73$	−10.3160	−1.20740	−0.603700	+	0.797212i	$0.706307\pi$
$74$	0	0
$75$	4.79034	0.553141
$76$	0	0
$77$	4.43076	0.504932
$78$	0	0
$79$	6.40327	0.720424	0.360212	−	0.932871i	$-0.382704\pi$
$79$	6.40327	0.720424	0.360212	+	0.932871i	$0.382704\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.9254	1.19922	0.599610	−	0.800293i	$-0.295323\pi$
$83$	10.9254	1.19922	0.599610	+	0.800293i	$0.295323\pi$
$84$	0	0
$85$	0.309883	0.0336115
$86$	0	0
$87$	−8.15542	−0.874352
$88$	0	0
$89$	−2.83435	−0.300441	−0.150220	−	0.988653i	$-0.547998\pi$
$89$	−2.83435	−0.300441	−0.150220	+	0.988653i	$0.547998\pi$
$90$	0	0
$91$	3.52458	0.369476
$92$	0	0
$93$	−1.46812	−0.152236
$94$	0	0
$95$	1.91742	0.196723
$96$	0	0
$97$	−6.73223	−0.683554	−0.341777	−	0.939781i	$-0.611029\pi$
$97$	−6.73223	−0.683554	−0.341777	+	0.939781i	$0.611029\pi$
$98$	0	0
$99$	1.86534	0.187474
$100$	0	0
$101$	−7.45223	−0.741524	−0.370762	−	0.928728i	$-0.620904\pi$
$101$	−7.45223	−0.741524	−0.370762	+	0.928728i	$0.620904\pi$
$102$	0	0
$103$	6.11552	0.602580	0.301290	−	0.953533i	$-0.402583\pi$
$103$	6.11552	0.602580	0.301290	+	0.953533i	$0.402583\pi$
$104$	0	0
$105$	1.08761	0.106140
$106$	0	0
$107$	13.0772	1.26422	0.632111	−	0.774878i	$-0.282189\pi$
$107$	13.0772	1.26422	0.632111	+	0.774878i	$0.282189\pi$
$108$	0	0
$109$	7.00238	0.670706	0.335353	−	0.942092i	$-0.391144\pi$
$109$	7.00238	0.670706	0.335353	+	0.942092i	$0.391144\pi$
$110$	0	0
$111$	8.61076	0.817297
$112$	0	0
$113$	−9.06929	−0.853167	−0.426584	−	0.904448i	$-0.640283\pi$
$113$	−9.06929	−0.853167	−0.426584	+	0.904448i	$0.640283\pi$
$114$	0	0
$115$	−0.733883	−0.0684349
$116$	0	0
$117$	1.48384	0.137181
$118$	0	0
$119$	−1.60754	−0.147363
$120$	0	0
$121$	−7.52049	−0.683681
$122$	0	0
$123$	12.1631	1.09671
$124$	0	0
$125$	4.48284	0.400957
$126$	0	0
$127$	−8.65569	−0.768068	−0.384034	−	0.923319i	$-0.625465\pi$
$127$	−8.65569	−0.768068	−0.384034	+	0.923319i	$0.625465\pi$
$128$	0	0
$129$	1.80872	0.159249
$130$	0	0
$131$	−8.05546	−0.703809	−0.351904	−	0.936036i	$-0.614466\pi$
$131$	−8.05546	−0.703809	−0.351904	+	0.936036i	$0.614466\pi$
$132$	0	0
$133$	−9.94676	−0.862493
$134$	0	0
$135$	0.457884	0.0394084
$136$	0	0
$137$	−5.88865	−0.503101	−0.251551	−	0.967844i	$-0.580940\pi$
$137$	−5.88865	−0.503101	−0.251551	+	0.967844i	$0.580940\pi$
$138$	0	0
$139$	−9.61934	−0.815901	−0.407951	−	0.913004i	$-0.633756\pi$
$139$	−9.61934	−0.815901	−0.407951	+	0.913004i	$0.633756\pi$
$140$	0	0
$141$	4.53272	0.381724
$142$	0	0
$143$	2.76788	0.231461
$144$	0	0
$145$	−3.73423	−0.310111
$146$	0	0
$147$	1.35793	0.112000
$148$	0	0
$149$	10.3048	0.844206	0.422103	−	0.906548i	$-0.361292\pi$
$149$	10.3048	0.844206	0.422103	+	0.906548i	$0.361292\pi$
$150$	0	0
$151$	9.91415	0.806802	0.403401	−	0.915023i	$-0.367828\pi$
$151$	9.91415	0.806802	0.403401	+	0.915023i	$0.367828\pi$
$152$	0	0
$153$	−0.676771	−0.0547137
$154$	0	0
$155$	−0.672226	−0.0539945
$156$	0	0
$157$	−15.1523	−1.20929	−0.604643	−	0.796497i	$-0.706684\pi$
$157$	−15.1523	−1.20929	−0.604643	+	0.796497i	$0.706684\pi$
$158$	0	0
$159$	7.07284	0.560912
$160$	0	0
$161$	3.80707	0.300039
$162$	0	0
$163$	−8.57960	−0.672006	−0.336003	−	0.941861i	$-0.609075\pi$
$163$	−8.57960	−0.672006	−0.336003	+	0.941861i	$0.609075\pi$
$164$	0	0
$165$	0.854111	0.0664925
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.7982	−0.830632
$170$	0	0
$171$	−4.18757	−0.320232
$172$	0	0
$173$	6.58645	0.500759	0.250379	−	0.968148i	$-0.419445\pi$
$173$	6.58645	0.500759	0.250379	+	0.968148i	$0.419445\pi$
$174$	0	0
$175$	−11.3785	−0.860136
$176$	0	0
$177$	15.0829	1.13370
$178$	0	0
$179$	17.1920	1.28499	0.642494	−	0.766290i	$-0.277900\pi$
$179$	17.1920	1.28499	0.642494	+	0.766290i	$0.277900\pi$
$180$	0	0
$181$	−24.8573	−1.84763	−0.923815	−	0.382838i	$-0.874947\pi$
$181$	−24.8573	−1.84763	−0.923815	+	0.382838i	$0.874947\pi$
$182$	0	0
$183$	−6.30014	−0.465720
$184$	0	0
$185$	3.94273	0.289875
$186$	0	0
$187$	−1.26241	−0.0923167
$188$	0	0
$189$	−2.37531	−0.172778
$190$	0	0
$191$	5.29302	0.382990	0.191495	−	0.981494i	$-0.438666\pi$
$191$	5.29302	0.382990	0.191495	+	0.981494i	$0.438666\pi$
$192$	0	0
$193$	3.18857	0.229519	0.114759	−	0.993393i	$-0.463390\pi$
$193$	3.18857	0.229519	0.114759	+	0.993393i	$0.463390\pi$
$194$	0	0
$195$	0.679427	0.0486548
$196$	0	0
$197$	−0.992148	−0.0706876	−0.0353438	−	0.999375i	$-0.511253\pi$
$197$	−0.992148	−0.0706876	−0.0353438	+	0.999375i	$0.511253\pi$
$198$	0	0
$199$	−7.00374	−0.496482	−0.248241	−	0.968698i	$-0.579852\pi$
$199$	−7.00374	−0.496482	−0.248241	+	0.968698i	$0.579852\pi$
$200$	0	0
$201$	−10.7228	−0.756331
$202$	0	0
$203$	19.3716	1.35962
$204$	0	0
$205$	5.56929	0.388976
$206$	0	0
$207$	1.60277	0.111400
$208$	0	0
$209$	−7.81126	−0.540316
$210$	0	0
$211$	10.7769	0.741914	0.370957	−	0.928650i	$-0.379030\pi$
$211$	10.7769	0.741914	0.370957	+	0.928650i	$0.379030\pi$
$212$	0	0
$213$	−1.23296	−0.0844810
$214$	0	0
$215$	0.828185	0.0564818
$216$	0	0
$217$	3.48722	0.236728
$218$	0	0
$219$	10.3160	0.697093
$220$	0	0
$221$	−1.00422	−0.0675512
$222$	0	0
$223$	1.32658	0.0888346	0.0444173	−	0.999013i	$-0.485857\pi$
$223$	1.32658	0.0888346	0.0444173	+	0.999013i	$0.485857\pi$
$224$	0	0
$225$	−4.79034	−0.319356
$226$	0	0
$227$	5.38477	0.357400	0.178700	−	0.983904i	$-0.442811\pi$
$227$	5.38477	0.357400	0.178700	+	0.983904i	$0.442811\pi$
$228$	0	0
$229$	−5.16299	−0.341180	−0.170590	−	0.985342i	$-0.554567\pi$
$229$	−5.16299	−0.341180	−0.170590	+	0.985342i	$0.554567\pi$
$230$	0	0
$231$	−4.43076	−0.291523
$232$	0	0
$233$	−6.17331	−0.404427	−0.202213	−	0.979341i	$-0.564813\pi$
$233$	−6.17331	−0.404427	−0.202213	+	0.979341i	$0.564813\pi$
$234$	0	0
$235$	2.07546	0.135388
$236$	0	0
$237$	−6.40327	−0.415937
$238$	0	0
$239$	−14.2571	−0.922215	−0.461108	−	0.887344i	$-0.652548\pi$
$239$	−14.2571	−0.922215	−0.461108	+	0.887344i	$0.652548\pi$
$240$	0	0
$241$	13.2484	0.853407	0.426704	−	0.904392i	$-0.359675\pi$
$241$	13.2484	0.853407	0.426704	+	0.904392i	$0.359675\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.621772	0.0397236
$246$	0	0
$247$	−6.21369	−0.395368
$248$	0	0
$249$	−10.9254	−0.692370
$250$	0	0
$251$	20.4505	1.29082	0.645411	−	0.763835i	$-0.276686\pi$
$251$	20.4505	1.29082	0.645411	+	0.763835i	$0.276686\pi$
$252$	0	0
$253$	2.98972	0.187962
$254$	0	0
$255$	−0.309883	−0.0194056
$256$	0	0
$257$	29.6854	1.85172	0.925861	−	0.377865i	$-0.123342\pi$
$257$	29.6854	1.85172	0.925861	+	0.377865i	$0.123342\pi$
$258$	0	0
$259$	−20.4532	−1.27090
$260$	0	0
$261$	8.15542	0.504808
$262$	0	0
$263$	19.1828	1.18286	0.591431	−	0.806356i	$-0.298563\pi$
$263$	19.1828	1.18286	0.591431	+	0.806356i	$0.298563\pi$
$264$	0	0
$265$	3.23854	0.198942
$266$	0	0
$267$	2.83435	0.173460
$268$	0	0
$269$	30.3063	1.84781	0.923904	−	0.382624i	$-0.124979\pi$
$269$	30.3063	1.84781	0.923904	+	0.382624i	$0.124979\pi$
$270$	0	0
$271$	−24.7437	−1.50307	−0.751537	−	0.659691i	$-0.770687\pi$
$271$	−24.7437	−1.50307	−0.751537	+	0.659691i	$0.770687\pi$
$272$	0	0
$273$	−3.52458	−0.213317
$274$	0	0
$275$	−8.93564	−0.538839
$276$	0	0
$277$	3.25833	0.195774	0.0978869	−	0.995198i	$-0.468792\pi$
$277$	3.25833	0.195774	0.0978869	+	0.995198i	$0.468792\pi$
$278$	0	0
$279$	1.46812	0.0878938
$280$	0	0
$281$	16.6904	0.995668	0.497834	−	0.867272i	$-0.334129\pi$
$281$	16.6904	0.995668	0.497834	+	0.867272i	$0.334129\pi$
$282$	0	0
$283$	6.95153	0.413225	0.206613	−	0.978423i	$-0.433756\pi$
$283$	6.95153	0.413225	0.206613	+	0.978423i	$0.433756\pi$
$284$	0	0
$285$	−1.91742	−0.113578
$286$	0	0
$287$	−28.8911	−1.70539
$288$	0	0
$289$	−16.5420	−0.973058
$290$	0	0
$291$	6.73223	0.394650
$292$	0	0
$293$	−8.01237	−0.468088	−0.234044	−	0.972226i	$-0.575196\pi$
$293$	−8.01237	−0.468088	−0.234044	+	0.972226i	$0.575196\pi$
$294$	0	0
$295$	6.90622	0.402096
$296$	0	0
$297$	−1.86534	−0.108238
$298$	0	0
$299$	2.37826	0.137538
$300$	0	0
$301$	−4.29627	−0.247633
$302$	0	0
$303$	7.45223	0.428119
$304$	0	0
$305$	−2.88473	−0.165179
$306$	0	0
$307$	−27.0713	−1.54504	−0.772519	−	0.634991i	$-0.781004\pi$
$307$	−27.0713	−1.54504	−0.772519	+	0.634991i	$0.781004\pi$
$308$	0	0
$309$	−6.11552	−0.347900
$310$	0	0
$311$	−24.6711	−1.39897	−0.699484	−	0.714648i	$-0.746587\pi$
$311$	−24.6711	−1.39897	−0.699484	+	0.714648i	$0.746587\pi$
$312$	0	0
$313$	−3.74346	−0.211593	−0.105797	−	0.994388i	$-0.533739\pi$
$313$	−3.74346	−0.211593	−0.105797	+	0.994388i	$0.533739\pi$
$314$	0	0
$315$	−1.08761	−0.0612801
$316$	0	0
$317$	34.0139	1.91041	0.955205	−	0.295944i	$-0.0956343\pi$
$317$	34.0139	1.91041	0.955205	+	0.295944i	$0.0956343\pi$
$318$	0	0
$319$	15.2127	0.851746
$320$	0	0
$321$	−13.0772	−0.729899
$322$	0	0
$323$	2.83403	0.157690
$324$	0	0
$325$	−7.10811	−0.394287
$326$	0	0
$327$	−7.00238	−0.387232
$328$	0	0
$329$	−10.7666	−0.593582
$330$	0	0
$331$	22.0014	1.20931	0.604653	−	0.796489i	$-0.293312\pi$
$331$	22.0014	1.20931	0.604653	+	0.796489i	$0.293312\pi$
$332$	0	0
$333$	−8.61076	−0.471867
$334$	0	0
$335$	−4.90981	−0.268252
$336$	0	0
$337$	−34.0326	−1.85387	−0.926936	−	0.375218i	$-0.877568\pi$
$337$	−34.0326	−1.85387	−0.926936	+	0.375218i	$0.877568\pi$
$338$	0	0
$339$	9.06929	0.492576
$340$	0	0
$341$	2.73854	0.148300
$342$	0	0
$343$	−19.8526	−1.07194
$344$	0	0
$345$	0.733883	0.0395109
$346$	0	0
$347$	−9.32656	−0.500676	−0.250338	−	0.968159i	$-0.580542\pi$
$347$	−9.32656	−0.500676	−0.250338	+	0.968159i	$0.580542\pi$
$348$	0	0
$349$	−12.5114	−0.669719	−0.334860	−	0.942268i	$-0.608689\pi$
$349$	−12.5114	−0.669719	−0.334860	+	0.942268i	$0.608689\pi$
$350$	0	0
$351$	−1.48384	−0.0792016
$352$	0	0
$353$	−18.8349	−1.00248	−0.501242	−	0.865307i	$-0.667123\pi$
$353$	−18.8349	−1.00248	−0.501242	+	0.865307i	$0.667123\pi$
$354$	0	0
$355$	−0.564552	−0.0299633
$356$	0	0
$357$	1.60754	0.0850800
$358$	0	0
$359$	−22.0405	−1.16325	−0.581626	−	0.813456i	$-0.697583\pi$
$359$	−22.0405	−1.16325	−0.581626	+	0.813456i	$0.697583\pi$
$360$	0	0
$361$	−1.46424	−0.0770655
$362$	0	0
$363$	7.52049	0.394723
$364$	0	0
$365$	4.72354	0.247242
$366$	0	0
$367$	24.4521	1.27639	0.638195	−	0.769874i	$-0.279681\pi$
$367$	24.4521	1.27639	0.638195	+	0.769874i	$0.279681\pi$
$368$	0	0
$369$	−12.1631	−0.633186
$370$	0	0
$371$	−16.8001	−0.872220
$372$	0	0
$373$	−25.9516	−1.34372	−0.671862	−	0.740676i	$-0.734505\pi$
$373$	−25.9516	−1.34372	−0.671862	+	0.740676i	$0.734505\pi$
$374$	0	0
$375$	−4.48284	−0.231493
$376$	0	0
$377$	12.1013	0.623251
$378$	0	0
$379$	−17.6988	−0.909128	−0.454564	−	0.890714i	$-0.650205\pi$
$379$	−17.6988	−0.909128	−0.454564	+	0.890714i	$0.650205\pi$
$380$	0	0
$381$	8.65569	0.443444
$382$	0	0
$383$	−29.3960	−1.50206	−0.751032	−	0.660266i	$-0.770443\pi$
$383$	−29.3960	−1.50206	−0.751032	+	0.660266i	$0.770443\pi$
$384$	0	0
$385$	−2.02877	−0.103396
$386$	0	0
$387$	−1.80872	−0.0919426
$388$	0	0
$389$	1.25391	0.0635759	0.0317880	−	0.999495i	$-0.489880\pi$
$389$	1.25391	0.0635759	0.0317880	+	0.999495i	$0.489880\pi$
$390$	0	0
$391$	−1.08471	−0.0548561
$392$	0	0
$393$	8.05546	0.406344
$394$	0	0
$395$	−2.93195	−0.147523
$396$	0	0
$397$	−14.3131	−0.718354	−0.359177	−	0.933269i	$-0.616943\pi$
$397$	−14.3131	−0.718354	−0.359177	+	0.933269i	$0.616943\pi$
$398$	0	0
$399$	9.94676	0.497961
$400$	0	0
$401$	−26.3545	−1.31608	−0.658040	−	0.752983i	$-0.728614\pi$
$401$	−26.3545	−1.31608	−0.658040	+	0.752983i	$0.728614\pi$
$402$	0	0
$403$	2.17845	0.108516
$404$	0	0
$405$	−0.457884	−0.0227524
$406$	0	0
$407$	−16.0620	−0.796165
$408$	0	0
$409$	−24.1880	−1.19602	−0.598009	−	0.801489i	$-0.704041\pi$
$409$	−24.1880	−1.19602	−0.598009	+	0.801489i	$0.704041\pi$
$410$	0	0
$411$	5.88865	0.290466
$412$	0	0
$413$	−35.8265	−1.76291
$414$	0	0
$415$	−5.00257	−0.245566
$416$	0	0
$417$	9.61934	0.471061
$418$	0	0
$419$	−36.5880	−1.78744	−0.893720	−	0.448625i	$-0.851914\pi$
$419$	−36.5880	−1.78744	−0.893720	+	0.448625i	$0.851914\pi$
$420$	0	0
$421$	7.82504	0.381369	0.190685	−	0.981651i	$-0.438929\pi$
$421$	7.82504	0.381369	0.190685	+	0.981651i	$0.438929\pi$
$422$	0	0
$423$	−4.53272	−0.220389
$424$	0	0
$425$	3.24197	0.157258
$426$	0	0
$427$	14.9648	0.724195
$428$	0	0
$429$	−2.76788	−0.133634
$430$	0	0
$431$	−28.1822	−1.35749	−0.678745	−	0.734374i	$-0.737476\pi$
$431$	−28.1822	−1.35749	−0.678745	+	0.734374i	$0.737476\pi$
$432$	0	0
$433$	21.7667	1.04604	0.523021	−	0.852320i	$-0.324805\pi$
$433$	21.7667	1.04604	0.523021	+	0.852320i	$0.324805\pi$
$434$	0	0
$435$	3.73423	0.179043
$436$	0	0
$437$	−6.71172	−0.321065
$438$	0	0
$439$	38.7660	1.85020	0.925099	−	0.379726i	$-0.123982\pi$
$439$	38.7660	1.85020	0.925099	+	0.379726i	$0.123982\pi$
$440$	0	0
$441$	−1.35793	−0.0646631
$442$	0	0
$443$	4.14294	0.196837	0.0984185	−	0.995145i	$-0.468622\pi$
$443$	4.14294	0.196837	0.0984185	+	0.995145i	$0.468622\pi$
$444$	0	0
$445$	1.29780	0.0615218
$446$	0	0
$447$	−10.3048	−0.487403
$448$	0	0
$449$	23.8195	1.12411	0.562055	−	0.827100i	$-0.310011\pi$
$449$	23.8195	1.12411	0.562055	+	0.827100i	$0.310011\pi$
$450$	0	0
$451$	−22.6884	−1.06835
$452$	0	0
$453$	−9.91415	−0.465807
$454$	0	0
$455$	−1.61385	−0.0756583
$456$	0	0
$457$	−23.7137	−1.10928	−0.554640	−	0.832090i	$-0.687144\pi$
$457$	−23.7137	−1.10928	−0.554640	+	0.832090i	$0.687144\pi$
$458$	0	0
$459$	0.676771	0.0315890
$460$	0	0
$461$	−11.4104	−0.531437	−0.265719	−	0.964051i	$-0.585609\pi$
$461$	−11.4104	−0.531437	−0.265719	+	0.964051i	$0.585609\pi$
$462$	0	0
$463$	−31.1257	−1.44653	−0.723267	−	0.690568i	$-0.757360\pi$
$463$	−31.1257	−1.44653	−0.723267	+	0.690568i	$0.757360\pi$
$464$	0	0
$465$	0.672226	0.0311738
$466$	0	0
$467$	−15.0522	−0.696532	−0.348266	−	0.937396i	$-0.613229\pi$
$467$	−15.0522	−0.696532	−0.348266	+	0.937396i	$0.613229\pi$
$468$	0	0
$469$	25.4700	1.17610
$470$	0	0
$471$	15.1523	0.698181
$472$	0	0
$473$	−3.37389	−0.155132
$474$	0	0
$475$	20.0599	0.920411
$476$	0	0
$477$	−7.07284	−0.323843
$478$	0	0
$479$	−9.31427	−0.425580	−0.212790	−	0.977098i	$-0.568255\pi$
$479$	−9.31427	−0.425580	−0.212790	+	0.977098i	$0.568255\pi$
$480$	0	0
$481$	−12.7770	−0.582581
$482$	0	0
$483$	−3.80707	−0.173228
$484$	0	0
$485$	3.08258	0.139973
$486$	0	0
$487$	24.4701	1.10884	0.554422	−	0.832235i	$-0.312939\pi$
$487$	24.4701	1.10884	0.554422	+	0.832235i	$0.312939\pi$
$488$	0	0
$489$	8.57960	0.387983
$490$	0	0
$491$	3.82613	0.172671	0.0863354	−	0.996266i	$-0.472484\pi$
$491$	3.82613	0.172671	0.0863354	+	0.996266i	$0.472484\pi$
$492$	0	0
$493$	−5.51935	−0.248579
$494$	0	0
$495$	−0.854111	−0.0383894
$496$	0	0
$497$	2.92865	0.131368
$498$	0	0
$499$	7.14511	0.319859	0.159929	−	0.987128i	$-0.448873\pi$
$499$	7.14511	0.319859	0.159929	+	0.987128i	$0.448873\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−1.70327	−0.0759451	−0.0379726	−	0.999279i	$-0.512090\pi$
$503$	−1.70327	−0.0759451	−0.0379726	+	0.999279i	$0.512090\pi$
$504$	0	0
$505$	3.41225	0.151843
$506$	0	0
$507$	10.7982	0.479566
$508$	0	0
$509$	−10.7859	−0.478074	−0.239037	−	0.971010i	$-0.576832\pi$
$509$	−10.7859	−0.478074	−0.239037	+	0.971010i	$0.576832\pi$
$510$	0	0
$511$	−24.5037	−1.08398
$512$	0	0
$513$	4.18757	0.184886
$514$	0	0
$515$	−2.80020	−0.123391
$516$	0	0
$517$	−8.45509	−0.371855
$518$	0	0
$519$	−6.58645	−0.289113
$520$	0	0
$521$	18.6019	0.814962	0.407481	−	0.913214i	$-0.366407\pi$
$521$	18.6019	0.814962	0.407481	+	0.913214i	$0.366407\pi$
$522$	0	0
$523$	−34.9400	−1.52782	−0.763910	−	0.645323i	$-0.776723\pi$
$523$	−34.9400	−1.52782	−0.763910	+	0.645323i	$0.776723\pi$
$524$	0	0
$525$	11.3785	0.496600
$526$	0	0
$527$	−0.993579	−0.0432810
$528$	0	0
$529$	−20.4311	−0.888310
$530$	0	0
$531$	−15.0829	−0.654542
$532$	0	0
$533$	−18.0481	−0.781751
$534$	0	0
$535$	−5.98785	−0.258877
$536$	0	0
$537$	−17.1920	−0.741889
$538$	0	0
$539$	−2.53300	−0.109104
$540$	0	0
$541$	5.89556	0.253470	0.126735	−	0.991937i	$-0.459550\pi$
$541$	5.89556	0.253470	0.126735	+	0.991937i	$0.459550\pi$
$542$	0	0
$543$	24.8573	1.06673
$544$	0	0
$545$	−3.20628	−0.137342
$546$	0	0
$547$	−32.6442	−1.39577	−0.697883	−	0.716212i	$-0.745874\pi$
$547$	−32.6442	−1.39577	−0.697883	+	0.716212i	$0.745874\pi$
$548$	0	0
$549$	6.30014	0.268883
$550$	0	0
$551$	−34.1514	−1.45490
$552$	0	0
$553$	15.2097	0.646783
$554$	0	0
$555$	−3.94273	−0.167359
$556$	0	0
$557$	−0.252911	−0.0107162	−0.00535810	−	0.999986i	$-0.501706\pi$
$557$	−0.252911	−0.0107162	−0.00535810	+	0.999986i	$0.501706\pi$
$558$	0	0
$559$	−2.68386	−0.113515
$560$	0	0
$561$	1.26241	0.0532991
$562$	0	0
$563$	43.2541	1.82294	0.911471	−	0.411364i	$-0.134948\pi$
$563$	43.2541	1.82294	0.911471	+	0.411364i	$0.134948\pi$
$564$	0	0
$565$	4.15268	0.174705
$566$	0	0
$567$	2.37531	0.0997534
$568$	0	0
$569$	0.698883	0.0292987	0.0146494	−	0.999893i	$-0.495337\pi$
$569$	0.698883	0.0292987	0.0146494	+	0.999893i	$0.495337\pi$
$570$	0	0
$571$	3.30453	0.138290	0.0691452	−	0.997607i	$-0.477973\pi$
$571$	3.30453	0.138290	0.0691452	+	0.997607i	$0.477973\pi$
$572$	0	0
$573$	−5.29302	−0.221119
$574$	0	0
$575$	−7.67782	−0.320187
$576$	0	0
$577$	−4.99066	−0.207764	−0.103882	−	0.994590i	$-0.533126\pi$
$577$	−4.99066	−0.207764	−0.103882	+	0.994590i	$0.533126\pi$
$578$	0	0
$579$	−3.18857	−0.132513
$580$	0	0
$581$	25.9512	1.07664
$582$	0	0
$583$	−13.1933	−0.546410
$584$	0	0
$585$	−0.679427	−0.0280908
$586$	0	0
$587$	20.3173	0.838586	0.419293	−	0.907851i	$-0.362278\pi$
$587$	20.3173	0.838586	0.419293	+	0.907851i	$0.362278\pi$
$588$	0	0
$589$	−6.14784	−0.253317
$590$	0	0
$591$	0.992148	0.0408115
$592$	0	0
$593$	38.9314	1.59872	0.799361	−	0.600851i	$-0.205172\pi$
$593$	38.9314	1.59872	0.799361	+	0.600851i	$0.205172\pi$
$594$	0	0
$595$	0.736066	0.0301758
$596$	0	0
$597$	7.00374	0.286644
$598$	0	0
$599$	31.7669	1.29796	0.648980	−	0.760805i	$-0.275196\pi$
$599$	31.7669	1.29796	0.648980	+	0.760805i	$0.275196\pi$
$600$	0	0
$601$	−18.8357	−0.768323	−0.384162	−	0.923266i	$-0.625509\pi$
$601$	−18.8357	−0.768323	−0.384162	+	0.923266i	$0.625509\pi$
$602$	0	0
$603$	10.7228	0.436668
$604$	0	0
$605$	3.44351	0.139999
$606$	0	0
$607$	−21.0801	−0.855614	−0.427807	−	0.903870i	$-0.640714\pi$
$607$	−21.0801	−0.855614	−0.427807	+	0.903870i	$0.640714\pi$
$608$	0	0
$609$	−19.3716	−0.784977
$610$	0	0
$611$	−6.72584	−0.272098
$612$	0	0
$613$	45.5832	1.84109	0.920543	−	0.390641i	$-0.127747\pi$
$613$	45.5832	1.84109	0.920543	+	0.390641i	$0.127747\pi$
$614$	0	0
$615$	−5.56929	−0.224575
$616$	0	0
$617$	−17.9426	−0.722341	−0.361170	−	0.932500i	$-0.617623\pi$
$617$	−17.9426	−0.722341	−0.361170	+	0.932500i	$0.617623\pi$
$618$	0	0
$619$	22.6408	0.910011	0.455005	−	0.890489i	$-0.349637\pi$
$619$	22.6408	0.910011	0.455005	+	0.890489i	$0.349637\pi$
$620$	0	0
$621$	−1.60277	−0.0643170
$622$	0	0
$623$	−6.73245	−0.269730
$624$	0	0
$625$	21.8991	0.875964
$626$	0	0
$627$	7.81126	0.311952
$628$	0	0
$629$	5.82751	0.232358
$630$	0	0
$631$	20.2768	0.807207	0.403604	−	0.914934i	$-0.367758\pi$
$631$	20.2768	0.807207	0.403604	+	0.914934i	$0.367758\pi$
$632$	0	0
$633$	−10.7769	−0.428344
$634$	0	0
$635$	3.96330	0.157279
$636$	0	0
$637$	−2.01495	−0.0798351
$638$	0	0
$639$	1.23296	0.0487751
$640$	0	0
$641$	14.6088	0.577012	0.288506	−	0.957478i	$-0.406841\pi$
$641$	14.6088	0.577012	0.288506	+	0.957478i	$0.406841\pi$
$642$	0	0
$643$	21.8745	0.862648	0.431324	−	0.902197i	$-0.358047\pi$
$643$	21.8745	0.862648	0.431324	+	0.902197i	$0.358047\pi$
$644$	0	0
$645$	−0.828185	−0.0326098
$646$	0	0
$647$	43.4270	1.70729	0.853646	−	0.520854i	$-0.174386\pi$
$647$	43.4270	1.70729	0.853646	+	0.520854i	$0.174386\pi$
$648$	0	0
$649$	−28.1348	−1.10439
$650$	0	0
$651$	−3.48722	−0.136675
$652$	0	0
$653$	12.5298	0.490328	0.245164	−	0.969482i	$-0.421158\pi$
$653$	12.5298	0.490328	0.245164	+	0.969482i	$0.421158\pi$
$654$	0	0
$655$	3.68846	0.144120
$656$	0	0
$657$	−10.3160	−0.402467
$658$	0	0
$659$	8.77130	0.341682	0.170841	−	0.985299i	$-0.445352\pi$
$659$	8.77130	0.341682	0.170841	+	0.985299i	$0.445352\pi$
$660$	0	0
$661$	−3.37077	−0.131108	−0.0655538	−	0.997849i	$-0.520881\pi$
$661$	−3.37077	−0.131108	−0.0655538	+	0.997849i	$0.520881\pi$
$662$	0	0
$663$	1.00422	0.0390007
$664$	0	0
$665$	4.55446	0.176614
$666$	0	0
$667$	13.0713	0.506121
$668$	0	0
$669$	−1.32658	−0.0512887
$670$	0	0
$671$	11.7519	0.453678
$672$	0	0
$673$	−41.0801	−1.58352	−0.791761	−	0.610831i	$-0.790836\pi$
$673$	−41.0801	−1.58352	−0.791761	+	0.610831i	$0.790836\pi$
$674$	0	0
$675$	4.79034	0.184380
$676$	0	0
$677$	−18.5374	−0.712448	−0.356224	−	0.934401i	$-0.615936\pi$
$677$	−18.5374	−0.712448	−0.356224	+	0.934401i	$0.615936\pi$
$678$	0	0
$679$	−15.9911	−0.613682
$680$	0	0
$681$	−5.38477	−0.206345
$682$	0	0
$683$	−35.2309	−1.34807	−0.674037	−	0.738697i	$-0.735441\pi$
$683$	−35.2309	−1.34807	−0.674037	+	0.738697i	$0.735441\pi$
$684$	0	0
$685$	2.69632	0.103021
$686$	0	0
$687$	5.16299	0.196980
$688$	0	0
$689$	−10.4950	−0.399826
$690$	0	0
$691$	−13.7748	−0.524018	−0.262009	−	0.965065i	$-0.584385\pi$
$691$	−13.7748	−0.524018	−0.262009	+	0.965065i	$0.584385\pi$
$692$	0	0
$693$	4.43076	0.168311
$694$	0	0
$695$	4.40454	0.167074
$696$	0	0
$697$	8.23164	0.311796
$698$	0	0
$699$	6.17331	0.233496
$700$	0	0
$701$	−34.3881	−1.29882	−0.649411	−	0.760438i	$-0.724985\pi$
$701$	−34.3881	−1.29882	−0.649411	+	0.760438i	$0.724985\pi$
$702$	0	0
$703$	36.0582	1.35996
$704$	0	0
$705$	−2.07546	−0.0781664
$706$	0	0
$707$	−17.7013	−0.665726
$708$	0	0
$709$	31.5151	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$709$	31.5151	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$710$	0	0
$711$	6.40327	0.240141
$712$	0	0
$713$	2.35305	0.0881225
$714$	0	0
$715$	−1.26737	−0.0473968
$716$	0	0
$717$	14.2571	0.532441
$718$	0	0
$719$	−33.6542	−1.25509	−0.627546	−	0.778580i	$-0.715940\pi$
$719$	−33.6542	−1.25509	−0.627546	+	0.778580i	$0.715940\pi$
$720$	0	0
$721$	14.5262	0.540985
$722$	0	0
$723$	−13.2484	−0.492715
$724$	0	0
$725$	−39.0672	−1.45092
$726$	0	0
$727$	−39.7782	−1.47529	−0.737645	−	0.675188i	$-0.764062\pi$
$727$	−39.7782	−1.47529	−0.737645	+	0.675188i	$0.764062\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.22409	0.0452747
$732$	0	0
$733$	−35.1183	−1.29712	−0.648562	−	0.761162i	$-0.724629\pi$
$733$	−35.1183	−1.29712	−0.648562	+	0.761162i	$0.724629\pi$
$734$	0	0
$735$	−0.621772	−0.0229344
$736$	0	0
$737$	20.0018	0.736775
$738$	0	0
$739$	−43.7065	−1.60777	−0.803885	−	0.594785i	$-0.797237\pi$
$739$	−43.7065	−1.60777	−0.803885	+	0.594785i	$0.797237\pi$
$740$	0	0
$741$	6.21369	0.228266
$742$	0	0
$743$	26.8588	0.985353	0.492676	−	0.870213i	$-0.336019\pi$
$743$	26.8588	0.985353	0.492676	+	0.870213i	$0.336019\pi$
$744$	0	0
$745$	−4.71842	−0.172870
$746$	0	0
$747$	10.9254	0.399740
$748$	0	0
$749$	31.0624	1.13499
$750$	0	0
$751$	−19.2922	−0.703983	−0.351992	−	0.936003i	$-0.614495\pi$
$751$	−19.2922	−0.703983	−0.351992	+	0.936003i	$0.614495\pi$
$752$	0	0
$753$	−20.4505	−0.745257
$754$	0	0
$755$	−4.53953	−0.165210
$756$	0	0
$757$	−15.9118	−0.578323	−0.289161	−	0.957280i	$-0.593376\pi$
$757$	−15.9118	−0.578323	−0.289161	+	0.957280i	$0.593376\pi$
$758$	0	0
$759$	−2.98972	−0.108520
$760$	0	0
$761$	18.2200	0.660473	0.330237	−	0.943898i	$-0.392871\pi$
$761$	18.2200	0.660473	0.330237	+	0.943898i	$0.392871\pi$
$762$	0	0
$763$	16.6328	0.602147
$764$	0	0
$765$	0.309883	0.0112038
$766$	0	0
$767$	−22.3806	−0.808118
$768$	0	0
$769$	−19.9699	−0.720134	−0.360067	−	0.932926i	$-0.617246\pi$
$769$	−19.9699	−0.720134	−0.360067	+	0.932926i	$0.617246\pi$
$770$	0	0
$771$	−29.6854	−1.06909
$772$	0	0
$773$	40.7753	1.46659	0.733294	−	0.679912i	$-0.237982\pi$
$773$	40.7753	1.46659	0.733294	+	0.679912i	$0.237982\pi$
$774$	0	0
$775$	−7.03278	−0.252625
$776$	0	0
$777$	20.4532	0.733754
$778$	0	0
$779$	50.9339	1.82490
$780$	0	0
$781$	2.29989	0.0822967
$782$	0	0
$783$	−8.15542	−0.291451
$784$	0	0
$785$	6.93799	0.247628
$786$	0	0
$787$	−0.273881	−0.00976282	−0.00488141	−	0.999988i	$-0.501554\pi$
$787$	−0.273881	−0.00976282	−0.00488141	+	0.999988i	$0.501554\pi$
$788$	0	0
$789$	−19.1828	−0.682925
$790$	0	0
$791$	−21.5423	−0.765957
$792$	0	0
$793$	9.34841	0.331972
$794$	0	0
$795$	−3.23854	−0.114859
$796$	0	0
$797$	36.2528	1.28414	0.642070	−	0.766646i	$-0.278076\pi$
$797$	36.2528	1.28414	0.642070	+	0.766646i	$0.278076\pi$
$798$	0	0
$799$	3.06762	0.108525
$800$	0	0
$801$	−2.83435	−0.100147
$802$	0	0
$803$	−19.2429	−0.679069
$804$	0	0
$805$	−1.74320	−0.0614396
$806$	0	0
$807$	−30.3063	−1.06683
$808$	0	0
$809$	−44.0893	−1.55010	−0.775049	−	0.631902i	$-0.782275\pi$
$809$	−44.0893	−1.55010	−0.775049	+	0.631902i	$0.782275\pi$
$810$	0	0
$811$	−34.4312	−1.20904	−0.604522	−	0.796589i	$-0.706636\pi$
$811$	−34.4312	−1.20904	−0.604522	+	0.796589i	$0.706636\pi$
$812$	0	0
$813$	24.7437	0.867800
$814$	0	0
$815$	3.92846	0.137608
$816$	0	0
$817$	7.57416	0.264986
$818$	0	0
$819$	3.52458	0.123159
$820$	0	0
$821$	28.6715	1.00064	0.500321	−	0.865840i	$-0.333215\pi$
$821$	28.6715	1.00064	0.500321	+	0.865840i	$0.333215\pi$
$822$	0	0
$823$	−53.2814	−1.85727	−0.928636	−	0.370991i	$-0.879018\pi$
$823$	−53.2814	−1.85727	−0.928636	+	0.370991i	$0.879018\pi$
$824$	0	0
$825$	8.93564	0.311099
$826$	0	0
$827$	−19.0653	−0.662964	−0.331482	−	0.943462i	$-0.607549\pi$
$827$	−19.0653	−0.662964	−0.331482	+	0.943462i	$0.607549\pi$
$828$	0	0
$829$	−49.5445	−1.72075	−0.860375	−	0.509662i	$-0.829771\pi$
$829$	−49.5445	−1.72075	−0.860375	+	0.509662i	$0.829771\pi$
$830$	0	0
$831$	−3.25833	−0.113030
$832$	0	0
$833$	0.919005	0.0318417
$834$	0	0
$835$	0.457884	0.0158457
$836$	0	0
$837$	−1.46812	−0.0507455
$838$	0	0
$839$	−19.2378	−0.664162	−0.332081	−	0.943251i	$-0.607751\pi$
$839$	−19.2378	−0.664162	−0.332081	+	0.943251i	$0.607751\pi$
$840$	0	0
$841$	37.5108	1.29348
$842$	0	0
$843$	−16.6904	−0.574849
$844$	0	0
$845$	4.94433	0.170090
$846$	0	0
$847$	−17.8635	−0.613796
$848$	0	0
$849$	−6.95153	−0.238576
$850$	0	0
$851$	−13.8011	−0.473095
$852$	0	0
$853$	12.1982	0.417657	0.208829	−	0.977952i	$-0.433035\pi$
$853$	12.1982	0.417657	0.208829	+	0.977952i	$0.433035\pi$
$854$	0	0
$855$	1.91742	0.0655744
$856$	0	0
$857$	26.7473	0.913671	0.456836	−	0.889551i	$-0.348983\pi$
$857$	26.7473	0.913671	0.456836	+	0.889551i	$0.348983\pi$
$858$	0	0
$859$	32.5008	1.10891	0.554457	−	0.832213i	$-0.312926\pi$
$859$	32.5008	1.10891	0.554457	+	0.832213i	$0.312926\pi$
$860$	0	0
$861$	28.8911	0.984606
$862$	0	0
$863$	−30.9910	−1.05495	−0.527473	−	0.849572i	$-0.676860\pi$
$863$	−30.9910	−1.05495	−0.527473	+	0.849572i	$0.676860\pi$
$864$	0	0
$865$	−3.01583	−0.102541
$866$	0	0
$867$	16.5420	0.561795
$868$	0	0
$869$	11.9443	0.405183
$870$	0	0
$871$	15.9110	0.539123
$872$	0	0
$873$	−6.73223	−0.227851
$874$	0	0
$875$	10.6481	0.359972
$876$	0	0
$877$	−11.1380	−0.376103	−0.188052	−	0.982159i	$-0.560217\pi$
$877$	−11.1380	−0.376103	−0.188052	+	0.982159i	$0.560217\pi$
$878$	0	0
$879$	8.01237	0.270251
$880$	0	0
$881$	−20.1387	−0.678491	−0.339245	−	0.940698i	$-0.610172\pi$
$881$	−20.1387	−0.678491	−0.339245	+	0.940698i	$0.610172\pi$
$882$	0	0
$883$	54.0866	1.82016	0.910079	−	0.414435i	$-0.136021\pi$
$883$	54.0866	1.82016	0.910079	+	0.414435i	$0.136021\pi$
$884$	0	0
$885$	−6.90622	−0.232150
$886$	0	0
$887$	−9.52921	−0.319960	−0.159980	−	0.987120i	$-0.551143\pi$
$887$	−9.52921	−0.319960	−0.159980	+	0.987120i	$0.551143\pi$
$888$	0	0
$889$	−20.5599	−0.689557
$890$	0	0
$891$	1.86534	0.0624914
$892$	0	0
$893$	18.9811	0.635179
$894$	0	0
$895$	−7.87193	−0.263130
$896$	0	0
$897$	−2.37826	−0.0794077
$898$	0	0
$899$	11.9731	0.399325
$900$	0	0
$901$	4.78669	0.159468
$902$	0	0
$903$	4.29627	0.142971
$904$	0	0
$905$	11.3818	0.378343
$906$	0	0
$907$	−6.16421	−0.204679	−0.102340	−	0.994750i	$-0.532633\pi$
$907$	−6.16421	−0.204679	−0.102340	+	0.994750i	$0.532633\pi$
$908$	0	0
$909$	−7.45223	−0.247175
$910$	0	0
$911$	34.4827	1.14246	0.571231	−	0.820790i	$-0.306466\pi$
$911$	34.4827	1.14246	0.571231	+	0.820790i	$0.306466\pi$
$912$	0	0
$913$	20.3797	0.674468
$914$	0	0
$915$	2.88473	0.0953663
$916$	0	0
$917$	−19.1342	−0.631866
$918$	0	0
$919$	−6.45550	−0.212947	−0.106474	−	0.994316i	$-0.533956\pi$
$919$	−6.45550	−0.212947	−0.106474	+	0.994316i	$0.533956\pi$
$920$	0	0
$921$	27.0713	0.892029
$922$	0	0
$923$	1.82952	0.0602192
$924$	0	0
$925$	41.2485	1.35624
$926$	0	0
$927$	6.11552	0.200860
$928$	0	0
$929$	−2.22817	−0.0731037	−0.0365519	−	0.999332i	$-0.511637\pi$
$929$	−2.22817	−0.0731037	−0.0365519	+	0.999332i	$0.511637\pi$
$930$	0	0
$931$	5.68641	0.186365
$932$	0	0
$933$	24.6711	0.807694
$934$	0	0
$935$	0.578038	0.0189039
$936$	0	0
$937$	36.3491	1.18747	0.593737	−	0.804659i	$-0.297652\pi$
$937$	36.3491	1.18747	0.593737	+	0.804659i	$0.297652\pi$
$938$	0	0
$939$	3.74346	0.122163
$940$	0	0
$941$	−20.8809	−0.680696	−0.340348	−	0.940299i	$-0.610545\pi$
$941$	−20.8809	−0.680696	−0.340348	+	0.940299i	$0.610545\pi$
$942$	0	0
$943$	−19.4947	−0.634834
$944$	0	0
$945$	1.08761	0.0353801
$946$	0	0
$947$	25.8019	0.838448	0.419224	−	0.907883i	$-0.362302\pi$
$947$	25.8019	0.838448	0.419224	+	0.907883i	$0.362302\pi$
$948$	0	0
$949$	−15.3073	−0.496897
$950$	0	0
$951$	−34.0139	−1.10298
$952$	0	0
$953$	−14.8439	−0.480841	−0.240421	−	0.970669i	$-0.577285\pi$
$953$	−14.8439	−0.480841	−0.240421	+	0.970669i	$0.577285\pi$
$954$	0	0
$955$	−2.42359	−0.0784255
$956$	0	0
$957$	−15.2127	−0.491756
$958$	0	0
$959$	−13.9873	−0.451675
$960$	0	0
$961$	−28.8446	−0.930472
$962$	0	0
$963$	13.0772	0.421408
$964$	0	0
$965$	−1.46000	−0.0469989
$966$	0	0
$967$	34.4866	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$967$	34.4866	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$968$	0	0
$969$	−2.83403	−0.0910421
$970$	0	0
$971$	−19.1329	−0.614004	−0.307002	−	0.951709i	$-0.599326\pi$
$971$	−19.1329	−0.614004	−0.307002	+	0.951709i	$0.599326\pi$
$972$	0	0
$973$	−22.8489	−0.732501
$974$	0	0
$975$	7.10811	0.227642
$976$	0	0
$977$	−19.0244	−0.608646	−0.304323	−	0.952569i	$-0.598430\pi$
$977$	−19.0244	−0.608646	−0.304323	+	0.952569i	$0.598430\pi$
$978$	0	0
$979$	−5.28704	−0.168975
$980$	0	0
$981$	7.00238	0.223569
$982$	0	0
$983$	−57.6794	−1.83969	−0.919843	−	0.392286i	$-0.871684\pi$
$983$	−57.6794	−1.83969	−0.919843	+	0.392286i	$0.871684\pi$
$984$	0	0
$985$	0.454288	0.0144748
$986$	0	0
$987$	10.7666	0.342705
$988$	0	0
$989$	−2.89897	−0.0921818
$990$	0	0
$991$	4.02410	0.127830	0.0639149	−	0.997955i	$-0.479641\pi$
$991$	4.02410	0.127830	0.0639149	+	0.997955i	$0.479641\pi$
$992$	0	0
$993$	−22.0014	−0.698194
$994$	0	0
$995$	3.20690	0.101665
$996$	0	0
$997$	−12.0269	−0.380895	−0.190447	−	0.981697i	$-0.560994\pi$
$997$	−12.0269	−0.380895	−0.190447	+	0.981697i	$0.560994\pi$
$998$	0	0
$999$	8.61076	0.272432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.s.1.3		5
4.3	odd	2		4008.2.a.f.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.f.1.3	✓	5	4.3	odd	2
8016.2.a.s.1.3		5	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.457884 q^{5} +2.37531 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.457884 q^{5} +2.37531 q^{7} +1.00000 q^{9} +1.86534 q^{11} +1.48384 q^{13} +0.457884 q^{15} -0.676771 q^{17} -4.18757 q^{19} -2.37531 q^{21} +1.60277 q^{23} -4.79034 q^{25} -1.00000 q^{27} +8.15542 q^{29} +1.46812 q^{31} -1.86534 q^{33} -1.08761 q^{35} -8.61076 q^{37} -1.48384 q^{39} -12.1631 q^{41} -1.80872 q^{43} -0.457884 q^{45} -4.53272 q^{47} -1.35793 q^{49} +0.676771 q^{51} -7.07284 q^{53} -0.854111 q^{55} +4.18757 q^{57} -15.0829 q^{59} +6.30014 q^{61} +2.37531 q^{63} -0.679427 q^{65} +10.7228 q^{67} -1.60277 q^{69} +1.23296 q^{71} -10.3160 q^{73} +4.79034 q^{75} +4.43076 q^{77} +6.40327 q^{79} +1.00000 q^{81} +10.9254 q^{83} +0.309883 q^{85} -8.15542 q^{87} -2.83435 q^{89} +3.52458 q^{91} -1.46812 q^{93} +1.91742 q^{95} -6.73223 q^{97} +1.86534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * q^99

Introduction

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