LMFDB - Embedded newform 8008.2.a.y.1.9

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.374242$ 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)

$f(q)$	$=$	$q+0.374242 q^{3} +3.66821 q^{5} +1.00000 q^{7} -2.85994 q^{9} +O(q^{10})$	$q+0.374242 q^{3} +3.66821 q^{5} +1.00000 q^{7} -2.85994 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.37280 q^{15} +1.83730 q^{17} -7.56300 q^{19} +0.374242 q^{21} -5.64869 q^{23} +8.45573 q^{25} -2.19303 q^{27} -3.73357 q^{29} +1.74855 q^{31} -0.374242 q^{33} +3.66821 q^{35} +5.77028 q^{37} -0.374242 q^{39} -12.4153 q^{41} +4.48406 q^{43} -10.4909 q^{45} -10.1674 q^{47} +1.00000 q^{49} +0.687595 q^{51} -0.412386 q^{53} -3.66821 q^{55} -2.83039 q^{57} -11.7860 q^{59} +5.17539 q^{61} -2.85994 q^{63} -3.66821 q^{65} -13.9379 q^{67} -2.11398 q^{69} +4.90321 q^{71} +14.5424 q^{73} +3.16449 q^{75} -1.00000 q^{77} +9.93053 q^{79} +7.75910 q^{81} +2.71855 q^{83} +6.73960 q^{85} -1.39726 q^{87} -17.7954 q^{89} -1.00000 q^{91} +0.654382 q^{93} -27.7426 q^{95} +10.2664 q^{97} +2.85994 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	0.374242	0.216069	0.108034	−	0.994147i	$-0.465544\pi$
$3$	0.374242	0.216069	0.108034	+	0.994147i	$0.465544\pi$
$4$	0	0
$5$	3.66821	1.64047	0.820236	−	0.572026i	$-0.193842\pi$
$5$	3.66821	1.64047	0.820236	+	0.572026i	$0.193842\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.85994	−0.953314
$10$	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
$15$	1.37280	0.354454
$16$	0	0
$17$	1.83730	0.445611	0.222805	−	0.974863i	$-0.428479\pi$
$17$	1.83730	0.445611	0.222805	+	0.974863i	$0.428479\pi$
$18$	0	0
$19$	−7.56300	−1.73507	−0.867536	−	0.497375i	$-0.834297\pi$
$19$	−7.56300	−1.73507	−0.867536	+	0.497375i	$0.834297\pi$
$20$	0	0
$21$	0.374242	0.0816662
$22$	0	0
$23$	−5.64869	−1.17783	−0.588917	−	0.808193i	$-0.700446\pi$
$23$	−5.64869	−1.17783	−0.588917	+	0.808193i	$0.700446\pi$
$24$	0	0
$25$	8.45573	1.69115
$26$	0	0
$27$	−2.19303	−0.422050
$28$	0	0
$29$	−3.73357	−0.693307	−0.346653	−	0.937993i	$-0.612682\pi$
$29$	−3.73357	−0.693307	−0.346653	+	0.937993i	$0.612682\pi$
$30$	0	0
$31$	1.74855	0.314050	0.157025	−	0.987595i	$-0.449810\pi$
$31$	1.74855	0.314050	0.157025	+	0.987595i	$0.449810\pi$
$32$	0	0
$33$	−0.374242	−0.0651471
$34$	0	0
$35$	3.66821	0.620040
$36$	0	0
$37$	5.77028	0.948629	0.474314	−	0.880356i	$-0.342696\pi$
$37$	5.77028	0.948629	0.474314	+	0.880356i	$0.342696\pi$
$38$	0	0
$39$	−0.374242	−0.0599266
$40$	0	0
$41$	−12.4153	−1.93895	−0.969475	−	0.245190i	$-0.921150\pi$
$41$	−12.4153	−1.93895	−0.969475	+	0.245190i	$0.921150\pi$
$42$	0	0
$43$	4.48406	0.683813	0.341906	−	0.939734i	$-0.388927\pi$
$43$	4.48406	0.683813	0.341906	+	0.939734i	$0.388927\pi$
$44$	0	0
$45$	−10.4909	−1.56388
$46$	0	0
$47$	−10.1674	−1.48306	−0.741531	−	0.670918i	$-0.765900\pi$
$47$	−10.1674	−1.48306	−0.741531	+	0.670918i	$0.765900\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.687595	0.0962825
$52$	0	0
$53$	−0.412386	−0.0566456	−0.0283228	−	0.999599i	$-0.509017\pi$
$53$	−0.412386	−0.0566456	−0.0283228	+	0.999599i	$0.509017\pi$
$54$	0	0
$55$	−3.66821	−0.494621
$56$	0	0
$57$	−2.83039	−0.374894
$58$	0	0
$59$	−11.7860	−1.53441	−0.767206	−	0.641401i	$-0.778353\pi$
$59$	−11.7860	−1.53441	−0.767206	+	0.641401i	$0.778353\pi$
$60$	0	0
$61$	5.17539	0.662640	0.331320	−	0.943518i	$-0.392506\pi$
$61$	5.17539	0.662640	0.331320	+	0.943518i	$0.392506\pi$
$62$	0	0
$63$	−2.85994	−0.360319
$64$	0	0
$65$	−3.66821	−0.454985
$66$	0	0
$67$	−13.9379	−1.70279	−0.851393	−	0.524529i	$-0.824242\pi$
$67$	−13.9379	−1.70279	−0.851393	+	0.524529i	$0.824242\pi$
$68$	0	0
$69$	−2.11398	−0.254493
$70$	0	0
$71$	4.90321	0.581904	0.290952	−	0.956738i	$-0.406028\pi$
$71$	4.90321	0.581904	0.290952	+	0.956738i	$0.406028\pi$
$72$	0	0
$73$	14.5424	1.70206	0.851032	−	0.525115i	$-0.175978\pi$
$73$	14.5424	1.70206	0.851032	+	0.525115i	$0.175978\pi$
$74$	0	0
$75$	3.16449	0.365403
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.93053	1.11727	0.558636	−	0.829413i	$-0.311325\pi$
$79$	9.93053	1.11727	0.558636	+	0.829413i	$0.311325\pi$
$80$	0	0
$81$	7.75910	0.862123
$82$	0	0
$83$	2.71855	0.298400	0.149200	−	0.988807i	$-0.452330\pi$
$83$	2.71855	0.298400	0.149200	+	0.988807i	$0.452330\pi$
$84$	0	0
$85$	6.73960	0.731012
$86$	0	0
$87$	−1.39726	−0.149802
$88$	0	0
$89$	−17.7954	−1.88631	−0.943157	−	0.332348i	$-0.892159\pi$
$89$	−17.7954	−1.88631	−0.943157	+	0.332348i	$0.892159\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0.654382	0.0678562
$94$	0	0
$95$	−27.7426	−2.84634
$96$	0	0
$97$	10.2664	1.04239	0.521196	−	0.853437i	$-0.325486\pi$
$97$	10.2664	1.04239	0.521196	+	0.853437i	$0.325486\pi$
$98$	0	0
$99$	2.85994	0.287435
$100$	0	0
$101$	−17.3328	−1.72468	−0.862338	−	0.506333i	$-0.831001\pi$
$101$	−17.3328	−1.72468	−0.862338	+	0.506333i	$0.831001\pi$
$102$	0	0
$103$	0.615058	0.0606035	0.0303017	−	0.999541i	$-0.490353\pi$
$103$	0.615058	0.0606035	0.0303017	+	0.999541i	$0.490353\pi$
$104$	0	0
$105$	1.37280	0.133971
$106$	0	0
$107$	10.1390	0.980175	0.490088	−	0.871673i	$-0.336965\pi$
$107$	10.1390	0.980175	0.490088	+	0.871673i	$0.336965\pi$
$108$	0	0
$109$	−12.9298	−1.23845	−0.619227	−	0.785212i	$-0.712554\pi$
$109$	−12.9298	−1.23845	−0.619227	+	0.785212i	$0.712554\pi$
$110$	0	0
$111$	2.15948	0.204969
$112$	0	0
$113$	−5.48052	−0.515564	−0.257782	−	0.966203i	$-0.582992\pi$
$113$	−5.48052	−0.515564	−0.257782	+	0.966203i	$0.582992\pi$
$114$	0	0
$115$	−20.7206	−1.93220
$116$	0	0
$117$	2.85994	0.264402
$118$	0	0
$119$	1.83730	0.168425
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.64634	−0.418946
$124$	0	0
$125$	12.6763	1.13381
$126$	0	0
$127$	−0.347348	−0.0308221	−0.0154111	−	0.999881i	$-0.504906\pi$
$127$	−0.347348	−0.0308221	−0.0154111	+	0.999881i	$0.504906\pi$
$128$	0	0
$129$	1.67812	0.147750
$130$	0	0
$131$	5.97212	0.521787	0.260893	−	0.965368i	$-0.415983\pi$
$131$	5.97212	0.521787	0.260893	+	0.965368i	$0.415983\pi$
$132$	0	0
$133$	−7.56300	−0.655796
$134$	0	0
$135$	−8.04450	−0.692361
$136$	0	0
$137$	−6.72558	−0.574605	−0.287303	−	0.957840i	$-0.592759\pi$
$137$	−6.72558	−0.574605	−0.287303	+	0.957840i	$0.592759\pi$
$138$	0	0
$139$	−12.2916	−1.04256	−0.521282	−	0.853385i	$-0.674546\pi$
$139$	−12.2916	−1.04256	−0.521282	+	0.853385i	$0.674546\pi$
$140$	0	0
$141$	−3.80505	−0.320443
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−13.6955	−1.13735
$146$	0	0
$147$	0.374242	0.0308669
$148$	0	0
$149$	−5.93356	−0.486096	−0.243048	−	0.970014i	$-0.578147\pi$
$149$	−5.93356	−0.486096	−0.243048	+	0.970014i	$0.578147\pi$
$150$	0	0
$151$	−19.9900	−1.62676	−0.813381	−	0.581732i	$-0.802375\pi$
$151$	−19.9900	−1.62676	−0.813381	+	0.581732i	$0.802375\pi$
$152$	0	0
$153$	−5.25458	−0.424807
$154$	0	0
$155$	6.41406	0.515189
$156$	0	0
$157$	−8.42600	−0.672468	−0.336234	−	0.941778i	$-0.609153\pi$
$157$	−8.42600	−0.672468	−0.336234	+	0.941778i	$0.609153\pi$
$158$	0	0
$159$	−0.154332	−0.0122393
$160$	0	0
$161$	−5.64869	−0.445179
$162$	0	0
$163$	−7.69740	−0.602907	−0.301454	−	0.953481i	$-0.597472\pi$
$163$	−7.69740	−0.602907	−0.301454	+	0.953481i	$0.597472\pi$
$164$	0	0
$165$	−1.37280	−0.106872
$166$	0	0
$167$	−17.1263	−1.32527	−0.662636	−	0.748941i	$-0.730562\pi$
$167$	−17.1263	−1.32527	−0.662636	+	0.748941i	$0.730562\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	21.6298	1.65407
$172$	0	0
$173$	−3.00911	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$173$	−3.00911	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$174$	0	0
$175$	8.45573	0.639193
$176$	0	0
$177$	−4.41083	−0.331538
$178$	0	0
$179$	−13.9691	−1.04410	−0.522051	−	0.852914i	$-0.674833\pi$
$179$	−13.9691	−1.04410	−0.522051	+	0.852914i	$0.674833\pi$
$180$	0	0
$181$	4.00172	0.297445	0.148723	−	0.988879i	$-0.452484\pi$
$181$	4.00172	0.297445	0.148723	+	0.988879i	$0.452484\pi$
$182$	0	0
$183$	1.93685	0.143176
$184$	0	0
$185$	21.1666	1.55620
$186$	0	0
$187$	−1.83730	−0.134357
$188$	0	0
$189$	−2.19303	−0.159520
$190$	0	0
$191$	5.68270	0.411186	0.205593	−	0.978638i	$-0.434088\pi$
$191$	5.68270	0.411186	0.205593	+	0.978638i	$0.434088\pi$
$192$	0	0
$193$	3.69748	0.266151	0.133075	−	0.991106i	$-0.457515\pi$
$193$	3.69748	0.266151	0.133075	+	0.991106i	$0.457515\pi$
$194$	0	0
$195$	−1.37280	−0.0983079
$196$	0	0
$197$	15.6567	1.11550	0.557748	−	0.830010i	$-0.311666\pi$
$197$	15.6567	1.11550	0.557748	+	0.830010i	$0.311666\pi$
$198$	0	0
$199$	−10.7878	−0.764725	−0.382362	−	0.924012i	$-0.624889\pi$
$199$	−10.7878	−0.764725	−0.382362	+	0.924012i	$0.624889\pi$
$200$	0	0
$201$	−5.21614	−0.367918
$202$	0	0
$203$	−3.73357	−0.262045
$204$	0	0
$205$	−45.5420	−3.18079
$206$	0	0
$207$	16.1549	1.12285
$208$	0	0
$209$	7.56300	0.523144
$210$	0	0
$211$	16.0456	1.10462	0.552312	−	0.833637i	$-0.313746\pi$
$211$	16.0456	1.10462	0.552312	+	0.833637i	$0.313746\pi$
$212$	0	0
$213$	1.83499	0.125731
$214$	0	0
$215$	16.4485	1.12178
$216$	0	0
$217$	1.74855	0.118700
$218$	0	0
$219$	5.44238	0.367762
$220$	0	0
$221$	−1.83730	−0.123590
$222$	0	0
$223$	7.61843	0.510168	0.255084	−	0.966919i	$-0.417897\pi$
$223$	7.61843	0.510168	0.255084	+	0.966919i	$0.417897\pi$
$224$	0	0
$225$	−24.1829	−1.61219
$226$	0	0
$227$	8.73535	0.579786	0.289893	−	0.957059i	$-0.406380\pi$
$227$	8.73535	0.579786	0.289893	+	0.957059i	$0.406380\pi$
$228$	0	0
$229$	−6.25431	−0.413296	−0.206648	−	0.978415i	$-0.566256\pi$
$229$	−6.25431	−0.413296	−0.206648	+	0.978415i	$0.566256\pi$
$230$	0	0
$231$	−0.374242	−0.0246233
$232$	0	0
$233$	28.0190	1.83558	0.917791	−	0.397063i	$-0.129971\pi$
$233$	28.0190	1.83558	0.917791	+	0.397063i	$0.129971\pi$
$234$	0	0
$235$	−37.2960	−2.43292
$236$	0	0
$237$	3.71642	0.241407
$238$	0	0
$239$	10.4528	0.676135	0.338067	−	0.941122i	$-0.390227\pi$
$239$	10.4528	0.676135	0.338067	+	0.941122i	$0.390227\pi$
$240$	0	0
$241$	24.8323	1.59959	0.799793	−	0.600276i	$-0.204942\pi$
$241$	24.8323	1.59959	0.799793	+	0.600276i	$0.204942\pi$
$242$	0	0
$243$	9.48288	0.608327
$244$	0	0
$245$	3.66821	0.234353
$246$	0	0
$247$	7.56300	0.481222
$248$	0	0
$249$	1.01740	0.0644748
$250$	0	0
$251$	14.5513	0.918467	0.459233	−	0.888316i	$-0.348124\pi$
$251$	14.5513	0.918467	0.459233	+	0.888316i	$0.348124\pi$
$252$	0	0
$253$	5.64869	0.355130
$254$	0	0
$255$	2.52224	0.157949
$256$	0	0
$257$	3.41006	0.212714	0.106357	−	0.994328i	$-0.466081\pi$
$257$	3.41006	0.212714	0.106357	+	0.994328i	$0.466081\pi$
$258$	0	0
$259$	5.77028	0.358548
$260$	0	0
$261$	10.6778	0.660939
$262$	0	0
$263$	3.13868	0.193539	0.0967696	−	0.995307i	$-0.469149\pi$
$263$	3.13868	0.193539	0.0967696	+	0.995307i	$0.469149\pi$
$264$	0	0
$265$	−1.51272	−0.0929255
$266$	0	0
$267$	−6.65980	−0.407573
$268$	0	0
$269$	16.5827	1.01106	0.505531	−	0.862808i	$-0.331297\pi$
$269$	16.5827	1.01106	0.505531	+	0.862808i	$0.331297\pi$
$270$	0	0
$271$	−19.3736	−1.17686	−0.588432	−	0.808546i	$-0.700255\pi$
$271$	−19.3736	−1.17686	−0.588432	+	0.808546i	$0.700255\pi$
$272$	0	0
$273$	−0.374242	−0.0226501
$274$	0	0
$275$	−8.45573	−0.509900
$276$	0	0
$277$	19.6335	1.17966	0.589831	−	0.807526i	$-0.299194\pi$
$277$	19.6335	1.17966	0.589831	+	0.807526i	$0.299194\pi$
$278$	0	0
$279$	−5.00077	−0.299388
$280$	0	0
$281$	−28.8706	−1.72228	−0.861138	−	0.508372i	$-0.830248\pi$
$281$	−28.8706	−1.72228	−0.861138	+	0.508372i	$0.830248\pi$
$282$	0	0
$283$	11.0960	0.659587	0.329793	−	0.944053i	$-0.393021\pi$
$283$	11.0960	0.659587	0.329793	+	0.944053i	$0.393021\pi$
$284$	0	0
$285$	−10.3825	−0.615004
$286$	0	0
$287$	−12.4153	−0.732854
$288$	0	0
$289$	−13.6243	−0.801431
$290$	0	0
$291$	3.84210	0.225228
$292$	0	0
$293$	29.3282	1.71337	0.856685	−	0.515839i	$-0.172520\pi$
$293$	29.3282	1.71337	0.856685	+	0.515839i	$0.172520\pi$
$294$	0	0
$295$	−43.2336	−2.51716
$296$	0	0
$297$	2.19303	0.127253
$298$	0	0
$299$	5.64869	0.326672
$300$	0	0
$301$	4.48406	0.258457
$302$	0	0
$303$	−6.48665	−0.372648
$304$	0	0
$305$	18.9844	1.08704
$306$	0	0
$307$	0.588014	0.0335597	0.0167799	−	0.999859i	$-0.494659\pi$
$307$	0.588014	0.0335597	0.0167799	+	0.999859i	$0.494659\pi$
$308$	0	0
$309$	0.230180	0.0130945
$310$	0	0
$311$	−1.85579	−0.105232	−0.0526160	−	0.998615i	$-0.516756\pi$
$311$	−1.85579	−0.105232	−0.0526160	+	0.998615i	$0.516756\pi$
$312$	0	0
$313$	5.23568	0.295938	0.147969	−	0.988992i	$-0.452726\pi$
$313$	5.23568	0.295938	0.147969	+	0.988992i	$0.452726\pi$
$314$	0	0
$315$	−10.4909	−0.591093
$316$	0	0
$317$	−7.83296	−0.439943	−0.219971	−	0.975506i	$-0.570596\pi$
$317$	−7.83296	−0.439943	−0.219971	+	0.975506i	$0.570596\pi$
$318$	0	0
$319$	3.73357	0.209040
$320$	0	0
$321$	3.79444	0.211785
$322$	0	0
$323$	−13.8955	−0.773167
$324$	0	0
$325$	−8.45573	−0.469040
$326$	0	0
$327$	−4.83888	−0.267591
$328$	0	0
$329$	−10.1674	−0.560545
$330$	0	0
$331$	−25.1488	−1.38230	−0.691152	−	0.722710i	$-0.742896\pi$
$331$	−25.1488	−1.38230	−0.691152	+	0.722710i	$0.742896\pi$
$332$	0	0
$333$	−16.5027	−0.904341
$334$	0	0
$335$	−51.1271	−2.79337
$336$	0	0
$337$	−2.03102	−0.110636	−0.0553182	−	0.998469i	$-0.517617\pi$
$337$	−2.03102	−0.110636	−0.0553182	+	0.998469i	$0.517617\pi$
$338$	0	0
$339$	−2.05104	−0.111397
$340$	0	0
$341$	−1.74855	−0.0946895
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.75450	−0.417488
$346$	0	0
$347$	−11.0112	−0.591113	−0.295557	−	0.955325i	$-0.595505\pi$
$347$	−11.0112	−0.591113	−0.295557	+	0.955325i	$0.595505\pi$
$348$	0	0
$349$	−13.3381	−0.713970	−0.356985	−	0.934110i	$-0.616195\pi$
$349$	−13.3381	−0.713970	−0.356985	+	0.934110i	$0.616195\pi$
$350$	0	0
$351$	2.19303	0.117056
$352$	0	0
$353$	22.8646	1.21696	0.608479	−	0.793570i	$-0.291780\pi$
$353$	22.8646	1.21696	0.608479	+	0.793570i	$0.291780\pi$
$354$	0	0
$355$	17.9860	0.954597
$356$	0	0
$357$	0.687595	0.0363914
$358$	0	0
$359$	6.63030	0.349934	0.174967	−	0.984574i	$-0.444018\pi$
$359$	6.63030	0.349934	0.174967	+	0.984574i	$0.444018\pi$
$360$	0	0
$361$	38.1990	2.01047
$362$	0	0
$363$	0.374242	0.0196426
$364$	0	0
$365$	53.3446	2.79219
$366$	0	0
$367$	−8.04286	−0.419834	−0.209917	−	0.977719i	$-0.567319\pi$
$367$	−8.04286	−0.419834	−0.209917	+	0.977719i	$0.567319\pi$
$368$	0	0
$369$	35.5072	1.84843
$370$	0	0
$371$	−0.412386	−0.0214100
$372$	0	0
$373$	−19.3644	−1.00265	−0.501324	−	0.865260i	$-0.667154\pi$
$373$	−19.3644	−1.00265	−0.501324	+	0.865260i	$0.667154\pi$
$374$	0	0
$375$	4.74401	0.244980
$376$	0	0
$377$	3.73357	0.192289
$378$	0	0
$379$	−6.71205	−0.344775	−0.172387	−	0.985029i	$-0.555148\pi$
$379$	−6.71205	−0.344775	−0.172387	+	0.985029i	$0.555148\pi$
$380$	0	0
$381$	−0.129992	−0.00665969
$382$	0	0
$383$	−25.5152	−1.30377	−0.651883	−	0.758319i	$-0.726021\pi$
$383$	−25.5152	−1.30377	−0.651883	+	0.758319i	$0.726021\pi$
$384$	0	0
$385$	−3.66821	−0.186949
$386$	0	0
$387$	−12.8242	−0.651889
$388$	0	0
$389$	−3.87362	−0.196400	−0.0982001	−	0.995167i	$-0.531309\pi$
$389$	−3.87362	−0.196400	−0.0982001	+	0.995167i	$0.531309\pi$
$390$	0	0
$391$	−10.3784	−0.524856
$392$	0	0
$393$	2.23502	0.112742
$394$	0	0
$395$	36.4272	1.83285
$396$	0	0
$397$	22.8561	1.14712	0.573558	−	0.819165i	$-0.305563\pi$
$397$	22.8561	1.14712	0.573558	+	0.819165i	$0.305563\pi$
$398$	0	0
$399$	−2.83039	−0.141697
$400$	0	0
$401$	−16.1528	−0.806632	−0.403316	−	0.915061i	$-0.632142\pi$
$401$	−16.1528	−0.806632	−0.403316	+	0.915061i	$0.632142\pi$
$402$	0	0
$403$	−1.74855	−0.0871017
$404$	0	0
$405$	28.4620	1.41429
$406$	0	0
$407$	−5.77028	−0.286022
$408$	0	0
$409$	25.7234	1.27194	0.635970	−	0.771714i	$-0.280600\pi$
$409$	25.7234	1.27194	0.635970	+	0.771714i	$0.280600\pi$
$410$	0	0
$411$	−2.51699	−0.124154
$412$	0	0
$413$	−11.7860	−0.579953
$414$	0	0
$415$	9.97221	0.489516
$416$	0	0
$417$	−4.60005	−0.225265
$418$	0	0
$419$	−19.1677	−0.936403	−0.468202	−	0.883622i	$-0.655098\pi$
$419$	−19.1677	−0.936403	−0.468202	+	0.883622i	$0.655098\pi$
$420$	0	0
$421$	−6.79107	−0.330977	−0.165488	−	0.986212i	$-0.552920\pi$
$421$	−6.79107	−0.330977	−0.165488	+	0.986212i	$0.552920\pi$
$422$	0	0
$423$	29.0781	1.41382
$424$	0	0
$425$	15.5357	0.753593
$426$	0	0
$427$	5.17539	0.250455
$428$	0	0
$429$	0.374242	0.0180686
$430$	0	0
$431$	−2.17751	−0.104887	−0.0524435	−	0.998624i	$-0.516701\pi$
$431$	−2.17751	−0.104887	−0.0524435	+	0.998624i	$0.516701\pi$
$432$	0	0
$433$	28.8359	1.38576	0.692882	−	0.721051i	$-0.256341\pi$
$433$	28.8359	1.38576	0.692882	+	0.721051i	$0.256341\pi$
$434$	0	0
$435$	−5.12543	−0.245745
$436$	0	0
$437$	42.7211	2.04363
$438$	0	0
$439$	−18.0395	−0.860980	−0.430490	−	0.902595i	$-0.641659\pi$
$439$	−18.0395	−0.860980	−0.430490	+	0.902595i	$0.641659\pi$
$440$	0	0
$441$	−2.85994	−0.136188
$442$	0	0
$443$	−18.6299	−0.885135	−0.442567	−	0.896735i	$-0.645932\pi$
$443$	−18.6299	−0.885135	−0.442567	+	0.896735i	$0.645932\pi$
$444$	0	0
$445$	−65.2773	−3.09444
$446$	0	0
$447$	−2.22059	−0.105030
$448$	0	0
$449$	35.2493	1.66352	0.831760	−	0.555136i	$-0.187334\pi$
$449$	35.2493	1.66352	0.831760	+	0.555136i	$0.187334\pi$
$450$	0	0
$451$	12.4153	0.584615
$452$	0	0
$453$	−7.48108	−0.351492
$454$	0	0
$455$	−3.66821	−0.171968
$456$	0	0
$457$	19.7593	0.924301	0.462150	−	0.886802i	$-0.347078\pi$
$457$	19.7593	0.924301	0.462150	+	0.886802i	$0.347078\pi$
$458$	0	0
$459$	−4.02927	−0.188070
$460$	0	0
$461$	−18.6821	−0.870113	−0.435056	−	0.900403i	$-0.643272\pi$
$461$	−18.6821	−0.870113	−0.435056	+	0.900403i	$0.643272\pi$
$462$	0	0
$463$	33.6272	1.56279	0.781394	−	0.624038i	$-0.214509\pi$
$463$	33.6272	1.56279	0.781394	+	0.624038i	$0.214509\pi$
$464$	0	0
$465$	2.40041	0.111316
$466$	0	0
$467$	−24.2611	−1.12267	−0.561335	−	0.827589i	$-0.689712\pi$
$467$	−24.2611	−1.12267	−0.561335	+	0.827589i	$0.689712\pi$
$468$	0	0
$469$	−13.9379	−0.643593
$470$	0	0
$471$	−3.15336	−0.145299
$472$	0	0
$473$	−4.48406	−0.206177
$474$	0	0
$475$	−63.9507	−2.93426
$476$	0	0
$477$	1.17940	0.0540011
$478$	0	0
$479$	−17.2943	−0.790197	−0.395099	−	0.918639i	$-0.629290\pi$
$479$	−17.2943	−0.790197	−0.395099	+	0.918639i	$0.629290\pi$
$480$	0	0
$481$	−5.77028	−0.263102
$482$	0	0
$483$	−2.11398	−0.0961893
$484$	0	0
$485$	37.6591	1.71001
$486$	0	0
$487$	−27.7873	−1.25916	−0.629581	−	0.776935i	$-0.716774\pi$
$487$	−27.7873	−1.25916	−0.629581	+	0.776935i	$0.716774\pi$
$488$	0	0
$489$	−2.88069	−0.130269
$490$	0	0
$491$	31.5679	1.42464	0.712320	−	0.701855i	$-0.247645\pi$
$491$	31.5679	1.42464	0.712320	+	0.701855i	$0.247645\pi$
$492$	0	0
$493$	−6.85969	−0.308945
$494$	0	0
$495$	10.4909	0.471529
$496$	0	0
$497$	4.90321	0.219939
$498$	0	0
$499$	12.5374	0.561252	0.280626	−	0.959817i	$-0.409458\pi$
$499$	12.5374	0.561252	0.280626	+	0.959817i	$0.409458\pi$
$500$	0	0
$501$	−6.40937	−0.286350
$502$	0	0
$503$	−24.3600	−1.08616	−0.543080	−	0.839681i	$-0.682742\pi$
$503$	−24.3600	−1.08616	−0.543080	+	0.839681i	$0.682742\pi$
$504$	0	0
$505$	−63.5802	−2.82928
$506$	0	0
$507$	0.374242	0.0166207
$508$	0	0
$509$	−29.3745	−1.30200	−0.651000	−	0.759078i	$-0.725650\pi$
$509$	−29.3745	−1.30200	−0.651000	+	0.759078i	$0.725650\pi$
$510$	0	0
$511$	14.5424	0.643319
$512$	0	0
$513$	16.5859	0.732287
$514$	0	0
$515$	2.25616	0.0994182
$516$	0	0
$517$	10.1674	0.447160
$518$	0	0
$519$	−1.12613	−0.0494318
$520$	0	0
$521$	−15.2997	−0.670293	−0.335146	−	0.942166i	$-0.608786\pi$
$521$	−15.2997	−0.670293	−0.335146	+	0.942166i	$0.608786\pi$
$522$	0	0
$523$	43.9372	1.92124	0.960620	−	0.277865i	$-0.0896267\pi$
$523$	43.9372	1.92124	0.960620	+	0.277865i	$0.0896267\pi$
$524$	0	0
$525$	3.16449	0.138110
$526$	0	0
$527$	3.21262	0.139944
$528$	0	0
$529$	8.90774	0.387293
$530$	0	0
$531$	33.7074	1.46278
$532$	0	0
$533$	12.4153	0.537768
$534$	0	0
$535$	37.1920	1.60795
$536$	0	0
$537$	−5.22783	−0.225598
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	31.5557	1.35668	0.678342	−	0.734746i	$-0.262699\pi$
$541$	31.5557	1.35668	0.678342	+	0.734746i	$0.262699\pi$
$542$	0	0
$543$	1.49761	0.0642686
$544$	0	0
$545$	−47.4293	−2.03165
$546$	0	0
$547$	−10.9325	−0.467441	−0.233721	−	0.972304i	$-0.575090\pi$
$547$	−10.9325	−0.467441	−0.233721	+	0.972304i	$0.575090\pi$
$548$	0	0
$549$	−14.8013	−0.631705
$550$	0	0
$551$	28.2370	1.20294
$552$	0	0
$553$	9.93053	0.422289
$554$	0	0
$555$	7.92142	0.336245
$556$	0	0
$557$	1.33827	0.0567044	0.0283522	−	0.999598i	$-0.490974\pi$
$557$	1.33827	0.0567044	0.0283522	+	0.999598i	$0.490974\pi$
$558$	0	0
$559$	−4.48406	−0.189656
$560$	0	0
$561$	−0.687595	−0.0290303
$562$	0	0
$563$	29.9813	1.26356	0.631780	−	0.775148i	$-0.282325\pi$
$563$	29.9813	1.26356	0.631780	+	0.775148i	$0.282325\pi$
$564$	0	0
$565$	−20.1037	−0.845768
$566$	0	0
$567$	7.75910	0.325852
$568$	0	0
$569$	−24.2600	−1.01703	−0.508515	−	0.861053i	$-0.669805\pi$
$569$	−24.2600	−1.01703	−0.508515	+	0.861053i	$0.669805\pi$
$570$	0	0
$571$	13.0856	0.547614	0.273807	−	0.961785i	$-0.411717\pi$
$571$	13.0856	0.547614	0.273807	+	0.961785i	$0.411717\pi$
$572$	0	0
$573$	2.12670	0.0888444
$574$	0	0
$575$	−47.7638	−1.99189
$576$	0	0
$577$	−13.0671	−0.543991	−0.271995	−	0.962299i	$-0.587684\pi$
$577$	−13.0671	−0.543991	−0.271995	+	0.962299i	$0.587684\pi$
$578$	0	0
$579$	1.38375	0.0575068
$580$	0	0
$581$	2.71855	0.112785
$582$	0	0
$583$	0.412386	0.0170793
$584$	0	0
$585$	10.4909	0.433744
$586$	0	0
$587$	−7.35049	−0.303387	−0.151694	−	0.988428i	$-0.548473\pi$
$587$	−7.35049	−0.303387	−0.151694	+	0.988428i	$0.548473\pi$
$588$	0	0
$589$	−13.2243	−0.544899
$590$	0	0
$591$	5.85940	0.241024
$592$	0	0
$593$	40.6531	1.66942	0.834712	−	0.550686i	$-0.185634\pi$
$593$	40.6531	1.66942	0.834712	+	0.550686i	$0.185634\pi$
$594$	0	0
$595$	6.73960	0.276297
$596$	0	0
$597$	−4.03723	−0.165233
$598$	0	0
$599$	−7.16907	−0.292920	−0.146460	−	0.989217i	$-0.546788\pi$
$599$	−7.16907	−0.292920	−0.146460	+	0.989217i	$0.546788\pi$
$600$	0	0
$601$	−20.1446	−0.821715	−0.410858	−	0.911700i	$-0.634771\pi$
$601$	−20.1446	−0.821715	−0.410858	+	0.911700i	$0.634771\pi$
$602$	0	0
$603$	39.8616	1.62329
$604$	0	0
$605$	3.66821	0.149134
$606$	0	0
$607$	−46.0048	−1.86728	−0.933639	−	0.358216i	$-0.883385\pi$
$607$	−46.0048	−1.86728	−0.933639	+	0.358216i	$0.883385\pi$
$608$	0	0
$609$	−1.39726	−0.0566197
$610$	0	0
$611$	10.1674	0.411328
$612$	0	0
$613$	3.33233	0.134592	0.0672958	−	0.997733i	$-0.478563\pi$
$613$	3.33233	0.134592	0.0672958	+	0.997733i	$0.478563\pi$
$614$	0	0
$615$	−17.0437	−0.687269
$616$	0	0
$617$	−29.4619	−1.18609	−0.593045	−	0.805169i	$-0.702075\pi$
$617$	−29.4619	−1.18609	−0.593045	+	0.805169i	$0.702075\pi$
$618$	0	0
$619$	−6.10549	−0.245401	−0.122700	−	0.992444i	$-0.539155\pi$
$619$	−6.10549	−0.245401	−0.122700	+	0.992444i	$0.539155\pi$
$620$	0	0
$621$	12.3878	0.497105
$622$	0	0
$623$	−17.7954	−0.712959
$624$	0	0
$625$	4.22072	0.168829
$626$	0	0
$627$	2.83039	0.113035
$628$	0	0
$629$	10.6017	0.422719
$630$	0	0
$631$	−8.17489	−0.325437	−0.162719	−	0.986673i	$-0.552026\pi$
$631$	−8.17489	−0.325437	−0.162719	+	0.986673i	$0.552026\pi$
$632$	0	0
$633$	6.00493	0.238675
$634$	0	0
$635$	−1.27414	−0.0505628
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−14.0229	−0.554738
$640$	0	0
$641$	5.11505	0.202032	0.101016	−	0.994885i	$-0.467791\pi$
$641$	5.11505	0.202032	0.101016	+	0.994885i	$0.467791\pi$
$642$	0	0
$643$	23.5056	0.926972	0.463486	−	0.886104i	$-0.346598\pi$
$643$	23.5056	0.926972	0.463486	+	0.886104i	$0.346598\pi$
$644$	0	0
$645$	6.15570	0.242380
$646$	0	0
$647$	45.7401	1.79823	0.899114	−	0.437715i	$-0.144212\pi$
$647$	45.7401	1.79823	0.899114	+	0.437715i	$0.144212\pi$
$648$	0	0
$649$	11.7860	0.462642
$650$	0	0
$651$	0.654382	0.0256472
$652$	0	0
$653$	−8.04928	−0.314993	−0.157496	−	0.987520i	$-0.550342\pi$
$653$	−8.04928	−0.314993	−0.157496	+	0.987520i	$0.550342\pi$
$654$	0	0
$655$	21.9070	0.855976
$656$	0	0
$657$	−41.5905	−1.62260
$658$	0	0
$659$	13.1745	0.513205	0.256603	−	0.966517i	$-0.417397\pi$
$659$	13.1745	0.513205	0.256603	+	0.966517i	$0.417397\pi$
$660$	0	0
$661$	20.0583	0.780176	0.390088	−	0.920778i	$-0.372445\pi$
$661$	20.0583	0.780176	0.390088	+	0.920778i	$0.372445\pi$
$662$	0	0
$663$	−0.687595	−0.0267040
$664$	0	0
$665$	−27.7426	−1.07581
$666$	0	0
$667$	21.0898	0.816600
$668$	0	0
$669$	2.85113	0.110231
$670$	0	0
$671$	−5.17539	−0.199794
$672$	0	0
$673$	−17.0122	−0.655771	−0.327886	−	0.944717i	$-0.606336\pi$
$673$	−17.0122	−0.655771	−0.327886	+	0.944717i	$0.606336\pi$
$674$	0	0
$675$	−18.5437	−0.713748
$676$	0	0
$677$	−7.83416	−0.301091	−0.150546	−	0.988603i	$-0.548103\pi$
$677$	−7.83416	−0.301091	−0.150546	+	0.988603i	$0.548103\pi$
$678$	0	0
$679$	10.2664	0.393987
$680$	0	0
$681$	3.26913	0.125273
$682$	0	0
$683$	37.9105	1.45060	0.725302	−	0.688431i	$-0.241700\pi$
$683$	37.9105	1.45060	0.725302	+	0.688431i	$0.241700\pi$
$684$	0	0
$685$	−24.6708	−0.942623
$686$	0	0
$687$	−2.34062	−0.0893004
$688$	0	0
$689$	0.412386	0.0157107
$690$	0	0
$691$	−46.5888	−1.77232	−0.886162	−	0.463376i	$-0.846638\pi$
$691$	−46.5888	−1.77232	−0.886162	+	0.463376i	$0.846638\pi$
$692$	0	0
$693$	2.85994	0.108640
$694$	0	0
$695$	−45.0883	−1.71030
$696$	0	0
$697$	−22.8107	−0.864017
$698$	0	0
$699$	10.4859	0.396612
$700$	0	0
$701$	−36.6628	−1.38474	−0.692368	−	0.721545i	$-0.743432\pi$
$701$	−36.6628	−1.38474	−0.692368	+	0.721545i	$0.743432\pi$
$702$	0	0
$703$	−43.6407	−1.64594
$704$	0	0
$705$	−13.9577	−0.525678
$706$	0	0
$707$	−17.3328	−0.651866
$708$	0	0
$709$	22.5551	0.847076	0.423538	−	0.905878i	$-0.360788\pi$
$709$	22.5551	0.847076	0.423538	+	0.905878i	$0.360788\pi$
$710$	0	0
$711$	−28.4008	−1.06511
$712$	0	0
$713$	−9.87705	−0.369898
$714$	0	0
$715$	3.66821	0.137183
$716$	0	0
$717$	3.91187	0.146091
$718$	0	0
$719$	1.92314	0.0717210	0.0358605	−	0.999357i	$-0.488583\pi$
$719$	1.92314	0.0717210	0.0358605	+	0.999357i	$0.488583\pi$
$720$	0	0
$721$	0.615058	0.0229060
$722$	0	0
$723$	9.29326	0.345620
$724$	0	0
$725$	−31.5701	−1.17248
$726$	0	0
$727$	−2.35686	−0.0874110	−0.0437055	−	0.999044i	$-0.513916\pi$
$727$	−2.35686	−0.0874110	−0.0437055	+	0.999044i	$0.513916\pi$
$728$	0	0
$729$	−19.7284	−0.730682
$730$	0	0
$731$	8.23857	0.304714
$732$	0	0
$733$	5.72337	0.211397	0.105699	−	0.994398i	$-0.466292\pi$
$733$	5.72337	0.211397	0.105699	+	0.994398i	$0.466292\pi$
$734$	0	0
$735$	1.37280	0.0506363
$736$	0	0
$737$	13.9379	0.513409
$738$	0	0
$739$	7.87657	0.289744	0.144872	−	0.989450i	$-0.453723\pi$
$739$	7.87657	0.289744	0.144872	+	0.989450i	$0.453723\pi$
$740$	0	0
$741$	2.83039	0.103977
$742$	0	0
$743$	−4.34939	−0.159564	−0.0797818	−	0.996812i	$-0.525422\pi$
$743$	−4.34939	−0.159564	−0.0797818	+	0.996812i	$0.525422\pi$
$744$	0	0
$745$	−21.7655	−0.797427
$746$	0	0
$747$	−7.77490	−0.284469
$748$	0	0
$749$	10.1390	0.370471
$750$	0	0
$751$	−47.0045	−1.71522	−0.857609	−	0.514303i	$-0.828051\pi$
$751$	−47.0045	−1.71522	−0.857609	+	0.514303i	$0.828051\pi$
$752$	0	0
$753$	5.44568	0.198452
$754$	0	0
$755$	−73.3273	−2.66865
$756$	0	0
$757$	−40.1138	−1.45796	−0.728981	−	0.684534i	$-0.760006\pi$
$757$	−40.1138	−1.45796	−0.728981	+	0.684534i	$0.760006\pi$
$758$	0	0
$759$	2.11398	0.0767325
$760$	0	0
$761$	−44.7843	−1.62343	−0.811715	−	0.584054i	$-0.801465\pi$
$761$	−44.7843	−1.62343	−0.811715	+	0.584054i	$0.801465\pi$
$762$	0	0
$763$	−12.9298	−0.468091
$764$	0	0
$765$	−19.2749	−0.696884
$766$	0	0
$767$	11.7860	0.425569
$768$	0	0
$769$	5.96549	0.215121	0.107560	−	0.994199i	$-0.465696\pi$
$769$	5.96549	0.215121	0.107560	+	0.994199i	$0.465696\pi$
$770$	0	0
$771$	1.27619	0.0459608
$772$	0	0
$773$	35.1293	1.26351	0.631757	−	0.775166i	$-0.282334\pi$
$773$	35.1293	1.26351	0.631757	+	0.775166i	$0.282334\pi$
$774$	0	0
$775$	14.7853	0.531104
$776$	0	0
$777$	2.15948	0.0774709
$778$	0	0
$779$	93.8972	3.36422
$780$	0	0
$781$	−4.90321	−0.175451
$782$	0	0
$783$	8.18785	0.292610
$784$	0	0
$785$	−30.9083	−1.10316
$786$	0	0
$787$	40.3870	1.43964	0.719821	−	0.694160i	$-0.244224\pi$
$787$	40.3870	1.43964	0.719821	+	0.694160i	$0.244224\pi$
$788$	0	0
$789$	1.17462	0.0418177
$790$	0	0
$791$	−5.48052	−0.194865
$792$	0	0
$793$	−5.17539	−0.183783
$794$	0	0
$795$	−0.566122	−0.0200783
$796$	0	0
$797$	−23.3042	−0.825478	−0.412739	−	0.910849i	$-0.635428\pi$
$797$	−23.3042	−0.825478	−0.412739	+	0.910849i	$0.635428\pi$
$798$	0	0
$799$	−18.6805	−0.660869
$800$	0	0
$801$	50.8940	1.79825
$802$	0	0
$803$	−14.5424	−0.513191
$804$	0	0
$805$	−20.7206	−0.730304
$806$	0	0
$807$	6.20592	0.218459
$808$	0	0
$809$	0.269364	0.00947033	0.00473516	−	0.999989i	$-0.498493\pi$
$809$	0.269364	0.00947033	0.00473516	+	0.999989i	$0.498493\pi$
$810$	0	0
$811$	−32.8329	−1.15292	−0.576459	−	0.817126i	$-0.695566\pi$
$811$	−32.8329	−1.15292	−0.576459	+	0.817126i	$0.695566\pi$
$812$	0	0
$813$	−7.25042	−0.254283
$814$	0	0
$815$	−28.2357	−0.989052
$816$	0	0
$817$	−33.9130	−1.18646
$818$	0	0
$819$	2.85994	0.0999345
$820$	0	0
$821$	39.0940	1.36439	0.682195	−	0.731170i	$-0.261026\pi$
$821$	39.0940	1.36439	0.682195	+	0.731170i	$0.261026\pi$
$822$	0	0
$823$	38.6365	1.34678	0.673392	−	0.739285i	$-0.264836\pi$
$823$	38.6365	1.34678	0.673392	+	0.739285i	$0.264836\pi$
$824$	0	0
$825$	−3.16449	−0.110173
$826$	0	0
$827$	2.02062	0.0702638	0.0351319	−	0.999383i	$-0.488815\pi$
$827$	2.02062	0.0702638	0.0351319	+	0.999383i	$0.488815\pi$
$828$	0	0
$829$	−21.6630	−0.752386	−0.376193	−	0.926541i	$-0.622767\pi$
$829$	−21.6630	−0.752386	−0.376193	+	0.926541i	$0.622767\pi$
$830$	0	0
$831$	7.34768	0.254888
$832$	0	0
$833$	1.83730	0.0636587
$834$	0	0
$835$	−62.8228	−2.17407
$836$	0	0
$837$	−3.83464	−0.132545
$838$	0	0
$839$	−42.9752	−1.48367	−0.741835	−	0.670583i	$-0.766044\pi$
$839$	−42.9752	−1.48367	−0.741835	+	0.670583i	$0.766044\pi$
$840$	0	0
$841$	−15.0605	−0.519326
$842$	0	0
$843$	−10.8046	−0.372130
$844$	0	0
$845$	3.66821	0.126190
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	4.15257	0.142516
$850$	0	0
$851$	−32.5946	−1.11733
$852$	0	0
$853$	24.4508	0.837178	0.418589	−	0.908176i	$-0.362525\pi$
$853$	24.4508	0.837178	0.418589	+	0.908176i	$0.362525\pi$
$854$	0	0
$855$	79.3424	2.71345
$856$	0	0
$857$	−34.5716	−1.18094	−0.590471	−	0.807059i	$-0.701058\pi$
$857$	−34.5716	−1.18094	−0.590471	+	0.807059i	$0.701058\pi$
$858$	0	0
$859$	−32.5539	−1.11073	−0.555363	−	0.831608i	$-0.687421\pi$
$859$	−32.5539	−1.11073	−0.555363	+	0.831608i	$0.687421\pi$
$860$	0	0
$861$	−4.64634	−0.158347
$862$	0	0
$863$	−26.7865	−0.911822	−0.455911	−	0.890025i	$-0.650686\pi$
$863$	−26.7865	−0.911822	−0.455911	+	0.890025i	$0.650686\pi$
$864$	0	0
$865$	−11.0380	−0.375304
$866$	0	0
$867$	−5.09879	−0.173164
$868$	0	0
$869$	−9.93053	−0.336870
$870$	0	0
$871$	13.9379	0.472268
$872$	0	0
$873$	−29.3612	−0.993727
$874$	0	0
$875$	12.6763	0.428538
$876$	0	0
$877$	25.9852	0.877458	0.438729	−	0.898619i	$-0.355429\pi$
$877$	25.9852	0.877458	0.438729	+	0.898619i	$0.355429\pi$
$878$	0	0
$879$	10.9758	0.370206
$880$	0	0
$881$	37.9108	1.27725	0.638623	−	0.769520i	$-0.279504\pi$
$881$	37.9108	1.27725	0.638623	+	0.769520i	$0.279504\pi$
$882$	0	0
$883$	25.5544	0.859974	0.429987	−	0.902835i	$-0.358518\pi$
$883$	25.5544	0.859974	0.429987	+	0.902835i	$0.358518\pi$
$884$	0	0
$885$	−16.1798	−0.543879
$886$	0	0
$887$	−36.8767	−1.23820	−0.619100	−	0.785312i	$-0.712502\pi$
$887$	−36.8767	−1.23820	−0.619100	+	0.785312i	$0.712502\pi$
$888$	0	0
$889$	−0.347348	−0.0116497
$890$	0	0
$891$	−7.75910	−0.259940
$892$	0	0
$893$	76.8958	2.57322
$894$	0	0
$895$	−51.2416	−1.71282
$896$	0	0
$897$	2.11398	0.0705836
$898$	0	0
$899$	−6.52835	−0.217733
$900$	0	0
$901$	−0.757678	−0.0252419
$902$	0	0
$903$	1.67812	0.0558444
$904$	0	0
$905$	14.6791	0.487951
$906$	0	0
$907$	−19.0484	−0.632493	−0.316246	−	0.948677i	$-0.602423\pi$
$907$	−19.0484	−0.632493	−0.316246	+	0.948677i	$0.602423\pi$
$908$	0	0
$909$	49.5708	1.64416
$910$	0	0
$911$	49.6121	1.64372	0.821862	−	0.569687i	$-0.192936\pi$
$911$	49.6121	1.64372	0.821862	+	0.569687i	$0.192936\pi$
$912$	0	0
$913$	−2.71855	−0.0899709
$914$	0	0
$915$	7.10475	0.234876
$916$	0	0
$917$	5.97212	0.197217
$918$	0	0
$919$	4.79980	0.158331	0.0791654	−	0.996861i	$-0.474774\pi$
$919$	4.79980	0.158331	0.0791654	+	0.996861i	$0.474774\pi$
$920$	0	0
$921$	0.220059	0.00725120
$922$	0	0
$923$	−4.90321	−0.161391
$924$	0	0
$925$	48.7920	1.60427
$926$	0	0
$927$	−1.75903	−0.0577742
$928$	0	0
$929$	37.7339	1.23801	0.619004	−	0.785388i	$-0.287537\pi$
$929$	37.7339	1.23801	0.619004	+	0.785388i	$0.287537\pi$
$930$	0	0
$931$	−7.56300	−0.247867
$932$	0	0
$933$	−0.694513	−0.0227373
$934$	0	0
$935$	−6.73960	−0.220408
$936$	0	0
$937$	56.2845	1.83874	0.919368	−	0.393399i	$-0.128701\pi$
$937$	56.2845	1.83874	0.919368	+	0.393399i	$0.128701\pi$
$938$	0	0
$939$	1.95941	0.0639429
$940$	0	0
$941$	−33.0073	−1.07601	−0.538003	−	0.842943i	$-0.680821\pi$
$941$	−33.0073	−1.07601	−0.538003	+	0.842943i	$0.680821\pi$
$942$	0	0
$943$	70.1304	2.28376
$944$	0	0
$945$	−8.04450	−0.261688
$946$	0	0
$947$	8.43995	0.274261	0.137131	−	0.990553i	$-0.456212\pi$
$947$	8.43995	0.274261	0.137131	+	0.990553i	$0.456212\pi$
$948$	0	0
$949$	−14.5424	−0.472067
$950$	0	0
$951$	−2.93142	−0.0950578
$952$	0	0
$953$	4.97118	0.161032	0.0805162	−	0.996753i	$-0.474343\pi$
$953$	4.97118	0.161032	0.0805162	+	0.996753i	$0.474343\pi$
$954$	0	0
$955$	20.8453	0.674539
$956$	0	0
$957$	1.39726	0.0451669
$958$	0	0
$959$	−6.72558	−0.217180
$960$	0	0
$961$	−27.9426	−0.901373
$962$	0	0
$963$	−28.9970	−0.934415
$964$	0	0
$965$	13.5631	0.436613
$966$	0	0
$967$	−5.35840	−0.172315	−0.0861573	−	0.996282i	$-0.527459\pi$
$967$	−5.35840	−0.172315	−0.0861573	+	0.996282i	$0.527459\pi$
$968$	0	0
$969$	−5.20028	−0.167057
$970$	0	0
$971$	6.57700	0.211066	0.105533	−	0.994416i	$-0.466345\pi$
$971$	6.57700	0.211066	0.105533	+	0.994416i	$0.466345\pi$
$972$	0	0
$973$	−12.2916	−0.394052
$974$	0	0
$975$	−3.16449	−0.101345
$976$	0	0
$977$	−19.3743	−0.619837	−0.309919	−	0.950763i	$-0.600302\pi$
$977$	−19.3743	−0.619837	−0.309919	+	0.950763i	$0.600302\pi$
$978$	0	0
$979$	17.7954	0.568745
$980$	0	0
$981$	36.9786	1.18064
$982$	0	0
$983$	−17.8748	−0.570118	−0.285059	−	0.958510i	$-0.592013\pi$
$983$	−17.8748	−0.570118	−0.285059	+	0.958510i	$0.592013\pi$
$984$	0	0
$985$	57.4321	1.82994
$986$	0	0
$987$	−3.80505	−0.121116
$988$	0	0
$989$	−25.3291	−0.805418
$990$	0	0
$991$	37.7034	1.19769	0.598843	−	0.800866i	$-0.295627\pi$
$991$	37.7034	1.19769	0.598843	+	0.800866i	$0.295627\pi$
$992$	0	0
$993$	−9.41173	−0.298672
$994$	0	0
$995$	−39.5718	−1.25451
$996$	0	0
$997$	20.3448	0.644326	0.322163	−	0.946684i	$-0.395590\pi$
$997$	20.3448	0.644326	0.322163	+	0.946684i	$0.395590\pi$
$998$	0	0
$999$	−12.6544	−0.400369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.9	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.9	✓	14	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)

\(f(q)\)	\(=\)	\(q+0.374242 q^{3} +3.66821 q^{5} +1.00000 q^{7} -2.85994 q^{9} +O(q^{10})\)	\(q+0.374242 q^{3} +3.66821 q^{5} +1.00000 q^{7} -2.85994 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.37280 q^{15} +1.83730 q^{17} -7.56300 q^{19} +0.374242 q^{21} -5.64869 q^{23} +8.45573 q^{25} -2.19303 q^{27} -3.73357 q^{29} +1.74855 q^{31} -0.374242 q^{33} +3.66821 q^{35} +5.77028 q^{37} -0.374242 q^{39} -12.4153 q^{41} +4.48406 q^{43} -10.4909 q^{45} -10.1674 q^{47} +1.00000 q^{49} +0.687595 q^{51} -0.412386 q^{53} -3.66821 q^{55} -2.83039 q^{57} -11.7860 q^{59} +5.17539 q^{61} -2.85994 q^{63} -3.66821 q^{65} -13.9379 q^{67} -2.11398 q^{69} +4.90321 q^{71} +14.5424 q^{73} +3.16449 q^{75} -1.00000 q^{77} +9.93053 q^{79} +7.75910 q^{81} +2.71855 q^{83} +6.73960 q^{85} -1.39726 q^{87} -17.7954 q^{89} -1.00000 q^{91} +0.654382 q^{93} -27.7426 q^{95} +10.2664 q^{97} +2.85994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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