LMFDB - Embedded newform 8008.2.a.y.1.1

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			1.1
Root			$3.38813$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38813 q^{3} -4.16385 q^{5} +1.00000 q^{7} +8.47945 q^{9} +O(q^{10})$	$q-3.38813 q^{3} -4.16385 q^{5} +1.00000 q^{7} +8.47945 q^{9} -1.00000 q^{11} -1.00000 q^{13} +14.1077 q^{15} -4.56251 q^{17} -0.578129 q^{19} -3.38813 q^{21} -6.33270 q^{23} +12.3376 q^{25} -18.5651 q^{27} +2.97225 q^{29} +5.14700 q^{31} +3.38813 q^{33} -4.16385 q^{35} +3.64891 q^{37} +3.38813 q^{39} -8.27358 q^{41} -10.1989 q^{43} -35.3071 q^{45} -3.73854 q^{47} +1.00000 q^{49} +15.4584 q^{51} -7.91971 q^{53} +4.16385 q^{55} +1.95878 q^{57} +14.8556 q^{59} -2.44437 q^{61} +8.47945 q^{63} +4.16385 q^{65} -1.18087 q^{67} +21.4560 q^{69} +13.4792 q^{71} +1.09880 q^{73} -41.8016 q^{75} -1.00000 q^{77} +9.93914 q^{79} +37.4627 q^{81} -15.7239 q^{83} +18.9976 q^{85} -10.0704 q^{87} +8.11272 q^{89} -1.00000 q^{91} -17.4387 q^{93} +2.40724 q^{95} +10.1163 q^{97} -8.47945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.38813	−1.95614	−0.978070	−	0.208278i	$-0.933214\pi$
$3$	−3.38813	−1.95614	−0.978070	+	0.208278i	$0.933214\pi$
$4$	0	0
$5$	−4.16385	−1.86213	−0.931065	−	0.364853i	$-0.881119\pi$
$5$	−4.16385	−1.86213	−0.931065	+	0.364853i	$0.881119\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.47945	2.82648
$10$	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
$15$	14.1077	3.64259
$16$	0	0
$17$	−4.56251	−1.10657	−0.553285	−	0.832992i	$-0.686626\pi$
$17$	−4.56251	−1.10657	−0.553285	+	0.832992i	$0.686626\pi$
$18$	0	0
$19$	−0.578129	−0.132632	−0.0663159	−	0.997799i	$-0.521125\pi$
$19$	−0.578129	−0.132632	−0.0663159	+	0.997799i	$0.521125\pi$
$20$	0	0
$21$	−3.38813	−0.739351
$22$	0	0
$23$	−6.33270	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$23$	−6.33270	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$24$	0	0
$25$	12.3376	2.46753
$26$	0	0
$27$	−18.5651	−3.57285
$28$	0	0
$29$	2.97225	0.551933	0.275967	−	0.961167i	$-0.411002\pi$
$29$	2.97225	0.551933	0.275967	+	0.961167i	$0.411002\pi$
$30$	0	0
$31$	5.14700	0.924429	0.462214	−	0.886768i	$-0.347055\pi$
$31$	5.14700	0.924429	0.462214	+	0.886768i	$0.347055\pi$
$32$	0	0
$33$	3.38813	0.589798
$34$	0	0
$35$	−4.16385	−0.703819
$36$	0	0
$37$	3.64891	0.599878	0.299939	−	0.953958i	$-0.403034\pi$
$37$	3.64891	0.599878	0.299939	+	0.953958i	$0.403034\pi$
$38$	0	0
$39$	3.38813	0.542535
$40$	0	0
$41$	−8.27358	−1.29212	−0.646058	−	0.763288i	$-0.723584\pi$
$41$	−8.27358	−1.29212	−0.646058	+	0.763288i	$0.723584\pi$
$42$	0	0
$43$	−10.1989	−1.55532	−0.777660	−	0.628685i	$-0.783594\pi$
$43$	−10.1989	−1.55532	−0.777660	+	0.628685i	$0.783594\pi$
$44$	0	0
$45$	−35.3071	−5.26328
$46$	0	0
$47$	−3.73854	−0.545322	−0.272661	−	0.962110i	$-0.587904\pi$
$47$	−3.73854	−0.545322	−0.272661	+	0.962110i	$0.587904\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.4584	2.16461
$52$	0	0
$53$	−7.91971	−1.08786	−0.543928	−	0.839132i	$-0.683063\pi$
$53$	−7.91971	−1.08786	−0.543928	+	0.839132i	$0.683063\pi$
$54$	0	0
$55$	4.16385	0.561453
$56$	0	0
$57$	1.95878	0.259446
$58$	0	0
$59$	14.8556	1.93404	0.967020	−	0.254701i	$-0.0819771\pi$
$59$	14.8556	1.93404	0.967020	+	0.254701i	$0.0819771\pi$
$60$	0	0
$61$	−2.44437	−0.312970	−0.156485	−	0.987680i	$-0.550016\pi$
$61$	−2.44437	−0.312970	−0.156485	+	0.987680i	$0.550016\pi$
$62$	0	0
$63$	8.47945	1.06831
$64$	0	0
$65$	4.16385	0.516462
$66$	0	0
$67$	−1.18087	−0.144266	−0.0721332	−	0.997395i	$-0.522981\pi$
$67$	−1.18087	−0.144266	−0.0721332	+	0.997395i	$0.522981\pi$
$68$	0	0
$69$	21.4560	2.58300
$70$	0	0
$71$	13.4792	1.59969	0.799844	−	0.600208i	$-0.204916\pi$
$71$	13.4792	1.59969	0.799844	+	0.600208i	$0.204916\pi$
$72$	0	0
$73$	1.09880	0.128605	0.0643024	−	0.997930i	$-0.479518\pi$
$73$	1.09880	0.128605	0.0643024	+	0.997930i	$0.479518\pi$
$74$	0	0
$75$	−41.8016	−4.82683
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.93914	1.11824	0.559120	−	0.829087i	$-0.311139\pi$
$79$	9.93914	1.11824	0.559120	+	0.829087i	$0.311139\pi$
$80$	0	0
$81$	37.4627	4.16252
$82$	0	0
$83$	−15.7239	−1.72592	−0.862960	−	0.505272i	$-0.831392\pi$
$83$	−15.7239	−1.72592	−0.862960	+	0.505272i	$0.831392\pi$
$84$	0	0
$85$	18.9976	2.06058
$86$	0	0
$87$	−10.0704	−1.07966
$88$	0	0
$89$	8.11272	0.859947	0.429973	−	0.902842i	$-0.358523\pi$
$89$	8.11272	0.859947	0.429973	+	0.902842i	$0.358523\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−17.4387	−1.80831
$94$	0	0
$95$	2.40724	0.246978
$96$	0	0
$97$	10.1163	1.02715	0.513575	−	0.858045i	$-0.328321\pi$
$97$	10.1163	1.02715	0.513575	+	0.858045i	$0.328321\pi$
$98$	0	0
$99$	−8.47945	−0.852216
$100$	0	0
$101$	12.5130	1.24509	0.622544	−	0.782585i	$-0.286099\pi$
$101$	12.5130	1.24509	0.622544	+	0.782585i	$0.286099\pi$
$102$	0	0
$103$	−6.10720	−0.601760	−0.300880	−	0.953662i	$-0.597280\pi$
$103$	−6.10720	−0.601760	−0.300880	+	0.953662i	$0.597280\pi$
$104$	0	0
$105$	14.1077	1.37677
$106$	0	0
$107$	4.32012	0.417642	0.208821	−	0.977954i	$-0.433037\pi$
$107$	4.32012	0.417642	0.208821	+	0.977954i	$0.433037\pi$
$108$	0	0
$109$	4.94475	0.473621	0.236810	−	0.971556i	$-0.423898\pi$
$109$	4.94475	0.473621	0.236810	+	0.971556i	$0.423898\pi$
$110$	0	0
$111$	−12.3630	−1.17344
$112$	0	0
$113$	13.5510	1.27477	0.637385	−	0.770545i	$-0.280016\pi$
$113$	13.5510	1.27477	0.637385	+	0.770545i	$0.280016\pi$
$114$	0	0
$115$	26.3684	2.45887
$116$	0	0
$117$	−8.47945	−0.783925
$118$	0	0
$119$	−4.56251	−0.418244
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	28.0320	2.52756
$124$	0	0
$125$	−30.5529	−2.73273
$126$	0	0
$127$	−5.22621	−0.463751	−0.231875	−	0.972745i	$-0.574486\pi$
$127$	−5.22621	−0.463751	−0.231875	+	0.972745i	$0.574486\pi$
$128$	0	0
$129$	34.5553	3.04242
$130$	0	0
$131$	7.90049	0.690269	0.345135	−	0.938553i	$-0.387833\pi$
$131$	7.90049	0.690269	0.345135	+	0.938553i	$0.387833\pi$
$132$	0	0
$133$	−0.578129	−0.0501301
$134$	0	0
$135$	77.3022	6.65312
$136$	0	0
$137$	12.1583	1.03876	0.519379	−	0.854544i	$-0.326163\pi$
$137$	12.1583	1.03876	0.519379	+	0.854544i	$0.326163\pi$
$138$	0	0
$139$	5.65024	0.479247	0.239624	−	0.970866i	$-0.422976\pi$
$139$	5.65024	0.479247	0.239624	+	0.970866i	$0.422976\pi$
$140$	0	0
$141$	12.6667	1.06673
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−12.3760	−1.02777
$146$	0	0
$147$	−3.38813	−0.279448
$148$	0	0
$149$	−8.36077	−0.684941	−0.342470	−	0.939529i	$-0.611264\pi$
$149$	−8.36077	−0.684941	−0.342470	+	0.939529i	$0.611264\pi$
$150$	0	0
$151$	1.72322	0.140233	0.0701167	−	0.997539i	$-0.477663\pi$
$151$	1.72322	0.140233	0.0701167	+	0.997539i	$0.477663\pi$
$152$	0	0
$153$	−38.6875	−3.12770
$154$	0	0
$155$	−21.4313	−1.72141
$156$	0	0
$157$	20.6885	1.65112	0.825561	−	0.564312i	$-0.190859\pi$
$157$	20.6885	1.65112	0.825561	+	0.564312i	$0.190859\pi$
$158$	0	0
$159$	26.8330	2.12800
$160$	0	0
$161$	−6.33270	−0.499087
$162$	0	0
$163$	−4.82463	−0.377894	−0.188947	−	0.981987i	$-0.560507\pi$
$163$	−4.82463	−0.377894	−0.188947	+	0.981987i	$0.560507\pi$
$164$	0	0
$165$	−14.1077	−1.09828
$166$	0	0
$167$	−0.413591	−0.0320046	−0.0160023	−	0.999872i	$-0.505094\pi$
$167$	−0.413591	−0.0320046	−0.0160023	+	0.999872i	$0.505094\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.90221	−0.374882
$172$	0	0
$173$	−6.38148	−0.485175	−0.242588	−	0.970129i	$-0.577996\pi$
$173$	−6.38148	−0.485175	−0.242588	+	0.970129i	$0.577996\pi$
$174$	0	0
$175$	12.3376	0.932638
$176$	0	0
$177$	−50.3329	−3.78325
$178$	0	0
$179$	−21.3150	−1.59316	−0.796580	−	0.604533i	$-0.793360\pi$
$179$	−21.3150	−1.59316	−0.796580	+	0.604533i	$0.793360\pi$
$180$	0	0
$181$	−10.9922	−0.817044	−0.408522	−	0.912749i	$-0.633956\pi$
$181$	−10.9922	−0.817044	−0.408522	+	0.912749i	$0.633956\pi$
$182$	0	0
$183$	8.28186	0.612212
$184$	0	0
$185$	−15.1935	−1.11705
$186$	0	0
$187$	4.56251	0.333644
$188$	0	0
$189$	−18.5651	−1.35041
$190$	0	0
$191$	−14.4677	−1.04684	−0.523422	−	0.852074i	$-0.675345\pi$
$191$	−14.4677	−1.04684	−0.523422	+	0.852074i	$0.675345\pi$
$192$	0	0
$193$	17.8784	1.28692	0.643459	−	0.765481i	$-0.277499\pi$
$193$	17.8784	1.28692	0.643459	+	0.765481i	$0.277499\pi$
$194$	0	0
$195$	−14.1077	−1.01027
$196$	0	0
$197$	13.7107	0.976844	0.488422	−	0.872608i	$-0.337573\pi$
$197$	13.7107	0.976844	0.488422	+	0.872608i	$0.337573\pi$
$198$	0	0
$199$	−1.64030	−0.116277	−0.0581387	−	0.998309i	$-0.518517\pi$
$199$	−1.64030	−0.116277	−0.0581387	+	0.998309i	$0.518517\pi$
$200$	0	0
$201$	4.00095	0.282205
$202$	0	0
$203$	2.97225	0.208611
$204$	0	0
$205$	34.4500	2.40609
$206$	0	0
$207$	−53.6978	−3.73226
$208$	0	0
$209$	0.578129	0.0399900
$210$	0	0
$211$	25.1395	1.73068	0.865338	−	0.501188i	$-0.167104\pi$
$211$	25.1395	1.73068	0.865338	+	0.501188i	$0.167104\pi$
$212$	0	0
$213$	−45.6694	−3.12921
$214$	0	0
$215$	42.4668	2.89621
$216$	0	0
$217$	5.14700	0.349401
$218$	0	0
$219$	−3.72288	−0.251569
$220$	0	0
$221$	4.56251	0.306907
$222$	0	0
$223$	4.50169	0.301456	0.150728	−	0.988575i	$-0.451838\pi$
$223$	4.50169	0.301456	0.150728	+	0.988575i	$0.451838\pi$
$224$	0	0
$225$	104.616	6.97443
$226$	0	0
$227$	6.91855	0.459201	0.229600	−	0.973285i	$-0.426258\pi$
$227$	6.91855	0.459201	0.229600	+	0.973285i	$0.426258\pi$
$228$	0	0
$229$	22.2406	1.46970	0.734851	−	0.678229i	$-0.237252\pi$
$229$	22.2406	1.46970	0.734851	+	0.678229i	$0.237252\pi$
$230$	0	0
$231$	3.38813	0.222923
$232$	0	0
$233$	−4.54086	−0.297482	−0.148741	−	0.988876i	$-0.547522\pi$
$233$	−4.54086	−0.297482	−0.148741	+	0.988876i	$0.547522\pi$
$234$	0	0
$235$	15.5667	1.01546
$236$	0	0
$237$	−33.6751	−2.18743
$238$	0	0
$239$	−23.6439	−1.52940	−0.764700	−	0.644387i	$-0.777113\pi$
$239$	−23.6439	−1.52940	−0.764700	+	0.644387i	$0.777113\pi$
$240$	0	0
$241$	18.5134	1.19255	0.596277	−	0.802779i	$-0.296646\pi$
$241$	18.5134	1.19255	0.596277	+	0.802779i	$0.296646\pi$
$242$	0	0
$243$	−71.2332	−4.56961
$244$	0	0
$245$	−4.16385	−0.266019
$246$	0	0
$247$	0.578129	0.0367855
$248$	0	0
$249$	53.2746	3.37614
$250$	0	0
$251$	0.345313	0.0217960	0.0108980	−	0.999941i	$-0.496531\pi$
$251$	0.345313	0.0217960	0.0108980	+	0.999941i	$0.496531\pi$
$252$	0	0
$253$	6.33270	0.398134
$254$	0	0
$255$	−64.3664	−4.03078
$256$	0	0
$257$	6.59696	0.411507	0.205754	−	0.978604i	$-0.434035\pi$
$257$	6.59696	0.411507	0.205754	+	0.978604i	$0.434035\pi$
$258$	0	0
$259$	3.64891	0.226732
$260$	0	0
$261$	25.2030	1.56003
$262$	0	0
$263$	18.8188	1.16042	0.580210	−	0.814467i	$-0.302971\pi$
$263$	18.8188	1.16042	0.580210	+	0.814467i	$0.302971\pi$
$264$	0	0
$265$	32.9765	2.02573
$266$	0	0
$267$	−27.4870	−1.68218
$268$	0	0
$269$	−18.8982	−1.15224	−0.576121	−	0.817364i	$-0.695434\pi$
$269$	−18.8982	−1.15224	−0.576121	+	0.817364i	$0.695434\pi$
$270$	0	0
$271$	−30.2005	−1.83455	−0.917274	−	0.398257i	$-0.869615\pi$
$271$	−30.2005	−1.83455	−0.917274	+	0.398257i	$0.869615\pi$
$272$	0	0
$273$	3.38813	0.205059
$274$	0	0
$275$	−12.3376	−0.743988
$276$	0	0
$277$	22.0964	1.32764	0.663821	−	0.747892i	$-0.268934\pi$
$277$	22.0964	1.32764	0.663821	+	0.747892i	$0.268934\pi$
$278$	0	0
$279$	43.6437	2.61288
$280$	0	0
$281$	−28.1117	−1.67700	−0.838500	−	0.544901i	$-0.816567\pi$
$281$	−28.1117	−1.67700	−0.838500	+	0.544901i	$0.816567\pi$
$282$	0	0
$283$	−7.10241	−0.422194	−0.211097	−	0.977465i	$-0.567704\pi$
$283$	−7.10241	−0.422194	−0.211097	+	0.977465i	$0.567704\pi$
$284$	0	0
$285$	−8.15606	−0.483123
$286$	0	0
$287$	−8.27358	−0.488374
$288$	0	0
$289$	3.81647	0.224499
$290$	0	0
$291$	−34.2752	−2.00925
$292$	0	0
$293$	−7.05751	−0.412304	−0.206152	−	0.978520i	$-0.566094\pi$
$293$	−7.05751	−0.412304	−0.206152	+	0.978520i	$0.566094\pi$
$294$	0	0
$295$	−61.8567	−3.60143
$296$	0	0
$297$	18.5651	1.07726
$298$	0	0
$299$	6.33270	0.366230
$300$	0	0
$301$	−10.1989	−0.587856
$302$	0	0
$303$	−42.3956	−2.43556
$304$	0	0
$305$	10.1780	0.582790
$306$	0	0
$307$	9.77610	0.557951	0.278976	−	0.960298i	$-0.410005\pi$
$307$	9.77610	0.557951	0.278976	+	0.960298i	$0.410005\pi$
$308$	0	0
$309$	20.6920	1.17713
$310$	0	0
$311$	−2.63596	−0.149472	−0.0747358	−	0.997203i	$-0.523811\pi$
$311$	−2.63596	−0.149472	−0.0747358	+	0.997203i	$0.523811\pi$
$312$	0	0
$313$	3.11565	0.176107	0.0880534	−	0.996116i	$-0.471935\pi$
$313$	3.11565	0.176107	0.0880534	+	0.996116i	$0.471935\pi$
$314$	0	0
$315$	−35.3071	−1.98933
$316$	0	0
$317$	26.3543	1.48020	0.740102	−	0.672495i	$-0.234777\pi$
$317$	26.3543	1.48020	0.740102	+	0.672495i	$0.234777\pi$
$318$	0	0
$319$	−2.97225	−0.166414
$320$	0	0
$321$	−14.6371	−0.816966
$322$	0	0
$323$	2.63772	0.146767
$324$	0	0
$325$	−12.3376	−0.684369
$326$	0	0
$327$	−16.7535	−0.926469
$328$	0	0
$329$	−3.73854	−0.206113
$330$	0	0
$331$	−32.3193	−1.77643	−0.888215	−	0.459428i	$-0.848054\pi$
$331$	−32.3193	−1.77643	−0.888215	+	0.459428i	$0.848054\pi$
$332$	0	0
$333$	30.9408	1.69554
$334$	0	0
$335$	4.91697	0.268643
$336$	0	0
$337$	24.9855	1.36105	0.680525	−	0.732725i	$-0.261752\pi$
$337$	24.9855	1.36105	0.680525	+	0.732725i	$0.261752\pi$
$338$	0	0
$339$	−45.9126	−2.49363
$340$	0	0
$341$	−5.14700	−0.278726
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−89.3397	−4.80989
$346$	0	0
$347$	0.828424	0.0444721	0.0222361	−	0.999753i	$-0.492921\pi$
$347$	0.828424	0.0444721	0.0222361	+	0.999753i	$0.492921\pi$
$348$	0	0
$349$	−1.41117	−0.0755381	−0.0377691	−	0.999286i	$-0.512025\pi$
$349$	−1.41117	−0.0755381	−0.0377691	+	0.999286i	$0.512025\pi$
$350$	0	0
$351$	18.5651	0.990931
$352$	0	0
$353$	13.0527	0.694725	0.347363	−	0.937731i	$-0.387077\pi$
$353$	13.0527	0.694725	0.347363	+	0.937731i	$0.387077\pi$
$354$	0	0
$355$	−56.1254	−2.97883
$356$	0	0
$357$	15.4584	0.818144
$358$	0	0
$359$	−4.03934	−0.213188	−0.106594	−	0.994303i	$-0.533995\pi$
$359$	−4.03934	−0.213188	−0.106594	+	0.994303i	$0.533995\pi$
$360$	0	0
$361$	−18.6658	−0.982409
$362$	0	0
$363$	−3.38813	−0.177831
$364$	0	0
$365$	−4.57524	−0.239479
$366$	0	0
$367$	−25.1989	−1.31537	−0.657686	−	0.753292i	$-0.728465\pi$
$367$	−25.1989	−1.31537	−0.657686	+	0.753292i	$0.728465\pi$
$368$	0	0
$369$	−70.1554	−3.65214
$370$	0	0
$371$	−7.91971	−0.411171
$372$	0	0
$373$	−10.8836	−0.563530	−0.281765	−	0.959483i	$-0.590920\pi$
$373$	−10.8836	−0.563530	−0.281765	+	0.959483i	$0.590920\pi$
$374$	0	0
$375$	103.517	5.34560
$376$	0	0
$377$	−2.97225	−0.153079
$378$	0	0
$379$	−0.232332	−0.0119341	−0.00596706	−	0.999982i	$-0.501899\pi$
$379$	−0.232332	−0.0119341	−0.00596706	+	0.999982i	$0.501899\pi$
$380$	0	0
$381$	17.7071	0.907162
$382$	0	0
$383$	−34.9499	−1.78586	−0.892928	−	0.450200i	$-0.851353\pi$
$383$	−34.9499	−1.78586	−0.892928	+	0.450200i	$0.851353\pi$
$384$	0	0
$385$	4.16385	0.212209
$386$	0	0
$387$	−86.4812	−4.39608
$388$	0	0
$389$	3.22161	0.163342	0.0816710	−	0.996659i	$-0.473974\pi$
$389$	3.22161	0.163342	0.0816710	+	0.996659i	$0.473974\pi$
$390$	0	0
$391$	28.8930	1.46118
$392$	0	0
$393$	−26.7679	−1.35026
$394$	0	0
$395$	−41.3851	−2.08231
$396$	0	0
$397$	−5.31581	−0.266793	−0.133396	−	0.991063i	$-0.542588\pi$
$397$	−5.31581	−0.266793	−0.133396	+	0.991063i	$0.542588\pi$
$398$	0	0
$399$	1.95878	0.0980615
$400$	0	0
$401$	−1.24669	−0.0622566	−0.0311283	−	0.999515i	$-0.509910\pi$
$401$	−1.24669	−0.0622566	−0.0311283	+	0.999515i	$0.509910\pi$
$402$	0	0
$403$	−5.14700	−0.256390
$404$	0	0
$405$	−155.989	−7.75115
$406$	0	0
$407$	−3.64891	−0.180870
$408$	0	0
$409$	15.8690	0.784670	0.392335	−	0.919822i	$-0.371667\pi$
$409$	15.8690	0.784670	0.392335	+	0.919822i	$0.371667\pi$
$410$	0	0
$411$	−41.1941	−2.03195
$412$	0	0
$413$	14.8556	0.730998
$414$	0	0
$415$	65.4719	3.21389
$416$	0	0
$417$	−19.1438	−0.937474
$418$	0	0
$419$	11.4891	0.561282	0.280641	−	0.959813i	$-0.409453\pi$
$419$	11.4891	0.561282	0.280641	+	0.959813i	$0.409453\pi$
$420$	0	0
$421$	−22.7327	−1.10792	−0.553962	−	0.832542i	$-0.686885\pi$
$421$	−22.7327	−1.10792	−0.553962	+	0.832542i	$0.686885\pi$
$422$	0	0
$423$	−31.7008	−1.54134
$424$	0	0
$425$	−56.2906	−2.73050
$426$	0	0
$427$	−2.44437	−0.118291
$428$	0	0
$429$	−3.38813	−0.163581
$430$	0	0
$431$	26.4519	1.27414	0.637071	−	0.770805i	$-0.280146\pi$
$431$	26.4519	1.27414	0.637071	+	0.770805i	$0.280146\pi$
$432$	0	0
$433$	−27.1072	−1.30269	−0.651344	−	0.758782i	$-0.725795\pi$
$433$	−27.1072	−1.30269	−0.651344	+	0.758782i	$0.725795\pi$
$434$	0	0
$435$	41.9316	2.01046
$436$	0	0
$437$	3.66112	0.175135
$438$	0	0
$439$	18.9418	0.904043	0.452022	−	0.892007i	$-0.350703\pi$
$439$	18.9418	0.904043	0.452022	+	0.892007i	$0.350703\pi$
$440$	0	0
$441$	8.47945	0.403783
$442$	0	0
$443$	−34.3524	−1.63213	−0.816065	−	0.577960i	$-0.803849\pi$
$443$	−34.3524	−1.63213	−0.816065	+	0.577960i	$0.803849\pi$
$444$	0	0
$445$	−33.7802	−1.60133
$446$	0	0
$447$	28.3274	1.33984
$448$	0	0
$449$	−11.5809	−0.546535	−0.273267	−	0.961938i	$-0.588104\pi$
$449$	−11.5809	−0.546535	−0.273267	+	0.961938i	$0.588104\pi$
$450$	0	0
$451$	8.27358	0.389588
$452$	0	0
$453$	−5.83849	−0.274316
$454$	0	0
$455$	4.16385	0.195204
$456$	0	0
$457$	15.5678	0.728230	0.364115	−	0.931354i	$-0.381372\pi$
$457$	15.5678	0.728230	0.364115	+	0.931354i	$0.381372\pi$
$458$	0	0
$459$	84.7034	3.95361
$460$	0	0
$461$	13.2297	0.616169	0.308084	−	0.951359i	$-0.400312\pi$
$461$	13.2297	0.616169	0.308084	+	0.951359i	$0.400312\pi$
$462$	0	0
$463$	−23.1636	−1.07651	−0.538253	−	0.842783i	$-0.680915\pi$
$463$	−23.1636	−1.07651	−0.538253	+	0.842783i	$0.680915\pi$
$464$	0	0
$465$	72.6122	3.36731
$466$	0	0
$467$	−22.0130	−1.01864	−0.509319	−	0.860578i	$-0.670103\pi$
$467$	−22.0130	−1.01864	−0.509319	+	0.860578i	$0.670103\pi$
$468$	0	0
$469$	−1.18087	−0.0545276
$470$	0	0
$471$	−70.0954	−3.22983
$472$	0	0
$473$	10.1989	0.468947
$474$	0	0
$475$	−7.13275	−0.327273
$476$	0	0
$477$	−67.1547	−3.07480
$478$	0	0
$479$	−4.80906	−0.219731	−0.109866	−	0.993946i	$-0.535042\pi$
$479$	−4.80906	−0.219731	−0.109866	+	0.993946i	$0.535042\pi$
$480$	0	0
$481$	−3.64891	−0.166376
$482$	0	0
$483$	21.4560	0.976284
$484$	0	0
$485$	−42.1226	−1.91269
$486$	0	0
$487$	16.4710	0.746373	0.373187	−	0.927756i	$-0.378265\pi$
$487$	16.4710	0.746373	0.373187	+	0.927756i	$0.378265\pi$
$488$	0	0
$489$	16.3465	0.739213
$490$	0	0
$491$	−18.1439	−0.818825	−0.409412	−	0.912349i	$-0.634266\pi$
$491$	−18.1439	−0.818825	−0.409412	+	0.912349i	$0.634266\pi$
$492$	0	0
$493$	−13.5609	−0.610753
$494$	0	0
$495$	35.3071	1.58694
$496$	0	0
$497$	13.4792	0.604625
$498$	0	0
$499$	−16.6272	−0.744334	−0.372167	−	0.928166i	$-0.621385\pi$
$499$	−16.6272	−0.744334	−0.372167	+	0.928166i	$0.621385\pi$
$500$	0	0
$501$	1.40130	0.0626055
$502$	0	0
$503$	−31.8164	−1.41862	−0.709311	−	0.704896i	$-0.750994\pi$
$503$	−31.8164	−1.41862	−0.709311	+	0.704896i	$0.750994\pi$
$504$	0	0
$505$	−52.1021	−2.31851
$506$	0	0
$507$	−3.38813	−0.150472
$508$	0	0
$509$	24.8265	1.10042	0.550209	−	0.835027i	$-0.314548\pi$
$509$	24.8265	1.10042	0.550209	+	0.835027i	$0.314548\pi$
$510$	0	0
$511$	1.09880	0.0486081
$512$	0	0
$513$	10.7330	0.473874
$514$	0	0
$515$	25.4295	1.12056
$516$	0	0
$517$	3.73854	0.164421
$518$	0	0
$519$	21.6213	0.949071
$520$	0	0
$521$	20.2143	0.885605	0.442803	−	0.896619i	$-0.353984\pi$
$521$	20.2143	0.885605	0.442803	+	0.896619i	$0.353984\pi$
$522$	0	0
$523$	32.5433	1.42302	0.711509	−	0.702677i	$-0.248012\pi$
$523$	32.5433	1.42302	0.711509	+	0.702677i	$0.248012\pi$
$524$	0	0
$525$	−41.8016	−1.82437
$526$	0	0
$527$	−23.4832	−1.02295
$528$	0	0
$529$	17.1031	0.743614
$530$	0	0
$531$	125.968	5.46653
$532$	0	0
$533$	8.27358	0.358369
$534$	0	0
$535$	−17.9883	−0.777704
$536$	0	0
$537$	72.2181	3.11644
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	0.876603	0.0376881	0.0188440	−	0.999822i	$-0.494001\pi$
$541$	0.876603	0.0376881	0.0188440	+	0.999822i	$0.494001\pi$
$542$	0	0
$543$	37.2430	1.59825
$544$	0	0
$545$	−20.5892	−0.881944
$546$	0	0
$547$	−10.7936	−0.461502	−0.230751	−	0.973013i	$-0.574118\pi$
$547$	−10.7936	−0.461502	−0.230751	+	0.973013i	$0.574118\pi$
$548$	0	0
$549$	−20.7269	−0.884603
$550$	0	0
$551$	−1.71834	−0.0732039
$552$	0	0
$553$	9.93914	0.422655
$554$	0	0
$555$	51.4777	2.18511
$556$	0	0
$557$	28.3459	1.20105	0.600526	−	0.799605i	$-0.294958\pi$
$557$	28.3459	1.20105	0.600526	+	0.799605i	$0.294958\pi$
$558$	0	0
$559$	10.1989	0.431368
$560$	0	0
$561$	−15.4584	−0.652653
$562$	0	0
$563$	18.2804	0.770428	0.385214	−	0.922827i	$-0.374128\pi$
$563$	18.2804	0.770428	0.385214	+	0.922827i	$0.374128\pi$
$564$	0	0
$565$	−56.4243	−2.37379
$566$	0	0
$567$	37.4627	1.57328
$568$	0	0
$569$	−29.4714	−1.23551	−0.617753	−	0.786372i	$-0.711957\pi$
$569$	−29.4714	−1.23551	−0.617753	+	0.786372i	$0.711957\pi$
$570$	0	0
$571$	−45.3596	−1.89824	−0.949120	−	0.314915i	$-0.898024\pi$
$571$	−45.3596	−1.89824	−0.949120	+	0.314915i	$0.898024\pi$
$572$	0	0
$573$	49.0184	2.04777
$574$	0	0
$575$	−78.1306	−3.25827
$576$	0	0
$577$	−14.8665	−0.618901	−0.309450	−	0.950916i	$-0.600145\pi$
$577$	−14.8665	−0.618901	−0.309450	+	0.950916i	$0.600145\pi$
$578$	0	0
$579$	−60.5745	−2.51739
$580$	0	0
$581$	−15.7239	−0.652337
$582$	0	0
$583$	7.91971	0.328001
$584$	0	0
$585$	35.3071	1.45977
$586$	0	0
$587$	19.3463	0.798509	0.399254	−	0.916840i	$-0.369269\pi$
$587$	19.3463	0.798509	0.399254	+	0.916840i	$0.369269\pi$
$588$	0	0
$589$	−2.97563	−0.122609
$590$	0	0
$591$	−46.4535	−1.91084
$592$	0	0
$593$	−11.1479	−0.457789	−0.228895	−	0.973451i	$-0.573511\pi$
$593$	−11.1479	−0.457789	−0.228895	+	0.973451i	$0.573511\pi$
$594$	0	0
$595$	18.9976	0.778826
$596$	0	0
$597$	5.55754	0.227455
$598$	0	0
$599$	13.1330	0.536601	0.268301	−	0.963335i	$-0.413538\pi$
$599$	13.1330	0.536601	0.268301	+	0.963335i	$0.413538\pi$
$600$	0	0
$601$	−27.5749	−1.12480	−0.562401	−	0.826864i	$-0.690122\pi$
$601$	−27.5749	−1.12480	−0.562401	+	0.826864i	$0.690122\pi$
$602$	0	0
$603$	−10.0131	−0.407767
$604$	0	0
$605$	−4.16385	−0.169285
$606$	0	0
$607$	33.5137	1.36028	0.680140	−	0.733082i	$-0.261919\pi$
$607$	33.5137	1.36028	0.680140	+	0.733082i	$0.261919\pi$
$608$	0	0
$609$	−10.0704	−0.408073
$610$	0	0
$611$	3.73854	0.151245
$612$	0	0
$613$	40.8801	1.65113	0.825566	−	0.564305i	$-0.190856\pi$
$613$	40.8801	1.65113	0.825566	+	0.564305i	$0.190856\pi$
$614$	0	0
$615$	−116.721	−4.70665
$616$	0	0
$617$	−24.4563	−0.984572	−0.492286	−	0.870433i	$-0.663839\pi$
$617$	−24.4563	−0.984572	−0.492286	+	0.870433i	$0.663839\pi$
$618$	0	0
$619$	10.8443	0.435870	0.217935	−	0.975963i	$-0.430068\pi$
$619$	10.8443	0.435870	0.217935	+	0.975963i	$0.430068\pi$
$620$	0	0
$621$	117.567	4.71781
$622$	0	0
$623$	8.11272	0.325029
$624$	0	0
$625$	65.5293	2.62117
$626$	0	0
$627$	−1.95878	−0.0782260
$628$	0	0
$629$	−16.6482	−0.663807
$630$	0	0
$631$	−24.5019	−0.975405	−0.487702	−	0.873010i	$-0.662165\pi$
$631$	−24.5019	−0.975405	−0.487702	+	0.873010i	$0.662165\pi$
$632$	0	0
$633$	−85.1761	−3.38544
$634$	0	0
$635$	21.7611	0.863565
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	114.296	4.52149
$640$	0	0
$641$	−36.0544	−1.42406	−0.712031	−	0.702148i	$-0.752225\pi$
$641$	−36.0544	−1.42406	−0.712031	+	0.702148i	$0.752225\pi$
$642$	0	0
$643$	−40.1974	−1.58523	−0.792616	−	0.609721i	$-0.791281\pi$
$643$	−40.1974	−1.58523	−0.792616	+	0.609721i	$0.791281\pi$
$644$	0	0
$645$	−143.883	−5.66539
$646$	0	0
$647$	25.7746	1.01330	0.506652	−	0.862151i	$-0.330883\pi$
$647$	25.7746	1.01330	0.506652	+	0.862151i	$0.330883\pi$
$648$	0	0
$649$	−14.8556	−0.583135
$650$	0	0
$651$	−17.4387	−0.683477
$652$	0	0
$653$	23.7994	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$653$	23.7994	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$654$	0	0
$655$	−32.8965	−1.28537
$656$	0	0
$657$	9.31722	0.363499
$658$	0	0
$659$	11.0585	0.430780	0.215390	−	0.976528i	$-0.430898\pi$
$659$	11.0585	0.430780	0.215390	+	0.976528i	$0.430898\pi$
$660$	0	0
$661$	9.20167	0.357904	0.178952	−	0.983858i	$-0.442729\pi$
$661$	9.20167	0.357904	0.178952	+	0.983858i	$0.442729\pi$
$662$	0	0
$663$	−15.4584	−0.600354
$664$	0	0
$665$	2.40724	0.0933488
$666$	0	0
$667$	−18.8224	−0.728806
$668$	0	0
$669$	−15.2523	−0.589689
$670$	0	0
$671$	2.44437	0.0943639
$672$	0	0
$673$	26.7574	1.03142	0.515711	−	0.856762i	$-0.327528\pi$
$673$	26.7574	1.03142	0.515711	+	0.856762i	$0.327528\pi$
$674$	0	0
$675$	−229.049	−8.81612
$676$	0	0
$677$	46.1583	1.77401	0.887004	−	0.461761i	$-0.152782\pi$
$677$	46.1583	1.77401	0.887004	+	0.461761i	$0.152782\pi$
$678$	0	0
$679$	10.1163	0.388226
$680$	0	0
$681$	−23.4410	−0.898260
$682$	0	0
$683$	23.2642	0.890179	0.445090	−	0.895486i	$-0.353172\pi$
$683$	23.2642	0.890179	0.445090	+	0.895486i	$0.353172\pi$
$684$	0	0
$685$	−50.6255	−1.93430
$686$	0	0
$687$	−75.3542	−2.87494
$688$	0	0
$689$	7.91971	0.301717
$690$	0	0
$691$	−3.09564	−0.117764	−0.0588819	−	0.998265i	$-0.518754\pi$
$691$	−3.09564	−0.117764	−0.0588819	+	0.998265i	$0.518754\pi$
$692$	0	0
$693$	−8.47945	−0.322107
$694$	0	0
$695$	−23.5267	−0.892420
$696$	0	0
$697$	37.7483	1.42982
$698$	0	0
$699$	15.3850	0.581916
$700$	0	0
$701$	31.7833	1.20044	0.600220	−	0.799835i	$-0.295080\pi$
$701$	31.7833	1.20044	0.600220	+	0.799835i	$0.295080\pi$
$702$	0	0
$703$	−2.10954	−0.0795629
$704$	0	0
$705$	−52.7422	−1.98638
$706$	0	0
$707$	12.5130	0.470599
$708$	0	0
$709$	−37.4506	−1.40649	−0.703244	−	0.710948i	$-0.748266\pi$
$709$	−37.4506	−1.40649	−0.703244	+	0.710948i	$0.748266\pi$
$710$	0	0
$711$	84.2784	3.16069
$712$	0	0
$713$	−32.5944	−1.22067
$714$	0	0
$715$	−4.16385	−0.155719
$716$	0	0
$717$	80.1088	2.99172
$718$	0	0
$719$	29.0398	1.08300	0.541500	−	0.840700i	$-0.317856\pi$
$719$	29.0398	1.08300	0.541500	+	0.840700i	$0.317856\pi$
$720$	0	0
$721$	−6.10720	−0.227444
$722$	0	0
$723$	−62.7259	−2.33280
$724$	0	0
$725$	36.6706	1.36191
$726$	0	0
$727$	17.5015	0.649094	0.324547	−	0.945870i	$-0.394788\pi$
$727$	17.5015	0.649094	0.324547	+	0.945870i	$0.394788\pi$
$728$	0	0
$729$	128.960	4.77628
$730$	0	0
$731$	46.5326	1.72107
$732$	0	0
$733$	−30.7054	−1.13413	−0.567066	−	0.823673i	$-0.691921\pi$
$733$	−30.7054	−1.13413	−0.567066	+	0.823673i	$0.691921\pi$
$734$	0	0
$735$	14.1077	0.520370
$736$	0	0
$737$	1.18087	0.0434980
$738$	0	0
$739$	−15.9903	−0.588211	−0.294106	−	0.955773i	$-0.595022\pi$
$739$	−15.9903	−0.588211	−0.294106	+	0.955773i	$0.595022\pi$
$740$	0	0
$741$	−1.95878	−0.0719575
$742$	0	0
$743$	35.6528	1.30798	0.653988	−	0.756505i	$-0.273095\pi$
$743$	35.6528	1.30798	0.653988	+	0.756505i	$0.273095\pi$
$744$	0	0
$745$	34.8130	1.27545
$746$	0	0
$747$	−133.330	−4.87828
$748$	0	0
$749$	4.32012	0.157854
$750$	0	0
$751$	7.26334	0.265043	0.132521	−	0.991180i	$-0.457693\pi$
$751$	7.26334	0.265043	0.132521	+	0.991180i	$0.457693\pi$
$752$	0	0
$753$	−1.16997	−0.0426359
$754$	0	0
$755$	−7.17522	−0.261133
$756$	0	0
$757$	48.8018	1.77373	0.886865	−	0.462028i	$-0.152878\pi$
$757$	48.8018	1.77373	0.886865	+	0.462028i	$0.152878\pi$
$758$	0	0
$759$	−21.4560	−0.778805
$760$	0	0
$761$	−37.9565	−1.37592	−0.687961	−	0.725748i	$-0.741494\pi$
$761$	−37.9565	−1.37592	−0.687961	+	0.725748i	$0.741494\pi$
$762$	0	0
$763$	4.94475	0.179012
$764$	0	0
$765$	161.089	5.82419
$766$	0	0
$767$	−14.8556	−0.536406
$768$	0	0
$769$	−52.9477	−1.90934	−0.954672	−	0.297660i	$-0.903794\pi$
$769$	−52.9477	−1.90934	−0.954672	+	0.297660i	$0.903794\pi$
$770$	0	0
$771$	−22.3514	−0.804965
$772$	0	0
$773$	−35.4012	−1.27329	−0.636647	−	0.771155i	$-0.719679\pi$
$773$	−35.4012	−1.27329	−0.636647	+	0.771155i	$0.719679\pi$
$774$	0	0
$775$	63.5019	2.28105
$776$	0	0
$777$	−12.3630	−0.443520
$778$	0	0
$779$	4.78320	0.171376
$780$	0	0
$781$	−13.4792	−0.482324
$782$	0	0
$783$	−55.1801	−1.97198
$784$	0	0
$785$	−86.1438	−3.07461
$786$	0	0
$787$	−6.57573	−0.234400	−0.117200	−	0.993108i	$-0.537392\pi$
$787$	−6.57573	−0.234400	−0.117200	+	0.993108i	$0.537392\pi$
$788$	0	0
$789$	−63.7607	−2.26994
$790$	0	0
$791$	13.5510	0.481818
$792$	0	0
$793$	2.44437	0.0868022
$794$	0	0
$795$	−111.729	−3.96261
$796$	0	0
$797$	37.3458	1.32286	0.661429	−	0.750008i	$-0.269950\pi$
$797$	37.3458	1.32286	0.661429	+	0.750008i	$0.269950\pi$
$798$	0	0
$799$	17.0571	0.603438
$800$	0	0
$801$	68.7914	2.43062
$802$	0	0
$803$	−1.09880	−0.0387758
$804$	0	0
$805$	26.3684	0.929365
$806$	0	0
$807$	64.0295	2.25395
$808$	0	0
$809$	0.109808	0.00386065	0.00193032	−	0.999998i	$-0.499386\pi$
$809$	0.109808	0.00386065	0.00193032	+	0.999998i	$0.499386\pi$
$810$	0	0
$811$	37.4524	1.31513	0.657566	−	0.753397i	$-0.271586\pi$
$811$	37.4524	1.31513	0.657566	+	0.753397i	$0.271586\pi$
$812$	0	0
$813$	102.323	3.58863
$814$	0	0
$815$	20.0890	0.703688
$816$	0	0
$817$	5.89629	0.206285
$818$	0	0
$819$	−8.47945	−0.296296
$820$	0	0
$821$	−52.5790	−1.83502	−0.917510	−	0.397712i	$-0.869804\pi$
$821$	−52.5790	−1.83502	−0.917510	+	0.397712i	$0.869804\pi$
$822$	0	0
$823$	−44.9995	−1.56858	−0.784292	−	0.620392i	$-0.786973\pi$
$823$	−44.9995	−1.56858	−0.784292	+	0.620392i	$0.786973\pi$
$824$	0	0
$825$	41.8016	1.45534
$826$	0	0
$827$	−23.4245	−0.814550	−0.407275	−	0.913306i	$-0.633521\pi$
$827$	−23.4245	−0.814550	−0.407275	+	0.913306i	$0.633521\pi$
$828$	0	0
$829$	−5.98216	−0.207769	−0.103885	−	0.994589i	$-0.533127\pi$
$829$	−5.98216	−0.207769	−0.103885	+	0.994589i	$0.533127\pi$
$830$	0	0
$831$	−74.8654	−2.59705
$832$	0	0
$833$	−4.56251	−0.158082
$834$	0	0
$835$	1.72213	0.0595968
$836$	0	0
$837$	−95.5545	−3.30285
$838$	0	0
$839$	20.9921	0.724728	0.362364	−	0.932037i	$-0.381970\pi$
$839$	20.9921	0.724728	0.362364	+	0.932037i	$0.381970\pi$
$840$	0	0
$841$	−20.1657	−0.695370
$842$	0	0
$843$	95.2460	3.28045
$844$	0	0
$845$	−4.16385	−0.143241
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	24.0639	0.825871
$850$	0	0
$851$	−23.1075	−0.792114
$852$	0	0
$853$	10.9772	0.375853	0.187926	−	0.982183i	$-0.439823\pi$
$853$	10.9772	0.375853	0.187926	+	0.982183i	$0.439823\pi$
$854$	0	0
$855$	20.4121	0.698078
$856$	0	0
$857$	10.1129	0.345449	0.172725	−	0.984970i	$-0.444743\pi$
$857$	10.1129	0.345449	0.172725	+	0.984970i	$0.444743\pi$
$858$	0	0
$859$	−25.4516	−0.868396	−0.434198	−	0.900817i	$-0.642968\pi$
$859$	−25.4516	−0.868396	−0.434198	+	0.900817i	$0.642968\pi$
$860$	0	0
$861$	28.0320	0.955328
$862$	0	0
$863$	0.579248	0.0197178	0.00985892	−	0.999951i	$-0.496862\pi$
$863$	0.579248	0.0197178	0.00985892	+	0.999951i	$0.496862\pi$
$864$	0	0
$865$	26.5715	0.903460
$866$	0	0
$867$	−12.9307	−0.439150
$868$	0	0
$869$	−9.93914	−0.337162
$870$	0	0
$871$	1.18087	0.0400123
$872$	0	0
$873$	85.7803	2.90322
$874$	0	0
$875$	−30.5529	−1.03288
$876$	0	0
$877$	18.5639	0.626860	0.313430	−	0.949611i	$-0.398522\pi$
$877$	18.5639	0.626860	0.313430	+	0.949611i	$0.398522\pi$
$878$	0	0
$879$	23.9118	0.806525
$880$	0	0
$881$	11.8188	0.398187	0.199094	−	0.979980i	$-0.436200\pi$
$881$	11.8188	0.398187	0.199094	+	0.979980i	$0.436200\pi$
$882$	0	0
$883$	41.9213	1.41076	0.705382	−	0.708827i	$-0.250775\pi$
$883$	41.9213	1.41076	0.705382	+	0.708827i	$0.250775\pi$
$884$	0	0
$885$	209.579	7.04491
$886$	0	0
$887$	18.8509	0.632950	0.316475	−	0.948601i	$-0.397501\pi$
$887$	18.8509	0.632950	0.316475	+	0.948601i	$0.397501\pi$
$888$	0	0
$889$	−5.22621	−0.175281
$890$	0	0
$891$	−37.4627	−1.25505
$892$	0	0
$893$	2.16136	0.0723271
$894$	0	0
$895$	88.7526	2.96667
$896$	0	0
$897$	−21.4560	−0.716396
$898$	0	0
$899$	15.2982	0.510223
$900$	0	0
$901$	36.1337	1.20379
$902$	0	0
$903$	34.5553	1.14993
$904$	0	0
$905$	45.7699	1.52144
$906$	0	0
$907$	−47.6986	−1.58381	−0.791903	−	0.610647i	$-0.790910\pi$
$907$	−47.6986	−1.58381	−0.791903	+	0.610647i	$0.790910\pi$
$908$	0	0
$909$	106.103	3.51922
$910$	0	0
$911$	−18.1443	−0.601148	−0.300574	−	0.953758i	$-0.597178\pi$
$911$	−18.1443	−0.601148	−0.300574	+	0.953758i	$0.597178\pi$
$912$	0	0
$913$	15.7239	0.520385
$914$	0	0
$915$	−34.4844	−1.14002
$916$	0	0
$917$	7.90049	0.260897
$918$	0	0
$919$	20.8502	0.687786	0.343893	−	0.939009i	$-0.388254\pi$
$919$	20.8502	0.687786	0.343893	+	0.939009i	$0.388254\pi$
$920$	0	0
$921$	−33.1227	−1.09143
$922$	0	0
$923$	−13.4792	−0.443674
$924$	0	0
$925$	45.0190	1.48022
$926$	0	0
$927$	−51.7857	−1.70086
$928$	0	0
$929$	22.6593	0.743429	0.371714	−	0.928347i	$-0.378770\pi$
$929$	22.6593	0.743429	0.371714	+	0.928347i	$0.378770\pi$
$930$	0	0
$931$	−0.578129	−0.0189474
$932$	0	0
$933$	8.93099	0.292387
$934$	0	0
$935$	−18.9976	−0.621288
$936$	0	0
$937$	−2.69027	−0.0878873	−0.0439437	−	0.999034i	$-0.513992\pi$
$937$	−2.69027	−0.0878873	−0.0439437	+	0.999034i	$0.513992\pi$
$938$	0	0
$939$	−10.5562	−0.344489
$940$	0	0
$941$	−35.6464	−1.16204	−0.581020	−	0.813889i	$-0.697346\pi$
$941$	−35.6464	−1.16204	−0.581020	+	0.813889i	$0.697346\pi$
$942$	0	0
$943$	52.3941	1.70619
$944$	0	0
$945$	77.3022	2.51464
$946$	0	0
$947$	11.8507	0.385097	0.192548	−	0.981287i	$-0.438325\pi$
$947$	11.8507	0.385097	0.192548	+	0.981287i	$0.438325\pi$
$948$	0	0
$949$	−1.09880	−0.0356686
$950$	0	0
$951$	−89.2918	−2.89549
$952$	0	0
$953$	−30.4931	−0.987768	−0.493884	−	0.869528i	$-0.664423\pi$
$953$	−30.4931	−0.987768	−0.493884	+	0.869528i	$0.664423\pi$
$954$	0	0
$955$	60.2412	1.94936
$956$	0	0
$957$	10.0704	0.325529
$958$	0	0
$959$	12.1583	0.392613
$960$	0	0
$961$	−4.50839	−0.145432
$962$	0	0
$963$	36.6322	1.18046
$964$	0	0
$965$	−74.4431	−2.39641
$966$	0	0
$967$	8.05417	0.259005	0.129502	−	0.991579i	$-0.458662\pi$
$967$	8.05417	0.259005	0.129502	+	0.991579i	$0.458662\pi$
$968$	0	0
$969$	−8.93694	−0.287096
$970$	0	0
$971$	40.5052	1.29987	0.649936	−	0.759989i	$-0.274796\pi$
$971$	40.5052	1.29987	0.649936	+	0.759989i	$0.274796\pi$
$972$	0	0
$973$	5.65024	0.181138
$974$	0	0
$975$	41.8016	1.33872
$976$	0	0
$977$	−15.4470	−0.494194	−0.247097	−	0.968991i	$-0.579477\pi$
$977$	−15.4470	−0.494194	−0.247097	+	0.968991i	$0.579477\pi$
$978$	0	0
$979$	−8.11272	−0.259284
$980$	0	0
$981$	41.9287	1.33868
$982$	0	0
$983$	1.70663	0.0544332	0.0272166	−	0.999630i	$-0.491336\pi$
$983$	1.70663	0.0544332	0.0272166	+	0.999630i	$0.491336\pi$
$984$	0	0
$985$	−57.0891	−1.81901
$986$	0	0
$987$	12.6667	0.403185
$988$	0	0
$989$	64.5867	2.05374
$990$	0	0
$991$	27.0360	0.858828	0.429414	−	0.903108i	$-0.358720\pi$
$991$	27.0360	0.858828	0.429414	+	0.903108i	$0.358720\pi$
$992$	0	0
$993$	109.502	3.47494
$994$	0	0
$995$	6.82995	0.216524
$996$	0	0
$997$	−60.0098	−1.90053	−0.950265	−	0.311441i	$-0.899188\pi$
$997$	−60.0098	−1.90053	−0.950265	+	0.311441i	$0.899188\pi$
$998$	0	0
$999$	−67.7424	−2.14327

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.38813 q^{3} -4.16385 q^{5} +1.00000 q^{7} +8.47945 q^{9} +O(q^{10})\)	\(q-3.38813 q^{3} -4.16385 q^{5} +1.00000 q^{7} +8.47945 q^{9} -1.00000 q^{11} -1.00000 q^{13} +14.1077 q^{15} -4.56251 q^{17} -0.578129 q^{19} -3.38813 q^{21} -6.33270 q^{23} +12.3376 q^{25} -18.5651 q^{27} +2.97225 q^{29} +5.14700 q^{31} +3.38813 q^{33} -4.16385 q^{35} +3.64891 q^{37} +3.38813 q^{39} -8.27358 q^{41} -10.1989 q^{43} -35.3071 q^{45} -3.73854 q^{47} +1.00000 q^{49} +15.4584 q^{51} -7.91971 q^{53} +4.16385 q^{55} +1.95878 q^{57} +14.8556 q^{59} -2.44437 q^{61} +8.47945 q^{63} +4.16385 q^{65} -1.18087 q^{67} +21.4560 q^{69} +13.4792 q^{71} +1.09880 q^{73} -41.8016 q^{75} -1.00000 q^{77} +9.93914 q^{79} +37.4627 q^{81} -15.7239 q^{83} +18.9976 q^{85} -10.0704 q^{87} +8.11272 q^{89} -1.00000 q^{91} -17.4387 q^{93} +2.40724 q^{95} +10.1163 q^{97} -8.47945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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