LMFDB - Embedded newform 1336.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 5x^{4} + 25x^{3} - 4x^{2} - 17x + 1 $ x^7 - x^6 - 9x^5 + 5x^4 + 25x^3 - 4x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.0583180$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.941682 q^{3} -2.86037 q^{5} +1.50273 q^{7} -2.11324 q^{9} +O(q^{10})$	$q-0.941682 q^{3} -2.86037 q^{5} +1.50273 q^{7} -2.11324 q^{9} +4.17608 q^{11} +2.74374 q^{13} +2.69356 q^{15} -0.166811 q^{17} -3.61310 q^{19} -1.41509 q^{21} -4.35078 q^{23} +3.18173 q^{25} +4.81504 q^{27} +0.193366 q^{29} +2.48681 q^{31} -3.93254 q^{33} -4.29837 q^{35} +1.39549 q^{37} -2.58373 q^{39} -3.49554 q^{41} -3.12610 q^{43} +6.04464 q^{45} -9.20902 q^{47} -4.74180 q^{49} +0.157083 q^{51} -6.24906 q^{53} -11.9452 q^{55} +3.40239 q^{57} -10.2002 q^{59} +3.14524 q^{61} -3.17562 q^{63} -7.84811 q^{65} -10.1979 q^{67} +4.09705 q^{69} -4.35631 q^{71} +0.531341 q^{73} -2.99618 q^{75} +6.27553 q^{77} -16.4470 q^{79} +1.80547 q^{81} +0.923810 q^{83} +0.477142 q^{85} -0.182089 q^{87} +2.34490 q^{89} +4.12310 q^{91} -2.34178 q^{93} +10.3348 q^{95} +7.23922 q^{97} -8.82505 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9	$ 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) $ 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * 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
$3$	−0.941682	−0.543680	−0.271840	−	0.962342i	$-0.587632\pi$
$3$	−0.941682	−0.543680	−0.271840	+	0.962342i	$0.587632\pi$
$4$	0	0
$5$	−2.86037	−1.27920	−0.639599	−	0.768709i	$-0.720900\pi$
$5$	−2.86037	−1.27920	−0.639599	+	0.768709i	$0.720900\pi$
$6$	0	0
$7$	1.50273	0.567979	0.283989	−	0.958827i	$-0.408342\pi$
$7$	1.50273	0.567979	0.283989	+	0.958827i	$0.408342\pi$
$8$	0	0
$9$	−2.11324	−0.704412
$10$	0	0
$11$	4.17608	1.25914	0.629568	−	0.776945i	$-0.283232\pi$
$11$	4.17608	1.25914	0.629568	+	0.776945i	$0.283232\pi$
$12$	0	0
$13$	2.74374	0.760976	0.380488	−	0.924786i	$-0.375756\pi$
$13$	2.74374	0.760976	0.380488	+	0.924786i	$0.375756\pi$
$14$	0	0
$15$	2.69356	0.695475
$16$	0	0
$17$	−0.166811	−0.0404577	−0.0202288	−	0.999795i	$-0.506439\pi$
$17$	−0.166811	−0.0404577	−0.0202288	+	0.999795i	$0.506439\pi$
$18$	0	0
$19$	−3.61310	−0.828902	−0.414451	−	0.910072i	$-0.636026\pi$
$19$	−3.61310	−0.828902	−0.414451	+	0.910072i	$0.636026\pi$
$20$	0	0
$21$	−1.41509	−0.308799
$22$	0	0
$23$	−4.35078	−0.907200	−0.453600	−	0.891205i	$-0.649861\pi$
$23$	−4.35078	−0.907200	−0.453600	+	0.891205i	$0.649861\pi$
$24$	0	0
$25$	3.18173	0.636347
$26$	0	0
$27$	4.81504	0.926655
$28$	0	0
$29$	0.193366	0.0359071	0.0179536	−	0.999839i	$-0.494285\pi$
$29$	0.193366	0.0359071	0.0179536	+	0.999839i	$0.494285\pi$
$30$	0	0
$31$	2.48681	0.446643	0.223322	−	0.974745i	$-0.428310\pi$
$31$	2.48681	0.446643	0.223322	+	0.974745i	$0.428310\pi$
$32$	0	0
$33$	−3.93254	−0.684568
$34$	0	0
$35$	−4.29837	−0.726557
$36$	0	0
$37$	1.39549	0.229418	0.114709	−	0.993399i	$-0.463406\pi$
$37$	1.39549	0.229418	0.114709	+	0.993399i	$0.463406\pi$
$38$	0	0
$39$	−2.58373	−0.413728
$40$	0	0
$41$	−3.49554	−0.545912	−0.272956	−	0.962027i	$-0.588001\pi$
$41$	−3.49554	−0.545912	−0.272956	+	0.962027i	$0.588001\pi$
$42$	0	0
$43$	−3.12610	−0.476725	−0.238363	−	0.971176i	$-0.576611\pi$
$43$	−3.12610	−0.476725	−0.238363	+	0.971176i	$0.576611\pi$
$44$	0	0
$45$	6.04464	0.901082
$46$	0	0
$47$	−9.20902	−1.34327	−0.671637	−	0.740881i	$-0.734408\pi$
$47$	−9.20902	−1.34327	−0.671637	+	0.740881i	$0.734408\pi$
$48$	0	0
$49$	−4.74180	−0.677400
$50$	0	0
$51$	0.157083	0.0219960
$52$	0	0
$53$	−6.24906	−0.858375	−0.429187	−	0.903215i	$-0.641200\pi$
$53$	−6.24906	−0.858375	−0.429187	+	0.903215i	$0.641200\pi$
$54$	0	0
$55$	−11.9452	−1.61069
$56$	0	0
$57$	3.40239	0.450658
$58$	0	0
$59$	−10.2002	−1.32795	−0.663974	−	0.747756i	$-0.731131\pi$
$59$	−10.2002	−1.32795	−0.663974	+	0.747756i	$0.731131\pi$
$60$	0	0
$61$	3.14524	0.402706	0.201353	−	0.979519i	$-0.435466\pi$
$61$	3.14524	0.402706	0.201353	+	0.979519i	$0.435466\pi$
$62$	0	0
$63$	−3.17562	−0.400091
$64$	0	0
$65$	−7.84811	−0.973438
$66$	0	0
$67$	−10.1979	−1.24587	−0.622937	−	0.782272i	$-0.714061\pi$
$67$	−10.1979	−1.24587	−0.622937	+	0.782272i	$0.714061\pi$
$68$	0	0
$69$	4.09705	0.493227
$70$	0	0
$71$	−4.35631	−0.516999	−0.258500	−	0.966011i	$-0.583228\pi$
$71$	−4.35631	−0.516999	−0.258500	+	0.966011i	$0.583228\pi$
$72$	0	0
$73$	0.531341	0.0621888	0.0310944	−	0.999516i	$-0.490101\pi$
$73$	0.531341	0.0621888	0.0310944	+	0.999516i	$0.490101\pi$
$74$	0	0
$75$	−2.99618	−0.345969
$76$	0	0
$77$	6.27553	0.715163
$78$	0	0
$79$	−16.4470	−1.85043	−0.925215	−	0.379443i	$-0.876116\pi$
$79$	−16.4470	−1.85043	−0.925215	+	0.379443i	$0.876116\pi$
$80$	0	0
$81$	1.80547	0.200607
$82$	0	0
$83$	0.923810	0.101401	0.0507007	−	0.998714i	$-0.483855\pi$
$83$	0.923810	0.101401	0.0507007	+	0.998714i	$0.483855\pi$
$84$	0	0
$85$	0.477142	0.0517534
$86$	0	0
$87$	−0.182089	−0.0195220
$88$	0	0
$89$	2.34490	0.248559	0.124280	−	0.992247i	$-0.460338\pi$
$89$	2.34490	0.248559	0.124280	+	0.992247i	$0.460338\pi$
$90$	0	0
$91$	4.12310	0.432218
$92$	0	0
$93$	−2.34178	−0.242831
$94$	0	0
$95$	10.3348	1.06033
$96$	0	0
$97$	7.23922	0.735032	0.367516	−	0.930017i	$-0.380208\pi$
$97$	7.23922	0.735032	0.367516	+	0.930017i	$0.380208\pi$
$98$	0	0
$99$	−8.82505	−0.886951
$100$	0	0
$101$	5.73720	0.570873	0.285436	−	0.958398i	$-0.407861\pi$
$101$	5.73720	0.570873	0.285436	+	0.958398i	$0.407861\pi$
$102$	0	0
$103$	−3.13868	−0.309264	−0.154632	−	0.987972i	$-0.549419\pi$
$103$	−3.13868	−0.309264	−0.154632	+	0.987972i	$0.549419\pi$
$104$	0	0
$105$	4.04770	0.395015
$106$	0	0
$107$	−3.33396	−0.322306	−0.161153	−	0.986929i	$-0.551521\pi$
$107$	−3.33396	−0.322306	−0.161153	+	0.986929i	$0.551521\pi$
$108$	0	0
$109$	18.7745	1.79827	0.899134	−	0.437673i	$-0.144197\pi$
$109$	18.7745	1.79827	0.899134	+	0.437673i	$0.144197\pi$
$110$	0	0
$111$	−1.31411	−0.124730
$112$	0	0
$113$	−18.7690	−1.76564	−0.882820	−	0.469711i	$-0.844358\pi$
$113$	−18.7690	−1.76564	−0.882820	+	0.469711i	$0.844358\pi$
$114$	0	0
$115$	12.4448	1.16049
$116$	0	0
$117$	−5.79816	−0.536040
$118$	0	0
$119$	−0.250672	−0.0229791
$120$	0	0
$121$	6.43969	0.585426
$122$	0	0
$123$	3.29169	0.296802
$124$	0	0
$125$	5.20092	0.465185
$126$	0	0
$127$	7.09768	0.629818	0.314909	−	0.949122i	$-0.398026\pi$
$127$	7.09768	0.629818	0.314909	+	0.949122i	$0.398026\pi$
$128$	0	0
$129$	2.94379	0.259186
$130$	0	0
$131$	−10.2337	−0.894126	−0.447063	−	0.894502i	$-0.647530\pi$
$131$	−10.2337	−0.894126	−0.447063	+	0.894502i	$0.647530\pi$
$132$	0	0
$133$	−5.42952	−0.470799
$134$	0	0
$135$	−13.7728	−1.18538
$136$	0	0
$137$	−14.0232	−1.19808	−0.599039	−	0.800720i	$-0.704451\pi$
$137$	−14.0232	−1.19808	−0.599039	+	0.800720i	$0.704451\pi$
$138$	0	0
$139$	4.10301	0.348013	0.174006	−	0.984745i	$-0.444329\pi$
$139$	4.10301	0.348013	0.174006	+	0.984745i	$0.444329\pi$
$140$	0	0
$141$	8.67197	0.730311
$142$	0	0
$143$	11.4581	0.958173
$144$	0	0
$145$	−0.553098	−0.0459323
$146$	0	0
$147$	4.46527	0.368289
$148$	0	0
$149$	−22.7985	−1.86772	−0.933861	−	0.357635i	$-0.883583\pi$
$149$	−22.7985	−1.86772	−0.933861	+	0.357635i	$0.883583\pi$
$150$	0	0
$151$	−21.2721	−1.73110	−0.865550	−	0.500823i	$-0.833031\pi$
$151$	−21.2721	−1.73110	−0.865550	+	0.500823i	$0.833031\pi$
$152$	0	0
$153$	0.352511	0.0284989
$154$	0	0
$155$	−7.11319	−0.571345
$156$	0	0
$157$	15.3039	1.22139	0.610693	−	0.791867i	$-0.290891\pi$
$157$	15.3039	1.22139	0.610693	+	0.791867i	$0.290891\pi$
$158$	0	0
$159$	5.88463	0.466682
$160$	0	0
$161$	−6.53805	−0.515270
$162$	0	0
$163$	12.0051	0.940314	0.470157	−	0.882583i	$-0.344197\pi$
$163$	12.0051	0.940314	0.470157	+	0.882583i	$0.344197\pi$
$164$	0	0
$165$	11.2485	0.875698
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.47191	−0.420916
$170$	0	0
$171$	7.63533	0.583888
$172$	0	0
$173$	0.927523	0.0705183	0.0352591	−	0.999378i	$-0.488774\pi$
$173$	0.927523	0.0705183	0.0352591	+	0.999378i	$0.488774\pi$
$174$	0	0
$175$	4.78129	0.361431
$176$	0	0
$177$	9.60530	0.721979
$178$	0	0
$179$	−1.80260	−0.134733	−0.0673665	−	0.997728i	$-0.521460\pi$
$179$	−1.80260	−0.134733	−0.0673665	+	0.997728i	$0.521460\pi$
$180$	0	0
$181$	21.7433	1.61617	0.808084	−	0.589067i	$-0.200505\pi$
$181$	21.7433	1.61617	0.808084	+	0.589067i	$0.200505\pi$
$182$	0	0
$183$	−2.96181	−0.218943
$184$	0	0
$185$	−3.99163	−0.293470
$186$	0	0
$187$	−0.696618	−0.0509418
$188$	0	0
$189$	7.23571	0.526320
$190$	0	0
$191$	4.34797	0.314608	0.157304	−	0.987550i	$-0.449720\pi$
$191$	4.34797	0.314608	0.157304	+	0.987550i	$0.449720\pi$
$192$	0	0
$193$	−1.18508	−0.0853040	−0.0426520	−	0.999090i	$-0.513581\pi$
$193$	−1.18508	−0.0853040	−0.0426520	+	0.999090i	$0.513581\pi$
$194$	0	0
$195$	7.39042	0.529239
$196$	0	0
$197$	14.0931	1.00409	0.502047	−	0.864840i	$-0.332581\pi$
$197$	14.0931	1.00409	0.502047	+	0.864840i	$0.332581\pi$
$198$	0	0
$199$	1.77562	0.125871	0.0629354	−	0.998018i	$-0.479954\pi$
$199$	1.77562	0.125871	0.0629354	+	0.998018i	$0.479954\pi$
$200$	0	0
$201$	9.60320	0.677357
$202$	0	0
$203$	0.290577	0.0203945
$204$	0	0
$205$	9.99855	0.698329
$206$	0	0
$207$	9.19422	0.639042
$208$	0	0
$209$	−15.0886	−1.04370
$210$	0	0
$211$	3.27131	0.225206	0.112603	−	0.993640i	$-0.464081\pi$
$211$	3.27131	0.225206	0.112603	+	0.993640i	$0.464081\pi$
$212$	0	0
$213$	4.10226	0.281082
$214$	0	0
$215$	8.94180	0.609826
$216$	0	0
$217$	3.73700	0.253684
$218$	0	0
$219$	−0.500355	−0.0338108
$220$	0	0
$221$	−0.457686	−0.0307873
$222$	0	0
$223$	−4.58639	−0.307127	−0.153564	−	0.988139i	$-0.549075\pi$
$223$	−4.58639	−0.307127	−0.153564	+	0.988139i	$0.549075\pi$
$224$	0	0
$225$	−6.72375	−0.448250
$226$	0	0
$227$	4.78649	0.317691	0.158845	−	0.987303i	$-0.449223\pi$
$227$	4.78649	0.317691	0.158845	+	0.987303i	$0.449223\pi$
$228$	0	0
$229$	−8.81130	−0.582267	−0.291134	−	0.956682i	$-0.594032\pi$
$229$	−8.81130	−0.582267	−0.291134	+	0.956682i	$0.594032\pi$
$230$	0	0
$231$	−5.90955	−0.388820
$232$	0	0
$233$	−8.77074	−0.574590	−0.287295	−	0.957842i	$-0.592756\pi$
$233$	−8.77074	−0.574590	−0.287295	+	0.957842i	$0.592756\pi$
$234$	0	0
$235$	26.3412	1.71831
$236$	0	0
$237$	15.4878	1.00604
$238$	0	0
$239$	13.2642	0.857992	0.428996	−	0.903306i	$-0.358867\pi$
$239$	13.2642	0.857992	0.428996	+	0.903306i	$0.358867\pi$
$240$	0	0
$241$	−7.05189	−0.454252	−0.227126	−	0.973865i	$-0.572933\pi$
$241$	−7.05189	−0.454252	−0.227126	+	0.973865i	$0.572933\pi$
$242$	0	0
$243$	−16.1453	−1.03572
$244$	0	0
$245$	13.5633	0.866529
$246$	0	0
$247$	−9.91340	−0.630775
$248$	0	0
$249$	−0.869936	−0.0551299
$250$	0	0
$251$	0.538352	0.0339805	0.0169903	−	0.999856i	$-0.494592\pi$
$251$	0.538352	0.0339805	0.0169903	+	0.999856i	$0.494592\pi$
$252$	0	0
$253$	−18.1692	−1.14229
$254$	0	0
$255$	−0.449316	−0.0281373
$256$	0	0
$257$	−26.5112	−1.65372	−0.826861	−	0.562406i	$-0.809876\pi$
$257$	−26.5112	−1.65372	−0.826861	+	0.562406i	$0.809876\pi$
$258$	0	0
$259$	2.09705	0.130304
$260$	0	0
$261$	−0.408627	−0.0252934
$262$	0	0
$263$	4.40738	0.271770	0.135885	−	0.990725i	$-0.456612\pi$
$263$	4.40738	0.271770	0.135885	+	0.990725i	$0.456612\pi$
$264$	0	0
$265$	17.8747	1.09803
$266$	0	0
$267$	−2.20815	−0.135137
$268$	0	0
$269$	10.9342	0.666669	0.333335	−	0.942809i	$-0.391826\pi$
$269$	10.9342	0.666669	0.333335	+	0.942809i	$0.391826\pi$
$270$	0	0
$271$	3.20705	0.194814	0.0974071	−	0.995245i	$-0.468945\pi$
$271$	3.20705	0.194814	0.0974071	+	0.995245i	$0.468945\pi$
$272$	0	0
$273$	−3.88265	−0.234988
$274$	0	0
$275$	13.2872	0.801247
$276$	0	0
$277$	21.4187	1.28693	0.643463	−	0.765477i	$-0.277497\pi$
$277$	21.4187	1.28693	0.643463	+	0.765477i	$0.277497\pi$
$278$	0	0
$279$	−5.25520	−0.314621
$280$	0	0
$281$	3.37238	0.201179	0.100590	−	0.994928i	$-0.467927\pi$
$281$	3.37238	0.201179	0.100590	+	0.994928i	$0.467927\pi$
$282$	0	0
$283$	11.9697	0.711527	0.355763	−	0.934576i	$-0.384221\pi$
$283$	11.9697	0.711527	0.355763	+	0.934576i	$0.384221\pi$
$284$	0	0
$285$	−9.73211	−0.576481
$286$	0	0
$287$	−5.25286	−0.310066
$288$	0	0
$289$	−16.9722	−0.998363
$290$	0	0
$291$	−6.81705	−0.399622
$292$	0	0
$293$	−15.5030	−0.905696	−0.452848	−	0.891588i	$-0.649592\pi$
$293$	−15.5030	−0.905696	−0.452848	+	0.891588i	$0.649592\pi$
$294$	0	0
$295$	29.1762	1.69871
$296$	0	0
$297$	20.1080	1.16679
$298$	0	0
$299$	−11.9374	−0.690357
$300$	0	0
$301$	−4.69768	−0.270770
$302$	0	0
$303$	−5.40262	−0.310372
$304$	0	0
$305$	−8.99655	−0.515141
$306$	0	0
$307$	16.3457	0.932898	0.466449	−	0.884548i	$-0.345533\pi$
$307$	16.3457	0.932898	0.466449	+	0.884548i	$0.345533\pi$
$308$	0	0
$309$	2.95564	0.168141
$310$	0	0
$311$	−5.49380	−0.311525	−0.155762	−	0.987795i	$-0.549783\pi$
$311$	−5.49380	−0.311525	−0.155762	+	0.987795i	$0.549783\pi$
$312$	0	0
$313$	−3.65695	−0.206703	−0.103351	−	0.994645i	$-0.532957\pi$
$313$	−3.65695	−0.206703	−0.103351	+	0.994645i	$0.532957\pi$
$314$	0	0
$315$	9.08346	0.511795
$316$	0	0
$317$	5.46747	0.307083	0.153542	−	0.988142i	$-0.450932\pi$
$317$	5.46747	0.307083	0.153542	+	0.988142i	$0.450932\pi$
$318$	0	0
$319$	0.807512	0.0452120
$320$	0	0
$321$	3.13953	0.175231
$322$	0	0
$323$	0.602706	0.0335355
$324$	0	0
$325$	8.72984	0.484244
$326$	0	0
$327$	−17.6796	−0.977683
$328$	0	0
$329$	−13.8387	−0.762951
$330$	0	0
$331$	−14.9278	−0.820505	−0.410252	−	0.911972i	$-0.634559\pi$
$331$	−14.9278	−0.820505	−0.410252	+	0.911972i	$0.634559\pi$
$332$	0	0
$333$	−2.94900	−0.161604
$334$	0	0
$335$	29.1699	1.59372
$336$	0	0
$337$	10.1662	0.553791	0.276895	−	0.960900i	$-0.410694\pi$
$337$	10.1662	0.553791	0.276895	+	0.960900i	$0.410694\pi$
$338$	0	0
$339$	17.6744	0.959944
$340$	0	0
$341$	10.3851	0.562385
$342$	0	0
$343$	−17.6448	−0.952728
$344$	0	0
$345$	−11.7191	−0.630935
$346$	0	0
$347$	−32.0148	−1.71864	−0.859322	−	0.511434i	$-0.829114\pi$
$347$	−32.0148	−1.71864	−0.859322	+	0.511434i	$0.829114\pi$
$348$	0	0
$349$	0.923682	0.0494436	0.0247218	−	0.999694i	$-0.492130\pi$
$349$	0.923682	0.0494436	0.0247218	+	0.999694i	$0.492130\pi$
$350$	0	0
$351$	13.2112	0.705162
$352$	0	0
$353$	28.6145	1.52299	0.761497	−	0.648168i	$-0.224465\pi$
$353$	28.6145	1.52299	0.761497	+	0.648168i	$0.224465\pi$
$354$	0	0
$355$	12.4607	0.661344
$356$	0	0
$357$	0.236054	0.0124933
$358$	0	0
$359$	−6.01599	−0.317512	−0.158756	−	0.987318i	$-0.550748\pi$
$359$	−6.01599	−0.317512	−0.158756	+	0.987318i	$0.550748\pi$
$360$	0	0
$361$	−5.94550	−0.312921
$362$	0	0
$363$	−6.06414	−0.318285
$364$	0	0
$365$	−1.51983	−0.0795518
$366$	0	0
$367$	10.1299	0.528777	0.264388	−	0.964416i	$-0.414830\pi$
$367$	10.1299	0.528777	0.264388	+	0.964416i	$0.414830\pi$
$368$	0	0
$369$	7.38690	0.384547
$370$	0	0
$371$	−9.39066	−0.487539
$372$	0	0
$373$	22.8600	1.18364	0.591822	−	0.806069i	$-0.298409\pi$
$373$	22.8600	1.18364	0.591822	+	0.806069i	$0.298409\pi$
$374$	0	0
$375$	−4.89761	−0.252912
$376$	0	0
$377$	0.530545	0.0273244
$378$	0	0
$379$	−12.5766	−0.646015	−0.323007	−	0.946396i	$-0.604694\pi$
$379$	−12.5766	−0.646015	−0.323007	+	0.946396i	$0.604694\pi$
$380$	0	0
$381$	−6.68376	−0.342419
$382$	0	0
$383$	6.74996	0.344907	0.172453	−	0.985018i	$-0.444831\pi$
$383$	6.74996	0.344907	0.172453	+	0.985018i	$0.444831\pi$
$384$	0	0
$385$	−17.9504	−0.914835
$386$	0	0
$387$	6.60618	0.335811
$388$	0	0
$389$	11.8833	0.602510	0.301255	−	0.953544i	$-0.402595\pi$
$389$	11.8833	0.602510	0.301255	+	0.953544i	$0.402595\pi$
$390$	0	0
$391$	0.725759	0.0367032
$392$	0	0
$393$	9.63693	0.486119
$394$	0	0
$395$	47.0445	2.36707
$396$	0	0
$397$	−17.8940	−0.898074	−0.449037	−	0.893513i	$-0.648233\pi$
$397$	−17.8940	−0.898074	−0.449037	+	0.893513i	$0.648233\pi$
$398$	0	0
$399$	5.11288	0.255964
$400$	0	0
$401$	5.88067	0.293666	0.146833	−	0.989161i	$-0.453092\pi$
$401$	5.88067	0.293666	0.146833	+	0.989161i	$0.453092\pi$
$402$	0	0
$403$	6.82314	0.339885
$404$	0	0
$405$	−5.16431	−0.256617
$406$	0	0
$407$	5.82770	0.288868
$408$	0	0
$409$	−1.49760	−0.0740518	−0.0370259	−	0.999314i	$-0.511788\pi$
$409$	−1.49760	−0.0740518	−0.0370259	+	0.999314i	$0.511788\pi$
$410$	0	0
$411$	13.2054	0.651372
$412$	0	0
$413$	−15.3281	−0.754246
$414$	0	0
$415$	−2.64244	−0.129712
$416$	0	0
$417$	−3.86373	−0.189208
$418$	0	0
$419$	−21.8653	−1.06819	−0.534095	−	0.845425i	$-0.679347\pi$
$419$	−21.8653	−1.06819	−0.534095	+	0.845425i	$0.679347\pi$
$420$	0	0
$421$	−31.2968	−1.52531	−0.762656	−	0.646805i	$-0.776105\pi$
$421$	−31.2968	−1.52531	−0.762656	+	0.646805i	$0.776105\pi$
$422$	0	0
$423$	19.4608	0.946217
$424$	0	0
$425$	−0.530749	−0.0257451
$426$	0	0
$427$	4.72644	0.228728
$428$	0	0
$429$	−10.7899	−0.520940
$430$	0	0
$431$	15.3434	0.739065	0.369533	−	0.929218i	$-0.379518\pi$
$431$	15.3434	0.739065	0.369533	+	0.929218i	$0.379518\pi$
$432$	0	0
$433$	31.9672	1.53625	0.768123	−	0.640302i	$-0.221191\pi$
$433$	31.9672	1.53625	0.768123	+	0.640302i	$0.221191\pi$
$434$	0	0
$435$	0.520843	0.0249725
$436$	0	0
$437$	15.7198	0.751980
$438$	0	0
$439$	−16.0603	−0.766514	−0.383257	−	0.923642i	$-0.625198\pi$
$439$	−16.0603	−0.766514	−0.383257	+	0.923642i	$0.625198\pi$
$440$	0	0
$441$	10.0205	0.477169
$442$	0	0
$443$	26.5411	1.26100	0.630502	−	0.776187i	$-0.282849\pi$
$443$	26.5411	1.26100	0.630502	+	0.776187i	$0.282849\pi$
$444$	0	0
$445$	−6.70729	−0.317956
$446$	0	0
$447$	21.4689	1.01544
$448$	0	0
$449$	9.48125	0.447448	0.223724	−	0.974653i	$-0.428179\pi$
$449$	9.48125	0.447448	0.223724	+	0.974653i	$0.428179\pi$
$450$	0	0
$451$	−14.5977	−0.687378
$452$	0	0
$453$	20.0316	0.941165
$454$	0	0
$455$	−11.7936	−0.552892
$456$	0	0
$457$	28.4238	1.32961	0.664805	−	0.747017i	$-0.268515\pi$
$457$	28.4238	1.32961	0.664805	+	0.747017i	$0.268515\pi$
$458$	0	0
$459$	−0.803203	−0.0374903
$460$	0	0
$461$	−0.0713275	−0.00332205	−0.00166103	−	0.999999i	$-0.500529\pi$
$461$	−0.0713275	−0.00332205	−0.00166103	+	0.999999i	$0.500529\pi$
$462$	0	0
$463$	10.9252	0.507739	0.253870	−	0.967238i	$-0.418297\pi$
$463$	10.9252	0.507739	0.253870	+	0.967238i	$0.418297\pi$
$464$	0	0
$465$	6.69836	0.310629
$466$	0	0
$467$	−0.846191	−0.0391570	−0.0195785	−	0.999808i	$-0.506232\pi$
$467$	−0.846191	−0.0391570	−0.0195785	+	0.999808i	$0.506232\pi$
$468$	0	0
$469$	−15.3247	−0.707630
$470$	0	0
$471$	−14.4114	−0.664044
$472$	0	0
$473$	−13.0548	−0.600262
$474$	0	0
$475$	−11.4959	−0.527469
$476$	0	0
$477$	13.2057	0.604649
$478$	0	0
$479$	−20.9042	−0.955138	−0.477569	−	0.878594i	$-0.658482\pi$
$479$	−20.9042	−0.955138	−0.477569	+	0.878594i	$0.658482\pi$
$480$	0	0
$481$	3.82887	0.174581
$482$	0	0
$483$	6.15676	0.280142
$484$	0	0
$485$	−20.7069	−0.940251
$486$	0	0
$487$	−15.0623	−0.682536	−0.341268	−	0.939966i	$-0.610856\pi$
$487$	−15.0623	−0.682536	−0.341268	+	0.939966i	$0.610856\pi$
$488$	0	0
$489$	−11.3050	−0.511230
$490$	0	0
$491$	−22.6048	−1.02014	−0.510069	−	0.860133i	$-0.670380\pi$
$491$	−22.6048	−1.02014	−0.510069	+	0.860133i	$0.670380\pi$
$492$	0	0
$493$	−0.0322556	−0.00145272
$494$	0	0
$495$	25.2429	1.13459
$496$	0	0
$497$	−6.54636	−0.293645
$498$	0	0
$499$	−25.4851	−1.14087	−0.570435	−	0.821343i	$-0.693225\pi$
$499$	−25.4851	−1.14087	−0.570435	+	0.821343i	$0.693225\pi$
$500$	0	0
$501$	0.941682	0.0420712
$502$	0	0
$503$	−0.510520	−0.0227630	−0.0113815	−	0.999935i	$-0.503623\pi$
$503$	−0.510520	−0.0227630	−0.0113815	+	0.999935i	$0.503623\pi$
$504$	0	0
$505$	−16.4105	−0.730259
$506$	0	0
$507$	5.15280	0.228844
$508$	0	0
$509$	−14.3404	−0.635628	−0.317814	−	0.948153i	$-0.602949\pi$
$509$	−14.3404	−0.635628	−0.317814	+	0.948153i	$0.602949\pi$
$510$	0	0
$511$	0.798463	0.0353219
$512$	0	0
$513$	−17.3972	−0.768107
$514$	0	0
$515$	8.97780	0.395609
$516$	0	0
$517$	−38.4576	−1.69137
$518$	0	0
$519$	−0.873431	−0.0383394
$520$	0	0
$521$	31.4336	1.37713	0.688564	−	0.725175i	$-0.258241\pi$
$521$	31.4336	1.37713	0.688564	+	0.725175i	$0.258241\pi$
$522$	0	0
$523$	18.0137	0.787684	0.393842	−	0.919178i	$-0.371146\pi$
$523$	18.0137	0.787684	0.393842	+	0.919178i	$0.371146\pi$
$524$	0	0
$525$	−4.50245	−0.196503
$526$	0	0
$527$	−0.414827	−0.0180702
$528$	0	0
$529$	−4.07073	−0.176988
$530$	0	0
$531$	21.5553	0.935421
$532$	0	0
$533$	−9.59085	−0.415426
$534$	0	0
$535$	9.53637	0.412293
$536$	0	0
$537$	1.69748	0.0732516
$538$	0	0
$539$	−19.8022	−0.852940
$540$	0	0
$541$	37.1272	1.59622	0.798112	−	0.602509i	$-0.205832\pi$
$541$	37.1272	1.59622	0.798112	+	0.602509i	$0.205832\pi$
$542$	0	0
$543$	−20.4753	−0.878679
$544$	0	0
$545$	−53.7020	−2.30034
$546$	0	0
$547$	−20.5010	−0.876561	−0.438280	−	0.898838i	$-0.644412\pi$
$547$	−20.5010	−0.876561	−0.438280	+	0.898838i	$0.644412\pi$
$548$	0	0
$549$	−6.64662	−0.283671
$550$	0	0
$551$	−0.698650	−0.0297635
$552$	0	0
$553$	−24.7154	−1.05100
$554$	0	0
$555$	3.75885	0.159554
$556$	0	0
$557$	−10.7686	−0.456281	−0.228140	−	0.973628i	$-0.573265\pi$
$557$	−10.7686	−0.456281	−0.228140	+	0.973628i	$0.573265\pi$
$558$	0	0
$559$	−8.57719	−0.362776
$560$	0	0
$561$	0.655993	0.0276960
$562$	0	0
$563$	−31.6001	−1.33178	−0.665892	−	0.746048i	$-0.731949\pi$
$563$	−31.6001	−1.33178	−0.665892	+	0.746048i	$0.731949\pi$
$564$	0	0
$565$	53.6864	2.25860
$566$	0	0
$567$	2.71313	0.113941
$568$	0	0
$569$	−18.3203	−0.768028	−0.384014	−	0.923327i	$-0.625459\pi$
$569$	−18.3203	−0.768028	−0.384014	+	0.923327i	$0.625459\pi$
$570$	0	0
$571$	−23.7060	−0.992065	−0.496032	−	0.868304i	$-0.665210\pi$
$571$	−23.7060	−0.992065	−0.496032	+	0.868304i	$0.665210\pi$
$572$	0	0
$573$	−4.09440	−0.171046
$574$	0	0
$575$	−13.8430	−0.577294
$576$	0	0
$577$	−22.3526	−0.930551	−0.465275	−	0.885166i	$-0.654045\pi$
$577$	−22.3526	−0.930551	−0.465275	+	0.885166i	$0.654045\pi$
$578$	0	0
$579$	1.11597	0.0463781
$580$	0	0
$581$	1.38824	0.0575938
$582$	0	0
$583$	−26.0966	−1.08081
$584$	0	0
$585$	16.5849	0.685701
$586$	0	0
$587$	−38.0989	−1.57251	−0.786255	−	0.617902i	$-0.787983\pi$
$587$	−38.0989	−1.57251	−0.786255	+	0.617902i	$0.787983\pi$
$588$	0	0
$589$	−8.98508	−0.370224
$590$	0	0
$591$	−13.2712	−0.545906
$592$	0	0
$593$	22.1551	0.909801	0.454900	−	0.890542i	$-0.349675\pi$
$593$	22.1551	0.909801	0.454900	+	0.890542i	$0.349675\pi$
$594$	0	0
$595$	0.717016	0.0293948
$596$	0	0
$597$	−1.67207	−0.0684334
$598$	0	0
$599$	46.8099	1.91260	0.956301	−	0.292384i	$-0.0944486\pi$
$599$	46.8099	1.91260	0.956301	+	0.292384i	$0.0944486\pi$
$600$	0	0
$601$	3.19069	0.130151	0.0650756	−	0.997880i	$-0.479271\pi$
$601$	3.19069	0.130151	0.0650756	+	0.997880i	$0.479271\pi$
$602$	0	0
$603$	21.5506	0.877608
$604$	0	0
$605$	−18.4199	−0.748875
$606$	0	0
$607$	46.1213	1.87200	0.936002	−	0.351995i	$-0.114496\pi$
$607$	46.1213	1.87200	0.936002	+	0.351995i	$0.114496\pi$
$608$	0	0
$609$	−0.273631	−0.0110881
$610$	0	0
$611$	−25.2671	−1.02220
$612$	0	0
$613$	21.3620	0.862803	0.431402	−	0.902160i	$-0.358019\pi$
$613$	21.3620	0.862803	0.431402	+	0.902160i	$0.358019\pi$
$614$	0	0
$615$	−9.41546	−0.379668
$616$	0	0
$617$	−44.4595	−1.78987	−0.894935	−	0.446196i	$-0.852778\pi$
$617$	−44.4595	−1.78987	−0.894935	+	0.446196i	$0.852778\pi$
$618$	0	0
$619$	−36.5659	−1.46971	−0.734855	−	0.678225i	$-0.762750\pi$
$619$	−36.5659	−1.46971	−0.734855	+	0.678225i	$0.762750\pi$
$620$	0	0
$621$	−20.9492	−0.840662
$622$	0	0
$623$	3.52376	0.141176
$624$	0	0
$625$	−30.7852	−1.23141
$626$	0	0
$627$	14.2087	0.567440
$628$	0	0
$629$	−0.232784	−0.00928170
$630$	0	0
$631$	−23.1530	−0.921706	−0.460853	−	0.887477i	$-0.652456\pi$
$631$	−23.1530	−0.921706	−0.460853	+	0.887477i	$0.652456\pi$
$632$	0	0
$633$	−3.08053	−0.122440
$634$	0	0
$635$	−20.3020	−0.805661
$636$	0	0
$637$	−13.0103	−0.515485
$638$	0	0
$639$	9.20591	0.364180
$640$	0	0
$641$	16.3508	0.645816	0.322908	−	0.946430i	$-0.395340\pi$
$641$	16.3508	0.645816	0.322908	+	0.946430i	$0.395340\pi$
$642$	0	0
$643$	−42.2056	−1.66443	−0.832213	−	0.554456i	$-0.812926\pi$
$643$	−42.2056	−1.66443	−0.832213	+	0.554456i	$0.812926\pi$
$644$	0	0
$645$	−8.42033	−0.331550
$646$	0	0
$647$	24.1688	0.950174	0.475087	−	0.879939i	$-0.342417\pi$
$647$	24.1688	0.950174	0.475087	+	0.879939i	$0.342417\pi$
$648$	0	0
$649$	−42.5967	−1.67207
$650$	0	0
$651$	−3.51906	−0.137923
$652$	0	0
$653$	15.6061	0.610716	0.305358	−	0.952238i	$-0.401224\pi$
$653$	15.6061	0.610716	0.305358	+	0.952238i	$0.401224\pi$
$654$	0	0
$655$	29.2723	1.14376
$656$	0	0
$657$	−1.12285	−0.0438065
$658$	0	0
$659$	11.5667	0.450574	0.225287	−	0.974292i	$-0.427668\pi$
$659$	11.5667	0.450574	0.225287	+	0.974292i	$0.427668\pi$
$660$	0	0
$661$	−7.20837	−0.280373	−0.140187	−	0.990125i	$-0.544770\pi$
$661$	−7.20837	−0.280373	−0.140187	+	0.990125i	$0.544770\pi$
$662$	0	0
$663$	0.430995	0.0167385
$664$	0	0
$665$	15.5304	0.602245
$666$	0	0
$667$	−0.841291	−0.0325749
$668$	0	0
$669$	4.31892	0.166979
$670$	0	0
$671$	13.1348	0.507062
$672$	0	0
$673$	19.3032	0.744084	0.372042	−	0.928216i	$-0.378658\pi$
$673$	19.3032	0.744084	0.372042	+	0.928216i	$0.378658\pi$
$674$	0	0
$675$	15.3202	0.589674
$676$	0	0
$677$	13.7259	0.527529	0.263765	−	0.964587i	$-0.415036\pi$
$677$	13.7259	0.527529	0.263765	+	0.964587i	$0.415036\pi$
$678$	0	0
$679$	10.8786	0.417482
$680$	0	0
$681$	−4.50735	−0.172722
$682$	0	0
$683$	−14.6340	−0.559954	−0.279977	−	0.960007i	$-0.590327\pi$
$683$	−14.6340	−0.559954	−0.279977	+	0.960007i	$0.590327\pi$
$684$	0	0
$685$	40.1114	1.53258
$686$	0	0
$687$	8.29744	0.316567
$688$	0	0
$689$	−17.1458	−0.653202
$690$	0	0
$691$	31.2308	1.18808	0.594038	−	0.804437i	$-0.297533\pi$
$691$	31.2308	1.18808	0.594038	+	0.804437i	$0.297533\pi$
$692$	0	0
$693$	−13.2617	−0.503769
$694$	0	0
$695$	−11.7361	−0.445177
$696$	0	0
$697$	0.583096	0.0220863
$698$	0	0
$699$	8.25925	0.312394
$700$	0	0
$701$	−38.0984	−1.43896	−0.719479	−	0.694515i	$-0.755619\pi$
$701$	−38.0984	−1.43896	−0.719479	+	0.694515i	$0.755619\pi$
$702$	0	0
$703$	−5.04206	−0.190165
$704$	0	0
$705$	−24.8051	−0.934212
$706$	0	0
$707$	8.62147	0.324244
$708$	0	0
$709$	−36.0667	−1.35451	−0.677256	−	0.735748i	$-0.736831\pi$
$709$	−36.0667	−1.35451	−0.677256	+	0.735748i	$0.736831\pi$
$710$	0	0
$711$	34.7563	1.30346
$712$	0	0
$713$	−10.8195	−0.405195
$714$	0	0
$715$	−32.7744	−1.22569
$716$	0	0
$717$	−12.4907	−0.466473
$718$	0	0
$719$	26.7033	0.995863	0.497932	−	0.867216i	$-0.334093\pi$
$719$	26.7033	0.995863	0.497932	+	0.867216i	$0.334093\pi$
$720$	0	0
$721$	−4.71659	−0.175655
$722$	0	0
$723$	6.64064	0.246968
$724$	0	0
$725$	0.615238	0.0228494
$726$	0	0
$727$	16.2515	0.602735	0.301367	−	0.953508i	$-0.402557\pi$
$727$	16.2515	0.602735	0.301367	+	0.953508i	$0.402557\pi$
$728$	0	0
$729$	9.78734	0.362494
$730$	0	0
$731$	0.521468	0.0192872
$732$	0	0
$733$	12.7314	0.470245	0.235123	−	0.971966i	$-0.424451\pi$
$733$	12.7314	0.470245	0.235123	+	0.971966i	$0.424451\pi$
$734$	0	0
$735$	−12.7723	−0.471115
$736$	0	0
$737$	−42.5874	−1.56873
$738$	0	0
$739$	7.72921	0.284324	0.142162	−	0.989843i	$-0.454595\pi$
$739$	7.72921	0.284324	0.142162	+	0.989843i	$0.454595\pi$
$740$	0	0
$741$	9.33527	0.342940
$742$	0	0
$743$	35.3892	1.29830	0.649152	−	0.760659i	$-0.275124\pi$
$743$	35.3892	1.29830	0.649152	+	0.760659i	$0.275124\pi$
$744$	0	0
$745$	65.2121	2.38919
$746$	0	0
$747$	−1.95223	−0.0714283
$748$	0	0
$749$	−5.01004	−0.183063
$750$	0	0
$751$	−16.5945	−0.605541	−0.302771	−	0.953063i	$-0.597912\pi$
$751$	−16.5945	−0.605541	−0.302771	+	0.953063i	$0.597912\pi$
$752$	0	0
$753$	−0.506957	−0.0184745
$754$	0	0
$755$	60.8461	2.21442
$756$	0	0
$757$	−15.2447	−0.554077	−0.277038	−	0.960859i	$-0.589353\pi$
$757$	−15.2447	−0.554077	−0.277038	+	0.960859i	$0.589353\pi$
$758$	0	0
$759$	17.1096	0.621040
$760$	0	0
$761$	11.9025	0.431466	0.215733	−	0.976452i	$-0.430786\pi$
$761$	11.9025	0.431466	0.215733	+	0.976452i	$0.430786\pi$
$762$	0	0
$763$	28.2130	1.02138
$764$	0	0
$765$	−1.00831	−0.0364557
$766$	0	0
$767$	−27.9865	−1.01054
$768$	0	0
$769$	28.5126	1.02819	0.514095	−	0.857733i	$-0.328128\pi$
$769$	28.5126	1.02819	0.514095	+	0.857733i	$0.328128\pi$
$770$	0	0
$771$	24.9651	0.899096
$772$	0	0
$773$	−2.52031	−0.0906493	−0.0453247	−	0.998972i	$-0.514432\pi$
$773$	−2.52031	−0.0906493	−0.0453247	+	0.998972i	$0.514432\pi$
$774$	0	0
$775$	7.91235	0.284220
$776$	0	0
$777$	−1.97475	−0.0708439
$778$	0	0
$779$	12.6297	0.452508
$780$	0	0
$781$	−18.1923	−0.650973
$782$	0	0
$783$	0.931064	0.0332735
$784$	0	0
$785$	−43.7749	−1.56239
$786$	0	0
$787$	11.9075	0.424458	0.212229	−	0.977220i	$-0.431928\pi$
$787$	11.9075	0.424458	0.212229	+	0.977220i	$0.431928\pi$
$788$	0	0
$789$	−4.15035	−0.147756
$790$	0	0
$791$	−28.2048	−1.00285
$792$	0	0
$793$	8.62970	0.306450
$794$	0	0
$795$	−16.8322	−0.596978
$796$	0	0
$797$	55.9231	1.98090	0.990449	−	0.137880i	$-0.0440290\pi$
$797$	55.9231	1.98090	0.990449	+	0.137880i	$0.0440290\pi$
$798$	0	0
$799$	1.53617	0.0543457
$800$	0	0
$801$	−4.95533	−0.175088
$802$	0	0
$803$	2.21893	0.0783042
$804$	0	0
$805$	18.7012	0.659132
$806$	0	0
$807$	−10.2965	−0.362455
$808$	0	0
$809$	−49.1704	−1.72874	−0.864370	−	0.502857i	$-0.832282\pi$
$809$	−49.1704	−1.72874	−0.864370	+	0.502857i	$0.832282\pi$
$810$	0	0
$811$	25.2602	0.887004	0.443502	−	0.896273i	$-0.353736\pi$
$811$	25.2602	0.887004	0.443502	+	0.896273i	$0.353736\pi$
$812$	0	0
$813$	−3.02002	−0.105917
$814$	0	0
$815$	−34.3391	−1.20285
$816$	0	0
$817$	11.2949	0.395159
$818$	0	0
$819$	−8.71307	−0.304459
$820$	0	0
$821$	37.8924	1.32245	0.661226	−	0.750187i	$-0.270036\pi$
$821$	37.8924	1.32245	0.661226	+	0.750187i	$0.270036\pi$
$822$	0	0
$823$	−10.1792	−0.354824	−0.177412	−	0.984137i	$-0.556772\pi$
$823$	−10.1792	−0.354824	−0.177412	+	0.984137i	$0.556772\pi$
$824$	0	0
$825$	−12.5123	−0.435623
$826$	0	0
$827$	49.8099	1.73206	0.866031	−	0.499991i	$-0.166663\pi$
$827$	49.8099	1.73206	0.866031	+	0.499991i	$0.166663\pi$
$828$	0	0
$829$	−34.0020	−1.18094	−0.590470	−	0.807060i	$-0.701058\pi$
$829$	−34.0020	−1.18094	−0.590470	+	0.807060i	$0.701058\pi$
$830$	0	0
$831$	−20.1696	−0.699677
$832$	0	0
$833$	0.790986	0.0274060
$834$	0	0
$835$	2.86037	0.0989873
$836$	0	0
$837$	11.9741	0.413884
$838$	0	0
$839$	−22.4819	−0.776163	−0.388081	−	0.921625i	$-0.626862\pi$
$839$	−22.4819	−0.776163	−0.388081	+	0.921625i	$0.626862\pi$
$840$	0	0
$841$	−28.9626	−0.998711
$842$	0	0
$843$	−3.17571	−0.109377
$844$	0	0
$845$	15.6517	0.538435
$846$	0	0
$847$	9.67711	0.332509
$848$	0	0
$849$	−11.2717	−0.386843
$850$	0	0
$851$	−6.07148	−0.208128
$852$	0	0
$853$	−12.1189	−0.414943	−0.207471	−	0.978241i	$-0.566523\pi$
$853$	−12.1189	−0.414943	−0.207471	+	0.978241i	$0.566523\pi$
$854$	0	0
$855$	−21.8399	−0.746909
$856$	0	0
$857$	4.17555	0.142634	0.0713171	−	0.997454i	$-0.477280\pi$
$857$	4.17555	0.142634	0.0713171	+	0.997454i	$0.477280\pi$
$858$	0	0
$859$	12.9222	0.440901	0.220450	−	0.975398i	$-0.429247\pi$
$859$	12.9222	0.440901	0.220450	+	0.975398i	$0.429247\pi$
$860$	0	0
$861$	4.94652	0.168577
$862$	0	0
$863$	2.58477	0.0879865	0.0439932	−	0.999032i	$-0.485992\pi$
$863$	2.58477	0.0879865	0.0439932	+	0.999032i	$0.485992\pi$
$864$	0	0
$865$	−2.65306	−0.0902068
$866$	0	0
$867$	15.9824	0.542790
$868$	0	0
$869$	−68.6840	−2.32995
$870$	0	0
$871$	−27.9804	−0.948080
$872$	0	0
$873$	−15.2982	−0.517765
$874$	0	0
$875$	7.81558	0.264215
$876$	0	0
$877$	40.5302	1.36861	0.684303	−	0.729198i	$-0.260106\pi$
$877$	40.5302	1.36861	0.684303	+	0.729198i	$0.260106\pi$
$878$	0	0
$879$	14.5989	0.492409
$880$	0	0
$881$	−23.5378	−0.793007	−0.396504	−	0.918033i	$-0.629777\pi$
$881$	−23.5378	−0.793007	−0.396504	+	0.918033i	$0.629777\pi$
$882$	0	0
$883$	7.29455	0.245481	0.122741	−	0.992439i	$-0.460832\pi$
$883$	7.29455	0.245481	0.122741	+	0.992439i	$0.460832\pi$
$884$	0	0
$885$	−27.4747	−0.923554
$886$	0	0
$887$	−15.9064	−0.534083	−0.267042	−	0.963685i	$-0.586046\pi$
$887$	−15.9064	−0.534083	−0.267042	+	0.963685i	$0.586046\pi$
$888$	0	0
$889$	10.6659	0.357723
$890$	0	0
$891$	7.53979	0.252592
$892$	0	0
$893$	33.2731	1.11344
$894$	0	0
$895$	5.15612	0.172350
$896$	0	0
$897$	11.2412	0.375334
$898$	0	0
$899$	0.480863	0.0160377
$900$	0	0
$901$	1.04241	0.0347279
$902$	0	0
$903$	4.42372	0.147212
$904$	0	0
$905$	−62.1940	−2.06740
$906$	0	0
$907$	−18.8191	−0.624879	−0.312439	−	0.949938i	$-0.601146\pi$
$907$	−18.8191	−0.624879	−0.312439	+	0.949938i	$0.601146\pi$
$908$	0	0
$909$	−12.1241	−0.402130
$910$	0	0
$911$	12.7529	0.422522	0.211261	−	0.977430i	$-0.432243\pi$
$911$	12.7529	0.422522	0.211261	+	0.977430i	$0.432243\pi$
$912$	0	0
$913$	3.85791	0.127678
$914$	0	0
$915$	8.47189	0.280072
$916$	0	0
$917$	−15.3786	−0.507845
$918$	0	0
$919$	−41.8179	−1.37945	−0.689723	−	0.724073i	$-0.742268\pi$
$919$	−41.8179	−1.37945	−0.689723	+	0.724073i	$0.742268\pi$
$920$	0	0
$921$	−15.3924	−0.507198
$922$	0	0
$923$	−11.9526	−0.393424
$924$	0	0
$925$	4.44009	0.145989
$926$	0	0
$927$	6.63278	0.217849
$928$	0	0
$929$	15.1866	0.498257	0.249129	−	0.968470i	$-0.419856\pi$
$929$	15.1866	0.498257	0.249129	+	0.968470i	$0.419856\pi$
$930$	0	0
$931$	17.1326	0.561499
$932$	0	0
$933$	5.17341	0.169370
$934$	0	0
$935$	1.99259	0.0651646
$936$	0	0
$937$	−13.9755	−0.456561	−0.228280	−	0.973595i	$-0.573310\pi$
$937$	−13.9755	−0.456561	−0.228280	+	0.973595i	$0.573310\pi$
$938$	0	0
$939$	3.44368	0.112380
$940$	0	0
$941$	16.0241	0.522371	0.261186	−	0.965289i	$-0.415887\pi$
$941$	16.0241	0.522371	0.261186	+	0.965289i	$0.415887\pi$
$942$	0	0
$943$	15.2083	0.495251
$944$	0	0
$945$	−20.6968	−0.673268
$946$	0	0
$947$	−16.3790	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$947$	−16.3790	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$948$	0	0
$949$	1.45786	0.0473242
$950$	0	0
$951$	−5.14861	−0.166955
$952$	0	0
$953$	−37.1875	−1.20462	−0.602311	−	0.798261i	$-0.705753\pi$
$953$	−37.1875	−1.20462	−0.602311	+	0.798261i	$0.705753\pi$
$954$	0	0
$955$	−12.4368	−0.402446
$956$	0	0
$957$	−0.760419	−0.0245809
$958$	0	0
$959$	−21.0730	−0.680483
$960$	0	0
$961$	−24.8158	−0.800510
$962$	0	0
$963$	7.04544	0.227036
$964$	0	0
$965$	3.38977	0.109121
$966$	0	0
$967$	25.4262	0.817652	0.408826	−	0.912612i	$-0.365938\pi$
$967$	25.4262	0.817652	0.408826	+	0.912612i	$0.365938\pi$
$968$	0	0
$969$	−0.567557	−0.0182326
$970$	0	0
$971$	−41.4103	−1.32892	−0.664460	−	0.747324i	$-0.731339\pi$
$971$	−41.4103	−1.32892	−0.664460	+	0.747324i	$0.731339\pi$
$972$	0	0
$973$	6.16572	0.197664
$974$	0	0
$975$	−8.22073	−0.263274
$976$	0	0
$977$	15.6390	0.500336	0.250168	−	0.968202i	$-0.419514\pi$
$977$	15.6390	0.500336	0.250168	+	0.968202i	$0.419514\pi$
$978$	0	0
$979$	9.79251	0.312970
$980$	0	0
$981$	−39.6749	−1.26672
$982$	0	0
$983$	−43.4041	−1.38437	−0.692187	−	0.721718i	$-0.743353\pi$
$983$	−43.4041	−1.38437	−0.692187	+	0.721718i	$0.743353\pi$
$984$	0	0
$985$	−40.3116	−1.28443
$986$	0	0
$987$	13.0316	0.414801
$988$	0	0
$989$	13.6010	0.432485
$990$	0	0
$991$	−62.0836	−1.97215	−0.986075	−	0.166301i	$-0.946818\pi$
$991$	−62.0836	−1.97215	−0.986075	+	0.166301i	$0.946818\pi$
$992$	0	0
$993$	14.0572	0.446092
$994$	0	0
$995$	−5.07895	−0.161014
$996$	0	0
$997$	51.4906	1.63072	0.815362	−	0.578952i	$-0.196538\pi$
$997$	51.4906	1.63072	0.815362	+	0.578952i	$0.196538\pi$
$998$	0	0
$999$	6.71936	0.212591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.b.1.4	✓	7
4.3	odd	2		2672.2.a.l.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.b.1.4	✓	7	1.1	even	1	trivial
2672.2.a.l.1.4		7	4.3	odd	2

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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-0.941682 q^{3} -2.86037 q^{5} +1.50273 q^{7} -2.11324 q^{9} +O(q^{10})\)	\(q-0.941682 q^{3} -2.86037 q^{5} +1.50273 q^{7} -2.11324 q^{9} +4.17608 q^{11} +2.74374 q^{13} +2.69356 q^{15} -0.166811 q^{17} -3.61310 q^{19} -1.41509 q^{21} -4.35078 q^{23} +3.18173 q^{25} +4.81504 q^{27} +0.193366 q^{29} +2.48681 q^{31} -3.93254 q^{33} -4.29837 q^{35} +1.39549 q^{37} -2.58373 q^{39} -3.49554 q^{41} -3.12610 q^{43} +6.04464 q^{45} -9.20902 q^{47} -4.74180 q^{49} +0.157083 q^{51} -6.24906 q^{53} -11.9452 q^{55} +3.40239 q^{57} -10.2002 q^{59} +3.14524 q^{61} -3.17562 q^{63} -7.84811 q^{65} -10.1979 q^{67} +4.09705 q^{69} -4.35631 q^{71} +0.531341 q^{73} -2.99618 q^{75} +6.27553 q^{77} -16.4470 q^{79} +1.80547 q^{81} +0.923810 q^{83} +0.477142 q^{85} -0.182089 q^{87} +2.34490 q^{89} +4.12310 q^{91} -2.34178 q^{93} +10.3348 q^{95} +7.23922 q^{97} -8.82505 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9	\( 7 q - 6 q^{3} - q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 9 q^{15} - 9 q^{17} - 3 q^{19} + 7 q^{21} - 10 q^{23} + 3 q^{25} - 12 q^{27} - 5 q^{29} - 21 q^{31} + 8 q^{33} - 12 q^{35} + 19 q^{37} - 27 q^{39} - 22 q^{41} - 19 q^{43} + 13 q^{45} - 13 q^{47} - 14 q^{49} + 4 q^{51} + 5 q^{53} - 17 q^{55} + 5 q^{57} - 18 q^{59} + 26 q^{61} - 20 q^{63} - 20 q^{65} - 27 q^{67} - 3 q^{69} - 46 q^{71} - 25 q^{73} - 19 q^{75} - 19 q^{77} - 22 q^{79} - 9 q^{81} + q^{83} - 11 q^{85} + 9 q^{87} - 3 q^{89} + 33 q^{93} - 40 q^{95} + 11 q^{97} - 21 q^{99}+O(q^{100}) \) 7 * q - 6 * q^3 - q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 9 * q^15 - 9 * q^17 - 3 * q^19 + 7 * q^21 - 10 * q^23 + 3 * q^25 - 12 * q^27 - 5 * q^29 - 21 * q^31 + 8 * q^33 - 12 * q^35 + 19 * q^37 - 27 * q^39 - 22 * q^41 - 19 * q^43 + 13 * q^45 - 13 * q^47 - 14 * q^49 + 4 * q^51 + 5 * q^53 - 17 * q^55 + 5 * q^57 - 18 * q^59 + 26 * q^61 - 20 * q^63 - 20 * q^65 - 27 * q^67 - 3 * q^69 - 46 * q^71 - 25 * q^73 - 19 * q^75 - 19 * q^77 - 22 * q^79 - 9 * q^81 + q^83 - 11 * q^85 + 9 * q^87 - 3 * q^89 + 33 * q^93 - 40 * q^95 + 11 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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