LMFDB - Embedded newform 8019.2.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35184 q^{2} +3.53116 q^{4} +1.70161 q^{5} -2.57467 q^{7} -3.60105 q^{8} +O(q^{10})$	\(q-2.35184 q^{2} +3.53116 q^{4} +1.70161 q^{5} -2.57467 q^{7} -3.60105 q^{8} -4.00192 q^{10} +1.00000 q^{11} -5.85163 q^{13} +6.05522 q^{14} +1.40678 q^{16} -5.66583 q^{17} +7.15847 q^{19} +6.00866 q^{20} -2.35184 q^{22} +2.88734 q^{23} -2.10453 q^{25} +13.7621 q^{26} -9.09158 q^{28} -6.75645 q^{29} +8.10135 q^{31} +3.89357 q^{32} +13.3251 q^{34} -4.38108 q^{35} +3.94863 q^{37} -16.8356 q^{38} -6.12759 q^{40} +1.08017 q^{41} +3.32415 q^{43} +3.53116 q^{44} -6.79056 q^{46} -1.14715 q^{47} -0.371069 q^{49} +4.94951 q^{50} -20.6631 q^{52} -3.73339 q^{53} +1.70161 q^{55} +9.27153 q^{56} +15.8901 q^{58} +1.47114 q^{59} -3.32762 q^{61} -19.0531 q^{62} -11.9706 q^{64} -9.95720 q^{65} +8.77367 q^{67} -20.0070 q^{68} +10.3036 q^{70} -6.47549 q^{71} -9.91186 q^{73} -9.28655 q^{74} +25.2777 q^{76} -2.57467 q^{77} +13.2145 q^{79} +2.39380 q^{80} -2.54040 q^{82} -10.3863 q^{83} -9.64104 q^{85} -7.81787 q^{86} -3.60105 q^{88} -10.1108 q^{89} +15.0660 q^{91} +10.1957 q^{92} +2.69792 q^{94} +12.1809 q^{95} -10.8677 q^{97} +0.872695 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

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 $

$2$	−2.35184	−1.66300	−0.831502	−	0.555522i	$-0.812518\pi$
$2$	−2.35184	−1.66300	−0.831502	+	0.555522i	$0.812518\pi$
$3$	0	0
$3$	0	0
$4$	3.53116	1.76558
$5$	1.70161	0.760983	0.380491	−	0.924784i	$-0.375755\pi$
$5$	1.70161	0.760983	0.380491	+	0.924784i	$0.375755\pi$
$6$	0	0
$7$	−2.57467	−0.973134	−0.486567	−	0.873643i	$-0.661751\pi$
$7$	−2.57467	−0.973134	−0.486567	+	0.873643i	$0.661751\pi$
$8$	−3.60105	−1.27316
$9$	0	0
$10$	−4.00192	−1.26552
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.85163	−1.62295	−0.811476	−	0.584386i	$-0.801335\pi$
$13$	−5.85163	−1.62295	−0.811476	+	0.584386i	$0.801335\pi$
$14$	6.05522	1.61833
$15$	0	0
$16$	1.40678	0.351696
$17$	−5.66583	−1.37417	−0.687083	−	0.726579i	$-0.741109\pi$
$17$	−5.66583	−1.37417	−0.687083	+	0.726579i	$0.741109\pi$
$18$	0	0
$19$	7.15847	1.64227	0.821133	−	0.570736i	$-0.193342\pi$
$19$	7.15847	1.64227	0.821133	+	0.570736i	$0.193342\pi$
$20$	6.00866	1.34358
$21$	0	0
$22$	−2.35184	−0.501414
$23$	2.88734	0.602052	0.301026	−	0.953616i	$-0.402671\pi$
$23$	2.88734	0.602052	0.301026	+	0.953616i	$0.402671\pi$
$24$	0	0
$25$	−2.10453	−0.420905
$26$	13.7621	2.69897
$27$	0	0
$28$	−9.09158	−1.71815
$29$	−6.75645	−1.25464	−0.627321	−	0.778761i	$-0.715849\pi$
$29$	−6.75645	−1.25464	−0.627321	+	0.778761i	$0.715849\pi$
$30$	0	0
$31$	8.10135	1.45504	0.727522	−	0.686084i	$-0.240672\pi$
$31$	8.10135	1.45504	0.727522	+	0.686084i	$0.240672\pi$
$32$	3.89357	0.688293
$33$	0	0
$34$	13.3251	2.28524
$35$	−4.38108	−0.740538
$36$	0	0
$37$	3.94863	0.649150	0.324575	−	0.945860i	$-0.394779\pi$
$37$	3.94863	0.649150	0.324575	+	0.945860i	$0.394779\pi$
$38$	−16.8356	−2.73110
$39$	0	0
$40$	−6.12759	−0.968856
$41$	1.08017	0.168695	0.0843474	−	0.996436i	$-0.473119\pi$
$41$	1.08017	0.168695	0.0843474	+	0.996436i	$0.473119\pi$
$42$	0	0
$43$	3.32415	0.506928	0.253464	−	0.967345i	$-0.418430\pi$
$43$	3.32415	0.506928	0.253464	+	0.967345i	$0.418430\pi$
$44$	3.53116	0.532343
$45$	0	0
$46$	−6.79056	−1.00121
$47$	−1.14715	−0.167329	−0.0836647	−	0.996494i	$-0.526662\pi$
$47$	−1.14715	−0.167329	−0.0836647	+	0.996494i	$0.526662\pi$
$48$	0	0
$49$	−0.371069	−0.0530098
$50$	4.94951	0.699967
$51$	0	0
$52$	−20.6631	−2.86545
$53$	−3.73339	−0.512820	−0.256410	−	0.966568i	$-0.582540\pi$
$53$	−3.73339	−0.512820	−0.256410	+	0.966568i	$0.582540\pi$
$54$	0	0
$55$	1.70161	0.229445
$56$	9.27153	1.23896
$57$	0	0
$58$	15.8901	2.08647
$59$	1.47114	0.191526	0.0957629	−	0.995404i	$-0.469471\pi$
$59$	1.47114	0.191526	0.0957629	+	0.995404i	$0.469471\pi$
$60$	0	0
$61$	−3.32762	−0.426059	−0.213029	−	0.977046i	$-0.568333\pi$
$61$	−3.32762	−0.426059	−0.213029	+	0.977046i	$0.568333\pi$
$62$	−19.0531	−2.41974
$63$	0	0
$64$	−11.9706	−1.49633
$65$	−9.95720	−1.23504
$66$	0	0
$67$	8.77367	1.07187	0.535937	−	0.844258i	$-0.319958\pi$
$67$	8.77367	1.07187	0.535937	+	0.844258i	$0.319958\pi$
$68$	−20.0070	−2.42620
$69$	0	0
$70$	10.3036	1.23152
$71$	−6.47549	−0.768499	−0.384250	−	0.923229i	$-0.625540\pi$
$71$	−6.47549	−0.768499	−0.384250	+	0.923229i	$0.625540\pi$
$72$	0	0
$73$	−9.91186	−1.16009	−0.580047	−	0.814583i	$-0.696966\pi$
$73$	−9.91186	−1.16009	−0.580047	+	0.814583i	$0.696966\pi$
$74$	−9.28655	−1.07954
$75$	0	0
$76$	25.2777	2.89956
$77$	−2.57467	−0.293411
$78$	0	0
$79$	13.2145	1.48675	0.743374	−	0.668876i	$-0.233224\pi$
$79$	13.2145	1.48675	0.743374	+	0.668876i	$0.233224\pi$
$80$	2.39380	0.267635
$81$	0	0
$82$	−2.54040	−0.280540
$83$	−10.3863	−1.14004	−0.570020	−	0.821631i	$-0.693065\pi$
$83$	−10.3863	−1.14004	−0.570020	+	0.821631i	$0.693065\pi$
$84$	0	0
$85$	−9.64104	−1.04572
$86$	−7.81787	−0.843023
$87$	0	0
$88$	−3.60105	−0.383874
$89$	−10.1108	−1.07175	−0.535873	−	0.844299i	$-0.680017\pi$
$89$	−10.1108	−1.07175	−0.535873	+	0.844299i	$0.680017\pi$
$90$	0	0
$91$	15.0660	1.57935
$92$	10.1957	1.06297
$93$	0	0
$94$	2.69792	0.278269
$95$	12.1809	1.24974
$96$	0	0
$97$	−10.8677	−1.10345	−0.551725	−	0.834026i	$-0.686030\pi$
$97$	−10.8677	−1.10345	−0.551725	+	0.834026i	$0.686030\pi$
$98$	0.872695	0.0881555
$99$	0	0
$100$	−7.43142	−0.743142
$101$	6.83986	0.680591	0.340296	−	0.940318i	$-0.389473\pi$
$101$	6.83986	0.680591	0.340296	+	0.940318i	$0.389473\pi$
$102$	0	0
$103$	6.36164	0.626831	0.313415	−	0.949616i	$-0.398527\pi$
$103$	6.36164	0.626831	0.313415	+	0.949616i	$0.398527\pi$
$104$	21.0720	2.06628
$105$	0	0
$106$	8.78034	0.852822
$107$	−14.4100	−1.39307	−0.696535	−	0.717523i	$-0.745276\pi$
$107$	−14.4100	−1.39307	−0.696535	+	0.717523i	$0.745276\pi$
$108$	0	0
$109$	−0.326093	−0.0312341	−0.0156170	−	0.999878i	$-0.504971\pi$
$109$	−0.326093	−0.0312341	−0.0156170	+	0.999878i	$0.504971\pi$
$110$	−4.00192	−0.381568
$111$	0	0
$112$	−3.62201	−0.342247
$113$	0.460936	0.0433612	0.0216806	−	0.999765i	$-0.493098\pi$
$113$	0.460936	0.0433612	0.0216806	+	0.999765i	$0.493098\pi$
$114$	0	0
$115$	4.91312	0.458151
$116$	−23.8581	−2.21517
$117$	0	0
$118$	−3.45988	−0.318508
$119$	14.5877	1.33725
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.82605	0.708537
$123$	0	0
$124$	28.6072	2.56900
$125$	−12.0891	−1.08128
$126$	0	0
$127$	18.9800	1.68420	0.842101	−	0.539320i	$-0.181318\pi$
$127$	18.9800	1.68420	0.842101	+	0.539320i	$0.181318\pi$
$128$	20.3659	1.80011
$129$	0	0
$130$	23.4178	2.05387
$131$	1.54770	0.135223	0.0676115	−	0.997712i	$-0.478462\pi$
$131$	1.54770	0.135223	0.0676115	+	0.997712i	$0.478462\pi$
$132$	0	0
$133$	−18.4307	−1.59815
$134$	−20.6343	−1.78253
$135$	0	0
$136$	20.4030	1.74954
$137$	3.93068	0.335820	0.167910	−	0.985802i	$-0.446298\pi$
$137$	3.93068	0.335820	0.167910	+	0.985802i	$0.446298\pi$
$138$	0	0
$139$	−9.74923	−0.826919	−0.413460	−	0.910522i	$-0.635680\pi$
$139$	−9.74923	−0.826919	−0.413460	+	0.910522i	$0.635680\pi$
$140$	−15.4703	−1.30748
$141$	0	0
$142$	15.2293	1.27802
$143$	−5.85163	−0.489338
$144$	0	0
$145$	−11.4968	−0.954761
$146$	23.3111	1.92924
$147$	0	0
$148$	13.9432	1.14613
$149$	−7.42561	−0.608330	−0.304165	−	0.952619i	$-0.598377\pi$
$149$	−7.42561	−0.608330	−0.304165	+	0.952619i	$0.598377\pi$
$150$	0	0
$151$	−8.58439	−0.698587	−0.349294	−	0.937013i	$-0.613578\pi$
$151$	−8.58439	−0.698587	−0.349294	+	0.937013i	$0.613578\pi$
$152$	−25.7780	−2.09088
$153$	0	0
$154$	6.05522	0.487944
$155$	13.7853	1.10726
$156$	0	0
$157$	22.4456	1.79136	0.895678	−	0.444704i	$-0.146691\pi$
$157$	22.4456	1.79136	0.895678	+	0.444704i	$0.146691\pi$
$158$	−31.0784	−2.47247
$159$	0	0
$160$	6.62534	0.523779
$161$	−7.43395	−0.585877
$162$	0	0
$163$	8.47721	0.663986	0.331993	−	0.943282i	$-0.392279\pi$
$163$	8.47721	0.663986	0.331993	+	0.943282i	$0.392279\pi$
$164$	3.81427	0.297844
$165$	0	0
$166$	24.4268	1.89589
$167$	−20.1997	−1.56310	−0.781549	−	0.623844i	$-0.785570\pi$
$167$	−20.1997	−1.56310	−0.781549	+	0.623844i	$0.785570\pi$
$168$	0	0
$169$	21.2416	1.63397
$170$	22.6742	1.73903
$171$	0	0
$172$	11.7381	0.895022
$173$	−8.19963	−0.623407	−0.311703	−	0.950179i	$-0.600899\pi$
$173$	−8.19963	−0.623407	−0.311703	+	0.950179i	$0.600899\pi$
$174$	0	0
$175$	5.41846	0.409597
$176$	1.40678	0.106040
$177$	0	0
$178$	23.7791	1.78232
$179$	18.5046	1.38310	0.691549	−	0.722329i	$-0.256928\pi$
$179$	18.5046	1.38310	0.691549	+	0.722329i	$0.256928\pi$
$180$	0	0
$181$	−2.76297	−0.205370	−0.102685	−	0.994714i	$-0.532743\pi$
$181$	−2.76297	−0.205370	−0.102685	+	0.994714i	$0.532743\pi$
$182$	−35.4329	−2.62646
$183$	0	0
$184$	−10.3975	−0.766511
$185$	6.71902	0.493992
$186$	0	0
$187$	−5.66583	−0.414327
$188$	−4.05078	−0.295434
$189$	0	0
$190$	−28.6476	−2.07832
$191$	−2.66385	−0.192749	−0.0963747	−	0.995345i	$-0.530725\pi$
$191$	−2.66385	−0.192749	−0.0963747	+	0.995345i	$0.530725\pi$
$192$	0	0
$193$	−3.30632	−0.237994	−0.118997	−	0.992895i	$-0.537968\pi$
$193$	−3.30632	−0.237994	−0.118997	+	0.992895i	$0.537968\pi$
$194$	25.5592	1.83504
$195$	0	0
$196$	−1.31030	−0.0935931
$197$	7.39454	0.526839	0.263419	−	0.964681i	$-0.415150\pi$
$197$	7.39454	0.526839	0.263419	+	0.964681i	$0.415150\pi$
$198$	0	0
$199$	18.2916	1.29666	0.648329	−	0.761360i	$-0.275468\pi$
$199$	18.2916	1.29666	0.648329	+	0.761360i	$0.275468\pi$
$200$	7.57851	0.535881
$201$	0	0
$202$	−16.0863	−1.13183
$203$	17.3956	1.22094
$204$	0	0
$205$	1.83803	0.128374
$206$	−14.9616	−1.04242
$207$	0	0
$208$	−8.23199	−0.570786
$209$	7.15847	0.495162
$210$	0	0
$211$	−1.37294	−0.0945172	−0.0472586	−	0.998883i	$-0.515048\pi$
$211$	−1.37294	−0.0945172	−0.0472586	+	0.998883i	$0.515048\pi$
$212$	−13.1832	−0.905425
$213$	0	0
$214$	33.8901	2.31668
$215$	5.65640	0.385763
$216$	0	0
$217$	−20.8583	−1.41595
$218$	0.766920	0.0519424
$219$	0	0
$220$	6.00866	0.405104
$221$	33.1544	2.23021
$222$	0	0
$223$	−14.2999	−0.957595	−0.478797	−	0.877925i	$-0.658927\pi$
$223$	−14.2999	−0.957595	−0.478797	+	0.877925i	$0.658927\pi$
$224$	−10.0247	−0.669801
$225$	0	0
$226$	−1.08405	−0.0721098
$227$	29.2773	1.94320	0.971600	−	0.236627i	$-0.0760421\pi$
$227$	29.2773	1.94320	0.971600	+	0.236627i	$0.0760421\pi$
$228$	0	0
$229$	14.7283	0.973272	0.486636	−	0.873605i	$-0.338224\pi$
$229$	14.7283	0.973272	0.486636	+	0.873605i	$0.338224\pi$
$230$	−11.5549	−0.761907
$231$	0	0
$232$	24.3303	1.59737
$233$	18.7783	1.23021	0.615103	−	0.788447i	$-0.289114\pi$
$233$	18.7783	1.23021	0.615103	+	0.788447i	$0.289114\pi$
$234$	0	0
$235$	−1.95201	−0.127335
$236$	5.19482	0.338154
$237$	0	0
$238$	−34.3079	−2.22385
$239$	8.07417	0.522275	0.261137	−	0.965302i	$-0.415902\pi$
$239$	8.07417	0.522275	0.261137	+	0.965302i	$0.415902\pi$
$240$	0	0
$241$	26.3675	1.69848	0.849240	−	0.528008i	$-0.177061\pi$
$241$	26.3675	1.69848	0.849240	+	0.528008i	$0.177061\pi$
$242$	−2.35184	−0.151182
$243$	0	0
$244$	−11.7504	−0.752241
$245$	−0.631414	−0.0403396
$246$	0	0
$247$	−41.8888	−2.66532
$248$	−29.1734	−1.85251
$249$	0	0
$250$	28.4317	1.79818
$251$	14.8526	0.937488	0.468744	−	0.883334i	$-0.344707\pi$
$251$	14.8526	0.937488	0.468744	+	0.883334i	$0.344707\pi$
$252$	0	0
$253$	2.88734	0.181525
$254$	−44.6380	−2.80083
$255$	0	0
$256$	−23.9561	−1.49726
$257$	−9.85842	−0.614951	−0.307476	−	0.951556i	$-0.599484\pi$
$257$	−9.85842	−0.614951	−0.307476	+	0.951556i	$0.599484\pi$
$258$	0	0
$259$	−10.1664	−0.631710
$260$	−35.1605	−2.18056
$261$	0	0
$262$	−3.63994	−0.224876
$263$	−12.7125	−0.783884	−0.391942	−	0.919990i	$-0.628197\pi$
$263$	−12.7125	−0.783884	−0.391942	+	0.919990i	$0.628197\pi$
$264$	0	0
$265$	−6.35277	−0.390247
$266$	43.3461	2.65772
$267$	0	0
$268$	30.9813	1.89248
$269$	9.78834	0.596806	0.298403	−	0.954440i	$-0.403546\pi$
$269$	9.78834	0.596806	0.298403	+	0.954440i	$0.403546\pi$
$270$	0	0
$271$	2.37582	0.144321	0.0721604	−	0.997393i	$-0.477011\pi$
$271$	2.37582	0.144321	0.0721604	+	0.997393i	$0.477011\pi$
$272$	−7.97060	−0.483289
$273$	0	0
$274$	−9.24433	−0.558470
$275$	−2.10453	−0.126908
$276$	0	0
$277$	−15.4417	−0.927804	−0.463902	−	0.885887i	$-0.653551\pi$
$277$	−15.4417	−0.927804	−0.463902	+	0.885887i	$0.653551\pi$
$278$	22.9287	1.37517
$279$	0	0
$280$	15.7765	0.942827
$281$	26.3959	1.57465	0.787323	−	0.616541i	$-0.211466\pi$
$281$	26.3959	1.57465	0.787323	+	0.616541i	$0.211466\pi$
$282$	0	0
$283$	29.7988	1.77135	0.885677	−	0.464302i	$-0.153695\pi$
$283$	29.7988	1.77135	0.885677	+	0.464302i	$0.153695\pi$
$284$	−22.8660	−1.35685
$285$	0	0
$286$	13.7621	0.813771
$287$	−2.78109	−0.164163
$288$	0	0
$289$	15.1017	0.888333
$290$	27.0388	1.58777
$291$	0	0
$292$	−35.0004	−2.04824
$293$	−13.7914	−0.805704	−0.402852	−	0.915265i	$-0.631981\pi$
$293$	−13.7914	−0.805704	−0.402852	+	0.915265i	$0.631981\pi$
$294$	0	0
$295$	2.50330	0.145748
$296$	−14.2192	−0.826475
$297$	0	0
$298$	17.4639	1.01165
$299$	−16.8956	−0.977100
$300$	0	0
$301$	−8.55858	−0.493309
$302$	20.1891	1.16175
$303$	0	0
$304$	10.0704	0.577579
$305$	−5.66232	−0.324223
$306$	0	0
$307$	−18.0347	−1.02930	−0.514648	−	0.857401i	$-0.672078\pi$
$307$	−18.0347	−1.02930	−0.514648	+	0.857401i	$0.672078\pi$
$308$	−9.09158	−0.518041
$309$	0	0
$310$	−32.4209	−1.84138
$311$	23.3840	1.32598	0.662992	−	0.748626i	$-0.269286\pi$
$311$	23.3840	1.32598	0.662992	+	0.748626i	$0.269286\pi$
$312$	0	0
$313$	14.7477	0.833591	0.416795	−	0.909000i	$-0.363153\pi$
$313$	14.7477	0.833591	0.416795	+	0.909000i	$0.363153\pi$
$314$	−52.7886	−2.97903
$315$	0	0
$316$	46.6626	2.62497
$317$	−16.2712	−0.913881	−0.456940	−	0.889497i	$-0.651055\pi$
$317$	−16.2712	−0.913881	−0.456940	+	0.889497i	$0.651055\pi$
$318$	0	0
$319$	−6.75645	−0.378289
$320$	−20.3693	−1.13868
$321$	0	0
$322$	17.4835	0.974316
$323$	−40.5587	−2.25675
$324$	0	0
$325$	12.3149	0.683108
$326$	−19.9371	−1.10421
$327$	0	0
$328$	−3.88976	−0.214776
$329$	2.95354	0.162834
$330$	0	0
$331$	−27.2928	−1.50015	−0.750074	−	0.661354i	$-0.769982\pi$
$331$	−27.2928	−1.50015	−0.750074	+	0.661354i	$0.769982\pi$
$332$	−36.6756	−2.01283
$333$	0	0
$334$	47.5064	2.59944
$335$	14.9294	0.815678
$336$	0	0
$337$	−27.8954	−1.51956	−0.759779	−	0.650182i	$-0.774693\pi$
$337$	−27.8954	−1.51956	−0.759779	+	0.650182i	$0.774693\pi$
$338$	−49.9569	−2.71730
$339$	0	0
$340$	−34.0441	−1.84630
$341$	8.10135	0.438712
$342$	0	0
$343$	18.9781	1.02472
$344$	−11.9704	−0.645402
$345$	0	0
$346$	19.2842	1.03673
$347$	33.5419	1.80062	0.900311	−	0.435248i	$-0.143339\pi$
$347$	33.5419	1.80062	0.900311	+	0.435248i	$0.143339\pi$
$348$	0	0
$349$	15.6295	0.836626	0.418313	−	0.908303i	$-0.362622\pi$
$349$	15.6295	0.836626	0.418313	+	0.908303i	$0.362622\pi$
$350$	−12.7434	−0.681161
$351$	0	0
$352$	3.89357	0.207528
$353$	−23.4003	−1.24547	−0.622737	−	0.782431i	$-0.713979\pi$
$353$	−23.4003	−1.24547	−0.622737	+	0.782431i	$0.713979\pi$
$354$	0	0
$355$	−11.0188	−0.584815
$356$	−35.7030	−1.89225
$357$	0	0
$358$	−43.5199	−2.30010
$359$	−21.0272	−1.10978	−0.554888	−	0.831925i	$-0.687239\pi$
$359$	−21.0272	−1.10978	−0.554888	+	0.831925i	$0.687239\pi$
$360$	0	0
$361$	32.2438	1.69704
$362$	6.49807	0.341531
$363$	0	0
$364$	53.2006	2.78847
$365$	−16.8661	−0.882812
$366$	0	0
$367$	14.6955	0.767097	0.383549	−	0.923521i	$-0.374702\pi$
$367$	14.6955	0.767097	0.383549	+	0.923521i	$0.374702\pi$
$368$	4.06186	0.211739
$369$	0	0
$370$	−15.8021	−0.821511
$371$	9.61224	0.499043
$372$	0	0
$373$	−13.3967	−0.693653	−0.346826	−	0.937929i	$-0.612741\pi$
$373$	−13.3967	−0.693653	−0.346826	+	0.937929i	$0.612741\pi$
$374$	13.3251	0.689027
$375$	0	0
$376$	4.13096	0.213038
$377$	39.5363	2.03622
$378$	0	0
$379$	2.34860	0.120639	0.0603197	−	0.998179i	$-0.480788\pi$
$379$	2.34860	0.120639	0.0603197	+	0.998179i	$0.480788\pi$
$380$	43.0128	2.20651
$381$	0	0
$382$	6.26496	0.320543
$383$	19.5881	1.00091	0.500454	−	0.865763i	$-0.333167\pi$
$383$	19.5881	1.00091	0.500454	+	0.865763i	$0.333167\pi$
$384$	0	0
$385$	−4.38108	−0.223281
$386$	7.77594	0.395785
$387$	0	0
$388$	−38.3757	−1.94823
$389$	−19.1153	−0.969184	−0.484592	−	0.874740i	$-0.661032\pi$
$389$	−19.1153	−0.969184	−0.484592	+	0.874740i	$0.661032\pi$
$390$	0	0
$391$	−16.3592	−0.827319
$392$	1.33624	0.0674902
$393$	0	0
$394$	−17.3908	−0.876135
$395$	22.4859	1.13139
$396$	0	0
$397$	−27.5024	−1.38031	−0.690153	−	0.723664i	$-0.742457\pi$
$397$	−27.5024	−1.38031	−0.690153	+	0.723664i	$0.742457\pi$
$398$	−43.0190	−2.15635
$399$	0	0
$400$	−2.96061	−0.148031
$401$	15.5087	0.774469	0.387234	−	0.921981i	$-0.373430\pi$
$401$	15.5087	0.774469	0.387234	+	0.921981i	$0.373430\pi$
$402$	0	0
$403$	−47.4061	−2.36147
$404$	24.1527	1.20164
$405$	0	0
$406$	−40.9118	−2.03042
$407$	3.94863	0.195726
$408$	0	0
$409$	0.884306	0.0437261	0.0218631	−	0.999761i	$-0.493040\pi$
$409$	0.884306	0.0437261	0.0218631	+	0.999761i	$0.493040\pi$
$410$	−4.32277	−0.213486
$411$	0	0
$412$	22.4640	1.10672
$413$	−3.78769	−0.186380
$414$	0	0
$415$	−17.6734	−0.867551
$416$	−22.7838	−1.11707
$417$	0	0
$418$	−16.8356	−0.823456
$419$	1.13942	0.0556645	0.0278322	−	0.999613i	$-0.491140\pi$
$419$	1.13942	0.0556645	0.0278322	+	0.999613i	$0.491140\pi$
$420$	0	0
$421$	10.6341	0.518272	0.259136	−	0.965841i	$-0.416562\pi$
$421$	10.6341	0.518272	0.259136	+	0.965841i	$0.416562\pi$
$422$	3.22894	0.157183
$423$	0	0
$424$	13.4441	0.652904
$425$	11.9239	0.578394
$426$	0	0
$427$	8.56754	0.414612
$428$	−50.8841	−2.45958
$429$	0	0
$430$	−13.3030	−0.641526
$431$	−1.38140	−0.0665398	−0.0332699	−	0.999446i	$-0.510592\pi$
$431$	−1.38140	−0.0665398	−0.0332699	+	0.999446i	$0.510592\pi$
$432$	0	0
$433$	−10.7613	−0.517154	−0.258577	−	0.965991i	$-0.583254\pi$
$433$	−10.7613	−0.517154	−0.258577	+	0.965991i	$0.583254\pi$
$434$	49.0554	2.35474
$435$	0	0
$436$	−1.15149	−0.0551463
$437$	20.6689	0.988729
$438$	0	0
$439$	23.8354	1.13760	0.568801	−	0.822475i	$-0.307407\pi$
$439$	23.8354	1.13760	0.568801	+	0.822475i	$0.307407\pi$
$440$	−6.12759	−0.292121
$441$	0	0
$442$	−77.9739	−3.70884
$443$	18.5626	0.881935	0.440968	−	0.897523i	$-0.354635\pi$
$443$	18.5626	0.881935	0.440968	+	0.897523i	$0.354635\pi$
$444$	0	0
$445$	−17.2047	−0.815580
$446$	33.6312	1.59248
$447$	0	0
$448$	30.8204	1.45613
$449$	10.9058	0.514675	0.257338	−	0.966322i	$-0.417155\pi$
$449$	10.9058	0.514675	0.257338	+	0.966322i	$0.417155\pi$
$450$	0	0
$451$	1.08017	0.0508634
$452$	1.62764	0.0765577
$453$	0	0
$454$	−68.8555	−3.23155
$455$	25.6365	1.20186
$456$	0	0
$457$	5.25948	0.246028	0.123014	−	0.992405i	$-0.460744\pi$
$457$	5.25948	0.246028	0.123014	+	0.992405i	$0.460744\pi$
$458$	−34.6386	−1.61855
$459$	0	0
$460$	17.3490	0.808903
$461$	19.6038	0.913042	0.456521	−	0.889713i	$-0.349095\pi$
$461$	19.6038	0.913042	0.456521	+	0.889713i	$0.349095\pi$
$462$	0	0
$463$	−0.305931	−0.0142178	−0.00710891	−	0.999975i	$-0.502263\pi$
$463$	−0.305931	−0.0142178	−0.00710891	+	0.999975i	$0.502263\pi$
$464$	−9.50487	−0.441253
$465$	0	0
$466$	−44.1636	−2.04584
$467$	2.92864	0.135521	0.0677606	−	0.997702i	$-0.478415\pi$
$467$	2.92864	0.135521	0.0677606	+	0.997702i	$0.478415\pi$
$468$	0	0
$469$	−22.5893	−1.04308
$470$	4.59081	0.211758
$471$	0	0
$472$	−5.29764	−0.243844
$473$	3.32415	0.152844
$474$	0	0
$475$	−15.0652	−0.691238
$476$	51.5114	2.36102
$477$	0	0
$478$	−18.9892	−0.868545
$479$	23.3444	1.06663	0.533316	−	0.845916i	$-0.320946\pi$
$479$	23.3444	1.06663	0.533316	+	0.845916i	$0.320946\pi$
$480$	0	0
$481$	−23.1059	−1.05354
$482$	−62.0122	−2.82458
$483$	0	0
$484$	3.53116	0.160507
$485$	−18.4926	−0.839707
$486$	0	0
$487$	23.7200	1.07486	0.537429	−	0.843309i	$-0.319396\pi$
$487$	23.7200	1.07486	0.537429	+	0.843309i	$0.319396\pi$
$488$	11.9830	0.542443
$489$	0	0
$490$	1.48499	0.0670848
$491$	21.9901	0.992398	0.496199	−	0.868209i	$-0.334729\pi$
$491$	21.9901	0.992398	0.496199	+	0.868209i	$0.334729\pi$
$492$	0	0
$493$	38.2809	1.72409
$494$	98.5158	4.43244
$495$	0	0
$496$	11.3968	0.511733
$497$	16.6723	0.747853
$498$	0	0
$499$	27.0238	1.20975	0.604875	−	0.796320i	$-0.293223\pi$
$499$	27.0238	1.20975	0.604875	+	0.796320i	$0.293223\pi$
$500$	−42.6887	−1.90910
$501$	0	0
$502$	−34.9310	−1.55905
$503$	−4.48721	−0.200075	−0.100037	−	0.994984i	$-0.531896\pi$
$503$	−4.48721	−0.200075	−0.100037	+	0.994984i	$0.531896\pi$
$504$	0	0
$505$	11.6388	0.517918
$506$	−6.79056	−0.301877
$507$	0	0
$508$	67.0215	2.97360
$509$	−0.980116	−0.0434429	−0.0217214	−	0.999764i	$-0.506915\pi$
$509$	−0.980116	−0.0434429	−0.0217214	+	0.999764i	$0.506915\pi$
$510$	0	0
$511$	25.5198	1.12893
$512$	15.6092	0.689837
$513$	0	0
$514$	23.1854	1.02267
$515$	10.8250	0.477007
$516$	0	0
$517$	−1.14715	−0.0504517
$518$	23.9098	1.05054
$519$	0	0
$520$	35.8564	1.57241
$521$	−42.4134	−1.85816	−0.929082	−	0.369875i	$-0.879401\pi$
$521$	−42.4134	−1.85816	−0.929082	+	0.369875i	$0.879401\pi$
$522$	0	0
$523$	4.55646	0.199240	0.0996200	−	0.995026i	$-0.468237\pi$
$523$	4.55646	0.199240	0.0996200	+	0.995026i	$0.468237\pi$
$524$	5.46518	0.238747
$525$	0	0
$526$	29.8977	1.30360
$527$	−45.9009	−1.99947
$528$	0	0
$529$	−14.6633	−0.637534
$530$	14.9407	0.648983
$531$	0	0
$532$	−65.0819	−2.82166
$533$	−6.32078	−0.273783
$534$	0	0
$535$	−24.5202	−1.06010
$536$	−31.5945	−1.36467
$537$	0	0
$538$	−23.0206	−0.992490
$539$	−0.371069	−0.0159831
$540$	0	0
$541$	−26.4371	−1.13662	−0.568311	−	0.822814i	$-0.692403\pi$
$541$	−26.4371	−1.13662	−0.568311	+	0.822814i	$0.692403\pi$
$542$	−5.58755	−0.240006
$543$	0	0
$544$	−22.0603	−0.945829
$545$	−0.554883	−0.0237686
$546$	0	0
$547$	35.6754	1.52537	0.762685	−	0.646770i	$-0.223880\pi$
$547$	35.6754	1.52537	0.762685	+	0.646770i	$0.223880\pi$
$548$	13.8799	0.592918
$549$	0	0
$550$	4.94951	0.211048
$551$	−48.3659	−2.06046
$552$	0	0
$553$	−34.0230	−1.44680
$554$	36.3165	1.54294
$555$	0	0
$556$	−34.4261	−1.45999
$557$	2.35000	0.0995728	0.0497864	−	0.998760i	$-0.484146\pi$
$557$	2.35000	0.0995728	0.0497864	+	0.998760i	$0.484146\pi$
$558$	0	0
$559$	−19.4517	−0.822719
$560$	−6.16324	−0.260444
$561$	0	0
$562$	−62.0789	−2.61864
$563$	2.44207	0.102921	0.0514606	−	0.998675i	$-0.483612\pi$
$563$	2.44207	0.102921	0.0514606	+	0.998675i	$0.483612\pi$
$564$	0	0
$565$	0.784333	0.0329971
$566$	−70.0820	−2.94577
$567$	0	0
$568$	23.3186	0.978426
$569$	−11.9477	−0.500876	−0.250438	−	0.968133i	$-0.580575\pi$
$569$	−11.9477	−0.500876	−0.250438	+	0.968133i	$0.580575\pi$
$570$	0	0
$571$	−33.0899	−1.38477	−0.692384	−	0.721529i	$-0.743440\pi$
$571$	−33.0899	−1.38477	−0.692384	+	0.721529i	$0.743440\pi$
$572$	−20.6631	−0.863966
$573$	0	0
$574$	6.54069	0.273003
$575$	−6.07648	−0.253407
$576$	0	0
$577$	23.9257	0.996042	0.498021	−	0.867165i	$-0.334060\pi$
$577$	23.9257	0.996042	0.498021	+	0.867165i	$0.334060\pi$
$578$	−35.5167	−1.47730
$579$	0	0
$580$	−40.5972	−1.68571
$581$	26.7412	1.10941
$582$	0	0
$583$	−3.73339	−0.154621
$584$	35.6931	1.47699
$585$	0	0
$586$	32.4353	1.33989
$587$	−13.0295	−0.537784	−0.268892	−	0.963170i	$-0.586657\pi$
$587$	−13.0295	−0.537784	−0.268892	+	0.963170i	$0.586657\pi$
$588$	0	0
$589$	57.9933	2.38957
$590$	−5.88737	−0.242379
$591$	0	0
$592$	5.55486	0.228304
$593$	37.1326	1.52485	0.762427	−	0.647074i	$-0.224007\pi$
$593$	37.1326	1.52485	0.762427	+	0.647074i	$0.224007\pi$
$594$	0	0
$595$	24.8225	1.01762
$596$	−26.2210	−1.07406
$597$	0	0
$598$	39.7359	1.62492
$599$	−11.9459	−0.488097	−0.244048	−	0.969763i	$-0.578476\pi$
$599$	−11.9459	−0.488097	−0.244048	+	0.969763i	$0.578476\pi$
$600$	0	0
$601$	32.9848	1.34548	0.672739	−	0.739880i	$-0.265118\pi$
$601$	32.9848	1.34548	0.672739	+	0.739880i	$0.265118\pi$
$602$	20.1284	0.820374
$603$	0	0
$604$	−30.3129	−1.23341
$605$	1.70161	0.0691803
$606$	0	0
$607$	−5.11814	−0.207739	−0.103869	−	0.994591i	$-0.533122\pi$
$607$	−5.11814	−0.207739	−0.103869	+	0.994591i	$0.533122\pi$
$608$	27.8720	1.13036
$609$	0	0
$610$	13.3169	0.539185
$611$	6.71272	0.271567
$612$	0	0
$613$	1.26749	0.0511934	0.0255967	−	0.999672i	$-0.491851\pi$
$613$	1.26749	0.0511934	0.0255967	+	0.999672i	$0.491851\pi$
$614$	42.4149	1.71172
$615$	0	0
$616$	9.27153	0.373560
$617$	−23.1904	−0.933612	−0.466806	−	0.884360i	$-0.654595\pi$
$617$	−23.1904	−0.933612	−0.466806	+	0.884360i	$0.654595\pi$
$618$	0	0
$619$	20.2856	0.815347	0.407673	−	0.913128i	$-0.366340\pi$
$619$	20.2856	0.815347	0.407673	+	0.913128i	$0.366340\pi$
$620$	48.6782	1.95496
$621$	0	0
$622$	−54.9955	−2.20512
$623$	26.0321	1.04295
$624$	0	0
$625$	−10.0483	−0.401934
$626$	−34.6843	−1.38626
$627$	0	0
$628$	79.2591	3.16278
$629$	−22.3723	−0.892040
$630$	0	0
$631$	−4.19753	−0.167101	−0.0835505	−	0.996504i	$-0.526626\pi$
$631$	−4.19753	−0.167101	−0.0835505	+	0.996504i	$0.526626\pi$
$632$	−47.5861	−1.89287
$633$	0	0
$634$	38.2673	1.51979
$635$	32.2965	1.28165
$636$	0	0
$637$	2.17136	0.0860323
$638$	15.8901	0.629096
$639$	0	0
$640$	34.6548	1.36985
$641$	−15.8038	−0.624212	−0.312106	−	0.950047i	$-0.601034\pi$
$641$	−15.8038	−0.624212	−0.312106	+	0.950047i	$0.601034\pi$
$642$	0	0
$643$	−21.9514	−0.865679	−0.432840	−	0.901471i	$-0.642488\pi$
$643$	−21.9514	−0.865679	−0.432840	+	0.901471i	$0.642488\pi$
$644$	−26.2505	−1.03441
$645$	0	0
$646$	95.3877	3.75298
$647$	−7.06675	−0.277823	−0.138911	−	0.990305i	$-0.544360\pi$
$647$	−7.06675	−0.277823	−0.138911	+	0.990305i	$0.544360\pi$
$648$	0	0
$649$	1.47114	0.0577472
$650$	−28.9627	−1.13601
$651$	0	0
$652$	29.9344	1.17232
$653$	−33.8861	−1.32606	−0.663032	−	0.748591i	$-0.730731\pi$
$653$	−33.8861	−1.32606	−0.663032	+	0.748591i	$0.730731\pi$
$654$	0	0
$655$	2.63358	0.102902
$656$	1.51957	0.0593293
$657$	0	0
$658$	−6.94626	−0.270793
$659$	−10.8597	−0.423032	−0.211516	−	0.977375i	$-0.567840\pi$
$659$	−10.8597	−0.423032	−0.211516	+	0.977375i	$0.567840\pi$
$660$	0	0
$661$	18.1852	0.707322	0.353661	−	0.935374i	$-0.384937\pi$
$661$	18.1852	0.707322	0.353661	+	0.935374i	$0.384937\pi$
$662$	64.1883	2.49475
$663$	0	0
$664$	37.4015	1.45146
$665$	−31.3619	−1.21616
$666$	0	0
$667$	−19.5082	−0.755359
$668$	−71.3283	−2.75977
$669$	0	0
$670$	−35.1115	−1.35648
$671$	−3.32762	−0.128461
$672$	0	0
$673$	−19.0693	−0.735067	−0.367534	−	0.930010i	$-0.619798\pi$
$673$	−19.0693	−0.735067	−0.367534	+	0.930010i	$0.619798\pi$
$674$	65.6055	2.52703
$675$	0	0
$676$	75.0076	2.88491
$677$	15.4032	0.591993	0.295997	−	0.955189i	$-0.404348\pi$
$677$	15.4032	0.591993	0.295997	+	0.955189i	$0.404348\pi$
$678$	0	0
$679$	27.9808	1.07381
$680$	34.7179	1.33137
$681$	0	0
$682$	−19.0531	−0.729580
$683$	45.5326	1.74226	0.871128	−	0.491056i	$-0.163389\pi$
$683$	45.5326	1.74226	0.871128	+	0.491056i	$0.163389\pi$
$684$	0	0
$685$	6.68848	0.255554
$686$	−44.6334	−1.70411
$687$	0	0
$688$	4.67636	0.178284
$689$	21.8464	0.832282
$690$	0	0
$691$	−1.29716	−0.0493463	−0.0246731	−	0.999696i	$-0.507855\pi$
$691$	−1.29716	−0.0493463	−0.0246731	+	0.999696i	$0.507855\pi$
$692$	−28.9542	−1.10068
$693$	0	0
$694$	−78.8852	−2.99444
$695$	−16.5894	−0.629271
$696$	0	0
$697$	−6.12008	−0.231815
$698$	−36.7580	−1.39131
$699$	0	0
$700$	19.1335	0.723177
$701$	37.1819	1.40434	0.702171	−	0.712008i	$-0.252214\pi$
$701$	37.1819	1.40434	0.702171	+	0.712008i	$0.252214\pi$
$702$	0	0
$703$	28.2661	1.06608
$704$	−11.9706	−0.451160
$705$	0	0
$706$	55.0339	2.07123
$707$	−17.6104	−0.662307
$708$	0	0
$709$	16.6489	0.625262	0.312631	−	0.949875i	$-0.398790\pi$
$709$	16.6489	0.625262	0.312631	+	0.949875i	$0.398790\pi$
$710$	25.9144	0.972549
$711$	0	0
$712$	36.4096	1.36451
$713$	23.3913	0.876012
$714$	0	0
$715$	−9.95720	−0.372378
$716$	65.3428	2.44197
$717$	0	0
$718$	49.4528	1.84556
$719$	22.0252	0.821400	0.410700	−	0.911771i	$-0.365284\pi$
$719$	22.0252	0.821400	0.410700	+	0.911771i	$0.365284\pi$
$720$	0	0
$721$	−16.3791	−0.609990
$722$	−75.8322	−2.82218
$723$	0	0
$724$	−9.75649	−0.362597
$725$	14.2191	0.528085
$726$	0	0
$727$	24.1071	0.894081	0.447041	−	0.894514i	$-0.352478\pi$
$727$	24.1071	0.894081	0.447041	+	0.894514i	$0.352478\pi$
$728$	−54.2536	−2.01077
$729$	0	0
$730$	39.6664	1.46812
$731$	−18.8341	−0.696603
$732$	0	0
$733$	20.8793	0.771194	0.385597	−	0.922667i	$-0.373996\pi$
$733$	20.8793	0.771194	0.385597	+	0.922667i	$0.373996\pi$
$734$	−34.5614	−1.27569
$735$	0	0
$736$	11.2421	0.414388
$737$	8.77367	0.323182
$738$	0	0
$739$	33.9168	1.24765	0.623824	−	0.781565i	$-0.285578\pi$
$739$	33.9168	1.24765	0.623824	+	0.781565i	$0.285578\pi$
$740$	23.7259	0.872183
$741$	0	0
$742$	−22.6065	−0.829910
$743$	10.2442	0.375824	0.187912	−	0.982186i	$-0.439828\pi$
$743$	10.2442	0.375824	0.187912	+	0.982186i	$0.439828\pi$
$744$	0	0
$745$	−12.6355	−0.462929
$746$	31.5068	1.15355
$747$	0	0
$748$	−20.0070	−0.731528
$749$	37.1011	1.35564
$750$	0	0
$751$	28.9933	1.05798	0.528990	−	0.848628i	$-0.322571\pi$
$751$	28.9933	1.05798	0.528990	+	0.848628i	$0.322571\pi$
$752$	−1.61380	−0.0588491
$753$	0	0
$754$	−92.9831	−3.38625
$755$	−14.6073	−0.531613
$756$	0	0
$757$	−40.5317	−1.47315	−0.736575	−	0.676356i	$-0.763558\pi$
$757$	−40.5317	−1.47315	−0.736575	+	0.676356i	$0.763558\pi$
$758$	−5.52354	−0.200624
$759$	0	0
$760$	−43.8642	−1.59112
$761$	7.10878	0.257693	0.128846	−	0.991665i	$-0.458873\pi$
$761$	7.10878	0.257693	0.128846	+	0.991665i	$0.458873\pi$
$762$	0	0
$763$	0.839583	0.0303949
$764$	−9.40649	−0.340315
$765$	0	0
$766$	−46.0682	−1.66451
$767$	−8.60856	−0.310837
$768$	0	0
$769$	2.64798	0.0954887	0.0477444	−	0.998860i	$-0.484797\pi$
$769$	2.64798	0.0954887	0.0477444	+	0.998860i	$0.484797\pi$
$770$	10.3036	0.371317
$771$	0	0
$772$	−11.6751	−0.420198
$773$	46.0512	1.65635	0.828173	−	0.560472i	$-0.189380\pi$
$773$	46.0512	1.65635	0.828173	+	0.560472i	$0.189380\pi$
$774$	0	0
$775$	−17.0495	−0.612436
$776$	39.1353	1.40487
$777$	0	0
$778$	44.9562	1.61176
$779$	7.73240	0.277042
$780$	0	0
$781$	−6.47549	−0.231711
$782$	38.4742	1.37583
$783$	0	0
$784$	−0.522014	−0.0186433
$785$	38.1937	1.36319
$786$	0	0
$787$	25.8038	0.919805	0.459902	−	0.887969i	$-0.347884\pi$
$787$	25.8038	0.919805	0.459902	+	0.887969i	$0.347884\pi$
$788$	26.1113	0.930177
$789$	0	0
$790$	−52.8833	−1.88150
$791$	−1.18676	−0.0421963
$792$	0	0
$793$	19.4720	0.691472
$794$	64.6813	2.29545
$795$	0	0
$796$	64.5906	2.28935
$797$	9.36363	0.331677	0.165838	−	0.986153i	$-0.446967\pi$
$797$	9.36363	0.331677	0.165838	+	0.986153i	$0.446967\pi$
$798$	0	0
$799$	6.49957	0.229938
$800$	−8.19412	−0.289706
$801$	0	0
$802$	−36.4741	−1.28794
$803$	−9.91186	−0.349782
$804$	0	0
$805$	−12.6497	−0.445842
$806$	111.492	3.92713
$807$	0	0
$808$	−24.6307	−0.866505
$809$	−8.31054	−0.292183	−0.146092	−	0.989271i	$-0.546669\pi$
$809$	−8.31054	−0.292183	−0.146092	+	0.989271i	$0.546669\pi$
$810$	0	0
$811$	33.3882	1.17242	0.586209	−	0.810160i	$-0.300620\pi$
$811$	33.3882	1.17242	0.586209	+	0.810160i	$0.300620\pi$
$812$	61.4269	2.15566
$813$	0	0
$814$	−9.28655	−0.325493
$815$	14.4249	0.505282
$816$	0	0
$817$	23.7958	0.832510
$818$	−2.07975	−0.0727167
$819$	0	0
$820$	6.49040	0.226654
$821$	−46.5341	−1.62405	−0.812025	−	0.583622i	$-0.801635\pi$
$821$	−46.5341	−1.62405	−0.812025	+	0.583622i	$0.801635\pi$
$822$	0	0
$823$	0.939018	0.0327321	0.0163661	−	0.999866i	$-0.494790\pi$
$823$	0.939018	0.0327321	0.0163661	+	0.999866i	$0.494790\pi$
$824$	−22.9086	−0.798059
$825$	0	0
$826$	8.90806	0.309951
$827$	−52.1918	−1.81489	−0.907443	−	0.420176i	$-0.861968\pi$
$827$	−52.1918	−1.81489	−0.907443	+	0.420176i	$0.861968\pi$
$828$	0	0
$829$	42.3895	1.47225	0.736124	−	0.676847i	$-0.236654\pi$
$829$	42.3895	1.47225	0.736124	+	0.676847i	$0.236654\pi$
$830$	41.5650	1.44274
$831$	0	0
$832$	70.0478	2.42847
$833$	2.10241	0.0728443
$834$	0	0
$835$	−34.3719	−1.18949
$836$	25.2777	0.874249
$837$	0	0
$838$	−2.67974	−0.0925702
$839$	−29.4430	−1.01648	−0.508242	−	0.861214i	$-0.669704\pi$
$839$	−29.4430	−1.01648	−0.508242	+	0.861214i	$0.669704\pi$
$840$	0	0
$841$	16.6497	0.574127
$842$	−25.0096	−0.861889
$843$	0	0
$844$	−4.84808	−0.166878
$845$	36.1449	1.24342
$846$	0	0
$847$	−2.57467	−0.0884667
$848$	−5.25207	−0.180357
$849$	0	0
$850$	−28.0431	−0.961871
$851$	11.4010	0.390822
$852$	0	0
$853$	33.1026	1.13341	0.566705	−	0.823921i	$-0.308218\pi$
$853$	33.1026	1.13341	0.566705	+	0.823921i	$0.308218\pi$
$854$	−20.1495	−0.689502
$855$	0	0
$856$	51.8912	1.77361
$857$	30.7484	1.05035	0.525173	−	0.850996i	$-0.324001\pi$
$857$	30.7484	1.05035	0.525173	+	0.850996i	$0.324001\pi$
$858$	0	0
$859$	31.3053	1.06812	0.534061	−	0.845446i	$-0.320665\pi$
$859$	31.3053	1.06812	0.534061	+	0.845446i	$0.320665\pi$
$860$	19.9737	0.681096
$861$	0	0
$862$	3.24884	0.110656
$863$	38.4963	1.31043	0.655214	−	0.755443i	$-0.272578\pi$
$863$	38.4963	1.31043	0.655214	+	0.755443i	$0.272578\pi$
$864$	0	0
$865$	−13.9526	−0.474402
$866$	25.3088	0.860029
$867$	0	0
$868$	−73.6541	−2.49998
$869$	13.2145	0.448271
$870$	0	0
$871$	−51.3403	−1.73960
$872$	1.17428	0.0397661
$873$	0	0
$874$	−48.6101	−1.64426
$875$	31.1255	1.05223
$876$	0	0
$877$	19.3399	0.653061	0.326531	−	0.945187i	$-0.394120\pi$
$877$	19.3399	0.653061	0.326531	+	0.945187i	$0.394120\pi$
$878$	−56.0571	−1.89184
$879$	0	0
$880$	2.39380	0.0806949
$881$	−18.0552	−0.608294	−0.304147	−	0.952625i	$-0.598371\pi$
$881$	−18.0552	−0.608294	−0.304147	+	0.952625i	$0.598371\pi$
$882$	0	0
$883$	28.4019	0.955800	0.477900	−	0.878414i	$-0.341398\pi$
$883$	28.4019	0.955800	0.477900	+	0.878414i	$0.341398\pi$
$884$	117.074	3.93761
$885$	0	0
$886$	−43.6563	−1.46666
$887$	16.1879	0.543537	0.271768	−	0.962363i	$-0.412392\pi$
$887$	16.1879	0.543537	0.271768	+	0.962363i	$0.412392\pi$
$888$	0	0
$889$	−48.8673	−1.63895
$890$	40.4627	1.35631
$891$	0	0
$892$	−50.4954	−1.69071
$893$	−8.21186	−0.274799
$894$	0	0
$895$	31.4876	1.05251
$896$	−52.4355	−1.75175
$897$	0	0
$898$	−25.6487	−0.855907
$899$	−54.7364	−1.82556
$900$	0	0
$901$	21.1527	0.704700
$902$	−2.54040	−0.0845860
$903$	0	0
$904$	−1.65985	−0.0552059
$905$	−4.70149	−0.156283
$906$	0	0
$907$	−41.9096	−1.39159	−0.695794	−	0.718242i	$-0.744947\pi$
$907$	−41.9096	−1.39159	−0.695794	+	0.718242i	$0.744947\pi$
$908$	103.383	3.43088
$909$	0	0
$910$	−60.2930	−1.99869
$911$	−28.2987	−0.937576	−0.468788	−	0.883311i	$-0.655309\pi$
$911$	−28.2987	−0.937576	−0.468788	+	0.883311i	$0.655309\pi$
$912$	0	0
$913$	−10.3863	−0.343735
$914$	−12.3695	−0.409146
$915$	0	0
$916$	52.0079	1.71839
$917$	−3.98482	−0.131590
$918$	0	0
$919$	29.5149	0.973606	0.486803	−	0.873512i	$-0.338163\pi$
$919$	29.5149	0.973606	0.486803	+	0.873512i	$0.338163\pi$
$920$	−17.6924	−0.583302
$921$	0	0
$922$	−46.1051	−1.51839
$923$	37.8922	1.24724
$924$	0	0
$925$	−8.30998	−0.273231
$926$	0.719502	0.0236443
$927$	0	0
$928$	−26.3067	−0.863561
$929$	−17.4409	−0.572218	−0.286109	−	0.958197i	$-0.592362\pi$
$929$	−17.4409	−0.572218	−0.286109	+	0.958197i	$0.592362\pi$
$930$	0	0
$931$	−2.65629	−0.0870563
$932$	66.3092	2.17203
$933$	0	0
$934$	−6.88769	−0.225372
$935$	−9.64104	−0.315296
$936$	0	0
$937$	16.4011	0.535800	0.267900	−	0.963447i	$-0.413670\pi$
$937$	16.4011	0.535800	0.267900	+	0.963447i	$0.413670\pi$
$938$	53.1265	1.73464
$939$	0	0
$940$	−6.89285	−0.224820
$941$	−42.1717	−1.37476	−0.687379	−	0.726299i	$-0.741239\pi$
$941$	−42.1717	−1.37476	−0.687379	+	0.726299i	$0.741239\pi$
$942$	0	0
$943$	3.11883	0.101563
$944$	2.06957	0.0673588
$945$	0	0
$946$	−7.81787	−0.254181
$947$	32.8926	1.06887	0.534434	−	0.845210i	$-0.320525\pi$
$947$	32.8926	1.06887	0.534434	+	0.845210i	$0.320525\pi$
$948$	0	0
$949$	58.0005	1.88278
$950$	35.4310	1.14953
$951$	0	0
$952$	−52.5309	−1.70254
$953$	28.5559	0.925018	0.462509	−	0.886615i	$-0.346949\pi$
$953$	28.5559	0.925018	0.462509	+	0.886615i	$0.346949\pi$
$954$	0	0
$955$	−4.53283	−0.146679
$956$	28.5112	0.922119
$957$	0	0
$958$	−54.9024	−1.77381
$959$	−10.1202	−0.326798
$960$	0	0
$961$	34.6318	1.11716
$962$	54.3415	1.75204
$963$	0	0
$964$	93.1079	2.99880
$965$	−5.62606	−0.181109
$966$	0	0
$967$	−2.89079	−0.0929614	−0.0464807	−	0.998919i	$-0.514801\pi$
$967$	−2.89079	−0.0929614	−0.0464807	+	0.998919i	$0.514801\pi$
$968$	−3.60105	−0.115742
$969$	0	0
$970$	43.4918	1.39644
$971$	18.3633	0.589307	0.294654	−	0.955604i	$-0.404796\pi$
$971$	18.3633	0.589307	0.294654	+	0.955604i	$0.404796\pi$
$972$	0	0
$973$	25.1011	0.804703
$974$	−55.7858	−1.78749
$975$	0	0
$976$	−4.68125	−0.149843
$977$	43.6452	1.39633	0.698167	−	0.715935i	$-0.253999\pi$
$977$	43.6452	1.39633	0.698167	+	0.715935i	$0.253999\pi$
$978$	0	0
$979$	−10.1108	−0.323144
$980$	−2.22963	−0.0712228
$981$	0	0
$982$	−51.7172	−1.65036
$983$	43.5907	1.39033	0.695164	−	0.718851i	$-0.255332\pi$
$983$	43.5907	1.39033	0.695164	+	0.718851i	$0.255332\pi$
$984$	0	0
$985$	12.5826	0.400915
$986$	−90.0307	−2.86716
$987$	0	0
$988$	−147.916	−4.70584
$989$	9.59794	0.305197
$990$	0	0
$991$	−10.4399	−0.331635	−0.165817	−	0.986156i	$-0.553026\pi$
$991$	−10.4399	−0.331635	−0.165817	+	0.986156i	$0.553026\pi$
$992$	31.5432	1.00150
$993$	0	0
$994$	−39.2105	−1.24368
$995$	31.1252	0.986734
$996$	0	0
$997$	−45.8695	−1.45270	−0.726350	−	0.687325i	$-0.758785\pi$
$997$	−45.8695	−1.45270	−0.726350	+	0.687325i	$0.758785\pi$
$998$	−63.5556	−2.01182
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.7		51
3.2	odd	2		8019.2.a.k.1.45		51
27.5	odd	18		891.2.j.c.397.15		102
27.11	odd	18		891.2.j.c.496.15		102
27.16	even	9		297.2.j.c.67.3	✓	102
27.22	even	9		297.2.j.c.133.3	yes	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.3	✓	102	27.16	even	9
297.2.j.c.133.3	yes	102	27.22	even	9
891.2.j.c.397.15		102	27.5	odd	18
891.2.j.c.496.15		102	27.11	odd	18
8019.2.a.k.1.45		51	3.2	odd	2
8019.2.a.l.1.7		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35184 q^{2} +3.53116 q^{4} +1.70161 q^{5} -2.57467 q^{7} -3.60105 q^{8} +O(q^{10})\)	\(q-2.35184 q^{2} +3.53116 q^{4} +1.70161 q^{5} -2.57467 q^{7} -3.60105 q^{8} -4.00192 q^{10} +1.00000 q^{11} -5.85163 q^{13} +6.05522 q^{14} +1.40678 q^{16} -5.66583 q^{17} +7.15847 q^{19} +6.00866 q^{20} -2.35184 q^{22} +2.88734 q^{23} -2.10453 q^{25} +13.7621 q^{26} -9.09158 q^{28} -6.75645 q^{29} +8.10135 q^{31} +3.89357 q^{32} +13.3251 q^{34} -4.38108 q^{35} +3.94863 q^{37} -16.8356 q^{38} -6.12759 q^{40} +1.08017 q^{41} +3.32415 q^{43} +3.53116 q^{44} -6.79056 q^{46} -1.14715 q^{47} -0.371069 q^{49} +4.94951 q^{50} -20.6631 q^{52} -3.73339 q^{53} +1.70161 q^{55} +9.27153 q^{56} +15.8901 q^{58} +1.47114 q^{59} -3.32762 q^{61} -19.0531 q^{62} -11.9706 q^{64} -9.95720 q^{65} +8.77367 q^{67} -20.0070 q^{68} +10.3036 q^{70} -6.47549 q^{71} -9.91186 q^{73} -9.28655 q^{74} +25.2777 q^{76} -2.57467 q^{77} +13.2145 q^{79} +2.39380 q^{80} -2.54040 q^{82} -10.3863 q^{83} -9.64104 q^{85} -7.81787 q^{86} -3.60105 q^{88} -10.1108 q^{89} +15.0660 q^{91} +10.1957 q^{92} +2.69792 q^{94} +12.1809 q^{95} -10.8677 q^{97} +0.872695 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more