LMFDB - Embedded newform 6044.2.a.a.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, 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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69210 q^{3} -3.14690 q^{5} -4.59556 q^{7} -0.136814 q^{9} +O(q^{10})$	$q-1.69210 q^{3} -3.14690 q^{5} -4.59556 q^{7} -0.136814 q^{9} -5.69483 q^{11} +3.92959 q^{13} +5.32485 q^{15} -2.33507 q^{17} +1.08583 q^{19} +7.77612 q^{21} +4.85256 q^{23} +4.90297 q^{25} +5.30779 q^{27} +4.08338 q^{29} -9.61535 q^{31} +9.63619 q^{33} +14.4618 q^{35} +3.31752 q^{37} -6.64925 q^{39} +6.45302 q^{41} -9.23341 q^{43} +0.430541 q^{45} +1.80366 q^{47} +14.1192 q^{49} +3.95117 q^{51} -8.50944 q^{53} +17.9210 q^{55} -1.83733 q^{57} +0.592149 q^{59} +8.40726 q^{61} +0.628739 q^{63} -12.3660 q^{65} +7.92906 q^{67} -8.21099 q^{69} +12.6221 q^{71} +5.92521 q^{73} -8.29630 q^{75} +26.1709 q^{77} -0.779830 q^{79} -8.57084 q^{81} +1.12808 q^{83} +7.34824 q^{85} -6.90947 q^{87} -5.30578 q^{89} -18.0587 q^{91} +16.2701 q^{93} -3.41700 q^{95} -6.82457 q^{97} +0.779134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.69210	−0.976932	−0.488466	−	0.872583i	$-0.662443\pi$
$3$	−1.69210	−0.976932	−0.488466	+	0.872583i	$0.662443\pi$
$4$	0	0
$5$	−3.14690	−1.40734	−0.703668	−	0.710529i	$-0.748456\pi$
$5$	−3.14690	−1.40734	−0.703668	+	0.710529i	$0.748456\pi$
$6$	0	0
$7$	−4.59556	−1.73696	−0.868479	−	0.495726i	$-0.834902\pi$
$7$	−4.59556	−1.73696	−0.868479	+	0.495726i	$0.834902\pi$
$8$	0	0
$9$	−0.136814	−0.0456048
$10$	0	0
$11$	−5.69483	−1.71706	−0.858528	−	0.512767i	$-0.828620\pi$
$11$	−5.69483	−1.71706	−0.858528	+	0.512767i	$0.828620\pi$
$12$	0	0
$13$	3.92959	1.08987	0.544937	−	0.838477i	$-0.316554\pi$
$13$	3.92959	1.08987	0.544937	+	0.838477i	$0.316554\pi$
$14$	0	0
$15$	5.32485	1.37487
$16$	0	0
$17$	−2.33507	−0.566338	−0.283169	−	0.959070i	$-0.591386\pi$
$17$	−2.33507	−0.566338	−0.283169	+	0.959070i	$0.591386\pi$
$18$	0	0
$19$	1.08583	0.249107	0.124553	−	0.992213i	$-0.460250\pi$
$19$	1.08583	0.249107	0.124553	+	0.992213i	$0.460250\pi$
$20$	0	0
$21$	7.77612	1.69689
$22$	0	0
$23$	4.85256	1.01183	0.505914	−	0.862584i	$-0.331155\pi$
$23$	4.85256	1.01183	0.505914	+	0.862584i	$0.331155\pi$
$24$	0	0
$25$	4.90297	0.980595
$26$	0	0
$27$	5.30779	1.02148
$28$	0	0
$29$	4.08338	0.758265	0.379132	−	0.925342i	$-0.376223\pi$
$29$	4.08338	0.758265	0.379132	+	0.925342i	$0.376223\pi$
$30$	0	0
$31$	−9.61535	−1.72697	−0.863484	−	0.504377i	$-0.831722\pi$
$31$	−9.61535	−1.72697	−0.863484	+	0.504377i	$0.831722\pi$
$32$	0	0
$33$	9.63619	1.67745
$34$	0	0
$35$	14.4618	2.44448
$36$	0	0
$37$	3.31752	0.545397	0.272699	−	0.962100i	$-0.412084\pi$
$37$	3.31752	0.545397	0.272699	+	0.962100i	$0.412084\pi$
$38$	0	0
$39$	−6.64925	−1.06473
$40$	0	0
$41$	6.45302	1.00779	0.503896	−	0.863764i	$-0.331899\pi$
$41$	6.45302	1.00779	0.503896	+	0.863764i	$0.331899\pi$
$42$	0	0
$43$	−9.23341	−1.40808	−0.704041	−	0.710159i	$-0.748623\pi$
$43$	−9.23341	−1.40808	−0.704041	+	0.710159i	$0.748623\pi$
$44$	0	0
$45$	0.430541	0.0641813
$46$	0	0
$47$	1.80366	0.263091	0.131545	−	0.991310i	$-0.458006\pi$
$47$	1.80366	0.263091	0.131545	+	0.991310i	$0.458006\pi$
$48$	0	0
$49$	14.1192	2.01702
$50$	0	0
$51$	3.95117	0.553274
$52$	0	0
$53$	−8.50944	−1.16886	−0.584431	−	0.811444i	$-0.698682\pi$
$53$	−8.50944	−1.16886	−0.584431	+	0.811444i	$0.698682\pi$
$54$	0	0
$55$	17.9210	2.41647
$56$	0	0
$57$	−1.83733	−0.243360
$58$	0	0
$59$	0.592149	0.0770912	0.0385456	−	0.999257i	$-0.487728\pi$
$59$	0.592149	0.0770912	0.0385456	+	0.999257i	$0.487728\pi$
$60$	0	0
$61$	8.40726	1.07644	0.538220	−	0.842805i	$-0.319097\pi$
$61$	8.40726	1.07644	0.538220	+	0.842805i	$0.319097\pi$
$62$	0	0
$63$	0.628739	0.0792136
$64$	0	0
$65$	−12.3660	−1.53382
$66$	0	0
$67$	7.92906	0.968688	0.484344	−	0.874878i	$-0.339058\pi$
$67$	7.92906	0.968688	0.484344	+	0.874878i	$0.339058\pi$
$68$	0	0
$69$	−8.21099	−0.988487
$70$	0	0
$71$	12.6221	1.49797	0.748987	−	0.662585i	$-0.230541\pi$
$71$	12.6221	1.49797	0.748987	+	0.662585i	$0.230541\pi$
$72$	0	0
$73$	5.92521	0.693493	0.346746	−	0.937959i	$-0.387286\pi$
$73$	5.92521	0.693493	0.346746	+	0.937959i	$0.387286\pi$
$74$	0	0
$75$	−8.29630	−0.957974
$76$	0	0
$77$	26.1709	2.98245
$78$	0	0
$79$	−0.779830	−0.0877377	−0.0438688	−	0.999037i	$-0.513968\pi$
$79$	−0.779830	−0.0877377	−0.0438688	+	0.999037i	$0.513968\pi$
$80$	0	0
$81$	−8.57084	−0.952315
$82$	0	0
$83$	1.12808	0.123823	0.0619114	−	0.998082i	$-0.480280\pi$
$83$	1.12808	0.123823	0.0619114	+	0.998082i	$0.480280\pi$
$84$	0	0
$85$	7.34824	0.797028
$86$	0	0
$87$	−6.90947	−0.740773
$88$	0	0
$89$	−5.30578	−0.562411	−0.281206	−	0.959648i	$-0.590734\pi$
$89$	−5.30578	−0.562411	−0.281206	+	0.959648i	$0.590734\pi$
$90$	0	0
$91$	−18.0587	−1.89306
$92$	0	0
$93$	16.2701	1.68713
$94$	0	0
$95$	−3.41700	−0.350577
$96$	0	0
$97$	−6.82457	−0.692930	−0.346465	−	0.938063i	$-0.612618\pi$
$97$	−6.82457	−0.692930	−0.346465	+	0.938063i	$0.612618\pi$
$98$	0	0
$99$	0.779134	0.0783059
$100$	0	0
$101$	9.83229	0.978349	0.489175	−	0.872186i	$-0.337298\pi$
$101$	9.83229	0.978349	0.489175	+	0.872186i	$0.337298\pi$
$102$	0	0
$103$	−12.3958	−1.22140	−0.610699	−	0.791863i	$-0.709112\pi$
$103$	−12.3958	−1.22140	−0.610699	+	0.791863i	$0.709112\pi$
$104$	0	0
$105$	−24.4707	−2.38809
$106$	0	0
$107$	4.92440	0.476060	0.238030	−	0.971258i	$-0.423498\pi$
$107$	4.92440	0.476060	0.238030	+	0.971258i	$0.423498\pi$
$108$	0	0
$109$	−0.00736550	−0.000705487 0	−0.000352744	−	1.00000i	$-0.500112\pi$
$109$	−0.00736550	−0.000705487 0	−0.000352744		1.00000i	$0.500112\pi$
$110$	0	0
$111$	−5.61356	−0.532816
$112$	0	0
$113$	2.04716	0.192580	0.0962901	−	0.995353i	$-0.469302\pi$
$113$	2.04716	0.192580	0.0962901	+	0.995353i	$0.469302\pi$
$114$	0	0
$115$	−15.2705	−1.42398
$116$	0	0
$117$	−0.537625	−0.0497034
$118$	0	0
$119$	10.7310	0.983706
$120$	0	0
$121$	21.4311	1.94828
$122$	0	0
$123$	−10.9191	−0.984545
$124$	0	0
$125$	0.305333	0.0273098
$126$	0	0
$127$	−16.4190	−1.45695	−0.728477	−	0.685071i	$-0.759771\pi$
$127$	−16.4190	−1.45695	−0.728477	+	0.685071i	$0.759771\pi$
$128$	0	0
$129$	15.6238	1.37560
$130$	0	0
$131$	−1.34708	−0.117695	−0.0588476	−	0.998267i	$-0.518743\pi$
$131$	−1.34708	−0.117695	−0.0588476	+	0.998267i	$0.518743\pi$
$132$	0	0
$133$	−4.99000	−0.432688
$134$	0	0
$135$	−16.7031	−1.43757
$136$	0	0
$137$	−13.3729	−1.14253	−0.571263	−	0.820767i	$-0.693546\pi$
$137$	−13.3729	−1.14253	−0.571263	+	0.820767i	$0.693546\pi$
$138$	0	0
$139$	−19.1789	−1.62673	−0.813367	−	0.581750i	$-0.802368\pi$
$139$	−19.1789	−1.62673	−0.813367	+	0.581750i	$0.802368\pi$
$140$	0	0
$141$	−3.05196	−0.257021
$142$	0	0
$143$	−22.3784	−1.87137
$144$	0	0
$145$	−12.8500	−1.06713
$146$	0	0
$147$	−23.8910	−1.97049
$148$	0	0
$149$	0.953182	0.0780877	0.0390439	−	0.999237i	$-0.487569\pi$
$149$	0.953182	0.0780877	0.0390439	+	0.999237i	$0.487569\pi$
$150$	0	0
$151$	9.91434	0.806817	0.403409	−	0.915020i	$-0.367825\pi$
$151$	9.91434	0.806817	0.403409	+	0.915020i	$0.367825\pi$
$152$	0	0
$153$	0.319471	0.0258277
$154$	0	0
$155$	30.2585	2.43042
$156$	0	0
$157$	−19.4114	−1.54920	−0.774598	−	0.632454i	$-0.782048\pi$
$157$	−19.4114	−1.54920	−0.774598	+	0.632454i	$0.782048\pi$
$158$	0	0
$159$	14.3988	1.14190
$160$	0	0
$161$	−22.3002	−1.75750
$162$	0	0
$163$	20.0599	1.57122	0.785608	−	0.618725i	$-0.212351\pi$
$163$	20.0599	1.57122	0.785608	+	0.618725i	$0.212351\pi$
$164$	0	0
$165$	−30.3241	−2.36073
$166$	0	0
$167$	5.03069	0.389287	0.194643	−	0.980874i	$-0.437645\pi$
$167$	5.03069	0.389287	0.194643	+	0.980874i	$0.437645\pi$
$168$	0	0
$169$	2.44171	0.187824
$170$	0	0
$171$	−0.148557	−0.0113605
$172$	0	0
$173$	20.7174	1.57512	0.787558	−	0.616241i	$-0.211345\pi$
$173$	20.7174	1.57512	0.787558	+	0.616241i	$0.211345\pi$
$174$	0	0
$175$	−22.5319	−1.70325
$176$	0	0
$177$	−1.00197	−0.0753128
$178$	0	0
$179$	−22.2961	−1.66649	−0.833245	−	0.552904i	$-0.813520\pi$
$179$	−22.2961	−1.66649	−0.833245	+	0.552904i	$0.813520\pi$
$180$	0	0
$181$	0.185538	0.0137909	0.00689547	−	0.999976i	$-0.497805\pi$
$181$	0.185538	0.0137909	0.00689547	+	0.999976i	$0.497805\pi$
$182$	0	0
$183$	−14.2259	−1.05161
$184$	0	0
$185$	−10.4399	−0.767557
$186$	0	0
$187$	13.2978	0.972434
$188$	0	0
$189$	−24.3923	−1.77428
$190$	0	0
$191$	7.76061	0.561538	0.280769	−	0.959775i	$-0.409411\pi$
$191$	7.76061	0.561538	0.280769	+	0.959775i	$0.409411\pi$
$192$	0	0
$193$	25.8010	1.85719	0.928597	−	0.371090i	$-0.121016\pi$
$193$	25.8010	1.85719	0.928597	+	0.371090i	$0.121016\pi$
$194$	0	0
$195$	20.9245	1.49843
$196$	0	0
$197$	−14.1393	−1.00738	−0.503692	−	0.863884i	$-0.668025\pi$
$197$	−14.1393	−1.00738	−0.503692	+	0.863884i	$0.668025\pi$
$198$	0	0
$199$	2.50667	0.177693	0.0888467	−	0.996045i	$-0.471682\pi$
$199$	2.50667	0.177693	0.0888467	+	0.996045i	$0.471682\pi$
$200$	0	0
$201$	−13.4167	−0.946342
$202$	0	0
$203$	−18.7654	−1.31707
$204$	0	0
$205$	−20.3070	−1.41830
$206$	0	0
$207$	−0.663900	−0.0461442
$208$	0	0
$209$	−6.18362	−0.427730
$210$	0	0
$211$	−0.118771	−0.00817653	−0.00408827	−	0.999992i	$-0.501301\pi$
$211$	−0.118771	−0.00817653	−0.00408827	+	0.999992i	$0.501301\pi$
$212$	0	0
$213$	−21.3579	−1.46342
$214$	0	0
$215$	29.0566	1.98164
$216$	0	0
$217$	44.1879	2.99967
$218$	0	0
$219$	−10.0260	−0.677495
$220$	0	0
$221$	−9.17589	−0.617237
$222$	0	0
$223$	−5.30992	−0.355578	−0.177789	−	0.984069i	$-0.556894\pi$
$223$	−5.30992	−0.355578	−0.177789	+	0.984069i	$0.556894\pi$
$224$	0	0
$225$	−0.670797	−0.0447198
$226$	0	0
$227$	20.4601	1.35798	0.678992	−	0.734146i	$-0.262417\pi$
$227$	20.4601	1.35798	0.678992	+	0.734146i	$0.262417\pi$
$228$	0	0
$229$	−24.0178	−1.58714	−0.793571	−	0.608477i	$-0.791781\pi$
$229$	−24.0178	−1.58714	−0.793571	+	0.608477i	$0.791781\pi$
$230$	0	0
$231$	−44.2837	−2.91365
$232$	0	0
$233$	24.8085	1.62526	0.812631	−	0.582779i	$-0.198035\pi$
$233$	24.8085	1.62526	0.812631	+	0.582779i	$0.198035\pi$
$234$	0	0
$235$	−5.67593	−0.370257
$236$	0	0
$237$	1.31955	0.0857137
$238$	0	0
$239$	22.7778	1.47337	0.736685	−	0.676236i	$-0.236390\pi$
$239$	22.7778	1.47337	0.736685	+	0.676236i	$0.236390\pi$
$240$	0	0
$241$	13.8617	0.892913	0.446456	−	0.894805i	$-0.352686\pi$
$241$	13.8617	0.892913	0.446456	+	0.894805i	$0.352686\pi$
$242$	0	0
$243$	−1.42069	−0.0911373
$244$	0	0
$245$	−44.4316	−2.83863
$246$	0	0
$247$	4.26687	0.271495
$248$	0	0
$249$	−1.90882	−0.120966
$250$	0	0
$251$	−29.4608	−1.85955	−0.929776	−	0.368126i	$-0.880000\pi$
$251$	−29.4608	−1.85955	−0.929776	+	0.368126i	$0.880000\pi$
$252$	0	0
$253$	−27.6345	−1.73737
$254$	0	0
$255$	−12.4339	−0.778642
$256$	0	0
$257$	8.96323	0.559111	0.279555	−	0.960130i	$-0.409813\pi$
$257$	8.96323	0.559111	0.279555	+	0.960130i	$0.409813\pi$
$258$	0	0
$259$	−15.2459	−0.947332
$260$	0	0
$261$	−0.558665	−0.0345805
$262$	0	0
$263$	9.43491	0.581781	0.290891	−	0.956756i	$-0.406048\pi$
$263$	9.43491	0.581781	0.290891	+	0.956756i	$0.406048\pi$
$264$	0	0
$265$	26.7783	1.64498
$266$	0	0
$267$	8.97788	0.549437
$268$	0	0
$269$	16.4934	1.00562	0.502809	−	0.864398i	$-0.332300\pi$
$269$	16.4934	1.00562	0.502809	+	0.864398i	$0.332300\pi$
$270$	0	0
$271$	−20.5270	−1.24693	−0.623463	−	0.781853i	$-0.714275\pi$
$271$	−20.5270	−1.24693	−0.623463	+	0.781853i	$0.714275\pi$
$272$	0	0
$273$	30.5570	1.84939
$274$	0	0
$275$	−27.9216	−1.68374
$276$	0	0
$277$	−32.1996	−1.93469	−0.967343	−	0.253471i	$-0.918428\pi$
$277$	−32.1996	−1.93469	−0.967343	+	0.253471i	$0.918428\pi$
$278$	0	0
$279$	1.31552	0.0787580
$280$	0	0
$281$	5.99197	0.357451	0.178725	−	0.983899i	$-0.442803\pi$
$281$	5.99197	0.357451	0.178725	+	0.983899i	$0.442803\pi$
$282$	0	0
$283$	−4.41773	−0.262607	−0.131303	−	0.991342i	$-0.541916\pi$
$283$	−4.41773	−0.262607	−0.131303	+	0.991342i	$0.541916\pi$
$284$	0	0
$285$	5.78189	0.342490
$286$	0	0
$287$	−29.6553	−1.75049
$288$	0	0
$289$	−11.5474	−0.679261
$290$	0	0
$291$	11.5478	0.676945
$292$	0	0
$293$	18.5419	1.08323	0.541613	−	0.840628i	$-0.317814\pi$
$293$	18.5419	1.08323	0.541613	+	0.840628i	$0.317814\pi$
$294$	0	0
$295$	−1.86343	−0.108493
$296$	0	0
$297$	−30.2269	−1.75395
$298$	0	0
$299$	19.0686	1.10276
$300$	0	0
$301$	42.4327	2.44578
$302$	0	0
$303$	−16.6372	−0.955780
$304$	0	0
$305$	−26.4568	−1.51491
$306$	0	0
$307$	−1.87316	−0.106907	−0.0534533	−	0.998570i	$-0.517023\pi$
$307$	−1.87316	−0.106907	−0.0534533	+	0.998570i	$0.517023\pi$
$308$	0	0
$309$	20.9749	1.19322
$310$	0	0
$311$	17.9051	1.01530	0.507652	−	0.861562i	$-0.330513\pi$
$311$	17.9051	1.01530	0.507652	+	0.861562i	$0.330513\pi$
$312$	0	0
$313$	0.974340	0.0550730	0.0275365	−	0.999621i	$-0.491234\pi$
$313$	0.974340	0.0550730	0.0275365	+	0.999621i	$0.491234\pi$
$314$	0	0
$315$	−1.97858	−0.111480
$316$	0	0
$317$	0.932531	0.0523761	0.0261881	−	0.999657i	$-0.491663\pi$
$317$	0.932531	0.0523761	0.0261881	+	0.999657i	$0.491663\pi$
$318$	0	0
$319$	−23.2541	−1.30198
$320$	0	0
$321$	−8.33255	−0.465078
$322$	0	0
$323$	−2.53549	−0.141079
$324$	0	0
$325$	19.2667	1.06872
$326$	0	0
$327$	0.0124631	0.000689213 0
$328$	0	0
$329$	−8.28882	−0.456977
$330$	0	0
$331$	0.783393	0.0430592	0.0215296	−	0.999768i	$-0.493146\pi$
$331$	0.783393	0.0430592	0.0215296	+	0.999768i	$0.493146\pi$
$332$	0	0
$333$	−0.453884	−0.0248727
$334$	0	0
$335$	−24.9519	−1.36327
$336$	0	0
$337$	21.7470	1.18463	0.592317	−	0.805705i	$-0.298213\pi$
$337$	21.7470	1.18463	0.592317	+	0.805705i	$0.298213\pi$
$338$	0	0
$339$	−3.46398	−0.188138
$340$	0	0
$341$	54.7578	2.96530
$342$	0	0
$343$	−32.7166	−1.76653
$344$	0	0
$345$	25.8392	1.39113
$346$	0	0
$347$	1.59377	0.0855580	0.0427790	−	0.999085i	$-0.486379\pi$
$347$	1.59377	0.0855580	0.0427790	+	0.999085i	$0.486379\pi$
$348$	0	0
$349$	8.66352	0.463748	0.231874	−	0.972746i	$-0.425514\pi$
$349$	8.66352	0.463748	0.231874	+	0.972746i	$0.425514\pi$
$350$	0	0
$351$	20.8575	1.11329
$352$	0	0
$353$	−28.2226	−1.50214	−0.751069	−	0.660224i	$-0.770461\pi$
$353$	−28.2226	−1.50214	−0.751069	+	0.660224i	$0.770461\pi$
$354$	0	0
$355$	−39.7206	−2.10815
$356$	0	0
$357$	−18.1578	−0.961014
$358$	0	0
$359$	6.57586	0.347061	0.173530	−	0.984829i	$-0.444483\pi$
$359$	6.57586	0.347061	0.173530	+	0.984829i	$0.444483\pi$
$360$	0	0
$361$	−17.8210	−0.937946
$362$	0	0
$363$	−36.2634	−1.90334
$364$	0	0
$365$	−18.6460	−0.975977
$366$	0	0
$367$	11.1926	0.584250	0.292125	−	0.956380i	$-0.405638\pi$
$367$	11.1926	0.584250	0.292125	+	0.956380i	$0.405638\pi$
$368$	0	0
$369$	−0.882866	−0.0459602
$370$	0	0
$371$	39.1056	2.03026
$372$	0	0
$373$	−26.0435	−1.34848	−0.674240	−	0.738513i	$-0.735529\pi$
$373$	−26.0435	−1.34848	−0.674240	+	0.738513i	$0.735529\pi$
$374$	0	0
$375$	−0.516652	−0.0266798
$376$	0	0
$377$	16.0460	0.826412
$378$	0	0
$379$	−17.9379	−0.921407	−0.460704	−	0.887554i	$-0.652403\pi$
$379$	−17.9379	−0.921407	−0.460704	+	0.887554i	$0.652403\pi$
$380$	0	0
$381$	27.7826	1.42334
$382$	0	0
$383$	18.9563	0.968621	0.484310	−	0.874896i	$-0.339071\pi$
$383$	18.9563	0.968621	0.484310	+	0.874896i	$0.339071\pi$
$384$	0	0
$385$	−82.3573	−4.19731
$386$	0	0
$387$	1.26326	0.0642153
$388$	0	0
$389$	−6.46208	−0.327640	−0.163820	−	0.986490i	$-0.552382\pi$
$389$	−6.46208	−0.327640	−0.163820	+	0.986490i	$0.552382\pi$
$390$	0	0
$391$	−11.3311	−0.573037
$392$	0	0
$393$	2.27939	0.114980
$394$	0	0
$395$	2.45405	0.123476
$396$	0	0
$397$	7.23140	0.362933	0.181467	−	0.983397i	$-0.441916\pi$
$397$	7.23140	0.362933	0.181467	+	0.983397i	$0.441916\pi$
$398$	0	0
$399$	8.44356	0.422707
$400$	0	0
$401$	−7.59144	−0.379098	−0.189549	−	0.981871i	$-0.560703\pi$
$401$	−7.59144	−0.379098	−0.189549	+	0.981871i	$0.560703\pi$
$402$	0	0
$403$	−37.7844	−1.88218
$404$	0	0
$405$	26.9716	1.34023
$406$	0	0
$407$	−18.8927	−0.936477
$408$	0	0
$409$	4.09486	0.202478	0.101239	−	0.994862i	$-0.467719\pi$
$409$	4.09486	0.202478	0.101239	+	0.994862i	$0.467719\pi$
$410$	0	0
$411$	22.6283	1.11617
$412$	0	0
$413$	−2.72126	−0.133904
$414$	0	0
$415$	−3.54995	−0.174260
$416$	0	0
$417$	32.4526	1.58921
$418$	0	0
$419$	−19.2867	−0.942219	−0.471109	−	0.882075i	$-0.656146\pi$
$419$	−19.2867	−0.942219	−0.471109	+	0.882075i	$0.656146\pi$
$420$	0	0
$421$	30.7625	1.49927	0.749637	−	0.661849i	$-0.230228\pi$
$421$	30.7625	1.49927	0.749637	+	0.661849i	$0.230228\pi$
$422$	0	0
$423$	−0.246766	−0.0119982
$424$	0	0
$425$	−11.4488	−0.555348
$426$	0	0
$427$	−38.6361	−1.86973
$428$	0	0
$429$	37.8663	1.82820
$430$	0	0
$431$	17.7863	0.856737	0.428369	−	0.903604i	$-0.359088\pi$
$431$	17.7863	0.856737	0.428369	+	0.903604i	$0.359088\pi$
$432$	0	0
$433$	12.3098	0.591573	0.295786	−	0.955254i	$-0.404418\pi$
$433$	12.3098	0.591573	0.295786	+	0.955254i	$0.404418\pi$
$434$	0	0
$435$	21.7434	1.04252
$436$	0	0
$437$	5.26906	0.252053
$438$	0	0
$439$	11.0190	0.525909	0.262954	−	0.964808i	$-0.415303\pi$
$439$	11.0190	0.525909	0.262954	+	0.964808i	$0.415303\pi$
$440$	0	0
$441$	−1.93170	−0.0919860
$442$	0	0
$443$	16.1366	0.766674	0.383337	−	0.923609i	$-0.374775\pi$
$443$	16.1366	0.766674	0.383337	+	0.923609i	$0.374775\pi$
$444$	0	0
$445$	16.6967	0.791502
$446$	0	0
$447$	−1.61287	−0.0762863
$448$	0	0
$449$	4.39561	0.207442	0.103721	−	0.994606i	$-0.466925\pi$
$449$	4.39561	0.207442	0.103721	+	0.994606i	$0.466925\pi$
$450$	0	0
$451$	−36.7489	−1.73044
$452$	0	0
$453$	−16.7760	−0.788205
$454$	0	0
$455$	56.8289	2.66418
$456$	0	0
$457$	−8.60279	−0.402421	−0.201211	−	0.979548i	$-0.564488\pi$
$457$	−8.60279	−0.402421	−0.201211	+	0.979548i	$0.564488\pi$
$458$	0	0
$459$	−12.3941	−0.578506
$460$	0	0
$461$	32.2090	1.50012	0.750062	−	0.661367i	$-0.230024\pi$
$461$	32.2090	1.50012	0.750062	+	0.661367i	$0.230024\pi$
$462$	0	0
$463$	37.4715	1.74145	0.870724	−	0.491772i	$-0.163651\pi$
$463$	37.4715	1.74145	0.870724	+	0.491772i	$0.163651\pi$
$464$	0	0
$465$	−51.2003	−2.37436
$466$	0	0
$467$	−9.02962	−0.417841	−0.208920	−	0.977933i	$-0.566995\pi$
$467$	−9.02962	−0.417841	−0.208920	+	0.977933i	$0.566995\pi$
$468$	0	0
$469$	−36.4385	−1.68257
$470$	0	0
$471$	32.8459	1.51346
$472$	0	0
$473$	52.5827	2.41775
$474$	0	0
$475$	5.32380	0.244273
$476$	0	0
$477$	1.16421	0.0533057
$478$	0	0
$479$	−22.7722	−1.04049	−0.520244	−	0.854018i	$-0.674159\pi$
$479$	−22.7722	−1.04049	−0.520244	+	0.854018i	$0.674159\pi$
$480$	0	0
$481$	13.0365	0.594414
$482$	0	0
$483$	37.7341	1.71696
$484$	0	0
$485$	21.4762	0.975185
$486$	0	0
$487$	−20.1539	−0.913260	−0.456630	−	0.889657i	$-0.650944\pi$
$487$	−20.1539	−0.913260	−0.456630	+	0.889657i	$0.650944\pi$
$488$	0	0
$489$	−33.9433	−1.53497
$490$	0	0
$491$	−1.37481	−0.0620444	−0.0310222	−	0.999519i	$-0.509876\pi$
$491$	−1.37481	−0.0620444	−0.0310222	+	0.999519i	$0.509876\pi$
$492$	0	0
$493$	−9.53499	−0.429434
$494$	0	0
$495$	−2.45186	−0.110203
$496$	0	0
$497$	−58.0058	−2.60192
$498$	0	0
$499$	34.1920	1.53064	0.765321	−	0.643648i	$-0.222580\pi$
$499$	34.1920	1.53064	0.765321	+	0.643648i	$0.222580\pi$
$500$	0	0
$501$	−8.51241	−0.380307
$502$	0	0
$503$	−25.7273	−1.14712	−0.573562	−	0.819162i	$-0.694439\pi$
$503$	−25.7273	−1.14712	−0.573562	+	0.819162i	$0.694439\pi$
$504$	0	0
$505$	−30.9412	−1.37687
$506$	0	0
$507$	−4.13160	−0.183491
$508$	0	0
$509$	−15.8511	−0.702589	−0.351295	−	0.936265i	$-0.614258\pi$
$509$	−15.8511	−0.702589	−0.351295	+	0.936265i	$0.614258\pi$
$510$	0	0
$511$	−27.2296	−1.20457
$512$	0	0
$513$	5.76336	0.254459
$514$	0	0
$515$	39.0085	1.71892
$516$	0	0
$517$	−10.2715	−0.451741
$518$	0	0
$519$	−35.0558	−1.53878
$520$	0	0
$521$	−25.8557	−1.13276	−0.566380	−	0.824145i	$-0.691656\pi$
$521$	−25.8557	−1.13276	−0.566380	+	0.824145i	$0.691656\pi$
$522$	0	0
$523$	18.7520	0.819968	0.409984	−	0.912093i	$-0.365534\pi$
$523$	18.7520	0.819968	0.409984	+	0.912093i	$0.365534\pi$
$524$	0	0
$525$	38.1261	1.66396
$526$	0	0
$527$	22.4525	0.978048
$528$	0	0
$529$	0.547319	0.0237965
$530$	0	0
$531$	−0.0810145	−0.00351573
$532$	0	0
$533$	25.3578	1.09837
$534$	0	0
$535$	−15.4966	−0.669976
$536$	0	0
$537$	37.7272	1.62805
$538$	0	0
$539$	−80.4062	−3.46334
$540$	0	0
$541$	−26.9779	−1.15987	−0.579935	−	0.814663i	$-0.696922\pi$
$541$	−26.9779	−1.15987	−0.579935	+	0.814663i	$0.696922\pi$
$542$	0	0
$543$	−0.313948	−0.0134728
$544$	0	0
$545$	0.0231785	0.000992857 0
$546$	0	0
$547$	−12.3083	−0.526265	−0.263132	−	0.964760i	$-0.584756\pi$
$547$	−12.3083	−0.526265	−0.263132	+	0.964760i	$0.584756\pi$
$548$	0	0
$549$	−1.15023	−0.0490908
$550$	0	0
$551$	4.43386	0.188889
$552$	0	0
$553$	3.58375	0.152397
$554$	0	0
$555$	17.6653	0.749851
$556$	0	0
$557$	12.3883	0.524908	0.262454	−	0.964945i	$-0.415468\pi$
$557$	12.3883	0.524908	0.262454	+	0.964945i	$0.415468\pi$
$558$	0	0
$559$	−36.2835	−1.53463
$560$	0	0
$561$	−22.5012	−0.950002
$562$	0	0
$563$	37.6637	1.58733	0.793667	−	0.608352i	$-0.208169\pi$
$563$	37.6637	1.58733	0.793667	+	0.608352i	$0.208169\pi$
$564$	0	0
$565$	−6.44219	−0.271025
$566$	0	0
$567$	39.3878	1.65413
$568$	0	0
$569$	30.4760	1.27762	0.638810	−	0.769365i	$-0.279427\pi$
$569$	30.4760	1.27762	0.638810	+	0.769365i	$0.279427\pi$
$570$	0	0
$571$	−30.0572	−1.25785	−0.628927	−	0.777464i	$-0.716506\pi$
$571$	−30.0572	−1.25785	−0.628927	+	0.777464i	$0.716506\pi$
$572$	0	0
$573$	−13.1317	−0.548584
$574$	0	0
$575$	23.7920	0.992193
$576$	0	0
$577$	−26.5982	−1.10730	−0.553648	−	0.832751i	$-0.686765\pi$
$577$	−26.5982	−1.10730	−0.553648	+	0.832751i	$0.686765\pi$
$578$	0	0
$579$	−43.6577	−1.81435
$580$	0	0
$581$	−5.18415	−0.215075
$582$	0	0
$583$	48.4598	2.00700
$584$	0	0
$585$	1.69185	0.0699494
$586$	0	0
$587$	21.0950	0.870682	0.435341	−	0.900266i	$-0.356628\pi$
$587$	21.0950	0.870682	0.435341	+	0.900266i	$0.356628\pi$
$588$	0	0
$589$	−10.4406	−0.430199
$590$	0	0
$591$	23.9250	0.984144
$592$	0	0
$593$	0.903545	0.0371042	0.0185521	−	0.999828i	$-0.494094\pi$
$593$	0.903545	0.0371042	0.0185521	+	0.999828i	$0.494094\pi$
$594$	0	0
$595$	−33.7693	−1.38441
$596$	0	0
$597$	−4.24153	−0.173594
$598$	0	0
$599$	−22.4974	−0.919219	−0.459609	−	0.888121i	$-0.652011\pi$
$599$	−22.4974	−0.919219	−0.459609	+	0.888121i	$0.652011\pi$
$600$	0	0
$601$	21.2701	0.867624	0.433812	−	0.901003i	$-0.357168\pi$
$601$	21.2701	0.867624	0.433812	+	0.901003i	$0.357168\pi$
$602$	0	0
$603$	−1.08481	−0.0441768
$604$	0	0
$605$	−67.4414	−2.74188
$606$	0	0
$607$	34.4773	1.39939	0.699694	−	0.714442i	$-0.253320\pi$
$607$	34.4773	1.39939	0.699694	+	0.714442i	$0.253320\pi$
$608$	0	0
$609$	31.7529	1.28669
$610$	0	0
$611$	7.08764	0.286735
$612$	0	0
$613$	−23.8354	−0.962703	−0.481352	−	0.876528i	$-0.659854\pi$
$613$	−23.8354	−0.962703	−0.481352	+	0.876528i	$0.659854\pi$
$614$	0	0
$615$	34.3614	1.38559
$616$	0	0
$617$	10.7763	0.433838	0.216919	−	0.976190i	$-0.430399\pi$
$617$	10.7763	0.433838	0.216919	+	0.976190i	$0.430399\pi$
$618$	0	0
$619$	−9.40006	−0.377820	−0.188910	−	0.981994i	$-0.560496\pi$
$619$	−9.40006	−0.377820	−0.188910	+	0.981994i	$0.560496\pi$
$620$	0	0
$621$	25.7563	1.03357
$622$	0	0
$623$	24.3830	0.976885
$624$	0	0
$625$	−25.4757	−1.01903
$626$	0	0
$627$	10.4633	0.417863
$628$	0	0
$629$	−7.74665	−0.308879
$630$	0	0
$631$	−12.9393	−0.515107	−0.257554	−	0.966264i	$-0.582916\pi$
$631$	−12.9393	−0.515107	−0.257554	+	0.966264i	$0.582916\pi$
$632$	0	0
$633$	0.200972	0.00798791
$634$	0	0
$635$	51.6690	2.05042
$636$	0	0
$637$	55.4826	2.19830
$638$	0	0
$639$	−1.72689	−0.0683147
$640$	0	0
$641$	33.0587	1.30574	0.652870	−	0.757470i	$-0.273565\pi$
$641$	33.0587	1.30574	0.652870	+	0.757470i	$0.273565\pi$
$642$	0	0
$643$	14.0151	0.552702	0.276351	−	0.961057i	$-0.410875\pi$
$643$	14.0151	0.552702	0.276351	+	0.961057i	$0.410875\pi$
$644$	0	0
$645$	−49.1665	−1.93593
$646$	0	0
$647$	−18.5787	−0.730406	−0.365203	−	0.930928i	$-0.619000\pi$
$647$	−18.5787	−0.730406	−0.365203	+	0.930928i	$0.619000\pi$
$648$	0	0
$649$	−3.37219	−0.132370
$650$	0	0
$651$	−74.7701	−2.93047
$652$	0	0
$653$	−28.9360	−1.13235	−0.566176	−	0.824285i	$-0.691578\pi$
$653$	−28.9360	−1.13235	−0.566176	+	0.824285i	$0.691578\pi$
$654$	0	0
$655$	4.23914	0.165637
$656$	0	0
$657$	−0.810653	−0.0316266
$658$	0	0
$659$	−39.6658	−1.54516	−0.772580	−	0.634918i	$-0.781034\pi$
$659$	−39.6658	−1.54516	−0.772580	+	0.634918i	$0.781034\pi$
$660$	0	0
$661$	31.2335	1.21484	0.607422	−	0.794380i	$-0.292204\pi$
$661$	31.2335	1.21484	0.607422	+	0.794380i	$0.292204\pi$
$662$	0	0
$663$	15.5265	0.602998
$664$	0	0
$665$	15.7030	0.608937
$666$	0	0
$667$	19.8148	0.767234
$668$	0	0
$669$	8.98488	0.347376
$670$	0	0
$671$	−47.8779	−1.84831
$672$	0	0
$673$	31.7035	1.22208	0.611041	−	0.791599i	$-0.290751\pi$
$673$	31.7035	1.22208	0.611041	+	0.791599i	$0.290751\pi$
$674$	0	0
$675$	26.0239	1.00166
$676$	0	0
$677$	−24.6884	−0.948851	−0.474425	−	0.880296i	$-0.657344\pi$
$677$	−24.6884	−0.948851	−0.474425	+	0.880296i	$0.657344\pi$
$678$	0	0
$679$	31.3627	1.20359
$680$	0	0
$681$	−34.6204	−1.32666
$682$	0	0
$683$	−23.2114	−0.888160	−0.444080	−	0.895987i	$-0.646469\pi$
$683$	−23.2114	−0.888160	−0.444080	+	0.895987i	$0.646469\pi$
$684$	0	0
$685$	42.0832	1.60792
$686$	0	0
$687$	40.6405	1.55053
$688$	0	0
$689$	−33.4386	−1.27391
$690$	0	0
$691$	33.6128	1.27869	0.639345	−	0.768920i	$-0.279205\pi$
$691$	33.6128	1.27869	0.639345	+	0.768920i	$0.279205\pi$
$692$	0	0
$693$	−3.58056	−0.136014
$694$	0	0
$695$	60.3541	2.28936
$696$	0	0
$697$	−15.0683	−0.570752
$698$	0	0
$699$	−41.9784	−1.58777
$700$	0	0
$701$	−31.6502	−1.19541	−0.597706	−	0.801715i	$-0.703921\pi$
$701$	−31.6502	−1.19541	−0.597706	+	0.801715i	$0.703921\pi$
$702$	0	0
$703$	3.60227	0.135862
$704$	0	0
$705$	9.60421	0.361716
$706$	0	0
$707$	−45.1849	−1.69935
$708$	0	0
$709$	−12.7600	−0.479211	−0.239605	−	0.970870i	$-0.577018\pi$
$709$	−12.7600	−0.479211	−0.239605	+	0.970870i	$0.577018\pi$
$710$	0	0
$711$	0.106692	0.00400126
$712$	0	0
$713$	−46.6590	−1.74739
$714$	0	0
$715$	70.4224	2.63365
$716$	0	0
$717$	−38.5421	−1.43938
$718$	0	0
$719$	15.6344	0.583066	0.291533	−	0.956561i	$-0.405835\pi$
$719$	15.6344	0.583066	0.291533	+	0.956561i	$0.405835\pi$
$720$	0	0
$721$	56.9658	2.12152
$722$	0	0
$723$	−23.4554	−0.872315
$724$	0	0
$725$	20.0207	0.743550
$726$	0	0
$727$	19.5645	0.725607	0.362804	−	0.931866i	$-0.381820\pi$
$727$	19.5645	0.725607	0.362804	+	0.931866i	$0.381820\pi$
$728$	0	0
$729$	28.1165	1.04135
$730$	0	0
$731$	21.5607	0.797451
$732$	0	0
$733$	−32.2321	−1.19052	−0.595260	−	0.803533i	$-0.702951\pi$
$733$	−32.2321	−1.19052	−0.595260	+	0.803533i	$0.702951\pi$
$734$	0	0
$735$	75.1825	2.77315
$736$	0	0
$737$	−45.1546	−1.66329
$738$	0	0
$739$	−34.3566	−1.26383	−0.631914	−	0.775039i	$-0.717730\pi$
$739$	−34.3566	−1.26383	−0.631914	+	0.775039i	$0.717730\pi$
$740$	0	0
$741$	−7.21996	−0.265232
$742$	0	0
$743$	−6.52763	−0.239475	−0.119738	−	0.992806i	$-0.538205\pi$
$743$	−6.52763	−0.239475	−0.119738	+	0.992806i	$0.538205\pi$
$744$	0	0
$745$	−2.99957	−0.109896
$746$	0	0
$747$	−0.154337	−0.00564691
$748$	0	0
$749$	−22.6304	−0.826896
$750$	0	0
$751$	−39.5317	−1.44253	−0.721266	−	0.692658i	$-0.756439\pi$
$751$	−39.5317	−1.44253	−0.721266	+	0.692658i	$0.756439\pi$
$752$	0	0
$753$	49.8505	1.81665
$754$	0	0
$755$	−31.1994	−1.13546
$756$	0	0
$757$	45.4056	1.65030	0.825148	−	0.564917i	$-0.191092\pi$
$757$	45.4056	1.65030	0.825148	+	0.564917i	$0.191092\pi$
$758$	0	0
$759$	46.7602	1.69729
$760$	0	0
$761$	9.34636	0.338805	0.169403	−	0.985547i	$-0.445816\pi$
$761$	9.34636	0.338805	0.169403	+	0.985547i	$0.445816\pi$
$762$	0	0
$763$	0.0338486	0.00122540
$764$	0	0
$765$	−1.00534	−0.0363483
$766$	0	0
$767$	2.32690	0.0840197
$768$	0	0
$769$	9.58409	0.345611	0.172805	−	0.984956i	$-0.444717\pi$
$769$	9.58409	0.345611	0.172805	+	0.984956i	$0.444717\pi$
$770$	0	0
$771$	−15.1666	−0.546213
$772$	0	0
$773$	42.3846	1.52447	0.762235	−	0.647301i	$-0.224102\pi$
$773$	42.3846	1.52447	0.762235	+	0.647301i	$0.224102\pi$
$774$	0	0
$775$	−47.1438	−1.69345
$776$	0	0
$777$	25.7975	0.925478
$778$	0	0
$779$	7.00689	0.251048
$780$	0	0
$781$	−71.8810	−2.57210
$782$	0	0
$783$	21.6737	0.774555
$784$	0	0
$785$	61.0856	2.18024
$786$	0	0
$787$	37.5259	1.33766	0.668828	−	0.743417i	$-0.266796\pi$
$787$	37.5259	1.33766	0.668828	+	0.743417i	$0.266796\pi$
$788$	0	0
$789$	−15.9648	−0.568361
$790$	0	0
$791$	−9.40782	−0.334504
$792$	0	0
$793$	33.0371	1.17318
$794$	0	0
$795$	−45.3115	−1.60703
$796$	0	0
$797$	9.67031	0.342540	0.171270	−	0.985224i	$-0.445213\pi$
$797$	9.67031	0.342540	0.171270	+	0.985224i	$0.445213\pi$
$798$	0	0
$799$	−4.21167	−0.148998
$800$	0	0
$801$	0.725906	0.0256486
$802$	0	0
$803$	−33.7430	−1.19077
$804$	0	0
$805$	70.1765	2.47340
$806$	0	0
$807$	−27.9083	−0.982420
$808$	0	0
$809$	−6.30000	−0.221496	−0.110748	−	0.993849i	$-0.535325\pi$
$809$	−6.30000	−0.221496	−0.110748	+	0.993849i	$0.535325\pi$
$810$	0	0
$811$	−10.0930	−0.354413	−0.177207	−	0.984174i	$-0.556706\pi$
$811$	−10.0930	−0.354413	−0.177207	+	0.984174i	$0.556706\pi$
$812$	0	0
$813$	34.7336	1.21816
$814$	0	0
$815$	−63.1266	−2.21123
$816$	0	0
$817$	−10.0259	−0.350763
$818$	0	0
$819$	2.47069	0.0863328
$820$	0	0
$821$	−15.9742	−0.557505	−0.278752	−	0.960363i	$-0.589921\pi$
$821$	−15.9742	−0.557505	−0.278752	+	0.960363i	$0.589921\pi$
$822$	0	0
$823$	9.46196	0.329823	0.164912	−	0.986308i	$-0.447266\pi$
$823$	9.46196	0.329823	0.164912	+	0.986308i	$0.447266\pi$
$824$	0	0
$825$	47.2460	1.64489
$826$	0	0
$827$	21.9899	0.764663	0.382331	−	0.924025i	$-0.375121\pi$
$827$	21.9899	0.764663	0.382331	+	0.924025i	$0.375121\pi$
$828$	0	0
$829$	−23.3581	−0.811259	−0.405629	−	0.914038i	$-0.632948\pi$
$829$	−23.3581	−0.811259	−0.405629	+	0.914038i	$0.632948\pi$
$830$	0	0
$831$	54.4848	1.89006
$832$	0	0
$833$	−32.9693	−1.14232
$834$	0	0
$835$	−15.8311	−0.547857
$836$	0	0
$837$	−51.0362	−1.76407
$838$	0	0
$839$	−42.1584	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$839$	−42.1584	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$840$	0	0
$841$	−12.3260	−0.425035
$842$	0	0
$843$	−10.1390	−0.349205
$844$	0	0
$845$	−7.68380	−0.264331
$846$	0	0
$847$	−98.4878	−3.38408
$848$	0	0
$849$	7.47521	0.256549
$850$	0	0
$851$	16.0985	0.551848
$852$	0	0
$853$	28.3356	0.970193	0.485097	−	0.874461i	$-0.338784\pi$
$853$	28.3356	0.970193	0.485097	+	0.874461i	$0.338784\pi$
$854$	0	0
$855$	0.467495	0.0159880
$856$	0	0
$857$	−12.6407	−0.431800	−0.215900	−	0.976416i	$-0.569268\pi$
$857$	−12.6407	−0.431800	−0.215900	+	0.976416i	$0.569268\pi$
$858$	0	0
$859$	−8.49113	−0.289714	−0.144857	−	0.989453i	$-0.546272\pi$
$859$	−8.49113	−0.289714	−0.144857	+	0.989453i	$0.546272\pi$
$860$	0	0
$861$	50.1795	1.71011
$862$	0	0
$863$	−58.2820	−1.98394	−0.991971	−	0.126464i	$-0.959637\pi$
$863$	−58.2820	−1.98394	−0.991971	+	0.126464i	$0.959637\pi$
$864$	0	0
$865$	−65.1956	−2.21672
$866$	0	0
$867$	19.5394	0.663591
$868$	0	0
$869$	4.44100	0.150650
$870$	0	0
$871$	31.1580	1.05575
$872$	0	0
$873$	0.933699	0.0316009
$874$	0	0
$875$	−1.40317	−0.0474359
$876$	0	0
$877$	−22.7129	−0.766961	−0.383480	−	0.923549i	$-0.625275\pi$
$877$	−22.7129	−0.766961	−0.383480	+	0.923549i	$0.625275\pi$
$878$	0	0
$879$	−31.3746	−1.05824
$880$	0	0
$881$	−27.0766	−0.912233	−0.456117	−	0.889920i	$-0.650760\pi$
$881$	−27.0766	−0.912233	−0.456117	+	0.889920i	$0.650760\pi$
$882$	0	0
$883$	−34.9690	−1.17680	−0.588400	−	0.808570i	$-0.700242\pi$
$883$	−34.9690	−1.17680	−0.588400	+	0.808570i	$0.700242\pi$
$884$	0	0
$885$	3.15311	0.105990
$886$	0	0
$887$	−19.5367	−0.655979	−0.327990	−	0.944681i	$-0.606371\pi$
$887$	−19.5367	−0.655979	−0.327990	+	0.944681i	$0.606371\pi$
$888$	0	0
$889$	75.4546	2.53067
$890$	0	0
$891$	48.8095	1.63518
$892$	0	0
$893$	1.95847	0.0655376
$894$	0	0
$895$	70.1636	2.34531
$896$	0	0
$897$	−32.2659	−1.07733
$898$	0	0
$899$	−39.2631	−1.30950
$900$	0	0
$901$	19.8702	0.661971
$902$	0	0
$903$	−71.8001	−2.38936
$904$	0	0
$905$	−0.583870	−0.0194085
$906$	0	0
$907$	−28.2224	−0.937108	−0.468554	−	0.883435i	$-0.655225\pi$
$907$	−28.2224	−0.937108	−0.468554	+	0.883435i	$0.655225\pi$
$908$	0	0
$909$	−1.34520	−0.0446174
$910$	0	0
$911$	35.0715	1.16197	0.580986	−	0.813914i	$-0.302667\pi$
$911$	35.0715	1.16197	0.580986	+	0.813914i	$0.302667\pi$
$912$	0	0
$913$	−6.42422	−0.212611
$914$	0	0
$915$	44.7674	1.47996
$916$	0	0
$917$	6.19061	0.204432
$918$	0	0
$919$	−6.03558	−0.199095	−0.0995477	−	0.995033i	$-0.531740\pi$
$919$	−6.03558	−0.199095	−0.0995477	+	0.995033i	$0.531740\pi$
$920$	0	0
$921$	3.16956	0.104440
$922$	0	0
$923$	49.5999	1.63260
$924$	0	0
$925$	16.2657	0.534813
$926$	0	0
$927$	1.69593	0.0557016
$928$	0	0
$929$	35.3941	1.16124	0.580622	−	0.814173i	$-0.302810\pi$
$929$	35.3941	1.16124	0.580622	+	0.814173i	$0.302810\pi$
$930$	0	0
$931$	15.3310	0.502454
$932$	0	0
$933$	−30.2971	−0.991883
$934$	0	0
$935$	−41.8470	−1.36854
$936$	0	0
$937$	43.3352	1.41570	0.707849	−	0.706364i	$-0.249666\pi$
$937$	43.3352	1.41570	0.707849	+	0.706364i	$0.249666\pi$
$938$	0	0
$939$	−1.64868	−0.0538025
$940$	0	0
$941$	37.3659	1.21809	0.609047	−	0.793134i	$-0.291552\pi$
$941$	37.3659	1.21809	0.609047	+	0.793134i	$0.291552\pi$
$942$	0	0
$943$	31.3137	1.01971
$944$	0	0
$945$	76.7600	2.49700
$946$	0	0
$947$	−4.93807	−0.160466	−0.0802329	−	0.996776i	$-0.525566\pi$
$947$	−4.93807	−0.160466	−0.0802329	+	0.996776i	$0.525566\pi$
$948$	0	0
$949$	23.2837	0.755819
$950$	0	0
$951$	−1.57793	−0.0511679
$952$	0	0
$953$	18.9905	0.615163	0.307581	−	0.951522i	$-0.400480\pi$
$953$	18.9905	0.615163	0.307581	+	0.951522i	$0.400480\pi$
$954$	0	0
$955$	−24.4219	−0.790273
$956$	0	0
$957$	39.3482	1.27195
$958$	0	0
$959$	61.4561	1.98452
$960$	0	0
$961$	61.4549	1.98242
$962$	0	0
$963$	−0.673729	−0.0217106
$964$	0	0
$965$	−81.1930	−2.61370
$966$	0	0
$967$	3.62083	0.116438	0.0582190	−	0.998304i	$-0.481458\pi$
$967$	3.62083	0.116438	0.0582190	+	0.998304i	$0.481458\pi$
$968$	0	0
$969$	4.29030	0.137824
$970$	0	0
$971$	17.0623	0.547557	0.273778	−	0.961793i	$-0.411727\pi$
$971$	17.0623	0.547557	0.273778	+	0.961793i	$0.411727\pi$
$972$	0	0
$973$	88.1379	2.82557
$974$	0	0
$975$	−32.6011	−1.04407
$976$	0	0
$977$	61.1954	1.95782	0.978908	−	0.204303i	$-0.0654928\pi$
$977$	61.1954	1.95782	0.978908	+	0.204303i	$0.0654928\pi$
$978$	0	0
$979$	30.2155	0.965691
$980$	0	0
$981$	0.00100771	3.21736e−5 0
$982$	0	0
$983$	−48.5723	−1.54922	−0.774608	−	0.632442i	$-0.782053\pi$
$983$	−48.5723	−1.54922	−0.774608	+	0.632442i	$0.782053\pi$
$984$	0	0
$985$	44.4949	1.41773
$986$	0	0
$987$	14.0255	0.446436
$988$	0	0
$989$	−44.8057	−1.42474
$990$	0	0
$991$	−19.5476	−0.620950	−0.310475	−	0.950581i	$-0.600488\pi$
$991$	−19.5476	−0.620950	−0.310475	+	0.950581i	$0.600488\pi$
$992$	0	0
$993$	−1.32558	−0.0420659
$994$	0	0
$995$	−7.88825	−0.250074
$996$	0	0
$997$	52.6915	1.66876	0.834378	−	0.551193i	$-0.185827\pi$
$997$	52.6915	1.66876	0.834378	+	0.551193i	$0.185827\pi$
$998$	0	0
$999$	17.6087	0.557115

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.16	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.16	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69210 q^{3} -3.14690 q^{5} -4.59556 q^{7} -0.136814 q^{9} +O(q^{10})\)	\(q-1.69210 q^{3} -3.14690 q^{5} -4.59556 q^{7} -0.136814 q^{9} -5.69483 q^{11} +3.92959 q^{13} +5.32485 q^{15} -2.33507 q^{17} +1.08583 q^{19} +7.77612 q^{21} +4.85256 q^{23} +4.90297 q^{25} +5.30779 q^{27} +4.08338 q^{29} -9.61535 q^{31} +9.63619 q^{33} +14.4618 q^{35} +3.31752 q^{37} -6.64925 q^{39} +6.45302 q^{41} -9.23341 q^{43} +0.430541 q^{45} +1.80366 q^{47} +14.1192 q^{49} +3.95117 q^{51} -8.50944 q^{53} +17.9210 q^{55} -1.83733 q^{57} +0.592149 q^{59} +8.40726 q^{61} +0.628739 q^{63} -12.3660 q^{65} +7.92906 q^{67} -8.21099 q^{69} +12.6221 q^{71} +5.92521 q^{73} -8.29630 q^{75} +26.1709 q^{77} -0.779830 q^{79} -8.57084 q^{81} +1.12808 q^{83} +7.34824 q^{85} -6.90947 q^{87} -5.30578 q^{89} -18.0587 q^{91} +16.2701 q^{93} -3.41700 q^{95} -6.82457 q^{97} +0.779134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more