LMFDB - Embedded newform 4028.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, 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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 27 x^{17} + 161 x^{16} + 253 x^{15} - 2103 x^{14} - 683 x^{13} + 14442 x^{12} + \cdots - 4088 $ x^19 - 5x^18 - 27x^17 + 161x^16 + 253x^15 - 2103x^14 - 683x^13 + 14442x^12 - 4144x^11 - 56325x^10 + 37245x^9 + 124233x^8 - 117486x^7 - 140498x^6 + 176987x^5 + 54044x^4 - 115318x^3 + 16295x^2 + 17097x - 4088
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.02356$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02356 q^{3} +1.31787 q^{5} +3.08066 q^{7} +6.14194 q^{9} +O(q^{10})$	$q-3.02356 q^{3} +1.31787 q^{5} +3.08066 q^{7} +6.14194 q^{9} +4.77491 q^{11} -4.41677 q^{13} -3.98467 q^{15} +2.25843 q^{17} +1.00000 q^{19} -9.31458 q^{21} -9.04358 q^{23} -3.26321 q^{25} -9.49987 q^{27} -5.83195 q^{29} -4.13809 q^{31} -14.4373 q^{33} +4.05992 q^{35} +8.70476 q^{37} +13.3544 q^{39} -7.87841 q^{41} -8.02037 q^{43} +8.09430 q^{45} +4.20950 q^{47} +2.49047 q^{49} -6.82850 q^{51} -1.00000 q^{53} +6.29273 q^{55} -3.02356 q^{57} -7.86310 q^{59} -11.1525 q^{61} +18.9212 q^{63} -5.82075 q^{65} -12.2824 q^{67} +27.3438 q^{69} -10.5153 q^{71} +12.9381 q^{73} +9.86653 q^{75} +14.7099 q^{77} -2.04728 q^{79} +10.2977 q^{81} -3.37292 q^{83} +2.97632 q^{85} +17.6333 q^{87} +12.8519 q^{89} -13.6066 q^{91} +12.5118 q^{93} +1.31787 q^{95} -0.0108226 q^{97} +29.3272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9	$ 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9 - 3 * q^11 - 23 * q^13 - 18 * q^15 - 7 * q^17 + 19 * q^19 - 4 * q^21 - 6 * q^23 + 18 * q^25 - 17 * q^27 - 4 * q^29 - 30 * q^31 - 10 * q^33 - q^35 - 31 * q^37 + 5 * q^39 - 15 * q^41 - 29 * q^43 + 6 * q^45 - 18 * q^47 + 23 * q^49 - 5 * q^51 - 19 * q^53 - 19 * q^55 - 5 * q^57 + 8 * q^59 - 4 * q^61 - 64 * q^63 - 26 * q^65 - 62 * q^67 + 3 * q^69 - 17 * q^71 + q^73 - 40 * q^75 - 14 * q^77 - 28 * q^79 + 11 * q^81 + 4 * q^83 - 31 * q^85 - 20 * q^87 + 33 * q^89 - 29 * q^91 - 59 * q^93 - 5 * q^95 + 5 * q^97 - 48 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.02356	−1.74566	−0.872828	−	0.488028i	$-0.837716\pi$
$3$	−3.02356	−1.74566	−0.872828	+	0.488028i	$0.837716\pi$
$4$	0	0
$5$	1.31787	0.589371	0.294685	−	0.955594i	$-0.404785\pi$
$5$	1.31787	0.589371	0.294685	+	0.955594i	$0.404785\pi$
$6$	0	0
$7$	3.08066	1.16438	0.582190	−	0.813053i	$-0.302196\pi$
$7$	3.08066	1.16438	0.582190	+	0.813053i	$0.302196\pi$
$8$	0	0
$9$	6.14194	2.04731
$10$	0	0
$11$	4.77491	1.43969	0.719845	−	0.694135i	$-0.244213\pi$
$11$	4.77491	1.43969	0.719845	+	0.694135i	$0.244213\pi$
$12$	0	0
$13$	−4.41677	−1.22499	−0.612496	−	0.790473i	$-0.709835\pi$
$13$	−4.41677	−1.22499	−0.612496	+	0.790473i	$0.709835\pi$
$14$	0	0
$15$	−3.98467	−1.02884
$16$	0	0
$17$	2.25843	0.547749	0.273874	−	0.961765i	$-0.411695\pi$
$17$	2.25843	0.547749	0.273874	+	0.961765i	$0.411695\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−9.31458	−2.03261
$22$	0	0
$23$	−9.04358	−1.88572	−0.942858	−	0.333194i	$-0.891873\pi$
$23$	−9.04358	−1.88572	−0.942858	+	0.333194i	$0.891873\pi$
$24$	0	0
$25$	−3.26321	−0.652642
$26$	0	0
$27$	−9.49987	−1.82825
$28$	0	0
$29$	−5.83195	−1.08297	−0.541483	−	0.840712i	$-0.682137\pi$
$29$	−5.83195	−1.08297	−0.541483	+	0.840712i	$0.682137\pi$
$30$	0	0
$31$	−4.13809	−0.743223	−0.371612	−	0.928388i	$-0.621195\pi$
$31$	−4.13809	−0.743223	−0.371612	+	0.928388i	$0.621195\pi$
$32$	0	0
$33$	−14.4373	−2.51320
$34$	0	0
$35$	4.05992	0.686251
$36$	0	0
$37$	8.70476	1.43105	0.715527	−	0.698585i	$-0.246187\pi$
$37$	8.70476	1.43105	0.715527	+	0.698585i	$0.246187\pi$
$38$	0	0
$39$	13.3544	2.13842
$40$	0	0
$41$	−7.87841	−1.23040	−0.615201	−	0.788371i	$-0.710925\pi$
$41$	−7.87841	−1.23040	−0.615201	+	0.788371i	$0.710925\pi$
$42$	0	0
$43$	−8.02037	−1.22309	−0.611547	−	0.791208i	$-0.709453\pi$
$43$	−8.02037	−1.22309	−0.611547	+	0.791208i	$0.709453\pi$
$44$	0	0
$45$	8.09430	1.20663
$46$	0	0
$47$	4.20950	0.614018	0.307009	−	0.951707i	$-0.400672\pi$
$47$	4.20950	0.614018	0.307009	+	0.951707i	$0.400672\pi$
$48$	0	0
$49$	2.49047	0.355781
$50$	0	0
$51$	−6.82850	−0.956181
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	6.29273	0.848511
$56$	0	0
$57$	−3.02356	−0.400481
$58$	0	0
$59$	−7.86310	−1.02369	−0.511844	−	0.859078i	$-0.671037\pi$
$59$	−7.86310	−1.02369	−0.511844	+	0.859078i	$0.671037\pi$
$60$	0	0
$61$	−11.1525	−1.42794	−0.713968	−	0.700178i	$-0.753104\pi$
$61$	−11.1525	−1.42794	−0.713968	+	0.700178i	$0.753104\pi$
$62$	0	0
$63$	18.9212	2.38385
$64$	0	0
$65$	−5.82075	−0.721975
$66$	0	0
$67$	−12.2824	−1.50054	−0.750269	−	0.661132i	$-0.770076\pi$
$67$	−12.2824	−1.50054	−0.750269	+	0.661132i	$0.770076\pi$
$68$	0	0
$69$	27.3438	3.29181
$70$	0	0
$71$	−10.5153	−1.24794	−0.623969	−	0.781449i	$-0.714481\pi$
$71$	−10.5153	−1.24794	−0.623969	+	0.781449i	$0.714481\pi$
$72$	0	0
$73$	12.9381	1.51428	0.757142	−	0.653250i	$-0.226595\pi$
$73$	12.9381	1.51428	0.757142	+	0.653250i	$0.226595\pi$
$74$	0	0
$75$	9.86653	1.13929
$76$	0	0
$77$	14.7099	1.67635
$78$	0	0
$79$	−2.04728	−0.230338	−0.115169	−	0.993346i	$-0.536741\pi$
$79$	−2.04728	−0.230338	−0.115169	+	0.993346i	$0.536741\pi$
$80$	0	0
$81$	10.2977	1.14418
$82$	0	0
$83$	−3.37292	−0.370226	−0.185113	−	0.982717i	$-0.559265\pi$
$83$	−3.37292	−0.370226	−0.185113	+	0.982717i	$0.559265\pi$
$84$	0	0
$85$	2.97632	0.322827
$86$	0	0
$87$	17.6333	1.89049
$88$	0	0
$89$	12.8519	1.36229	0.681147	−	0.732147i	$-0.261482\pi$
$89$	12.8519	1.36229	0.681147	+	0.732147i	$0.261482\pi$
$90$	0	0
$91$	−13.6066	−1.42636
$92$	0	0
$93$	12.5118	1.29741
$94$	0	0
$95$	1.31787	0.135211
$96$	0	0
$97$	−0.0108226	−0.00109887	−0.000549434	−	1.00000i	$-0.500175\pi$
$97$	−0.0108226	−0.00109887	−0.000549434		1.00000i	$0.500175\pi$
$98$	0	0
$99$	29.3272	2.94750
$100$	0	0
$101$	17.9755	1.78862	0.894312	−	0.447443i	$-0.147665\pi$
$101$	17.9755	1.78862	0.894312	+	0.447443i	$0.147665\pi$
$102$	0	0
$103$	−1.95888	−0.193014	−0.0965070	−	0.995332i	$-0.530767\pi$
$103$	−1.95888	−0.193014	−0.0965070	+	0.995332i	$0.530767\pi$
$104$	0	0
$105$	−12.2754	−1.19796
$106$	0	0
$107$	−10.7168	−1.03604	−0.518018	−	0.855370i	$-0.673330\pi$
$107$	−10.7168	−1.03604	−0.518018	+	0.855370i	$0.673330\pi$
$108$	0	0
$109$	20.4001	1.95397	0.976986	−	0.213302i	$-0.0684218\pi$
$109$	20.4001	1.95397	0.976986	+	0.213302i	$0.0684218\pi$
$110$	0	0
$111$	−26.3194	−2.49813
$112$	0	0
$113$	15.8255	1.48874	0.744368	−	0.667769i	$-0.232751\pi$
$113$	15.8255	1.48874	0.744368	+	0.667769i	$0.232751\pi$
$114$	0	0
$115$	−11.9183	−1.11139
$116$	0	0
$117$	−27.1276	−2.50795
$118$	0	0
$119$	6.95744	0.637788
$120$	0	0
$121$	11.7998	1.07271
$122$	0	0
$123$	23.8209	2.14786
$124$	0	0
$125$	−10.8899	−0.974019
$126$	0	0
$127$	−7.65248	−0.679048	−0.339524	−	0.940597i	$-0.610266\pi$
$127$	−7.65248	−0.679048	−0.339524	+	0.940597i	$0.610266\pi$
$128$	0	0
$129$	24.2501	2.13510
$130$	0	0
$131$	0.184881	0.0161531	0.00807657	−	0.999967i	$-0.497429\pi$
$131$	0.184881	0.0161531	0.00807657	+	0.999967i	$0.497429\pi$
$132$	0	0
$133$	3.08066	0.267127
$134$	0	0
$135$	−12.5196	−1.07752
$136$	0	0
$137$	−3.23224	−0.276149	−0.138074	−	0.990422i	$-0.544091\pi$
$137$	−3.23224	−0.276149	−0.138074	+	0.990422i	$0.544091\pi$
$138$	0	0
$139$	−0.438252	−0.0371721	−0.0185860	−	0.999827i	$-0.505916\pi$
$139$	−0.438252	−0.0371721	−0.0185860	+	0.999827i	$0.505916\pi$
$140$	0	0
$141$	−12.7277	−1.07186
$142$	0	0
$143$	−21.0897	−1.76361
$144$	0	0
$145$	−7.68577	−0.638268
$146$	0	0
$147$	−7.53008	−0.621071
$148$	0	0
$149$	−16.8007	−1.37636	−0.688182	−	0.725538i	$-0.741591\pi$
$149$	−16.8007	−1.37636	−0.688182	+	0.725538i	$0.741591\pi$
$150$	0	0
$151$	−1.36401	−0.111002	−0.0555008	−	0.998459i	$-0.517676\pi$
$151$	−1.36401	−0.111002	−0.0555008	+	0.998459i	$0.517676\pi$
$152$	0	0
$153$	13.8711	1.12141
$154$	0	0
$155$	−5.45348	−0.438034
$156$	0	0
$157$	−23.4681	−1.87296	−0.936478	−	0.350727i	$-0.885934\pi$
$157$	−23.4681	−1.87296	−0.936478	+	0.350727i	$0.885934\pi$
$158$	0	0
$159$	3.02356	0.239784
$160$	0	0
$161$	−27.8602	−2.19569
$162$	0	0
$163$	−21.2716	−1.66612	−0.833061	−	0.553182i	$-0.813413\pi$
$163$	−21.2716	−1.66612	−0.833061	+	0.553182i	$0.813413\pi$
$164$	0	0
$165$	−19.0265	−1.48121
$166$	0	0
$167$	−20.7194	−1.60332	−0.801658	−	0.597783i	$-0.796048\pi$
$167$	−20.7194	−1.60332	−0.801658	+	0.597783i	$0.796048\pi$
$168$	0	0
$169$	6.50790	0.500607
$170$	0	0
$171$	6.14194	0.469686
$172$	0	0
$173$	−13.2765	−1.00939	−0.504696	−	0.863297i	$-0.668395\pi$
$173$	−13.2765	−1.00939	−0.504696	+	0.863297i	$0.668395\pi$
$174$	0	0
$175$	−10.0528	−0.759924
$176$	0	0
$177$	23.7746	1.78701
$178$	0	0
$179$	−7.44709	−0.556622	−0.278311	−	0.960491i	$-0.589775\pi$
$179$	−7.44709	−0.556622	−0.278311	+	0.960491i	$0.589775\pi$
$180$	0	0
$181$	16.3400	1.21454	0.607272	−	0.794494i	$-0.292264\pi$
$181$	16.3400	1.21454	0.607272	+	0.794494i	$0.292264\pi$
$182$	0	0
$183$	33.7204	2.49269
$184$	0	0
$185$	11.4718	0.843421
$186$	0	0
$187$	10.7838	0.788588
$188$	0	0
$189$	−29.2659	−2.12878
$190$	0	0
$191$	20.6374	1.49327	0.746634	−	0.665235i	$-0.231669\pi$
$191$	20.6374	1.49327	0.746634	+	0.665235i	$0.231669\pi$
$192$	0	0
$193$	3.60066	0.259181	0.129591	−	0.991568i	$-0.458634\pi$
$193$	3.60066	0.259181	0.129591	+	0.991568i	$0.458634\pi$
$194$	0	0
$195$	17.5994	1.26032
$196$	0	0
$197$	−23.7183	−1.68986	−0.844930	−	0.534877i	$-0.820358\pi$
$197$	−23.7183	−1.68986	−0.844930	+	0.534877i	$0.820358\pi$
$198$	0	0
$199$	7.66810	0.543577	0.271789	−	0.962357i	$-0.412385\pi$
$199$	7.66810	0.543577	0.271789	+	0.962357i	$0.412385\pi$
$200$	0	0
$201$	37.1368	2.61942
$202$	0	0
$203$	−17.9663	−1.26098
$204$	0	0
$205$	−10.3827	−0.725162
$206$	0	0
$207$	−55.5452	−3.86066
$208$	0	0
$209$	4.77491	0.330288
$210$	0	0
$211$	−3.65129	−0.251365	−0.125683	−	0.992071i	$-0.540112\pi$
$211$	−3.65129	−0.251365	−0.125683	+	0.992071i	$0.540112\pi$
$212$	0	0
$213$	31.7938	2.17847
$214$	0	0
$215$	−10.5698	−0.720856
$216$	0	0
$217$	−12.7480	−0.865394
$218$	0	0
$219$	−39.1191	−2.64342
$220$	0	0
$221$	−9.97496	−0.670988
$222$	0	0
$223$	5.42400	0.363218	0.181609	−	0.983371i	$-0.441870\pi$
$223$	5.42400	0.363218	0.181609	+	0.983371i	$0.441870\pi$
$224$	0	0
$225$	−20.0425	−1.33616
$226$	0	0
$227$	14.3846	0.954740	0.477370	−	0.878702i	$-0.341590\pi$
$227$	14.3846	0.954740	0.477370	+	0.878702i	$0.341590\pi$
$228$	0	0
$229$	5.44505	0.359819	0.179909	−	0.983683i	$-0.442420\pi$
$229$	5.44505	0.359819	0.179909	+	0.983683i	$0.442420\pi$
$230$	0	0
$231$	−44.4763	−2.92632
$232$	0	0
$233$	1.53001	0.100234	0.0501170	−	0.998743i	$-0.484041\pi$
$233$	1.53001	0.100234	0.0501170	+	0.998743i	$0.484041\pi$
$234$	0	0
$235$	5.54758	0.361884
$236$	0	0
$237$	6.19010	0.402090
$238$	0	0
$239$	2.01211	0.130152	0.0650762	−	0.997880i	$-0.479271\pi$
$239$	2.01211	0.130152	0.0650762	+	0.997880i	$0.479271\pi$
$240$	0	0
$241$	12.8239	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$241$	12.8239	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$242$	0	0
$243$	−2.63600	−0.169099
$244$	0	0
$245$	3.28212	0.209687
$246$	0	0
$247$	−4.41677	−0.281033
$248$	0	0
$249$	10.1982	0.646288
$250$	0	0
$251$	−12.6773	−0.800183	−0.400092	−	0.916475i	$-0.631022\pi$
$251$	−12.6773	−0.800183	−0.400092	+	0.916475i	$0.631022\pi$
$252$	0	0
$253$	−43.1823	−2.71485
$254$	0	0
$255$	−8.99909	−0.563545
$256$	0	0
$257$	6.55238	0.408726	0.204363	−	0.978895i	$-0.434488\pi$
$257$	6.55238	0.408726	0.204363	+	0.978895i	$0.434488\pi$
$258$	0	0
$259$	26.8164	1.66629
$260$	0	0
$261$	−35.8195	−2.21717
$262$	0	0
$263$	−7.47320	−0.460817	−0.230409	−	0.973094i	$-0.574006\pi$
$263$	−7.47320	−0.460817	−0.230409	+	0.973094i	$0.574006\pi$
$264$	0	0
$265$	−1.31787	−0.0809563
$266$	0	0
$267$	−38.8584	−2.37810
$268$	0	0
$269$	14.8875	0.907710	0.453855	−	0.891076i	$-0.350048\pi$
$269$	14.8875	0.907710	0.453855	+	0.891076i	$0.350048\pi$
$270$	0	0
$271$	27.8093	1.68929	0.844647	−	0.535324i	$-0.179810\pi$
$271$	27.8093	1.68929	0.844647	+	0.535324i	$0.179810\pi$
$272$	0	0
$273$	41.1404	2.48993
$274$	0	0
$275$	−15.5815	−0.939603
$276$	0	0
$277$	−24.2379	−1.45632	−0.728158	−	0.685409i	$-0.759623\pi$
$277$	−24.2379	−1.45632	−0.728158	+	0.685409i	$0.759623\pi$
$278$	0	0
$279$	−25.4159	−1.52161
$280$	0	0
$281$	9.12430	0.544310	0.272155	−	0.962253i	$-0.412264\pi$
$281$	9.12430	0.544310	0.272155	+	0.962253i	$0.412264\pi$
$282$	0	0
$283$	17.5388	1.04257	0.521286	−	0.853382i	$-0.325453\pi$
$283$	17.5388	1.04257	0.521286	+	0.853382i	$0.325453\pi$
$284$	0	0
$285$	−3.98467	−0.236032
$286$	0	0
$287$	−24.2707	−1.43265
$288$	0	0
$289$	−11.8995	−0.699971
$290$	0	0
$291$	0.0327228	0.00191825
$292$	0	0
$293$	−10.2903	−0.601166	−0.300583	−	0.953756i	$-0.597181\pi$
$293$	−10.2903	−0.601166	−0.300583	+	0.953756i	$0.597181\pi$
$294$	0	0
$295$	−10.3626	−0.603332
$296$	0	0
$297$	−45.3611	−2.63212
$298$	0	0
$299$	39.9434	2.30999
$300$	0	0
$301$	−24.7080	−1.42415
$302$	0	0
$303$	−54.3500	−3.12232
$304$	0	0
$305$	−14.6976	−0.841583
$306$	0	0
$307$	−24.8044	−1.41566	−0.707832	−	0.706381i	$-0.750327\pi$
$307$	−24.8044	−1.41566	−0.707832	+	0.706381i	$0.750327\pi$
$308$	0	0
$309$	5.92280	0.336936
$310$	0	0
$311$	−2.32099	−0.131611	−0.0658056	−	0.997832i	$-0.520962\pi$
$311$	−2.32099	−0.131611	−0.0658056	+	0.997832i	$0.520962\pi$
$312$	0	0
$313$	−20.7599	−1.17342	−0.586710	−	0.809797i	$-0.699577\pi$
$313$	−20.7599	−1.17342	−0.586710	+	0.809797i	$0.699577\pi$
$314$	0	0
$315$	24.9358	1.40497
$316$	0	0
$317$	−20.0569	−1.12651	−0.563254	−	0.826284i	$-0.690451\pi$
$317$	−20.0569	−1.12651	−0.563254	+	0.826284i	$0.690451\pi$
$318$	0	0
$319$	−27.8470	−1.55913
$320$	0	0
$321$	32.4030	1.80856
$322$	0	0
$323$	2.25843	0.125662
$324$	0	0
$325$	14.4129	0.799482
$326$	0	0
$327$	−61.6809	−3.41096
$328$	0	0
$329$	12.9680	0.714951
$330$	0	0
$331$	6.88650	0.378516	0.189258	−	0.981927i	$-0.439392\pi$
$331$	6.88650	0.378516	0.189258	+	0.981927i	$0.439392\pi$
$332$	0	0
$333$	53.4642	2.92982
$334$	0	0
$335$	−16.1867	−0.884373
$336$	0	0
$337$	21.4533	1.16864	0.584319	−	0.811524i	$-0.301362\pi$
$337$	21.4533	1.16864	0.584319	+	0.811524i	$0.301362\pi$
$338$	0	0
$339$	−47.8494	−2.59882
$340$	0	0
$341$	−19.7590	−1.07001
$342$	0	0
$343$	−13.8923	−0.750116
$344$	0	0
$345$	36.0357	1.94010
$346$	0	0
$347$	30.4339	1.63378	0.816889	−	0.576795i	$-0.195697\pi$
$347$	30.4339	1.63378	0.816889	+	0.576795i	$0.195697\pi$
$348$	0	0
$349$	−6.36311	−0.340609	−0.170305	−	0.985391i	$-0.554475\pi$
$349$	−6.36311	−0.340609	−0.170305	+	0.985391i	$0.554475\pi$
$350$	0	0
$351$	41.9588	2.23960
$352$	0	0
$353$	20.8691	1.11075	0.555374	−	0.831601i	$-0.312575\pi$
$353$	20.8691	1.11075	0.555374	+	0.831601i	$0.312575\pi$
$354$	0	0
$355$	−13.8579	−0.735499
$356$	0	0
$357$	−21.0363	−1.11336
$358$	0	0
$359$	18.0539	0.952846	0.476423	−	0.879216i	$-0.341933\pi$
$359$	18.0539	0.952846	0.476423	+	0.879216i	$0.341933\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−35.6774	−1.87258
$364$	0	0
$365$	17.0507	0.892475
$366$	0	0
$367$	32.2995	1.68602	0.843011	−	0.537897i	$-0.180781\pi$
$367$	32.2995	1.68602	0.843011	+	0.537897i	$0.180781\pi$
$368$	0	0
$369$	−48.3888	−2.51902
$370$	0	0
$371$	−3.08066	−0.159940
$372$	0	0
$373$	−0.216499	−0.0112099	−0.00560496	−	0.999984i	$-0.501784\pi$
$373$	−0.216499	−0.0112099	−0.00560496	+	0.999984i	$0.501784\pi$
$374$	0	0
$375$	32.9262	1.70030
$376$	0	0
$377$	25.7584	1.32663
$378$	0	0
$379$	6.84016	0.351355	0.175678	−	0.984448i	$-0.443788\pi$
$379$	6.84016	0.351355	0.175678	+	0.984448i	$0.443788\pi$
$380$	0	0
$381$	23.1378	1.18538
$382$	0	0
$383$	−27.0814	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$383$	−27.0814	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$384$	0	0
$385$	19.3857	0.987989
$386$	0	0
$387$	−49.2607	−2.50406
$388$	0	0
$389$	25.7831	1.30726	0.653628	−	0.756816i	$-0.273246\pi$
$389$	25.7831	1.30726	0.653628	+	0.756816i	$0.273246\pi$
$390$	0	0
$391$	−20.4242	−1.03290
$392$	0	0
$393$	−0.559000	−0.0281978
$394$	0	0
$395$	−2.69806	−0.135754
$396$	0	0
$397$	−22.2611	−1.11725	−0.558627	−	0.829419i	$-0.688672\pi$
$397$	−22.2611	−1.11725	−0.558627	+	0.829419i	$0.688672\pi$
$398$	0	0
$399$	−9.31458	−0.466312
$400$	0	0
$401$	−10.4605	−0.522374	−0.261187	−	0.965288i	$-0.584114\pi$
$401$	−10.4605	−0.522374	−0.261187	+	0.965288i	$0.584114\pi$
$402$	0	0
$403$	18.2770	0.910443
$404$	0	0
$405$	13.5710	0.674348
$406$	0	0
$407$	41.5645	2.06027
$408$	0	0
$409$	7.48137	0.369930	0.184965	−	0.982745i	$-0.440783\pi$
$409$	7.48137	0.369930	0.184965	+	0.982745i	$0.440783\pi$
$410$	0	0
$411$	9.77289	0.482061
$412$	0	0
$413$	−24.2235	−1.19196
$414$	0	0
$415$	−4.44508	−0.218200
$416$	0	0
$417$	1.32508	0.0648896
$418$	0	0
$419$	9.97110	0.487120	0.243560	−	0.969886i	$-0.421685\pi$
$419$	9.97110	0.487120	0.243560	+	0.969886i	$0.421685\pi$
$420$	0	0
$421$	−16.1753	−0.788336	−0.394168	−	0.919038i	$-0.628967\pi$
$421$	−16.1753	−0.788336	−0.394168	+	0.919038i	$0.628967\pi$
$422$	0	0
$423$	25.8545	1.25709
$424$	0	0
$425$	−7.36972	−0.357484
$426$	0	0
$427$	−34.3572	−1.66266
$428$	0	0
$429$	63.7661	3.07866
$430$	0	0
$431$	0.752304	0.0362372	0.0181186	−	0.999836i	$-0.494232\pi$
$431$	0.752304	0.0362372	0.0181186	+	0.999836i	$0.494232\pi$
$432$	0	0
$433$	−10.2141	−0.490856	−0.245428	−	0.969415i	$-0.578929\pi$
$433$	−10.2141	−0.490856	−0.245428	+	0.969415i	$0.578929\pi$
$434$	0	0
$435$	23.2384	1.11420
$436$	0	0
$437$	−9.04358	−0.432613
$438$	0	0
$439$	29.0099	1.38456	0.692282	−	0.721627i	$-0.256605\pi$
$439$	29.0099	1.38456	0.692282	+	0.721627i	$0.256605\pi$
$440$	0	0
$441$	15.2963	0.728395
$442$	0	0
$443$	−25.6786	−1.22003	−0.610013	−	0.792392i	$-0.708836\pi$
$443$	−25.6786	−1.22003	−0.610013	+	0.792392i	$0.708836\pi$
$444$	0	0
$445$	16.9371	0.802896
$446$	0	0
$447$	50.7979	2.40266
$448$	0	0
$449$	28.6669	1.35288	0.676438	−	0.736500i	$-0.263523\pi$
$449$	28.6669	1.35288	0.676438	+	0.736500i	$0.263523\pi$
$450$	0	0
$451$	−37.6187	−1.77140
$452$	0	0
$453$	4.12418	0.193771
$454$	0	0
$455$	−17.9317	−0.840653
$456$	0	0
$457$	−3.55798	−0.166435	−0.0832176	−	0.996531i	$-0.526520\pi$
$457$	−3.55798	−0.166435	−0.0832176	+	0.996531i	$0.526520\pi$
$458$	0	0
$459$	−21.4548	−1.00142
$460$	0	0
$461$	16.1424	0.751826	0.375913	−	0.926655i	$-0.377329\pi$
$461$	16.1424	0.751826	0.375913	+	0.926655i	$0.377329\pi$
$462$	0	0
$463$	2.80264	0.130250	0.0651249	−	0.997877i	$-0.479255\pi$
$463$	2.80264	0.130250	0.0651249	+	0.997877i	$0.479255\pi$
$464$	0	0
$465$	16.4889	0.764656
$466$	0	0
$467$	19.4816	0.901501	0.450751	−	0.892650i	$-0.351156\pi$
$467$	19.4816	0.901501	0.450751	+	0.892650i	$0.351156\pi$
$468$	0	0
$469$	−37.8380	−1.74720
$470$	0	0
$471$	70.9572	3.26954
$472$	0	0
$473$	−38.2966	−1.76088
$474$	0	0
$475$	−3.26321	−0.149726
$476$	0	0
$477$	−6.14194	−0.281220
$478$	0	0
$479$	−37.5574	−1.71604	−0.858020	−	0.513616i	$-0.828305\pi$
$479$	−37.5574	−1.71604	−0.858020	+	0.513616i	$0.828305\pi$
$480$	0	0
$481$	−38.4470	−1.75303
$482$	0	0
$483$	84.2371	3.83292
$484$	0	0
$485$	−0.0142628	−0.000647640 0
$486$	0	0
$487$	−26.3562	−1.19431	−0.597156	−	0.802125i	$-0.703703\pi$
$487$	−26.3562	−1.19431	−0.597156	+	0.802125i	$0.703703\pi$
$488$	0	0
$489$	64.3161	2.90847
$490$	0	0
$491$	−7.33025	−0.330810	−0.165405	−	0.986226i	$-0.552893\pi$
$491$	−7.33025	−0.330810	−0.165405	+	0.986226i	$0.552893\pi$
$492$	0	0
$493$	−13.1710	−0.593193
$494$	0	0
$495$	38.6496	1.73717
$496$	0	0
$497$	−32.3941	−1.45308
$498$	0	0
$499$	−35.8125	−1.60319	−0.801594	−	0.597868i	$-0.796014\pi$
$499$	−35.8125	−1.60319	−0.801594	+	0.597868i	$0.796014\pi$
$500$	0	0
$501$	62.6465	2.79884
$502$	0	0
$503$	19.7911	0.882440	0.441220	−	0.897399i	$-0.354546\pi$
$503$	19.7911	0.882440	0.441220	+	0.897399i	$0.354546\pi$
$504$	0	0
$505$	23.6894	1.05416
$506$	0	0
$507$	−19.6770	−0.873888
$508$	0	0
$509$	20.1686	0.893957	0.446979	−	0.894545i	$-0.352500\pi$
$509$	20.1686	0.893957	0.446979	+	0.894545i	$0.352500\pi$
$510$	0	0
$511$	39.8577	1.76320
$512$	0	0
$513$	−9.49987	−0.419430
$514$	0	0
$515$	−2.58155	−0.113757
$516$	0	0
$517$	20.1000	0.883996
$518$	0	0
$519$	40.1423	1.76205
$520$	0	0
$521$	−25.1077	−1.09999	−0.549993	−	0.835169i	$-0.685370\pi$
$521$	−25.1077	−1.09999	−0.549993	+	0.835169i	$0.685370\pi$
$522$	0	0
$523$	−22.5825	−0.987465	−0.493732	−	0.869614i	$-0.664368\pi$
$523$	−22.5825	−0.987465	−0.493732	+	0.869614i	$0.664368\pi$
$524$	0	0
$525$	30.3954	1.32657
$526$	0	0
$527$	−9.34557	−0.407099
$528$	0	0
$529$	58.7863	2.55593
$530$	0	0
$531$	−48.2947	−2.09581
$532$	0	0
$533$	34.7972	1.50723
$534$	0	0
$535$	−14.1234	−0.610609
$536$	0	0
$537$	22.5168	0.971670
$538$	0	0
$539$	11.8918	0.512214
$540$	0	0
$541$	−1.01763	−0.0437515	−0.0218758	−	0.999761i	$-0.506964\pi$
$541$	−1.01763	−0.0437515	−0.0218758	+	0.999761i	$0.506964\pi$
$542$	0	0
$543$	−49.4051	−2.12018
$544$	0	0
$545$	26.8847	1.15161
$546$	0	0
$547$	−32.2381	−1.37840	−0.689201	−	0.724571i	$-0.742038\pi$
$547$	−32.2381	−1.37840	−0.689201	+	0.724571i	$0.742038\pi$
$548$	0	0
$549$	−68.4983	−2.92343
$550$	0	0
$551$	−5.83195	−0.248449
$552$	0	0
$553$	−6.30699	−0.268200
$554$	0	0
$555$	−34.6856	−1.47232
$556$	0	0
$557$	−35.3267	−1.49684	−0.748420	−	0.663225i	$-0.769187\pi$
$557$	−35.3267	−1.49684	−0.748420	+	0.663225i	$0.769187\pi$
$558$	0	0
$559$	35.4242	1.49828
$560$	0	0
$561$	−32.6055	−1.37660
$562$	0	0
$563$	17.1629	0.723329	0.361665	−	0.932308i	$-0.382209\pi$
$563$	17.1629	0.723329	0.361665	+	0.932308i	$0.382209\pi$
$564$	0	0
$565$	20.8560	0.877417
$566$	0	0
$567$	31.7236	1.33226
$568$	0	0
$569$	46.4004	1.94520	0.972602	−	0.232476i	$-0.0746826\pi$
$569$	46.4004	1.94520	0.972602	+	0.232476i	$0.0746826\pi$
$570$	0	0
$571$	−21.5410	−0.901461	−0.450730	−	0.892660i	$-0.648836\pi$
$571$	−21.5410	−0.901461	−0.450730	+	0.892660i	$0.648836\pi$
$572$	0	0
$573$	−62.3984	−2.60673
$574$	0	0
$575$	29.5111	1.23070
$576$	0	0
$577$	−6.80341	−0.283230	−0.141615	−	0.989922i	$-0.545229\pi$
$577$	−6.80341	−0.283230	−0.141615	+	0.989922i	$0.545229\pi$
$578$	0	0
$579$	−10.8868	−0.452441
$580$	0	0
$581$	−10.3908	−0.431084
$582$	0	0
$583$	−4.77491	−0.197757
$584$	0	0
$585$	−35.7507	−1.47811
$586$	0	0
$587$	7.64981	0.315741	0.157871	−	0.987460i	$-0.449537\pi$
$587$	7.64981	0.315741	0.157871	+	0.987460i	$0.449537\pi$
$588$	0	0
$589$	−4.13809	−0.170507
$590$	0	0
$591$	71.7138	2.94991
$592$	0	0
$593$	−19.3386	−0.794142	−0.397071	−	0.917788i	$-0.629973\pi$
$593$	−19.3386	−0.794142	−0.397071	+	0.917788i	$0.629973\pi$
$594$	0	0
$595$	9.16902	0.375893
$596$	0	0
$597$	−23.1850	−0.948899
$598$	0	0
$599$	−20.8336	−0.851237	−0.425619	−	0.904903i	$-0.639943\pi$
$599$	−20.8336	−0.851237	−0.425619	+	0.904903i	$0.639943\pi$
$600$	0	0
$601$	−15.2740	−0.623040	−0.311520	−	0.950240i	$-0.600838\pi$
$601$	−15.2740	−0.623040	−0.311520	+	0.950240i	$0.600838\pi$
$602$	0	0
$603$	−75.4381	−3.07208
$604$	0	0
$605$	15.5506	0.632222
$606$	0	0
$607$	−3.34532	−0.135782	−0.0678912	−	0.997693i	$-0.521627\pi$
$607$	−3.34532	−0.135782	−0.0678912	+	0.997693i	$0.521627\pi$
$608$	0	0
$609$	54.3221	2.20124
$610$	0	0
$611$	−18.5924	−0.752168
$612$	0	0
$613$	−28.2172	−1.13968	−0.569840	−	0.821756i	$-0.692995\pi$
$613$	−28.2172	−1.13968	−0.569840	+	0.821756i	$0.692995\pi$
$614$	0	0
$615$	31.3929	1.26588
$616$	0	0
$617$	33.1577	1.33488	0.667440	−	0.744663i	$-0.267390\pi$
$617$	33.1577	1.33488	0.667440	+	0.744663i	$0.267390\pi$
$618$	0	0
$619$	−0.771845	−0.0310231	−0.0155115	−	0.999880i	$-0.504938\pi$
$619$	−0.771845	−0.0310231	−0.0155115	+	0.999880i	$0.504938\pi$
$620$	0	0
$621$	85.9129	3.44756
$622$	0	0
$623$	39.5922	1.58623
$624$	0	0
$625$	1.96461	0.0785842
$626$	0	0
$627$	−14.4373	−0.576568
$628$	0	0
$629$	19.6591	0.783858
$630$	0	0
$631$	47.0814	1.87428	0.937140	−	0.348953i	$-0.113463\pi$
$631$	47.0814	1.87428	0.937140	+	0.348953i	$0.113463\pi$
$632$	0	0
$633$	11.0399	0.438797
$634$	0	0
$635$	−10.0850	−0.400211
$636$	0	0
$637$	−10.9998	−0.435829
$638$	0	0
$639$	−64.5845	−2.55492
$640$	0	0
$641$	−14.1913	−0.560523	−0.280261	−	0.959924i	$-0.590421\pi$
$641$	−14.1913	−0.560523	−0.280261	+	0.959924i	$0.590421\pi$
$642$	0	0
$643$	23.6246	0.931662	0.465831	−	0.884874i	$-0.345756\pi$
$643$	23.6246	0.931662	0.465831	+	0.884874i	$0.345756\pi$
$644$	0	0
$645$	31.9586	1.25837
$646$	0	0
$647$	13.7258	0.539618	0.269809	−	0.962914i	$-0.413039\pi$
$647$	13.7258	0.539618	0.269809	+	0.962914i	$0.413039\pi$
$648$	0	0
$649$	−37.5456	−1.47379
$650$	0	0
$651$	38.5446	1.51068
$652$	0	0
$653$	−19.0134	−0.744053	−0.372026	−	0.928222i	$-0.621337\pi$
$653$	−19.0134	−0.744053	−0.372026	+	0.928222i	$0.621337\pi$
$654$	0	0
$655$	0.243650	0.00952019
$656$	0	0
$657$	79.4648	3.10022
$658$	0	0
$659$	8.49782	0.331028	0.165514	−	0.986207i	$-0.447072\pi$
$659$	8.49782	0.331028	0.165514	+	0.986207i	$0.447072\pi$
$660$	0	0
$661$	−33.2513	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$661$	−33.2513	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$662$	0	0
$663$	30.1599	1.17131
$664$	0	0
$665$	4.05992	0.157437
$666$	0	0
$667$	52.7417	2.04217
$668$	0	0
$669$	−16.3998	−0.634053
$670$	0	0
$671$	−53.2524	−2.05579
$672$	0	0
$673$	−29.3151	−1.13001	−0.565007	−	0.825086i	$-0.691127\pi$
$673$	−29.3151	−1.13001	−0.565007	+	0.825086i	$0.691127\pi$
$674$	0	0
$675$	31.0001	1.19319
$676$	0	0
$677$	27.9638	1.07474	0.537369	−	0.843347i	$-0.319418\pi$
$677$	27.9638	1.07474	0.537369	+	0.843347i	$0.319418\pi$
$678$	0	0
$679$	−0.0333407	−0.00127950
$680$	0	0
$681$	−43.4928	−1.66665
$682$	0	0
$683$	6.84444	0.261895	0.130948	−	0.991389i	$-0.458198\pi$
$683$	6.84444	0.261895	0.130948	+	0.991389i	$0.458198\pi$
$684$	0	0
$685$	−4.25968	−0.162754
$686$	0	0
$687$	−16.4634	−0.628120
$688$	0	0
$689$	4.41677	0.168266
$690$	0	0
$691$	−14.8882	−0.566374	−0.283187	−	0.959065i	$-0.591392\pi$
$691$	−14.8882	−0.566374	−0.283187	+	0.959065i	$0.591392\pi$
$692$	0	0
$693$	90.3473	3.43201
$694$	0	0
$695$	−0.577561	−0.0219081
$696$	0	0
$697$	−17.7928	−0.673951
$698$	0	0
$699$	−4.62608	−0.174974
$700$	0	0
$701$	−4.45547	−0.168281	−0.0841403	−	0.996454i	$-0.526814\pi$
$701$	−4.45547	−0.168281	−0.0841403	+	0.996454i	$0.526814\pi$
$702$	0	0
$703$	8.70476	0.328306
$704$	0	0
$705$	−16.7735	−0.631726
$706$	0	0
$707$	55.3763	2.08264
$708$	0	0
$709$	4.80518	0.180462	0.0902312	−	0.995921i	$-0.471239\pi$
$709$	4.80518	0.180462	0.0902312	+	0.995921i	$0.471239\pi$
$710$	0	0
$711$	−12.5743	−0.471573
$712$	0	0
$713$	37.4231	1.40151
$714$	0	0
$715$	−27.7935	−1.03942
$716$	0	0
$717$	−6.08374	−0.227201
$718$	0	0
$719$	−2.37064	−0.0884099	−0.0442049	−	0.999022i	$-0.514075\pi$
$719$	−2.37064	−0.0884099	−0.0442049	+	0.999022i	$0.514075\pi$
$720$	0	0
$721$	−6.03464	−0.224742
$722$	0	0
$723$	−38.7740	−1.44202
$724$	0	0
$725$	19.0309	0.706789
$726$	0	0
$727$	−16.4340	−0.609503	−0.304751	−	0.952432i	$-0.598573\pi$
$727$	−16.4340	−0.609503	−0.304751	+	0.952432i	$0.598573\pi$
$728$	0	0
$729$	−22.9228	−0.848994
$730$	0	0
$731$	−18.1134	−0.669949
$732$	0	0
$733$	28.0316	1.03537	0.517686	−	0.855571i	$-0.326793\pi$
$733$	28.0316	1.03537	0.517686	+	0.855571i	$0.326793\pi$
$734$	0	0
$735$	−9.92369	−0.366041
$736$	0	0
$737$	−58.6476	−2.16031
$738$	0	0
$739$	−38.8386	−1.42870	−0.714351	−	0.699788i	$-0.753278\pi$
$739$	−38.8386	−1.42870	−0.714351	+	0.699788i	$0.753278\pi$
$740$	0	0
$741$	13.3544	0.490586
$742$	0	0
$743$	16.3925	0.601382	0.300691	−	0.953722i	$-0.402783\pi$
$743$	16.3925	0.601382	0.300691	+	0.953722i	$0.402783\pi$
$744$	0	0
$745$	−22.1411	−0.811189
$746$	0	0
$747$	−20.7163	−0.757970
$748$	0	0
$749$	−33.0149	−1.20634
$750$	0	0
$751$	−28.2860	−1.03217	−0.516085	−	0.856537i	$-0.672611\pi$
$751$	−28.2860	−1.03217	−0.516085	+	0.856537i	$0.672611\pi$
$752$	0	0
$753$	38.3306	1.39684
$754$	0	0
$755$	−1.79759	−0.0654211
$756$	0	0
$757$	−35.3492	−1.28479	−0.642395	−	0.766374i	$-0.722059\pi$
$757$	−35.3492	−1.28479	−0.642395	+	0.766374i	$0.722059\pi$
$758$	0	0
$759$	130.564	4.73919
$760$	0	0
$761$	23.3508	0.846465	0.423233	−	0.906021i	$-0.360895\pi$
$761$	23.3508	0.846465	0.423233	+	0.906021i	$0.360895\pi$
$762$	0	0
$763$	62.8457	2.27517
$764$	0	0
$765$	18.2804	0.660928
$766$	0	0
$767$	34.7295	1.25401
$768$	0	0
$769$	−24.7978	−0.894232	−0.447116	−	0.894476i	$-0.647549\pi$
$769$	−24.7978	−0.894232	−0.447116	+	0.894476i	$0.647549\pi$
$770$	0	0
$771$	−19.8115	−0.713495
$772$	0	0
$773$	21.2831	0.765501	0.382750	−	0.923852i	$-0.374977\pi$
$773$	21.2831	0.765501	0.382750	+	0.923852i	$0.374977\pi$
$774$	0	0
$775$	13.5035	0.485059
$776$	0	0
$777$	−81.0812	−2.90877
$778$	0	0
$779$	−7.87841	−0.282273
$780$	0	0
$781$	−50.2097	−1.79665
$782$	0	0
$783$	55.4028	1.97993
$784$	0	0
$785$	−30.9279	−1.10386
$786$	0	0
$787$	−4.74193	−0.169032	−0.0845158	−	0.996422i	$-0.526934\pi$
$787$	−4.74193	−0.169032	−0.0845158	+	0.996422i	$0.526934\pi$
$788$	0	0
$789$	22.5957	0.804428
$790$	0	0
$791$	48.7529	1.73345
$792$	0	0
$793$	49.2582	1.74921
$794$	0	0
$795$	3.98467	0.141322
$796$	0	0
$797$	26.8742	0.951933	0.475967	−	0.879463i	$-0.342098\pi$
$797$	26.8742	0.951933	0.475967	+	0.879463i	$0.342098\pi$
$798$	0	0
$799$	9.50684	0.336328
$800$	0	0
$801$	78.9354	2.78904
$802$	0	0
$803$	61.7781	2.18010
$804$	0	0
$805$	−36.7162	−1.29408
$806$	0	0
$807$	−45.0135	−1.58455
$808$	0	0
$809$	46.8623	1.64759	0.823796	−	0.566887i	$-0.191852\pi$
$809$	46.8623	1.64759	0.823796	+	0.566887i	$0.191852\pi$
$810$	0	0
$811$	33.2858	1.16882	0.584411	−	0.811458i	$-0.301326\pi$
$811$	33.2858	1.16882	0.584411	+	0.811458i	$0.301326\pi$
$812$	0	0
$813$	−84.0832	−2.94893
$814$	0	0
$815$	−28.0333	−0.981963
$816$	0	0
$817$	−8.02037	−0.280597
$818$	0	0
$819$	−83.5709	−2.92020
$820$	0	0
$821$	52.2873	1.82484	0.912419	−	0.409257i	$-0.134212\pi$
$821$	52.2873	1.82484	0.912419	+	0.409257i	$0.134212\pi$
$822$	0	0
$823$	50.5498	1.76206	0.881028	−	0.473064i	$-0.156852\pi$
$823$	50.5498	1.76206	0.881028	+	0.473064i	$0.156852\pi$
$824$	0	0
$825$	47.1118	1.64022
$826$	0	0
$827$	18.5837	0.646218	0.323109	−	0.946362i	$-0.395272\pi$
$827$	18.5837	0.646218	0.323109	+	0.946362i	$0.395272\pi$
$828$	0	0
$829$	52.7163	1.83091	0.915456	−	0.402419i	$-0.131830\pi$
$829$	52.7163	1.83091	0.915456	+	0.402419i	$0.131830\pi$
$830$	0	0
$831$	73.2850	2.54223
$832$	0	0
$833$	5.62453	0.194878
$834$	0	0
$835$	−27.3055	−0.944947
$836$	0	0
$837$	39.3113	1.35880
$838$	0	0
$839$	−36.1036	−1.24643	−0.623217	−	0.782049i	$-0.714175\pi$
$839$	−36.1036	−1.24643	−0.623217	+	0.782049i	$0.714175\pi$
$840$	0	0
$841$	5.01163	0.172815
$842$	0	0
$843$	−27.5879	−0.950179
$844$	0	0
$845$	8.57658	0.295043
$846$	0	0
$847$	36.3511	1.24904
$848$	0	0
$849$	−53.0296	−1.81997
$850$	0	0
$851$	−78.7222	−2.69856
$852$	0	0
$853$	−33.2557	−1.13865	−0.569327	−	0.822111i	$-0.692796\pi$
$853$	−33.2557	−1.13865	−0.569327	+	0.822111i	$0.692796\pi$
$854$	0	0
$855$	8.09430	0.276819
$856$	0	0
$857$	1.64528	0.0562018	0.0281009	−	0.999605i	$-0.491054\pi$
$857$	1.64528	0.0562018	0.0281009	+	0.999605i	$0.491054\pi$
$858$	0	0
$859$	−34.2237	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$859$	−34.2237	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$860$	0	0
$861$	73.3841	2.50092
$862$	0	0
$863$	1.15052	0.0391641	0.0195821	−	0.999808i	$-0.493766\pi$
$863$	1.15052	0.0391641	0.0195821	+	0.999808i	$0.493766\pi$
$864$	0	0
$865$	−17.4967	−0.594906
$866$	0	0
$867$	35.9790	1.22191
$868$	0	0
$869$	−9.77560	−0.331615
$870$	0	0
$871$	54.2488	1.83815
$872$	0	0
$873$	−0.0664718	−0.00224973
$874$	0	0
$875$	−33.5480	−1.13413
$876$	0	0
$877$	−6.61231	−0.223282	−0.111641	−	0.993749i	$-0.535611\pi$
$877$	−6.61231	−0.223282	−0.111641	+	0.993749i	$0.535611\pi$
$878$	0	0
$879$	31.1134	1.04943
$880$	0	0
$881$	−15.7835	−0.531761	−0.265881	−	0.964006i	$-0.585663\pi$
$881$	−15.7835	−0.531761	−0.265881	+	0.964006i	$0.585663\pi$
$882$	0	0
$883$	−36.1773	−1.21746	−0.608732	−	0.793376i	$-0.708321\pi$
$883$	−36.1773	−1.21746	−0.608732	+	0.793376i	$0.708321\pi$
$884$	0	0
$885$	31.3319	1.05321
$886$	0	0
$887$	6.75185	0.226705	0.113353	−	0.993555i	$-0.463841\pi$
$887$	6.75185	0.226705	0.113353	+	0.993555i	$0.463841\pi$
$888$	0	0
$889$	−23.5747	−0.790669
$890$	0	0
$891$	49.1704	1.64727
$892$	0	0
$893$	4.20950	0.140865
$894$	0	0
$895$	−9.81432	−0.328056
$896$	0	0
$897$	−120.772	−4.03245
$898$	0	0
$899$	24.1331	0.804885
$900$	0	0
$901$	−2.25843	−0.0752391
$902$	0	0
$903$	74.7063	2.48607
$904$	0	0
$905$	21.5341	0.715817
$906$	0	0
$907$	42.0658	1.39677	0.698386	−	0.715721i	$-0.253902\pi$
$907$	42.0658	1.39677	0.698386	+	0.715721i	$0.253902\pi$
$908$	0	0
$909$	110.404	3.66188
$910$	0	0
$911$	−39.6841	−1.31479	−0.657396	−	0.753545i	$-0.728342\pi$
$911$	−39.6841	−1.31479	−0.657396	+	0.753545i	$0.728342\pi$
$912$	0	0
$913$	−16.1054	−0.533011
$914$	0	0
$915$	44.4392	1.46912
$916$	0	0
$917$	0.569556	0.0188084
$918$	0	0
$919$	−52.9043	−1.74515	−0.872577	−	0.488477i	$-0.837553\pi$
$919$	−52.9043	−1.74515	−0.872577	+	0.488477i	$0.837553\pi$
$920$	0	0
$921$	74.9978	2.47126
$922$	0	0
$923$	46.4438	1.52872
$924$	0	0
$925$	−28.4055	−0.933967
$926$	0	0
$927$	−12.0313	−0.395161
$928$	0	0
$929$	38.2477	1.25487	0.627433	−	0.778671i	$-0.284106\pi$
$929$	38.2477	1.25487	0.627433	+	0.778671i	$0.284106\pi$
$930$	0	0
$931$	2.49047	0.0816217
$932$	0	0
$933$	7.01766	0.229748
$934$	0	0
$935$	14.2117	0.464771
$936$	0	0
$937$	29.3959	0.960322	0.480161	−	0.877180i	$-0.340578\pi$
$937$	29.3959	0.960322	0.480161	+	0.877180i	$0.340578\pi$
$938$	0	0
$939$	62.7690	2.04839
$940$	0	0
$941$	−23.2537	−0.758049	−0.379025	−	0.925387i	$-0.623740\pi$
$941$	−23.2537	−0.758049	−0.379025	+	0.925387i	$0.623740\pi$
$942$	0	0
$943$	71.2490	2.32019
$944$	0	0
$945$	−38.5687	−1.25464
$946$	0	0
$947$	25.8822	0.841057	0.420529	−	0.907279i	$-0.361845\pi$
$947$	25.8822	0.841057	0.420529	+	0.907279i	$0.361845\pi$
$948$	0	0
$949$	−57.1445	−1.85499
$950$	0	0
$951$	60.6434	1.96650
$952$	0	0
$953$	24.3429	0.788543	0.394271	−	0.918994i	$-0.370997\pi$
$953$	24.3429	0.788543	0.394271	+	0.918994i	$0.370997\pi$
$954$	0	0
$955$	27.1974	0.880088
$956$	0	0
$957$	84.1973	2.72171
$958$	0	0
$959$	−9.95743	−0.321542
$960$	0	0
$961$	−13.8762	−0.447620
$962$	0	0
$963$	−65.8222	−2.12109
$964$	0	0
$965$	4.74521	0.152754
$966$	0	0
$967$	−49.6378	−1.59624	−0.798122	−	0.602496i	$-0.794173\pi$
$967$	−49.6378	−1.59624	−0.798122	+	0.602496i	$0.794173\pi$
$968$	0	0
$969$	−6.82850	−0.219363
$970$	0	0
$971$	58.6138	1.88101	0.940503	−	0.339786i	$-0.110355\pi$
$971$	58.6138	1.88101	0.940503	+	0.339786i	$0.110355\pi$
$972$	0	0
$973$	−1.35011	−0.0432824
$974$	0	0
$975$	−43.5782	−1.39562
$976$	0	0
$977$	−35.5807	−1.13833	−0.569163	−	0.822224i	$-0.692733\pi$
$977$	−35.5807	−1.13833	−0.569163	+	0.822224i	$0.692733\pi$
$978$	0	0
$979$	61.3664	1.96128
$980$	0	0
$981$	125.296	4.00040
$982$	0	0
$983$	−28.4626	−0.907814	−0.453907	−	0.891049i	$-0.649970\pi$
$983$	−28.4626	−0.907814	−0.453907	+	0.891049i	$0.649970\pi$
$984$	0	0
$985$	−31.2577	−0.995954
$986$	0	0
$987$	−39.2097	−1.24806
$988$	0	0
$989$	72.5328	2.30641
$990$	0	0
$991$	−28.3720	−0.901267	−0.450633	−	0.892709i	$-0.648802\pi$
$991$	−28.3720	−0.901267	−0.450633	+	0.892709i	$0.648802\pi$
$992$	0	0
$993$	−20.8218	−0.660759
$994$	0	0
$995$	10.1056	0.320369
$996$	0	0
$997$	2.22420	0.0704412	0.0352206	−	0.999380i	$-0.488787\pi$
$997$	2.22420	0.0704412	0.0352206	+	0.999380i	$0.488787\pi$
$998$	0	0
$999$	−82.6942	−2.61633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.c.1.3	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.c.1.3	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q-3.02356 q^{3} +1.31787 q^{5} +3.08066 q^{7} +6.14194 q^{9} +O(q^{10})\)	\(q-3.02356 q^{3} +1.31787 q^{5} +3.08066 q^{7} +6.14194 q^{9} +4.77491 q^{11} -4.41677 q^{13} -3.98467 q^{15} +2.25843 q^{17} +1.00000 q^{19} -9.31458 q^{21} -9.04358 q^{23} -3.26321 q^{25} -9.49987 q^{27} -5.83195 q^{29} -4.13809 q^{31} -14.4373 q^{33} +4.05992 q^{35} +8.70476 q^{37} +13.3544 q^{39} -7.87841 q^{41} -8.02037 q^{43} +8.09430 q^{45} +4.20950 q^{47} +2.49047 q^{49} -6.82850 q^{51} -1.00000 q^{53} +6.29273 q^{55} -3.02356 q^{57} -7.86310 q^{59} -11.1525 q^{61} +18.9212 q^{63} -5.82075 q^{65} -12.2824 q^{67} +27.3438 q^{69} -10.5153 q^{71} +12.9381 q^{73} +9.86653 q^{75} +14.7099 q^{77} -2.04728 q^{79} +10.2977 q^{81} -3.37292 q^{83} +2.97632 q^{85} +17.6333 q^{87} +12.8519 q^{89} -13.6066 q^{91} +12.5118 q^{93} +1.31787 q^{95} -0.0108226 q^{97} +29.3272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9} - 3 q^{11} - 23 q^{13} - 18 q^{15} - 7 q^{17} + 19 q^{19} - 4 q^{21} - 6 q^{23} + 18 q^{25} - 17 q^{27} - 4 q^{29} - 30 q^{31} - 10 q^{33} - q^{35} - 31 q^{37} + 5 q^{39} - 15 q^{41} - 29 q^{43} + 6 q^{45} - 18 q^{47} + 23 q^{49} - 5 q^{51} - 19 q^{53} - 19 q^{55} - 5 q^{57} + 8 q^{59} - 4 q^{61} - 64 q^{63} - 26 q^{65} - 62 q^{67} + 3 q^{69} - 17 q^{71} + q^{73} - 40 q^{75} - 14 q^{77} - 28 q^{79} + 11 q^{81} + 4 q^{83} - 31 q^{85} - 20 q^{87} + 33 q^{89} - 29 q^{91} - 59 q^{93} - 5 q^{95} + 5 q^{97} - 48 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 - 5 * q^5 - 10 * q^7 + 22 * q^9 - 3 * q^11 - 23 * q^13 - 18 * q^15 - 7 * q^17 + 19 * q^19 - 4 * q^21 - 6 * q^23 + 18 * q^25 - 17 * q^27 - 4 * q^29 - 30 * q^31 - 10 * q^33 - q^35 - 31 * q^37 + 5 * q^39 - 15 * q^41 - 29 * q^43 + 6 * q^45 - 18 * q^47 + 23 * q^49 - 5 * q^51 - 19 * q^53 - 19 * q^55 - 5 * q^57 + 8 * q^59 - 4 * q^61 - 64 * q^63 - 26 * q^65 - 62 * q^67 + 3 * q^69 - 17 * q^71 + q^73 - 40 * q^75 - 14 * q^77 - 28 * q^79 + 11 * q^81 + 4 * q^83 - 31 * q^85 - 20 * q^87 + 33 * q^89 - 29 * q^91 - 59 * q^93 - 5 * q^95 + 5 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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