LMFDB - Embedded newform 4028.2.a.c.1.2

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			1.2
Root			$3.10589$ 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)

$f(q)$	$=$	$q-3.10589 q^{3} -2.95976 q^{5} -2.66591 q^{7} +6.64656 q^{9} +O(q^{10})$	$q-3.10589 q^{3} -2.95976 q^{5} -2.66591 q^{7} +6.64656 q^{9} -3.12881 q^{11} +0.107241 q^{13} +9.19269 q^{15} -5.45908 q^{17} +1.00000 q^{19} +8.28003 q^{21} -3.74885 q^{23} +3.76018 q^{25} -11.3258 q^{27} +8.50772 q^{29} +6.30276 q^{31} +9.71776 q^{33} +7.89046 q^{35} +0.101395 q^{37} -0.333079 q^{39} +11.4554 q^{41} -5.02268 q^{43} -19.6722 q^{45} +3.92122 q^{47} +0.107082 q^{49} +16.9553 q^{51} -1.00000 q^{53} +9.26054 q^{55} -3.10589 q^{57} +5.15291 q^{59} -5.66639 q^{61} -17.7191 q^{63} -0.317407 q^{65} -12.7626 q^{67} +11.6435 q^{69} +7.58556 q^{71} +4.93858 q^{73} -11.6787 q^{75} +8.34114 q^{77} +3.68621 q^{79} +15.2371 q^{81} -1.48357 q^{83} +16.1576 q^{85} -26.4241 q^{87} +14.5590 q^{89} -0.285895 q^{91} -19.5757 q^{93} -2.95976 q^{95} +16.8886 q^{97} -20.7959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.10589	−1.79319	−0.896594	−	0.442854i	$-0.853966\pi$
$3$	−3.10589	−1.79319	−0.896594	+	0.442854i	$0.853966\pi$
$4$	0	0
$5$	−2.95976	−1.32364	−0.661822	−	0.749661i	$-0.730217\pi$
$5$	−2.95976	−1.32364	−0.661822	+	0.749661i	$0.730217\pi$
$6$	0	0
$7$	−2.66591	−1.00762	−0.503810	−	0.863815i	$-0.668069\pi$
$7$	−2.66591	−1.00762	−0.503810	+	0.863815i	$0.668069\pi$
$8$	0	0
$9$	6.64656	2.21552
$10$	0	0
$11$	−3.12881	−0.943373	−0.471687	−	0.881766i	$-0.656355\pi$
$11$	−3.12881	−0.943373	−0.471687	+	0.881766i	$0.656355\pi$
$12$	0	0
$13$	0.107241	0.0297433	0.0148716	−	0.999889i	$-0.495266\pi$
$13$	0.107241	0.0297433	0.0148716	+	0.999889i	$0.495266\pi$
$14$	0	0
$15$	9.19269	2.37354
$16$	0	0
$17$	−5.45908	−1.32402	−0.662011	−	0.749494i	$-0.730297\pi$
$17$	−5.45908	−1.32402	−0.662011	+	0.749494i	$0.730297\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.28003	1.80685
$22$	0	0
$23$	−3.74885	−0.781690	−0.390845	−	0.920457i	$-0.627817\pi$
$23$	−3.74885	−0.781690	−0.390845	+	0.920457i	$0.627817\pi$
$24$	0	0
$25$	3.76018	0.752036
$26$	0	0
$27$	−11.3258	−2.17965
$28$	0	0
$29$	8.50772	1.57984	0.789922	−	0.613207i	$-0.210121\pi$
$29$	8.50772	1.57984	0.789922	+	0.613207i	$0.210121\pi$
$30$	0	0
$31$	6.30276	1.13201	0.566005	−	0.824402i	$-0.308488\pi$
$31$	6.30276	1.13201	0.566005	+	0.824402i	$0.308488\pi$
$32$	0	0
$33$	9.71776	1.69164
$34$	0	0
$35$	7.89046	1.33373
$36$	0	0
$37$	0.101395	0.0166693	0.00833463	−	0.999965i	$-0.497347\pi$
$37$	0.101395	0.0166693	0.00833463	+	0.999965i	$0.497347\pi$
$38$	0	0
$39$	−0.333079	−0.0533353
$40$	0	0
$41$	11.4554	1.78903	0.894515	−	0.447037i	$-0.147521\pi$
$41$	11.4554	1.78903	0.894515	+	0.447037i	$0.147521\pi$
$42$	0	0
$43$	−5.02268	−0.765951	−0.382975	−	0.923759i	$-0.625101\pi$
$43$	−5.02268	−0.765951	−0.382975	+	0.923759i	$0.625101\pi$
$44$	0	0
$45$	−19.6722	−2.93256
$46$	0	0
$47$	3.92122	0.571968	0.285984	−	0.958234i	$-0.407680\pi$
$47$	3.92122	0.571968	0.285984	+	0.958234i	$0.407680\pi$
$48$	0	0
$49$	0.107082	0.0152974
$50$	0	0
$51$	16.9553	2.37422
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	9.26054	1.24869
$56$	0	0
$57$	−3.10589	−0.411385
$58$	0	0
$59$	5.15291	0.670852	0.335426	−	0.942067i	$-0.391120\pi$
$59$	5.15291	0.670852	0.335426	+	0.942067i	$0.391120\pi$
$60$	0	0
$61$	−5.66639	−0.725507	−0.362754	−	0.931885i	$-0.618163\pi$
$61$	−5.66639	−0.725507	−0.362754	+	0.931885i	$0.618163\pi$
$62$	0	0
$63$	−17.7191	−2.23240
$64$	0	0
$65$	−0.317407	−0.0393695
$66$	0	0
$67$	−12.7626	−1.55920	−0.779598	−	0.626280i	$-0.784577\pi$
$67$	−12.7626	−1.55920	−0.779598	+	0.626280i	$0.784577\pi$
$68$	0	0
$69$	11.6435	1.40172
$70$	0	0
$71$	7.58556	0.900241	0.450120	−	0.892968i	$-0.351381\pi$
$71$	7.58556	0.900241	0.450120	+	0.892968i	$0.351381\pi$
$72$	0	0
$73$	4.93858	0.578017	0.289008	−	0.957327i	$-0.406674\pi$
$73$	4.93858	0.578017	0.289008	+	0.957327i	$0.406674\pi$
$74$	0	0
$75$	−11.6787	−1.34854
$76$	0	0
$77$	8.34114	0.950561
$78$	0	0
$79$	3.68621	0.414731	0.207366	−	0.978264i	$-0.433511\pi$
$79$	3.68621	0.414731	0.207366	+	0.978264i	$0.433511\pi$
$80$	0	0
$81$	15.2371	1.69301
$82$	0	0
$83$	−1.48357	−0.162843	−0.0814217	−	0.996680i	$-0.525946\pi$
$83$	−1.48357	−0.162843	−0.0814217	+	0.996680i	$0.525946\pi$
$84$	0	0
$85$	16.1576	1.75254
$86$	0	0
$87$	−26.4241	−2.83296
$88$	0	0
$89$	14.5590	1.54325	0.771625	−	0.636078i	$-0.219444\pi$
$89$	14.5590	1.54325	0.771625	+	0.636078i	$0.219444\pi$
$90$	0	0
$91$	−0.285895	−0.0299699
$92$	0	0
$93$	−19.5757	−2.02990
$94$	0	0
$95$	−2.95976	−0.303665
$96$	0	0
$97$	16.8886	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$97$	16.8886	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$98$	0	0
$99$	−20.7959	−2.09006
$100$	0	0
$101$	7.83282	0.779394	0.389697	−	0.920943i	$-0.372580\pi$
$101$	7.83282	0.779394	0.389697	+	0.920943i	$0.372580\pi$
$102$	0	0
$103$	−4.36838	−0.430430	−0.215215	−	0.976567i	$-0.569045\pi$
$103$	−4.36838	−0.430430	−0.215215	+	0.976567i	$0.569045\pi$
$104$	0	0
$105$	−24.5069	−2.39163
$106$	0	0
$107$	7.50777	0.725804	0.362902	−	0.931827i	$-0.381786\pi$
$107$	7.50777	0.725804	0.362902	+	0.931827i	$0.381786\pi$
$108$	0	0
$109$	4.33146	0.414878	0.207439	−	0.978248i	$-0.433487\pi$
$109$	4.33146	0.414878	0.207439	+	0.978248i	$0.433487\pi$
$110$	0	0
$111$	−0.314922	−0.0298911
$112$	0	0
$113$	8.89312	0.836594	0.418297	−	0.908310i	$-0.362627\pi$
$113$	8.89312	0.836594	0.418297	+	0.908310i	$0.362627\pi$
$114$	0	0
$115$	11.0957	1.03468
$116$	0	0
$117$	0.712783	0.0658968
$118$	0	0
$119$	14.5534	1.33411
$120$	0	0
$121$	−1.21052	−0.110047
$122$	0	0
$123$	−35.5792	−3.20807
$124$	0	0
$125$	3.66957	0.328216
$126$	0	0
$127$	−13.9511	−1.23796	−0.618981	−	0.785406i	$-0.712454\pi$
$127$	−13.9511	−1.23796	−0.618981	+	0.785406i	$0.712454\pi$
$128$	0	0
$129$	15.5999	1.37349
$130$	0	0
$131$	9.06046	0.791616	0.395808	−	0.918333i	$-0.370465\pi$
$131$	9.06046	0.791616	0.395808	+	0.918333i	$0.370465\pi$
$132$	0	0
$133$	−2.66591	−0.231164
$134$	0	0
$135$	33.5217	2.88509
$136$	0	0
$137$	−10.9266	−0.933525	−0.466763	−	0.884383i	$-0.654580\pi$
$137$	−10.9266	−0.933525	−0.466763	+	0.884383i	$0.654580\pi$
$138$	0	0
$139$	−4.31154	−0.365700	−0.182850	−	0.983141i	$-0.558532\pi$
$139$	−4.31154	−0.365700	−0.182850	+	0.983141i	$0.558532\pi$
$140$	0	0
$141$	−12.1789	−1.02565
$142$	0	0
$143$	−0.335537	−0.0280590
$144$	0	0
$145$	−25.1808	−2.09115
$146$	0	0
$147$	−0.332585	−0.0274311
$148$	0	0
$149$	−14.6462	−1.19986	−0.599932	−	0.800051i	$-0.704806\pi$
$149$	−14.6462	−1.19986	−0.599932	+	0.800051i	$0.704806\pi$
$150$	0	0
$151$	−10.6307	−0.865116	−0.432558	−	0.901606i	$-0.642389\pi$
$151$	−10.6307	−0.865116	−0.432558	+	0.901606i	$0.642389\pi$
$152$	0	0
$153$	−36.2841	−2.93340
$154$	0	0
$155$	−18.6547	−1.49838
$156$	0	0
$157$	−2.73357	−0.218163	−0.109081	−	0.994033i	$-0.534791\pi$
$157$	−2.73357	−0.218163	−0.109081	+	0.994033i	$0.534791\pi$
$158$	0	0
$159$	3.10589	0.246313
$160$	0	0
$161$	9.99411	0.787646
$162$	0	0
$163$	−12.3037	−0.963701	−0.481850	−	0.876253i	$-0.660035\pi$
$163$	−12.3037	−0.963701	−0.481850	+	0.876253i	$0.660035\pi$
$164$	0	0
$165$	−28.7622	−2.23914
$166$	0	0
$167$	15.6261	1.20918	0.604591	−	0.796536i	$-0.293336\pi$
$167$	15.6261	1.20918	0.604591	+	0.796536i	$0.293336\pi$
$168$	0	0
$169$	−12.9885	−0.999115
$170$	0	0
$171$	6.64656	0.508275
$172$	0	0
$173$	−10.0006	−0.760330	−0.380165	−	0.924919i	$-0.624133\pi$
$173$	−10.0006	−0.760330	−0.380165	+	0.924919i	$0.624133\pi$
$174$	0	0
$175$	−10.0243	−0.757766
$176$	0	0
$177$	−16.0044	−1.20296
$178$	0	0
$179$	−23.7336	−1.77393	−0.886966	−	0.461834i	$-0.847192\pi$
$179$	−23.7336	−1.77393	−0.886966	+	0.461834i	$0.847192\pi$
$180$	0	0
$181$	−18.2872	−1.35927	−0.679637	−	0.733549i	$-0.737863\pi$
$181$	−18.2872	−1.35927	−0.679637	+	0.733549i	$0.737863\pi$
$182$	0	0
$183$	17.5992	1.30097
$184$	0	0
$185$	−0.300105	−0.0220642
$186$	0	0
$187$	17.0805	1.24905
$188$	0	0
$189$	30.1936	2.19626
$190$	0	0
$191$	5.72146	0.413990	0.206995	−	0.978342i	$-0.433632\pi$
$191$	5.72146	0.413990	0.206995	+	0.978342i	$0.433632\pi$
$192$	0	0
$193$	13.4859	0.970735	0.485368	−	0.874310i	$-0.338686\pi$
$193$	13.4859	0.970735	0.485368	+	0.874310i	$0.338686\pi$
$194$	0	0
$195$	0.985833	0.0705969
$196$	0	0
$197$	16.0697	1.14492	0.572460	−	0.819932i	$-0.305989\pi$
$197$	16.0697	1.14492	0.572460	+	0.819932i	$0.305989\pi$
$198$	0	0
$199$	−24.4745	−1.73495	−0.867475	−	0.497482i	$-0.834258\pi$
$199$	−24.4745	−1.73495	−0.867475	+	0.497482i	$0.834258\pi$
$200$	0	0
$201$	39.6391	2.79593
$202$	0	0
$203$	−22.6808	−1.59188
$204$	0	0
$205$	−33.9052	−2.36804
$206$	0	0
$207$	−24.9170	−1.73185
$208$	0	0
$209$	−3.12881	−0.216425
$210$	0	0
$211$	−19.7878	−1.36225	−0.681124	−	0.732168i	$-0.738508\pi$
$211$	−19.7878	−1.36225	−0.681124	+	0.732168i	$0.738508\pi$
$212$	0	0
$213$	−23.5599	−1.61430
$214$	0	0
$215$	14.8659	1.01385
$216$	0	0
$217$	−16.8026	−1.14063
$218$	0	0
$219$	−15.3387	−1.03649
$220$	0	0
$221$	−0.585437	−0.0393808
$222$	0	0
$223$	7.18495	0.481140	0.240570	−	0.970632i	$-0.422666\pi$
$223$	7.18495	0.481140	0.240570	+	0.970632i	$0.422666\pi$
$224$	0	0
$225$	24.9923	1.66615
$226$	0	0
$227$	10.3790	0.688878	0.344439	−	0.938809i	$-0.388069\pi$
$227$	10.3790	0.688878	0.344439	+	0.938809i	$0.388069\pi$
$228$	0	0
$229$	−4.66969	−0.308581	−0.154291	−	0.988025i	$-0.549309\pi$
$229$	−4.66969	−0.308581	−0.154291	+	0.988025i	$0.549309\pi$
$230$	0	0
$231$	−25.9067	−1.70453
$232$	0	0
$233$	−18.3181	−1.20006	−0.600030	−	0.799977i	$-0.704845\pi$
$233$	−18.3181	−1.20006	−0.600030	+	0.799977i	$0.704845\pi$
$234$	0	0
$235$	−11.6059	−0.757083
$236$	0	0
$237$	−11.4490	−0.743691
$238$	0	0
$239$	−1.74886	−0.113124	−0.0565621	−	0.998399i	$-0.518014\pi$
$239$	−1.74886	−0.113124	−0.0565621	+	0.998399i	$0.518014\pi$
$240$	0	0
$241$	−0.551095	−0.0354992	−0.0177496	−	0.999842i	$-0.505650\pi$
$241$	−0.551095	−0.0354992	−0.0177496	+	0.999842i	$0.505650\pi$
$242$	0	0
$243$	−13.3473	−0.856227
$244$	0	0
$245$	−0.316937	−0.0202484
$246$	0	0
$247$	0.107241	0.00682357
$248$	0	0
$249$	4.60782	0.292009
$250$	0	0
$251$	−10.7264	−0.677044	−0.338522	−	0.940958i	$-0.609927\pi$
$251$	−10.7264	−0.677044	−0.338522	+	0.940958i	$0.609927\pi$
$252$	0	0
$253$	11.7295	0.737425
$254$	0	0
$255$	−50.1837	−3.14262
$256$	0	0
$257$	22.8553	1.42567	0.712837	−	0.701330i	$-0.247410\pi$
$257$	22.8553	1.42567	0.712837	+	0.701330i	$0.247410\pi$
$258$	0	0
$259$	−0.270310	−0.0167963
$260$	0	0
$261$	56.5471	3.50018
$262$	0	0
$263$	24.7444	1.52581	0.762903	−	0.646513i	$-0.223773\pi$
$263$	24.7444	1.52581	0.762903	+	0.646513i	$0.223773\pi$
$264$	0	0
$265$	2.95976	0.181817
$266$	0	0
$267$	−45.2187	−2.76734
$268$	0	0
$269$	9.50839	0.579737	0.289868	−	0.957066i	$-0.406388\pi$
$269$	9.50839	0.579737	0.289868	+	0.957066i	$0.406388\pi$
$270$	0	0
$271$	25.0322	1.52060	0.760299	−	0.649573i	$-0.225052\pi$
$271$	25.0322	1.52060	0.760299	+	0.649573i	$0.225052\pi$
$272$	0	0
$273$	0.887958	0.0537417
$274$	0	0
$275$	−11.7649	−0.709450
$276$	0	0
$277$	13.2776	0.797772	0.398886	−	0.917000i	$-0.369397\pi$
$277$	13.2776	0.797772	0.398886	+	0.917000i	$0.369397\pi$
$278$	0	0
$279$	41.8917	2.50799
$280$	0	0
$281$	−32.8758	−1.96121	−0.980604	−	0.195999i	$-0.937205\pi$
$281$	−32.8758	−1.96121	−0.980604	+	0.195999i	$0.937205\pi$
$282$	0	0
$283$	−12.5520	−0.746138	−0.373069	−	0.927804i	$-0.621694\pi$
$283$	−12.5520	−0.746138	−0.373069	+	0.927804i	$0.621694\pi$
$284$	0	0
$285$	9.19269	0.544528
$286$	0	0
$287$	−30.5390	−1.80266
$288$	0	0
$289$	12.8016	0.753035
$290$	0	0
$291$	−52.4541	−3.07491
$292$	0	0
$293$	−3.49998	−0.204471	−0.102235	−	0.994760i	$-0.532599\pi$
$293$	−3.49998	−0.204471	−0.102235	+	0.994760i	$0.532599\pi$
$294$	0	0
$295$	−15.2514	−0.887970
$296$	0	0
$297$	35.4364	2.05623
$298$	0	0
$299$	−0.402030	−0.0232500
$300$	0	0
$301$	13.3900	0.771787
$302$	0	0
$303$	−24.3279	−1.39760
$304$	0	0
$305$	16.7712	0.960314
$306$	0	0
$307$	22.2753	1.27132	0.635659	−	0.771970i	$-0.280728\pi$
$307$	22.2753	1.27132	0.635659	+	0.771970i	$0.280728\pi$
$308$	0	0
$309$	13.5677	0.771841
$310$	0	0
$311$	−6.20327	−0.351755	−0.175878	−	0.984412i	$-0.556276\pi$
$311$	−6.20327	−0.351755	−0.175878	+	0.984412i	$0.556276\pi$
$312$	0	0
$313$	−1.32225	−0.0747383	−0.0373691	−	0.999302i	$-0.511898\pi$
$313$	−1.32225	−0.0747383	−0.0373691	+	0.999302i	$0.511898\pi$
$314$	0	0
$315$	52.4444	2.95491
$316$	0	0
$317$	23.5210	1.32107	0.660536	−	0.750795i	$-0.270329\pi$
$317$	23.5210	1.32107	0.660536	+	0.750795i	$0.270329\pi$
$318$	0	0
$319$	−26.6191	−1.49038
$320$	0	0
$321$	−23.3183	−1.30150
$322$	0	0
$323$	−5.45908	−0.303752
$324$	0	0
$325$	0.403245	0.0223680
$326$	0	0
$327$	−13.4530	−0.743954
$328$	0	0
$329$	−10.4536	−0.576327
$330$	0	0
$331$	−15.4213	−0.847633	−0.423816	−	0.905748i	$-0.639310\pi$
$331$	−15.4213	−0.847633	−0.423816	+	0.905748i	$0.639310\pi$
$332$	0	0
$333$	0.673929	0.0369311
$334$	0	0
$335$	37.7741	2.06382
$336$	0	0
$337$	4.70124	0.256093	0.128047	−	0.991768i	$-0.459129\pi$
$337$	4.70124	0.256093	0.128047	+	0.991768i	$0.459129\pi$
$338$	0	0
$339$	−27.6210	−1.50017
$340$	0	0
$341$	−19.7202	−1.06791
$342$	0	0
$343$	18.3759	0.992206
$344$	0	0
$345$	−34.4620	−1.85537
$346$	0	0
$347$	2.49256	0.133808	0.0669038	−	0.997759i	$-0.478688\pi$
$347$	2.49256	0.133808	0.0669038	+	0.997759i	$0.478688\pi$
$348$	0	0
$349$	1.95888	0.104857	0.0524283	−	0.998625i	$-0.483304\pi$
$349$	1.95888	0.104857	0.0524283	+	0.998625i	$0.483304\pi$
$350$	0	0
$351$	−1.21459	−0.0648301
$352$	0	0
$353$	−13.7123	−0.729830	−0.364915	−	0.931041i	$-0.618902\pi$
$353$	−13.7123	−0.729830	−0.364915	+	0.931041i	$0.618902\pi$
$354$	0	0
$355$	−22.4515	−1.19160
$356$	0	0
$357$	−45.2014	−2.39231
$358$	0	0
$359$	−21.9749	−1.15979	−0.579896	−	0.814690i	$-0.696907\pi$
$359$	−21.9749	−1.15979	−0.579896	+	0.814690i	$0.696907\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.75975	0.197335
$364$	0	0
$365$	−14.6170	−0.765089
$366$	0	0
$367$	0.436354	0.0227775	0.0113887	−	0.999935i	$-0.496375\pi$
$367$	0.436354	0.0227775	0.0113887	+	0.999935i	$0.496375\pi$
$368$	0	0
$369$	76.1389	3.96363
$370$	0	0
$371$	2.66591	0.138407
$372$	0	0
$373$	−13.7232	−0.710561	−0.355281	−	0.934760i	$-0.615615\pi$
$373$	−13.7232	−0.710561	−0.355281	+	0.934760i	$0.615615\pi$
$374$	0	0
$375$	−11.3973	−0.588553
$376$	0	0
$377$	0.912376	0.0469897
$378$	0	0
$379$	−12.6178	−0.648131	−0.324066	−	0.946035i	$-0.605050\pi$
$379$	−12.6178	−0.648131	−0.324066	+	0.946035i	$0.605050\pi$
$380$	0	0
$381$	43.3306	2.21990
$382$	0	0
$383$	−22.8139	−1.16574	−0.582869	−	0.812566i	$-0.698070\pi$
$383$	−22.8139	−1.16574	−0.582869	+	0.812566i	$0.698070\pi$
$384$	0	0
$385$	−24.6878	−1.25821
$386$	0	0
$387$	−33.3835	−1.69698
$388$	0	0
$389$	−5.52584	−0.280171	−0.140086	−	0.990139i	$-0.544738\pi$
$389$	−5.52584	−0.280171	−0.140086	+	0.990139i	$0.544738\pi$
$390$	0	0
$391$	20.4653	1.03497
$392$	0	0
$393$	−28.1408	−1.41952
$394$	0	0
$395$	−10.9103	−0.548957
$396$	0	0
$397$	−3.37736	−0.169505	−0.0847525	−	0.996402i	$-0.527010\pi$
$397$	−3.37736	−0.169505	−0.0847525	+	0.996402i	$0.527010\pi$
$398$	0	0
$399$	8.28003	0.414520
$400$	0	0
$401$	32.2325	1.60961	0.804807	−	0.593537i	$-0.202269\pi$
$401$	32.2325	1.60961	0.804807	+	0.593537i	$0.202269\pi$
$402$	0	0
$403$	0.675914	0.0336697
$404$	0	0
$405$	−45.0981	−2.24094
$406$	0	0
$407$	−0.317246	−0.0157253
$408$	0	0
$409$	17.1893	0.849955	0.424978	−	0.905204i	$-0.360282\pi$
$409$	17.1893	0.849955	0.424978	+	0.905204i	$0.360282\pi$
$410$	0	0
$411$	33.9369	1.67399
$412$	0	0
$413$	−13.7372	−0.675963
$414$	0	0
$415$	4.39102	0.215547
$416$	0	0
$417$	13.3912	0.655768
$418$	0	0
$419$	24.3976	1.19190	0.595951	−	0.803021i	$-0.296775\pi$
$419$	24.3976	1.19190	0.595951	+	0.803021i	$0.296775\pi$
$420$	0	0
$421$	3.59765	0.175339	0.0876695	−	0.996150i	$-0.472058\pi$
$421$	3.59765	0.175339	0.0876695	+	0.996150i	$0.472058\pi$
$422$	0	0
$423$	26.0626	1.26721
$424$	0	0
$425$	−20.5271	−0.995712
$426$	0	0
$427$	15.1061	0.731036
$428$	0	0
$429$	1.04214	0.0503150
$430$	0	0
$431$	−30.6455	−1.47614	−0.738070	−	0.674724i	$-0.764263\pi$
$431$	−30.6455	−1.47614	−0.738070	+	0.674724i	$0.764263\pi$
$432$	0	0
$433$	35.5713	1.70945	0.854723	−	0.519084i	$-0.173727\pi$
$433$	35.5713	1.70945	0.854723	+	0.519084i	$0.173727\pi$
$434$	0	0
$435$	78.2089	3.74983
$436$	0	0
$437$	−3.74885	−0.179332
$438$	0	0
$439$	9.25900	0.441908	0.220954	−	0.975284i	$-0.429083\pi$
$439$	9.25900	0.441908	0.220954	+	0.975284i	$0.429083\pi$
$440$	0	0
$441$	0.711727	0.0338917
$442$	0	0
$443$	27.7389	1.31792	0.658959	−	0.752179i	$-0.270997\pi$
$443$	27.7389	1.31792	0.658959	+	0.752179i	$0.270997\pi$
$444$	0	0
$445$	−43.0911	−2.04272
$446$	0	0
$447$	45.4895	2.15158
$448$	0	0
$449$	18.7860	0.886566	0.443283	−	0.896382i	$-0.353814\pi$
$449$	18.7860	0.886566	0.443283	+	0.896382i	$0.353814\pi$
$450$	0	0
$451$	−35.8418	−1.68772
$452$	0	0
$453$	33.0179	1.55132
$454$	0	0
$455$	0.846180	0.0396695
$456$	0	0
$457$	−9.09973	−0.425668	−0.212834	−	0.977088i	$-0.568269\pi$
$457$	−9.09973	−0.425668	−0.212834	+	0.977088i	$0.568269\pi$
$458$	0	0
$459$	61.8286	2.88591
$460$	0	0
$461$	13.3191	0.620331	0.310165	−	0.950683i	$-0.399616\pi$
$461$	13.3191	0.620331	0.310165	+	0.950683i	$0.399616\pi$
$462$	0	0
$463$	31.6329	1.47010	0.735052	−	0.678010i	$-0.237158\pi$
$463$	31.6329	1.47010	0.735052	+	0.678010i	$0.237158\pi$
$464$	0	0
$465$	57.9393	2.68687
$466$	0	0
$467$	−13.8667	−0.641675	−0.320837	−	0.947134i	$-0.603964\pi$
$467$	−13.8667	−0.641675	−0.320837	+	0.947134i	$0.603964\pi$
$468$	0	0
$469$	34.0239	1.57108
$470$	0	0
$471$	8.49017	0.391206
$472$	0	0
$473$	15.7150	0.722577
$474$	0	0
$475$	3.76018	0.172529
$476$	0	0
$477$	−6.64656	−0.304325
$478$	0	0
$479$	−29.2763	−1.33767	−0.668833	−	0.743412i	$-0.733206\pi$
$479$	−29.2763	−1.33767	−0.668833	+	0.743412i	$0.733206\pi$
$480$	0	0
$481$	0.0108737	0.000495798 0
$482$	0	0
$483$	−31.0406	−1.41240
$484$	0	0
$485$	−49.9861	−2.26975
$486$	0	0
$487$	27.1566	1.23058	0.615292	−	0.788299i	$-0.289038\pi$
$487$	27.1566	1.23058	0.615292	+	0.788299i	$0.289038\pi$
$488$	0	0
$489$	38.2140	1.72810
$490$	0	0
$491$	16.4249	0.741244	0.370622	−	0.928784i	$-0.379145\pi$
$491$	16.4249	0.741244	0.370622	+	0.928784i	$0.379145\pi$
$492$	0	0
$493$	−46.4444	−2.09175
$494$	0	0
$495$	61.5507	2.76650
$496$	0	0
$497$	−20.2224	−0.907100
$498$	0	0
$499$	39.1877	1.75428	0.877141	−	0.480233i	$-0.159448\pi$
$499$	39.1877	1.75428	0.877141	+	0.480233i	$0.159448\pi$
$500$	0	0
$501$	−48.5329	−2.16829
$502$	0	0
$503$	−10.2152	−0.455471	−0.227736	−	0.973723i	$-0.573132\pi$
$503$	−10.2152	−0.455471	−0.227736	+	0.973723i	$0.573132\pi$
$504$	0	0
$505$	−23.1833	−1.03164
$506$	0	0
$507$	40.3409	1.79160
$508$	0	0
$509$	−6.46678	−0.286635	−0.143317	−	0.989677i	$-0.545777\pi$
$509$	−6.46678	−0.286635	−0.143317	+	0.989677i	$0.545777\pi$
$510$	0	0
$511$	−13.1658	−0.582421
$512$	0	0
$513$	−11.3258	−0.500047
$514$	0	0
$515$	12.9294	0.569736
$516$	0	0
$517$	−12.2688	−0.539580
$518$	0	0
$519$	31.0607	1.36341
$520$	0	0
$521$	−34.2954	−1.50251	−0.751254	−	0.660013i	$-0.770551\pi$
$521$	−34.2954	−1.50251	−0.751254	+	0.660013i	$0.770551\pi$
$522$	0	0
$523$	−13.0603	−0.571088	−0.285544	−	0.958366i	$-0.592174\pi$
$523$	−13.0603	−0.571088	−0.285544	+	0.958366i	$0.592174\pi$
$524$	0	0
$525$	31.1344	1.35882
$526$	0	0
$527$	−34.4073	−1.49881
$528$	0	0
$529$	−8.94611	−0.388961
$530$	0	0
$531$	34.2491	1.48629
$532$	0	0
$533$	1.22849	0.0532116
$534$	0	0
$535$	−22.2212	−0.960706
$536$	0	0
$537$	73.7140	3.18099
$538$	0	0
$539$	−0.335040	−0.0144312
$540$	0	0
$541$	−6.50430	−0.279642	−0.139821	−	0.990177i	$-0.544653\pi$
$541$	−6.50430	−0.279642	−0.139821	+	0.990177i	$0.544653\pi$
$542$	0	0
$543$	56.7979	2.43743
$544$	0	0
$545$	−12.8201	−0.549151
$546$	0	0
$547$	−19.6093	−0.838431	−0.419216	−	0.907887i	$-0.637695\pi$
$547$	−19.6093	−0.838431	−0.419216	+	0.907887i	$0.637695\pi$
$548$	0	0
$549$	−37.6620	−1.60738
$550$	0	0
$551$	8.50772	0.362441
$552$	0	0
$553$	−9.82711	−0.417891
$554$	0	0
$555$	0.932094	0.0395652
$556$	0	0
$557$	−34.9944	−1.48276	−0.741381	−	0.671084i	$-0.765829\pi$
$557$	−34.9944	−1.48276	−0.741381	+	0.671084i	$0.765829\pi$
$558$	0	0
$559$	−0.538636	−0.0227819
$560$	0	0
$561$	−53.0501	−2.23977
$562$	0	0
$563$	17.8852	0.753771	0.376885	−	0.926260i	$-0.376995\pi$
$563$	17.8852	0.753771	0.376885	+	0.926260i	$0.376995\pi$
$564$	0	0
$565$	−26.3215	−1.10735
$566$	0	0
$567$	−40.6207	−1.70591
$568$	0	0
$569$	26.0571	1.09237	0.546185	−	0.837665i	$-0.316080\pi$
$569$	26.0571	1.09237	0.546185	+	0.837665i	$0.316080\pi$
$570$	0	0
$571$	−25.4963	−1.06699	−0.533493	−	0.845805i	$-0.679121\pi$
$571$	−25.4963	−1.06699	−0.533493	+	0.845805i	$0.679121\pi$
$572$	0	0
$573$	−17.7702	−0.742362
$574$	0	0
$575$	−14.0964	−0.587859
$576$	0	0
$577$	17.7040	0.737027	0.368514	−	0.929622i	$-0.379867\pi$
$577$	17.7040	0.737027	0.368514	+	0.929622i	$0.379867\pi$
$578$	0	0
$579$	−41.8857	−1.74071
$580$	0	0
$581$	3.95508	0.164084
$582$	0	0
$583$	3.12881	0.129582
$584$	0	0
$585$	−2.10967	−0.0872240
$586$	0	0
$587$	21.9233	0.904873	0.452437	−	0.891797i	$-0.350555\pi$
$587$	21.9233	0.904873	0.452437	+	0.891797i	$0.350555\pi$
$588$	0	0
$589$	6.30276	0.259701
$590$	0	0
$591$	−49.9108	−2.05306
$592$	0	0
$593$	−1.75148	−0.0719245	−0.0359623	−	0.999353i	$-0.511450\pi$
$593$	−1.75148	−0.0719245	−0.0359623	+	0.999353i	$0.511450\pi$
$594$	0	0
$595$	−43.0747	−1.76589
$596$	0	0
$597$	76.0150	3.11109
$598$	0	0
$599$	9.33931	0.381594	0.190797	−	0.981630i	$-0.438893\pi$
$599$	9.33931	0.381594	0.190797	+	0.981630i	$0.438893\pi$
$600$	0	0
$601$	48.6588	1.98484	0.992418	−	0.122910i	$-0.0392225\pi$
$601$	48.6588	1.98484	0.992418	+	0.122910i	$0.0392225\pi$
$602$	0	0
$603$	−84.8272	−3.45443
$604$	0	0
$605$	3.58285	0.145664
$606$	0	0
$607$	−43.9924	−1.78560	−0.892798	−	0.450456i	$-0.851261\pi$
$607$	−43.9924	−1.78560	−0.892798	+	0.450456i	$0.851261\pi$
$608$	0	0
$609$	70.4442	2.85454
$610$	0	0
$611$	0.420515	0.0170122
$612$	0	0
$613$	16.8791	0.681739	0.340870	−	0.940111i	$-0.389279\pi$
$613$	16.8791	0.681739	0.340870	+	0.940111i	$0.389279\pi$
$614$	0	0
$615$	105.306	4.24634
$616$	0	0
$617$	−29.4843	−1.18699	−0.593497	−	0.804836i	$-0.702253\pi$
$617$	−29.4843	−1.18699	−0.593497	+	0.804836i	$0.702253\pi$
$618$	0	0
$619$	23.7393	0.954164	0.477082	−	0.878859i	$-0.341694\pi$
$619$	23.7393	0.954164	0.477082	+	0.878859i	$0.341694\pi$
$620$	0	0
$621$	42.4588	1.70381
$622$	0	0
$623$	−38.8130	−1.55501
$624$	0	0
$625$	−29.6619	−1.18648
$626$	0	0
$627$	9.71776	0.388090
$628$	0	0
$629$	−0.553524	−0.0220705
$630$	0	0
$631$	−22.1990	−0.883729	−0.441864	−	0.897082i	$-0.645683\pi$
$631$	−22.1990	−0.883729	−0.441864	+	0.897082i	$0.645683\pi$
$632$	0	0
$633$	61.4587	2.44276
$634$	0	0
$635$	41.2919	1.63862
$636$	0	0
$637$	0.0114836	0.000454995 0
$638$	0	0
$639$	50.4179	1.99450
$640$	0	0
$641$	34.6718	1.36945	0.684727	−	0.728800i	$-0.259922\pi$
$641$	34.6718	1.36945	0.684727	+	0.728800i	$0.259922\pi$
$642$	0	0
$643$	−14.0242	−0.553059	−0.276530	−	0.961005i	$-0.589184\pi$
$643$	−14.0242	−0.553059	−0.276530	+	0.961005i	$0.589184\pi$
$644$	0	0
$645$	−46.1719	−1.81802
$646$	0	0
$647$	−6.46362	−0.254111	−0.127056	−	0.991896i	$-0.540553\pi$
$647$	−6.46362	−0.254111	−0.127056	+	0.991896i	$0.540553\pi$
$648$	0	0
$649$	−16.1225	−0.632863
$650$	0	0
$651$	52.1870	2.04537
$652$	0	0
$653$	−36.6240	−1.43321	−0.716604	−	0.697480i	$-0.754305\pi$
$653$	−36.6240	−1.43321	−0.716604	+	0.697480i	$0.754305\pi$
$654$	0	0
$655$	−26.8168	−1.04782
$656$	0	0
$657$	32.8245	1.28061
$658$	0	0
$659$	18.0580	0.703438	0.351719	−	0.936106i	$-0.385597\pi$
$659$	18.0580	0.703438	0.351719	+	0.936106i	$0.385597\pi$
$660$	0	0
$661$	−21.1624	−0.823121	−0.411561	−	0.911382i	$-0.635016\pi$
$661$	−21.1624	−0.823121	−0.411561	+	0.911382i	$0.635016\pi$
$662$	0	0
$663$	1.81830	0.0706171
$664$	0	0
$665$	7.89046	0.305979
$666$	0	0
$667$	−31.8942	−1.23495
$668$	0	0
$669$	−22.3157	−0.862774
$670$	0	0
$671$	17.7291	0.684424
$672$	0	0
$673$	50.2536	1.93714	0.968568	−	0.248751i	$-0.0800200\pi$
$673$	50.2536	1.93714	0.968568	+	0.248751i	$0.0800200\pi$
$674$	0	0
$675$	−42.5871	−1.63918
$676$	0	0
$677$	−24.8048	−0.953326	−0.476663	−	0.879086i	$-0.658154\pi$
$677$	−24.8048	−0.953326	−0.476663	+	0.879086i	$0.658154\pi$
$678$	0	0
$679$	−45.0234	−1.72784
$680$	0	0
$681$	−32.2360	−1.23529
$682$	0	0
$683$	9.66410	0.369787	0.184893	−	0.982759i	$-0.440806\pi$
$683$	9.66410	0.369787	0.184893	+	0.982759i	$0.440806\pi$
$684$	0	0
$685$	32.3402	1.23566
$686$	0	0
$687$	14.5035	0.553344
$688$	0	0
$689$	−0.107241	−0.00408555
$690$	0	0
$691$	−25.4335	−0.967537	−0.483768	−	0.875196i	$-0.660732\pi$
$691$	−25.4335	−0.967537	−0.483768	+	0.875196i	$0.660732\pi$
$692$	0	0
$693$	55.4399	2.10599
$694$	0	0
$695$	12.7611	0.484057
$696$	0	0
$697$	−62.5359	−2.36872
$698$	0	0
$699$	56.8941	2.15193
$700$	0	0
$701$	19.8043	0.747997	0.373999	−	0.927429i	$-0.377986\pi$
$701$	19.8043	0.747997	0.373999	+	0.927429i	$0.377986\pi$
$702$	0	0
$703$	0.101395	0.00382419
$704$	0	0
$705$	36.0465	1.35759
$706$	0	0
$707$	−20.8816	−0.785333
$708$	0	0
$709$	19.1797	0.720310	0.360155	−	0.932892i	$-0.382724\pi$
$709$	19.1797	0.720310	0.360155	+	0.932892i	$0.382724\pi$
$710$	0	0
$711$	24.5006	0.918845
$712$	0	0
$713$	−23.6281	−0.884880
$714$	0	0
$715$	0.993109	0.0371402
$716$	0	0
$717$	5.43177	0.202853
$718$	0	0
$719$	1.87160	0.0697988	0.0348994	−	0.999391i	$-0.488889\pi$
$719$	1.87160	0.0697988	0.0348994	+	0.999391i	$0.488889\pi$
$720$	0	0
$721$	11.6457	0.433709
$722$	0	0
$723$	1.71164	0.0636566
$724$	0	0
$725$	31.9906	1.18810
$726$	0	0
$727$	13.9342	0.516790	0.258395	−	0.966039i	$-0.416806\pi$
$727$	13.9342	0.516790	0.258395	+	0.966039i	$0.416806\pi$
$728$	0	0
$729$	−4.25612	−0.157634
$730$	0	0
$731$	27.4192	1.01414
$732$	0	0
$733$	−23.6072	−0.871953	−0.435977	−	0.899958i	$-0.643597\pi$
$733$	−23.6072	−0.871953	−0.435977	+	0.899958i	$0.643597\pi$
$734$	0	0
$735$	0.984371	0.0363091
$736$	0	0
$737$	39.9317	1.47090
$738$	0	0
$739$	1.76342	0.0648685	0.0324343	−	0.999474i	$-0.489674\pi$
$739$	1.76342	0.0648685	0.0324343	+	0.999474i	$0.489674\pi$
$740$	0	0
$741$	−0.333079	−0.0122359
$742$	0	0
$743$	16.3962	0.601520	0.300760	−	0.953700i	$-0.402760\pi$
$743$	16.3962	0.601520	0.300760	+	0.953700i	$0.402760\pi$
$744$	0	0
$745$	43.3493	1.58819
$746$	0	0
$747$	−9.86067	−0.360783
$748$	0	0
$749$	−20.0150	−0.731334
$750$	0	0
$751$	47.2598	1.72454	0.862268	−	0.506453i	$-0.169043\pi$
$751$	47.2598	1.72454	0.862268	+	0.506453i	$0.169043\pi$
$752$	0	0
$753$	33.3150	1.21407
$754$	0	0
$755$	31.4644	1.14511
$756$	0	0
$757$	18.7025	0.679755	0.339877	−	0.940470i	$-0.389614\pi$
$757$	18.7025	0.679755	0.339877	+	0.940470i	$0.389614\pi$
$758$	0	0
$759$	−36.4304	−1.32234
$760$	0	0
$761$	−45.0775	−1.63406	−0.817029	−	0.576597i	$-0.804380\pi$
$761$	−45.0775	−1.63406	−0.817029	+	0.576597i	$0.804380\pi$
$762$	0	0
$763$	−11.5473	−0.418039
$764$	0	0
$765$	107.392	3.88278
$766$	0	0
$767$	0.552603	0.0199533
$768$	0	0
$769$	28.6657	1.03371	0.516857	−	0.856072i	$-0.327102\pi$
$769$	28.6657	1.03371	0.516857	+	0.856072i	$0.327102\pi$
$770$	0	0
$771$	−70.9860	−2.55650
$772$	0	0
$773$	11.9805	0.430910	0.215455	−	0.976514i	$-0.430877\pi$
$773$	11.9805	0.430910	0.215455	+	0.976514i	$0.430877\pi$
$774$	0	0
$775$	23.6995	0.851312
$776$	0	0
$777$	0.839555	0.0301189
$778$	0	0
$779$	11.4554	0.410432
$780$	0	0
$781$	−23.7338	−0.849263
$782$	0	0
$783$	−96.3569	−3.44351
$784$	0	0
$785$	8.09071	0.288770
$786$	0	0
$787$	−38.6172	−1.37656	−0.688278	−	0.725447i	$-0.741633\pi$
$787$	−38.6172	−1.37656	−0.688278	+	0.725447i	$0.741633\pi$
$788$	0	0
$789$	−76.8535	−2.73606
$790$	0	0
$791$	−23.7083	−0.842969
$792$	0	0
$793$	−0.607669	−0.0215790
$794$	0	0
$795$	−9.19269	−0.326031
$796$	0	0
$797$	−37.1425	−1.31566	−0.657828	−	0.753168i	$-0.728525\pi$
$797$	−37.1425	−1.31566	−0.657828	+	0.753168i	$0.728525\pi$
$798$	0	0
$799$	−21.4063	−0.757299
$800$	0	0
$801$	96.7672	3.41910
$802$	0	0
$803$	−15.4519	−0.545285
$804$	0	0
$805$	−29.5802	−1.04256
$806$	0	0
$807$	−29.5320	−1.03958
$808$	0	0
$809$	−12.4076	−0.436229	−0.218114	−	0.975923i	$-0.569991\pi$
$809$	−12.4076	−0.436229	−0.218114	+	0.975923i	$0.569991\pi$
$810$	0	0
$811$	2.83937	0.0997038	0.0498519	−	0.998757i	$-0.484125\pi$
$811$	2.83937	0.0997038	0.0498519	+	0.998757i	$0.484125\pi$
$812$	0	0
$813$	−77.7473	−2.72672
$814$	0	0
$815$	36.4160	1.27560
$816$	0	0
$817$	−5.02268	−0.175721
$818$	0	0
$819$	−1.90022	−0.0663989
$820$	0	0
$821$	−25.4646	−0.888720	−0.444360	−	0.895848i	$-0.646569\pi$
$821$	−25.4646	−0.888720	−0.444360	+	0.895848i	$0.646569\pi$
$822$	0	0
$823$	33.8682	1.18057	0.590285	−	0.807195i	$-0.299015\pi$
$823$	33.8682	1.18057	0.590285	+	0.807195i	$0.299015\pi$
$824$	0	0
$825$	36.5405	1.27218
$826$	0	0
$827$	−0.858097	−0.0298390	−0.0149195	−	0.999889i	$-0.504749\pi$
$827$	−0.858097	−0.0298390	−0.0149195	+	0.999889i	$0.504749\pi$
$828$	0	0
$829$	−42.6388	−1.48091	−0.740454	−	0.672107i	$-0.765389\pi$
$829$	−42.6388	−1.48091	−0.740454	+	0.672107i	$0.765389\pi$
$830$	0	0
$831$	−41.2387	−1.43056
$832$	0	0
$833$	−0.584569	−0.0202541
$834$	0	0
$835$	−46.2495	−1.60053
$836$	0	0
$837$	−71.3839	−2.46739
$838$	0	0
$839$	−1.39245	−0.0480726	−0.0240363	−	0.999711i	$-0.507652\pi$
$839$	−1.39245	−0.0480726	−0.0240363	+	0.999711i	$0.507652\pi$
$840$	0	0
$841$	43.3813	1.49591
$842$	0	0
$843$	102.109	3.51681
$844$	0	0
$845$	38.4428	1.32247
$846$	0	0
$847$	3.22714	0.110886
$848$	0	0
$849$	38.9851	1.33796
$850$	0	0
$851$	−0.380115	−0.0130302
$852$	0	0
$853$	−35.8163	−1.22633	−0.613164	−	0.789955i	$-0.710104\pi$
$853$	−35.8163	−1.22633	−0.613164	+	0.789955i	$0.710104\pi$
$854$	0	0
$855$	−19.6722	−0.672776
$856$	0	0
$857$	15.8693	0.542086	0.271043	−	0.962567i	$-0.412632\pi$
$857$	15.8693	0.542086	0.271043	+	0.962567i	$0.412632\pi$
$858$	0	0
$859$	13.9644	0.476458	0.238229	−	0.971209i	$-0.423433\pi$
$859$	13.9644	0.476458	0.238229	+	0.971209i	$0.423433\pi$
$860$	0	0
$861$	94.8509	3.23251
$862$	0	0
$863$	49.5152	1.68552	0.842758	−	0.538293i	$-0.180931\pi$
$863$	49.5152	1.68552	0.842758	+	0.538293i	$0.180931\pi$
$864$	0	0
$865$	29.5993	1.00641
$866$	0	0
$867$	−39.7604	−1.35033
$868$	0	0
$869$	−11.5335	−0.391246
$870$	0	0
$871$	−1.36867	−0.0463756
$872$	0	0
$873$	112.251	3.79912
$874$	0	0
$875$	−9.78275	−0.330717
$876$	0	0
$877$	−0.985452	−0.0332764	−0.0166382	−	0.999862i	$-0.505296\pi$
$877$	−0.985452	−0.0332764	−0.0166382	+	0.999862i	$0.505296\pi$
$878$	0	0
$879$	10.8705	0.366654
$880$	0	0
$881$	−40.6521	−1.36960	−0.684802	−	0.728729i	$-0.740111\pi$
$881$	−40.6521	−1.36960	−0.684802	+	0.728729i	$0.740111\pi$
$882$	0	0
$883$	1.75081	0.0589193	0.0294597	−	0.999566i	$-0.490621\pi$
$883$	1.75081	0.0589193	0.0294597	+	0.999566i	$0.490621\pi$
$884$	0	0
$885$	47.3691	1.59230
$886$	0	0
$887$	10.6487	0.357547	0.178773	−	0.983890i	$-0.442787\pi$
$887$	10.6487	0.357547	0.178773	+	0.983890i	$0.442787\pi$
$888$	0	0
$889$	37.1924	1.24739
$890$	0	0
$891$	−47.6740	−1.59714
$892$	0	0
$893$	3.92122	0.131219
$894$	0	0
$895$	70.2458	2.34806
$896$	0	0
$897$	1.24866	0.0416916
$898$	0	0
$899$	53.6221	1.78840
$900$	0	0
$901$	5.45908	0.181868
$902$	0	0
$903$	−41.5879	−1.38396
$904$	0	0
$905$	54.1256	1.79920
$906$	0	0
$907$	−7.56077	−0.251051	−0.125526	−	0.992090i	$-0.540062\pi$
$907$	−7.56077	−0.251051	−0.125526	+	0.992090i	$0.540062\pi$
$908$	0	0
$909$	52.0613	1.72676
$910$	0	0
$911$	15.8233	0.524249	0.262125	−	0.965034i	$-0.415577\pi$
$911$	15.8233	0.524249	0.262125	+	0.965034i	$0.415577\pi$
$912$	0	0
$913$	4.64183	0.153622
$914$	0	0
$915$	−52.0894	−1.72202
$916$	0	0
$917$	−24.1544	−0.797648
$918$	0	0
$919$	−17.8330	−0.588256	−0.294128	−	0.955766i	$-0.595029\pi$
$919$	−17.8330	−0.588256	−0.294128	+	0.955766i	$0.595029\pi$
$920$	0	0
$921$	−69.1846	−2.27971
$922$	0	0
$923$	0.813483	0.0267761
$924$	0	0
$925$	0.381264	0.0125359
$926$	0	0
$927$	−29.0347	−0.953625
$928$	0	0
$929$	54.4460	1.78632	0.893158	−	0.449744i	$-0.148485\pi$
$929$	54.4460	1.78632	0.893158	+	0.449744i	$0.148485\pi$
$930$	0	0
$931$	0.107082	0.00350947
$932$	0	0
$933$	19.2667	0.630763
$934$	0	0
$935$	−50.5541	−1.65329
$936$	0	0
$937$	−0.232867	−0.00760742	−0.00380371	−	0.999993i	$-0.501211\pi$
$937$	−0.232867	−0.00760742	−0.00380371	+	0.999993i	$0.501211\pi$
$938$	0	0
$939$	4.10678	0.134020
$940$	0	0
$941$	46.3420	1.51071	0.755354	−	0.655317i	$-0.227465\pi$
$941$	46.3420	1.51071	0.755354	+	0.655317i	$0.227465\pi$
$942$	0	0
$943$	−42.9445	−1.39847
$944$	0	0
$945$	−89.3659	−2.90707
$946$	0	0
$947$	−9.34013	−0.303513	−0.151757	−	0.988418i	$-0.548493\pi$
$947$	−9.34013	−0.303513	−0.151757	+	0.988418i	$0.548493\pi$
$948$	0	0
$949$	0.529617	0.0171921
$950$	0	0
$951$	−73.0537	−2.36893
$952$	0	0
$953$	40.0701	1.29800	0.648999	−	0.760789i	$-0.275188\pi$
$953$	40.0701	1.29800	0.648999	+	0.760789i	$0.275188\pi$
$954$	0	0
$955$	−16.9342	−0.547976
$956$	0	0
$957$	82.6760	2.67253
$958$	0	0
$959$	29.1294	0.940638
$960$	0	0
$961$	8.72479	0.281445
$962$	0	0
$963$	49.9008	1.60803
$964$	0	0
$965$	−39.9150	−1.28491
$966$	0	0
$967$	−2.93883	−0.0945064	−0.0472532	−	0.998883i	$-0.515047\pi$
$967$	−2.93883	−0.0945064	−0.0472532	+	0.998883i	$0.515047\pi$
$968$	0	0
$969$	16.9553	0.544683
$970$	0	0
$971$	38.4469	1.23382	0.616910	−	0.787033i	$-0.288384\pi$
$971$	38.4469	1.23382	0.616910	+	0.787033i	$0.288384\pi$
$972$	0	0
$973$	11.4942	0.368486
$974$	0	0
$975$	−1.25244	−0.0401100
$976$	0	0
$977$	−18.4928	−0.591637	−0.295818	−	0.955244i	$-0.595592\pi$
$977$	−18.4928	−0.591637	−0.295818	+	0.955244i	$0.595592\pi$
$978$	0	0
$979$	−45.5524	−1.45586
$980$	0	0
$981$	28.7893	0.919171
$982$	0	0
$983$	−56.6752	−1.80766	−0.903830	−	0.427893i	$-0.859256\pi$
$983$	−56.6752	−1.80766	−0.903830	+	0.427893i	$0.859256\pi$
$984$	0	0
$985$	−47.5625	−1.51547
$986$	0	0
$987$	32.4678	1.03346
$988$	0	0
$989$	18.8293	0.598736
$990$	0	0
$991$	−18.0109	−0.572134	−0.286067	−	0.958210i	$-0.592348\pi$
$991$	−18.0109	−0.572134	−0.286067	+	0.958210i	$0.592348\pi$
$992$	0	0
$993$	47.8970	1.51996
$994$	0	0
$995$	72.4386	2.29646
$996$	0	0
$997$	−11.7459	−0.371997	−0.185998	−	0.982550i	$-0.559552\pi$
$997$	−11.7459	−0.371997	−0.185998	+	0.982550i	$0.559552\pi$
$998$	0	0
$999$	−1.14838	−0.0363332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.c.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.c.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-3.10589 q^{3} -2.95976 q^{5} -2.66591 q^{7} +6.64656 q^{9} +O(q^{10})\)	\(q-3.10589 q^{3} -2.95976 q^{5} -2.66591 q^{7} +6.64656 q^{9} -3.12881 q^{11} +0.107241 q^{13} +9.19269 q^{15} -5.45908 q^{17} +1.00000 q^{19} +8.28003 q^{21} -3.74885 q^{23} +3.76018 q^{25} -11.3258 q^{27} +8.50772 q^{29} +6.30276 q^{31} +9.71776 q^{33} +7.89046 q^{35} +0.101395 q^{37} -0.333079 q^{39} +11.4554 q^{41} -5.02268 q^{43} -19.6722 q^{45} +3.92122 q^{47} +0.107082 q^{49} +16.9553 q^{51} -1.00000 q^{53} +9.26054 q^{55} -3.10589 q^{57} +5.15291 q^{59} -5.66639 q^{61} -17.7191 q^{63} -0.317407 q^{65} -12.7626 q^{67} +11.6435 q^{69} +7.58556 q^{71} +4.93858 q^{73} -11.6787 q^{75} +8.34114 q^{77} +3.68621 q^{79} +15.2371 q^{81} -1.48357 q^{83} +16.1576 q^{85} -26.4241 q^{87} +14.5590 q^{89} -0.285895 q^{91} -19.5757 q^{93} -2.95976 q^{95} +16.8886 q^{97} -20.7959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 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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