LMFDB - Embedded newform 4028.2.a.c.1.13

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			1.13
Root			$-1.34880$ 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)

$f(q)$	$=$	$q+1.34880 q^{3} +0.901606 q^{5} +0.205950 q^{7} -1.18075 q^{9} +O(q^{10})$	$q+1.34880 q^{3} +0.901606 q^{5} +0.205950 q^{7} -1.18075 q^{9} -0.815233 q^{11} -0.909710 q^{13} +1.21608 q^{15} -7.34199 q^{17} +1.00000 q^{19} +0.277784 q^{21} -0.663932 q^{23} -4.18711 q^{25} -5.63898 q^{27} +6.35656 q^{29} +0.965308 q^{31} -1.09958 q^{33} +0.185685 q^{35} +8.11109 q^{37} -1.22701 q^{39} -11.6773 q^{41} -3.30007 q^{43} -1.06457 q^{45} +6.76840 q^{47} -6.95758 q^{49} -9.90285 q^{51} -1.00000 q^{53} -0.735019 q^{55} +1.34880 q^{57} -10.7162 q^{59} -5.28028 q^{61} -0.243174 q^{63} -0.820200 q^{65} -13.2974 q^{67} -0.895510 q^{69} +1.77560 q^{71} -15.3700 q^{73} -5.64756 q^{75} -0.167897 q^{77} +4.10769 q^{79} -4.06359 q^{81} +13.4496 q^{83} -6.61958 q^{85} +8.57371 q^{87} +3.72075 q^{89} -0.187354 q^{91} +1.30200 q^{93} +0.901606 q^{95} +3.36795 q^{97} +0.962584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.34880	0.778728	0.389364	−	0.921084i	$-0.372695\pi$
$3$	1.34880	0.778728	0.389364	+	0.921084i	$0.372695\pi$
$4$	0	0
$5$	0.901606	0.403210	0.201605	−	0.979467i	$-0.435384\pi$
$5$	0.901606	0.403210	0.201605	+	0.979467i	$0.435384\pi$
$6$	0	0
$7$	0.205950	0.0778416	0.0389208	−	0.999242i	$-0.487608\pi$
$7$	0.205950	0.0778416	0.0389208	+	0.999242i	$0.487608\pi$
$8$	0	0
$9$	−1.18075	−0.393582
$10$	0	0
$11$	−0.815233	−0.245802	−0.122901	−	0.992419i	$-0.539220\pi$
$11$	−0.815233	−0.245802	−0.122901	+	0.992419i	$0.539220\pi$
$12$	0	0
$13$	−0.909710	−0.252308	−0.126154	−	0.992011i	$-0.540263\pi$
$13$	−0.909710	−0.252308	−0.126154	+	0.992011i	$0.540263\pi$
$14$	0	0
$15$	1.21608	0.313991
$16$	0	0
$17$	−7.34199	−1.78069	−0.890347	−	0.455282i	$-0.849538\pi$
$17$	−7.34199	−1.78069	−0.890347	+	0.455282i	$0.849538\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.277784	0.0606175
$22$	0	0
$23$	−0.663932	−0.138439	−0.0692197	−	0.997601i	$-0.522051\pi$
$23$	−0.663932	−0.138439	−0.0692197	+	0.997601i	$0.522051\pi$
$24$	0	0
$25$	−4.18711	−0.837421
$26$	0	0
$27$	−5.63898	−1.08522
$28$	0	0
$29$	6.35656	1.18038	0.590192	−	0.807263i	$-0.299052\pi$
$29$	6.35656	1.18038	0.590192	+	0.807263i	$0.299052\pi$
$30$	0	0
$31$	0.965308	0.173374	0.0866872	−	0.996236i	$-0.472372\pi$
$31$	0.965308	0.173374	0.0866872	+	0.996236i	$0.472372\pi$
$32$	0	0
$33$	−1.09958	−0.191413
$34$	0	0
$35$	0.185685	0.0313865
$36$	0	0
$37$	8.11109	1.33345	0.666727	−	0.745302i	$-0.267695\pi$
$37$	8.11109	1.33345	0.666727	+	0.745302i	$0.267695\pi$
$38$	0	0
$39$	−1.22701	−0.196479
$40$	0	0
$41$	−11.6773	−1.82369	−0.911847	−	0.410530i	$-0.865344\pi$
$41$	−11.6773	−1.82369	−0.911847	+	0.410530i	$0.865344\pi$
$42$	0	0
$43$	−3.30007	−0.503256	−0.251628	−	0.967824i	$-0.580966\pi$
$43$	−3.30007	−0.503256	−0.251628	+	0.967824i	$0.580966\pi$
$44$	0	0
$45$	−1.06457	−0.158696
$46$	0	0
$47$	6.76840	0.987273	0.493636	−	0.869668i	$-0.335667\pi$
$47$	6.76840	0.987273	0.493636	+	0.869668i	$0.335667\pi$
$48$	0	0
$49$	−6.95758	−0.993941
$50$	0	0
$51$	−9.90285	−1.38668
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	−0.735019	−0.0991099
$56$	0	0
$57$	1.34880	0.178652
$58$	0	0
$59$	−10.7162	−1.39514	−0.697568	−	0.716518i	$-0.745735\pi$
$59$	−10.7162	−1.39514	−0.697568	+	0.716518i	$0.745735\pi$
$60$	0	0
$61$	−5.28028	−0.676071	−0.338035	−	0.941133i	$-0.609762\pi$
$61$	−5.28028	−0.676071	−0.338035	+	0.941133i	$0.609762\pi$
$62$	0	0
$63$	−0.243174	−0.0306371
$64$	0	0
$65$	−0.820200	−0.101733
$66$	0	0
$67$	−13.2974	−1.62453	−0.812267	−	0.583286i	$-0.801767\pi$
$67$	−13.2974	−1.62453	−0.812267	+	0.583286i	$0.801767\pi$
$68$	0	0
$69$	−0.895510	−0.107807
$70$	0	0
$71$	1.77560	0.210725	0.105362	−	0.994434i	$-0.466400\pi$
$71$	1.77560	0.210725	0.105362	+	0.994434i	$0.466400\pi$
$72$	0	0
$73$	−15.3700	−1.79892	−0.899459	−	0.437005i	$-0.856039\pi$
$73$	−15.3700	−1.79892	−0.899459	+	0.437005i	$0.856039\pi$
$74$	0	0
$75$	−5.64756	−0.652124
$76$	0	0
$77$	−0.167897	−0.0191336
$78$	0	0
$79$	4.10769	0.462152	0.231076	−	0.972936i	$-0.425775\pi$
$79$	4.10769	0.462152	0.231076	+	0.972936i	$0.425775\pi$
$80$	0	0
$81$	−4.06359	−0.451510
$82$	0	0
$83$	13.4496	1.47628	0.738142	−	0.674645i	$-0.235703\pi$
$83$	13.4496	1.47628	0.738142	+	0.674645i	$0.235703\pi$
$84$	0	0
$85$	−6.61958	−0.717994
$86$	0	0
$87$	8.57371	0.919198
$88$	0	0
$89$	3.72075	0.394399	0.197199	−	0.980363i	$-0.436815\pi$
$89$	3.72075	0.394399	0.197199	+	0.980363i	$0.436815\pi$
$90$	0	0
$91$	−0.187354	−0.0196401
$92$	0	0
$93$	1.30200	0.135012
$94$	0	0
$95$	0.901606	0.0925028
$96$	0	0
$97$	3.36795	0.341963	0.170982	−	0.985274i	$-0.445306\pi$
$97$	3.36795	0.341963	0.170982	+	0.985274i	$0.445306\pi$
$98$	0	0
$99$	0.962584	0.0967433
$100$	0	0
$101$	−14.3574	−1.42862	−0.714308	−	0.699832i	$-0.753258\pi$
$101$	−14.3574	−1.42862	−0.714308	+	0.699832i	$0.753258\pi$
$102$	0	0
$103$	−10.3511	−1.01992	−0.509961	−	0.860197i	$-0.670340\pi$
$103$	−10.3511	−1.01992	−0.509961	+	0.860197i	$0.670340\pi$
$104$	0	0
$105$	0.250452	0.0244416
$106$	0	0
$107$	16.0583	1.55241	0.776207	−	0.630478i	$-0.217141\pi$
$107$	16.0583	1.55241	0.776207	+	0.630478i	$0.217141\pi$
$108$	0	0
$109$	15.3605	1.47127	0.735633	−	0.677380i	$-0.236885\pi$
$109$	15.3605	1.47127	0.735633	+	0.677380i	$0.236885\pi$
$110$	0	0
$111$	10.9402	1.03840
$112$	0	0
$113$	0.729446	0.0686205	0.0343103	−	0.999411i	$-0.489077\pi$
$113$	0.729446	0.0686205	0.0343103	+	0.999411i	$0.489077\pi$
$114$	0	0
$115$	−0.598605	−0.0558202
$116$	0	0
$117$	1.07414	0.0993040
$118$	0	0
$119$	−1.51208	−0.138612
$120$	0	0
$121$	−10.3354	−0.939581
$122$	0	0
$123$	−15.7504	−1.42016
$124$	0	0
$125$	−8.28315	−0.740867
$126$	0	0
$127$	−19.6078	−1.73991	−0.869956	−	0.493130i	$-0.835853\pi$
$127$	−19.6078	−1.73991	−0.869956	+	0.493130i	$0.835853\pi$
$128$	0	0
$129$	−4.45112	−0.391899
$130$	0	0
$131$	0.215942	0.0188670	0.00943349	−	0.999956i	$-0.496997\pi$
$131$	0.215942	0.0188670	0.00943349	+	0.999956i	$0.496997\pi$
$132$	0	0
$133$	0.205950	0.0178581
$134$	0	0
$135$	−5.08414	−0.437573
$136$	0	0
$137$	−5.31289	−0.453911	−0.226956	−	0.973905i	$-0.572877\pi$
$137$	−5.31289	−0.453911	−0.226956	+	0.973905i	$0.572877\pi$
$138$	0	0
$139$	14.1061	1.19646	0.598231	−	0.801324i	$-0.295871\pi$
$139$	14.1061	1.19646	0.598231	+	0.801324i	$0.295871\pi$
$140$	0	0
$141$	9.12920	0.768817
$142$	0	0
$143$	0.741625	0.0620178
$144$	0	0
$145$	5.73111	0.475943
$146$	0	0
$147$	−9.38437	−0.774010
$148$	0	0
$149$	8.80570	0.721391	0.360696	−	0.932684i	$-0.382539\pi$
$149$	8.80570	0.721391	0.360696	+	0.932684i	$0.382539\pi$
$150$	0	0
$151$	−2.28323	−0.185807	−0.0929033	−	0.995675i	$-0.529615\pi$
$151$	−2.28323	−0.185807	−0.0929033	+	0.995675i	$0.529615\pi$
$152$	0	0
$153$	8.66904	0.700850
$154$	0	0
$155$	0.870327	0.0699064
$156$	0	0
$157$	4.72959	0.377462	0.188731	−	0.982029i	$-0.439563\pi$
$157$	4.72959	0.377462	0.188731	+	0.982029i	$0.439563\pi$
$158$	0	0
$159$	−1.34880	−0.106967
$160$	0	0
$161$	−0.136737	−0.0107763
$162$	0	0
$163$	−7.63179	−0.597768	−0.298884	−	0.954289i	$-0.596614\pi$
$163$	−7.63179	−0.597768	−0.298884	+	0.954289i	$0.596614\pi$
$164$	0	0
$165$	−0.991391	−0.0771797
$166$	0	0
$167$	−7.28406	−0.563658	−0.281829	−	0.959465i	$-0.590941\pi$
$167$	−7.28406	−0.563658	−0.281829	+	0.959465i	$0.590941\pi$
$168$	0	0
$169$	−12.1724	−0.936341
$170$	0	0
$171$	−1.18075	−0.0902940
$172$	0	0
$173$	18.1159	1.37733	0.688664	−	0.725081i	$-0.258198\pi$
$173$	18.1159	1.37733	0.688664	+	0.725081i	$0.258198\pi$
$174$	0	0
$175$	−0.862333	−0.0651862
$176$	0	0
$177$	−14.4540	−1.08643
$178$	0	0
$179$	−16.1934	−1.21035	−0.605176	−	0.796092i	$-0.706897\pi$
$179$	−16.1934	−1.21035	−0.605176	+	0.796092i	$0.706897\pi$
$180$	0	0
$181$	24.3126	1.80714	0.903571	−	0.428438i	$-0.140936\pi$
$181$	24.3126	1.80714	0.903571	+	0.428438i	$0.140936\pi$
$182$	0	0
$183$	−7.12203	−0.526475
$184$	0	0
$185$	7.31300	0.537663
$186$	0	0
$187$	5.98543	0.437698
$188$	0	0
$189$	−1.16135	−0.0844754
$190$	0	0
$191$	−4.39312	−0.317875	−0.158938	−	0.987289i	$-0.550807\pi$
$191$	−4.39312	−0.317875	−0.158938	+	0.987289i	$0.550807\pi$
$192$	0	0
$193$	−12.2396	−0.881026	−0.440513	−	0.897746i	$-0.645203\pi$
$193$	−12.2396	−0.881026	−0.440513	+	0.897746i	$0.645203\pi$
$194$	0	0
$195$	−1.10628	−0.0792226
$196$	0	0
$197$	9.87404	0.703496	0.351748	−	0.936095i	$-0.385587\pi$
$197$	9.87404	0.703496	0.351748	+	0.936095i	$0.385587\pi$
$198$	0	0
$199$	−7.09039	−0.502624	−0.251312	−	0.967906i	$-0.580862\pi$
$199$	−7.09039	−0.502624	−0.251312	+	0.967906i	$0.580862\pi$
$200$	0	0
$201$	−17.9355	−1.26507
$202$	0	0
$203$	1.30913	0.0918830
$204$	0	0
$205$	−10.5284	−0.735332
$206$	0	0
$207$	0.783936	0.0544873
$208$	0	0
$209$	−0.815233	−0.0563908
$210$	0	0
$211$	4.27513	0.294312	0.147156	−	0.989113i	$-0.452988\pi$
$211$	4.27513	0.294312	0.147156	+	0.989113i	$0.452988\pi$
$212$	0	0
$213$	2.39492	0.164097
$214$	0	0
$215$	−2.97536	−0.202918
$216$	0	0
$217$	0.198805	0.0134957
$218$	0	0
$219$	−20.7310	−1.40087
$220$	0	0
$221$	6.67908	0.449284
$222$	0	0
$223$	21.7584	1.45705	0.728525	−	0.685019i	$-0.240206\pi$
$223$	21.7584	1.45705	0.728525	+	0.685019i	$0.240206\pi$
$224$	0	0
$225$	4.94392	0.329594
$226$	0	0
$227$	13.3146	0.883721	0.441861	−	0.897084i	$-0.354319\pi$
$227$	13.3146	0.883721	0.441861	+	0.897084i	$0.354319\pi$
$228$	0	0
$229$	−2.45560	−0.162271	−0.0811354	−	0.996703i	$-0.525855\pi$
$229$	−2.45560	−0.162271	−0.0811354	+	0.996703i	$0.525855\pi$
$230$	0	0
$231$	−0.226459	−0.0148999
$232$	0	0
$233$	−21.6280	−1.41690	−0.708449	−	0.705762i	$-0.750605\pi$
$233$	−21.6280	−1.41690	−0.708449	+	0.705762i	$0.750605\pi$
$234$	0	0
$235$	6.10243	0.398079
$236$	0	0
$237$	5.54044	0.359890
$238$	0	0
$239$	−24.2981	−1.57171	−0.785857	−	0.618408i	$-0.787778\pi$
$239$	−24.2981	−1.57171	−0.785857	+	0.618408i	$0.787778\pi$
$240$	0	0
$241$	16.0074	1.03112	0.515562	−	0.856852i	$-0.327583\pi$
$241$	16.0074	1.03112	0.515562	+	0.856852i	$0.327583\pi$
$242$	0	0
$243$	11.4360	0.733618
$244$	0	0
$245$	−6.27300	−0.400767
$246$	0	0
$247$	−0.909710	−0.0578835
$248$	0	0
$249$	18.1408	1.14962
$250$	0	0
$251$	18.7193	1.18155	0.590774	−	0.806837i	$-0.298822\pi$
$251$	18.7193	1.18155	0.590774	+	0.806837i	$0.298822\pi$
$252$	0	0
$253$	0.541259	0.0340287
$254$	0	0
$255$	−8.92847	−0.559122
$256$	0	0
$257$	12.9215	0.806018	0.403009	−	0.915196i	$-0.367964\pi$
$257$	12.9215	0.806018	0.403009	+	0.915196i	$0.367964\pi$
$258$	0	0
$259$	1.67047	0.103798
$260$	0	0
$261$	−7.50549	−0.464578
$262$	0	0
$263$	−5.94778	−0.366756	−0.183378	−	0.983042i	$-0.558703\pi$
$263$	−5.94778	−0.366756	−0.183378	+	0.983042i	$0.558703\pi$
$264$	0	0
$265$	−0.901606	−0.0553852
$266$	0	0
$267$	5.01854	0.307129
$268$	0	0
$269$	−19.4632	−1.18669	−0.593346	−	0.804948i	$-0.702193\pi$
$269$	−19.4632	−1.18669	−0.593346	+	0.804948i	$0.702193\pi$
$270$	0	0
$271$	−12.7772	−0.776162	−0.388081	−	0.921625i	$-0.626862\pi$
$271$	−12.7772	−0.776162	−0.388081	+	0.921625i	$0.626862\pi$
$272$	0	0
$273$	−0.252703	−0.0152943
$274$	0	0
$275$	3.41347	0.205840
$276$	0	0
$277$	−5.56265	−0.334227	−0.167114	−	0.985938i	$-0.553445\pi$
$277$	−5.56265	−0.334227	−0.167114	+	0.985938i	$0.553445\pi$
$278$	0	0
$279$	−1.13978	−0.0682371
$280$	0	0
$281$	21.4374	1.27885	0.639423	−	0.768855i	$-0.279173\pi$
$281$	21.4374	1.27885	0.639423	+	0.768855i	$0.279173\pi$
$282$	0	0
$283$	−15.5952	−0.927038	−0.463519	−	0.886087i	$-0.653413\pi$
$283$	−15.5952	−0.927038	−0.463519	+	0.886087i	$0.653413\pi$
$284$	0	0
$285$	1.21608	0.0720345
$286$	0	0
$287$	−2.40494	−0.141959
$288$	0	0
$289$	36.9048	2.17087
$290$	0	0
$291$	4.54268	0.266296
$292$	0	0
$293$	2.03187	0.118703	0.0593516	−	0.998237i	$-0.481097\pi$
$293$	2.03187	0.118703	0.0593516	+	0.998237i	$0.481097\pi$
$294$	0	0
$295$	−9.66183	−0.562534
$296$	0	0
$297$	4.59708	0.266750
$298$	0	0
$299$	0.603986	0.0349294
$300$	0	0
$301$	−0.679647	−0.0391742
$302$	0	0
$303$	−19.3652	−1.11250
$304$	0	0
$305$	−4.76073	−0.272599
$306$	0	0
$307$	14.8066	0.845059	0.422530	−	0.906349i	$-0.361142\pi$
$307$	14.8066	0.845059	0.422530	+	0.906349i	$0.361142\pi$
$308$	0	0
$309$	−13.9615	−0.794243
$310$	0	0
$311$	−4.43636	−0.251563	−0.125781	−	0.992058i	$-0.540144\pi$
$311$	−4.43636	−0.251563	−0.125781	+	0.992058i	$0.540144\pi$
$312$	0	0
$313$	−11.3036	−0.638920	−0.319460	−	0.947600i	$-0.603502\pi$
$313$	−11.3036	−0.638920	−0.319460	+	0.947600i	$0.603502\pi$
$314$	0	0
$315$	−0.219247	−0.0123532
$316$	0	0
$317$	3.07360	0.172631	0.0863154	−	0.996268i	$-0.472491\pi$
$317$	3.07360	0.172631	0.0863154	+	0.996268i	$0.472491\pi$
$318$	0	0
$319$	−5.18208	−0.290141
$320$	0	0
$321$	21.6594	1.20891
$322$	0	0
$323$	−7.34199	−0.408519
$324$	0	0
$325$	3.80905	0.211288
$326$	0	0
$327$	20.7182	1.14572
$328$	0	0
$329$	1.39395	0.0768509
$330$	0	0
$331$	30.2424	1.66227	0.831137	−	0.556068i	$-0.187691\pi$
$331$	30.2424	1.66227	0.831137	+	0.556068i	$0.187691\pi$
$332$	0	0
$333$	−9.57714	−0.524824
$334$	0	0
$335$	−11.9890	−0.655029
$336$	0	0
$337$	−10.4794	−0.570848	−0.285424	−	0.958401i	$-0.592134\pi$
$337$	−10.4794	−0.570848	−0.285424	+	0.958401i	$0.592134\pi$
$338$	0	0
$339$	0.983875	0.0534367
$340$	0	0
$341$	−0.786951	−0.0426158
$342$	0	0
$343$	−2.87456	−0.155212
$344$	0	0
$345$	−0.807397	−0.0434688
$346$	0	0
$347$	−20.1408	−1.08122	−0.540608	−	0.841275i	$-0.681806\pi$
$347$	−20.1408	−1.08122	−0.540608	+	0.841275i	$0.681806\pi$
$348$	0	0
$349$	3.76926	0.201764	0.100882	−	0.994898i	$-0.467834\pi$
$349$	3.76926	0.201764	0.100882	+	0.994898i	$0.467834\pi$
$350$	0	0
$351$	5.12983	0.273810
$352$	0	0
$353$	14.0208	0.746253	0.373126	−	0.927781i	$-0.378286\pi$
$353$	14.0208	0.746253	0.373126	+	0.927781i	$0.378286\pi$
$354$	0	0
$355$	1.60089	0.0849665
$356$	0	0
$357$	−2.03949	−0.107941
$358$	0	0
$359$	2.63686	0.139168	0.0695841	−	0.997576i	$-0.477833\pi$
$359$	2.63686	0.139168	0.0695841	+	0.997576i	$0.477833\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.9403	−0.731679
$364$	0	0
$365$	−13.8576	−0.725342
$366$	0	0
$367$	−22.3934	−1.16893	−0.584464	−	0.811420i	$-0.698695\pi$
$367$	−22.3934	−1.16893	−0.584464	+	0.811420i	$0.698695\pi$
$368$	0	0
$369$	13.7880	0.717774
$370$	0	0
$371$	−0.205950	−0.0106924
$372$	0	0
$373$	−28.0122	−1.45041	−0.725207	−	0.688530i	$-0.758256\pi$
$373$	−28.0122	−1.45041	−0.725207	+	0.688530i	$0.758256\pi$
$374$	0	0
$375$	−11.1723	−0.576934
$376$	0	0
$377$	−5.78263	−0.297820
$378$	0	0
$379$	−21.4513	−1.10188	−0.550939	−	0.834546i	$-0.685730\pi$
$379$	−21.4513	−1.10188	−0.550939	+	0.834546i	$0.685730\pi$
$380$	0	0
$381$	−26.4469	−1.35492
$382$	0	0
$383$	−4.76812	−0.243640	−0.121820	−	0.992552i	$-0.538873\pi$
$383$	−4.76812	−0.243640	−0.121820	+	0.992552i	$0.538873\pi$
$384$	0	0
$385$	−0.151377	−0.00771487
$386$	0	0
$387$	3.89655	0.198073
$388$	0	0
$389$	34.3614	1.74219	0.871096	−	0.491112i	$-0.163410\pi$
$389$	34.3614	1.74219	0.871096	+	0.491112i	$0.163410\pi$
$390$	0	0
$391$	4.87458	0.246518
$392$	0	0
$393$	0.291263	0.0146923
$394$	0	0
$395$	3.70352	0.186344
$396$	0	0
$397$	−35.0900	−1.76112	−0.880558	−	0.473938i	$-0.842832\pi$
$397$	−35.0900	−1.76112	−0.880558	+	0.473938i	$0.842832\pi$
$398$	0	0
$399$	0.277784	0.0139066
$400$	0	0
$401$	10.1506	0.506895	0.253448	−	0.967349i	$-0.418435\pi$
$401$	10.1506	0.506895	0.253448	+	0.967349i	$0.418435\pi$
$402$	0	0
$403$	−0.878150	−0.0437438
$404$	0	0
$405$	−3.66376	−0.182054
$406$	0	0
$407$	−6.61242	−0.327766
$408$	0	0
$409$	−10.0315	−0.496027	−0.248014	−	0.968757i	$-0.579778\pi$
$409$	−10.0315	−0.496027	−0.248014	+	0.968757i	$0.579778\pi$
$410$	0	0
$411$	−7.16601	−0.353473
$412$	0	0
$413$	−2.20701	−0.108600
$414$	0	0
$415$	12.1262	0.595253
$416$	0	0
$417$	19.0262	0.931719
$418$	0	0
$419$	31.0615	1.51745	0.758727	−	0.651408i	$-0.225822\pi$
$419$	31.0615	1.51745	0.758727	+	0.651408i	$0.225822\pi$
$420$	0	0
$421$	40.8279	1.98983	0.994915	−	0.100716i	$-0.0321135\pi$
$421$	40.8279	1.98983	0.994915	+	0.100716i	$0.0321135\pi$
$422$	0	0
$423$	−7.99177	−0.388573
$424$	0	0
$425$	30.7417	1.49119
$426$	0	0
$427$	−1.08747	−0.0526264
$428$	0	0
$429$	1.00030	0.0482950
$430$	0	0
$431$	19.4980	0.939188	0.469594	−	0.882882i	$-0.344400\pi$
$431$	19.4980	0.939188	0.469594	+	0.882882i	$0.344400\pi$
$432$	0	0
$433$	20.1716	0.969386	0.484693	−	0.874684i	$-0.338931\pi$
$433$	20.1716	0.969386	0.484693	+	0.874684i	$0.338931\pi$
$434$	0	0
$435$	7.73010	0.370630
$436$	0	0
$437$	−0.663932	−0.0317602
$438$	0	0
$439$	−12.0106	−0.573233	−0.286616	−	0.958045i	$-0.592531\pi$
$439$	−12.0106	−0.573233	−0.286616	+	0.958045i	$0.592531\pi$
$440$	0	0
$441$	8.21515	0.391198
$442$	0	0
$443$	−30.4628	−1.44733	−0.723665	−	0.690151i	$-0.757544\pi$
$443$	−30.4628	−1.44733	−0.723665	+	0.690151i	$0.757544\pi$
$444$	0	0
$445$	3.35465	0.159026
$446$	0	0
$447$	11.8771	0.561768
$448$	0	0
$449$	−25.8389	−1.21941	−0.609707	−	0.792627i	$-0.708713\pi$
$449$	−25.8389	−1.21941	−0.609707	+	0.792627i	$0.708713\pi$
$450$	0	0
$451$	9.51975	0.448267
$452$	0	0
$453$	−3.07961	−0.144693
$454$	0	0
$455$	−0.168920	−0.00791908
$456$	0	0
$457$	18.8732	0.882853	0.441426	−	0.897297i	$-0.354473\pi$
$457$	18.8732	0.882853	0.441426	+	0.897297i	$0.354473\pi$
$458$	0	0
$459$	41.4013	1.93245
$460$	0	0
$461$	−13.4504	−0.626446	−0.313223	−	0.949680i	$-0.601409\pi$
$461$	−13.4504	−0.626446	−0.313223	+	0.949680i	$0.601409\pi$
$462$	0	0
$463$	−30.9327	−1.43756	−0.718781	−	0.695236i	$-0.755300\pi$
$463$	−30.9327	−1.43756	−0.718781	+	0.695236i	$0.755300\pi$
$464$	0	0
$465$	1.17389	0.0544381
$466$	0	0
$467$	5.14249	0.237966	0.118983	−	0.992896i	$-0.462037\pi$
$467$	5.14249	0.237966	0.118983	+	0.992896i	$0.462037\pi$
$468$	0	0
$469$	−2.73859	−0.126456
$470$	0	0
$471$	6.37925	0.293940
$472$	0	0
$473$	2.69032	0.123701
$474$	0	0
$475$	−4.18711	−0.192118
$476$	0	0
$477$	1.18075	0.0540627
$478$	0	0
$479$	12.6593	0.578416	0.289208	−	0.957266i	$-0.406608\pi$
$479$	12.6593	0.578416	0.289208	+	0.957266i	$0.406608\pi$
$480$	0	0
$481$	−7.37874	−0.336441
$482$	0	0
$483$	−0.184430	−0.00839185
$484$	0	0
$485$	3.03656	0.137883
$486$	0	0
$487$	31.6602	1.43466	0.717330	−	0.696733i	$-0.245364\pi$
$487$	31.6602	1.43466	0.717330	+	0.696733i	$0.245364\pi$
$488$	0	0
$489$	−10.2937	−0.465499
$490$	0	0
$491$	−39.1894	−1.76859	−0.884296	−	0.466926i	$-0.845361\pi$
$491$	−39.1894	−1.76859	−0.884296	+	0.466926i	$0.845361\pi$
$492$	0	0
$493$	−46.6698	−2.10190
$494$	0	0
$495$	0.867871	0.0390079
$496$	0	0
$497$	0.365684	0.0164032
$498$	0	0
$499$	−23.2881	−1.04252	−0.521259	−	0.853398i	$-0.674538\pi$
$499$	−23.2881	−1.04252	−0.521259	+	0.853398i	$0.674538\pi$
$500$	0	0
$501$	−9.82472	−0.438936
$502$	0	0
$503$	10.3026	0.459372	0.229686	−	0.973265i	$-0.426230\pi$
$503$	10.3026	0.459372	0.229686	+	0.973265i	$0.426230\pi$
$504$	0	0
$505$	−12.9447	−0.576032
$506$	0	0
$507$	−16.4181	−0.729155
$508$	0	0
$509$	7.48813	0.331906	0.165953	−	0.986134i	$-0.446930\pi$
$509$	7.48813	0.331906	0.165953	+	0.986134i	$0.446930\pi$
$510$	0	0
$511$	−3.16544	−0.140031
$512$	0	0
$513$	−5.63898	−0.248967
$514$	0	0
$515$	−9.33260	−0.411243
$516$	0	0
$517$	−5.51782	−0.242674
$518$	0	0
$519$	24.4347	1.07256
$520$	0	0
$521$	−13.9473	−0.611042	−0.305521	−	0.952185i	$-0.598831\pi$
$521$	−13.9473	−0.611042	−0.305521	+	0.952185i	$0.598831\pi$
$522$	0	0
$523$	9.10564	0.398162	0.199081	−	0.979983i	$-0.436204\pi$
$523$	9.10564	0.398162	0.199081	+	0.979983i	$0.436204\pi$
$524$	0	0
$525$	−1.16311	−0.0507624
$526$	0	0
$527$	−7.08728	−0.308727
$528$	0	0
$529$	−22.5592	−0.980835
$530$	0	0
$531$	12.6532	0.549101
$532$	0	0
$533$	10.6230	0.460133
$534$	0	0
$535$	14.4783	0.625950
$536$	0	0
$537$	−21.8416	−0.942535
$538$	0	0
$539$	5.67205	0.244313
$540$	0	0
$541$	38.7392	1.66553	0.832763	−	0.553629i	$-0.186757\pi$
$541$	38.7392	1.66553	0.832763	+	0.553629i	$0.186757\pi$
$542$	0	0
$543$	32.7928	1.40727
$544$	0	0
$545$	13.8491	0.593230
$546$	0	0
$547$	−39.4695	−1.68760	−0.843798	−	0.536661i	$-0.819685\pi$
$547$	−39.4695	−1.68760	−0.843798	+	0.536661i	$0.819685\pi$
$548$	0	0
$549$	6.23468	0.266090
$550$	0	0
$551$	6.35656	0.270799
$552$	0	0
$553$	0.845977	0.0359746
$554$	0	0
$555$	9.86376	0.418693
$556$	0	0
$557$	−6.56300	−0.278083	−0.139042	−	0.990287i	$-0.544402\pi$
$557$	−6.56300	−0.278083	−0.139042	+	0.990287i	$0.544402\pi$
$558$	0	0
$559$	3.00210	0.126975
$560$	0	0
$561$	8.07313	0.340848
$562$	0	0
$563$	17.5000	0.737539	0.368769	−	0.929521i	$-0.379779\pi$
$563$	17.5000	0.737539	0.368769	+	0.929521i	$0.379779\pi$
$564$	0	0
$565$	0.657673	0.0276685
$566$	0	0
$567$	−0.836895	−0.0351463
$568$	0	0
$569$	−5.77677	−0.242175	−0.121087	−	0.992642i	$-0.538638\pi$
$569$	−5.77677	−0.242175	−0.121087	+	0.992642i	$0.538638\pi$
$570$	0	0
$571$	18.7693	0.785472	0.392736	−	0.919651i	$-0.371529\pi$
$571$	18.7693	0.785472	0.392736	+	0.919651i	$0.371529\pi$
$572$	0	0
$573$	−5.92543	−0.247538
$574$	0	0
$575$	2.77996	0.115932
$576$	0	0
$577$	2.12878	0.0886223	0.0443112	−	0.999018i	$-0.485891\pi$
$577$	2.12878	0.0886223	0.0443112	+	0.999018i	$0.485891\pi$
$578$	0	0
$579$	−16.5087	−0.686080
$580$	0	0
$581$	2.76994	0.114916
$582$	0	0
$583$	0.815233	0.0337635
$584$	0	0
$585$	0.968449	0.0400404
$586$	0	0
$587$	45.6652	1.88481	0.942403	−	0.334480i	$-0.108561\pi$
$587$	45.6652	1.88481	0.942403	+	0.334480i	$0.108561\pi$
$588$	0	0
$589$	0.965308	0.0397748
$590$	0	0
$591$	13.3181	0.547832
$592$	0	0
$593$	21.1126	0.866991	0.433495	−	0.901156i	$-0.357280\pi$
$593$	21.1126	0.866991	0.433495	+	0.901156i	$0.357280\pi$
$594$	0	0
$595$	−1.36330	−0.0558898
$596$	0	0
$597$	−9.56350	−0.391408
$598$	0	0
$599$	−23.1006	−0.943864	−0.471932	−	0.881635i	$-0.656443\pi$
$599$	−23.1006	−0.943864	−0.471932	+	0.881635i	$0.656443\pi$
$600$	0	0
$601$	−31.9959	−1.30514	−0.652571	−	0.757728i	$-0.726310\pi$
$601$	−31.9959	−1.30514	−0.652571	+	0.757728i	$0.726310\pi$
$602$	0	0
$603$	15.7009	0.639388
$604$	0	0
$605$	−9.31845	−0.378849
$606$	0	0
$607$	−5.68708	−0.230831	−0.115416	−	0.993317i	$-0.536820\pi$
$607$	−5.68708	−0.230831	−0.115416	+	0.993317i	$0.536820\pi$
$608$	0	0
$609$	1.76575	0.0715518
$610$	0	0
$611$	−6.15728	−0.249097
$612$	0	0
$613$	15.1817	0.613182	0.306591	−	0.951841i	$-0.400812\pi$
$613$	15.1817	0.613182	0.306591	+	0.951841i	$0.400812\pi$
$614$	0	0
$615$	−14.2006	−0.572624
$616$	0	0
$617$	−5.56257	−0.223941	−0.111970	−	0.993712i	$-0.535716\pi$
$617$	−5.56257	−0.223941	−0.111970	+	0.993712i	$0.535716\pi$
$618$	0	0
$619$	−6.80898	−0.273676	−0.136838	−	0.990593i	$-0.543694\pi$
$619$	−6.80898	−0.273676	−0.136838	+	0.990593i	$0.543694\pi$
$620$	0	0
$621$	3.74390	0.150237
$622$	0	0
$623$	0.766287	0.0307006
$624$	0	0
$625$	13.4674	0.538696
$626$	0	0
$627$	−1.09958	−0.0439131
$628$	0	0
$629$	−59.5515	−2.37448
$630$	0	0
$631$	2.92463	0.116428	0.0582139	−	0.998304i	$-0.481459\pi$
$631$	2.92463	0.116428	0.0582139	+	0.998304i	$0.481459\pi$
$632$	0	0
$633$	5.76628	0.229189
$634$	0	0
$635$	−17.6785	−0.701550
$636$	0	0
$637$	6.32938	0.250779
$638$	0	0
$639$	−2.09653	−0.0829376
$640$	0	0
$641$	−36.0644	−1.42446	−0.712230	−	0.701946i	$-0.752315\pi$
$641$	−36.0644	−1.42446	−0.712230	+	0.701946i	$0.752315\pi$
$642$	0	0
$643$	−29.3962	−1.15927	−0.579637	−	0.814875i	$-0.696805\pi$
$643$	−29.3962	−1.15927	−0.579637	+	0.814875i	$0.696805\pi$
$644$	0	0
$645$	−4.01316	−0.158018
$646$	0	0
$647$	−25.0382	−0.984351	−0.492176	−	0.870496i	$-0.663798\pi$
$647$	−25.0382	−0.984351	−0.492176	+	0.870496i	$0.663798\pi$
$648$	0	0
$649$	8.73624	0.342927
$650$	0	0
$651$	0.268147	0.0105095
$652$	0	0
$653$	33.2213	1.30005	0.650024	−	0.759914i	$-0.274759\pi$
$653$	33.2213	1.30005	0.650024	+	0.759914i	$0.274759\pi$
$654$	0	0
$655$	0.194695	0.00760736
$656$	0	0
$657$	18.1480	0.708022
$658$	0	0
$659$	−0.338846	−0.0131996	−0.00659978	−	0.999978i	$-0.502101\pi$
$659$	−0.338846	−0.0131996	−0.00659978	+	0.999978i	$0.502101\pi$
$660$	0	0
$661$	−11.8728	−0.461800	−0.230900	−	0.972977i	$-0.574167\pi$
$661$	−11.8728	−0.461800	−0.230900	+	0.972977i	$0.574167\pi$
$662$	0	0
$663$	9.00872	0.349870
$664$	0	0
$665$	0.185685	0.00720057
$666$	0	0
$667$	−4.22032	−0.163412
$668$	0	0
$669$	29.3477	1.13465
$670$	0	0
$671$	4.30466	0.166179
$672$	0	0
$673$	17.3736	0.669701	0.334851	−	0.942271i	$-0.391314\pi$
$673$	17.3736	0.669701	0.334851	+	0.942271i	$0.391314\pi$
$674$	0	0
$675$	23.6110	0.908788
$676$	0	0
$677$	7.70210	0.296016	0.148008	−	0.988986i	$-0.452714\pi$
$677$	7.70210	0.296016	0.148008	+	0.988986i	$0.452714\pi$
$678$	0	0
$679$	0.693627	0.0266190
$680$	0	0
$681$	17.9587	0.688179
$682$	0	0
$683$	−8.11596	−0.310549	−0.155274	−	0.987871i	$-0.549626\pi$
$683$	−8.11596	−0.310549	−0.155274	+	0.987871i	$0.549626\pi$
$684$	0	0
$685$	−4.79014	−0.183022
$686$	0	0
$687$	−3.31211	−0.126365
$688$	0	0
$689$	0.909710	0.0346572
$690$	0	0
$691$	1.60018	0.0608739	0.0304369	−	0.999537i	$-0.490310\pi$
$691$	1.60018	0.0608739	0.0304369	+	0.999537i	$0.490310\pi$
$692$	0	0
$693$	0.198244	0.00753065
$694$	0	0
$695$	12.7181	0.482426
$696$	0	0
$697$	85.7349	3.24744
$698$	0	0
$699$	−29.1718	−1.10338
$700$	0	0
$701$	28.8951	1.09135	0.545677	−	0.837996i	$-0.316273\pi$
$701$	28.8951	1.09135	0.545677	+	0.837996i	$0.316273\pi$
$702$	0	0
$703$	8.11109	0.305915
$704$	0	0
$705$	8.23094	0.309995
$706$	0	0
$707$	−2.95690	−0.111206
$708$	0	0
$709$	−9.91093	−0.372213	−0.186106	−	0.982530i	$-0.559587\pi$
$709$	−9.91093	−0.372213	−0.186106	+	0.982530i	$0.559587\pi$
$710$	0	0
$711$	−4.85015	−0.181895
$712$	0	0
$713$	−0.640899	−0.0240019
$714$	0	0
$715$	0.668654	0.0250062
$716$	0	0
$717$	−32.7732	−1.22394
$718$	0	0
$719$	2.21068	0.0824444	0.0412222	−	0.999150i	$-0.486875\pi$
$719$	2.21068	0.0824444	0.0412222	+	0.999150i	$0.486875\pi$
$720$	0	0
$721$	−2.13180	−0.0793924
$722$	0	0
$723$	21.5907	0.802966
$724$	0	0
$725$	−26.6156	−0.988478
$726$	0	0
$727$	24.4508	0.906832	0.453416	−	0.891299i	$-0.350205\pi$
$727$	24.4508	0.906832	0.453416	+	0.891299i	$0.350205\pi$
$728$	0	0
$729$	27.6156	1.02280
$730$	0	0
$731$	24.2291	0.896144
$732$	0	0
$733$	11.4307	0.422203	0.211101	−	0.977464i	$-0.432295\pi$
$733$	11.4307	0.422203	0.211101	+	0.977464i	$0.432295\pi$
$734$	0	0
$735$	−8.46100	−0.312089
$736$	0	0
$737$	10.8405	0.399314
$738$	0	0
$739$	12.0871	0.444632	0.222316	−	0.974975i	$-0.428638\pi$
$739$	12.0871	0.444632	0.222316	+	0.974975i	$0.428638\pi$
$740$	0	0
$741$	−1.22701	−0.0450755
$742$	0	0
$743$	11.0178	0.404203	0.202102	−	0.979365i	$-0.435223\pi$
$743$	11.0178	0.404203	0.202102	+	0.979365i	$0.435223\pi$
$744$	0	0
$745$	7.93927	0.290873
$746$	0	0
$747$	−15.8806	−0.581040
$748$	0	0
$749$	3.30720	0.120842
$750$	0	0
$751$	−43.1699	−1.57529	−0.787647	−	0.616127i	$-0.788701\pi$
$751$	−43.1699	−1.57529	−0.787647	+	0.616127i	$0.788701\pi$
$752$	0	0
$753$	25.2485	0.920105
$754$	0	0
$755$	−2.05857	−0.0749191
$756$	0	0
$757$	36.4535	1.32492	0.662462	−	0.749095i	$-0.269511\pi$
$757$	36.4535	1.32492	0.662462	+	0.749095i	$0.269511\pi$
$758$	0	0
$759$	0.730049	0.0264991
$760$	0	0
$761$	34.2895	1.24299	0.621497	−	0.783417i	$-0.286525\pi$
$761$	34.2895	1.24299	0.621497	+	0.783417i	$0.286525\pi$
$762$	0	0
$763$	3.16348	0.114526
$764$	0	0
$765$	7.81605	0.282590
$766$	0	0
$767$	9.74868	0.352004
$768$	0	0
$769$	29.3343	1.05782	0.528911	−	0.848677i	$-0.322601\pi$
$769$	29.3343	1.05782	0.528911	+	0.848677i	$0.322601\pi$
$770$	0	0
$771$	17.4284	0.627669
$772$	0	0
$773$	48.0940	1.72982	0.864910	−	0.501927i	$-0.167375\pi$
$773$	48.0940	1.72982	0.864910	+	0.501927i	$0.167375\pi$
$774$	0	0
$775$	−4.04185	−0.145187
$776$	0	0
$777$	2.25313	0.0808306
$778$	0	0
$779$	−11.6773	−0.418384
$780$	0	0
$781$	−1.44753	−0.0517966
$782$	0	0
$783$	−35.8445	−1.28098
$784$	0	0
$785$	4.26422	0.152197
$786$	0	0
$787$	−6.54754	−0.233395	−0.116697	−	0.993168i	$-0.537231\pi$
$787$	−6.54754	−0.233395	−0.116697	+	0.993168i	$0.537231\pi$
$788$	0	0
$789$	−8.02235	−0.285603
$790$	0	0
$791$	0.150229	0.00534153
$792$	0	0
$793$	4.80352	0.170578
$794$	0	0
$795$	−1.21608	−0.0431300
$796$	0	0
$797$	39.4505	1.39741	0.698705	−	0.715410i	$-0.253760\pi$
$797$	39.4505	1.39741	0.698705	+	0.715410i	$0.253760\pi$
$798$	0	0
$799$	−49.6935	−1.75803
$800$	0	0
$801$	−4.39327	−0.155228
$802$	0	0
$803$	12.5301	0.442178
$804$	0	0
$805$	−0.123282	−0.00434514
$806$	0	0
$807$	−26.2519	−0.924110
$808$	0	0
$809$	−49.6187	−1.74450	−0.872251	−	0.489059i	$-0.837340\pi$
$809$	−49.6187	−1.74450	−0.872251	+	0.489059i	$0.837340\pi$
$810$	0	0
$811$	23.0566	0.809627	0.404814	−	0.914399i	$-0.367336\pi$
$811$	23.0566	0.809627	0.404814	+	0.914399i	$0.367336\pi$
$812$	0	0
$813$	−17.2339	−0.604419
$814$	0	0
$815$	−6.88086	−0.241026
$816$	0	0
$817$	−3.30007	−0.115455
$818$	0	0
$819$	0.221218	0.00772999
$820$	0	0
$821$	37.5732	1.31131	0.655657	−	0.755059i	$-0.272392\pi$
$821$	37.5732	1.31131	0.655657	+	0.755059i	$0.272392\pi$
$822$	0	0
$823$	−45.6617	−1.59167	−0.795833	−	0.605516i	$-0.792967\pi$
$823$	−45.6617	−1.59167	−0.795833	+	0.605516i	$0.792967\pi$
$824$	0	0
$825$	4.60407	0.160293
$826$	0	0
$827$	−18.1671	−0.631732	−0.315866	−	0.948804i	$-0.602295\pi$
$827$	−18.1671	−0.631732	−0.315866	+	0.948804i	$0.602295\pi$
$828$	0	0
$829$	−39.0896	−1.35764	−0.678818	−	0.734306i	$-0.737508\pi$
$829$	−39.0896	−1.35764	−0.678818	+	0.734306i	$0.737508\pi$
$830$	0	0
$831$	−7.50288	−0.260272
$832$	0	0
$833$	51.0825	1.76990
$834$	0	0
$835$	−6.56735	−0.227273
$836$	0	0
$837$	−5.44335	−0.188150
$838$	0	0
$839$	−17.8442	−0.616052	−0.308026	−	0.951378i	$-0.599668\pi$
$839$	−17.8442	−0.616052	−0.308026	+	0.951378i	$0.599668\pi$
$840$	0	0
$841$	11.4059	0.393305
$842$	0	0
$843$	28.9147	0.995874
$844$	0	0
$845$	−10.9747	−0.377542
$846$	0	0
$847$	−2.12857	−0.0731385
$848$	0	0
$849$	−21.0347	−0.721910
$850$	0	0
$851$	−5.38521	−0.184603
$852$	0	0
$853$	16.4617	0.563639	0.281819	−	0.959468i	$-0.409062\pi$
$853$	16.4617	0.563639	0.281819	+	0.959468i	$0.409062\pi$
$854$	0	0
$855$	−1.06457	−0.0364075
$856$	0	0
$857$	−27.3347	−0.933735	−0.466867	−	0.884327i	$-0.654617\pi$
$857$	−27.3347	−0.933735	−0.466867	+	0.884327i	$0.654617\pi$
$858$	0	0
$859$	−22.2923	−0.760605	−0.380302	−	0.924862i	$-0.624180\pi$
$859$	−22.2923	−0.760605	−0.380302	+	0.924862i	$0.624180\pi$
$860$	0	0
$861$	−3.24378	−0.110548
$862$	0	0
$863$	−49.5819	−1.68779	−0.843894	−	0.536509i	$-0.819743\pi$
$863$	−49.5819	−1.68779	−0.843894	+	0.536509i	$0.819743\pi$
$864$	0	0
$865$	16.3334	0.555353
$866$	0	0
$867$	49.7771	1.69052
$868$	0	0
$869$	−3.34873	−0.113598
$870$	0	0
$871$	12.0968	0.409883
$872$	0	0
$873$	−3.97669	−0.134591
$874$	0	0
$875$	−1.70591	−0.0576703
$876$	0	0
$877$	57.3664	1.93713	0.968563	−	0.248769i	$-0.0800259\pi$
$877$	57.3664	1.93713	0.968563	+	0.248769i	$0.0800259\pi$
$878$	0	0
$879$	2.74058	0.0924375
$880$	0	0
$881$	5.26802	0.177484	0.0887420	−	0.996055i	$-0.471715\pi$
$881$	5.26802	0.177484	0.0887420	+	0.996055i	$0.471715\pi$
$882$	0	0
$883$	5.07132	0.170663	0.0853317	−	0.996353i	$-0.472805\pi$
$883$	5.07132	0.170663	0.0853317	+	0.996353i	$0.472805\pi$
$884$	0	0
$885$	−13.0318	−0.438061
$886$	0	0
$887$	53.1241	1.78373	0.891866	−	0.452300i	$-0.149396\pi$
$887$	53.1241	1.78373	0.891866	+	0.452300i	$0.149396\pi$
$888$	0	0
$889$	−4.03822	−0.135438
$890$	0	0
$891$	3.31278	0.110982
$892$	0	0
$893$	6.76840	0.226496
$894$	0	0
$895$	−14.6001	−0.488026
$896$	0	0
$897$	0.814654	0.0272005
$898$	0	0
$899$	6.13604	0.204648
$900$	0	0
$901$	7.34199	0.244597
$902$	0	0
$903$	−0.916706	−0.0305061
$904$	0	0
$905$	21.9204	0.728658
$906$	0	0
$907$	−49.5061	−1.64382	−0.821911	−	0.569616i	$-0.807092\pi$
$907$	−49.5061	−1.64382	−0.821911	+	0.569616i	$0.807092\pi$
$908$	0	0
$909$	16.9525	0.562278
$910$	0	0
$911$	24.3579	0.807013	0.403506	−	0.914977i	$-0.367791\pi$
$911$	24.3579	0.807013	0.403506	+	0.914977i	$0.367791\pi$
$912$	0	0
$913$	−10.9645	−0.362874
$914$	0	0
$915$	−6.42126	−0.212280
$916$	0	0
$917$	0.0444733	0.00146864
$918$	0	0
$919$	21.8767	0.721645	0.360823	−	0.932634i	$-0.382496\pi$
$919$	21.8767	0.721645	0.360823	+	0.932634i	$0.382496\pi$
$920$	0	0
$921$	19.9711	0.658072
$922$	0	0
$923$	−1.61528	−0.0531676
$924$	0	0
$925$	−33.9620	−1.11666
$926$	0	0
$927$	12.2220	0.401424
$928$	0	0
$929$	−22.9653	−0.753468	−0.376734	−	0.926322i	$-0.622953\pi$
$929$	−22.9653	−0.753468	−0.376734	+	0.926322i	$0.622953\pi$
$930$	0	0
$931$	−6.95758	−0.228026
$932$	0	0
$933$	−5.98375	−0.195899
$934$	0	0
$935$	5.39650	0.176484
$936$	0	0
$937$	17.9101	0.585097	0.292549	−	0.956251i	$-0.405497\pi$
$937$	17.9101	0.585097	0.292549	+	0.956251i	$0.405497\pi$
$938$	0	0
$939$	−15.2463	−0.497545
$940$	0	0
$941$	37.8607	1.23422	0.617111	−	0.786876i	$-0.288303\pi$
$941$	37.8607	1.23422	0.617111	+	0.786876i	$0.288303\pi$
$942$	0	0
$943$	7.75296	0.252471
$944$	0	0
$945$	−1.04708	−0.0340614
$946$	0	0
$947$	−59.6831	−1.93944	−0.969719	−	0.244222i	$-0.921468\pi$
$947$	−59.6831	−1.93944	−0.969719	+	0.244222i	$0.921468\pi$
$948$	0	0
$949$	13.9822	0.453882
$950$	0	0
$951$	4.14567	0.134433
$952$	0	0
$953$	−7.44469	−0.241157	−0.120579	−	0.992704i	$-0.538475\pi$
$953$	−7.44469	−0.241157	−0.120579	+	0.992704i	$0.538475\pi$
$954$	0	0
$955$	−3.96086	−0.128171
$956$	0	0
$957$	−6.98957	−0.225941
$958$	0	0
$959$	−1.09419	−0.0353332
$960$	0	0
$961$	−30.0682	−0.969941
$962$	0	0
$963$	−18.9608	−0.611003
$964$	0	0
$965$	−11.0353	−0.355239
$966$	0	0
$967$	7.73084	0.248607	0.124303	−	0.992244i	$-0.460330\pi$
$967$	7.73084	0.248607	0.124303	+	0.992244i	$0.460330\pi$
$968$	0	0
$969$	−9.90285	−0.318125
$970$	0	0
$971$	−9.96000	−0.319632	−0.159816	−	0.987147i	$-0.551090\pi$
$971$	−9.96000	−0.319632	−0.159816	+	0.987147i	$0.551090\pi$
$972$	0	0
$973$	2.90514	0.0931345
$974$	0	0
$975$	5.13764	0.164536
$976$	0	0
$977$	−37.5937	−1.20273	−0.601365	−	0.798975i	$-0.705376\pi$
$977$	−37.5937	−1.20273	−0.601365	+	0.798975i	$0.705376\pi$
$978$	0	0
$979$	−3.03328	−0.0969440
$980$	0	0
$981$	−18.1368	−0.579065
$982$	0	0
$983$	−4.41185	−0.140716	−0.0703581	−	0.997522i	$-0.522414\pi$
$983$	−4.41185	−0.140716	−0.0703581	+	0.997522i	$0.522414\pi$
$984$	0	0
$985$	8.90249	0.283657
$986$	0	0
$987$	1.88015	0.0598460
$988$	0	0
$989$	2.19102	0.0696704
$990$	0	0
$991$	−51.1144	−1.62370	−0.811851	−	0.583864i	$-0.801540\pi$
$991$	−51.1144	−1.62370	−0.811851	+	0.583864i	$0.801540\pi$
$992$	0	0
$993$	40.7909	1.29446
$994$	0	0
$995$	−6.39274	−0.202663
$996$	0	0
$997$	6.17988	0.195719	0.0978593	−	0.995200i	$-0.468800\pi$
$997$	6.17988	0.195719	0.0978593	+	0.995200i	$0.468800\pi$
$998$	0	0
$999$	−45.7382	−1.44709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.c.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.c.1.13	✓	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)

\(f(q)\)	\(=\)	\(q+1.34880 q^{3} +0.901606 q^{5} +0.205950 q^{7} -1.18075 q^{9} +O(q^{10})\)	\(q+1.34880 q^{3} +0.901606 q^{5} +0.205950 q^{7} -1.18075 q^{9} -0.815233 q^{11} -0.909710 q^{13} +1.21608 q^{15} -7.34199 q^{17} +1.00000 q^{19} +0.277784 q^{21} -0.663932 q^{23} -4.18711 q^{25} -5.63898 q^{27} +6.35656 q^{29} +0.965308 q^{31} -1.09958 q^{33} +0.185685 q^{35} +8.11109 q^{37} -1.22701 q^{39} -11.6773 q^{41} -3.30007 q^{43} -1.06457 q^{45} +6.76840 q^{47} -6.95758 q^{49} -9.90285 q^{51} -1.00000 q^{53} -0.735019 q^{55} +1.34880 q^{57} -10.7162 q^{59} -5.28028 q^{61} -0.243174 q^{63} -0.820200 q^{65} -13.2974 q^{67} -0.895510 q^{69} +1.77560 q^{71} -15.3700 q^{73} -5.64756 q^{75} -0.167897 q^{77} +4.10769 q^{79} -4.06359 q^{81} +13.4496 q^{83} -6.61958 q^{85} +8.57371 q^{87} +3.72075 q^{89} -0.187354 q^{91} +1.30200 q^{93} +0.901606 q^{95} +3.36795 q^{97} +0.962584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} - 5 q^{5} - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 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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