LMFDB - Embedded newform 4028.2.a.e.1.18

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:	$0$
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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$3.01880$ 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)

$f(q)$	$=$	$q+3.01880 q^{3} -3.20894 q^{5} +2.67289 q^{7} +6.11314 q^{9} +O(q^{10})$	$q+3.01880 q^{3} -3.20894 q^{5} +2.67289 q^{7} +6.11314 q^{9} -0.155633 q^{11} +0.231564 q^{13} -9.68714 q^{15} -4.83108 q^{17} +1.00000 q^{19} +8.06892 q^{21} +6.81201 q^{23} +5.29729 q^{25} +9.39794 q^{27} +0.627008 q^{29} +3.31172 q^{31} -0.469825 q^{33} -8.57715 q^{35} +2.21910 q^{37} +0.699044 q^{39} +4.43432 q^{41} +8.41063 q^{43} -19.6167 q^{45} -2.94189 q^{47} +0.144357 q^{49} -14.5841 q^{51} +1.00000 q^{53} +0.499417 q^{55} +3.01880 q^{57} +1.63058 q^{59} +5.80719 q^{61} +16.3398 q^{63} -0.743074 q^{65} +1.73258 q^{67} +20.5641 q^{69} +2.95365 q^{71} +4.75410 q^{73} +15.9914 q^{75} -0.415990 q^{77} +16.4477 q^{79} +10.0311 q^{81} +3.89842 q^{83} +15.5027 q^{85} +1.89281 q^{87} -14.1067 q^{89} +0.618945 q^{91} +9.99742 q^{93} -3.20894 q^{95} +2.93455 q^{97} -0.951406 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * 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
$3$	3.01880	1.74290	0.871452	−	0.490481i	$-0.163179\pi$
$3$	3.01880	1.74290	0.871452	+	0.490481i	$0.163179\pi$
$4$	0	0
$5$	−3.20894	−1.43508	−0.717541	−	0.696517i	$-0.754732\pi$
$5$	−3.20894	−1.43508	−0.717541	+	0.696517i	$0.754732\pi$
$6$	0	0
$7$	2.67289	1.01026	0.505129	−	0.863044i	$-0.331445\pi$
$7$	2.67289	1.01026	0.505129	+	0.863044i	$0.331445\pi$
$8$	0	0
$9$	6.11314	2.03771
$10$	0	0
$11$	−0.155633	−0.0469251	−0.0234626	−	0.999725i	$-0.507469\pi$
$11$	−0.155633	−0.0469251	−0.0234626	+	0.999725i	$0.507469\pi$
$12$	0	0
$13$	0.231564	0.0642242	0.0321121	−	0.999484i	$-0.489777\pi$
$13$	0.231564	0.0642242	0.0321121	+	0.999484i	$0.489777\pi$
$14$	0	0
$15$	−9.68714	−2.50121
$16$	0	0
$17$	−4.83108	−1.17171	−0.585855	−	0.810416i	$-0.699241\pi$
$17$	−4.83108	−1.17171	−0.585855	+	0.810416i	$0.699241\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.06892	1.76078
$22$	0	0
$23$	6.81201	1.42040	0.710201	−	0.703999i	$-0.248604\pi$
$23$	6.81201	1.42040	0.710201	+	0.703999i	$0.248604\pi$
$24$	0	0
$25$	5.29729	1.05946
$26$	0	0
$27$	9.39794	1.80863
$28$	0	0
$29$	0.627008	0.116433	0.0582163	−	0.998304i	$-0.481459\pi$
$29$	0.627008	0.116433	0.0582163	+	0.998304i	$0.481459\pi$
$30$	0	0
$31$	3.31172	0.594803	0.297401	−	0.954753i	$-0.403880\pi$
$31$	3.31172	0.594803	0.297401	+	0.954753i	$0.403880\pi$
$32$	0	0
$33$	−0.469825	−0.0817859
$34$	0	0
$35$	−8.57715	−1.44980
$36$	0	0
$37$	2.21910	0.364817	0.182409	−	0.983223i	$-0.441611\pi$
$37$	2.21910	0.364817	0.182409	+	0.983223i	$0.441611\pi$
$38$	0	0
$39$	0.699044	0.111937
$40$	0	0
$41$	4.43432	0.692525	0.346262	−	0.938138i	$-0.387451\pi$
$41$	4.43432	0.692525	0.346262	+	0.938138i	$0.387451\pi$
$42$	0	0
$43$	8.41063	1.28261	0.641305	−	0.767286i	$-0.278393\pi$
$43$	8.41063	1.28261	0.641305	+	0.767286i	$0.278393\pi$
$44$	0	0
$45$	−19.6167	−2.92428
$46$	0	0
$47$	−2.94189	−0.429119	−0.214559	−	0.976711i	$-0.568832\pi$
$47$	−2.94189	−0.429119	−0.214559	+	0.976711i	$0.568832\pi$
$48$	0	0
$49$	0.144357	0.0206224
$50$	0	0
$51$	−14.5841	−2.04218
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	0.499417	0.0673413
$56$	0	0
$57$	3.01880	0.399850
$58$	0	0
$59$	1.63058	0.212284	0.106142	−	0.994351i	$-0.466150\pi$
$59$	1.63058	0.212284	0.106142	+	0.994351i	$0.466150\pi$
$60$	0	0
$61$	5.80719	0.743534	0.371767	−	0.928326i	$-0.378752\pi$
$61$	5.80719	0.743534	0.371767	+	0.928326i	$0.378752\pi$
$62$	0	0
$63$	16.3398	2.05862
$64$	0	0
$65$	−0.743074	−0.0921670
$66$	0	0
$67$	1.73258	0.211669	0.105834	−	0.994384i	$-0.466249\pi$
$67$	1.73258	0.211669	0.105834	+	0.994384i	$0.466249\pi$
$68$	0	0
$69$	20.5641	2.47562
$70$	0	0
$71$	2.95365	0.350534	0.175267	−	0.984521i	$-0.443921\pi$
$71$	2.95365	0.350534	0.175267	+	0.984521i	$0.443921\pi$
$72$	0	0
$73$	4.75410	0.556426	0.278213	−	0.960519i	$-0.410258\pi$
$73$	4.75410	0.556426	0.278213	+	0.960519i	$0.410258\pi$
$74$	0	0
$75$	15.9914	1.84653
$76$	0	0
$77$	−0.415990	−0.0474065
$78$	0	0
$79$	16.4477	1.85051	0.925256	−	0.379345i	$-0.123850\pi$
$79$	16.4477	1.85051	0.925256	+	0.379345i	$0.123850\pi$
$80$	0	0
$81$	10.0311	1.11456
$82$	0	0
$83$	3.89842	0.427907	0.213954	−	0.976844i	$-0.431366\pi$
$83$	3.89842	0.427907	0.213954	+	0.976844i	$0.431366\pi$
$84$	0	0
$85$	15.5027	1.68150
$86$	0	0
$87$	1.89281	0.202931
$88$	0	0
$89$	−14.1067	−1.49530	−0.747651	−	0.664091i	$-0.768819\pi$
$89$	−14.1067	−1.49530	−0.747651	+	0.664091i	$0.768819\pi$
$90$	0	0
$91$	0.618945	0.0648831
$92$	0	0
$93$	9.99742	1.03668
$94$	0	0
$95$	−3.20894	−0.329230
$96$	0	0
$97$	2.93455	0.297958	0.148979	−	0.988840i	$-0.452401\pi$
$97$	2.93455	0.297958	0.148979	+	0.988840i	$0.452401\pi$
$98$	0	0
$99$	−0.951406	−0.0956199
$100$	0	0
$101$	8.26979	0.822875	0.411437	−	0.911438i	$-0.365027\pi$
$101$	8.26979	0.822875	0.411437	+	0.911438i	$0.365027\pi$
$102$	0	0
$103$	−4.33551	−0.427191	−0.213595	−	0.976922i	$-0.568517\pi$
$103$	−4.33551	−0.427191	−0.213595	+	0.976922i	$0.568517\pi$
$104$	0	0
$105$	−25.8927	−2.52687
$106$	0	0
$107$	−4.76560	−0.460708	−0.230354	−	0.973107i	$-0.573988\pi$
$107$	−4.76560	−0.460708	−0.230354	+	0.973107i	$0.573988\pi$
$108$	0	0
$109$	−1.74939	−0.167561	−0.0837806	−	0.996484i	$-0.526700\pi$
$109$	−1.74939	−0.167561	−0.0837806	+	0.996484i	$0.526700\pi$
$110$	0	0
$111$	6.69900	0.635841
$112$	0	0
$113$	−3.00670	−0.282847	−0.141423	−	0.989949i	$-0.545168\pi$
$113$	−3.00670	−0.282847	−0.141423	+	0.989949i	$0.545168\pi$
$114$	0	0
$115$	−21.8593	−2.03839
$116$	0	0
$117$	1.41558	0.130871
$118$	0	0
$119$	−12.9130	−1.18373
$120$	0	0
$121$	−10.9758	−0.997798
$122$	0	0
$123$	13.3863	1.20700
$124$	0	0
$125$	−0.953986	−0.0853271
$126$	0	0
$127$	−18.8069	−1.66884	−0.834420	−	0.551130i	$-0.814197\pi$
$127$	−18.8069	−1.66884	−0.834420	+	0.551130i	$0.814197\pi$
$128$	0	0
$129$	25.3900	2.23546
$130$	0	0
$131$	−8.79961	−0.768826	−0.384413	−	0.923161i	$-0.625596\pi$
$131$	−8.79961	−0.768826	−0.384413	+	0.923161i	$0.625596\pi$
$132$	0	0
$133$	2.67289	0.231769
$134$	0	0
$135$	−30.1574	−2.59554
$136$	0	0
$137$	1.49188	0.127460	0.0637301	−	0.997967i	$-0.479700\pi$
$137$	1.49188	0.127460	0.0637301	+	0.997967i	$0.479700\pi$
$138$	0	0
$139$	−9.31582	−0.790157	−0.395079	−	0.918647i	$-0.629283\pi$
$139$	−9.31582	−0.790157	−0.395079	+	0.918647i	$0.629283\pi$
$140$	0	0
$141$	−8.88097	−0.747912
$142$	0	0
$143$	−0.0360390	−0.00301373
$144$	0	0
$145$	−2.01203	−0.167090
$146$	0	0
$147$	0.435784	0.0359428
$148$	0	0
$149$	20.6194	1.68920	0.844602	−	0.535394i	$-0.179837\pi$
$149$	20.6194	1.68920	0.844602	+	0.535394i	$0.179837\pi$
$150$	0	0
$151$	15.9897	1.30123	0.650613	−	0.759409i	$-0.274512\pi$
$151$	15.9897	1.30123	0.650613	+	0.759409i	$0.274512\pi$
$152$	0	0
$153$	−29.5331	−2.38761
$154$	0	0
$155$	−10.6271	−0.853590
$156$	0	0
$157$	6.27204	0.500563	0.250282	−	0.968173i	$-0.419477\pi$
$157$	6.27204	0.500563	0.250282	+	0.968173i	$0.419477\pi$
$158$	0	0
$159$	3.01880	0.239406
$160$	0	0
$161$	18.2078	1.43497
$162$	0	0
$163$	9.82509	0.769560	0.384780	−	0.923008i	$-0.374277\pi$
$163$	9.82509	0.769560	0.384780	+	0.923008i	$0.374277\pi$
$164$	0	0
$165$	1.50764	0.117369
$166$	0	0
$167$	−2.84752	−0.220348	−0.110174	−	0.993912i	$-0.535141\pi$
$167$	−2.84752	−0.220348	−0.110174	+	0.993912i	$0.535141\pi$
$168$	0	0
$169$	−12.9464	−0.995875
$170$	0	0
$171$	6.11314	0.467483
$172$	0	0
$173$	−1.57284	−0.119581	−0.0597903	−	0.998211i	$-0.519043\pi$
$173$	−1.57284	−0.119581	−0.0597903	+	0.998211i	$0.519043\pi$
$174$	0	0
$175$	14.1591	1.07033
$176$	0	0
$177$	4.92240	0.369990
$178$	0	0
$179$	−13.1055	−0.979548	−0.489774	−	0.871849i	$-0.662921\pi$
$179$	−13.1055	−0.979548	−0.489774	+	0.871849i	$0.662921\pi$
$180$	0	0
$181$	16.5376	1.22923	0.614616	−	0.788826i	$-0.289311\pi$
$181$	16.5376	1.22923	0.614616	+	0.788826i	$0.289311\pi$
$182$	0	0
$183$	17.5307	1.29591
$184$	0	0
$185$	−7.12094	−0.523542
$186$	0	0
$187$	0.751876	0.0549826
$188$	0	0
$189$	25.1197	1.82719
$190$	0	0
$191$	−1.86873	−0.135217	−0.0676083	−	0.997712i	$-0.521537\pi$
$191$	−1.86873	−0.135217	−0.0676083	+	0.997712i	$0.521537\pi$
$192$	0	0
$193$	2.79937	0.201503	0.100752	−	0.994912i	$-0.467875\pi$
$193$	2.79937	0.201503	0.100752	+	0.994912i	$0.467875\pi$
$194$	0	0
$195$	−2.24319	−0.160638
$196$	0	0
$197$	−0.649981	−0.0463092	−0.0231546	−	0.999732i	$-0.507371\pi$
$197$	−0.649981	−0.0463092	−0.0231546	+	0.999732i	$0.507371\pi$
$198$	0	0
$199$	−3.27874	−0.232424	−0.116212	−	0.993224i	$-0.537075\pi$
$199$	−3.27874	−0.232424	−0.116212	+	0.993224i	$0.537075\pi$
$200$	0	0
$201$	5.23032	0.368918
$202$	0	0
$203$	1.67593	0.117627
$204$	0	0
$205$	−14.2295	−0.993829
$206$	0	0
$207$	41.6428	2.89437
$208$	0	0
$209$	−0.155633	−0.0107654
$210$	0	0
$211$	−10.4774	−0.721296	−0.360648	−	0.932702i	$-0.617444\pi$
$211$	−10.4774	−0.721296	−0.360648	+	0.932702i	$0.617444\pi$
$212$	0	0
$213$	8.91648	0.610947
$214$	0	0
$215$	−26.9892	−1.84065
$216$	0	0
$217$	8.85188	0.600905
$218$	0	0
$219$	14.3517	0.969797
$220$	0	0
$221$	−1.11870	−0.0752522
$222$	0	0
$223$	16.8126	1.12585	0.562926	−	0.826507i	$-0.309675\pi$
$223$	16.8126	1.12585	0.562926	+	0.826507i	$0.309675\pi$
$224$	0	0
$225$	32.3831	2.15887
$226$	0	0
$227$	−18.9700	−1.25909	−0.629543	−	0.776966i	$-0.716758\pi$
$227$	−18.9700	−1.25909	−0.629543	+	0.776966i	$0.716758\pi$
$228$	0	0
$229$	−5.84638	−0.386339	−0.193170	−	0.981165i	$-0.561877\pi$
$229$	−5.84638	−0.386339	−0.193170	+	0.981165i	$0.561877\pi$
$230$	0	0
$231$	−1.25579	−0.0826250
$232$	0	0
$233$	2.58069	0.169067	0.0845333	−	0.996421i	$-0.473060\pi$
$233$	2.58069	0.169067	0.0845333	+	0.996421i	$0.473060\pi$
$234$	0	0
$235$	9.44034	0.615820
$236$	0	0
$237$	49.6523	3.22526
$238$	0	0
$239$	−8.36741	−0.541243	−0.270621	−	0.962686i	$-0.587229\pi$
$239$	−8.36741	−0.541243	−0.270621	+	0.962686i	$0.587229\pi$
$240$	0	0
$241$	28.9935	1.86764	0.933818	−	0.357749i	$-0.116456\pi$
$241$	28.9935	1.86764	0.933818	+	0.357749i	$0.116456\pi$
$242$	0	0
$243$	2.08791	0.133940
$244$	0	0
$245$	−0.463232	−0.0295948
$246$	0	0
$247$	0.231564	0.0147340
$248$	0	0
$249$	11.7685	0.745801
$250$	0	0
$251$	−21.8278	−1.37776	−0.688879	−	0.724876i	$-0.741897\pi$
$251$	−21.8278	−1.37776	−0.688879	+	0.724876i	$0.741897\pi$
$252$	0	0
$253$	−1.06017	−0.0666525
$254$	0	0
$255$	46.7994	2.93069
$256$	0	0
$257$	−2.72805	−0.170171	−0.0850856	−	0.996374i	$-0.527116\pi$
$257$	−2.72805	−0.170171	−0.0850856	+	0.996374i	$0.527116\pi$
$258$	0	0
$259$	5.93141	0.368560
$260$	0	0
$261$	3.83299	0.237256
$262$	0	0
$263$	3.60010	0.221992	0.110996	−	0.993821i	$-0.464596\pi$
$263$	3.60010	0.221992	0.110996	+	0.993821i	$0.464596\pi$
$264$	0	0
$265$	−3.20894	−0.197124
$266$	0	0
$267$	−42.5851	−2.60617
$268$	0	0
$269$	11.3782	0.693741	0.346871	−	0.937913i	$-0.387244\pi$
$269$	11.3782	0.693741	0.346871	+	0.937913i	$0.387244\pi$
$270$	0	0
$271$	−19.7704	−1.20097	−0.600484	−	0.799637i	$-0.705025\pi$
$271$	−19.7704	−1.20097	−0.600484	+	0.799637i	$0.705025\pi$
$272$	0	0
$273$	1.86847	0.113085
$274$	0	0
$275$	−0.824433	−0.0497152
$276$	0	0
$277$	22.1246	1.32934	0.664671	−	0.747136i	$-0.268572\pi$
$277$	22.1246	1.32934	0.664671	+	0.747136i	$0.268572\pi$
$278$	0	0
$279$	20.2450	1.21204
$280$	0	0
$281$	20.2776	1.20966	0.604830	−	0.796354i	$-0.293241\pi$
$281$	20.2776	1.20966	0.604830	+	0.796354i	$0.293241\pi$
$282$	0	0
$283$	−9.73095	−0.578445	−0.289222	−	0.957262i	$-0.593397\pi$
$283$	−9.73095	−0.578445	−0.289222	+	0.957262i	$0.593397\pi$
$284$	0	0
$285$	−9.68714	−0.573817
$286$	0	0
$287$	11.8525	0.699629
$288$	0	0
$289$	6.33937	0.372904
$290$	0	0
$291$	8.85880	0.519312
$292$	0	0
$293$	−12.2458	−0.715407	−0.357703	−	0.933835i	$-0.616440\pi$
$293$	−12.2458	−0.715407	−0.357703	+	0.933835i	$0.616440\pi$
$294$	0	0
$295$	−5.23244	−0.304645
$296$	0	0
$297$	−1.46263	−0.0848704
$298$	0	0
$299$	1.57741	0.0912242
$300$	0	0
$301$	22.4807	1.29577
$302$	0	0
$303$	24.9648	1.43419
$304$	0	0
$305$	−18.6349	−1.06703
$306$	0	0
$307$	1.96945	0.112403	0.0562013	−	0.998419i	$-0.482101\pi$
$307$	1.96945	0.112403	0.0562013	+	0.998419i	$0.482101\pi$
$308$	0	0
$309$	−13.0880	−0.744552
$310$	0	0
$311$	−26.7938	−1.51934	−0.759668	−	0.650311i	$-0.774639\pi$
$311$	−26.7938	−1.51934	−0.759668	+	0.650311i	$0.774639\pi$
$312$	0	0
$313$	28.2347	1.59592	0.797959	−	0.602712i	$-0.205913\pi$
$313$	28.2347	1.59592	0.797959	+	0.602712i	$0.205913\pi$
$314$	0	0
$315$	−52.4333	−2.95428
$316$	0	0
$317$	0.996239	0.0559544	0.0279772	−	0.999609i	$-0.491093\pi$
$317$	0.996239	0.0559544	0.0279772	+	0.999609i	$0.491093\pi$
$318$	0	0
$319$	−0.0975832	−0.00546361
$320$	0	0
$321$	−14.3864	−0.802970
$322$	0	0
$323$	−4.83108	−0.268809
$324$	0	0
$325$	1.22666	0.0680429
$326$	0	0
$327$	−5.28106	−0.292043
$328$	0	0
$329$	−7.86335	−0.433521
$330$	0	0
$331$	25.5300	1.40326	0.701628	−	0.712543i	$-0.252457\pi$
$331$	25.5300	1.40326	0.701628	+	0.712543i	$0.252457\pi$
$332$	0	0
$333$	13.5656	0.743393
$334$	0	0
$335$	−5.55976	−0.303762
$336$	0	0
$337$	−24.9666	−1.36002	−0.680010	−	0.733203i	$-0.738025\pi$
$337$	−24.9666	−1.36002	−0.680010	+	0.733203i	$0.738025\pi$
$338$	0	0
$339$	−9.07663	−0.492975
$340$	0	0
$341$	−0.515413	−0.0279112
$342$	0	0
$343$	−18.3244	−0.989425
$344$	0	0
$345$	−65.9889	−3.55272
$346$	0	0
$347$	23.8875	1.28235	0.641174	−	0.767396i	$-0.278448\pi$
$347$	23.8875	1.28235	0.641174	+	0.767396i	$0.278448\pi$
$348$	0	0
$349$	9.80673	0.524942	0.262471	−	0.964940i	$-0.415463\pi$
$349$	9.80673	0.524942	0.262471	+	0.964940i	$0.415463\pi$
$350$	0	0
$351$	2.17622	0.116158
$352$	0	0
$353$	−8.99297	−0.478647	−0.239324	−	0.970940i	$-0.576926\pi$
$353$	−8.99297	−0.478647	−0.239324	+	0.970940i	$0.576926\pi$
$354$	0	0
$355$	−9.47810	−0.503045
$356$	0	0
$357$	−38.9816	−2.06313
$358$	0	0
$359$	−19.7883	−1.04439	−0.522194	−	0.852827i	$-0.674886\pi$
$359$	−19.7883	−1.04439	−0.522194	+	0.852827i	$0.674886\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−33.1337	−1.73907
$364$	0	0
$365$	−15.2556	−0.798516
$366$	0	0
$367$	−15.4906	−0.808604	−0.404302	−	0.914626i	$-0.632485\pi$
$367$	−15.4906	−0.808604	−0.404302	+	0.914626i	$0.632485\pi$
$368$	0	0
$369$	27.1076	1.41117
$370$	0	0
$371$	2.67289	0.138770
$372$	0	0
$373$	−17.2935	−0.895423	−0.447711	−	0.894178i	$-0.647761\pi$
$373$	−17.2935	−0.895423	−0.447711	+	0.894178i	$0.647761\pi$
$374$	0	0
$375$	−2.87989	−0.148717
$376$	0	0
$377$	0.145192	0.00747779
$378$	0	0
$379$	−21.2638	−1.09225	−0.546123	−	0.837705i	$-0.683897\pi$
$379$	−21.2638	−1.09225	−0.546123	+	0.837705i	$0.683897\pi$
$380$	0	0
$381$	−56.7741	−2.90863
$382$	0	0
$383$	5.18115	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$383$	5.18115	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$384$	0	0
$385$	1.33489	0.0680322
$386$	0	0
$387$	51.4154	2.61359
$388$	0	0
$389$	−3.14411	−0.159413	−0.0797063	−	0.996818i	$-0.525398\pi$
$389$	−3.14411	−0.159413	−0.0797063	+	0.996818i	$0.525398\pi$
$390$	0	0
$391$	−32.9094	−1.66430
$392$	0	0
$393$	−26.5642	−1.33999
$394$	0	0
$395$	−52.7797	−2.65563
$396$	0	0
$397$	−0.0843299	−0.00423240	−0.00211620	−	0.999998i	$-0.500674\pi$
$397$	−0.0843299	−0.00423240	−0.00211620	+	0.999998i	$0.500674\pi$
$398$	0	0
$399$	8.06892	0.403951
$400$	0	0
$401$	14.8474	0.741443	0.370722	−	0.928744i	$-0.379110\pi$
$401$	14.8474	0.741443	0.370722	+	0.928744i	$0.379110\pi$
$402$	0	0
$403$	0.766875	0.0382007
$404$	0	0
$405$	−32.1890	−1.59949
$406$	0	0
$407$	−0.345365	−0.0171191
$408$	0	0
$409$	−18.2089	−0.900374	−0.450187	−	0.892934i	$-0.648643\pi$
$409$	−18.2089	−0.900374	−0.450187	+	0.892934i	$0.648643\pi$
$410$	0	0
$411$	4.50369	0.222151
$412$	0	0
$413$	4.35837	0.214462
$414$	0	0
$415$	−12.5098	−0.614082
$416$	0	0
$417$	−28.1226	−1.37717
$418$	0	0
$419$	−14.1915	−0.693303	−0.346651	−	0.937994i	$-0.612681\pi$
$419$	−14.1915	−0.693303	−0.346651	+	0.937994i	$0.612681\pi$
$420$	0	0
$421$	−17.4860	−0.852214	−0.426107	−	0.904673i	$-0.640115\pi$
$421$	−17.4860	−0.852214	−0.426107	+	0.904673i	$0.640115\pi$
$422$	0	0
$423$	−17.9842	−0.874421
$424$	0	0
$425$	−25.5917	−1.24138
$426$	0	0
$427$	15.5220	0.751162
$428$	0	0
$429$	−0.108794	−0.00525264
$430$	0	0
$431$	−20.9424	−1.00876	−0.504381	−	0.863481i	$-0.668279\pi$
$431$	−20.9424	−1.00876	−0.504381	+	0.863481i	$0.668279\pi$
$432$	0	0
$433$	−5.15596	−0.247780	−0.123890	−	0.992296i	$-0.539537\pi$
$433$	−5.15596	−0.247780	−0.123890	+	0.992296i	$0.539537\pi$
$434$	0	0
$435$	−6.07392	−0.291222
$436$	0	0
$437$	6.81201	0.325863
$438$	0	0
$439$	−23.8471	−1.13816	−0.569081	−	0.822282i	$-0.692701\pi$
$439$	−23.8471	−1.13816	−0.569081	+	0.822282i	$0.692701\pi$
$440$	0	0
$441$	0.882473	0.0420225
$442$	0	0
$443$	−7.53299	−0.357903	−0.178952	−	0.983858i	$-0.557271\pi$
$443$	−7.53299	−0.357903	−0.178952	+	0.983858i	$0.557271\pi$
$444$	0	0
$445$	45.2674	2.14588
$446$	0	0
$447$	62.2457	2.94412
$448$	0	0
$449$	7.06110	0.333234	0.166617	−	0.986022i	$-0.446716\pi$
$449$	7.06110	0.333234	0.166617	+	0.986022i	$0.446716\pi$
$450$	0	0
$451$	−0.690127	−0.0324968
$452$	0	0
$453$	48.2698	2.26791
$454$	0	0
$455$	−1.98616	−0.0931125
$456$	0	0
$457$	−38.5242	−1.80209	−0.901044	−	0.433729i	$-0.857198\pi$
$457$	−38.5242	−1.80209	−0.901044	+	0.433729i	$0.857198\pi$
$458$	0	0
$459$	−45.4022	−2.11919
$460$	0	0
$461$	8.52894	0.397232	0.198616	−	0.980077i	$-0.436355\pi$
$461$	8.52894	0.397232	0.198616	+	0.980077i	$0.436355\pi$
$462$	0	0
$463$	12.9410	0.601420	0.300710	−	0.953716i	$-0.402776\pi$
$463$	12.9410	0.601420	0.300710	+	0.953716i	$0.402776\pi$
$464$	0	0
$465$	−32.0811	−1.48773
$466$	0	0
$467$	17.4592	0.807917	0.403959	−	0.914777i	$-0.367634\pi$
$467$	17.4592	0.807917	0.403959	+	0.914777i	$0.367634\pi$
$468$	0	0
$469$	4.63101	0.213840
$470$	0	0
$471$	18.9340	0.872434
$472$	0	0
$473$	−1.30897	−0.0601866
$474$	0	0
$475$	5.29729	0.243056
$476$	0	0
$477$	6.11314	0.279901
$478$	0	0
$479$	−13.2423	−0.605056	−0.302528	−	0.953141i	$-0.597831\pi$
$479$	−13.2423	−0.605056	−0.302528	+	0.953141i	$0.597831\pi$
$480$	0	0
$481$	0.513862	0.0234301
$482$	0	0
$483$	54.9656	2.50102
$484$	0	0
$485$	−9.41678	−0.427594
$486$	0	0
$487$	−7.96451	−0.360906	−0.180453	−	0.983584i	$-0.557756\pi$
$487$	−7.96451	−0.360906	−0.180453	+	0.983584i	$0.557756\pi$
$488$	0	0
$489$	29.6600	1.34127
$490$	0	0
$491$	15.7284	0.709814	0.354907	−	0.934902i	$-0.384513\pi$
$491$	15.7284	0.709814	0.354907	+	0.934902i	$0.384513\pi$
$492$	0	0
$493$	−3.02913	−0.136425
$494$	0	0
$495$	3.05300	0.137222
$496$	0	0
$497$	7.89480	0.354130
$498$	0	0
$499$	20.2848	0.908072	0.454036	−	0.890983i	$-0.349984\pi$
$499$	20.2848	0.908072	0.454036	+	0.890983i	$0.349984\pi$
$500$	0	0
$501$	−8.59609	−0.384045
$502$	0	0
$503$	9.05229	0.403622	0.201811	−	0.979425i	$-0.435317\pi$
$503$	9.05229	0.403622	0.201811	+	0.979425i	$0.435317\pi$
$504$	0	0
$505$	−26.5372	−1.18089
$506$	0	0
$507$	−39.0825	−1.73571
$508$	0	0
$509$	−22.0692	−0.978202	−0.489101	−	0.872227i	$-0.662675\pi$
$509$	−22.0692	−0.978202	−0.489101	+	0.872227i	$0.662675\pi$
$510$	0	0
$511$	12.7072	0.562134
$512$	0	0
$513$	9.39794	0.414929
$514$	0	0
$515$	13.9124	0.613053
$516$	0	0
$517$	0.457855	0.0201364
$518$	0	0
$519$	−4.74808	−0.208417
$520$	0	0
$521$	−35.4901	−1.55485	−0.777425	−	0.628975i	$-0.783475\pi$
$521$	−35.4901	−1.55485	−0.777425	+	0.628975i	$0.783475\pi$
$522$	0	0
$523$	25.7518	1.12605	0.563023	−	0.826441i	$-0.309638\pi$
$523$	25.7518	1.12605	0.563023	+	0.826441i	$0.309638\pi$
$524$	0	0
$525$	42.7434	1.86548
$526$	0	0
$527$	−15.9992	−0.696936
$528$	0	0
$529$	23.4035	1.01754
$530$	0	0
$531$	9.96798	0.432574
$532$	0	0
$533$	1.02683	0.0444768
$534$	0	0
$535$	15.2925	0.661153
$536$	0	0
$537$	−39.5627	−1.70726
$538$	0	0
$539$	−0.0224667	−0.000967708 0
$540$	0	0
$541$	−7.96561	−0.342469	−0.171234	−	0.985230i	$-0.554775\pi$
$541$	−7.96561	−0.342469	−0.171234	+	0.985230i	$0.554775\pi$
$542$	0	0
$543$	49.9238	2.14243
$544$	0	0
$545$	5.61369	0.240464
$546$	0	0
$547$	33.7358	1.44244	0.721219	−	0.692707i	$-0.243582\pi$
$547$	33.7358	1.44244	0.721219	+	0.692707i	$0.243582\pi$
$548$	0	0
$549$	35.5001	1.51511
$550$	0	0
$551$	0.627008	0.0267115
$552$	0	0
$553$	43.9629	1.86949
$554$	0	0
$555$	−21.4967	−0.912484
$556$	0	0
$557$	−16.7876	−0.711314	−0.355657	−	0.934616i	$-0.615743\pi$
$557$	−16.7876	−0.711314	−0.355657	+	0.934616i	$0.615743\pi$
$558$	0	0
$559$	1.94760	0.0823746
$560$	0	0
$561$	2.26976	0.0958294
$562$	0	0
$563$	18.9719	0.799572	0.399786	−	0.916609i	$-0.369084\pi$
$563$	18.9719	0.799572	0.399786	+	0.916609i	$0.369084\pi$
$564$	0	0
$565$	9.64833	0.405908
$566$	0	0
$567$	26.8119	1.12600
$568$	0	0
$569$	−43.0540	−1.80492	−0.902459	−	0.430776i	$-0.858240\pi$
$569$	−43.0540	−1.80492	−0.902459	+	0.430776i	$0.858240\pi$
$570$	0	0
$571$	−40.5338	−1.69629	−0.848144	−	0.529766i	$-0.822280\pi$
$571$	−40.5338	−1.69629	−0.848144	+	0.529766i	$0.822280\pi$
$572$	0	0
$573$	−5.64132	−0.235670
$574$	0	0
$575$	36.0852	1.50486
$576$	0	0
$577$	−36.5489	−1.52155	−0.760774	−	0.649016i	$-0.775181\pi$
$577$	−36.5489	−1.52155	−0.760774	+	0.649016i	$0.775181\pi$
$578$	0	0
$579$	8.45074	0.351201
$580$	0	0
$581$	10.4201	0.432297
$582$	0	0
$583$	−0.155633	−0.00644566
$584$	0	0
$585$	−4.54251	−0.187810
$586$	0	0
$587$	−17.0443	−0.703493	−0.351747	−	0.936095i	$-0.614412\pi$
$587$	−17.0443	−0.703493	−0.351747	+	0.936095i	$0.614412\pi$
$588$	0	0
$589$	3.31172	0.136457
$590$	0	0
$591$	−1.96216	−0.0807125
$592$	0	0
$593$	−43.6536	−1.79264	−0.896320	−	0.443409i	$-0.853769\pi$
$593$	−43.6536	−1.79264	−0.896320	+	0.443409i	$0.853769\pi$
$594$	0	0
$595$	41.4369	1.69875
$596$	0	0
$597$	−9.89785	−0.405092
$598$	0	0
$599$	0.898235	0.0367009	0.0183504	−	0.999832i	$-0.494159\pi$
$599$	0.898235	0.0367009	0.0183504	+	0.999832i	$0.494159\pi$
$600$	0	0
$601$	2.55624	0.104271	0.0521355	−	0.998640i	$-0.483397\pi$
$601$	2.55624	0.104271	0.0521355	+	0.998640i	$0.483397\pi$
$602$	0	0
$603$	10.5915	0.431320
$604$	0	0
$605$	35.2206	1.43192
$606$	0	0
$607$	28.9498	1.17503	0.587517	−	0.809211i	$-0.300105\pi$
$607$	28.9498	1.17503	0.587517	+	0.809211i	$0.300105\pi$
$608$	0	0
$609$	5.05928	0.205012
$610$	0	0
$611$	−0.681235	−0.0275598
$612$	0	0
$613$	−32.3143	−1.30516	−0.652581	−	0.757719i	$-0.726314\pi$
$613$	−32.3143	−1.30516	−0.652581	+	0.757719i	$0.726314\pi$
$614$	0	0
$615$	−42.9559	−1.73215
$616$	0	0
$617$	−12.7286	−0.512435	−0.256217	−	0.966619i	$-0.582476\pi$
$617$	−12.7286	−0.512435	−0.256217	+	0.966619i	$0.582476\pi$
$618$	0	0
$619$	19.5916	0.787453	0.393727	−	0.919228i	$-0.371186\pi$
$619$	19.5916	0.787453	0.393727	+	0.919228i	$0.371186\pi$
$620$	0	0
$621$	64.0188	2.56899
$622$	0	0
$623$	−37.7056	−1.51064
$624$	0	0
$625$	−23.4252	−0.937007
$626$	0	0
$627$	−0.469825	−0.0187630
$628$	0	0
$629$	−10.7206	−0.427460
$630$	0	0
$631$	47.9136	1.90741	0.953705	−	0.300744i	$-0.0972350\pi$
$631$	47.9136	1.90741	0.953705	+	0.300744i	$0.0972350\pi$
$632$	0	0
$633$	−31.6292	−1.25715
$634$	0	0
$635$	60.3501	2.39492
$636$	0	0
$637$	0.0334278	0.00132446
$638$	0	0
$639$	18.0561	0.714288
$640$	0	0
$641$	9.57508	0.378193	0.189096	−	0.981959i	$-0.439444\pi$
$641$	9.57508	0.378193	0.189096	+	0.981959i	$0.439444\pi$
$642$	0	0
$643$	−4.37493	−0.172530	−0.0862651	−	0.996272i	$-0.527493\pi$
$643$	−4.37493	−0.172530	−0.0862651	+	0.996272i	$0.527493\pi$
$644$	0	0
$645$	−81.4750	−3.20807
$646$	0	0
$647$	−44.5592	−1.75180	−0.875902	−	0.482490i	$-0.839733\pi$
$647$	−44.5592	−1.75180	−0.875902	+	0.482490i	$0.839733\pi$
$648$	0	0
$649$	−0.253773	−0.00996144
$650$	0	0
$651$	26.7220	1.04732
$652$	0	0
$653$	17.5559	0.687016	0.343508	−	0.939150i	$-0.388385\pi$
$653$	17.5559	0.687016	0.343508	+	0.939150i	$0.388385\pi$
$654$	0	0
$655$	28.2374	1.10333
$656$	0	0
$657$	29.0625	1.13384
$658$	0	0
$659$	−31.4701	−1.22590	−0.612951	−	0.790121i	$-0.710018\pi$
$659$	−31.4701	−1.22590	−0.612951	+	0.790121i	$0.710018\pi$
$660$	0	0
$661$	−10.3900	−0.404125	−0.202062	−	0.979373i	$-0.564764\pi$
$661$	−10.3900	−0.404125	−0.202062	+	0.979373i	$0.564764\pi$
$662$	0	0
$663$	−3.37714	−0.131157
$664$	0	0
$665$	−8.57715	−0.332608
$666$	0	0
$667$	4.27119	0.165381
$668$	0	0
$669$	50.7537	1.96225
$670$	0	0
$671$	−0.903790	−0.0348904
$672$	0	0
$673$	−10.3583	−0.399284	−0.199642	−	0.979869i	$-0.563978\pi$
$673$	−10.3583	−0.399284	−0.199642	+	0.979869i	$0.563978\pi$
$674$	0	0
$675$	49.7836	1.91617
$676$	0	0
$677$	−13.3755	−0.514064	−0.257032	−	0.966403i	$-0.582745\pi$
$677$	−13.3755	−0.514064	−0.257032	+	0.966403i	$0.582745\pi$
$678$	0	0
$679$	7.84373	0.301015
$680$	0	0
$681$	−57.2667	−2.19447
$682$	0	0
$683$	−2.34677	−0.0897968	−0.0448984	−	0.998992i	$-0.514296\pi$
$683$	−2.34677	−0.0897968	−0.0448984	+	0.998992i	$0.514296\pi$
$684$	0	0
$685$	−4.78736	−0.182916
$686$	0	0
$687$	−17.6490	−0.673352
$688$	0	0
$689$	0.231564	0.00882187
$690$	0	0
$691$	−9.45319	−0.359616	−0.179808	−	0.983702i	$-0.557548\pi$
$691$	−9.45319	−0.359616	−0.179808	+	0.983702i	$0.557548\pi$
$692$	0	0
$693$	−2.54301	−0.0966008
$694$	0	0
$695$	29.8939	1.13394
$696$	0	0
$697$	−21.4226	−0.811438
$698$	0	0
$699$	7.79058	0.294667
$700$	0	0
$701$	35.1738	1.32850	0.664248	−	0.747513i	$-0.268752\pi$
$701$	35.1738	1.32850	0.664248	+	0.747513i	$0.268752\pi$
$702$	0	0
$703$	2.21910	0.0836948
$704$	0	0
$705$	28.4985	1.07331
$706$	0	0
$707$	22.1043	0.831316
$708$	0	0
$709$	50.3917	1.89250	0.946249	−	0.323438i	$-0.104839\pi$
$709$	50.3917	1.89250	0.946249	+	0.323438i	$0.104839\pi$
$710$	0	0
$711$	100.547	3.77081
$712$	0	0
$713$	22.5595	0.844859
$714$	0	0
$715$	0.115647	0.00432495
$716$	0	0
$717$	−25.2595	−0.943334
$718$	0	0
$719$	−20.3760	−0.759896	−0.379948	−	0.925008i	$-0.624058\pi$
$719$	−20.3760	−0.759896	−0.379948	+	0.925008i	$0.624058\pi$
$720$	0	0
$721$	−11.5884	−0.431573
$722$	0	0
$723$	87.5255	3.25511
$724$	0	0
$725$	3.32144	0.123355
$726$	0	0
$727$	20.6751	0.766796	0.383398	−	0.923583i	$-0.374754\pi$
$727$	20.6751	0.766796	0.383398	+	0.923583i	$0.374754\pi$
$728$	0	0
$729$	−23.7902	−0.881118
$730$	0	0
$731$	−40.6325	−1.50285
$732$	0	0
$733$	21.9324	0.810092	0.405046	−	0.914296i	$-0.367256\pi$
$733$	21.9324	0.810092	0.405046	+	0.914296i	$0.367256\pi$
$734$	0	0
$735$	−1.39840	−0.0515809
$736$	0	0
$737$	−0.269647	−0.00993258
$738$	0	0
$739$	6.36556	0.234161	0.117080	−	0.993122i	$-0.462646\pi$
$739$	6.36556	0.234161	0.117080	+	0.993122i	$0.462646\pi$
$740$	0	0
$741$	0.699044	0.0256800
$742$	0	0
$743$	−26.0455	−0.955516	−0.477758	−	0.878491i	$-0.658550\pi$
$743$	−26.0455	−0.955516	−0.477758	+	0.878491i	$0.658550\pi$
$744$	0	0
$745$	−66.1663	−2.42415
$746$	0	0
$747$	23.8316	0.871952
$748$	0	0
$749$	−12.7379	−0.465434
$750$	0	0
$751$	27.3853	0.999304	0.499652	−	0.866226i	$-0.333461\pi$
$751$	27.3853	0.999304	0.499652	+	0.866226i	$0.333461\pi$
$752$	0	0
$753$	−65.8937	−2.40130
$754$	0	0
$755$	−51.3101	−1.86737
$756$	0	0
$757$	19.3810	0.704414	0.352207	−	0.935922i	$-0.385431\pi$
$757$	19.3810	0.704414	0.352207	+	0.935922i	$0.385431\pi$
$758$	0	0
$759$	−3.20045	−0.116169
$760$	0	0
$761$	53.9131	1.95435	0.977174	−	0.212442i	$-0.0681416\pi$
$761$	53.9131	1.95435	0.977174	+	0.212442i	$0.0681416\pi$
$762$	0	0
$763$	−4.67594	−0.169280
$764$	0	0
$765$	94.7699	3.42641
$766$	0	0
$767$	0.377584	0.0136338
$768$	0	0
$769$	−33.0353	−1.19128	−0.595642	−	0.803250i	$-0.703102\pi$
$769$	−33.0353	−1.19128	−0.595642	+	0.803250i	$0.703102\pi$
$770$	0	0
$771$	−8.23544	−0.296592
$772$	0	0
$773$	54.9394	1.97603	0.988017	−	0.154347i	$-0.0493273\pi$
$773$	54.9394	1.97603	0.988017	+	0.154347i	$0.0493273\pi$
$774$	0	0
$775$	17.5431	0.630169
$776$	0	0
$777$	17.9057	0.642364
$778$	0	0
$779$	4.43432	0.158876
$780$	0	0
$781$	−0.459686	−0.0164489
$782$	0	0
$783$	5.89259	0.210584
$784$	0	0
$785$	−20.1266	−0.718349
$786$	0	0
$787$	−34.5860	−1.23286	−0.616429	−	0.787411i	$-0.711421\pi$
$787$	−34.5860	−1.23286	−0.616429	+	0.787411i	$0.711421\pi$
$788$	0	0
$789$	10.8680	0.386910
$790$	0	0
$791$	−8.03659	−0.285748
$792$	0	0
$793$	1.34473	0.0477529
$794$	0	0
$795$	−9.68714	−0.343567
$796$	0	0
$797$	−26.8662	−0.951650	−0.475825	−	0.879540i	$-0.657851\pi$
$797$	−26.8662	−0.951650	−0.475825	+	0.879540i	$0.657851\pi$
$798$	0	0
$799$	14.2125	0.502802
$800$	0	0
$801$	−86.2360	−3.04700
$802$	0	0
$803$	−0.739896	−0.0261103
$804$	0	0
$805$	−58.4276	−2.05930
$806$	0	0
$807$	34.3485	1.20912
$808$	0	0
$809$	−26.2516	−0.922956	−0.461478	−	0.887152i	$-0.652681\pi$
$809$	−26.2516	−0.922956	−0.461478	+	0.887152i	$0.652681\pi$
$810$	0	0
$811$	−39.5697	−1.38948	−0.694741	−	0.719260i	$-0.744481\pi$
$811$	−39.5697	−1.38948	−0.694741	+	0.719260i	$0.744481\pi$
$812$	0	0
$813$	−59.6829	−2.09317
$814$	0	0
$815$	−31.5281	−1.10438
$816$	0	0
$817$	8.41063	0.294251
$818$	0	0
$819$	3.78370	0.132213
$820$	0	0
$821$	−9.19247	−0.320819	−0.160410	−	0.987051i	$-0.551282\pi$
$821$	−9.19247	−0.320819	−0.160410	+	0.987051i	$0.551282\pi$
$822$	0	0
$823$	33.4556	1.16619	0.583095	−	0.812404i	$-0.301842\pi$
$823$	33.4556	1.16619	0.583095	+	0.812404i	$0.301842\pi$
$824$	0	0
$825$	−2.48880	−0.0866488
$826$	0	0
$827$	50.6566	1.76150	0.880752	−	0.473579i	$-0.157038\pi$
$827$	50.6566	1.76150	0.880752	+	0.473579i	$0.157038\pi$
$828$	0	0
$829$	−6.64406	−0.230758	−0.115379	−	0.993322i	$-0.536808\pi$
$829$	−6.64406	−0.230758	−0.115379	+	0.993322i	$0.536808\pi$
$830$	0	0
$831$	66.7898	2.31691
$832$	0	0
$833$	−0.697400	−0.0241635
$834$	0	0
$835$	9.13752	0.316217
$836$	0	0
$837$	31.1234	1.07578
$838$	0	0
$839$	−32.5076	−1.12229	−0.561143	−	0.827719i	$-0.689638\pi$
$839$	−32.5076	−1.12229	−0.561143	+	0.827719i	$0.689638\pi$
$840$	0	0
$841$	−28.6069	−0.986443
$842$	0	0
$843$	61.2140	2.10832
$844$	0	0
$845$	41.5441	1.42916
$846$	0	0
$847$	−29.3371	−1.00803
$848$	0	0
$849$	−29.3758	−1.00817
$850$	0	0
$851$	15.1165	0.518187
$852$	0	0
$853$	−17.1071	−0.585736	−0.292868	−	0.956153i	$-0.594610\pi$
$853$	−17.1071	−0.585736	−0.292868	+	0.956153i	$0.594610\pi$
$854$	0	0
$855$	−19.6167	−0.670877
$856$	0	0
$857$	−17.7175	−0.605217	−0.302609	−	0.953115i	$-0.597858\pi$
$857$	−17.7175	−0.605217	−0.302609	+	0.953115i	$0.597858\pi$
$858$	0	0
$859$	−17.6079	−0.600773	−0.300387	−	0.953818i	$-0.597116\pi$
$859$	−17.6079	−0.600773	−0.300387	+	0.953818i	$0.597116\pi$
$860$	0	0
$861$	35.7802	1.21939
$862$	0	0
$863$	−14.8275	−0.504733	−0.252366	−	0.967632i	$-0.581209\pi$
$863$	−14.8275	−0.504733	−0.252366	+	0.967632i	$0.581209\pi$
$864$	0	0
$865$	5.04714	0.171608
$866$	0	0
$867$	19.1373	0.649936
$868$	0	0
$869$	−2.55980	−0.0868354
$870$	0	0
$871$	0.401204	0.0135943
$872$	0	0
$873$	17.9393	0.607153
$874$	0	0
$875$	−2.54990	−0.0862024
$876$	0	0
$877$	−6.30116	−0.212775	−0.106388	−	0.994325i	$-0.533928\pi$
$877$	−6.30116	−0.212775	−0.106388	+	0.994325i	$0.533928\pi$
$878$	0	0
$879$	−36.9676	−1.24689
$880$	0	0
$881$	−4.32722	−0.145788	−0.0728939	−	0.997340i	$-0.523223\pi$
$881$	−4.32722	−0.145788	−0.0728939	+	0.997340i	$0.523223\pi$
$882$	0	0
$883$	−15.6830	−0.527774	−0.263887	−	0.964554i	$-0.585005\pi$
$883$	−15.6830	−0.527774	−0.263887	+	0.964554i	$0.585005\pi$
$884$	0	0
$885$	−15.7957	−0.530966
$886$	0	0
$887$	−32.8228	−1.10208	−0.551040	−	0.834479i	$-0.685769\pi$
$887$	−32.8228	−1.10208	−0.551040	+	0.834479i	$0.685769\pi$
$888$	0	0
$889$	−50.2687	−1.68596
$890$	0	0
$891$	−1.56116	−0.0523009
$892$	0	0
$893$	−2.94189	−0.0984465
$894$	0	0
$895$	42.0546	1.40573
$896$	0	0
$897$	4.76189	0.158995
$898$	0	0
$899$	2.07648	0.0692544
$900$	0	0
$901$	−4.83108	−0.160947
$902$	0	0
$903$	67.8647	2.25840
$904$	0	0
$905$	−53.0683	−1.76405
$906$	0	0
$907$	−22.1331	−0.734916	−0.367458	−	0.930040i	$-0.619772\pi$
$907$	−22.1331	−0.734916	−0.367458	+	0.930040i	$0.619772\pi$
$908$	0	0
$909$	50.5544	1.67678
$910$	0	0
$911$	40.8662	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$911$	40.8662	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$912$	0	0
$913$	−0.606723	−0.0200796
$914$	0	0
$915$	−56.2550	−1.85973
$916$	0	0
$917$	−23.5204	−0.776713
$918$	0	0
$919$	16.2666	0.536587	0.268293	−	0.963337i	$-0.413540\pi$
$919$	16.2666	0.536587	0.268293	+	0.963337i	$0.413540\pi$
$920$	0	0
$921$	5.94538	0.195907
$922$	0	0
$923$	0.683959	0.0225128
$924$	0	0
$925$	11.7552	0.386508
$926$	0	0
$927$	−26.5036	−0.870492
$928$	0	0
$929$	55.7241	1.82825	0.914125	−	0.405433i	$-0.132879\pi$
$929$	55.7241	1.82825	0.914125	+	0.405433i	$0.132879\pi$
$930$	0	0
$931$	0.144357	0.00473110
$932$	0	0
$933$	−80.8851	−2.64806
$934$	0	0
$935$	−2.41272	−0.0789045
$936$	0	0
$937$	−14.2577	−0.465779	−0.232890	−	0.972503i	$-0.574818\pi$
$937$	−14.2577	−0.465779	−0.232890	+	0.972503i	$0.574818\pi$
$938$	0	0
$939$	85.2348	2.78153
$940$	0	0
$941$	19.4923	0.635431	0.317715	−	0.948186i	$-0.397084\pi$
$941$	19.4923	0.635431	0.317715	+	0.948186i	$0.397084\pi$
$942$	0	0
$943$	30.2066	0.983663
$944$	0	0
$945$	−80.6075	−2.62216
$946$	0	0
$947$	34.9526	1.13581	0.567903	−	0.823095i	$-0.307755\pi$
$947$	34.9526	1.13581	0.567903	+	0.823095i	$0.307755\pi$
$948$	0	0
$949$	1.10088	0.0357360
$950$	0	0
$951$	3.00745	0.0975231
$952$	0	0
$953$	−15.2146	−0.492850	−0.246425	−	0.969162i	$-0.579256\pi$
$953$	−15.2146	−0.492850	−0.246425	+	0.969162i	$0.579256\pi$
$954$	0	0
$955$	5.99664	0.194047
$956$	0	0
$957$	−0.294584	−0.00952254
$958$	0	0
$959$	3.98764	0.128768
$960$	0	0
$961$	−20.0325	−0.646210
$962$	0	0
$963$	−29.1328	−0.938791
$964$	0	0
$965$	−8.98302	−0.289174
$966$	0	0
$967$	−42.3167	−1.36081	−0.680406	−	0.732835i	$-0.738197\pi$
$967$	−42.3167	−1.36081	−0.680406	+	0.732835i	$0.738197\pi$
$968$	0	0
$969$	−14.5841	−0.468508
$970$	0	0
$971$	27.2751	0.875300	0.437650	−	0.899145i	$-0.355811\pi$
$971$	27.2751	0.875300	0.437650	+	0.899145i	$0.355811\pi$
$972$	0	0
$973$	−24.9002	−0.798263
$974$	0	0
$975$	3.70304	0.118592
$976$	0	0
$977$	−36.8970	−1.18044	−0.590220	−	0.807243i	$-0.700959\pi$
$977$	−36.8970	−1.18044	−0.590220	+	0.807243i	$0.700959\pi$
$978$	0	0
$979$	2.19546	0.0701672
$980$	0	0
$981$	−10.6943	−0.341442
$982$	0	0
$983$	44.5855	1.42206	0.711028	−	0.703164i	$-0.248230\pi$
$983$	44.5855	1.42206	0.711028	+	0.703164i	$0.248230\pi$
$984$	0	0
$985$	2.08575	0.0664575
$986$	0	0
$987$	−23.7379	−0.755585
$988$	0	0
$989$	57.2933	1.82182
$990$	0	0
$991$	−36.9389	−1.17340	−0.586702	−	0.809803i	$-0.699574\pi$
$991$	−36.9389	−1.17340	−0.586702	+	0.809803i	$0.699574\pi$
$992$	0	0
$993$	77.0699	2.44574
$994$	0	0
$995$	10.5213	0.333547
$996$	0	0
$997$	−51.0148	−1.61566	−0.807828	−	0.589418i	$-0.799357\pi$
$997$	−51.0148	−1.61566	−0.807828	+	0.589418i	$0.799357\pi$
$998$	0	0
$999$	20.8549	0.659821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.18	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.18	✓	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)

\(f(q)\)	\(=\)	\(q+3.01880 q^{3} -3.20894 q^{5} +2.67289 q^{7} +6.11314 q^{9} +O(q^{10})\)	\(q+3.01880 q^{3} -3.20894 q^{5} +2.67289 q^{7} +6.11314 q^{9} -0.155633 q^{11} +0.231564 q^{13} -9.68714 q^{15} -4.83108 q^{17} +1.00000 q^{19} +8.06892 q^{21} +6.81201 q^{23} +5.29729 q^{25} +9.39794 q^{27} +0.627008 q^{29} +3.31172 q^{31} -0.469825 q^{33} -8.57715 q^{35} +2.21910 q^{37} +0.699044 q^{39} +4.43432 q^{41} +8.41063 q^{43} -19.6167 q^{45} -2.94189 q^{47} +0.144357 q^{49} -14.5841 q^{51} +1.00000 q^{53} +0.499417 q^{55} +3.01880 q^{57} +1.63058 q^{59} +5.80719 q^{61} +16.3398 q^{63} -0.743074 q^{65} +1.73258 q^{67} +20.5641 q^{69} +2.95365 q^{71} +4.75410 q^{73} +15.9914 q^{75} -0.415990 q^{77} +16.4477 q^{79} +10.0311 q^{81} +3.89842 q^{83} +15.5027 q^{85} +1.89281 q^{87} -14.1067 q^{89} +0.618945 q^{91} +9.99742 q^{93} -3.20894 q^{95} +2.93455 q^{97} -0.951406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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