LMFDB - Embedded newform 4028.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.44269$ 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)

$f(q)$	$=$	$q-2.44269 q^{3} -3.59133 q^{5} +0.459217 q^{7} +2.96674 q^{9} +O(q^{10})$	$q-2.44269 q^{3} -3.59133 q^{5} +0.459217 q^{7} +2.96674 q^{9} -3.29838 q^{11} -5.41041 q^{13} +8.77250 q^{15} +6.52731 q^{17} -1.00000 q^{19} -1.12173 q^{21} +0.522155 q^{23} +7.89763 q^{25} +0.0812467 q^{27} -0.139943 q^{29} +2.51077 q^{31} +8.05692 q^{33} -1.64920 q^{35} +5.52082 q^{37} +13.2160 q^{39} +0.693306 q^{41} -2.18127 q^{43} -10.6545 q^{45} +2.32738 q^{47} -6.78912 q^{49} -15.9442 q^{51} +1.00000 q^{53} +11.8456 q^{55} +2.44269 q^{57} +9.55995 q^{59} +6.77344 q^{61} +1.36238 q^{63} +19.4306 q^{65} -2.42099 q^{67} -1.27546 q^{69} -3.70484 q^{71} +0.506311 q^{73} -19.2915 q^{75} -1.51467 q^{77} +1.41051 q^{79} -9.09868 q^{81} +5.81322 q^{83} -23.4417 q^{85} +0.341839 q^{87} -0.997992 q^{89} -2.48455 q^{91} -6.13304 q^{93} +3.59133 q^{95} +0.119394 q^{97} -9.78542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	−2.44269	−1.41029	−0.705144	−	0.709064i	$-0.749118\pi$
$3$	−2.44269	−1.41029	−0.705144	+	0.709064i	$0.749118\pi$
$4$	0	0
$5$	−3.59133	−1.60609	−0.803045	−	0.595918i	$-0.796788\pi$
$5$	−3.59133	−1.60609	−0.803045	+	0.595918i	$0.796788\pi$
$6$	0	0
$7$	0.459217	0.173568	0.0867839	−	0.996227i	$-0.472341\pi$
$7$	0.459217	0.173568	0.0867839	+	0.996227i	$0.472341\pi$
$8$	0	0
$9$	2.96674	0.988913
$10$	0	0
$11$	−3.29838	−0.994498	−0.497249	−	0.867608i	$-0.665656\pi$
$11$	−3.29838	−0.994498	−0.497249	+	0.867608i	$0.665656\pi$
$12$	0	0
$13$	−5.41041	−1.50058	−0.750289	−	0.661110i	$-0.770086\pi$
$13$	−5.41041	−1.50058	−0.750289	+	0.661110i	$0.770086\pi$
$14$	0	0
$15$	8.77250	2.26505
$16$	0	0
$17$	6.52731	1.58311	0.791553	−	0.611100i	$-0.209273\pi$
$17$	6.52731	1.58311	0.791553	+	0.611100i	$0.209273\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.12173	−0.244781
$22$	0	0
$23$	0.522155	0.108877	0.0544384	−	0.998517i	$-0.482663\pi$
$23$	0.522155	0.108877	0.0544384	+	0.998517i	$0.482663\pi$
$24$	0	0
$25$	7.89763	1.57953
$26$	0	0
$27$	0.0812467	0.0156359
$28$	0	0
$29$	−0.139943	−0.0259869	−0.0129934	−	0.999916i	$-0.504136\pi$
$29$	−0.139943	−0.0259869	−0.0129934	+	0.999916i	$0.504136\pi$
$30$	0	0
$31$	2.51077	0.450948	0.225474	−	0.974249i	$-0.427607\pi$
$31$	2.51077	0.450948	0.225474	+	0.974249i	$0.427607\pi$
$32$	0	0
$33$	8.05692	1.40253
$34$	0	0
$35$	−1.64920	−0.278765
$36$	0	0
$37$	5.52082	0.907617	0.453809	−	0.891099i	$-0.350065\pi$
$37$	5.52082	0.907617	0.453809	+	0.891099i	$0.350065\pi$
$38$	0	0
$39$	13.2160	2.11625
$40$	0	0
$41$	0.693306	0.108276	0.0541381	−	0.998533i	$-0.482759\pi$
$41$	0.693306	0.108276	0.0541381	+	0.998533i	$0.482759\pi$
$42$	0	0
$43$	−2.18127	−0.332641	−0.166321	−	0.986072i	$-0.553189\pi$
$43$	−2.18127	−0.332641	−0.166321	+	0.986072i	$0.553189\pi$
$44$	0	0
$45$	−10.6545	−1.58828
$46$	0	0
$47$	2.32738	0.339484	0.169742	−	0.985489i	$-0.445707\pi$
$47$	2.32738	0.339484	0.169742	+	0.985489i	$0.445707\pi$
$48$	0	0
$49$	−6.78912	−0.969874
$50$	0	0
$51$	−15.9442	−2.23264
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	11.8456	1.59725
$56$	0	0
$57$	2.44269	0.323542
$58$	0	0
$59$	9.55995	1.24460	0.622300	−	0.782779i	$-0.286198\pi$
$59$	9.55995	1.24460	0.622300	+	0.782779i	$0.286198\pi$
$60$	0	0
$61$	6.77344	0.867250	0.433625	−	0.901093i	$-0.357234\pi$
$61$	6.77344	0.867250	0.433625	+	0.901093i	$0.357234\pi$
$62$	0	0
$63$	1.36238	0.171643
$64$	0	0
$65$	19.4306	2.41006
$66$	0	0
$67$	−2.42099	−0.295771	−0.147885	−	0.989005i	$-0.547247\pi$
$67$	−2.42099	−0.295771	−0.147885	+	0.989005i	$0.547247\pi$
$68$	0	0
$69$	−1.27546	−0.153548
$70$	0	0
$71$	−3.70484	−0.439683	−0.219842	−	0.975536i	$-0.570554\pi$
$71$	−3.70484	−0.439683	−0.219842	+	0.975536i	$0.570554\pi$
$72$	0	0
$73$	0.506311	0.0592592	0.0296296	−	0.999561i	$-0.490567\pi$
$73$	0.506311	0.0592592	0.0296296	+	0.999561i	$0.490567\pi$
$74$	0	0
$75$	−19.2915	−2.22759
$76$	0	0
$77$	−1.51467	−0.172613
$78$	0	0
$79$	1.41051	0.158695	0.0793476	−	0.996847i	$-0.474716\pi$
$79$	1.41051	0.158695	0.0793476	+	0.996847i	$0.474716\pi$
$80$	0	0
$81$	−9.09868	−1.01096
$82$	0	0
$83$	5.81322	0.638084	0.319042	−	0.947741i	$-0.396639\pi$
$83$	5.81322	0.638084	0.319042	+	0.947741i	$0.396639\pi$
$84$	0	0
$85$	−23.4417	−2.54261
$86$	0	0
$87$	0.341839	0.0366490
$88$	0	0
$89$	−0.997992	−0.105787	−0.0528935	−	0.998600i	$-0.516844\pi$
$89$	−0.997992	−0.105787	−0.0528935	+	0.998600i	$0.516844\pi$
$90$	0	0
$91$	−2.48455	−0.260452
$92$	0	0
$93$	−6.13304	−0.635967
$94$	0	0
$95$	3.59133	0.368462
$96$	0	0
$97$	0.119394	0.0121226	0.00606131	−	0.999982i	$-0.498071\pi$
$97$	0.119394	0.0121226	0.00606131	+	0.999982i	$0.498071\pi$
$98$	0	0
$99$	−9.78542	−0.983472
$100$	0	0
$101$	14.7246	1.46515	0.732577	−	0.680684i	$-0.238317\pi$
$101$	14.7246	1.46515	0.732577	+	0.680684i	$0.238317\pi$
$102$	0	0
$103$	8.60036	0.847419	0.423709	−	0.905798i	$-0.360728\pi$
$103$	8.60036	0.847419	0.423709	+	0.905798i	$0.360728\pi$
$104$	0	0
$105$	4.02848	0.393140
$106$	0	0
$107$	−8.84427	−0.855007	−0.427504	−	0.904014i	$-0.640607\pi$
$107$	−8.84427	−0.855007	−0.427504	+	0.904014i	$0.640607\pi$
$108$	0	0
$109$	5.17303	0.495486	0.247743	−	0.968826i	$-0.420311\pi$
$109$	5.17303	0.495486	0.247743	+	0.968826i	$0.420311\pi$
$110$	0	0
$111$	−13.4857	−1.28000
$112$	0	0
$113$	2.27378	0.213899	0.106949	−	0.994264i	$-0.465892\pi$
$113$	2.27378	0.213899	0.106949	+	0.994264i	$0.465892\pi$
$114$	0	0
$115$	−1.87523	−0.174866
$116$	0	0
$117$	−16.0513	−1.48394
$118$	0	0
$119$	2.99745	0.274776
$120$	0	0
$121$	−0.120708	−0.0109734
$122$	0	0
$123$	−1.69353	−0.152701
$124$	0	0
$125$	−10.4063	−0.930770
$126$	0	0
$127$	0.423104	0.0375444	0.0187722	−	0.999824i	$-0.494024\pi$
$127$	0.423104	0.0375444	0.0187722	+	0.999824i	$0.494024\pi$
$128$	0	0
$129$	5.32818	0.469120
$130$	0	0
$131$	−13.6472	−1.19236	−0.596181	−	0.802850i	$-0.703316\pi$
$131$	−13.6472	−1.19236	−0.596181	+	0.802850i	$0.703316\pi$
$132$	0	0
$133$	−0.459217	−0.0398192
$134$	0	0
$135$	−0.291784	−0.0251127
$136$	0	0
$137$	15.7066	1.34191	0.670953	−	0.741500i	$-0.265885\pi$
$137$	15.7066	1.34191	0.670953	+	0.741500i	$0.265885\pi$
$138$	0	0
$139$	−1.85836	−0.157624	−0.0788120	−	0.996889i	$-0.525113\pi$
$139$	−1.85836	−0.157624	−0.0788120	+	0.996889i	$0.525113\pi$
$140$	0	0
$141$	−5.68508	−0.478770
$142$	0	0
$143$	17.8456	1.49232
$144$	0	0
$145$	0.502583	0.0417372
$146$	0	0
$147$	16.5837	1.36780
$148$	0	0
$149$	−11.3101	−0.926562	−0.463281	−	0.886211i	$-0.653328\pi$
$149$	−11.3101	−0.926562	−0.463281	+	0.886211i	$0.653328\pi$
$150$	0	0
$151$	−6.98244	−0.568223	−0.284111	−	0.958791i	$-0.591699\pi$
$151$	−6.98244	−0.568223	−0.284111	+	0.958791i	$0.591699\pi$
$152$	0	0
$153$	19.3648	1.56555
$154$	0	0
$155$	−9.01700	−0.724263
$156$	0	0
$157$	−22.1303	−1.76619	−0.883097	−	0.469190i	$-0.844546\pi$
$157$	−22.1303	−1.76619	−0.883097	+	0.469190i	$0.844546\pi$
$158$	0	0
$159$	−2.44269	−0.193718
$160$	0	0
$161$	0.239783	0.0188975
$162$	0	0
$163$	−13.9225	−1.09049	−0.545246	−	0.838276i	$-0.683564\pi$
$163$	−13.9225	−1.09049	−0.545246	+	0.838276i	$0.683564\pi$
$164$	0	0
$165$	−28.9350	−2.25259
$166$	0	0
$167$	−12.6856	−0.981639	−0.490820	−	0.871261i	$-0.663303\pi$
$167$	−12.6856	−0.981639	−0.490820	+	0.871261i	$0.663303\pi$
$168$	0	0
$169$	16.2726	1.25174
$170$	0	0
$171$	−2.96674	−0.226872
$172$	0	0
$173$	16.3626	1.24403	0.622014	−	0.783006i	$-0.286315\pi$
$173$	16.3626	1.24403	0.622014	+	0.783006i	$0.286315\pi$
$174$	0	0
$175$	3.62673	0.274155
$176$	0	0
$177$	−23.3520	−1.75524
$178$	0	0
$179$	19.7991	1.47986	0.739928	−	0.672686i	$-0.234860\pi$
$179$	19.7991	1.47986	0.739928	+	0.672686i	$0.234860\pi$
$180$	0	0
$181$	−0.832069	−0.0618472	−0.0309236	−	0.999522i	$-0.509845\pi$
$181$	−0.832069	−0.0618472	−0.0309236	+	0.999522i	$0.509845\pi$
$182$	0	0
$183$	−16.5454	−1.22307
$184$	0	0
$185$	−19.8271	−1.45772
$186$	0	0
$187$	−21.5295	−1.57440
$188$	0	0
$189$	0.0373099	0.00271390
$190$	0	0
$191$	−20.6072	−1.49109	−0.745544	−	0.666457i	$-0.767810\pi$
$191$	−20.6072	−1.49109	−0.745544	+	0.666457i	$0.767810\pi$
$192$	0	0
$193$	−14.5116	−1.04457	−0.522283	−	0.852772i	$-0.674920\pi$
$193$	−14.5116	−1.04457	−0.522283	+	0.852772i	$0.674920\pi$
$194$	0	0
$195$	−47.4628	−3.39889
$196$	0	0
$197$	−11.1698	−0.795816	−0.397908	−	0.917425i	$-0.630264\pi$
$197$	−11.1698	−0.795816	−0.397908	+	0.917425i	$0.630264\pi$
$198$	0	0
$199$	12.6912	0.899656	0.449828	−	0.893115i	$-0.351485\pi$
$199$	12.6912	0.899656	0.449828	+	0.893115i	$0.351485\pi$
$200$	0	0
$201$	5.91372	0.417122
$202$	0	0
$203$	−0.0642644	−0.00451048
$204$	0	0
$205$	−2.48989	−0.173901
$206$	0	0
$207$	1.54910	0.107670
$208$	0	0
$209$	3.29838	0.228154
$210$	0	0
$211$	14.0065	0.964250	0.482125	−	0.876103i	$-0.339865\pi$
$211$	14.0065	0.964250	0.482125	+	0.876103i	$0.339865\pi$
$212$	0	0
$213$	9.04977	0.620080
$214$	0	0
$215$	7.83367	0.534252
$216$	0	0
$217$	1.15299	0.0782700
$218$	0	0
$219$	−1.23676	−0.0835726
$220$	0	0
$221$	−35.3155	−2.37557
$222$	0	0
$223$	−23.6750	−1.58539	−0.792697	−	0.609616i	$-0.791324\pi$
$223$	−23.6750	−1.58539	−0.792697	+	0.609616i	$0.791324\pi$
$224$	0	0
$225$	23.4302	1.56201
$226$	0	0
$227$	−21.0646	−1.39810	−0.699052	−	0.715070i	$-0.746395\pi$
$227$	−21.0646	−1.39810	−0.699052	+	0.715070i	$0.746395\pi$
$228$	0	0
$229$	3.48274	0.230146	0.115073	−	0.993357i	$-0.463290\pi$
$229$	3.48274	0.230146	0.115073	+	0.993357i	$0.463290\pi$
$230$	0	0
$231$	3.69987	0.243434
$232$	0	0
$233$	26.4470	1.73260	0.866301	−	0.499522i	$-0.166491\pi$
$233$	26.4470	1.73260	0.866301	+	0.499522i	$0.166491\pi$
$234$	0	0
$235$	−8.35840	−0.545242
$236$	0	0
$237$	−3.44545	−0.223806
$238$	0	0
$239$	−9.16490	−0.592828	−0.296414	−	0.955060i	$-0.595791\pi$
$239$	−9.16490	−0.592828	−0.296414	+	0.955060i	$0.595791\pi$
$240$	0	0
$241$	18.5407	1.19431	0.597156	−	0.802125i	$-0.296297\pi$
$241$	18.5407	1.19431	0.597156	+	0.802125i	$0.296297\pi$
$242$	0	0
$243$	21.9815	1.41011
$244$	0	0
$245$	24.3819	1.55771
$246$	0	0
$247$	5.41041	0.344256
$248$	0	0
$249$	−14.1999	−0.899882
$250$	0	0
$251$	10.2376	0.646190	0.323095	−	0.946367i	$-0.395277\pi$
$251$	10.2376	0.646190	0.323095	+	0.946367i	$0.395277\pi$
$252$	0	0
$253$	−1.72226	−0.108278
$254$	0	0
$255$	57.2609	3.58581
$256$	0	0
$257$	14.4460	0.901118	0.450559	−	0.892747i	$-0.351225\pi$
$257$	14.4460	0.901118	0.450559	+	0.892747i	$0.351225\pi$
$258$	0	0
$259$	2.53526	0.157533
$260$	0	0
$261$	−0.415176	−0.0256987
$262$	0	0
$263$	−12.2924	−0.757979	−0.378989	−	0.925401i	$-0.623728\pi$
$263$	−12.2924	−0.757979	−0.378989	+	0.925401i	$0.623728\pi$
$264$	0	0
$265$	−3.59133	−0.220613
$266$	0	0
$267$	2.43779	0.149190
$268$	0	0
$269$	−15.5365	−0.947276	−0.473638	−	0.880720i	$-0.657060\pi$
$269$	−15.5365	−0.947276	−0.473638	+	0.880720i	$0.657060\pi$
$270$	0	0
$271$	−25.2889	−1.53619	−0.768094	−	0.640337i	$-0.778795\pi$
$271$	−25.2889	−1.53619	−0.768094	+	0.640337i	$0.778795\pi$
$272$	0	0
$273$	6.06900	0.367312
$274$	0	0
$275$	−26.0494	−1.57084
$276$	0	0
$277$	−22.6407	−1.36035	−0.680173	−	0.733052i	$-0.738095\pi$
$277$	−22.6407	−1.36035	−0.680173	+	0.733052i	$0.738095\pi$
$278$	0	0
$279$	7.44881	0.445948
$280$	0	0
$281$	11.2599	0.671709	0.335854	−	0.941914i	$-0.390975\pi$
$281$	11.2599	0.671709	0.335854	+	0.941914i	$0.390975\pi$
$282$	0	0
$283$	−32.1904	−1.91352	−0.956760	−	0.290879i	$-0.906052\pi$
$283$	−32.1904	−1.91352	−0.956760	+	0.290879i	$0.906052\pi$
$284$	0	0
$285$	−8.77250	−0.519638
$286$	0	0
$287$	0.318378	0.0187933
$288$	0	0
$289$	25.6058	1.50623
$290$	0	0
$291$	−0.291642	−0.0170964
$292$	0	0
$293$	−7.03529	−0.411006	−0.205503	−	0.978656i	$-0.565883\pi$
$293$	−7.03529	−0.411006	−0.205503	+	0.978656i	$0.565883\pi$
$294$	0	0
$295$	−34.3329	−1.99894
$296$	0	0
$297$	−0.267982	−0.0155499
$298$	0	0
$299$	−2.82507	−0.163378
$300$	0	0
$301$	−1.00168	−0.0577358
$302$	0	0
$303$	−35.9677	−2.06629
$304$	0	0
$305$	−24.3256	−1.39288
$306$	0	0
$307$	−4.19538	−0.239443	−0.119721	−	0.992808i	$-0.538200\pi$
$307$	−4.19538	−0.239443	−0.119721	+	0.992808i	$0.538200\pi$
$308$	0	0
$309$	−21.0080	−1.19510
$310$	0	0
$311$	−15.1362	−0.858296	−0.429148	−	0.903234i	$-0.641186\pi$
$311$	−15.1362	−0.858296	−0.429148	+	0.903234i	$0.641186\pi$
$312$	0	0
$313$	−2.26357	−0.127944	−0.0639722	−	0.997952i	$-0.520377\pi$
$313$	−2.26357	−0.127944	−0.0639722	+	0.997952i	$0.520377\pi$
$314$	0	0
$315$	−4.89274	−0.275675
$316$	0	0
$317$	8.24703	0.463199	0.231600	−	0.972811i	$-0.425604\pi$
$317$	8.24703	0.463199	0.231600	+	0.972811i	$0.425604\pi$
$318$	0	0
$319$	0.461586	0.0258439
$320$	0	0
$321$	21.6038	1.20581
$322$	0	0
$323$	−6.52731	−0.363189
$324$	0	0
$325$	−42.7294	−2.37020
$326$	0	0
$327$	−12.6361	−0.698778
$328$	0	0
$329$	1.06877	0.0589234
$330$	0	0
$331$	−28.0021	−1.53913	−0.769567	−	0.638567i	$-0.779528\pi$
$331$	−28.0021	−1.53913	−0.769567	+	0.638567i	$0.779528\pi$
$332$	0	0
$333$	16.3788	0.897555
$334$	0	0
$335$	8.69455	0.475034
$336$	0	0
$337$	−24.9915	−1.36137	−0.680686	−	0.732576i	$-0.738318\pi$
$337$	−24.9915	−1.36137	−0.680686	+	0.732576i	$0.738318\pi$
$338$	0	0
$339$	−5.55413	−0.301659
$340$	0	0
$341$	−8.28147	−0.448467
$342$	0	0
$343$	−6.33220	−0.341907
$344$	0	0
$345$	4.58061	0.246612
$346$	0	0
$347$	−2.47834	−0.133044	−0.0665222	−	0.997785i	$-0.521190\pi$
$347$	−2.47834	−0.133044	−0.0665222	+	0.997785i	$0.521190\pi$
$348$	0	0
$349$	30.3845	1.62645	0.813223	−	0.581952i	$-0.197711\pi$
$349$	30.3845	1.62645	0.813223	+	0.581952i	$0.197711\pi$
$350$	0	0
$351$	−0.439578	−0.0234630
$352$	0	0
$353$	−9.45770	−0.503382	−0.251691	−	0.967808i	$-0.580987\pi$
$353$	−9.45770	−0.503382	−0.251691	+	0.967808i	$0.580987\pi$
$354$	0	0
$355$	13.3053	0.706171
$356$	0	0
$357$	−7.32185	−0.387514
$358$	0	0
$359$	−6.37708	−0.336569	−0.168285	−	0.985738i	$-0.553823\pi$
$359$	−6.37708	−0.336569	−0.168285	+	0.985738i	$0.553823\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.294852	0.0154757
$364$	0	0
$365$	−1.81833	−0.0951756
$366$	0	0
$367$	−5.25534	−0.274327	−0.137163	−	0.990548i	$-0.543799\pi$
$367$	−5.25534	−0.274327	−0.137163	+	0.990548i	$0.543799\pi$
$368$	0	0
$369$	2.05686	0.107076
$370$	0	0
$371$	0.459217	0.0238414
$372$	0	0
$373$	15.2815	0.791248	0.395624	−	0.918413i	$-0.370528\pi$
$373$	15.2815	0.791248	0.395624	+	0.918413i	$0.370528\pi$
$374$	0	0
$375$	25.4194	1.31265
$376$	0	0
$377$	0.757152	0.0389953
$378$	0	0
$379$	−15.9601	−0.819818	−0.409909	−	0.912127i	$-0.634439\pi$
$379$	−15.9601	−0.819818	−0.409909	+	0.912127i	$0.634439\pi$
$380$	0	0
$381$	−1.03351	−0.0529485
$382$	0	0
$383$	2.18080	0.111434	0.0557169	−	0.998447i	$-0.482256\pi$
$383$	2.18080	0.111434	0.0557169	+	0.998447i	$0.482256\pi$
$384$	0	0
$385$	5.43968	0.277232
$386$	0	0
$387$	−6.47127	−0.328953
$388$	0	0
$389$	21.8895	1.10984	0.554921	−	0.831903i	$-0.312749\pi$
$389$	21.8895	1.10984	0.554921	+	0.831903i	$0.312749\pi$
$390$	0	0
$391$	3.40827	0.172364
$392$	0	0
$393$	33.3359	1.68157
$394$	0	0
$395$	−5.06562	−0.254879
$396$	0	0
$397$	27.0175	1.35597	0.677984	−	0.735076i	$-0.262854\pi$
$397$	27.0175	1.35597	0.677984	+	0.735076i	$0.262854\pi$
$398$	0	0
$399$	1.12173	0.0561565
$400$	0	0
$401$	24.2691	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$401$	24.2691	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$402$	0	0
$403$	−13.5843	−0.676683
$404$	0	0
$405$	32.6763	1.62370
$406$	0	0
$407$	−18.2097	−0.902624
$408$	0	0
$409$	15.4805	0.765463	0.382731	−	0.923860i	$-0.374983\pi$
$409$	15.4805	0.765463	0.382731	+	0.923860i	$0.374983\pi$
$410$	0	0
$411$	−38.3664	−1.89247
$412$	0	0
$413$	4.39009	0.216022
$414$	0	0
$415$	−20.8772	−1.02482
$416$	0	0
$417$	4.53940	0.222295
$418$	0	0
$419$	21.6618	1.05825	0.529124	−	0.848545i	$-0.322521\pi$
$419$	21.6618	1.05825	0.529124	+	0.848545i	$0.322521\pi$
$420$	0	0
$421$	−31.1808	−1.51966	−0.759830	−	0.650122i	$-0.774718\pi$
$421$	−31.1808	−1.51966	−0.759830	+	0.650122i	$0.774718\pi$
$422$	0	0
$423$	6.90474	0.335720
$424$	0	0
$425$	51.5503	2.50056
$426$	0	0
$427$	3.11048	0.150527
$428$	0	0
$429$	−43.5912	−2.10460
$430$	0	0
$431$	14.4428	0.695683	0.347841	−	0.937553i	$-0.386915\pi$
$431$	14.4428	0.695683	0.347841	+	0.937553i	$0.386915\pi$
$432$	0	0
$433$	13.4118	0.644529	0.322264	−	0.946650i	$-0.395556\pi$
$433$	13.4118	0.644529	0.322264	+	0.946650i	$0.395556\pi$
$434$	0	0
$435$	−1.22765	−0.0588615
$436$	0	0
$437$	−0.522155	−0.0249781
$438$	0	0
$439$	10.6117	0.506469	0.253235	−	0.967405i	$-0.418506\pi$
$439$	10.6117	0.506469	0.253235	+	0.967405i	$0.418506\pi$
$440$	0	0
$441$	−20.1415	−0.959121
$442$	0	0
$443$	31.0643	1.47591	0.737955	−	0.674850i	$-0.235792\pi$
$443$	31.0643	1.47591	0.737955	+	0.674850i	$0.235792\pi$
$444$	0	0
$445$	3.58411	0.169903
$446$	0	0
$447$	27.6272	1.30672
$448$	0	0
$449$	10.0955	0.476438	0.238219	−	0.971212i	$-0.423436\pi$
$449$	10.0955	0.476438	0.238219	+	0.971212i	$0.423436\pi$
$450$	0	0
$451$	−2.28679	−0.107681
$452$	0	0
$453$	17.0559	0.801358
$454$	0	0
$455$	8.92285	0.418309
$456$	0	0
$457$	−20.2946	−0.949341	−0.474671	−	0.880164i	$-0.657433\pi$
$457$	−20.2946	−0.949341	−0.474671	+	0.880164i	$0.657433\pi$
$458$	0	0
$459$	0.530323	0.0247534
$460$	0	0
$461$	7.08506	0.329984	0.164992	−	0.986295i	$-0.447240\pi$
$461$	7.08506	0.329984	0.164992	+	0.986295i	$0.447240\pi$
$462$	0	0
$463$	−13.5240	−0.628512	−0.314256	−	0.949338i	$-0.601755\pi$
$463$	−13.5240	−0.628512	−0.314256	+	0.949338i	$0.601755\pi$
$464$	0	0
$465$	22.0258	1.02142
$466$	0	0
$467$	23.4060	1.08310	0.541550	−	0.840668i	$-0.317838\pi$
$467$	23.4060	1.08310	0.541550	+	0.840668i	$0.317838\pi$
$468$	0	0
$469$	−1.11176	−0.0513362
$470$	0	0
$471$	54.0576	2.49084
$472$	0	0
$473$	7.19467	0.330811
$474$	0	0
$475$	−7.89763	−0.362368
$476$	0	0
$477$	2.96674	0.135838
$478$	0	0
$479$	−8.39605	−0.383625	−0.191813	−	0.981432i	$-0.561437\pi$
$479$	−8.39605	−0.383625	−0.191813	+	0.981432i	$0.561437\pi$
$480$	0	0
$481$	−29.8699	−1.36195
$482$	0	0
$483$	−0.585715	−0.0266509
$484$	0	0
$485$	−0.428782	−0.0194700
$486$	0	0
$487$	−16.8844	−0.765106	−0.382553	−	0.923934i	$-0.624955\pi$
$487$	−16.8844	−0.765106	−0.382553	+	0.923934i	$0.624955\pi$
$488$	0	0
$489$	34.0083	1.53791
$490$	0	0
$491$	18.2612	0.824115	0.412057	−	0.911158i	$-0.364810\pi$
$491$	18.2612	0.824115	0.412057	+	0.911158i	$0.364810\pi$
$492$	0	0
$493$	−0.913455	−0.0411399
$494$	0	0
$495$	35.1427	1.57954
$496$	0	0
$497$	−1.70132	−0.0763148
$498$	0	0
$499$	17.2036	0.770140	0.385070	−	0.922887i	$-0.374177\pi$
$499$	17.2036	0.770140	0.385070	+	0.922887i	$0.374177\pi$
$500$	0	0
$501$	30.9869	1.38439
$502$	0	0
$503$	29.7989	1.32867	0.664333	−	0.747436i	$-0.268715\pi$
$503$	29.7989	1.32867	0.664333	+	0.747436i	$0.268715\pi$
$504$	0	0
$505$	−52.8809	−2.35317
$506$	0	0
$507$	−39.7488	−1.76531
$508$	0	0
$509$	7.79995	0.345727	0.172863	−	0.984946i	$-0.444698\pi$
$509$	7.79995	0.345727	0.172863	+	0.984946i	$0.444698\pi$
$510$	0	0
$511$	0.232507	0.0102855
$512$	0	0
$513$	−0.0812467	−0.00358713
$514$	0	0
$515$	−30.8867	−1.36103
$516$	0	0
$517$	−7.67659	−0.337616
$518$	0	0
$519$	−39.9689	−1.75444
$520$	0	0
$521$	6.85304	0.300237	0.150119	−	0.988668i	$-0.452034\pi$
$521$	6.85304	0.300237	0.150119	+	0.988668i	$0.452034\pi$
$522$	0	0
$523$	−15.0537	−0.658254	−0.329127	−	0.944286i	$-0.606755\pi$
$523$	−15.0537	−0.658254	−0.329127	+	0.944286i	$0.606755\pi$
$524$	0	0
$525$	−8.85897	−0.386637
$526$	0	0
$527$	16.3886	0.713899
$528$	0	0
$529$	−22.7274	−0.988146
$530$	0	0
$531$	28.3619	1.23080
$532$	0	0
$533$	−3.75107	−0.162477
$534$	0	0
$535$	31.7626	1.37322
$536$	0	0
$537$	−48.3631	−2.08702
$538$	0	0
$539$	22.3931	0.964538
$540$	0	0
$541$	28.6664	1.23246	0.616232	−	0.787564i	$-0.288658\pi$
$541$	28.6664	1.23246	0.616232	+	0.787564i	$0.288658\pi$
$542$	0	0
$543$	2.03249	0.0872223
$544$	0	0
$545$	−18.5780	−0.795796
$546$	0	0
$547$	−2.68722	−0.114897	−0.0574486	−	0.998348i	$-0.518297\pi$
$547$	−2.68722	−0.114897	−0.0574486	+	0.998348i	$0.518297\pi$
$548$	0	0
$549$	20.0950	0.857635
$550$	0	0
$551$	0.139943	0.00596179
$552$	0	0
$553$	0.647732	0.0275444
$554$	0	0
$555$	48.4314	2.05580
$556$	0	0
$557$	−14.3606	−0.608476	−0.304238	−	0.952596i	$-0.598402\pi$
$557$	−14.3606	−0.608476	−0.304238	+	0.952596i	$0.598402\pi$
$558$	0	0
$559$	11.8016	0.499154
$560$	0	0
$561$	52.5900	2.22035
$562$	0	0
$563$	31.2562	1.31729	0.658646	−	0.752453i	$-0.271130\pi$
$563$	31.2562	1.31729	0.658646	+	0.752453i	$0.271130\pi$
$564$	0	0
$565$	−8.16588	−0.343541
$566$	0	0
$567$	−4.17827	−0.175471
$568$	0	0
$569$	42.7820	1.79351	0.896756	−	0.442525i	$-0.145917\pi$
$569$	42.7820	1.79351	0.896756	+	0.442525i	$0.145917\pi$
$570$	0	0
$571$	−26.1120	−1.09275	−0.546377	−	0.837540i	$-0.683993\pi$
$571$	−26.1120	−1.09275	−0.546377	+	0.837540i	$0.683993\pi$
$572$	0	0
$573$	50.3371	2.10286
$574$	0	0
$575$	4.12379	0.171974
$576$	0	0
$577$	21.4036	0.891045	0.445522	−	0.895271i	$-0.353018\pi$
$577$	21.4036	0.891045	0.445522	+	0.895271i	$0.353018\pi$
$578$	0	0
$579$	35.4473	1.47314
$580$	0	0
$581$	2.66953	0.110751
$582$	0	0
$583$	−3.29838	−0.136605
$584$	0	0
$585$	57.6454	2.38334
$586$	0	0
$587$	37.4229	1.54461	0.772304	−	0.635254i	$-0.219104\pi$
$587$	37.4229	1.54461	0.772304	+	0.635254i	$0.219104\pi$
$588$	0	0
$589$	−2.51077	−0.103455
$590$	0	0
$591$	27.2844	1.12233
$592$	0	0
$593$	8.00459	0.328709	0.164355	−	0.986401i	$-0.447446\pi$
$593$	8.00459	0.328709	0.164355	+	0.986401i	$0.447446\pi$
$594$	0	0
$595$	−10.7648	−0.441315
$596$	0	0
$597$	−31.0007	−1.26877
$598$	0	0
$599$	0.810623	0.0331212	0.0165606	−	0.999863i	$-0.494728\pi$
$599$	0.810623	0.0331212	0.0165606	+	0.999863i	$0.494728\pi$
$600$	0	0
$601$	32.5115	1.32617	0.663086	−	0.748544i	$-0.269247\pi$
$601$	32.5115	1.32617	0.663086	+	0.748544i	$0.269247\pi$
$602$	0	0
$603$	−7.18243	−0.292491
$604$	0	0
$605$	0.433501	0.0176243
$606$	0	0
$607$	−1.17275	−0.0476003	−0.0238002	−	0.999717i	$-0.507577\pi$
$607$	−1.17275	−0.0476003	−0.0238002	+	0.999717i	$0.507577\pi$
$608$	0	0
$609$	0.156978	0.00636108
$610$	0	0
$611$	−12.5921	−0.509422
$612$	0	0
$613$	−35.1142	−1.41825	−0.709124	−	0.705084i	$-0.750909\pi$
$613$	−35.1142	−1.41825	−0.709124	+	0.705084i	$0.750909\pi$
$614$	0	0
$615$	6.08203	0.245251
$616$	0	0
$617$	−39.2551	−1.58035	−0.790176	−	0.612880i	$-0.790011\pi$
$617$	−39.2551	−1.58035	−0.790176	+	0.612880i	$0.790011\pi$
$618$	0	0
$619$	14.4939	0.582557	0.291279	−	0.956638i	$-0.405919\pi$
$619$	14.4939	0.582557	0.291279	+	0.956638i	$0.405919\pi$
$620$	0	0
$621$	0.0424234	0.00170239
$622$	0	0
$623$	−0.458295	−0.0183612
$624$	0	0
$625$	−2.11562	−0.0846248
$626$	0	0
$627$	−8.05692	−0.321762
$628$	0	0
$629$	36.0361	1.43685
$630$	0	0
$631$	−35.0764	−1.39637	−0.698184	−	0.715918i	$-0.746008\pi$
$631$	−35.0764	−1.39637	−0.698184	+	0.715918i	$0.746008\pi$
$632$	0	0
$633$	−34.2136	−1.35987
$634$	0	0
$635$	−1.51951	−0.0602997
$636$	0	0
$637$	36.7319	1.45537
$638$	0	0
$639$	−10.9913	−0.434808
$640$	0	0
$641$	8.53787	0.337225	0.168613	−	0.985682i	$-0.446071\pi$
$641$	8.53787	0.337225	0.168613	+	0.985682i	$0.446071\pi$
$642$	0	0
$643$	21.7819	0.858993	0.429496	−	0.903069i	$-0.358691\pi$
$643$	21.7819	0.858993	0.429496	+	0.903069i	$0.358691\pi$
$644$	0	0
$645$	−19.1352	−0.753449
$646$	0	0
$647$	−23.7341	−0.933085	−0.466543	−	0.884499i	$-0.654501\pi$
$647$	−23.7341	−0.933085	−0.466543	+	0.884499i	$0.654501\pi$
$648$	0	0
$649$	−31.5323	−1.23775
$650$	0	0
$651$	−2.81640	−0.110383
$652$	0	0
$653$	−29.9648	−1.17261	−0.586306	−	0.810089i	$-0.699419\pi$
$653$	−29.9648	−1.17261	−0.586306	+	0.810089i	$0.699419\pi$
$654$	0	0
$655$	49.0115	1.91504
$656$	0	0
$657$	1.50209	0.0586022
$658$	0	0
$659$	−27.3446	−1.06520	−0.532598	−	0.846368i	$-0.678784\pi$
$659$	−27.3446	−1.06520	−0.532598	+	0.846368i	$0.678784\pi$
$660$	0	0
$661$	−30.3894	−1.18201	−0.591005	−	0.806668i	$-0.701269\pi$
$661$	−30.3894	−1.18201	−0.591005	+	0.806668i	$0.701269\pi$
$662$	0	0
$663$	86.2648	3.35025
$664$	0	0
$665$	1.64920	0.0639532
$666$	0	0
$667$	−0.0730722	−0.00282937
$668$	0	0
$669$	57.8307	2.23586
$670$	0	0
$671$	−22.3414	−0.862479
$672$	0	0
$673$	32.5992	1.25661	0.628303	−	0.777969i	$-0.283750\pi$
$673$	32.5992	1.25661	0.628303	+	0.777969i	$0.283750\pi$
$674$	0	0
$675$	0.641656	0.0246974
$676$	0	0
$677$	−6.02569	−0.231586	−0.115793	−	0.993273i	$-0.536941\pi$
$677$	−6.02569	−0.231586	−0.115793	+	0.993273i	$0.536941\pi$
$678$	0	0
$679$	0.0548277	0.00210409
$680$	0	0
$681$	51.4542	1.97173
$682$	0	0
$683$	−9.87214	−0.377747	−0.188874	−	0.982001i	$-0.560484\pi$
$683$	−9.87214	−0.377747	−0.188874	+	0.982001i	$0.560484\pi$
$684$	0	0
$685$	−56.4075	−2.15522
$686$	0	0
$687$	−8.50726	−0.324572
$688$	0	0
$689$	−5.41041	−0.206120
$690$	0	0
$691$	−33.1098	−1.25955	−0.629777	−	0.776776i	$-0.716854\pi$
$691$	−33.1098	−1.25955	−0.629777	+	0.776776i	$0.716854\pi$
$692$	0	0
$693$	−4.49363	−0.170699
$694$	0	0
$695$	6.67397	0.253158
$696$	0	0
$697$	4.52543	0.171413
$698$	0	0
$699$	−64.6019	−2.44347
$700$	0	0
$701$	37.5585	1.41857	0.709283	−	0.704924i	$-0.249019\pi$
$701$	37.5585	1.41857	0.709283	+	0.704924i	$0.249019\pi$
$702$	0	0
$703$	−5.52082	−0.208222
$704$	0	0
$705$	20.4170	0.768948
$706$	0	0
$707$	6.76180	0.254303
$708$	0	0
$709$	19.3947	0.728385	0.364192	−	0.931324i	$-0.381345\pi$
$709$	19.3947	0.728385	0.364192	+	0.931324i	$0.381345\pi$
$710$	0	0
$711$	4.18463	0.156936
$712$	0	0
$713$	1.31101	0.0490978
$714$	0	0
$715$	−64.0893	−2.39680
$716$	0	0
$717$	22.3870	0.836058
$718$	0	0
$719$	−24.8440	−0.926525	−0.463263	−	0.886221i	$-0.653321\pi$
$719$	−24.8440	−0.926525	−0.463263	+	0.886221i	$0.653321\pi$
$720$	0	0
$721$	3.94943	0.147085
$722$	0	0
$723$	−45.2892	−1.68432
$724$	0	0
$725$	−1.10522	−0.0410469
$726$	0	0
$727$	−24.2408	−0.899041	−0.449520	−	0.893270i	$-0.648405\pi$
$727$	−24.2408	−0.899041	−0.449520	+	0.893270i	$0.648405\pi$
$728$	0	0
$729$	−26.3980	−0.977704
$730$	0	0
$731$	−14.2379	−0.526606
$732$	0	0
$733$	10.4327	0.385340	0.192670	−	0.981264i	$-0.438285\pi$
$733$	10.4327	0.385340	0.192670	+	0.981264i	$0.438285\pi$
$734$	0	0
$735$	−59.5576	−2.19681
$736$	0	0
$737$	7.98533	0.294143
$738$	0	0
$739$	−7.54830	−0.277669	−0.138834	−	0.990316i	$-0.544336\pi$
$739$	−7.54830	−0.277669	−0.138834	+	0.990316i	$0.544336\pi$
$740$	0	0
$741$	−13.2160	−0.485501
$742$	0	0
$743$	−43.4918	−1.59556	−0.797779	−	0.602950i	$-0.793992\pi$
$743$	−43.4918	−1.59556	−0.797779	+	0.602950i	$0.793992\pi$
$744$	0	0
$745$	40.6184	1.48814
$746$	0	0
$747$	17.2463	0.631010
$748$	0	0
$749$	−4.06144	−0.148402
$750$	0	0
$751$	−38.7891	−1.41544	−0.707718	−	0.706495i	$-0.750275\pi$
$751$	−38.7891	−1.41544	−0.707718	+	0.706495i	$0.750275\pi$
$752$	0	0
$753$	−25.0072	−0.911314
$754$	0	0
$755$	25.0762	0.912617
$756$	0	0
$757$	−28.2065	−1.02518	−0.512592	−	0.858632i	$-0.671315\pi$
$757$	−28.2065	−1.02518	−0.512592	+	0.858632i	$0.671315\pi$
$758$	0	0
$759$	4.20696	0.152703
$760$	0	0
$761$	6.67415	0.241938	0.120969	−	0.992656i	$-0.461400\pi$
$761$	6.67415	0.241938	0.120969	+	0.992656i	$0.461400\pi$
$762$	0	0
$763$	2.37554	0.0860004
$764$	0	0
$765$	−69.5455	−2.51442
$766$	0	0
$767$	−51.7233	−1.86762
$768$	0	0
$769$	−2.63193	−0.0949098	−0.0474549	−	0.998873i	$-0.515111\pi$
$769$	−2.63193	−0.0949098	−0.0474549	+	0.998873i	$0.515111\pi$
$770$	0	0
$771$	−35.2871	−1.27084
$772$	0	0
$773$	4.21113	0.151464	0.0757319	−	0.997128i	$-0.475871\pi$
$773$	4.21113	0.151464	0.0757319	+	0.997128i	$0.475871\pi$
$774$	0	0
$775$	19.8291	0.712284
$776$	0	0
$777$	−6.19285	−0.222167
$778$	0	0
$779$	−0.693306	−0.0248403
$780$	0	0
$781$	12.2199	0.437264
$782$	0	0
$783$	−0.0113700	−0.000406329 0
$784$	0	0
$785$	79.4773	2.83667
$786$	0	0
$787$	−3.68760	−0.131449	−0.0657245	−	0.997838i	$-0.520936\pi$
$787$	−3.68760	−0.131449	−0.0657245	+	0.997838i	$0.520936\pi$
$788$	0	0
$789$	30.0264	1.06897
$790$	0	0
$791$	1.04416	0.0371260
$792$	0	0
$793$	−36.6471	−1.30138
$794$	0	0
$795$	8.77250	0.311129
$796$	0	0
$797$	12.8926	0.456680	0.228340	−	0.973581i	$-0.426670\pi$
$797$	12.8926	0.456680	0.228340	+	0.973581i	$0.426670\pi$
$798$	0	0
$799$	15.1916	0.537439
$800$	0	0
$801$	−2.96078	−0.104614
$802$	0	0
$803$	−1.67000	−0.0589332
$804$	0	0
$805$	−0.861138	−0.0303511
$806$	0	0
$807$	37.9508	1.33593
$808$	0	0
$809$	14.9880	0.526950	0.263475	−	0.964666i	$-0.415131\pi$
$809$	14.9880	0.526950	0.263475	+	0.964666i	$0.415131\pi$
$810$	0	0
$811$	−8.76181	−0.307669	−0.153834	−	0.988097i	$-0.549162\pi$
$811$	−8.76181	−0.307669	−0.153834	+	0.988097i	$0.549162\pi$
$812$	0	0
$813$	61.7729	2.16647
$814$	0	0
$815$	50.0002	1.75143
$816$	0	0
$817$	2.18127	0.0763131
$818$	0	0
$819$	−7.37102	−0.257564
$820$	0	0
$821$	−42.0284	−1.46680	−0.733401	−	0.679796i	$-0.762068\pi$
$821$	−42.0284	−1.46680	−0.733401	+	0.679796i	$0.762068\pi$
$822$	0	0
$823$	−41.5337	−1.44777	−0.723886	−	0.689919i	$-0.757646\pi$
$823$	−41.5337	−1.44777	−0.723886	+	0.689919i	$0.757646\pi$
$824$	0	0
$825$	63.6305	2.21533
$826$	0	0
$827$	−24.2613	−0.843648	−0.421824	−	0.906678i	$-0.638610\pi$
$827$	−24.2613	−0.843648	−0.421824	+	0.906678i	$0.638610\pi$
$828$	0	0
$829$	−20.2229	−0.702371	−0.351185	−	0.936306i	$-0.614221\pi$
$829$	−20.2229	−0.702371	−0.351185	+	0.936306i	$0.614221\pi$
$830$	0	0
$831$	55.3041	1.91848
$832$	0	0
$833$	−44.3147	−1.53541
$834$	0	0
$835$	45.5581	1.57660
$836$	0	0
$837$	0.203992	0.00705100
$838$	0	0
$839$	44.1482	1.52417	0.762083	−	0.647480i	$-0.224177\pi$
$839$	44.1482	1.52417	0.762083	+	0.647480i	$0.224177\pi$
$840$	0	0
$841$	−28.9804	−0.999325
$842$	0	0
$843$	−27.5044	−0.947303
$844$	0	0
$845$	−58.4401	−2.01040
$846$	0	0
$847$	−0.0554311	−0.00190464
$848$	0	0
$849$	78.6312	2.69861
$850$	0	0
$851$	2.88273	0.0988186
$852$	0	0
$853$	15.1577	0.518988	0.259494	−	0.965745i	$-0.416444\pi$
$853$	15.1577	0.518988	0.259494	+	0.965745i	$0.416444\pi$
$854$	0	0
$855$	10.6545	0.364377
$856$	0	0
$857$	−56.4588	−1.92859	−0.964297	−	0.264822i	$-0.914687\pi$
$857$	−56.4588	−1.92859	−0.964297	+	0.264822i	$0.914687\pi$
$858$	0	0
$859$	57.1254	1.94909	0.974547	−	0.224182i	$-0.0719709\pi$
$859$	57.1254	1.94909	0.974547	+	0.224182i	$0.0719709\pi$
$860$	0	0
$861$	−0.777699	−0.0265039
$862$	0	0
$863$	−42.5571	−1.44866	−0.724330	−	0.689453i	$-0.757851\pi$
$863$	−42.5571	−1.44866	−0.724330	+	0.689453i	$0.757851\pi$
$864$	0	0
$865$	−58.7636	−1.99802
$866$	0	0
$867$	−62.5471	−2.12421
$868$	0	0
$869$	−4.65241	−0.157822
$870$	0	0
$871$	13.0985	0.443827
$872$	0	0
$873$	0.354210	0.0119882
$874$	0	0
$875$	−4.77876	−0.161552
$876$	0	0
$877$	35.6540	1.20395	0.601974	−	0.798515i	$-0.294381\pi$
$877$	35.6540	1.20395	0.601974	+	0.798515i	$0.294381\pi$
$878$	0	0
$879$	17.1850	0.579637
$880$	0	0
$881$	37.3422	1.25809	0.629045	−	0.777369i	$-0.283446\pi$
$881$	37.3422	1.25809	0.629045	+	0.777369i	$0.283446\pi$
$882$	0	0
$883$	−45.0379	−1.51565	−0.757823	−	0.652460i	$-0.773737\pi$
$883$	−45.0379	−1.51565	−0.757823	+	0.652460i	$0.773737\pi$
$884$	0	0
$885$	83.8647	2.81908
$886$	0	0
$887$	−14.1403	−0.474785	−0.237392	−	0.971414i	$-0.576293\pi$
$887$	−14.1403	−0.474785	−0.237392	+	0.971414i	$0.576293\pi$
$888$	0	0
$889$	0.194297	0.00651650
$890$	0	0
$891$	30.0109	1.00540
$892$	0	0
$893$	−2.32738	−0.0778829
$894$	0	0
$895$	−71.1051	−2.37678
$896$	0	0
$897$	6.90078	0.230410
$898$	0	0
$899$	−0.351366	−0.0117187
$900$	0	0
$901$	6.52731	0.217456
$902$	0	0
$903$	2.44679	0.0814241
$904$	0	0
$905$	2.98823	0.0993321
$906$	0	0
$907$	24.6723	0.819231	0.409616	−	0.912258i	$-0.365663\pi$
$907$	24.6723	0.819231	0.409616	+	0.912258i	$0.365663\pi$
$908$	0	0
$909$	43.6841	1.44891
$910$	0	0
$911$	−11.1526	−0.369502	−0.184751	−	0.982785i	$-0.559148\pi$
$911$	−11.1526	−0.369502	−0.184751	+	0.982785i	$0.559148\pi$
$912$	0	0
$913$	−19.1742	−0.634573
$914$	0	0
$915$	59.4200	1.96437
$916$	0	0
$917$	−6.26703	−0.206955
$918$	0	0
$919$	−9.65827	−0.318597	−0.159299	−	0.987230i	$-0.550923\pi$
$919$	−9.65827	−0.318597	−0.159299	+	0.987230i	$0.550923\pi$
$920$	0	0
$921$	10.2480	0.337684
$922$	0	0
$923$	20.0447	0.659779
$924$	0	0
$925$	43.6014	1.43360
$926$	0	0
$927$	25.5150	0.838023
$928$	0	0
$929$	−18.7420	−0.614906	−0.307453	−	0.951563i	$-0.599477\pi$
$929$	−18.7420	−0.614906	−0.307453	+	0.951563i	$0.599477\pi$
$930$	0	0
$931$	6.78912	0.222504
$932$	0	0
$933$	36.9731	1.21044
$934$	0	0
$935$	77.3196	2.52862
$936$	0	0
$937$	−53.7374	−1.75552	−0.877762	−	0.479096i	$-0.840965\pi$
$937$	−53.7374	−1.75552	−0.877762	+	0.479096i	$0.840965\pi$
$938$	0	0
$939$	5.52919	0.180438
$940$	0	0
$941$	51.6184	1.68271	0.841356	−	0.540481i	$-0.181758\pi$
$941$	51.6184	1.68271	0.841356	+	0.540481i	$0.181758\pi$
$942$	0	0
$943$	0.362014	0.0117888
$944$	0	0
$945$	−0.133992	−0.00435876
$946$	0	0
$947$	−10.9639	−0.356278	−0.178139	−	0.984005i	$-0.557008\pi$
$947$	−10.9639	−0.356278	−0.178139	+	0.984005i	$0.557008\pi$
$948$	0	0
$949$	−2.73935	−0.0889231
$950$	0	0
$951$	−20.1449	−0.653244
$952$	0	0
$953$	−28.3189	−0.917340	−0.458670	−	0.888607i	$-0.651674\pi$
$953$	−28.3189	−0.917340	−0.458670	+	0.888607i	$0.651674\pi$
$954$	0	0
$955$	74.0073	2.39482
$956$	0	0
$957$	−1.12751	−0.0364473
$958$	0	0
$959$	7.21274	0.232912
$960$	0	0
$961$	−24.6960	−0.796646
$962$	0	0
$963$	−26.2386	−0.845528
$964$	0	0
$965$	52.1158	1.67767
$966$	0	0
$967$	−4.90853	−0.157848	−0.0789239	−	0.996881i	$-0.525148\pi$
$967$	−4.90853	−0.157848	−0.0789239	+	0.996881i	$0.525148\pi$
$968$	0	0
$969$	15.9442	0.512202
$970$	0	0
$971$	−11.6759	−0.374698	−0.187349	−	0.982293i	$-0.559990\pi$
$971$	−11.6759	−0.374698	−0.187349	+	0.982293i	$0.559990\pi$
$972$	0	0
$973$	−0.853390	−0.0273584
$974$	0	0
$975$	104.375	3.34267
$976$	0	0
$977$	14.9786	0.479208	0.239604	−	0.970871i	$-0.422982\pi$
$977$	14.9786	0.479208	0.239604	+	0.970871i	$0.422982\pi$
$978$	0	0
$979$	3.29175	0.105205
$980$	0	0
$981$	15.3470	0.489993
$982$	0	0
$983$	55.5582	1.77203	0.886016	−	0.463655i	$-0.153462\pi$
$983$	55.5582	1.77203	0.886016	+	0.463655i	$0.153462\pi$
$984$	0	0
$985$	40.1144	1.27815
$986$	0	0
$987$	−2.61069	−0.0830990
$988$	0	0
$989$	−1.13896	−0.0362169
$990$	0	0
$991$	−25.8769	−0.822007	−0.411004	−	0.911634i	$-0.634822\pi$
$991$	−25.8769	−0.822007	−0.411004	+	0.911634i	$0.634822\pi$
$992$	0	0
$993$	68.4004	2.17062
$994$	0	0
$995$	−45.5783	−1.44493
$996$	0	0
$997$	29.0385	0.919659	0.459830	−	0.888007i	$-0.347910\pi$
$997$	29.0385	0.919659	0.459830	+	0.888007i	$0.347910\pi$
$998$	0	0
$999$	0.448549	0.0141915

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.3	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.44269 q^{3} -3.59133 q^{5} +0.459217 q^{7} +2.96674 q^{9} +O(q^{10})\)	\(q-2.44269 q^{3} -3.59133 q^{5} +0.459217 q^{7} +2.96674 q^{9} -3.29838 q^{11} -5.41041 q^{13} +8.77250 q^{15} +6.52731 q^{17} -1.00000 q^{19} -1.12173 q^{21} +0.522155 q^{23} +7.89763 q^{25} +0.0812467 q^{27} -0.139943 q^{29} +2.51077 q^{31} +8.05692 q^{33} -1.64920 q^{35} +5.52082 q^{37} +13.2160 q^{39} +0.693306 q^{41} -2.18127 q^{43} -10.6545 q^{45} +2.32738 q^{47} -6.78912 q^{49} -15.9442 q^{51} +1.00000 q^{53} +11.8456 q^{55} +2.44269 q^{57} +9.55995 q^{59} +6.77344 q^{61} +1.36238 q^{63} +19.4306 q^{65} -2.42099 q^{67} -1.27546 q^{69} -3.70484 q^{71} +0.506311 q^{73} -19.2915 q^{75} -1.51467 q^{77} +1.41051 q^{79} -9.09868 q^{81} +5.81322 q^{83} -23.4417 q^{85} +0.341839 q^{87} -0.997992 q^{89} -2.48455 q^{91} -6.13304 q^{93} +3.59133 q^{95} +0.119394 q^{97} -9.78542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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