LMFDB - Embedded newform 4028.2.a.e.1.6

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			1.6
Root			$-1.30887$ 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)

$f(q)$	$=$	$q-1.30887 q^{3} +3.64710 q^{5} +1.67706 q^{7} -1.28687 q^{9} +O(q^{10})$	$q-1.30887 q^{3} +3.64710 q^{5} +1.67706 q^{7} -1.28687 q^{9} +1.55095 q^{11} -0.417445 q^{13} -4.77357 q^{15} -1.15945 q^{17} +1.00000 q^{19} -2.19505 q^{21} -6.03406 q^{23} +8.30135 q^{25} +5.61094 q^{27} -3.72313 q^{29} +7.84462 q^{31} -2.02999 q^{33} +6.11640 q^{35} +3.51435 q^{37} +0.546380 q^{39} +6.10255 q^{41} +0.837110 q^{43} -4.69333 q^{45} +4.08638 q^{47} -4.18748 q^{49} +1.51757 q^{51} +1.00000 q^{53} +5.65647 q^{55} -1.30887 q^{57} -6.71588 q^{59} +1.25269 q^{61} -2.15815 q^{63} -1.52246 q^{65} +14.6692 q^{67} +7.89779 q^{69} +13.1745 q^{71} +1.35505 q^{73} -10.8654 q^{75} +2.60103 q^{77} +14.5554 q^{79} -3.48337 q^{81} +14.8244 q^{83} -4.22865 q^{85} +4.87309 q^{87} -5.94514 q^{89} -0.700079 q^{91} -10.2676 q^{93} +3.64710 q^{95} +6.59511 q^{97} -1.99587 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.30887	−0.755675	−0.377837	−	0.925872i	$-0.623332\pi$
$3$	−1.30887	−0.755675	−0.377837	+	0.925872i	$0.623332\pi$
$4$	0	0
$5$	3.64710	1.63103	0.815517	−	0.578733i	$-0.196453\pi$
$5$	3.64710	1.63103	0.815517	+	0.578733i	$0.196453\pi$
$6$	0	0
$7$	1.67706	0.633868	0.316934	−	0.948448i	$-0.397347\pi$
$7$	1.67706	0.633868	0.316934	+	0.948448i	$0.397347\pi$
$8$	0	0
$9$	−1.28687	−0.428956
$10$	0	0
$11$	1.55095	0.467629	0.233815	−	0.972281i	$-0.424879\pi$
$11$	1.55095	0.467629	0.233815	+	0.972281i	$0.424879\pi$
$12$	0	0
$13$	−0.417445	−0.115778	−0.0578892	−	0.998323i	$-0.518437\pi$
$13$	−0.417445	−0.115778	−0.0578892	+	0.998323i	$0.518437\pi$
$14$	0	0
$15$	−4.77357	−1.23253
$16$	0	0
$17$	−1.15945	−0.281209	−0.140604	−	0.990066i	$-0.544905\pi$
$17$	−1.15945	−0.281209	−0.140604	+	0.990066i	$0.544905\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.19505	−0.478998
$22$	0	0
$23$	−6.03406	−1.25819	−0.629095	−	0.777329i	$-0.716574\pi$
$23$	−6.03406	−1.25819	−0.629095	+	0.777329i	$0.716574\pi$
$24$	0	0
$25$	8.30135	1.66027
$26$	0	0
$27$	5.61094	1.07983
$28$	0	0
$29$	−3.72313	−0.691369	−0.345684	−	0.938351i	$-0.612353\pi$
$29$	−3.72313	−0.691369	−0.345684	+	0.938351i	$0.612353\pi$
$30$	0	0
$31$	7.84462	1.40893	0.704467	−	0.709736i	$-0.251186\pi$
$31$	7.84462	1.40893	0.704467	+	0.709736i	$0.251186\pi$
$32$	0	0
$33$	−2.02999	−0.353375
$34$	0	0
$35$	6.11640	1.03386
$36$	0	0
$37$	3.51435	0.577755	0.288878	−	0.957366i	$-0.406718\pi$
$37$	3.51435	0.577755	0.288878	+	0.957366i	$0.406718\pi$
$38$	0	0
$39$	0.546380	0.0874908
$40$	0	0
$41$	6.10255	0.953057	0.476529	−	0.879159i	$-0.341895\pi$
$41$	6.10255	0.953057	0.476529	+	0.879159i	$0.341895\pi$
$42$	0	0
$43$	0.837110	0.127658	0.0638290	−	0.997961i	$-0.479669\pi$
$43$	0.837110	0.127658	0.0638290	+	0.997961i	$0.479669\pi$
$44$	0	0
$45$	−4.69333	−0.699641
$46$	0	0
$47$	4.08638	0.596060	0.298030	−	0.954557i	$-0.403671\pi$
$47$	4.08638	0.596060	0.298030	+	0.954557i	$0.403671\pi$
$48$	0	0
$49$	−4.18748	−0.598211
$50$	0	0
$51$	1.51757	0.212502
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	5.65647	0.762719
$56$	0	0
$57$	−1.30887	−0.173364
$58$	0	0
$59$	−6.71588	−0.874333	−0.437167	−	0.899381i	$-0.644018\pi$
$59$	−6.71588	−0.874333	−0.437167	+	0.899381i	$0.644018\pi$
$60$	0	0
$61$	1.25269	0.160391	0.0801953	−	0.996779i	$-0.474446\pi$
$61$	1.25269	0.160391	0.0801953	+	0.996779i	$0.474446\pi$
$62$	0	0
$63$	−2.15815	−0.271901
$64$	0	0
$65$	−1.52246	−0.188838
$66$	0	0
$67$	14.6692	1.79213	0.896066	−	0.443920i	$-0.146413\pi$
$67$	14.6692	1.79213	0.896066	+	0.443920i	$0.146413\pi$
$68$	0	0
$69$	7.89779	0.950782
$70$	0	0
$71$	13.1745	1.56353	0.781766	−	0.623572i	$-0.214319\pi$
$71$	13.1745	1.56353	0.781766	+	0.623572i	$0.214319\pi$
$72$	0	0
$73$	1.35505	0.158597	0.0792983	−	0.996851i	$-0.474732\pi$
$73$	1.35505	0.158597	0.0792983	+	0.996851i	$0.474732\pi$
$74$	0	0
$75$	−10.8654	−1.25462
$76$	0	0
$77$	2.60103	0.296415
$78$	0	0
$79$	14.5554	1.63761	0.818805	−	0.574072i	$-0.194637\pi$
$79$	14.5554	1.63761	0.818805	+	0.574072i	$0.194637\pi$
$80$	0	0
$81$	−3.48337	−0.387041
$82$	0	0
$83$	14.8244	1.62719	0.813593	−	0.581434i	$-0.197508\pi$
$83$	14.8244	1.62719	0.813593	+	0.581434i	$0.197508\pi$
$84$	0	0
$85$	−4.22865	−0.458661
$86$	0	0
$87$	4.87309	0.522450
$88$	0	0
$89$	−5.94514	−0.630184	−0.315092	−	0.949061i	$-0.602035\pi$
$89$	−5.94514	−0.630184	−0.315092	+	0.949061i	$0.602035\pi$
$90$	0	0
$91$	−0.700079	−0.0733882
$92$	0	0
$93$	−10.2676	−1.06470
$94$	0	0
$95$	3.64710	0.374185
$96$	0	0
$97$	6.59511	0.669632	0.334816	−	0.942283i	$-0.391326\pi$
$97$	6.59511	0.669632	0.334816	+	0.942283i	$0.391326\pi$
$98$	0	0
$99$	−1.99587	−0.200592
$100$	0	0
$101$	−9.93660	−0.988729	−0.494364	−	0.869255i	$-0.664599\pi$
$101$	−9.93660	−0.988729	−0.494364	+	0.869255i	$0.664599\pi$
$102$	0	0
$103$	−9.30268	−0.916620	−0.458310	−	0.888792i	$-0.651545\pi$
$103$	−9.30268	−0.916620	−0.458310	+	0.888792i	$0.651545\pi$
$104$	0	0
$105$	−8.00556	−0.781262
$106$	0	0
$107$	−11.8566	−1.14622	−0.573109	−	0.819479i	$-0.694263\pi$
$107$	−11.8566	−1.14622	−0.573109	+	0.819479i	$0.694263\pi$
$108$	0	0
$109$	8.42981	0.807429	0.403715	−	0.914885i	$-0.367719\pi$
$109$	8.42981	0.807429	0.403715	+	0.914885i	$0.367719\pi$
$110$	0	0
$111$	−4.59981	−0.436595
$112$	0	0
$113$	−11.6031	−1.09153	−0.545766	−	0.837938i	$-0.683761\pi$
$113$	−11.6031	−1.09153	−0.545766	+	0.837938i	$0.683761\pi$
$114$	0	0
$115$	−22.0068	−2.05215
$116$	0	0
$117$	0.537196	0.0496638
$118$	0	0
$119$	−1.94447	−0.178249
$120$	0	0
$121$	−8.59455	−0.781323
$122$	0	0
$123$	−7.98742	−0.720201
$124$	0	0
$125$	12.0404	1.07692
$126$	0	0
$127$	1.50600	0.133636	0.0668179	−	0.997765i	$-0.478715\pi$
$127$	1.50600	0.133636	0.0668179	+	0.997765i	$0.478715\pi$
$128$	0	0
$129$	−1.09567	−0.0964679
$130$	0	0
$131$	−16.7778	−1.46588	−0.732940	−	0.680293i	$-0.761852\pi$
$131$	−16.7778	−1.46588	−0.732940	+	0.680293i	$0.761852\pi$
$132$	0	0
$133$	1.67706	0.145419
$134$	0	0
$135$	20.4637	1.76123
$136$	0	0
$137$	1.79733	0.153556	0.0767781	−	0.997048i	$-0.475537\pi$
$137$	1.79733	0.153556	0.0767781	+	0.997048i	$0.475537\pi$
$138$	0	0
$139$	19.2495	1.63272	0.816360	−	0.577543i	$-0.195988\pi$
$139$	19.2495	1.63272	0.816360	+	0.577543i	$0.195988\pi$
$140$	0	0
$141$	−5.34853	−0.450427
$142$	0	0
$143$	−0.647436	−0.0541413
$144$	0	0
$145$	−13.5786	−1.12765
$146$	0	0
$147$	5.48085	0.452053
$148$	0	0
$149$	13.4185	1.09929	0.549643	−	0.835399i	$-0.314764\pi$
$149$	13.4185	1.09929	0.549643	+	0.835399i	$0.314764\pi$
$150$	0	0
$151$	−16.8484	−1.37111	−0.685553	−	0.728023i	$-0.740440\pi$
$151$	−16.8484	−1.37111	−0.685553	+	0.728023i	$0.740440\pi$
$152$	0	0
$153$	1.49206	0.120626
$154$	0	0
$155$	28.6101	2.29802
$156$	0	0
$157$	9.18584	0.733110	0.366555	−	0.930396i	$-0.380537\pi$
$157$	9.18584	0.733110	0.366555	+	0.930396i	$0.380537\pi$
$158$	0	0
$159$	−1.30887	−0.103800
$160$	0	0
$161$	−10.1195	−0.797526
$162$	0	0
$163$	−10.4251	−0.816557	−0.408279	−	0.912857i	$-0.633871\pi$
$163$	−10.4251	−0.816557	−0.408279	+	0.912857i	$0.633871\pi$
$164$	0	0
$165$	−7.40357	−0.576367
$166$	0	0
$167$	19.0823	1.47663	0.738317	−	0.674454i	$-0.235621\pi$
$167$	19.0823	1.47663	0.738317	+	0.674454i	$0.235621\pi$
$168$	0	0
$169$	−12.8257	−0.986595
$170$	0	0
$171$	−1.28687	−0.0984092
$172$	0	0
$173$	−3.07133	−0.233509	−0.116755	−	0.993161i	$-0.537249\pi$
$173$	−3.07133	−0.233509	−0.116755	+	0.993161i	$0.537249\pi$
$174$	0	0
$175$	13.9218	1.05239
$176$	0	0
$177$	8.79019	0.660711
$178$	0	0
$179$	−3.76918	−0.281722	−0.140861	−	0.990029i	$-0.544987\pi$
$179$	−3.76918	−0.281722	−0.140861	+	0.990029i	$0.544987\pi$
$180$	0	0
$181$	−19.6005	−1.45690	−0.728448	−	0.685101i	$-0.759758\pi$
$181$	−19.6005	−1.45690	−0.728448	+	0.685101i	$0.759758\pi$
$182$	0	0
$183$	−1.63961	−0.121203
$184$	0	0
$185$	12.8172	0.942338
$186$	0	0
$187$	−1.79826	−0.131501
$188$	0	0
$189$	9.40987	0.684467
$190$	0	0
$191$	12.1553	0.879530	0.439765	−	0.898113i	$-0.355062\pi$
$191$	12.1553	0.879530	0.439765	+	0.898113i	$0.355062\pi$
$192$	0	0
$193$	16.2449	1.16933	0.584667	−	0.811274i	$-0.301225\pi$
$193$	16.2449	1.16933	0.584667	+	0.811274i	$0.301225\pi$
$194$	0	0
$195$	1.99270	0.142700
$196$	0	0
$197$	−18.6425	−1.32822	−0.664112	−	0.747633i	$-0.731190\pi$
$197$	−18.6425	−1.32822	−0.664112	+	0.747633i	$0.731190\pi$
$198$	0	0
$199$	6.15096	0.436030	0.218015	−	0.975945i	$-0.430042\pi$
$199$	6.15096	0.436030	0.218015	+	0.975945i	$0.430042\pi$
$200$	0	0
$201$	−19.2001	−1.35427
$202$	0	0
$203$	−6.24391	−0.438237
$204$	0	0
$205$	22.2566	1.55447
$206$	0	0
$207$	7.76504	0.539707
$208$	0	0
$209$	1.55095	0.107281
$210$	0	0
$211$	−2.34718	−0.161587	−0.0807934	−	0.996731i	$-0.525745\pi$
$211$	−2.34718	−0.161587	−0.0807934	+	0.996731i	$0.525745\pi$
$212$	0	0
$213$	−17.2437	−1.18152
$214$	0	0
$215$	3.05302	0.208215
$216$	0	0
$217$	13.1559	0.893079
$218$	0	0
$219$	−1.77358	−0.119847
$220$	0	0
$221$	0.484008	0.0325579
$222$	0	0
$223$	12.2920	0.823131	0.411566	−	0.911380i	$-0.364982\pi$
$223$	12.2920	0.823131	0.411566	+	0.911380i	$0.364982\pi$
$224$	0	0
$225$	−10.6827	−0.712182
$226$	0	0
$227$	27.4180	1.81980	0.909898	−	0.414832i	$-0.136160\pi$
$227$	27.4180	1.81980	0.909898	+	0.414832i	$0.136160\pi$
$228$	0	0
$229$	1.53568	0.101481	0.0507403	−	0.998712i	$-0.483842\pi$
$229$	1.53568	0.101481	0.0507403	+	0.998712i	$0.483842\pi$
$230$	0	0
$231$	−3.40441	−0.223993
$232$	0	0
$233$	18.0891	1.18506	0.592528	−	0.805550i	$-0.298130\pi$
$233$	18.0891	1.18506	0.592528	+	0.805550i	$0.298130\pi$
$234$	0	0
$235$	14.9034	0.972193
$236$	0	0
$237$	−19.0511	−1.23750
$238$	0	0
$239$	7.89683	0.510803	0.255402	−	0.966835i	$-0.417792\pi$
$239$	7.89683	0.510803	0.255402	+	0.966835i	$0.417792\pi$
$240$	0	0
$241$	29.4478	1.89690	0.948451	−	0.316925i	$-0.102650\pi$
$241$	29.4478	1.89690	0.948451	+	0.316925i	$0.102650\pi$
$242$	0	0
$243$	−12.2735	−0.787348
$244$	0	0
$245$	−15.2722	−0.975702
$246$	0	0
$247$	−0.417445	−0.0265614
$248$	0	0
$249$	−19.4031	−1.22962
$250$	0	0
$251$	2.65764	0.167749	0.0838745	−	0.996476i	$-0.473271\pi$
$251$	2.65764	0.167749	0.0838745	+	0.996476i	$0.473271\pi$
$252$	0	0
$253$	−9.35853	−0.588366
$254$	0	0
$255$	5.53474	0.346599
$256$	0	0
$257$	14.8689	0.927495	0.463748	−	0.885967i	$-0.346504\pi$
$257$	14.8689	0.927495	0.463748	+	0.885967i	$0.346504\pi$
$258$	0	0
$259$	5.89376	0.366221
$260$	0	0
$261$	4.79118	0.296566
$262$	0	0
$263$	−30.0156	−1.85084	−0.925420	−	0.378943i	$-0.876288\pi$
$263$	−30.0156	−1.85084	−0.925420	+	0.378943i	$0.876288\pi$
$264$	0	0
$265$	3.64710	0.224040
$266$	0	0
$267$	7.78140	0.476214
$268$	0	0
$269$	11.8407	0.721943	0.360972	−	0.932577i	$-0.382445\pi$
$269$	11.8407	0.721943	0.360972	+	0.932577i	$0.382445\pi$
$270$	0	0
$271$	20.1381	1.22330	0.611652	−	0.791127i	$-0.290505\pi$
$271$	20.1381	1.22330	0.611652	+	0.791127i	$0.290505\pi$
$272$	0	0
$273$	0.916310	0.0554576
$274$	0	0
$275$	12.8750	0.776391
$276$	0	0
$277$	1.73415	0.104195	0.0520974	−	0.998642i	$-0.483409\pi$
$277$	1.73415	0.104195	0.0520974	+	0.998642i	$0.483409\pi$
$278$	0	0
$279$	−10.0950	−0.604371
$280$	0	0
$281$	15.6692	0.934744	0.467372	−	0.884061i	$-0.345201\pi$
$281$	15.6692	0.934744	0.467372	+	0.884061i	$0.345201\pi$
$282$	0	0
$283$	9.86213	0.586243	0.293121	−	0.956075i	$-0.405306\pi$
$283$	9.86213	0.586243	0.293121	+	0.956075i	$0.405306\pi$
$284$	0	0
$285$	−4.77357	−0.282762
$286$	0	0
$287$	10.2343	0.604113
$288$	0	0
$289$	−15.6557	−0.920922
$290$	0	0
$291$	−8.63213	−0.506024
$292$	0	0
$293$	8.87234	0.518328	0.259164	−	0.965833i	$-0.416553\pi$
$293$	8.87234	0.518328	0.259164	+	0.965833i	$0.416553\pi$
$294$	0	0
$295$	−24.4935	−1.42607
$296$	0	0
$297$	8.70229	0.504958
$298$	0	0
$299$	2.51889	0.145671
$300$	0	0
$301$	1.40388	0.0809184
$302$	0	0
$303$	13.0057	0.747157
$304$	0	0
$305$	4.56869	0.261603
$306$	0	0
$307$	2.43006	0.138691	0.0693455	−	0.997593i	$-0.477909\pi$
$307$	2.43006	0.138691	0.0693455	+	0.997593i	$0.477909\pi$
$308$	0	0
$309$	12.1760	0.692667
$310$	0	0
$311$	32.5898	1.84800	0.923998	−	0.382397i	$-0.124901\pi$
$311$	32.5898	1.84800	0.923998	+	0.382397i	$0.124901\pi$
$312$	0	0
$313$	6.10929	0.345318	0.172659	−	0.984982i	$-0.444764\pi$
$313$	6.10929	0.345318	0.172659	+	0.984982i	$0.444764\pi$
$314$	0	0
$315$	−7.87099	−0.443480
$316$	0	0
$317$	−1.77557	−0.0997262	−0.0498631	−	0.998756i	$-0.515879\pi$
$317$	−1.77557	−0.0997262	−0.0498631	+	0.998756i	$0.515879\pi$
$318$	0	0
$319$	−5.77439	−0.323304
$320$	0	0
$321$	15.5187	0.866167
$322$	0	0
$323$	−1.15945	−0.0645137
$324$	0	0
$325$	−3.46536	−0.192223
$326$	0	0
$327$	−11.0335	−0.610154
$328$	0	0
$329$	6.85309	0.377823
$330$	0	0
$331$	−25.6743	−1.41119	−0.705593	−	0.708617i	$-0.749319\pi$
$331$	−25.6743	−1.41119	−0.705593	+	0.708617i	$0.749319\pi$
$332$	0	0
$333$	−4.52250	−0.247831
$334$	0	0
$335$	53.5002	2.92303
$336$	0	0
$337$	−35.6223	−1.94047	−0.970236	−	0.242161i	$-0.922144\pi$
$337$	−35.6223	−1.94047	−0.970236	+	0.242161i	$0.922144\pi$
$338$	0	0
$339$	15.1870	0.824843
$340$	0	0
$341$	12.1666	0.658859
$342$	0	0
$343$	−18.7620	−1.01306
$344$	0	0
$345$	28.8040	1.55076
$346$	0	0
$347$	−2.52329	−0.135457	−0.0677286	−	0.997704i	$-0.521575\pi$
$347$	−2.52329	−0.135457	−0.0677286	+	0.997704i	$0.521575\pi$
$348$	0	0
$349$	−34.8446	−1.86519	−0.932594	−	0.360927i	$-0.882460\pi$
$349$	−34.8446	−1.86519	−0.932594	+	0.360927i	$0.882460\pi$
$350$	0	0
$351$	−2.34226	−0.125020
$352$	0	0
$353$	−17.5359	−0.933342	−0.466671	−	0.884431i	$-0.654547\pi$
$353$	−17.5359	−0.933342	−0.466671	+	0.884431i	$0.654547\pi$
$354$	0	0
$355$	48.0489	2.55017
$356$	0	0
$357$	2.54505	0.134699
$358$	0	0
$359$	−9.06924	−0.478656	−0.239328	−	0.970939i	$-0.576927\pi$
$359$	−9.06924	−0.478656	−0.239328	+	0.970939i	$0.576927\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.2491	0.590426
$364$	0	0
$365$	4.94200	0.258676
$366$	0	0
$367$	−32.9811	−1.72160	−0.860799	−	0.508945i	$-0.830036\pi$
$367$	−32.9811	−1.72160	−0.860799	+	0.508945i	$0.830036\pi$
$368$	0	0
$369$	−7.85316	−0.408819
$370$	0	0
$371$	1.67706	0.0870685
$372$	0	0
$373$	−24.0537	−1.24545	−0.622726	−	0.782440i	$-0.713975\pi$
$373$	−24.0537	−1.24545	−0.622726	+	0.782440i	$0.713975\pi$
$374$	0	0
$375$	−15.7592	−0.813804
$376$	0	0
$377$	1.55420	0.0800455
$378$	0	0
$379$	30.1773	1.55010	0.775052	−	0.631898i	$-0.217724\pi$
$379$	30.1773	1.55010	0.775052	+	0.631898i	$0.217724\pi$
$380$	0	0
$381$	−1.97115	−0.100985
$382$	0	0
$383$	31.4562	1.60733	0.803667	−	0.595079i	$-0.202879\pi$
$383$	31.4562	1.60733	0.803667	+	0.595079i	$0.202879\pi$
$384$	0	0
$385$	9.48623	0.483463
$386$	0	0
$387$	−1.07725	−0.0547596
$388$	0	0
$389$	−34.0067	−1.72421	−0.862104	−	0.506732i	$-0.830853\pi$
$389$	−34.0067	−1.72421	−0.862104	+	0.506732i	$0.830853\pi$
$390$	0	0
$391$	6.99622	0.353814
$392$	0	0
$393$	21.9599	1.10773
$394$	0	0
$395$	53.0850	2.67100
$396$	0	0
$397$	27.8863	1.39957	0.699787	−	0.714352i	$-0.253278\pi$
$397$	27.8863	1.39957	0.699787	+	0.714352i	$0.253278\pi$
$398$	0	0
$399$	−2.19505	−0.109890
$400$	0	0
$401$	6.17263	0.308246	0.154123	−	0.988052i	$-0.450745\pi$
$401$	6.17263	0.308246	0.154123	+	0.988052i	$0.450745\pi$
$402$	0	0
$403$	−3.27469	−0.163124
$404$	0	0
$405$	−12.7042	−0.631278
$406$	0	0
$407$	5.45058	0.270175
$408$	0	0
$409$	−17.8663	−0.883433	−0.441717	−	0.897155i	$-0.645630\pi$
$409$	−17.8663	−0.883433	−0.441717	+	0.897155i	$0.645630\pi$
$410$	0	0
$411$	−2.35246	−0.116039
$412$	0	0
$413$	−11.2629	−0.554212
$414$	0	0
$415$	54.0660	2.65400
$416$	0	0
$417$	−25.1950	−1.23381
$418$	0	0
$419$	−20.9775	−1.02482	−0.512408	−	0.858742i	$-0.671247\pi$
$419$	−20.9775	−1.02482	−0.512408	+	0.858742i	$0.671247\pi$
$420$	0	0
$421$	−0.565178	−0.0275451	−0.0137725	−	0.999905i	$-0.504384\pi$
$421$	−0.565178	−0.0275451	−0.0137725	+	0.999905i	$0.504384\pi$
$422$	0	0
$423$	−5.25863	−0.255683
$424$	0	0
$425$	−9.62504	−0.466883
$426$	0	0
$427$	2.10084	0.101667
$428$	0	0
$429$	0.847408	0.0409132
$430$	0	0
$431$	−0.795023	−0.0382949	−0.0191475	−	0.999817i	$-0.506095\pi$
$431$	−0.795023	−0.0382949	−0.0191475	+	0.999817i	$0.506095\pi$
$432$	0	0
$433$	11.5558	0.555336	0.277668	−	0.960677i	$-0.410438\pi$
$433$	11.5558	0.555336	0.277668	+	0.960677i	$0.410438\pi$
$434$	0	0
$435$	17.7726	0.852133
$436$	0	0
$437$	−6.03406	−0.288648
$438$	0	0
$439$	15.6812	0.748423	0.374211	−	0.927343i	$-0.377913\pi$
$439$	15.6812	0.748423	0.374211	+	0.927343i	$0.377913\pi$
$440$	0	0
$441$	5.38873	0.256606
$442$	0	0
$443$	25.0369	1.18954	0.594769	−	0.803897i	$-0.297243\pi$
$443$	25.0369	1.18954	0.594769	+	0.803897i	$0.297243\pi$
$444$	0	0
$445$	−21.6825	−1.02785
$446$	0	0
$447$	−17.5630	−0.830703
$448$	0	0
$449$	−17.3993	−0.821123	−0.410562	−	0.911833i	$-0.634667\pi$
$449$	−17.3993	−0.821123	−0.410562	+	0.911833i	$0.634667\pi$
$450$	0	0
$451$	9.46474	0.445677
$452$	0	0
$453$	22.0524	1.03611
$454$	0	0
$455$	−2.55326	−0.119699
$456$	0	0
$457$	15.3900	0.719912	0.359956	−	0.932969i	$-0.382792\pi$
$457$	15.3900	0.719912	0.359956	+	0.932969i	$0.382792\pi$
$458$	0	0
$459$	−6.50562	−0.303657
$460$	0	0
$461$	−26.6922	−1.24318	−0.621589	−	0.783344i	$-0.713513\pi$
$461$	−26.6922	−1.24318	−0.621589	+	0.783344i	$0.713513\pi$
$462$	0	0
$463$	−4.32248	−0.200883	−0.100441	−	0.994943i	$-0.532025\pi$
$463$	−4.32248	−0.200883	−0.100441	+	0.994943i	$0.532025\pi$
$464$	0	0
$465$	−37.4468	−1.73656
$466$	0	0
$467$	31.6559	1.46486	0.732431	−	0.680841i	$-0.238386\pi$
$467$	31.6559	1.46486	0.732431	+	0.680841i	$0.238386\pi$
$468$	0	0
$469$	24.6012	1.13598
$470$	0	0
$471$	−12.0230	−0.553993
$472$	0	0
$473$	1.29832	0.0596966
$474$	0	0
$475$	8.30135	0.380892
$476$	0	0
$477$	−1.28687	−0.0589216
$478$	0	0
$479$	−35.1055	−1.60401	−0.802005	−	0.597317i	$-0.796233\pi$
$479$	−35.1055	−1.60401	−0.802005	+	0.597317i	$0.796233\pi$
$480$	0	0
$481$	−1.46705	−0.0668915
$482$	0	0
$483$	13.2450	0.602670
$484$	0	0
$485$	24.0531	1.09219
$486$	0	0
$487$	17.2877	0.783380	0.391690	−	0.920097i	$-0.371891\pi$
$487$	17.2877	0.783380	0.391690	+	0.920097i	$0.371891\pi$
$488$	0	0
$489$	13.6451	0.617052
$490$	0	0
$491$	−1.19161	−0.0537766	−0.0268883	−	0.999638i	$-0.508560\pi$
$491$	−1.19161	−0.0537766	−0.0268883	+	0.999638i	$0.508560\pi$
$492$	0	0
$493$	4.31680	0.194419
$494$	0	0
$495$	−7.27913	−0.327172
$496$	0	0
$497$	22.0945	0.991073
$498$	0	0
$499$	−9.51371	−0.425892	−0.212946	−	0.977064i	$-0.568306\pi$
$499$	−9.51371	−0.425892	−0.212946	+	0.977064i	$0.568306\pi$
$500$	0	0
$501$	−24.9762	−1.11585
$502$	0	0
$503$	−5.55604	−0.247732	−0.123866	−	0.992299i	$-0.539529\pi$
$503$	−5.55604	−0.247732	−0.123866	+	0.992299i	$0.539529\pi$
$504$	0	0
$505$	−36.2398	−1.61265
$506$	0	0
$507$	16.7872	0.745545
$508$	0	0
$509$	16.6016	0.735852	0.367926	−	0.929855i	$-0.380068\pi$
$509$	16.6016	0.735852	0.367926	+	0.929855i	$0.380068\pi$
$510$	0	0
$511$	2.27250	0.100529
$512$	0	0
$513$	5.61094	0.247729
$514$	0	0
$515$	−33.9278	−1.49504
$516$	0	0
$517$	6.33777	0.278735
$518$	0	0
$519$	4.01997	0.176457
$520$	0	0
$521$	−18.9374	−0.829664	−0.414832	−	0.909898i	$-0.636160\pi$
$521$	−18.9374	−0.829664	−0.414832	+	0.909898i	$0.636160\pi$
$522$	0	0
$523$	−44.6173	−1.95098	−0.975489	−	0.220049i	$-0.929378\pi$
$523$	−44.6173	−1.95098	−0.975489	+	0.220049i	$0.929378\pi$
$524$	0	0
$525$	−18.2218	−0.795267
$526$	0	0
$527$	−9.09547	−0.396205
$528$	0	0
$529$	13.4099	0.583040
$530$	0	0
$531$	8.64244	0.375050
$532$	0	0
$533$	−2.54748	−0.110343
$534$	0	0
$535$	−43.2421	−1.86952
$536$	0	0
$537$	4.93336	0.212890
$538$	0	0
$539$	−6.49457	−0.279741
$540$	0	0
$541$	−1.73925	−0.0747762	−0.0373881	−	0.999301i	$-0.511904\pi$
$541$	−1.73925	−0.0747762	−0.0373881	+	0.999301i	$0.511904\pi$
$542$	0	0
$543$	25.6545	1.10094
$544$	0	0
$545$	30.7444	1.31694
$546$	0	0
$547$	25.9090	1.10779	0.553894	−	0.832588i	$-0.313141\pi$
$547$	25.9090	1.10779	0.553894	+	0.832588i	$0.313141\pi$
$548$	0	0
$549$	−1.61205	−0.0688005
$550$	0	0
$551$	−3.72313	−0.158611
$552$	0	0
$553$	24.4102	1.03803
$554$	0	0
$555$	−16.7760	−0.712101
$556$	0	0
$557$	−12.7163	−0.538807	−0.269404	−	0.963027i	$-0.586827\pi$
$557$	−12.7163	−0.538807	−0.269404	+	0.963027i	$0.586827\pi$
$558$	0	0
$559$	−0.349447	−0.0147800
$560$	0	0
$561$	2.35368	0.0993723
$562$	0	0
$563$	−2.22329	−0.0937005	−0.0468502	−	0.998902i	$-0.514918\pi$
$563$	−2.22329	−0.0937005	−0.0468502	+	0.998902i	$0.514918\pi$
$564$	0	0
$565$	−42.3178	−1.78033
$566$	0	0
$567$	−5.84182	−0.245333
$568$	0	0
$569$	6.04734	0.253517	0.126759	−	0.991934i	$-0.459543\pi$
$569$	6.04734	0.253517	0.126759	+	0.991934i	$0.459543\pi$
$570$	0	0
$571$	36.3287	1.52031	0.760154	−	0.649743i	$-0.225124\pi$
$571$	36.3287	1.52031	0.760154	+	0.649743i	$0.225124\pi$
$572$	0	0
$573$	−15.9097	−0.664639
$574$	0	0
$575$	−50.0909	−2.08893
$576$	0	0
$577$	12.0553	0.501870	0.250935	−	0.968004i	$-0.419262\pi$
$577$	12.0553	0.501870	0.250935	+	0.968004i	$0.419262\pi$
$578$	0	0
$579$	−21.2624	−0.883636
$580$	0	0
$581$	24.8613	1.03142
$582$	0	0
$583$	1.55095	0.0642338
$584$	0	0
$585$	1.95921	0.0810033
$586$	0	0
$587$	21.6681	0.894339	0.447169	−	0.894449i	$-0.352432\pi$
$587$	21.6681	0.894339	0.447169	+	0.894449i	$0.352432\pi$
$588$	0	0
$589$	7.84462	0.323232
$590$	0	0
$591$	24.4006	1.00371
$592$	0	0
$593$	−13.2960	−0.546003	−0.273001	−	0.962014i	$-0.588016\pi$
$593$	−13.2960	−0.546003	−0.273001	+	0.962014i	$0.588016\pi$
$594$	0	0
$595$	−7.09168	−0.290731
$596$	0	0
$597$	−8.05078	−0.329497
$598$	0	0
$599$	4.76045	0.194507	0.0972533	−	0.995260i	$-0.468994\pi$
$599$	4.76045	0.194507	0.0972533	+	0.995260i	$0.468994\pi$
$600$	0	0
$601$	−24.7845	−1.01098	−0.505491	−	0.862832i	$-0.668689\pi$
$601$	−24.7845	−1.01098	−0.505491	+	0.862832i	$0.668689\pi$
$602$	0	0
$603$	−18.8774	−0.768745
$604$	0	0
$605$	−31.3452	−1.27436
$606$	0	0
$607$	−1.12664	−0.0457290	−0.0228645	−	0.999739i	$-0.507279\pi$
$607$	−1.12664	−0.0457290	−0.0228645	+	0.999739i	$0.507279\pi$
$608$	0	0
$609$	8.17245	0.331164
$610$	0	0
$611$	−1.70584	−0.0690108
$612$	0	0
$613$	15.1716	0.612774	0.306387	−	0.951907i	$-0.400880\pi$
$613$	15.1716	0.612774	0.306387	+	0.951907i	$0.400880\pi$
$614$	0	0
$615$	−29.1309	−1.17467
$616$	0	0
$617$	−11.1394	−0.448456	−0.224228	−	0.974537i	$-0.571986\pi$
$617$	−11.1394	−0.448456	−0.224228	+	0.974537i	$0.571986\pi$
$618$	0	0
$619$	25.1684	1.01160	0.505802	−	0.862649i	$-0.331196\pi$
$619$	25.1684	1.01160	0.505802	+	0.862649i	$0.331196\pi$
$620$	0	0
$621$	−33.8568	−1.35862
$622$	0	0
$623$	−9.97035	−0.399454
$624$	0	0
$625$	2.40569	0.0962277
$626$	0	0
$627$	−2.02999	−0.0810699
$628$	0	0
$629$	−4.07472	−0.162470
$630$	0	0
$631$	−27.2279	−1.08393	−0.541963	−	0.840402i	$-0.682319\pi$
$631$	−27.2279	−1.08393	−0.541963	+	0.840402i	$0.682319\pi$
$632$	0	0
$633$	3.07215	0.122107
$634$	0	0
$635$	5.49253	0.217964
$636$	0	0
$637$	1.74804	0.0692599
$638$	0	0
$639$	−16.9539	−0.670685
$640$	0	0
$641$	36.0635	1.42442	0.712211	−	0.701965i	$-0.247694\pi$
$641$	36.0635	1.42442	0.712211	+	0.701965i	$0.247694\pi$
$642$	0	0
$643$	−4.45152	−0.175551	−0.0877755	−	0.996140i	$-0.527976\pi$
$643$	−4.45152	−0.175551	−0.0877755	+	0.996140i	$0.527976\pi$
$644$	0	0
$645$	−3.99600	−0.157342
$646$	0	0
$647$	−39.8040	−1.56486	−0.782429	−	0.622740i	$-0.786019\pi$
$647$	−39.8040	−1.56486	−0.782429	+	0.622740i	$0.786019\pi$
$648$	0	0
$649$	−10.4160	−0.408864
$650$	0	0
$651$	−17.2193	−0.674877
$652$	0	0
$653$	−46.1136	−1.80457	−0.902283	−	0.431145i	$-0.858110\pi$
$653$	−46.1136	−1.80457	−0.902283	+	0.431145i	$0.858110\pi$
$654$	0	0
$655$	−61.1902	−2.39090
$656$	0	0
$657$	−1.74377	−0.0680309
$658$	0	0
$659$	−30.7506	−1.19787	−0.598937	−	0.800796i	$-0.704410\pi$
$659$	−30.7506	−1.19787	−0.598937	+	0.800796i	$0.704410\pi$
$660$	0	0
$661$	5.02457	0.195433	0.0977165	−	0.995214i	$-0.468846\pi$
$661$	5.02457	0.195433	0.0977165	+	0.995214i	$0.468846\pi$
$662$	0	0
$663$	−0.633502	−0.0246032
$664$	0	0
$665$	6.11640	0.237184
$666$	0	0
$667$	22.4656	0.869872
$668$	0	0
$669$	−16.0886	−0.622019
$670$	0	0
$671$	1.94286	0.0750033
$672$	0	0
$673$	36.0210	1.38851	0.694253	−	0.719731i	$-0.255735\pi$
$673$	36.0210	1.38851	0.694253	+	0.719731i	$0.255735\pi$
$674$	0	0
$675$	46.5784	1.79280
$676$	0	0
$677$	−15.5240	−0.596638	−0.298319	−	0.954466i	$-0.596426\pi$
$677$	−15.5240	−0.596638	−0.298319	+	0.954466i	$0.596426\pi$
$678$	0	0
$679$	11.0604	0.424459
$680$	0	0
$681$	−35.8865	−1.37517
$682$	0	0
$683$	−4.41734	−0.169025	−0.0845124	−	0.996422i	$-0.526933\pi$
$683$	−4.41734	−0.169025	−0.0845124	+	0.996422i	$0.526933\pi$
$684$	0	0
$685$	6.55504	0.250455
$686$	0	0
$687$	−2.01000	−0.0766863
$688$	0	0
$689$	−0.417445	−0.0159034
$690$	0	0
$691$	22.8363	0.868734	0.434367	−	0.900736i	$-0.356972\pi$
$691$	22.8363	0.868734	0.434367	+	0.900736i	$0.356972\pi$
$692$	0	0
$693$	−3.34718	−0.127149
$694$	0	0
$695$	70.2049	2.66302
$696$	0	0
$697$	−7.07562	−0.268008
$698$	0	0
$699$	−23.6762	−0.895517
$700$	0	0
$701$	−24.9776	−0.943392	−0.471696	−	0.881761i	$-0.656358\pi$
$701$	−24.9776	−0.943392	−0.471696	+	0.881761i	$0.656358\pi$
$702$	0	0
$703$	3.51435	0.132546
$704$	0	0
$705$	−19.5066	−0.734662
$706$	0	0
$707$	−16.6643	−0.626724
$708$	0	0
$709$	20.9983	0.788609	0.394304	−	0.918980i	$-0.370986\pi$
$709$	20.9983	0.788609	0.394304	+	0.918980i	$0.370986\pi$
$710$	0	0
$711$	−18.7308	−0.702462
$712$	0	0
$713$	−47.3349	−1.77271
$714$	0	0
$715$	−2.36127	−0.0883063
$716$	0	0
$717$	−10.3359	−0.386001
$718$	0	0
$719$	−12.7628	−0.475972	−0.237986	−	0.971269i	$-0.576487\pi$
$719$	−12.7628	−0.475972	−0.237986	+	0.971269i	$0.576487\pi$
$720$	0	0
$721$	−15.6011	−0.581017
$722$	0	0
$723$	−38.5433	−1.43344
$724$	0	0
$725$	−30.9070	−1.14786
$726$	0	0
$727$	−49.3583	−1.83060	−0.915299	−	0.402774i	$-0.868046\pi$
$727$	−49.3583	−1.83060	−0.915299	+	0.402774i	$0.868046\pi$
$728$	0	0
$729$	26.5146	0.982021
$730$	0	0
$731$	−0.970590	−0.0358986
$732$	0	0
$733$	−0.454292	−0.0167797	−0.00838983	−	0.999965i	$-0.502671\pi$
$733$	−0.454292	−0.0167797	−0.00838983	+	0.999965i	$0.502671\pi$
$734$	0	0
$735$	19.9892	0.737314
$736$	0	0
$737$	22.7513	0.838053
$738$	0	0
$739$	−6.18019	−0.227342	−0.113671	−	0.993518i	$-0.536261\pi$
$739$	−6.18019	−0.227342	−0.113671	+	0.993518i	$0.536261\pi$
$740$	0	0
$741$	0.546380	0.0200718
$742$	0	0
$743$	−10.9062	−0.400109	−0.200055	−	0.979785i	$-0.564112\pi$
$743$	−10.9062	−0.400109	−0.200055	+	0.979785i	$0.564112\pi$
$744$	0	0
$745$	48.9387	1.79297
$746$	0	0
$747$	−19.0770	−0.697991
$748$	0	0
$749$	−19.8841	−0.726551
$750$	0	0
$751$	−42.6285	−1.55554	−0.777768	−	0.628552i	$-0.783648\pi$
$751$	−42.6285	−1.55554	−0.777768	+	0.628552i	$0.783648\pi$
$752$	0	0
$753$	−3.47850	−0.126764
$754$	0	0
$755$	−61.4480	−2.23632
$756$	0	0
$757$	−50.3598	−1.83036	−0.915179	−	0.403047i	$-0.867951\pi$
$757$	−50.3598	−1.83036	−0.915179	+	0.403047i	$0.867951\pi$
$758$	0	0
$759$	12.2491	0.444613
$760$	0	0
$761$	−21.1417	−0.766385	−0.383192	−	0.923669i	$-0.625175\pi$
$761$	−21.1417	−0.766385	−0.383192	+	0.923669i	$0.625175\pi$
$762$	0	0
$763$	14.1373	0.511804
$764$	0	0
$765$	5.44170	0.196745
$766$	0	0
$767$	2.80351	0.101229
$768$	0	0
$769$	2.95456	0.106544	0.0532721	−	0.998580i	$-0.483035\pi$
$769$	2.95456	0.106544	0.0532721	+	0.998580i	$0.483035\pi$
$770$	0	0
$771$	−19.4614	−0.700885
$772$	0	0
$773$	43.7316	1.57292	0.786458	−	0.617643i	$-0.211912\pi$
$773$	43.7316	1.57292	0.786458	+	0.617643i	$0.211912\pi$
$774$	0	0
$775$	65.1209	2.33921
$776$	0	0
$777$	−7.71415	−0.276744
$778$	0	0
$779$	6.10255	0.218646
$780$	0	0
$781$	20.4331	0.731153
$782$	0	0
$783$	−20.8903	−0.746558
$784$	0	0
$785$	33.5017	1.19573
$786$	0	0
$787$	20.2274	0.721029	0.360514	−	0.932754i	$-0.382601\pi$
$787$	20.2274	0.721029	0.360514	+	0.932754i	$0.382601\pi$
$788$	0	0
$789$	39.2864	1.39863
$790$	0	0
$791$	−19.4591	−0.691887
$792$	0	0
$793$	−0.522929	−0.0185698
$794$	0	0
$795$	−4.77357	−0.169301
$796$	0	0
$797$	6.56535	0.232557	0.116278	−	0.993217i	$-0.462904\pi$
$797$	6.56535	0.232557	0.116278	+	0.993217i	$0.462904\pi$
$798$	0	0
$799$	−4.73797	−0.167617
$800$	0	0
$801$	7.65061	0.270321
$802$	0	0
$803$	2.10161	0.0741643
$804$	0	0
$805$	−36.9067	−1.30079
$806$	0	0
$807$	−15.4980	−0.545554
$808$	0	0
$809$	−51.7824	−1.82057	−0.910286	−	0.413979i	$-0.864139\pi$
$809$	−51.7824	−1.82057	−0.910286	+	0.413979i	$0.864139\pi$
$810$	0	0
$811$	0.311321	0.0109320	0.00546598	−	0.999985i	$-0.498260\pi$
$811$	0.311321	0.0109320	0.00546598	+	0.999985i	$0.498260\pi$
$812$	0	0
$813$	−26.3581	−0.924421
$814$	0	0
$815$	−38.0214	−1.33183
$816$	0	0
$817$	0.837110	0.0292868
$818$	0	0
$819$	0.900908	0.0314803
$820$	0	0
$821$	−23.5108	−0.820533	−0.410266	−	0.911966i	$-0.634564\pi$
$821$	−23.5108	−0.820533	−0.410266	+	0.911966i	$0.634564\pi$
$822$	0	0
$823$	−23.9089	−0.833412	−0.416706	−	0.909041i	$-0.636816\pi$
$823$	−23.9089	−0.833412	−0.416706	+	0.909041i	$0.636816\pi$
$824$	0	0
$825$	−16.8516	−0.586699
$826$	0	0
$827$	−4.39201	−0.152725	−0.0763626	−	0.997080i	$-0.524331\pi$
$827$	−4.39201	−0.152725	−0.0763626	+	0.997080i	$0.524331\pi$
$828$	0	0
$829$	23.1477	0.803954	0.401977	−	0.915650i	$-0.368323\pi$
$829$	23.1477	0.803954	0.401977	+	0.915650i	$0.368323\pi$
$830$	0	0
$831$	−2.26977	−0.0787374
$832$	0	0
$833$	4.85519	0.168222
$834$	0	0
$835$	69.5951	2.40844
$836$	0	0
$837$	44.0157	1.52140
$838$	0	0
$839$	13.4974	0.465983	0.232991	−	0.972479i	$-0.425149\pi$
$839$	13.4974	0.465983	0.232991	+	0.972479i	$0.425149\pi$
$840$	0	0
$841$	−15.1383	−0.522010
$842$	0	0
$843$	−20.5088	−0.706362
$844$	0	0
$845$	−46.7768	−1.60917
$846$	0	0
$847$	−14.4136	−0.495256
$848$	0	0
$849$	−12.9082	−0.443009
$850$	0	0
$851$	−21.2058	−0.726925
$852$	0	0
$853$	46.9259	1.60671	0.803356	−	0.595499i	$-0.203046\pi$
$853$	46.9259	1.60671	0.803356	+	0.595499i	$0.203046\pi$
$854$	0	0
$855$	−4.69333	−0.160509
$856$	0	0
$857$	−14.5919	−0.498449	−0.249225	−	0.968446i	$-0.580176\pi$
$857$	−14.5919	−0.498449	−0.249225	+	0.968446i	$0.580176\pi$
$858$	0	0
$859$	−20.9677	−0.715408	−0.357704	−	0.933835i	$-0.616440\pi$
$859$	−20.9677	−0.715408	−0.357704	+	0.933835i	$0.616440\pi$
$860$	0	0
$861$	−13.3954	−0.456513
$862$	0	0
$863$	−28.7700	−0.979341	−0.489670	−	0.871908i	$-0.662883\pi$
$863$	−28.7700	−0.979341	−0.489670	+	0.871908i	$0.662883\pi$
$864$	0	0
$865$	−11.2015	−0.380861
$866$	0	0
$867$	20.4912	0.695917
$868$	0	0
$869$	22.5747	0.765794
$870$	0	0
$871$	−6.12360	−0.207490
$872$	0	0
$873$	−8.48703	−0.287243
$874$	0	0
$875$	20.1924	0.682628
$876$	0	0
$877$	58.0006	1.95854	0.979270	−	0.202558i	$-0.0649254\pi$
$877$	58.0006	1.95854	0.979270	+	0.202558i	$0.0649254\pi$
$878$	0	0
$879$	−11.6127	−0.391687
$880$	0	0
$881$	25.5837	0.861937	0.430969	−	0.902367i	$-0.358172\pi$
$881$	25.5837	0.861937	0.430969	+	0.902367i	$0.358172\pi$
$882$	0	0
$883$	−15.0999	−0.508154	−0.254077	−	0.967184i	$-0.581772\pi$
$883$	−15.0999	−0.508154	−0.254077	+	0.967184i	$0.581772\pi$
$884$	0	0
$885$	32.0587	1.07764
$886$	0	0
$887$	−55.5634	−1.86564	−0.932818	−	0.360348i	$-0.882658\pi$
$887$	−55.5634	−1.86564	−0.932818	+	0.360348i	$0.882658\pi$
$888$	0	0
$889$	2.52565	0.0847075
$890$	0	0
$891$	−5.40254	−0.180992
$892$	0	0
$893$	4.08638	0.136745
$894$	0	0
$895$	−13.7466	−0.459498
$896$	0	0
$897$	−3.29689	−0.110080
$898$	0	0
$899$	−29.2066	−0.974093
$900$	0	0
$901$	−1.15945	−0.0386270
$902$	0	0
$903$	−1.83749	−0.0611480
$904$	0	0
$905$	−71.4851	−2.37625
$906$	0	0
$907$	−13.0191	−0.432292	−0.216146	−	0.976361i	$-0.569349\pi$
$907$	−13.0191	−0.432292	−0.216146	+	0.976361i	$0.569349\pi$
$908$	0	0
$909$	12.7871	0.424121
$910$	0	0
$911$	−1.39069	−0.0460757	−0.0230379	−	0.999735i	$-0.507334\pi$
$911$	−1.39069	−0.0460757	−0.0230379	+	0.999735i	$0.507334\pi$
$912$	0	0
$913$	22.9919	0.760920
$914$	0	0
$915$	−5.97981	−0.197686
$916$	0	0
$917$	−28.1373	−0.929175
$918$	0	0
$919$	−34.4635	−1.13685	−0.568423	−	0.822737i	$-0.692446\pi$
$919$	−34.4635	−1.13685	−0.568423	+	0.822737i	$0.692446\pi$
$920$	0	0
$921$	−3.18063	−0.104805
$922$	0	0
$923$	−5.49965	−0.181023
$924$	0	0
$925$	29.1738	0.959230
$926$	0	0
$927$	11.9713	0.393189
$928$	0	0
$929$	−21.7090	−0.712250	−0.356125	−	0.934438i	$-0.615902\pi$
$929$	−21.7090	−0.712250	−0.356125	+	0.934438i	$0.615902\pi$
$930$	0	0
$931$	−4.18748	−0.137239
$932$	0	0
$933$	−42.6557	−1.39648
$934$	0	0
$935$	−6.55842	−0.214483
$936$	0	0
$937$	−42.6150	−1.39217	−0.696086	−	0.717958i	$-0.745077\pi$
$937$	−42.6150	−1.39217	−0.696086	+	0.717958i	$0.745077\pi$
$938$	0	0
$939$	−7.99625	−0.260948
$940$	0	0
$941$	−31.0087	−1.01085	−0.505427	−	0.862870i	$-0.668665\pi$
$941$	−31.0087	−1.01085	−0.505427	+	0.862870i	$0.668665\pi$
$942$	0	0
$943$	−36.8231	−1.19913
$944$	0	0
$945$	34.3187	1.11639
$946$	0	0
$947$	−10.5219	−0.341917	−0.170958	−	0.985278i	$-0.554686\pi$
$947$	−10.5219	−0.341917	−0.170958	+	0.985278i	$0.554686\pi$
$948$	0	0
$949$	−0.565658	−0.0183620
$950$	0	0
$951$	2.32399	0.0753606
$952$	0	0
$953$	−42.9885	−1.39253	−0.696266	−	0.717783i	$-0.745157\pi$
$953$	−42.9885	−1.39253	−0.696266	+	0.717783i	$0.745157\pi$
$954$	0	0
$955$	44.3318	1.43454
$956$	0	0
$957$	7.55791	0.244313
$958$	0	0
$959$	3.01422	0.0973344
$960$	0	0
$961$	30.5380	0.985098
$962$	0	0
$963$	15.2578	0.491676
$964$	0	0
$965$	59.2468	1.90722
$966$	0	0
$967$	42.3021	1.36035	0.680173	−	0.733052i	$-0.261905\pi$
$967$	42.3021	1.36035	0.680173	+	0.733052i	$0.261905\pi$
$968$	0	0
$969$	1.51757	0.0487514
$970$	0	0
$971$	28.4053	0.911568	0.455784	−	0.890090i	$-0.349359\pi$
$971$	28.4053	0.911568	0.455784	+	0.890090i	$0.349359\pi$
$972$	0	0
$973$	32.2825	1.03493
$974$	0	0
$975$	4.53569	0.145258
$976$	0	0
$977$	−6.89033	−0.220441	−0.110221	−	0.993907i	$-0.535156\pi$
$977$	−6.89033	−0.220441	−0.110221	+	0.993907i	$0.535156\pi$
$978$	0	0
$979$	−9.22062	−0.294692
$980$	0	0
$981$	−10.8480	−0.346351
$982$	0	0
$983$	−60.2960	−1.92314	−0.961571	−	0.274556i	$-0.911469\pi$
$983$	−60.2960	−1.92314	−0.961571	+	0.274556i	$0.911469\pi$
$984$	0	0
$985$	−67.9912	−2.16638
$986$	0	0
$987$	−8.96979	−0.285512
$988$	0	0
$989$	−5.05117	−0.160618
$990$	0	0
$991$	−3.80719	−0.120939	−0.0604697	−	0.998170i	$-0.519260\pi$
$991$	−3.80719	−0.120939	−0.0604697	+	0.998170i	$0.519260\pi$
$992$	0	0
$993$	33.6042	1.06640
$994$	0	0
$995$	22.4332	0.711179
$996$	0	0
$997$	7.73884	0.245092	0.122546	−	0.992463i	$-0.460894\pi$
$997$	7.73884	0.245092	0.122546	+	0.992463i	$0.460894\pi$
$998$	0	0
$999$	19.7188	0.623875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.6	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.30887 q^{3} +3.64710 q^{5} +1.67706 q^{7} -1.28687 q^{9} +O(q^{10})\)	\(q-1.30887 q^{3} +3.64710 q^{5} +1.67706 q^{7} -1.28687 q^{9} +1.55095 q^{11} -0.417445 q^{13} -4.77357 q^{15} -1.15945 q^{17} +1.00000 q^{19} -2.19505 q^{21} -6.03406 q^{23} +8.30135 q^{25} +5.61094 q^{27} -3.72313 q^{29} +7.84462 q^{31} -2.02999 q^{33} +6.11640 q^{35} +3.51435 q^{37} +0.546380 q^{39} +6.10255 q^{41} +0.837110 q^{43} -4.69333 q^{45} +4.08638 q^{47} -4.18748 q^{49} +1.51757 q^{51} +1.00000 q^{53} +5.65647 q^{55} -1.30887 q^{57} -6.71588 q^{59} +1.25269 q^{61} -2.15815 q^{63} -1.52246 q^{65} +14.6692 q^{67} +7.89779 q^{69} +13.1745 q^{71} +1.35505 q^{73} -10.8654 q^{75} +2.60103 q^{77} +14.5554 q^{79} -3.48337 q^{81} +14.8244 q^{83} -4.22865 q^{85} +4.87309 q^{87} -5.94514 q^{89} -0.700079 q^{91} -10.2676 q^{93} +3.64710 q^{95} +6.59511 q^{97} -1.99587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 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

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