LMFDB - Embedded newform 8036.2.a.t.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$1.53485$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53485 q^{3} +1.74990 q^{5} -0.644239 q^{9} +O(q^{10})$	$q+1.53485 q^{3} +1.74990 q^{5} -0.644239 q^{9} +1.47075 q^{11} -0.836260 q^{13} +2.68583 q^{15} +4.36833 q^{17} -0.873796 q^{19} +3.62063 q^{23} -1.93785 q^{25} -5.59336 q^{27} +5.63878 q^{29} +9.19128 q^{31} +2.25737 q^{33} +2.15332 q^{37} -1.28353 q^{39} -1.00000 q^{41} -1.50136 q^{43} -1.12735 q^{45} +8.53276 q^{47} +6.70473 q^{51} +7.12226 q^{53} +2.57366 q^{55} -1.34115 q^{57} -12.9801 q^{59} -12.0540 q^{61} -1.46337 q^{65} -1.52821 q^{67} +5.55711 q^{69} -4.82901 q^{71} -0.0120510 q^{73} -2.97431 q^{75} +2.55280 q^{79} -6.65224 q^{81} +11.5377 q^{83} +7.64414 q^{85} +8.65467 q^{87} +3.91059 q^{89} +14.1072 q^{93} -1.52906 q^{95} -13.1173 q^{97} -0.947512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

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
$3$	1.53485	0.886145	0.443073	−	0.896486i	$-0.353888\pi$
$3$	1.53485	0.886145	0.443073	+	0.896486i	$0.353888\pi$
$4$	0	0
$5$	1.74990	0.782579	0.391289	−	0.920268i	$-0.372029\pi$
$5$	1.74990	0.782579	0.391289	+	0.920268i	$0.372029\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.644239	−0.214746
$10$	0	0
$11$	1.47075	0.443447	0.221723	−	0.975110i	$-0.428832\pi$
$11$	1.47075	0.443447	0.221723	+	0.975110i	$0.428832\pi$
$12$	0	0
$13$	−0.836260	−0.231937	−0.115968	−	0.993253i	$-0.536997\pi$
$13$	−0.836260	−0.231937	−0.115968	+	0.993253i	$0.536997\pi$
$14$	0	0
$15$	2.68583	0.693479
$16$	0	0
$17$	4.36833	1.05948	0.529738	−	0.848161i	$-0.322290\pi$
$17$	4.36833	1.05948	0.529738	+	0.848161i	$0.322290\pi$
$18$	0	0
$19$	−0.873796	−0.200463	−0.100231	−	0.994964i	$-0.531958\pi$
$19$	−0.873796	−0.200463	−0.100231	+	0.994964i	$0.531958\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.62063	0.754953	0.377476	−	0.926019i	$-0.376792\pi$
$23$	3.62063	0.754953	0.377476	+	0.926019i	$0.376792\pi$
$24$	0	0
$25$	−1.93785	−0.387570
$26$	0	0
$27$	−5.59336	−1.07644
$28$	0	0
$29$	5.63878	1.04709	0.523547	−	0.851997i	$-0.324608\pi$
$29$	5.63878	1.04709	0.523547	+	0.851997i	$0.324608\pi$
$30$	0	0
$31$	9.19128	1.65080	0.825401	−	0.564547i	$-0.190949\pi$
$31$	9.19128	1.65080	0.825401	+	0.564547i	$0.190949\pi$
$32$	0	0
$33$	2.25737	0.392958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.15332	0.354004	0.177002	−	0.984210i	$-0.443360\pi$
$37$	2.15332	0.354004	0.177002	+	0.984210i	$0.443360\pi$
$38$	0	0
$39$	−1.28353	−0.205530
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−1.50136	−0.228955	−0.114477	−	0.993426i	$-0.536519\pi$
$43$	−1.50136	−0.228955	−0.114477	+	0.993426i	$0.536519\pi$
$44$	0	0
$45$	−1.12735	−0.168056
$46$	0	0
$47$	8.53276	1.24463	0.622315	−	0.782767i	$-0.286192\pi$
$47$	8.53276	1.24463	0.622315	+	0.782767i	$0.286192\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.70473	0.938850
$52$	0	0
$53$	7.12226	0.978318	0.489159	−	0.872195i	$-0.337304\pi$
$53$	7.12226	0.978318	0.489159	+	0.872195i	$0.337304\pi$
$54$	0	0
$55$	2.57366	0.347032
$56$	0	0
$57$	−1.34115	−0.177639
$58$	0	0
$59$	−12.9801	−1.68986	−0.844930	−	0.534878i	$-0.820358\pi$
$59$	−12.9801	−1.68986	−0.844930	+	0.534878i	$0.820358\pi$
$60$	0	0
$61$	−12.0540	−1.54335	−0.771675	−	0.636017i	$-0.780581\pi$
$61$	−12.0540	−1.54335	−0.771675	+	0.636017i	$0.780581\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.46337	−0.181509
$66$	0	0
$67$	−1.52821	−0.186701	−0.0933506	−	0.995633i	$-0.529758\pi$
$67$	−1.52821	−0.186701	−0.0933506	+	0.995633i	$0.529758\pi$
$68$	0	0
$69$	5.55711	0.668998
$70$	0	0
$71$	−4.82901	−0.573098	−0.286549	−	0.958066i	$-0.592508\pi$
$71$	−4.82901	−0.573098	−0.286549	+	0.958066i	$0.592508\pi$
$72$	0	0
$73$	−0.0120510	−0.00141047	−0.000705233	−	1.00000i	$-0.500224\pi$
$73$	−0.0120510	−0.00141047	−0.000705233		1.00000i	$0.500224\pi$
$74$	0	0
$75$	−2.97431	−0.343444
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.55280	0.287212	0.143606	−	0.989635i	$-0.454130\pi$
$79$	2.55280	0.287212	0.143606	+	0.989635i	$0.454130\pi$
$80$	0	0
$81$	−6.65224	−0.739138
$82$	0	0
$83$	11.5377	1.26643	0.633215	−	0.773976i	$-0.281735\pi$
$83$	11.5377	1.26643	0.633215	+	0.773976i	$0.281735\pi$
$84$	0	0
$85$	7.64414	0.829124
$86$	0	0
$87$	8.65467	0.927878
$88$	0	0
$89$	3.91059	0.414522	0.207261	−	0.978286i	$-0.433545\pi$
$89$	3.91059	0.414522	0.207261	+	0.978286i	$0.433545\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.1072	1.46285
$94$	0	0
$95$	−1.52906	−0.156878
$96$	0	0
$97$	−13.1173	−1.33186	−0.665931	−	0.746014i	$-0.731965\pi$
$97$	−13.1173	−1.33186	−0.665931	+	0.746014i	$0.731965\pi$
$98$	0	0
$99$	−0.947512	−0.0952285
$100$	0	0
$101$	15.6521	1.55744	0.778720	−	0.627371i	$-0.215869\pi$
$101$	15.6521	1.55744	0.778720	+	0.627371i	$0.215869\pi$
$102$	0	0
$103$	1.93280	0.190445	0.0952225	−	0.995456i	$-0.469644\pi$
$103$	1.93280	0.190445	0.0952225	+	0.995456i	$0.469644\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.99673	0.289705	0.144852	−	0.989453i	$-0.453729\pi$
$107$	2.99673	0.289705	0.144852	+	0.989453i	$0.453729\pi$
$108$	0	0
$109$	4.99040	0.477994	0.238997	−	0.971020i	$-0.423181\pi$
$109$	4.99040	0.477994	0.238997	+	0.971020i	$0.423181\pi$
$110$	0	0
$111$	3.30502	0.313699
$112$	0	0
$113$	11.7124	1.10181	0.550905	−	0.834568i	$-0.314283\pi$
$113$	11.7124	1.10181	0.550905	+	0.834568i	$0.314283\pi$
$114$	0	0
$115$	6.33573	0.590810
$116$	0	0
$117$	0.538751	0.0498076
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.83691	−0.803355
$122$	0	0
$123$	−1.53485	−0.138393
$124$	0	0
$125$	−12.1405	−1.08588
$126$	0	0
$127$	19.8201	1.75875	0.879376	−	0.476128i	$-0.157960\pi$
$127$	19.8201	1.75875	0.879376	+	0.476128i	$0.157960\pi$
$128$	0	0
$129$	−2.30436	−0.202887
$130$	0	0
$131$	−1.67196	−0.146080	−0.0730400	−	0.997329i	$-0.523270\pi$
$131$	−1.67196	−0.146080	−0.0730400	+	0.997329i	$0.523270\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.78781	−0.842401
$136$	0	0
$137$	8.48383	0.724823	0.362411	−	0.932018i	$-0.381954\pi$
$137$	8.48383	0.724823	0.362411	+	0.932018i	$0.381954\pi$
$138$	0	0
$139$	−7.87576	−0.668013	−0.334006	−	0.942571i	$-0.608401\pi$
$139$	−7.87576	−0.668013	−0.334006	+	0.942571i	$0.608401\pi$
$140$	0	0
$141$	13.0965	1.10292
$142$	0	0
$143$	−1.22993	−0.102852
$144$	0	0
$145$	9.86729	0.819434
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.57393	0.784327	0.392163	−	0.919896i	$-0.371727\pi$
$149$	9.57393	0.784327	0.392163	+	0.919896i	$0.371727\pi$
$150$	0	0
$151$	−9.19441	−0.748230	−0.374115	−	0.927382i	$-0.622054\pi$
$151$	−9.19441	−0.748230	−0.374115	+	0.927382i	$0.622054\pi$
$152$	0	0
$153$	−2.81425	−0.227519
$154$	0	0
$155$	16.0838	1.29188
$156$	0	0
$157$	9.21209	0.735205	0.367603	−	0.929983i	$-0.380179\pi$
$157$	9.21209	0.735205	0.367603	+	0.929983i	$0.380179\pi$
$158$	0	0
$159$	10.9316	0.866932
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.0758	1.57246	0.786229	−	0.617935i	$-0.212031\pi$
$163$	20.0758	1.57246	0.786229	+	0.617935i	$0.212031\pi$
$164$	0	0
$165$	3.95018	0.307521
$166$	0	0
$167$	1.62426	0.125689	0.0628445	−	0.998023i	$-0.479983\pi$
$167$	1.62426	0.125689	0.0628445	+	0.998023i	$0.479983\pi$
$168$	0	0
$169$	−12.3007	−0.946205
$170$	0	0
$171$	0.562934	0.0430486
$172$	0	0
$173$	23.7696	1.80717	0.903586	−	0.428407i	$-0.140925\pi$
$173$	23.7696	1.80717	0.903586	+	0.428407i	$0.140925\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−19.9224	−1.49746
$178$	0	0
$179$	−23.1111	−1.72741	−0.863704	−	0.504000i	$-0.831861\pi$
$179$	−23.1111	−1.72741	−0.863704	+	0.504000i	$0.831861\pi$
$180$	0	0
$181$	−0.910960	−0.0677111	−0.0338556	−	0.999427i	$-0.510779\pi$
$181$	−0.910960	−0.0677111	−0.0338556	+	0.999427i	$0.510779\pi$
$182$	0	0
$183$	−18.5010	−1.36763
$184$	0	0
$185$	3.76810	0.277036
$186$	0	0
$187$	6.42471	0.469821
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.21085	0.159972	0.0799858	−	0.996796i	$-0.474513\pi$
$191$	2.21085	0.159972	0.0799858	+	0.996796i	$0.474513\pi$
$192$	0	0
$193$	−14.2409	−1.02508	−0.512540	−	0.858664i	$-0.671295\pi$
$193$	−14.2409	−1.02508	−0.512540	+	0.858664i	$0.671295\pi$
$194$	0	0
$195$	−2.24605	−0.160843
$196$	0	0
$197$	−6.33659	−0.451463	−0.225732	−	0.974190i	$-0.572477\pi$
$197$	−6.33659	−0.451463	−0.225732	+	0.974190i	$0.572477\pi$
$198$	0	0
$199$	−7.50825	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$199$	−7.50825	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$200$	0	0
$201$	−2.34558	−0.165444
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.74990	−0.122218
$206$	0	0
$207$	−2.33255	−0.162123
$208$	0	0
$209$	−1.28513	−0.0888945
$210$	0	0
$211$	10.8136	0.744436	0.372218	−	0.928145i	$-0.378597\pi$
$211$	10.8136	0.744436	0.372218	+	0.928145i	$0.378597\pi$
$212$	0	0
$213$	−7.41180	−0.507848
$214$	0	0
$215$	−2.62723	−0.179175
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.0184965	−0.00124988
$220$	0	0
$221$	−3.65306	−0.245732
$222$	0	0
$223$	−5.08072	−0.340230	−0.170115	−	0.985424i	$-0.554414\pi$
$223$	−5.08072	−0.340230	−0.170115	+	0.985424i	$0.554414\pi$
$224$	0	0
$225$	1.24844	0.0832293
$226$	0	0
$227$	16.5185	1.09637	0.548186	−	0.836357i	$-0.315319\pi$
$227$	16.5185	1.09637	0.548186	+	0.836357i	$0.315319\pi$
$228$	0	0
$229$	15.0319	0.993335	0.496668	−	0.867941i	$-0.334557\pi$
$229$	15.0319	0.993335	0.496668	+	0.867941i	$0.334557\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.6539	0.697960	0.348980	−	0.937130i	$-0.386528\pi$
$233$	10.6539	0.697960	0.348980	+	0.937130i	$0.386528\pi$
$234$	0	0
$235$	14.9315	0.974022
$236$	0	0
$237$	3.91816	0.254512
$238$	0	0
$239$	3.32385	0.215002	0.107501	−	0.994205i	$-0.465715\pi$
$239$	3.32385	0.215002	0.107501	+	0.994205i	$0.465715\pi$
$240$	0	0
$241$	4.59832	0.296204	0.148102	−	0.988972i	$-0.452684\pi$
$241$	4.59832	0.296204	0.148102	+	0.988972i	$0.452684\pi$
$242$	0	0
$243$	6.56989	0.421458
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.730721	0.0464947
$248$	0	0
$249$	17.7087	1.12224
$250$	0	0
$251$	24.7116	1.55978	0.779892	−	0.625914i	$-0.215274\pi$
$251$	24.7116	1.55978	0.779892	+	0.625914i	$0.215274\pi$
$252$	0	0
$253$	5.32502	0.334781
$254$	0	0
$255$	11.7326	0.734724
$256$	0	0
$257$	6.69307	0.417502	0.208751	−	0.977969i	$-0.433060\pi$
$257$	6.69307	0.417502	0.208751	+	0.977969i	$0.433060\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.63272	−0.224860
$262$	0	0
$263$	−3.51374	−0.216667	−0.108333	−	0.994115i	$-0.534551\pi$
$263$	−3.51374	−0.216667	−0.108333	+	0.994115i	$0.534551\pi$
$264$	0	0
$265$	12.4632	0.765611
$266$	0	0
$267$	6.00217	0.367327
$268$	0	0
$269$	13.9825	0.852527	0.426264	−	0.904599i	$-0.359830\pi$
$269$	13.9825	0.852527	0.426264	+	0.904599i	$0.359830\pi$
$270$	0	0
$271$	18.9801	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$271$	18.9801	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.85009	−0.171867
$276$	0	0
$277$	−24.6215	−1.47936	−0.739682	−	0.672956i	$-0.765024\pi$
$277$	−24.6215	−1.47936	−0.739682	+	0.672956i	$0.765024\pi$
$278$	0	0
$279$	−5.92138	−0.354504
$280$	0	0
$281$	23.6723	1.41217	0.706086	−	0.708126i	$-0.250459\pi$
$281$	23.6723	1.41217	0.706086	+	0.708126i	$0.250459\pi$
$282$	0	0
$283$	2.70659	0.160890	0.0804452	−	0.996759i	$-0.474366\pi$
$283$	2.70659	0.160890	0.0804452	+	0.996759i	$0.474366\pi$
$284$	0	0
$285$	−2.34687	−0.139017
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.08233	0.122490
$290$	0	0
$291$	−20.1331	−1.18022
$292$	0	0
$293$	−16.9056	−0.987636	−0.493818	−	0.869565i	$-0.664399\pi$
$293$	−16.9056	−0.987636	−0.493818	+	0.869565i	$0.664399\pi$
$294$	0	0
$295$	−22.7138	−1.32245
$296$	0	0
$297$	−8.22640	−0.477344
$298$	0	0
$299$	−3.02778	−0.175101
$300$	0	0
$301$	0	0
$302$	0	0
$303$	24.0236	1.38012
$304$	0	0
$305$	−21.0932	−1.20779
$306$	0	0
$307$	−22.6447	−1.29240	−0.646200	−	0.763168i	$-0.723643\pi$
$307$	−22.6447	−1.29240	−0.646200	+	0.763168i	$0.723643\pi$
$308$	0	0
$309$	2.96656	0.168762
$310$	0	0
$311$	−26.2813	−1.49028	−0.745138	−	0.666910i	$-0.767617\pi$
$311$	−26.2813	−1.49028	−0.745138	+	0.666910i	$0.767617\pi$
$312$	0	0
$313$	14.2905	0.807744	0.403872	−	0.914815i	$-0.367664\pi$
$313$	14.2905	0.807744	0.403872	+	0.914815i	$0.367664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.3808	0.863872	0.431936	−	0.901904i	$-0.357831\pi$
$317$	15.3808	0.863872	0.431936	+	0.901904i	$0.357831\pi$
$318$	0	0
$319$	8.29321	0.464330
$320$	0	0
$321$	4.59952	0.256720
$322$	0	0
$323$	−3.81703	−0.212385
$324$	0	0
$325$	1.62055	0.0898918
$326$	0	0
$327$	7.65952	0.423572
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7.41289	0.407449	0.203725	−	0.979028i	$-0.434695\pi$
$331$	7.41289	0.407449	0.203725	+	0.979028i	$0.434695\pi$
$332$	0	0
$333$	−1.38725	−0.0760211
$334$	0	0
$335$	−2.67422	−0.146108
$336$	0	0
$337$	−19.1830	−1.04497	−0.522484	−	0.852649i	$-0.674994\pi$
$337$	−19.1830	−1.04497	−0.522484	+	0.852649i	$0.674994\pi$
$338$	0	0
$339$	17.9768	0.976364
$340$	0	0
$341$	13.5180	0.732043
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.72439	0.523544
$346$	0	0
$347$	12.0308	0.645848	0.322924	−	0.946425i	$-0.395334\pi$
$347$	12.0308	0.645848	0.322924	+	0.946425i	$0.395334\pi$
$348$	0	0
$349$	14.4115	0.771429	0.385714	−	0.922618i	$-0.373955\pi$
$349$	14.4115	0.771429	0.385714	+	0.922618i	$0.373955\pi$
$350$	0	0
$351$	4.67750	0.249667
$352$	0	0
$353$	12.2429	0.651622	0.325811	−	0.945435i	$-0.394363\pi$
$353$	12.2429	0.651622	0.325811	+	0.945435i	$0.394363\pi$
$354$	0	0
$355$	−8.45028	−0.448494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.58537	0.242007	0.121003	−	0.992652i	$-0.461389\pi$
$359$	4.58537	0.242007	0.121003	+	0.992652i	$0.461389\pi$
$360$	0	0
$361$	−18.2365	−0.959815
$362$	0	0
$363$	−13.5633	−0.711889
$364$	0	0
$365$	−0.0210881	−0.00110380
$366$	0	0
$367$	−29.7496	−1.55292	−0.776459	−	0.630168i	$-0.782986\pi$
$367$	−29.7496	−1.55292	−0.776459	+	0.630168i	$0.782986\pi$
$368$	0	0
$369$	0.644239	0.0335377
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.6714	−1.48455	−0.742274	−	0.670097i	$-0.766252\pi$
$373$	−28.6714	−1.48455	−0.742274	+	0.670097i	$0.766252\pi$
$374$	0	0
$375$	−18.6339	−0.962250
$376$	0	0
$377$	−4.71548	−0.242860
$378$	0	0
$379$	−12.6935	−0.652022	−0.326011	−	0.945366i	$-0.605705\pi$
$379$	−12.6935	−0.652022	−0.326011	+	0.945366i	$0.605705\pi$
$380$	0	0
$381$	30.4209	1.55851
$382$	0	0
$383$	15.7683	0.805724	0.402862	−	0.915261i	$-0.368015\pi$
$383$	15.7683	0.805724	0.402862	+	0.915261i	$0.368015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.967233	0.0491672
$388$	0	0
$389$	−23.6678	−1.20000	−0.600002	−	0.799999i	$-0.704833\pi$
$389$	−23.6678	−1.20000	−0.600002	+	0.799999i	$0.704833\pi$
$390$	0	0
$391$	15.8161	0.799854
$392$	0	0
$393$	−2.56621	−0.129448
$394$	0	0
$395$	4.46714	0.224766
$396$	0	0
$397$	25.3815	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$397$	25.3815	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.3440	−1.11581	−0.557903	−	0.829906i	$-0.688394\pi$
$401$	−22.3440	−1.11581	−0.557903	+	0.829906i	$0.688394\pi$
$402$	0	0
$403$	−7.68630	−0.382882
$404$	0	0
$405$	−11.6408	−0.578434
$406$	0	0
$407$	3.16699	0.156982
$408$	0	0
$409$	17.5318	0.866893	0.433447	−	0.901179i	$-0.357297\pi$
$409$	17.5318	0.866893	0.433447	+	0.901179i	$0.357297\pi$
$410$	0	0
$411$	13.0214	0.642298
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.1899	0.991081
$416$	0	0
$417$	−12.0881	−0.591957
$418$	0	0
$419$	12.1476	0.593448	0.296724	−	0.954963i	$-0.404106\pi$
$419$	12.1476	0.593448	0.296724	+	0.954963i	$0.404106\pi$
$420$	0	0
$421$	−35.1116	−1.71123	−0.855617	−	0.517609i	$-0.826822\pi$
$421$	−35.1116	−1.71123	−0.855617	+	0.517609i	$0.826822\pi$
$422$	0	0
$423$	−5.49714	−0.267280
$424$	0	0
$425$	−8.46518	−0.410622
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.88775	−0.0911415
$430$	0	0
$431$	−10.7760	−0.519063	−0.259532	−	0.965735i	$-0.583568\pi$
$431$	−10.7760	−0.519063	−0.259532	+	0.965735i	$0.583568\pi$
$432$	0	0
$433$	27.9881	1.34502	0.672511	−	0.740087i	$-0.265216\pi$
$433$	27.9881	1.34502	0.672511	+	0.740087i	$0.265216\pi$
$434$	0	0
$435$	15.1448	0.726138
$436$	0	0
$437$	−3.16369	−0.151340
$438$	0	0
$439$	9.33745	0.445652	0.222826	−	0.974858i	$-0.428472\pi$
$439$	9.33745	0.445652	0.222826	+	0.974858i	$0.428472\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.3580	0.682170	0.341085	−	0.940033i	$-0.389206\pi$
$443$	14.3580	0.682170	0.341085	+	0.940033i	$0.389206\pi$
$444$	0	0
$445$	6.84314	0.324396
$446$	0	0
$447$	14.6945	0.695027
$448$	0	0
$449$	27.7795	1.31100	0.655498	−	0.755197i	$-0.272459\pi$
$449$	27.7795	1.31100	0.655498	+	0.755197i	$0.272459\pi$
$450$	0	0
$451$	−1.47075	−0.0692547
$452$	0	0
$453$	−14.1120	−0.663041
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.8732	−0.602182	−0.301091	−	0.953595i	$-0.597351\pi$
$457$	−12.8732	−0.602182	−0.301091	+	0.953595i	$0.597351\pi$
$458$	0	0
$459$	−24.4336	−1.14046
$460$	0	0
$461$	−13.0210	−0.606447	−0.303224	−	0.952919i	$-0.598063\pi$
$461$	−13.0210	−0.606447	−0.303224	+	0.952919i	$0.598063\pi$
$462$	0	0
$463$	−31.6222	−1.46961	−0.734805	−	0.678278i	$-0.762726\pi$
$463$	−31.6222	−1.46961	−0.734805	+	0.678278i	$0.762726\pi$
$464$	0	0
$465$	24.6862	1.14480
$466$	0	0
$467$	−25.3972	−1.17524	−0.587621	−	0.809136i	$-0.699935\pi$
$467$	−25.3972	−1.17524	−0.587621	+	0.809136i	$0.699935\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.1392	0.651499
$472$	0	0
$473$	−2.20812	−0.101529
$474$	0	0
$475$	1.69329	0.0776934
$476$	0	0
$477$	−4.58844	−0.210090
$478$	0	0
$479$	−38.1819	−1.74457	−0.872287	−	0.488994i	$-0.837364\pi$
$479$	−38.1819	−1.74457	−0.872287	+	0.488994i	$0.837364\pi$
$480$	0	0
$481$	−1.80074	−0.0821066
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.9540	−1.04229
$486$	0	0
$487$	−27.5359	−1.24777	−0.623886	−	0.781516i	$-0.714447\pi$
$487$	−27.5359	−1.24777	−0.623886	+	0.781516i	$0.714447\pi$
$488$	0	0
$489$	30.8133	1.39343
$490$	0	0
$491$	−10.7959	−0.487211	−0.243606	−	0.969874i	$-0.578330\pi$
$491$	−10.7959	−0.487211	−0.243606	+	0.969874i	$0.578330\pi$
$492$	0	0
$493$	24.6320	1.10937
$494$	0	0
$495$	−1.65805	−0.0745238
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.62488	0.162272	0.0811360	−	0.996703i	$-0.474145\pi$
$499$	3.62488	0.162272	0.0811360	+	0.996703i	$0.474145\pi$
$500$	0	0
$501$	2.49299	0.111379
$502$	0	0
$503$	3.23617	0.144294	0.0721469	−	0.997394i	$-0.477015\pi$
$503$	3.23617	0.144294	0.0721469	+	0.997394i	$0.477015\pi$
$504$	0	0
$505$	27.3896	1.21882
$506$	0	0
$507$	−18.8797	−0.838475
$508$	0	0
$509$	26.9095	1.19274	0.596372	−	0.802708i	$-0.296608\pi$
$509$	26.9095	1.19274	0.596372	+	0.802708i	$0.296608\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.88745	0.215786
$514$	0	0
$515$	3.38221	0.149038
$516$	0	0
$517$	12.5495	0.551927
$518$	0	0
$519$	36.4828	1.60142
$520$	0	0
$521$	−14.1349	−0.619262	−0.309631	−	0.950857i	$-0.600206\pi$
$521$	−14.1349	−0.619262	−0.309631	+	0.950857i	$0.600206\pi$
$522$	0	0
$523$	10.7558	0.470318	0.235159	−	0.971957i	$-0.424439\pi$
$523$	10.7558	0.470318	0.235159	+	0.971957i	$0.424439\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.1506	1.74899
$528$	0	0
$529$	−9.89107	−0.430047
$530$	0	0
$531$	8.36226	0.362891
$532$	0	0
$533$	0.836260	0.0362224
$534$	0	0
$535$	5.24397	0.226717
$536$	0	0
$537$	−35.4721	−1.53073
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.6645	−1.23238	−0.616191	−	0.787597i	$-0.711325\pi$
$541$	−28.6645	−1.23238	−0.616191	+	0.787597i	$0.711325\pi$
$542$	0	0
$543$	−1.39819	−0.0600019
$544$	0	0
$545$	8.73271	0.374068
$546$	0	0
$547$	−41.8343	−1.78870	−0.894352	−	0.447363i	$-0.852363\pi$
$547$	−41.8343	−1.78870	−0.894352	+	0.447363i	$0.852363\pi$
$548$	0	0
$549$	7.76563	0.331429
$550$	0	0
$551$	−4.92714	−0.209903
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.78346	0.245494
$556$	0	0
$557$	12.8237	0.543357	0.271678	−	0.962388i	$-0.412421\pi$
$557$	12.8237	0.543357	0.271678	+	0.962388i	$0.412421\pi$
$558$	0	0
$559$	1.25553	0.0531031
$560$	0	0
$561$	9.86095	0.416330
$562$	0	0
$563$	−38.7389	−1.63265	−0.816325	−	0.577593i	$-0.803992\pi$
$563$	−38.7389	−1.63265	−0.816325	+	0.577593i	$0.803992\pi$
$564$	0	0
$565$	20.4955	0.862254
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.1097	−1.05266	−0.526328	−	0.850282i	$-0.676431\pi$
$569$	−25.1097	−1.05266	−0.526328	+	0.850282i	$0.676431\pi$
$570$	0	0
$571$	2.81230	0.117691	0.0588456	−	0.998267i	$-0.481258\pi$
$571$	2.81230	0.117691	0.0588456	+	0.998267i	$0.481258\pi$
$572$	0	0
$573$	3.39332	0.141758
$574$	0	0
$575$	−7.01623	−0.292597
$576$	0	0
$577$	−1.58233	−0.0658732	−0.0329366	−	0.999457i	$-0.510486\pi$
$577$	−1.58233	−0.0658732	−0.0329366	+	0.999457i	$0.510486\pi$
$578$	0	0
$579$	−21.8576	−0.908369
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.4750	0.433832
$584$	0	0
$585$	0.942761	0.0389784
$586$	0	0
$587$	9.92623	0.409699	0.204850	−	0.978793i	$-0.434329\pi$
$587$	9.92623	0.409699	0.204850	+	0.978793i	$0.434329\pi$
$588$	0	0
$589$	−8.03131	−0.330924
$590$	0	0
$591$	−9.72571	−0.400062
$592$	0	0
$593$	1.35477	0.0556337	0.0278168	−	0.999613i	$-0.491144\pi$
$593$	1.35477	0.0556337	0.0278168	+	0.999613i	$0.491144\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.5240	−0.471647
$598$	0	0
$599$	−24.0465	−0.982511	−0.491256	−	0.871015i	$-0.663462\pi$
$599$	−24.0465	−0.982511	−0.491256	+	0.871015i	$0.663462\pi$
$600$	0	0
$601$	−14.8462	−0.605587	−0.302794	−	0.953056i	$-0.597919\pi$
$601$	−14.8462	−0.605587	−0.302794	+	0.953056i	$0.597919\pi$
$602$	0	0
$603$	0.984535	0.0400934
$604$	0	0
$605$	−15.4637	−0.628689
$606$	0	0
$607$	−15.7007	−0.637271	−0.318635	−	0.947877i	$-0.603225\pi$
$607$	−15.7007	−0.637271	−0.318635	+	0.947877i	$0.603225\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.13561	−0.288676
$612$	0	0
$613$	−26.1916	−1.05787	−0.528935	−	0.848662i	$-0.677408\pi$
$613$	−26.1916	−1.05787	−0.528935	+	0.848662i	$0.677408\pi$
$614$	0	0
$615$	−2.68583	−0.108303
$616$	0	0
$617$	3.85660	0.155261	0.0776304	−	0.996982i	$-0.475265\pi$
$617$	3.85660	0.155261	0.0776304	+	0.996982i	$0.475265\pi$
$618$	0	0
$619$	11.6126	0.466751	0.233376	−	0.972387i	$-0.425023\pi$
$619$	11.6126	0.466751	0.233376	+	0.972387i	$0.425023\pi$
$620$	0	0
$621$	−20.2514	−0.812663
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.5555	−0.462219
$626$	0	0
$627$	−1.97248	−0.0787734
$628$	0	0
$629$	9.40643	0.375059
$630$	0	0
$631$	−13.3241	−0.530425	−0.265212	−	0.964190i	$-0.585442\pi$
$631$	−13.3241	−0.530425	−0.265212	+	0.964190i	$0.585442\pi$
$632$	0	0
$633$	16.5972	0.659678
$634$	0	0
$635$	34.6832	1.37636
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.11104	0.123071
$640$	0	0
$641$	30.9769	1.22352	0.611758	−	0.791045i	$-0.290463\pi$
$641$	30.9769	1.22352	0.611758	+	0.791045i	$0.290463\pi$
$642$	0	0
$643$	−11.8317	−0.466598	−0.233299	−	0.972405i	$-0.574952\pi$
$643$	−11.8317	−0.466598	−0.233299	+	0.972405i	$0.574952\pi$
$644$	0	0
$645$	−4.03240	−0.158775
$646$	0	0
$647$	−45.1197	−1.77384	−0.886919	−	0.461924i	$-0.847159\pi$
$647$	−45.1197	−1.77384	−0.886919	+	0.461924i	$0.847159\pi$
$648$	0	0
$649$	−19.0904	−0.749362
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8017	0.461838	0.230919	−	0.972973i	$-0.425827\pi$
$653$	11.8017	0.461838	0.230919	+	0.972973i	$0.425827\pi$
$654$	0	0
$655$	−2.92576	−0.114319
$656$	0	0
$657$	0.00776374	0.000302892 0
$658$	0	0
$659$	−15.6377	−0.609158	−0.304579	−	0.952487i	$-0.598516\pi$
$659$	−15.6377	−0.609158	−0.304579	+	0.952487i	$0.598516\pi$
$660$	0	0
$661$	21.5749	0.839168	0.419584	−	0.907717i	$-0.362176\pi$
$661$	21.5749	0.839168	0.419584	+	0.907717i	$0.362176\pi$
$662$	0	0
$663$	−5.60690	−0.217754
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.4159	0.790507
$668$	0	0
$669$	−7.79814	−0.301494
$670$	0	0
$671$	−17.7283	−0.684393
$672$	0	0
$673$	37.3954	1.44149	0.720743	−	0.693202i	$-0.243801\pi$
$673$	37.3954	1.44149	0.720743	+	0.693202i	$0.243801\pi$
$674$	0	0
$675$	10.8391	0.417197
$676$	0	0
$677$	21.3496	0.820532	0.410266	−	0.911966i	$-0.365436\pi$
$677$	21.3496	0.820532	0.410266	+	0.911966i	$0.365436\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	25.3534	0.971545
$682$	0	0
$683$	−25.4024	−0.971998	−0.485999	−	0.873959i	$-0.661544\pi$
$683$	−25.4024	−0.971998	−0.485999	+	0.873959i	$0.661544\pi$
$684$	0	0
$685$	14.8459	0.567231
$686$	0	0
$687$	23.0717	0.880239
$688$	0	0
$689$	−5.95606	−0.226908
$690$	0	0
$691$	18.1327	0.689802	0.344901	−	0.938639i	$-0.387913\pi$
$691$	18.1327	0.689802	0.344901	+	0.938639i	$0.387913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.7818	−0.522773
$696$	0	0
$697$	−4.36833	−0.165462
$698$	0	0
$699$	16.3521	0.618494
$700$	0	0
$701$	−20.3965	−0.770364	−0.385182	−	0.922841i	$-0.625861\pi$
$701$	−20.3965	−0.770364	−0.385182	+	0.922841i	$0.625861\pi$
$702$	0	0
$703$	−1.88157	−0.0709646
$704$	0	0
$705$	22.9176	0.863125
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.6785	0.551264	0.275632	−	0.961263i	$-0.411113\pi$
$709$	14.6785	0.551264	0.275632	+	0.961263i	$0.411113\pi$
$710$	0	0
$711$	−1.64461	−0.0616777
$712$	0	0
$713$	33.2782	1.24628
$714$	0	0
$715$	−2.15225	−0.0804895
$716$	0	0
$717$	5.10161	0.190523
$718$	0	0
$719$	38.3806	1.43135	0.715677	−	0.698431i	$-0.246118\pi$
$719$	38.3806	1.43135	0.715677	+	0.698431i	$0.246118\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.05772	0.262480
$724$	0	0
$725$	−10.9271	−0.405823
$726$	0	0
$727$	11.6526	0.432172	0.216086	−	0.976374i	$-0.430671\pi$
$727$	11.6526	0.432172	0.216086	+	0.976374i	$0.430671\pi$
$728$	0	0
$729$	30.0405	1.11261
$730$	0	0
$731$	−6.55843	−0.242572
$732$	0	0
$733$	−9.35519	−0.345542	−0.172771	−	0.984962i	$-0.555272\pi$
$733$	−9.35519	−0.345542	−0.172771	+	0.984962i	$0.555272\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.24762	−0.0827920
$738$	0	0
$739$	7.70686	0.283501	0.141751	−	0.989902i	$-0.454727\pi$
$739$	7.70686	0.283501	0.141751	+	0.989902i	$0.454727\pi$
$740$	0	0
$741$	1.12155	0.0412010
$742$	0	0
$743$	−6.50488	−0.238641	−0.119320	−	0.992856i	$-0.538072\pi$
$743$	−6.50488	−0.238641	−0.119320	+	0.992856i	$0.538072\pi$
$744$	0	0
$745$	16.7534	0.613797
$746$	0	0
$747$	−7.43305	−0.271961
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.2720	1.17762	0.588811	−	0.808270i	$-0.299596\pi$
$751$	32.2720	1.17762	0.588811	+	0.808270i	$0.299596\pi$
$752$	0	0
$753$	37.9286	1.38220
$754$	0	0
$755$	−16.0893	−0.585549
$756$	0	0
$757$	13.6668	0.496730	0.248365	−	0.968667i	$-0.420107\pi$
$757$	13.6668	0.496730	0.248365	+	0.968667i	$0.420107\pi$
$758$	0	0
$759$	8.17310	0.296665
$760$	0	0
$761$	−13.7686	−0.499113	−0.249556	−	0.968360i	$-0.580285\pi$
$761$	−13.7686	−0.499113	−0.249556	+	0.968360i	$0.580285\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.92465	−0.178051
$766$	0	0
$767$	10.8547	0.391941
$768$	0	0
$769$	−27.2814	−0.983793	−0.491897	−	0.870654i	$-0.663696\pi$
$769$	−27.2814	−0.983793	−0.491897	+	0.870654i	$0.663696\pi$
$770$	0	0
$771$	10.2728	0.369968
$772$	0	0
$773$	−10.5768	−0.380421	−0.190210	−	0.981743i	$-0.560917\pi$
$773$	−10.5768	−0.380421	−0.190210	+	0.981743i	$0.560917\pi$
$774$	0	0
$775$	−17.8113	−0.639802
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.873796	0.0313070
$780$	0	0
$781$	−7.10224	−0.254138
$782$	0	0
$783$	−31.5397	−1.12714
$784$	0	0
$785$	16.1202	0.575356
$786$	0	0
$787$	53.0929	1.89256	0.946280	−	0.323349i	$-0.104809\pi$
$787$	53.0929	1.89256	0.946280	+	0.323349i	$0.104809\pi$
$788$	0	0
$789$	−5.39307	−0.191998
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0802	0.357960
$794$	0	0
$795$	19.1292	0.678443
$796$	0	0
$797$	20.6355	0.730946	0.365473	−	0.930822i	$-0.380907\pi$
$797$	20.6355	0.730946	0.365473	+	0.930822i	$0.380907\pi$
$798$	0	0
$799$	37.2739	1.31866
$800$	0	0
$801$	−2.51936	−0.0890171
$802$	0	0
$803$	−0.0177240	−0.000625466 0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.4610	0.755463
$808$	0	0
$809$	−12.5891	−0.442609	−0.221304	−	0.975205i	$-0.571031\pi$
$809$	−12.5891	−0.442609	−0.221304	+	0.975205i	$0.571031\pi$
$810$	0	0
$811$	−3.08769	−0.108423	−0.0542117	−	0.998529i	$-0.517265\pi$
$811$	−3.08769	−0.108423	−0.0542117	+	0.998529i	$0.517265\pi$
$812$	0	0
$813$	29.1315	1.02169
$814$	0	0
$815$	35.1306	1.23057
$816$	0	0
$817$	1.31188	0.0458969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.0668	0.386235	0.193117	−	0.981176i	$-0.438140\pi$
$821$	11.0668	0.386235	0.193117	+	0.981176i	$0.438140\pi$
$822$	0	0
$823$	11.4780	0.400100	0.200050	−	0.979786i	$-0.435890\pi$
$823$	11.4780	0.400100	0.200050	+	0.979786i	$0.435890\pi$
$824$	0	0
$825$	−4.37445	−0.152299
$826$	0	0
$827$	−37.8562	−1.31639	−0.658195	−	0.752848i	$-0.728680\pi$
$827$	−37.8562	−1.31639	−0.658195	+	0.752848i	$0.728680\pi$
$828$	0	0
$829$	−4.82053	−0.167424	−0.0837120	−	0.996490i	$-0.526678\pi$
$829$	−4.82053	−0.167424	−0.0837120	+	0.996490i	$0.526678\pi$
$830$	0	0
$831$	−37.7903	−1.31093
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.84229	0.0983616
$836$	0	0
$837$	−51.4101	−1.77699
$838$	0	0
$839$	−12.4951	−0.431379	−0.215690	−	0.976462i	$-0.569200\pi$
$839$	−12.4951	−0.431379	−0.215690	+	0.976462i	$0.569200\pi$
$840$	0	0
$841$	2.79579	0.0964065
$842$	0	0
$843$	36.3334	1.25139
$844$	0	0
$845$	−21.5249	−0.740480
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.15421	0.142572
$850$	0	0
$851$	7.79637	0.267256
$852$	0	0
$853$	40.7247	1.39439	0.697194	−	0.716882i	$-0.254432\pi$
$853$	40.7247	1.39439	0.697194	+	0.716882i	$0.254432\pi$
$854$	0	0
$855$	0.985077	0.0336889
$856$	0	0
$857$	−10.9278	−0.373287	−0.186643	−	0.982428i	$-0.559761\pi$
$857$	−10.9278	−0.373287	−0.186643	+	0.982428i	$0.559761\pi$
$858$	0	0
$859$	−24.9462	−0.851154	−0.425577	−	0.904922i	$-0.639929\pi$
$859$	−24.9462	−0.851154	−0.425577	+	0.904922i	$0.639929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.5081	−1.31083	−0.655416	−	0.755268i	$-0.727507\pi$
$863$	−38.5081	−1.31083	−0.655416	+	0.755268i	$0.727507\pi$
$864$	0	0
$865$	41.5945	1.41425
$866$	0	0
$867$	3.19606	0.108544
$868$	0	0
$869$	3.75451	0.127363
$870$	0	0
$871$	1.27799	0.0433029
$872$	0	0
$873$	8.45068	0.286012
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.9021	−0.638278	−0.319139	−	0.947708i	$-0.603394\pi$
$877$	−18.9021	−0.638278	−0.319139	+	0.947708i	$0.603394\pi$
$878$	0	0
$879$	−25.9475	−0.875189
$880$	0	0
$881$	11.7741	0.396681	0.198340	−	0.980133i	$-0.436445\pi$
$881$	11.7741	0.396681	0.198340	+	0.980133i	$0.436445\pi$
$882$	0	0
$883$	−23.9761	−0.806860	−0.403430	−	0.915010i	$-0.632182\pi$
$883$	−23.9761	−0.806860	−0.403430	+	0.915010i	$0.632182\pi$
$884$	0	0
$885$	−34.8622	−1.17188
$886$	0	0
$887$	0.850580	0.0285597	0.0142798	−	0.999898i	$-0.495454\pi$
$887$	0.850580	0.0285597	0.0142798	+	0.999898i	$0.495454\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.78375	−0.327768
$892$	0	0
$893$	−7.45590	−0.249502
$894$	0	0
$895$	−40.4422	−1.35183
$896$	0	0
$897$	−4.64719	−0.155165
$898$	0	0
$899$	51.8276	1.72855
$900$	0	0
$901$	31.1124	1.03650
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.59409	−0.0529893
$906$	0	0
$907$	−0.892050	−0.0296200	−0.0148100	−	0.999890i	$-0.504714\pi$
$907$	−0.892050	−0.0296200	−0.0148100	+	0.999890i	$0.504714\pi$
$908$	0	0
$909$	−10.0837	−0.334455
$910$	0	0
$911$	31.4136	1.04078	0.520389	−	0.853929i	$-0.325787\pi$
$911$	31.4136	1.04078	0.520389	+	0.853929i	$0.325787\pi$
$912$	0	0
$913$	16.9691	0.561594
$914$	0	0
$915$	−32.3749	−1.07028
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−30.9838	−1.02206	−0.511031	−	0.859562i	$-0.670736\pi$
$919$	−30.9838	−1.02206	−0.511031	+	0.859562i	$0.670736\pi$
$920$	0	0
$921$	−34.7562	−1.14525
$922$	0	0
$923$	4.03831	0.132923
$924$	0	0
$925$	−4.17282	−0.137201
$926$	0	0
$927$	−1.24519	−0.0408973
$928$	0	0
$929$	7.29045	0.239192	0.119596	−	0.992823i	$-0.461840\pi$
$929$	7.29045	0.239192	0.119596	+	0.992823i	$0.461840\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−40.3378	−1.32060
$934$	0	0
$935$	11.2426	0.367672
$936$	0	0
$937$	39.2582	1.28251	0.641255	−	0.767328i	$-0.278414\pi$
$937$	39.2582	1.28251	0.641255	+	0.767328i	$0.278414\pi$
$938$	0	0
$939$	21.9337	0.715779
$940$	0	0
$941$	−39.7133	−1.29462	−0.647308	−	0.762229i	$-0.724105\pi$
$941$	−39.7133	−1.29462	−0.647308	+	0.762229i	$0.724105\pi$
$942$	0	0
$943$	−3.62063	−0.117904
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0389	1.04112	0.520562	−	0.853824i	$-0.325722\pi$
$947$	32.0389	1.04112	0.520562	+	0.853824i	$0.325722\pi$
$948$	0	0
$949$	0.0100778	0.000327139 0
$950$	0	0
$951$	23.6072	0.765516
$952$	0	0
$953$	24.7950	0.803190	0.401595	−	0.915817i	$-0.368456\pi$
$953$	24.7950	0.803190	0.401595	+	0.915817i	$0.368456\pi$
$954$	0	0
$955$	3.86877	0.125190
$956$	0	0
$957$	12.7288	0.411464
$958$	0	0
$959$	0	0
$960$	0	0
$961$	53.4796	1.72515
$962$	0	0
$963$	−1.93061	−0.0622130
$964$	0	0
$965$	−24.9201	−0.802205
$966$	0	0
$967$	−33.1966	−1.06753	−0.533765	−	0.845633i	$-0.679223\pi$
$967$	−33.1966	−1.06753	−0.533765	+	0.845633i	$0.679223\pi$
$968$	0	0
$969$	−5.85857	−0.188204
$970$	0	0
$971$	−37.8578	−1.21492	−0.607458	−	0.794352i	$-0.707811\pi$
$971$	−37.8578	−1.21492	−0.607458	+	0.794352i	$0.707811\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.48730	0.0796572
$976$	0	0
$977$	−27.4977	−0.879729	−0.439864	−	0.898064i	$-0.644973\pi$
$977$	−27.4977	−0.879729	−0.439864	+	0.898064i	$0.644973\pi$
$978$	0	0
$979$	5.75149	0.183818
$980$	0	0
$981$	−3.21501	−0.102647
$982$	0	0
$983$	−42.9383	−1.36952	−0.684759	−	0.728770i	$-0.740092\pi$
$983$	−42.9383	−1.36952	−0.684759	+	0.728770i	$0.740092\pi$
$984$	0	0
$985$	−11.0884	−0.353306
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.43586	−0.172850
$990$	0	0
$991$	8.57637	0.272437	0.136219	−	0.990679i	$-0.456505\pi$
$991$	8.57637	0.272437	0.136219	+	0.990679i	$0.456505\pi$
$992$	0	0
$993$	11.3777	0.361059
$994$	0	0
$995$	−13.1387	−0.416524
$996$	0	0
$997$	−40.4669	−1.28160	−0.640800	−	0.767708i	$-0.721397\pi$
$997$	−40.4669	−1.28160	−0.640800	+	0.767708i	$0.721397\pi$
$998$	0	0
$999$	−12.0443	−0.381065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.15	yes	20
7.6	odd	2		8036.2.a.s.1.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.6	✓	20	7.6	odd	2
8036.2.a.t.1.15	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53485 q^{3} +1.74990 q^{5} -0.644239 q^{9} +O(q^{10})\)	\(q+1.53485 q^{3} +1.74990 q^{5} -0.644239 q^{9} +1.47075 q^{11} -0.836260 q^{13} +2.68583 q^{15} +4.36833 q^{17} -0.873796 q^{19} +3.62063 q^{23} -1.93785 q^{25} -5.59336 q^{27} +5.63878 q^{29} +9.19128 q^{31} +2.25737 q^{33} +2.15332 q^{37} -1.28353 q^{39} -1.00000 q^{41} -1.50136 q^{43} -1.12735 q^{45} +8.53276 q^{47} +6.70473 q^{51} +7.12226 q^{53} +2.57366 q^{55} -1.34115 q^{57} -12.9801 q^{59} -12.0540 q^{61} -1.46337 q^{65} -1.52821 q^{67} +5.55711 q^{69} -4.82901 q^{71} -0.0120510 q^{73} -2.97431 q^{75} +2.55280 q^{79} -6.65224 q^{81} +11.5377 q^{83} +7.64414 q^{85} +8.65467 q^{87} +3.91059 q^{89} +14.1072 q^{93} -1.52906 q^{95} -13.1173 q^{97} -0.947512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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