LMFDB - Embedded newform 8036.2.a.r.1.8

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.515082$ 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)

$f(q)$	$=$	$q+0.515082 q^{3} -2.19964 q^{5} -2.73469 q^{9} +O(q^{10})$	$q+0.515082 q^{3} -2.19964 q^{5} -2.73469 q^{9} +4.22160 q^{11} -3.63941 q^{13} -1.13299 q^{15} +7.98698 q^{17} -1.14014 q^{19} -7.43446 q^{23} -0.161604 q^{25} -2.95383 q^{27} +6.08068 q^{29} +1.33243 q^{31} +2.17447 q^{33} -2.87776 q^{37} -1.87460 q^{39} +1.00000 q^{41} +5.14701 q^{43} +6.01532 q^{45} -8.12473 q^{47} +4.11395 q^{51} -1.63075 q^{53} -9.28598 q^{55} -0.587265 q^{57} +2.97978 q^{59} -10.2708 q^{61} +8.00538 q^{65} +2.46808 q^{67} -3.82935 q^{69} +13.0326 q^{71} -11.5199 q^{73} -0.0832393 q^{75} +9.06941 q^{79} +6.68261 q^{81} -7.51957 q^{83} -17.5684 q^{85} +3.13205 q^{87} +17.9819 q^{89} +0.686309 q^{93} +2.50789 q^{95} -13.2426 q^{97} -11.5448 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * 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
$3$	0.515082	0.297383	0.148691	−	0.988884i	$-0.452494\pi$
$3$	0.515082	0.297383	0.148691	+	0.988884i	$0.452494\pi$
$4$	0	0
$5$	−2.19964	−0.983707	−0.491853	−	0.870678i	$-0.663680\pi$
$5$	−2.19964	−0.983707	−0.491853	+	0.870678i	$0.663680\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.73469	−0.911564
$10$	0	0
$11$	4.22160	1.27286	0.636430	−	0.771335i	$-0.280410\pi$
$11$	4.22160	1.27286	0.636430	+	0.771335i	$0.280410\pi$
$12$	0	0
$13$	−3.63941	−1.00939	−0.504696	−	0.863297i	$-0.668395\pi$
$13$	−3.63941	−1.00939	−0.504696	+	0.863297i	$0.668395\pi$
$14$	0	0
$15$	−1.13299	−0.292537
$16$	0	0
$17$	7.98698	1.93713	0.968563	−	0.248768i	$-0.0800256\pi$
$17$	7.98698	1.93713	0.968563	+	0.248768i	$0.0800256\pi$
$18$	0	0
$19$	−1.14014	−0.261566	−0.130783	−	0.991411i	$-0.541749\pi$
$19$	−1.14014	−0.261566	−0.130783	+	0.991411i	$0.541749\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.43446	−1.55019	−0.775096	−	0.631844i	$-0.782298\pi$
$23$	−7.43446	−1.55019	−0.775096	+	0.631844i	$0.782298\pi$
$24$	0	0
$25$	−0.161604	−0.0323208
$26$	0	0
$27$	−2.95383	−0.568466
$28$	0	0
$29$	6.08068	1.12915	0.564577	−	0.825380i	$-0.309039\pi$
$29$	6.08068	1.12915	0.564577	+	0.825380i	$0.309039\pi$
$30$	0	0
$31$	1.33243	0.239311	0.119655	−	0.992815i	$-0.461821\pi$
$31$	1.33243	0.239311	0.119655	+	0.992815i	$0.461821\pi$
$32$	0	0
$33$	2.17447	0.378526
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.87776	−0.473101	−0.236551	−	0.971619i	$-0.576017\pi$
$37$	−2.87776	−0.473101	−0.236551	+	0.971619i	$0.576017\pi$
$38$	0	0
$39$	−1.87460	−0.300176
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.14701	0.784912	0.392456	−	0.919771i	$-0.371626\pi$
$43$	5.14701	0.784912	0.392456	+	0.919771i	$0.371626\pi$
$44$	0	0
$45$	6.01532	0.896711
$46$	0	0
$47$	−8.12473	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$47$	−8.12473	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.11395	0.576068
$52$	0	0
$53$	−1.63075	−0.224001	−0.112001	−	0.993708i	$-0.535726\pi$
$53$	−1.63075	−0.224001	−0.112001	+	0.993708i	$0.535726\pi$
$54$	0	0
$55$	−9.28598	−1.25212
$56$	0	0
$57$	−0.587265	−0.0777852
$58$	0	0
$59$	2.97978	0.387935	0.193967	−	0.981008i	$-0.437864\pi$
$59$	2.97978	0.387935	0.193967	+	0.981008i	$0.437864\pi$
$60$	0	0
$61$	−10.2708	−1.31505	−0.657524	−	0.753434i	$-0.728396\pi$
$61$	−10.2708	−1.31505	−0.657524	+	0.753434i	$0.728396\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.00538	0.992946
$66$	0	0
$67$	2.46808	0.301525	0.150762	−	0.988570i	$-0.451827\pi$
$67$	2.46808	0.301525	0.150762	+	0.988570i	$0.451827\pi$
$68$	0	0
$69$	−3.82935	−0.461000
$70$	0	0
$71$	13.0326	1.54668	0.773342	−	0.633989i	$-0.218584\pi$
$71$	13.0326	1.54668	0.773342	+	0.633989i	$0.218584\pi$
$72$	0	0
$73$	−11.5199	−1.34831	−0.674154	−	0.738591i	$-0.735491\pi$
$73$	−11.5199	−1.34831	−0.674154	+	0.738591i	$0.735491\pi$
$74$	0	0
$75$	−0.0832393	−0.00961165
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.06941	1.02039	0.510194	−	0.860059i	$-0.329573\pi$
$79$	9.06941	1.02039	0.510194	+	0.860059i	$0.329573\pi$
$80$	0	0
$81$	6.68261	0.742512
$82$	0	0
$83$	−7.51957	−0.825380	−0.412690	−	0.910872i	$-0.635411\pi$
$83$	−7.51957	−0.825380	−0.412690	+	0.910872i	$0.635411\pi$
$84$	0	0
$85$	−17.5684	−1.90556
$86$	0	0
$87$	3.13205	0.335791
$88$	0	0
$89$	17.9819	1.90608	0.953041	−	0.302840i	$-0.0979348\pi$
$89$	17.9819	1.90608	0.953041	+	0.302840i	$0.0979348\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.686309	0.0711669
$94$	0	0
$95$	2.50789	0.257304
$96$	0	0
$97$	−13.2426	−1.34459	−0.672293	−	0.740285i	$-0.734691\pi$
$97$	−13.2426	−1.34459	−0.672293	+	0.740285i	$0.734691\pi$
$98$	0	0
$99$	−11.5448	−1.16029
$100$	0	0
$101$	−5.90610	−0.587678	−0.293839	−	0.955855i	$-0.594933\pi$
$101$	−5.90610	−0.587678	−0.293839	+	0.955855i	$0.594933\pi$
$102$	0	0
$103$	4.32873	0.426522	0.213261	−	0.976995i	$-0.431591\pi$
$103$	4.32873	0.426522	0.213261	+	0.976995i	$0.431591\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6786	−1.12901	−0.564504	−	0.825430i	$-0.690933\pi$
$107$	−11.6786	−1.12901	−0.564504	+	0.825430i	$0.690933\pi$
$108$	0	0
$109$	15.2466	1.46036	0.730178	−	0.683257i	$-0.239437\pi$
$109$	15.2466	1.46036	0.730178	+	0.683257i	$0.239437\pi$
$110$	0	0
$111$	−1.48228	−0.140692
$112$	0	0
$113$	17.2122	1.61919	0.809594	−	0.586990i	$-0.199687\pi$
$113$	17.2122	1.61919	0.809594	+	0.586990i	$0.199687\pi$
$114$	0	0
$115$	16.3531	1.52493
$116$	0	0
$117$	9.95267	0.920125
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.82189	0.620172
$122$	0	0
$123$	0.515082	0.0464434
$124$	0	0
$125$	11.3536	1.01550
$126$	0	0
$127$	−8.42630	−0.747713	−0.373857	−	0.927487i	$-0.621965\pi$
$127$	−8.42630	−0.747713	−0.373857	+	0.927487i	$0.621965\pi$
$128$	0	0
$129$	2.65113	0.233419
$130$	0	0
$131$	11.8466	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$131$	11.8466	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.49736	0.559204
$136$	0	0
$137$	−17.6176	−1.50518	−0.752589	−	0.658491i	$-0.771195\pi$
$137$	−17.6176	−1.50518	−0.752589	+	0.658491i	$0.771195\pi$
$138$	0	0
$139$	3.43664	0.291492	0.145746	−	0.989322i	$-0.453442\pi$
$139$	3.43664	0.291492	0.145746	+	0.989322i	$0.453442\pi$
$140$	0	0
$141$	−4.18490	−0.352432
$142$	0	0
$143$	−15.3641	−1.28481
$144$	0	0
$145$	−13.3753	−1.11076
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.2855	1.66185	0.830926	−	0.556383i	$-0.187811\pi$
$149$	20.2855	1.66185	0.830926	+	0.556383i	$0.187811\pi$
$150$	0	0
$151$	0.792622	0.0645027	0.0322513	−	0.999480i	$-0.489732\pi$
$151$	0.792622	0.0645027	0.0322513	+	0.999480i	$0.489732\pi$
$152$	0	0
$153$	−21.8419	−1.76581
$154$	0	0
$155$	−2.93085	−0.235412
$156$	0	0
$157$	−5.29832	−0.422852	−0.211426	−	0.977394i	$-0.567811\pi$
$157$	−5.29832	−0.422852	−0.211426	+	0.977394i	$0.567811\pi$
$158$	0	0
$159$	−0.839972	−0.0666141
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.33217	0.730952	0.365476	−	0.930821i	$-0.380906\pi$
$163$	9.33217	0.730952	0.365476	+	0.930821i	$0.380906\pi$
$164$	0	0
$165$	−4.78304	−0.372359
$166$	0	0
$167$	−5.22005	−0.403939	−0.201970	−	0.979392i	$-0.564734\pi$
$167$	−5.22005	−0.403939	−0.201970	+	0.979392i	$0.564734\pi$
$168$	0	0
$169$	0.245336	0.0188720
$170$	0	0
$171$	3.11793	0.238434
$172$	0	0
$173$	0.187866	0.0142832	0.00714158	−	0.999974i	$-0.497727\pi$
$173$	0.187866	0.0142832	0.00714158	+	0.999974i	$0.497727\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.53483	0.115365
$178$	0	0
$179$	13.3865	1.00055	0.500277	−	0.865865i	$-0.333231\pi$
$179$	13.3865	1.00055	0.500277	+	0.865865i	$0.333231\pi$
$180$	0	0
$181$	0.303123	0.0225309	0.0112655	−	0.999937i	$-0.496414\pi$
$181$	0.303123	0.0225309	0.0112655	+	0.999937i	$0.496414\pi$
$182$	0	0
$183$	−5.29033	−0.391072
$184$	0	0
$185$	6.33003	0.465393
$186$	0	0
$187$	33.7178	2.46569
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.01783	−0.0736479	−0.0368239	−	0.999322i	$-0.511724\pi$
$191$	−1.01783	−0.0736479	−0.0368239	+	0.999322i	$0.511724\pi$
$192$	0	0
$193$	17.1199	1.23231	0.616157	−	0.787623i	$-0.288689\pi$
$193$	17.1199	1.23231	0.616157	+	0.787623i	$0.288689\pi$
$194$	0	0
$195$	4.12343	0.295285
$196$	0	0
$197$	−4.81082	−0.342757	−0.171379	−	0.985205i	$-0.554822\pi$
$197$	−4.81082	−0.342757	−0.171379	+	0.985205i	$0.554822\pi$
$198$	0	0
$199$	12.5535	0.889892	0.444946	−	0.895557i	$-0.353223\pi$
$199$	12.5535	0.889892	0.444946	+	0.895557i	$0.353223\pi$
$200$	0	0
$201$	1.27127	0.0896681
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.19964	−0.153629
$206$	0	0
$207$	20.3309	1.41310
$208$	0	0
$209$	−4.81321	−0.332937
$210$	0	0
$211$	20.7515	1.42859	0.714297	−	0.699843i	$-0.246747\pi$
$211$	20.7515	1.42859	0.714297	+	0.699843i	$0.246747\pi$
$212$	0	0
$213$	6.71285	0.459957
$214$	0	0
$215$	−11.3215	−0.772123
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.93371	−0.400963
$220$	0	0
$221$	−29.0679	−1.95532
$222$	0	0
$223$	11.6034	0.777022	0.388511	−	0.921444i	$-0.372990\pi$
$223$	11.6034	0.777022	0.388511	+	0.921444i	$0.372990\pi$
$224$	0	0
$225$	0.441937	0.0294625
$226$	0	0
$227$	0.694596	0.0461020	0.0230510	−	0.999734i	$-0.492662\pi$
$227$	0.694596	0.0461020	0.0230510	+	0.999734i	$0.492662\pi$
$228$	0	0
$229$	−25.1712	−1.66336	−0.831681	−	0.555254i	$-0.812621\pi$
$229$	−25.1712	−1.66336	−0.831681	+	0.555254i	$0.812621\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.94397	0.192866	0.0964330	−	0.995339i	$-0.469257\pi$
$233$	2.94397	0.192866	0.0964330	+	0.995339i	$0.469257\pi$
$234$	0	0
$235$	17.8714	1.16580
$236$	0	0
$237$	4.67149	0.303446
$238$	0	0
$239$	7.56259	0.489183	0.244592	−	0.969626i	$-0.421346\pi$
$239$	7.56259	0.489183	0.244592	+	0.969626i	$0.421346\pi$
$240$	0	0
$241$	7.59060	0.488953	0.244477	−	0.969655i	$-0.421384\pi$
$241$	7.59060	0.488953	0.244477	+	0.969655i	$0.421384\pi$
$242$	0	0
$243$	12.3036	0.789276
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.14944	0.264023
$248$	0	0
$249$	−3.87319	−0.245454
$250$	0	0
$251$	23.7549	1.49940	0.749698	−	0.661780i	$-0.230199\pi$
$251$	23.7549	1.49940	0.749698	+	0.661780i	$0.230199\pi$
$252$	0	0
$253$	−31.3853	−1.97318
$254$	0	0
$255$	−9.04918	−0.566682
$256$	0	0
$257$	1.53771	0.0959198	0.0479599	−	0.998849i	$-0.484728\pi$
$257$	1.53771	0.0959198	0.0479599	+	0.998849i	$0.484728\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−16.6288	−1.02930
$262$	0	0
$263$	16.3727	1.00958	0.504791	−	0.863242i	$-0.331570\pi$
$263$	16.3727	1.00958	0.504791	+	0.863242i	$0.331570\pi$
$264$	0	0
$265$	3.58707	0.220352
$266$	0	0
$267$	9.26217	0.566836
$268$	0	0
$269$	−18.6581	−1.13760	−0.568801	−	0.822475i	$-0.692593\pi$
$269$	−18.6581	−1.13760	−0.568801	+	0.822475i	$0.692593\pi$
$270$	0	0
$271$	−16.3271	−0.991800	−0.495900	−	0.868380i	$-0.665162\pi$
$271$	−16.3271	−0.991800	−0.495900	+	0.868380i	$0.665162\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.682228	−0.0411399
$276$	0	0
$277$	−19.2494	−1.15659	−0.578293	−	0.815829i	$-0.696281\pi$
$277$	−19.2494	−1.15659	−0.578293	+	0.815829i	$0.696281\pi$
$278$	0	0
$279$	−3.64378	−0.218147
$280$	0	0
$281$	25.4587	1.51874	0.759369	−	0.650660i	$-0.225508\pi$
$281$	25.4587	1.51874	0.759369	+	0.650660i	$0.225508\pi$
$282$	0	0
$283$	6.72536	0.399781	0.199891	−	0.979818i	$-0.435941\pi$
$283$	6.72536	0.399781	0.199891	+	0.979818i	$0.435941\pi$
$284$	0	0
$285$	1.29177	0.0765178
$286$	0	0
$287$	0	0
$288$	0	0
$289$	46.7918	2.75246
$290$	0	0
$291$	−6.82104	−0.399857
$292$	0	0
$293$	14.0817	0.822660	0.411330	−	0.911487i	$-0.365064\pi$
$293$	14.0817	0.822660	0.411330	+	0.911487i	$0.365064\pi$
$294$	0	0
$295$	−6.55444	−0.381614
$296$	0	0
$297$	−12.4699	−0.723577
$298$	0	0
$299$	27.0571	1.56475
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.04212	−0.174765
$304$	0	0
$305$	22.5921	1.29362
$306$	0	0
$307$	22.4646	1.28212	0.641062	−	0.767489i	$-0.278494\pi$
$307$	22.4646	1.28212	0.641062	+	0.767489i	$0.278494\pi$
$308$	0	0
$309$	2.22965	0.126840
$310$	0	0
$311$	15.1252	0.857671	0.428835	−	0.903383i	$-0.358924\pi$
$311$	15.1252	0.857671	0.428835	+	0.903383i	$0.358924\pi$
$312$	0	0
$313$	19.4123	1.09725	0.548625	−	0.836068i	$-0.315151\pi$
$313$	19.4123	1.09725	0.548625	+	0.836068i	$0.315151\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.76362	0.0990546	0.0495273	−	0.998773i	$-0.484229\pi$
$317$	1.76362	0.0990546	0.0495273	+	0.998773i	$0.484229\pi$
$318$	0	0
$319$	25.6702	1.43725
$320$	0	0
$321$	−6.01541	−0.335747
$322$	0	0
$323$	−9.10627	−0.506686
$324$	0	0
$325$	0.588144	0.0326244
$326$	0	0
$327$	7.85322	0.434284
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.1176	0.666045	0.333023	−	0.942919i	$-0.391931\pi$
$331$	12.1176	0.666045	0.333023	+	0.942919i	$0.391931\pi$
$332$	0	0
$333$	7.86979	0.431262
$334$	0	0
$335$	−5.42889	−0.296612
$336$	0	0
$337$	5.78905	0.315349	0.157675	−	0.987491i	$-0.449600\pi$
$337$	5.78905	0.315349	0.157675	+	0.987491i	$0.449600\pi$
$338$	0	0
$339$	8.86569	0.481518
$340$	0	0
$341$	5.62497	0.304609
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.42318	0.453489
$346$	0	0
$347$	11.9344	0.640672	0.320336	−	0.947304i	$-0.396204\pi$
$347$	11.9344	0.640672	0.320336	+	0.947304i	$0.396204\pi$
$348$	0	0
$349$	3.41253	0.182669	0.0913344	−	0.995820i	$-0.470887\pi$
$349$	3.41253	0.182669	0.0913344	+	0.995820i	$0.470887\pi$
$350$	0	0
$351$	10.7502	0.573805
$352$	0	0
$353$	3.72027	0.198010	0.0990049	−	0.995087i	$-0.468434\pi$
$353$	3.72027	0.198010	0.0990049	+	0.995087i	$0.468434\pi$
$354$	0	0
$355$	−28.6669	−1.52148
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.3819	1.33961	0.669803	−	0.742539i	$-0.266379\pi$
$359$	25.3819	1.33961	0.669803	+	0.742539i	$0.266379\pi$
$360$	0	0
$361$	−17.7001	−0.931583
$362$	0	0
$363$	3.51383	0.184428
$364$	0	0
$365$	25.3397	1.32634
$366$	0	0
$367$	7.66006	0.399852	0.199926	−	0.979811i	$-0.435930\pi$
$367$	7.66006	0.399852	0.199926	+	0.979811i	$0.435930\pi$
$368$	0	0
$369$	−2.73469	−0.142362
$370$	0	0
$371$	0	0
$372$	0	0
$373$	32.1886	1.66666	0.833331	−	0.552774i	$-0.186431\pi$
$373$	32.1886	1.66666	0.833331	+	0.552774i	$0.186431\pi$
$374$	0	0
$375$	5.84806	0.301992
$376$	0	0
$377$	−22.1301	−1.13976
$378$	0	0
$379$	−4.38797	−0.225395	−0.112697	−	0.993629i	$-0.535949\pi$
$379$	−4.38797	−0.225395	−0.112697	+	0.993629i	$0.535949\pi$
$380$	0	0
$381$	−4.34023	−0.222357
$382$	0	0
$383$	−5.35384	−0.273568	−0.136784	−	0.990601i	$-0.543677\pi$
$383$	−5.35384	−0.273568	−0.136784	+	0.990601i	$0.543677\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.0755	−0.715497
$388$	0	0
$389$	1.67620	0.0849866	0.0424933	−	0.999097i	$-0.486470\pi$
$389$	1.67620	0.0849866	0.0424933	+	0.999097i	$0.486470\pi$
$390$	0	0
$391$	−59.3788	−3.00292
$392$	0	0
$393$	6.10198	0.307804
$394$	0	0
$395$	−19.9494	−1.00376
$396$	0	0
$397$	−7.85249	−0.394105	−0.197053	−	0.980393i	$-0.563137\pi$
$397$	−7.85249	−0.394105	−0.197053	+	0.980393i	$0.563137\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.10274	0.254819	0.127409	−	0.991850i	$-0.459334\pi$
$401$	5.10274	0.254819	0.127409	+	0.991850i	$0.459334\pi$
$402$	0	0
$403$	−4.84925	−0.241559
$404$	0	0
$405$	−14.6993	−0.730414
$406$	0	0
$407$	−12.1488	−0.602192
$408$	0	0
$409$	−21.6135	−1.06872	−0.534360	−	0.845257i	$-0.679447\pi$
$409$	−21.6135	−1.06872	−0.534360	+	0.845257i	$0.679447\pi$
$410$	0	0
$411$	−9.07453	−0.447614
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.5403	0.811932
$416$	0	0
$417$	1.77015	0.0866848
$418$	0	0
$419$	29.2736	1.43011	0.715054	−	0.699069i	$-0.246402\pi$
$419$	29.2736	1.43011	0.715054	+	0.699069i	$0.246402\pi$
$420$	0	0
$421$	−1.32720	−0.0646837	−0.0323419	−	0.999477i	$-0.510297\pi$
$421$	−1.32720	−0.0646837	−0.0323419	+	0.999477i	$0.510297\pi$
$422$	0	0
$423$	22.2186	1.08031
$424$	0	0
$425$	−1.29073	−0.0626095
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.91379	−0.382081
$430$	0	0
$431$	−13.2691	−0.639151	−0.319575	−	0.947561i	$-0.603540\pi$
$431$	−13.2691	−0.639151	−0.319575	+	0.947561i	$0.603540\pi$
$432$	0	0
$433$	20.4388	0.982225	0.491113	−	0.871096i	$-0.336590\pi$
$433$	20.4388	0.982225	0.491113	+	0.871096i	$0.336590\pi$
$434$	0	0
$435$	−6.88936	−0.330320
$436$	0	0
$437$	8.47632	0.405477
$438$	0	0
$439$	−1.64250	−0.0783922	−0.0391961	−	0.999232i	$-0.512480\pi$
$439$	−1.64250	−0.0783922	−0.0391961	+	0.999232i	$0.512480\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.3859	−0.778518	−0.389259	−	0.921128i	$-0.627269\pi$
$443$	−16.3859	−0.778518	−0.389259	+	0.921128i	$0.627269\pi$
$444$	0	0
$445$	−39.5537	−1.87503
$446$	0	0
$447$	10.4487	0.494206
$448$	0	0
$449$	39.6265	1.87009	0.935045	−	0.354528i	$-0.115358\pi$
$449$	39.6265	1.87009	0.935045	+	0.354528i	$0.115358\pi$
$450$	0	0
$451$	4.22160	0.198787
$452$	0	0
$453$	0.408265	0.0191820
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.21966	0.337721	0.168861	−	0.985640i	$-0.445991\pi$
$457$	7.21966	0.337721	0.168861	+	0.985640i	$0.445991\pi$
$458$	0	0
$459$	−23.5922	−1.10119
$460$	0	0
$461$	−19.4564	−0.906176	−0.453088	−	0.891466i	$-0.649678\pi$
$461$	−19.4564	−0.906176	−0.453088	+	0.891466i	$0.649678\pi$
$462$	0	0
$463$	−17.9014	−0.831950	−0.415975	−	0.909376i	$-0.636560\pi$
$463$	−17.9014	−0.831950	−0.415975	+	0.909376i	$0.636560\pi$
$464$	0	0
$465$	−1.50963	−0.0700074
$466$	0	0
$467$	1.60602	0.0743176	0.0371588	−	0.999309i	$-0.488169\pi$
$467$	1.60602	0.0743176	0.0371588	+	0.999309i	$0.488169\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.72907	−0.125749
$472$	0	0
$473$	21.7286	0.999082
$474$	0	0
$475$	0.184251	0.00845403
$476$	0	0
$477$	4.45961	0.204191
$478$	0	0
$479$	−1.93345	−0.0883415	−0.0441708	−	0.999024i	$-0.514065\pi$
$479$	−1.93345	−0.0883415	−0.0441708	+	0.999024i	$0.514065\pi$
$480$	0	0
$481$	10.4734	0.477545
$482$	0	0
$483$	0	0
$484$	0	0
$485$	29.1290	1.32268
$486$	0	0
$487$	31.5340	1.42894	0.714471	−	0.699665i	$-0.246667\pi$
$487$	31.5340	1.42894	0.714471	+	0.699665i	$0.246667\pi$
$488$	0	0
$489$	4.80683	0.217372
$490$	0	0
$491$	−41.6704	−1.88056	−0.940279	−	0.340404i	$-0.889436\pi$
$491$	−41.6704	−1.88056	−0.940279	+	0.340404i	$0.889436\pi$
$492$	0	0
$493$	48.5663	2.18731
$494$	0	0
$495$	25.3943	1.14139
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.75198	0.257494	0.128747	−	0.991677i	$-0.458904\pi$
$499$	5.75198	0.257494	0.128747	+	0.991677i	$0.458904\pi$
$500$	0	0
$501$	−2.68875	−0.120125
$502$	0	0
$503$	−40.6067	−1.81057	−0.905283	−	0.424810i	$-0.860341\pi$
$503$	−40.6067	−1.81057	−0.905283	+	0.424810i	$0.860341\pi$
$504$	0	0
$505$	12.9913	0.578103
$506$	0	0
$507$	0.126368	0.00561220
$508$	0	0
$509$	13.1720	0.583837	0.291919	−	0.956443i	$-0.405706\pi$
$509$	13.1720	0.583837	0.291919	+	0.956443i	$0.405706\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.36778	0.148691
$514$	0	0
$515$	−9.52163	−0.419573
$516$	0	0
$517$	−34.2993	−1.50848
$518$	0	0
$519$	0.0967662	0.00424757
$520$	0	0
$521$	−34.1957	−1.49814	−0.749070	−	0.662491i	$-0.769499\pi$
$521$	−34.1957	−1.49814	−0.749070	+	0.662491i	$0.769499\pi$
$522$	0	0
$523$	−35.1253	−1.53592	−0.767961	−	0.640497i	$-0.778729\pi$
$523$	−35.1253	−1.53592	−0.767961	+	0.640497i	$0.778729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.6421	0.463576
$528$	0	0
$529$	32.2711	1.40309
$530$	0	0
$531$	−8.14879	−0.353627
$532$	0	0
$533$	−3.63941	−0.157641
$534$	0	0
$535$	25.6886	1.11061
$536$	0	0
$537$	6.89515	0.297548
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.0445	1.03376	0.516878	−	0.856059i	$-0.327094\pi$
$541$	24.0445	1.03376	0.516878	+	0.856059i	$0.327094\pi$
$542$	0	0
$543$	0.156133	0.00670031
$544$	0	0
$545$	−33.5369	−1.43656
$546$	0	0
$547$	−34.5404	−1.47684	−0.738421	−	0.674340i	$-0.764428\pi$
$547$	−34.5404	−1.47684	−0.738421	+	0.674340i	$0.764428\pi$
$548$	0	0
$549$	28.0876	1.19875
$550$	0	0
$551$	−6.93283	−0.295348
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.26048	0.138400
$556$	0	0
$557$	26.6174	1.12782	0.563908	−	0.825837i	$-0.309297\pi$
$557$	26.6174	1.12782	0.563908	+	0.825837i	$0.309297\pi$
$558$	0	0
$559$	−18.7321	−0.792283
$560$	0	0
$561$	17.3674	0.733253
$562$	0	0
$563$	38.4714	1.62137	0.810687	−	0.585479i	$-0.199094\pi$
$563$	38.4714	1.62137	0.810687	+	0.585479i	$0.199094\pi$
$564$	0	0
$565$	−37.8606	−1.59281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.1979	−0.930583	−0.465291	−	0.885158i	$-0.654050\pi$
$569$	−22.1979	−0.930583	−0.465291	+	0.885158i	$0.654050\pi$
$570$	0	0
$571$	−32.3436	−1.35354	−0.676769	−	0.736196i	$-0.736620\pi$
$571$	−32.3436	−1.35354	−0.676769	+	0.736196i	$0.736620\pi$
$572$	0	0
$573$	−0.524268	−0.0219016
$574$	0	0
$575$	1.20144	0.0501035
$576$	0	0
$577$	−23.4107	−0.974599	−0.487299	−	0.873235i	$-0.662018\pi$
$577$	−23.4107	−0.974599	−0.487299	+	0.873235i	$0.662018\pi$
$578$	0	0
$579$	8.81813	0.366469
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.88439	−0.285122
$584$	0	0
$585$	−21.8923	−0.905133
$586$	0	0
$587$	6.45509	0.266430	0.133215	−	0.991087i	$-0.457470\pi$
$587$	6.45509	0.266430	0.133215	+	0.991087i	$0.457470\pi$
$588$	0	0
$589$	−1.51915	−0.0625956
$590$	0	0
$591$	−2.47797	−0.101930
$592$	0	0
$593$	38.1095	1.56497	0.782484	−	0.622671i	$-0.213952\pi$
$593$	38.1095	1.56497	0.782484	+	0.622671i	$0.213952\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.46606	0.264638
$598$	0	0
$599$	−16.4950	−0.673967	−0.336983	−	0.941511i	$-0.609407\pi$
$599$	−16.4950	−0.673967	−0.336983	+	0.941511i	$0.609407\pi$
$600$	0	0
$601$	−4.14054	−0.168896	−0.0844482	−	0.996428i	$-0.526913\pi$
$601$	−4.14054	−0.168896	−0.0844482	+	0.996428i	$0.526913\pi$
$602$	0	0
$603$	−6.74945	−0.274859
$604$	0	0
$605$	−15.0057	−0.610067
$606$	0	0
$607$	40.8860	1.65951	0.829756	−	0.558127i	$-0.188480\pi$
$607$	40.8860	1.65951	0.829756	+	0.558127i	$0.188480\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.5693	1.19624
$612$	0	0
$613$	38.5223	1.55590	0.777951	−	0.628325i	$-0.216259\pi$
$613$	38.5223	1.55590	0.777951	+	0.628325i	$0.216259\pi$
$614$	0	0
$615$	−1.13299	−0.0456867
$616$	0	0
$617$	−26.2855	−1.05822	−0.529108	−	0.848555i	$-0.677473\pi$
$617$	−26.2855	−1.05822	−0.529108	+	0.848555i	$0.677473\pi$
$618$	0	0
$619$	47.3181	1.90188	0.950938	−	0.309383i	$-0.100122\pi$
$619$	47.3181	1.90188	0.950938	+	0.309383i	$0.100122\pi$
$620$	0	0
$621$	21.9602	0.881231
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.1659	−0.966635
$626$	0	0
$627$	−2.47920	−0.0990096
$628$	0	0
$629$	−22.9846	−0.916457
$630$	0	0
$631$	4.48510	0.178549	0.0892746	−	0.996007i	$-0.471545\pi$
$631$	4.48510	0.178549	0.0892746	+	0.996007i	$0.471545\pi$
$632$	0	0
$633$	10.6887	0.424839
$634$	0	0
$635$	18.5348	0.735531
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−35.6401	−1.40990
$640$	0	0
$641$	−35.7692	−1.41280	−0.706399	−	0.707814i	$-0.749682\pi$
$641$	−35.7692	−1.41280	−0.706399	+	0.707814i	$0.749682\pi$
$642$	0	0
$643$	−30.7547	−1.21285	−0.606424	−	0.795142i	$-0.707397\pi$
$643$	−30.7547	−1.21285	−0.606424	+	0.795142i	$0.707397\pi$
$644$	0	0
$645$	−5.83152	−0.229616
$646$	0	0
$647$	−44.6457	−1.75520	−0.877602	−	0.479391i	$-0.840858\pi$
$647$	−44.6457	−1.75520	−0.877602	+	0.479391i	$0.840858\pi$
$648$	0	0
$649$	12.5795	0.493787
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29.4217	−1.15136	−0.575680	−	0.817675i	$-0.695263\pi$
$653$	−29.4217	−1.15136	−0.575680	+	0.817675i	$0.695263\pi$
$654$	0	0
$655$	−26.0583	−1.01818
$656$	0	0
$657$	31.5035	1.22907
$658$	0	0
$659$	39.7367	1.54792	0.773961	−	0.633234i	$-0.218273\pi$
$659$	39.7367	1.54792	0.773961	+	0.633234i	$0.218273\pi$
$660$	0	0
$661$	9.58987	0.373003	0.186501	−	0.982455i	$-0.440285\pi$
$661$	9.58987	0.373003	0.186501	+	0.982455i	$0.440285\pi$
$662$	0	0
$663$	−14.9724	−0.581478
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−45.2066	−1.75040
$668$	0	0
$669$	5.97670	0.231073
$670$	0	0
$671$	−43.3594	−1.67387
$672$	0	0
$673$	28.1956	1.08686	0.543431	−	0.839454i	$-0.317125\pi$
$673$	28.1956	1.08686	0.543431	+	0.839454i	$0.317125\pi$
$674$	0	0
$675$	0.477352	0.0183733
$676$	0	0
$677$	9.29576	0.357265	0.178633	−	0.983916i	$-0.442833\pi$
$677$	9.29576	0.357265	0.178633	+	0.983916i	$0.442833\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.357774	0.0137099
$682$	0	0
$683$	13.2127	0.505570	0.252785	−	0.967522i	$-0.418653\pi$
$683$	13.2127	0.505570	0.252785	+	0.967522i	$0.418653\pi$
$684$	0	0
$685$	38.7524	1.48065
$686$	0	0
$687$	−12.9652	−0.494655
$688$	0	0
$689$	5.93499	0.226105
$690$	0	0
$691$	6.77498	0.257732	0.128866	−	0.991662i	$-0.458866\pi$
$691$	6.77498	0.257732	0.128866	+	0.991662i	$0.458866\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.55937	−0.286743
$696$	0	0
$697$	7.98698	0.302528
$698$	0	0
$699$	1.51639	0.0573550
$700$	0	0
$701$	29.8871	1.12882	0.564410	−	0.825494i	$-0.309104\pi$
$701$	29.8871	1.12882	0.564410	+	0.825494i	$0.309104\pi$
$702$	0	0
$703$	3.28105	0.123747
$704$	0	0
$705$	9.20525	0.346690
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.95367	0.186039	0.0930194	−	0.995664i	$-0.470348\pi$
$709$	4.95367	0.186039	0.0930194	+	0.995664i	$0.470348\pi$
$710$	0	0
$711$	−24.8020	−0.930149
$712$	0	0
$713$	−9.90587	−0.370978
$714$	0	0
$715$	33.7955	1.26388
$716$	0	0
$717$	3.89535	0.145475
$718$	0	0
$719$	18.8348	0.702418	0.351209	−	0.936297i	$-0.385771\pi$
$719$	18.8348	0.702418	0.351209	+	0.936297i	$0.385771\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.90978	0.145406
$724$	0	0
$725$	−0.982663	−0.0364952
$726$	0	0
$727$	42.1079	1.56169	0.780847	−	0.624722i	$-0.214788\pi$
$727$	42.1079	1.56169	0.780847	+	0.624722i	$0.214788\pi$
$728$	0	0
$729$	−13.7105	−0.507795
$730$	0	0
$731$	41.1090	1.52047
$732$	0	0
$733$	−15.6455	−0.577881	−0.288940	−	0.957347i	$-0.593303\pi$
$733$	−15.6455	−0.577881	−0.288940	+	0.957347i	$0.593303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.4193	0.383798
$738$	0	0
$739$	32.1517	1.18272	0.591360	−	0.806407i	$-0.298591\pi$
$739$	32.1517	1.18272	0.591360	+	0.806407i	$0.298591\pi$
$740$	0	0
$741$	2.13730	0.0785157
$742$	0	0
$743$	−3.81171	−0.139838	−0.0699190	−	0.997553i	$-0.522274\pi$
$743$	−3.81171	−0.139838	−0.0699190	+	0.997553i	$0.522274\pi$
$744$	0	0
$745$	−44.6207	−1.63478
$746$	0	0
$747$	20.5637	0.752386
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−19.1017	−0.697030	−0.348515	−	0.937303i	$-0.613314\pi$
$751$	−19.1017	−0.697030	−0.348515	+	0.937303i	$0.613314\pi$
$752$	0	0
$753$	12.2357	0.445894
$754$	0	0
$755$	−1.74348	−0.0634517
$756$	0	0
$757$	30.1926	1.09737	0.548684	−	0.836030i	$-0.315129\pi$
$757$	30.1926	1.09737	0.548684	+	0.836030i	$0.315129\pi$
$758$	0	0
$759$	−16.1660	−0.586788
$760$	0	0
$761$	−51.0772	−1.85155	−0.925774	−	0.378076i	$-0.876586\pi$
$761$	−51.0772	−1.85155	−0.925774	+	0.378076i	$0.876586\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	48.0442	1.73704
$766$	0	0
$767$	−10.8447	−0.391578
$768$	0	0
$769$	30.8036	1.11081	0.555404	−	0.831581i	$-0.312564\pi$
$769$	30.8036	1.11081	0.555404	+	0.831581i	$0.312564\pi$
$770$	0	0
$771$	0.792047	0.0285249
$772$	0	0
$773$	−31.1901	−1.12183	−0.560916	−	0.827873i	$-0.689551\pi$
$773$	−31.1901	−1.12183	−0.560916	+	0.827873i	$0.689551\pi$
$774$	0	0
$775$	−0.215326	−0.00773473
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.14014	−0.0408498
$780$	0	0
$781$	55.0184	1.96871
$782$	0	0
$783$	−17.9613	−0.641885
$784$	0	0
$785$	11.6544	0.415963
$786$	0	0
$787$	−12.4267	−0.442963	−0.221481	−	0.975165i	$-0.571089\pi$
$787$	−12.4267	−0.442963	−0.221481	+	0.975165i	$0.571089\pi$
$788$	0	0
$789$	8.43326	0.300232
$790$	0	0
$791$	0	0
$792$	0	0
$793$	37.3799	1.32740
$794$	0	0
$795$	1.84763	0.0655287
$796$	0	0
$797$	−17.3519	−0.614636	−0.307318	−	0.951607i	$-0.599432\pi$
$797$	−17.3519	−0.614636	−0.307318	+	0.951607i	$0.599432\pi$
$798$	0	0
$799$	−64.8920	−2.29571
$800$	0	0
$801$	−49.1751	−1.73752
$802$	0	0
$803$	−48.6326	−1.71621
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.61043	−0.338303
$808$	0	0
$809$	26.0371	0.915416	0.457708	−	0.889103i	$-0.348670\pi$
$809$	26.0371	0.915416	0.457708	+	0.889103i	$0.348670\pi$
$810$	0	0
$811$	−16.2483	−0.570557	−0.285278	−	0.958445i	$-0.592086\pi$
$811$	−16.2483	−0.570557	−0.285278	+	0.958445i	$0.592086\pi$
$812$	0	0
$813$	−8.40978	−0.294944
$814$	0	0
$815$	−20.5274	−0.719042
$816$	0	0
$817$	−5.86831	−0.205306
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.5449	1.06603	0.533013	−	0.846107i	$-0.321060\pi$
$821$	30.5449	1.06603	0.533013	+	0.846107i	$0.321060\pi$
$822$	0	0
$823$	−47.3618	−1.65093	−0.825464	−	0.564455i	$-0.809086\pi$
$823$	−47.3618	−1.65093	−0.825464	+	0.564455i	$0.809086\pi$
$824$	0	0
$825$	−0.351403	−0.0122343
$826$	0	0
$827$	−49.3914	−1.71751	−0.858754	−	0.512388i	$-0.828761\pi$
$827$	−49.3914	−1.71751	−0.858754	+	0.512388i	$0.828761\pi$
$828$	0	0
$829$	1.74976	0.0607717	0.0303859	−	0.999538i	$-0.490326\pi$
$829$	1.74976	0.0607717	0.0303859	+	0.999538i	$0.490326\pi$
$830$	0	0
$831$	−9.91502	−0.343948
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.4822	0.397358
$836$	0	0
$837$	−3.93577	−0.136040
$838$	0	0
$839$	−48.9756	−1.69083	−0.845413	−	0.534113i	$-0.820646\pi$
$839$	−48.9756	−1.69083	−0.845413	+	0.534113i	$0.820646\pi$
$840$	0	0
$841$	7.97468	0.274989
$842$	0	0
$843$	13.1133	0.451646
$844$	0	0
$845$	−0.539649	−0.0185645
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.46411	0.118888
$850$	0	0
$851$	21.3946	0.733397
$852$	0	0
$853$	26.3270	0.901420	0.450710	−	0.892670i	$-0.351171\pi$
$853$	26.3270	0.901420	0.450710	+	0.892670i	$0.351171\pi$
$854$	0	0
$855$	−6.85831	−0.234549
$856$	0	0
$857$	41.2096	1.40769	0.703847	−	0.710352i	$-0.251464\pi$
$857$	41.2096	1.40769	0.703847	+	0.710352i	$0.251464\pi$
$858$	0	0
$859$	13.1281	0.447925	0.223962	−	0.974598i	$-0.428101\pi$
$859$	13.1281	0.447925	0.223962	+	0.974598i	$0.428101\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.7054	−1.14735	−0.573673	−	0.819084i	$-0.694482\pi$
$863$	−33.7054	−1.14735	−0.573673	+	0.819084i	$0.694482\pi$
$864$	0	0
$865$	−0.413236	−0.0140505
$866$	0	0
$867$	24.1016	0.818533
$868$	0	0
$869$	38.2874	1.29881
$870$	0	0
$871$	−8.98238	−0.304356
$872$	0	0
$873$	36.2145	1.22568
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.0469	0.406795	0.203398	−	0.979096i	$-0.434802\pi$
$877$	12.0469	0.406795	0.203398	+	0.979096i	$0.434802\pi$
$878$	0	0
$879$	7.25321	0.244645
$880$	0	0
$881$	25.8023	0.869302	0.434651	−	0.900599i	$-0.356872\pi$
$881$	25.8023	0.869302	0.434651	+	0.900599i	$0.356872\pi$
$882$	0	0
$883$	55.2747	1.86014	0.930070	−	0.367381i	$-0.119746\pi$
$883$	55.2747	1.86014	0.930070	+	0.367381i	$0.119746\pi$
$884$	0	0
$885$	−3.37607	−0.113485
$886$	0	0
$887$	−52.5771	−1.76536	−0.882682	−	0.469970i	$-0.844265\pi$
$887$	−52.5771	−1.76536	−0.882682	+	0.469970i	$0.844265\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	28.2113	0.945113
$892$	0	0
$893$	9.26333	0.309985
$894$	0	0
$895$	−29.4454	−0.984252
$896$	0	0
$897$	13.9366	0.465330
$898$	0	0
$899$	8.10206	0.270219
$900$	0	0
$901$	−13.0248	−0.433919
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.666759	−0.0221638
$906$	0	0
$907$	−35.7961	−1.18859	−0.594294	−	0.804248i	$-0.702569\pi$
$907$	−35.7961	−1.18859	−0.594294	+	0.804248i	$0.702569\pi$
$908$	0	0
$909$	16.1513	0.535706
$910$	0	0
$911$	5.12715	0.169870	0.0849350	−	0.996386i	$-0.472932\pi$
$911$	5.12715	0.169870	0.0849350	+	0.996386i	$0.472932\pi$
$912$	0	0
$913$	−31.7446	−1.05059
$914$	0	0
$915$	11.6368	0.384700
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.73713	0.0902897	0.0451449	−	0.998980i	$-0.485625\pi$
$919$	2.73713	0.0902897	0.0451449	+	0.998980i	$0.485625\pi$
$920$	0	0
$921$	11.5711	0.381281
$922$	0	0
$923$	−47.4310	−1.56121
$924$	0	0
$925$	0.465058	0.0152910
$926$	0	0
$927$	−11.8377	−0.388802
$928$	0	0
$929$	−13.6670	−0.448399	−0.224199	−	0.974543i	$-0.571977\pi$
$929$	−13.6670	−0.448399	−0.224199	+	0.974543i	$0.571977\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	7.79071	0.255056
$934$	0	0
$935$	−74.1669	−2.42552
$936$	0	0
$937$	−6.96389	−0.227500	−0.113750	−	0.993509i	$-0.536286\pi$
$937$	−6.96389	−0.227500	−0.113750	+	0.993509i	$0.536286\pi$
$938$	0	0
$939$	9.99895	0.326303
$940$	0	0
$941$	22.5498	0.735102	0.367551	−	0.930003i	$-0.380196\pi$
$941$	22.5498	0.735102	0.367551	+	0.930003i	$0.380196\pi$
$942$	0	0
$943$	−7.43446	−0.242099
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.9769	−1.42906	−0.714528	−	0.699607i	$-0.753358\pi$
$947$	−43.9769	−1.42906	−0.714528	+	0.699607i	$0.753358\pi$
$948$	0	0
$949$	41.9258	1.36097
$950$	0	0
$951$	0.908407	0.0294571
$952$	0	0
$953$	6.89729	0.223425	0.111713	−	0.993741i	$-0.464366\pi$
$953$	6.89729	0.223425	0.111713	+	0.993741i	$0.464366\pi$
$954$	0	0
$955$	2.23886	0.0724479
$956$	0	0
$957$	13.2222	0.427415
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.2246	−0.942730
$962$	0	0
$963$	31.9372	1.02916
$964$	0	0
$965$	−37.6574	−1.21224
$966$	0	0
$967$	24.7226	0.795025	0.397512	−	0.917597i	$-0.369874\pi$
$967$	24.7226	0.795025	0.397512	+	0.917597i	$0.369874\pi$
$968$	0	0
$969$	−4.69047	−0.150680
$970$	0	0
$971$	−51.8477	−1.66387	−0.831937	−	0.554871i	$-0.812768\pi$
$971$	−51.8477	−1.66387	−0.831937	+	0.554871i	$0.812768\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.302942	0.00970192
$976$	0	0
$977$	9.99891	0.319893	0.159947	−	0.987126i	$-0.448868\pi$
$977$	9.99891	0.319893	0.159947	+	0.987126i	$0.448868\pi$
$978$	0	0
$979$	75.9126	2.42618
$980$	0	0
$981$	−41.6946	−1.33121
$982$	0	0
$983$	−1.22767	−0.0391566	−0.0195783	−	0.999808i	$-0.506232\pi$
$983$	−1.22767	−0.0391566	−0.0195783	+	0.999808i	$0.506232\pi$
$984$	0	0
$985$	10.5821	0.337172
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−38.2652	−1.21676
$990$	0	0
$991$	43.7198	1.38881	0.694403	−	0.719586i	$-0.255669\pi$
$991$	43.7198	1.38881	0.694403	+	0.719586i	$0.255669\pi$
$992$	0	0
$993$	6.24157	0.198070
$994$	0	0
$995$	−27.6131	−0.875393
$996$	0	0
$997$	−5.79205	−0.183436	−0.0917180	−	0.995785i	$-0.529236\pi$
$997$	−5.79205	−0.183436	−0.0917180	+	0.995785i	$0.529236\pi$
$998$	0	0
$999$	8.50044	0.268942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.8		15
7.3	odd	6		1148.2.i.e.821.8	yes	30
7.5	odd	6		1148.2.i.e.165.8	✓	30
7.6	odd	2		8036.2.a.q.1.8		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.8	✓	30	7.5	odd	6
1148.2.i.e.821.8	yes	30	7.3	odd	6
8036.2.a.q.1.8		15	7.6	odd	2
8036.2.a.r.1.8		15	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)

\(f(q)\)	\(=\)	\(q+0.515082 q^{3} -2.19964 q^{5} -2.73469 q^{9} +O(q^{10})\)	\(q+0.515082 q^{3} -2.19964 q^{5} -2.73469 q^{9} +4.22160 q^{11} -3.63941 q^{13} -1.13299 q^{15} +7.98698 q^{17} -1.14014 q^{19} -7.43446 q^{23} -0.161604 q^{25} -2.95383 q^{27} +6.08068 q^{29} +1.33243 q^{31} +2.17447 q^{33} -2.87776 q^{37} -1.87460 q^{39} +1.00000 q^{41} +5.14701 q^{43} +6.01532 q^{45} -8.12473 q^{47} +4.11395 q^{51} -1.63075 q^{53} -9.28598 q^{55} -0.587265 q^{57} +2.97978 q^{59} -10.2708 q^{61} +8.00538 q^{65} +2.46808 q^{67} -3.82935 q^{69} +13.0326 q^{71} -11.5199 q^{73} -0.0832393 q^{75} +9.06941 q^{79} +6.68261 q^{81} -7.51957 q^{83} -17.5684 q^{85} +3.13205 q^{87} +17.9819 q^{89} +0.686309 q^{93} +2.50789 q^{95} -13.2426 q^{97} -11.5448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

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