LMFDB - Embedded newform 8036.2.a.q.1.1

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			1.1
Root			$3.42372$ 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)

$f(q)$	$=$	$q-3.42372 q^{3} +2.35740 q^{5} +8.72183 q^{9} +O(q^{10})$	$q-3.42372 q^{3} +2.35740 q^{5} +8.72183 q^{9} -0.142921 q^{11} -2.89147 q^{13} -8.07106 q^{15} -1.39444 q^{17} -7.23396 q^{19} -4.70073 q^{23} +0.557331 q^{25} -19.5899 q^{27} +2.50126 q^{29} -2.29868 q^{31} +0.489322 q^{33} -8.95769 q^{37} +9.89958 q^{39} -1.00000 q^{41} +10.0638 q^{43} +20.5608 q^{45} -0.577516 q^{47} +4.77417 q^{51} -2.67866 q^{53} -0.336922 q^{55} +24.7670 q^{57} -5.00779 q^{59} +10.1819 q^{61} -6.81635 q^{65} +4.81589 q^{67} +16.0940 q^{69} -10.7912 q^{71} -2.06872 q^{73} -1.90814 q^{75} +2.74538 q^{79} +40.9048 q^{81} +8.58858 q^{83} -3.28725 q^{85} -8.56361 q^{87} +11.3580 q^{89} +7.87002 q^{93} -17.0533 q^{95} -4.73305 q^{97} -1.24653 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.42372	−1.97668	−0.988341	−	0.152254i	$-0.951347\pi$
$3$	−3.42372	−1.97668	−0.988341	+	0.152254i	$0.951347\pi$
$4$	0	0
$5$	2.35740	1.05426	0.527130	−	0.849784i	$-0.323268\pi$
$5$	2.35740	1.05426	0.527130	+	0.849784i	$0.323268\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.72183	2.90728
$10$	0	0
$11$	−0.142921	−0.0430924	−0.0215462	−	0.999768i	$-0.506859\pi$
$11$	−0.142921	−0.0430924	−0.0215462	+	0.999768i	$0.506859\pi$
$12$	0	0
$13$	−2.89147	−0.801950	−0.400975	−	0.916089i	$-0.631329\pi$
$13$	−2.89147	−0.801950	−0.400975	+	0.916089i	$0.631329\pi$
$14$	0	0
$15$	−8.07106	−2.08394
$16$	0	0
$17$	−1.39444	−0.338202	−0.169101	−	0.985599i	$-0.554086\pi$
$17$	−1.39444	−0.338202	−0.169101	+	0.985599i	$0.554086\pi$
$18$	0	0
$19$	−7.23396	−1.65958	−0.829792	−	0.558073i	$-0.811541\pi$
$19$	−7.23396	−1.65958	−0.829792	+	0.558073i	$0.811541\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.70073	−0.980170	−0.490085	−	0.871675i	$-0.663034\pi$
$23$	−4.70073	−0.980170	−0.490085	+	0.871675i	$0.663034\pi$
$24$	0	0
$25$	0.557331	0.111466
$26$	0	0
$27$	−19.5899	−3.77008
$28$	0	0
$29$	2.50126	0.464473	0.232236	−	0.972659i	$-0.425396\pi$
$29$	2.50126	0.464473	0.232236	+	0.972659i	$0.425396\pi$
$30$	0	0
$31$	−2.29868	−0.412855	−0.206427	−	0.978462i	$-0.566184\pi$
$31$	−2.29868	−0.412855	−0.206427	+	0.978462i	$0.566184\pi$
$32$	0	0
$33$	0.489322	0.0851799
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.95769	−1.47263	−0.736317	−	0.676636i	$-0.763437\pi$
$37$	−8.95769	−1.47263	−0.736317	+	0.676636i	$0.763437\pi$
$38$	0	0
$39$	9.89958	1.58520
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	10.0638	1.53471	0.767355	−	0.641223i	$-0.221573\pi$
$43$	10.0638	1.53471	0.767355	+	0.641223i	$0.221573\pi$
$44$	0	0
$45$	20.5608	3.06503
$46$	0	0
$47$	−0.577516	−0.0842393	−0.0421197	−	0.999113i	$-0.513411\pi$
$47$	−0.577516	−0.0842393	−0.0421197	+	0.999113i	$0.513411\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.77417	0.668517
$52$	0	0
$53$	−2.67866	−0.367942	−0.183971	−	0.982932i	$-0.558895\pi$
$53$	−2.67866	−0.367942	−0.183971	+	0.982932i	$0.558895\pi$
$54$	0	0
$55$	−0.336922	−0.0454306
$56$	0	0
$57$	24.7670	3.28047
$58$	0	0
$59$	−5.00779	−0.651959	−0.325979	−	0.945377i	$-0.605694\pi$
$59$	−5.00779	−0.651959	−0.325979	+	0.945377i	$0.605694\pi$
$60$	0	0
$61$	10.1819	1.30366	0.651832	−	0.758364i	$-0.274001\pi$
$61$	10.1819	1.30366	0.651832	+	0.758364i	$0.274001\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.81635	−0.845465
$66$	0	0
$67$	4.81589	0.588354	0.294177	−	0.955751i	$-0.404954\pi$
$67$	4.81589	0.588354	0.294177	+	0.955751i	$0.404954\pi$
$68$	0	0
$69$	16.0940	1.93749
$70$	0	0
$71$	−10.7912	−1.28068	−0.640339	−	0.768092i	$-0.721206\pi$
$71$	−10.7912	−1.28068	−0.640339	+	0.768092i	$0.721206\pi$
$72$	0	0
$73$	−2.06872	−0.242125	−0.121063	−	0.992645i	$-0.538630\pi$
$73$	−2.06872	−0.242125	−0.121063	+	0.992645i	$0.538630\pi$
$74$	0	0
$75$	−1.90814	−0.220333
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.74538	0.308880	0.154440	−	0.988002i	$-0.450643\pi$
$79$	2.74538	0.308880	0.154440	+	0.988002i	$0.450643\pi$
$80$	0	0
$81$	40.9048	4.54497
$82$	0	0
$83$	8.58858	0.942720	0.471360	−	0.881941i	$-0.343763\pi$
$83$	8.58858	0.942720	0.471360	+	0.881941i	$0.343763\pi$
$84$	0	0
$85$	−3.28725	−0.356553
$86$	0	0
$87$	−8.56361	−0.918116
$88$	0	0
$89$	11.3580	1.20394	0.601970	−	0.798518i	$-0.294383\pi$
$89$	11.3580	1.20394	0.601970	+	0.798518i	$0.294383\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.87002	0.816083
$94$	0	0
$95$	−17.0533	−1.74963
$96$	0	0
$97$	−4.73305	−0.480568	−0.240284	−	0.970703i	$-0.577241\pi$
$97$	−4.73305	−0.480568	−0.240284	+	0.970703i	$0.577241\pi$
$98$	0	0
$99$	−1.24653	−0.125281
$100$	0	0
$101$	7.85182	0.781285	0.390643	−	0.920542i	$-0.372253\pi$
$101$	7.85182	0.781285	0.390643	+	0.920542i	$0.372253\pi$
$102$	0	0
$103$	−15.3743	−1.51488	−0.757438	−	0.652907i	$-0.773549\pi$
$103$	−15.3743	−1.51488	−0.757438	+	0.652907i	$0.773549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.8820	1.72872	0.864361	−	0.502871i	$-0.167723\pi$
$107$	17.8820	1.72872	0.864361	+	0.502871i	$0.167723\pi$
$108$	0	0
$109$	12.7126	1.21764	0.608822	−	0.793307i	$-0.291642\pi$
$109$	12.7126	1.21764	0.608822	+	0.793307i	$0.291642\pi$
$110$	0	0
$111$	30.6686	2.91093
$112$	0	0
$113$	8.71167	0.819525	0.409763	−	0.912192i	$-0.365612\pi$
$113$	8.71167	0.819525	0.409763	+	0.912192i	$0.365612\pi$
$114$	0	0
$115$	−11.0815	−1.03336
$116$	0	0
$117$	−25.2189	−2.33149
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9796	−0.998143
$122$	0	0
$123$	3.42372	0.308706
$124$	0	0
$125$	−10.4731	−0.936746
$126$	0	0
$127$	−5.87321	−0.521163	−0.260581	−	0.965452i	$-0.583914\pi$
$127$	−5.87321	−0.521163	−0.260581	+	0.965452i	$0.583914\pi$
$128$	0	0
$129$	−34.4555	−3.03363
$130$	0	0
$131$	11.1197	0.971533	0.485766	−	0.874089i	$-0.338541\pi$
$131$	11.1197	0.971533	0.485766	+	0.874089i	$0.338541\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−46.1812	−3.97465
$136$	0	0
$137$	11.7117	1.00060	0.500298	−	0.865853i	$-0.333224\pi$
$137$	11.7117	1.00060	0.500298	+	0.865853i	$0.333224\pi$
$138$	0	0
$139$	6.11570	0.518727	0.259364	−	0.965780i	$-0.416487\pi$
$139$	6.11570	0.518727	0.259364	+	0.965780i	$0.416487\pi$
$140$	0	0
$141$	1.97725	0.166514
$142$	0	0
$143$	0.413253	0.0345579
$144$	0	0
$145$	5.89648	0.489676
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0820	1.23557	0.617784	−	0.786348i	$-0.288031\pi$
$149$	15.0820	1.23557	0.617784	+	0.786348i	$0.288031\pi$
$150$	0	0
$151$	15.2951	1.24470	0.622349	−	0.782740i	$-0.286178\pi$
$151$	15.2951	1.24470	0.622349	+	0.782740i	$0.286178\pi$
$152$	0	0
$153$	−12.1621	−0.983245
$154$	0	0
$155$	−5.41890	−0.435256
$156$	0	0
$157$	−13.0031	−1.03776	−0.518881	−	0.854847i	$-0.673651\pi$
$157$	−13.0031	−1.03776	−0.518881	+	0.854847i	$0.673651\pi$
$158$	0	0
$159$	9.17096	0.727304
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.252349	−0.0197655	−0.00988275	−	0.999951i	$-0.503146\pi$
$163$	−0.252349	−0.0197655	−0.00988275	+	0.999951i	$0.503146\pi$
$164$	0	0
$165$	1.15353	0.0898019
$166$	0	0
$167$	−21.3675	−1.65347	−0.826733	−	0.562595i	$-0.809803\pi$
$167$	−21.3675	−1.65347	−0.826733	+	0.562595i	$0.809803\pi$
$168$	0	0
$169$	−4.63939	−0.356876
$170$	0	0
$171$	−63.0933	−4.82487
$172$	0	0
$173$	1.58550	0.120543	0.0602715	−	0.998182i	$-0.480803\pi$
$173$	1.58550	0.120543	0.0602715	+	0.998182i	$0.480803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.1452	1.28872
$178$	0	0
$179$	−16.9472	−1.26670	−0.633348	−	0.773867i	$-0.718320\pi$
$179$	−16.9472	−1.26670	−0.633348	+	0.773867i	$0.718320\pi$
$180$	0	0
$181$	18.3755	1.36584	0.682920	−	0.730493i	$-0.260710\pi$
$181$	18.3755	1.36584	0.682920	+	0.730493i	$0.260710\pi$
$182$	0	0
$183$	−34.8601	−2.57693
$184$	0	0
$185$	−21.1168	−1.55254
$186$	0	0
$187$	0.199295	0.0145739
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.5422	−1.84817	−0.924083	−	0.382191i	$-0.875170\pi$
$191$	−25.5422	−1.84817	−0.924083	+	0.382191i	$0.875170\pi$
$192$	0	0
$193$	−4.00444	−0.288246	−0.144123	−	0.989560i	$-0.546036\pi$
$193$	−4.00444	−0.288246	−0.144123	+	0.989560i	$0.546036\pi$
$194$	0	0
$195$	23.3373	1.67122
$196$	0	0
$197$	12.0879	0.861228	0.430614	−	0.902536i	$-0.358297\pi$
$197$	12.0879	0.861228	0.430614	+	0.902536i	$0.358297\pi$
$198$	0	0
$199$	−19.5784	−1.38788	−0.693939	−	0.720034i	$-0.744126\pi$
$199$	−19.5784	−1.38788	−0.693939	+	0.720034i	$0.744126\pi$
$200$	0	0
$201$	−16.4882	−1.16299
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.35740	−0.164648
$206$	0	0
$207$	−40.9990	−2.84963
$208$	0	0
$209$	1.03389	0.0715154
$210$	0	0
$211$	14.0250	0.965521	0.482761	−	0.875752i	$-0.339634\pi$
$211$	14.0250	0.965521	0.482761	+	0.875752i	$0.339634\pi$
$212$	0	0
$213$	36.9460	2.53149
$214$	0	0
$215$	23.7243	1.61798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.08271	0.478605
$220$	0	0
$221$	4.03199	0.271221
$222$	0	0
$223$	−14.7762	−0.989486	−0.494743	−	0.869039i	$-0.664738\pi$
$223$	−14.7762	−0.989486	−0.494743	+	0.869039i	$0.664738\pi$
$224$	0	0
$225$	4.86095	0.324063
$226$	0	0
$227$	−18.2506	−1.21134	−0.605668	−	0.795718i	$-0.707094\pi$
$227$	−18.2506	−1.21134	−0.605668	+	0.795718i	$0.707094\pi$
$228$	0	0
$229$	−5.92817	−0.391745	−0.195872	−	0.980629i	$-0.562754\pi$
$229$	−5.92817	−0.391745	−0.195872	+	0.980629i	$0.562754\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.39823	0.550187	0.275093	−	0.961418i	$-0.411291\pi$
$233$	8.39823	0.550187	0.275093	+	0.961418i	$0.411291\pi$
$234$	0	0
$235$	−1.36144	−0.0888102
$236$	0	0
$237$	−9.39941	−0.610557
$238$	0	0
$239$	27.2326	1.76153	0.880766	−	0.473552i	$-0.157028\pi$
$239$	27.2326	1.76153	0.880766	+	0.473552i	$0.157028\pi$
$240$	0	0
$241$	0.162129	0.0104437	0.00522183	−	0.999986i	$-0.498338\pi$
$241$	0.162129	0.0104437	0.00522183	+	0.999986i	$0.498338\pi$
$242$	0	0
$243$	−81.2766	−5.21389
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.9168	1.33090
$248$	0	0
$249$	−29.4049	−1.86346
$250$	0	0
$251$	27.1939	1.71646	0.858232	−	0.513261i	$-0.171563\pi$
$251$	27.1939	1.71646	0.858232	+	0.513261i	$0.171563\pi$
$252$	0	0
$253$	0.671834	0.0422379
$254$	0	0
$255$	11.2546	0.704792
$256$	0	0
$257$	22.0781	1.37719	0.688597	−	0.725144i	$-0.258227\pi$
$257$	22.0781	1.37719	0.688597	+	0.725144i	$0.258227\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.8156	1.35035
$262$	0	0
$263$	15.3850	0.948679	0.474339	−	0.880342i	$-0.342687\pi$
$263$	15.3850	0.948679	0.474339	+	0.880342i	$0.342687\pi$
$264$	0	0
$265$	−6.31466	−0.387907
$266$	0	0
$267$	−38.8864	−2.37981
$268$	0	0
$269$	20.7321	1.26406	0.632028	−	0.774946i	$-0.282223\pi$
$269$	20.7321	1.26406	0.632028	+	0.774946i	$0.282223\pi$
$270$	0	0
$271$	12.7232	0.772880	0.386440	−	0.922315i	$-0.373705\pi$
$271$	12.7232	0.772880	0.386440	+	0.922315i	$0.373705\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.0796545	−0.00480334
$276$	0	0
$277$	12.0748	0.725502	0.362751	−	0.931886i	$-0.381838\pi$
$277$	12.0748	0.725502	0.362751	+	0.931886i	$0.381838\pi$
$278$	0	0
$279$	−20.0487	−1.20028
$280$	0	0
$281$	0.165786	0.00988995	0.00494497	−	0.999988i	$-0.498426\pi$
$281$	0.165786	0.00988995	0.00494497	+	0.999988i	$0.498426\pi$
$282$	0	0
$283$	8.94687	0.531836	0.265918	−	0.963996i	$-0.414325\pi$
$283$	8.94687	0.531836	0.265918	+	0.963996i	$0.414325\pi$
$284$	0	0
$285$	58.3857	3.45847
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0555	−0.885620
$290$	0	0
$291$	16.2046	0.949931
$292$	0	0
$293$	17.2016	1.00493	0.502465	−	0.864597i	$-0.332427\pi$
$293$	17.2016	1.00493	0.502465	+	0.864597i	$0.332427\pi$
$294$	0	0
$295$	−11.8054	−0.687334
$296$	0	0
$297$	2.79981	0.162462
$298$	0	0
$299$	13.5920	0.786048
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−26.8824	−1.54435
$304$	0	0
$305$	24.0029	1.37440
$306$	0	0
$307$	33.1945	1.89451	0.947256	−	0.320478i	$-0.103843\pi$
$307$	33.1945	1.89451	0.947256	+	0.320478i	$0.103843\pi$
$308$	0	0
$309$	52.6373	2.99443
$310$	0	0
$311$	10.9294	0.619748	0.309874	−	0.950778i	$-0.399713\pi$
$311$	10.9294	0.619748	0.309874	+	0.950778i	$0.399713\pi$
$312$	0	0
$313$	−10.5812	−0.598087	−0.299043	−	0.954240i	$-0.596667\pi$
$313$	−10.5812	−0.598087	−0.299043	+	0.954240i	$0.596667\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.9547	−1.73859	−0.869295	−	0.494294i	$-0.835427\pi$
$317$	−30.9547	−1.73859	−0.869295	+	0.494294i	$0.835427\pi$
$318$	0	0
$319$	−0.357484	−0.0200152
$320$	0	0
$321$	−61.2230	−3.41714
$322$	0	0
$323$	10.0873	0.561274
$324$	0	0
$325$	−1.61151	−0.0893904
$326$	0	0
$327$	−43.5242	−2.40690
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.2861	−1.38985	−0.694925	−	0.719082i	$-0.744563\pi$
$331$	−25.2861	−1.38985	−0.694925	+	0.719082i	$0.744563\pi$
$332$	0	0
$333$	−78.1274	−4.28135
$334$	0	0
$335$	11.3530	0.620279
$336$	0	0
$337$	5.87859	0.320227	0.160114	−	0.987099i	$-0.448814\pi$
$337$	5.87859	0.320227	0.160114	+	0.987099i	$0.448814\pi$
$338$	0	0
$339$	−29.8263	−1.61994
$340$	0	0
$341$	0.328530	0.0177909
$342$	0	0
$343$	0	0
$344$	0	0
$345$	37.9399	2.04262
$346$	0	0
$347$	7.35540	0.394858	0.197429	−	0.980317i	$-0.436741\pi$
$347$	7.35540	0.394858	0.197429	+	0.980317i	$0.436741\pi$
$348$	0	0
$349$	−4.99991	−0.267639	−0.133820	−	0.991006i	$-0.542724\pi$
$349$	−4.99991	−0.267639	−0.133820	+	0.991006i	$0.542724\pi$
$350$	0	0
$351$	56.6436	3.02341
$352$	0	0
$353$	8.91487	0.474491	0.237245	−	0.971450i	$-0.423755\pi$
$353$	8.91487	0.474491	0.237245	+	0.971450i	$0.423755\pi$
$354$	0	0
$355$	−25.4391	−1.35017
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.4212	0.919457	0.459728	−	0.888060i	$-0.347947\pi$
$359$	17.4212	0.919457	0.459728	+	0.888060i	$0.347947\pi$
$360$	0	0
$361$	33.3302	1.75422
$362$	0	0
$363$	37.5909	1.97301
$364$	0	0
$365$	−4.87680	−0.255263
$366$	0	0
$367$	0.663350	0.0346266	0.0173133	−	0.999850i	$-0.494489\pi$
$367$	0.663350	0.0346266	0.0173133	+	0.999850i	$0.494489\pi$
$368$	0	0
$369$	−8.72183	−0.454040
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.3767	−0.899733	−0.449866	−	0.893096i	$-0.648528\pi$
$373$	−17.3767	−0.899733	−0.449866	+	0.893096i	$0.648528\pi$
$374$	0	0
$375$	35.8571	1.85165
$376$	0	0
$377$	−7.23233	−0.372484
$378$	0	0
$379$	18.2083	0.935297	0.467648	−	0.883915i	$-0.345101\pi$
$379$	18.2083	0.935297	0.467648	+	0.883915i	$0.345101\pi$
$380$	0	0
$381$	20.1082	1.03017
$382$	0	0
$383$	−2.90905	−0.148645	−0.0743227	−	0.997234i	$-0.523679\pi$
$383$	−2.90905	−0.148645	−0.0743227	+	0.997234i	$0.523679\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	87.7744	4.46182
$388$	0	0
$389$	−18.3040	−0.928049	−0.464024	−	0.885822i	$-0.653595\pi$
$389$	−18.3040	−0.928049	−0.464024	+	0.885822i	$0.653595\pi$
$390$	0	0
$391$	6.55489	0.331495
$392$	0	0
$393$	−38.0707	−1.92041
$394$	0	0
$395$	6.47196	0.325640
$396$	0	0
$397$	−23.3639	−1.17260	−0.586300	−	0.810094i	$-0.699416\pi$
$397$	−23.3639	−1.17260	−0.586300	+	0.810094i	$0.699416\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.02683	0.151153	0.0755764	−	0.997140i	$-0.475920\pi$
$401$	3.02683	0.151153	0.0755764	+	0.997140i	$0.475920\pi$
$402$	0	0
$403$	6.64656	0.331089
$404$	0	0
$405$	96.4289	4.79159
$406$	0	0
$407$	1.28024	0.0634593
$408$	0	0
$409$	−7.53066	−0.372367	−0.186184	−	0.982515i	$-0.559612\pi$
$409$	−7.53066	−0.372367	−0.186184	+	0.982515i	$0.559612\pi$
$410$	0	0
$411$	−40.0975	−1.97786
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.2467	0.993872
$416$	0	0
$417$	−20.9384	−1.02536
$418$	0	0
$419$	40.5782	1.98238	0.991188	−	0.132463i	$-0.0422886\pi$
$419$	40.5782	1.98238	0.991188	+	0.132463i	$0.0422886\pi$
$420$	0	0
$421$	8.44073	0.411376	0.205688	−	0.978618i	$-0.434057\pi$
$421$	8.44073	0.411376	0.205688	+	0.978618i	$0.434057\pi$
$422$	0	0
$423$	−5.03699	−0.244907
$424$	0	0
$425$	−0.777166	−0.0376981
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.41486	−0.0683100
$430$	0	0
$431$	16.4807	0.793845	0.396923	−	0.917852i	$-0.370078\pi$
$431$	16.4807	0.793845	0.396923	+	0.917852i	$0.370078\pi$
$432$	0	0
$433$	−8.61281	−0.413905	−0.206953	−	0.978351i	$-0.566355\pi$
$433$	−8.61281	−0.413905	−0.206953	+	0.978351i	$0.566355\pi$
$434$	0	0
$435$	−20.1879	−0.967933
$436$	0	0
$437$	34.0049	1.62668
$438$	0	0
$439$	−32.4539	−1.54894	−0.774470	−	0.632610i	$-0.781984\pi$
$439$	−32.4539	−1.54894	−0.774470	+	0.632610i	$0.781984\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.8043	1.55858	0.779290	−	0.626664i	$-0.215580\pi$
$443$	32.8043	1.55858	0.779290	+	0.626664i	$0.215580\pi$
$444$	0	0
$445$	26.7752	1.26927
$446$	0	0
$447$	−51.6365	−2.44232
$448$	0	0
$449$	4.28694	0.202313	0.101157	−	0.994871i	$-0.467746\pi$
$449$	4.28694	0.202313	0.101157	+	0.994871i	$0.467746\pi$
$450$	0	0
$451$	0.142921	0.00672990
$452$	0	0
$453$	−52.3661	−2.46037
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.1645	0.475474	0.237737	−	0.971330i	$-0.423594\pi$
$457$	10.1645	0.475474	0.237737	+	0.971330i	$0.423594\pi$
$458$	0	0
$459$	27.3170	1.27505
$460$	0	0
$461$	−26.9754	−1.25637	−0.628184	−	0.778065i	$-0.716201\pi$
$461$	−26.9754	−1.25637	−0.628184	+	0.778065i	$0.716201\pi$
$462$	0	0
$463$	−26.2945	−1.22201	−0.611005	−	0.791627i	$-0.709234\pi$
$463$	−26.2945	−1.22201	−0.611005	+	0.791627i	$0.709234\pi$
$464$	0	0
$465$	18.5528	0.860364
$466$	0	0
$467$	26.2776	1.21598	0.607990	−	0.793945i	$-0.291976\pi$
$467$	26.2776	1.21598	0.607990	+	0.793945i	$0.291976\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	44.5190	2.05133
$472$	0	0
$473$	−1.43833	−0.0661343
$474$	0	0
$475$	−4.03171	−0.184988
$476$	0	0
$477$	−23.3628	−1.06971
$478$	0	0
$479$	10.5456	0.481839	0.240919	−	0.970545i	$-0.422551\pi$
$479$	10.5456	0.481839	0.240919	+	0.970545i	$0.422551\pi$
$480$	0	0
$481$	25.9009	1.18098
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.1577	−0.506644
$486$	0	0
$487$	41.8841	1.89795	0.948974	−	0.315354i	$-0.102123\pi$
$487$	41.8841	1.89795	0.948974	+	0.315354i	$0.102123\pi$
$488$	0	0
$489$	0.863971	0.0390701
$490$	0	0
$491$	−9.42364	−0.425283	−0.212641	−	0.977130i	$-0.568207\pi$
$491$	−9.42364	−0.425283	−0.212641	+	0.977130i	$0.568207\pi$
$492$	0	0
$493$	−3.48786	−0.157085
$494$	0	0
$495$	−2.93858	−0.132079
$496$	0	0
$497$	0	0
$498$	0	0
$499$	39.6921	1.77686	0.888430	−	0.459011i	$-0.151796\pi$
$499$	39.6921	1.77686	0.888430	+	0.459011i	$0.151796\pi$
$500$	0	0
$501$	73.1562	3.26838
$502$	0	0
$503$	−14.7881	−0.659371	−0.329685	−	0.944091i	$-0.606943\pi$
$503$	−14.7881	−0.659371	−0.329685	+	0.944091i	$0.606943\pi$
$504$	0	0
$505$	18.5099	0.823678
$506$	0	0
$507$	15.8840	0.705431
$508$	0	0
$509$	−3.37315	−0.149512	−0.0747561	−	0.997202i	$-0.523818\pi$
$509$	−3.37315	−0.149512	−0.0747561	+	0.997202i	$0.523818\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	141.713	6.25676
$514$	0	0
$515$	−36.2434	−1.59708
$516$	0	0
$517$	0.0825392	0.00363007
$518$	0	0
$519$	−5.42829	−0.238275
$520$	0	0
$521$	34.0110	1.49005	0.745025	−	0.667037i	$-0.232438\pi$
$521$	34.0110	1.49005	0.745025	+	0.667037i	$0.232438\pi$
$522$	0	0
$523$	32.9876	1.44245	0.721223	−	0.692703i	$-0.243580\pi$
$523$	32.9876	1.44245	0.721223	+	0.692703i	$0.243580\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.20537	0.139628
$528$	0	0
$529$	−0.903114	−0.0392658
$530$	0	0
$531$	−43.6771	−1.89542
$532$	0	0
$533$	2.89147	0.125244
$534$	0	0
$535$	42.1551	1.82253
$536$	0	0
$537$	58.0225	2.50386
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.0357	−0.474460	−0.237230	−	0.971453i	$-0.576240\pi$
$541$	−11.0357	−0.474460	−0.237230	+	0.971453i	$0.576240\pi$
$542$	0	0
$543$	−62.9125	−2.69983
$544$	0	0
$545$	29.9686	1.28371
$546$	0	0
$547$	−1.82860	−0.0781851	−0.0390926	−	0.999236i	$-0.512447\pi$
$547$	−1.82860	−0.0781851	−0.0390926	+	0.999236i	$0.512447\pi$
$548$	0	0
$549$	88.8051	3.79011
$550$	0	0
$551$	−18.0940	−0.770832
$552$	0	0
$553$	0	0
$554$	0	0
$555$	72.2981	3.06888
$556$	0	0
$557$	−33.4105	−1.41565	−0.707825	−	0.706388i	$-0.750323\pi$
$557$	−33.4105	−1.41565	−0.707825	+	0.706388i	$0.750323\pi$
$558$	0	0
$559$	−29.0991	−1.23076
$560$	0	0
$561$	−0.682330	−0.0288080
$562$	0	0
$563$	−15.0651	−0.634919	−0.317459	−	0.948272i	$-0.602830\pi$
$563$	−15.0651	−0.634919	−0.317459	+	0.948272i	$0.602830\pi$
$564$	0	0
$565$	20.5369	0.863993
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.0813	−1.63837	−0.819186	−	0.573528i	$-0.805574\pi$
$569$	−39.0813	−1.63837	−0.819186	+	0.573528i	$0.805574\pi$
$570$	0	0
$571$	7.81585	0.327083	0.163542	−	0.986536i	$-0.447708\pi$
$571$	7.81585	0.327083	0.163542	+	0.986536i	$0.447708\pi$
$572$	0	0
$573$	87.4491	3.65324
$574$	0	0
$575$	−2.61987	−0.109256
$576$	0	0
$577$	6.42279	0.267384	0.133692	−	0.991023i	$-0.457317\pi$
$577$	6.42279	0.267384	0.133692	+	0.991023i	$0.457317\pi$
$578$	0	0
$579$	13.7100	0.569770
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.382837	0.0158555
$584$	0	0
$585$	−59.4510	−2.45800
$586$	0	0
$587$	−8.49043	−0.350438	−0.175219	−	0.984530i	$-0.556063\pi$
$587$	−8.49043	−0.350438	−0.175219	+	0.984530i	$0.556063\pi$
$588$	0	0
$589$	16.6285	0.685167
$590$	0	0
$591$	−41.3856	−1.70237
$592$	0	0
$593$	−5.85832	−0.240572	−0.120286	−	0.992739i	$-0.538381\pi$
$593$	−5.85832	−0.240572	−0.120286	+	0.992739i	$0.538381\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	67.0310	2.74340
$598$	0	0
$599$	−21.4262	−0.875450	−0.437725	−	0.899109i	$-0.644216\pi$
$599$	−21.4262	−0.875450	−0.437725	+	0.899109i	$0.644216\pi$
$600$	0	0
$601$	40.7091	1.66056	0.830279	−	0.557347i	$-0.188181\pi$
$601$	40.7091	1.66056	0.830279	+	0.557347i	$0.188181\pi$
$602$	0	0
$603$	42.0033	1.71051
$604$	0	0
$605$	−25.8832	−1.05230
$606$	0	0
$607$	31.5300	1.27976	0.639882	−	0.768474i	$-0.278983\pi$
$607$	31.5300	1.27976	0.639882	+	0.768474i	$0.278983\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.66987	0.0675557
$612$	0	0
$613$	−22.5191	−0.909538	−0.454769	−	0.890609i	$-0.650278\pi$
$613$	−22.5191	−0.909538	−0.454769	+	0.890609i	$0.650278\pi$
$614$	0	0
$615$	8.07106	0.325457
$616$	0	0
$617$	8.08398	0.325449	0.162724	−	0.986672i	$-0.447972\pi$
$617$	8.08398	0.325449	0.162724	+	0.986672i	$0.447972\pi$
$618$	0	0
$619$	17.8816	0.718720	0.359360	−	0.933199i	$-0.382995\pi$
$619$	17.8816	0.718720	0.359360	+	0.933199i	$0.382995\pi$
$620$	0	0
$621$	92.0869	3.69532
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.4760	−1.09904
$626$	0	0
$627$	−3.53973	−0.141363
$628$	0	0
$629$	12.4910	0.498047
$630$	0	0
$631$	−2.38709	−0.0950284	−0.0475142	−	0.998871i	$-0.515130\pi$
$631$	−2.38709	−0.0950284	−0.0475142	+	0.998871i	$0.515130\pi$
$632$	0	0
$633$	−48.0176	−1.90853
$634$	0	0
$635$	−13.8455	−0.549442
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−94.1189	−3.72328
$640$	0	0
$641$	7.11567	0.281052	0.140526	−	0.990077i	$-0.455121\pi$
$641$	7.11567	0.281052	0.140526	+	0.990077i	$0.455121\pi$
$642$	0	0
$643$	−3.43177	−0.135336	−0.0676679	−	0.997708i	$-0.521556\pi$
$643$	−3.43177	−0.135336	−0.0676679	+	0.997708i	$0.521556\pi$
$644$	0	0
$645$	−81.2253	−3.19824
$646$	0	0
$647$	−8.56016	−0.336535	−0.168267	−	0.985741i	$-0.553817\pi$
$647$	−8.56016	−0.336535	−0.168267	+	0.985741i	$0.553817\pi$
$648$	0	0
$649$	0.715719	0.0280944
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.3539	−0.483447	−0.241724	−	0.970345i	$-0.577713\pi$
$653$	−12.3539	−0.483447	−0.241724	+	0.970345i	$0.577713\pi$
$654$	0	0
$655$	26.2136	1.02425
$656$	0	0
$657$	−18.0430	−0.703925
$658$	0	0
$659$	−48.6368	−1.89462	−0.947310	−	0.320320i	$-0.896210\pi$
$659$	−48.6368	−1.89462	−0.947310	+	0.320320i	$0.896210\pi$
$660$	0	0
$661$	12.3667	0.481007	0.240504	−	0.970648i	$-0.422687\pi$
$661$	12.3667	0.481007	0.240504	+	0.970648i	$0.422687\pi$
$662$	0	0
$663$	−13.8044	−0.536117
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.7578	−0.455263
$668$	0	0
$669$	50.5894	1.95590
$670$	0	0
$671$	−1.45521	−0.0561780
$672$	0	0
$673$	−44.1413	−1.70152	−0.850760	−	0.525554i	$-0.823858\pi$
$673$	−44.1413	−1.70152	−0.850760	+	0.525554i	$0.823858\pi$
$674$	0	0
$675$	−10.9181	−0.420237
$676$	0	0
$677$	−10.3534	−0.397912	−0.198956	−	0.980008i	$-0.563755\pi$
$677$	−10.3534	−0.397912	−0.198956	+	0.980008i	$0.563755\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	62.4849	2.39443
$682$	0	0
$683$	−23.0424	−0.881693	−0.440846	−	0.897583i	$-0.645322\pi$
$683$	−23.0424	−0.881693	−0.440846	+	0.897583i	$0.645322\pi$
$684$	0	0
$685$	27.6091	1.05489
$686$	0	0
$687$	20.2964	0.774355
$688$	0	0
$689$	7.74526	0.295071
$690$	0	0
$691$	10.9576	0.416847	0.208423	−	0.978039i	$-0.433167\pi$
$691$	10.9576	0.416847	0.208423	+	0.978039i	$0.433167\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.4172	0.546874
$696$	0	0
$697$	1.39444	0.0528182
$698$	0	0
$699$	−28.7532	−1.08754
$700$	0	0
$701$	3.05919	0.115544	0.0577721	−	0.998330i	$-0.481600\pi$
$701$	3.05919	0.115544	0.0577721	+	0.998330i	$0.481600\pi$
$702$	0	0
$703$	64.7995	2.44396
$704$	0	0
$705$	4.66117	0.175550
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.2009	−0.908884	−0.454442	−	0.890776i	$-0.650161\pi$
$709$	−24.2009	−0.908884	−0.454442	+	0.890776i	$0.650161\pi$
$710$	0	0
$711$	23.9448	0.897998
$712$	0	0
$713$	10.8055	0.404668
$714$	0	0
$715$	0.974201	0.0364331
$716$	0	0
$717$	−93.2368	−3.48199
$718$	0	0
$719$	0.874589	0.0326167	0.0163083	−	0.999867i	$-0.494809\pi$
$719$	0.874589	0.0326167	0.0163083	+	0.999867i	$0.494809\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.555084	−0.0206438
$724$	0	0
$725$	1.39403	0.0517731
$726$	0	0
$727$	21.7602	0.807040	0.403520	−	0.914971i	$-0.367787\pi$
$727$	21.7602	0.807040	0.403520	+	0.914971i	$0.367787\pi$
$728$	0	0
$729$	155.554	5.76124
$730$	0	0
$731$	−14.0333	−0.519041
$732$	0	0
$733$	9.33091	0.344645	0.172323	−	0.985041i	$-0.444873\pi$
$733$	9.33091	0.344645	0.172323	+	0.985041i	$0.444873\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.688293	−0.0253536
$738$	0	0
$739$	35.0351	1.28879	0.644394	−	0.764693i	$-0.277110\pi$
$739$	35.0351	1.28879	0.644394	+	0.764693i	$0.277110\pi$
$740$	0	0
$741$	−71.6131	−2.63077
$742$	0	0
$743$	−30.6618	−1.12487	−0.562436	−	0.826841i	$-0.690136\pi$
$743$	−30.6618	−1.12487	−0.562436	+	0.826841i	$0.690136\pi$
$744$	0	0
$745$	35.5543	1.30261
$746$	0	0
$747$	74.9081	2.74075
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.72070	−0.135770	−0.0678851	−	0.997693i	$-0.521625\pi$
$751$	−3.72070	−0.135770	−0.0678851	+	0.997693i	$0.521625\pi$
$752$	0	0
$753$	−93.1042	−3.39291
$754$	0	0
$755$	36.0567	1.31224
$756$	0	0
$757$	32.8896	1.19539	0.597697	−	0.801722i	$-0.296082\pi$
$757$	32.8896	1.19539	0.597697	+	0.801722i	$0.296082\pi$
$758$	0	0
$759$	−2.30017	−0.0834909
$760$	0	0
$761$	1.06473	0.0385963	0.0192982	−	0.999814i	$-0.493857\pi$
$761$	1.06473	0.0385963	0.0192982	+	0.999814i	$0.493857\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−28.6709	−1.03660
$766$	0	0
$767$	14.4799	0.522838
$768$	0	0
$769$	51.5858	1.86023	0.930116	−	0.367267i	$-0.119706\pi$
$769$	51.5858	1.86023	0.930116	+	0.367267i	$0.119706\pi$
$770$	0	0
$771$	−75.5891	−2.72228
$772$	0	0
$773$	36.3187	1.30629	0.653147	−	0.757231i	$-0.273449\pi$
$773$	36.3187	1.30629	0.653147	+	0.757231i	$0.273449\pi$
$774$	0	0
$775$	−1.28112	−0.0460194
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.23396	0.259183
$780$	0	0
$781$	1.54229	0.0551875
$782$	0	0
$783$	−48.9995	−1.75110
$784$	0	0
$785$	−30.6535	−1.09407
$786$	0	0
$787$	−9.90378	−0.353032	−0.176516	−	0.984298i	$-0.556483\pi$
$787$	−9.90378	−0.353032	−0.176516	+	0.984298i	$0.556483\pi$
$788$	0	0
$789$	−52.6738	−1.87524
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29.4408	−1.04547
$794$	0	0
$795$	21.6196	0.766768
$796$	0	0
$797$	7.09187	0.251207	0.125603	−	0.992081i	$-0.459913\pi$
$797$	7.09187	0.251207	0.125603	+	0.992081i	$0.459913\pi$
$798$	0	0
$799$	0.805312	0.0284899
$800$	0	0
$801$	99.0621	3.50019
$802$	0	0
$803$	0.295664	0.0104338
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−70.9806	−2.49864
$808$	0	0
$809$	37.8692	1.33141	0.665706	−	0.746214i	$-0.268131\pi$
$809$	37.8692	1.33141	0.665706	+	0.746214i	$0.268131\pi$
$810$	0	0
$811$	−27.2005	−0.955140	−0.477570	−	0.878594i	$-0.658482\pi$
$811$	−27.2005	−0.955140	−0.477570	+	0.878594i	$0.658482\pi$
$812$	0	0
$813$	−43.5607	−1.52774
$814$	0	0
$815$	−0.594887	−0.0208380
$816$	0	0
$817$	−72.8009	−2.54698
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.1330	0.388545	0.194272	−	0.980948i	$-0.437765\pi$
$821$	11.1330	0.388545	0.194272	+	0.980948i	$0.437765\pi$
$822$	0	0
$823$	−46.5787	−1.62363	−0.811815	−	0.583915i	$-0.801520\pi$
$823$	−46.5787	−1.62363	−0.811815	+	0.583915i	$0.801520\pi$
$824$	0	0
$825$	0.272714	0.00949469
$826$	0	0
$827$	−22.1049	−0.768662	−0.384331	−	0.923195i	$-0.625568\pi$
$827$	−22.1049	−0.768662	−0.384331	+	0.923195i	$0.625568\pi$
$828$	0	0
$829$	−0.0353550	−0.00122793	−0.000613965	−	1.00000i	$-0.500195\pi$
$829$	−0.0353550	−0.00122793	−0.000613965		1.00000i	$0.500195\pi$
$830$	0	0
$831$	−41.3405	−1.43409
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−50.3717	−1.74318
$836$	0	0
$837$	45.0309	1.55649
$838$	0	0
$839$	19.0702	0.658375	0.329188	−	0.944265i	$-0.393225\pi$
$839$	19.0702	0.658375	0.329188	+	0.944265i	$0.393225\pi$
$840$	0	0
$841$	−22.7437	−0.784265
$842$	0	0
$843$	−0.567603	−0.0195493
$844$	0	0
$845$	−10.9369	−0.376241
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.6315	−1.05127
$850$	0	0
$851$	42.1077	1.44343
$852$	0	0
$853$	33.9594	1.16275	0.581373	−	0.813637i	$-0.302516\pi$
$853$	33.9594	1.16275	0.581373	+	0.813637i	$0.302516\pi$
$854$	0	0
$855$	−148.736	−5.08667
$856$	0	0
$857$	8.30473	0.283684	0.141842	−	0.989889i	$-0.454697\pi$
$857$	8.30473	0.283684	0.141842	+	0.989889i	$0.454697\pi$
$858$	0	0
$859$	47.3641	1.61604	0.808021	−	0.589154i	$-0.200539\pi$
$859$	47.3641	1.61604	0.808021	+	0.589154i	$0.200539\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.83698	0.232733	0.116367	−	0.993206i	$-0.462875\pi$
$863$	6.83698	0.232733	0.116367	+	0.993206i	$0.462875\pi$
$864$	0	0
$865$	3.73765	0.127084
$866$	0	0
$867$	51.5459	1.75059
$868$	0	0
$869$	−0.392373	−0.0133104
$870$	0	0
$871$	−13.9250	−0.471831
$872$	0	0
$873$	−41.2808	−1.39714
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.9935	−1.41802	−0.709011	−	0.705198i	$-0.750858\pi$
$877$	−41.9935	−1.41802	−0.709011	+	0.705198i	$0.750858\pi$
$878$	0	0
$879$	−58.8935	−1.98643
$880$	0	0
$881$	−26.1100	−0.879670	−0.439835	−	0.898079i	$-0.644963\pi$
$881$	−26.1100	−0.879670	−0.439835	+	0.898079i	$0.644963\pi$
$882$	0	0
$883$	−24.5630	−0.826611	−0.413305	−	0.910593i	$-0.635626\pi$
$883$	−24.5630	−0.826611	−0.413305	+	0.910593i	$0.635626\pi$
$884$	0	0
$885$	40.4182	1.35864
$886$	0	0
$887$	48.1527	1.61681	0.808405	−	0.588626i	$-0.200331\pi$
$887$	48.1527	1.61681	0.808405	+	0.588626i	$0.200331\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.84616	−0.195854
$892$	0	0
$893$	4.17772	0.139802
$894$	0	0
$895$	−39.9514	−1.33543
$896$	0	0
$897$	−46.5353	−1.55377
$898$	0	0
$899$	−5.74960	−0.191760
$900$	0	0
$901$	3.73523	0.124439
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43.3184	1.43995
$906$	0	0
$907$	47.0075	1.56086	0.780429	−	0.625244i	$-0.215001\pi$
$907$	47.0075	1.56086	0.780429	+	0.625244i	$0.215001\pi$
$908$	0	0
$909$	68.4822	2.27141
$910$	0	0
$911$	9.28031	0.307470	0.153735	−	0.988112i	$-0.450870\pi$
$911$	9.28031	0.307470	0.153735	+	0.988112i	$0.450870\pi$
$912$	0	0
$913$	−1.22749	−0.0406240
$914$	0	0
$915$	−82.1791	−2.71676
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.9220	0.756126	0.378063	−	0.925780i	$-0.376590\pi$
$919$	22.9220	0.756126	0.378063	+	0.925780i	$0.376590\pi$
$920$	0	0
$921$	−113.649	−3.74485
$922$	0	0
$923$	31.2024	1.02704
$924$	0	0
$925$	−4.99240	−0.164149
$926$	0	0
$927$	−134.092	−4.40416
$928$	0	0
$929$	−6.33924	−0.207984	−0.103992	−	0.994578i	$-0.533162\pi$
$929$	−6.33924	−0.207984	−0.103992	+	0.994578i	$0.533162\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−37.4191	−1.22505
$934$	0	0
$935$	0.469818	0.0153647
$936$	0	0
$937$	−37.6996	−1.23159	−0.615797	−	0.787905i	$-0.711166\pi$
$937$	−37.6996	−1.23159	−0.615797	+	0.787905i	$0.711166\pi$
$938$	0	0
$939$	36.2271	1.18223
$940$	0	0
$941$	4.58289	0.149398	0.0746990	−	0.997206i	$-0.476200\pi$
$941$	4.58289	0.149398	0.0746990	+	0.997206i	$0.476200\pi$
$942$	0	0
$943$	4.70073	0.153077
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.8988	−0.971581	−0.485791	−	0.874075i	$-0.661468\pi$
$947$	−29.8988	−0.971581	−0.485791	+	0.874075i	$0.661468\pi$
$948$	0	0
$949$	5.98164	0.194172
$950$	0	0
$951$	105.980	3.43664
$952$	0	0
$953$	14.5225	0.470429	0.235214	−	0.971944i	$-0.424421\pi$
$953$	14.5225	0.470429	0.235214	+	0.971944i	$0.424421\pi$
$954$	0	0
$955$	−60.2131	−1.94845
$956$	0	0
$957$	1.22392	0.0395638
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.7161	−0.829551
$962$	0	0
$963$	155.964	5.02587
$964$	0	0
$965$	−9.44006	−0.303886
$966$	0	0
$967$	−11.0380	−0.354957	−0.177479	−	0.984125i	$-0.556794\pi$
$967$	−11.0380	−0.354957	−0.177479	+	0.984125i	$0.556794\pi$
$968$	0	0
$969$	−34.5361	−1.10946
$970$	0	0
$971$	−30.7755	−0.987633	−0.493817	−	0.869566i	$-0.664399\pi$
$971$	−30.7755	−0.987633	−0.493817	+	0.869566i	$0.664399\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5.51734	0.176696
$976$	0	0
$977$	−31.3186	−1.00197	−0.500985	−	0.865456i	$-0.667029\pi$
$977$	−31.3186	−1.00197	−0.500985	+	0.865456i	$0.667029\pi$
$978$	0	0
$979$	−1.62329	−0.0518807
$980$	0	0
$981$	110.877	3.54003
$982$	0	0
$983$	−2.27332	−0.0725075	−0.0362538	−	0.999343i	$-0.511542\pi$
$983$	−2.27332	−0.0725075	−0.0362538	+	0.999343i	$0.511542\pi$
$984$	0	0
$985$	28.4960	0.907959
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−47.3071	−1.50428
$990$	0	0
$991$	−1.16054	−0.0368658	−0.0184329	−	0.999830i	$-0.505868\pi$
$991$	−1.16054	−0.0368658	−0.0184329	+	0.999830i	$0.505868\pi$
$992$	0	0
$993$	86.5724	2.74729
$994$	0	0
$995$	−46.1542	−1.46319
$996$	0	0
$997$	1.53162	0.0485069	0.0242534	−	0.999706i	$-0.492279\pi$
$997$	1.53162	0.0485069	0.0242534	+	0.999706i	$0.492279\pi$
$998$	0	0
$999$	175.480	5.55195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.q.1.1		15
7.2	even	3		1148.2.i.e.165.15	✓	30
7.4	even	3		1148.2.i.e.821.15	yes	30
7.6	odd	2		8036.2.a.r.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.15	✓	30	7.2	even	3
1148.2.i.e.821.15	yes	30	7.4	even	3
8036.2.a.q.1.1		15	1.1	even	1	trivial
8036.2.a.r.1.15		15	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.42372 q^{3} +2.35740 q^{5} +8.72183 q^{9} +O(q^{10})\)	\(q-3.42372 q^{3} +2.35740 q^{5} +8.72183 q^{9} -0.142921 q^{11} -2.89147 q^{13} -8.07106 q^{15} -1.39444 q^{17} -7.23396 q^{19} -4.70073 q^{23} +0.557331 q^{25} -19.5899 q^{27} +2.50126 q^{29} -2.29868 q^{31} +0.489322 q^{33} -8.95769 q^{37} +9.89958 q^{39} -1.00000 q^{41} +10.0638 q^{43} +20.5608 q^{45} -0.577516 q^{47} +4.77417 q^{51} -2.67866 q^{53} -0.336922 q^{55} +24.7670 q^{57} -5.00779 q^{59} +10.1819 q^{61} -6.81635 q^{65} +4.81589 q^{67} +16.0940 q^{69} -10.7912 q^{71} -2.06872 q^{73} -1.90814 q^{75} +2.74538 q^{79} +40.9048 q^{81} +8.58858 q^{83} -3.28725 q^{85} -8.56361 q^{87} +11.3580 q^{89} +7.87002 q^{93} -17.0533 q^{95} -4.73305 q^{97} -1.24653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) \) 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

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