LMFDB - Embedded newform 8036.2.a.t.1.16

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			1.16
Root			$2.15119$ 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)

$f(q)$	$=$	$q+2.15119 q^{3} -3.75581 q^{5} +1.62763 q^{9} +O(q^{10})$	$q+2.15119 q^{3} -3.75581 q^{5} +1.62763 q^{9} +4.89580 q^{11} +1.80162 q^{13} -8.07948 q^{15} +4.27358 q^{17} +5.31507 q^{19} +0.795860 q^{23} +9.10613 q^{25} -2.95223 q^{27} +0.466132 q^{29} -2.71538 q^{31} +10.5318 q^{33} -10.1055 q^{37} +3.87562 q^{39} -1.00000 q^{41} -3.98724 q^{43} -6.11308 q^{45} +2.29800 q^{47} +9.19329 q^{51} -1.15028 q^{53} -18.3877 q^{55} +11.4338 q^{57} +6.87226 q^{59} +8.97900 q^{61} -6.76653 q^{65} -3.33631 q^{67} +1.71205 q^{69} +0.810480 q^{71} -2.12535 q^{73} +19.5890 q^{75} -6.00504 q^{79} -11.2337 q^{81} -4.93699 q^{83} -16.0507 q^{85} +1.00274 q^{87} +13.0344 q^{89} -5.84130 q^{93} -19.9624 q^{95} +12.0806 q^{97} +7.96856 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.15119	1.24199	0.620996	−	0.783814i	$-0.286728\pi$
$3$	2.15119	1.24199	0.620996	+	0.783814i	$0.286728\pi$
$4$	0	0
$5$	−3.75581	−1.67965	−0.839825	−	0.542857i	$-0.817343\pi$
$5$	−3.75581	−1.67965	−0.839825	+	0.542857i	$0.817343\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.62763	0.542544
$10$	0	0
$11$	4.89580	1.47614	0.738069	−	0.674725i	$-0.235738\pi$
$11$	4.89580	1.47614	0.738069	+	0.674725i	$0.235738\pi$
$12$	0	0
$13$	1.80162	0.499678	0.249839	−	0.968287i	$-0.419622\pi$
$13$	1.80162	0.499678	0.249839	+	0.968287i	$0.419622\pi$
$14$	0	0
$15$	−8.07948	−2.08611
$16$	0	0
$17$	4.27358	1.03649	0.518247	−	0.855231i	$-0.326585\pi$
$17$	4.27358	1.03649	0.518247	+	0.855231i	$0.326585\pi$
$18$	0	0
$19$	5.31507	1.21936	0.609681	−	0.792647i	$-0.291298\pi$
$19$	5.31507	1.21936	0.609681	+	0.792647i	$0.291298\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.795860	0.165948	0.0829742	−	0.996552i	$-0.473558\pi$
$23$	0.795860	0.165948	0.0829742	+	0.996552i	$0.473558\pi$
$24$	0	0
$25$	9.10613	1.82123
$26$	0	0
$27$	−2.95223	−0.568157
$28$	0	0
$29$	0.466132	0.0865585	0.0432792	−	0.999063i	$-0.486219\pi$
$29$	0.466132	0.0865585	0.0432792	+	0.999063i	$0.486219\pi$
$30$	0	0
$31$	−2.71538	−0.487696	−0.243848	−	0.969813i	$-0.578410\pi$
$31$	−2.71538	−0.487696	−0.243848	+	0.969813i	$0.578410\pi$
$32$	0	0
$33$	10.5318	1.83335
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.1055	−1.66133	−0.830667	−	0.556769i	$-0.812041\pi$
$37$	−10.1055	−1.66133	−0.830667	+	0.556769i	$0.812041\pi$
$38$	0	0
$39$	3.87562	0.620597
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.98724	−0.608048	−0.304024	−	0.952664i	$-0.598330\pi$
$43$	−3.98724	−0.608048	−0.304024	+	0.952664i	$0.598330\pi$
$44$	0	0
$45$	−6.11308	−0.911284
$46$	0	0
$47$	2.29800	0.335197	0.167599	−	0.985855i	$-0.446399\pi$
$47$	2.29800	0.335197	0.167599	+	0.985855i	$0.446399\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	9.19329	1.28732
$52$	0	0
$53$	−1.15028	−0.158003	−0.0790015	−	0.996874i	$-0.525173\pi$
$53$	−1.15028	−0.158003	−0.0790015	+	0.996874i	$0.525173\pi$
$54$	0	0
$55$	−18.3877	−2.47940
$56$	0	0
$57$	11.4338	1.51444
$58$	0	0
$59$	6.87226	0.894692	0.447346	−	0.894361i	$-0.352369\pi$
$59$	6.87226	0.894692	0.447346	+	0.894361i	$0.352369\pi$
$60$	0	0
$61$	8.97900	1.14964	0.574821	−	0.818279i	$-0.305072\pi$
$61$	8.97900	1.14964	0.574821	+	0.818279i	$0.305072\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.76653	−0.839285
$66$	0	0
$67$	−3.33631	−0.407595	−0.203797	−	0.979013i	$-0.565328\pi$
$67$	−3.33631	−0.407595	−0.203797	+	0.979013i	$0.565328\pi$
$68$	0	0
$69$	1.71205	0.206107
$70$	0	0
$71$	0.810480	0.0961863	0.0480931	−	0.998843i	$-0.484686\pi$
$71$	0.810480	0.0961863	0.0480931	+	0.998843i	$0.484686\pi$
$72$	0	0
$73$	−2.12535	−0.248753	−0.124376	−	0.992235i	$-0.539693\pi$
$73$	−2.12535	−0.248753	−0.124376	+	0.992235i	$0.539693\pi$
$74$	0	0
$75$	19.5890	2.26195
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.00504	−0.675619	−0.337810	−	0.941214i	$-0.609686\pi$
$79$	−6.00504	−0.675619	−0.337810	+	0.941214i	$0.609686\pi$
$80$	0	0
$81$	−11.2337	−1.24819
$82$	0	0
$83$	−4.93699	−0.541905	−0.270952	−	0.962593i	$-0.587339\pi$
$83$	−4.93699	−0.541905	−0.270952	+	0.962593i	$0.587339\pi$
$84$	0	0
$85$	−16.0507	−1.74095
$86$	0	0
$87$	1.00274	0.107505
$88$	0	0
$89$	13.0344	1.38165	0.690824	−	0.723023i	$-0.257248\pi$
$89$	13.0344	1.38165	0.690824	+	0.723023i	$0.257248\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.84130	−0.605715
$94$	0	0
$95$	−19.9624	−2.04810
$96$	0	0
$97$	12.0806	1.22660	0.613298	−	0.789852i	$-0.289843\pi$
$97$	12.0806	1.22660	0.613298	+	0.789852i	$0.289843\pi$
$98$	0	0
$99$	7.96856	0.800870
$100$	0	0
$101$	16.5725	1.64903	0.824513	−	0.565843i	$-0.191449\pi$
$101$	16.5725	1.64903	0.824513	+	0.565843i	$0.191449\pi$
$102$	0	0
$103$	3.30586	0.325736	0.162868	−	0.986648i	$-0.447926\pi$
$103$	3.30586	0.325736	0.162868	+	0.986648i	$0.447926\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.64267	0.642171	0.321085	−	0.947050i	$-0.395952\pi$
$107$	6.64267	0.642171	0.321085	+	0.947050i	$0.395952\pi$
$108$	0	0
$109$	−2.65846	−0.254634	−0.127317	−	0.991862i	$-0.540637\pi$
$109$	−2.65846	−0.254634	−0.127317	+	0.991862i	$0.540637\pi$
$110$	0	0
$111$	−21.7389	−2.06336
$112$	0	0
$113$	−4.59350	−0.432120	−0.216060	−	0.976380i	$-0.569321\pi$
$113$	−4.59350	−0.432120	−0.216060	+	0.976380i	$0.569321\pi$
$114$	0	0
$115$	−2.98910	−0.278735
$116$	0	0
$117$	2.93237	0.271098
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.9688	1.17899
$122$	0	0
$123$	−2.15119	−0.193967
$124$	0	0
$125$	−15.4218	−1.37937
$126$	0	0
$127$	15.9912	1.41899	0.709494	−	0.704711i	$-0.248924\pi$
$127$	15.9912	1.41899	0.709494	+	0.704711i	$0.248924\pi$
$128$	0	0
$129$	−8.57731	−0.755190
$130$	0	0
$131$	5.20712	0.454948	0.227474	−	0.973784i	$-0.426953\pi$
$131$	5.20712	0.454948	0.227474	+	0.973784i	$0.426953\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.0880	0.954304
$136$	0	0
$137$	4.66661	0.398696	0.199348	−	0.979929i	$-0.436118\pi$
$137$	4.66661	0.398696	0.199348	+	0.979929i	$0.436118\pi$
$138$	0	0
$139$	6.46110	0.548023	0.274012	−	0.961726i	$-0.411649\pi$
$139$	6.46110	0.548023	0.274012	+	0.961726i	$0.411649\pi$
$140$	0	0
$141$	4.94343	0.416312
$142$	0	0
$143$	8.82035	0.737595
$144$	0	0
$145$	−1.75070	−0.145388
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.41639	−0.607574	−0.303787	−	0.952740i	$-0.598251\pi$
$149$	−7.41639	−0.607574	−0.303787	+	0.952740i	$0.598251\pi$
$150$	0	0
$151$	6.87902	0.559806	0.279903	−	0.960028i	$-0.409698\pi$
$151$	6.87902	0.559806	0.279903	+	0.960028i	$0.409698\pi$
$152$	0	0
$153$	6.95581	0.562344
$154$	0	0
$155$	10.1985	0.819159
$156$	0	0
$157$	−13.9637	−1.11443	−0.557214	−	0.830369i	$-0.688130\pi$
$157$	−13.9637	−1.11443	−0.557214	+	0.830369i	$0.688130\pi$
$158$	0	0
$159$	−2.47447	−0.196239
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.5670	1.68926	0.844628	−	0.535353i	$-0.179822\pi$
$163$	21.5670	1.68926	0.844628	+	0.535353i	$0.179822\pi$
$164$	0	0
$165$	−39.5555	−3.07939
$166$	0	0
$167$	−13.2690	−1.02679	−0.513394	−	0.858153i	$-0.671612\pi$
$167$	−13.2690	−1.02679	−0.513394	+	0.858153i	$0.671612\pi$
$168$	0	0
$169$	−9.75418	−0.750321
$170$	0	0
$171$	8.65099	0.661557
$172$	0	0
$173$	−3.58266	−0.272384	−0.136192	−	0.990682i	$-0.543486\pi$
$173$	−3.58266	−0.272384	−0.136192	+	0.990682i	$0.543486\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.7836	1.11120
$178$	0	0
$179$	−0.682622	−0.0510216	−0.0255108	−	0.999675i	$-0.508121\pi$
$179$	−0.682622	−0.0510216	−0.0255108	+	0.999675i	$0.508121\pi$
$180$	0	0
$181$	6.20632	0.461312	0.230656	−	0.973035i	$-0.425913\pi$
$181$	6.20632	0.461312	0.230656	+	0.973035i	$0.425913\pi$
$182$	0	0
$183$	19.3156	1.42785
$184$	0	0
$185$	37.9544	2.79046
$186$	0	0
$187$	20.9226	1.53001
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.9415	−1.15349	−0.576745	−	0.816924i	$-0.695677\pi$
$191$	−15.9415	−1.15349	−0.576745	+	0.816924i	$0.695677\pi$
$192$	0	0
$193$	19.9740	1.43776	0.718880	−	0.695134i	$-0.244655\pi$
$193$	19.9740	1.43776	0.718880	+	0.695134i	$0.244655\pi$
$194$	0	0
$195$	−14.5561	−1.04239
$196$	0	0
$197$	8.46343	0.602994	0.301497	−	0.953467i	$-0.402514\pi$
$197$	8.46343	0.602994	0.301497	+	0.953467i	$0.402514\pi$
$198$	0	0
$199$	26.7937	1.89935	0.949677	−	0.313231i	$-0.101411\pi$
$199$	26.7937	1.89935	0.949677	+	0.313231i	$0.101411\pi$
$200$	0	0
$201$	−7.17704	−0.506229
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.75581	0.262317
$206$	0	0
$207$	1.29537	0.0900343
$208$	0	0
$209$	26.0215	1.79995
$210$	0	0
$211$	9.86401	0.679066	0.339533	−	0.940594i	$-0.389731\pi$
$211$	9.86401	0.679066	0.339533	+	0.940594i	$0.389731\pi$
$212$	0	0
$213$	1.74350	0.119463
$214$	0	0
$215$	14.9753	1.02131
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.57203	−0.308949
$220$	0	0
$221$	7.69934	0.517914
$222$	0	0
$223$	−28.6007	−1.91524	−0.957622	−	0.288028i	$-0.907000\pi$
$223$	−28.6007	−1.91524	−0.957622	+	0.288028i	$0.907000\pi$
$224$	0	0
$225$	14.8214	0.988095
$226$	0	0
$227$	−15.9504	−1.05867	−0.529333	−	0.848414i	$-0.677558\pi$
$227$	−15.9504	−1.05867	−0.529333	+	0.848414i	$0.677558\pi$
$228$	0	0
$229$	20.0736	1.32650	0.663249	−	0.748399i	$-0.269177\pi$
$229$	20.0736	1.32650	0.663249	+	0.748399i	$0.269177\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.0841	−1.83985	−0.919926	−	0.392093i	$-0.871751\pi$
$233$	−28.0841	−1.83985	−0.919926	+	0.392093i	$0.871751\pi$
$234$	0	0
$235$	−8.63084	−0.563014
$236$	0	0
$237$	−12.9180	−0.839114
$238$	0	0
$239$	4.35261	0.281547	0.140773	−	0.990042i	$-0.455041\pi$
$239$	4.35261	0.281547	0.140773	+	0.990042i	$0.455041\pi$
$240$	0	0
$241$	20.0638	1.29243	0.646213	−	0.763157i	$-0.276352\pi$
$241$	20.0638	1.29243	0.646213	+	0.763157i	$0.276352\pi$
$242$	0	0
$243$	−15.3092	−0.982085
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.57572	0.609289
$248$	0	0
$249$	−10.6204	−0.673041
$250$	0	0
$251$	14.6468	0.924495	0.462248	−	0.886751i	$-0.347043\pi$
$251$	14.6468	0.924495	0.462248	+	0.886751i	$0.347043\pi$
$252$	0	0
$253$	3.89637	0.244963
$254$	0	0
$255$	−34.5283	−2.16224
$256$	0	0
$257$	−31.3870	−1.95787	−0.978935	−	0.204174i	$-0.934549\pi$
$257$	−31.3870	−1.95787	−0.978935	+	0.204174i	$0.934549\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.758691	0.0469618
$262$	0	0
$263$	6.23273	0.384327	0.192163	−	0.981363i	$-0.438450\pi$
$263$	6.23273	0.384327	0.192163	+	0.981363i	$0.438450\pi$
$264$	0	0
$265$	4.32024	0.265390
$266$	0	0
$267$	28.0396	1.71600
$268$	0	0
$269$	−13.7736	−0.839793	−0.419897	−	0.907572i	$-0.637934\pi$
$269$	−13.7736	−0.839793	−0.419897	+	0.907572i	$0.637934\pi$
$270$	0	0
$271$	24.4987	1.48819	0.744096	−	0.668073i	$-0.232880\pi$
$271$	24.4987	1.48819	0.744096	+	0.668073i	$0.232880\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	44.5818	2.68838
$276$	0	0
$277$	28.3534	1.70359	0.851794	−	0.523876i	$-0.175515\pi$
$277$	28.3534	1.70359	0.851794	+	0.523876i	$0.175515\pi$
$278$	0	0
$279$	−4.41964	−0.264597
$280$	0	0
$281$	−16.7300	−0.998026	−0.499013	−	0.866595i	$-0.666304\pi$
$281$	−16.7300	−0.998026	−0.499013	+	0.866595i	$0.666304\pi$
$282$	0	0
$283$	0.592314	0.0352094	0.0176047	−	0.999845i	$-0.494396\pi$
$283$	0.592314	0.0352094	0.0176047	+	0.999845i	$0.494396\pi$
$284$	0	0
$285$	−42.9430	−2.54373
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.26345	0.0743206
$290$	0	0
$291$	25.9876	1.52342
$292$	0	0
$293$	8.47237	0.494961	0.247480	−	0.968893i	$-0.420397\pi$
$293$	8.47237	0.494961	0.247480	+	0.968893i	$0.420397\pi$
$294$	0	0
$295$	−25.8109	−1.50277
$296$	0	0
$297$	−14.4535	−0.838678
$298$	0	0
$299$	1.43384	0.0829208
$300$	0	0
$301$	0	0
$302$	0	0
$303$	35.6507	2.04808
$304$	0	0
$305$	−33.7234	−1.93100
$306$	0	0
$307$	24.7488	1.41249	0.706245	−	0.707967i	$-0.250388\pi$
$307$	24.7488	1.41249	0.706245	+	0.707967i	$0.250388\pi$
$308$	0	0
$309$	7.11154	0.404561
$310$	0	0
$311$	28.6463	1.62438	0.812190	−	0.583393i	$-0.198275\pi$
$311$	28.6463	1.62438	0.812190	+	0.583393i	$0.198275\pi$
$312$	0	0
$313$	−7.17226	−0.405400	−0.202700	−	0.979241i	$-0.564972\pi$
$313$	−7.17226	−0.405400	−0.202700	+	0.979241i	$0.564972\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.8823	−1.79069	−0.895343	−	0.445377i	$-0.853070\pi$
$317$	−31.8823	−1.79069	−0.895343	+	0.445377i	$0.853070\pi$
$318$	0	0
$319$	2.28209	0.127772
$320$	0	0
$321$	14.2897	0.797571
$322$	0	0
$323$	22.7144	1.26386
$324$	0	0
$325$	16.4057	0.910027
$326$	0	0
$327$	−5.71886	−0.316254
$328$	0	0
$329$	0	0
$330$	0	0
$331$	36.1504	1.98700	0.993501	−	0.113819i	$-0.0363084\pi$
$331$	36.1504	1.98700	0.993501	+	0.113819i	$0.0363084\pi$
$332$	0	0
$333$	−16.4480	−0.901347
$334$	0	0
$335$	12.5305	0.684617
$336$	0	0
$337$	−6.03769	−0.328894	−0.164447	−	0.986386i	$-0.552584\pi$
$337$	−6.03769	−0.328894	−0.164447	+	0.986386i	$0.552584\pi$
$338$	0	0
$339$	−9.88150	−0.536689
$340$	0	0
$341$	−13.2939	−0.719907
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.43014	−0.346187
$346$	0	0
$347$	−1.13405	−0.0608790	−0.0304395	−	0.999537i	$-0.509691\pi$
$347$	−1.13405	−0.0608790	−0.0304395	+	0.999537i	$0.509691\pi$
$348$	0	0
$349$	−30.5719	−1.63648	−0.818239	−	0.574878i	$-0.805049\pi$
$349$	−30.5719	−1.63648	−0.818239	+	0.574878i	$0.805049\pi$
$350$	0	0
$351$	−5.31878	−0.283896
$352$	0	0
$353$	−5.97272	−0.317896	−0.158948	−	0.987287i	$-0.550810\pi$
$353$	−5.97272	−0.317896	−0.158948	+	0.987287i	$0.550810\pi$
$354$	0	0
$355$	−3.04401	−0.161559
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.4602	0.974290	0.487145	−	0.873321i	$-0.338038\pi$
$359$	18.4602	0.974290	0.487145	+	0.873321i	$0.338038\pi$
$360$	0	0
$361$	9.25001	0.486843
$362$	0	0
$363$	27.8985	1.46429
$364$	0	0
$365$	7.98240	0.417818
$366$	0	0
$367$	−13.6502	−0.712536	−0.356268	−	0.934384i	$-0.615951\pi$
$367$	−13.6502	−0.712536	−0.356268	+	0.934384i	$0.615951\pi$
$368$	0	0
$369$	−1.62763	−0.0847311
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.5757	0.858255	0.429128	−	0.903244i	$-0.358821\pi$
$373$	16.5757	0.858255	0.429128	+	0.903244i	$0.358821\pi$
$374$	0	0
$375$	−33.1754	−1.71317
$376$	0	0
$377$	0.839790	0.0432514
$378$	0	0
$379$	20.3549	1.04556	0.522781	−	0.852467i	$-0.324895\pi$
$379$	20.3549	1.04556	0.522781	+	0.852467i	$0.324895\pi$
$380$	0	0
$381$	34.4001	1.76237
$382$	0	0
$383$	−1.13617	−0.0580554	−0.0290277	−	0.999579i	$-0.509241\pi$
$383$	−1.13617	−0.0580554	−0.0290277	+	0.999579i	$0.509241\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.48975	−0.329893
$388$	0	0
$389$	−34.8476	−1.76684	−0.883421	−	0.468580i	$-0.844766\pi$
$389$	−34.8476	−1.76684	−0.883421	+	0.468580i	$0.844766\pi$
$390$	0	0
$391$	3.40117	0.172005
$392$	0	0
$393$	11.2015	0.565042
$394$	0	0
$395$	22.5538	1.13480
$396$	0	0
$397$	35.0264	1.75792	0.878962	−	0.476892i	$-0.158237\pi$
$397$	35.0264	1.75792	0.878962	+	0.476892i	$0.158237\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.52844	0.0763268	0.0381634	−	0.999272i	$-0.487849\pi$
$401$	1.52844	0.0763268	0.0381634	+	0.999272i	$0.487849\pi$
$402$	0	0
$403$	−4.89207	−0.243691
$404$	0	0
$405$	42.1917	2.09652
$406$	0	0
$407$	−49.4745	−2.45236
$408$	0	0
$409$	27.0186	1.33599	0.667993	−	0.744168i	$-0.267154\pi$
$409$	27.0186	1.33599	0.667993	+	0.744168i	$0.267154\pi$
$410$	0	0
$411$	10.0388	0.495177
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.5424	0.910211
$416$	0	0
$417$	13.8991	0.680640
$418$	0	0
$419$	30.0427	1.46768	0.733841	−	0.679321i	$-0.237726\pi$
$419$	30.0427	1.46768	0.733841	+	0.679321i	$0.237726\pi$
$420$	0	0
$421$	7.97506	0.388681	0.194340	−	0.980934i	$-0.437743\pi$
$421$	7.97506	0.388681	0.194340	+	0.980934i	$0.437743\pi$
$422$	0	0
$423$	3.74029	0.181859
$424$	0	0
$425$	38.9157	1.88769
$426$	0	0
$427$	0	0
$428$	0	0
$429$	18.9743	0.916087
$430$	0	0
$431$	10.0717	0.485136	0.242568	−	0.970134i	$-0.422010\pi$
$431$	10.0717	0.485136	0.242568	+	0.970134i	$0.422010\pi$
$432$	0	0
$433$	19.6295	0.943332	0.471666	−	0.881777i	$-0.343653\pi$
$433$	19.6295	0.943332	0.471666	+	0.881777i	$0.343653\pi$
$434$	0	0
$435$	−3.76610	−0.180571
$436$	0	0
$437$	4.23006	0.202351
$438$	0	0
$439$	−19.3190	−0.922048	−0.461024	−	0.887388i	$-0.652518\pi$
$439$	−19.3190	−0.922048	−0.461024	+	0.887388i	$0.652518\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.5851	−1.07305	−0.536526	−	0.843884i	$-0.680264\pi$
$443$	−22.5851	−1.07305	−0.536526	+	0.843884i	$0.680264\pi$
$444$	0	0
$445$	−48.9549	−2.32068
$446$	0	0
$447$	−15.9541	−0.754602
$448$	0	0
$449$	24.3860	1.15085	0.575424	−	0.817855i	$-0.304837\pi$
$449$	24.3860	1.15085	0.575424	+	0.817855i	$0.304837\pi$
$450$	0	0
$451$	−4.89580	−0.230534
$452$	0	0
$453$	14.7981	0.695275
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6503	1.19987	0.599935	−	0.800049i	$-0.295193\pi$
$457$	25.6503	1.19987	0.599935	+	0.800049i	$0.295193\pi$
$458$	0	0
$459$	−12.6166	−0.588891
$460$	0	0
$461$	−33.4123	−1.55617	−0.778084	−	0.628161i	$-0.783808\pi$
$461$	−33.4123	−1.55617	−0.778084	+	0.628161i	$0.783808\pi$
$462$	0	0
$463$	12.7774	0.593816	0.296908	−	0.954906i	$-0.404044\pi$
$463$	12.7774	0.593816	0.296908	+	0.954906i	$0.404044\pi$
$464$	0	0
$465$	21.9388	1.01739
$466$	0	0
$467$	29.1333	1.34813	0.674063	−	0.738674i	$-0.264547\pi$
$467$	29.1333	1.34813	0.674063	+	0.738674i	$0.264547\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−30.0387	−1.38411
$472$	0	0
$473$	−19.5207	−0.897563
$474$	0	0
$475$	48.3997	2.22073
$476$	0	0
$477$	−1.87223	−0.0857236
$478$	0	0
$479$	32.5993	1.48950	0.744751	−	0.667343i	$-0.232568\pi$
$479$	32.5993	1.48950	0.744751	+	0.667343i	$0.232568\pi$
$480$	0	0
$481$	−18.2062	−0.830133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−45.3723	−2.06025
$486$	0	0
$487$	−9.16540	−0.415324	−0.207662	−	0.978201i	$-0.566585\pi$
$487$	−9.16540	−0.415324	−0.207662	+	0.978201i	$0.566585\pi$
$488$	0	0
$489$	46.3947	2.09804
$490$	0	0
$491$	10.8127	0.487970	0.243985	−	0.969779i	$-0.421545\pi$
$491$	10.8127	0.487970	0.243985	+	0.969779i	$0.421545\pi$
$492$	0	0
$493$	1.99205	0.0897174
$494$	0	0
$495$	−29.9284	−1.34518
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−41.0860	−1.83926	−0.919632	−	0.392782i	$-0.871513\pi$
$499$	−41.0860	−1.83926	−0.919632	+	0.392782i	$0.871513\pi$
$500$	0	0
$501$	−28.5442	−1.27526
$502$	0	0
$503$	40.6226	1.81127	0.905637	−	0.424054i	$-0.139393\pi$
$503$	40.6226	1.81127	0.905637	+	0.424054i	$0.139393\pi$
$504$	0	0
$505$	−62.2432	−2.76979
$506$	0	0
$507$	−20.9831	−0.931893
$508$	0	0
$509$	−11.0824	−0.491218	−0.245609	−	0.969369i	$-0.578988\pi$
$509$	−11.0824	−0.491218	−0.245609	+	0.969369i	$0.578988\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.6913	−0.692788
$514$	0	0
$515$	−12.4162	−0.547122
$516$	0	0
$517$	11.2505	0.494798
$518$	0	0
$519$	−7.70699	−0.338299
$520$	0	0
$521$	−5.97583	−0.261806	−0.130903	−	0.991395i	$-0.541788\pi$
$521$	−5.97583	−0.261806	−0.130903	+	0.991395i	$0.541788\pi$
$522$	0	0
$523$	35.8554	1.56785	0.783925	−	0.620856i	$-0.213215\pi$
$523$	35.8554	1.56785	0.783925	+	0.620856i	$0.213215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.6044	−0.505494
$528$	0	0
$529$	−22.3666	−0.972461
$530$	0	0
$531$	11.1855	0.485410
$532$	0	0
$533$	−1.80162	−0.0780367
$534$	0	0
$535$	−24.9486	−1.07862
$536$	0	0
$537$	−1.46845	−0.0633684
$538$	0	0
$539$	0	0
$540$	0	0
$541$	43.4848	1.86956	0.934780	−	0.355228i	$-0.115597\pi$
$541$	43.4848	1.86956	0.934780	+	0.355228i	$0.115597\pi$
$542$	0	0
$543$	13.3510	0.572946
$544$	0	0
$545$	9.98468	0.427697
$546$	0	0
$547$	−20.0946	−0.859183	−0.429592	−	0.903023i	$-0.641342\pi$
$547$	−20.0946	−0.859183	−0.429592	+	0.903023i	$0.641342\pi$
$548$	0	0
$549$	14.6145	0.623732
$550$	0	0
$551$	2.47752	0.105546
$552$	0	0
$553$	0	0
$554$	0	0
$555$	81.6472	3.46573
$556$	0	0
$557$	10.9467	0.463826	0.231913	−	0.972736i	$-0.425501\pi$
$557$	10.9467	0.463826	0.231913	+	0.972736i	$0.425501\pi$
$558$	0	0
$559$	−7.18347	−0.303828
$560$	0	0
$561$	45.0085	1.90026
$562$	0	0
$563$	−12.1109	−0.510413	−0.255207	−	0.966887i	$-0.582144\pi$
$563$	−12.1109	−0.510413	−0.255207	+	0.966887i	$0.582144\pi$
$564$	0	0
$565$	17.2523	0.725810
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.9522	−1.33950	−0.669752	−	0.742584i	$-0.733600\pi$
$569$	−31.9522	−1.33950	−0.669752	+	0.742584i	$0.733600\pi$
$570$	0	0
$571$	−18.7625	−0.785185	−0.392593	−	0.919712i	$-0.628422\pi$
$571$	−18.7625	−0.785185	−0.392593	+	0.919712i	$0.628422\pi$
$572$	0	0
$573$	−34.2933	−1.43262
$574$	0	0
$575$	7.24721	0.302229
$576$	0	0
$577$	−2.53985	−0.105736	−0.0528678	−	0.998602i	$-0.516836\pi$
$577$	−2.53985	−0.105736	−0.0528678	+	0.998602i	$0.516836\pi$
$578$	0	0
$579$	42.9679	1.78569
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.63154	−0.233235
$584$	0	0
$585$	−11.0134	−0.455349
$586$	0	0
$587$	−39.3984	−1.62615	−0.813073	−	0.582162i	$-0.802207\pi$
$587$	−39.3984	−1.62615	−0.813073	+	0.582162i	$0.802207\pi$
$588$	0	0
$589$	−14.4324	−0.594678
$590$	0	0
$591$	18.2065	0.748914
$592$	0	0
$593$	19.8084	0.813434	0.406717	−	0.913554i	$-0.366674\pi$
$593$	19.8084	0.813434	0.406717	+	0.913554i	$0.366674\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	57.6384	2.35898
$598$	0	0
$599$	−8.58948	−0.350957	−0.175478	−	0.984483i	$-0.556147\pi$
$599$	−8.58948	−0.350957	−0.175478	+	0.984483i	$0.556147\pi$
$600$	0	0
$601$	23.9096	0.975295	0.487647	−	0.873041i	$-0.337855\pi$
$601$	23.9096	0.975295	0.487647	+	0.873041i	$0.337855\pi$
$602$	0	0
$603$	−5.43028	−0.221138
$604$	0	0
$605$	−48.7086	−1.98028
$606$	0	0
$607$	−37.4089	−1.51838	−0.759190	−	0.650869i	$-0.774405\pi$
$607$	−37.4089	−1.51838	−0.759190	+	0.650869i	$0.774405\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.14011	0.167491
$612$	0	0
$613$	−42.5834	−1.71993	−0.859964	−	0.510355i	$-0.829514\pi$
$613$	−42.5834	−1.71993	−0.859964	+	0.510355i	$0.829514\pi$
$614$	0	0
$615$	8.07948	0.325796
$616$	0	0
$617$	−20.0073	−0.805463	−0.402731	−	0.915318i	$-0.631939\pi$
$617$	−20.0073	−0.805463	−0.402731	+	0.915318i	$0.631939\pi$
$618$	0	0
$619$	−36.3030	−1.45914	−0.729571	−	0.683905i	$-0.760280\pi$
$619$	−36.3030	−1.45914	−0.729571	+	0.683905i	$0.760280\pi$
$620$	0	0
$621$	−2.34956	−0.0942847
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12.3909	0.495637
$626$	0	0
$627$	55.9773	2.23552
$628$	0	0
$629$	−43.1866	−1.72196
$630$	0	0
$631$	38.2166	1.52138	0.760690	−	0.649115i	$-0.224861\pi$
$631$	38.2166	1.52138	0.760690	+	0.649115i	$0.224861\pi$
$632$	0	0
$633$	21.2194	0.843395
$634$	0	0
$635$	−60.0599	−2.38340
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.31916	0.0521853
$640$	0	0
$641$	3.65170	0.144233	0.0721167	−	0.997396i	$-0.477025\pi$
$641$	3.65170	0.144233	0.0721167	+	0.997396i	$0.477025\pi$
$642$	0	0
$643$	−12.9103	−0.509132	−0.254566	−	0.967055i	$-0.581933\pi$
$643$	−12.9103	−0.509132	−0.254566	+	0.967055i	$0.581933\pi$
$644$	0	0
$645$	32.2148	1.26846
$646$	0	0
$647$	35.5774	1.39869	0.699345	−	0.714784i	$-0.253475\pi$
$647$	35.5774	1.39869	0.699345	+	0.714784i	$0.253475\pi$
$648$	0	0
$649$	33.6452	1.32069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.3469	−0.991899	−0.495950	−	0.868351i	$-0.665180\pi$
$653$	−25.3469	−0.991899	−0.495950	+	0.868351i	$0.665180\pi$
$654$	0	0
$655$	−19.5570	−0.764153
$656$	0	0
$657$	−3.45928	−0.134959
$658$	0	0
$659$	−14.6083	−0.569059	−0.284530	−	0.958667i	$-0.591837\pi$
$659$	−14.6083	−0.569059	−0.284530	+	0.958667i	$0.591837\pi$
$660$	0	0
$661$	−23.5466	−0.915855	−0.457928	−	0.888989i	$-0.651408\pi$
$661$	−23.5466	−0.915855	−0.457928	+	0.888989i	$0.651408\pi$
$662$	0	0
$663$	16.5628	0.643245
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.370976	0.0143642
$668$	0	0
$669$	−61.5256	−2.37872
$670$	0	0
$671$	43.9594	1.69703
$672$	0	0
$673$	17.9476	0.691827	0.345914	−	0.938266i	$-0.387569\pi$
$673$	17.9476	0.691827	0.345914	+	0.938266i	$0.387569\pi$
$674$	0	0
$675$	−26.8834	−1.03474
$676$	0	0
$677$	−1.41360	−0.0543291	−0.0271645	−	0.999631i	$-0.508648\pi$
$677$	−1.41360	−0.0543291	−0.0271645	+	0.999631i	$0.508648\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−34.3124	−1.31485
$682$	0	0
$683$	−26.7811	−1.02475	−0.512374	−	0.858762i	$-0.671234\pi$
$683$	−26.7811	−1.02475	−0.512374	+	0.858762i	$0.671234\pi$
$684$	0	0
$685$	−17.5269	−0.669669
$686$	0	0
$687$	43.1821	1.64750
$688$	0	0
$689$	−2.07236	−0.0789507
$690$	0	0
$691$	−30.5061	−1.16050	−0.580252	−	0.814437i	$-0.697046\pi$
$691$	−30.5061	−1.16050	−0.580252	+	0.814437i	$0.697046\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.2667	−0.920487
$696$	0	0
$697$	−4.27358	−0.161873
$698$	0	0
$699$	−60.4143	−2.28508
$700$	0	0
$701$	0.524271	0.0198014	0.00990072	−	0.999951i	$-0.496848\pi$
$701$	0.524271	0.0198014	0.00990072	+	0.999951i	$0.496848\pi$
$702$	0	0
$703$	−53.7115	−2.02577
$704$	0	0
$705$	−18.5666	−0.699259
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−42.2384	−1.58630	−0.793148	−	0.609029i	$-0.791559\pi$
$709$	−42.2384	−1.58630	−0.793148	+	0.609029i	$0.791559\pi$
$710$	0	0
$711$	−9.77399	−0.366553
$712$	0	0
$713$	−2.16106	−0.0809324
$714$	0	0
$715$	−33.1276	−1.23890
$716$	0	0
$717$	9.36330	0.349679
$718$	0	0
$719$	−10.6669	−0.397807	−0.198904	−	0.980019i	$-0.563738\pi$
$719$	−10.6669	−0.397807	−0.198904	+	0.980019i	$0.563738\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	43.1612	1.60518
$724$	0	0
$725$	4.24465	0.157643
$726$	0	0
$727$	−23.1863	−0.859931	−0.429965	−	0.902845i	$-0.641474\pi$
$727$	−23.1863	−0.859931	−0.429965	+	0.902845i	$0.641474\pi$
$728$	0	0
$729$	0.768094	0.0284479
$730$	0	0
$731$	−17.0398	−0.630238
$732$	0	0
$733$	−0.143539	−0.00530173	−0.00265086	−	0.999996i	$-0.500844\pi$
$733$	−0.143539	−0.00530173	−0.00265086	+	0.999996i	$0.500844\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.3339	−0.601667
$738$	0	0
$739$	21.0099	0.772863	0.386431	−	0.922318i	$-0.373708\pi$
$739$	21.0099	0.772863	0.386431	+	0.922318i	$0.373708\pi$
$740$	0	0
$741$	20.5992	0.756732
$742$	0	0
$743$	5.99529	0.219946	0.109973	−	0.993935i	$-0.464924\pi$
$743$	5.99529	0.219946	0.109973	+	0.993935i	$0.464924\pi$
$744$	0	0
$745$	27.8546	1.02051
$746$	0	0
$747$	−8.03560	−0.294007
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.5741	−0.933213	−0.466607	−	0.884465i	$-0.654524\pi$
$751$	−25.5741	−0.933213	−0.466607	+	0.884465i	$0.654524\pi$
$752$	0	0
$753$	31.5080	1.14822
$754$	0	0
$755$	−25.8363	−0.940279
$756$	0	0
$757$	−42.9799	−1.56213	−0.781066	−	0.624449i	$-0.785324\pi$
$757$	−42.9799	−1.56213	−0.781066	+	0.624449i	$0.785324\pi$
$758$	0	0
$759$	8.38185	0.304242
$760$	0	0
$761$	20.1201	0.729352	0.364676	−	0.931134i	$-0.381180\pi$
$761$	20.1201	0.729352	0.364676	+	0.931134i	$0.381180\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−26.1247	−0.944541
$766$	0	0
$767$	12.3812	0.447058
$768$	0	0
$769$	5.03102	0.181423	0.0907116	−	0.995877i	$-0.471086\pi$
$769$	5.03102	0.181423	0.0907116	+	0.995877i	$0.471086\pi$
$770$	0	0
$771$	−67.5196	−2.43166
$772$	0	0
$773$	13.5637	0.487851	0.243925	−	0.969794i	$-0.421565\pi$
$773$	13.5637	0.487851	0.243925	+	0.969794i	$0.421565\pi$
$774$	0	0
$775$	−24.7266	−0.888205
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.31507	−0.190432
$780$	0	0
$781$	3.96795	0.141984
$782$	0	0
$783$	−1.37613	−0.0491788
$784$	0	0
$785$	52.4452	1.87185
$786$	0	0
$787$	9.03110	0.321924	0.160962	−	0.986961i	$-0.448540\pi$
$787$	9.03110	0.321924	0.160962	+	0.986961i	$0.448540\pi$
$788$	0	0
$789$	13.4078	0.477331
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.1767	0.574452
$794$	0	0
$795$	9.29366	0.329612
$796$	0	0
$797$	−5.18087	−0.183516	−0.0917579	−	0.995781i	$-0.529249\pi$
$797$	−5.18087	−0.183516	−0.0917579	+	0.995781i	$0.529249\pi$
$798$	0	0
$799$	9.82066	0.347430
$800$	0	0
$801$	21.2153	0.749605
$802$	0	0
$803$	−10.4053	−0.367194
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29.6297	−1.04302
$808$	0	0
$809$	−44.1674	−1.55284	−0.776421	−	0.630214i	$-0.782967\pi$
$809$	−44.1674	−1.55284	−0.776421	+	0.630214i	$0.782967\pi$
$810$	0	0
$811$	−34.6464	−1.21660	−0.608300	−	0.793707i	$-0.708148\pi$
$811$	−34.6464	−1.21660	−0.608300	+	0.793707i	$0.708148\pi$
$812$	0	0
$813$	52.7015	1.84832
$814$	0	0
$815$	−81.0015	−2.83736
$816$	0	0
$817$	−21.1925	−0.741430
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.0119	−1.36152	−0.680762	−	0.732505i	$-0.738351\pi$
$821$	−39.0119	−1.36152	−0.680762	+	0.732505i	$0.738351\pi$
$822$	0	0
$823$	18.9152	0.659343	0.329672	−	0.944096i	$-0.393062\pi$
$823$	18.9152	0.659343	0.329672	+	0.944096i	$0.393062\pi$
$824$	0	0
$825$	95.9040	3.33895
$826$	0	0
$827$	−18.9323	−0.658339	−0.329169	−	0.944271i	$-0.606769\pi$
$827$	−18.9323	−0.658339	−0.329169	+	0.944271i	$0.606769\pi$
$828$	0	0
$829$	−14.3111	−0.497046	−0.248523	−	0.968626i	$-0.579945\pi$
$829$	−14.3111	−0.497046	−0.248523	+	0.968626i	$0.579945\pi$
$830$	0	0
$831$	60.9936	2.11584
$832$	0	0
$833$	0	0
$834$	0	0
$835$	49.8360	1.72464
$836$	0	0
$837$	8.01642	0.277088
$838$	0	0
$839$	−51.1176	−1.76478	−0.882388	−	0.470522i	$-0.844066\pi$
$839$	−51.1176	−1.76478	−0.882388	+	0.470522i	$0.844066\pi$
$840$	0	0
$841$	−28.7827	−0.992508
$842$	0	0
$843$	−35.9894	−1.23954
$844$	0	0
$845$	36.6349	1.26028
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.27418	0.0437298
$850$	0	0
$851$	−8.04257	−0.275696
$852$	0	0
$853$	49.6010	1.69831	0.849153	−	0.528147i	$-0.177113\pi$
$853$	49.6010	1.69831	0.849153	+	0.528147i	$0.177113\pi$
$854$	0	0
$855$	−32.4915	−1.11119
$856$	0	0
$857$	−51.3659	−1.75463	−0.877313	−	0.479919i	$-0.840666\pi$
$857$	−51.3659	−1.75463	−0.877313	+	0.479919i	$0.840666\pi$
$858$	0	0
$859$	40.1818	1.37099	0.685493	−	0.728080i	$-0.259587\pi$
$859$	40.1818	1.37099	0.685493	+	0.728080i	$0.259587\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.8413	−0.641367	−0.320683	−	0.947186i	$-0.603913\pi$
$863$	−18.8413	−0.641367	−0.320683	+	0.947186i	$0.603913\pi$
$864$	0	0
$865$	13.4558	0.457511
$866$	0	0
$867$	2.71792	0.0923056
$868$	0	0
$869$	−29.3994	−0.997308
$870$	0	0
$871$	−6.01075	−0.203666
$872$	0	0
$873$	19.6627	0.665482
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.08184	−0.205369	−0.102685	−	0.994714i	$-0.532743\pi$
$877$	−6.08184	−0.205369	−0.102685	+	0.994714i	$0.532743\pi$
$878$	0	0
$879$	18.2257	0.614738
$880$	0	0
$881$	−38.8074	−1.30745	−0.653727	−	0.756731i	$-0.726795\pi$
$881$	−38.8074	−1.30745	−0.653727	+	0.756731i	$0.726795\pi$
$882$	0	0
$883$	−43.4610	−1.46258	−0.731289	−	0.682068i	$-0.761081\pi$
$883$	−43.4610	−1.46258	−0.731289	+	0.682068i	$0.761081\pi$
$884$	0	0
$885$	−55.5243	−1.86643
$886$	0	0
$887$	7.92458	0.266081	0.133041	−	0.991111i	$-0.457526\pi$
$887$	7.92458	0.266081	0.133041	+	0.991111i	$0.457526\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−54.9980	−1.84250
$892$	0	0
$893$	12.2140	0.408727
$894$	0	0
$895$	2.56380	0.0856984
$896$	0	0
$897$	3.08446	0.102987
$898$	0	0
$899$	−1.26572	−0.0422143
$900$	0	0
$901$	−4.91581	−0.163769
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.3098	−0.774843
$906$	0	0
$907$	24.7983	0.823415	0.411707	−	0.911316i	$-0.364933\pi$
$907$	24.7983	0.823415	0.411707	+	0.911316i	$0.364933\pi$
$908$	0	0
$909$	26.9739	0.894669
$910$	0	0
$911$	−19.4131	−0.643185	−0.321592	−	0.946878i	$-0.604218\pi$
$911$	−19.4131	−0.643185	−0.321592	+	0.946878i	$0.604218\pi$
$912$	0	0
$913$	−24.1705	−0.799927
$914$	0	0
$915$	−72.5456	−2.39828
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.6399	0.548898	0.274449	−	0.961602i	$-0.411505\pi$
$919$	16.6399	0.548898	0.274449	+	0.961602i	$0.411505\pi$
$920$	0	0
$921$	53.2395	1.75430
$922$	0	0
$923$	1.46017	0.0480622
$924$	0	0
$925$	−92.0220	−3.02566
$926$	0	0
$927$	5.38072	0.176726
$928$	0	0
$929$	40.3751	1.32467	0.662333	−	0.749210i	$-0.269567\pi$
$929$	40.3751	1.32467	0.662333	+	0.749210i	$0.269567\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	61.6236	2.01747
$934$	0	0
$935$	−78.5812	−2.56988
$936$	0	0
$937$	−31.0533	−1.01447	−0.507234	−	0.861809i	$-0.669332\pi$
$937$	−31.0533	−1.01447	−0.507234	+	0.861809i	$0.669332\pi$
$938$	0	0
$939$	−15.4289	−0.503504
$940$	0	0
$941$	38.3305	1.24954	0.624769	−	0.780810i	$-0.285193\pi$
$941$	38.3305	1.24954	0.624769	+	0.780810i	$0.285193\pi$
$942$	0	0
$943$	−0.795860	−0.0259168
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.77100	0.155037	0.0775183	−	0.996991i	$-0.475300\pi$
$947$	4.77100	0.155037	0.0775183	+	0.996991i	$0.475300\pi$
$948$	0	0
$949$	−3.82906	−0.124297
$950$	0	0
$951$	−68.5849	−2.22402
$952$	0	0
$953$	−6.73083	−0.218033	−0.109016	−	0.994040i	$-0.534770\pi$
$953$	−6.73083	−0.218033	−0.109016	+	0.994040i	$0.534770\pi$
$954$	0	0
$955$	59.8734	1.93746
$956$	0	0
$957$	4.90921	0.158692
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.6267	−0.762152
$962$	0	0
$963$	10.8118	0.348406
$964$	0	0
$965$	−75.0186	−2.41493
$966$	0	0
$967$	−41.8582	−1.34607	−0.673034	−	0.739611i	$-0.735009\pi$
$967$	−41.8582	−1.34607	−0.673034	+	0.739611i	$0.735009\pi$
$968$	0	0
$969$	48.8630	1.56971
$970$	0	0
$971$	32.8108	1.05295	0.526475	−	0.850191i	$-0.323513\pi$
$971$	32.8108	1.05295	0.526475	+	0.850191i	$0.323513\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	35.2919	1.13025
$976$	0	0
$977$	20.6724	0.661368	0.330684	−	0.943741i	$-0.392720\pi$
$977$	20.6724	0.661368	0.330684	+	0.943741i	$0.392720\pi$
$978$	0	0
$979$	63.8140	2.03950
$980$	0	0
$981$	−4.32700	−0.138150
$982$	0	0
$983$	−2.71927	−0.0867313	−0.0433657	−	0.999059i	$-0.513808\pi$
$983$	−2.71927	−0.0867313	−0.0433657	+	0.999059i	$0.513808\pi$
$984$	0	0
$985$	−31.7870	−1.01282
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.17328	−0.100905
$990$	0	0
$991$	8.32195	0.264356	0.132178	−	0.991226i	$-0.457803\pi$
$991$	8.32195	0.264356	0.132178	+	0.991226i	$0.457803\pi$
$992$	0	0
$993$	77.7664	2.46784
$994$	0	0
$995$	−100.632	−3.19025
$996$	0	0
$997$	−4.74009	−0.150120	−0.0750600	−	0.997179i	$-0.523915\pi$
$997$	−4.74009	−0.150120	−0.0750600	+	0.997179i	$0.523915\pi$
$998$	0	0
$999$	29.8338	0.943898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.16	yes	20
7.6	odd	2		8036.2.a.s.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.5	✓	20	7.6	odd	2
8036.2.a.t.1.16	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)

\(f(q)\)	\(=\)	\(q+2.15119 q^{3} -3.75581 q^{5} +1.62763 q^{9} +O(q^{10})\)	\(q+2.15119 q^{3} -3.75581 q^{5} +1.62763 q^{9} +4.89580 q^{11} +1.80162 q^{13} -8.07948 q^{15} +4.27358 q^{17} +5.31507 q^{19} +0.795860 q^{23} +9.10613 q^{25} -2.95223 q^{27} +0.466132 q^{29} -2.71538 q^{31} +10.5318 q^{33} -10.1055 q^{37} +3.87562 q^{39} -1.00000 q^{41} -3.98724 q^{43} -6.11308 q^{45} +2.29800 q^{47} +9.19329 q^{51} -1.15028 q^{53} -18.3877 q^{55} +11.4338 q^{57} +6.87226 q^{59} +8.97900 q^{61} -6.76653 q^{65} -3.33631 q^{67} +1.71205 q^{69} +0.810480 q^{71} -2.12535 q^{73} +19.5890 q^{75} -6.00504 q^{79} -11.2337 q^{81} -4.93699 q^{83} -16.0507 q^{85} +1.00274 q^{87} +13.0344 q^{89} -5.84130 q^{93} -19.9624 q^{95} +12.0806 q^{97} +7.96856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 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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