LMFDB - Embedded newform 8036.2.a.s.1.11

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:	$1$
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			1.11
Root			$-0.0954148$ 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)

$f(q)$	$=$	$q+0.0954148 q^{3} +0.581124 q^{5} -2.99090 q^{9} +O(q^{10})$	$q+0.0954148 q^{3} +0.581124 q^{5} -2.99090 q^{9} -4.59295 q^{11} +5.44659 q^{13} +0.0554478 q^{15} +0.983634 q^{17} -0.599838 q^{19} +5.72519 q^{23} -4.66230 q^{25} -0.571620 q^{27} +6.69584 q^{29} -1.57820 q^{31} -0.438235 q^{33} -8.89231 q^{37} +0.519685 q^{39} +1.00000 q^{41} -5.24063 q^{43} -1.73808 q^{45} +7.66590 q^{47} +0.0938533 q^{51} -11.3390 q^{53} -2.66907 q^{55} -0.0572334 q^{57} -7.53338 q^{59} -5.90355 q^{61} +3.16514 q^{65} +2.45712 q^{67} +0.546268 q^{69} -8.01542 q^{71} +15.2262 q^{73} -0.444852 q^{75} +13.2660 q^{79} +8.91815 q^{81} +15.3819 q^{83} +0.571613 q^{85} +0.638882 q^{87} -10.0913 q^{89} -0.150584 q^{93} -0.348580 q^{95} -1.44854 q^{97} +13.7370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0954148	0.0550878	0.0275439	−	0.999621i	$-0.491231\pi$
$3$	0.0954148	0.0550878	0.0275439	+	0.999621i	$0.491231\pi$
$4$	0	0
$5$	0.581124	0.259886	0.129943	−	0.991521i	$-0.458521\pi$
$5$	0.581124	0.259886	0.129943	+	0.991521i	$0.458521\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.99090	−0.996965
$10$	0	0
$11$	−4.59295	−1.38483	−0.692413	−	0.721501i	$-0.743452\pi$
$11$	−4.59295	−1.38483	−0.692413	+	0.721501i	$0.743452\pi$
$12$	0	0
$13$	5.44659	1.51061	0.755306	−	0.655373i	$-0.227488\pi$
$13$	5.44659	1.51061	0.755306	+	0.655373i	$0.227488\pi$
$14$	0	0
$15$	0.0554478	0.0143166
$16$	0	0
$17$	0.983634	0.238566	0.119283	−	0.992860i	$-0.461940\pi$
$17$	0.983634	0.238566	0.119283	+	0.992860i	$0.461940\pi$
$18$	0	0
$19$	−0.599838	−0.137612	−0.0688061	−	0.997630i	$-0.521919\pi$
$19$	−0.599838	−0.137612	−0.0688061	+	0.997630i	$0.521919\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.72519	1.19379	0.596893	−	0.802321i	$-0.296402\pi$
$23$	5.72519	1.19379	0.596893	+	0.802321i	$0.296402\pi$
$24$	0	0
$25$	−4.66230	−0.932459
$26$	0	0
$27$	−0.571620	−0.110008
$28$	0	0
$29$	6.69584	1.24339	0.621693	−	0.783261i	$-0.286445\pi$
$29$	6.69584	1.24339	0.621693	+	0.783261i	$0.286445\pi$
$30$	0	0
$31$	−1.57820	−0.283454	−0.141727	−	0.989906i	$-0.545265\pi$
$31$	−1.57820	−0.283454	−0.141727	+	0.989906i	$0.545265\pi$
$32$	0	0
$33$	−0.438235	−0.0762870
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.89231	−1.46189	−0.730944	−	0.682438i	$-0.760920\pi$
$37$	−8.89231	−1.46189	−0.730944	+	0.682438i	$0.760920\pi$
$38$	0	0
$39$	0.519685	0.0832162
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−5.24063	−0.799188	−0.399594	−	0.916692i	$-0.630849\pi$
$43$	−5.24063	−0.799188	−0.399594	+	0.916692i	$0.630849\pi$
$44$	0	0
$45$	−1.73808	−0.259098
$46$	0	0
$47$	7.66590	1.11819	0.559093	−	0.829105i	$-0.311149\pi$
$47$	7.66590	1.11819	0.559093	+	0.829105i	$0.311149\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.0938533	0.0131421
$52$	0	0
$53$	−11.3390	−1.55753	−0.778767	−	0.627313i	$-0.784155\pi$
$53$	−11.3390	−1.55753	−0.778767	+	0.627313i	$0.784155\pi$
$54$	0	0
$55$	−2.66907	−0.359897
$56$	0	0
$57$	−0.0572334	−0.00758075
$58$	0	0
$59$	−7.53338	−0.980763	−0.490381	−	0.871508i	$-0.663142\pi$
$59$	−7.53338	−0.980763	−0.490381	+	0.871508i	$0.663142\pi$
$60$	0	0
$61$	−5.90355	−0.755872	−0.377936	−	0.925832i	$-0.623366\pi$
$61$	−5.90355	−0.755872	−0.377936	+	0.925832i	$0.623366\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.16514	0.392587
$66$	0	0
$67$	2.45712	0.300184	0.150092	−	0.988672i	$-0.452043\pi$
$67$	2.45712	0.300184	0.150092	+	0.988672i	$0.452043\pi$
$68$	0	0
$69$	0.546268	0.0657630
$70$	0	0
$71$	−8.01542	−0.951255	−0.475628	−	0.879647i	$-0.657779\pi$
$71$	−8.01542	−0.951255	−0.475628	+	0.879647i	$0.657779\pi$
$72$	0	0
$73$	15.2262	1.78209	0.891045	−	0.453915i	$-0.149973\pi$
$73$	15.2262	1.78209	0.891045	+	0.453915i	$0.149973\pi$
$74$	0	0
$75$	−0.444852	−0.0513671
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.2660	1.49254	0.746269	−	0.665644i	$-0.231843\pi$
$79$	13.2660	1.49254	0.746269	+	0.665644i	$0.231843\pi$
$80$	0	0
$81$	8.91815	0.990905
$82$	0	0
$83$	15.3819	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$83$	15.3819	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$84$	0	0
$85$	0.571613	0.0620001
$86$	0	0
$87$	0.638882	0.0684954
$88$	0	0
$89$	−10.0913	−1.06967	−0.534835	−	0.844956i	$-0.679626\pi$
$89$	−10.0913	−1.06967	−0.534835	+	0.844956i	$0.679626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.150584	−0.0156148
$94$	0	0
$95$	−0.348580	−0.0357636
$96$	0	0
$97$	−1.44854	−0.147077	−0.0735383	−	0.997292i	$-0.523429\pi$
$97$	−1.44854	−0.147077	−0.0735383	+	0.997292i	$0.523429\pi$
$98$	0	0
$99$	13.7370	1.38062
$100$	0	0
$101$	−6.29563	−0.626438	−0.313219	−	0.949681i	$-0.601407\pi$
$101$	−6.29563	−0.626438	−0.313219	+	0.949681i	$0.601407\pi$
$102$	0	0
$103$	6.62534	0.652814	0.326407	−	0.945229i	$-0.394162\pi$
$103$	6.62534	0.652814	0.326407	+	0.945229i	$0.394162\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.55352	0.730226	0.365113	−	0.930963i	$-0.381030\pi$
$107$	7.55352	0.730226	0.365113	+	0.930963i	$0.381030\pi$
$108$	0	0
$109$	−11.8097	−1.13116	−0.565581	−	0.824693i	$-0.691348\pi$
$109$	−11.8097	−1.13116	−0.565581	+	0.824693i	$0.691348\pi$
$110$	0	0
$111$	−0.848458	−0.0805321
$112$	0	0
$113$	−14.8380	−1.39584	−0.697921	−	0.716175i	$-0.745891\pi$
$113$	−14.8380	−1.39584	−0.697921	+	0.716175i	$0.745891\pi$
$114$	0	0
$115$	3.32705	0.310249
$116$	0	0
$117$	−16.2902	−1.50603
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.0952	0.917743
$122$	0	0
$123$	0.0954148	0.00860326
$124$	0	0
$125$	−5.61499	−0.502220
$126$	0	0
$127$	0.395980	0.0351375	0.0175688	−	0.999846i	$-0.494407\pi$
$127$	0.395980	0.0351375	0.0175688	+	0.999846i	$0.494407\pi$
$128$	0	0
$129$	−0.500033	−0.0440255
$130$	0	0
$131$	−20.4474	−1.78650	−0.893249	−	0.449562i	$-0.851580\pi$
$131$	−20.4474	−1.78650	−0.893249	+	0.449562i	$0.851580\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.332182	−0.0285897
$136$	0	0
$137$	−16.0149	−1.36825	−0.684123	−	0.729367i	$-0.739815\pi$
$137$	−16.0149	−1.36825	−0.684123	+	0.729367i	$0.739815\pi$
$138$	0	0
$139$	−19.5142	−1.65517	−0.827587	−	0.561338i	$-0.810287\pi$
$139$	−19.5142	−1.65517	−0.827587	+	0.561338i	$0.810287\pi$
$140$	0	0
$141$	0.731440	0.0615984
$142$	0	0
$143$	−25.0159	−2.09193
$144$	0	0
$145$	3.89111	0.323139
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.6265	0.952480	0.476240	−	0.879315i	$-0.341999\pi$
$149$	11.6265	0.952480	0.476240	+	0.879315i	$0.341999\pi$
$150$	0	0
$151$	13.2062	1.07470	0.537352	−	0.843358i	$-0.319425\pi$
$151$	13.2062	1.07470	0.537352	+	0.843358i	$0.319425\pi$
$152$	0	0
$153$	−2.94195	−0.237842
$154$	0	0
$155$	−0.917131	−0.0736657
$156$	0	0
$157$	−5.52975	−0.441322	−0.220661	−	0.975351i	$-0.570822\pi$
$157$	−5.52975	−0.441322	−0.220661	+	0.975351i	$0.570822\pi$
$158$	0	0
$159$	−1.08191	−0.0858010
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.36982	−0.655575	−0.327787	−	0.944752i	$-0.606303\pi$
$163$	−8.36982	−0.655575	−0.327787	+	0.944752i	$0.606303\pi$
$164$	0	0
$165$	−0.254669	−0.0198259
$166$	0	0
$167$	−17.4012	−1.34654	−0.673272	−	0.739395i	$-0.735112\pi$
$167$	−17.4012	−1.34654	−0.673272	+	0.739395i	$0.735112\pi$
$168$	0	0
$169$	16.6653	1.28195
$170$	0	0
$171$	1.79405	0.137195
$172$	0	0
$173$	−21.5364	−1.63738	−0.818691	−	0.574234i	$-0.805300\pi$
$173$	−21.5364	−1.63738	−0.818691	+	0.574234i	$0.805300\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.718796	−0.0540280
$178$	0	0
$179$	11.8043	0.882292	0.441146	−	0.897435i	$-0.354572\pi$
$179$	11.8043	0.882292	0.441146	+	0.897435i	$0.354572\pi$
$180$	0	0
$181$	−20.4162	−1.51753	−0.758764	−	0.651366i	$-0.774196\pi$
$181$	−20.4162	−1.51753	−0.758764	+	0.651366i	$0.774196\pi$
$182$	0	0
$183$	−0.563286	−0.0416393
$184$	0	0
$185$	−5.16753	−0.379925
$186$	0	0
$187$	−4.51778	−0.330373
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.1007	−0.803221	−0.401610	−	0.915811i	$-0.631549\pi$
$191$	−11.1007	−0.803221	−0.401610	+	0.915811i	$0.631549\pi$
$192$	0	0
$193$	−11.9320	−0.858887	−0.429443	−	0.903094i	$-0.641290\pi$
$193$	−11.9320	−0.858887	−0.429443	+	0.903094i	$0.641290\pi$
$194$	0	0
$195$	0.302001	0.0216268
$196$	0	0
$197$	21.0210	1.49768	0.748841	−	0.662750i	$-0.230611\pi$
$197$	21.0210	1.49768	0.748841	+	0.662750i	$0.230611\pi$
$198$	0	0
$199$	15.4861	1.09778	0.548890	−	0.835895i	$-0.315051\pi$
$199$	15.4861	1.09778	0.548890	+	0.835895i	$0.315051\pi$
$200$	0	0
$201$	0.234445	0.0165365
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.581124	0.0405874
$206$	0	0
$207$	−17.1235	−1.19016
$208$	0	0
$209$	2.75503	0.190569
$210$	0	0
$211$	1.51633	0.104389	0.0521943	−	0.998637i	$-0.483378\pi$
$211$	1.51633	0.104389	0.0521943	+	0.998637i	$0.483378\pi$
$212$	0	0
$213$	−0.764789	−0.0524025
$214$	0	0
$215$	−3.04545	−0.207698
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.45280	0.0981714
$220$	0	0
$221$	5.35745	0.360381
$222$	0	0
$223$	−6.28641	−0.420969	−0.210485	−	0.977597i	$-0.567504\pi$
$223$	−6.28641	−0.420969	−0.210485	+	0.977597i	$0.567504\pi$
$224$	0	0
$225$	13.9444	0.929629
$226$	0	0
$227$	0.493741	0.0327707	0.0163854	−	0.999866i	$-0.494784\pi$
$227$	0.493741	0.0327707	0.0163854	+	0.999866i	$0.494784\pi$
$228$	0	0
$229$	−7.56972	−0.500221	−0.250111	−	0.968217i	$-0.580467\pi$
$229$	−7.56972	−0.500221	−0.250111	+	0.968217i	$0.580467\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.34824	0.415887	0.207943	−	0.978141i	$-0.433323\pi$
$233$	6.34824	0.415887	0.207943	+	0.978141i	$0.433323\pi$
$234$	0	0
$235$	4.45483	0.290601
$236$	0	0
$237$	1.26577	0.0822206
$238$	0	0
$239$	6.25165	0.404385	0.202193	−	0.979346i	$-0.435193\pi$
$239$	6.25165	0.404385	0.202193	+	0.979346i	$0.435193\pi$
$240$	0	0
$241$	−24.6273	−1.58638	−0.793192	−	0.608972i	$-0.791582\pi$
$241$	−24.6273	−1.58638	−0.793192	+	0.608972i	$0.791582\pi$
$242$	0	0
$243$	2.56578	0.164595
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.26707	−0.207879
$248$	0	0
$249$	1.46766	0.0930091
$250$	0	0
$251$	27.0165	1.70527	0.852634	−	0.522508i	$-0.175004\pi$
$251$	27.0165	1.70527	0.852634	+	0.522508i	$0.175004\pi$
$252$	0	0
$253$	−26.2955	−1.65319
$254$	0	0
$255$	0.0545403	0.00341545
$256$	0	0
$257$	−18.7011	−1.16654	−0.583272	−	0.812277i	$-0.698228\pi$
$257$	−18.7011	−1.16654	−0.583272	+	0.812277i	$0.698228\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−20.0266	−1.23961
$262$	0	0
$263$	−13.8753	−0.855589	−0.427795	−	0.903876i	$-0.640709\pi$
$263$	−13.8753	−0.855589	−0.427795	+	0.903876i	$0.640709\pi$
$264$	0	0
$265$	−6.58937	−0.404782
$266$	0	0
$267$	−0.962855	−0.0589257
$268$	0	0
$269$	−3.83150	−0.233610	−0.116805	−	0.993155i	$-0.537265\pi$
$269$	−3.83150	−0.233610	−0.116805	+	0.993155i	$0.537265\pi$
$270$	0	0
$271$	−8.34801	−0.507105	−0.253553	−	0.967322i	$-0.581599\pi$
$271$	−8.34801	−0.507105	−0.253553	+	0.967322i	$0.581599\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.4137	1.29129
$276$	0	0
$277$	−26.9207	−1.61751	−0.808755	−	0.588146i	$-0.799858\pi$
$277$	−26.9207	−1.61751	−0.808755	+	0.588146i	$0.799858\pi$
$278$	0	0
$279$	4.72024	0.282593
$280$	0	0
$281$	0.388605	0.0231822	0.0115911	−	0.999933i	$-0.496310\pi$
$281$	0.388605	0.0231822	0.0115911	+	0.999933i	$0.496310\pi$
$282$	0	0
$283$	−24.3878	−1.44970	−0.724851	−	0.688906i	$-0.758091\pi$
$283$	−24.3878	−1.44970	−0.724851	+	0.688906i	$0.758091\pi$
$284$	0	0
$285$	−0.0332597	−0.00197013
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.0325	−0.943086
$290$	0	0
$291$	−0.138212	−0.00810213
$292$	0	0
$293$	−5.78762	−0.338116	−0.169058	−	0.985606i	$-0.554073\pi$
$293$	−5.78762	−0.338116	−0.169058	+	0.985606i	$0.554073\pi$
$294$	0	0
$295$	−4.37782	−0.254887
$296$	0	0
$297$	2.62542	0.152342
$298$	0	0
$299$	31.1828	1.80335
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.600696	−0.0345091
$304$	0	0
$305$	−3.43069	−0.196441
$306$	0	0
$307$	4.26739	0.243553	0.121776	−	0.992558i	$-0.461141\pi$
$307$	4.26739	0.243553	0.121776	+	0.992558i	$0.461141\pi$
$308$	0	0
$309$	0.632155	0.0359621
$310$	0	0
$311$	16.1879	0.917929	0.458965	−	0.888455i	$-0.348220\pi$
$311$	16.1879	0.917929	0.458965	+	0.888455i	$0.348220\pi$
$312$	0	0
$313$	27.0694	1.53005	0.765026	−	0.644000i	$-0.222726\pi$
$313$	27.0694	1.53005	0.765026	+	0.644000i	$0.222726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.39340	0.359089	0.179545	−	0.983750i	$-0.442538\pi$
$317$	6.39340	0.359089	0.179545	+	0.983750i	$0.442538\pi$
$318$	0	0
$319$	−30.7537	−1.72187
$320$	0	0
$321$	0.720717	0.0402265
$322$	0	0
$323$	−0.590021	−0.0328297
$324$	0	0
$325$	−25.3936	−1.40858
$326$	0	0
$327$	−1.12682	−0.0623132
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.0413	1.54129	0.770645	−	0.637265i	$-0.219934\pi$
$331$	28.0413	1.54129	0.770645	+	0.637265i	$0.219934\pi$
$332$	0	0
$333$	26.5960	1.45745
$334$	0	0
$335$	1.42789	0.0780138
$336$	0	0
$337$	−7.07825	−0.385577	−0.192788	−	0.981240i	$-0.561753\pi$
$337$	−7.07825	−0.385577	−0.192788	+	0.981240i	$0.561753\pi$
$338$	0	0
$339$	−1.41576	−0.0768938
$340$	0	0
$341$	7.24860	0.392534
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.317449	0.0170909
$346$	0	0
$347$	−28.8184	−1.54705	−0.773527	−	0.633763i	$-0.781509\pi$
$347$	−28.8184	−1.54705	−0.773527	+	0.633763i	$0.781509\pi$
$348$	0	0
$349$	−12.9849	−0.695068	−0.347534	−	0.937667i	$-0.612981\pi$
$349$	−12.9849	−0.695068	−0.347534	+	0.937667i	$0.612981\pi$
$350$	0	0
$351$	−3.11338	−0.166180
$352$	0	0
$353$	4.77530	0.254163	0.127082	−	0.991892i	$-0.459439\pi$
$353$	4.77530	0.254163	0.127082	+	0.991892i	$0.459439\pi$
$354$	0	0
$355$	−4.65795	−0.247218
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.81936	−0.412690	−0.206345	−	0.978479i	$-0.566157\pi$
$359$	−7.81936	−0.412690	−0.206345	+	0.978479i	$0.566157\pi$
$360$	0	0
$361$	−18.6402	−0.981063
$362$	0	0
$363$	0.963229	0.0505564
$364$	0	0
$365$	8.84830	0.463141
$366$	0	0
$367$	−20.6849	−1.07974	−0.539872	−	0.841747i	$-0.681527\pi$
$367$	−20.6849	−1.07974	−0.539872	+	0.841747i	$0.681527\pi$
$368$	0	0
$369$	−2.99090	−0.155700
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.2162	0.736087	0.368043	−	0.929809i	$-0.380028\pi$
$373$	14.2162	0.736087	0.368043	+	0.929809i	$0.380028\pi$
$374$	0	0
$375$	−0.535753	−0.0276662
$376$	0	0
$377$	36.4695	1.87827
$378$	0	0
$379$	18.8430	0.967898	0.483949	−	0.875096i	$-0.339202\pi$
$379$	18.8430	0.967898	0.483949	+	0.875096i	$0.339202\pi$
$380$	0	0
$381$	0.0377823	0.00193565
$382$	0	0
$383$	−30.4597	−1.55642	−0.778209	−	0.628005i	$-0.783872\pi$
$383$	−30.4597	−1.55642	−0.778209	+	0.628005i	$0.783872\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.6742	0.796763
$388$	0	0
$389$	−19.2411	−0.975562	−0.487781	−	0.872966i	$-0.662194\pi$
$389$	−19.2411	−0.975562	−0.487781	+	0.872966i	$0.662194\pi$
$390$	0	0
$391$	5.63150	0.284797
$392$	0	0
$393$	−1.95098	−0.0984142
$394$	0	0
$395$	7.70917	0.387890
$396$	0	0
$397$	−17.0342	−0.854920	−0.427460	−	0.904034i	$-0.640591\pi$
$397$	−17.0342	−0.854920	−0.427460	+	0.904034i	$0.640591\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8576	0.741953	0.370976	−	0.928642i	$-0.379023\pi$
$401$	14.8576	0.741953	0.370976	+	0.928642i	$0.379023\pi$
$402$	0	0
$403$	−8.59582	−0.428188
$404$	0	0
$405$	5.18255	0.257523
$406$	0	0
$407$	40.8419	2.02446
$408$	0	0
$409$	24.0336	1.18838	0.594191	−	0.804324i	$-0.297472\pi$
$409$	24.0336	1.18838	0.594191	+	0.804324i	$0.297472\pi$
$410$	0	0
$411$	−1.52806	−0.0753736
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.93877	0.438787
$416$	0	0
$417$	−1.86194	−0.0911798
$418$	0	0
$419$	33.2915	1.62640	0.813199	−	0.581986i	$-0.197724\pi$
$419$	33.2915	1.62640	0.813199	+	0.581986i	$0.197724\pi$
$420$	0	0
$421$	36.2319	1.76583	0.882917	−	0.469528i	$-0.155576\pi$
$421$	36.2319	1.76583	0.882917	+	0.469528i	$0.155576\pi$
$422$	0	0
$423$	−22.9279	−1.11479
$424$	0	0
$425$	−4.58599	−0.222453
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.38689	−0.115240
$430$	0	0
$431$	−31.1137	−1.49869	−0.749347	−	0.662178i	$-0.769632\pi$
$431$	−31.1137	−1.49869	−0.749347	+	0.662178i	$0.769632\pi$
$432$	0	0
$433$	7.86622	0.378026	0.189013	−	0.981975i	$-0.439471\pi$
$433$	7.86622	0.378026	0.189013	+	0.981975i	$0.439471\pi$
$434$	0	0
$435$	0.371270	0.0178010
$436$	0	0
$437$	−3.43419	−0.164280
$438$	0	0
$439$	−18.6709	−0.891115	−0.445558	−	0.895253i	$-0.646995\pi$
$439$	−18.6709	−0.891115	−0.445558	+	0.895253i	$0.646995\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.4653	−0.497222	−0.248611	−	0.968603i	$-0.579974\pi$
$443$	−10.4653	−0.497222	−0.248611	+	0.968603i	$0.579974\pi$
$444$	0	0
$445$	−5.86426	−0.277993
$446$	0	0
$447$	1.10934	0.0524700
$448$	0	0
$449$	20.9301	0.987754	0.493877	−	0.869532i	$-0.335579\pi$
$449$	20.9301	0.987754	0.493877	+	0.869532i	$0.335579\pi$
$450$	0	0
$451$	−4.59295	−0.216274
$452$	0	0
$453$	1.26007	0.0592030
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.8341	1.06813	0.534066	−	0.845443i	$-0.320663\pi$
$457$	22.8341	1.06813	0.534066	+	0.845443i	$0.320663\pi$
$458$	0	0
$459$	−0.562265	−0.0262443
$460$	0	0
$461$	−40.8607	−1.90307	−0.951535	−	0.307540i	$-0.900494\pi$
$461$	−40.8607	−1.90307	−0.951535	+	0.307540i	$0.900494\pi$
$462$	0	0
$463$	−17.7860	−0.826587	−0.413294	−	0.910598i	$-0.635622\pi$
$463$	−17.7860	−0.826587	−0.413294	+	0.910598i	$0.635622\pi$
$464$	0	0
$465$	−0.0875078	−0.00405808
$466$	0	0
$467$	20.9022	0.967237	0.483618	−	0.875279i	$-0.339322\pi$
$467$	20.9022	0.967237	0.483618	+	0.875279i	$0.339322\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.527620	−0.0243115
$472$	0	0
$473$	24.0699	1.10674
$474$	0	0
$475$	2.79662	0.128318
$476$	0	0
$477$	33.9138	1.55281
$478$	0	0
$479$	12.7888	0.584336	0.292168	−	0.956367i	$-0.405623\pi$
$479$	12.7888	0.584336	0.292168	+	0.956367i	$0.405623\pi$
$480$	0	0
$481$	−48.4328	−2.20834
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.841779	−0.0382232
$486$	0	0
$487$	−23.5318	−1.06633	−0.533165	−	0.846011i	$-0.678997\pi$
$487$	−23.5318	−1.06633	−0.533165	+	0.846011i	$0.678997\pi$
$488$	0	0
$489$	−0.798605	−0.0361141
$490$	0	0
$491$	−24.3633	−1.09950	−0.549749	−	0.835330i	$-0.685277\pi$
$491$	−24.3633	−1.09950	−0.549749	+	0.835330i	$0.685277\pi$
$492$	0	0
$493$	6.58626	0.296630
$494$	0	0
$495$	7.98291	0.358805
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.7231	−0.614331	−0.307166	−	0.951656i	$-0.599381\pi$
$499$	−13.7231	−0.614331	−0.307166	+	0.951656i	$0.599381\pi$
$500$	0	0
$501$	−1.66033	−0.0741781
$502$	0	0
$503$	−9.60601	−0.428311	−0.214155	−	0.976800i	$-0.568700\pi$
$503$	−9.60601	−0.428311	−0.214155	+	0.976800i	$0.568700\pi$
$504$	0	0
$505$	−3.65854	−0.162803
$506$	0	0
$507$	1.59012	0.0706196
$508$	0	0
$509$	32.6017	1.44505	0.722523	−	0.691347i	$-0.242982\pi$
$509$	32.6017	1.44505	0.722523	+	0.691347i	$0.242982\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.342880	0.0151385
$514$	0	0
$515$	3.85014	0.169657
$516$	0	0
$517$	−35.2091	−1.54849
$518$	0	0
$519$	−2.05489	−0.0901997
$520$	0	0
$521$	4.97451	0.217937	0.108969	−	0.994045i	$-0.465245\pi$
$521$	4.97451	0.217937	0.108969	+	0.994045i	$0.465245\pi$
$522$	0	0
$523$	−31.7275	−1.38735	−0.693674	−	0.720289i	$-0.744009\pi$
$523$	−31.7275	−1.38735	−0.693674	+	0.720289i	$0.744009\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.55237	−0.0676225
$528$	0	0
$529$	9.77785	0.425124
$530$	0	0
$531$	22.5316	0.977786
$532$	0	0
$533$	5.44659	0.235918
$534$	0	0
$535$	4.38953	0.189776
$536$	0	0
$537$	1.12630	0.0486035
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−41.1725	−1.77014	−0.885071	−	0.465456i	$-0.845890\pi$
$541$	−41.1725	−1.77014	−0.885071	+	0.465456i	$0.845890\pi$
$542$	0	0
$543$	−1.94801	−0.0835972
$544$	0	0
$545$	−6.86288	−0.293974
$546$	0	0
$547$	−41.3456	−1.76781	−0.883906	−	0.467665i	$-0.845096\pi$
$547$	−41.3456	−1.76781	−0.883906	+	0.467665i	$0.845096\pi$
$548$	0	0
$549$	17.6569	0.753578
$550$	0	0
$551$	−4.01642	−0.171105
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.493059	−0.0209292
$556$	0	0
$557$	6.17315	0.261565	0.130782	−	0.991411i	$-0.458251\pi$
$557$	6.17315	0.261565	0.130782	+	0.991411i	$0.458251\pi$
$558$	0	0
$559$	−28.5435	−1.20726
$560$	0	0
$561$	−0.431063	−0.0181995
$562$	0	0
$563$	26.7733	1.12836	0.564179	−	0.825652i	$-0.309193\pi$
$563$	26.7733	1.12836	0.564179	+	0.825652i	$0.309193\pi$
$564$	0	0
$565$	−8.62271	−0.362760
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.05626	−0.337736	−0.168868	−	0.985639i	$-0.554011\pi$
$569$	−8.05626	−0.337736	−0.168868	+	0.985639i	$0.554011\pi$
$570$	0	0
$571$	18.7571	0.784961	0.392481	−	0.919760i	$-0.371617\pi$
$571$	18.7571	0.784961	0.392481	+	0.919760i	$0.371617\pi$
$572$	0	0
$573$	−1.05917	−0.0442476
$574$	0	0
$575$	−26.6925	−1.11316
$576$	0	0
$577$	−25.0101	−1.04118	−0.520591	−	0.853806i	$-0.674288\pi$
$577$	−25.0101	−1.04118	−0.520591	+	0.853806i	$0.674288\pi$
$578$	0	0
$579$	−1.13849	−0.0473141
$580$	0	0
$581$	0	0
$582$	0	0
$583$	52.0795	2.15691
$584$	0	0
$585$	−9.46661	−0.391396
$586$	0	0
$587$	36.6357	1.51212	0.756059	−	0.654503i	$-0.227122\pi$
$587$	36.6357	1.51212	0.756059	+	0.654503i	$0.227122\pi$
$588$	0	0
$589$	0.946666	0.0390067
$590$	0	0
$591$	2.00571	0.0825039
$592$	0	0
$593$	14.6459	0.601435	0.300717	−	0.953713i	$-0.402774\pi$
$593$	14.6459	0.601435	0.300717	+	0.953713i	$0.402774\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.47760	0.0604742
$598$	0	0
$599$	−3.88764	−0.158845	−0.0794223	−	0.996841i	$-0.525308\pi$
$599$	−3.88764	−0.158845	−0.0794223	+	0.996841i	$0.525308\pi$
$600$	0	0
$601$	12.5556	0.512154	0.256077	−	0.966656i	$-0.417570\pi$
$601$	12.5556	0.512154	0.256077	+	0.966656i	$0.417570\pi$
$602$	0	0
$603$	−7.34898	−0.299273
$604$	0	0
$605$	5.86654	0.238509
$606$	0	0
$607$	−46.4201	−1.88413	−0.942066	−	0.335427i	$-0.891120\pi$
$607$	−46.4201	−1.88413	−0.942066	+	0.335427i	$0.891120\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	41.7530	1.68914
$612$	0	0
$613$	47.3316	1.91170	0.955852	−	0.293848i	$-0.0949361\pi$
$613$	47.3316	1.91170	0.955852	+	0.293848i	$0.0949361\pi$
$614$	0	0
$615$	0.0554478	0.00223587
$616$	0	0
$617$	−9.61953	−0.387268	−0.193634	−	0.981074i	$-0.562027\pi$
$617$	−9.61953	−0.387268	−0.193634	+	0.981074i	$0.562027\pi$
$618$	0	0
$619$	−26.4825	−1.06442	−0.532211	−	0.846612i	$-0.678639\pi$
$619$	−26.4825	−1.06442	−0.532211	+	0.846612i	$0.678639\pi$
$620$	0	0
$621$	−3.27264	−0.131326
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.0485	0.801939
$626$	0	0
$627$	0.262870	0.0104980
$628$	0	0
$629$	−8.74679	−0.348757
$630$	0	0
$631$	22.5890	0.899256	0.449628	−	0.893216i	$-0.351557\pi$
$631$	22.5890	0.899256	0.449628	+	0.893216i	$0.351557\pi$
$632$	0	0
$633$	0.144680	0.00575053
$634$	0	0
$635$	0.230113	0.00913177
$636$	0	0
$637$	0	0
$638$	0	0
$639$	23.9733	0.948368
$640$	0	0
$641$	−9.40462	−0.371460	−0.185730	−	0.982601i	$-0.559465\pi$
$641$	−9.40462	−0.371460	−0.185730	+	0.982601i	$0.559465\pi$
$642$	0	0
$643$	−27.6967	−1.09225	−0.546126	−	0.837703i	$-0.683898\pi$
$643$	−27.6967	−1.09225	−0.546126	+	0.837703i	$0.683898\pi$
$644$	0	0
$645$	−0.290581	−0.0114416
$646$	0	0
$647$	−3.51502	−0.138190	−0.0690949	−	0.997610i	$-0.522011\pi$
$647$	−3.51502	−0.138190	−0.0690949	+	0.997610i	$0.522011\pi$
$648$	0	0
$649$	34.6004	1.35819
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.7146	0.849758	0.424879	−	0.905250i	$-0.360317\pi$
$653$	21.7146	0.849758	0.424879	+	0.905250i	$0.360317\pi$
$654$	0	0
$655$	−11.8825	−0.464286
$656$	0	0
$657$	−45.5399	−1.77668
$658$	0	0
$659$	25.4011	0.989488	0.494744	−	0.869039i	$-0.335262\pi$
$659$	25.4011	0.989488	0.494744	+	0.869039i	$0.335262\pi$
$660$	0	0
$661$	−14.1958	−0.552154	−0.276077	−	0.961135i	$-0.589035\pi$
$661$	−14.1958	−0.552154	−0.276077	+	0.961135i	$0.589035\pi$
$662$	0	0
$663$	0.511180	0.0198526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	38.3350	1.48434
$668$	0	0
$669$	−0.599816	−0.0231902
$670$	0	0
$671$	27.1147	1.04675
$672$	0	0
$673$	−22.0625	−0.850447	−0.425223	−	0.905088i	$-0.639804\pi$
$673$	−22.0625	−0.850447	−0.425223	+	0.905088i	$0.639804\pi$
$674$	0	0
$675$	2.66506	0.102578
$676$	0	0
$677$	1.95203	0.0750225	0.0375113	−	0.999296i	$-0.488057\pi$
$677$	1.95203	0.0750225	0.0375113	+	0.999296i	$0.488057\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.0471102	0.00180526
$682$	0	0
$683$	34.2307	1.30980	0.654901	−	0.755715i	$-0.272710\pi$
$683$	34.2307	1.30980	0.654901	+	0.755715i	$0.272710\pi$
$684$	0	0
$685$	−9.30664	−0.355588
$686$	0	0
$687$	−0.722263	−0.0275561
$688$	0	0
$689$	−61.7590	−2.35283
$690$	0	0
$691$	15.3176	0.582708	0.291354	−	0.956615i	$-0.405894\pi$
$691$	15.3176	0.582708	0.291354	+	0.956615i	$0.405894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.3402	−0.430157
$696$	0	0
$697$	0.983634	0.0372578
$698$	0	0
$699$	0.605716	0.0229103
$700$	0	0
$701$	−0.218657	−0.00825856	−0.00412928	−	0.999991i	$-0.501314\pi$
$701$	−0.218657	−0.00825856	−0.00412928	+	0.999991i	$0.501314\pi$
$702$	0	0
$703$	5.33395	0.201174
$704$	0	0
$705$	0.425057	0.0160086
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−47.6398	−1.78915	−0.894576	−	0.446916i	$-0.852522\pi$
$709$	−47.6398	−1.78915	−0.894576	+	0.446916i	$0.852522\pi$
$710$	0	0
$711$	−39.6771	−1.48801
$712$	0	0
$713$	−9.03552	−0.338383
$714$	0	0
$715$	−14.5373	−0.543665
$716$	0	0
$717$	0.596499	0.0222767
$718$	0	0
$719$	44.2096	1.64874	0.824371	−	0.566050i	$-0.191529\pi$
$719$	44.2096	1.64874	0.824371	+	0.566050i	$0.191529\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.34981	−0.0873903
$724$	0	0
$725$	−31.2180	−1.15941
$726$	0	0
$727$	40.9714	1.51955	0.759773	−	0.650189i	$-0.225310\pi$
$727$	40.9714	1.51955	0.759773	+	0.650189i	$0.225310\pi$
$728$	0	0
$729$	−26.5096	−0.981838
$730$	0	0
$731$	−5.15486	−0.190659
$732$	0	0
$733$	−49.0440	−1.81148	−0.905741	−	0.423831i	$-0.860685\pi$
$733$	−49.0440	−1.81148	−0.905741	+	0.423831i	$0.860685\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.2854	−0.415703
$738$	0	0
$739$	45.0783	1.65823	0.829116	−	0.559077i	$-0.188844\pi$
$739$	45.0783	1.65823	0.829116	+	0.559077i	$0.188844\pi$
$740$	0	0
$741$	−0.311727	−0.0114516
$742$	0	0
$743$	0.752868	0.0276200	0.0138100	−	0.999905i	$-0.495604\pi$
$743$	0.752868	0.0276200	0.0138100	+	0.999905i	$0.495604\pi$
$744$	0	0
$745$	6.75643	0.247536
$746$	0	0
$747$	−46.0056	−1.68326
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3.88864	0.141898	0.0709492	−	0.997480i	$-0.477397\pi$
$751$	3.88864	0.141898	0.0709492	+	0.997480i	$0.477397\pi$
$752$	0	0
$753$	2.57778	0.0939394
$754$	0	0
$755$	7.67442	0.279301
$756$	0	0
$757$	−1.51890	−0.0552054	−0.0276027	−	0.999619i	$-0.508787\pi$
$757$	−1.51890	−0.0552054	−0.0276027	+	0.999619i	$0.508787\pi$
$758$	0	0
$759$	−2.50898	−0.0910703
$760$	0	0
$761$	−4.91919	−0.178321	−0.0891603	−	0.996017i	$-0.528418\pi$
$761$	−4.91919	−0.178321	−0.0891603	+	0.996017i	$0.528418\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.70964	−0.0618120
$766$	0	0
$767$	−41.0312	−1.48155
$768$	0	0
$769$	−42.8410	−1.54489	−0.772443	−	0.635084i	$-0.780965\pi$
$769$	−42.8410	−1.54489	−0.772443	+	0.635084i	$0.780965\pi$
$770$	0	0
$771$	−1.78436	−0.0642622
$772$	0	0
$773$	−14.7391	−0.530128	−0.265064	−	0.964231i	$-0.585393\pi$
$773$	−14.7391	−0.530128	−0.265064	+	0.964231i	$0.585393\pi$
$774$	0	0
$775$	7.35805	0.264309
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.599838	−0.0214914
$780$	0	0
$781$	36.8144	1.31732
$782$	0	0
$783$	−3.82748	−0.136783
$784$	0	0
$785$	−3.21347	−0.114694
$786$	0	0
$787$	0.810739	0.0288997	0.0144499	−	0.999896i	$-0.495400\pi$
$787$	0.810739	0.0288997	0.0144499	+	0.999896i	$0.495400\pi$
$788$	0	0
$789$	−1.32391	−0.0471325
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−32.1542	−1.14183
$794$	0	0
$795$	−0.628723	−0.0222985
$796$	0	0
$797$	24.7544	0.876844	0.438422	−	0.898769i	$-0.355537\pi$
$797$	24.7544	0.876844	0.438422	+	0.898769i	$0.355537\pi$
$798$	0	0
$799$	7.54044	0.266762
$800$	0	0
$801$	30.1819	1.06642
$802$	0	0
$803$	−69.9331	−2.46789
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.365581	−0.0128691
$808$	0	0
$809$	37.5304	1.31950	0.659750	−	0.751485i	$-0.270662\pi$
$809$	37.5304	1.31950	0.659750	+	0.751485i	$0.270662\pi$
$810$	0	0
$811$	−43.5412	−1.52894	−0.764470	−	0.644660i	$-0.776999\pi$
$811$	−43.5412	−1.52894	−0.764470	+	0.644660i	$0.776999\pi$
$812$	0	0
$813$	−0.796523	−0.0279353
$814$	0	0
$815$	−4.86390	−0.170375
$816$	0	0
$817$	3.14353	0.109978
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.9523	−0.556739	−0.278370	−	0.960474i	$-0.589794\pi$
$821$	−15.9523	−0.556739	−0.278370	+	0.960474i	$0.589794\pi$
$822$	0	0
$823$	−53.2301	−1.85549	−0.927743	−	0.373220i	$-0.878254\pi$
$823$	−53.2301	−1.85549	−0.927743	+	0.373220i	$0.878254\pi$
$824$	0	0
$825$	2.04318	0.0711345
$826$	0	0
$827$	31.9051	1.10945	0.554724	−	0.832034i	$-0.312824\pi$
$827$	31.9051	1.10945	0.554724	+	0.832034i	$0.312824\pi$
$828$	0	0
$829$	−2.57219	−0.0893357	−0.0446679	−	0.999002i	$-0.514223\pi$
$829$	−2.57219	−0.0893357	−0.0446679	+	0.999002i	$0.514223\pi$
$830$	0	0
$831$	−2.56864	−0.0891050
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10.1122	−0.349948
$836$	0	0
$837$	0.902132	0.0311823
$838$	0	0
$839$	10.5911	0.365644	0.182822	−	0.983146i	$-0.441477\pi$
$839$	10.5911	0.365644	0.182822	+	0.983146i	$0.441477\pi$
$840$	0	0
$841$	15.8343	0.546010
$842$	0	0
$843$	0.0370787	0.00127706
$844$	0	0
$845$	9.68461	0.333161
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.32695	−0.0798608
$850$	0	0
$851$	−50.9102	−1.74518
$852$	0	0
$853$	28.8643	0.988296	0.494148	−	0.869378i	$-0.335480\pi$
$853$	28.8643	0.988296	0.494148	+	0.869378i	$0.335480\pi$
$854$	0	0
$855$	1.04257	0.0356550
$856$	0	0
$857$	21.0012	0.717386	0.358693	−	0.933456i	$-0.383222\pi$
$857$	21.0012	0.717386	0.358693	+	0.933456i	$0.383222\pi$
$858$	0	0
$859$	20.3172	0.693214	0.346607	−	0.938010i	$-0.387334\pi$
$859$	20.3172	0.693214	0.346607	+	0.938010i	$0.387334\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.50558	0.187412	0.0937061	−	0.995600i	$-0.470129\pi$
$863$	5.50558	0.187412	0.0937061	+	0.995600i	$0.470129\pi$
$864$	0	0
$865$	−12.5153	−0.425533
$866$	0	0
$867$	−1.52973	−0.0519525
$868$	0	0
$869$	−60.9299	−2.06691
$870$	0	0
$871$	13.3829	0.453462
$872$	0	0
$873$	4.33243	0.146630
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.96003	0.336326	0.168163	−	0.985759i	$-0.446216\pi$
$877$	9.96003	0.336326	0.168163	+	0.985759i	$0.446216\pi$
$878$	0	0
$879$	−0.552224	−0.0186261
$880$	0	0
$881$	−30.1577	−1.01604	−0.508019	−	0.861346i	$-0.669622\pi$
$881$	−30.1577	−1.01604	−0.508019	+	0.861346i	$0.669622\pi$
$882$	0	0
$883$	−30.2088	−1.01661	−0.508304	−	0.861178i	$-0.669727\pi$
$883$	−30.2088	−1.01661	−0.508304	+	0.861178i	$0.669727\pi$
$884$	0	0
$885$	−0.417709	−0.0140411
$886$	0	0
$887$	12.4675	0.418616	0.209308	−	0.977850i	$-0.432879\pi$
$887$	12.4675	0.418616	0.209308	+	0.977850i	$0.432879\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−40.9606	−1.37223
$892$	0	0
$893$	−4.59830	−0.153876
$894$	0	0
$895$	6.85974	0.229296
$896$	0	0
$897$	2.97530	0.0993423
$898$	0	0
$899$	−10.5674	−0.352442
$900$	0	0
$901$	−11.1534	−0.371575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.8644	−0.394385
$906$	0	0
$907$	28.5295	0.947306	0.473653	−	0.880712i	$-0.342935\pi$
$907$	28.5295	0.947306	0.473653	+	0.880712i	$0.342935\pi$
$908$	0	0
$909$	18.8296	0.624537
$910$	0	0
$911$	28.1381	0.932258	0.466129	−	0.884717i	$-0.345648\pi$
$911$	28.1381	0.932258	0.466129	+	0.884717i	$0.345648\pi$
$912$	0	0
$913$	−70.6482	−2.33811
$914$	0	0
$915$	−0.327339	−0.0108215
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.4022	−0.640021	−0.320011	−	0.947414i	$-0.603686\pi$
$919$	−19.4022	−0.640021	−0.320011	+	0.947414i	$0.603686\pi$
$920$	0	0
$921$	0.407172	0.0134168
$922$	0	0
$923$	−43.6567	−1.43698
$924$	0	0
$925$	41.4586	1.36315
$926$	0	0
$927$	−19.8157	−0.650833
$928$	0	0
$929$	4.26837	0.140041	0.0700203	−	0.997546i	$-0.477694\pi$
$929$	4.26837	0.140041	0.0700203	+	0.997546i	$0.477694\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.54456	0.0505667
$934$	0	0
$935$	−2.62539	−0.0858594
$936$	0	0
$937$	31.3745	1.02496	0.512481	−	0.858699i	$-0.328727\pi$
$937$	31.3745	1.02496	0.512481	+	0.858699i	$0.328727\pi$
$938$	0	0
$939$	2.58282	0.0842871
$940$	0	0
$941$	34.2915	1.11787	0.558935	−	0.829212i	$-0.311210\pi$
$941$	34.2915	1.11787	0.558935	+	0.829212i	$0.311210\pi$
$942$	0	0
$943$	5.72519	0.186438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.3857	−0.987403	−0.493701	−	0.869632i	$-0.664356\pi$
$947$	−30.3857	−0.987403	−0.493701	+	0.869632i	$0.664356\pi$
$948$	0	0
$949$	82.9308	2.69205
$950$	0	0
$951$	0.610025	0.0197814
$952$	0	0
$953$	−30.1336	−0.976124	−0.488062	−	0.872809i	$-0.662296\pi$
$953$	−30.1336	−0.976124	−0.488062	+	0.872809i	$0.662296\pi$
$954$	0	0
$955$	−6.45090	−0.208746
$956$	0	0
$957$	−2.93435	−0.0948542
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.5093	−0.919654
$962$	0	0
$963$	−22.5918	−0.728010
$964$	0	0
$965$	−6.93399	−0.223213
$966$	0	0
$967$	6.17238	0.198490	0.0992451	−	0.995063i	$-0.468357\pi$
$967$	6.17238	0.198490	0.0992451	+	0.995063i	$0.468357\pi$
$968$	0	0
$969$	−0.0562968	−0.00180851
$970$	0	0
$971$	50.8413	1.63157	0.815787	−	0.578353i	$-0.196304\pi$
$971$	50.8413	1.63157	0.815787	+	0.578353i	$0.196304\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.42292	−0.0775957
$976$	0	0
$977$	31.6934	1.01396	0.506981	−	0.861957i	$-0.330761\pi$
$977$	31.6934	1.01396	0.506981	+	0.861957i	$0.330761\pi$
$978$	0	0
$979$	46.3486	1.48131
$980$	0	0
$981$	35.3215	1.12773
$982$	0	0
$983$	0.0998845	0.00318582	0.00159291	−	0.999999i	$-0.499493\pi$
$983$	0.0998845	0.00318582	0.00159291	+	0.999999i	$0.499493\pi$
$984$	0	0
$985$	12.2158	0.389227
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.0036	−0.954059
$990$	0	0
$991$	−7.29957	−0.231879	−0.115939	−	0.993256i	$-0.536988\pi$
$991$	−7.29957	−0.231879	−0.115939	+	0.993256i	$0.536988\pi$
$992$	0	0
$993$	2.67555	0.0849062
$994$	0	0
$995$	8.99933	0.285298
$996$	0	0
$997$	−16.3456	−0.517669	−0.258834	−	0.965922i	$-0.583338\pi$
$997$	−16.3456	−0.517669	−0.258834	+	0.965922i	$0.583338\pi$
$998$	0	0
$999$	5.08303	0.160820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.11	✓	20
7.6	odd	2		8036.2.a.t.1.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.11	✓	20	1.1	even	1	trivial
8036.2.a.t.1.10	yes	20	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)

\(f(q)\)	\(=\)	\(q+0.0954148 q^{3} +0.581124 q^{5} -2.99090 q^{9} +O(q^{10})\)	\(q+0.0954148 q^{3} +0.581124 q^{5} -2.99090 q^{9} -4.59295 q^{11} +5.44659 q^{13} +0.0554478 q^{15} +0.983634 q^{17} -0.599838 q^{19} +5.72519 q^{23} -4.66230 q^{25} -0.571620 q^{27} +6.69584 q^{29} -1.57820 q^{31} -0.438235 q^{33} -8.89231 q^{37} +0.519685 q^{39} +1.00000 q^{41} -5.24063 q^{43} -1.73808 q^{45} +7.66590 q^{47} +0.0938533 q^{51} -11.3390 q^{53} -2.66907 q^{55} -0.0572334 q^{57} -7.53338 q^{59} -5.90355 q^{61} +3.16514 q^{65} +2.45712 q^{67} +0.546268 q^{69} -8.01542 q^{71} +15.2262 q^{73} -0.444852 q^{75} +13.2660 q^{79} +8.91815 q^{81} +15.3819 q^{83} +0.571613 q^{85} +0.638882 q^{87} -10.0913 q^{89} -0.150584 q^{93} -0.348580 q^{95} -1.44854 q^{97} +13.7370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 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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