LMFDB - Embedded newform 7220.2.a.w.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7220,2,Mod(1,7220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7220, 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("7220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7220 = 2^{2} \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7220.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:	$57.6519902594$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 12x^{7} + 33x^{6} + 51x^{5} - 117x^{4} - 86x^{3} + 162x^{2} + 45x - 73 $ x^9 - 3x^8 - 12x^7 + 33x^6 + 51x^5 - 117x^4 - 86x^3 + 162x^2 + 45x - 73
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.67739$ of defining polynomial
Character	$\chi$	$=$	7220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67739 q^{3} +1.00000 q^{5} -0.461272 q^{7} -0.186377 q^{9} +O(q^{10})$	$q+1.67739 q^{3} +1.00000 q^{5} -0.461272 q^{7} -0.186377 q^{9} +1.28436 q^{11} -5.94694 q^{13} +1.67739 q^{15} +5.56326 q^{17} -0.773731 q^{21} +2.18815 q^{23} +1.00000 q^{25} -5.34478 q^{27} -9.55254 q^{29} -9.23857 q^{31} +2.15436 q^{33} -0.461272 q^{35} +8.90443 q^{37} -9.97531 q^{39} -4.82173 q^{41} -3.77632 q^{43} -0.186377 q^{45} -6.85092 q^{47} -6.78723 q^{49} +9.33174 q^{51} -7.90836 q^{53} +1.28436 q^{55} -5.35984 q^{59} -10.1384 q^{61} +0.0859705 q^{63} -5.94694 q^{65} +6.60140 q^{67} +3.67036 q^{69} +6.89386 q^{71} +13.0905 q^{73} +1.67739 q^{75} -0.592438 q^{77} +1.38838 q^{79} -8.40613 q^{81} +0.132099 q^{83} +5.56326 q^{85} -16.0233 q^{87} -3.08032 q^{89} +2.74316 q^{91} -15.4966 q^{93} +0.584577 q^{97} -0.239375 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + 9 q^{5} + 6 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + 9 * q^5 + 6 * q^9	$ 9 q - 3 q^{3} + 9 q^{5} + 6 q^{9} - 15 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 18 q^{27} - 15 q^{29} - 6 q^{31} - 9 q^{33} - 18 q^{37} - 6 q^{39} - 18 q^{41} - 6 q^{43} + 6 q^{45} - 15 q^{47} + 3 q^{49} + 18 q^{51} - 33 q^{53} + 3 q^{59} - 15 q^{63} - 15 q^{65} - 12 q^{67} - 27 q^{69} - 6 q^{71} + 3 q^{73} - 3 q^{75} - 18 q^{77} - 15 q^{79} - 15 q^{81} + 6 q^{83} + 3 q^{85} - 21 q^{87} - 3 q^{89} - 33 q^{91} - 9 q^{93} - 45 q^{97} + 42 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + 9 * q^5 + 6 * q^9 - 15 * q^13 - 3 * q^15 + 3 * q^17 - 12 * q^21 + 6 * q^23 + 9 * q^25 - 18 * q^27 - 15 * q^29 - 6 * q^31 - 9 * q^33 - 18 * q^37 - 6 * q^39 - 18 * q^41 - 6 * q^43 + 6 * q^45 - 15 * q^47 + 3 * q^49 + 18 * q^51 - 33 * q^53 + 3 * q^59 - 15 * q^63 - 15 * q^65 - 12 * q^67 - 27 * q^69 - 6 * q^71 + 3 * q^73 - 3 * q^75 - 18 * q^77 - 15 * q^79 - 15 * q^81 + 6 * q^83 + 3 * q^85 - 21 * q^87 - 3 * q^89 - 33 * q^91 - 9 * q^93 - 45 * q^97 + 42 * 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
$3$	1.67739	0.968439	0.484220	−	0.874947i	$-0.339104\pi$
$3$	1.67739	0.968439	0.484220	+	0.874947i	$0.339104\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.461272	−0.174344	−0.0871722	−	0.996193i	$-0.527783\pi$
$7$	−0.461272	−0.174344	−0.0871722	+	0.996193i	$0.527783\pi$
$8$	0	0
$9$	−0.186377	−0.0621257
$10$	0	0
$11$	1.28436	0.387248	0.193624	−	0.981076i	$-0.437976\pi$
$11$	1.28436	0.387248	0.193624	+	0.981076i	$0.437976\pi$
$12$	0	0
$13$	−5.94694	−1.64938	−0.824692	−	0.565582i	$-0.808651\pi$
$13$	−5.94694	−1.64938	−0.824692	+	0.565582i	$0.808651\pi$
$14$	0	0
$15$	1.67739	0.433099
$16$	0	0
$17$	5.56326	1.34929	0.674645	−	0.738142i	$-0.264297\pi$
$17$	5.56326	1.34929	0.674645	+	0.738142i	$0.264297\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.773731	−0.168842
$22$	0	0
$23$	2.18815	0.456260	0.228130	−	0.973631i	$-0.426739\pi$
$23$	2.18815	0.456260	0.228130	+	0.973631i	$0.426739\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.34478	−1.02860
$28$	0	0
$29$	−9.55254	−1.77386	−0.886931	−	0.461901i	$-0.847167\pi$
$29$	−9.55254	−1.77386	−0.886931	+	0.461901i	$0.847167\pi$
$30$	0	0
$31$	−9.23857	−1.65930	−0.829648	−	0.558286i	$-0.811459\pi$
$31$	−9.23857	−1.65930	−0.829648	+	0.558286i	$0.811459\pi$
$32$	0	0
$33$	2.15436	0.375027
$34$	0	0
$35$	−0.461272	−0.0779692
$36$	0	0
$37$	8.90443	1.46388	0.731940	−	0.681369i	$-0.238615\pi$
$37$	8.90443	1.46388	0.731940	+	0.681369i	$0.238615\pi$
$38$	0	0
$39$	−9.97531	−1.59733
$40$	0	0
$41$	−4.82173	−0.753028	−0.376514	−	0.926411i	$-0.622877\pi$
$41$	−4.82173	−0.753028	−0.376514	+	0.926411i	$0.622877\pi$
$42$	0	0
$43$	−3.77632	−0.575883	−0.287942	−	0.957648i	$-0.592971\pi$
$43$	−3.77632	−0.575883	−0.287942	+	0.957648i	$0.592971\pi$
$44$	0	0
$45$	−0.186377	−0.0277834
$46$	0	0
$47$	−6.85092	−0.999310	−0.499655	−	0.866225i	$-0.666540\pi$
$47$	−6.85092	−0.999310	−0.499655	+	0.866225i	$0.666540\pi$
$48$	0	0
$49$	−6.78723	−0.969604
$50$	0	0
$51$	9.33174	1.30671
$52$	0	0
$53$	−7.90836	−1.08630	−0.543148	−	0.839637i	$-0.682768\pi$
$53$	−7.90836	−1.08630	−0.543148	+	0.839637i	$0.682768\pi$
$54$	0	0
$55$	1.28436	0.173183
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.35984	−0.697792	−0.348896	−	0.937161i	$-0.613443\pi$
$59$	−5.35984	−0.697792	−0.348896	+	0.937161i	$0.613443\pi$
$60$	0	0
$61$	−10.1384	−1.29808	−0.649042	−	0.760752i	$-0.724830\pi$
$61$	−10.1384	−1.29808	−0.649042	+	0.760752i	$0.724830\pi$
$62$	0	0
$63$	0.0859705	0.0108313
$64$	0	0
$65$	−5.94694	−0.737627
$66$	0	0
$67$	6.60140	0.806489	0.403245	−	0.915092i	$-0.367882\pi$
$67$	6.60140	0.806489	0.403245	+	0.915092i	$0.367882\pi$
$68$	0	0
$69$	3.67036	0.441860
$70$	0	0
$71$	6.89386	0.818150	0.409075	−	0.912501i	$-0.365851\pi$
$71$	6.89386	0.818150	0.409075	+	0.912501i	$0.365851\pi$
$72$	0	0
$73$	13.0905	1.53212	0.766062	−	0.642766i	$-0.222213\pi$
$73$	13.0905	1.53212	0.766062	+	0.642766i	$0.222213\pi$
$74$	0	0
$75$	1.67739	0.193688
$76$	0	0
$77$	−0.592438	−0.0675146
$78$	0	0
$79$	1.38838	0.156205	0.0781023	−	0.996945i	$-0.475114\pi$
$79$	1.38838	0.156205	0.0781023	+	0.996945i	$0.475114\pi$
$80$	0	0
$81$	−8.40613	−0.934015
$82$	0	0
$83$	0.132099	0.0144998	0.00724990	−	0.999974i	$-0.497692\pi$
$83$	0.132099	0.0144998	0.00724990	+	0.999974i	$0.497692\pi$
$84$	0	0
$85$	5.56326	0.603421
$86$	0	0
$87$	−16.0233	−1.71788
$88$	0	0
$89$	−3.08032	−0.326514	−0.163257	−	0.986584i	$-0.552200\pi$
$89$	−3.08032	−0.326514	−0.163257	+	0.986584i	$0.552200\pi$
$90$	0	0
$91$	2.74316	0.287561
$92$	0	0
$93$	−15.4966	−1.60693
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.584577	0.0593548	0.0296774	−	0.999560i	$-0.490552\pi$
$97$	0.584577	0.0593548	0.0296774	+	0.999560i	$0.490552\pi$
$98$	0	0
$99$	−0.239375	−0.0240581
$100$	0	0
$101$	5.38530	0.535857	0.267929	−	0.963439i	$-0.413661\pi$
$101$	5.38530	0.535857	0.267929	+	0.963439i	$0.413661\pi$
$102$	0	0
$103$	2.52984	0.249272	0.124636	−	0.992203i	$-0.460224\pi$
$103$	2.52984	0.249272	0.124636	+	0.992203i	$0.460224\pi$
$104$	0	0
$105$	−0.773731	−0.0755084
$106$	0	0
$107$	7.27102	0.702916	0.351458	−	0.936204i	$-0.385686\pi$
$107$	7.27102	0.702916	0.351458	+	0.936204i	$0.385686\pi$
$108$	0	0
$109$	−6.89578	−0.660496	−0.330248	−	0.943894i	$-0.607132\pi$
$109$	−6.89578	−0.660496	−0.330248	+	0.943894i	$0.607132\pi$
$110$	0	0
$111$	14.9362	1.41768
$112$	0	0
$113$	9.85398	0.926984	0.463492	−	0.886101i	$-0.346596\pi$
$113$	9.85398	0.926984	0.463492	+	0.886101i	$0.346596\pi$
$114$	0	0
$115$	2.18815	0.204046
$116$	0	0
$117$	1.10837	0.102469
$118$	0	0
$119$	−2.56618	−0.235241
$120$	0	0
$121$	−9.35042	−0.850039
$122$	0	0
$123$	−8.08791	−0.729262
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.3173	−1.71413	−0.857065	−	0.515209i	$-0.827714\pi$
$127$	−19.3173	−1.71413	−0.857065	+	0.515209i	$0.827714\pi$
$128$	0	0
$129$	−6.33434	−0.557708
$130$	0	0
$131$	−9.86595	−0.861992	−0.430996	−	0.902354i	$-0.641838\pi$
$131$	−9.86595	−0.861992	−0.430996	+	0.902354i	$0.641838\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.34478	−0.460006
$136$	0	0
$137$	10.6640	0.911084	0.455542	−	0.890214i	$-0.349446\pi$
$137$	10.6640	0.911084	0.455542	+	0.890214i	$0.349446\pi$
$138$	0	0
$139$	1.31694	0.111702	0.0558509	−	0.998439i	$-0.482213\pi$
$139$	1.31694	0.111702	0.0558509	+	0.998439i	$0.482213\pi$
$140$	0	0
$141$	−11.4916	−0.967771
$142$	0	0
$143$	−7.63800	−0.638721
$144$	0	0
$145$	−9.55254	−0.793296
$146$	0	0
$147$	−11.3848	−0.939002
$148$	0	0
$149$	16.8876	1.38349	0.691745	−	0.722142i	$-0.256842\pi$
$149$	16.8876	1.38349	0.691745	+	0.722142i	$0.256842\pi$
$150$	0	0
$151$	21.5094	1.75041	0.875206	−	0.483751i	$-0.160726\pi$
$151$	21.5094	1.75041	0.875206	+	0.483751i	$0.160726\pi$
$152$	0	0
$153$	−1.03686	−0.0838256
$154$	0	0
$155$	−9.23857	−0.742060
$156$	0	0
$157$	−0.535621	−0.0427472	−0.0213736	−	0.999772i	$-0.506804\pi$
$157$	−0.535621	−0.0427472	−0.0213736	+	0.999772i	$0.506804\pi$
$158$	0	0
$159$	−13.2654	−1.05201
$160$	0	0
$161$	−1.00933	−0.0795464
$162$	0	0
$163$	−4.71197	−0.369070	−0.184535	−	0.982826i	$-0.559078\pi$
$163$	−4.71197	−0.369070	−0.184535	+	0.982826i	$0.559078\pi$
$164$	0	0
$165$	2.15436	0.167717
$166$	0	0
$167$	−15.3884	−1.19079	−0.595395	−	0.803433i	$-0.703005\pi$
$167$	−15.3884	−1.19079	−0.595395	+	0.803433i	$0.703005\pi$
$168$	0	0
$169$	22.3661	1.72047
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.0257	−1.52252	−0.761262	−	0.648445i	$-0.775420\pi$
$173$	−20.0257	−1.52252	−0.761262	+	0.648445i	$0.775420\pi$
$174$	0	0
$175$	−0.461272	−0.0348689
$176$	0	0
$177$	−8.99052	−0.675769
$178$	0	0
$179$	−12.9417	−0.967307	−0.483653	−	0.875260i	$-0.660690\pi$
$179$	−12.9417	−0.967307	−0.483653	+	0.875260i	$0.660690\pi$
$180$	0	0
$181$	7.06253	0.524954	0.262477	−	0.964938i	$-0.415461\pi$
$181$	7.06253	0.524954	0.262477	+	0.964938i	$0.415461\pi$
$182$	0	0
$183$	−17.0060	−1.25712
$184$	0	0
$185$	8.90443	0.654667
$186$	0	0
$187$	7.14522	0.522510
$188$	0	0
$189$	2.46540	0.179331
$190$	0	0
$191$	−23.1066	−1.67194	−0.835968	−	0.548778i	$-0.815093\pi$
$191$	−23.1066	−1.67194	−0.835968	+	0.548778i	$0.815093\pi$
$192$	0	0
$193$	−7.56845	−0.544789	−0.272395	−	0.962186i	$-0.587816\pi$
$193$	−7.56845	−0.544789	−0.272395	+	0.962186i	$0.587816\pi$
$194$	0	0
$195$	−9.97531	−0.714347
$196$	0	0
$197$	−2.08763	−0.148737	−0.0743687	−	0.997231i	$-0.523694\pi$
$197$	−2.08763	−0.148737	−0.0743687	+	0.997231i	$0.523694\pi$
$198$	0	0
$199$	−10.6750	−0.756729	−0.378364	−	0.925657i	$-0.623513\pi$
$199$	−10.6750	−0.756729	−0.378364	+	0.925657i	$0.623513\pi$
$200$	0	0
$201$	11.0731	0.781036
$202$	0	0
$203$	4.40632	0.309263
$204$	0	0
$205$	−4.82173	−0.336764
$206$	0	0
$207$	−0.407820	−0.0283455
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.51330	−0.173023	−0.0865114	−	0.996251i	$-0.527572\pi$
$211$	−2.51330	−0.173023	−0.0865114	+	0.996251i	$0.527572\pi$
$212$	0	0
$213$	11.5637	0.792329
$214$	0	0
$215$	−3.77632	−0.257543
$216$	0	0
$217$	4.26149	0.289289
$218$	0	0
$219$	21.9578	1.48377
$220$	0	0
$221$	−33.0844	−2.22550
$222$	0	0
$223$	−5.72683	−0.383497	−0.191748	−	0.981444i	$-0.561416\pi$
$223$	−5.72683	−0.383497	−0.191748	+	0.981444i	$0.561416\pi$
$224$	0	0
$225$	−0.186377	−0.0124251
$226$	0	0
$227$	10.9014	0.723551	0.361775	−	0.932265i	$-0.382171\pi$
$227$	10.9014	0.723551	0.361775	+	0.932265i	$0.382171\pi$
$228$	0	0
$229$	0.764616	0.0505272	0.0252636	−	0.999681i	$-0.491957\pi$
$229$	0.764616	0.0505272	0.0252636	+	0.999681i	$0.491957\pi$
$230$	0	0
$231$	−0.993748	−0.0653838
$232$	0	0
$233$	−12.8472	−0.841649	−0.420825	−	0.907142i	$-0.638259\pi$
$233$	−12.8472	−0.841649	−0.420825	+	0.907142i	$0.638259\pi$
$234$	0	0
$235$	−6.85092	−0.446905
$236$	0	0
$237$	2.32884	0.151275
$238$	0	0
$239$	−9.75384	−0.630924	−0.315462	−	0.948938i	$-0.602159\pi$
$239$	−9.75384	−0.630924	−0.315462	+	0.948938i	$0.602159\pi$
$240$	0	0
$241$	9.69643	0.624602	0.312301	−	0.949983i	$-0.398900\pi$
$241$	9.69643	0.624602	0.312301	+	0.949983i	$0.398900\pi$
$242$	0	0
$243$	1.93402	0.124068
$244$	0	0
$245$	−6.78723	−0.433620
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.221582	0.0140422
$250$	0	0
$251$	14.3109	0.903299	0.451649	−	0.892195i	$-0.350836\pi$
$251$	14.3109	0.903299	0.451649	+	0.892195i	$0.350836\pi$
$252$	0	0
$253$	2.81036	0.176686
$254$	0	0
$255$	9.33174	0.584376
$256$	0	0
$257$	−16.3907	−1.02242	−0.511211	−	0.859455i	$-0.670803\pi$
$257$	−16.3907	−1.02242	−0.511211	+	0.859455i	$0.670803\pi$
$258$	0	0
$259$	−4.10737	−0.255219
$260$	0	0
$261$	1.78037	0.110202
$262$	0	0
$263$	−30.1636	−1.85997	−0.929983	−	0.367602i	$-0.880179\pi$
$263$	−30.1636	−1.85997	−0.929983	+	0.367602i	$0.880179\pi$
$264$	0	0
$265$	−7.90836	−0.485807
$266$	0	0
$267$	−5.16689	−0.316209
$268$	0	0
$269$	10.6645	0.650226	0.325113	−	0.945675i	$-0.394598\pi$
$269$	10.6645	0.650226	0.325113	+	0.945675i	$0.394598\pi$
$270$	0	0
$271$	29.0141	1.76248	0.881241	−	0.472666i	$-0.156708\pi$
$271$	29.0141	1.76248	0.881241	+	0.472666i	$0.156708\pi$
$272$	0	0
$273$	4.60133	0.278485
$274$	0	0
$275$	1.28436	0.0774497
$276$	0	0
$277$	−16.2311	−0.975233	−0.487617	−	0.873058i	$-0.662134\pi$
$277$	−16.2311	−0.975233	−0.487617	+	0.873058i	$0.662134\pi$
$278$	0	0
$279$	1.72186	0.103085
$280$	0	0
$281$	1.95222	0.116460	0.0582299	−	0.998303i	$-0.481454\pi$
$281$	1.95222	0.116460	0.0582299	+	0.998303i	$0.481454\pi$
$282$	0	0
$283$	20.0909	1.19428	0.597141	−	0.802136i	$-0.296303\pi$
$283$	20.0909	1.19428	0.597141	+	0.802136i	$0.296303\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.22413	0.131286
$288$	0	0
$289$	13.9499	0.820583
$290$	0	0
$291$	0.980561	0.0574815
$292$	0	0
$293$	−19.3065	−1.12790	−0.563948	−	0.825810i	$-0.690718\pi$
$293$	−19.3065	−1.12790	−0.563948	+	0.825810i	$0.690718\pi$
$294$	0	0
$295$	−5.35984	−0.312062
$296$	0	0
$297$	−6.86461	−0.398325
$298$	0	0
$299$	−13.0128	−0.752547
$300$	0	0
$301$	1.74191	0.100402
$302$	0	0
$303$	9.03322	0.518945
$304$	0	0
$305$	−10.1384	−0.580521
$306$	0	0
$307$	10.8754	0.620690	0.310345	−	0.950624i	$-0.399555\pi$
$307$	10.8754	0.620690	0.310345	+	0.950624i	$0.399555\pi$
$308$	0	0
$309$	4.24351	0.241405
$310$	0	0
$311$	−17.2424	−0.977728	−0.488864	−	0.872360i	$-0.662589\pi$
$311$	−17.2424	−0.977728	−0.488864	+	0.872360i	$0.662589\pi$
$312$	0	0
$313$	7.83520	0.442872	0.221436	−	0.975175i	$-0.428926\pi$
$313$	7.83520	0.442872	0.221436	+	0.975175i	$0.428926\pi$
$314$	0	0
$315$	0.0859705	0.00484389
$316$	0	0
$317$	10.2980	0.578394	0.289197	−	0.957270i	$-0.406612\pi$
$317$	10.2980	0.578394	0.289197	+	0.957270i	$0.406612\pi$
$318$	0	0
$319$	−12.2689	−0.686926
$320$	0	0
$321$	12.1963	0.680731
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5.94694	−0.329877
$326$	0	0
$327$	−11.5669	−0.639650
$328$	0	0
$329$	3.16014	0.174224
$330$	0	0
$331$	−33.4855	−1.84053	−0.920266	−	0.391294i	$-0.872027\pi$
$331$	−33.4855	−1.84053	−0.920266	+	0.391294i	$0.872027\pi$
$332$	0	0
$333$	−1.65958	−0.0909445
$334$	0	0
$335$	6.60140	0.360673
$336$	0	0
$337$	−24.8923	−1.35597	−0.677985	−	0.735075i	$-0.737147\pi$
$337$	−24.8923	−1.35597	−0.677985	+	0.735075i	$0.737147\pi$
$338$	0	0
$339$	16.5289	0.897728
$340$	0	0
$341$	−11.8656	−0.642560
$342$	0	0
$343$	6.35966	0.343389
$344$	0	0
$345$	3.67036	0.197606
$346$	0	0
$347$	25.2328	1.35457	0.677283	−	0.735723i	$-0.263157\pi$
$347$	25.2328	1.35457	0.677283	+	0.735723i	$0.263157\pi$
$348$	0	0
$349$	28.4048	1.52047	0.760236	−	0.649647i	$-0.225083\pi$
$349$	28.4048	1.52047	0.760236	+	0.649647i	$0.225083\pi$
$350$	0	0
$351$	31.7851	1.69656
$352$	0	0
$353$	−30.1212	−1.60319	−0.801596	−	0.597866i	$-0.796015\pi$
$353$	−30.1212	−1.60319	−0.801596	+	0.597866i	$0.796015\pi$
$354$	0	0
$355$	6.89386	0.365888
$356$	0	0
$357$	−4.30447	−0.227817
$358$	0	0
$359$	−34.1282	−1.80122	−0.900608	−	0.434632i	$-0.856878\pi$
$359$	−34.1282	−1.80122	−0.900608	+	0.434632i	$0.856878\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−15.6843	−0.823211
$364$	0	0
$365$	13.0905	0.685187
$366$	0	0
$367$	22.5991	1.17966	0.589832	−	0.807526i	$-0.299194\pi$
$367$	22.5991	1.17966	0.589832	+	0.807526i	$0.299194\pi$
$368$	0	0
$369$	0.898660	0.0467824
$370$	0	0
$371$	3.64790	0.189390
$372$	0	0
$373$	12.1191	0.627503	0.313752	−	0.949505i	$-0.398414\pi$
$373$	12.1191	0.627503	0.313752	+	0.949505i	$0.398414\pi$
$374$	0	0
$375$	1.67739	0.0866198
$376$	0	0
$377$	56.8084	2.92578
$378$	0	0
$379$	34.8034	1.78773	0.893866	−	0.448335i	$-0.147983\pi$
$379$	34.8034	1.78773	0.893866	+	0.448335i	$0.147983\pi$
$380$	0	0
$381$	−32.4025	−1.66003
$382$	0	0
$383$	−27.6062	−1.41061	−0.705305	−	0.708904i	$-0.749190\pi$
$383$	−27.6062	−1.41061	−0.705305	+	0.708904i	$0.749190\pi$
$384$	0	0
$385$	−0.592438	−0.0301935
$386$	0	0
$387$	0.703819	0.0357771
$388$	0	0
$389$	21.7658	1.10357	0.551785	−	0.833986i	$-0.313947\pi$
$389$	21.7658	1.10357	0.551785	+	0.833986i	$0.313947\pi$
$390$	0	0
$391$	12.1732	0.615627
$392$	0	0
$393$	−16.5490	−0.834787
$394$	0	0
$395$	1.38838	0.0698568
$396$	0	0
$397$	0.697862	0.0350247	0.0175124	−	0.999847i	$-0.494425\pi$
$397$	0.697862	0.0350247	0.0175124	+	0.999847i	$0.494425\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.5594	0.527311	0.263655	−	0.964617i	$-0.415072\pi$
$401$	10.5594	0.527311	0.263655	+	0.964617i	$0.415072\pi$
$402$	0	0
$403$	54.9412	2.73682
$404$	0	0
$405$	−8.40613	−0.417704
$406$	0	0
$407$	11.4365	0.566885
$408$	0	0
$409$	12.2184	0.604160	0.302080	−	0.953283i	$-0.402319\pi$
$409$	12.2184	0.604160	0.302080	+	0.953283i	$0.402319\pi$
$410$	0	0
$411$	17.8876	0.882329
$412$	0	0
$413$	2.47235	0.121656
$414$	0	0
$415$	0.132099	0.00648451
$416$	0	0
$417$	2.20902	0.108176
$418$	0	0
$419$	26.6471	1.30180	0.650899	−	0.759165i	$-0.274392\pi$
$419$	26.6471	1.30180	0.650899	+	0.759165i	$0.274392\pi$
$420$	0	0
$421$	15.3651	0.748851	0.374425	−	0.927257i	$-0.377840\pi$
$421$	15.3651	0.748851	0.374425	+	0.927257i	$0.377840\pi$
$422$	0	0
$423$	1.27685	0.0620828
$424$	0	0
$425$	5.56326	0.269858
$426$	0	0
$427$	4.67655	0.226314
$428$	0	0
$429$	−12.8119	−0.618563
$430$	0	0
$431$	38.3315	1.84637	0.923183	−	0.384361i	$-0.125578\pi$
$431$	38.3315	1.84637	0.923183	+	0.384361i	$0.125578\pi$
$432$	0	0
$433$	−32.9081	−1.58146	−0.790731	−	0.612163i	$-0.790299\pi$
$433$	−32.9081	−1.58146	−0.790731	+	0.612163i	$0.790299\pi$
$434$	0	0
$435$	−16.0233	−0.768258
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−23.6527	−1.12888	−0.564442	−	0.825473i	$-0.690909\pi$
$439$	−23.6527	−1.12888	−0.564442	+	0.825473i	$0.690909\pi$
$440$	0	0
$441$	1.26498	0.0602373
$442$	0	0
$443$	−16.5423	−0.785950	−0.392975	−	0.919549i	$-0.628554\pi$
$443$	−16.5423	−0.785950	−0.392975	+	0.919549i	$0.628554\pi$
$444$	0	0
$445$	−3.08032	−0.146021
$446$	0	0
$447$	28.3271	1.33983
$448$	0	0
$449$	−29.3503	−1.38513	−0.692563	−	0.721357i	$-0.743519\pi$
$449$	−29.3503	−1.38513	−0.692563	+	0.721357i	$0.743519\pi$
$450$	0	0
$451$	−6.19283	−0.291609
$452$	0	0
$453$	36.0796	1.69517
$454$	0	0
$455$	2.74316	0.128601
$456$	0	0
$457$	−5.48753	−0.256696	−0.128348	−	0.991729i	$-0.540967\pi$
$457$	−5.48753	−0.256696	−0.128348	+	0.991729i	$0.540967\pi$
$458$	0	0
$459$	−29.7344	−1.38789
$460$	0	0
$461$	−33.1907	−1.54584	−0.772922	−	0.634502i	$-0.781205\pi$
$461$	−33.1907	−1.54584	−0.772922	+	0.634502i	$0.781205\pi$
$462$	0	0
$463$	−12.1404	−0.564212	−0.282106	−	0.959383i	$-0.591033\pi$
$463$	−12.1404	−0.564212	−0.282106	+	0.959383i	$0.591033\pi$
$464$	0	0
$465$	−15.4966	−0.718640
$466$	0	0
$467$	20.4184	0.944850	0.472425	−	0.881371i	$-0.343379\pi$
$467$	20.4184	0.944850	0.472425	+	0.881371i	$0.343379\pi$
$468$	0	0
$469$	−3.04504	−0.140607
$470$	0	0
$471$	−0.898443	−0.0413981
$472$	0	0
$473$	−4.85014	−0.223010
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.47394	0.0674869
$478$	0	0
$479$	11.4520	0.523254	0.261627	−	0.965169i	$-0.415741\pi$
$479$	11.4520	0.523254	0.261627	+	0.965169i	$0.415741\pi$
$480$	0	0
$481$	−52.9541	−2.41450
$482$	0	0
$483$	−1.69304	−0.0770358
$484$	0	0
$485$	0.584577	0.0265443
$486$	0	0
$487$	−12.9204	−0.585481	−0.292740	−	0.956192i	$-0.594567\pi$
$487$	−12.9204	−0.585481	−0.292740	+	0.956192i	$0.594567\pi$
$488$	0	0
$489$	−7.90379	−0.357422
$490$	0	0
$491$	−32.0662	−1.44713	−0.723564	−	0.690257i	$-0.757498\pi$
$491$	−32.0662	−1.44713	−0.723564	+	0.690257i	$0.757498\pi$
$492$	0	0
$493$	−53.1433	−2.39346
$494$	0	0
$495$	−0.239375	−0.0107591
$496$	0	0
$497$	−3.17994	−0.142640
$498$	0	0
$499$	−1.95330	−0.0874417	−0.0437209	−	0.999044i	$-0.513921\pi$
$499$	−1.95330	−0.0874417	−0.0437209	+	0.999044i	$0.513921\pi$
$500$	0	0
$501$	−25.8123	−1.15321
$502$	0	0
$503$	27.1934	1.21249	0.606247	−	0.795276i	$-0.292674\pi$
$503$	27.1934	1.21249	0.606247	+	0.795276i	$0.292674\pi$
$504$	0	0
$505$	5.38530	0.239643
$506$	0	0
$507$	37.5165	1.66617
$508$	0	0
$509$	24.6322	1.09180	0.545901	−	0.837850i	$-0.316188\pi$
$509$	24.6322	1.09180	0.545901	+	0.837850i	$0.316188\pi$
$510$	0	0
$511$	−6.03827	−0.267117
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.52984	0.111478
$516$	0	0
$517$	−8.79904	−0.386981
$518$	0	0
$519$	−33.5908	−1.47447
$520$	0	0
$521$	−1.60095	−0.0701390	−0.0350695	−	0.999385i	$-0.511165\pi$
$521$	−1.60095	−0.0701390	−0.0350695	+	0.999385i	$0.511165\pi$
$522$	0	0
$523$	−6.02327	−0.263379	−0.131690	−	0.991291i	$-0.542040\pi$
$523$	−6.02327	−0.263379	−0.131690	+	0.991291i	$0.542040\pi$
$524$	0	0
$525$	−0.773731	−0.0337684
$526$	0	0
$527$	−51.3966	−2.23887
$528$	0	0
$529$	−18.2120	−0.791827
$530$	0	0
$531$	0.998952	0.0433508
$532$	0	0
$533$	28.6745	1.24203
$534$	0	0
$535$	7.27102	0.314354
$536$	0	0
$537$	−21.7082	−0.936778
$538$	0	0
$539$	−8.71723	−0.375478
$540$	0	0
$541$	−23.1729	−0.996280	−0.498140	−	0.867097i	$-0.665983\pi$
$541$	−23.1729	−0.996280	−0.498140	+	0.867097i	$0.665983\pi$
$542$	0	0
$543$	11.8466	0.508386
$544$	0	0
$545$	−6.89578	−0.295383
$546$	0	0
$547$	24.4795	1.04667	0.523335	−	0.852127i	$-0.324688\pi$
$547$	24.4795	1.04667	0.523335	+	0.852127i	$0.324688\pi$
$548$	0	0
$549$	1.88956	0.0806444
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−0.640420	−0.0272334
$554$	0	0
$555$	14.9362	0.634005
$556$	0	0
$557$	22.1712	0.939425	0.469712	−	0.882819i	$-0.344358\pi$
$557$	22.1712	0.939425	0.469712	+	0.882819i	$0.344358\pi$
$558$	0	0
$559$	22.4575	0.949852
$560$	0	0
$561$	11.9853	0.506020
$562$	0	0
$563$	0.662808	0.0279340	0.0139670	−	0.999902i	$-0.495554\pi$
$563$	0.662808	0.0279340	0.0139670	+	0.999902i	$0.495554\pi$
$564$	0	0
$565$	9.85398	0.414560
$566$	0	0
$567$	3.87751	0.162840
$568$	0	0
$569$	2.63574	0.110496	0.0552480	−	0.998473i	$-0.482405\pi$
$569$	2.63574	0.110496	0.0552480	+	0.998473i	$0.482405\pi$
$570$	0	0
$571$	−38.1084	−1.59479	−0.797393	−	0.603461i	$-0.793788\pi$
$571$	−38.1084	−1.59479	−0.797393	+	0.603461i	$0.793788\pi$
$572$	0	0
$573$	−38.7587	−1.61917
$574$	0	0
$575$	2.18815	0.0912520
$576$	0	0
$577$	43.6751	1.81822	0.909109	−	0.416559i	$-0.136764\pi$
$577$	43.6751	1.81822	0.909109	+	0.416559i	$0.136764\pi$
$578$	0	0
$579$	−12.6952	−0.527595
$580$	0	0
$581$	−0.0609338	−0.00252796
$582$	0	0
$583$	−10.1572	−0.420667
$584$	0	0
$585$	1.10837	0.0458256
$586$	0	0
$587$	36.7384	1.51636	0.758179	−	0.652047i	$-0.226089\pi$
$587$	36.7384	1.51636	0.758179	+	0.652047i	$0.226089\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−3.50176	−0.144043
$592$	0	0
$593$	−8.56125	−0.351568	−0.175784	−	0.984429i	$-0.556246\pi$
$593$	−8.56125	−0.351568	−0.175784	+	0.984429i	$0.556246\pi$
$594$	0	0
$595$	−2.56618	−0.105203
$596$	0	0
$597$	−17.9061	−0.732846
$598$	0	0
$599$	21.3337	0.871673	0.435837	−	0.900026i	$-0.356453\pi$
$599$	21.3337	0.871673	0.435837	+	0.900026i	$0.356453\pi$
$600$	0	0
$601$	−10.5424	−0.430034	−0.215017	−	0.976610i	$-0.568981\pi$
$601$	−10.5424	−0.430034	−0.215017	+	0.976610i	$0.568981\pi$
$602$	0	0
$603$	−1.23035	−0.0501037
$604$	0	0
$605$	−9.35042	−0.380149
$606$	0	0
$607$	−5.51930	−0.224022	−0.112011	−	0.993707i	$-0.535729\pi$
$607$	−5.51930	−0.224022	−0.112011	+	0.993707i	$0.535729\pi$
$608$	0	0
$609$	7.39110	0.299502
$610$	0	0
$611$	40.7420	1.64825
$612$	0	0
$613$	−21.0387	−0.849746	−0.424873	−	0.905253i	$-0.639681\pi$
$613$	−21.0387	−0.849746	−0.424873	+	0.905253i	$0.639681\pi$
$614$	0	0
$615$	−8.08791	−0.326136
$616$	0	0
$617$	9.11210	0.366839	0.183420	−	0.983035i	$-0.441283\pi$
$617$	9.11210	0.366839	0.183420	+	0.983035i	$0.441283\pi$
$618$	0	0
$619$	11.2940	0.453943	0.226971	−	0.973901i	$-0.427118\pi$
$619$	11.2940	0.453943	0.226971	+	0.973901i	$0.427118\pi$
$620$	0	0
$621$	−11.6952	−0.469311
$622$	0	0
$623$	1.42087	0.0569258
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	49.5377	1.97520
$630$	0	0
$631$	−6.20698	−0.247096	−0.123548	−	0.992339i	$-0.539427\pi$
$631$	−6.20698	−0.247096	−0.123548	+	0.992339i	$0.539427\pi$
$632$	0	0
$633$	−4.21578	−0.167562
$634$	0	0
$635$	−19.3173	−0.766582
$636$	0	0
$637$	40.3632	1.59925
$638$	0	0
$639$	−1.28486	−0.0508281
$640$	0	0
$641$	−16.8913	−0.667168	−0.333584	−	0.942720i	$-0.608258\pi$
$641$	−16.8913	−0.667168	−0.333584	+	0.942720i	$0.608258\pi$
$642$	0	0
$643$	22.9899	0.906633	0.453317	−	0.891350i	$-0.350241\pi$
$643$	22.9899	0.906633	0.453317	+	0.891350i	$0.350241\pi$
$644$	0	0
$645$	−6.33434	−0.249415
$646$	0	0
$647$	12.9497	0.509105	0.254553	−	0.967059i	$-0.418072\pi$
$647$	12.9497	0.509105	0.254553	+	0.967059i	$0.418072\pi$
$648$	0	0
$649$	−6.88396	−0.270219
$650$	0	0
$651$	7.14817	0.280159
$652$	0	0
$653$	10.3111	0.403506	0.201753	−	0.979436i	$-0.435336\pi$
$653$	10.3111	0.403506	0.201753	+	0.979436i	$0.435336\pi$
$654$	0	0
$655$	−9.86595	−0.385495
$656$	0	0
$657$	−2.43976	−0.0951843
$658$	0	0
$659$	−18.8753	−0.735277	−0.367639	−	0.929969i	$-0.619834\pi$
$659$	−18.8753	−0.735277	−0.367639	+	0.929969i	$0.619834\pi$
$660$	0	0
$661$	−26.1360	−1.01657	−0.508287	−	0.861188i	$-0.669721\pi$
$661$	−26.1360	−1.01657	−0.508287	+	0.861188i	$0.669721\pi$
$662$	0	0
$663$	−55.4953	−2.15526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.9024	−0.809342
$668$	0	0
$669$	−9.60610	−0.371393
$670$	0	0
$671$	−13.0213	−0.502681
$672$	0	0
$673$	−4.88006	−0.188112	−0.0940562	−	0.995567i	$-0.529983\pi$
$673$	−4.88006	−0.188112	−0.0940562	+	0.995567i	$0.529983\pi$
$674$	0	0
$675$	−5.34478	−0.205721
$676$	0	0
$677$	−11.8220	−0.454356	−0.227178	−	0.973853i	$-0.572950\pi$
$677$	−11.8220	−0.454356	−0.227178	+	0.973853i	$0.572950\pi$
$678$	0	0
$679$	−0.269649	−0.0103482
$680$	0	0
$681$	18.2858	0.700715
$682$	0	0
$683$	−1.10522	−0.0422901	−0.0211450	−	0.999776i	$-0.506731\pi$
$683$	−1.10522	−0.0422901	−0.0211450	+	0.999776i	$0.506731\pi$
$684$	0	0
$685$	10.6640	0.407449
$686$	0	0
$687$	1.28256	0.0489325
$688$	0	0
$689$	47.0305	1.79172
$690$	0	0
$691$	−13.4805	−0.512823	−0.256412	−	0.966568i	$-0.582540\pi$
$691$	−13.4805	−0.512823	−0.256412	+	0.966568i	$0.582540\pi$
$692$	0	0
$693$	0.110417	0.00419439
$694$	0	0
$695$	1.31694	0.0499546
$696$	0	0
$697$	−26.8246	−1.01605
$698$	0	0
$699$	−21.5497	−0.815086
$700$	0	0
$701$	24.8533	0.938695	0.469347	−	0.883014i	$-0.344489\pi$
$701$	24.8533	0.938695	0.469347	+	0.883014i	$0.344489\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−11.4916	−0.432800
$706$	0	0
$707$	−2.48409	−0.0934237
$708$	0	0
$709$	1.82542	0.0685552	0.0342776	−	0.999412i	$-0.489087\pi$
$709$	1.82542	0.0685552	0.0342776	+	0.999412i	$0.489087\pi$
$710$	0	0
$711$	−0.258762	−0.00970432
$712$	0	0
$713$	−20.2153	−0.757070
$714$	0	0
$715$	−7.63800	−0.285645
$716$	0	0
$717$	−16.3610	−0.611011
$718$	0	0
$719$	32.6663	1.21825	0.609123	−	0.793076i	$-0.291522\pi$
$719$	32.6663	1.21825	0.609123	+	0.793076i	$0.291522\pi$
$720$	0	0
$721$	−1.16694	−0.0434592
$722$	0	0
$723$	16.2647	0.604889
$724$	0	0
$725$	−9.55254	−0.354773
$726$	0	0
$727$	15.9255	0.590643	0.295322	−	0.955398i	$-0.404573\pi$
$727$	15.9255	0.590643	0.295322	+	0.955398i	$0.404573\pi$
$728$	0	0
$729$	28.4625	1.05417
$730$	0	0
$731$	−21.0087	−0.777033
$732$	0	0
$733$	12.2930	0.454054	0.227027	−	0.973888i	$-0.427099\pi$
$733$	12.2930	0.454054	0.227027	+	0.973888i	$0.427099\pi$
$734$	0	0
$735$	−11.3848	−0.419935
$736$	0	0
$737$	8.47856	0.312312
$738$	0	0
$739$	26.7251	0.983099	0.491550	−	0.870850i	$-0.336431\pi$
$739$	26.7251	0.983099	0.491550	+	0.870850i	$0.336431\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.2554	−0.522980	−0.261490	−	0.965206i	$-0.584214\pi$
$743$	−14.2554	−0.522980	−0.261490	+	0.965206i	$0.584214\pi$
$744$	0	0
$745$	16.8876	0.618716
$746$	0	0
$747$	−0.0246203	−0.000900810 0
$748$	0	0
$749$	−3.35392	−0.122549
$750$	0	0
$751$	−11.1013	−0.405093	−0.202547	−	0.979273i	$-0.564922\pi$
$751$	−11.1013	−0.405093	−0.202547	+	0.979273i	$0.564922\pi$
$752$	0	0
$753$	24.0050	0.874790
$754$	0	0
$755$	21.5094	0.782808
$756$	0	0
$757$	2.21820	0.0806220	0.0403110	−	0.999187i	$-0.487165\pi$
$757$	2.21820	0.0806220	0.0403110	+	0.999187i	$0.487165\pi$
$758$	0	0
$759$	4.71406	0.171110
$760$	0	0
$761$	−49.1007	−1.77990	−0.889950	−	0.456058i	$-0.849261\pi$
$761$	−49.1007	−1.77990	−0.889950	+	0.456058i	$0.849261\pi$
$762$	0	0
$763$	3.18083	0.115154
$764$	0	0
$765$	−1.03686	−0.0374879
$766$	0	0
$767$	31.8746	1.15093
$768$	0	0
$769$	32.2429	1.16271	0.581355	−	0.813650i	$-0.302523\pi$
$769$	32.2429	1.16271	0.581355	+	0.813650i	$0.302523\pi$
$770$	0	0
$771$	−27.4935	−0.990154
$772$	0	0
$773$	−6.12488	−0.220297	−0.110148	−	0.993915i	$-0.535133\pi$
$773$	−6.12488	−0.220297	−0.110148	+	0.993915i	$0.535133\pi$
$774$	0	0
$775$	−9.23857	−0.331859
$776$	0	0
$777$	−6.88964	−0.247164
$778$	0	0
$779$	0	0
$780$	0	0
$781$	8.85418	0.316827
$782$	0	0
$783$	51.0563	1.82460
$784$	0	0
$785$	−0.535621	−0.0191171
$786$	0	0
$787$	15.1085	0.538561	0.269281	−	0.963062i	$-0.413214\pi$
$787$	15.1085	0.538561	0.269281	+	0.963062i	$0.413214\pi$
$788$	0	0
$789$	−50.5960	−1.80126
$790$	0	0
$791$	−4.54536	−0.161615
$792$	0	0
$793$	60.2922	2.14104
$794$	0	0
$795$	−13.2654	−0.470474
$796$	0	0
$797$	34.3148	1.21549	0.607747	−	0.794131i	$-0.292074\pi$
$797$	34.3148	1.21549	0.607747	+	0.794131i	$0.292074\pi$
$798$	0	0
$799$	−38.1135	−1.34836
$800$	0	0
$801$	0.574102	0.0202849
$802$	0	0
$803$	16.8129	0.593313
$804$	0	0
$805$	−1.00933	−0.0355742
$806$	0	0
$807$	17.8885	0.629704
$808$	0	0
$809$	22.8317	0.802721	0.401361	−	0.915920i	$-0.368537\pi$
$809$	22.8317	0.802721	0.401361	+	0.915920i	$0.368537\pi$
$810$	0	0
$811$	39.9023	1.40116	0.700581	−	0.713573i	$-0.252924\pi$
$811$	39.9023	1.40116	0.700581	+	0.713573i	$0.252924\pi$
$812$	0	0
$813$	48.6679	1.70686
$814$	0	0
$815$	−4.71197	−0.165053
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.511261	−0.0178649
$820$	0	0
$821$	−18.7296	−0.653667	−0.326834	−	0.945082i	$-0.605982\pi$
$821$	−18.7296	−0.653667	−0.326834	+	0.945082i	$0.605982\pi$
$822$	0	0
$823$	4.88681	0.170343	0.0851717	−	0.996366i	$-0.472856\pi$
$823$	4.88681	0.170343	0.0851717	+	0.996366i	$0.472856\pi$
$824$	0	0
$825$	2.15436	0.0750053
$826$	0	0
$827$	−19.7615	−0.687174	−0.343587	−	0.939121i	$-0.611642\pi$
$827$	−19.7615	−0.687174	−0.343587	+	0.939121i	$0.611642\pi$
$828$	0	0
$829$	−9.15167	−0.317851	−0.158925	−	0.987291i	$-0.550803\pi$
$829$	−9.15167	−0.317851	−0.158925	+	0.987291i	$0.550803\pi$
$830$	0	0
$831$	−27.2258	−0.944454
$832$	0	0
$833$	−37.7591	−1.30828
$834$	0	0
$835$	−15.3884	−0.532538
$836$	0	0
$837$	49.3782	1.70676
$838$	0	0
$839$	48.3693	1.66989	0.834946	−	0.550331i	$-0.185499\pi$
$839$	48.3693	1.66989	0.834946	+	0.550331i	$0.185499\pi$
$840$	0	0
$841$	62.2511	2.14659
$842$	0	0
$843$	3.27463	0.112784
$844$	0	0
$845$	22.3661	0.769416
$846$	0	0
$847$	4.31309	0.148200
$848$	0	0
$849$	33.7002	1.15659
$850$	0	0
$851$	19.4842	0.667909
$852$	0	0
$853$	−4.04655	−0.138551	−0.0692756	−	0.997598i	$-0.522069\pi$
$853$	−4.04655	−0.138551	−0.0692756	+	0.997598i	$0.522069\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.3997	0.696840	0.348420	−	0.937339i	$-0.386718\pi$
$857$	20.3997	0.696840	0.348420	+	0.937339i	$0.386718\pi$
$858$	0	0
$859$	−51.8278	−1.76834	−0.884172	−	0.467162i	$-0.845276\pi$
$859$	−51.8278	−1.76834	−0.884172	+	0.467162i	$0.845276\pi$
$860$	0	0
$861$	3.73072	0.127143
$862$	0	0
$863$	18.9008	0.643390	0.321695	−	0.946843i	$-0.395747\pi$
$863$	18.9008	0.643390	0.321695	+	0.946843i	$0.395747\pi$
$864$	0	0
$865$	−20.0257	−0.680893
$866$	0	0
$867$	23.3994	0.794685
$868$	0	0
$869$	1.78317	0.0604900
$870$	0	0
$871$	−39.2581	−1.33021
$872$	0	0
$873$	−0.108952	−0.00368746
$874$	0	0
$875$	−0.461272	−0.0155938
$876$	0	0
$877$	−12.5196	−0.422758	−0.211379	−	0.977404i	$-0.567795\pi$
$877$	−12.5196	−0.422758	−0.211379	+	0.977404i	$0.567795\pi$
$878$	0	0
$879$	−32.3844	−1.09230
$880$	0	0
$881$	16.3990	0.552495	0.276248	−	0.961086i	$-0.410909\pi$
$881$	16.3990	0.552495	0.276248	+	0.961086i	$0.410909\pi$
$882$	0	0
$883$	−21.9835	−0.739804	−0.369902	−	0.929071i	$-0.620609\pi$
$883$	−21.9835	−0.739804	−0.369902	+	0.929071i	$0.620609\pi$
$884$	0	0
$885$	−8.99052	−0.302213
$886$	0	0
$887$	−11.3757	−0.381958	−0.190979	−	0.981594i	$-0.561166\pi$
$887$	−11.3757	−0.381958	−0.190979	+	0.981594i	$0.561166\pi$
$888$	0	0
$889$	8.91051	0.298849
$890$	0	0
$891$	−10.7965	−0.361696
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−12.9417	−0.432593
$896$	0	0
$897$	−21.8274	−0.728796
$898$	0	0
$899$	88.2519	2.94336
$900$	0	0
$901$	−43.9963	−1.46573
$902$	0	0
$903$	2.92186	0.0972333
$904$	0	0
$905$	7.06253	0.234767
$906$	0	0
$907$	49.9863	1.65977	0.829883	−	0.557937i	$-0.188407\pi$
$907$	49.9863	1.65977	0.829883	+	0.557937i	$0.188407\pi$
$908$	0	0
$909$	−1.00370	−0.0332905
$910$	0	0
$911$	13.9854	0.463358	0.231679	−	0.972792i	$-0.425578\pi$
$911$	13.9854	0.463358	0.231679	+	0.972792i	$0.425578\pi$
$912$	0	0
$913$	0.169663	0.00561503
$914$	0	0
$915$	−17.0060	−0.562199
$916$	0	0
$917$	4.55089	0.150283
$918$	0	0
$919$	−9.95395	−0.328351	−0.164175	−	0.986431i	$-0.552496\pi$
$919$	−9.95395	−0.328351	−0.164175	+	0.986431i	$0.552496\pi$
$920$	0	0
$921$	18.2422	0.601100
$922$	0	0
$923$	−40.9973	−1.34944
$924$	0	0
$925$	8.90443	0.292776
$926$	0	0
$927$	−0.471503	−0.0154862
$928$	0	0
$929$	7.59496	0.249183	0.124591	−	0.992208i	$-0.460238\pi$
$929$	7.59496	0.249183	0.124591	+	0.992208i	$0.460238\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−28.9222	−0.946870
$934$	0	0
$935$	7.14522	0.233674
$936$	0	0
$937$	9.72383	0.317664	0.158832	−	0.987306i	$-0.449227\pi$
$937$	9.72383	0.317664	0.158832	+	0.987306i	$0.449227\pi$
$938$	0	0
$939$	13.1427	0.428894
$940$	0	0
$941$	−16.6453	−0.542620	−0.271310	−	0.962492i	$-0.587457\pi$
$941$	−16.6453	−0.542620	−0.271310	+	0.962492i	$0.587457\pi$
$942$	0	0
$943$	−10.5507	−0.343576
$944$	0	0
$945$	2.46540	0.0801994
$946$	0	0
$947$	−52.3594	−1.70145	−0.850725	−	0.525611i	$-0.823837\pi$
$947$	−52.3594	−1.70145	−0.850725	+	0.525611i	$0.823837\pi$
$948$	0	0
$949$	−77.8482	−2.52706
$950$	0	0
$951$	17.2737	0.560139
$952$	0	0
$953$	−43.3549	−1.40440	−0.702202	−	0.711978i	$-0.747800\pi$
$953$	−43.3549	−1.40440	−0.702202	+	0.711978i	$0.747800\pi$
$954$	0	0
$955$	−23.1066	−0.747713
$956$	0	0
$957$	−20.5797	−0.665246
$958$	0	0
$959$	−4.91899	−0.158842
$960$	0	0
$961$	54.3512	1.75326
$962$	0	0
$963$	−1.35515	−0.0436691
$964$	0	0
$965$	−7.56845	−0.243637
$966$	0	0
$967$	17.5569	0.564592	0.282296	−	0.959327i	$-0.408904\pi$
$967$	17.5569	0.564592	0.282296	+	0.959327i	$0.408904\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.6341	1.78538	0.892692	−	0.450667i	$-0.148814\pi$
$971$	55.6341	1.78538	0.892692	+	0.450667i	$0.148814\pi$
$972$	0	0
$973$	−0.607470	−0.0194746
$974$	0	0
$975$	−9.97531	−0.319466
$976$	0	0
$977$	−5.64127	−0.180480	−0.0902401	−	0.995920i	$-0.528763\pi$
$977$	−5.64127	−0.180480	−0.0902401	+	0.995920i	$0.528763\pi$
$978$	0	0
$979$	−3.95624	−0.126442
$980$	0	0
$981$	1.28522	0.0410338
$982$	0	0
$983$	15.3397	0.489260	0.244630	−	0.969616i	$-0.421333\pi$
$983$	15.3397	0.489260	0.244630	+	0.969616i	$0.421333\pi$
$984$	0	0
$985$	−2.08763	−0.0665174
$986$	0	0
$987$	5.30077	0.168725
$988$	0	0
$989$	−8.26313	−0.262752
$990$	0	0
$991$	43.1688	1.37130	0.685651	−	0.727930i	$-0.259518\pi$
$991$	43.1688	1.37130	0.685651	+	0.727930i	$0.259518\pi$
$992$	0	0
$993$	−56.1681	−1.78244
$994$	0	0
$995$	−10.6750	−0.338419
$996$	0	0
$997$	−9.71898	−0.307803	−0.153902	−	0.988086i	$-0.549184\pi$
$997$	−9.71898	−0.307803	−0.153902	+	0.988086i	$0.549184\pi$
$998$	0	0
$999$	−47.5923	−1.50575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.w.1.8		9
19.4	even	9		380.2.u.b.301.1	yes	18
19.5	even	9		380.2.u.b.101.1	✓	18
19.18	odd	2		7220.2.a.y.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.u.b.101.1	✓	18	19.5	even	9
380.2.u.b.301.1	yes	18	19.4	even	9
7220.2.a.w.1.8		9	1.1	even	1	trivial
7220.2.a.y.1.2		9	19.18	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:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67739 q^{3} +1.00000 q^{5} -0.461272 q^{7} -0.186377 q^{9} +O(q^{10})\)	\(q+1.67739 q^{3} +1.00000 q^{5} -0.461272 q^{7} -0.186377 q^{9} +1.28436 q^{11} -5.94694 q^{13} +1.67739 q^{15} +5.56326 q^{17} -0.773731 q^{21} +2.18815 q^{23} +1.00000 q^{25} -5.34478 q^{27} -9.55254 q^{29} -9.23857 q^{31} +2.15436 q^{33} -0.461272 q^{35} +8.90443 q^{37} -9.97531 q^{39} -4.82173 q^{41} -3.77632 q^{43} -0.186377 q^{45} -6.85092 q^{47} -6.78723 q^{49} +9.33174 q^{51} -7.90836 q^{53} +1.28436 q^{55} -5.35984 q^{59} -10.1384 q^{61} +0.0859705 q^{63} -5.94694 q^{65} +6.60140 q^{67} +3.67036 q^{69} +6.89386 q^{71} +13.0905 q^{73} +1.67739 q^{75} -0.592438 q^{77} +1.38838 q^{79} -8.40613 q^{81} +0.132099 q^{83} +5.56326 q^{85} -16.0233 q^{87} -3.08032 q^{89} +2.74316 q^{91} -15.4966 q^{93} +0.584577 q^{97} -0.239375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + 9 q^{5} + 6 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + 9 * q^5 + 6 * q^9	\( 9 q - 3 q^{3} + 9 q^{5} + 6 q^{9} - 15 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 18 q^{27} - 15 q^{29} - 6 q^{31} - 9 q^{33} - 18 q^{37} - 6 q^{39} - 18 q^{41} - 6 q^{43} + 6 q^{45} - 15 q^{47} + 3 q^{49} + 18 q^{51} - 33 q^{53} + 3 q^{59} - 15 q^{63} - 15 q^{65} - 12 q^{67} - 27 q^{69} - 6 q^{71} + 3 q^{73} - 3 q^{75} - 18 q^{77} - 15 q^{79} - 15 q^{81} + 6 q^{83} + 3 q^{85} - 21 q^{87} - 3 q^{89} - 33 q^{91} - 9 q^{93} - 45 q^{97} + 42 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + 9 * q^5 + 6 * q^9 - 15 * q^13 - 3 * q^15 + 3 * q^17 - 12 * q^21 + 6 * q^23 + 9 * q^25 - 18 * q^27 - 15 * q^29 - 6 * q^31 - 9 * q^33 - 18 * q^37 - 6 * q^39 - 18 * q^41 - 6 * q^43 + 6 * q^45 - 15 * q^47 + 3 * q^49 + 18 * q^51 - 33 * q^53 + 3 * q^59 - 15 * q^63 - 15 * q^65 - 12 * q^67 - 27 * q^69 - 6 * q^71 + 3 * q^73 - 3 * q^75 - 18 * q^77 - 15 * q^79 - 15 * q^81 + 6 * q^83 + 3 * q^85 - 21 * q^87 - 3 * q^89 - 33 * q^91 - 9 * q^93 - 45 * q^97 + 42 * q^99

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