LMFDB - Embedded newform 4788.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.00000 q^{5} +1.00000 q^{7} -0.791288 q^{11} -1.00000 q^{13} +0.791288 q^{17} +1.00000 q^{19} +4.58258 q^{23} +4.00000 q^{25} +0.791288 q^{29} -6.37386 q^{31} -3.00000 q^{35} +5.00000 q^{37} -0.791288 q^{41} +2.00000 q^{43} -1.41742 q^{47} +1.00000 q^{49} -5.37386 q^{53} +2.37386 q^{55} +6.16515 q^{59} -1.00000 q^{61} +3.00000 q^{65} +4.37386 q^{67} -6.16515 q^{71} +2.62614 q^{73} -0.791288 q^{77} -10.0000 q^{79} -0.626136 q^{83} -2.37386 q^{85} +1.58258 q^{89} -1.00000 q^{91} -3.00000 q^{95} -7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 6 * q^5 + 2 * q^7	$ 2 q - 6 q^{5} + 2 q^{7} + 3 q^{11} - 2 q^{13} - 3 q^{17} + 2 q^{19} + 8 q^{25} - 3 q^{29} + q^{31} - 6 q^{35} + 10 q^{37} + 3 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 3 q^{53} - 9 q^{55} - 6 q^{59} - 2 q^{61} + 6 q^{65} - 5 q^{67} + 6 q^{71} + 19 q^{73} + 3 q^{77} - 20 q^{79} - 15 q^{83} + 9 q^{85} - 6 q^{89} - 2 q^{91} - 6 q^{95} - 14 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 2 * q^7 + 3 * q^11 - 2 * q^13 - 3 * q^17 + 2 * q^19 + 8 * q^25 - 3 * q^29 + q^31 - 6 * q^35 + 10 * q^37 + 3 * q^41 + 4 * q^43 - 12 * q^47 + 2 * q^49 + 3 * q^53 - 9 * q^55 - 6 * q^59 - 2 * q^61 + 6 * q^65 - 5 * q^67 + 6 * q^71 + 19 * q^73 + 3 * q^77 - 20 * q^79 - 15 * q^83 + 9 * q^85 - 6 * q^89 - 2 * q^91 - 6 * q^95 - 14 * 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	0
$3$	0	0
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.791288	−0.238582	−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288	−0.238582	−0.119291	+	0.992859i	$0.538062\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.791288	0.191915	0.0959577	−	0.995385i	$-0.469409\pi$
$17$	0.791288	0.191915	0.0959577	+	0.995385i	$0.469409\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.58258	0.955533	0.477767	−	0.878487i	$-0.341446\pi$
$23$	4.58258	0.955533	0.477767	+	0.878487i	$0.341446\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.791288	0.146938	0.0734692	−	0.997297i	$-0.476593\pi$
$29$	0.791288	0.146938	0.0734692	+	0.997297i	$0.476593\pi$
$30$	0	0
$31$	−6.37386	−1.14478	−0.572390	−	0.819982i	$-0.693984\pi$
$31$	−6.37386	−1.14478	−0.572390	+	0.819982i	$0.693984\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.791288	−0.123578	−0.0617892	−	0.998089i	$-0.519681\pi$
$41$	−0.791288	−0.123578	−0.0617892	+	0.998089i	$0.519681\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.37386	−0.738157	−0.369078	−	0.929398i	$-0.620327\pi$
$53$	−5.37386	−0.738157	−0.369078	+	0.929398i	$0.620327\pi$
$54$	0	0
$55$	2.37386	0.320092
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.16515	0.802634	0.401317	−	0.915939i	$-0.368553\pi$
$59$	6.16515	0.802634	0.401317	+	0.915939i	$0.368553\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	4.37386	0.534352	0.267176	−	0.963648i	$-0.413909\pi$
$67$	4.37386	0.534352	0.267176	+	0.963648i	$0.413909\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.16515	−0.731669	−0.365834	−	0.930680i	$-0.619216\pi$
$71$	−6.16515	−0.731669	−0.365834	+	0.930680i	$0.619216\pi$
$72$	0	0
$73$	2.62614	0.307366	0.153683	−	0.988120i	$-0.450887\pi$
$73$	2.62614	0.307366	0.153683	+	0.988120i	$0.450887\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.791288	−0.0901756
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.626136	−0.0687274	−0.0343637	−	0.999409i	$-0.510940\pi$
$83$	−0.626136	−0.0687274	−0.0343637	+	0.999409i	$0.510940\pi$
$84$	0	0
$85$	−2.37386	−0.257482
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.58258	0.167753	0.0838763	−	0.996476i	$-0.473270\pi$
$89$	1.58258	0.167753	0.0838763	+	0.996476i	$0.473270\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.7477	−1.96497	−0.982486	−	0.186336i	$-0.940339\pi$
$101$	−19.7477	−1.96497	−0.982486	+	0.186336i	$0.940339\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.1652	1.75609	0.878046	−	0.478577i	$-0.158847\pi$
$107$	18.1652	1.75609	0.878046	+	0.478577i	$0.158847\pi$
$108$	0	0
$109$	15.7477	1.50836	0.754179	−	0.656668i	$-0.228035\pi$
$109$	15.7477	1.50836	0.754179	+	0.656668i	$0.228035\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.37386	0.505531	0.252765	−	0.967528i	$-0.418660\pi$
$113$	5.37386	0.505531	0.252765	+	0.967528i	$0.418660\pi$
$114$	0	0
$115$	−13.7477	−1.28198
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.791288	0.0725372
$120$	0	0
$121$	−10.3739	−0.943079
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−20.7477	−1.84106	−0.920532	−	0.390668i	$-0.872244\pi$
$127$	−20.7477	−1.84106	−0.920532	+	0.390668i	$0.872244\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.1216	−1.67066	−0.835331	−	0.549748i	$-0.814724\pi$
$131$	−19.1216	−1.67066	−0.835331	+	0.549748i	$0.814724\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−17.7477	−1.50534	−0.752671	−	0.658396i	$-0.771235\pi$
$139$	−17.7477	−1.50534	−0.752671	+	0.658396i	$0.771235\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.791288	0.0661708
$144$	0	0
$145$	−2.37386	−0.197139
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−0.373864	−0.0304246	−0.0152123	−	0.999884i	$-0.504842\pi$
$151$	−0.373864	−0.0304246	−0.0152123	+	0.999884i	$0.504842\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19.1216	1.53588
$156$	0	0
$157$	21.1216	1.68569	0.842843	−	0.538159i	$-0.180880\pi$
$157$	21.1216	1.68569	0.842843	+	0.538159i	$0.180880\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.58258	0.361158
$162$	0	0
$163$	−9.37386	−0.734218	−0.367109	−	0.930178i	$-0.619652\pi$
$163$	−9.37386	−0.734218	−0.367109	+	0.930178i	$0.619652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.5826	−1.74749	−0.873746	−	0.486382i	$-0.838316\pi$
$167$	−22.5826	−1.74749	−0.873746	+	0.486382i	$0.838316\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.74773	0.360963	0.180482	−	0.983578i	$-0.442234\pi$
$173$	4.74773	0.360963	0.180482	+	0.983578i	$0.442234\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.2087	0.837778	0.418889	−	0.908037i	$-0.362420\pi$
$179$	11.2087	0.837778	0.418889	+	0.908037i	$0.362420\pi$
$180$	0	0
$181$	4.37386	0.325107	0.162553	−	0.986700i	$-0.448027\pi$
$181$	4.37386	0.325107	0.162553	+	0.986700i	$0.448027\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.0000	−1.10282
$186$	0	0
$187$	−0.626136	−0.0457876
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.53901	0.617861	0.308931	−	0.951085i	$-0.400029\pi$
$191$	8.53901	0.617861	0.308931	+	0.951085i	$0.400029\pi$
$192$	0	0
$193$	−6.37386	−0.458801	−0.229400	−	0.973332i	$-0.573677\pi$
$193$	−6.37386	−0.458801	−0.229400	+	0.973332i	$0.573677\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.3739	−0.810354	−0.405177	−	0.914238i	$-0.632790\pi$
$197$	−11.3739	−0.810354	−0.405177	+	0.914238i	$0.632790\pi$
$198$	0	0
$199$	3.74773	0.265669	0.132835	−	0.991138i	$-0.457592\pi$
$199$	3.74773	0.265669	0.132835	+	0.991138i	$0.457592\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.791288	0.0555375
$204$	0	0
$205$	2.37386	0.165798
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.791288	−0.0547345
$210$	0	0
$211$	−10.6261	−0.731533	−0.365767	−	0.930707i	$-0.619193\pi$
$211$	−10.6261	−0.731533	−0.365767	+	0.930707i	$0.619193\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−6.37386	−0.432686
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.791288	−0.0532278
$222$	0	0
$223$	3.74773	0.250966	0.125483	−	0.992096i	$-0.459952\pi$
$223$	3.74773	0.250966	0.125483	+	0.992096i	$0.459952\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.791288	−0.0525196	−0.0262598	−	0.999655i	$-0.508360\pi$
$227$	−0.791288	−0.0525196	−0.0262598	+	0.999655i	$0.508360\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.1216	−0.859624	−0.429812	−	0.902918i	$-0.641420\pi$
$233$	−13.1216	−0.859624	−0.429812	+	0.902918i	$0.641420\pi$
$234$	0	0
$235$	4.25227	0.277388
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.7477	−1.27737	−0.638687	−	0.769467i	$-0.720522\pi$
$239$	−19.7477	−1.27737	−0.638687	+	0.769467i	$0.720522\pi$
$240$	0	0
$241$	18.7477	1.20765	0.603824	−	0.797118i	$-0.293643\pi$
$241$	18.7477	1.20765	0.603824	+	0.797118i	$0.293643\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.20871	−0.328771	−0.164385	−	0.986396i	$-0.552564\pi$
$251$	−5.20871	−0.328771	−0.164385	+	0.986396i	$0.552564\pi$
$252$	0	0
$253$	−3.62614	−0.227973
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.7913	−0.985033	−0.492517	−	0.870303i	$-0.663923\pi$
$257$	−15.7913	−0.985033	−0.492517	+	0.870303i	$0.663923\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.20871	−0.136195	−0.0680975	−	0.997679i	$-0.521693\pi$
$263$	−2.20871	−0.136195	−0.0680975	+	0.997679i	$0.521693\pi$
$264$	0	0
$265$	16.1216	0.990341
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.9564	−0.789968	−0.394984	−	0.918688i	$-0.629250\pi$
$269$	−12.9564	−0.789968	−0.394984	+	0.918688i	$0.629250\pi$
$270$	0	0
$271$	27.1216	1.64752	0.823760	−	0.566939i	$-0.191873\pi$
$271$	27.1216	1.64752	0.823760	+	0.566939i	$0.191873\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.16515	−0.190866
$276$	0	0
$277$	30.7477	1.84745	0.923726	−	0.383054i	$-0.125128\pi$
$277$	30.7477	1.84745	0.923726	+	0.383054i	$0.125128\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.5826	−1.70509	−0.852547	−	0.522651i	$-0.824943\pi$
$281$	−28.5826	−1.70509	−0.852547	+	0.522651i	$0.824943\pi$
$282$	0	0
$283$	7.37386	0.438331	0.219165	−	0.975688i	$-0.429667\pi$
$283$	7.37386	0.438331	0.219165	+	0.975688i	$0.429667\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.791288	−0.0467082
$288$	0	0
$289$	−16.3739	−0.963168
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.74773	0.452627	0.226314	−	0.974055i	$-0.427333\pi$
$293$	7.74773	0.452627	0.226314	+	0.974055i	$0.427333\pi$
$294$	0	0
$295$	−18.4955	−1.07685
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.58258	−0.265017
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.00000	0.171780
$306$	0	0
$307$	−13.6261	−0.777685	−0.388842	−	0.921304i	$-0.627125\pi$
$307$	−13.6261	−0.777685	−0.388842	+	0.921304i	$0.627125\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.7913	−1.74601	−0.873007	−	0.487708i	$-0.837833\pi$
$311$	−30.7913	−1.74601	−0.873007	+	0.487708i	$0.837833\pi$
$312$	0	0
$313$	−14.7477	−0.833591	−0.416795	−	0.909000i	$-0.636847\pi$
$313$	−14.7477	−0.833591	−0.416795	+	0.909000i	$0.636847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.1652	−1.86274	−0.931370	−	0.364073i	$-0.881386\pi$
$317$	−33.1652	−1.86274	−0.931370	+	0.364073i	$0.881386\pi$
$318$	0	0
$319$	−0.626136	−0.0350569
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.791288	0.0440284
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.41742	−0.0781451
$330$	0	0
$331$	14.6261	0.803925	0.401963	−	0.915656i	$-0.368328\pi$
$331$	14.6261	0.803925	0.401963	+	0.915656i	$0.368328\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.1216	−0.716909
$336$	0	0
$337$	−12.3739	−0.674047	−0.337024	−	0.941496i	$-0.609420\pi$
$337$	−12.3739	−0.674047	−0.337024	+	0.941496i	$0.609420\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.04356	0.273124
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.79129	0.203527	0.101763	−	0.994809i	$-0.467552\pi$
$347$	3.79129	0.203527	0.101763	+	0.994809i	$0.467552\pi$
$348$	0	0
$349$	−15.3739	−0.822944	−0.411472	−	0.911422i	$-0.634985\pi$
$349$	−15.3739	−0.822944	−0.411472	+	0.911422i	$0.634985\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.5390	−1.25286	−0.626428	−	0.779480i	$-0.715484\pi$
$353$	−23.5390	−1.25286	−0.626428	+	0.779480i	$0.715484\pi$
$354$	0	0
$355$	18.4955	0.981637
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.3739	0.758624	0.379312	−	0.925269i	$-0.376161\pi$
$359$	14.3739	0.758624	0.379312	+	0.925269i	$0.376161\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.87841	−0.412375
$366$	0	0
$367$	32.4955	1.69625	0.848124	−	0.529797i	$-0.177732\pi$
$367$	32.4955	1.69625	0.848124	+	0.529797i	$0.177732\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.37386	−0.278997
$372$	0	0
$373$	2.62614	0.135976	0.0679881	−	0.997686i	$-0.478342\pi$
$373$	2.62614	0.135976	0.0679881	+	0.997686i	$0.478342\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.791288	−0.0407534
$378$	0	0
$379$	−5.74773	−0.295241	−0.147620	−	0.989044i	$-0.547161\pi$
$379$	−5.74773	−0.295241	−0.147620	+	0.989044i	$0.547161\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	2.37386	0.120983
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.3739	1.64142	0.820710	−	0.571345i	$-0.193578\pi$
$389$	32.3739	1.64142	0.820710	+	0.571345i	$0.193578\pi$
$390$	0	0
$391$	3.62614	0.183382
$392$	0	0
$393$	0	0
$394$	0	0
$395$	30.0000	1.50946
$396$	0	0
$397$	−32.7477	−1.64356	−0.821781	−	0.569804i	$-0.807019\pi$
$397$	−32.7477	−1.64356	−0.821781	+	0.569804i	$0.807019\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.3739	1.31705	0.658524	−	0.752560i	$-0.271181\pi$
$401$	26.3739	1.31705	0.658524	+	0.752560i	$0.271181\pi$
$402$	0	0
$403$	6.37386	0.317505
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.95644	−0.196113
$408$	0	0
$409$	−14.1216	−0.698268	−0.349134	−	0.937073i	$-0.613524\pi$
$409$	−14.1216	−0.698268	−0.349134	+	0.937073i	$0.613524\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.16515	0.303367
$414$	0	0
$415$	1.87841	0.0922075
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.74773	−0.0853821	−0.0426910	−	0.999088i	$-0.513593\pi$
$419$	−1.74773	−0.0853821	−0.0426910	+	0.999088i	$0.513593\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.16515	0.153532
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	−17.7477	−0.852901	−0.426451	−	0.904511i	$-0.640236\pi$
$433$	−17.7477	−0.852901	−0.426451	+	0.904511i	$0.640236\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.58258	0.219214
$438$	0	0
$439$	−13.4955	−0.644103	−0.322051	−	0.946722i	$-0.604372\pi$
$439$	−13.4955	−0.644103	−0.322051	+	0.946722i	$0.604372\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.9564	1.75585	0.877927	−	0.478795i	$-0.158926\pi$
$443$	36.9564	1.75585	0.877927	+	0.478795i	$0.158926\pi$
$444$	0	0
$445$	−4.74773	−0.225064
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.1216	0.760825	0.380412	−	0.924817i	$-0.375782\pi$
$449$	16.1216	0.760825	0.380412	+	0.924817i	$0.375782\pi$
$450$	0	0
$451$	0.626136	0.0294836
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.00000	0.140642
$456$	0	0
$457$	28.8693	1.35045	0.675225	−	0.737612i	$-0.264047\pi$
$457$	28.8693	1.35045	0.675225	+	0.737612i	$0.264047\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.5390	−0.956597	−0.478299	−	0.878197i	$-0.658746\pi$
$461$	−20.5390	−0.956597	−0.478299	+	0.878197i	$0.658746\pi$
$462$	0	0
$463$	20.4955	0.952505	0.476252	−	0.879309i	$-0.341995\pi$
$463$	20.4955	0.952505	0.476252	+	0.879309i	$0.341995\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.2867	1.17013	0.585065	−	0.810986i	$-0.301069\pi$
$467$	25.2867	1.17013	0.585065	+	0.810986i	$0.301069\pi$
$468$	0	0
$469$	4.37386	0.201966
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.58258	−0.0727669
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.3739	−1.20505	−0.602526	−	0.798099i	$-0.705839\pi$
$479$	−26.3739	−1.20505	−0.602526	+	0.798099i	$0.705839\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.0000	0.953561
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.16515	−0.142841	−0.0714206	−	0.997446i	$-0.522753\pi$
$491$	−3.16515	−0.142841	−0.0714206	+	0.997446i	$0.522753\pi$
$492$	0	0
$493$	0.626136	0.0281998
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.16515	−0.276545
$498$	0	0
$499$	31.3739	1.40449	0.702244	−	0.711937i	$-0.252182\pi$
$499$	31.3739	1.40449	0.702244	+	0.711937i	$0.252182\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.3303	−1.35236	−0.676181	−	0.736736i	$-0.736366\pi$
$503$	−30.3303	−1.35236	−0.676181	+	0.736736i	$0.736366\pi$
$504$	0	0
$505$	59.2432	2.63629
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.3303	1.61031	0.805156	−	0.593063i	$-0.202081\pi$
$509$	36.3303	1.61031	0.805156	+	0.593063i	$0.202081\pi$
$510$	0	0
$511$	2.62614	0.116173
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39.0000	1.71855
$516$	0	0
$517$	1.12159	0.0493275
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.5826	0.989361	0.494680	−	0.869075i	$-0.335285\pi$
$521$	22.5826	0.989361	0.494680	+	0.869075i	$0.335285\pi$
$522$	0	0
$523$	27.7477	1.21332	0.606662	−	0.794960i	$-0.292508\pi$
$523$	27.7477	1.21332	0.606662	+	0.794960i	$0.292508\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.04356	−0.219701
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.791288	0.0342745
$534$	0	0
$535$	−54.4955	−2.35604
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.791288	−0.0340832
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−47.2432	−2.02368
$546$	0	0
$547$	1.37386	0.0587422	0.0293711	−	0.999569i	$-0.490650\pi$
$547$	1.37386	0.0587422	0.0293711	+	0.999569i	$0.490650\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.791288	0.0337100
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.3739	−0.609040	−0.304520	−	0.952506i	$-0.598496\pi$
$557$	−14.3739	−0.609040	−0.304520	+	0.952506i	$0.598496\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.16515	−0.259830	−0.129915	−	0.991525i	$-0.541470\pi$
$563$	−6.16515	−0.259830	−0.129915	+	0.991525i	$0.541470\pi$
$564$	0	0
$565$	−16.1216	−0.678240
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.8348	−0.496143	−0.248071	−	0.968742i	$-0.579797\pi$
$569$	−11.8348	−0.496143	−0.248071	+	0.968742i	$0.579797\pi$
$570$	0	0
$571$	−22.4955	−0.941405	−0.470703	−	0.882292i	$-0.656000\pi$
$571$	−22.4955	−0.941405	−0.470703	+	0.882292i	$0.656000\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.3303	0.764426
$576$	0	0
$577$	18.1216	0.754412	0.377206	−	0.926129i	$-0.376885\pi$
$577$	18.1216	0.754412	0.377206	+	0.926129i	$0.376885\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.626136	−0.0259765
$582$	0	0
$583$	4.25227	0.176111
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.16515	−0.378286	−0.189143	−	0.981950i	$-0.560571\pi$
$587$	−9.16515	−0.378286	−0.189143	+	0.981950i	$0.560571\pi$
$588$	0	0
$589$	−6.37386	−0.262630
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.9129	−0.694529	−0.347264	−	0.937767i	$-0.612889\pi$
$593$	−16.9129	−0.694529	−0.347264	+	0.937767i	$0.612889\pi$
$594$	0	0
$595$	−2.37386	−0.0973189
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.62614	0.270737	0.135368	−	0.990795i	$-0.456778\pi$
$599$	6.62614	0.270737	0.135368	+	0.990795i	$0.456778\pi$
$600$	0	0
$601$	25.3739	1.03502	0.517511	−	0.855677i	$-0.326859\pi$
$601$	25.3739	1.03502	0.517511	+	0.855677i	$0.326859\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	31.1216	1.26527
$606$	0	0
$607$	−28.4955	−1.15659	−0.578297	−	0.815826i	$-0.696283\pi$
$607$	−28.4955	−1.15659	−0.578297	+	0.815826i	$0.696283\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.41742	0.0573428
$612$	0	0
$613$	20.6261	0.833082	0.416541	−	0.909117i	$-0.363242\pi$
$613$	20.6261	0.833082	0.416541	+	0.909117i	$0.363242\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.9564	−0.642382	−0.321191	−	0.947014i	$-0.604083\pi$
$617$	−15.9564	−0.642382	−0.321191	+	0.947014i	$0.604083\pi$
$618$	0	0
$619$	−9.37386	−0.376767	−0.188384	−	0.982096i	$-0.560325\pi$
$619$	−9.37386	−0.376767	−0.188384	+	0.982096i	$0.560325\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.58258	0.0634046
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.95644	0.157754
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	62.2432	2.47005
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.7913	0.979197	0.489598	−	0.871948i	$-0.337143\pi$
$641$	24.7913	0.979197	0.489598	+	0.871948i	$0.337143\pi$
$642$	0	0
$643$	23.0000	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.5826	0.887813	0.443906	−	0.896073i	$-0.353592\pi$
$647$	22.5826	0.887813	0.443906	+	0.896073i	$0.353592\pi$
$648$	0	0
$649$	−4.87841	−0.191494
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.5826	−1.11852	−0.559261	−	0.828991i	$-0.688915\pi$
$653$	−28.5826	−1.11852	−0.559261	+	0.828991i	$0.688915\pi$
$654$	0	0
$655$	57.3648	2.24143
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.0436	0.547059	0.273530	−	0.961864i	$-0.411809\pi$
$659$	14.0436	0.547059	0.273530	+	0.961864i	$0.411809\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	3.62614	0.140405
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.791288	0.0305473
$672$	0	0
$673$	24.1216	0.929819	0.464909	−	0.885358i	$-0.346087\pi$
$673$	24.1216	0.929819	0.464909	+	0.885358i	$0.346087\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.956439	−0.0367589	−0.0183795	−	0.999831i	$-0.505851\pi$
$677$	−0.956439	−0.0367589	−0.0183795	+	0.999831i	$0.505851\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17.8348	0.682432	0.341216	−	0.939985i	$-0.389161\pi$
$683$	17.8348	0.682432	0.341216	+	0.939985i	$0.389161\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.37386	0.204728
$690$	0	0
$691$	21.2523	0.808475	0.404237	−	0.914654i	$-0.367537\pi$
$691$	21.2523	0.808475	0.404237	+	0.914654i	$0.367537\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	53.2432	2.01963
$696$	0	0
$697$	−0.626136	−0.0237166
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.66970	−0.214142	−0.107071	−	0.994251i	$-0.534147\pi$
$701$	−5.66970	−0.214142	−0.107071	+	0.994251i	$0.534147\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.7477	−0.742690
$708$	0	0
$709$	−51.2432	−1.92448	−0.962239	−	0.272206i	$-0.912247\pi$
$709$	−51.2432	−1.92448	−0.962239	+	0.272206i	$0.912247\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−29.2087	−1.09387
$714$	0	0
$715$	−2.37386	−0.0887775
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.16515	0.117551
$726$	0	0
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.58258	0.0585337
$732$	0	0
$733$	−13.4955	−0.498466	−0.249233	−	0.968444i	$-0.580178\pi$
$733$	−13.4955	−0.498466	−0.249233	+	0.968444i	$0.580178\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.46099	−0.127487
$738$	0	0
$739$	−8.25227	−0.303565	−0.151782	−	0.988414i	$-0.548501\pi$
$739$	−8.25227	−0.303565	−0.151782	+	0.988414i	$0.548501\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.3303	0.562414	0.281207	−	0.959647i	$-0.409265\pi$
$743$	15.3303	0.562414	0.281207	+	0.959647i	$0.409265\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.1652	0.663740
$750$	0	0
$751$	25.3739	0.925905	0.462953	−	0.886383i	$-0.346790\pi$
$751$	25.3739	0.925905	0.462953	+	0.886383i	$0.346790\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.12159	0.0408189
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.66970	0.0967764	0.0483882	−	0.998829i	$-0.484592\pi$
$761$	2.66970	0.0967764	0.0483882	+	0.998829i	$0.484592\pi$
$762$	0	0
$763$	15.7477	0.570106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.16515	−0.222611
$768$	0	0
$769$	41.4955	1.49636	0.748182	−	0.663493i	$-0.230927\pi$
$769$	41.4955	1.49636	0.748182	+	0.663493i	$0.230927\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.7477	0.386569	0.193284	−	0.981143i	$-0.438086\pi$
$773$	10.7477	0.386569	0.193284	+	0.981143i	$0.438086\pi$
$774$	0	0
$775$	−25.4955	−0.915824
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.791288	−0.0283508
$780$	0	0
$781$	4.87841	0.174563
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−63.3648	−2.26159
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.37386	0.191073
$792$	0	0
$793$	1.00000	0.0355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.8693	−1.27056	−0.635278	−	0.772283i	$-0.719115\pi$
$797$	−35.8693	−1.27056	−0.635278	+	0.772283i	$0.719115\pi$
$798$	0	0
$799$	−1.12159	−0.0396790
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.07803	−0.0733321
$804$	0	0
$805$	−13.7477	−0.484544
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.9129	−0.805574	−0.402787	−	0.915294i	$-0.631958\pi$
$809$	−22.9129	−0.805574	−0.402787	+	0.915294i	$0.631958\pi$
$810$	0	0
$811$	23.4955	0.825037	0.412518	−	0.910949i	$-0.364649\pi$
$811$	23.4955	0.825037	0.412518	+	0.910949i	$0.364649\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.1216	0.985056
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.66970	−0.197874	−0.0989369	−	0.995094i	$-0.531544\pi$
$821$	−5.66970	−0.197874	−0.0989369	+	0.995094i	$0.531544\pi$
$822$	0	0
$823$	28.2432	0.984495	0.492248	−	0.870455i	$-0.336175\pi$
$823$	28.2432	0.984495	0.492248	+	0.870455i	$0.336175\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.6606	1.17049	0.585247	−	0.810855i	$-0.300998\pi$
$827$	33.6606	1.17049	0.585247	+	0.810855i	$0.300998\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.791288	0.0274165
$834$	0	0
$835$	67.7477	2.34451
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.1652	−0.937845	−0.468923	−	0.883239i	$-0.655358\pi$
$839$	−27.1652	−0.937845	−0.468923	+	0.883239i	$0.655358\pi$
$840$	0	0
$841$	−28.3739	−0.978409
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36.0000	1.23844
$846$	0	0
$847$	−10.3739	−0.356450
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.9129	0.785443
$852$	0	0
$853$	−32.1216	−1.09982	−0.549911	−	0.835223i	$-0.685338\pi$
$853$	−32.1216	−1.09982	−0.549911	+	0.835223i	$0.685338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.7042	−1.32211	−0.661055	−	0.750338i	$-0.729891\pi$
$857$	−38.7042	−1.32211	−0.661055	+	0.750338i	$0.729891\pi$
$858$	0	0
$859$	−6.37386	−0.217473	−0.108737	−	0.994071i	$-0.534681\pi$
$859$	−6.37386	−0.217473	−0.108737	+	0.994071i	$0.534681\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.7042	−0.398414	−0.199207	−	0.979957i	$-0.563837\pi$
$863$	−11.7042	−0.398414	−0.199207	+	0.979957i	$0.563837\pi$
$864$	0	0
$865$	−14.2432	−0.484283
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.91288	0.268426
$870$	0	0
$871$	−4.37386	−0.148203
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−11.2523	−0.379962	−0.189981	−	0.981788i	$-0.560843\pi$
$877$	−11.2523	−0.379962	−0.189981	+	0.981788i	$0.560843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.3739	0.383195	0.191598	−	0.981474i	$-0.438633\pi$
$881$	11.3739	0.383195	0.191598	+	0.981474i	$0.438633\pi$
$882$	0	0
$883$	51.7477	1.74145	0.870725	−	0.491771i	$-0.163650\pi$
$883$	51.7477	1.74145	0.870725	+	0.491771i	$0.163650\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.58258	0.153868	0.0769339	−	0.997036i	$-0.475487\pi$
$887$	4.58258	0.153868	0.0769339	+	0.997036i	$0.475487\pi$
$888$	0	0
$889$	−20.7477	−0.695856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.41742	−0.0474323
$894$	0	0
$895$	−33.6261	−1.12400
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.04356	−0.168212
$900$	0	0
$901$	−4.25227	−0.141664
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.1216	−0.436176
$906$	0	0
$907$	−51.2432	−1.70150	−0.850751	−	0.525569i	$-0.823852\pi$
$907$	−51.2432	−1.70150	−0.850751	+	0.525569i	$0.823852\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.4955	1.70612	0.853060	−	0.521812i	$-0.174744\pi$
$911$	51.4955	1.70612	0.853060	+	0.521812i	$0.174744\pi$
$912$	0	0
$913$	0.495454	0.0163971
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.1216	−0.631451
$918$	0	0
$919$	20.4955	0.676083	0.338041	−	0.941131i	$-0.390236\pi$
$919$	20.4955	0.676083	0.338041	+	0.941131i	$0.390236\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.16515	0.202928
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.8693	0.684700	0.342350	−	0.939572i	$-0.388777\pi$
$929$	20.8693	0.684700	0.342350	+	0.939572i	$0.388777\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.87841	0.0614306
$936$	0	0
$937$	−1.62614	−0.0531236	−0.0265618	−	0.999647i	$-0.508456\pi$
$937$	−1.62614	−0.0531236	−0.0265618	+	0.999647i	$0.508456\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.4955	0.896326	0.448163	−	0.893952i	$-0.352078\pi$
$941$	27.4955	0.896326	0.448163	+	0.893952i	$0.352078\pi$
$942$	0	0
$943$	−3.62614	−0.118083
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.6170	−1.41736	−0.708682	−	0.705528i	$-0.750710\pi$
$947$	−43.6170	−1.41736	−0.708682	+	0.705528i	$0.750710\pi$
$948$	0	0
$949$	−2.62614	−0.0852480
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−53.5390	−1.73430	−0.867149	−	0.498048i	$-0.834050\pi$
$953$	−53.5390	−1.73430	−0.867149	+	0.498048i	$0.834050\pi$
$954$	0	0
$955$	−25.6170	−0.828948
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	9.62614	0.310521
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.1216	0.615546
$966$	0	0
$967$	−29.1216	−0.936487	−0.468244	−	0.883599i	$-0.655113\pi$
$967$	−29.1216	−0.936487	−0.468244	+	0.883599i	$0.655113\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	−17.7477	−0.568966
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.5826	1.39433	0.697165	−	0.716911i	$-0.254444\pi$
$977$	43.5826	1.39433	0.697165	+	0.716911i	$0.254444\pi$
$978$	0	0
$979$	−1.25227	−0.0400228
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.3303	−1.44581	−0.722906	−	0.690946i	$-0.757194\pi$
$983$	−45.3303	−1.44581	−0.722906	+	0.690946i	$0.757194\pi$
$984$	0	0
$985$	34.1216	1.08720
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.16515	0.291435
$990$	0	0
$991$	18.7477	0.595541	0.297771	−	0.954637i	$-0.403757\pi$
$991$	18.7477	0.595541	0.297771	+	0.954637i	$0.403757\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.2432	−0.356433
$996$	0	0
$997$	−44.1216	−1.39734	−0.698672	−	0.715442i	$-0.746225\pi$
$997$	−44.1216	−1.39734	−0.698672	+	0.715442i	$0.746225\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.g.1.1		2
3.2	odd	2		532.2.a.c.1.2	✓	2
12.11	even	2		2128.2.a.k.1.1		2
21.20	even	2		3724.2.a.e.1.1		2
24.5	odd	2		8512.2.a.t.1.1		2
24.11	even	2		8512.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.c.1.2	✓	2	3.2	odd	2
2128.2.a.k.1.1		2	12.11	even	2
3724.2.a.e.1.1		2	21.20	even	2
4788.2.a.g.1.1		2	1.1	even	1	trivial
8512.2.a.m.1.2		2	24.11	even	2
8512.2.a.t.1.1		2	24.5	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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.00000 q^{5} +1.00000 q^{7} -0.791288 q^{11} -1.00000 q^{13} +0.791288 q^{17} +1.00000 q^{19} +4.58258 q^{23} +4.00000 q^{25} +0.791288 q^{29} -6.37386 q^{31} -3.00000 q^{35} +5.00000 q^{37} -0.791288 q^{41} +2.00000 q^{43} -1.41742 q^{47} +1.00000 q^{49} -5.37386 q^{53} +2.37386 q^{55} +6.16515 q^{59} -1.00000 q^{61} +3.00000 q^{65} +4.37386 q^{67} -6.16515 q^{71} +2.62614 q^{73} -0.791288 q^{77} -10.0000 q^{79} -0.626136 q^{83} -2.37386 q^{85} +1.58258 q^{89} -1.00000 q^{91} -3.00000 q^{95} -7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 6 * q^5 + 2 * q^7	\( 2 q - 6 q^{5} + 2 q^{7} + 3 q^{11} - 2 q^{13} - 3 q^{17} + 2 q^{19} + 8 q^{25} - 3 q^{29} + q^{31} - 6 q^{35} + 10 q^{37} + 3 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 3 q^{53} - 9 q^{55} - 6 q^{59} - 2 q^{61} + 6 q^{65} - 5 q^{67} + 6 q^{71} + 19 q^{73} + 3 q^{77} - 20 q^{79} - 15 q^{83} + 9 q^{85} - 6 q^{89} - 2 q^{91} - 6 q^{95} - 14 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 2 * q^7 + 3 * q^11 - 2 * q^13 - 3 * q^17 + 2 * q^19 + 8 * q^25 - 3 * q^29 + q^31 - 6 * q^35 + 10 * q^37 + 3 * q^41 + 4 * q^43 - 12 * q^47 + 2 * q^49 + 3 * q^53 - 9 * q^55 - 6 * q^59 - 2 * q^61 + 6 * q^65 - 5 * q^67 + 6 * q^71 + 19 * q^73 + 3 * q^77 - 20 * q^79 - 15 * q^83 + 9 * q^85 - 6 * q^89 - 2 * q^91 - 6 * q^95 - 14 * q^97

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