LMFDB - Embedded newform 2288.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(1,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, 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("2288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.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:	$18.2697719825$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	2288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} +2.00000 q^{5} -0.208712 q^{7} +4.79129 q^{9} +O(q^{10})$	$q-2.79129 q^{3} +2.00000 q^{5} -0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} -1.00000 q^{13} -5.58258 q^{15} -4.00000 q^{17} -1.20871 q^{19} +0.582576 q^{21} -0.208712 q^{23} -1.00000 q^{25} -5.00000 q^{27} -3.58258 q^{29} -2.79129 q^{33} -0.417424 q^{35} +4.00000 q^{37} +2.79129 q^{39} +1.79129 q^{41} +7.16515 q^{43} +9.58258 q^{45} -3.58258 q^{47} -6.95644 q^{49} +11.1652 q^{51} +4.37386 q^{53} +2.00000 q^{55} +3.37386 q^{57} +5.58258 q^{59} +2.41742 q^{61} -1.00000 q^{63} -2.00000 q^{65} -11.1652 q^{67} +0.582576 q^{69} -10.7913 q^{73} +2.79129 q^{75} -0.208712 q^{77} -12.7477 q^{79} -0.417424 q^{81} -0.626136 q^{83} -8.00000 q^{85} +10.0000 q^{87} +9.58258 q^{89} +0.208712 q^{91} -2.41742 q^{95} +18.7477 q^{97} +4.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	$ 2 q - q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 8 q^{17} - 7 q^{19} - 8 q^{21} - 5 q^{23} - 2 q^{25} - 10 q^{27} + 2 q^{29} - q^{33} - 10 q^{35} + 8 q^{37} + q^{39} - q^{41} - 4 q^{43} + 10 q^{45} + 2 q^{47} + 9 q^{49} + 4 q^{51} - 5 q^{53} + 4 q^{55} - 7 q^{57} + 2 q^{59} + 14 q^{61} - 2 q^{63} - 4 q^{65} - 4 q^{67} - 8 q^{69} - 17 q^{73} + q^{75} - 5 q^{77} + 2 q^{79} - 10 q^{81} - 15 q^{83} - 16 q^{85} + 20 q^{87} + 10 q^{89} + 5 q^{91} - 14 q^{95} + 10 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 8 * q^17 - 7 * q^19 - 8 * q^21 - 5 * q^23 - 2 * q^25 - 10 * q^27 + 2 * q^29 - q^33 - 10 * q^35 + 8 * q^37 + q^39 - q^41 - 4 * q^43 + 10 * q^45 + 2 * q^47 + 9 * q^49 + 4 * q^51 - 5 * q^53 + 4 * q^55 - 7 * q^57 + 2 * q^59 + 14 * q^61 - 2 * q^63 - 4 * q^65 - 4 * q^67 - 8 * q^69 - 17 * q^73 + q^75 - 5 * q^77 + 2 * q^79 - 10 * q^81 - 15 * q^83 - 16 * q^85 + 20 * q^87 + 10 * q^89 + 5 * q^91 - 14 * q^95 + 10 * q^97 + 5 * 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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−0.208712	−0.0788858	−0.0394429	−	0.999222i	$-0.512558\pi$
$7$	−0.208712	−0.0788858	−0.0394429	+	0.999222i	$0.512558\pi$
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−5.58258	−1.44141
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−1.20871	−0.277298	−0.138649	−	0.990342i	$-0.544276\pi$
$19$	−1.20871	−0.277298	−0.138649	+	0.990342i	$0.544276\pi$
$20$	0	0
$21$	0.582576	0.127128
$22$	0	0
$23$	−0.208712	−0.0435195	−0.0217597	−	0.999763i	$-0.506927\pi$
$23$	−0.208712	−0.0435195	−0.0217597	+	0.999763i	$0.506927\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−3.58258	−0.665268	−0.332634	−	0.943056i	$-0.607937\pi$
$29$	−3.58258	−0.665268	−0.332634	+	0.943056i	$0.607937\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−2.79129	−0.485901
$34$	0	0
$35$	−0.417424	−0.0705576
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	2.79129	0.446964
$40$	0	0
$41$	1.79129	0.279752	0.139876	−	0.990169i	$-0.455330\pi$
$41$	1.79129	0.279752	0.139876	+	0.990169i	$0.455330\pi$
$42$	0	0
$43$	7.16515	1.09268	0.546338	−	0.837565i	$-0.316022\pi$
$43$	7.16515	1.09268	0.546338	+	0.837565i	$0.316022\pi$
$44$	0	0
$45$	9.58258	1.42849
$46$	0	0
$47$	−3.58258	−0.522572	−0.261286	−	0.965261i	$-0.584147\pi$
$47$	−3.58258	−0.522572	−0.261286	+	0.965261i	$0.584147\pi$
$48$	0	0
$49$	−6.95644	−0.993777
$50$	0	0
$51$	11.1652	1.56343
$52$	0	0
$53$	4.37386	0.600796	0.300398	−	0.953814i	$-0.402880\pi$
$53$	4.37386	0.600796	0.300398	+	0.953814i	$0.402880\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	3.37386	0.446879
$58$	0	0
$59$	5.58258	0.726789	0.363395	−	0.931635i	$-0.381618\pi$
$59$	5.58258	0.726789	0.363395	+	0.931635i	$0.381618\pi$
$60$	0	0
$61$	2.41742	0.309519	0.154760	−	0.987952i	$-0.450540\pi$
$61$	2.41742	0.309519	0.154760	+	0.987952i	$0.450540\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−11.1652	−1.36404	−0.682020	−	0.731333i	$-0.738898\pi$
$67$	−11.1652	−1.36404	−0.682020	+	0.731333i	$0.738898\pi$
$68$	0	0
$69$	0.582576	0.0701339
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−10.7913	−1.26302	−0.631512	−	0.775366i	$-0.717565\pi$
$73$	−10.7913	−1.26302	−0.631512	+	0.775366i	$0.717565\pi$
$74$	0	0
$75$	2.79129	0.322310
$76$	0	0
$77$	−0.208712	−0.0237850
$78$	0	0
$79$	−12.7477	−1.43423	−0.717116	−	0.696954i	$-0.754538\pi$
$79$	−12.7477	−1.43423	−0.717116	+	0.696954i	$0.754538\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	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
$85$	−8.00000	−0.867722
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	9.58258	1.01575	0.507875	−	0.861430i	$-0.330431\pi$
$89$	9.58258	1.01575	0.507875	+	0.861430i	$0.330431\pi$
$90$	0	0
$91$	0.208712	0.0218790
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.41742	−0.248023
$96$	0	0
$97$	18.7477	1.90354	0.951772	−	0.306807i	$-0.0992607\pi$
$97$	18.7477	1.90354	0.951772	+	0.306807i	$0.0992607\pi$
$98$	0	0
$99$	4.79129	0.481543
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	0.208712	0.0205650	0.0102825	−	0.999947i	$-0.496727\pi$
$103$	0.208712	0.0205650	0.0102825	+	0.999947i	$0.496727\pi$
$104$	0	0
$105$	1.16515	0.113707
$106$	0	0
$107$	−9.58258	−0.926383	−0.463191	−	0.886258i	$-0.653296\pi$
$107$	−9.58258	−0.926383	−0.463191	+	0.886258i	$0.653296\pi$
$108$	0	0
$109$	−1.20871	−0.115774	−0.0578868	−	0.998323i	$-0.518436\pi$
$109$	−1.20871	−0.115774	−0.0578868	+	0.998323i	$0.518436\pi$
$110$	0	0
$111$	−11.1652	−1.05975
$112$	0	0
$113$	−21.1216	−1.98695	−0.993476	−	0.114041i	$-0.963621\pi$
$113$	−21.1216	−1.98695	−0.993476	+	0.114041i	$0.963621\pi$
$114$	0	0
$115$	−0.417424	−0.0389250
$116$	0	0
$117$	−4.79129	−0.442955
$118$	0	0
$119$	0.834849	0.0765304
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.00000	−0.450835
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−17.5826	−1.56020	−0.780101	−	0.625654i	$-0.784832\pi$
$127$	−17.5826	−1.56020	−0.780101	+	0.625654i	$0.784832\pi$
$128$	0	0
$129$	−20.0000	−1.76090
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	0.252273	0.0218748
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	−23.1652	−1.97913	−0.989566	−	0.144079i	$-0.953978\pi$
$137$	−23.1652	−1.97913	−0.989566	+	0.144079i	$0.953978\pi$
$138$	0	0
$139$	19.5826	1.66097	0.830486	−	0.557039i	$-0.188063\pi$
$139$	19.5826	1.66097	0.830486	+	0.557039i	$0.188063\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−7.16515	−0.595033
$146$	0	0
$147$	19.4174	1.60152
$148$	0	0
$149$	1.37386	0.112551	0.0562756	−	0.998415i	$-0.482077\pi$
$149$	1.37386	0.112551	0.0562756	+	0.998415i	$0.482077\pi$
$150$	0	0
$151$	−7.16515	−0.583092	−0.291546	−	0.956557i	$-0.594170\pi$
$151$	−7.16515	−0.583092	−0.291546	+	0.956557i	$0.594170\pi$
$152$	0	0
$153$	−19.1652	−1.54941
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.5390	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$157$	−12.5390	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$158$	0	0
$159$	−12.2087	−0.968214
$160$	0	0
$161$	0.0435608	0.00343307
$162$	0	0
$163$	1.16515	0.0912617	0.0456309	−	0.998958i	$-0.485470\pi$
$163$	1.16515	0.0912617	0.0456309	+	0.998958i	$0.485470\pi$
$164$	0	0
$165$	−5.58258	−0.434603
$166$	0	0
$167$	−15.5390	−1.20245	−0.601223	−	0.799082i	$-0.705319\pi$
$167$	−15.5390	−1.20245	−0.601223	+	0.799082i	$0.705319\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.79129	−0.442871
$172$	0	0
$173$	−15.5826	−1.18472	−0.592361	−	0.805673i	$-0.701804\pi$
$173$	−15.5826	−1.18472	−0.592361	+	0.805673i	$0.701804\pi$
$174$	0	0
$175$	0.208712	0.0157772
$176$	0	0
$177$	−15.5826	−1.17126
$178$	0	0
$179$	−3.16515	−0.236575	−0.118287	−	0.992979i	$-0.537740\pi$
$179$	−3.16515	−0.236575	−0.118287	+	0.992979i	$0.537740\pi$
$180$	0	0
$181$	12.3739	0.919742	0.459871	−	0.887986i	$-0.347896\pi$
$181$	12.3739	0.919742	0.459871	+	0.887986i	$0.347896\pi$
$182$	0	0
$183$	−6.74773	−0.498806
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	1.04356	0.0759079
$190$	0	0
$191$	2.04356	0.147867	0.0739334	−	0.997263i	$-0.476445\pi$
$191$	2.04356	0.147867	0.0739334	+	0.997263i	$0.476445\pi$
$192$	0	0
$193$	5.37386	0.386819	0.193410	−	0.981118i	$-0.438045\pi$
$193$	5.37386	0.386819	0.193410	+	0.981118i	$0.438045\pi$
$194$	0	0
$195$	5.58258	0.399777
$196$	0	0
$197$	−24.3739	−1.73657	−0.868283	−	0.496069i	$-0.834776\pi$
$197$	−24.3739	−1.73657	−0.868283	+	0.496069i	$0.834776\pi$
$198$	0	0
$199$	−1.20871	−0.0856833	−0.0428417	−	0.999082i	$-0.513641\pi$
$199$	−1.20871	−0.0856833	−0.0428417	+	0.999082i	$0.513641\pi$
$200$	0	0
$201$	31.1652	2.19822
$202$	0	0
$203$	0.747727	0.0524802
$204$	0	0
$205$	3.58258	0.250218
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.20871	−0.0836084
$210$	0	0
$211$	−13.5826	−0.935063	−0.467532	−	0.883976i	$-0.654857\pi$
$211$	−13.5826	−0.935063	−0.467532	+	0.883976i	$0.654857\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.3303	0.977319
$216$	0	0
$217$	0	0
$218$	0	0
$219$	30.1216	2.03543
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−18.4174	−1.23332	−0.616661	−	0.787229i	$-0.711515\pi$
$223$	−18.4174	−1.23332	−0.616661	+	0.787229i	$0.711515\pi$
$224$	0	0
$225$	−4.79129	−0.319419
$226$	0	0
$227$	3.62614	0.240675	0.120338	−	0.992733i	$-0.461602\pi$
$227$	3.62614	0.240675	0.120338	+	0.992733i	$0.461602\pi$
$228$	0	0
$229$	9.16515	0.605650	0.302825	−	0.953046i	$-0.402070\pi$
$229$	9.16515	0.605650	0.302825	+	0.953046i	$0.402070\pi$
$230$	0	0
$231$	0.582576	0.0383307
$232$	0	0
$233$	6.74773	0.442058	0.221029	−	0.975267i	$-0.429058\pi$
$233$	6.74773	0.442058	0.221029	+	0.975267i	$0.429058\pi$
$234$	0	0
$235$	−7.16515	−0.467403
$236$	0	0
$237$	35.5826	2.31134
$238$	0	0
$239$	11.5390	0.746397	0.373198	−	0.927752i	$-0.378261\pi$
$239$	11.5390	0.746397	0.373198	+	0.927752i	$0.378261\pi$
$240$	0	0
$241$	−13.7913	−0.888375	−0.444187	−	0.895934i	$-0.646508\pi$
$241$	−13.7913	−0.888375	−0.444187	+	0.895934i	$0.646508\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	−13.9129	−0.888861
$246$	0	0
$247$	1.20871	0.0769085
$248$	0	0
$249$	1.74773	0.110758
$250$	0	0
$251$	15.7913	0.996737	0.498369	−	0.866965i	$-0.333933\pi$
$251$	15.7913	0.996737	0.498369	+	0.866965i	$0.333933\pi$
$252$	0	0
$253$	−0.208712	−0.0131216
$254$	0	0
$255$	22.3303	1.39838
$256$	0	0
$257$	−9.79129	−0.610764	−0.305382	−	0.952230i	$-0.598784\pi$
$257$	−9.79129	−0.610764	−0.305382	+	0.952230i	$0.598784\pi$
$258$	0	0
$259$	−0.834849	−0.0518750
$260$	0	0
$261$	−17.1652	−1.06250
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	8.74773	0.537369
$266$	0	0
$267$	−26.7477	−1.63693
$268$	0	0
$269$	23.7042	1.44527	0.722634	−	0.691231i	$-0.242931\pi$
$269$	23.7042	1.44527	0.722634	+	0.691231i	$0.242931\pi$
$270$	0	0
$271$	−27.5390	−1.67288	−0.836438	−	0.548062i	$-0.815366\pi$
$271$	−27.5390	−1.67288	−0.836438	+	0.548062i	$0.815366\pi$
$272$	0	0
$273$	−0.582576	−0.0352591
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	20.0000	1.20168	0.600842	−	0.799368i	$-0.294832\pi$
$277$	20.0000	1.20168	0.600842	+	0.799368i	$0.294832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.7042	1.59304	0.796519	−	0.604614i	$-0.206673\pi$
$281$	26.7042	1.59304	0.796519	+	0.604614i	$0.206673\pi$
$282$	0	0
$283$	−27.4955	−1.63444	−0.817218	−	0.576329i	$-0.804485\pi$
$283$	−27.4955	−1.63444	−0.817218	+	0.576329i	$0.804485\pi$
$284$	0	0
$285$	6.74773	0.399701
$286$	0	0
$287$	−0.373864	−0.0220685
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−52.3303	−3.06766
$292$	0	0
$293$	17.1652	1.00280	0.501399	−	0.865216i	$-0.332819\pi$
$293$	17.1652	1.00280	0.501399	+	0.865216i	$0.332819\pi$
$294$	0	0
$295$	11.1652	0.650060
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	0.208712	0.0120701
$300$	0	0
$301$	−1.49545	−0.0861965
$302$	0	0
$303$	33.4955	1.92426
$304$	0	0
$305$	4.83485	0.276843
$306$	0	0
$307$	−18.3303	−1.04617	−0.523083	−	0.852282i	$-0.675218\pi$
$307$	−18.3303	−1.04617	−0.523083	+	0.852282i	$0.675218\pi$
$308$	0	0
$309$	−0.582576	−0.0331416
$310$	0	0
$311$	−19.5390	−1.10796	−0.553978	−	0.832531i	$-0.686891\pi$
$311$	−19.5390	−1.10796	−0.553978	+	0.832531i	$0.686891\pi$
$312$	0	0
$313$	2.20871	0.124844	0.0624219	−	0.998050i	$-0.480118\pi$
$313$	2.20871	0.124844	0.0624219	+	0.998050i	$0.480118\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	−3.58258	−0.200586
$320$	0	0
$321$	26.7477	1.49291
$322$	0	0
$323$	4.83485	0.269018
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	3.37386	0.186575
$328$	0	0
$329$	0.747727	0.0412235
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	19.1652	1.05024
$334$	0	0
$335$	−22.3303	−1.22003
$336$	0	0
$337$	−32.3303	−1.76114	−0.880572	−	0.473913i	$-0.842841\pi$
$337$	−32.3303	−1.76114	−0.880572	+	0.473913i	$0.842841\pi$
$338$	0	0
$339$	58.9564	3.20207
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.91288	0.157281
$344$	0	0
$345$	1.16515	0.0627296
$346$	0	0
$347$	−14.8348	−0.796376	−0.398188	−	0.917304i	$-0.630361\pi$
$347$	−14.8348	−0.796376	−0.398188	+	0.917304i	$0.630361\pi$
$348$	0	0
$349$	20.1216	1.07708	0.538542	−	0.842599i	$-0.318975\pi$
$349$	20.1216	1.07708	0.538542	+	0.842599i	$0.318975\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	10.4174	0.554464	0.277232	−	0.960803i	$-0.410583\pi$
$353$	10.4174	0.554464	0.277232	+	0.960803i	$0.410583\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.33030	−0.123333
$358$	0	0
$359$	25.7042	1.35661	0.678307	−	0.734779i	$-0.262714\pi$
$359$	25.7042	1.35661	0.678307	+	0.734779i	$0.262714\pi$
$360$	0	0
$361$	−17.5390	−0.923106
$362$	0	0
$363$	−2.79129	−0.146505
$364$	0	0
$365$	−21.5826	−1.12968
$366$	0	0
$367$	−2.04356	−0.106673	−0.0533365	−	0.998577i	$-0.516986\pi$
$367$	−2.04356	−0.106673	−0.0533365	+	0.998577i	$0.516986\pi$
$368$	0	0
$369$	8.58258	0.446791
$370$	0	0
$371$	−0.912878	−0.0473943
$372$	0	0
$373$	3.58258	0.185499	0.0927494	−	0.995689i	$-0.470434\pi$
$373$	3.58258	0.185499	0.0927494	+	0.995689i	$0.470434\pi$
$374$	0	0
$375$	33.4955	1.72970
$376$	0	0
$377$	3.58258	0.184512
$378$	0	0
$379$	6.74773	0.346607	0.173304	−	0.984868i	$-0.444556\pi$
$379$	6.74773	0.346607	0.173304	+	0.984868i	$0.444556\pi$
$380$	0	0
$381$	49.0780	2.51434
$382$	0	0
$383$	29.9129	1.52848	0.764238	−	0.644934i	$-0.223115\pi$
$383$	29.9129	1.52848	0.764238	+	0.644934i	$0.223115\pi$
$384$	0	0
$385$	−0.417424	−0.0212739
$386$	0	0
$387$	34.3303	1.74511
$388$	0	0
$389$	−26.7042	−1.35395	−0.676977	−	0.736004i	$-0.736711\pi$
$389$	−26.7042	−1.35395	−0.676977	+	0.736004i	$0.736711\pi$
$390$	0	0
$391$	0.834849	0.0422201
$392$	0	0
$393$	−16.7477	−0.844811
$394$	0	0
$395$	−25.4955	−1.28282
$396$	0	0
$397$	33.1652	1.66451	0.832256	−	0.554392i	$-0.187049\pi$
$397$	33.1652	1.66451	0.832256	+	0.554392i	$0.187049\pi$
$398$	0	0
$399$	−0.704166	−0.0352524
$400$	0	0
$401$	15.1652	0.757312	0.378656	−	0.925538i	$-0.376386\pi$
$401$	15.1652	0.757312	0.378656	+	0.925538i	$0.376386\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.834849	−0.0414840
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	14.8348	0.733536	0.366768	−	0.930312i	$-0.380464\pi$
$409$	14.8348	0.733536	0.366768	+	0.930312i	$0.380464\pi$
$410$	0	0
$411$	64.6606	3.18947
$412$	0	0
$413$	−1.16515	−0.0573334
$414$	0	0
$415$	−1.25227	−0.0614717
$416$	0	0
$417$	−54.6606	−2.67674
$418$	0	0
$419$	−7.37386	−0.360237	−0.180118	−	0.983645i	$-0.557648\pi$
$419$	−7.37386	−0.360237	−0.180118	+	0.983645i	$0.557648\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	−17.1652	−0.834598
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−0.504546	−0.0244167
$428$	0	0
$429$	2.79129	0.134765
$430$	0	0
$431$	16.6261	0.800853	0.400426	−	0.916329i	$-0.368862\pi$
$431$	16.6261	0.800853	0.400426	+	0.916329i	$0.368862\pi$
$432$	0	0
$433$	−15.6261	−0.750944	−0.375472	−	0.926834i	$-0.622519\pi$
$433$	−15.6261	−0.750944	−0.375472	+	0.926834i	$0.622519\pi$
$434$	0	0
$435$	20.0000	0.958927
$436$	0	0
$437$	0.252273	0.0120679
$438$	0	0
$439$	14.8348	0.708029	0.354014	−	0.935240i	$-0.384816\pi$
$439$	14.8348	0.708029	0.354014	+	0.935240i	$0.384816\pi$
$440$	0	0
$441$	−33.3303	−1.58716
$442$	0	0
$443$	36.4519	1.73188	0.865941	−	0.500146i	$-0.166720\pi$
$443$	36.4519	1.73188	0.865941	+	0.500146i	$0.166720\pi$
$444$	0	0
$445$	19.1652	0.908515
$446$	0	0
$447$	−3.83485	−0.181382
$448$	0	0
$449$	−9.16515	−0.432530	−0.216265	−	0.976335i	$-0.569388\pi$
$449$	−9.16515	−0.432530	−0.216265	+	0.976335i	$0.569388\pi$
$450$	0	0
$451$	1.79129	0.0843485
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	0.417424	0.0195692
$456$	0	0
$457$	−7.62614	−0.356736	−0.178368	−	0.983964i	$-0.557082\pi$
$457$	−7.62614	−0.356736	−0.178368	+	0.983964i	$0.557082\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	12.3739	0.576308	0.288154	−	0.957584i	$-0.406958\pi$
$461$	12.3739	0.576308	0.288154	+	0.957584i	$0.406958\pi$
$462$	0	0
$463$	39.1652	1.82016	0.910079	−	0.414434i	$-0.136020\pi$
$463$	39.1652	1.82016	0.910079	+	0.414434i	$0.136020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.3303	−1.03332	−0.516662	−	0.856190i	$-0.672825\pi$
$467$	−22.3303	−1.03332	−0.516662	+	0.856190i	$0.672825\pi$
$468$	0	0
$469$	2.33030	0.107603
$470$	0	0
$471$	35.0000	1.61271
$472$	0	0
$473$	7.16515	0.329454
$474$	0	0
$475$	1.20871	0.0554595
$476$	0	0
$477$	20.9564	0.959529
$478$	0	0
$479$	39.1652	1.78950	0.894751	−	0.446566i	$-0.147353\pi$
$479$	39.1652	1.78950	0.894751	+	0.446566i	$0.147353\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−0.121591	−0.00553257
$484$	0	0
$485$	37.4955	1.70258
$486$	0	0
$487$	−17.5826	−0.796743	−0.398371	−	0.917224i	$-0.630424\pi$
$487$	−17.5826	−0.796743	−0.398371	+	0.917224i	$0.630424\pi$
$488$	0	0
$489$	−3.25227	−0.147073
$490$	0	0
$491$	−21.1652	−0.955170	−0.477585	−	0.878586i	$-0.658488\pi$
$491$	−21.1652	−0.955170	−0.477585	+	0.878586i	$0.658488\pi$
$492$	0	0
$493$	14.3303	0.645404
$494$	0	0
$495$	9.58258	0.430705
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	43.3739	1.93780
$502$	0	0
$503$	−21.4955	−0.958435	−0.479217	−	0.877696i	$-0.659079\pi$
$503$	−21.4955	−0.958435	−0.479217	+	0.877696i	$0.659079\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0	0
$507$	−2.79129	−0.123965
$508$	0	0
$509$	−19.5826	−0.867982	−0.433991	−	0.900917i	$-0.642895\pi$
$509$	−19.5826	−0.867982	−0.433991	+	0.900917i	$0.642895\pi$
$510$	0	0
$511$	2.25227	0.0996347
$512$	0	0
$513$	6.04356	0.266830
$514$	0	0
$515$	0.417424	0.0183939
$516$	0	0
$517$	−3.58258	−0.157561
$518$	0	0
$519$	43.4955	1.90924
$520$	0	0
$521$	2.62614	0.115053	0.0575266	−	0.998344i	$-0.481679\pi$
$521$	2.62614	0.115053	0.0575266	+	0.998344i	$0.481679\pi$
$522$	0	0
$523$	−25.1652	−1.10040	−0.550198	−	0.835034i	$-0.685448\pi$
$523$	−25.1652	−1.10040	−0.550198	+	0.835034i	$0.685448\pi$
$524$	0	0
$525$	−0.582576	−0.0254257
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.9564	−0.998106
$530$	0	0
$531$	26.7477	1.16075
$532$	0	0
$533$	−1.79129	−0.0775893
$534$	0	0
$535$	−19.1652	−0.828582
$536$	0	0
$537$	8.83485	0.381252
$538$	0	0
$539$	−6.95644	−0.299635
$540$	0	0
$541$	−8.53901	−0.367121	−0.183560	−	0.983008i	$-0.558762\pi$
$541$	−8.53901	−0.367121	−0.183560	+	0.983008i	$0.558762\pi$
$542$	0	0
$543$	−34.5390	−1.48221
$544$	0	0
$545$	−2.41742	−0.103551
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	11.5826	0.494332
$550$	0	0
$551$	4.33030	0.184477
$552$	0	0
$553$	2.66061	0.113140
$554$	0	0
$555$	−22.3303	−0.947869
$556$	0	0
$557$	38.8693	1.64695	0.823473	−	0.567356i	$-0.192033\pi$
$557$	38.8693	1.64695	0.823473	+	0.567356i	$0.192033\pi$
$558$	0	0
$559$	−7.16515	−0.303054
$560$	0	0
$561$	11.1652	0.471393
$562$	0	0
$563$	19.5826	0.825307	0.412654	−	0.910888i	$-0.364602\pi$
$563$	19.5826	0.825307	0.412654	+	0.910888i	$0.364602\pi$
$564$	0	0
$565$	−42.2432	−1.77718
$566$	0	0
$567$	0.0871215	0.00365876
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−14.7477	−0.617173	−0.308587	−	0.951196i	$-0.599856\pi$
$571$	−14.7477	−0.617173	−0.308587	+	0.951196i	$0.599856\pi$
$572$	0	0
$573$	−5.70417	−0.238295
$574$	0	0
$575$	0.208712	0.00870390
$576$	0	0
$577$	−21.5826	−0.898494	−0.449247	−	0.893408i	$-0.648308\pi$
$577$	−21.5826	−0.898494	−0.449247	+	0.893408i	$0.648308\pi$
$578$	0	0
$579$	−15.0000	−0.623379
$580$	0	0
$581$	0.130682	0.00542161
$582$	0	0
$583$	4.37386	0.181147
$584$	0	0
$585$	−9.58258	−0.396191
$586$	0	0
$587$	28.4174	1.17291	0.586456	−	0.809981i	$-0.300523\pi$
$587$	28.4174	1.17291	0.586456	+	0.809981i	$0.300523\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	68.0345	2.79856
$592$	0	0
$593$	29.3739	1.20624	0.603120	−	0.797650i	$-0.293924\pi$
$593$	29.3739	1.20624	0.603120	+	0.797650i	$0.293924\pi$
$594$	0	0
$595$	1.66970	0.0684509
$596$	0	0
$597$	3.37386	0.138083
$598$	0	0
$599$	−6.79129	−0.277484	−0.138742	−	0.990329i	$-0.544306\pi$
$599$	−6.79129	−0.277484	−0.138742	+	0.990329i	$0.544306\pi$
$600$	0	0
$601$	−20.4174	−0.832844	−0.416422	−	0.909171i	$-0.636716\pi$
$601$	−20.4174	−0.832844	−0.416422	+	0.909171i	$0.636716\pi$
$602$	0	0
$603$	−53.4955	−2.17850
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	5.25227	0.213183	0.106592	−	0.994303i	$-0.466006\pi$
$607$	5.25227	0.213183	0.106592	+	0.994303i	$0.466006\pi$
$608$	0	0
$609$	−2.08712	−0.0845744
$610$	0	0
$611$	3.58258	0.144935
$612$	0	0
$613$	−9.12159	−0.368418	−0.184209	−	0.982887i	$-0.558972\pi$
$613$	−9.12159	−0.368418	−0.184209	+	0.982887i	$0.558972\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−22.7477	−0.915789	−0.457895	−	0.889007i	$-0.651396\pi$
$617$	−22.7477	−0.915789	−0.457895	+	0.889007i	$0.651396\pi$
$618$	0	0
$619$	0.834849	0.0335554	0.0167777	−	0.999859i	$-0.494659\pi$
$619$	0.834849	0.0335554	0.0167777	+	0.999859i	$0.494659\pi$
$620$	0	0
$621$	1.04356	0.0418767
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	3.37386	0.134739
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	0	0
$633$	37.9129	1.50690
$634$	0	0
$635$	−35.1652	−1.39549
$636$	0	0
$637$	6.95644	0.275624
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.9564	0.511749	0.255874	−	0.966710i	$-0.417637\pi$
$641$	12.9564	0.511749	0.255874	+	0.966710i	$0.417637\pi$
$642$	0	0
$643$	5.91288	0.233181	0.116591	−	0.993180i	$-0.462803\pi$
$643$	5.91288	0.233181	0.116591	+	0.993180i	$0.462803\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	0	0
$647$	−14.0436	−0.552109	−0.276055	−	0.961142i	$-0.589027\pi$
$647$	−14.0436	−0.552109	−0.276055	+	0.961142i	$0.589027\pi$
$648$	0	0
$649$	5.58258	0.219135
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−51.7042	−2.01717
$658$	0	0
$659$	23.4955	0.915253	0.457626	−	0.889145i	$-0.348700\pi$
$659$	23.4955	0.915253	0.457626	+	0.889145i	$0.348700\pi$
$660$	0	0
$661$	−0.0871215	−0.00338863	−0.00169432	−	0.999999i	$-0.500539\pi$
$661$	−0.0871215	−0.00338863	−0.00169432	+	0.999999i	$0.500539\pi$
$662$	0	0
$663$	−11.1652	−0.433619
$664$	0	0
$665$	0.504546	0.0195654
$666$	0	0
$667$	0.747727	0.0289521
$668$	0	0
$669$	51.4083	1.98756
$670$	0	0
$671$	2.41742	0.0933236
$672$	0	0
$673$	37.4955	1.44534	0.722672	−	0.691191i	$-0.242914\pi$
$673$	37.4955	1.44534	0.722672	+	0.691191i	$0.242914\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−6.33030	−0.243293	−0.121647	−	0.992573i	$-0.538817\pi$
$677$	−6.33030	−0.243293	−0.121647	+	0.992573i	$0.538817\pi$
$678$	0	0
$679$	−3.91288	−0.150162
$680$	0	0
$681$	−10.1216	−0.387860
$682$	0	0
$683$	−38.0000	−1.45403	−0.727015	−	0.686622i	$-0.759093\pi$
$683$	−38.0000	−1.45403	−0.727015	+	0.686622i	$0.759093\pi$
$684$	0	0
$685$	−46.3303	−1.77019
$686$	0	0
$687$	−25.5826	−0.976036
$688$	0	0
$689$	−4.37386	−0.166631
$690$	0	0
$691$	13.9129	0.529271	0.264635	−	0.964349i	$-0.414748\pi$
$691$	13.9129	0.529271	0.264635	+	0.964349i	$0.414748\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	39.1652	1.48562
$696$	0	0
$697$	−7.16515	−0.271399
$698$	0	0
$699$	−18.8348	−0.712399
$700$	0	0
$701$	−29.9129	−1.12979	−0.564897	−	0.825161i	$-0.691084\pi$
$701$	−29.9129	−1.12979	−0.564897	+	0.825161i	$0.691084\pi$
$702$	0	0
$703$	−4.83485	−0.182350
$704$	0	0
$705$	20.0000	0.753244
$706$	0	0
$707$	2.50455	0.0941931
$708$	0	0
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	−61.0780	−2.29061
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	−32.2087	−1.20286
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−0.0435608	−0.00162229
$722$	0	0
$723$	38.4955	1.43166
$724$	0	0
$725$	3.58258	0.133054
$726$	0	0
$727$	−38.1216	−1.41385	−0.706926	−	0.707288i	$-0.749919\pi$
$727$	−38.1216	−1.41385	−0.706926	+	0.707288i	$0.749919\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	−28.6606	−1.06005
$732$	0	0
$733$	−15.0436	−0.555647	−0.277823	−	0.960632i	$-0.589613\pi$
$733$	−15.0436	−0.555647	−0.277823	+	0.960632i	$0.589613\pi$
$734$	0	0
$735$	38.8348	1.43244
$736$	0	0
$737$	−11.1652	−0.411274
$738$	0	0
$739$	−34.9564	−1.28589	−0.642947	−	0.765911i	$-0.722288\pi$
$739$	−34.9564	−1.28589	−0.642947	+	0.765911i	$0.722288\pi$
$740$	0	0
$741$	−3.37386	−0.123942
$742$	0	0
$743$	15.1652	0.556355	0.278178	−	0.960530i	$-0.410270\pi$
$743$	15.1652	0.556355	0.278178	+	0.960530i	$0.410270\pi$
$744$	0	0
$745$	2.74773	0.100669
$746$	0	0
$747$	−3.00000	−0.109764
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	41.8693	1.52783	0.763917	−	0.645315i	$-0.223274\pi$
$751$	41.8693	1.52783	0.763917	+	0.645315i	$0.223274\pi$
$752$	0	0
$753$	−44.0780	−1.60629
$754$	0	0
$755$	−14.3303	−0.521533
$756$	0	0
$757$	−0.956439	−0.0347624	−0.0173812	−	0.999849i	$-0.505533\pi$
$757$	−0.956439	−0.0347624	−0.0173812	+	0.999849i	$0.505533\pi$
$758$	0	0
$759$	0.582576	0.0211462
$760$	0	0
$761$	1.37386	0.0498025	0.0249013	−	0.999690i	$-0.492073\pi$
$761$	1.37386	0.0498025	0.0249013	+	0.999690i	$0.492073\pi$
$762$	0	0
$763$	0.252273	0.00913289
$764$	0	0
$765$	−38.3303	−1.38584
$766$	0	0
$767$	−5.58258	−0.201575
$768$	0	0
$769$	−46.4519	−1.67510	−0.837549	−	0.546362i	$-0.816012\pi$
$769$	−46.4519	−1.67510	−0.837549	+	0.546362i	$0.816012\pi$
$770$	0	0
$771$	27.3303	0.984277
$772$	0	0
$773$	−9.49545	−0.341528	−0.170764	−	0.985312i	$-0.554624\pi$
$773$	−9.49545	−0.341528	−0.170764	+	0.985312i	$0.554624\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.33030	0.0835991
$778$	0	0
$779$	−2.16515	−0.0775746
$780$	0	0
$781$	0	0
$782$	0	0
$783$	17.9129	0.640154
$784$	0	0
$785$	−25.0780	−0.895073
$786$	0	0
$787$	17.7042	0.631085	0.315543	−	0.948911i	$-0.397813\pi$
$787$	17.7042	0.631085	0.315543	+	0.948911i	$0.397813\pi$
$788$	0	0
$789$	50.2432	1.78870
$790$	0	0
$791$	4.40833	0.156742
$792$	0	0
$793$	−2.41742	−0.0858453
$794$	0	0
$795$	−24.4174	−0.865997
$796$	0	0
$797$	−13.1652	−0.466334	−0.233167	−	0.972437i	$-0.574909\pi$
$797$	−13.1652	−0.466334	−0.233167	+	0.972437i	$0.574909\pi$
$798$	0	0
$799$	14.3303	0.506970
$800$	0	0
$801$	45.9129	1.62225
$802$	0	0
$803$	−10.7913	−0.380816
$804$	0	0
$805$	0.0871215	0.00307063
$806$	0	0
$807$	−66.1652	−2.32912
$808$	0	0
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−6.95644	−0.244274	−0.122137	−	0.992513i	$-0.538975\pi$
$811$	−6.95644	−0.244274	−0.122137	+	0.992513i	$0.538975\pi$
$812$	0	0
$813$	76.8693	2.69592
$814$	0	0
$815$	2.33030	0.0816269
$816$	0	0
$817$	−8.66061	−0.302996
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−51.4955	−1.79720	−0.898602	−	0.438765i	$-0.855416\pi$
$821$	−51.4955	−1.79720	−0.898602	+	0.438765i	$0.855416\pi$
$822$	0	0
$823$	−5.87841	−0.204908	−0.102454	−	0.994738i	$-0.532670\pi$
$823$	−5.87841	−0.204908	−0.102454	+	0.994738i	$0.532670\pi$
$824$	0	0
$825$	2.79129	0.0971802
$826$	0	0
$827$	−12.2867	−0.427252	−0.213626	−	0.976916i	$-0.568527\pi$
$827$	−12.2867	−0.427252	−0.213626	+	0.976916i	$0.568527\pi$
$828$	0	0
$829$	12.1216	0.421000	0.210500	−	0.977594i	$-0.432491\pi$
$829$	12.1216	0.421000	0.210500	+	0.977594i	$0.432491\pi$
$830$	0	0
$831$	−55.8258	−1.93657
$832$	0	0
$833$	27.8258	0.964105
$834$	0	0
$835$	−31.0780	−1.07550
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.16515	0.0402255	0.0201127	−	0.999798i	$-0.493597\pi$
$839$	1.16515	0.0402255	0.0201127	+	0.999798i	$0.493597\pi$
$840$	0	0
$841$	−16.1652	−0.557419
$842$	0	0
$843$	−74.5390	−2.56726
$844$	0	0
$845$	2.00000	0.0688021
$846$	0	0
$847$	−0.208712	−0.00717143
$848$	0	0
$849$	76.7477	2.63398
$850$	0	0
$851$	−0.834849	−0.0286182
$852$	0	0
$853$	−33.1216	−1.13406	−0.567031	−	0.823697i	$-0.691908\pi$
$853$	−33.1216	−1.13406	−0.567031	+	0.823697i	$0.691908\pi$
$854$	0	0
$855$	−11.5826	−0.396116
$856$	0	0
$857$	25.4955	0.870908	0.435454	−	0.900211i	$-0.356588\pi$
$857$	25.4955	0.870908	0.435454	+	0.900211i	$0.356588\pi$
$858$	0	0
$859$	−19.6261	−0.669635	−0.334818	−	0.942283i	$-0.608675\pi$
$859$	−19.6261	−0.669635	−0.334818	+	0.942283i	$0.608675\pi$
$860$	0	0
$861$	1.04356	0.0355645
$862$	0	0
$863$	30.6606	1.04370	0.521850	−	0.853038i	$-0.325242\pi$
$863$	30.6606	1.04370	0.521850	+	0.853038i	$0.325242\pi$
$864$	0	0
$865$	−31.1652	−1.05965
$866$	0	0
$867$	2.79129	0.0947971
$868$	0	0
$869$	−12.7477	−0.432437
$870$	0	0
$871$	11.1652	0.378317
$872$	0	0
$873$	89.8258	3.04014
$874$	0	0
$875$	2.50455	0.0846691
$876$	0	0
$877$	41.9564	1.41677	0.708384	−	0.705827i	$-0.249424\pi$
$877$	41.9564	1.41677	0.708384	+	0.705827i	$0.249424\pi$
$878$	0	0
$879$	−47.9129	−1.61606
$880$	0	0
$881$	14.0436	0.473140	0.236570	−	0.971614i	$-0.423977\pi$
$881$	14.0436	0.473140	0.236570	+	0.971614i	$0.423977\pi$
$882$	0	0
$883$	−44.2867	−1.49037	−0.745184	−	0.666859i	$-0.767638\pi$
$883$	−44.2867	−1.49037	−0.745184	+	0.666859i	$0.767638\pi$
$884$	0	0
$885$	−31.1652	−1.04761
$886$	0	0
$887$	35.0780	1.17780	0.588902	−	0.808204i	$-0.299560\pi$
$887$	35.0780	1.17780	0.588902	+	0.808204i	$0.299560\pi$
$888$	0	0
$889$	3.66970	0.123078
$890$	0	0
$891$	−0.417424	−0.0139842
$892$	0	0
$893$	4.33030	0.144908
$894$	0	0
$895$	−6.33030	−0.211599
$896$	0	0
$897$	−0.582576	−0.0194516
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−17.4955	−0.582858
$902$	0	0
$903$	4.17424	0.138910
$904$	0	0
$905$	24.7477	0.822642
$906$	0	0
$907$	−12.8693	−0.427319	−0.213659	−	0.976908i	$-0.568538\pi$
$907$	−12.8693	−0.427319	−0.213659	+	0.976908i	$0.568538\pi$
$908$	0	0
$909$	−57.4955	−1.90700
$910$	0	0
$911$	32.6261	1.08095	0.540476	−	0.841359i	$-0.318244\pi$
$911$	32.6261	1.08095	0.540476	+	0.841359i	$0.318244\pi$
$912$	0	0
$913$	−0.626136	−0.0207221
$914$	0	0
$915$	−13.4955	−0.446146
$916$	0	0
$917$	−1.25227	−0.0413537
$918$	0	0
$919$	35.4955	1.17089	0.585443	−	0.810713i	$-0.300920\pi$
$919$	35.4955	1.17089	0.585443	+	0.810713i	$0.300920\pi$
$920$	0	0
$921$	51.1652	1.68595
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	1.00000	0.0328443
$928$	0	0
$929$	9.49545	0.311536	0.155768	−	0.987794i	$-0.450215\pi$
$929$	9.49545	0.311536	0.155768	+	0.987794i	$0.450215\pi$
$930$	0	0
$931$	8.40833	0.275572
$932$	0	0
$933$	54.5390	1.78553
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	57.8258	1.88909	0.944543	−	0.328389i	$-0.106506\pi$
$937$	57.8258	1.88909	0.944543	+	0.328389i	$0.106506\pi$
$938$	0	0
$939$	−6.16515	−0.201192
$940$	0	0
$941$	−33.9564	−1.10695	−0.553474	−	0.832866i	$-0.686698\pi$
$941$	−33.9564	−1.10695	−0.553474	+	0.832866i	$0.686698\pi$
$942$	0	0
$943$	−0.373864	−0.0121747
$944$	0	0
$945$	2.08712	0.0678941
$946$	0	0
$947$	−17.0780	−0.554961	−0.277481	−	0.960731i	$-0.589499\pi$
$947$	−17.0780	−0.554961	−0.277481	+	0.960731i	$0.589499\pi$
$948$	0	0
$949$	10.7913	0.350300
$950$	0	0
$951$	−22.3303	−0.724110
$952$	0	0
$953$	−23.0780	−0.747571	−0.373785	−	0.927515i	$-0.621940\pi$
$953$	−23.0780	−0.747571	−0.373785	+	0.927515i	$0.621940\pi$
$954$	0	0
$955$	4.08712	0.132256
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	4.83485	0.156125
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−45.9129	−1.47952
$964$	0	0
$965$	10.7477	0.345982
$966$	0	0
$967$	2.79129	0.0897618	0.0448809	−	0.998992i	$-0.485709\pi$
$967$	2.79129	0.0897618	0.0448809	+	0.998992i	$0.485709\pi$
$968$	0	0
$969$	−13.4955	−0.433536
$970$	0	0
$971$	35.4519	1.13771	0.568853	−	0.822439i	$-0.307387\pi$
$971$	35.4519	1.13771	0.568853	+	0.822439i	$0.307387\pi$
$972$	0	0
$973$	−4.08712	−0.131027
$974$	0	0
$975$	−2.79129	−0.0893928
$976$	0	0
$977$	5.66970	0.181390	0.0906948	−	0.995879i	$-0.471091\pi$
$977$	5.66970	0.181390	0.0906948	+	0.995879i	$0.471091\pi$
$978$	0	0
$979$	9.58258	0.306260
$980$	0	0
$981$	−5.79129	−0.184902
$982$	0	0
$983$	7.16515	0.228533	0.114266	−	0.993450i	$-0.463548\pi$
$983$	7.16515	0.228533	0.114266	+	0.993450i	$0.463548\pi$
$984$	0	0
$985$	−48.7477	−1.55323
$986$	0	0
$987$	−2.08712	−0.0664338
$988$	0	0
$989$	−1.49545	−0.0475527
$990$	0	0
$991$	36.3739	1.15545	0.577727	−	0.816230i	$-0.303940\pi$
$991$	36.3739	1.15545	0.577727	+	0.816230i	$0.303940\pi$
$992$	0	0
$993$	78.1561	2.48021
$994$	0	0
$995$	−2.41742	−0.0766375
$996$	0	0
$997$	−13.5826	−0.430164	−0.215082	−	0.976596i	$-0.569002\pi$
$997$	−13.5826	−0.430164	−0.215082	+	0.976596i	$0.569002\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.a.n.1.1		2
4.3	odd	2		572.2.a.d.1.2	✓	2
8.3	odd	2		9152.2.a.bl.1.1		2
8.5	even	2		9152.2.a.bn.1.2		2
12.11	even	2		5148.2.a.g.1.1		2
44.43	even	2		6292.2.a.o.1.2		2
52.51	odd	2		7436.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.d.1.2	✓	2	4.3	odd	2
2288.2.a.n.1.1		2	1.1	even	1	trivial
5148.2.a.g.1.1		2	12.11	even	2
6292.2.a.o.1.2		2	44.43	even	2
7436.2.a.h.1.2		2	52.51	odd	2
9152.2.a.bl.1.1		2	8.3	odd	2
9152.2.a.bn.1.2		2	8.5	even	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:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} +2.00000 q^{5} -0.208712 q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q-2.79129 q^{3} +2.00000 q^{5} -0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} -1.00000 q^{13} -5.58258 q^{15} -4.00000 q^{17} -1.20871 q^{19} +0.582576 q^{21} -0.208712 q^{23} -1.00000 q^{25} -5.00000 q^{27} -3.58258 q^{29} -2.79129 q^{33} -0.417424 q^{35} +4.00000 q^{37} +2.79129 q^{39} +1.79129 q^{41} +7.16515 q^{43} +9.58258 q^{45} -3.58258 q^{47} -6.95644 q^{49} +11.1652 q^{51} +4.37386 q^{53} +2.00000 q^{55} +3.37386 q^{57} +5.58258 q^{59} +2.41742 q^{61} -1.00000 q^{63} -2.00000 q^{65} -11.1652 q^{67} +0.582576 q^{69} -10.7913 q^{73} +2.79129 q^{75} -0.208712 q^{77} -12.7477 q^{79} -0.417424 q^{81} -0.626136 q^{83} -8.00000 q^{85} +10.0000 q^{87} +9.58258 q^{89} +0.208712 q^{91} -2.41742 q^{95} +18.7477 q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	\( 2 q - q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 8 q^{17} - 7 q^{19} - 8 q^{21} - 5 q^{23} - 2 q^{25} - 10 q^{27} + 2 q^{29} - q^{33} - 10 q^{35} + 8 q^{37} + q^{39} - q^{41} - 4 q^{43} + 10 q^{45} + 2 q^{47} + 9 q^{49} + 4 q^{51} - 5 q^{53} + 4 q^{55} - 7 q^{57} + 2 q^{59} + 14 q^{61} - 2 q^{63} - 4 q^{65} - 4 q^{67} - 8 q^{69} - 17 q^{73} + q^{75} - 5 q^{77} + 2 q^{79} - 10 q^{81} - 15 q^{83} - 16 q^{85} + 20 q^{87} + 10 q^{89} + 5 q^{91} - 14 q^{95} + 10 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 8 * q^17 - 7 * q^19 - 8 * q^21 - 5 * q^23 - 2 * q^25 - 10 * q^27 + 2 * q^29 - q^33 - 10 * q^35 + 8 * q^37 + q^39 - q^41 - 4 * q^43 + 10 * q^45 + 2 * q^47 + 9 * q^49 + 4 * q^51 - 5 * q^53 + 4 * q^55 - 7 * q^57 + 2 * q^59 + 14 * q^61 - 2 * q^63 - 4 * q^65 - 4 * q^67 - 8 * q^69 - 17 * q^73 + q^75 - 5 * q^77 + 2 * q^79 - 10 * q^81 - 15 * q^83 - 16 * q^85 + 20 * q^87 + 10 * q^89 + 5 * q^91 - 14 * q^95 + 10 * q^97 + 5 * q^99

Introduction

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