LMFDB - Embedded newform 1521.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 1 $ x^4 - 2x^3 - 7x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.35008$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} +2.23607 q^{5} -3.46410 q^{7} +2.23607 q^{8} +5.00000 q^{10} +4.47214 q^{11} -7.74597 q^{14} -1.00000 q^{16} +3.87298 q^{17} +3.46410 q^{19} +6.70820 q^{20} +10.0000 q^{22} +7.74597 q^{23} -10.3923 q^{28} +3.87298 q^{29} -6.70820 q^{32} +8.66025 q^{34} -7.74597 q^{35} -1.73205 q^{37} +7.74597 q^{38} +5.00000 q^{40} +2.23607 q^{41} -2.00000 q^{43} +13.4164 q^{44} +17.3205 q^{46} +4.47214 q^{47} +5.00000 q^{49} -11.6190 q^{53} +10.0000 q^{55} -7.74597 q^{56} +8.66025 q^{58} -8.94427 q^{59} -7.00000 q^{61} -13.0000 q^{64} -3.46410 q^{67} +11.6190 q^{68} -17.3205 q^{70} -4.47214 q^{71} -15.5885 q^{73} -3.87298 q^{74} +10.3923 q^{76} -15.4919 q^{77} +8.00000 q^{79} -2.23607 q^{80} +5.00000 q^{82} -4.47214 q^{83} +8.66025 q^{85} -4.47214 q^{86} +10.0000 q^{88} +4.47214 q^{89} +23.2379 q^{92} +10.0000 q^{94} +7.74597 q^{95} -6.92820 q^{97} +11.1803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4} + 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} - 8 q^{43} + 20 q^{49} + 40 q^{55} - 28 q^{61} - 52 q^{64} + 32 q^{79} + 20 q^{82} + 40 q^{88} + 40 q^{94}+O(q^{100}) $ 4 * q + 12 * q^4 + 20 * q^10 - 4 * q^16 + 40 * q^22 + 20 * q^40 - 8 * q^43 + 20 * q^49 + 40 * q^55 - 28 * q^61 - 52 * q^64 + 32 * q^79 + 20 * q^82 + 40 * q^88 + 40 * q^94

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	5.00000	1.58114
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−7.74597	−2.07020
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.87298	0.939336	0.469668	−	0.882843i	$-0.344374\pi$
$17$	3.87298	0.939336	0.469668	+	0.882843i	$0.344374\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	6.70820	1.50000
$21$	0	0
$22$	10.0000	2.13201
$23$	7.74597	1.61515	0.807573	−	0.589768i	$-0.200781\pi$
$23$	7.74597	1.61515	0.807573	+	0.589768i	$0.200781\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−10.3923	−1.96396
$29$	3.87298	0.719195	0.359597	−	0.933108i	$-0.382914\pi$
$29$	3.87298	0.719195	0.359597	+	0.933108i	$0.382914\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	8.66025	1.48522
$35$	−7.74597	−1.30931
$36$	0	0
$37$	−1.73205	−0.284747	−0.142374	−	0.989813i	$-0.545473\pi$
$37$	−1.73205	−0.284747	−0.142374	+	0.989813i	$0.545473\pi$
$38$	7.74597	1.25656
$39$	0	0
$40$	5.00000	0.790569
$41$	2.23607	0.349215	0.174608	−	0.984638i	$-0.444134\pi$
$41$	2.23607	0.349215	0.174608	+	0.984638i	$0.444134\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	13.4164	2.02260
$45$	0	0
$46$	17.3205	2.55377
$47$	4.47214	0.652328	0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214	0.652328	0.326164	+	0.945313i	$0.394244\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.6190	−1.59599	−0.797993	−	0.602667i	$-0.794105\pi$
$53$	−11.6190	−1.59599	−0.797993	+	0.602667i	$0.794105\pi$
$54$	0	0
$55$	10.0000	1.34840
$56$	−7.74597	−1.03510
$57$	0	0
$58$	8.66025	1.13715
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	11.6190	1.40900
$69$	0	0
$70$	−17.3205	−2.07020
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	−15.5885	−1.82449	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	−15.5885	−1.82449	−0.912245	+	0.409644i	$0.865653\pi$
$74$	−3.87298	−0.450225
$75$	0	0
$76$	10.3923	1.19208
$77$	−15.4919	−1.76547
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.23607	−0.250000
$81$	0	0
$82$	5.00000	0.552158
$83$	−4.47214	−0.490881	−0.245440	−	0.969412i	$-0.578933\pi$
$83$	−4.47214	−0.490881	−0.245440	+	0.969412i	$0.578933\pi$
$84$	0	0
$85$	8.66025	0.939336
$86$	−4.47214	−0.482243
$87$	0	0
$88$	10.0000	1.06600
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	0	0
$92$	23.2379	2.42272
$93$	0	0
$94$	10.0000	1.03142
$95$	7.74597	0.794719
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	11.1803	1.12938
$99$	0	0
$100$	0	0
$101$	−3.87298	−0.385376	−0.192688	−	0.981260i	$-0.561721\pi$
$101$	−3.87298	−0.385376	−0.192688	+	0.981260i	$0.561721\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	0	0
$106$	−25.9808	−2.52347
$107$	−7.74597	−0.748831	−0.374415	−	0.927261i	$-0.622157\pi$
$107$	−7.74597	−0.748831	−0.374415	+	0.927261i	$0.622157\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	22.3607	2.13201
$111$	0	0
$112$	3.46410	0.327327
$113$	19.3649	1.82170	0.910849	−	0.412740i	$-0.135428\pi$
$113$	19.3649	1.82170	0.910849	+	0.412740i	$0.135428\pi$
$114$	0	0
$115$	17.3205	1.61515
$116$	11.6190	1.07879
$117$	0	0
$118$	−20.0000	−1.84115
$119$	−13.4164	−1.22988
$120$	0	0
$121$	9.00000	0.818182
$122$	−15.6525	−1.41711
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	−7.74597	−0.669150
$135$	0	0
$136$	8.66025	0.742611
$137$	−15.6525	−1.33728	−0.668641	−	0.743586i	$-0.733124\pi$
$137$	−15.6525	−1.33728	−0.668641	+	0.743586i	$0.733124\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−23.2379	−1.96396
$141$	0	0
$142$	−10.0000	−0.839181
$143$	0	0
$144$	0	0
$145$	8.66025	0.719195
$146$	−34.8569	−2.88477
$147$	0	0
$148$	−5.19615	−0.427121
$149$	−11.1803	−0.915929	−0.457965	−	0.888970i	$-0.651421\pi$
$149$	−11.1803	−0.915929	−0.457965	+	0.888970i	$0.651421\pi$
$150$	0	0
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	7.74597	0.628281
$153$	0	0
$154$	−34.6410	−2.79145
$155$	0	0
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	17.8885	1.42314
$159$	0	0
$160$	−15.0000	−1.18585
$161$	−26.8328	−2.11472
$162$	0	0
$163$	−13.8564	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$163$	−13.8564	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$164$	6.70820	0.523823
$165$	0	0
$166$	−10.0000	−0.776151
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	0	0
$170$	19.3649	1.48522
$171$	0	0
$172$	−6.00000	−0.457496
$173$	15.4919	1.17783	0.588915	−	0.808195i	$-0.299555\pi$
$173$	15.4919	1.17783	0.588915	+	0.808195i	$0.299555\pi$
$174$	0	0
$175$	0	0
$176$	−4.47214	−0.337100
$177$	0	0
$178$	10.0000	0.749532
$179$	7.74597	0.578961	0.289480	−	0.957184i	$-0.406518\pi$
$179$	7.74597	0.578961	0.289480	+	0.957184i	$0.406518\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	0	0
$184$	17.3205	1.27688
$185$	−3.87298	−0.284747
$186$	0	0
$187$	17.3205	1.26660
$188$	13.4164	0.978492
$189$	0	0
$190$	17.3205	1.25656
$191$	−15.4919	−1.12096	−0.560478	−	0.828169i	$-0.689383\pi$
$191$	−15.4919	−1.12096	−0.560478	+	0.828169i	$0.689383\pi$
$192$	0	0
$193$	−1.73205	−0.124676	−0.0623379	−	0.998055i	$-0.519856\pi$
$193$	−1.73205	−0.124676	−0.0623379	+	0.998055i	$0.519856\pi$
$194$	−15.4919	−1.11226
$195$	0	0
$196$	15.0000	1.07143
$197$	−4.47214	−0.318626	−0.159313	−	0.987228i	$-0.550928\pi$
$197$	−4.47214	−0.318626	−0.159313	+	0.987228i	$0.550928\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	0	0
$202$	−8.66025	−0.609333
$203$	−13.4164	−0.941647
$204$	0	0
$205$	5.00000	0.349215
$206$	−4.47214	−0.311588
$207$	0	0
$208$	0	0
$209$	15.4919	1.07160
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−34.8569	−2.39398
$213$	0	0
$214$	−17.3205	−1.18401
$215$	−4.47214	−0.304997
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	30.0000	2.02260
$221$	0	0
$222$	0	0
$223$	27.7128	1.85579	0.927894	−	0.372845i	$-0.121618\pi$
$223$	27.7128	1.85579	0.927894	+	0.372845i	$0.121618\pi$
$224$	23.2379	1.55265
$225$	0	0
$226$	43.3013	2.88036
$227$	22.3607	1.48413	0.742065	−	0.670328i	$-0.233846\pi$
$227$	22.3607	1.48413	0.742065	+	0.670328i	$0.233846\pi$
$228$	0	0
$229$	−20.7846	−1.37349	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	−20.7846	−1.37349	−0.686743	+	0.726900i	$0.740960\pi$
$230$	38.7298	2.55377
$231$	0	0
$232$	8.66025	0.568574
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	−26.8328	−1.74667
$237$	0	0
$238$	−30.0000	−1.94461
$239$	−4.47214	−0.289278	−0.144639	−	0.989484i	$-0.546202\pi$
$239$	−4.47214	−0.289278	−0.144639	+	0.989484i	$0.546202\pi$
$240$	0	0
$241$	−19.0526	−1.22728	−0.613642	−	0.789585i	$-0.710296\pi$
$241$	−19.0526	−1.22728	−0.613642	+	0.789585i	$0.710296\pi$
$242$	20.1246	1.29366
$243$	0	0
$244$	−21.0000	−1.34439
$245$	11.1803	0.714286
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−25.0000	−1.58114
$251$	15.4919	0.977842	0.488921	−	0.872328i	$-0.337391\pi$
$251$	15.4919	0.977842	0.488921	+	0.872328i	$0.337391\pi$
$252$	0	0
$253$	34.6410	2.17786
$254$	8.94427	0.561214
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−3.87298	−0.241590	−0.120795	−	0.992677i	$-0.538544\pi$
$257$	−3.87298	−0.241590	−0.120795	+	0.992677i	$0.538544\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.74597	−0.477637	−0.238818	−	0.971064i	$-0.576760\pi$
$263$	−7.74597	−0.477637	−0.238818	+	0.971064i	$0.576760\pi$
$264$	0	0
$265$	−25.9808	−1.59599
$266$	−26.8328	−1.64523
$267$	0	0
$268$	−10.3923	−0.634811
$269$	−15.4919	−0.944560	−0.472280	−	0.881449i	$-0.656569\pi$
$269$	−15.4919	−0.944560	−0.472280	+	0.881449i	$0.656569\pi$
$270$	0	0
$271$	−6.92820	−0.420858	−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820	−0.420858	−0.210429	+	0.977609i	$0.567486\pi$
$272$	−3.87298	−0.234834
$273$	0	0
$274$	−35.0000	−2.11443
$275$	0	0
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	17.8885	1.07288
$279$	0	0
$280$	−17.3205	−1.03510
$281$	−15.6525	−0.933748	−0.466874	−	0.884324i	$-0.654620\pi$
$281$	−15.6525	−0.933748	−0.466874	+	0.884324i	$0.654620\pi$
$282$	0	0
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−13.4164	−0.796117
$285$	0	0
$286$	0	0
$287$	−7.74597	−0.457230
$288$	0	0
$289$	−2.00000	−0.117647
$290$	19.3649	1.13715
$291$	0	0
$292$	−46.7654	−2.73674
$293$	−2.23607	−0.130632	−0.0653162	−	0.997865i	$-0.520806\pi$
$293$	−2.23607	−0.130632	−0.0653162	+	0.997865i	$0.520806\pi$
$294$	0	0
$295$	−20.0000	−1.16445
$296$	−3.87298	−0.225113
$297$	0	0
$298$	−25.0000	−1.44821
$299$	0	0
$300$	0	0
$301$	6.92820	0.399335
$302$	23.2379	1.33719
$303$	0	0
$304$	−3.46410	−0.198680
$305$	−15.6525	−0.896258
$306$	0	0
$307$	−10.3923	−0.593120	−0.296560	−	0.955014i	$-0.595840\pi$
$307$	−10.3923	−0.593120	−0.296560	+	0.955014i	$0.595840\pi$
$308$	−46.4758	−2.64820
$309$	0	0
$310$	0	0
$311$	23.2379	1.31770	0.658850	−	0.752274i	$-0.271043\pi$
$311$	23.2379	1.31770	0.658850	+	0.752274i	$0.271043\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	24.5967	1.38807
$315$	0	0
$316$	24.0000	1.35011
$317$	29.0689	1.63267	0.816336	−	0.577578i	$-0.196002\pi$
$317$	29.0689	1.63267	0.816336	+	0.577578i	$0.196002\pi$
$318$	0	0
$319$	17.3205	0.969762
$320$	−29.0689	−1.62500
$321$	0	0
$322$	−60.0000	−3.34367
$323$	13.4164	0.746509
$324$	0	0
$325$	0	0
$326$	−30.9839	−1.71604
$327$	0	0
$328$	5.00000	0.276079
$329$	−15.4919	−0.854098
$330$	0	0
$331$	−27.7128	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$331$	−27.7128	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$332$	−13.4164	−0.736321
$333$	0	0
$334$	−20.0000	−1.09435
$335$	−7.74597	−0.423207
$336$	0	0
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	0	0
$340$	25.9808	1.40900
$341$	0	0
$342$	0	0
$343$	6.92820	0.374088
$344$	−4.47214	−0.241121
$345$	0	0
$346$	34.6410	1.86231
$347$	7.74597	0.415825	0.207913	−	0.978147i	$-0.433333\pi$
$347$	7.74597	0.415825	0.207913	+	0.978147i	$0.433333\pi$
$348$	0	0
$349$	13.8564	0.741716	0.370858	−	0.928689i	$-0.379064\pi$
$349$	13.8564	0.741716	0.370858	+	0.928689i	$0.379064\pi$
$350$	0	0
$351$	0	0
$352$	−30.0000	−1.59901
$353$	2.23607	0.119014	0.0595069	−	0.998228i	$-0.481047\pi$
$353$	2.23607	0.119014	0.0595069	+	0.998228i	$0.481047\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	13.4164	0.711068
$357$	0	0
$358$	17.3205	0.915417
$359$	31.3050	1.65221	0.826106	−	0.563515i	$-0.190551\pi$
$359$	31.3050	1.65221	0.826106	+	0.563515i	$0.190551\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	42.4853	2.23298
$363$	0	0
$364$	0	0
$365$	−34.8569	−1.82449
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−7.74597	−0.403786
$369$	0	0
$370$	−8.66025	−0.450225
$371$	40.2492	2.08964
$372$	0	0
$373$	−37.0000	−1.91579	−0.957894	−	0.287123i	$-0.907301\pi$
$373$	−37.0000	−1.91579	−0.957894	+	0.287123i	$0.907301\pi$
$374$	38.7298	2.00267
$375$	0	0
$376$	10.0000	0.515711
$377$	0	0
$378$	0	0
$379$	−13.8564	−0.711756	−0.355878	−	0.934532i	$-0.615818\pi$
$379$	−13.8564	−0.711756	−0.355878	+	0.934532i	$0.615818\pi$
$380$	23.2379	1.19208
$381$	0	0
$382$	−34.6410	−1.77239
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	−34.6410	−1.76547
$386$	−3.87298	−0.197130
$387$	0	0
$388$	−20.7846	−1.05518
$389$	11.6190	0.589104	0.294552	−	0.955635i	$-0.404830\pi$
$389$	11.6190	0.589104	0.294552	+	0.955635i	$0.404830\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	11.1803	0.564692
$393$	0	0
$394$	−10.0000	−0.503793
$395$	17.8885	0.900070
$396$	0	0
$397$	−13.8564	−0.695433	−0.347717	−	0.937600i	$-0.613043\pi$
$397$	−13.8564	−0.695433	−0.347717	+	0.937600i	$0.613043\pi$
$398$	49.1935	2.46585
$399$	0	0
$400$	0	0
$401$	−29.0689	−1.45163	−0.725815	−	0.687890i	$-0.758537\pi$
$401$	−29.0689	−1.45163	−0.725815	+	0.687890i	$0.758537\pi$
$402$	0	0
$403$	0	0
$404$	−11.6190	−0.578064
$405$	0	0
$406$	−30.0000	−1.48888
$407$	−7.74597	−0.383953
$408$	0	0
$409$	8.66025	0.428222	0.214111	−	0.976809i	$-0.431315\pi$
$409$	8.66025	0.428222	0.214111	+	0.976809i	$0.431315\pi$
$410$	11.1803	0.552158
$411$	0	0
$412$	−6.00000	−0.295599
$413$	30.9839	1.52462
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	0	0
$418$	34.6410	1.69435
$419$	15.4919	0.756830	0.378415	−	0.925636i	$-0.376469\pi$
$419$	15.4919	0.756830	0.378415	+	0.925636i	$0.376469\pi$
$420$	0	0
$421$	15.5885	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$421$	15.5885	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$422$	17.8885	0.870801
$423$	0	0
$424$	−25.9808	−1.26174
$425$	0	0
$426$	0	0
$427$	24.2487	1.17348
$428$	−23.2379	−1.12325
$429$	0	0
$430$	−10.0000	−0.482243
$431$	−4.47214	−0.215415	−0.107708	−	0.994183i	$-0.534351\pi$
$431$	−4.47214	−0.215415	−0.107708	+	0.994183i	$0.534351\pi$
$432$	0	0
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.8328	1.28359
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	22.3607	1.06600
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	61.9677	2.93426
$447$	0	0
$448$	45.0333	2.12762
$449$	4.47214	0.211053	0.105527	−	0.994416i	$-0.466347\pi$
$449$	4.47214	0.211053	0.105527	+	0.994416i	$0.466347\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	58.0948	2.73255
$453$	0	0
$454$	50.0000	2.34662
$455$	0	0
$456$	0	0
$457$	12.1244	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$457$	12.1244	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$458$	−46.4758	−2.17167
$459$	0	0
$460$	51.9615	2.42272
$461$	2.23607	0.104144	0.0520720	−	0.998643i	$-0.483417\pi$
$461$	2.23607	0.104144	0.0520720	+	0.998643i	$0.483417\pi$
$462$	0	0
$463$	−10.3923	−0.482971	−0.241486	−	0.970404i	$-0.577635\pi$
$463$	−10.3923	−0.482971	−0.241486	+	0.970404i	$0.577635\pi$
$464$	−3.87298	−0.179799
$465$	0	0
$466$	0	0
$467$	−23.2379	−1.07532	−0.537661	−	0.843161i	$-0.680692\pi$
$467$	−23.2379	−1.07532	−0.537661	+	0.843161i	$0.680692\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	22.3607	1.03142
$471$	0	0
$472$	−20.0000	−0.920575
$473$	−8.94427	−0.411258
$474$	0	0
$475$	0	0
$476$	−40.2492	−1.84482
$477$	0	0
$478$	−10.0000	−0.457389
$479$	17.8885	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$480$	0	0
$481$	0	0
$482$	−42.6028	−1.94051
$483$	0	0
$484$	27.0000	1.22727
$485$	−15.4919	−0.703452
$486$	0	0
$487$	−38.1051	−1.72671	−0.863354	−	0.504599i	$-0.831640\pi$
$487$	−38.1051	−1.72671	−0.863354	+	0.504599i	$0.831640\pi$
$488$	−15.6525	−0.708554
$489$	0	0
$490$	25.0000	1.12938
$491$	−38.7298	−1.74785	−0.873926	−	0.486058i	$-0.838434\pi$
$491$	−38.7298	−1.74785	−0.873926	+	0.486058i	$0.838434\pi$
$492$	0	0
$493$	15.0000	0.675566
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.4919	0.694908
$498$	0	0
$499$	20.7846	0.930447	0.465223	−	0.885193i	$-0.345974\pi$
$499$	20.7846	0.930447	0.465223	+	0.885193i	$0.345974\pi$
$500$	−33.5410	−1.50000
$501$	0	0
$502$	34.6410	1.54610
$503$	7.74597	0.345376	0.172688	−	0.984977i	$-0.444755\pi$
$503$	7.74597	0.345376	0.172688	+	0.984977i	$0.444755\pi$
$504$	0	0
$505$	−8.66025	−0.385376
$506$	77.4597	3.44350
$507$	0	0
$508$	12.0000	0.532414
$509$	29.0689	1.28846	0.644228	−	0.764834i	$-0.277179\pi$
$509$	29.0689	1.28846	0.644228	+	0.764834i	$0.277179\pi$
$510$	0	0
$511$	54.0000	2.38882
$512$	11.1803	0.494106
$513$	0	0
$514$	−8.66025	−0.381987
$515$	−4.47214	−0.197066
$516$	0	0
$517$	20.0000	0.879599
$518$	13.4164	0.589483
$519$	0	0
$520$	0	0
$521$	11.6190	0.509035	0.254518	−	0.967068i	$-0.418083\pi$
$521$	11.6190	0.509035	0.254518	+	0.967068i	$0.418083\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	−17.3205	−0.755210
$527$	0	0
$528$	0	0
$529$	37.0000	1.60870
$530$	−58.0948	−2.52347
$531$	0	0
$532$	−36.0000	−1.56080
$533$	0	0
$534$	0	0
$535$	−17.3205	−0.748831
$536$	−7.74597	−0.334575
$537$	0	0
$538$	−34.6410	−1.49348
$539$	22.3607	0.963143
$540$	0	0
$541$	25.9808	1.11700	0.558500	−	0.829504i	$-0.311377\pi$
$541$	25.9808	1.11700	0.558500	+	0.829504i	$0.311377\pi$
$542$	−15.4919	−0.665436
$543$	0	0
$544$	−25.9808	−1.11392
$545$	0	0
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	−46.9574	−2.00592
$549$	0	0
$550$	0	0
$551$	13.4164	0.571558
$552$	0	0
$553$	−27.7128	−1.17847
$554$	−24.5967	−1.04502
$555$	0	0
$556$	24.0000	1.01783
$557$	−29.0689	−1.23169	−0.615844	−	0.787868i	$-0.711185\pi$
$557$	−29.0689	−1.23169	−0.615844	+	0.787868i	$0.711185\pi$
$558$	0	0
$559$	0	0
$560$	7.74597	0.327327
$561$	0	0
$562$	−35.0000	−1.47639
$563$	−30.9839	−1.30581	−0.652907	−	0.757438i	$-0.726451\pi$
$563$	−30.9839	−1.30581	−0.652907	+	0.757438i	$0.726451\pi$
$564$	0	0
$565$	43.3013	1.82170
$566$	22.3607	0.939889
$567$	0	0
$568$	−10.0000	−0.419591
$569$	−15.4919	−0.649456	−0.324728	−	0.945808i	$-0.605273\pi$
$569$	−15.4919	−0.649456	−0.324728	+	0.945808i	$0.605273\pi$
$570$	0	0
$571$	34.0000	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$571$	34.0000	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$572$	0	0
$573$	0	0
$574$	−17.3205	−0.722944
$575$	0	0
$576$	0	0
$577$	5.19615	0.216319	0.108159	−	0.994134i	$-0.465504\pi$
$577$	5.19615	0.216319	0.108159	+	0.994134i	$0.465504\pi$
$578$	−4.47214	−0.186016
$579$	0	0
$580$	25.9808	1.07879
$581$	15.4919	0.642714
$582$	0	0
$583$	−51.9615	−2.15203
$584$	−34.8569	−1.44239
$585$	0	0
$586$	−5.00000	−0.206548
$587$	35.7771	1.47668	0.738339	−	0.674430i	$-0.235610\pi$
$587$	35.7771	1.47668	0.738339	+	0.674430i	$0.235610\pi$
$588$	0	0
$589$	0	0
$590$	−44.7214	−1.84115
$591$	0	0
$592$	1.73205	0.0711868
$593$	−2.23607	−0.0918243	−0.0459122	−	0.998945i	$-0.514619\pi$
$593$	−2.23607	−0.0918243	−0.0459122	+	0.998945i	$0.514619\pi$
$594$	0	0
$595$	−30.0000	−1.22988
$596$	−33.5410	−1.37389
$597$	0	0
$598$	0	0
$599$	46.4758	1.89895	0.949475	−	0.313843i	$-0.101617\pi$
$599$	46.4758	1.89895	0.949475	+	0.313843i	$0.101617\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	15.4919	0.631404
$603$	0	0
$604$	31.1769	1.26857
$605$	20.1246	0.818182
$606$	0	0
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−23.2379	−0.942421
$609$	0	0
$610$	−35.0000	−1.41711
$611$	0	0
$612$	0	0
$613$	1.73205	0.0699569	0.0349784	−	0.999388i	$-0.488864\pi$
$613$	1.73205	0.0699569	0.0349784	+	0.999388i	$0.488864\pi$
$614$	−23.2379	−0.937805
$615$	0	0
$616$	−34.6410	−1.39573
$617$	29.0689	1.17027	0.585135	−	0.810936i	$-0.301042\pi$
$617$	29.0689	1.17027	0.585135	+	0.810936i	$0.301042\pi$
$618$	0	0
$619$	20.7846	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$620$	0	0
$621$	0	0
$622$	51.9615	2.08347
$623$	−15.4919	−0.620671
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−49.1935	−1.96617
$627$	0	0
$628$	33.0000	1.31684
$629$	−6.70820	−0.267474
$630$	0	0
$631$	6.92820	0.275807	0.137904	−	0.990446i	$-0.455964\pi$
$631$	6.92820	0.275807	0.137904	+	0.990446i	$0.455964\pi$
$632$	17.8885	0.711568
$633$	0	0
$634$	65.0000	2.58148
$635$	8.94427	0.354943
$636$	0	0
$637$	0	0
$638$	38.7298	1.53333
$639$	0	0
$640$	−35.0000	−1.38350
$641$	−19.3649	−0.764868	−0.382434	−	0.923983i	$-0.624914\pi$
$641$	−19.3649	−0.764868	−0.382434	+	0.923983i	$0.624914\pi$
$642$	0	0
$643$	−27.7128	−1.09289	−0.546443	−	0.837496i	$-0.684019\pi$
$643$	−27.7128	−1.09289	−0.546443	+	0.837496i	$0.684019\pi$
$644$	−80.4984	−3.17208
$645$	0	0
$646$	30.0000	1.18033
$647$	30.9839	1.21810	0.609051	−	0.793131i	$-0.291551\pi$
$647$	30.9839	1.21810	0.609051	+	0.793131i	$0.291551\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	0	0
$652$	−41.5692	−1.62798
$653$	15.4919	0.606246	0.303123	−	0.952951i	$-0.401971\pi$
$653$	15.4919	0.606246	0.303123	+	0.952951i	$0.401971\pi$
$654$	0	0
$655$	0	0
$656$	−2.23607	−0.0873038
$657$	0	0
$658$	−34.6410	−1.35045
$659$	30.9839	1.20696	0.603480	−	0.797378i	$-0.293780\pi$
$659$	30.9839	1.20696	0.603480	+	0.797378i	$0.293780\pi$
$660$	0	0
$661$	19.0526	0.741059	0.370529	−	0.928821i	$-0.379176\pi$
$661$	19.0526	0.741059	0.370529	+	0.928821i	$0.379176\pi$
$662$	−61.9677	−2.40844
$663$	0	0
$664$	−10.0000	−0.388075
$665$	−26.8328	−1.04053
$666$	0	0
$667$	30.0000	1.16160
$668$	−26.8328	−1.03819
$669$	0	0
$670$	−17.3205	−0.669150
$671$	−31.3050	−1.20851
$672$	0	0
$673$	43.0000	1.65753	0.828764	−	0.559598i	$-0.189045\pi$
$673$	43.0000	1.65753	0.828764	+	0.559598i	$0.189045\pi$
$674$	−24.5967	−0.947431
$675$	0	0
$676$	0	0
$677$	46.4758	1.78621	0.893105	−	0.449848i	$-0.148522\pi$
$677$	46.4758	1.78621	0.893105	+	0.449848i	$0.148522\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	19.3649	0.742611
$681$	0	0
$682$	0	0
$683$	−35.7771	−1.36897	−0.684486	−	0.729026i	$-0.739973\pi$
$683$	−35.7771	−1.36897	−0.684486	+	0.729026i	$0.739973\pi$
$684$	0	0
$685$	−35.0000	−1.33728
$686$	15.4919	0.591485
$687$	0	0
$688$	2.00000	0.0762493
$689$	0	0
$690$	0	0
$691$	−24.2487	−0.922464	−0.461232	−	0.887279i	$-0.652592\pi$
$691$	−24.2487	−0.922464	−0.461232	+	0.887279i	$0.652592\pi$
$692$	46.4758	1.76674
$693$	0	0
$694$	17.3205	0.657477
$695$	17.8885	0.678551
$696$	0	0
$697$	8.66025	0.328031
$698$	30.9839	1.17276
$699$	0	0
$700$	0	0
$701$	46.4758	1.75537	0.877683	−	0.479241i	$-0.159088\pi$
$701$	46.4758	1.75537	0.877683	+	0.479241i	$0.159088\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−58.1378	−2.19115
$705$	0	0
$706$	5.00000	0.188177
$707$	13.4164	0.504576
$708$	0	0
$709$	12.1244	0.455340	0.227670	−	0.973738i	$-0.426889\pi$
$709$	12.1244	0.455340	0.227670	+	0.973738i	$0.426889\pi$
$710$	−22.3607	−0.839181
$711$	0	0
$712$	10.0000	0.374766
$713$	0	0
$714$	0	0
$715$	0	0
$716$	23.2379	0.868441
$717$	0	0
$718$	70.0000	2.61238
$719$	−30.9839	−1.15550	−0.577752	−	0.816213i	$-0.696070\pi$
$719$	−30.9839	−1.15550	−0.577752	+	0.816213i	$0.696070\pi$
$720$	0	0
$721$	6.92820	0.258020
$722$	−15.6525	−0.582525
$723$	0	0
$724$	57.0000	2.11839
$725$	0	0
$726$	0	0
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	0	0
$729$	0	0
$730$	−77.9423	−2.88477
$731$	−7.74597	−0.286495
$732$	0	0
$733$	−36.3731	−1.34347	−0.671735	−	0.740792i	$-0.734451\pi$
$733$	−36.3731	−1.34347	−0.671735	+	0.740792i	$0.734451\pi$
$734$	31.3050	1.15549
$735$	0	0
$736$	−51.9615	−1.91533
$737$	−15.4919	−0.570653
$738$	0	0
$739$	34.6410	1.27429	0.637145	−	0.770744i	$-0.280115\pi$
$739$	34.6410	1.27429	0.637145	+	0.770744i	$0.280115\pi$
$740$	−11.6190	−0.427121
$741$	0	0
$742$	90.0000	3.30400
$743$	8.94427	0.328134	0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427	0.328134	0.164067	+	0.986449i	$0.447539\pi$
$744$	0	0
$745$	−25.0000	−0.915929
$746$	−82.7345	−3.02913
$747$	0	0
$748$	51.9615	1.89990
$749$	26.8328	0.980450
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	−4.47214	−0.163082
$753$	0	0
$754$	0	0
$755$	23.2379	0.845714
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	−30.9839	−1.12538
$759$	0	0
$760$	17.3205	0.628281
$761$	31.3050	1.13480	0.567402	−	0.823441i	$-0.307949\pi$
$761$	31.3050	1.13480	0.567402	+	0.823441i	$0.307949\pi$
$762$	0	0
$763$	0	0
$764$	−46.4758	−1.68144
$765$	0	0
$766$	−40.0000	−1.44526
$767$	0	0
$768$	0	0
$769$	48.4974	1.74886	0.874431	−	0.485150i	$-0.161235\pi$
$769$	48.4974	1.74886	0.874431	+	0.485150i	$0.161235\pi$
$770$	−77.4597	−2.79145
$771$	0	0
$772$	−5.19615	−0.187014
$773$	−4.47214	−0.160852	−0.0804258	−	0.996761i	$-0.525628\pi$
$773$	−4.47214	−0.160852	−0.0804258	+	0.996761i	$0.525628\pi$
$774$	0	0
$775$	0	0
$776$	−15.4919	−0.556128
$777$	0	0
$778$	25.9808	0.931455
$779$	7.74597	0.277528
$780$	0	0
$781$	−20.0000	−0.715656
$782$	67.0820	2.39885
$783$	0	0
$784$	−5.00000	−0.178571
$785$	24.5967	0.877896
$786$	0	0
$787$	27.7128	0.987855	0.493928	−	0.869503i	$-0.335561\pi$
$787$	27.7128	0.987855	0.493928	+	0.869503i	$0.335561\pi$
$788$	−13.4164	−0.477940
$789$	0	0
$790$	40.0000	1.42314
$791$	−67.0820	−2.38516
$792$	0	0
$793$	0	0
$794$	−30.9839	−1.09958
$795$	0	0
$796$	66.0000	2.33931
$797$	−30.9839	−1.09750	−0.548752	−	0.835985i	$-0.684897\pi$
$797$	−30.9839	−1.09750	−0.548752	+	0.835985i	$0.684897\pi$
$798$	0	0
$799$	17.3205	0.612756
$800$	0	0
$801$	0	0
$802$	−65.0000	−2.29523
$803$	−69.7137	−2.46014
$804$	0	0
$805$	−60.0000	−2.11472
$806$	0	0
$807$	0	0
$808$	−8.66025	−0.304667
$809$	−42.6028	−1.49784	−0.748918	−	0.662663i	$-0.769426\pi$
$809$	−42.6028	−1.49784	−0.748918	+	0.662663i	$0.769426\pi$
$810$	0	0
$811$	−41.5692	−1.45969	−0.729846	−	0.683611i	$-0.760408\pi$
$811$	−41.5692	−1.45969	−0.729846	+	0.683611i	$0.760408\pi$
$812$	−40.2492	−1.41247
$813$	0	0
$814$	−17.3205	−0.607083
$815$	−30.9839	−1.08532
$816$	0	0
$817$	−6.92820	−0.242387
$818$	19.3649	0.677078
$819$	0	0
$820$	15.0000	0.523823
$821$	−4.47214	−0.156079	−0.0780393	−	0.996950i	$-0.524866\pi$
$821$	−4.47214	−0.156079	−0.0780393	+	0.996950i	$0.524866\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−4.47214	−0.155794
$825$	0	0
$826$	69.2820	2.41063
$827$	17.8885	0.622046	0.311023	−	0.950402i	$-0.399328\pi$
$827$	17.8885	0.622046	0.311023	+	0.950402i	$0.399328\pi$
$828$	0	0
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	−22.3607	−0.776151
$831$	0	0
$832$	0	0
$833$	19.3649	0.670955
$834$	0	0
$835$	−20.0000	−0.692129
$836$	46.4758	1.60740
$837$	0	0
$838$	34.6410	1.19665
$839$	17.8885	0.617581	0.308791	−	0.951130i	$-0.400076\pi$
$839$	17.8885	0.617581	0.308791	+	0.951130i	$0.400076\pi$
$840$	0	0
$841$	−14.0000	−0.482759
$842$	34.8569	1.20125
$843$	0	0
$844$	24.0000	0.826114
$845$	0	0
$846$	0	0
$847$	−31.1769	−1.07125
$848$	11.6190	0.398996
$849$	0	0
$850$	0	0
$851$	−13.4164	−0.459909
$852$	0	0
$853$	5.19615	0.177913	0.0889564	−	0.996036i	$-0.471647\pi$
$853$	5.19615	0.177913	0.0889564	+	0.996036i	$0.471647\pi$
$854$	54.2218	1.85543
$855$	0	0
$856$	−17.3205	−0.592003
$857$	−34.8569	−1.19069	−0.595344	−	0.803471i	$-0.702984\pi$
$857$	−34.8569	−1.19069	−0.595344	+	0.803471i	$0.702984\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	−13.4164	−0.457496
$861$	0	0
$862$	−10.0000	−0.340601
$863$	−31.3050	−1.06563	−0.532816	−	0.846231i	$-0.678866\pi$
$863$	−31.3050	−1.06563	−0.532816	+	0.846231i	$0.678866\pi$
$864$	0	0
$865$	34.6410	1.17783
$866$	−78.2624	−2.65946
$867$	0	0
$868$	0	0
$869$	35.7771	1.21365
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	60.0000	2.02953
$875$	38.7298	1.30931
$876$	0	0
$877$	29.4449	0.994282	0.497141	−	0.867670i	$-0.334383\pi$
$877$	29.4449	0.994282	0.497141	+	0.867670i	$0.334383\pi$
$878$	−4.47214	−0.150927
$879$	0	0
$880$	−10.0000	−0.337100
$881$	−27.1109	−0.913389	−0.456694	−	0.889624i	$-0.650967\pi$
$881$	−27.1109	−0.913389	−0.456694	+	0.889624i	$0.650967\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.4919	0.520168	0.260084	−	0.965586i	$-0.416250\pi$
$887$	15.4919	0.520168	0.260084	+	0.965586i	$0.416250\pi$
$888$	0	0
$889$	−13.8564	−0.464729
$890$	22.3607	0.749532
$891$	0	0
$892$	83.1384	2.78368
$893$	15.4919	0.518418
$894$	0	0
$895$	17.3205	0.578961
$896$	54.2218	1.81142
$897$	0	0
$898$	10.0000	0.333704
$899$	0	0
$900$	0	0
$901$	−45.0000	−1.49917
$902$	22.3607	0.744529
$903$	0	0
$904$	43.3013	1.44018
$905$	42.4853	1.41226
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	67.0820	2.22620
$909$	0	0
$910$	0	0
$911$	−46.4758	−1.53981	−0.769906	−	0.638157i	$-0.779697\pi$
$911$	−46.4758	−1.53981	−0.769906	+	0.638157i	$0.779697\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	27.1109	0.896748
$915$	0	0
$916$	−62.3538	−2.06023
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	38.7298	1.27688
$921$	0	0
$922$	5.00000	0.164666
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−23.2379	−0.763645
$927$	0	0
$928$	−25.9808	−0.852860
$929$	55.9017	1.83408	0.917038	−	0.398801i	$-0.130573\pi$
$929$	55.9017	1.83408	0.917038	+	0.398801i	$0.130573\pi$
$930$	0	0
$931$	17.3205	0.567657
$932$	0	0
$933$	0	0
$934$	−51.9615	−1.70023
$935$	38.7298	1.26660
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	26.8328	0.876122
$939$	0	0
$940$	30.0000	0.978492
$941$	−4.47214	−0.145787	−0.0728937	−	0.997340i	$-0.523223\pi$
$941$	−4.47214	−0.145787	−0.0728937	+	0.997340i	$0.523223\pi$
$942$	0	0
$943$	17.3205	0.564033
$944$	8.94427	0.291111
$945$	0	0
$946$	−20.0000	−0.650256
$947$	17.8885	0.581300	0.290650	−	0.956830i	$-0.406129\pi$
$947$	17.8885	0.581300	0.290650	+	0.956830i	$0.406129\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−30.0000	−0.972306
$953$	15.4919	0.501833	0.250916	−	0.968009i	$-0.419268\pi$
$953$	15.4919	0.501833	0.250916	+	0.968009i	$0.419268\pi$
$954$	0	0
$955$	−34.6410	−1.12096
$956$	−13.4164	−0.433918
$957$	0	0
$958$	40.0000	1.29234
$959$	54.2218	1.75091
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	−57.1577	−1.84092
$965$	−3.87298	−0.124676
$966$	0	0
$967$	10.3923	0.334194	0.167097	−	0.985940i	$-0.446561\pi$
$967$	10.3923	0.334194	0.167097	+	0.985940i	$0.446561\pi$
$968$	20.1246	0.646830
$969$	0	0
$970$	−34.6410	−1.11226
$971$	30.9839	0.994320	0.497160	−	0.867659i	$-0.334376\pi$
$971$	30.9839	0.994320	0.497160	+	0.867659i	$0.334376\pi$
$972$	0	0
$973$	−27.7128	−0.888432
$974$	−85.2056	−2.73016
$975$	0	0
$976$	7.00000	0.224065
$977$	−11.1803	−0.357691	−0.178845	−	0.983877i	$-0.557236\pi$
$977$	−11.1803	−0.357691	−0.178845	+	0.983877i	$0.557236\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	33.5410	1.07143
$981$	0	0
$982$	−86.6025	−2.76360
$983$	−8.94427	−0.285278	−0.142639	−	0.989775i	$-0.545559\pi$
$983$	−8.94427	−0.285278	−0.142639	+	0.989775i	$0.545559\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	33.5410	1.06816
$987$	0	0
$988$	0	0
$989$	−15.4919	−0.492615
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	0	0
$994$	34.6410	1.09875
$995$	49.1935	1.55954
$996$	0	0
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	46.4758	1.47117
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.u.1.3		4
3.2	odd	2	inner	1521.2.a.u.1.1		4
13.2	odd	12		117.2.q.d.82.2	yes	4
13.5	odd	4		1521.2.b.g.1351.1		4
13.7	odd	12		117.2.q.d.10.2	yes	4
13.8	odd	4		1521.2.b.g.1351.4		4
13.12	even	2	inner	1521.2.a.u.1.2		4
39.2	even	12		117.2.q.d.82.1	yes	4
39.5	even	4		1521.2.b.g.1351.3		4
39.8	even	4		1521.2.b.g.1351.2		4
39.20	even	12		117.2.q.d.10.1	✓	4
39.38	odd	2	inner	1521.2.a.u.1.4		4
52.7	even	12		1872.2.by.l.1297.2		4
52.15	even	12		1872.2.by.l.433.1		4
156.59	odd	12		1872.2.by.l.1297.1		4
156.119	odd	12		1872.2.by.l.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.q.d.10.1	✓	4	39.20	even	12
117.2.q.d.10.2	yes	4	13.7	odd	12
117.2.q.d.82.1	yes	4	39.2	even	12
117.2.q.d.82.2	yes	4	13.2	odd	12
1521.2.a.u.1.1		4	3.2	odd	2	inner
1521.2.a.u.1.2		4	13.12	even	2	inner
1521.2.a.u.1.3		4	1.1	even	1	trivial
1521.2.a.u.1.4		4	39.38	odd	2	inner
1521.2.b.g.1351.1		4	13.5	odd	4
1521.2.b.g.1351.2		4	39.8	even	4
1521.2.b.g.1351.3		4	39.5	even	4
1521.2.b.g.1351.4		4	13.8	odd	4
1872.2.by.l.433.1		4	52.15	even	12
1872.2.by.l.433.2		4	156.119	odd	12
1872.2.by.l.1297.1		4	156.59	odd	12
1872.2.by.l.1297.2		4	52.7	even	12

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} +2.23607 q^{5} -3.46410 q^{7} +2.23607 q^{8} +5.00000 q^{10} +4.47214 q^{11} -7.74597 q^{14} -1.00000 q^{16} +3.87298 q^{17} +3.46410 q^{19} +6.70820 q^{20} +10.0000 q^{22} +7.74597 q^{23} -10.3923 q^{28} +3.87298 q^{29} -6.70820 q^{32} +8.66025 q^{34} -7.74597 q^{35} -1.73205 q^{37} +7.74597 q^{38} +5.00000 q^{40} +2.23607 q^{41} -2.00000 q^{43} +13.4164 q^{44} +17.3205 q^{46} +4.47214 q^{47} +5.00000 q^{49} -11.6190 q^{53} +10.0000 q^{55} -7.74597 q^{56} +8.66025 q^{58} -8.94427 q^{59} -7.00000 q^{61} -13.0000 q^{64} -3.46410 q^{67} +11.6190 q^{68} -17.3205 q^{70} -4.47214 q^{71} -15.5885 q^{73} -3.87298 q^{74} +10.3923 q^{76} -15.4919 q^{77} +8.00000 q^{79} -2.23607 q^{80} +5.00000 q^{82} -4.47214 q^{83} +8.66025 q^{85} -4.47214 q^{86} +10.0000 q^{88} +4.47214 q^{89} +23.2379 q^{92} +10.0000 q^{94} +7.74597 q^{95} -6.92820 q^{97} +11.1803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} + 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} - 8 q^{43} + 20 q^{49} + 40 q^{55} - 28 q^{61} - 52 q^{64} + 32 q^{79} + 20 q^{82} + 40 q^{88} + 40 q^{94}+O(q^{100}) \) 4 * q + 12 * q^4 + 20 * q^10 - 4 * q^16 + 40 * q^22 + 20 * q^40 - 8 * q^43 + 20 * q^49 + 40 * q^55 - 28 * q^61 - 52 * q^64 + 32 * q^79 + 20 * q^82 + 40 * q^88 + 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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