LMFDB - Embedded newform 4761.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, 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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1587)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{2} +5.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -9.46410 q^{8} +O(q^{10})$	$q-2.73205 q^{2} +5.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -9.46410 q^{8} +7.46410 q^{10} +1.26795 q^{11} +1.00000 q^{13} +0.732051 q^{14} +14.9282 q^{16} +6.73205 q^{17} +1.46410 q^{19} -14.9282 q^{20} -3.46410 q^{22} +2.46410 q^{25} -2.73205 q^{26} -1.46410 q^{28} -7.66025 q^{29} -2.00000 q^{31} -21.8564 q^{32} -18.3923 q^{34} +0.732051 q^{35} -7.19615 q^{37} -4.00000 q^{38} +25.8564 q^{40} -2.53590 q^{41} -6.26795 q^{43} +6.92820 q^{44} +8.19615 q^{47} -6.92820 q^{49} -6.73205 q^{50} +5.46410 q^{52} -8.92820 q^{53} -3.46410 q^{55} +2.53590 q^{56} +20.9282 q^{58} +9.66025 q^{59} -3.19615 q^{61} +5.46410 q^{62} +29.8564 q^{64} -2.73205 q^{65} +10.2679 q^{67} +36.7846 q^{68} -2.00000 q^{70} -4.19615 q^{71} +6.92820 q^{73} +19.6603 q^{74} +8.00000 q^{76} -0.339746 q^{77} -4.00000 q^{79} -40.7846 q^{80} +6.92820 q^{82} +6.00000 q^{83} -18.3923 q^{85} +17.1244 q^{86} -12.0000 q^{88} +2.19615 q^{89} -0.267949 q^{91} -22.3923 q^{94} -4.00000 q^{95} +14.9282 q^{97} +18.9282 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} + 8 q^{10} + 6 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} - 16 q^{20} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32} - 16 q^{34} - 2 q^{35} - 4 q^{37} - 8 q^{38} + 24 q^{40} - 12 q^{41} - 16 q^{43} + 6 q^{47} - 10 q^{50} + 4 q^{52} - 4 q^{53} + 12 q^{56} + 28 q^{58} + 2 q^{59} + 4 q^{61} + 4 q^{62} + 32 q^{64} - 2 q^{65} + 24 q^{67} + 32 q^{68} - 4 q^{70} + 2 q^{71} + 22 q^{74} + 16 q^{76} - 18 q^{77} - 8 q^{79} - 40 q^{80} + 12 q^{83} - 16 q^{85} + 10 q^{86} - 24 q^{88} - 6 q^{89} - 4 q^{91} - 24 q^{94} - 8 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 + 8 * q^10 + 6 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 - 16 * q^20 - 2 * q^25 - 2 * q^26 + 4 * q^28 + 2 * q^29 - 4 * q^31 - 16 * q^32 - 16 * q^34 - 2 * q^35 - 4 * q^37 - 8 * q^38 + 24 * q^40 - 12 * q^41 - 16 * q^43 + 6 * q^47 - 10 * q^50 + 4 * q^52 - 4 * q^53 + 12 * q^56 + 28 * q^58 + 2 * q^59 + 4 * q^61 + 4 * q^62 + 32 * q^64 - 2 * q^65 + 24 * q^67 + 32 * q^68 - 4 * q^70 + 2 * q^71 + 22 * q^74 + 16 * q^76 - 18 * q^77 - 8 * q^79 - 40 * q^80 + 12 * q^83 - 16 * q^85 + 10 * q^86 - 24 * q^88 - 6 * q^89 - 4 * q^91 - 24 * q^94 - 8 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	−2.73205	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−2.73205	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	−0.267949	−0.101275	−0.0506376	−	0.998717i	$-0.516125\pi$
$7$	−0.267949	−0.101275	−0.0506376	+	0.998717i	$0.516125\pi$
$8$	−9.46410	−3.34607
$9$	0	0
$10$	7.46410	2.36036
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0.732051	0.195649
$15$	0	0
$16$	14.9282	3.73205
$17$	6.73205	1.63276	0.816381	−	0.577514i	$-0.195977\pi$
$17$	6.73205	1.63276	0.816381	+	0.577514i	$0.195977\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	−14.9282	−3.33805
$21$	0	0
$22$	−3.46410	−0.738549
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.46410	0.492820
$26$	−2.73205	−0.535799
$27$	0	0
$28$	−1.46410	−0.276689
$29$	−7.66025	−1.42247	−0.711237	−	0.702953i	$-0.751865\pi$
$29$	−7.66025	−1.42247	−0.711237	+	0.702953i	$0.751865\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−21.8564	−3.86370
$33$	0	0
$34$	−18.3923	−3.15425
$35$	0.732051	0.123739
$36$	0	0
$37$	−7.19615	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$37$	−7.19615	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	25.8564	4.08826
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	−6.26795	−0.955853	−0.477927	−	0.878400i	$-0.658612\pi$
$43$	−6.26795	−0.955853	−0.477927	+	0.878400i	$0.658612\pi$
$44$	6.92820	1.04447
$45$	0	0
$46$	0	0
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	−6.73205	−0.952056
$51$	0	0
$52$	5.46410	0.757735
$53$	−8.92820	−1.22638	−0.613192	−	0.789934i	$-0.710115\pi$
$53$	−8.92820	−1.22638	−0.613192	+	0.789934i	$0.710115\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	2.53590	0.338874
$57$	0	0
$58$	20.9282	2.74801
$59$	9.66025	1.25766	0.628829	−	0.777544i	$-0.283535\pi$
$59$	9.66025	1.25766	0.628829	+	0.777544i	$0.283535\pi$
$60$	0	0
$61$	−3.19615	−0.409225	−0.204613	−	0.978843i	$-0.565593\pi$
$61$	−3.19615	−0.409225	−0.204613	+	0.978843i	$0.565593\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	29.8564	3.73205
$65$	−2.73205	−0.338869
$66$	0	0
$67$	10.2679	1.25443	0.627215	−	0.778846i	$-0.284195\pi$
$67$	10.2679	1.25443	0.627215	+	0.778846i	$0.284195\pi$
$68$	36.7846	4.46079
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−4.19615	−0.497992	−0.248996	−	0.968505i	$-0.580101\pi$
$71$	−4.19615	−0.497992	−0.248996	+	0.968505i	$0.580101\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	19.6603	2.28546
$75$	0	0
$76$	8.00000	0.917663
$77$	−0.339746	−0.0387176
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−40.7846	−4.55986
$81$	0	0
$82$	6.92820	0.765092
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−18.3923	−1.99493
$86$	17.1244	1.84657
$87$	0	0
$88$	−12.0000	−1.27920
$89$	2.19615	0.232792	0.116396	−	0.993203i	$-0.462866\pi$
$89$	2.19615	0.232792	0.116396	+	0.993203i	$0.462866\pi$
$90$	0	0
$91$	−0.267949	−0.0280887
$92$	0	0
$93$	0	0
$94$	−22.3923	−2.30959
$95$	−4.00000	−0.410391
$96$	0	0
$97$	14.9282	1.51573	0.757865	−	0.652412i	$-0.226243\pi$
$97$	14.9282	1.51573	0.757865	+	0.652412i	$0.226243\pi$
$98$	18.9282	1.91204
$99$	0	0
$100$	13.4641	1.34641
$101$	−0.339746	−0.0338060	−0.0169030	−	0.999857i	$-0.505381\pi$
$101$	−0.339746	−0.0338060	−0.0169030	+	0.999857i	$0.505381\pi$
$102$	0	0
$103$	17.5885	1.73304	0.866521	−	0.499140i	$-0.166351\pi$
$103$	17.5885	1.73304	0.866521	+	0.499140i	$0.166351\pi$
$104$	−9.46410	−0.928032
$105$	0	0
$106$	24.3923	2.36919
$107$	11.8564	1.14620	0.573101	−	0.819485i	$-0.305740\pi$
$107$	11.8564	1.14620	0.573101	+	0.819485i	$0.305740\pi$
$108$	0	0
$109$	7.46410	0.714931	0.357466	−	0.933926i	$-0.383641\pi$
$109$	7.46410	0.714931	0.357466	+	0.933926i	$0.383641\pi$
$110$	9.46410	0.902367
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−15.8564	−1.49165	−0.745823	−	0.666145i	$-0.767943\pi$
$113$	−15.8564	−1.49165	−0.745823	+	0.666145i	$0.767943\pi$
$114$	0	0
$115$	0	0
$116$	−41.8564	−3.88627
$117$	0	0
$118$	−26.3923	−2.42961
$119$	−1.80385	−0.165358
$120$	0	0
$121$	−9.39230	−0.853846
$122$	8.73205	0.790563
$123$	0	0
$124$	−10.9282	−0.981382
$125$	6.92820	0.619677
$126$	0	0
$127$	9.39230	0.833432	0.416716	−	0.909037i	$-0.363181\pi$
$127$	9.39230	0.833432	0.416716	+	0.909037i	$0.363181\pi$
$128$	−37.8564	−3.34607
$129$	0	0
$130$	7.46410	0.654645
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	−0.392305	−0.0340171
$134$	−28.0526	−2.42337
$135$	0	0
$136$	−63.7128	−5.46333
$137$	−7.26795	−0.620943	−0.310471	−	0.950583i	$-0.600487\pi$
$137$	−7.26795	−0.620943	−0.310471	+	0.950583i	$0.600487\pi$
$138$	0	0
$139$	−15.5359	−1.31774	−0.658869	−	0.752258i	$-0.728965\pi$
$139$	−15.5359	−1.31774	−0.658869	+	0.752258i	$0.728965\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	11.4641	0.962046
$143$	1.26795	0.106031
$144$	0	0
$145$	20.9282	1.73799
$146$	−18.9282	−1.56651
$147$	0	0
$148$	−39.3205	−3.23213
$149$	20.9282	1.71451	0.857253	−	0.514896i	$-0.172169\pi$
$149$	20.9282	1.71451	0.857253	+	0.514896i	$0.172169\pi$
$150$	0	0
$151$	−22.8564	−1.86003	−0.930014	−	0.367524i	$-0.880206\pi$
$151$	−22.8564	−1.86003	−0.930014	+	0.367524i	$0.880206\pi$
$152$	−13.8564	−1.12390
$153$	0	0
$154$	0.928203	0.0747967
$155$	5.46410	0.438887
$156$	0	0
$157$	0.535898	0.0427693	0.0213847	−	0.999771i	$-0.493193\pi$
$157$	0.535898	0.0427693	0.0213847	+	0.999771i	$0.493193\pi$
$158$	10.9282	0.869401
$159$	0	0
$160$	59.7128	4.72071
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−13.8564	−1.08200
$165$	0	0
$166$	−16.3923	−1.27229
$167$	−0.535898	−0.0414691	−0.0207345	−	0.999785i	$-0.506600\pi$
$167$	−0.535898	−0.0414691	−0.0207345	+	0.999785i	$0.506600\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	50.2487	3.85390
$171$	0	0
$172$	−34.2487	−2.61144
$173$	3.12436	0.237540	0.118770	−	0.992922i	$-0.462105\pi$
$173$	3.12436	0.237540	0.118770	+	0.992922i	$0.462105\pi$
$174$	0	0
$175$	−0.660254	−0.0499105
$176$	18.9282	1.42677
$177$	0	0
$178$	−6.00000	−0.449719
$179$	5.80385	0.433800	0.216900	−	0.976194i	$-0.430405\pi$
$179$	5.80385	0.433800	0.216900	+	0.976194i	$0.430405\pi$
$180$	0	0
$181$	−0.535898	−0.0398330	−0.0199165	−	0.999802i	$-0.506340\pi$
$181$	−0.535898	−0.0398330	−0.0199165	+	0.999802i	$0.506340\pi$
$182$	0.732051	0.0542632
$183$	0	0
$184$	0	0
$185$	19.6603	1.44545
$186$	0	0
$187$	8.53590	0.624207
$188$	44.7846	3.26625
$189$	0	0
$190$	10.9282	0.792815
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	−40.7846	−2.92816
$195$	0	0
$196$	−37.8564	−2.70403
$197$	8.39230	0.597927	0.298963	−	0.954265i	$-0.403359\pi$
$197$	8.39230	0.597927	0.298963	+	0.954265i	$0.403359\pi$
$198$	0	0
$199$	−17.7321	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$199$	−17.7321	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$200$	−23.3205	−1.64901
$201$	0	0
$202$	0.928203	0.0653082
$203$	2.05256	0.144061
$204$	0	0
$205$	6.92820	0.483887
$206$	−48.0526	−3.34798
$207$	0	0
$208$	14.9282	1.03508
$209$	1.85641	0.128410
$210$	0	0
$211$	−1.53590	−0.105736	−0.0528678	−	0.998602i	$-0.516836\pi$
$211$	−1.53590	−0.105736	−0.0528678	+	0.998602i	$0.516836\pi$
$212$	−48.7846	−3.35054
$213$	0	0
$214$	−32.3923	−2.21429
$215$	17.1244	1.16787
$216$	0	0
$217$	0.535898	0.0363792
$218$	−20.3923	−1.38114
$219$	0	0
$220$	−18.9282	−1.27614
$221$	6.73205	0.452847
$222$	0	0
$223$	−19.9282	−1.33449	−0.667246	−	0.744838i	$-0.732527\pi$
$223$	−19.9282	−1.33449	−0.667246	+	0.744838i	$0.732527\pi$
$224$	5.85641	0.391298
$225$	0	0
$226$	43.3205	2.88164
$227$	4.73205	0.314077	0.157039	−	0.987592i	$-0.449805\pi$
$227$	4.73205	0.314077	0.157039	+	0.987592i	$0.449805\pi$
$228$	0	0
$229$	−4.80385	−0.317447	−0.158724	−	0.987323i	$-0.550738\pi$
$229$	−4.80385	−0.317447	−0.158724	+	0.987323i	$0.550738\pi$
$230$	0	0
$231$	0	0
$232$	72.4974	4.75969
$233$	−10.9282	−0.715930	−0.357965	−	0.933735i	$-0.616529\pi$
$233$	−10.9282	−0.715930	−0.357965	+	0.933735i	$0.616529\pi$
$234$	0	0
$235$	−22.3923	−1.46071
$236$	52.7846	3.43599
$237$	0	0
$238$	4.92820	0.319448
$239$	−24.1962	−1.56512	−0.782559	−	0.622576i	$-0.786086\pi$
$239$	−24.1962	−1.56512	−0.782559	+	0.622576i	$0.786086\pi$
$240$	0	0
$241$	8.80385	0.567106	0.283553	−	0.958957i	$-0.408487\pi$
$241$	8.80385	0.567106	0.283553	+	0.958957i	$0.408487\pi$
$242$	25.6603	1.64950
$243$	0	0
$244$	−17.4641	−1.11802
$245$	18.9282	1.20928
$246$	0	0
$247$	1.46410	0.0931586
$248$	18.9282	1.20194
$249$	0	0
$250$	−18.9282	−1.19712
$251$	24.9282	1.57345	0.786727	−	0.617301i	$-0.211774\pi$
$251$	24.9282	1.57345	0.786727	+	0.617301i	$0.211774\pi$
$252$	0	0
$253$	0	0
$254$	−25.6603	−1.61007
$255$	0	0
$256$	43.7128	2.73205
$257$	−26.3923	−1.64631	−0.823153	−	0.567819i	$-0.807787\pi$
$257$	−26.3923	−1.64631	−0.823153	+	0.567819i	$0.807787\pi$
$258$	0	0
$259$	1.92820	0.119813
$260$	−14.9282	−0.925808
$261$	0	0
$262$	13.8564	0.856052
$263$	−27.1244	−1.67256	−0.836280	−	0.548303i	$-0.815274\pi$
$263$	−27.1244	−1.67256	−0.836280	+	0.548303i	$0.815274\pi$
$264$	0	0
$265$	24.3923	1.49841
$266$	1.07180	0.0657161
$267$	0	0
$268$	56.1051	3.42717
$269$	−3.46410	−0.211210	−0.105605	−	0.994408i	$-0.533678\pi$
$269$	−3.46410	−0.211210	−0.105605	+	0.994408i	$0.533678\pi$
$270$	0	0
$271$	−2.60770	−0.158406	−0.0792031	−	0.996858i	$-0.525238\pi$
$271$	−2.60770	−0.158406	−0.0792031	+	0.996858i	$0.525238\pi$
$272$	100.497	6.09355
$273$	0	0
$274$	19.8564	1.19957
$275$	3.12436	0.188406
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	42.4449	2.54567
$279$	0	0
$280$	−6.92820	−0.414039
$281$	−22.2487	−1.32725	−0.663623	−	0.748067i	$-0.730982\pi$
$281$	−22.2487	−1.32725	−0.663623	+	0.748067i	$0.730982\pi$
$282$	0	0
$283$	28.1244	1.67182	0.835910	−	0.548867i	$-0.184941\pi$
$283$	28.1244	1.67182	0.835910	+	0.548867i	$0.184941\pi$
$284$	−22.9282	−1.36054
$285$	0	0
$286$	−3.46410	−0.204837
$287$	0.679492	0.0401091
$288$	0	0
$289$	28.3205	1.66591
$290$	−57.1769	−3.35754
$291$	0	0
$292$	37.8564	2.21538
$293$	−13.2679	−0.775122	−0.387561	−	0.921844i	$-0.626682\pi$
$293$	−13.2679	−0.775122	−0.387561	+	0.921844i	$0.626682\pi$
$294$	0	0
$295$	−26.3923	−1.53662
$296$	68.1051	3.95853
$297$	0	0
$298$	−57.1769	−3.31217
$299$	0	0
$300$	0	0
$301$	1.67949	0.0968043
$302$	62.4449	3.59330
$303$	0	0
$304$	21.8564	1.25355
$305$	8.73205	0.499996
$306$	0	0
$307$	−4.60770	−0.262975	−0.131488	−	0.991318i	$-0.541975\pi$
$307$	−4.60770	−0.262975	−0.131488	+	0.991318i	$0.541975\pi$
$308$	−1.85641	−0.105779
$309$	0	0
$310$	−14.9282	−0.847865
$311$	9.60770	0.544802	0.272401	−	0.962184i	$-0.412182\pi$
$311$	9.60770	0.544802	0.272401	+	0.962184i	$0.412182\pi$
$312$	0	0
$313$	−20.1244	−1.13750	−0.568748	−	0.822512i	$-0.692572\pi$
$313$	−20.1244	−1.13750	−0.568748	+	0.822512i	$0.692572\pi$
$314$	−1.46410	−0.0826240
$315$	0	0
$316$	−21.8564	−1.22952
$317$	−15.8564	−0.890585	−0.445292	−	0.895385i	$-0.646900\pi$
$317$	−15.8564	−0.890585	−0.445292	+	0.895385i	$0.646900\pi$
$318$	0	0
$319$	−9.71281	−0.543813
$320$	−81.5692	−4.55986
$321$	0	0
$322$	0	0
$323$	9.85641	0.548425
$324$	0	0
$325$	2.46410	0.136684
$326$	2.73205	0.151314
$327$	0	0
$328$	24.0000	1.32518
$329$	−2.19615	−0.121078
$330$	0	0
$331$	−12.8564	−0.706652	−0.353326	−	0.935500i	$-0.614949\pi$
$331$	−12.8564	−0.706652	−0.353326	+	0.935500i	$0.614949\pi$
$332$	32.7846	1.79929
$333$	0	0
$334$	1.46410	0.0801121
$335$	−28.0526	−1.53268
$336$	0	0
$337$	23.7321	1.29277	0.646384	−	0.763013i	$-0.276281\pi$
$337$	23.7321	1.29277	0.646384	+	0.763013i	$0.276281\pi$
$338$	32.7846	1.78325
$339$	0	0
$340$	−100.497	−5.45024
$341$	−2.53590	−0.137327
$342$	0	0
$343$	3.73205	0.201512
$344$	59.3205	3.19835
$345$	0	0
$346$	−8.53590	−0.458893
$347$	−16.7321	−0.898224	−0.449112	−	0.893476i	$-0.648260\pi$
$347$	−16.7321	−0.898224	−0.449112	+	0.893476i	$0.648260\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	1.80385	0.0964197
$351$	0	0
$352$	−27.7128	−1.47710
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	11.4641	0.608451
$356$	12.0000	0.635999
$357$	0	0
$358$	−15.8564	−0.838037
$359$	10.9282	0.576769	0.288384	−	0.957515i	$-0.406882\pi$
$359$	10.9282	0.576769	0.288384	+	0.957515i	$0.406882\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	1.46410	0.0769515
$363$	0	0
$364$	−1.46410	−0.0767398
$365$	−18.9282	−0.990747
$366$	0	0
$367$	−33.4641	−1.74681	−0.873406	−	0.486993i	$-0.838094\pi$
$367$	−33.4641	−1.74681	−0.873406	+	0.486993i	$0.838094\pi$
$368$	0	0
$369$	0	0
$370$	−53.7128	−2.79240
$371$	2.39230	0.124202
$372$	0	0
$373$	−19.1962	−0.993939	−0.496970	−	0.867768i	$-0.665554\pi$
$373$	−19.1962	−0.993939	−0.496970	+	0.867768i	$0.665554\pi$
$374$	−23.3205	−1.20587
$375$	0	0
$376$	−77.5692	−4.00033
$377$	−7.66025	−0.394523
$378$	0	0
$379$	31.1962	1.60244	0.801219	−	0.598371i	$-0.204185\pi$
$379$	31.1962	1.60244	0.801219	+	0.598371i	$0.204185\pi$
$380$	−21.8564	−1.12121
$381$	0	0
$382$	−5.46410	−0.279568
$383$	1.07180	0.0547663	0.0273831	−	0.999625i	$-0.491283\pi$
$383$	1.07180	0.0547663	0.0273831	+	0.999625i	$0.491283\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	8.19615	0.417173
$387$	0	0
$388$	81.5692	4.14105
$389$	−13.5167	−0.685322	−0.342661	−	0.939459i	$-0.611328\pi$
$389$	−13.5167	−0.685322	−0.342661	+	0.939459i	$0.611328\pi$
$390$	0	0
$391$	0	0
$392$	65.5692	3.31175
$393$	0	0
$394$	−22.9282	−1.15511
$395$	10.9282	0.549858
$396$	0	0
$397$	20.4641	1.02706	0.513532	−	0.858070i	$-0.328337\pi$
$397$	20.4641	1.02706	0.513532	+	0.858070i	$0.328337\pi$
$398$	48.4449	2.42832
$399$	0	0
$400$	36.7846	1.83923
$401$	2.53590	0.126637	0.0633184	−	0.997993i	$-0.479832\pi$
$401$	2.53590	0.126637	0.0633184	+	0.997993i	$0.479832\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	−1.85641	−0.0923597
$405$	0	0
$406$	−5.60770	−0.278305
$407$	−9.12436	−0.452278
$408$	0	0
$409$	18.8564	0.932389	0.466195	−	0.884682i	$-0.345625\pi$
$409$	18.8564	0.932389	0.466195	+	0.884682i	$0.345625\pi$
$410$	−18.9282	−0.934797
$411$	0	0
$412$	96.1051	4.73476
$413$	−2.58846	−0.127370
$414$	0	0
$415$	−16.3923	−0.804667
$416$	−21.8564	−1.07160
$417$	0	0
$418$	−5.07180	−0.248070
$419$	−4.73205	−0.231176	−0.115588	−	0.993297i	$-0.536875\pi$
$419$	−4.73205	−0.231176	−0.115588	+	0.993297i	$0.536875\pi$
$420$	0	0
$421$	−1.33975	−0.0652952	−0.0326476	−	0.999467i	$-0.510394\pi$
$421$	−1.33975	−0.0652952	−0.0326476	+	0.999467i	$0.510394\pi$
$422$	4.19615	0.204266
$423$	0	0
$424$	84.4974	4.10356
$425$	16.5885	0.804658
$426$	0	0
$427$	0.856406	0.0414444
$428$	64.7846	3.13148
$429$	0	0
$430$	−46.7846	−2.25615
$431$	−24.3923	−1.17494	−0.587468	−	0.809247i	$-0.699875\pi$
$431$	−24.3923	−1.17494	−0.587468	+	0.809247i	$0.699875\pi$
$432$	0	0
$433$	−3.19615	−0.153597	−0.0767986	−	0.997047i	$-0.524470\pi$
$433$	−3.19615	−0.153597	−0.0767986	+	0.997047i	$0.524470\pi$
$434$	−1.46410	−0.0702791
$435$	0	0
$436$	40.7846	1.95323
$437$	0	0
$438$	0	0
$439$	−31.7846	−1.51700	−0.758498	−	0.651675i	$-0.774067\pi$
$439$	−31.7846	−1.51700	−0.758498	+	0.651675i	$0.774067\pi$
$440$	32.7846	1.56294
$441$	0	0
$442$	−18.3923	−0.874833
$443$	−23.6603	−1.12413	−0.562066	−	0.827092i	$-0.689993\pi$
$443$	−23.6603	−1.12413	−0.562066	+	0.827092i	$0.689993\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	54.4449	2.57804
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−23.6603	−1.11660	−0.558298	−	0.829640i	$-0.688545\pi$
$449$	−23.6603	−1.11660	−0.558298	+	0.829640i	$0.688545\pi$
$450$	0	0
$451$	−3.21539	−0.151407
$452$	−86.6410	−4.07525
$453$	0	0
$454$	−12.9282	−0.606751
$455$	0.732051	0.0343191
$456$	0	0
$457$	−13.1962	−0.617290	−0.308645	−	0.951177i	$-0.599876\pi$
$457$	−13.1962	−0.617290	−0.308645	+	0.951177i	$0.599876\pi$
$458$	13.1244	0.613261
$459$	0	0
$460$	0	0
$461$	−22.0526	−1.02709	−0.513545	−	0.858063i	$-0.671668\pi$
$461$	−22.0526	−1.02709	−0.513545	+	0.858063i	$0.671668\pi$
$462$	0	0
$463$	6.07180	0.282180	0.141090	−	0.989997i	$-0.454939\pi$
$463$	6.07180	0.282180	0.141090	+	0.989997i	$0.454939\pi$
$464$	−114.354	−5.30874
$465$	0	0
$466$	29.8564	1.38307
$467$	25.6603	1.18741	0.593707	−	0.804681i	$-0.297664\pi$
$467$	25.6603	1.18741	0.593707	+	0.804681i	$0.297664\pi$
$468$	0	0
$469$	−2.75129	−0.127043
$470$	61.1769	2.82188
$471$	0	0
$472$	−91.4256	−4.20821
$473$	−7.94744	−0.365424
$474$	0	0
$475$	3.60770	0.165532
$476$	−9.85641	−0.451768
$477$	0	0
$478$	66.1051	3.02358
$479$	10.3923	0.474837	0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923	0.474837	0.237418	+	0.971408i	$0.423699\pi$
$480$	0	0
$481$	−7.19615	−0.328116
$482$	−24.0526	−1.09556
$483$	0	0
$484$	−51.3205	−2.33275
$485$	−40.7846	−1.85193
$486$	0	0
$487$	−16.3205	−0.739553	−0.369776	−	0.929121i	$-0.620566\pi$
$487$	−16.3205	−0.739553	−0.369776	+	0.929121i	$0.620566\pi$
$488$	30.2487	1.36929
$489$	0	0
$490$	−51.7128	−2.33615
$491$	−10.0526	−0.453666	−0.226833	−	0.973934i	$-0.572837\pi$
$491$	−10.0526	−0.453666	−0.226833	+	0.973934i	$0.572837\pi$
$492$	0	0
$493$	−51.5692	−2.32256
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−29.8564	−1.34059
$497$	1.12436	0.0504342
$498$	0	0
$499$	26.3205	1.17827	0.589134	−	0.808035i	$-0.299469\pi$
$499$	26.3205	1.17827	0.589134	+	0.808035i	$0.299469\pi$
$500$	37.8564	1.69299
$501$	0	0
$502$	−68.1051	−3.03968
$503$	−12.7321	−0.567694	−0.283847	−	0.958870i	$-0.591611\pi$
$503$	−12.7321	−0.567694	−0.283847	+	0.958870i	$0.591611\pi$
$504$	0	0
$505$	0.928203	0.0413045
$506$	0	0
$507$	0	0
$508$	51.3205	2.27698
$509$	−17.8564	−0.791471	−0.395736	−	0.918364i	$-0.629510\pi$
$509$	−17.8564	−0.791471	−0.395736	+	0.918364i	$0.629510\pi$
$510$	0	0
$511$	−1.85641	−0.0821226
$512$	−43.7128	−1.93185
$513$	0	0
$514$	72.1051	3.18042
$515$	−48.0526	−2.11745
$516$	0	0
$517$	10.3923	0.457053
$518$	−5.26795	−0.231460
$519$	0	0
$520$	25.8564	1.13388
$521$	−19.8564	−0.869925	−0.434962	−	0.900449i	$-0.643238\pi$
$521$	−19.8564	−0.869925	−0.434962	+	0.900449i	$0.643238\pi$
$522$	0	0
$523$	21.4641	0.938560	0.469280	−	0.883050i	$-0.344514\pi$
$523$	21.4641	0.938560	0.469280	+	0.883050i	$0.344514\pi$
$524$	−27.7128	−1.21064
$525$	0	0
$526$	74.1051	3.23114
$527$	−13.4641	−0.586505
$528$	0	0
$529$	0	0
$530$	−66.6410	−2.89470
$531$	0	0
$532$	−2.14359	−0.0929366
$533$	−2.53590	−0.109842
$534$	0	0
$535$	−32.3923	−1.40044
$536$	−97.1769	−4.19740
$537$	0	0
$538$	9.46410	0.408026
$539$	−8.78461	−0.378380
$540$	0	0
$541$	13.4641	0.578867	0.289433	−	0.957198i	$-0.406533\pi$
$541$	13.4641	0.578867	0.289433	+	0.957198i	$0.406533\pi$
$542$	7.12436	0.306017
$543$	0	0
$544$	−147.138	−6.30851
$545$	−20.3923	−0.873510
$546$	0	0
$547$	−38.1051	−1.62926	−0.814629	−	0.579983i	$-0.803059\pi$
$547$	−38.1051	−1.62926	−0.814629	+	0.579983i	$0.803059\pi$
$548$	−39.7128	−1.69645
$549$	0	0
$550$	−8.53590	−0.363972
$551$	−11.2154	−0.477792
$552$	0	0
$553$	1.07180	0.0455774
$554$	−62.8372	−2.66970
$555$	0	0
$556$	−84.8897	−3.60013
$557$	−15.4641	−0.655235	−0.327618	−	0.944810i	$-0.606246\pi$
$557$	−15.4641	−0.655235	−0.327618	+	0.944810i	$0.606246\pi$
$558$	0	0
$559$	−6.26795	−0.265106
$560$	10.9282	0.461801
$561$	0	0
$562$	60.7846	2.56404
$563$	−8.53590	−0.359745	−0.179873	−	0.983690i	$-0.557569\pi$
$563$	−8.53590	−0.359745	−0.179873	+	0.983690i	$0.557569\pi$
$564$	0	0
$565$	43.3205	1.82251
$566$	−76.8372	−3.22971
$567$	0	0
$568$	39.7128	1.66631
$569$	−10.3397	−0.433465	−0.216732	−	0.976231i	$-0.569540\pi$
$569$	−10.3397	−0.433465	−0.216732	+	0.976231i	$0.569540\pi$
$570$	0	0
$571$	21.7321	0.909458	0.454729	−	0.890630i	$-0.349736\pi$
$571$	21.7321	0.909458	0.454729	+	0.890630i	$0.349736\pi$
$572$	6.92820	0.289683
$573$	0	0
$574$	−1.85641	−0.0774849
$575$	0	0
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−77.3731	−3.21830
$579$	0	0
$580$	114.354	4.74828
$581$	−1.60770	−0.0666984
$582$	0	0
$583$	−11.3205	−0.468848
$584$	−65.5692	−2.71327
$585$	0	0
$586$	36.2487	1.49742
$587$	35.6603	1.47186	0.735928	−	0.677060i	$-0.236746\pi$
$587$	35.6603	1.47186	0.735928	+	0.677060i	$0.236746\pi$
$588$	0	0
$589$	−2.92820	−0.120655
$590$	72.1051	2.96852
$591$	0	0
$592$	−107.426	−4.41517
$593$	11.1244	0.456823	0.228411	−	0.973565i	$-0.426647\pi$
$593$	11.1244	0.456823	0.228411	+	0.973565i	$0.426647\pi$
$594$	0	0
$595$	4.92820	0.202037
$596$	114.354	4.68412
$597$	0	0
$598$	0	0
$599$	−22.5885	−0.922939	−0.461470	−	0.887156i	$-0.652678\pi$
$599$	−22.5885	−0.922939	−0.461470	+	0.887156i	$0.652678\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	−4.58846	−0.187012
$603$	0	0
$604$	−124.890	−5.08169
$605$	25.6603	1.04324
$606$	0	0
$607$	20.6077	0.836441	0.418220	−	0.908346i	$-0.362654\pi$
$607$	20.6077	0.836441	0.418220	+	0.908346i	$0.362654\pi$
$608$	−32.0000	−1.29777
$609$	0	0
$610$	−23.8564	−0.965918
$611$	8.19615	0.331581
$612$	0	0
$613$	−4.94744	−0.199825	−0.0999126	−	0.994996i	$-0.531856\pi$
$613$	−4.94744	−0.199825	−0.0999126	+	0.994996i	$0.531856\pi$
$614$	12.5885	0.508029
$615$	0	0
$616$	3.21539	0.129552
$617$	−12.7846	−0.514689	−0.257345	−	0.966320i	$-0.582848\pi$
$617$	−12.7846	−0.514689	−0.257345	+	0.966320i	$0.582848\pi$
$618$	0	0
$619$	25.0526	1.00695	0.503474	−	0.864011i	$-0.332055\pi$
$619$	25.0526	1.00695	0.503474	+	0.864011i	$0.332055\pi$
$620$	29.8564	1.19906
$621$	0	0
$622$	−26.2487	−1.05248
$623$	−0.588457	−0.0235760
$624$	0	0
$625$	−31.2487	−1.24995
$626$	54.9808	2.19747
$627$	0	0
$628$	2.92820	0.116848
$629$	−48.4449	−1.93162
$630$	0	0
$631$	−47.5885	−1.89447	−0.947233	−	0.320545i	$-0.896134\pi$
$631$	−47.5885	−1.89447	−0.947233	+	0.320545i	$0.896134\pi$
$632$	37.8564	1.50585
$633$	0	0
$634$	43.3205	1.72048
$635$	−25.6603	−1.01830
$636$	0	0
$637$	−6.92820	−0.274505
$638$	26.5359	1.05057
$639$	0	0
$640$	103.426	4.08826
$641$	−4.53590	−0.179157	−0.0895786	−	0.995980i	$-0.528552\pi$
$641$	−4.53590	−0.179157	−0.0895786	+	0.995980i	$0.528552\pi$
$642$	0	0
$643$	31.0526	1.22459	0.612297	−	0.790628i	$-0.290246\pi$
$643$	31.0526	1.22459	0.612297	+	0.790628i	$0.290246\pi$
$644$	0	0
$645$	0	0
$646$	−26.9282	−1.05948
$647$	2.87564	0.113053	0.0565266	−	0.998401i	$-0.481997\pi$
$647$	2.87564	0.113053	0.0565266	+	0.998401i	$0.481997\pi$
$648$	0	0
$649$	12.2487	0.480804
$650$	−6.73205	−0.264053
$651$	0	0
$652$	−5.46410	−0.213991
$653$	−13.0718	−0.511539	−0.255769	−	0.966738i	$-0.582329\pi$
$653$	−13.0718	−0.511539	−0.255769	+	0.966738i	$0.582329\pi$
$654$	0	0
$655$	13.8564	0.541415
$656$	−37.8564	−1.47804
$657$	0	0
$658$	6.00000	0.233904
$659$	−39.2679	−1.52966	−0.764831	−	0.644231i	$-0.777178\pi$
$659$	−39.2679	−1.52966	−0.764831	+	0.644231i	$0.777178\pi$
$660$	0	0
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	35.1244	1.36515
$663$	0	0
$664$	−56.7846	−2.20367
$665$	1.07180	0.0415625
$666$	0	0
$667$	0	0
$668$	−2.92820	−0.113296
$669$	0	0
$670$	76.6410	2.96090
$671$	−4.05256	−0.156447
$672$	0	0
$673$	43.7846	1.68777	0.843886	−	0.536522i	$-0.180262\pi$
$673$	43.7846	1.68777	0.843886	+	0.536522i	$0.180262\pi$
$674$	−64.8372	−2.49743
$675$	0	0
$676$	−65.5692	−2.52189
$677$	11.2679	0.433062	0.216531	−	0.976276i	$-0.430526\pi$
$677$	11.2679	0.433062	0.216531	+	0.976276i	$0.430526\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	174.067	6.67515
$681$	0	0
$682$	6.92820	0.265295
$683$	−44.1962	−1.69112	−0.845559	−	0.533881i	$-0.820733\pi$
$683$	−44.1962	−1.69112	−0.845559	+	0.533881i	$0.820733\pi$
$684$	0	0
$685$	19.8564	0.758674
$686$	−10.1962	−0.389291
$687$	0	0
$688$	−93.5692	−3.56729
$689$	−8.92820	−0.340137
$690$	0	0
$691$	8.46410	0.321990	0.160995	−	0.986955i	$-0.448530\pi$
$691$	8.46410	0.321990	0.160995	+	0.986955i	$0.448530\pi$
$692$	17.0718	0.648972
$693$	0	0
$694$	45.7128	1.73523
$695$	42.4449	1.61003
$696$	0	0
$697$	−17.0718	−0.646640
$698$	30.0526	1.13751
$699$	0	0
$700$	−3.60770	−0.136358
$701$	−4.67949	−0.176742	−0.0883710	−	0.996088i	$-0.528166\pi$
$701$	−4.67949	−0.176742	−0.0883710	+	0.996088i	$0.528166\pi$
$702$	0	0
$703$	−10.5359	−0.397369
$704$	37.8564	1.42677
$705$	0	0
$706$	−16.3923	−0.616933
$707$	0.0910347	0.00342371
$708$	0	0
$709$	−15.8756	−0.596222	−0.298111	−	0.954531i	$-0.596357\pi$
$709$	−15.8756	−0.596222	−0.298111	+	0.954531i	$0.596357\pi$
$710$	−31.3205	−1.17544
$711$	0	0
$712$	−20.7846	−0.778936
$713$	0	0
$714$	0	0
$715$	−3.46410	−0.129550
$716$	31.7128	1.18516
$717$	0	0
$718$	−29.8564	−1.11423
$719$	−15.8038	−0.589384	−0.294692	−	0.955592i	$-0.595217\pi$
$719$	−15.8038	−0.589384	−0.294692	+	0.955592i	$0.595217\pi$
$720$	0	0
$721$	−4.71281	−0.175514
$722$	46.0526	1.71390
$723$	0	0
$724$	−2.92820	−0.108826
$725$	−18.8756	−0.701024
$726$	0	0
$727$	30.9282	1.14706	0.573532	−	0.819183i	$-0.305573\pi$
$727$	30.9282	1.14706	0.573532	+	0.819183i	$0.305573\pi$
$728$	2.53590	0.0939866
$729$	0	0
$730$	51.7128	1.91398
$731$	−42.1962	−1.56068
$732$	0	0
$733$	−17.7321	−0.654948	−0.327474	−	0.944860i	$-0.606197\pi$
$733$	−17.7321	−0.654948	−0.327474	+	0.944860i	$0.606197\pi$
$734$	91.4256	3.37458
$735$	0	0
$736$	0	0
$737$	13.0192	0.479570
$738$	0	0
$739$	−1.67949	−0.0617811	−0.0308906	−	0.999523i	$-0.509834\pi$
$739$	−1.67949	−0.0617811	−0.0308906	+	0.999523i	$0.509834\pi$
$740$	107.426	3.94904
$741$	0	0
$742$	−6.53590	−0.239940
$743$	−19.6077	−0.719337	−0.359668	−	0.933080i	$-0.617110\pi$
$743$	−19.6077	−0.719337	−0.359668	+	0.933080i	$0.617110\pi$
$744$	0	0
$745$	−57.1769	−2.09480
$746$	52.4449	1.92014
$747$	0	0
$748$	46.6410	1.70536
$749$	−3.17691	−0.116082
$750$	0	0
$751$	29.4641	1.07516	0.537580	−	0.843213i	$-0.319339\pi$
$751$	29.4641	1.07516	0.537580	+	0.843213i	$0.319339\pi$
$752$	122.354	4.46179
$753$	0	0
$754$	20.9282	0.762160
$755$	62.4449	2.27260
$756$	0	0
$757$	−15.1769	−0.551614	−0.275807	−	0.961213i	$-0.588945\pi$
$757$	−15.1769	−0.551614	−0.275807	+	0.961213i	$0.588945\pi$
$758$	−85.2295	−3.09567
$759$	0	0
$760$	37.8564	1.37320
$761$	29.6603	1.07518	0.537592	−	0.843205i	$-0.319334\pi$
$761$	29.6603	1.07518	0.537592	+	0.843205i	$0.319334\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	10.9282	0.395369
$765$	0	0
$766$	−2.92820	−0.105800
$767$	9.66025	0.348812
$768$	0	0
$769$	22.1244	0.797825	0.398912	−	0.916989i	$-0.369388\pi$
$769$	22.1244	0.797825	0.398912	+	0.916989i	$0.369388\pi$
$770$	−2.53590	−0.0913874
$771$	0	0
$772$	−16.3923	−0.589972
$773$	37.0333	1.33200	0.665998	−	0.745954i	$-0.268006\pi$
$773$	37.0333	1.33200	0.665998	+	0.745954i	$0.268006\pi$
$774$	0	0
$775$	−4.92820	−0.177026
$776$	−141.282	−5.07173
$777$	0	0
$778$	36.9282	1.32394
$779$	−3.71281	−0.133025
$780$	0	0
$781$	−5.32051	−0.190383
$782$	0	0
$783$	0	0
$784$	−103.426	−3.69377
$785$	−1.46410	−0.0522560
$786$	0	0
$787$	−49.1769	−1.75297	−0.876484	−	0.481431i	$-0.840117\pi$
$787$	−49.1769	−1.75297	−0.876484	+	0.481431i	$0.840117\pi$
$788$	45.8564	1.63357
$789$	0	0
$790$	−29.8564	−1.06224
$791$	4.24871	0.151067
$792$	0	0
$793$	−3.19615	−0.113499
$794$	−55.9090	−1.98413
$795$	0	0
$796$	−96.8897	−3.43417
$797$	−22.0526	−0.781142	−0.390571	−	0.920573i	$-0.627722\pi$
$797$	−22.0526	−0.781142	−0.390571	+	0.920573i	$0.627722\pi$
$798$	0	0
$799$	55.1769	1.95202
$800$	−53.8564	−1.90411
$801$	0	0
$802$	−6.92820	−0.244643
$803$	8.78461	0.310002
$804$	0	0
$805$	0	0
$806$	5.46410	0.192465
$807$	0	0
$808$	3.21539	0.113117
$809$	−0.483340	−0.0169933	−0.00849666	−	0.999964i	$-0.502705\pi$
$809$	−0.483340	−0.0169933	−0.00849666	+	0.999964i	$0.502705\pi$
$810$	0	0
$811$	−19.5359	−0.685998	−0.342999	−	0.939336i	$-0.611443\pi$
$811$	−19.5359	−0.685998	−0.342999	+	0.939336i	$0.611443\pi$
$812$	11.2154	0.393583
$813$	0	0
$814$	24.9282	0.873733
$815$	2.73205	0.0956996
$816$	0	0
$817$	−9.17691	−0.321060
$818$	−51.5167	−1.80124
$819$	0	0
$820$	37.8564	1.32200
$821$	13.4641	0.469900	0.234950	−	0.972007i	$-0.424507\pi$
$821$	13.4641	0.469900	0.234950	+	0.972007i	$0.424507\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−166.459	−5.79887
$825$	0	0
$826$	7.07180	0.246059
$827$	39.7128	1.38095	0.690475	−	0.723356i	$-0.257402\pi$
$827$	39.7128	1.38095	0.690475	+	0.723356i	$0.257402\pi$
$828$	0	0
$829$	30.7846	1.06919	0.534597	−	0.845107i	$-0.320463\pi$
$829$	30.7846	1.06919	0.534597	+	0.845107i	$0.320463\pi$
$830$	44.7846	1.55450
$831$	0	0
$832$	29.8564	1.03508
$833$	−46.6410	−1.61602
$834$	0	0
$835$	1.46410	0.0506673
$836$	10.1436	0.350824
$837$	0	0
$838$	12.9282	0.446597
$839$	−38.5359	−1.33041	−0.665203	−	0.746662i	$-0.731655\pi$
$839$	−38.5359	−1.33041	−0.665203	+	0.746662i	$0.731655\pi$
$840$	0	0
$841$	29.6795	1.02343
$842$	3.66025	0.126141
$843$	0	0
$844$	−8.39230	−0.288875
$845$	32.7846	1.12782
$846$	0	0
$847$	2.51666	0.0864735
$848$	−133.282	−4.57692
$849$	0	0
$850$	−45.3205	−1.55448
$851$	0	0
$852$	0	0
$853$	33.9282	1.16168	0.580840	−	0.814018i	$-0.302724\pi$
$853$	33.9282	1.16168	0.580840	+	0.814018i	$0.302724\pi$
$854$	−2.33975	−0.0800645
$855$	0	0
$856$	−112.210	−3.83527
$857$	−43.5167	−1.48650	−0.743250	−	0.669013i	$-0.766717\pi$
$857$	−43.5167	−1.48650	−0.743250	+	0.669013i	$0.766717\pi$
$858$	0	0
$859$	−29.9282	−1.02114	−0.510569	−	0.859837i	$-0.670565\pi$
$859$	−29.9282	−1.02114	−0.510569	+	0.859837i	$0.670565\pi$
$860$	93.5692	3.19068
$861$	0	0
$862$	66.6410	2.26980
$863$	48.9282	1.66554	0.832768	−	0.553623i	$-0.186755\pi$
$863$	48.9282	1.66554	0.832768	+	0.553623i	$0.186755\pi$
$864$	0	0
$865$	−8.53590	−0.290229
$866$	8.73205	0.296727
$867$	0	0
$868$	2.92820	0.0993897
$869$	−5.07180	−0.172049
$870$	0	0
$871$	10.2679	0.347916
$872$	−70.6410	−2.39221
$873$	0	0
$874$	0	0
$875$	−1.85641	−0.0627580
$876$	0	0
$877$	−53.4256	−1.80406	−0.902028	−	0.431678i	$-0.857922\pi$
$877$	−53.4256	−1.80406	−0.902028	+	0.431678i	$0.857922\pi$
$878$	86.8372	2.93061
$879$	0	0
$880$	−51.7128	−1.74324
$881$	25.1769	0.848232	0.424116	−	0.905608i	$-0.360585\pi$
$881$	25.1769	0.848232	0.424116	+	0.905608i	$0.360585\pi$
$882$	0	0
$883$	55.5692	1.87005	0.935027	−	0.354578i	$-0.115375\pi$
$883$	55.5692	1.87005	0.935027	+	0.354578i	$0.115375\pi$
$884$	36.7846	1.23720
$885$	0	0
$886$	64.6410	2.17166
$887$	−13.0718	−0.438908	−0.219454	−	0.975623i	$-0.570428\pi$
$887$	−13.0718	−0.438908	−0.219454	+	0.975623i	$0.570428\pi$
$888$	0	0
$889$	−2.51666	−0.0844061
$890$	16.3923	0.549471
$891$	0	0
$892$	−108.890	−3.64590
$893$	12.0000	0.401565
$894$	0	0
$895$	−15.8564	−0.530021
$896$	10.1436	0.338874
$897$	0	0
$898$	64.6410	2.15710
$899$	15.3205	0.510968
$900$	0	0
$901$	−60.1051	−2.00239
$902$	8.78461	0.292496
$903$	0	0
$904$	150.067	4.99114
$905$	1.46410	0.0486684
$906$	0	0
$907$	12.5167	0.415609	0.207804	−	0.978170i	$-0.433368\pi$
$907$	12.5167	0.415609	0.207804	+	0.978170i	$0.433368\pi$
$908$	25.8564	0.858075
$909$	0	0
$910$	−2.00000	−0.0662994
$911$	−49.3205	−1.63406	−0.817031	−	0.576594i	$-0.804381\pi$
$911$	−49.3205	−1.63406	−0.817031	+	0.576594i	$0.804381\pi$
$912$	0	0
$913$	7.60770	0.251778
$914$	36.0526	1.19251
$915$	0	0
$916$	−26.2487	−0.867282
$917$	1.35898	0.0448776
$918$	0	0
$919$	−33.8372	−1.11619	−0.558093	−	0.829779i	$-0.688467\pi$
$919$	−33.8372	−1.11619	−0.558093	+	0.829779i	$0.688467\pi$
$920$	0	0
$921$	0	0
$922$	60.2487	1.98419
$923$	−4.19615	−0.138118
$924$	0	0
$925$	−17.7321	−0.583026
$926$	−16.5885	−0.545131
$927$	0	0
$928$	167.426	5.49602
$929$	39.8038	1.30592	0.652961	−	0.757392i	$-0.273527\pi$
$929$	39.8038	1.30592	0.652961	+	0.757392i	$0.273527\pi$
$930$	0	0
$931$	−10.1436	−0.332443
$932$	−59.7128	−1.95596
$933$	0	0
$934$	−70.1051	−2.29391
$935$	−23.3205	−0.762662
$936$	0	0
$937$	−5.98076	−0.195383	−0.0976915	−	0.995217i	$-0.531146\pi$
$937$	−5.98076	−0.195383	−0.0976915	+	0.995217i	$0.531146\pi$
$938$	7.51666	0.245428
$939$	0	0
$940$	−122.354	−3.99074
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	0	0
$943$	0	0
$944$	144.210	4.69364
$945$	0	0
$946$	21.7128	0.705944
$947$	−32.5885	−1.05898	−0.529491	−	0.848315i	$-0.677617\pi$
$947$	−32.5885	−1.05898	−0.529491	+	0.848315i	$0.677617\pi$
$948$	0	0
$949$	6.92820	0.224899
$950$	−9.85641	−0.319784
$951$	0	0
$952$	17.0718	0.553300
$953$	−26.0526	−0.843925	−0.421963	−	0.906613i	$-0.638659\pi$
$953$	−26.0526	−0.843925	−0.421963	+	0.906613i	$0.638659\pi$
$954$	0	0
$955$	−5.46410	−0.176814
$956$	−132.210	−4.27598
$957$	0	0
$958$	−28.3923	−0.917314
$959$	1.94744	0.0628862
$960$	0	0
$961$	−27.0000	−0.870968
$962$	19.6603	0.633872
$963$	0	0
$964$	48.1051	1.54936
$965$	8.19615	0.263843
$966$	0	0
$967$	27.2487	0.876259	0.438130	−	0.898912i	$-0.355641\pi$
$967$	27.2487	0.876259	0.438130	+	0.898912i	$0.355641\pi$
$968$	88.8897	2.85702
$969$	0	0
$970$	111.426	3.57766
$971$	−8.14359	−0.261340	−0.130670	−	0.991426i	$-0.541713\pi$
$971$	−8.14359	−0.261340	−0.130670	+	0.991426i	$0.541713\pi$
$972$	0	0
$973$	4.16283	0.133454
$974$	44.5885	1.42871
$975$	0	0
$976$	−47.7128	−1.52725
$977$	17.0718	0.546175	0.273088	−	0.961989i	$-0.411955\pi$
$977$	17.0718	0.546175	0.273088	+	0.961989i	$0.411955\pi$
$978$	0	0
$979$	2.78461	0.0889965
$980$	103.426	3.30381
$981$	0	0
$982$	27.4641	0.876415
$983$	41.8038	1.33334	0.666668	−	0.745355i	$-0.267720\pi$
$983$	41.8038	1.33334	0.666668	+	0.745355i	$0.267720\pi$
$984$	0	0
$985$	−22.9282	−0.730553
$986$	140.890	4.48684
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−25.6077	−0.813455	−0.406728	−	0.913549i	$-0.633330\pi$
$991$	−25.6077	−0.813455	−0.406728	+	0.913549i	$0.633330\pi$
$992$	43.7128	1.38788
$993$	0	0
$994$	−3.07180	−0.0974315
$995$	48.4449	1.53581
$996$	0	0
$997$	39.0000	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$997$	39.0000	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$998$	−71.9090	−2.27624
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.l.1.1		2
3.2	odd	2		1587.2.a.l.1.2	yes	2
23.22	odd	2		4761.2.a.m.1.1		2
69.68	even	2		1587.2.a.k.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1587.2.a.k.1.2	✓	2	69.68	even	2
1587.2.a.l.1.2	yes	2	3.2	odd	2
4761.2.a.l.1.1		2	1.1	even	1	trivial
4761.2.a.m.1.1		2	23.22	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} +5.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -9.46410 q^{8} +O(q^{10})\)	\(q-2.73205 q^{2} +5.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -9.46410 q^{8} +7.46410 q^{10} +1.26795 q^{11} +1.00000 q^{13} +0.732051 q^{14} +14.9282 q^{16} +6.73205 q^{17} +1.46410 q^{19} -14.9282 q^{20} -3.46410 q^{22} +2.46410 q^{25} -2.73205 q^{26} -1.46410 q^{28} -7.66025 q^{29} -2.00000 q^{31} -21.8564 q^{32} -18.3923 q^{34} +0.732051 q^{35} -7.19615 q^{37} -4.00000 q^{38} +25.8564 q^{40} -2.53590 q^{41} -6.26795 q^{43} +6.92820 q^{44} +8.19615 q^{47} -6.92820 q^{49} -6.73205 q^{50} +5.46410 q^{52} -8.92820 q^{53} -3.46410 q^{55} +2.53590 q^{56} +20.9282 q^{58} +9.66025 q^{59} -3.19615 q^{61} +5.46410 q^{62} +29.8564 q^{64} -2.73205 q^{65} +10.2679 q^{67} +36.7846 q^{68} -2.00000 q^{70} -4.19615 q^{71} +6.92820 q^{73} +19.6603 q^{74} +8.00000 q^{76} -0.339746 q^{77} -4.00000 q^{79} -40.7846 q^{80} +6.92820 q^{82} +6.00000 q^{83} -18.3923 q^{85} +17.1244 q^{86} -12.0000 q^{88} +2.19615 q^{89} -0.267949 q^{91} -22.3923 q^{94} -4.00000 q^{95} +14.9282 q^{97} +18.9282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} + 8 q^{10} + 6 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} - 16 q^{20} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32} - 16 q^{34} - 2 q^{35} - 4 q^{37} - 8 q^{38} + 24 q^{40} - 12 q^{41} - 16 q^{43} + 6 q^{47} - 10 q^{50} + 4 q^{52} - 4 q^{53} + 12 q^{56} + 28 q^{58} + 2 q^{59} + 4 q^{61} + 4 q^{62} + 32 q^{64} - 2 q^{65} + 24 q^{67} + 32 q^{68} - 4 q^{70} + 2 q^{71} + 22 q^{74} + 16 q^{76} - 18 q^{77} - 8 q^{79} - 40 q^{80} + 12 q^{83} - 16 q^{85} + 10 q^{86} - 24 q^{88} - 6 q^{89} - 4 q^{91} - 24 q^{94} - 8 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 + 8 * q^10 + 6 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 - 16 * q^20 - 2 * q^25 - 2 * q^26 + 4 * q^28 + 2 * q^29 - 4 * q^31 - 16 * q^32 - 16 * q^34 - 2 * q^35 - 4 * q^37 - 8 * q^38 + 24 * q^40 - 12 * q^41 - 16 * q^43 + 6 * q^47 - 10 * q^50 + 4 * q^52 - 4 * q^53 + 12 * q^56 + 28 * q^58 + 2 * q^59 + 4 * q^61 + 4 * q^62 + 32 * q^64 - 2 * q^65 + 24 * q^67 + 32 * q^68 - 4 * q^70 + 2 * q^71 + 22 * q^74 + 16 * q^76 - 18 * q^77 - 8 * q^79 - 40 * q^80 + 12 * q^83 - 16 * q^85 + 10 * q^86 - 24 * q^88 - 6 * q^89 - 4 * q^91 - 24 * q^94 - 8 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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