LMFDB - Embedded newform 4923.2.a.l.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, 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("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.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:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.59964$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59964 q^{2} +4.75812 q^{4} +3.02323 q^{5} -0.561390 q^{7} +7.17011 q^{8} +O(q^{10})$	\(q+2.59964 q^{2} +4.75812 q^{4} +3.02323 q^{5} -0.561390 q^{7} +7.17011 q^{8} +7.85930 q^{10} -4.23111 q^{11} -4.87964 q^{13} -1.45941 q^{14} +9.12345 q^{16} +6.39965 q^{17} +6.29628 q^{19} +14.3849 q^{20} -10.9994 q^{22} +9.21357 q^{23} +4.13991 q^{25} -12.6853 q^{26} -2.67116 q^{28} -1.14750 q^{29} -7.52167 q^{31} +9.37744 q^{32} +16.6368 q^{34} -1.69721 q^{35} +8.74347 q^{37} +16.3680 q^{38} +21.6769 q^{40} -2.81890 q^{41} -1.57023 q^{43} -20.1321 q^{44} +23.9520 q^{46} +4.08126 q^{47} -6.68484 q^{49} +10.7623 q^{50} -23.2179 q^{52} +4.00434 q^{53} -12.7916 q^{55} -4.02522 q^{56} -2.98307 q^{58} +12.2872 q^{59} -0.473603 q^{61} -19.5536 q^{62} +6.13107 q^{64} -14.7523 q^{65} +1.80684 q^{67} +30.4503 q^{68} -4.41213 q^{70} +1.76871 q^{71} +0.611682 q^{73} +22.7298 q^{74} +29.9584 q^{76} +2.37530 q^{77} -2.49642 q^{79} +27.5823 q^{80} -7.32811 q^{82} -8.29336 q^{83} +19.3476 q^{85} -4.08204 q^{86} -30.3375 q^{88} +14.8521 q^{89} +2.73938 q^{91} +43.8393 q^{92} +10.6098 q^{94} +19.0351 q^{95} -7.36588 q^{97} -17.3782 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * 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 $

$2$	2.59964	1.83822	0.919111	−	0.393999i	$-0.128909\pi$
$2$	2.59964	1.83822	0.919111	+	0.393999i	$0.128909\pi$
$3$	0	0
$3$	0	0
$4$	4.75812	2.37906
$5$	3.02323	1.35203	0.676015	−	0.736888i	$-0.263706\pi$
$5$	3.02323	1.35203	0.676015	+	0.736888i	$0.263706\pi$
$6$	0	0
$7$	−0.561390	−0.212185	−0.106093	−	0.994356i	$-0.533834\pi$
$7$	−0.561390	−0.212185	−0.106093	+	0.994356i	$0.533834\pi$
$8$	7.17011	2.53502
$9$	0	0
$10$	7.85930	2.48533
$11$	−4.23111	−1.27573	−0.637864	−	0.770149i	$-0.720182\pi$
$11$	−4.23111	−1.27573	−0.637864	+	0.770149i	$0.720182\pi$
$12$	0	0
$13$	−4.87964	−1.35337	−0.676684	−	0.736274i	$-0.736584\pi$
$13$	−4.87964	−1.35337	−0.676684	+	0.736274i	$0.736584\pi$
$14$	−1.45941	−0.390044
$15$	0	0
$16$	9.12345	2.28086
$17$	6.39965	1.55214	0.776071	−	0.630645i	$-0.217210\pi$
$17$	6.39965	1.55214	0.776071	+	0.630645i	$0.217210\pi$
$18$	0	0
$19$	6.29628	1.44447	0.722233	−	0.691650i	$-0.243116\pi$
$19$	6.29628	1.44447	0.722233	+	0.691650i	$0.243116\pi$
$20$	14.3849	3.21656
$21$	0	0
$22$	−10.9994	−2.34507
$23$	9.21357	1.92116	0.960581	−	0.277999i	$-0.0896713\pi$
$23$	9.21357	1.92116	0.960581	+	0.277999i	$0.0896713\pi$
$24$	0	0
$25$	4.13991	0.827983
$26$	−12.6853	−2.48779
$27$	0	0
$28$	−2.67116	−0.504801
$29$	−1.14750	−0.213085	−0.106542	−	0.994308i	$-0.533978\pi$
$29$	−1.14750	−0.213085	−0.106542	+	0.994308i	$0.533978\pi$
$30$	0	0
$31$	−7.52167	−1.35093	−0.675466	−	0.737391i	$-0.736058\pi$
$31$	−7.52167	−1.35093	−0.675466	+	0.737391i	$0.736058\pi$
$32$	9.37744	1.65771
$33$	0	0
$34$	16.6368	2.85318
$35$	−1.69721	−0.286881
$36$	0	0
$37$	8.74347	1.43742	0.718708	−	0.695312i	$-0.244734\pi$
$37$	8.74347	1.43742	0.718708	+	0.695312i	$0.244734\pi$
$38$	16.3680	2.65525
$39$	0	0
$40$	21.6769	3.42741
$41$	−2.81890	−0.440238	−0.220119	−	0.975473i	$-0.570645\pi$
$41$	−2.81890	−0.440238	−0.220119	+	0.975473i	$0.570645\pi$
$42$	0	0
$43$	−1.57023	−0.239458	−0.119729	−	0.992807i	$-0.538203\pi$
$43$	−1.57023	−0.239458	−0.119729	+	0.992807i	$0.538203\pi$
$44$	−20.1321	−3.03503
$45$	0	0
$46$	23.9520	3.53152
$47$	4.08126	0.595313	0.297657	−	0.954673i	$-0.403795\pi$
$47$	4.08126	0.595313	0.297657	+	0.954673i	$0.403795\pi$
$48$	0	0
$49$	−6.68484	−0.954977
$50$	10.7623	1.52202
$51$	0	0
$52$	−23.2179	−3.21974
$53$	4.00434	0.550039	0.275019	−	0.961439i	$-0.411316\pi$
$53$	4.00434	0.550039	0.275019	+	0.961439i	$0.411316\pi$
$54$	0	0
$55$	−12.7916	−1.72482
$56$	−4.02522	−0.537893
$57$	0	0
$58$	−2.98307	−0.391697
$59$	12.2872	1.59966	0.799828	−	0.600229i	$-0.204924\pi$
$59$	12.2872	1.59966	0.799828	+	0.600229i	$0.204924\pi$
$60$	0	0
$61$	−0.473603	−0.0606386	−0.0303193	−	0.999540i	$-0.509652\pi$
$61$	−0.473603	−0.0606386	−0.0303193	+	0.999540i	$0.509652\pi$
$62$	−19.5536	−2.48331
$63$	0	0
$64$	6.13107	0.766383
$65$	−14.7523	−1.82979
$66$	0	0
$67$	1.80684	0.220740	0.110370	−	0.993891i	$-0.464796\pi$
$67$	1.80684	0.220740	0.110370	+	0.993891i	$0.464796\pi$
$68$	30.4503	3.69264
$69$	0	0
$70$	−4.41213	−0.527350
$71$	1.76871	0.209907	0.104954	−	0.994477i	$-0.466531\pi$
$71$	1.76871	0.209907	0.104954	+	0.994477i	$0.466531\pi$
$72$	0	0
$73$	0.611682	0.0715919	0.0357960	−	0.999359i	$-0.488603\pi$
$73$	0.611682	0.0715919	0.0357960	+	0.999359i	$0.488603\pi$
$74$	22.7298	2.64229
$75$	0	0
$76$	29.9584	3.43647
$77$	2.37530	0.270691
$78$	0	0
$79$	−2.49642	−0.280869	−0.140434	−	0.990090i	$-0.544850\pi$
$79$	−2.49642	−0.280869	−0.140434	+	0.990090i	$0.544850\pi$
$80$	27.5823	3.08379
$81$	0	0
$82$	−7.32811	−0.809254
$83$	−8.29336	−0.910315	−0.455158	−	0.890411i	$-0.650417\pi$
$83$	−8.29336	−0.910315	−0.455158	+	0.890411i	$0.650417\pi$
$84$	0	0
$85$	19.3476	2.09854
$86$	−4.08204	−0.440178
$87$	0	0
$88$	−30.3375	−3.23399
$89$	14.8521	1.57432	0.787162	−	0.616746i	$-0.211550\pi$
$89$	14.8521	1.57432	0.787162	+	0.616746i	$0.211550\pi$
$90$	0	0
$91$	2.73938	0.287165
$92$	43.8393	4.57056
$93$	0	0
$94$	10.6098	1.09432
$95$	19.0351	1.95296
$96$	0	0
$97$	−7.36588	−0.747891	−0.373946	−	0.927451i	$-0.621995\pi$
$97$	−7.36588	−0.747891	−0.373946	+	0.927451i	$0.621995\pi$
$98$	−17.3782	−1.75546
$99$	0	0
$100$	19.6982	1.96982
$101$	−5.05569	−0.503060	−0.251530	−	0.967849i	$-0.580934\pi$
$101$	−5.05569	−0.503060	−0.251530	+	0.967849i	$0.580934\pi$
$102$	0	0
$103$	1.90289	0.187497	0.0937486	−	0.995596i	$-0.470115\pi$
$103$	1.90289	0.187497	0.0937486	+	0.995596i	$0.470115\pi$
$104$	−34.9875	−3.43081
$105$	0	0
$106$	10.4098	1.01109
$107$	−11.8307	−1.14371	−0.571856	−	0.820354i	$-0.693776\pi$
$107$	−11.8307	−1.14371	−0.571856	+	0.820354i	$0.693776\pi$
$108$	0	0
$109$	4.76603	0.456503	0.228251	−	0.973602i	$-0.426699\pi$
$109$	4.76603	0.456503	0.228251	+	0.973602i	$0.426699\pi$
$110$	−33.2536	−3.17060
$111$	0	0
$112$	−5.12181	−0.483965
$113$	−12.6049	−1.18577	−0.592886	−	0.805287i	$-0.702011\pi$
$113$	−12.6049	−1.18577	−0.592886	+	0.805287i	$0.702011\pi$
$114$	0	0
$115$	27.8547	2.59747
$116$	−5.45992	−0.506941
$117$	0	0
$118$	31.9423	2.94052
$119$	−3.59270	−0.329342
$120$	0	0
$121$	6.90230	0.627482
$122$	−1.23120	−0.111467
$123$	0	0
$124$	−35.7890	−3.21395
$125$	−2.60024	−0.232572
$126$	0	0
$127$	−20.3142	−1.80260	−0.901298	−	0.433199i	$-0.857385\pi$
$127$	−20.3142	−1.80260	−0.901298	+	0.433199i	$0.857385\pi$
$128$	−2.81634	−0.248931
$129$	0	0
$130$	−38.3505	−3.36356
$131$	−13.5167	−1.18096	−0.590478	−	0.807054i	$-0.701061\pi$
$131$	−13.5167	−1.18096	−0.590478	+	0.807054i	$0.701061\pi$
$132$	0	0
$133$	−3.53467	−0.306494
$134$	4.69712	0.405770
$135$	0	0
$136$	45.8861	3.93470
$137$	−8.32726	−0.711446	−0.355723	−	0.934591i	$-0.615765\pi$
$137$	−8.32726	−0.711446	−0.355723	+	0.934591i	$0.615765\pi$
$138$	0	0
$139$	−16.6457	−1.41187	−0.705936	−	0.708275i	$-0.749474\pi$
$139$	−16.6457	−1.41187	−0.705936	+	0.708275i	$0.749474\pi$
$140$	−8.07552	−0.682506
$141$	0	0
$142$	4.59801	0.385856
$143$	20.6463	1.72653
$144$	0	0
$145$	−3.46914	−0.288096
$146$	1.59015	0.131602
$147$	0	0
$148$	41.6024	3.41970
$149$	−9.40634	−0.770598	−0.385299	−	0.922792i	$-0.625902\pi$
$149$	−9.40634	−0.770598	−0.385299	+	0.922792i	$0.625902\pi$
$150$	0	0
$151$	−20.3010	−1.65207	−0.826035	−	0.563619i	$-0.809409\pi$
$151$	−20.3010	−1.65207	−0.826035	+	0.563619i	$0.809409\pi$
$152$	45.1450	3.66174
$153$	0	0
$154$	6.17492	0.497590
$155$	−22.7397	−1.82650
$156$	0	0
$157$	−6.07853	−0.485120	−0.242560	−	0.970136i	$-0.577987\pi$
$157$	−6.07853	−0.485120	−0.242560	+	0.970136i	$0.577987\pi$
$158$	−6.48978	−0.516299
$159$	0	0
$160$	28.3502	2.24128
$161$	−5.17240	−0.407643
$162$	0	0
$163$	7.36070	0.576535	0.288267	−	0.957550i	$-0.406921\pi$
$163$	7.36070	0.576535	0.288267	+	0.957550i	$0.406921\pi$
$164$	−13.4126	−1.04735
$165$	0	0
$166$	−21.5597	−1.67336
$167$	−7.50992	−0.581135	−0.290568	−	0.956854i	$-0.593844\pi$
$167$	−7.50992	−0.581135	−0.290568	+	0.956854i	$0.593844\pi$
$168$	0	0
$169$	10.8108	0.831604
$170$	50.2967	3.85758
$171$	0	0
$172$	−7.47136	−0.569686
$173$	−5.25063	−0.399198	−0.199599	−	0.979878i	$-0.563964\pi$
$173$	−5.25063	−0.399198	−0.199599	+	0.979878i	$0.563964\pi$
$174$	0	0
$175$	−2.32410	−0.175686
$176$	−38.6023	−2.90976
$177$	0	0
$178$	38.6102	2.89396
$179$	9.57025	0.715314	0.357657	−	0.933853i	$-0.383576\pi$
$179$	9.57025	0.715314	0.357657	+	0.933853i	$0.383576\pi$
$180$	0	0
$181$	−1.66933	−0.124081	−0.0620403	−	0.998074i	$-0.519761\pi$
$181$	−1.66933	−0.124081	−0.0620403	+	0.998074i	$0.519761\pi$
$182$	7.12139	0.527872
$183$	0	0
$184$	66.0623	4.87018
$185$	26.4335	1.94343
$186$	0	0
$187$	−27.0776	−1.98011
$188$	19.4191	1.41629
$189$	0	0
$190$	49.4843	3.58997
$191$	−26.7101	−1.93268	−0.966339	−	0.257272i	$-0.917176\pi$
$191$	−26.7101	−1.93268	−0.966339	+	0.257272i	$0.917176\pi$
$192$	0	0
$193$	17.1594	1.23516	0.617582	−	0.786506i	$-0.288112\pi$
$193$	17.1594	1.23516	0.617582	+	0.786506i	$0.288112\pi$
$194$	−19.1486	−1.37479
$195$	0	0
$196$	−31.8073	−2.27195
$197$	5.98278	0.426255	0.213128	−	0.977024i	$-0.431635\pi$
$197$	5.98278	0.426255	0.213128	+	0.977024i	$0.431635\pi$
$198$	0	0
$199$	16.0373	1.13685	0.568426	−	0.822734i	$-0.307552\pi$
$199$	16.0373	1.13685	0.568426	+	0.822734i	$0.307552\pi$
$200$	29.6836	2.09895
$201$	0	0
$202$	−13.1430	−0.924737
$203$	0.644192	0.0452134
$204$	0	0
$205$	−8.52217	−0.595214
$206$	4.94682	0.344661
$207$	0	0
$208$	−44.5191	−3.08684
$209$	−26.6403	−1.84274
$210$	0	0
$211$	11.5087	0.792291	0.396146	−	0.918188i	$-0.370348\pi$
$211$	11.5087	0.792291	0.396146	+	0.918188i	$0.370348\pi$
$212$	19.0531	1.30857
$213$	0	0
$214$	−30.7554	−2.10240
$215$	−4.74718	−0.323755
$216$	0	0
$217$	4.22259	0.286648
$218$	12.3900	0.839154
$219$	0	0
$220$	−60.8640	−4.10345
$221$	−31.2279	−2.10062
$222$	0	0
$223$	4.40148	0.294745	0.147372	−	0.989081i	$-0.452918\pi$
$223$	4.40148	0.294745	0.147372	+	0.989081i	$0.452918\pi$
$224$	−5.26440	−0.351742
$225$	0	0
$226$	−32.7682	−2.17971
$227$	−14.9252	−0.990618	−0.495309	−	0.868717i	$-0.664945\pi$
$227$	−14.9252	−0.990618	−0.495309	+	0.868717i	$0.664945\pi$
$228$	0	0
$229$	4.19125	0.276966	0.138483	−	0.990365i	$-0.455777\pi$
$229$	4.19125	0.276966	0.138483	+	0.990365i	$0.455777\pi$
$230$	72.4122	4.77472
$231$	0	0
$232$	−8.22766	−0.540173
$233$	−1.32186	−0.0865980	−0.0432990	−	0.999062i	$-0.513787\pi$
$233$	−1.32186	−0.0865980	−0.0432990	+	0.999062i	$0.513787\pi$
$234$	0	0
$235$	12.3386	0.804881
$236$	58.4639	3.80568
$237$	0	0
$238$	−9.33971	−0.605403
$239$	3.57501	0.231248	0.115624	−	0.993293i	$-0.463113\pi$
$239$	3.57501	0.231248	0.115624	+	0.993293i	$0.463113\pi$
$240$	0	0
$241$	14.5198	0.935301	0.467650	−	0.883914i	$-0.345101\pi$
$241$	14.5198	0.935301	0.467650	+	0.883914i	$0.345101\pi$
$242$	17.9435	1.15345
$243$	0	0
$244$	−2.25346	−0.144263
$245$	−20.2098	−1.29116
$246$	0	0
$247$	−30.7235	−1.95489
$248$	−53.9312	−3.42464
$249$	0	0
$250$	−6.75968	−0.427520
$251$	−5.30634	−0.334933	−0.167467	−	0.985878i	$-0.553559\pi$
$251$	−5.30634	−0.334933	−0.167467	+	0.985878i	$0.553559\pi$
$252$	0	0
$253$	−38.9836	−2.45088
$254$	−52.8097	−3.31357
$255$	0	0
$256$	−19.5836	−1.22397
$257$	2.08054	0.129781	0.0648903	−	0.997892i	$-0.479330\pi$
$257$	2.08054	0.129781	0.0648903	+	0.997892i	$0.479330\pi$
$258$	0	0
$259$	−4.90849	−0.304999
$260$	−70.1930	−4.35318
$261$	0	0
$262$	−35.1384	−2.17086
$263$	−16.6250	−1.02514	−0.512570	−	0.858645i	$-0.671306\pi$
$263$	−16.6250	−1.02514	−0.512570	+	0.858645i	$0.671306\pi$
$264$	0	0
$265$	12.1060	0.743668
$266$	−9.18885	−0.563405
$267$	0	0
$268$	8.59714	0.525154
$269$	7.61200	0.464112	0.232056	−	0.972702i	$-0.425455\pi$
$269$	7.61200	0.464112	0.232056	+	0.972702i	$0.425455\pi$
$270$	0	0
$271$	4.15827	0.252597	0.126298	−	0.991992i	$-0.459690\pi$
$271$	4.15827	0.252597	0.126298	+	0.991992i	$0.459690\pi$
$272$	58.3868	3.54022
$273$	0	0
$274$	−21.6479	−1.30779
$275$	−17.5164	−1.05628
$276$	0	0
$277$	9.42089	0.566047	0.283023	−	0.959113i	$-0.408663\pi$
$277$	9.42089	0.566047	0.283023	+	0.959113i	$0.408663\pi$
$278$	−43.2729	−2.59533
$279$	0	0
$280$	−12.1692	−0.727247
$281$	23.1796	1.38278	0.691391	−	0.722481i	$-0.256998\pi$
$281$	23.1796	1.38278	0.691391	+	0.722481i	$0.256998\pi$
$282$	0	0
$283$	−24.6282	−1.46399	−0.731997	−	0.681308i	$-0.761412\pi$
$283$	−24.6282	−1.46399	−0.731997	+	0.681308i	$0.761412\pi$
$284$	8.41573	0.499382
$285$	0	0
$286$	53.6729	3.17374
$287$	1.58250	0.0934120
$288$	0	0
$289$	23.9555	1.40915
$290$	−9.01851	−0.529585
$291$	0	0
$292$	2.91045	0.170321
$293$	−23.3449	−1.36382	−0.681910	−	0.731436i	$-0.738851\pi$
$293$	−23.3449	−1.36382	−0.681910	+	0.731436i	$0.738851\pi$
$294$	0	0
$295$	37.1470	2.16278
$296$	62.6916	3.64387
$297$	0	0
$298$	−24.4531	−1.41653
$299$	−44.9589	−2.60004
$300$	0	0
$301$	0.881513	0.0508096
$302$	−52.7752	−3.03687
$303$	0	0
$304$	57.4438	3.29463
$305$	−1.43181	−0.0819851
$306$	0	0
$307$	−13.1646	−0.751342	−0.375671	−	0.926753i	$-0.622588\pi$
$307$	−13.1646	−0.751342	−0.375671	+	0.926753i	$0.622588\pi$
$308$	11.3020	0.643989
$309$	0	0
$310$	−59.1151	−3.35751
$311$	18.1219	1.02760	0.513800	−	0.857910i	$-0.328237\pi$
$311$	18.1219	1.02760	0.513800	+	0.857910i	$0.328237\pi$
$312$	0	0
$313$	14.2295	0.804299	0.402150	−	0.915574i	$-0.368263\pi$
$313$	14.2295	0.804299	0.402150	+	0.915574i	$0.368263\pi$
$314$	−15.8020	−0.891758
$315$	0	0
$316$	−11.8783	−0.668204
$317$	−0.706042	−0.0396553	−0.0198276	−	0.999803i	$-0.506312\pi$
$317$	−0.706042	−0.0396553	−0.0198276	+	0.999803i	$0.506312\pi$
$318$	0	0
$319$	4.85518	0.271838
$320$	18.5356	1.03617
$321$	0	0
$322$	−13.4464	−0.749337
$323$	40.2940	2.24202
$324$	0	0
$325$	−20.2013	−1.12056
$326$	19.1352	1.05980
$327$	0	0
$328$	−20.2118	−1.11601
$329$	−2.29118	−0.126317
$330$	0	0
$331$	−1.50165	−0.0825381	−0.0412691	−	0.999148i	$-0.513140\pi$
$331$	−1.50165	−0.0825381	−0.0412691	+	0.999148i	$0.513140\pi$
$332$	−39.4608	−2.16569
$333$	0	0
$334$	−19.5231	−1.06825
$335$	5.46248	0.298447
$336$	0	0
$337$	−10.5820	−0.576437	−0.288219	−	0.957565i	$-0.593063\pi$
$337$	−10.5820	−0.576437	−0.288219	+	0.957565i	$0.593063\pi$
$338$	28.1043	1.52867
$339$	0	0
$340$	92.0581	4.99255
$341$	31.8250	1.72342
$342$	0	0
$343$	7.68253	0.414818
$344$	−11.2587	−0.607031
$345$	0	0
$346$	−13.6497	−0.733815
$347$	21.9609	1.17892	0.589461	−	0.807797i	$-0.299340\pi$
$347$	21.9609	1.17892	0.589461	+	0.807797i	$0.299340\pi$
$348$	0	0
$349$	−13.7632	−0.736727	−0.368364	−	0.929682i	$-0.620082\pi$
$349$	−13.7632	−0.736727	−0.368364	+	0.929682i	$0.620082\pi$
$350$	−6.04183	−0.322949
$351$	0	0
$352$	−39.6770	−2.11479
$353$	33.1723	1.76558	0.882791	−	0.469766i	$-0.155662\pi$
$353$	33.1723	1.76558	0.882791	+	0.469766i	$0.155662\pi$
$354$	0	0
$355$	5.34722	0.283801
$356$	70.6682	3.74541
$357$	0	0
$358$	24.8792	1.31491
$359$	18.0716	0.953782	0.476891	−	0.878962i	$-0.341764\pi$
$359$	18.0716	0.953782	0.476891	+	0.878962i	$0.341764\pi$
$360$	0	0
$361$	20.6431	1.08648
$362$	−4.33966	−0.228087
$363$	0	0
$364$	13.0343	0.683182
$365$	1.84925	0.0967944
$366$	0	0
$367$	−32.7727	−1.71072	−0.855361	−	0.518033i	$-0.826665\pi$
$367$	−32.7727	−1.71072	−0.855361	+	0.518033i	$0.826665\pi$
$368$	84.0595	4.38191
$369$	0	0
$370$	68.7175	3.57245
$371$	−2.24800	−0.116710
$372$	0	0
$373$	3.25311	0.168440	0.0842199	−	0.996447i	$-0.473160\pi$
$373$	3.25311	0.168440	0.0842199	+	0.996447i	$0.473160\pi$
$374$	−70.3920	−3.63988
$375$	0	0
$376$	29.2631	1.50913
$377$	5.59936	0.288382
$378$	0	0
$379$	−19.1603	−0.984198	−0.492099	−	0.870539i	$-0.663770\pi$
$379$	−19.1603	−0.984198	−0.492099	+	0.870539i	$0.663770\pi$
$380$	90.5712	4.64620
$381$	0	0
$382$	−69.4367	−3.55269
$383$	−22.6686	−1.15831	−0.579155	−	0.815217i	$-0.696617\pi$
$383$	−22.6686	−1.15831	−0.579155	+	0.815217i	$0.696617\pi$
$384$	0	0
$385$	7.18108	0.365982
$386$	44.6084	2.27051
$387$	0	0
$388$	−35.0477	−1.77928
$389$	39.0808	1.98148	0.990738	−	0.135788i	$-0.0433565\pi$
$389$	39.0808	1.98148	0.990738	+	0.135788i	$0.0433565\pi$
$390$	0	0
$391$	58.9636	2.98192
$392$	−47.9310	−2.42088
$393$	0	0
$394$	15.5531	0.783551
$395$	−7.54724	−0.379743
$396$	0	0
$397$	17.4671	0.876649	0.438324	−	0.898817i	$-0.355572\pi$
$397$	17.4671	0.876649	0.438324	+	0.898817i	$0.355572\pi$
$398$	41.6911	2.08979
$399$	0	0
$400$	37.7703	1.88851
$401$	−7.16631	−0.357868	−0.178934	−	0.983861i	$-0.557265\pi$
$401$	−7.16631	−0.357868	−0.178934	+	0.983861i	$0.557265\pi$
$402$	0	0
$403$	36.7030	1.82831
$404$	−24.0556	−1.19681
$405$	0	0
$406$	1.67467	0.0831123
$407$	−36.9946	−1.83375
$408$	0	0
$409$	−20.3350	−1.00550	−0.502750	−	0.864432i	$-0.667678\pi$
$409$	−20.3350	−1.00550	−0.502750	+	0.864432i	$0.667678\pi$
$410$	−22.1546	−1.09414
$411$	0	0
$412$	9.05417	0.446067
$413$	−6.89791	−0.339424
$414$	0	0
$415$	−25.0727	−1.23077
$416$	−45.7585	−2.24350
$417$	0	0
$418$	−69.2550	−3.38737
$419$	28.3481	1.38490	0.692448	−	0.721468i	$-0.256532\pi$
$419$	28.3481	1.38490	0.692448	+	0.721468i	$0.256532\pi$
$420$	0	0
$421$	−2.21872	−0.108134	−0.0540670	−	0.998537i	$-0.517218\pi$
$421$	−2.21872	−0.108134	−0.0540670	+	0.998537i	$0.517218\pi$
$422$	29.9184	1.45641
$423$	0	0
$424$	28.7116	1.39436
$425$	26.4940	1.28515
$426$	0	0
$427$	0.265876	0.0128666
$428$	−56.2916	−2.72096
$429$	0	0
$430$	−12.3409	−0.595133
$431$	1.89601	0.0913274	0.0456637	−	0.998957i	$-0.485460\pi$
$431$	1.89601	0.0913274	0.0456637	+	0.998957i	$0.485460\pi$
$432$	0	0
$433$	−33.8222	−1.62539	−0.812697	−	0.582687i	$-0.802001\pi$
$433$	−33.8222	−1.62539	−0.812697	+	0.582687i	$0.802001\pi$
$434$	10.9772	0.526923
$435$	0	0
$436$	22.6773	1.08605
$437$	58.0112	2.77505
$438$	0	0
$439$	6.31659	0.301474	0.150737	−	0.988574i	$-0.451835\pi$
$439$	6.31659	0.301474	0.150737	+	0.988574i	$0.451835\pi$
$440$	−91.7173	−4.37245
$441$	0	0
$442$	−81.1814	−3.86140
$443$	−19.5315	−0.927967	−0.463984	−	0.885844i	$-0.653580\pi$
$443$	−19.5315	−0.927967	−0.463984	+	0.885844i	$0.653580\pi$
$444$	0	0
$445$	44.9014	2.12853
$446$	11.4423	0.541807
$447$	0	0
$448$	−3.44192	−0.162615
$449$	15.3252	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$449$	15.3252	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$450$	0	0
$451$	11.9271	0.561623
$452$	−59.9757	−2.82102
$453$	0	0
$454$	−38.8000	−1.82098
$455$	8.28176	0.388255
$456$	0	0
$457$	−14.7763	−0.691207	−0.345603	−	0.938381i	$-0.612326\pi$
$457$	−14.7763	−0.691207	−0.345603	+	0.938381i	$0.612326\pi$
$458$	10.8957	0.509124
$459$	0	0
$460$	132.536	6.17953
$461$	28.1311	1.31020	0.655098	−	0.755544i	$-0.272627\pi$
$461$	28.1311	1.31020	0.655098	+	0.755544i	$0.272627\pi$
$462$	0	0
$463$	16.7165	0.776880	0.388440	−	0.921474i	$-0.373014\pi$
$463$	16.7165	0.776880	0.388440	+	0.921474i	$0.373014\pi$
$464$	−10.4691	−0.486016
$465$	0	0
$466$	−3.43636	−0.159186
$467$	22.4105	1.03703	0.518517	−	0.855067i	$-0.326484\pi$
$467$	22.4105	1.03703	0.518517	+	0.855067i	$0.326484\pi$
$468$	0	0
$469$	−1.01434	−0.0468379
$470$	32.0759	1.47955
$471$	0	0
$472$	88.1005	4.05515
$473$	6.64383	0.305484
$474$	0	0
$475$	26.0660	1.19599
$476$	−17.0945	−0.783524
$477$	0	0
$478$	9.29372	0.425085
$479$	15.0630	0.688247	0.344123	−	0.938924i	$-0.388176\pi$
$479$	15.0630	0.688247	0.344123	+	0.938924i	$0.388176\pi$
$480$	0	0
$481$	−42.6649	−1.94535
$482$	37.7462	1.71929
$483$	0	0
$484$	32.8420	1.49282
$485$	−22.2687	−1.01117
$486$	0	0
$487$	39.0354	1.76886	0.884432	−	0.466669i	$-0.154546\pi$
$487$	39.0354	1.76886	0.884432	+	0.466669i	$0.154546\pi$
$488$	−3.39578	−0.153720
$489$	0	0
$490$	−52.5382	−2.37343
$491$	−21.8721	−0.987075	−0.493537	−	0.869725i	$-0.664296\pi$
$491$	−21.8721	−0.987075	−0.493537	+	0.869725i	$0.664296\pi$
$492$	0	0
$493$	−7.34356	−0.330737
$494$	−79.8701	−3.59353
$495$	0	0
$496$	−68.6236	−3.08129
$497$	−0.992936	−0.0445393
$498$	0	0
$499$	20.4160	0.913947	0.456973	−	0.889480i	$-0.348933\pi$
$499$	20.4160	0.913947	0.456973	+	0.889480i	$0.348933\pi$
$500$	−12.3722	−0.553304
$501$	0	0
$502$	−13.7946	−0.615681
$503$	17.2592	0.769551	0.384776	−	0.923010i	$-0.374279\pi$
$503$	17.2592	0.769551	0.384776	+	0.923010i	$0.374279\pi$
$504$	0	0
$505$	−15.2845	−0.680152
$506$	−101.343	−4.50526
$507$	0	0
$508$	−96.6575	−4.28848
$509$	29.7166	1.31717	0.658584	−	0.752508i	$-0.271156\pi$
$509$	29.7166	1.31717	0.658584	+	0.752508i	$0.271156\pi$
$510$	0	0
$511$	−0.343392	−0.0151908
$512$	−45.2776	−2.00100
$513$	0	0
$514$	5.40865	0.238566
$515$	5.75287	0.253502
$516$	0	0
$517$	−17.2683	−0.759458
$518$	−12.7603	−0.560655
$519$	0	0
$520$	−105.775	−4.63855
$521$	−32.4429	−1.42135	−0.710675	−	0.703520i	$-0.751610\pi$
$521$	−32.4429	−1.42135	−0.710675	+	0.703520i	$0.751610\pi$
$522$	0	0
$523$	5.45155	0.238380	0.119190	−	0.992871i	$-0.461970\pi$
$523$	5.45155	0.238380	0.119190	+	0.992871i	$0.461970\pi$
$524$	−64.3138	−2.80956
$525$	0	0
$526$	−43.2189	−1.88443
$527$	−48.1361	−2.09684
$528$	0	0
$529$	61.8899	2.69087
$530$	31.4713	1.36703
$531$	0	0
$532$	−16.8184	−0.729168
$533$	13.7552	0.595803
$534$	0	0
$535$	−35.7668	−1.54633
$536$	12.9552	0.559580
$537$	0	0
$538$	19.7884	0.853140
$539$	28.2843	1.21829
$540$	0	0
$541$	31.1191	1.33792	0.668958	−	0.743301i	$-0.266741\pi$
$541$	31.1191	1.33792	0.668958	+	0.743301i	$0.266741\pi$
$542$	10.8100	0.464329
$543$	0	0
$544$	60.0123	2.57301
$545$	14.4088	0.617205
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−39.6221	−1.69257
$549$	0	0
$550$	−45.5364	−1.94168
$551$	−7.22495	−0.307793
$552$	0	0
$553$	1.40146	0.0595963
$554$	24.4909	1.04052
$555$	0	0
$556$	−79.2023	−3.35893
$557$	−41.0931	−1.74117	−0.870585	−	0.492017i	$-0.836260\pi$
$557$	−41.0931	−1.74117	−0.870585	+	0.492017i	$0.836260\pi$
$558$	0	0
$559$	7.66217	0.324075
$560$	−15.4844	−0.654335
$561$	0	0
$562$	60.2587	2.54186
$563$	−6.83856	−0.288211	−0.144105	−	0.989562i	$-0.546030\pi$
$563$	−6.83856	−0.288211	−0.144105	+	0.989562i	$0.546030\pi$
$564$	0	0
$565$	−38.1076	−1.60320
$566$	−64.0244	−2.69115
$567$	0	0
$568$	12.6818	0.532118
$569$	27.4209	1.14955	0.574773	−	0.818313i	$-0.305091\pi$
$569$	27.4209	1.14955	0.574773	+	0.818313i	$0.305091\pi$
$570$	0	0
$571$	−34.1032	−1.42718	−0.713588	−	0.700566i	$-0.752931\pi$
$571$	−34.1032	−1.42718	−0.713588	+	0.700566i	$0.752931\pi$
$572$	98.2374	4.10751
$573$	0	0
$574$	4.11392	0.171712
$575$	38.1434	1.59069
$576$	0	0
$577$	−33.6417	−1.40052	−0.700260	−	0.713888i	$-0.746933\pi$
$577$	−33.6417	−1.40052	−0.700260	+	0.713888i	$0.746933\pi$
$578$	62.2756	2.59032
$579$	0	0
$580$	−16.5066	−0.685398
$581$	4.65581	0.193155
$582$	0	0
$583$	−16.9428	−0.701700
$584$	4.38582	0.181487
$585$	0	0
$586$	−60.6882	−2.50701
$587$	13.2895	0.548516	0.274258	−	0.961656i	$-0.411568\pi$
$587$	13.2895	0.548516	0.274258	+	0.961656i	$0.411568\pi$
$588$	0	0
$589$	−47.3586	−1.95138
$590$	96.5688	3.97567
$591$	0	0
$592$	79.7705	3.27855
$593$	44.2951	1.81898	0.909491	−	0.415724i	$-0.136472\pi$
$593$	44.2951	1.81898	0.909491	+	0.415724i	$0.136472\pi$
$594$	0	0
$595$	−10.8615	−0.445280
$596$	−44.7565	−1.83330
$597$	0	0
$598$	−116.877	−4.77945
$599$	−5.59011	−0.228406	−0.114203	−	0.993457i	$-0.536431\pi$
$599$	−5.59011	−0.228406	−0.114203	+	0.993457i	$0.536431\pi$
$600$	0	0
$601$	25.4727	1.03905	0.519526	−	0.854454i	$-0.326108\pi$
$601$	25.4727	1.03905	0.519526	+	0.854454i	$0.326108\pi$
$602$	2.29161	0.0933992
$603$	0	0
$604$	−96.5944	−3.93037
$605$	20.8672	0.848374
$606$	0	0
$607$	−27.4291	−1.11331	−0.556657	−	0.830742i	$-0.687916\pi$
$607$	−27.4291	−1.11331	−0.556657	+	0.830742i	$0.687916\pi$
$608$	59.0430	2.39451
$609$	0	0
$610$	−3.72218	−0.150707
$611$	−19.9151	−0.805678
$612$	0	0
$613$	16.8113	0.679000	0.339500	−	0.940606i	$-0.389742\pi$
$613$	16.8113	0.679000	0.339500	+	0.940606i	$0.389742\pi$
$614$	−34.2231	−1.38113
$615$	0	0
$616$	17.0312	0.686205
$617$	−12.2317	−0.492429	−0.246215	−	0.969215i	$-0.579187\pi$
$617$	−12.2317	−0.492429	−0.246215	+	0.969215i	$0.579187\pi$
$618$	0	0
$619$	−17.1538	−0.689471	−0.344736	−	0.938700i	$-0.612031\pi$
$619$	−17.1538	−0.689471	−0.344736	+	0.938700i	$0.612031\pi$
$620$	−108.198	−4.34535
$621$	0	0
$622$	47.1105	1.88896
$623$	−8.33784	−0.334048
$624$	0	0
$625$	−28.5607	−1.14243
$626$	36.9916	1.47848
$627$	0	0
$628$	−28.9224	−1.15413
$629$	55.9551	2.23108
$630$	0	0
$631$	−6.57839	−0.261882	−0.130941	−	0.991390i	$-0.541800\pi$
$631$	−6.57839	−0.261882	−0.130941	+	0.991390i	$0.541800\pi$
$632$	−17.8996	−0.712007
$633$	0	0
$634$	−1.83545	−0.0728952
$635$	−61.4146	−2.43716
$636$	0	0
$637$	32.6196	1.29244
$638$	12.6217	0.499698
$639$	0	0
$640$	−8.51443	−0.336562
$641$	−2.33872	−0.0923737	−0.0461868	−	0.998933i	$-0.514707\pi$
$641$	−2.33872	−0.0923737	−0.0461868	+	0.998933i	$0.514707\pi$
$642$	0	0
$643$	−17.3754	−0.685218	−0.342609	−	0.939478i	$-0.611311\pi$
$643$	−17.3754	−0.685218	−0.342609	+	0.939478i	$0.611311\pi$
$644$	−24.6109	−0.969805
$645$	0	0
$646$	104.750	4.12132
$647$	−38.9519	−1.53136	−0.765679	−	0.643223i	$-0.777597\pi$
$647$	−38.9519	−1.53136	−0.765679	+	0.643223i	$0.777597\pi$
$648$	0	0
$649$	−51.9885	−2.04073
$650$	−52.5160	−2.05985
$651$	0	0
$652$	35.0231	1.37161
$653$	7.75059	0.303304	0.151652	−	0.988434i	$-0.451541\pi$
$653$	7.75059	0.303304	0.151652	+	0.988434i	$0.451541\pi$
$654$	0	0
$655$	−40.8639	−1.59669
$656$	−25.7180	−1.00412
$657$	0	0
$658$	−5.95624	−0.232198
$659$	22.3674	0.871309	0.435655	−	0.900114i	$-0.356517\pi$
$659$	22.3674	0.871309	0.435655	+	0.900114i	$0.356517\pi$
$660$	0	0
$661$	30.2526	1.17669	0.588345	−	0.808610i	$-0.299780\pi$
$661$	30.2526	1.17669	0.588345	+	0.808610i	$0.299780\pi$
$662$	−3.90375	−0.151723
$663$	0	0
$664$	−59.4643	−2.30766
$665$	−10.6861	−0.414389
$666$	0	0
$667$	−10.5725	−0.409370
$668$	−35.7331	−1.38255
$669$	0	0
$670$	14.2005	0.548612
$671$	2.00386	0.0773583
$672$	0	0
$673$	11.7416	0.452604	0.226302	−	0.974057i	$-0.427336\pi$
$673$	11.7416	0.452604	0.226302	+	0.974057i	$0.427336\pi$
$674$	−27.5093	−1.05962
$675$	0	0
$676$	51.4393	1.97843
$677$	30.4285	1.16946	0.584732	−	0.811227i	$-0.301200\pi$
$677$	30.4285	1.16946	0.584732	+	0.811227i	$0.301200\pi$
$678$	0	0
$679$	4.13513	0.158692
$680$	138.724	5.31984
$681$	0	0
$682$	82.7336	3.16803
$683$	11.6989	0.447647	0.223823	−	0.974630i	$-0.428146\pi$
$683$	11.6989	0.447647	0.223823	+	0.974630i	$0.428146\pi$
$684$	0	0
$685$	−25.1752	−0.961895
$686$	19.9718	0.762527
$687$	0	0
$688$	−14.3259	−0.546171
$689$	−19.5397	−0.744404
$690$	0	0
$691$	27.2076	1.03502	0.517512	−	0.855676i	$-0.326858\pi$
$691$	27.2076	1.03502	0.517512	+	0.855676i	$0.326858\pi$
$692$	−24.9831	−0.949716
$693$	0	0
$694$	57.0903	2.16712
$695$	−50.3239	−1.90889
$696$	0	0
$697$	−18.0399	−0.683311
$698$	−35.7794	−1.35427
$699$	0	0
$700$	−11.0584	−0.417967
$701$	21.2354	0.802049	0.401024	−	0.916067i	$-0.368654\pi$
$701$	21.2354	0.802049	0.401024	+	0.916067i	$0.368654\pi$
$702$	0	0
$703$	55.0513	2.07630
$704$	−25.9412	−0.977697
$705$	0	0
$706$	86.2359	3.24553
$707$	2.83821	0.106742
$708$	0	0
$709$	16.8807	0.633967	0.316983	−	0.948431i	$-0.397330\pi$
$709$	16.8807	0.633967	0.316983	+	0.948431i	$0.397330\pi$
$710$	13.9008	0.521689
$711$	0	0
$712$	106.491	3.99094
$713$	−69.3015	−2.59536
$714$	0	0
$715$	62.4184	2.33432
$716$	45.5364	1.70177
$717$	0	0
$718$	46.9796	1.75326
$719$	20.3235	0.757941	0.378970	−	0.925409i	$-0.376278\pi$
$719$	20.3235	0.757941	0.378970	+	0.925409i	$0.376278\pi$
$720$	0	0
$721$	−1.06826	−0.0397842
$722$	53.6646	1.99719
$723$	0	0
$724$	−7.94288	−0.295195
$725$	−4.75053	−0.176430
$726$	0	0
$727$	−9.13646	−0.338853	−0.169426	−	0.985543i	$-0.554191\pi$
$727$	−9.13646	−0.338853	−0.169426	+	0.985543i	$0.554191\pi$
$728$	19.6416	0.727967
$729$	0	0
$730$	4.80739	0.177929
$731$	−10.0489	−0.371674
$732$	0	0
$733$	−6.55001	−0.241930	−0.120965	−	0.992657i	$-0.538599\pi$
$733$	−6.55001	−0.241930	−0.120965	+	0.992657i	$0.538599\pi$
$734$	−85.1972	−3.14469
$735$	0	0
$736$	86.3997	3.18474
$737$	−7.64493	−0.281605
$738$	0	0
$739$	−37.2949	−1.37192	−0.685958	−	0.727641i	$-0.740617\pi$
$739$	−37.2949	−1.37192	−0.685958	+	0.727641i	$0.740617\pi$
$740$	125.774	4.62353
$741$	0	0
$742$	−5.84398	−0.214539
$743$	45.7404	1.67805	0.839026	−	0.544092i	$-0.183126\pi$
$743$	45.7404	1.67805	0.839026	+	0.544092i	$0.183126\pi$
$744$	0	0
$745$	−28.4375	−1.04187
$746$	8.45692	0.309630
$747$	0	0
$748$	−128.838	−4.71080
$749$	6.64161	0.242679
$750$	0	0
$751$	−38.4995	−1.40487	−0.702434	−	0.711749i	$-0.747903\pi$
$751$	−38.4995	−1.40487	−0.702434	+	0.711749i	$0.747903\pi$
$752$	37.2352	1.35783
$753$	0	0
$754$	14.5563	0.530109
$755$	−61.3745	−2.23365
$756$	0	0
$757$	20.5989	0.748681	0.374340	−	0.927291i	$-0.377869\pi$
$757$	20.5989	0.748681	0.374340	+	0.927291i	$0.377869\pi$
$758$	−49.8098	−1.80917
$759$	0	0
$760$	136.484	4.95078
$761$	−18.6360	−0.675555	−0.337778	−	0.941226i	$-0.609675\pi$
$761$	−18.6360	−0.675555	−0.337778	+	0.941226i	$0.609675\pi$
$762$	0	0
$763$	−2.67560	−0.0968632
$764$	−127.090	−4.59795
$765$	0	0
$766$	−58.9301	−2.12923
$767$	−59.9571	−2.16492
$768$	0	0
$769$	32.6001	1.17559	0.587795	−	0.809010i	$-0.299996\pi$
$769$	32.6001	1.17559	0.587795	+	0.809010i	$0.299996\pi$
$770$	18.6682	0.672756
$771$	0	0
$772$	81.6467	2.93853
$773$	50.6910	1.82323	0.911614	−	0.411047i	$-0.134837\pi$
$773$	50.6910	1.82323	0.911614	+	0.411047i	$0.134837\pi$
$774$	0	0
$775$	−31.1391	−1.11855
$776$	−52.8141	−1.89592
$777$	0	0
$778$	101.596	3.64239
$779$	−17.7486	−0.635908
$780$	0	0
$781$	−7.48361	−0.267785
$782$	153.284	5.48143
$783$	0	0
$784$	−60.9888	−2.17817
$785$	−18.3768	−0.655896
$786$	0	0
$787$	17.5394	0.625213	0.312607	−	0.949883i	$-0.398798\pi$
$787$	17.5394	0.625213	0.312607	+	0.949883i	$0.398798\pi$
$788$	28.4668	1.01409
$789$	0	0
$790$	−19.6201	−0.698052
$791$	7.07627	0.251603
$792$	0	0
$793$	2.31101	0.0820663
$794$	45.4081	1.61147
$795$	0	0
$796$	76.3072	2.70464
$797$	11.2655	0.399044	0.199522	−	0.979893i	$-0.436061\pi$
$797$	11.2655	0.399044	0.199522	+	0.979893i	$0.436061\pi$
$798$	0	0
$799$	26.1186	0.924011
$800$	38.8218	1.37256
$801$	0	0
$802$	−18.6298	−0.657841
$803$	−2.58809	−0.0913318
$804$	0	0
$805$	−15.6374	−0.551145
$806$	95.4146	3.36084
$807$	0	0
$808$	−36.2499	−1.27527
$809$	1.43789	0.0505534	0.0252767	−	0.999680i	$-0.491953\pi$
$809$	1.43789	0.0505534	0.0252767	+	0.999680i	$0.491953\pi$
$810$	0	0
$811$	−39.2119	−1.37692	−0.688458	−	0.725276i	$-0.741712\pi$
$811$	−39.2119	−1.37692	−0.688458	+	0.725276i	$0.741712\pi$
$812$	3.06514	0.107565
$813$	0	0
$814$	−96.1725	−3.37084
$815$	22.2531	0.779492
$816$	0	0
$817$	−9.88663	−0.345889
$818$	−52.8636	−1.84833
$819$	0	0
$820$	−40.5495	−1.41605
$821$	−9.41950	−0.328743	−0.164371	−	0.986399i	$-0.552560\pi$
$821$	−9.41950	−0.328743	−0.164371	+	0.986399i	$0.552560\pi$
$822$	0	0
$823$	−28.7149	−1.00094	−0.500470	−	0.865754i	$-0.666839\pi$
$823$	−28.7149	−1.00094	−0.500470	+	0.865754i	$0.666839\pi$
$824$	13.6439	0.475308
$825$	0	0
$826$	−17.9321	−0.623936
$827$	−32.2145	−1.12021	−0.560104	−	0.828422i	$-0.689239\pi$
$827$	−32.2145	−1.12021	−0.560104	+	0.828422i	$0.689239\pi$
$828$	0	0
$829$	−41.3853	−1.43737	−0.718685	−	0.695335i	$-0.755256\pi$
$829$	−41.3853	−1.43737	−0.718685	+	0.695335i	$0.755256\pi$
$830$	−65.1800	−2.26243
$831$	0	0
$832$	−29.9174	−1.03720
$833$	−42.7806	−1.48226
$834$	0	0
$835$	−22.7042	−0.785711
$836$	−126.757	−4.38400
$837$	0	0
$838$	73.6948	2.54575
$839$	−31.6092	−1.09127	−0.545635	−	0.838023i	$-0.683711\pi$
$839$	−31.6092	−1.09127	−0.545635	+	0.838023i	$0.683711\pi$
$840$	0	0
$841$	−27.6833	−0.954595
$842$	−5.76788	−0.198774
$843$	0	0
$844$	54.7597	1.88491
$845$	32.6837	1.12435
$846$	0	0
$847$	−3.87488	−0.133142
$848$	36.5334	1.25456
$849$	0	0
$850$	68.8748	2.36238
$851$	80.5585	2.76151
$852$	0	0
$853$	26.9128	0.921477	0.460739	−	0.887536i	$-0.347585\pi$
$853$	26.9128	0.921477	0.460739	+	0.887536i	$0.347585\pi$
$854$	0.691180	0.0236517
$855$	0	0
$856$	−84.8270	−2.89933
$857$	−25.4512	−0.869398	−0.434699	−	0.900576i	$-0.643145\pi$
$857$	−25.4512	−0.869398	−0.434699	+	0.900576i	$0.643145\pi$
$858$	0	0
$859$	33.5688	1.14535	0.572676	−	0.819781i	$-0.305905\pi$
$859$	33.5688	1.14535	0.572676	+	0.819781i	$0.305905\pi$
$860$	−22.5876	−0.770231
$861$	0	0
$862$	4.92893	0.167880
$863$	−21.3439	−0.726554	−0.363277	−	0.931681i	$-0.618342\pi$
$863$	−21.3439	−0.726554	−0.363277	+	0.931681i	$0.618342\pi$
$864$	0	0
$865$	−15.8739	−0.539728
$866$	−87.9256	−2.98783
$867$	0	0
$868$	20.0916	0.681953
$869$	10.5626	0.358312
$870$	0	0
$871$	−8.81671	−0.298743
$872$	34.1729	1.15724
$873$	0	0
$874$	150.808	5.10116
$875$	1.45975	0.0493485
$876$	0	0
$877$	−32.2943	−1.09050	−0.545251	−	0.838273i	$-0.683566\pi$
$877$	−32.2943	−1.09050	−0.545251	+	0.838273i	$0.683566\pi$
$878$	16.4208	0.554176
$879$	0	0
$880$	−116.704	−3.93408
$881$	−16.3456	−0.550698	−0.275349	−	0.961344i	$-0.588793\pi$
$881$	−16.3456	−0.550698	−0.275349	+	0.961344i	$0.588793\pi$
$882$	0	0
$883$	7.21171	0.242693	0.121347	−	0.992610i	$-0.461279\pi$
$883$	7.21171	0.242693	0.121347	+	0.992610i	$0.461279\pi$
$884$	−148.586	−4.99750
$885$	0	0
$886$	−50.7747	−1.70581
$887$	−35.6173	−1.19591	−0.597956	−	0.801529i	$-0.704020\pi$
$887$	−35.6173	−1.19591	−0.597956	+	0.801529i	$0.704020\pi$
$888$	0	0
$889$	11.4042	0.382485
$890$	116.727	3.91271
$891$	0	0
$892$	20.9428	0.701216
$893$	25.6968	0.859910
$894$	0	0
$895$	28.9330	0.967125
$896$	1.58106	0.0528196
$897$	0	0
$898$	39.8401	1.32948
$899$	8.63109	0.287863
$900$	0	0
$901$	25.6264	0.853738
$902$	31.0060	1.03239
$903$	0	0
$904$	−90.3786	−3.00595
$905$	−5.04677	−0.167760
$906$	0	0
$907$	−49.0472	−1.62859	−0.814293	−	0.580454i	$-0.802875\pi$
$907$	−49.0472	−1.62859	−0.814293	+	0.580454i	$0.802875\pi$
$908$	−71.0157	−2.35674
$909$	0	0
$910$	21.5296	0.713699
$911$	45.6848	1.51361	0.756803	−	0.653643i	$-0.226760\pi$
$911$	45.6848	1.51361	0.756803	+	0.653643i	$0.226760\pi$
$912$	0	0
$913$	35.0901	1.16131
$914$	−38.4131	−1.27059
$915$	0	0
$916$	19.9425	0.658917
$917$	7.58811	0.250581
$918$	0	0
$919$	−12.9459	−0.427045	−0.213522	−	0.976938i	$-0.568494\pi$
$919$	−12.9459	−0.427045	−0.213522	+	0.976938i	$0.568494\pi$
$920$	199.721	6.58462
$921$	0	0
$922$	73.1307	2.40843
$923$	−8.63067	−0.284082
$924$	0	0
$925$	36.1972	1.19016
$926$	43.4568	1.42808
$927$	0	0
$928$	−10.7606	−0.353233
$929$	4.49235	0.147389	0.0736946	−	0.997281i	$-0.476521\pi$
$929$	4.49235	0.147389	0.0736946	+	0.997281i	$0.476521\pi$
$930$	0	0
$931$	−42.0896	−1.37943
$932$	−6.28957	−0.206022
$933$	0	0
$934$	58.2592	1.90630
$935$	−81.8618	−2.67717
$936$	0	0
$937$	−1.90994	−0.0623951	−0.0311975	−	0.999513i	$-0.509932\pi$
$937$	−1.90994	−0.0623951	−0.0311975	+	0.999513i	$0.509932\pi$
$938$	−2.63692	−0.0860984
$939$	0	0
$940$	58.7085	1.91486
$941$	1.34900	0.0439761	0.0219881	−	0.999758i	$-0.493000\pi$
$941$	1.34900	0.0439761	0.0219881	+	0.999758i	$0.493000\pi$
$942$	0	0
$943$	−25.9721	−0.845768
$944$	112.102	3.64860
$945$	0	0
$946$	17.2716	0.561547
$947$	23.5579	0.765530	0.382765	−	0.923846i	$-0.374972\pi$
$947$	23.5579	0.765530	0.382765	+	0.923846i	$0.374972\pi$
$948$	0	0
$949$	−2.98478	−0.0968902
$950$	67.7623	2.19850
$951$	0	0
$952$	−25.7600	−0.834887
$953$	−44.6075	−1.44498	−0.722489	−	0.691383i	$-0.757002\pi$
$953$	−44.6075	−1.44498	−0.722489	+	0.691383i	$0.757002\pi$
$954$	0	0
$955$	−80.7509	−2.61304
$956$	17.0103	0.550152
$957$	0	0
$958$	39.1584	1.26515
$959$	4.67484	0.150958
$960$	0	0
$961$	25.5756	0.825019
$962$	−110.913	−3.57599
$963$	0	0
$964$	69.0868	2.22514
$965$	51.8769	1.66998
$966$	0	0
$967$	−8.74032	−0.281070	−0.140535	−	0.990076i	$-0.544882\pi$
$967$	−8.74032	−0.281070	−0.140535	+	0.990076i	$0.544882\pi$
$968$	49.4902	1.59068
$969$	0	0
$970$	−57.8906	−1.85876
$971$	−36.7867	−1.18054	−0.590271	−	0.807205i	$-0.700979\pi$
$971$	−36.7867	−1.18054	−0.590271	+	0.807205i	$0.700979\pi$
$972$	0	0
$973$	9.34474	0.299579
$974$	101.478	3.25156
$975$	0	0
$976$	−4.32089	−0.138308
$977$	17.6031	0.563172	0.281586	−	0.959536i	$-0.409140\pi$
$977$	17.6031	0.563172	0.281586	+	0.959536i	$0.409140\pi$
$978$	0	0
$979$	−62.8411	−2.00841
$980$	−96.1606	−3.07174
$981$	0	0
$982$	−56.8596	−1.81446
$983$	50.6464	1.61537	0.807684	−	0.589615i	$-0.200721\pi$
$983$	50.6464	1.61537	0.807684	+	0.589615i	$0.200721\pi$
$984$	0	0
$985$	18.0873	0.576309
$986$	−19.0906	−0.607969
$987$	0	0
$988$	−146.186	−4.65080
$989$	−14.4675	−0.460039
$990$	0	0
$991$	4.79288	0.152251	0.0761254	−	0.997098i	$-0.475745\pi$
$991$	4.79288	0.152251	0.0761254	+	0.997098i	$0.475745\pi$
$992$	−70.5341	−2.23946
$993$	0	0
$994$	−2.58127	−0.0818730
$995$	48.4844	1.53706
$996$	0	0
$997$	12.0839	0.382700	0.191350	−	0.981522i	$-0.438714\pi$
$997$	12.0839	0.382700	0.191350	+	0.981522i	$0.438714\pi$
$998$	53.0743	1.68004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.17		18
3.2	odd	2		547.2.a.b.1.2	✓	18
12.11	even	2		8752.2.a.s.1.3		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.2	✓	18	3.2	odd	2
4923.2.a.l.1.17		18	1.1	even	1	trivial
8752.2.a.s.1.3		18	12.11	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:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59964 q^{2} +4.75812 q^{4} +3.02323 q^{5} -0.561390 q^{7} +7.17011 q^{8} +O(q^{10})\)	\(q+2.59964 q^{2} +4.75812 q^{4} +3.02323 q^{5} -0.561390 q^{7} +7.17011 q^{8} +7.85930 q^{10} -4.23111 q^{11} -4.87964 q^{13} -1.45941 q^{14} +9.12345 q^{16} +6.39965 q^{17} +6.29628 q^{19} +14.3849 q^{20} -10.9994 q^{22} +9.21357 q^{23} +4.13991 q^{25} -12.6853 q^{26} -2.67116 q^{28} -1.14750 q^{29} -7.52167 q^{31} +9.37744 q^{32} +16.6368 q^{34} -1.69721 q^{35} +8.74347 q^{37} +16.3680 q^{38} +21.6769 q^{40} -2.81890 q^{41} -1.57023 q^{43} -20.1321 q^{44} +23.9520 q^{46} +4.08126 q^{47} -6.68484 q^{49} +10.7623 q^{50} -23.2179 q^{52} +4.00434 q^{53} -12.7916 q^{55} -4.02522 q^{56} -2.98307 q^{58} +12.2872 q^{59} -0.473603 q^{61} -19.5536 q^{62} +6.13107 q^{64} -14.7523 q^{65} +1.80684 q^{67} +30.4503 q^{68} -4.41213 q^{70} +1.76871 q^{71} +0.611682 q^{73} +22.7298 q^{74} +29.9584 q^{76} +2.37530 q^{77} -2.49642 q^{79} +27.5823 q^{80} -7.32811 q^{82} -8.29336 q^{83} +19.3476 q^{85} -4.08204 q^{86} -30.3375 q^{88} +14.8521 q^{89} +2.73938 q^{91} +43.8393 q^{92} +10.6098 q^{94} +19.0351 q^{95} -7.36588 q^{97} -17.3782 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

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