LMFDB - Embedded newform 4527.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4527,2,Mod(1,4527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4527, 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("4527.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4527 = 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4527.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:	$36.1482769950$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4527.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81497 q^{2} +1.29413 q^{4} -2.10638 q^{5} +1.73516 q^{7} +1.28114 q^{8} +O(q^{10})$	\(q-1.81497 q^{2} +1.29413 q^{4} -2.10638 q^{5} +1.73516 q^{7} +1.28114 q^{8} +3.82302 q^{10} -6.36270 q^{11} -2.18456 q^{13} -3.14927 q^{14} -4.91349 q^{16} +0.631406 q^{17} -7.04772 q^{19} -2.72593 q^{20} +11.5481 q^{22} -6.00543 q^{23} -0.563171 q^{25} +3.96493 q^{26} +2.24552 q^{28} -7.71180 q^{29} +0.467852 q^{31} +6.35558 q^{32} -1.14599 q^{34} -3.65490 q^{35} +2.99418 q^{37} +12.7914 q^{38} -2.69855 q^{40} +8.67325 q^{41} -11.0553 q^{43} -8.23417 q^{44} +10.8997 q^{46} -4.33189 q^{47} -3.98923 q^{49} +1.02214 q^{50} -2.82711 q^{52} +8.09634 q^{53} +13.4023 q^{55} +2.22297 q^{56} +13.9967 q^{58} -7.20052 q^{59} +5.93532 q^{61} -0.849139 q^{62} -1.70824 q^{64} +4.60152 q^{65} -2.21237 q^{67} +0.817122 q^{68} +6.63355 q^{70} +5.83367 q^{71} -9.72628 q^{73} -5.43437 q^{74} -9.12067 q^{76} -11.0403 q^{77} +1.62156 q^{79} +10.3497 q^{80} -15.7417 q^{82} -1.72808 q^{83} -1.32998 q^{85} +20.0650 q^{86} -8.15148 q^{88} +11.1083 q^{89} -3.79056 q^{91} -7.77181 q^{92} +7.86227 q^{94} +14.8452 q^{95} +7.97564 q^{97} +7.24034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{2} + 36 q^{4} - 9 q^{5} + 11 q^{7} - 18 q^{8}+O(q^{10}) $ 26 * q - 4 * q^2 + 36 * q^4 - 9 * q^5 + 11 * q^7 - 18 * q^8	$ 26 q - 4 q^{2} + 36 q^{4} - 9 q^{5} + 11 q^{7} - 18 q^{8} + 4 q^{10} + 17 q^{11} + 14 q^{13} - q^{14} + 48 q^{16} - 17 q^{17} - 22 q^{19} + 19 q^{20} + 38 q^{22} - 27 q^{23} + 93 q^{25} - q^{26} - 9 q^{28} - 13 q^{29} + 26 q^{31} - 5 q^{32} - 32 q^{34} + 22 q^{35} + 55 q^{37} + 24 q^{38} - 7 q^{40} - 24 q^{41} + 20 q^{43} + 27 q^{44} + 6 q^{46} + 25 q^{47} + 65 q^{49} + 16 q^{50} + 32 q^{52} - 30 q^{53} + 25 q^{55} - 3 q^{56} + 58 q^{58} + 26 q^{59} + 15 q^{61} + 12 q^{62} + 44 q^{64} - 20 q^{65} - 20 q^{67} + 4 q^{68} + 2 q^{70} + 35 q^{71} + 38 q^{73} + 59 q^{74} - 42 q^{76} + 6 q^{77} + 21 q^{79} + 100 q^{80} - 59 q^{82} + 48 q^{83} + 6 q^{85} + 7 q^{86} + 106 q^{88} + 5 q^{89} - 24 q^{91} - 26 q^{92} - 22 q^{94} - 43 q^{95} + 142 q^{97} + 38 q^{98}+O(q^{100}) $ 26 * q - 4 * q^2 + 36 * q^4 - 9 * q^5 + 11 * q^7 - 18 * q^8 + 4 * q^10 + 17 * q^11 + 14 * q^13 - q^14 + 48 * q^16 - 17 * q^17 - 22 * q^19 + 19 * q^20 + 38 * q^22 - 27 * q^23 + 93 * q^25 - q^26 - 9 * q^28 - 13 * q^29 + 26 * q^31 - 5 * q^32 - 32 * q^34 + 22 * q^35 + 55 * q^37 + 24 * q^38 - 7 * q^40 - 24 * q^41 + 20 * q^43 + 27 * q^44 + 6 * q^46 + 25 * q^47 + 65 * q^49 + 16 * q^50 + 32 * q^52 - 30 * q^53 + 25 * q^55 - 3 * q^56 + 58 * q^58 + 26 * q^59 + 15 * q^61 + 12 * q^62 + 44 * q^64 - 20 * q^65 - 20 * q^67 + 4 * q^68 + 2 * q^70 + 35 * q^71 + 38 * q^73 + 59 * q^74 - 42 * q^76 + 6 * q^77 + 21 * q^79 + 100 * q^80 - 59 * q^82 + 48 * q^83 + 6 * q^85 + 7 * q^86 + 106 * q^88 + 5 * q^89 - 24 * q^91 - 26 * q^92 - 22 * q^94 - 43 * q^95 + 142 * q^97 + 38 * 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 $

$2$	−1.81497	−1.28338	−0.641690	−	0.766964i	$-0.721766\pi$
$2$	−1.81497	−1.28338	−0.641690	+	0.766964i	$0.721766\pi$
$3$	0	0
$3$	0	0
$4$	1.29413	0.647065
$5$	−2.10638	−0.942001	−0.471000	−	0.882133i	$-0.656107\pi$
$5$	−2.10638	−0.942001	−0.471000	+	0.882133i	$0.656107\pi$
$6$	0	0
$7$	1.73516	0.655828	0.327914	−	0.944708i	$-0.393654\pi$
$7$	1.73516	0.655828	0.327914	+	0.944708i	$0.393654\pi$
$8$	1.28114	0.452950
$9$	0	0
$10$	3.82302	1.20895
$11$	−6.36270	−1.91843	−0.959213	−	0.282683i	$-0.908776\pi$
$11$	−6.36270	−1.91843	−0.959213	+	0.282683i	$0.908776\pi$
$12$	0	0
$13$	−2.18456	−0.605889	−0.302945	−	0.953008i	$-0.597970\pi$
$13$	−2.18456	−0.605889	−0.302945	+	0.953008i	$0.597970\pi$
$14$	−3.14927	−0.841677
$15$	0	0
$16$	−4.91349	−1.22837
$17$	0.631406	0.153139	0.0765693	−	0.997064i	$-0.475603\pi$
$17$	0.631406	0.153139	0.0765693	+	0.997064i	$0.475603\pi$
$18$	0	0
$19$	−7.04772	−1.61686	−0.808429	−	0.588594i	$-0.799682\pi$
$19$	−7.04772	−1.61686	−0.808429	+	0.588594i	$0.799682\pi$
$20$	−2.72593	−0.609536
$21$	0	0
$22$	11.5481	2.46207
$23$	−6.00543	−1.25222	−0.626109	−	0.779735i	$-0.715354\pi$
$23$	−6.00543	−1.25222	−0.626109	+	0.779735i	$0.715354\pi$
$24$	0	0
$25$	−0.563171	−0.112634
$26$	3.96493	0.777586
$27$	0	0
$28$	2.24552	0.424364
$29$	−7.71180	−1.43205	−0.716023	−	0.698077i	$-0.754039\pi$
$29$	−7.71180	−1.43205	−0.716023	+	0.698077i	$0.754039\pi$
$30$	0	0
$31$	0.467852	0.0840286	0.0420143	−	0.999117i	$-0.486622\pi$
$31$	0.467852	0.0840286	0.0420143	+	0.999117i	$0.486622\pi$
$32$	6.35558	1.12352
$33$	0	0
$34$	−1.14599	−0.196535
$35$	−3.65490	−0.617791
$36$	0	0
$37$	2.99418	0.492241	0.246120	−	0.969239i	$-0.420844\pi$
$37$	2.99418	0.492241	0.246120	+	0.969239i	$0.420844\pi$
$38$	12.7914	2.07504
$39$	0	0
$40$	−2.69855	−0.426679
$41$	8.67325	1.35453	0.677267	−	0.735738i	$-0.263164\pi$
$41$	8.67325	1.35453	0.677267	+	0.735738i	$0.263164\pi$
$42$	0	0
$43$	−11.0553	−1.68591	−0.842956	−	0.537983i	$-0.819186\pi$
$43$	−11.0553	−1.68591	−0.842956	+	0.537983i	$0.819186\pi$
$44$	−8.23417	−1.24135
$45$	0	0
$46$	10.8997	1.60707
$47$	−4.33189	−0.631871	−0.315936	−	0.948781i	$-0.602318\pi$
$47$	−4.33189	−0.631871	−0.315936	+	0.948781i	$0.602318\pi$
$48$	0	0
$49$	−3.98923	−0.569889
$50$	1.02214	0.144553
$51$	0	0
$52$	−2.82711	−0.392050
$53$	8.09634	1.11212	0.556059	−	0.831143i	$-0.312313\pi$
$53$	8.09634	1.11212	0.556059	+	0.831143i	$0.312313\pi$
$54$	0	0
$55$	13.4023	1.80716
$56$	2.22297	0.297057
$57$	0	0
$58$	13.9967	1.83786
$59$	−7.20052	−0.937428	−0.468714	−	0.883350i	$-0.655282\pi$
$59$	−7.20052	−0.937428	−0.468714	+	0.883350i	$0.655282\pi$
$60$	0	0
$61$	5.93532	0.759940	0.379970	−	0.924999i	$-0.375934\pi$
$61$	5.93532	0.759940	0.379970	+	0.924999i	$0.375934\pi$
$62$	−0.849139	−0.107841
$63$	0	0
$64$	−1.70824	−0.213530
$65$	4.60152	0.570748
$66$	0	0
$67$	−2.21237	−0.270283	−0.135142	−	0.990826i	$-0.543149\pi$
$67$	−2.21237	−0.270283	−0.135142	+	0.990826i	$0.543149\pi$
$68$	0.817122	0.0990906
$69$	0	0
$70$	6.63355	0.792861
$71$	5.83367	0.692330	0.346165	−	0.938174i	$-0.387484\pi$
$71$	5.83367	0.692330	0.346165	+	0.938174i	$0.387484\pi$
$72$	0	0
$73$	−9.72628	−1.13837	−0.569187	−	0.822208i	$-0.692742\pi$
$73$	−9.72628	−1.13837	−0.569187	+	0.822208i	$0.692742\pi$
$74$	−5.43437	−0.631732
$75$	0	0
$76$	−9.12067	−1.04621
$77$	−11.0403	−1.25816
$78$	0	0
$79$	1.62156	0.182439	0.0912197	−	0.995831i	$-0.470923\pi$
$79$	1.62156	0.182439	0.0912197	+	0.995831i	$0.470923\pi$
$80$	10.3497	1.15713
$81$	0	0
$82$	−15.7417	−1.73838
$83$	−1.72808	−0.189682	−0.0948409	−	0.995492i	$-0.530234\pi$
$83$	−1.72808	−0.189682	−0.0948409	+	0.995492i	$0.530234\pi$
$84$	0	0
$85$	−1.32998	−0.144257
$86$	20.0650	2.16367
$87$	0	0
$88$	−8.15148	−0.868951
$89$	11.1083	1.17748	0.588740	−	0.808322i	$-0.299624\pi$
$89$	11.1083	1.17748	0.588740	+	0.808322i	$0.299624\pi$
$90$	0	0
$91$	−3.79056	−0.397359
$92$	−7.77181	−0.810267
$93$	0	0
$94$	7.86227	0.810931
$95$	14.8452	1.52308
$96$	0	0
$97$	7.97564	0.809803	0.404902	−	0.914360i	$-0.367306\pi$
$97$	7.97564	0.809803	0.404902	+	0.914360i	$0.367306\pi$
$98$	7.24034	0.731385
$99$	0	0
$100$	−0.728817	−0.0728817
$101$	5.37750	0.535081	0.267541	−	0.963547i	$-0.413789\pi$
$101$	5.37750	0.535081	0.267541	+	0.963547i	$0.413789\pi$
$102$	0	0
$103$	5.38662	0.530759	0.265380	−	0.964144i	$-0.414503\pi$
$103$	5.38662	0.530759	0.265380	+	0.964144i	$0.414503\pi$
$104$	−2.79872	−0.274437
$105$	0	0
$106$	−14.6946	−1.42727
$107$	−15.8750	−1.53470	−0.767348	−	0.641231i	$-0.778424\pi$
$107$	−15.8750	−1.53470	−0.767348	+	0.641231i	$0.778424\pi$
$108$	0	0
$109$	−0.871591	−0.0834832	−0.0417416	−	0.999128i	$-0.513291\pi$
$109$	−0.871591	−0.0834832	−0.0417416	+	0.999128i	$0.513291\pi$
$110$	−24.3247	−2.31927
$111$	0	0
$112$	−8.52568	−0.805601
$113$	−12.6631	−1.19125	−0.595624	−	0.803264i	$-0.703095\pi$
$113$	−12.6631	−1.19125	−0.595624	+	0.803264i	$0.703095\pi$
$114$	0	0
$115$	12.6497	1.17959
$116$	−9.98008	−0.926627
$117$	0	0
$118$	13.0688	1.20308
$119$	1.09559	0.100433
$120$	0	0
$121$	29.4840	2.68036
$122$	−10.7725	−0.975293
$123$	0	0
$124$	0.605461	0.0543720
$125$	11.7181	1.04810
$126$	0	0
$127$	−14.0605	−1.24767	−0.623835	−	0.781556i	$-0.714426\pi$
$127$	−14.0605	−1.24767	−0.623835	+	0.781556i	$0.714426\pi$
$128$	−9.61075	−0.849479
$129$	0	0
$130$	−8.35163	−0.732487
$131$	7.90255	0.690449	0.345224	−	0.938520i	$-0.387803\pi$
$131$	7.90255	0.690449	0.345224	+	0.938520i	$0.387803\pi$
$132$	0	0
$133$	−12.2289	−1.06038
$134$	4.01538	0.346876
$135$	0	0
$136$	0.808917	0.0693640
$137$	−11.2138	−0.958057	−0.479029	−	0.877799i	$-0.659011\pi$
$137$	−11.2138	−0.958057	−0.479029	+	0.877799i	$0.659011\pi$
$138$	0	0
$139$	−8.13599	−0.690086	−0.345043	−	0.938587i	$-0.612136\pi$
$139$	−8.13599	−0.690086	−0.345043	+	0.938587i	$0.612136\pi$
$140$	−4.72992	−0.399751
$141$	0	0
$142$	−10.5880	−0.888522
$143$	13.8997	1.16235
$144$	0	0
$145$	16.2440	1.34899
$146$	17.6529	1.46097
$147$	0	0
$148$	3.87486	0.318512
$149$	−15.4639	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$149$	−15.4639	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$150$	0	0
$151$	−13.8861	−1.13003	−0.565015	−	0.825080i	$-0.691130\pi$
$151$	−13.8861	−1.13003	−0.565015	+	0.825080i	$0.691130\pi$
$152$	−9.02908	−0.732355
$153$	0	0
$154$	20.0379	1.61470
$155$	−0.985473	−0.0791551
$156$	0	0
$157$	15.0454	1.20075	0.600377	−	0.799717i	$-0.295017\pi$
$157$	15.0454	1.20075	0.600377	+	0.799717i	$0.295017\pi$
$158$	−2.94308	−0.234139
$159$	0	0
$160$	−13.3873	−1.05836
$161$	−10.4204	−0.821240
$162$	0	0
$163$	−14.6205	−1.14517	−0.572584	−	0.819846i	$-0.694059\pi$
$163$	−14.6205	−1.14517	−0.572584	+	0.819846i	$0.694059\pi$
$164$	11.2243	0.876471
$165$	0	0
$166$	3.13643	0.243434
$167$	−4.21559	−0.326212	−0.163106	−	0.986609i	$-0.552151\pi$
$167$	−4.21559	−0.326212	−0.163106	+	0.986609i	$0.552151\pi$
$168$	0	0
$169$	−8.22768	−0.632898
$170$	2.41388	0.185136
$171$	0	0
$172$	−14.3069	−1.09089
$173$	−19.3646	−1.47226	−0.736132	−	0.676838i	$-0.763350\pi$
$173$	−19.3646	−1.47226	−0.736132	+	0.676838i	$0.763350\pi$
$174$	0	0
$175$	−0.977191	−0.0738687
$176$	31.2631	2.35654
$177$	0	0
$178$	−20.1613	−1.51116
$179$	−2.65395	−0.198366	−0.0991829	−	0.995069i	$-0.531623\pi$
$179$	−2.65395	−0.198366	−0.0991829	+	0.995069i	$0.531623\pi$
$180$	0	0
$181$	19.7622	1.46891	0.734457	−	0.678656i	$-0.237437\pi$
$181$	19.7622	1.46891	0.734457	+	0.678656i	$0.237437\pi$
$182$	6.87978	0.509963
$183$	0	0
$184$	−7.69376	−0.567192
$185$	−6.30688	−0.463691
$186$	0	0
$187$	−4.01745	−0.293785
$188$	−5.60603	−0.408862
$189$	0	0
$190$	−26.9436	−1.95469
$191$	−11.4647	−0.829557	−0.414779	−	0.909922i	$-0.636141\pi$
$191$	−11.4647	−0.829557	−0.414779	+	0.909922i	$0.636141\pi$
$192$	0	0
$193$	20.2977	1.46106	0.730531	−	0.682880i	$-0.239273\pi$
$193$	20.2977	1.46106	0.730531	+	0.682880i	$0.239273\pi$
$194$	−14.4756	−1.03929
$195$	0	0
$196$	−5.16258	−0.368756
$197$	2.51938	0.179498	0.0897490	−	0.995964i	$-0.471394\pi$
$197$	2.51938	0.179498	0.0897490	+	0.995964i	$0.471394\pi$
$198$	0	0
$199$	2.93963	0.208385	0.104193	−	0.994557i	$-0.466774\pi$
$199$	2.93963	0.208385	0.104193	+	0.994557i	$0.466774\pi$
$200$	−0.721498	−0.0510176
$201$	0	0
$202$	−9.76002	−0.686713
$203$	−13.3812	−0.939176
$204$	0	0
$205$	−18.2691	−1.27597
$206$	−9.77657	−0.681166
$207$	0	0
$208$	10.7338	0.744257
$209$	44.8425	3.10182
$210$	0	0
$211$	13.6787	0.941679	0.470840	−	0.882219i	$-0.343951\pi$
$211$	13.6787	0.941679	0.470840	+	0.882219i	$0.343951\pi$
$212$	10.4777	0.719613
$213$	0	0
$214$	28.8127	1.96960
$215$	23.2866	1.58813
$216$	0	0
$217$	0.811797	0.0551084
$218$	1.58191	0.107141
$219$	0	0
$220$	17.3443	1.16935
$221$	−1.37935	−0.0927849
$222$	0	0
$223$	2.06965	0.138594	0.0692969	−	0.997596i	$-0.477924\pi$
$223$	2.06965	0.138594	0.0692969	+	0.997596i	$0.477924\pi$
$224$	11.0279	0.736835
$225$	0	0
$226$	22.9833	1.52882
$227$	16.1707	1.07328	0.536642	−	0.843810i	$-0.319693\pi$
$227$	16.1707	1.07328	0.536642	+	0.843810i	$0.319693\pi$
$228$	0	0
$229$	24.1008	1.59262	0.796312	−	0.604886i	$-0.206781\pi$
$229$	24.1008	1.59262	0.796312	+	0.604886i	$0.206781\pi$
$230$	−22.9589	−1.51386
$231$	0	0
$232$	−9.87986	−0.648645
$233$	5.26985	0.345240	0.172620	−	0.984989i	$-0.444777\pi$
$233$	5.26985	0.345240	0.172620	+	0.984989i	$0.444777\pi$
$234$	0	0
$235$	9.12460	0.595223
$236$	−9.31841	−0.606577
$237$	0	0
$238$	−1.98847	−0.128893
$239$	−5.69129	−0.368139	−0.184069	−	0.982913i	$-0.558927\pi$
$239$	−5.69129	−0.368139	−0.184069	+	0.982913i	$0.558927\pi$
$240$	0	0
$241$	23.7241	1.52820	0.764100	−	0.645097i	$-0.223183\pi$
$241$	23.7241	1.52820	0.764100	+	0.645097i	$0.223183\pi$
$242$	−53.5127	−3.43992
$243$	0	0
$244$	7.68108	0.491731
$245$	8.40282	0.536836
$246$	0	0
$247$	15.3962	0.979636
$248$	0.599381	0.0380607
$249$	0	0
$250$	−21.2681	−1.34511
$251$	−24.6823	−1.55793	−0.778966	−	0.627066i	$-0.784256\pi$
$251$	−24.6823	−1.55793	−0.778966	+	0.627066i	$0.784256\pi$
$252$	0	0
$253$	38.2107	2.40229
$254$	25.5195	1.60123
$255$	0	0
$256$	20.8597	1.30373
$257$	12.5279	0.781466	0.390733	−	0.920504i	$-0.372222\pi$
$257$	12.5279	0.781466	0.390733	+	0.920504i	$0.372222\pi$
$258$	0	0
$259$	5.19538	0.322825
$260$	5.95496	0.369311
$261$	0	0
$262$	−14.3429	−0.886108
$263$	27.6584	1.70549	0.852745	−	0.522327i	$-0.174936\pi$
$263$	27.6584	1.70549	0.852745	+	0.522327i	$0.174936\pi$
$264$	0	0
$265$	−17.0540	−1.04762
$266$	22.1951	1.36087
$267$	0	0
$268$	−2.86309	−0.174891
$269$	1.14164	0.0696069	0.0348034	−	0.999394i	$-0.488919\pi$
$269$	1.14164	0.0696069	0.0348034	+	0.999394i	$0.488919\pi$
$270$	0	0
$271$	−14.2437	−0.865241	−0.432621	−	0.901576i	$-0.642411\pi$
$271$	−14.2437	−0.865241	−0.432621	+	0.901576i	$0.642411\pi$
$272$	−3.10241	−0.188111
$273$	0	0
$274$	20.3527	1.22955
$275$	3.58329	0.216080
$276$	0	0
$277$	9.26397	0.556618	0.278309	−	0.960492i	$-0.410226\pi$
$277$	9.26397	0.556618	0.278309	+	0.960492i	$0.410226\pi$
$278$	14.7666	0.885643
$279$	0	0
$280$	−4.68242	−0.279828
$281$	−19.7177	−1.17626	−0.588128	−	0.808768i	$-0.700135\pi$
$281$	−19.7177	−1.17626	−0.588128	+	0.808768i	$0.700135\pi$
$282$	0	0
$283$	−8.14971	−0.484450	−0.242225	−	0.970220i	$-0.577877\pi$
$283$	−8.14971	−0.484450	−0.242225	+	0.970220i	$0.577877\pi$
$284$	7.54953	0.447982
$285$	0	0
$286$	−25.2276	−1.49174
$287$	15.0495	0.888341
$288$	0	0
$289$	−16.6013	−0.976549
$290$	−29.4824	−1.73127
$291$	0	0
$292$	−12.5871	−0.736603
$293$	−7.02797	−0.410579	−0.205289	−	0.978701i	$-0.565814\pi$
$293$	−7.02797	−0.410579	−0.205289	+	0.978701i	$0.565814\pi$
$294$	0	0
$295$	15.1670	0.883058
$296$	3.83595	0.222960
$297$	0	0
$298$	28.0666	1.62586
$299$	13.1192	0.758705
$300$	0	0
$301$	−19.1826	−1.10567
$302$	25.2028	1.45026
$303$	0	0
$304$	34.6289	1.98610
$305$	−12.5020	−0.715865
$306$	0	0
$307$	−9.82004	−0.560459	−0.280230	−	0.959933i	$-0.590411\pi$
$307$	−9.82004	−0.560459	−0.280230	+	0.959933i	$0.590411\pi$
$308$	−14.2876	−0.814111
$309$	0	0
$310$	1.78861	0.101586
$311$	−2.70208	−0.153221	−0.0766105	−	0.997061i	$-0.524410\pi$
$311$	−2.70208	−0.153221	−0.0766105	+	0.997061i	$0.524410\pi$
$312$	0	0
$313$	−10.9934	−0.621386	−0.310693	−	0.950510i	$-0.600561\pi$
$313$	−10.9934	−0.621386	−0.310693	+	0.950510i	$0.600561\pi$
$314$	−27.3070	−1.54102
$315$	0	0
$316$	2.09851	0.118050
$317$	−23.7918	−1.33628	−0.668140	−	0.744036i	$-0.732909\pi$
$317$	−23.7918	−1.33628	−0.668140	+	0.744036i	$0.732909\pi$
$318$	0	0
$319$	49.0679	2.74728
$320$	3.59820	0.201145
$321$	0	0
$322$	18.9127	1.05396
$323$	−4.44997	−0.247603
$324$	0	0
$325$	1.23028	0.0682438
$326$	26.5359	1.46969
$327$	0	0
$328$	11.1116	0.613535
$329$	−7.51652	−0.414399
$330$	0	0
$331$	−11.6897	−0.642523	−0.321261	−	0.946991i	$-0.604107\pi$
$331$	−11.6897	−0.642523	−0.321261	+	0.946991i	$0.604107\pi$
$332$	−2.23637	−0.122737
$333$	0	0
$334$	7.65119	0.418654
$335$	4.66008	0.254607
$336$	0	0
$337$	1.78759	0.0973762	0.0486881	−	0.998814i	$-0.484496\pi$
$337$	1.78759	0.0973762	0.0486881	+	0.998814i	$0.484496\pi$
$338$	14.9330	0.812250
$339$	0	0
$340$	−1.72117	−0.0933434
$341$	−2.97680	−0.161203
$342$	0	0
$343$	−19.0680	−1.02958
$344$	−14.1633	−0.763633
$345$	0	0
$346$	35.1463	1.88948
$347$	−11.9985	−0.644116	−0.322058	−	0.946720i	$-0.604375\pi$
$347$	−11.9985	−0.644116	−0.322058	+	0.946720i	$0.604375\pi$
$348$	0	0
$349$	−14.2527	−0.762928	−0.381464	−	0.924384i	$-0.624580\pi$
$349$	−14.2527	−0.762928	−0.381464	+	0.924384i	$0.624580\pi$
$350$	1.77358	0.0948016
$351$	0	0
$352$	−40.4387	−2.15539
$353$	27.6920	1.47390	0.736948	−	0.675950i	$-0.236266\pi$
$353$	27.6920	1.47390	0.736948	+	0.675950i	$0.236266\pi$
$354$	0	0
$355$	−12.2879	−0.652175
$356$	14.3756	0.761907
$357$	0	0
$358$	4.81686	0.254579
$359$	23.2691	1.22809	0.614047	−	0.789270i	$-0.289541\pi$
$359$	23.2691	1.22809	0.614047	+	0.789270i	$0.289541\pi$
$360$	0	0
$361$	30.6703	1.61423
$362$	−35.8679	−1.88517
$363$	0	0
$364$	−4.90548	−0.257117
$365$	20.4872	1.07235
$366$	0	0
$367$	28.3461	1.47965	0.739827	−	0.672797i	$-0.234907\pi$
$367$	28.3461	1.47965	0.739827	+	0.672797i	$0.234907\pi$
$368$	29.5076	1.53819
$369$	0	0
$370$	11.4468	0.595092
$371$	14.0484	0.729358
$372$	0	0
$373$	17.8836	0.925980	0.462990	−	0.886363i	$-0.346777\pi$
$373$	17.8836	0.925980	0.462990	+	0.886363i	$0.346777\pi$
$374$	7.29157	0.377038
$375$	0	0
$376$	−5.54974	−0.286206
$377$	16.8469	0.867661
$378$	0	0
$379$	21.7991	1.11974	0.559872	−	0.828579i	$-0.310850\pi$
$379$	21.7991	1.11974	0.559872	+	0.828579i	$0.310850\pi$
$380$	19.2116	0.985533
$381$	0	0
$382$	20.8081	1.06464
$383$	35.4957	1.81375	0.906874	−	0.421402i	$-0.138462\pi$
$383$	35.4957	1.81375	0.906874	+	0.421402i	$0.138462\pi$
$384$	0	0
$385$	23.2550	1.18519
$386$	−36.8398	−1.87510
$387$	0	0
$388$	10.3215	0.523996
$389$	12.2599	0.621602	0.310801	−	0.950475i	$-0.399403\pi$
$389$	12.2599	0.621602	0.310801	+	0.950475i	$0.399403\pi$
$390$	0	0
$391$	−3.79186	−0.191763
$392$	−5.11074	−0.258131
$393$	0	0
$394$	−4.57260	−0.230364
$395$	−3.41561	−0.171858
$396$	0	0
$397$	−36.3867	−1.82620	−0.913098	−	0.407740i	$-0.866317\pi$
$397$	−36.3867	−1.82620	−0.913098	+	0.407740i	$0.866317\pi$
$398$	−5.33536	−0.267437
$399$	0	0
$400$	2.76713	0.138357
$401$	−30.2605	−1.51114	−0.755568	−	0.655070i	$-0.772639\pi$
$401$	−30.2605	−1.51114	−0.755568	+	0.655070i	$0.772639\pi$
$402$	0	0
$403$	−1.02205	−0.0509120
$404$	6.95918	0.346232
$405$	0	0
$406$	24.2865	1.20532
$407$	−19.0511	−0.944328
$408$	0	0
$409$	−28.7574	−1.42196	−0.710982	−	0.703210i	$-0.751749\pi$
$409$	−28.7574	−1.42196	−0.710982	+	0.703210i	$0.751749\pi$
$410$	33.1580	1.63756
$411$	0	0
$412$	6.97099	0.343436
$413$	−12.4940	−0.614791
$414$	0	0
$415$	3.64000	0.178681
$416$	−13.8842	−0.680728
$417$	0	0
$418$	−81.3880	−3.98082
$419$	20.0041	0.977264	0.488632	−	0.872490i	$-0.337496\pi$
$419$	20.0041	0.977264	0.488632	+	0.872490i	$0.337496\pi$
$420$	0	0
$421$	−21.9536	−1.06995	−0.534976	−	0.844867i	$-0.679679\pi$
$421$	−21.9536	−1.06995	−0.534976	+	0.844867i	$0.679679\pi$
$422$	−24.8264	−1.20853
$423$	0	0
$424$	10.3725	0.503733
$425$	−0.355590	−0.0172486
$426$	0	0
$427$	10.2987	0.498390
$428$	−20.5443	−0.993048
$429$	0	0
$430$	−42.2645	−2.03817
$431$	25.5771	1.23200	0.616002	−	0.787745i	$-0.288751\pi$
$431$	25.5771	1.23200	0.616002	+	0.787745i	$0.288751\pi$
$432$	0	0
$433$	−2.27003	−0.109091	−0.0545454	−	0.998511i	$-0.517371\pi$
$433$	−2.27003	−0.109091	−0.0545454	+	0.998511i	$0.517371\pi$
$434$	−1.47339	−0.0707250
$435$	0	0
$436$	−1.12795	−0.0540191
$437$	42.3246	2.02466
$438$	0	0
$439$	−16.9124	−0.807187	−0.403593	−	0.914938i	$-0.632239\pi$
$439$	−16.9124	−0.807187	−0.403593	+	0.914938i	$0.632239\pi$
$440$	17.1701	0.818552
$441$	0	0
$442$	2.50348	0.119078
$443$	−34.5988	−1.64384	−0.821920	−	0.569603i	$-0.807097\pi$
$443$	−34.5988	−1.64384	−0.821920	+	0.569603i	$0.807097\pi$
$444$	0	0
$445$	−23.3983	−1.10919
$446$	−3.75635	−0.177869
$447$	0	0
$448$	−2.96407	−0.140039
$449$	−13.0996	−0.618208	−0.309104	−	0.951028i	$-0.600029\pi$
$449$	−13.0996	−0.618208	−0.309104	+	0.951028i	$0.600029\pi$
$450$	0	0
$451$	−55.1853	−2.59857
$452$	−16.3877	−0.770815
$453$	0	0
$454$	−29.3493	−1.37743
$455$	7.98436	0.374313
$456$	0	0
$457$	8.01557	0.374952	0.187476	−	0.982269i	$-0.439969\pi$
$457$	8.01557	0.374952	0.187476	+	0.982269i	$0.439969\pi$
$458$	−43.7423	−2.04394
$459$	0	0
$460$	16.3704	0.763272
$461$	15.1868	0.707320	0.353660	−	0.935374i	$-0.384937\pi$
$461$	15.1868	0.707320	0.353660	+	0.935374i	$0.384937\pi$
$462$	0	0
$463$	34.6450	1.61009	0.805046	−	0.593212i	$-0.202141\pi$
$463$	34.6450	1.61009	0.805046	+	0.593212i	$0.202141\pi$
$464$	37.8919	1.75909
$465$	0	0
$466$	−9.56464	−0.443074
$467$	37.1749	1.72025	0.860125	−	0.510083i	$-0.170385\pi$
$467$	37.1749	1.72025	0.860125	+	0.510083i	$0.170385\pi$
$468$	0	0
$469$	−3.83880	−0.177259
$470$	−16.5609	−0.763898
$471$	0	0
$472$	−9.22483	−0.424607
$473$	70.3413	3.23430
$474$	0	0
$475$	3.96907	0.182113
$476$	1.41784	0.0649864
$477$	0	0
$478$	10.3295	0.472462
$479$	−28.4871	−1.30161	−0.650805	−	0.759245i	$-0.725569\pi$
$479$	−28.4871	−1.30161	−0.650805	+	0.759245i	$0.725569\pi$
$480$	0	0
$481$	−6.54099	−0.298243
$482$	−43.0585	−1.96126
$483$	0	0
$484$	38.1561	1.73437
$485$	−16.7997	−0.762836
$486$	0	0
$487$	14.9167	0.675941	0.337971	−	0.941157i	$-0.390260\pi$
$487$	14.9167	0.675941	0.337971	+	0.941157i	$0.390260\pi$
$488$	7.60395	0.344215
$489$	0	0
$490$	−15.2509	−0.688965
$491$	33.6713	1.51956	0.759782	−	0.650178i	$-0.225306\pi$
$491$	33.6713	1.51956	0.759782	+	0.650178i	$0.225306\pi$
$492$	0	0
$493$	−4.86928	−0.219301
$494$	−27.9437	−1.25725
$495$	0	0
$496$	−2.29878	−0.103218
$497$	10.1223	0.454049
$498$	0	0
$499$	−21.4249	−0.959112	−0.479556	−	0.877511i	$-0.659202\pi$
$499$	−21.4249	−0.959112	−0.479556	+	0.877511i	$0.659202\pi$
$500$	15.1648	0.678191
$501$	0	0
$502$	44.7977	1.99942
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−11.3270	−0.504047
$506$	−69.3515	−3.08305
$507$	0	0
$508$	−18.1962	−0.807324
$509$	20.3267	0.900965	0.450482	−	0.892785i	$-0.351252\pi$
$509$	20.3267	0.900965	0.450482	+	0.892785i	$0.351252\pi$
$510$	0	0
$511$	−16.8766	−0.746578
$512$	−18.6384	−0.823708
$513$	0	0
$514$	−22.7377	−1.00292
$515$	−11.3463	−0.499976
$516$	0	0
$517$	27.5625	1.21220
$518$	−9.42948	−0.414308
$519$	0	0
$520$	5.89517	0.258520
$521$	18.4511	0.808359	0.404179	−	0.914680i	$-0.367557\pi$
$521$	18.4511	0.808359	0.404179	+	0.914680i	$0.367557\pi$
$522$	0	0
$523$	−9.91110	−0.433382	−0.216691	−	0.976240i	$-0.569526\pi$
$523$	−9.91110	−0.433382	−0.216691	+	0.976240i	$0.569526\pi$
$524$	10.2269	0.446765
$525$	0	0
$526$	−50.1993	−2.18879
$527$	0.295404	0.0128680
$528$	0	0
$529$	13.0652	0.568050
$530$	30.9525	1.34449
$531$	0	0
$532$	−15.8258	−0.686135
$533$	−18.9473	−0.820697
$534$	0	0
$535$	33.4388	1.44568
$536$	−2.83434	−0.122425
$537$	0	0
$538$	−2.07204	−0.0893321
$539$	25.3823	1.09329
$540$	0	0
$541$	−7.37029	−0.316874	−0.158437	−	0.987369i	$-0.550645\pi$
$541$	−7.37029	−0.316874	−0.158437	+	0.987369i	$0.550645\pi$
$542$	25.8519	1.11043
$543$	0	0
$544$	4.01295	0.172054
$545$	1.83590	0.0786413
$546$	0	0
$547$	16.8670	0.721179	0.360590	−	0.932725i	$-0.382575\pi$
$547$	16.8670	0.721179	0.360590	+	0.932725i	$0.382575\pi$
$548$	−14.5121	−0.619925
$549$	0	0
$550$	−6.50358	−0.277313
$551$	54.3506	2.31541
$552$	0	0
$553$	2.81366	0.119649
$554$	−16.8139	−0.714353
$555$	0	0
$556$	−10.5290	−0.446531
$557$	21.6892	0.919002	0.459501	−	0.888177i	$-0.348028\pi$
$557$	21.6892	0.919002	0.459501	+	0.888177i	$0.348028\pi$
$558$	0	0
$559$	24.1509	1.02148
$560$	17.9583	0.758877
$561$	0	0
$562$	35.7870	1.50958
$563$	−22.2548	−0.937929	−0.468964	−	0.883217i	$-0.655373\pi$
$563$	−22.2548	−0.937929	−0.468964	+	0.883217i	$0.655373\pi$
$564$	0	0
$565$	26.6733	1.12216
$566$	14.7915	0.621734
$567$	0	0
$568$	7.47372	0.313591
$569$	−11.2599	−0.472039	−0.236020	−	0.971748i	$-0.575843\pi$
$569$	−11.2599	−0.472039	−0.236020	+	0.971748i	$0.575843\pi$
$570$	0	0
$571$	−31.1893	−1.30523	−0.652616	−	0.757689i	$-0.726328\pi$
$571$	−31.1893	−1.30523	−0.652616	+	0.757689i	$0.726328\pi$
$572$	17.9881	0.752119
$573$	0	0
$574$	−27.3144	−1.14008
$575$	3.38208	0.141043
$576$	0	0
$577$	2.51403	0.104660	0.0523302	−	0.998630i	$-0.483335\pi$
$577$	2.51403	0.104660	0.0523302	+	0.998630i	$0.483335\pi$
$578$	30.1310	1.25328
$579$	0	0
$580$	21.0218	0.872884
$581$	−2.99850	−0.124399
$582$	0	0
$583$	−51.5146	−2.13352
$584$	−12.4607	−0.515626
$585$	0	0
$586$	12.7556	0.526929
$587$	16.1608	0.667026	0.333513	−	0.942746i	$-0.391766\pi$
$587$	16.1608	0.667026	0.333513	+	0.942746i	$0.391766\pi$
$588$	0	0
$589$	−3.29729	−0.135862
$590$	−27.5277	−1.13330
$591$	0	0
$592$	−14.7119	−0.604655
$593$	11.4918	0.471912	0.235956	−	0.971764i	$-0.424178\pi$
$593$	11.4918	0.471912	0.235956	+	0.971764i	$0.424178\pi$
$594$	0	0
$595$	−2.30773	−0.0946076
$596$	−20.0124	−0.819738
$597$	0	0
$598$	−23.8111	−0.973707
$599$	24.5532	1.00322	0.501608	−	0.865095i	$-0.332742\pi$
$599$	24.5532	1.00322	0.501608	+	0.865095i	$0.332742\pi$
$600$	0	0
$601$	−46.5706	−1.89965	−0.949827	−	0.312777i	$-0.898741\pi$
$601$	−46.5706	−1.89965	−0.949827	+	0.312777i	$0.898741\pi$
$602$	34.8160	1.41899
$603$	0	0
$604$	−17.9704	−0.731204
$605$	−62.1044	−2.52490
$606$	0	0
$607$	−27.3430	−1.10982	−0.554909	−	0.831911i	$-0.687247\pi$
$607$	−27.3430	−1.10982	−0.554909	+	0.831911i	$0.687247\pi$
$608$	−44.7923	−1.81657
$609$	0	0
$610$	22.6909	0.918727
$611$	9.46329	0.382844
$612$	0	0
$613$	37.3092	1.50691	0.753453	−	0.657502i	$-0.228387\pi$
$613$	37.3092	1.50691	0.753453	+	0.657502i	$0.228387\pi$
$614$	17.8231	0.719282
$615$	0	0
$616$	−14.1441	−0.569882
$617$	−36.4668	−1.46810	−0.734049	−	0.679096i	$-0.762372\pi$
$617$	−36.4668	−1.46810	−0.734049	+	0.679096i	$0.762372\pi$
$618$	0	0
$619$	43.5639	1.75098	0.875491	−	0.483234i	$-0.160538\pi$
$619$	43.5639	1.75098	0.875491	+	0.483234i	$0.160538\pi$
$620$	−1.27533	−0.0512185
$621$	0	0
$622$	4.90421	0.196641
$623$	19.2747	0.772225
$624$	0	0
$625$	−21.8670	−0.874679
$626$	19.9528	0.797474
$627$	0	0
$628$	19.4707	0.776967
$629$	1.89055	0.0753810
$630$	0	0
$631$	42.4843	1.69127	0.845637	−	0.533759i	$-0.179221\pi$
$631$	42.4843	1.69127	0.845637	+	0.533759i	$0.179221\pi$
$632$	2.07743	0.0826358
$633$	0	0
$634$	43.1815	1.71496
$635$	29.6168	1.17531
$636$	0	0
$637$	8.71472	0.345290
$638$	−89.0570	−3.52580
$639$	0	0
$640$	20.2439	0.800210
$641$	7.90513	0.312234	0.156117	−	0.987739i	$-0.450102\pi$
$641$	7.90513	0.312234	0.156117	+	0.987739i	$0.450102\pi$
$642$	0	0
$643$	−36.7190	−1.44806	−0.724028	−	0.689771i	$-0.757711\pi$
$643$	−36.7190	−1.44806	−0.724028	+	0.689771i	$0.757711\pi$
$644$	−13.4853	−0.531396
$645$	0	0
$646$	8.07659	0.317769
$647$	37.2473	1.46434	0.732172	−	0.681120i	$-0.238507\pi$
$647$	37.2473	1.46434	0.732172	+	0.681120i	$0.238507\pi$
$648$	0	0
$649$	45.8147	1.79839
$650$	−2.23293	−0.0875828
$651$	0	0
$652$	−18.9209	−0.740998
$653$	8.25286	0.322959	0.161480	−	0.986876i	$-0.448373\pi$
$653$	8.25286	0.322959	0.161480	+	0.986876i	$0.448373\pi$
$654$	0	0
$655$	−16.6458	−0.650403
$656$	−42.6159	−1.66387
$657$	0	0
$658$	13.6423	0.531831
$659$	1.05840	0.0412295	0.0206147	−	0.999787i	$-0.493438\pi$
$659$	1.05840	0.0412295	0.0206147	+	0.999787i	$0.493438\pi$
$660$	0	0
$661$	17.5566	0.682872	0.341436	−	0.939905i	$-0.389087\pi$
$661$	17.5566	0.682872	0.341436	+	0.939905i	$0.389087\pi$
$662$	21.2165	0.824601
$663$	0	0
$664$	−2.21391	−0.0859163
$665$	25.7587	0.998880
$666$	0	0
$667$	46.3127	1.79323
$668$	−5.45552	−0.211081
$669$	0	0
$670$	−8.45792	−0.326758
$671$	−37.7647	−1.45789
$672$	0	0
$673$	8.97715	0.346044	0.173022	−	0.984918i	$-0.444647\pi$
$673$	8.97715	0.346044	0.173022	+	0.984918i	$0.444647\pi$
$674$	−3.24443	−0.124971
$675$	0	0
$676$	−10.6477	−0.409527
$677$	−1.53938	−0.0591633	−0.0295817	−	0.999562i	$-0.509418\pi$
$677$	−1.53938	−0.0591633	−0.0295817	+	0.999562i	$0.509418\pi$
$678$	0	0
$679$	13.8390	0.531092
$680$	−1.70388	−0.0653410
$681$	0	0
$682$	5.40282	0.206885
$683$	23.4907	0.898848	0.449424	−	0.893319i	$-0.351629\pi$
$683$	23.4907	0.898848	0.449424	+	0.893319i	$0.351629\pi$
$684$	0	0
$685$	23.6204	0.902491
$686$	34.6080	1.32134
$687$	0	0
$688$	54.3199	2.07093
$689$	−17.6870	−0.673820
$690$	0	0
$691$	−8.94518	−0.340291	−0.170145	−	0.985419i	$-0.554424\pi$
$691$	−8.94518	−0.340291	−0.170145	+	0.985419i	$0.554424\pi$
$692$	−25.0603	−0.952651
$693$	0	0
$694$	21.7770	0.826645
$695$	17.1375	0.650062
$696$	0	0
$697$	5.47634	0.207431
$698$	25.8682	0.979126
$699$	0	0
$700$	−1.26461	−0.0477978
$701$	−2.19846	−0.0830347	−0.0415174	−	0.999138i	$-0.513219\pi$
$701$	−2.19846	−0.0830347	−0.0415174	+	0.999138i	$0.513219\pi$
$702$	0	0
$703$	−21.1022	−0.795883
$704$	10.8690	0.409642
$705$	0	0
$706$	−50.2602	−1.89157
$707$	9.33081	0.350921
$708$	0	0
$709$	−33.8056	−1.26960	−0.634799	−	0.772677i	$-0.718917\pi$
$709$	−33.8056	−1.26960	−0.634799	+	0.772677i	$0.718917\pi$
$710$	22.3023	0.836989
$711$	0	0
$712$	14.2313	0.533339
$713$	−2.80965	−0.105222
$714$	0	0
$715$	−29.2781	−1.09494
$716$	−3.43456	−0.128356
$717$	0	0
$718$	−42.2327	−1.57611
$719$	−15.9502	−0.594843	−0.297422	−	0.954746i	$-0.596127\pi$
$719$	−15.9502	−0.594843	−0.297422	+	0.954746i	$0.596127\pi$
$720$	0	0
$721$	9.34664	0.348087
$722$	−55.6658	−2.07167
$723$	0	0
$724$	25.5749	0.950483
$725$	4.34306	0.161297
$726$	0	0
$727$	8.83593	0.327707	0.163853	−	0.986485i	$-0.447608\pi$
$727$	8.83593	0.327707	0.163853	+	0.986485i	$0.447608\pi$
$728$	−4.85622	−0.179984
$729$	0	0
$730$	−37.1838	−1.37623
$731$	−6.98036	−0.258178
$732$	0	0
$733$	1.49567	0.0552437	0.0276219	−	0.999618i	$-0.491207\pi$
$733$	1.49567	0.0552437	0.0276219	+	0.999618i	$0.491207\pi$
$734$	−51.4474	−1.89896
$735$	0	0
$736$	−38.1680	−1.40689
$737$	14.0766	0.518519
$738$	0	0
$739$	−2.77630	−0.102128	−0.0510639	−	0.998695i	$-0.516261\pi$
$739$	−2.77630	−0.102128	−0.0510639	+	0.998695i	$0.516261\pi$
$740$	−8.16193	−0.300038
$741$	0	0
$742$	−25.4975	−0.936044
$743$	−44.6921	−1.63959	−0.819797	−	0.572654i	$-0.805914\pi$
$743$	−44.6921	−1.63959	−0.819797	+	0.572654i	$0.805914\pi$
$744$	0	0
$745$	32.5729	1.19338
$746$	−32.4584	−1.18838
$747$	0	0
$748$	−5.19910	−0.190098
$749$	−27.5457	−1.00650
$750$	0	0
$751$	−0.221622	−0.00808710	−0.00404355	−	0.999992i	$-0.501287\pi$
$751$	−0.221622	−0.00808710	−0.00404355	+	0.999992i	$0.501287\pi$
$752$	21.2847	0.776173
$753$	0	0
$754$	−30.5767	−1.11354
$755$	29.2493	1.06449
$756$	0	0
$757$	−4.76369	−0.173139	−0.0865696	−	0.996246i	$-0.527590\pi$
$757$	−4.76369	−0.173139	−0.0865696	+	0.996246i	$0.527590\pi$
$758$	−39.5648	−1.43706
$759$	0	0
$760$	19.0187	0.689879
$761$	−12.3929	−0.449243	−0.224621	−	0.974446i	$-0.572115\pi$
$761$	−12.3929	−0.449243	−0.224621	+	0.974446i	$0.572115\pi$
$762$	0	0
$763$	−1.51235	−0.0547507
$764$	−14.8368	−0.536777
$765$	0	0
$766$	−64.4239	−2.32773
$767$	15.7300	0.567977
$768$	0	0
$769$	49.4167	1.78201	0.891005	−	0.453993i	$-0.150001\pi$
$769$	49.4167	1.78201	0.891005	+	0.453993i	$0.150001\pi$
$770$	−42.2073	−1.52105
$771$	0	0
$772$	26.2679	0.945402
$773$	−4.87037	−0.175175	−0.0875876	−	0.996157i	$-0.527916\pi$
$773$	−4.87037	−0.175175	−0.0875876	+	0.996157i	$0.527916\pi$
$774$	0	0
$775$	−0.263480	−0.00946450
$776$	10.2179	0.366800
$777$	0	0
$778$	−22.2514	−0.797752
$779$	−61.1266	−2.19009
$780$	0	0
$781$	−37.1179	−1.32818
$782$	6.88213	0.246105
$783$	0	0
$784$	19.6010	0.700036
$785$	−31.6913	−1.13111
$786$	0	0
$787$	15.7824	0.562582	0.281291	−	0.959622i	$-0.409237\pi$
$787$	15.7824	0.562582	0.281291	+	0.959622i	$0.409237\pi$
$788$	3.26040	0.116147
$789$	0	0
$790$	6.19924	0.220559
$791$	−21.9725	−0.781253
$792$	0	0
$793$	−12.9661	−0.460440
$794$	66.0409	2.34370
$795$	0	0
$796$	3.80427	0.134839
$797$	−19.4658	−0.689515	−0.344757	−	0.938692i	$-0.612039\pi$
$797$	−19.4658	−0.689515	−0.344757	+	0.938692i	$0.612039\pi$
$798$	0	0
$799$	−2.73518	−0.0967638
$800$	−3.57928	−0.126547
$801$	0	0
$802$	54.9220	1.93936
$803$	61.8854	2.18389
$804$	0	0
$805$	21.9492	0.773609
$806$	1.85500	0.0653395
$807$	0	0
$808$	6.88930	0.242365
$809$	−0.500555	−0.0175986	−0.00879929	−	0.999961i	$-0.502801\pi$
$809$	−0.500555	−0.0175986	−0.00879929	+	0.999961i	$0.502801\pi$
$810$	0	0
$811$	17.1329	0.601618	0.300809	−	0.953684i	$-0.402743\pi$
$811$	17.1329	0.601618	0.300809	+	0.953684i	$0.402743\pi$
$812$	−17.3170	−0.607708
$813$	0	0
$814$	34.5772	1.21193
$815$	30.7963	1.07875
$816$	0	0
$817$	77.9144	2.72588
$818$	52.1940	1.82492
$819$	0	0
$820$	−23.6426	−0.825637
$821$	−2.73557	−0.0954722	−0.0477361	−	0.998860i	$-0.515201\pi$
$821$	−2.73557	−0.0954722	−0.0477361	+	0.998860i	$0.515201\pi$
$822$	0	0
$823$	−12.9005	−0.449683	−0.224841	−	0.974395i	$-0.572186\pi$
$823$	−12.9005	−0.449683	−0.224841	+	0.974395i	$0.572186\pi$
$824$	6.90099	0.240407
$825$	0	0
$826$	22.6764	0.789011
$827$	−0.345192	−0.0120035	−0.00600174	−	0.999982i	$-0.501910\pi$
$827$	−0.345192	−0.0120035	−0.00600174	+	0.999982i	$0.501910\pi$
$828$	0	0
$829$	−38.8024	−1.34766	−0.673832	−	0.738885i	$-0.735353\pi$
$829$	−38.8024	−1.34766	−0.673832	+	0.738885i	$0.735353\pi$
$830$	−6.60650	−0.229315
$831$	0	0
$832$	3.73176	0.129375
$833$	−2.51882	−0.0872720
$834$	0	0
$835$	8.87963	0.307292
$836$	58.0321	2.00708
$837$	0	0
$838$	−36.3069	−1.25420
$839$	6.45512	0.222855	0.111428	−	0.993773i	$-0.464458\pi$
$839$	6.45512	0.222855	0.111428	+	0.993773i	$0.464458\pi$
$840$	0	0
$841$	30.4719	1.05076
$842$	39.8452	1.37316
$843$	0	0
$844$	17.7020	0.609328
$845$	17.3306	0.596191
$846$	0	0
$847$	51.1594	1.75786
$848$	−39.7813	−1.36609
$849$	0	0
$850$	0.645386	0.0221366
$851$	−17.9814	−0.616393
$852$	0	0
$853$	2.58663	0.0885647	0.0442823	−	0.999019i	$-0.485900\pi$
$853$	2.58663	0.0885647	0.0442823	+	0.999019i	$0.485900\pi$
$854$	−18.6919	−0.639624
$855$	0	0
$856$	−20.3380	−0.695140
$857$	−25.1394	−0.858745	−0.429373	−	0.903127i	$-0.641265\pi$
$857$	−25.1394	−0.858745	−0.429373	+	0.903127i	$0.641265\pi$
$858$	0	0
$859$	−49.8581	−1.70113	−0.850567	−	0.525866i	$-0.823741\pi$
$859$	−49.8581	−1.70113	−0.850567	+	0.525866i	$0.823741\pi$
$860$	30.1358	1.02762
$861$	0	0
$862$	−46.4217	−1.58113
$863$	−29.5886	−1.00721	−0.503605	−	0.863934i	$-0.667993\pi$
$863$	−29.5886	−1.00721	−0.503605	+	0.863934i	$0.667993\pi$
$864$	0	0
$865$	40.7892	1.38687
$866$	4.12005	0.140005
$867$	0	0
$868$	1.05057	0.0356587
$869$	−10.3175	−0.349997
$870$	0	0
$871$	4.83305	0.163762
$872$	−1.11663	−0.0378137
$873$	0	0
$874$	−76.8180	−2.59841
$875$	20.3328	0.687375
$876$	0	0
$877$	−11.0963	−0.374697	−0.187348	−	0.982294i	$-0.559989\pi$
$877$	−11.0963	−0.374697	−0.187348	+	0.982294i	$0.559989\pi$
$878$	30.6956	1.03593
$879$	0	0
$880$	−65.8518	−2.21986
$881$	−9.44078	−0.318068	−0.159034	−	0.987273i	$-0.550838\pi$
$881$	−9.44078	−0.318068	−0.159034	+	0.987273i	$0.550838\pi$
$882$	0	0
$883$	−5.46071	−0.183767	−0.0918837	−	0.995770i	$-0.529289\pi$
$883$	−5.46071	−0.183767	−0.0918837	+	0.995770i	$0.529289\pi$
$884$	−1.78506	−0.0600379
$885$	0	0
$886$	62.7960	2.10967
$887$	−38.6891	−1.29905	−0.649527	−	0.760338i	$-0.725033\pi$
$887$	−38.6891	−1.29905	−0.649527	+	0.760338i	$0.725033\pi$
$888$	0	0
$889$	−24.3972	−0.818257
$890$	42.4674	1.42351
$891$	0	0
$892$	2.67839	0.0896792
$893$	30.5299	1.02165
$894$	0	0
$895$	5.59023	0.186861
$896$	−16.6762	−0.557112
$897$	0	0
$898$	23.7754	0.793396
$899$	−3.60798	−0.120333
$900$	0	0
$901$	5.11208	0.170308
$902$	100.160	3.33496
$903$	0	0
$904$	−16.2232	−0.539575
$905$	−41.6267	−1.38372
$906$	0	0
$907$	−7.01063	−0.232784	−0.116392	−	0.993203i	$-0.537133\pi$
$907$	−7.01063	−0.232784	−0.116392	+	0.993203i	$0.537133\pi$
$908$	20.9269	0.694485
$909$	0	0
$910$	−14.4914	−0.480386
$911$	29.9470	0.992187	0.496093	−	0.868269i	$-0.334767\pi$
$911$	29.9470	0.992187	0.496093	+	0.868269i	$0.334767\pi$
$912$	0	0
$913$	10.9953	0.363891
$914$	−14.5480	−0.481207
$915$	0	0
$916$	31.1896	1.03053
$917$	13.7122	0.452816
$918$	0	0
$919$	0.754207	0.0248790	0.0124395	−	0.999923i	$-0.496040\pi$
$919$	0.754207	0.0248790	0.0124395	+	0.999923i	$0.496040\pi$
$920$	16.2060	0.534295
$921$	0	0
$922$	−27.5637	−0.907761
$923$	−12.7440	−0.419475
$924$	0	0
$925$	−1.68624	−0.0554431
$926$	−62.8799	−2.06636
$927$	0	0
$928$	−49.0130	−1.60893
$929$	−24.4277	−0.801448	−0.400724	−	0.916199i	$-0.631241\pi$
$929$	−24.4277	−0.801448	−0.400724	+	0.916199i	$0.631241\pi$
$930$	0	0
$931$	28.1149	0.921430
$932$	6.81988	0.223392
$933$	0	0
$934$	−67.4715	−2.20774
$935$	8.46227	0.276746
$936$	0	0
$937$	44.1224	1.44141	0.720707	−	0.693239i	$-0.243817\pi$
$937$	44.1224	1.44141	0.720707	+	0.693239i	$0.243817\pi$
$938$	6.96733	0.227491
$939$	0	0
$940$	11.8084	0.385148
$941$	−2.98022	−0.0971523	−0.0485762	−	0.998819i	$-0.515468\pi$
$941$	−2.98022	−0.0971523	−0.0485762	+	0.998819i	$0.515468\pi$
$942$	0	0
$943$	−52.0866	−1.69617
$944$	35.3797	1.15151
$945$	0	0
$946$	−127.668	−4.15083
$947$	−16.6297	−0.540394	−0.270197	−	0.962805i	$-0.587089\pi$
$947$	−16.6297	−0.540394	−0.270197	+	0.962805i	$0.587089\pi$
$948$	0	0
$949$	21.2477	0.689729
$950$	−7.20376	−0.233721
$951$	0	0
$952$	1.40360	0.0454909
$953$	20.8876	0.676615	0.338307	−	0.941036i	$-0.390146\pi$
$953$	20.8876	0.676615	0.338307	+	0.941036i	$0.390146\pi$
$954$	0	0
$955$	24.1490	0.781444
$956$	−7.36526	−0.238210
$957$	0	0
$958$	51.7034	1.67046
$959$	−19.4577	−0.628321
$960$	0	0
$961$	−30.7811	−0.992939
$962$	11.8717	0.382760
$963$	0	0
$964$	30.7020	0.988845
$965$	−42.7547	−1.37632
$966$	0	0
$967$	8.51761	0.273908	0.136954	−	0.990577i	$-0.456269\pi$
$967$	8.51761	0.273908	0.136954	+	0.990577i	$0.456269\pi$
$968$	37.7730	1.21407
$969$	0	0
$970$	30.4910	0.979008
$971$	5.67027	0.181968	0.0909838	−	0.995852i	$-0.470999\pi$
$971$	5.67027	0.181968	0.0909838	+	0.995852i	$0.470999\pi$
$972$	0	0
$973$	−14.1172	−0.452578
$974$	−27.0735	−0.867490
$975$	0	0
$976$	−29.1631	−0.933489
$977$	−16.0162	−0.512404	−0.256202	−	0.966623i	$-0.582471\pi$
$977$	−16.0162	−0.512404	−0.256202	+	0.966623i	$0.582471\pi$
$978$	0	0
$979$	−70.6790	−2.25891
$980$	10.8743	0.347368
$981$	0	0
$982$	−61.1125	−1.95018
$983$	−56.1584	−1.79117	−0.895587	−	0.444886i	$-0.853244\pi$
$983$	−56.1584	−1.79117	−0.895587	+	0.444886i	$0.853244\pi$
$984$	0	0
$985$	−5.30676	−0.169087
$986$	8.83762	0.281447
$987$	0	0
$988$	19.9247	0.633888
$989$	66.3916	2.11113
$990$	0	0
$991$	4.22053	0.134069	0.0670347	−	0.997751i	$-0.478646\pi$
$991$	4.22053	0.134069	0.0670347	+	0.997751i	$0.478646\pi$
$992$	2.97347	0.0944077
$993$	0	0
$994$	−18.3718	−0.582718
$995$	−6.19198	−0.196299
$996$	0	0
$997$	−7.81561	−0.247523	−0.123761	−	0.992312i	$-0.539496\pi$
$997$	−7.81561	−0.247523	−0.123761	+	0.992312i	$0.539496\pi$
$998$	38.8857	1.23091
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.o.1.8		26
3.2	odd	2		503.2.a.f.1.19	✓	26
12.11	even	2		8048.2.a.u.1.12		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.19	✓	26	3.2	odd	2
4527.2.a.o.1.8		26	1.1	even	1	trivial
8048.2.a.u.1.12		26	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:	\( N \)	\(=\)	\( 4527 = 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4527.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81497 q^{2} +1.29413 q^{4} -2.10638 q^{5} +1.73516 q^{7} +1.28114 q^{8} +O(q^{10})\)	\(q-1.81497 q^{2} +1.29413 q^{4} -2.10638 q^{5} +1.73516 q^{7} +1.28114 q^{8} +3.82302 q^{10} -6.36270 q^{11} -2.18456 q^{13} -3.14927 q^{14} -4.91349 q^{16} +0.631406 q^{17} -7.04772 q^{19} -2.72593 q^{20} +11.5481 q^{22} -6.00543 q^{23} -0.563171 q^{25} +3.96493 q^{26} +2.24552 q^{28} -7.71180 q^{29} +0.467852 q^{31} +6.35558 q^{32} -1.14599 q^{34} -3.65490 q^{35} +2.99418 q^{37} +12.7914 q^{38} -2.69855 q^{40} +8.67325 q^{41} -11.0553 q^{43} -8.23417 q^{44} +10.8997 q^{46} -4.33189 q^{47} -3.98923 q^{49} +1.02214 q^{50} -2.82711 q^{52} +8.09634 q^{53} +13.4023 q^{55} +2.22297 q^{56} +13.9967 q^{58} -7.20052 q^{59} +5.93532 q^{61} -0.849139 q^{62} -1.70824 q^{64} +4.60152 q^{65} -2.21237 q^{67} +0.817122 q^{68} +6.63355 q^{70} +5.83367 q^{71} -9.72628 q^{73} -5.43437 q^{74} -9.12067 q^{76} -11.0403 q^{77} +1.62156 q^{79} +10.3497 q^{80} -15.7417 q^{82} -1.72808 q^{83} -1.32998 q^{85} +20.0650 q^{86} -8.15148 q^{88} +11.1083 q^{89} -3.79056 q^{91} -7.77181 q^{92} +7.86227 q^{94} +14.8452 q^{95} +7.97564 q^{97} +7.24034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{2} + 36 q^{4} - 9 q^{5} + 11 q^{7} - 18 q^{8}+O(q^{10}) \) 26 * q - 4 * q^2 + 36 * q^4 - 9 * q^5 + 11 * q^7 - 18 * q^8	\( 26 q - 4 q^{2} + 36 q^{4} - 9 q^{5} + 11 q^{7} - 18 q^{8} + 4 q^{10} + 17 q^{11} + 14 q^{13} - q^{14} + 48 q^{16} - 17 q^{17} - 22 q^{19} + 19 q^{20} + 38 q^{22} - 27 q^{23} + 93 q^{25} - q^{26} - 9 q^{28} - 13 q^{29} + 26 q^{31} - 5 q^{32} - 32 q^{34} + 22 q^{35} + 55 q^{37} + 24 q^{38} - 7 q^{40} - 24 q^{41} + 20 q^{43} + 27 q^{44} + 6 q^{46} + 25 q^{47} + 65 q^{49} + 16 q^{50} + 32 q^{52} - 30 q^{53} + 25 q^{55} - 3 q^{56} + 58 q^{58} + 26 q^{59} + 15 q^{61} + 12 q^{62} + 44 q^{64} - 20 q^{65} - 20 q^{67} + 4 q^{68} + 2 q^{70} + 35 q^{71} + 38 q^{73} + 59 q^{74} - 42 q^{76} + 6 q^{77} + 21 q^{79} + 100 q^{80} - 59 q^{82} + 48 q^{83} + 6 q^{85} + 7 q^{86} + 106 q^{88} + 5 q^{89} - 24 q^{91} - 26 q^{92} - 22 q^{94} - 43 q^{95} + 142 q^{97} + 38 q^{98}+O(q^{100}) \) 26 * q - 4 * q^2 + 36 * q^4 - 9 * q^5 + 11 * q^7 - 18 * q^8 + 4 * q^10 + 17 * q^11 + 14 * q^13 - q^14 + 48 * q^16 - 17 * q^17 - 22 * q^19 + 19 * q^20 + 38 * q^22 - 27 * q^23 + 93 * q^25 - q^26 - 9 * q^28 - 13 * q^29 + 26 * q^31 - 5 * q^32 - 32 * q^34 + 22 * q^35 + 55 * q^37 + 24 * q^38 - 7 * q^40 - 24 * q^41 + 20 * q^43 + 27 * q^44 + 6 * q^46 + 25 * q^47 + 65 * q^49 + 16 * q^50 + 32 * q^52 - 30 * q^53 + 25 * q^55 - 3 * q^56 + 58 * q^58 + 26 * q^59 + 15 * q^61 + 12 * q^62 + 44 * q^64 - 20 * q^65 - 20 * q^67 + 4 * q^68 + 2 * q^70 + 35 * q^71 + 38 * q^73 + 59 * q^74 - 42 * q^76 + 6 * q^77 + 21 * q^79 + 100 * q^80 - 59 * q^82 + 48 * q^83 + 6 * q^85 + 7 * q^86 + 106 * q^88 + 5 * q^89 - 24 * q^91 - 26 * q^92 - 22 * q^94 - 43 * q^95 + 142 * q^97 + 38 * q^98

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