LMFDB - Embedded newform 5445.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.737640$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +O(q^{10})$	$q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +2.45589 q^{10} +0.313133 q^{13} -8.07211 q^{14} +4.18926 q^{16} +5.00000 q^{17} +7.45408 q^{19} +4.03138 q^{20} -1.07392 q^{23} +1.00000 q^{25} +0.769020 q^{26} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} +0.310680 q^{32} +12.2794 q^{34} -3.28684 q^{35} +2.63428 q^{37} +18.3064 q^{38} +4.98884 q^{40} +10.8472 q^{41} +5.51468 q^{43} -2.63743 q^{46} +11.9982 q^{47} +3.80333 q^{49} +2.45589 q^{50} +1.26236 q^{52} -4.93543 q^{53} -16.3975 q^{56} -12.3690 q^{58} +9.16409 q^{59} -9.18431 q^{61} +8.47033 q^{62} -7.61553 q^{64} +0.313133 q^{65} -15.2739 q^{67} +20.1569 q^{68} -8.07211 q^{70} -3.07211 q^{71} +8.65269 q^{73} +6.46950 q^{74} +30.0502 q^{76} -5.41446 q^{79} +4.18926 q^{80} +26.6395 q^{82} +16.2454 q^{83} +5.00000 q^{85} +13.5434 q^{86} -1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} +29.4662 q^{94} +7.45408 q^{95} +0.224082 q^{97} +9.34054 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) $ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8	$ 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100}) $ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 + 5 * q^10 - 3 * q^13 + 5 * q^14 + 15 * q^16 + 20 * q^17 - 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 - 3 * q^28 + 5 * q^29 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 - 7 * q^37 - q^38 + 15 * q^40 + 20 * q^41 + 2 * q^43 - 7 * q^46 + 20 * q^47 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 + 7 * q^61 - 12 * q^62 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 + 5 * q^70 + 25 * q^71 - 23 * q^73 - 7 * q^74 + 7 * q^76 + 15 * q^80 + 11 * q^82 + 33 * q^83 + 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 + 17 * q^94 - 3 * q^95 + 25 * 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 $

$2$	2.45589	1.73657	0.868287	−	0.496062i	$-0.165221\pi$
$2$	2.45589	1.73657	0.868287	+	0.496062i	$0.165221\pi$
$3$	0	0
$3$	0	0
$4$	4.03138	2.01569
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.28684	−1.24231	−0.621155	−	0.783688i	$-0.713336\pi$
$7$	−3.28684	−1.24231	−0.621155	+	0.783688i	$0.713336\pi$
$8$	4.98884	1.76382
$9$	0	0
$10$	2.45589	0.776620
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.313133	0.0868476	0.0434238	−	0.999057i	$-0.486173\pi$
$13$	0.313133	0.0868476	0.0434238	+	0.999057i	$0.486173\pi$
$14$	−8.07211	−2.15736
$15$	0	0
$16$	4.18926	1.04732
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	7.45408	1.71008	0.855041	−	0.518560i	$-0.173532\pi$
$19$	7.45408	1.71008	0.855041	+	0.518560i	$0.173532\pi$
$20$	4.03138	0.901444
$21$	0	0
$22$	0	0
$23$	−1.07392	−0.223928	−0.111964	−	0.993712i	$-0.535714\pi$
$23$	−1.07392	−0.223928	−0.111964	+	0.993712i	$0.535714\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.769020	0.150817
$27$	0	0
$28$	−13.2505	−2.50411
$29$	−5.03647	−0.935249	−0.467624	−	0.883927i	$-0.654890\pi$
$29$	−5.03647	−0.935249	−0.467624	+	0.883927i	$0.654890\pi$
$30$	0	0
$31$	3.44899	0.619457	0.309728	−	0.950825i	$-0.399762\pi$
$31$	3.44899	0.619457	0.309728	+	0.950825i	$0.399762\pi$
$32$	0.310680	0.0549210
$33$	0	0
$34$	12.2794	2.10591
$35$	−3.28684	−0.555578
$36$	0	0
$37$	2.63428	0.433073	0.216537	−	0.976274i	$-0.430524\pi$
$37$	2.63428	0.433073	0.216537	+	0.976274i	$0.430524\pi$
$38$	18.3064	2.96969
$39$	0	0
$40$	4.98884	0.788805
$41$	10.8472	1.69405	0.847024	−	0.531554i	$-0.178392\pi$
$41$	10.8472	1.69405	0.847024	+	0.531554i	$0.178392\pi$
$42$	0	0
$43$	5.51468	0.840980	0.420490	−	0.907297i	$-0.361858\pi$
$43$	5.51468	0.840980	0.420490	+	0.907297i	$0.361858\pi$
$44$	0	0
$45$	0	0
$46$	−2.63743	−0.388868
$47$	11.9982	1.75012	0.875058	−	0.484018i	$-0.160823\pi$
$47$	11.9982	1.75012	0.875058	+	0.484018i	$0.160823\pi$
$48$	0	0
$49$	3.80333	0.543333
$50$	2.45589	0.347315
$51$	0	0
$52$	1.26236	0.175058
$53$	−4.93543	−0.677934	−0.338967	−	0.940798i	$-0.610077\pi$
$53$	−4.93543	−0.677934	−0.338967	+	0.940798i	$0.610077\pi$
$54$	0	0
$55$	0	0
$56$	−16.3975	−2.19121
$57$	0	0
$58$	−12.3690	−1.62413
$59$	9.16409	1.19306	0.596531	−	0.802590i	$-0.296545\pi$
$59$	9.16409	1.19306	0.596531	+	0.802590i	$0.296545\pi$
$60$	0	0
$61$	−9.18431	−1.17593	−0.587965	−	0.808886i	$-0.700071\pi$
$61$	−9.18431	−1.17593	−0.587965	+	0.808886i	$0.700071\pi$
$62$	8.47033	1.07573
$63$	0	0
$64$	−7.61553	−0.951942
$65$	0.313133	0.0388394
$66$	0	0
$67$	−15.2739	−1.86600	−0.933000	−	0.359876i	$-0.882819\pi$
$67$	−15.2739	−1.86600	−0.933000	+	0.359876i	$0.882819\pi$
$68$	20.1569	2.44438
$69$	0	0
$70$	−8.07211	−0.964802
$71$	−3.07211	−0.364593	−0.182296	−	0.983244i	$-0.558353\pi$
$71$	−3.07211	−0.364593	−0.182296	+	0.983244i	$0.558353\pi$
$72$	0	0
$73$	8.65269	1.01272	0.506361	−	0.862322i	$-0.330991\pi$
$73$	8.65269	1.01272	0.506361	+	0.862322i	$0.330991\pi$
$74$	6.46950	0.752064
$75$	0	0
$76$	30.0502	3.44700
$77$	0	0
$78$	0	0
$79$	−5.41446	−0.609175	−0.304587	−	0.952484i	$-0.598519\pi$
$79$	−5.41446	−0.609175	−0.304587	+	0.952484i	$0.598519\pi$
$80$	4.18926	0.468374
$81$	0	0
$82$	26.6395	2.94184
$83$	16.2454	1.78317	0.891583	−	0.452857i	$-0.149595\pi$
$83$	16.2454	1.78317	0.891583	+	0.452857i	$0.149595\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	13.5434	1.46042
$87$	0	0
$88$	0	0
$89$	−1.62118	−0.171845	−0.0859223	−	0.996302i	$-0.527384\pi$
$89$	−1.62118	−0.171845	−0.0859223	+	0.996302i	$0.527384\pi$
$90$	0	0
$91$	−1.02922	−0.107892
$92$	−4.32938	−0.451369
$93$	0	0
$94$	29.4662	3.03921
$95$	7.45408	0.764772
$96$	0	0
$97$	0.224082	0.0227521	0.0113760	−	0.999935i	$-0.496379\pi$
$97$	0.224082	0.0227521	0.0113760	+	0.999935i	$0.496379\pi$
$98$	9.34054	0.943537
$99$	0	0
$100$	4.03138	0.403138
$101$	−0.505326	−0.0502818	−0.0251409	−	0.999684i	$-0.508003\pi$
$101$	−0.505326	−0.0502818	−0.0251409	+	0.999684i	$0.508003\pi$
$102$	0	0
$103$	−6.40197	−0.630805	−0.315402	−	0.948958i	$-0.602139\pi$
$103$	−6.40197	−0.630805	−0.315402	+	0.948958i	$0.602139\pi$
$104$	1.56217	0.153184
$105$	0	0
$106$	−12.1209	−1.17728
$107$	2.09249	0.202289	0.101144	−	0.994872i	$-0.467750\pi$
$107$	2.09249	0.202289	0.101144	+	0.994872i	$0.467750\pi$
$108$	0	0
$109$	6.69278	0.641052	0.320526	−	0.947240i	$-0.396140\pi$
$109$	6.69278	0.641052	0.320526	+	0.947240i	$0.396140\pi$
$110$	0	0
$111$	0	0
$112$	−13.7694	−1.30109
$113$	10.7941	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$113$	10.7941	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$114$	0	0
$115$	−1.07392	−0.100144
$116$	−20.3039	−1.88517
$117$	0	0
$118$	22.5060	2.07184
$119$	−16.4342	−1.50652
$120$	0	0
$121$	0	0
$122$	−22.5556	−2.04209
$123$	0	0
$124$	13.9042	1.24863
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.0033	1.50880	0.754398	−	0.656417i	$-0.227929\pi$
$127$	17.0033	1.50880	0.754398	+	0.656417i	$0.227929\pi$
$128$	−19.3242	−1.70804
$129$	0	0
$130$	0.769020	0.0674475
$131$	−0.0430508	−0.00376136	−0.00188068	−	0.999998i	$-0.500599\pi$
$131$	−0.0430508	−0.00376136	−0.00188068	+	0.999998i	$0.500599\pi$
$132$	0	0
$133$	−24.5004	−2.12445
$134$	−37.5109	−3.24045
$135$	0	0
$136$	24.9442	2.13895
$137$	−7.36257	−0.629027	−0.314513	−	0.949253i	$-0.601841\pi$
$137$	−7.36257	−0.629027	−0.314513	+	0.949253i	$0.601841\pi$
$138$	0	0
$139$	−13.2393	−1.12295	−0.561473	−	0.827495i	$-0.689765\pi$
$139$	−13.2393	−1.12295	−0.561473	+	0.827495i	$0.689765\pi$
$140$	−13.2505	−1.11987
$141$	0	0
$142$	−7.54476	−0.633142
$143$	0	0
$144$	0	0
$145$	−5.03647	−0.418256
$146$	21.2500	1.75867
$147$	0	0
$148$	10.6198	0.872942
$149$	−5.87858	−0.481592	−0.240796	−	0.970576i	$-0.577409\pi$
$149$	−5.87858	−0.481592	−0.240796	+	0.970576i	$0.577409\pi$
$150$	0	0
$151$	−7.62821	−0.620775	−0.310387	−	0.950610i	$-0.600459\pi$
$151$	−7.62821	−0.620775	−0.310387	+	0.950610i	$0.600459\pi$
$152$	37.1872	3.01628
$153$	0	0
$154$	0	0
$155$	3.44899	0.277029
$156$	0	0
$157$	−10.1332	−0.808719	−0.404360	−	0.914600i	$-0.632506\pi$
$157$	−10.1332	−0.808719	−0.404360	+	0.914600i	$0.632506\pi$
$158$	−13.2973	−1.05788
$159$	0	0
$160$	0.310680	0.0245614
$161$	3.52981	0.278188
$162$	0	0
$163$	−5.02906	−0.393906	−0.196953	−	0.980413i	$-0.563105\pi$
$163$	−5.02906	−0.393906	−0.196953	+	0.980413i	$0.563105\pi$
$164$	43.7292	3.41468
$165$	0	0
$166$	39.8969	3.09660
$167$	5.79105	0.448125	0.224062	−	0.974575i	$-0.428068\pi$
$167$	5.79105	0.448125	0.224062	+	0.974575i	$0.428068\pi$
$168$	0	0
$169$	−12.9019	−0.992457
$170$	12.2794	0.941790
$171$	0	0
$172$	22.2318	1.69516
$173$	16.0652	1.22142	0.610708	−	0.791856i	$-0.290885\pi$
$173$	16.0652	1.22142	0.610708	+	0.791856i	$0.290885\pi$
$174$	0	0
$175$	−3.28684	−0.248462
$176$	0	0
$177$	0	0
$178$	−3.98143	−0.298421
$179$	8.30309	0.620602	0.310301	−	0.950638i	$-0.399570\pi$
$179$	8.30309	0.620602	0.310301	+	0.950638i	$0.399570\pi$
$180$	0	0
$181$	−6.46425	−0.480484	−0.240242	−	0.970713i	$-0.577227\pi$
$181$	−6.46425	−0.480484	−0.240242	+	0.970713i	$0.577227\pi$
$182$	−2.52765	−0.187362
$183$	0	0
$184$	−5.35762	−0.394969
$185$	2.63428	0.193676
$186$	0	0
$187$	0	0
$188$	48.3693	3.52769
$189$	0	0
$190$	18.3064	1.32808
$191$	−15.3693	−1.11208	−0.556041	−	0.831155i	$-0.687680\pi$
$191$	−15.3693	−1.11208	−0.556041	+	0.831155i	$0.687680\pi$
$192$	0	0
$193$	−15.7518	−1.13384	−0.566919	−	0.823773i	$-0.691865\pi$
$193$	−15.7518	−1.13384	−0.566919	+	0.823773i	$0.691865\pi$
$194$	0.550320	0.0395107
$195$	0	0
$196$	15.3327	1.09519
$197$	16.3940	1.16802	0.584010	−	0.811746i	$-0.301483\pi$
$197$	16.3940	1.16802	0.584010	+	0.811746i	$0.301483\pi$
$198$	0	0
$199$	6.96500	0.493736	0.246868	−	0.969049i	$-0.420599\pi$
$199$	6.96500	0.493736	0.246868	+	0.969049i	$0.420599\pi$
$200$	4.98884	0.352764
$201$	0	0
$202$	−1.24102	−0.0873180
$203$	16.5541	1.16187
$204$	0	0
$205$	10.8472	0.757602
$206$	−15.7225	−1.09544
$207$	0	0
$208$	1.31180	0.0909569
$209$	0	0
$210$	0	0
$211$	19.9531	1.37363	0.686814	−	0.726833i	$-0.259008\pi$
$211$	19.9531	1.37363	0.686814	+	0.726833i	$0.259008\pi$
$212$	−19.8966	−1.36650
$213$	0	0
$214$	5.13892	0.351289
$215$	5.51468	0.376098
$216$	0	0
$217$	−11.3363	−0.769557
$218$	16.4367	1.11323
$219$	0	0
$220$	0	0
$221$	1.56567	0.105318
$222$	0	0
$223$	−20.1466	−1.34912	−0.674559	−	0.738221i	$-0.735666\pi$
$223$	−20.1466	−1.34912	−0.674559	+	0.738221i	$0.735666\pi$
$224$	−1.02116	−0.0682289
$225$	0	0
$226$	26.5091	1.76336
$227$	0.533937	0.0354386	0.0177193	−	0.999843i	$-0.494359\pi$
$227$	0.533937	0.0354386	0.0177193	+	0.999843i	$0.494359\pi$
$228$	0	0
$229$	−22.1931	−1.46656	−0.733279	−	0.679928i	$-0.762011\pi$
$229$	−22.1931	−1.46656	−0.733279	+	0.679928i	$0.762011\pi$
$230$	−2.63743	−0.173907
$231$	0	0
$232$	−25.1261	−1.64961
$233$	−4.56567	−0.299107	−0.149553	−	0.988754i	$-0.547784\pi$
$233$	−4.56567	−0.299107	−0.149553	+	0.988754i	$0.547784\pi$
$234$	0	0
$235$	11.9982	0.782676
$236$	36.9439	2.40485
$237$	0	0
$238$	−40.3606	−2.61619
$239$	−5.86053	−0.379086	−0.189543	−	0.981872i	$-0.560701\pi$
$239$	−5.86053	−0.379086	−0.189543	+	0.981872i	$0.560701\pi$
$240$	0	0
$241$	−9.96074	−0.641628	−0.320814	−	0.947142i	$-0.603956\pi$
$241$	−9.96074	−0.641628	−0.320814	+	0.947142i	$0.603956\pi$
$242$	0	0
$243$	0	0
$244$	−37.0254	−2.37031
$245$	3.80333	0.242986
$246$	0	0
$247$	2.33412	0.148517
$248$	17.2065	1.09261
$249$	0	0
$250$	2.45589	0.155324
$251$	16.8788	1.06538	0.532690	−	0.846310i	$-0.321181\pi$
$251$	16.8788	1.06538	0.532690	+	0.846310i	$0.321181\pi$
$252$	0	0
$253$	0	0
$254$	41.7581	2.62014
$255$	0	0
$256$	−32.2271	−2.01419
$257$	−11.1436	−0.695117	−0.347559	−	0.937658i	$-0.612989\pi$
$257$	−11.1436	−0.695117	−0.347559	+	0.937658i	$0.612989\pi$
$258$	0	0
$259$	−8.65847	−0.538011
$260$	1.26236	0.0782882
$261$	0	0
$262$	−0.105728	−0.00653189
$263$	−26.8726	−1.65704	−0.828519	−	0.559961i	$-0.810816\pi$
$263$	−26.8726	−1.65704	−0.828519	+	0.559961i	$0.810816\pi$
$264$	0	0
$265$	−4.93543	−0.303181
$266$	−60.1701	−3.68927
$267$	0	0
$268$	−61.5748	−3.76128
$269$	10.0629	0.613545	0.306773	−	0.951783i	$-0.400751\pi$
$269$	10.0629	0.613545	0.306773	+	0.951783i	$0.400751\pi$
$270$	0	0
$271$	−10.5441	−0.640509	−0.320255	−	0.947331i	$-0.603768\pi$
$271$	−10.5441	−0.640509	−0.320255	+	0.947331i	$0.603768\pi$
$272$	20.9463	1.27006
$273$	0	0
$274$	−18.0816	−1.09235
$275$	0	0
$276$	0	0
$277$	17.9376	1.07777	0.538883	−	0.842381i	$-0.318847\pi$
$277$	17.9376	1.07777	0.538883	+	0.842381i	$0.318847\pi$
$278$	−32.5143	−1.95008
$279$	0	0
$280$	−16.3975	−0.979939
$281$	7.41103	0.442105	0.221052	−	0.975262i	$-0.429051\pi$
$281$	7.41103	0.442105	0.221052	+	0.975262i	$0.429051\pi$
$282$	0	0
$283$	−5.38684	−0.320214	−0.160107	−	0.987100i	$-0.551184\pi$
$283$	−5.38684	−0.320214	−0.160107	+	0.987100i	$0.551184\pi$
$284$	−12.3848	−0.734905
$285$	0	0
$286$	0	0
$287$	−35.6530	−2.10453
$288$	0	0
$289$	8.00000	0.470588
$290$	−12.3690	−0.726332
$291$	0	0
$292$	34.8823	2.04133
$293$	−1.74006	−0.101655	−0.0508277	−	0.998707i	$-0.516186\pi$
$293$	−1.74006	−0.101655	−0.0508277	+	0.998707i	$0.516186\pi$
$294$	0	0
$295$	9.16409	0.533554
$296$	13.1420	0.763864
$297$	0	0
$298$	−14.4371	−0.836321
$299$	−0.336280	−0.0194476
$300$	0	0
$301$	−18.1259	−1.04476
$302$	−18.7340	−1.07802
$303$	0	0
$304$	31.2271	1.79100
$305$	−9.18431	−0.525892
$306$	0	0
$307$	−21.3566	−1.21889	−0.609444	−	0.792829i	$-0.708607\pi$
$307$	−21.3566	−1.21889	−0.609444	+	0.792829i	$0.708607\pi$
$308$	0	0
$309$	0	0
$310$	8.47033	0.481082
$311$	−32.8096	−1.86046	−0.930231	−	0.366975i	$-0.880393\pi$
$311$	−32.8096	−1.86046	−0.930231	+	0.366975i	$0.880393\pi$
$312$	0	0
$313$	−3.45852	−0.195487	−0.0977436	−	0.995212i	$-0.531163\pi$
$313$	−3.45852	−0.195487	−0.0977436	+	0.995212i	$0.531163\pi$
$314$	−24.8860	−1.40440
$315$	0	0
$316$	−21.8278	−1.22791
$317$	2.87566	0.161513	0.0807565	−	0.996734i	$-0.474266\pi$
$317$	2.87566	0.161513	0.0807565	+	0.996734i	$0.474266\pi$
$318$	0	0
$319$	0	0
$320$	−7.61553	−0.425721
$321$	0	0
$322$	8.66881	0.483094
$323$	37.2704	2.07378
$324$	0	0
$325$	0.313133	0.0173695
$326$	−12.3508	−0.684048
$327$	0	0
$328$	54.1150	2.98800
$329$	−39.4362	−2.17419
$330$	0	0
$331$	−14.1221	−0.776219	−0.388109	−	0.921613i	$-0.626872\pi$
$331$	−14.1221	−0.776219	−0.388109	+	0.921613i	$0.626872\pi$
$332$	65.4915	3.59431
$333$	0	0
$334$	14.2222	0.778202
$335$	−15.2739	−0.834501
$336$	0	0
$337$	−15.9490	−0.868796	−0.434398	−	0.900721i	$-0.643039\pi$
$337$	−15.9490	−0.868796	−0.434398	+	0.900721i	$0.643039\pi$
$338$	−31.6857	−1.72348
$339$	0	0
$340$	20.1569	1.09316
$341$	0	0
$342$	0	0
$343$	10.5070	0.567322
$344$	27.5118	1.48334
$345$	0	0
$346$	39.4543	2.12108
$347$	29.6801	1.59331	0.796656	−	0.604433i	$-0.206600\pi$
$347$	29.6801	1.59331	0.796656	+	0.604433i	$0.206600\pi$
$348$	0	0
$349$	−31.6937	−1.69653	−0.848263	−	0.529574i	$-0.822352\pi$
$349$	−31.6937	−1.69653	−0.848263	+	0.529574i	$0.822352\pi$
$350$	−8.07211	−0.431472
$351$	0	0
$352$	0	0
$353$	−1.20189	−0.0639703	−0.0319852	−	0.999488i	$-0.510183\pi$
$353$	−1.20189	−0.0639703	−0.0319852	+	0.999488i	$0.510183\pi$
$354$	0	0
$355$	−3.07211	−0.163051
$356$	−6.53559	−0.346385
$357$	0	0
$358$	20.3915	1.07772
$359$	−11.6591	−0.615343	−0.307671	−	0.951493i	$-0.599550\pi$
$359$	−11.6591	−0.615343	−0.307671	+	0.951493i	$0.599550\pi$
$360$	0	0
$361$	36.5633	1.92438
$362$	−15.8755	−0.834396
$363$	0	0
$364$	−4.14918	−0.217476
$365$	8.65269	0.452903
$366$	0	0
$367$	15.9860	0.834465	0.417232	−	0.908800i	$-0.363000\pi$
$367$	15.9860	0.834465	0.417232	+	0.908800i	$0.363000\pi$
$368$	−4.49894	−0.234523
$369$	0	0
$370$	6.46950	0.336333
$371$	16.2220	0.842203
$372$	0	0
$373$	−0.321975	−0.0166712	−0.00833561	−	0.999965i	$-0.502653\pi$
$373$	−0.321975	−0.0166712	−0.00833561	+	0.999965i	$0.502653\pi$
$374$	0	0
$375$	0	0
$376$	59.8570	3.08689
$377$	−1.57709	−0.0812241
$378$	0	0
$379$	11.4174	0.586475	0.293237	−	0.956040i	$-0.405267\pi$
$379$	11.4174	0.586475	0.293237	+	0.956040i	$0.405267\pi$
$380$	30.0502	1.54154
$381$	0	0
$382$	−37.7452	−1.93121
$383$	−28.3673	−1.44950	−0.724750	−	0.689012i	$-0.758045\pi$
$383$	−28.3673	−1.44950	−0.724750	+	0.689012i	$0.758045\pi$
$384$	0	0
$385$	0	0
$386$	−38.6846	−1.96899
$387$	0	0
$388$	0.903359	0.0458611
$389$	−15.1802	−0.769666	−0.384833	−	0.922986i	$-0.625741\pi$
$389$	−15.1802	−0.769666	−0.384833	+	0.922986i	$0.625741\pi$
$390$	0	0
$391$	−5.36960	−0.271553
$392$	18.9742	0.958341
$393$	0	0
$394$	40.2617	2.02835
$395$	−5.41446	−0.272431
$396$	0	0
$397$	5.22461	0.262216	0.131108	−	0.991368i	$-0.458147\pi$
$397$	5.22461	0.262216	0.131108	+	0.991368i	$0.458147\pi$
$398$	17.1053	0.857409
$399$	0	0
$400$	4.18926	0.209463
$401$	−14.0007	−0.699160	−0.349580	−	0.936907i	$-0.613676\pi$
$401$	−14.0007	−0.699160	−0.349580	+	0.936907i	$0.613676\pi$
$402$	0	0
$403$	1.07999	0.0537983
$404$	−2.03716	−0.101352
$405$	0	0
$406$	40.6549	2.01767
$407$	0	0
$408$	0	0
$409$	−33.3112	−1.64713	−0.823567	−	0.567218i	$-0.808020\pi$
$409$	−33.3112	−1.64713	−0.823567	+	0.567218i	$0.808020\pi$
$410$	26.6395	1.31563
$411$	0	0
$412$	−25.8088	−1.27151
$413$	−30.1209	−1.48215
$414$	0	0
$415$	16.2454	0.797456
$416$	0.0972843	0.00476975
$417$	0	0
$418$	0	0
$419$	5.28460	0.258170	0.129085	−	0.991634i	$-0.458796\pi$
$419$	5.28460	0.258170	0.129085	+	0.991634i	$0.458796\pi$
$420$	0	0
$421$	−30.7810	−1.50017	−0.750087	−	0.661340i	$-0.769988\pi$
$421$	−30.7810	−1.50017	−0.750087	+	0.661340i	$0.769988\pi$
$422$	49.0026	2.38541
$423$	0	0
$424$	−24.6221	−1.19575
$425$	5.00000	0.242536
$426$	0	0
$427$	30.1874	1.46087
$428$	8.43562	0.407751
$429$	0	0
$430$	13.5434	0.653122
$431$	12.3506	0.594910	0.297455	−	0.954736i	$-0.403862\pi$
$431$	12.3506	0.594910	0.297455	+	0.954736i	$0.403862\pi$
$432$	0	0
$433$	1.41287	0.0678983	0.0339491	−	0.999424i	$-0.489192\pi$
$433$	1.41287	0.0678983	0.0339491	+	0.999424i	$0.489192\pi$
$434$	−27.8406	−1.33639
$435$	0	0
$436$	26.9811	1.29216
$437$	−8.00509	−0.382935
$438$	0	0
$439$	7.58532	0.362028	0.181014	−	0.983481i	$-0.442062\pi$
$439$	7.58532	0.362028	0.181014	+	0.983481i	$0.442062\pi$
$440$	0	0
$441$	0	0
$442$	3.84510	0.182893
$443$	−11.0662	−0.525771	−0.262885	−	0.964827i	$-0.584674\pi$
$443$	−11.0662	−0.525771	−0.262885	+	0.964827i	$0.584674\pi$
$444$	0	0
$445$	−1.62118	−0.0768512
$446$	−49.4779	−2.34284
$447$	0	0
$448$	25.0311	1.18261
$449$	−6.32856	−0.298663	−0.149332	−	0.988787i	$-0.547712\pi$
$449$	−6.32856	−0.298663	−0.149332	+	0.988787i	$0.547712\pi$
$450$	0	0
$451$	0	0
$452$	43.5152	2.04678
$453$	0	0
$454$	1.31129	0.0615418
$455$	−1.02922	−0.0482506
$456$	0	0
$457$	0.189579	0.00886814	0.00443407	−	0.999990i	$-0.498589\pi$
$457$	0.189579	0.00886814	0.00443407	+	0.999990i	$0.498589\pi$
$458$	−54.5036	−2.54679
$459$	0	0
$460$	−4.32938	−0.201859
$461$	−26.6198	−1.23981	−0.619904	−	0.784678i	$-0.712828\pi$
$461$	−26.6198	−1.23981	−0.619904	+	0.784678i	$0.712828\pi$
$462$	0	0
$463$	20.9935	0.975652	0.487826	−	0.872941i	$-0.337790\pi$
$463$	20.9935	0.975652	0.487826	+	0.872941i	$0.337790\pi$
$464$	−21.0991	−0.979501
$465$	0	0
$466$	−11.2128	−0.519421
$467$	−7.89989	−0.365563	−0.182782	−	0.983154i	$-0.558510\pi$
$467$	−7.89989	−0.365563	−0.182782	+	0.983154i	$0.558510\pi$
$468$	0	0
$469$	50.2028	2.31815
$470$	29.4662	1.35917
$471$	0	0
$472$	45.7182	2.10435
$473$	0	0
$474$	0	0
$475$	7.45408	0.342017
$476$	−66.2525	−3.03668
$477$	0	0
$478$	−14.3928	−0.658311
$479$	39.9728	1.82640	0.913201	−	0.407509i	$-0.133602\pi$
$479$	39.9728	1.82640	0.913201	+	0.407509i	$0.133602\pi$
$480$	0	0
$481$	0.824882	0.0376114
$482$	−24.4624	−1.11423
$483$	0	0
$484$	0	0
$485$	0.224082	0.0101750
$486$	0	0
$487$	−9.93556	−0.450223	−0.225112	−	0.974333i	$-0.572275\pi$
$487$	−9.93556	−0.450223	−0.225112	+	0.974333i	$0.572275\pi$
$488$	−45.8190	−2.07413
$489$	0	0
$490$	9.34054	0.421963
$491$	4.97349	0.224451	0.112225	−	0.993683i	$-0.464202\pi$
$491$	4.97349	0.224451	0.112225	+	0.993683i	$0.464202\pi$
$492$	0	0
$493$	−25.1823	−1.13416
$494$	5.73234	0.257910
$495$	0	0
$496$	14.4487	0.648767
$497$	10.0975	0.452937
$498$	0	0
$499$	43.7757	1.95967	0.979834	−	0.199812i	$-0.0640332\pi$
$499$	43.7757	1.95967	0.979834	+	0.199812i	$0.0640332\pi$
$500$	4.03138	0.180289
$501$	0	0
$502$	41.4524	1.85011
$503$	6.28236	0.280117	0.140058	−	0.990143i	$-0.455271\pi$
$503$	6.28236	0.280117	0.140058	+	0.990143i	$0.455271\pi$
$504$	0	0
$505$	−0.505326	−0.0224867
$506$	0	0
$507$	0	0
$508$	68.5467	3.04127
$509$	−24.8381	−1.10093	−0.550465	−	0.834858i	$-0.685550\pi$
$509$	−24.8381	−1.10093	−0.550465	+	0.834858i	$0.685550\pi$
$510$	0	0
$511$	−28.4400	−1.25811
$512$	−40.4976	−1.78976
$513$	0	0
$514$	−27.3674	−1.20712
$515$	−6.40197	−0.282104
$516$	0	0
$517$	0	0
$518$	−21.2642	−0.934296
$519$	0	0
$520$	1.56217	0.0685058
$521$	6.94869	0.304428	0.152214	−	0.988348i	$-0.451360\pi$
$521$	6.94869	0.304428	0.152214	+	0.988348i	$0.451360\pi$
$522$	0	0
$523$	26.7510	1.16974	0.584869	−	0.811128i	$-0.301146\pi$
$523$	26.7510	1.16974	0.584869	+	0.811128i	$0.301146\pi$
$524$	−0.173554	−0.00758175
$525$	0	0
$526$	−65.9962	−2.87757
$527$	17.2449	0.751202
$528$	0	0
$529$	−21.8467	−0.949856
$530$	−12.1209	−0.526496
$531$	0	0
$532$	−98.7703	−4.28224
$533$	3.39662	0.147124
$534$	0	0
$535$	2.09249	0.0904662
$536$	−76.1989	−3.29129
$537$	0	0
$538$	24.7133	1.06547
$539$	0	0
$540$	0	0
$541$	−14.5084	−0.623767	−0.311883	−	0.950120i	$-0.600960\pi$
$541$	−14.5084	−0.623767	−0.311883	+	0.950120i	$0.600960\pi$
$542$	−25.8951	−1.11229
$543$	0	0
$544$	1.55340	0.0666015
$545$	6.69278	0.286687
$546$	0	0
$547$	26.7346	1.14309	0.571543	−	0.820572i	$-0.306345\pi$
$547$	26.7346	1.14309	0.571543	+	0.820572i	$0.306345\pi$
$548$	−29.6813	−1.26792
$549$	0	0
$550$	0	0
$551$	−37.5422	−1.59935
$552$	0	0
$553$	17.7965	0.756784
$554$	44.0527	1.87162
$555$	0	0
$556$	−53.3728	−2.26351
$557$	17.2444	0.730670	0.365335	−	0.930876i	$-0.380954\pi$
$557$	17.2444	0.730670	0.365335	+	0.930876i	$0.380954\pi$
$558$	0	0
$559$	1.72683	0.0730371
$560$	−13.7694	−0.581865
$561$	0	0
$562$	18.2006	0.767748
$563$	−0.831914	−0.0350610	−0.0175305	−	0.999846i	$-0.505580\pi$
$563$	−0.831914	−0.0350610	−0.0175305	+	0.999846i	$0.505580\pi$
$564$	0	0
$565$	10.7941	0.454112
$566$	−13.2295	−0.556076
$567$	0	0
$568$	−15.3263	−0.643076
$569$	11.5961	0.486132	0.243066	−	0.970010i	$-0.421847\pi$
$569$	11.5961	0.486132	0.243066	+	0.970010i	$0.421847\pi$
$570$	0	0
$571$	−21.8414	−0.914034	−0.457017	−	0.889458i	$-0.651082\pi$
$571$	−21.8414	−0.914034	−0.457017	+	0.889458i	$0.651082\pi$
$572$	0	0
$573$	0	0
$574$	−87.5598	−3.65468
$575$	−1.07392	−0.0447856
$576$	0	0
$577$	9.74587	0.405726	0.202863	−	0.979207i	$-0.434975\pi$
$577$	9.74587	0.405726	0.202863	+	0.979207i	$0.434975\pi$
$578$	19.6471	0.817211
$579$	0	0
$580$	−20.3039	−0.843074
$581$	−53.3961	−2.21524
$582$	0	0
$583$	0	0
$584$	43.1669	1.78626
$585$	0	0
$586$	−4.27339	−0.176532
$587$	−22.9441	−0.947005	−0.473502	−	0.880793i	$-0.657010\pi$
$587$	−22.9441	−0.947005	−0.473502	+	0.880793i	$0.657010\pi$
$588$	0	0
$589$	25.7090	1.05932
$590$	22.5060	0.926556
$591$	0	0
$592$	11.0357	0.453565
$593$	28.7819	1.18193	0.590965	−	0.806697i	$-0.298747\pi$
$593$	28.7819	1.18193	0.590965	+	0.806697i	$0.298747\pi$
$594$	0	0
$595$	−16.4342	−0.673737
$596$	−23.6988	−0.970741
$597$	0	0
$598$	−0.825867	−0.0337722
$599$	29.1951	1.19288	0.596440	−	0.802657i	$-0.296581\pi$
$599$	29.1951	1.19288	0.596440	+	0.802657i	$0.296581\pi$
$600$	0	0
$601$	6.68087	0.272518	0.136259	−	0.990673i	$-0.456492\pi$
$601$	6.68087	0.272518	0.136259	+	0.990673i	$0.456492\pi$
$602$	−44.5151	−1.81430
$603$	0	0
$604$	−30.7522	−1.25129
$605$	0	0
$606$	0	0
$607$	42.6108	1.72952	0.864759	−	0.502188i	$-0.167471\pi$
$607$	42.6108	1.72952	0.864759	+	0.502188i	$0.167471\pi$
$608$	2.31583	0.0939194
$609$	0	0
$610$	−22.5556	−0.913251
$611$	3.75703	0.151993
$612$	0	0
$613$	5.83156	0.235535	0.117767	−	0.993041i	$-0.462426\pi$
$613$	5.83156	0.235535	0.117767	+	0.993041i	$0.462426\pi$
$614$	−52.4495	−2.11669
$615$	0	0
$616$	0	0
$617$	33.6386	1.35424	0.677119	−	0.735874i	$-0.263228\pi$
$617$	33.6386	1.35424	0.677119	+	0.735874i	$0.263228\pi$
$618$	0	0
$619$	42.5616	1.71070	0.855349	−	0.518053i	$-0.173343\pi$
$619$	42.5616	1.71070	0.855349	+	0.518053i	$0.173343\pi$
$620$	13.9042	0.558405
$621$	0	0
$622$	−80.5766	−3.23083
$623$	5.32856	0.213484
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.49374	−0.339478
$627$	0	0
$628$	−40.8509	−1.63013
$629$	13.1714	0.525179
$630$	0	0
$631$	−8.89989	−0.354299	−0.177149	−	0.984184i	$-0.556688\pi$
$631$	−8.89989	−0.354299	−0.177149	+	0.984184i	$0.556688\pi$
$632$	−27.0119	−1.07448
$633$	0	0
$634$	7.06228	0.280479
$635$	17.0033	0.674755
$636$	0	0
$637$	1.19095	0.0471871
$638$	0	0
$639$	0	0
$640$	−19.3242	−0.763858
$641$	6.16806	0.243624	0.121812	−	0.992553i	$-0.461130\pi$
$641$	6.16806	0.243624	0.121812	+	0.992553i	$0.461130\pi$
$642$	0	0
$643$	4.35335	0.171680	0.0858398	−	0.996309i	$-0.472643\pi$
$643$	4.35335	0.171680	0.0858398	+	0.996309i	$0.472643\pi$
$644$	14.2300	0.560740
$645$	0	0
$646$	91.5318	3.60127
$647$	−13.4933	−0.530478	−0.265239	−	0.964183i	$-0.585451\pi$
$647$	−13.4933	−0.530478	−0.265239	+	0.964183i	$0.585451\pi$
$648$	0	0
$649$	0	0
$650$	0.769020	0.0301635
$651$	0	0
$652$	−20.2741	−0.793993
$653$	−27.4481	−1.07413	−0.537064	−	0.843541i	$-0.680467\pi$
$653$	−27.4481	−1.07413	−0.537064	+	0.843541i	$0.680467\pi$
$654$	0	0
$655$	−0.0430508	−0.00168213
$656$	45.4418	1.77420
$657$	0	0
$658$	−96.8507	−3.77563
$659$	−18.7768	−0.731441	−0.365721	−	0.930725i	$-0.619177\pi$
$659$	−18.7768	−0.731441	−0.365721	+	0.930725i	$0.619177\pi$
$660$	0	0
$661$	−21.6525	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$661$	−21.6525	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$662$	−34.6822	−1.34796
$663$	0	0
$664$	81.0458	3.14519
$665$	−24.5004	−0.950084
$666$	0	0
$667$	5.40877	0.209428
$668$	23.3459	0.903281
$669$	0	0
$670$	−37.5109	−1.44917
$671$	0	0
$672$	0	0
$673$	−23.3021	−0.898232	−0.449116	−	0.893474i	$-0.648261\pi$
$673$	−23.3021	−0.898232	−0.449116	+	0.893474i	$0.648261\pi$
$674$	−39.1689	−1.50873
$675$	0	0
$676$	−52.0127	−2.00049
$677$	33.2808	1.27909	0.639543	−	0.768756i	$-0.279124\pi$
$677$	33.2808	1.27909	0.639543	+	0.768756i	$0.279124\pi$
$678$	0	0
$679$	−0.736522	−0.0282651
$680$	24.9442	0.956566
$681$	0	0
$682$	0	0
$683$	−16.9244	−0.647593	−0.323796	−	0.946127i	$-0.604959\pi$
$683$	−16.9244	−0.647593	−0.323796	+	0.946127i	$0.604959\pi$
$684$	0	0
$685$	−7.36257	−0.281309
$686$	25.8039	0.985197
$687$	0	0
$688$	23.1024	0.880772
$689$	−1.54545	−0.0588769
$690$	0	0
$691$	−48.4335	−1.84250	−0.921249	−	0.388973i	$-0.872830\pi$
$691$	−48.4335	−1.84250	−0.921249	+	0.388973i	$0.872830\pi$
$692$	64.7650	2.46200
$693$	0	0
$694$	72.8910	2.76690
$695$	−13.2393	−0.502197
$696$	0	0
$697$	54.2360	2.05434
$698$	−77.8362	−2.94614
$699$	0	0
$700$	−13.2505	−0.500822
$701$	−45.4161	−1.71534	−0.857672	−	0.514197i	$-0.828090\pi$
$701$	−45.4161	−1.71534	−0.857672	+	0.514197i	$0.828090\pi$
$702$	0	0
$703$	19.6361	0.740591
$704$	0	0
$705$	0	0
$706$	−2.95171	−0.111089
$707$	1.66093	0.0624655
$708$	0	0
$709$	−1.76497	−0.0662848	−0.0331424	−	0.999451i	$-0.510551\pi$
$709$	−1.76497	−0.0662848	−0.0331424	+	0.999451i	$0.510551\pi$
$710$	−7.54476	−0.283150
$711$	0	0
$712$	−8.08780	−0.303103
$713$	−3.70394	−0.138714
$714$	0	0
$715$	0	0
$716$	33.4729	1.25094
$717$	0	0
$718$	−28.6334	−1.06859
$719$	3.29998	0.123069	0.0615343	−	0.998105i	$-0.480401\pi$
$719$	3.29998	0.123069	0.0615343	+	0.998105i	$0.480401\pi$
$720$	0	0
$721$	21.0423	0.783655
$722$	89.7952	3.34183
$723$	0	0
$724$	−26.0599	−0.968507
$725$	−5.03647	−0.187050
$726$	0	0
$727$	11.7838	0.437037	0.218519	−	0.975833i	$-0.429878\pi$
$727$	11.7838	0.437037	0.218519	+	0.975833i	$0.429878\pi$
$728$	−5.13461	−0.190301
$729$	0	0
$730$	21.2500	0.786499
$731$	27.5734	1.01984
$732$	0	0
$733$	−5.73108	−0.211682	−0.105841	−	0.994383i	$-0.533754\pi$
$733$	−5.73108	−0.211682	−0.105841	+	0.994383i	$0.533754\pi$
$734$	39.2599	1.44911
$735$	0	0
$736$	−0.333646	−0.0122983
$737$	0	0
$738$	0	0
$739$	21.0551	0.774524	0.387262	−	0.921970i	$-0.373421\pi$
$739$	21.0551	0.774524	0.387262	+	0.921970i	$0.373421\pi$
$740$	10.6198	0.390391
$741$	0	0
$742$	39.8393	1.46255
$743$	13.1283	0.481630	0.240815	−	0.970571i	$-0.422585\pi$
$743$	13.1283	0.481630	0.240815	+	0.970571i	$0.422585\pi$
$744$	0	0
$745$	−5.87858	−0.215375
$746$	−0.790734	−0.0289508
$747$	0	0
$748$	0	0
$749$	−6.87768	−0.251305
$750$	0	0
$751$	25.6251	0.935073	0.467537	−	0.883974i	$-0.345142\pi$
$751$	25.6251	0.935073	0.467537	+	0.883974i	$0.345142\pi$
$752$	50.2636	1.83292
$753$	0	0
$754$	−3.87315	−0.141052
$755$	−7.62821	−0.277619
$756$	0	0
$757$	31.1970	1.13387	0.566936	−	0.823762i	$-0.308129\pi$
$757$	31.1970	1.13387	0.566936	+	0.823762i	$0.308129\pi$
$758$	28.0400	1.01846
$759$	0	0
$760$	37.1872	1.34892
$761$	11.3761	0.412382	0.206191	−	0.978512i	$-0.433893\pi$
$761$	11.3761	0.412382	0.206191	+	0.978512i	$0.433893\pi$
$762$	0	0
$763$	−21.9981	−0.796385
$764$	−61.9594	−2.24161
$765$	0	0
$766$	−69.6668	−2.51717
$767$	2.86958	0.103615
$768$	0	0
$769$	−10.3938	−0.374811	−0.187405	−	0.982283i	$-0.560008\pi$
$769$	−10.3938	−0.374811	−0.187405	+	0.982283i	$0.560008\pi$
$770$	0	0
$771$	0	0
$772$	−63.5014	−2.28547
$773$	−14.0348	−0.504796	−0.252398	−	0.967623i	$-0.581219\pi$
$773$	−14.0348	−0.504796	−0.252398	+	0.967623i	$0.581219\pi$
$774$	0	0
$775$	3.44899	0.123891
$776$	1.11791	0.0401306
$777$	0	0
$778$	−37.2808	−1.33658
$779$	80.8559	2.89696
$780$	0	0
$781$	0	0
$782$	−13.1871	−0.471571
$783$	0	0
$784$	15.9331	0.569041
$785$	−10.1332	−0.361670
$786$	0	0
$787$	13.8176	0.492545	0.246273	−	0.969201i	$-0.420794\pi$
$787$	13.8176	0.492545	0.246273	+	0.969201i	$0.420794\pi$
$788$	66.0902	2.35437
$789$	0	0
$790$	−13.2973	−0.473097
$791$	−35.4785	−1.26147
$792$	0	0
$793$	−2.87591	−0.102127
$794$	12.8311	0.455357
$795$	0	0
$796$	28.0786	0.995218
$797$	5.38594	0.190780	0.0953898	−	0.995440i	$-0.469590\pi$
$797$	5.38594	0.190780	0.0953898	+	0.995440i	$0.469590\pi$
$798$	0	0
$799$	59.9910	2.12233
$800$	0.310680	0.0109842
$801$	0	0
$802$	−34.3840	−1.21414
$803$	0	0
$804$	0	0
$805$	3.52981	0.124409
$806$	2.65234	0.0934248
$807$	0	0
$808$	−2.52099	−0.0886880
$809$	23.7748	0.835876	0.417938	−	0.908476i	$-0.362753\pi$
$809$	23.7748	0.835876	0.417938	+	0.908476i	$0.362753\pi$
$810$	0	0
$811$	−9.46335	−0.332303	−0.166152	−	0.986100i	$-0.553134\pi$
$811$	−9.46335	−0.332303	−0.166152	+	0.986100i	$0.553134\pi$
$812$	66.7358	2.34197
$813$	0	0
$814$	0	0
$815$	−5.02906	−0.176160
$816$	0	0
$817$	41.1068	1.43815
$818$	−81.8086	−2.86037
$819$	0	0
$820$	43.7292	1.52709
$821$	−10.4189	−0.363622	−0.181811	−	0.983334i	$-0.558196\pi$
$821$	−10.4189	−0.363622	−0.181811	+	0.983334i	$0.558196\pi$
$822$	0	0
$823$	24.3540	0.848928	0.424464	−	0.905445i	$-0.360463\pi$
$823$	24.3540	0.848928	0.424464	+	0.905445i	$0.360463\pi$
$824$	−31.9384	−1.11263
$825$	0	0
$826$	−73.9736	−2.57387
$827$	−22.0006	−0.765037	−0.382519	−	0.923948i	$-0.624943\pi$
$827$	−22.0006	−0.765037	−0.382519	+	0.923948i	$0.624943\pi$
$828$	0	0
$829$	3.03289	0.105337	0.0526683	−	0.998612i	$-0.483227\pi$
$829$	3.03289	0.105337	0.0526683	+	0.998612i	$0.483227\pi$
$830$	39.8969	1.38484
$831$	0	0
$832$	−2.38468	−0.0826738
$833$	19.0166	0.658888
$834$	0	0
$835$	5.79105	0.200408
$836$	0	0
$837$	0	0
$838$	12.9784	0.448331
$839$	48.5383	1.67573	0.837864	−	0.545879i	$-0.183804\pi$
$839$	48.5383	1.67573	0.837864	+	0.545879i	$0.183804\pi$
$840$	0	0
$841$	−3.63399	−0.125310
$842$	−75.5946	−2.60516
$843$	0	0
$844$	80.4386	2.76881
$845$	−12.9019	−0.443840
$846$	0	0
$847$	0	0
$848$	−20.6758	−0.710011
$849$	0	0
$850$	12.2794	0.421181
$851$	−2.82901	−0.0969773
$852$	0	0
$853$	11.7632	0.402766	0.201383	−	0.979513i	$-0.435456\pi$
$853$	11.7632	0.402766	0.201383	+	0.979513i	$0.435456\pi$
$854$	74.1368	2.53691
$855$	0	0
$856$	10.4391	0.356801
$857$	1.61311	0.0551026	0.0275513	−	0.999620i	$-0.491229\pi$
$857$	1.61311	0.0551026	0.0275513	+	0.999620i	$0.491229\pi$
$858$	0	0
$859$	47.3263	1.61475	0.807376	−	0.590038i	$-0.200887\pi$
$859$	47.3263	1.61475	0.807376	+	0.590038i	$0.200887\pi$
$860$	22.2318	0.758097
$861$	0	0
$862$	30.3318	1.03310
$863$	9.97233	0.339462	0.169731	−	0.985490i	$-0.445710\pi$
$863$	9.97233	0.339462	0.169731	+	0.985490i	$0.445710\pi$
$864$	0	0
$865$	16.0652	0.546234
$866$	3.46985	0.117910
$867$	0	0
$868$	−45.7009	−1.55119
$869$	0	0
$870$	0	0
$871$	−4.78276	−0.162058
$872$	33.3892	1.13070
$873$	0	0
$874$	−19.6596	−0.664996
$875$	−3.28684	−0.111116
$876$	0	0
$877$	−26.0057	−0.878151	−0.439076	−	0.898450i	$-0.644694\pi$
$877$	−26.0057	−0.878151	−0.439076	+	0.898450i	$0.644694\pi$
$878$	18.6287	0.628688
$879$	0	0
$880$	0	0
$881$	−10.4081	−0.350657	−0.175329	−	0.984510i	$-0.556099\pi$
$881$	−10.4081	−0.350657	−0.175329	+	0.984510i	$0.556099\pi$
$882$	0	0
$883$	−53.7283	−1.80810	−0.904051	−	0.427424i	$-0.859421\pi$
$883$	−53.7283	−1.80810	−0.904051	+	0.427424i	$0.859421\pi$
$884$	6.31180	0.212289
$885$	0	0
$886$	−27.1773	−0.913040
$887$	−40.6246	−1.36404	−0.682021	−	0.731333i	$-0.738899\pi$
$887$	−40.6246	−1.36404	−0.682021	+	0.731333i	$0.738899\pi$
$888$	0	0
$889$	−55.8871	−1.87439
$890$	−3.98143	−0.133458
$891$	0	0
$892$	−81.2188	−2.71941
$893$	89.4354	2.99284
$894$	0	0
$895$	8.30309	0.277542
$896$	63.5157	2.12191
$897$	0	0
$898$	−15.5422	−0.518651
$899$	−17.3707	−0.579346
$900$	0	0
$901$	−24.6772	−0.822115
$902$	0	0
$903$	0	0
$904$	53.8501	1.79103
$905$	−6.46425	−0.214879
$906$	0	0
$907$	−19.4070	−0.644398	−0.322199	−	0.946672i	$-0.604422\pi$
$907$	−19.4070	−0.644398	−0.322199	+	0.946672i	$0.604422\pi$
$908$	2.15250	0.0714333
$909$	0	0
$910$	−2.52765	−0.0837907
$911$	−10.7208	−0.355195	−0.177597	−	0.984103i	$-0.556832\pi$
$911$	−10.7208	−0.355195	−0.177597	+	0.984103i	$0.556832\pi$
$912$	0	0
$913$	0	0
$914$	0.465585	0.0154002
$915$	0	0
$916$	−89.4686	−2.95613
$917$	0.141501	0.00467278
$918$	0	0
$919$	23.1310	0.763021	0.381511	−	0.924364i	$-0.375404\pi$
$919$	23.1310	0.763021	0.381511	+	0.924364i	$0.375404\pi$
$920$	−5.35762	−0.176635
$921$	0	0
$922$	−65.3752	−2.15302
$923$	−0.961981	−0.0316640
$924$	0	0
$925$	2.63428	0.0866147
$926$	51.5577	1.69429
$927$	0	0
$928$	−1.56473	−0.0513648
$929$	−26.2273	−0.860489	−0.430245	−	0.902712i	$-0.641573\pi$
$929$	−26.2273	−0.860489	−0.430245	+	0.902712i	$0.641573\pi$
$930$	0	0
$931$	28.3503	0.929144
$932$	−18.4059	−0.602907
$933$	0	0
$934$	−19.4012	−0.634828
$935$	0	0
$936$	0	0
$937$	−23.4011	−0.764480	−0.382240	−	0.924063i	$-0.624847\pi$
$937$	−23.4011	−0.764480	−0.382240	+	0.924063i	$0.624847\pi$
$938$	123.292	4.02564
$939$	0	0
$940$	48.3693	1.57763
$941$	−10.9687	−0.357570	−0.178785	−	0.983888i	$-0.557217\pi$
$941$	−10.9687	−0.357570	−0.178785	+	0.983888i	$0.557217\pi$
$942$	0	0
$943$	−11.6490	−0.379345
$944$	38.3908	1.24951
$945$	0	0
$946$	0	0
$947$	13.3652	0.434310	0.217155	−	0.976137i	$-0.430322\pi$
$947$	13.3652	0.434310	0.217155	+	0.976137i	$0.430322\pi$
$948$	0	0
$949$	2.70945	0.0879524
$950$	18.3064	0.593937
$951$	0	0
$952$	−81.9876	−2.65723
$953$	−21.8242	−0.706956	−0.353478	−	0.935443i	$-0.615001\pi$
$953$	−21.8242	−0.706956	−0.353478	+	0.935443i	$0.615001\pi$
$954$	0	0
$955$	−15.3693	−0.497339
$956$	−23.6260	−0.764120
$957$	0	0
$958$	98.1686	3.17168
$959$	24.1996	0.781446
$960$	0	0
$961$	−19.1045	−0.616273
$962$	2.02582	0.0653150
$963$	0	0
$964$	−40.1555	−1.29332
$965$	−15.7518	−0.507068
$966$	0	0
$967$	−16.6600	−0.535750	−0.267875	−	0.963454i	$-0.586321\pi$
$967$	−16.6600	−0.535750	−0.267875	+	0.963454i	$0.586321\pi$
$968$	0	0
$969$	0	0
$970$	0.550320	0.0176697
$971$	−11.1032	−0.356320	−0.178160	−	0.984002i	$-0.557014\pi$
$971$	−11.1032	−0.356320	−0.178160	+	0.984002i	$0.557014\pi$
$972$	0	0
$973$	43.5156	1.39505
$974$	−24.4006	−0.781846
$975$	0	0
$976$	−38.4755	−1.23157
$977$	−18.8144	−0.601926	−0.300963	−	0.953636i	$-0.597308\pi$
$977$	−18.8144	−0.601926	−0.300963	+	0.953636i	$0.597308\pi$
$978$	0	0
$979$	0	0
$980$	15.3327	0.489784
$981$	0	0
$982$	12.2143	0.389775
$983$	1.37848	0.0439667	0.0219833	−	0.999758i	$-0.493002\pi$
$983$	1.37848	0.0439667	0.0219833	+	0.999758i	$0.493002\pi$
$984$	0	0
$985$	16.3940	0.522355
$986$	−61.8450	−1.96955
$987$	0	0
$988$	9.40973	0.299363
$989$	−5.92233	−0.188319
$990$	0	0
$991$	−46.3186	−1.47136	−0.735680	−	0.677329i	$-0.763137\pi$
$991$	−46.3186	−1.47136	−0.735680	+	0.677329i	$0.763137\pi$
$992$	1.07153	0.0340212
$993$	0	0
$994$	24.7984	0.786558
$995$	6.96500	0.220805
$996$	0	0
$997$	−14.5470	−0.460709	−0.230355	−	0.973107i	$-0.573989\pi$
$997$	−14.5470	−0.460709	−0.230355	+	0.973107i	$0.573989\pi$
$998$	107.508	3.40311
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bv.1.3		4
3.2	odd	2		1815.2.a.o.1.2		4
11.7	odd	10		495.2.n.d.181.1		8
11.8	odd	10		495.2.n.d.361.1		8
11.10	odd	2		5445.2.a.be.1.2		4
15.14	odd	2		9075.2.a.dj.1.3		4
33.8	even	10		165.2.m.a.31.2	yes	8
33.29	even	10		165.2.m.a.16.2	✓	8
33.32	even	2		1815.2.a.x.1.3		4
165.8	odd	20		825.2.bx.h.724.1		16
165.29	even	10		825.2.n.k.676.1		8
165.62	odd	20		825.2.bx.h.49.1		16
165.74	even	10		825.2.n.k.526.1		8
165.107	odd	20		825.2.bx.h.724.4		16
165.128	odd	20		825.2.bx.h.49.4		16
165.164	even	2		9075.2.a.cl.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.16.2	✓	8	33.29	even	10
165.2.m.a.31.2	yes	8	33.8	even	10
495.2.n.d.181.1		8	11.7	odd	10
495.2.n.d.361.1		8	11.8	odd	10
825.2.n.k.526.1		8	165.74	even	10
825.2.n.k.676.1		8	165.29	even	10
825.2.bx.h.49.1		16	165.62	odd	20
825.2.bx.h.49.4		16	165.128	odd	20
825.2.bx.h.724.1		16	165.8	odd	20
825.2.bx.h.724.4		16	165.107	odd	20
1815.2.a.o.1.2		4	3.2	odd	2
1815.2.a.x.1.3		4	33.32	even	2
5445.2.a.be.1.2		4	11.10	odd	2
5445.2.a.bv.1.3		4	1.1	even	1	trivial
9075.2.a.cl.1.2		4	165.164	even	2
9075.2.a.dj.1.3		4	15.14	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +O(q^{10})\)	\(q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +2.45589 q^{10} +0.313133 q^{13} -8.07211 q^{14} +4.18926 q^{16} +5.00000 q^{17} +7.45408 q^{19} +4.03138 q^{20} -1.07392 q^{23} +1.00000 q^{25} +0.769020 q^{26} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} +0.310680 q^{32} +12.2794 q^{34} -3.28684 q^{35} +2.63428 q^{37} +18.3064 q^{38} +4.98884 q^{40} +10.8472 q^{41} +5.51468 q^{43} -2.63743 q^{46} +11.9982 q^{47} +3.80333 q^{49} +2.45589 q^{50} +1.26236 q^{52} -4.93543 q^{53} -16.3975 q^{56} -12.3690 q^{58} +9.16409 q^{59} -9.18431 q^{61} +8.47033 q^{62} -7.61553 q^{64} +0.313133 q^{65} -15.2739 q^{67} +20.1569 q^{68} -8.07211 q^{70} -3.07211 q^{71} +8.65269 q^{73} +6.46950 q^{74} +30.0502 q^{76} -5.41446 q^{79} +4.18926 q^{80} +26.6395 q^{82} +16.2454 q^{83} +5.00000 q^{85} +13.5434 q^{86} -1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} +29.4662 q^{94} +7.45408 q^{95} +0.224082 q^{97} +9.34054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) \) 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8	\( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100}) \) 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 + 5 * q^10 - 3 * q^13 + 5 * q^14 + 15 * q^16 + 20 * q^17 - 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 - 3 * q^28 + 5 * q^29 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 - 7 * q^37 - q^38 + 15 * q^40 + 20 * q^41 + 2 * q^43 - 7 * q^46 + 20 * q^47 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 + 7 * q^61 - 12 * q^62 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 + 5 * q^70 + 25 * q^71 - 23 * q^73 - 7 * q^74 + 7 * q^76 + 15 * q^80 + 11 * q^82 + 33 * q^83 + 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 + 17 * q^94 - 3 * q^95 + 25 * q^98

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