LMFDB - Embedded newform 5445.2.a.ba.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.39138$ 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)

$f(q)$	$=$	$q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} +5.11009 q^{7} -4.11009 q^{8} +O(q^{10})$	$q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} +5.11009 q^{7} -4.11009 q^{8} +2.39138 q^{10} -5.43742 q^{13} -12.2202 q^{14} +2.39138 q^{16} +1.32733 q^{17} -1.67267 q^{19} -3.71871 q^{20} +2.11009 q^{23} +1.00000 q^{25} +13.0029 q^{26} +19.0029 q^{28} +0.782765 q^{29} -4.43742 q^{31} +2.50147 q^{32} -3.17415 q^{34} -5.11009 q^{35} -11.3303 q^{37} +4.00000 q^{38} +4.11009 q^{40} +11.5655 q^{41} +4.78276 q^{43} -5.04604 q^{46} -9.45544 q^{47} +19.1130 q^{49} -2.39138 q^{50} -20.2202 q^{52} +8.23820 q^{53} -21.0029 q^{56} -1.87189 q^{58} -10.0921 q^{59} +0.779816 q^{61} +10.6116 q^{62} -10.7647 q^{64} +5.43742 q^{65} -4.45544 q^{67} +4.93594 q^{68} +12.2202 q^{70} -12.9109 q^{71} -2.32733 q^{73} +27.0950 q^{74} -6.22018 q^{76} -1.12811 q^{79} -2.39138 q^{80} -27.6576 q^{82} -4.00000 q^{83} -1.32733 q^{85} -11.4374 q^{86} -14.7828 q^{89} -27.7857 q^{91} +7.84682 q^{92} +22.6116 q^{94} +1.67267 q^{95} +4.45544 q^{97} -45.7066 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7	$ 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + q^{10} - 4 q^{13} - 12 q^{14} + q^{16} + 4 q^{17} - 5 q^{19} - 5 q^{20} - 6 q^{23} + 3 q^{25} + 2 q^{26} + 20 q^{28} - 10 q^{29} - q^{31} - 11 q^{32} + 9 q^{34} - 3 q^{35} + 3 q^{37} + 12 q^{38} + 10 q^{41} + 2 q^{43} - 9 q^{46} - 16 q^{47} + 8 q^{49} - q^{50} - 36 q^{52} - 26 q^{56} - 18 q^{58} - 18 q^{59} + 27 q^{61} + q^{62} - 20 q^{64} + 4 q^{65} - q^{67} + 21 q^{68} + 12 q^{70} - 14 q^{71} - 7 q^{73} + 32 q^{74} + 6 q^{76} + 9 q^{79} - q^{80} - 46 q^{82} - 12 q^{83} - 4 q^{85} - 22 q^{86} - 32 q^{89} - 34 q^{91} + 5 q^{92} + 37 q^{94} + 5 q^{95} + q^{97} - 57 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + q^10 - 4 * q^13 - 12 * q^14 + q^16 + 4 * q^17 - 5 * q^19 - 5 * q^20 - 6 * q^23 + 3 * q^25 + 2 * q^26 + 20 * q^28 - 10 * q^29 - q^31 - 11 * q^32 + 9 * q^34 - 3 * q^35 + 3 * q^37 + 12 * q^38 + 10 * q^41 + 2 * q^43 - 9 * q^46 - 16 * q^47 + 8 * q^49 - q^50 - 36 * q^52 - 26 * q^56 - 18 * q^58 - 18 * q^59 + 27 * q^61 + q^62 - 20 * q^64 + 4 * q^65 - q^67 + 21 * q^68 + 12 * q^70 - 14 * q^71 - 7 * q^73 + 32 * q^74 + 6 * q^76 + 9 * q^79 - q^80 - 46 * q^82 - 12 * q^83 - 4 * q^85 - 22 * q^86 - 32 * q^89 - 34 * q^91 + 5 * q^92 + 37 * q^94 + 5 * q^95 + q^97 - 57 * 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 $

$2$	−2.39138	−1.69096	−0.845481	−	0.534005i	$-0.820686\pi$
$2$	−2.39138	−1.69096	−0.845481	+	0.534005i	$0.820686\pi$
$3$	0	0
$3$	0	0
$4$	3.71871	1.85935
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	5.11009	1.93143	0.965717	−	0.259599i	$-0.0835902\pi$
$7$	5.11009	1.93143	0.965717	+	0.259599i	$0.0835902\pi$
$8$	−4.11009	−1.45314
$9$	0	0
$10$	2.39138	0.756222
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.43742	−1.50807	−0.754034	−	0.656835i	$-0.771895\pi$
$13$	−5.43742	−1.50807	−0.754034	+	0.656835i	$0.771895\pi$
$14$	−12.2202	−3.26598
$15$	0	0
$16$	2.39138	0.597846
$17$	1.32733	0.321924	0.160962	−	0.986961i	$-0.448540\pi$
$17$	1.32733	0.321924	0.160962	+	0.986961i	$0.448540\pi$
$18$	0	0
$19$	−1.67267	−0.383737	−0.191869	−	0.981421i	$-0.561455\pi$
$19$	−1.67267	−0.383737	−0.191869	+	0.981421i	$0.561455\pi$
$20$	−3.71871	−0.831529
$21$	0	0
$22$	0	0
$23$	2.11009	0.439985	0.219992	−	0.975502i	$-0.429397\pi$
$23$	2.11009	0.439985	0.219992	+	0.975502i	$0.429397\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	13.0029	2.55009
$27$	0	0
$28$	19.0029	3.59122
$29$	0.782765	0.145356	0.0726779	−	0.997355i	$-0.476845\pi$
$29$	0.782765	0.145356	0.0726779	+	0.997355i	$0.476845\pi$
$30$	0	0
$31$	−4.43742	−0.796984	−0.398492	−	0.917172i	$-0.630466\pi$
$31$	−4.43742	−0.796984	−0.398492	+	0.917172i	$0.630466\pi$
$32$	2.50147	0.442202
$33$	0	0
$34$	−3.17415	−0.544362
$35$	−5.11009	−0.863763
$36$	0	0
$37$	−11.3303	−1.86269	−0.931343	−	0.364143i	$-0.881362\pi$
$37$	−11.3303	−1.86269	−0.931343	+	0.364143i	$0.881362\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	4.11009	0.649863
$41$	11.5655	1.80623	0.903116	−	0.429396i	$-0.141274\pi$
$41$	11.5655	1.80623	0.903116	+	0.429396i	$0.141274\pi$
$42$	0	0
$43$	4.78276	0.729365	0.364682	−	0.931132i	$-0.381178\pi$
$43$	4.78276	0.729365	0.364682	+	0.931132i	$0.381178\pi$
$44$	0	0
$45$	0	0
$46$	−5.04604	−0.743998
$47$	−9.45544	−1.37922	−0.689609	−	0.724182i	$-0.742217\pi$
$47$	−9.45544	−1.37922	−0.689609	+	0.724182i	$0.742217\pi$
$48$	0	0
$49$	19.1130	2.73043
$50$	−2.39138	−0.338193
$51$	0	0
$52$	−20.2202	−2.80404
$53$	8.23820	1.13160	0.565802	−	0.824541i	$-0.308567\pi$
$53$	8.23820	1.13160	0.565802	+	0.824541i	$0.308567\pi$
$54$	0	0
$55$	0	0
$56$	−21.0029	−2.80664
$57$	0	0
$58$	−1.87189	−0.245791
$59$	−10.0921	−1.31388	−0.656938	−	0.753945i	$-0.728149\pi$
$59$	−10.0921	−1.31388	−0.656938	+	0.753945i	$0.728149\pi$
$60$	0	0
$61$	0.779816	0.0998452	0.0499226	−	0.998753i	$-0.484103\pi$
$61$	0.779816	0.0998452	0.0499226	+	0.998753i	$0.484103\pi$
$62$	10.6116	1.34767
$63$	0	0
$64$	−10.7647	−1.34559
$65$	5.43742	0.674429
$66$	0	0
$67$	−4.45544	−0.544318	−0.272159	−	0.962252i	$-0.587738\pi$
$67$	−4.45544	−0.544318	−0.272159	+	0.962252i	$0.587738\pi$
$68$	4.93594	0.598571
$69$	0	0
$70$	12.2202	1.46059
$71$	−12.9109	−1.53224	−0.766119	−	0.642698i	$-0.777815\pi$
$71$	−12.9109	−1.53224	−0.766119	+	0.642698i	$0.777815\pi$
$72$	0	0
$73$	−2.32733	−0.272393	−0.136197	−	0.990682i	$-0.543488\pi$
$73$	−2.32733	−0.272393	−0.136197	+	0.990682i	$0.543488\pi$
$74$	27.0950	3.14973
$75$	0	0
$76$	−6.22018	−0.713504
$77$	0	0
$78$	0	0
$79$	−1.12811	−0.126922	−0.0634612	−	0.997984i	$-0.520214\pi$
$79$	−1.12811	−0.126922	−0.0634612	+	0.997984i	$0.520214\pi$
$80$	−2.39138	−0.267365
$81$	0	0
$82$	−27.6576	−3.05427
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−1.32733	−0.143969
$86$	−11.4374	−1.23333
$87$	0	0
$88$	0	0
$89$	−14.7828	−1.56697	−0.783485	−	0.621411i	$-0.786560\pi$
$89$	−14.7828	−1.56697	−0.783485	+	0.621411i	$0.786560\pi$
$90$	0	0
$91$	−27.7857	−2.91273
$92$	7.84682	0.818088
$93$	0	0
$94$	22.6116	2.33221
$95$	1.67267	0.171613
$96$	0	0
$97$	4.45544	0.452381	0.226191	−	0.974083i	$-0.427373\pi$
$97$	4.45544	0.452381	0.226191	+	0.974083i	$0.427373\pi$
$98$	−45.7066	−4.61706
$99$	0	0
$100$	3.71871	0.371871
$101$	−6.87484	−0.684072	−0.342036	−	0.939687i	$-0.611116\pi$
$101$	−6.87484	−0.684072	−0.342036	+	0.939687i	$0.611116\pi$
$102$	0	0
$103$	10.5835	1.04283	0.521414	−	0.853304i	$-0.325405\pi$
$103$	10.5835	1.04283	0.521414	+	0.853304i	$0.325405\pi$
$104$	22.3483	2.19143
$105$	0	0
$106$	−19.7007	−1.91350
$107$	−1.41940	−0.137219	−0.0686094	−	0.997644i	$-0.521856\pi$
$107$	−1.41940	−0.137219	−0.0686094	+	0.997644i	$0.521856\pi$
$108$	0	0
$109$	−15.2022	−1.45610	−0.728052	−	0.685522i	$-0.759574\pi$
$109$	−15.2022	−1.45610	−0.728052	+	0.685522i	$0.759574\pi$
$110$	0	0
$111$	0	0
$112$	12.2202	1.15470
$113$	−2.01802	−0.189839	−0.0949196	−	0.995485i	$-0.530259\pi$
$113$	−2.01802	−0.189839	−0.0949196	+	0.995485i	$0.530259\pi$
$114$	0	0
$115$	−2.11009	−0.196767
$116$	2.91087	0.270268
$117$	0	0
$118$	24.1340	2.22172
$119$	6.78276	0.621775
$120$	0	0
$121$	0	0
$122$	−1.86484	−0.168834
$123$	0	0
$124$	−16.5015	−1.48188
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.98493	0.886019	0.443010	−	0.896517i	$-0.353911\pi$
$127$	9.98493	0.886019	0.443010	+	0.896517i	$0.353911\pi$
$128$	20.7397	1.83315
$129$	0	0
$130$	−13.0029	−1.14043
$131$	−4.87484	−0.425917	−0.212958	−	0.977061i	$-0.568310\pi$
$131$	−4.87484	−0.425917	−0.212958	+	0.977061i	$0.568310\pi$
$132$	0	0
$133$	−8.54751	−0.741163
$134$	10.6547	0.920422
$135$	0	0
$136$	−5.45544	−0.467800
$137$	−19.7677	−1.68887	−0.844434	−	0.535659i	$-0.820063\pi$
$137$	−19.7677	−1.68887	−0.844434	+	0.535659i	$0.820063\pi$
$138$	0	0
$139$	6.76475	0.573778	0.286889	−	0.957964i	$-0.407379\pi$
$139$	6.76475	0.573778	0.286889	+	0.957964i	$0.407379\pi$
$140$	−19.0029	−1.60604
$141$	0	0
$142$	30.8748	2.59096
$143$	0	0
$144$	0	0
$145$	−0.782765	−0.0650051
$146$	5.56553	0.460607
$147$	0	0
$148$	−42.1340	−3.46339
$149$	0.128110	0.0104952	0.00524760	−	0.999986i	$-0.498330\pi$
$149$	0.128110	0.0104952	0.00524760	+	0.999986i	$0.498330\pi$
$150$	0	0
$151$	−5.45544	−0.443957	−0.221979	−	0.975052i	$-0.571252\pi$
$151$	−5.45544	−0.443957	−0.221979	+	0.975052i	$0.571252\pi$
$152$	6.87484	0.557623
$153$	0	0
$154$	0	0
$155$	4.43742	0.356422
$156$	0	0
$157$	11.9849	0.956502	0.478251	−	0.878223i	$-0.341271\pi$
$157$	11.9849	0.956502	0.478251	+	0.878223i	$0.341271\pi$
$158$	2.69774	0.214621
$159$	0	0
$160$	−2.50147	−0.197759
$161$	10.7828	0.849801
$162$	0	0
$163$	9.98493	0.782080	0.391040	−	0.920374i	$-0.372115\pi$
$163$	9.98493	0.782080	0.391040	+	0.920374i	$0.372115\pi$
$164$	43.0088	3.35843
$165$	0	0
$166$	9.56553	0.742429
$167$	7.67562	0.593957	0.296979	−	0.954884i	$-0.404021\pi$
$167$	7.67562	0.593957	0.296979	+	0.954884i	$0.404021\pi$
$168$	0	0
$169$	16.5655	1.27427
$170$	3.17415	0.243446
$171$	0	0
$172$	17.7857	1.35615
$173$	−4.69069	−0.356627	−0.178313	−	0.983974i	$-0.557064\pi$
$173$	−4.69069	−0.356627	−0.178313	+	0.983974i	$0.557064\pi$
$174$	0	0
$175$	5.11009	0.386287
$176$	0	0
$177$	0	0
$178$	35.3512	2.64969
$179$	−11.4374	−0.854873	−0.427436	−	0.904045i	$-0.640583\pi$
$179$	−11.4374	−0.854873	−0.427436	+	0.904045i	$0.640583\pi$
$180$	0	0
$181$	−8.11304	−0.603038	−0.301519	−	0.953460i	$-0.597494\pi$
$181$	−8.11304	−0.603038	−0.301519	+	0.953460i	$0.597494\pi$
$182$	66.4463	4.92532
$183$	0	0
$184$	−8.67267	−0.639358
$185$	11.3303	0.833018
$186$	0	0
$187$	0	0
$188$	−35.1620	−2.56445
$189$	0	0
$190$	−4.00000	−0.290191
$191$	1.00295	0.0725708	0.0362854	−	0.999341i	$-0.488447\pi$
$191$	1.00295	0.0725708	0.0362854	+	0.999341i	$0.488447\pi$
$192$	0	0
$193$	−15.9849	−1.15062	−0.575310	−	0.817935i	$-0.695119\pi$
$193$	−15.9849	−1.15062	−0.575310	+	0.817935i	$0.695119\pi$
$194$	−10.6547	−0.764960
$195$	0	0
$196$	71.0759	5.07685
$197$	16.2202	1.15564	0.577820	−	0.816164i	$-0.303904\pi$
$197$	16.2202	1.15564	0.577820	+	0.816164i	$0.303904\pi$
$198$	0	0
$199$	−1.34240	−0.0951600	−0.0475800	−	0.998867i	$-0.515151\pi$
$199$	−1.34240	−0.0951600	−0.0475800	+	0.998867i	$0.515151\pi$
$200$	−4.11009	−0.290627
$201$	0	0
$202$	16.4404	1.15674
$203$	4.00000	0.280745
$204$	0	0
$205$	−11.5655	−0.807772
$206$	−25.3093	−1.76338
$207$	0	0
$208$	−13.0029	−0.901592
$209$	0	0
$210$	0	0
$211$	−12.2231	−0.841475	−0.420738	−	0.907182i	$-0.638229\pi$
$211$	−12.2231	−0.841475	−0.420738	+	0.907182i	$0.638229\pi$
$212$	30.6355	2.10405
$213$	0	0
$214$	3.39433	0.232032
$215$	−4.78276	−0.326182
$216$	0	0
$217$	−22.6756	−1.53932
$218$	36.3542	2.46222
$219$	0	0
$220$	0	0
$221$	−7.21724	−0.485484
$222$	0	0
$223$	11.2022	0.750153	0.375076	−	0.926994i	$-0.377617\pi$
$223$	11.2022	0.750153	0.375076	+	0.926994i	$0.377617\pi$
$224$	12.7828	0.854084
$225$	0	0
$226$	4.82585	0.321011
$227$	−14.5446	−0.965357	−0.482678	−	0.875798i	$-0.660336\pi$
$227$	−14.5446	−0.965357	−0.482678	+	0.875798i	$0.660336\pi$
$228$	0	0
$229$	−14.9289	−0.986529	−0.493265	−	0.869879i	$-0.664197\pi$
$229$	−14.9289	−0.986529	−0.493265	+	0.869879i	$0.664197\pi$
$230$	5.04604	0.332726
$231$	0	0
$232$	−3.21724	−0.211222
$233$	27.5475	1.80470	0.902349	−	0.431007i	$-0.141841\pi$
$233$	27.5475	1.80470	0.902349	+	0.431007i	$0.141841\pi$
$234$	0	0
$235$	9.45544	0.616805
$236$	−37.5295	−2.44296
$237$	0	0
$238$	−16.2202	−1.05140
$239$	3.21724	0.208106	0.104053	−	0.994572i	$-0.466819\pi$
$239$	3.21724	0.208106	0.104053	+	0.994572i	$0.466819\pi$
$240$	0	0
$241$	−21.5835	−1.39032	−0.695159	−	0.718856i	$-0.744666\pi$
$241$	−21.5835	−1.39032	−0.695159	+	0.718856i	$0.744666\pi$
$242$	0	0
$243$	0	0
$244$	2.89991	0.185648
$245$	−19.1130	−1.22109
$246$	0	0
$247$	9.09502	0.578702
$248$	18.2382	1.15813
$249$	0	0
$250$	2.39138	0.151244
$251$	5.43742	0.343207	0.171603	−	0.985166i	$-0.445105\pi$
$251$	5.43742	0.343207	0.171603	+	0.985166i	$0.445105\pi$
$252$	0	0
$253$	0	0
$254$	−23.8778	−1.49823
$255$	0	0
$256$	−28.0670	−1.75419
$257$	12.2022	0.761150	0.380575	−	0.924750i	$-0.375726\pi$
$257$	12.2022	0.761150	0.380575	+	0.924750i	$0.375726\pi$
$258$	0	0
$259$	−57.8988	−3.59765
$260$	20.2202	1.25400
$261$	0	0
$262$	11.6576	0.720209
$263$	1.85387	0.114315	0.0571573	−	0.998365i	$-0.481796\pi$
$263$	1.85387	0.114315	0.0571573	+	0.998365i	$0.481796\pi$
$264$	0	0
$265$	−8.23820	−0.506069
$266$	20.4404	1.25328
$267$	0	0
$268$	−16.5685	−1.01208
$269$	9.74968	0.594448	0.297224	−	0.954808i	$-0.403939\pi$
$269$	9.74968	0.594448	0.297224	+	0.954808i	$0.403939\pi$
$270$	0	0
$271$	15.2051	0.923645	0.461822	−	0.886972i	$-0.347196\pi$
$271$	15.2051	0.923645	0.461822	+	0.886972i	$0.347196\pi$
$272$	3.17415	0.192461
$273$	0	0
$274$	47.2721	2.85581
$275$	0	0
$276$	0	0
$277$	−12.3332	−0.741032	−0.370516	−	0.928826i	$-0.620819\pi$
$277$	−12.3332	−0.741032	−0.370516	+	0.928826i	$0.620819\pi$
$278$	−16.1771	−0.970238
$279$	0	0
$280$	21.0029	1.25517
$281$	−14.7828	−0.881866	−0.440933	−	0.897540i	$-0.645352\pi$
$281$	−14.7828	−0.881866	−0.440933	+	0.897540i	$0.645352\pi$
$282$	0	0
$283$	29.0239	1.72529	0.862646	−	0.505808i	$-0.168805\pi$
$283$	29.0239	1.72529	0.862646	+	0.505808i	$0.168805\pi$
$284$	−48.0118	−2.84898
$285$	0	0
$286$	0	0
$287$	59.1009	3.48862
$288$	0	0
$289$	−15.2382	−0.896365
$290$	1.87189	0.109921
$291$	0	0
$292$	−8.65465	−0.506475
$293$	−9.32733	−0.544908	−0.272454	−	0.962169i	$-0.587835\pi$
$293$	−9.32733	−0.544908	−0.272454	+	0.962169i	$0.587835\pi$
$294$	0	0
$295$	10.0921	0.587583
$296$	46.5685	2.70674
$297$	0	0
$298$	−0.306360	−0.0177470
$299$	−11.4735	−0.663527
$300$	0	0
$301$	24.4404	1.40872
$302$	13.0460	0.750715
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−0.779816	−0.0446521
$306$	0	0
$307$	−13.2382	−0.755544	−0.377772	−	0.925899i	$-0.623310\pi$
$307$	−13.2382	−0.755544	−0.377772	+	0.925899i	$0.623310\pi$
$308$	0	0
$309$	0	0
$310$	−10.6116	−0.602696
$311$	−2.78276	−0.157796	−0.0788981	−	0.996883i	$-0.525140\pi$
$311$	−2.78276	−0.157796	−0.0788981	+	0.996883i	$0.525140\pi$
$312$	0	0
$313$	3.87189	0.218852	0.109426	−	0.993995i	$-0.465099\pi$
$313$	3.87189	0.218852	0.109426	+	0.993995i	$0.465099\pi$
$314$	−28.6606	−1.61741
$315$	0	0
$316$	−4.19511	−0.235994
$317$	9.10714	0.511508	0.255754	−	0.966742i	$-0.417676\pi$
$317$	9.10714	0.511508	0.255754	+	0.966742i	$0.417676\pi$
$318$	0	0
$319$	0	0
$320$	10.7647	0.601768
$321$	0	0
$322$	−25.7857	−1.43698
$323$	−2.22018	−0.123534
$324$	0	0
$325$	−5.43742	−0.301614
$326$	−23.8778	−1.32247
$327$	0	0
$328$	−47.5354	−2.62470
$329$	−48.3182	−2.66387
$330$	0	0
$331$	−6.21724	−0.341730	−0.170865	−	0.985294i	$-0.554656\pi$
$331$	−6.21724	−0.341730	−0.170865	+	0.985294i	$0.554656\pi$
$332$	−14.8748	−0.816363
$333$	0	0
$334$	−18.3553	−1.00436
$335$	4.45544	0.243427
$336$	0	0
$337$	−26.9879	−1.47012	−0.735062	−	0.678000i	$-0.762847\pi$
$337$	−26.9879	−1.47012	−0.735062	+	0.678000i	$0.762847\pi$
$338$	−39.6145	−2.15475
$339$	0	0
$340$	−4.93594	−0.267689
$341$	0	0
$342$	0	0
$343$	61.8988	3.34222
$344$	−19.6576	−1.05987
$345$	0	0
$346$	11.2172	0.603042
$347$	−12.1160	−0.650420	−0.325210	−	0.945642i	$-0.605435\pi$
$347$	−12.1160	−0.650420	−0.325210	+	0.945642i	$0.605435\pi$
$348$	0	0
$349$	−23.4764	−1.25666	−0.628332	−	0.777946i	$-0.716262\pi$
$349$	−23.4764	−1.25666	−0.628332	+	0.777946i	$0.716262\pi$
$350$	−12.2202	−0.653196
$351$	0	0
$352$	0	0
$353$	−17.5115	−0.932042	−0.466021	−	0.884774i	$-0.654313\pi$
$353$	−17.5115	−0.932042	−0.466021	+	0.884774i	$0.654313\pi$
$354$	0	0
$355$	12.9109	0.685238
$356$	−54.9728	−2.91355
$357$	0	0
$358$	27.3512	1.44556
$359$	12.8388	0.677606	0.338803	−	0.940857i	$-0.389978\pi$
$359$	12.8388	0.677606	0.338803	+	0.940857i	$0.389978\pi$
$360$	0	0
$361$	−16.2022	−0.852746
$362$	19.4014	1.01971
$363$	0	0
$364$	−103.327	−5.41581
$365$	2.32733	0.121818
$366$	0	0
$367$	−4.81880	−0.251539	−0.125770	−	0.992059i	$-0.540140\pi$
$367$	−4.81880	−0.251539	−0.125770	+	0.992059i	$0.540140\pi$
$368$	5.04604	0.263043
$369$	0	0
$370$	−27.0950	−1.40860
$371$	42.0980	2.18562
$372$	0	0
$373$	−16.1071	−0.833996	−0.416998	−	0.908907i	$-0.636918\pi$
$373$	−16.1071	−0.833996	−0.416998	+	0.908907i	$0.636918\pi$
$374$	0	0
$375$	0	0
$376$	38.8627	2.00419
$377$	−4.25622	−0.219206
$378$	0	0
$379$	8.36631	0.429749	0.214874	−	0.976642i	$-0.431066\pi$
$379$	8.36631	0.429749	0.214874	+	0.976642i	$0.431066\pi$
$380$	6.22018	0.319089
$381$	0	0
$382$	−2.39843	−0.122715
$383$	11.3453	0.579720	0.289860	−	0.957069i	$-0.406391\pi$
$383$	11.3453	0.579720	0.289860	+	0.957069i	$0.406391\pi$
$384$	0	0
$385$	0	0
$386$	38.2261	1.94566
$387$	0	0
$388$	16.5685	0.841137
$389$	−5.18120	−0.262697	−0.131349	−	0.991336i	$-0.541931\pi$
$389$	−5.18120	−0.262697	−0.131349	+	0.991336i	$0.541931\pi$
$390$	0	0
$391$	2.80078	0.141642
$392$	−78.5564	−3.96770
$393$	0	0
$394$	−38.7887	−1.95414
$395$	1.12811	0.0567614
$396$	0	0
$397$	0.767696	0.0385295	0.0192648	−	0.999814i	$-0.493867\pi$
$397$	0.767696	0.0385295	0.0192648	+	0.999814i	$0.493867\pi$
$398$	3.21018	0.160912
$399$	0	0
$400$	2.39138	0.119569
$401$	−18.0360	−0.900677	−0.450338	−	0.892858i	$-0.648697\pi$
$401$	−18.0360	−0.900677	−0.450338	+	0.892858i	$0.648697\pi$
$402$	0	0
$403$	24.1281	1.20191
$404$	−25.5655	−1.27193
$405$	0	0
$406$	−9.56553	−0.474729
$407$	0	0
$408$	0	0
$409$	2.55963	0.126566	0.0632828	−	0.997996i	$-0.479843\pi$
$409$	2.55963	0.126566	0.0632828	+	0.997996i	$0.479843\pi$
$410$	27.6576	1.36591
$411$	0	0
$412$	39.3571	1.93899
$413$	−51.5714	−2.53766
$414$	0	0
$415$	4.00000	0.196352
$416$	−13.6016	−0.666872
$417$	0	0
$418$	0	0
$419$	−11.0950	−0.542027	−0.271014	−	0.962575i	$-0.587359\pi$
$419$	−11.0950	−0.542027	−0.271014	+	0.962575i	$0.587359\pi$
$420$	0	0
$421$	15.3332	0.747296	0.373648	−	0.927571i	$-0.378107\pi$
$421$	15.3332	0.747296	0.373648	+	0.927571i	$0.378107\pi$
$422$	29.2302	1.42290
$423$	0	0
$424$	−33.8598	−1.64438
$425$	1.32733	0.0643848
$426$	0	0
$427$	3.98493	0.192844
$428$	−5.27834	−0.255138
$429$	0	0
$430$	11.4374	0.551561
$431$	−3.68774	−0.177632	−0.0888161	−	0.996048i	$-0.528308\pi$
$431$	−3.68774	−0.177632	−0.0888161	+	0.996048i	$0.528308\pi$
$432$	0	0
$433$	27.9489	1.34314	0.671569	−	0.740942i	$-0.265621\pi$
$433$	27.9489	1.34314	0.671569	+	0.740942i	$0.265621\pi$
$434$	54.2261	2.60294
$435$	0	0
$436$	−56.5324	−2.70741
$437$	−3.52949	−0.168839
$438$	0	0
$439$	24.0029	1.14560	0.572799	−	0.819696i	$-0.305858\pi$
$439$	24.0029	1.14560	0.572799	+	0.819696i	$0.305858\pi$
$440$	0	0
$441$	0	0
$442$	17.2592	0.820935
$443$	6.22018	0.295530	0.147765	−	0.989023i	$-0.452792\pi$
$443$	6.22018	0.295530	0.147765	+	0.989023i	$0.452792\pi$
$444$	0	0
$445$	14.7828	0.700770
$446$	−26.7887	−1.26848
$447$	0	0
$448$	−55.0088	−2.59892
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	0	0
$452$	−7.50442	−0.352978
$453$	0	0
$454$	34.7816	1.63238
$455$	27.7857	1.30261
$456$	0	0
$457$	16.8748	0.789372	0.394686	−	0.918816i	$-0.370853\pi$
$457$	16.8748	0.789372	0.394686	+	0.918816i	$0.370853\pi$
$458$	35.7007	1.66818
$459$	0	0
$460$	−7.84682	−0.365860
$461$	23.8719	1.11182	0.555912	−	0.831241i	$-0.312369\pi$
$461$	23.8719	1.11182	0.555912	+	0.831241i	$0.312369\pi$
$462$	0	0
$463$	26.2261	1.21883	0.609415	−	0.792852i	$-0.291404\pi$
$463$	26.2261	1.21883	0.609415	+	0.792852i	$0.291404\pi$
$464$	1.87189	0.0869003
$465$	0	0
$466$	−65.8766	−3.05168
$467$	34.5505	1.59880	0.799402	−	0.600796i	$-0.205150\pi$
$467$	34.5505	1.59880	0.799402	+	0.600796i	$0.205150\pi$
$468$	0	0
$469$	−22.7677	−1.05131
$470$	−22.6116	−1.04299
$471$	0	0
$472$	41.4794	1.90924
$473$	0	0
$474$	0	0
$475$	−1.67267	−0.0767475
$476$	25.2231	1.15610
$477$	0	0
$478$	−7.69364	−0.351899
$479$	−35.2290	−1.60966	−0.804828	−	0.593508i	$-0.797742\pi$
$479$	−35.2290	−1.60966	−0.804828	+	0.593508i	$0.797742\pi$
$480$	0	0
$481$	61.6075	2.80906
$482$	51.6145	2.35098
$483$	0	0
$484$	0	0
$485$	−4.45544	−0.202311
$486$	0	0
$487$	−19.6576	−0.890771	−0.445386	−	0.895339i	$-0.646933\pi$
$487$	−19.6576	−0.890771	−0.445386	+	0.895339i	$0.646933\pi$
$488$	−3.20511	−0.145089
$489$	0	0
$490$	45.7066	2.06481
$491$	−0.470507	−0.0212337	−0.0106168	−	0.999944i	$-0.503380\pi$
$491$	−0.470507	−0.0212337	−0.0106168	+	0.999944i	$0.503380\pi$
$492$	0	0
$493$	1.03899	0.0467935
$494$	−21.7497	−0.978564
$495$	0	0
$496$	−10.6116	−0.476473
$497$	−65.9758	−2.95942
$498$	0	0
$499$	−29.8568	−1.33657	−0.668287	−	0.743903i	$-0.732972\pi$
$499$	−29.8568	−1.33657	−0.668287	+	0.743903i	$0.732972\pi$
$500$	−3.71871	−0.166306
$501$	0	0
$502$	−13.0029	−0.580350
$503$	25.8598	1.15303	0.576515	−	0.817087i	$-0.304412\pi$
$503$	25.8598	1.15303	0.576515	+	0.817087i	$0.304412\pi$
$504$	0	0
$505$	6.87484	0.305926
$506$	0	0
$507$	0	0
$508$	37.1311	1.64742
$509$	25.4433	1.12776	0.563878	−	0.825858i	$-0.309309\pi$
$509$	25.4433	1.12776	0.563878	+	0.825858i	$0.309309\pi$
$510$	0	0
$511$	−11.8929	−0.526109
$512$	25.6396	1.13312
$513$	0	0
$514$	−29.1800	−1.28708
$515$	−10.5835	−0.466367
$516$	0	0
$517$	0	0
$518$	138.458	6.08350
$519$	0	0
$520$	−22.3483	−0.980038
$521$	40.5383	1.77602	0.888008	−	0.459827i	$-0.152089\pi$
$521$	40.5383	1.77602	0.888008	+	0.459827i	$0.152089\pi$
$522$	0	0
$523$	−35.3303	−1.54489	−0.772443	−	0.635085i	$-0.780965\pi$
$523$	−35.3303	−1.54489	−0.772443	+	0.635085i	$0.780965\pi$
$524$	−18.1281	−0.791930
$525$	0	0
$526$	−4.43332	−0.193302
$527$	−5.88991	−0.256568
$528$	0	0
$529$	−18.5475	−0.806414
$530$	19.7007	0.855743
$531$	0	0
$532$	−31.7857	−1.37809
$533$	−62.8866	−2.72392
$534$	0	0
$535$	1.41940	0.0613661
$536$	18.3123	0.790969
$537$	0	0
$538$	−23.3152	−1.00519
$539$	0	0
$540$	0	0
$541$	22.6966	0.975803	0.487901	−	0.872899i	$-0.337763\pi$
$541$	22.6966	0.975803	0.487901	+	0.872899i	$0.337763\pi$
$542$	−36.3612	−1.56185
$543$	0	0
$544$	3.32028	0.142356
$545$	15.2022	0.651189
$546$	0	0
$547$	−7.47346	−0.319542	−0.159771	−	0.987154i	$-0.551076\pi$
$547$	−7.47346	−0.319542	−0.159771	+	0.987154i	$0.551076\pi$
$548$	−73.5103	−3.14021
$549$	0	0
$550$	0	0
$551$	−1.30931	−0.0557784
$552$	0	0
$553$	−5.76475	−0.245142
$554$	29.4935	1.25306
$555$	0	0
$556$	25.1561	1.06686
$557$	−25.8037	−1.09334	−0.546670	−	0.837348i	$-0.684105\pi$
$557$	−25.8037	−1.09334	−0.546670	+	0.837348i	$0.684105\pi$
$558$	0	0
$559$	−26.0059	−1.09993
$560$	−12.2202	−0.516397
$561$	0	0
$562$	35.3512	1.49120
$563$	−9.74968	−0.410900	−0.205450	−	0.978668i	$-0.565866\pi$
$563$	−9.74968	−0.410900	−0.205450	+	0.978668i	$0.565866\pi$
$564$	0	0
$565$	2.01802	0.0848987
$566$	−69.4073	−2.91741
$567$	0	0
$568$	53.0649	2.22655
$569$	−21.2231	−0.889720	−0.444860	−	0.895600i	$-0.646747\pi$
$569$	−21.2231	−0.889720	−0.444860	+	0.895600i	$0.646747\pi$
$570$	0	0
$571$	1.81880	0.0761144	0.0380572	−	0.999276i	$-0.487883\pi$
$571$	1.81880	0.0761144	0.0380572	+	0.999276i	$0.487883\pi$
$572$	0	0
$573$	0	0
$574$	−141.333	−5.89912
$575$	2.11009	0.0879969
$576$	0	0
$577$	−39.4643	−1.64292	−0.821460	−	0.570266i	$-0.806840\pi$
$577$	−39.4643	−1.64292	−0.821460	+	0.570266i	$0.806840\pi$
$578$	36.4404	1.51572
$579$	0	0
$580$	−2.91087	−0.120868
$581$	−20.4404	−0.848009
$582$	0	0
$583$	0	0
$584$	9.56553	0.395824
$585$	0	0
$586$	22.3052	0.921420
$587$	−29.6396	−1.22336	−0.611678	−	0.791107i	$-0.709505\pi$
$587$	−29.6396	−1.22336	−0.611678	+	0.791107i	$0.709505\pi$
$588$	0	0
$589$	7.42235	0.305833
$590$	−24.1340	−0.993581
$591$	0	0
$592$	−27.0950	−1.11360
$593$	−29.1311	−1.19627	−0.598135	−	0.801396i	$-0.704091\pi$
$593$	−29.1311	−1.19627	−0.598135	+	0.801396i	$0.704091\pi$
$594$	0	0
$595$	−6.78276	−0.278066
$596$	0.476404	0.0195143
$597$	0	0
$598$	27.4374	1.12200
$599$	−44.1039	−1.80204	−0.901018	−	0.433782i	$-0.857179\pi$
$599$	−44.1039	−1.80204	−0.901018	+	0.433782i	$0.857179\pi$
$600$	0	0
$601$	34.5174	1.40799	0.703997	−	0.710203i	$-0.251397\pi$
$601$	34.5174	1.40799	0.703997	+	0.710203i	$0.251397\pi$
$602$	−58.4463	−2.38209
$603$	0	0
$604$	−20.2872	−0.825474
$605$	0	0
$606$	0	0
$607$	−2.99705	−0.121647	−0.0608233	−	0.998149i	$-0.519373\pi$
$607$	−2.99705	−0.121647	−0.0608233	+	0.998149i	$0.519373\pi$
$608$	−4.18415	−0.169690
$609$	0	0
$610$	1.86484	0.0755051
$611$	51.4132	2.07995
$612$	0	0
$613$	0.761798	0.0307687	0.0153844	−	0.999882i	$-0.495103\pi$
$613$	0.761798	0.0307687	0.0153844	+	0.999882i	$0.495103\pi$
$614$	31.6576	1.27760
$615$	0	0
$616$	0	0
$617$	23.8217	0.959028	0.479514	−	0.877534i	$-0.340813\pi$
$617$	23.8217	0.959028	0.479514	+	0.877534i	$0.340813\pi$
$618$	0	0
$619$	−30.4463	−1.22374	−0.611869	−	0.790959i	$-0.709582\pi$
$619$	−30.4463	−1.22374	−0.611869	+	0.790959i	$0.709582\pi$
$620$	16.5015	0.662715
$621$	0	0
$622$	6.65465	0.266827
$623$	−75.5413	−3.02650
$624$	0	0
$625$	1.00000	0.0400000
$626$	−9.25917	−0.370071
$627$	0	0
$628$	44.5685	1.77848
$629$	−15.0390	−0.599644
$630$	0	0
$631$	−10.5446	−0.419772	−0.209886	−	0.977726i	$-0.567309\pi$
$631$	−10.5446	−0.419772	−0.209886	+	0.977726i	$0.567309\pi$
$632$	4.63664	0.184435
$633$	0	0
$634$	−21.7787	−0.864941
$635$	−9.98493	−0.396240
$636$	0	0
$637$	−103.926	−4.11768
$638$	0	0
$639$	0	0
$640$	−20.7397	−0.819808
$641$	−17.4014	−0.687313	−0.343657	−	0.939095i	$-0.611666\pi$
$641$	−17.4014	−0.687313	−0.343657	+	0.939095i	$0.611666\pi$
$642$	0	0
$643$	−32.3693	−1.27652	−0.638260	−	0.769821i	$-0.720345\pi$
$643$	−32.3693	−1.27652	−0.638260	+	0.769821i	$0.720345\pi$
$644$	40.0980	1.58008
$645$	0	0
$646$	5.30931	0.208892
$647$	−38.3362	−1.50715	−0.753575	−	0.657362i	$-0.771672\pi$
$647$	−38.3362	−1.50715	−0.753575	+	0.657362i	$0.771672\pi$
$648$	0	0
$649$	0	0
$650$	13.0029	0.510018
$651$	0	0
$652$	37.1311	1.45416
$653$	−3.74968	−0.146736	−0.0733681	−	0.997305i	$-0.523375\pi$
$653$	−3.74968	−0.146736	−0.0733681	+	0.997305i	$0.523375\pi$
$654$	0	0
$655$	4.87484	0.190476
$656$	27.6576	1.07985
$657$	0	0
$658$	115.547	4.50450
$659$	−15.9079	−0.619685	−0.309842	−	0.950788i	$-0.600276\pi$
$659$	−15.9079	−0.619685	−0.309842	+	0.950788i	$0.600276\pi$
$660$	0	0
$661$	17.2382	0.670488	0.335244	−	0.942131i	$-0.391181\pi$
$661$	17.2382	0.670488	0.335244	+	0.942131i	$0.391181\pi$
$662$	14.8678	0.577853
$663$	0	0
$664$	16.4404	0.638010
$665$	8.54751	0.331458
$666$	0	0
$667$	1.65171	0.0639543
$668$	28.5434	1.10438
$669$	0	0
$670$	−10.6547	−0.411625
$671$	0	0
$672$	0	0
$673$	−21.2942	−0.820833	−0.410416	−	0.911898i	$-0.634617\pi$
$673$	−21.2942	−0.820833	−0.410416	+	0.911898i	$0.634617\pi$
$674$	64.5383	2.48592
$675$	0	0
$676$	61.6024	2.36932
$677$	−38.0419	−1.46207	−0.731035	−	0.682340i	$-0.760962\pi$
$677$	−38.0419	−1.46207	−0.731035	+	0.682340i	$0.760962\pi$
$678$	0	0
$679$	22.7677	0.873744
$680$	5.45544	0.209206
$681$	0	0
$682$	0	0
$683$	−33.0950	−1.26635	−0.633173	−	0.774010i	$-0.718248\pi$
$683$	−33.0950	−1.26635	−0.633173	+	0.774010i	$0.718248\pi$
$684$	0	0
$685$	19.7677	0.755285
$686$	−148.024	−5.65157
$687$	0	0
$688$	11.4374	0.436048
$689$	−44.7946	−1.70654
$690$	0	0
$691$	−32.8778	−1.25073	−0.625365	−	0.780332i	$-0.715050\pi$
$691$	−32.8778	−1.25073	−0.625365	+	0.780332i	$0.715050\pi$
$692$	−17.4433	−0.663095
$693$	0	0
$694$	28.9740	1.09984
$695$	−6.76475	−0.256601
$696$	0	0
$697$	15.3512	0.581470
$698$	56.1411	2.12497
$699$	0	0
$700$	19.0029	0.718244
$701$	−35.2231	−1.33036	−0.665180	−	0.746683i	$-0.731645\pi$
$701$	−35.2231	−1.33036	−0.665180	+	0.746683i	$0.731645\pi$
$702$	0	0
$703$	18.9518	0.714782
$704$	0	0
$705$	0	0
$706$	41.8766	1.57605
$707$	−35.1311	−1.32124
$708$	0	0
$709$	1.14318	0.0429330	0.0214665	−	0.999770i	$-0.493166\pi$
$709$	1.14318	0.0429330	0.0214665	+	0.999770i	$0.493166\pi$
$710$	−30.8748	−1.15871
$711$	0	0
$712$	60.7585	2.27702
$713$	−9.36336	−0.350661
$714$	0	0
$715$	0	0
$716$	−42.5324	−1.58951
$717$	0	0
$718$	−30.7025	−1.14581
$719$	−1.16120	−0.0433054	−0.0216527	−	0.999766i	$-0.506893\pi$
$719$	−1.16120	−0.0433054	−0.0216527	+	0.999766i	$0.506893\pi$
$720$	0	0
$721$	54.0829	2.01415
$722$	38.7456	1.44196
$723$	0	0
$724$	−30.1700	−1.12126
$725$	0.782765	0.0290712
$726$	0	0
$727$	6.65465	0.246807	0.123404	−	0.992357i	$-0.460619\pi$
$727$	6.65465	0.246807	0.123404	+	0.992357i	$0.460619\pi$
$728$	114.202	4.23260
$729$	0	0
$730$	−5.56553	−0.205989
$731$	6.34829	0.234800
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	11.5236	0.425344
$735$	0	0
$736$	5.27834	0.194562
$737$	0	0
$738$	0	0
$739$	−4.47346	−0.164559	−0.0822794	−	0.996609i	$-0.526220\pi$
$739$	−4.47346	−0.164559	−0.0822794	+	0.996609i	$0.526220\pi$
$740$	42.1340	1.54888
$741$	0	0
$742$	−100.672	−3.69580
$743$	24.3303	0.892591	0.446296	−	0.894886i	$-0.352743\pi$
$743$	24.3303	0.892591	0.446296	+	0.894886i	$0.352743\pi$
$744$	0	0
$745$	−0.128110	−0.00469359
$746$	38.5183	1.41026
$747$	0	0
$748$	0	0
$749$	−7.25327	−0.265029
$750$	0	0
$751$	32.6635	1.19191	0.595954	−	0.803019i	$-0.296774\pi$
$751$	32.6635	1.19191	0.595954	+	0.803019i	$0.296774\pi$
$752$	−22.6116	−0.824559
$753$	0	0
$754$	10.1783	0.370670
$755$	5.45544	0.198544
$756$	0	0
$757$	−12.0711	−0.438732	−0.219366	−	0.975643i	$-0.570399\pi$
$757$	−12.0711	−0.438732	−0.219366	+	0.975643i	$0.570399\pi$
$758$	−20.0071	−0.726689
$759$	0	0
$760$	−6.87484	−0.249377
$761$	16.3123	0.591319	0.295659	−	0.955293i	$-0.404461\pi$
$761$	16.3123	0.591319	0.295659	+	0.955293i	$0.404461\pi$
$762$	0	0
$763$	−77.6845	−2.81237
$764$	3.72968	0.134935
$765$	0	0
$766$	−27.1311	−0.980285
$767$	54.8748	1.98142
$768$	0	0
$769$	14.6016	0.526546	0.263273	−	0.964721i	$-0.415198\pi$
$769$	14.6016	0.526546	0.263273	+	0.964721i	$0.415198\pi$
$770$	0	0
$771$	0	0
$772$	−59.4433	−2.13941
$773$	26.2742	0.945019	0.472509	−	0.881326i	$-0.343348\pi$
$773$	26.2742	0.945019	0.472509	+	0.881326i	$0.343348\pi$
$774$	0	0
$775$	−4.43742	−0.159397
$776$	−18.3123	−0.657372
$777$	0	0
$778$	12.3902	0.444211
$779$	−19.3453	−0.693119
$780$	0	0
$781$	0	0
$782$	−6.69774	−0.239511
$783$	0	0
$784$	45.7066	1.63238
$785$	−11.9849	−0.427761
$786$	0	0
$787$	−44.8807	−1.59983	−0.799913	−	0.600116i	$-0.795121\pi$
$787$	−44.8807	−1.59983	−0.799913	+	0.600116i	$0.795121\pi$
$788$	60.3182	2.14875
$789$	0	0
$790$	−2.69774	−0.0959814
$791$	−10.3123	−0.366662
$792$	0	0
$793$	−4.24019	−0.150573
$794$	−1.83585	−0.0651520
$795$	0	0
$796$	−4.99198	−0.176936
$797$	38.5124	1.36418	0.682090	−	0.731268i	$-0.261071\pi$
$797$	38.5124	1.36418	0.682090	+	0.731268i	$0.261071\pi$
$798$	0	0
$799$	−12.5505	−0.444003
$800$	2.50147	0.0884405
$801$	0	0
$802$	43.1311	1.52301
$803$	0	0
$804$	0	0
$805$	−10.7828	−0.380043
$806$	−57.6995	−2.03238
$807$	0	0
$808$	28.2562	0.994050
$809$	9.09502	0.319764	0.159882	−	0.987136i	$-0.448889\pi$
$809$	9.09502	0.319764	0.159882	+	0.987136i	$0.448889\pi$
$810$	0	0
$811$	16.7297	0.587458	0.293729	−	0.955889i	$-0.405104\pi$
$811$	16.7297	0.587458	0.293729	+	0.955889i	$0.405104\pi$
$812$	14.8748	0.522005
$813$	0	0
$814$	0	0
$815$	−9.98493	−0.349757
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−6.12106	−0.214018
$819$	0	0
$820$	−43.0088	−1.50193
$821$	−7.38138	−0.257612	−0.128806	−	0.991670i	$-0.541114\pi$
$821$	−7.38138	−0.257612	−0.128806	+	0.991670i	$0.541114\pi$
$822$	0	0
$823$	−38.0410	−1.32602	−0.663012	−	0.748608i	$-0.730722\pi$
$823$	−38.0410	−1.32602	−0.663012	+	0.748608i	$0.730722\pi$
$824$	−43.4994	−1.51537
$825$	0	0
$826$	123.327	4.29110
$827$	46.1900	1.60619	0.803093	−	0.595854i	$-0.203186\pi$
$827$	46.1900	1.60619	0.803093	+	0.595854i	$0.203186\pi$
$828$	0	0
$829$	8.55963	0.297288	0.148644	−	0.988891i	$-0.452509\pi$
$829$	8.55963	0.297288	0.148644	+	0.988891i	$0.452509\pi$
$830$	−9.56553	−0.332024
$831$	0	0
$832$	58.5324	2.02925
$833$	25.3693	0.878993
$834$	0	0
$835$	−7.67562	−0.265626
$836$	0	0
$837$	0	0
$838$	26.5324	0.916548
$839$	−55.7556	−1.92490	−0.962448	−	0.271466i	$-0.912492\pi$
$839$	−55.7556	−1.92490	−0.962448	+	0.271466i	$0.912492\pi$
$840$	0	0
$841$	−28.3873	−0.978872
$842$	−36.6676	−1.26365
$843$	0	0
$844$	−45.4543	−1.56460
$845$	−16.5655	−0.569872
$846$	0	0
$847$	0	0
$848$	19.7007	0.676525
$849$	0	0
$850$	−3.17415	−0.108872
$851$	−23.9079	−0.819553
$852$	0	0
$853$	12.7258	0.435722	0.217861	−	0.975980i	$-0.430092\pi$
$853$	12.7258	0.435722	0.217861	+	0.975980i	$0.430092\pi$
$854$	−9.52949	−0.326093
$855$	0	0
$856$	5.83387	0.199398
$857$	13.5835	0.464005	0.232003	−	0.972715i	$-0.425472\pi$
$857$	13.5835	0.464005	0.232003	+	0.972715i	$0.425472\pi$
$858$	0	0
$859$	11.0239	0.376131	0.188066	−	0.982156i	$-0.439778\pi$
$859$	11.0239	0.376131	0.188066	+	0.982156i	$0.439778\pi$
$860$	−17.7857	−0.606488
$861$	0	0
$862$	8.81880	0.300370
$863$	46.1900	1.57233	0.786164	−	0.618018i	$-0.212064\pi$
$863$	46.1900	1.57233	0.786164	+	0.618018i	$0.212064\pi$
$864$	0	0
$865$	4.69069	0.159488
$866$	−66.8365	−2.27120
$867$	0	0
$868$	−84.3241	−2.86214
$869$	0	0
$870$	0	0
$871$	24.2261	0.820869
$872$	62.4823	2.11592
$873$	0	0
$874$	8.44037	0.285500
$875$	−5.11009	−0.172753
$876$	0	0
$877$	22.3634	0.755157	0.377579	−	0.925978i	$-0.376757\pi$
$877$	22.3634	0.755157	0.377579	+	0.925978i	$0.376757\pi$
$878$	−57.4002	−1.93716
$879$	0	0
$880$	0	0
$881$	−36.5124	−1.23014	−0.615068	−	0.788474i	$-0.710871\pi$
$881$	−36.5124	−1.23014	−0.615068	+	0.788474i	$0.710871\pi$
$882$	0	0
$883$	−42.4672	−1.42914	−0.714568	−	0.699566i	$-0.753377\pi$
$883$	−42.4672	−1.42914	−0.714568	+	0.699566i	$0.753377\pi$
$884$	−26.8388	−0.902687
$885$	0	0
$886$	−14.8748	−0.499730
$887$	17.5354	0.588781	0.294390	−	0.955685i	$-0.404883\pi$
$887$	17.5354	0.588781	0.294390	+	0.955685i	$0.404883\pi$
$888$	0	0
$889$	51.0239	1.71129
$890$	−35.3512	−1.18498
$891$	0	0
$892$	41.6576	1.39480
$893$	15.8159	0.529257
$894$	0	0
$895$	11.4374	0.382311
$896$	105.982	3.54060
$897$	0	0
$898$	52.6104	1.75563
$899$	−3.47346	−0.115846
$900$	0	0
$901$	10.9348	0.364291
$902$	0	0
$903$	0	0
$904$	8.29424	0.275862
$905$	8.11304	0.269687
$906$	0	0
$907$	−24.5835	−0.816283	−0.408142	−	0.912919i	$-0.633823\pi$
$907$	−24.5835	−0.816283	−0.408142	+	0.912919i	$0.633823\pi$
$908$	−54.0870	−1.79494
$909$	0	0
$910$	−66.4463	−2.20267
$911$	42.5685	1.41036	0.705178	−	0.709030i	$-0.250867\pi$
$911$	42.5685	1.41036	0.705178	+	0.709030i	$0.250867\pi$
$912$	0	0
$913$	0	0
$914$	−40.3542	−1.33480
$915$	0	0
$916$	−55.5162	−1.83431
$917$	−24.9109	−0.822630
$918$	0	0
$919$	−32.1491	−1.06050	−0.530250	−	0.847841i	$-0.677902\pi$
$919$	−32.1491	−1.06050	−0.530250	+	0.847841i	$0.677902\pi$
$920$	8.67267	0.285930
$921$	0	0
$922$	−57.0868	−1.88005
$923$	70.2018	2.31072
$924$	0	0
$925$	−11.3303	−0.372537
$926$	−62.7166	−2.06100
$927$	0	0
$928$	1.95807	0.0642767
$929$	−8.17825	−0.268320	−0.134160	−	0.990960i	$-0.542834\pi$
$929$	−8.17825	−0.268320	−0.134160	+	0.990960i	$0.542834\pi$
$930$	0	0
$931$	−31.9699	−1.04777
$932$	102.441	3.35557
$933$	0	0
$934$	−82.6234	−2.70352
$935$	0	0
$936$	0	0
$937$	19.0741	0.623122	0.311561	−	0.950226i	$-0.399148\pi$
$937$	19.0741	0.623122	0.311561	+	0.950226i	$0.399148\pi$
$938$	54.4463	1.77773
$939$	0	0
$940$	35.1620	1.14686
$941$	−31.0029	−1.01067	−0.505334	−	0.862924i	$-0.668631\pi$
$941$	−31.0029	−1.01067	−0.505334	+	0.862924i	$0.668631\pi$
$942$	0	0
$943$	24.4043	0.794714
$944$	−24.1340	−0.785495
$945$	0	0
$946$	0	0
$947$	−14.3604	−0.466651	−0.233325	−	0.972399i	$-0.574961\pi$
$947$	−14.3604	−0.466651	−0.233325	+	0.972399i	$0.574961\pi$
$948$	0	0
$949$	12.6547	0.410787
$950$	4.00000	0.129777
$951$	0	0
$952$	−27.8778	−0.903524
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	0	0
$955$	−1.00295	−0.0324547
$956$	11.9640	0.386942
$957$	0	0
$958$	84.2461	2.72187
$959$	−101.015	−3.26194
$960$	0	0
$961$	−11.3093	−0.364816
$962$	−147.327	−4.75001
$963$	0	0
$964$	−80.2629	−2.58510
$965$	15.9849	0.514573
$966$	0	0
$967$	21.4165	0.688707	0.344353	−	0.938840i	$-0.388098\pi$
$967$	21.4165	0.688707	0.344353	+	0.938840i	$0.388098\pi$
$968$	0	0
$969$	0	0
$970$	10.6547	0.342100
$971$	3.61567	0.116032	0.0580162	−	0.998316i	$-0.481523\pi$
$971$	3.61567	0.116032	0.0580162	+	0.998316i	$0.481523\pi$
$972$	0	0
$973$	34.5685	1.10821
$974$	47.0088	1.50626
$975$	0	0
$976$	1.86484	0.0596920
$977$	−20.2022	−0.646325	−0.323162	−	0.946344i	$-0.604746\pi$
$977$	−20.2022	−0.646325	−0.323162	+	0.946344i	$0.604746\pi$
$978$	0	0
$979$	0	0
$980$	−71.0759	−2.27043
$981$	0	0
$982$	1.12516	0.0359053
$983$	46.8067	1.49290	0.746451	−	0.665441i	$-0.231756\pi$
$983$	46.8067	1.49290	0.746451	+	0.665441i	$0.231756\pi$
$984$	0	0
$985$	−16.2202	−0.516818
$986$	−2.48461	−0.0791261
$987$	0	0
$988$	33.8217	1.07601
$989$	10.0921	0.320909
$990$	0	0
$991$	−47.3893	−1.50537	−0.752685	−	0.658381i	$-0.771242\pi$
$991$	−47.3893	−1.50537	−0.752685	+	0.658381i	$0.771242\pi$
$992$	−11.1001	−0.352428
$993$	0	0
$994$	157.773	5.00426
$995$	1.34240	0.0425568
$996$	0	0
$997$	−30.4554	−0.964533	−0.482267	−	0.876024i	$-0.660186\pi$
$997$	−30.4554	−0.964533	−0.482267	+	0.876024i	$0.660186\pi$
$998$	71.3991	2.26010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.ba.1.1		3
3.2	odd	2		1815.2.a.n.1.3	yes	3
11.10	odd	2		5445.2.a.bc.1.3		3
15.14	odd	2		9075.2.a.ce.1.1		3
33.32	even	2		1815.2.a.l.1.1	✓	3
165.164	even	2		9075.2.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.l.1.1	✓	3	33.32	even	2
1815.2.a.n.1.3	yes	3	3.2	odd	2
5445.2.a.ba.1.1		3	1.1	even	1	trivial
5445.2.a.bc.1.3		3	11.10	odd	2
9075.2.a.ce.1.1		3	15.14	odd	2
9075.2.a.ci.1.3		3	165.164	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} +5.11009 q^{7} -4.11009 q^{8} +O(q^{10})\)	\(q-2.39138 q^{2} +3.71871 q^{4} -1.00000 q^{5} +5.11009 q^{7} -4.11009 q^{8} +2.39138 q^{10} -5.43742 q^{13} -12.2202 q^{14} +2.39138 q^{16} +1.32733 q^{17} -1.67267 q^{19} -3.71871 q^{20} +2.11009 q^{23} +1.00000 q^{25} +13.0029 q^{26} +19.0029 q^{28} +0.782765 q^{29} -4.43742 q^{31} +2.50147 q^{32} -3.17415 q^{34} -5.11009 q^{35} -11.3303 q^{37} +4.00000 q^{38} +4.11009 q^{40} +11.5655 q^{41} +4.78276 q^{43} -5.04604 q^{46} -9.45544 q^{47} +19.1130 q^{49} -2.39138 q^{50} -20.2202 q^{52} +8.23820 q^{53} -21.0029 q^{56} -1.87189 q^{58} -10.0921 q^{59} +0.779816 q^{61} +10.6116 q^{62} -10.7647 q^{64} +5.43742 q^{65} -4.45544 q^{67} +4.93594 q^{68} +12.2202 q^{70} -12.9109 q^{71} -2.32733 q^{73} +27.0950 q^{74} -6.22018 q^{76} -1.12811 q^{79} -2.39138 q^{80} -27.6576 q^{82} -4.00000 q^{83} -1.32733 q^{85} -11.4374 q^{86} -14.7828 q^{89} -27.7857 q^{91} +7.84682 q^{92} +22.6116 q^{94} +1.67267 q^{95} +4.45544 q^{97} -45.7066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7	\( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + q^{10} - 4 q^{13} - 12 q^{14} + q^{16} + 4 q^{17} - 5 q^{19} - 5 q^{20} - 6 q^{23} + 3 q^{25} + 2 q^{26} + 20 q^{28} - 10 q^{29} - q^{31} - 11 q^{32} + 9 q^{34} - 3 q^{35} + 3 q^{37} + 12 q^{38} + 10 q^{41} + 2 q^{43} - 9 q^{46} - 16 q^{47} + 8 q^{49} - q^{50} - 36 q^{52} - 26 q^{56} - 18 q^{58} - 18 q^{59} + 27 q^{61} + q^{62} - 20 q^{64} + 4 q^{65} - q^{67} + 21 q^{68} + 12 q^{70} - 14 q^{71} - 7 q^{73} + 32 q^{74} + 6 q^{76} + 9 q^{79} - q^{80} - 46 q^{82} - 12 q^{83} - 4 q^{85} - 22 q^{86} - 32 q^{89} - 34 q^{91} + 5 q^{92} + 37 q^{94} + 5 q^{95} + q^{97} - 57 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + q^10 - 4 * q^13 - 12 * q^14 + q^16 + 4 * q^17 - 5 * q^19 - 5 * q^20 - 6 * q^23 + 3 * q^25 + 2 * q^26 + 20 * q^28 - 10 * q^29 - q^31 - 11 * q^32 + 9 * q^34 - 3 * q^35 + 3 * q^37 + 12 * q^38 + 10 * q^41 + 2 * q^43 - 9 * q^46 - 16 * q^47 + 8 * q^49 - q^50 - 36 * q^52 - 26 * q^56 - 18 * q^58 - 18 * q^59 + 27 * q^61 + q^62 - 20 * q^64 + 4 * q^65 - q^67 + 21 * q^68 + 12 * q^70 - 14 * q^71 - 7 * q^73 + 32 * q^74 + 6 * q^76 + 9 * q^79 - q^80 - 46 * q^82 - 12 * q^83 - 4 * q^85 - 22 * q^86 - 32 * q^89 - 34 * q^91 + 5 * q^92 + 37 * q^94 + 5 * q^95 + q^97 - 57 * q^98

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