LMFDB - Embedded newform 5445.2.a.s.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} -5.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.00000 q^{20} -6.92820 q^{23} +1.00000 q^{25} +9.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -10.9282 q^{31} +5.19615 q^{32} -2.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} +1.73205 q^{40} +3.46410 q^{41} +4.92820 q^{43} +12.0000 q^{46} +6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -5.46410 q^{52} -0.928203 q^{53} -3.46410 q^{56} +6.00000 q^{58} +6.92820 q^{59} -2.00000 q^{61} +18.9282 q^{62} +1.00000 q^{64} -5.46410 q^{65} +8.00000 q^{67} +3.46410 q^{70} -13.8564 q^{71} +8.39230 q^{73} +8.53590 q^{74} -5.46410 q^{76} +6.53590 q^{79} -5.00000 q^{80} -6.00000 q^{82} +8.53590 q^{83} -8.53590 q^{86} -0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} -12.0000 q^{94} -5.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7} - 4 q^{13} - 10 q^{16} - 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} - 4 q^{28} - 8 q^{31} - 4 q^{35} + 4 q^{37} + 12 q^{38} - 4 q^{43} + 24 q^{46} - 6 q^{49} - 4 q^{52} + 12 q^{53} + 12 q^{58} - 4 q^{61} + 24 q^{62} + 2 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{73} + 24 q^{74} - 4 q^{76} + 20 q^{79} - 10 q^{80} - 12 q^{82} + 24 q^{83} - 24 q^{86} + 12 q^{89} + 8 q^{91} - 24 q^{94} - 4 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7 - 4 * q^13 - 10 * q^16 - 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 - 4 * q^28 - 8 * q^31 - 4 * q^35 + 4 * q^37 + 12 * q^38 - 4 * q^43 + 24 * q^46 - 6 * q^49 - 4 * q^52 + 12 * q^53 + 12 * q^58 - 4 * q^61 + 24 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 - 4 * q^73 + 24 * q^74 - 4 * q^76 + 20 * q^79 - 10 * q^80 - 12 * q^82 + 24 * q^83 - 24 * q^86 + 12 * q^89 + 8 * q^91 - 24 * q^94 - 4 * q^95 - 20 * q^97

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−1.73205	−0.547723
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.46410	1.85606
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−10.9282	−1.96276	−0.981382	−	0.192068i	$-0.938481\pi$
$31$	−10.9282	−1.96276	−0.981382	+	0.192068i	$0.938481\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−4.92820	−0.810192	−0.405096	−	0.914274i	$-0.632762\pi$
$37$	−4.92820	−0.810192	−0.405096	+	0.914274i	$0.632762\pi$
$38$	9.46410	1.53528
$39$	0	0
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	0	0
$46$	12.0000	1.76930
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.73205	−0.244949
$51$	0	0
$52$	−5.46410	−0.757735
$53$	−0.928203	−0.127499	−0.0637493	−	0.997966i	$-0.520306\pi$
$53$	−0.928203	−0.127499	−0.0637493	+	0.997966i	$0.520306\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	6.00000	0.787839
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	18.9282	2.40388
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.46410	−0.677738
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	3.46410	0.414039
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	8.39230	0.982245	0.491122	−	0.871091i	$-0.336587\pi$
$73$	8.39230	0.982245	0.491122	+	0.871091i	$0.336587\pi$
$74$	8.53590	0.992278
$75$	0	0
$76$	−5.46410	−0.626775
$77$	0	0
$78$	0	0
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	−5.00000	−0.559017
$81$	0	0
$82$	−6.00000	−0.662589
$83$	8.53590	0.936937	0.468468	−	0.883480i	$-0.344806\pi$
$83$	8.53590	0.936937	0.468468	+	0.883480i	$0.344806\pi$
$84$	0	0
$85$	0	0
$86$	−8.53590	−0.920450
$87$	0	0
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	−6.92820	−0.722315
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−5.46410	−0.560605
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	5.19615	0.524891
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−9.46410	−0.928032
$105$	0	0
$106$	1.60770	0.156153
$107$	8.53590	0.825196	0.412598	−	0.910913i	$-0.364621\pi$
$107$	8.53590	0.825196	0.412598	+	0.910913i	$0.364621\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	10.0000	0.944911
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	−6.92820	−0.646058
$116$	−3.46410	−0.321634
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	0	0
$122$	3.46410	0.313625
$123$	0	0
$124$	−10.9282	−0.981382
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.92820	−0.792250	−0.396125	−	0.918197i	$-0.629645\pi$
$127$	−8.92820	−0.792250	−0.396125	+	0.918197i	$0.629645\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	9.46410	0.830057
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	10.9282	0.947595
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	24.0000	2.01404
$143$	0	0
$144$	0	0
$145$	−3.46410	−0.287678
$146$	−14.5359	−1.20300
$147$	0	0
$148$	−4.92820	−0.405096
$149$	−15.4641	−1.26687	−0.633434	−	0.773796i	$-0.718355\pi$
$149$	−15.4641	−1.26687	−0.633434	+	0.773796i	$0.718355\pi$
$150$	0	0
$151$	20.3923	1.65950	0.829751	−	0.558134i	$-0.188482\pi$
$151$	20.3923	1.65950	0.829751	+	0.558134i	$0.188482\pi$
$152$	−9.46410	−0.767640
$153$	0	0
$154$	0	0
$155$	−10.9282	−0.877774
$156$	0	0
$157$	−3.07180	−0.245156	−0.122578	−	0.992459i	$-0.539116\pi$
$157$	−3.07180	−0.245156	−0.122578	+	0.992459i	$0.539116\pi$
$158$	−11.3205	−0.900611
$159$	0	0
$160$	5.19615	0.410792
$161$	13.8564	1.09204
$162$	0	0
$163$	9.85641	0.772013	0.386007	−	0.922496i	$-0.373854\pi$
$163$	9.85641	0.772013	0.386007	+	0.922496i	$0.373854\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	4.92820	0.375772
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	1.60770	0.120502
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	15.8564	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$181$	15.8564	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$182$	−18.9282	−1.40305
$183$	0	0
$184$	−12.0000	−0.884652
$185$	−4.92820	−0.362329
$186$	0	0
$187$	0	0
$188$	6.92820	0.505291
$189$	0	0
$190$	9.46410	0.686598
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	−24.3923	−1.75580	−0.877898	−	0.478847i	$-0.841055\pi$
$193$	−24.3923	−1.75580	−0.877898	+	0.478847i	$0.841055\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	18.0000	1.26648
$203$	6.92820	0.486265
$204$	0	0
$205$	3.46410	0.241943
$206$	−13.8564	−0.965422
$207$	0	0
$208$	27.3205	1.89434
$209$	0	0
$210$	0	0
$211$	8.39230	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$211$	8.39230	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$212$	−0.928203	−0.0637493
$213$	0	0
$214$	−14.7846	−1.01066
$215$	4.92820	0.336101
$216$	0	0
$217$	21.8564	1.48371
$218$	−17.3205	−1.17309
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.85641	0.660034	0.330017	−	0.943975i	$-0.392946\pi$
$223$	9.85641	0.660034	0.330017	+	0.943975i	$0.392946\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	−22.3923	−1.48951
$227$	15.4641	1.02639	0.513194	−	0.858272i	$-0.328462\pi$
$227$	15.4641	1.02639	0.513194	+	0.858272i	$0.328462\pi$
$228$	0	0
$229$	−23.8564	−1.57648	−0.788238	−	0.615371i	$-0.789006\pi$
$229$	−23.8564	−1.57648	−0.788238	+	0.615371i	$0.789006\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	−6.00000	−0.393919
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	6.92820	0.451946
$236$	6.92820	0.450988
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−0.143594	−0.00924967	−0.00462484	−	0.999989i	$-0.501472\pi$
$241$	−0.143594	−0.00924967	−0.00462484	+	0.999989i	$0.501472\pi$
$242$	0	0
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−3.00000	−0.191663
$246$	0	0
$247$	29.8564	1.89972
$248$	−18.9282	−1.20194
$249$	0	0
$250$	−1.73205	−0.109545
$251$	1.85641	0.117175	0.0585877	−	0.998282i	$-0.481340\pi$
$251$	1.85641	0.117175	0.0585877	+	0.998282i	$0.481340\pi$
$252$	0	0
$253$	0	0
$254$	15.4641	0.970304
$255$	0	0
$256$	19.0000	1.18750
$257$	19.8564	1.23861	0.619304	−	0.785151i	$-0.287415\pi$
$257$	19.8564	1.23861	0.619304	+	0.785151i	$0.287415\pi$
$258$	0	0
$259$	9.85641	0.612447
$260$	−5.46410	−0.338869
$261$	0	0
$262$	−32.7846	−2.02544
$263$	20.5359	1.26630	0.633149	−	0.774030i	$-0.281762\pi$
$263$	20.5359	1.26630	0.633149	+	0.774030i	$0.281762\pi$
$264$	0	0
$265$	−0.928203	−0.0570191
$266$	−18.9282	−1.16056
$267$	0	0
$268$	8.00000	0.488678
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	0	0
$271$	11.6077	0.705117	0.352559	−	0.935790i	$-0.385312\pi$
$271$	11.6077	0.705117	0.352559	+	0.935790i	$0.385312\pi$
$272$	0	0
$273$	0	0
$274$	−31.1769	−1.88347
$275$	0	0
$276$	0	0
$277$	−29.4641	−1.77033	−0.885163	−	0.465281i	$-0.845953\pi$
$277$	−29.4641	−1.77033	−0.885163	+	0.465281i	$0.845953\pi$
$278$	21.4641	1.28733
$279$	0	0
$280$	−3.46410	−0.207020
$281$	3.46410	0.206651	0.103325	−	0.994648i	$-0.467052\pi$
$281$	3.46410	0.206651	0.103325	+	0.994648i	$0.467052\pi$
$282$	0	0
$283$	4.92820	0.292951	0.146476	−	0.989214i	$-0.453207\pi$
$283$	4.92820	0.292951	0.146476	+	0.989214i	$0.453207\pi$
$284$	−13.8564	−0.822226
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	0	0
$292$	8.39230	0.491122
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	6.92820	0.403376
$296$	−8.53590	−0.496139
$297$	0	0
$298$	26.7846	1.55159
$299$	37.8564	2.18929
$300$	0	0
$301$	−9.85641	−0.568114
$302$	−35.3205	−2.03247
$303$	0	0
$304$	27.3205	1.56694
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	0	0
$310$	18.9282	1.07505
$311$	−5.07180	−0.287595	−0.143798	−	0.989607i	$-0.545931\pi$
$311$	−5.07180	−0.287595	−0.143798	+	0.989607i	$0.545931\pi$
$312$	0	0
$313$	20.9282	1.18293	0.591466	−	0.806330i	$-0.298549\pi$
$313$	20.9282	1.18293	0.591466	+	0.806330i	$0.298549\pi$
$314$	5.32051	0.300254
$315$	0	0
$316$	6.53590	0.367673
$317$	24.9282	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$317$	24.9282	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	−24.0000	−1.33747
$323$	0	0
$324$	0	0
$325$	−5.46410	−0.303094
$326$	−17.0718	−0.945519
$327$	0	0
$328$	6.00000	0.331295
$329$	−13.8564	−0.763928
$330$	0	0
$331$	9.85641	0.541757	0.270879	−	0.962614i	$-0.412686\pi$
$331$	9.85641	0.541757	0.270879	+	0.962614i	$0.412686\pi$
$332$	8.53590	0.468468
$333$	0	0
$334$	18.0000	0.984916
$335$	8.00000	0.437087
$336$	0	0
$337$	−33.1769	−1.80726	−0.903631	−	0.428312i	$-0.859108\pi$
$337$	−33.1769	−1.80726	−0.903631	+	0.428312i	$0.859108\pi$
$338$	−29.1962	−1.58806
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	8.53590	0.460225
$345$	0	0
$346$	20.7846	1.11739
$347$	22.3923	1.20208	0.601041	−	0.799218i	$-0.294753\pi$
$347$	22.3923	1.20208	0.601041	+	0.799218i	$0.294753\pi$
$348$	0	0
$349$	8.14359	0.435917	0.217958	−	0.975958i	$-0.430060\pi$
$349$	8.14359	0.435917	0.217958	+	0.975958i	$0.430060\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	−13.8564	−0.735422
$356$	−0.928203	−0.0491947
$357$	0	0
$358$	12.0000	0.634220
$359$	−20.7846	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−27.4641	−1.44348
$363$	0	0
$364$	10.9282	0.572793
$365$	8.39230	0.439273
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	34.6410	1.80579
$369$	0	0
$370$	8.53590	0.443760
$371$	1.85641	0.0963798
$372$	0	0
$373$	20.3923	1.05587	0.527937	−	0.849284i	$-0.322966\pi$
$373$	20.3923	1.05587	0.527937	+	0.849284i	$0.322966\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	18.9282	0.974852
$378$	0	0
$379$	−17.8564	−0.917222	−0.458611	−	0.888637i	$-0.651653\pi$
$379$	−17.8564	−0.917222	−0.458611	+	0.888637i	$0.651653\pi$
$380$	−5.46410	−0.280302
$381$	0	0
$382$	−32.7846	−1.67741
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	0	0
$386$	42.2487	2.15040
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−11.0718	−0.561362	−0.280681	−	0.959801i	$-0.590560\pi$
$389$	−11.0718	−0.561362	−0.280681	+	0.959801i	$0.590560\pi$
$390$	0	0
$391$	0	0
$392$	−5.19615	−0.262445
$393$	0	0
$394$	20.7846	1.04711
$395$	6.53590	0.328857
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	42.9282	2.15180
$399$	0	0
$400$	−5.00000	−0.250000
$401$	7.85641	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$401$	7.85641	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$402$	0	0
$403$	59.7128	2.97451
$404$	−10.3923	−0.517036
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	6.78461	0.335477	0.167739	−	0.985831i	$-0.446354\pi$
$409$	6.78461	0.335477	0.167739	+	0.985831i	$0.446354\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	8.00000	0.394132
$413$	−13.8564	−0.681829
$414$	0	0
$415$	8.53590	0.419011
$416$	−28.3923	−1.39205
$417$	0	0
$418$	0	0
$419$	−30.9282	−1.51094	−0.755471	−	0.655182i	$-0.772592\pi$
$419$	−30.9282	−1.51094	−0.755471	+	0.655182i	$0.772592\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−14.5359	−0.707596
$423$	0	0
$424$	−1.60770	−0.0780766
$425$	0	0
$426$	0	0
$427$	4.00000	0.193574
$428$	8.53590	0.412598
$429$	0	0
$430$	−8.53590	−0.411638
$431$	8.78461	0.423140	0.211570	−	0.977363i	$-0.432142\pi$
$431$	8.78461	0.423140	0.211570	+	0.977363i	$0.432142\pi$
$432$	0	0
$433$	0.143594	0.00690067	0.00345033	−	0.999994i	$-0.498902\pi$
$433$	0.143594	0.00690067	0.00345033	+	0.999994i	$0.498902\pi$
$434$	−37.8564	−1.81717
$435$	0	0
$436$	10.0000	0.478913
$437$	37.8564	1.81092
$438$	0	0
$439$	−33.1769	−1.58345	−0.791724	−	0.610879i	$-0.790816\pi$
$439$	−33.1769	−1.58345	−0.791724	+	0.610879i	$0.790816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	−17.0718	−0.808373
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	26.7846	1.26404	0.632022	−	0.774950i	$-0.282225\pi$
$449$	26.7846	1.26404	0.632022	+	0.774950i	$0.282225\pi$
$450$	0	0
$451$	0	0
$452$	12.9282	0.608092
$453$	0	0
$454$	−26.7846	−1.25706
$455$	10.9282	0.512322
$456$	0	0
$457$	−12.3923	−0.579688	−0.289844	−	0.957074i	$-0.593603\pi$
$457$	−12.3923	−0.579688	−0.289844	+	0.957074i	$0.593603\pi$
$458$	41.3205	1.93078
$459$	0	0
$460$	−6.92820	−0.323029
$461$	36.2487	1.68827	0.844135	−	0.536130i	$-0.180114\pi$
$461$	36.2487	1.68827	0.844135	+	0.536130i	$0.180114\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	−20.7846	−0.962828
$467$	−5.07180	−0.234695	−0.117347	−	0.993091i	$-0.537439\pi$
$467$	−5.07180	−0.234695	−0.117347	+	0.993091i	$0.537439\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	−12.0000	−0.553519
$471$	0	0
$472$	12.0000	0.552345
$473$	0	0
$474$	0	0
$475$	−5.46410	−0.250710
$476$	0	0
$477$	0	0
$478$	20.7846	0.950666
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	26.9282	1.22782
$482$	0.248711	0.0113285
$483$	0	0
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−31.7128	−1.43704	−0.718522	−	0.695504i	$-0.755181\pi$
$487$	−31.7128	−1.43704	−0.718522	+	0.695504i	$0.755181\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	5.19615	0.234738
$491$	−30.9282	−1.39577	−0.697885	−	0.716210i	$-0.745875\pi$
$491$	−30.9282	−1.39577	−0.697885	+	0.716210i	$0.745875\pi$
$492$	0	0
$493$	0	0
$494$	−51.7128	−2.32667
$495$	0	0
$496$	54.6410	2.45345
$497$	27.7128	1.24309
$498$	0	0
$499$	28.7846	1.28858	0.644288	−	0.764783i	$-0.277154\pi$
$499$	28.7846	1.28858	0.644288	+	0.764783i	$0.277154\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−3.21539	−0.143510
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	0	0
$507$	0	0
$508$	−8.92820	−0.396125
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−16.7846	−0.742507
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−34.3923	−1.51698
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	−17.0718	−0.750092
$519$	0	0
$520$	−9.46410	−0.415028
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	18.9282	0.826882
$525$	0	0
$526$	−35.5692	−1.55089
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	1.60770	0.0698338
$531$	0	0
$532$	10.9282	0.473798
$533$	−18.9282	−0.819871
$534$	0	0
$535$	8.53590	0.369039
$536$	13.8564	0.598506
$537$	0	0
$538$	34.3923	1.48276
$539$	0	0
$540$	0	0
$541$	−27.8564	−1.19764	−0.598820	−	0.800883i	$-0.704364\pi$
$541$	−27.8564	−1.19764	−0.598820	+	0.800883i	$0.704364\pi$
$542$	−20.1051	−0.863589
$543$	0	0
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	−13.0718	−0.555869
$554$	51.0333	2.16820
$555$	0	0
$556$	−12.3923	−0.525551
$557$	3.21539	0.136240	0.0681202	−	0.997677i	$-0.478300\pi$
$557$	3.21539	0.136240	0.0681202	+	0.997677i	$0.478300\pi$
$558$	0	0
$559$	−26.9282	−1.13894
$560$	10.0000	0.422577
$561$	0	0
$562$	−6.00000	−0.253095
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	−8.53590	−0.358791
$567$	0	0
$568$	−24.0000	−1.00702
$569$	5.32051	0.223047	0.111524	−	0.993762i	$-0.464427\pi$
$569$	5.32051	0.223047	0.111524	+	0.993762i	$0.464427\pi$
$570$	0	0
$571$	−3.60770	−0.150977	−0.0754887	−	0.997147i	$-0.524052\pi$
$571$	−3.60770	−0.150977	−0.0754887	+	0.997147i	$0.524052\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	−6.92820	−0.288926
$576$	0	0
$577$	−18.7846	−0.782014	−0.391007	−	0.920388i	$-0.627873\pi$
$577$	−18.7846	−0.782014	−0.391007	+	0.920388i	$0.627873\pi$
$578$	29.4449	1.22474
$579$	0	0
$580$	−3.46410	−0.143839
$581$	−17.0718	−0.708257
$582$	0	0
$583$	0	0
$584$	14.5359	0.601500
$585$	0	0
$586$	24.0000	0.991431
$587$	18.9282	0.781251	0.390625	−	0.920550i	$-0.372259\pi$
$587$	18.9282	0.781251	0.390625	+	0.920550i	$0.372259\pi$
$588$	0	0
$589$	59.7128	2.46042
$590$	−12.0000	−0.494032
$591$	0	0
$592$	24.6410	1.01274
$593$	8.78461	0.360741	0.180370	−	0.983599i	$-0.442270\pi$
$593$	8.78461	0.360741	0.180370	+	0.983599i	$0.442270\pi$
$594$	0	0
$595$	0	0
$596$	−15.4641	−0.633434
$597$	0	0
$598$	−65.5692	−2.68132
$599$	37.8564	1.54677	0.773385	−	0.633936i	$-0.218562\pi$
$599$	37.8564	1.54677	0.773385	+	0.633936i	$0.218562\pi$
$600$	0	0
$601$	32.6410	1.33145	0.665727	−	0.746195i	$-0.268121\pi$
$601$	32.6410	1.33145	0.665727	+	0.746195i	$0.268121\pi$
$602$	17.0718	0.695794
$603$	0	0
$604$	20.3923	0.829751
$605$	0	0
$606$	0	0
$607$	18.7846	0.762444	0.381222	−	0.924484i	$-0.375503\pi$
$607$	18.7846	0.762444	0.381222	+	0.924484i	$0.375503\pi$
$608$	−28.3923	−1.15146
$609$	0	0
$610$	3.46410	0.140257
$611$	−37.8564	−1.53151
$612$	0	0
$613$	20.3923	0.823637	0.411819	−	0.911266i	$-0.364894\pi$
$613$	20.3923	0.823637	0.411819	+	0.911266i	$0.364894\pi$
$614$	24.2487	0.978598
$615$	0	0
$616$	0	0
$617$	36.9282	1.48667	0.743337	−	0.668917i	$-0.233242\pi$
$617$	36.9282	1.48667	0.743337	+	0.668917i	$0.233242\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−10.9282	−0.438887
$621$	0	0
$622$	8.78461	0.352231
$623$	1.85641	0.0743754
$624$	0	0
$625$	1.00000	0.0400000
$626$	−36.2487	−1.44879
$627$	0	0
$628$	−3.07180	−0.122578
$629$	0	0
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	11.3205	0.450306
$633$	0	0
$634$	−43.1769	−1.71477
$635$	−8.92820	−0.354305
$636$	0	0
$637$	16.3923	0.649487
$638$	0	0
$639$	0	0
$640$	−12.1244	−0.479257
$641$	12.9282	0.510633	0.255317	−	0.966857i	$-0.417820\pi$
$641$	12.9282	0.510633	0.255317	+	0.966857i	$0.417820\pi$
$642$	0	0
$643$	37.5692	1.48159	0.740793	−	0.671734i	$-0.234450\pi$
$643$	37.5692	1.48159	0.740793	+	0.671734i	$0.234450\pi$
$644$	13.8564	0.546019
$645$	0	0
$646$	0	0
$647$	−27.7128	−1.08950	−0.544752	−	0.838597i	$-0.683376\pi$
$647$	−27.7128	−1.08950	−0.544752	+	0.838597i	$0.683376\pi$
$648$	0	0
$649$	0	0
$650$	9.46410	0.371213
$651$	0	0
$652$	9.85641	0.386007
$653$	−19.8564	−0.777041	−0.388521	−	0.921440i	$-0.627014\pi$
$653$	−19.8564	−0.777041	−0.388521	+	0.921440i	$0.627014\pi$
$654$	0	0
$655$	18.9282	0.739586
$656$	−17.3205	−0.676252
$657$	0	0
$658$	24.0000	0.935617
$659$	−15.7128	−0.612084	−0.306042	−	0.952018i	$-0.599005\pi$
$659$	−15.7128	−0.612084	−0.306042	+	0.952018i	$0.599005\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−17.0718	−0.663514
$663$	0	0
$664$	14.7846	0.573754
$665$	10.9282	0.423778
$666$	0	0
$667$	24.0000	0.929284
$668$	−10.3923	−0.402090
$669$	0	0
$670$	−13.8564	−0.535320
$671$	0	0
$672$	0	0
$673$	3.32051	0.127996	0.0639981	−	0.997950i	$-0.479615\pi$
$673$	3.32051	0.127996	0.0639981	+	0.997950i	$0.479615\pi$
$674$	57.4641	2.21343
$675$	0	0
$676$	16.8564	0.648323
$677$	8.78461	0.337620	0.168810	−	0.985649i	$-0.446008\pi$
$677$	8.78461	0.337620	0.168810	+	0.985649i	$0.446008\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.7846	1.25447	0.627234	−	0.778831i	$-0.284187\pi$
$683$	32.7846	1.25447	0.627234	+	0.778831i	$0.284187\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	−34.6410	−1.32260
$687$	0	0
$688$	−24.6410	−0.939430
$689$	5.07180	0.193220
$690$	0	0
$691$	47.7128	1.81508	0.907540	−	0.419965i	$-0.137958\pi$
$691$	47.7128	1.81508	0.907540	+	0.419965i	$0.137958\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−38.7846	−1.47224
$695$	−12.3923	−0.470067
$696$	0	0
$697$	0	0
$698$	−14.1051	−0.533887
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	39.4641	1.49054	0.745269	−	0.666764i	$-0.232321\pi$
$701$	39.4641	1.49054	0.745269	+	0.666764i	$0.232321\pi$
$702$	0	0
$703$	26.9282	1.01562
$704$	0	0
$705$	0	0
$706$	22.3923	0.842746
$707$	20.7846	0.781686
$708$	0	0
$709$	−11.8564	−0.445277	−0.222638	−	0.974901i	$-0.571467\pi$
$709$	−11.8564	−0.445277	−0.222638	+	0.974901i	$0.571467\pi$
$710$	24.0000	0.900704
$711$	0	0
$712$	−1.60770	−0.0602509
$713$	75.7128	2.83547
$714$	0	0
$715$	0	0
$716$	−6.92820	−0.258919
$717$	0	0
$718$	36.0000	1.34351
$719$	5.07180	0.189146	0.0945731	−	0.995518i	$-0.469851\pi$
$719$	5.07180	0.189146	0.0945731	+	0.995518i	$0.469851\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−18.8038	−0.699807
$723$	0	0
$724$	15.8564	0.589299
$725$	−3.46410	−0.128654
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	18.9282	0.701526
$729$	0	0
$730$	−14.5359	−0.537998
$731$	0	0
$732$	0	0
$733$	−53.9615	−1.99311	−0.996557	−	0.0829082i	$-0.973579\pi$
$733$	−53.9615	−1.99311	−0.996557	+	0.0829082i	$0.973579\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	−36.0000	−1.32698
$737$	0	0
$738$	0	0
$739$	−17.4641	−0.642427	−0.321214	−	0.947007i	$-0.604091\pi$
$739$	−17.4641	−0.642427	−0.321214	+	0.947007i	$0.604091\pi$
$740$	−4.92820	−0.181164
$741$	0	0
$742$	−3.21539	−0.118041
$743$	25.6077	0.939455	0.469728	−	0.882811i	$-0.344352\pi$
$743$	25.6077	0.939455	0.469728	+	0.882811i	$0.344352\pi$
$744$	0	0
$745$	−15.4641	−0.566561
$746$	−35.3205	−1.29318
$747$	0	0
$748$	0	0
$749$	−17.0718	−0.623790
$750$	0	0
$751$	26.9282	0.982624	0.491312	−	0.870984i	$-0.336517\pi$
$751$	26.9282	0.982624	0.491312	+	0.870984i	$0.336517\pi$
$752$	−34.6410	−1.26323
$753$	0	0
$754$	−32.7846	−1.19395
$755$	20.3923	0.742152
$756$	0	0
$757$	34.7846	1.26427	0.632134	−	0.774859i	$-0.282179\pi$
$757$	34.7846	1.26427	0.632134	+	0.774859i	$0.282179\pi$
$758$	30.9282	1.12336
$759$	0	0
$760$	−9.46410	−0.343299
$761$	−32.5359	−1.17943	−0.589713	−	0.807613i	$-0.700759\pi$
$761$	−32.5359	−1.17943	−0.589713	+	0.807613i	$0.700759\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	18.9282	0.684798
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−37.8564	−1.36692
$768$	0	0
$769$	−50.4974	−1.82098	−0.910492	−	0.413527i	$-0.864297\pi$
$769$	−50.4974	−1.82098	−0.910492	+	0.413527i	$0.864297\pi$
$770$	0	0
$771$	0	0
$772$	−24.3923	−0.877898
$773$	4.14359	0.149035	0.0745174	−	0.997220i	$-0.476258\pi$
$773$	4.14359	0.149035	0.0745174	+	0.997220i	$0.476258\pi$
$774$	0	0
$775$	−10.9282	−0.392553
$776$	−17.3205	−0.621770
$777$	0	0
$778$	19.1769	0.687526
$779$	−18.9282	−0.678173
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	15.0000	0.535714
$785$	−3.07180	−0.109637
$786$	0	0
$787$	−22.7846	−0.812184	−0.406092	−	0.913832i	$-0.633109\pi$
$787$	−22.7846	−0.812184	−0.406092	+	0.913832i	$0.633109\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	−11.3205	−0.402766
$791$	−25.8564	−0.919348
$792$	0	0
$793$	10.9282	0.388072
$794$	−3.46410	−0.122936
$795$	0	0
$796$	−24.7846	−0.878467
$797$	52.6410	1.86464	0.932320	−	0.361634i	$-0.117781\pi$
$797$	52.6410	1.86464	0.932320	+	0.361634i	$0.117781\pi$
$798$	0	0
$799$	0	0
$800$	5.19615	0.183712
$801$	0	0
$802$	−13.6077	−0.480504
$803$	0	0
$804$	0	0
$805$	13.8564	0.488374
$806$	−103.426	−3.64301
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−15.4641	−0.543689	−0.271844	−	0.962341i	$-0.587634\pi$
$809$	−15.4641	−0.543689	−0.271844	+	0.962341i	$0.587634\pi$
$810$	0	0
$811$	−12.3923	−0.435153	−0.217576	−	0.976043i	$-0.569815\pi$
$811$	−12.3923	−0.435153	−0.217576	+	0.976043i	$0.569815\pi$
$812$	6.92820	0.243132
$813$	0	0
$814$	0	0
$815$	9.85641	0.345255
$816$	0	0
$817$	−26.9282	−0.942099
$818$	−11.7513	−0.410874
$819$	0	0
$820$	3.46410	0.120972
$821$	20.5359	0.716708	0.358354	−	0.933586i	$-0.383338\pi$
$821$	20.5359	0.716708	0.358354	+	0.933586i	$0.383338\pi$
$822$	0	0
$823$	−33.5692	−1.17015	−0.585075	−	0.810979i	$-0.698935\pi$
$823$	−33.5692	−1.17015	−0.585075	+	0.810979i	$0.698935\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	22.3923	0.778657	0.389328	−	0.921099i	$-0.372707\pi$
$827$	22.3923	0.778657	0.389328	+	0.921099i	$0.372707\pi$
$828$	0	0
$829$	29.7128	1.03197	0.515984	−	0.856598i	$-0.327426\pi$
$829$	29.7128	1.03197	0.515984	+	0.856598i	$0.327426\pi$
$830$	−14.7846	−0.513181
$831$	0	0
$832$	−5.46410	−0.189434
$833$	0	0
$834$	0	0
$835$	−10.3923	−0.359641
$836$	0	0
$837$	0	0
$838$	53.5692	1.85052
$839$	−56.7846	−1.96042	−0.980211	−	0.197954i	$-0.936570\pi$
$839$	−56.7846	−1.96042	−0.980211	+	0.197954i	$0.936570\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−3.46410	−0.119381
$843$	0	0
$844$	8.39230	0.288875
$845$	16.8564	0.579878
$846$	0	0
$847$	0	0
$848$	4.64102	0.159373
$849$	0	0
$850$	0	0
$851$	34.1436	1.17043
$852$	0	0
$853$	−3.60770	−0.123525	−0.0617626	−	0.998091i	$-0.519672\pi$
$853$	−3.60770	−0.123525	−0.0617626	+	0.998091i	$0.519672\pi$
$854$	−6.92820	−0.237078
$855$	0	0
$856$	14.7846	0.505328
$857$	−37.8564	−1.29315	−0.646575	−	0.762850i	$-0.723799\pi$
$857$	−37.8564	−1.29315	−0.646575	+	0.762850i	$0.723799\pi$
$858$	0	0
$859$	−7.71281	−0.263158	−0.131579	−	0.991306i	$-0.542005\pi$
$859$	−7.71281	−0.263158	−0.131579	+	0.991306i	$0.542005\pi$
$860$	4.92820	0.168050
$861$	0	0
$862$	−15.2154	−0.518238
$863$	37.8564	1.28865	0.644324	−	0.764753i	$-0.277139\pi$
$863$	37.8564	1.28865	0.644324	+	0.764753i	$0.277139\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	−0.248711	−0.00845155
$867$	0	0
$868$	21.8564	0.741855
$869$	0	0
$870$	0	0
$871$	−43.7128	−1.48115
$872$	17.3205	0.586546
$873$	0	0
$874$	−65.5692	−2.21791
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	34.2487	1.15650	0.578248	−	0.815861i	$-0.303736\pi$
$877$	34.2487	1.15650	0.578248	+	0.815861i	$0.303736\pi$
$878$	57.4641	1.93932
$879$	0	0
$880$	0	0
$881$	0.928203	0.0312720	0.0156360	−	0.999878i	$-0.495023\pi$
$881$	0.928203	0.0312720	0.0156360	+	0.999878i	$0.495023\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	0	0
$886$	20.7846	0.698273
$887$	12.2487	0.411271	0.205636	−	0.978629i	$-0.434074\pi$
$887$	12.2487	0.411271	0.205636	+	0.978629i	$0.434074\pi$
$888$	0	0
$889$	17.8564	0.598885
$890$	1.60770	0.0538901
$891$	0	0
$892$	9.85641	0.330017
$893$	−37.8564	−1.26682
$894$	0	0
$895$	−6.92820	−0.231584
$896$	24.2487	0.810093
$897$	0	0
$898$	−46.3923	−1.54813
$899$	37.8564	1.26258
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	22.3923	0.744757
$905$	15.8564	0.527085
$906$	0	0
$907$	18.1436	0.602448	0.301224	−	0.953553i	$-0.402605\pi$
$907$	18.1436	0.602448	0.301224	+	0.953553i	$0.402605\pi$
$908$	15.4641	0.513194
$909$	0	0
$910$	−18.9282	−0.627464
$911$	−18.9282	−0.627119	−0.313560	−	0.949568i	$-0.601522\pi$
$911$	−18.9282	−0.627119	−0.313560	+	0.949568i	$0.601522\pi$
$912$	0	0
$913$	0	0
$914$	21.4641	0.709969
$915$	0	0
$916$	−23.8564	−0.788238
$917$	−37.8564	−1.25013
$918$	0	0
$919$	32.3923	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$919$	32.3923	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$920$	−12.0000	−0.395628
$921$	0	0
$922$	−62.7846	−2.06770
$923$	75.7128	2.49212
$924$	0	0
$925$	−4.92820	−0.162038
$926$	48.4974	1.59372
$927$	0	0
$928$	−18.0000	−0.590879
$929$	−2.78461	−0.0913601	−0.0456800	−	0.998956i	$-0.514545\pi$
$929$	−2.78461	−0.0913601	−0.0456800	+	0.998956i	$0.514545\pi$
$930$	0	0
$931$	16.3923	0.537236
$932$	12.0000	0.393073
$933$	0	0
$934$	8.78461	0.287441
$935$	0	0
$936$	0	0
$937$	20.3923	0.666188	0.333094	−	0.942894i	$-0.391907\pi$
$937$	20.3923	0.666188	0.333094	+	0.942894i	$0.391907\pi$
$938$	27.7128	0.904855
$939$	0	0
$940$	6.92820	0.225973
$941$	27.4641	0.895304	0.447652	−	0.894208i	$-0.352260\pi$
$941$	27.4641	0.895304	0.447652	+	0.894208i	$0.352260\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−34.6410	−1.12747
$945$	0	0
$946$	0	0
$947$	18.9282	0.615084	0.307542	−	0.951535i	$-0.400494\pi$
$947$	18.9282	0.615084	0.307542	+	0.951535i	$0.400494\pi$
$948$	0	0
$949$	−45.8564	−1.48856
$950$	9.46410	0.307056
$951$	0	0
$952$	0	0
$953$	−3.21539	−0.104157	−0.0520784	−	0.998643i	$-0.516585\pi$
$953$	−3.21539	−0.104157	−0.0520784	+	0.998643i	$0.516585\pi$
$954$	0	0
$955$	18.9282	0.612502
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−20.7846	−0.671520
$959$	−36.0000	−1.16250
$960$	0	0
$961$	88.4256	2.85244
$962$	−46.6410	−1.50377
$963$	0	0
$964$	−0.143594	−0.00462484
$965$	−24.3923	−0.785216
$966$	0	0
$967$	−22.7846	−0.732704	−0.366352	−	0.930476i	$-0.619393\pi$
$967$	−22.7846	−0.732704	−0.366352	+	0.930476i	$0.619393\pi$
$968$	0	0
$969$	0	0
$970$	17.3205	0.556128
$971$	−25.8564	−0.829772	−0.414886	−	0.909873i	$-0.636178\pi$
$971$	−25.8564	−0.829772	−0.414886	+	0.909873i	$0.636178\pi$
$972$	0	0
$973$	24.7846	0.794558
$974$	54.9282	1.76001
$975$	0	0
$976$	10.0000	0.320092
$977$	−47.5692	−1.52187	−0.760937	−	0.648826i	$-0.775260\pi$
$977$	−47.5692	−1.52187	−0.760937	+	0.648826i	$0.775260\pi$
$978$	0	0
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	53.5692	1.70946
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	29.8564	0.949859
$989$	−34.1436	−1.08570
$990$	0	0
$991$	−7.21539	−0.229204	−0.114602	−	0.993411i	$-0.536559\pi$
$991$	−7.21539	−0.229204	−0.114602	+	0.993411i	$0.536559\pi$
$992$	−56.7846	−1.80291
$993$	0	0
$994$	−48.0000	−1.52247
$995$	−24.7846	−0.785725
$996$	0	0
$997$	−27.6077	−0.874344	−0.437172	−	0.899378i	$-0.644020\pi$
$997$	−27.6077	−0.874344	−0.437172	+	0.899378i	$0.644020\pi$
$998$	−49.8564	−1.57818
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.s.1.1		2
3.2	odd	2		1815.2.a.i.1.2		2
11.10	odd	2		495.2.a.c.1.2		2
15.14	odd	2		9075.2.a.bh.1.1		2
33.32	even	2		165.2.a.b.1.1	✓	2
44.43	even	2		7920.2.a.bz.1.2		2
55.32	even	4		2475.2.c.n.199.4		4
55.43	even	4		2475.2.c.n.199.1		4
55.54	odd	2		2475.2.a.r.1.1		2
132.131	odd	2		2640.2.a.x.1.2		2
165.32	odd	4		825.2.c.c.199.1		4
165.98	odd	4		825.2.c.c.199.4		4
165.164	even	2		825.2.a.e.1.2		2
231.230	odd	2		8085.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.1	✓	2	33.32	even	2
495.2.a.c.1.2		2	11.10	odd	2
825.2.a.e.1.2		2	165.164	even	2
825.2.c.c.199.1		4	165.32	odd	4
825.2.c.c.199.4		4	165.98	odd	4
1815.2.a.i.1.2		2	3.2	odd	2
2475.2.a.r.1.1		2	55.54	odd	2
2475.2.c.n.199.1		4	55.43	even	4
2475.2.c.n.199.4		4	55.32	even	4
2640.2.a.x.1.2		2	132.131	odd	2
5445.2.a.s.1.1		2	1.1	even	1	trivial
7920.2.a.bz.1.2		2	44.43	even	2
8085.2.a.bd.1.1		2	231.230	odd	2
9075.2.a.bh.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} -5.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.00000 q^{20} -6.92820 q^{23} +1.00000 q^{25} +9.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -10.9282 q^{31} +5.19615 q^{32} -2.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} +1.73205 q^{40} +3.46410 q^{41} +4.92820 q^{43} +12.0000 q^{46} +6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -5.46410 q^{52} -0.928203 q^{53} -3.46410 q^{56} +6.00000 q^{58} +6.92820 q^{59} -2.00000 q^{61} +18.9282 q^{62} +1.00000 q^{64} -5.46410 q^{65} +8.00000 q^{67} +3.46410 q^{70} -13.8564 q^{71} +8.39230 q^{73} +8.53590 q^{74} -5.46410 q^{76} +6.53590 q^{79} -5.00000 q^{80} -6.00000 q^{82} +8.53590 q^{83} -8.53590 q^{86} -0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} -12.0000 q^{94} -5.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7} - 4 q^{13} - 10 q^{16} - 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} - 4 q^{28} - 8 q^{31} - 4 q^{35} + 4 q^{37} + 12 q^{38} - 4 q^{43} + 24 q^{46} - 6 q^{49} - 4 q^{52} + 12 q^{53} + 12 q^{58} - 4 q^{61} + 24 q^{62} + 2 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{73} + 24 q^{74} - 4 q^{76} + 20 q^{79} - 10 q^{80} - 12 q^{82} + 24 q^{83} - 24 q^{86} + 12 q^{89} + 8 q^{91} - 24 q^{94} - 4 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 - 4 * q^7 - 4 * q^13 - 10 * q^16 - 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 - 4 * q^28 - 8 * q^31 - 4 * q^35 + 4 * q^37 + 12 * q^38 - 4 * q^43 + 24 * q^46 - 6 * q^49 - 4 * q^52 + 12 * q^53 + 12 * q^58 - 4 * q^61 + 24 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 - 4 * q^73 + 24 * q^74 - 4 * q^76 + 20 * q^79 - 10 * q^80 - 12 * q^82 + 24 * q^83 - 24 * q^86 + 12 * q^89 + 8 * q^91 - 24 * q^94 - 4 * q^95 - 20 * q^97

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