LMFDB - Embedded newform 5445.2.a.r.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 605)
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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.46410 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.46410 q^{7} +1.73205 q^{8} +1.73205 q^{10} -6.00000 q^{14} -5.00000 q^{16} +6.92820 q^{17} -6.92820 q^{19} -1.00000 q^{20} -6.00000 q^{23} +1.00000 q^{25} +3.46410 q^{28} +4.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -3.46410 q^{35} +10.0000 q^{37} +12.0000 q^{38} -1.73205 q^{40} -6.92820 q^{41} -3.46410 q^{43} +10.3923 q^{46} +6.00000 q^{47} +5.00000 q^{49} -1.73205 q^{50} +6.00000 q^{53} +6.00000 q^{56} +6.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} +10.0000 q^{67} +6.92820 q^{68} +6.00000 q^{70} -6.92820 q^{73} -17.3205 q^{74} -6.92820 q^{76} -6.92820 q^{79} +5.00000 q^{80} +12.0000 q^{82} +17.3205 q^{83} -6.92820 q^{85} +6.00000 q^{86} +6.00000 q^{89} -6.00000 q^{92} -10.3923 q^{94} +6.92820 q^{95} -10.0000 q^{97} -8.66025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5	$ 2 q + 2 q^{4} - 2 q^{5} - 12 q^{14} - 10 q^{16} - 2 q^{20} - 12 q^{23} + 2 q^{25} + 8 q^{31} - 24 q^{34} + 20 q^{37} + 24 q^{38} + 12 q^{47} + 10 q^{49} + 12 q^{53} + 12 q^{56} + 2 q^{64} + 20 q^{67} + 12 q^{70} + 10 q^{80} + 24 q^{82} + 12 q^{86} + 12 q^{89} - 12 q^{92} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 12 * q^14 - 10 * q^16 - 2 * q^20 - 12 * q^23 + 2 * q^25 + 8 * q^31 - 24 * q^34 + 20 * q^37 + 24 * q^38 + 12 * q^47 + 10 * q^49 + 12 * q^53 + 12 * q^56 + 2 * q^64 + 20 * q^67 + 12 * q^70 + 10 * q^80 + 24 * q^82 + 12 * q^86 + 12 * q^89 - 12 * q^92 - 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
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	1.73205	0.547723
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−5.00000	−1.25000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	−6.92820	−1.58944	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820	−1.58944	−0.794719	+	0.606977i	$0.792382\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	3.46410	0.654654
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	−3.46410	−0.585540
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	12.0000	1.94666
$39$	0	0
$40$	−1.73205	−0.273861
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0	0
$46$	10.3923	1.53226
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	−1.73205	−0.244949
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.92820	0.887066	0.443533	−	0.896258i	$-0.353725\pi$
$61$	6.92820	0.887066	0.443533	+	0.896258i	$0.353725\pi$
$62$	−6.92820	−0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	6.00000	0.717137
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−6.92820	−0.810885	−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	−17.3205	−2.01347
$75$	0	0
$76$	−6.92820	−0.794719
$77$	0	0
$78$	0	0
$79$	−6.92820	−0.779484	−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820	−0.779484	−0.389742	+	0.920924i	$0.627436\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	12.0000	1.32518
$83$	17.3205	1.90117	0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205	1.90117	0.950586	+	0.310460i	$0.100483\pi$
$84$	0	0
$85$	−6.92820	−0.751469
$86$	6.00000	0.646997
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−10.3923	−1.07188
$95$	6.92820	0.710819
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−8.66025	−0.874818
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	0	0
$106$	−10.3923	−1.00939
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	0	0
$112$	−17.3205	−1.63663
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	0	0
$122$	−12.0000	−1.08643
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.3923	0.922168	0.461084	−	0.887357i	$-0.347461\pi$
$127$	10.3923	0.922168	0.461084	+	0.887357i	$0.347461\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	−6.92820	−0.605320	−0.302660	−	0.953099i	$-0.597875\pi$
$131$	−6.92820	−0.605320	−0.302660	+	0.953099i	$0.597875\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	−17.3205	−1.49626
$135$	0	0
$136$	12.0000	1.02899
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	13.8564	1.17529	0.587643	−	0.809121i	$-0.300056\pi$
$139$	13.8564	1.17529	0.587643	+	0.809121i	$0.300056\pi$
$140$	−3.46410	−0.292770
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	10.0000	0.821995
$149$	−6.92820	−0.567581	−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820	−0.567581	−0.283790	+	0.958886i	$0.591592\pi$
$150$	0	0
$151$	6.92820	0.563809	0.281905	−	0.959442i	$-0.409034\pi$
$151$	6.92820	0.563809	0.281905	+	0.959442i	$0.409034\pi$
$152$	−12.0000	−0.973329
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	−5.19615	−0.410792
$161$	−20.7846	−1.63806
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	−30.0000	−2.32845
$167$	−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
$169$	−13.0000	−1.00000
$170$	12.0000	0.920358
$171$	0	0
$172$	−3.46410	−0.264135
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	3.46410	0.261861
$176$	0	0
$177$	0	0
$178$	−10.3923	−0.778936
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	−10.3923	−0.766131
$185$	−10.0000	−0.735215
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	−12.0000	−0.870572
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−6.92820	−0.498703	−0.249351	−	0.968413i	$-0.580217\pi$
$193$	−6.92820	−0.498703	−0.249351	+	0.968413i	$0.580217\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	5.00000	0.357143
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	24.0000	1.68863
$203$	0	0
$204$	0	0
$205$	6.92820	0.483887
$206$	−3.46410	−0.241355
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.7846	−1.43087	−0.715436	−	0.698679i	$-0.753772\pi$
$211$	−20.7846	−1.43087	−0.715436	+	0.698679i	$0.753772\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	3.46410	0.236250
$216$	0	0
$217$	13.8564	0.940634
$218$	−12.0000	−0.812743
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	18.0000	1.20268
$225$	0	0
$226$	10.3923	0.691286
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−10.3923	−0.685248
$231$	0	0
$232$	0	0
$233$	−6.92820	−0.453882	−0.226941	−	0.973909i	$-0.572872\pi$
$233$	−6.92820	−0.453882	−0.226941	+	0.973909i	$0.572872\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	−41.5692	−2.69453
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	−20.7846	−1.33885	−0.669427	−	0.742878i	$-0.733460\pi$
$241$	−20.7846	−1.33885	−0.669427	+	0.742878i	$0.733460\pi$
$242$	0	0
$243$	0	0
$244$	6.92820	0.443533
$245$	−5.00000	−0.319438
$246$	0	0
$247$	0	0
$248$	6.92820	0.439941
$249$	0	0
$250$	1.73205	0.109545
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−18.0000	−1.12942
$255$	0	0
$256$	19.0000	1.18750
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	34.6410	2.15249
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	−3.46410	−0.213606	−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410	−0.213606	−0.106803	+	0.994280i	$0.534061\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	41.5692	2.54877
$267$	0	0
$268$	10.0000	0.610847
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	6.92820	0.420858	0.210429	−	0.977609i	$-0.432514\pi$
$271$	6.92820	0.420858	0.210429	+	0.977609i	$0.432514\pi$
$272$	−34.6410	−2.10042
$273$	0	0
$274$	10.3923	0.627822
$275$	0	0
$276$	0	0
$277$	27.7128	1.66510	0.832551	−	0.553949i	$-0.186880\pi$
$277$	27.7128	1.66510	0.832551	+	0.553949i	$0.186880\pi$
$278$	−24.0000	−1.43942
$279$	0	0
$280$	−6.00000	−0.358569
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	24.2487	1.44144	0.720718	−	0.693228i	$-0.243812\pi$
$283$	24.2487	1.44144	0.720718	+	0.693228i	$0.243812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.0000	−1.41668
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	−6.92820	−0.405442
$293$	−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
$295$	0	0
$296$	17.3205	1.00673
$297$	0	0
$298$	12.0000	0.695141
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	−12.0000	−0.690522
$303$	0	0
$304$	34.6410	1.98680
$305$	−6.92820	−0.396708
$306$	0	0
$307$	−3.46410	−0.197707	−0.0988534	−	0.995102i	$-0.531517\pi$
$307$	−3.46410	−0.197707	−0.0988534	+	0.995102i	$0.531517\pi$
$308$	0	0
$309$	0	0
$310$	6.92820	0.393496
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	3.46410	0.195491
$315$	0	0
$316$	−6.92820	−0.389742
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	36.0000	2.00620
$323$	−48.0000	−2.67079
$324$	0	0
$325$	0	0
$326$	−3.46410	−0.191859
$327$	0	0
$328$	−12.0000	−0.662589
$329$	20.7846	1.14589
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	17.3205	0.950586
$333$	0	0
$334$	18.0000	0.984916
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−20.7846	−1.13221	−0.566105	−	0.824333i	$-0.691550\pi$
$337$	−20.7846	−1.13221	−0.566105	+	0.824333i	$0.691550\pi$
$338$	22.5167	1.22474
$339$	0	0
$340$	−6.92820	−0.375735
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	−6.00000	−0.323498
$345$	0	0
$346$	0	0
$347$	−24.2487	−1.30174	−0.650870	−	0.759190i	$-0.725596\pi$
$347$	−24.2487	−1.30174	−0.650870	+	0.759190i	$0.725596\pi$
$348$	0	0
$349$	−13.8564	−0.741716	−0.370858	−	0.928689i	$-0.620936\pi$
$349$	−13.8564	−0.741716	−0.370858	+	0.928689i	$0.620936\pi$
$350$	−6.00000	−0.320713
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−20.7846	−1.09850
$359$	34.6410	1.82828	0.914141	−	0.405395i	$-0.132866\pi$
$359$	34.6410	1.82828	0.914141	+	0.405395i	$0.132866\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	−24.2487	−1.27448
$363$	0	0
$364$	0	0
$365$	6.92820	0.362639
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	30.0000	1.56386
$369$	0	0
$370$	17.3205	0.900450
$371$	20.7846	1.07908
$372$	0	0
$373$	27.7128	1.43492	0.717458	−	0.696602i	$-0.245306\pi$
$373$	27.7128	1.43492	0.717458	+	0.696602i	$0.245306\pi$
$374$	0	0
$375$	0	0
$376$	10.3923	0.535942
$377$	0	0
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	6.92820	0.355409
$381$	0	0
$382$	−20.7846	−1.06343
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	8.66025	0.437409
$393$	0	0
$394$	0	0
$395$	6.92820	0.348596
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−6.92820	−0.347279
$399$	0	0
$400$	−5.00000	−0.250000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	−13.8564	−0.689382
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	20.7846	1.02773	0.513866	−	0.857870i	$-0.328213\pi$
$409$	20.7846	1.02773	0.513866	+	0.857870i	$0.328213\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	2.00000	0.0985329
$413$	0	0
$414$	0	0
$415$	−17.3205	−0.850230
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	36.0000	1.75245
$423$	0	0
$424$	10.3923	0.504695
$425$	6.92820	0.336067
$426$	0	0
$427$	24.0000	1.16144
$428$	3.46410	0.167444
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−27.7128	−1.33488	−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128	−1.33488	−0.667440	+	0.744664i	$0.732610\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−24.0000	−1.15204
$435$	0	0
$436$	6.92820	0.331801
$437$	41.5692	1.98853
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−24.2487	−1.14821
$447$	0	0
$448$	3.46410	0.163663
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	3.46410	0.161867
$459$	0	0
$460$	6.00000	0.279751
$461$	27.7128	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$461$	27.7128	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	0	0
$466$	12.0000	0.555889
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	34.6410	1.59957
$470$	10.3923	0.479361
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.92820	−0.317888
$476$	24.0000	1.10004
$477$	0	0
$478$	−12.0000	−0.548867
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	0	0
$482$	36.0000	1.63976
$483$	0	0
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	12.0000	0.543214
$489$	0	0
$490$	8.66025	0.391230
$491$	−13.8564	−0.625331	−0.312665	−	0.949863i	$-0.601222\pi$
$491$	−13.8564	−0.625331	−0.312665	+	0.949863i	$0.601222\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−20.0000	−0.898027
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	20.7846	0.927663
$503$	17.3205	0.772283	0.386142	−	0.922440i	$-0.373808\pi$
$503$	17.3205	0.772283	0.386142	+	0.922440i	$0.373808\pi$
$504$	0	0
$505$	13.8564	0.616602
$506$	0	0
$507$	0	0
$508$	10.3923	0.461084
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−10.3923	−0.458385
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	0	0
$518$	−60.0000	−2.63625
$519$	0	0
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	−24.2487	−1.06032	−0.530161	−	0.847897i	$-0.677869\pi$
$523$	−24.2487	−1.06032	−0.530161	+	0.847897i	$0.677869\pi$
$524$	−6.92820	−0.302660
$525$	0	0
$526$	6.00000	0.261612
$527$	27.7128	1.20719
$528$	0	0
$529$	13.0000	0.565217
$530$	10.3923	0.451413
$531$	0	0
$532$	−24.0000	−1.04053
$533$	0	0
$534$	0	0
$535$	−3.46410	−0.149766
$536$	17.3205	0.748132
$537$	0	0
$538$	−51.9615	−2.24022
$539$	0	0
$540$	0	0
$541$	−13.8564	−0.595733	−0.297867	−	0.954607i	$-0.596275\pi$
$541$	−13.8564	−0.595733	−0.297867	+	0.954607i	$0.596275\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	36.0000	1.54349
$545$	−6.92820	−0.296772
$546$	0	0
$547$	17.3205	0.740571	0.370286	−	0.928918i	$-0.379260\pi$
$547$	17.3205	0.740571	0.370286	+	0.928918i	$0.379260\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−24.0000	−1.02058
$554$	−48.0000	−2.03932
$555$	0	0
$556$	13.8564	0.587643
$557$	27.7128	1.17423	0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128	1.17423	0.587115	+	0.809504i	$0.300264\pi$
$558$	0	0
$559$	0	0
$560$	17.3205	0.731925
$561$	0	0
$562$	−12.0000	−0.506189
$563$	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
$565$	6.00000	0.252422
$566$	−42.0000	−1.76539
$567$	0	0
$568$	0	0
$569$	6.92820	0.290445	0.145223	−	0.989399i	$-0.453610\pi$
$569$	6.92820	0.290445	0.145223	+	0.989399i	$0.453610\pi$
$570$	0	0
$571$	27.7128	1.15975	0.579873	−	0.814707i	$-0.303102\pi$
$571$	27.7128	1.15975	0.579873	+	0.814707i	$0.303102\pi$
$572$	0	0
$573$	0	0
$574$	41.5692	1.73507
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−53.6936	−2.23336
$579$	0	0
$580$	0	0
$581$	60.0000	2.48922
$582$	0	0
$583$	0	0
$584$	−12.0000	−0.496564
$585$	0	0
$586$	24.0000	0.991431
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	−27.7128	−1.14189
$590$	0	0
$591$	0	0
$592$	−50.0000	−2.05499
$593$	6.92820	0.284507	0.142254	−	0.989830i	$-0.454565\pi$
$593$	6.92820	0.284507	0.142254	+	0.989830i	$0.454565\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	−6.92820	−0.283790
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−48.4974	−1.97825	−0.989126	−	0.147074i	$-0.953015\pi$
$601$	−48.4974	−1.97825	−0.989126	+	0.147074i	$0.953015\pi$
$602$	20.7846	0.847117
$603$	0	0
$604$	6.92820	0.281905
$605$	0	0
$606$	0	0
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	−36.0000	−1.45999
$609$	0	0
$610$	12.0000	0.485866
$611$	0	0
$612$	0	0
$613$	27.7128	1.11931	0.559655	−	0.828726i	$-0.310934\pi$
$613$	27.7128	1.11931	0.559655	+	0.828726i	$0.310934\pi$
$614$	6.00000	0.242140
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−41.5692	−1.66677
$623$	20.7846	0.832718
$624$	0	0
$625$	1.00000	0.0400000
$626$	−58.8897	−2.35371
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	69.2820	2.76246
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−31.1769	−1.23819
$635$	−10.3923	−0.412406
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	12.1244	0.479257
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	−20.7846	−0.819028
$645$	0	0
$646$	83.1384	3.27104
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	2.00000	0.0783260
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	6.92820	0.270707
$656$	34.6410	1.35250
$657$	0	0
$658$	−36.0000	−1.40343
$659$	27.7128	1.07954	0.539769	−	0.841813i	$-0.318512\pi$
$659$	27.7128	1.07954	0.539769	+	0.841813i	$0.318512\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−13.8564	−0.538545
$663$	0	0
$664$	30.0000	1.16423
$665$	24.0000	0.930680
$666$	0	0
$667$	0	0
$668$	−10.3923	−0.402090
$669$	0	0
$670$	17.3205	0.669150
$671$	0	0
$672$	0	0
$673$	−34.6410	−1.33531	−0.667657	−	0.744469i	$-0.732703\pi$
$673$	−34.6410	−1.33531	−0.667657	+	0.744469i	$0.732703\pi$
$674$	36.0000	1.38667
$675$	0	0
$676$	−13.0000	−0.500000
$677$	13.8564	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$677$	13.8564	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$678$	0	0
$679$	−34.6410	−1.32940
$680$	−12.0000	−0.460179
$681$	0	0
$682$	0	0
$683$	42.0000	1.60709	0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000	1.60709	0.803543	+	0.595247i	$0.202946\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	12.0000	0.458162
$687$	0	0
$688$	17.3205	0.660338
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	0	0
$694$	42.0000	1.59430
$695$	−13.8564	−0.525603
$696$	0	0
$697$	−48.0000	−1.81813
$698$	24.0000	0.908413
$699$	0	0
$700$	3.46410	0.130931
$701$	−20.7846	−0.785024	−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846	−0.785024	−0.392512	+	0.919747i	$0.628394\pi$
$702$	0	0
$703$	−69.2820	−2.61302
$704$	0	0
$705$	0	0
$706$	−10.3923	−0.391120
$707$	−48.0000	−1.80523
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	10.3923	0.389468
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−60.0000	−2.23918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	6.92820	0.258020
$722$	−50.2295	−1.86935
$723$	0	0
$724$	14.0000	0.520306
$725$	0	0
$726$	0	0
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	0	0
$729$	0	0
$730$	−12.0000	−0.444140
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−41.5692	−1.53539	−0.767697	−	0.640813i	$-0.778597\pi$
$733$	−41.5692	−1.53539	−0.767697	+	0.640813i	$0.778597\pi$
$734$	−3.46410	−0.127862
$735$	0	0
$736$	−31.1769	−1.14920
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−31.1769	−1.14377	−0.571885	−	0.820334i	$-0.693788\pi$
$743$	−31.1769	−1.14377	−0.571885	+	0.820334i	$0.693788\pi$
$744$	0	0
$745$	6.92820	0.253830
$746$	−48.0000	−1.75740
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−30.0000	−1.09399
$753$	0	0
$754$	0	0
$755$	−6.92820	−0.252143
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−55.4256	−2.01315
$759$	0	0
$760$	12.0000	0.435286
$761$	−13.8564	−0.502294	−0.251147	−	0.967949i	$-0.580808\pi$
$761$	−13.8564	−0.502294	−0.251147	+	0.967949i	$0.580808\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	12.0000	0.434145
$765$	0	0
$766$	31.1769	1.12647
$767$	0	0
$768$	0	0
$769$	41.5692	1.49902	0.749512	−	0.661991i	$-0.230288\pi$
$769$	41.5692	1.49902	0.749512	+	0.661991i	$0.230288\pi$
$770$	0	0
$771$	0	0
$772$	−6.92820	−0.249351
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−17.3205	−0.621770
$777$	0	0
$778$	10.3923	0.372582
$779$	48.0000	1.71978
$780$	0	0
$781$	0	0
$782$	72.0000	2.57471
$783$	0	0
$784$	−25.0000	−0.892857
$785$	2.00000	0.0713831
$786$	0	0
$787$	−45.0333	−1.60526	−0.802632	−	0.596474i	$-0.796568\pi$
$787$	−45.0333	−1.60526	−0.802632	+	0.596474i	$0.796568\pi$
$788$	0	0
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−20.7846	−0.739016
$792$	0	0
$793$	0	0
$794$	58.8897	2.08992
$795$	0	0
$796$	4.00000	0.141776
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	41.5692	1.47061
$800$	5.19615	0.183712
$801$	0	0
$802$	−31.1769	−1.10090
$803$	0	0
$804$	0	0
$805$	20.7846	0.732561
$806$	0	0
$807$	0	0
$808$	−24.0000	−0.844317
$809$	−13.8564	−0.487165	−0.243583	−	0.969880i	$-0.578323\pi$
$809$	−13.8564	−0.487165	−0.243583	+	0.969880i	$0.578323\pi$
$810$	0	0
$811$	−13.8564	−0.486564	−0.243282	−	0.969956i	$-0.578224\pi$
$811$	−13.8564	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	24.0000	0.839654
$818$	−36.0000	−1.25871
$819$	0	0
$820$	6.92820	0.241943
$821$	48.4974	1.69257	0.846286	−	0.532729i	$-0.178834\pi$
$821$	48.4974	1.69257	0.846286	+	0.532729i	$0.178834\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	3.46410	0.120678
$825$	0	0
$826$	0	0
$827$	17.3205	0.602293	0.301147	−	0.953578i	$-0.402631\pi$
$827$	17.3205	0.602293	0.301147	+	0.953578i	$0.402631\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	30.0000	1.04132
$831$	0	0
$832$	0	0
$833$	34.6410	1.20024
$834$	0	0
$835$	10.3923	0.359641
$836$	0	0
$837$	0	0
$838$	−20.7846	−0.717992
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	45.0333	1.55195
$843$	0	0
$844$	−20.7846	−0.715436
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	−30.0000	−1.03020
$849$	0	0
$850$	−12.0000	−0.411597
$851$	−60.0000	−2.05677
$852$	0	0
$853$	41.5692	1.42330	0.711651	−	0.702533i	$-0.247948\pi$
$853$	41.5692	1.42330	0.711651	+	0.702533i	$0.247948\pi$
$854$	−41.5692	−1.42247
$855$	0	0
$856$	6.00000	0.205076
$857$	−20.7846	−0.709989	−0.354994	−	0.934868i	$-0.615517\pi$
$857$	−20.7846	−0.709989	−0.354994	+	0.934868i	$0.615517\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	3.46410	0.118125
$861$	0	0
$862$	48.0000	1.63489
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	0	0
$866$	58.8897	2.00115
$867$	0	0
$868$	13.8564	0.470317
$869$	0	0
$870$	0	0
$871$	0	0
$872$	12.0000	0.406371
$873$	0	0
$874$	−72.0000	−2.43544
$875$	−3.46410	−0.117108
$876$	0	0
$877$	27.7128	0.935795	0.467898	−	0.883783i	$-0.345012\pi$
$877$	27.7128	0.935795	0.467898	+	0.883783i	$0.345012\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	0	0
$886$	−10.3923	−0.349136
$887$	17.3205	0.581566	0.290783	−	0.956789i	$-0.406084\pi$
$887$	17.3205	0.581566	0.290783	+	0.956789i	$0.406084\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	10.3923	0.348351
$891$	0	0
$892$	14.0000	0.468755
$893$	−41.5692	−1.39106
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−42.0000	−1.40312
$897$	0	0
$898$	51.9615	1.73398
$899$	0	0
$900$	0	0
$901$	41.5692	1.38487
$902$	0	0
$903$	0	0
$904$	−10.3923	−0.345643
$905$	−14.0000	−0.465376
$906$	0	0
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	−3.46410	−0.114960
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	0	0
$914$	−36.0000	−1.19077
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	−24.0000	−0.792550
$918$	0	0
$919$	−13.8564	−0.457081	−0.228540	−	0.973534i	$-0.573395\pi$
$919$	−13.8564	−0.457081	−0.228540	+	0.973534i	$0.573395\pi$
$920$	10.3923	0.342624
$921$	0	0
$922$	−48.0000	−1.58080
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	−24.2487	−0.796862
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−34.6410	−1.13531
$932$	−6.92820	−0.226941
$933$	0	0
$934$	31.1769	1.02014
$935$	0	0
$936$	0	0
$937$	−34.6410	−1.13167	−0.565836	−	0.824518i	$-0.691447\pi$
$937$	−34.6410	−1.13167	−0.565836	+	0.824518i	$0.691447\pi$
$938$	−60.0000	−1.95907
$939$	0	0
$940$	−6.00000	−0.195698
$941$	13.8564	0.451706	0.225853	−	0.974161i	$-0.427483\pi$
$941$	13.8564	0.451706	0.225853	+	0.974161i	$0.427483\pi$
$942$	0	0
$943$	41.5692	1.35368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	0	0
$949$	0	0
$950$	12.0000	0.389331
$951$	0	0
$952$	41.5692	1.34727
$953$	−34.6410	−1.12213	−0.561066	−	0.827771i	$-0.689609\pi$
$953$	−34.6410	−1.12213	−0.561066	+	0.827771i	$0.689609\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	6.92820	0.224074
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	−20.7846	−0.669427
$965$	6.92820	0.223027
$966$	0	0
$967$	24.2487	0.779786	0.389893	−	0.920860i	$-0.372512\pi$
$967$	24.2487	0.779786	0.389893	+	0.920860i	$0.372512\pi$
$968$	0	0
$969$	0	0
$970$	−17.3205	−0.556128
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	48.0000	1.53881
$974$	24.2487	0.776979
$975$	0	0
$976$	−34.6410	−1.10883
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	−5.00000	−0.159719
$981$	0	0
$982$	24.0000	0.765871
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.7846	0.660912
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	20.7846	0.659912
$993$	0	0
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	41.5692	1.31651	0.658255	−	0.752795i	$-0.271295\pi$
$997$	41.5692	1.31651	0.658255	+	0.752795i	$0.271295\pi$
$998$	−27.7128	−0.877234
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.r.1.1		2
3.2	odd	2		605.2.a.f.1.2	yes	2
11.10	odd	2	inner	5445.2.a.r.1.2		2
12.11	even	2		9680.2.a.bg.1.1		2
15.14	odd	2		3025.2.a.j.1.1		2
33.2	even	10		605.2.g.h.81.2		8
33.5	odd	10		605.2.g.h.366.1		8
33.8	even	10		605.2.g.h.251.1		8
33.14	odd	10		605.2.g.h.251.2		8
33.17	even	10		605.2.g.h.366.2		8
33.20	odd	10		605.2.g.h.81.1		8
33.26	odd	10		605.2.g.h.511.2		8
33.29	even	10		605.2.g.h.511.1		8
33.32	even	2		605.2.a.f.1.1	✓	2
132.131	odd	2		9680.2.a.bg.1.2		2
165.164	even	2		3025.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.f.1.1	✓	2	33.32	even	2
605.2.a.f.1.2	yes	2	3.2	odd	2
605.2.g.h.81.1		8	33.20	odd	10
605.2.g.h.81.2		8	33.2	even	10
605.2.g.h.251.1		8	33.8	even	10
605.2.g.h.251.2		8	33.14	odd	10
605.2.g.h.366.1		8	33.5	odd	10
605.2.g.h.366.2		8	33.17	even	10
605.2.g.h.511.1		8	33.29	even	10
605.2.g.h.511.2		8	33.26	odd	10
3025.2.a.j.1.1		2	15.14	odd	2
3025.2.a.j.1.2		2	165.164	even	2
5445.2.a.r.1.1		2	1.1	even	1	trivial
5445.2.a.r.1.2		2	11.10	odd	2	inner
9680.2.a.bg.1.1		2	12.11	even	2
9680.2.a.bg.1.2		2	132.131	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.46410 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.46410 q^{7} +1.73205 q^{8} +1.73205 q^{10} -6.00000 q^{14} -5.00000 q^{16} +6.92820 q^{17} -6.92820 q^{19} -1.00000 q^{20} -6.00000 q^{23} +1.00000 q^{25} +3.46410 q^{28} +4.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -3.46410 q^{35} +10.0000 q^{37} +12.0000 q^{38} -1.73205 q^{40} -6.92820 q^{41} -3.46410 q^{43} +10.3923 q^{46} +6.00000 q^{47} +5.00000 q^{49} -1.73205 q^{50} +6.00000 q^{53} +6.00000 q^{56} +6.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} +10.0000 q^{67} +6.92820 q^{68} +6.00000 q^{70} -6.92820 q^{73} -17.3205 q^{74} -6.92820 q^{76} -6.92820 q^{79} +5.00000 q^{80} +12.0000 q^{82} +17.3205 q^{83} -6.92820 q^{85} +6.00000 q^{86} +6.00000 q^{89} -6.00000 q^{92} -10.3923 q^{94} +6.92820 q^{95} -10.0000 q^{97} -8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5	\( 2 q + 2 q^{4} - 2 q^{5} - 12 q^{14} - 10 q^{16} - 2 q^{20} - 12 q^{23} + 2 q^{25} + 8 q^{31} - 24 q^{34} + 20 q^{37} + 24 q^{38} + 12 q^{47} + 10 q^{49} + 12 q^{53} + 12 q^{56} + 2 q^{64} + 20 q^{67} + 12 q^{70} + 10 q^{80} + 24 q^{82} + 12 q^{86} + 12 q^{89} - 12 q^{92} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 12 * q^14 - 10 * q^16 - 2 * q^20 - 12 * q^23 + 2 * q^25 + 8 * q^31 - 24 * q^34 + 20 * q^37 + 24 * q^38 + 12 * q^47 + 10 * q^49 + 12 * q^53 + 12 * q^56 + 2 * q^64 + 20 * q^67 + 12 * q^70 + 10 * q^80 + 24 * q^82 + 12 * q^86 + 12 * q^89 - 12 * q^92 - 20 * q^97

Introduction

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