LMFDB - Embedded newform 5445.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 6x + 11 $ x^4 - x^3 - 8x^2 + 6x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.55157$ 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)

$f(q)$	$=$	$q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +O(q^{10})$	$q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +0.933531 q^{10} -1.44843 q^{13} -1.90541 q^{14} -0.469405 q^{16} -0.867063 q^{17} +3.12852 q^{19} -1.12852 q^{20} +4.70547 q^{23} +1.00000 q^{25} -1.35216 q^{26} +2.30340 q^{28} -2.03835 q^{29} +10.6136 q^{31} +5.40294 q^{32} -0.809430 q^{34} -2.04108 q^{35} +4.15664 q^{37} +2.92057 q^{38} -2.92057 q^{40} +0.805012 q^{41} -2.34089 q^{43} +4.39271 q^{46} -10.3803 q^{47} -2.83399 q^{49} +0.933531 q^{50} +1.63459 q^{52} -7.21596 q^{53} +5.96112 q^{56} -1.90286 q^{58} +8.32351 q^{59} -8.76752 q^{61} +9.90814 q^{62} +5.98262 q^{64} -1.44843 q^{65} -3.15664 q^{67} +0.978497 q^{68} -1.90541 q^{70} -12.8707 q^{71} -14.7184 q^{73} +3.88035 q^{74} -3.53059 q^{76} -16.9992 q^{79} -0.469405 q^{80} +0.751504 q^{82} -6.14148 q^{83} -0.867063 q^{85} -2.18529 q^{86} -3.77194 q^{89} +2.95637 q^{91} -5.31022 q^{92} -9.69031 q^{94} +3.12852 q^{95} -1.93795 q^{97} -2.64562 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} + 4 q^{5} - 8 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 9 * q^4 + 4 * q^5 - 8 * q^7 - 3 * q^8	$ 4 q - q^{2} + 9 q^{4} + 4 q^{5} - 8 q^{7} - 3 q^{8} - q^{10} - 15 q^{13} - 7 q^{14} + 7 q^{16} + 6 q^{17} - q^{19} + 9 q^{20} + q^{23} + 4 q^{25} + 18 q^{26} - 31 q^{28} - 17 q^{29} + 15 q^{31} + 8 q^{32} - 35 q^{34} - 8 q^{35} - q^{37} + 3 q^{38} - 3 q^{40} + 12 q^{41} - 14 q^{43} - 9 q^{46} - 14 q^{47} + 20 q^{49} - q^{50} - 39 q^{52} - 2 q^{53} + 12 q^{56} - 11 q^{58} + 11 q^{59} + q^{61} + 30 q^{62} - 3 q^{64} - 15 q^{65} + 5 q^{67} + 19 q^{68} - 7 q^{70} + 3 q^{71} - 45 q^{73} - 29 q^{74} - 23 q^{76} + 7 q^{80} + 11 q^{82} - 15 q^{83} + 6 q^{85} + 10 q^{86} - 2 q^{89} + 16 q^{91} - 34 q^{92} - 29 q^{94} - q^{95} - 26 q^{97} + q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 + 4 * q^5 - 8 * q^7 - 3 * q^8 - q^10 - 15 * q^13 - 7 * q^14 + 7 * q^16 + 6 * q^17 - q^19 + 9 * q^20 + q^23 + 4 * q^25 + 18 * q^26 - 31 * q^28 - 17 * q^29 + 15 * q^31 + 8 * q^32 - 35 * q^34 - 8 * q^35 - q^37 + 3 * q^38 - 3 * q^40 + 12 * q^41 - 14 * q^43 - 9 * q^46 - 14 * q^47 + 20 * q^49 - q^50 - 39 * q^52 - 2 * q^53 + 12 * q^56 - 11 * q^58 + 11 * q^59 + q^61 + 30 * q^62 - 3 * q^64 - 15 * q^65 + 5 * q^67 + 19 * q^68 - 7 * q^70 + 3 * q^71 - 45 * q^73 - 29 * q^74 - 23 * q^76 + 7 * q^80 + 11 * q^82 - 15 * q^83 + 6 * q^85 + 10 * q^86 - 2 * q^89 + 16 * q^91 - 34 * q^92 - 29 * q^94 - q^95 - 26 * q^97 + 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 $

$2$	0.933531	0.660106	0.330053	−	0.943962i	$-0.392933\pi$
$2$	0.933531	0.660106	0.330053	+	0.943962i	$0.392933\pi$
$3$	0	0
$3$	0	0
$4$	−1.12852	−0.564260
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.04108	−0.771456	−0.385728	−	0.922613i	$-0.626050\pi$
$7$	−2.04108	−0.771456	−0.385728	+	0.922613i	$0.626050\pi$
$8$	−2.92057	−1.03258
$9$	0	0
$10$	0.933531	0.295209
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.44843	−0.401724	−0.200862	−	0.979620i	$-0.564374\pi$
$13$	−1.44843	−0.401724	−0.200862	+	0.979620i	$0.564374\pi$
$14$	−1.90541	−0.509243
$15$	0	0
$16$	−0.469405	−0.117351
$17$	−0.867063	−0.210294	−0.105147	−	0.994457i	$-0.533531\pi$
$17$	−0.867063	−0.210294	−0.105147	+	0.994457i	$0.533531\pi$
$18$	0	0
$19$	3.12852	0.717732	0.358866	−	0.933389i	$-0.383164\pi$
$19$	3.12852	0.717732	0.358866	+	0.933389i	$0.383164\pi$
$20$	−1.12852	−0.252345
$21$	0	0
$22$	0	0
$23$	4.70547	0.981159	0.490580	−	0.871396i	$-0.336785\pi$
$23$	4.70547	0.981159	0.490580	+	0.871396i	$0.336785\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.35216	−0.265180
$27$	0	0
$28$	2.30340	0.435301
$29$	−2.03835	−0.378512	−0.189256	−	0.981928i	$-0.560608\pi$
$29$	−2.03835	−0.378512	−0.189256	+	0.981928i	$0.560608\pi$
$30$	0	0
$31$	10.6136	1.90626	0.953131	−	0.302558i	$-0.0978407\pi$
$31$	10.6136	1.90626	0.953131	+	0.302558i	$0.0978407\pi$
$32$	5.40294	0.955113
$33$	0	0
$34$	−0.809430	−0.138816
$35$	−2.04108	−0.345005
$36$	0	0
$37$	4.15664	0.683347	0.341674	−	0.939819i	$-0.389006\pi$
$37$	4.15664	0.683347	0.341674	+	0.939819i	$0.389006\pi$
$38$	2.92057	0.473779
$39$	0	0
$40$	−2.92057	−0.461783
$41$	0.805012	0.125722	0.0628609	−	0.998022i	$-0.479978\pi$
$41$	0.805012	0.125722	0.0628609	+	0.998022i	$0.479978\pi$
$42$	0	0
$43$	−2.34089	−0.356982	−0.178491	−	0.983942i	$-0.557122\pi$
$43$	−2.34089	−0.356982	−0.178491	+	0.983942i	$0.557122\pi$
$44$	0	0
$45$	0	0
$46$	4.39271	0.647669
$47$	−10.3803	−1.51412	−0.757060	−	0.653346i	$-0.773365\pi$
$47$	−10.3803	−1.51412	−0.757060	+	0.653346i	$0.773365\pi$
$48$	0	0
$49$	−2.83399	−0.404856
$50$	0.933531	0.132021
$51$	0	0
$52$	1.63459	0.226676
$53$	−7.21596	−0.991188	−0.495594	−	0.868554i	$-0.665050\pi$
$53$	−7.21596	−0.991188	−0.495594	+	0.868554i	$0.665050\pi$
$54$	0	0
$55$	0	0
$56$	5.96112	0.796588
$57$	0	0
$58$	−1.90286	−0.249858
$59$	8.32351	1.08363	0.541814	−	0.840498i	$-0.317738\pi$
$59$	8.32351	1.08363	0.541814	+	0.840498i	$0.317738\pi$
$60$	0	0
$61$	−8.76752	−1.12257	−0.561283	−	0.827624i	$-0.689692\pi$
$61$	−8.76752	−1.12257	−0.561283	+	0.827624i	$0.689692\pi$
$62$	9.90814	1.25834
$63$	0	0
$64$	5.98262	0.747828
$65$	−1.44843	−0.179656
$66$	0	0
$67$	−3.15664	−0.385645	−0.192822	−	0.981234i	$-0.561764\pi$
$67$	−3.15664	−0.385645	−0.192822	+	0.981234i	$0.561764\pi$
$68$	0.978497	0.118660
$69$	0	0
$70$	−1.90541	−0.227740
$71$	−12.8707	−1.52747	−0.763733	−	0.645532i	$-0.776636\pi$
$71$	−12.8707	−1.52747	−0.763733	+	0.645532i	$0.776636\pi$
$72$	0	0
$73$	−14.7184	−1.72266	−0.861331	−	0.508044i	$-0.830369\pi$
$73$	−14.7184	−1.72266	−0.861331	+	0.508044i	$0.830369\pi$
$74$	3.88035	0.451082
$75$	0	0
$76$	−3.53059	−0.404987
$77$	0	0
$78$	0	0
$79$	−16.9992	−1.91256	−0.956278	−	0.292458i	$-0.905527\pi$
$79$	−16.9992	−1.91256	−0.956278	+	0.292458i	$0.905527\pi$
$80$	−0.469405	−0.0524811
$81$	0	0
$82$	0.751504	0.0829897
$83$	−6.14148	−0.674115	−0.337058	−	0.941484i	$-0.609432\pi$
$83$	−6.14148	−0.674115	−0.337058	+	0.941484i	$0.609432\pi$
$84$	0	0
$85$	−0.867063	−0.0940461
$86$	−2.18529	−0.235646
$87$	0	0
$88$	0	0
$89$	−3.77194	−0.399825	−0.199913	−	0.979814i	$-0.564066\pi$
$89$	−3.77194	−0.399825	−0.199913	+	0.979814i	$0.564066\pi$
$90$	0	0
$91$	2.95637	0.309912
$92$	−5.31022	−0.553628
$93$	0	0
$94$	−9.69031	−0.999480
$95$	3.12852	0.320979
$96$	0	0
$97$	−1.93795	−0.196769	−0.0983845	−	0.995148i	$-0.531367\pi$
$97$	−1.93795	−0.196769	−0.0983845	+	0.995148i	$0.531367\pi$
$98$	−2.64562	−0.267248
$99$	0	0
$100$	−1.12852	−0.112852
$101$	5.25176	0.522570	0.261285	−	0.965262i	$-0.415854\pi$
$101$	5.25176	0.522570	0.261285	+	0.965262i	$0.415854\pi$
$102$	0	0
$103$	−13.6007	−1.34011	−0.670056	−	0.742310i	$-0.733730\pi$
$103$	−13.6007	−1.34011	−0.670056	+	0.742310i	$0.733730\pi$
$104$	4.23026	0.414811
$105$	0	0
$106$	−6.73632	−0.654290
$107$	9.99063	0.965831	0.482915	−	0.875667i	$-0.339578\pi$
$107$	9.99063	0.965831	0.482915	+	0.875667i	$0.339578\pi$
$108$	0	0
$109$	−6.75183	−0.646708	−0.323354	−	0.946278i	$-0.604811\pi$
$109$	−6.75183	−0.646708	−0.323354	+	0.946278i	$0.604811\pi$
$110$	0	0
$111$	0	0
$112$	0.958094	0.0905314
$113$	5.24461	0.493371	0.246686	−	0.969096i	$-0.420658\pi$
$113$	5.24461	0.493371	0.246686	+	0.969096i	$0.420658\pi$
$114$	0	0
$115$	4.70547	0.438788
$116$	2.30032	0.213579
$117$	0	0
$118$	7.77025	0.715310
$119$	1.76974	0.162232
$120$	0	0
$121$	0	0
$122$	−8.18476	−0.741013
$123$	0	0
$124$	−11.9777	−1.07563
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.6751	−1.83462	−0.917311	−	0.398172i	$-0.869645\pi$
$127$	−20.6751	−1.83462	−0.917311	+	0.398172i	$0.869645\pi$
$128$	−5.22091	−0.461468
$129$	0	0
$130$	−1.35216	−0.118592
$131$	−20.3089	−1.77439	−0.887197	−	0.461392i	$-0.847350\pi$
$131$	−20.3089	−1.77439	−0.887197	+	0.461392i	$0.847350\pi$
$132$	0	0
$133$	−6.38556	−0.553698
$134$	−2.94682	−0.254567
$135$	0	0
$136$	2.53232	0.217144
$137$	3.47982	0.297301	0.148650	−	0.988890i	$-0.452507\pi$
$137$	3.47982	0.297301	0.148650	+	0.988890i	$0.452507\pi$
$138$	0	0
$139$	2.73086	0.231629	0.115814	−	0.993271i	$-0.463052\pi$
$139$	2.73086	0.231629	0.115814	+	0.993271i	$0.463052\pi$
$140$	2.30340	0.194673
$141$	0	0
$142$	−12.0152	−1.00829
$143$	0	0
$144$	0	0
$145$	−2.03835	−0.169276
$146$	−13.7401	−1.13714
$147$	0	0
$148$	−4.69085	−0.385585
$149$	9.10841	0.746190	0.373095	−	0.927793i	$-0.378297\pi$
$149$	9.10841	0.746190	0.373095	+	0.927793i	$0.378297\pi$
$150$	0	0
$151$	21.5744	1.75570	0.877852	−	0.478933i	$-0.158976\pi$
$151$	21.5744	1.75570	0.877852	+	0.478933i	$0.158976\pi$
$152$	−9.13706	−0.741114
$153$	0	0
$154$	0	0
$155$	10.6136	0.852506
$156$	0	0
$157$	−23.7940	−1.89896	−0.949482	−	0.313821i	$-0.898391\pi$
$157$	−23.7940	−1.89896	−0.949482	+	0.313821i	$0.898391\pi$
$158$	−15.8693	−1.26249
$159$	0	0
$160$	5.40294	0.427140
$161$	−9.60425	−0.756921
$162$	0	0
$163$	−1.80332	−0.141247	−0.0706236	−	0.997503i	$-0.522499\pi$
$163$	−1.80332	−0.141247	−0.0706236	+	0.997503i	$0.522499\pi$
$164$	−0.908472	−0.0709397
$165$	0	0
$166$	−5.73326	−0.444988
$167$	24.5874	1.90263	0.951315	−	0.308220i	$-0.0997333\pi$
$167$	24.5874	1.90263	0.951315	+	0.308220i	$0.0997333\pi$
$168$	0	0
$169$	−10.9020	−0.838618
$170$	−0.809430	−0.0620804
$171$	0	0
$172$	2.64174	0.201430
$173$	16.2115	1.23254	0.616270	−	0.787535i	$-0.288643\pi$
$173$	16.2115	1.23254	0.616270	+	0.787535i	$0.288643\pi$
$174$	0	0
$175$	−2.04108	−0.154291
$176$	0	0
$177$	0	0
$178$	−3.52123	−0.263927
$179$	−16.9697	−1.26837	−0.634186	−	0.773181i	$-0.718665\pi$
$179$	−16.9697	−1.26837	−0.634186	+	0.773181i	$0.718665\pi$
$180$	0	0
$181$	18.3508	1.36400	0.682002	−	0.731350i	$-0.261109\pi$
$181$	18.3508	1.36400	0.682002	+	0.731350i	$0.261109\pi$
$182$	2.75986	0.204575
$183$	0	0
$184$	−13.7427	−1.01312
$185$	4.15664	0.305602
$186$	0	0
$187$	0	0
$188$	11.7143	0.854356
$189$	0	0
$190$	2.92057	0.211880
$191$	15.3985	1.11420	0.557099	−	0.830446i	$-0.311914\pi$
$191$	15.3985	1.11420	0.557099	+	0.830446i	$0.311914\pi$
$192$	0	0
$193$	−7.37942	−0.531182	−0.265591	−	0.964086i	$-0.585567\pi$
$193$	−7.37942	−0.531182	−0.265591	+	0.964086i	$0.585567\pi$
$194$	−1.80914	−0.129888
$195$	0	0
$196$	3.19822	0.228444
$197$	−3.14676	−0.224197	−0.112099	−	0.993697i	$-0.535757\pi$
$197$	−3.14676	−0.224197	−0.112099	+	0.993697i	$0.535757\pi$
$198$	0	0
$199$	−3.25757	−0.230923	−0.115462	−	0.993312i	$-0.536835\pi$
$199$	−3.25757	−0.230923	−0.115462	+	0.993312i	$0.536835\pi$
$200$	−2.92057	−0.206516
$201$	0	0
$202$	4.90268	0.344951
$203$	4.16043	0.292005
$204$	0	0
$205$	0.805012	0.0562245
$206$	−12.6966	−0.884617
$207$	0	0
$208$	0.679903	0.0471428
$209$	0	0
$210$	0	0
$211$	19.9242	1.37164	0.685818	−	0.727773i	$-0.259445\pi$
$211$	19.9242	1.37164	0.685818	+	0.727773i	$0.259445\pi$
$212$	8.14335	0.559288
$213$	0	0
$214$	9.32657	0.637551
$215$	−2.34089	−0.159647
$216$	0	0
$217$	−21.6632	−1.47060
$218$	−6.30305	−0.426896
$219$	0	0
$220$	0	0
$221$	1.25588	0.0844799
$222$	0	0
$223$	10.4801	0.701802	0.350901	−	0.936412i	$-0.385875\pi$
$223$	10.4801	0.701802	0.350901	+	0.936412i	$0.385875\pi$
$224$	−11.0278	−0.736828
$225$	0	0
$226$	4.89601	0.325678
$227$	−17.6731	−1.17301	−0.586503	−	0.809947i	$-0.699496\pi$
$227$	−17.6731	−1.17301	−0.586503	+	0.809947i	$0.699496\pi$
$228$	0	0
$229$	4.12967	0.272897	0.136448	−	0.990647i	$-0.456431\pi$
$229$	4.12967	0.272897	0.136448	+	0.990647i	$0.456431\pi$
$230$	4.39271	0.289646
$231$	0	0
$232$	5.95314	0.390843
$233$	−13.5990	−0.890898	−0.445449	−	0.895307i	$-0.646956\pi$
$233$	−13.5990	−0.890898	−0.445449	+	0.895307i	$0.646956\pi$
$234$	0	0
$235$	−10.3803	−0.677135
$236$	−9.39324	−0.611448
$237$	0	0
$238$	1.65211	0.107090
$239$	−8.00632	−0.517886	−0.258943	−	0.965893i	$-0.583374\pi$
$239$	−8.00632	−0.517886	−0.258943	+	0.965893i	$0.583374\pi$
$240$	0	0
$241$	−0.501943	−0.0323330	−0.0161665	−	0.999869i	$-0.505146\pi$
$241$	−0.501943	−0.0323330	−0.0161665	+	0.999869i	$0.505146\pi$
$242$	0	0
$243$	0	0
$244$	9.89432	0.633419
$245$	−2.83399	−0.181057
$246$	0	0
$247$	−4.53146	−0.288330
$248$	−30.9978	−1.96836
$249$	0	0
$250$	0.933531	0.0590417
$251$	18.5405	1.17027	0.585133	−	0.810937i	$-0.301042\pi$
$251$	18.5405	1.17027	0.585133	+	0.810937i	$0.301042\pi$
$252$	0	0
$253$	0	0
$254$	−19.3009	−1.21105
$255$	0	0
$256$	−16.8391	−1.05245
$257$	−7.38010	−0.460358	−0.230179	−	0.973148i	$-0.573931\pi$
$257$	−7.38010	−0.460358	−0.230179	+	0.973148i	$0.573931\pi$
$258$	0	0
$259$	−8.48403	−0.527172
$260$	1.63459	0.101373
$261$	0	0
$262$	−18.9590	−1.17129
$263$	3.05950	0.188657	0.0943285	−	0.995541i	$-0.469930\pi$
$263$	3.05950	0.188657	0.0943285	+	0.995541i	$0.469930\pi$
$264$	0	0
$265$	−7.21596	−0.443273
$266$	−5.96112	−0.365500
$267$	0	0
$268$	3.56233	0.217604
$269$	17.6801	1.07797	0.538987	−	0.842314i	$-0.318807\pi$
$269$	17.6801	1.07797	0.538987	+	0.842314i	$0.318807\pi$
$270$	0	0
$271$	−15.2992	−0.929358	−0.464679	−	0.885479i	$-0.653830\pi$
$271$	−15.2992	−0.929358	−0.464679	+	0.885479i	$0.653830\pi$
$272$	0.407004	0.0246782
$273$	0	0
$274$	3.24852	0.196250
$275$	0	0
$276$	0	0
$277$	−17.8238	−1.07093	−0.535463	−	0.844559i	$-0.679863\pi$
$277$	−17.8238	−1.07093	−0.535463	+	0.844559i	$0.679863\pi$
$278$	2.54935	0.152900
$279$	0	0
$280$	5.96112	0.356245
$281$	17.2990	1.03197	0.515985	−	0.856597i	$-0.327426\pi$
$281$	17.2990	1.03197	0.515985	+	0.856597i	$0.327426\pi$
$282$	0	0
$283$	−25.7764	−1.53225	−0.766124	−	0.642693i	$-0.777817\pi$
$283$	−25.7764	−1.53225	−0.766124	+	0.642693i	$0.777817\pi$
$284$	14.5248	0.861887
$285$	0	0
$286$	0	0
$287$	−1.64309	−0.0969888
$288$	0	0
$289$	−16.2482	−0.955777
$290$	−1.90286	−0.111740
$291$	0	0
$292$	16.6100	0.972029
$293$	−13.5306	−0.790468	−0.395234	−	0.918581i	$-0.629336\pi$
$293$	−13.5306	−0.790468	−0.395234	+	0.918581i	$0.629336\pi$
$294$	0	0
$295$	8.32351	0.484613
$296$	−12.1398	−0.705609
$297$	0	0
$298$	8.50299	0.492565
$299$	−6.81557	−0.394155
$300$	0	0
$301$	4.77794	0.275396
$302$	20.1404	1.15895
$303$	0	0
$304$	−1.46854	−0.0842268
$305$	−8.76752	−0.502027
$306$	0	0
$307$	−13.0268	−0.743478	−0.371739	−	0.928337i	$-0.621238\pi$
$307$	−13.0268	−0.743478	−0.371739	+	0.928337i	$0.621238\pi$
$308$	0	0
$309$	0	0
$310$	9.90814	0.562745
$311$	−1.90319	−0.107920	−0.0539601	−	0.998543i	$-0.517184\pi$
$311$	−1.90319	−0.107920	−0.0539601	+	0.998543i	$0.517184\pi$
$312$	0	0
$313$	10.9324	0.617934	0.308967	−	0.951073i	$-0.400017\pi$
$313$	10.9324	0.617934	0.308967	+	0.951073i	$0.400017\pi$
$314$	−22.2124	−1.25352
$315$	0	0
$316$	19.1839	1.07918
$317$	16.0723	0.902708	0.451354	−	0.892345i	$-0.350941\pi$
$317$	16.0723	0.902708	0.451354	+	0.892345i	$0.350941\pi$
$318$	0	0
$319$	0	0
$320$	5.98262	0.334439
$321$	0	0
$322$	−8.96587	−0.499648
$323$	−2.71262	−0.150934
$324$	0	0
$325$	−1.44843	−0.0803447
$326$	−1.68346	−0.0932382
$327$	0	0
$328$	−2.35109	−0.129817
$329$	21.1870	1.16808
$330$	0	0
$331$	1.39579	0.0767195	0.0383597	−	0.999264i	$-0.487787\pi$
$331$	1.39579	0.0767195	0.0383597	+	0.999264i	$0.487787\pi$
$332$	6.93078	0.380376
$333$	0	0
$334$	22.9531	1.25594
$335$	−3.15664	−0.172466
$336$	0	0
$337$	−2.76340	−0.150532	−0.0752660	−	0.997163i	$-0.523981\pi$
$337$	−2.76340	−0.150532	−0.0752660	+	0.997163i	$0.523981\pi$
$338$	−10.1774	−0.553577
$339$	0	0
$340$	0.978497	0.0530664
$341$	0	0
$342$	0	0
$343$	20.0720	1.08378
$344$	6.83672	0.368611
$345$	0	0
$346$	15.1340	0.813608
$347$	3.12683	0.167857	0.0839286	−	0.996472i	$-0.473253\pi$
$347$	3.12683	0.167857	0.0839286	+	0.996472i	$0.473253\pi$
$348$	0	0
$349$	13.1277	0.702709	0.351355	−	0.936242i	$-0.385721\pi$
$349$	13.1277	0.702709	0.351355	+	0.936242i	$0.385721\pi$
$350$	−1.90541	−0.101849
$351$	0	0
$352$	0	0
$353$	−10.7984	−0.574739	−0.287370	−	0.957820i	$-0.592781\pi$
$353$	−10.7984	−0.574739	−0.287370	+	0.957820i	$0.592781\pi$
$354$	0	0
$355$	−12.8707	−0.683103
$356$	4.25671	0.225605
$357$	0	0
$358$	−15.8417	−0.837260
$359$	−23.9716	−1.26517	−0.632585	−	0.774491i	$-0.718006\pi$
$359$	−23.9716	−1.26517	−0.632585	+	0.774491i	$0.718006\pi$
$360$	0	0
$361$	−9.21237	−0.484861
$362$	17.1310	0.900388
$363$	0	0
$364$	−3.33632	−0.174871
$365$	−14.7184	−0.770398
$366$	0	0
$367$	8.87287	0.463160	0.231580	−	0.972816i	$-0.425610\pi$
$367$	8.87287	0.463160	0.231580	+	0.972816i	$0.425610\pi$
$368$	−2.20877	−0.115140
$369$	0	0
$370$	3.88035	0.201730
$371$	14.7283	0.764658
$372$	0	0
$373$	−26.3389	−1.36378	−0.681888	−	0.731456i	$-0.738841\pi$
$373$	−26.3389	−1.36378	−0.681888	+	0.731456i	$0.738841\pi$
$374$	0	0
$375$	0	0
$376$	30.3163	1.56345
$377$	2.95242	0.152057
$378$	0	0
$379$	4.69610	0.241223	0.120611	−	0.992700i	$-0.461515\pi$
$379$	4.69610	0.241223	0.120611	+	0.992700i	$0.461515\pi$
$380$	−3.53059	−0.181116
$381$	0	0
$382$	14.3750	0.735489
$383$	6.89931	0.352538	0.176269	−	0.984342i	$-0.443597\pi$
$383$	6.89931	0.352538	0.176269	+	0.984342i	$0.443597\pi$
$384$	0	0
$385$	0	0
$386$	−6.88892	−0.350637
$387$	0	0
$388$	2.18701	0.111029
$389$	−2.40462	−0.121919	−0.0609596	−	0.998140i	$-0.519416\pi$
$389$	−2.40462	−0.121919	−0.0609596	+	0.998140i	$0.519416\pi$
$390$	0	0
$391$	−4.07994	−0.206331
$392$	8.27688	0.418045
$393$	0	0
$394$	−2.93760	−0.147994
$395$	−16.9992	−0.855321
$396$	0	0
$397$	37.6715	1.89068	0.945338	−	0.326091i	$-0.105732\pi$
$397$	37.6715	1.89068	0.945338	+	0.326091i	$0.105732\pi$
$398$	−3.04104	−0.152434
$399$	0	0
$400$	−0.469405	−0.0234703
$401$	12.5168	0.625060	0.312530	−	0.949908i	$-0.398824\pi$
$401$	12.5168	0.625060	0.312530	+	0.949908i	$0.398824\pi$
$402$	0	0
$403$	−15.3731	−0.765790
$404$	−5.92671	−0.294865
$405$	0	0
$406$	3.88390	0.192754
$407$	0	0
$408$	0	0
$409$	−26.4622	−1.30847	−0.654237	−	0.756290i	$-0.727010\pi$
$409$	−26.4622	−1.30847	−0.654237	+	0.756290i	$0.727010\pi$
$410$	0.751504	0.0371141
$411$	0	0
$412$	15.3486	0.756171
$413$	−16.9889	−0.835971
$414$	0	0
$415$	−6.14148	−0.301473
$416$	−7.82580	−0.383691
$417$	0	0
$418$	0	0
$419$	−30.9711	−1.51304	−0.756518	−	0.653973i	$-0.773101\pi$
$419$	−30.9711	−1.51304	−0.756518	+	0.653973i	$0.773101\pi$
$420$	0	0
$421$	18.2435	0.889132	0.444566	−	0.895746i	$-0.353358\pi$
$421$	18.2435	0.889132	0.444566	+	0.895746i	$0.353358\pi$
$422$	18.5998	0.905426
$423$	0	0
$424$	21.0747	1.02348
$425$	−0.867063	−0.0420587
$426$	0	0
$427$	17.8952	0.866010
$428$	−11.2746	−0.544979
$429$	0	0
$430$	−2.18529	−0.105384
$431$	−30.3488	−1.46185	−0.730925	−	0.682458i	$-0.760911\pi$
$431$	−30.3488	−1.46185	−0.730925	+	0.682458i	$0.760911\pi$
$432$	0	0
$433$	12.7012	0.610382	0.305191	−	0.952291i	$-0.401280\pi$
$433$	12.7012	0.610382	0.305191	+	0.952291i	$0.401280\pi$
$434$	−20.2233	−0.970750
$435$	0	0
$436$	7.61957	0.364911
$437$	14.7212	0.704209
$438$	0	0
$439$	−25.6564	−1.22451	−0.612257	−	0.790659i	$-0.709738\pi$
$439$	−25.6564	−1.22451	−0.612257	+	0.790659i	$0.709738\pi$
$440$	0	0
$441$	0	0
$442$	1.17241	0.0557657
$443$	26.2148	1.24550	0.622751	−	0.782420i	$-0.286015\pi$
$443$	26.2148	1.24550	0.622751	+	0.782420i	$0.286015\pi$
$444$	0	0
$445$	−3.77194	−0.178807
$446$	9.78354	0.463264
$447$	0	0
$448$	−12.2110	−0.576916
$449$	12.0061	0.566605	0.283302	−	0.959031i	$-0.408570\pi$
$449$	12.0061	0.566605	0.283302	+	0.959031i	$0.408570\pi$
$450$	0	0
$451$	0	0
$452$	−5.91864	−0.278390
$453$	0	0
$454$	−16.4984	−0.774309
$455$	2.95637	0.138597
$456$	0	0
$457$	13.7890	0.645022	0.322511	−	0.946566i	$-0.395473\pi$
$457$	13.7890	0.645022	0.322511	+	0.946566i	$0.395473\pi$
$458$	3.85518	0.180141
$459$	0	0
$460$	−5.31022	−0.247590
$461$	−11.3262	−0.527515	−0.263758	−	0.964589i	$-0.584962\pi$
$461$	−11.3262	−0.527515	−0.263758	+	0.964589i	$0.584962\pi$
$462$	0	0
$463$	6.18784	0.287573	0.143787	−	0.989609i	$-0.454072\pi$
$463$	6.18784	0.287573	0.143787	+	0.989609i	$0.454072\pi$
$464$	0.956812	0.0444189
$465$	0	0
$466$	−12.6951	−0.588087
$467$	−8.48288	−0.392541	−0.196270	−	0.980550i	$-0.562883\pi$
$467$	−8.48288	−0.392541	−0.196270	+	0.980550i	$0.562883\pi$
$468$	0	0
$469$	6.44295	0.297508
$470$	−9.69031	−0.446981
$471$	0	0
$472$	−24.3094	−1.11893
$473$	0	0
$474$	0	0
$475$	3.12852	0.143546
$476$	−1.99719	−0.0915411
$477$	0	0
$478$	−7.47415	−0.341860
$479$	−9.23554	−0.421982	−0.210991	−	0.977488i	$-0.567669\pi$
$479$	−9.23554	−0.421982	−0.210991	+	0.977488i	$0.567669\pi$
$480$	0	0
$481$	−6.02062	−0.274517
$482$	−0.468579	−0.0213432
$483$	0	0
$484$	0	0
$485$	−1.93795	−0.0879977
$486$	0	0
$487$	20.1708	0.914024	0.457012	−	0.889460i	$-0.348920\pi$
$487$	20.1708	0.914024	0.457012	+	0.889460i	$0.348920\pi$
$488$	25.6062	1.15914
$489$	0	0
$490$	−2.64562	−0.119517
$491$	5.33594	0.240807	0.120404	−	0.992725i	$-0.461581\pi$
$491$	5.33594	0.240807	0.120404	+	0.992725i	$0.461581\pi$
$492$	0	0
$493$	1.76738	0.0795986
$494$	−4.23026	−0.190328
$495$	0	0
$496$	−4.98209	−0.223702
$497$	26.2700	1.17837
$498$	0	0
$499$	−39.2816	−1.75849	−0.879243	−	0.476374i	$-0.841951\pi$
$499$	−39.2816	−1.75849	−0.879243	+	0.476374i	$0.841951\pi$
$500$	−1.12852	−0.0504689
$501$	0	0
$502$	17.3081	0.772500
$503$	−32.4181	−1.44545	−0.722727	−	0.691134i	$-0.757111\pi$
$503$	−32.4181	−1.44545	−0.722727	+	0.691134i	$0.757111\pi$
$504$	0	0
$505$	5.25176	0.233700
$506$	0	0
$507$	0	0
$508$	23.3323	1.03520
$509$	17.0656	0.756421	0.378211	−	0.925720i	$-0.376539\pi$
$509$	17.0656	0.756421	0.378211	+	0.925720i	$0.376539\pi$
$510$	0	0
$511$	30.0415	1.32896
$512$	−5.27803	−0.233258
$513$	0	0
$514$	−6.88955	−0.303885
$515$	−13.6007	−0.599316
$516$	0	0
$517$	0	0
$518$	−7.92011	−0.347990
$519$	0	0
$520$	4.23026	0.185509
$521$	15.0471	0.659224	0.329612	−	0.944116i	$-0.393082\pi$
$521$	15.0471	0.659224	0.329612	+	0.944116i	$0.393082\pi$
$522$	0	0
$523$	−37.0416	−1.61972	−0.809859	−	0.586624i	$-0.800456\pi$
$523$	−37.0416	−1.61972	−0.809859	+	0.586624i	$0.800456\pi$
$524$	22.9189	1.00122
$525$	0	0
$526$	2.85614	0.124534
$527$	−9.20267	−0.400875
$528$	0	0
$529$	−0.858520	−0.0373270
$530$	−6.73632	−0.292607
$531$	0	0
$532$	7.20623	0.312430
$533$	−1.16601	−0.0505054
$534$	0	0
$535$	9.99063	0.431933
$536$	9.21919	0.398208
$537$	0	0
$538$	16.5049	0.711577
$539$	0	0
$540$	0	0
$541$	0.177099	0.00761407	0.00380704	−	0.999993i	$-0.498788\pi$
$541$	0.177099	0.00761407	0.00380704	+	0.999993i	$0.498788\pi$
$542$	−14.2822	−0.613475
$543$	0	0
$544$	−4.68468	−0.200854
$545$	−6.75183	−0.289217
$546$	0	0
$547$	1.21270	0.0518511	0.0259256	−	0.999664i	$-0.491747\pi$
$547$	1.21270	0.0518511	0.0259256	+	0.999664i	$0.491747\pi$
$548$	−3.92704	−0.167755
$549$	0	0
$550$	0	0
$551$	−6.37702	−0.271670
$552$	0	0
$553$	34.6967	1.47545
$554$	−16.6390	−0.706925
$555$	0	0
$556$	−3.08183	−0.130699
$557$	−1.93846	−0.0821352	−0.0410676	−	0.999156i	$-0.513076\pi$
$557$	−1.93846	−0.0821352	−0.0410676	+	0.999156i	$0.513076\pi$
$558$	0	0
$559$	3.39062	0.143408
$560$	0.958094	0.0404869
$561$	0	0
$562$	16.1491	0.681210
$563$	20.9958	0.884866	0.442433	−	0.896802i	$-0.354115\pi$
$563$	20.9958	0.884866	0.442433	+	0.896802i	$0.354115\pi$
$564$	0	0
$565$	5.24461	0.220642
$566$	−24.0631	−1.01145
$567$	0	0
$568$	37.5897	1.57723
$569$	39.3757	1.65071	0.825357	−	0.564612i	$-0.190974\pi$
$569$	39.3757	1.65071	0.825357	+	0.564612i	$0.190974\pi$
$570$	0	0
$571$	30.6796	1.28390	0.641950	−	0.766746i	$-0.278126\pi$
$571$	30.6796	1.28390	0.641950	+	0.766746i	$0.278126\pi$
$572$	0	0
$573$	0	0
$574$	−1.53388	−0.0640229
$575$	4.70547	0.196232
$576$	0	0
$577$	−34.7038	−1.44474	−0.722369	−	0.691508i	$-0.756947\pi$
$577$	−34.7038	−1.44474	−0.722369	+	0.691508i	$0.756947\pi$
$578$	−15.1682	−0.630914
$579$	0	0
$580$	2.30032	0.0955155
$581$	12.5353	0.520050
$582$	0	0
$583$	0	0
$584$	42.9862	1.77878
$585$	0	0
$586$	−12.6313	−0.521793
$587$	−8.05710	−0.332552	−0.166276	−	0.986079i	$-0.553174\pi$
$587$	−8.05710	−0.332552	−0.166276	+	0.986079i	$0.553174\pi$
$588$	0	0
$589$	33.2049	1.36818
$590$	7.77025	0.319896
$591$	0	0
$592$	−1.95115	−0.0801917
$593$	28.1409	1.15561	0.577805	−	0.816175i	$-0.303909\pi$
$593$	28.1409	1.15561	0.577805	+	0.816175i	$0.303909\pi$
$594$	0	0
$595$	1.76974	0.0725524
$596$	−10.2790	−0.421045
$597$	0	0
$598$	−6.36255	−0.260184
$599$	28.5988	1.16851	0.584257	−	0.811568i	$-0.301386\pi$
$599$	28.5988	1.16851	0.584257	+	0.811568i	$0.301386\pi$
$600$	0	0
$601$	−7.27610	−0.296799	−0.148399	−	0.988928i	$-0.547412\pi$
$601$	−7.27610	−0.296799	−0.148399	+	0.988928i	$0.547412\pi$
$602$	4.46035	0.181790
$603$	0	0
$604$	−24.3472	−0.990672
$605$	0	0
$606$	0	0
$607$	−5.34785	−0.217063	−0.108531	−	0.994093i	$-0.534615\pi$
$607$	−5.34785	−0.217063	−0.108531	+	0.994093i	$0.534615\pi$
$608$	16.9032	0.685515
$609$	0	0
$610$	−8.18476	−0.331391
$611$	15.0352	0.608257
$612$	0	0
$613$	−4.64099	−0.187448	−0.0937238	−	0.995598i	$-0.529877\pi$
$613$	−4.64099	−0.187448	−0.0937238	+	0.995598i	$0.529877\pi$
$614$	−12.1609	−0.490774
$615$	0	0
$616$	0	0
$617$	−24.5843	−0.989727	−0.494864	−	0.868971i	$-0.664782\pi$
$617$	−24.5843	−0.989727	−0.494864	+	0.868971i	$0.664782\pi$
$618$	0	0
$619$	−6.13789	−0.246703	−0.123351	−	0.992363i	$-0.539364\pi$
$619$	−6.13789	−0.246703	−0.123351	+	0.992363i	$0.539364\pi$
$620$	−11.9777	−0.481035
$621$	0	0
$622$	−1.77669	−0.0712387
$623$	7.69884	0.308447
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.2057	0.407902
$627$	0	0
$628$	26.8519	1.07151
$629$	−3.60407	−0.143704
$630$	0	0
$631$	−17.7085	−0.704966	−0.352483	−	0.935818i	$-0.614662\pi$
$631$	−17.7085	−0.704966	−0.352483	+	0.935818i	$0.614662\pi$
$632$	49.6473	1.97486
$633$	0	0
$634$	15.0040	0.595883
$635$	−20.6751	−0.820468
$636$	0	0
$637$	4.10485	0.162640
$638$	0	0
$639$	0	0
$640$	−5.22091	−0.206375
$641$	−39.8621	−1.57446	−0.787230	−	0.616659i	$-0.788486\pi$
$641$	−39.8621	−1.57446	−0.787230	+	0.616659i	$0.788486\pi$
$642$	0	0
$643$	−17.8662	−0.704576	−0.352288	−	0.935892i	$-0.614596\pi$
$643$	−17.8662	−0.704576	−0.352288	+	0.935892i	$0.614596\pi$
$644$	10.8386	0.427100
$645$	0	0
$646$	−2.53232	−0.0996327
$647$	−5.04075	−0.198172	−0.0990862	−	0.995079i	$-0.531592\pi$
$647$	−5.04075	−0.198172	−0.0990862	+	0.995079i	$0.531592\pi$
$648$	0	0
$649$	0	0
$650$	−1.35216	−0.0530360
$651$	0	0
$652$	2.03509	0.0797001
$653$	−40.7212	−1.59354	−0.796771	−	0.604281i	$-0.793460\pi$
$653$	−40.7212	−1.59354	−0.796771	+	0.604281i	$0.793460\pi$
$654$	0	0
$655$	−20.3089	−0.793533
$656$	−0.377877	−0.0147536
$657$	0	0
$658$	19.7787	0.771054
$659$	−48.7556	−1.89925	−0.949624	−	0.313390i	$-0.898535\pi$
$659$	−48.7556	−1.89925	−0.949624	+	0.313390i	$0.898535\pi$
$660$	0	0
$661$	−41.0061	−1.59495	−0.797477	−	0.603350i	$-0.793832\pi$
$661$	−41.0061	−1.59495	−0.797477	+	0.603350i	$0.793832\pi$
$662$	1.30301	0.0506430
$663$	0	0
$664$	17.9366	0.696076
$665$	−6.38556	−0.247621
$666$	0	0
$667$	−9.59140	−0.371380
$668$	−27.7474	−1.07358
$669$	0	0
$670$	−2.94682	−0.113846
$671$	0	0
$672$	0	0
$673$	−30.4936	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$673$	−30.4936	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$674$	−2.57972	−0.0993671
$675$	0	0
$676$	12.3032	0.473198
$677$	43.4424	1.66963	0.834813	−	0.550534i	$-0.185576\pi$
$677$	43.4424	1.66963	0.834813	+	0.550534i	$0.185576\pi$
$678$	0	0
$679$	3.95551	0.151799
$680$	2.53232	0.0971099
$681$	0	0
$682$	0	0
$683$	43.5970	1.66819	0.834096	−	0.551619i	$-0.185990\pi$
$683$	43.5970	1.66819	0.834096	+	0.551619i	$0.185990\pi$
$684$	0	0
$685$	3.47982	0.132957
$686$	18.7378	0.715413
$687$	0	0
$688$	1.09882	0.0418923
$689$	10.4518	0.398184
$690$	0	0
$691$	−8.30897	−0.316088	−0.158044	−	0.987432i	$-0.550519\pi$
$691$	−8.30897	−0.316088	−0.158044	+	0.987432i	$0.550519\pi$
$692$	−18.2950	−0.695473
$693$	0	0
$694$	2.91900	0.110804
$695$	2.73086	0.103588
$696$	0	0
$697$	−0.697996	−0.0264385
$698$	12.2551	0.463863
$699$	0	0
$700$	2.30340	0.0870603
$701$	−9.08744	−0.343228	−0.171614	−	0.985164i	$-0.554898\pi$
$701$	−9.08744	−0.343228	−0.171614	+	0.985164i	$0.554898\pi$
$702$	0	0
$703$	13.0041	0.490460
$704$	0	0
$705$	0	0
$706$	−10.0806	−0.379389
$707$	−10.7193	−0.403139
$708$	0	0
$709$	−4.85568	−0.182359	−0.0911794	−	0.995834i	$-0.529064\pi$
$709$	−4.85568	−0.182359	−0.0911794	+	0.995834i	$0.529064\pi$
$710$	−12.0152	−0.450921
$711$	0	0
$712$	11.0162	0.412850
$713$	49.9421	1.87035
$714$	0	0
$715$	0	0
$716$	19.1506	0.715691
$717$	0	0
$718$	−22.3782	−0.835147
$719$	−23.9034	−0.891448	−0.445724	−	0.895171i	$-0.647054\pi$
$719$	−23.9034	−0.891448	−0.445724	+	0.895171i	$0.647054\pi$
$720$	0	0
$721$	27.7600	1.03384
$722$	−8.60003	−0.320060
$723$	0	0
$724$	−20.7092	−0.769653
$725$	−2.03835	−0.0757024
$726$	0	0
$727$	−11.3674	−0.421592	−0.210796	−	0.977530i	$-0.567606\pi$
$727$	−11.3674	−0.421592	−0.210796	+	0.977530i	$0.567606\pi$
$728$	−8.63429	−0.320008
$729$	0	0
$730$	−13.7401	−0.508545
$731$	2.02969	0.0750710
$732$	0	0
$733$	−34.1178	−1.26017	−0.630085	−	0.776526i	$-0.716980\pi$
$733$	−34.1178	−1.26017	−0.630085	+	0.776526i	$0.716980\pi$
$734$	8.28311	0.305735
$735$	0	0
$736$	25.4234	0.937118
$737$	0	0
$738$	0	0
$739$	2.33848	0.0860225	0.0430113	−	0.999075i	$-0.486305\pi$
$739$	2.33848	0.0860225	0.0430113	+	0.999075i	$0.486305\pi$
$740$	−4.69085	−0.172439
$741$	0	0
$742$	13.7494	0.504755
$743$	−14.4885	−0.531533	−0.265767	−	0.964037i	$-0.585625\pi$
$743$	−14.4885	−0.531533	−0.265767	+	0.964037i	$0.585625\pi$
$744$	0	0
$745$	9.10841	0.333706
$746$	−24.5882	−0.900238
$747$	0	0
$748$	0	0
$749$	−20.3917	−0.745096
$750$	0	0
$751$	33.3203	1.21587	0.607937	−	0.793985i	$-0.291997\pi$
$751$	33.3203	1.21587	0.607937	+	0.793985i	$0.291997\pi$
$752$	4.87256	0.177684
$753$	0	0
$754$	2.75617	0.100374
$755$	21.5744	0.785174
$756$	0	0
$757$	40.3778	1.46755	0.733777	−	0.679390i	$-0.237756\pi$
$757$	40.3778	1.46755	0.733777	+	0.679390i	$0.237756\pi$
$758$	4.38396	0.159233
$759$	0	0
$760$	−9.13706	−0.331436
$761$	49.8971	1.80877	0.904384	−	0.426720i	$-0.140331\pi$
$761$	49.8971	1.80877	0.904384	+	0.426720i	$0.140331\pi$
$762$	0	0
$763$	13.7810	0.498907
$764$	−17.3775	−0.628697
$765$	0	0
$766$	6.44072	0.232713
$767$	−12.0561	−0.435319
$768$	0	0
$769$	4.93932	0.178116	0.0890582	−	0.996026i	$-0.471614\pi$
$769$	4.93932	0.178116	0.0890582	+	0.996026i	$0.471614\pi$
$770$	0	0
$771$	0	0
$772$	8.32781	0.299725
$773$	33.4710	1.20387	0.601934	−	0.798546i	$-0.294397\pi$
$773$	33.4710	1.20387	0.601934	+	0.798546i	$0.294397\pi$
$774$	0	0
$775$	10.6136	0.381252
$776$	5.65992	0.203179
$777$	0	0
$778$	−2.24479	−0.0804797
$779$	2.51850	0.0902345
$780$	0	0
$781$	0	0
$782$	−3.80875	−0.136201
$783$	0	0
$784$	1.33029	0.0475104
$785$	−23.7940	−0.849243
$786$	0	0
$787$	25.9400	0.924661	0.462331	−	0.886708i	$-0.347013\pi$
$787$	25.9400	0.924661	0.462331	+	0.886708i	$0.347013\pi$
$788$	3.55118	0.126506
$789$	0	0
$790$	−15.8693	−0.564603
$791$	−10.7047	−0.380614
$792$	0	0
$793$	12.6992	0.450961
$794$	35.1675	1.24805
$795$	0	0
$796$	3.67623	0.130301
$797$	−9.86264	−0.349353	−0.174676	−	0.984626i	$-0.555888\pi$
$797$	−9.86264	−0.349353	−0.174676	+	0.984626i	$0.555888\pi$
$798$	0	0
$799$	9.00035	0.318410
$800$	5.40294	0.191023
$801$	0	0
$802$	11.6848	0.412606
$803$	0	0
$804$	0	0
$805$	−9.60425	−0.338505
$806$	−14.3513	−0.505503
$807$	0	0
$808$	−15.3381	−0.539594
$809$	33.0590	1.16229	0.581146	−	0.813799i	$-0.302604\pi$
$809$	33.0590	1.16229	0.581146	+	0.813799i	$0.302604\pi$
$810$	0	0
$811$	5.21312	0.183057	0.0915286	−	0.995802i	$-0.470825\pi$
$811$	5.21312	0.183057	0.0915286	+	0.995802i	$0.470825\pi$
$812$	−4.69513	−0.164767
$813$	0	0
$814$	0	0
$815$	−1.80332	−0.0631677
$816$	0	0
$817$	−7.32351	−0.256217
$818$	−24.7033	−0.863731
$819$	0	0
$820$	−0.908472	−0.0317252
$821$	1.63933	0.0572131	0.0286066	−	0.999591i	$-0.490893\pi$
$821$	1.63933	0.0572131	0.0286066	+	0.999591i	$0.490893\pi$
$822$	0	0
$823$	−17.8894	−0.623586	−0.311793	−	0.950150i	$-0.600930\pi$
$823$	−17.8894	−0.623586	−0.311793	+	0.950150i	$0.600930\pi$
$824$	39.7217	1.38377
$825$	0	0
$826$	−15.8597	−0.551830
$827$	−35.5449	−1.23602	−0.618009	−	0.786171i	$-0.712061\pi$
$827$	−35.5449	−1.23602	−0.618009	+	0.786171i	$0.712061\pi$
$828$	0	0
$829$	−26.9512	−0.936053	−0.468026	−	0.883715i	$-0.655035\pi$
$829$	−26.9512	−0.936053	−0.468026	+	0.883715i	$0.655035\pi$
$830$	−5.73326	−0.199004
$831$	0	0
$832$	−8.66544	−0.300420
$833$	2.45725	0.0851386
$834$	0	0
$835$	24.5874	0.850882
$836$	0	0
$837$	0	0
$838$	−28.9124	−0.998764
$839$	11.8373	0.408667	0.204334	−	0.978901i	$-0.434497\pi$
$839$	11.8373	0.408667	0.204334	+	0.978901i	$0.434497\pi$
$840$	0	0
$841$	−24.8451	−0.856729
$842$	17.0308	0.586921
$843$	0	0
$844$	−22.4848	−0.773959
$845$	−10.9020	−0.375041
$846$	0	0
$847$	0	0
$848$	3.38721	0.116317
$849$	0	0
$850$	−0.809430	−0.0277632
$851$	19.5590	0.670472
$852$	0	0
$853$	−30.3043	−1.03760	−0.518799	−	0.854896i	$-0.673621\pi$
$853$	−30.3043	−1.03760	−0.518799	+	0.854896i	$0.673621\pi$
$854$	16.7057	0.571659
$855$	0	0
$856$	−29.1783	−0.997295
$857$	29.2318	0.998540	0.499270	−	0.866446i	$-0.333602\pi$
$857$	29.2318	0.998540	0.499270	+	0.866446i	$0.333602\pi$
$858$	0	0
$859$	−36.7151	−1.25270	−0.626351	−	0.779541i	$-0.715452\pi$
$859$	−36.7151	−1.25270	−0.626351	+	0.779541i	$0.715452\pi$
$860$	2.64174	0.0900824
$861$	0	0
$862$	−28.3315	−0.964976
$863$	−43.5758	−1.48334	−0.741669	−	0.670766i	$-0.765966\pi$
$863$	−43.5758	−1.48334	−0.741669	+	0.670766i	$0.765966\pi$
$864$	0	0
$865$	16.2115	0.551209
$866$	11.8570	0.402917
$867$	0	0
$868$	24.4474	0.829798
$869$	0	0
$870$	0	0
$871$	4.57219	0.154923
$872$	19.7192	0.667777
$873$	0	0
$874$	13.7427	0.464853
$875$	−2.04108	−0.0690011
$876$	0	0
$877$	−11.1173	−0.375404	−0.187702	−	0.982226i	$-0.560104\pi$
$877$	−11.1173	−0.375404	−0.187702	+	0.982226i	$0.560104\pi$
$878$	−23.9511	−0.808309
$879$	0	0
$880$	0	0
$881$	39.1155	1.31783	0.658917	−	0.752216i	$-0.271015\pi$
$881$	39.1155	1.31783	0.658917	+	0.752216i	$0.271015\pi$
$882$	0	0
$883$	19.5187	0.656857	0.328428	−	0.944529i	$-0.393481\pi$
$883$	19.5187	0.656857	0.328428	+	0.944529i	$0.393481\pi$
$884$	−1.41729	−0.0476686
$885$	0	0
$886$	24.4723	0.822164
$887$	5.19108	0.174299	0.0871497	−	0.996195i	$-0.472224\pi$
$887$	5.19108	0.174299	0.0871497	+	0.996195i	$0.472224\pi$
$888$	0	0
$889$	42.1996	1.41533
$890$	−3.52123	−0.118032
$891$	0	0
$892$	−11.8270	−0.395999
$893$	−32.4749	−1.08673
$894$	0	0
$895$	−16.9697	−0.567233
$896$	10.6563	0.356002
$897$	0	0
$898$	11.2081	0.374019
$899$	−21.6343	−0.721543
$900$	0	0
$901$	6.25669	0.208440
$902$	0	0
$903$	0	0
$904$	−15.3173	−0.509444
$905$	18.3508	0.610001
$906$	0	0
$907$	−42.4379	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$907$	−42.4379	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$908$	19.9445	0.661880
$909$	0	0
$910$	2.75986	0.0914886
$911$	19.7499	0.654344	0.327172	−	0.944965i	$-0.393904\pi$
$911$	19.7499	0.654344	0.327172	+	0.944965i	$0.393904\pi$
$912$	0	0
$913$	0	0
$914$	12.8725	0.425783
$915$	0	0
$916$	−4.66042	−0.153985
$917$	41.4520	1.36887
$918$	0	0
$919$	−24.4853	−0.807696	−0.403848	−	0.914826i	$-0.632328\pi$
$919$	−24.4853	−0.807696	−0.403848	+	0.914826i	$0.632328\pi$
$920$	−13.7427	−0.453082
$921$	0	0
$922$	−10.5734	−0.348216
$923$	18.6423	0.613619
$924$	0	0
$925$	4.15664	0.136669
$926$	5.77654	0.189829
$927$	0	0
$928$	−11.0131	−0.361522
$929$	6.05289	0.198589	0.0992944	−	0.995058i	$-0.468341\pi$
$929$	6.05289	0.198589	0.0992944	+	0.995058i	$0.468341\pi$
$930$	0	0
$931$	−8.86620	−0.290578
$932$	15.3467	0.502698
$933$	0	0
$934$	−7.91903	−0.259119
$935$	0	0
$936$	0	0
$937$	6.83865	0.223409	0.111704	−	0.993741i	$-0.464369\pi$
$937$	6.83865	0.223409	0.111704	+	0.993741i	$0.464369\pi$
$938$	6.01470	0.196387
$939$	0	0
$940$	11.7143	0.382080
$941$	3.95417	0.128902	0.0644512	−	0.997921i	$-0.479470\pi$
$941$	3.95417	0.128902	0.0644512	+	0.997921i	$0.479470\pi$
$942$	0	0
$943$	3.78796	0.123353
$944$	−3.90710	−0.127165
$945$	0	0
$946$	0	0
$947$	40.1742	1.30549	0.652743	−	0.757579i	$-0.273618\pi$
$947$	40.1742	1.30549	0.652743	+	0.757579i	$0.273618\pi$
$948$	0	0
$949$	21.3187	0.692034
$950$	2.92057	0.0947558
$951$	0	0
$952$	−5.16866	−0.167517
$953$	41.2470	1.33612	0.668061	−	0.744106i	$-0.267124\pi$
$953$	41.2470	1.33612	0.668061	+	0.744106i	$0.267124\pi$
$954$	0	0
$955$	15.3985	0.498284
$956$	9.03529	0.292222
$957$	0	0
$958$	−8.62166	−0.278553
$959$	−7.10258	−0.229354
$960$	0	0
$961$	81.6488	2.63383
$962$	−5.62044	−0.181210
$963$	0	0
$964$	0.566452	0.0182442
$965$	−7.37942	−0.237552
$966$	0	0
$967$	9.16826	0.294831	0.147416	−	0.989075i	$-0.452904\pi$
$967$	9.16826	0.294831	0.147416	+	0.989075i	$0.452904\pi$
$968$	0	0
$969$	0	0
$970$	−1.80914	−0.0580879
$971$	3.71264	0.119144	0.0595722	−	0.998224i	$-0.481026\pi$
$971$	3.71264	0.119144	0.0595722	+	0.998224i	$0.481026\pi$
$972$	0	0
$973$	−5.57391	−0.178691
$974$	18.8300	0.603353
$975$	0	0
$976$	4.11552	0.131735
$977$	−12.3932	−0.396495	−0.198247	−	0.980152i	$-0.563525\pi$
$977$	−12.3932	−0.396495	−0.198247	+	0.980152i	$0.563525\pi$
$978$	0	0
$979$	0	0
$980$	3.19822	0.102163
$981$	0	0
$982$	4.98126	0.158958
$983$	−23.2949	−0.742992	−0.371496	−	0.928435i	$-0.621155\pi$
$983$	−23.2949	−0.742992	−0.371496	+	0.928435i	$0.621155\pi$
$984$	0	0
$985$	−3.14676	−0.100264
$986$	1.64990	0.0525436
$987$	0	0
$988$	5.11384	0.162693
$989$	−11.0150	−0.350256
$990$	0	0
$991$	−37.7826	−1.20020	−0.600101	−	0.799924i	$-0.704873\pi$
$991$	−37.7826	−1.20020	−0.600101	+	0.799924i	$0.704873\pi$
$992$	57.3447	1.82070
$993$	0	0
$994$	24.5239	0.777851
$995$	−3.25757	−0.103272
$996$	0	0
$997$	7.87590	0.249432	0.124716	−	0.992192i	$-0.460198\pi$
$997$	7.87590	0.249432	0.124716	+	0.992192i	$0.460198\pi$
$998$	−36.6706	−1.16079
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bk.1.3		4
3.2	odd	2		1815.2.a.v.1.2		4
11.3	even	5		495.2.n.b.361.2		8
11.4	even	5		495.2.n.b.181.2		8
11.10	odd	2		5445.2.a.br.1.2		4
15.14	odd	2		9075.2.a.cq.1.3		4
33.14	odd	10		165.2.m.b.31.1	yes	8
33.26	odd	10		165.2.m.b.16.1	✓	8
33.32	even	2		1815.2.a.r.1.3		4
165.14	odd	10		825.2.n.i.526.2		8
165.47	even	20		825.2.bx.g.724.2		16
165.59	odd	10		825.2.n.i.676.2		8
165.92	even	20		825.2.bx.g.49.3		16
165.113	even	20		825.2.bx.g.724.3		16
165.158	even	20		825.2.bx.g.49.2		16
165.164	even	2		9075.2.a.dg.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.b.16.1	✓	8	33.26	odd	10
165.2.m.b.31.1	yes	8	33.14	odd	10
495.2.n.b.181.2		8	11.4	even	5
495.2.n.b.361.2		8	11.3	even	5
825.2.n.i.526.2		8	165.14	odd	10
825.2.n.i.676.2		8	165.59	odd	10
825.2.bx.g.49.2		16	165.158	even	20
825.2.bx.g.49.3		16	165.92	even	20
825.2.bx.g.724.2		16	165.47	even	20
825.2.bx.g.724.3		16	165.113	even	20
1815.2.a.r.1.3		4	33.32	even	2
1815.2.a.v.1.2		4	3.2	odd	2
5445.2.a.bk.1.3		4	1.1	even	1	trivial
5445.2.a.br.1.2		4	11.10	odd	2
9075.2.a.cq.1.3		4	15.14	odd	2
9075.2.a.dg.1.2		4	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)

\(f(q)\)	\(=\)	\(q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +O(q^{10})\)	\(q+0.933531 q^{2} -1.12852 q^{4} +1.00000 q^{5} -2.04108 q^{7} -2.92057 q^{8} +0.933531 q^{10} -1.44843 q^{13} -1.90541 q^{14} -0.469405 q^{16} -0.867063 q^{17} +3.12852 q^{19} -1.12852 q^{20} +4.70547 q^{23} +1.00000 q^{25} -1.35216 q^{26} +2.30340 q^{28} -2.03835 q^{29} +10.6136 q^{31} +5.40294 q^{32} -0.809430 q^{34} -2.04108 q^{35} +4.15664 q^{37} +2.92057 q^{38} -2.92057 q^{40} +0.805012 q^{41} -2.34089 q^{43} +4.39271 q^{46} -10.3803 q^{47} -2.83399 q^{49} +0.933531 q^{50} +1.63459 q^{52} -7.21596 q^{53} +5.96112 q^{56} -1.90286 q^{58} +8.32351 q^{59} -8.76752 q^{61} +9.90814 q^{62} +5.98262 q^{64} -1.44843 q^{65} -3.15664 q^{67} +0.978497 q^{68} -1.90541 q^{70} -12.8707 q^{71} -14.7184 q^{73} +3.88035 q^{74} -3.53059 q^{76} -16.9992 q^{79} -0.469405 q^{80} +0.751504 q^{82} -6.14148 q^{83} -0.867063 q^{85} -2.18529 q^{86} -3.77194 q^{89} +2.95637 q^{91} -5.31022 q^{92} -9.69031 q^{94} +3.12852 q^{95} -1.93795 q^{97} -2.64562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} + 4 q^{5} - 8 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 9 * q^4 + 4 * q^5 - 8 * q^7 - 3 * q^8	\( 4 q - q^{2} + 9 q^{4} + 4 q^{5} - 8 q^{7} - 3 q^{8} - q^{10} - 15 q^{13} - 7 q^{14} + 7 q^{16} + 6 q^{17} - q^{19} + 9 q^{20} + q^{23} + 4 q^{25} + 18 q^{26} - 31 q^{28} - 17 q^{29} + 15 q^{31} + 8 q^{32} - 35 q^{34} - 8 q^{35} - q^{37} + 3 q^{38} - 3 q^{40} + 12 q^{41} - 14 q^{43} - 9 q^{46} - 14 q^{47} + 20 q^{49} - q^{50} - 39 q^{52} - 2 q^{53} + 12 q^{56} - 11 q^{58} + 11 q^{59} + q^{61} + 30 q^{62} - 3 q^{64} - 15 q^{65} + 5 q^{67} + 19 q^{68} - 7 q^{70} + 3 q^{71} - 45 q^{73} - 29 q^{74} - 23 q^{76} + 7 q^{80} + 11 q^{82} - 15 q^{83} + 6 q^{85} + 10 q^{86} - 2 q^{89} + 16 q^{91} - 34 q^{92} - 29 q^{94} - q^{95} - 26 q^{97} + q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 + 4 * q^5 - 8 * q^7 - 3 * q^8 - q^10 - 15 * q^13 - 7 * q^14 + 7 * q^16 + 6 * q^17 - q^19 + 9 * q^20 + q^23 + 4 * q^25 + 18 * q^26 - 31 * q^28 - 17 * q^29 + 15 * q^31 + 8 * q^32 - 35 * q^34 - 8 * q^35 - q^37 + 3 * q^38 - 3 * q^40 + 12 * q^41 - 14 * q^43 - 9 * q^46 - 14 * q^47 + 20 * q^49 - q^50 - 39 * q^52 - 2 * q^53 + 12 * q^56 - 11 * q^58 + 11 * q^59 + q^61 + 30 * q^62 - 3 * q^64 - 15 * q^65 + 5 * q^67 + 19 * q^68 - 7 * q^70 + 3 * q^71 - 45 * q^73 - 29 * q^74 - 23 * q^76 + 7 * q^80 + 11 * q^82 - 15 * q^83 + 6 * q^85 + 10 * q^86 - 2 * q^89 + 16 * q^91 - 34 * q^92 - 29 * q^94 - q^95 - 26 * q^97 + q^98

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