LMFDB - Embedded newform 5220.2.a.z.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 22x^{5} + 38x^{4} + 81x^{3} - 75x^{2} - 42x + 18 $ x^7 - 3x^6 - 22x^5 + 38x^4 + 81x^3 - 75x^2 - 42x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.72639$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +5.05487 q^{7} -1.88250 q^{11} +0.442075 q^{13} +2.34074 q^{17} -3.14544 q^{19} -7.67601 q^{23} +1.00000 q^{25} +1.00000 q^{29} -0.738641 q^{31} +5.05487 q^{35} +5.06321 q^{37} +3.06321 q^{41} +12.9306 q^{43} -5.53035 q^{47} +18.5518 q^{49} +8.58586 q^{53} -1.88250 q^{55} +13.9938 q^{59} +4.40680 q^{61} +0.442075 q^{65} +8.17072 q^{67} +2.40680 q^{71} -13.4784 q^{73} -9.51578 q^{77} +12.4761 q^{79} -15.5434 q^{83} +2.34074 q^{85} +14.3168 q^{89} +2.23464 q^{91} -3.14544 q^{95} -7.30663 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} + 3 q^{11} + 2 q^{13} + 8 q^{17} + 6 q^{19} + q^{23} + 7 q^{25} + 7 q^{29} - 2 q^{31} + 15 q^{37} + q^{41} + 9 q^{43} + 4 q^{47} + 29 q^{49} + 17 q^{53} + 3 q^{55} - 4 q^{59} + 6 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 + 3 * q^11 + 2 * q^13 + 8 * q^17 + 6 * q^19 + q^23 + 7 * q^25 + 7 * q^29 - 2 * q^31 + 15 * q^37 + q^41 + 9 * q^43 + 4 * q^47 + 29 * q^49 + 17 * q^53 + 3 * q^55 - 4 * q^59 + 6 * q^61 + 2 * q^65 + 24 * q^67 - 8 * q^71 + 29 * q^73 + 8 * q^79 + 7 * q^83 + 8 * q^85 - 4 * q^89 + 26 * q^91 + 6 * q^95 + 15 * q^97

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	5.05487	1.91056	0.955282	−	0.295698i	$-0.0955521\pi$
$7$	5.05487	1.91056	0.955282	+	0.295698i	$0.0955521\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.88250	−0.567594	−0.283797	−	0.958884i	$-0.591594\pi$
$11$	−1.88250	−0.567594	−0.283797	+	0.958884i	$0.591594\pi$
$12$	0	0
$13$	0.442075	0.122610	0.0613048	−	0.998119i	$-0.480474\pi$
$13$	0.442075	0.122610	0.0613048	+	0.998119i	$0.480474\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.34074	0.567712	0.283856	−	0.958867i	$-0.408386\pi$
$17$	2.34074	0.567712	0.283856	+	0.958867i	$0.408386\pi$
$18$	0	0
$19$	−3.14544	−0.721614	−0.360807	−	0.932640i	$-0.617499\pi$
$19$	−3.14544	−0.721614	−0.360807	+	0.932640i	$0.617499\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.67601	−1.60056	−0.800279	−	0.599627i	$-0.795316\pi$
$23$	−7.67601	−1.60056	−0.800279	+	0.599627i	$0.795316\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.738641	−0.132664	−0.0663319	−	0.997798i	$-0.521130\pi$
$31$	−0.738641	−0.132664	−0.0663319	+	0.997798i	$0.521130\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.05487	0.854430
$36$	0	0
$37$	5.06321	0.832387	0.416193	−	0.909276i	$-0.363364\pi$
$37$	5.06321	0.832387	0.416193	+	0.909276i	$0.363364\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.06321	0.478393	0.239197	−	0.970971i	$-0.423116\pi$
$41$	3.06321	0.478393	0.239197	+	0.970971i	$0.423116\pi$
$42$	0	0
$43$	12.9306	1.97190	0.985951	−	0.167036i	$-0.0534196\pi$
$43$	12.9306	1.97190	0.985951	+	0.167036i	$0.0534196\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.53035	−0.806685	−0.403343	−	0.915049i	$-0.632152\pi$
$47$	−5.53035	−0.806685	−0.403343	+	0.915049i	$0.632152\pi$
$48$	0	0
$49$	18.5518	2.65025
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.58586	1.17936	0.589680	−	0.807637i	$-0.299254\pi$
$53$	8.58586	1.17936	0.589680	+	0.807637i	$0.299254\pi$
$54$	0	0
$55$	−1.88250	−0.253836
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.9938	1.82184	0.910921	−	0.412581i	$-0.135373\pi$
$59$	13.9938	1.82184	0.910921	+	0.412581i	$0.135373\pi$
$60$	0	0
$61$	4.40680	0.564233	0.282117	−	0.959380i	$-0.408963\pi$
$61$	4.40680	0.564233	0.282117	+	0.959380i	$0.408963\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.442075	0.0548327
$66$	0	0
$67$	8.17072	0.998213	0.499106	−	0.866541i	$-0.333662\pi$
$67$	8.17072	0.998213	0.499106	+	0.866541i	$0.333662\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.40680	0.285635	0.142817	−	0.989749i	$-0.454384\pi$
$71$	2.40680	0.285635	0.142817	+	0.989749i	$0.454384\pi$
$72$	0	0
$73$	−13.4784	−1.57753	−0.788765	−	0.614694i	$-0.789279\pi$
$73$	−13.4784	−1.57753	−0.788765	+	0.614694i	$0.789279\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.51578	−1.08442
$78$	0	0
$79$	12.4761	1.40367	0.701833	−	0.712341i	$-0.252365\pi$
$79$	12.4761	1.40367	0.701833	+	0.712341i	$0.252365\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.5434	−1.70611	−0.853056	−	0.521819i	$-0.825254\pi$
$83$	−15.5434	−1.70611	−0.853056	+	0.521819i	$0.825254\pi$
$84$	0	0
$85$	2.34074	0.253889
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.3168	1.51758	0.758790	−	0.651336i	$-0.225791\pi$
$89$	14.3168	1.51758	0.758790	+	0.651336i	$0.225791\pi$
$90$	0	0
$91$	2.23464	0.234253
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.14544	−0.322716
$96$	0	0
$97$	−7.30663	−0.741876	−0.370938	−	0.928658i	$-0.620964\pi$
$97$	−7.30663	−0.741876	−0.370938	+	0.928658i	$0.620964\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0817	−1.00317	−0.501586	−	0.865108i	$-0.667250\pi$
$101$	−10.0817	−1.00317	−0.501586	+	0.865108i	$0.667250\pi$
$102$	0	0
$103$	−2.88228	−0.284000	−0.142000	−	0.989867i	$-0.545353\pi$
$103$	−2.88228	−0.284000	−0.142000	+	0.989867i	$0.545353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.5184	−1.69356	−0.846782	−	0.531941i	$-0.821463\pi$
$107$	−17.5184	−1.69356	−0.846782	+	0.531941i	$0.821463\pi$
$108$	0	0
$109$	−1.64261	−0.157333	−0.0786666	−	0.996901i	$-0.525066\pi$
$109$	−1.64261	−0.157333	−0.0786666	+	0.996901i	$0.525066\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.9026	1.40192	0.700960	−	0.713200i	$-0.252755\pi$
$113$	14.9026	1.40192	0.700960	+	0.713200i	$0.252755\pi$
$114$	0	0
$115$	−7.67601	−0.715792
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.8321	1.08465
$120$	0	0
$121$	−7.45621	−0.677837
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.84234	−0.163482	−0.0817408	−	0.996654i	$-0.526048\pi$
$127$	−1.84234	−0.163482	−0.0817408	+	0.996654i	$0.526048\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.81928	−0.421063	−0.210531	−	0.977587i	$-0.567519\pi$
$131$	−4.81928	−0.421063	−0.210531	+	0.977587i	$0.567519\pi$
$132$	0	0
$133$	−15.8998	−1.37869
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.7061	1.93991	0.969957	−	0.243278i	$-0.0782226\pi$
$137$	22.7061	1.93991	0.969957	+	0.243278i	$0.0782226\pi$
$138$	0	0
$139$	−1.91209	−0.162181	−0.0810907	−	0.996707i	$-0.525840\pi$
$139$	−1.91209	−0.162181	−0.0810907	+	0.996707i	$0.525840\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.832205	−0.0695924
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.88409	−0.645889	−0.322945	−	0.946418i	$-0.604673\pi$
$149$	−7.88409	−0.645889	−0.322945	+	0.946418i	$0.604673\pi$
$150$	0	0
$151$	11.9688	0.974004	0.487002	−	0.873401i	$-0.338090\pi$
$151$	11.9688	0.974004	0.487002	+	0.873401i	$0.338090\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.738641	−0.0593291
$156$	0	0
$157$	17.8877	1.42760	0.713798	−	0.700351i	$-0.246973\pi$
$157$	17.8877	1.42760	0.713798	+	0.700351i	$0.246973\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−38.8013	−3.05797
$162$	0	0
$163$	22.4424	1.75782	0.878911	−	0.476985i	$-0.158270\pi$
$163$	22.4424	1.75782	0.878911	+	0.476985i	$0.158270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.4693	0.810140	0.405070	−	0.914286i	$-0.367247\pi$
$167$	10.4693	0.810140	0.405070	+	0.914286i	$0.367247\pi$
$168$	0	0
$169$	−12.8046	−0.984967
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.6336	−1.11257	−0.556285	−	0.830992i	$-0.687774\pi$
$173$	−14.6336	−1.11257	−0.556285	+	0.830992i	$0.687774\pi$
$174$	0	0
$175$	5.05487	0.382113
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.4058	1.37572	0.687858	−	0.725845i	$-0.258551\pi$
$179$	18.4058	1.37572	0.687858	+	0.725845i	$0.258551\pi$
$180$	0	0
$181$	−0.723558	−0.0537816	−0.0268908	−	0.999638i	$-0.508561\pi$
$181$	−0.723558	−0.0537816	−0.0268908	+	0.999638i	$0.508561\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.06321	0.372255
$186$	0	0
$187$	−4.40643	−0.322230
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.23013	0.523154	0.261577	−	0.965183i	$-0.415758\pi$
$191$	7.23013	0.523154	0.261577	+	0.965183i	$0.415758\pi$
$192$	0	0
$193$	13.7301	0.988314	0.494157	−	0.869373i	$-0.335477\pi$
$193$	13.7301	0.988314	0.494157	+	0.869373i	$0.335477\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.9092	−0.848492	−0.424246	−	0.905547i	$-0.639461\pi$
$197$	−11.9092	−0.848492	−0.424246	+	0.905547i	$0.639461\pi$
$198$	0	0
$199$	−10.0891	−0.715195	−0.357598	−	0.933876i	$-0.616404\pi$
$199$	−10.0891	−0.715195	−0.357598	+	0.933876i	$0.616404\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.05487	0.354783
$204$	0	0
$205$	3.06321	0.213944
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.92128	0.409584
$210$	0	0
$211$	17.9987	1.23908	0.619541	−	0.784964i	$-0.287319\pi$
$211$	17.9987	1.23908	0.619541	+	0.784964i	$0.287319\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.9306	0.881861
$216$	0	0
$217$	−3.73374	−0.253463
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.03478	0.0696070
$222$	0	0
$223$	−19.9222	−1.33409	−0.667045	−	0.745017i	$-0.732441\pi$
$223$	−19.9222	−1.33409	−0.667045	+	0.745017i	$0.732441\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.4897	−0.895342	−0.447671	−	0.894198i	$-0.647746\pi$
$227$	−13.4897	−0.895342	−0.447671	+	0.894198i	$0.647746\pi$
$228$	0	0
$229$	3.84124	0.253836	0.126918	−	0.991913i	$-0.459491\pi$
$229$	3.84124	0.253836	0.126918	+	0.991913i	$0.459491\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.99792	−0.130888	−0.0654441	−	0.997856i	$-0.520846\pi$
$233$	−1.99792	−0.130888	−0.0654441	+	0.997856i	$0.520846\pi$
$234$	0	0
$235$	−5.53035	−0.360761
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.9938	−1.61671	−0.808356	−	0.588694i	$-0.799642\pi$
$239$	−24.9938	−1.61671	−0.808356	+	0.588694i	$0.799642\pi$
$240$	0	0
$241$	4.48382	0.288828	0.144414	−	0.989517i	$-0.453870\pi$
$241$	4.48382	0.288828	0.144414	+	0.989517i	$0.453870\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.5518	1.18523
$246$	0	0
$247$	−1.39052	−0.0884769
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.97652	−0.503474	−0.251737	−	0.967796i	$-0.581002\pi$
$251$	−7.97652	−0.503474	−0.251737	+	0.967796i	$0.581002\pi$
$252$	0	0
$253$	14.4501	0.908467
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.61058	0.537113	0.268557	−	0.963264i	$-0.413453\pi$
$257$	8.61058	0.537113	0.268557	+	0.963264i	$0.413453\pi$
$258$	0	0
$259$	25.5939	1.59033
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.17173	−0.318902	−0.159451	−	0.987206i	$-0.550972\pi$
$263$	−5.17173	−0.318902	−0.159451	+	0.987206i	$0.550972\pi$
$264$	0	0
$265$	8.58586	0.527425
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.4919	1.31038	0.655191	−	0.755464i	$-0.272588\pi$
$269$	21.4919	1.31038	0.655191	+	0.755464i	$0.272588\pi$
$270$	0	0
$271$	−6.06926	−0.368681	−0.184341	−	0.982862i	$-0.559015\pi$
$271$	−6.06926	−0.368681	−0.184341	+	0.982862i	$0.559015\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.88250	−0.113519
$276$	0	0
$277$	3.81073	0.228965	0.114482	−	0.993425i	$-0.463479\pi$
$277$	3.81073	0.228965	0.114482	+	0.993425i	$0.463479\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.3030	−0.912903	−0.456452	−	0.889748i	$-0.650880\pi$
$281$	−15.3030	−0.912903	−0.456452	+	0.889748i	$0.650880\pi$
$282$	0	0
$283$	−13.6995	−0.814350	−0.407175	−	0.913350i	$-0.633486\pi$
$283$	−13.6995	−0.814350	−0.407175	+	0.913350i	$0.633486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.4842	0.914001
$288$	0	0
$289$	−11.5209	−0.677703
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.0745	−0.588557	−0.294279	−	0.955720i	$-0.595079\pi$
$293$	−10.0745	−0.588557	−0.294279	+	0.955720i	$0.595079\pi$
$294$	0	0
$295$	13.9938	0.814753
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.39338	−0.196244
$300$	0	0
$301$	65.3627	3.76744
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.40680	0.252333
$306$	0	0
$307$	−10.9380	−0.624262	−0.312131	−	0.950039i	$-0.601043\pi$
$307$	−10.9380	−0.624262	−0.312131	+	0.950039i	$0.601043\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.4481	−0.875983	−0.437991	−	0.898979i	$-0.644310\pi$
$311$	−15.4481	−0.875983	−0.437991	+	0.898979i	$0.644310\pi$
$312$	0	0
$313$	8.10941	0.458371	0.229185	−	0.973383i	$-0.426394\pi$
$313$	8.10941	0.458371	0.229185	+	0.973383i	$0.426394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.1210	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$317$	−19.1210	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$318$	0	0
$319$	−1.88250	−0.105399
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.36266	−0.409669
$324$	0	0
$325$	0.442075	0.0245219
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−27.9553	−1.54122
$330$	0	0
$331$	16.3090	0.896424	0.448212	−	0.893927i	$-0.352061\pi$
$331$	16.3090	0.896424	0.448212	+	0.893927i	$0.352061\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.17072	0.446414
$336$	0	0
$337$	−13.3025	−0.724636	−0.362318	−	0.932055i	$-0.618015\pi$
$337$	−13.3025	−0.724636	−0.362318	+	0.932055i	$0.618015\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.39049	0.0752992
$342$	0	0
$343$	58.3927	3.15291
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9722	−1.12584	−0.562922	−	0.826510i	$-0.690323\pi$
$347$	−20.9722	−1.12584	−0.562922	+	0.826510i	$0.690323\pi$
$348$	0	0
$349$	25.8758	1.38510	0.692551	−	0.721369i	$-0.256487\pi$
$349$	25.8758	1.38510	0.692551	+	0.721369i	$0.256487\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.8779	−0.845096	−0.422548	−	0.906340i	$-0.638864\pi$
$353$	−15.8779	−0.845096	−0.422548	+	0.906340i	$0.638864\pi$
$354$	0	0
$355$	2.40680	0.127740
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.4114	−1.65783	−0.828916	−	0.559374i	$-0.811042\pi$
$359$	−31.4114	−1.65783	−0.828916	+	0.559374i	$0.811042\pi$
$360$	0	0
$361$	−9.10618	−0.479273
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.4784	−0.705493
$366$	0	0
$367$	12.8686	0.671734	0.335867	−	0.941909i	$-0.390971\pi$
$367$	12.8686	0.671734	0.335867	+	0.941909i	$0.390971\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	43.4005	2.25324
$372$	0	0
$373$	38.6023	1.99875	0.999377	−	0.0353037i	$-0.0112399\pi$
$373$	38.6023	1.99875	0.999377	+	0.0353037i	$0.0112399\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.442075	0.0227680
$378$	0	0
$379$	−20.4005	−1.04790	−0.523952	−	0.851748i	$-0.675543\pi$
$379$	−20.4005	−1.04790	−0.523952	+	0.851748i	$0.675543\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.1502	0.723040	0.361520	−	0.932364i	$-0.382258\pi$
$383$	14.1502	0.723040	0.361520	+	0.932364i	$0.382258\pi$
$384$	0	0
$385$	−9.51578	−0.484969
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.3800	0.576990	0.288495	−	0.957481i	$-0.406845\pi$
$389$	11.3800	0.576990	0.288495	+	0.957481i	$0.406845\pi$
$390$	0	0
$391$	−17.9675	−0.908657
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.4761	0.627739
$396$	0	0
$397$	23.2256	1.16566	0.582830	−	0.812594i	$-0.301945\pi$
$397$	23.2256	1.16566	0.582830	+	0.812594i	$0.301945\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.2196	−1.25940	−0.629702	−	0.776836i	$-0.716823\pi$
$401$	−25.2196	−1.25940	−0.629702	+	0.776836i	$0.716823\pi$
$402$	0	0
$403$	−0.326535	−0.0162659
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.53147	−0.472457
$408$	0	0
$409$	−18.0955	−0.894767	−0.447383	−	0.894342i	$-0.647644\pi$
$409$	−18.0955	−0.894767	−0.447383	+	0.894342i	$0.647644\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	70.7371	3.48074
$414$	0	0
$415$	−15.5434	−0.762997
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.6323	1.10566	0.552831	−	0.833293i	$-0.313548\pi$
$419$	22.6323	1.10566	0.552831	+	0.833293i	$0.313548\pi$
$420$	0	0
$421$	−6.53323	−0.318410	−0.159205	−	0.987246i	$-0.550893\pi$
$421$	−6.53323	−0.318410	−0.159205	+	0.987246i	$0.550893\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.34074	0.113542
$426$	0	0
$427$	22.2758	1.07800
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.2241	−0.685150	−0.342575	−	0.939491i	$-0.611299\pi$
$431$	−14.2241	−0.685150	−0.342575	+	0.939491i	$0.611299\pi$
$432$	0	0
$433$	−16.7527	−0.805082	−0.402541	−	0.915402i	$-0.631873\pi$
$433$	−16.7527	−0.805082	−0.402541	+	0.915402i	$0.631873\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.1445	1.15499
$438$	0	0
$439$	14.5209	0.693047	0.346523	−	0.938041i	$-0.387362\pi$
$439$	14.5209	0.693047	0.346523	+	0.938041i	$0.387362\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.8334	1.94005	0.970026	−	0.243001i	$-0.0781320\pi$
$443$	40.8334	1.94005	0.970026	+	0.243001i	$0.0781320\pi$
$444$	0	0
$445$	14.3168	0.678682
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.0851	0.570329	0.285165	−	0.958479i	$-0.407952\pi$
$449$	12.0851	0.570329	0.285165	+	0.958479i	$0.407952\pi$
$450$	0	0
$451$	−5.76648	−0.271533
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.23464	0.104761
$456$	0	0
$457$	14.2285	0.665580	0.332790	−	0.943001i	$-0.392010\pi$
$457$	14.2285	0.665580	0.332790	+	0.943001i	$0.392010\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.4001	1.18300	0.591500	−	0.806305i	$-0.298536\pi$
$461$	25.4001	1.18300	0.591500	+	0.806305i	$0.298536\pi$
$462$	0	0
$463$	−19.1813	−0.891431	−0.445715	−	0.895175i	$-0.647051\pi$
$463$	−19.1813	−0.891431	−0.445715	+	0.895175i	$0.647051\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.0749	−0.928953	−0.464477	−	0.885585i	$-0.653758\pi$
$467$	−20.0749	−0.928953	−0.464477	+	0.885585i	$0.653758\pi$
$468$	0	0
$469$	41.3020	1.90715
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.3418	−1.11924
$474$	0	0
$475$	−3.14544	−0.144323
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.1360	1.05711	0.528555	−	0.848899i	$-0.322734\pi$
$479$	23.1360	1.05711	0.528555	+	0.848899i	$0.322734\pi$
$480$	0	0
$481$	2.23832	0.102059
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.30663	−0.331777
$486$	0	0
$487$	15.0624	0.682545	0.341272	−	0.939964i	$-0.389142\pi$
$487$	15.0624	0.682545	0.341272	+	0.939964i	$0.389142\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.17175	−0.323656	−0.161828	−	0.986819i	$-0.551739\pi$
$491$	−7.17175	−0.323656	−0.161828	+	0.986819i	$0.551739\pi$
$492$	0	0
$493$	2.34074	0.105422
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.1661	0.545724
$498$	0	0
$499$	−3.00446	−0.134498	−0.0672490	−	0.997736i	$-0.521422\pi$
$499$	−3.00446	−0.134498	−0.0672490	+	0.997736i	$0.521422\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.3262	0.817123	0.408562	−	0.912731i	$-0.366031\pi$
$503$	18.3262	0.817123	0.408562	+	0.912731i	$0.366031\pi$
$504$	0	0
$505$	−10.0817	−0.448632
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12.9294	0.573085	0.286543	−	0.958068i	$-0.407494\pi$
$509$	12.9294	0.573085	0.286543	+	0.958068i	$0.407494\pi$
$510$	0	0
$511$	−68.1318	−3.01397
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.88228	−0.127009
$516$	0	0
$517$	10.4109	0.457869
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.1490	−1.18942	−0.594710	−	0.803940i	$-0.702733\pi$
$521$	−27.1490	−1.18942	−0.594710	+	0.803940i	$0.702733\pi$
$522$	0	0
$523$	−27.7464	−1.21327	−0.606633	−	0.794982i	$-0.707480\pi$
$523$	−27.7464	−1.21327	−0.606633	+	0.794982i	$0.707480\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.72897	−0.0753149
$528$	0	0
$529$	35.9211	1.56179
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.35417	0.0586556
$534$	0	0
$535$	−17.5184	−0.757385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−34.9236	−1.50427
$540$	0	0
$541$	−12.5395	−0.539117	−0.269559	−	0.962984i	$-0.586878\pi$
$541$	−12.5395	−0.539117	−0.269559	+	0.962984i	$0.586878\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.64261	−0.0703615
$546$	0	0
$547$	−40.8712	−1.74752	−0.873762	−	0.486353i	$-0.838327\pi$
$547$	−40.8712	−1.74752	−0.873762	+	0.486353i	$0.838327\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.14544	−0.134000
$552$	0	0
$553$	63.0649	2.68179
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.2900	1.24106	0.620529	−	0.784183i	$-0.286918\pi$
$557$	29.2900	1.24106	0.620529	+	0.784183i	$0.286918\pi$
$558$	0	0
$559$	5.71631	0.241774
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.9463	1.97855	0.989275	−	0.146063i	$-0.0466602\pi$
$563$	46.9463	1.97855	0.989275	+	0.146063i	$0.0466602\pi$
$564$	0	0
$565$	14.9026	0.626958
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.1401	0.425096	0.212548	−	0.977151i	$-0.431824\pi$
$569$	10.1401	0.425096	0.212548	+	0.977151i	$0.431824\pi$
$570$	0	0
$571$	6.25849	0.261910	0.130955	−	0.991388i	$-0.458196\pi$
$571$	6.25849	0.261910	0.130955	+	0.991388i	$0.458196\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.67601	−0.320112
$576$	0	0
$577$	29.0988	1.21140	0.605699	−	0.795694i	$-0.292893\pi$
$577$	29.0988	1.21140	0.605699	+	0.795694i	$0.292893\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−78.5701	−3.25964
$582$	0	0
$583$	−16.1628	−0.669397
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.8834	−1.56361	−0.781807	−	0.623520i	$-0.785702\pi$
$587$	−37.8834	−1.56361	−0.781807	+	0.623520i	$0.785702\pi$
$588$	0	0
$589$	2.32335	0.0957322
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.70392	0.234232	0.117116	−	0.993118i	$-0.462635\pi$
$593$	5.70392	0.234232	0.117116	+	0.993118i	$0.462635\pi$
$594$	0	0
$595$	11.8321	0.485070
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.4894	−0.878033	−0.439017	−	0.898479i	$-0.644673\pi$
$599$	−21.4894	−0.878033	−0.439017	+	0.898479i	$0.644673\pi$
$600$	0	0
$601$	−30.0582	−1.22610	−0.613050	−	0.790044i	$-0.710058\pi$
$601$	−30.0582	−1.22610	−0.613050	+	0.790044i	$0.710058\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.45621	−0.303138
$606$	0	0
$607$	−34.0116	−1.38049	−0.690245	−	0.723576i	$-0.742497\pi$
$607$	−34.0116	−1.38049	−0.690245	+	0.723576i	$0.742497\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.44483	−0.0989074
$612$	0	0
$613$	13.7404	0.554968	0.277484	−	0.960730i	$-0.410499\pi$
$613$	13.7404	0.554968	0.277484	+	0.960730i	$0.410499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.3466	1.58403	0.792017	−	0.610499i	$-0.209031\pi$
$617$	39.3466	1.58403	0.792017	+	0.610499i	$0.209031\pi$
$618$	0	0
$619$	−25.8592	−1.03937	−0.519685	−	0.854358i	$-0.673951\pi$
$619$	−25.8592	−1.03937	−0.519685	+	0.854358i	$0.673951\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	72.3697	2.89943
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.8517	0.472556
$630$	0	0
$631$	21.0500	0.837988	0.418994	−	0.907989i	$-0.362383\pi$
$631$	21.0500	0.837988	0.418994	+	0.907989i	$0.362383\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.84234	−0.0731112
$636$	0	0
$637$	8.20127	0.324946
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.54375	−0.218965	−0.109483	−	0.993989i	$-0.534919\pi$
$641$	−5.54375	−0.218965	−0.109483	+	0.993989i	$0.534919\pi$
$642$	0	0
$643$	−32.4462	−1.27955	−0.639777	−	0.768560i	$-0.720973\pi$
$643$	−32.4462	−1.27955	−0.639777	+	0.768560i	$0.720973\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.2558	−1.30742	−0.653711	−	0.756744i	$-0.726789\pi$
$647$	−33.2558	−1.30742	−0.653711	+	0.756744i	$0.726789\pi$
$648$	0	0
$649$	−26.3433	−1.03407
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.3645	1.03172	0.515862	−	0.856672i	$-0.327472\pi$
$653$	26.3645	1.03172	0.515862	+	0.856672i	$0.327472\pi$
$654$	0	0
$655$	−4.81928	−0.188305
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.8337	−1.62961	−0.814804	−	0.579736i	$-0.803156\pi$
$659$	−41.8337	−1.62961	−0.814804	+	0.579736i	$0.803156\pi$
$660$	0	0
$661$	28.5436	1.11022	0.555108	−	0.831778i	$-0.312677\pi$
$661$	28.5436	1.11022	0.555108	+	0.831778i	$0.312677\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.8998	−0.616569
$666$	0	0
$667$	−7.67601	−0.297216
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.29578	−0.320255
$672$	0	0
$673$	−33.2892	−1.28320	−0.641602	−	0.767038i	$-0.721730\pi$
$673$	−33.2892	−1.28320	−0.641602	+	0.767038i	$0.721730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.846740	−0.0325429	−0.0162714	−	0.999868i	$-0.505180\pi$
$677$	−0.846740	−0.0325429	−0.0162714	+	0.999868i	$0.505180\pi$
$678$	0	0
$679$	−36.9341	−1.41740
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.60547	−0.137959	−0.0689797	−	0.997618i	$-0.521974\pi$
$683$	−3.60547	−0.137959	−0.0689797	+	0.997618i	$0.521974\pi$
$684$	0	0
$685$	22.7061	0.867556
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.79560	0.144601
$690$	0	0
$691$	−30.5133	−1.16078	−0.580391	−	0.814338i	$-0.697100\pi$
$691$	−30.5133	−1.16078	−0.580391	+	0.814338i	$0.697100\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.91209	−0.0725297
$696$	0	0
$697$	7.17018	0.271590
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.10358	0.306068	0.153034	−	0.988221i	$-0.451096\pi$
$701$	8.10358	0.306068	0.153034	+	0.988221i	$0.451096\pi$
$702$	0	0
$703$	−15.9260	−0.600662
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.9620	−1.91662
$708$	0	0
$709$	−32.5630	−1.22293	−0.611465	−	0.791271i	$-0.709420\pi$
$709$	−32.5630	−1.22293	−0.611465	+	0.791271i	$0.709420\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.66982	0.212336
$714$	0	0
$715$	−0.832205	−0.0311227
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.0732	−0.972365	−0.486182	−	0.873857i	$-0.661611\pi$
$719$	−26.0732	−0.972365	−0.486182	+	0.873857i	$0.661611\pi$
$720$	0	0
$721$	−14.5696	−0.542599
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−47.3945	−1.75777	−0.878883	−	0.477037i	$-0.841711\pi$
$727$	−47.3945	−1.75777	−0.878883	+	0.477037i	$0.841711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30.2672	1.11947
$732$	0	0
$733$	−20.0282	−0.739758	−0.369879	−	0.929080i	$-0.620601\pi$
$733$	−20.0282	−0.739758	−0.369879	+	0.929080i	$0.620601\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.3813	−0.566579
$738$	0	0
$739$	−34.1405	−1.25588	−0.627939	−	0.778262i	$-0.716101\pi$
$739$	−34.1405	−1.25588	−0.627939	+	0.778262i	$0.716101\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.5853	−0.425022	−0.212511	−	0.977159i	$-0.568164\pi$
$743$	−11.5853	−0.425022	−0.212511	+	0.977159i	$0.568164\pi$
$744$	0	0
$745$	−7.88409	−0.288851
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−88.5531	−3.23566
$750$	0	0
$751$	−37.4832	−1.36778	−0.683891	−	0.729585i	$-0.739713\pi$
$751$	−37.4832	−1.36778	−0.683891	+	0.729585i	$0.739713\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.9688	0.435588
$756$	0	0
$757$	4.61469	0.167724	0.0838618	−	0.996477i	$-0.473275\pi$
$757$	4.61469	0.167724	0.0838618	+	0.996477i	$0.473275\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.06064	0.147198	0.0735991	−	0.997288i	$-0.476551\pi$
$761$	4.06064	0.147198	0.0735991	+	0.997288i	$0.476551\pi$
$762$	0	0
$763$	−8.30317	−0.300595
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.18633	0.223375
$768$	0	0
$769$	33.4576	1.20651	0.603256	−	0.797547i	$-0.293870\pi$
$769$	33.4576	1.20651	0.603256	+	0.797547i	$0.293870\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.3745	1.20040	0.600199	−	0.799851i	$-0.295088\pi$
$773$	33.3745	1.20040	0.600199	+	0.799851i	$0.295088\pi$
$774$	0	0
$775$	−0.738641	−0.0265328
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.63516	−0.345215
$780$	0	0
$781$	−4.53079	−0.162125
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.8877	0.638441
$786$	0	0
$787$	18.5816	0.662363	0.331182	−	0.943567i	$-0.392553\pi$
$787$	18.5816	0.662363	0.331182	+	0.943567i	$0.392553\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	75.3309	2.67846
$792$	0	0
$793$	1.94814	0.0691804
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.0136	1.13398	0.566991	−	0.823724i	$-0.308108\pi$
$797$	32.0136	1.13398	0.566991	+	0.823724i	$0.308108\pi$
$798$	0	0
$799$	−12.9451	−0.457965
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.3731	0.895396
$804$	0	0
$805$	−38.8013	−1.36757
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.50155	−0.193424	−0.0967121	−	0.995312i	$-0.530833\pi$
$809$	−5.50155	−0.193424	−0.0967121	+	0.995312i	$0.530833\pi$
$810$	0	0
$811$	43.8798	1.54083	0.770414	−	0.637544i	$-0.220050\pi$
$811$	43.8798	1.54083	0.770414	+	0.637544i	$0.220050\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.4424	0.786122
$816$	0	0
$817$	−40.6726	−1.42295
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.18639	−0.250807	−0.125403	−	0.992106i	$-0.540023\pi$
$821$	−7.18639	−0.250807	−0.125403	+	0.992106i	$0.540023\pi$
$822$	0	0
$823$	−28.1741	−0.982088	−0.491044	−	0.871135i	$-0.663385\pi$
$823$	−28.1741	−0.982088	−0.491044	+	0.871135i	$0.663385\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.7979	−0.862309	−0.431154	−	0.902278i	$-0.641894\pi$
$827$	−24.7979	−0.862309	−0.431154	+	0.902278i	$0.641894\pi$
$828$	0	0
$829$	−2.41731	−0.0839567	−0.0419783	−	0.999119i	$-0.513366\pi$
$829$	−2.41731	−0.0839567	−0.0419783	+	0.999119i	$0.513366\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	43.4248	1.50458
$834$	0	0
$835$	10.4693	0.362306
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.1123	−1.73007	−0.865034	−	0.501713i	$-0.832704\pi$
$839$	−50.1123	−1.73007	−0.865034	+	0.501713i	$0.832704\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.8046	−0.440491
$846$	0	0
$847$	−37.6902	−1.29505
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−38.8653	−1.33228
$852$	0	0
$853$	−16.7797	−0.574524	−0.287262	−	0.957852i	$-0.592745\pi$
$853$	−16.7797	−0.574524	−0.287262	+	0.957852i	$0.592745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.1226	−1.33640	−0.668200	−	0.743981i	$-0.732935\pi$
$857$	−39.1226	−1.33640	−0.668200	+	0.743981i	$0.732935\pi$
$858$	0	0
$859$	34.0463	1.16164	0.580822	−	0.814030i	$-0.302731\pi$
$859$	34.0463	1.16164	0.580822	+	0.814030i	$0.302731\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.2187	1.81159	0.905793	−	0.423720i	$-0.139276\pi$
$863$	53.2187	1.81159	0.905793	+	0.423720i	$0.139276\pi$
$864$	0	0
$865$	−14.6336	−0.497556
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.4861	−0.796712
$870$	0	0
$871$	3.61208	0.122391
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.05487	0.170886
$876$	0	0
$877$	−1.79289	−0.0605415	−0.0302707	−	0.999542i	$-0.509637\pi$
$877$	−1.79289	−0.0605415	−0.0302707	+	0.999542i	$0.509637\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51.9826	−1.75134	−0.875670	−	0.482911i	$-0.839580\pi$
$881$	−51.9826	−1.75134	−0.875670	+	0.482911i	$0.839580\pi$
$882$	0	0
$883$	−4.82566	−0.162396	−0.0811981	−	0.996698i	$-0.525875\pi$
$883$	−4.82566	−0.162396	−0.0811981	+	0.996698i	$0.525875\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.9192	1.03816	0.519082	−	0.854724i	$-0.326274\pi$
$887$	30.9192	1.03816	0.519082	+	0.854724i	$0.326274\pi$
$888$	0	0
$889$	−9.31282	−0.312342
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.3954	0.582116
$894$	0	0
$895$	18.4058	0.615239
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.738641	−0.0246351
$900$	0	0
$901$	20.0973	0.669537
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.723558	−0.0240519
$906$	0	0
$907$	−2.20780	−0.0733089	−0.0366545	−	0.999328i	$-0.511670\pi$
$907$	−2.20780	−0.0733089	−0.0366545	+	0.999328i	$0.511670\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.99196	−0.331048	−0.165524	−	0.986206i	$-0.552932\pi$
$911$	−9.99196	−0.331048	−0.165524	+	0.986206i	$0.552932\pi$
$912$	0	0
$913$	29.2604	0.968378
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.3609	−0.804467
$918$	0	0
$919$	20.2582	0.668257	0.334128	−	0.942528i	$-0.391558\pi$
$919$	20.2582	0.668257	0.334128	+	0.942528i	$0.391558\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.06399	0.0350216
$924$	0	0
$925$	5.06321	0.166477
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25.8525	−0.848192	−0.424096	−	0.905617i	$-0.639408\pi$
$929$	−25.8525	−0.848192	−0.424096	+	0.905617i	$0.639408\pi$
$930$	0	0
$931$	−58.3535	−1.91246
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.40643	−0.144106
$936$	0	0
$937$	25.0030	0.816813	0.408407	−	0.912800i	$-0.366085\pi$
$937$	25.0030	0.816813	0.408407	+	0.912800i	$0.366085\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.17179	−0.201195	−0.100597	−	0.994927i	$-0.532075\pi$
$941$	−6.17179	−0.201195	−0.100597	+	0.994927i	$0.532075\pi$
$942$	0	0
$943$	−23.5132	−0.765697
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.9491	0.583267	0.291633	−	0.956530i	$-0.405801\pi$
$947$	17.9491	0.583267	0.291633	+	0.956530i	$0.405801\pi$
$948$	0	0
$949$	−5.95848	−0.193420
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.05951	0.0991073	0.0495537	−	0.998771i	$-0.484220\pi$
$953$	3.05951	0.0991073	0.0495537	+	0.998771i	$0.484220\pi$
$954$	0	0
$955$	7.23013	0.233961
$956$	0	0
$957$	0	0
$958$	0	0
$959$	114.777	3.70633
$960$	0	0
$961$	−30.4544	−0.982400
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.7301	0.441987
$966$	0	0
$967$	−37.2342	−1.19737	−0.598685	−	0.800985i	$-0.704310\pi$
$967$	−37.2342	−1.19737	−0.598685	+	0.800985i	$0.704310\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.6522	1.46505	0.732524	−	0.680741i	$-0.238342\pi$
$971$	45.6522	1.46505	0.732524	+	0.680741i	$0.238342\pi$
$972$	0	0
$973$	−9.66537	−0.309858
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.4021	1.51653	0.758264	−	0.651948i	$-0.226048\pi$
$977$	47.4021	1.51653	0.758264	+	0.651948i	$0.226048\pi$
$978$	0	0
$979$	−26.9513	−0.861368
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.4737	0.844380	0.422190	−	0.906507i	$-0.361261\pi$
$983$	26.4737	0.844380	0.422190	+	0.906507i	$0.361261\pi$
$984$	0	0
$985$	−11.9092	−0.379457
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−99.2556	−3.15614
$990$	0	0
$991$	−42.4199	−1.34751	−0.673757	−	0.738953i	$-0.735320\pi$
$991$	−42.4199	−1.34751	−0.673757	+	0.738953i	$0.735320\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.0891	−0.319845
$996$	0	0
$997$	−26.8974	−0.851849	−0.425924	−	0.904759i	$-0.640051\pi$
$997$	−26.8974	−0.851849	−0.425924	+	0.904759i	$0.640051\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.z.1.7	yes	7
3.2	odd	2		5220.2.a.y.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5220.2.a.y.1.7	✓	7	3.2	odd	2
5220.2.a.z.1.7	yes	7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +5.05487 q^{7} -1.88250 q^{11} +0.442075 q^{13} +2.34074 q^{17} -3.14544 q^{19} -7.67601 q^{23} +1.00000 q^{25} +1.00000 q^{29} -0.738641 q^{31} +5.05487 q^{35} +5.06321 q^{37} +3.06321 q^{41} +12.9306 q^{43} -5.53035 q^{47} +18.5518 q^{49} +8.58586 q^{53} -1.88250 q^{55} +13.9938 q^{59} +4.40680 q^{61} +0.442075 q^{65} +8.17072 q^{67} +2.40680 q^{71} -13.4784 q^{73} -9.51578 q^{77} +12.4761 q^{79} -15.5434 q^{83} +2.34074 q^{85} +14.3168 q^{89} +2.23464 q^{91} -3.14544 q^{95} -7.30663 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} + 3 q^{11} + 2 q^{13} + 8 q^{17} + 6 q^{19} + q^{23} + 7 q^{25} + 7 q^{29} - 2 q^{31} + 15 q^{37} + q^{41} + 9 q^{43} + 4 q^{47} + 29 q^{49} + 17 q^{53} + 3 q^{55} - 4 q^{59} + 6 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 + 3 * q^11 + 2 * q^13 + 8 * q^17 + 6 * q^19 + q^23 + 7 * q^25 + 7 * q^29 - 2 * q^31 + 15 * q^37 + q^41 + 9 * q^43 + 4 * q^47 + 29 * q^49 + 17 * q^53 + 3 * q^55 - 4 * q^59 + 6 * q^61 + 2 * q^65 + 24 * q^67 - 8 * q^71 + 29 * q^73 + 8 * q^79 + 7 * q^83 + 8 * q^85 - 4 * q^89 + 26 * q^91 + 6 * q^95 + 15 * q^97

Introduction

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