LMFDB - Embedded newform 6084.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6084,2,Mod(1,6084)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6084, 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("6084.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6084.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:	$48.5809845897$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6084.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{5} +3.46410 q^{7} +O(q^{10})$	$q+1.73205 q^{5} +3.46410 q^{7} +3.46410 q^{11} +3.00000 q^{17} +3.46410 q^{19} +6.00000 q^{23} -2.00000 q^{25} -9.00000 q^{29} +6.00000 q^{35} +5.19615 q^{37} +8.66025 q^{41} -2.00000 q^{43} +3.46410 q^{47} +5.00000 q^{49} +9.00000 q^{53} +6.00000 q^{55} -13.8564 q^{59} -11.0000 q^{61} +10.3923 q^{67} +10.3923 q^{71} -5.19615 q^{73} +12.0000 q^{77} -8.00000 q^{79} +3.46410 q^{83} +5.19615 q^{85} -6.92820 q^{89} +6.00000 q^{95} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 6 q^{17} + 12 q^{23} - 4 q^{25} - 18 q^{29} + 12 q^{35} - 4 q^{43} + 10 q^{49} + 18 q^{53} + 12 q^{55} - 22 q^{61} + 24 q^{77} - 16 q^{79} + 12 q^{95}+O(q^{100}) $ 2 * q + 6 * q^17 + 12 * q^23 - 4 * q^25 - 18 * q^29 + 12 * q^35 - 4 * q^43 + 10 * q^49 + 18 * q^53 + 12 * q^55 - 22 * q^61 + 24 * q^77 - 16 * q^79 + 12 * q^95

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	5.19615	0.854242	0.427121	−	0.904194i	$-0.359528\pi$
$37$	5.19615	0.854242	0.427121	+	0.904194i	$0.359528\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.66025	1.35250	0.676252	−	0.736670i	$-0.263603\pi$
$41$	8.66025	1.35250	0.676252	+	0.736670i	$0.263603\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.3923	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	−5.19615	−0.608164	−0.304082	−	0.952646i	$-0.598350\pi$
$73$	−5.19615	−0.608164	−0.304082	+	0.952646i	$0.598350\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	5.19615	0.563602
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	10.3923	0.969087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.3923	0.952661
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.5885	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$137$	−15.5885	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−15.5885	−1.29455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.66025	−0.709476	−0.354738	−	0.934966i	$-0.615430\pi$
$149$	−8.66025	−0.709476	−0.354738	+	0.934966i	$0.615430\pi$
$150$	0	0
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.7846	1.63806
$162$	0	0
$163$	−13.8564	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$163$	−13.8564	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−6.92820	−0.523723
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	13.0000	0.966282	0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000	0.966282	0.483141	+	0.875542i	$0.339496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	10.3923	0.759961
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1.73205	−0.124676	−0.0623379	−	0.998055i	$-0.519856\pi$
$193$	−1.73205	−0.124676	−0.0623379	+	0.998055i	$0.519856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−31.1769	−2.18819
$204$	0	0
$205$	15.0000	1.04765
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.8564	0.927894	0.463947	−	0.885863i	$-0.346433\pi$
$223$	13.8564	0.927894	0.463947	+	0.885863i	$0.346433\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	25.9808	1.67357	0.836784	−	0.547533i	$-0.184433\pi$
$241$	25.9808	1.67357	0.836784	+	0.547533i	$0.184433\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.66025	0.553283
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	20.7846	1.30672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	18.0000	1.11847
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	15.5885	0.957591
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−6.92820	−0.420858	−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820	−0.420858	−0.210429	+	0.977609i	$0.567486\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.92820	−0.417786
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.4449	−1.75653	−0.878267	−	0.478171i	$-0.841300\pi$
$281$	−29.4449	−1.75653	−0.878267	+	0.478171i	$0.841300\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.0000	1.77084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.73205	−0.101187	−0.0505937	−	0.998719i	$-0.516111\pi$
$293$	−1.73205	−0.101187	−0.0505937	+	0.998719i	$0.516111\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.92820	−0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.0526	−1.09095
$306$	0	0
$307$	−17.3205	−0.988534	−0.494267	−	0.869310i	$-0.664563\pi$
$307$	−17.3205	−0.988534	−0.494267	+	0.869310i	$0.664563\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.73205	−0.0972817	−0.0486408	−	0.998816i	$-0.515489\pi$
$317$	−1.73205	−0.0972817	−0.0486408	+	0.998816i	$0.515489\pi$
$318$	0	0
$319$	−31.1769	−1.74557
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−13.8564	−0.761617	−0.380808	−	0.924654i	$-0.624354\pi$
$331$	−13.8564	−0.761617	−0.380808	+	0.924654i	$0.624354\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.0000	0.983445
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	13.8564	0.741716	0.370858	−	0.928689i	$-0.379064\pi$
$349$	13.8564	0.741716	0.370858	+	0.928689i	$0.379064\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.9808	1.38282	0.691408	−	0.722464i	$-0.256991\pi$
$353$	25.9808	1.38282	0.691408	+	0.722464i	$0.256991\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.46410	−0.182828	−0.0914141	−	0.995813i	$-0.529139\pi$
$359$	−3.46410	−0.182828	−0.0914141	+	0.995813i	$0.529139\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.00000	−0.471082
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.1769	1.61862
$372$	0	0
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	13.8564	0.711756	0.355878	−	0.934532i	$-0.384182\pi$
$379$	13.8564	0.711756	0.355878	+	0.934532i	$0.384182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	20.7846	1.05928
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.8564	−0.697191
$396$	0	0
$397$	−27.7128	−1.39087	−0.695433	−	0.718591i	$-0.744787\pi$
$397$	−27.7128	−1.39087	−0.695433	+	0.718591i	$0.744787\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.3731	1.81638	0.908192	−	0.418554i	$-0.137463\pi$
$401$	36.3731	1.81638	0.908192	+	0.418554i	$0.137463\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	12.1244	0.599511	0.299755	−	0.954016i	$-0.403095\pi$
$409$	12.1244	0.599511	0.299755	+	0.954016i	$0.403095\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	12.1244	0.590905	0.295452	−	0.955357i	$-0.404530\pi$
$421$	12.1244	0.590905	0.295452	+	0.955357i	$0.404530\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−38.1051	−1.84404
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.1769	1.50174	0.750870	−	0.660451i	$-0.229635\pi$
$431$	31.1769	1.50174	0.750870	+	0.660451i	$0.229635\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.7846	0.994263
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.92820	0.326962	0.163481	−	0.986546i	$-0.447728\pi$
$449$	6.92820	0.326962	0.163481	+	0.986546i	$0.447728\pi$
$450$	0	0
$451$	30.0000	1.41264
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.66025	0.405110	0.202555	−	0.979271i	$-0.435076\pi$
$457$	8.66025	0.405110	0.202555	+	0.979271i	$0.435076\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.19615	0.242009	0.121004	−	0.992652i	$-0.461388\pi$
$461$	5.19615	0.242009	0.121004	+	0.992652i	$0.461388\pi$
$462$	0	0
$463$	3.46410	0.160990	0.0804952	−	0.996755i	$-0.474350\pi$
$463$	3.46410	0.160990	0.0804952	+	0.996755i	$0.474350\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.92820	−0.318559
$474$	0	0
$475$	−6.92820	−0.317888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.92820	0.316558	0.158279	−	0.987394i	$-0.449406\pi$
$479$	6.92820	0.316558	0.158279	+	0.987394i	$0.449406\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	17.3205	0.784867	0.392434	−	0.919780i	$-0.371633\pi$
$487$	17.3205	0.784867	0.392434	+	0.919780i	$0.371633\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.0000	1.61482
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	5.19615	0.231226
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.73205	−0.0767718	−0.0383859	−	0.999263i	$-0.512222\pi$
$509$	−1.73205	−0.0767718	−0.0383859	+	0.999263i	$0.512222\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−24.2487	−1.06853
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	31.1769	1.34790
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.3205	0.746047
$540$	0	0
$541$	1.73205	0.0744667	0.0372333	−	0.999307i	$-0.488146\pi$
$541$	1.73205	0.0744667	0.0372333	+	0.999307i	$0.488146\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.1769	−1.32818
$552$	0	0
$553$	−27.7128	−1.17847
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.73205	0.0733893	0.0366947	−	0.999327i	$-0.488317\pi$
$557$	1.73205	0.0733893	0.0366947	+	0.999327i	$0.488317\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−5.19615	−0.218604
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−8.66025	−0.360531	−0.180266	−	0.983618i	$-0.557696\pi$
$577$	−8.66025	−0.360531	−0.180266	+	0.983618i	$0.557696\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	31.1769	1.29122
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.1244	−0.497888	−0.248944	−	0.968518i	$-0.580083\pi$
$593$	−12.1244	−0.497888	−0.248944	+	0.968518i	$0.580083\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.73205	0.0704179
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−5.19615	−0.209871	−0.104935	−	0.994479i	$-0.533464\pi$
$613$	−5.19615	−0.209871	−0.104935	+	0.994479i	$0.533464\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.9808	−1.04595	−0.522973	−	0.852349i	$-0.675177\pi$
$617$	−25.9808	−1.04595	−0.522973	+	0.852349i	$0.675177\pi$
$618$	0	0
$619$	13.8564	0.556936	0.278468	−	0.960446i	$-0.410173\pi$
$619$	13.8564	0.556936	0.278468	+	0.960446i	$0.410173\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.5885	0.621552
$630$	0	0
$631$	27.7128	1.10323	0.551615	−	0.834099i	$-0.314012\pi$
$631$	27.7128	1.10323	0.551615	+	0.834099i	$0.314012\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.7128	−1.09975
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	0	0
$643$	6.92820	0.273222	0.136611	−	0.990625i	$-0.456379\pi$
$643$	6.92820	0.273222	0.136611	+	0.990625i	$0.456379\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	20.7846	0.812122
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	12.1244	0.471583	0.235791	−	0.971804i	$-0.424232\pi$
$661$	12.1244	0.471583	0.235791	+	0.971804i	$0.424232\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.7846	0.805993
$666$	0	0
$667$	−54.0000	−2.09089
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−38.1051	−1.47103
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.8564	0.530201	0.265100	−	0.964221i	$-0.414595\pi$
$683$	13.8564	0.530201	0.265100	+	0.964221i	$0.414595\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−10.3923	−0.395342	−0.197671	−	0.980268i	$-0.563338\pi$
$691$	−10.3923	−0.395342	−0.197671	+	0.980268i	$0.563338\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.92820	−0.262802
$696$	0	0
$697$	25.9808	0.984092
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.3923	0.390843
$708$	0	0
$709$	15.5885	0.585437	0.292718	−	0.956199i	$-0.405440\pi$
$709$	15.5885	0.585437	0.292718	+	0.956199i	$0.405440\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−48.4974	−1.80614
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.0000	0.668503
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.00000	−0.221918
$732$	0	0
$733$	36.3731	1.34347	0.671735	−	0.740792i	$-0.265549\pi$
$733$	36.3731	1.34347	0.671735	+	0.740792i	$0.265549\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	−27.7128	−1.01943	−0.509716	−	0.860343i	$-0.670250\pi$
$739$	−27.7128	−1.01943	−0.509716	+	0.860343i	$0.670250\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.5692	−1.52503	−0.762513	−	0.646972i	$-0.776035\pi$
$743$	−41.5692	−1.52503	−0.762513	+	0.646972i	$0.776035\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	62.3538	2.27836
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.0000	−1.09181
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.7846	−0.753442	−0.376721	−	0.926327i	$-0.622948\pi$
$761$	−20.7846	−0.753442	−0.376721	+	0.926327i	$0.622948\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−34.6410	−1.24919	−0.624593	−	0.780950i	$-0.714735\pi$
$769$	−34.6410	−1.24919	−0.624593	+	0.780950i	$0.714735\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30.0000	1.07486
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	39.8372	1.42185
$786$	0	0
$787$	−41.5692	−1.48178	−0.740891	−	0.671625i	$-0.765597\pi$
$787$	−41.5692	−1.48178	−0.740891	+	0.671625i	$0.765597\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	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
$799$	10.3923	0.367653
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	36.0000	1.26883
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.7128	−0.967184	−0.483592	−	0.875294i	$-0.660668\pi$
$821$	−27.7128	−0.967184	−0.483592	+	0.875294i	$0.660668\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.7128	0.963669	0.481834	−	0.876262i	$-0.339971\pi$
$827$	27.7128	0.963669	0.481834	+	0.876262i	$0.339971\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.0000	0.519719
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.5692	−1.43513	−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692	−1.43513	−0.717564	+	0.696492i	$0.754743\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.46410	0.119028
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.1769	1.06873
$852$	0	0
$853$	−43.3013	−1.48261	−0.741304	−	0.671170i	$-0.765792\pi$
$853$	−43.3013	−1.48261	−0.741304	+	0.671170i	$0.765792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1051	1.29711	0.648557	−	0.761166i	$-0.275373\pi$
$863$	38.1051	1.29711	0.648557	+	0.761166i	$0.275373\pi$
$864$	0	0
$865$	31.1769	1.06005
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−27.7128	−0.940093
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−42.0000	−1.41986
$876$	0	0
$877$	36.3731	1.22823	0.614116	−	0.789216i	$-0.289513\pi$
$877$	36.3731	1.22823	0.614116	+	0.789216i	$0.289513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−55.4256	−1.85892
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	31.1769	1.04213
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.5167	0.748479
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.5692	1.37274
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.3923	−0.341697
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−57.1577	−1.87528	−0.937641	−	0.347604i	$-0.886995\pi$
$929$	−57.1577	−1.87528	−0.937641	+	0.347604i	$0.886995\pi$
$930$	0	0
$931$	17.3205	0.567657
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.0000	0.588663
$936$	0	0
$937$	−41.0000	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	51.9615	1.69210
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.8564	−0.450273	−0.225136	−	0.974327i	$-0.572283\pi$
$947$	−13.8564	−0.450273	−0.225136	+	0.974327i	$0.572283\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−54.0000	−1.74375
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.00000	−0.0965734
$966$	0	0
$967$	−45.0333	−1.44817	−0.724087	−	0.689709i	$-0.757739\pi$
$967$	−45.0333	−1.44817	−0.724087	+	0.689709i	$0.757739\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−13.8564	−0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.66025	0.277066	0.138533	−	0.990358i	$-0.455761\pi$
$977$	8.66025	0.277066	0.138533	+	0.990358i	$0.455761\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.3205	−0.549097
$996$	0	0
$997$	−55.0000	−1.74187	−0.870934	−	0.491400i	$-0.836485\pi$
$997$	−55.0000	−1.74187	−0.870934	+	0.491400i	$0.836485\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6084.2.a.u.1.2		2
3.2	odd	2		2028.2.a.h.1.1		2
12.11	even	2		8112.2.a.bt.1.1		2
13.2	odd	12		468.2.t.c.433.1		2
13.5	odd	4		6084.2.b.c.4393.1		2
13.7	odd	12		468.2.t.c.361.1		2
13.8	odd	4		6084.2.b.c.4393.2		2
13.12	even	2	inner	6084.2.a.u.1.1		2
39.2	even	12		156.2.q.a.121.1	yes	2
39.5	even	4		2028.2.b.b.337.2		2
39.8	even	4		2028.2.b.b.337.1		2
39.11	even	12		2028.2.q.a.1837.1		2
39.17	odd	6		2028.2.i.h.2005.2		4
39.20	even	12		156.2.q.a.49.1	✓	2
39.23	odd	6		2028.2.i.h.529.2		4
39.29	odd	6		2028.2.i.h.529.1		4
39.32	even	12		2028.2.q.a.361.1		2
39.35	odd	6		2028.2.i.h.2005.1		4
39.38	odd	2		2028.2.a.h.1.2		2
52.7	even	12		1872.2.by.b.1297.1		2
52.15	even	12		1872.2.by.b.433.1		2
156.59	odd	12		624.2.bv.a.49.1		2
156.119	odd	12		624.2.bv.a.433.1		2
156.155	even	2		8112.2.a.bt.1.2		2
195.2	odd	12		3900.2.bw.e.2149.1		4
195.59	even	12		3900.2.cd.a.2701.1		2
195.98	odd	12		3900.2.bw.e.49.1		4
195.119	even	12		3900.2.cd.a.901.1		2
195.137	odd	12		3900.2.bw.e.49.2		4
195.158	odd	12		3900.2.bw.e.2149.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.a.49.1	✓	2	39.20	even	12
156.2.q.a.121.1	yes	2	39.2	even	12
468.2.t.c.361.1		2	13.7	odd	12
468.2.t.c.433.1		2	13.2	odd	12
624.2.bv.a.49.1		2	156.59	odd	12
624.2.bv.a.433.1		2	156.119	odd	12
1872.2.by.b.433.1		2	52.15	even	12
1872.2.by.b.1297.1		2	52.7	even	12
2028.2.a.h.1.1		2	3.2	odd	2
2028.2.a.h.1.2		2	39.38	odd	2
2028.2.b.b.337.1		2	39.8	even	4
2028.2.b.b.337.2		2	39.5	even	4
2028.2.i.h.529.1		4	39.29	odd	6
2028.2.i.h.529.2		4	39.23	odd	6
2028.2.i.h.2005.1		4	39.35	odd	6
2028.2.i.h.2005.2		4	39.17	odd	6
2028.2.q.a.361.1		2	39.32	even	12
2028.2.q.a.1837.1		2	39.11	even	12
3900.2.bw.e.49.1		4	195.98	odd	12
3900.2.bw.e.49.2		4	195.137	odd	12
3900.2.bw.e.2149.1		4	195.2	odd	12
3900.2.bw.e.2149.2		4	195.158	odd	12
3900.2.cd.a.901.1		2	195.119	even	12
3900.2.cd.a.2701.1		2	195.59	even	12
6084.2.a.u.1.1		2	13.12	even	2	inner
6084.2.a.u.1.2		2	1.1	even	1	trivial
6084.2.b.c.4393.1		2	13.5	odd	4
6084.2.b.c.4393.2		2	13.8	odd	4
8112.2.a.bt.1.1		2	12.11	even	2
8112.2.a.bt.1.2		2	156.155	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:	\( N \)	\(=\)	\( 6084 = 2^{2} \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6084.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{5} +3.46410 q^{7} +O(q^{10})\)	\(q+1.73205 q^{5} +3.46410 q^{7} +3.46410 q^{11} +3.00000 q^{17} +3.46410 q^{19} +6.00000 q^{23} -2.00000 q^{25} -9.00000 q^{29} +6.00000 q^{35} +5.19615 q^{37} +8.66025 q^{41} -2.00000 q^{43} +3.46410 q^{47} +5.00000 q^{49} +9.00000 q^{53} +6.00000 q^{55} -13.8564 q^{59} -11.0000 q^{61} +10.3923 q^{67} +10.3923 q^{71} -5.19615 q^{73} +12.0000 q^{77} -8.00000 q^{79} +3.46410 q^{83} +5.19615 q^{85} -6.92820 q^{89} +6.00000 q^{95} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 6 q^{17} + 12 q^{23} - 4 q^{25} - 18 q^{29} + 12 q^{35} - 4 q^{43} + 10 q^{49} + 18 q^{53} + 12 q^{55} - 22 q^{61} + 24 q^{77} - 16 q^{79} + 12 q^{95}+O(q^{100}) \) 2 * q + 6 * q^17 + 12 * q^23 - 4 * q^25 - 18 * q^29 + 12 * q^35 - 4 * q^43 + 10 * q^49 + 18 * q^53 + 12 * q^55 - 22 * q^61 + 24 * q^77 - 16 * q^79 + 12 * q^95

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