LMFDB - Embedded newform 5733.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199374400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 10x^{2} - 1 $ x^6 - 8x^4 + 10x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 819)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.330837$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69180 q^{2} +5.24581 q^{4} -1.40467 q^{5} -8.73709 q^{8} +O(q^{10})$	$q-2.69180 q^{2} +5.24581 q^{4} -1.40467 q^{5} -8.73709 q^{8} +3.78109 q^{10} -6.04528 q^{11} -1.00000 q^{13} +13.0269 q^{16} -6.70695 q^{17} -1.53528 q^{19} -7.36863 q^{20} +16.2727 q^{22} -0.742996 q^{23} -3.02690 q^{25} +2.69180 q^{26} +6.12660 q^{29} -10.0269 q^{31} -17.5917 q^{32} +18.0538 q^{34} -6.49162 q^{37} +4.13268 q^{38} +12.2727 q^{40} -7.53127 q^{41} -1.53528 q^{43} -31.7124 q^{44} +2.00000 q^{46} +5.30229 q^{47} +8.14783 q^{50} -5.24581 q^{52} +5.96396 q^{53} +8.49162 q^{55} -16.4916 q^{58} -10.1055 q^{59} -10.0000 q^{61} +26.9905 q^{62} +21.2996 q^{64} +1.40467 q^{65} -4.49162 q^{67} -35.1834 q^{68} +3.23594 q^{71} -8.02690 q^{73} +17.4742 q^{74} -8.05381 q^{76} -10.5185 q^{79} -18.2985 q^{80} +20.2727 q^{82} -8.11162 q^{83} +9.42105 q^{85} +4.13268 q^{86} +52.8181 q^{88} +1.24202 q^{89} -3.89762 q^{92} -14.2727 q^{94} +2.15657 q^{95} +0.464716 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4}+O(q^{10}) $ 6 * q + 12 * q^4	$ 6 q + 12 q^{4} - 8 q^{10} - 6 q^{13} + 28 q^{16} + 2 q^{19} + 28 q^{22} + 32 q^{25} - 10 q^{31} + 8 q^{34} + 4 q^{40} + 2 q^{43} + 12 q^{46} - 12 q^{52} + 12 q^{55} - 60 q^{58} - 60 q^{61} + 8 q^{64} + 12 q^{67} + 2 q^{73} + 52 q^{76} + 26 q^{79} + 52 q^{82} + 40 q^{85} + 108 q^{88} - 16 q^{94} + 14 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 - 8 * q^10 - 6 * q^13 + 28 * q^16 + 2 * q^19 + 28 * q^22 + 32 * q^25 - 10 * q^31 + 8 * q^34 + 4 * q^40 + 2 * q^43 + 12 * q^46 - 12 * q^52 + 12 * q^55 - 60 * q^58 - 60 * q^61 + 8 * q^64 + 12 * q^67 + 2 * q^73 + 52 * q^76 + 26 * q^79 + 52 * q^82 + 40 * q^85 + 108 * q^88 - 16 * q^94 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.69180	−1.90339	−0.951697	−	0.307040i	$-0.900661\pi$
$2$	−2.69180	−1.90339	−0.951697	+	0.307040i	$0.900661\pi$
$3$	0	0
$3$	0	0
$4$	5.24581	2.62291
$5$	−1.40467	−0.628187	−0.314094	−	0.949392i	$-0.601701\pi$
$5$	−1.40467	−0.628187	−0.314094	+	0.949392i	$0.601701\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.73709	−3.08903
$9$	0	0
$10$	3.78109	1.19569
$11$	−6.04528	−1.82272	−0.911360	−	0.411609i	$-0.864967\pi$
$11$	−6.04528	−1.82272	−0.911360	+	0.411609i	$0.864967\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	13.0269	3.25673
$17$	−6.70695	−1.62668	−0.813338	−	0.581792i	$-0.802352\pi$
$17$	−6.70695	−1.62668	−0.813338	+	0.581792i	$0.802352\pi$
$18$	0	0
$19$	−1.53528	−0.352218	−0.176109	−	0.984371i	$-0.556351\pi$
$19$	−1.53528	−0.352218	−0.176109	+	0.984371i	$0.556351\pi$
$20$	−7.36863	−1.64768
$21$	0	0
$22$	16.2727	3.46935
$23$	−0.742996	−0.154925	−0.0774627	−	0.996995i	$-0.524682\pi$
$23$	−0.742996	−0.154925	−0.0774627	+	0.996995i	$0.524682\pi$
$24$	0	0
$25$	−3.02690	−0.605381
$26$	2.69180	0.527906
$27$	0	0
$28$	0	0
$29$	6.12660	1.13768	0.568841	−	0.822448i	$-0.307392\pi$
$29$	6.12660	1.13768	0.568841	+	0.822448i	$0.307392\pi$
$30$	0	0
$31$	−10.0269	−1.80089	−0.900443	−	0.434975i	$-0.856757\pi$
$31$	−10.0269	−1.80089	−0.900443	+	0.434975i	$0.856757\pi$
$32$	−17.5917	−3.10980
$33$	0	0
$34$	18.0538	3.09620
$35$	0	0
$36$	0	0
$37$	−6.49162	−1.06722	−0.533608	−	0.845732i	$-0.679164\pi$
$37$	−6.49162	−1.06722	−0.533608	+	0.845732i	$0.679164\pi$
$38$	4.13268	0.670410
$39$	0	0
$40$	12.2727	1.94049
$41$	−7.53127	−1.17619	−0.588094	−	0.808793i	$-0.700121\pi$
$41$	−7.53127	−1.17619	−0.588094	+	0.808793i	$0.700121\pi$
$42$	0	0
$43$	−1.53528	−0.234129	−0.117064	−	0.993124i	$-0.537348\pi$
$43$	−1.53528	−0.234129	−0.117064	+	0.993124i	$0.537348\pi$
$44$	−31.7124	−4.78082
$45$	0	0
$46$	2.00000	0.294884
$47$	5.30229	0.773418	0.386709	−	0.922202i	$-0.373612\pi$
$47$	5.30229	0.773418	0.386709	+	0.922202i	$0.373612\pi$
$48$	0	0
$49$	0	0
$50$	8.14783	1.15228
$51$	0	0
$52$	−5.24581	−0.727463
$53$	5.96396	0.819213	0.409606	−	0.912262i	$-0.365666\pi$
$53$	5.96396	0.819213	0.409606	+	0.912262i	$0.365666\pi$
$54$	0	0
$55$	8.49162	1.14501
$56$	0	0
$57$	0	0
$58$	−16.4916	−2.16546
$59$	−10.1055	−1.31563	−0.657815	−	0.753180i	$-0.728519\pi$
$59$	−10.1055	−1.31563	−0.657815	+	0.753180i	$0.728519\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	26.9905	3.42779
$63$	0	0
$64$	21.2996	2.66245
$65$	1.40467	0.174228
$66$	0	0
$67$	−4.49162	−0.548739	−0.274369	−	0.961624i	$-0.588469\pi$
$67$	−4.49162	−0.548739	−0.274369	+	0.961624i	$0.588469\pi$
$68$	−35.1834	−4.26662
$69$	0	0
$70$	0	0
$71$	3.23594	0.384036	0.192018	−	0.981391i	$-0.438497\pi$
$71$	3.23594	0.384036	0.192018	+	0.981391i	$0.438497\pi$
$72$	0	0
$73$	−8.02690	−0.939478	−0.469739	−	0.882805i	$-0.655652\pi$
$73$	−8.02690	−0.939478	−0.469739	+	0.882805i	$0.655652\pi$
$74$	17.4742	2.03133
$75$	0	0
$76$	−8.05381	−0.923835
$77$	0	0
$78$	0	0
$79$	−10.5185	−1.18343	−0.591713	−	0.806149i	$-0.701548\pi$
$79$	−10.5185	−1.18343	−0.591713	+	0.806149i	$0.701548\pi$
$80$	−18.2985	−2.04583
$81$	0	0
$82$	20.2727	2.23875
$83$	−8.11162	−0.890366	−0.445183	−	0.895440i	$-0.646861\pi$
$83$	−8.11162	−0.890366	−0.445183	+	0.895440i	$0.646861\pi$
$84$	0	0
$85$	9.42105	1.02186
$86$	4.13268	0.445639
$87$	0	0
$88$	52.8181	5.63043
$89$	1.24202	0.131654	0.0658271	−	0.997831i	$-0.479031\pi$
$89$	1.24202	0.131654	0.0658271	+	0.997831i	$0.479031\pi$
$90$	0	0
$91$	0	0
$92$	−3.89762	−0.406355
$93$	0	0
$94$	−14.2727	−1.47212
$95$	2.15657	0.221259
$96$	0	0
$97$	0.464716	0.0471847	0.0235924	−	0.999722i	$-0.492490\pi$
$97$	0.464716	0.0471847	0.0235924	+	0.999722i	$0.492490\pi$
$98$	0	0
$99$	0	0
$100$	−15.8786	−1.58786
$101$	−14.6648	−1.45921	−0.729603	−	0.683871i	$-0.760295\pi$
$101$	−14.6648	−1.45921	−0.729603	+	0.683871i	$0.760295\pi$
$102$	0	0
$103$	11.5622	1.13926	0.569628	−	0.821903i	$-0.307087\pi$
$103$	11.5622	1.13926	0.569628	+	0.821903i	$0.307087\pi$
$104$	8.73709	0.856742
$105$	0	0
$106$	−16.0538	−1.55928
$107$	5.22096	0.504729	0.252365	−	0.967632i	$-0.418792\pi$
$107$	5.22096	0.504729	0.252365	+	0.967632i	$0.418792\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−22.8578	−2.17940
$111$	0	0
$112$	0	0
$113$	−6.12660	−0.576342	−0.288171	−	0.957579i	$-0.593047\pi$
$113$	−6.12660	−0.576342	−0.288171	+	0.957579i	$0.593047\pi$
$114$	0	0
$115$	1.04366	0.0973221
$116$	32.1390	2.98403
$117$	0	0
$118$	27.2021	2.50416
$119$	0	0
$120$	0	0
$121$	25.5454	2.32231
$122$	26.9180	2.43705
$123$	0	0
$124$	−52.5992	−4.72355
$125$	11.2751	1.00848
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−22.1510	−1.95789
$129$	0	0
$130$	−3.78109	−0.331624
$131$	4.29533	0.375285	0.187642	−	0.982237i	$-0.439915\pi$
$131$	4.29533	0.375285	0.187642	+	0.982237i	$0.439915\pi$
$132$	0	0
$133$	0	0
$134$	12.0906	1.04447
$135$	0	0
$136$	58.5992	5.02484
$137$	2.14767	0.183487	0.0917437	−	0.995783i	$-0.470756\pi$
$137$	2.14767	0.183487	0.0917437	+	0.995783i	$0.470756\pi$
$138$	0	0
$139$	−21.4749	−1.82147	−0.910737	−	0.412987i	$-0.864486\pi$
$139$	−21.4749	−1.82147	−0.910737	+	0.412987i	$0.864486\pi$
$140$	0	0
$141$	0	0
$142$	−8.71053	−0.730971
$143$	6.04528	0.505532
$144$	0	0
$145$	−8.60585	−0.714677
$146$	21.6069	1.78820
$147$	0	0
$148$	−34.0538	−2.79921
$149$	−15.7242	−1.28818	−0.644089	−	0.764950i	$-0.722763\pi$
$149$	−15.7242	−1.28818	−0.644089	+	0.764950i	$0.722763\pi$
$150$	0	0
$151$	11.5622	0.940918	0.470459	−	0.882422i	$-0.344088\pi$
$151$	11.5622	0.940918	0.470459	+	0.882422i	$0.344088\pi$
$152$	13.4139	1.08801
$153$	0	0
$154$	0	0
$155$	14.0845	1.13129
$156$	0	0
$157$	−1.07057	−0.0854406	−0.0427203	−	0.999087i	$-0.513602\pi$
$157$	−1.07057	−0.0854406	−0.0427203	+	0.999087i	$0.513602\pi$
$158$	28.3138	2.25253
$159$	0	0
$160$	24.7105	1.95354
$161$	0	0
$162$	0	0
$163$	20.9832	1.64353	0.821767	−	0.569823i	$-0.192988\pi$
$163$	20.9832	1.64353	0.821767	+	0.569823i	$0.192988\pi$
$164$	−39.5076	−3.08503
$165$	0	0
$166$	21.8349	1.69472
$167$	8.27427	0.640282	0.320141	−	0.947370i	$-0.396270\pi$
$167$	8.27427	0.640282	0.320141	+	0.947370i	$0.396270\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−25.3596	−1.94499
$171$	0	0
$172$	−8.05381	−0.614097
$173$	0.235068	0.0178719	0.00893594	−	0.999960i	$-0.497156\pi$
$173$	0.235068	0.0178719	0.00893594	+	0.999960i	$0.497156\pi$
$174$	0	0
$175$	0	0
$176$	−78.7513	−5.93610
$177$	0	0
$178$	−3.34328	−0.250590
$179$	0.580350	0.0433774	0.0216887	−	0.999765i	$-0.493096\pi$
$179$	0.580350	0.0433774	0.0216887	+	0.999765i	$0.493096\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	6.49162	0.478569
$185$	9.11858	0.670411
$186$	0	0
$187$	40.5454	2.96498
$188$	27.8148	2.02860
$189$	0	0
$190$	−5.80505	−0.421143
$191$	−20.2835	−1.46766	−0.733832	−	0.679331i	$-0.762270\pi$
$191$	−20.2835	−1.46766	−0.733832	+	0.679331i	$0.762270\pi$
$192$	0	0
$193$	−21.5622	−1.55208	−0.776040	−	0.630684i	$-0.782775\pi$
$193$	−21.5622	−1.55208	−0.776040	+	0.630684i	$0.782775\pi$
$194$	−1.25092	−0.0898111
$195$	0	0
$196$	0	0
$197$	10.1055	0.719990	0.359995	−	0.932954i	$-0.382778\pi$
$197$	10.1055	0.719990	0.359995	+	0.932954i	$0.382778\pi$
$198$	0	0
$199$	−0.983241	−0.0697001	−0.0348501	−	0.999393i	$-0.511095\pi$
$199$	−0.983241	−0.0697001	−0.0348501	+	0.999393i	$0.511095\pi$
$200$	26.4463	1.87004
$201$	0	0
$202$	39.4749	2.77744
$203$	0	0
$204$	0	0
$205$	10.5789	0.738866
$206$	−31.1231	−2.16845
$207$	0	0
$208$	−13.0269	−0.903253
$209$	9.28122	0.641996
$210$	0	0
$211$	−2.46472	−0.169678	−0.0848390	−	0.996395i	$-0.527038\pi$
$211$	−2.46472	−0.169678	−0.0848390	+	0.996395i	$0.527038\pi$
$212$	31.2858	2.14872
$213$	0	0
$214$	−14.0538	−0.960699
$215$	2.15657	0.147077
$216$	0	0
$217$	0	0
$218$	−26.9180	−1.82312
$219$	0	0
$220$	44.5454	3.00325
$221$	6.70695	0.451159
$222$	0	0
$223$	−2.46472	−0.165050	−0.0825248	−	0.996589i	$-0.526298\pi$
$223$	−2.46472	−0.165050	−0.0825248	+	0.996589i	$0.526298\pi$
$224$	0	0
$225$	0	0
$226$	16.4916	1.09701
$227$	−15.7242	−1.04365	−0.521827	−	0.853052i	$-0.674749\pi$
$227$	−15.7242	−1.04365	−0.521827	+	0.853052i	$0.674749\pi$
$228$	0	0
$229$	13.5622	0.896215	0.448107	−	0.893980i	$-0.352098\pi$
$229$	13.5622	0.896215	0.448107	+	0.893980i	$0.352098\pi$
$230$	−2.80934	−0.185242
$231$	0	0
$232$	−53.5287	−3.51433
$233$	−23.2031	−1.52008	−0.760042	−	0.649874i	$-0.774821\pi$
$233$	−23.2031	−1.52008	−0.760042	+	0.649874i	$0.774821\pi$
$234$	0	0
$235$	−7.44796	−0.485851
$236$	−53.0118	−3.45077
$237$	0	0
$238$	0	0
$239$	16.6499	1.07699	0.538495	−	0.842629i	$-0.318993\pi$
$239$	16.6499	1.07699	0.538495	+	0.842629i	$0.318993\pi$
$240$	0	0
$241$	24.0807	1.55118	0.775588	−	0.631240i	$-0.217454\pi$
$241$	24.0807	1.55118	0.775588	+	0.631240i	$0.217454\pi$
$242$	−68.7633	−4.42027
$243$	0	0
$244$	−52.4581	−3.35829
$245$	0	0
$246$	0	0
$247$	1.53528	0.0976878
$248$	87.6059	5.56298
$249$	0	0
$250$	−30.3505	−1.91953
$251$	−5.45603	−0.344382	−0.172191	−	0.985064i	$-0.555085\pi$
$251$	−5.45603	−0.344382	−0.172191	+	0.985064i	$0.555085\pi$
$252$	0	0
$253$	4.49162	0.282386
$254$	−10.7672	−0.675595
$255$	0	0
$256$	17.0269	1.06418
$257$	−3.89762	−0.243127	−0.121563	−	0.992584i	$-0.538791\pi$
$257$	−3.89762	−0.243127	−0.121563	+	0.992584i	$0.538791\pi$
$258$	0	0
$259$	0	0
$260$	7.36863	0.456983
$261$	0	0
$262$	−11.5622	−0.714314
$263$	4.87568	0.300647	0.150324	−	0.988637i	$-0.451968\pi$
$263$	4.87568	0.300647	0.150324	+	0.988637i	$0.451968\pi$
$264$	0	0
$265$	−8.37739	−0.514619
$266$	0	0
$267$	0	0
$268$	−23.5622	−1.43929
$269$	8.03030	0.489616	0.244808	−	0.969572i	$-0.421275\pi$
$269$	8.03030	0.489616	0.244808	+	0.969572i	$0.421275\pi$
$270$	0	0
$271$	−32.5454	−1.97699	−0.988497	−	0.151240i	$-0.951673\pi$
$271$	−32.5454	−1.97699	−0.988497	+	0.151240i	$0.951673\pi$
$272$	−87.3709	−5.29764
$273$	0	0
$274$	−5.78109	−0.349249
$275$	18.2985	1.10344
$276$	0	0
$277$	11.5353	0.693088	0.346544	−	0.938034i	$-0.387355\pi$
$277$	11.5353	0.693088	0.346544	+	0.938034i	$0.387355\pi$
$278$	57.8061	3.46698
$279$	0	0
$280$	0	0
$281$	−4.48687	−0.267664	−0.133832	−	0.991004i	$-0.542728\pi$
$281$	−4.48687	−0.267664	−0.133832	+	0.991004i	$0.542728\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	16.9751	1.00729
$285$	0	0
$286$	−16.2727	−0.962226
$287$	0	0
$288$	0	0
$289$	27.9832	1.64607
$290$	23.1653	1.36031
$291$	0	0
$292$	−42.1076	−2.46416
$293$	22.9391	1.34012	0.670058	−	0.742308i	$-0.266269\pi$
$293$	22.9391	1.34012	0.670058	+	0.742308i	$0.266269\pi$
$294$	0	0
$295$	14.1949	0.826462
$296$	56.7178	3.29666
$297$	0	0
$298$	42.3265	2.45191
$299$	0.742996	0.0429686
$300$	0	0
$301$	0	0
$302$	−31.1231	−1.79094
$303$	0	0
$304$	−20.0000	−1.14708
$305$	14.0467	0.804311
$306$	0	0
$307$	12.6059	0.719454	0.359727	−	0.933058i	$-0.382870\pi$
$307$	12.6059	0.719454	0.359727	+	0.933058i	$0.382870\pi$
$308$	0	0
$309$	0	0
$310$	−37.9127	−2.15330
$311$	18.7251	1.06180	0.530901	−	0.847434i	$-0.321853\pi$
$311$	18.7251	1.06180	0.530901	+	0.847434i	$0.321853\pi$
$312$	0	0
$313$	−30.9832	−1.75128	−0.875638	−	0.482968i	$-0.839559\pi$
$313$	−30.9832	−1.75128	−0.875638	+	0.482968i	$0.839559\pi$
$314$	2.88176	0.162627
$315$	0	0
$316$	−55.1782	−3.10402
$317$	−5.11965	−0.287548	−0.143774	−	0.989611i	$-0.545924\pi$
$317$	−5.11965	−0.287548	−0.143774	+	0.989611i	$0.545924\pi$
$318$	0	0
$319$	−37.0371	−2.07368
$320$	−29.9189	−1.67252
$321$	0	0
$322$	0	0
$323$	10.2971	0.572945
$324$	0	0
$325$	3.02690	0.167902
$326$	−56.4828	−3.12829
$327$	0	0
$328$	65.8014	3.63327
$329$	0	0
$330$	0	0
$331$	−16.0538	−0.882397	−0.441199	−	0.897410i	$-0.645447\pi$
$331$	−16.0538	−0.882397	−0.441199	+	0.897410i	$0.645447\pi$
$332$	−42.5520	−2.33535
$333$	0	0
$334$	−22.2727	−1.21871
$335$	6.30924	0.344711
$336$	0	0
$337$	2.11423	0.115170	0.0575848	−	0.998341i	$-0.481660\pi$
$337$	2.11423	0.115170	0.0575848	+	0.998341i	$0.481660\pi$
$338$	−2.69180	−0.146415
$339$	0	0
$340$	49.4211	2.68023
$341$	60.6155	3.28251
$342$	0	0
$343$	0	0
$344$	13.4139	0.723230
$345$	0	0
$346$	−0.632757	−0.0340172
$347$	−9.51629	−0.510861	−0.255431	−	0.966827i	$-0.582217\pi$
$347$	−9.51629	−0.510861	−0.255431	+	0.966827i	$0.582217\pi$
$348$	0	0
$349$	−30.4312	−1.62894	−0.814472	−	0.580202i	$-0.802973\pi$
$349$	−30.4312	−1.62894	−0.814472	+	0.580202i	$0.802973\pi$
$350$	0	0
$351$	0	0
$352$	106.347	5.66830
$353$	−33.8890	−1.80373	−0.901864	−	0.432021i	$-0.857801\pi$
$353$	−33.8890	−1.80373	−0.901864	+	0.432021i	$0.857801\pi$
$354$	0	0
$355$	−4.54543	−0.241246
$356$	6.51542	0.345316
$357$	0	0
$358$	−1.56219	−0.0825642
$359$	−9.70783	−0.512360	−0.256180	−	0.966629i	$-0.582464\pi$
$359$	−9.70783	−0.512360	−0.256180	+	0.966629i	$0.582464\pi$
$360$	0	0
$361$	−16.6429	−0.875942
$362$	26.9180	1.41478
$363$	0	0
$364$	0	0
$365$	11.2751	0.590168
$366$	0	0
$367$	20.4916	1.06965	0.534827	−	0.844962i	$-0.320377\pi$
$367$	20.4916	1.06965	0.534827	+	0.844962i	$0.320377\pi$
$368$	−9.67894	−0.504550
$369$	0	0
$370$	−24.5454	−1.27606
$371$	0	0
$372$	0	0
$373$	−1.56219	−0.0808871	−0.0404435	−	0.999182i	$-0.512877\pi$
$373$	−1.56219	−0.0808871	−0.0404435	+	0.999182i	$0.512877\pi$
$374$	−109.140	−5.64351
$375$	0	0
$376$	−46.3265	−2.38911
$377$	−6.12660	−0.315536
$378$	0	0
$379$	−28.5992	−1.46904	−0.734522	−	0.678585i	$-0.762594\pi$
$379$	−28.5992	−1.46904	−0.734522	+	0.678585i	$0.762594\pi$
$380$	11.3129	0.580341
$381$	0	0
$382$	54.5992	2.79354
$383$	−2.31031	−0.118051	−0.0590257	−	0.998256i	$-0.518799\pi$
$383$	−2.31031	−0.118051	−0.0590257	+	0.998256i	$0.518799\pi$
$384$	0	0
$385$	0	0
$386$	58.0412	2.95422
$387$	0	0
$388$	2.43781	0.123761
$389$	21.5344	1.09184	0.545920	−	0.837838i	$-0.316180\pi$
$389$	21.5344	1.09184	0.545920	+	0.837838i	$0.316180\pi$
$390$	0	0
$391$	4.98324	0.252013
$392$	0	0
$393$	0	0
$394$	−27.2021	−1.37042
$395$	14.7750	0.743413
$396$	0	0
$397$	13.0101	0.652960	0.326480	−	0.945204i	$-0.394137\pi$
$397$	13.0101	0.652960	0.326480	+	0.945204i	$0.394137\pi$
$398$	2.64669	0.132667
$399$	0	0
$400$	−39.4312	−1.97156
$401$	−5.11965	−0.255663	−0.127832	−	0.991796i	$-0.540802\pi$
$401$	−5.11965	−0.255663	−0.127832	+	0.991796i	$0.540802\pi$
$402$	0	0
$403$	10.0269	0.499476
$404$	−76.9289	−3.82736
$405$	0	0
$406$	0	0
$407$	39.2437	1.94524
$408$	0	0
$409$	−16.5185	−0.816788	−0.408394	−	0.912806i	$-0.633911\pi$
$409$	−16.5185	−0.816788	−0.408394	+	0.912806i	$0.633911\pi$
$410$	−28.4765	−1.40635
$411$	0	0
$412$	60.6530	2.98816
$413$	0	0
$414$	0	0
$415$	11.3941	0.559317
$416$	17.5917	0.862504
$417$	0	0
$418$	−24.9832	−1.22197
$419$	−18.3998	−0.898889	−0.449445	−	0.893308i	$-0.648378\pi$
$419$	−18.3998	−0.898889	−0.449445	+	0.893308i	$0.648378\pi$
$420$	0	0
$421$	−35.4749	−1.72894	−0.864469	−	0.502685i	$-0.832345\pi$
$421$	−35.4749	−1.72894	−0.864469	+	0.502685i	$0.832345\pi$
$422$	6.63453	0.322964
$423$	0	0
$424$	−52.1076	−2.53057
$425$	20.3013	0.984758
$426$	0	0
$427$	0	0
$428$	27.3882	1.32386
$429$	0	0
$430$	−5.80505	−0.279945
$431$	−37.1684	−1.79034	−0.895170	−	0.445725i	$-0.852946\pi$
$431$	−37.1684	−1.79034	−0.895170	+	0.445725i	$0.852946\pi$
$432$	0	0
$433$	−1.07057	−0.0514482	−0.0257241	−	0.999669i	$-0.508189\pi$
$433$	−1.07057	−0.0514482	−0.0257241	+	0.999669i	$0.508189\pi$
$434$	0	0
$435$	0	0
$436$	52.4581	2.51229
$437$	1.14071	0.0545676
$438$	0	0
$439$	13.4749	0.643120	0.321560	−	0.946889i	$-0.395793\pi$
$439$	13.4749	0.643120	0.321560	+	0.946889i	$0.395793\pi$
$440$	−74.1920	−3.53697
$441$	0	0
$442$	−18.0538	−0.858732
$443$	−22.9680	−1.09124	−0.545621	−	0.838032i	$-0.683706\pi$
$443$	−22.9680	−1.09124	−0.545621	+	0.838032i	$0.683706\pi$
$444$	0	0
$445$	−1.74463	−0.0827035
$446$	6.63453	0.314154
$447$	0	0
$448$	0	0
$449$	−12.5896	−0.594139	−0.297070	−	0.954856i	$-0.596009\pi$
$449$	−12.5896	−0.594139	−0.297070	+	0.954856i	$0.596009\pi$
$450$	0	0
$451$	45.5287	2.14386
$452$	−32.1390	−1.51169
$453$	0	0
$454$	42.3265	1.98648
$455$	0	0
$456$	0	0
$457$	−40.0203	−1.87207	−0.936035	−	0.351907i	$-0.885533\pi$
$457$	−40.0203	−1.87207	−0.936035	+	0.351907i	$0.885533\pi$
$458$	−36.5068	−1.70585
$459$	0	0
$460$	5.47486	0.255267
$461$	7.20598	0.335616	0.167808	−	0.985820i	$-0.446331\pi$
$461$	7.20598	0.335616	0.167808	+	0.985820i	$0.446331\pi$
$462$	0	0
$463$	0.491620	0.0228475	0.0114238	−	0.999935i	$-0.496364\pi$
$463$	0.491620	0.0228475	0.0114238	+	0.999935i	$0.496364\pi$
$464$	79.8107	3.70512
$465$	0	0
$466$	62.4581	2.89332
$467$	21.3718	0.988968	0.494484	−	0.869187i	$-0.335357\pi$
$467$	21.3718	0.988968	0.494484	+	0.869187i	$0.335357\pi$
$468$	0	0
$469$	0	0
$470$	20.0484	0.924766
$471$	0	0
$472$	88.2930	4.06402
$473$	9.28122	0.426751
$474$	0	0
$475$	4.64716	0.213226
$476$	0	0
$477$	0	0
$478$	−44.8181	−2.04993
$479$	24.4975	1.11932	0.559660	−	0.828722i	$-0.310932\pi$
$479$	24.4975	1.11932	0.559660	+	0.828722i	$0.310932\pi$
$480$	0	0
$481$	6.49162	0.295992
$482$	−64.8206	−2.95250
$483$	0	0
$484$	134.006	6.09120
$485$	−0.652772	−0.0296409
$486$	0	0
$487$	17.4749	0.791862	0.395931	−	0.918280i	$-0.370422\pi$
$487$	17.4749	0.791862	0.395931	+	0.918280i	$0.370422\pi$
$488$	87.3709	3.95509
$489$	0	0
$490$	0	0
$491$	−26.7554	−1.20745	−0.603727	−	0.797191i	$-0.706318\pi$
$491$	−26.7554	−1.20745	−0.603727	+	0.797191i	$0.706318\pi$
$492$	0	0
$493$	−41.0909	−1.85064
$494$	−4.13268	−0.185938
$495$	0	0
$496$	−130.620	−5.86499
$497$	0	0
$498$	0	0
$499$	2.14114	0.0958504	0.0479252	−	0.998851i	$-0.484739\pi$
$499$	2.14114	0.0958504	0.0479252	+	0.998851i	$0.484739\pi$
$500$	59.1473	2.64515
$501$	0	0
$502$	14.6866	0.655494
$503$	37.7577	1.68353	0.841766	−	0.539843i	$-0.181516\pi$
$503$	37.7577	1.68353	0.841766	+	0.539843i	$0.181516\pi$
$504$	0	0
$505$	20.5992	0.916654
$506$	−12.0906	−0.537491
$507$	0	0
$508$	20.9832	0.930981
$509$	−11.0112	−0.488062	−0.244031	−	0.969767i	$-0.578470\pi$
$509$	−11.0112	−0.488062	−0.244031	+	0.969767i	$0.578470\pi$
$510$	0	0
$511$	0	0
$512$	−1.53110	−0.0676659
$513$	0	0
$514$	10.4916	0.462766
$515$	−16.2410	−0.715666
$516$	0	0
$517$	−32.0538	−1.40972
$518$	0	0
$519$	0	0
$520$	−12.2727	−0.538194
$521$	24.7415	1.08394	0.541972	−	0.840396i	$-0.317678\pi$
$521$	24.7415	1.08394	0.541972	+	0.840396i	$0.317678\pi$
$522$	0	0
$523$	11.5084	0.503226	0.251613	−	0.967828i	$-0.419039\pi$
$523$	11.5084	0.503226	0.251613	+	0.967828i	$0.419039\pi$
$524$	22.5325	0.984336
$525$	0	0
$526$	−13.1244	−0.572250
$527$	67.2500	2.92946
$528$	0	0
$529$	−22.4480	−0.975998
$530$	22.5503	0.979522
$531$	0	0
$532$	0	0
$533$	7.53127	0.326216
$534$	0	0
$535$	−7.33372	−0.317065
$536$	39.2437	1.69507
$537$	0	0
$538$	−21.6160	−0.931932
$539$	0	0
$540$	0	0
$541$	38.4916	1.65488	0.827442	−	0.561551i	$-0.189795\pi$
$541$	38.4916	1.65488	0.827442	+	0.561551i	$0.189795\pi$
$542$	87.6059	3.76300
$543$	0	0
$544$	117.987	5.05864
$545$	−14.0467	−0.601694
$546$	0	0
$547$	40.4312	1.72871	0.864357	−	0.502879i	$-0.167726\pi$
$547$	40.4312	1.72871	0.864357	+	0.502879i	$0.167726\pi$
$548$	11.2662	0.481270
$549$	0	0
$550$	−49.2560	−2.10028
$551$	−9.40608	−0.400712
$552$	0	0
$553$	0	0
$554$	−31.0507	−1.31922
$555$	0	0
$556$	−112.653	−4.77755
$557$	26.0035	1.10180	0.550902	−	0.834570i	$-0.314284\pi$
$557$	26.0035	1.10180	0.550902	+	0.834570i	$0.314284\pi$
$558$	0	0
$559$	1.53528	0.0649356
$560$	0	0
$561$	0	0
$562$	12.0778	0.509470
$563$	−37.9203	−1.59815	−0.799076	−	0.601231i	$-0.794677\pi$
$563$	−37.9203	−1.59815	−0.799076	+	0.601231i	$0.794677\pi$
$564$	0	0
$565$	8.60585	0.362051
$566$	−10.7672	−0.452580
$567$	0	0
$568$	−28.2727	−1.18630
$569$	−12.9060	−0.541047	−0.270523	−	0.962713i	$-0.587197\pi$
$569$	−12.9060	−0.541047	−0.270523	+	0.962713i	$0.587197\pi$
$570$	0	0
$571$	−6.51853	−0.272792	−0.136396	−	0.990654i	$-0.543552\pi$
$571$	−6.51853	−0.272792	−0.136396	+	0.990654i	$0.543552\pi$
$572$	31.7124	1.32596
$573$	0	0
$574$	0	0
$575$	2.24898	0.0937889
$576$	0	0
$577$	6.43781	0.268010	0.134005	−	0.990981i	$-0.457216\pi$
$577$	6.43781	0.268010	0.134005	+	0.990981i	$0.457216\pi$
$578$	−75.3254	−3.13312
$579$	0	0
$580$	−45.1447	−1.87453
$581$	0	0
$582$	0	0
$583$	−36.0538	−1.49320
$584$	70.1318	2.90207
$585$	0	0
$586$	−61.7476	−2.55077
$587$	−1.00696	−0.0415615	−0.0207807	−	0.999784i	$-0.506615\pi$
$587$	−1.00696	−0.0415615	−0.0207807	+	0.999784i	$0.506615\pi$
$588$	0	0
$589$	15.3941	0.634305
$590$	−38.2100	−1.57308
$591$	0	0
$592$	−84.5657	−3.47563
$593$	−21.1278	−0.867616	−0.433808	−	0.901005i	$-0.642830\pi$
$593$	−21.1278	−0.867616	−0.433808	+	0.901005i	$0.642830\pi$
$594$	0	0
$595$	0	0
$596$	−82.4863	−3.37877
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	37.8679	1.54724	0.773620	−	0.633650i	$-0.218444\pi$
$599$	37.8679	1.54724	0.773620	+	0.633650i	$0.218444\pi$
$600$	0	0
$601$	44.9497	1.83354	0.916769	−	0.399418i	$-0.130788\pi$
$601$	44.9497	1.83354	0.916769	+	0.399418i	$0.130788\pi$
$602$	0	0
$603$	0	0
$604$	60.6530	2.46794
$605$	−35.8829	−1.45885
$606$	0	0
$607$	26.6328	1.08099	0.540495	−	0.841347i	$-0.318237\pi$
$607$	26.6328	1.08099	0.540495	+	0.841347i	$0.318237\pi$
$608$	27.0083	1.09533
$609$	0	0
$610$	−37.8109	−1.53092
$611$	−5.30229	−0.214508
$612$	0	0
$613$	35.0371	1.41513	0.707567	−	0.706647i	$-0.249793\pi$
$613$	35.0371	1.41513	0.707567	+	0.706647i	$0.249793\pi$
$614$	−33.9325	−1.36940
$615$	0	0
$616$	0	0
$617$	41.5540	1.67290	0.836450	−	0.548043i	$-0.184627\pi$
$617$	41.5540	1.67290	0.836450	+	0.548043i	$0.184627\pi$
$618$	0	0
$619$	−5.42105	−0.217890	−0.108945	−	0.994048i	$-0.534747\pi$
$619$	−5.42105	−0.217890	−0.108945	+	0.994048i	$0.534747\pi$
$620$	73.8845	2.96727
$621$	0	0
$622$	−50.4043	−2.02103
$623$	0	0
$624$	0	0
$625$	−0.703325	−0.0281330
$626$	83.4008	3.33337
$627$	0	0
$628$	−5.61600	−0.224103
$629$	43.5390	1.73601
$630$	0	0
$631$	22.8959	0.911472	0.455736	−	0.890115i	$-0.349376\pi$
$631$	22.8959	0.911472	0.455736	+	0.890115i	$0.349376\pi$
$632$	91.9013	3.65564
$633$	0	0
$634$	13.7811	0.547317
$635$	−5.61868	−0.222970
$636$	0	0
$637$	0	0
$638$	99.6965	3.94702
$639$	0	0
$640$	31.1148	1.22992
$641$	12.2732	0.484762	0.242381	−	0.970181i	$-0.422072\pi$
$641$	12.2732	0.484762	0.242381	+	0.970181i	$0.422072\pi$
$642$	0	0
$643$	19.0168	0.749948	0.374974	−	0.927035i	$-0.377652\pi$
$643$	19.0168	0.749948	0.374974	+	0.927035i	$0.377652\pi$
$644$	0	0
$645$	0	0
$646$	−27.7177	−1.09054
$647$	−1.48599	−0.0584204	−0.0292102	−	0.999573i	$-0.509299\pi$
$647$	−1.48599	−0.0584204	−0.0292102	+	0.999573i	$0.509299\pi$
$648$	0	0
$649$	61.0909	2.39803
$650$	−8.14783	−0.319584
$651$	0	0
$652$	110.074	4.31083
$653$	−39.7138	−1.55412	−0.777061	−	0.629426i	$-0.783290\pi$
$653$	−39.7138	−1.55412	−0.777061	+	0.629426i	$0.783290\pi$
$654$	0	0
$655$	−6.03352	−0.235749
$656$	−98.1092	−3.83052
$657$	0	0
$658$	0	0
$659$	−0.580350	−0.0226072	−0.0113036	−	0.999936i	$-0.503598\pi$
$659$	−0.580350	−0.0226072	−0.0113036	+	0.999936i	$0.503598\pi$
$660$	0	0
$661$	35.6429	1.38635	0.693174	−	0.720770i	$-0.256212\pi$
$661$	35.6429	1.38635	0.693174	+	0.720770i	$0.256212\pi$
$662$	43.2137	1.67955
$663$	0	0
$664$	70.8720	2.75037
$665$	0	0
$666$	0	0
$667$	−4.55204	−0.176256
$668$	43.4052	1.67940
$669$	0	0
$670$	−16.9832	−0.656120
$671$	60.4528	2.33376
$672$	0	0
$673$	−16.5185	−0.636742	−0.318371	−	0.947966i	$-0.603136\pi$
$673$	−16.5185	−0.636742	−0.318371	+	0.947966i	$0.603136\pi$
$674$	−5.69110	−0.219213
$675$	0	0
$676$	5.24581	0.201762
$677$	16.4583	0.632544	0.316272	−	0.948668i	$-0.397569\pi$
$677$	16.4583	0.632544	0.316272	+	0.948668i	$0.397569\pi$
$678$	0	0
$679$	0	0
$680$	−82.3125	−3.15654
$681$	0	0
$682$	−163.165	−6.24791
$683$	35.5376	1.35981	0.679904	−	0.733301i	$-0.262021\pi$
$683$	35.5376	1.35981	0.679904	+	0.733301i	$0.262021\pi$
$684$	0	0
$685$	−3.01676	−0.115264
$686$	0	0
$687$	0	0
$688$	−20.0000	−0.762493
$689$	−5.96396	−0.227209
$690$	0	0
$691$	18.9563	0.721133	0.360567	−	0.932734i	$-0.382583\pi$
$691$	18.9563	0.721133	0.360567	+	0.932734i	$0.382583\pi$
$692$	1.23312	0.0468763
$693$	0	0
$694$	25.6160	0.972370
$695$	30.1651	1.14423
$696$	0	0
$697$	50.5119	1.91328
$698$	81.9148	3.10052
$699$	0	0
$700$	0	0
$701$	16.4059	0.619642	0.309821	−	0.950795i	$-0.399731\pi$
$701$	16.4059	0.619642	0.309821	+	0.950795i	$0.399731\pi$
$702$	0	0
$703$	9.96648	0.375893
$704$	−128.762	−4.85291
$705$	0	0
$706$	91.2224	3.43320
$707$	0	0
$708$	0	0
$709$	16.1411	0.606193	0.303097	−	0.952960i	$-0.401980\pi$
$709$	16.1411	0.606193	0.303097	+	0.952960i	$0.401980\pi$
$710$	12.2354	0.459187
$711$	0	0
$712$	−10.8517	−0.406683
$713$	7.44995	0.279003
$714$	0	0
$715$	−8.49162	−0.317569
$716$	3.04441	0.113775
$717$	0	0
$718$	26.1316	0.975222
$719$	10.9299	0.407615	0.203808	−	0.979011i	$-0.434668\pi$
$719$	10.9299	0.407615	0.203808	+	0.979011i	$0.434668\pi$
$720$	0	0
$721$	0	0
$722$	44.7994	1.66726
$723$	0	0
$724$	−52.4581	−1.94959
$725$	−18.5446	−0.688731
$726$	0	0
$727$	8.92943	0.331174	0.165587	−	0.986195i	$-0.447048\pi$
$727$	8.92943	0.331174	0.165587	+	0.986195i	$0.447048\pi$
$728$	0	0
$729$	0	0
$730$	−30.3505	−1.12332
$731$	10.2971	0.380851
$732$	0	0
$733$	−21.0101	−0.776027	−0.388014	−	0.921654i	$-0.626839\pi$
$733$	−21.0101	−0.776027	−0.388014	+	0.921654i	$0.626839\pi$
$734$	−55.1594	−2.03597
$735$	0	0
$736$	13.0706	0.481788
$737$	27.1531	1.00020
$738$	0	0
$739$	11.6160	0.427301	0.213651	−	0.976910i	$-0.431465\pi$
$739$	11.6160	0.427301	0.213651	+	0.976910i	$0.431465\pi$
$740$	47.8343	1.75843
$741$	0	0
$742$	0	0
$743$	11.8266	0.433876	0.216938	−	0.976185i	$-0.430393\pi$
$743$	11.8266	0.433876	0.216938	+	0.976185i	$0.430393\pi$
$744$	0	0
$745$	22.0873	0.809217
$746$	4.20511	0.153960
$747$	0	0
$748$	212.694	7.77685
$749$	0	0
$750$	0	0
$751$	−15.3941	−0.561740	−0.280870	−	0.959746i	$-0.590623\pi$
$751$	−15.3941	−0.561740	−0.280870	+	0.959746i	$0.590623\pi$
$752$	69.0724	2.51881
$753$	0	0
$754$	16.4916	0.600589
$755$	−16.2410	−0.591072
$756$	0	0
$757$	2.81520	0.102320	0.0511601	−	0.998690i	$-0.483708\pi$
$757$	2.81520	0.102320	0.0511601	+	0.998690i	$0.483708\pi$
$758$	76.9836	2.79617
$759$	0	0
$760$	−18.8421	−0.683475
$761$	−35.4998	−1.28687	−0.643433	−	0.765502i	$-0.722491\pi$
$761$	−35.4998	−1.28687	−0.643433	+	0.765502i	$0.722491\pi$
$762$	0	0
$763$	0	0
$764$	−106.403	−3.84954
$765$	0	0
$766$	6.21891	0.224698
$767$	10.1055	0.364890
$768$	0	0
$769$	−4.41091	−0.159061	−0.0795307	−	0.996832i	$-0.525342\pi$
$769$	−4.41091	−0.159061	−0.0795307	+	0.996832i	$0.525342\pi$
$770$	0	0
$771$	0	0
$772$	−113.111	−4.07096
$773$	−2.38273	−0.0857010	−0.0428505	−	0.999081i	$-0.513644\pi$
$773$	−2.38273	−0.0857010	−0.0428505	+	0.999081i	$0.513644\pi$
$774$	0	0
$775$	30.3505	1.09022
$776$	−4.06026	−0.145755
$777$	0	0
$778$	−57.9665	−2.07820
$779$	11.5626	0.414275
$780$	0	0
$781$	−19.5622	−0.699990
$782$	−13.4139	−0.479680
$783$	0	0
$784$	0	0
$785$	1.50379	0.0536727
$786$	0	0
$787$	−8.11423	−0.289241	−0.144621	−	0.989487i	$-0.546196\pi$
$787$	−8.11423	−0.289241	−0.144621	+	0.989487i	$0.546196\pi$
$788$	53.0118	1.88847
$789$	0	0
$790$	−39.7715	−1.41501
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	−35.0208	−1.24284
$795$	0	0
$796$	−5.15790	−0.182817
$797$	13.4863	0.477710	0.238855	−	0.971055i	$-0.423228\pi$
$797$	13.4863	0.477710	0.238855	+	0.971055i	$0.423228\pi$
$798$	0	0
$799$	−35.5622	−1.25810
$800$	53.2484	1.88262
$801$	0	0
$802$	13.7811	0.486627
$803$	48.5249	1.71241
$804$	0	0
$805$	0	0
$806$	−26.9905	−0.950699
$807$	0	0
$808$	128.128	4.50752
$809$	−17.7292	−0.623327	−0.311663	−	0.950193i	$-0.600886\pi$
$809$	−17.7292	−0.623327	−0.311663	+	0.950193i	$0.600886\pi$
$810$	0	0
$811$	−5.42105	−0.190359	−0.0951794	−	0.995460i	$-0.530342\pi$
$811$	−5.42105	−0.190359	−0.0951794	+	0.995460i	$0.530342\pi$
$812$	0	0
$813$	0	0
$814$	−105.636	−3.70255
$815$	−29.4745	−1.03245
$816$	0	0
$817$	2.35710	0.0824644
$818$	44.4646	1.55467
$819$	0	0
$820$	55.4952	1.93797
$821$	−31.6400	−1.10424	−0.552121	−	0.833764i	$-0.686182\pi$
$821$	−31.6400	−1.10424	−0.552121	+	0.833764i	$0.686182\pi$
$822$	0	0
$823$	−39.1244	−1.36379	−0.681895	−	0.731450i	$-0.738844\pi$
$823$	−39.1244	−1.36379	−0.681895	+	0.731450i	$0.738844\pi$
$824$	−101.020	−3.51919
$825$	0	0
$826$	0	0
$827$	−23.2666	−0.809058	−0.404529	−	0.914525i	$-0.632565\pi$
$827$	−23.2666	−0.809058	−0.404529	+	0.914525i	$0.632565\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	−30.6708	−1.06460
$831$	0	0
$832$	−21.2996	−0.738431
$833$	0	0
$834$	0	0
$835$	−11.6226	−0.402217
$836$	48.6875	1.68389
$837$	0	0
$838$	49.5287	1.71094
$839$	−37.2587	−1.28631	−0.643156	−	0.765735i	$-0.722375\pi$
$839$	−37.2587	−1.28631	−0.643156	+	0.765735i	$0.722375\pi$
$840$	0	0
$841$	8.53528	0.294320
$842$	95.4914	3.29085
$843$	0	0
$844$	−12.9294	−0.445049
$845$	−1.40467	−0.0483221
$846$	0	0
$847$	0	0
$848$	77.6919	2.66795
$849$	0	0
$850$	−54.6472	−1.87438
$851$	4.82325	0.165339
$852$	0	0
$853$	−32.0269	−1.09658	−0.548290	−	0.836288i	$-0.684721\pi$
$853$	−32.0269	−1.09658	−0.548290	+	0.836288i	$0.684721\pi$
$854$	0	0
$855$	0	0
$856$	−45.6160	−1.55912
$857$	9.98643	0.341130	0.170565	−	0.985346i	$-0.445441\pi$
$857$	9.98643	0.341130	0.170565	+	0.985346i	$0.445441\pi$
$858$	0	0
$859$	−48.1614	−1.64325	−0.821623	−	0.570031i	$-0.806931\pi$
$859$	−48.1614	−1.64325	−0.821623	+	0.570031i	$0.806931\pi$
$860$	11.3129	0.385768
$861$	0	0
$862$	100.050	3.40772
$863$	17.9732	0.611815	0.305907	−	0.952061i	$-0.401040\pi$
$863$	17.9732	0.611815	0.305907	+	0.952061i	$0.401040\pi$
$864$	0	0
$865$	−0.330193	−0.0112269
$866$	2.88176	0.0979262
$867$	0	0
$868$	0	0
$869$	63.5874	2.15706
$870$	0	0
$871$	4.49162	0.152193
$872$	−87.3709	−2.95875
$873$	0	0
$874$	−3.07057	−0.103864
$875$	0	0
$876$	0	0
$877$	18.9294	0.639201	0.319601	−	0.947552i	$-0.396451\pi$
$877$	18.9294	0.639201	0.319601	+	0.947552i	$0.396451\pi$
$878$	−36.2717	−1.22411
$879$	0	0
$880$	110.620	3.72898
$881$	−38.5207	−1.29779	−0.648897	−	0.760876i	$-0.724769\pi$
$881$	−38.5207	−1.29779	−0.648897	+	0.760876i	$0.724769\pi$
$882$	0	0
$883$	22.1411	0.745109	0.372554	−	0.928010i	$-0.378482\pi$
$883$	22.1411	0.745109	0.372554	+	0.928010i	$0.378482\pi$
$884$	35.1834	1.18335
$885$	0	0
$886$	61.8253	2.07706
$887$	−27.6232	−0.927498	−0.463749	−	0.885967i	$-0.653496\pi$
$887$	−27.6232	−0.927498	−0.463749	+	0.885967i	$0.653496\pi$
$888$	0	0
$889$	0	0
$890$	4.69621	0.157417
$891$	0	0
$892$	−12.9294	−0.432909
$893$	−8.14051	−0.272412
$894$	0	0
$895$	−0.815200	−0.0272491
$896$	0	0
$897$	0	0
$898$	33.8887	1.13088
$899$	−61.4309	−2.04883
$900$	0	0
$901$	−40.0000	−1.33259
$902$	−122.554	−4.08061
$903$	0	0
$904$	53.5287	1.78034
$905$	14.0467	0.466928
$906$	0	0
$907$	−39.7302	−1.31922	−0.659610	−	0.751608i	$-0.729279\pi$
$907$	−39.7302	−1.31922	−0.659610	+	0.751608i	$0.729279\pi$
$908$	−82.4863	−2.73740
$909$	0	0
$910$	0	0
$911$	17.2915	0.572894	0.286447	−	0.958096i	$-0.407526\pi$
$911$	17.2915	0.572894	0.286447	+	0.958096i	$0.407526\pi$
$912$	0	0
$913$	49.0371	1.62289
$914$	107.727	3.56329
$915$	0	0
$916$	71.1447	2.35069
$917$	0	0
$918$	0	0
$919$	34.1411	1.12621	0.563106	−	0.826385i	$-0.309606\pi$
$919$	34.1411	1.12621	0.563106	+	0.826385i	$0.309606\pi$
$920$	−9.11858	−0.300631
$921$	0	0
$922$	−19.3971	−0.638809
$923$	−3.23594	−0.106512
$924$	0	0
$925$	19.6495	0.646072
$926$	−1.32335	−0.0434879
$927$	0	0
$928$	−107.777	−3.53797
$929$	−40.1782	−1.31820	−0.659102	−	0.752053i	$-0.729064\pi$
$929$	−40.1782	−1.31820	−0.659102	+	0.752053i	$0.729064\pi$
$930$	0	0
$931$	0	0
$932$	−121.719	−3.98703
$933$	0	0
$934$	−57.5287	−1.88240
$935$	−56.9529	−1.86256
$936$	0	0
$937$	8.89591	0.290617	0.145308	−	0.989386i	$-0.453583\pi$
$937$	8.89591	0.290617	0.145308	+	0.989386i	$0.453583\pi$
$938$	0	0
$939$	0	0
$940$	−39.0706	−1.27434
$941$	−10.3784	−0.338326	−0.169163	−	0.985588i	$-0.554106\pi$
$941$	−10.3784	−0.338326	−0.169163	+	0.985588i	$0.554106\pi$
$942$	0	0
$943$	5.59571	0.182221
$944$	−131.644	−4.28465
$945$	0	0
$946$	−24.9832	−0.812275
$947$	−7.36863	−0.239448	−0.119724	−	0.992807i	$-0.538201\pi$
$947$	−7.36863	−0.239448	−0.119724	+	0.992807i	$0.538201\pi$
$948$	0	0
$949$	8.02690	0.260564
$950$	−12.5092	−0.405853
$951$	0	0
$952$	0	0
$953$	11.7453	0.380467	0.190233	−	0.981739i	$-0.439075\pi$
$953$	11.7453	0.380467	0.190233	+	0.981739i	$0.439075\pi$
$954$	0	0
$955$	28.4916	0.921967
$956$	87.3420	2.82484
$957$	0	0
$958$	−65.9425	−2.13051
$959$	0	0
$960$	0	0
$961$	69.5388	2.24319
$962$	−17.4742	−0.563390
$963$	0	0
$964$	126.323	4.06859
$965$	30.2877	0.974997
$966$	0	0
$967$	28.0538	0.902150	0.451075	−	0.892486i	$-0.351041\pi$
$967$	28.0538	0.902150	0.451075	+	0.892486i	$0.351041\pi$
$968$	−223.193	−7.17368
$969$	0	0
$970$	1.75713	0.0564182
$971$	29.1092	0.934160	0.467080	−	0.884215i	$-0.345306\pi$
$971$	29.1092	0.934160	0.467080	+	0.884215i	$0.345306\pi$
$972$	0	0
$973$	0	0
$974$	−47.0389	−1.50722
$975$	0	0
$976$	−130.269	−4.16981
$977$	−2.83823	−0.0908030	−0.0454015	−	0.998969i	$-0.514457\pi$
$977$	−2.83823	−0.0908030	−0.0454015	+	0.998969i	$0.514457\pi$
$978$	0	0
$979$	−7.50838	−0.239969
$980$	0	0
$981$	0	0
$982$	72.0203	2.29826
$983$	−36.5881	−1.16698	−0.583489	−	0.812121i	$-0.698313\pi$
$983$	−36.5881	−1.16698	−0.583489	+	0.812121i	$0.698313\pi$
$984$	0	0
$985$	−14.1949	−0.452289
$986$	110.609	3.52249
$987$	0	0
$988$	8.05381	0.256226
$989$	1.14071	0.0362725
$990$	0	0
$991$	−32.9832	−1.04775	−0.523874	−	0.851796i	$-0.675514\pi$
$991$	−32.9832	−1.04775	−0.523874	+	0.851796i	$0.675514\pi$
$992$	176.390	5.60040
$993$	0	0
$994$	0	0
$995$	1.38113	0.0437847
$996$	0	0
$997$	−60.0203	−1.90086	−0.950431	−	0.310936i	$-0.899358\pi$
$997$	−60.0203	−1.90086	−0.950431	+	0.310936i	$0.899358\pi$
$998$	−5.76352	−0.182441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bv.1.1		6
3.2	odd	2	inner	5733.2.a.bv.1.6		6
7.6	odd	2		819.2.a.m.1.1	✓	6
21.20	even	2		819.2.a.m.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.a.m.1.1	✓	6	7.6	odd	2
819.2.a.m.1.6	yes	6	21.20	even	2
5733.2.a.bv.1.1		6	1.1	even	1	trivial
5733.2.a.bv.1.6		6	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69180 q^{2} +5.24581 q^{4} -1.40467 q^{5} -8.73709 q^{8} +O(q^{10})\)	\(q-2.69180 q^{2} +5.24581 q^{4} -1.40467 q^{5} -8.73709 q^{8} +3.78109 q^{10} -6.04528 q^{11} -1.00000 q^{13} +13.0269 q^{16} -6.70695 q^{17} -1.53528 q^{19} -7.36863 q^{20} +16.2727 q^{22} -0.742996 q^{23} -3.02690 q^{25} +2.69180 q^{26} +6.12660 q^{29} -10.0269 q^{31} -17.5917 q^{32} +18.0538 q^{34} -6.49162 q^{37} +4.13268 q^{38} +12.2727 q^{40} -7.53127 q^{41} -1.53528 q^{43} -31.7124 q^{44} +2.00000 q^{46} +5.30229 q^{47} +8.14783 q^{50} -5.24581 q^{52} +5.96396 q^{53} +8.49162 q^{55} -16.4916 q^{58} -10.1055 q^{59} -10.0000 q^{61} +26.9905 q^{62} +21.2996 q^{64} +1.40467 q^{65} -4.49162 q^{67} -35.1834 q^{68} +3.23594 q^{71} -8.02690 q^{73} +17.4742 q^{74} -8.05381 q^{76} -10.5185 q^{79} -18.2985 q^{80} +20.2727 q^{82} -8.11162 q^{83} +9.42105 q^{85} +4.13268 q^{86} +52.8181 q^{88} +1.24202 q^{89} -3.89762 q^{92} -14.2727 q^{94} +2.15657 q^{95} +0.464716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4}+O(q^{10}) \) 6 * q + 12 * q^4	\( 6 q + 12 q^{4} - 8 q^{10} - 6 q^{13} + 28 q^{16} + 2 q^{19} + 28 q^{22} + 32 q^{25} - 10 q^{31} + 8 q^{34} + 4 q^{40} + 2 q^{43} + 12 q^{46} - 12 q^{52} + 12 q^{55} - 60 q^{58} - 60 q^{61} + 8 q^{64} + 12 q^{67} + 2 q^{73} + 52 q^{76} + 26 q^{79} + 52 q^{82} + 40 q^{85} + 108 q^{88} - 16 q^{94} + 14 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 - 8 * q^10 - 6 * q^13 + 28 * q^16 + 2 * q^19 + 28 * q^22 + 32 * q^25 - 10 * q^31 + 8 * q^34 + 4 * q^40 + 2 * q^43 + 12 * q^46 - 12 * q^52 + 12 * q^55 - 60 * q^58 - 60 * q^61 + 8 * q^64 + 12 * q^67 + 2 * q^73 + 52 * q^76 + 26 * q^79 + 52 * q^82 + 40 * q^85 + 108 * q^88 - 16 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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