LMFDB - Embedded newform 2088.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2088,2,Mod(1,2088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2088, 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("2088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2088 = 2^{3} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2088.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:	$16.6727639420$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	2088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.62620 q^{5} +O(q^{10})$	$q+1.62620 q^{5} -4.49084 q^{11} +0.103084 q^{13} -2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} -1.00000 q^{29} -6.76156 q^{31} +5.25240 q^{37} -5.79383 q^{41} -10.0140 q^{43} -11.5371 q^{47} -7.00000 q^{49} +7.14931 q^{53} -7.30299 q^{55} +1.52311 q^{59} +9.04623 q^{61} +0.167635 q^{65} +15.0462 q^{67} +12.0279 q^{71} -1.79383 q^{73} -1.98605 q^{79} -6.47689 q^{83} -3.25240 q^{85} -12.7110 q^{89} -11.7938 q^{95} +1.25240 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5}+O(q^{10}) $ 3 * q - 4 * q^5	$ 3 q - 4 q^{5} - 2 q^{11} + 4 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} - 3 q^{29} - 14 q^{31} - 2 q^{37} - 10 q^{41} - 6 q^{43} + 2 q^{47} - 21 q^{49} - 26 q^{55} - 8 q^{59} + 2 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 2 q^{73} - 30 q^{79} - 32 q^{83} + 8 q^{85} - 10 q^{89} - 28 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 2 * q^11 + 4 * q^13 - 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 - 3 * q^29 - 14 * q^31 - 2 * q^37 - 10 * q^41 - 6 * q^43 + 2 * q^47 - 21 * q^49 - 26 * q^55 - 8 * q^59 + 2 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 2 * q^73 - 30 * q^79 - 32 * q^83 + 8 * q^85 - 10 * q^89 - 28 * q^95 - 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.62620	0.727258	0.363629	−	0.931544i	$-0.381538\pi$
$5$	1.62620	0.727258	0.363629	+	0.931544i	$0.381538\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.49084	−1.35404	−0.677019	−	0.735965i	$-0.736729\pi$
$11$	−4.49084	−1.35404	−0.677019	+	0.735965i	$0.736729\pi$
$12$	0	0
$13$	0.103084	0.0285903	0.0142951	−	0.999898i	$-0.495450\pi$
$13$	0.103084	0.0285903	0.0142951	+	0.999898i	$0.495450\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−7.25240	−1.66381	−0.831907	−	0.554915i	$-0.812751\pi$
$19$	−7.25240	−1.66381	−0.831907	+	0.554915i	$0.812751\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.52311	1.15165	0.575824	−	0.817573i	$-0.304681\pi$
$23$	5.52311	1.15165	0.575824	+	0.817573i	$0.304681\pi$
$24$	0	0
$25$	−2.35548	−0.471096
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.76156	−1.21441	−0.607206	−	0.794545i	$-0.707710\pi$
$31$	−6.76156	−1.21441	−0.607206	+	0.794545i	$0.707710\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.25240	0.863489	0.431744	−	0.901996i	$-0.357898\pi$
$37$	5.25240	0.863489	0.431744	+	0.901996i	$0.357898\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.79383	−0.904845	−0.452422	−	0.891804i	$-0.649440\pi$
$41$	−5.79383	−0.904845	−0.452422	+	0.891804i	$0.649440\pi$
$42$	0	0
$43$	−10.0140	−1.52711	−0.763557	−	0.645741i	$-0.776549\pi$
$43$	−10.0140	−1.52711	−0.763557	+	0.645741i	$0.776549\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.5371	−1.68285	−0.841427	−	0.540371i	$-0.818284\pi$
$47$	−11.5371	−1.68285	−0.841427	+	0.540371i	$0.818284\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.14931	0.982034	0.491017	−	0.871150i	$-0.336625\pi$
$53$	7.14931	0.982034	0.491017	+	0.871150i	$0.336625\pi$
$54$	0	0
$55$	−7.30299	−0.984735
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.52311	0.198293	0.0991463	−	0.995073i	$-0.468389\pi$
$59$	1.52311	0.198293	0.0991463	+	0.995073i	$0.468389\pi$
$60$	0	0
$61$	9.04623	1.15825	0.579125	−	0.815238i	$-0.303394\pi$
$61$	9.04623	1.15825	0.579125	+	0.815238i	$0.303394\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.167635	0.0207925
$66$	0	0
$67$	15.0462	1.83819	0.919095	−	0.394037i	$-0.128922\pi$
$67$	15.0462	1.83819	0.919095	+	0.394037i	$0.128922\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0279	1.42745	0.713725	−	0.700426i	$-0.247007\pi$
$71$	12.0279	1.42745	0.713725	+	0.700426i	$0.247007\pi$
$72$	0	0
$73$	−1.79383	−0.209952	−0.104976	−	0.994475i	$-0.533477\pi$
$73$	−1.79383	−0.209952	−0.104976	+	0.994475i	$0.533477\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.98605	−0.223448	−0.111724	−	0.993739i	$-0.535637\pi$
$79$	−1.98605	−0.223448	−0.111724	+	0.993739i	$0.535637\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.47689	−0.710931	−0.355465	−	0.934689i	$-0.615678\pi$
$83$	−6.47689	−0.710931	−0.355465	+	0.934689i	$0.615678\pi$
$84$	0	0
$85$	−3.25240	−0.352772
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.7110	−1.34736	−0.673680	−	0.739024i	$-0.735287\pi$
$89$	−12.7110	−1.34736	−0.673680	+	0.739024i	$0.735287\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.7938	−1.21002
$96$	0	0
$97$	1.25240	0.127162	0.0635808	−	0.997977i	$-0.479748\pi$
$97$	1.25240	0.127162	0.0635808	+	0.997977i	$0.479748\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.2986	−1.62177	−0.810887	−	0.585203i	$-0.801015\pi$
$101$	−16.2986	−1.62177	−0.810887	+	0.585203i	$0.801015\pi$
$102$	0	0
$103$	−5.52311	−0.544209	−0.272104	−	0.962268i	$-0.587720\pi$
$103$	−5.52311	−0.544209	−0.272104	+	0.962268i	$0.587720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.541436	−0.0523426	−0.0261713	−	0.999657i	$-0.508332\pi$
$107$	−0.541436	−0.0523426	−0.0261713	+	0.999657i	$0.508332\pi$
$108$	0	0
$109$	15.9248	1.52532	0.762661	−	0.646799i	$-0.223893\pi$
$109$	15.9248	1.52532	0.762661	+	0.646799i	$0.223893\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.5048	1.17635	0.588176	−	0.808733i	$-0.299846\pi$
$113$	12.5048	1.17635	0.588176	+	0.808733i	$0.299846\pi$
$114$	0	0
$115$	8.98168	0.837546
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.16763	0.833421
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.9615	−1.06987
$126$	0	0
$127$	−0.206167	−0.0182944	−0.00914720	−	0.999958i	$-0.502912\pi$
$127$	−0.206167	−0.0182944	−0.00914720	+	0.999958i	$0.502912\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.2524	−1.33261	−0.666304	−	0.745680i	$-0.732125\pi$
$131$	−15.2524	−1.33261	−0.666304	+	0.745680i	$0.732125\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.25240	0.448742	0.224371	−	0.974504i	$-0.427967\pi$
$137$	5.25240	0.448742	0.224371	+	0.974504i	$0.427967\pi$
$138$	0	0
$139$	−12.5693	−1.06612	−0.533059	−	0.846078i	$-0.678958\pi$
$139$	−12.5693	−1.06612	−0.533059	+	0.846078i	$0.678958\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.462932	−0.0387123
$144$	0	0
$145$	−1.62620	−0.135048
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.309251	0.0253348	0.0126674	−	0.999920i	$-0.495968\pi$
$149$	0.309251	0.0253348	0.0126674	+	0.999920i	$0.495968\pi$
$150$	0	0
$151$	−11.4586	−0.932485	−0.466242	−	0.884657i	$-0.654393\pi$
$151$	−11.4586	−0.932485	−0.466242	+	0.884657i	$0.654393\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.9956	−0.883190
$156$	0	0
$157$	13.2524	1.05766	0.528828	−	0.848729i	$-0.322632\pi$
$157$	13.2524	1.05766	0.528828	+	0.848729i	$0.322632\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.17389	−0.248598	−0.124299	−	0.992245i	$-0.539668\pi$
$163$	−3.17389	−0.248598	−0.124299	+	0.992245i	$0.539668\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0279	0.930747	0.465374	−	0.885114i	$-0.345920\pi$
$167$	12.0279	0.930747	0.465374	+	0.885114i	$0.345920\pi$
$168$	0	0
$169$	−12.9894	−0.999183
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.5048	−0.950722	−0.475361	−	0.879791i	$-0.657682\pi$
$173$	−12.5048	−0.950722	−0.475361	+	0.879791i	$0.657682\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.02791	−0.600034	−0.300017	−	0.953934i	$-0.596992\pi$
$179$	−8.02791	−0.600034	−0.300017	+	0.953934i	$0.596992\pi$
$180$	0	0
$181$	−11.9248	−0.886365	−0.443183	−	0.896431i	$-0.646151\pi$
$181$	−11.9248	−0.886365	−0.443183	+	0.896431i	$0.646151\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.54144	0.627979
$186$	0	0
$187$	8.98168	0.656805
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.71096	−0.485588	−0.242794	−	0.970078i	$-0.578064\pi$
$191$	−6.71096	−0.485588	−0.242794	+	0.970078i	$0.578064\pi$
$192$	0	0
$193$	12.2986	0.885274	0.442637	−	0.896701i	$-0.354043\pi$
$193$	12.2986	0.885274	0.442637	+	0.896701i	$0.354043\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.50479	−0.320953	−0.160477	−	0.987040i	$-0.551303\pi$
$197$	−4.50479	−0.320953	−0.160477	+	0.987040i	$0.551303\pi$
$198$	0	0
$199$	−19.4586	−1.37938	−0.689690	−	0.724104i	$-0.742253\pi$
$199$	−19.4586	−1.37938	−0.689690	+	0.724104i	$0.742253\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.42192	−0.658055
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.5693	2.25287
$210$	0	0
$211$	1.77988	0.122532	0.0612660	−	0.998121i	$-0.480486\pi$
$211$	1.77988	0.122532	0.0612660	+	0.998121i	$0.480486\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.2847	−1.11061
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.206167	−0.0138683
$222$	0	0
$223$	26.5327	1.77676	0.888380	−	0.459108i	$-0.151831\pi$
$223$	26.5327	1.77676	0.888380	+	0.459108i	$0.151831\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.39401	0.358013	0.179007	−	0.983848i	$-0.442712\pi$
$227$	5.39401	0.358013	0.179007	+	0.983848i	$0.442712\pi$
$228$	0	0
$229$	16.2986	1.07704	0.538522	−	0.842612i	$-0.318983\pi$
$229$	16.2986	1.07704	0.538522	+	0.842612i	$0.318983\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.64452	−0.566321	−0.283161	−	0.959072i	$-0.591383\pi$
$233$	−8.64452	−0.566321	−0.283161	+	0.959072i	$0.591383\pi$
$234$	0	0
$235$	−18.7616	−1.22387
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.5231	−0.874738	−0.437369	−	0.899282i	$-0.644090\pi$
$239$	−13.5231	−0.874738	−0.437369	+	0.899282i	$0.644090\pi$
$240$	0	0
$241$	8.30925	0.535246	0.267623	−	0.963524i	$-0.413762\pi$
$241$	8.30925	0.535246	0.267623	+	0.963524i	$0.413762\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−11.3834	−0.727258
$246$	0	0
$247$	−0.747604	−0.0475689
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7432	1.24618	0.623091	−	0.782149i	$-0.285877\pi$
$251$	19.7432	1.24618	0.623091	+	0.782149i	$0.285877\pi$
$252$	0	0
$253$	−24.8034	−1.55938
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.4200	−1.33614	−0.668072	−	0.744096i	$-0.732880\pi$
$257$	−21.4200	−1.33614	−0.668072	+	0.744096i	$0.732880\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.2847	−0.757505	−0.378753	−	0.925498i	$-0.623647\pi$
$263$	−12.2847	−0.757505	−0.378753	+	0.925498i	$0.623647\pi$
$264$	0	0
$265$	11.6262	0.714192
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.5972	1.37778	0.688889	−	0.724867i	$-0.258099\pi$
$269$	22.5972	1.37778	0.688889	+	0.724867i	$0.258099\pi$
$270$	0	0
$271$	−23.7432	−1.44230	−0.721149	−	0.692780i	$-0.756386\pi$
$271$	−23.7432	−1.44230	−0.721149	+	0.692780i	$0.756386\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.5781	0.637882
$276$	0	0
$277$	−1.58767	−0.0953936	−0.0476968	−	0.998862i	$-0.515188\pi$
$277$	−1.58767	−0.0953936	−0.0476968	+	0.998862i	$0.515188\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.8969	1.18695	0.593475	−	0.804852i	$-0.297755\pi$
$281$	19.8969	1.18695	0.593475	+	0.804852i	$0.297755\pi$
$282$	0	0
$283$	20.9817	1.24723	0.623616	−	0.781731i	$-0.285663\pi$
$283$	20.9817	1.24723	0.623616	+	0.781731i	$0.285663\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.50479	−0.263173	−0.131586	−	0.991305i	$-0.542007\pi$
$293$	−4.50479	−0.263173	−0.131586	+	0.991305i	$0.542007\pi$
$294$	0	0
$295$	2.47689	0.144210
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.569343	0.0329260
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.7110	0.842347
$306$	0	0
$307$	−9.60162	−0.547993	−0.273997	−	0.961731i	$-0.588346\pi$
$307$	−9.60162	−0.547993	−0.273997	+	0.961731i	$0.588346\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.2986	1.71808	0.859039	−	0.511911i	$-0.171062\pi$
$311$	30.2986	1.71808	0.859039	+	0.511911i	$0.171062\pi$
$312$	0	0
$313$	2.37380	0.134175	0.0670876	−	0.997747i	$-0.478629\pi$
$313$	2.37380	0.134175	0.0670876	+	0.997747i	$0.478629\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.5510	0.648770	0.324385	−	0.945925i	$-0.394843\pi$
$317$	11.5510	0.648770	0.324385	+	0.945925i	$0.394843\pi$
$318$	0	0
$319$	4.49084	0.251439
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.5048	0.807068
$324$	0	0
$325$	−0.242812	−0.0134688
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.3126	−0.676761	−0.338380	−	0.941009i	$-0.609879\pi$
$331$	−12.3126	−0.676761	−0.338380	+	0.941009i	$0.609879\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	24.4681	1.33684
$336$	0	0
$337$	6.95377	0.378796	0.189398	−	0.981900i	$-0.439346\pi$
$337$	6.95377	0.378796	0.189398	+	0.981900i	$0.439346\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.3651	1.64436
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4402	0.882558	0.441279	−	0.897370i	$-0.354525\pi$
$347$	16.4402	0.882558	0.441279	+	0.897370i	$0.354525\pi$
$348$	0	0
$349$	22.9431	1.22812	0.614059	−	0.789260i	$-0.289536\pi$
$349$	22.9431	1.22812	0.614059	+	0.789260i	$0.289536\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.09246	−0.430718	−0.215359	−	0.976535i	$-0.569092\pi$
$353$	−8.09246	−0.430718	−0.215359	+	0.976535i	$0.569092\pi$
$354$	0	0
$355$	19.5598	1.03812
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.5187	−1.08294	−0.541469	−	0.840721i	$-0.682132\pi$
$359$	−20.5187	−1.08294	−0.541469	+	0.840721i	$0.682132\pi$
$360$	0	0
$361$	33.5972	1.76828
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.91713	−0.152689
$366$	0	0
$367$	1.75719	0.0917245	0.0458622	−	0.998948i	$-0.485396\pi$
$367$	1.75719	0.0917245	0.0458622	+	0.998948i	$0.485396\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	28.1310	1.45657	0.728284	−	0.685276i	$-0.240318\pi$
$373$	28.1310	1.45657	0.728284	+	0.685276i	$0.240318\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.103084	−0.00530908
$378$	0	0
$379$	24.8034	1.27407	0.637033	−	0.770837i	$-0.280161\pi$
$379$	24.8034	1.27407	0.637033	+	0.770837i	$0.280161\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.5048	−1.55872	−0.779361	−	0.626575i	$-0.784456\pi$
$383$	−30.5048	−1.55872	−0.779361	+	0.626575i	$0.784456\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.8401	1.66506	0.832529	−	0.553982i	$-0.186892\pi$
$389$	32.8401	1.66506	0.832529	+	0.553982i	$0.186892\pi$
$390$	0	0
$391$	−11.0462	−0.558632
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.22971	−0.162504
$396$	0	0
$397$	−28.3372	−1.42220	−0.711101	−	0.703090i	$-0.751803\pi$
$397$	−28.3372	−1.42220	−0.711101	+	0.703090i	$0.751803\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.94315	0.346724	0.173362	−	0.984858i	$-0.444537\pi$
$401$	6.94315	0.346724	0.173362	+	0.984858i	$0.444537\pi$
$402$	0	0
$403$	−0.697006	−0.0347204
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.5877	−1.16920
$408$	0	0
$409$	−39.1387	−1.93528	−0.967642	−	0.252328i	$-0.918804\pi$
$409$	−39.1387	−1.93528	−0.967642	+	0.252328i	$0.918804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.5327	−0.517030
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.6156	−0.762871	−0.381435	−	0.924396i	$-0.624570\pi$
$419$	−15.6156	−0.762871	−0.381435	+	0.924396i	$0.624570\pi$
$420$	0	0
$421$	−8.50479	−0.414498	−0.207249	−	0.978288i	$-0.566451\pi$
$421$	−8.50479	−0.414498	−0.207249	+	0.978288i	$0.566451\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.71096	0.228515
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.60599	0.318199	0.159100	−	0.987263i	$-0.449141\pi$
$431$	6.60599	0.318199	0.159100	+	0.987263i	$0.449141\pi$
$432$	0	0
$433$	−2.20617	−0.106022	−0.0530108	−	0.998594i	$-0.516882\pi$
$433$	−2.20617	−0.106022	−0.0530108	+	0.998594i	$0.516882\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−40.0558	−1.91613
$438$	0	0
$439$	4.44024	0.211921	0.105961	−	0.994370i	$-0.466208\pi$
$439$	4.44024	0.211921	0.105961	+	0.994370i	$0.466208\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.7572	1.03372	0.516858	−	0.856071i	$-0.327102\pi$
$443$	21.7572	1.03372	0.516858	+	0.856071i	$0.327102\pi$
$444$	0	0
$445$	−20.6705	−0.979877
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.5972	1.63275	0.816373	−	0.577526i	$-0.195982\pi$
$449$	34.5972	1.63275	0.816373	+	0.577526i	$0.195982\pi$
$450$	0	0
$451$	26.0192	1.22519
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.0925	−1.31411	−0.657055	−	0.753843i	$-0.728198\pi$
$457$	−28.0925	−1.31411	−0.657055	+	0.753843i	$0.728198\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.91713	0.415312	0.207656	−	0.978202i	$-0.433417\pi$
$461$	8.91713	0.415312	0.207656	+	0.978202i	$0.433417\pi$
$462$	0	0
$463$	3.55976	0.165436	0.0827180	−	0.996573i	$-0.473640\pi$
$463$	3.55976	0.165436	0.0827180	+	0.996573i	$0.473640\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.0968	0.513500	0.256750	−	0.966478i	$-0.417348\pi$
$467$	11.0968	0.513500	0.256750	+	0.966478i	$0.417348\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.9711	2.06777
$474$	0	0
$475$	17.0829	0.783816
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.77988	0.264089	0.132045	−	0.991244i	$-0.457846\pi$
$479$	5.77988	0.264089	0.132045	+	0.991244i	$0.457846\pi$
$480$	0	0
$481$	0.541436	0.0246874
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.03664	0.0924792
$486$	0	0
$487$	11.0462	0.500552	0.250276	−	0.968174i	$-0.419479\pi$
$487$	11.0462	0.500552	0.250276	+	0.968174i	$0.419479\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.03228	0.407621	0.203810	−	0.979010i	$-0.434667\pi$
$491$	9.03228	0.407621	0.203810	+	0.979010i	$0.434667\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−39.6156	−1.77344	−0.886718	−	0.462310i	$-0.847021\pi$
$499$	−39.6156	−1.77344	−0.886718	+	0.462310i	$0.847021\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.7895	1.19448	0.597242	−	0.802061i	$-0.296263\pi$
$503$	26.7895	1.19448	0.597242	+	0.802061i	$0.296263\pi$
$504$	0	0
$505$	−26.5048	−1.17945
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.1127	0.758506	0.379253	−	0.925293i	$-0.376181\pi$
$509$	17.1127	0.758506	0.379253	+	0.925293i	$0.376181\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.98168	−0.395780
$516$	0	0
$517$	51.8111	2.27865
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.5896	−0.858234	−0.429117	−	0.903249i	$-0.641175\pi$
$521$	−19.5896	−0.858234	−0.429117	+	0.903249i	$0.641175\pi$
$522$	0	0
$523$	15.0462	0.657926	0.328963	−	0.944343i	$-0.393301\pi$
$523$	15.0462	0.657926	0.328963	+	0.944343i	$0.393301\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.5231	0.589076
$528$	0	0
$529$	7.50479	0.326295
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.597250	−0.0258698
$534$	0	0
$535$	−0.880483	−0.0380666
$536$	0	0
$537$	0	0
$538$	0	0
$539$	31.4359	1.35404
$540$	0	0
$541$	31.2158	1.34207	0.671035	−	0.741426i	$-0.265850\pi$
$541$	31.2158	1.34207	0.671035	+	0.741426i	$0.265850\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.8969	1.10930
$546$	0	0
$547$	10.5048	0.449152	0.224576	−	0.974457i	$-0.427900\pi$
$547$	10.5048	0.449152	0.224576	+	0.974457i	$0.427900\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.25240	0.308962
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.6801	−1.17284	−0.586422	−	0.810006i	$-0.699464\pi$
$557$	−27.6801	−1.17284	−0.586422	+	0.810006i	$0.699464\pi$
$558$	0	0
$559$	−1.03228	−0.0436606
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.6508	−0.575312	−0.287656	−	0.957734i	$-0.592876\pi$
$563$	−13.6508	−0.575312	−0.287656	+	0.957734i	$0.592876\pi$
$564$	0	0
$565$	20.3353	0.855511
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.1387	−1.13771	−0.568856	−	0.822437i	$-0.692614\pi$
$569$	−27.1387	−1.13771	−0.568856	+	0.822437i	$0.692614\pi$
$570$	0	0
$571$	21.9634	0.919138	0.459569	−	0.888142i	$-0.348004\pi$
$571$	21.9634	0.919138	0.459569	+	0.888142i	$0.348004\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.0096	−0.542537
$576$	0	0
$577$	−10.5414	−0.438846	−0.219423	−	0.975630i	$-0.570417\pi$
$577$	−10.5414	−0.438846	−0.219423	+	0.975630i	$0.570417\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−32.1064	−1.32971
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.9450	−0.616848	−0.308424	−	0.951249i	$-0.599801\pi$
$587$	−14.9450	−0.616848	−0.308424	+	0.951249i	$0.599801\pi$
$588$	0	0
$589$	49.0375	2.02055
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.8015	−1.01848	−0.509238	−	0.860626i	$-0.670073\pi$
$593$	−24.8015	−1.01848	−0.509238	+	0.860626i	$0.670073\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.4446	0.549332	0.274666	−	0.961540i	$-0.411433\pi$
$599$	13.4446	0.549332	0.274666	+	0.961540i	$0.411433\pi$
$600$	0	0
$601$	25.2524	1.03007	0.515033	−	0.857170i	$-0.327780\pi$
$601$	25.2524	1.03007	0.515033	+	0.857170i	$0.327780\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.9084	0.606112
$606$	0	0
$607$	−2.78946	−0.113221	−0.0566104	−	0.998396i	$-0.518029\pi$
$607$	−2.78946	−0.113221	−0.0566104	+	0.998396i	$0.518029\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.18928	−0.0481133
$612$	0	0
$613$	−43.1772	−1.74391	−0.871956	−	0.489585i	$-0.837148\pi$
$613$	−43.1772	−1.74391	−0.871956	+	0.489585i	$0.837148\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.7476	−0.432682	−0.216341	−	0.976318i	$-0.569412\pi$
$617$	−10.7476	−0.432682	−0.216341	+	0.976318i	$0.569412\pi$
$618$	0	0
$619$	30.3771	1.22096	0.610480	−	0.792032i	$-0.290977\pi$
$619$	30.3771	1.22096	0.610480	+	0.792032i	$0.290977\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.67432	−0.306973
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.5048	−0.418853
$630$	0	0
$631$	−13.5231	−0.538347	−0.269173	−	0.963092i	$-0.586750\pi$
$631$	−13.5231	−0.538347	−0.269173	+	0.963092i	$0.586750\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.335269	−0.0133047
$636$	0	0
$637$	−0.721586	−0.0285903
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.6339	0.656999	0.328500	−	0.944504i	$-0.393457\pi$
$641$	16.6339	0.656999	0.328500	+	0.944504i	$0.393457\pi$
$642$	0	0
$643$	24.9538	0.984081	0.492040	−	0.870572i	$-0.336251\pi$
$643$	24.9538	0.984081	0.492040	+	0.870572i	$0.336251\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.0646	−0.395678	−0.197839	−	0.980234i	$-0.563392\pi$
$647$	−10.0646	−0.395678	−0.197839	+	0.980234i	$0.563392\pi$
$648$	0	0
$649$	−6.84006	−0.268496
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.2620	−1.65384	−0.826920	−	0.562320i	$-0.809909\pi$
$653$	−42.2620	−1.65384	−0.826920	+	0.562320i	$0.809909\pi$
$654$	0	0
$655$	−24.8034	−0.969150
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.3867	1.69011	0.845053	−	0.534682i	$-0.179569\pi$
$659$	43.3867	1.69011	0.845053	+	0.534682i	$0.179569\pi$
$660$	0	0
$661$	−0.0924575	−0.00359618	−0.00179809	−	0.999998i	$-0.500572\pi$
$661$	−0.0924575	−0.00359618	−0.00179809	+	0.999998i	$0.500572\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.52311	−0.213856
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−40.6252	−1.56832
$672$	0	0
$673$	1.86027	0.0717082	0.0358541	−	0.999357i	$-0.488585\pi$
$673$	1.86027	0.0717082	0.0358541	+	0.999357i	$0.488585\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.1387	0.428094	0.214047	−	0.976823i	$-0.431335\pi$
$677$	11.1387	0.428094	0.214047	+	0.976823i	$0.431335\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.54144	0.326829	0.163414	−	0.986558i	$-0.447749\pi$
$683$	8.54144	0.326829	0.163414	+	0.986558i	$0.447749\pi$
$684$	0	0
$685$	8.54144	0.326352
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.736978	0.0280766
$690$	0	0
$691$	−24.5972	−0.935723	−0.467862	−	0.883802i	$-0.654975\pi$
$691$	−24.5972	−0.935723	−0.467862	+	0.883802i	$0.654975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.4402	−0.775343
$696$	0	0
$697$	11.5877	0.438914
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.56165	−0.134522	−0.0672608	−	0.997735i	$-0.521426\pi$
$701$	−3.56165	−0.134522	−0.0672608	+	0.997735i	$0.521426\pi$
$702$	0	0
$703$	−38.0925	−1.43668
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.3651	−0.915049	−0.457525	−	0.889197i	$-0.651264\pi$
$709$	−24.3651	−0.915049	−0.457525	+	0.889197i	$0.651264\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−37.3449	−1.39858
$714$	0	0
$715$	−0.752820	−0.0281539
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.89881	−0.294576	−0.147288	−	0.989094i	$-0.547054\pi$
$719$	−7.89881	−0.294576	−0.147288	+	0.989094i	$0.547054\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.35548	0.0874803
$726$	0	0
$727$	33.8130	1.25405	0.627027	−	0.778997i	$-0.284271\pi$
$727$	33.8130	1.25405	0.627027	+	0.778997i	$0.284271\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.0279	0.740759
$732$	0	0
$733$	45.7205	1.68873	0.844363	−	0.535771i	$-0.179979\pi$
$733$	45.7205	1.68873	0.844363	+	0.535771i	$0.179979\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−67.5702	−2.48898
$738$	0	0
$739$	−8.28467	−0.304757	−0.152378	−	0.988322i	$-0.548693\pi$
$739$	−8.28467	−0.304757	−0.152378	+	0.988322i	$0.548693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.70138	0.0624174	0.0312087	−	0.999513i	$-0.490064\pi$
$743$	1.70138	0.0624174	0.0312087	+	0.999513i	$0.490064\pi$
$744$	0	0
$745$	0.502904	0.0184250
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.4277	0.380513	0.190257	−	0.981734i	$-0.439068\pi$
$751$	10.4277	0.380513	0.190257	+	0.981734i	$0.439068\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.6339	−0.678157
$756$	0	0
$757$	−45.9267	−1.66923	−0.834617	−	0.550830i	$-0.814311\pi$
$757$	−45.9267	−1.66923	−0.834617	+	0.550830i	$0.814311\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.58767	−0.0575528	−0.0287764	−	0.999586i	$-0.509161\pi$
$761$	−1.58767	−0.0575528	−0.0287764	+	0.999586i	$0.509161\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.157008	0.00566924
$768$	0	0
$769$	−8.37569	−0.302035	−0.151018	−	0.988531i	$-0.548255\pi$
$769$	−8.37569	−0.302035	−0.151018	+	0.988531i	$0.548255\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.2062	−0.367090	−0.183545	−	0.983011i	$-0.558757\pi$
$773$	−10.2062	−0.367090	−0.183545	+	0.983011i	$0.558757\pi$
$774$	0	0
$775$	15.9267	0.572104
$776$	0	0
$777$	0	0
$778$	0	0
$779$	42.0192	1.50549
$780$	0	0
$781$	−54.0154	−1.93282
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.5510	0.769189
$786$	0	0
$787$	26.6618	0.950391	0.475195	−	0.879880i	$-0.342377\pi$
$787$	26.6618	0.950391	0.475195	+	0.879880i	$0.342377\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.932519	0.0331147
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.0462	−0.887183	−0.443591	−	0.896229i	$-0.646296\pi$
$797$	−25.0462	−0.887183	−0.443591	+	0.896229i	$0.646296\pi$
$798$	0	0
$799$	23.0741	0.816304
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.05581	0.284283
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.1695	−1.13102	−0.565510	−	0.824741i	$-0.691321\pi$
$809$	−32.1695	−1.13102	−0.565510	+	0.824741i	$0.691321\pi$
$810$	0	0
$811$	26.5048	0.930709	0.465355	−	0.885124i	$-0.345927\pi$
$811$	26.5048	0.930709	0.465355	+	0.885124i	$0.345927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.16138	−0.180795
$816$	0	0
$817$	72.6252	2.54083
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.2051	−0.949465	−0.474733	−	0.880130i	$-0.657455\pi$
$821$	−27.2051	−0.949465	−0.474733	+	0.880130i	$0.657455\pi$
$822$	0	0
$823$	8.61850	0.300422	0.150211	−	0.988654i	$-0.452005\pi$
$823$	8.61850	0.300422	0.150211	+	0.988654i	$0.452005\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.8636	1.31665	0.658323	−	0.752735i	$-0.271266\pi$
$827$	37.8636	1.31665	0.658323	+	0.752735i	$0.271266\pi$
$828$	0	0
$829$	−51.9634	−1.80476	−0.902381	−	0.430939i	$-0.858182\pi$
$829$	−51.9634	−1.80476	−0.902381	+	0.430939i	$0.858182\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	19.5598	0.676893
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.0785	−1.10747	−0.553736	−	0.832692i	$-0.686799\pi$
$839$	−32.0785	−1.10747	−0.553736	+	0.832692i	$0.686799\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−21.1233	−0.726663
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	29.0096	0.994436
$852$	0	0
$853$	−40.5048	−1.38686	−0.693429	−	0.720525i	$-0.743901\pi$
$853$	−40.5048	−1.38686	−0.693429	+	0.720525i	$0.743901\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.3834	−1.07204	−0.536018	−	0.844207i	$-0.680072\pi$
$857$	−31.3834	−1.07204	−0.536018	+	0.844207i	$0.680072\pi$
$858$	0	0
$859$	32.5187	1.10953	0.554763	−	0.832009i	$-0.312809\pi$
$859$	32.5187	1.10953	0.554763	+	0.832009i	$0.312809\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.4036	1.57960	0.789798	−	0.613367i	$-0.210185\pi$
$863$	46.4036	1.57960	0.789798	+	0.613367i	$0.210185\pi$
$864$	0	0
$865$	−20.3353	−0.691420
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.91902	0.302557
$870$	0	0
$871$	1.55102	0.0525543
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−47.1493	−1.59212	−0.796060	−	0.605218i	$-0.793086\pi$
$877$	−47.1493	−1.59212	−0.796060	+	0.605218i	$0.793086\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.95377	0.0995151	0.0497575	−	0.998761i	$-0.484155\pi$
$881$	2.95377	0.0995151	0.0497575	+	0.998761i	$0.484155\pi$
$882$	0	0
$883$	−14.6339	−0.492470	−0.246235	−	0.969210i	$-0.579193\pi$
$883$	−14.6339	−0.492470	−0.246235	+	0.969210i	$0.579193\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.3955	1.12131	0.560655	−	0.828050i	$-0.310549\pi$
$887$	33.3955	1.12131	0.560655	+	0.828050i	$0.310549\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	83.6714	2.79996
$894$	0	0
$895$	−13.0550	−0.436379
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.76156	0.225511
$900$	0	0
$901$	−14.2986	−0.476356
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.3921	−0.644616
$906$	0	0
$907$	7.25240	0.240812	0.120406	−	0.992725i	$-0.461580\pi$
$907$	7.25240	0.240812	0.120406	+	0.992725i	$0.461580\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.6604	0.485719	0.242860	−	0.970061i	$-0.421915\pi$
$911$	14.6604	0.485719	0.242860	+	0.970061i	$0.421915\pi$
$912$	0	0
$913$	29.0867	0.962628
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−42.1204	−1.38942	−0.694711	−	0.719289i	$-0.744468\pi$
$919$	−42.1204	−1.38942	−0.694711	+	0.719289i	$0.744468\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.23988	0.0408112
$924$	0	0
$925$	−12.3719	−0.406786
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.5144	1.36204	0.681021	−	0.732264i	$-0.261536\pi$
$929$	41.5144	1.36204	0.681021	+	0.732264i	$0.261536\pi$
$930$	0	0
$931$	50.7668	1.66381
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.6060	0.477667
$936$	0	0
$937$	−43.4219	−1.41853	−0.709266	−	0.704941i	$-0.750974\pi$
$937$	−43.4219	−1.41853	−0.709266	+	0.704941i	$0.750974\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.0848	0.426551	0.213276	−	0.976992i	$-0.431587\pi$
$941$	13.0848	0.426551	0.213276	+	0.976992i	$0.431587\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.9128	−1.62195	−0.810973	−	0.585083i	$-0.801062\pi$
$947$	−49.9128	−1.62195	−0.810973	+	0.585083i	$0.801062\pi$
$948$	0	0
$949$	−0.184915	−0.00600259
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.6175	−1.28334	−0.641668	−	0.766983i	$-0.721757\pi$
$953$	−39.6175	−1.28334	−0.641668	+	0.766983i	$0.721757\pi$
$954$	0	0
$955$	−10.9133	−0.353148
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	14.7187	0.474795
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20.0000	0.643823
$966$	0	0
$967$	−37.6574	−1.21098	−0.605491	−	0.795852i	$-0.707023\pi$
$967$	−37.6574	−1.21098	−0.605491	+	0.795852i	$0.707023\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.15994	0.293956	0.146978	−	0.989140i	$-0.453045\pi$
$971$	9.15994	0.293956	0.146978	+	0.989140i	$0.453045\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45.4758	−1.45490	−0.727451	−	0.686160i	$-0.759295\pi$
$977$	−45.4758	−1.45490	−0.727451	+	0.686160i	$0.759295\pi$
$978$	0	0
$979$	57.0829	1.82438
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.9494	−0.891447	−0.445724	−	0.895171i	$-0.647054\pi$
$983$	−27.9494	−0.891447	−0.445724	+	0.895171i	$0.647054\pi$
$984$	0	0
$985$	−7.32568	−0.233416
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−55.3082	−1.75870
$990$	0	0
$991$	13.1108	0.416478	0.208239	−	0.978078i	$-0.433227\pi$
$991$	13.1108	0.416478	0.208239	+	0.978078i	$0.433227\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31.6435	−1.00317
$996$	0	0
$997$	28.8401	0.913374	0.456687	−	0.889627i	$-0.349036\pi$
$997$	28.8401	0.913374	0.456687	+	0.889627i	$0.349036\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2088.2.a.s.1.3		3
3.2	odd	2		232.2.a.d.1.3	✓	3
4.3	odd	2		4176.2.a.bu.1.3		3
12.11	even	2		464.2.a.j.1.1		3
15.14	odd	2		5800.2.a.p.1.1		3
24.5	odd	2		1856.2.a.x.1.1		3
24.11	even	2		1856.2.a.y.1.3		3
87.86	odd	2		6728.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.d.1.3	✓	3	3.2	odd	2
464.2.a.j.1.1		3	12.11	even	2
1856.2.a.x.1.1		3	24.5	odd	2
1856.2.a.y.1.3		3	24.11	even	2
2088.2.a.s.1.3		3	1.1	even	1	trivial
4176.2.a.bu.1.3		3	4.3	odd	2
5800.2.a.p.1.1		3	15.14	odd	2
6728.2.a.j.1.1		3	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2088.a (trivial)

\(f(q)\)	\(=\)	\(q+1.62620 q^{5} +O(q^{10})\)	\(q+1.62620 q^{5} -4.49084 q^{11} +0.103084 q^{13} -2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} -1.00000 q^{29} -6.76156 q^{31} +5.25240 q^{37} -5.79383 q^{41} -10.0140 q^{43} -11.5371 q^{47} -7.00000 q^{49} +7.14931 q^{53} -7.30299 q^{55} +1.52311 q^{59} +9.04623 q^{61} +0.167635 q^{65} +15.0462 q^{67} +12.0279 q^{71} -1.79383 q^{73} -1.98605 q^{79} -6.47689 q^{83} -3.25240 q^{85} -12.7110 q^{89} -11.7938 q^{95} +1.25240 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5}+O(q^{10}) \) 3 * q - 4 * q^5	\( 3 q - 4 q^{5} - 2 q^{11} + 4 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} - 3 q^{29} - 14 q^{31} - 2 q^{37} - 10 q^{41} - 6 q^{43} + 2 q^{47} - 21 q^{49} - 26 q^{55} - 8 q^{59} + 2 q^{61} + 2 q^{65} + 20 q^{67} - 12 q^{71} + 2 q^{73} - 30 q^{79} - 32 q^{83} + 8 q^{85} - 10 q^{89} - 28 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 2 * q^11 + 4 * q^13 - 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 - 3 * q^29 - 14 * q^31 - 2 * q^37 - 10 * q^41 - 6 * q^43 + 2 * q^47 - 21 * q^49 - 26 * q^55 - 8 * q^59 + 2 * q^61 + 2 * q^65 + 20 * q^67 - 12 * q^71 + 2 * q^73 - 30 * q^79 - 32 * q^83 + 8 * q^85 - 10 * q^89 - 28 * q^95 - 14 * q^97

Introduction

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