LMFDB - Embedded newform 1856.2.a.y.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1856,2,Mod(1,1856)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1856.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,-4,0,0,0,5,0,2,0,-4,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.8202346151$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	1856.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +4.49084 q^{11} -0.103084 q^{13} +4.49084 q^{15} +2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} +4.49084 q^{27} -1.00000 q^{29} +6.76156 q^{31} +12.4017 q^{33} -5.25240 q^{37} -0.284672 q^{39} +5.79383 q^{41} -10.0140 q^{43} +7.52311 q^{45} -11.5371 q^{47} -7.00000 q^{49} +5.52311 q^{51} +7.14931 q^{53} +7.30299 q^{55} -20.0279 q^{57} -1.52311 q^{59} -9.04623 q^{61} -0.167635 q^{65} +15.0462 q^{67} +15.2524 q^{69} +12.0279 q^{71} -1.79383 q^{73} -6.50479 q^{75} +1.98605 q^{79} -1.47689 q^{81} +6.47689 q^{83} +3.25240 q^{85} -2.76156 q^{87} +12.7110 q^{89} +18.6724 q^{93} -11.7938 q^{95} +1.25240 q^{97} +20.7755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43}+ \cdots + 32 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 4 * q^5 + 5 * q^9 + 2 * q^11 - 4 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 + 2 * q^27 - 3 * q^29 + 14 * q^31 - 2 * q^33 + 2 * q^37 + 18 * q^39 + 10 * q^41 - 6 * q^43 + 10 * q^45 + 2 * q^47 - 21 * q^49 + 4 * q^51 + 26 * q^55 - 12 * q^57 + 8 * q^59 - 2 * q^61 - 2 * q^65 + 20 * q^67 + 28 * q^69 - 12 * q^71 + 2 * q^73 + 16 * q^75 + 30 * q^79 - 17 * q^81 + 32 * q^83 - 8 * q^85 - 2 * q^87 + 10 * q^89 + 22 * q^93 - 28 * q^95 - 14 * q^97 + 32 * q^99

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
$3$	2.76156	1.59439	0.797193	−	0.603725i	$-0.206317\pi$
$3$	2.76156	1.59439	0.797193	+	0.603725i	$0.206317\pi$
$4$	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
$9$	4.62620	1.54207
$10$	0	0
$11$	4.49084	1.35404	0.677019	−	0.735965i	$-0.263271\pi$
$11$	4.49084	1.35404	0.677019	+	0.735965i	$0.263271\pi$
$12$	0	0
$13$	−0.103084	−0.0285903	−0.0142951	−	0.999898i	$-0.504550\pi$
$13$	−0.103084	−0.0285903	−0.0142951	+	0.999898i	$0.504550\pi$
$14$	0	0
$15$	4.49084	1.15953
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$27$	4.49084	0.864262
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.76156	1.21441	0.607206	−	0.794545i	$-0.292290\pi$
$31$	6.76156	1.21441	0.607206	+	0.794545i	$0.292290\pi$
$32$	0	0
$33$	12.4017	2.15886
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.25240	−0.863489	−0.431744	−	0.901996i	$-0.642102\pi$
$37$	−5.25240	−0.863489	−0.431744	+	0.901996i	$0.642102\pi$
$38$	0	0
$39$	−0.284672	−0.0455839
$40$	0	0
$41$	5.79383	0.904845	0.452422	−	0.891804i	$-0.350560\pi$
$41$	5.79383	0.904845	0.452422	+	0.891804i	$0.350560\pi$
$42$	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
$45$	7.52311	1.12148
$46$	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
$51$	5.52311	0.773391
$52$	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
$57$	−20.0279	−2.65276
$58$	0	0
$59$	−1.52311	−0.198293	−0.0991463	−	0.995073i	$-0.531611\pi$
$59$	−1.52311	−0.198293	−0.0991463	+	0.995073i	$0.531611\pi$
$60$	0	0
$61$	−9.04623	−1.15825	−0.579125	−	0.815238i	$-0.696606\pi$
$61$	−9.04623	−1.15825	−0.579125	+	0.815238i	$0.696606\pi$
$62$	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
$69$	15.2524	1.83617
$70$	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
$75$	−6.50479	−0.751109
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.98605	0.223448	0.111724	−	0.993739i	$-0.464363\pi$
$79$	1.98605	0.223448	0.111724	+	0.993739i	$0.464363\pi$
$80$	0	0
$81$	−1.47689	−0.164098
$82$	0	0
$83$	6.47689	0.710931	0.355465	−	0.934689i	$-0.384322\pi$
$83$	6.47689	0.710931	0.355465	+	0.934689i	$0.384322\pi$
$84$	0	0
$85$	3.25240	0.352772
$86$	0	0
$87$	−2.76156	−0.296070
$88$	0	0
$89$	12.7110	1.34736	0.673680	−	0.739024i	$-0.264713\pi$
$89$	12.7110	1.34736	0.673680	+	0.739024i	$0.264713\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	18.6724	1.93624
$94$	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
$99$	20.7755	2.08802
$100$	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	$-0.412280\pi$
$103$	5.52311	0.544209	0.272104	+	0.962268i	$0.412280\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.541436	0.0523426	0.0261713	−	0.999657i	$-0.491668\pi$
$107$	0.541436	0.0523426	0.0261713	+	0.999657i	$0.491668\pi$
$108$	0	0
$109$	−15.9248	−1.52532	−0.762661	−	0.646799i	$-0.776107\pi$
$109$	−15.9248	−1.52532	−0.762661	+	0.646799i	$0.776107\pi$
$110$	0	0
$111$	−14.5048	−1.37673
$112$	0	0
$113$	−12.5048	−1.17635	−0.588176	−	0.808733i	$-0.700154\pi$
$113$	−12.5048	−1.17635	−0.588176	+	0.808733i	$0.700154\pi$
$114$	0	0
$115$	8.98168	0.837546
$116$	0	0
$117$	−0.476886	−0.0440881
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.16763	0.833421
$122$	0	0
$123$	16.0000	1.44267
$124$	0	0
$125$	−11.9615	−1.06987
$126$	0	0
$127$	0.206167	0.0182944	0.00914720	−	0.999958i	$-0.497088\pi$
$127$	0.206167	0.0182944	0.00914720	+	0.999958i	$0.497088\pi$
$128$	0	0
$129$	−27.6541	−2.43481
$130$	0	0
$131$	15.2524	1.33261	0.666304	−	0.745680i	$-0.267875\pi$
$131$	15.2524	1.33261	0.666304	+	0.745680i	$0.267875\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.30299	0.628542
$136$	0	0
$137$	−5.25240	−0.448742	−0.224371	−	0.974504i	$-0.572033\pi$
$137$	−5.25240	−0.448742	−0.224371	+	0.974504i	$0.572033\pi$
$138$	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
$141$	−31.8603	−2.68312
$142$	0	0
$143$	−0.462932	−0.0387123
$144$	0	0
$145$	−1.62620	−0.135048
$146$	0	0
$147$	−19.3309	−1.59439
$148$	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	$-0.345607\pi$
$151$	11.4586	0.932485	0.466242	+	0.884657i	$0.345607\pi$
$152$	0	0
$153$	9.25240	0.748012
$154$	0	0
$155$	10.9956	0.883190
$156$	0	0
$157$	−13.2524	−1.05766	−0.528828	−	0.848729i	$-0.677368\pi$
$157$	−13.2524	−1.05766	−0.528828	+	0.848729i	$0.677368\pi$
$158$	0	0
$159$	19.7432	1.56574
$160$	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
$165$	20.1676	1.57005
$166$	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
$171$	−33.5510	−2.56571
$172$	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
$177$	−4.20617	−0.316155
$178$	0	0
$179$	8.02791	0.600034	0.300017	−	0.953934i	$-0.403008\pi$
$179$	8.02791	0.600034	0.300017	+	0.953934i	$0.403008\pi$
$180$	0	0
$181$	11.9248	0.886365	0.443183	−	0.896431i	$-0.353849\pi$
$181$	11.9248	0.886365	0.443183	+	0.896431i	$0.353849\pi$
$182$	0	0
$183$	−24.9817	−1.84670
$184$	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
$195$	−0.462932	−0.0331513
$196$	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	$-0.257747\pi$
$199$	19.4586	1.37938	0.689690	+	0.724104i	$0.257747\pi$
$200$	0	0
$201$	41.5510	2.93078
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.42192	0.658055
$206$	0	0
$207$	25.5510	1.77592
$208$	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
$213$	33.2158	2.27591
$214$	0	0
$215$	−16.2847	−1.11061
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.95377	−0.334745
$220$	0	0
$221$	−0.206167	−0.0138683
$222$	0	0
$223$	−26.5327	−1.77676	−0.888380	−	0.459108i	$-0.848169\pi$
$223$	−26.5327	−1.77676	−0.888380	+	0.459108i	$0.848169\pi$
$224$	0	0
$225$	−10.8969	−0.726461
$226$	0	0
$227$	−5.39401	−0.358013	−0.179007	−	0.983848i	$-0.557288\pi$
$227$	−5.39401	−0.358013	−0.179007	+	0.983848i	$0.557288\pi$
$228$	0	0
$229$	−16.2986	−1.07704	−0.538522	−	0.842612i	$-0.681017\pi$
$229$	−16.2986	−1.07704	−0.538522	+	0.842612i	$0.681017\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.64452	0.566321	0.283161	−	0.959072i	$-0.408617\pi$
$233$	8.64452	0.566321	0.283161	+	0.959072i	$0.408617\pi$
$234$	0	0
$235$	−18.7616	−1.22387
$236$	0	0
$237$	5.48458	0.356262
$238$	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
$243$	−17.5510	−1.12590
$244$	0	0
$245$	−11.3834	−0.727258
$246$	0	0
$247$	0.747604	0.0475689
$248$	0	0
$249$	17.8863	1.13350
$250$	0	0
$251$	−19.7432	−1.24618	−0.623091	−	0.782149i	$-0.714123\pi$
$251$	−19.7432	−1.24618	−0.623091	+	0.782149i	$0.714123\pi$
$252$	0	0
$253$	24.8034	1.55938
$254$	0	0
$255$	8.98168	0.562454
$256$	0	0
$257$	21.4200	1.33614	0.668072	−	0.744096i	$-0.267120\pi$
$257$	21.4200	1.33614	0.668072	+	0.744096i	$0.267120\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.62620	−0.286354
$262$	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
$267$	35.1020	2.14821
$268$	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	$-0.243614\pi$
$271$	23.7432	1.44230	0.721149	+	0.692780i	$0.243614\pi$
$272$	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	$-0.484812\pi$
$277$	1.58767	0.0953936	0.0476968	+	0.998862i	$0.484812\pi$
$278$	0	0
$279$	31.2803	1.87270
$280$	0	0
$281$	−19.8969	−1.18695	−0.593475	−	0.804852i	$-0.702245\pi$
$281$	−19.8969	−1.18695	−0.593475	+	0.804852i	$0.702245\pi$
$282$	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
$285$	−32.5693	−1.92924
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	3.45856	0.202745
$292$	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
$297$	20.1676	1.17024
$298$	0	0
$299$	−0.569343	−0.0329260
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−45.0096	−2.58573
$304$	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
$309$	15.2524	0.867679
$310$	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
$321$	1.49521	0.0834544
$322$	0	0
$323$	−14.5048	−0.807068
$324$	0	0
$325$	0.242812	0.0134688
$326$	0	0
$327$	−43.9773	−2.43195
$328$	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
$333$	−24.2986	−1.33156
$334$	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
$339$	−34.5327	−1.87556
$340$	0	0
$341$	30.3651	1.64436
$342$	0	0
$343$	0	0
$344$	0	0
$345$	24.8034	1.33537
$346$	0	0
$347$	−16.4402	−0.882558	−0.441279	−	0.897370i	$-0.645475\pi$
$347$	−16.4402	−0.882558	−0.441279	+	0.897370i	$0.645475\pi$
$348$	0	0
$349$	−22.9431	−1.22812	−0.614059	−	0.789260i	$-0.710464\pi$
$349$	−22.9431	−1.22812	−0.614059	+	0.789260i	$0.710464\pi$
$350$	0	0
$351$	−0.462932	−0.0247095
$352$	0	0
$353$	8.09246	0.430718	0.215359	−	0.976535i	$-0.430908\pi$
$353$	8.09246	0.430718	0.215359	+	0.976535i	$0.430908\pi$
$354$	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
$363$	25.3169	1.32880
$364$	0	0
$365$	−2.91713	−0.152689
$366$	0	0
$367$	−1.75719	−0.0917245	−0.0458622	−	0.998948i	$-0.514604\pi$
$367$	−1.75719	−0.0917245	−0.0458622	+	0.998948i	$0.514604\pi$
$368$	0	0
$369$	26.8034	1.39533
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.1310	−1.45657	−0.728284	−	0.685276i	$-0.759682\pi$
$373$	−28.1310	−1.45657	−0.728284	+	0.685276i	$0.759682\pi$
$374$	0	0
$375$	−33.0323	−1.70578
$376$	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
$381$	0.569343	0.0291683
$382$	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
$387$	−46.3265	−2.35491
$388$	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
$393$	42.1204	2.12469
$394$	0	0
$395$	3.22971	0.162504
$396$	0	0
$397$	28.3372	1.42220	0.711101	−	0.703090i	$-0.248197\pi$
$397$	28.3372	1.42220	0.711101	+	0.703090i	$0.248197\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.94315	−0.346724	−0.173362	−	0.984858i	$-0.555463\pi$
$401$	−6.94315	−0.346724	−0.173362	+	0.984858i	$0.555463\pi$
$402$	0	0
$403$	−0.697006	−0.0347204
$404$	0	0
$405$	−2.40171	−0.119342
$406$	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
$411$	−14.5048	−0.715469
$412$	0	0
$413$	0	0
$414$	0	0
$415$	10.5327	0.517030
$416$	0	0
$417$	−34.7110	−1.69980
$418$	0	0
$419$	15.6156	0.762871	0.381435	−	0.924396i	$-0.375430\pi$
$419$	15.6156	0.762871	0.381435	+	0.924396i	$0.375430\pi$
$420$	0	0
$421$	8.50479	0.414498	0.207249	−	0.978288i	$-0.433549\pi$
$421$	8.50479	0.414498	0.207249	+	0.978288i	$0.433549\pi$
$422$	0	0
$423$	−53.3728	−2.59507
$424$	0	0
$425$	−4.71096	−0.228515
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.27841	−0.0617224
$430$	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
$435$	−4.49084	−0.215319
$436$	0	0
$437$	−40.0558	−1.91613
$438$	0	0
$439$	−4.44024	−0.211921	−0.105961	−	0.994370i	$-0.533792\pi$
$439$	−4.44024	−0.211921	−0.105961	+	0.994370i	$0.533792\pi$
$440$	0	0
$441$	−32.3834	−1.54207
$442$	0	0
$443$	−21.7572	−1.03372	−0.516858	−	0.856071i	$-0.672898\pi$
$443$	−21.7572	−1.03372	−0.516858	+	0.856071i	$0.672898\pi$
$444$	0	0
$445$	20.6705	0.979877
$446$	0	0
$447$	0.854015	0.0403935
$448$	0	0
$449$	−34.5972	−1.63275	−0.816373	−	0.577526i	$-0.804018\pi$
$449$	−34.5972	−1.63275	−0.816373	+	0.577526i	$0.804018\pi$
$450$	0	0
$451$	26.0192	1.22519
$452$	0	0
$453$	31.6435	1.48674
$454$	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
$459$	8.98168	0.419229
$460$	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	$-0.526360\pi$
$463$	−3.55976	−0.165436	−0.0827180	+	0.996573i	$0.526360\pi$
$464$	0	0
$465$	30.3651	1.40815
$466$	0	0
$467$	−11.0968	−0.513500	−0.256750	−	0.966478i	$-0.582652\pi$
$467$	−11.0968	−0.513500	−0.256750	+	0.966478i	$0.582652\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−36.5972	−1.68631
$472$	0	0
$473$	−44.9711	−2.06777
$474$	0	0
$475$	17.0829	0.783816
$476$	0	0
$477$	33.0741	1.51436
$478$	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	$-0.580521\pi$
$487$	−11.0462	−0.500552	−0.250276	+	0.968174i	$0.580521\pi$
$488$	0	0
$489$	−8.76488	−0.396362
$490$	0	0
$491$	−9.03228	−0.407621	−0.203810	−	0.979010i	$-0.565333\pi$
$491$	−9.03228	−0.407621	−0.203810	+	0.979010i	$0.565333\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	33.7851	1.51853
$496$	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
$501$	33.2158	1.48397
$502$	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
$507$	−35.8709	−1.59308
$508$	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
$513$	−32.5693	−1.43797
$514$	0	0
$515$	8.98168	0.395780
$516$	0	0
$517$	−51.8111	−2.27865
$518$	0	0
$519$	−34.5327	−1.51582
$520$	0	0
$521$	19.5896	0.858234	0.429117	−	0.903249i	$-0.358825\pi$
$521$	19.5896	0.858234	0.429117	+	0.903249i	$0.358825\pi$
$522$	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
$531$	−7.04623	−0.305780
$532$	0	0
$533$	−0.597250	−0.0258698
$534$	0	0
$535$	0.880483	0.0380666
$536$	0	0
$537$	22.1695	0.956686
$538$	0	0
$539$	−31.4359	−1.35404
$540$	0	0
$541$	−31.2158	−1.34207	−0.671035	−	0.741426i	$-0.734150\pi$
$541$	−31.2158	−1.34207	−0.671035	+	0.741426i	$0.734150\pi$
$542$	0	0
$543$	32.9311	1.41321
$544$	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
$549$	−41.8496	−1.78610
$550$	0	0
$551$	7.25240	0.308962
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−23.5877	−1.00124
$556$	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
$561$	24.8034	1.04720
$562$	0	0
$563$	13.6508	0.575312	0.287656	−	0.957734i	$-0.407124\pi$
$563$	13.6508	0.575312	0.287656	+	0.957734i	$0.407124\pi$
$564$	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	$-0.307386\pi$
$569$	27.1387	1.13771	0.568856	+	0.822437i	$0.307386\pi$
$570$	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
$573$	−18.5327	−0.774215
$574$	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
$579$	33.9634	1.41147
$580$	0	0
$581$	0	0
$582$	0	0
$583$	32.1064	1.32971
$584$	0	0
$585$	−0.775511	−0.0320634
$586$	0	0
$587$	14.9450	0.616848	0.308424	−	0.951249i	$-0.400199\pi$
$587$	14.9450	0.616848	0.308424	+	0.951249i	$0.400199\pi$
$588$	0	0
$589$	−49.0375	−2.02055
$590$	0	0
$591$	−12.4402	−0.511723
$592$	0	0
$593$	24.8015	1.01848	0.509238	−	0.860626i	$-0.329927\pi$
$593$	24.8015	1.01848	0.509238	+	0.860626i	$0.329927\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	53.7359	2.19927
$598$	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
$603$	69.6068	2.83461
$604$	0	0
$605$	14.9084	0.606112
$606$	0	0
$607$	2.78946	0.113221	0.0566104	−	0.998396i	$-0.481971\pi$
$607$	2.78946	0.113221	0.0566104	+	0.998396i	$0.481971\pi$
$608$	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	$-0.162852\pi$
$613$	43.1772	1.74391	0.871956	+	0.489585i	$0.162852\pi$
$614$	0	0
$615$	26.0192	1.04919
$616$	0	0
$617$	10.7476	0.432682	0.216341	−	0.976318i	$-0.430588\pi$
$617$	10.7476	0.432682	0.216341	+	0.976318i	$0.430588\pi$
$618$	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
$621$	24.8034	0.995327
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.67432	−0.306973
$626$	0	0
$627$	−89.9421	−3.59194
$628$	0	0
$629$	−10.5048	−0.418853
$630$	0	0
$631$	13.5231	0.538347	0.269173	−	0.963092i	$-0.413250\pi$
$631$	13.5231	0.538347	0.269173	+	0.963092i	$0.413250\pi$
$632$	0	0
$633$	4.91524	0.195363
$634$	0	0
$635$	0.335269	0.0133047
$636$	0	0
$637$	0.721586	0.0285903
$638$	0	0
$639$	55.6435	2.20122
$640$	0	0
$641$	−16.6339	−0.656999	−0.328500	−	0.944504i	$-0.606543\pi$
$641$	−16.6339	−0.656999	−0.328500	+	0.944504i	$0.606543\pi$
$642$	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
$645$	−44.9711	−1.77073
$646$	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
$657$	−8.29862	−0.323760
$658$	0	0
$659$	−43.3867	−1.69011	−0.845053	−	0.534682i	$-0.820431\pi$
$659$	−43.3867	−1.69011	−0.845053	+	0.534682i	$0.820431\pi$
$660$	0	0
$661$	0.0924575	0.00359618	0.00179809	−	0.999998i	$-0.499428\pi$
$661$	0.0924575	0.00359618	0.00179809	+	0.999998i	$0.499428\pi$
$662$	0	0
$663$	−0.569343	−0.0221115
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.52311	−0.213856
$668$	0	0
$669$	−73.2716	−2.83284
$670$	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
$675$	−10.5781	−0.407151
$676$	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
$681$	−14.8959	−0.570811
$682$	0	0
$683$	−8.54144	−0.326829	−0.163414	−	0.986558i	$-0.552251\pi$
$683$	−8.54144	−0.326829	−0.163414	+	0.986558i	$0.552251\pi$
$684$	0	0
$685$	−8.54144	−0.326352
$686$	0	0
$687$	−45.0096	−1.71722
$688$	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
$699$	23.8723	0.902935
$700$	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
$705$	−51.8111	−1.95132
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24.3651	0.915049	0.457525	−	0.889197i	$-0.348736\pi$
$709$	24.3651	0.915049	0.457525	+	0.889197i	$0.348736\pi$
$710$	0	0
$711$	9.18785	0.344571
$712$	0	0
$713$	37.3449	1.39858
$714$	0	0
$715$	−0.752820	−0.0281539
$716$	0	0
$717$	−37.3449	−1.39467
$718$	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
$723$	22.9465	0.853388
$724$	0	0
$725$	2.35548	0.0874803
$726$	0	0
$727$	−33.8130	−1.25405	−0.627027	−	0.778997i	$-0.715729\pi$
$727$	−33.8130	−1.25405	−0.627027	+	0.778997i	$0.715729\pi$
$728$	0	0
$729$	−44.0375	−1.63102
$730$	0	0
$731$	−20.0279	−0.740759
$732$	0	0
$733$	−45.7205	−1.68873	−0.844363	−	0.535771i	$-0.820021\pi$
$733$	−45.7205	−1.68873	−0.844363	+	0.535771i	$0.820021\pi$
$734$	0	0
$735$	−31.4359	−1.15953
$736$	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
$741$	2.06455	0.0758432
$742$	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
$747$	29.9634	1.09630
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.4277	−0.380513	−0.190257	−	0.981734i	$-0.560932\pi$
$751$	−10.4277	−0.380513	−0.190257	+	0.981734i	$0.560932\pi$
$752$	0	0
$753$	−54.5221	−1.98689
$754$	0	0
$755$	18.6339	0.678157
$756$	0	0
$757$	45.9267	1.66923	0.834617	−	0.550830i	$-0.185689\pi$
$757$	45.9267	1.66923	0.834617	+	0.550830i	$0.185689\pi$
$758$	0	0
$759$	68.4961	2.48625
$760$	0	0
$761$	1.58767	0.0575528	0.0287764	−	0.999586i	$-0.490839\pi$
$761$	1.58767	0.0575528	0.0287764	+	0.999586i	$0.490839\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	15.0462	0.543998
$766$	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
$771$	59.1526	2.13033
$772$	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
$783$	−4.49084	−0.160489
$784$	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
$789$	−33.9248	−1.20776
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.932519	0.0331147
$794$	0	0
$795$	32.1064	1.13870
$796$	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
$801$	58.8034	2.07772
$802$	0	0
$803$	−8.05581	−0.284283
$804$	0	0
$805$	0	0
$806$	0	0
$807$	62.4036	2.19671
$808$	0	0
$809$	32.1695	1.13102	0.565510	−	0.824741i	$-0.308679\pi$
$809$	32.1695	1.13102	0.565510	+	0.824741i	$0.308679\pi$
$810$	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
$813$	65.5683	2.29958
$814$	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	$-0.547995\pi$
$823$	−8.61850	−0.300422	−0.150211	+	0.988654i	$0.547995\pi$
$824$	0	0
$825$	−29.2120	−1.01703
$826$	0	0
$827$	−37.8636	−1.31665	−0.658323	−	0.752735i	$-0.728734\pi$
$827$	−37.8636	−1.31665	−0.658323	+	0.752735i	$0.728734\pi$
$828$	0	0
$829$	51.9634	1.80476	0.902381	−	0.430939i	$-0.141818\pi$
$829$	51.9634	1.80476	0.902381	+	0.430939i	$0.141818\pi$
$830$	0	0
$831$	4.38443	0.152094
$832$	0	0
$833$	−14.0000	−0.485071
$834$	0	0
$835$	19.5598	0.676893
$836$	0	0
$837$	30.3651	1.04957
$838$	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
$843$	−54.9465	−1.89246
$844$	0	0
$845$	−21.1233	−0.726663
$846$	0	0
$847$	0	0
$848$	0	0
$849$	57.9421	1.98857
$850$	0	0
$851$	−29.0096	−0.994436
$852$	0	0
$853$	40.5048	1.38686	0.693429	−	0.720525i	$-0.256099\pi$
$853$	40.5048	1.38686	0.693429	+	0.720525i	$0.256099\pi$
$854$	0	0
$855$	−54.5606	−1.86593
$856$	0	0
$857$	31.3834	1.07204	0.536018	−	0.844207i	$-0.319928\pi$
$857$	31.3834	1.07204	0.536018	+	0.844207i	$0.319928\pi$
$858$	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
$867$	−35.9002	−1.21924
$868$	0	0
$869$	8.91902	0.302557
$870$	0	0
$871$	−1.55102	−0.0525543
$872$	0	0
$873$	5.79383	0.196092
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.1493	1.59212	0.796060	−	0.605218i	$-0.206914\pi$
$877$	47.1493	1.59212	0.796060	+	0.605218i	$0.206914\pi$
$878$	0	0
$879$	−12.4402	−0.419599
$880$	0	0
$881$	−2.95377	−0.0995151	−0.0497575	−	0.998761i	$-0.515845\pi$
$881$	−2.95377	−0.0995151	−0.0497575	+	0.998761i	$0.515845\pi$
$882$	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
$885$	−6.84006	−0.229926
$886$	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
$891$	−6.63246	−0.222196
$892$	0	0
$893$	83.6714	2.79996
$894$	0	0
$895$	13.0550	0.436379
$896$	0	0
$897$	−1.57227	−0.0524967
$898$	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
$909$	−75.4007	−2.50088
$910$	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
$915$	−40.6252	−1.34303
$916$	0	0
$917$	0	0
$918$	0	0
$919$	42.1204	1.38942	0.694711	−	0.719289i	$-0.255532\pi$
$919$	42.1204	1.38942	0.694711	+	0.719289i	$0.255532\pi$
$920$	0	0
$921$	−26.5154	−0.873713
$922$	0	0
$923$	−1.23988	−0.0408112
$924$	0	0
$925$	12.3719	0.406786
$926$	0	0
$927$	25.5510	0.839206
$928$	0	0
$929$	−41.5144	−1.36204	−0.681021	−	0.732264i	$-0.738464\pi$
$929$	−41.5144	−1.36204	−0.681021	+	0.732264i	$0.738464\pi$
$930$	0	0
$931$	50.7668	1.66381
$932$	0	0
$933$	83.6714	2.73928
$934$	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
$939$	6.55539	0.213927
$940$	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	$-0.198938\pi$
$947$	49.9128	1.62195	0.810973	+	0.585083i	$0.198938\pi$
$948$	0	0
$949$	0.184915	0.00600259
$950$	0	0
$951$	31.8988	1.03439
$952$	0	0
$953$	39.6175	1.28334	0.641668	−	0.766983i	$-0.278243\pi$
$953$	39.6175	1.28334	0.641668	+	0.766983i	$0.278243\pi$
$954$	0	0
$955$	−10.9133	−0.353148
$956$	0	0
$957$	−12.4017	−0.400890
$958$	0	0
$959$	0	0
$960$	0	0
$961$	14.7187	0.474795
$962$	0	0
$963$	2.50479	0.0807158
$964$	0	0
$965$	20.0000	0.643823
$966$	0	0
$967$	37.6574	1.21098	0.605491	−	0.795852i	$-0.292977\pi$
$967$	37.6574	1.21098	0.605491	+	0.795852i	$0.292977\pi$
$968$	0	0
$969$	−40.0558	−1.28678
$970$	0	0
$971$	−9.15994	−0.293956	−0.146978	−	0.989140i	$-0.546955\pi$
$971$	−9.15994	−0.293956	−0.146978	+	0.989140i	$0.546955\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.670538	0.0214744
$976$	0	0
$977$	45.4758	1.45490	0.727451	−	0.686160i	$-0.240705\pi$
$977$	45.4758	1.45490	0.727451	+	0.686160i	$0.240705\pi$
$978$	0	0
$979$	57.0829	1.82438
$980$	0	0
$981$	−73.6714	−2.35215
$982$	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	$-0.566773\pi$
$991$	−13.1108	−0.416478	−0.208239	+	0.978078i	$0.566773\pi$
$992$	0	0
$993$	−34.0019	−1.07902
$994$	0	0
$995$	31.6435	1.00317
$996$	0	0
$997$	−28.8401	−0.913374	−0.456687	−	0.889627i	$-0.650964\pi$
$997$	−28.8401	−0.913374	−0.456687	+	0.889627i	$0.650964\pi$
$998$	0	0
$999$	−23.5877	−0.746281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.y.1.3		3
4.3	odd	2		1856.2.a.x.1.1		3
8.3	odd	2		232.2.a.d.1.3	✓	3
8.5	even	2		464.2.a.j.1.1		3
24.5	odd	2		4176.2.a.bu.1.3		3
24.11	even	2		2088.2.a.s.1.3		3
40.19	odd	2		5800.2.a.p.1.1		3
232.115	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	8.3	odd	2
464.2.a.j.1.1		3	8.5	even	2
1856.2.a.x.1.1		3	4.3	odd	2
1856.2.a.y.1.3		3	1.1	even	1	trivial
2088.2.a.s.1.3		3	24.11	even	2
4176.2.a.bu.1.3		3	24.5	odd	2
5800.2.a.p.1.1		3	40.19	odd	2
6728.2.a.j.1.1		3	232.115	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76156 q^{3} +1.62620 q^{5} +4.62620 q^{9} +4.49084 q^{11} -0.103084 q^{13} +4.49084 q^{15} +2.00000 q^{17} -7.25240 q^{19} +5.52311 q^{23} -2.35548 q^{25} +4.49084 q^{27} -1.00000 q^{29} +6.76156 q^{31} +12.4017 q^{33} -5.25240 q^{37} -0.284672 q^{39} +5.79383 q^{41} -10.0140 q^{43} +7.52311 q^{45} -11.5371 q^{47} -7.00000 q^{49} +5.52311 q^{51} +7.14931 q^{53} +7.30299 q^{55} -20.0279 q^{57} -1.52311 q^{59} -9.04623 q^{61} -0.167635 q^{65} +15.0462 q^{67} +15.2524 q^{69} +12.0279 q^{71} -1.79383 q^{73} -6.50479 q^{75} +1.98605 q^{79} -1.47689 q^{81} +6.47689 q^{83} +3.25240 q^{85} -2.76156 q^{87} +12.7110 q^{89} +18.6724 q^{93} -11.7938 q^{95} +1.25240 q^{97} +20.7755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43}+ \cdots + 32 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 4 * q^5 + 5 * q^9 + 2 * q^11 - 4 * q^13 + 2 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^23 + 7 * q^25 + 2 * q^27 - 3 * q^29 + 14 * q^31 - 2 * q^33 + 2 * q^37 + 18 * q^39 + 10 * q^41 - 6 * q^43 + 10 * q^45 + 2 * q^47 - 21 * q^49 + 4 * q^51 + 26 * q^55 - 12 * q^57 + 8 * q^59 - 2 * q^61 - 2 * q^65 + 20 * q^67 + 28 * q^69 - 12 * q^71 + 2 * q^73 + 16 * q^75 + 30 * q^79 - 17 * q^81 + 32 * q^83 - 8 * q^85 - 2 * q^87 + 10 * q^89 + 22 * q^93 - 28 * q^95 - 14 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more