LMFDB - Embedded newform 1456.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, 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("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +1.56155 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +1.56155 q^{5} +1.00000 q^{7} +3.56155 q^{9} -4.56155 q^{11} +1.00000 q^{13} -4.00000 q^{15} -4.00000 q^{17} +6.68466 q^{19} -2.56155 q^{21} -5.00000 q^{23} -2.56155 q^{25} -1.43845 q^{27} +5.56155 q^{29} -6.12311 q^{31} +11.6847 q^{33} +1.56155 q^{35} +5.43845 q^{37} -2.56155 q^{39} -8.56155 q^{41} +11.8078 q^{43} +5.56155 q^{45} -3.00000 q^{47} +1.00000 q^{49} +10.2462 q^{51} -8.68466 q^{53} -7.12311 q^{55} -17.1231 q^{57} -5.12311 q^{59} -13.6847 q^{61} +3.56155 q^{63} +1.56155 q^{65} -4.56155 q^{67} +12.8078 q^{69} +2.24621 q^{71} +11.2462 q^{73} +6.56155 q^{75} -4.56155 q^{77} -10.1231 q^{79} -7.00000 q^{81} -5.56155 q^{83} -6.24621 q^{85} -14.2462 q^{87} -8.43845 q^{89} +1.00000 q^{91} +15.6847 q^{93} +10.4384 q^{95} -10.3693 q^{97} -16.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9	$ 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 8 q^{15} - 8 q^{17} + q^{19} - q^{21} - 10 q^{23} - q^{25} - 7 q^{27} + 7 q^{29} - 4 q^{31} + 11 q^{33} - q^{35} + 15 q^{37} - q^{39} - 13 q^{41} + 3 q^{43} + 7 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} - 5 q^{53} - 6 q^{55} - 26 q^{57} - 2 q^{59} - 15 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} + 5 q^{69} - 12 q^{71} + 6 q^{73} + 9 q^{75} - 5 q^{77} - 12 q^{79} - 14 q^{81} - 7 q^{83} + 4 q^{85} - 12 q^{87} - 21 q^{89} + 2 q^{91} + 19 q^{93} + 25 q^{95} + 4 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 8 * q^15 - 8 * q^17 + q^19 - q^21 - 10 * q^23 - q^25 - 7 * q^27 + 7 * q^29 - 4 * q^31 + 11 * q^33 - q^35 + 15 * q^37 - q^39 - 13 * q^41 + 3 * q^43 + 7 * q^45 - 6 * q^47 + 2 * q^49 + 4 * q^51 - 5 * q^53 - 6 * q^55 - 26 * q^57 - 2 * q^59 - 15 * q^61 + 3 * q^63 - q^65 - 5 * q^67 + 5 * q^69 - 12 * q^71 + 6 * q^73 + 9 * q^75 - 5 * q^77 - 12 * q^79 - 14 * q^81 - 7 * q^83 + 4 * q^85 - 12 * q^87 - 21 * q^89 + 2 * q^91 + 19 * q^93 + 25 * q^95 + 4 * q^97 - 16 * q^99

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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−4.56155	−1.37536	−0.687680	−	0.726014i	$-0.741371\pi$
$11$	−4.56155	−1.37536	−0.687680	+	0.726014i	$0.741371\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.68466	1.53357	0.766783	−	0.641907i	$-0.221856\pi$
$19$	6.68466	1.53357	0.766783	+	0.641907i	$0.221856\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	5.56155	1.03275	0.516377	−	0.856361i	$-0.327280\pi$
$29$	5.56155	1.03275	0.516377	+	0.856361i	$0.327280\pi$
$30$	0	0
$31$	−6.12311	−1.09974	−0.549871	−	0.835250i	$-0.685323\pi$
$31$	−6.12311	−1.09974	−0.549871	+	0.835250i	$0.685323\pi$
$32$	0	0
$33$	11.6847	2.03404
$34$	0	0
$35$	1.56155	0.263951
$36$	0	0
$37$	5.43845	0.894075	0.447038	−	0.894515i	$-0.352479\pi$
$37$	5.43845	0.894075	0.447038	+	0.894515i	$0.352479\pi$
$38$	0	0
$39$	−2.56155	−0.410177
$40$	0	0
$41$	−8.56155	−1.33709	−0.668545	−	0.743672i	$-0.733083\pi$
$41$	−8.56155	−1.33709	−0.668545	+	0.743672i	$0.733083\pi$
$42$	0	0
$43$	11.8078	1.80067	0.900334	−	0.435200i	$-0.143323\pi$
$43$	11.8078	1.80067	0.900334	+	0.435200i	$0.143323\pi$
$44$	0	0
$45$	5.56155	0.829067
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.2462	1.43476
$52$	0	0
$53$	−8.68466	−1.19293	−0.596465	−	0.802639i	$-0.703428\pi$
$53$	−8.68466	−1.19293	−0.596465	+	0.802639i	$0.703428\pi$
$54$	0	0
$55$	−7.12311	−0.960479
$56$	0	0
$57$	−17.1231	−2.26801
$58$	0	0
$59$	−5.12311	−0.666972	−0.333486	−	0.942755i	$-0.608225\pi$
$59$	−5.12311	−0.666972	−0.333486	+	0.942755i	$0.608225\pi$
$60$	0	0
$61$	−13.6847	−1.75214	−0.876070	−	0.482183i	$-0.839844\pi$
$61$	−13.6847	−1.75214	−0.876070	+	0.482183i	$0.839844\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	0	0
$65$	1.56155	0.193687
$66$	0	0
$67$	−4.56155	−0.557282	−0.278641	−	0.960395i	$-0.589884\pi$
$67$	−4.56155	−0.557282	−0.278641	+	0.960395i	$0.589884\pi$
$68$	0	0
$69$	12.8078	1.54187
$70$	0	0
$71$	2.24621	0.266576	0.133288	−	0.991077i	$-0.457446\pi$
$71$	2.24621	0.266576	0.133288	+	0.991077i	$0.457446\pi$
$72$	0	0
$73$	11.2462	1.31627	0.658135	−	0.752900i	$-0.271346\pi$
$73$	11.2462	1.31627	0.658135	+	0.752900i	$0.271346\pi$
$74$	0	0
$75$	6.56155	0.757663
$76$	0	0
$77$	−4.56155	−0.519837
$78$	0	0
$79$	−10.1231	−1.13894	−0.569469	−	0.822013i	$-0.692851\pi$
$79$	−10.1231	−1.13894	−0.569469	+	0.822013i	$0.692851\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−5.56155	−0.610460	−0.305230	−	0.952279i	$-0.598733\pi$
$83$	−5.56155	−0.610460	−0.305230	+	0.952279i	$0.598733\pi$
$84$	0	0
$85$	−6.24621	−0.677497
$86$	0	0
$87$	−14.2462	−1.52735
$88$	0	0
$89$	−8.43845	−0.894474	−0.447237	−	0.894416i	$-0.647592\pi$
$89$	−8.43845	−0.894474	−0.447237	+	0.894416i	$0.647592\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	15.6847	1.62642
$94$	0	0
$95$	10.4384	1.07096
$96$	0	0
$97$	−10.3693	−1.05284	−0.526422	−	0.850223i	$-0.676467\pi$
$97$	−10.3693	−1.05284	−0.526422	+	0.850223i	$0.676467\pi$
$98$	0	0
$99$	−16.2462	−1.63281
$100$	0	0
$101$	−12.5616	−1.24992	−0.624961	−	0.780656i	$-0.714885\pi$
$101$	−12.5616	−1.24992	−0.624961	+	0.780656i	$0.714885\pi$
$102$	0	0
$103$	−13.1231	−1.29306	−0.646529	−	0.762889i	$-0.723780\pi$
$103$	−13.1231	−1.29306	−0.646529	+	0.762889i	$0.723780\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	−13.9309	−1.32226
$112$	0	0
$113$	7.24621	0.681666	0.340833	−	0.940124i	$-0.389291\pi$
$113$	7.24621	0.681666	0.340833	+	0.940124i	$0.389291\pi$
$114$	0	0
$115$	−7.80776	−0.728078
$116$	0	0
$117$	3.56155	0.329266
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	9.80776	0.891615
$122$	0	0
$123$	21.9309	1.97744
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	15.6847	1.39179	0.695894	−	0.718144i	$-0.255008\pi$
$127$	15.6847	1.39179	0.695894	+	0.718144i	$0.255008\pi$
$128$	0	0
$129$	−30.2462	−2.66303
$130$	0	0
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	0	0
$133$	6.68466	0.579633
$134$	0	0
$135$	−2.24621	−0.193323
$136$	0	0
$137$	−7.36932	−0.629603	−0.314802	−	0.949157i	$-0.601938\pi$
$137$	−7.36932	−0.629603	−0.314802	+	0.949157i	$0.601938\pi$
$138$	0	0
$139$	−8.87689	−0.752928	−0.376464	−	0.926431i	$-0.622860\pi$
$139$	−8.87689	−0.752928	−0.376464	+	0.926431i	$0.622860\pi$
$140$	0	0
$141$	7.68466	0.647165
$142$	0	0
$143$	−4.56155	−0.381456
$144$	0	0
$145$	8.68466	0.721222
$146$	0	0
$147$	−2.56155	−0.211273
$148$	0	0
$149$	2.31534	0.189680	0.0948401	−	0.995493i	$-0.469766\pi$
$149$	2.31534	0.189680	0.0948401	+	0.995493i	$0.469766\pi$
$150$	0	0
$151$	5.75379	0.468237	0.234118	−	0.972208i	$-0.424780\pi$
$151$	5.75379	0.468237	0.234118	+	0.972208i	$0.424780\pi$
$152$	0	0
$153$	−14.2462	−1.15174
$154$	0	0
$155$	−9.56155	−0.768002
$156$	0	0
$157$	−12.8078	−1.02217	−0.511085	−	0.859530i	$-0.670756\pi$
$157$	−12.8078	−1.02217	−0.511085	+	0.859530i	$0.670756\pi$
$158$	0	0
$159$	22.2462	1.76424
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	−18.2462	−1.42915	−0.714577	−	0.699557i	$-0.753381\pi$
$163$	−18.2462	−1.42915	−0.714577	+	0.699557i	$0.753381\pi$
$164$	0	0
$165$	18.2462	1.42047
$166$	0	0
$167$	−2.43845	−0.188693	−0.0943464	−	0.995539i	$-0.530076\pi$
$167$	−2.43845	−0.188693	−0.0943464	+	0.995539i	$0.530076\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	23.8078	1.82063
$172$	0	0
$173$	−0.630683	−0.0479499	−0.0239750	−	0.999713i	$-0.507632\pi$
$173$	−0.630683	−0.0479499	−0.0239750	+	0.999713i	$0.507632\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	0	0
$177$	13.1231	0.986393
$178$	0	0
$179$	20.4384	1.52764	0.763821	−	0.645429i	$-0.223321\pi$
$179$	20.4384	1.52764	0.763821	+	0.645429i	$0.223321\pi$
$180$	0	0
$181$	−6.80776	−0.506017	−0.253009	−	0.967464i	$-0.581420\pi$
$181$	−6.80776	−0.506017	−0.253009	+	0.967464i	$0.581420\pi$
$182$	0	0
$183$	35.0540	2.59126
$184$	0	0
$185$	8.49242	0.624375
$186$	0	0
$187$	18.2462	1.33430
$188$	0	0
$189$	−1.43845	−0.104632
$190$	0	0
$191$	12.4924	0.903920	0.451960	−	0.892038i	$-0.350725\pi$
$191$	12.4924	0.903920	0.451960	+	0.892038i	$0.350725\pi$
$192$	0	0
$193$	8.87689	0.638973	0.319486	−	0.947591i	$-0.396490\pi$
$193$	8.87689	0.638973	0.319486	+	0.947591i	$0.396490\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	10.8078	0.770021	0.385011	−	0.922912i	$-0.374198\pi$
$197$	10.8078	0.770021	0.385011	+	0.922912i	$0.374198\pi$
$198$	0	0
$199$	14.2462	1.00989	0.504944	−	0.863152i	$-0.331513\pi$
$199$	14.2462	1.00989	0.504944	+	0.863152i	$0.331513\pi$
$200$	0	0
$201$	11.6847	0.824172
$202$	0	0
$203$	5.56155	0.390344
$204$	0	0
$205$	−13.3693	−0.933754
$206$	0	0
$207$	−17.8078	−1.23773
$208$	0	0
$209$	−30.4924	−2.10920
$210$	0	0
$211$	−17.8078	−1.22594	−0.612969	−	0.790107i	$-0.710025\pi$
$211$	−17.8078	−1.22594	−0.612969	+	0.790107i	$0.710025\pi$
$212$	0	0
$213$	−5.75379	−0.394243
$214$	0	0
$215$	18.4384	1.25749
$216$	0	0
$217$	−6.12311	−0.415663
$218$	0	0
$219$	−28.8078	−1.94665
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−26.3693	−1.76582	−0.882910	−	0.469542i	$-0.844419\pi$
$223$	−26.3693	−1.76582	−0.882910	+	0.469542i	$0.844419\pi$
$224$	0	0
$225$	−9.12311	−0.608207
$226$	0	0
$227$	16.4924	1.09464	0.547320	−	0.836923i	$-0.315648\pi$
$227$	16.4924	1.09464	0.547320	+	0.836923i	$0.315648\pi$
$228$	0	0
$229$	23.1231	1.52802	0.764009	−	0.645206i	$-0.223228\pi$
$229$	23.1231	1.52802	0.764009	+	0.645206i	$0.223228\pi$
$230$	0	0
$231$	11.6847	0.768794
$232$	0	0
$233$	9.24621	0.605739	0.302870	−	0.953032i	$-0.402055\pi$
$233$	9.24621	0.605739	0.302870	+	0.953032i	$0.402055\pi$
$234$	0	0
$235$	−4.68466	−0.305593
$236$	0	0
$237$	25.9309	1.68439
$238$	0	0
$239$	−11.3693	−0.735420	−0.367710	−	0.929941i	$-0.619858\pi$
$239$	−11.3693	−0.735420	−0.367710	+	0.929941i	$0.619858\pi$
$240$	0	0
$241$	−8.93087	−0.575288	−0.287644	−	0.957737i	$-0.592872\pi$
$241$	−8.93087	−0.575288	−0.287644	+	0.957737i	$0.592872\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	1.56155	0.0997639
$246$	0	0
$247$	6.68466	0.425335
$248$	0	0
$249$	14.2462	0.902817
$250$	0	0
$251$	6.80776	0.429702	0.214851	−	0.976647i	$-0.431073\pi$
$251$	6.80776	0.429702	0.214851	+	0.976647i	$0.431073\pi$
$252$	0	0
$253$	22.8078	1.43391
$254$	0	0
$255$	16.0000	1.00196
$256$	0	0
$257$	−13.3693	−0.833955	−0.416978	−	0.908917i	$-0.636911\pi$
$257$	−13.3693	−0.833955	−0.416978	+	0.908917i	$0.636911\pi$
$258$	0	0
$259$	5.43845	0.337929
$260$	0	0
$261$	19.8078	1.22607
$262$	0	0
$263$	22.9309	1.41398	0.706989	−	0.707225i	$-0.250053\pi$
$263$	22.9309	1.41398	0.706989	+	0.707225i	$0.250053\pi$
$264$	0	0
$265$	−13.5616	−0.833080
$266$	0	0
$267$	21.6155	1.32285
$268$	0	0
$269$	7.19224	0.438518	0.219259	−	0.975667i	$-0.429636\pi$
$269$	7.19224	0.438518	0.219259	+	0.975667i	$0.429636\pi$
$270$	0	0
$271$	−20.3153	−1.23407	−0.617035	−	0.786936i	$-0.711666\pi$
$271$	−20.3153	−1.23407	−0.617035	+	0.786936i	$0.711666\pi$
$272$	0	0
$273$	−2.56155	−0.155032
$274$	0	0
$275$	11.6847	0.704611
$276$	0	0
$277$	21.8078	1.31030	0.655151	−	0.755498i	$-0.272605\pi$
$277$	21.8078	1.31030	0.655151	+	0.755498i	$0.272605\pi$
$278$	0	0
$279$	−21.8078	−1.30560
$280$	0	0
$281$	−0.876894	−0.0523111	−0.0261556	−	0.999658i	$-0.508327\pi$
$281$	−0.876894	−0.0523111	−0.0261556	+	0.999658i	$0.508327\pi$
$282$	0	0
$283$	5.43845	0.323282	0.161641	−	0.986850i	$-0.448321\pi$
$283$	5.43845	0.323282	0.161641	+	0.986850i	$0.448321\pi$
$284$	0	0
$285$	−26.7386	−1.58386
$286$	0	0
$287$	−8.56155	−0.505372
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	26.5616	1.55707
$292$	0	0
$293$	6.68466	0.390522	0.195261	−	0.980751i	$-0.437445\pi$
$293$	6.68466	0.390522	0.195261	+	0.980751i	$0.437445\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	6.56155	0.380740
$298$	0	0
$299$	−5.00000	−0.289157
$300$	0	0
$301$	11.8078	0.680588
$302$	0	0
$303$	32.1771	1.84853
$304$	0	0
$305$	−21.3693	−1.22360
$306$	0	0
$307$	33.1771	1.89352	0.946758	−	0.321946i	$-0.104337\pi$
$307$	33.1771	1.89352	0.946758	+	0.321946i	$0.104337\pi$
$308$	0	0
$309$	33.6155	1.91232
$310$	0	0
$311$	4.24621	0.240781	0.120390	−	0.992727i	$-0.461585\pi$
$311$	4.24621	0.240781	0.120390	+	0.992727i	$0.461585\pi$
$312$	0	0
$313$	10.4924	0.593067	0.296533	−	0.955022i	$-0.404169\pi$
$313$	10.4924	0.593067	0.296533	+	0.955022i	$0.404169\pi$
$314$	0	0
$315$	5.56155	0.313358
$316$	0	0
$317$	−11.0540	−0.620853	−0.310427	−	0.950597i	$-0.600472\pi$
$317$	−11.0540	−0.620853	−0.310427	+	0.950597i	$0.600472\pi$
$318$	0	0
$319$	−25.3693	−1.42041
$320$	0	0
$321$	10.2462	0.571888
$322$	0	0
$323$	−26.7386	−1.48778
$324$	0	0
$325$	−2.56155	−0.142089
$326$	0	0
$327$	21.1231	1.16811
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−33.0540	−1.81681	−0.908405	−	0.418090i	$-0.862700\pi$
$331$	−33.0540	−1.81681	−0.908405	+	0.418090i	$0.862700\pi$
$332$	0	0
$333$	19.3693	1.06143
$334$	0	0
$335$	−7.12311	−0.389177
$336$	0	0
$337$	1.00000	0.0544735	0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000	0.0544735	0.0272367	+	0.999629i	$0.491329\pi$
$338$	0	0
$339$	−18.5616	−1.00813
$340$	0	0
$341$	27.9309	1.51254
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	20.0000	1.07676
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	1.31534	0.0704086	0.0352043	−	0.999380i	$-0.488792\pi$
$349$	1.31534	0.0704086	0.0352043	+	0.999380i	$0.488792\pi$
$350$	0	0
$351$	−1.43845	−0.0767786
$352$	0	0
$353$	−4.56155	−0.242787	−0.121393	−	0.992604i	$-0.538736\pi$
$353$	−4.56155	−0.242787	−0.121393	+	0.992604i	$0.538736\pi$
$354$	0	0
$355$	3.50758	0.186163
$356$	0	0
$357$	10.2462	0.542287
$358$	0	0
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	0	0
$361$	25.6847	1.35182
$362$	0	0
$363$	−25.1231	−1.31862
$364$	0	0
$365$	17.5616	0.919214
$366$	0	0
$367$	11.6155	0.606326	0.303163	−	0.952939i	$-0.401957\pi$
$367$	11.6155	0.606326	0.303163	+	0.952939i	$0.401957\pi$
$368$	0	0
$369$	−30.4924	−1.58737
$370$	0	0
$371$	−8.68466	−0.450885
$372$	0	0
$373$	27.1231	1.40438	0.702191	−	0.711989i	$-0.252205\pi$
$373$	27.1231	1.40438	0.702191	+	0.711989i	$0.252205\pi$
$374$	0	0
$375$	30.2462	1.56191
$376$	0	0
$377$	5.56155	0.286435
$378$	0	0
$379$	−37.8617	−1.94483	−0.972413	−	0.233264i	$-0.925059\pi$
$379$	−37.8617	−1.94483	−0.972413	+	0.233264i	$0.925059\pi$
$380$	0	0
$381$	−40.1771	−2.05833
$382$	0	0
$383$	−10.5616	−0.539670	−0.269835	−	0.962907i	$-0.586969\pi$
$383$	−10.5616	−0.539670	−0.269835	+	0.962907i	$0.586969\pi$
$384$	0	0
$385$	−7.12311	−0.363027
$386$	0	0
$387$	42.0540	2.13772
$388$	0	0
$389$	31.6155	1.60297	0.801485	−	0.598014i	$-0.204043\pi$
$389$	31.6155	1.60297	0.801485	+	0.598014i	$0.204043\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	0	0
$393$	−4.49242	−0.226613
$394$	0	0
$395$	−15.8078	−0.795375
$396$	0	0
$397$	12.1922	0.611911	0.305955	−	0.952046i	$-0.401024\pi$
$397$	12.1922	0.611911	0.305955	+	0.952046i	$0.401024\pi$
$398$	0	0
$399$	−17.1231	−0.857227
$400$	0	0
$401$	−28.4924	−1.42284	−0.711422	−	0.702765i	$-0.751948\pi$
$401$	−28.4924	−1.42284	−0.711422	+	0.702765i	$0.751948\pi$
$402$	0	0
$403$	−6.12311	−0.305014
$404$	0	0
$405$	−10.9309	−0.543159
$406$	0	0
$407$	−24.8078	−1.22968
$408$	0	0
$409$	22.3002	1.10267	0.551337	−	0.834283i	$-0.314118\pi$
$409$	22.3002	1.10267	0.551337	+	0.834283i	$0.314118\pi$
$410$	0	0
$411$	18.8769	0.931129
$412$	0	0
$413$	−5.12311	−0.252092
$414$	0	0
$415$	−8.68466	−0.426313
$416$	0	0
$417$	22.7386	1.11352
$418$	0	0
$419$	−30.1771	−1.47425	−0.737123	−	0.675758i	$-0.763816\pi$
$419$	−30.1771	−1.47425	−0.737123	+	0.675758i	$0.763816\pi$
$420$	0	0
$421$	17.3002	0.843160	0.421580	−	0.906791i	$-0.361476\pi$
$421$	17.3002	0.843160	0.421580	+	0.906791i	$0.361476\pi$
$422$	0	0
$423$	−10.6847	−0.519506
$424$	0	0
$425$	10.2462	0.497014
$426$	0	0
$427$	−13.6847	−0.662247
$428$	0	0
$429$	11.6847	0.564141
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	1.12311	0.0539730	0.0269865	−	0.999636i	$-0.491409\pi$
$433$	1.12311	0.0539730	0.0269865	+	0.999636i	$0.491409\pi$
$434$	0	0
$435$	−22.2462	−1.06662
$436$	0	0
$437$	−33.4233	−1.59885
$438$	0	0
$439$	24.2462	1.15721	0.578604	−	0.815608i	$-0.303598\pi$
$439$	24.2462	1.15721	0.578604	+	0.815608i	$0.303598\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	−22.9309	−1.08948	−0.544739	−	0.838605i	$-0.683371\pi$
$443$	−22.9309	−1.08948	−0.544739	+	0.838605i	$0.683371\pi$
$444$	0	0
$445$	−13.1771	−0.624654
$446$	0	0
$447$	−5.93087	−0.280521
$448$	0	0
$449$	−38.1080	−1.79843	−0.899213	−	0.437512i	$-0.855860\pi$
$449$	−38.1080	−1.79843	−0.899213	+	0.437512i	$0.855860\pi$
$450$	0	0
$451$	39.0540	1.83898
$452$	0	0
$453$	−14.7386	−0.692481
$454$	0	0
$455$	1.56155	0.0732067
$456$	0	0
$457$	22.8769	1.07014	0.535068	−	0.844809i	$-0.320286\pi$
$457$	22.8769	1.07014	0.535068	+	0.844809i	$0.320286\pi$
$458$	0	0
$459$	5.75379	0.268564
$460$	0	0
$461$	39.6155	1.84508	0.922540	−	0.385903i	$-0.126110\pi$
$461$	39.6155	1.84508	0.922540	+	0.385903i	$0.126110\pi$
$462$	0	0
$463$	7.12311	0.331039	0.165519	−	0.986207i	$-0.447070\pi$
$463$	7.12311	0.331039	0.165519	+	0.986207i	$0.447070\pi$
$464$	0	0
$465$	24.4924	1.13581
$466$	0	0
$467$	−33.8617	−1.56693	−0.783467	−	0.621433i	$-0.786551\pi$
$467$	−33.8617	−1.56693	−0.783467	+	0.621433i	$0.786551\pi$
$468$	0	0
$469$	−4.56155	−0.210633
$470$	0	0
$471$	32.8078	1.51170
$472$	0	0
$473$	−53.8617	−2.47657
$474$	0	0
$475$	−17.1231	−0.785662
$476$	0	0
$477$	−30.9309	−1.41623
$478$	0	0
$479$	−2.43845	−0.111415	−0.0557077	−	0.998447i	$-0.517742\pi$
$479$	−2.43845	−0.111415	−0.0557077	+	0.998447i	$0.517742\pi$
$480$	0	0
$481$	5.43845	0.247972
$482$	0	0
$483$	12.8078	0.582773
$484$	0	0
$485$	−16.1922	−0.735252
$486$	0	0
$487$	32.7386	1.48353	0.741765	−	0.670660i	$-0.233989\pi$
$487$	32.7386	1.48353	0.741765	+	0.670660i	$0.233989\pi$
$488$	0	0
$489$	46.7386	2.11359
$490$	0	0
$491$	17.7538	0.801217	0.400609	−	0.916249i	$-0.368799\pi$
$491$	17.7538	0.801217	0.400609	+	0.916249i	$0.368799\pi$
$492$	0	0
$493$	−22.2462	−1.00192
$494$	0	0
$495$	−25.3693	−1.14027
$496$	0	0
$497$	2.24621	0.100756
$498$	0	0
$499$	−6.06913	−0.271692	−0.135846	−	0.990730i	$-0.543375\pi$
$499$	−6.06913	−0.271692	−0.135846	+	0.990730i	$0.543375\pi$
$500$	0	0
$501$	6.24621	0.279060
$502$	0	0
$503$	−27.1231	−1.20936	−0.604680	−	0.796469i	$-0.706699\pi$
$503$	−27.1231	−1.20936	−0.604680	+	0.796469i	$0.706699\pi$
$504$	0	0
$505$	−19.6155	−0.872880
$506$	0	0
$507$	−2.56155	−0.113763
$508$	0	0
$509$	−29.8078	−1.32121	−0.660603	−	0.750735i	$-0.729699\pi$
$509$	−29.8078	−1.32121	−0.660603	+	0.750735i	$0.729699\pi$
$510$	0	0
$511$	11.2462	0.497503
$512$	0	0
$513$	−9.61553	−0.424536
$514$	0	0
$515$	−20.4924	−0.903004
$516$	0	0
$517$	13.6847	0.601851
$518$	0	0
$519$	1.61553	0.0709138
$520$	0	0
$521$	−41.1231	−1.80164	−0.900818	−	0.434197i	$-0.857032\pi$
$521$	−41.1231	−1.80164	−0.900818	+	0.434197i	$0.857032\pi$
$522$	0	0
$523$	−9.19224	−0.401948	−0.200974	−	0.979597i	$-0.564411\pi$
$523$	−9.19224	−0.401948	−0.200974	+	0.979597i	$0.564411\pi$
$524$	0	0
$525$	6.56155	0.286370
$526$	0	0
$527$	24.4924	1.06691
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	−18.2462	−0.791818
$532$	0	0
$533$	−8.56155	−0.370842
$534$	0	0
$535$	−6.24621	−0.270047
$536$	0	0
$537$	−52.3542	−2.25925
$538$	0	0
$539$	−4.56155	−0.196480
$540$	0	0
$541$	14.8769	0.639608	0.319804	−	0.947484i	$-0.396383\pi$
$541$	14.8769	0.639608	0.319804	+	0.947484i	$0.396383\pi$
$542$	0	0
$543$	17.4384	0.748355
$544$	0	0
$545$	−12.8769	−0.551586
$546$	0	0
$547$	−9.56155	−0.408822	−0.204411	−	0.978885i	$-0.565528\pi$
$547$	−9.56155	−0.408822	−0.204411	+	0.978885i	$0.565528\pi$
$548$	0	0
$549$	−48.7386	−2.08011
$550$	0	0
$551$	37.1771	1.58380
$552$	0	0
$553$	−10.1231	−0.430478
$554$	0	0
$555$	−21.7538	−0.923397
$556$	0	0
$557$	11.6847	0.495095	0.247547	−	0.968876i	$-0.420375\pi$
$557$	11.6847	0.495095	0.247547	+	0.968876i	$0.420375\pi$
$558$	0	0
$559$	11.8078	0.499415
$560$	0	0
$561$	−46.7386	−1.97331
$562$	0	0
$563$	17.9309	0.755696	0.377848	−	0.925868i	$-0.376664\pi$
$563$	17.9309	0.755696	0.377848	+	0.925868i	$0.376664\pi$
$564$	0	0
$565$	11.3153	0.476040
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	34.8617	1.46148	0.730740	−	0.682656i	$-0.239175\pi$
$569$	34.8617	1.46148	0.730740	+	0.682656i	$0.239175\pi$
$570$	0	0
$571$	10.1922	0.426532	0.213266	−	0.976994i	$-0.431590\pi$
$571$	10.1922	0.426532	0.213266	+	0.976994i	$0.431590\pi$
$572$	0	0
$573$	−32.0000	−1.33682
$574$	0	0
$575$	12.8078	0.534121
$576$	0	0
$577$	22.9848	0.956872	0.478436	−	0.878123i	$-0.341204\pi$
$577$	22.9848	0.956872	0.478436	+	0.878123i	$0.341204\pi$
$578$	0	0
$579$	−22.7386	−0.944985
$580$	0	0
$581$	−5.56155	−0.230732
$582$	0	0
$583$	39.6155	1.64071
$584$	0	0
$585$	5.56155	0.229942
$586$	0	0
$587$	32.5464	1.34333	0.671667	−	0.740853i	$-0.265579\pi$
$587$	32.5464	1.34333	0.671667	+	0.740853i	$0.265579\pi$
$588$	0	0
$589$	−40.9309	−1.68653
$590$	0	0
$591$	−27.6847	−1.13879
$592$	0	0
$593$	4.93087	0.202487	0.101243	−	0.994862i	$-0.467718\pi$
$593$	4.93087	0.202487	0.101243	+	0.994862i	$0.467718\pi$
$594$	0	0
$595$	−6.24621	−0.256070
$596$	0	0
$597$	−36.4924	−1.49354
$598$	0	0
$599$	−14.6155	−0.597174	−0.298587	−	0.954382i	$-0.596515\pi$
$599$	−14.6155	−0.597174	−0.298587	+	0.954382i	$0.596515\pi$
$600$	0	0
$601$	−1.50758	−0.0614954	−0.0307477	−	0.999527i	$-0.509789\pi$
$601$	−1.50758	−0.0614954	−0.0307477	+	0.999527i	$0.509789\pi$
$602$	0	0
$603$	−16.2462	−0.661597
$604$	0	0
$605$	15.3153	0.622657
$606$	0	0
$607$	−38.9848	−1.58235	−0.791173	−	0.611592i	$-0.790529\pi$
$607$	−38.9848	−1.58235	−0.791173	+	0.611592i	$0.790529\pi$
$608$	0	0
$609$	−14.2462	−0.577286
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	26.5616	1.07281	0.536406	−	0.843960i	$-0.319782\pi$
$613$	26.5616	1.07281	0.536406	+	0.843960i	$0.319782\pi$
$614$	0	0
$615$	34.2462	1.38094
$616$	0	0
$617$	41.2311	1.65990	0.829950	−	0.557838i	$-0.188369\pi$
$617$	41.2311	1.65990	0.829950	+	0.557838i	$0.188369\pi$
$618$	0	0
$619$	−33.1231	−1.33133	−0.665665	−	0.746251i	$-0.731852\pi$
$619$	−33.1231	−1.33133	−0.665665	+	0.746251i	$0.731852\pi$
$620$	0	0
$621$	7.19224	0.288614
$622$	0	0
$623$	−8.43845	−0.338079
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	78.1080	3.11933
$628$	0	0
$629$	−21.7538	−0.867380
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	0	0
$633$	45.6155	1.81305
$634$	0	0
$635$	24.4924	0.971952
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	36.8617	1.45595	0.727976	−	0.685603i	$-0.240461\pi$
$641$	36.8617	1.45595	0.727976	+	0.685603i	$0.240461\pi$
$642$	0	0
$643$	13.7538	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$643$	13.7538	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$644$	0	0
$645$	−47.2311	−1.85972
$646$	0	0
$647$	−9.36932	−0.368346	−0.184173	−	0.982894i	$-0.558961\pi$
$647$	−9.36932	−0.368346	−0.184173	+	0.982894i	$0.558961\pi$
$648$	0	0
$649$	23.3693	0.917326
$650$	0	0
$651$	15.6847	0.614730
$652$	0	0
$653$	−35.1231	−1.37447	−0.687237	−	0.726434i	$-0.741176\pi$
$653$	−35.1231	−1.37447	−0.687237	+	0.726434i	$0.741176\pi$
$654$	0	0
$655$	2.73863	0.107007
$656$	0	0
$657$	40.0540	1.56265
$658$	0	0
$659$	46.7926	1.82278	0.911391	−	0.411542i	$-0.135010\pi$
$659$	46.7926	1.82278	0.911391	+	0.411542i	$0.135010\pi$
$660$	0	0
$661$	−1.06913	−0.0415843	−0.0207922	−	0.999784i	$-0.506619\pi$
$661$	−1.06913	−0.0415843	−0.0207922	+	0.999784i	$0.506619\pi$
$662$	0	0
$663$	10.2462	0.397930
$664$	0	0
$665$	10.4384	0.404786
$666$	0	0
$667$	−27.8078	−1.07672
$668$	0	0
$669$	67.5464	2.61149
$670$	0	0
$671$	62.4233	2.40982
$672$	0	0
$673$	28.8617	1.11254	0.556269	−	0.831002i	$-0.312232\pi$
$673$	28.8617	1.11254	0.556269	+	0.831002i	$0.312232\pi$
$674$	0	0
$675$	3.68466	0.141823
$676$	0	0
$677$	40.4233	1.55359	0.776797	−	0.629751i	$-0.216843\pi$
$677$	40.4233	1.55359	0.776797	+	0.629751i	$0.216843\pi$
$678$	0	0
$679$	−10.3693	−0.397938
$680$	0	0
$681$	−42.2462	−1.61888
$682$	0	0
$683$	27.0540	1.03519	0.517596	−	0.855625i	$-0.326827\pi$
$683$	27.0540	1.03519	0.517596	+	0.855625i	$0.326827\pi$
$684$	0	0
$685$	−11.5076	−0.439682
$686$	0	0
$687$	−59.2311	−2.25981
$688$	0	0
$689$	−8.68466	−0.330859
$690$	0	0
$691$	26.9309	1.02450	0.512249	−	0.858837i	$-0.328812\pi$
$691$	26.9309	1.02450	0.512249	+	0.858837i	$0.328812\pi$
$692$	0	0
$693$	−16.2462	−0.617143
$694$	0	0
$695$	−13.8617	−0.525806
$696$	0	0
$697$	34.2462	1.29717
$698$	0	0
$699$	−23.6847	−0.895836
$700$	0	0
$701$	−9.06913	−0.342536	−0.171268	−	0.985224i	$-0.554786\pi$
$701$	−9.06913	−0.342536	−0.171268	+	0.985224i	$0.554786\pi$
$702$	0	0
$703$	36.3542	1.37112
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	−12.5616	−0.472426
$708$	0	0
$709$	6.17708	0.231985	0.115993	−	0.993250i	$-0.462995\pi$
$709$	6.17708	0.231985	0.115993	+	0.993250i	$0.462995\pi$
$710$	0	0
$711$	−36.0540	−1.35213
$712$	0	0
$713$	30.6155	1.14656
$714$	0	0
$715$	−7.12311	−0.266389
$716$	0	0
$717$	29.1231	1.08762
$718$	0	0
$719$	−32.2462	−1.20258	−0.601290	−	0.799031i	$-0.705347\pi$
$719$	−32.2462	−1.20258	−0.601290	+	0.799031i	$0.705347\pi$
$720$	0	0
$721$	−13.1231	−0.488730
$722$	0	0
$723$	22.8769	0.850801
$724$	0	0
$725$	−14.2462	−0.529091
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−47.2311	−1.74690
$732$	0	0
$733$	−21.0691	−0.778206	−0.389103	−	0.921194i	$-0.627215\pi$
$733$	−21.0691	−0.778206	−0.389103	+	0.921194i	$0.627215\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	20.8078	0.766464
$738$	0	0
$739$	25.3693	0.933225	0.466613	−	0.884462i	$-0.345474\pi$
$739$	25.3693	0.933225	0.466613	+	0.884462i	$0.345474\pi$
$740$	0	0
$741$	−17.1231	−0.629033
$742$	0	0
$743$	−6.73863	−0.247216	−0.123608	−	0.992331i	$-0.539447\pi$
$743$	−6.73863	−0.247216	−0.123608	+	0.992331i	$0.539447\pi$
$744$	0	0
$745$	3.61553	0.132463
$746$	0	0
$747$	−19.8078	−0.724728
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	0	0
$753$	−17.4384	−0.635492
$754$	0	0
$755$	8.98485	0.326992
$756$	0	0
$757$	−21.4233	−0.778643	−0.389321	−	0.921102i	$-0.627290\pi$
$757$	−21.4233	−0.778643	−0.389321	+	0.921102i	$0.627290\pi$
$758$	0	0
$759$	−58.4233	−2.12063
$760$	0	0
$761$	−3.38447	−0.122687	−0.0613435	−	0.998117i	$-0.519539\pi$
$761$	−3.38447	−0.122687	−0.0613435	+	0.998117i	$0.519539\pi$
$762$	0	0
$763$	−8.24621	−0.298533
$764$	0	0
$765$	−22.2462	−0.804313
$766$	0	0
$767$	−5.12311	−0.184985
$768$	0	0
$769$	−50.6155	−1.82524	−0.912621	−	0.408806i	$-0.865945\pi$
$769$	−50.6155	−1.82524	−0.912621	+	0.408806i	$0.865945\pi$
$770$	0	0
$771$	34.2462	1.23335
$772$	0	0
$773$	40.7386	1.46527	0.732633	−	0.680623i	$-0.238291\pi$
$773$	40.7386	1.46527	0.732633	+	0.680623i	$0.238291\pi$
$774$	0	0
$775$	15.6847	0.563410
$776$	0	0
$777$	−13.9309	−0.499767
$778$	0	0
$779$	−57.2311	−2.05052
$780$	0	0
$781$	−10.2462	−0.366638
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	27.8078	0.991240	0.495620	−	0.868540i	$-0.334941\pi$
$787$	27.8078	0.991240	0.495620	+	0.868540i	$0.334941\pi$
$788$	0	0
$789$	−58.7386	−2.09115
$790$	0	0
$791$	7.24621	0.257646
$792$	0	0
$793$	−13.6847	−0.485956
$794$	0	0
$795$	34.7386	1.23205
$796$	0	0
$797$	−16.5616	−0.586640	−0.293320	−	0.956014i	$-0.594760\pi$
$797$	−16.5616	−0.586640	−0.293320	+	0.956014i	$0.594760\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−30.0540	−1.06191
$802$	0	0
$803$	−51.3002	−1.81034
$804$	0	0
$805$	−7.80776	−0.275188
$806$	0	0
$807$	−18.4233	−0.648531
$808$	0	0
$809$	−51.6695	−1.81660	−0.908302	−	0.418316i	$-0.862620\pi$
$809$	−51.6695	−1.81660	−0.908302	+	0.418316i	$0.862620\pi$
$810$	0	0
$811$	16.4924	0.579127	0.289564	−	0.957159i	$-0.406490\pi$
$811$	16.4924	0.579127	0.289564	+	0.957159i	$0.406490\pi$
$812$	0	0
$813$	52.0388	1.82508
$814$	0	0
$815$	−28.4924	−0.998046
$816$	0	0
$817$	78.9309	2.76144
$818$	0	0
$819$	3.56155	0.124451
$820$	0	0
$821$	1.50758	0.0526148	0.0263074	−	0.999654i	$-0.491625\pi$
$821$	1.50758	0.0526148	0.0263074	+	0.999654i	$0.491625\pi$
$822$	0	0
$823$	−43.0540	−1.50077	−0.750384	−	0.661003i	$-0.770131\pi$
$823$	−43.0540	−1.50077	−0.750384	+	0.661003i	$0.770131\pi$
$824$	0	0
$825$	−29.9309	−1.04206
$826$	0	0
$827$	46.7386	1.62526	0.812631	−	0.582779i	$-0.198035\pi$
$827$	46.7386	1.62526	0.812631	+	0.582779i	$0.198035\pi$
$828$	0	0
$829$	−11.7538	−0.408226	−0.204113	−	0.978947i	$-0.565431\pi$
$829$	−11.7538	−0.408226	−0.204113	+	0.978947i	$0.565431\pi$
$830$	0	0
$831$	−55.8617	−1.93782
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	−3.80776	−0.131773
$836$	0	0
$837$	8.80776	0.304441
$838$	0	0
$839$	16.1771	0.558495	0.279247	−	0.960219i	$-0.409915\pi$
$839$	16.1771	0.558495	0.279247	+	0.960219i	$0.409915\pi$
$840$	0	0
$841$	1.93087	0.0665817
$842$	0	0
$843$	2.24621	0.0773636
$844$	0	0
$845$	1.56155	0.0537190
$846$	0	0
$847$	9.80776	0.336999
$848$	0	0
$849$	−13.9309	−0.478106
$850$	0	0
$851$	−27.1922	−0.932138
$852$	0	0
$853$	5.06913	0.173564	0.0867819	−	0.996227i	$-0.472342\pi$
$853$	5.06913	0.173564	0.0867819	+	0.996227i	$0.472342\pi$
$854$	0	0
$855$	37.1771	1.27143
$856$	0	0
$857$	32.2462	1.10151	0.550755	−	0.834667i	$-0.314340\pi$
$857$	32.2462	1.10151	0.550755	+	0.834667i	$0.314340\pi$
$858$	0	0
$859$	30.6695	1.04643	0.523215	−	0.852201i	$-0.324732\pi$
$859$	30.6695	1.04643	0.523215	+	0.852201i	$0.324732\pi$
$860$	0	0
$861$	21.9309	0.747402
$862$	0	0
$863$	−14.2462	−0.484947	−0.242473	−	0.970158i	$-0.577959\pi$
$863$	−14.2462	−0.484947	−0.242473	+	0.970158i	$0.577959\pi$
$864$	0	0
$865$	−0.984845	−0.0334857
$866$	0	0
$867$	2.56155	0.0869949
$868$	0	0
$869$	46.1771	1.56645
$870$	0	0
$871$	−4.56155	−0.154562
$872$	0	0
$873$	−36.9309	−1.24992
$874$	0	0
$875$	−11.8078	−0.399175
$876$	0	0
$877$	−20.1771	−0.681332	−0.340666	−	0.940184i	$-0.610652\pi$
$877$	−20.1771	−0.681332	−0.340666	+	0.940184i	$0.610652\pi$
$878$	0	0
$879$	−17.1231	−0.577548
$880$	0	0
$881$	−22.9848	−0.774379	−0.387190	−	0.922000i	$-0.626554\pi$
$881$	−22.9848	−0.774379	−0.387190	+	0.922000i	$0.626554\pi$
$882$	0	0
$883$	−25.1231	−0.845460	−0.422730	−	0.906256i	$-0.638928\pi$
$883$	−25.1231	−0.845460	−0.422730	+	0.906256i	$0.638928\pi$
$884$	0	0
$885$	20.4924	0.688845
$886$	0	0
$887$	−12.6307	−0.424097	−0.212048	−	0.977259i	$-0.568013\pi$
$887$	−12.6307	−0.424097	−0.212048	+	0.977259i	$0.568013\pi$
$888$	0	0
$889$	15.6847	0.526047
$890$	0	0
$891$	31.9309	1.06972
$892$	0	0
$893$	−20.0540	−0.671081
$894$	0	0
$895$	31.9157	1.06682
$896$	0	0
$897$	12.8078	0.427639
$898$	0	0
$899$	−34.0540	−1.13576
$900$	0	0
$901$	34.7386	1.15731
$902$	0	0
$903$	−30.2462	−1.00653
$904$	0	0
$905$	−10.6307	−0.353376
$906$	0	0
$907$	−28.5464	−0.947868	−0.473934	−	0.880560i	$-0.657166\pi$
$907$	−28.5464	−0.947868	−0.473934	+	0.880560i	$0.657166\pi$
$908$	0	0
$909$	−44.7386	−1.48389
$910$	0	0
$911$	9.17708	0.304050	0.152025	−	0.988377i	$-0.451421\pi$
$911$	9.17708	0.304050	0.152025	+	0.988377i	$0.451421\pi$
$912$	0	0
$913$	25.3693	0.839602
$914$	0	0
$915$	54.7386	1.80960
$916$	0	0
$917$	1.75379	0.0579152
$918$	0	0
$919$	−38.9157	−1.28371	−0.641855	−	0.766826i	$-0.721835\pi$
$919$	−38.9157	−1.28371	−0.641855	+	0.766826i	$0.721835\pi$
$920$	0	0
$921$	−84.9848	−2.80035
$922$	0	0
$923$	2.24621	0.0739349
$924$	0	0
$925$	−13.9309	−0.458044
$926$	0	0
$927$	−46.7386	−1.53510
$928$	0	0
$929$	−57.3542	−1.88173	−0.940865	−	0.338783i	$-0.889985\pi$
$929$	−57.3542	−1.88173	−0.940865	+	0.338783i	$0.889985\pi$
$930$	0	0
$931$	6.68466	0.219081
$932$	0	0
$933$	−10.8769	−0.356094
$934$	0	0
$935$	28.4924	0.931802
$936$	0	0
$937$	−10.7386	−0.350816	−0.175408	−	0.984496i	$-0.556124\pi$
$937$	−10.7386	−0.350816	−0.175408	+	0.984496i	$0.556124\pi$
$938$	0	0
$939$	−26.8769	−0.877094
$940$	0	0
$941$	−51.6695	−1.68438	−0.842189	−	0.539183i	$-0.818733\pi$
$941$	−51.6695	−1.68438	−0.842189	+	0.539183i	$0.818733\pi$
$942$	0	0
$943$	42.8078	1.39401
$944$	0	0
$945$	−2.24621	−0.0730693
$946$	0	0
$947$	−3.12311	−0.101487	−0.0507436	−	0.998712i	$-0.516159\pi$
$947$	−3.12311	−0.101487	−0.0507436	+	0.998712i	$0.516159\pi$
$948$	0	0
$949$	11.2462	0.365067
$950$	0	0
$951$	28.3153	0.918188
$952$	0	0
$953$	37.3153	1.20876	0.604381	−	0.796695i	$-0.293420\pi$
$953$	37.3153	1.20876	0.604381	+	0.796695i	$0.293420\pi$
$954$	0	0
$955$	19.5076	0.631250
$956$	0	0
$957$	64.9848	2.10066
$958$	0	0
$959$	−7.36932	−0.237968
$960$	0	0
$961$	6.49242	0.209433
$962$	0	0
$963$	−14.2462	−0.459078
$964$	0	0
$965$	13.8617	0.446225
$966$	0	0
$967$	39.7538	1.27840	0.639198	−	0.769042i	$-0.279267\pi$
$967$	39.7538	1.27840	0.639198	+	0.769042i	$0.279267\pi$
$968$	0	0
$969$	68.4924	2.20029
$970$	0	0
$971$	−2.80776	−0.0901054	−0.0450527	−	0.998985i	$-0.514346\pi$
$971$	−2.80776	−0.0901054	−0.0450527	+	0.998985i	$0.514346\pi$
$972$	0	0
$973$	−8.87689	−0.284580
$974$	0	0
$975$	6.56155	0.210138
$976$	0	0
$977$	−14.4924	−0.463654	−0.231827	−	0.972757i	$-0.574470\pi$
$977$	−14.4924	−0.463654	−0.231827	+	0.972757i	$0.574470\pi$
$978$	0	0
$979$	38.4924	1.23022
$980$	0	0
$981$	−29.3693	−0.937690
$982$	0	0
$983$	16.3002	0.519895	0.259948	−	0.965623i	$-0.416295\pi$
$983$	16.3002	0.519895	0.259948	+	0.965623i	$0.416295\pi$
$984$	0	0
$985$	16.8769	0.537743
$986$	0	0
$987$	7.68466	0.244605
$988$	0	0
$989$	−59.0388	−1.87733
$990$	0	0
$991$	19.5464	0.620912	0.310456	−	0.950588i	$-0.399518\pi$
$991$	19.5464	0.620912	0.310456	+	0.950588i	$0.399518\pi$
$992$	0	0
$993$	84.6695	2.68691
$994$	0	0
$995$	22.2462	0.705252
$996$	0	0
$997$	1.43845	0.0455561	0.0227780	−	0.999741i	$-0.492749\pi$
$997$	1.43845	0.0455561	0.0227780	+	0.999741i	$0.492749\pi$
$998$	0	0
$999$	−7.82292	−0.247506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.o.1.1		2
4.3	odd	2		728.2.a.g.1.2	✓	2
8.3	odd	2		5824.2.a.bj.1.1		2
8.5	even	2		5824.2.a.bo.1.2		2
12.11	even	2		6552.2.a.bg.1.1		2
28.27	even	2		5096.2.a.n.1.1		2
52.51	odd	2		9464.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.g.1.2	✓	2	4.3	odd	2
1456.2.a.o.1.1		2	1.1	even	1	trivial
5096.2.a.n.1.1		2	28.27	even	2
5824.2.a.bj.1.1		2	8.3	odd	2
5824.2.a.bo.1.2		2	8.5	even	2
6552.2.a.bg.1.1		2	12.11	even	2
9464.2.a.r.1.2		2	52.51	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +1.56155 q^{5} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +1.56155 q^{5} +1.00000 q^{7} +3.56155 q^{9} -4.56155 q^{11} +1.00000 q^{13} -4.00000 q^{15} -4.00000 q^{17} +6.68466 q^{19} -2.56155 q^{21} -5.00000 q^{23} -2.56155 q^{25} -1.43845 q^{27} +5.56155 q^{29} -6.12311 q^{31} +11.6847 q^{33} +1.56155 q^{35} +5.43845 q^{37} -2.56155 q^{39} -8.56155 q^{41} +11.8078 q^{43} +5.56155 q^{45} -3.00000 q^{47} +1.00000 q^{49} +10.2462 q^{51} -8.68466 q^{53} -7.12311 q^{55} -17.1231 q^{57} -5.12311 q^{59} -13.6847 q^{61} +3.56155 q^{63} +1.56155 q^{65} -4.56155 q^{67} +12.8078 q^{69} +2.24621 q^{71} +11.2462 q^{73} +6.56155 q^{75} -4.56155 q^{77} -10.1231 q^{79} -7.00000 q^{81} -5.56155 q^{83} -6.24621 q^{85} -14.2462 q^{87} -8.43845 q^{89} +1.00000 q^{91} +15.6847 q^{93} +10.4384 q^{95} -10.3693 q^{97} -16.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9	\( 2 q - q^{3} - q^{5} + 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 8 q^{15} - 8 q^{17} + q^{19} - q^{21} - 10 q^{23} - q^{25} - 7 q^{27} + 7 q^{29} - 4 q^{31} + 11 q^{33} - q^{35} + 15 q^{37} - q^{39} - 13 q^{41} + 3 q^{43} + 7 q^{45} - 6 q^{47} + 2 q^{49} + 4 q^{51} - 5 q^{53} - 6 q^{55} - 26 q^{57} - 2 q^{59} - 15 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} + 5 q^{69} - 12 q^{71} + 6 q^{73} + 9 q^{75} - 5 q^{77} - 12 q^{79} - 14 q^{81} - 7 q^{83} + 4 q^{85} - 12 q^{87} - 21 q^{89} + 2 q^{91} + 19 q^{93} + 25 q^{95} + 4 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 8 * q^15 - 8 * q^17 + q^19 - q^21 - 10 * q^23 - q^25 - 7 * q^27 + 7 * q^29 - 4 * q^31 + 11 * q^33 - q^35 + 15 * q^37 - q^39 - 13 * q^41 + 3 * q^43 + 7 * q^45 - 6 * q^47 + 2 * q^49 + 4 * q^51 - 5 * q^53 - 6 * q^55 - 26 * q^57 - 2 * q^59 - 15 * q^61 + 3 * q^63 - q^65 - 5 * q^67 + 5 * q^69 - 12 * q^71 + 6 * q^73 + 9 * q^75 - 5 * q^77 - 12 * q^79 - 14 * q^81 - 7 * q^83 + 4 * q^85 - 12 * q^87 - 21 * q^89 + 2 * q^91 + 19 * q^93 + 25 * q^95 + 4 * q^97 - 16 * q^99

Introduction

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