LMFDB - Embedded newform 456.6.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,6,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("456.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	456.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:	$73.1350218347$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 $ x^6 - 2x^5 - 4725x^4 + 92430x^3 + 1610577x^2 - 16081740*x - 24661341
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$9.09727$ of defining polynomial
Character	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +68.0602 q^{5} -181.508 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +68.0602 q^{5} -181.508 q^{7} +81.0000 q^{9} +56.2292 q^{11} -183.089 q^{13} -612.542 q^{15} +2031.56 q^{17} +361.000 q^{19} +1633.57 q^{21} -4003.43 q^{23} +1507.19 q^{25} -729.000 q^{27} -4073.95 q^{29} +6141.15 q^{31} -506.063 q^{33} -12353.4 q^{35} +3425.06 q^{37} +1647.80 q^{39} -6813.12 q^{41} -2183.21 q^{43} +5512.87 q^{45} -8348.99 q^{47} +16138.0 q^{49} -18284.0 q^{51} +39002.6 q^{53} +3826.97 q^{55} -3249.00 q^{57} -30218.1 q^{59} +12329.1 q^{61} -14702.1 q^{63} -12461.0 q^{65} -8132.92 q^{67} +36030.9 q^{69} -11559.2 q^{71} -28897.5 q^{73} -13564.7 q^{75} -10206.0 q^{77} +164.115 q^{79} +6561.00 q^{81} +13470.3 q^{83} +138268. q^{85} +36665.5 q^{87} -103567. q^{89} +33232.0 q^{91} -55270.3 q^{93} +24569.7 q^{95} -34490.7 q^{97} +4554.57 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	68.0602	1.21750	0.608749	−	0.793363i	$-0.291672\pi$
$5$	68.0602	1.21750	0.608749	+	0.793363i	$0.291672\pi$
$6$	0	0
$7$	−181.508	−1.40007	−0.700035	−	0.714108i	$-0.746832\pi$
$7$	−181.508	−1.40007	−0.700035	+	0.714108i	$0.746832\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	56.2292	0.140114	0.0700568	−	0.997543i	$-0.477682\pi$
$11$	56.2292	0.140114	0.0700568	+	0.997543i	$0.477682\pi$
$12$	0	0
$13$	−183.089	−0.300471	−0.150236	−	0.988650i	$-0.548003\pi$
$13$	−183.089	−0.300471	−0.150236	+	0.988650i	$0.548003\pi$
$14$	0	0
$15$	−612.542	−0.702922
$16$	0	0
$17$	2031.56	1.70493	0.852465	−	0.522785i	$-0.175107\pi$
$17$	2031.56	1.70493	0.852465	+	0.522785i	$0.175107\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	1633.57	0.808331
$22$	0	0
$23$	−4003.43	−1.57802	−0.789010	−	0.614380i	$-0.789406\pi$
$23$	−4003.43	−1.57802	−0.789010	+	0.614380i	$0.789406\pi$
$24$	0	0
$25$	1507.19	0.482300
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4073.95	−0.899540	−0.449770	−	0.893144i	$-0.648494\pi$
$29$	−4073.95	−0.899540	−0.449770	+	0.893144i	$0.648494\pi$
$30$	0	0
$31$	6141.15	1.14774	0.573872	−	0.818945i	$-0.305441\pi$
$31$	6141.15	1.14774	0.573872	+	0.818945i	$0.305441\pi$
$32$	0	0
$33$	−506.063	−0.0808947
$34$	0	0
$35$	−12353.4	−1.70458
$36$	0	0
$37$	3425.06	0.411305	0.205653	−	0.978625i	$-0.434068\pi$
$37$	3425.06	0.411305	0.205653	+	0.978625i	$0.434068\pi$
$38$	0	0
$39$	1647.80	0.173477
$40$	0	0
$41$	−6813.12	−0.632974	−0.316487	−	0.948597i	$-0.602503\pi$
$41$	−6813.12	−0.632974	−0.316487	+	0.948597i	$0.602503\pi$
$42$	0	0
$43$	−2183.21	−0.180063	−0.0900315	−	0.995939i	$-0.528697\pi$
$43$	−2183.21	−0.180063	−0.0900315	+	0.995939i	$0.528697\pi$
$44$	0	0
$45$	5512.87	0.405832
$46$	0	0
$47$	−8348.99	−0.551302	−0.275651	−	0.961258i	$-0.588893\pi$
$47$	−8348.99	−0.551302	−0.275651	+	0.961258i	$0.588893\pi$
$48$	0	0
$49$	16138.0	0.960198
$50$	0	0
$51$	−18284.0	−0.984341
$52$	0	0
$53$	39002.6	1.90723	0.953616	−	0.301027i	$-0.0973293\pi$
$53$	39002.6	1.90723	0.953616	+	0.301027i	$0.0973293\pi$
$54$	0	0
$55$	3826.97	0.170588
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−30218.1	−1.13015	−0.565076	−	0.825039i	$-0.691153\pi$
$59$	−30218.1	−1.13015	−0.565076	+	0.825039i	$0.691153\pi$
$60$	0	0
$61$	12329.1	0.424237	0.212118	−	0.977244i	$-0.431964\pi$
$61$	12329.1	0.424237	0.212118	+	0.977244i	$0.431964\pi$
$62$	0	0
$63$	−14702.1	−0.466690
$64$	0	0
$65$	−12461.0	−0.365823
$66$	0	0
$67$	−8132.92	−0.221340	−0.110670	−	0.993857i	$-0.535300\pi$
$67$	−8132.92	−0.221340	−0.110670	+	0.993857i	$0.535300\pi$
$68$	0	0
$69$	36030.9	0.911071
$70$	0	0
$71$	−11559.2	−0.272134	−0.136067	−	0.990700i	$-0.543446\pi$
$71$	−11559.2	−0.272134	−0.136067	+	0.990700i	$0.543446\pi$
$72$	0	0
$73$	−28897.5	−0.634677	−0.317338	−	0.948312i	$-0.602789\pi$
$73$	−28897.5	−0.634677	−0.317338	+	0.948312i	$0.602789\pi$
$74$	0	0
$75$	−13564.7	−0.278456
$76$	0	0
$77$	−10206.0	−0.196169
$78$	0	0
$79$	164.115	0.00295856	0.00147928	−	0.999999i	$-0.499529\pi$
$79$	164.115	0.00295856	0.00147928	+	0.999999i	$0.499529\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	13470.3	0.214625	0.107313	−	0.994225i	$-0.465775\pi$
$83$	13470.3	0.214625	0.107313	+	0.994225i	$0.465775\pi$
$84$	0	0
$85$	138268.	2.07575
$86$	0	0
$87$	36665.5	0.519350
$88$	0	0
$89$	−103567.	−1.38595	−0.692975	−	0.720961i	$-0.743700\pi$
$89$	−103567.	−1.38595	−0.692975	+	0.720961i	$0.743700\pi$
$90$	0	0
$91$	33232.0	0.420681
$92$	0	0
$93$	−55270.3	−0.662651
$94$	0	0
$95$	24569.7	0.279313
$96$	0	0
$97$	−34490.7	−0.372197	−0.186099	−	0.982531i	$-0.559584\pi$
$97$	−34490.7	−0.372197	−0.186099	+	0.982531i	$0.559584\pi$
$98$	0	0
$99$	4554.57	0.0467046
$100$	0	0
$101$	−72989.7	−0.711964	−0.355982	−	0.934493i	$-0.615854\pi$
$101$	−72989.7	−0.711964	−0.355982	+	0.934493i	$0.615854\pi$
$102$	0	0
$103$	−15482.1	−0.143793	−0.0718964	−	0.997412i	$-0.522905\pi$
$103$	−15482.1	−0.143793	−0.0718964	+	0.997412i	$0.522905\pi$
$104$	0	0
$105$	111181.	0.984141
$106$	0	0
$107$	−48215.9	−0.407128	−0.203564	−	0.979062i	$-0.565252\pi$
$107$	−48215.9	−0.407128	−0.203564	+	0.979062i	$0.565252\pi$
$108$	0	0
$109$	−151242.	−1.21929	−0.609643	−	0.792676i	$-0.708687\pi$
$109$	−151242.	−1.21929	−0.609643	+	0.792676i	$0.708687\pi$
$110$	0	0
$111$	−30825.6	−0.237467
$112$	0	0
$113$	−233254.	−1.71843	−0.859217	−	0.511612i	$-0.829049\pi$
$113$	−233254.	−1.71843	−0.859217	+	0.511612i	$0.829049\pi$
$114$	0	0
$115$	−272474.	−1.92124
$116$	0	0
$117$	−14830.2	−0.100157
$118$	0	0
$119$	−368743.	−2.38702
$120$	0	0
$121$	−157889.	−0.980368
$122$	0	0
$123$	61318.0	0.365448
$124$	0	0
$125$	−110109.	−0.630299
$126$	0	0
$127$	−200268.	−1.10180	−0.550900	−	0.834571i	$-0.685715\pi$
$127$	−200268.	−1.10180	−0.550900	+	0.834571i	$0.685715\pi$
$128$	0	0
$129$	19648.9	0.103959
$130$	0	0
$131$	−4663.95	−0.0237452	−0.0118726	−	0.999930i	$-0.503779\pi$
$131$	−4663.95	−0.0237452	−0.0118726	+	0.999930i	$0.503779\pi$
$132$	0	0
$133$	−65524.3	−0.321198
$134$	0	0
$135$	−49615.9	−0.234307
$136$	0	0
$137$	−25589.9	−0.116484	−0.0582421	−	0.998302i	$-0.518550\pi$
$137$	−25589.9	−0.116484	−0.0582421	+	0.998302i	$0.518550\pi$
$138$	0	0
$139$	−67288.8	−0.295397	−0.147698	−	0.989032i	$-0.547187\pi$
$139$	−67288.8	−0.295397	−0.147698	+	0.989032i	$0.547187\pi$
$140$	0	0
$141$	75140.9	0.318294
$142$	0	0
$143$	−10294.9	−0.0421002
$144$	0	0
$145$	−277274.	−1.09519
$146$	0	0
$147$	−145242.	−0.554370
$148$	0	0
$149$	−236573.	−0.872969	−0.436485	−	0.899712i	$-0.643777\pi$
$149$	−236573.	−0.872969	−0.436485	+	0.899712i	$0.643777\pi$
$150$	0	0
$151$	−96930.5	−0.345954	−0.172977	−	0.984926i	$-0.555339\pi$
$151$	−96930.5	−0.345954	−0.172977	+	0.984926i	$0.555339\pi$
$152$	0	0
$153$	164556.	0.568310
$154$	0	0
$155$	417968.	1.39738
$156$	0	0
$157$	−51875.0	−0.167961	−0.0839806	−	0.996467i	$-0.526763\pi$
$157$	−51875.0	−0.167961	−0.0839806	+	0.996467i	$0.526763\pi$
$158$	0	0
$159$	−351023.	−1.10114
$160$	0	0
$161$	726653.	2.20934
$162$	0	0
$163$	−194876.	−0.574499	−0.287250	−	0.957856i	$-0.592741\pi$
$163$	−194876.	−0.574499	−0.287250	+	0.957856i	$0.592741\pi$
$164$	0	0
$165$	−34442.8	−0.0984891
$166$	0	0
$167$	512937.	1.42322	0.711611	−	0.702573i	$-0.247966\pi$
$167$	512937.	1.42322	0.711611	+	0.702573i	$0.247966\pi$
$168$	0	0
$169$	−337772.	−0.909717
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	344170.	0.874295	0.437148	−	0.899390i	$-0.355989\pi$
$173$	344170.	0.874295	0.437148	+	0.899390i	$0.355989\pi$
$174$	0	0
$175$	−273566.	−0.675254
$176$	0	0
$177$	271963.	0.652494
$178$	0	0
$179$	−535947.	−1.25023	−0.625114	−	0.780534i	$-0.714947\pi$
$179$	−535947.	−1.25023	−0.625114	+	0.780534i	$0.714947\pi$
$180$	0	0
$181$	717175.	1.62715	0.813577	−	0.581457i	$-0.197517\pi$
$181$	717175.	1.62715	0.813577	+	0.581457i	$0.197517\pi$
$182$	0	0
$183$	−110962.	−0.244933
$184$	0	0
$185$	233110.	0.500763
$186$	0	0
$187$	114233.	0.238884
$188$	0	0
$189$	132319.	0.269444
$190$	0	0
$191$	−616524.	−1.22283	−0.611416	−	0.791310i	$-0.709400\pi$
$191$	−616524.	−1.22283	−0.611416	+	0.791310i	$0.709400\pi$
$192$	0	0
$193$	−297746.	−0.575377	−0.287688	−	0.957724i	$-0.592887\pi$
$193$	−297746.	−0.575377	−0.287688	+	0.957724i	$0.592887\pi$
$194$	0	0
$195$	112149.	0.211208
$196$	0	0
$197$	750295.	1.37742	0.688710	−	0.725037i	$-0.258177\pi$
$197$	750295.	1.37742	0.688710	+	0.725037i	$0.258177\pi$
$198$	0	0
$199$	−447059.	−0.800262	−0.400131	−	0.916458i	$-0.631035\pi$
$199$	−447059.	−0.800262	−0.400131	+	0.916458i	$0.631035\pi$
$200$	0	0
$201$	73196.3	0.127791
$202$	0	0
$203$	739453.	1.25942
$204$	0	0
$205$	−463702.	−0.770645
$206$	0	0
$207$	−324278.	−0.526007
$208$	0	0
$209$	20298.8	0.0321443
$210$	0	0
$211$	765080.	1.18304	0.591522	−	0.806289i	$-0.298527\pi$
$211$	765080.	1.18304	0.591522	+	0.806289i	$0.298527\pi$
$212$	0	0
$213$	104033.	0.157117
$214$	0	0
$215$	−148590.	−0.219226
$216$	0	0
$217$	−1.11467e6	−1.60692
$218$	0	0
$219$	260077.	0.366431
$220$	0	0
$221$	−371955.	−0.512283
$222$	0	0
$223$	624803.	0.841358	0.420679	−	0.907210i	$-0.361792\pi$
$223$	624803.	0.841358	0.420679	+	0.907210i	$0.361792\pi$
$224$	0	0
$225$	122082.	0.160767
$226$	0	0
$227$	−8488.97	−0.0109343	−0.00546714	−	0.999985i	$-0.501740\pi$
$227$	−8488.97	−0.0109343	−0.00546714	+	0.999985i	$0.501740\pi$
$228$	0	0
$229$	502157.	0.632777	0.316388	−	0.948630i	$-0.397530\pi$
$229$	502157.	0.632777	0.316388	+	0.948630i	$0.397530\pi$
$230$	0	0
$231$	91854.4	0.113258
$232$	0	0
$233$	−319233.	−0.385228	−0.192614	−	0.981275i	$-0.561696\pi$
$233$	−319233.	−0.385228	−0.192614	+	0.981275i	$0.561696\pi$
$234$	0	0
$235$	−568234.	−0.671209
$236$	0	0
$237$	−1477.04	−0.00170813
$238$	0	0
$239$	−756618.	−0.856805	−0.428402	−	0.903588i	$-0.640923\pi$
$239$	−756618.	−0.856805	−0.428402	+	0.903588i	$0.640923\pi$
$240$	0	0
$241$	−645205.	−0.715575	−0.357788	−	0.933803i	$-0.616469\pi$
$241$	−645205.	−0.715575	−0.357788	+	0.933803i	$0.616469\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.09836e6	1.16904
$246$	0	0
$247$	−66095.0	−0.0689329
$248$	0	0
$249$	−121232.	−0.123914
$250$	0	0
$251$	−1.69048e6	−1.69366	−0.846829	−	0.531866i	$-0.821491\pi$
$251$	−1.69048e6	−1.69366	−0.846829	+	0.531866i	$0.821491\pi$
$252$	0	0
$253$	−225110.	−0.221102
$254$	0	0
$255$	−1.24441e6	−1.19843
$256$	0	0
$257$	−200205.	−0.189079	−0.0945393	−	0.995521i	$-0.530138\pi$
$257$	−200205.	−0.189079	−0.0945393	+	0.995521i	$0.530138\pi$
$258$	0	0
$259$	−621675.	−0.575857
$260$	0	0
$261$	−329990.	−0.299847
$262$	0	0
$263$	−791636.	−0.705726	−0.352863	−	0.935675i	$-0.614792\pi$
$263$	−791636.	−0.705726	−0.352863	+	0.935675i	$0.614792\pi$
$264$	0	0
$265$	2.65452e6	2.32205
$266$	0	0
$267$	932106.	0.800179
$268$	0	0
$269$	679190.	0.572283	0.286141	−	0.958187i	$-0.407627\pi$
$269$	679190.	0.572283	0.286141	+	0.958187i	$0.407627\pi$
$270$	0	0
$271$	−1.62480e6	−1.34393	−0.671965	−	0.740583i	$-0.734549\pi$
$271$	−1.62480e6	−1.34393	−0.671965	+	0.740583i	$0.734549\pi$
$272$	0	0
$273$	−299088.	−0.242880
$274$	0	0
$275$	84748.0	0.0675768
$276$	0	0
$277$	−2.46645e6	−1.93141	−0.965703	−	0.259651i	$-0.916392\pi$
$277$	−2.46645e6	−1.93141	−0.965703	+	0.259651i	$0.916392\pi$
$278$	0	0
$279$	497433.	0.382582
$280$	0	0
$281$	955572.	0.721934	0.360967	−	0.932579i	$-0.382447\pi$
$281$	955572.	0.721934	0.360967	+	0.932579i	$0.382447\pi$
$282$	0	0
$283$	−152472.	−0.113168	−0.0565841	−	0.998398i	$-0.518021\pi$
$283$	−152472.	−0.113168	−0.0565841	+	0.998398i	$0.518021\pi$
$284$	0	0
$285$	−221127.	−0.161261
$286$	0	0
$287$	1.23663e6	0.886209
$288$	0	0
$289$	2.70736e6	1.90678
$290$	0	0
$291$	310417.	0.214888
$292$	0	0
$293$	2.54402e6	1.73121	0.865607	−	0.500724i	$-0.166933\pi$
$293$	2.54402e6	1.73121	0.865607	+	0.500724i	$0.166933\pi$
$294$	0	0
$295$	−2.05665e6	−1.37596
$296$	0	0
$297$	−40991.1	−0.0269649
$298$	0	0
$299$	732983.	0.474150
$300$	0	0
$301$	396270.	0.252101
$302$	0	0
$303$	656907.	0.411053
$304$	0	0
$305$	839124.	0.516507
$306$	0	0
$307$	1.13060e6	0.684644	0.342322	−	0.939583i	$-0.388787\pi$
$307$	1.13060e6	0.684644	0.342322	+	0.939583i	$0.388787\pi$
$308$	0	0
$309$	139339.	0.0830188
$310$	0	0
$311$	3.39201e6	1.98864	0.994320	−	0.106428i	$-0.0339415\pi$
$311$	3.39201e6	1.98864	0.994320	+	0.106428i	$0.0339415\pi$
$312$	0	0
$313$	430684.	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$313$	430684.	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$314$	0	0
$315$	−1.00063e6	−0.568194
$316$	0	0
$317$	−3.42810e6	−1.91604	−0.958022	−	0.286695i	$-0.907443\pi$
$317$	−3.42810e6	−1.91604	−0.958022	+	0.286695i	$0.907443\pi$
$318$	0	0
$319$	−229075.	−0.126038
$320$	0	0
$321$	433943.	0.235055
$322$	0	0
$323$	733392.	0.391138
$324$	0	0
$325$	−275949.	−0.144917
$326$	0	0
$327$	1.36118e6	0.703955
$328$	0	0
$329$	1.51541e6	0.771862
$330$	0	0
$331$	−21956.4	−0.0110152	−0.00550758	−	0.999985i	$-0.501753\pi$
$331$	−21956.4	−0.0110152	−0.00550758	+	0.999985i	$0.501753\pi$
$332$	0	0
$333$	277430.	0.137102
$334$	0	0
$335$	−553528.	−0.269481
$336$	0	0
$337$	2.54253e6	1.21953	0.609764	−	0.792583i	$-0.291264\pi$
$337$	2.54253e6	1.21953	0.609764	+	0.792583i	$0.291264\pi$
$338$	0	0
$339$	2.09928e6	0.992138
$340$	0	0
$341$	345312.	0.160815
$342$	0	0
$343$	121421.	0.0557262
$344$	0	0
$345$	2.45227e6	1.10923
$346$	0	0
$347$	−1.25363e6	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$347$	−1.25363e6	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$348$	0	0
$349$	1.36606e6	0.600352	0.300176	−	0.953884i	$-0.402955\pi$
$349$	1.36606e6	0.600352	0.300176	+	0.953884i	$0.402955\pi$
$350$	0	0
$351$	133472.	0.0578258
$352$	0	0
$353$	3.16721e6	1.35282	0.676411	−	0.736525i	$-0.263534\pi$
$353$	3.16721e6	1.35282	0.676411	+	0.736525i	$0.263534\pi$
$354$	0	0
$355$	−786723.	−0.331322
$356$	0	0
$357$	3.31869e6	1.37815
$358$	0	0
$359$	2.40859e6	0.986342	0.493171	−	0.869932i	$-0.335838\pi$
$359$	2.40859e6	0.986342	0.493171	+	0.869932i	$0.335838\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	1.42100e6	0.566016
$364$	0	0
$365$	−1.96677e6	−0.772717
$366$	0	0
$367$	2.29575e6	0.889734	0.444867	−	0.895597i	$-0.353251\pi$
$367$	2.29575e6	0.889734	0.444867	+	0.895597i	$0.353251\pi$
$368$	0	0
$369$	−551862.	−0.210991
$370$	0	0
$371$	−7.07926e6	−2.67026
$372$	0	0
$373$	−4.59398e6	−1.70969	−0.854844	−	0.518885i	$-0.826347\pi$
$373$	−4.59398e6	−1.70969	−0.854844	+	0.518885i	$0.826347\pi$
$374$	0	0
$375$	990978.	0.363903
$376$	0	0
$377$	745894.	0.270286
$378$	0	0
$379$	−1.51884e6	−0.543143	−0.271572	−	0.962418i	$-0.587543\pi$
$379$	−1.51884e6	−0.543143	−0.271572	+	0.962418i	$0.587543\pi$
$380$	0	0
$381$	1.80241e6	0.636125
$382$	0	0
$383$	796963.	0.277614	0.138807	−	0.990319i	$-0.455673\pi$
$383$	796963.	0.277614	0.138807	+	0.990319i	$0.455673\pi$
$384$	0	0
$385$	−694625.	−0.238835
$386$	0	0
$387$	−176840.	−0.0600210
$388$	0	0
$389$	1.08768e6	0.364440	0.182220	−	0.983258i	$-0.441672\pi$
$389$	1.08768e6	0.364440	0.182220	+	0.983258i	$0.441672\pi$
$390$	0	0
$391$	−8.13319e6	−2.69041
$392$	0	0
$393$	41975.5	0.0137093
$394$	0	0
$395$	11169.7	0.00360204
$396$	0	0
$397$	−90856.2	−0.0289320	−0.0144660	−	0.999895i	$-0.504605\pi$
$397$	−90856.2	−0.0289320	−0.0144660	+	0.999895i	$0.504605\pi$
$398$	0	0
$399$	589718.	0.185444
$400$	0	0
$401$	−3.42123e6	−1.06248	−0.531241	−	0.847221i	$-0.678274\pi$
$401$	−3.42123e6	−1.06248	−0.531241	+	0.847221i	$0.678274\pi$
$402$	0	0
$403$	−1.12437e6	−0.344864
$404$	0	0
$405$	446543.	0.135277
$406$	0	0
$407$	192589.	0.0576295
$408$	0	0
$409$	−1.71976e6	−0.508346	−0.254173	−	0.967159i	$-0.581803\pi$
$409$	−1.71976e6	−0.508346	−0.254173	+	0.967159i	$0.581803\pi$
$410$	0	0
$411$	230309.	0.0672522
$412$	0	0
$413$	5.48481e6	1.58229
$414$	0	0
$415$	916789.	0.261306
$416$	0	0
$417$	605599.	0.170547
$418$	0	0
$419$	6.64727e6	1.84973	0.924864	−	0.380297i	$-0.124178\pi$
$419$	6.64727e6	1.84973	0.924864	+	0.380297i	$0.124178\pi$
$420$	0	0
$421$	4.48576e6	1.23348	0.616739	−	0.787168i	$-0.288454\pi$
$421$	4.48576e6	1.23348	0.616739	+	0.787168i	$0.288454\pi$
$422$	0	0
$423$	−676268.	−0.183767
$424$	0	0
$425$	3.06193e6	0.822287
$426$	0	0
$427$	−2.23783e6	−0.593961
$428$	0	0
$429$	92654.5	0.0243065
$430$	0	0
$431$	4.50573e6	1.16835	0.584174	−	0.811629i	$-0.301419\pi$
$431$	4.50573e6	1.16835	0.584174	+	0.811629i	$0.301419\pi$
$432$	0	0
$433$	6.74569e6	1.72905	0.864524	−	0.502592i	$-0.167620\pi$
$433$	6.74569e6	1.72905	0.864524	+	0.502592i	$0.167620\pi$
$434$	0	0
$435$	2.49546e6	0.632307
$436$	0	0
$437$	−1.44524e6	−0.362023
$438$	0	0
$439$	−5.54556e6	−1.37336	−0.686679	−	0.726960i	$-0.740932\pi$
$439$	−5.54556e6	−1.37336	−0.686679	+	0.726960i	$0.740932\pi$
$440$	0	0
$441$	1.30718e6	0.320066
$442$	0	0
$443$	1.36139e6	0.329589	0.164795	−	0.986328i	$-0.447304\pi$
$443$	1.36139e6	0.329589	0.164795	+	0.986328i	$0.447304\pi$
$444$	0	0
$445$	−7.04881e6	−1.68739
$446$	0	0
$447$	2.12915e6	0.504009
$448$	0	0
$449$	−577806.	−0.135259	−0.0676295	−	0.997711i	$-0.521544\pi$
$449$	−577806.	−0.135259	−0.0676295	+	0.997711i	$0.521544\pi$
$450$	0	0
$451$	−383096.	−0.0886884
$452$	0	0
$453$	872375.	0.199737
$454$	0	0
$455$	2.26178e6	0.512178
$456$	0	0
$457$	4.53412e6	1.01555	0.507777	−	0.861489i	$-0.330468\pi$
$457$	4.53412e6	1.01555	0.507777	+	0.861489i	$0.330468\pi$
$458$	0	0
$459$	−1.48100e6	−0.328114
$460$	0	0
$461$	−7.58278e6	−1.66179	−0.830895	−	0.556429i	$-0.812171\pi$
$461$	−7.58278e6	−1.66179	−0.830895	+	0.556429i	$0.812171\pi$
$462$	0	0
$463$	−8.74510e6	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$463$	−8.74510e6	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$464$	0	0
$465$	−3.76171e6	−0.806775
$466$	0	0
$467$	3.25842e6	0.691378	0.345689	−	0.938349i	$-0.387645\pi$
$467$	3.25842e6	0.691378	0.345689	+	0.938349i	$0.387645\pi$
$468$	0	0
$469$	1.47619e6	0.309891
$470$	0	0
$471$	466875.	0.0969724
$472$	0	0
$473$	−122760.	−0.0252293
$474$	0	0
$475$	544095.	0.110647
$476$	0	0
$477$	3.15921e6	0.635744
$478$	0	0
$479$	−481496.	−0.0958857	−0.0479429	−	0.998850i	$-0.515267\pi$
$479$	−481496.	−0.0958857	−0.0479429	+	0.998850i	$0.515267\pi$
$480$	0	0
$481$	−627091.	−0.123586
$482$	0	0
$483$	−6.53988e6	−1.27556
$484$	0	0
$485$	−2.34745e6	−0.453149
$486$	0	0
$487$	4.42000e6	0.844500	0.422250	−	0.906479i	$-0.361240\pi$
$487$	4.42000e6	0.844500	0.422250	+	0.906479i	$0.361240\pi$
$488$	0	0
$489$	1.75388e6	0.331687
$490$	0	0
$491$	−257981.	−0.0482930	−0.0241465	−	0.999708i	$-0.507687\pi$
$491$	−257981.	−0.0482930	−0.0241465	+	0.999708i	$0.507687\pi$
$492$	0	0
$493$	−8.27645e6	−1.53365
$494$	0	0
$495$	309985.	0.0568627
$496$	0	0
$497$	2.09809e6	0.381007
$498$	0	0
$499$	−1.08106e7	−1.94356	−0.971782	−	0.235882i	$-0.924202\pi$
$499$	−1.08106e7	−1.94356	−0.971782	+	0.235882i	$0.924202\pi$
$500$	0	0
$501$	−4.61643e6	−0.821698
$502$	0	0
$503$	1.13024e7	1.99183	0.995915	−	0.0902925i	$-0.0287802\pi$
$503$	1.13024e7	1.99183	0.995915	+	0.0902925i	$0.0287802\pi$
$504$	0	0
$505$	−4.96769e6	−0.866814
$506$	0	0
$507$	3.03994e6	0.525225
$508$	0	0
$509$	5.04265e6	0.862708	0.431354	−	0.902183i	$-0.358036\pi$
$509$	5.04265e6	0.862708	0.431354	+	0.902183i	$0.358036\pi$
$510$	0	0
$511$	5.24511e6	0.888592
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	−1.05372e6	−0.175067
$516$	0	0
$517$	−469458.	−0.0772449
$518$	0	0
$519$	−3.09753e6	−0.504775
$520$	0	0
$521$	−5.51864e6	−0.890713	−0.445357	−	0.895353i	$-0.646923\pi$
$521$	−5.51864e6	−0.890713	−0.445357	+	0.895353i	$0.646923\pi$
$522$	0	0
$523$	5.84641e6	0.934621	0.467310	−	0.884093i	$-0.345223\pi$
$523$	5.84641e6	0.934621	0.467310	+	0.884093i	$0.345223\pi$
$524$	0	0
$525$	2.46209e6	0.389858
$526$	0	0
$527$	1.24761e7	1.95682
$528$	0	0
$529$	9.59111e6	1.49015
$530$	0	0
$531$	−2.44766e6	−0.376717
$532$	0	0
$533$	1.24740e6	0.190191
$534$	0	0
$535$	−3.28158e6	−0.495677
$536$	0	0
$537$	4.82352e6	0.721819
$538$	0	0
$539$	907430.	0.134537
$540$	0	0
$541$	−2.21955e6	−0.326041	−0.163020	−	0.986623i	$-0.552124\pi$
$541$	−2.21955e6	−0.326041	−0.163020	+	0.986623i	$0.552124\pi$
$542$	0	0
$543$	−6.45458e6	−0.939438
$544$	0	0
$545$	−1.02935e7	−1.48448
$546$	0	0
$547$	−1.37703e6	−0.196777	−0.0983883	−	0.995148i	$-0.531369\pi$
$547$	−1.37703e6	−0.196777	−0.0983883	+	0.995148i	$0.531369\pi$
$548$	0	0
$549$	998661.	0.141412
$550$	0	0
$551$	−1.47070e6	−0.206369
$552$	0	0
$553$	−29788.2	−0.00414220
$554$	0	0
$555$	−2.09799e6	−0.289116
$556$	0	0
$557$	−4.61215e6	−0.629891	−0.314946	−	0.949110i	$-0.601986\pi$
$557$	−4.61215e6	−0.629891	−0.314946	+	0.949110i	$0.601986\pi$
$558$	0	0
$559$	399721.	0.0541038
$560$	0	0
$561$	−1.02810e6	−0.137920
$562$	0	0
$563$	−1.39621e7	−1.85643	−0.928214	−	0.372046i	$-0.878657\pi$
$563$	−1.39621e7	−1.85643	−0.928214	+	0.372046i	$0.878657\pi$
$564$	0	0
$565$	−1.58753e7	−2.09219
$566$	0	0
$567$	−1.19087e6	−0.155563
$568$	0	0
$569$	−1.25243e7	−1.62170	−0.810852	−	0.585251i	$-0.800996\pi$
$569$	−1.25243e7	−1.62170	−0.810852	+	0.585251i	$0.800996\pi$
$570$	0	0
$571$	−8.03755e6	−1.03165	−0.515826	−	0.856693i	$-0.672515\pi$
$571$	−8.03755e6	−1.03165	−0.515826	+	0.856693i	$0.672515\pi$
$572$	0	0
$573$	5.54872e6	0.706002
$574$	0	0
$575$	−6.03392e6	−0.761079
$576$	0	0
$577$	1.33502e7	1.66935	0.834674	−	0.550744i	$-0.185656\pi$
$577$	1.33502e7	1.66935	0.834674	+	0.550744i	$0.185656\pi$
$578$	0	0
$579$	2.67971e6	0.332194
$580$	0	0
$581$	−2.44496e6	−0.300491
$582$	0	0
$583$	2.19308e6	0.267229
$584$	0	0
$585$	−1.00934e6	−0.121941
$586$	0	0
$587$	−4.01353e6	−0.480763	−0.240382	−	0.970678i	$-0.577273\pi$
$587$	−4.01353e6	−0.480763	−0.240382	+	0.970678i	$0.577273\pi$
$588$	0	0
$589$	2.21695e6	0.263311
$590$	0	0
$591$	−6.75265e6	−0.795254
$592$	0	0
$593$	8.78650e6	1.02608	0.513038	−	0.858366i	$-0.328520\pi$
$593$	8.78650e6	1.02608	0.513038	+	0.858366i	$0.328520\pi$
$594$	0	0
$595$	−2.50967e7	−2.90619
$596$	0	0
$597$	4.02353e6	0.462031
$598$	0	0
$599$	1.36952e7	1.55955	0.779777	−	0.626058i	$-0.215333\pi$
$599$	1.36952e7	1.55955	0.779777	+	0.626058i	$0.215333\pi$
$600$	0	0
$601$	−1.47348e7	−1.66402	−0.832010	−	0.554760i	$-0.812810\pi$
$601$	−1.47348e7	−1.66402	−0.832010	+	0.554760i	$0.812810\pi$
$602$	0	0
$603$	−658767.	−0.0737800
$604$	0	0
$605$	−1.07460e7	−1.19360
$606$	0	0
$607$	7.98391e6	0.879516	0.439758	−	0.898116i	$-0.355064\pi$
$607$	7.98391e6	0.879516	0.439758	+	0.898116i	$0.355064\pi$
$608$	0	0
$609$	−6.65508e6	−0.727126
$610$	0	0
$611$	1.52861e6	0.165650
$612$	0	0
$613$	−1.43938e7	−1.54712	−0.773558	−	0.633725i	$-0.781525\pi$
$613$	−1.43938e7	−1.54712	−0.773558	+	0.633725i	$0.781525\pi$
$614$	0	0
$615$	4.17332e6	0.444932
$616$	0	0
$617$	−1.35120e7	−1.42892	−0.714459	−	0.699677i	$-0.753327\pi$
$617$	−1.35120e7	−1.42892	−0.714459	+	0.699677i	$0.753327\pi$
$618$	0	0
$619$	1.80703e6	0.189557	0.0947783	−	0.995498i	$-0.469786\pi$
$619$	1.80703e6	0.189557	0.0947783	+	0.995498i	$0.469786\pi$
$620$	0	0
$621$	2.91850e6	0.303690
$622$	0	0
$623$	1.87983e7	1.94043
$624$	0	0
$625$	−1.22040e7	−1.24969
$626$	0	0
$627$	−182689.	−0.0185585
$628$	0	0
$629$	6.95821e6	0.701247
$630$	0	0
$631$	1.73492e7	1.73463	0.867313	−	0.497762i	$-0.165845\pi$
$631$	1.73492e7	1.73463	0.867313	+	0.497762i	$0.165845\pi$
$632$	0	0
$633$	−6.88572e6	−0.683030
$634$	0	0
$635$	−1.36303e7	−1.34144
$636$	0	0
$637$	−2.95469e6	−0.288512
$638$	0	0
$639$	−936297.	−0.0907113
$640$	0	0
$641$	−5.83239e6	−0.560662	−0.280331	−	0.959903i	$-0.590444\pi$
$641$	−5.83239e6	−0.560662	−0.280331	+	0.959903i	$0.590444\pi$
$642$	0	0
$643$	−1.08231e7	−1.03235	−0.516173	−	0.856484i	$-0.672644\pi$
$643$	−1.08231e7	−1.03235	−0.516173	+	0.856484i	$0.672644\pi$
$644$	0	0
$645$	1.33731e6	0.126570
$646$	0	0
$647$	1.06311e7	0.998433	0.499217	−	0.866477i	$-0.333621\pi$
$647$	1.06311e7	0.998433	0.499217	+	0.866477i	$0.333621\pi$
$648$	0	0
$649$	−1.69914e6	−0.158350
$650$	0	0
$651$	1.00320e7	0.927758
$652$	0	0
$653$	1.53877e7	1.41219	0.706093	−	0.708119i	$-0.250456\pi$
$653$	1.53877e7	1.41219	0.706093	+	0.708119i	$0.250456\pi$
$654$	0	0
$655$	−317429.	−0.0289097
$656$	0	0
$657$	−2.34070e6	−0.211559
$658$	0	0
$659$	1.66297e6	0.149166	0.0745832	−	0.997215i	$-0.476237\pi$
$659$	1.66297e6	0.149166	0.0745832	+	0.997215i	$0.476237\pi$
$660$	0	0
$661$	3.97953e6	0.354265	0.177133	−	0.984187i	$-0.443318\pi$
$661$	3.97953e6	0.354265	0.177133	+	0.984187i	$0.443318\pi$
$662$	0	0
$663$	3.34759e6	0.295766
$664$	0	0
$665$	−4.45959e6	−0.391058
$666$	0	0
$667$	1.63098e7	1.41949
$668$	0	0
$669$	−5.62323e6	−0.485758
$670$	0	0
$671$	693259.	0.0594414
$672$	0	0
$673$	7.83424e6	0.666744	0.333372	−	0.942795i	$-0.391814\pi$
$673$	7.83424e6	0.666744	0.333372	+	0.942795i	$0.391814\pi$
$674$	0	0
$675$	−1.09874e6	−0.0928186
$676$	0	0
$677$	−7.68877e6	−0.644741	−0.322370	−	0.946614i	$-0.604480\pi$
$677$	−7.68877e6	−0.644741	−0.322370	+	0.946614i	$0.604480\pi$
$678$	0	0
$679$	6.26033e6	0.521102
$680$	0	0
$681$	76400.7	0.00631291
$682$	0	0
$683$	1.48293e7	1.21638	0.608190	−	0.793792i	$-0.291896\pi$
$683$	1.48293e7	1.21638	0.608190	+	0.793792i	$0.291896\pi$
$684$	0	0
$685$	−1.74165e6	−0.141819
$686$	0	0
$687$	−4.51941e6	−0.365334
$688$	0	0
$689$	−7.14093e6	−0.573069
$690$	0	0
$691$	−2.34870e7	−1.87125	−0.935626	−	0.352992i	$-0.885164\pi$
$691$	−2.34870e7	−1.87125	−0.935626	+	0.352992i	$0.885164\pi$
$692$	0	0
$693$	−826689.	−0.0653897
$694$	0	0
$695$	−4.57969e6	−0.359645
$696$	0	0
$697$	−1.38412e7	−1.07918
$698$	0	0
$699$	2.87309e6	0.222411
$700$	0	0
$701$	−3.22661e6	−0.248000	−0.124000	−	0.992282i	$-0.539572\pi$
$701$	−3.22661e6	−0.248000	−0.124000	+	0.992282i	$0.539572\pi$
$702$	0	0
$703$	1.23645e6	0.0943599
$704$	0	0
$705$	5.11411e6	0.387522
$706$	0	0
$707$	1.32482e7	0.996800
$708$	0	0
$709$	2.35520e6	0.175959	0.0879796	−	0.996122i	$-0.471959\pi$
$709$	2.35520e6	0.175959	0.0879796	+	0.996122i	$0.471959\pi$
$710$	0	0
$711$	13293.3	0.000986188 0
$712$	0	0
$713$	−2.45857e7	−1.81116
$714$	0	0
$715$	−700675.	−0.0512568
$716$	0	0
$717$	6.80956e6	0.494677
$718$	0	0
$719$	−2.13336e7	−1.53901	−0.769507	−	0.638638i	$-0.779498\pi$
$719$	−2.13336e7	−1.53901	−0.769507	+	0.638638i	$0.779498\pi$
$720$	0	0
$721$	2.81012e6	0.201320
$722$	0	0
$723$	5.80685e6	0.413138
$724$	0	0
$725$	−6.14020e6	−0.433848
$726$	0	0
$727$	1.88770e6	0.132464	0.0662319	−	0.997804i	$-0.478902\pi$
$727$	1.88770e6	0.132464	0.0662319	+	0.997804i	$0.478902\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.43531e6	−0.306995
$732$	0	0
$733$	−899424.	−0.0618308	−0.0309154	−	0.999522i	$-0.509842\pi$
$733$	−899424.	−0.0618308	−0.0309154	+	0.999522i	$0.509842\pi$
$734$	0	0
$735$	−9.88522e6	−0.674944
$736$	0	0
$737$	−457308.	−0.0310127
$738$	0	0
$739$	−157022.	−0.0105767	−0.00528833	−	0.999986i	$-0.501683\pi$
$739$	−157022.	−0.0105767	−0.00528833	+	0.999986i	$0.501683\pi$
$740$	0	0
$741$	594855.	0.0397984
$742$	0	0
$743$	1.95058e7	1.29626	0.648128	−	0.761531i	$-0.275552\pi$
$743$	1.95058e7	1.29626	0.648128	+	0.761531i	$0.275552\pi$
$744$	0	0
$745$	−1.61012e7	−1.06284
$746$	0	0
$747$	1.09109e6	0.0715418
$748$	0	0
$749$	8.75156e6	0.570008
$750$	0	0
$751$	1.45467e7	0.941163	0.470582	−	0.882356i	$-0.344044\pi$
$751$	1.45467e7	0.941163	0.470582	+	0.882356i	$0.344044\pi$
$752$	0	0
$753$	1.52143e7	0.977834
$754$	0	0
$755$	−6.59711e6	−0.421198
$756$	0	0
$757$	−1.20959e7	−0.767183	−0.383592	−	0.923503i	$-0.625313\pi$
$757$	−1.20959e7	−0.767183	−0.383592	+	0.923503i	$0.625313\pi$
$758$	0	0
$759$	2.02599e6	0.127653
$760$	0	0
$761$	−1.04786e7	−0.655905	−0.327952	−	0.944694i	$-0.606359\pi$
$761$	−1.04786e7	−0.655905	−0.327952	+	0.944694i	$0.606359\pi$
$762$	0	0
$763$	2.74515e7	1.70709
$764$	0	0
$765$	1.11997e7	0.691916
$766$	0	0
$767$	5.53259e6	0.339578
$768$	0	0
$769$	−1.61815e6	−0.0986739	−0.0493370	−	0.998782i	$-0.515711\pi$
$769$	−1.61815e6	−0.0986739	−0.0493370	+	0.998782i	$0.515711\pi$
$770$	0	0
$771$	1.80185e6	0.109165
$772$	0	0
$773$	2.19651e7	1.32216	0.661080	−	0.750315i	$-0.270098\pi$
$773$	2.19651e7	1.32216	0.661080	+	0.750315i	$0.270098\pi$
$774$	0	0
$775$	9.25586e6	0.553557
$776$	0	0
$777$	5.59508e6	0.332471
$778$	0	0
$779$	−2.45953e6	−0.145214
$780$	0	0
$781$	−649966.	−0.0381297
$782$	0	0
$783$	2.96991e6	0.173117
$784$	0	0
$785$	−3.53062e6	−0.204492
$786$	0	0
$787$	1.02291e7	0.588707	0.294354	−	0.955697i	$-0.404896\pi$
$787$	1.02291e7	0.588707	0.294354	+	0.955697i	$0.404896\pi$
$788$	0	0
$789$	7.12472e6	0.407451
$790$	0	0
$791$	4.23374e7	2.40593
$792$	0	0
$793$	−2.25733e6	−0.127471
$794$	0	0
$795$	−2.38907e7	−1.34064
$796$	0	0
$797$	−2.15124e7	−1.19962	−0.599808	−	0.800144i	$-0.704756\pi$
$797$	−2.15124e7	−1.19962	−0.599808	+	0.800144i	$0.704756\pi$
$798$	0	0
$799$	−1.69614e7	−0.939931
$800$	0	0
$801$	−8.38895e6	−0.461983
$802$	0	0
$803$	−1.62488e6	−0.0889269
$804$	0	0
$805$	4.94562e7	2.68987
$806$	0	0
$807$	−6.11271e6	−0.330408
$808$	0	0
$809$	−5.72326e6	−0.307448	−0.153724	−	0.988114i	$-0.549127\pi$
$809$	−5.72326e6	−0.307448	−0.153724	+	0.988114i	$0.549127\pi$
$810$	0	0
$811$	2.78783e7	1.48838	0.744191	−	0.667967i	$-0.232835\pi$
$811$	2.78783e7	1.48838	0.744191	+	0.667967i	$0.232835\pi$
$812$	0	0
$813$	1.46232e7	0.775918
$814$	0	0
$815$	−1.32633e7	−0.699451
$816$	0	0
$817$	−788139.	−0.0413093
$818$	0	0
$819$	2.69179e6	0.140227
$820$	0	0
$821$	6.54662e6	0.338969	0.169484	−	0.985533i	$-0.445790\pi$
$821$	6.54662e6	0.338969	0.169484	+	0.985533i	$0.445790\pi$
$822$	0	0
$823$	−1.03390e7	−0.532082	−0.266041	−	0.963962i	$-0.585716\pi$
$823$	−1.03390e7	−0.532082	−0.266041	+	0.963962i	$0.585716\pi$
$824$	0	0
$825$	−762732.	−0.0390155
$826$	0	0
$827$	−1.19751e7	−0.608858	−0.304429	−	0.952535i	$-0.598466\pi$
$827$	−1.19751e7	−0.608858	−0.304429	+	0.952535i	$0.598466\pi$
$828$	0	0
$829$	−1.39945e7	−0.707249	−0.353625	−	0.935387i	$-0.615051\pi$
$829$	−1.39945e7	−0.707249	−0.353625	+	0.935387i	$0.615051\pi$
$830$	0	0
$831$	2.21981e7	1.11510
$832$	0	0
$833$	3.27853e7	1.63707
$834$	0	0
$835$	3.49106e7	1.73277
$836$	0	0
$837$	−4.47690e6	−0.220884
$838$	0	0
$839$	−2.12669e7	−1.04303	−0.521517	−	0.853241i	$-0.674634\pi$
$839$	−2.12669e7	−1.04303	−0.521517	+	0.853241i	$0.674634\pi$
$840$	0	0
$841$	−3.91410e6	−0.190828
$842$	0	0
$843$	−8.60015e6	−0.416809
$844$	0	0
$845$	−2.29888e7	−1.10758
$846$	0	0
$847$	2.86581e7	1.37258
$848$	0	0
$849$	1.37225e6	0.0653377
$850$	0	0
$851$	−1.37120e7	−0.649048
$852$	0	0
$853$	−4.31526e6	−0.203065	−0.101532	−	0.994832i	$-0.532375\pi$
$853$	−4.31526e6	−0.203065	−0.101532	+	0.994832i	$0.532375\pi$
$854$	0	0
$855$	1.99015e6	0.0931044
$856$	0	0
$857$	1.59971e6	0.0744029	0.0372015	−	0.999308i	$-0.488156\pi$
$857$	1.59971e6	0.0744029	0.0372015	+	0.999308i	$0.488156\pi$
$858$	0	0
$859$	2.97990e7	1.37790	0.688951	−	0.724808i	$-0.258072\pi$
$859$	2.97990e7	1.37790	0.688951	+	0.724808i	$0.258072\pi$
$860$	0	0
$861$	−1.11297e7	−0.511653
$862$	0	0
$863$	−1.17818e7	−0.538498	−0.269249	−	0.963071i	$-0.586775\pi$
$863$	−1.17818e7	−0.538498	−0.269249	+	0.963071i	$0.586775\pi$
$864$	0	0
$865$	2.34243e7	1.06445
$866$	0	0
$867$	−2.43662e7	−1.10088
$868$	0	0
$869$	9228.07	0.000414535 0
$870$	0	0
$871$	1.48905e6	0.0665063
$872$	0	0
$873$	−2.79375e6	−0.124066
$874$	0	0
$875$	1.99856e7	0.882462
$876$	0	0
$877$	−1.60831e7	−0.706108	−0.353054	−	0.935603i	$-0.614857\pi$
$877$	−1.60831e7	−0.706108	−0.353054	+	0.935603i	$0.614857\pi$
$878$	0	0
$879$	−2.28961e7	−0.999517
$880$	0	0
$881$	−2.06461e7	−0.896187	−0.448093	−	0.893987i	$-0.647897\pi$
$881$	−2.06461e7	−0.896187	−0.448093	+	0.893987i	$0.647897\pi$
$882$	0	0
$883$	−2.44612e7	−1.05579	−0.527894	−	0.849310i	$-0.677018\pi$
$883$	−2.44612e7	−1.05579	−0.527894	+	0.849310i	$0.677018\pi$
$884$	0	0
$885$	1.85098e7	0.794409
$886$	0	0
$887$	2.49921e7	1.06658	0.533290	−	0.845932i	$-0.320955\pi$
$887$	2.49921e7	1.06658	0.533290	+	0.845932i	$0.320955\pi$
$888$	0	0
$889$	3.63502e7	1.54260
$890$	0	0
$891$	368920.	0.0155682
$892$	0	0
$893$	−3.01399e6	−0.126477
$894$	0	0
$895$	−3.64766e7	−1.52215
$896$	0	0
$897$	−6.59685e6	−0.273751
$898$	0	0
$899$	−2.50187e7	−1.03244
$900$	0	0
$901$	7.92359e7	3.25169
$902$	0	0
$903$	−3.56643e6	−0.145551
$904$	0	0
$905$	4.88111e7	1.98106
$906$	0	0
$907$	1.33799e6	0.0540051	0.0270026	−	0.999635i	$-0.491404\pi$
$907$	1.33799e6	0.0540051	0.0270026	+	0.999635i	$0.491404\pi$
$908$	0	0
$909$	−5.91216e6	−0.237321
$910$	0	0
$911$	−4.41172e6	−0.176121	−0.0880606	−	0.996115i	$-0.528067\pi$
$911$	−4.41172e6	−0.176121	−0.0880606	+	0.996115i	$0.528067\pi$
$912$	0	0
$913$	757423.	0.0300719
$914$	0	0
$915$	−7.55211e6	−0.298206
$916$	0	0
$917$	846542.	0.0332449
$918$	0	0
$919$	−1.72978e7	−0.675619	−0.337810	−	0.941214i	$-0.609686\pi$
$919$	−1.72978e7	−0.675619	−0.337810	+	0.941214i	$0.609686\pi$
$920$	0	0
$921$	−1.01754e7	−0.395279
$922$	0	0
$923$	2.11636e6	0.0817685
$924$	0	0
$925$	5.16221e6	0.198373
$926$	0	0
$927$	−1.25405e6	−0.0479309
$928$	0	0
$929$	−1.09081e7	−0.414676	−0.207338	−	0.978269i	$-0.566480\pi$
$929$	−1.09081e7	−0.414676	−0.207338	+	0.978269i	$0.566480\pi$
$930$	0	0
$931$	5.82583e6	0.220284
$932$	0	0
$933$	−3.05281e7	−1.14814
$934$	0	0
$935$	7.77471e6	0.290841
$936$	0	0
$937$	3.85474e7	1.43432	0.717159	−	0.696909i	$-0.245442\pi$
$937$	3.85474e7	1.43432	0.717159	+	0.696909i	$0.245442\pi$
$938$	0	0
$939$	−3.87616e6	−0.143462
$940$	0	0
$941$	2.02392e6	0.0745110	0.0372555	−	0.999306i	$-0.488138\pi$
$941$	2.02392e6	0.0745110	0.0372555	+	0.999306i	$0.488138\pi$
$942$	0	0
$943$	2.72758e7	0.998847
$944$	0	0
$945$	9.00566e6	0.328047
$946$	0	0
$947$	1.58028e7	0.572611	0.286306	−	0.958138i	$-0.407573\pi$
$947$	1.58028e7	0.572611	0.286306	+	0.958138i	$0.407573\pi$
$948$	0	0
$949$	5.29080e6	0.190702
$950$	0	0
$951$	3.08529e7	1.10623
$952$	0	0
$953$	−4.55765e7	−1.62558	−0.812791	−	0.582555i	$-0.802053\pi$
$953$	−4.55765e7	−1.62558	−0.812791	+	0.582555i	$0.802053\pi$
$954$	0	0
$955$	−4.19607e7	−1.48879
$956$	0	0
$957$	2.06168e6	0.0727680
$958$	0	0
$959$	4.64477e6	0.163086
$960$	0	0
$961$	9.08453e6	0.317318
$962$	0	0
$963$	−3.90549e6	−0.135709
$964$	0	0
$965$	−2.02646e7	−0.700520
$966$	0	0
$967$	−4.49937e7	−1.54734	−0.773669	−	0.633589i	$-0.781581\pi$
$967$	−4.49937e7	−1.54734	−0.773669	+	0.633589i	$0.781581\pi$
$968$	0	0
$969$	−6.60052e6	−0.225823
$970$	0	0
$971$	4.31949e7	1.47023	0.735114	−	0.677944i	$-0.237129\pi$
$971$	4.31949e7	1.47023	0.735114	+	0.677944i	$0.237129\pi$
$972$	0	0
$973$	1.22134e7	0.413577
$974$	0	0
$975$	2.48354e6	0.0836681
$976$	0	0
$977$	8.76839e6	0.293889	0.146945	−	0.989145i	$-0.453056\pi$
$977$	8.76839e6	0.293889	0.146945	+	0.989145i	$0.453056\pi$
$978$	0	0
$979$	−5.82351e6	−0.194191
$980$	0	0
$981$	−1.22506e7	−0.406428
$982$	0	0
$983$	−4.18236e7	−1.38051	−0.690253	−	0.723568i	$-0.742501\pi$
$983$	−4.18236e7	−1.38051	−0.690253	+	0.723568i	$0.742501\pi$
$984$	0	0
$985$	5.10652e7	1.67700
$986$	0	0
$987$	−1.36387e7	−0.445634
$988$	0	0
$989$	8.74033e6	0.284143
$990$	0	0
$991$	2.87743e7	0.930723	0.465362	−	0.885121i	$-0.345924\pi$
$991$	2.87743e7	0.930723	0.465362	+	0.885121i	$0.345924\pi$
$992$	0	0
$993$	197607.	0.00635960
$994$	0	0
$995$	−3.04269e7	−0.974317
$996$	0	0
$997$	−9.50959e6	−0.302987	−0.151493	−	0.988458i	$-0.548408\pi$
$997$	−9.50959e6	−0.302987	−0.151493	+	0.988458i	$0.548408\pi$
$998$	0	0
$999$	−2.49687e6	−0.0791558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.6.a.f.1.6	✓	6
4.3	odd	2		912.6.a.y.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.6.a.f.1.6	✓	6	1.1	even	1	trivial
912.6.a.y.1.6		6	4.3	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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +68.0602 q^{5} -181.508 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +68.0602 q^{5} -181.508 q^{7} +81.0000 q^{9} +56.2292 q^{11} -183.089 q^{13} -612.542 q^{15} +2031.56 q^{17} +361.000 q^{19} +1633.57 q^{21} -4003.43 q^{23} +1507.19 q^{25} -729.000 q^{27} -4073.95 q^{29} +6141.15 q^{31} -506.063 q^{33} -12353.4 q^{35} +3425.06 q^{37} +1647.80 q^{39} -6813.12 q^{41} -2183.21 q^{43} +5512.87 q^{45} -8348.99 q^{47} +16138.0 q^{49} -18284.0 q^{51} +39002.6 q^{53} +3826.97 q^{55} -3249.00 q^{57} -30218.1 q^{59} +12329.1 q^{61} -14702.1 q^{63} -12461.0 q^{65} -8132.92 q^{67} +36030.9 q^{69} -11559.2 q^{71} -28897.5 q^{73} -13564.7 q^{75} -10206.0 q^{77} +164.115 q^{79} +6561.00 q^{81} +13470.3 q^{83} +138268. q^{85} +36665.5 q^{87} -103567. q^{89} +33232.0 q^{91} -55270.3 q^{93} +24569.7 q^{95} -34490.7 q^{97} +4554.57 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * q^99

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